TPTP Problem File: ITP217^4.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : ITP217^4 : TPTP v8.2.0. Released v8.0.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer problem Time_Reasoning 00330_012302
% 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 : 0029_Time_Reasoning_00330_012302 [Des22]
% Status : Theorem
% Rating : 1.00 v8.1.0
% Syntax : Number of formulae : 9619 (3180 unt; 757 typ; 0 def)
% Number of atoms : 26518 (10232 equ; 5 cnn)
% Maximal formula atoms : 48 ( 2 avg)
% Number of connectives : 166065 (2141 ~; 297 |;1648 &;150562 @)
% ( 0 <=>;11417 =>; 0 <=; 0 <~>)
% Maximal formula depth : 34 ( 7 avg)
% Number of types : 10 ( 9 usr)
% Number of type conns : 6235 (6235 >; 0 *; 0 +; 0 <<)
% Number of symbols : 752 ( 748 usr; 13 con; 0-10 aty)
% Number of variables : 31625 (4405 ^;25789 !; 628 ?;31625 :)
% ( 803 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% from the van Emde Boas Trees session in the Archive of Formal
% proofs -
% www.isa-afp.org/browser_info/current/AFP/Van_Emde_Boas_Trees
% 2022-02-17 16:31:57.529
%------------------------------------------------------------------------------
% Could-be-implicit typings (20)
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_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_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 (737)
thf(sy_cl_Lattices_Obounded__lattice__top,type,
bounded_lattice_top:
!>[A: $tType] : $o ).
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : $o ).
thf(sy_cl_Nat_Osize,type,
size:
!>[A: $tType] : $o ).
thf(sy_cl_Heap_Oheap,type,
heap:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Odvd,type,
dvd:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oone,type,
one:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oidom,type,
idom:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oring,type,
ring:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oplus,type,
plus:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ozero,type,
zero:
!>[A: $tType] : $o ).
thf(sy_cl_Num_Onumeral,type,
numeral:
!>[A: $tType] : $o ).
thf(sy_cl_Power_Opower,type,
power:
!>[A: $tType] : $o ).
thf(sy_cl_Fields_Ofield,type,
field:
!>[A: $tType] : $o ).
thf(sy_cl_GCD_Oring__gcd,type,
ring_gcd:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_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_Rings_Osemidom,type,
semidom:
!>[A: $tType] : $o ).
thf(sy_cl_Fields_Oinverse,type,
inverse:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemiring,type,
semiring:
!>[A: $tType] : $o ).
thf(sy_cl_Nat_Oring__char__0,type,
ring_char_0:
!>[A: $tType] : $o ).
thf(sy_cl_Num_Oneg__numeral,type,
neg_numeral:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oorder,type,
order:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Ocomm__ring,type,
comm_ring:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Omult__zero,type,
mult_zero:
!>[A: $tType] : $o ).
thf(sy_cl_GCD_Osemiring__Gcd,type,
semiring_Gcd:
!>[A: $tType] : $o ).
thf(sy_cl_GCD_Osemiring__gcd,type,
semiring_gcd:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ogroup__add,type,
group_add:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Olattice,type,
lattice:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Ono__bot,type,
no_bot:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Ono__top,type,
no_top:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemiring__0,type,
semiring_0:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemiring__1,type,
semiring_1:
!>[A: $tType] : $o ).
thf(sy_cl_Finite__Set_Ofinite,type,
finite_finite:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Omonoid__add,type,
monoid_add:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Ocomm__ring__1,type,
comm_ring_1:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oidom__divide,type,
idom_divide:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Omonoid__mult,type,
monoid_mult:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Opreorder,type,
preorder:
!>[A: $tType] : $o ).
thf(sy_cl_Parity_Oring__parity,type,
ring_parity:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oidom__abs__sgn,type,
idom_abs_sgn:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oordered__ring,type,
ordered_ring:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Ozero__neq__one,type,
zero_neq_one:
!>[A: $tType] : $o ).
thf(sy_cl_Fields_Ofield__char__0,type,
field_char_0:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oab__group__add,type,
ab_group_add:
!>[A: $tType] : $o ).
thf(sy_cl_Nat_Osemiring__char__0,type,
semiring_char_0:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oorder__bot,type,
order_bot:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oorder__top,type,
order_top:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Owellorder,type,
wellorder:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Ocomm__semiring,type,
comm_semiring:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Ozero__less__one,type,
zero_less_one:
!>[A: $tType] : $o ).
thf(sy_cl_Fields_Odivision__ring,type,
division_ring:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Osemigroup__add,type,
semigroup_add:
!>[A: $tType] : $o ).
thf(sy_cl_Num_Osemiring__numeral,type,
semiring_numeral:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemidom__divide,type,
semidom_divide:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemidom__modulo,type,
semidom_modulo:
!>[A: $tType] : $o ).
thf(sy_cl_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_Oab__semigroup__add,type,
ab_semigroup_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocomm__monoid__diff,type,
comm_monoid_diff:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocomm__monoid__mult,type,
comm_monoid_mult:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oalgebraic__semidom,type,
algebraic_semidom:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemiring__1__cancel,type,
semiring_1_cancel:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oab__semigroup__mult,type,
ab_semigroup_mult:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Obounded__lattice,type,
bounded_lattice:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Odistrib__lattice,type,
distrib_lattice:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Osemilattice__inf,type,
semilattice_inf:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Osemilattice__sup,type,
semilattice_sup:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Odense__linorder,type,
dense_linorder:
!>[A: $tType] : $o ).
thf(sy_cl_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_Rings_Ocomm__semiring__1__cancel,type,
comm_s4317794764714335236cancel:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__ring__strict,type,
linord4710134922213307826strict:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocancel__comm__monoid__add,type,
cancel1802427076303600483id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oring__1__no__zero__divisors,type,
ring_15535105094025558882visors:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocancel__ab__semigroup__add,type,
cancel2418104881723323429up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Olinordered__ab__group__add,type,
linord5086331880401160121up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__comm__monoid__add,type,
ordere6911136660526730532id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Archimedean__Field_Ofloor__ceiling,type,
archim2362893244070406136eiling:
!>[A: $tType] : $o ).
thf(sy_cl_GCD_Osemiring__gcd__mult__normalize,type,
semiri6843258321239162965malize:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__group__add__abs,type,
ordere166539214618696060dd_abs:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__semigroup__add,type,
ordere6658533253407199908up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemiring__no__zero__divisors,type,
semiri3467727345109120633visors:
!>[A: $tType] : $o ).
thf(sy_cl_Boolean__Algebras_Oboolean__algebra,type,
boolea8198339166811842893lgebra:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__semiring__strict,type,
linord8928482502909563296strict:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_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_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_Euclidean__Division_Ounique__euclidean__ring__with__nat,type,
euclid8789492081693882211th_nat:
!>[A: $tType] : $o ).
thf(sy_cl_Euclidean__Division_Ounique__euclidean__semiring__with__nat,type,
euclid5411537665997757685th_nat:
!>[A: $tType] : $o ).
thf(sy_cl_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,type,
semiri1453513574482234551roduct:
!>[A: $tType] : $o ).
thf(sy_cl_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations,type,
bit_un5681908812861735899ations:
!>[A: $tType] : $o ).
thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__lattice,type,
condit1219197933456340205attice:
!>[A: $tType] : $o ).
thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__linorder,type,
condit6923001295902523014norder:
!>[A: $tType] : $o ).
thf(sy_c_Archimedean__Field_Oceiling,type,
archimedean_ceiling:
!>[A: $tType] : ( A > int ) ).
thf(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor,type,
archim6421214686448440834_floor:
!>[A: $tType] : ( A > int ) ).
thf(sy_c_Archimedean__Field_Ofrac,type,
archimedean_frac:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Archimedean__Field_Oround,type,
archimedean_round:
!>[A: $tType] : ( A > int ) ).
thf(sy_c_Array__Time_Oalloc,type,
array_alloc:
!>[A: $tType] : ( ( list @ A ) > ( heap_ext @ product_unit ) > ( product_prod @ ( array @ A ) @ ( heap_ext @ product_unit ) ) ) ).
thf(sy_c_Array__Time_Ofreeze,type,
array_freeze:
!>[A: $tType] : ( ( array @ A ) > ( heap_Time_Heap @ ( list @ A ) ) ) ).
thf(sy_c_Array__Time_Oget,type,
array_get:
!>[A: $tType] : ( ( heap_ext @ product_unit ) > ( array @ A ) > ( list @ A ) ) ).
thf(sy_c_Array__Time_Olen,type,
array_len:
!>[A: $tType] : ( ( array @ A ) > ( heap_Time_Heap @ nat ) ) ).
thf(sy_c_Array__Time_Olen_H,type,
array_len2:
!>[B: $tType,A: $tType] : ( ( array @ B ) > ( heap_Time_Heap @ A ) ) ).
thf(sy_c_Array__Time_Olength,type,
array_length:
!>[A: $tType] : ( ( heap_ext @ product_unit ) > ( array @ A ) > nat ) ).
thf(sy_c_Array__Time_Omake,type,
array_make:
!>[A: $tType] : ( nat > ( nat > A ) > ( heap_Time_Heap @ ( array @ A ) ) ) ).
thf(sy_c_Array__Time_Omake_H,type,
array_make2:
!>[B: $tType,A: $tType] : ( code_integer > ( B > A ) > ( heap_Time_Heap @ ( array @ A ) ) ) ).
thf(sy_c_Array__Time_Omap__entry,type,
array_map_entry:
!>[A: $tType] : ( nat > ( A > A ) > ( array @ A ) > ( heap_Time_Heap @ ( array @ A ) ) ) ).
thf(sy_c_Array__Time_Onew,type,
array_new:
!>[A: $tType] : ( nat > A > ( heap_Time_Heap @ ( array @ A ) ) ) ).
thf(sy_c_Array__Time_Onth,type,
array_nth:
!>[A: $tType] : ( ( array @ A ) > nat > ( heap_Time_Heap @ A ) ) ).
thf(sy_c_Array__Time_Oof__list,type,
array_of_list:
!>[A: $tType] : ( ( list @ A ) > ( heap_Time_Heap @ ( array @ A ) ) ) ).
thf(sy_c_Array__Time_Opresent,type,
array_present:
!>[A: $tType] : ( ( heap_ext @ product_unit ) > ( array @ A ) > $o ) ).
thf(sy_c_Array__Time_Oswap,type,
array_swap:
!>[A: $tType] : ( nat > A > ( array @ A ) > ( heap_Time_Heap @ A ) ) ).
thf(sy_c_Array__Time_Oupd,type,
array_upd:
!>[A: $tType] : ( nat > A > ( array @ A ) > ( heap_Time_Heap @ ( array @ A ) ) ) ).
thf(sy_c_Array__Time_Oupd_H,type,
array_upd2:
!>[A: $tType] : ( ( array @ A ) > code_integer > A > ( heap_Time_Heap @ product_unit ) ) ).
thf(sy_c_Array__Time_Oupdate,type,
array_update:
!>[A: $tType] : ( ( array @ A ) > nat > A > ( heap_ext @ product_unit ) > ( heap_ext @ product_unit ) ) ).
thf(sy_c_BNF__Cardinal__Order__Relation_OnatLess,type,
bNF_Ca8459412986667044542atLess: set @ ( product_prod @ nat @ nat ) ).
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_Oconvol,type,
bNF_convol:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > B ) > ( A > C ) > A > ( product_prod @ B @ C ) ) ).
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__Def_Ovimage2p,type,
bNF_vimage2p:
!>[A: $tType,D: $tType,B: $tType,E: $tType,C: $tType] : ( ( A > D ) > ( B > E ) > ( D > E > C ) > A > B > C ) ).
thf(sy_c_BNF__Greatest__Fixpoint_OShift,type,
bNF_Greatest_Shift:
!>[A: $tType] : ( ( set @ ( list @ A ) ) > A > ( set @ ( list @ A ) ) ) ).
thf(sy_c_BNF__Greatest__Fixpoint_OSucc,type,
bNF_Greatest_Succ:
!>[A: $tType] : ( ( set @ ( list @ A ) ) > ( list @ A ) > ( set @ A ) ) ).
thf(sy_c_BNF__Greatest__Fixpoint_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_Oimage2p,type,
bNF_Greatest_image2p:
!>[C: $tType,A: $tType,D: $tType,B: $tType] : ( ( C > A ) > ( D > B ) > ( C > D > $o ) > A > B > $o ) ).
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_Oshift,type,
bNF_Greatest_shift:
!>[A: $tType,B: $tType] : ( ( ( list @ A ) > B ) > A > ( list @ A ) > B ) ).
thf(sy_c_BNF__Greatest__Fixpoint_Ouniv,type,
bNF_Greatest_univ:
!>[B: $tType,A: $tType] : ( ( B > A ) > ( set @ B ) > A ) ).
thf(sy_c_BNF__Wellorder__Constructions_OFunc,type,
bNF_Wellorder_Func:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( set @ B ) > ( set @ ( A > B ) ) ) ).
thf(sy_c_BNF__Wellorder__Constructions_Ocurr,type,
bNF_Wellorder_curr:
!>[A: $tType,B: $tType,C: $tType] : ( ( set @ A ) > ( ( product_prod @ A @ B ) > C ) > A > B > C ) ).
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_Ofsts,type,
basic_fsts:
!>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > ( set @ A ) ) ).
thf(sy_c_Basic__BNFs_Opred__fun,type,
basic_pred_fun:
!>[A: $tType,B: $tType] : ( ( A > $o ) > ( B > $o ) > ( A > B ) > $o ) ).
thf(sy_c_Basic__BNFs_Opred__prod,type,
basic_pred_prod:
!>[A: $tType,B: $tType] : ( ( A > $o ) > ( B > $o ) > ( product_prod @ A @ B ) > $o ) ).
thf(sy_c_Basic__BNFs_Orel__prod,type,
basic_rel_prod:
!>[A: $tType,B: $tType,C: $tType,D: $tType] : ( ( A > B > $o ) > ( C > D > $o ) > ( product_prod @ A @ C ) > ( product_prod @ B @ D ) > $o ) ).
thf(sy_c_Basic__BNFs_Osnds,type,
basic_snds:
!>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > ( set @ B ) ) ).
thf(sy_c_Binomial_Obinomial,type,
binomial: nat > nat > nat ).
thf(sy_c_Binomial_Ogbinomial,type,
gbinomial:
!>[A: $tType] : ( A > nat > A ) ).
thf(sy_c_Bit__Operations_Oand__int__rel,type,
bit_and_int_rel: ( product_prod @ int @ int ) > ( product_prod @ int @ int ) > $o ).
thf(sy_c_Bit__Operations_Oand__not__num,type,
bit_and_not_num: num > num > ( option @ num ) ).
thf(sy_c_Bit__Operations_Oand__not__num__rel,type,
bit_and_not_num_rel: ( product_prod @ num @ num ) > ( product_prod @ num @ num ) > $o ).
thf(sy_c_Bit__Operations_Oconcat__bit,type,
bit_concat_bit: nat > int > int > int ).
thf(sy_c_Bit__Operations_Oor__not__num__neg,type,
bit_or_not_num_neg: num > num > num ).
thf(sy_c_Bit__Operations_Oring__bit__operations__class_Onot,type,
bit_ri4277139882892585799ns_not:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Bit__Operations_Oring__bit__operations__class_Osigned__take__bit,type,
bit_ri4674362597316999326ke_bit:
!>[A: $tType] : ( nat > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand,type,
bit_se5824344872417868541ns_and:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Odrop__bit,type,
bit_se4197421643247451524op_bit:
!>[A: $tType] : ( nat > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit,type,
bit_se8732182000553998342ip_bit:
!>[A: $tType] : ( nat > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Omask,type,
bit_se2239418461657761734s_mask:
!>[A: $tType] : ( nat > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oor,type,
bit_se1065995026697491101ons_or:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Opush__bit,type,
bit_se4730199178511100633sh_bit:
!>[A: $tType] : ( nat > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit,type,
bit_se5668285175392031749et_bit:
!>[A: $tType] : ( nat > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Otake__bit,type,
bit_se2584673776208193580ke_bit:
!>[A: $tType] : ( nat > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit,type,
bit_se2638667681897837118et_bit:
!>[A: $tType] : ( nat > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oxor,type,
bit_se5824344971392196577ns_xor:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bits__class_Obit,type,
bit_se5641148757651400278ts_bit:
!>[A: $tType] : ( A > nat > $o ) ).
thf(sy_c_Bit__Operations_Osemiring__bits__class_Opossible__bit,type,
bit_se6407376104438227557le_bit:
!>[A: $tType] : ( ( itself @ A ) > nat > $o ) ).
thf(sy_c_Bit__Operations_Otake__bit__num,type,
bit_take_bit_num: nat > num > ( option @ num ) ).
thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oand__num,type,
bit_un7362597486090784418nd_num: num > num > ( option @ num ) ).
thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oand__num__rel,type,
bit_un4731106466462545111um_rel: ( product_prod @ num @ num ) > ( product_prod @ num @ num ) > $o ).
thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oxor__num,type,
bit_un2480387367778600638or_num: num > num > ( option @ num ) ).
thf(sy_c_Boolean__Algebras_Oabstract__boolean__algebra,type,
boolea2506097494486148201lgebra:
!>[A: $tType] : ( ( A > A > A ) > ( A > A > A ) > ( A > A ) > A > A > $o ) ).
thf(sy_c_Boolean__Algebras_Oabstract__boolean__algebra__sym__diff,type,
boolea3799213064322606851m_diff:
!>[A: $tType] : ( ( A > A > A ) > ( A > A > A ) > ( A > A ) > A > A > ( A > A > A ) > $o ) ).
thf(sy_c_Code__Numeral_ONat,type,
code_Nat: code_integer > code_natural ).
thf(sy_c_Code__Numeral_ONeg,type,
code_Neg: num > code_integer ).
thf(sy_c_Code__Numeral_OSuc,type,
code_Suc: code_natural > code_natural ).
thf(sy_c_Code__Numeral_Obit__cut__integer,type,
code_bit_cut_integer: code_integer > ( product_prod @ code_integer @ $o ) ).
thf(sy_c_Code__Numeral_Odivmod__abs,type,
code_divmod_abs: code_integer > code_integer > ( product_prod @ code_integer @ code_integer ) ).
thf(sy_c_Code__Numeral_Odivmod__integer,type,
code_divmod_integer: code_integer > code_integer > ( product_prod @ code_integer @ code_integer ) ).
thf(sy_c_Code__Numeral_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__nat,type,
code_integer_of_nat: nat > code_integer ).
thf(sy_c_Code__Numeral_Ointeger__of__natural,type,
code_i5400310926305786745atural: code_natural > 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_Ocase__natural,type,
code_case_natural:
!>[T: $tType] : ( T > ( code_natural > T ) > code_natural > T ) ).
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_Onatural_Orec__natural,type,
code_rec_natural:
!>[T: $tType] : ( T > ( code_natural > T > T ) > code_natural > T ) ).
thf(sy_c_Code__Numeral_Onatural_Orec__set__natural,type,
code_rec_set_natural:
!>[T: $tType] : ( T > ( code_natural > T > T ) > code_natural > T > $o ) ).
thf(sy_c_Code__Numeral_Onatural__of__integer,type,
code_n4118661773612635043nteger: code_integer > code_natural ).
thf(sy_c_Code__Numeral_Onegative,type,
code_negative: num > code_integer ).
thf(sy_c_Code__Numeral_Onum__of__integer,type,
code_num_of_integer: code_integer > num ).
thf(sy_c_Code__Numeral_Opcr__integer,type,
code_pcr_integer: int > code_integer > $o ).
thf(sy_c_Code__Numeral_Opositive,type,
code_positive: num > code_integer ).
thf(sy_c_Code__Numeral_Osize__nat,type,
code_size_nat: code_natural > nat ).
thf(sy_c_Code__Numeral_Osub,type,
code_sub: num > num > code_integer ).
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_Conditionally__Complete__Lattices_Opreorder__class_Obdd__above,type,
condit941137186595557371_above:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__below,type,
condit1013018076250108175_below:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Divides_Oadjust__div,type,
adjust_div: ( product_prod @ int @ int ) > int ).
thf(sy_c_Divides_Oadjust__mod,type,
adjust_mod: 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_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_Ofiltermap,type,
filtermap:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( filter @ A ) > ( filter @ B ) ) ).
thf(sy_c_Filter_Ofinite__subsets__at__top,type,
finite5375528669736107172at_top:
!>[A: $tType] : ( ( set @ A ) > ( filter @ ( set @ A ) ) ) ).
thf(sy_c_Filter_Omap__filter__on,type,
map_filter_on:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B ) > ( filter @ A ) > ( filter @ B ) ) ).
thf(sy_c_Filter_Oprincipal,type,
principal:
!>[A: $tType] : ( ( set @ A ) > ( filter @ A ) ) ).
thf(sy_c_Filter_Oprod__filter,type,
prod_filter:
!>[A: $tType,B: $tType] : ( ( filter @ A ) > ( filter @ B ) > ( filter @ ( product_prod @ A @ B ) ) ) ).
thf(sy_c_Filter_Orel__filter,type,
rel_filter:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > ( filter @ A ) > ( filter @ B ) > $o ) ).
thf(sy_c_Finite__Set_Ocard,type,
finite_card:
!>[B: $tType] : ( ( set @ B ) > nat ) ).
thf(sy_c_Finite__Set_Ocomp__fun__commute,type,
finite6289374366891150609ommute:
!>[A: $tType,B: $tType] : ( ( A > B > B ) > $o ) ).
thf(sy_c_Finite__Set_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_Ofolding__on_OF,type,
finite_folding_F:
!>[A: $tType,B: $tType] : ( ( A > B > B ) > B > ( set @ A ) > B ) ).
thf(sy_c_Fun_Obij__betw,type,
bij_betw:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) > $o ) ).
thf(sy_c_Fun_Ocomp,type,
comp:
!>[B: $tType,C: $tType,A: $tType] : ( ( B > C ) > ( A > B ) > A > C ) ).
thf(sy_c_Fun_Ofun__upd,type,
fun_upd:
!>[A: $tType,B: $tType] : ( ( A > B ) > A > B > A > B ) ).
thf(sy_c_Fun_Oid,type,
id:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Fun_Oinj__on,type,
inj_on:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > $o ) ).
thf(sy_c_Fun_Omap__fun,type,
map_fun:
!>[C: $tType,A: $tType,B: $tType,D: $tType] : ( ( C > A ) > ( B > D ) > ( A > B ) > C > D ) ).
thf(sy_c_Fun_Ooverride__on,type,
override_on:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( A > B ) > ( set @ A ) > A > B ) ).
thf(sy_c_Fun_Othe__inv__into,type,
the_inv_into:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B ) > B > A ) ).
thf(sy_c_Fun__Def_Ois__measure,type,
fun_is_measure:
!>[A: $tType] : ( ( A > nat ) > $o ) ).
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_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_Obounded__quasi__semilattice__set,type,
bounde6485984586167503788ce_set:
!>[A: $tType] : ( ( A > A > A ) > A > A > ( A > A ) > $o ) ).
thf(sy_c_GCD_Obounded__quasi__semilattice__set_OF,type,
bounde2362111253966948842tice_F:
!>[A: $tType] : ( ( A > A > A ) > A > A > ( set @ A ) > A ) ).
thf(sy_c_GCD_Ogcd__class_Ogcd,type,
gcd_gcd:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_GCD_Ogcd__class_Olcm,type,
gcd_lcm:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_GCD_Ogcd__nat__rel,type,
gcd_nat_rel: ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) > $o ).
thf(sy_c_GCD_Osemiring__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_OEx1,type,
ex1:
!>[A: $tType] : ( ( A > $o ) > $o ) ).
thf(sy_c_HOL_ONO__MATCH,type,
nO_MATCH:
!>[A: $tType,B: $tType] : ( A > B > $o ) ).
thf(sy_c_HOL_OThe,type,
the:
!>[A: $tType] : ( ( A > $o ) > A ) ).
thf(sy_c_HOL_OUniq,type,
uniq:
!>[A: $tType] : ( ( A > $o ) > $o ) ).
thf(sy_c_HOL_Oundefined,type,
undefined:
!>[A: $tType] : A ).
thf(sy_c_Heap__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_Osize__Heap,type,
heap_Time_size_Heap:
!>[A: $tType] : ( ( A > nat ) > ( heap_Time_Heap @ A ) > nat ) ).
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_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_Ocr__int,type,
cr_int: ( product_prod @ nat @ nat ) > int > $o ).
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__max,type,
lattices_ord_arg_max:
!>[B: $tType,A: $tType] : ( ( B > A ) > ( B > $o ) > B ) ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__max__on,type,
lattic1883929316492267755max_on:
!>[B: $tType,A: $tType] : ( ( B > A ) > ( set @ B ) > B ) ).
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_Oord__class_Ois__arg__max,type,
lattic501386751176901750rg_max:
!>[B: $tType,A: $tType] : ( ( B > A ) > ( B > $o ) > B > $o ) ).
thf(sy_c_Lattices__Big_Oord__class_Ois__arg__min,type,
lattic501386751177426532rg_min:
!>[B: $tType,A: $tType] : ( ( B > A ) > ( B > $o ) > B > $o ) ).
thf(sy_c_Lattices__Big_Osemilattice__inf_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_OQuotient,type,
quotient:
!>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( A > B ) > ( B > A ) > ( A > B > $o ) > $o ) ).
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_Ogen__length,type,
gen_length:
!>[A: $tType] : ( nat > ( list @ A ) > nat ) ).
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_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_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_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_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_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_Osorted__wrt,type,
sorted_wrt:
!>[A: $tType] : ( ( A > A > $o ) > ( 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_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__comp,type,
map_comp:
!>[B: $tType,C: $tType,A: $tType] : ( ( B > ( option @ C ) ) > ( A > ( option @ B ) ) > A > ( option @ C ) ) ).
thf(sy_c_Map_Omap__le,type,
map_le:
!>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( A > ( option @ B ) ) > $o ) ).
thf(sy_c_Map_Omap__of,type,
map_of:
!>[A: $tType,B: $tType] : ( ( list @ ( product_prod @ A @ B ) ) > A > ( option @ B ) ) ).
thf(sy_c_Map_Omap__upds,type,
map_upds:
!>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( list @ A ) > ( list @ B ) > A > ( option @ B ) ) ).
thf(sy_c_Map_Oran,type,
ran:
!>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( set @ B ) ) ).
thf(sy_c_Map_Orestrict__map,type,
restrict_map:
!>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( set @ A ) > A > ( option @ B ) ) ).
thf(sy_c_Misc_OCODE__ABORT,type,
cODE_ABORT:
!>[A: $tType] : ( ( product_unit > A ) > A ) ).
thf(sy_c_Misc_OEps__Opt,type,
eps_Opt:
!>[A: $tType] : ( ( A > $o ) > ( option @ A ) ) ).
thf(sy_c_Misc_Obijective,type,
bijective:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > $o ) ).
thf(sy_c_Misc_Obrk__rel,type,
brk_rel:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ ( product_prod @ ( product_prod @ $o @ A ) @ ( product_prod @ $o @ B ) ) ) ) ).
thf(sy_c_Misc_Odflt__None__set,type,
dflt_None_set:
!>[A: $tType] : ( ( set @ A ) > ( option @ ( set @ A ) ) ) ).
thf(sy_c_Misc_Ofilter__rev,type,
filter_rev:
!>[A: $tType] : ( ( A > $o ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_Misc_Ofilter__rev__aux,type,
filter_rev_aux:
!>[A: $tType] : ( ( list @ A ) > ( A > $o ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_Misc_Ofun__of__rel,type,
fun_of_rel:
!>[B: $tType,A: $tType] : ( ( set @ ( product_prod @ B @ A ) ) > B > A ) ).
thf(sy_c_Misc_Oinv__on,type,
inv_on:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > B > A ) ).
thf(sy_c_Misc_Olist__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__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_Osum__mset,type,
comm_monoid_sum_mset:
!>[A: $tType] : ( ( A > A > A ) > A > ( multiset @ A ) > 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_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_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_ONats,type,
semiring_1_Nats:
!>[A: $tType] : ( set @ A ) ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat,type,
semiring_1_of_nat:
!>[A: $tType] : ( nat > A ) ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux,type,
semiri8178284476397505188at_aux:
!>[A: $tType] : ( ( A > A ) > nat > A > A ) ).
thf(sy_c_Nat_Osize__class_Osize,type,
size_size:
!>[A: $tType] : ( A > nat ) ).
thf(sy_c_Nat__Bijection_Olist__decode,type,
nat_list_decode: nat > ( list @ nat ) ).
thf(sy_c_Nat__Bijection_Olist__decode__rel,type,
nat_list_decode_rel: nat > nat > $o ).
thf(sy_c_Nat__Bijection_Olist__encode,type,
nat_list_encode: ( list @ nat ) > nat ).
thf(sy_c_Nat__Bijection_Olist__encode__rel,type,
nat_list_encode_rel: ( list @ nat ) > ( list @ nat ) > $o ).
thf(sy_c_Nat__Bijection_Oprod__decode,type,
nat_prod_decode: nat > ( product_prod @ nat @ nat ) ).
thf(sy_c_Nat__Bijection_Oprod__decode__aux,type,
nat_prod_decode_aux: nat > nat > ( product_prod @ nat @ nat ) ).
thf(sy_c_Nat__Bijection_Oprod__decode__aux__rel,type,
nat_pr5047031295181774490ux_rel: ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) > $o ).
thf(sy_c_Nat__Bijection_Oprod__encode,type,
nat_prod_encode: ( product_prod @ nat @ nat ) > nat ).
thf(sy_c_Nat__Bijection_Oset__decode,type,
nat_set_decode: nat > ( set @ nat ) ).
thf(sy_c_Nat__Bijection_Oset__encode,type,
nat_set_encode: ( set @ nat ) > nat ).
thf(sy_c_Nat__Bijection_Otriangle,type,
nat_triangle: nat > nat ).
thf(sy_c_Num_OBitM,type,
bitM: num > num ).
thf(sy_c_Num_Oinc,type,
inc: num > num ).
thf(sy_c_Num_Oneg__numeral__class_Odbl,type,
neg_numeral_dbl:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__dec,type,
neg_numeral_dbl_dec:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__inc,type,
neg_numeral_dbl_inc:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Num_Oneg__numeral__class_Osub,type,
neg_numeral_sub:
!>[A: $tType] : ( num > num > A ) ).
thf(sy_c_Num_Onum_OBit0,type,
bit0: num > num ).
thf(sy_c_Num_Onum_OBit1,type,
bit1: num > num ).
thf(sy_c_Num_Onum_OOne,type,
one2: num ).
thf(sy_c_Num_Onum_Ocase__num,type,
case_num:
!>[A: $tType] : ( A > ( num > A ) > ( num > A ) > num > A ) ).
thf(sy_c_Num_Onum_Orec__num,type,
rec_num:
!>[A: $tType] : ( A > ( num > A > A ) > ( num > A > A ) > num > A ) ).
thf(sy_c_Num_Onum_Osize__num,type,
size_num: num > nat ).
thf(sy_c_Num_Onum__of__nat,type,
num_of_nat: nat > num ).
thf(sy_c_Num_Onumeral__class_Onumeral,type,
numeral_numeral:
!>[A: $tType] : ( num > A ) ).
thf(sy_c_Num_Opow,type,
pow: num > num > num ).
thf(sy_c_Num_Opred__numeral,type,
pred_numeral: num > nat ).
thf(sy_c_Num_Oring__1__class_Oiszero,type,
ring_1_iszero:
!>[A: $tType] : ( A > $o ) ).
thf(sy_c_Num_Osqr,type,
sqr: num > num ).
thf(sy_c_Option_Ooption_ONone,type,
none:
!>[A: $tType] : ( option @ A ) ).
thf(sy_c_Option_Ooption_OSome,type,
some:
!>[A: $tType] : ( A > ( option @ A ) ) ).
thf(sy_c_Option_Ooption_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_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_Orelation__of,type,
order_relation_of:
!>[A: $tType] : ( ( A > A > $o ) > ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) ) ).
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_Opower,type,
power2:
!>[A: $tType] : ( A > ( A > A > A ) > A > nat > 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_Omap__prod,type,
product_map_prod:
!>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( A > C ) > ( B > D ) > ( product_prod @ A @ B ) > ( product_prod @ C @ D ) ) ).
thf(sy_c_Product__Type_Oprod_Ocase__prod,type,
product_case_prod:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
thf(sy_c_Product__Type_Oprod_Ofst,type,
product_fst:
!>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > A ) ).
thf(sy_c_Product__Type_Oprod_Osnd,type,
product_snd:
!>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > B ) ).
thf(sy_c_Product__Type_Oproduct,type,
product_product:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( set @ B ) > ( set @ ( product_prod @ A @ B ) ) ) ).
thf(sy_c_Product__Type_Oscomp,type,
product_scomp:
!>[A: $tType,B: $tType,C: $tType,D: $tType] : ( ( A > ( product_prod @ B @ C ) ) > ( B > C > D ) > A > D ) ).
thf(sy_c_Pure_Otype,type,
type2:
!>[A: $tType] : ( itself @ A ) ).
thf(sy_c_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_Orat_OAbs__rat,type,
abs_rat: ( set @ ( product_prod @ int @ int ) ) > rat ).
thf(sy_c_Rat_Orat_ORep__rat,type,
rep_rat: rat > ( set @ ( 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_Refine__Imp__Hol_Oassert_H,type,
refine_Imp_assert: $o > ( heap_Time_Heap @ product_unit ) ).
thf(sy_c_Refine__Imp__Hol_Orefines,type,
refine_Imp_refines:
!>[A: $tType] : ( ( heap_Time_Heap @ A ) > ( heap_Time_Heap @ A ) > $o ) ).
thf(sy_c_Relation_ODomainp,type,
domainp:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > A > $o ) ).
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_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_Oirrefl,type,
irrefl:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Relation_Oirreflp,type,
irreflp:
!>[A: $tType] : ( ( A > A > $o ) > $o ) ).
thf(sy_c_Relation_Orelcomp,type,
relcomp:
!>[A: $tType,B: $tType,C: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ ( product_prod @ B @ C ) ) > ( set @ ( product_prod @ A @ C ) ) ) ).
thf(sy_c_Relation_Orelcompp,type,
relcompp:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > B > $o ) > ( B > C > $o ) > A > C > $o ) ).
thf(sy_c_Relation_Osingle__valued,type,
single_valued:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > $o ) ).
thf(sy_c_Relation_Osingle__valuedp,type,
single_valuedp:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > $o ) ).
thf(sy_c_Relation_Otrans,type,
trans:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Relation_Otransp,type,
transp:
!>[A: $tType] : ( ( A > A > $o ) > $o ) ).
thf(sy_c_Rings_Oalgebraic__semidom__class_Ocoprime,type,
algebr8660921524188924756oprime:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Rings_Odivide__class_Odivide,type,
divide_divide:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Rings_Odvd__class_Odvd,type,
dvd_dvd:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Rings_Omodulo__class_Omodulo,type,
modulo_modulo:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool,type,
zero_neq_one_of_bool:
!>[A: $tType] : ( $o > A ) ).
thf(sy_c_Set_OBall,type,
ball:
!>[A: $tType] : ( ( set @ A ) > ( A > $o ) > $o ) ).
thf(sy_c_Set_OBex,type,
bex:
!>[A: $tType] : ( ( set @ A ) > ( A > $o ) > $o ) ).
thf(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_Set_OPow,type,
pow2:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( set @ A ) ) ) ).
thf(sy_c_Set_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_Othe__elem,type,
the_elem:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_Set_Ovimage,type,
vimage:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ B ) > ( set @ A ) ) ).
thf(sy_c_Set__Interval_Ofold__atLeastAtMost__nat,type,
set_fo6178422350223883121st_nat:
!>[A: $tType] : ( ( nat > A > A ) > nat > nat > A > A ) ).
thf(sy_c_Set__Interval_Ofold__atLeastAtMost__nat__rel,type,
set_fo1817059534552279752at_rel:
!>[A: $tType] : ( ( product_prod @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) ) > ( product_prod @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) ) > $o ) ).
thf(sy_c_Set__Interval_Oord__class_OatLeast,type,
set_ord_atLeast:
!>[A: $tType] : ( A > ( set @ A ) ) ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost,type,
set_or1337092689740270186AtMost:
!>[A: $tType] : ( A > A > ( set @ A ) ) ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan,type,
set_or7035219750837199246ssThan:
!>[A: $tType] : ( A > A > ( set @ A ) ) ).
thf(sy_c_Set__Interval_Oord__class_OatMost,type,
set_ord_atMost:
!>[A: $tType] : ( A > ( set @ A ) ) ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThan,type,
set_ord_greaterThan:
!>[A: $tType] : ( A > ( set @ A ) ) ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost,type,
set_or3652927894154168847AtMost:
!>[A: $tType] : ( A > A > ( set @ A ) ) ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan,type,
set_or5935395276787703475ssThan:
!>[A: $tType] : ( A > A > ( set @ A ) ) ).
thf(sy_c_Set__Interval_Oord__class_OlessThan,type,
set_ord_lessThan:
!>[A: $tType] : ( A > ( set @ A ) ) ).
thf(sy_c_String_Ochar_OChar,type,
char2: $o > $o > $o > $o > $o > $o > $o > $o > char ).
thf(sy_c_String_Ochar_Osize__char,type,
size_char: char > nat ).
thf(sy_c_String_Ochar__of__integer,type,
char_of_integer: code_integer > char ).
thf(sy_c_String_Ocomm__semiring__1__class_Oof__char,type,
comm_s6883823935334413003f_char:
!>[A: $tType] : ( char > A ) ).
thf(sy_c_String_Ointeger__of__char,type,
integer_of_char: char > code_integer ).
thf(sy_c_String_Ounique__euclidean__semiring__with__bit__operations__class_Ochar__of,type,
unique5772411509450598832har_of:
!>[A: $tType] : ( A > char ) ).
thf(sy_c_Time__Reasoning_OEQ,type,
time_EQ:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Time__Reasoning_OTBOUND,type,
time_TBOUND:
!>[A: $tType] : ( ( heap_Time_Heap @ A ) > nat > $o ) ).
thf(sy_c_Time__Reasoning_Ofails,type,
time_fails:
!>[A: $tType] : ( ( heap_Time_Heap @ A ) > ( heap_ext @ product_unit ) > $o ) ).
thf(sy_c_Time__Reasoning_Othe__heap,type,
time_the_heap:
!>[A: $tType] : ( ( heap_Time_Heap @ A ) > ( heap_ext @ product_unit ) > ( heap_ext @ product_unit ) ) ).
thf(sy_c_Time__Reasoning_Othe__res,type,
time_the_res:
!>[A: $tType] : ( ( heap_Time_Heap @ A ) > ( heap_ext @ product_unit ) > A ) ).
thf(sy_c_Time__Reasoning_Otime,type,
time_time:
!>[A: $tType] : ( ( heap_Time_Heap @ A ) > ( heap_ext @ product_unit ) > nat ) ).
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_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_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_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_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_fChoice,type,
fChoice:
!>[A: $tType] : ( ( A > $o ) > A ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_P,type,
p: $o ).
thf(sy_v_h,type,
h: heap_ext @ product_unit ).
thf(sy_v_m,type,
m: heap_Time_Heap @ a ).
thf(sy_v_t,type,
t: nat ).
% Relevant facts (8180)
thf(fact_0__092_060open_062_092_060And_062h_O_AP_A_092_060Longrightarrow_062_Atime_Am_Ah_A_092_060le_062_At_092_060close_062,axiom,
! [H: heap_ext @ product_unit] :
( p
=> ( ord_less_eq @ nat @ ( time_time @ a @ m @ H ) @ t ) ) ).
% \<open>\<And>h. P \<Longrightarrow> time m h \<le> t\<close>
thf(fact_1_assms,axiom,
( p
=> ( time_TBOUND @ a @ m @ t ) ) ).
% assms
thf(fact_2_Heap__Time__Monad_Obind__bind,axiom,
! [C: $tType,B: $tType,A: $tType,F: heap_Time_Heap @ A,G: A > ( heap_Time_Heap @ C ),K: C > ( heap_Time_Heap @ B )] :
( ( heap_Time_bind @ C @ B @ ( heap_Time_bind @ A @ C @ F @ G ) @ K )
= ( heap_Time_bind @ A @ B @ F
@ ^ [X: A] : ( heap_Time_bind @ C @ B @ ( G @ X ) @ K ) ) ) ).
% Heap_Time_Monad.bind_bind
thf(fact_3_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A] : ( ord_less_eq @ A @ X2 @ X2 ) ) ).
% order_refl
thf(fact_4_dual__order_Orefl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% dual_order.refl
thf(fact_5_distrib__if__bind,axiom,
! [A: $tType,B: $tType,B2: $o,C2: heap_Time_Heap @ B,D2: heap_Time_Heap @ B,F: B > ( heap_Time_Heap @ A )] :
( ( B2
=> ( ( heap_Time_bind @ B @ A @ ( if @ ( heap_Time_Heap @ B ) @ B2 @ C2 @ D2 ) @ F )
= ( heap_Time_bind @ B @ A @ C2 @ F ) ) )
& ( ~ B2
=> ( ( heap_Time_bind @ B @ A @ ( if @ ( heap_Time_Heap @ B ) @ B2 @ C2 @ D2 ) @ F )
= ( heap_Time_bind @ B @ A @ D2 @ F ) ) ) ) ).
% distrib_if_bind
thf(fact_6_TBOUNDD,axiom,
! [A: $tType,M: heap_Time_Heap @ A,T2: nat,H: heap_ext @ product_unit] :
( ( time_TBOUND @ A @ M @ T2 )
=> ( ord_less_eq @ nat @ ( time_time @ A @ M @ H ) @ T2 ) ) ).
% TBOUNDD
thf(fact_7_TBOUNDI,axiom,
! [A: $tType,M: heap_Time_Heap @ A,T2: nat] :
( ! [H2: heap_ext @ product_unit] : ( ord_less_eq @ nat @ ( time_time @ A @ M @ H2 ) @ T2 )
=> ( time_TBOUND @ A @ M @ T2 ) ) ).
% TBOUNDI
thf(fact_8_TBOUND__def,axiom,
! [A: $tType] :
( ( time_TBOUND @ A )
= ( ^ [M2: heap_Time_Heap @ A,T3: nat] :
! [H3: heap_ext @ product_unit] : ( ord_less_eq @ nat @ ( time_time @ A @ M2 @ H3 ) @ T3 ) ) ) ).
% TBOUND_def
thf(fact_9_le__refl,axiom,
! [N: nat] : ( ord_less_eq @ nat @ N @ N ) ).
% le_refl
thf(fact_10_le__trans,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( ord_less_eq @ nat @ J @ K )
=> ( ord_less_eq @ nat @ I @ K ) ) ) ).
% le_trans
thf(fact_11_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq @ nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_12_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( ord_less_eq @ nat @ N @ M )
=> ( M = N ) ) ) ).
% le_antisym
thf(fact_13_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
| ( ord_less_eq @ nat @ N @ M ) ) ).
% nat_le_linear
thf(fact_14_TBOUND__mono,axiom,
! [A: $tType,C2: heap_Time_Heap @ A,T2: nat,T4: nat] :
( ( time_TBOUND @ A @ C2 @ T2 )
=> ( ( ord_less_eq @ nat @ T2 @ T4 )
=> ( time_TBOUND @ A @ C2 @ T4 ) ) ) ).
% TBOUND_mono
thf(fact_15_TBOUND__eqI,axiom,
! [A: $tType,M: heap_Time_Heap @ A,T2: nat] :
( ! [H2: heap_ext @ product_unit] :
( ( time_time @ A @ M @ H2 )
= T2 )
=> ( time_TBOUND @ A @ M @ T2 ) ) ).
% TBOUND_eqI
thf(fact_16_order__antisym__conv,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y: A,X2: A] :
( ( ord_less_eq @ A @ Y @ X2 )
=> ( ( ord_less_eq @ A @ X2 @ Y )
= ( X2 = Y ) ) ) ) ).
% order_antisym_conv
thf(fact_17_linorder__le__cases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y: A] :
( ~ ( ord_less_eq @ A @ X2 @ Y )
=> ( ord_less_eq @ A @ Y @ X2 ) ) ) ).
% linorder_le_cases
thf(fact_18_ord__le__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A2: A,B2: A,F: A > B,C2: B] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X3: A,Y2: A] :
( ( ord_less_eq @ A @ X3 @ Y2 )
=> ( ord_less_eq @ B @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq @ B @ ( F @ A2 ) @ C2 ) ) ) ) ) ).
% ord_le_eq_subst
thf(fact_19_ord__eq__le__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A2: A,F: B > A,B2: B,C2: B] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq @ B @ B2 @ C2 )
=> ( ! [X3: B,Y2: B] :
( ( ord_less_eq @ B @ X3 @ Y2 )
=> ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F @ C2 ) ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_20_linorder__linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y: A] :
( ( ord_less_eq @ A @ X2 @ Y )
| ( ord_less_eq @ A @ Y @ X2 ) ) ) ).
% linorder_linear
thf(fact_21_order__eq__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A,Y: A] :
( ( X2 = Y )
=> ( ord_less_eq @ A @ X2 @ Y ) ) ) ).
% order_eq_refl
thf(fact_22_order__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A2: A,B2: A,F: A > C,C2: C] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ C @ ( F @ B2 ) @ C2 )
=> ( ! [X3: A,Y2: A] :
( ( ord_less_eq @ A @ X3 @ Y2 )
=> ( ord_less_eq @ C @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq @ C @ ( F @ A2 ) @ C2 ) ) ) ) ) ).
% order_subst2
thf(fact_23_order__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A2: A,F: B > A,B2: B,C2: B] :
( ( ord_less_eq @ A @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq @ B @ B2 @ C2 )
=> ( ! [X3: B,Y2: B] :
( ( ord_less_eq @ B @ X3 @ Y2 )
=> ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F @ C2 ) ) ) ) ) ) ).
% order_subst1
thf(fact_24_Orderings_Oorder__eq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y3: A,Z: A] : Y3 = Z )
= ( ^ [A3: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
& ( ord_less_eq @ A @ B3 @ A3 ) ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_25_le__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ( ( ord_less_eq @ ( A > B ) )
= ( ^ [F2: A > B,G2: A > B] :
! [X: A] : ( ord_less_eq @ B @ ( F2 @ X ) @ ( G2 @ X ) ) ) ) ) ).
% le_fun_def
thf(fact_26_le__funI,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F: A > B,G: A > B] :
( ! [X3: A] : ( ord_less_eq @ B @ ( F @ X3 ) @ ( G @ X3 ) )
=> ( ord_less_eq @ ( A > B ) @ F @ G ) ) ) ).
% le_funI
thf(fact_27_le__funE,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F: A > B,G: A > B,X2: A] :
( ( ord_less_eq @ ( A > B ) @ F @ G )
=> ( ord_less_eq @ B @ ( F @ X2 ) @ ( G @ X2 ) ) ) ) ).
% le_funE
thf(fact_28_le__funD,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F: A > B,G: A > B,X2: A] :
( ( ord_less_eq @ ( A > B ) @ F @ G )
=> ( ord_less_eq @ B @ ( F @ X2 ) @ ( G @ X2 ) ) ) ) ).
% le_funD
thf(fact_29_antisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ) ).
% antisym
thf(fact_30_dual__order_Otrans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ C2 @ B2 )
=> ( ord_less_eq @ A @ C2 @ A2 ) ) ) ) ).
% dual_order.trans
thf(fact_31_dual__order_Oantisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ) ).
% dual_order.antisym
thf(fact_32_dual__order_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y3: A,Z: A] : Y3 = Z )
= ( ^ [A3: A,B3: A] :
( ( ord_less_eq @ A @ B3 @ A3 )
& ( ord_less_eq @ A @ A3 @ B3 ) ) ) ) ) ).
% dual_order.eq_iff
thf(fact_33_linorder__wlog,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > A > $o,A2: A,B2: A] :
( ! [A4: A,B4: A] :
( ( ord_less_eq @ A @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: A,B4: A] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A2 @ B2 ) ) ) ) ).
% linorder_wlog
thf(fact_34_order__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A,Y: A,Z2: A] :
( ( ord_less_eq @ A @ X2 @ Y )
=> ( ( ord_less_eq @ A @ Y @ Z2 )
=> ( ord_less_eq @ A @ X2 @ Z2 ) ) ) ) ).
% order_trans
thf(fact_35_order_Otrans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ C2 )
=> ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% order.trans
thf(fact_36_order__antisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X2: A,Y: A] :
( ( ord_less_eq @ A @ X2 @ Y )
=> ( ( ord_less_eq @ A @ Y @ X2 )
=> ( X2 = Y ) ) ) ) ).
% order_antisym
thf(fact_37_ord__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( B2 = C2 )
=> ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% ord_le_eq_trans
thf(fact_38_ord__eq__le__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( A2 = B2 )
=> ( ( ord_less_eq @ A @ B2 @ C2 )
=> ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% ord_eq_le_trans
thf(fact_39_order__class_Oorder__eq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y3: A,Z: A] : Y3 = Z )
= ( ^ [X: A,Y4: A] :
( ( ord_less_eq @ A @ X @ Y4 )
& ( ord_less_eq @ A @ Y4 @ X ) ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_40_le__cases3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y: A,Z2: A] :
( ( ( ord_less_eq @ A @ X2 @ Y )
=> ~ ( ord_less_eq @ A @ Y @ Z2 ) )
=> ( ( ( ord_less_eq @ A @ Y @ X2 )
=> ~ ( ord_less_eq @ A @ X2 @ Z2 ) )
=> ( ( ( ord_less_eq @ A @ X2 @ Z2 )
=> ~ ( ord_less_eq @ A @ Z2 @ Y ) )
=> ( ( ( ord_less_eq @ A @ Z2 @ Y )
=> ~ ( ord_less_eq @ A @ Y @ X2 ) )
=> ( ( ( ord_less_eq @ A @ Y @ Z2 )
=> ~ ( ord_less_eq @ A @ Z2 @ X2 ) )
=> ~ ( ( ord_less_eq @ A @ Z2 @ X2 )
=> ~ ( ord_less_eq @ A @ X2 @ Y ) ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_41_nle__le,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A] :
( ( ~ ( ord_less_eq @ A @ A2 @ B2 ) )
= ( ( ord_less_eq @ A @ B2 @ A2 )
& ( B2 != A2 ) ) ) ) ).
% nle_le
thf(fact_42_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B2: nat] :
( ( P @ K )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq @ nat @ Y2 @ B2 ) )
=> ? [X3: nat] :
( ( P @ X3 )
& ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq @ nat @ Y5 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_43_raise__bind,axiom,
! [B: $tType,A: $tType,E2: list @ char,F: B > ( heap_Time_Heap @ A )] :
( ( heap_Time_bind @ B @ A @ ( heap_Time_raise @ B @ E2 ) @ F )
= ( heap_Time_raise @ A @ E2 ) ) ).
% raise_bind
thf(fact_44_Greatest__equality,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P: A > $o,X2: A] :
( ( P @ X2 )
=> ( ! [Y2: A] :
( ( P @ Y2 )
=> ( ord_less_eq @ A @ Y2 @ X2 ) )
=> ( ( order_Greatest @ A @ P )
= X2 ) ) ) ) ).
% Greatest_equality
thf(fact_45_mem__Collect__eq,axiom,
! [A: $tType,A2: A,P: A > $o] :
( ( member @ A @ A2 @ ( collect @ A @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A: $tType,A5: set @ A] :
( ( collect @ A
@ ^ [X: A] : ( member @ A @ X @ 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,F: A > B,G: A > B] :
( ! [X3: A] :
( ( F @ X3 )
= ( G @ X3 ) )
=> ( F = G ) ) ).
% ext
thf(fact_49_GreatestI2__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P: A > $o,X2: A,Q: A > $o] :
( ( P @ X2 )
=> ( ! [Y2: A] :
( ( P @ Y2 )
=> ( ord_less_eq @ A @ Y2 @ X2 ) )
=> ( ! [X3: A] :
( ( P @ X3 )
=> ( ! [Y5: A] :
( ( P @ Y5 )
=> ( ord_less_eq @ A @ Y5 @ X3 ) )
=> ( Q @ X3 ) ) )
=> ( Q @ ( order_Greatest @ A @ P ) ) ) ) ) ) ).
% GreatestI2_order
thf(fact_50_TBOUND__assert_H__bind,axiom,
! [A: $tType,P: $o,P2: $o,M: heap_Time_Heap @ A,T2: nat] :
( ( time_EQ @ $o @ P @ P2 )
=> ( ( P
=> ( time_TBOUND @ A @ M @ T2 ) )
=> ( time_TBOUND @ A
@ ( heap_Time_bind @ product_unit @ A @ ( refine_Imp_assert @ P )
@ ^ [Uu: product_unit] : M )
@ ( if @ nat @ P2 @ T2 @ ( zero_zero @ nat ) ) ) ) ) ).
% TBOUND_assert'_bind
thf(fact_51_ex__has__least__nat,axiom,
! [A: $tType,P: A > $o,K: A,M: A > nat] :
( ( P @ K )
=> ? [X3: A] :
( ( P @ X3 )
& ! [Y5: A] :
( ( P @ Y5 )
=> ( ord_less_eq @ nat @ ( M @ X3 ) @ ( M @ Y5 ) ) ) ) ) ).
% ex_has_least_nat
thf(fact_52_bounded__Max__nat,axiom,
! [P: nat > $o,X2: nat,M3: nat] :
( ( P @ X2 )
=> ( ! [X3: nat] :
( ( P @ X3 )
=> ( ord_less_eq @ nat @ X3 @ M3 ) )
=> ~ ! [M4: nat] :
( ( P @ M4 )
=> ~ ! [X4: nat] :
( ( P @ X4 )
=> ( ord_less_eq @ nat @ X4 @ M4 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_53_le__rel__bool__arg__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_less_eq @ ( $o > A ) )
= ( ^ [X5: $o > A,Y6: $o > A] :
( ( ord_less_eq @ A @ ( X5 @ $false ) @ ( Y6 @ $false ) )
& ( ord_less_eq @ A @ ( X5 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_54_verit__la__disequality,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A] :
( ( A2 = B2 )
| ~ ( ord_less_eq @ A @ A2 @ B2 )
| ~ ( ord_less_eq @ A @ B2 @ A2 ) ) ) ).
% verit_la_disequality
thf(fact_55_ord__eq__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( A2 = B2 )
=> ( ( ord_less_eq @ A @ B2 @ C2 )
=> ( ( C2 = D2 )
=> ( ord_less_eq @ A @ A2 @ D2 ) ) ) ) ) ).
% ord_eq_le_eq_trans
thf(fact_56_verit__comp__simplify1_I2_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% verit_comp_simplify1(2)
thf(fact_57_refines__assert_H__bind,axiom,
! [A: $tType,P3: heap_Time_Heap @ A,Q2: heap_Time_Heap @ A,Phi: $o] :
( ( refine_Imp_refines @ A @ P3 @ Q2 )
=> ( refine_Imp_refines @ A @ P3
@ ( heap_Time_bind @ product_unit @ A @ ( refine_Imp_assert @ Phi )
@ ^ [Uu: product_unit] : Q2 ) ) ) ).
% refines_assert'_bind
thf(fact_58_ureturn__bind,axiom,
! [A: $tType,B: $tType,X2: B,F: B > ( heap_Time_Heap @ A )] :
( ( heap_Time_bind @ B @ A @ ( heap_Time_ureturn @ B @ X2 ) @ F )
= ( F @ X2 ) ) ).
% ureturn_bind
thf(fact_59_bot__nat__0_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ A2 ) ).
% bot_nat_0.extremum
thf(fact_60_le0,axiom,
! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% le0
thf(fact_61_refines__refl,axiom,
! [A: $tType,P3: heap_Time_Heap @ A] : ( refine_Imp_refines @ A @ P3 @ P3 ) ).
% refines_refl
thf(fact_62_refines__If,axiom,
! [A: $tType,B2: $o,T2: heap_Time_Heap @ A,T4: heap_Time_Heap @ A,E2: heap_Time_Heap @ A,E3: heap_Time_Heap @ A] :
( ( B2
=> ( refine_Imp_refines @ A @ T2 @ T4 ) )
=> ( ( ~ B2
=> ( refine_Imp_refines @ A @ E2 @ E3 ) )
=> ( refine_Imp_refines @ A @ ( if @ ( heap_Time_Heap @ A ) @ B2 @ T2 @ E2 ) @ ( if @ ( heap_Time_Heap @ A ) @ B2 @ T4 @ E3 ) ) ) ) ).
% refines_If
thf(fact_63_EQD,axiom,
! [A: $tType,A2: A,B2: A] :
( ( time_EQ @ A @ A2 @ B2 )
=> ( A2 = B2 ) ) ).
% EQD
thf(fact_64_EQI,axiom,
! [A: $tType,A2: A,B2: A] :
( ( A2 = B2 )
=> ( time_EQ @ A @ A2 @ B2 ) ) ).
% EQI
thf(fact_65_EQ__def,axiom,
! [A: $tType] :
( ( time_EQ @ A )
= ( ^ [Y3: A,Z: A] : Y3 = Z ) ) ).
% EQ_def
thf(fact_66_EQ__refl,axiom,
! [A: $tType,A2: A] : ( time_EQ @ A @ A2 @ A2 ) ).
% EQ_refl
thf(fact_67_time__raise,axiom,
! [A: $tType,Msg: list @ char,H: heap_ext @ product_unit] :
( ( time_time @ A @ ( heap_Time_raise @ A @ Msg ) @ H )
= ( zero_zero @ nat ) ) ).
% time_raise
thf(fact_68_refines__bind,axiom,
! [B: $tType,A: $tType,M: heap_Time_Heap @ A,M5: heap_Time_Heap @ A,F: A > ( heap_Time_Heap @ B ),F3: A > ( heap_Time_Heap @ B )] :
( ( refine_Imp_refines @ A @ M @ M5 )
=> ( ! [X3: A] : ( refine_Imp_refines @ B @ ( F @ X3 ) @ ( F3 @ X3 ) )
=> ( refine_Imp_refines @ B @ ( heap_Time_bind @ A @ B @ M @ F ) @ ( heap_Time_bind @ A @ B @ M5 @ F3 ) ) ) ) ).
% refines_bind
thf(fact_69_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% less_eq_nat.simps(1)
thf(fact_70_bot__nat__0_Oextremum__unique,axiom,
! [A2: nat] :
( ( ord_less_eq @ nat @ A2 @ ( zero_zero @ nat ) )
= ( A2
= ( zero_zero @ nat ) ) ) ).
% bot_nat_0.extremum_unique
thf(fact_71_bot__nat__0_Oextremum__uniqueI,axiom,
! [A2: nat] :
( ( ord_less_eq @ nat @ A2 @ ( zero_zero @ nat ) )
=> ( A2
= ( zero_zero @ nat ) ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_72_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq @ nat @ N @ ( zero_zero @ nat ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% le_0_eq
thf(fact_73_time__assert_H,axiom,
! [P: $o,H: heap_ext @ product_unit] :
( ( time_time @ product_unit @ ( refine_Imp_assert @ P ) @ H )
= ( zero_zero @ nat ) ) ).
% time_assert'
thf(fact_74_TBOUND__refines,axiom,
! [A: $tType,C2: heap_Time_Heap @ A,T2: nat,D2: heap_Time_Heap @ A] :
( ( time_TBOUND @ A @ C2 @ T2 )
=> ( ( refine_Imp_refines @ A @ C2 @ D2 )
=> ( time_TBOUND @ A @ D2 @ T2 ) ) ) ).
% TBOUND_refines
thf(fact_75_GreatestI__nat,axiom,
! [P: nat > $o,K: nat,B2: nat] :
( ( P @ K )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq @ nat @ Y2 @ B2 ) )
=> ( P @ ( order_Greatest @ nat @ P ) ) ) ) ).
% GreatestI_nat
thf(fact_76_Greatest__le__nat,axiom,
! [P: nat > $o,K: nat,B2: nat] :
( ( P @ K )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq @ nat @ Y2 @ B2 ) )
=> ( ord_less_eq @ nat @ K @ ( order_Greatest @ nat @ P ) ) ) ) ).
% Greatest_le_nat
thf(fact_77_GreatestI__ex__nat,axiom,
! [P: nat > $o,B2: nat] :
( ? [X_1: nat] : ( P @ X_1 )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq @ nat @ Y2 @ B2 ) )
=> ( P @ ( order_Greatest @ nat @ P ) ) ) ) ).
% GreatestI_ex_nat
thf(fact_78_bind__ureturn,axiom,
! [A: $tType,F: heap_Time_Heap @ A] :
( ( heap_Time_bind @ A @ A @ F @ ( heap_Time_ureturn @ A ) )
= F ) ).
% bind_ureturn
thf(fact_79_le__zero__eq,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [N: A] :
( ( ord_less_eq @ A @ N @ ( zero_zero @ A ) )
= ( N
= ( zero_zero @ A ) ) ) ) ).
% le_zero_eq
thf(fact_80_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_81_zero__le,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [X2: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X2 ) ) ).
% zero_le
thf(fact_82_time__refines,axiom,
! [A: $tType,C2: heap_Time_Heap @ A,C3: heap_Time_Heap @ A,H: heap_ext @ product_unit] :
( ( refine_Imp_refines @ A @ C2 @ C3 )
=> ( ~ ( time_fails @ A @ C3 @ H )
=> ( ord_less_eq @ nat @ ( time_time @ A @ C2 @ H ) @ ( time_time @ A @ C3 @ H ) ) ) ) ).
% time_refines
thf(fact_83_fun__cong__unused__0,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( zero @ B )
=> ! [F: ( A > B ) > C,G: C] :
( ( F
= ( ^ [X: A > B] : G ) )
=> ( ( F
@ ^ [X: A] : ( zero_zero @ B ) )
= G ) ) ) ).
% fun_cong_unused_0
thf(fact_84_is__arg__max__linorder,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ( ( lattic501386751176901750rg_max @ A @ B )
= ( ^ [F2: A > B,P4: A > $o,X: A] :
( ( P4 @ X )
& ! [Y4: A] :
( ( P4 @ Y4 )
=> ( ord_less_eq @ B @ ( F2 @ Y4 ) @ ( F2 @ X ) ) ) ) ) ) ) ).
% is_arg_max_linorder
thf(fact_85_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_86_zero__natural_Orsp,axiom,
( ( zero_zero @ nat )
= ( zero_zero @ nat ) ) ).
% zero_natural.rsp
thf(fact_87_zero__reorient,axiom,
! [A: $tType] :
( ( zero @ A )
=> ! [X2: A] :
( ( ( zero_zero @ A )
= X2 )
= ( X2
= ( zero_zero @ A ) ) ) ) ).
% zero_reorient
thf(fact_88_of__nat__aux_Osimps_I1_J,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [Inc: A > A,I: A] :
( ( semiri8178284476397505188at_aux @ A @ Inc @ ( zero_zero @ nat ) @ I )
= I ) ) ).
% of_nat_aux.simps(1)
thf(fact_89_arg__max__equality,axiom,
! [A: $tType,C: $tType] :
( ( order @ A )
=> ! [P: C > $o,K: C,F: C > A] :
( ( P @ K )
=> ( ! [X3: C] :
( ( P @ X3 )
=> ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ K ) ) )
=> ( ( F @ ( lattices_ord_arg_max @ C @ A @ F @ P ) )
= ( F @ K ) ) ) ) ) ).
% arg_max_equality
thf(fact_90_of__nat__eq__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [M: nat,N: nat] :
( ( ( semiring_1_of_nat @ A @ M )
= ( semiring_1_of_nat @ A @ N ) )
= ( M = N ) ) ) ).
% of_nat_eq_iff
thf(fact_91_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_92_of__nat__0__eq__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [N: nat] :
( ( ( zero_zero @ A )
= ( semiring_1_of_nat @ A @ N ) )
= ( ( zero_zero @ nat )
= N ) ) ) ).
% of_nat_0_eq_iff
thf(fact_93_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_94_of__nat__le__iff,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [M: nat,N: nat] :
( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) )
= ( ord_less_eq @ nat @ M @ N ) ) ) ).
% of_nat_le_iff
thf(fact_95_int__ops_I1_J,axiom,
( ( semiring_1_of_nat @ int @ ( zero_zero @ nat ) )
= ( zero_zero @ int ) ) ).
% int_ops(1)
thf(fact_96_nat__int__comparison_I3_J,axiom,
( ( ord_less_eq @ nat )
= ( ^ [A3: nat,B3: nat] : ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ A3 ) @ ( semiring_1_of_nat @ int @ B3 ) ) ) ) ).
% nat_int_comparison(3)
thf(fact_97_nat__leq__as__int,axiom,
( ( ord_less_eq @ nat )
= ( ^ [A3: nat,B3: nat] : ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ A3 ) @ ( semiring_1_of_nat @ int @ B3 ) ) ) ) ).
% nat_leq_as_int
thf(fact_98_of__nat__0__le__iff,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [N: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( semiring_1_of_nat @ A @ N ) ) ) ).
% of_nat_0_le_iff
thf(fact_99_of__nat__mono,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [I: nat,J: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ I ) @ ( semiring_1_of_nat @ A @ J ) ) ) ) ).
% of_nat_mono
thf(fact_100_fails__refines,axiom,
! [A: $tType,C2: heap_Time_Heap @ A,C3: heap_Time_Heap @ A,H: heap_ext @ product_unit] :
( ( refine_Imp_refines @ A @ C2 @ C3 )
=> ( ( time_fails @ A @ C2 @ H )
=> ( time_fails @ A @ C3 @ H ) ) ) ).
% fails_refines
thf(fact_101_fails__assert_H,axiom,
! [P: $o,H: heap_ext @ product_unit] :
( ( time_fails @ product_unit @ ( refine_Imp_assert @ P ) @ H )
= ~ P ) ).
% fails_assert'
thf(fact_102_arg__max__on__def,axiom,
! [A: $tType,B: $tType] :
( ( ord @ A )
=> ( ( lattic1883929316492267755max_on @ B @ A )
= ( ^ [F2: B > A,S: set @ B] :
( lattices_ord_arg_max @ B @ A @ F2
@ ^ [X: B] : ( member @ B @ X @ S ) ) ) ) ) ).
% arg_max_on_def
thf(fact_103_real__arch__simple,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X2: A] :
? [N2: nat] : ( ord_less_eq @ A @ X2 @ ( semiring_1_of_nat @ A @ N2 ) ) ) ).
% real_arch_simple
thf(fact_104_time__wait,axiom,
! [N: nat,H: heap_ext @ product_unit] :
( ( time_time @ product_unit @ ( heap_Time_wait @ N ) @ H )
= N ) ).
% time_wait
thf(fact_105_arg__max__def,axiom,
! [A: $tType,B: $tType] :
( ( ord @ A )
=> ( ( lattices_ord_arg_max @ B @ A )
= ( ^ [F2: B > A,P4: B > $o] : ( fChoice @ B @ ( lattic501386751176901750rg_max @ B @ A @ F2 @ P4 ) ) ) ) ) ).
% arg_max_def
thf(fact_106_zle__int,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ M ) @ ( semiring_1_of_nat @ int @ N ) )
= ( ord_less_eq @ nat @ M @ N ) ) ).
% zle_int
thf(fact_107_dbl__simps_I2_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( neg_numeral_dbl @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% dbl_simps(2)
thf(fact_108_fails__bind,axiom,
! [A: $tType,B: $tType,M: heap_Time_Heap @ B,F: B > ( heap_Time_Heap @ A ),H: heap_ext @ product_unit] :
( ( time_fails @ A @ ( heap_Time_bind @ B @ A @ M @ F ) @ H )
= ( ~ ( time_fails @ B @ M @ H )
=> ( time_fails @ A @ ( F @ ( time_the_res @ B @ M @ H ) ) @ ( time_the_heap @ B @ M @ H ) ) ) ) ).
% fails_bind
thf(fact_109_Heap_Osize__neq,axiom,
! [A: $tType,X2: heap_Time_Heap @ A] :
( ( size_size @ ( heap_Time_Heap @ A ) @ X2 )
!= ( zero_zero @ nat ) ) ).
% Heap.size_neq
thf(fact_110_TBOUND__bind,axiom,
! [A: $tType,B: $tType,M: heap_Time_Heap @ A,T_1: nat,F: A > ( heap_Time_Heap @ B ),T_2: nat] :
( ( time_TBOUND @ A @ M @ T_1 )
=> ( ! [X3: A,H2: heap_ext @ product_unit] :
( ( X3
= ( time_the_res @ A @ M @ H2 ) )
=> ( ~ ( time_fails @ A @ M @ H2 )
=> ( time_TBOUND @ B @ ( F @ X3 ) @ T_2 ) ) )
=> ( time_TBOUND @ B @ ( heap_Time_bind @ A @ B @ M @ F ) @ ( plus_plus @ nat @ T_1 @ T_2 ) ) ) ) ).
% TBOUND_bind
thf(fact_111_timeFrame__zero,axiom,
! [A: $tType,H: option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )] :
( ( heap_Time_timeFrame @ A @ ( zero_zero @ nat ) @ H )
= H ) ).
% timeFrame_zero
thf(fact_112_arg__min__equality,axiom,
! [A: $tType,C: $tType] :
( ( order @ A )
=> ! [P: C > $o,K: C,F: C > A] :
( ( P @ K )
=> ( ! [X3: C] :
( ( P @ X3 )
=> ( ord_less_eq @ A @ ( F @ K ) @ ( F @ X3 ) ) )
=> ( ( F @ ( lattices_ord_arg_min @ C @ A @ F @ P ) )
= ( F @ K ) ) ) ) ) ).
% arg_min_equality
thf(fact_113_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_114_add__left__cancel,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ( plus_plus @ A @ A2 @ B2 )
= ( plus_plus @ A @ A2 @ C2 ) )
= ( B2 = C2 ) ) ) ).
% add_left_cancel
thf(fact_115_add__right__cancel,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( ( plus_plus @ A @ B2 @ A2 )
= ( plus_plus @ A @ C2 @ A2 ) )
= ( B2 = C2 ) ) ) ).
% add_right_cancel
thf(fact_116_add__le__cancel__left,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ C2 @ A2 ) @ ( plus_plus @ A @ C2 @ B2 ) )
= ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% add_le_cancel_left
thf(fact_117_add__le__cancel__right,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) )
= ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% add_le_cancel_right
thf(fact_118_double__eq__0__iff,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A2: A] :
( ( ( plus_plus @ A @ A2 @ A2 )
= ( zero_zero @ A ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% double_eq_0_iff
thf(fact_119_add__0,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ! [A2: A] :
( ( plus_plus @ A @ ( zero_zero @ A ) @ A2 )
= A2 ) ) ).
% add_0
thf(fact_120_zero__eq__add__iff__both__eq__0,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [X2: A,Y: A] :
( ( ( zero_zero @ A )
= ( plus_plus @ A @ X2 @ Y ) )
= ( ( X2
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_121_add__eq__0__iff__both__eq__0,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [X2: A,Y: A] :
( ( ( plus_plus @ A @ X2 @ Y )
= ( zero_zero @ A ) )
= ( ( X2
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_122_add__cancel__right__right,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [A2: A,B2: A] :
( ( A2
= ( plus_plus @ A @ A2 @ B2 ) )
= ( B2
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_right_right
thf(fact_123_add__cancel__right__left,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [A2: A,B2: A] :
( ( A2
= ( plus_plus @ A @ B2 @ A2 ) )
= ( B2
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_right_left
thf(fact_124_add__cancel__left__right,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [A2: A,B2: A] :
( ( ( plus_plus @ A @ A2 @ B2 )
= A2 )
= ( B2
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_left_right
thf(fact_125_add__cancel__left__left,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [B2: A,A2: A] :
( ( ( plus_plus @ A @ B2 @ A2 )
= A2 )
= ( B2
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_left_left
thf(fact_126_double__zero__sym,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A2: A] :
( ( ( zero_zero @ A )
= ( plus_plus @ A @ A2 @ A2 ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% double_zero_sym
thf(fact_127_add_Oright__neutral,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ! [A2: A] :
( ( plus_plus @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% add.right_neutral
thf(fact_128_add__is__0,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus @ nat @ M @ N )
= ( zero_zero @ nat ) )
= ( ( M
= ( zero_zero @ nat ) )
& ( N
= ( zero_zero @ nat ) ) ) ) ).
% add_is_0
thf(fact_129_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus @ nat @ M @ ( zero_zero @ nat ) )
= M ) ).
% Nat.add_0_right
thf(fact_130_nat__add__left__cancel__le,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ K @ M ) @ ( plus_plus @ nat @ K @ N ) )
= ( ord_less_eq @ nat @ M @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_131_timeFrame__assoc,axiom,
! [A: $tType,N: nat,N3: nat,F: option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )] :
( ( heap_Time_timeFrame @ A @ N @ ( heap_Time_timeFrame @ A @ N3 @ F ) )
= ( heap_Time_timeFrame @ A @ ( plus_plus @ nat @ N @ N3 ) @ F ) ) ).
% timeFrame_assoc
thf(fact_132_add__le__same__cancel1,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ B2 @ A2 ) @ B2 )
= ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% add_le_same_cancel1
thf(fact_133_add__le__same__cancel2,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ B2 ) @ B2 )
= ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% add_le_same_cancel2
thf(fact_134_le__add__same__cancel1,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ ( plus_plus @ A @ A2 @ B2 ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) ) ).
% le_add_same_cancel1
thf(fact_135_le__add__same__cancel2,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ ( plus_plus @ A @ B2 @ A2 ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) ) ).
% le_add_same_cancel2
thf(fact_136_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ A2 ) @ ( zero_zero @ A ) )
= ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_137_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ A2 ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_138_of__nat__add,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [M: nat,N: nat] :
( ( semiring_1_of_nat @ A @ ( plus_plus @ nat @ M @ N ) )
= ( plus_plus @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% of_nat_add
thf(fact_139_less__eq__int__code_I1_J,axiom,
ord_less_eq @ int @ ( zero_zero @ int ) @ ( zero_zero @ int ) ).
% less_eq_int_code(1)
thf(fact_140_nonneg__int__cases,axiom,
! [K: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
=> ~ ! [N2: nat] :
( K
!= ( semiring_1_of_nat @ int @ N2 ) ) ) ).
% nonneg_int_cases
thf(fact_141_zero__le__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
=> ? [N2: nat] :
( K
= ( semiring_1_of_nat @ int @ N2 ) ) ) ).
% zero_le_imp_eq_int
thf(fact_142_int__if,axiom,
! [P: $o,A2: nat,B2: nat] :
( ( P
=> ( ( semiring_1_of_nat @ int @ ( if @ nat @ P @ A2 @ B2 ) )
= ( semiring_1_of_nat @ int @ A2 ) ) )
& ( ~ P
=> ( ( semiring_1_of_nat @ int @ ( if @ nat @ P @ A2 @ B2 ) )
= ( semiring_1_of_nat @ int @ B2 ) ) ) ) ).
% int_if
thf(fact_143_zero__integer_Orsp,axiom,
( ( zero_zero @ int )
= ( zero_zero @ int ) ) ).
% zero_integer.rsp
thf(fact_144_nat__int__comparison_I1_J,axiom,
( ( ^ [Y3: nat,Z: nat] : Y3 = Z )
= ( ^ [A3: nat,B3: nat] :
( ( semiring_1_of_nat @ int @ A3 )
= ( semiring_1_of_nat @ int @ B3 ) ) ) ) ).
% nat_int_comparison(1)
thf(fact_145_is__num__normalize_I1_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 )
= ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% is_num_normalize(1)
thf(fact_146_add__mono__thms__linordered__semiring_I4_J,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus @ A @ I @ K )
= ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_147_group__cancel_Oadd1,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A5: A,K: A,A2: A,B2: A] :
( ( A5
= ( plus_plus @ A @ K @ A2 ) )
=> ( ( plus_plus @ A @ A5 @ B2 )
= ( plus_plus @ A @ K @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).
% group_cancel.add1
thf(fact_148_group__cancel_Oadd2,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [B5: A,K: A,B2: A,A2: A] :
( ( B5
= ( plus_plus @ A @ K @ B2 ) )
=> ( ( plus_plus @ A @ A2 @ B5 )
= ( plus_plus @ A @ K @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).
% group_cancel.add2
thf(fact_149_dbl__def,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( neg_numeral_dbl @ A )
= ( ^ [X: A] : ( plus_plus @ A @ X @ X ) ) ) ) ).
% dbl_def
thf(fact_150_add_Oassoc,axiom,
! [A: $tType] :
( ( semigroup_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 )
= ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% add.assoc
thf(fact_151_add_Oleft__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ( plus_plus @ A @ A2 @ B2 )
= ( plus_plus @ A @ A2 @ C2 ) )
= ( B2 = C2 ) ) ) ).
% add.left_cancel
thf(fact_152_add_Oright__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( ( plus_plus @ A @ B2 @ A2 )
= ( plus_plus @ A @ C2 @ A2 ) )
= ( B2 = C2 ) ) ) ).
% add.right_cancel
thf(fact_153_ab__semigroup__add__class_Oadd_Ocommute,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A )
=> ( ( plus_plus @ A )
= ( ^ [A3: A,B3: A] : ( plus_plus @ A @ B3 @ A3 ) ) ) ) ).
% ab_semigroup_add_class.add.commute
thf(fact_154_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( plus_plus @ A @ B2 @ ( plus_plus @ A @ A2 @ C2 ) )
= ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% ab_semigroup_add_class.add.left_commute
thf(fact_155_add__left__imp__eq,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ( plus_plus @ A @ A2 @ B2 )
= ( plus_plus @ A @ A2 @ C2 ) )
=> ( B2 = C2 ) ) ) ).
% add_left_imp_eq
thf(fact_156_add__right__imp__eq,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( ( plus_plus @ A @ B2 @ A2 )
= ( plus_plus @ A @ C2 @ A2 ) )
=> ( B2 = C2 ) ) ) ).
% add_right_imp_eq
thf(fact_157_int__int__eq,axiom,
! [M: nat,N: nat] :
( ( ( semiring_1_of_nat @ int @ M )
= ( semiring_1_of_nat @ int @ N ) )
= ( M = N ) ) ).
% int_int_eq
thf(fact_158_verit__la__generic,axiom,
! [A2: int,X2: int] :
( ( ord_less_eq @ int @ A2 @ X2 )
| ( A2 = X2 )
| ( ord_less_eq @ int @ X2 @ A2 ) ) ).
% verit_la_generic
thf(fact_159_fails__wait,axiom,
! [X2: nat,H: heap_ext @ product_unit] :
~ ( time_fails @ product_unit @ ( heap_Time_wait @ X2 ) @ H ) ).
% fails_wait
thf(fact_160_verit__sko__forall__indirect2,axiom,
! [A: $tType,X2: A,P: A > $o,P2: A > $o] :
( ( X2
= ( fChoice @ A
@ ^ [X: A] :
~ ( P @ X ) ) )
=> ( ! [X3: A] :
( ( P @ X3 )
= ( P2 @ X3 ) )
=> ( ( ! [X5: A] : ( P2 @ X5 ) )
= ( P @ X2 ) ) ) ) ).
% verit_sko_forall_indirect2
thf(fact_161_verit__sko__forall__indirect,axiom,
! [A: $tType,X2: A,P: A > $o] :
( ( X2
= ( fChoice @ A
@ ^ [X: A] :
~ ( P @ X ) ) )
=> ( ( ! [X5: A] : ( P @ X5 ) )
= ( P @ X2 ) ) ) ).
% verit_sko_forall_indirect
thf(fact_162_verit__sko__ex__indirect2,axiom,
! [A: $tType,X2: A,P: A > $o,P2: A > $o] :
( ( X2
= ( fChoice @ A @ P ) )
=> ( ! [X3: A] :
( ( P @ X3 )
= ( P2 @ X3 ) )
=> ( ( ? [X5: A] : ( P2 @ X5 ) )
= ( P @ X2 ) ) ) ) ).
% verit_sko_ex_indirect2
thf(fact_163_verit__sko__ex__indirect,axiom,
! [A: $tType,X2: A,P: A > $o] :
( ( X2
= ( fChoice @ A @ P ) )
=> ( ( ? [X5: A] : ( P @ X5 ) )
= ( P @ X2 ) ) ) ).
% verit_sko_ex_indirect
thf(fact_164_verit__sko__forall_H_H,axiom,
! [A: $tType,B5: A,A5: A,P: A > $o] :
( ( B5 = A5 )
=> ( ( ( fChoice @ A @ P )
= A5 )
= ( ( fChoice @ A @ P )
= B5 ) ) ) ).
% verit_sko_forall''
thf(fact_165_verit__sko__forall_H,axiom,
! [A: $tType,P: A > $o,A5: $o] :
( ( ( P
@ ( fChoice @ A
@ ^ [X: A] :
~ ( P @ X ) ) )
= A5 )
=> ( ( ! [X5: A] : ( P @ X5 ) )
= A5 ) ) ).
% verit_sko_forall'
thf(fact_166_verit__sko__forall,axiom,
! [A: $tType] :
( ( ^ [P5: A > $o] :
! [X6: A] : ( P5 @ X6 ) )
= ( ^ [P4: A > $o] :
( P4
@ ( fChoice @ A
@ ^ [X: A] :
~ ( P4 @ X ) ) ) ) ) ).
% verit_sko_forall
thf(fact_167_verit__sko__ex_H,axiom,
! [A: $tType,P: A > $o,A5: $o] :
( ( ( P @ ( fChoice @ A @ P ) )
= A5 )
=> ( ( ? [X5: A] : ( P @ X5 ) )
= A5 ) ) ).
% verit_sko_ex'
thf(fact_168_size__neq__size__imp__neq,axiom,
! [A: $tType] :
( ( size @ A )
=> ! [X2: A,Y: A] :
( ( ( size_size @ A @ X2 )
!= ( size_size @ A @ Y ) )
=> ( X2 != Y ) ) ) ).
% size_neq_size_imp_neq
thf(fact_169_add__mono__thms__linordered__semiring_I3_J,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( ord_less_eq @ A @ I @ J )
& ( K = L ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_170_add__mono__thms__linordered__semiring_I2_J,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( I = J )
& ( ord_less_eq @ A @ K @ L ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_171_add__mono__thms__linordered__semiring_I1_J,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( ord_less_eq @ A @ I @ J )
& ( ord_less_eq @ A @ K @ L ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_172_add__mono,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ D2 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ D2 ) ) ) ) ) ).
% add_mono
thf(fact_173_add__left__mono,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ C2 @ A2 ) @ ( plus_plus @ A @ C2 @ B2 ) ) ) ) ).
% add_left_mono
thf(fact_174_less__eqE,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ~ ! [C4: A] :
( B2
!= ( plus_plus @ A @ A2 @ C4 ) ) ) ) ).
% less_eqE
thf(fact_175_add__right__mono,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% add_right_mono
thf(fact_176_le__iff__add,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A3: A,B3: A] :
? [C5: A] :
( B3
= ( plus_plus @ A @ A3 @ C5 ) ) ) ) ) ).
% le_iff_add
thf(fact_177_add__le__imp__le__left,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ C2 @ A2 ) @ ( plus_plus @ A @ C2 @ B2 ) )
=> ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% add_le_imp_le_left
thf(fact_178_add__le__imp__le__right,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) )
=> ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% add_le_imp_le_right
thf(fact_179_verit__sum__simplify,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [A2: A] :
( ( plus_plus @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% verit_sum_simplify
thf(fact_180_add_Ogroup__left__neutral,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A] :
( ( plus_plus @ A @ ( zero_zero @ A ) @ A2 )
= A2 ) ) ).
% add.group_left_neutral
thf(fact_181_add_Ocomm__neutral,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A2: A] :
( ( plus_plus @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% add.comm_neutral
thf(fact_182_comm__monoid__add__class_Oadd__0,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A2: A] :
( ( plus_plus @ A @ ( zero_zero @ A ) @ A2 )
= A2 ) ) ).
% comm_monoid_add_class.add_0
thf(fact_183_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus @ nat @ ( zero_zero @ nat ) @ N )
= N ) ).
% plus_nat.add_0
thf(fact_184_add__eq__self__zero,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus @ nat @ M @ N )
= M )
=> ( N
= ( zero_zero @ nat ) ) ) ).
% add_eq_self_zero
thf(fact_185_add__leE,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ M @ K ) @ N )
=> ~ ( ( ord_less_eq @ nat @ M @ N )
=> ~ ( ord_less_eq @ nat @ K @ N ) ) ) ).
% add_leE
thf(fact_186_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq @ nat @ N @ ( plus_plus @ nat @ N @ M ) ) ).
% le_add1
thf(fact_187_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq @ nat @ N @ ( plus_plus @ nat @ M @ N ) ) ).
% le_add2
thf(fact_188_add__leD1,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ M @ K ) @ N )
=> ( ord_less_eq @ nat @ M @ N ) ) ).
% add_leD1
thf(fact_189_add__leD2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ M @ K ) @ N )
=> ( ord_less_eq @ nat @ K @ N ) ) ).
% add_leD2
thf(fact_190_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq @ nat @ K @ L )
=> ? [N2: nat] :
( L
= ( plus_plus @ nat @ K @ N2 ) ) ) ).
% le_Suc_ex
thf(fact_191_add__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( ord_less_eq @ nat @ K @ L )
=> ( ord_less_eq @ nat @ ( plus_plus @ nat @ I @ K ) @ ( plus_plus @ nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_192_add__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ord_less_eq @ nat @ ( plus_plus @ nat @ I @ K ) @ ( plus_plus @ nat @ J @ K ) ) ) ).
% add_le_mono1
thf(fact_193_trans__le__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ord_less_eq @ nat @ I @ ( plus_plus @ nat @ J @ M ) ) ) ).
% trans_le_add1
thf(fact_194_trans__le__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ord_less_eq @ nat @ I @ ( plus_plus @ nat @ M @ J ) ) ) ).
% trans_le_add2
thf(fact_195_nat__le__iff__add,axiom,
( ( ord_less_eq @ nat )
= ( ^ [M2: nat,N4: nat] :
? [K2: nat] :
( N4
= ( plus_plus @ nat @ M2 @ K2 ) ) ) ) ).
% nat_le_iff_add
thf(fact_196_add__decreasing,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ C2 @ B2 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C2 ) @ B2 ) ) ) ) ).
% add_decreasing
thf(fact_197_add__increasing,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ B2 @ C2 )
=> ( ord_less_eq @ A @ B2 @ ( plus_plus @ A @ A2 @ C2 ) ) ) ) ) ).
% add_increasing
thf(fact_198_add__decreasing2,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C2 ) @ B2 ) ) ) ) ).
% add_decreasing2
thf(fact_199_add__increasing2,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [C2: A,B2: A,A2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ord_less_eq @ A @ B2 @ ( plus_plus @ A @ A2 @ C2 ) ) ) ) ) ).
% add_increasing2
thf(fact_200_add__nonneg__nonneg,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_201_add__nonpos__nonpos,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% add_nonpos_nonpos
thf(fact_202_add__nonneg__eq__0__iff,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [X2: A,Y: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ( plus_plus @ A @ X2 @ Y )
= ( zero_zero @ A ) )
= ( ( X2
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_203_add__nonpos__eq__0__iff,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [X2: A,Y: A] :
( ( ord_less_eq @ A @ X2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ Y @ ( zero_zero @ A ) )
=> ( ( ( plus_plus @ A @ X2 @ Y )
= ( zero_zero @ A ) )
= ( ( X2
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_204_arg__min__nat__le,axiom,
! [A: $tType,P: A > $o,X2: A,M: A > nat] :
( ( P @ X2 )
=> ( ord_less_eq @ nat @ ( M @ ( lattices_ord_arg_min @ A @ nat @ M @ P ) ) @ ( M @ X2 ) ) ) ).
% arg_min_nat_le
thf(fact_205_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 ) )
& ! [Y5: A] :
( ( P @ Y5 )
=> ( ord_less_eq @ nat @ ( M @ ( lattices_ord_arg_min @ A @ nat @ M @ P ) ) @ ( M @ Y5 ) ) ) ) ) ).
% arg_min_nat_lemma
thf(fact_206_time__bind,axiom,
! [A: $tType,B: $tType,M: heap_Time_Heap @ B,H: heap_ext @ product_unit,F: B > ( heap_Time_Heap @ A )] :
( ( ( ~ ( time_fails @ B @ M @ H )
& ~ ( time_fails @ A @ ( F @ ( time_the_res @ B @ M @ H ) ) @ ( time_the_heap @ B @ M @ H ) ) )
=> ( ( time_time @ A @ ( heap_Time_bind @ B @ A @ M @ F ) @ H )
= ( plus_plus @ nat @ ( time_time @ B @ M @ H ) @ ( time_time @ A @ ( F @ ( time_the_res @ B @ M @ H ) ) @ ( time_the_heap @ B @ M @ H ) ) ) ) )
& ( ~ ( ~ ( time_fails @ B @ M @ H )
& ~ ( time_fails @ A @ ( F @ ( time_the_res @ B @ M @ H ) ) @ ( time_the_heap @ B @ M @ H ) ) )
=> ( ( time_time @ A @ ( heap_Time_bind @ B @ A @ M @ F ) @ H )
= ( zero_zero @ nat ) ) ) ) ).
% time_bind
thf(fact_207_some__sym__eq__trivial,axiom,
! [A: $tType,X2: A] :
( ( fChoice @ A
@ ( ^ [Y3: A,Z: A] : Y3 = Z
@ X2 ) )
= X2 ) ).
% some_sym_eq_trivial
thf(fact_208_some__eq__trivial,axiom,
! [A: $tType,X2: A] :
( ( fChoice @ A
@ ^ [Y4: A] : Y4 = X2 )
= X2 ) ).
% some_eq_trivial
thf(fact_209_some__equality,axiom,
! [A: $tType,P: A > $o,A2: A] :
( ( P @ A2 )
=> ( ! [X3: A] :
( ( P @ X3 )
=> ( X3 = A2 ) )
=> ( ( fChoice @ A @ P )
= A2 ) ) ) ).
% some_equality
thf(fact_210_rec__nat__0__imp,axiom,
! [A: $tType,F: nat > A,F1: A,F22: nat > A > A] :
( ( F
= ( rec_nat @ A @ F1 @ F22 ) )
=> ( ( F @ ( zero_zero @ nat ) )
= F1 ) ) ).
% rec_nat_0_imp
thf(fact_211_imp__le__cong,axiom,
! [X2: int,X7: int,P: $o,P2: $o] :
( ( X2 = X7 )
=> ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X7 )
=> ( P = P2 ) )
=> ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X2 )
=> P )
= ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X7 )
=> P2 ) ) ) ) ).
% imp_le_cong
thf(fact_212_conj__le__cong,axiom,
! [X2: int,X7: int,P: $o,P2: $o] :
( ( X2 = X7 )
=> ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X7 )
=> ( P = P2 ) )
=> ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X2 )
& P )
= ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X7 )
& P2 ) ) ) ) ).
% conj_le_cong
thf(fact_213_Euclid__induct,axiom,
! [P: nat > nat > $o,A2: nat,B2: nat] :
( ! [A4: nat,B4: nat] :
( ( P @ A4 @ B4 )
= ( P @ B4 @ A4 ) )
=> ( ! [A4: nat] : ( P @ A4 @ ( zero_zero @ nat ) )
=> ( ! [A4: nat,B4: nat] :
( ( P @ A4 @ B4 )
=> ( P @ A4 @ ( plus_plus @ nat @ A4 @ B4 ) ) )
=> ( P @ A2 @ B2 ) ) ) ) ).
% Euclid_induct
thf(fact_214_add__0__iff,axiom,
! [A: $tType] :
( ( semiri1453513574482234551roduct @ A )
=> ! [B2: A,A2: A] :
( ( B2
= ( plus_plus @ A @ B2 @ A2 ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% add_0_iff
thf(fact_215_arg__min__def,axiom,
! [A: $tType,B: $tType] :
( ( ord @ A )
=> ( ( lattices_ord_arg_min @ B @ A )
= ( ^ [F2: B > A,P4: B > $o] : ( fChoice @ B @ ( lattic501386751177426532rg_min @ B @ A @ F2 @ P4 ) ) ) ) ) ).
% arg_min_def
thf(fact_216_of__nat__code,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( semiring_1_of_nat @ A )
= ( ^ [N4: nat] :
( semiri8178284476397505188at_aux @ A
@ ^ [I2: A] : ( plus_plus @ A @ I2 @ ( one_one @ A ) )
@ N4
@ ( zero_zero @ A ) ) ) ) ) ).
% of_nat_code
thf(fact_217_some1__equality,axiom,
! [A: $tType,P: A > $o,A2: A] :
( ? [X4: A] :
( ( P @ X4 )
& ! [Y2: A] :
( ( P @ Y2 )
=> ( Y2 = X4 ) ) )
=> ( ( P @ A2 )
=> ( ( fChoice @ A @ P )
= A2 ) ) ) ).
% some1_equality
thf(fact_218_some__eq__ex,axiom,
! [A: $tType,P: A > $o] :
( ( P @ ( fChoice @ A @ P ) )
= ( ? [X5: A] : ( P @ X5 ) ) ) ).
% some_eq_ex
thf(fact_219_of__nat__eq__1__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [N: nat] :
( ( ( semiring_1_of_nat @ A @ N )
= ( one_one @ A ) )
= ( N
= ( one_one @ nat ) ) ) ) ).
% of_nat_eq_1_iff
thf(fact_220_of__nat__1__eq__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [N: nat] :
( ( ( one_one @ A )
= ( semiring_1_of_nat @ A @ N ) )
= ( N
= ( one_one @ nat ) ) ) ) ).
% of_nat_1_eq_iff
thf(fact_221_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_222_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_223_one__reorient,axiom,
! [A: $tType] :
( ( one @ A )
=> ! [X2: A] :
( ( ( one_one @ A )
= X2 )
= ( X2
= ( one_one @ A ) ) ) ) ).
% one_reorient
thf(fact_224_odd__nonzero,axiom,
! [Z2: int] :
( ( plus_plus @ int @ ( plus_plus @ int @ ( one_one @ int ) @ Z2 ) @ Z2 )
!= ( zero_zero @ int ) ) ).
% odd_nonzero
thf(fact_225_int__ge__induct,axiom,
! [K: int,I: int,P: int > $o] :
( ( ord_less_eq @ int @ K @ I )
=> ( ( P @ K )
=> ( ! [I3: int] :
( ( ord_less_eq @ int @ K @ I3 )
=> ( ( P @ I3 )
=> ( P @ ( plus_plus @ int @ I3 @ ( one_one @ int ) ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_ge_induct
thf(fact_226_is__arg__min__arg__min__nat,axiom,
! [A: $tType,P: A > $o,X2: A,M: A > nat] :
( ( P @ X2 )
=> ( lattic501386751177426532rg_min @ A @ nat @ M @ P @ ( lattices_ord_arg_min @ A @ nat @ M @ P ) ) ) ).
% is_arg_min_arg_min_nat
thf(fact_227_int__ops_I2_J,axiom,
( ( semiring_1_of_nat @ int @ ( one_one @ nat ) )
= ( one_one @ int ) ) ).
% int_ops(2)
thf(fact_228_plus__int__code_I2_J,axiom,
! [L: int] :
( ( plus_plus @ int @ ( zero_zero @ int ) @ L )
= L ) ).
% plus_int_code(2)
thf(fact_229_plus__int__code_I1_J,axiom,
! [K: int] :
( ( plus_plus @ int @ K @ ( zero_zero @ int ) )
= K ) ).
% plus_int_code(1)
thf(fact_230_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_231_zadd__int__left,axiom,
! [M: nat,N: nat,Z2: int] :
( ( plus_plus @ int @ ( semiring_1_of_nat @ int @ M ) @ ( plus_plus @ int @ ( semiring_1_of_nat @ int @ N ) @ Z2 ) )
= ( plus_plus @ int @ ( semiring_1_of_nat @ int @ ( plus_plus @ nat @ M @ N ) ) @ Z2 ) ) ).
% zadd_int_left
thf(fact_232_int__ops_I5_J,axiom,
! [A2: nat,B2: nat] :
( ( semiring_1_of_nat @ int @ ( plus_plus @ nat @ A2 @ B2 ) )
= ( plus_plus @ int @ ( semiring_1_of_nat @ int @ A2 ) @ ( semiring_1_of_nat @ int @ B2 ) ) ) ).
% int_ops(5)
thf(fact_233_int__plus,axiom,
! [N: nat,M: nat] :
( ( semiring_1_of_nat @ int @ ( plus_plus @ nat @ N @ M ) )
= ( plus_plus @ int @ ( semiring_1_of_nat @ int @ N ) @ ( semiring_1_of_nat @ int @ M ) ) ) ).
% int_plus
thf(fact_234_zle__iff__zadd,axiom,
( ( ord_less_eq @ int )
= ( ^ [W: int,Z3: int] :
? [N4: nat] :
( Z3
= ( plus_plus @ int @ W @ ( semiring_1_of_nat @ int @ N4 ) ) ) ) ) ).
% zle_iff_zadd
thf(fact_235_nat__geq__1__eq__neqz,axiom,
! [X2: nat] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ X2 )
= ( X2
!= ( zero_zero @ nat ) ) ) ).
% nat_geq_1_eq_neqz
thf(fact_236_is__arg__min__linorder,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ( ( lattic501386751177426532rg_min @ A @ B )
= ( ^ [F2: A > B,P4: A > $o,X: A] :
( ( P4 @ X )
& ! [Y4: A] :
( ( P4 @ Y4 )
=> ( ord_less_eq @ B @ ( F2 @ X ) @ ( F2 @ Y4 ) ) ) ) ) ) ) ).
% is_arg_min_linorder
thf(fact_237_is__arg__min__antimono,axiom,
! [B: $tType,A: $tType] :
( ( order @ B )
=> ! [F: A > B,P: A > $o,X2: A,Y: A] :
( ( lattic501386751177426532rg_min @ A @ B @ F @ P @ X2 )
=> ( ( ord_less_eq @ B @ ( F @ Y ) @ ( F @ X2 ) )
=> ( ( P @ Y )
=> ( lattic501386751177426532rg_min @ A @ B @ F @ P @ Y ) ) ) ) ) ).
% is_arg_min_antimono
thf(fact_238_someI,axiom,
! [A: $tType,P: A > $o,X2: A] :
( ( P @ X2 )
=> ( P @ ( fChoice @ A @ P ) ) ) ).
% someI
thf(fact_239_Eps__cong,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X3: A] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( fChoice @ A @ P )
= ( fChoice @ A @ Q ) ) ) ).
% Eps_cong
thf(fact_240_tfl__some,axiom,
! [A: $tType,P6: A > $o,X4: A] :
( ( P6 @ X4 )
=> ( P6 @ ( fChoice @ A @ P6 ) ) ) ).
% tfl_some
thf(fact_241_some__eq__imp,axiom,
! [A: $tType,P: A > $o,A2: A,B2: A] :
( ( ( fChoice @ A @ P )
= A2 )
=> ( ( P @ B2 )
=> ( P @ A2 ) ) ) ).
% some_eq_imp
thf(fact_242_someI2,axiom,
! [A: $tType,P: A > $o,A2: A,Q: A > $o] :
( ( P @ A2 )
=> ( ! [X3: A] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( Q @ ( fChoice @ A @ P ) ) ) ) ).
% someI2
thf(fact_243_someI__ex,axiom,
! [A: $tType,P: A > $o] :
( ? [X_1: A] : ( P @ X_1 )
=> ( P @ ( fChoice @ A @ P ) ) ) ).
% someI_ex
thf(fact_244_someI2__ex,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ? [X_1: A] : ( P @ X_1 )
=> ( ! [X3: A] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( Q @ ( fChoice @ A @ P ) ) ) ) ).
% someI2_ex
thf(fact_245_someI2__bex,axiom,
! [A: $tType,A5: set @ A,P: A > $o,Q: A > $o] :
( ? [X4: A] :
( ( member @ A @ X4 @ A5 )
& ( P @ X4 ) )
=> ( ! [X3: A] :
( ( ( member @ A @ X3 @ A5 )
& ( P @ X3 ) )
=> ( Q @ X3 ) )
=> ( Q
@ ( fChoice @ A
@ ^ [X: A] :
( ( member @ A @ X @ A5 )
& ( P @ X ) ) ) ) ) ) ).
% someI2_bex
thf(fact_246_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_247_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_248_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_249_size__union,axiom,
! [A: $tType,M3: multiset @ A,N5: multiset @ A] :
( ( size_size @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ M3 @ N5 ) )
= ( plus_plus @ nat @ ( size_size @ ( multiset @ A ) @ M3 ) @ ( size_size @ ( multiset @ A ) @ N5 ) ) ) ).
% size_union
thf(fact_250_size__eq__0__iff__empty,axiom,
! [A: $tType,M3: multiset @ A] :
( ( ( size_size @ ( multiset @ A ) @ M3 )
= ( zero_zero @ nat ) )
= ( M3
= ( zero_zero @ ( multiset @ A ) ) ) ) ).
% size_eq_0_iff_empty
thf(fact_251_size__empty,axiom,
! [A: $tType] :
( ( size_size @ ( multiset @ A ) @ ( zero_zero @ ( multiset @ A ) ) )
= ( zero_zero @ nat ) ) ).
% size_empty
thf(fact_252_zero__neq__one,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ( ( zero_zero @ A )
!= ( one_one @ A ) ) ) ).
% zero_neq_one
thf(fact_253_return__bind,axiom,
! [B: $tType,A: $tType,X2: B,F: B > ( heap_Time_Heap @ A )] :
( ( heap_Time_bind @ B @ A @ ( heap_Time_return @ B @ X2 ) @ F )
= ( heap_Time_bind @ product_unit @ A @ ( heap_Time_wait @ ( one_one @ nat ) )
@ ^ [Uu: product_unit] : ( F @ X2 ) ) ) ).
% return_bind
thf(fact_254_bind__return,axiom,
! [A: $tType,F: heap_Time_Heap @ A] :
( ( heap_Time_bind @ A @ A @ F @ ( heap_Time_return @ A ) )
= ( heap_Time_bind @ product_unit @ A @ ( heap_Time_wait @ ( one_one @ nat ) )
@ ^ [Uu: product_unit] : F ) ) ).
% bind_return
thf(fact_255_time__array__nth,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [P3: array @ A,I: nat,H: heap_ext @ product_unit] :
( ( ( time_fails @ A @ ( array_nth @ A @ P3 @ I ) @ H )
=> ( ( time_time @ A @ ( array_nth @ A @ P3 @ I ) @ H )
= ( zero_zero @ nat ) ) )
& ( ~ ( time_fails @ A @ ( array_nth @ A @ P3 @ I ) @ H )
=> ( ( time_time @ A @ ( array_nth @ A @ P3 @ I ) @ H )
= ( one_one @ nat ) ) ) ) ) ).
% time_array_nth
thf(fact_256_dbl__inc__simps_I2_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( neg_numeral_dbl_inc @ A @ ( zero_zero @ A ) )
= ( one_one @ A ) ) ) ).
% dbl_inc_simps(2)
thf(fact_257_union__eq__empty,axiom,
! [A: $tType,M3: multiset @ A,N5: multiset @ A] :
( ( ( plus_plus @ ( multiset @ A ) @ M3 @ N5 )
= ( zero_zero @ ( multiset @ A ) ) )
= ( ( M3
= ( zero_zero @ ( multiset @ A ) ) )
& ( N5
= ( zero_zero @ ( multiset @ A ) ) ) ) ) ).
% union_eq_empty
thf(fact_258_empty__eq__union,axiom,
! [A: $tType,M3: multiset @ A,N5: multiset @ A] :
( ( ( zero_zero @ ( multiset @ A ) )
= ( plus_plus @ ( multiset @ A ) @ M3 @ N5 ) )
= ( ( M3
= ( zero_zero @ ( multiset @ A ) ) )
& ( N5
= ( zero_zero @ ( multiset @ A ) ) ) ) ) ).
% empty_eq_union
thf(fact_259_subset__mset_Ozero__eq__add__iff__both__eq__0,axiom,
! [A: $tType,X2: multiset @ A,Y: multiset @ A] :
( ( ( zero_zero @ ( multiset @ A ) )
= ( plus_plus @ ( multiset @ A ) @ X2 @ Y ) )
= ( ( X2
= ( zero_zero @ ( multiset @ A ) ) )
& ( Y
= ( zero_zero @ ( multiset @ A ) ) ) ) ) ).
% subset_mset.zero_eq_add_iff_both_eq_0
thf(fact_260_subset__mset_Oadd__eq__0__iff__both__eq__0,axiom,
! [A: $tType,X2: multiset @ A,Y: multiset @ A] :
( ( ( plus_plus @ ( multiset @ A ) @ X2 @ Y )
= ( zero_zero @ ( multiset @ A ) ) )
= ( ( X2
= ( zero_zero @ ( multiset @ A ) ) )
& ( Y
= ( zero_zero @ ( multiset @ A ) ) ) ) ) ).
% subset_mset.add_eq_0_iff_both_eq_0
thf(fact_261_empty__neutral_I1_J,axiom,
! [A: $tType,X2: multiset @ A] :
( ( plus_plus @ ( multiset @ A ) @ ( zero_zero @ ( multiset @ A ) ) @ X2 )
= X2 ) ).
% empty_neutral(1)
thf(fact_262_empty__neutral_I2_J,axiom,
! [A: $tType,X2: multiset @ A] :
( ( plus_plus @ ( multiset @ A ) @ X2 @ ( zero_zero @ ( multiset @ A ) ) )
= X2 ) ).
% empty_neutral(2)
thf(fact_263_multi__union__self__other__eq,axiom,
! [A: $tType,A5: multiset @ A,X8: multiset @ A,Y7: multiset @ A] :
( ( ( plus_plus @ ( multiset @ A ) @ A5 @ X8 )
= ( plus_plus @ ( multiset @ A ) @ A5 @ Y7 ) )
=> ( X8 = Y7 ) ) ).
% multi_union_self_other_eq
thf(fact_264_union__right__cancel,axiom,
! [A: $tType,M3: multiset @ A,K3: multiset @ A,N5: multiset @ A] :
( ( ( plus_plus @ ( multiset @ A ) @ M3 @ K3 )
= ( plus_plus @ ( multiset @ A ) @ N5 @ K3 ) )
= ( M3 = N5 ) ) ).
% union_right_cancel
thf(fact_265_union__left__cancel,axiom,
! [A: $tType,K3: multiset @ A,M3: multiset @ A,N5: multiset @ A] :
( ( ( plus_plus @ ( multiset @ A ) @ K3 @ M3 )
= ( plus_plus @ ( multiset @ A ) @ K3 @ N5 ) )
= ( M3 = N5 ) ) ).
% union_left_cancel
thf(fact_266_union__commute,axiom,
! [A: $tType] :
( ( plus_plus @ ( multiset @ A ) )
= ( ^ [M6: multiset @ A,N6: multiset @ A] : ( plus_plus @ ( multiset @ A ) @ N6 @ M6 ) ) ) ).
% union_commute
thf(fact_267_union__lcomm,axiom,
! [A: $tType,M3: multiset @ A,N5: multiset @ A,K3: multiset @ A] :
( ( plus_plus @ ( multiset @ A ) @ M3 @ ( plus_plus @ ( multiset @ A ) @ N5 @ K3 ) )
= ( plus_plus @ ( multiset @ A ) @ N5 @ ( plus_plus @ ( multiset @ A ) @ M3 @ K3 ) ) ) ).
% union_lcomm
thf(fact_268_union__assoc,axiom,
! [A: $tType,M3: multiset @ A,N5: multiset @ A,K3: multiset @ A] :
( ( plus_plus @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ M3 @ N5 ) @ K3 )
= ( plus_plus @ ( multiset @ A ) @ M3 @ ( plus_plus @ ( multiset @ A ) @ N5 @ K3 ) ) ) ).
% union_assoc
thf(fact_269_mset__distrib,axiom,
! [A: $tType,A5: multiset @ A,B5: multiset @ A,M3: multiset @ A,N5: multiset @ A] :
( ( ( plus_plus @ ( multiset @ A ) @ A5 @ B5 )
= ( plus_plus @ ( multiset @ A ) @ M3 @ N5 ) )
=> ~ ! [Am: multiset @ A,An: multiset @ A] :
( ( A5
= ( plus_plus @ ( multiset @ A ) @ Am @ An ) )
=> ! [Bm: multiset @ A,Bn: multiset @ A] :
( ( B5
= ( plus_plus @ ( multiset @ A ) @ Bm @ Bn ) )
=> ( ( M3
= ( plus_plus @ ( multiset @ A ) @ Am @ Bm ) )
=> ( N5
!= ( plus_plus @ ( multiset @ A ) @ An @ Bn ) ) ) ) ) ) ).
% mset_distrib
thf(fact_270_one__natural_Orsp,axiom,
( ( one_one @ nat )
= ( one_one @ nat ) ) ).
% one_natural.rsp
thf(fact_271_one__integer_Orsp,axiom,
( ( one_one @ int )
= ( one_one @ int ) ) ).
% one_integer.rsp
thf(fact_272_fold__if__return,axiom,
! [A: $tType,B2: $o,C2: A,D2: A] :
( ( B2
=> ( ( heap_Time_return @ A @ C2 )
= ( heap_Time_return @ A @ ( if @ A @ B2 @ C2 @ D2 ) ) ) )
& ( ~ B2
=> ( ( heap_Time_return @ A @ D2 )
= ( heap_Time_return @ A @ ( if @ A @ B2 @ C2 @ D2 ) ) ) ) ) ).
% fold_if_return
thf(fact_273_time__return,axiom,
! [A: $tType,X2: A,H: heap_ext @ product_unit] :
( ( time_time @ A @ ( heap_Time_return @ A @ X2 ) @ H )
= ( one_one @ nat ) ) ).
% time_return
thf(fact_274_fails__return,axiom,
! [A: $tType,X2: A,H: heap_ext @ product_unit] :
~ ( time_fails @ A @ ( heap_Time_return @ A @ X2 ) @ H ) ).
% fails_return
thf(fact_275_dbl__inc__def,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( neg_numeral_dbl_inc @ A )
= ( ^ [X: A] : ( plus_plus @ A @ ( plus_plus @ A @ X @ X ) @ ( one_one @ A ) ) ) ) ) ).
% dbl_inc_def
thf(fact_276_bind__lift,axiom,
! [A: $tType,B: $tType,F: heap_Time_Heap @ B,G: B > A] :
( ( heap_Time_bind @ B @ A @ F @ ( heap_Time_lift @ B @ A @ G ) )
= ( heap_Time_bind @ B @ A @ F
@ ^ [X: B] : ( heap_Time_return @ A @ ( G @ X ) ) ) ) ).
% bind_lift
thf(fact_277_lift__collapse,axiom,
! [A: $tType,B: $tType] :
( ( heap_Time_lift @ B @ A )
= ( ^ [F2: B > A,X: B] : ( heap_Time_return @ A @ ( F2 @ X ) ) ) ) ).
% lift_collapse
thf(fact_278_lookup__chain,axiom,
! [B: $tType,A: $tType] :
( ( heap @ B )
=> ! [R: ref @ B,F: heap_Time_Heap @ A] :
( ( heap_Time_bind @ B @ A @ ( ref_lookup @ B @ R )
@ ^ [Uu: B] : F )
= ( heap_Time_bind @ product_unit @ A @ ( heap_Time_wait @ ( one_one @ nat ) )
@ ^ [Uu: product_unit] : F ) ) ) ).
% lookup_chain
thf(fact_279_univ__def,axiom,
! [A: $tType,B: $tType] :
( ( bNF_Greatest_univ @ B @ A )
= ( ^ [F2: B > A,X5: set @ B] :
( F2
@ ( fChoice @ B
@ ^ [X: B] : ( member @ B @ X @ X5 ) ) ) ) ) ).
% univ_def
thf(fact_280_ceiling__add__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A] :
( ( archimedean_ceiling @ A @ ( plus_plus @ A @ X2 @ ( one_one @ A ) ) )
= ( plus_plus @ int @ ( archimedean_ceiling @ A @ X2 ) @ ( one_one @ int ) ) ) ) ).
% ceiling_add_one
thf(fact_281_ceiling__le__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A] :
( ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X2 ) @ ( one_one @ int ) )
= ( ord_less_eq @ A @ X2 @ ( one_one @ A ) ) ) ) ).
% ceiling_le_one
thf(fact_282_ceiling__le__zero,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A] :
( ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X2 ) @ ( zero_zero @ int ) )
= ( ord_less_eq @ A @ X2 @ ( zero_zero @ A ) ) ) ) ).
% ceiling_le_zero
thf(fact_283_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_284_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_285_size__multiset__union,axiom,
! [A: $tType,F: A > nat,M3: multiset @ A,N5: multiset @ A] :
( ( size_multiset @ A @ F @ ( plus_plus @ ( multiset @ A ) @ M3 @ N5 ) )
= ( plus_plus @ nat @ ( size_multiset @ A @ F @ M3 ) @ ( size_multiset @ A @ F @ N5 ) ) ) ).
% size_multiset_union
thf(fact_286_of__int__eq__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [W2: int,Z2: int] :
( ( ( ring_1_of_int @ A @ W2 )
= ( ring_1_of_int @ A @ Z2 ) )
= ( W2 = Z2 ) ) ) ).
% of_int_eq_iff
thf(fact_287_ceiling__of__int,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z2: int] :
( ( archimedean_ceiling @ A @ ( ring_1_of_int @ A @ Z2 ) )
= Z2 ) ) ).
% ceiling_of_int
thf(fact_288_of__int__0,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( ( ring_1_of_int @ A @ ( zero_zero @ int ) )
= ( zero_zero @ A ) ) ) ).
% of_int_0
thf(fact_289_of__int__0__eq__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [Z2: int] :
( ( ( zero_zero @ A )
= ( ring_1_of_int @ A @ Z2 ) )
= ( Z2
= ( zero_zero @ int ) ) ) ) ).
% of_int_0_eq_iff
thf(fact_290_of__int__eq__0__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [Z2: int] :
( ( ( ring_1_of_int @ A @ Z2 )
= ( zero_zero @ A ) )
= ( Z2
= ( zero_zero @ int ) ) ) ) ).
% of_int_eq_0_iff
thf(fact_291_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_292_of__int__eq__1__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [Z2: int] :
( ( ( ring_1_of_int @ A @ Z2 )
= ( one_one @ A ) )
= ( Z2
= ( one_one @ int ) ) ) ) ).
% of_int_eq_1_iff
thf(fact_293_of__int__1,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( ( ring_1_of_int @ A @ ( one_one @ int ) )
= ( one_one @ A ) ) ) ).
% of_int_1
thf(fact_294_of__int__add,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [W2: int,Z2: int] :
( ( ring_1_of_int @ A @ ( plus_plus @ int @ W2 @ Z2 ) )
= ( plus_plus @ A @ ( ring_1_of_int @ A @ W2 ) @ ( ring_1_of_int @ A @ Z2 ) ) ) ) ).
% of_int_add
thf(fact_295_ceiling__zero,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archimedean_ceiling @ A @ ( zero_zero @ A ) )
= ( zero_zero @ int ) ) ) ).
% ceiling_zero
thf(fact_296_of__int__of__nat__eq,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [N: nat] :
( ( ring_1_of_int @ A @ ( semiring_1_of_nat @ int @ N ) )
= ( semiring_1_of_nat @ A @ N ) ) ) ).
% of_int_of_nat_eq
thf(fact_297_ceiling__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archimedean_ceiling @ A @ ( one_one @ A ) )
= ( one_one @ int ) ) ) ).
% ceiling_one
thf(fact_298_ceiling__of__nat,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [N: nat] :
( ( archimedean_ceiling @ A @ ( semiring_1_of_nat @ A @ N ) )
= ( semiring_1_of_nat @ int @ N ) ) ) ).
% ceiling_of_nat
thf(fact_299_size__multiset__eq__0__iff__empty,axiom,
! [A: $tType,F: A > nat,M3: multiset @ A] :
( ( ( size_multiset @ A @ F @ M3 )
= ( zero_zero @ nat ) )
= ( M3
= ( zero_zero @ ( multiset @ A ) ) ) ) ).
% size_multiset_eq_0_iff_empty
thf(fact_300_size__multiset__empty,axiom,
! [A: $tType,F: A > nat] :
( ( size_multiset @ A @ F @ ( zero_zero @ ( multiset @ A ) ) )
= ( zero_zero @ nat ) ) ).
% size_multiset_empty
thf(fact_301_ceiling__add__of__int,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A,Z2: int] :
( ( archimedean_ceiling @ A @ ( plus_plus @ A @ X2 @ ( ring_1_of_int @ A @ Z2 ) ) )
= ( plus_plus @ int @ ( archimedean_ceiling @ A @ X2 ) @ Z2 ) ) ) ).
% ceiling_add_of_int
thf(fact_302_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_303_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_304_le__of__int__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A] : ( ord_less_eq @ A @ X2 @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ X2 ) ) ) ) ).
% le_of_int_ceiling
thf(fact_305_ceiling__le__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A,Z2: int] :
( ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X2 ) @ Z2 )
= ( ord_less_eq @ A @ X2 @ ( ring_1_of_int @ A @ Z2 ) ) ) ) ).
% ceiling_le_iff
thf(fact_306_ex__le__of__int,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X2: A] :
? [Z4: int] : ( ord_less_eq @ A @ X2 @ ( ring_1_of_int @ A @ Z4 ) ) ) ).
% ex_le_of_int
thf(fact_307_ceiling__mono,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Y: A,X2: A] :
( ( ord_less_eq @ A @ Y @ X2 )
=> ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ Y ) @ ( archimedean_ceiling @ A @ X2 ) ) ) ) ).
% ceiling_mono
thf(fact_308_ceiling__add__le,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A,Y: A] : ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ ( plus_plus @ A @ X2 @ Y ) ) @ ( plus_plus @ int @ ( archimedean_ceiling @ A @ X2 ) @ ( archimedean_ceiling @ A @ Y ) ) ) ) ).
% ceiling_add_le
thf(fact_309_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_310_size__multiset__overloaded__def,axiom,
! [B: $tType] :
( ( size_size @ ( multiset @ B ) )
= ( size_multiset @ B
@ ^ [Uu: B] : ( zero_zero @ nat ) ) ) ).
% size_multiset_overloaded_def
thf(fact_311_change__def,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( ( ref_change @ A )
= ( ^ [F2: A > A,R2: ref @ A] :
( heap_Time_bind @ A @ A @ ( ref_lookup @ A @ R2 )
@ ^ [X: A] :
( heap_Time_bind @ product_unit @ A @ ( ref_update @ A @ R2 @ ( F2 @ X ) )
@ ^ [Uu: product_unit] : ( heap_Time_return @ A @ ( F2 @ X ) ) ) ) ) ) ) ).
% change_def
thf(fact_312_one__le__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A] :
( ( ord_less_eq @ int @ ( one_one @ int ) @ ( archimedean_ceiling @ A @ X2 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ X2 ) ) ) ).
% one_le_ceiling
thf(fact_313_ceiling__less__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A] :
( ( ord_less @ int @ ( archimedean_ceiling @ A @ X2 ) @ ( one_one @ int ) )
= ( ord_less_eq @ A @ X2 @ ( zero_zero @ A ) ) ) ) ).
% ceiling_less_one
thf(fact_314_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_315_one__le__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A] :
( ( ord_less_eq @ int @ ( one_one @ int ) @ ( archim6421214686448440834_floor @ A @ X2 ) )
= ( ord_less_eq @ A @ ( one_one @ A ) @ X2 ) ) ) ).
% one_le_floor
thf(fact_316_arg__min__on__def,axiom,
! [A: $tType,B: $tType] :
( ( ord @ A )
=> ( ( lattic7623131987881927897min_on @ B @ A )
= ( ^ [F2: B > A,S: set @ B] :
( lattices_ord_arg_min @ B @ A @ F2
@ ^ [X: B] : ( member @ B @ X @ S ) ) ) ) ) ).
% arg_min_on_def
thf(fact_317_power__int__decreasing,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [N: int,N5: int,A2: A] :
( ( ord_less_eq @ int @ N @ N5 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ A2 @ ( one_one @ A ) )
=> ( ( ( A2
!= ( zero_zero @ A ) )
| ( N5
!= ( zero_zero @ int ) )
| ( N
= ( zero_zero @ int ) ) )
=> ( ord_less_eq @ A @ ( power_int @ A @ A2 @ N5 ) @ ( power_int @ A @ A2 @ N ) ) ) ) ) ) ) ).
% power_int_decreasing
thf(fact_318_power__int__le__one,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X2: A,N: int] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X2 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N )
=> ( ( ord_less_eq @ A @ X2 @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( power_int @ A @ X2 @ N ) @ ( one_one @ A ) ) ) ) ) ) ).
% power_int_le_one
thf(fact_319_zero__le__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( archim6421214686448440834_floor @ A @ X2 ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ X2 ) ) ) ).
% zero_le_floor
thf(fact_320_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_321_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_322_bot__nat__0_Onot__eq__extremum,axiom,
! [A2: nat] :
( ( A2
!= ( zero_zero @ nat ) )
= ( ord_less @ nat @ ( zero_zero @ nat ) @ A2 ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_323_neq0__conv,axiom,
! [N: nat] :
( ( N
!= ( zero_zero @ nat ) )
= ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% neq0_conv
thf(fact_324_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% less_nat_zero_code
thf(fact_325_nat__add__left__cancel__less,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less @ nat @ ( plus_plus @ nat @ K @ M ) @ ( plus_plus @ nat @ K @ N ) )
= ( ord_less @ nat @ M @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_326_not__gr__zero,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [N: A] :
( ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ N ) )
= ( N
= ( zero_zero @ A ) ) ) ) ).
% not_gr_zero
thf(fact_327_add__less__cancel__left,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ C2 @ A2 ) @ ( plus_plus @ A @ C2 @ B2 ) )
= ( ord_less @ A @ A2 @ B2 ) ) ) ).
% add_less_cancel_left
thf(fact_328_add__less__cancel__right,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) )
= ( ord_less @ A @ A2 @ B2 ) ) ) ).
% add_less_cancel_right
thf(fact_329_of__nat__less__iff,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [M: nat,N: nat] :
( ( ord_less @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) )
= ( ord_less @ nat @ M @ N ) ) ) ).
% of_nat_less_iff
thf(fact_330_add__gr__0,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( plus_plus @ nat @ M @ N ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
| ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% add_gr_0
thf(fact_331_less__one,axiom,
! [N: nat] :
( ( ord_less @ nat @ N @ ( one_one @ nat ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% less_one
thf(fact_332_power__int__1__left,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [N: int] :
( ( power_int @ A @ ( one_one @ A ) @ N )
= ( one_one @ A ) ) ) ).
% power_int_1_left
thf(fact_333_nat__int,axiom,
! [N: nat] :
( ( nat2 @ ( semiring_1_of_nat @ int @ N ) )
= N ) ).
% nat_int
thf(fact_334_power__int__1__right,axiom,
! [A: $tType] :
( ( ( inverse @ A )
& ( monoid_mult @ A ) )
=> ! [Y: A] :
( ( power_int @ A @ Y @ ( one_one @ int ) )
= Y ) ) ).
% power_int_1_right
thf(fact_335_floor__of__int,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z2: int] :
( ( archim6421214686448440834_floor @ A @ ( ring_1_of_int @ A @ Z2 ) )
= Z2 ) ) ).
% floor_of_int
thf(fact_336_length__rotate1,axiom,
! [A: $tType,Xs: list @ A] :
( ( size_size @ ( list @ A ) @ ( rotate1 @ A @ Xs ) )
= ( size_size @ ( list @ A ) @ Xs ) ) ).
% length_rotate1
thf(fact_337_add__less__same__cancel1,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [B2: A,A2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ B2 @ A2 ) @ B2 )
= ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% add_less_same_cancel1
thf(fact_338_add__less__same__cancel2,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A2 @ B2 ) @ B2 )
= ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% add_less_same_cancel2
thf(fact_339_less__add__same__cancel1,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ ( plus_plus @ A @ A2 @ B2 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ).
% less_add_same_cancel1
thf(fact_340_less__add__same__cancel2,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ ( plus_plus @ A @ B2 @ A2 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ).
% less_add_same_cancel2
thf(fact_341_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A2 @ A2 ) @ ( zero_zero @ A ) )
= ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_342_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ A2 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_343_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_344_power__int__eq__0__iff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X2: A,N: int] :
( ( ( power_int @ A @ X2 @ N )
= ( zero_zero @ A ) )
= ( ( X2
= ( zero_zero @ A ) )
& ( N
!= ( zero_zero @ int ) ) ) ) ) ).
% power_int_eq_0_iff
thf(fact_345_power__int__0__left,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [M: int] :
( ( M
!= ( zero_zero @ int ) )
=> ( ( power_int @ A @ ( zero_zero @ A ) @ M )
= ( zero_zero @ A ) ) ) ) ).
% power_int_0_left
thf(fact_346_floor__zero,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archim6421214686448440834_floor @ A @ ( zero_zero @ A ) )
= ( zero_zero @ int ) ) ) ).
% floor_zero
thf(fact_347_power__int__0__right,axiom,
! [B: $tType] :
( ( ( inverse @ B )
& ( power @ B ) )
=> ! [X2: B] :
( ( power_int @ B @ X2 @ ( zero_zero @ int ) )
= ( one_one @ B ) ) ) ).
% power_int_0_right
thf(fact_348_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_349_floor__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archim6421214686448440834_floor @ A @ ( one_one @ A ) )
= ( one_one @ int ) ) ) ).
% floor_one
thf(fact_350_floor__of__nat,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [N: nat] :
( ( archim6421214686448440834_floor @ A @ ( semiring_1_of_nat @ A @ N ) )
= ( semiring_1_of_nat @ int @ N ) ) ) ).
% floor_of_nat
thf(fact_351_of__nat__0__less__iff,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( semiring_1_of_nat @ A @ N ) )
= ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% of_nat_0_less_iff
thf(fact_352_nat__0__iff,axiom,
! [I: int] :
( ( ( nat2 @ I )
= ( zero_zero @ nat ) )
= ( ord_less_eq @ int @ I @ ( zero_zero @ int ) ) ) ).
% nat_0_iff
thf(fact_353_nat__le__0,axiom,
! [Z2: int] :
( ( ord_less_eq @ int @ Z2 @ ( zero_zero @ int ) )
=> ( ( nat2 @ Z2 )
= ( zero_zero @ nat ) ) ) ).
% nat_le_0
thf(fact_354_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_355_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_356_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_357_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_358_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_359_power__int__mono__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,N: int] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ N )
=> ( ( ord_less_eq @ A @ ( power_int @ A @ A2 @ N ) @ ( power_int @ A @ B2 @ N ) )
= ( ord_less_eq @ A @ A2 @ B2 ) ) ) ) ) ) ).
% power_int_mono_iff
thf(fact_360_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_361_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_362_floor__less__zero,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A] :
( ( ord_less @ int @ ( archim6421214686448440834_floor @ A @ X2 ) @ ( zero_zero @ int ) )
= ( ord_less @ A @ X2 @ ( zero_zero @ A ) ) ) ) ).
% floor_less_zero
thf(fact_363_zero__less__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A] :
( ( ord_less @ int @ ( zero_zero @ int ) @ ( archim6421214686448440834_floor @ A @ X2 ) )
= ( ord_less_eq @ A @ ( one_one @ A ) @ X2 ) ) ) ).
% zero_less_floor
thf(fact_364_floor__le__zero,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A] :
( ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X2 ) @ ( zero_zero @ int ) )
= ( ord_less @ A @ X2 @ ( one_one @ A ) ) ) ) ).
% floor_le_zero
thf(fact_365_zero__less__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A] :
( ( ord_less @ int @ ( zero_zero @ int ) @ ( archimedean_ceiling @ A @ X2 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ X2 ) ) ) ).
% zero_less_ceiling
thf(fact_366_floor__less__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A] :
( ( ord_less @ int @ ( archim6421214686448440834_floor @ A @ X2 ) @ ( one_one @ int ) )
= ( ord_less @ A @ X2 @ ( one_one @ A ) ) ) ) ).
% floor_less_one
thf(fact_367_one__less__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A] :
( ( ord_less @ int @ ( one_one @ int ) @ ( archimedean_ceiling @ A @ X2 ) )
= ( ord_less @ A @ ( one_one @ A ) @ X2 ) ) ) ).
% one_less_ceiling
thf(fact_368_neq__if__length__neq,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs )
!= ( size_size @ ( list @ A ) @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_369_Ex__list__of__length,axiom,
! [A: $tType,N: nat] :
? [Xs2: list @ A] :
( ( size_size @ ( list @ A ) @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_370_length__induct,axiom,
! [A: $tType,P: ( list @ A ) > $o,Xs: list @ A] :
( ! [Xs2: list @ A] :
( ! [Ys2: list @ A] :
( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Ys2 ) @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( P @ Ys2 ) )
=> ( P @ Xs2 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_371_floor__less__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A,Z2: int] :
( ( ord_less @ int @ ( archim6421214686448440834_floor @ A @ X2 ) @ Z2 )
= ( ord_less @ A @ X2 @ ( ring_1_of_int @ A @ Z2 ) ) ) ) ).
% floor_less_iff
thf(fact_372_minf_I11_J,axiom,
! [C: $tType,D: $tType] :
( ( ord @ C )
=> ! [F4: D] :
? [Z4: C] :
! [X4: C] :
( ( ord_less @ C @ X4 @ Z4 )
=> ( F4 = F4 ) ) ) ).
% minf(11)
thf(fact_373_minf_I7_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ X4 @ Z4 )
=> ~ ( ord_less @ A @ T2 @ X4 ) ) ) ).
% minf(7)
thf(fact_374_minf_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ X4 @ Z4 )
=> ( ord_less @ A @ X4 @ T2 ) ) ) ).
% minf(5)
thf(fact_375_minf_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ X4 @ Z4 )
=> ( X4 != T2 ) ) ) ).
% minf(4)
thf(fact_376_minf_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ X4 @ Z4 )
=> ( X4 != T2 ) ) ) ).
% minf(3)
thf(fact_377_minf_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o,P2: A > $o,Q: A > $o,Q3: A > $o] :
( ? [Z5: A] :
! [X3: A] :
( ( ord_less @ A @ X3 @ Z5 )
=> ( ( P @ X3 )
= ( P2 @ X3 ) ) )
=> ( ? [Z5: A] :
! [X3: A] :
( ( ord_less @ A @ X3 @ Z5 )
=> ( ( Q @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ X4 @ Z4 )
=> ( ( ( P @ X4 )
| ( Q @ X4 ) )
= ( ( P2 @ X4 )
| ( Q3 @ X4 ) ) ) ) ) ) ) ).
% minf(2)
thf(fact_378_minf_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o,P2: A > $o,Q: A > $o,Q3: A > $o] :
( ? [Z5: A] :
! [X3: A] :
( ( ord_less @ A @ X3 @ Z5 )
=> ( ( P @ X3 )
= ( P2 @ X3 ) ) )
=> ( ? [Z5: A] :
! [X3: A] :
( ( ord_less @ A @ X3 @ Z5 )
=> ( ( Q @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ X4 @ Z4 )
=> ( ( ( P @ X4 )
& ( Q @ X4 ) )
= ( ( P2 @ X4 )
& ( Q3 @ X4 ) ) ) ) ) ) ) ).
% minf(1)
thf(fact_379_pinf_I11_J,axiom,
! [C: $tType,D: $tType] :
( ( ord @ C )
=> ! [F4: D] :
? [Z4: C] :
! [X4: C] :
( ( ord_less @ C @ Z4 @ X4 )
=> ( F4 = F4 ) ) ) ).
% pinf(11)
thf(fact_380_pinf_I7_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ Z4 @ X4 )
=> ( ord_less @ A @ T2 @ X4 ) ) ) ).
% pinf(7)
thf(fact_381_pinf_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ Z4 @ X4 )
=> ~ ( ord_less @ A @ X4 @ T2 ) ) ) ).
% pinf(5)
thf(fact_382_pinf_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ Z4 @ X4 )
=> ( X4 != T2 ) ) ) ).
% pinf(4)
thf(fact_383_pinf_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ Z4 @ X4 )
=> ( X4 != T2 ) ) ) ).
% pinf(3)
thf(fact_384_pinf_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o,P2: A > $o,Q: A > $o,Q3: A > $o] :
( ? [Z5: A] :
! [X3: A] :
( ( ord_less @ A @ Z5 @ X3 )
=> ( ( P @ X3 )
= ( P2 @ X3 ) ) )
=> ( ? [Z5: A] :
! [X3: A] :
( ( ord_less @ A @ Z5 @ X3 )
=> ( ( Q @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ Z4 @ X4 )
=> ( ( ( P @ X4 )
| ( Q @ X4 ) )
= ( ( P2 @ X4 )
| ( Q3 @ X4 ) ) ) ) ) ) ) ).
% pinf(2)
thf(fact_385_pinf_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o,P2: A > $o,Q: A > $o,Q3: A > $o] :
( ? [Z5: A] :
! [X3: A] :
( ( ord_less @ A @ Z5 @ X3 )
=> ( ( P @ X3 )
= ( P2 @ X3 ) ) )
=> ( ? [Z5: A] :
! [X3: A] :
( ( ord_less @ A @ Z5 @ X3 )
=> ( ( Q @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ Z4 @ X4 )
=> ( ( ( P @ X4 )
& ( Q @ X4 ) )
= ( ( P2 @ X4 )
& ( Q3 @ X4 ) ) ) ) ) ) ) ).
% pinf(1)
thf(fact_386_lt__ex,axiom,
! [A: $tType] :
( ( no_bot @ A )
=> ! [X2: A] :
? [Y2: A] : ( ord_less @ A @ Y2 @ X2 ) ) ).
% lt_ex
thf(fact_387_gt__ex,axiom,
! [A: $tType] :
( ( no_top @ A )
=> ! [X2: A] :
? [X_12: A] : ( ord_less @ A @ X2 @ X_12 ) ) ).
% gt_ex
thf(fact_388_dense,axiom,
! [A: $tType] :
( ( dense_order @ A )
=> ! [X2: A,Y: A] :
( ( ord_less @ A @ X2 @ Y )
=> ? [Z4: A] :
( ( ord_less @ A @ X2 @ Z4 )
& ( ord_less @ A @ Z4 @ Y ) ) ) ) ).
% dense
thf(fact_389_less__imp__neq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X2: A,Y: A] :
( ( ord_less @ A @ X2 @ Y )
=> ( X2 != Y ) ) ) ).
% less_imp_neq
thf(fact_390_order_Oasym,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ~ ( ord_less @ A @ B2 @ A2 ) ) ) ).
% order.asym
thf(fact_391_ord__eq__less__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( A2 = B2 )
=> ( ( ord_less @ A @ B2 @ C2 )
=> ( ord_less @ A @ A2 @ C2 ) ) ) ) ).
% ord_eq_less_trans
thf(fact_392_ord__less__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( B2 = C2 )
=> ( ord_less @ A @ A2 @ C2 ) ) ) ) ).
% ord_less_eq_trans
thf(fact_393_less__induct,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P: A > $o,A2: A] :
( ! [X3: A] :
( ! [Y5: A] :
( ( ord_less @ A @ Y5 @ X3 )
=> ( P @ Y5 ) )
=> ( P @ X3 ) )
=> ( P @ A2 ) ) ) ).
% less_induct
thf(fact_394_antisym__conv3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Y: A,X2: A] :
( ~ ( ord_less @ A @ Y @ X2 )
=> ( ( ~ ( ord_less @ A @ X2 @ Y ) )
= ( X2 = Y ) ) ) ) ).
% antisym_conv3
thf(fact_395_linorder__cases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y: A] :
( ~ ( ord_less @ A @ X2 @ Y )
=> ( ( X2 != Y )
=> ( ord_less @ A @ Y @ X2 ) ) ) ) ).
% linorder_cases
thf(fact_396_dual__order_Oasym,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B2: A,A2: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ~ ( ord_less @ A @ A2 @ B2 ) ) ) ).
% dual_order.asym
thf(fact_397_dual__order_Oirrefl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A] :
~ ( ord_less @ A @ A2 @ A2 ) ) ).
% dual_order.irrefl
thf(fact_398_exists__least__iff,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ( ( ^ [P5: A > $o] :
? [X6: A] : ( P5 @ X6 ) )
= ( ^ [P4: A > $o] :
? [N4: A] :
( ( P4 @ N4 )
& ! [M2: A] :
( ( ord_less @ A @ M2 @ N4 )
=> ~ ( P4 @ M2 ) ) ) ) ) ) ).
% exists_least_iff
thf(fact_399_linorder__less__wlog,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > A > $o,A2: A,B2: A] :
( ! [A4: A,B4: A] :
( ( ord_less @ A @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: A] : ( P @ A4 @ A4 )
=> ( ! [A4: A,B4: A] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A2 @ B2 ) ) ) ) ) ).
% linorder_less_wlog
thf(fact_400_order_Ostrict__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ B2 @ C2 )
=> ( ord_less @ A @ A2 @ C2 ) ) ) ) ).
% order.strict_trans
thf(fact_401_not__less__iff__gr__or__eq,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y: A] :
( ( ~ ( ord_less @ A @ X2 @ Y ) )
= ( ( ord_less @ A @ Y @ X2 )
| ( X2 = Y ) ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_402_dual__order_Ostrict__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ( ( ord_less @ A @ C2 @ B2 )
=> ( ord_less @ A @ C2 @ A2 ) ) ) ) ).
% dual_order.strict_trans
thf(fact_403_order_Ostrict__implies__not__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( A2 != B2 ) ) ) ).
% order.strict_implies_not_eq
thf(fact_404_dual__order_Ostrict__implies__not__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B2: A,A2: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ( A2 != B2 ) ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_405_measure__induct,axiom,
! [B: $tType,A: $tType] :
( ( wellorder @ B )
=> ! [F: A > B,P: A > $o,A2: A] :
( ! [X3: A] :
( ! [Y5: A] :
( ( ord_less @ B @ ( F @ Y5 ) @ ( F @ X3 ) )
=> ( P @ Y5 ) )
=> ( P @ X3 ) )
=> ( P @ A2 ) ) ) ).
% measure_induct
thf(fact_406_measure__induct__rule,axiom,
! [B: $tType,A: $tType] :
( ( wellorder @ B )
=> ! [F: A > B,P: A > $o,A2: A] :
( ! [X3: A] :
( ! [Y5: A] :
( ( ord_less @ B @ ( F @ Y5 ) @ ( F @ X3 ) )
=> ( P @ Y5 ) )
=> ( P @ X3 ) )
=> ( P @ A2 ) ) ) ).
% measure_induct_rule
thf(fact_407_linorder__neqE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y: A] :
( ( X2 != Y )
=> ( ~ ( ord_less @ A @ X2 @ Y )
=> ( ord_less @ A @ Y @ X2 ) ) ) ) ).
% linorder_neqE
thf(fact_408_order__less__asym,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A,Y: A] :
( ( ord_less @ A @ X2 @ Y )
=> ~ ( ord_less @ A @ Y @ X2 ) ) ) ).
% order_less_asym
thf(fact_409_linorder__neq__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y: A] :
( ( X2 != Y )
= ( ( ord_less @ A @ X2 @ Y )
| ( ord_less @ A @ Y @ X2 ) ) ) ) ).
% linorder_neq_iff
thf(fact_410_order__less__asym_H,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ~ ( ord_less @ A @ B2 @ A2 ) ) ) ).
% order_less_asym'
thf(fact_411_order__less__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A,Y: A,Z2: A] :
( ( ord_less @ A @ X2 @ Y )
=> ( ( ord_less @ A @ Y @ Z2 )
=> ( ord_less @ A @ X2 @ Z2 ) ) ) ) ).
% order_less_trans
thf(fact_412_ord__eq__less__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A2: A,F: B > A,B2: B,C2: B] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less @ B @ B2 @ C2 )
=> ( ! [X3: B,Y2: B] :
( ( ord_less @ B @ X3 @ Y2 )
=> ( ord_less @ A @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less @ A @ A2 @ ( F @ C2 ) ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_413_ord__less__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A2: A,B2: A,F: A > B,C2: B] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X3: A,Y2: A] :
( ( ord_less @ A @ X3 @ Y2 )
=> ( ord_less @ B @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less @ B @ ( F @ A2 ) @ C2 ) ) ) ) ) ).
% ord_less_eq_subst
thf(fact_414_order__less__irrefl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A] :
~ ( ord_less @ A @ X2 @ X2 ) ) ).
% order_less_irrefl
thf(fact_415_order__less__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A2: A,F: B > A,B2: B,C2: B] :
( ( ord_less @ A @ A2 @ ( F @ B2 ) )
=> ( ( ord_less @ B @ B2 @ C2 )
=> ( ! [X3: B,Y2: B] :
( ( ord_less @ B @ X3 @ Y2 )
=> ( ord_less @ A @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less @ A @ A2 @ ( F @ C2 ) ) ) ) ) ) ).
% order_less_subst1
thf(fact_416_order__less__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A2: A,B2: A,F: A > C,C2: C] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ C @ ( F @ B2 ) @ C2 )
=> ( ! [X3: A,Y2: A] :
( ( ord_less @ A @ X3 @ Y2 )
=> ( ord_less @ C @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less @ C @ ( F @ A2 ) @ C2 ) ) ) ) ) ).
% order_less_subst2
thf(fact_417_order__less__not__sym,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A,Y: A] :
( ( ord_less @ A @ X2 @ Y )
=> ~ ( ord_less @ A @ Y @ X2 ) ) ) ).
% order_less_not_sym
thf(fact_418_order__less__imp__triv,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A,Y: A,P: $o] :
( ( ord_less @ A @ X2 @ Y )
=> ( ( ord_less @ A @ Y @ X2 )
=> P ) ) ) ).
% order_less_imp_triv
thf(fact_419_linorder__less__linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y: A] :
( ( ord_less @ A @ X2 @ Y )
| ( X2 = Y )
| ( ord_less @ A @ Y @ X2 ) ) ) ).
% linorder_less_linear
thf(fact_420_order__less__imp__not__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X2: A,Y: A] :
( ( ord_less @ A @ X2 @ Y )
=> ( X2 != Y ) ) ) ).
% order_less_imp_not_eq
thf(fact_421_order__less__imp__not__eq2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X2: A,Y: A] :
( ( ord_less @ A @ X2 @ Y )
=> ( Y != X2 ) ) ) ).
% order_less_imp_not_eq2
thf(fact_422_order__less__imp__not__less,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A,Y: A] :
( ( ord_less @ A @ X2 @ Y )
=> ~ ( ord_less @ A @ Y @ X2 ) ) ) ).
% order_less_imp_not_less
thf(fact_423_verit__comp__simplify1_I1_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A] :
~ ( ord_less @ A @ A2 @ A2 ) ) ).
% verit_comp_simplify1(1)
thf(fact_424_floor__less__cancel,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A,Y: A] :
( ( ord_less @ int @ ( archim6421214686448440834_floor @ A @ X2 ) @ ( archim6421214686448440834_floor @ A @ Y ) )
=> ( ord_less @ A @ X2 @ Y ) ) ) ).
% floor_less_cancel
thf(fact_425_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_426_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_427_power__int__strict__increasing,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [N: int,N5: int,A2: A] :
( ( ord_less @ int @ N @ N5 )
=> ( ( ord_less @ A @ ( one_one @ A ) @ A2 )
=> ( ord_less @ A @ ( power_int @ A @ A2 @ N ) @ ( power_int @ A @ A2 @ N5 ) ) ) ) ) ).
% power_int_strict_increasing
thf(fact_428_zero__less__power__int,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X2: A,N: int] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X2 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( power_int @ A @ X2 @ N ) ) ) ) ).
% zero_less_power_int
thf(fact_429_less__imp__of__nat__less,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ( ord_less @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% less_imp_of_nat_less
thf(fact_430_of__nat__less__imp__less,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [M: nat,N: nat] :
( ( ord_less @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) )
=> ( ord_less @ nat @ M @ N ) ) ) ).
% of_nat_less_imp_less
thf(fact_431_nat__int__comparison_I2_J,axiom,
( ( ord_less @ nat )
= ( ^ [A3: nat,B3: nat] : ( ord_less @ int @ ( semiring_1_of_nat @ int @ A3 ) @ ( semiring_1_of_nat @ int @ B3 ) ) ) ) ).
% nat_int_comparison(2)
thf(fact_432_power__int__strict__decreasing,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [N: int,N5: int,A2: A] :
( ( ord_less @ int @ N @ N5 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ A2 @ ( one_one @ A ) )
=> ( ord_less @ A @ ( power_int @ A @ A2 @ N5 ) @ ( power_int @ A @ A2 @ N ) ) ) ) ) ) ).
% power_int_strict_decreasing
thf(fact_433_power__int__strict__antimono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,N: int] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ int @ N @ ( zero_zero @ int ) )
=> ( ord_less @ A @ ( power_int @ A @ B2 @ N ) @ ( power_int @ A @ A2 @ N ) ) ) ) ) ) ).
% power_int_strict_antimono
thf(fact_434_ceiling__less__cancel,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A,Y: A] :
( ( ord_less @ int @ ( archimedean_ceiling @ A @ X2 ) @ ( archimedean_ceiling @ A @ Y ) )
=> ( ord_less @ A @ X2 @ Y ) ) ) ).
% ceiling_less_cancel
thf(fact_435_one__less__power__int,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,N: int] :
( ( ord_less @ A @ ( one_one @ A ) @ A2 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ N )
=> ( ord_less @ A @ ( one_one @ A ) @ ( power_int @ A @ A2 @ N ) ) ) ) ) ).
% one_less_power_int
thf(fact_436_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_437_nat__less__as__int,axiom,
( ( ord_less @ nat )
= ( ^ [A3: nat,B3: nat] : ( ord_less @ int @ ( semiring_1_of_nat @ int @ A3 ) @ ( semiring_1_of_nat @ int @ B3 ) ) ) ) ).
% nat_less_as_int
thf(fact_438_power__int__antimono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,N: int] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ int @ N @ ( zero_zero @ int ) )
=> ( ord_less_eq @ A @ ( power_int @ A @ B2 @ N ) @ ( power_int @ A @ A2 @ N ) ) ) ) ) ) ).
% power_int_antimono
thf(fact_439_power__int__strict__mono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,N: int] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ N )
=> ( ord_less @ A @ ( power_int @ A @ A2 @ N ) @ ( power_int @ A @ B2 @ N ) ) ) ) ) ) ).
% power_int_strict_mono
thf(fact_440_power__int__le__imp__less__exp,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X2: A,M: int,N: int] :
( ( ord_less @ A @ ( one_one @ A ) @ X2 )
=> ( ( ord_less @ A @ ( power_int @ A @ X2 @ M ) @ ( power_int @ A @ X2 @ N ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N )
=> ( ord_less @ int @ M @ N ) ) ) ) ) ).
% power_int_le_imp_less_exp
thf(fact_441_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_442_less__ceiling__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z2: int,X2: A] :
( ( ord_less @ int @ Z2 @ ( archimedean_ceiling @ A @ X2 ) )
= ( ord_less @ A @ ( ring_1_of_int @ A @ Z2 ) @ X2 ) ) ) ).
% less_ceiling_iff
thf(fact_443_verit__comp__simplify1_I3_J,axiom,
! [B: $tType] :
( ( linorder @ B )
=> ! [B6: B,A6: B] :
( ( ~ ( ord_less_eq @ B @ B6 @ A6 ) )
= ( ord_less @ B @ A6 @ B6 ) ) ) ).
% verit_comp_simplify1(3)
thf(fact_444_leD,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y: A,X2: A] :
( ( ord_less_eq @ A @ Y @ X2 )
=> ~ ( ord_less @ A @ X2 @ Y ) ) ) ).
% leD
thf(fact_445_leI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y: A] :
( ~ ( ord_less @ A @ X2 @ Y )
=> ( ord_less_eq @ A @ Y @ X2 ) ) ) ).
% leI
thf(fact_446_nless__le,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B2: A] :
( ( ~ ( ord_less @ A @ A2 @ B2 ) )
= ( ~ ( ord_less_eq @ A @ A2 @ B2 )
| ( A2 = B2 ) ) ) ) ).
% nless_le
thf(fact_447_antisym__conv1,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X2: A,Y: A] :
( ~ ( ord_less @ A @ X2 @ Y )
=> ( ( ord_less_eq @ A @ X2 @ Y )
= ( X2 = Y ) ) ) ) ).
% antisym_conv1
thf(fact_448_antisym__conv2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X2: A,Y: A] :
( ( ord_less_eq @ A @ X2 @ Y )
=> ( ( ~ ( ord_less @ A @ X2 @ Y ) )
= ( X2 = Y ) ) ) ) ).
% antisym_conv2
thf(fact_449_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_450_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_451_less__le__not__le,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( ord_less @ A )
= ( ^ [X: A,Y4: A] :
( ( ord_less_eq @ A @ X @ Y4 )
& ~ ( ord_less_eq @ A @ Y4 @ X ) ) ) ) ) ).
% less_le_not_le
thf(fact_452_not__le__imp__less,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Y: A,X2: A] :
( ~ ( ord_less_eq @ A @ Y @ X2 )
=> ( ord_less @ A @ X2 @ Y ) ) ) ).
% not_le_imp_less
thf(fact_453_order_Oorder__iff__strict,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A3: A,B3: A] :
( ( ord_less @ A @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ) ).
% order.order_iff_strict
thf(fact_454_order_Ostrict__iff__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less @ A )
= ( ^ [A3: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ) ).
% order.strict_iff_order
thf(fact_455_order_Ostrict__trans1,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ B2 @ C2 )
=> ( ord_less @ A @ A2 @ C2 ) ) ) ) ).
% order.strict_trans1
thf(fact_456_order_Ostrict__trans2,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ C2 )
=> ( ord_less @ A @ A2 @ C2 ) ) ) ) ).
% order.strict_trans2
thf(fact_457_order_Ostrict__iff__not,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( ord_less @ A )
= ( ^ [A3: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
& ~ ( ord_less_eq @ A @ B3 @ A3 ) ) ) ) ) ).
% order.strict_iff_not
thf(fact_458_dense__ge__bounded,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [Z2: A,X2: A,Y: A] :
( ( ord_less @ A @ Z2 @ X2 )
=> ( ! [W3: A] :
( ( ord_less @ A @ Z2 @ W3 )
=> ( ( ord_less @ A @ W3 @ X2 )
=> ( ord_less_eq @ A @ Y @ W3 ) ) )
=> ( ord_less_eq @ A @ Y @ Z2 ) ) ) ) ).
% dense_ge_bounded
thf(fact_459_dense__le__bounded,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [X2: A,Y: A,Z2: A] :
( ( ord_less @ A @ X2 @ Y )
=> ( ! [W3: A] :
( ( ord_less @ A @ X2 @ W3 )
=> ( ( ord_less @ A @ W3 @ Y )
=> ( ord_less_eq @ A @ W3 @ Z2 ) ) )
=> ( ord_less_eq @ A @ Y @ Z2 ) ) ) ) ).
% dense_le_bounded
thf(fact_460_dual__order_Oorder__iff__strict,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B3: A,A3: A] :
( ( ord_less @ A @ B3 @ A3 )
| ( A3 = B3 ) ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_461_dual__order_Ostrict__iff__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less @ A )
= ( ^ [B3: A,A3: A] :
( ( ord_less_eq @ A @ B3 @ A3 )
& ( A3 != B3 ) ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_462_dual__order_Ostrict__trans1,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less @ A @ C2 @ B2 )
=> ( ord_less @ A @ C2 @ A2 ) ) ) ) ).
% dual_order.strict_trans1
thf(fact_463_dual__order_Ostrict__trans2,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ C2 @ B2 )
=> ( ord_less @ A @ C2 @ A2 ) ) ) ) ).
% dual_order.strict_trans2
thf(fact_464_dual__order_Ostrict__iff__not,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( ord_less @ A )
= ( ^ [B3: A,A3: A] :
( ( ord_less_eq @ A @ B3 @ A3 )
& ~ ( ord_less_eq @ A @ A3 @ B3 ) ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_465_order_Ostrict__implies__order,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% order.strict_implies_order
thf(fact_466_dual__order_Ostrict__implies__order,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B2: A,A2: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ( ord_less_eq @ A @ B2 @ A2 ) ) ) ).
% dual_order.strict_implies_order
thf(fact_467_order__le__less,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [X: A,Y4: A] :
( ( ord_less @ A @ X @ Y4 )
| ( X = Y4 ) ) ) ) ) ).
% order_le_less
thf(fact_468_order__less__le,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less @ A )
= ( ^ [X: A,Y4: A] :
( ( ord_less_eq @ A @ X @ Y4 )
& ( X != Y4 ) ) ) ) ) ).
% order_less_le
thf(fact_469_linorder__not__le,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y: A] :
( ( ~ ( ord_less_eq @ A @ X2 @ Y ) )
= ( ord_less @ A @ Y @ X2 ) ) ) ).
% linorder_not_le
thf(fact_470_linorder__not__less,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y: A] :
( ( ~ ( ord_less @ A @ X2 @ Y ) )
= ( ord_less_eq @ A @ Y @ X2 ) ) ) ).
% linorder_not_less
thf(fact_471_order__less__imp__le,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A,Y: A] :
( ( ord_less @ A @ X2 @ Y )
=> ( ord_less_eq @ A @ X2 @ Y ) ) ) ).
% order_less_imp_le
thf(fact_472_order__le__neq__trans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less @ A @ A2 @ B2 ) ) ) ) ).
% order_le_neq_trans
thf(fact_473_order__neq__le__trans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B2: A] :
( ( A2 != B2 )
=> ( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ord_less @ A @ A2 @ B2 ) ) ) ) ).
% order_neq_le_trans
thf(fact_474_order__le__less__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A,Y: A,Z2: A] :
( ( ord_less_eq @ A @ X2 @ Y )
=> ( ( ord_less @ A @ Y @ Z2 )
=> ( ord_less @ A @ X2 @ Z2 ) ) ) ) ).
% order_le_less_trans
thf(fact_475_order__less__le__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A,Y: A,Z2: A] :
( ( ord_less @ A @ X2 @ Y )
=> ( ( ord_less_eq @ A @ Y @ Z2 )
=> ( ord_less @ A @ X2 @ Z2 ) ) ) ) ).
% order_less_le_trans
thf(fact_476_order__le__less__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A2: A,F: B > A,B2: B,C2: B] :
( ( ord_less_eq @ A @ A2 @ ( F @ B2 ) )
=> ( ( ord_less @ B @ B2 @ C2 )
=> ( ! [X3: B,Y2: B] :
( ( ord_less @ B @ X3 @ Y2 )
=> ( ord_less @ A @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less @ A @ A2 @ ( F @ C2 ) ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_477_order__le__less__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A2: A,B2: A,F: A > C,C2: C] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less @ C @ ( F @ B2 ) @ C2 )
=> ( ! [X3: A,Y2: A] :
( ( ord_less_eq @ A @ X3 @ Y2 )
=> ( ord_less_eq @ C @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less @ C @ ( F @ A2 ) @ C2 ) ) ) ) ) ).
% order_le_less_subst2
thf(fact_478_order__less__le__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A2: A,F: B > A,B2: B,C2: B] :
( ( ord_less @ A @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq @ B @ B2 @ C2 )
=> ( ! [X3: B,Y2: B] :
( ( ord_less_eq @ B @ X3 @ Y2 )
=> ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less @ A @ A2 @ ( F @ C2 ) ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_479_order__less__le__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A2: A,B2: A,F: A > C,C2: C] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ C @ ( F @ B2 ) @ C2 )
=> ( ! [X3: A,Y2: A] :
( ( ord_less @ A @ X3 @ Y2 )
=> ( ord_less @ C @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less @ C @ ( F @ A2 ) @ C2 ) ) ) ) ) ).
% order_less_le_subst2
thf(fact_480_linorder__le__less__linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y: A] :
( ( ord_less_eq @ A @ X2 @ Y )
| ( ord_less @ A @ Y @ X2 ) ) ) ).
% linorder_le_less_linear
thf(fact_481_order__le__imp__less__or__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X2: A,Y: A] :
( ( ord_less_eq @ A @ X2 @ Y )
=> ( ( ord_less @ A @ X2 @ Y )
| ( X2 = Y ) ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_482_minf_I8_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ X4 @ Z4 )
=> ~ ( ord_less_eq @ A @ T2 @ X4 ) ) ) ).
% minf(8)
thf(fact_483_minf_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ X4 @ Z4 )
=> ( ord_less_eq @ A @ X4 @ T2 ) ) ) ).
% minf(6)
thf(fact_484_pinf_I8_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ Z4 @ X4 )
=> ( ord_less_eq @ A @ T2 @ X4 ) ) ) ).
% pinf(8)
thf(fact_485_pinf_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ Z4 @ X4 )
=> ~ ( ord_less_eq @ A @ X4 @ T2 ) ) ) ).
% pinf(6)
thf(fact_486_zero__less__iff__neq__zero,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [N: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ N )
= ( N
!= ( zero_zero @ A ) ) ) ) ).
% zero_less_iff_neq_zero
thf(fact_487_gr__implies__not__zero,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [M: A,N: A] :
( ( ord_less @ A @ M @ N )
=> ( N
!= ( zero_zero @ A ) ) ) ) ).
% gr_implies_not_zero
thf(fact_488_not__less__zero,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [N: A] :
~ ( ord_less @ A @ N @ ( zero_zero @ A ) ) ) ).
% not_less_zero
thf(fact_489_gr__zeroI,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [N: A] :
( ( N
!= ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ N ) ) ) ).
% gr_zeroI
thf(fact_490_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_491_add__mono__thms__linordered__field_I5_J,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( ord_less @ A @ I @ J )
& ( ord_less @ A @ K @ L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_492_add__mono__thms__linordered__field_I2_J,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( I = J )
& ( ord_less @ A @ K @ L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_493_add__mono__thms__linordered__field_I1_J,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( ord_less @ A @ I @ J )
& ( K = L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_494_add__strict__mono,axiom,
! [A: $tType] :
( ( strict9044650504122735259up_add @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ C2 @ D2 )
=> ( ord_less @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ D2 ) ) ) ) ) ).
% add_strict_mono
thf(fact_495_add__strict__left__mono,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ord_less @ A @ ( plus_plus @ A @ C2 @ A2 ) @ ( plus_plus @ A @ C2 @ B2 ) ) ) ) ).
% add_strict_left_mono
thf(fact_496_add__strict__right__mono,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ord_less @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% add_strict_right_mono
thf(fact_497_add__less__imp__less__left,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ C2 @ A2 ) @ ( plus_plus @ A @ C2 @ B2 ) )
=> ( ord_less @ A @ A2 @ B2 ) ) ) ).
% add_less_imp_less_left
thf(fact_498_add__less__imp__less__right,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) )
=> ( ord_less @ A @ A2 @ B2 ) ) ) ).
% add_less_imp_less_right
thf(fact_499_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_500_bot__nat__0_Oextremum__strict,axiom,
! [A2: nat] :
~ ( ord_less @ nat @ A2 @ ( zero_zero @ nat ) ) ).
% bot_nat_0.extremum_strict
thf(fact_501_gr0I,axiom,
! [N: nat] :
( ( N
!= ( zero_zero @ nat ) )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% gr0I
thf(fact_502_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% not_gr0
thf(fact_503_not__less0,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% not_less0
thf(fact_504_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% less_zeroE
thf(fact_505_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ( N
!= ( zero_zero @ nat ) ) ) ).
% gr_implies_not0
thf(fact_506_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ ( zero_zero @ nat ) )
=> ( ! [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ~ ( P @ N2 )
=> ? [M7: nat] :
( ( ord_less @ nat @ M7 @ N2 )
& ~ ( P @ M7 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_507_infinite__descent0__measure,axiom,
! [A: $tType,V: A > nat,P: A > $o,X2: A] :
( ! [X3: A] :
( ( ( V @ X3 )
= ( zero_zero @ nat ) )
=> ( P @ X3 ) )
=> ( ! [X3: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( V @ X3 ) )
=> ( ~ ( P @ X3 )
=> ? [Y5: A] :
( ( ord_less @ nat @ ( V @ Y5 ) @ ( V @ X3 ) )
& ~ ( P @ Y5 ) ) ) )
=> ( P @ X2 ) ) ) ).
% infinite_descent0_measure
thf(fact_508_less__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ( ( ord_less @ ( A > B ) )
= ( ^ [F2: A > B,G2: A > B] :
( ( ord_less_eq @ ( A > B ) @ F2 @ G2 )
& ~ ( ord_less_eq @ ( A > B ) @ G2 @ F2 ) ) ) ) ) ).
% less_fun_def
thf(fact_509_reals__Archimedean2,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X2: A] :
? [N2: nat] : ( ord_less @ A @ X2 @ ( semiring_1_of_nat @ A @ N2 ) ) ) ).
% reals_Archimedean2
thf(fact_510_less__int__code_I1_J,axiom,
~ ( ord_less @ int @ ( zero_zero @ int ) @ ( zero_zero @ int ) ) ).
% less_int_code(1)
thf(fact_511_exists__leI,axiom,
! [N: nat,P: nat > $o] :
( ( ! [N7: nat] :
( ( ord_less @ nat @ N7 @ N )
=> ~ ( P @ N7 ) )
=> ( P @ N ) )
=> ? [N8: nat] :
( ( ord_less_eq @ nat @ N8 @ N )
& ( P @ N8 ) ) ) ).
% exists_leI
thf(fact_512_Lattices__Big_Oex__has__greatest__nat,axiom,
! [A: $tType,P: A > $o,K: A,F: A > nat,B2: nat] :
( ( P @ K )
=> ( ! [Y2: A] :
( ( P @ Y2 )
=> ( ord_less @ nat @ ( F @ Y2 ) @ B2 ) )
=> ? [X3: A] :
( ( P @ X3 )
& ! [Y5: A] :
( ( P @ Y5 )
=> ( ord_less_eq @ nat @ ( F @ Y5 ) @ ( F @ X3 ) ) ) ) ) ) ).
% Lattices_Big.ex_has_greatest_nat
thf(fact_513_nat__less__le,axiom,
( ( ord_less @ nat )
= ( ^ [M2: nat,N4: nat] :
( ( ord_less_eq @ nat @ M2 @ N4 )
& ( M2 != N4 ) ) ) ) ).
% nat_less_le
thf(fact_514_less__imp__le__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ( ord_less_eq @ nat @ M @ N ) ) ).
% less_imp_le_nat
thf(fact_515_le__eq__less__or__eq,axiom,
( ( ord_less_eq @ nat )
= ( ^ [M2: nat,N4: nat] :
( ( ord_less @ nat @ M2 @ N4 )
| ( M2 = N4 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_516_less__or__eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( ( ord_less @ nat @ M @ N )
| ( M = N ) )
=> ( ord_less_eq @ nat @ M @ N ) ) ).
% less_or_eq_imp_le
thf(fact_517_le__neq__implies__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( M != N )
=> ( ord_less @ nat @ M @ N ) ) ) ).
% le_neq_implies_less
thf(fact_518_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I3: nat,J2: nat] :
( ( ord_less @ nat @ I3 @ J2 )
=> ( ord_less @ nat @ ( F @ I3 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq @ nat @ I @ J )
=> ( ord_less_eq @ nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_519_arg__max__nat__le,axiom,
! [A: $tType,P: A > $o,X2: A,F: A > nat,B2: nat] :
( ( P @ X2 )
=> ( ! [Y2: A] :
( ( P @ Y2 )
=> ( ord_less @ nat @ ( F @ Y2 ) @ B2 ) )
=> ( ord_less_eq @ nat @ ( F @ X2 ) @ ( F @ ( lattices_ord_arg_max @ A @ nat @ F @ P ) ) ) ) ) ).
% arg_max_nat_le
thf(fact_520_arg__max__nat__lemma,axiom,
! [A: $tType,P: A > $o,K: A,F: A > nat,B2: nat] :
( ( P @ K )
=> ( ! [Y2: A] :
( ( P @ Y2 )
=> ( ord_less @ nat @ ( F @ Y2 ) @ B2 ) )
=> ( ( P @ ( lattices_ord_arg_max @ A @ nat @ F @ P ) )
& ! [Y5: A] :
( ( P @ Y5 )
=> ( ord_less_eq @ nat @ ( F @ Y5 ) @ ( F @ ( lattices_ord_arg_max @ A @ nat @ F @ P ) ) ) ) ) ) ) ).
% arg_max_nat_lemma
thf(fact_521_add__lessD1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less @ nat @ ( plus_plus @ nat @ I @ J ) @ K )
=> ( ord_less @ nat @ I @ K ) ) ).
% add_lessD1
thf(fact_522_add__less__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less @ nat @ I @ J )
=> ( ( ord_less @ nat @ K @ L )
=> ( ord_less @ nat @ ( plus_plus @ nat @ I @ K ) @ ( plus_plus @ nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_523_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less @ nat @ ( plus_plus @ nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_524_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less @ nat @ ( plus_plus @ nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_525_add__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less @ nat @ I @ J )
=> ( ord_less @ nat @ ( plus_plus @ nat @ I @ K ) @ ( plus_plus @ nat @ J @ K ) ) ) ).
% add_less_mono1
thf(fact_526_trans__less__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less @ nat @ I @ J )
=> ( ord_less @ nat @ I @ ( plus_plus @ nat @ J @ M ) ) ) ).
% trans_less_add1
thf(fact_527_trans__less__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less @ nat @ I @ J )
=> ( ord_less @ nat @ I @ ( plus_plus @ nat @ M @ J ) ) ) ).
% trans_less_add2
thf(fact_528_less__add__eq__less,axiom,
! [K: nat,L: nat,M: nat,N: nat] :
( ( ord_less @ nat @ K @ L )
=> ( ( ( plus_plus @ nat @ M @ L )
= ( plus_plus @ nat @ K @ N ) )
=> ( ord_less @ nat @ M @ N ) ) ) ).
% less_add_eq_less
thf(fact_529_union__less__mono,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A5: multiset @ A,C6: multiset @ A,B5: multiset @ A,D3: multiset @ A] :
( ( ord_less @ ( multiset @ A ) @ A5 @ C6 )
=> ( ( ord_less @ ( multiset @ A ) @ B5 @ D3 )
=> ( ord_less @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ A5 @ B5 ) @ ( plus_plus @ ( multiset @ A ) @ C6 @ D3 ) ) ) ) ) ).
% union_less_mono
thf(fact_530_union__le__mono2,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B5: multiset @ A,D3: multiset @ A,C6: multiset @ A] :
( ( ord_less @ ( multiset @ A ) @ B5 @ D3 )
=> ( ord_less @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ C6 @ B5 ) @ ( plus_plus @ ( multiset @ A ) @ C6 @ D3 ) ) ) ) ).
% union_le_mono2
thf(fact_531_union__le__mono1,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B5: multiset @ A,D3: multiset @ A,C6: multiset @ A] :
( ( ord_less @ ( multiset @ A ) @ B5 @ D3 )
=> ( ord_less @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ B5 @ C6 ) @ ( plus_plus @ ( multiset @ A ) @ D3 @ C6 ) ) ) ) ).
% union_le_mono1
thf(fact_532_ex__less__of__int,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X2: A] :
? [Z4: int] : ( ord_less @ A @ X2 @ ( ring_1_of_int @ A @ Z4 ) ) ) ).
% ex_less_of_int
thf(fact_533_ex__of__int__less,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X2: A] :
? [Z4: int] : ( ord_less @ A @ ( ring_1_of_int @ A @ Z4 ) @ X2 ) ) ).
% ex_of_int_less
thf(fact_534_of__nat__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [R: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ R )
=> ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ ( nat2 @ ( archim6421214686448440834_floor @ A @ R ) ) ) @ R ) ) ) ).
% of_nat_floor
thf(fact_535_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_536_split__nat,axiom,
! [P: nat > $o,I: int] :
( ( P @ ( nat2 @ I ) )
= ( ! [N4: nat] :
( ( I
= ( semiring_1_of_nat @ int @ N4 ) )
=> ( P @ N4 ) )
& ( ( ord_less @ int @ I @ ( zero_zero @ int ) )
=> ( P @ ( zero_zero @ nat ) ) ) ) ) ).
% split_nat
thf(fact_537_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_538_zero__less__imp__eq__int,axiom,
! [K: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ K )
=> ? [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
& ( K
= ( semiring_1_of_nat @ int @ N2 ) ) ) ) ).
% zero_less_imp_eq_int
thf(fact_539_pos__int__cases,axiom,
! [K: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ K )
=> ~ ! [N2: nat] :
( ( K
= ( semiring_1_of_nat @ int @ N2 ) )
=> ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ).
% pos_int_cases
thf(fact_540_of__nat__less__of__int__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: nat,X2: int] :
( ( ord_less @ A @ ( semiring_1_of_nat @ A @ N ) @ ( ring_1_of_int @ A @ X2 ) )
= ( ord_less @ int @ ( semiring_1_of_nat @ int @ N ) @ X2 ) ) ) ).
% of_nat_less_of_int_iff
thf(fact_541_arg__minI,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [P: A > $o,X2: A,F: A > B,Q: A > $o] :
( ( P @ X2 )
=> ( ! [Y2: A] :
( ( P @ Y2 )
=> ~ ( ord_less @ B @ ( F @ Y2 ) @ ( F @ X2 ) ) )
=> ( ! [X3: A] :
( ( P @ X3 )
=> ( ! [Y5: A] :
( ( P @ Y5 )
=> ~ ( ord_less @ B @ ( F @ Y5 ) @ ( F @ X3 ) ) )
=> ( Q @ X3 ) ) )
=> ( Q @ ( lattices_ord_arg_min @ A @ B @ F @ P ) ) ) ) ) ) ).
% arg_minI
thf(fact_542_arg__maxI,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [P: A > $o,X2: A,F: A > B,Q: A > $o] :
( ( P @ X2 )
=> ( ! [Y2: A] :
( ( P @ Y2 )
=> ~ ( ord_less @ B @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ! [X3: A] :
( ( P @ X3 )
=> ( ! [Y5: A] :
( ( P @ Y5 )
=> ~ ( ord_less @ B @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( Q @ X3 ) ) )
=> ( Q @ ( lattices_ord_arg_max @ A @ B @ F @ P ) ) ) ) ) ) ).
% arg_maxI
thf(fact_543_is__arg__min__def,axiom,
! [A: $tType,B: $tType] :
( ( ord @ A )
=> ( ( lattic501386751177426532rg_min @ B @ A )
= ( ^ [F2: B > A,P4: B > $o,X: B] :
( ( P4 @ X )
& ~ ? [Y4: B] :
( ( P4 @ Y4 )
& ( ord_less @ A @ ( F2 @ Y4 ) @ ( F2 @ X ) ) ) ) ) ) ) ).
% is_arg_min_def
thf(fact_544_zero__le__power__int,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X2: A,N: int] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X2 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_int @ A @ X2 @ N ) ) ) ) ).
% zero_le_power_int
thf(fact_545_power__int__le__imp__le__exp,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X2: A,M: int,N: int] :
( ( ord_less @ A @ ( one_one @ A ) @ X2 )
=> ( ( ord_less_eq @ A @ ( power_int @ A @ X2 @ M ) @ ( power_int @ A @ X2 @ N ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N )
=> ( ord_less_eq @ int @ M @ N ) ) ) ) ) ).
% power_int_le_imp_le_exp
thf(fact_546_floor__split,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [P: int > $o,T2: A] :
( ( P @ ( archim6421214686448440834_floor @ A @ T2 ) )
= ( ! [I2: int] :
( ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ I2 ) @ T2 )
& ( ord_less @ A @ T2 @ ( plus_plus @ A @ ( ring_1_of_int @ A @ I2 ) @ ( one_one @ A ) ) ) )
=> ( P @ I2 ) ) ) ) ) ).
% floor_split
thf(fact_547_floor__eq__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A,A2: int] :
( ( ( archim6421214686448440834_floor @ A @ X2 )
= A2 )
= ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ A2 ) @ X2 )
& ( ord_less @ A @ X2 @ ( plus_plus @ A @ ( ring_1_of_int @ A @ A2 ) @ ( one_one @ A ) ) ) ) ) ) ).
% floor_eq_iff
thf(fact_548_floor__unique,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z2: int,X2: A] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z2 ) @ X2 )
=> ( ( ord_less @ A @ X2 @ ( plus_plus @ A @ ( ring_1_of_int @ A @ Z2 ) @ ( one_one @ A ) ) )
=> ( ( archim6421214686448440834_floor @ A @ X2 )
= Z2 ) ) ) ) ).
% floor_unique
thf(fact_549_less__floor__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z2: int,X2: A] :
( ( ord_less @ int @ Z2 @ ( archim6421214686448440834_floor @ A @ X2 ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ ( ring_1_of_int @ A @ Z2 ) @ ( one_one @ A ) ) @ X2 ) ) ) ).
% less_floor_iff
thf(fact_550_power__int__not__zero,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X2: A,N: int] :
( ( ( X2
!= ( zero_zero @ A ) )
| ( N
= ( zero_zero @ int ) ) )
=> ( ( power_int @ A @ X2 @ N )
!= ( zero_zero @ A ) ) ) ) ).
% power_int_not_zero
thf(fact_551_floor__le__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A,Z2: int] :
( ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X2 ) @ Z2 )
= ( ord_less @ A @ X2 @ ( plus_plus @ A @ ( ring_1_of_int @ A @ Z2 ) @ ( one_one @ A ) ) ) ) ) ).
% floor_le_iff
thf(fact_552_floor__correct,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ X2 ) ) @ X2 )
& ( ord_less @ A @ X2 @ ( ring_1_of_int @ A @ ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X2 ) @ ( one_one @ int ) ) ) ) ) ) ).
% floor_correct
thf(fact_553_floor__mono,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A,Y: A] :
( ( ord_less_eq @ A @ X2 @ Y )
=> ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X2 ) @ ( archim6421214686448440834_floor @ A @ Y ) ) ) ) ).
% floor_mono
thf(fact_554_nat__zero__as__int,axiom,
( ( zero_zero @ nat )
= ( nat2 @ ( zero_zero @ int ) ) ) ).
% nat_zero_as_int
thf(fact_555_of__int__floor__le,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A] : ( ord_less_eq @ A @ ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ X2 ) ) @ X2 ) ) ).
% of_int_floor_le
thf(fact_556_nat__mono,axiom,
! [X2: int,Y: int] :
( ( ord_less_eq @ int @ X2 @ Y )
=> ( ord_less_eq @ nat @ ( nat2 @ X2 ) @ ( nat2 @ Y ) ) ) ).
% nat_mono
thf(fact_557_ex__nat,axiom,
( ( ^ [P5: nat > $o] :
? [X6: nat] : ( P5 @ X6 ) )
= ( ^ [P4: nat > $o] :
? [X: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
& ( P4 @ ( nat2 @ X ) ) ) ) ) ).
% ex_nat
thf(fact_558_all__nat,axiom,
( ( ^ [P5: nat > $o] :
! [X6: nat] : ( P5 @ X6 ) )
= ( ^ [P4: nat > $o] :
! [X: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
=> ( P4 @ ( nat2 @ X ) ) ) ) ) ).
% all_nat
thf(fact_559_eq__nat__nat__iff,axiom,
! [Z2: int,Z6: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z2 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z6 )
=> ( ( ( nat2 @ Z2 )
= ( nat2 @ Z6 ) )
= ( Z2 = Z6 ) ) ) ) ).
% eq_nat_nat_iff
thf(fact_560_is__arg__max__def,axiom,
! [A: $tType,B: $tType] :
( ( ord @ A )
=> ( ( lattic501386751176901750rg_max @ B @ A )
= ( ^ [F2: B > A,P4: B > $o,X: B] :
( ( P4 @ X )
& ~ ? [Y4: B] :
( ( P4 @ Y4 )
& ( ord_less @ A @ ( F2 @ X ) @ ( F2 @ Y4 ) ) ) ) ) ) ) ).
% is_arg_max_def
thf(fact_561_nat__one__as__int,axiom,
( ( one_one @ nat )
= ( nat2 @ ( one_one @ int ) ) ) ).
% nat_one_as_int
thf(fact_562_add__less__le__mono,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ D2 )
=> ( ord_less @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ D2 ) ) ) ) ) ).
% add_less_le_mono
thf(fact_563_add__le__less__mono,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ C2 @ D2 )
=> ( ord_less @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ D2 ) ) ) ) ) ).
% add_le_less_mono
thf(fact_564_add__mono__thms__linordered__field_I3_J,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( ord_less @ A @ I @ J )
& ( ord_less_eq @ A @ K @ L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_565_add__mono__thms__linordered__field_I4_J,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( ord_less_eq @ A @ I @ J )
& ( ord_less @ A @ K @ L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_566_add__less__zeroD,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X2: A,Y: A] :
( ( ord_less @ A @ ( plus_plus @ A @ X2 @ Y ) @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ X2 @ ( zero_zero @ A ) )
| ( ord_less @ A @ Y @ ( zero_zero @ A ) ) ) ) ) ).
% add_less_zeroD
thf(fact_567_add__neg__neg,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% add_neg_neg
thf(fact_568_add__pos__pos,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).
% add_pos_pos
thf(fact_569_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ~ ! [C4: A] :
( ( B2
= ( plus_plus @ A @ A2 @ C4 ) )
=> ( C4
= ( zero_zero @ A ) ) ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_570_pos__add__strict,axiom,
! [A: $tType] :
( ( strict7427464778891057005id_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ B2 @ C2 )
=> ( ord_less @ A @ B2 @ ( plus_plus @ A @ A2 @ C2 ) ) ) ) ) ).
% pos_add_strict
thf(fact_571_zero__less__one,axiom,
! [A: $tType] :
( ( zero_less_one @ A )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).
% zero_less_one
thf(fact_572_not__one__less__zero,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ~ ( ord_less @ A @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ).
% not_one_less_zero
thf(fact_573_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_574_less__add__one,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A] : ( ord_less @ A @ A2 @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) ) ) ).
% less_add_one
thf(fact_575_add__mono1,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ord_less @ A @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) @ ( plus_plus @ A @ B2 @ ( one_one @ A ) ) ) ) ) ).
% add_mono1
thf(fact_576_floor__le__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A] : ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X2 ) @ ( archimedean_ceiling @ A @ X2 ) ) ) ).
% floor_le_ceiling
thf(fact_577_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_578_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ ( zero_zero @ nat ) )
=> ? [K4: nat] :
( ( ord_less_eq @ nat @ K4 @ N )
& ! [I4: nat] :
( ( ord_less @ nat @ I4 @ K4 )
=> ~ ( P @ I4 ) )
& ( P @ K4 ) ) ) ) ).
% ex_least_nat_le
thf(fact_579_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less @ nat @ I @ J )
=> ? [K4: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K4 )
& ( ( plus_plus @ nat @ I @ K4 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_580_ex__has__greatest__nat__lemma,axiom,
! [A: $tType,P: A > $o,K: A,F: A > nat,N: nat] :
( ( P @ K )
=> ( ! [X3: A] :
( ( P @ X3 )
=> ? [Y5: A] :
( ( P @ Y5 )
& ~ ( ord_less_eq @ nat @ ( F @ Y5 ) @ ( F @ X3 ) ) ) )
=> ? [Y2: A] :
( ( P @ Y2 )
& ~ ( ord_less @ nat @ ( F @ Y2 ) @ ( plus_plus @ nat @ ( F @ K ) @ N ) ) ) ) ) ).
% ex_has_greatest_nat_lemma
thf(fact_581_mono__nat__linear__lb,axiom,
! [F: nat > nat,M: nat,K: nat] :
( ! [M4: nat,N2: nat] :
( ( ord_less @ nat @ M4 @ N2 )
=> ( ord_less @ nat @ ( F @ M4 ) @ ( F @ N2 ) ) )
=> ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus @ nat @ M @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_582_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_583_int__gr__induct,axiom,
! [K: int,I: int,P: int > $o] :
( ( ord_less @ int @ K @ I )
=> ( ( 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 @ I ) ) ) ) ).
% int_gr_induct
thf(fact_584_power__int__0__left__If,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [M: int] :
( ( ( M
= ( zero_zero @ int ) )
=> ( ( power_int @ A @ ( zero_zero @ A ) @ M )
= ( one_one @ A ) ) )
& ( ( M
!= ( zero_zero @ int ) )
=> ( ( power_int @ A @ ( zero_zero @ A ) @ M )
= ( zero_zero @ A ) ) ) ) ) ).
% power_int_0_left_If
thf(fact_585_power__int__increasing,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [N: int,N5: int,A2: A] :
( ( ord_less_eq @ int @ N @ N5 )
=> ( ( ord_less_eq @ A @ ( one_one @ A ) @ A2 )
=> ( ord_less_eq @ A @ ( power_int @ A @ A2 @ N ) @ ( power_int @ A @ A2 @ N5 ) ) ) ) ) ).
% power_int_increasing
thf(fact_586_le__floor__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z2: int,X2: A] :
( ( ord_less_eq @ int @ Z2 @ ( archim6421214686448440834_floor @ A @ X2 ) )
= ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z2 ) @ X2 ) ) ) ).
% le_floor_iff
thf(fact_587_le__floor__add,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A,Y: A] : ( ord_less_eq @ int @ ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X2 ) @ ( archim6421214686448440834_floor @ A @ Y ) ) @ ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X2 @ Y ) ) ) ) ).
% le_floor_add
thf(fact_588_int__add__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z2: int,X2: A] :
( ( plus_plus @ int @ Z2 @ ( archim6421214686448440834_floor @ A @ X2 ) )
= ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ ( ring_1_of_int @ A @ Z2 ) @ X2 ) ) ) ) ).
% int_add_floor
thf(fact_589_floor__add__int,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A,Z2: int] :
( ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X2 ) @ Z2 )
= ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X2 @ ( ring_1_of_int @ A @ Z2 ) ) ) ) ) ).
% floor_add_int
thf(fact_590_nat__le__iff,axiom,
! [X2: int,N: nat] :
( ( ord_less_eq @ nat @ ( nat2 @ X2 ) @ N )
= ( ord_less_eq @ int @ X2 @ ( semiring_1_of_nat @ int @ N ) ) ) ).
% nat_le_iff
thf(fact_591_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_592_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_593_of__nat__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [R: A] : ( ord_less_eq @ A @ R @ ( semiring_1_of_nat @ A @ ( nat2 @ ( archimedean_ceiling @ A @ R ) ) ) ) ) ).
% of_nat_ceiling
thf(fact_594_nat__int__add,axiom,
! [A2: nat,B2: nat] :
( ( nat2 @ ( plus_plus @ int @ ( semiring_1_of_nat @ int @ A2 ) @ ( semiring_1_of_nat @ int @ B2 ) ) )
= ( plus_plus @ nat @ A2 @ B2 ) ) ).
% nat_int_add
thf(fact_595_add__neg__nonpos,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% add_neg_nonpos
thf(fact_596_add__nonneg__pos,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).
% add_nonneg_pos
thf(fact_597_add__nonpos__neg,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% add_nonpos_neg
thf(fact_598_add__pos__nonneg,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).
% add_pos_nonneg
thf(fact_599_add__strict__increasing,axiom,
! [A: $tType] :
( ( ordere8940638589300402666id_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ B2 @ C2 )
=> ( ord_less @ A @ B2 @ ( plus_plus @ A @ A2 @ C2 ) ) ) ) ) ).
% add_strict_increasing
thf(fact_600_add__strict__increasing2,axiom,
! [A: $tType] :
( ( ordere8940638589300402666id_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ B2 @ C2 )
=> ( ord_less @ A @ B2 @ ( plus_plus @ A @ A2 @ C2 ) ) ) ) ) ).
% add_strict_increasing2
thf(fact_601_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_602_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_603_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_604_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_605_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_606_nonempty__has__size,axiom,
! [A: $tType,S2: multiset @ A] :
( ( S2
!= ( zero_zero @ ( multiset @ A ) ) )
= ( ord_less @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( multiset @ A ) @ S2 ) ) ) ).
% nonempty_has_size
thf(fact_607_nat__plus__as__int,axiom,
( ( plus_plus @ nat )
= ( ^ [A3: nat,B3: nat] : ( nat2 @ ( plus_plus @ int @ ( semiring_1_of_nat @ int @ A3 ) @ ( semiring_1_of_nat @ int @ B3 ) ) ) ) ) ).
% nat_plus_as_int
thf(fact_608_power__int__mono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X2: A,Y: A,N: int] :
( ( ord_less_eq @ A @ X2 @ Y )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X2 )
=> ( ord_less_eq @ A @ ( power_int @ A @ X2 @ N ) @ ( power_int @ A @ Y @ N ) ) ) ) ) ) ).
% power_int_mono
thf(fact_609_one__le__power__int,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X2: A,N: int] :
( ( ord_less_eq @ A @ ( one_one @ A ) @ X2 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ ( power_int @ A @ X2 @ N ) ) ) ) ) ).
% one_le_power_int
thf(fact_610_one__add__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A] :
( ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X2 ) @ ( one_one @ int ) )
= ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X2 @ ( one_one @ A ) ) ) ) ) ).
% one_add_floor
thf(fact_611_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_612_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_613_le__nat__iff,axiom,
! [K: int,N: nat] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
=> ( ( ord_less_eq @ nat @ N @ ( nat2 @ K ) )
= ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ N ) @ K ) ) ) ).
% le_nat_iff
thf(fact_614_nat__add__distrib,axiom,
! [Z2: int,Z6: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z2 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z6 )
=> ( ( nat2 @ ( plus_plus @ int @ Z2 @ Z6 ) )
= ( plus_plus @ nat @ ( nat2 @ Z2 ) @ ( nat2 @ Z6 ) ) ) ) ) ).
% nat_add_distrib
thf(fact_615_ceiling__altdef,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archimedean_ceiling @ A )
= ( ^ [X: A] :
( if @ int
@ ( X
= ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ X ) ) )
@ ( archim6421214686448440834_floor @ A @ X )
@ ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( one_one @ int ) ) ) ) ) ) ).
% ceiling_altdef
thf(fact_616_floor__exists,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X2: A] :
? [Z4: int] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z4 ) @ X2 )
& ( ord_less @ A @ X2 @ ( ring_1_of_int @ A @ ( plus_plus @ int @ Z4 @ ( one_one @ int ) ) ) ) ) ) ).
% floor_exists
thf(fact_617_floor__exists1,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X2: A] :
? [X3: int] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ X3 ) @ X2 )
& ( ord_less @ A @ X2 @ ( ring_1_of_int @ A @ ( plus_plus @ int @ X3 @ ( one_one @ int ) ) ) )
& ! [Y5: int] :
( ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Y5 ) @ X2 )
& ( ord_less @ A @ X2 @ ( ring_1_of_int @ A @ ( plus_plus @ int @ Y5 @ ( one_one @ int ) ) ) ) )
=> ( Y5 = X3 ) ) ) ) ).
% floor_exists1
thf(fact_618_discrete,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ( ( ord_less @ A )
= ( ^ [A3: A] : ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) ) ) ) ) ).
% discrete
thf(fact_619_field__le__epsilon,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X2: A,Y: A] :
( ! [E4: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ E4 )
=> ( ord_less_eq @ A @ X2 @ ( plus_plus @ A @ Y @ E4 ) ) )
=> ( ord_less_eq @ A @ X2 @ Y ) ) ) ).
% field_le_epsilon
thf(fact_620_floor__add,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A,Y: A] :
( ( ( ord_less @ A @ ( plus_plus @ A @ ( archimedean_frac @ A @ X2 ) @ ( archimedean_frac @ A @ Y ) ) @ ( one_one @ A ) )
=> ( ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X2 @ Y ) )
= ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X2 ) @ ( archim6421214686448440834_floor @ A @ Y ) ) ) )
& ( ~ ( ord_less @ A @ ( plus_plus @ A @ ( archimedean_frac @ A @ X2 ) @ ( archimedean_frac @ A @ Y ) ) @ ( one_one @ A ) )
=> ( ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X2 @ Y ) )
= ( plus_plus @ int @ ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X2 ) @ ( archim6421214686448440834_floor @ A @ Y ) ) @ ( one_one @ int ) ) ) ) ) ) ).
% floor_add
thf(fact_621_complete__interval,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [A2: A,B2: A,P: A > $o] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( P @ A2 )
=> ( ~ ( P @ B2 )
=> ? [C4: A] :
( ( ord_less_eq @ A @ A2 @ C4 )
& ( ord_less_eq @ A @ C4 @ B2 )
& ! [X4: A] :
( ( ( ord_less_eq @ A @ A2 @ X4 )
& ( ord_less @ A @ X4 @ C4 ) )
=> ( P @ X4 ) )
& ! [D4: A] :
( ! [X3: A] :
( ( ( ord_less_eq @ A @ A2 @ X3 )
& ( ord_less @ A @ X3 @ D4 ) )
=> ( P @ X3 ) )
=> ( ord_less_eq @ A @ D4 @ C4 ) ) ) ) ) ) ) ).
% complete_interval
thf(fact_622_zero__le__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( archimedean_ceiling @ A @ X2 ) )
= ( ord_less @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ X2 ) ) ) ).
% zero_le_ceiling
thf(fact_623_ceiling__less__zero,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A] :
( ( ord_less @ int @ ( archimedean_ceiling @ A @ X2 ) @ ( zero_zero @ int ) )
= ( ord_less_eq @ A @ X2 @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ).
% ceiling_less_zero
thf(fact_624_time__array__of__list,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [Xs: list @ A,H: heap_ext @ product_unit] :
( ( time_time @ ( array @ A ) @ ( array_of_list @ A @ Xs ) @ H )
= ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) ) ) ) ).
% time_array_of_list
thf(fact_625_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_626_gen__length__def,axiom,
! [A: $tType] :
( ( gen_length @ A )
= ( ^ [N4: nat,Xs3: list @ A] : ( plus_plus @ nat @ N4 @ ( size_size @ ( list @ A ) @ Xs3 ) ) ) ) ).
% gen_length_def
thf(fact_627_verit__minus__simplify_I4_J,axiom,
! [B: $tType] :
( ( group_add @ B )
=> ! [B2: B] :
( ( uminus_uminus @ B @ ( uminus_uminus @ B @ B2 ) )
= B2 ) ) ).
% verit_minus_simplify(4)
thf(fact_628_neg__equal__iff__equal,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A] :
( ( ( uminus_uminus @ A @ A2 )
= ( uminus_uminus @ A @ B2 ) )
= ( A2 = B2 ) ) ) ).
% neg_equal_iff_equal
thf(fact_629_add_Oinverse__inverse,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A] :
( ( uminus_uminus @ A @ ( uminus_uminus @ A @ A2 ) )
= A2 ) ) ).
% add.inverse_inverse
thf(fact_630_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_631_nat_Oinject,axiom,
! [X22: nat,Y22: nat] :
( ( ( suc @ X22 )
= ( suc @ Y22 ) )
= ( X22 = Y22 ) ) ).
% nat.inject
thf(fact_632_frac__frac,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A] :
( ( archimedean_frac @ A @ ( archimedean_frac @ A @ X2 ) )
= ( archimedean_frac @ A @ X2 ) ) ) ).
% frac_frac
thf(fact_633_neg__le__iff__le,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A2 ) )
= ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% neg_le_iff_le
thf(fact_634_add_Oinverse__neutral,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ( ( uminus_uminus @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% add.inverse_neutral
thf(fact_635_neg__0__equal__iff__equal,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A] :
( ( ( zero_zero @ A )
= ( uminus_uminus @ A @ A2 ) )
= ( ( zero_zero @ A )
= A2 ) ) ) ).
% neg_0_equal_iff_equal
thf(fact_636_neg__equal__0__iff__equal,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A] :
( ( ( uminus_uminus @ A @ A2 )
= ( zero_zero @ A ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% neg_equal_0_iff_equal
thf(fact_637_equal__neg__zero,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A2: A] :
( ( A2
= ( uminus_uminus @ A @ A2 ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% equal_neg_zero
thf(fact_638_neg__equal__zero,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A2: A] :
( ( ( uminus_uminus @ A @ A2 )
= A2 )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% neg_equal_zero
thf(fact_639_neg__less__iff__less,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [B2: A,A2: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A2 ) )
= ( ord_less @ A @ A2 @ B2 ) ) ) ).
% neg_less_iff_less
thf(fact_640_minus__add__distrib,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A2: A,B2: A] :
( ( uminus_uminus @ A @ ( plus_plus @ A @ A2 @ B2 ) )
= ( plus_plus @ A @ ( uminus_uminus @ A @ A2 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ).
% minus_add_distrib
thf(fact_641_minus__add__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ A2 ) @ ( plus_plus @ A @ A2 @ B2 ) )
= B2 ) ) ).
% minus_add_cancel
thf(fact_642_add__minus__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A] :
( ( plus_plus @ A @ A2 @ ( plus_plus @ A @ ( uminus_uminus @ A @ A2 ) @ B2 ) )
= B2 ) ) ).
% add_minus_cancel
thf(fact_643_lessI,axiom,
! [N: nat] : ( ord_less @ nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_644_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ( ord_less @ nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_645_Suc__less__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ ( suc @ M ) @ ( suc @ N ) )
= ( ord_less @ nat @ M @ N ) ) ).
% Suc_less_eq
thf(fact_646_Suc__le__mono,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq @ nat @ ( suc @ N ) @ ( suc @ M ) )
= ( ord_less_eq @ nat @ N @ M ) ) ).
% Suc_le_mono
thf(fact_647_add__Suc__right,axiom,
! [M: nat,N: nat] :
( ( plus_plus @ nat @ M @ ( suc @ N ) )
= ( suc @ ( plus_plus @ nat @ M @ N ) ) ) ).
% add_Suc_right
thf(fact_648_negative__eq__positive,axiom,
! [N: nat,M: nat] :
( ( ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N ) )
= ( semiring_1_of_nat @ int @ M ) )
= ( ( N
= ( zero_zero @ nat ) )
& ( M
= ( zero_zero @ nat ) ) ) ) ).
% negative_eq_positive
thf(fact_649_of__int__minus,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [Z2: int] :
( ( ring_1_of_int @ A @ ( uminus_uminus @ int @ Z2 ) )
= ( uminus_uminus @ A @ ( ring_1_of_int @ A @ Z2 ) ) ) ) ).
% of_int_minus
thf(fact_650_negative__zle,axiom,
! [N: nat,M: nat] : ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N ) ) @ ( semiring_1_of_nat @ int @ M ) ) ).
% negative_zle
thf(fact_651_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_652_neg__less__eq__nonneg,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ A2 )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% neg_less_eq_nonneg
thf(fact_653_less__eq__neg__nonpos,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ A2 @ ( uminus_uminus @ A @ A2 ) )
= ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% less_eq_neg_nonpos
thf(fact_654_neg__le__0__iff__le,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ ( zero_zero @ A ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% neg_le_0_iff_le
thf(fact_655_neg__0__le__iff__le,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ A2 ) )
= ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% neg_0_le_iff_le
thf(fact_656_neg__less__0__iff__less,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ A2 ) @ ( zero_zero @ A ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% neg_less_0_iff_less
thf(fact_657_neg__0__less__iff__less,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ A2 ) )
= ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% neg_0_less_iff_less
thf(fact_658_neg__less__pos,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A2: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ A2 ) @ A2 )
= ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% neg_less_pos
thf(fact_659_less__neg__neg,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A2: A] :
( ( ord_less @ A @ A2 @ ( uminus_uminus @ A @ A2 ) )
= ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% less_neg_neg
thf(fact_660_ab__left__minus,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ A2 ) @ A2 )
= ( zero_zero @ A ) ) ) ).
% ab_left_minus
thf(fact_661_add_Oright__inverse,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A] :
( ( plus_plus @ A @ A2 @ ( uminus_uminus @ A @ A2 ) )
= ( zero_zero @ A ) ) ) ).
% add.right_inverse
thf(fact_662_zero__less__Suc,axiom,
! [N: nat] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_663_less__Suc0,axiom,
! [N: nat] :
( ( ord_less @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% less_Suc0
thf(fact_664_negative__zless,axiom,
! [N: nat,M: nat] : ( ord_less @ int @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ ( suc @ N ) ) ) @ ( semiring_1_of_nat @ int @ M ) ) ).
% negative_zless
thf(fact_665_nat__zminus__int,axiom,
! [N: nat] :
( ( nat2 @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N ) ) )
= ( zero_zero @ nat ) ) ).
% nat_zminus_int
thf(fact_666_floor__uminus__of__int,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z2: int] :
( ( archim6421214686448440834_floor @ A @ ( uminus_uminus @ A @ ( ring_1_of_int @ A @ Z2 ) ) )
= ( uminus_uminus @ int @ Z2 ) ) ) ).
% floor_uminus_of_int
thf(fact_667_frac__of__int,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z2: int] :
( ( archimedean_frac @ A @ ( ring_1_of_int @ A @ Z2 ) )
= ( zero_zero @ A ) ) ) ).
% frac_of_int
thf(fact_668_dbl__inc__simps_I4_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( neg_numeral_dbl_inc @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% dbl_inc_simps(4)
thf(fact_669_add__neg__numeral__special_I8_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( one_one @ A ) )
= ( zero_zero @ A ) ) ) ).
% add_neg_numeral_special(8)
thf(fact_670_add__neg__numeral__special_I7_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( plus_plus @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( zero_zero @ A ) ) ) ).
% add_neg_numeral_special(7)
thf(fact_671_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_672_nat__1,axiom,
( ( nat2 @ ( one_one @ int ) )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% nat_1
thf(fact_673_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_674_arg__max__natI,axiom,
! [A: $tType,P: A > $o,K: A,F: A > nat,B2: nat] :
( ( P @ K )
=> ( ! [Y2: A] :
( ( P @ Y2 )
=> ( ord_less @ nat @ ( F @ Y2 ) @ B2 ) )
=> ( P @ ( lattices_ord_arg_max @ A @ nat @ F @ P ) ) ) ) ).
% arg_max_natI
thf(fact_675_Nat_OlessE,axiom,
! [I: nat,K: nat] :
( ( ord_less @ nat @ I @ K )
=> ( ( K
!= ( suc @ I ) )
=> ~ ! [J2: nat] :
( ( ord_less @ nat @ I @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ) ).
% Nat.lessE
thf(fact_676_Suc__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ ( suc @ M ) @ N )
=> ( ord_less @ nat @ M @ N ) ) ).
% Suc_lessD
thf(fact_677_Suc__lessE,axiom,
! [I: nat,K: nat] :
( ( ord_less @ nat @ ( suc @ I ) @ K )
=> ~ ! [J2: nat] :
( ( ord_less @ nat @ I @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ).
% Suc_lessE
thf(fact_678_Suc__lessI,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ( ( ( suc @ M )
!= N )
=> ( ord_less @ nat @ ( suc @ M ) @ N ) ) ) ).
% Suc_lessI
thf(fact_679_less__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less @ nat @ M @ N )
=> ( M = N ) ) ) ).
% less_SucE
thf(fact_680_less__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ( ord_less @ nat @ M @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_681_Ex__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I2: nat] :
( ( ord_less @ nat @ I2 @ ( suc @ N ) )
& ( P @ I2 ) ) )
= ( ( P @ N )
| ? [I2: nat] :
( ( ord_less @ nat @ I2 @ N )
& ( P @ I2 ) ) ) ) ).
% Ex_less_Suc
thf(fact_682_less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ ( suc @ N ) )
= ( ( ord_less @ nat @ M @ N )
| ( M = N ) ) ) ).
% less_Suc_eq
thf(fact_683_nat__neq__iff,axiom,
! [M: nat,N: nat] :
( ( M != N )
= ( ( ord_less @ nat @ M @ N )
| ( ord_less @ nat @ N @ M ) ) ) ).
% nat_neq_iff
thf(fact_684_not__less__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less @ nat @ M @ N ) )
= ( ord_less @ nat @ N @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_685_All__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( suc @ N ) )
=> ( P @ I2 ) ) )
= ( ( P @ N )
& ! [I2: nat] :
( ( ord_less @ nat @ I2 @ N )
=> ( P @ I2 ) ) ) ) ).
% All_less_Suc
thf(fact_686_Suc__less__eq2,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ ( suc @ N ) @ M )
= ( ? [M8: nat] :
( ( M
= ( suc @ M8 ) )
& ( ord_less @ nat @ N @ M8 ) ) ) ) ).
% Suc_less_eq2
thf(fact_687_less__antisym,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less @ nat @ N @ M )
=> ( ( ord_less @ nat @ N @ ( suc @ M ) )
=> ( M = N ) ) ) ).
% less_antisym
thf(fact_688_Suc__less__SucD,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ ( suc @ M ) @ ( suc @ N ) )
=> ( ord_less @ nat @ M @ N ) ) ).
% Suc_less_SucD
thf(fact_689_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ N ) ).
% less_not_refl
thf(fact_690_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_691_less__not__refl3,axiom,
! [S3: nat,T2: nat] :
( ( ord_less @ nat @ S3 @ T2 )
=> ( S3 != T2 ) ) ).
% less_not_refl3
thf(fact_692_less__trans__Suc,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less @ nat @ I @ J )
=> ( ( ord_less @ nat @ J @ K )
=> ( ord_less @ nat @ ( suc @ I ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_693_less__Suc__induct,axiom,
! [I: nat,J: nat,P: nat > nat > $o] :
( ( ord_less @ nat @ I @ J )
=> ( ! [I3: nat] : ( P @ I3 @ ( suc @ I3 ) )
=> ( ! [I3: nat,J2: nat,K4: nat] :
( ( ord_less @ nat @ I3 @ J2 )
=> ( ( ord_less @ nat @ J2 @ K4 )
=> ( ( P @ I3 @ J2 )
=> ( ( P @ J2 @ K4 )
=> ( P @ I3 @ K4 ) ) ) ) )
=> ( P @ I @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_694_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_695_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M7: nat] :
( ( ord_less @ nat @ M7 @ N2 )
=> ( P @ M7 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_696_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ~ ( P @ N2 )
=> ? [M7: nat] :
( ( ord_less @ nat @ M7 @ N2 )
& ~ ( P @ M7 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_697_linorder__neqE__nat,axiom,
! [X2: nat,Y: nat] :
( ( X2 != Y )
=> ( ~ ( ord_less @ nat @ X2 @ Y )
=> ( ord_less @ nat @ Y @ X2 ) ) ) ).
% linorder_neqE_nat
thf(fact_698_strict__inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less @ nat @ I @ J )
=> ( ! [I3: nat] :
( ( J
= ( suc @ I3 ) )
=> ( P @ I3 ) )
=> ( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ J )
=> ( ( P @ ( suc @ I3 ) )
=> ( P @ I3 ) ) )
=> ( P @ I ) ) ) ) ).
% strict_inc_induct
thf(fact_699_not__less__less__Suc__eq,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less @ nat @ N @ M )
=> ( ( ord_less @ nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% not_less_less_Suc_eq
thf(fact_700_infinite__descent__measure,axiom,
! [A: $tType,P: A > $o,V: A > nat,X2: A] :
( ! [X3: A] :
( ~ ( P @ X3 )
=> ? [Y5: A] :
( ( ord_less @ nat @ ( V @ Y5 ) @ ( V @ X3 ) )
& ~ ( P @ Y5 ) ) )
=> ( P @ X2 ) ) ).
% infinite_descent_measure
thf(fact_701_verit__negate__coefficient_I3_J,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A] :
( ( A2 = B2 )
=> ( ( uminus_uminus @ A @ A2 )
= ( uminus_uminus @ A @ B2 ) ) ) ) ).
% verit_negate_coefficient(3)
thf(fact_702_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_703_Suc__inject,axiom,
! [X2: nat,Y: nat] :
( ( ( suc @ X2 )
= ( suc @ Y ) )
=> ( X2 = Y ) ) ).
% Suc_inject
thf(fact_704_minus__equation__iff,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A] :
( ( ( uminus_uminus @ A @ A2 )
= B2 )
= ( ( uminus_uminus @ A @ B2 )
= A2 ) ) ) ).
% minus_equation_iff
thf(fact_705_equation__minus__iff,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A] :
( ( A2
= ( uminus_uminus @ A @ B2 ) )
= ( B2
= ( uminus_uminus @ A @ A2 ) ) ) ) ).
% equation_minus_iff
thf(fact_706_fails__array__of__list,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [Xs: list @ A,H: heap_ext @ product_unit] :
~ ( time_fails @ ( array @ A ) @ ( array_of_list @ A @ Xs ) @ H ) ) ).
% fails_array_of_list
thf(fact_707_int__of__nat__induct,axiom,
! [P: int > $o,Z2: int] :
( ! [N2: nat] : ( P @ ( semiring_1_of_nat @ int @ N2 ) )
=> ( ! [N2: nat] : ( P @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ ( suc @ N2 ) ) ) )
=> ( P @ Z2 ) ) ) ).
% int_of_nat_induct
thf(fact_708_int__cases,axiom,
! [Z2: int] :
( ! [N2: nat] :
( Z2
!= ( semiring_1_of_nat @ int @ N2 ) )
=> ~ ! [N2: nat] :
( Z2
!= ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ ( suc @ N2 ) ) ) ) ) ).
% int_cases
thf(fact_709_ceiling__def,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archimedean_ceiling @ A )
= ( ^ [X: A] : ( uminus_uminus @ int @ ( archim6421214686448440834_floor @ A @ ( uminus_uminus @ A @ X ) ) ) ) ) ) ).
% ceiling_def
thf(fact_710_floor__minus,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A] :
( ( archim6421214686448440834_floor @ A @ ( uminus_uminus @ A @ X2 ) )
= ( uminus_uminus @ int @ ( archimedean_ceiling @ A @ X2 ) ) ) ) ).
% floor_minus
thf(fact_711_ceiling__minus,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A] :
( ( archimedean_ceiling @ A @ ( uminus_uminus @ A @ X2 ) )
= ( uminus_uminus @ int @ ( archim6421214686448440834_floor @ A @ X2 ) ) ) ) ).
% ceiling_minus
thf(fact_712_le__imp__neg__le,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A2 ) ) ) ) ).
% le_imp_neg_le
thf(fact_713_minus__le__iff,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ B2 )
= ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ A2 ) ) ) ).
% minus_le_iff
thf(fact_714_le__minus__iff,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ ( uminus_uminus @ A @ B2 ) )
= ( ord_less_eq @ A @ B2 @ ( uminus_uminus @ A @ A2 ) ) ) ) ).
% le_minus_iff
thf(fact_715_verit__negate__coefficient_I2_J,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A2 ) ) ) ) ).
% verit_negate_coefficient(2)
thf(fact_716_minus__less__iff,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ A2 ) @ B2 )
= ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ A2 ) ) ) ).
% minus_less_iff
thf(fact_717_less__minus__iff,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ ( uminus_uminus @ A @ B2 ) )
= ( ord_less @ A @ B2 @ ( uminus_uminus @ A @ A2 ) ) ) ) ).
% less_minus_iff
thf(fact_718_add_Oinverse__distrib__swap,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A] :
( ( uminus_uminus @ A @ ( plus_plus @ A @ A2 @ B2 ) )
= ( plus_plus @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A2 ) ) ) ) ).
% add.inverse_distrib_swap
thf(fact_719_group__cancel_Oneg1,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A5: A,K: A,A2: A] :
( ( A5
= ( plus_plus @ A @ K @ A2 ) )
=> ( ( uminus_uminus @ A @ A5 )
= ( plus_plus @ A @ ( uminus_uminus @ A @ K ) @ ( uminus_uminus @ A @ A2 ) ) ) ) ) ).
% group_cancel.neg1
thf(fact_720_is__num__normalize_I8_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [A2: A,B2: A] :
( ( uminus_uminus @ A @ ( plus_plus @ A @ A2 @ B2 ) )
= ( plus_plus @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A2 ) ) ) ) ).
% is_num_normalize(8)
thf(fact_721_one__neq__neg__one,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ( ( one_one @ A )
!= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% one_neq_neg_one
thf(fact_722_lift__Suc__mono__less__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F: nat > A,N: nat,M: nat] :
( ! [N2: nat] : ( ord_less @ A @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less @ A @ ( F @ N ) @ ( F @ M ) )
= ( ord_less @ nat @ N @ M ) ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_723_lift__Suc__mono__less,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F: nat > A,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less @ A @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less @ nat @ N @ N3 )
=> ( ord_less @ A @ ( F @ N ) @ ( F @ N3 ) ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_724_dependent__nat__choice,axiom,
! [A: $tType,P: nat > A > $o,Q: nat > A > A > $o] :
( ? [X_1: A] : ( P @ ( zero_zero @ nat ) @ X_1 )
=> ( ! [X3: A,N2: nat] :
( ( P @ N2 @ X3 )
=> ? [Y5: A] :
( ( P @ ( suc @ N2 ) @ Y5 )
& ( Q @ N2 @ X3 @ Y5 ) ) )
=> ? [F5: nat > A] :
! [N9: nat] :
( ( P @ N9 @ ( F5 @ N9 ) )
& ( Q @ N9 @ ( F5 @ N9 ) @ ( F5 @ ( suc @ N9 ) ) ) ) ) ) ).
% dependent_nat_choice
thf(fact_725_not0__implies__Suc,axiom,
! [N: nat] :
( ( N
!= ( zero_zero @ nat ) )
=> ? [M4: nat] :
( N
= ( suc @ M4 ) ) ) ).
% not0_implies_Suc
thf(fact_726_Zero__not__Suc,axiom,
! [M: nat] :
( ( zero_zero @ nat )
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_727_Zero__neq__Suc,axiom,
! [M: nat] :
( ( zero_zero @ nat )
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_728_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= ( zero_zero @ nat ) ) ).
% Suc_neq_Zero
thf(fact_729_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ ( zero_zero @ nat ) ) ) ) ).
% zero_induct
thf(fact_730_diff__induct,axiom,
! [P: nat > nat > $o,M: nat,N: nat] :
( ! [X3: nat] : ( P @ X3 @ ( zero_zero @ nat ) )
=> ( ! [Y2: nat] : ( P @ ( zero_zero @ nat ) @ ( suc @ Y2 ) )
=> ( ! [X3: nat,Y2: nat] :
( ( P @ X3 @ Y2 )
=> ( P @ ( suc @ X3 ) @ ( suc @ Y2 ) ) )
=> ( P @ M @ N ) ) ) ) ).
% diff_induct
thf(fact_731_nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ ( zero_zero @ nat ) )
=> ( ! [N2: nat] :
( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) )
=> ( P @ N ) ) ) ).
% nat_induct
thf(fact_732_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y
!= ( zero_zero @ nat ) )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_733_nat_OdiscI,axiom,
! [Nat: nat,X22: nat] :
( ( Nat
= ( suc @ X22 ) )
=> ( Nat
!= ( zero_zero @ nat ) ) ) ).
% nat.discI
thf(fact_734_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( ( zero_zero @ nat )
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_735_old_Onat_Odistinct_I2_J,axiom,
! [Nat4: nat] :
( ( suc @ Nat4 )
!= ( zero_zero @ nat ) ) ).
% old.nat.distinct(2)
thf(fact_736_nat_Odistinct_I1_J,axiom,
! [X22: nat] :
( ( zero_zero @ nat )
!= ( suc @ X22 ) ) ).
% nat.distinct(1)
thf(fact_737_exists__least__lemma,axiom,
! [P: nat > $o] :
( ~ ( P @ ( zero_zero @ nat ) )
=> ( ? [X_1: nat] : ( P @ X_1 )
=> ? [N2: nat] :
( ~ ( P @ N2 )
& ( P @ ( suc @ N2 ) ) ) ) ) ).
% exists_least_lemma
thf(fact_738_less__Suc__eq__0__disj,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ ( suc @ N ) )
= ( ( M
= ( zero_zero @ nat ) )
| ? [J3: nat] :
( ( M
= ( suc @ J3 ) )
& ( ord_less @ nat @ J3 @ N ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_739_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ? [M4: nat] :
( N
= ( suc @ M4 ) ) ) ).
% gr0_implies_Suc
thf(fact_740_All__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( suc @ N ) )
=> ( P @ I2 ) ) )
= ( ( P @ ( zero_zero @ nat ) )
& ! [I2: nat] :
( ( ord_less @ nat @ I2 @ N )
=> ( P @ ( suc @ I2 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_741_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
= ( ? [M2: nat] :
( N
= ( suc @ M2 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_742_Ex__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I2: nat] :
( ( ord_less @ nat @ I2 @ ( suc @ N ) )
& ( P @ I2 ) ) )
= ( ( P @ ( zero_zero @ nat ) )
| ? [I2: nat] :
( ( ord_less @ nat @ I2 @ N )
& ( P @ ( suc @ I2 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_743_uminus__int__code_I1_J,axiom,
( ( uminus_uminus @ int @ ( zero_zero @ int ) )
= ( zero_zero @ int ) ) ).
% uminus_int_code(1)
thf(fact_744_transitive__stepwise__le,axiom,
! [M: nat,N: nat,R3: nat > nat > $o] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ! [X3: nat] : ( R3 @ X3 @ X3 )
=> ( ! [X3: nat,Y2: nat,Z4: nat] :
( ( R3 @ X3 @ Y2 )
=> ( ( R3 @ Y2 @ Z4 )
=> ( R3 @ X3 @ Z4 ) ) )
=> ( ! [N2: nat] : ( R3 @ N2 @ ( suc @ N2 ) )
=> ( R3 @ M @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_745_nat__induct__at__least,axiom,
! [M: nat,N: nat,P: nat > $o] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( P @ M )
=> ( ! [N2: nat] :
( ( ord_less_eq @ nat @ M @ N2 )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_746_full__nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M7: nat] :
( ( ord_less_eq @ nat @ ( suc @ M7 ) @ N2 )
=> ( P @ M7 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% full_nat_induct
thf(fact_747_not__less__eq__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_eq @ nat @ M @ N ) )
= ( ord_less_eq @ nat @ ( suc @ N ) @ M ) ) ).
% not_less_eq_eq
thf(fact_748_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq @ nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_749_le__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ ( suc @ N ) )
= ( ( ord_less_eq @ nat @ M @ N )
| ( M
= ( suc @ N ) ) ) ) ).
% le_Suc_eq
thf(fact_750_Suc__le__D,axiom,
! [N: nat,M5: nat] :
( ( ord_less_eq @ nat @ ( suc @ N ) @ M5 )
=> ? [M4: nat] :
( M5
= ( suc @ M4 ) ) ) ).
% Suc_le_D
thf(fact_751_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ord_less_eq @ nat @ M @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_752_le__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_eq @ nat @ M @ N )
=> ( M
= ( suc @ N ) ) ) ) ).
% le_SucE
thf(fact_753_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ ( suc @ M ) @ N )
=> ( ord_less_eq @ nat @ M @ N ) ) ).
% Suc_leD
thf(fact_754_Suc__leI,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ( ord_less_eq @ nat @ ( suc @ M ) @ N ) ) ).
% Suc_leI
thf(fact_755_Suc__le__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ ( suc @ M ) @ N )
= ( ord_less @ nat @ M @ N ) ) ).
% Suc_le_eq
thf(fact_756_dec__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( P @ I )
=> ( ! [N2: nat] :
( ( ord_less_eq @ nat @ I @ N2 )
=> ( ( ord_less @ nat @ N2 @ J )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) ) )
=> ( P @ J ) ) ) ) ).
% dec_induct
thf(fact_757_inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( P @ J )
=> ( ! [N2: nat] :
( ( ord_less_eq @ nat @ I @ N2 )
=> ( ( ord_less @ nat @ N2 @ J )
=> ( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) ) ) )
=> ( P @ I ) ) ) ) ).
% inc_induct
thf(fact_758_Suc__le__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ ( suc @ M ) @ N )
=> ( ord_less @ nat @ M @ N ) ) ).
% Suc_le_lessD
thf(fact_759_le__less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( ord_less @ nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% le_less_Suc_eq
thf(fact_760_less__Suc__eq__le,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ ( suc @ N ) )
= ( ord_less_eq @ nat @ M @ N ) ) ).
% less_Suc_eq_le
thf(fact_761_less__eq__Suc__le,axiom,
( ( ord_less @ nat )
= ( ^ [N4: nat] : ( ord_less_eq @ nat @ ( suc @ N4 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_762_le__imp__less__Suc,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ord_less @ nat @ M @ ( suc @ N ) ) ) ).
% le_imp_less_Suc
thf(fact_763_nat__in__between__eq_I1_J,axiom,
! [A2: nat,B2: nat] :
( ( ( ord_less @ nat @ A2 @ B2 )
& ( ord_less_eq @ nat @ B2 @ ( suc @ A2 ) ) )
= ( B2
= ( suc @ A2 ) ) ) ).
% nat_in_between_eq(1)
thf(fact_764_nat__in__between__eq_I2_J,axiom,
! [A2: nat,B2: nat] :
( ( ( ord_less_eq @ nat @ A2 @ B2 )
& ( ord_less @ nat @ B2 @ ( suc @ A2 ) ) )
= ( B2 = A2 ) ) ).
% nat_in_between_eq(2)
thf(fact_765_add__Suc__shift,axiom,
! [M: nat,N: nat] :
( ( plus_plus @ nat @ ( suc @ M ) @ N )
= ( plus_plus @ nat @ M @ ( suc @ N ) ) ) ).
% add_Suc_shift
thf(fact_766_add__Suc,axiom,
! [M: nat,N: nat] :
( ( plus_plus @ nat @ ( suc @ M ) @ N )
= ( suc @ ( plus_plus @ nat @ M @ N ) ) ) ).
% add_Suc
thf(fact_767_nat__arith_Osuc1,axiom,
! [A5: nat,K: nat,A2: nat] :
( ( A5
= ( plus_plus @ nat @ K @ A2 ) )
=> ( ( suc @ A5 )
= ( plus_plus @ nat @ K @ ( suc @ A2 ) ) ) ) ).
% nat_arith.suc1
thf(fact_768_less__imp__Suc__add,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ? [K4: nat] :
( N
= ( suc @ ( plus_plus @ nat @ M @ K4 ) ) ) ) ).
% less_imp_Suc_add
thf(fact_769_less__iff__Suc__add,axiom,
( ( ord_less @ nat )
= ( ^ [M2: nat,N4: nat] :
? [K2: nat] :
( N4
= ( suc @ ( plus_plus @ nat @ M2 @ K2 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_770_less__add__Suc2,axiom,
! [I: nat,M: nat] : ( ord_less @ nat @ I @ ( suc @ ( plus_plus @ nat @ M @ I ) ) ) ).
% less_add_Suc2
thf(fact_771_less__add__Suc1,axiom,
! [I: nat,M: nat] : ( ord_less @ nat @ I @ ( suc @ ( plus_plus @ nat @ I @ M ) ) ) ).
% less_add_Suc1
thf(fact_772_less__natE,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ~ ! [Q4: nat] :
( N
!= ( suc @ ( plus_plus @ nat @ M @ Q4 ) ) ) ) ).
% less_natE
thf(fact_773_int__cases2,axiom,
! [Z2: int] :
( ! [N2: nat] :
( Z2
!= ( semiring_1_of_nat @ int @ N2 ) )
=> ~ ! [N2: nat] :
( Z2
!= ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N2 ) ) ) ) ).
% int_cases2
thf(fact_774_not__zle__0__negative,axiom,
! [N: nat] :
~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ ( suc @ N ) ) ) ) ).
% not_zle_0_negative
thf(fact_775_negD,axiom,
! [X2: int] :
( ( ord_less @ int @ X2 @ ( zero_zero @ int ) )
=> ? [N2: nat] :
( X2
= ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ ( suc @ N2 ) ) ) ) ) ).
% negD
thf(fact_776_negative__zless__0,axiom,
! [N: nat] : ( ord_less @ int @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ ( suc @ N ) ) ) @ ( zero_zero @ int ) ) ).
% negative_zless_0
thf(fact_777_of__nat__aux_Osimps_I2_J,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [Inc: A > A,N: nat,I: A] :
( ( semiri8178284476397505188at_aux @ A @ Inc @ ( suc @ N ) @ I )
= ( semiri8178284476397505188at_aux @ A @ Inc @ N @ ( Inc @ I ) ) ) ) ).
% of_nat_aux.simps(2)
thf(fact_778_rec__nat__Suc__imp,axiom,
! [A: $tType,F: nat > A,F1: A,F22: nat > A > A,N: nat] :
( ( F
= ( rec_nat @ A @ F1 @ F22 ) )
=> ( ( F @ ( suc @ N ) )
= ( F22 @ N @ ( F @ N ) ) ) ) ).
% rec_nat_Suc_imp
thf(fact_779_neg__eq__iff__add__eq__0,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A] :
( ( ( uminus_uminus @ A @ A2 )
= B2 )
= ( ( plus_plus @ A @ A2 @ B2 )
= ( zero_zero @ A ) ) ) ) ).
% neg_eq_iff_add_eq_0
thf(fact_780_eq__neg__iff__add__eq__0,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A] :
( ( A2
= ( uminus_uminus @ A @ B2 ) )
= ( ( plus_plus @ A @ A2 @ B2 )
= ( zero_zero @ A ) ) ) ) ).
% eq_neg_iff_add_eq_0
thf(fact_781_add_Oinverse__unique,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A] :
( ( ( plus_plus @ A @ A2 @ B2 )
= ( zero_zero @ A ) )
=> ( ( uminus_uminus @ A @ A2 )
= B2 ) ) ) ).
% add.inverse_unique
thf(fact_782_ab__group__add__class_Oab__left__minus,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A2: A] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ A2 ) @ A2 )
= ( zero_zero @ A ) ) ) ).
% ab_group_add_class.ab_left_minus
thf(fact_783_add__eq__0__iff,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A] :
( ( ( plus_plus @ A @ A2 @ B2 )
= ( zero_zero @ A ) )
= ( B2
= ( uminus_uminus @ A @ A2 ) ) ) ) ).
% add_eq_0_iff
thf(fact_784_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_785_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_786_zero__neq__neg__one,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ( ( zero_zero @ A )
!= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% zero_neq_neg_one
thf(fact_787_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_788_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_789_lift__Suc__antimono__le,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F: nat > A,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_eq @ A @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
=> ( ( ord_less_eq @ nat @ N @ N3 )
=> ( ord_less_eq @ A @ ( F @ N3 ) @ ( F @ N ) ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_790_lift__Suc__mono__le,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F: nat > A,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_eq @ A @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq @ nat @ N @ N3 )
=> ( ord_less_eq @ A @ ( F @ N ) @ ( F @ N3 ) ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_791_of__nat__neq__0,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [N: nat] :
( ( semiring_1_of_nat @ A @ ( suc @ N ) )
!= ( zero_zero @ A ) ) ) ).
% of_nat_neq_0
thf(fact_792_nat__compl__induct_H,axiom,
! [P: nat > $o,N: nat] :
( ( P @ ( zero_zero @ nat ) )
=> ( ! [N2: nat] :
( ! [Nn: nat] :
( ( ord_less_eq @ nat @ Nn @ N2 )
=> ( P @ Nn ) )
=> ( P @ ( suc @ N2 ) ) )
=> ( P @ N ) ) ) ).
% nat_compl_induct'
thf(fact_793_nat__compl__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ ( zero_zero @ nat ) )
=> ( ! [N2: nat] :
( ! [Nn: nat] :
( ( ord_less_eq @ nat @ Nn @ N2 )
=> ( P @ Nn ) )
=> ( P @ ( suc @ N2 ) ) )
=> ( P @ N ) ) ) ).
% nat_compl_induct
thf(fact_794_ex__least__nat__less,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ ( zero_zero @ nat ) )
=> ? [K4: nat] :
( ( ord_less @ nat @ K4 @ N )
& ! [I4: nat] :
( ( ord_less_eq @ nat @ I4 @ K4 )
=> ~ ( P @ I4 ) )
& ( P @ ( suc @ K4 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_795_one__is__add,axiom,
! [M: nat,N: nat] :
( ( ( suc @ ( zero_zero @ nat ) )
= ( plus_plus @ nat @ M @ N ) )
= ( ( ( M
= ( suc @ ( zero_zero @ nat ) ) )
& ( N
= ( zero_zero @ nat ) ) )
| ( ( M
= ( zero_zero @ nat ) )
& ( N
= ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ).
% one_is_add
thf(fact_796_add__is__1,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus @ nat @ M @ N )
= ( suc @ ( zero_zero @ nat ) ) )
= ( ( ( M
= ( suc @ ( zero_zero @ nat ) ) )
& ( N
= ( zero_zero @ nat ) ) )
| ( ( M
= ( zero_zero @ nat ) )
& ( N
= ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ).
% add_is_1
thf(fact_797_One__nat__def,axiom,
( ( one_one @ nat )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% One_nat_def
thf(fact_798_nat__induct__non__zero,axiom,
! [N: nat,P: nat > $o] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( P @ ( one_one @ nat ) )
=> ( ! [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_non_zero
thf(fact_799_int__cases4,axiom,
! [M: int] :
( ! [N2: nat] :
( M
!= ( semiring_1_of_nat @ int @ N2 ) )
=> ~ ! [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( M
!= ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N2 ) ) ) ) ) ).
% int_cases4
thf(fact_800_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus @ nat @ ( one_one @ nat ) ) ) ).
% Suc_eq_plus1_left
thf(fact_801_plus__1__eq__Suc,axiom,
( ( plus_plus @ nat @ ( one_one @ nat ) )
= suc ) ).
% plus_1_eq_Suc
thf(fact_802_Suc__eq__plus1,axiom,
( suc
= ( ^ [N4: nat] : ( plus_plus @ nat @ N4 @ ( one_one @ nat ) ) ) ) ).
% Suc_eq_plus1
thf(fact_803_not__int__zless__negative,axiom,
! [N: nat,M: nat] :
~ ( ord_less @ int @ ( semiring_1_of_nat @ int @ N ) @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ M ) ) ) ).
% not_int_zless_negative
thf(fact_804_of__int__of__nat,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( ( ring_1_of_int @ A )
= ( ^ [K2: int] : ( if @ A @ ( ord_less @ int @ K2 @ ( zero_zero @ int ) ) @ ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ ( nat2 @ ( uminus_uminus @ int @ K2 ) ) ) ) @ ( semiring_1_of_nat @ A @ ( nat2 @ K2 ) ) ) ) ) ) ).
% of_int_of_nat
thf(fact_805_frac__ge__0,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( archimedean_frac @ A @ X2 ) ) ) ).
% frac_ge_0
thf(fact_806_frac__lt__1,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A] : ( ord_less @ A @ ( archimedean_frac @ A @ X2 ) @ ( one_one @ A ) ) ) ).
% frac_lt_1
thf(fact_807_frac__1__eq,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A] :
( ( archimedean_frac @ A @ ( plus_plus @ A @ X2 @ ( one_one @ A ) ) )
= ( archimedean_frac @ A @ X2 ) ) ) ).
% frac_1_eq
thf(fact_808_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_809_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_810_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_811_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_812_int__cases3,axiom,
! [K: int] :
( ( K
!= ( zero_zero @ int ) )
=> ( ! [N2: nat] :
( ( K
= ( semiring_1_of_nat @ int @ N2 ) )
=> ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) )
=> ~ ! [N2: nat] :
( ( K
= ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N2 ) ) )
=> ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ).
% int_cases3
thf(fact_813_int__zle__neg,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ N ) @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ M ) ) )
= ( ( N
= ( zero_zero @ nat ) )
& ( M
= ( zero_zero @ nat ) ) ) ) ).
% int_zle_neg
thf(fact_814_nonpos__int__cases,axiom,
! [K: int] :
( ( ord_less_eq @ int @ K @ ( zero_zero @ int ) )
=> ~ ! [N2: nat] :
( K
!= ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N2 ) ) ) ) ).
% nonpos_int_cases
thf(fact_815_negative__zle__0,axiom,
! [N: nat] : ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N ) ) @ ( zero_zero @ int ) ) ).
% negative_zle_0
thf(fact_816_zless__iff__Suc__zadd,axiom,
( ( ord_less @ int )
= ( ^ [W: int,Z3: int] :
? [N4: nat] :
( Z3
= ( plus_plus @ int @ W @ ( semiring_1_of_nat @ int @ ( suc @ N4 ) ) ) ) ) ) ).
% zless_iff_Suc_zadd
thf(fact_817_int__Suc,axiom,
! [N: nat] :
( ( semiring_1_of_nat @ int @ ( suc @ N ) )
= ( plus_plus @ int @ ( semiring_1_of_nat @ int @ N ) @ ( one_one @ int ) ) ) ).
% int_Suc
thf(fact_818_int__ops_I4_J,axiom,
! [A2: nat] :
( ( semiring_1_of_nat @ int @ ( suc @ A2 ) )
= ( plus_plus @ int @ ( semiring_1_of_nat @ int @ A2 ) @ ( one_one @ int ) ) ) ).
% int_ops(4)
thf(fact_819_neg__int__cases,axiom,
! [K: int] :
( ( ord_less @ int @ K @ ( zero_zero @ int ) )
=> ~ ! [N2: nat] :
( ( K
= ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N2 ) ) )
=> ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ).
% neg_int_cases
thf(fact_820_Suc__as__int,axiom,
( suc
= ( ^ [A3: nat] : ( nat2 @ ( plus_plus @ int @ ( semiring_1_of_nat @ int @ A3 ) @ ( one_one @ int ) ) ) ) ) ).
% Suc_as_int
thf(fact_821_frac__eq,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A] :
( ( ( archimedean_frac @ A @ X2 )
= X2 )
= ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X2 )
& ( ord_less @ A @ X2 @ ( one_one @ A ) ) ) ) ) ).
% frac_eq
thf(fact_822_length__code,axiom,
! [A: $tType] :
( ( size_size @ ( list @ A ) )
= ( gen_length @ A @ ( zero_zero @ nat ) ) ) ).
% length_code
thf(fact_823_compl__le__compl__iff,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X2: A,Y: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ X2 ) @ ( uminus_uminus @ A @ Y ) )
= ( ord_less_eq @ A @ Y @ X2 ) ) ) ).
% compl_le_compl_iff
thf(fact_824_power__int__minus__one__minus,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [N: int] :
( ( power_int @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( uminus_uminus @ int @ N ) )
= ( power_int @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N ) ) ) ).
% power_int_minus_one_minus
thf(fact_825_dbl__dec__simps_I2_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( neg_numeral_dbl_dec @ A @ ( zero_zero @ A ) )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% dbl_dec_simps(2)
thf(fact_826_nat__descend__induct,axiom,
! [N: nat,P: nat > $o,M: nat] :
( ! [K4: nat] :
( ( ord_less @ nat @ N @ K4 )
=> ( P @ K4 ) )
=> ( ! [K4: nat] :
( ( ord_less_eq @ nat @ K4 @ N )
=> ( ! [I4: nat] :
( ( ord_less @ nat @ K4 @ I4 )
=> ( P @ I4 ) )
=> ( P @ K4 ) ) )
=> ( P @ M ) ) ) ).
% nat_descend_induct
thf(fact_827_list__decode_Ocases,axiom,
! [X2: nat] :
( ( X2
!= ( zero_zero @ nat ) )
=> ~ ! [N2: nat] :
( X2
!= ( suc @ N2 ) ) ) ).
% list_decode.cases
thf(fact_828_time__array__new,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [N: nat,X2: A,H: heap_ext @ product_unit] :
( ( time_time @ ( array @ A ) @ ( array_new @ A @ N @ X2 ) @ H )
= ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) ) ).
% time_array_new
thf(fact_829_compl__le__swap2,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [Y: A,X2: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ Y ) @ X2 )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ X2 ) @ Y ) ) ) ).
% compl_le_swap2
thf(fact_830_compl__le__swap1,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [Y: A,X2: A] :
( ( ord_less_eq @ A @ Y @ ( uminus_uminus @ A @ X2 ) )
=> ( ord_less_eq @ A @ X2 @ ( uminus_uminus @ A @ Y ) ) ) ) ).
% compl_le_swap1
thf(fact_831_compl__mono,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X2: A,Y: A] :
( ( ord_less_eq @ A @ X2 @ Y )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ Y ) @ ( uminus_uminus @ A @ X2 ) ) ) ) ).
% compl_mono
thf(fact_832_dbl__dec__simps_I3_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( neg_numeral_dbl_dec @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% dbl_dec_simps(3)
thf(fact_833_fails__array__new,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [N: nat,X2: A,H: heap_ext @ product_unit] :
~ ( time_fails @ ( array @ A ) @ ( array_new @ A @ N @ X2 ) @ H ) ) ).
% fails_array_new
thf(fact_834_Nitpick_OEps__psimp,axiom,
! [A: $tType,P: A > $o,X2: A,Y: A] :
( ( P @ X2 )
=> ( ~ ( P @ Y )
=> ( ( ( fChoice @ A @ P )
= Y )
=> ( ( fChoice @ A @ P )
= X2 ) ) ) ) ).
% Nitpick.Eps_psimp
thf(fact_835_time__array__upd,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [I: nat,X2: A,P3: array @ A,H: heap_ext @ product_unit] :
( ( ( time_fails @ ( array @ A ) @ ( array_upd @ A @ I @ X2 @ P3 ) @ H )
=> ( ( time_time @ ( array @ A ) @ ( array_upd @ A @ I @ X2 @ P3 ) @ H )
= ( zero_zero @ nat ) ) )
& ( ~ ( time_fails @ ( array @ A ) @ ( array_upd @ A @ I @ X2 @ P3 ) @ H )
=> ( ( time_time @ ( array @ A ) @ ( array_upd @ A @ I @ X2 @ P3 ) @ H )
= ( one_one @ nat ) ) ) ) ) ).
% time_array_upd
thf(fact_836_triangle__Suc,axiom,
! [N: nat] :
( ( nat_triangle @ ( suc @ N ) )
= ( plus_plus @ nat @ ( nat_triangle @ N ) @ ( suc @ N ) ) ) ).
% triangle_Suc
thf(fact_837_time__array__make,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [N: nat,F: nat > A,H: heap_ext @ product_unit] :
( ( time_time @ ( array @ A ) @ ( array_make @ A @ N @ F ) @ H )
= ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) ) ).
% time_array_make
thf(fact_838_bot__nat__0_Oordering__top__axioms,axiom,
( ordering_top @ nat
@ ^ [X: nat,Y4: nat] : ( ord_less_eq @ nat @ Y4 @ X )
@ ^ [X: nat,Y4: nat] : ( ord_less @ nat @ Y4 @ X )
@ ( zero_zero @ nat ) ) ).
% bot_nat_0.ordering_top_axioms
thf(fact_839_power__int__add__1_H,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X2: A,M: int] :
( ( ( X2
!= ( zero_zero @ A ) )
| ( M
!= ( uminus_uminus @ int @ ( one_one @ int ) ) ) )
=> ( ( power_int @ A @ X2 @ ( plus_plus @ int @ M @ ( one_one @ int ) ) )
= ( times_times @ A @ X2 @ ( power_int @ A @ X2 @ M ) ) ) ) ) ).
% power_int_add_1'
thf(fact_840_power__int__add__1,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X2: A,M: int] :
( ( ( X2
!= ( zero_zero @ A ) )
| ( M
!= ( uminus_uminus @ int @ ( one_one @ int ) ) ) )
=> ( ( power_int @ A @ X2 @ ( plus_plus @ int @ M @ ( one_one @ int ) ) )
= ( times_times @ A @ ( power_int @ A @ X2 @ M ) @ X2 ) ) ) ) ).
% power_int_add_1
thf(fact_841_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_842_foldr__length,axiom,
! [A: $tType,L: list @ A] :
( ( foldr @ A @ nat
@ ^ [X: A] : suc
@ L
@ ( zero_zero @ nat ) )
= ( size_size @ ( list @ A ) @ L ) ) ).
% foldr_length
thf(fact_843_le__ceiling__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z2: int,X2: A] :
( ( ord_less_eq @ int @ Z2 @ ( archimedean_ceiling @ A @ X2 ) )
= ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ Z2 ) @ ( one_one @ A ) ) @ X2 ) ) ) ).
% le_ceiling_iff
thf(fact_844_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus @ nat @ M @ M )
= ( zero_zero @ nat ) ) ).
% diff_self_eq_0
thf(fact_845_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus @ nat @ ( zero_zero @ nat ) @ N )
= ( zero_zero @ nat ) ) ).
% diff_0_eq_0
thf(fact_846_mult__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ( times_times @ nat @ M @ K )
= ( times_times @ nat @ N @ K ) )
= ( ( M = N )
| ( K
= ( zero_zero @ nat ) ) ) ) ).
% mult_cancel2
thf(fact_847_mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times @ nat @ K @ M )
= ( times_times @ nat @ K @ N ) )
= ( ( M = N )
| ( K
= ( zero_zero @ nat ) ) ) ) ).
% mult_cancel1
thf(fact_848_mult__0__right,axiom,
! [M: nat] :
( ( times_times @ nat @ M @ ( zero_zero @ nat ) )
= ( zero_zero @ nat ) ) ).
% mult_0_right
thf(fact_849_mult__is__0,axiom,
! [M: nat,N: nat] :
( ( ( times_times @ nat @ M @ N )
= ( zero_zero @ nat ) )
= ( ( M
= ( zero_zero @ nat ) )
| ( N
= ( zero_zero @ nat ) ) ) ) ).
% mult_is_0
thf(fact_850_Suc__diff__diff,axiom,
! [M: nat,N: nat,K: nat] :
( ( minus_minus @ nat @ ( minus_minus @ nat @ ( suc @ M ) @ N ) @ ( suc @ K ) )
= ( minus_minus @ nat @ ( minus_minus @ nat @ M @ N ) @ K ) ) ).
% Suc_diff_diff
thf(fact_851_diff__Suc__Suc,axiom,
! [M: nat,N: nat] :
( ( minus_minus @ nat @ ( suc @ M ) @ ( suc @ N ) )
= ( minus_minus @ nat @ M @ N ) ) ).
% diff_Suc_Suc
thf(fact_852_diff__diff__cancel,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq @ nat @ I @ N )
=> ( ( minus_minus @ nat @ N @ ( minus_minus @ nat @ N @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_853_diff__diff__left,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus @ nat @ ( minus_minus @ nat @ I @ J ) @ K )
= ( minus_minus @ nat @ I @ ( plus_plus @ nat @ J @ K ) ) ) ).
% diff_diff_left
thf(fact_854_diff__diff__add__mset,axiom,
! [A: $tType,M3: multiset @ A,N5: multiset @ A,P: multiset @ A] :
( ( minus_minus @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ M3 @ N5 ) @ P )
= ( minus_minus @ ( multiset @ A ) @ M3 @ ( plus_plus @ ( multiset @ A ) @ N5 @ P ) ) ) ).
% diff_diff_add_mset
thf(fact_855_nat__1__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ( one_one @ nat )
= ( times_times @ nat @ M @ N ) )
= ( ( M
= ( one_one @ nat ) )
& ( N
= ( one_one @ nat ) ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_856_nat__mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( ( times_times @ nat @ M @ N )
= ( one_one @ nat ) )
= ( ( M
= ( one_one @ nat ) )
& ( N
= ( one_one @ nat ) ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_857_mult__zero__left,axiom,
! [A: $tType] :
( ( mult_zero @ A )
=> ! [A2: A] :
( ( times_times @ A @ ( zero_zero @ A ) @ A2 )
= ( zero_zero @ A ) ) ) ).
% mult_zero_left
thf(fact_858_mult__zero__right,axiom,
! [A: $tType] :
( ( mult_zero @ A )
=> ! [A2: A] :
( ( times_times @ A @ A2 @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% mult_zero_right
thf(fact_859_mult__eq__0__iff,axiom,
! [A: $tType] :
( ( semiri3467727345109120633visors @ A )
=> ! [A2: A,B2: A] :
( ( ( times_times @ A @ A2 @ B2 )
= ( zero_zero @ A ) )
= ( ( A2
= ( zero_zero @ A ) )
| ( B2
= ( zero_zero @ A ) ) ) ) ) ).
% mult_eq_0_iff
thf(fact_860_mult__cancel__left,axiom,
! [A: $tType] :
( ( semiri6575147826004484403cancel @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ( times_times @ A @ C2 @ A2 )
= ( times_times @ A @ C2 @ B2 ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( A2 = B2 ) ) ) ) ).
% mult_cancel_left
thf(fact_861_mult__cancel__right,axiom,
! [A: $tType] :
( ( semiri6575147826004484403cancel @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( ( times_times @ A @ A2 @ C2 )
= ( times_times @ A @ B2 @ C2 ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( A2 = B2 ) ) ) ) ).
% mult_cancel_right
thf(fact_862_diff__self,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A] :
( ( minus_minus @ A @ A2 @ A2 )
= ( zero_zero @ A ) ) ) ).
% diff_self
thf(fact_863_diff__0__right,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A] :
( ( minus_minus @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% diff_0_right
thf(fact_864_zero__diff,axiom,
! [A: $tType] :
( ( comm_monoid_diff @ A )
=> ! [A2: A] :
( ( minus_minus @ A @ ( zero_zero @ A ) @ A2 )
= ( zero_zero @ A ) ) ) ).
% zero_diff
thf(fact_865_diff__zero,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [A2: A] :
( ( minus_minus @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% diff_zero
thf(fact_866_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [A2: A] :
( ( minus_minus @ A @ A2 @ A2 )
= ( zero_zero @ A ) ) ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_867_mult__1,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A2: A] :
( ( times_times @ A @ ( one_one @ A ) @ A2 )
= A2 ) ) ).
% mult_1
thf(fact_868_mult_Oright__neutral,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A2: A] :
( ( times_times @ A @ A2 @ ( one_one @ A ) )
= A2 ) ) ).
% mult.right_neutral
thf(fact_869_add__diff__cancel__right_H,axiom,
! [A: $tType] :
( ( cancel2418104881723323429up_add @ A )
=> ! [A2: A,B2: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ B2 )
= A2 ) ) ).
% add_diff_cancel_right'
thf(fact_870_add__diff__cancel__right,axiom,
! [A: $tType] :
( ( cancel2418104881723323429up_add @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) )
= ( minus_minus @ A @ A2 @ B2 ) ) ) ).
% add_diff_cancel_right
thf(fact_871_add__diff__cancel__left_H,axiom,
! [A: $tType] :
( ( cancel2418104881723323429up_add @ A )
=> ! [A2: A,B2: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ A2 )
= B2 ) ) ).
% add_diff_cancel_left'
thf(fact_872_add__diff__cancel__left,axiom,
! [A: $tType] :
( ( cancel2418104881723323429up_add @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ C2 @ A2 ) @ ( plus_plus @ A @ C2 @ B2 ) )
= ( minus_minus @ A @ A2 @ B2 ) ) ) ).
% add_diff_cancel_left
thf(fact_873_diff__add__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A] :
( ( plus_plus @ A @ ( minus_minus @ A @ A2 @ B2 ) @ B2 )
= A2 ) ) ).
% diff_add_cancel
thf(fact_874_add__diff__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ B2 )
= A2 ) ) ).
% add_diff_cancel
thf(fact_875_minus__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A] :
( ( uminus_uminus @ A @ ( minus_minus @ A @ A2 @ B2 ) )
= ( minus_minus @ A @ B2 @ A2 ) ) ) ).
% minus_diff_eq
thf(fact_876_one__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ( suc @ ( zero_zero @ nat ) )
= ( times_times @ nat @ M @ N ) )
= ( ( M
= ( suc @ ( zero_zero @ nat ) ) )
& ( N
= ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% one_eq_mult_iff
thf(fact_877_mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( ( times_times @ nat @ M @ N )
= ( suc @ ( zero_zero @ nat ) ) )
= ( ( M
= ( suc @ ( zero_zero @ nat ) ) )
& ( N
= ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% mult_eq_1_iff
thf(fact_878_zero__less__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N @ M ) )
= ( ord_less @ nat @ M @ N ) ) ).
% zero_less_diff
thf(fact_879_nat__0__less__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( times_times @ nat @ M @ N ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% nat_0_less_mult_iff
thf(fact_880_mult__less__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less @ nat @ ( times_times @ nat @ M @ K ) @ ( times_times @ nat @ N @ K ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
& ( ord_less @ nat @ M @ N ) ) ) ).
% mult_less_cancel2
thf(fact_881_of__nat__mult,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [M: nat,N: nat] :
( ( semiring_1_of_nat @ A @ ( times_times @ nat @ M @ N ) )
= ( times_times @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% of_nat_mult
thf(fact_882_diff__is__0__eq,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus @ nat @ M @ N )
= ( zero_zero @ nat ) )
= ( ord_less_eq @ nat @ M @ N ) ) ).
% diff_is_0_eq
thf(fact_883_diff__is__0__eq_H,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( minus_minus @ nat @ M @ N )
= ( zero_zero @ nat ) ) ) ).
% diff_is_0_eq'
thf(fact_884_mult__Suc__right,axiom,
! [M: nat,N: nat] :
( ( times_times @ nat @ M @ ( suc @ N ) )
= ( plus_plus @ nat @ M @ ( times_times @ nat @ M @ N ) ) ) ).
% mult_Suc_right
thf(fact_885_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq @ nat @ K @ J )
=> ( ( plus_plus @ nat @ I @ ( minus_minus @ nat @ J @ K ) )
= ( minus_minus @ nat @ ( plus_plus @ nat @ I @ J ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_886_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq @ nat @ K @ J )
=> ( ( plus_plus @ nat @ ( minus_minus @ nat @ J @ K ) @ I )
= ( minus_minus @ nat @ ( plus_plus @ nat @ J @ I ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_887_Nat_Odiff__diff__right,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq @ nat @ K @ J )
=> ( ( minus_minus @ nat @ I @ ( minus_minus @ nat @ J @ K ) )
= ( minus_minus @ nat @ ( plus_plus @ nat @ I @ K ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_888_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_889_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus @ nat @ ( suc @ N ) @ ( one_one @ nat ) )
= N ) ).
% diff_Suc_1
thf(fact_890_of__int__diff,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [W2: int,Z2: int] :
( ( ring_1_of_int @ A @ ( minus_minus @ int @ W2 @ Z2 ) )
= ( minus_minus @ A @ ( ring_1_of_int @ A @ W2 ) @ ( ring_1_of_int @ A @ Z2 ) ) ) ) ).
% of_int_diff
thf(fact_891_triangle__0,axiom,
( ( nat_triangle @ ( zero_zero @ nat ) )
= ( zero_zero @ nat ) ) ).
% triangle_0
thf(fact_892_diff__ge__0__iff__ge,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A2 @ B2 ) )
= ( ord_less_eq @ A @ B2 @ A2 ) ) ) ).
% diff_ge_0_iff_ge
thf(fact_893_zero__comp__diff__simps_I1_J,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A2 @ B2 ) )
= ( ord_less_eq @ A @ B2 @ A2 ) ) ) ).
% zero_comp_diff_simps(1)
thf(fact_894_diff__gt__0__iff__gt,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A2 @ B2 ) )
= ( ord_less @ A @ B2 @ A2 ) ) ) ).
% diff_gt_0_iff_gt
thf(fact_895_zero__comp__diff__simps_I2_J,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A2 @ B2 ) )
= ( ord_less @ A @ B2 @ A2 ) ) ) ).
% zero_comp_diff_simps(2)
thf(fact_896_le__add__diff__inverse,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( plus_plus @ A @ B2 @ ( minus_minus @ A @ A2 @ B2 ) )
= A2 ) ) ) ).
% le_add_diff_inverse
thf(fact_897_le__add__diff__inverse2,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( plus_plus @ A @ ( minus_minus @ A @ A2 @ B2 ) @ B2 )
= A2 ) ) ) ).
% le_add_diff_inverse2
thf(fact_898_mult__cancel__right2,axiom,
! [A: $tType] :
( ( ring_15535105094025558882visors @ A )
=> ! [A2: A,C2: A] :
( ( ( times_times @ A @ A2 @ C2 )
= C2 )
= ( ( C2
= ( zero_zero @ A ) )
| ( A2
= ( one_one @ A ) ) ) ) ) ).
% mult_cancel_right2
thf(fact_899_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_900_mult__cancel__left2,axiom,
! [A: $tType] :
( ( ring_15535105094025558882visors @ A )
=> ! [C2: A,A2: A] :
( ( ( times_times @ A @ C2 @ A2 )
= C2 )
= ( ( C2
= ( zero_zero @ A ) )
| ( A2
= ( one_one @ A ) ) ) ) ) ).
% mult_cancel_left2
thf(fact_901_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_902_diff__add__zero,axiom,
! [A: $tType] :
( ( comm_monoid_diff @ A )
=> ! [A2: A,B2: A] :
( ( minus_minus @ A @ A2 @ ( plus_plus @ A @ A2 @ B2 ) )
= ( zero_zero @ A ) ) ) ).
% diff_add_zero
thf(fact_903_diff__numeral__special_I9_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( minus_minus @ A @ ( one_one @ A ) @ ( one_one @ A ) )
= ( zero_zero @ A ) ) ) ).
% diff_numeral_special(9)
thf(fact_904_diff__0,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A] :
( ( minus_minus @ A @ ( zero_zero @ A ) @ A2 )
= ( uminus_uminus @ A @ A2 ) ) ) ).
% diff_0
thf(fact_905_verit__minus__simplify_I3_J,axiom,
! [B: $tType] :
( ( group_add @ B )
=> ! [B2: B] :
( ( minus_minus @ B @ ( zero_zero @ B ) @ B2 )
= ( uminus_uminus @ B @ B2 ) ) ) ).
% verit_minus_simplify(3)
thf(fact_906_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_907_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_908_uminus__add__conv__diff,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A2: A,B2: A] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ A2 ) @ B2 )
= ( minus_minus @ A @ B2 @ A2 ) ) ) ).
% uminus_add_conv_diff
thf(fact_909_diff__minus__eq__add,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A] :
( ( minus_minus @ A @ A2 @ ( uminus_uminus @ A @ B2 ) )
= ( plus_plus @ A @ A2 @ B2 ) ) ) ).
% diff_minus_eq_add
thf(fact_910_Suc__pred,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( suc @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) )
= N ) ) ).
% Suc_pred
thf(fact_911_length__0__conv,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( zero_zero @ nat ) )
= ( Xs
= ( nil @ A ) ) ) ).
% length_0_conv
thf(fact_912_one__le__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( times_times @ nat @ M @ N ) )
= ( ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ M )
& ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N ) ) ) ).
% one_le_mult_iff
thf(fact_913_mult__le__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq @ nat @ ( times_times @ nat @ M @ K ) @ ( times_times @ nat @ N @ K ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ord_less_eq @ nat @ M @ N ) ) ) ).
% mult_le_cancel2
thf(fact_914_diff__Suc__diff__eq1,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq @ nat @ K @ J )
=> ( ( minus_minus @ nat @ I @ ( suc @ ( minus_minus @ nat @ J @ K ) ) )
= ( minus_minus @ nat @ ( plus_plus @ nat @ I @ K ) @ ( suc @ J ) ) ) ) ).
% diff_Suc_diff_eq1
thf(fact_915_diff__Suc__diff__eq2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq @ nat @ K @ J )
=> ( ( minus_minus @ nat @ ( suc @ ( minus_minus @ nat @ J @ K ) ) @ I )
= ( minus_minus @ nat @ ( suc @ J ) @ ( plus_plus @ nat @ K @ I ) ) ) ) ).
% diff_Suc_diff_eq2
thf(fact_916_Suc__diff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ M )
=> ( ( suc @ ( minus_minus @ nat @ N @ M ) )
= ( minus_minus @ nat @ N @ ( minus_minus @ nat @ M @ ( one_one @ nat ) ) ) ) ) ) ).
% Suc_diff
thf(fact_917_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_918_floor__diff__of__int,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A,Z2: int] :
( ( archim6421214686448440834_floor @ A @ ( minus_minus @ A @ X2 @ ( ring_1_of_int @ A @ Z2 ) ) )
= ( minus_minus @ int @ ( archim6421214686448440834_floor @ A @ X2 ) @ Z2 ) ) ) ).
% floor_diff_of_int
thf(fact_919_ceiling__diff__of__int,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A,Z2: int] :
( ( archimedean_ceiling @ A @ ( minus_minus @ A @ X2 @ ( ring_1_of_int @ A @ Z2 ) ) )
= ( minus_minus @ int @ ( archimedean_ceiling @ A @ X2 ) @ Z2 ) ) ) ).
% ceiling_diff_of_int
thf(fact_920_diff__numeral__special_I12_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( minus_minus @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( zero_zero @ A ) ) ) ).
% diff_numeral_special(12)
thf(fact_921_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_922_Suc__diff__1,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( suc @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) )
= N ) ) ).
% Suc_diff_1
thf(fact_923_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_924_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_925_floor__diff__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A] :
( ( archim6421214686448440834_floor @ A @ ( minus_minus @ A @ X2 @ ( one_one @ A ) ) )
= ( minus_minus @ int @ ( archim6421214686448440834_floor @ A @ X2 ) @ ( one_one @ int ) ) ) ) ).
% floor_diff_one
thf(fact_926_ceiling__diff__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A] :
( ( archimedean_ceiling @ A @ ( minus_minus @ A @ X2 @ ( one_one @ A ) ) )
= ( minus_minus @ int @ ( archimedean_ceiling @ A @ X2 ) @ ( one_one @ int ) ) ) ) ).
% ceiling_diff_one
thf(fact_927_int__ops_I7_J,axiom,
! [A2: nat,B2: nat] :
( ( semiring_1_of_nat @ int @ ( times_times @ nat @ A2 @ B2 ) )
= ( times_times @ int @ ( semiring_1_of_nat @ int @ A2 ) @ ( semiring_1_of_nat @ int @ B2 ) ) ) ).
% int_ops(7)
thf(fact_928_ordering__top_Oextremum__uniqueI,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,Top: A,A2: A] :
( ( ordering_top @ A @ Less_eq @ Less @ Top )
=> ( ( Less_eq @ Top @ A2 )
=> ( A2 = Top ) ) ) ).
% ordering_top.extremum_uniqueI
thf(fact_929_ordering__top_Onot__eq__extremum,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,Top: A,A2: A] :
( ( ordering_top @ A @ Less_eq @ Less @ Top )
=> ( ( A2 != Top )
= ( Less @ A2 @ Top ) ) ) ).
% ordering_top.not_eq_extremum
thf(fact_930_ordering__top_Oextremum__unique,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,Top: A,A2: A] :
( ( ordering_top @ A @ Less_eq @ Less @ Top )
=> ( ( Less_eq @ Top @ A2 )
= ( A2 = Top ) ) ) ).
% ordering_top.extremum_unique
thf(fact_931_ordering__top_Oextremum__strict,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,Top: A,A2: A] :
( ( ordering_top @ A @ Less_eq @ Less @ Top )
=> ~ ( Less @ Top @ A2 ) ) ).
% ordering_top.extremum_strict
thf(fact_932_ordering__top_Oextremum,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,Top: A,A2: A] :
( ( ordering_top @ A @ Less_eq @ Less @ Top )
=> ( Less_eq @ A2 @ Top ) ) ).
% ordering_top.extremum
thf(fact_933_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: $tType] :
( ( cancel2418104881723323429up_add @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( minus_minus @ A @ ( minus_minus @ A @ A2 @ C2 ) @ B2 )
= ( minus_minus @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C2 ) ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_934_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
! [A: $tType] :
( ( ab_semigroup_mult @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( times_times @ A @ B2 @ ( times_times @ A @ A2 @ C2 ) )
= ( times_times @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% ab_semigroup_mult_class.mult.left_commute
thf(fact_935_ab__semigroup__mult__class_Omult_Ocommute,axiom,
! [A: $tType] :
( ( ab_semigroup_mult @ A )
=> ( ( times_times @ A )
= ( ^ [A3: A,B3: A] : ( times_times @ A @ B3 @ A3 ) ) ) ) ).
% ab_semigroup_mult_class.mult.commute
thf(fact_936_mult_Oassoc,axiom,
! [A: $tType] :
( ( semigroup_mult @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( times_times @ A @ ( times_times @ A @ A2 @ B2 ) @ C2 )
= ( times_times @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% mult.assoc
thf(fact_937_diff__eq__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( ( minus_minus @ A @ A2 @ B2 )
= ( minus_minus @ A @ C2 @ D2 ) )
=> ( ( A2 = B2 )
= ( C2 = D2 ) ) ) ) ).
% diff_eq_diff_eq
thf(fact_938_inf__period_I1_J,axiom,
! [A: $tType] :
( ( ( comm_ring @ A )
& ( dvd @ A ) )
=> ! [P: A > $o,D3: A,Q: A > $o] :
( ! [X3: A,K4: A] :
( ( P @ X3 )
= ( P @ ( minus_minus @ A @ X3 @ ( times_times @ A @ K4 @ D3 ) ) ) )
=> ( ! [X3: A,K4: A] :
( ( Q @ X3 )
= ( Q @ ( minus_minus @ A @ X3 @ ( times_times @ A @ K4 @ D3 ) ) ) )
=> ! [X4: A,K5: A] :
( ( ( P @ X4 )
& ( Q @ X4 ) )
= ( ( P @ ( minus_minus @ A @ X4 @ ( times_times @ A @ K5 @ D3 ) ) )
& ( Q @ ( minus_minus @ A @ X4 @ ( times_times @ A @ K5 @ D3 ) ) ) ) ) ) ) ) ).
% inf_period(1)
thf(fact_939_inf__period_I2_J,axiom,
! [A: $tType] :
( ( ( comm_ring @ A )
& ( dvd @ A ) )
=> ! [P: A > $o,D3: A,Q: A > $o] :
( ! [X3: A,K4: A] :
( ( P @ X3 )
= ( P @ ( minus_minus @ A @ X3 @ ( times_times @ A @ K4 @ D3 ) ) ) )
=> ( ! [X3: A,K4: A] :
( ( Q @ X3 )
= ( Q @ ( minus_minus @ A @ X3 @ ( times_times @ A @ K4 @ D3 ) ) ) )
=> ! [X4: A,K5: A] :
( ( ( P @ X4 )
| ( Q @ X4 ) )
= ( ( P @ ( minus_minus @ A @ X4 @ ( times_times @ A @ K5 @ D3 ) ) )
| ( Q @ ( minus_minus @ A @ X4 @ ( times_times @ A @ K5 @ D3 ) ) ) ) ) ) ) ) ).
% inf_period(2)
thf(fact_940_square__diff__square__factored,axiom,
! [A: $tType] :
( ( comm_ring @ A )
=> ! [X2: A,Y: A] :
( ( minus_minus @ A @ ( times_times @ A @ X2 @ X2 ) @ ( times_times @ A @ Y @ Y ) )
= ( times_times @ A @ ( plus_plus @ A @ X2 @ Y ) @ ( minus_minus @ A @ X2 @ Y ) ) ) ) ).
% square_diff_square_factored
thf(fact_941_eq__add__iff2,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A2: A,E2: A,C2: A,B2: A,D2: A] :
( ( ( plus_plus @ A @ ( times_times @ A @ A2 @ E2 ) @ C2 )
= ( plus_plus @ A @ ( times_times @ A @ B2 @ E2 ) @ D2 ) )
= ( C2
= ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ B2 @ A2 ) @ E2 ) @ D2 ) ) ) ) ).
% eq_add_iff2
thf(fact_942_eq__add__iff1,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A2: A,E2: A,C2: A,B2: A,D2: A] :
( ( ( plus_plus @ A @ ( times_times @ A @ A2 @ E2 ) @ C2 )
= ( plus_plus @ A @ ( times_times @ A @ B2 @ E2 ) @ D2 ) )
= ( ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ A2 @ B2 ) @ E2 ) @ C2 )
= D2 ) ) ) ).
% eq_add_iff1
thf(fact_943_le__add__iff1,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [A2: A,E2: A,C2: A,B2: A,D2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ A2 @ E2 ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ B2 @ E2 ) @ D2 ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ A2 @ B2 ) @ E2 ) @ C2 ) @ D2 ) ) ) ).
% le_add_iff1
thf(fact_944_le__add__iff2,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [A2: A,E2: A,C2: A,B2: A,D2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ A2 @ E2 ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ B2 @ E2 ) @ D2 ) )
= ( ord_less_eq @ A @ C2 @ ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ B2 @ A2 ) @ E2 ) @ D2 ) ) ) ) ).
% le_add_iff2
thf(fact_945_less__add__iff2,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [A2: A,E2: A,C2: A,B2: A,D2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ A2 @ E2 ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ B2 @ E2 ) @ D2 ) )
= ( ord_less @ A @ C2 @ ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ B2 @ A2 ) @ E2 ) @ D2 ) ) ) ) ).
% less_add_iff2
thf(fact_946_less__add__iff1,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [A2: A,E2: A,C2: A,B2: A,D2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ A2 @ E2 ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ B2 @ E2 ) @ D2 ) )
= ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ A2 @ B2 ) @ E2 ) @ C2 ) @ D2 ) ) ) ).
% less_add_iff1
thf(fact_947_square__diff__one__factored,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X2: A] :
( ( minus_minus @ A @ ( times_times @ A @ X2 @ X2 ) @ ( one_one @ A ) )
= ( times_times @ A @ ( plus_plus @ A @ X2 @ ( one_one @ A ) ) @ ( minus_minus @ A @ X2 @ ( one_one @ A ) ) ) ) ) ).
% square_diff_one_factored
thf(fact_948_of__nat__diff,axiom,
! [A: $tType] :
( ( semiring_1_cancel @ A )
=> ! [N: nat,M: nat] :
( ( ord_less_eq @ nat @ N @ M )
=> ( ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ M @ N ) )
= ( minus_minus @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ) ).
% of_nat_diff
thf(fact_949_minusinfinity,axiom,
! [D2: int,P1: int > $o,P: int > $o] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D2 )
=> ( ! [X3: int,K4: int] :
( ( P1 @ X3 )
= ( P1 @ ( minus_minus @ int @ X3 @ ( times_times @ int @ K4 @ D2 ) ) ) )
=> ( ? [Z5: int] :
! [X3: int] :
( ( ord_less @ int @ X3 @ Z5 )
=> ( ( P @ X3 )
= ( P1 @ X3 ) ) )
=> ( ? [X_1: int] : ( P1 @ X_1 )
=> ? [X_12: int] : ( P @ X_12 ) ) ) ) ) ).
% minusinfinity
thf(fact_950_plusinfinity,axiom,
! [D2: int,P2: int > $o,P: int > $o] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D2 )
=> ( ! [X3: int,K4: int] :
( ( P2 @ X3 )
= ( P2 @ ( minus_minus @ int @ X3 @ ( times_times @ int @ K4 @ D2 ) ) ) )
=> ( ? [Z5: int] :
! [X3: int] :
( ( ord_less @ int @ Z5 @ X3 )
=> ( ( P @ X3 )
= ( P2 @ X3 ) ) )
=> ( ? [X_1: int] : ( P2 @ X_1 )
=> ? [X_12: int] : ( P @ X_12 ) ) ) ) ) ).
% plusinfinity
thf(fact_951_int__minus,axiom,
! [N: nat,M: nat] :
( ( semiring_1_of_nat @ int @ ( minus_minus @ nat @ N @ M ) )
= ( semiring_1_of_nat @ int @ ( nat2 @ ( minus_minus @ int @ ( semiring_1_of_nat @ int @ N ) @ ( semiring_1_of_nat @ int @ M ) ) ) ) ) ).
% int_minus
thf(fact_952_diff__size__le__size__Diff,axiom,
! [A: $tType,M3: multiset @ A,M9: multiset @ A] : ( ord_less_eq @ nat @ ( minus_minus @ nat @ ( size_size @ ( multiset @ A ) @ M3 ) @ ( size_size @ ( multiset @ A ) @ M9 ) ) @ ( size_size @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ M3 @ M9 ) ) ) ).
% diff_size_le_size_Diff
thf(fact_953_diff__eq__diff__less__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( ( minus_minus @ A @ A2 @ B2 )
= ( minus_minus @ A @ C2 @ D2 ) )
=> ( ( ord_less_eq @ A @ A2 @ B2 )
= ( ord_less_eq @ A @ C2 @ D2 ) ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_954_diff__right__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ A2 @ C2 ) @ ( minus_minus @ A @ B2 @ C2 ) ) ) ) ).
% diff_right_mono
thf(fact_955_diff__left__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ C2 @ A2 ) @ ( minus_minus @ A @ C2 @ B2 ) ) ) ) ).
% diff_left_mono
thf(fact_956_diff__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A,D2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ D2 @ C2 )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ A2 @ C2 ) @ ( minus_minus @ A @ B2 @ D2 ) ) ) ) ) ).
% diff_mono
thf(fact_957_eq__iff__diff__eq__0,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ( ( ^ [Y3: A,Z: A] : Y3 = Z )
= ( ^ [A3: A,B3: A] :
( ( minus_minus @ A @ A3 @ B3 )
= ( zero_zero @ A ) ) ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_958_diff__strict__right__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ord_less @ A @ ( minus_minus @ A @ A2 @ C2 ) @ ( minus_minus @ A @ B2 @ C2 ) ) ) ) ).
% diff_strict_right_mono
thf(fact_959_diff__strict__left__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ( ord_less @ A @ ( minus_minus @ A @ C2 @ A2 ) @ ( minus_minus @ A @ C2 @ B2 ) ) ) ) ).
% diff_strict_left_mono
thf(fact_960_diff__eq__diff__less,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( ( minus_minus @ A @ A2 @ B2 )
= ( minus_minus @ A @ C2 @ D2 ) )
=> ( ( ord_less @ A @ A2 @ B2 )
= ( ord_less @ A @ C2 @ D2 ) ) ) ) ).
% diff_eq_diff_less
thf(fact_961_diff__strict__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A,D2: A,C2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ D2 @ C2 )
=> ( ord_less @ A @ ( minus_minus @ A @ A2 @ C2 ) @ ( minus_minus @ A @ B2 @ D2 ) ) ) ) ) ).
% diff_strict_mono
thf(fact_962_diff__diff__eq,axiom,
! [A: $tType] :
( ( cancel2418104881723323429up_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( minus_minus @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C2 )
= ( minus_minus @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% diff_diff_eq
thf(fact_963_add__implies__diff,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [C2: A,B2: A,A2: A] :
( ( ( plus_plus @ A @ C2 @ B2 )
= A2 )
=> ( C2
= ( minus_minus @ A @ A2 @ B2 ) ) ) ) ).
% add_implies_diff
thf(fact_964_diff__add__eq__diff__diff__swap,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( minus_minus @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) )
= ( minus_minus @ A @ ( minus_minus @ A @ A2 @ C2 ) @ B2 ) ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_965_diff__add__eq,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( plus_plus @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C2 )
= ( minus_minus @ A @ ( plus_plus @ A @ A2 @ C2 ) @ B2 ) ) ) ).
% diff_add_eq
thf(fact_966_diff__diff__eq2,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( minus_minus @ A @ A2 @ ( minus_minus @ A @ B2 @ C2 ) )
= ( minus_minus @ A @ ( plus_plus @ A @ A2 @ C2 ) @ B2 ) ) ) ).
% diff_diff_eq2
thf(fact_967_add__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( plus_plus @ A @ A2 @ ( minus_minus @ A @ B2 @ C2 ) )
= ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 ) ) ) ).
% add_diff_eq
thf(fact_968_eq__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( A2
= ( minus_minus @ A @ C2 @ B2 ) )
= ( ( plus_plus @ A @ A2 @ B2 )
= C2 ) ) ) ).
% eq_diff_eq
thf(fact_969_diff__eq__eq,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ( minus_minus @ A @ A2 @ B2 )
= C2 )
= ( A2
= ( plus_plus @ A @ C2 @ B2 ) ) ) ) ).
% diff_eq_eq
thf(fact_970_group__cancel_Osub1,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A5: A,K: A,A2: A,B2: A] :
( ( A5
= ( plus_plus @ A @ K @ A2 ) )
=> ( ( minus_minus @ A @ A5 @ B2 )
= ( plus_plus @ A @ K @ ( minus_minus @ A @ A2 @ B2 ) ) ) ) ) ).
% group_cancel.sub1
thf(fact_971_minus__diff__commute,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [B2: A,A2: A] :
( ( minus_minus @ A @ ( uminus_uminus @ A @ B2 ) @ A2 )
= ( minus_minus @ A @ ( uminus_uminus @ A @ A2 ) @ B2 ) ) ) ).
% minus_diff_commute
thf(fact_972_mult__not__zero,axiom,
! [A: $tType] :
( ( mult_zero @ A )
=> ! [A2: A,B2: A] :
( ( ( times_times @ A @ A2 @ B2 )
!= ( zero_zero @ A ) )
=> ( ( A2
!= ( zero_zero @ A ) )
& ( B2
!= ( zero_zero @ A ) ) ) ) ) ).
% mult_not_zero
thf(fact_973_divisors__zero,axiom,
! [A: $tType] :
( ( semiri3467727345109120633visors @ A )
=> ! [A2: A,B2: A] :
( ( ( times_times @ A @ A2 @ B2 )
= ( zero_zero @ A ) )
=> ( ( A2
= ( zero_zero @ A ) )
| ( B2
= ( zero_zero @ A ) ) ) ) ) ).
% divisors_zero
thf(fact_974_no__zero__divisors,axiom,
! [A: $tType] :
( ( semiri3467727345109120633visors @ A )
=> ! [A2: A,B2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ A2 @ B2 )
!= ( zero_zero @ A ) ) ) ) ) ).
% no_zero_divisors
thf(fact_975_mult__left__cancel,axiom,
! [A: $tType] :
( ( semiri6575147826004484403cancel @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( ( times_times @ A @ C2 @ A2 )
= ( times_times @ A @ C2 @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% mult_left_cancel
thf(fact_976_mult__right__cancel,axiom,
! [A: $tType] :
( ( semiri6575147826004484403cancel @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( ( times_times @ A @ A2 @ C2 )
= ( times_times @ A @ B2 @ C2 ) )
= ( A2 = B2 ) ) ) ) ).
% mult_right_cancel
thf(fact_977_crossproduct__eq,axiom,
! [A: $tType] :
( ( semiri1453513574482234551roduct @ A )
=> ! [W2: A,Y: A,X2: A,Z2: A] :
( ( ( plus_plus @ A @ ( times_times @ A @ W2 @ Y ) @ ( times_times @ A @ X2 @ Z2 ) )
= ( plus_plus @ A @ ( times_times @ A @ W2 @ Z2 ) @ ( times_times @ A @ X2 @ Y ) ) )
= ( ( W2 = X2 )
| ( Y = Z2 ) ) ) ) ).
% crossproduct_eq
thf(fact_978_crossproduct__noteq,axiom,
! [A: $tType] :
( ( semiri1453513574482234551roduct @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( ( A2 != B2 )
& ( C2 != D2 ) )
= ( ( plus_plus @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ D2 ) )
!= ( plus_plus @ A @ ( times_times @ A @ A2 @ D2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ) ).
% crossproduct_noteq
thf(fact_979_ring__class_Oring__distribs_I2_J,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( times_times @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 )
= ( plus_plus @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_980_ring__class_Oring__distribs_I1_J,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( times_times @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) )
= ( plus_plus @ A @ ( times_times @ A @ A2 @ B2 ) @ ( times_times @ A @ A2 @ C2 ) ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_981_comm__semiring__class_Odistrib,axiom,
! [A: $tType] :
( ( comm_semiring @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( times_times @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 )
= ( plus_plus @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% comm_semiring_class.distrib
thf(fact_982_distrib__left,axiom,
! [A: $tType] :
( ( semiring @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( times_times @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) )
= ( plus_plus @ A @ ( times_times @ A @ A2 @ B2 ) @ ( times_times @ A @ A2 @ C2 ) ) ) ) ).
% distrib_left
thf(fact_983_distrib__right,axiom,
! [A: $tType] :
( ( semiring @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( times_times @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 )
= ( plus_plus @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% distrib_right
thf(fact_984_combine__common__factor,axiom,
! [A: $tType] :
( ( semiring @ A )
=> ! [A2: A,E2: A,B2: A,C2: A] :
( ( plus_plus @ A @ ( times_times @ A @ A2 @ E2 ) @ ( plus_plus @ A @ ( times_times @ A @ B2 @ E2 ) @ C2 ) )
= ( plus_plus @ A @ ( times_times @ A @ ( plus_plus @ A @ A2 @ B2 ) @ E2 ) @ C2 ) ) ) ).
% combine_common_factor
thf(fact_985_mult_Ocomm__neutral,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: A] :
( ( times_times @ A @ A2 @ ( one_one @ A ) )
= A2 ) ) ).
% mult.comm_neutral
thf(fact_986_comm__monoid__mult__class_Omult__1,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: A] :
( ( times_times @ A @ ( one_one @ A ) @ A2 )
= A2 ) ) ).
% comm_monoid_mult_class.mult_1
thf(fact_987_diffs0__imp__equal,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus @ nat @ M @ N )
= ( zero_zero @ nat ) )
=> ( ( ( minus_minus @ nat @ N @ M )
= ( zero_zero @ nat ) )
=> ( M = N ) ) ) ).
% diffs0_imp_equal
thf(fact_988_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus @ nat @ M @ ( zero_zero @ nat ) )
= M ) ).
% minus_nat.diff_0
thf(fact_989_minus__int__code_I1_J,axiom,
! [K: int] :
( ( minus_minus @ int @ K @ ( zero_zero @ int ) )
= K ) ).
% minus_int_code(1)
thf(fact_990_zero__induct__lemma,axiom,
! [P: nat > $o,K: nat,I: nat] :
( ( P @ K )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ ( minus_minus @ nat @ K @ I ) ) ) ) ).
% zero_induct_lemma
thf(fact_991_diff__less__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less @ nat @ M @ N )
=> ( ( ord_less @ nat @ M @ L )
=> ( ord_less @ nat @ ( minus_minus @ nat @ L @ N ) @ ( minus_minus @ nat @ L @ M ) ) ) ) ).
% diff_less_mono2
thf(fact_992_less__imp__diff__less,axiom,
! [J: nat,K: nat,N: nat] :
( ( ord_less @ nat @ J @ K )
=> ( ord_less @ nat @ ( minus_minus @ nat @ J @ N ) @ K ) ) ).
% less_imp_diff_less
thf(fact_993_diff__le__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ord_less_eq @ nat @ ( minus_minus @ nat @ L @ N ) @ ( minus_minus @ nat @ L @ M ) ) ) ).
% diff_le_mono2
thf(fact_994_le__diff__iff_H,axiom,
! [A2: nat,C2: nat,B2: nat] :
( ( ord_less_eq @ nat @ A2 @ C2 )
=> ( ( ord_less_eq @ nat @ B2 @ C2 )
=> ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ C2 @ A2 ) @ ( minus_minus @ nat @ C2 @ B2 ) )
= ( ord_less_eq @ nat @ B2 @ A2 ) ) ) ) ).
% le_diff_iff'
thf(fact_995_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_996_diff__le__mono,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ L ) @ ( minus_minus @ nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_997_Nat_Odiff__diff__eq,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ M )
=> ( ( ord_less_eq @ nat @ K @ N )
=> ( ( minus_minus @ nat @ ( minus_minus @ nat @ M @ K ) @ ( minus_minus @ nat @ N @ K ) )
= ( minus_minus @ nat @ M @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_998_le__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ M )
=> ( ( ord_less_eq @ nat @ K @ N )
=> ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ K ) @ ( minus_minus @ nat @ N @ K ) )
= ( ord_less_eq @ nat @ M @ N ) ) ) ) ).
% le_diff_iff
thf(fact_999_eq__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ M )
=> ( ( ord_less_eq @ nat @ K @ N )
=> ( ( ( minus_minus @ nat @ M @ K )
= ( minus_minus @ nat @ N @ K ) )
= ( M = N ) ) ) ) ).
% eq_diff_iff
thf(fact_1000_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N ) @ N )
= M ) ).
% diff_add_inverse2
thf(fact_1001_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus @ nat @ ( plus_plus @ nat @ N @ M ) @ N )
= M ) ).
% diff_add_inverse
thf(fact_1002_diff__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( minus_minus @ nat @ ( plus_plus @ nat @ M @ K ) @ ( plus_plus @ nat @ N @ K ) )
= ( minus_minus @ nat @ M @ N ) ) ).
% diff_cancel2
thf(fact_1003_Nat_Odiff__cancel,axiom,
! [K: nat,M: nat,N: nat] :
( ( minus_minus @ nat @ ( plus_plus @ nat @ K @ M ) @ ( plus_plus @ nat @ K @ N ) )
= ( minus_minus @ nat @ M @ N ) ) ).
% Nat.diff_cancel
thf(fact_1004_Multiset_Odiff__add,axiom,
! [A: $tType,M3: multiset @ A,N5: multiset @ A,Q: multiset @ A] :
( ( minus_minus @ ( multiset @ A ) @ M3 @ ( plus_plus @ ( multiset @ A ) @ N5 @ Q ) )
= ( minus_minus @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ M3 @ N5 ) @ Q ) ) ).
% Multiset.diff_add
thf(fact_1005_diff__union__cancelL,axiom,
! [A: $tType,N5: multiset @ A,M3: multiset @ A] :
( ( minus_minus @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ N5 @ M3 ) @ N5 )
= M3 ) ).
% diff_union_cancelL
thf(fact_1006_diff__union__cancelR,axiom,
! [A: $tType,M3: multiset @ A,N5: multiset @ A] :
( ( minus_minus @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ M3 @ N5 ) @ N5 )
= M3 ) ).
% diff_union_cancelR
thf(fact_1007_mult__0,axiom,
! [N: nat] :
( ( times_times @ nat @ ( zero_zero @ nat ) @ N )
= ( zero_zero @ nat ) ) ).
% mult_0
thf(fact_1008_Suc__mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times @ nat @ ( suc @ K ) @ M )
= ( times_times @ nat @ ( suc @ K ) @ N ) )
= ( M = N ) ) ).
% Suc_mult_cancel1
thf(fact_1009_mult__of__nat__commute,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [X2: nat,Y: A] :
( ( times_times @ A @ ( semiring_1_of_nat @ A @ X2 ) @ Y )
= ( times_times @ A @ Y @ ( semiring_1_of_nat @ A @ X2 ) ) ) ) ).
% mult_of_nat_commute
thf(fact_1010_times__int__code_I2_J,axiom,
! [L: int] :
( ( times_times @ int @ ( zero_zero @ int ) @ L )
= ( zero_zero @ int ) ) ).
% times_int_code(2)
thf(fact_1011_times__int__code_I1_J,axiom,
! [K: int] :
( ( times_times @ int @ K @ ( zero_zero @ int ) )
= ( zero_zero @ int ) ) ).
% times_int_code(1)
thf(fact_1012_mult__le__mono2,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ord_less_eq @ nat @ ( times_times @ nat @ K @ I ) @ ( times_times @ nat @ K @ J ) ) ) ).
% mult_le_mono2
thf(fact_1013_mult__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ord_less_eq @ nat @ ( times_times @ nat @ I @ K ) @ ( times_times @ nat @ J @ K ) ) ) ).
% mult_le_mono1
thf(fact_1014_mult__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( ord_less_eq @ nat @ K @ L )
=> ( ord_less_eq @ nat @ ( times_times @ nat @ I @ K ) @ ( times_times @ nat @ J @ L ) ) ) ) ).
% mult_le_mono
thf(fact_1015_le__square,axiom,
! [M: nat] : ( ord_less_eq @ nat @ M @ ( times_times @ nat @ M @ M ) ) ).
% le_square
thf(fact_1016_le__cube,axiom,
! [M: nat] : ( ord_less_eq @ nat @ M @ ( times_times @ nat @ M @ ( times_times @ nat @ M @ M ) ) ) ).
% le_cube
thf(fact_1017_int__diff__cases,axiom,
! [Z2: int] :
~ ! [M4: nat,N2: nat] :
( Z2
!= ( minus_minus @ int @ ( semiring_1_of_nat @ int @ M4 ) @ ( semiring_1_of_nat @ int @ N2 ) ) ) ).
% int_diff_cases
thf(fact_1018_add__mult__distrib2,axiom,
! [K: nat,M: nat,N: nat] :
( ( times_times @ nat @ K @ ( plus_plus @ nat @ M @ N ) )
= ( plus_plus @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N ) ) ) ).
% add_mult_distrib2
thf(fact_1019_add__mult__distrib,axiom,
! [M: nat,N: nat,K: nat] :
( ( times_times @ nat @ ( plus_plus @ nat @ M @ N ) @ K )
= ( plus_plus @ nat @ ( times_times @ nat @ M @ K ) @ ( times_times @ nat @ N @ K ) ) ) ).
% add_mult_distrib
thf(fact_1020_nat__minus__as__int,axiom,
( ( minus_minus @ nat )
= ( ^ [A3: nat,B3: nat] : ( nat2 @ ( minus_minus @ int @ ( semiring_1_of_nat @ int @ A3 ) @ ( semiring_1_of_nat @ int @ B3 ) ) ) ) ) ).
% nat_minus_as_int
thf(fact_1021_nat__mult__1,axiom,
! [N: nat] :
( ( times_times @ nat @ ( one_one @ nat ) @ N )
= N ) ).
% nat_mult_1
thf(fact_1022_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times @ nat @ N @ ( one_one @ nat ) )
= N ) ).
% nat_mult_1_right
thf(fact_1023_mult__of__int__commute,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X2: int,Y: A] :
( ( times_times @ A @ ( ring_1_of_int @ A @ X2 ) @ Y )
= ( times_times @ A @ Y @ ( ring_1_of_int @ A @ X2 ) ) ) ) ).
% mult_of_int_commute
thf(fact_1024_int__distrib_I2_J,axiom,
! [W2: int,Z1: int,Z22: int] :
( ( times_times @ int @ W2 @ ( plus_plus @ int @ Z1 @ Z22 ) )
= ( plus_plus @ int @ ( times_times @ int @ W2 @ Z1 ) @ ( times_times @ int @ W2 @ Z22 ) ) ) ).
% int_distrib(2)
thf(fact_1025_int__distrib_I1_J,axiom,
! [Z1: int,Z22: int,W2: int] :
( ( times_times @ int @ ( plus_plus @ int @ Z1 @ Z22 ) @ W2 )
= ( plus_plus @ int @ ( times_times @ int @ Z1 @ W2 ) @ ( times_times @ int @ Z22 @ W2 ) ) ) ).
% int_distrib(1)
thf(fact_1026_nat__times__as__int,axiom,
( ( times_times @ nat )
= ( ^ [A3: nat,B3: nat] : ( nat2 @ ( times_times @ int @ ( semiring_1_of_nat @ int @ A3 ) @ ( semiring_1_of_nat @ int @ B3 ) ) ) ) ) ).
% nat_times_as_int
thf(fact_1027_power__int__mult__distrib,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [X2: A,Y: A,M: int] :
( ( power_int @ A @ ( times_times @ A @ X2 @ Y ) @ M )
= ( times_times @ A @ ( power_int @ A @ X2 @ M ) @ ( power_int @ A @ Y @ M ) ) ) ) ).
% power_int_mult_distrib
thf(fact_1028_power__int__commutes,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X2: A,N: int] :
( ( times_times @ A @ ( power_int @ A @ X2 @ N ) @ X2 )
= ( times_times @ A @ X2 @ ( power_int @ A @ X2 @ N ) ) ) ) ).
% power_int_commutes
thf(fact_1029_power__int__mult,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X2: A,M: int,N: int] :
( ( power_int @ A @ X2 @ ( times_times @ int @ M @ N ) )
= ( power_int @ A @ ( power_int @ A @ X2 @ M ) @ N ) ) ) ).
% power_int_mult
thf(fact_1030_int__ops_I6_J,axiom,
! [A2: nat,B2: nat] :
( ( ( ord_less @ int @ ( semiring_1_of_nat @ int @ A2 ) @ ( semiring_1_of_nat @ int @ B2 ) )
=> ( ( semiring_1_of_nat @ int @ ( minus_minus @ nat @ A2 @ B2 ) )
= ( zero_zero @ int ) ) )
& ( ~ ( ord_less @ int @ ( semiring_1_of_nat @ int @ A2 ) @ ( semiring_1_of_nat @ int @ B2 ) )
=> ( ( semiring_1_of_nat @ int @ ( minus_minus @ nat @ A2 @ B2 ) )
= ( minus_minus @ int @ ( semiring_1_of_nat @ int @ A2 ) @ ( semiring_1_of_nat @ int @ B2 ) ) ) ) ) ).
% int_ops(6)
thf(fact_1031_mult__eq__if,axiom,
( ( times_times @ nat )
= ( ^ [M2: nat,N4: nat] :
( if @ nat
@ ( M2
= ( zero_zero @ nat ) )
@ ( zero_zero @ nat )
@ ( plus_plus @ nat @ N4 @ ( times_times @ nat @ ( minus_minus @ nat @ M2 @ ( one_one @ nat ) ) @ N4 ) ) ) ) ) ).
% mult_eq_if
thf(fact_1032_decr__mult__lemma,axiom,
! [D2: int,P: int > $o,K: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D2 )
=> ( ! [X3: int] :
( ( P @ X3 )
=> ( P @ ( minus_minus @ int @ X3 @ D2 ) ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
=> ! [X4: int] :
( ( P @ X4 )
=> ( P @ ( minus_minus @ int @ X4 @ ( times_times @ int @ K @ D2 ) ) ) ) ) ) ) ).
% decr_mult_lemma
thf(fact_1033_nat__diff__distrib_H,axiom,
! [X2: int,Y: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X2 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y )
=> ( ( nat2 @ ( minus_minus @ int @ X2 @ Y ) )
= ( minus_minus @ nat @ ( nat2 @ X2 ) @ ( nat2 @ Y ) ) ) ) ) ).
% nat_diff_distrib'
thf(fact_1034_nat__diff__distrib,axiom,
! [Z6: int,Z2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z6 )
=> ( ( ord_less_eq @ int @ Z6 @ Z2 )
=> ( ( nat2 @ ( minus_minus @ int @ Z2 @ Z6 ) )
= ( minus_minus @ nat @ ( nat2 @ Z2 ) @ ( nat2 @ Z6 ) ) ) ) ) ).
% nat_diff_distrib
thf(fact_1035_lambda__zero,axiom,
! [A: $tType] :
( ( mult_zero @ A )
=> ( ( ^ [H3: A] : ( zero_zero @ A ) )
= ( times_times @ A @ ( zero_zero @ A ) ) ) ) ).
% lambda_zero
thf(fact_1036_lambda__one,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ( ( ^ [X: A] : X )
= ( times_times @ A @ ( one_one @ A ) ) ) ) ).
% lambda_one
thf(fact_1037_nat__mult__distrib,axiom,
! [Z2: int,Z6: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z2 )
=> ( ( nat2 @ ( times_times @ int @ Z2 @ Z6 ) )
= ( times_times @ nat @ ( nat2 @ Z2 ) @ ( nat2 @ Z6 ) ) ) ) ).
% nat_mult_distrib
thf(fact_1038_zdiff__int__split,axiom,
! [P: int > $o,X2: nat,Y: nat] :
( ( P @ ( semiring_1_of_nat @ int @ ( minus_minus @ nat @ X2 @ Y ) ) )
= ( ( ( ord_less_eq @ nat @ Y @ X2 )
=> ( P @ ( minus_minus @ int @ ( semiring_1_of_nat @ int @ X2 ) @ ( semiring_1_of_nat @ int @ Y ) ) ) )
& ( ( ord_less @ nat @ X2 @ Y )
=> ( P @ ( zero_zero @ int ) ) ) ) ) ).
% zdiff_int_split
thf(fact_1039_le__iff__diff__le__0,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A3: A,B3: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ A3 @ B3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% le_iff_diff_le_0
thf(fact_1040_power__int__minus__mult,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [X2: A,N: int] :
( ( ( X2
!= ( zero_zero @ A ) )
| ( N
!= ( zero_zero @ int ) ) )
=> ( ( times_times @ A @ ( power_int @ A @ X2 @ ( minus_minus @ int @ N @ ( one_one @ int ) ) ) @ X2 )
= ( power_int @ A @ X2 @ N ) ) ) ) ).
% power_int_minus_mult
thf(fact_1041_less__iff__diff__less__0,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ( ( ord_less @ A )
= ( ^ [A3: A,B3: A] : ( ord_less @ A @ ( minus_minus @ A @ A3 @ B3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% less_iff_diff_less_0
thf(fact_1042_le__mult__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less_eq @ int @ ( times_times @ int @ ( archim6421214686448440834_floor @ A @ A2 ) @ ( archim6421214686448440834_floor @ A @ B2 ) ) @ ( archim6421214686448440834_floor @ A @ ( times_times @ A @ A2 @ B2 ) ) ) ) ) ) ).
% le_mult_floor
thf(fact_1043_diff__le__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C2 )
= ( ord_less_eq @ A @ A2 @ ( plus_plus @ A @ C2 @ B2 ) ) ) ) ).
% diff_le_eq
thf(fact_1044_le__diff__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ ( minus_minus @ A @ C2 @ B2 ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 ) ) ) ).
% le_diff_eq
thf(fact_1045_ordered__cancel__comm__monoid__diff__class_Odiff__add,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( plus_plus @ A @ ( minus_minus @ A @ B2 @ A2 ) @ A2 )
= B2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add
thf(fact_1046_le__add__diff,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ord_less_eq @ A @ C2 @ ( minus_minus @ A @ ( plus_plus @ A @ B2 @ C2 ) @ A2 ) ) ) ) ).
% le_add_diff
thf(fact_1047_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ ( minus_minus @ A @ B2 @ A2 ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ C2 @ A2 ) @ B2 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_1048_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( plus_plus @ A @ C2 @ ( minus_minus @ A @ B2 @ A2 ) )
= ( minus_minus @ A @ ( plus_plus @ A @ C2 @ B2 ) @ A2 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_1049_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( minus_minus @ A @ ( plus_plus @ A @ C2 @ B2 ) @ A2 )
= ( plus_plus @ A @ C2 @ ( minus_minus @ A @ B2 @ A2 ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_1050_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( plus_plus @ A @ ( minus_minus @ A @ B2 @ A2 ) @ C2 )
= ( minus_minus @ A @ ( plus_plus @ A @ B2 @ C2 ) @ A2 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_1051_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( minus_minus @ A @ ( plus_plus @ A @ B2 @ C2 ) @ A2 )
= ( plus_plus @ A @ ( minus_minus @ A @ B2 @ A2 ) @ C2 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_1052_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( minus_minus @ A @ C2 @ ( minus_minus @ A @ B2 @ A2 ) )
= ( minus_minus @ A @ ( plus_plus @ A @ C2 @ A2 ) @ B2 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_1053_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( plus_plus @ A @ A2 @ ( minus_minus @ A @ B2 @ A2 ) )
= B2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_1054_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ( minus_minus @ A @ B2 @ A2 )
= C2 )
= ( B2
= ( plus_plus @ A @ C2 @ A2 ) ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_1055_add__le__imp__le__diff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [I: A,K: A,N: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ N )
=> ( ord_less_eq @ A @ I @ ( minus_minus @ A @ N @ K ) ) ) ) ).
% add_le_imp_le_diff
thf(fact_1056_add__le__add__imp__diff__le,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [I: A,K: A,N: A,J: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ N )
=> ( ( ord_less_eq @ A @ N @ ( plus_plus @ A @ J @ K ) )
=> ( ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ N )
=> ( ( ord_less_eq @ A @ N @ ( plus_plus @ A @ J @ K ) )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ N @ K ) @ J ) ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_1057_mult__ceiling__le,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ ( times_times @ A @ A2 @ B2 ) ) @ ( times_times @ int @ ( archimedean_ceiling @ A @ A2 ) @ ( archimedean_ceiling @ A @ B2 ) ) ) ) ) ) ).
% mult_ceiling_le
thf(fact_1058_less__diff__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( ord_less @ A @ A2 @ ( minus_minus @ A @ C2 @ B2 ) )
= ( ord_less @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 ) ) ) ).
% less_diff_eq
thf(fact_1059_diff__less__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C2 )
= ( ord_less @ A @ A2 @ ( plus_plus @ A @ C2 @ B2 ) ) ) ) ).
% diff_less_eq
thf(fact_1060_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,B2: A] :
( ~ ( ord_less @ A @ A2 @ B2 )
=> ( ( plus_plus @ A @ B2 @ ( minus_minus @ A @ A2 @ B2 ) )
= A2 ) ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_1061_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ( ( minus_minus @ A )
= ( ^ [A3: A,B3: A] : ( plus_plus @ A @ A3 @ ( uminus_uminus @ A @ B3 ) ) ) ) ) ).
% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_1062_diff__conv__add__uminus,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ( ( minus_minus @ A )
= ( ^ [A3: A,B3: A] : ( plus_plus @ A @ A3 @ ( uminus_uminus @ A @ B3 ) ) ) ) ) ).
% diff_conv_add_uminus
thf(fact_1063_group__cancel_Osub2,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [B5: A,K: A,B2: A,A2: A] :
( ( B5
= ( plus_plus @ A @ K @ B2 ) )
=> ( ( minus_minus @ A @ A2 @ B5 )
= ( plus_plus @ A @ ( uminus_uminus @ A @ K ) @ ( minus_minus @ A @ A2 @ B2 ) ) ) ) ) ).
% group_cancel.sub2
thf(fact_1064_nat__mult__distrib__neg,axiom,
! [Z2: int,Z6: int] :
( ( ord_less_eq @ int @ Z2 @ ( zero_zero @ int ) )
=> ( ( nat2 @ ( times_times @ int @ Z2 @ Z6 ) )
= ( times_times @ nat @ ( nat2 @ ( uminus_uminus @ int @ Z2 ) ) @ ( nat2 @ ( uminus_uminus @ int @ Z6 ) ) ) ) ) ).
% nat_mult_distrib_neg
thf(fact_1065_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: $tType] :
( ( ordere2520102378445227354miring @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_1066_zero__le__mult__iff,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) )
| ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) ) ) ) ) ).
% zero_le_mult_iff
thf(fact_1067_mult__nonneg__nonpos2,axiom,
! [A: $tType] :
( ( ordered_semiring_0 @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ B2 @ A2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_1068_mult__nonpos__nonneg,axiom,
! [A: $tType] :
( ( ordered_semiring_0 @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A2 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% mult_nonpos_nonneg
thf(fact_1069_mult__nonneg__nonpos,axiom,
! [A: $tType] :
( ( ordered_semiring_0 @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ A2 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% mult_nonneg_nonpos
thf(fact_1070_mult__nonneg__nonneg,axiom,
! [A: $tType] :
( ( ordered_semiring_0 @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_1071_split__mult__neg__le,axiom,
! [A: $tType] :
( ( ordered_semiring_0 @ A )
=> ! [A2: A,B2: A] :
( ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
& ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) )
| ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ A2 @ B2 ) @ ( zero_zero @ A ) ) ) ) ).
% split_mult_neg_le
thf(fact_1072_mult__le__0__iff,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ A2 @ B2 ) @ ( zero_zero @ A ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
& ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) )
| ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) ) ) ) ).
% mult_le_0_iff
thf(fact_1073_mult__right__mono,axiom,
! [A: $tType] :
( ( ordered_semiring @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ) ).
% mult_right_mono
thf(fact_1074_mult__right__mono__neg,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ) ).
% mult_right_mono_neg
thf(fact_1075_mult__left__mono,axiom,
! [A: $tType] :
( ( ordered_semiring @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) ) ) ) ) ).
% mult_left_mono
thf(fact_1076_mult__nonpos__nonpos,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) ) ) ) ) ).
% mult_nonpos_nonpos
thf(fact_1077_mult__left__mono__neg,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) ) ) ) ) ).
% mult_left_mono_neg
thf(fact_1078_split__mult__pos__le,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [A2: A,B2: A] :
( ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) )
| ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) ) ) ) ).
% split_mult_pos_le
thf(fact_1079_zero__le__square,axiom,
! [A: $tType] :
( ( linordered_ring @ A )
=> ! [A2: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ A2 ) ) ) ).
% zero_le_square
thf(fact_1080_mult__mono_H,axiom,
! [A: $tType] :
( ( ordered_semiring @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ D2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ D2 ) ) ) ) ) ) ) ).
% mult_mono'
thf(fact_1081_mult__mono,axiom,
! [A: $tType] :
( ( ordered_semiring @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ D2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ D2 ) ) ) ) ) ) ) ).
% mult_mono
thf(fact_1082_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: $tType] :
( ( linord2810124833399127020strict @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_1083_mult__less__cancel__right__disj,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
& ( ord_less @ A @ A2 @ B2 ) )
| ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
& ( ord_less @ A @ B2 @ A2 ) ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_1084_mult__strict__right__mono,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ) ).
% mult_strict_right_mono
thf(fact_1085_mult__strict__right__mono__neg,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_1086_mult__less__cancel__left__disj,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
& ( ord_less @ A @ A2 @ B2 ) )
| ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
& ( ord_less @ A @ B2 @ A2 ) ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_1087_mult__strict__left__mono,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) ) ) ) ) ).
% mult_strict_left_mono
thf(fact_1088_mult__strict__left__mono__neg,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_1089_mult__less__cancel__left__pos,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( ord_less @ A @ A2 @ B2 ) ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_1090_mult__less__cancel__left__neg,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( ord_less @ A @ B2 @ A2 ) ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_1091_zero__less__mult__pos2,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [B2: A,A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ B2 @ A2 ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ) ).
% zero_less_mult_pos2
thf(fact_1092_zero__less__mult__pos,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ) ).
% zero_less_mult_pos
thf(fact_1093_zero__less__mult__iff,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
& ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) )
| ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
& ( ord_less @ A @ B2 @ ( zero_zero @ A ) ) ) ) ) ) ).
% zero_less_mult_iff
thf(fact_1094_mult__pos__neg2,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ B2 @ A2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% mult_pos_neg2
thf(fact_1095_mult__pos__pos,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) ) ) ) ) ).
% mult_pos_pos
thf(fact_1096_mult__pos__neg,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ A2 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% mult_pos_neg
thf(fact_1097_mult__neg__pos,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less @ A @ ( times_times @ A @ A2 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% mult_neg_pos
thf(fact_1098_mult__less__0__iff,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( times_times @ A @ A2 @ B2 ) @ ( zero_zero @ A ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
& ( ord_less @ A @ B2 @ ( zero_zero @ A ) ) )
| ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
& ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ) ) ).
% mult_less_0_iff
thf(fact_1099_not__square__less__zero,axiom,
! [A: $tType] :
( ( linordered_ring @ A )
=> ! [A2: A] :
~ ( ord_less @ A @ ( times_times @ A @ A2 @ A2 ) @ ( zero_zero @ A ) ) ) ).
% not_square_less_zero
thf(fact_1100_mult__neg__neg,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) ) ) ) ) ).
% mult_neg_neg
thf(fact_1101_add__scale__eq__noteq,axiom,
! [A: $tType] :
( ( semiri1453513574482234551roduct @ A )
=> ! [R: A,A2: A,B2: A,C2: A,D2: A] :
( ( R
!= ( zero_zero @ A ) )
=> ( ( ( A2 = B2 )
& ( C2 != D2 ) )
=> ( ( plus_plus @ A @ A2 @ ( times_times @ A @ R @ C2 ) )
!= ( plus_plus @ A @ B2 @ ( times_times @ A @ R @ D2 ) ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_1102_less__1__mult,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [M: A,N: A] :
( ( ord_less @ A @ ( one_one @ A ) @ M )
=> ( ( ord_less @ A @ ( one_one @ A ) @ N )
=> ( ord_less @ A @ ( one_one @ A ) @ ( times_times @ A @ M @ N ) ) ) ) ) ).
% less_1_mult
thf(fact_1103_Suc__to__right,axiom,
! [N: nat,M: nat] :
( ( ( suc @ N )
= M )
=> ( N
= ( minus_minus @ nat @ M @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% Suc_to_right
thf(fact_1104_diff__less,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( ord_less @ nat @ ( minus_minus @ nat @ M @ N ) @ M ) ) ) ).
% diff_less
thf(fact_1105_Suc__diff__Suc,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ N @ M )
=> ( ( suc @ ( minus_minus @ nat @ M @ ( suc @ N ) ) )
= ( minus_minus @ nat @ M @ N ) ) ) ).
% Suc_diff_Suc
thf(fact_1106_diff__less__Suc,axiom,
! [M: nat,N: nat] : ( ord_less @ nat @ ( minus_minus @ nat @ M @ N ) @ ( suc @ M ) ) ).
% diff_less_Suc
thf(fact_1107_square__eq__1__iff,axiom,
! [A: $tType] :
( ( ring_15535105094025558882visors @ A )
=> ! [X2: A] :
( ( ( times_times @ A @ X2 @ X2 )
= ( one_one @ A ) )
= ( ( X2
= ( one_one @ A ) )
| ( X2
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ) ).
% square_eq_1_iff
thf(fact_1108_Suc__diff__le,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq @ nat @ N @ M )
=> ( ( minus_minus @ nat @ ( suc @ M ) @ N )
= ( suc @ ( minus_minus @ nat @ M @ N ) ) ) ) ).
% Suc_diff_le
thf(fact_1109_diff__add__0,axiom,
! [N: nat,M: nat] :
( ( minus_minus @ nat @ N @ ( plus_plus @ nat @ N @ M ) )
= ( zero_zero @ nat ) ) ).
% diff_add_0
thf(fact_1110_less__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ M )
=> ( ( ord_less_eq @ nat @ K @ N )
=> ( ( ord_less @ nat @ ( minus_minus @ nat @ M @ K ) @ ( minus_minus @ nat @ N @ K ) )
= ( ord_less @ nat @ M @ N ) ) ) ) ).
% less_diff_iff
thf(fact_1111_diff__less__mono,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( ord_less @ nat @ A2 @ B2 )
=> ( ( ord_less_eq @ nat @ C2 @ A2 )
=> ( ord_less @ nat @ ( minus_minus @ nat @ A2 @ C2 ) @ ( minus_minus @ nat @ B2 @ C2 ) ) ) ) ).
% diff_less_mono
thf(fact_1112_add__diff__inverse__nat,axiom,
! [M: nat,N: nat] :
( ~ ( ord_less @ nat @ M @ N )
=> ( ( plus_plus @ nat @ N @ ( minus_minus @ nat @ M @ N ) )
= M ) ) ).
% add_diff_inverse_nat
thf(fact_1113_less__diff__conv,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less @ nat @ I @ ( minus_minus @ nat @ J @ K ) )
= ( ord_less @ nat @ ( plus_plus @ nat @ I @ K ) @ J ) ) ).
% less_diff_conv
thf(fact_1114_union__diff__assoc,axiom,
! [A: $tType,C6: multiset @ A,B5: multiset @ A,A5: multiset @ A] :
( ( ( minus_minus @ ( multiset @ A ) @ C6 @ B5 )
= ( zero_zero @ ( multiset @ A ) ) )
=> ( ( minus_minus @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ A5 @ B5 ) @ C6 )
= ( plus_plus @ ( multiset @ A ) @ A5 @ ( minus_minus @ ( multiset @ A ) @ B5 @ C6 ) ) ) ) ).
% union_diff_assoc
thf(fact_1115_le__diff__conv,axiom,
! [J: nat,K: nat,I: nat] :
( ( ord_less_eq @ nat @ ( minus_minus @ nat @ J @ K ) @ I )
= ( ord_less_eq @ nat @ J @ ( plus_plus @ nat @ I @ K ) ) ) ).
% le_diff_conv
thf(fact_1116_Nat_Ole__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq @ nat @ K @ J )
=> ( ( ord_less_eq @ nat @ I @ ( minus_minus @ nat @ J @ K ) )
= ( ord_less_eq @ nat @ ( plus_plus @ nat @ I @ K ) @ J ) ) ) ).
% Nat.le_diff_conv2
thf(fact_1117_Nat_Odiff__add__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq @ nat @ K @ J )
=> ( ( minus_minus @ nat @ ( plus_plus @ nat @ I @ J ) @ K )
= ( plus_plus @ nat @ I @ ( minus_minus @ nat @ J @ K ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_1118_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq @ nat @ K @ J )
=> ( ( minus_minus @ nat @ ( plus_plus @ nat @ J @ I ) @ K )
= ( plus_plus @ nat @ ( minus_minus @ nat @ J @ K ) @ I ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_1119_Nat_Ole__imp__diff__is__add,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( ( minus_minus @ nat @ J @ I )
= K )
= ( J
= ( plus_plus @ nat @ K @ I ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_1120_minus__int__code_I2_J,axiom,
! [L: int] :
( ( minus_minus @ int @ ( zero_zero @ int ) @ L )
= ( uminus_uminus @ int @ L ) ) ).
% minus_int_code(2)
thf(fact_1121_mult__less__mono2,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less @ nat @ I @ J )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ord_less @ nat @ ( times_times @ nat @ K @ I ) @ ( times_times @ nat @ K @ J ) ) ) ) ).
% mult_less_mono2
thf(fact_1122_mult__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less @ nat @ I @ J )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ord_less @ nat @ ( times_times @ nat @ I @ K ) @ ( times_times @ nat @ J @ K ) ) ) ) ).
% mult_less_mono1
thf(fact_1123_int__le__induct,axiom,
! [I: int,K: int,P: int > $o] :
( ( ord_less_eq @ int @ I @ K )
=> ( ( P @ K )
=> ( ! [I3: int] :
( ( ord_less_eq @ int @ I3 @ K )
=> ( ( P @ I3 )
=> ( P @ ( minus_minus @ int @ I3 @ ( one_one @ int ) ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_le_induct
thf(fact_1124_Suc__mult__less__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less @ nat @ ( times_times @ nat @ ( suc @ K ) @ M ) @ ( times_times @ nat @ ( suc @ K ) @ N ) )
= ( ord_less @ nat @ M @ N ) ) ).
% Suc_mult_less_cancel1
thf(fact_1125_diff__Suc__eq__diff__pred,axiom,
! [M: nat,N: nat] :
( ( minus_minus @ nat @ M @ ( suc @ N ) )
= ( minus_minus @ nat @ ( minus_minus @ nat @ M @ ( one_one @ nat ) ) @ N ) ) ).
% diff_Suc_eq_diff_pred
thf(fact_1126_diff__nat__eq__if,axiom,
! [Z6: int,Z2: int] :
( ( ( ord_less @ int @ Z6 @ ( zero_zero @ int ) )
=> ( ( minus_minus @ nat @ ( nat2 @ Z2 ) @ ( nat2 @ Z6 ) )
= ( nat2 @ Z2 ) ) )
& ( ~ ( ord_less @ int @ Z6 @ ( zero_zero @ int ) )
=> ( ( minus_minus @ nat @ ( nat2 @ Z2 ) @ ( nat2 @ Z6 ) )
= ( if @ nat @ ( ord_less @ int @ ( minus_minus @ int @ Z2 @ Z6 ) @ ( zero_zero @ int ) ) @ ( zero_zero @ nat ) @ ( nat2 @ ( minus_minus @ int @ Z2 @ Z6 ) ) ) ) ) ) ).
% diff_nat_eq_if
thf(fact_1127_int__less__induct,axiom,
! [I: int,K: int,P: int > $o] :
( ( ord_less @ int @ I @ 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 @ I ) ) ) ) ).
% int_less_induct
thf(fact_1128_Suc__mult__le__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ ( times_times @ nat @ ( suc @ K ) @ M ) @ ( times_times @ nat @ ( suc @ K ) @ N ) )
= ( ord_less_eq @ nat @ M @ N ) ) ).
% Suc_mult_le_cancel1
thf(fact_1129_mult__Suc,axiom,
! [M: nat,N: nat] :
( ( times_times @ nat @ ( suc @ M ) @ N )
= ( plus_plus @ nat @ N @ ( times_times @ nat @ M @ N ) ) ) ).
% mult_Suc
thf(fact_1130_mlex__snd__decrI,axiom,
! [A2: nat,A6: nat,B2: nat,B6: nat,N5: nat] :
( ( A2 = A6 )
=> ( ( ord_less @ nat @ B2 @ B6 )
=> ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ A2 @ N5 ) @ B2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ A6 @ N5 ) @ B6 ) ) ) ) ).
% mlex_snd_decrI
thf(fact_1131_mlex__fst__decrI,axiom,
! [A2: nat,A6: nat,B2: nat,N5: nat,B6: nat] :
( ( ord_less @ nat @ A2 @ A6 )
=> ( ( ord_less @ nat @ B2 @ N5 )
=> ( ( ord_less @ nat @ B6 @ N5 )
=> ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ A2 @ N5 ) @ B2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ A6 @ N5 ) @ B6 ) ) ) ) ) ).
% mlex_fst_decrI
thf(fact_1132_mlex__bound,axiom,
! [A2: nat,A5: nat,B2: nat,N5: nat] :
( ( ord_less @ nat @ A2 @ A5 )
=> ( ( ord_less @ nat @ B2 @ N5 )
=> ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ A2 @ N5 ) @ B2 ) @ ( times_times @ nat @ A5 @ N5 ) ) ) ) ).
% mlex_bound
thf(fact_1133_mlex__leI,axiom,
! [A2: nat,A6: nat,B2: nat,B6: nat,N5: nat] :
( ( ord_less_eq @ nat @ A2 @ A6 )
=> ( ( ord_less_eq @ nat @ B2 @ B6 )
=> ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ A2 @ N5 ) @ B2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ A6 @ N5 ) @ B6 ) ) ) ) ).
% mlex_leI
thf(fact_1134_mult__eq__self__implies__10,axiom,
! [M: nat,N: nat] :
( ( M
= ( times_times @ nat @ M @ N ) )
=> ( ( N
= ( one_one @ nat ) )
| ( M
= ( zero_zero @ nat ) ) ) ) ).
% mult_eq_self_implies_10
thf(fact_1135_zmult__zless__mono2,axiom,
! [I: int,J: int,K: int] :
( ( ord_less @ int @ I @ J )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ K )
=> ( ord_less @ int @ ( times_times @ int @ K @ I ) @ ( times_times @ int @ K @ J ) ) ) ) ).
% zmult_zless_mono2
thf(fact_1136_zmult__eq__1__iff,axiom,
! [M: int,N: int] :
( ( ( times_times @ int @ M @ N )
= ( one_one @ int ) )
= ( ( ( M
= ( one_one @ int ) )
& ( N
= ( one_one @ int ) ) )
| ( ( M
= ( uminus_uminus @ int @ ( one_one @ int ) ) )
& ( N
= ( uminus_uminus @ int @ ( one_one @ int ) ) ) ) ) ) ).
% zmult_eq_1_iff
thf(fact_1137_pos__zmult__eq__1__iff__lemma,axiom,
! [M: int,N: int] :
( ( ( times_times @ int @ M @ N )
= ( one_one @ int ) )
=> ( ( M
= ( one_one @ int ) )
| ( M
= ( uminus_uminus @ int @ ( one_one @ int ) ) ) ) ) ).
% pos_zmult_eq_1_iff_lemma
thf(fact_1138_list_Osize_I3_J,axiom,
! [A: $tType] :
( ( size_size @ ( list @ A ) @ ( nil @ A ) )
= ( zero_zero @ nat ) ) ).
% list.size(3)
thf(fact_1139_len__greater__imp__nonempty,axiom,
! [A: $tType,X2: nat,L: list @ A] :
( ( ord_less @ nat @ X2 @ ( size_size @ ( list @ A ) @ L ) )
=> ( L
!= ( nil @ A ) ) ) ).
% len_greater_imp_nonempty
thf(fact_1140_fails__array__make,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [N: nat,F: nat > A,H: heap_ext @ product_unit] :
~ ( time_fails @ ( array @ A ) @ ( array_make @ A @ N @ F ) @ H ) ) ).
% fails_array_make
thf(fact_1141_mult__less__le__imp__less,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ D2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ D2 ) ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_1142_mult__le__less__imp__less,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ C2 @ D2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ D2 ) ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_1143_mult__right__le__imp__le,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ A2 @ B2 ) ) ) ) ).
% mult_right_le_imp_le
thf(fact_1144_mult__left__le__imp__le,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ A2 @ B2 ) ) ) ) ).
% mult_left_le_imp_le
thf(fact_1145_mult__le__cancel__left__pos,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( ord_less_eq @ A @ A2 @ B2 ) ) ) ) ).
% mult_le_cancel_left_pos
thf(fact_1146_mult__le__cancel__left__neg,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ).
% mult_le_cancel_left_neg
thf(fact_1147_mult__less__cancel__right,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ A2 @ B2 ) )
& ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B2 @ A2 ) ) ) ) ) ).
% mult_less_cancel_right
thf(fact_1148_mult__strict__mono_H,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ C2 @ D2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ D2 ) ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_1149_mult__right__less__imp__less,axiom,
! [A: $tType] :
( ( linordered_semiring @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ A2 @ B2 ) ) ) ) ).
% mult_right_less_imp_less
thf(fact_1150_mult__less__cancel__left,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ A2 @ B2 ) )
& ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B2 @ A2 ) ) ) ) ) ).
% mult_less_cancel_left
thf(fact_1151_mult__strict__mono,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ C2 @ D2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ D2 ) ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_1152_mult__left__less__imp__less,axiom,
! [A: $tType] :
( ( linordered_semiring @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ A2 @ B2 ) ) ) ) ).
% mult_left_less_imp_less
thf(fact_1153_mult__le__cancel__right,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ A2 @ B2 ) )
& ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ) ).
% mult_le_cancel_right
thf(fact_1154_mult__le__cancel__left,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ A2 @ B2 ) )
& ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ) ).
% mult_le_cancel_left
thf(fact_1155_sum__squares__ge__zero,axiom,
! [A: $tType] :
( ( linordered_ring @ A )
=> ! [X2: A,Y: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( times_times @ A @ X2 @ X2 ) @ ( times_times @ A @ Y @ Y ) ) ) ) ).
% sum_squares_ge_zero
thf(fact_1156_mult__left__le,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [C2: A,A2: A] :
( ( ord_less_eq @ A @ C2 @ ( one_one @ A ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ A2 ) ) ) ) ).
% mult_left_le
thf(fact_1157_mult__le__one,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ ( 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 @ A2 @ B2 ) @ ( one_one @ A ) ) ) ) ) ) ).
% mult_le_one
thf(fact_1158_mult__right__le__one__le,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X2: A,Y: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ord_less_eq @ A @ Y @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ X2 @ Y ) @ X2 ) ) ) ) ) ).
% mult_right_le_one_le
thf(fact_1159_mult__left__le__one__le,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X2: A,Y: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ord_less_eq @ A @ Y @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ Y @ X2 ) @ X2 ) ) ) ) ) ).
% mult_left_le_one_le
thf(fact_1160_not__sum__squares__lt__zero,axiom,
! [A: $tType] :
( ( linordered_ring @ A )
=> ! [X2: A,Y: A] :
~ ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ X2 @ X2 ) @ ( times_times @ A @ Y @ Y ) ) @ ( zero_zero @ A ) ) ) ).
% not_sum_squares_lt_zero
thf(fact_1161_ex__less__of__nat__mult,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X2: A,Y: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X2 )
=> ? [N2: nat] : ( ord_less @ A @ Y @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N2 ) @ X2 ) ) ) ) ).
% ex_less_of_nat_mult
thf(fact_1162_diff__Suc__less,axiom,
! [N: nat,I: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less @ nat @ ( minus_minus @ nat @ N @ ( suc @ I ) ) @ N ) ) ).
% diff_Suc_less
thf(fact_1163_nat__diff__split__asm,axiom,
! [P: nat > $o,A2: nat,B2: nat] :
( ( P @ ( minus_minus @ nat @ A2 @ B2 ) )
= ( ~ ( ( ( ord_less @ nat @ A2 @ B2 )
& ~ ( P @ ( zero_zero @ nat ) ) )
| ? [D5: nat] :
( ( A2
= ( plus_plus @ nat @ B2 @ D5 ) )
& ~ ( P @ D5 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_1164_nat__diff__split,axiom,
! [P: nat > $o,A2: nat,B2: nat] :
( ( P @ ( minus_minus @ nat @ A2 @ B2 ) )
= ( ( ( ord_less @ nat @ A2 @ B2 )
=> ( P @ ( zero_zero @ nat ) ) )
& ! [D5: nat] :
( ( A2
= ( plus_plus @ nat @ B2 @ D5 ) )
=> ( P @ D5 ) ) ) ) ).
% nat_diff_split
thf(fact_1165_less__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq @ nat @ K @ J )
=> ( ( ord_less @ nat @ ( minus_minus @ nat @ J @ K ) @ I )
= ( ord_less @ nat @ J @ ( plus_plus @ nat @ I @ K ) ) ) ) ).
% less_diff_conv2
thf(fact_1166_power__int__minus__one__diff__commute,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: int,B2: int] :
( ( power_int @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( minus_minus @ int @ A2 @ B2 ) )
= ( power_int @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( minus_minus @ int @ B2 @ A2 ) ) ) ) ).
% power_int_minus_one_diff_commute
thf(fact_1167_n__less__n__mult__m,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ M )
=> ( ord_less @ nat @ N @ ( times_times @ nat @ N @ M ) ) ) ) ).
% n_less_n_mult_m
thf(fact_1168_n__less__m__mult__n,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ M )
=> ( ord_less @ nat @ N @ ( times_times @ nat @ M @ N ) ) ) ) ).
% n_less_m_mult_n
thf(fact_1169_one__less__mult,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N )
=> ( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ M )
=> ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( times_times @ nat @ M @ N ) ) ) ) ).
% one_less_mult
thf(fact_1170_int__induct,axiom,
! [P: int > $o,K: int,I: 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 @ I ) ) ) ) ).
% int_induct
thf(fact_1171_pos__zmult__eq__1__iff,axiom,
! [M: int,N: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ M )
=> ( ( ( times_times @ int @ M @ N )
= ( one_one @ int ) )
= ( ( M
= ( one_one @ int ) )
& ( N
= ( one_one @ int ) ) ) ) ) ).
% pos_zmult_eq_1_iff
thf(fact_1172_frac__def,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archimedean_frac @ A )
= ( ^ [X: A] : ( minus_minus @ A @ X @ ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ X ) ) ) ) ) ) ).
% frac_def
thf(fact_1173_dbl__dec__def,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( neg_numeral_dbl_dec @ A )
= ( ^ [X: A] : ( minus_minus @ A @ ( plus_plus @ A @ X @ X ) @ ( one_one @ A ) ) ) ) ) ).
% dbl_dec_def
thf(fact_1174_field__le__mult__one__interval,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X2: A,Y: A] :
( ! [Z4: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Z4 )
=> ( ( ord_less @ A @ Z4 @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ Z4 @ X2 ) @ Y ) ) )
=> ( ord_less_eq @ A @ X2 @ Y ) ) ) ).
% field_le_mult_one_interval
thf(fact_1175_mult__less__cancel__right2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,C2: A] :
( ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ C2 )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ A2 @ ( one_one @ A ) ) )
& ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( one_one @ A ) @ A2 ) ) ) ) ) ).
% mult_less_cancel_right2
thf(fact_1176_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_1177_mult__less__cancel__left2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C2: A,A2: A] :
( ( ord_less @ A @ ( times_times @ A @ C2 @ A2 ) @ C2 )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ A2 @ ( one_one @ A ) ) )
& ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( one_one @ A ) @ A2 ) ) ) ) ) ).
% mult_less_cancel_left2
thf(fact_1178_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_1179_mult__le__cancel__right2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,C2: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ C2 )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ A2 @ ( one_one @ A ) ) )
& ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ A2 ) ) ) ) ) ).
% mult_le_cancel_right2
thf(fact_1180_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_1181_mult__le__cancel__left2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C2: A,A2: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A2 ) @ C2 )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ A2 @ ( one_one @ A ) ) )
& ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ A2 ) ) ) ) ) ).
% mult_le_cancel_left2
thf(fact_1182_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_1183_convex__bound__le,axiom,
! [A: $tType] :
( ( linord6961819062388156250ring_1 @ A )
=> ! [X2: A,A2: A,Y: A,U: A,V2: A] :
( ( ord_less_eq @ A @ X2 @ A2 )
=> ( ( ord_less_eq @ A @ Y @ A2 )
=> ( ( 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 @ X2 ) @ ( times_times @ A @ V2 @ Y ) ) @ A2 ) ) ) ) ) ) ) ).
% convex_bound_le
thf(fact_1184_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_1185_Suc__diff__eq__diff__pred,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( minus_minus @ nat @ ( suc @ M ) @ N )
= ( minus_minus @ nat @ M @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ).
% Suc_diff_eq_diff_pred
thf(fact_1186_Suc__pred_H,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( N
= ( suc @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ).
% Suc_pred'
thf(fact_1187_frac__add,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A,Y: A] :
( ( ( ord_less @ A @ ( plus_plus @ A @ ( archimedean_frac @ A @ X2 ) @ ( archimedean_frac @ A @ Y ) ) @ ( one_one @ A ) )
=> ( ( archimedean_frac @ A @ ( plus_plus @ A @ X2 @ Y ) )
= ( plus_plus @ A @ ( archimedean_frac @ A @ X2 ) @ ( archimedean_frac @ A @ Y ) ) ) )
& ( ~ ( ord_less @ A @ ( plus_plus @ A @ ( archimedean_frac @ A @ X2 ) @ ( archimedean_frac @ A @ Y ) ) @ ( one_one @ A ) )
=> ( ( archimedean_frac @ A @ ( plus_plus @ A @ X2 @ Y ) )
= ( minus_minus @ A @ ( plus_plus @ A @ ( archimedean_frac @ A @ X2 ) @ ( archimedean_frac @ A @ Y ) ) @ ( one_one @ A ) ) ) ) ) ) ).
% frac_add
thf(fact_1188_add__eq__if,axiom,
( ( plus_plus @ nat )
= ( ^ [M2: nat,N4: nat] :
( if @ nat
@ ( M2
= ( zero_zero @ nat ) )
@ N4
@ ( suc @ ( plus_plus @ nat @ ( minus_minus @ nat @ M2 @ ( one_one @ nat ) ) @ N4 ) ) ) ) ) ).
% add_eq_if
thf(fact_1189_Suc__n__minus__m__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( ord_less @ nat @ ( one_one @ nat ) @ M )
=> ( ( suc @ ( minus_minus @ nat @ N @ M ) )
= ( minus_minus @ nat @ N @ ( minus_minus @ nat @ M @ ( one_one @ nat ) ) ) ) ) ) ).
% Suc_n_minus_m_eq
thf(fact_1190_foldr__length__aux,axiom,
! [A: $tType,L: list @ A,A2: nat] :
( ( foldr @ A @ nat
@ ^ [X: A] : suc
@ L
@ A2 )
= ( plus_plus @ nat @ A2 @ ( size_size @ ( list @ A ) @ L ) ) ) ).
% foldr_length_aux
thf(fact_1191_zmult__zless__mono2__lemma,axiom,
! [I: int,J: int,K: nat] :
( ( ord_less @ int @ I @ J )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ord_less @ int @ ( times_times @ int @ ( semiring_1_of_nat @ int @ K ) @ I ) @ ( times_times @ int @ ( semiring_1_of_nat @ int @ K ) @ J ) ) ) ) ).
% zmult_zless_mono2_lemma
thf(fact_1192_power__int__add,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X2: A,M: int,N: int] :
( ( ( X2
!= ( zero_zero @ A ) )
| ( ( plus_plus @ int @ M @ N )
!= ( zero_zero @ int ) ) )
=> ( ( power_int @ A @ X2 @ ( plus_plus @ int @ M @ N ) )
= ( times_times @ A @ ( power_int @ A @ X2 @ M ) @ ( power_int @ A @ X2 @ N ) ) ) ) ) ).
% power_int_add
thf(fact_1193_unique__quotient__lemma__neg,axiom,
! [B2: int,Q5: int,R4: int,Q2: int,R: int] :
( ( ord_less_eq @ int @ ( plus_plus @ int @ ( times_times @ int @ B2 @ Q5 ) @ R4 ) @ ( plus_plus @ int @ ( times_times @ int @ B2 @ Q2 ) @ R ) )
=> ( ( ord_less_eq @ int @ R @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ B2 @ R )
=> ( ( ord_less @ int @ B2 @ R4 )
=> ( ord_less_eq @ int @ Q2 @ Q5 ) ) ) ) ) ).
% unique_quotient_lemma_neg
thf(fact_1194_unique__quotient__lemma,axiom,
! [B2: int,Q5: int,R4: int,Q2: int,R: int] :
( ( ord_less_eq @ int @ ( plus_plus @ int @ ( times_times @ int @ B2 @ Q5 ) @ R4 ) @ ( plus_plus @ int @ ( times_times @ int @ B2 @ Q2 ) @ R ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R4 )
=> ( ( ord_less @ int @ R4 @ B2 )
=> ( ( ord_less @ int @ R @ B2 )
=> ( ord_less_eq @ int @ Q5 @ Q2 ) ) ) ) ) ).
% unique_quotient_lemma
thf(fact_1195_zdiv__mono2__neg__lemma,axiom,
! [B2: int,Q2: int,R: int,B6: int,Q5: int,R4: int] :
( ( ( plus_plus @ int @ ( times_times @ int @ B2 @ Q2 ) @ R )
= ( plus_plus @ int @ ( times_times @ int @ B6 @ Q5 ) @ R4 ) )
=> ( ( ord_less @ int @ ( plus_plus @ int @ ( times_times @ int @ B6 @ Q5 ) @ R4 ) @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ R @ B2 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R4 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B6 )
=> ( ( ord_less_eq @ int @ B6 @ B2 )
=> ( ord_less_eq @ int @ Q5 @ Q2 ) ) ) ) ) ) ) ).
% zdiv_mono2_neg_lemma
thf(fact_1196_zdiv__mono2__lemma,axiom,
! [B2: int,Q2: int,R: int,B6: int,Q5: int,R4: int] :
( ( ( plus_plus @ int @ ( times_times @ int @ B2 @ Q2 ) @ R )
= ( plus_plus @ int @ ( times_times @ int @ B6 @ Q5 ) @ R4 ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( plus_plus @ int @ ( times_times @ int @ B6 @ Q5 ) @ R4 ) )
=> ( ( ord_less @ int @ R4 @ B6 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B6 )
=> ( ( ord_less_eq @ int @ B6 @ B2 )
=> ( ord_less_eq @ int @ Q2 @ Q5 ) ) ) ) ) ) ) ).
% zdiv_mono2_lemma
thf(fact_1197_q__pos__lemma,axiom,
! [B6: int,Q5: int,R4: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( plus_plus @ int @ ( times_times @ int @ B6 @ Q5 ) @ R4 ) )
=> ( ( ord_less @ int @ R4 @ B6 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B6 )
=> ( ord_less_eq @ int @ ( zero_zero @ int ) @ Q5 ) ) ) ) ).
% q_pos_lemma
thf(fact_1198_incr__mult__lemma,axiom,
! [D2: int,P: int > $o,K: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D2 )
=> ( ! [X3: int] :
( ( P @ X3 )
=> ( P @ ( plus_plus @ int @ X3 @ D2 ) ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
=> ! [X4: int] :
( ( P @ X4 )
=> ( P @ ( plus_plus @ int @ X4 @ ( times_times @ int @ K @ D2 ) ) ) ) ) ) ) ).
% incr_mult_lemma
thf(fact_1199_ceiling__diff__floor__le__1,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A] : ( ord_less_eq @ int @ ( minus_minus @ int @ ( archimedean_ceiling @ A @ X2 ) @ ( archim6421214686448440834_floor @ A @ X2 ) ) @ ( one_one @ int ) ) ) ).
% ceiling_diff_floor_le_1
thf(fact_1200_convex__bound__lt,axiom,
! [A: $tType] :
( ( linord715952674999750819strict @ A )
=> ! [X2: A,A2: A,Y: A,U: A,V2: A] :
( ( ord_less @ A @ X2 @ A2 )
=> ( ( ord_less @ A @ Y @ A2 )
=> ( ( 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 @ X2 ) @ ( times_times @ A @ V2 @ Y ) ) @ A2 ) ) ) ) ) ) ) ).
% convex_bound_lt
thf(fact_1201_ceiling__correct,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A] :
( ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ X2 ) ) @ ( one_one @ A ) ) @ X2 )
& ( ord_less_eq @ A @ X2 @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ X2 ) ) ) ) ) ).
% ceiling_correct
thf(fact_1202_ceiling__unique,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z2: int,X2: A] :
( ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ Z2 ) @ ( one_one @ A ) ) @ X2 )
=> ( ( ord_less_eq @ A @ X2 @ ( ring_1_of_int @ A @ Z2 ) )
=> ( ( archimedean_ceiling @ A @ X2 )
= Z2 ) ) ) ) ).
% ceiling_unique
thf(fact_1203_ceiling__eq__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A,A2: int] :
( ( ( archimedean_ceiling @ A @ X2 )
= A2 )
= ( ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ A2 ) @ ( one_one @ A ) ) @ X2 )
& ( ord_less_eq @ A @ X2 @ ( ring_1_of_int @ A @ A2 ) ) ) ) ) ).
% ceiling_eq_iff
thf(fact_1204_ceiling__split,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [P: int > $o,T2: A] :
( ( P @ ( archimedean_ceiling @ A @ T2 ) )
= ( ! [I2: int] :
( ( ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ I2 ) @ ( one_one @ A ) ) @ T2 )
& ( ord_less_eq @ A @ T2 @ ( ring_1_of_int @ A @ I2 ) ) )
=> ( P @ I2 ) ) ) ) ) ).
% ceiling_split
thf(fact_1205_ceiling__less__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A,Z2: int] :
( ( ord_less @ int @ ( archimedean_ceiling @ A @ X2 ) @ Z2 )
= ( ord_less_eq @ A @ X2 @ ( minus_minus @ A @ ( ring_1_of_int @ A @ Z2 ) @ ( one_one @ A ) ) ) ) ) ).
% ceiling_less_iff
thf(fact_1206_nat__mult__le__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ord_less_eq @ nat @ M @ N ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_1207_nat__mult__less__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
& ( ord_less @ nat @ M @ N ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_1208_sum__squares__gt__zero__iff,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [X2: A,Y: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( times_times @ A @ X2 @ X2 ) @ ( times_times @ A @ Y @ Y ) ) )
= ( ( X2
!= ( zero_zero @ A ) )
| ( Y
!= ( zero_zero @ A ) ) ) ) ) ).
% sum_squares_gt_zero_iff
thf(fact_1209_sum__squares__le__zero__iff,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [X2: A,Y: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ X2 @ X2 ) @ ( times_times @ A @ Y @ Y ) ) @ ( zero_zero @ A ) )
= ( ( X2
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ).
% sum_squares_le_zero_iff
thf(fact_1210_mult__le__cancel__iff2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z2: A,X2: A,Y: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Z2 )
=> ( ( ord_less_eq @ A @ ( times_times @ A @ Z2 @ X2 ) @ ( times_times @ A @ Z2 @ Y ) )
= ( ord_less_eq @ A @ X2 @ Y ) ) ) ) ).
% mult_le_cancel_iff2
thf(fact_1211_mult__le__cancel__iff1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z2: A,X2: A,Y: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Z2 )
=> ( ( ord_less_eq @ A @ ( times_times @ A @ X2 @ Z2 ) @ ( times_times @ A @ Y @ Z2 ) )
= ( ord_less_eq @ A @ X2 @ Y ) ) ) ) ).
% mult_le_cancel_iff1
thf(fact_1212_array__make,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( ( array_new @ A )
= ( ^ [N4: nat,X: A] :
( array_make @ A @ N4
@ ^ [Uu: nat] : X ) ) ) ) ).
% array_make
thf(fact_1213_nat__less__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N ) )
= ( ord_less @ nat @ M @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% nat_less_add_iff2
thf(fact_1214_int__distrib_I3_J,axiom,
! [Z1: int,Z22: int,W2: int] :
( ( times_times @ int @ ( minus_minus @ int @ Z1 @ Z22 ) @ W2 )
= ( minus_minus @ int @ ( times_times @ int @ Z1 @ W2 ) @ ( times_times @ int @ Z22 @ W2 ) ) ) ).
% int_distrib(3)
thf(fact_1215_int__distrib_I4_J,axiom,
! [W2: int,Z1: int,Z22: int] :
( ( times_times @ int @ W2 @ ( minus_minus @ int @ Z1 @ Z22 ) )
= ( minus_minus @ int @ ( times_times @ int @ W2 @ Z1 ) @ ( times_times @ int @ W2 @ Z22 ) ) ) ).
% int_distrib(4)
thf(fact_1216_diff__mult__distrib2,axiom,
! [K: nat,M: nat,N: nat] :
( ( times_times @ nat @ K @ ( minus_minus @ nat @ M @ N ) )
= ( minus_minus @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N ) ) ) ).
% diff_mult_distrib2
thf(fact_1217_diff__mult__distrib,axiom,
! [M: nat,N: nat,K: nat] :
( ( times_times @ nat @ ( minus_minus @ nat @ M @ N ) @ K )
= ( minus_minus @ nat @ ( times_times @ nat @ M @ K ) @ ( times_times @ nat @ N @ K ) ) ) ).
% diff_mult_distrib
thf(fact_1218_diff__commute,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus @ nat @ ( minus_minus @ nat @ I @ J ) @ K )
= ( minus_minus @ nat @ ( minus_minus @ nat @ I @ K ) @ J ) ) ).
% diff_commute
thf(fact_1219_nat__mult__eq__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times @ nat @ K @ M )
= ( times_times @ nat @ K @ N ) )
= ( ( K
= ( zero_zero @ nat ) )
| ( M = N ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_1220_left__add__mult__distrib,axiom,
! [I: nat,U: nat,J: nat,K: nat] :
( ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ K ) )
= ( plus_plus @ nat @ ( times_times @ nat @ ( plus_plus @ nat @ I @ J ) @ U ) @ K ) ) ).
% left_add_mult_distrib
thf(fact_1221_upd__ureturn,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [I: nat,X2: A,A2: array @ A] :
( ( heap_Time_bind @ ( array @ A ) @ ( array @ A ) @ ( array_upd @ A @ I @ X2 @ A2 )
@ ^ [Uu: array @ A] : ( heap_Time_ureturn @ ( array @ A ) @ A2 ) )
= ( array_upd @ A @ I @ X2 @ A2 ) ) ) ).
% upd_ureturn
thf(fact_1222_nat__mult__less__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ( ord_less @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N ) )
= ( ord_less @ nat @ M @ N ) ) ) ).
% nat_mult_less_cancel1
thf(fact_1223_nat__mult__eq__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ( ( times_times @ nat @ K @ M )
= ( times_times @ nat @ K @ N ) )
= ( M = N ) ) ) ).
% nat_mult_eq_cancel1
thf(fact_1224_nat__diff__add__eq2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( minus_minus @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N ) )
= ( minus_minus @ nat @ M @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% nat_diff_add_eq2
thf(fact_1225_nat__diff__add__eq1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ J @ I )
=> ( ( minus_minus @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N ) )
= ( minus_minus @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% nat_diff_add_eq1
thf(fact_1226_nat__le__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N ) )
= ( ord_less_eq @ nat @ M @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% nat_le_add_iff2
thf(fact_1227_nat__le__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ J @ I )
=> ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N ) )
= ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% nat_le_add_iff1
thf(fact_1228_nat__eq__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M )
= ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N ) )
= ( M
= ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% nat_eq_add_iff2
thf(fact_1229_nat__eq__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ J @ I )
=> ( ( ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M )
= ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N ) )
= ( ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ I @ J ) @ U ) @ M )
= N ) ) ) ).
% nat_eq_add_iff1
thf(fact_1230_mult__less__iff1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z2: A,X2: A,Y: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Z2 )
=> ( ( ord_less @ A @ ( times_times @ A @ X2 @ Z2 ) @ ( times_times @ A @ Y @ Z2 ) )
= ( ord_less @ A @ X2 @ Y ) ) ) ) ).
% mult_less_iff1
thf(fact_1231_sum__squares__eq__zero__iff,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [X2: A,Y: A] :
( ( ( plus_plus @ A @ ( times_times @ A @ X2 @ X2 ) @ ( times_times @ A @ Y @ Y ) )
= ( zero_zero @ A ) )
= ( ( X2
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ).
% sum_squares_eq_zero_iff
thf(fact_1232_nat__mult__le__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ( ord_less_eq @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N ) )
= ( ord_less_eq @ nat @ M @ N ) ) ) ).
% nat_mult_le_cancel1
thf(fact_1233_nat__less__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ J @ I )
=> ( ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N ) )
= ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% nat_less_add_iff1
thf(fact_1234_nat0__intermed__int__val,axiom,
! [N: nat,F: nat > int,K: int] :
( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ N )
=> ( ord_less_eq @ int @ ( abs_abs @ int @ ( minus_minus @ int @ ( F @ ( plus_plus @ nat @ I3 @ ( one_one @ nat ) ) ) @ ( F @ I3 ) ) ) @ ( one_one @ int ) ) )
=> ( ( ord_less_eq @ int @ ( F @ ( zero_zero @ nat ) ) @ K )
=> ( ( ord_less_eq @ int @ K @ ( F @ N ) )
=> ? [I3: nat] :
( ( ord_less_eq @ nat @ I3 @ N )
& ( ( F @ I3 )
= K ) ) ) ) ) ).
% nat0_intermed_int_val
thf(fact_1235_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_1236_power__int__minus__left__distrib,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( division_ring @ A )
& ( one @ B )
& ( uminus @ B ) )
=> ! [X2: C,A2: A,N: int] :
( ( nO_MATCH @ B @ C @ ( uminus_uminus @ B @ ( one_one @ B ) ) @ X2 )
=> ( ( power_int @ A @ ( uminus_uminus @ A @ A2 ) @ N )
= ( times_times @ A @ ( power_int @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N ) @ ( power_int @ A @ A2 @ N ) ) ) ) ) ).
% power_int_minus_left_distrib
thf(fact_1237_ceiling__divide__lower,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Q2: A,P3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Q2 )
=> ( ord_less @ A @ ( times_times @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ ( divide_divide @ A @ P3 @ Q2 ) ) ) @ ( one_one @ A ) ) @ Q2 ) @ P3 ) ) ) ).
% ceiling_divide_lower
thf(fact_1238_floor__divide__upper,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Q2: A,P3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Q2 )
=> ( ord_less @ A @ P3 @ ( times_times @ A @ ( plus_plus @ A @ ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ ( divide_divide @ A @ P3 @ Q2 ) ) ) @ ( one_one @ A ) ) @ Q2 ) ) ) ) ).
% floor_divide_upper
thf(fact_1239_nat__ivt__aux,axiom,
! [N: nat,F: nat > int,K: int] :
( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ N )
=> ( ord_less_eq @ int @ ( abs_abs @ int @ ( minus_minus @ int @ ( F @ ( suc @ I3 ) ) @ ( F @ I3 ) ) ) @ ( one_one @ int ) ) )
=> ( ( ord_less_eq @ int @ ( F @ ( zero_zero @ nat ) ) @ K )
=> ( ( ord_less_eq @ int @ K @ ( F @ N ) )
=> ? [I3: nat] :
( ( ord_less_eq @ nat @ I3 @ N )
& ( ( F @ I3 )
= K ) ) ) ) ) ).
% nat_ivt_aux
thf(fact_1240_mult__ceiling__le__Ints,axiom,
! [A: $tType,B: $tType] :
( ( ( archim2362893244070406136eiling @ B )
& ( linordered_idom @ A ) )
=> ! [A2: B,B2: B] :
( ( ord_less_eq @ B @ ( zero_zero @ B ) @ A2 )
=> ( ( member @ B @ A2 @ ( ring_1_Ints @ B ) )
=> ( ord_less_eq @ A @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ B @ ( times_times @ B @ A2 @ B2 ) ) ) @ ( ring_1_of_int @ A @ ( times_times @ int @ ( archimedean_ceiling @ B @ A2 ) @ ( archimedean_ceiling @ B @ B2 ) ) ) ) ) ) ) ).
% mult_ceiling_le_Ints
thf(fact_1241_le__mult__floor__Ints,axiom,
! [A: $tType,B: $tType] :
( ( ( archim2362893244070406136eiling @ B )
& ( linordered_idom @ A ) )
=> ! [A2: B,B2: B] :
( ( ord_less_eq @ B @ ( zero_zero @ B ) @ A2 )
=> ( ( member @ B @ A2 @ ( ring_1_Ints @ B ) )
=> ( ord_less_eq @ A @ ( ring_1_of_int @ A @ ( times_times @ int @ ( archim6421214686448440834_floor @ B @ A2 ) @ ( archim6421214686448440834_floor @ B @ B2 ) ) ) @ ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ B @ ( times_times @ B @ A2 @ B2 ) ) ) ) ) ) ) ).
% le_mult_floor_Ints
thf(fact_1242_frac__unique__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A,A2: A] :
( ( ( archimedean_frac @ A @ X2 )
= A2 )
= ( ( member @ A @ ( minus_minus @ A @ X2 @ A2 ) @ ( ring_1_Ints @ A ) )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
& ( ord_less @ A @ A2 @ ( one_one @ A ) ) ) ) ) ).
% frac_unique_iff
thf(fact_1243_abs__idempotent,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A] :
( ( abs_abs @ A @ ( abs_abs @ A @ A2 ) )
= ( abs_abs @ A @ A2 ) ) ) ).
% abs_idempotent
thf(fact_1244_divide__eq__0__iff,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A2: A,B2: A] :
( ( ( divide_divide @ A @ A2 @ B2 )
= ( zero_zero @ A ) )
= ( ( A2
= ( zero_zero @ A ) )
| ( B2
= ( zero_zero @ A ) ) ) ) ) ).
% divide_eq_0_iff
thf(fact_1245_divide__cancel__left,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ( divide_divide @ A @ C2 @ A2 )
= ( divide_divide @ A @ C2 @ B2 ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( A2 = B2 ) ) ) ) ).
% divide_cancel_left
thf(fact_1246_divide__cancel__right,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( ( divide_divide @ A @ A2 @ C2 )
= ( divide_divide @ A @ B2 @ C2 ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( A2 = B2 ) ) ) ) ).
% divide_cancel_right
thf(fact_1247_division__ring__divide__zero,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A] :
( ( divide_divide @ A @ A2 @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% division_ring_divide_zero
thf(fact_1248_div__0,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A2: A] :
( ( divide_divide @ A @ ( zero_zero @ A ) @ A2 )
= ( zero_zero @ A ) ) ) ).
% div_0
thf(fact_1249_div__by__0,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A2: A] :
( ( divide_divide @ A @ A2 @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% div_by_0
thf(fact_1250_div__by__1,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A2: A] :
( ( divide_divide @ A @ A2 @ ( one_one @ A ) )
= A2 ) ) ).
% div_by_1
thf(fact_1251_abs__0__eq,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A] :
( ( ( zero_zero @ A )
= ( abs_abs @ A @ A2 ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% abs_0_eq
thf(fact_1252_abs__eq__0,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A] :
( ( ( abs_abs @ A @ A2 )
= ( zero_zero @ A ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% abs_eq_0
thf(fact_1253_abs__zero,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ( ( abs_abs @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% abs_zero
thf(fact_1254_abs__0,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ( ( abs_abs @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% abs_0
thf(fact_1255_abs__add__abs,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A,B2: A] :
( ( abs_abs @ A @ ( plus_plus @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B2 ) ) )
= ( plus_plus @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B2 ) ) ) ) ).
% abs_add_abs
thf(fact_1256_abs__1,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ( ( abs_abs @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% abs_1
thf(fact_1257_nat__mult__div__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( K
= ( zero_zero @ nat ) )
=> ( ( divide_divide @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N ) )
= ( zero_zero @ nat ) ) )
& ( ( K
!= ( zero_zero @ nat ) )
=> ( ( divide_divide @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N ) )
= ( divide_divide @ nat @ M @ N ) ) ) ) ).
% nat_mult_div_cancel_disj
thf(fact_1258_abs__minus__cancel,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A] :
( ( abs_abs @ A @ ( uminus_uminus @ A @ A2 ) )
= ( abs_abs @ A @ A2 ) ) ) ).
% abs_minus_cancel
thf(fact_1259_abs__of__nat,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: nat] :
( ( abs_abs @ A @ ( semiring_1_of_nat @ A @ N ) )
= ( semiring_1_of_nat @ A @ N ) ) ) ).
% abs_of_nat
thf(fact_1260_slice__eq__bounds__empty,axiom,
! [A: $tType,I: nat,Xs: list @ A] :
( ( slice @ A @ I @ I @ Xs )
= ( nil @ A ) ) ).
% slice_eq_bounds_empty
thf(fact_1261_slice__Nil,axiom,
! [A: $tType,Begin: nat,End: nat] :
( ( slice @ A @ Begin @ End @ ( nil @ A ) )
= ( nil @ A ) ) ).
% slice_Nil
thf(fact_1262_frac__in__Ints__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A] :
( ( member @ A @ ( archimedean_frac @ A @ X2 ) @ ( ring_1_Ints @ A ) )
= ( member @ A @ X2 @ ( ring_1_Ints @ A ) ) ) ) ).
% frac_in_Ints_iff
thf(fact_1263_nonzero__mult__div__cancel__left,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A2: A,B2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ A2 @ B2 ) @ A2 )
= B2 ) ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_1264_nonzero__mult__div__cancel__right,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [B2: A,A2: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ A2 @ B2 ) @ B2 )
= A2 ) ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_1265_nonzero__mult__divide__mult__cancel__right2,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( divide_divide @ A @ A2 @ B2 ) ) ) ) ).
% nonzero_mult_divide_mult_cancel_right2
thf(fact_1266_nonzero__mult__divide__mult__cancel__right,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) )
= ( divide_divide @ A @ A2 @ B2 ) ) ) ) ).
% nonzero_mult_divide_mult_cancel_right
thf(fact_1267_nonzero__mult__divide__mult__cancel__left2,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ B2 @ C2 ) )
= ( divide_divide @ A @ A2 @ B2 ) ) ) ) ).
% nonzero_mult_divide_mult_cancel_left2
thf(fact_1268_nonzero__mult__divide__mult__cancel__left,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( divide_divide @ A @ A2 @ B2 ) ) ) ) ).
% nonzero_mult_divide_mult_cancel_left
thf(fact_1269_mult__divide__mult__cancel__left__if,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ( C2
= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( zero_zero @ A ) ) )
& ( ( C2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( divide_divide @ A @ A2 @ B2 ) ) ) ) ) ).
% mult_divide_mult_cancel_left_if
thf(fact_1270_zero__eq__1__divide__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A] :
( ( ( zero_zero @ A )
= ( divide_divide @ A @ ( one_one @ A ) @ A2 ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% zero_eq_1_divide_iff
thf(fact_1271_one__divide__eq__0__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A] :
( ( ( divide_divide @ A @ ( one_one @ A ) @ A2 )
= ( zero_zero @ A ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% one_divide_eq_0_iff
thf(fact_1272_eq__divide__eq__1,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,A2: A] :
( ( ( one_one @ A )
= ( divide_divide @ A @ B2 @ A2 ) )
= ( ( A2
!= ( zero_zero @ A ) )
& ( A2 = B2 ) ) ) ) ).
% eq_divide_eq_1
thf(fact_1273_divide__eq__eq__1,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,A2: A] :
( ( ( divide_divide @ A @ B2 @ A2 )
= ( one_one @ A ) )
= ( ( A2
!= ( zero_zero @ A ) )
& ( A2 = B2 ) ) ) ) ).
% divide_eq_eq_1
thf(fact_1274_divide__self__if,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A] :
( ( ( A2
= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ A2 @ A2 )
= ( zero_zero @ A ) ) )
& ( ( A2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ A2 @ A2 )
= ( one_one @ A ) ) ) ) ) ).
% divide_self_if
thf(fact_1275_divide__self,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ A2 @ A2 )
= ( one_one @ A ) ) ) ) ).
% divide_self
thf(fact_1276_one__eq__divide__iff,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A2: A,B2: A] :
( ( ( one_one @ A )
= ( divide_divide @ A @ A2 @ B2 ) )
= ( ( B2
!= ( zero_zero @ A ) )
& ( A2 = B2 ) ) ) ) ).
% one_eq_divide_iff
thf(fact_1277_divide__eq__1__iff,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A2: A,B2: A] :
( ( ( divide_divide @ A @ A2 @ B2 )
= ( one_one @ A ) )
= ( ( B2
!= ( zero_zero @ A ) )
& ( A2 = B2 ) ) ) ) ).
% divide_eq_1_iff
thf(fact_1278_div__self,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ A2 @ A2 )
= ( one_one @ A ) ) ) ) ).
% div_self
thf(fact_1279_abs__le__zero__iff,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ A2 ) @ ( zero_zero @ A ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% abs_le_zero_iff
thf(fact_1280_abs__le__self__iff,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ A2 ) @ A2 )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% abs_le_self_iff
thf(fact_1281_abs__of__nonneg,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( abs_abs @ A @ A2 )
= A2 ) ) ) ).
% abs_of_nonneg
thf(fact_1282_zero__less__abs__iff,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( abs_abs @ A @ A2 ) )
= ( A2
!= ( zero_zero @ A ) ) ) ) ).
% zero_less_abs_iff
thf(fact_1283_divide__minus1,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [X2: A] :
( ( divide_divide @ A @ X2 @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( uminus_uminus @ A @ X2 ) ) ) ).
% divide_minus1
thf(fact_1284_abs__neg__one,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( ( abs_abs @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( one_one @ A ) ) ) ).
% abs_neg_one
thf(fact_1285_abs__power__int__minus,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,N: int] :
( ( abs_abs @ A @ ( power_int @ A @ ( uminus_uminus @ A @ A2 ) @ N ) )
= ( abs_abs @ A @ ( power_int @ A @ A2 @ N ) ) ) ) ).
% abs_power_int_minus
thf(fact_1286_of__int__abs,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X2: int] :
( ( ring_1_of_int @ A @ ( abs_abs @ int @ X2 ) )
= ( abs_abs @ A @ ( ring_1_of_int @ A @ X2 ) ) ) ) ).
% of_int_abs
thf(fact_1287_frac__eq__0__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A] :
( ( ( archimedean_frac @ A @ X2 )
= ( zero_zero @ A ) )
= ( member @ A @ X2 @ ( ring_1_Ints @ A ) ) ) ) ).
% frac_eq_0_iff
thf(fact_1288_slice__complete,axiom,
! [A: $tType,Xs: list @ A] :
( ( slice @ A @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) @ Xs )
= Xs ) ).
% slice_complete
thf(fact_1289_zero__le__divide__1__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ ( one_one @ A ) @ A2 ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% zero_le_divide_1_iff
thf(fact_1290_divide__le__0__1__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ ( one_one @ A ) @ A2 ) @ ( zero_zero @ A ) )
= ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% divide_le_0_1_iff
thf(fact_1291_zero__less__divide__1__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ ( one_one @ A ) @ A2 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% zero_less_divide_1_iff
thf(fact_1292_less__divide__eq__1__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B2 @ A2 ) )
= ( ord_less @ A @ A2 @ B2 ) ) ) ) ).
% less_divide_eq_1_pos
thf(fact_1293_less__divide__eq__1__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B2 @ A2 ) )
= ( ord_less @ A @ B2 @ A2 ) ) ) ) ).
% less_divide_eq_1_neg
thf(fact_1294_divide__less__eq__1__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ ( divide_divide @ A @ B2 @ A2 ) @ ( one_one @ A ) )
= ( ord_less @ A @ B2 @ A2 ) ) ) ) ).
% divide_less_eq_1_pos
thf(fact_1295_divide__less__eq__1__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( divide_divide @ A @ B2 @ A2 ) @ ( one_one @ A ) )
= ( ord_less @ A @ A2 @ B2 ) ) ) ) ).
% divide_less_eq_1_neg
thf(fact_1296_divide__less__0__1__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A] :
( ( ord_less @ A @ ( divide_divide @ A @ ( one_one @ A ) @ A2 ) @ ( zero_zero @ A ) )
= ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% divide_less_0_1_iff
thf(fact_1297_nonzero__divide__mult__cancel__right,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [B2: A,A2: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ B2 @ ( times_times @ A @ A2 @ B2 ) )
= ( divide_divide @ A @ ( one_one @ A ) @ A2 ) ) ) ) ).
% nonzero_divide_mult_cancel_right
thf(fact_1298_nonzero__divide__mult__cancel__left,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A2: A,B2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ A2 @ ( times_times @ A @ A2 @ B2 ) )
= ( divide_divide @ A @ ( one_one @ A ) @ B2 ) ) ) ) ).
% nonzero_divide_mult_cancel_left
thf(fact_1299_abs__of__nonpos,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( abs_abs @ A @ A2 )
= ( uminus_uminus @ A @ A2 ) ) ) ) ).
% abs_of_nonpos
thf(fact_1300_zero__le__divide__abs__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A2 @ ( abs_abs @ A @ B2 ) ) )
= ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
| ( B2
= ( zero_zero @ A ) ) ) ) ) ).
% zero_le_divide_abs_iff
thf(fact_1301_divide__le__0__abs__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ A2 @ ( abs_abs @ A @ B2 ) ) @ ( zero_zero @ A ) )
= ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
| ( B2
= ( zero_zero @ A ) ) ) ) ) ).
% divide_le_0_abs_iff
thf(fact_1302_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_1303_floor__add2,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A,Y: A] :
( ( ( member @ A @ X2 @ ( ring_1_Ints @ A ) )
| ( member @ A @ Y @ ( ring_1_Ints @ A ) ) )
=> ( ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X2 @ Y ) )
= ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X2 ) @ ( archim6421214686448440834_floor @ A @ Y ) ) ) ) ) ).
% floor_add2
thf(fact_1304_frac__gt__0__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( archimedean_frac @ A @ X2 ) )
= ( ~ ( member @ A @ X2 @ ( ring_1_Ints @ A ) ) ) ) ) ).
% frac_gt_0_iff
thf(fact_1305_le__divide__eq__1__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B2 @ A2 ) )
= ( ord_less_eq @ A @ A2 @ B2 ) ) ) ) ).
% le_divide_eq_1_pos
thf(fact_1306_le__divide__eq__1__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B2 @ A2 ) )
= ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ).
% le_divide_eq_1_neg
thf(fact_1307_divide__le__eq__1__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ A2 ) @ ( one_one @ A ) )
= ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ).
% divide_le_eq_1_pos
thf(fact_1308_divide__le__eq__1__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ A2 ) @ ( one_one @ A ) )
= ( ord_less_eq @ A @ A2 @ B2 ) ) ) ) ).
% divide_le_eq_1_neg
thf(fact_1309_div__abs__eq__div__nat,axiom,
! [K: int,L: int] :
( ( divide_divide @ int @ ( abs_abs @ int @ K ) @ ( abs_abs @ int @ L ) )
= ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ ( nat2 @ ( abs_abs @ int @ K ) ) @ ( nat2 @ ( abs_abs @ int @ L ) ) ) ) ) ).
% div_abs_eq_div_nat
thf(fact_1310_zdiv__int,axiom,
! [A2: nat,B2: nat] :
( ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ A2 @ B2 ) )
= ( divide_divide @ int @ ( semiring_1_of_nat @ int @ A2 ) @ ( semiring_1_of_nat @ int @ B2 ) ) ) ).
% zdiv_int
thf(fact_1311_Ints__abs,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A] :
( ( member @ A @ A2 @ ( ring_1_Ints @ A ) )
=> ( member @ A @ ( abs_abs @ A @ A2 ) @ ( ring_1_Ints @ A ) ) ) ) ).
% Ints_abs
thf(fact_1312_nonzero__abs__divide,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,A2: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( abs_abs @ A @ ( divide_divide @ A @ A2 @ B2 ) )
= ( divide_divide @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B2 ) ) ) ) ) ).
% nonzero_abs_divide
thf(fact_1313_abs__div__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y: A,X2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Y )
=> ( ( divide_divide @ A @ ( abs_abs @ A @ X2 ) @ Y )
= ( abs_abs @ A @ ( divide_divide @ A @ X2 @ Y ) ) ) ) ) ).
% abs_div_pos
thf(fact_1314_abs__ge__self,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A] : ( ord_less_eq @ A @ A2 @ ( abs_abs @ A @ A2 ) ) ) ).
% abs_ge_self
thf(fact_1315_abs__le__D1,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ A2 ) @ B2 )
=> ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% abs_le_D1
thf(fact_1316_abs__eq__0__iff,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A2: A] :
( ( ( abs_abs @ A @ A2 )
= ( zero_zero @ A ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% abs_eq_0_iff
thf(fact_1317_floor__divide__of__int__eq,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [K: int,L: int] :
( ( archim6421214686448440834_floor @ A @ ( divide_divide @ A @ ( ring_1_of_int @ A @ K ) @ ( ring_1_of_int @ A @ L ) ) )
= ( divide_divide @ int @ K @ L ) ) ) ).
% floor_divide_of_int_eq
thf(fact_1318_abs__one,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( ( abs_abs @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% abs_one
thf(fact_1319_abs__minus__commute,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A,B2: A] :
( ( abs_abs @ A @ ( minus_minus @ A @ A2 @ B2 ) )
= ( abs_abs @ A @ ( minus_minus @ A @ B2 @ A2 ) ) ) ) ).
% abs_minus_commute
thf(fact_1320_add__divide__distrib,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( divide_divide @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 )
= ( plus_plus @ A @ ( divide_divide @ A @ A2 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ).
% add_divide_distrib
thf(fact_1321_power__int__abs,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,N: int] :
( ( abs_abs @ A @ ( power_int @ A @ A2 @ N ) )
= ( power_int @ A @ ( abs_abs @ A @ A2 ) @ N ) ) ) ).
% power_int_abs
thf(fact_1322_nat__div__as__int,axiom,
( ( divide_divide @ nat )
= ( ^ [A3: nat,B3: nat] : ( nat2 @ ( divide_divide @ int @ ( semiring_1_of_nat @ int @ A3 ) @ ( semiring_1_of_nat @ int @ B3 ) ) ) ) ) ).
% nat_div_as_int
thf(fact_1323_distrib__right__NO__MATCH,axiom,
! [B: $tType,A: $tType] :
( ( semiring @ A )
=> ! [X2: B,Y: B,C2: A,A2: A,B2: A] :
( ( nO_MATCH @ B @ A @ ( divide_divide @ B @ X2 @ Y ) @ C2 )
=> ( ( times_times @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 )
= ( plus_plus @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ) ).
% distrib_right_NO_MATCH
thf(fact_1324_distrib__left__NO__MATCH,axiom,
! [B: $tType,A: $tType] :
( ( semiring @ A )
=> ! [X2: B,Y: B,A2: A,B2: A,C2: A] :
( ( nO_MATCH @ B @ A @ ( divide_divide @ B @ X2 @ Y ) @ A2 )
=> ( ( times_times @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) )
= ( plus_plus @ A @ ( times_times @ A @ A2 @ B2 ) @ ( times_times @ A @ A2 @ C2 ) ) ) ) ) ).
% distrib_left_NO_MATCH
thf(fact_1325_power__int__divide__distrib,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [X2: A,Y: A,M: int] :
( ( power_int @ A @ ( divide_divide @ A @ X2 @ Y ) @ M )
= ( divide_divide @ A @ ( power_int @ A @ X2 @ M ) @ ( power_int @ A @ Y @ M ) ) ) ) ).
% power_int_divide_distrib
thf(fact_1326_Ints__0,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( member @ A @ ( zero_zero @ A ) @ ( ring_1_Ints @ A ) ) ) ).
% Ints_0
thf(fact_1327_Ints__add,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [A2: A,B2: A] :
( ( member @ A @ A2 @ ( ring_1_Ints @ A ) )
=> ( ( member @ A @ B2 @ ( ring_1_Ints @ A ) )
=> ( member @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( ring_1_Ints @ A ) ) ) ) ) ).
% Ints_add
thf(fact_1328_Ints__mult,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [A2: A,B2: A] :
( ( member @ A @ A2 @ ( ring_1_Ints @ A ) )
=> ( ( member @ A @ B2 @ ( ring_1_Ints @ A ) )
=> ( member @ A @ ( times_times @ A @ A2 @ B2 ) @ ( ring_1_Ints @ A ) ) ) ) ) ).
% Ints_mult
thf(fact_1329_Ints__1,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( member @ A @ ( one_one @ A ) @ ( ring_1_Ints @ A ) ) ) ).
% Ints_1
thf(fact_1330_Ints__diff,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [A2: A,B2: A] :
( ( member @ A @ A2 @ ( ring_1_Ints @ A ) )
=> ( ( member @ A @ B2 @ ( ring_1_Ints @ A ) )
=> ( member @ A @ ( minus_minus @ A @ A2 @ B2 ) @ ( ring_1_Ints @ A ) ) ) ) ) ).
% Ints_diff
thf(fact_1331_minus__in__Ints__iff,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X2: A] :
( ( member @ A @ ( uminus_uminus @ A @ X2 ) @ ( ring_1_Ints @ A ) )
= ( member @ A @ X2 @ ( ring_1_Ints @ A ) ) ) ) ).
% minus_in_Ints_iff
thf(fact_1332_Ints__minus,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [A2: A] :
( ( member @ A @ A2 @ ( ring_1_Ints @ A ) )
=> ( member @ A @ ( uminus_uminus @ A @ A2 ) @ ( ring_1_Ints @ A ) ) ) ) ).
% Ints_minus
thf(fact_1333_Ints__of__nat,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [N: nat] : ( member @ A @ ( semiring_1_of_nat @ A @ N ) @ ( ring_1_Ints @ A ) ) ) ).
% Ints_of_nat
thf(fact_1334_Ints__of__int,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [Z2: int] : ( member @ A @ ( ring_1_of_int @ A @ Z2 ) @ ( ring_1_Ints @ A ) ) ) ).
% Ints_of_int
thf(fact_1335_Ints__induct,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [Q2: A,P: A > $o] :
( ( member @ A @ Q2 @ ( ring_1_Ints @ A ) )
=> ( ! [Z4: int] : ( P @ ( ring_1_of_int @ A @ Z4 ) )
=> ( P @ Q2 ) ) ) ) ).
% Ints_induct
thf(fact_1336_Ints__cases,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [Q2: A] :
( ( member @ A @ Q2 @ ( ring_1_Ints @ A ) )
=> ~ ! [Z4: int] :
( Q2
!= ( ring_1_of_int @ A @ Z4 ) ) ) ) ).
% Ints_cases
thf(fact_1337_Ints__nonzero__abs__ge1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X2: A] :
( ( member @ A @ X2 @ ( ring_1_Ints @ A ) )
=> ( ( X2
!= ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ ( abs_abs @ A @ X2 ) ) ) ) ) ).
% Ints_nonzero_abs_ge1
thf(fact_1338_Ints__nonzero__abs__less1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X2: A] :
( ( member @ A @ X2 @ ( ring_1_Ints @ A ) )
=> ( ( ord_less @ A @ ( abs_abs @ A @ X2 ) @ ( one_one @ A ) )
=> ( X2
= ( zero_zero @ A ) ) ) ) ) ).
% Ints_nonzero_abs_less1
thf(fact_1339_Ints__eq__abs__less1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X2: A,Y: A] :
( ( member @ A @ X2 @ ( ring_1_Ints @ A ) )
=> ( ( member @ A @ Y @ ( ring_1_Ints @ A ) )
=> ( ( X2 = Y )
= ( ord_less @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X2 @ Y ) ) @ ( one_one @ A ) ) ) ) ) ) ).
% Ints_eq_abs_less1
thf(fact_1340_floor__divide__of__nat__eq,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [M: nat,N: nat] :
( ( archim6421214686448440834_floor @ A @ ( divide_divide @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) ) )
= ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ M @ N ) ) ) ) ).
% floor_divide_of_nat_eq
thf(fact_1341_abs__ge__zero,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( abs_abs @ A @ A2 ) ) ) ).
% abs_ge_zero
thf(fact_1342_abs__not__less__zero,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A] :
~ ( ord_less @ A @ ( abs_abs @ A @ A2 ) @ ( zero_zero @ A ) ) ) ).
% abs_not_less_zero
thf(fact_1343_abs__of__pos,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( abs_abs @ A @ A2 )
= A2 ) ) ) ).
% abs_of_pos
thf(fact_1344_abs__triangle__ineq,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A,B2: A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( plus_plus @ A @ A2 @ B2 ) ) @ ( plus_plus @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B2 ) ) ) ) ).
% abs_triangle_ineq
thf(fact_1345_abs__triangle__ineq2__sym,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A,B2: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B2 ) ) @ ( abs_abs @ A @ ( minus_minus @ A @ B2 @ A2 ) ) ) ) ).
% abs_triangle_ineq2_sym
thf(fact_1346_abs__triangle__ineq3,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A,B2: A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B2 ) ) ) @ ( abs_abs @ A @ ( minus_minus @ A @ A2 @ B2 ) ) ) ) ).
% abs_triangle_ineq3
thf(fact_1347_abs__triangle__ineq2,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A,B2: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B2 ) ) @ ( abs_abs @ A @ ( minus_minus @ A @ A2 @ B2 ) ) ) ) ).
% abs_triangle_ineq2
thf(fact_1348_abs__leI,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ B2 )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ A2 ) @ B2 ) ) ) ) ).
% abs_leI
thf(fact_1349_abs__le__D2,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ A2 ) @ B2 )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ B2 ) ) ) ).
% abs_le_D2
thf(fact_1350_abs__le__iff,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ A2 ) @ B2 )
= ( ( ord_less_eq @ A @ A2 @ B2 )
& ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ B2 ) ) ) ) ).
% abs_le_iff
thf(fact_1351_abs__ge__minus__self,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ ( abs_abs @ A @ A2 ) ) ) ).
% abs_ge_minus_self
thf(fact_1352_divide__right__mono__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ C2 ) @ ( divide_divide @ A @ A2 @ C2 ) ) ) ) ) ).
% divide_right_mono_neg
thf(fact_1353_divide__nonpos__nonpos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X2: A,Y: A] :
( ( ord_less_eq @ A @ X2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ Y @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X2 @ Y ) ) ) ) ) ).
% divide_nonpos_nonpos
thf(fact_1354_divide__nonpos__nonneg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X2: A,Y: A] :
( ( ord_less_eq @ A @ X2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ X2 @ Y ) @ ( zero_zero @ A ) ) ) ) ) ).
% divide_nonpos_nonneg
thf(fact_1355_divide__nonneg__nonpos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X2: A,Y: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X2 )
=> ( ( ord_less_eq @ A @ Y @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ X2 @ Y ) @ ( zero_zero @ A ) ) ) ) ) ).
% divide_nonneg_nonpos
thf(fact_1356_divide__nonneg__nonneg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X2: A,Y: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X2 @ Y ) ) ) ) ) ).
% divide_nonneg_nonneg
thf(fact_1357_zero__le__divide__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A2 @ B2 ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) )
| ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) ) ) ) ) ).
% zero_le_divide_iff
thf(fact_1358_divide__right__mono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ A2 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ) ).
% divide_right_mono
thf(fact_1359_divide__le__0__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ A2 @ B2 ) @ ( zero_zero @ A ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
& ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) )
| ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) ) ) ) ).
% divide_le_0_iff
thf(fact_1360_divide__neg__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X2: A,Y: A] :
( ( ord_less @ A @ X2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ Y @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X2 @ Y ) ) ) ) ) ).
% divide_neg_neg
thf(fact_1361_divide__neg__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X2: A,Y: A] :
( ( ord_less @ A @ X2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ Y )
=> ( ord_less @ A @ ( divide_divide @ A @ X2 @ Y ) @ ( zero_zero @ A ) ) ) ) ) ).
% divide_neg_pos
thf(fact_1362_divide__pos__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X2: A,Y: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X2 )
=> ( ( ord_less @ A @ Y @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( divide_divide @ A @ X2 @ Y ) @ ( zero_zero @ A ) ) ) ) ) ).
% divide_pos_neg
thf(fact_1363_divide__pos__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X2: A,Y: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ Y )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X2 @ Y ) ) ) ) ) ).
% divide_pos_pos
thf(fact_1364_divide__less__0__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( divide_divide @ A @ A2 @ B2 ) @ ( zero_zero @ A ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
& ( ord_less @ A @ B2 @ ( zero_zero @ A ) ) )
| ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
& ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ) ) ).
% divide_less_0_iff
thf(fact_1365_divide__less__cancel,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( ord_less @ A @ ( divide_divide @ A @ A2 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ A2 @ B2 ) )
& ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B2 @ A2 ) )
& ( C2
!= ( zero_zero @ A ) ) ) ) ) ).
% divide_less_cancel
thf(fact_1366_zero__less__divide__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A2 @ B2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
& ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) )
| ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
& ( ord_less @ A @ B2 @ ( zero_zero @ A ) ) ) ) ) ) ).
% zero_less_divide_iff
thf(fact_1367_divide__strict__right__mono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( divide_divide @ A @ A2 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ) ).
% divide_strict_right_mono
thf(fact_1368_divide__strict__right__mono__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( divide_divide @ A @ A2 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ) ).
% divide_strict_right_mono_neg
thf(fact_1369_nonzero__eq__divide__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( A2
= ( divide_divide @ A @ B2 @ C2 ) )
= ( ( times_times @ A @ A2 @ C2 )
= B2 ) ) ) ) ).
% nonzero_eq_divide_eq
thf(fact_1370_nonzero__divide__eq__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [C2: A,B2: A,A2: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( ( divide_divide @ A @ B2 @ C2 )
= A2 )
= ( B2
= ( times_times @ A @ A2 @ C2 ) ) ) ) ) ).
% nonzero_divide_eq_eq
thf(fact_1371_eq__divide__imp,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( ( times_times @ A @ A2 @ C2 )
= B2 )
=> ( A2
= ( divide_divide @ A @ B2 @ C2 ) ) ) ) ) ).
% eq_divide_imp
thf(fact_1372_divide__eq__imp,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [C2: A,B2: A,A2: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( B2
= ( times_times @ A @ A2 @ C2 ) )
=> ( ( divide_divide @ A @ B2 @ C2 )
= A2 ) ) ) ) ).
% divide_eq_imp
thf(fact_1373_eq__divide__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( A2
= ( divide_divide @ A @ B2 @ C2 ) )
= ( ( ( C2
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ A2 @ C2 )
= B2 ) )
& ( ( C2
= ( zero_zero @ A ) )
=> ( A2
= ( zero_zero @ A ) ) ) ) ) ) ).
% eq_divide_eq
thf(fact_1374_divide__eq__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,C2: A,A2: A] :
( ( ( divide_divide @ A @ B2 @ C2 )
= A2 )
= ( ( ( C2
!= ( zero_zero @ A ) )
=> ( B2
= ( times_times @ A @ A2 @ C2 ) ) )
& ( ( C2
= ( zero_zero @ A ) )
=> ( A2
= ( zero_zero @ A ) ) ) ) ) ) ).
% divide_eq_eq
thf(fact_1375_frac__eq__eq,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [Y: A,Z2: A,X2: A,W2: A] :
( ( Y
!= ( zero_zero @ A ) )
=> ( ( Z2
!= ( zero_zero @ A ) )
=> ( ( ( divide_divide @ A @ X2 @ Y )
= ( divide_divide @ A @ W2 @ Z2 ) )
= ( ( times_times @ A @ X2 @ Z2 )
= ( times_times @ A @ W2 @ Y ) ) ) ) ) ) ).
% frac_eq_eq
thf(fact_1376_right__inverse__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,A2: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( ( divide_divide @ A @ A2 @ B2 )
= ( one_one @ A ) )
= ( A2 = B2 ) ) ) ) ).
% right_inverse_eq
thf(fact_1377_nonzero__minus__divide__right,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,A2: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( uminus_uminus @ A @ ( divide_divide @ A @ A2 @ B2 ) )
= ( divide_divide @ A @ A2 @ ( uminus_uminus @ A @ B2 ) ) ) ) ) ).
% nonzero_minus_divide_right
thf(fact_1378_nonzero__minus__divide__divide,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,A2: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( uminus_uminus @ A @ A2 ) @ ( uminus_uminus @ A @ B2 ) )
= ( divide_divide @ A @ A2 @ B2 ) ) ) ) ).
% nonzero_minus_divide_divide
thf(fact_1379_abs__zmult__eq__1,axiom,
! [M: int,N: int] :
( ( ( abs_abs @ int @ ( times_times @ int @ M @ N ) )
= ( one_one @ int ) )
=> ( ( abs_abs @ int @ M )
= ( one_one @ int ) ) ) ).
% abs_zmult_eq_1
thf(fact_1380_div__neg__pos__less0,axiom,
! [A2: int,B2: int] :
( ( ord_less @ int @ A2 @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ord_less @ int @ ( divide_divide @ int @ A2 @ B2 ) @ ( zero_zero @ int ) ) ) ) ).
% div_neg_pos_less0
thf(fact_1381_neg__imp__zdiv__neg__iff,axiom,
! [B2: int,A2: int] :
( ( ord_less @ int @ B2 @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ ( divide_divide @ int @ A2 @ B2 ) @ ( zero_zero @ int ) )
= ( ord_less @ int @ ( zero_zero @ int ) @ A2 ) ) ) ).
% neg_imp_zdiv_neg_iff
thf(fact_1382_pos__imp__zdiv__neg__iff,axiom,
! [B2: int,A2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ( ord_less @ int @ ( divide_divide @ int @ A2 @ B2 ) @ ( zero_zero @ int ) )
= ( ord_less @ int @ A2 @ ( zero_zero @ int ) ) ) ) ).
% pos_imp_zdiv_neg_iff
thf(fact_1383_power__int__one__over,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X2: A,N: int] :
( ( power_int @ A @ ( divide_divide @ A @ ( one_one @ A ) @ X2 ) @ N )
= ( divide_divide @ A @ ( one_one @ A ) @ ( power_int @ A @ X2 @ N ) ) ) ) ).
% power_int_one_over
thf(fact_1384_Ints__double__eq__0__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [A2: A] :
( ( member @ A @ A2 @ ( ring_1_Ints @ A ) )
=> ( ( ( plus_plus @ A @ A2 @ A2 )
= ( zero_zero @ A ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ) ).
% Ints_double_eq_0_iff
thf(fact_1385_dense__eq0__I,axiom,
! [A: $tType] :
( ( ( ordere166539214618696060dd_abs @ A )
& ( dense_linorder @ A ) )
=> ! [X2: A] :
( ! [E4: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ E4 )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ X2 ) @ E4 ) )
=> ( X2
= ( zero_zero @ A ) ) ) ) ).
% dense_eq0_I
thf(fact_1386_abs__eq__mult,axiom,
! [A: $tType] :
( ( ordered_ring_abs @ A )
=> ! [A2: A,B2: A] :
( ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
| ( ord_less_eq @ A @ A2 @ ( 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 @ A2 @ B2 ) )
= ( times_times @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B2 ) ) ) ) ) ).
% abs_eq_mult
thf(fact_1387_abs__mult__pos,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X2: A,Y: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X2 )
=> ( ( times_times @ A @ ( abs_abs @ A @ Y ) @ X2 )
= ( abs_abs @ A @ ( times_times @ A @ Y @ X2 ) ) ) ) ) ).
% abs_mult_pos
thf(fact_1388_abs__minus__le__zero,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( abs_abs @ A @ A2 ) ) @ ( zero_zero @ A ) ) ) ).
% abs_minus_le_zero
thf(fact_1389_eq__abs__iff_H,axiom,
! [A: $tType] :
( ( linordered_ring @ A )
=> ! [A2: A,B2: A] :
( ( A2
= ( abs_abs @ A @ B2 ) )
= ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
& ( ( B2 = A2 )
| ( B2
= ( uminus_uminus @ A @ A2 ) ) ) ) ) ) ).
% eq_abs_iff'
thf(fact_1390_abs__eq__iff_H,axiom,
! [A: $tType] :
( ( linordered_ring @ A )
=> ! [A2: A,B2: A] :
( ( ( abs_abs @ A @ A2 )
= B2 )
= ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
& ( ( A2 = B2 )
| ( A2
= ( uminus_uminus @ A @ B2 ) ) ) ) ) ) ).
% abs_eq_iff'
thf(fact_1391_abs__diff__triangle__ineq,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( plus_plus @ A @ C2 @ D2 ) ) ) @ ( plus_plus @ A @ ( abs_abs @ A @ ( minus_minus @ A @ A2 @ C2 ) ) @ ( abs_abs @ A @ ( minus_minus @ A @ B2 @ D2 ) ) ) ) ) ).
% abs_diff_triangle_ineq
thf(fact_1392_abs__triangle__ineq4,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A,B2: A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ A2 @ B2 ) ) @ ( plus_plus @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B2 ) ) ) ) ).
% abs_triangle_ineq4
thf(fact_1393_abs__diff__le__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X2: A,A2: A,R: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X2 @ A2 ) ) @ R )
= ( ( ord_less_eq @ A @ ( minus_minus @ A @ A2 @ R ) @ X2 )
& ( ord_less_eq @ A @ X2 @ ( plus_plus @ A @ A2 @ R ) ) ) ) ) ).
% abs_diff_le_iff
thf(fact_1394_abs__if,axiom,
! [A: $tType] :
( ( abs_if @ A )
=> ( ( abs_abs @ A )
= ( ^ [A3: A] : ( if @ A @ ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) @ ( uminus_uminus @ A @ A3 ) @ A3 ) ) ) ) ).
% abs_if
thf(fact_1395_abs__of__neg,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A] :
( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( abs_abs @ A @ A2 )
= ( uminus_uminus @ A @ A2 ) ) ) ) ).
% abs_of_neg
thf(fact_1396_abs__if__raw,axiom,
! [A: $tType] :
( ( abs_if @ A )
=> ( ( abs_abs @ A )
= ( ^ [A3: A] : ( if @ A @ ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) @ ( uminus_uminus @ A @ A3 ) @ A3 ) ) ) ) ).
% abs_if_raw
thf(fact_1397_abs__diff__less__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X2: A,A2: A,R: A] :
( ( ord_less @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X2 @ A2 ) ) @ R )
= ( ( ord_less @ A @ ( minus_minus @ A @ A2 @ R ) @ X2 )
& ( ord_less @ A @ X2 @ ( plus_plus @ A @ A2 @ R ) ) ) ) ) ).
% abs_diff_less_iff
thf(fact_1398_div__positive,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [B2: A,A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A2 @ B2 ) ) ) ) ) ).
% div_positive
thf(fact_1399_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ A2 @ B2 )
=> ( ( divide_divide @ A @ A2 @ B2 )
= ( zero_zero @ A ) ) ) ) ) ).
% unique_euclidean_semiring_numeral_class.div_less
thf(fact_1400_divide__nonpos__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X2: A,Y: A] :
( ( ord_less_eq @ A @ X2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ Y )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ X2 @ Y ) @ ( zero_zero @ A ) ) ) ) ) ).
% divide_nonpos_pos
thf(fact_1401_divide__nonpos__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X2: A,Y: A] :
( ( ord_less_eq @ A @ X2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ Y @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X2 @ Y ) ) ) ) ) ).
% divide_nonpos_neg
thf(fact_1402_divide__nonneg__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X2: A,Y: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ Y )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X2 @ Y ) ) ) ) ) ).
% divide_nonneg_pos
thf(fact_1403_divide__nonneg__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X2: A,Y: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X2 )
=> ( ( ord_less @ A @ Y @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ X2 @ Y ) @ ( zero_zero @ A ) ) ) ) ) ).
% divide_nonneg_neg
thf(fact_1404_divide__le__cancel,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ A2 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ A2 @ B2 ) )
& ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ) ).
% divide_le_cancel
thf(fact_1405_frac__less2,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X2: A,Y: A,W2: A,Z2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X2 )
=> ( ( ord_less_eq @ A @ X2 @ Y )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ W2 )
=> ( ( ord_less @ A @ W2 @ Z2 )
=> ( ord_less @ A @ ( divide_divide @ A @ X2 @ Z2 ) @ ( divide_divide @ A @ Y @ W2 ) ) ) ) ) ) ) ).
% frac_less2
thf(fact_1406_frac__less,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X2: A,Y: A,W2: A,Z2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X2 )
=> ( ( ord_less @ A @ X2 @ Y )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ W2 )
=> ( ( ord_less_eq @ A @ W2 @ Z2 )
=> ( ord_less @ A @ ( divide_divide @ A @ X2 @ Z2 ) @ ( divide_divide @ A @ Y @ W2 ) ) ) ) ) ) ) ).
% frac_less
thf(fact_1407_frac__le,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y: A,X2: A,W2: A,Z2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ord_less_eq @ A @ X2 @ Y )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ W2 )
=> ( ( ord_less_eq @ A @ W2 @ Z2 )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ X2 @ Z2 ) @ ( divide_divide @ A @ Y @ W2 ) ) ) ) ) ) ) ).
% frac_le
thf(fact_1408_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( divide_divide @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) )
= ( divide_divide @ A @ ( divide_divide @ A @ A2 @ B2 ) @ C2 ) ) ) ) ).
% unique_euclidean_semiring_numeral_class.div_mult2_eq
thf(fact_1409_divide__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,C2: A,A2: A] :
( ( ord_less @ A @ ( divide_divide @ A @ B2 @ C2 ) @ A2 )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ B2 @ ( times_times @ A @ A2 @ C2 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ B2 ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ) ) ) ) ).
% divide_less_eq
thf(fact_1410_less__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less @ A @ A2 @ ( divide_divide @ A @ B2 @ C2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ B2 ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B2 @ ( times_times @ A @ A2 @ C2 ) ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% less_divide_eq
thf(fact_1411_neg__divide__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,B2: A,A2: A] :
( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( divide_divide @ A @ B2 @ C2 ) @ A2 )
= ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ B2 ) ) ) ) ).
% neg_divide_less_eq
thf(fact_1412_neg__less__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ A2 @ ( divide_divide @ A @ B2 @ C2 ) )
= ( ord_less @ A @ B2 @ ( times_times @ A @ A2 @ C2 ) ) ) ) ) ).
% neg_less_divide_eq
thf(fact_1413_pos__divide__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,B2: A,A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less @ A @ ( divide_divide @ A @ B2 @ C2 ) @ A2 )
= ( ord_less @ A @ B2 @ ( times_times @ A @ A2 @ C2 ) ) ) ) ) ).
% pos_divide_less_eq
thf(fact_1414_pos__less__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less @ A @ A2 @ ( divide_divide @ A @ B2 @ C2 ) )
= ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ B2 ) ) ) ) ).
% pos_less_divide_eq
thf(fact_1415_mult__imp__div__pos__less,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y: A,X2: A,Z2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ord_less @ A @ X2 @ ( times_times @ A @ Z2 @ Y ) )
=> ( ord_less @ A @ ( divide_divide @ A @ X2 @ Y ) @ Z2 ) ) ) ) ).
% mult_imp_div_pos_less
thf(fact_1416_mult__imp__less__div__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y: A,Z2: A,X2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ord_less @ A @ ( times_times @ A @ Z2 @ Y ) @ X2 )
=> ( ord_less @ A @ Z2 @ ( divide_divide @ A @ X2 @ Y ) ) ) ) ) ).
% mult_imp_less_div_pos
thf(fact_1417_divide__strict__left__mono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) )
=> ( ord_less @ A @ ( divide_divide @ A @ C2 @ A2 ) @ ( divide_divide @ A @ C2 @ B2 ) ) ) ) ) ) ).
% divide_strict_left_mono
thf(fact_1418_divide__strict__left__mono__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) )
=> ( ord_less @ A @ ( divide_divide @ A @ C2 @ A2 ) @ ( divide_divide @ A @ C2 @ B2 ) ) ) ) ) ) ).
% divide_strict_left_mono_neg
thf(fact_1419_less__divide__eq__1,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,A2: A] :
( ( ord_less @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B2 @ A2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
& ( ord_less @ A @ A2 @ B2 ) )
| ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
& ( ord_less @ A @ B2 @ A2 ) ) ) ) ) ).
% less_divide_eq_1
thf(fact_1420_divide__less__eq__1,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,A2: A] :
( ( ord_less @ A @ ( divide_divide @ A @ B2 @ A2 ) @ ( one_one @ A ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
& ( ord_less @ A @ B2 @ A2 ) )
| ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
& ( ord_less @ A @ A2 @ B2 ) )
| ( A2
= ( zero_zero @ A ) ) ) ) ) ).
% divide_less_eq_1
thf(fact_1421_add__divide__eq__if__simps_I2_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z2: A,A2: A,B2: A] :
( ( ( Z2
= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( divide_divide @ A @ A2 @ Z2 ) @ B2 )
= B2 ) )
& ( ( Z2
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( divide_divide @ A @ A2 @ Z2 ) @ B2 )
= ( divide_divide @ A @ ( plus_plus @ A @ A2 @ ( times_times @ A @ B2 @ Z2 ) ) @ Z2 ) ) ) ) ) ).
% add_divide_eq_if_simps(2)
thf(fact_1422_add__divide__eq__if__simps_I1_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z2: A,A2: A,B2: A] :
( ( ( Z2
= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ A2 @ ( divide_divide @ A @ B2 @ Z2 ) )
= A2 ) )
& ( ( Z2
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ A2 @ ( divide_divide @ A @ B2 @ Z2 ) )
= ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ A2 @ Z2 ) @ B2 ) @ Z2 ) ) ) ) ) ).
% add_divide_eq_if_simps(1)
thf(fact_1423_add__frac__eq,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [Y: A,Z2: A,X2: A,W2: A] :
( ( Y
!= ( zero_zero @ A ) )
=> ( ( Z2
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( divide_divide @ A @ X2 @ Y ) @ ( divide_divide @ A @ W2 @ Z2 ) )
= ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ X2 @ Z2 ) @ ( times_times @ A @ W2 @ Y ) ) @ ( times_times @ A @ Y @ Z2 ) ) ) ) ) ) ).
% add_frac_eq
thf(fact_1424_add__frac__num,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [Y: A,X2: A,Z2: A] :
( ( Y
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( divide_divide @ A @ X2 @ Y ) @ Z2 )
= ( divide_divide @ A @ ( plus_plus @ A @ X2 @ ( times_times @ A @ Z2 @ Y ) ) @ Y ) ) ) ) ).
% add_frac_num
thf(fact_1425_add__num__frac,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [Y: A,Z2: A,X2: A] :
( ( Y
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ Z2 @ ( divide_divide @ A @ X2 @ Y ) )
= ( divide_divide @ A @ ( plus_plus @ A @ X2 @ ( times_times @ A @ Z2 @ Y ) ) @ Y ) ) ) ) ).
% add_num_frac
thf(fact_1426_add__divide__eq__iff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z2: A,X2: A,Y: A] :
( ( Z2
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ X2 @ ( divide_divide @ A @ Y @ Z2 ) )
= ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ X2 @ Z2 ) @ Y ) @ Z2 ) ) ) ) ).
% add_divide_eq_iff
thf(fact_1427_divide__add__eq__iff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z2: A,X2: A,Y: A] :
( ( Z2
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( divide_divide @ A @ X2 @ Z2 ) @ Y )
= ( divide_divide @ A @ ( plus_plus @ A @ X2 @ ( times_times @ A @ Y @ Z2 ) ) @ Z2 ) ) ) ) ).
% divide_add_eq_iff
thf(fact_1428_divide__diff__eq__iff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z2: A,X2: A,Y: A] :
( ( Z2
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( divide_divide @ A @ X2 @ Z2 ) @ Y )
= ( divide_divide @ A @ ( minus_minus @ A @ X2 @ ( times_times @ A @ Y @ Z2 ) ) @ Z2 ) ) ) ) ).
% divide_diff_eq_iff
thf(fact_1429_diff__divide__eq__iff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z2: A,X2: A,Y: A] :
( ( Z2
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ X2 @ ( divide_divide @ A @ Y @ Z2 ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ X2 @ Z2 ) @ Y ) @ Z2 ) ) ) ) ).
% diff_divide_eq_iff
thf(fact_1430_diff__frac__eq,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [Y: A,Z2: A,X2: A,W2: A] :
( ( Y
!= ( zero_zero @ A ) )
=> ( ( Z2
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( divide_divide @ A @ X2 @ Y ) @ ( divide_divide @ A @ W2 @ Z2 ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ X2 @ Z2 ) @ ( times_times @ A @ W2 @ Y ) ) @ ( times_times @ A @ Y @ Z2 ) ) ) ) ) ) ).
% diff_frac_eq
thf(fact_1431_add__divide__eq__if__simps_I4_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z2: A,A2: A,B2: A] :
( ( ( Z2
= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ A2 @ ( divide_divide @ A @ B2 @ Z2 ) )
= A2 ) )
& ( ( Z2
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ A2 @ ( divide_divide @ A @ B2 @ Z2 ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ A2 @ Z2 ) @ B2 ) @ Z2 ) ) ) ) ) ).
% add_divide_eq_if_simps(4)
thf(fact_1432_less__half__sum,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ord_less @ A @ A2 @ ( divide_divide @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( plus_plus @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) ) ) ) ).
% less_half_sum
thf(fact_1433_gt__half__sum,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ord_less @ A @ ( divide_divide @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( plus_plus @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) @ B2 ) ) ) ).
% gt_half_sum
thf(fact_1434_eq__minus__divide__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( A2
= ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) )
= ( ( ( C2
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ A2 @ C2 )
= ( uminus_uminus @ A @ B2 ) ) )
& ( ( C2
= ( zero_zero @ A ) )
=> ( A2
= ( zero_zero @ A ) ) ) ) ) ) ).
% eq_minus_divide_eq
thf(fact_1435_minus__divide__eq__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,C2: A,A2: A] :
( ( ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) )
= A2 )
= ( ( ( C2
!= ( zero_zero @ A ) )
=> ( ( uminus_uminus @ A @ B2 )
= ( times_times @ A @ A2 @ C2 ) ) )
& ( ( C2
= ( zero_zero @ A ) )
=> ( A2
= ( zero_zero @ A ) ) ) ) ) ) ).
% minus_divide_eq_eq
thf(fact_1436_nonzero__neg__divide__eq__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( ( uminus_uminus @ A @ ( divide_divide @ A @ A2 @ B2 ) )
= C2 )
= ( ( uminus_uminus @ A @ A2 )
= ( times_times @ A @ C2 @ B2 ) ) ) ) ) ).
% nonzero_neg_divide_eq_eq
thf(fact_1437_nonzero__neg__divide__eq__eq2,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,C2: A,A2: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( C2
= ( uminus_uminus @ A @ ( divide_divide @ A @ A2 @ B2 ) ) )
= ( ( times_times @ A @ C2 @ B2 )
= ( uminus_uminus @ A @ A2 ) ) ) ) ) ).
% nonzero_neg_divide_eq_eq2
thf(fact_1438_divide__eq__minus__1__iff,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A2: A,B2: A] :
( ( ( divide_divide @ A @ A2 @ B2 )
= ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( ( B2
!= ( zero_zero @ A ) )
& ( A2
= ( uminus_uminus @ A @ B2 ) ) ) ) ) ).
% divide_eq_minus_1_iff
thf(fact_1439_zabs__def,axiom,
( ( abs_abs @ int )
= ( ^ [I2: int] : ( if @ int @ ( ord_less @ int @ I2 @ ( zero_zero @ int ) ) @ ( uminus_uminus @ int @ I2 ) @ I2 ) ) ) ).
% zabs_def
thf(fact_1440_nat__mult__div__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ( divide_divide @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N ) )
= ( divide_divide @ nat @ M @ N ) ) ) ).
% nat_mult_div_cancel1
thf(fact_1441_nat__abs__mult__distrib,axiom,
! [W2: int,Z2: int] :
( ( nat2 @ ( abs_abs @ int @ ( times_times @ int @ W2 @ Z2 ) ) )
= ( times_times @ nat @ ( nat2 @ ( abs_abs @ int @ W2 ) ) @ ( nat2 @ ( abs_abs @ int @ Z2 ) ) ) ) ).
% nat_abs_mult_distrib
thf(fact_1442_nonneg1__imp__zdiv__pos__iff,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A2 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ A2 @ B2 ) )
= ( ( ord_less_eq @ int @ B2 @ A2 )
& ( ord_less @ int @ ( zero_zero @ int ) @ B2 ) ) ) ) ).
% nonneg1_imp_zdiv_pos_iff
thf(fact_1443_pos__imp__zdiv__nonneg__iff,axiom,
! [B2: int,A2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ A2 @ B2 ) )
= ( ord_less_eq @ int @ ( zero_zero @ int ) @ A2 ) ) ) ).
% pos_imp_zdiv_nonneg_iff
thf(fact_1444_neg__imp__zdiv__nonneg__iff,axiom,
! [B2: int,A2: int] :
( ( ord_less @ int @ B2 @ ( zero_zero @ int ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ A2 @ B2 ) )
= ( ord_less_eq @ int @ A2 @ ( zero_zero @ int ) ) ) ) ).
% neg_imp_zdiv_nonneg_iff
thf(fact_1445_pos__imp__zdiv__pos__iff,axiom,
! [K: int,I: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ K )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ I @ K ) )
= ( ord_less_eq @ int @ K @ I ) ) ) ).
% pos_imp_zdiv_pos_iff
thf(fact_1446_div__nonpos__pos__le0,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq @ int @ A2 @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ord_less_eq @ int @ ( divide_divide @ int @ A2 @ B2 ) @ ( zero_zero @ int ) ) ) ) ).
% div_nonpos_pos_le0
thf(fact_1447_div__nonneg__neg__le0,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A2 )
=> ( ( ord_less @ int @ B2 @ ( zero_zero @ int ) )
=> ( ord_less_eq @ int @ ( divide_divide @ int @ A2 @ B2 ) @ ( zero_zero @ int ) ) ) ) ).
% div_nonneg_neg_le0
thf(fact_1448_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_1449_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_1450_zdiv__mono2__neg,axiom,
! [A2: int,B6: int,B2: int] :
( ( ord_less @ int @ A2 @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B6 )
=> ( ( ord_less_eq @ int @ B6 @ B2 )
=> ( ord_less_eq @ int @ ( divide_divide @ int @ A2 @ B6 ) @ ( divide_divide @ int @ A2 @ B2 ) ) ) ) ) ).
% zdiv_mono2_neg
thf(fact_1451_zdiv__mono1__neg,axiom,
! [A2: int,A6: int,B2: int] :
( ( ord_less_eq @ int @ A2 @ A6 )
=> ( ( ord_less @ int @ B2 @ ( zero_zero @ int ) )
=> ( ord_less_eq @ int @ ( divide_divide @ int @ A6 @ B2 ) @ ( divide_divide @ int @ A2 @ B2 ) ) ) ) ).
% zdiv_mono1_neg
thf(fact_1452_zdiv__eq__0__iff,axiom,
! [I: int,K: int] :
( ( ( divide_divide @ int @ I @ K )
= ( zero_zero @ int ) )
= ( ( K
= ( zero_zero @ int ) )
| ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ I )
& ( ord_less @ int @ I @ K ) )
| ( ( ord_less_eq @ int @ I @ ( zero_zero @ int ) )
& ( ord_less @ int @ K @ I ) ) ) ) ).
% zdiv_eq_0_iff
thf(fact_1453_zdiv__mono2,axiom,
! [A2: int,B6: int,B2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A2 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B6 )
=> ( ( ord_less_eq @ int @ B6 @ B2 )
=> ( ord_less_eq @ int @ ( divide_divide @ int @ A2 @ B2 ) @ ( divide_divide @ int @ A2 @ B6 ) ) ) ) ) ).
% zdiv_mono2
thf(fact_1454_zdiv__mono1,axiom,
! [A2: int,A6: int,B2: int] :
( ( ord_less_eq @ int @ A2 @ A6 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ord_less_eq @ int @ ( divide_divide @ int @ A2 @ B2 ) @ ( divide_divide @ int @ A6 @ B2 ) ) ) ) ).
% zdiv_mono1
thf(fact_1455_int__div__less__self,axiom,
! [X2: int,K: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ X2 )
=> ( ( ord_less @ int @ ( one_one @ int ) @ K )
=> ( ord_less @ int @ ( divide_divide @ int @ X2 @ K ) @ X2 ) ) ) ).
% int_div_less_self
thf(fact_1456_power__int__diff,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [X2: A,M: int,N: int] :
( ( ( X2
!= ( zero_zero @ A ) )
| ( M != N ) )
=> ( ( power_int @ A @ X2 @ ( minus_minus @ int @ M @ N ) )
= ( divide_divide @ A @ ( power_int @ A @ X2 @ M ) @ ( power_int @ A @ X2 @ N ) ) ) ) ) ).
% power_int_diff
thf(fact_1457_Ints__odd__nonzero,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [A2: A] :
( ( member @ A @ A2 @ ( ring_1_Ints @ A ) )
=> ( ( plus_plus @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A2 ) @ A2 )
!= ( zero_zero @ A ) ) ) ) ).
% Ints_odd_nonzero
thf(fact_1458_abs__add__one__gt__zero,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X2: A] : ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( one_one @ A ) @ ( abs_abs @ A @ X2 ) ) ) ) ).
% abs_add_one_gt_zero
thf(fact_1459_of__int__leD,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: int,X2: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ ( ring_1_of_int @ A @ N ) ) @ X2 )
=> ( ( N
= ( zero_zero @ int ) )
| ( ord_less_eq @ A @ ( one_one @ A ) @ X2 ) ) ) ) ).
% of_int_leD
thf(fact_1460_of__int__lessD,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: int,X2: A] :
( ( ord_less @ A @ ( abs_abs @ A @ ( ring_1_of_int @ A @ N ) ) @ X2 )
=> ( ( N
= ( zero_zero @ int ) )
| ( ord_less @ A @ ( one_one @ A ) @ X2 ) ) ) ) ).
% of_int_lessD
thf(fact_1461_divide__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,C2: A,A2: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ C2 ) @ A2 )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ B2 @ ( times_times @ A @ A2 @ C2 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ B2 ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ) ) ) ) ).
% divide_le_eq
thf(fact_1462_le__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ ( divide_divide @ A @ B2 @ C2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ B2 ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B2 @ ( times_times @ A @ A2 @ C2 ) ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% le_divide_eq
thf(fact_1463_divide__left__mono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ C2 @ A2 ) @ ( divide_divide @ A @ C2 @ B2 ) ) ) ) ) ) ).
% divide_left_mono
thf(fact_1464_neg__divide__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,B2: A,A2: A] :
( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ C2 ) @ A2 )
= ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ B2 ) ) ) ) ).
% neg_divide_le_eq
thf(fact_1465_neg__le__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ A2 @ ( divide_divide @ A @ B2 @ C2 ) )
= ( ord_less_eq @ A @ B2 @ ( times_times @ A @ A2 @ C2 ) ) ) ) ) ).
% neg_le_divide_eq
thf(fact_1466_pos__divide__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,B2: A,A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ C2 ) @ A2 )
= ( ord_less_eq @ A @ B2 @ ( times_times @ A @ A2 @ C2 ) ) ) ) ) ).
% pos_divide_le_eq
thf(fact_1467_pos__le__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less_eq @ A @ A2 @ ( divide_divide @ A @ B2 @ C2 ) )
= ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ B2 ) ) ) ) ).
% pos_le_divide_eq
thf(fact_1468_mult__imp__div__pos__le,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y: A,X2: A,Z2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ord_less_eq @ A @ X2 @ ( times_times @ A @ Z2 @ Y ) )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ X2 @ Y ) @ Z2 ) ) ) ) ).
% mult_imp_div_pos_le
thf(fact_1469_mult__imp__le__div__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y: A,Z2: A,X2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ord_less_eq @ A @ ( times_times @ A @ Z2 @ Y ) @ X2 )
=> ( ord_less_eq @ A @ Z2 @ ( divide_divide @ A @ X2 @ Y ) ) ) ) ) ).
% mult_imp_le_div_pos
thf(fact_1470_divide__left__mono__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ C2 @ A2 ) @ ( divide_divide @ A @ C2 @ B2 ) ) ) ) ) ) ).
% divide_left_mono_neg
thf(fact_1471_le__divide__eq__1,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B2 @ A2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
& ( ord_less_eq @ A @ A2 @ B2 ) )
| ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ) ).
% le_divide_eq_1
thf(fact_1472_divide__le__eq__1,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ A2 ) @ ( one_one @ A ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
& ( ord_less_eq @ A @ B2 @ A2 ) )
| ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ A2 @ B2 ) )
| ( A2
= ( zero_zero @ A ) ) ) ) ) ).
% divide_le_eq_1
thf(fact_1473_frac__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y: A,Z2: A,X2: A,W2: A] :
( ( Y
!= ( zero_zero @ A ) )
=> ( ( Z2
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( divide_divide @ A @ X2 @ Y ) @ ( divide_divide @ A @ W2 @ Z2 ) )
= ( ord_less_eq @ A @ ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ X2 @ Z2 ) @ ( times_times @ A @ W2 @ Y ) ) @ ( times_times @ A @ Y @ Z2 ) ) @ ( zero_zero @ A ) ) ) ) ) ) ).
% frac_le_eq
thf(fact_1474_frac__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y: A,Z2: A,X2: A,W2: A] :
( ( Y
!= ( zero_zero @ A ) )
=> ( ( Z2
!= ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( divide_divide @ A @ X2 @ Y ) @ ( divide_divide @ A @ W2 @ Z2 ) )
= ( ord_less @ A @ ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ X2 @ Z2 ) @ ( times_times @ A @ W2 @ Y ) ) @ ( times_times @ A @ Y @ Z2 ) ) @ ( zero_zero @ A ) ) ) ) ) ) ).
% frac_less_eq
thf(fact_1475_pos__minus__divide__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,B2: A,A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) @ A2 )
= ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A2 @ C2 ) ) ) ) ) ).
% pos_minus_divide_less_eq
thf(fact_1476_pos__less__minus__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less @ A @ A2 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) )
= ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ) ).
% pos_less_minus_divide_eq
thf(fact_1477_neg__minus__divide__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,B2: A,A2: A] :
( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) @ A2 )
= ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ) ).
% neg_minus_divide_less_eq
thf(fact_1478_neg__less__minus__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ A2 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) )
= ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A2 @ C2 ) ) ) ) ) ).
% neg_less_minus_divide_eq
thf(fact_1479_minus__divide__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,C2: A,A2: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) @ A2 )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A2 @ C2 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ ( uminus_uminus @ A @ B2 ) ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ) ) ) ) ).
% minus_divide_less_eq
thf(fact_1480_less__minus__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less @ A @ A2 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ A2 @ 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 @ A2 @ C2 ) ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% less_minus_divide_eq
thf(fact_1481_minus__divide__add__eq__iff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z2: A,X2: A,Y: A] :
( ( Z2
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ X2 @ Z2 ) ) @ Y )
= ( divide_divide @ A @ ( plus_plus @ A @ ( uminus_uminus @ A @ X2 ) @ ( times_times @ A @ Y @ Z2 ) ) @ Z2 ) ) ) ) ).
% minus_divide_add_eq_iff
thf(fact_1482_add__divide__eq__if__simps_I3_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z2: A,A2: A,B2: A] :
( ( ( Z2
= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ A2 @ Z2 ) ) @ B2 )
= B2 ) )
& ( ( Z2
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ A2 @ Z2 ) ) @ B2 )
= ( divide_divide @ A @ ( plus_plus @ A @ ( uminus_uminus @ A @ A2 ) @ ( times_times @ A @ B2 @ Z2 ) ) @ Z2 ) ) ) ) ) ).
% add_divide_eq_if_simps(3)
thf(fact_1483_add__divide__eq__if__simps_I6_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z2: A,A2: A,B2: A] :
( ( ( Z2
= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ A2 @ Z2 ) ) @ B2 )
= ( uminus_uminus @ A @ B2 ) ) )
& ( ( Z2
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ A2 @ Z2 ) ) @ B2 )
= ( divide_divide @ A @ ( minus_minus @ A @ ( uminus_uminus @ A @ A2 ) @ ( times_times @ A @ B2 @ Z2 ) ) @ Z2 ) ) ) ) ) ).
% add_divide_eq_if_simps(6)
thf(fact_1484_add__divide__eq__if__simps_I5_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z2: A,A2: A,B2: A] :
( ( ( Z2
= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( divide_divide @ A @ A2 @ Z2 ) @ B2 )
= ( uminus_uminus @ A @ B2 ) ) )
& ( ( Z2
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( divide_divide @ A @ A2 @ Z2 ) @ B2 )
= ( divide_divide @ A @ ( minus_minus @ A @ A2 @ ( times_times @ A @ B2 @ Z2 ) ) @ Z2 ) ) ) ) ) ).
% add_divide_eq_if_simps(5)
thf(fact_1485_minus__divide__diff__eq__iff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z2: A,X2: A,Y: A] :
( ( Z2
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ X2 @ Z2 ) ) @ Y )
= ( divide_divide @ A @ ( minus_minus @ A @ ( uminus_uminus @ A @ X2 ) @ ( times_times @ A @ Y @ Z2 ) ) @ Z2 ) ) ) ) ).
% minus_divide_diff_eq_iff
thf(fact_1486_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_1487_div__geq,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ~ ( ord_less @ nat @ M @ N )
=> ( ( divide_divide @ nat @ M @ N )
= ( suc @ ( divide_divide @ nat @ ( minus_minus @ nat @ M @ N ) @ N ) ) ) ) ) ).
% div_geq
thf(fact_1488_verit__less__mono__div__int2,axiom,
! [A5: int,B5: int,N: int] :
( ( ord_less_eq @ int @ A5 @ B5 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ ( uminus_uminus @ int @ N ) )
=> ( ord_less_eq @ int @ ( divide_divide @ int @ B5 @ N ) @ ( divide_divide @ int @ A5 @ N ) ) ) ) ).
% verit_less_mono_div_int2
thf(fact_1489_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_1490_Ints__odd__less__0,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A] :
( ( member @ A @ A2 @ ( ring_1_Ints @ A ) )
=> ( ( ord_less @ A @ ( plus_plus @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A2 ) @ A2 ) @ ( zero_zero @ A ) )
= ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ) ).
% Ints_odd_less_0
thf(fact_1491_scaling__mono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [U: A,V2: A,R: A,S3: A] :
( ( ord_less_eq @ A @ U @ V2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ R )
=> ( ( ord_less_eq @ A @ R @ S3 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ U @ ( divide_divide @ A @ ( times_times @ A @ R @ ( minus_minus @ A @ V2 @ U ) ) @ S3 ) ) @ V2 ) ) ) ) ) ).
% scaling_mono
thf(fact_1492_pos__minus__divide__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,B2: A,A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) @ A2 )
= ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A2 @ C2 ) ) ) ) ) ).
% pos_minus_divide_le_eq
thf(fact_1493_pos__le__minus__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less_eq @ A @ A2 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) )
= ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ) ).
% pos_le_minus_divide_eq
thf(fact_1494_neg__minus__divide__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,B2: A,A2: A] :
( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) @ A2 )
= ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ) ).
% neg_minus_divide_le_eq
thf(fact_1495_neg__le__minus__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ A2 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) )
= ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A2 @ C2 ) ) ) ) ) ).
% neg_le_minus_divide_eq
thf(fact_1496_minus__divide__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,C2: A,A2: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) @ A2 )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A2 @ C2 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ ( uminus_uminus @ A @ B2 ) ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ) ) ) ) ).
% minus_divide_le_eq
thf(fact_1497_le__minus__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A2 @ 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 @ A2 @ C2 ) ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% le_minus_divide_eq
thf(fact_1498_nat__approx__posE,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [E2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ E2 )
=> ~ ! [N2: nat] :
~ ( ord_less @ A @ ( divide_divide @ A @ ( one_one @ A ) @ ( semiring_1_of_nat @ A @ ( suc @ N2 ) ) ) @ E2 ) ) ) ).
% nat_approx_posE
thf(fact_1499_nat__abs__int__diff,axiom,
! [A2: nat,B2: nat] :
( ( ( ord_less_eq @ nat @ A2 @ B2 )
=> ( ( nat2 @ ( abs_abs @ int @ ( minus_minus @ int @ ( semiring_1_of_nat @ int @ A2 ) @ ( semiring_1_of_nat @ int @ B2 ) ) ) )
= ( minus_minus @ nat @ B2 @ A2 ) ) )
& ( ~ ( ord_less_eq @ nat @ A2 @ B2 )
=> ( ( nat2 @ ( abs_abs @ int @ ( minus_minus @ int @ ( semiring_1_of_nat @ int @ A2 ) @ ( semiring_1_of_nat @ int @ B2 ) ) ) )
= ( minus_minus @ nat @ A2 @ B2 ) ) ) ) ).
% nat_abs_int_diff
thf(fact_1500_split__zdiv,axiom,
! [P: int > $o,N: int,K: int] :
( ( P @ ( divide_divide @ int @ N @ K ) )
= ( ( ( K
= ( zero_zero @ int ) )
=> ( P @ ( zero_zero @ int ) ) )
& ( ( ord_less @ int @ ( zero_zero @ int ) @ K )
=> ! [I2: int,J3: int] :
( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ J3 )
& ( ord_less @ int @ J3 @ K )
& ( N
= ( plus_plus @ int @ ( times_times @ int @ K @ I2 ) @ J3 ) ) )
=> ( P @ I2 ) ) )
& ( ( ord_less @ int @ K @ ( zero_zero @ int ) )
=> ! [I2: int,J3: int] :
( ( ( ord_less @ int @ K @ J3 )
& ( ord_less_eq @ int @ J3 @ ( zero_zero @ int ) )
& ( N
= ( plus_plus @ int @ ( times_times @ int @ K @ I2 ) @ J3 ) ) )
=> ( P @ I2 ) ) ) ) ) ).
% split_zdiv
thf(fact_1501_int__div__neg__eq,axiom,
! [A2: int,B2: int,Q2: int,R: int] :
( ( A2
= ( plus_plus @ int @ ( times_times @ int @ B2 @ Q2 ) @ R ) )
=> ( ( ord_less_eq @ int @ R @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ B2 @ R )
=> ( ( divide_divide @ int @ A2 @ B2 )
= Q2 ) ) ) ) ).
% int_div_neg_eq
thf(fact_1502_int__div__pos__eq,axiom,
! [A2: int,B2: int,Q2: int,R: int] :
( ( A2
= ( plus_plus @ int @ ( times_times @ int @ B2 @ Q2 ) @ R ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R )
=> ( ( ord_less @ int @ R @ B2 )
=> ( ( divide_divide @ int @ A2 @ B2 )
= Q2 ) ) ) ) ).
% int_div_pos_eq
thf(fact_1503_frac__neg,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A] :
( ( ( member @ A @ X2 @ ( ring_1_Ints @ A ) )
=> ( ( archimedean_frac @ A @ ( uminus_uminus @ A @ X2 ) )
= ( zero_zero @ A ) ) )
& ( ~ ( member @ A @ X2 @ ( ring_1_Ints @ A ) )
=> ( ( archimedean_frac @ A @ ( uminus_uminus @ A @ X2 ) )
= ( minus_minus @ A @ ( one_one @ A ) @ ( archimedean_frac @ A @ X2 ) ) ) ) ) ) ).
% frac_neg
thf(fact_1504_nat__intermed__int__val,axiom,
! [M: nat,N: nat,F: nat > int,K: int] :
( ! [I3: nat] :
( ( ( ord_less_eq @ nat @ M @ I3 )
& ( ord_less @ nat @ I3 @ N ) )
=> ( ord_less_eq @ int @ ( abs_abs @ int @ ( minus_minus @ int @ ( F @ ( suc @ I3 ) ) @ ( F @ I3 ) ) ) @ ( one_one @ int ) ) )
=> ( ( ord_less_eq @ nat @ M @ N )
=> ( ( ord_less_eq @ int @ ( F @ M ) @ K )
=> ( ( ord_less_eq @ int @ K @ ( F @ N ) )
=> ? [I3: nat] :
( ( ord_less_eq @ nat @ M @ I3 )
& ( ord_less_eq @ nat @ I3 @ N )
& ( ( F @ I3 )
= K ) ) ) ) ) ) ).
% nat_intermed_int_val
thf(fact_1505_floor__divide__lower,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Q2: A,P3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Q2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ ( divide_divide @ A @ P3 @ Q2 ) ) ) @ Q2 ) @ P3 ) ) ) ).
% floor_divide_lower
thf(fact_1506_ceiling__divide__upper,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Q2: A,P3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Q2 )
=> ( ord_less_eq @ A @ P3 @ ( times_times @ A @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ ( divide_divide @ A @ P3 @ Q2 ) ) ) @ Q2 ) ) ) ) ).
% ceiling_divide_upper
thf(fact_1507_decr__lemma,axiom,
! [D2: int,X2: int,Z2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D2 )
=> ( ord_less @ int @ ( minus_minus @ int @ X2 @ ( times_times @ int @ ( plus_plus @ int @ ( abs_abs @ int @ ( minus_minus @ int @ X2 @ Z2 ) ) @ ( one_one @ int ) ) @ D2 ) ) @ Z2 ) ) ).
% decr_lemma
thf(fact_1508_incr__lemma,axiom,
! [D2: int,Z2: int,X2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D2 )
=> ( ord_less @ int @ Z2 @ ( plus_plus @ int @ X2 @ ( times_times @ int @ ( plus_plus @ int @ ( abs_abs @ int @ ( minus_minus @ int @ X2 @ Z2 ) ) @ ( one_one @ int ) ) @ D2 ) ) ) ) ).
% incr_lemma
thf(fact_1509_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_1510_div__add__self1__no__field,axiom,
! [B: $tType,A: $tType] :
( ( ( euclid4440199948858584721cancel @ A )
& ( field @ B ) )
=> ! [X2: B,B2: A,A2: A] :
( ( nO_MATCH @ B @ A @ X2 @ B2 )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ B2 @ A2 ) @ B2 )
= ( plus_plus @ A @ ( divide_divide @ A @ A2 @ B2 ) @ ( one_one @ A ) ) ) ) ) ) ).
% div_add_self1_no_field
thf(fact_1511_div__add__self2__no__field,axiom,
! [B: $tType,A: $tType] :
( ( ( euclid4440199948858584721cancel @ A )
& ( field @ B ) )
=> ! [X2: B,B2: A,A2: A] :
( ( nO_MATCH @ B @ A @ X2 @ B2 )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A2 @ B2 ) @ B2 )
= ( plus_plus @ A @ ( divide_divide @ A @ A2 @ B2 ) @ ( one_one @ A ) ) ) ) ) ) ).
% div_add_self2_no_field
thf(fact_1512_div__mult__self1,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A2 @ ( times_times @ A @ C2 @ B2 ) ) @ B2 )
= ( plus_plus @ A @ C2 @ ( divide_divide @ A @ A2 @ B2 ) ) ) ) ) ).
% div_mult_self1
thf(fact_1513_div__mult__self2,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) ) @ B2 )
= ( plus_plus @ A @ C2 @ ( divide_divide @ A @ A2 @ B2 ) ) ) ) ) ).
% div_mult_self2
thf(fact_1514_div__mult__self3,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B2: A,C2: A,A2: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ C2 @ B2 ) @ A2 ) @ B2 )
= ( plus_plus @ A @ C2 @ ( divide_divide @ A @ A2 @ B2 ) ) ) ) ) ).
% div_mult_self3
thf(fact_1515_div__mult__self4,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B2: A,C2: A,A2: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ B2 @ C2 ) @ A2 ) @ B2 )
= ( plus_plus @ A @ C2 @ ( divide_divide @ A @ A2 @ B2 ) ) ) ) ) ).
% div_mult_self4
thf(fact_1516_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_1517_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_1518_div__mult__mult1__if,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ( C2
= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( zero_zero @ A ) ) )
& ( ( C2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( divide_divide @ A @ A2 @ B2 ) ) ) ) ) ).
% div_mult_mult1_if
thf(fact_1519_div__mult__mult2,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) )
= ( divide_divide @ A @ A2 @ B2 ) ) ) ) ).
% div_mult_mult2
thf(fact_1520_div__mult__mult1,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( divide_divide @ A @ A2 @ B2 ) ) ) ) ).
% div_mult_mult1
thf(fact_1521_div__minus1__right,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A2: A] :
( ( divide_divide @ A @ A2 @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( uminus_uminus @ A @ A2 ) ) ) ).
% div_minus1_right
thf(fact_1522_div__by__Suc__0,axiom,
! [M: nat] :
( ( divide_divide @ nat @ M @ ( suc @ ( zero_zero @ nat ) ) )
= M ) ).
% div_by_Suc_0
thf(fact_1523_div__less,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ( ( divide_divide @ nat @ M @ N )
= ( zero_zero @ nat ) ) ) ).
% div_less
thf(fact_1524_div__mult__self__is__m,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( divide_divide @ nat @ ( times_times @ nat @ M @ N ) @ N )
= M ) ) ).
% div_mult_self_is_m
thf(fact_1525_div__mult__self1__is__m,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( divide_divide @ nat @ ( times_times @ nat @ N @ M ) @ N )
= M ) ) ).
% div_mult_self1_is_m
thf(fact_1526_div__le__dividend,axiom,
! [M: nat,N: nat] : ( ord_less_eq @ nat @ ( divide_divide @ nat @ M @ N ) @ M ) ).
% div_le_dividend
thf(fact_1527_div__le__mono,axiom,
! [M: nat,N: nat,K: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ord_less_eq @ nat @ ( divide_divide @ nat @ M @ K ) @ ( divide_divide @ nat @ N @ K ) ) ) ).
% div_le_mono
thf(fact_1528_Euclidean__Division_Odiv__eq__0__iff,axiom,
! [M: nat,N: nat] :
( ( ( divide_divide @ nat @ M @ N )
= ( zero_zero @ nat ) )
= ( ( ord_less @ nat @ M @ N )
| ( N
= ( zero_zero @ nat ) ) ) ) ).
% Euclidean_Division.div_eq_0_iff
thf(fact_1529_Suc__div__le__mono,axiom,
! [M: nat,N: nat] : ( ord_less_eq @ nat @ ( divide_divide @ nat @ M @ N ) @ ( divide_divide @ nat @ ( suc @ M ) @ N ) ) ).
% Suc_div_le_mono
thf(fact_1530_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [M: nat,N: nat] :
( ( semiring_1_of_nat @ A @ ( divide_divide @ nat @ M @ N ) )
= ( divide_divide @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% unique_euclidean_semiring_with_nat_class.of_nat_div
thf(fact_1531_div__times__less__eq__dividend,axiom,
! [M: nat,N: nat] : ( ord_less_eq @ nat @ ( times_times @ nat @ ( divide_divide @ nat @ M @ N ) @ N ) @ M ) ).
% div_times_less_eq_dividend
thf(fact_1532_times__div__less__eq__dividend,axiom,
! [N: nat,M: nat] : ( ord_less_eq @ nat @ ( times_times @ nat @ N @ ( divide_divide @ nat @ M @ N ) ) @ M ) ).
% times_div_less_eq_dividend
thf(fact_1533_div__mult2__eq_H,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A2: A,M: nat,N: nat] :
( ( divide_divide @ A @ A2 @ ( times_times @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) ) )
= ( divide_divide @ A @ ( divide_divide @ A @ A2 @ ( semiring_1_of_nat @ A @ M ) ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% div_mult2_eq'
thf(fact_1534_div__le__mono2,axiom,
! [M: nat,N: nat,K: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( ( ord_less_eq @ nat @ M @ N )
=> ( ord_less_eq @ nat @ ( divide_divide @ nat @ K @ N ) @ ( divide_divide @ nat @ K @ M ) ) ) ) ).
% div_le_mono2
thf(fact_1535_div__greater__zero__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( divide_divide @ nat @ M @ N ) )
= ( ( ord_less_eq @ nat @ N @ M )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% div_greater_zero_iff
thf(fact_1536_div__less__iff__less__mult,axiom,
! [Q2: nat,M: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ Q2 )
=> ( ( ord_less @ nat @ ( divide_divide @ nat @ M @ Q2 ) @ N )
= ( ord_less @ nat @ M @ ( times_times @ nat @ N @ Q2 ) ) ) ) ).
% div_less_iff_less_mult
thf(fact_1537_div__eq__dividend__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( ( ( divide_divide @ nat @ M @ N )
= M )
= ( N
= ( one_one @ nat ) ) ) ) ).
% div_eq_dividend_iff
thf(fact_1538_div__less__dividend,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ ( one_one @ nat ) @ N )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( ord_less @ nat @ ( divide_divide @ nat @ M @ N ) @ M ) ) ) ).
% div_less_dividend
thf(fact_1539_zdiv__zmult2__eq,axiom,
! [C2: int,A2: int,B2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ C2 )
=> ( ( divide_divide @ int @ A2 @ ( times_times @ int @ B2 @ C2 ) )
= ( divide_divide @ int @ ( divide_divide @ int @ A2 @ B2 ) @ C2 ) ) ) ).
% zdiv_zmult2_eq
thf(fact_1540_nat__div__distrib,axiom,
! [X2: int,Y: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X2 )
=> ( ( nat2 @ ( divide_divide @ int @ X2 @ Y ) )
= ( divide_divide @ nat @ ( nat2 @ X2 ) @ ( nat2 @ Y ) ) ) ) ).
% nat_div_distrib
thf(fact_1541_nat__div__distrib_H,axiom,
! [Y: int,X2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y )
=> ( ( nat2 @ ( divide_divide @ int @ X2 @ Y ) )
= ( divide_divide @ nat @ ( nat2 @ X2 ) @ ( nat2 @ Y ) ) ) ) ).
% nat_div_distrib'
thf(fact_1542_div__add__self2,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B2: A,A2: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A2 @ B2 ) @ B2 )
= ( plus_plus @ A @ ( divide_divide @ A @ A2 @ B2 ) @ ( one_one @ A ) ) ) ) ) ).
% div_add_self2
thf(fact_1543_div__add__self1,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B2: A,A2: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ B2 @ A2 ) @ B2 )
= ( plus_plus @ A @ ( divide_divide @ A @ A2 @ B2 ) @ ( one_one @ A ) ) ) ) ) ).
% div_add_self1
thf(fact_1544_div__if,axiom,
( ( divide_divide @ nat )
= ( ^ [M2: nat,N4: nat] :
( if @ nat
@ ( ( ord_less @ nat @ M2 @ N4 )
| ( N4
= ( zero_zero @ nat ) ) )
@ ( zero_zero @ nat )
@ ( suc @ ( divide_divide @ nat @ ( minus_minus @ nat @ M2 @ N4 ) @ N4 ) ) ) ) ) ).
% div_if
thf(fact_1545_less__eq__div__iff__mult__less__eq,axiom,
! [Q2: nat,M: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ Q2 )
=> ( ( ord_less_eq @ nat @ M @ ( divide_divide @ nat @ N @ Q2 ) )
= ( ord_less_eq @ nat @ ( times_times @ nat @ M @ Q2 ) @ N ) ) ) ).
% less_eq_div_iff_mult_less_eq
thf(fact_1546_div__nat__eqI,axiom,
! [N: nat,Q2: nat,M: nat] :
( ( ord_less_eq @ nat @ ( times_times @ nat @ N @ Q2 ) @ M )
=> ( ( ord_less @ nat @ M @ ( times_times @ nat @ N @ ( suc @ Q2 ) ) )
=> ( ( divide_divide @ nat @ M @ N )
= Q2 ) ) ) ).
% div_nat_eqI
thf(fact_1547_split__div,axiom,
! [P: nat > $o,M: nat,N: nat] :
( ( P @ ( divide_divide @ nat @ M @ N ) )
= ( ( ( N
= ( zero_zero @ nat ) )
=> ( P @ ( zero_zero @ nat ) ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ! [I2: nat,J3: nat] :
( ( ord_less @ nat @ J3 @ N )
=> ( ( M
= ( plus_plus @ nat @ ( times_times @ nat @ N @ I2 ) @ J3 ) )
=> ( P @ I2 ) ) ) ) ) ) ).
% split_div
thf(fact_1548_dividend__less__div__times,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less @ nat @ M @ ( plus_plus @ nat @ N @ ( times_times @ nat @ ( divide_divide @ nat @ M @ N ) @ N ) ) ) ) ).
% dividend_less_div_times
thf(fact_1549_dividend__less__times__div,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less @ nat @ M @ ( plus_plus @ nat @ N @ ( times_times @ nat @ N @ ( divide_divide @ nat @ M @ N ) ) ) ) ) ).
% dividend_less_times_div
thf(fact_1550_le__div__geq,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ nat @ N @ M )
=> ( ( divide_divide @ nat @ M @ N )
= ( suc @ ( divide_divide @ nat @ ( minus_minus @ nat @ M @ N ) @ N ) ) ) ) ) ).
% le_div_geq
thf(fact_1551_split__div_H,axiom,
! [P: nat > $o,M: nat,N: nat] :
( ( P @ ( divide_divide @ nat @ M @ N ) )
= ( ( ( N
= ( zero_zero @ nat ) )
& ( P @ ( zero_zero @ nat ) ) )
| ? [Q6: nat] :
( ( ord_less_eq @ nat @ ( times_times @ nat @ N @ Q6 ) @ M )
& ( ord_less @ nat @ M @ ( times_times @ nat @ N @ ( suc @ Q6 ) ) )
& ( P @ Q6 ) ) ) ) ).
% split_div'
thf(fact_1552_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_1553_bits__div__by__1,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A2: A] :
( ( divide_divide @ A @ A2 @ ( one_one @ A ) )
= A2 ) ) ).
% bits_div_by_1
thf(fact_1554_bits__div__by__0,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A2: A] :
( ( divide_divide @ A @ A2 @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% bits_div_by_0
thf(fact_1555_bits__div__0,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A2: A] :
( ( divide_divide @ A @ ( zero_zero @ A ) @ A2 )
= ( zero_zero @ A ) ) ) ).
% bits_div_0
thf(fact_1556_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_1557_gbinomial__absorption_H,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,A2: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ( gbinomial @ A @ A2 @ K )
= ( times_times @ A @ ( divide_divide @ A @ A2 @ ( semiring_1_of_nat @ A @ K ) ) @ ( gbinomial @ A @ ( minus_minus @ A @ A2 @ ( one_one @ A ) ) @ ( minus_minus @ nat @ K @ ( one_one @ nat ) ) ) ) ) ) ) ).
% gbinomial_absorption'
thf(fact_1558_pochhammer__absorb__comp,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [R: A,K: nat] :
( ( times_times @ A @ ( minus_minus @ A @ R @ ( semiring_1_of_nat @ A @ K ) ) @ ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ R ) @ K ) )
= ( times_times @ A @ R @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ ( uminus_uminus @ A @ R ) @ ( one_one @ A ) ) @ K ) ) ) ) ).
% pochhammer_absorb_comp
thf(fact_1559_power_Opower__eq__if,axiom,
! [A: $tType] :
( ( power2 @ A )
= ( ^ [One: A,Times: A > A > A,P7: A,M2: nat] :
( if @ A
@ ( M2
= ( zero_zero @ nat ) )
@ One
@ ( Times @ P7 @ ( power2 @ A @ One @ Times @ P7 @ ( minus_minus @ nat @ M2 @ ( one_one @ nat ) ) ) ) ) ) ) ).
% power.power_eq_if
thf(fact_1560_fact__reduce,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( semiring_char_0_fact @ A @ N )
= ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ) ) ).
% fact_reduce
thf(fact_1561_power__one,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [N: nat] :
( ( power_power @ A @ ( one_one @ A ) @ N )
= ( one_one @ A ) ) ) ).
% power_one
thf(fact_1562_of__nat__power__eq__of__nat__cancel__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [X2: nat,B2: nat,W2: nat] :
( ( ( semiring_1_of_nat @ A @ X2 )
= ( power_power @ A @ ( semiring_1_of_nat @ A @ B2 ) @ W2 ) )
= ( X2
= ( power_power @ nat @ B2 @ W2 ) ) ) ) ).
% of_nat_power_eq_of_nat_cancel_iff
thf(fact_1563_of__nat__eq__of__nat__power__cancel__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [B2: nat,W2: nat,X2: nat] :
( ( ( power_power @ A @ ( semiring_1_of_nat @ A @ B2 ) @ W2 )
= ( semiring_1_of_nat @ A @ X2 ) )
= ( ( power_power @ nat @ B2 @ W2 )
= X2 ) ) ) ).
% of_nat_eq_of_nat_power_cancel_iff
thf(fact_1564_of__nat__power,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [M: nat,N: nat] :
( ( semiring_1_of_nat @ A @ ( power_power @ nat @ M @ N ) )
= ( power_power @ A @ ( semiring_1_of_nat @ A @ M ) @ N ) ) ) ).
% of_nat_power
thf(fact_1565_power__one__right,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A2: A] :
( ( power_power @ A @ A2 @ ( one_one @ nat ) )
= A2 ) ) ).
% power_one_right
thf(fact_1566_of__nat__fact,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [N: nat] :
( ( semiring_1_of_nat @ A @ ( semiring_char_0_fact @ nat @ N ) )
= ( semiring_char_0_fact @ A @ N ) ) ) ).
% of_nat_fact
thf(fact_1567_gbinomial__1,axiom,
! [A: $tType] :
( ( ( semiring_char_0 @ A )
& ( semidom_divide @ A ) )
=> ! [A2: A] :
( ( gbinomial @ A @ A2 @ ( one_one @ nat ) )
= A2 ) ) ).
% gbinomial_1
thf(fact_1568_pochhammer__1,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A2: A] :
( ( comm_s3205402744901411588hammer @ A @ A2 @ ( one_one @ nat ) )
= A2 ) ) ).
% pochhammer_1
thf(fact_1569_power__inject__exp,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,M: nat,N: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A2 )
=> ( ( ( power_power @ A @ A2 @ M )
= ( power_power @ A @ A2 @ N ) )
= ( M = N ) ) ) ) ).
% power_inject_exp
thf(fact_1570_power__0__Suc,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [N: nat] :
( ( power_power @ A @ ( zero_zero @ A ) @ ( suc @ N ) )
= ( zero_zero @ A ) ) ) ).
% power_0_Suc
thf(fact_1571_power__Suc0__right,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A2: A] :
( ( power_power @ A @ A2 @ ( suc @ ( zero_zero @ nat ) ) )
= A2 ) ) ).
% power_Suc0_right
thf(fact_1572_gbinomial__0_I2_J,axiom,
! [B: $tType] :
( ( ( semiring_char_0 @ B )
& ( semidom_divide @ B ) )
=> ! [K: nat] :
( ( gbinomial @ B @ ( zero_zero @ B ) @ ( suc @ K ) )
= ( zero_zero @ B ) ) ) ).
% gbinomial_0(2)
thf(fact_1573_fact__0,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiring_char_0_fact @ A @ ( zero_zero @ nat ) )
= ( one_one @ A ) ) ) ).
% fact_0
thf(fact_1574_gbinomial__0_I1_J,axiom,
! [A: $tType] :
( ( ( semiring_char_0 @ A )
& ( semidom_divide @ A ) )
=> ! [A2: A] :
( ( gbinomial @ A @ A2 @ ( zero_zero @ nat ) )
= ( one_one @ A ) ) ) ).
% gbinomial_0(1)
thf(fact_1575_gbinomial__Suc0,axiom,
! [A: $tType] :
( ( ( semiring_char_0 @ A )
& ( semidom_divide @ A ) )
=> ! [A2: A] :
( ( gbinomial @ A @ A2 @ ( suc @ ( zero_zero @ nat ) ) )
= A2 ) ) ).
% gbinomial_Suc0
thf(fact_1576_pochhammer__0,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A2: A] :
( ( comm_s3205402744901411588hammer @ A @ A2 @ ( zero_zero @ nat ) )
= ( one_one @ A ) ) ) ).
% pochhammer_0
thf(fact_1577_fact__1,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiring_char_0_fact @ A @ ( one_one @ nat ) )
= ( one_one @ A ) ) ) ).
% fact_1
thf(fact_1578_pochhammer__Suc0,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A2: A] :
( ( comm_s3205402744901411588hammer @ A @ A2 @ ( suc @ ( zero_zero @ nat ) ) )
= A2 ) ) ).
% pochhammer_Suc0
thf(fact_1579_power__int__of__nat,axiom,
! [A: $tType] :
( ( ( inverse @ A )
& ( power @ A ) )
=> ! [X2: A,N: nat] :
( ( power_int @ A @ X2 @ ( semiring_1_of_nat @ int @ N ) )
= ( power_power @ A @ X2 @ N ) ) ) ).
% power_int_of_nat
thf(fact_1580_of__int__power,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [Z2: int,N: nat] :
( ( ring_1_of_int @ A @ ( power_power @ int @ Z2 @ N ) )
= ( power_power @ A @ ( ring_1_of_int @ A @ Z2 ) @ N ) ) ) ).
% of_int_power
thf(fact_1581_of__int__eq__of__int__power__cancel__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [B2: int,W2: nat,X2: int] :
( ( ( power_power @ A @ ( ring_1_of_int @ A @ B2 ) @ W2 )
= ( ring_1_of_int @ A @ X2 ) )
= ( ( power_power @ int @ B2 @ W2 )
= X2 ) ) ) ).
% of_int_eq_of_int_power_cancel_iff
thf(fact_1582_of__int__power__eq__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [X2: int,B2: int,W2: nat] :
( ( ( ring_1_of_int @ A @ X2 )
= ( power_power @ A @ ( ring_1_of_int @ A @ B2 ) @ W2 ) )
= ( X2
= ( power_power @ int @ B2 @ W2 ) ) ) ) ).
% of_int_power_eq_of_int_cancel_iff
thf(fact_1583_power__strict__increasing__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B2: A,X2: nat,Y: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ B2 )
=> ( ( ord_less @ A @ ( power_power @ A @ B2 @ X2 ) @ ( power_power @ A @ B2 @ Y ) )
= ( ord_less @ nat @ X2 @ Y ) ) ) ) ).
% power_strict_increasing_iff
thf(fact_1584_minus__one__mult__self,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [N: nat] :
( ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N ) @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N ) )
= ( one_one @ A ) ) ) ).
% minus_one_mult_self
thf(fact_1585_left__minus__one__mult__self,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [N: nat,A2: A] :
( ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N ) @ ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N ) @ A2 ) )
= A2 ) ) ).
% left_minus_one_mult_self
thf(fact_1586_power__eq__0__iff,axiom,
! [A: $tType] :
( ( semiri2026040879449505780visors @ A )
=> ! [A2: A,N: nat] :
( ( ( power_power @ A @ A2 @ N )
= ( zero_zero @ A ) )
= ( ( A2
= ( zero_zero @ A ) )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% power_eq_0_iff
thf(fact_1587_of__nat__less__of__nat__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B2: nat,W2: nat,X2: nat] :
( ( ord_less @ A @ ( power_power @ A @ ( semiring_1_of_nat @ A @ B2 ) @ W2 ) @ ( semiring_1_of_nat @ A @ X2 ) )
= ( ord_less @ nat @ ( power_power @ nat @ B2 @ W2 ) @ X2 ) ) ) ).
% of_nat_less_of_nat_power_cancel_iff
thf(fact_1588_of__nat__power__less__of__nat__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X2: nat,B2: nat,W2: nat] :
( ( ord_less @ A @ ( semiring_1_of_nat @ A @ X2 ) @ ( power_power @ A @ ( semiring_1_of_nat @ A @ B2 ) @ W2 ) )
= ( ord_less @ nat @ X2 @ ( power_power @ nat @ B2 @ W2 ) ) ) ) ).
% of_nat_power_less_of_nat_cancel_iff
thf(fact_1589_of__nat__power__le__of__nat__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X2: nat,B2: nat,W2: nat] :
( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ X2 ) @ ( power_power @ A @ ( semiring_1_of_nat @ A @ B2 ) @ W2 ) )
= ( ord_less_eq @ nat @ X2 @ ( power_power @ nat @ B2 @ W2 ) ) ) ) ).
% of_nat_power_le_of_nat_cancel_iff
thf(fact_1590_of__nat__le__of__nat__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B2: nat,W2: nat,X2: nat] :
( ( ord_less_eq @ A @ ( power_power @ A @ ( semiring_1_of_nat @ A @ B2 ) @ W2 ) @ ( semiring_1_of_nat @ A @ X2 ) )
= ( ord_less_eq @ nat @ ( power_power @ nat @ B2 @ W2 ) @ X2 ) ) ) ).
% of_nat_le_of_nat_power_cancel_iff
thf(fact_1591_fact__Suc__0,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiring_char_0_fact @ A @ ( suc @ ( zero_zero @ nat ) ) )
= ( one_one @ A ) ) ) ).
% fact_Suc_0
thf(fact_1592_fact__Suc,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [N: nat] :
( ( semiring_char_0_fact @ A @ ( suc @ N ) )
= ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ N ) ) @ ( semiring_char_0_fact @ A @ N ) ) ) ) ).
% fact_Suc
thf(fact_1593_power__strict__decreasing__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B2: A,M: nat,N: 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 @ N ) )
= ( ord_less @ nat @ N @ M ) ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_1594_power__mono__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,B2: A,N: nat] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ A @ ( power_power @ A @ A2 @ N ) @ ( power_power @ A @ B2 @ N ) )
= ( ord_less_eq @ A @ A2 @ B2 ) ) ) ) ) ) ).
% power_mono_iff
thf(fact_1595_power__increasing__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B2: A,X2: nat,Y: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ B2 )
=> ( ( ord_less_eq @ A @ ( power_power @ A @ B2 @ X2 ) @ ( power_power @ A @ B2 @ Y ) )
= ( ord_less_eq @ nat @ X2 @ Y ) ) ) ) ).
% power_increasing_iff
thf(fact_1596_zero__less__power__abs__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ ( abs_abs @ A @ A2 ) @ N ) )
= ( ( A2
!= ( zero_zero @ A ) )
| ( N
= ( zero_zero @ nat ) ) ) ) ) ).
% zero_less_power_abs_iff
thf(fact_1597_of__int__le__of__int__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [B2: int,W2: nat,X2: int] :
( ( ord_less_eq @ A @ ( power_power @ A @ ( ring_1_of_int @ A @ B2 ) @ W2 ) @ ( ring_1_of_int @ A @ X2 ) )
= ( ord_less_eq @ int @ ( power_power @ int @ B2 @ W2 ) @ X2 ) ) ) ).
% of_int_le_of_int_power_cancel_iff
thf(fact_1598_of__int__power__le__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X2: int,B2: int,W2: nat] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ X2 ) @ ( power_power @ A @ ( ring_1_of_int @ A @ B2 ) @ W2 ) )
= ( ord_less_eq @ int @ X2 @ ( power_power @ int @ B2 @ W2 ) ) ) ) ).
% of_int_power_le_of_int_cancel_iff
thf(fact_1599_of__int__less__of__int__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [B2: int,W2: nat,X2: int] :
( ( ord_less @ A @ ( power_power @ A @ ( ring_1_of_int @ A @ B2 ) @ W2 ) @ ( ring_1_of_int @ A @ X2 ) )
= ( ord_less @ int @ ( power_power @ int @ B2 @ W2 ) @ X2 ) ) ) ).
% of_int_less_of_int_power_cancel_iff
thf(fact_1600_of__int__power__less__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X2: int,B2: int,W2: nat] :
( ( ord_less @ A @ ( ring_1_of_int @ A @ X2 ) @ ( power_power @ A @ ( ring_1_of_int @ A @ B2 ) @ W2 ) )
= ( ord_less @ int @ X2 @ ( power_power @ int @ B2 @ W2 ) ) ) ) ).
% of_int_power_less_of_int_cancel_iff
thf(fact_1601_power__decreasing__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B2: A,M: nat,N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less @ A @ B2 @ ( one_one @ A ) )
=> ( ( ord_less_eq @ A @ ( power_power @ A @ B2 @ M ) @ ( power_power @ A @ B2 @ N ) )
= ( ord_less_eq @ nat @ N @ M ) ) ) ) ) ).
% power_decreasing_iff
thf(fact_1602_of__nat__zero__less__power__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X2: nat,N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ ( semiring_1_of_nat @ A @ X2 ) @ N ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ X2 )
| ( N
= ( zero_zero @ nat ) ) ) ) ) ).
% of_nat_zero_less_power_iff
thf(fact_1603_pochhammer__of__nat,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [X2: nat,N: nat] :
( ( comm_s3205402744901411588hammer @ A @ ( semiring_1_of_nat @ A @ X2 ) @ N )
= ( semiring_1_of_nat @ A @ ( comm_s3205402744901411588hammer @ nat @ X2 @ N ) ) ) ) ).
% pochhammer_of_nat
thf(fact_1604_pochhammer__fact,axiom,
! [A: $tType] :
( ( ( semiring_char_0 @ A )
& ( comm_semiring_1 @ A ) )
=> ( ( semiring_char_0_fact @ A )
= ( comm_s3205402744901411588hammer @ A @ ( one_one @ A ) ) ) ) ).
% pochhammer_fact
thf(fact_1605_of__nat__gbinomial,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [N: nat,K: nat] :
( ( semiring_1_of_nat @ A @ ( gbinomial @ nat @ N @ K ) )
= ( gbinomial @ A @ ( semiring_1_of_nat @ A @ N ) @ K ) ) ) ).
% of_nat_gbinomial
thf(fact_1606_gbinomial__pochhammer,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( ( gbinomial @ A )
= ( ^ [A3: A,K2: nat] : ( divide_divide @ A @ ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K2 ) @ ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ A3 ) @ K2 ) ) @ ( semiring_char_0_fact @ A @ K2 ) ) ) ) ) ).
% gbinomial_pochhammer
thf(fact_1607_gbinomial__ge__n__over__k__pow__k,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [K: nat,A2: A] :
( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ K ) @ A2 )
=> ( ord_less_eq @ A @ ( power_power @ A @ ( divide_divide @ A @ A2 @ ( semiring_1_of_nat @ A @ K ) ) @ K ) @ ( gbinomial @ A @ A2 @ K ) ) ) ) ).
% gbinomial_ge_n_over_k_pow_k
thf(fact_1608_fact__mono__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ord_less_eq @ nat @ ( semiring_char_0_fact @ nat @ M ) @ ( semiring_char_0_fact @ nat @ N ) ) ) ).
% fact_mono_nat
thf(fact_1609_fact__ge__self,axiom,
! [N: nat] : ( ord_less_eq @ nat @ N @ ( semiring_char_0_fact @ nat @ N ) ) ).
% fact_ge_self
thf(fact_1610_power__not__zero,axiom,
! [A: $tType] :
( ( semiri2026040879449505780visors @ A )
=> ! [A2: A,N: nat] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( power_power @ A @ A2 @ N )
!= ( zero_zero @ A ) ) ) ) ).
% power_not_zero
thf(fact_1611_fact__nonzero,axiom,
! [A: $tType] :
( ( ( semiring_char_0 @ A )
& ( semiri3467727345109120633visors @ A ) )
=> ! [N: nat] :
( ( semiring_char_0_fact @ A @ N )
!= ( zero_zero @ A ) ) ) ).
% fact_nonzero
thf(fact_1612_pochhammer__same,axiom,
! [A: $tType] :
( ( ( semiring_char_0 @ A )
& ( comm_ring_1 @ A )
& ( semiri3467727345109120633visors @ A ) )
=> ! [N: nat] :
( ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ N ) ) @ N )
= ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N ) @ ( semiring_char_0_fact @ A @ N ) ) ) ) ).
% pochhammer_same
thf(fact_1613_Ints__power,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [A2: A,N: nat] :
( ( member @ A @ A2 @ ( ring_1_Ints @ A ) )
=> ( member @ A @ ( power_power @ A @ A2 @ N ) @ ( ring_1_Ints @ A ) ) ) ) ).
% Ints_power
thf(fact_1614_gbinomial__pochhammer_H,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( ( gbinomial @ A )
= ( ^ [A3: A,K2: nat] : ( divide_divide @ A @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ ( minus_minus @ A @ A3 @ ( semiring_1_of_nat @ A @ K2 ) ) @ ( one_one @ A ) ) @ K2 ) @ ( semiring_char_0_fact @ A @ K2 ) ) ) ) ) ).
% gbinomial_pochhammer'
thf(fact_1615_gbinomial__index__swap,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,N: nat] :
( ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K ) @ ( gbinomial @ A @ ( minus_minus @ A @ ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ N ) ) @ ( one_one @ A ) ) @ K ) )
= ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N ) @ ( gbinomial @ A @ ( minus_minus @ A @ ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ K ) ) @ ( one_one @ A ) ) @ N ) ) ) ) ).
% gbinomial_index_swap
thf(fact_1616_gbinomial__negated__upper,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( ( gbinomial @ A )
= ( ^ [A3: A,K2: nat] : ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K2 ) @ ( gbinomial @ A @ ( minus_minus @ A @ ( minus_minus @ A @ ( semiring_1_of_nat @ A @ K2 ) @ A3 ) @ ( one_one @ A ) ) @ K2 ) ) ) ) ) ).
% gbinomial_negated_upper
thf(fact_1617_fact__less__mono__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( ( ord_less @ nat @ M @ N )
=> ( ord_less @ nat @ ( semiring_char_0_fact @ nat @ M ) @ ( semiring_char_0_fact @ nat @ N ) ) ) ) ).
% fact_less_mono_nat
thf(fact_1618_zero__le__power,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,N: nat] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ N ) ) ) ) ).
% zero_le_power
thf(fact_1619_power__mono,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,B2: A,N: nat] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ord_less_eq @ A @ ( power_power @ A @ A2 @ N ) @ ( power_power @ A @ B2 @ N ) ) ) ) ) ).
% power_mono
thf(fact_1620_zero__less__power,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ N ) ) ) ) ).
% zero_less_power
thf(fact_1621_gbinomial__Suc__Suc,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A2: A,K: nat] :
( ( gbinomial @ A @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) @ ( suc @ K ) )
= ( plus_plus @ A @ ( gbinomial @ A @ A2 @ K ) @ ( gbinomial @ A @ A2 @ ( suc @ K ) ) ) ) ) ).
% gbinomial_Suc_Suc
thf(fact_1622_one__le__power,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,N: nat] :
( ( ord_less_eq @ A @ ( one_one @ A ) @ A2 )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ ( power_power @ A @ A2 @ N ) ) ) ) ).
% one_le_power
thf(fact_1623_left__right__inverse__power,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [X2: A,Y: A,N: nat] :
( ( ( times_times @ A @ X2 @ Y )
= ( one_one @ A ) )
=> ( ( times_times @ A @ ( power_power @ A @ X2 @ N ) @ ( power_power @ A @ Y @ N ) )
= ( one_one @ A ) ) ) ) ).
% left_right_inverse_power
thf(fact_1624_power__0,axiom,
! [A: $tType] :
( ( power @ A )
=> ! [A2: A] :
( ( power_power @ A @ A2 @ ( zero_zero @ nat ) )
= ( one_one @ A ) ) ) ).
% power_0
thf(fact_1625_power__one__over,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A,N: nat] :
( ( power_power @ A @ ( divide_divide @ A @ ( one_one @ A ) @ A2 ) @ N )
= ( divide_divide @ A @ ( one_one @ A ) @ ( power_power @ A @ A2 @ N ) ) ) ) ).
% power_one_over
thf(fact_1626_power__add,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A2: A,M: nat,N: nat] :
( ( power_power @ A @ A2 @ ( plus_plus @ nat @ M @ N ) )
= ( times_times @ A @ ( power_power @ A @ A2 @ M ) @ ( power_power @ A @ A2 @ N ) ) ) ) ).
% power_add
thf(fact_1627_fact__ge__zero,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( semiring_char_0_fact @ A @ N ) ) ) ).
% fact_ge_zero
thf(fact_1628_gbinomial__of__nat__symmetric,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ N )
=> ( ( gbinomial @ A @ ( semiring_1_of_nat @ A @ N ) @ K )
= ( gbinomial @ A @ ( semiring_1_of_nat @ A @ N ) @ ( minus_minus @ nat @ N @ K ) ) ) ) ) ).
% gbinomial_of_nat_symmetric
thf(fact_1629_fact__gt__zero,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N: nat] : ( ord_less @ A @ ( zero_zero @ A ) @ ( semiring_char_0_fact @ A @ N ) ) ) ).
% fact_gt_zero
thf(fact_1630_fact__not__neg,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N: nat] :
~ ( ord_less @ A @ ( semiring_char_0_fact @ A @ N ) @ ( zero_zero @ A ) ) ) ).
% fact_not_neg
thf(fact_1631_fact__ge__1,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N: nat] : ( ord_less_eq @ A @ ( one_one @ A ) @ ( semiring_char_0_fact @ A @ N ) ) ) ).
% fact_ge_1
thf(fact_1632_fact__mono,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ord_less_eq @ A @ ( semiring_char_0_fact @ A @ M ) @ ( semiring_char_0_fact @ A @ N ) ) ) ) ).
% fact_mono
thf(fact_1633_pochhammer__pos,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X2: A,N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X2 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( comm_s3205402744901411588hammer @ A @ X2 @ N ) ) ) ) ).
% pochhammer_pos
thf(fact_1634_pochhammer__neq__0__mono,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A2: A,M: nat,N: nat] :
( ( ( comm_s3205402744901411588hammer @ A @ A2 @ M )
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ nat @ N @ M )
=> ( ( comm_s3205402744901411588hammer @ A @ A2 @ N )
!= ( zero_zero @ A ) ) ) ) ) ).
% pochhammer_neq_0_mono
thf(fact_1635_pochhammer__eq__0__mono,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A2: A,N: nat,M: nat] :
( ( ( comm_s3205402744901411588hammer @ A @ A2 @ N )
= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ nat @ N @ M )
=> ( ( comm_s3205402744901411588hammer @ A @ A2 @ M )
= ( zero_zero @ A ) ) ) ) ) ).
% pochhammer_eq_0_mono
thf(fact_1636_gbinomial__minus,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A2: A,K: nat] :
( ( gbinomial @ A @ ( uminus_uminus @ A @ A2 ) @ K )
= ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K ) @ ( gbinomial @ A @ ( minus_minus @ A @ ( plus_plus @ A @ A2 @ ( semiring_1_of_nat @ A @ K ) ) @ ( one_one @ A ) ) @ K ) ) ) ) ).
% gbinomial_minus
thf(fact_1637_power_Opower_Opower__0,axiom,
! [A: $tType,One2: A,Times2: A > A > A,A2: A] :
( ( power2 @ A @ One2 @ Times2 @ A2 @ ( zero_zero @ nat ) )
= One2 ) ).
% power.power.power_0
thf(fact_1638_gbinomial__addition__formula,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A2: A,K: nat] :
( ( gbinomial @ A @ A2 @ ( suc @ K ) )
= ( plus_plus @ A @ ( gbinomial @ A @ ( minus_minus @ A @ A2 @ ( one_one @ A ) ) @ ( suc @ K ) ) @ ( gbinomial @ A @ ( minus_minus @ A @ A2 @ ( one_one @ A ) ) @ K ) ) ) ) ).
% gbinomial_addition_formula
thf(fact_1639_gbinomial__absorb__comp,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A2: A,K: nat] :
( ( times_times @ A @ ( minus_minus @ A @ A2 @ ( semiring_1_of_nat @ A @ K ) ) @ ( gbinomial @ A @ A2 @ K ) )
= ( times_times @ A @ A2 @ ( gbinomial @ A @ ( minus_minus @ A @ A2 @ ( one_one @ A ) ) @ K ) ) ) ) ).
% gbinomial_absorb_comp
thf(fact_1640_gbinomial__mult__1_H,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A2: A,K: nat] :
( ( times_times @ A @ ( gbinomial @ A @ A2 @ K ) @ A2 )
= ( plus_plus @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ K ) @ ( gbinomial @ A @ A2 @ K ) ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ K ) ) @ ( gbinomial @ A @ A2 @ ( suc @ K ) ) ) ) ) ) ).
% gbinomial_mult_1'
thf(fact_1641_gbinomial__mult__1,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A2: A,K: nat] :
( ( times_times @ A @ A2 @ ( gbinomial @ A @ A2 @ K ) )
= ( plus_plus @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ K ) @ ( gbinomial @ A @ A2 @ K ) ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ K ) ) @ ( gbinomial @ A @ A2 @ ( suc @ K ) ) ) ) ) ) ).
% gbinomial_mult_1
thf(fact_1642_fact__ge__Suc__0__nat,axiom,
! [N: nat] : ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( semiring_char_0_fact @ nat @ N ) ) ).
% fact_ge_Suc_0_nat
thf(fact_1643_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_1644_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_1645_power__less__imp__less__base,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,N: nat,B2: A] :
( ( ord_less @ A @ ( power_power @ A @ A2 @ N ) @ ( power_power @ A @ B2 @ N ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less @ A @ A2 @ B2 ) ) ) ) ).
% power_less_imp_less_base
thf(fact_1646_power__le__one,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,N: nat] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ A2 @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( power_power @ A @ A2 @ N ) @ ( one_one @ A ) ) ) ) ) ).
% power_le_one
thf(fact_1647_power__le__imp__le__base,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,N: nat,B2: A] :
( ( ord_less_eq @ A @ ( power_power @ A @ A2 @ ( suc @ N ) ) @ ( power_power @ A @ B2 @ ( suc @ N ) ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less_eq @ A @ A2 @ B2 ) ) ) ) ).
% power_le_imp_le_base
thf(fact_1648_power__inject__base,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,N: nat,B2: A] :
( ( ( power_power @ A @ A2 @ ( suc @ N ) )
= ( power_power @ A @ B2 @ ( suc @ N ) ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( A2 = B2 ) ) ) ) ) ).
% power_inject_base
thf(fact_1649_power__gt1__lemma,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,N: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A2 )
=> ( ord_less @ A @ ( one_one @ A ) @ ( times_times @ A @ A2 @ ( power_power @ A @ A2 @ N ) ) ) ) ) ).
% power_gt1_lemma
thf(fact_1650_power__less__power__Suc,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,N: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A2 )
=> ( ord_less @ A @ ( power_power @ A @ A2 @ N ) @ ( times_times @ A @ A2 @ ( power_power @ A @ A2 @ N ) ) ) ) ) ).
% power_less_power_Suc
thf(fact_1651_power__0__left,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [N: nat] :
( ( ( N
= ( zero_zero @ nat ) )
=> ( ( power_power @ A @ ( zero_zero @ A ) @ N )
= ( one_one @ A ) ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ( ( power_power @ A @ ( zero_zero @ A ) @ N )
= ( zero_zero @ A ) ) ) ) ) ).
% power_0_left
thf(fact_1652_power__gt1,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,N: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A2 )
=> ( ord_less @ A @ ( one_one @ A ) @ ( power_power @ A @ A2 @ ( suc @ N ) ) ) ) ) ).
% power_gt1
thf(fact_1653_power__less__imp__less__exp,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,M: nat,N: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A2 )
=> ( ( ord_less @ A @ ( power_power @ A @ A2 @ M ) @ ( power_power @ A @ A2 @ N ) )
=> ( ord_less @ nat @ M @ N ) ) ) ) ).
% power_less_imp_less_exp
thf(fact_1654_power__strict__increasing,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N: nat,N5: nat,A2: A] :
( ( ord_less @ nat @ N @ N5 )
=> ( ( ord_less @ A @ ( one_one @ A ) @ A2 )
=> ( ord_less @ A @ ( power_power @ A @ A2 @ N ) @ ( power_power @ A @ A2 @ N5 ) ) ) ) ) ).
% power_strict_increasing
thf(fact_1655_power__minus,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [A2: A,N: nat] :
( ( power_power @ A @ ( uminus_uminus @ A @ A2 ) @ N )
= ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N ) @ ( power_power @ A @ A2 @ N ) ) ) ) ).
% power_minus
thf(fact_1656_power__increasing,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N: nat,N5: nat,A2: A] :
( ( ord_less_eq @ nat @ N @ N5 )
=> ( ( ord_less_eq @ A @ ( one_one @ A ) @ A2 )
=> ( ord_less_eq @ A @ ( power_power @ A @ A2 @ N ) @ ( power_power @ A @ A2 @ N5 ) ) ) ) ) ).
% power_increasing
thf(fact_1657_zero__power,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( power_power @ A @ ( zero_zero @ A ) @ N )
= ( zero_zero @ A ) ) ) ) ).
% zero_power
thf(fact_1658_zero__le__power__abs,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,N: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ ( abs_abs @ A @ A2 ) @ N ) ) ) ).
% zero_le_power_abs
thf(fact_1659_fact__less__mono,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [M: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( ( ord_less @ nat @ M @ N )
=> ( ord_less @ A @ ( semiring_char_0_fact @ A @ M ) @ ( semiring_char_0_fact @ A @ N ) ) ) ) ) ).
% fact_less_mono
thf(fact_1660_pochhammer__nonneg,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X2: A,N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X2 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( comm_s3205402744901411588hammer @ A @ X2 @ N ) ) ) ) ).
% pochhammer_nonneg
thf(fact_1661_pochhammer__0__left,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [N: nat] :
( ( ( N
= ( zero_zero @ nat ) )
=> ( ( comm_s3205402744901411588hammer @ A @ ( zero_zero @ A ) @ N )
= ( one_one @ A ) ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ( ( comm_s3205402744901411588hammer @ A @ ( zero_zero @ A ) @ N )
= ( zero_zero @ A ) ) ) ) ) ).
% pochhammer_0_left
thf(fact_1662_Suc__times__gbinomial,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,A2: A] :
( ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ K ) ) @ ( gbinomial @ A @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) @ ( suc @ K ) ) )
= ( times_times @ A @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) @ ( gbinomial @ A @ A2 @ K ) ) ) ) ).
% Suc_times_gbinomial
thf(fact_1663_gbinomial__absorption,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,A2: A] :
( ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ K ) ) @ ( gbinomial @ A @ A2 @ ( suc @ K ) ) )
= ( times_times @ A @ A2 @ ( gbinomial @ A @ ( minus_minus @ A @ A2 @ ( one_one @ A ) ) @ K ) ) ) ) ).
% gbinomial_absorption
thf(fact_1664_gbinomial__trinomial__revision,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,M: nat,A2: A] :
( ( ord_less_eq @ nat @ K @ M )
=> ( ( times_times @ A @ ( gbinomial @ A @ A2 @ M ) @ ( gbinomial @ A @ ( semiring_1_of_nat @ A @ M ) @ K ) )
= ( times_times @ A @ ( gbinomial @ A @ A2 @ K ) @ ( gbinomial @ A @ ( minus_minus @ A @ A2 @ ( semiring_1_of_nat @ A @ K ) ) @ ( minus_minus @ nat @ M @ K ) ) ) ) ) ) ).
% gbinomial_trinomial_revision
thf(fact_1665_power__Suc__less,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ A2 @ ( one_one @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ A2 @ ( power_power @ A @ A2 @ N ) ) @ ( power_power @ A @ A2 @ N ) ) ) ) ) ).
% power_Suc_less
thf(fact_1666_power__Suc__le__self,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,N: nat] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ A2 @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( power_power @ A @ A2 @ ( suc @ N ) ) @ A2 ) ) ) ) ).
% power_Suc_le_self
thf(fact_1667_power__Suc__less__one,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ A2 @ ( one_one @ A ) )
=> ( ord_less @ A @ ( power_power @ A @ A2 @ ( suc @ N ) ) @ ( one_one @ A ) ) ) ) ) ).
% power_Suc_less_one
thf(fact_1668_power__strict__decreasing,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N: nat,N5: nat,A2: A] :
( ( ord_less @ nat @ N @ N5 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ A2 @ ( one_one @ A ) )
=> ( ord_less @ A @ ( power_power @ A @ A2 @ N5 ) @ ( power_power @ A @ A2 @ N ) ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_1669_power__decreasing,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N: nat,N5: nat,A2: A] :
( ( ord_less_eq @ nat @ N @ N5 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ A2 @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( power_power @ A @ A2 @ N5 ) @ ( power_power @ A @ A2 @ N ) ) ) ) ) ) ).
% power_decreasing
thf(fact_1670_power__eq__imp__eq__base,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,N: nat,B2: A] :
( ( ( power_power @ A @ A2 @ N )
= ( power_power @ A @ B2 @ N ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( A2 = B2 ) ) ) ) ) ) ).
% power_eq_imp_eq_base
thf(fact_1671_power__eq__iff__eq__base,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N: nat,A2: A,B2: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ( power_power @ A @ A2 @ N )
= ( power_power @ A @ B2 @ N ) )
= ( A2 = B2 ) ) ) ) ) ) ).
% power_eq_iff_eq_base
thf(fact_1672_power__le__imp__le__exp,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,M: nat,N: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ ( power_power @ A @ A2 @ M ) @ ( power_power @ A @ A2 @ N ) )
=> ( ord_less_eq @ nat @ M @ N ) ) ) ) ).
% power_le_imp_le_exp
thf(fact_1673_self__le__power,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,N: nat] :
( ( ord_less_eq @ A @ ( one_one @ A ) @ A2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less_eq @ A @ A2 @ ( power_power @ A @ A2 @ N ) ) ) ) ) ).
% self_le_power
thf(fact_1674_one__less__power,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,N: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less @ A @ ( one_one @ A ) @ ( power_power @ A @ A2 @ N ) ) ) ) ) ).
% one_less_power
thf(fact_1675_power__diff,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A2: A,N: nat,M: nat] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ nat @ N @ M )
=> ( ( power_power @ A @ A2 @ ( minus_minus @ nat @ M @ N ) )
= ( divide_divide @ A @ ( power_power @ A @ A2 @ M ) @ ( power_power @ A @ A2 @ N ) ) ) ) ) ) ).
% power_diff
thf(fact_1676_pochhammer__rec,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A2: A,N: nat] :
( ( comm_s3205402744901411588hammer @ A @ A2 @ ( suc @ N ) )
= ( times_times @ A @ A2 @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) @ N ) ) ) ) ).
% pochhammer_rec
thf(fact_1677_pochhammer__Suc,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A2: A,N: nat] :
( ( comm_s3205402744901411588hammer @ A @ A2 @ ( suc @ N ) )
= ( times_times @ A @ ( comm_s3205402744901411588hammer @ A @ A2 @ N ) @ ( plus_plus @ A @ A2 @ ( semiring_1_of_nat @ A @ N ) ) ) ) ) ).
% pochhammer_Suc
thf(fact_1678_pochhammer__rec_H,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [Z2: A,N: nat] :
( ( comm_s3205402744901411588hammer @ A @ Z2 @ ( suc @ N ) )
= ( times_times @ A @ ( plus_plus @ A @ Z2 @ ( semiring_1_of_nat @ A @ N ) ) @ ( comm_s3205402744901411588hammer @ A @ Z2 @ N ) ) ) ) ).
% pochhammer_rec'
thf(fact_1679_pochhammer__eq__0__iff,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A2: A,N: nat] :
( ( ( comm_s3205402744901411588hammer @ A @ A2 @ N )
= ( zero_zero @ A ) )
= ( ? [K2: nat] :
( ( ord_less @ nat @ K2 @ N )
& ( A2
= ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ K2 ) ) ) ) ) ) ) ).
% pochhammer_eq_0_iff
thf(fact_1680_pochhammer__of__nat__eq__0__iff,axiom,
! [A: $tType] :
( ( ( ring_char_0 @ A )
& ( idom @ A ) )
=> ! [N: nat,K: nat] :
( ( ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ N ) ) @ K )
= ( zero_zero @ A ) )
= ( ord_less @ nat @ N @ K ) ) ) ).
% pochhammer_of_nat_eq_0_iff
thf(fact_1681_pochhammer__of__nat__eq__0__lemma,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [N: nat,K: nat] :
( ( ord_less @ nat @ N @ K )
=> ( ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ N ) ) @ K )
= ( zero_zero @ A ) ) ) ) ).
% pochhammer_of_nat_eq_0_lemma
thf(fact_1682_gbinomial__rec,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A2: A,K: nat] :
( ( gbinomial @ A @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) @ ( suc @ K ) )
= ( times_times @ A @ ( gbinomial @ A @ A2 @ K ) @ ( divide_divide @ A @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) @ ( semiring_1_of_nat @ A @ ( suc @ K ) ) ) ) ) ) ).
% gbinomial_rec
thf(fact_1683_gbinomial__factors,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A2: A,K: nat] :
( ( gbinomial @ A @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) @ ( suc @ K ) )
= ( times_times @ A @ ( divide_divide @ A @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) @ ( semiring_1_of_nat @ A @ ( suc @ K ) ) ) @ ( gbinomial @ A @ A2 @ K ) ) ) ) ).
% gbinomial_factors
thf(fact_1684_pochhammer__of__nat__eq__0__lemma_H,axiom,
! [A: $tType] :
( ( ( ring_char_0 @ A )
& ( idom @ A ) )
=> ! [K: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ N )
=> ( ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ N ) ) @ K )
!= ( zero_zero @ A ) ) ) ) ).
% pochhammer_of_nat_eq_0_lemma'
thf(fact_1685_pochhammer__product_H,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [Z2: A,N: nat,M: nat] :
( ( comm_s3205402744901411588hammer @ A @ Z2 @ ( plus_plus @ nat @ N @ M ) )
= ( times_times @ A @ ( comm_s3205402744901411588hammer @ A @ Z2 @ N ) @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ Z2 @ ( semiring_1_of_nat @ A @ N ) ) @ M ) ) ) ) ).
% pochhammer_product'
thf(fact_1686_power__minus_H,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X2: A,N: nat] :
( ( nO_MATCH @ A @ A @ ( one_one @ A ) @ X2 )
=> ( ( power_power @ A @ ( uminus_uminus @ A @ X2 ) @ N )
= ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N ) @ ( power_power @ A @ X2 @ N ) ) ) ) ) ).
% power_minus'
thf(fact_1687_power__strict__mono,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,B2: A,N: nat] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less @ A @ ( power_power @ A @ A2 @ N ) @ ( power_power @ A @ B2 @ N ) ) ) ) ) ) ).
% power_strict_mono
thf(fact_1688_power__diff__power__eq,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A2: A,N: nat,M: nat] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( ( ord_less_eq @ nat @ N @ M )
=> ( ( divide_divide @ A @ ( power_power @ A @ A2 @ M ) @ ( power_power @ A @ A2 @ N ) )
= ( power_power @ A @ A2 @ ( minus_minus @ nat @ M @ N ) ) ) )
& ( ~ ( ord_less_eq @ nat @ N @ M )
=> ( ( divide_divide @ A @ ( power_power @ A @ A2 @ M ) @ ( power_power @ A @ A2 @ N ) )
= ( divide_divide @ A @ ( one_one @ A ) @ ( power_power @ A @ A2 @ ( minus_minus @ nat @ N @ M ) ) ) ) ) ) ) ) ).
% power_diff_power_eq
thf(fact_1689_power__eq__if,axiom,
! [A: $tType] :
( ( power @ A )
=> ( ( power_power @ A )
= ( ^ [P7: A,M2: nat] :
( if @ A
@ ( M2
= ( zero_zero @ nat ) )
@ ( one_one @ A )
@ ( times_times @ A @ P7 @ ( power_power @ A @ P7 @ ( minus_minus @ nat @ M2 @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% power_eq_if
thf(fact_1690_neg__one__power__add__eq__neg__one__power__diff,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [K: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ N )
=> ( ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( plus_plus @ nat @ N @ K ) )
= ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( minus_minus @ nat @ N @ K ) ) ) ) ) ).
% neg_one_power_add_eq_neg_one_power_diff
thf(fact_1691_power__minus__mult,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [N: nat,A2: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( times_times @ A @ ( power_power @ A @ A2 @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) @ A2 )
= ( power_power @ A @ A2 @ N ) ) ) ) ).
% power_minus_mult
thf(fact_1692_gbinomial__reduce__nat,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,A2: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ( gbinomial @ A @ A2 @ K )
= ( plus_plus @ A @ ( gbinomial @ A @ ( minus_minus @ A @ A2 @ ( one_one @ A ) ) @ ( minus_minus @ nat @ K @ ( one_one @ nat ) ) ) @ ( gbinomial @ A @ ( minus_minus @ A @ A2 @ ( one_one @ A ) ) @ K ) ) ) ) ) ).
% gbinomial_reduce_nat
thf(fact_1693_pochhammer__product,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [M: nat,N: nat,Z2: A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( comm_s3205402744901411588hammer @ A @ Z2 @ N )
= ( times_times @ A @ ( comm_s3205402744901411588hammer @ A @ Z2 @ M ) @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ Z2 @ ( semiring_1_of_nat @ A @ M ) ) @ ( minus_minus @ nat @ N @ M ) ) ) ) ) ) ).
% pochhammer_product
thf(fact_1694_fact__num__eq__if,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiring_char_0_fact @ A )
= ( ^ [M2: nat] :
( if @ A
@ ( M2
= ( zero_zero @ nat ) )
@ ( one_one @ A )
@ ( times_times @ A @ ( semiring_1_of_nat @ A @ M2 ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ M2 @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% fact_num_eq_if
thf(fact_1695_gbinomial__code,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( ( gbinomial @ A )
= ( ^ [A3: A,K2: nat] :
( if @ A
@ ( K2
= ( zero_zero @ nat ) )
@ ( one_one @ A )
@ ( divide_divide @ A
@ ( set_fo6178422350223883121st_nat @ A
@ ^ [L2: nat] : ( times_times @ A @ ( minus_minus @ A @ A3 @ ( semiring_1_of_nat @ A @ L2 ) ) )
@ ( zero_zero @ nat )
@ ( minus_minus @ nat @ K2 @ ( one_one @ nat ) )
@ ( one_one @ A ) )
@ ( semiring_char_0_fact @ A @ K2 ) ) ) ) ) ) ).
% gbinomial_code
thf(fact_1696_pochhammer__code,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ( ( comm_s3205402744901411588hammer @ A )
= ( ^ [A3: A,N4: nat] :
( if @ A
@ ( N4
= ( zero_zero @ nat ) )
@ ( one_one @ A )
@ ( set_fo6178422350223883121st_nat @ A
@ ^ [O: nat] : ( times_times @ A @ ( plus_plus @ A @ A3 @ ( semiring_1_of_nat @ A @ O ) ) )
@ ( zero_zero @ nat )
@ ( minus_minus @ nat @ N4 @ ( one_one @ nat ) )
@ ( one_one @ A ) ) ) ) ) ) ).
% pochhammer_code
thf(fact_1697_slice__nth,axiom,
! [A: $tType,From: nat,To: nat,Xs: list @ A,I: nat] :
( ( ord_less @ nat @ From @ To )
=> ( ( ord_less_eq @ nat @ To @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ord_less @ nat @ I @ ( minus_minus @ nat @ To @ From ) )
=> ( ( nth @ A @ ( slice @ A @ From @ To @ Xs ) @ I )
= ( nth @ A @ Xs @ ( plus_plus @ nat @ From @ I ) ) ) ) ) ) ).
% slice_nth
thf(fact_1698_split__pos__lemma,axiom,
! [K: int,P: int > int > $o,N: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ K )
=> ( ( P @ ( divide_divide @ int @ N @ K ) @ ( modulo_modulo @ int @ N @ K ) )
= ( ! [I2: int,J3: int] :
( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ J3 )
& ( ord_less @ int @ J3 @ K )
& ( N
= ( plus_plus @ int @ ( times_times @ int @ K @ I2 ) @ J3 ) ) )
=> ( P @ I2 @ J3 ) ) ) ) ) ).
% split_pos_lemma
thf(fact_1699_split__neg__lemma,axiom,
! [K: int,P: int > int > $o,N: int] :
( ( ord_less @ int @ K @ ( zero_zero @ int ) )
=> ( ( P @ ( divide_divide @ int @ N @ K ) @ ( modulo_modulo @ int @ N @ K ) )
= ( ! [I2: int,J3: int] :
( ( ( ord_less @ int @ K @ J3 )
& ( ord_less_eq @ int @ J3 @ ( zero_zero @ int ) )
& ( N
= ( plus_plus @ int @ ( times_times @ int @ K @ I2 ) @ J3 ) ) )
=> ( P @ I2 @ J3 ) ) ) ) ) ).
% split_neg_lemma
thf(fact_1700_verit__le__mono__div__int,axiom,
! [A5: int,B5: int,N: int] :
( ( ord_less @ int @ A5 @ B5 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ N )
=> ( ord_less_eq @ int
@ ( plus_plus @ int @ ( divide_divide @ int @ A5 @ N )
@ ( if @ int
@ ( ( modulo_modulo @ int @ B5 @ N )
= ( zero_zero @ int ) )
@ ( one_one @ int )
@ ( zero_zero @ int ) ) )
@ ( divide_divide @ int @ B5 @ N ) ) ) ) ).
% verit_le_mono_div_int
thf(fact_1701_verit__le__mono__div,axiom,
! [A5: nat,B5: nat,N: nat] :
( ( ord_less @ nat @ A5 @ B5 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less_eq @ nat
@ ( plus_plus @ nat @ ( divide_divide @ nat @ A5 @ N )
@ ( if @ nat
@ ( ( modulo_modulo @ nat @ B5 @ N )
= ( zero_zero @ nat ) )
@ ( one_one @ nat )
@ ( zero_zero @ nat ) ) )
@ ( divide_divide @ nat @ B5 @ N ) ) ) ) ).
% verit_le_mono_div
thf(fact_1702_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_1703_numeral__eq__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [M: num,N: num] :
( ( ( numeral_numeral @ A @ M )
= ( numeral_numeral @ A @ N ) )
= ( M = N ) ) ) ).
% numeral_eq_iff
thf(fact_1704_int__eq__iff__numeral,axiom,
! [M: nat,V2: num] :
( ( ( semiring_1_of_nat @ int @ M )
= ( numeral_numeral @ int @ V2 ) )
= ( M
= ( numeral_numeral @ nat @ V2 ) ) ) ).
% int_eq_iff_numeral
thf(fact_1705_nat__numeral,axiom,
! [K: num] :
( ( nat2 @ ( numeral_numeral @ int @ K ) )
= ( numeral_numeral @ nat @ K ) ) ).
% nat_numeral
thf(fact_1706_power__int__mult__numeral,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X2: A,M: num,N: num] :
( ( power_int @ A @ ( power_int @ A @ X2 @ ( numeral_numeral @ int @ M ) ) @ ( numeral_numeral @ int @ N ) )
= ( power_int @ A @ X2 @ ( numeral_numeral @ int @ ( times_times @ num @ M @ N ) ) ) ) ) ).
% power_int_mult_numeral
thf(fact_1707_numeral__le__iff,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [M: num,N: num] :
( ( ord_less_eq @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N ) )
= ( ord_less_eq @ num @ M @ N ) ) ) ).
% numeral_le_iff
thf(fact_1708_numeral__less__iff,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [M: num,N: num] :
( ( ord_less @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N ) )
= ( ord_less @ num @ M @ N ) ) ) ).
% numeral_less_iff
thf(fact_1709_numeral__plus__numeral,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ! [M: num,N: num] :
( ( plus_plus @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N ) )
= ( numeral_numeral @ A @ ( plus_plus @ num @ M @ N ) ) ) ) ).
% numeral_plus_numeral
thf(fact_1710_add__numeral__left,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ! [V2: num,W2: num,Z2: A] :
( ( plus_plus @ A @ ( numeral_numeral @ A @ V2 ) @ ( plus_plus @ A @ ( numeral_numeral @ A @ W2 ) @ Z2 ) )
= ( plus_plus @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ V2 @ W2 ) ) @ Z2 ) ) ) ).
% add_numeral_left
thf(fact_1711_numeral__times__numeral,axiom,
! [A: $tType] :
( ( semiring_numeral @ A )
=> ! [M: num,N: num] :
( ( times_times @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N ) )
= ( numeral_numeral @ A @ ( times_times @ num @ M @ N ) ) ) ) ).
% numeral_times_numeral
thf(fact_1712_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_1713_neg__numeral__eq__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [M: num,N: num] :
( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( M = N ) ) ) ).
% neg_numeral_eq_iff
thf(fact_1714_power__zero__numeral,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [K: num] :
( ( power_power @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ nat @ K ) )
= ( zero_zero @ A ) ) ) ).
% power_zero_numeral
thf(fact_1715_mod__self,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [A2: A] :
( ( modulo_modulo @ A @ A2 @ A2 )
= ( zero_zero @ A ) ) ) ).
% mod_self
thf(fact_1716_mod__by__0,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [A2: A] :
( ( modulo_modulo @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% mod_by_0
thf(fact_1717_mod__0,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [A2: A] :
( ( modulo_modulo @ A @ ( zero_zero @ A ) @ A2 )
= ( zero_zero @ A ) ) ) ).
% mod_0
thf(fact_1718_bits__mod__0,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A2: A] :
( ( modulo_modulo @ A @ ( zero_zero @ A ) @ A2 )
= ( zero_zero @ A ) ) ) ).
% bits_mod_0
thf(fact_1719_power__add__numeral,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A2: A,M: num,N: num] :
( ( times_times @ A @ ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ M ) ) @ ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ N ) ) )
= ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ ( plus_plus @ num @ M @ N ) ) ) ) ) ).
% power_add_numeral
thf(fact_1720_power__add__numeral2,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A2: A,M: num,N: num,B2: A] :
( ( times_times @ A @ ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ M ) ) @ ( times_times @ A @ ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ N ) ) @ B2 ) )
= ( times_times @ A @ ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ ( plus_plus @ num @ M @ N ) ) ) @ B2 ) ) ) ).
% power_add_numeral2
thf(fact_1721_of__nat__numeral,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [N: num] :
( ( semiring_1_of_nat @ A @ ( numeral_numeral @ nat @ N ) )
= ( numeral_numeral @ A @ N ) ) ) ).
% of_nat_numeral
thf(fact_1722_mod__add__self1,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B2: A,A2: A] :
( ( modulo_modulo @ A @ ( plus_plus @ A @ B2 @ A2 ) @ B2 )
= ( modulo_modulo @ A @ A2 @ B2 ) ) ) ).
% mod_add_self1
thf(fact_1723_mod__add__self2,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A2: A,B2: A] :
( ( modulo_modulo @ A @ ( plus_plus @ A @ A2 @ B2 ) @ B2 )
= ( modulo_modulo @ A @ A2 @ B2 ) ) ) ).
% mod_add_self2
thf(fact_1724_abs__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: num] :
( ( abs_abs @ A @ ( numeral_numeral @ A @ N ) )
= ( numeral_numeral @ A @ N ) ) ) ).
% abs_numeral
thf(fact_1725_of__int__eq__numeral__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [Z2: int,N: num] :
( ( ( ring_1_of_int @ A @ Z2 )
= ( numeral_numeral @ A @ N ) )
= ( Z2
= ( numeral_numeral @ int @ N ) ) ) ) ).
% of_int_eq_numeral_iff
thf(fact_1726_of__int__numeral,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [K: num] :
( ( ring_1_of_int @ A @ ( numeral_numeral @ int @ K ) )
= ( numeral_numeral @ A @ K ) ) ) ).
% of_int_numeral
thf(fact_1727_power__int__add__numeral,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X2: A,M: num,N: num] :
( ( times_times @ A @ ( power_int @ A @ X2 @ ( numeral_numeral @ int @ M ) ) @ ( power_int @ A @ X2 @ ( numeral_numeral @ int @ N ) ) )
= ( power_int @ A @ X2 @ ( numeral_numeral @ int @ ( plus_plus @ num @ M @ N ) ) ) ) ) ).
% power_int_add_numeral
thf(fact_1728_power__int__add__numeral2,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X2: A,M: num,N: num,B2: A] :
( ( times_times @ A @ ( power_int @ A @ X2 @ ( numeral_numeral @ int @ M ) ) @ ( times_times @ A @ ( power_int @ A @ X2 @ ( numeral_numeral @ int @ N ) ) @ B2 ) )
= ( times_times @ A @ ( power_int @ A @ X2 @ ( numeral_numeral @ int @ ( plus_plus @ num @ M @ N ) ) ) @ B2 ) ) ) ).
% power_int_add_numeral2
thf(fact_1729_floor__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V2: num] :
( ( archim6421214686448440834_floor @ A @ ( numeral_numeral @ A @ V2 ) )
= ( numeral_numeral @ int @ V2 ) ) ) ).
% floor_numeral
thf(fact_1730_ceiling__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V2: num] :
( ( archimedean_ceiling @ A @ ( numeral_numeral @ A @ V2 ) )
= ( numeral_numeral @ int @ V2 ) ) ) ).
% ceiling_numeral
thf(fact_1731_power__int__numeral,axiom,
! [A: $tType] :
( ( ( inverse @ A )
& ( power @ A ) )
=> ! [X2: A,N: num] :
( ( power_int @ A @ X2 @ ( numeral_numeral @ int @ N ) )
= ( power_power @ A @ X2 @ ( numeral_numeral @ nat @ N ) ) ) ) ).
% power_int_numeral
thf(fact_1732_distrib__right__numeral,axiom,
! [A: $tType] :
( ( ( numeral @ A )
& ( semiring @ A ) )
=> ! [A2: A,B2: A,V2: num] :
( ( times_times @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( numeral_numeral @ A @ V2 ) )
= ( plus_plus @ A @ ( times_times @ A @ A2 @ ( numeral_numeral @ A @ V2 ) ) @ ( times_times @ A @ B2 @ ( numeral_numeral @ A @ V2 ) ) ) ) ) ).
% distrib_right_numeral
thf(fact_1733_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_1734_neg__numeral__le__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M: num,N: num] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( ord_less_eq @ num @ N @ M ) ) ) ).
% neg_numeral_le_iff
thf(fact_1735_left__diff__distrib__numeral,axiom,
! [A: $tType] :
( ( ( numeral @ A )
& ( ring @ A ) )
=> ! [A2: A,B2: A,V2: num] :
( ( times_times @ A @ ( minus_minus @ A @ A2 @ B2 ) @ ( numeral_numeral @ A @ V2 ) )
= ( minus_minus @ A @ ( times_times @ A @ A2 @ ( numeral_numeral @ A @ V2 ) ) @ ( times_times @ A @ B2 @ ( numeral_numeral @ A @ V2 ) ) ) ) ) ).
% left_diff_distrib_numeral
thf(fact_1736_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_1737_neg__numeral__less__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M: num,N: num] :
( ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( ord_less @ num @ N @ M ) ) ) ).
% neg_numeral_less_iff
thf(fact_1738_semiring__norm_I168_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [V2: num,W2: num,Y: A] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) @ ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ Y ) )
= ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ V2 @ W2 ) ) ) @ Y ) ) ) ).
% semiring_norm(168)
thf(fact_1739_add__neg__numeral__simps_I3_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M: num,N: num] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( uminus_uminus @ A @ ( plus_plus @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N ) ) ) ) ) ).
% add_neg_numeral_simps(3)
thf(fact_1740_mult__neg__numeral__simps_I3_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [M: num,N: num] :
( ( times_times @ A @ ( numeral_numeral @ A @ M ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( times_times @ num @ M @ N ) ) ) ) ) ).
% mult_neg_numeral_simps(3)
thf(fact_1741_mult__neg__numeral__simps_I2_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [M: num,N: num] :
( ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( numeral_numeral @ A @ N ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( times_times @ num @ M @ N ) ) ) ) ) ).
% mult_neg_numeral_simps(2)
thf(fact_1742_mult__neg__numeral__simps_I1_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [M: num,N: num] :
( ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( numeral_numeral @ A @ ( times_times @ num @ M @ N ) ) ) ) ).
% mult_neg_numeral_simps(1)
thf(fact_1743_diff__numeral__simps_I3_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M: num,N: num] :
( ( minus_minus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( numeral_numeral @ A @ N ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ M @ N ) ) ) ) ) ).
% diff_numeral_simps(3)
thf(fact_1744_diff__numeral__simps_I2_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M: num,N: num] :
( ( minus_minus @ A @ ( numeral_numeral @ A @ M ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( numeral_numeral @ A @ ( plus_plus @ num @ M @ N ) ) ) ) ).
% diff_numeral_simps(2)
thf(fact_1745_mod__mult__self2__is__0,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A2: A,B2: A] :
( ( modulo_modulo @ A @ ( times_times @ A @ A2 @ B2 ) @ B2 )
= ( zero_zero @ A ) ) ) ).
% mod_mult_self2_is_0
thf(fact_1746_mod__mult__self1__is__0,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B2: A,A2: A] :
( ( modulo_modulo @ A @ ( times_times @ A @ B2 @ A2 ) @ B2 )
= ( zero_zero @ A ) ) ) ).
% mod_mult_self1_is_0
thf(fact_1747_mod__by__1,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [A2: A] :
( ( modulo_modulo @ A @ A2 @ ( one_one @ A ) )
= ( zero_zero @ A ) ) ) ).
% mod_by_1
thf(fact_1748_bits__mod__by__1,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A2: A] :
( ( modulo_modulo @ A @ A2 @ ( one_one @ A ) )
= ( zero_zero @ A ) ) ) ).
% bits_mod_by_1
thf(fact_1749_mod__mult__self1,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( modulo_modulo @ A @ ( plus_plus @ A @ A2 @ ( times_times @ A @ C2 @ B2 ) ) @ B2 )
= ( modulo_modulo @ A @ A2 @ B2 ) ) ) ).
% mod_mult_self1
thf(fact_1750_mod__mult__self2,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( modulo_modulo @ A @ ( plus_plus @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) ) @ B2 )
= ( modulo_modulo @ A @ A2 @ B2 ) ) ) ).
% mod_mult_self2
thf(fact_1751_mod__mult__self3,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [C2: A,B2: A,A2: A] :
( ( modulo_modulo @ A @ ( plus_plus @ A @ ( times_times @ A @ C2 @ B2 ) @ A2 ) @ B2 )
= ( modulo_modulo @ A @ A2 @ B2 ) ) ) ).
% mod_mult_self3
thf(fact_1752_mod__mult__self4,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B2: A,C2: A,A2: A] :
( ( modulo_modulo @ A @ ( plus_plus @ A @ ( times_times @ A @ B2 @ C2 ) @ A2 ) @ B2 )
= ( modulo_modulo @ A @ A2 @ B2 ) ) ) ).
% mod_mult_self4
thf(fact_1753_mod__div__trivial,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A2: A,B2: A] :
( ( divide_divide @ A @ ( modulo_modulo @ A @ A2 @ B2 ) @ B2 )
= ( zero_zero @ A ) ) ) ).
% mod_div_trivial
thf(fact_1754_bits__mod__div__trivial,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A2: A,B2: A] :
( ( divide_divide @ A @ ( modulo_modulo @ A @ A2 @ B2 ) @ B2 )
= ( zero_zero @ A ) ) ) ).
% bits_mod_div_trivial
thf(fact_1755_abs__neg__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: num] :
( ( abs_abs @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( numeral_numeral @ A @ N ) ) ) ).
% abs_neg_numeral
thf(fact_1756_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_1757_power__int__mult__distrib__numeral2,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [X2: A,W2: num,M: int] :
( ( power_int @ A @ ( times_times @ A @ X2 @ ( numeral_numeral @ A @ W2 ) ) @ M )
= ( times_times @ A @ ( power_int @ A @ X2 @ M ) @ ( power_int @ A @ ( numeral_numeral @ A @ W2 ) @ M ) ) ) ) ).
% power_int_mult_distrib_numeral2
thf(fact_1758_mod__by__Suc__0,axiom,
! [M: nat] :
( ( modulo_modulo @ nat @ M @ ( suc @ ( zero_zero @ nat ) ) )
= ( zero_zero @ nat ) ) ).
% mod_by_Suc_0
thf(fact_1759_nat__neg__numeral,axiom,
! [K: num] :
( ( nat2 @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) )
= ( zero_zero @ nat ) ) ).
% nat_neg_numeral
thf(fact_1760_nat__power__eq__Suc__0__iff,axiom,
! [X2: nat,M: nat] :
( ( ( power_power @ nat @ X2 @ M )
= ( suc @ ( zero_zero @ nat ) ) )
= ( ( M
= ( zero_zero @ nat ) )
| ( X2
= ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% nat_power_eq_Suc_0_iff
thf(fact_1761_power__Suc__0,axiom,
! [N: nat] :
( ( power_power @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% power_Suc_0
thf(fact_1762_nat__zero__less__power__iff,axiom,
! [X2: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( power_power @ nat @ X2 @ N ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ X2 )
| ( N
= ( zero_zero @ nat ) ) ) ) ).
% nat_zero_less_power_iff
thf(fact_1763_diff__nat__numeral,axiom,
! [V2: num,V3: num] :
( ( minus_minus @ nat @ ( numeral_numeral @ nat @ V2 ) @ ( numeral_numeral @ nat @ V3 ) )
= ( nat2 @ ( minus_minus @ int @ ( numeral_numeral @ int @ V2 ) @ ( numeral_numeral @ int @ V3 ) ) ) ) ).
% diff_nat_numeral
thf(fact_1764_nat__eq__numeral__power__cancel__iff,axiom,
! [Y: int,X2: num,N: nat] :
( ( ( nat2 @ Y )
= ( power_power @ nat @ ( numeral_numeral @ nat @ X2 ) @ N ) )
= ( Y
= ( power_power @ int @ ( numeral_numeral @ int @ X2 ) @ N ) ) ) ).
% nat_eq_numeral_power_cancel_iff
thf(fact_1765_numeral__power__eq__nat__cancel__iff,axiom,
! [X2: num,N: nat,Y: int] :
( ( ( power_power @ nat @ ( numeral_numeral @ nat @ X2 ) @ N )
= ( nat2 @ Y ) )
= ( ( power_power @ int @ ( numeral_numeral @ int @ X2 ) @ N )
= Y ) ) ).
% numeral_power_eq_nat_cancel_iff
thf(fact_1766_dbl__simps_I1_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [K: num] :
( ( neg_numeral_dbl @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ K ) ) )
= ( uminus_uminus @ A @ ( neg_numeral_dbl @ A @ ( numeral_numeral @ A @ K ) ) ) ) ) ).
% dbl_simps(1)
thf(fact_1767_le__divide__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,W2: num] :
( ( ord_less_eq @ A @ A2 @ ( divide_divide @ A @ B2 @ ( numeral_numeral @ A @ W2 ) ) )
= ( ord_less_eq @ A @ ( times_times @ A @ A2 @ ( numeral_numeral @ A @ W2 ) ) @ B2 ) ) ) ).
% le_divide_eq_numeral1(1)
thf(fact_1768_divide__le__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,W2: num,A2: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ ( numeral_numeral @ A @ W2 ) ) @ A2 )
= ( ord_less_eq @ A @ B2 @ ( times_times @ A @ A2 @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ).
% divide_le_eq_numeral1(1)
thf(fact_1769_eq__divide__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A,B2: A,W2: num] :
( ( A2
= ( divide_divide @ A @ B2 @ ( numeral_numeral @ A @ W2 ) ) )
= ( ( ( ( numeral_numeral @ A @ W2 )
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ A2 @ ( numeral_numeral @ A @ W2 ) )
= B2 ) )
& ( ( ( numeral_numeral @ A @ W2 )
= ( zero_zero @ A ) )
=> ( A2
= ( zero_zero @ A ) ) ) ) ) ) ).
% eq_divide_eq_numeral1(1)
thf(fact_1770_divide__eq__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,W2: num,A2: A] :
( ( ( divide_divide @ A @ B2 @ ( numeral_numeral @ A @ W2 ) )
= A2 )
= ( ( ( ( numeral_numeral @ A @ W2 )
!= ( zero_zero @ A ) )
=> ( B2
= ( times_times @ A @ A2 @ ( numeral_numeral @ A @ W2 ) ) ) )
& ( ( ( numeral_numeral @ A @ W2 )
= ( zero_zero @ A ) )
=> ( A2
= ( zero_zero @ A ) ) ) ) ) ) ).
% divide_eq_eq_numeral1(1)
thf(fact_1771_divide__less__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,W2: num,A2: A] :
( ( ord_less @ A @ ( divide_divide @ A @ B2 @ ( numeral_numeral @ A @ W2 ) ) @ A2 )
= ( ord_less @ A @ B2 @ ( times_times @ A @ A2 @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ).
% divide_less_eq_numeral1(1)
thf(fact_1772_less__divide__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,W2: num] :
( ( ord_less @ A @ A2 @ ( divide_divide @ A @ B2 @ ( numeral_numeral @ A @ W2 ) ) )
= ( ord_less @ A @ ( times_times @ A @ A2 @ ( numeral_numeral @ A @ W2 ) ) @ B2 ) ) ) ).
% less_divide_eq_numeral1(1)
thf(fact_1773_mod__minus1__right,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A2: A] :
( ( modulo_modulo @ A @ A2 @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( zero_zero @ A ) ) ) ).
% mod_minus1_right
thf(fact_1774_of__int__numeral__le__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: num,Z2: int] :
( ( ord_less_eq @ A @ ( numeral_numeral @ A @ N ) @ ( ring_1_of_int @ A @ Z2 ) )
= ( ord_less_eq @ int @ ( numeral_numeral @ int @ N ) @ Z2 ) ) ) ).
% of_int_numeral_le_iff
thf(fact_1775_of__int__le__numeral__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z2: int,N: num] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z2 ) @ ( numeral_numeral @ A @ N ) )
= ( ord_less_eq @ int @ Z2 @ ( numeral_numeral @ int @ N ) ) ) ) ).
% of_int_le_numeral_iff
thf(fact_1776_of__int__numeral__less__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: num,Z2: int] :
( ( ord_less @ A @ ( numeral_numeral @ A @ N ) @ ( ring_1_of_int @ A @ Z2 ) )
= ( ord_less @ int @ ( numeral_numeral @ int @ N ) @ Z2 ) ) ) ).
% of_int_numeral_less_iff
thf(fact_1777_of__int__less__numeral__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z2: int,N: num] :
( ( ord_less @ A @ ( ring_1_of_int @ A @ Z2 ) @ ( numeral_numeral @ A @ N ) )
= ( ord_less @ int @ Z2 @ ( numeral_numeral @ int @ N ) ) ) ) ).
% of_int_less_numeral_iff
thf(fact_1778_numeral__le__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V2: num,X2: A] :
( ( ord_less_eq @ int @ ( numeral_numeral @ int @ V2 ) @ ( archim6421214686448440834_floor @ A @ X2 ) )
= ( ord_less_eq @ A @ ( numeral_numeral @ A @ V2 ) @ X2 ) ) ) ).
% numeral_le_floor
thf(fact_1779_floor__less__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A,V2: num] :
( ( ord_less @ int @ ( archim6421214686448440834_floor @ A @ X2 ) @ ( numeral_numeral @ int @ V2 ) )
= ( ord_less @ A @ X2 @ ( numeral_numeral @ A @ V2 ) ) ) ) ).
% floor_less_numeral
thf(fact_1780_ceiling__le__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A,V2: num] :
( ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X2 ) @ ( numeral_numeral @ int @ V2 ) )
= ( ord_less_eq @ A @ X2 @ ( numeral_numeral @ A @ V2 ) ) ) ) ).
% ceiling_le_numeral
thf(fact_1781_numeral__less__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V2: num,X2: A] :
( ( ord_less @ int @ ( numeral_numeral @ int @ V2 ) @ ( archimedean_ceiling @ A @ X2 ) )
= ( ord_less @ A @ ( numeral_numeral @ A @ V2 ) @ X2 ) ) ) ).
% numeral_less_ceiling
thf(fact_1782_numeral__power__eq__of__nat__cancel__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [X2: num,N: nat,Y: nat] :
( ( ( power_power @ A @ ( numeral_numeral @ A @ X2 ) @ N )
= ( semiring_1_of_nat @ A @ Y ) )
= ( ( power_power @ nat @ ( numeral_numeral @ nat @ X2 ) @ N )
= Y ) ) ) ).
% numeral_power_eq_of_nat_cancel_iff
thf(fact_1783_real__of__nat__eq__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [Y: nat,X2: num,N: nat] :
( ( ( semiring_1_of_nat @ A @ Y )
= ( power_power @ A @ ( numeral_numeral @ A @ X2 ) @ N ) )
= ( Y
= ( power_power @ nat @ ( numeral_numeral @ nat @ X2 ) @ N ) ) ) ) ).
% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_1784_Suc__mod__mult__self1,axiom,
! [M: nat,K: nat,N: nat] :
( ( modulo_modulo @ nat @ ( suc @ ( plus_plus @ nat @ M @ ( times_times @ nat @ K @ N ) ) ) @ N )
= ( modulo_modulo @ nat @ ( suc @ M ) @ N ) ) ).
% Suc_mod_mult_self1
thf(fact_1785_Suc__mod__mult__self2,axiom,
! [M: nat,N: nat,K: nat] :
( ( modulo_modulo @ nat @ ( suc @ ( plus_plus @ nat @ M @ ( times_times @ nat @ N @ K ) ) ) @ N )
= ( modulo_modulo @ nat @ ( suc @ M ) @ N ) ) ).
% Suc_mod_mult_self2
thf(fact_1786_Suc__mod__mult__self3,axiom,
! [K: nat,N: nat,M: nat] :
( ( modulo_modulo @ nat @ ( suc @ ( plus_plus @ nat @ ( times_times @ nat @ K @ N ) @ M ) ) @ N )
= ( modulo_modulo @ nat @ ( suc @ M ) @ N ) ) ).
% Suc_mod_mult_self3
thf(fact_1787_Suc__mod__mult__self4,axiom,
! [N: nat,K: nat,M: nat] :
( ( modulo_modulo @ nat @ ( suc @ ( plus_plus @ nat @ ( times_times @ nat @ N @ K ) @ M ) ) @ N )
= ( modulo_modulo @ nat @ ( suc @ M ) @ N ) ) ).
% Suc_mod_mult_self4
thf(fact_1788_floor__neg__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V2: num] :
( ( archim6421214686448440834_floor @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) )
= ( uminus_uminus @ int @ ( numeral_numeral @ int @ V2 ) ) ) ) ).
% floor_neg_numeral
thf(fact_1789_ceiling__add__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A,V2: num] :
( ( archimedean_ceiling @ A @ ( plus_plus @ A @ X2 @ ( numeral_numeral @ A @ V2 ) ) )
= ( plus_plus @ int @ ( archimedean_ceiling @ A @ X2 ) @ ( numeral_numeral @ int @ V2 ) ) ) ) ).
% ceiling_add_numeral
thf(fact_1790_floor__diff__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A,V2: num] :
( ( archim6421214686448440834_floor @ A @ ( minus_minus @ A @ X2 @ ( numeral_numeral @ A @ V2 ) ) )
= ( minus_minus @ int @ ( archim6421214686448440834_floor @ A @ X2 ) @ ( numeral_numeral @ int @ V2 ) ) ) ) ).
% floor_diff_numeral
thf(fact_1791_ceiling__neg__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V2: num] :
( ( archimedean_ceiling @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) )
= ( uminus_uminus @ int @ ( numeral_numeral @ int @ V2 ) ) ) ) ).
% ceiling_neg_numeral
thf(fact_1792_Suc__times__numeral__mod__eq,axiom,
! [K: num,N: nat] :
( ( ( numeral_numeral @ nat @ K )
!= ( one_one @ nat ) )
=> ( ( modulo_modulo @ nat @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ K ) @ N ) ) @ ( numeral_numeral @ nat @ K ) )
= ( one_one @ nat ) ) ) ).
% Suc_times_numeral_mod_eq
thf(fact_1793_ceiling__diff__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A,V2: num] :
( ( archimedean_ceiling @ A @ ( minus_minus @ A @ X2 @ ( numeral_numeral @ A @ V2 ) ) )
= ( minus_minus @ int @ ( archimedean_ceiling @ A @ X2 ) @ ( numeral_numeral @ int @ V2 ) ) ) ) ).
% ceiling_diff_numeral
thf(fact_1794_of__int__eq__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [Y: int,X2: num,N: nat] :
( ( ( ring_1_of_int @ A @ Y )
= ( power_power @ A @ ( numeral_numeral @ A @ X2 ) @ N ) )
= ( Y
= ( power_power @ int @ ( numeral_numeral @ int @ X2 ) @ N ) ) ) ) ).
% of_int_eq_numeral_power_cancel_iff
thf(fact_1795_numeral__power__eq__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [X2: num,N: nat,Y: int] :
( ( ( power_power @ A @ ( numeral_numeral @ A @ X2 ) @ N )
= ( ring_1_of_int @ A @ Y ) )
= ( ( power_power @ int @ ( numeral_numeral @ int @ X2 ) @ N )
= Y ) ) ) ).
% numeral_power_eq_of_int_cancel_iff
thf(fact_1796_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_1797_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_1798_le__divide__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,W2: num] :
( ( ord_less_eq @ A @ A2 @ ( divide_divide @ A @ B2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) )
= ( ord_less_eq @ A @ B2 @ ( times_times @ A @ A2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ) ).
% le_divide_eq_numeral1(2)
thf(fact_1799_divide__le__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,W2: num,A2: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) @ A2 )
= ( ord_less_eq @ A @ ( times_times @ A @ A2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) @ B2 ) ) ) ).
% divide_le_eq_numeral1(2)
thf(fact_1800_eq__divide__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A,B2: A,W2: num] :
( ( A2
= ( divide_divide @ A @ B2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) )
= ( ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) )
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ A2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) )
= B2 ) )
& ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) )
= ( zero_zero @ A ) )
=> ( A2
= ( zero_zero @ A ) ) ) ) ) ) ).
% eq_divide_eq_numeral1(2)
thf(fact_1801_divide__eq__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,W2: num,A2: A] :
( ( ( divide_divide @ A @ B2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) )
= A2 )
= ( ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) )
!= ( zero_zero @ A ) )
=> ( B2
= ( times_times @ A @ A2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) ) )
& ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) )
= ( zero_zero @ A ) )
=> ( A2
= ( zero_zero @ A ) ) ) ) ) ) ).
% divide_eq_eq_numeral1(2)
thf(fact_1802_less__divide__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,W2: num] :
( ( ord_less @ A @ A2 @ ( divide_divide @ A @ B2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) )
= ( ord_less @ A @ B2 @ ( times_times @ A @ A2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ) ).
% less_divide_eq_numeral1(2)
thf(fact_1803_divide__less__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,W2: num,A2: A] :
( ( ord_less @ A @ ( divide_divide @ A @ B2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) @ A2 )
= ( ord_less @ A @ ( times_times @ A @ A2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) @ B2 ) ) ) ).
% divide_less_eq_numeral1(2)
thf(fact_1804_dbl__dec__simps_I1_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [K: num] :
( ( neg_numeral_dbl_dec @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ K ) ) )
= ( uminus_uminus @ A @ ( neg_numeral_dbl_inc @ A @ ( numeral_numeral @ A @ K ) ) ) ) ) ).
% dbl_dec_simps(1)
thf(fact_1805_dbl__inc__simps_I1_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [K: num] :
( ( neg_numeral_dbl_inc @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ K ) ) )
= ( uminus_uminus @ A @ ( neg_numeral_dbl_dec @ A @ ( numeral_numeral @ A @ K ) ) ) ) ) ).
% dbl_inc_simps(1)
thf(fact_1806_nat__numeral__diff__1,axiom,
! [V2: num] :
( ( minus_minus @ nat @ ( numeral_numeral @ nat @ V2 ) @ ( one_one @ nat ) )
= ( nat2 @ ( minus_minus @ int @ ( numeral_numeral @ int @ V2 ) @ ( one_one @ int ) ) ) ) ).
% nat_numeral_diff_1
thf(fact_1807_numeral__power__less__nat__cancel__iff,axiom,
! [X2: num,N: nat,A2: int] :
( ( ord_less @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ X2 ) @ N ) @ ( nat2 @ A2 ) )
= ( ord_less @ int @ ( power_power @ int @ ( numeral_numeral @ int @ X2 ) @ N ) @ A2 ) ) ).
% numeral_power_less_nat_cancel_iff
thf(fact_1808_nat__less__numeral__power__cancel__iff,axiom,
! [A2: int,X2: num,N: nat] :
( ( ord_less @ nat @ ( nat2 @ A2 ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ X2 ) @ N ) )
= ( ord_less @ int @ A2 @ ( power_power @ int @ ( numeral_numeral @ int @ X2 ) @ N ) ) ) ).
% nat_less_numeral_power_cancel_iff
thf(fact_1809_nat__le__numeral__power__cancel__iff,axiom,
! [A2: int,X2: num,N: nat] :
( ( ord_less_eq @ nat @ ( nat2 @ A2 ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ X2 ) @ N ) )
= ( ord_less_eq @ int @ A2 @ ( power_power @ int @ ( numeral_numeral @ int @ X2 ) @ N ) ) ) ).
% nat_le_numeral_power_cancel_iff
thf(fact_1810_numeral__power__le__nat__cancel__iff,axiom,
! [X2: num,N: nat,A2: int] :
( ( ord_less_eq @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ X2 ) @ N ) @ ( nat2 @ A2 ) )
= ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ X2 ) @ N ) @ A2 ) ) ).
% numeral_power_le_nat_cancel_iff
thf(fact_1811_numeral__less__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V2: num,X2: A] :
( ( ord_less @ int @ ( numeral_numeral @ int @ V2 ) @ ( archim6421214686448440834_floor @ A @ X2 ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ ( numeral_numeral @ A @ V2 ) @ ( one_one @ A ) ) @ X2 ) ) ) ).
% numeral_less_floor
thf(fact_1812_floor__le__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A,V2: num] :
( ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X2 ) @ ( numeral_numeral @ int @ V2 ) )
= ( ord_less @ A @ X2 @ ( plus_plus @ A @ ( numeral_numeral @ A @ V2 ) @ ( one_one @ A ) ) ) ) ) ).
% floor_le_numeral
thf(fact_1813_ceiling__less__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A,V2: num] :
( ( ord_less @ int @ ( archimedean_ceiling @ A @ X2 ) @ ( numeral_numeral @ int @ V2 ) )
= ( ord_less_eq @ A @ X2 @ ( minus_minus @ A @ ( numeral_numeral @ A @ V2 ) @ ( one_one @ A ) ) ) ) ) ).
% ceiling_less_numeral
thf(fact_1814_numeral__le__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V2: num,X2: A] :
( ( ord_less_eq @ int @ ( numeral_numeral @ int @ V2 ) @ ( archimedean_ceiling @ A @ X2 ) )
= ( ord_less @ A @ ( minus_minus @ A @ ( numeral_numeral @ A @ V2 ) @ ( one_one @ A ) ) @ X2 ) ) ) ).
% numeral_le_ceiling
thf(fact_1815_numeral__power__less__of__nat__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [I: num,N: nat,X2: nat] :
( ( ord_less @ A @ ( power_power @ A @ ( numeral_numeral @ A @ I ) @ N ) @ ( semiring_1_of_nat @ A @ X2 ) )
= ( ord_less @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ I ) @ N ) @ X2 ) ) ) ).
% numeral_power_less_of_nat_cancel_iff
thf(fact_1816_of__nat__less__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X2: nat,I: num,N: nat] :
( ( ord_less @ A @ ( semiring_1_of_nat @ A @ X2 ) @ ( power_power @ A @ ( numeral_numeral @ A @ I ) @ N ) )
= ( ord_less @ nat @ X2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ I ) @ N ) ) ) ) ).
% of_nat_less_numeral_power_cancel_iff
thf(fact_1817_of__nat__le__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X2: nat,I: num,N: nat] :
( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ X2 ) @ ( power_power @ A @ ( numeral_numeral @ A @ I ) @ N ) )
= ( ord_less_eq @ nat @ X2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ I ) @ N ) ) ) ) ).
% of_nat_le_numeral_power_cancel_iff
thf(fact_1818_numeral__power__le__of__nat__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [I: num,N: nat,X2: nat] :
( ( ord_less_eq @ A @ ( power_power @ A @ ( numeral_numeral @ A @ I ) @ N ) @ ( semiring_1_of_nat @ A @ X2 ) )
= ( ord_less_eq @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ I ) @ N ) @ X2 ) ) ) ).
% numeral_power_le_of_nat_cancel_iff
thf(fact_1819_neg__numeral__le__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V2: num,X2: A] :
( ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V2 ) ) @ ( archim6421214686448440834_floor @ A @ X2 ) )
= ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) @ X2 ) ) ) ).
% neg_numeral_le_floor
thf(fact_1820_floor__less__neg__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A,V2: num] :
( ( ord_less @ int @ ( archim6421214686448440834_floor @ A @ X2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V2 ) ) )
= ( ord_less @ A @ X2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) ) ) ) ).
% floor_less_neg_numeral
thf(fact_1821_ceiling__le__neg__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A,V2: num] :
( ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V2 ) ) )
= ( ord_less_eq @ A @ X2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) ) ) ) ).
% ceiling_le_neg_numeral
thf(fact_1822_neg__numeral__less__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V2: num,X2: A] :
( ( ord_less @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V2 ) ) @ ( archimedean_ceiling @ A @ X2 ) )
= ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) @ X2 ) ) ) ).
% neg_numeral_less_ceiling
thf(fact_1823_of__int__le__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: int,X2: num,N: nat] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ A2 ) @ ( power_power @ A @ ( numeral_numeral @ A @ X2 ) @ N ) )
= ( ord_less_eq @ int @ A2 @ ( power_power @ int @ ( numeral_numeral @ int @ X2 ) @ N ) ) ) ) ).
% of_int_le_numeral_power_cancel_iff
thf(fact_1824_numeral__power__le__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X2: num,N: nat,A2: int] :
( ( ord_less_eq @ A @ ( power_power @ A @ ( numeral_numeral @ A @ X2 ) @ N ) @ ( ring_1_of_int @ A @ A2 ) )
= ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ X2 ) @ N ) @ A2 ) ) ) ).
% numeral_power_le_of_int_cancel_iff
thf(fact_1825_numeral__power__less__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X2: num,N: nat,A2: int] :
( ( ord_less @ A @ ( power_power @ A @ ( numeral_numeral @ A @ X2 ) @ N ) @ ( ring_1_of_int @ A @ A2 ) )
= ( ord_less @ int @ ( power_power @ int @ ( numeral_numeral @ int @ X2 ) @ N ) @ A2 ) ) ) ).
% numeral_power_less_of_int_cancel_iff
thf(fact_1826_of__int__less__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: int,X2: num,N: nat] :
( ( ord_less @ A @ ( ring_1_of_int @ A @ A2 ) @ ( power_power @ A @ ( numeral_numeral @ A @ X2 ) @ N ) )
= ( ord_less @ int @ A2 @ ( power_power @ int @ ( numeral_numeral @ int @ X2 ) @ N ) ) ) ) ).
% of_int_less_numeral_power_cancel_iff
thf(fact_1827_of__int__eq__neg__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [Y: int,X2: num,N: nat] :
( ( ( ring_1_of_int @ A @ Y )
= ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X2 ) ) @ N ) )
= ( Y
= ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X2 ) ) @ N ) ) ) ) ).
% of_int_eq_neg_numeral_power_cancel_iff
thf(fact_1828_neg__numeral__power__eq__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [X2: num,N: nat,Y: int] :
( ( ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X2 ) ) @ N )
= ( ring_1_of_int @ A @ Y ) )
= ( ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X2 ) ) @ N )
= Y ) ) ) ).
% neg_numeral_power_eq_of_int_cancel_iff
thf(fact_1829_neg__numeral__less__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V2: num,X2: A] :
( ( ord_less @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V2 ) ) @ ( archim6421214686448440834_floor @ A @ X2 ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) @ ( one_one @ A ) ) @ X2 ) ) ) ).
% neg_numeral_less_floor
thf(fact_1830_floor__le__neg__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A,V2: num] :
( ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V2 ) ) )
= ( ord_less @ A @ X2 @ ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) @ ( one_one @ A ) ) ) ) ) ).
% floor_le_neg_numeral
thf(fact_1831_ceiling__less__neg__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A,V2: num] :
( ( ord_less @ int @ ( archimedean_ceiling @ A @ X2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V2 ) ) )
= ( ord_less_eq @ A @ X2 @ ( minus_minus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) @ ( one_one @ A ) ) ) ) ) ).
% ceiling_less_neg_numeral
thf(fact_1832_neg__numeral__le__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V2: num,X2: A] :
( ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V2 ) ) @ ( archimedean_ceiling @ A @ X2 ) )
= ( ord_less @ A @ ( minus_minus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) @ ( one_one @ A ) ) @ X2 ) ) ) ).
% neg_numeral_le_ceiling
thf(fact_1833_of__int__le__neg__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: int,X2: num,N: nat] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ A2 ) @ ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X2 ) ) @ N ) )
= ( ord_less_eq @ int @ A2 @ ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X2 ) ) @ N ) ) ) ) ).
% of_int_le_neg_numeral_power_cancel_iff
thf(fact_1834_neg__numeral__power__le__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X2: num,N: nat,A2: int] :
( ( ord_less_eq @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X2 ) ) @ N ) @ ( ring_1_of_int @ A @ A2 ) )
= ( ord_less_eq @ int @ ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X2 ) ) @ N ) @ A2 ) ) ) ).
% neg_numeral_power_le_of_int_cancel_iff
thf(fact_1835_neg__numeral__power__less__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X2: num,N: nat,A2: int] :
( ( ord_less @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X2 ) ) @ N ) @ ( ring_1_of_int @ A @ A2 ) )
= ( ord_less @ int @ ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X2 ) ) @ N ) @ A2 ) ) ) ).
% neg_numeral_power_less_of_int_cancel_iff
thf(fact_1836_of__int__less__neg__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: int,X2: num,N: nat] :
( ( ord_less @ A @ ( ring_1_of_int @ A @ A2 ) @ ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X2 ) ) @ N ) )
= ( ord_less @ int @ A2 @ ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X2 ) ) @ N ) ) ) ) ).
% of_int_less_neg_numeral_power_cancel_iff
thf(fact_1837_zmod__int,axiom,
! [A2: nat,B2: nat] :
( ( semiring_1_of_nat @ int @ ( modulo_modulo @ nat @ A2 @ B2 ) )
= ( modulo_modulo @ int @ ( semiring_1_of_nat @ int @ A2 ) @ ( semiring_1_of_nat @ int @ B2 ) ) ) ).
% zmod_int
thf(fact_1838_nat__numeral__as__int,axiom,
( ( numeral_numeral @ nat )
= ( ^ [I2: num] : ( nat2 @ ( numeral_numeral @ int @ I2 ) ) ) ) ).
% nat_numeral_as_int
thf(fact_1839_int__ops_I3_J,axiom,
! [N: num] :
( ( semiring_1_of_nat @ int @ ( numeral_numeral @ nat @ N ) )
= ( numeral_numeral @ int @ N ) ) ).
% int_ops(3)
thf(fact_1840_of__nat__mod,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [M: nat,N: nat] :
( ( semiring_1_of_nat @ A @ ( modulo_modulo @ nat @ M @ N ) )
= ( modulo_modulo @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% of_nat_mod
thf(fact_1841_mod__add__eq,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( modulo_modulo @ A @ ( plus_plus @ A @ ( modulo_modulo @ A @ A2 @ C2 ) @ ( modulo_modulo @ A @ B2 @ C2 ) ) @ C2 )
= ( modulo_modulo @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 ) ) ) ).
% mod_add_eq
thf(fact_1842_mod__add__cong,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A2: A,C2: A,A6: A,B2: A,B6: A] :
( ( ( modulo_modulo @ A @ A2 @ C2 )
= ( modulo_modulo @ A @ A6 @ C2 ) )
=> ( ( ( modulo_modulo @ A @ B2 @ C2 )
= ( modulo_modulo @ A @ B6 @ C2 ) )
=> ( ( modulo_modulo @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 )
= ( modulo_modulo @ A @ ( plus_plus @ A @ A6 @ B6 ) @ C2 ) ) ) ) ) ).
% mod_add_cong
thf(fact_1843_mod__add__left__eq,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( modulo_modulo @ A @ ( plus_plus @ A @ ( modulo_modulo @ A @ A2 @ C2 ) @ B2 ) @ C2 )
= ( modulo_modulo @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 ) ) ) ).
% mod_add_left_eq
thf(fact_1844_mod__add__right__eq,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( modulo_modulo @ A @ ( plus_plus @ A @ A2 @ ( modulo_modulo @ A @ B2 @ C2 ) ) @ C2 )
= ( modulo_modulo @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 ) ) ) ).
% mod_add_right_eq
thf(fact_1845_zero__neq__numeral,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [N: num] :
( ( zero_zero @ A )
!= ( numeral_numeral @ A @ N ) ) ) ).
% zero_neq_numeral
thf(fact_1846_mod__less__eq__dividend,axiom,
! [M: nat,N: nat] : ( ord_less_eq @ nat @ ( modulo_modulo @ nat @ M @ N ) @ M ) ).
% mod_less_eq_dividend
thf(fact_1847_neg__numeral__neq__numeral,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [M: num,N: num] :
( ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) )
!= ( numeral_numeral @ A @ N ) ) ) ).
% neg_numeral_neq_numeral
thf(fact_1848_numeral__neq__neg__numeral,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [M: num,N: num] :
( ( numeral_numeral @ A @ M )
!= ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) ) ) ).
% numeral_neq_neg_numeral
thf(fact_1849_nat__mod__as__int,axiom,
( ( modulo_modulo @ nat )
= ( ^ [A3: nat,B3: nat] : ( nat2 @ ( modulo_modulo @ int @ ( semiring_1_of_nat @ int @ A3 ) @ ( semiring_1_of_nat @ int @ B3 ) ) ) ) ) ).
% nat_mod_as_int
thf(fact_1850_Ints__numeral,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [N: num] : ( member @ A @ ( numeral_numeral @ A @ N ) @ ( ring_1_Ints @ A ) ) ) ).
% Ints_numeral
thf(fact_1851_nat__mod__distrib,axiom,
! [X2: int,Y: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X2 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y )
=> ( ( nat2 @ ( modulo_modulo @ int @ X2 @ Y ) )
= ( modulo_modulo @ nat @ ( nat2 @ X2 ) @ ( nat2 @ Y ) ) ) ) ) ).
% nat_mod_distrib
thf(fact_1852_mod__abs__eq__div__nat,axiom,
! [K: int,L: int] :
( ( modulo_modulo @ int @ ( abs_abs @ int @ K ) @ ( abs_abs @ int @ L ) )
= ( semiring_1_of_nat @ int @ ( modulo_modulo @ nat @ ( nat2 @ ( abs_abs @ int @ K ) ) @ ( nat2 @ ( abs_abs @ int @ L ) ) ) ) ) ).
% mod_abs_eq_div_nat
thf(fact_1853_of__int__neg__numeral,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [K: num] :
( ( ring_1_of_int @ A @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ K ) ) ) ) ).
% of_int_neg_numeral
thf(fact_1854_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ord_less_eq @ A @ ( modulo_modulo @ A @ A2 @ B2 ) @ A2 ) ) ) ).
% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
thf(fact_1855_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [B2: A,A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less @ A @ ( modulo_modulo @ A @ A2 @ B2 ) @ B2 ) ) ) ).
% unique_euclidean_semiring_numeral_class.pos_mod_bound
thf(fact_1856_mod__eqE,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( ( modulo_modulo @ A @ A2 @ C2 )
= ( modulo_modulo @ A @ B2 @ C2 ) )
=> ~ ! [D6: A] :
( B2
!= ( plus_plus @ A @ A2 @ ( times_times @ A @ C2 @ D6 ) ) ) ) ) ).
% mod_eqE
thf(fact_1857_mod__eq__self__iff__div__eq__0,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [A2: A,B2: A] :
( ( ( modulo_modulo @ A @ A2 @ B2 )
= A2 )
= ( ( divide_divide @ A @ A2 @ B2 )
= ( zero_zero @ A ) ) ) ) ).
% mod_eq_self_iff_div_eq_0
thf(fact_1858_div__add1__eq,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( divide_divide @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 )
= ( plus_plus @ A @ ( plus_plus @ A @ ( divide_divide @ A @ A2 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) @ ( divide_divide @ A @ ( plus_plus @ A @ ( modulo_modulo @ A @ A2 @ C2 ) @ ( modulo_modulo @ A @ B2 @ C2 ) ) @ C2 ) ) ) ) ).
% div_add1_eq
thf(fact_1859_nth__rotate1,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ ( rotate1 @ A @ Xs ) @ N )
= ( nth @ A @ Xs @ ( modulo_modulo @ nat @ ( suc @ N ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) ).
% nth_rotate1
thf(fact_1860_list__eq__iff__nth__eq,axiom,
! [A: $tType] :
( ( ^ [Y3: list @ A,Z: list @ A] : Y3 = Z )
= ( ^ [Xs3: list @ A,Ys3: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs3 )
= ( size_size @ ( list @ A ) @ Ys3 ) )
& ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs3 ) )
=> ( ( nth @ A @ Xs3 @ I2 )
= ( nth @ A @ Ys3 @ I2 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_1861_Skolem__list__nth,axiom,
! [A: $tType,K: nat,P: nat > A > $o] :
( ( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ K )
=> ? [X5: A] : ( P @ I2 @ X5 ) ) )
= ( ? [Xs3: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs3 )
= K )
& ! [I2: nat] :
( ( ord_less @ nat @ I2 @ K )
=> ( P @ I2 @ ( nth @ A @ Xs3 @ I2 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_1862_nth__equalityI,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ A ) @ Ys ) )
=> ( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ Xs @ I3 )
= ( nth @ A @ Ys @ I3 ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_1863_obtain__list__from__elements,axiom,
! [A: $tType,N: nat,P: A > nat > $o] :
( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ N )
=> ? [Li: A] : ( P @ Li @ I3 ) )
=> ~ ! [L3: list @ A] :
( ( ( size_size @ ( list @ A ) @ L3 )
= N )
=> ~ ! [I4: nat] :
( ( ord_less @ nat @ I4 @ N )
=> ( P @ ( nth @ A @ L3 @ I4 ) @ I4 ) ) ) ) ).
% obtain_list_from_elements
thf(fact_1864_mod__Suc,axiom,
! [M: nat,N: nat] :
( ( ( ( suc @ ( modulo_modulo @ nat @ M @ N ) )
= N )
=> ( ( modulo_modulo @ nat @ ( suc @ M ) @ N )
= ( zero_zero @ nat ) ) )
& ( ( ( suc @ ( modulo_modulo @ nat @ M @ N ) )
!= N )
=> ( ( modulo_modulo @ nat @ ( suc @ M ) @ N )
= ( suc @ ( modulo_modulo @ nat @ M @ N ) ) ) ) ) ).
% mod_Suc
thf(fact_1865_mod__less__divisor,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less @ nat @ ( modulo_modulo @ nat @ M @ N ) @ N ) ) ).
% mod_less_divisor
thf(fact_1866_gcd__nat__induct,axiom,
! [P: nat > nat > $o,M: nat,N: nat] :
( ! [M4: nat] : ( P @ M4 @ ( zero_zero @ nat ) )
=> ( ! [M4: nat,N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( P @ N2 @ ( modulo_modulo @ nat @ M4 @ N2 ) )
=> ( P @ M4 @ N2 ) ) )
=> ( P @ M @ N ) ) ) ).
% gcd_nat_induct
thf(fact_1867_zero__le__numeral,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [N: num] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ N ) ) ) ).
% zero_le_numeral
thf(fact_1868_not__numeral__le__zero,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [N: num] :
~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ N ) @ ( zero_zero @ A ) ) ) ).
% not_numeral_le_zero
thf(fact_1869_mod__Suc__le__divisor,axiom,
! [M: nat,N: nat] : ( ord_less_eq @ nat @ ( modulo_modulo @ nat @ M @ ( suc @ N ) ) @ N ) ).
% mod_Suc_le_divisor
thf(fact_1870_zero__less__numeral,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [N: num] : ( ord_less @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ N ) ) ) ).
% zero_less_numeral
thf(fact_1871_not__numeral__less__zero,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [N: num] :
~ ( ord_less @ A @ ( numeral_numeral @ A @ N ) @ ( zero_zero @ A ) ) ) ).
% not_numeral_less_zero
thf(fact_1872_one__le__numeral,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [N: num] : ( ord_less_eq @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ N ) ) ) ).
% one_le_numeral
thf(fact_1873_nat__power__less__imp__less,axiom,
! [I: nat,M: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ I )
=> ( ( ord_less @ nat @ ( power_power @ nat @ I @ M ) @ ( power_power @ nat @ I @ N ) )
=> ( ord_less @ nat @ M @ N ) ) ) ).
% nat_power_less_imp_less
thf(fact_1874_mod__eq__0D,axiom,
! [M: nat,D2: nat] :
( ( ( modulo_modulo @ nat @ M @ D2 )
= ( zero_zero @ nat ) )
=> ? [Q4: nat] :
( M
= ( times_times @ nat @ D2 @ Q4 ) ) ) ).
% mod_eq_0D
thf(fact_1875_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_1876_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_1877_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_1878_not__numeral__less__one,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [N: num] :
~ ( ord_less @ A @ ( numeral_numeral @ A @ N ) @ ( one_one @ A ) ) ) ).
% not_numeral_less_one
thf(fact_1879_le__mod__geq,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq @ nat @ N @ M )
=> ( ( modulo_modulo @ nat @ M @ N )
= ( modulo_modulo @ nat @ ( minus_minus @ nat @ M @ N ) @ N ) ) ) ).
% le_mod_geq
thf(fact_1880_not__numeral__le__neg__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M: num,N: num] :
~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ M ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) ) ) ).
% not_numeral_le_neg_numeral
thf(fact_1881_neg__numeral__le__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M: num,N: num] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( numeral_numeral @ A @ N ) ) ) ).
% neg_numeral_le_numeral
thf(fact_1882_zero__neq__neg__numeral,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [N: num] :
( ( zero_zero @ A )
!= ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) ) ) ).
% zero_neq_neg_numeral
thf(fact_1883_one__plus__numeral__commute,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ! [X2: num] :
( ( plus_plus @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ X2 ) )
= ( plus_plus @ A @ ( numeral_numeral @ A @ X2 ) @ ( one_one @ A ) ) ) ) ).
% one_plus_numeral_commute
thf(fact_1884_zmod__eq__0D,axiom,
! [M: int,D2: int] :
( ( ( modulo_modulo @ int @ M @ D2 )
= ( zero_zero @ int ) )
=> ? [Q4: int] :
( M
= ( times_times @ int @ D2 @ Q4 ) ) ) ).
% zmod_eq_0D
thf(fact_1885_zmod__eq__0__iff,axiom,
! [M: int,D2: int] :
( ( ( modulo_modulo @ int @ M @ D2 )
= ( zero_zero @ int ) )
= ( ? [Q6: int] :
( M
= ( times_times @ int @ D2 @ Q6 ) ) ) ) ).
% zmod_eq_0_iff
thf(fact_1886_zmod__zminus2__not__zero,axiom,
! [K: int,L: int] :
( ( ( modulo_modulo @ int @ K @ ( uminus_uminus @ int @ L ) )
!= ( zero_zero @ int ) )
=> ( ( modulo_modulo @ int @ K @ L )
!= ( zero_zero @ int ) ) ) ).
% zmod_zminus2_not_zero
thf(fact_1887_zmod__zminus1__not__zero,axiom,
! [K: int,L: int] :
( ( ( modulo_modulo @ int @ ( uminus_uminus @ int @ K ) @ L )
!= ( zero_zero @ int ) )
=> ( ( modulo_modulo @ int @ K @ L )
!= ( zero_zero @ int ) ) ) ).
% zmod_zminus1_not_zero
thf(fact_1888_neg__numeral__less__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M: num,N: num] : ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( numeral_numeral @ A @ N ) ) ) ).
% neg_numeral_less_numeral
thf(fact_1889_not__numeral__less__neg__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M: num,N: num] :
~ ( ord_less @ A @ ( numeral_numeral @ A @ M ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) ) ) ).
% not_numeral_less_neg_numeral
thf(fact_1890_nat__mod__eq__iff,axiom,
! [X2: nat,N: nat,Y: nat] :
( ( ( modulo_modulo @ nat @ X2 @ N )
= ( modulo_modulo @ nat @ Y @ N ) )
= ( ? [Q1: nat,Q22: nat] :
( ( plus_plus @ nat @ X2 @ ( times_times @ nat @ N @ Q1 ) )
= ( plus_plus @ nat @ Y @ ( times_times @ nat @ N @ Q22 ) ) ) ) ) ).
% nat_mod_eq_iff
thf(fact_1891_numeral__neq__neg__one,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [N: num] :
( ( numeral_numeral @ A @ N )
!= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% numeral_neq_neg_one
thf(fact_1892_one__neq__neg__numeral,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [N: num] :
( ( one_one @ A )
!= ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) ) ) ).
% one_neq_neg_numeral
thf(fact_1893_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ A2 @ B2 )
=> ( ( modulo_modulo @ A @ A2 @ B2 )
= A2 ) ) ) ) ).
% unique_euclidean_semiring_numeral_class.mod_less
thf(fact_1894_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [B2: A,A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( modulo_modulo @ A @ A2 @ B2 ) ) ) ) ).
% unique_euclidean_semiring_numeral_class.pos_mod_sign
thf(fact_1895_div__mult1__eq,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( divide_divide @ A @ ( times_times @ A @ A2 @ B2 ) @ C2 )
= ( plus_plus @ A @ ( times_times @ A @ A2 @ ( divide_divide @ A @ B2 @ C2 ) ) @ ( divide_divide @ A @ ( times_times @ A @ A2 @ ( modulo_modulo @ A @ B2 @ C2 ) ) @ C2 ) ) ) ) ).
% div_mult1_eq
thf(fact_1896_cancel__div__mod__rules_I2_J,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ ( times_times @ A @ B2 @ ( divide_divide @ A @ A2 @ B2 ) ) @ ( modulo_modulo @ A @ A2 @ B2 ) ) @ C2 )
= ( plus_plus @ A @ A2 @ C2 ) ) ) ).
% cancel_div_mod_rules(2)
thf(fact_1897_cancel__div__mod__rules_I1_J,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ ( times_times @ A @ ( divide_divide @ A @ A2 @ B2 ) @ B2 ) @ ( modulo_modulo @ A @ A2 @ B2 ) ) @ C2 )
= ( plus_plus @ A @ A2 @ C2 ) ) ) ).
% cancel_div_mod_rules(1)
thf(fact_1898_mod__div__decomp,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [A2: A,B2: A] :
( A2
= ( plus_plus @ A @ ( times_times @ A @ ( divide_divide @ A @ A2 @ B2 ) @ B2 ) @ ( modulo_modulo @ A @ A2 @ B2 ) ) ) ) ).
% mod_div_decomp
thf(fact_1899_div__mult__mod__eq,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [A2: A,B2: A] :
( ( plus_plus @ A @ ( times_times @ A @ ( divide_divide @ A @ A2 @ B2 ) @ B2 ) @ ( modulo_modulo @ A @ A2 @ B2 ) )
= A2 ) ) ).
% div_mult_mod_eq
thf(fact_1900_mod__div__mult__eq,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [A2: A,B2: A] :
( ( plus_plus @ A @ ( modulo_modulo @ A @ A2 @ B2 ) @ ( times_times @ A @ ( divide_divide @ A @ A2 @ B2 ) @ B2 ) )
= A2 ) ) ).
% mod_div_mult_eq
thf(fact_1901_mod__mult__div__eq,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [A2: A,B2: A] :
( ( plus_plus @ A @ ( modulo_modulo @ A @ A2 @ B2 ) @ ( times_times @ A @ B2 @ ( divide_divide @ A @ A2 @ B2 ) ) )
= A2 ) ) ).
% mod_mult_div_eq
thf(fact_1902_mult__div__mod__eq,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [B2: A,A2: A] :
( ( plus_plus @ A @ ( times_times @ A @ B2 @ ( divide_divide @ A @ A2 @ B2 ) ) @ ( modulo_modulo @ A @ A2 @ B2 ) )
= A2 ) ) ).
% mult_div_mod_eq
thf(fact_1903_mod__le__divisor,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less_eq @ nat @ ( modulo_modulo @ nat @ M @ N ) @ N ) ) ).
% mod_le_divisor
thf(fact_1904_fact__mod,axiom,
! [A: $tType] :
( ( ( linordered_semidom @ A )
& ( semidom_modulo @ A ) )
=> ! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( modulo_modulo @ A @ ( semiring_char_0_fact @ A @ N ) @ ( semiring_char_0_fact @ A @ M ) )
= ( zero_zero @ A ) ) ) ) ).
% fact_mod
thf(fact_1905_neg__numeral__le__zero,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: num] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) @ ( zero_zero @ A ) ) ) ).
% neg_numeral_le_zero
thf(fact_1906_not__zero__le__neg__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: num] :
~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) ) ) ).
% not_zero_le_neg_numeral
thf(fact_1907_power__gt__expt,axiom,
! [N: nat,K: nat] :
( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N )
=> ( ord_less @ nat @ K @ ( power_power @ nat @ N @ K ) ) ) ).
% power_gt_expt
thf(fact_1908_not__zero__less__neg__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: num] :
~ ( ord_less @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) ) ) ).
% not_zero_less_neg_numeral
thf(fact_1909_neg__numeral__less__zero,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: num] : ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) @ ( zero_zero @ A ) ) ) ).
% neg_numeral_less_zero
thf(fact_1910_nat__one__le__power,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ I )
=> ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( power_power @ nat @ I @ N ) ) ) ).
% nat_one_le_power
thf(fact_1911_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_1912_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_1913_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_1914_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_1915_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_1916_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_1917_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_1918_neg__mod__conj,axiom,
! [B2: int,A2: int] :
( ( ord_less @ int @ B2 @ ( zero_zero @ int ) )
=> ( ( ord_less_eq @ int @ ( modulo_modulo @ int @ A2 @ B2 ) @ ( zero_zero @ int ) )
& ( ord_less @ int @ B2 @ ( modulo_modulo @ int @ A2 @ B2 ) ) ) ) ).
% neg_mod_conj
thf(fact_1919_pos__mod__conj,axiom,
! [B2: int,A2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( modulo_modulo @ int @ A2 @ B2 ) )
& ( ord_less @ int @ ( modulo_modulo @ int @ A2 @ B2 ) @ B2 ) ) ) ).
% pos_mod_conj
thf(fact_1920_zmod__trivial__iff,axiom,
! [I: int,K: int] :
( ( ( modulo_modulo @ int @ I @ K )
= I )
= ( ( K
= ( zero_zero @ int ) )
| ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ I )
& ( ord_less @ int @ I @ K ) )
| ( ( ord_less_eq @ int @ I @ ( zero_zero @ int ) )
& ( ord_less @ int @ K @ I ) ) ) ) ).
% zmod_trivial_iff
thf(fact_1921_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_1922_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_1923_nat__mod__eq__lemma,axiom,
! [X2: nat,N: nat,Y: nat] :
( ( ( modulo_modulo @ nat @ X2 @ N )
= ( modulo_modulo @ nat @ Y @ N ) )
=> ( ( ord_less_eq @ nat @ Y @ X2 )
=> ? [Q4: nat] :
( X2
= ( plus_plus @ nat @ Y @ ( times_times @ nat @ N @ Q4 ) ) ) ) ) ).
% nat_mod_eq_lemma
thf(fact_1924_mod__eq__nat2E,axiom,
! [M: nat,Q2: nat,N: nat] :
( ( ( modulo_modulo @ nat @ M @ Q2 )
= ( modulo_modulo @ nat @ N @ Q2 ) )
=> ( ( ord_less_eq @ nat @ M @ N )
=> ~ ! [S4: nat] :
( N
!= ( plus_plus @ nat @ M @ ( times_times @ nat @ Q2 @ S4 ) ) ) ) ) ).
% mod_eq_nat2E
thf(fact_1925_mod__eq__nat1E,axiom,
! [M: nat,Q2: nat,N: nat] :
( ( ( modulo_modulo @ nat @ M @ Q2 )
= ( modulo_modulo @ nat @ N @ Q2 ) )
=> ( ( ord_less_eq @ nat @ N @ M )
=> ~ ! [S4: nat] :
( M
!= ( plus_plus @ nat @ N @ ( times_times @ nat @ Q2 @ S4 ) ) ) ) ) ).
% mod_eq_nat1E
thf(fact_1926_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_1927_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_1928_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_1929_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_1930_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_1931_div__less__mono,axiom,
! [A5: nat,B5: nat,N: nat] :
( ( ord_less @ nat @ A5 @ B5 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ( modulo_modulo @ nat @ A5 @ N )
= ( zero_zero @ nat ) )
=> ( ( ( modulo_modulo @ nat @ B5 @ N )
= ( zero_zero @ nat ) )
=> ( ord_less @ nat @ ( divide_divide @ nat @ A5 @ N ) @ ( divide_divide @ nat @ B5 @ N ) ) ) ) ) ) ).
% div_less_mono
thf(fact_1932_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_1933_zmod__zminus1__eq__if,axiom,
! [A2: int,B2: int] :
( ( ( ( modulo_modulo @ int @ A2 @ B2 )
= ( zero_zero @ int ) )
=> ( ( modulo_modulo @ int @ ( uminus_uminus @ int @ A2 ) @ B2 )
= ( zero_zero @ int ) ) )
& ( ( ( modulo_modulo @ int @ A2 @ B2 )
!= ( zero_zero @ int ) )
=> ( ( modulo_modulo @ int @ ( uminus_uminus @ int @ A2 ) @ B2 )
= ( minus_minus @ int @ B2 @ ( modulo_modulo @ int @ A2 @ B2 ) ) ) ) ) ).
% zmod_zminus1_eq_if
thf(fact_1934_zmod__zminus2__eq__if,axiom,
! [A2: int,B2: int] :
( ( ( ( modulo_modulo @ int @ A2 @ B2 )
= ( zero_zero @ int ) )
=> ( ( modulo_modulo @ int @ A2 @ ( uminus_uminus @ int @ B2 ) )
= ( zero_zero @ int ) ) )
& ( ( ( modulo_modulo @ int @ A2 @ B2 )
!= ( zero_zero @ int ) )
=> ( ( modulo_modulo @ int @ A2 @ ( uminus_uminus @ int @ B2 ) )
= ( minus_minus @ int @ ( modulo_modulo @ int @ A2 @ B2 ) @ B2 ) ) ) ) ).
% zmod_zminus2_eq_if
thf(fact_1935_zdiv__mono__strict,axiom,
! [A5: int,B5: int,N: int] :
( ( ord_less @ int @ A5 @ B5 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ N )
=> ( ( ( modulo_modulo @ int @ A5 @ N )
= ( zero_zero @ int ) )
=> ( ( ( modulo_modulo @ int @ B5 @ N )
= ( zero_zero @ int ) )
=> ( ord_less @ int @ ( divide_divide @ int @ A5 @ N ) @ ( divide_divide @ int @ B5 @ N ) ) ) ) ) ) ).
% zdiv_mono_strict
thf(fact_1936_mod__mult2__eq,axiom,
! [M: nat,N: nat,Q2: nat] :
( ( modulo_modulo @ nat @ M @ ( times_times @ nat @ N @ Q2 ) )
= ( plus_plus @ nat @ ( times_times @ nat @ N @ ( modulo_modulo @ nat @ ( divide_divide @ nat @ M @ N ) @ Q2 ) ) @ ( modulo_modulo @ nat @ M @ N ) ) ) ).
% mod_mult2_eq
thf(fact_1937_div__mod__decomp,axiom,
! [A5: nat,N: nat] :
( A5
= ( plus_plus @ nat @ ( times_times @ nat @ ( divide_divide @ nat @ A5 @ N ) @ N ) @ ( modulo_modulo @ nat @ A5 @ N ) ) ) ).
% div_mod_decomp
thf(fact_1938_fact__le__power,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N: nat] : ( ord_less_eq @ A @ ( semiring_char_0_fact @ A @ N ) @ ( semiring_1_of_nat @ A @ ( power_power @ nat @ N @ N ) ) ) ) ).
% fact_le_power
thf(fact_1939_div__mod__decomp__int,axiom,
! [A5: int,N: int] :
( A5
= ( plus_plus @ int @ ( times_times @ int @ ( divide_divide @ int @ A5 @ N ) @ N ) @ ( modulo_modulo @ int @ A5 @ N ) ) ) ).
% div_mod_decomp_int
thf(fact_1940_fact__div__fact__le__pow,axiom,
! [R: nat,N: nat] :
( ( ord_less_eq @ nat @ R @ N )
=> ( ord_less_eq @ nat @ ( divide_divide @ nat @ ( semiring_char_0_fact @ nat @ N ) @ ( semiring_char_0_fact @ nat @ ( minus_minus @ nat @ N @ R ) ) ) @ ( power_power @ nat @ N @ R ) ) ) ).
% fact_div_fact_le_pow
thf(fact_1941_mod__mult2__eq_H,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A2: A,M: nat,N: nat] :
( ( modulo_modulo @ A @ A2 @ ( times_times @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) ) )
= ( plus_plus @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ M ) @ ( modulo_modulo @ A @ ( divide_divide @ A @ A2 @ ( semiring_1_of_nat @ A @ M ) ) @ ( semiring_1_of_nat @ A @ N ) ) ) @ ( modulo_modulo @ A @ A2 @ ( semiring_1_of_nat @ A @ M ) ) ) ) ) ).
% mod_mult2_eq'
thf(fact_1942_split__mod,axiom,
! [P: nat > $o,M: nat,N: nat] :
( ( P @ ( modulo_modulo @ nat @ M @ N ) )
= ( ( ( N
= ( zero_zero @ nat ) )
=> ( P @ M ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ! [I2: nat,J3: nat] :
( ( ord_less @ nat @ J3 @ N )
=> ( ( M
= ( plus_plus @ nat @ ( times_times @ nat @ N @ I2 ) @ J3 ) )
=> ( P @ J3 ) ) ) ) ) ) ).
% split_mod
thf(fact_1943_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_1944_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_1945_field__char__0__class_Oof__nat__div,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [M: nat,N: nat] :
( ( semiring_1_of_nat @ A @ ( divide_divide @ nat @ M @ N ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ ( modulo_modulo @ nat @ M @ N ) ) ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% field_char_0_class.of_nat_div
thf(fact_1946_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_1947_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_1948_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_1949_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_1950_array__of__list__make,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( ( array_of_list @ A )
= ( ^ [Xs3: list @ A] : ( array_make @ A @ ( size_size @ ( list @ A ) @ Xs3 ) @ ( nth @ A @ Xs3 ) ) ) ) ) ).
% array_of_list_make
thf(fact_1951_nat__power__eq,axiom,
! [Z2: int,N: nat] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z2 )
=> ( ( nat2 @ ( power_power @ int @ Z2 @ N ) )
= ( power_power @ nat @ ( nat2 @ Z2 ) @ N ) ) ) ).
% nat_power_eq
thf(fact_1952_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( modulo_modulo @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) )
= ( plus_plus @ A @ ( times_times @ A @ B2 @ ( modulo_modulo @ A @ ( divide_divide @ A @ A2 @ B2 ) @ C2 ) ) @ ( modulo_modulo @ A @ A2 @ B2 ) ) ) ) ) ).
% unique_euclidean_semiring_numeral_class.mod_mult2_eq
thf(fact_1953_fold__atLeastAtMost__nat_Osimps,axiom,
! [A: $tType] :
( ( set_fo6178422350223883121st_nat @ A )
= ( ^ [F2: nat > A > A,A3: nat,B3: nat,Acc: A] : ( if @ A @ ( ord_less @ nat @ B3 @ A3 ) @ Acc @ ( set_fo6178422350223883121st_nat @ A @ F2 @ ( plus_plus @ nat @ A3 @ ( one_one @ nat ) ) @ B3 @ ( F2 @ A3 @ Acc ) ) ) ) ) ).
% fold_atLeastAtMost_nat.simps
thf(fact_1954_fold__atLeastAtMost__nat_Oelims,axiom,
! [A: $tType,X2: nat > A > A,Xa: nat,Xb: nat,Xc: A,Y: A] :
( ( ( set_fo6178422350223883121st_nat @ A @ X2 @ Xa @ Xb @ Xc )
= Y )
=> ( ( ( ord_less @ nat @ Xb @ Xa )
=> ( Y = Xc ) )
& ( ~ ( ord_less @ nat @ Xb @ Xa )
=> ( Y
= ( set_fo6178422350223883121st_nat @ A @ X2 @ ( plus_plus @ nat @ Xa @ ( one_one @ nat ) ) @ Xb @ ( X2 @ Xa @ Xc ) ) ) ) ) ) ).
% fold_atLeastAtMost_nat.elims
thf(fact_1955_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_1956_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_1957_Suc__times__mod__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ M )
=> ( ( modulo_modulo @ nat @ ( suc @ ( times_times @ nat @ M @ N ) ) @ M )
= ( one_one @ nat ) ) ) ).
% Suc_times_mod_eq
thf(fact_1958_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_1959_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_1960_int__mod__pos__eq,axiom,
! [A2: int,B2: int,Q2: int,R: int] :
( ( A2
= ( plus_plus @ int @ ( times_times @ int @ B2 @ Q2 ) @ R ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R )
=> ( ( ord_less @ int @ R @ B2 )
=> ( ( modulo_modulo @ int @ A2 @ B2 )
= R ) ) ) ) ).
% int_mod_pos_eq
thf(fact_1961_int__mod__neg__eq,axiom,
! [A2: int,B2: int,Q2: int,R: int] :
( ( A2
= ( plus_plus @ int @ ( times_times @ int @ B2 @ Q2 ) @ R ) )
=> ( ( ord_less_eq @ int @ R @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ B2 @ R )
=> ( ( modulo_modulo @ int @ A2 @ B2 )
= R ) ) ) ) ).
% int_mod_neg_eq
thf(fact_1962_split__zmod,axiom,
! [P: int > $o,N: int,K: int] :
( ( P @ ( modulo_modulo @ int @ N @ K ) )
= ( ( ( K
= ( zero_zero @ int ) )
=> ( P @ N ) )
& ( ( ord_less @ int @ ( zero_zero @ int ) @ K )
=> ! [I2: int,J3: int] :
( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ J3 )
& ( ord_less @ int @ J3 @ K )
& ( N
= ( plus_plus @ int @ ( times_times @ int @ K @ I2 ) @ J3 ) ) )
=> ( P @ J3 ) ) )
& ( ( ord_less @ int @ K @ ( zero_zero @ int ) )
=> ! [I2: int,J3: int] :
( ( ( ord_less @ int @ K @ J3 )
& ( ord_less_eq @ int @ J3 @ ( zero_zero @ int ) )
& ( N
= ( plus_plus @ int @ ( times_times @ int @ K @ I2 ) @ J3 ) ) )
=> ( P @ J3 ) ) ) ) ) ).
% split_zmod
thf(fact_1963_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_1964_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_1965_zmod__zmult2__eq,axiom,
! [C2: int,A2: int,B2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ C2 )
=> ( ( modulo_modulo @ int @ A2 @ ( times_times @ int @ B2 @ C2 ) )
= ( plus_plus @ int @ ( times_times @ int @ B2 @ ( modulo_modulo @ int @ ( divide_divide @ int @ A2 @ B2 ) @ C2 ) ) @ ( modulo_modulo @ int @ A2 @ B2 ) ) ) ) ).
% zmod_zmult2_eq
thf(fact_1966_zdiv__zminus1__eq__if,axiom,
! [B2: int,A2: int] :
( ( B2
!= ( zero_zero @ int ) )
=> ( ( ( ( modulo_modulo @ int @ A2 @ B2 )
= ( zero_zero @ int ) )
=> ( ( divide_divide @ int @ ( uminus_uminus @ int @ A2 ) @ B2 )
= ( uminus_uminus @ int @ ( divide_divide @ int @ A2 @ B2 ) ) ) )
& ( ( ( modulo_modulo @ int @ A2 @ B2 )
!= ( zero_zero @ int ) )
=> ( ( divide_divide @ int @ ( uminus_uminus @ int @ A2 ) @ B2 )
= ( minus_minus @ int @ ( uminus_uminus @ int @ ( divide_divide @ int @ A2 @ B2 ) ) @ ( one_one @ int ) ) ) ) ) ) ).
% zdiv_zminus1_eq_if
thf(fact_1967_zdiv__zminus2__eq__if,axiom,
! [B2: int,A2: int] :
( ( B2
!= ( zero_zero @ int ) )
=> ( ( ( ( modulo_modulo @ int @ A2 @ B2 )
= ( zero_zero @ int ) )
=> ( ( divide_divide @ int @ A2 @ ( uminus_uminus @ int @ B2 ) )
= ( uminus_uminus @ int @ ( divide_divide @ int @ A2 @ B2 ) ) ) )
& ( ( ( modulo_modulo @ int @ A2 @ B2 )
!= ( zero_zero @ int ) )
=> ( ( divide_divide @ int @ A2 @ ( uminus_uminus @ int @ B2 ) )
= ( minus_minus @ int @ ( uminus_uminus @ int @ ( divide_divide @ int @ A2 @ B2 ) ) @ ( one_one @ int ) ) ) ) ) ) ).
% zdiv_zminus2_eq_if
thf(fact_1968_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_1969_sorted__in__between,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [I: nat,J: nat,L: list @ A,X2: A] :
( ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ I )
=> ( ( ord_less @ nat @ I @ J )
=> ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ L ) )
=> ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ L )
=> ( ( ord_less_eq @ A @ ( nth @ A @ L @ I ) @ X2 )
=> ( ( ord_less @ A @ X2 @ ( nth @ A @ L @ J ) )
=> ~ ! [K4: nat] :
( ( ord_less_eq @ nat @ I @ K4 )
=> ( ( ord_less @ nat @ K4 @ J )
=> ( ( ord_less_eq @ A @ ( nth @ A @ L @ K4 ) @ X2 )
=> ~ ( ord_less @ A @ X2 @ ( nth @ A @ L @ ( plus_plus @ nat @ K4 @ ( one_one @ nat ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% sorted_in_between
thf(fact_1970_diff__numeral__special_I6_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M: num] :
( ( minus_minus @ A @ ( numeral_numeral @ A @ M ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( numeral_numeral @ A @ ( inc @ M ) ) ) ) ).
% diff_numeral_special(6)
thf(fact_1971_diff__numeral__special_I5_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [N: num] :
( ( minus_minus @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ N ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( inc @ N ) ) ) ) ) ).
% diff_numeral_special(5)
thf(fact_1972_add__neg__numeral__special_I5_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [N: num] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( inc @ N ) ) ) ) ) ).
% add_neg_numeral_special(5)
thf(fact_1973_add__neg__numeral__special_I6_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M: num] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( inc @ M ) ) ) ) ) ).
% add_neg_numeral_special(6)
thf(fact_1974_nth__rotate,axiom,
! [A: $tType,N: nat,Xs: list @ A,M: nat] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ ( rotate @ A @ M @ Xs ) @ N )
= ( nth @ A @ Xs @ ( modulo_modulo @ nat @ ( plus_plus @ nat @ M @ N ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) ).
% nth_rotate
thf(fact_1975_nth__Cons__pos,axiom,
! [A: $tType,N: nat,X2: A,Xs: list @ A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( nth @ A @ ( cons @ A @ X2 @ Xs ) @ N )
= ( nth @ A @ Xs @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ).
% nth_Cons_pos
thf(fact_1976_length__rotate,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( size_size @ ( list @ A ) @ ( rotate @ A @ N @ Xs ) )
= ( size_size @ ( list @ A ) @ Xs ) ) ).
% length_rotate
thf(fact_1977_nth__Cons__0,axiom,
! [A: $tType,X2: A,Xs: list @ A] :
( ( nth @ A @ ( cons @ A @ X2 @ Xs ) @ ( zero_zero @ nat ) )
= X2 ) ).
% nth_Cons_0
thf(fact_1978_rotate__length01,axiom,
! [A: $tType,Xs: list @ A,N: nat] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) )
=> ( ( rotate @ A @ N @ Xs )
= Xs ) ) ).
% rotate_length01
thf(fact_1979_rotate__id,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( ( modulo_modulo @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
= ( zero_zero @ nat ) )
=> ( ( rotate @ A @ N @ Xs )
= Xs ) ) ).
% rotate_id
thf(fact_1980_nth__Cons__numeral,axiom,
! [A: $tType,X2: A,Xs: list @ A,V2: num] :
( ( nth @ A @ ( cons @ A @ X2 @ Xs ) @ ( numeral_numeral @ nat @ V2 ) )
= ( nth @ A @ Xs @ ( minus_minus @ nat @ ( numeral_numeral @ nat @ V2 ) @ ( one_one @ nat ) ) ) ) ).
% nth_Cons_numeral
thf(fact_1981_mult__inc,axiom,
! [X2: num,Y: num] :
( ( times_times @ num @ X2 @ ( inc @ Y ) )
= ( plus_plus @ num @ ( times_times @ num @ X2 @ Y ) @ X2 ) ) ).
% mult_inc
thf(fact_1982_sorted2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y: A,Zs: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( cons @ A @ X2 @ ( cons @ A @ Y @ Zs ) ) )
= ( ( ord_less_eq @ A @ X2 @ Y )
& ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( cons @ A @ Y @ Zs ) ) ) ) ) ).
% sorted2
thf(fact_1983_add__inc,axiom,
! [X2: num,Y: num] :
( ( plus_plus @ num @ X2 @ ( inc @ Y ) )
= ( inc @ ( plus_plus @ num @ X2 @ Y ) ) ) ).
% add_inc
thf(fact_1984_list__tail__coinc,axiom,
! [A: $tType,N1: A,R1: list @ A,N22: A,R22: list @ A] :
( ( ( cons @ A @ N1 @ R1 )
= ( cons @ A @ N22 @ R22 ) )
=> ( ( N1 = N22 )
& ( R1 = R22 ) ) ) ).
% list_tail_coinc
thf(fact_1985_sorted__wrt__true,axiom,
! [A: $tType,Xs: list @ A] :
( sorted_wrt @ A
@ ^ [Uu: A,Uv: A] : $true
@ Xs ) ).
% sorted_wrt_true
thf(fact_1986_sorted1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A] : ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( cons @ A @ X2 @ ( nil @ A ) ) ) ) ).
% sorted1
thf(fact_1987_mergesort__by__rel__merge__induct,axiom,
! [A: $tType,B: $tType,P: ( list @ A ) > ( list @ B ) > $o,R3: A > B > $o,Xs: list @ A,Ys: list @ B] :
( ! [Xs2: list @ A] : ( P @ Xs2 @ ( nil @ B ) )
=> ( ! [X_12: list @ B] : ( P @ ( nil @ A ) @ X_12 )
=> ( ! [X3: A,Xs2: list @ A,Y2: B,Ys4: list @ B] :
( ( R3 @ X3 @ Y2 )
=> ( ( P @ Xs2 @ ( cons @ B @ Y2 @ Ys4 ) )
=> ( P @ ( cons @ A @ X3 @ Xs2 ) @ ( cons @ B @ Y2 @ Ys4 ) ) ) )
=> ( ! [X3: A,Xs2: list @ A,Y2: B,Ys4: list @ B] :
( ~ ( R3 @ X3 @ Y2 )
=> ( ( P @ ( cons @ A @ X3 @ Xs2 ) @ Ys4 )
=> ( P @ ( cons @ A @ X3 @ Xs2 ) @ ( cons @ B @ Y2 @ Ys4 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% mergesort_by_rel_merge_induct
thf(fact_1988_list__induct__first2,axiom,
! [A: $tType,P: ( list @ A ) > $o,Xs: list @ A] :
( ( P @ ( nil @ A ) )
=> ( ! [X3: A] : ( P @ ( cons @ A @ X3 @ ( nil @ A ) ) )
=> ( ! [X1: A,X23: A,Xs2: list @ A] :
( ( P @ Xs2 )
=> ( P @ ( cons @ A @ X1 @ ( cons @ A @ X23 @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_induct_first2
thf(fact_1989_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 ) )
=> ( ! [E4: A,W12: list @ A,W23: list @ B] :
( ( P @ W12 @ W23 )
=> ( P @ ( cons @ A @ E4 @ W12 ) @ W23 ) )
=> ( ! [E4: B,W13: list @ A,W24: list @ B] :
( ( P @ W13 @ W24 )
=> ( P @ W13 @ ( cons @ B @ E4 @ W24 ) ) )
=> ( P @ W1 @ W22 ) ) ) ) ).
% list_2pre_induct
thf(fact_1990_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_1991_rotate__rotate,axiom,
! [A: $tType,M: nat,N: nat,Xs: list @ A] :
( ( rotate @ A @ M @ ( rotate @ A @ N @ Xs ) )
= ( rotate @ A @ ( plus_plus @ nat @ M @ N ) @ Xs ) ) ).
% rotate_rotate
thf(fact_1992_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_1993_sorted0,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( nil @ A ) ) ) ).
% sorted0
thf(fact_1994_sorted__wrt__less__idx,axiom,
! [Ns: list @ nat,I: nat] :
( ( sorted_wrt @ nat @ ( ord_less @ nat ) @ Ns )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ nat ) @ Ns ) )
=> ( ord_less_eq @ nat @ I @ ( nth @ nat @ Ns @ I ) ) ) ) ).
% sorted_wrt_less_idx
thf(fact_1995_list__induct4,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,Xs: list @ A,Ys: list @ B,Zs: list @ C,Ws: list @ D,P: ( list @ A ) > ( list @ B ) > ( list @ C ) > ( list @ D ) > $o] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( ( size_size @ ( list @ B ) @ Ys )
= ( size_size @ ( list @ C ) @ Zs ) )
=> ( ( ( size_size @ ( list @ C ) @ Zs )
= ( size_size @ ( list @ D ) @ Ws ) )
=> ( ( P @ ( nil @ A ) @ ( nil @ B ) @ ( nil @ C ) @ ( nil @ D ) )
=> ( ! [X3: A,Xs2: list @ A,Y2: B,Ys4: list @ B,Z4: C,Zs2: list @ C,W3: D,Ws2: list @ D] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ B ) @ Ys4 ) )
=> ( ( ( size_size @ ( list @ B ) @ Ys4 )
= ( size_size @ ( list @ C ) @ Zs2 ) )
=> ( ( ( size_size @ ( list @ C ) @ Zs2 )
= ( size_size @ ( list @ D ) @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys4 @ Zs2 @ Ws2 )
=> ( P @ ( cons @ A @ X3 @ Xs2 ) @ ( cons @ B @ Y2 @ Ys4 ) @ ( cons @ C @ Z4 @ Zs2 ) @ ( cons @ D @ W3 @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_1996_list__induct3,axiom,
! [B: $tType,A: $tType,C: $tType,Xs: list @ A,Ys: list @ B,Zs: list @ C,P: ( list @ A ) > ( list @ B ) > ( list @ C ) > $o] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( ( size_size @ ( list @ B ) @ Ys )
= ( size_size @ ( list @ C ) @ Zs ) )
=> ( ( P @ ( nil @ A ) @ ( nil @ B ) @ ( nil @ C ) )
=> ( ! [X3: A,Xs2: list @ A,Y2: B,Ys4: list @ B,Z4: C,Zs2: list @ C] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ B ) @ Ys4 ) )
=> ( ( ( size_size @ ( list @ B ) @ Ys4 )
= ( size_size @ ( list @ C ) @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys4 @ Zs2 )
=> ( P @ ( cons @ A @ X3 @ Xs2 ) @ ( cons @ B @ Y2 @ Ys4 ) @ ( cons @ C @ Z4 @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_1997_list__induct2,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B,P: ( list @ A ) > ( list @ B ) > $o] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( P @ ( nil @ A ) @ ( nil @ B ) )
=> ( ! [X3: A,Xs2: list @ A,Y2: B,Ys4: list @ B] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ B ) @ Ys4 ) )
=> ( ( P @ Xs2 @ Ys4 )
=> ( P @ ( cons @ A @ X3 @ Xs2 ) @ ( cons @ B @ Y2 @ Ys4 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_1998_length__Cons,axiom,
! [A: $tType,X2: A,Xs: list @ A] :
( ( size_size @ ( list @ A ) @ ( cons @ A @ X2 @ Xs ) )
= ( suc @ ( size_size @ ( list @ A ) @ Xs ) ) ) ).
% length_Cons
thf(fact_1999_Suc__length__conv,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( ( suc @ N )
= ( size_size @ ( list @ A ) @ Xs ) )
= ( ? [Y4: A,Ys3: list @ A] :
( ( Xs
= ( cons @ A @ Y4 @ Ys3 ) )
& ( ( size_size @ ( list @ A ) @ Ys3 )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_2000_length__Suc__conv,axiom,
! [A: $tType,Xs: list @ A,N: nat] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( suc @ N ) )
= ( ? [Y4: A,Ys3: list @ A] :
( ( Xs
= ( cons @ A @ Y4 @ Ys3 ) )
& ( ( size_size @ ( list @ A ) @ Ys3 )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_2001_impossible__Cons,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,X2: A] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ A ) @ Ys ) )
=> ( Xs
!= ( cons @ A @ X2 @ Ys ) ) ) ).
% impossible_Cons
thf(fact_2002_rotate__conv__mod,axiom,
! [A: $tType] :
( ( rotate @ A )
= ( ^ [N4: nat,Xs3: list @ A] : ( rotate @ A @ ( modulo_modulo @ nat @ N4 @ ( size_size @ ( list @ A ) @ Xs3 ) ) @ Xs3 ) ) ) ).
% rotate_conv_mod
thf(fact_2003_sorted__wrt__nth__less,axiom,
! [A: $tType,P: A > A > $o,Xs: list @ A,I: nat,J: nat] :
( ( sorted_wrt @ A @ P @ Xs )
=> ( ( ord_less @ nat @ I @ J )
=> ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( P @ ( nth @ A @ Xs @ I ) @ ( nth @ A @ Xs @ J ) ) ) ) ) ).
% sorted_wrt_nth_less
thf(fact_2004_sorted__wrt__iff__nth__less,axiom,
! [A: $tType] :
( ( sorted_wrt @ A )
= ( ^ [P4: A > A > $o,Xs3: list @ A] :
! [I2: nat,J3: nat] :
( ( ord_less @ nat @ I2 @ J3 )
=> ( ( ord_less @ nat @ J3 @ ( size_size @ ( list @ A ) @ Xs3 ) )
=> ( P4 @ ( nth @ A @ Xs3 @ I2 ) @ ( nth @ A @ Xs3 @ J3 ) ) ) ) ) ) ).
% sorted_wrt_iff_nth_less
thf(fact_2005_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_2006_length__compl__induct,axiom,
! [A: $tType,P: ( list @ A ) > $o,L: list @ A] :
( ( P @ ( nil @ A ) )
=> ( ! [E4: A,L3: list @ A] :
( ! [Ll: list @ A] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Ll ) @ ( size_size @ ( list @ A ) @ L3 ) )
=> ( P @ Ll ) )
=> ( P @ ( cons @ A @ E4 @ L3 ) ) )
=> ( P @ L ) ) ) ).
% length_compl_induct
thf(fact_2007_Suc__le__length__iff,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( ord_less_eq @ nat @ ( suc @ N ) @ ( size_size @ ( list @ A ) @ Xs ) )
= ( ? [X: A,Ys3: list @ A] :
( ( Xs
= ( cons @ A @ X @ Ys3 ) )
& ( ord_less_eq @ nat @ N @ ( size_size @ ( list @ A ) @ Ys3 ) ) ) ) ) ).
% Suc_le_length_iff
thf(fact_2008_list__decomp__1,axiom,
! [A: $tType,L: list @ A] :
( ( ( size_size @ ( list @ A ) @ L )
= ( one_one @ nat ) )
=> ? [A4: A] :
( L
= ( cons @ A @ A4 @ ( nil @ A ) ) ) ) ).
% list_decomp_1
thf(fact_2009_sorted__iff__nth__mono__less,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
= ( ! [I2: nat,J3: nat] :
( ( ord_less @ nat @ I2 @ J3 )
=> ( ( ord_less @ nat @ J3 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ord_less_eq @ A @ ( nth @ A @ Xs @ I2 ) @ ( nth @ A @ Xs @ J3 ) ) ) ) ) ) ) ).
% sorted_iff_nth_mono_less
thf(fact_2010_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_2011_numeral__inc,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ! [X2: num] :
( ( numeral_numeral @ A @ ( inc @ X2 ) )
= ( plus_plus @ A @ ( numeral_numeral @ A @ X2 ) @ ( one_one @ A ) ) ) ) ).
% numeral_inc
thf(fact_2012_list_Osize_I4_J,axiom,
! [A: $tType,X21: A,X222: list @ A] :
( ( size_size @ ( list @ A ) @ ( cons @ A @ X21 @ X222 ) )
= ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ X222 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% list.size(4)
thf(fact_2013_nth__Cons_H,axiom,
! [A: $tType,N: nat,X2: A,Xs: list @ A] :
( ( ( N
= ( zero_zero @ nat ) )
=> ( ( nth @ A @ ( cons @ A @ X2 @ Xs ) @ N )
= X2 ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ( ( nth @ A @ ( cons @ A @ X2 @ Xs ) @ N )
= ( nth @ A @ Xs @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ) ).
% nth_Cons'
thf(fact_2014_sorted__iff__nth__Suc,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
= ( ! [I2: nat] :
( ( ord_less @ nat @ ( suc @ I2 ) @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ord_less_eq @ A @ ( nth @ A @ Xs @ I2 ) @ ( nth @ A @ Xs @ ( suc @ I2 ) ) ) ) ) ) ) ).
% sorted_iff_nth_Suc
thf(fact_2015_sorted__iff__nth__mono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
= ( ! [I2: nat,J3: nat] :
( ( ord_less_eq @ nat @ I2 @ J3 )
=> ( ( ord_less @ nat @ J3 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ord_less_eq @ A @ ( nth @ A @ Xs @ I2 ) @ ( nth @ A @ Xs @ J3 ) ) ) ) ) ) ) ).
% sorted_iff_nth_mono
thf(fact_2016_sorted__nth__mono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,I: nat,J: nat] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( ( ord_less_eq @ nat @ I @ J )
=> ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ord_less_eq @ A @ ( nth @ A @ Xs @ I ) @ ( nth @ A @ Xs @ J ) ) ) ) ) ) ).
% sorted_nth_mono
thf(fact_2017_nth__non__equal__first__eq,axiom,
! [A: $tType,X2: A,Y: A,Xs: list @ A,N: nat] :
( ( X2 != Y )
=> ( ( ( nth @ A @ ( cons @ A @ X2 @ Xs ) @ N )
= Y )
= ( ( ( nth @ A @ Xs @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) )
= Y )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% nth_non_equal_first_eq
thf(fact_2018_slice__Cons,axiom,
! [A: $tType,Begin: nat,End: nat,X2: A,Xs: list @ A] :
( ( ( ( Begin
= ( zero_zero @ nat ) )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ End ) )
=> ( ( slice @ A @ Begin @ End @ ( cons @ A @ X2 @ Xs ) )
= ( cons @ A @ X2 @ ( 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 @ X2 @ Xs ) )
= ( slice @ A @ ( minus_minus @ nat @ Begin @ ( one_one @ nat ) ) @ ( minus_minus @ nat @ End @ ( one_one @ nat ) ) @ Xs ) ) ) ) ).
% slice_Cons
thf(fact_2019_upto__aux__rec,axiom,
( upto_aux
= ( ^ [I2: int,J3: int,Js: list @ int] : ( if @ ( list @ int ) @ ( ord_less @ int @ J3 @ I2 ) @ Js @ ( upto_aux @ I2 @ ( minus_minus @ int @ J3 @ ( one_one @ int ) ) @ ( cons @ int @ J3 @ Js ) ) ) ) ) ).
% upto_aux_rec
thf(fact_2020_round__neg__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [N: num] :
( ( archimedean_round @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) ) ) ).
% round_neg_numeral
thf(fact_2021_shift__def,axiom,
! [B: $tType,A: $tType] :
( ( bNF_Greatest_shift @ A @ B )
= ( ^ [Lab: ( list @ A ) > B,K2: A,Kl: list @ A] : ( Lab @ ( cons @ A @ K2 @ Kl ) ) ) ) ).
% shift_def
thf(fact_2022_upto__rec__numeral_I4_J,axiom,
! [M: num,N: num] :
( ( ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
=> ( ( upto @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
= ( 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 @ N ) ) ) ) ) )
& ( ~ ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
=> ( ( upto @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
= ( nil @ int ) ) ) ) ).
% upto_rec_numeral(4)
thf(fact_2023_upto__rec__numeral_I3_J,axiom,
! [M: num,N: num] :
( ( ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( numeral_numeral @ int @ N ) )
=> ( ( upto @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( numeral_numeral @ int @ N ) )
= ( 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 @ N ) ) ) ) )
& ( ~ ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( numeral_numeral @ int @ N ) )
=> ( ( upto @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( numeral_numeral @ int @ N ) )
= ( nil @ int ) ) ) ) ).
% upto_rec_numeral(3)
thf(fact_2024_upto__rec__numeral_I2_J,axiom,
! [M: num,N: num] :
( ( ( ord_less_eq @ int @ ( numeral_numeral @ int @ M ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
=> ( ( upto @ ( numeral_numeral @ int @ M ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
= ( cons @ int @ ( numeral_numeral @ int @ M ) @ ( upto @ ( plus_plus @ int @ ( numeral_numeral @ int @ M ) @ ( one_one @ int ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) ) ) ) )
& ( ~ ( ord_less_eq @ int @ ( numeral_numeral @ int @ M ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
=> ( ( upto @ ( numeral_numeral @ int @ M ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
= ( nil @ int ) ) ) ) ).
% upto_rec_numeral(2)
thf(fact_2025_diff__numeral__special_I4_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M: num] :
( ( minus_minus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( one_one @ A ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ M @ one2 ) ) ) ) ) ).
% diff_numeral_special(4)
thf(fact_2026_round__of__int,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [N: int] :
( ( archimedean_round @ A @ ( ring_1_of_int @ A @ N ) )
= N ) ) ).
% round_of_int
thf(fact_2027_one__eq__numeral__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [N: num] :
( ( ( one_one @ A )
= ( numeral_numeral @ A @ N ) )
= ( one2 = N ) ) ) ).
% one_eq_numeral_iff
thf(fact_2028_numeral__eq__one__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [N: num] :
( ( ( numeral_numeral @ A @ N )
= ( one_one @ A ) )
= ( N = one2 ) ) ) ).
% numeral_eq_one_iff
thf(fact_2029_round__0,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archimedean_round @ A @ ( zero_zero @ A ) )
= ( zero_zero @ int ) ) ) ).
% round_0
thf(fact_2030_round__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [N: num] :
( ( archimedean_round @ A @ ( numeral_numeral @ A @ N ) )
= ( numeral_numeral @ int @ N ) ) ) ).
% round_numeral
thf(fact_2031_round__1,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archimedean_round @ A @ ( one_one @ A ) )
= ( one_one @ int ) ) ) ).
% round_1
thf(fact_2032_round__of__nat,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [N: nat] :
( ( archimedean_round @ A @ ( semiring_1_of_nat @ A @ N ) )
= ( semiring_1_of_nat @ int @ N ) ) ) ).
% round_of_nat
thf(fact_2033_neg__one__eq__numeral__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [N: num] :
( ( ( uminus_uminus @ A @ ( one_one @ A ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( N = one2 ) ) ) ).
% neg_one_eq_numeral_iff
thf(fact_2034_numeral__eq__neg__one__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [N: num] :
( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) )
= ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( N = one2 ) ) ) ).
% numeral_eq_neg_one_iff
thf(fact_2035_Suc__numeral,axiom,
! [N: num] :
( ( suc @ ( numeral_numeral @ nat @ N ) )
= ( numeral_numeral @ nat @ ( plus_plus @ num @ N @ one2 ) ) ) ).
% Suc_numeral
thf(fact_2036_nth__upto,axiom,
! [I: int,K: nat,J: int] :
( ( ord_less_eq @ int @ ( plus_plus @ int @ I @ ( semiring_1_of_nat @ int @ K ) ) @ J )
=> ( ( nth @ int @ ( upto @ I @ J ) @ K )
= ( plus_plus @ int @ I @ ( semiring_1_of_nat @ int @ K ) ) ) ) ).
% nth_upto
thf(fact_2037_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_2038_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_2039_one__plus__numeral,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ! [N: num] :
( ( plus_plus @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ N ) )
= ( numeral_numeral @ A @ ( plus_plus @ num @ one2 @ N ) ) ) ) ).
% one_plus_numeral
thf(fact_2040_numeral__plus__one,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ! [N: num] :
( ( plus_plus @ A @ ( numeral_numeral @ A @ N ) @ ( one_one @ A ) )
= ( numeral_numeral @ A @ ( plus_plus @ num @ N @ one2 ) ) ) ) ).
% numeral_plus_one
thf(fact_2041_numeral__le__one__iff,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [N: num] :
( ( ord_less_eq @ A @ ( numeral_numeral @ A @ N ) @ ( one_one @ A ) )
= ( ord_less_eq @ num @ N @ one2 ) ) ) ).
% numeral_le_one_iff
thf(fact_2042_one__less__numeral__iff,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [N: num] :
( ( ord_less @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ N ) )
= ( ord_less @ num @ one2 @ N ) ) ) ).
% one_less_numeral_iff
thf(fact_2043_length__upto,axiom,
! [I: int,J: int] :
( ( size_size @ ( list @ int ) @ ( upto @ I @ J ) )
= ( nat2 @ ( plus_plus @ int @ ( minus_minus @ int @ J @ I ) @ ( one_one @ int ) ) ) ) ).
% length_upto
thf(fact_2044_diff__numeral__special_I3_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [N: num] :
( ( minus_minus @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( numeral_numeral @ A @ ( plus_plus @ num @ one2 @ N ) ) ) ) ).
% diff_numeral_special(3)
thf(fact_2045_upto__rec__numeral_I1_J,axiom,
! [M: num,N: num] :
( ( ( ord_less_eq @ int @ ( numeral_numeral @ int @ M ) @ ( numeral_numeral @ int @ N ) )
=> ( ( upto @ ( numeral_numeral @ int @ M ) @ ( numeral_numeral @ int @ N ) )
= ( cons @ int @ ( numeral_numeral @ int @ M ) @ ( upto @ ( plus_plus @ int @ ( numeral_numeral @ int @ M ) @ ( one_one @ int ) ) @ ( numeral_numeral @ int @ N ) ) ) ) )
& ( ~ ( ord_less_eq @ int @ ( numeral_numeral @ int @ M ) @ ( numeral_numeral @ int @ N ) )
=> ( ( upto @ ( numeral_numeral @ int @ M ) @ ( numeral_numeral @ int @ N ) )
= ( nil @ int ) ) ) ) ).
% upto_rec_numeral(1)
thf(fact_2046_num_Osize_I4_J,axiom,
( ( size_size @ num @ one2 )
= ( zero_zero @ nat ) ) ).
% num.size(4)
thf(fact_2047_sorted__upto,axiom,
! [M: int,N: int] : ( sorted_wrt @ int @ ( ord_less_eq @ int ) @ ( upto @ M @ N ) ) ).
% sorted_upto
thf(fact_2048_add__One__commute,axiom,
! [N: num] :
( ( plus_plus @ num @ one2 @ N )
= ( plus_plus @ num @ N @ one2 ) ) ).
% add_One_commute
thf(fact_2049_num__induct,axiom,
! [P: num > $o,X2: num] :
( ( P @ one2 )
=> ( ! [X3: num] :
( ( P @ X3 )
=> ( P @ ( inc @ X3 ) ) )
=> ( P @ X2 ) ) ) ).
% num_induct
thf(fact_2050_le__num__One__iff,axiom,
! [X2: num] :
( ( ord_less_eq @ num @ X2 @ one2 )
= ( X2 = one2 ) ) ).
% le_num_One_iff
thf(fact_2051_mult__numeral__1,axiom,
! [A: $tType] :
( ( semiring_numeral @ A )
=> ! [A2: A] :
( ( times_times @ A @ ( numeral_numeral @ A @ one2 ) @ A2 )
= A2 ) ) ).
% mult_numeral_1
thf(fact_2052_mult__numeral__1__right,axiom,
! [A: $tType] :
( ( semiring_numeral @ A )
=> ! [A2: A] :
( ( times_times @ A @ A2 @ ( numeral_numeral @ A @ one2 ) )
= A2 ) ) ).
% mult_numeral_1_right
thf(fact_2053_numeral__One,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ( ( numeral_numeral @ A @ one2 )
= ( one_one @ A ) ) ) ).
% numeral_One
thf(fact_2054_divide__numeral__1,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A2: A] :
( ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ one2 ) )
= A2 ) ) ).
% divide_numeral_1
thf(fact_2055_numerals_I1_J,axiom,
( ( numeral_numeral @ nat @ one2 )
= ( one_one @ nat ) ) ).
% numerals(1)
thf(fact_2056_add__One,axiom,
! [X2: num] :
( ( plus_plus @ num @ X2 @ one2 )
= ( inc @ X2 ) ) ).
% add_One
thf(fact_2057_round__mono,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A,Y: A] :
( ( ord_less_eq @ A @ X2 @ Y )
=> ( ord_less_eq @ int @ ( archimedean_round @ A @ X2 ) @ ( archimedean_round @ A @ Y ) ) ) ) ).
% round_mono
thf(fact_2058_floor__le__round,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A] : ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X2 ) @ ( archimedean_round @ A @ X2 ) ) ) ).
% floor_le_round
thf(fact_2059_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_2060_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_2061_uminus__numeral__One,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ one2 ) )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% uminus_numeral_One
thf(fact_2062_ceiling__ge__round,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A] : ( ord_less_eq @ int @ ( archimedean_round @ A @ X2 ) @ ( archimedean_ceiling @ A @ X2 ) ) ) ).
% ceiling_ge_round
thf(fact_2063_cong__exp__iff__simps_I1_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [N: num] :
( ( modulo_modulo @ A @ ( numeral_numeral @ A @ N ) @ ( numeral_numeral @ A @ one2 ) )
= ( zero_zero @ A ) ) ) ).
% cong_exp_iff_simps(1)
thf(fact_2064_numeral__1__eq__Suc__0,axiom,
( ( numeral_numeral @ nat @ one2 )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% numeral_1_eq_Suc_0
thf(fact_2065_upto__rec1,axiom,
! [I: int,J: int] :
( ( ord_less_eq @ int @ I @ J )
=> ( ( upto @ I @ J )
= ( cons @ int @ I @ ( upto @ ( plus_plus @ int @ I @ ( one_one @ int ) ) @ J ) ) ) ) ).
% upto_rec1
thf(fact_2066_upto_Osimps,axiom,
( upto
= ( ^ [I2: int,J3: int] : ( if @ ( list @ int ) @ ( ord_less_eq @ int @ I2 @ J3 ) @ ( cons @ int @ I2 @ ( upto @ ( plus_plus @ int @ I2 @ ( one_one @ int ) ) @ J3 ) ) @ ( nil @ int ) ) ) ) ).
% upto.simps
thf(fact_2067_upto_Oelims,axiom,
! [X2: int,Xa: int,Y: list @ int] :
( ( ( upto @ X2 @ Xa )
= Y )
=> ( ( ( ord_less_eq @ int @ X2 @ Xa )
=> ( Y
= ( cons @ int @ X2 @ ( upto @ ( plus_plus @ int @ X2 @ ( one_one @ int ) ) @ Xa ) ) ) )
& ( ~ ( ord_less_eq @ int @ X2 @ Xa )
=> ( Y
= ( nil @ int ) ) ) ) ) ).
% upto.elims
thf(fact_2068_Suc__nat__number__of__add,axiom,
! [V2: num,N: nat] :
( ( suc @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ V2 ) @ N ) )
= ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( plus_plus @ num @ V2 @ one2 ) ) @ N ) ) ).
% Suc_nat_number_of_add
thf(fact_2069_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_2070_pochhammer__double,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [Z2: A,N: nat] :
( ( comm_s3205402744901411588hammer @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Z2 ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
= ( times_times @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) @ ( comm_s3205402744901411588hammer @ A @ Z2 @ N ) ) @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ Z2 @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ N ) ) ) ) ).
% pochhammer_double
thf(fact_2071_fact__double,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [N: nat] :
( ( semiring_char_0_fact @ A @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
= ( times_times @ A @ ( times_times @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) @ ( comm_s3205402744901411588hammer @ A @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ N ) ) @ ( semiring_char_0_fact @ A @ N ) ) ) ) ).
% fact_double
thf(fact_2072_round__unique,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A,Y: int] :
( ( ord_less @ A @ ( minus_minus @ A @ X2 @ ( 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 @ X2 @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) )
=> ( ( archimedean_round @ A @ X2 )
= Y ) ) ) ) ).
% round_unique
thf(fact_2073_divmod__digit__1_I1_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ ( modulo_modulo @ A @ A2 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) )
=> ( ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A2 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) ) @ ( one_one @ A ) )
= ( divide_divide @ A @ A2 @ B2 ) ) ) ) ) ) ).
% divmod_digit_1(1)
thf(fact_2074_round__altdef,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archimedean_round @ A )
= ( ^ [X: A] : ( if @ int @ ( ord_less_eq @ A @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( archimedean_frac @ A @ X ) ) @ ( archimedean_ceiling @ A @ X ) @ ( archim6421214686448440834_floor @ A @ X ) ) ) ) ) ).
% round_altdef
thf(fact_2075_neg__zmod__mult__2,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq @ int @ A2 @ ( 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 ) ) @ A2 ) )
= ( minus_minus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( modulo_modulo @ int @ ( plus_plus @ int @ B2 @ ( one_one @ int ) ) @ A2 ) ) @ ( one_one @ int ) ) ) ) ).
% neg_zmod_mult_2
thf(fact_2076_round__unique_H,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A,N: int] :
( ( ord_less @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X2 @ ( ring_1_of_int @ A @ N ) ) ) @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
=> ( ( archimedean_round @ A @ X2 )
= N ) ) ) ).
% round_unique'
thf(fact_2077_verit__eq__simplify_I8_J,axiom,
! [X22: num,Y22: num] :
( ( ( bit0 @ X22 )
= ( bit0 @ Y22 ) )
= ( X22 = Y22 ) ) ).
% verit_eq_simplify(8)
thf(fact_2078_semiring__norm_I6_J,axiom,
! [M: num,N: num] :
( ( plus_plus @ num @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( bit0 @ ( plus_plus @ num @ M @ N ) ) ) ).
% semiring_norm(6)
thf(fact_2079_semiring__norm_I2_J,axiom,
( ( plus_plus @ num @ one2 @ one2 )
= ( bit0 @ one2 ) ) ).
% semiring_norm(2)
thf(fact_2080_num__double,axiom,
! [N: num] :
( ( times_times @ num @ ( bit0 @ one2 ) @ N )
= ( bit0 @ N ) ) ).
% num_double
thf(fact_2081_dbl__simps_I5_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [K: num] :
( ( neg_numeral_dbl @ A @ ( numeral_numeral @ A @ K ) )
= ( numeral_numeral @ A @ ( bit0 @ K ) ) ) ) ).
% dbl_simps(5)
thf(fact_2082_one__add__one,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ( ( plus_plus @ A @ ( one_one @ A ) @ ( one_one @ A ) )
= ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ).
% one_add_one
thf(fact_2083_one__mod__two__eq__one,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ( ( modulo_modulo @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( one_one @ A ) ) ) ).
% one_mod_two_eq_one
thf(fact_2084_bits__one__mod__two__eq__one,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ( ( modulo_modulo @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( one_one @ A ) ) ) ).
% bits_one_mod_two_eq_one
thf(fact_2085_zero__eq__power2,axiom,
! [A: $tType] :
( ( semiri2026040879449505780visors @ A )
=> ! [A2: A] :
( ( ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( zero_zero @ A ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% zero_eq_power2
thf(fact_2086_add__2__eq__Suc_H,axiom,
! [N: nat] :
( ( plus_plus @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( suc @ ( suc @ N ) ) ) ).
% add_2_eq_Suc'
thf(fact_2087_add__2__eq__Suc,axiom,
! [N: nat] :
( ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
= ( suc @ ( suc @ N ) ) ) ).
% add_2_eq_Suc
thf(fact_2088_Suc__1,axiom,
( ( suc @ ( one_one @ nat ) )
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ).
% Suc_1
thf(fact_2089_add__self__div__2,axiom,
! [M: nat] :
( ( divide_divide @ nat @ ( plus_plus @ nat @ M @ M ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= M ) ).
% add_self_div_2
thf(fact_2090_dbl__simps_I3_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( neg_numeral_dbl @ A @ ( one_one @ A ) )
= ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ).
% dbl_simps(3)
thf(fact_2091_one__div__two__eq__zero,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ( ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( zero_zero @ A ) ) ) ).
% one_div_two_eq_zero
thf(fact_2092_bits__1__div__2,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ( ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( zero_zero @ A ) ) ) ).
% bits_1_div_2
thf(fact_2093_add__neg__numeral__special_I9_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% add_neg_numeral_special(9)
thf(fact_2094_diff__numeral__special_I10_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( minus_minus @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( one_one @ A ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% diff_numeral_special(10)
thf(fact_2095_diff__numeral__special_I11_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( minus_minus @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ).
% diff_numeral_special(11)
thf(fact_2096_not__mod__2__eq__0__eq__1,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A2: A] :
( ( ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
!= ( zero_zero @ A ) )
= ( ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( one_one @ A ) ) ) ) ).
% not_mod_2_eq_0_eq_1
thf(fact_2097_not__mod__2__eq__1__eq__0,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A2: A] :
( ( ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
!= ( one_one @ A ) )
= ( ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( zero_zero @ A ) ) ) ) ).
% not_mod_2_eq_1_eq_0
thf(fact_2098_minus__1__div__2__eq,axiom,
! [A: $tType] :
( ( euclid8789492081693882211th_nat @ A )
=> ( ( divide_divide @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% minus_1_div_2_eq
thf(fact_2099_power2__less__eq__zero__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( zero_zero @ A ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% power2_less_eq_zero_iff
thf(fact_2100_power2__eq__iff__nonneg,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X2: A,Y: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ( power_power @ A @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( X2 = Y ) ) ) ) ) ).
% power2_eq_iff_nonneg
thf(fact_2101_zero__less__power2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( A2
!= ( zero_zero @ A ) ) ) ) ).
% zero_less_power2
thf(fact_2102_minus__1__mod__2__eq,axiom,
! [A: $tType] :
( ( euclid8789492081693882211th_nat @ A )
=> ( ( modulo_modulo @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( one_one @ A ) ) ) ).
% minus_1_mod_2_eq
thf(fact_2103_bits__minus__1__mod__2__eq,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( modulo_modulo @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( one_one @ A ) ) ) ).
% bits_minus_1_mod_2_eq
thf(fact_2104_not__mod2__eq__Suc__0__eq__0,axiom,
! [N: nat] :
( ( ( modulo_modulo @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
!= ( suc @ ( zero_zero @ nat ) ) )
= ( ( modulo_modulo @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( zero_zero @ nat ) ) ) ).
% not_mod2_eq_Suc_0_eq_0
thf(fact_2105_add__self__mod__2,axiom,
! [M: nat] :
( ( modulo_modulo @ nat @ ( plus_plus @ nat @ M @ M ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( zero_zero @ nat ) ) ).
% add_self_mod_2
thf(fact_2106_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_2107_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_2108_dbl__simps_I4_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( neg_numeral_dbl @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% dbl_simps(4)
thf(fact_2109_power__minus1__even,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [N: nat] :
( ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
= ( one_one @ A ) ) ) ).
% power_minus1_even
thf(fact_2110_one__less__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A] :
( ( ord_less @ int @ ( one_one @ int ) @ ( archim6421214686448440834_floor @ A @ X2 ) )
= ( ord_less_eq @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X2 ) ) ) ).
% one_less_floor
thf(fact_2111_floor__le__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A] :
( ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X2 ) @ ( one_one @ int ) )
= ( ord_less @ A @ X2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% floor_le_one
thf(fact_2112_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_2113_verit__eq__simplify_I10_J,axiom,
! [X22: num] :
( one2
!= ( bit0 @ X22 ) ) ).
% verit_eq_simplify(10)
thf(fact_2114_numeral__Bit0,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ! [N: num] :
( ( numeral_numeral @ A @ ( bit0 @ N ) )
= ( plus_plus @ A @ ( numeral_numeral @ A @ N ) @ ( numeral_numeral @ A @ N ) ) ) ) ).
% numeral_Bit0
thf(fact_2115_inc_Osimps_I1_J,axiom,
( ( inc @ one2 )
= ( bit0 @ one2 ) ) ).
% inc.simps(1)
thf(fact_2116_numeral__code_I2_J,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ! [N: num] :
( ( numeral_numeral @ A @ ( bit0 @ N ) )
= ( plus_plus @ A @ ( numeral_numeral @ A @ N ) @ ( numeral_numeral @ A @ N ) ) ) ) ).
% numeral_code(2)
thf(fact_2117_cong__exp__iff__simps_I2_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [N: num,Q2: num] :
( ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit0 @ N ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q2 ) ) )
= ( zero_zero @ A ) )
= ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ N ) @ ( numeral_numeral @ A @ Q2 ) )
= ( zero_zero @ A ) ) ) ) ).
% cong_exp_iff_simps(2)
thf(fact_2118_num_Osize_I5_J,axiom,
! [X22: num] :
( ( size_size @ num @ ( bit0 @ X22 ) )
= ( plus_plus @ nat @ ( size_size @ num @ X22 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% num.size(5)
thf(fact_2119_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_2120_left__add__twice,axiom,
! [A: $tType] :
( ( semiring_numeral @ A )
=> ! [A2: A,B2: A] :
( ( plus_plus @ A @ A2 @ ( plus_plus @ A @ A2 @ B2 ) )
= ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) @ B2 ) ) ) ).
% left_add_twice
thf(fact_2121_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_2122_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_2123_zero__power2,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( power_power @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( zero_zero @ A ) ) ) ).
% zero_power2
thf(fact_2124_numeral__2__eq__2,axiom,
( ( numeral_numeral @ nat @ ( bit0 @ one2 ) )
= ( suc @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% numeral_2_eq_2
thf(fact_2125_one__power2,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( power_power @ A @ ( one_one @ A ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( one_one @ A ) ) ) ).
% one_power2
thf(fact_2126_nat__1__add__1,axiom,
( ( plus_plus @ nat @ ( one_one @ nat ) @ ( one_one @ nat ) )
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ).
% nat_1_add_1
thf(fact_2127_power2__nat__le__imp__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ ( power_power @ nat @ M @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ N )
=> ( ord_less_eq @ nat @ M @ N ) ) ).
% power2_nat_le_imp_le
thf(fact_2128_power2__nat__le__eq__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ ( power_power @ nat @ M @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( ord_less_eq @ nat @ M @ N ) ) ).
% power2_nat_le_eq_le
thf(fact_2129_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_2130_half__gt__zero,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% half_gt_zero
thf(fact_2131_half__gt__zero__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% half_gt_zero_iff
thf(fact_2132_zero__le__power2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% zero_le_power2
thf(fact_2133_power2__eq__imp__eq,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X2: A,Y: A] :
( ( ( power_power @ A @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( X2 = Y ) ) ) ) ) ).
% power2_eq_imp_eq
thf(fact_2134_power2__le__imp__le,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X2: A,Y: A] :
( ( ord_less_eq @ A @ ( power_power @ A @ X2 @ ( 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 @ X2 @ Y ) ) ) ) ).
% power2_le_imp_le
thf(fact_2135_power2__less__0,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A] :
~ ( ord_less @ A @ ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( zero_zero @ A ) ) ) ).
% power2_less_0
thf(fact_2136_exp__add__not__zero__imp__right,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [M: nat,N: nat] :
( ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ M @ N ) )
!= ( zero_zero @ A ) )
=> ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N )
!= ( zero_zero @ A ) ) ) ) ).
% exp_add_not_zero_imp_right
thf(fact_2137_exp__add__not__zero__imp__left,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [M: nat,N: nat] :
( ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ M @ N ) )
!= ( zero_zero @ A ) )
=> ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M )
!= ( zero_zero @ A ) ) ) ) ).
% exp_add_not_zero_imp_left
thf(fact_2138_sum__power2__eq__zero__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X2: A,Y: A] :
( ( ( plus_plus @ A @ ( power_power @ A @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( zero_zero @ A ) )
= ( ( X2
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ).
% sum_power2_eq_zero_iff
thf(fact_2139_exp__not__zero__imp__exp__diff__not__zero,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [N: nat,M: nat] :
( ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N )
!= ( zero_zero @ A ) )
=> ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N @ M ) )
!= ( zero_zero @ A ) ) ) ) ).
% exp_not_zero_imp_exp_diff_not_zero
thf(fact_2140_list__decomp__2,axiom,
! [A: $tType,L: list @ A] :
( ( ( size_size @ ( list @ A ) @ L )
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
=> ? [A4: A,B4: A] :
( L
= ( cons @ A @ A4 @ ( cons @ A @ B4 @ ( nil @ A ) ) ) ) ) ).
% list_decomp_2
thf(fact_2141_div__exp__eq,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A2: A,M: nat,N: nat] :
( ( divide_divide @ A @ ( divide_divide @ A @ A2 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
= ( divide_divide @ A @ A2 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ M @ N ) ) ) ) ) ).
% div_exp_eq
thf(fact_2142_power2__eq__1__iff,axiom,
! [A: $tType] :
( ( ring_15535105094025558882visors @ A )
=> ! [A2: A] :
( ( ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( one_one @ A ) )
= ( ( A2
= ( one_one @ A ) )
| ( A2
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ) ).
% power2_eq_1_iff
thf(fact_2143_less__2__cases__iff,axiom,
! [N: nat] :
( ( ord_less @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( ( N
= ( zero_zero @ nat ) )
| ( N
= ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% less_2_cases_iff
thf(fact_2144_less__2__cases,axiom,
! [N: nat] :
( ( ord_less @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
=> ( ( N
= ( zero_zero @ nat ) )
| ( N
= ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% less_2_cases
thf(fact_2145_abs__le__square__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X2: A,Y: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ X2 ) @ ( abs_abs @ A @ Y ) )
= ( ord_less_eq @ A @ ( power_power @ A @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% abs_le_square_iff
thf(fact_2146_abs__square__eq__1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X2: A] :
( ( ( power_power @ A @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( one_one @ A ) )
= ( ( abs_abs @ A @ X2 )
= ( one_one @ A ) ) ) ) ).
% abs_square_eq_1
thf(fact_2147_nat__induct2,axiom,
! [P: nat > $o,N: nat] :
( ( P @ ( zero_zero @ nat ) )
=> ( ( P @ ( one_one @ nat ) )
=> ( ! [N2: nat] :
( ( P @ N2 )
=> ( P @ ( plus_plus @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct2
thf(fact_2148_nat__2,axiom,
( ( nat2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) )
= ( suc @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% nat_2
thf(fact_2149_diff__le__diff__pow,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K )
=> ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ N ) @ ( minus_minus @ nat @ ( power_power @ nat @ K @ M ) @ ( power_power @ nat @ K @ N ) ) ) ) ).
% diff_le_diff_pow
thf(fact_2150_not__exp__less__eq__0__int,axiom,
! [N: nat] :
~ ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) @ ( zero_zero @ int ) ) ).
% not_exp_less_eq_0_int
thf(fact_2151_divmod__digit__0_I2_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [B2: A,A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less @ A @ ( modulo_modulo @ A @ A2 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) @ B2 )
=> ( ( modulo_modulo @ A @ A2 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) )
= ( modulo_modulo @ A @ A2 @ B2 ) ) ) ) ) ).
% divmod_digit_0(2)
thf(fact_2152_power2__less__imp__less,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X2: A,Y: A] :
( ( ord_less @ A @ ( power_power @ A @ X2 @ ( 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 @ X2 @ Y ) ) ) ) ).
% power2_less_imp_less
thf(fact_2153_sum__power2__le__zero__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X2: A,Y: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ ( power_power @ A @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( zero_zero @ A ) )
= ( ( X2
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ).
% sum_power2_le_zero_iff
thf(fact_2154_sum__power2__ge__zero,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X2: A,Y: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( power_power @ A @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% sum_power2_ge_zero
thf(fact_2155_bits__stable__imp__add__self,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A2: A] :
( ( ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= A2 )
=> ( ( plus_plus @ A @ A2 @ ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
= ( zero_zero @ A ) ) ) ) ).
% bits_stable_imp_add_self
thf(fact_2156_sum__power2__gt__zero__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X2: A,Y: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( power_power @ A @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
= ( ( X2
!= ( zero_zero @ A ) )
| ( Y
!= ( zero_zero @ A ) ) ) ) ) ).
% sum_power2_gt_zero_iff
thf(fact_2157_not__sum__power2__lt__zero,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X2: A,Y: A] :
~ ( ord_less @ A @ ( plus_plus @ A @ ( power_power @ A @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( zero_zero @ A ) ) ) ).
% not_sum_power2_lt_zero
thf(fact_2158_power2__sum,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [X2: A,Y: A] :
( ( power_power @ A @ ( plus_plus @ A @ X2 @ Y ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( plus_plus @ A @ ( plus_plus @ A @ ( power_power @ A @ X2 @ ( 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 ) ) @ X2 ) @ Y ) ) ) ) ).
% power2_sum
thf(fact_2159_square__le__1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X2: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ X2 )
=> ( ( ord_less_eq @ A @ X2 @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( power_power @ A @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) ) ) ) ) ).
% square_le_1
thf(fact_2160_zero__le__even__power_H,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,N: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% zero_le_even_power'
thf(fact_2161_power2__le__iff__abs__le,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Y: A,X2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ord_less_eq @ A @ ( power_power @ A @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( ord_less_eq @ A @ ( abs_abs @ A @ X2 ) @ Y ) ) ) ) ).
% power2_le_iff_abs_le
thf(fact_2162_abs__square__le__1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X2: A] :
( ( ord_less_eq @ A @ ( power_power @ A @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) )
= ( ord_less_eq @ A @ ( abs_abs @ A @ X2 ) @ ( one_one @ A ) ) ) ) ).
% abs_square_le_1
thf(fact_2163_abs__square__less__1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X2: A] :
( ( ord_less @ A @ ( power_power @ A @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) )
= ( ord_less @ A @ ( abs_abs @ A @ X2 ) @ ( one_one @ A ) ) ) ) ).
% abs_square_less_1
thf(fact_2164_div__exp__mod__exp__eq,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A2: A,N: nat,M: nat] :
( ( modulo_modulo @ A @ ( divide_divide @ A @ A2 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) )
= ( divide_divide @ A @ ( modulo_modulo @ A @ A2 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ N @ M ) ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% div_exp_mod_exp_eq
thf(fact_2165_nat__bit__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ ( zero_zero @ nat ) )
=> ( ! [N2: nat] :
( ( P @ N2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( P @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) )
=> ( ! [N2: nat] :
( ( P @ N2 )
=> ( P @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_bit_induct
thf(fact_2166_div__2__gt__zero,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% div_2_gt_zero
thf(fact_2167_Suc__n__div__2__gt__zero,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ ( divide_divide @ nat @ ( suc @ N ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% Suc_n_div_2_gt_zero
thf(fact_2168_fact__code,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiring_char_0_fact @ A )
= ( ^ [N4: nat] : ( semiring_1_of_nat @ A @ ( set_fo6178422350223883121st_nat @ nat @ ( times_times @ nat ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N4 @ ( one_one @ nat ) ) ) ) ) ) ).
% fact_code
thf(fact_2169_minus__1__div__exp__eq__int,axiom,
! [N: nat] :
( ( divide_divide @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) )
= ( uminus_uminus @ int @ ( one_one @ int ) ) ) ).
% minus_1_div_exp_eq_int
thf(fact_2170_divmod__digit__0_I1_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [B2: A,A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less @ A @ ( modulo_modulo @ A @ A2 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) @ B2 )
=> ( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A2 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) )
= ( divide_divide @ A @ A2 @ B2 ) ) ) ) ) ).
% divmod_digit_0(1)
thf(fact_2171_power2__diff,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [X2: A,Y: A] :
( ( power_power @ A @ ( minus_minus @ A @ X2 @ Y ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( minus_minus @ A @ ( plus_plus @ A @ ( power_power @ A @ X2 @ ( 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 ) ) @ X2 ) @ Y ) ) ) ) ).
% power2_diff
thf(fact_2172_odd__0__le__power__imp__0__le,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,N: nat] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% odd_0_le_power_imp_0_le
thf(fact_2173_odd__power__less__zero,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,N: nat] :
( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( power_power @ A @ A2 @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) @ ( zero_zero @ A ) ) ) ) ).
% odd_power_less_zero
thf(fact_2174_mult__exp__mod__exp__eq,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [M: nat,N: nat,A2: A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( modulo_modulo @ A @ ( times_times @ A @ A2 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
= ( times_times @ A @ ( modulo_modulo @ A @ A2 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N @ M ) ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) ) ) ) ) ).
% mult_exp_mod_exp_eq
thf(fact_2175_power__minus1__odd,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [N: nat] :
( ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% power_minus1_odd
thf(fact_2176_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 )
=> ? [N2: nat] :
( ( ord_less @ nat @ ( power_power @ nat @ B2 @ N2 ) @ K )
& ( ord_less_eq @ nat @ K @ ( power_power @ nat @ B2 @ ( plus_plus @ nat @ N2 @ ( one_one @ nat ) ) ) ) ) ) ) ).
% ex_power_ivl2
thf(fact_2177_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 )
=> ? [N2: nat] :
( ( ord_less_eq @ nat @ ( power_power @ nat @ B2 @ N2 ) @ K )
& ( ord_less @ nat @ K @ ( power_power @ nat @ B2 @ ( plus_plus @ nat @ N2 @ ( one_one @ nat ) ) ) ) ) ) ) ).
% ex_power_ivl1
thf(fact_2178_int__bit__induct,axiom,
! [P: int > $o,K: int] :
( ( P @ ( zero_zero @ int ) )
=> ( ( P @ ( uminus_uminus @ int @ ( one_one @ int ) ) )
=> ( ! [K4: int] :
( ( P @ K4 )
=> ( ( K4
!= ( zero_zero @ int ) )
=> ( P @ ( times_times @ int @ K4 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) )
=> ( ! [K4: int] :
( ( P @ K4 )
=> ( ( K4
!= ( uminus_uminus @ int @ ( one_one @ int ) ) )
=> ( P @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ K4 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) )
=> ( P @ K ) ) ) ) ) ).
% int_bit_induct
thf(fact_2179_round__def,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archimedean_round @ A )
= ( ^ [X: A] : ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% round_def
thf(fact_2180_mod__double__modulus,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M: A,X2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ M )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X2 )
=> ( ( ( modulo_modulo @ A @ X2 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) )
= ( modulo_modulo @ A @ X2 @ M ) )
| ( ( modulo_modulo @ A @ X2 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) )
= ( plus_plus @ A @ ( modulo_modulo @ A @ X2 @ M ) @ M ) ) ) ) ) ) ).
% mod_double_modulus
thf(fact_2181_divmod__digit__1_I2_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ ( modulo_modulo @ A @ A2 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) )
=> ( ( minus_minus @ A @ ( modulo_modulo @ A @ A2 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) @ B2 )
= ( modulo_modulo @ A @ A2 @ B2 ) ) ) ) ) ) ).
% divmod_digit_1(2)
thf(fact_2182_of__int__round__le,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A] : ( ord_less_eq @ A @ ( ring_1_of_int @ A @ ( archimedean_round @ A @ X2 ) ) @ ( plus_plus @ A @ X2 @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% of_int_round_le
thf(fact_2183_of__int__round__ge,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ X2 @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( ring_1_of_int @ A @ ( archimedean_round @ A @ X2 ) ) ) ) ).
% of_int_round_ge
thf(fact_2184_of__int__round__gt,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A] : ( ord_less @ A @ ( minus_minus @ A @ X2 @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( ring_1_of_int @ A @ ( archimedean_round @ A @ X2 ) ) ) ) ).
% of_int_round_gt
thf(fact_2185_pos__zdiv__mult__2,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A2 )
=> ( ( 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 ) ) @ A2 ) )
= ( divide_divide @ int @ B2 @ A2 ) ) ) ).
% pos_zdiv_mult_2
thf(fact_2186_neg__zdiv__mult__2,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq @ int @ A2 @ ( 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 ) ) @ A2 ) )
= ( divide_divide @ int @ ( plus_plus @ int @ B2 @ ( one_one @ int ) ) @ A2 ) ) ) ).
% neg_zdiv_mult_2
thf(fact_2187_pos__zmod__mult__2,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A2 )
=> ( ( 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 ) ) @ A2 ) )
= ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( modulo_modulo @ int @ B2 @ A2 ) ) ) ) ) ).
% pos_zmod_mult_2
thf(fact_2188_of__int__round__abs__le,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X2: A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ ( archimedean_round @ A @ X2 ) ) @ X2 ) ) @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% of_int_round_abs_le
thf(fact_2189_set__bit__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A] :
( ( bit_se5668285175392031749et_bit @ A @ ( zero_zero @ nat ) @ A2 )
= ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% set_bit_0
thf(fact_2190_unset__bit__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A] :
( ( bit_se2638667681897837118et_bit @ A @ ( zero_zero @ nat ) @ A2 )
= ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% unset_bit_0
thf(fact_2191_set__bit__Suc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A2: A] :
( ( bit_se5668285175392031749et_bit @ A @ ( suc @ N ) @ A2 )
= ( plus_plus @ A @ ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5668285175392031749et_bit @ A @ N @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% set_bit_Suc
thf(fact_2192_flip__bit__Suc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A2: A] :
( ( bit_se8732182000553998342ip_bit @ A @ ( suc @ N ) @ A2 )
= ( plus_plus @ A @ ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se8732182000553998342ip_bit @ A @ N @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% flip_bit_Suc
thf(fact_2193_unset__bit__Suc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A2: A] :
( ( bit_se2638667681897837118et_bit @ A @ ( suc @ N ) @ A2 )
= ( plus_plus @ A @ ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se2638667681897837118et_bit @ A @ N @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% unset_bit_Suc
thf(fact_2194_signed__take__bit__rec,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( bit_ri4674362597316999326ke_bit @ A )
= ( ^ [N4: nat,A3: A] :
( if @ A
@ ( N4
= ( zero_zero @ nat ) )
@ ( uminus_uminus @ A @ ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
@ ( plus_plus @ A @ ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_ri4674362597316999326ke_bit @ A @ ( minus_minus @ nat @ N4 @ ( one_one @ nat ) ) @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% signed_take_bit_rec
thf(fact_2195_time__array__map__entry,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [I: nat,F: A > A,P3: array @ A,H: heap_ext @ product_unit] :
( ( ( time_fails @ ( array @ A ) @ ( array_map_entry @ A @ I @ F @ P3 ) @ H )
=> ( ( time_time @ ( array @ A ) @ ( array_map_entry @ A @ I @ F @ P3 ) @ H )
= ( zero_zero @ nat ) ) )
& ( ~ ( time_fails @ ( array @ A ) @ ( array_map_entry @ A @ I @ F @ P3 ) @ H )
=> ( ( time_time @ ( array @ A ) @ ( array_map_entry @ A @ I @ F @ P3 ) @ H )
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% time_array_map_entry
thf(fact_2196_unset__bit__nonnegative__int__iff,axiom,
! [N: nat,K: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se2638667681897837118et_bit @ int @ N @ K ) )
= ( ord_less_eq @ int @ ( zero_zero @ int ) @ K ) ) ).
% unset_bit_nonnegative_int_iff
thf(fact_2197_set__bit__nonnegative__int__iff,axiom,
! [N: nat,K: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se5668285175392031749et_bit @ int @ N @ K ) )
= ( ord_less_eq @ int @ ( zero_zero @ int ) @ K ) ) ).
% set_bit_nonnegative_int_iff
thf(fact_2198_flip__bit__nonnegative__int__iff,axiom,
! [N: nat,K: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se8732182000553998342ip_bit @ int @ N @ K ) )
= ( ord_less_eq @ int @ ( zero_zero @ int ) @ K ) ) ).
% flip_bit_nonnegative_int_iff
thf(fact_2199_unset__bit__negative__int__iff,axiom,
! [N: nat,K: int] :
( ( ord_less @ int @ ( bit_se2638667681897837118et_bit @ int @ N @ K ) @ ( zero_zero @ int ) )
= ( ord_less @ int @ K @ ( zero_zero @ int ) ) ) ).
% unset_bit_negative_int_iff
thf(fact_2200_set__bit__negative__int__iff,axiom,
! [N: nat,K: int] :
( ( ord_less @ int @ ( bit_se5668285175392031749et_bit @ int @ N @ K ) @ ( zero_zero @ int ) )
= ( ord_less @ int @ K @ ( zero_zero @ int ) ) ) ).
% set_bit_negative_int_iff
thf(fact_2201_flip__bit__negative__int__iff,axiom,
! [N: nat,K: int] :
( ( ord_less @ int @ ( bit_se8732182000553998342ip_bit @ int @ N @ K ) @ ( zero_zero @ int ) )
= ( ord_less @ int @ K @ ( zero_zero @ int ) ) ) ).
% flip_bit_negative_int_iff
thf(fact_2202_signed__take__bit__of__0,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: nat] :
( ( bit_ri4674362597316999326ke_bit @ A @ N @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% signed_take_bit_of_0
thf(fact_2203_signed__take__bit__of__minus__1,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: nat] :
( ( bit_ri4674362597316999326ke_bit @ A @ N @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% signed_take_bit_of_minus_1
thf(fact_2204_signed__take__bit__Suc__1,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: nat] :
( ( bit_ri4674362597316999326ke_bit @ A @ ( suc @ N ) @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% signed_take_bit_Suc_1
thf(fact_2205_signed__take__bit__numeral__of__1,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [K: num] :
( ( bit_ri4674362597316999326ke_bit @ A @ ( numeral_numeral @ nat @ K ) @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% signed_take_bit_numeral_of_1
thf(fact_2206_signed__take__bit__0,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A2: A] :
( ( bit_ri4674362597316999326ke_bit @ A @ ( zero_zero @ nat ) @ A2 )
= ( uminus_uminus @ A @ ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% signed_take_bit_0
thf(fact_2207_signed__take__bit__add,axiom,
! [N: nat,K: int,L: int] :
( ( bit_ri4674362597316999326ke_bit @ int @ N @ ( plus_plus @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K ) @ ( bit_ri4674362597316999326ke_bit @ int @ N @ L ) ) )
= ( bit_ri4674362597316999326ke_bit @ int @ N @ ( plus_plus @ int @ K @ L ) ) ) ).
% signed_take_bit_add
thf(fact_2208_unset__bit__nat__def,axiom,
( ( bit_se2638667681897837118et_bit @ nat )
= ( ^ [M2: nat,N4: nat] : ( nat2 @ ( bit_se2638667681897837118et_bit @ int @ M2 @ ( semiring_1_of_nat @ int @ N4 ) ) ) ) ) ).
% unset_bit_nat_def
thf(fact_2209_unset__bit__less__eq,axiom,
! [N: nat,K: int] : ( ord_less_eq @ int @ ( bit_se2638667681897837118et_bit @ int @ N @ K ) @ K ) ).
% unset_bit_less_eq
thf(fact_2210_set__bit__greater__eq,axiom,
! [K: int,N: nat] : ( ord_less_eq @ int @ K @ ( bit_se5668285175392031749et_bit @ int @ N @ K ) ) ).
% set_bit_greater_eq
thf(fact_2211_signed__take__bit__int__greater__eq__self__iff,axiom,
! [K: int,N: nat] :
( ( ord_less_eq @ int @ K @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K ) )
= ( ord_less @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ).
% signed_take_bit_int_greater_eq_self_iff
thf(fact_2212_signed__take__bit__int__less__self__iff,axiom,
! [N: nat,K: int] :
( ( ord_less @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K ) @ K )
= ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) @ K ) ) ).
% signed_take_bit_int_less_self_iff
thf(fact_2213_signed__take__bit__int__greater__eq__minus__exp,axiom,
! [N: nat,K: int] : ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K ) ) ).
% signed_take_bit_int_greater_eq_minus_exp
thf(fact_2214_signed__take__bit__int__less__eq__self__iff,axiom,
! [N: nat,K: int] :
( ( ord_less_eq @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K ) @ K )
= ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) @ K ) ) ).
% signed_take_bit_int_less_eq_self_iff
thf(fact_2215_signed__take__bit__int__less__eq,axiom,
! [N: nat,K: int] :
( ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) @ K )
=> ( ord_less_eq @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K ) @ ( minus_minus @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( suc @ N ) ) ) ) ) ).
% signed_take_bit_int_less_eq
thf(fact_2216_signed__take__bit__int__eq__self__iff,axiom,
! [N: nat,K: int] :
( ( ( bit_ri4674362597316999326ke_bit @ int @ N @ K )
= K )
= ( ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) @ K )
& ( ord_less @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% signed_take_bit_int_eq_self_iff
thf(fact_2217_signed__take__bit__int__eq__self,axiom,
! [N: nat,K: int] :
( ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) @ K )
=> ( ( ord_less @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( bit_ri4674362597316999326ke_bit @ int @ N @ K )
= K ) ) ) ).
% signed_take_bit_int_eq_self
thf(fact_2218_signed__take__bit__int__greater__eq,axiom,
! [K: int,N: nat] :
( ( ord_less @ int @ K @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) )
=> ( ord_less_eq @ int @ ( plus_plus @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( suc @ N ) ) ) @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K ) ) ) ).
% signed_take_bit_int_greater_eq
thf(fact_2219_signed__take__bit__Suc,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: nat,A2: A] :
( ( bit_ri4674362597316999326ke_bit @ A @ ( suc @ N ) @ A2 )
= ( plus_plus @ A @ ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_ri4674362597316999326ke_bit @ A @ N @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% signed_take_bit_Suc
thf(fact_2220_time__array__swap,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [I: nat,F: A > A,P3: array @ A,H: heap_ext @ product_unit,X2: A] :
( ( ( time_fails @ ( array @ A ) @ ( array_map_entry @ A @ I @ F @ P3 ) @ H )
=> ( ( time_time @ A @ ( array_swap @ A @ I @ X2 @ P3 ) @ H )
= ( zero_zero @ nat ) ) )
& ( ~ ( time_fails @ ( array @ A ) @ ( array_map_entry @ A @ I @ F @ P3 ) @ H )
=> ( ( time_time @ A @ ( array_swap @ A @ I @ X2 @ P3 ) @ H )
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% time_array_swap
thf(fact_2221_binomial__code,axiom,
( binomial
= ( ^ [N4: nat,K2: nat] : ( if @ nat @ ( ord_less @ nat @ N4 @ K2 ) @ ( zero_zero @ nat ) @ ( if @ nat @ ( ord_less @ nat @ N4 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K2 ) ) @ ( binomial @ N4 @ ( minus_minus @ nat @ N4 @ K2 ) ) @ ( divide_divide @ nat @ ( set_fo6178422350223883121st_nat @ nat @ ( times_times @ nat ) @ ( plus_plus @ nat @ ( minus_minus @ nat @ N4 @ K2 ) @ ( one_one @ nat ) ) @ N4 @ ( one_one @ nat ) ) @ ( semiring_char_0_fact @ nat @ K2 ) ) ) ) ) ) ).
% binomial_code
thf(fact_2222_signed__take__bit__Suc__minus__bit1,axiom,
! [N: nat,K: num] :
( ( bit_ri4674362597316999326ke_bit @ int @ ( suc @ N ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ K ) ) ) )
= ( plus_plus @ int @ ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N @ ( minus_minus @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) @ ( one_one @ int ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( one_one @ int ) ) ) ).
% signed_take_bit_Suc_minus_bit1
thf(fact_2223_concat__bit__Suc,axiom,
! [N: nat,K: int,L: int] :
( ( bit_concat_bit @ ( suc @ N ) @ K @ L )
= ( plus_plus @ int @ ( modulo_modulo @ int @ K @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_concat_bit @ N @ ( divide_divide @ int @ K @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ L ) ) ) ) ).
% concat_bit_Suc
thf(fact_2224_even__succ__mod__exp,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A2: A,N: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( modulo_modulo @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A2 ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
= ( plus_plus @ A @ ( one_one @ A ) @ ( modulo_modulo @ A @ A2 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ) ) ).
% even_succ_mod_exp
thf(fact_2225_even__succ__div__exp,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A2: A,N: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A2 ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
= ( divide_divide @ A @ A2 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ) ).
% even_succ_div_exp
thf(fact_2226_take__bit__rec,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se2584673776208193580ke_bit @ A )
= ( ^ [N4: nat,A3: A] :
( if @ A
@ ( N4
= ( zero_zero @ nat ) )
@ ( zero_zero @ A )
@ ( plus_plus @ A @ ( times_times @ A @ ( bit_se2584673776208193580ke_bit @ A @ ( minus_minus @ nat @ N4 @ ( one_one @ nat ) ) @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% take_bit_rec
thf(fact_2227_verit__eq__simplify_I9_J,axiom,
! [X32: num,Y32: num] :
( ( ( bit1 @ X32 )
= ( bit1 @ Y32 ) )
= ( X32 = Y32 ) ) ).
% verit_eq_simplify(9)
thf(fact_2228_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_2229_int__dvd__int__iff,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd @ int @ ( semiring_1_of_nat @ int @ M ) @ ( semiring_1_of_nat @ int @ N ) )
= ( dvd_dvd @ nat @ M @ N ) ) ).
% int_dvd_int_iff
thf(fact_2230_dvd__0__right,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A2: A] : ( dvd_dvd @ A @ A2 @ ( zero_zero @ A ) ) ) ).
% dvd_0_right
thf(fact_2231_dvd__0__left__iff,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A2: A] :
( ( dvd_dvd @ A @ ( zero_zero @ A ) @ A2 )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% dvd_0_left_iff
thf(fact_2232_dvd__add__triv__left__iff,axiom,
! [A: $tType] :
( ( comm_s4317794764714335236cancel @ A )
=> ! [A2: A,B2: A] :
( ( dvd_dvd @ A @ A2 @ ( plus_plus @ A @ A2 @ B2 ) )
= ( dvd_dvd @ A @ A2 @ B2 ) ) ) ).
% dvd_add_triv_left_iff
thf(fact_2233_dvd__add__triv__right__iff,axiom,
! [A: $tType] :
( ( comm_s4317794764714335236cancel @ A )
=> ! [A2: A,B2: A] :
( ( dvd_dvd @ A @ A2 @ ( plus_plus @ A @ B2 @ A2 ) )
= ( dvd_dvd @ A @ A2 @ B2 ) ) ) ).
% dvd_add_triv_right_iff
thf(fact_2234_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_2235_dvd__1__left,axiom,
! [K: nat] : ( dvd_dvd @ nat @ ( suc @ ( zero_zero @ nat ) ) @ K ) ).
% dvd_1_left
thf(fact_2236_take__bit__of__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat] :
( ( bit_se2584673776208193580ke_bit @ A @ N @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% take_bit_of_0
thf(fact_2237_nat__mult__dvd__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( dvd_dvd @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N ) )
= ( ( K
= ( zero_zero @ nat ) )
| ( dvd_dvd @ nat @ M @ N ) ) ) ).
% nat_mult_dvd_cancel_disj
thf(fact_2238_zdvd1__eq,axiom,
! [X2: int] :
( ( dvd_dvd @ int @ X2 @ ( one_one @ int ) )
= ( ( abs_abs @ int @ X2 )
= ( one_one @ int ) ) ) ).
% zdvd1_eq
thf(fact_2239_binomial__n__n,axiom,
! [N: nat] :
( ( binomial @ N @ N )
= ( one_one @ nat ) ) ).
% binomial_n_n
thf(fact_2240_concat__bit__0,axiom,
! [K: int,L: int] :
( ( bit_concat_bit @ ( zero_zero @ nat ) @ K @ L )
= L ) ).
% concat_bit_0
thf(fact_2241_concat__bit__of__zero__2,axiom,
! [N: nat,K: int] :
( ( bit_concat_bit @ N @ K @ ( zero_zero @ int ) )
= ( bit_se2584673776208193580ke_bit @ int @ N @ K ) ) ).
% concat_bit_of_zero_2
thf(fact_2242_dvd__mult__cancel__left,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( dvd_dvd @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( dvd_dvd @ A @ A2 @ B2 ) ) ) ) ).
% dvd_mult_cancel_left
thf(fact_2243_dvd__mult__cancel__right,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( dvd_dvd @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( dvd_dvd @ A @ A2 @ B2 ) ) ) ) ).
% dvd_mult_cancel_right
thf(fact_2244_dvd__times__left__cancel__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ ( times_times @ A @ A2 @ B2 ) @ ( times_times @ A @ A2 @ C2 ) )
= ( dvd_dvd @ A @ B2 @ C2 ) ) ) ) ).
% dvd_times_left_cancel_iff
thf(fact_2245_dvd__times__right__cancel__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ ( times_times @ A @ B2 @ A2 ) @ ( times_times @ A @ C2 @ A2 ) )
= ( dvd_dvd @ A @ B2 @ C2 ) ) ) ) ).
% dvd_times_right_cancel_iff
thf(fact_2246_dvd__add__times__triv__right__iff,axiom,
! [A: $tType] :
( ( comm_s4317794764714335236cancel @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A2 @ ( plus_plus @ A @ B2 @ ( times_times @ A @ C2 @ A2 ) ) )
= ( dvd_dvd @ A @ A2 @ B2 ) ) ) ).
% dvd_add_times_triv_right_iff
thf(fact_2247_dvd__add__times__triv__left__iff,axiom,
! [A: $tType] :
( ( comm_s4317794764714335236cancel @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( dvd_dvd @ A @ A2 @ ( plus_plus @ A @ ( times_times @ A @ C2 @ A2 ) @ B2 ) )
= ( dvd_dvd @ A @ A2 @ B2 ) ) ) ).
% dvd_add_times_triv_left_iff
thf(fact_2248_unit__prod,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,B2: A] :
( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( dvd_dvd @ A @ ( times_times @ A @ A2 @ B2 ) @ ( one_one @ A ) ) ) ) ) ).
% unit_prod
thf(fact_2249_div__add,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( dvd_dvd @ A @ C2 @ A2 )
=> ( ( dvd_dvd @ A @ C2 @ B2 )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 )
= ( plus_plus @ A @ ( divide_divide @ A @ A2 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ) ) ).
% div_add
thf(fact_2250_unit__div,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,B2: A] :
( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( dvd_dvd @ A @ ( divide_divide @ A @ A2 @ B2 ) @ ( one_one @ A ) ) ) ) ) ).
% unit_div
thf(fact_2251_unit__div__1__unit,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A] :
( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
=> ( dvd_dvd @ A @ ( divide_divide @ A @ ( one_one @ A ) @ A2 ) @ ( one_one @ A ) ) ) ) ).
% unit_div_1_unit
thf(fact_2252_unit__div__1__div__1,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A] :
( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
=> ( ( divide_divide @ A @ ( one_one @ A ) @ ( divide_divide @ A @ ( one_one @ A ) @ A2 ) )
= A2 ) ) ) ).
% unit_div_1_div_1
thf(fact_2253_dvd__imp__mod__0,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [A2: A,B2: A] :
( ( dvd_dvd @ A @ A2 @ B2 )
=> ( ( modulo_modulo @ A @ B2 @ A2 )
= ( zero_zero @ A ) ) ) ) ).
% dvd_imp_mod_0
thf(fact_2254_take__bit__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A] :
( ( bit_se2584673776208193580ke_bit @ A @ ( zero_zero @ nat ) @ A2 )
= ( zero_zero @ A ) ) ) ).
% take_bit_0
thf(fact_2255_take__bit__Suc__1,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N: nat] :
( ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N ) @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% take_bit_Suc_1
thf(fact_2256_take__bit__numeral__1,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [L: num] :
( ( bit_se2584673776208193580ke_bit @ A @ ( numeral_numeral @ nat @ L ) @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% take_bit_numeral_1
thf(fact_2257_binomial__1,axiom,
! [N: nat] :
( ( binomial @ N @ ( suc @ ( zero_zero @ nat ) ) )
= N ) ).
% binomial_1
thf(fact_2258_binomial__0__Suc,axiom,
! [K: nat] :
( ( binomial @ ( zero_zero @ nat ) @ ( suc @ K ) )
= ( zero_zero @ nat ) ) ).
% binomial_0_Suc
thf(fact_2259_binomial__eq__0__iff,axiom,
! [N: nat,K: nat] :
( ( ( binomial @ N @ K )
= ( zero_zero @ nat ) )
= ( ord_less @ nat @ N @ K ) ) ).
% binomial_eq_0_iff
thf(fact_2260_binomial__Suc__Suc,axiom,
! [N: nat,K: nat] :
( ( binomial @ ( suc @ N ) @ ( suc @ K ) )
= ( plus_plus @ nat @ ( binomial @ N @ K ) @ ( binomial @ N @ ( suc @ K ) ) ) ) ).
% binomial_Suc_Suc
thf(fact_2261_of__nat__nat__take__bit__eq,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [N: nat,K: int] :
( ( semiring_1_of_nat @ A @ ( nat2 @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) ) )
= ( ring_1_of_int @ A @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) ) ) ) ).
% of_nat_nat_take_bit_eq
thf(fact_2262_semiring__norm_I7_J,axiom,
! [M: num,N: num] :
( ( plus_plus @ num @ ( bit0 @ M ) @ ( bit1 @ N ) )
= ( bit1 @ ( plus_plus @ num @ M @ N ) ) ) ).
% semiring_norm(7)
thf(fact_2263_semiring__norm_I9_J,axiom,
! [M: num,N: num] :
( ( plus_plus @ num @ ( bit1 @ M ) @ ( bit0 @ N ) )
= ( bit1 @ ( plus_plus @ num @ M @ N ) ) ) ).
% semiring_norm(9)
thf(fact_2264_binomial__n__0,axiom,
! [N: nat] :
( ( binomial @ N @ ( zero_zero @ nat ) )
= ( one_one @ nat ) ) ).
% binomial_n_0
thf(fact_2265_dvd__nat__abs__iff,axiom,
! [N: nat,K: int] :
( ( dvd_dvd @ nat @ N @ ( nat2 @ ( abs_abs @ int @ K ) ) )
= ( dvd_dvd @ int @ ( semiring_1_of_nat @ int @ N ) @ K ) ) ).
% dvd_nat_abs_iff
thf(fact_2266_nat__abs__dvd__iff,axiom,
! [K: int,N: nat] :
( ( dvd_dvd @ nat @ ( nat2 @ ( abs_abs @ int @ K ) ) @ N )
= ( dvd_dvd @ int @ K @ ( semiring_1_of_nat @ int @ N ) ) ) ).
% nat_abs_dvd_iff
thf(fact_2267_concat__bit__nonnegative__iff,axiom,
! [N: nat,K: int,L: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_concat_bit @ N @ K @ L ) )
= ( ord_less_eq @ int @ ( zero_zero @ int ) @ L ) ) ).
% concat_bit_nonnegative_iff
thf(fact_2268_concat__bit__negative__iff,axiom,
! [N: nat,K: int,L: int] :
( ( ord_less @ int @ ( bit_concat_bit @ N @ K @ L ) @ ( zero_zero @ int ) )
= ( ord_less @ int @ L @ ( zero_zero @ int ) ) ) ).
% concat_bit_negative_iff
thf(fact_2269_dbl__inc__simps_I5_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [K: num] :
( ( neg_numeral_dbl_inc @ A @ ( numeral_numeral @ A @ K ) )
= ( numeral_numeral @ A @ ( bit1 @ K ) ) ) ) ).
% dbl_inc_simps(5)
thf(fact_2270_unit__mult__div__div,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,B2: A] :
( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
=> ( ( times_times @ A @ B2 @ ( divide_divide @ A @ ( one_one @ A ) @ A2 ) )
= ( divide_divide @ A @ B2 @ A2 ) ) ) ) ).
% unit_mult_div_div
thf(fact_2271_unit__div__mult__self,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,B2: A] :
( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
=> ( ( times_times @ A @ ( divide_divide @ A @ B2 @ A2 ) @ A2 )
= B2 ) ) ) ).
% unit_div_mult_self
thf(fact_2272_take__bit__of__1__eq__0__iff,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N: nat] :
( ( ( bit_se2584673776208193580ke_bit @ A @ N @ ( one_one @ A ) )
= ( zero_zero @ A ) )
= ( N
= ( zero_zero @ nat ) ) ) ) ).
% take_bit_of_1_eq_0_iff
thf(fact_2273_pow__divides__pow__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [N: nat,A2: A,B2: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( dvd_dvd @ A @ ( power_power @ A @ A2 @ N ) @ ( power_power @ A @ B2 @ N ) )
= ( dvd_dvd @ A @ A2 @ B2 ) ) ) ) ).
% pow_divides_pow_iff
thf(fact_2274_zero__less__binomial__iff,axiom,
! [N: nat,K: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( binomial @ N @ K ) )
= ( ord_less_eq @ nat @ K @ N ) ) ).
% zero_less_binomial_iff
thf(fact_2275_semiring__norm_I3_J,axiom,
! [N: num] :
( ( plus_plus @ num @ one2 @ ( bit0 @ N ) )
= ( bit1 @ N ) ) ).
% semiring_norm(3)
thf(fact_2276_semiring__norm_I4_J,axiom,
! [N: num] :
( ( plus_plus @ num @ one2 @ ( bit1 @ N ) )
= ( bit0 @ ( plus_plus @ num @ N @ one2 ) ) ) ).
% semiring_norm(4)
thf(fact_2277_semiring__norm_I5_J,axiom,
! [M: num] :
( ( plus_plus @ num @ ( bit0 @ M ) @ one2 )
= ( bit1 @ M ) ) ).
% semiring_norm(5)
thf(fact_2278_semiring__norm_I8_J,axiom,
! [M: num] :
( ( plus_plus @ num @ ( bit1 @ M ) @ one2 )
= ( bit0 @ ( plus_plus @ num @ M @ one2 ) ) ) ).
% semiring_norm(8)
thf(fact_2279_semiring__norm_I10_J,axiom,
! [M: num,N: num] :
( ( plus_plus @ num @ ( bit1 @ M ) @ ( bit1 @ N ) )
= ( bit0 @ ( plus_plus @ num @ ( plus_plus @ num @ M @ N ) @ one2 ) ) ) ).
% semiring_norm(10)
thf(fact_2280_semiring__norm_I16_J,axiom,
! [M: num,N: num] :
( ( times_times @ num @ ( bit1 @ M ) @ ( bit1 @ N ) )
= ( bit1 @ ( plus_plus @ num @ ( plus_plus @ num @ M @ N ) @ ( bit0 @ ( times_times @ num @ M @ N ) ) ) ) ) ).
% semiring_norm(16)
thf(fact_2281_odd__add,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A2: A,B2: A] :
( ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ A @ A2 @ B2 ) ) )
= ( ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) )
!= ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) ) ) ) ).
% odd_add
thf(fact_2282_even__add,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A2: A,B2: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ A @ A2 @ B2 ) )
= ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
= ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) ) ) ).
% even_add
thf(fact_2283_dbl__inc__simps_I3_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( neg_numeral_dbl_inc @ A @ ( one_one @ A ) )
= ( numeral_numeral @ A @ ( bit1 @ one2 ) ) ) ) ).
% dbl_inc_simps(3)
thf(fact_2284_even__plus__one__iff,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A2: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) )
= ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) ) ) ).
% even_plus_one_iff
thf(fact_2285_even__diff,axiom,
! [A: $tType] :
( ( ring_parity @ A )
=> ! [A2: A,B2: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( minus_minus @ A @ A2 @ B2 ) )
= ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ).
% even_diff
thf(fact_2286_zero__le__power__eq__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,W2: num] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ ( 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 ) @ A2 ) ) ) ) ) ).
% zero_le_power_eq_numeral
thf(fact_2287_power__less__zero__eq,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,N: nat] :
( ( ord_less @ A @ ( power_power @ A @ A2 @ N ) @ ( zero_zero @ A ) )
= ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
& ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ) ).
% power_less_zero_eq
thf(fact_2288_power__less__zero__eq__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,W2: num] :
( ( ord_less @ A @ ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ W2 ) ) @ ( zero_zero @ A ) )
= ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W2 ) )
& ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ) ).
% power_less_zero_eq_numeral
thf(fact_2289_neg__one__even__power,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [N: nat] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N )
= ( one_one @ A ) ) ) ) ).
% neg_one_even_power
thf(fact_2290_neg__one__odd__power,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [N: nat] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ).
% neg_one_odd_power
thf(fact_2291_even__of__nat,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [N: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ A @ N ) )
= ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ).
% even_of_nat
thf(fact_2292_even__take__bit__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A2: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se2584673776208193580ke_bit @ A @ N @ A2 ) )
= ( ( N
= ( zero_zero @ nat ) )
| ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) ) ) ).
% even_take_bit_eq
thf(fact_2293_odd__Suc__minus__one,axiom,
! [N: nat] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( suc @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) )
= N ) ) ).
% odd_Suc_minus_one
thf(fact_2294_even__diff__nat,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ M @ N ) )
= ( ( ord_less @ nat @ M @ N )
| ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ M @ N ) ) ) ) ).
% even_diff_nat
thf(fact_2295_Suc__mod__eq__add3__mod__numeral,axiom,
! [M: nat,V2: num] :
( ( modulo_modulo @ nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ ( numeral_numeral @ nat @ V2 ) )
= ( modulo_modulo @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) @ M ) @ ( numeral_numeral @ nat @ V2 ) ) ) ).
% Suc_mod_eq_add3_mod_numeral
thf(fact_2296_mod__Suc__eq__mod__add3,axiom,
! [M: nat,N: nat] :
( ( modulo_modulo @ nat @ M @ ( suc @ ( suc @ ( suc @ N ) ) ) )
= ( modulo_modulo @ nat @ M @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) @ N ) ) ) ).
% mod_Suc_eq_mod_add3
thf(fact_2297_Suc__div__eq__add3__div__numeral,axiom,
! [M: nat,V2: num] :
( ( divide_divide @ nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ ( numeral_numeral @ nat @ V2 ) )
= ( divide_divide @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) @ M ) @ ( numeral_numeral @ nat @ V2 ) ) ) ).
% Suc_div_eq_add3_div_numeral
thf(fact_2298_div__Suc__eq__div__add3,axiom,
! [M: nat,N: nat] :
( ( divide_divide @ nat @ M @ ( suc @ ( suc @ ( suc @ N ) ) ) )
= ( divide_divide @ nat @ M @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) @ N ) ) ) ).
% div_Suc_eq_div_add3
thf(fact_2299_dbl__dec__simps_I4_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( neg_numeral_dbl_dec @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( bit1 @ one2 ) ) ) ) ) ).
% dbl_dec_simps(4)
thf(fact_2300_even__succ__div__two,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A2: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% even_succ_div_two
thf(fact_2301_odd__succ__div__two,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A2: A] :
( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( plus_plus @ A @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) ) ) ) ) ).
% odd_succ_div_two
thf(fact_2302_even__succ__div__2,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A2: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A2 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% even_succ_div_2
thf(fact_2303_zero__less__power__eq__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,W2: num] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ W2 ) ) )
= ( ( ( numeral_numeral @ nat @ W2 )
= ( zero_zero @ nat ) )
| ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W2 ) )
& ( A2
!= ( zero_zero @ A ) ) )
| ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W2 ) )
& ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ) ) ).
% zero_less_power_eq_numeral
thf(fact_2304_even__power,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A2: A,N: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( power_power @ A @ A2 @ N ) )
= ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% even_power
thf(fact_2305_take__bit__Suc__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A] :
( ( bit_se2584673776208193580ke_bit @ A @ ( suc @ ( zero_zero @ nat ) ) @ A2 )
= ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% take_bit_Suc_0
thf(fact_2306_odd__two__times__div__two__nat,axiom,
! [N: nat] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ).
% odd_two_times_div_two_nat
thf(fact_2307_odd__two__times__div__two__succ,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A2: A] :
( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
=> ( ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( one_one @ A ) )
= A2 ) ) ) ).
% odd_two_times_div_two_succ
thf(fact_2308_semiring__parity__class_Oeven__mask__iff,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [N: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ A ) ) )
= ( N
= ( zero_zero @ nat ) ) ) ) ).
% semiring_parity_class.even_mask_iff
thf(fact_2309_power__le__zero__eq__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,W2: num] :
( ( ord_less_eq @ A @ ( power_power @ A @ A2 @ ( 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 @ A2 @ ( zero_zero @ A ) ) )
| ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W2 ) )
& ( A2
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% power_le_zero_eq_numeral
thf(fact_2310_zmod__numeral__Bit1,axiom,
! [V2: num,W2: num] :
( ( modulo_modulo @ int @ ( numeral_numeral @ int @ ( bit1 @ V2 ) ) @ ( numeral_numeral @ int @ ( bit0 @ W2 ) ) )
= ( plus_plus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( modulo_modulo @ int @ ( numeral_numeral @ int @ V2 ) @ ( numeral_numeral @ int @ W2 ) ) ) @ ( one_one @ int ) ) ) ).
% zmod_numeral_Bit1
thf(fact_2311_signed__take__bit__Suc__bit1,axiom,
! [N: nat,K: num] :
( ( bit_ri4674362597316999326ke_bit @ int @ ( suc @ N ) @ ( numeral_numeral @ int @ ( bit1 @ K ) ) )
= ( plus_plus @ int @ ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N @ ( numeral_numeral @ int @ K ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( one_one @ int ) ) ) ).
% signed_take_bit_Suc_bit1
thf(fact_2312_of__nat__dvd__iff,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [M: nat,N: nat] :
( ( dvd_dvd @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) )
= ( dvd_dvd @ nat @ M @ N ) ) ) ).
% of_nat_dvd_iff
thf(fact_2313_take__bit__of__nat,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,M: nat] :
( ( bit_se2584673776208193580ke_bit @ A @ N @ ( semiring_1_of_nat @ A @ M ) )
= ( semiring_1_of_nat @ A @ ( bit_se2584673776208193580ke_bit @ nat @ N @ M ) ) ) ) ).
% take_bit_of_nat
thf(fact_2314_subset__divisors__dvd,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [C5: A] : ( dvd_dvd @ A @ C5 @ A2 ) )
@ ( collect @ A
@ ^ [C5: A] : ( dvd_dvd @ A @ C5 @ B2 ) ) )
= ( dvd_dvd @ A @ A2 @ B2 ) ) ) ).
% subset_divisors_dvd
thf(fact_2315_strict__subset__divisors__dvd,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ ( set @ A )
@ ( collect @ A
@ ^ [C5: A] : ( dvd_dvd @ A @ C5 @ A2 ) )
@ ( collect @ A
@ ^ [C5: A] : ( dvd_dvd @ A @ C5 @ B2 ) ) )
= ( ( dvd_dvd @ A @ A2 @ B2 )
& ~ ( dvd_dvd @ A @ B2 @ A2 ) ) ) ) ).
% strict_subset_divisors_dvd
thf(fact_2316_take__bit__add,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N: nat,A2: A,B2: A] :
( ( bit_se2584673776208193580ke_bit @ A @ N @ ( plus_plus @ A @ ( bit_se2584673776208193580ke_bit @ A @ N @ A2 ) @ ( bit_se2584673776208193580ke_bit @ A @ N @ B2 ) ) )
= ( bit_se2584673776208193580ke_bit @ A @ N @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ).
% take_bit_add
thf(fact_2317_take__bit__tightened,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A2: A,B2: A,M: nat] :
( ( ( bit_se2584673776208193580ke_bit @ A @ N @ A2 )
= ( bit_se2584673776208193580ke_bit @ A @ N @ B2 ) )
=> ( ( ord_less_eq @ nat @ M @ N )
=> ( ( bit_se2584673776208193580ke_bit @ A @ M @ A2 )
= ( bit_se2584673776208193580ke_bit @ A @ M @ B2 ) ) ) ) ) ).
% take_bit_tightened
thf(fact_2318_take__bit__nat__less__eq__self,axiom,
! [N: nat,M: nat] : ( ord_less_eq @ nat @ ( bit_se2584673776208193580ke_bit @ nat @ N @ M ) @ M ) ).
% take_bit_nat_less_eq_self
thf(fact_2319_take__bit__tightened__less__eq__nat,axiom,
! [M: nat,N: nat,Q2: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ord_less_eq @ nat @ ( bit_se2584673776208193580ke_bit @ nat @ M @ Q2 ) @ ( bit_se2584673776208193580ke_bit @ nat @ N @ Q2 ) ) ) ).
% take_bit_tightened_less_eq_nat
thf(fact_2320_dvd__field__iff,axiom,
! [A: $tType] :
( ( field @ A )
=> ( ( dvd_dvd @ A )
= ( ^ [A3: A,B3: A] :
( ( A3
= ( zero_zero @ A ) )
=> ( B3
= ( zero_zero @ A ) ) ) ) ) ) ).
% dvd_field_iff
thf(fact_2321_dvd__0__left,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A2: A] :
( ( dvd_dvd @ A @ ( zero_zero @ A ) @ A2 )
=> ( A2
= ( zero_zero @ A ) ) ) ) ).
% dvd_0_left
thf(fact_2322_choose__one,axiom,
! [N: nat] :
( ( binomial @ N @ ( one_one @ nat ) )
= N ) ).
% choose_one
thf(fact_2323_dvd__add,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A2 @ B2 )
=> ( ( dvd_dvd @ A @ A2 @ C2 )
=> ( dvd_dvd @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ) ).
% dvd_add
thf(fact_2324_dvd__add__left__iff,axiom,
! [A: $tType] :
( ( comm_s4317794764714335236cancel @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( dvd_dvd @ A @ A2 @ C2 )
=> ( ( dvd_dvd @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) )
= ( dvd_dvd @ A @ A2 @ B2 ) ) ) ) ).
% dvd_add_left_iff
thf(fact_2325_dvd__add__right__iff,axiom,
! [A: $tType] :
( ( comm_s4317794764714335236cancel @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A2 @ B2 )
=> ( ( dvd_dvd @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) )
= ( dvd_dvd @ A @ A2 @ C2 ) ) ) ) ).
% dvd_add_right_iff
thf(fact_2326_division__decomp,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) )
=> ? [B7: A,C7: A] :
( ( A2
= ( times_times @ A @ B7 @ C7 ) )
& ( dvd_dvd @ A @ B7 @ B2 )
& ( dvd_dvd @ A @ C7 @ C2 ) ) ) ) ).
% division_decomp
thf(fact_2327_dvd__productE,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [P3: A,A2: A,B2: A] :
( ( dvd_dvd @ A @ P3 @ ( times_times @ A @ A2 @ B2 ) )
=> ~ ! [X3: A,Y2: A] :
( ( P3
= ( times_times @ A @ X3 @ Y2 ) )
=> ( ( dvd_dvd @ A @ X3 @ A2 )
=> ~ ( dvd_dvd @ A @ Y2 @ B2 ) ) ) ) ) ).
% dvd_productE
thf(fact_2328_one__dvd,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: A] : ( dvd_dvd @ A @ ( one_one @ A ) @ A2 ) ) ).
% one_dvd
thf(fact_2329_unit__imp__dvd,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A2: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( dvd_dvd @ A @ B2 @ A2 ) ) ) ).
% unit_imp_dvd
thf(fact_2330_dvd__unit__imp__unit,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,B2: A] :
( ( dvd_dvd @ A @ A2 @ B2 )
=> ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( dvd_dvd @ A @ A2 @ ( one_one @ A ) ) ) ) ) ).
% dvd_unit_imp_unit
thf(fact_2331_verit__eq__simplify_I14_J,axiom,
! [X22: num,X32: num] :
( ( bit0 @ X22 )
!= ( bit1 @ X32 ) ) ).
% verit_eq_simplify(14)
thf(fact_2332_verit__eq__simplify_I12_J,axiom,
! [X32: num] :
( one2
!= ( bit1 @ X32 ) ) ).
% verit_eq_simplify(12)
thf(fact_2333_gcd__nat_Oextremum,axiom,
! [A2: nat] : ( dvd_dvd @ nat @ A2 @ ( zero_zero @ nat ) ) ).
% gcd_nat.extremum
thf(fact_2334_gcd__nat_Oextremum__strict,axiom,
! [A2: nat] :
~ ( ( dvd_dvd @ nat @ ( zero_zero @ nat ) @ A2 )
& ( ( zero_zero @ nat )
!= A2 ) ) ).
% gcd_nat.extremum_strict
thf(fact_2335_gcd__nat_Oextremum__unique,axiom,
! [A2: nat] :
( ( dvd_dvd @ nat @ ( zero_zero @ nat ) @ A2 )
= ( A2
= ( zero_zero @ nat ) ) ) ).
% gcd_nat.extremum_unique
thf(fact_2336_gcd__nat_Onot__eq__extremum,axiom,
! [A2: nat] :
( ( A2
!= ( zero_zero @ nat ) )
= ( ( dvd_dvd @ nat @ A2 @ ( zero_zero @ nat ) )
& ( A2
!= ( zero_zero @ nat ) ) ) ) ).
% gcd_nat.not_eq_extremum
thf(fact_2337_gcd__nat_Oextremum__uniqueI,axiom,
! [A2: nat] :
( ( dvd_dvd @ nat @ ( zero_zero @ nat ) @ A2 )
=> ( A2
= ( zero_zero @ nat ) ) ) ).
% gcd_nat.extremum_uniqueI
thf(fact_2338_dvd__diff__nat,axiom,
! [K: nat,M: nat,N: nat] :
( ( dvd_dvd @ nat @ K @ M )
=> ( ( dvd_dvd @ nat @ K @ N )
=> ( dvd_dvd @ nat @ K @ ( minus_minus @ nat @ M @ N ) ) ) ) ).
% dvd_diff_nat
thf(fact_2339_uminus__dvd__conv_I2_J,axiom,
( ( dvd_dvd @ int )
= ( ^ [D5: int,T3: int] : ( dvd_dvd @ int @ D5 @ ( uminus_uminus @ int @ T3 ) ) ) ) ).
% uminus_dvd_conv(2)
thf(fact_2340_uminus__dvd__conv_I1_J,axiom,
( ( dvd_dvd @ int )
= ( ^ [D5: int] : ( dvd_dvd @ int @ ( uminus_uminus @ int @ D5 ) ) ) ) ).
% uminus_dvd_conv(1)
thf(fact_2341_zdvd__zdiffD,axiom,
! [K: int,M: int,N: int] :
( ( dvd_dvd @ int @ K @ ( minus_minus @ int @ M @ N ) )
=> ( ( dvd_dvd @ int @ K @ N )
=> ( dvd_dvd @ int @ K @ M ) ) ) ).
% zdvd_zdiffD
thf(fact_2342_zdvd__antisym__abs,axiom,
! [A2: int,B2: int] :
( ( dvd_dvd @ int @ A2 @ B2 )
=> ( ( dvd_dvd @ int @ B2 @ A2 )
=> ( ( abs_abs @ int @ A2 )
= ( abs_abs @ int @ B2 ) ) ) ) ).
% zdvd_antisym_abs
thf(fact_2343_take__bit__nat__eq,axiom,
! [K: int,N: nat] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
=> ( ( bit_se2584673776208193580ke_bit @ nat @ N @ ( nat2 @ K ) )
= ( nat2 @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) ) ) ) ).
% take_bit_nat_eq
thf(fact_2344_nat__take__bit__eq,axiom,
! [K: int,N: nat] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
=> ( ( nat2 @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) )
= ( bit_se2584673776208193580ke_bit @ nat @ N @ ( nat2 @ K ) ) ) ) ).
% nat_take_bit_eq
thf(fact_2345_concat__bit__assoc,axiom,
! [N: nat,K: int,M: nat,L: int,R: int] :
( ( bit_concat_bit @ N @ K @ ( bit_concat_bit @ M @ L @ R ) )
= ( bit_concat_bit @ ( plus_plus @ nat @ M @ N ) @ ( bit_concat_bit @ N @ K @ L ) @ R ) ) ).
% concat_bit_assoc
thf(fact_2346_binomial__eq__0,axiom,
! [N: nat,K: nat] :
( ( ord_less @ nat @ N @ K )
=> ( ( binomial @ N @ K )
= ( zero_zero @ nat ) ) ) ).
% binomial_eq_0
thf(fact_2347_binomial__symmetric,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ N )
=> ( ( binomial @ N @ K )
= ( binomial @ N @ ( minus_minus @ nat @ N @ K ) ) ) ) ).
% binomial_symmetric
thf(fact_2348_take__bit__tightened__less__eq__int,axiom,
! [M: nat,N: nat,K: int] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ord_less_eq @ int @ ( bit_se2584673776208193580ke_bit @ int @ M @ K ) @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) ) ) ).
% take_bit_tightened_less_eq_int
thf(fact_2349_take__bit__int__less__eq__self__iff,axiom,
! [N: nat,K: int] :
( ( ord_less_eq @ int @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) @ K )
= ( ord_less_eq @ int @ ( zero_zero @ int ) @ K ) ) ).
% take_bit_int_less_eq_self_iff
thf(fact_2350_take__bit__nonnegative,axiom,
! [N: nat,K: int] : ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) ) ).
% take_bit_nonnegative
thf(fact_2351_not__take__bit__negative,axiom,
! [N: nat,K: int] :
~ ( ord_less @ int @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) @ ( zero_zero @ int ) ) ).
% not_take_bit_negative
thf(fact_2352_take__bit__int__greater__self__iff,axiom,
! [K: int,N: nat] :
( ( ord_less @ int @ K @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) )
= ( ord_less @ int @ K @ ( zero_zero @ int ) ) ) ).
% take_bit_int_greater_self_iff
thf(fact_2353_choose__mult__lemma,axiom,
! [M: nat,R: nat,K: nat] :
( ( times_times @ nat @ ( binomial @ ( plus_plus @ nat @ ( plus_plus @ nat @ M @ R ) @ K ) @ ( plus_plus @ nat @ M @ K ) ) @ ( binomial @ ( plus_plus @ nat @ M @ K ) @ K ) )
= ( times_times @ nat @ ( binomial @ ( plus_plus @ nat @ ( plus_plus @ nat @ M @ R ) @ K ) @ K ) @ ( binomial @ ( plus_plus @ nat @ M @ R ) @ M ) ) ) ).
% choose_mult_lemma
thf(fact_2354_not__is__unit__0,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ~ ( dvd_dvd @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).
% not_is_unit_0
thf(fact_2355_signed__take__bit__take__bit,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [M: nat,N: nat,A2: A] :
( ( bit_ri4674362597316999326ke_bit @ A @ M @ ( bit_se2584673776208193580ke_bit @ A @ N @ A2 ) )
= ( if @ ( A > A ) @ ( ord_less_eq @ nat @ N @ M ) @ ( bit_se2584673776208193580ke_bit @ A @ N ) @ ( bit_ri4674362597316999326ke_bit @ A @ M ) @ A2 ) ) ) ).
% signed_take_bit_take_bit
thf(fact_2356_pinf_I9_J,axiom,
! [B: $tType] :
( ( ( plus @ B )
& ( linorder @ B )
& ( dvd @ B ) )
=> ! [D2: B,S3: B] :
? [Z4: B] :
! [X4: B] :
( ( ord_less @ B @ Z4 @ X4 )
=> ( ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X4 @ S3 ) )
= ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X4 @ S3 ) ) ) ) ) ).
% pinf(9)
thf(fact_2357_pinf_I10_J,axiom,
! [B: $tType] :
( ( ( plus @ B )
& ( linorder @ B )
& ( dvd @ B ) )
=> ! [D2: B,S3: B] :
? [Z4: B] :
! [X4: B] :
( ( ord_less @ B @ Z4 @ X4 )
=> ( ( ~ ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X4 @ S3 ) ) )
= ( ~ ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X4 @ S3 ) ) ) ) ) ) ).
% pinf(10)
thf(fact_2358_minf_I9_J,axiom,
! [B: $tType] :
( ( ( plus @ B )
& ( linorder @ B )
& ( dvd @ B ) )
=> ! [D2: B,S3: B] :
? [Z4: B] :
! [X4: B] :
( ( ord_less @ B @ X4 @ Z4 )
=> ( ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X4 @ S3 ) )
= ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X4 @ S3 ) ) ) ) ) ).
% minf(9)
thf(fact_2359_minf_I10_J,axiom,
! [B: $tType] :
( ( ( plus @ B )
& ( linorder @ B )
& ( dvd @ B ) )
=> ! [D2: B,S3: B] :
? [Z4: B] :
! [X4: B] :
( ( ord_less @ B @ X4 @ Z4 )
=> ( ( ~ ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X4 @ S3 ) ) )
= ( ~ ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X4 @ S3 ) ) ) ) ) ) ).
% minf(10)
thf(fact_2360_is__unit__mult__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,B2: A] :
( ( dvd_dvd @ A @ ( times_times @ A @ A2 @ B2 ) @ ( one_one @ A ) )
= ( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
& ( dvd_dvd @ A @ B2 @ ( one_one @ A ) ) ) ) ) ).
% is_unit_mult_iff
thf(fact_2361_dvd__mult__unit__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ A2 @ ( times_times @ A @ C2 @ B2 ) )
= ( dvd_dvd @ A @ A2 @ C2 ) ) ) ) ).
% dvd_mult_unit_iff
thf(fact_2362_mult__unit__dvd__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ ( times_times @ A @ A2 @ B2 ) @ C2 )
= ( dvd_dvd @ A @ A2 @ C2 ) ) ) ) ).
% mult_unit_dvd_iff
thf(fact_2363_dvd__mult__unit__iff_H,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) )
= ( dvd_dvd @ A @ A2 @ C2 ) ) ) ) ).
% dvd_mult_unit_iff'
thf(fact_2364_mult__unit__dvd__iff_H,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ ( times_times @ A @ A2 @ B2 ) @ C2 )
= ( dvd_dvd @ A @ B2 @ C2 ) ) ) ) ).
% mult_unit_dvd_iff'
thf(fact_2365_unit__mult__left__cancel,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
=> ( ( ( times_times @ A @ A2 @ B2 )
= ( times_times @ A @ A2 @ C2 ) )
= ( B2 = C2 ) ) ) ) ).
% unit_mult_left_cancel
thf(fact_2366_unit__mult__right__cancel,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
=> ( ( ( times_times @ A @ B2 @ A2 )
= ( times_times @ A @ C2 @ A2 ) )
= ( B2 = C2 ) ) ) ) ).
% unit_mult_right_cancel
thf(fact_2367_dvd__div__eq__0__iff,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [B2: A,A2: A] :
( ( dvd_dvd @ A @ B2 @ A2 )
=> ( ( ( divide_divide @ A @ A2 @ B2 )
= ( zero_zero @ A ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ) ).
% dvd_div_eq_0_iff
thf(fact_2368_div__plus__div__distrib__dvd__right,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [C2: A,B2: A,A2: A] :
( ( dvd_dvd @ A @ C2 @ B2 )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 )
= ( plus_plus @ A @ ( divide_divide @ A @ A2 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ) ).
% div_plus_div_distrib_dvd_right
thf(fact_2369_div__plus__div__distrib__dvd__left,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( dvd_dvd @ A @ C2 @ A2 )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 )
= ( plus_plus @ A @ ( divide_divide @ A @ A2 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ) ).
% div_plus_div_distrib_dvd_left
thf(fact_2370_unit__div__cancel,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
=> ( ( ( divide_divide @ A @ B2 @ A2 )
= ( divide_divide @ A @ C2 @ A2 ) )
= ( B2 = C2 ) ) ) ) ).
% unit_div_cancel
thf(fact_2371_div__unit__dvd__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ ( divide_divide @ A @ A2 @ B2 ) @ C2 )
= ( dvd_dvd @ A @ A2 @ C2 ) ) ) ) ).
% div_unit_dvd_iff
thf(fact_2372_dvd__div__unit__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ A2 @ ( divide_divide @ A @ C2 @ B2 ) )
= ( dvd_dvd @ A @ A2 @ C2 ) ) ) ) ).
% dvd_div_unit_iff
thf(fact_2373_mod__0__imp__dvd,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [A2: A,B2: A] :
( ( ( modulo_modulo @ A @ A2 @ B2 )
= ( zero_zero @ A ) )
=> ( dvd_dvd @ A @ B2 @ A2 ) ) ) ).
% mod_0_imp_dvd
thf(fact_2374_dvd__eq__mod__eq__0,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ( ( dvd_dvd @ A )
= ( ^ [A3: A,B3: A] :
( ( modulo_modulo @ A @ B3 @ A3 )
= ( zero_zero @ A ) ) ) ) ) ).
% dvd_eq_mod_eq_0
thf(fact_2375_mod__eq__0__iff__dvd,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [A2: A,B2: A] :
( ( ( modulo_modulo @ A @ A2 @ B2 )
= ( zero_zero @ A ) )
= ( dvd_dvd @ A @ B2 @ A2 ) ) ) ).
% mod_eq_0_iff_dvd
thf(fact_2376_num_Oexhaust,axiom,
! [Y: num] :
( ( Y != one2 )
=> ( ! [X23: num] :
( Y
!= ( bit0 @ X23 ) )
=> ~ ! [X33: num] :
( Y
!= ( bit1 @ X33 ) ) ) ) ).
% num.exhaust
thf(fact_2377_take__bit__unset__bit__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,M: nat,A2: A] :
( ( ( ord_less_eq @ nat @ N @ M )
=> ( ( bit_se2584673776208193580ke_bit @ A @ N @ ( bit_se2638667681897837118et_bit @ A @ M @ A2 ) )
= ( bit_se2584673776208193580ke_bit @ A @ N @ A2 ) ) )
& ( ~ ( ord_less_eq @ nat @ N @ M )
=> ( ( bit_se2584673776208193580ke_bit @ A @ N @ ( bit_se2638667681897837118et_bit @ A @ M @ A2 ) )
= ( bit_se2638667681897837118et_bit @ A @ M @ ( bit_se2584673776208193580ke_bit @ A @ N @ A2 ) ) ) ) ) ) ).
% take_bit_unset_bit_eq
thf(fact_2378_take__bit__set__bit__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,M: nat,A2: A] :
( ( ( ord_less_eq @ nat @ N @ M )
=> ( ( bit_se2584673776208193580ke_bit @ A @ N @ ( bit_se5668285175392031749et_bit @ A @ M @ A2 ) )
= ( bit_se2584673776208193580ke_bit @ A @ N @ A2 ) ) )
& ( ~ ( ord_less_eq @ nat @ N @ M )
=> ( ( bit_se2584673776208193580ke_bit @ A @ N @ ( bit_se5668285175392031749et_bit @ A @ M @ A2 ) )
= ( bit_se5668285175392031749et_bit @ A @ M @ ( bit_se2584673776208193580ke_bit @ A @ N @ A2 ) ) ) ) ) ) ).
% take_bit_set_bit_eq
thf(fact_2379_binomial__le__pow,axiom,
! [R: nat,N: nat] :
( ( ord_less_eq @ nat @ R @ N )
=> ( ord_less_eq @ nat @ ( binomial @ N @ R ) @ ( power_power @ nat @ N @ R ) ) ) ).
% binomial_le_pow
thf(fact_2380_take__bit__flip__bit__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,M: nat,A2: A] :
( ( ( ord_less_eq @ nat @ N @ M )
=> ( ( bit_se2584673776208193580ke_bit @ A @ N @ ( bit_se8732182000553998342ip_bit @ A @ M @ A2 ) )
= ( bit_se2584673776208193580ke_bit @ A @ N @ A2 ) ) )
& ( ~ ( ord_less_eq @ nat @ N @ M )
=> ( ( bit_se2584673776208193580ke_bit @ A @ N @ ( bit_se8732182000553998342ip_bit @ A @ M @ A2 ) )
= ( bit_se8732182000553998342ip_bit @ A @ M @ ( bit_se2584673776208193580ke_bit @ A @ N @ A2 ) ) ) ) ) ) ).
% take_bit_flip_bit_eq
thf(fact_2381_binomial__gbinomial,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [N: nat,K: nat] :
( ( semiring_1_of_nat @ A @ ( binomial @ N @ K ) )
= ( gbinomial @ A @ ( semiring_1_of_nat @ A @ N ) @ K ) ) ) ).
% binomial_gbinomial
thf(fact_2382_dvd__pos__nat,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( dvd_dvd @ nat @ M @ N )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ M ) ) ) ).
% dvd_pos_nat
thf(fact_2383_nat__dvd__not__less,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( ( ord_less @ nat @ M @ N )
=> ~ ( dvd_dvd @ nat @ N @ M ) ) ) ).
% nat_dvd_not_less
thf(fact_2384_dvd__power__le,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [X2: A,Y: A,N: nat,M: nat] :
( ( dvd_dvd @ A @ X2 @ Y )
=> ( ( ord_less_eq @ nat @ N @ M )
=> ( dvd_dvd @ A @ ( power_power @ A @ X2 @ N ) @ ( power_power @ A @ Y @ M ) ) ) ) ) ).
% dvd_power_le
thf(fact_2385_power__le__dvd,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A2: A,N: nat,B2: A,M: nat] :
( ( dvd_dvd @ A @ ( power_power @ A @ A2 @ N ) @ B2 )
=> ( ( ord_less_eq @ nat @ M @ N )
=> ( dvd_dvd @ A @ ( power_power @ A @ A2 @ M ) @ B2 ) ) ) ) ).
% power_le_dvd
thf(fact_2386_le__imp__power__dvd,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [M: nat,N: nat,A2: A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( dvd_dvd @ A @ ( power_power @ A @ A2 @ M ) @ ( power_power @ A @ A2 @ N ) ) ) ) ).
% le_imp_power_dvd
thf(fact_2387_dvd__minus__self,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd @ nat @ M @ ( minus_minus @ nat @ N @ M ) )
= ( ( ord_less @ nat @ N @ M )
| ( dvd_dvd @ nat @ M @ N ) ) ) ).
% dvd_minus_self
thf(fact_2388_zdvd__antisym__nonneg,axiom,
! [M: int,N: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ M )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N )
=> ( ( dvd_dvd @ int @ M @ N )
=> ( ( dvd_dvd @ int @ N @ M )
=> ( M = N ) ) ) ) ) ).
% zdvd_antisym_nonneg
thf(fact_2389_zdvd__not__zless,axiom,
! [M: int,N: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ M )
=> ( ( ord_less @ int @ M @ N )
=> ~ ( dvd_dvd @ int @ N @ M ) ) ) ).
% zdvd_not_zless
thf(fact_2390_less__eq__dvd__minus,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( dvd_dvd @ nat @ M @ N )
= ( dvd_dvd @ nat @ M @ ( minus_minus @ nat @ N @ M ) ) ) ) ).
% less_eq_dvd_minus
thf(fact_2391_dvd__diffD1,axiom,
! [K: nat,M: nat,N: nat] :
( ( dvd_dvd @ nat @ K @ ( minus_minus @ nat @ M @ N ) )
=> ( ( dvd_dvd @ nat @ K @ M )
=> ( ( ord_less_eq @ nat @ N @ M )
=> ( dvd_dvd @ nat @ K @ N ) ) ) ) ).
% dvd_diffD1
thf(fact_2392_dvd__diffD,axiom,
! [K: nat,M: nat,N: nat] :
( ( dvd_dvd @ nat @ K @ ( minus_minus @ nat @ M @ N ) )
=> ( ( dvd_dvd @ nat @ K @ N )
=> ( ( ord_less_eq @ nat @ N @ M )
=> ( dvd_dvd @ nat @ K @ M ) ) ) ) ).
% dvd_diffD
thf(fact_2393_zdvd__mult__cancel,axiom,
! [K: int,M: int,N: int] :
( ( dvd_dvd @ int @ ( times_times @ int @ K @ M ) @ ( times_times @ int @ K @ N ) )
=> ( ( K
!= ( zero_zero @ int ) )
=> ( dvd_dvd @ int @ M @ N ) ) ) ).
% zdvd_mult_cancel
thf(fact_2394_zdvd__mono,axiom,
! [K: int,M: int,T2: int] :
( ( K
!= ( zero_zero @ int ) )
=> ( ( dvd_dvd @ int @ M @ T2 )
= ( dvd_dvd @ int @ ( times_times @ int @ K @ M ) @ ( times_times @ int @ K @ T2 ) ) ) ) ).
% zdvd_mono
thf(fact_2395_bezout__add__nat,axiom,
! [A2: nat,B2: nat] :
? [D6: nat,X3: nat,Y2: nat] :
( ( dvd_dvd @ nat @ D6 @ A2 )
& ( dvd_dvd @ nat @ D6 @ B2 )
& ( ( ( times_times @ nat @ A2 @ X3 )
= ( plus_plus @ nat @ ( times_times @ nat @ B2 @ Y2 ) @ D6 ) )
| ( ( times_times @ nat @ B2 @ X3 )
= ( plus_plus @ nat @ ( times_times @ nat @ A2 @ Y2 ) @ D6 ) ) ) ) ).
% bezout_add_nat
thf(fact_2396_bezout__lemma__nat,axiom,
! [D2: nat,A2: nat,B2: nat,X2: nat,Y: nat] :
( ( dvd_dvd @ nat @ D2 @ A2 )
=> ( ( dvd_dvd @ nat @ D2 @ B2 )
=> ( ( ( ( times_times @ nat @ A2 @ X2 )
= ( plus_plus @ nat @ ( times_times @ nat @ B2 @ Y ) @ D2 ) )
| ( ( times_times @ nat @ B2 @ X2 )
= ( plus_plus @ nat @ ( times_times @ nat @ A2 @ Y ) @ D2 ) ) )
=> ? [X3: nat,Y2: nat] :
( ( dvd_dvd @ nat @ D2 @ A2 )
& ( dvd_dvd @ nat @ D2 @ ( plus_plus @ nat @ A2 @ B2 ) )
& ( ( ( times_times @ nat @ A2 @ X3 )
= ( plus_plus @ nat @ ( times_times @ nat @ ( plus_plus @ nat @ A2 @ B2 ) @ Y2 ) @ D2 ) )
| ( ( times_times @ nat @ ( plus_plus @ nat @ A2 @ B2 ) @ X3 )
= ( plus_plus @ nat @ ( times_times @ nat @ A2 @ Y2 ) @ D2 ) ) ) ) ) ) ) ).
% bezout_lemma_nat
thf(fact_2397_bezout1__nat,axiom,
! [A2: nat,B2: nat] :
? [D6: nat,X3: nat,Y2: nat] :
( ( dvd_dvd @ nat @ D6 @ A2 )
& ( dvd_dvd @ nat @ D6 @ B2 )
& ( ( ( minus_minus @ nat @ ( times_times @ nat @ A2 @ X3 ) @ ( times_times @ nat @ B2 @ Y2 ) )
= D6 )
| ( ( minus_minus @ nat @ ( times_times @ nat @ B2 @ X3 ) @ ( times_times @ nat @ A2 @ Y2 ) )
= D6 ) ) ) ).
% bezout1_nat
thf(fact_2398_zdvd__reduce,axiom,
! [K: int,N: int,M: int] :
( ( dvd_dvd @ int @ K @ ( plus_plus @ int @ N @ ( times_times @ int @ K @ M ) ) )
= ( dvd_dvd @ int @ K @ N ) ) ).
% zdvd_reduce
thf(fact_2399_zdvd__period,axiom,
! [A2: int,D2: int,X2: int,T2: int,C2: int] :
( ( dvd_dvd @ int @ A2 @ D2 )
=> ( ( dvd_dvd @ int @ A2 @ ( plus_plus @ int @ X2 @ T2 ) )
= ( dvd_dvd @ int @ A2 @ ( plus_plus @ int @ ( plus_plus @ int @ X2 @ ( times_times @ int @ C2 @ D2 ) ) @ T2 ) ) ) ) ).
% zdvd_period
thf(fact_2400_fact__dvd,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N: nat,M: nat] :
( ( ord_less_eq @ nat @ N @ M )
=> ( dvd_dvd @ A @ ( semiring_char_0_fact @ A @ N ) @ ( semiring_char_0_fact @ A @ M ) ) ) ) ).
% fact_dvd
thf(fact_2401_take__bit__eq__0__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A2: A] :
( ( ( bit_se2584673776208193580ke_bit @ A @ N @ A2 )
= ( zero_zero @ A ) )
= ( dvd_dvd @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) @ A2 ) ) ) ).
% take_bit_eq_0_iff
thf(fact_2402_inc_Osimps_I3_J,axiom,
! [X2: num] :
( ( inc @ ( bit1 @ X2 ) )
= ( bit0 @ ( inc @ X2 ) ) ) ).
% inc.simps(3)
thf(fact_2403_inc_Osimps_I2_J,axiom,
! [X2: num] :
( ( inc @ ( bit0 @ X2 ) )
= ( bit1 @ X2 ) ) ).
% inc.simps(2)
thf(fact_2404_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_2405_zero__less__binomial,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ N )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ ( binomial @ N @ K ) ) ) ).
% zero_less_binomial
thf(fact_2406_take__bit__signed__take__bit,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [M: nat,N: nat,A2: A] :
( ( ord_less_eq @ nat @ M @ ( suc @ N ) )
=> ( ( bit_se2584673776208193580ke_bit @ A @ M @ ( bit_ri4674362597316999326ke_bit @ A @ N @ A2 ) )
= ( bit_se2584673776208193580ke_bit @ A @ M @ A2 ) ) ) ) ).
% take_bit_signed_take_bit
thf(fact_2407_Suc__times__binomial__add,axiom,
! [A2: nat,B2: nat] :
( ( times_times @ nat @ ( suc @ A2 ) @ ( binomial @ ( suc @ ( plus_plus @ nat @ A2 @ B2 ) ) @ ( suc @ A2 ) ) )
= ( times_times @ nat @ ( suc @ B2 ) @ ( binomial @ ( suc @ ( plus_plus @ nat @ A2 @ B2 ) ) @ A2 ) ) ) ).
% Suc_times_binomial_add
thf(fact_2408_choose__mult,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ M )
=> ( ( ord_less_eq @ nat @ M @ N )
=> ( ( times_times @ nat @ ( binomial @ N @ M ) @ ( binomial @ M @ K ) )
= ( times_times @ nat @ ( binomial @ N @ K ) @ ( binomial @ ( minus_minus @ nat @ N @ K ) @ ( minus_minus @ nat @ M @ K ) ) ) ) ) ) ).
% choose_mult
thf(fact_2409_unity__coeff__ex,axiom,
! [A: $tType] :
( ( ( dvd @ A )
& ( semiring_0 @ A ) )
=> ! [P: A > $o,L: A] :
( ( ? [X: A] : ( P @ ( times_times @ A @ L @ X ) ) )
= ( ? [X: A] :
( ( dvd_dvd @ A @ L @ ( plus_plus @ A @ X @ ( zero_zero @ A ) ) )
& ( P @ X ) ) ) ) ) ).
% unity_coeff_ex
thf(fact_2410_unit__dvdE,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,B2: A] :
( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
=> ~ ( ( A2
!= ( zero_zero @ A ) )
=> ! [C4: A] :
( B2
!= ( times_times @ A @ A2 @ C4 ) ) ) ) ) ).
% unit_dvdE
thf(fact_2411_binomial__absorb__comp,axiom,
! [N: nat,K: nat] :
( ( times_times @ nat @ ( minus_minus @ nat @ N @ K ) @ ( binomial @ N @ K ) )
= ( times_times @ nat @ N @ ( binomial @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ K ) ) ) ).
% binomial_absorb_comp
thf(fact_2412_inf__period_I3_J,axiom,
! [A: $tType] :
( ( ( comm_ring @ A )
& ( dvd @ A ) )
=> ! [D2: A,D3: A,T2: A] :
( ( dvd_dvd @ A @ D2 @ D3 )
=> ! [X4: A,K5: A] :
( ( dvd_dvd @ A @ D2 @ ( plus_plus @ A @ X4 @ T2 ) )
= ( dvd_dvd @ A @ D2 @ ( plus_plus @ A @ ( minus_minus @ A @ X4 @ ( times_times @ A @ K5 @ D3 ) ) @ T2 ) ) ) ) ) ).
% inf_period(3)
thf(fact_2413_inf__period_I4_J,axiom,
! [A: $tType] :
( ( ( comm_ring @ A )
& ( dvd @ A ) )
=> ! [D2: A,D3: A,T2: A] :
( ( dvd_dvd @ A @ D2 @ D3 )
=> ! [X4: A,K5: A] :
( ( ~ ( dvd_dvd @ A @ D2 @ ( plus_plus @ A @ X4 @ T2 ) ) )
= ( ~ ( dvd_dvd @ A @ D2 @ ( plus_plus @ A @ ( minus_minus @ A @ X4 @ ( times_times @ A @ K5 @ D3 ) ) @ T2 ) ) ) ) ) ) ).
% inf_period(4)
thf(fact_2414_dvd__div__eq__mult,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ A2 @ B2 )
=> ( ( ( divide_divide @ A @ B2 @ A2 )
= C2 )
= ( B2
= ( times_times @ A @ C2 @ A2 ) ) ) ) ) ) ).
% dvd_div_eq_mult
thf(fact_2415_div__dvd__iff__mult,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ B2 @ A2 )
=> ( ( dvd_dvd @ A @ ( divide_divide @ A @ A2 @ B2 ) @ C2 )
= ( dvd_dvd @ A @ A2 @ ( times_times @ A @ C2 @ B2 ) ) ) ) ) ) ).
% div_dvd_iff_mult
thf(fact_2416_dvd__div__iff__mult,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C2: A,B2: A,A2: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ C2 @ B2 )
=> ( ( dvd_dvd @ A @ A2 @ ( divide_divide @ A @ B2 @ C2 ) )
= ( dvd_dvd @ A @ ( times_times @ A @ A2 @ C2 ) @ B2 ) ) ) ) ) ).
% dvd_div_iff_mult
thf(fact_2417_dvd__div__div__eq__mult,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,C2: A,B2: A,D2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( C2
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ A2 @ B2 )
=> ( ( dvd_dvd @ A @ C2 @ D2 )
=> ( ( ( divide_divide @ A @ B2 @ A2 )
= ( divide_divide @ A @ D2 @ C2 ) )
= ( ( times_times @ A @ B2 @ C2 )
= ( times_times @ A @ A2 @ D2 ) ) ) ) ) ) ) ) ).
% dvd_div_div_eq_mult
thf(fact_2418_take__bit__decr__eq,axiom,
! [N: nat,K: int] :
( ( ( bit_se2584673776208193580ke_bit @ int @ N @ K )
!= ( zero_zero @ int ) )
=> ( ( bit_se2584673776208193580ke_bit @ int @ N @ ( minus_minus @ int @ K @ ( one_one @ int ) ) )
= ( minus_minus @ int @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) @ ( one_one @ int ) ) ) ) ).
% take_bit_decr_eq
thf(fact_2419_unit__div__eq__0__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A2: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( ( divide_divide @ A @ A2 @ B2 )
= ( zero_zero @ A ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ) ).
% unit_div_eq_0_iff
thf(fact_2420_numeral__Bit1,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ! [N: num] :
( ( numeral_numeral @ A @ ( bit1 @ N ) )
= ( plus_plus @ A @ ( plus_plus @ A @ ( numeral_numeral @ A @ N ) @ ( numeral_numeral @ A @ N ) ) @ ( one_one @ A ) ) ) ) ).
% numeral_Bit1
thf(fact_2421_unit__eq__div1,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( ( divide_divide @ A @ A2 @ B2 )
= C2 )
= ( A2
= ( times_times @ A @ C2 @ B2 ) ) ) ) ) ).
% unit_eq_div1
thf(fact_2422_unit__eq__div2,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( A2
= ( divide_divide @ A @ C2 @ B2 ) )
= ( ( times_times @ A @ A2 @ B2 )
= C2 ) ) ) ) ).
% unit_eq_div2
thf(fact_2423_div__mult__unit2,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C2: A,B2: A,A2: A] :
( ( dvd_dvd @ A @ C2 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ B2 @ A2 )
=> ( ( divide_divide @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) )
= ( divide_divide @ A @ ( divide_divide @ A @ A2 @ B2 ) @ C2 ) ) ) ) ) ).
% div_mult_unit2
thf(fact_2424_unit__div__commute,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( times_times @ A @ ( divide_divide @ A @ A2 @ B2 ) @ C2 )
= ( divide_divide @ A @ ( times_times @ A @ A2 @ C2 ) @ B2 ) ) ) ) ).
% unit_div_commute
thf(fact_2425_unit__div__mult__swap,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( dvd_dvd @ A @ C2 @ ( one_one @ A ) )
=> ( ( times_times @ A @ A2 @ ( divide_divide @ A @ B2 @ C2 ) )
= ( divide_divide @ A @ ( times_times @ A @ A2 @ B2 ) @ C2 ) ) ) ) ).
% unit_div_mult_swap
thf(fact_2426_is__unit__div__mult2__eq,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,C2: A,A2: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ C2 @ ( one_one @ A ) )
=> ( ( divide_divide @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) )
= ( divide_divide @ A @ ( divide_divide @ A @ A2 @ B2 ) @ C2 ) ) ) ) ) ).
% is_unit_div_mult2_eq
thf(fact_2427_unit__imp__mod__eq__0,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [B2: A,A2: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( modulo_modulo @ A @ A2 @ B2 )
= ( zero_zero @ A ) ) ) ) ).
% unit_imp_mod_eq_0
thf(fact_2428_is__unit__power__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,N: nat] :
( ( dvd_dvd @ A @ ( power_power @ A @ A2 @ N ) @ ( one_one @ A ) )
= ( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
| ( N
= ( zero_zero @ nat ) ) ) ) ) ).
% is_unit_power_iff
thf(fact_2429_eval__nat__numeral_I3_J,axiom,
! [N: num] :
( ( numeral_numeral @ nat @ ( bit1 @ N ) )
= ( suc @ ( numeral_numeral @ nat @ ( bit0 @ N ) ) ) ) ).
% eval_nat_numeral(3)
thf(fact_2430_dvd__imp__le,axiom,
! [K: nat,N: nat] :
( ( dvd_dvd @ nat @ K @ N )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less_eq @ nat @ K @ N ) ) ) ).
% dvd_imp_le
thf(fact_2431_dvd__mult__cancel,axiom,
! [K: nat,M: nat,N: nat] :
( ( dvd_dvd @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( dvd_dvd @ nat @ M @ N ) ) ) ).
% dvd_mult_cancel
thf(fact_2432_nat__mult__dvd__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ( dvd_dvd @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N ) )
= ( dvd_dvd @ nat @ M @ N ) ) ) ).
% nat_mult_dvd_cancel1
thf(fact_2433_bezout__add__strong__nat,axiom,
! [A2: nat,B2: nat] :
( ( A2
!= ( zero_zero @ nat ) )
=> ? [D6: nat,X3: nat,Y2: nat] :
( ( dvd_dvd @ nat @ D6 @ A2 )
& ( dvd_dvd @ nat @ D6 @ B2 )
& ( ( times_times @ nat @ A2 @ X3 )
= ( plus_plus @ nat @ ( times_times @ nat @ B2 @ Y2 ) @ D6 ) ) ) ) ).
% bezout_add_strong_nat
thf(fact_2434_take__bit__Suc__bit1,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N: nat,K: num] :
( ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N ) @ ( numeral_numeral @ A @ ( bit1 @ K ) ) )
= ( plus_plus @ A @ ( times_times @ A @ ( bit_se2584673776208193580ke_bit @ A @ N @ ( numeral_numeral @ A @ K ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) ) ) ) ).
% take_bit_Suc_bit1
thf(fact_2435_zdvd__imp__le,axiom,
! [Z2: int,N: int] :
( ( dvd_dvd @ int @ Z2 @ N )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ N )
=> ( ord_less_eq @ int @ Z2 @ N ) ) ) ).
% zdvd_imp_le
thf(fact_2436_fact__fact__dvd__fact,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [K: nat,N: nat] : ( dvd_dvd @ A @ ( times_times @ A @ ( semiring_char_0_fact @ A @ K ) @ ( semiring_char_0_fact @ A @ N ) ) @ ( semiring_char_0_fact @ A @ ( plus_plus @ nat @ K @ N ) ) ) ) ).
% fact_fact_dvd_fact
thf(fact_2437_mod__greater__zero__iff__not__dvd,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( modulo_modulo @ nat @ M @ N ) )
= ( ~ ( dvd_dvd @ nat @ N @ M ) ) ) ).
% mod_greater_zero_iff_not_dvd
thf(fact_2438_dvd__imp__le__int,axiom,
! [I: int,D2: int] :
( ( I
!= ( zero_zero @ int ) )
=> ( ( dvd_dvd @ int @ D2 @ I )
=> ( ord_less_eq @ int @ ( abs_abs @ int @ D2 ) @ ( abs_abs @ int @ I ) ) ) ) ).
% dvd_imp_le_int
thf(fact_2439_mod__eq__dvd__iff__nat,axiom,
! [N: nat,M: nat,Q2: nat] :
( ( ord_less_eq @ nat @ N @ M )
=> ( ( ( modulo_modulo @ nat @ M @ Q2 )
= ( modulo_modulo @ nat @ N @ Q2 ) )
= ( dvd_dvd @ nat @ Q2 @ ( minus_minus @ nat @ M @ N ) ) ) ) ).
% mod_eq_dvd_iff_nat
thf(fact_2440_of__int__divide__in__Ints,axiom,
! [A: $tType] :
( ( idom_divide @ A )
=> ! [B2: int,A2: int] :
( ( dvd_dvd @ int @ B2 @ A2 )
=> ( member @ A @ ( divide_divide @ A @ ( ring_1_of_int @ A @ A2 ) @ ( ring_1_of_int @ A @ B2 ) ) @ ( ring_1_Ints @ A ) ) ) ) ).
% of_int_divide_in_Ints
thf(fact_2441_numeral__code_I3_J,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ! [N: num] :
( ( numeral_numeral @ A @ ( bit1 @ N ) )
= ( plus_plus @ A @ ( plus_plus @ A @ ( numeral_numeral @ A @ N ) @ ( numeral_numeral @ A @ N ) ) @ ( one_one @ A ) ) ) ) ).
% numeral_code(3)
thf(fact_2442_dvd__fact,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ M )
=> ( ( ord_less_eq @ nat @ M @ N )
=> ( dvd_dvd @ nat @ M @ ( semiring_char_0_fact @ nat @ N ) ) ) ) ).
% dvd_fact
thf(fact_2443_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_2444_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_2445_stable__imp__take__bit__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A,N: nat] :
( ( ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= A2 )
=> ( ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
=> ( ( bit_se2584673776208193580ke_bit @ A @ N @ A2 )
= ( zero_zero @ A ) ) )
& ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
=> ( ( bit_se2584673776208193580ke_bit @ A @ N @ A2 )
= ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ A ) ) ) ) ) ) ) ).
% stable_imp_take_bit_eq
thf(fact_2446_gcd__nat_Oordering__top__axioms,axiom,
( ordering_top @ nat @ ( dvd_dvd @ nat )
@ ^ [M2: nat,N4: nat] :
( ( dvd_dvd @ nat @ M2 @ N4 )
& ( M2 != N4 ) )
@ ( zero_zero @ nat ) ) ).
% gcd_nat.ordering_top_axioms
thf(fact_2447_binomial__absorption,axiom,
! [K: nat,N: nat] :
( ( times_times @ nat @ ( suc @ K ) @ ( binomial @ N @ ( suc @ K ) ) )
= ( times_times @ nat @ N @ ( binomial @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ K ) ) ) ).
% binomial_absorption
thf(fact_2448_even__zero,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( zero_zero @ A ) ) ) ).
% even_zero
thf(fact_2449_is__unitE,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,C2: A] :
( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
=> ~ ( ( A2
!= ( zero_zero @ A ) )
=> ! [B4: A] :
( ( B4
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ B4 @ ( one_one @ A ) )
=> ( ( ( divide_divide @ A @ ( one_one @ A ) @ A2 )
= B4 )
=> ( ( ( divide_divide @ A @ ( one_one @ A ) @ B4 )
= A2 )
=> ( ( ( times_times @ A @ A2 @ B4 )
= ( one_one @ A ) )
=> ( ( divide_divide @ A @ C2 @ A2 )
!= ( times_times @ A @ C2 @ B4 ) ) ) ) ) ) ) ) ) ) ).
% is_unitE
thf(fact_2450_is__unit__div__mult__cancel__left,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,B2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( divide_divide @ A @ A2 @ ( times_times @ A @ A2 @ B2 ) )
= ( divide_divide @ A @ ( one_one @ A ) @ B2 ) ) ) ) ) ).
% is_unit_div_mult_cancel_left
thf(fact_2451_is__unit__div__mult__cancel__right,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,B2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( divide_divide @ A @ A2 @ ( times_times @ A @ B2 @ A2 ) )
= ( divide_divide @ A @ ( one_one @ A ) @ B2 ) ) ) ) ) ).
% is_unit_div_mult_cancel_right
thf(fact_2452_odd__even__add,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A2: A,B2: A] :
( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
=> ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 )
=> ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).
% odd_even_add
thf(fact_2453_odd__one,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( one_one @ A ) ) ) ).
% odd_one
thf(fact_2454_odd__mod__4__div__2,axiom,
! [N: nat] :
( ( ( modulo_modulo @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) )
= ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) )
=> ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% odd_mod_4_div_2
thf(fact_2455_cong__exp__iff__simps_I3_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [N: num,Q2: num] :
( ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit1 @ N ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q2 ) ) )
!= ( zero_zero @ A ) ) ) ).
% cong_exp_iff_simps(3)
thf(fact_2456_dvd__power__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [X2: A,M: nat,N: nat] :
( ( X2
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ ( power_power @ A @ X2 @ M ) @ ( power_power @ A @ X2 @ N ) )
= ( ( dvd_dvd @ A @ X2 @ ( one_one @ A ) )
| ( ord_less_eq @ nat @ M @ N ) ) ) ) ) ).
% dvd_power_iff
thf(fact_2457_dvd__power,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [N: nat,X2: A] :
( ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
| ( X2
= ( one_one @ A ) ) )
=> ( dvd_dvd @ A @ X2 @ ( power_power @ A @ X2 @ N ) ) ) ) ).
% dvd_power
thf(fact_2458_numeral__3__eq__3,axiom,
( ( numeral_numeral @ nat @ ( bit1 @ one2 ) )
= ( suc @ ( suc @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% numeral_3_eq_3
thf(fact_2459_Suc3__eq__add__3,axiom,
! [N: nat] :
( ( suc @ ( suc @ ( suc @ N ) ) )
= ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) @ N ) ) ).
% Suc3_eq_add_3
thf(fact_2460_binomial__fact__lemma,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ N )
=> ( ( times_times @ nat @ ( times_times @ nat @ ( semiring_char_0_fact @ nat @ K ) @ ( semiring_char_0_fact @ nat @ ( minus_minus @ nat @ N @ K ) ) ) @ ( binomial @ N @ K ) )
= ( semiring_char_0_fact @ nat @ N ) ) ) ).
% binomial_fact_lemma
thf(fact_2461_dvd__mult__cancel2,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( ( dvd_dvd @ nat @ ( times_times @ nat @ N @ M ) @ M )
= ( N
= ( one_one @ nat ) ) ) ) ).
% dvd_mult_cancel2
thf(fact_2462_dvd__mult__cancel1,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( ( dvd_dvd @ nat @ ( times_times @ nat @ M @ N ) @ M )
= ( N
= ( one_one @ nat ) ) ) ) ).
% dvd_mult_cancel1
thf(fact_2463_choose__dvd,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [K: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ N )
=> ( dvd_dvd @ A @ ( times_times @ A @ ( semiring_char_0_fact @ A @ K ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ N @ K ) ) ) @ ( semiring_char_0_fact @ A @ N ) ) ) ) ).
% choose_dvd
thf(fact_2464_even__even__mod__4__iff,axiom,
! [N: nat] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
= ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( modulo_modulo @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ).
% even_even_mod_4_iff
thf(fact_2465_dvd__minus__add,axiom,
! [Q2: nat,N: nat,R: nat,M: nat] :
( ( ord_less_eq @ nat @ Q2 @ N )
=> ( ( ord_less_eq @ nat @ Q2 @ ( times_times @ nat @ R @ M ) )
=> ( ( dvd_dvd @ nat @ M @ ( minus_minus @ nat @ N @ Q2 ) )
= ( dvd_dvd @ nat @ M @ ( plus_plus @ nat @ N @ ( minus_minus @ nat @ ( times_times @ nat @ R @ M ) @ Q2 ) ) ) ) ) ) ).
% dvd_minus_add
thf(fact_2466_mod__nat__eqI,axiom,
! [R: nat,N: nat,M: nat] :
( ( ord_less @ nat @ R @ N )
=> ( ( ord_less_eq @ nat @ R @ M )
=> ( ( dvd_dvd @ nat @ N @ ( minus_minus @ nat @ M @ R ) )
=> ( ( modulo_modulo @ nat @ M @ N )
= R ) ) ) ) ).
% mod_nat_eqI
thf(fact_2467_zdvd__mult__cancel1,axiom,
! [M: int,N: int] :
( ( M
!= ( zero_zero @ int ) )
=> ( ( dvd_dvd @ int @ ( times_times @ int @ M @ N ) @ M )
= ( ( abs_abs @ int @ N )
= ( one_one @ int ) ) ) ) ).
% zdvd_mult_cancel1
thf(fact_2468_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_2469_power__dvd__imp__le,axiom,
! [I: nat,M: nat,N: nat] :
( ( dvd_dvd @ nat @ ( power_power @ nat @ I @ M ) @ ( power_power @ nat @ I @ N ) )
=> ( ( ord_less @ nat @ ( one_one @ nat ) @ I )
=> ( ord_less_eq @ nat @ M @ N ) ) ) ).
% power_dvd_imp_le
thf(fact_2470_take__bit__Suc__minus__bit1,axiom,
! [N: nat,K: num] :
( ( bit_se2584673776208193580ke_bit @ int @ ( suc @ N ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ K ) ) ) )
= ( plus_plus @ int @ ( times_times @ int @ ( bit_se2584673776208193580ke_bit @ int @ N @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( inc @ K ) ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( one_one @ int ) ) ) ).
% take_bit_Suc_minus_bit1
thf(fact_2471_num_Osize_I6_J,axiom,
! [X32: num] :
( ( size_size @ num @ ( bit1 @ X32 ) )
= ( plus_plus @ nat @ ( size_size @ num @ X32 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% num.size(6)
thf(fact_2472_binomial__ge__n__over__k__pow__k,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [K: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ N )
=> ( ord_less_eq @ A @ ( power_power @ A @ ( divide_divide @ A @ ( semiring_1_of_nat @ A @ N ) @ ( semiring_1_of_nat @ A @ K ) ) @ K ) @ ( semiring_1_of_nat @ A @ ( binomial @ N @ K ) ) ) ) ) ).
% binomial_ge_n_over_k_pow_k
thf(fact_2473_choose__reduce__nat,axiom,
! [N: nat,K: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ( binomial @ N @ K )
= ( plus_plus @ nat @ ( binomial @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ ( minus_minus @ nat @ K @ ( one_one @ nat ) ) ) @ ( binomial @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ K ) ) ) ) ) ).
% choose_reduce_nat
thf(fact_2474_times__binomial__minus1__eq,axiom,
! [K: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ( times_times @ nat @ K @ ( binomial @ N @ K ) )
= ( times_times @ nat @ N @ ( binomial @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ ( minus_minus @ nat @ K @ ( one_one @ nat ) ) ) ) ) ) ).
% times_binomial_minus1_eq
thf(fact_2475_binomial__le__pow2,axiom,
! [N: nat,K: nat] : ( ord_less_eq @ nat @ ( binomial @ N @ K ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ).
% binomial_le_pow2
thf(fact_2476_even__iff__mod__2__eq__zero,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A2: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
= ( ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( zero_zero @ A ) ) ) ) ).
% even_iff_mod_2_eq_zero
thf(fact_2477_cong__exp__iff__simps_I7_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [Q2: num,N: num] :
( ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ one2 ) @ ( numeral_numeral @ A @ ( bit0 @ Q2 ) ) )
= ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit1 @ N ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q2 ) ) ) )
= ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ N ) @ ( numeral_numeral @ A @ Q2 ) )
= ( zero_zero @ A ) ) ) ) ).
% cong_exp_iff_simps(7)
thf(fact_2478_cong__exp__iff__simps_I11_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M: num,Q2: num] :
( ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit1 @ M ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q2 ) ) )
= ( modulo_modulo @ A @ ( numeral_numeral @ A @ one2 ) @ ( numeral_numeral @ A @ ( bit0 @ Q2 ) ) ) )
= ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ Q2 ) )
= ( zero_zero @ A ) ) ) ) ).
% cong_exp_iff_simps(11)
thf(fact_2479_odd__iff__mod__2__eq__one,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A2: A] :
( ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) )
= ( ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( one_one @ A ) ) ) ) ).
% odd_iff_mod_2_eq_one
thf(fact_2480_power__mono__odd,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: nat,A2: A,B2: A] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ord_less_eq @ A @ ( power_power @ A @ A2 @ N ) @ ( power_power @ A @ B2 @ N ) ) ) ) ) ).
% power_mono_odd
thf(fact_2481_odd__pos,axiom,
! [N: nat] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% odd_pos
thf(fact_2482_binomial__altdef__nat,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ N )
=> ( ( binomial @ N @ K )
= ( divide_divide @ nat @ ( semiring_char_0_fact @ nat @ N ) @ ( times_times @ nat @ ( semiring_char_0_fact @ nat @ K ) @ ( semiring_char_0_fact @ nat @ ( minus_minus @ nat @ N @ K ) ) ) ) ) ) ).
% binomial_altdef_nat
thf(fact_2483_Suc__mod__eq__add3__mod,axiom,
! [M: nat,N: nat] :
( ( modulo_modulo @ nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ N )
= ( modulo_modulo @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) @ M ) @ N ) ) ).
% Suc_mod_eq_add3_mod
thf(fact_2484_Suc__div__eq__add3__div,axiom,
! [M: nat,N: nat] :
( ( divide_divide @ nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ N )
= ( divide_divide @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) @ M ) @ N ) ) ).
% Suc_div_eq_add3_div
thf(fact_2485_even__unset__bit__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,A2: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se2638667681897837118et_bit @ A @ M @ A2 ) )
= ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
| ( M
= ( zero_zero @ nat ) ) ) ) ) ).
% even_unset_bit_iff
thf(fact_2486_even__set__bit__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,A2: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5668285175392031749et_bit @ A @ M @ A2 ) )
= ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
& ( M
!= ( zero_zero @ nat ) ) ) ) ) ).
% even_set_bit_iff
thf(fact_2487_even__flip__bit__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,A2: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se8732182000553998342ip_bit @ A @ M @ A2 ) )
= ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
!= ( M
= ( zero_zero @ nat ) ) ) ) ) ).
% even_flip_bit_iff
thf(fact_2488_even__diff__iff,axiom,
! [K: int,L: int] :
( ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( minus_minus @ int @ K @ L ) )
= ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( plus_plus @ int @ K @ L ) ) ) ).
% even_diff_iff
thf(fact_2489_even__add__abs__iff,axiom,
! [K: int,L: int] :
( ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( plus_plus @ int @ K @ ( abs_abs @ int @ L ) ) )
= ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( plus_plus @ int @ K @ L ) ) ) ).
% even_add_abs_iff
thf(fact_2490_even__abs__add__iff,axiom,
! [K: int,L: int] :
( ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( plus_plus @ int @ ( abs_abs @ int @ K ) @ L ) )
= ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( plus_plus @ int @ K @ L ) ) ) ).
% even_abs_add_iff
thf(fact_2491_dvd__power__iff__le,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K )
=> ( ( dvd_dvd @ nat @ ( power_power @ nat @ K @ M ) @ ( power_power @ nat @ K @ N ) )
= ( ord_less_eq @ nat @ M @ N ) ) ) ).
% dvd_power_iff_le
thf(fact_2492_take__bit__nat__less__self__iff,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ ( bit_se2584673776208193580ke_bit @ nat @ N @ M ) @ M )
= ( ord_less_eq @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ M ) ) ).
% take_bit_nat_less_self_iff
thf(fact_2493_binomial__addition__formula,axiom,
! [N: nat,K: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( binomial @ N @ ( suc @ K ) )
= ( plus_plus @ nat @ ( binomial @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ ( suc @ K ) ) @ ( binomial @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ K ) ) ) ) ).
% binomial_addition_formula
thf(fact_2494_binomial__fact,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ N )
=> ( ( semiring_1_of_nat @ A @ ( binomial @ N @ K ) )
= ( divide_divide @ A @ ( semiring_char_0_fact @ A @ N ) @ ( times_times @ A @ ( semiring_char_0_fact @ A @ K ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ N @ K ) ) ) ) ) ) ) ).
% binomial_fact
thf(fact_2495_fact__binomial,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ N )
=> ( ( times_times @ A @ ( semiring_char_0_fact @ A @ K ) @ ( semiring_1_of_nat @ A @ ( binomial @ N @ K ) ) )
= ( divide_divide @ A @ ( semiring_char_0_fact @ A @ N ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ N @ K ) ) ) ) ) ) ).
% fact_binomial
thf(fact_2496_take__bit__int__less__self__iff,axiom,
! [N: nat,K: int] :
( ( ord_less @ int @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) @ K )
= ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) @ K ) ) ).
% take_bit_int_less_self_iff
thf(fact_2497_take__bit__int__greater__eq__self__iff,axiom,
! [K: int,N: nat] :
( ( ord_less_eq @ int @ K @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) )
= ( ord_less @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ).
% take_bit_int_greater_eq_self_iff
thf(fact_2498_oddE,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A2: A] :
( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
=> ~ ! [B4: A] :
( A2
!= ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B4 ) @ ( one_one @ A ) ) ) ) ) ).
% oddE
thf(fact_2499_mod2__eq__if,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A2: A] :
( ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
=> ( ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( zero_zero @ A ) ) )
& ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
=> ( ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( one_one @ A ) ) ) ) ) ).
% mod2_eq_if
thf(fact_2500_parity__cases,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A2: A] :
( ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
=> ( ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
!= ( zero_zero @ A ) ) )
=> ~ ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
=> ( ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
!= ( one_one @ A ) ) ) ) ) ).
% parity_cases
thf(fact_2501_zero__le__even__power,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: nat,A2: A] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ N ) ) ) ) ).
% zero_le_even_power
thf(fact_2502_zero__le__odd__power,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: nat,A2: A] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ N ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ) ).
% zero_le_odd_power
thf(fact_2503_zero__le__power__eq,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,N: nat] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ N ) )
= ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
| ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ) ) ).
% zero_le_power_eq
thf(fact_2504_minus__one__power__iff,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [N: nat] :
( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N )
= ( one_one @ A ) ) )
& ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ) ).
% minus_one_power_iff
thf(fact_2505_power__mono__even,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: nat,A2: A,B2: A] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( ord_less_eq @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B2 ) )
=> ( ord_less_eq @ A @ ( power_power @ A @ A2 @ N ) @ ( power_power @ A @ B2 @ N ) ) ) ) ) ).
% power_mono_even
thf(fact_2506_choose__two,axiom,
! [N: nat] :
( ( binomial @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( divide_divide @ nat @ ( times_times @ nat @ N @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% choose_two
thf(fact_2507_take__bit__int__eq__self__iff,axiom,
! [N: nat,K: int] :
( ( ( bit_se2584673776208193580ke_bit @ int @ N @ K )
= K )
= ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
& ( ord_less @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% take_bit_int_eq_self_iff
thf(fact_2508_take__bit__int__eq__self,axiom,
! [K: int,N: nat] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
=> ( ( ord_less @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( bit_se2584673776208193580ke_bit @ int @ N @ K )
= K ) ) ) ).
% take_bit_int_eq_self
thf(fact_2509_signed__take__bit__eq__take__bit__shift,axiom,
( ( bit_ri4674362597316999326ke_bit @ int )
= ( ^ [N4: nat,K2: int] : ( minus_minus @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( suc @ N4 ) @ ( plus_plus @ int @ K2 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N4 ) ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N4 ) ) ) ) ).
% signed_take_bit_eq_take_bit_shift
thf(fact_2510_take__bit__incr__eq,axiom,
! [N: nat,K: int] :
( ( ( bit_se2584673776208193580ke_bit @ int @ N @ K )
!= ( minus_minus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ int ) ) )
=> ( ( bit_se2584673776208193580ke_bit @ int @ N @ ( plus_plus @ int @ K @ ( one_one @ int ) ) )
= ( plus_plus @ int @ ( one_one @ int ) @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) ) ) ) ).
% take_bit_incr_eq
thf(fact_2511_zero__less__power__eq,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ N ) )
= ( ( N
= ( zero_zero @ nat ) )
| ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
& ( A2
!= ( zero_zero @ A ) ) )
| ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
& ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ) ) ).
% zero_less_power_eq
thf(fact_2512_take__bit__Suc__minus__1__eq,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: nat] :
( ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( suc @ N ) ) @ ( one_one @ A ) ) ) ) ).
% take_bit_Suc_minus_1_eq
thf(fact_2513_take__bit__Suc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A2: A] :
( ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N ) @ A2 )
= ( plus_plus @ A @ ( times_times @ A @ ( bit_se2584673776208193580ke_bit @ A @ N @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% take_bit_Suc
thf(fact_2514_take__bit__numeral__minus__1__eq,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [K: num] :
( ( bit_se2584673776208193580ke_bit @ A @ ( numeral_numeral @ nat @ K ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ K ) ) @ ( one_one @ A ) ) ) ) ).
% take_bit_numeral_minus_1_eq
thf(fact_2515_take__bit__int__less__eq,axiom,
! [N: nat,K: int] :
( ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) @ K )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less_eq @ int @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) @ ( minus_minus @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ).
% take_bit_int_less_eq
thf(fact_2516_take__bit__int__greater__eq,axiom,
! [K: int,N: nat] :
( ( ord_less @ int @ K @ ( zero_zero @ int ) )
=> ( ord_less_eq @ int @ ( plus_plus @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) ) ) ).
% take_bit_int_greater_eq
thf(fact_2517_even__mask__div__iff_H,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [M: nat,N: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) @ ( one_one @ A ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) )
= ( ord_less_eq @ nat @ M @ N ) ) ) ).
% even_mask_div_iff'
thf(fact_2518_power__le__zero__eq,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,N: nat] :
( ( ord_less_eq @ A @ ( power_power @ A @ A2 @ N ) @ ( zero_zero @ A ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
& ( ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
& ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) )
| ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
& ( A2
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% power_le_zero_eq
thf(fact_2519_even__mod__4__div__2,axiom,
! [N: nat] :
( ( ( modulo_modulo @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) )
= ( suc @ ( zero_zero @ nat ) ) )
=> ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% even_mod_4_div_2
thf(fact_2520_take__bit__minus__small__eq,axiom,
! [K: int,N: nat] :
( ( ord_less @ int @ ( zero_zero @ int ) @ K )
=> ( ( ord_less_eq @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( bit_se2584673776208193580ke_bit @ int @ N @ ( uminus_uminus @ int @ K ) )
= ( minus_minus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) @ K ) ) ) ) ).
% take_bit_minus_small_eq
thf(fact_2521_even__mask__div__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [M: nat,N: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) @ ( one_one @ A ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) )
= ( ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N )
= ( zero_zero @ A ) )
| ( ord_less_eq @ nat @ M @ N ) ) ) ) ).
% even_mask_div_iff
thf(fact_2522_even__mult__exp__div__exp__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A2: A,M: nat,N: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ ( times_times @ A @ A2 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) )
= ( ( ord_less @ nat @ N @ M )
| ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N )
= ( zero_zero @ A ) )
| ( ( ord_less_eq @ nat @ M @ N )
& ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A2 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N @ M ) ) ) ) ) ) ) ) ).
% even_mult_exp_div_exp_iff
thf(fact_2523_signed__take__bit__numeral__minus__bit1,axiom,
! [L: num,K: num] :
( ( bit_ri4674362597316999326ke_bit @ int @ ( numeral_numeral @ nat @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ K ) ) ) )
= ( plus_plus @ int @ ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( pred_numeral @ L ) @ ( minus_minus @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) @ ( one_one @ int ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( one_one @ int ) ) ) ).
% signed_take_bit_numeral_minus_bit1
thf(fact_2524_signed__take__bit__numeral__bit1,axiom,
! [L: num,K: num] :
( ( bit_ri4674362597316999326ke_bit @ int @ ( numeral_numeral @ nat @ L ) @ ( numeral_numeral @ int @ ( bit1 @ K ) ) )
= ( plus_plus @ int @ ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( pred_numeral @ L ) @ ( numeral_numeral @ int @ K ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( one_one @ int ) ) ) ).
% signed_take_bit_numeral_bit1
thf(fact_2525_take__bit__numeral__minus__bit1,axiom,
! [L: num,K: num] :
( ( bit_se2584673776208193580ke_bit @ int @ ( numeral_numeral @ nat @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ K ) ) ) )
= ( plus_plus @ int @ ( times_times @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( pred_numeral @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( inc @ K ) ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( one_one @ int ) ) ) ).
% take_bit_numeral_minus_bit1
thf(fact_2526_flip__bit__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A] :
( ( bit_se8732182000553998342ip_bit @ A @ ( zero_zero @ nat ) @ A2 )
= ( plus_plus @ A @ ( zero_neq_one_of_bool @ A @ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% flip_bit_0
thf(fact_2527_num_Osize__gen_I3_J,axiom,
! [X32: num] :
( ( size_num @ ( bit1 @ X32 ) )
= ( plus_plus @ nat @ ( size_num @ X32 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% num.size_gen(3)
thf(fact_2528_num_Osize__gen_I2_J,axiom,
! [X22: num] :
( ( size_num @ ( bit0 @ X22 ) )
= ( plus_plus @ nat @ ( size_num @ X22 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% num.size_gen(2)
thf(fact_2529_nat__of__bool,axiom,
! [P: $o] :
( ( nat2 @ ( zero_neq_one_of_bool @ int @ P ) )
= ( zero_neq_one_of_bool @ nat @ P ) ) ).
% nat_of_bool
thf(fact_2530_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_2531_of__bool__eq__0__iff,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ! [P: $o] :
( ( ( zero_neq_one_of_bool @ A @ P )
= ( zero_zero @ A ) )
= ~ P ) ) ).
% of_bool_eq_0_iff
thf(fact_2532_of__bool__eq_I1_J,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ( ( zero_neq_one_of_bool @ A @ $false )
= ( zero_zero @ A ) ) ) ).
% of_bool_eq(1)
thf(fact_2533_of__bool__eq__1__iff,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ! [P: $o] :
( ( ( zero_neq_one_of_bool @ A @ P )
= ( one_one @ A ) )
= P ) ) ).
% of_bool_eq_1_iff
thf(fact_2534_of__bool__eq_I2_J,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ( ( zero_neq_one_of_bool @ A @ $true )
= ( one_one @ A ) ) ) ).
% of_bool_eq(2)
thf(fact_2535_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_2536_of__int__of__bool,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [P: $o] :
( ( ring_1_of_int @ A @ ( zero_neq_one_of_bool @ int @ P ) )
= ( zero_neq_one_of_bool @ A @ P ) ) ) ).
% of_int_of_bool
thf(fact_2537_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_2538_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_2539_of__bool__not__iff,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [P: $o] :
( ( zero_neq_one_of_bool @ A @ ~ P )
= ( minus_minus @ A @ ( one_one @ A ) @ ( zero_neq_one_of_bool @ A @ P ) ) ) ) ).
% of_bool_not_iff
thf(fact_2540_Suc__0__mod__eq,axiom,
! [N: nat] :
( ( modulo_modulo @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N )
= ( zero_neq_one_of_bool @ nat
@ ( N
!= ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% Suc_0_mod_eq
thf(fact_2541_pred__numeral__simps_I1_J,axiom,
( ( pred_numeral @ one2 )
= ( zero_zero @ nat ) ) ).
% pred_numeral_simps(1)
thf(fact_2542_eq__numeral__Suc,axiom,
! [K: num,N: nat] :
( ( ( numeral_numeral @ nat @ K )
= ( suc @ N ) )
= ( ( pred_numeral @ K )
= N ) ) ).
% eq_numeral_Suc
thf(fact_2543_Suc__eq__numeral,axiom,
! [N: nat,K: num] :
( ( ( suc @ N )
= ( numeral_numeral @ nat @ K ) )
= ( N
= ( pred_numeral @ K ) ) ) ).
% Suc_eq_numeral
thf(fact_2544_pred__numeral__inc,axiom,
! [K: num] :
( ( pred_numeral @ ( inc @ K ) )
= ( numeral_numeral @ nat @ K ) ) ).
% pred_numeral_inc
thf(fact_2545_rec__nat__numeral,axiom,
! [A: $tType,A2: A,F: nat > A > A,V2: num] :
( ( rec_nat @ A @ A2 @ F @ ( numeral_numeral @ nat @ V2 ) )
= ( F @ ( pred_numeral @ V2 ) @ ( rec_nat @ A @ A2 @ F @ ( pred_numeral @ V2 ) ) ) ) ).
% rec_nat_numeral
thf(fact_2546_less__numeral__Suc,axiom,
! [K: num,N: nat] :
( ( ord_less @ nat @ ( numeral_numeral @ nat @ K ) @ ( suc @ N ) )
= ( ord_less @ nat @ ( pred_numeral @ K ) @ N ) ) ).
% less_numeral_Suc
thf(fact_2547_less__Suc__numeral,axiom,
! [N: nat,K: num] :
( ( ord_less @ nat @ ( suc @ N ) @ ( numeral_numeral @ nat @ K ) )
= ( ord_less @ nat @ N @ ( pred_numeral @ K ) ) ) ).
% less_Suc_numeral
thf(fact_2548_pred__numeral__simps_I3_J,axiom,
! [K: num] :
( ( pred_numeral @ ( bit1 @ K ) )
= ( numeral_numeral @ nat @ ( bit0 @ K ) ) ) ).
% pred_numeral_simps(3)
thf(fact_2549_le__numeral__Suc,axiom,
! [K: num,N: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ K ) @ ( suc @ N ) )
= ( ord_less_eq @ nat @ ( pred_numeral @ K ) @ N ) ) ).
% le_numeral_Suc
thf(fact_2550_le__Suc__numeral,axiom,
! [N: nat,K: num] :
( ( ord_less_eq @ nat @ ( suc @ N ) @ ( numeral_numeral @ nat @ K ) )
= ( ord_less_eq @ nat @ N @ ( pred_numeral @ K ) ) ) ).
% le_Suc_numeral
thf(fact_2551_diff__numeral__Suc,axiom,
! [K: num,N: nat] :
( ( minus_minus @ nat @ ( numeral_numeral @ nat @ K ) @ ( suc @ N ) )
= ( minus_minus @ nat @ ( pred_numeral @ K ) @ N ) ) ).
% diff_numeral_Suc
thf(fact_2552_diff__Suc__numeral,axiom,
! [N: nat,K: num] :
( ( minus_minus @ nat @ ( suc @ N ) @ ( numeral_numeral @ nat @ K ) )
= ( minus_minus @ nat @ N @ ( pred_numeral @ K ) ) ) ).
% diff_Suc_numeral
thf(fact_2553_take__bit__of__Suc__0,axiom,
! [N: nat] :
( ( bit_se2584673776208193580ke_bit @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) )
= ( zero_neq_one_of_bool @ nat @ ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% take_bit_of_Suc_0
thf(fact_2554_rec__nat__add__eq__if,axiom,
! [A: $tType,A2: A,F: nat > A > A,V2: num,N: nat] :
( ( rec_nat @ A @ A2 @ F @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ V2 ) @ N ) )
= ( F @ ( plus_plus @ nat @ ( pred_numeral @ V2 ) @ N ) @ ( rec_nat @ A @ A2 @ F @ ( plus_plus @ nat @ ( pred_numeral @ V2 ) @ N ) ) ) ) ).
% rec_nat_add_eq_if
thf(fact_2555_take__bit__of__1,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat] :
( ( bit_se2584673776208193580ke_bit @ A @ N @ ( one_one @ A ) )
= ( zero_neq_one_of_bool @ A @ ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% take_bit_of_1
thf(fact_2556_of__bool__half__eq__0,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [B2: $o] :
( ( divide_divide @ A @ ( zero_neq_one_of_bool @ A @ B2 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( zero_zero @ A ) ) ) ).
% of_bool_half_eq_0
thf(fact_2557_bits__1__div__exp,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [N: nat] :
( ( divide_divide @ A @ ( one_one @ A ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
= ( zero_neq_one_of_bool @ A
@ ( N
= ( zero_zero @ nat ) ) ) ) ) ).
% bits_1_div_exp
thf(fact_2558_one__div__2__pow__eq,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [N: nat] :
( ( divide_divide @ A @ ( one_one @ A ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
= ( zero_neq_one_of_bool @ A
@ ( N
= ( zero_zero @ nat ) ) ) ) ) ).
% one_div_2_pow_eq
thf(fact_2559_take__bit__of__2,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N: nat] :
( ( bit_se2584673776208193580ke_bit @ A @ N @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( times_times @ A @ ( zero_neq_one_of_bool @ A @ ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% take_bit_of_2
thf(fact_2560_one__mod__2__pow__eq,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [N: nat] :
( ( modulo_modulo @ A @ ( one_one @ A ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
= ( zero_neq_one_of_bool @ A @ ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% one_mod_2_pow_eq
thf(fact_2561_dvd__antisym,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd @ nat @ M @ N )
=> ( ( dvd_dvd @ nat @ N @ M )
=> ( M = N ) ) ) ).
% dvd_antisym
thf(fact_2562_gcd__nat_Onot__eq__order__implies__strict,axiom,
! [A2: nat,B2: nat] :
( ( A2 != B2 )
=> ( ( dvd_dvd @ nat @ A2 @ B2 )
=> ( ( dvd_dvd @ nat @ A2 @ B2 )
& ( A2 != B2 ) ) ) ) ).
% gcd_nat.not_eq_order_implies_strict
thf(fact_2563_gcd__nat_Ostrict__implies__not__eq,axiom,
! [A2: nat,B2: nat] :
( ( ( dvd_dvd @ nat @ A2 @ B2 )
& ( A2 != B2 ) )
=> ( A2 != B2 ) ) ).
% gcd_nat.strict_implies_not_eq
thf(fact_2564_gcd__nat_Ostrict__implies__order,axiom,
! [A2: nat,B2: nat] :
( ( ( dvd_dvd @ nat @ A2 @ B2 )
& ( A2 != B2 ) )
=> ( dvd_dvd @ nat @ A2 @ B2 ) ) ).
% gcd_nat.strict_implies_order
thf(fact_2565_gcd__nat_Ostrict__iff__order,axiom,
! [A2: nat,B2: nat] :
( ( ( dvd_dvd @ nat @ A2 @ B2 )
& ( A2 != B2 ) )
= ( ( dvd_dvd @ nat @ A2 @ B2 )
& ( A2 != B2 ) ) ) ).
% gcd_nat.strict_iff_order
thf(fact_2566_gcd__nat_Oorder__iff__strict,axiom,
( ( dvd_dvd @ nat )
= ( ^ [A3: nat,B3: nat] :
( ( ( dvd_dvd @ nat @ A3 @ B3 )
& ( A3 != B3 ) )
| ( A3 = B3 ) ) ) ) ).
% gcd_nat.order_iff_strict
thf(fact_2567_gcd__nat_Ostrict__iff__not,axiom,
! [A2: nat,B2: nat] :
( ( ( dvd_dvd @ nat @ A2 @ B2 )
& ( A2 != B2 ) )
= ( ( dvd_dvd @ nat @ A2 @ B2 )
& ~ ( dvd_dvd @ nat @ B2 @ A2 ) ) ) ).
% gcd_nat.strict_iff_not
thf(fact_2568_gcd__nat_Ostrict__trans2,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( ( dvd_dvd @ nat @ A2 @ B2 )
& ( A2 != B2 ) )
=> ( ( dvd_dvd @ nat @ B2 @ C2 )
=> ( ( dvd_dvd @ nat @ A2 @ C2 )
& ( A2 != C2 ) ) ) ) ).
% gcd_nat.strict_trans2
thf(fact_2569_gcd__nat_Ostrict__trans1,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( dvd_dvd @ nat @ A2 @ B2 )
=> ( ( ( dvd_dvd @ nat @ B2 @ C2 )
& ( B2 != C2 ) )
=> ( ( dvd_dvd @ nat @ A2 @ C2 )
& ( A2 != C2 ) ) ) ) ).
% gcd_nat.strict_trans1
thf(fact_2570_gcd__nat_Ostrict__trans,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( ( dvd_dvd @ nat @ A2 @ B2 )
& ( A2 != B2 ) )
=> ( ( ( dvd_dvd @ nat @ B2 @ C2 )
& ( B2 != C2 ) )
=> ( ( dvd_dvd @ nat @ A2 @ C2 )
& ( A2 != C2 ) ) ) ) ).
% gcd_nat.strict_trans
thf(fact_2571_gcd__nat_Oantisym,axiom,
! [A2: nat,B2: nat] :
( ( dvd_dvd @ nat @ A2 @ B2 )
=> ( ( dvd_dvd @ nat @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% gcd_nat.antisym
thf(fact_2572_gcd__nat_Oirrefl,axiom,
! [A2: nat] :
~ ( ( dvd_dvd @ nat @ A2 @ A2 )
& ( A2 != A2 ) ) ).
% gcd_nat.irrefl
thf(fact_2573_gcd__nat_Oeq__iff,axiom,
( ( ^ [Y3: nat,Z: nat] : Y3 = Z )
= ( ^ [A3: nat,B3: nat] :
( ( dvd_dvd @ nat @ A3 @ B3 )
& ( dvd_dvd @ nat @ B3 @ A3 ) ) ) ) ).
% gcd_nat.eq_iff
thf(fact_2574_gcd__nat_Otrans,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( dvd_dvd @ nat @ A2 @ B2 )
=> ( ( dvd_dvd @ nat @ B2 @ C2 )
=> ( dvd_dvd @ nat @ A2 @ C2 ) ) ) ).
% gcd_nat.trans
thf(fact_2575_gcd__nat_Orefl,axiom,
! [A2: nat] : ( dvd_dvd @ nat @ A2 @ A2 ) ).
% gcd_nat.refl
thf(fact_2576_gcd__nat_Oasym,axiom,
! [A2: nat,B2: nat] :
( ( ( dvd_dvd @ nat @ A2 @ B2 )
& ( A2 != B2 ) )
=> ~ ( ( dvd_dvd @ nat @ B2 @ A2 )
& ( B2 != A2 ) ) ) ).
% gcd_nat.asym
thf(fact_2577_subset__CollectI,axiom,
! [A: $tType,B5: set @ A,A5: set @ A,Q: A > $o,P: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ B5 @ A5 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ B5 )
=> ( ( Q @ X3 )
=> ( P @ X3 ) ) )
=> ( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [X: A] :
( ( member @ A @ X @ B5 )
& ( Q @ X ) ) )
@ ( collect @ A
@ ^ [X: A] :
( ( member @ A @ X @ A5 )
& ( P @ X ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_2578_subset__Collect__iff,axiom,
! [A: $tType,B5: set @ A,A5: set @ A,P: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ B5 @ A5 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5
@ ( collect @ A
@ ^ [X: A] :
( ( member @ A @ X @ A5 )
& ( P @ X ) ) ) )
= ( ! [X: A] :
( ( member @ A @ X @ B5 )
=> ( P @ X ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_2579_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_2580_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_2581_of__bool__def,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ( ( zero_neq_one_of_bool @ A )
= ( ^ [P7: $o] : ( if @ A @ P7 @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ) ) ).
% of_bool_def
thf(fact_2582_split__of__bool,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ! [P: A > $o,P3: $o] :
( ( P @ ( zero_neq_one_of_bool @ A @ P3 ) )
= ( ( P3
=> ( P @ ( one_one @ A ) ) )
& ( ~ P3
=> ( P @ ( zero_zero @ A ) ) ) ) ) ) ).
% split_of_bool
thf(fact_2583_split__of__bool__asm,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ! [P: A > $o,P3: $o] :
( ( P @ ( zero_neq_one_of_bool @ A @ P3 ) )
= ( ~ ( ( P3
& ~ ( P @ ( one_one @ A ) ) )
| ( ~ P3
& ~ ( P @ ( zero_zero @ A ) ) ) ) ) ) ) ).
% split_of_bool_asm
thf(fact_2584_numeral__eq__Suc,axiom,
( ( numeral_numeral @ nat )
= ( ^ [K2: num] : ( suc @ ( pred_numeral @ K2 ) ) ) ) ).
% numeral_eq_Suc
thf(fact_2585_pred__numeral__def,axiom,
( pred_numeral
= ( ^ [K2: num] : ( minus_minus @ nat @ ( numeral_numeral @ nat @ K2 ) @ ( one_one @ nat ) ) ) ) ).
% pred_numeral_def
thf(fact_2586_num_Osize__gen_I1_J,axiom,
( ( size_num @ one2 )
= ( zero_zero @ nat ) ) ).
% num.size_gen(1)
thf(fact_2587_bits__induct,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [P: A > $o,A2: A] :
( ! [A4: A] :
( ( ( divide_divide @ A @ A4 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= A4 )
=> ( P @ A4 ) )
=> ( ! [A4: A,B4: $o] :
( ( P @ A4 )
=> ( ( ( divide_divide @ A @ ( plus_plus @ A @ ( zero_neq_one_of_bool @ A @ B4 ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A4 ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= A4 )
=> ( P @ ( plus_plus @ A @ ( zero_neq_one_of_bool @ A @ B4 ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A4 ) ) ) ) )
=> ( P @ A2 ) ) ) ) ).
% bits_induct
thf(fact_2588_exp__div__exp__eq,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [M: nat,N: nat] :
( ( divide_divide @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
= ( times_times @ A
@ ( zero_neq_one_of_bool @ A
@ ( ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M )
!= ( zero_zero @ A ) )
& ( ord_less_eq @ nat @ N @ M ) ) )
@ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ M @ N ) ) ) ) ) ).
% exp_div_exp_eq
thf(fact_2589_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_2590_set__decode__0,axiom,
! [X2: nat] :
( ( member @ nat @ ( zero_zero @ nat ) @ ( nat_set_decode @ X2 ) )
= ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ X2 ) ) ) ).
% set_decode_0
thf(fact_2591_set__decode__def,axiom,
( nat_set_decode
= ( ^ [X: nat] :
( collect @ nat
@ ^ [N4: nat] :
~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N4 ) ) ) ) ) ) ).
% set_decode_def
thf(fact_2592_mask__numeral,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: num] :
( ( bit_se2239418461657761734s_mask @ A @ ( numeral_numeral @ nat @ N ) )
= ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se2239418461657761734s_mask @ A @ ( pred_numeral @ N ) ) ) ) ) ) ).
% mask_numeral
thf(fact_2593_and__int__unfold,axiom,
( ( bit_se5824344872417868541ns_and @ int )
= ( ^ [K2: int,L2: int] :
( if @ int
@ ( ( K2
= ( zero_zero @ int ) )
| ( L2
= ( zero_zero @ int ) ) )
@ ( zero_zero @ int )
@ ( if @ int
@ ( K2
= ( uminus_uminus @ int @ ( one_one @ int ) ) )
@ L2
@ ( if @ int
@ ( L2
= ( uminus_uminus @ int @ ( one_one @ int ) ) )
@ K2
@ ( plus_plus @ int @ ( times_times @ int @ ( modulo_modulo @ int @ K2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ int @ L2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( divide_divide @ int @ K2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ).
% and_int_unfold
thf(fact_2594_take__bit__eq__mask__iff__exp__dvd,axiom,
! [N: nat,K: int] :
( ( ( bit_se2584673776208193580ke_bit @ int @ N @ K )
= ( bit_se2239418461657761734s_mask @ int @ N ) )
= ( dvd_dvd @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) @ ( plus_plus @ int @ K @ ( one_one @ int ) ) ) ) ).
% take_bit_eq_mask_iff_exp_dvd
thf(fact_2595_modulo__int__def,axiom,
( ( modulo_modulo @ int )
= ( ^ [K2: int,L2: int] :
( if @ int
@ ( L2
= ( zero_zero @ int ) )
@ K2
@ ( if @ int
@ ( ( sgn_sgn @ int @ K2 )
= ( sgn_sgn @ int @ L2 ) )
@ ( times_times @ int @ ( sgn_sgn @ int @ L2 ) @ ( semiring_1_of_nat @ int @ ( modulo_modulo @ nat @ ( nat2 @ ( abs_abs @ int @ K2 ) ) @ ( nat2 @ ( abs_abs @ int @ L2 ) ) ) ) )
@ ( times_times @ int @ ( sgn_sgn @ int @ L2 )
@ ( minus_minus @ int
@ ( times_times @ int @ ( abs_abs @ int @ L2 )
@ ( zero_neq_one_of_bool @ int
@ ~ ( dvd_dvd @ int @ L2 @ K2 ) ) )
@ ( semiring_1_of_nat @ int @ ( modulo_modulo @ nat @ ( nat2 @ ( abs_abs @ int @ K2 ) ) @ ( nat2 @ ( abs_abs @ int @ L2 ) ) ) ) ) ) ) ) ) ) ).
% modulo_int_def
thf(fact_2596_mask__nat__positive__iff,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( bit_se2239418461657761734s_mask @ nat @ N ) )
= ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% mask_nat_positive_iff
thf(fact_2597_bit_Oconj__zero__right,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X2: A] :
( ( bit_se5824344872417868541ns_and @ A @ X2 @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% bit.conj_zero_right
thf(fact_2598_bit_Oconj__zero__left,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X2: A] :
( ( bit_se5824344872417868541ns_and @ A @ ( zero_zero @ A ) @ X2 )
= ( zero_zero @ A ) ) ) ).
% bit.conj_zero_left
thf(fact_2599_zero__and__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A] :
( ( bit_se5824344872417868541ns_and @ A @ ( zero_zero @ A ) @ A2 )
= ( zero_zero @ A ) ) ) ).
% zero_and_eq
thf(fact_2600_and__zero__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A] :
( ( bit_se5824344872417868541ns_and @ A @ A2 @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% and_zero_eq
thf(fact_2601_sgn__0,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ( ( sgn_sgn @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% sgn_0
thf(fact_2602_sgn__1,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ( ( sgn_sgn @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% sgn_1
thf(fact_2603_power__int__sgn,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,N: int] :
( ( sgn_sgn @ A @ ( power_int @ A @ A2 @ N ) )
= ( power_int @ A @ ( sgn_sgn @ A @ A2 ) @ N ) ) ) ).
% power_int_sgn
thf(fact_2604_sgn__greater,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( sgn_sgn @ A @ A2 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% sgn_greater
thf(fact_2605_sgn__less,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A] :
( ( ord_less @ A @ ( sgn_sgn @ A @ A2 ) @ ( zero_zero @ A ) )
= ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% sgn_less
thf(fact_2606_and_Oleft__neutral,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A2: A] :
( ( bit_se5824344872417868541ns_and @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ A2 )
= A2 ) ) ).
% and.left_neutral
thf(fact_2607_and_Oright__neutral,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A2: A] :
( ( bit_se5824344872417868541ns_and @ A @ A2 @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= A2 ) ) ).
% and.right_neutral
thf(fact_2608_bit_Oconj__one__right,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X2: A] :
( ( bit_se5824344872417868541ns_and @ A @ X2 @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= X2 ) ) ).
% bit.conj_one_right
thf(fact_2609_mask__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se2239418461657761734s_mask @ A @ ( zero_zero @ nat ) )
= ( zero_zero @ A ) ) ) ).
% mask_0
thf(fact_2610_mask__eq__0__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat] :
( ( ( bit_se2239418461657761734s_mask @ A @ N )
= ( zero_zero @ A ) )
= ( N
= ( zero_zero @ nat ) ) ) ) ).
% mask_eq_0_iff
thf(fact_2611_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_2612_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_2613_sgn__pos,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( sgn_sgn @ A @ A2 )
= ( one_one @ A ) ) ) ) ).
% sgn_pos
thf(fact_2614_and__numerals_I2_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [Y: num] :
( ( bit_se5824344872417868541ns_and @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit1 @ Y ) ) )
= ( one_one @ A ) ) ) ).
% and_numerals(2)
thf(fact_2615_and__numerals_I8_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X2: num] :
( ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ ( bit1 @ X2 ) ) @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% and_numerals(8)
thf(fact_2616_abs__sgn__eq__1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( abs_abs @ A @ ( sgn_sgn @ A @ A2 ) )
= ( one_one @ A ) ) ) ) ).
% abs_sgn_eq_1
thf(fact_2617_mask__Suc__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se2239418461657761734s_mask @ A @ ( suc @ ( zero_zero @ nat ) ) )
= ( one_one @ A ) ) ) ).
% mask_Suc_0
thf(fact_2618_sgn__mult__self__eq,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A] :
( ( times_times @ A @ ( sgn_sgn @ A @ A2 ) @ ( sgn_sgn @ A @ A2 ) )
= ( zero_neq_one_of_bool @ A
@ ( A2
!= ( zero_zero @ A ) ) ) ) ) ).
% sgn_mult_self_eq
thf(fact_2619_take__bit__minus__one__eq__mask,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: nat] :
( ( bit_se2584673776208193580ke_bit @ A @ N @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( bit_se2239418461657761734s_mask @ A @ N ) ) ) ).
% take_bit_minus_one_eq_mask
thf(fact_2620_idom__abs__sgn__class_Oabs__sgn,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A2: A] :
( ( sgn_sgn @ A @ ( abs_abs @ A @ A2 ) )
= ( zero_neq_one_of_bool @ A
@ ( A2
!= ( zero_zero @ A ) ) ) ) ) ).
% idom_abs_sgn_class.abs_sgn
thf(fact_2621_sgn__abs,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A2: A] :
( ( abs_abs @ A @ ( sgn_sgn @ A @ A2 ) )
= ( zero_neq_one_of_bool @ A
@ ( A2
!= ( zero_zero @ A ) ) ) ) ) ).
% sgn_abs
thf(fact_2622_sgn__mult__dvd__iff,axiom,
! [R: int,L: int,K: int] :
( ( dvd_dvd @ int @ ( times_times @ int @ ( sgn_sgn @ int @ R ) @ L ) @ K )
= ( ( dvd_dvd @ int @ L @ K )
& ( ( R
= ( zero_zero @ int ) )
=> ( K
= ( zero_zero @ int ) ) ) ) ) ).
% sgn_mult_dvd_iff
thf(fact_2623_mult__sgn__dvd__iff,axiom,
! [L: int,R: int,K: int] :
( ( dvd_dvd @ int @ ( times_times @ int @ L @ ( sgn_sgn @ int @ R ) ) @ K )
= ( ( dvd_dvd @ int @ L @ K )
& ( ( R
= ( zero_zero @ int ) )
=> ( K
= ( zero_zero @ int ) ) ) ) ) ).
% mult_sgn_dvd_iff
thf(fact_2624_dvd__sgn__mult__iff,axiom,
! [L: int,R: int,K: int] :
( ( dvd_dvd @ int @ L @ ( times_times @ int @ ( sgn_sgn @ int @ R ) @ K ) )
= ( ( dvd_dvd @ int @ L @ K )
| ( R
= ( zero_zero @ int ) ) ) ) ).
% dvd_sgn_mult_iff
thf(fact_2625_dvd__mult__sgn__iff,axiom,
! [L: int,K: int,R: int] :
( ( dvd_dvd @ int @ L @ ( times_times @ int @ K @ ( sgn_sgn @ int @ R ) ) )
= ( ( dvd_dvd @ int @ L @ K )
| ( R
= ( zero_zero @ int ) ) ) ) ).
% dvd_mult_sgn_iff
thf(fact_2626_and__numerals_I5_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X2: num] :
( ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ ( bit0 @ X2 ) ) @ ( one_one @ A ) )
= ( zero_zero @ A ) ) ) ).
% and_numerals(5)
thf(fact_2627_and__numerals_I1_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [Y: num] :
( ( bit_se5824344872417868541ns_and @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ Y ) ) )
= ( zero_zero @ A ) ) ) ).
% and_numerals(1)
thf(fact_2628_sgn__neg,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A] :
( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( sgn_sgn @ A @ A2 )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ).
% sgn_neg
thf(fact_2629_sgn__of__nat,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: nat] :
( ( sgn_sgn @ A @ ( semiring_1_of_nat @ A @ N ) )
= ( zero_neq_one_of_bool @ A @ ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% sgn_of_nat
thf(fact_2630_and__minus__numerals_I2_J,axiom,
! [N: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( one_one @ int ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) )
= ( one_one @ int ) ) ).
% and_minus_numerals(2)
thf(fact_2631_and__minus__numerals_I6_J,axiom,
! [N: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) @ ( one_one @ int ) )
= ( one_one @ int ) ) ).
% and_minus_numerals(6)
thf(fact_2632_and__minus__numerals_I5_J,axiom,
! [N: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) @ ( one_one @ int ) )
= ( zero_zero @ int ) ) ).
% and_minus_numerals(5)
thf(fact_2633_and__minus__numerals_I1_J,axiom,
! [N: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( one_one @ int ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) )
= ( zero_zero @ int ) ) ).
% and_minus_numerals(1)
thf(fact_2634_and__numerals_I7_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X2: num,Y: num] :
( ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ ( bit1 @ X2 ) ) @ ( 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 @ X2 ) @ ( numeral_numeral @ A @ Y ) ) ) ) ) ) ).
% and_numerals(7)
thf(fact_2635_of__nat__mask__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat] :
( ( semiring_1_of_nat @ A @ ( bit_se2239418461657761734s_mask @ nat @ N ) )
= ( bit_se2239418461657761734s_mask @ A @ N ) ) ) ).
% of_nat_mask_eq
thf(fact_2636_of__nat__and__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,N: nat] :
( ( semiring_1_of_nat @ A @ ( bit_se5824344872417868541ns_and @ nat @ M @ N ) )
= ( bit_se5824344872417868541ns_and @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% of_nat_and_eq
thf(fact_2637_sgn__0__0,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A] :
( ( ( sgn_sgn @ A @ A2 )
= ( zero_zero @ A ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% sgn_0_0
thf(fact_2638_sgn__eq__0__iff,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A2: A] :
( ( ( sgn_sgn @ A @ A2 )
= ( zero_zero @ A ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% sgn_eq_0_iff
thf(fact_2639_same__sgn__sgn__add,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [B2: A,A2: A] :
( ( ( sgn_sgn @ A @ B2 )
= ( sgn_sgn @ A @ A2 ) )
=> ( ( sgn_sgn @ A @ ( plus_plus @ A @ A2 @ B2 ) )
= ( sgn_sgn @ A @ A2 ) ) ) ) ).
% same_sgn_sgn_add
thf(fact_2640_less__eq__mask,axiom,
! [N: nat] : ( ord_less_eq @ nat @ N @ ( bit_se2239418461657761734s_mask @ nat @ N ) ) ).
% less_eq_mask
thf(fact_2641_subset__decode__imp__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ ( set @ nat ) @ ( nat_set_decode @ M ) @ ( nat_set_decode @ N ) )
=> ( ord_less_eq @ nat @ M @ N ) ) ).
% subset_decode_imp_le
thf(fact_2642_and__eq__minus__1__iff,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A2: A,B2: A] :
( ( ( bit_se5824344872417868541ns_and @ A @ A2 @ B2 )
= ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( ( A2
= ( uminus_uminus @ A @ ( one_one @ A ) ) )
& ( B2
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ) ).
% and_eq_minus_1_iff
thf(fact_2643_AND__upper2_H,axiom,
! [Y: int,Z2: int,X2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y )
=> ( ( ord_less_eq @ int @ Y @ Z2 )
=> ( ord_less_eq @ int @ ( bit_se5824344872417868541ns_and @ int @ X2 @ Y ) @ Z2 ) ) ) ).
% AND_upper2'
thf(fact_2644_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_2645_AND__upper2,axiom,
! [Y: int,X2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y )
=> ( ord_less_eq @ int @ ( bit_se5824344872417868541ns_and @ int @ X2 @ Y ) @ Y ) ) ).
% AND_upper2
thf(fact_2646_AND__upper1,axiom,
! [X2: int,Y: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X2 )
=> ( ord_less_eq @ int @ ( bit_se5824344872417868541ns_and @ int @ X2 @ Y ) @ X2 ) ) ).
% AND_upper1
thf(fact_2647_AND__lower,axiom,
! [X2: int,Y: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X2 )
=> ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se5824344872417868541ns_and @ int @ X2 @ Y ) ) ) ).
% AND_lower
thf(fact_2648_int__sgnE,axiom,
! [K: int] :
~ ! [N2: nat,L3: int] :
( K
!= ( times_times @ int @ ( sgn_sgn @ int @ L3 ) @ ( semiring_1_of_nat @ int @ N2 ) ) ) ).
% int_sgnE
thf(fact_2649_sgn__not__eq__imp,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [B2: A,A2: A] :
( ( ( sgn_sgn @ A @ B2 )
!= ( sgn_sgn @ A @ A2 ) )
=> ( ( ( sgn_sgn @ A @ A2 )
!= ( zero_zero @ A ) )
=> ( ( ( sgn_sgn @ A @ B2 )
!= ( zero_zero @ A ) )
=> ( ( sgn_sgn @ A @ A2 )
= ( uminus_uminus @ A @ ( sgn_sgn @ A @ B2 ) ) ) ) ) ) ) ).
% sgn_not_eq_imp
thf(fact_2650_sgn__minus__1,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ( ( sgn_sgn @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% sgn_minus_1
thf(fact_2651_same__sgn__abs__add,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [B2: A,A2: A] :
( ( ( sgn_sgn @ A @ B2 )
= ( sgn_sgn @ A @ A2 ) )
=> ( ( abs_abs @ A @ ( plus_plus @ A @ A2 @ B2 ) )
= ( plus_plus @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B2 ) ) ) ) ) ).
% same_sgn_abs_add
thf(fact_2652_mask__nonnegative__int,axiom,
! [N: nat] : ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se2239418461657761734s_mask @ int @ N ) ) ).
% mask_nonnegative_int
thf(fact_2653_not__mask__negative__int,axiom,
! [N: nat] :
~ ( ord_less @ int @ ( bit_se2239418461657761734s_mask @ int @ N ) @ ( zero_zero @ int ) ) ).
% not_mask_negative_int
thf(fact_2654_AND__upper2_H_H,axiom,
! [Y: int,Z2: int,X2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y )
=> ( ( ord_less @ int @ Y @ Z2 )
=> ( ord_less @ int @ ( bit_se5824344872417868541ns_and @ int @ X2 @ Y ) @ Z2 ) ) ) ).
% AND_upper2''
thf(fact_2655_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_2656_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_2657_sgn__mod,axiom,
! [L: int,K: int] :
( ( L
!= ( zero_zero @ int ) )
=> ( ~ ( dvd_dvd @ int @ L @ K )
=> ( ( sgn_sgn @ int @ ( modulo_modulo @ int @ K @ L ) )
= ( sgn_sgn @ int @ L ) ) ) ) ).
% sgn_mod
thf(fact_2658_sgn__1__pos,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A] :
( ( ( sgn_sgn @ A @ A2 )
= ( one_one @ A ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% sgn_1_pos
thf(fact_2659_abs__sgn__eq,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A] :
( ( ( A2
= ( zero_zero @ A ) )
=> ( ( abs_abs @ A @ ( sgn_sgn @ A @ A2 ) )
= ( zero_zero @ A ) ) )
& ( ( A2
!= ( zero_zero @ A ) )
=> ( ( abs_abs @ A @ ( sgn_sgn @ A @ A2 ) )
= ( one_one @ A ) ) ) ) ) ).
% abs_sgn_eq
thf(fact_2660_less__mask,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N )
=> ( ord_less @ nat @ N @ ( bit_se2239418461657761734s_mask @ nat @ N ) ) ) ).
% less_mask
thf(fact_2661_zsgn__def,axiom,
( ( sgn_sgn @ int )
= ( ^ [I2: int] :
( if @ int
@ ( I2
= ( zero_zero @ int ) )
@ ( zero_zero @ int )
@ ( if @ int @ ( ord_less @ int @ ( zero_zero @ int ) @ I2 ) @ ( one_one @ int ) @ ( uminus_uminus @ int @ ( one_one @ int ) ) ) ) ) ) ).
% zsgn_def
thf(fact_2662_div__sgn__abs__cancel,axiom,
! [V2: int,K: int,L: int] :
( ( V2
!= ( zero_zero @ int ) )
=> ( ( divide_divide @ int @ ( times_times @ int @ ( sgn_sgn @ int @ V2 ) @ ( abs_abs @ int @ K ) ) @ ( times_times @ int @ ( sgn_sgn @ int @ V2 ) @ ( abs_abs @ int @ L ) ) )
= ( divide_divide @ int @ ( abs_abs @ int @ K ) @ ( abs_abs @ int @ L ) ) ) ) ).
% div_sgn_abs_cancel
thf(fact_2663_sgn__if,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( ( sgn_sgn @ A )
= ( ^ [X: A] :
( if @ A
@ ( X
= ( zero_zero @ A ) )
@ ( zero_zero @ A )
@ ( if @ A @ ( ord_less @ A @ ( zero_zero @ A ) @ X ) @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ) ) ).
% sgn_if
thf(fact_2664_sgn__1__neg,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A] :
( ( ( sgn_sgn @ A @ A2 )
= ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% sgn_1_neg
thf(fact_2665_one__and__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A] :
( ( bit_se5824344872417868541ns_and @ A @ ( one_one @ A ) @ A2 )
= ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% one_and_eq
thf(fact_2666_and__one__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A] :
( ( bit_se5824344872417868541ns_and @ A @ A2 @ ( one_one @ A ) )
= ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% and_one_eq
thf(fact_2667_take__bit__eq__mask__iff,axiom,
! [N: nat,K: int] :
( ( ( bit_se2584673776208193580ke_bit @ int @ N @ K )
= ( bit_se2239418461657761734s_mask @ int @ N ) )
= ( ( bit_se2584673776208193580ke_bit @ int @ N @ ( plus_plus @ int @ K @ ( one_one @ int ) ) )
= ( zero_zero @ int ) ) ) ).
% take_bit_eq_mask_iff
thf(fact_2668_div__noneq__sgn__abs,axiom,
! [L: int,K: int] :
( ( L
!= ( zero_zero @ int ) )
=> ( ( ( sgn_sgn @ int @ K )
!= ( sgn_sgn @ int @ L ) )
=> ( ( divide_divide @ int @ K @ L )
= ( minus_minus @ int @ ( uminus_uminus @ int @ ( divide_divide @ int @ ( abs_abs @ int @ K ) @ ( abs_abs @ int @ L ) ) )
@ ( zero_neq_one_of_bool @ int
@ ~ ( dvd_dvd @ int @ L @ K ) ) ) ) ) ) ).
% div_noneq_sgn_abs
thf(fact_2669_semiring__bit__operations__class_Oeven__mask__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se2239418461657761734s_mask @ A @ N ) )
= ( N
= ( zero_zero @ nat ) ) ) ) ).
% semiring_bit_operations_class.even_mask_iff
thf(fact_2670_and__int__rec,axiom,
( ( bit_se5824344872417868541ns_and @ int )
= ( ^ [K2: int,L2: int] :
( plus_plus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K2 )
& ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L2 ) ) )
@ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( divide_divide @ int @ K2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% and_int_rec
thf(fact_2671_mask__half__int,axiom,
! [N: nat] :
( ( divide_divide @ int @ ( bit_se2239418461657761734s_mask @ int @ N ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) )
= ( bit_se2239418461657761734s_mask @ int @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ).
% mask_half_int
thf(fact_2672_mask__nat__def,axiom,
( ( bit_se2239418461657761734s_mask @ nat )
= ( ^ [N4: nat] : ( minus_minus @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N4 ) @ ( one_one @ nat ) ) ) ) ).
% mask_nat_def
thf(fact_2673_mask__int__def,axiom,
( ( bit_se2239418461657761734s_mask @ int )
= ( ^ [N4: nat] : ( minus_minus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N4 ) @ ( one_one @ int ) ) ) ) ).
% mask_int_def
thf(fact_2674_mask__eq__exp__minus__1,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se2239418461657761734s_mask @ A )
= ( ^ [N4: nat] : ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N4 ) @ ( one_one @ A ) ) ) ) ) ).
% mask_eq_exp_minus_1
thf(fact_2675_divide__int__unfold,axiom,
! [L: int,K: int,N: nat,M: nat] :
( ( ( ( ( sgn_sgn @ int @ L )
= ( zero_zero @ int ) )
| ( ( sgn_sgn @ int @ K )
= ( zero_zero @ int ) )
| ( N
= ( zero_zero @ nat ) ) )
=> ( ( divide_divide @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K ) @ ( semiring_1_of_nat @ int @ M ) ) @ ( times_times @ int @ ( sgn_sgn @ int @ L ) @ ( semiring_1_of_nat @ int @ N ) ) )
= ( zero_zero @ int ) ) )
& ( ~ ( ( ( sgn_sgn @ int @ L )
= ( zero_zero @ int ) )
| ( ( sgn_sgn @ int @ K )
= ( zero_zero @ int ) )
| ( N
= ( zero_zero @ nat ) ) )
=> ( ( ( ( sgn_sgn @ int @ K )
= ( sgn_sgn @ int @ L ) )
=> ( ( divide_divide @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K ) @ ( semiring_1_of_nat @ int @ M ) ) @ ( times_times @ int @ ( sgn_sgn @ int @ L ) @ ( semiring_1_of_nat @ int @ N ) ) )
= ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ M @ N ) ) ) )
& ( ( ( sgn_sgn @ int @ K )
!= ( sgn_sgn @ int @ L ) )
=> ( ( divide_divide @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K ) @ ( semiring_1_of_nat @ int @ M ) ) @ ( times_times @ int @ ( sgn_sgn @ int @ L ) @ ( semiring_1_of_nat @ int @ N ) ) )
= ( uminus_uminus @ int
@ ( semiring_1_of_nat @ int
@ ( plus_plus @ nat @ ( divide_divide @ nat @ M @ N )
@ ( zero_neq_one_of_bool @ nat
@ ~ ( dvd_dvd @ nat @ N @ M ) ) ) ) ) ) ) ) ) ) ).
% divide_int_unfold
thf(fact_2676_modulo__int__unfold,axiom,
! [L: int,K: int,N: nat,M: nat] :
( ( ( ( ( sgn_sgn @ int @ L )
= ( zero_zero @ int ) )
| ( ( sgn_sgn @ int @ K )
= ( zero_zero @ int ) )
| ( N
= ( zero_zero @ nat ) ) )
=> ( ( modulo_modulo @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K ) @ ( semiring_1_of_nat @ int @ M ) ) @ ( times_times @ int @ ( sgn_sgn @ int @ L ) @ ( semiring_1_of_nat @ int @ N ) ) )
= ( times_times @ int @ ( sgn_sgn @ int @ K ) @ ( semiring_1_of_nat @ int @ M ) ) ) )
& ( ~ ( ( ( sgn_sgn @ int @ L )
= ( zero_zero @ int ) )
| ( ( sgn_sgn @ int @ K )
= ( zero_zero @ int ) )
| ( N
= ( zero_zero @ nat ) ) )
=> ( ( ( ( sgn_sgn @ int @ K )
= ( sgn_sgn @ int @ L ) )
=> ( ( modulo_modulo @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K ) @ ( semiring_1_of_nat @ int @ M ) ) @ ( times_times @ int @ ( sgn_sgn @ int @ L ) @ ( semiring_1_of_nat @ int @ N ) ) )
= ( times_times @ int @ ( sgn_sgn @ int @ L ) @ ( semiring_1_of_nat @ int @ ( modulo_modulo @ nat @ M @ N ) ) ) ) )
& ( ( ( sgn_sgn @ int @ K )
!= ( sgn_sgn @ int @ L ) )
=> ( ( modulo_modulo @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K ) @ ( semiring_1_of_nat @ int @ M ) ) @ ( times_times @ int @ ( sgn_sgn @ int @ L ) @ ( semiring_1_of_nat @ int @ N ) ) )
= ( times_times @ int @ ( sgn_sgn @ int @ L )
@ ( minus_minus @ int
@ ( semiring_1_of_nat @ int
@ ( times_times @ nat @ N
@ ( zero_neq_one_of_bool @ nat
@ ~ ( dvd_dvd @ nat @ N @ M ) ) ) )
@ ( semiring_1_of_nat @ int @ ( modulo_modulo @ nat @ M @ N ) ) ) ) ) ) ) ) ) ).
% modulo_int_unfold
thf(fact_2677_divide__int__def,axiom,
( ( divide_divide @ int )
= ( ^ [K2: int,L2: int] :
( if @ int
@ ( L2
= ( zero_zero @ int ) )
@ ( zero_zero @ int )
@ ( if @ int
@ ( ( sgn_sgn @ int @ K2 )
= ( sgn_sgn @ int @ L2 ) )
@ ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ ( nat2 @ ( abs_abs @ int @ K2 ) ) @ ( nat2 @ ( abs_abs @ int @ L2 ) ) ) )
@ ( uminus_uminus @ int
@ ( semiring_1_of_nat @ int
@ ( plus_plus @ nat @ ( divide_divide @ nat @ ( nat2 @ ( abs_abs @ int @ K2 ) ) @ ( nat2 @ ( abs_abs @ int @ L2 ) ) )
@ ( zero_neq_one_of_bool @ nat
@ ~ ( dvd_dvd @ int @ L2 @ K2 ) ) ) ) ) ) ) ) ) ).
% divide_int_def
thf(fact_2678_take__bit__Suc__from__most,axiom,
! [N: nat,K: int] :
( ( bit_se2584673776208193580ke_bit @ int @ ( suc @ N ) @ K )
= ( plus_plus @ int @ ( times_times @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) @ ( zero_neq_one_of_bool @ int @ ( bit_se5641148757651400278ts_bit @ int @ K @ N ) ) ) @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) ) ) ).
% take_bit_Suc_from_most
thf(fact_2679_Suc__0__xor__eq,axiom,
! [N: nat] :
( ( bit_se5824344971392196577ns_xor @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N )
= ( minus_minus @ nat @ ( plus_plus @ nat @ N @ ( zero_neq_one_of_bool @ nat @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) )
@ ( zero_neq_one_of_bool @ nat
@ ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% Suc_0_xor_eq
thf(fact_2680_xor__Suc__0__eq,axiom,
! [N: nat] :
( ( bit_se5824344971392196577ns_xor @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) )
= ( minus_minus @ nat @ ( plus_plus @ nat @ N @ ( zero_neq_one_of_bool @ nat @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) )
@ ( zero_neq_one_of_bool @ nat
@ ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% xor_Suc_0_eq
thf(fact_2681_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_2682_push__bit__numeral__minus__1,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: num] :
( ( bit_se4730199178511100633sh_bit @ A @ ( numeral_numeral @ nat @ N ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( uminus_uminus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ N ) ) ) ) ) ).
% push_bit_numeral_minus_1
thf(fact_2683_xor__numerals_I6_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X2: num,Y: num] :
( ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ ( bit1 @ X2 ) ) @ ( 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 @ X2 ) @ ( numeral_numeral @ A @ Y ) ) ) ) ) ) ).
% xor_numerals(6)
thf(fact_2684_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_2685_push__bit__nonnegative__int__iff,axiom,
! [N: nat,K: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se4730199178511100633sh_bit @ int @ N @ K ) )
= ( ord_less_eq @ int @ ( zero_zero @ int ) @ K ) ) ).
% push_bit_nonnegative_int_iff
thf(fact_2686_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_2687_push__bit__negative__int__iff,axiom,
! [N: nat,K: int] :
( ( ord_less @ int @ ( bit_se4730199178511100633sh_bit @ int @ N @ K ) @ ( zero_zero @ int ) )
= ( ord_less @ int @ K @ ( zero_zero @ int ) ) ) ).
% push_bit_negative_int_iff
thf(fact_2688_bit_Oxor__self,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X2: A] :
( ( bit_se5824344971392196577ns_xor @ A @ X2 @ X2 )
= ( zero_zero @ A ) ) ) ).
% bit.xor_self
thf(fact_2689_xor__self__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A] :
( ( bit_se5824344971392196577ns_xor @ A @ A2 @ A2 )
= ( zero_zero @ A ) ) ) ).
% xor_self_eq
thf(fact_2690_xor_Oleft__neutral,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A] :
( ( bit_se5824344971392196577ns_xor @ A @ ( zero_zero @ A ) @ A2 )
= A2 ) ) ).
% xor.left_neutral
thf(fact_2691_xor_Oright__neutral,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A] :
( ( bit_se5824344971392196577ns_xor @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% xor.right_neutral
thf(fact_2692_push__bit__of__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat] :
( ( bit_se4730199178511100633sh_bit @ A @ N @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% push_bit_of_0
thf(fact_2693_push__bit__eq__0__iff,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N: nat,A2: A] :
( ( ( bit_se4730199178511100633sh_bit @ A @ N @ A2 )
= ( zero_zero @ A ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% push_bit_eq_0_iff
thf(fact_2694_push__bit__push__bit,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,N: nat,A2: A] :
( ( bit_se4730199178511100633sh_bit @ A @ M @ ( bit_se4730199178511100633sh_bit @ A @ N @ A2 ) )
= ( bit_se4730199178511100633sh_bit @ A @ ( plus_plus @ nat @ M @ N ) @ A2 ) ) ) ).
% push_bit_push_bit
thf(fact_2695_concat__bit__of__zero__1,axiom,
! [N: nat,L: int] :
( ( bit_concat_bit @ N @ ( zero_zero @ int ) @ L )
= ( bit_se4730199178511100633sh_bit @ int @ N @ L ) ) ).
% concat_bit_of_zero_1
thf(fact_2696_signed__take__bit__nonnegative__iff,axiom,
! [N: nat,K: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K ) )
= ( ~ ( bit_se5641148757651400278ts_bit @ int @ K @ N ) ) ) ).
% signed_take_bit_nonnegative_iff
thf(fact_2697_signed__take__bit__negative__iff,axiom,
! [N: nat,K: int] :
( ( ord_less @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K ) @ ( zero_zero @ int ) )
= ( bit_se5641148757651400278ts_bit @ int @ K @ N ) ) ).
% signed_take_bit_negative_iff
thf(fact_2698_xor__numerals_I8_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X2: num] :
( ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ ( bit1 @ X2 ) ) @ ( one_one @ A ) )
= ( numeral_numeral @ A @ ( bit0 @ X2 ) ) ) ) ).
% xor_numerals(8)
thf(fact_2699_xor__numerals_I5_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X2: num] :
( ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ ( bit0 @ X2 ) ) @ ( one_one @ A ) )
= ( numeral_numeral @ A @ ( bit1 @ X2 ) ) ) ) ).
% xor_numerals(5)
thf(fact_2700_xor__numerals_I2_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [Y: num] :
( ( bit_se5824344971392196577ns_xor @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit1 @ Y ) ) )
= ( numeral_numeral @ A @ ( bit0 @ Y ) ) ) ) ).
% xor_numerals(2)
thf(fact_2701_xor__numerals_I1_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [Y: num] :
( ( bit_se5824344971392196577ns_xor @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ Y ) ) )
= ( numeral_numeral @ A @ ( bit1 @ Y ) ) ) ) ).
% xor_numerals(1)
thf(fact_2702_and__nat__numerals_I1_J,axiom,
! [Y: num] :
( ( bit_se5824344872417868541ns_and @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( numeral_numeral @ nat @ ( bit0 @ Y ) ) )
= ( zero_zero @ nat ) ) ).
% and_nat_numerals(1)
thf(fact_2703_and__nat__numerals_I3_J,axiom,
! [X2: num] :
( ( bit_se5824344872417868541ns_and @ nat @ ( numeral_numeral @ nat @ ( bit0 @ X2 ) ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( zero_zero @ nat ) ) ).
% and_nat_numerals(3)
thf(fact_2704_bit__0,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A2: A] :
( ( bit_se5641148757651400278ts_bit @ A @ A2 @ ( zero_zero @ nat ) )
= ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) ) ) ).
% bit_0
thf(fact_2705_even__push__bit__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A2: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se4730199178511100633sh_bit @ A @ N @ A2 ) )
= ( ( N
!= ( zero_zero @ nat ) )
| ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) ) ) ).
% even_push_bit_iff
thf(fact_2706_push__bit__of__1,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat] :
( ( bit_se4730199178511100633sh_bit @ A @ N @ ( one_one @ A ) )
= ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) ) ).
% push_bit_of_1
thf(fact_2707_push__bit__of__Suc__0,axiom,
! [N: nat] :
( ( bit_se4730199178511100633sh_bit @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ).
% push_bit_of_Suc_0
thf(fact_2708_xor__nat__numerals_I4_J,axiom,
! [X2: num] :
( ( bit_se5824344971392196577ns_xor @ nat @ ( numeral_numeral @ nat @ ( bit1 @ X2 ) ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( numeral_numeral @ nat @ ( bit0 @ X2 ) ) ) ).
% xor_nat_numerals(4)
thf(fact_2709_xor__nat__numerals_I3_J,axiom,
! [X2: num] :
( ( bit_se5824344971392196577ns_xor @ nat @ ( numeral_numeral @ nat @ ( bit0 @ X2 ) ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( numeral_numeral @ nat @ ( bit1 @ X2 ) ) ) ).
% xor_nat_numerals(3)
thf(fact_2710_xor__nat__numerals_I2_J,axiom,
! [Y: num] :
( ( bit_se5824344971392196577ns_xor @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( numeral_numeral @ nat @ ( bit1 @ Y ) ) )
= ( numeral_numeral @ nat @ ( bit0 @ Y ) ) ) ).
% xor_nat_numerals(2)
thf(fact_2711_xor__nat__numerals_I1_J,axiom,
! [Y: num] :
( ( bit_se5824344971392196577ns_xor @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( numeral_numeral @ nat @ ( bit0 @ Y ) ) )
= ( numeral_numeral @ nat @ ( bit1 @ Y ) ) ) ).
% xor_nat_numerals(1)
thf(fact_2712_and__nat__numerals_I4_J,axiom,
! [X2: num] :
( ( bit_se5824344872417868541ns_and @ nat @ ( numeral_numeral @ nat @ ( bit1 @ X2 ) ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( one_one @ nat ) ) ).
% and_nat_numerals(4)
thf(fact_2713_and__nat__numerals_I2_J,axiom,
! [Y: num] :
( ( bit_se5824344872417868541ns_and @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( numeral_numeral @ nat @ ( bit1 @ Y ) ) )
= ( one_one @ nat ) ) ).
% and_nat_numerals(2)
thf(fact_2714_bit__mod__2__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A2: A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ N )
= ( ( N
= ( zero_zero @ nat ) )
& ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) ) ) ).
% bit_mod_2_iff
thf(fact_2715_and__Suc__0__eq,axiom,
! [N: nat] :
( ( bit_se5824344872417868541ns_and @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) )
= ( modulo_modulo @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% and_Suc_0_eq
thf(fact_2716_Suc__0__and__eq,axiom,
! [N: nat] :
( ( bit_se5824344872417868541ns_and @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N )
= ( modulo_modulo @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% Suc_0_and_eq
thf(fact_2717_xor__numerals_I4_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X2: num,Y: num] :
( ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ ( bit0 @ X2 ) ) @ ( 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 @ X2 ) @ ( numeral_numeral @ A @ Y ) ) ) ) ) ) ).
% xor_numerals(4)
thf(fact_2718_flip__bit__nat__def,axiom,
( ( bit_se8732182000553998342ip_bit @ nat )
= ( ^ [M2: nat,N4: nat] : ( bit_se5824344971392196577ns_xor @ nat @ N4 @ ( bit_se4730199178511100633sh_bit @ nat @ M2 @ ( one_one @ nat ) ) ) ) ) ).
% flip_bit_nat_def
thf(fact_2719_of__nat__xor__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,N: nat] :
( ( semiring_1_of_nat @ A @ ( bit_se5824344971392196577ns_xor @ nat @ M @ N ) )
= ( bit_se5824344971392196577ns_xor @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% of_nat_xor_eq
thf(fact_2720_of__nat__push__bit,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,N: nat] :
( ( semiring_1_of_nat @ A @ ( bit_se4730199178511100633sh_bit @ nat @ M @ N ) )
= ( bit_se4730199178511100633sh_bit @ A @ M @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% of_nat_push_bit
thf(fact_2721_push__bit__of__nat,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,M: nat] :
( ( bit_se4730199178511100633sh_bit @ A @ N @ ( semiring_1_of_nat @ A @ M ) )
= ( semiring_1_of_nat @ A @ ( bit_se4730199178511100633sh_bit @ nat @ N @ M ) ) ) ) ).
% push_bit_of_nat
thf(fact_2722_bit__of__nat__iff__bit,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [M: nat,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( semiring_1_of_nat @ A @ M ) @ N )
= ( bit_se5641148757651400278ts_bit @ nat @ M @ N ) ) ) ).
% bit_of_nat_iff_bit
thf(fact_2723_xor__nat__def,axiom,
( ( bit_se5824344971392196577ns_xor @ nat )
= ( ^ [M2: nat,N4: nat] : ( nat2 @ ( bit_se5824344971392196577ns_xor @ int @ ( semiring_1_of_nat @ int @ M2 ) @ ( semiring_1_of_nat @ int @ N4 ) ) ) ) ) ).
% xor_nat_def
thf(fact_2724_bit__push__bit__iff__int,axiom,
! [M: nat,K: int,N: nat] :
( ( bit_se5641148757651400278ts_bit @ int @ ( bit_se4730199178511100633sh_bit @ int @ M @ K ) @ N )
= ( ( ord_less_eq @ nat @ M @ N )
& ( bit_se5641148757651400278ts_bit @ int @ K @ ( minus_minus @ nat @ N @ M ) ) ) ) ).
% bit_push_bit_iff_int
thf(fact_2725_bit__disjunctive__add__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A2: A,B2: A,N: nat] :
( ! [N2: nat] :
( ~ ( bit_se5641148757651400278ts_bit @ A @ A2 @ N2 )
| ~ ( bit_se5641148757651400278ts_bit @ A @ B2 @ N2 ) )
=> ( ( bit_se5641148757651400278ts_bit @ A @ ( plus_plus @ A @ A2 @ B2 ) @ N )
= ( ( bit_se5641148757651400278ts_bit @ A @ A2 @ N )
| ( bit_se5641148757651400278ts_bit @ A @ B2 @ N ) ) ) ) ) ).
% bit_disjunctive_add_iff
thf(fact_2726_flip__bit__eq__xor,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se8732182000553998342ip_bit @ A )
= ( ^ [N4: nat,A3: A] : ( bit_se5824344971392196577ns_xor @ A @ A3 @ ( bit_se4730199178511100633sh_bit @ A @ N4 @ ( one_one @ A ) ) ) ) ) ) ).
% flip_bit_eq_xor
thf(fact_2727_flip__bit__int__def,axiom,
( ( bit_se8732182000553998342ip_bit @ int )
= ( ^ [N4: nat,K2: int] : ( bit_se5824344971392196577ns_xor @ int @ K2 @ ( bit_se4730199178511100633sh_bit @ int @ N4 @ ( one_one @ int ) ) ) ) ) ).
% flip_bit_int_def
thf(fact_2728_push__bit__add,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A2: A,B2: A] :
( ( bit_se4730199178511100633sh_bit @ A @ N @ ( plus_plus @ A @ A2 @ B2 ) )
= ( plus_plus @ A @ ( bit_se4730199178511100633sh_bit @ A @ N @ A2 ) @ ( bit_se4730199178511100633sh_bit @ A @ N @ B2 ) ) ) ) ).
% push_bit_add
thf(fact_2729_bit__iff__and__push__bit__not__eq__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se5641148757651400278ts_bit @ A )
= ( ^ [A3: A,N4: nat] :
( ( bit_se5824344872417868541ns_and @ A @ A3 @ ( bit_se4730199178511100633sh_bit @ A @ N4 @ ( one_one @ A ) ) )
!= ( zero_zero @ A ) ) ) ) ) ).
% bit_iff_and_push_bit_not_eq_0
thf(fact_2730_bit__1__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( one_one @ A ) @ N )
= ( N
= ( zero_zero @ nat ) ) ) ) ).
% bit_1_iff
thf(fact_2731_not__bit__1__Suc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat] :
~ ( bit_se5641148757651400278ts_bit @ A @ ( one_one @ A ) @ ( suc @ N ) ) ) ).
% not_bit_1_Suc
thf(fact_2732_bit__numeral__simps_I1_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N: num] :
~ ( bit_se5641148757651400278ts_bit @ A @ ( one_one @ A ) @ ( numeral_numeral @ nat @ N ) ) ) ).
% bit_numeral_simps(1)
thf(fact_2733_bit__of__bool__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [B2: $o,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( zero_neq_one_of_bool @ A @ B2 ) @ N )
= ( B2
& ( N
= ( zero_zero @ nat ) ) ) ) ) ).
% bit_of_bool_iff
thf(fact_2734_XOR__lower,axiom,
! [X2: int,Y: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X2 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y )
=> ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se5824344971392196577ns_xor @ int @ X2 @ Y ) ) ) ) ).
% XOR_lower
thf(fact_2735_push__bit__take__bit,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,N: nat,A2: A] :
( ( bit_se4730199178511100633sh_bit @ A @ M @ ( bit_se2584673776208193580ke_bit @ A @ N @ A2 ) )
= ( bit_se2584673776208193580ke_bit @ A @ ( plus_plus @ nat @ M @ N ) @ ( bit_se4730199178511100633sh_bit @ A @ M @ A2 ) ) ) ) ).
% push_bit_take_bit
thf(fact_2736_bit__horner__sum__bit__iff,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [Bs: list @ $o,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( groups4207007520872428315er_sum @ $o @ A @ ( zero_neq_one_of_bool @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Bs ) @ N )
= ( ( ord_less @ nat @ N @ ( size_size @ ( list @ $o ) @ Bs ) )
& ( nth @ $o @ Bs @ N ) ) ) ) ).
% bit_horner_sum_bit_iff
thf(fact_2737_bit__not__int__iff_H,axiom,
! [K: int,N: nat] :
( ( bit_se5641148757651400278ts_bit @ int @ ( minus_minus @ int @ ( uminus_uminus @ int @ K ) @ ( one_one @ int ) ) @ N )
= ( ~ ( bit_se5641148757651400278ts_bit @ int @ K @ N ) ) ) ).
% bit_not_int_iff'
thf(fact_2738_and__nat__def,axiom,
( ( bit_se5824344872417868541ns_and @ nat )
= ( ^ [M2: nat,N4: nat] : ( nat2 @ ( bit_se5824344872417868541ns_and @ int @ ( semiring_1_of_nat @ int @ M2 ) @ ( semiring_1_of_nat @ int @ N4 ) ) ) ) ) ).
% and_nat_def
thf(fact_2739_signed__take__bit__code,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( bit_ri4674362597316999326ke_bit @ A )
= ( ^ [N4: nat,A3: A] : ( if @ A @ ( bit_se5641148757651400278ts_bit @ A @ ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N4 ) @ A3 ) @ N4 ) @ ( plus_plus @ A @ ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N4 ) @ A3 ) @ ( bit_se4730199178511100633sh_bit @ A @ ( suc @ N4 ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) @ ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N4 ) @ A3 ) ) ) ) ) ).
% signed_take_bit_code
thf(fact_2740_concat__bit__eq,axiom,
( bit_concat_bit
= ( ^ [N4: nat,K2: int,L2: int] : ( plus_plus @ int @ ( bit_se2584673776208193580ke_bit @ int @ N4 @ K2 ) @ ( bit_se4730199178511100633sh_bit @ int @ N4 @ L2 ) ) ) ) ).
% concat_bit_eq
thf(fact_2741_bit__imp__take__bit__positive,axiom,
! [N: nat,M: nat,K: int] :
( ( ord_less @ nat @ N @ M )
=> ( ( bit_se5641148757651400278ts_bit @ int @ K @ N )
=> ( ord_less @ int @ ( zero_zero @ int ) @ ( bit_se2584673776208193580ke_bit @ int @ M @ K ) ) ) ) ).
% bit_imp_take_bit_positive
thf(fact_2742_bit__concat__bit__iff,axiom,
! [M: nat,K: int,L: int,N: nat] :
( ( bit_se5641148757651400278ts_bit @ int @ ( bit_concat_bit @ M @ K @ L ) @ N )
= ( ( ( ord_less @ nat @ N @ M )
& ( bit_se5641148757651400278ts_bit @ int @ K @ N ) )
| ( ( ord_less_eq @ nat @ M @ N )
& ( bit_se5641148757651400278ts_bit @ int @ L @ ( minus_minus @ nat @ N @ M ) ) ) ) ) ).
% bit_concat_bit_iff
thf(fact_2743_exp__eq__0__imp__not__bit,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [N: nat,A2: A] :
( ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N )
= ( zero_zero @ A ) )
=> ~ ( bit_se5641148757651400278ts_bit @ A @ A2 @ N ) ) ) ).
% exp_eq_0_imp_not_bit
thf(fact_2744_int__bit__bound,axiom,
! [K: int] :
~ ! [N2: nat] :
( ! [M7: nat] :
( ( ord_less_eq @ nat @ N2 @ M7 )
=> ( ( bit_se5641148757651400278ts_bit @ int @ K @ M7 )
= ( bit_se5641148757651400278ts_bit @ int @ K @ N2 ) ) )
=> ~ ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( bit_se5641148757651400278ts_bit @ int @ K @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) )
= ( ~ ( bit_se5641148757651400278ts_bit @ int @ K @ N2 ) ) ) ) ) ).
% int_bit_bound
thf(fact_2745_and__exp__eq__0__iff__not__bit,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A,N: nat] :
( ( ( bit_se5824344872417868541ns_and @ A @ A2 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
= ( zero_zero @ A ) )
= ( ~ ( bit_se5641148757651400278ts_bit @ A @ A2 @ N ) ) ) ) ).
% and_exp_eq_0_iff_not_bit
thf(fact_2746_push__bit__minus__one,axiom,
! [N: nat] :
( ( bit_se4730199178511100633sh_bit @ int @ N @ ( uminus_uminus @ int @ ( one_one @ int ) ) )
= ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ).
% push_bit_minus_one
thf(fact_2747_even__bit__succ__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A2: A,N: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
=> ( ( bit_se5641148757651400278ts_bit @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A2 ) @ N )
= ( ( bit_se5641148757651400278ts_bit @ A @ A2 @ N )
| ( N
= ( zero_zero @ nat ) ) ) ) ) ) ).
% even_bit_succ_iff
thf(fact_2748_odd__bit__iff__bit__pred,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A2: A,N: nat] :
( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
=> ( ( bit_se5641148757651400278ts_bit @ A @ A2 @ N )
= ( ( bit_se5641148757651400278ts_bit @ A @ ( minus_minus @ A @ A2 @ ( one_one @ A ) ) @ N )
| ( N
= ( zero_zero @ nat ) ) ) ) ) ) ).
% odd_bit_iff_bit_pred
thf(fact_2749_XOR__upper,axiom,
! [X2: int,N: nat,Y: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X2 )
=> ( ( ord_less @ int @ X2 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( ord_less @ int @ Y @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) )
=> ( ord_less @ int @ ( bit_se5824344971392196577ns_xor @ int @ X2 @ Y ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ).
% XOR_upper
thf(fact_2750_bit__sum__mult__2__cases,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A,B2: A,N: nat] :
( ! [J2: nat] :
~ ( bit_se5641148757651400278ts_bit @ A @ A2 @ ( suc @ J2 ) )
=> ( ( bit_se5641148757651400278ts_bit @ A @ ( plus_plus @ A @ A2 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) @ N )
= ( ( ( N
= ( zero_zero @ nat ) )
=> ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ( bit_se5641148757651400278ts_bit @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) @ N ) ) ) ) ) ) ).
% bit_sum_mult_2_cases
thf(fact_2751_bit__rec,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ( ( bit_se5641148757651400278ts_bit @ A )
= ( ^ [A3: A,N4: nat] :
( ( ( N4
= ( zero_zero @ nat ) )
=> ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) )
& ( ( N4
!= ( zero_zero @ nat ) )
=> ( bit_se5641148757651400278ts_bit @ A @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( minus_minus @ nat @ N4 @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% bit_rec
thf(fact_2752_and__nat__unfold,axiom,
( ( bit_se5824344872417868541ns_and @ nat )
= ( ^ [M2: nat,N4: nat] :
( if @ nat
@ ( ( M2
= ( zero_zero @ nat ) )
| ( N4
= ( zero_zero @ nat ) ) )
@ ( zero_zero @ nat )
@ ( plus_plus @ nat @ ( times_times @ nat @ ( modulo_modulo @ nat @ M2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ nat @ N4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ nat @ ( divide_divide @ nat @ M2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ).
% and_nat_unfold
thf(fact_2753_and__nat__rec,axiom,
( ( bit_se5824344872417868541ns_and @ nat )
= ( ^ [M2: nat,N4: nat] :
( plus_plus @ nat
@ ( zero_neq_one_of_bool @ nat
@ ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M2 )
& ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N4 ) ) )
@ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ nat @ ( divide_divide @ nat @ M2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% and_nat_rec
thf(fact_2754_xor__nat__unfold,axiom,
( ( bit_se5824344971392196577ns_xor @ nat )
= ( ^ [M2: nat,N4: nat] :
( if @ nat
@ ( M2
= ( zero_zero @ nat ) )
@ N4
@ ( if @ nat
@ ( N4
= ( zero_zero @ nat ) )
@ M2
@ ( plus_plus @ nat @ ( modulo_modulo @ nat @ ( plus_plus @ nat @ ( modulo_modulo @ nat @ M2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ nat @ N4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ nat @ ( divide_divide @ nat @ M2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% xor_nat_unfold
thf(fact_2755_xor__int__rec,axiom,
( ( bit_se5824344971392196577ns_xor @ int )
= ( ^ [K2: int,L2: int] :
( plus_plus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K2 ) )
!= ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L2 ) ) ) )
@ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ int @ ( divide_divide @ int @ K2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% xor_int_rec
thf(fact_2756_xor__nat__rec,axiom,
( ( bit_se5824344971392196577ns_xor @ nat )
= ( ^ [M2: nat,N4: nat] :
( plus_plus @ nat
@ ( zero_neq_one_of_bool @ nat
@ ( ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M2 ) )
!= ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N4 ) ) ) )
@ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ nat @ ( divide_divide @ nat @ M2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% xor_nat_rec
thf(fact_2757_xor__one__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A] :
( ( bit_se5824344971392196577ns_xor @ A @ A2 @ ( one_one @ A ) )
= ( minus_minus @ A @ ( plus_plus @ A @ A2 @ ( zero_neq_one_of_bool @ A @ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) )
@ ( zero_neq_one_of_bool @ A
@ ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) ) ) ) ).
% xor_one_eq
thf(fact_2758_one__xor__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A] :
( ( bit_se5824344971392196577ns_xor @ A @ ( one_one @ A ) @ A2 )
= ( minus_minus @ A @ ( plus_plus @ A @ A2 @ ( zero_neq_one_of_bool @ A @ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) )
@ ( zero_neq_one_of_bool @ A
@ ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) ) ) ) ).
% one_xor_eq
thf(fact_2759_set__bit__eq,axiom,
( ( bit_se5668285175392031749et_bit @ int )
= ( ^ [N4: nat,K2: int] :
( plus_plus @ int @ K2
@ ( times_times @ int
@ ( zero_neq_one_of_bool @ int
@ ~ ( bit_se5641148757651400278ts_bit @ int @ K2 @ N4 ) )
@ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N4 ) ) ) ) ) ).
% set_bit_eq
thf(fact_2760_horner__sum__simps_I2_J,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_0 @ A )
=> ! [F: B > A,A2: A,X2: B,Xs: list @ B] :
( ( groups4207007520872428315er_sum @ B @ A @ F @ A2 @ ( cons @ B @ X2 @ Xs ) )
= ( plus_plus @ A @ ( F @ X2 ) @ ( times_times @ A @ A2 @ ( groups4207007520872428315er_sum @ B @ A @ F @ A2 @ Xs ) ) ) ) ) ).
% horner_sum_simps(2)
thf(fact_2761_horner__sum__simps_I1_J,axiom,
! [B: $tType,A: $tType] :
( ( comm_semiring_0 @ A )
=> ! [F: B > A,A2: A] :
( ( groups4207007520872428315er_sum @ B @ A @ F @ A2 @ ( nil @ B ) )
= ( zero_zero @ A ) ) ) ).
% horner_sum_simps(1)
thf(fact_2762_horner__sum__foldr,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_0 @ A )
=> ( ( groups4207007520872428315er_sum @ B @ A )
= ( ^ [F2: B > A,A3: A,Xs3: list @ B] :
( foldr @ B @ A
@ ^ [X: B,B3: A] : ( plus_plus @ A @ ( F2 @ X ) @ ( times_times @ A @ A3 @ B3 ) )
@ Xs3
@ ( zero_zero @ A ) ) ) ) ) ).
% horner_sum_foldr
thf(fact_2763_xor__int__unfold,axiom,
( ( bit_se5824344971392196577ns_xor @ int )
= ( ^ [K2: int,L2: int] :
( if @ int
@ ( K2
= ( uminus_uminus @ int @ ( one_one @ int ) ) )
@ ( bit_ri4277139882892585799ns_not @ int @ L2 )
@ ( if @ int
@ ( L2
= ( uminus_uminus @ int @ ( one_one @ int ) ) )
@ ( bit_ri4277139882892585799ns_not @ int @ K2 )
@ ( if @ int
@ ( K2
= ( zero_zero @ int ) )
@ L2
@ ( if @ int
@ ( L2
= ( zero_zero @ int ) )
@ K2
@ ( plus_plus @ int @ ( abs_abs @ int @ ( minus_minus @ int @ ( modulo_modulo @ int @ K2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ int @ L2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ int @ ( divide_divide @ int @ K2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ).
% xor_int_unfold
thf(fact_2764_length__subseqs,axiom,
! [A: $tType,Xs: list @ A] :
( ( size_size @ ( list @ ( list @ A ) ) @ ( subseqs @ A @ Xs ) )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ).
% length_subseqs
thf(fact_2765_mask__eq__sum__exp__nat,axiom,
! [N: nat] :
( ( minus_minus @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( groups7311177749621191930dd_sum @ nat @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
@ ( collect @ nat
@ ^ [Q6: nat] : ( ord_less @ nat @ Q6 @ N ) ) ) ) ).
% mask_eq_sum_exp_nat
thf(fact_2766_Ints__sum,axiom,
! [A: $tType,B: $tType] :
( ( ring_1 @ B )
=> ! [A5: set @ A,F: A > B] :
( ! [X3: A] :
( ( member @ A @ X3 @ A5 )
=> ( member @ B @ ( F @ X3 ) @ ( ring_1_Ints @ B ) ) )
=> ( member @ B @ ( groups7311177749621191930dd_sum @ A @ B @ F @ A5 ) @ ( ring_1_Ints @ B ) ) ) ) ).
% Ints_sum
thf(fact_2767_of__int__sum,axiom,
! [A: $tType,B: $tType] :
( ( ring_1 @ A )
=> ! [F: B > int,A5: set @ B] :
( ( ring_1_of_int @ A @ ( groups7311177749621191930dd_sum @ B @ int @ F @ A5 ) )
= ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X: B] : ( ring_1_of_int @ A @ ( F @ X ) )
@ A5 ) ) ) ).
% of_int_sum
thf(fact_2768_bit_Oconj__cancel__right,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X2: A] :
( ( bit_se5824344872417868541ns_and @ A @ X2 @ ( bit_ri4277139882892585799ns_not @ A @ X2 ) )
= ( zero_zero @ A ) ) ) ).
% bit.conj_cancel_right
thf(fact_2769_bit_Oconj__cancel__left,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X2: A] :
( ( bit_se5824344872417868541ns_and @ A @ ( bit_ri4277139882892585799ns_not @ A @ X2 ) @ X2 )
= ( zero_zero @ A ) ) ) ).
% bit.conj_cancel_left
thf(fact_2770_of__nat__sum,axiom,
! [A: $tType,B: $tType] :
( ( semiring_1 @ A )
=> ! [F: B > nat,A5: set @ B] :
( ( semiring_1_of_nat @ A @ ( groups7311177749621191930dd_sum @ B @ nat @ F @ A5 ) )
= ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X: B] : ( semiring_1_of_nat @ A @ ( F @ X ) )
@ A5 ) ) ) ).
% of_nat_sum
thf(fact_2771_bit_Ocompl__zero,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( bit_ri4277139882892585799ns_not @ A @ ( zero_zero @ A ) )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% bit.compl_zero
thf(fact_2772_bit_Ocompl__one,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( bit_ri4277139882892585799ns_not @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( zero_zero @ A ) ) ) ).
% bit.compl_one
thf(fact_2773_bit_Oxor__one__left,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X2: A] :
( ( bit_se5824344971392196577ns_xor @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ X2 )
= ( bit_ri4277139882892585799ns_not @ A @ X2 ) ) ) ).
% bit.xor_one_left
thf(fact_2774_bit_Oxor__one__right,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X2: A] :
( ( bit_se5824344971392196577ns_xor @ A @ X2 @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( bit_ri4277139882892585799ns_not @ A @ X2 ) ) ) ).
% bit.xor_one_right
thf(fact_2775_bit_Oxor__cancel__left,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X2: A] :
( ( bit_se5824344971392196577ns_xor @ A @ ( bit_ri4277139882892585799ns_not @ A @ X2 ) @ X2 )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% bit.xor_cancel_left
thf(fact_2776_bit_Oxor__cancel__right,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X2: A] :
( ( bit_se5824344971392196577ns_xor @ A @ X2 @ ( bit_ri4277139882892585799ns_not @ A @ X2 ) )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% bit.xor_cancel_right
thf(fact_2777_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_2778_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_2779_push__bit__minus__one__eq__not__mask,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: nat] :
( ( bit_se4730199178511100633sh_bit @ A @ N @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( bit_ri4277139882892585799ns_not @ A @ ( bit_se2239418461657761734s_mask @ A @ N ) ) ) ) ).
% push_bit_minus_one_eq_not_mask
thf(fact_2780_not__one__eq,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( bit_ri4277139882892585799ns_not @ A @ ( one_one @ A ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% not_one_eq
thf(fact_2781_mod__sum__eq,axiom,
! [B: $tType,A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [F: B > A,A2: A,A5: set @ B] :
( ( modulo_modulo @ A
@ ( groups7311177749621191930dd_sum @ B @ A
@ ^ [I2: B] : ( modulo_modulo @ A @ ( F @ I2 ) @ A2 )
@ A5 )
@ A2 )
= ( modulo_modulo @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F @ A5 ) @ A2 ) ) ) ).
% mod_sum_eq
thf(fact_2782_bit__Suc__0__iff,axiom,
! [N: nat] :
( ( bit_se5641148757651400278ts_bit @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N )
= ( N
= ( zero_zero @ nat ) ) ) ).
% bit_Suc_0_iff
thf(fact_2783_not__bit__Suc__0__Suc,axiom,
! [N: nat] :
~ ( bit_se5641148757651400278ts_bit @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( suc @ N ) ) ).
% not_bit_Suc_0_Suc
thf(fact_2784_bit__push__bit__iff__nat,axiom,
! [M: nat,Q2: nat,N: nat] :
( ( bit_se5641148757651400278ts_bit @ nat @ ( bit_se4730199178511100633sh_bit @ nat @ M @ Q2 ) @ N )
= ( ( ord_less_eq @ nat @ M @ N )
& ( bit_se5641148757651400278ts_bit @ nat @ Q2 @ ( minus_minus @ nat @ N @ M ) ) ) ) ).
% bit_push_bit_iff_nat
thf(fact_2785_not__diff__distrib,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A2: A,B2: A] :
( ( bit_ri4277139882892585799ns_not @ A @ ( minus_minus @ A @ A2 @ B2 ) )
= ( plus_plus @ A @ ( bit_ri4277139882892585799ns_not @ A @ A2 ) @ B2 ) ) ) ).
% not_diff_distrib
thf(fact_2786_not__add__distrib,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A2: A,B2: A] :
( ( bit_ri4277139882892585799ns_not @ A @ ( plus_plus @ A @ A2 @ B2 ) )
= ( minus_minus @ A @ ( bit_ri4277139882892585799ns_not @ A @ A2 ) @ B2 ) ) ) ).
% not_add_distrib
thf(fact_2787_Misc_Ofoldr__Cons,axiom,
! [A: $tType,Xs: list @ A] :
( ( foldr @ A @ ( list @ A ) @ ( cons @ A ) @ Xs @ ( nil @ A ) )
= Xs ) ).
% Misc.foldr_Cons
thf(fact_2788_not__bit__Suc__0__numeral,axiom,
! [N: num] :
~ ( bit_se5641148757651400278ts_bit @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( numeral_numeral @ nat @ N ) ) ).
% not_bit_Suc_0_numeral
thf(fact_2789_minus__eq__not__plus__1,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( uminus_uminus @ A )
= ( ^ [A3: A] : ( plus_plus @ A @ ( bit_ri4277139882892585799ns_not @ A @ A3 ) @ ( one_one @ A ) ) ) ) ) ).
% minus_eq_not_plus_1
thf(fact_2790_not__eq__complement,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( bit_ri4277139882892585799ns_not @ A )
= ( ^ [A3: A] : ( minus_minus @ A @ ( uminus_uminus @ A @ A3 ) @ ( one_one @ A ) ) ) ) ) ).
% not_eq_complement
thf(fact_2791_minus__eq__not__minus__1,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( uminus_uminus @ A )
= ( ^ [A3: A] : ( bit_ri4277139882892585799ns_not @ A @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) ) ) ) ) ).
% minus_eq_not_minus_1
thf(fact_2792_not__int__def,axiom,
( ( bit_ri4277139882892585799ns_not @ int )
= ( ^ [K2: int] : ( minus_minus @ int @ ( uminus_uminus @ int @ K2 ) @ ( one_one @ int ) ) ) ) ).
% not_int_def
thf(fact_2793_and__not__numerals_I1_J,axiom,
( ( bit_se5824344872417868541ns_and @ int @ ( one_one @ int ) @ ( bit_ri4277139882892585799ns_not @ int @ ( one_one @ int ) ) )
= ( zero_zero @ int ) ) ).
% and_not_numerals(1)
thf(fact_2794_sum__power__add,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X2: A,M: nat,I5: set @ nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( power_power @ A @ X2 @ ( plus_plus @ nat @ M @ I2 ) )
@ I5 )
= ( times_times @ A @ ( power_power @ A @ X2 @ M ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X2 ) @ I5 ) ) ) ) ).
% sum_power_add
thf(fact_2795_unset__bit__int__def,axiom,
( ( bit_se2638667681897837118et_bit @ int )
= ( ^ [N4: nat,K2: int] : ( bit_se5824344872417868541ns_and @ int @ K2 @ ( bit_ri4277139882892585799ns_not @ int @ ( bit_se4730199178511100633sh_bit @ int @ N4 @ ( one_one @ int ) ) ) ) ) ) ).
% unset_bit_int_def
thf(fact_2796_and__not__numerals_I2_J,axiom,
! [N: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( one_one @ int ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) )
= ( one_one @ int ) ) ).
% and_not_numerals(2)
thf(fact_2797_and__not__numerals_I4_J,axiom,
! [M: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ ( bit0 @ M ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( one_one @ int ) ) )
= ( numeral_numeral @ int @ ( bit0 @ M ) ) ) ).
% and_not_numerals(4)
thf(fact_2798_bit__nat__iff,axiom,
! [K: int,N: nat] :
( ( bit_se5641148757651400278ts_bit @ nat @ ( nat2 @ K ) @ N )
= ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
& ( bit_se5641148757651400278ts_bit @ int @ K @ N ) ) ) ).
% bit_nat_iff
thf(fact_2799_bit__minus__int__iff,axiom,
! [K: int,N: nat] :
( ( bit_se5641148757651400278ts_bit @ int @ ( uminus_uminus @ int @ K ) @ N )
= ( bit_se5641148757651400278ts_bit @ int @ ( bit_ri4277139882892585799ns_not @ int @ ( minus_minus @ int @ K @ ( one_one @ int ) ) ) @ N ) ) ).
% bit_minus_int_iff
thf(fact_2800_take__bit__not__mask__eq__0,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( bit_se2584673776208193580ke_bit @ A @ M @ ( bit_ri4277139882892585799ns_not @ A @ ( bit_se2239418461657761734s_mask @ A @ N ) ) )
= ( zero_zero @ A ) ) ) ) ).
% take_bit_not_mask_eq_0
thf(fact_2801_push__bit__mask__eq,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [M: nat,N: nat] :
( ( bit_se4730199178511100633sh_bit @ A @ M @ ( bit_se2239418461657761734s_mask @ A @ N ) )
= ( bit_se5824344872417868541ns_and @ A @ ( bit_se2239418461657761734s_mask @ A @ ( plus_plus @ nat @ N @ M ) ) @ ( bit_ri4277139882892585799ns_not @ A @ ( bit_se2239418461657761734s_mask @ A @ M ) ) ) ) ) ).
% push_bit_mask_eq
thf(fact_2802_unset__bit__eq__and__not,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( bit_se2638667681897837118et_bit @ A )
= ( ^ [N4: nat,A3: A] : ( bit_se5824344872417868541ns_and @ A @ A3 @ ( bit_ri4277139882892585799ns_not @ A @ ( bit_se4730199178511100633sh_bit @ A @ N4 @ ( one_one @ A ) ) ) ) ) ) ) ).
% unset_bit_eq_and_not
thf(fact_2803_and__not__numerals_I7_J,axiom,
! [M: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ ( bit1 @ M ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( one_one @ int ) ) )
= ( numeral_numeral @ int @ ( bit0 @ M ) ) ) ).
% and_not_numerals(7)
thf(fact_2804_and__not__numerals_I3_J,axiom,
! [N: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( one_one @ int ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) )
= ( zero_zero @ int ) ) ).
% and_not_numerals(3)
thf(fact_2805_bit__not__iff__eq,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A2: A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( bit_ri4277139882892585799ns_not @ A @ A2 ) @ N )
= ( ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N )
!= ( zero_zero @ A ) )
& ~ ( bit_se5641148757651400278ts_bit @ A @ A2 @ N ) ) ) ) ).
% bit_not_iff_eq
thf(fact_2806_mask__eq__sum__exp,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [N: nat] :
( ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ A ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
@ ( collect @ nat
@ ^ [Q6: nat] : ( ord_less @ nat @ Q6 @ N ) ) ) ) ) ).
% mask_eq_sum_exp
thf(fact_2807_and__not__numerals_I8_J,axiom,
! [M: num,N: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ ( bit1 @ M ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) )
= ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ M ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N ) ) ) ) ) ) ).
% and_not_numerals(8)
thf(fact_2808_not__int__rec,axiom,
( ( bit_ri4277139882892585799ns_not @ int )
= ( ^ [K2: int] : ( plus_plus @ int @ ( zero_neq_one_of_bool @ int @ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K2 ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( divide_divide @ int @ K2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% not_int_rec
thf(fact_2809_sum__abs__ge__zero,axiom,
! [B: $tType,A: $tType] :
( ( ordere166539214618696060dd_abs @ B )
=> ! [F: A > B,A5: set @ A] :
( ord_less_eq @ B @ ( zero_zero @ B )
@ ( groups7311177749621191930dd_sum @ A @ B
@ ^ [I2: A] : ( abs_abs @ B @ ( F @ I2 ) )
@ A5 ) ) ) ).
% sum_abs_ge_zero
thf(fact_2810_sum__abs,axiom,
! [B: $tType,A: $tType] :
( ( ordere166539214618696060dd_abs @ B )
=> ! [F: A > B,A5: set @ A] :
( ord_less_eq @ B @ ( abs_abs @ B @ ( groups7311177749621191930dd_sum @ A @ B @ F @ A5 ) )
@ ( groups7311177749621191930dd_sum @ A @ B
@ ^ [I2: A] : ( abs_abs @ B @ ( F @ I2 ) )
@ A5 ) ) ) ).
% sum_abs
thf(fact_2811_convex__sum__bound__le,axiom,
! [A: $tType,B: $tType] :
( ( linordered_idom @ B )
=> ! [I5: set @ A,X2: A > B,A2: A > B,B2: B,Delta: B] :
( ! [I3: A] :
( ( member @ A @ I3 @ I5 )
=> ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( X2 @ I3 ) ) )
=> ( ( ( groups7311177749621191930dd_sum @ A @ B @ X2 @ I5 )
= ( one_one @ B ) )
=> ( ! [I3: A] :
( ( member @ A @ I3 @ I5 )
=> ( ord_less_eq @ B @ ( abs_abs @ B @ ( minus_minus @ B @ ( A2 @ I3 ) @ B2 ) ) @ Delta ) )
=> ( ord_less_eq @ B
@ ( abs_abs @ B
@ ( minus_minus @ B
@ ( groups7311177749621191930dd_sum @ A @ B
@ ^ [I2: A] : ( times_times @ B @ ( A2 @ I2 ) @ ( X2 @ I2 ) )
@ I5 )
@ B2 ) )
@ Delta ) ) ) ) ) ).
% convex_sum_bound_le
thf(fact_2812_abs__sum__abs,axiom,
! [B: $tType,A: $tType] :
( ( ordere166539214618696060dd_abs @ B )
=> ! [F: A > B,A5: set @ A] :
( ( abs_abs @ B
@ ( groups7311177749621191930dd_sum @ A @ B
@ ^ [A3: A] : ( abs_abs @ B @ ( F @ A3 ) )
@ A5 ) )
= ( groups7311177749621191930dd_sum @ A @ B
@ ^ [A3: A] : ( abs_abs @ B @ ( F @ A3 ) )
@ A5 ) ) ) ).
% abs_sum_abs
thf(fact_2813_sum_Oneutral__const,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A5: set @ B] :
( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [Uu: B] : ( zero_zero @ A )
@ A5 )
= ( zero_zero @ A ) ) ) ).
% sum.neutral_const
thf(fact_2814_sum__SucD,axiom,
! [A: $tType,F: A > nat,A5: set @ A,N: nat] :
( ( ( groups7311177749621191930dd_sum @ A @ nat @ F @ A5 )
= ( suc @ N ) )
=> ? [X3: A] :
( ( member @ A @ X3 @ A5 )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F @ X3 ) ) ) ) ).
% sum_SucD
thf(fact_2815_int__sum,axiom,
! [B: $tType,F: B > nat,A5: set @ B] :
( ( semiring_1_of_nat @ int @ ( groups7311177749621191930dd_sum @ B @ nat @ F @ A5 ) )
= ( groups7311177749621191930dd_sum @ B @ int
@ ^ [X: B] : ( semiring_1_of_nat @ int @ ( F @ X ) )
@ A5 ) ) ).
% int_sum
thf(fact_2816_sum_Oswap,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: B > C > A,B5: set @ C,A5: set @ B] :
( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [I2: B] : ( groups7311177749621191930dd_sum @ C @ A @ ( G @ I2 ) @ B5 )
@ A5 )
= ( groups7311177749621191930dd_sum @ C @ A
@ ^ [J3: C] :
( groups7311177749621191930dd_sum @ B @ A
@ ^ [I2: B] : ( G @ I2 @ J3 )
@ A5 )
@ B5 ) ) ) ).
% sum.swap
thf(fact_2817_sum_Oneutral,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A5: set @ B,G: B > A] :
( ! [X3: B] :
( ( member @ B @ X3 @ A5 )
=> ( ( G @ X3 )
= ( zero_zero @ A ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ A5 )
= ( zero_zero @ A ) ) ) ) ).
% sum.neutral
thf(fact_2818_sum_Onot__neutral__contains__not__neutral,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: B > A,A5: set @ B] :
( ( ( groups7311177749621191930dd_sum @ B @ A @ G @ A5 )
!= ( zero_zero @ A ) )
=> ~ ! [A4: B] :
( ( member @ B @ A4 @ A5 )
=> ( ( G @ A4 )
= ( zero_zero @ A ) ) ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_2819_sum__mono,axiom,
! [A: $tType,B: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [K3: set @ B,F: B > A,G: B > A] :
( ! [I3: B] :
( ( member @ B @ I3 @ K3 )
=> ( ord_less_eq @ A @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F @ K3 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ K3 ) ) ) ) ).
% sum_mono
thf(fact_2820_sum_Odistrib,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: B > A,H: B > A,A5: set @ B] :
( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X: B] : ( plus_plus @ A @ ( G @ X ) @ ( H @ X ) )
@ A5 )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G @ A5 ) @ ( groups7311177749621191930dd_sum @ B @ A @ H @ A5 ) ) ) ) ).
% sum.distrib
thf(fact_2821_sum__distrib__left,axiom,
! [A: $tType,B: $tType] :
( ( semiring_0 @ A )
=> ! [R: A,F: B > A,A5: set @ B] :
( ( times_times @ A @ R @ ( groups7311177749621191930dd_sum @ B @ A @ F @ A5 ) )
= ( groups7311177749621191930dd_sum @ B @ A
@ ^ [N4: B] : ( times_times @ A @ R @ ( F @ N4 ) )
@ A5 ) ) ) ).
% sum_distrib_left
thf(fact_2822_sum__distrib__right,axiom,
! [A: $tType,B: $tType] :
( ( semiring_0 @ A )
=> ! [F: B > A,A5: set @ B,R: A] :
( ( times_times @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F @ A5 ) @ R )
= ( groups7311177749621191930dd_sum @ B @ A
@ ^ [N4: B] : ( times_times @ A @ ( F @ N4 ) @ R )
@ A5 ) ) ) ).
% sum_distrib_right
thf(fact_2823_sum__product,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( semiring_0 @ B )
=> ! [F: A > B,A5: set @ A,G: C > B,B5: set @ C] :
( ( times_times @ B @ ( groups7311177749621191930dd_sum @ A @ B @ F @ A5 ) @ ( groups7311177749621191930dd_sum @ C @ B @ G @ B5 ) )
= ( groups7311177749621191930dd_sum @ A @ B
@ ^ [I2: A] :
( groups7311177749621191930dd_sum @ C @ B
@ ^ [J3: C] : ( times_times @ B @ ( F @ I2 ) @ ( G @ J3 ) )
@ B5 )
@ A5 ) ) ) ).
% sum_product
thf(fact_2824_sum__subtractf,axiom,
! [A: $tType,B: $tType] :
( ( ab_group_add @ A )
=> ! [F: B > A,G: B > A,A5: set @ B] :
( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X: B] : ( minus_minus @ A @ ( F @ X ) @ ( G @ X ) )
@ A5 )
= ( minus_minus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F @ A5 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ A5 ) ) ) ) ).
% sum_subtractf
thf(fact_2825_sum__negf,axiom,
! [A: $tType,B: $tType] :
( ( ab_group_add @ A )
=> ! [F: B > A,A5: set @ B] :
( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X: B] : ( uminus_uminus @ A @ ( F @ X ) )
@ A5 )
= ( uminus_uminus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F @ A5 ) ) ) ) ).
% sum_negf
thf(fact_2826_sum__divide__distrib,axiom,
! [A: $tType,B: $tType] :
( ( field @ A )
=> ! [F: B > A,A5: set @ B,R: A] :
( ( divide_divide @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F @ A5 ) @ R )
= ( groups7311177749621191930dd_sum @ B @ A
@ ^ [N4: B] : ( divide_divide @ A @ ( F @ N4 ) @ R )
@ A5 ) ) ) ).
% sum_divide_distrib
thf(fact_2827_sum__nonneg,axiom,
! [A: $tType,B: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A5: set @ B,F: B > A] :
( ! [X3: B] :
( ( member @ B @ X3 @ A5 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F @ X3 ) ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( groups7311177749621191930dd_sum @ B @ A @ F @ A5 ) ) ) ) ).
% sum_nonneg
thf(fact_2828_sum__nonpos,axiom,
! [B: $tType,A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A5: set @ B,F: B > A] :
( ! [X3: B] :
( ( member @ B @ X3 @ A5 )
=> ( ord_less_eq @ A @ ( F @ X3 ) @ ( zero_zero @ A ) ) )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F @ A5 ) @ ( zero_zero @ A ) ) ) ) ).
% sum_nonpos
thf(fact_2829_sum__cong__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A5: set @ nat,F: nat > A,G: nat > A] :
( ~ ( member @ nat @ ( zero_zero @ nat ) @ A5 )
=> ( ! [X3: nat] :
( ( member @ nat @ ( suc @ X3 ) @ A5 )
=> ( ( F @ ( suc @ X3 ) )
= ( G @ ( suc @ X3 ) ) ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ F @ A5 )
= ( groups7311177749621191930dd_sum @ nat @ A @ G @ A5 ) ) ) ) ) ).
% sum_cong_Suc
thf(fact_2830_sum__subtractf__nat,axiom,
! [A: $tType,A5: set @ A,G: A > nat,F: A > nat] :
( ! [X3: A] :
( ( member @ A @ X3 @ A5 )
=> ( ord_less_eq @ nat @ ( G @ X3 ) @ ( F @ X3 ) ) )
=> ( ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [X: A] : ( minus_minus @ nat @ ( F @ X ) @ ( G @ X ) )
@ A5 )
= ( minus_minus @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ F @ A5 ) @ ( groups7311177749621191930dd_sum @ A @ nat @ G @ A5 ) ) ) ) ).
% sum_subtractf_nat
thf(fact_2831_choose__odd__sum,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] :
( if @ A
@ ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I2 )
@ ( semiring_1_of_nat @ A @ ( binomial @ N @ I2 ) )
@ ( zero_zero @ A ) )
@ ( set_ord_atMost @ nat @ N ) ) )
= ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% choose_odd_sum
thf(fact_2832_choose__even__sum,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( if @ A @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I2 ) @ ( semiring_1_of_nat @ A @ ( binomial @ N @ I2 ) ) @ ( zero_zero @ A ) )
@ ( set_ord_atMost @ nat @ N ) ) )
= ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% choose_even_sum
thf(fact_2833_gbinomial__partial__row__sum,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A2: A,M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K2: nat] : ( times_times @ A @ ( gbinomial @ A @ A2 @ K2 ) @ ( minus_minus @ A @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( semiring_1_of_nat @ A @ K2 ) ) )
@ ( 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 @ A2 @ ( plus_plus @ nat @ M @ ( one_one @ nat ) ) ) ) ) ) ).
% gbinomial_partial_row_sum
thf(fact_2834_sum__gp,axiom,
! [A: $tType] :
( ( ( division_ring @ A )
& ( comm_ring @ A ) )
=> ! [N: nat,M: nat,X2: A] :
( ( ( ord_less @ nat @ N @ M )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X2 ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( zero_zero @ A ) ) )
& ( ~ ( ord_less @ nat @ N @ M )
=> ( ( ( X2
= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X2 ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) @ M ) ) ) )
& ( ( X2
!= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X2 ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ X2 @ M ) @ ( power_power @ A @ X2 @ ( suc @ N ) ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ X2 ) ) ) ) ) ) ) ) ).
% sum_gp
thf(fact_2835_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_2836_gchoose__row__sum__weighted,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [R: A,M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K2: nat] : ( times_times @ A @ ( gbinomial @ A @ R @ K2 ) @ ( minus_minus @ A @ ( divide_divide @ A @ R @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( semiring_1_of_nat @ A @ K2 ) ) )
@ ( 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 @ R @ ( suc @ M ) ) ) ) ) ).
% gchoose_row_sum_weighted
thf(fact_2837_atMost__eq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X2: A,Y: A] :
( ( ( set_ord_atMost @ A @ X2 )
= ( set_ord_atMost @ A @ Y ) )
= ( X2 = Y ) ) ) ).
% atMost_eq_iff
thf(fact_2838_atLeastAtMost__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [I: A,L: A,U: A] :
( ( member @ A @ I @ ( set_or1337092689740270186AtMost @ A @ L @ U ) )
= ( ( ord_less_eq @ A @ L @ I )
& ( ord_less_eq @ A @ I @ U ) ) ) ) ).
% atLeastAtMost_iff
thf(fact_2839_Icc__eq__Icc,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,H: A,L4: A,H4: A] :
( ( ( set_or1337092689740270186AtMost @ A @ L @ H )
= ( set_or1337092689740270186AtMost @ A @ L4 @ H4 ) )
= ( ( ( L = L4 )
& ( H = H4 ) )
| ( ~ ( ord_less_eq @ A @ L @ H )
& ~ ( ord_less_eq @ A @ L4 @ H4 ) ) ) ) ) ).
% Icc_eq_Icc
thf(fact_2840_atMost__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [I: A,K: A] :
( ( member @ A @ I @ ( set_ord_atMost @ A @ K ) )
= ( ord_less_eq @ A @ I @ K ) ) ) ).
% atMost_iff
thf(fact_2841_atLeastatMost__subset__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) @ ( set_or1337092689740270186AtMost @ A @ C2 @ D2 ) )
= ( ~ ( ord_less_eq @ A @ A2 @ B2 )
| ( ( ord_less_eq @ A @ C2 @ A2 )
& ( ord_less_eq @ A @ B2 @ D2 ) ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_2842_atMost__subset__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A,Y: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_ord_atMost @ A @ X2 ) @ ( set_ord_atMost @ A @ Y ) )
= ( ord_less_eq @ A @ X2 @ Y ) ) ) ).
% atMost_subset_iff
thf(fact_2843_Icc__subset__Iic__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [L: A,H: A,H4: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ H ) @ ( set_ord_atMost @ A @ H4 ) )
= ( ~ ( ord_less_eq @ A @ L @ H )
| ( ord_less_eq @ A @ H @ H4 ) ) ) ) ).
% Icc_subset_Iic_iff
thf(fact_2844_sum_OatMost__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_atMost @ nat @ ( suc @ N ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_atMost @ nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ).
% sum.atMost_Suc
thf(fact_2845_sum_Ocl__ivl__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [N: nat,M: nat,G: nat > A] :
( ( ( ord_less @ nat @ ( suc @ N ) @ M )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ ( suc @ N ) ) )
= ( zero_zero @ A ) ) )
& ( ~ ( ord_less @ nat @ ( suc @ N ) @ M )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ ( suc @ N ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ) ).
% sum.cl_ivl_Suc
thf(fact_2846_not__Iic__le__Icc,axiom,
! [A: $tType] :
( ( no_bot @ A )
=> ! [H: A,L4: A,H4: A] :
~ ( ord_less_eq @ ( set @ A ) @ ( set_ord_atMost @ A @ H ) @ ( set_or1337092689740270186AtMost @ A @ L4 @ H4 ) ) ) ).
% not_Iic_le_Icc
thf(fact_2847_not__Iic__eq__Icc,axiom,
! [A: $tType] :
( ( no_bot @ A )
=> ! [H4: A,L: A,H: A] :
( ( set_ord_atMost @ A @ H4 )
!= ( set_or1337092689740270186AtMost @ A @ L @ H ) ) ) ).
% not_Iic_eq_Icc
thf(fact_2848_atMost__atLeast0,axiom,
( ( set_ord_atMost @ nat )
= ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) ) ) ).
% atMost_atLeast0
thf(fact_2849_atMost__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( set_ord_atMost @ A )
= ( ^ [U2: A] :
( collect @ A
@ ^ [X: A] : ( ord_less_eq @ A @ X @ U2 ) ) ) ) ) ).
% atMost_def
thf(fact_2850_all__nat__less,axiom,
! [N: nat,P: nat > $o] :
( ( ! [M2: nat] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( P @ M2 ) ) )
= ( ! [X: nat] :
( ( member @ nat @ X @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
=> ( P @ X ) ) ) ) ).
% all_nat_less
thf(fact_2851_ex__nat__less,axiom,
! [N: nat,P: nat > $o] :
( ( ? [M2: nat] :
( ( ord_less_eq @ nat @ M2 @ N )
& ( P @ M2 ) ) )
= ( ? [X: nat] :
( ( member @ nat @ X @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
& ( P @ X ) ) ) ) ).
% ex_nat_less
thf(fact_2852_sum_Oshift__bounds__cl__Suc__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,M: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ ( suc @ N ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% sum.shift_bounds_cl_Suc_ivl
thf(fact_2853_sum_Oshift__bounds__cl__nat__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,M: nat,K: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ M @ K ) @ ( plus_plus @ nat @ N @ K ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( G @ ( plus_plus @ nat @ I2 @ K ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% sum.shift_bounds_cl_nat_ivl
thf(fact_2854_sum__power__shift,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [M: nat,N: nat,X2: A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X2 ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( times_times @ A @ ( power_power @ A @ X2 @ M ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X2 ) @ ( set_ord_atMost @ nat @ ( minus_minus @ nat @ N @ M ) ) ) ) ) ) ) ).
% sum_power_shift
thf(fact_2855_atLeastatMost__psubset__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( ord_less @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) @ ( set_or1337092689740270186AtMost @ A @ C2 @ D2 ) )
= ( ( ~ ( ord_less_eq @ A @ A2 @ B2 )
| ( ( ord_less_eq @ A @ C2 @ A2 )
& ( ord_less_eq @ A @ B2 @ D2 )
& ( ( ord_less @ A @ C2 @ A2 )
| ( ord_less @ A @ B2 @ D2 ) ) ) )
& ( ord_less_eq @ A @ C2 @ D2 ) ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_2856_sum_OatLeastAtMost__rev,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,N: nat,M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ N @ M ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( G @ ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N ) @ I2 ) )
@ ( set_or1337092689740270186AtMost @ nat @ N @ M ) ) ) ) ).
% sum.atLeastAtMost_rev
thf(fact_2857_sum__choose__upper,axiom,
! [M: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [K2: nat] : ( binomial @ K2 @ M )
@ ( set_ord_atMost @ nat @ N ) )
= ( binomial @ ( suc @ N ) @ ( suc @ M ) ) ) ).
% sum_choose_upper
thf(fact_2858_sum__shift__lb__Suc0__0,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [F: nat > A,K: nat] :
( ( ( F @ ( zero_zero @ nat ) )
= ( zero_zero @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ F @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ ( zero_zero @ nat ) ) @ K ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ F @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ K ) ) ) ) ) ).
% sum_shift_lb_Suc0_0
thf(fact_2859_sum_OatLeast0__atMost__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ).
% sum.atLeast0_atMost_Suc
thf(fact_2860_sum_Onat__ivl__Suc_H,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M: nat,N: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M @ ( suc @ N ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ ( suc @ N ) ) )
= ( plus_plus @ A @ ( G @ ( suc @ N ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ) ) ).
% sum.nat_ivl_Suc'
thf(fact_2861_sum_OatLeast__Suc__atMost,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M: nat,N: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( plus_plus @ A @ ( G @ M ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ N ) ) ) ) ) ) ).
% sum.atLeast_Suc_atMost
thf(fact_2862_sum_OatMost__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_atMost @ nat @ ( suc @ N ) ) )
= ( plus_plus @ A @ ( G @ ( zero_zero @ nat ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ) ).
% sum.atMost_Suc_shift
thf(fact_2863_sum__telescope,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [F: nat > A,I: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( minus_minus @ A @ ( F @ I2 ) @ ( F @ ( suc @ I2 ) ) )
@ ( set_ord_atMost @ nat @ I ) )
= ( minus_minus @ A @ ( F @ ( zero_zero @ nat ) ) @ ( F @ ( suc @ I ) ) ) ) ) ).
% sum_telescope
thf(fact_2864_sum_OSuc__reindex__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M: nat,N: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) )
= ( plus_plus @ A @ ( G @ M )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ) ) ).
% sum.Suc_reindex_ivl
thf(fact_2865_sum__Suc__diff,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [M: nat,N: nat,F: nat > A] :
( ( ord_less_eq @ nat @ M @ ( suc @ N ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( minus_minus @ A @ ( F @ ( suc @ I2 ) ) @ ( F @ I2 ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( minus_minus @ A @ ( F @ ( suc @ N ) ) @ ( F @ M ) ) ) ) ) ).
% sum_Suc_diff
thf(fact_2866_sum__choose__lower,axiom,
! [R: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [K2: nat] : ( binomial @ ( plus_plus @ nat @ R @ K2 ) @ K2 )
@ ( set_ord_atMost @ nat @ N ) )
= ( binomial @ ( suc @ ( plus_plus @ nat @ R @ N ) ) @ N ) ) ).
% sum_choose_lower
thf(fact_2867_choose__rising__sum_I1_J,axiom,
! [N: nat,M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [J3: nat] : ( binomial @ ( plus_plus @ nat @ N @ J3 ) @ N )
@ ( set_ord_atMost @ nat @ M ) )
= ( binomial @ ( plus_plus @ nat @ ( plus_plus @ nat @ N @ M ) @ ( one_one @ nat ) ) @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) ) ).
% choose_rising_sum(1)
thf(fact_2868_choose__rising__sum_I2_J,axiom,
! [N: nat,M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [J3: nat] : ( binomial @ ( plus_plus @ nat @ N @ J3 ) @ N )
@ ( set_ord_atMost @ nat @ M ) )
= ( binomial @ ( plus_plus @ nat @ ( plus_plus @ nat @ N @ M ) @ ( one_one @ nat ) ) @ M ) ) ).
% choose_rising_sum(2)
thf(fact_2869_sum__atLeastAtMost__code,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [F: nat > A,A2: nat,B2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ F @ ( set_or1337092689740270186AtMost @ nat @ A2 @ B2 ) )
= ( set_fo6178422350223883121st_nat @ A
@ ^ [A3: nat] : ( plus_plus @ A @ ( F @ A3 ) )
@ A2
@ B2
@ ( zero_zero @ A ) ) ) ) ).
% sum_atLeastAtMost_code
thf(fact_2870_sum_Oub__add__nat,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M: nat,N: nat,G: nat > A,P3: nat] :
( ( ord_less_eq @ nat @ M @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ ( plus_plus @ nat @ N @ P3 ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) @ ( plus_plus @ nat @ N @ P3 ) ) ) ) ) ) ) ).
% sum.ub_add_nat
thf(fact_2871_gbinomial__parallel__sum,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A2: A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K2: nat] : ( gbinomial @ A @ ( plus_plus @ A @ A2 @ ( semiring_1_of_nat @ A @ K2 ) ) @ K2 )
@ ( set_ord_atMost @ nat @ N ) )
= ( gbinomial @ A @ ( plus_plus @ A @ ( plus_plus @ A @ A2 @ ( semiring_1_of_nat @ A @ N ) ) @ ( one_one @ A ) ) @ N ) ) ) ).
% gbinomial_parallel_sum
thf(fact_2872_sum__choose__diagonal,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [K2: nat] : ( binomial @ ( minus_minus @ nat @ N @ K2 ) @ ( minus_minus @ nat @ M @ K2 ) )
@ ( set_ord_atMost @ nat @ M ) )
= ( binomial @ ( suc @ N ) @ M ) ) ) ).
% sum_choose_diagonal
thf(fact_2873_vandermonde,axiom,
! [M: nat,N: nat,R: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [K2: nat] : ( times_times @ nat @ ( binomial @ M @ K2 ) @ ( binomial @ N @ ( minus_minus @ nat @ R @ K2 ) ) )
@ ( set_ord_atMost @ nat @ R ) )
= ( binomial @ ( plus_plus @ nat @ M @ N ) @ R ) ) ).
% vandermonde
thf(fact_2874_sum__gp__basic,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X2: A,N: nat] :
( ( times_times @ A @ ( minus_minus @ A @ ( one_one @ A ) @ X2 ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X2 ) @ ( set_ord_atMost @ nat @ N ) ) )
= ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X2 @ ( suc @ N ) ) ) ) ) ).
% sum_gp_basic
thf(fact_2875_sum__natinterval__diff,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [M: nat,N: nat,F: nat > A] :
( ( ( ord_less_eq @ nat @ M @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K2: nat] : ( minus_minus @ A @ ( F @ K2 ) @ ( F @ ( plus_plus @ nat @ K2 @ ( one_one @ nat ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( minus_minus @ A @ ( F @ M ) @ ( F @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) ) ) )
& ( ~ ( ord_less_eq @ nat @ M @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K2: nat] : ( minus_minus @ A @ ( F @ K2 ) @ ( F @ ( plus_plus @ nat @ K2 @ ( one_one @ nat ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( zero_zero @ A ) ) ) ) ) ).
% sum_natinterval_diff
thf(fact_2876_sum__telescope_H_H,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [M: nat,N: nat,F: nat > A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K2: nat] : ( minus_minus @ A @ ( F @ K2 ) @ ( F @ ( minus_minus @ nat @ K2 @ ( one_one @ nat ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ N ) )
= ( minus_minus @ A @ ( F @ N ) @ ( F @ M ) ) ) ) ) ).
% sum_telescope''
thf(fact_2877_choose__row__sum,axiom,
! [N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat @ ( binomial @ N ) @ ( set_ord_atMost @ nat @ N ) )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ).
% choose_row_sum
thf(fact_2878_binomial,axiom,
! [A2: nat,B2: nat,N: nat] :
( ( power_power @ nat @ ( plus_plus @ nat @ A2 @ B2 ) @ N )
= ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [K2: nat] : ( times_times @ nat @ ( times_times @ nat @ ( semiring_1_of_nat @ nat @ ( binomial @ N @ K2 ) ) @ ( power_power @ nat @ A2 @ K2 ) ) @ ( power_power @ nat @ B2 @ ( minus_minus @ nat @ N @ K2 ) ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ).
% binomial
thf(fact_2879_sum_Oin__pairs__0,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_atMost @ nat @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( plus_plus @ A @ ( G @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I2 ) ) @ ( G @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I2 ) ) ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% sum.in_pairs_0
thf(fact_2880_gbinomial__sum__lower__neg,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A2: A,M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K2: nat] : ( times_times @ A @ ( gbinomial @ A @ A2 @ K2 ) @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K2 ) )
@ ( set_ord_atMost @ nat @ M ) )
= ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ M ) @ ( gbinomial @ A @ ( minus_minus @ A @ A2 @ ( one_one @ A ) ) @ M ) ) ) ) ).
% gbinomial_sum_lower_neg
thf(fact_2881_binomial__ring,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A2: A,B2: A,N: nat] :
( ( power_power @ A @ ( plus_plus @ A @ A2 @ B2 ) @ N )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K2: nat] : ( times_times @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( binomial @ N @ K2 ) ) @ ( power_power @ A @ A2 @ K2 ) ) @ ( power_power @ A @ B2 @ ( minus_minus @ nat @ N @ K2 ) ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% binomial_ring
thf(fact_2882_sum__gp__multiplied,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [M: nat,N: nat,X2: A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( times_times @ A @ ( minus_minus @ A @ ( one_one @ A ) @ X2 ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X2 ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) )
= ( minus_minus @ A @ ( power_power @ A @ X2 @ M ) @ ( power_power @ A @ X2 @ ( suc @ N ) ) ) ) ) ) ).
% sum_gp_multiplied
thf(fact_2883_pochhammer__binomial__sum,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [A2: A,B2: A,N: nat] :
( ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ A2 @ B2 ) @ N )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K2: nat] : ( times_times @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( binomial @ N @ K2 ) ) @ ( comm_s3205402744901411588hammer @ A @ A2 @ K2 ) ) @ ( comm_s3205402744901411588hammer @ A @ B2 @ ( minus_minus @ nat @ N @ K2 ) ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% pochhammer_binomial_sum
thf(fact_2884_sum_Oin__pairs,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,M: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( plus_plus @ A @ ( G @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I2 ) ) @ ( G @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I2 ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% sum.in_pairs
thf(fact_2885_choose__square__sum,axiom,
! [N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [K2: nat] : ( power_power @ nat @ ( binomial @ N @ K2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( binomial @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ N ) ) ).
% choose_square_sum
thf(fact_2886_sum_Ozero__middle,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [P3: nat,K: nat,G: 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 ) @ ( G @ 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 ) @ ( G @ J3 ) @ ( H @ J3 ) )
@ ( set_ord_atMost @ nat @ ( minus_minus @ nat @ P3 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) ) ).
% sum.zero_middle
thf(fact_2887_gbinomial__partial__sum__poly,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [M: nat,A2: A,X2: A,Y: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K2: nat] : ( times_times @ A @ ( times_times @ A @ ( gbinomial @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ M ) @ A2 ) @ K2 ) @ ( power_power @ A @ X2 @ K2 ) ) @ ( power_power @ A @ Y @ ( minus_minus @ nat @ M @ K2 ) ) )
@ ( set_ord_atMost @ nat @ M ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K2: nat] : ( times_times @ A @ ( times_times @ A @ ( gbinomial @ A @ ( uminus_uminus @ A @ A2 ) @ K2 ) @ ( power_power @ A @ ( uminus_uminus @ A @ X2 ) @ K2 ) ) @ ( power_power @ A @ ( plus_plus @ A @ X2 @ Y ) @ ( minus_minus @ nat @ M @ K2 ) ) )
@ ( set_ord_atMost @ nat @ M ) ) ) ) ).
% gbinomial_partial_sum_poly
thf(fact_2888_gbinomial__sum__up__index,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J3: nat] : ( gbinomial @ A @ ( semiring_1_of_nat @ A @ J3 ) @ K )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
= ( gbinomial @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N ) @ ( one_one @ A ) ) @ ( plus_plus @ nat @ K @ ( one_one @ nat ) ) ) ) ) ).
% gbinomial_sum_up_index
thf(fact_2889_gauss__sum__nat,axiom,
! [N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [X: nat] : X
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
= ( divide_divide @ nat @ ( times_times @ nat @ N @ ( suc @ N ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% gauss_sum_nat
thf(fact_2890_sum__gp0,axiom,
! [A: $tType] :
( ( ( division_ring @ A )
& ( comm_ring @ A ) )
=> ! [X2: A,N: nat] :
( ( ( X2
= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X2 ) @ ( set_ord_atMost @ nat @ N ) )
= ( semiring_1_of_nat @ A @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) ) )
& ( ( X2
!= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X2 ) @ ( set_ord_atMost @ nat @ N ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X2 @ ( suc @ N ) ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ X2 ) ) ) ) ) ) ).
% sum_gp0
thf(fact_2891_choose__alternating__linear__sum,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [N: nat] :
( ( N
!= ( one_one @ nat ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( times_times @ A @ ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ I2 ) @ ( semiring_1_of_nat @ A @ I2 ) ) @ ( semiring_1_of_nat @ A @ ( binomial @ N @ I2 ) ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) ) ) ) ).
% choose_alternating_linear_sum
thf(fact_2892_gbinomial__partial__sum__poly__xpos,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [M: nat,A2: A,X2: A,Y: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K2: nat] : ( times_times @ A @ ( times_times @ A @ ( gbinomial @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ M ) @ A2 ) @ K2 ) @ ( power_power @ A @ X2 @ K2 ) ) @ ( power_power @ A @ Y @ ( minus_minus @ nat @ M @ K2 ) ) )
@ ( set_ord_atMost @ nat @ M ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K2: nat] : ( times_times @ A @ ( times_times @ A @ ( gbinomial @ A @ ( minus_minus @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ K2 ) @ A2 ) @ ( one_one @ A ) ) @ K2 ) @ ( power_power @ A @ X2 @ K2 ) ) @ ( power_power @ A @ ( plus_plus @ A @ X2 @ Y ) @ ( minus_minus @ nat @ M @ K2 ) ) )
@ ( set_ord_atMost @ nat @ M ) ) ) ) ).
% gbinomial_partial_sum_poly_xpos
thf(fact_2893_gbinomial__sum__nat__pow2,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K2: nat] : ( divide_divide @ A @ ( gbinomial @ A @ ( semiring_1_of_nat @ A @ ( plus_plus @ nat @ M @ K2 ) ) @ K2 ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ K2 ) )
@ ( set_ord_atMost @ nat @ M ) )
= ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) ) ) ).
% gbinomial_sum_nat_pow2
thf(fact_2894_double__arith__series,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A2: A,D2: A,N: nat] :
( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( plus_plus @ A @ A2 @ ( times_times @ A @ ( semiring_1_of_nat @ A @ I2 ) @ D2 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) )
= ( times_times @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N ) @ ( one_one @ A ) ) @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ D2 ) ) ) ) ) ).
% double_arith_series
thf(fact_2895_double__gauss__sum,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [N: nat] :
( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( semiring_1_of_nat @ A ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) )
= ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N ) @ ( one_one @ A ) ) ) ) ) ).
% double_gauss_sum
thf(fact_2896_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_2897_choose__linear__sum,axiom,
! [N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [I2: nat] : ( times_times @ nat @ I2 @ ( binomial @ N @ I2 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( times_times @ nat @ N @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ).
% choose_linear_sum
thf(fact_2898_arith__series__nat,axiom,
! [A2: nat,D2: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [I2: nat] : ( plus_plus @ nat @ A2 @ ( times_times @ nat @ I2 @ D2 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
= ( divide_divide @ nat @ ( times_times @ nat @ ( suc @ N ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ A2 ) @ ( times_times @ nat @ N @ D2 ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% arith_series_nat
thf(fact_2899_Sum__Icc__nat,axiom,
! [M: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [X: nat] : X
@ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( divide_divide @ nat @ ( minus_minus @ nat @ ( times_times @ nat @ N @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) @ ( times_times @ nat @ M @ ( minus_minus @ nat @ M @ ( one_one @ nat ) ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% Sum_Icc_nat
thf(fact_2900_choose__alternating__sum,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ I2 ) @ ( semiring_1_of_nat @ A @ ( binomial @ N @ I2 ) ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) ) ) ) ).
% choose_alternating_sum
thf(fact_2901_double__gauss__sum__from__Suc__0,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [N: nat] :
( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( semiring_1_of_nat @ A ) @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N ) ) )
= ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N ) @ ( one_one @ A ) ) ) ) ) ).
% double_gauss_sum_from_Suc_0
thf(fact_2902_gauss__sum,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ ( semiring_1_of_nat @ A ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
= ( divide_divide @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N ) @ ( one_one @ A ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% gauss_sum
thf(fact_2903_arith__series,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A2: A,D2: A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( plus_plus @ A @ A2 @ ( times_times @ A @ ( semiring_1_of_nat @ A @ I2 ) @ D2 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
= ( divide_divide @ A @ ( times_times @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N ) @ ( one_one @ A ) ) @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ D2 ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% arith_series
thf(fact_2904_sum__gp__offset,axiom,
! [A: $tType] :
( ( ( division_ring @ A )
& ( comm_ring @ A ) )
=> ! [X2: A,M: nat,N: nat] :
( ( ( X2
= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X2 ) @ ( set_or1337092689740270186AtMost @ nat @ M @ ( plus_plus @ nat @ M @ N ) ) )
= ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N ) @ ( one_one @ A ) ) ) )
& ( ( X2
!= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X2 ) @ ( set_or1337092689740270186AtMost @ nat @ M @ ( plus_plus @ nat @ M @ N ) ) )
= ( divide_divide @ A @ ( times_times @ A @ ( power_power @ A @ X2 @ M ) @ ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X2 @ ( suc @ N ) ) ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ X2 ) ) ) ) ) ) ).
% sum_gp_offset
thf(fact_2905_gauss__sum__from__Suc__0,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ ( semiring_1_of_nat @ A ) @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N ) )
= ( divide_divide @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N ) @ ( one_one @ A ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% gauss_sum_from_Suc_0
thf(fact_2906_pochhammer__times__pochhammer__half,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [Z2: A,N: nat] :
( ( times_times @ A @ ( comm_s3205402744901411588hammer @ A @ Z2 @ ( suc @ N ) ) @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ Z2 @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( suc @ N ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [K2: nat] : ( plus_plus @ A @ Z2 @ ( divide_divide @ A @ ( semiring_1_of_nat @ A @ K2 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ nat ) ) ) ) ) ) ).
% pochhammer_times_pochhammer_half
thf(fact_2907_Sum__Icc__int,axiom,
! [M: int,N: int] :
( ( ord_less_eq @ int @ M @ N )
=> ( ( groups7311177749621191930dd_sum @ int @ int
@ ^ [X: int] : X
@ ( set_or1337092689740270186AtMost @ int @ M @ N ) )
= ( divide_divide @ int @ ( minus_minus @ int @ ( times_times @ int @ N @ ( plus_plus @ int @ N @ ( one_one @ int ) ) ) @ ( times_times @ int @ M @ ( minus_minus @ int @ M @ ( one_one @ int ) ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ).
% Sum_Icc_int
thf(fact_2908_Sum__Ico__nat,axiom,
! [M: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [X: nat] : X
@ ( set_or7035219750837199246ssThan @ nat @ M @ N ) )
= ( divide_divide @ nat @ ( minus_minus @ nat @ ( times_times @ nat @ N @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) @ ( times_times @ nat @ M @ ( minus_minus @ nat @ M @ ( one_one @ nat ) ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% Sum_Ico_nat
thf(fact_2909_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_2910_gbinomial__Suc,axiom,
! [A: $tType] :
( ( ( semiring_char_0 @ A )
& ( semidom_divide @ A ) )
=> ! [A2: A,K: nat] :
( ( gbinomial @ A @ A2 @ ( suc @ K ) )
= ( divide_divide @ A
@ ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I2: nat] : ( minus_minus @ A @ A2 @ ( semiring_1_of_nat @ A @ I2 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ K ) )
@ ( semiring_char_0_fact @ A @ ( suc @ K ) ) ) ) ) ).
% gbinomial_Suc
thf(fact_2911_ivl__diff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [I: A,N: A,M: A] :
( ( ord_less_eq @ A @ I @ N )
=> ( ( minus_minus @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ I @ M ) @ ( set_or7035219750837199246ssThan @ A @ I @ N ) )
= ( set_or7035219750837199246ssThan @ A @ N @ M ) ) ) ) ).
% ivl_diff
thf(fact_2912_Ints__prod,axiom,
! [A: $tType,B: $tType] :
( ( ( comm_monoid_mult @ B )
& ( ring_1 @ B ) )
=> ! [A5: set @ A,F: A > B] :
( ! [X3: A] :
( ( member @ A @ X3 @ A5 )
=> ( member @ B @ ( F @ X3 ) @ ( ring_1_Ints @ B ) ) )
=> ( member @ B @ ( groups7121269368397514597t_prod @ A @ B @ F @ A5 ) @ ( ring_1_Ints @ B ) ) ) ) ).
% Ints_prod
thf(fact_2913_prod_Oneutral__const,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A5: set @ B] :
( ( groups7121269368397514597t_prod @ B @ A
@ ^ [Uu: B] : ( one_one @ A )
@ A5 )
= ( one_one @ A ) ) ) ).
% prod.neutral_const
thf(fact_2914_of__nat__prod,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_1 @ A )
=> ! [F: B > nat,A5: set @ B] :
( ( semiring_1_of_nat @ A @ ( groups7121269368397514597t_prod @ B @ nat @ F @ A5 ) )
= ( groups7121269368397514597t_prod @ B @ A
@ ^ [X: B] : ( semiring_1_of_nat @ A @ ( F @ X ) )
@ A5 ) ) ) ).
% of_nat_prod
thf(fact_2915_of__int__prod,axiom,
! [A: $tType,B: $tType] :
( ( comm_ring_1 @ A )
=> ! [F: B > int,A5: set @ B] :
( ( ring_1_of_int @ A @ ( groups7121269368397514597t_prod @ B @ int @ F @ A5 ) )
= ( groups7121269368397514597t_prod @ B @ A
@ ^ [X: B] : ( ring_1_of_int @ A @ ( F @ X ) )
@ A5 ) ) ) ).
% of_int_prod
thf(fact_2916_atLeastLessThan__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [I: A,L: A,U: A] :
( ( member @ A @ I @ ( set_or7035219750837199246ssThan @ A @ L @ U ) )
= ( ( ord_less_eq @ A @ L @ I )
& ( ord_less @ A @ I @ U ) ) ) ) ).
% atLeastLessThan_iff
thf(fact_2917_ivl__subset,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [I: A,J: A,M: A,N: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ I @ J ) @ ( set_or7035219750837199246ssThan @ A @ M @ N ) )
= ( ( ord_less_eq @ A @ J @ I )
| ( ( ord_less_eq @ A @ M @ I )
& ( ord_less_eq @ A @ J @ N ) ) ) ) ) ).
% ivl_subset
thf(fact_2918_prod_OatMost__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_atMost @ nat @ ( suc @ N ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_atMost @ nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ).
% prod.atMost_Suc
thf(fact_2919_sum_Oop__ivl__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [N: nat,M: nat,G: nat > A] :
( ( ( ord_less @ nat @ N @ M )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ ( suc @ N ) ) )
= ( zero_zero @ A ) ) )
& ( ~ ( ord_less @ nat @ N @ M )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ ( suc @ N ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) @ ( G @ N ) ) ) ) ) ) ).
% sum.op_ivl_Suc
thf(fact_2920_prod_Ocl__ivl__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [N: nat,M: nat,G: nat > A] :
( ( ( ord_less @ nat @ ( suc @ N ) @ M )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ ( suc @ N ) ) )
= ( one_one @ A ) ) )
& ( ~ ( ord_less @ nat @ ( suc @ N ) @ M )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ ( suc @ N ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ) ).
% prod.cl_ivl_Suc
thf(fact_2921_prod_Oop__ivl__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [N: nat,M: nat,G: nat > A] :
( ( ( ord_less @ nat @ N @ M )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ ( suc @ N ) ) )
= ( one_one @ A ) ) )
& ( ~ ( ord_less @ nat @ N @ M )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ ( suc @ N ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) @ ( G @ N ) ) ) ) ) ) ).
% prod.op_ivl_Suc
thf(fact_2922_prod_Oshift__bounds__nat__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,M: nat,K: nat,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( plus_plus @ nat @ M @ K ) @ ( plus_plus @ nat @ N @ K ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I2: nat] : ( G @ ( plus_plus @ nat @ I2 @ K ) )
@ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ).
% prod.shift_bounds_nat_ivl
thf(fact_2923_prod_Oshift__bounds__Suc__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,M: nat,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M ) @ ( suc @ N ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
@ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ).
% prod.shift_bounds_Suc_ivl
thf(fact_2924_prod_Oivl__cong,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( comm_monoid_mult @ A ) )
=> ! [A2: B,C2: B,B2: B,D2: B,G: B > A,H: B > A] :
( ( A2 = C2 )
=> ( ( B2 = D2 )
=> ( ! [X3: B] :
( ( ord_less_eq @ B @ C2 @ X3 )
=> ( ( ord_less @ B @ X3 @ D2 )
=> ( ( G @ X3 )
= ( H @ X3 ) ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( set_or7035219750837199246ssThan @ B @ A2 @ B2 ) )
= ( groups7121269368397514597t_prod @ B @ A @ H @ ( set_or7035219750837199246ssThan @ B @ C2 @ D2 ) ) ) ) ) ) ) ).
% prod.ivl_cong
thf(fact_2925_prod_OatLeastLessThan__concat,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M: nat,N: nat,P3: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( ord_less_eq @ nat @ N @ P3 )
=> ( ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ N @ P3 ) ) )
= ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ P3 ) ) ) ) ) ) ).
% prod.atLeastLessThan_concat
thf(fact_2926_prod_Oswap,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: B > C > A,B5: set @ C,A5: set @ B] :
( ( groups7121269368397514597t_prod @ B @ A
@ ^ [I2: B] : ( groups7121269368397514597t_prod @ C @ A @ ( G @ I2 ) @ B5 )
@ A5 )
= ( groups7121269368397514597t_prod @ C @ A
@ ^ [J3: C] :
( groups7121269368397514597t_prod @ B @ A
@ ^ [I2: B] : ( G @ I2 @ J3 )
@ A5 )
@ B5 ) ) ) ).
% prod.swap
thf(fact_2927_prod_Onot__neutral__contains__not__neutral,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: B > A,A5: set @ B] :
( ( ( groups7121269368397514597t_prod @ B @ A @ G @ A5 )
!= ( one_one @ A ) )
=> ~ ! [A4: B] :
( ( member @ B @ A4 @ A5 )
=> ( ( G @ A4 )
= ( one_one @ A ) ) ) ) ) ).
% prod.not_neutral_contains_not_neutral
thf(fact_2928_prod_Oneutral,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A5: set @ B,G: B > A] :
( ! [X3: B] :
( ( member @ B @ X3 @ A5 )
=> ( ( G @ X3 )
= ( one_one @ A ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ A5 )
= ( one_one @ A ) ) ) ) ).
% prod.neutral
thf(fact_2929_atLeastLessThan__inj_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( ( set_or7035219750837199246ssThan @ A @ A2 @ B2 )
= ( set_or7035219750837199246ssThan @ A @ C2 @ D2 ) )
=> ( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ C2 @ D2 )
=> ( B2 = D2 ) ) ) ) ) ).
% atLeastLessThan_inj(2)
thf(fact_2930_atLeastLessThan__inj_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( ( set_or7035219750837199246ssThan @ A @ A2 @ B2 )
= ( set_or7035219750837199246ssThan @ A @ C2 @ D2 ) )
=> ( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ C2 @ D2 )
=> ( A2 = C2 ) ) ) ) ) ).
% atLeastLessThan_inj(1)
thf(fact_2931_atLeastLessThan__eq__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ C2 @ D2 )
=> ( ( ( set_or7035219750837199246ssThan @ A @ A2 @ B2 )
= ( set_or7035219750837199246ssThan @ A @ C2 @ D2 ) )
= ( ( A2 = C2 )
& ( B2 = D2 ) ) ) ) ) ) ).
% atLeastLessThan_eq_iff
thf(fact_2932_prod_Odistrib,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: B > A,H: B > A,A5: set @ B] :
( ( groups7121269368397514597t_prod @ B @ A
@ ^ [X: B] : ( times_times @ A @ ( G @ X ) @ ( H @ X ) )
@ A5 )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G @ A5 ) @ ( groups7121269368397514597t_prod @ B @ A @ H @ A5 ) ) ) ) ).
% prod.distrib
thf(fact_2933_prod_OatLeast0__lessThan__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) @ ( G @ N ) ) ) ) ).
% prod.atLeast0_lessThan_Suc
thf(fact_2934_prod_OatLeast__Suc__lessThan,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M: nat,N: nat,G: nat > A] :
( ( ord_less @ nat @ M @ N )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) )
= ( times_times @ A @ ( G @ M ) @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M ) @ N ) ) ) ) ) ) ).
% prod.atLeast_Suc_lessThan
thf(fact_2935_prod_OatLeastLessThan__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: nat,B2: nat,G: nat > A] :
( ( ord_less_eq @ nat @ A2 @ B2 )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ A2 @ ( suc @ B2 ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ A2 @ B2 ) ) @ ( G @ B2 ) ) ) ) ) ).
% prod.atLeastLessThan_Suc
thf(fact_2936_prod__dividef,axiom,
! [A: $tType,B: $tType] :
( ( field @ A )
=> ! [F: B > A,G: B > A,A5: set @ B] :
( ( groups7121269368397514597t_prod @ B @ A
@ ^ [X: B] : ( divide_divide @ A @ ( F @ X ) @ ( G @ X ) )
@ A5 )
= ( divide_divide @ A @ ( groups7121269368397514597t_prod @ B @ A @ F @ A5 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ A5 ) ) ) ) ).
% prod_dividef
thf(fact_2937_prod__power__distrib,axiom,
! [B: $tType,A: $tType] :
( ( comm_semiring_1 @ B )
=> ! [F: A > B,A5: set @ A,N: nat] :
( ( power_power @ B @ ( groups7121269368397514597t_prod @ A @ B @ F @ A5 ) @ N )
= ( groups7121269368397514597t_prod @ A @ B
@ ^ [X: A] : ( power_power @ B @ ( F @ X ) @ N )
@ A5 ) ) ) ).
% prod_power_distrib
thf(fact_2938_mod__prod__eq,axiom,
! [B: $tType,A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [F: B > A,A2: A,A5: set @ B] :
( ( modulo_modulo @ A
@ ( groups7121269368397514597t_prod @ B @ A
@ ^ [I2: B] : ( modulo_modulo @ A @ ( F @ I2 ) @ A2 )
@ A5 )
@ A2 )
= ( modulo_modulo @ A @ ( groups7121269368397514597t_prod @ B @ A @ F @ A5 ) @ A2 ) ) ) ).
% mod_prod_eq
thf(fact_2939_prod_Olast__plus,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M: nat,N: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( times_times @ A @ ( G @ N ) @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ) ) ).
% prod.last_plus
thf(fact_2940_abs__prod,axiom,
! [A: $tType,B: $tType] :
( ( linordered_field @ A )
=> ! [F: B > A,A5: set @ B] :
( ( abs_abs @ A @ ( groups7121269368397514597t_prod @ B @ A @ F @ A5 ) )
= ( groups7121269368397514597t_prod @ B @ A
@ ^ [X: B] : ( abs_abs @ A @ ( F @ X ) )
@ A5 ) ) ) ).
% abs_prod
thf(fact_2941_prod__Suc__Suc__fact,axiom,
! [N: nat] :
( ( groups7121269368397514597t_prod @ nat @ nat @ suc @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N ) )
= ( semiring_char_0_fact @ nat @ N ) ) ).
% prod_Suc_Suc_fact
thf(fact_2942_prod__Suc__fact,axiom,
! [N: nat] :
( ( groups7121269368397514597t_prod @ nat @ nat @ suc @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) )
= ( semiring_char_0_fact @ nat @ N ) ) ).
% prod_Suc_fact
thf(fact_2943_prod_OatLeastLessThan__rev,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N: nat,M: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ N @ M ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I2: nat] : ( G @ ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N ) @ ( suc @ I2 ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ N @ M ) ) ) ) ).
% prod.atLeastLessThan_rev
thf(fact_2944_prod_Onested__swap,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: nat > nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I2: nat] : ( groups7121269368397514597t_prod @ nat @ A @ ( A2 @ I2 ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ I2 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [J3: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I2: nat] : ( A2 @ I2 @ J3 )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J3 ) @ N ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% prod.nested_swap
thf(fact_2945_prod_Ohead__if,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [N: nat,M: nat,G: nat > A] :
( ( ( ord_less @ nat @ N @ M )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( one_one @ A ) ) )
& ( ~ ( ord_less @ nat @ N @ M )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) @ ( G @ N ) ) ) ) ) ) ).
% prod.head_if
thf(fact_2946_prod__mono,axiom,
! [A: $tType,B: $tType] :
( ( linordered_semidom @ A )
=> ! [A5: set @ B,F: B > A,G: B > A] :
( ! [I3: B] :
( ( member @ B @ I3 @ A5 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F @ I3 ) )
& ( ord_less_eq @ A @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
=> ( ord_less_eq @ A @ ( groups7121269368397514597t_prod @ B @ A @ F @ A5 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ A5 ) ) ) ) ).
% prod_mono
thf(fact_2947_prod__nonneg,axiom,
! [A: $tType,B: $tType] :
( ( linordered_semidom @ A )
=> ! [A5: set @ B,F: B > A] :
( ! [X3: B] :
( ( member @ B @ X3 @ A5 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F @ X3 ) ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( groups7121269368397514597t_prod @ B @ A @ F @ A5 ) ) ) ) ).
% prod_nonneg
thf(fact_2948_prod__pos,axiom,
! [A: $tType,B: $tType] :
( ( linordered_semidom @ A )
=> ! [A5: set @ B,F: B > A] :
( ! [X3: B] :
( ( member @ B @ X3 @ A5 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( F @ X3 ) ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( groups7121269368397514597t_prod @ B @ A @ F @ A5 ) ) ) ) ).
% prod_pos
thf(fact_2949_prod__ge__1,axiom,
! [A: $tType,B: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [A5: set @ B,F: B > A] :
( ! [X3: B] :
( ( member @ B @ X3 @ A5 )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ ( F @ X3 ) ) )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ ( groups7121269368397514597t_prod @ B @ A @ F @ A5 ) ) ) ) ).
% prod_ge_1
thf(fact_2950_fact__prod__Suc,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiring_char_0_fact @ A )
= ( ^ [N4: nat] : ( semiring_1_of_nat @ A @ ( groups7121269368397514597t_prod @ nat @ nat @ suc @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N4 ) ) ) ) ) ) ).
% fact_prod_Suc
thf(fact_2951_atLeastLessThan__subset__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ A2 @ B2 ) @ ( set_or7035219750837199246ssThan @ A @ C2 @ D2 ) )
=> ( ( ord_less_eq @ A @ B2 @ A2 )
| ( ( ord_less_eq @ A @ C2 @ A2 )
& ( ord_less_eq @ A @ B2 @ D2 ) ) ) ) ) ).
% atLeastLessThan_subset_iff
thf(fact_2952_all__nat__less__eq,axiom,
! [N: nat,P: nat > $o] :
( ( ! [M2: nat] :
( ( ord_less @ nat @ M2 @ N )
=> ( P @ M2 ) ) )
= ( ! [X: nat] :
( ( member @ nat @ X @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) )
=> ( P @ X ) ) ) ) ).
% all_nat_less_eq
thf(fact_2953_ex__nat__less__eq,axiom,
! [N: nat,P: nat > $o] :
( ( ? [M2: nat] :
( ( ord_less @ nat @ M2 @ N )
& ( P @ M2 ) ) )
= ( ? [X: nat] :
( ( member @ nat @ X @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) )
& ( P @ X ) ) ) ) ).
% ex_nat_less_eq
thf(fact_2954_prod_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N: nat,M: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ N @ M ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I2: nat] : ( G @ ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N ) @ I2 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ N ) @ M ) ) ) ) ).
% prod.atLeastLessThan_rev_at_least_Suc_atMost
thf(fact_2955_pochhammer__prod,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ( ( comm_s3205402744901411588hammer @ A )
= ( ^ [A3: A,N4: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I2: nat] : ( plus_plus @ A @ A3 @ ( semiring_1_of_nat @ A @ I2 ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N4 ) ) ) ) ) ).
% pochhammer_prod
thf(fact_2956_fact__prod__rev,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiring_char_0_fact @ A )
= ( ^ [N4: nat] : ( semiring_1_of_nat @ A @ ( groups7121269368397514597t_prod @ nat @ nat @ ( minus_minus @ nat @ N4 ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N4 ) ) ) ) ) ) ).
% fact_prod_rev
thf(fact_2957_atLeastLessThanSuc__atLeastAtMost,axiom,
! [L: nat,U: nat] :
( ( set_or7035219750837199246ssThan @ nat @ L @ ( suc @ U ) )
= ( set_or1337092689740270186AtMost @ nat @ L @ U ) ) ).
% atLeastLessThanSuc_atLeastAtMost
thf(fact_2958_prod_Oshift__bounds__cl__Suc__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,M: nat,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ ( suc @ N ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% prod.shift_bounds_cl_Suc_ivl
thf(fact_2959_power__sum,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [C2: A,F: B > nat,A5: set @ B] :
( ( power_power @ A @ C2 @ ( groups7311177749621191930dd_sum @ B @ nat @ F @ A5 ) )
= ( groups7121269368397514597t_prod @ B @ A
@ ^ [A3: B] : ( power_power @ A @ C2 @ ( F @ A3 ) )
@ A5 ) ) ) ).
% power_sum
thf(fact_2960_prod_Oshift__bounds__cl__nat__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,M: nat,K: nat,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ M @ K ) @ ( plus_plus @ nat @ N @ K ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I2: nat] : ( G @ ( plus_plus @ nat @ I2 @ K ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% prod.shift_bounds_cl_nat_ivl
thf(fact_2961_sum_Oshift__bounds__Suc__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,M: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M ) @ ( suc @ N ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
@ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ).
% sum.shift_bounds_Suc_ivl
thf(fact_2962_sum_Oshift__bounds__nat__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,M: nat,K: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( plus_plus @ nat @ M @ K ) @ ( plus_plus @ nat @ N @ K ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( G @ ( plus_plus @ nat @ I2 @ K ) )
@ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ).
% sum.shift_bounds_nat_ivl
thf(fact_2963_prod__le__1,axiom,
! [B: $tType,A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [A5: set @ B,F: B > A] :
( ! [X3: B] :
( ( member @ B @ X3 @ A5 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F @ X3 ) )
& ( ord_less_eq @ A @ ( F @ X3 ) @ ( one_one @ A ) ) ) )
=> ( ord_less_eq @ A @ ( groups7121269368397514597t_prod @ B @ A @ F @ A5 ) @ ( one_one @ A ) ) ) ) ).
% prod_le_1
thf(fact_2964_fact__split,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [K: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ N )
=> ( ( semiring_char_0_fact @ A @ N )
= ( times_times @ A @ ( semiring_1_of_nat @ A @ ( groups7121269368397514597t_prod @ nat @ nat @ suc @ ( set_or7035219750837199246ssThan @ nat @ ( minus_minus @ nat @ N @ K ) @ N ) ) ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ N @ K ) ) ) ) ) ) ).
% fact_split
thf(fact_2965_sum_Oivl__cong,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( comm_monoid_add @ A ) )
=> ! [A2: B,C2: B,B2: B,D2: B,G: B > A,H: B > A] :
( ( A2 = C2 )
=> ( ( B2 = D2 )
=> ( ! [X3: B] :
( ( ord_less_eq @ B @ C2 @ X3 )
=> ( ( ord_less @ B @ X3 @ D2 )
=> ( ( G @ X3 )
= ( H @ X3 ) ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( set_or7035219750837199246ssThan @ B @ A2 @ B2 ) )
= ( groups7311177749621191930dd_sum @ B @ A @ H @ ( set_or7035219750837199246ssThan @ B @ C2 @ D2 ) ) ) ) ) ) ) ).
% sum.ivl_cong
thf(fact_2966_gbinomial__altdef__of__nat,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( ( gbinomial @ A )
= ( ^ [A3: A,K2: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I2: nat] : ( divide_divide @ A @ ( minus_minus @ A @ A3 @ ( semiring_1_of_nat @ A @ I2 ) ) @ ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ K2 @ I2 ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K2 ) ) ) ) ) ).
% gbinomial_altdef_of_nat
thf(fact_2967_binomial__altdef__of__nat,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ N )
=> ( ( semiring_1_of_nat @ A @ ( binomial @ N @ K ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I2: nat] : ( divide_divide @ A @ ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ N @ I2 ) ) @ ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ K @ I2 ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K ) ) ) ) ) ).
% binomial_altdef_of_nat
thf(fact_2968_gbinomial__mult__fact_H,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A2: A,K: nat] :
( ( times_times @ A @ ( gbinomial @ A @ A2 @ K ) @ ( semiring_char_0_fact @ A @ K ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I2: nat] : ( minus_minus @ A @ A2 @ ( semiring_1_of_nat @ A @ I2 ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K ) ) ) ) ).
% gbinomial_mult_fact'
thf(fact_2969_gbinomial__mult__fact,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,A2: A] :
( ( times_times @ A @ ( semiring_char_0_fact @ A @ K ) @ ( gbinomial @ A @ A2 @ K ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I2: nat] : ( minus_minus @ A @ A2 @ ( semiring_1_of_nat @ A @ I2 ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K ) ) ) ) ).
% gbinomial_mult_fact
thf(fact_2970_gbinomial__prod__rev,axiom,
! [A: $tType] :
( ( ( semiring_char_0 @ A )
& ( semidom_divide @ A ) )
=> ( ( gbinomial @ A )
= ( ^ [A3: A,K2: nat] :
( divide_divide @ A
@ ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I2: nat] : ( minus_minus @ A @ A3 @ ( semiring_1_of_nat @ A @ I2 ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K2 ) )
@ ( semiring_char_0_fact @ A @ K2 ) ) ) ) ) ).
% gbinomial_prod_rev
thf(fact_2971_sum_OatLeastLessThan__concat,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M: nat,N: nat,P3: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( ord_less_eq @ nat @ N @ P3 )
=> ( ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ N @ P3 ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ P3 ) ) ) ) ) ) ).
% sum.atLeastLessThan_concat
thf(fact_2972_sum__diff__nat__ivl,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [M: nat,N: nat,P3: nat,F: nat > A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( ord_less_eq @ nat @ N @ P3 )
=> ( ( minus_minus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F @ ( set_or7035219750837199246ssThan @ nat @ M @ P3 ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ F @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ F @ ( set_or7035219750837199246ssThan @ nat @ N @ P3 ) ) ) ) ) ) ).
% sum_diff_nat_ivl
thf(fact_2973_bset_I1_J,axiom,
! [D3: int,B5: set @ int,P: int > $o,Q: int > $o] :
( ! [X3: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D3 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ B5 )
=> ( X3
!= ( plus_plus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ( P @ X3 )
=> ( P @ ( minus_minus @ int @ X3 @ D3 ) ) ) )
=> ( ! [X3: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D3 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ B5 )
=> ( X3
!= ( plus_plus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ( Q @ X3 )
=> ( Q @ ( minus_minus @ int @ X3 @ D3 ) ) ) )
=> ! [X4: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D3 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ B5 )
=> ( X4
!= ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ( P @ X4 )
& ( Q @ X4 ) )
=> ( ( P @ ( minus_minus @ int @ X4 @ D3 ) )
& ( Q @ ( minus_minus @ int @ X4 @ D3 ) ) ) ) ) ) ) ).
% bset(1)
thf(fact_2974_bset_I2_J,axiom,
! [D3: int,B5: set @ int,P: int > $o,Q: int > $o] :
( ! [X3: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D3 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ B5 )
=> ( X3
!= ( plus_plus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ( P @ X3 )
=> ( P @ ( minus_minus @ int @ X3 @ D3 ) ) ) )
=> ( ! [X3: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D3 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ B5 )
=> ( X3
!= ( plus_plus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ( Q @ X3 )
=> ( Q @ ( minus_minus @ int @ X3 @ D3 ) ) ) )
=> ! [X4: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D3 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ B5 )
=> ( X4
!= ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ( P @ X4 )
| ( Q @ X4 ) )
=> ( ( P @ ( minus_minus @ int @ X4 @ D3 ) )
| ( Q @ ( minus_minus @ int @ X4 @ D3 ) ) ) ) ) ) ) ).
% bset(2)
thf(fact_2975_aset_I1_J,axiom,
! [D3: int,A5: set @ int,P: int > $o,Q: int > $o] :
( ! [X3: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D3 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ A5 )
=> ( X3
!= ( minus_minus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ( P @ X3 )
=> ( P @ ( plus_plus @ int @ X3 @ D3 ) ) ) )
=> ( ! [X3: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D3 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ A5 )
=> ( X3
!= ( minus_minus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ( Q @ X3 )
=> ( Q @ ( plus_plus @ int @ X3 @ D3 ) ) ) )
=> ! [X4: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D3 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ A5 )
=> ( X4
!= ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ( P @ X4 )
& ( Q @ X4 ) )
=> ( ( P @ ( plus_plus @ int @ X4 @ D3 ) )
& ( Q @ ( plus_plus @ int @ X4 @ D3 ) ) ) ) ) ) ) ).
% aset(1)
thf(fact_2976_aset_I2_J,axiom,
! [D3: int,A5: set @ int,P: int > $o,Q: int > $o] :
( ! [X3: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D3 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ A5 )
=> ( X3
!= ( minus_minus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ( P @ X3 )
=> ( P @ ( plus_plus @ int @ X3 @ D3 ) ) ) )
=> ( ! [X3: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D3 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ A5 )
=> ( X3
!= ( minus_minus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ( Q @ X3 )
=> ( Q @ ( plus_plus @ int @ X3 @ D3 ) ) ) )
=> ! [X4: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D3 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ A5 )
=> ( X4
!= ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ( P @ X4 )
| ( Q @ X4 ) )
=> ( ( P @ ( plus_plus @ int @ X4 @ D3 ) )
| ( Q @ ( plus_plus @ int @ X4 @ D3 ) ) ) ) ) ) ) ).
% aset(2)
thf(fact_2977_prod_OatLeastAtMost__rev,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N: nat,M: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ N @ M ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I2: nat] : ( G @ ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N ) @ I2 ) )
@ ( set_or1337092689740270186AtMost @ nat @ N @ M ) ) ) ) ).
% prod.atLeastAtMost_rev
thf(fact_2978_prod_OatLeast0__atMost__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ).
% prod.atLeast0_atMost_Suc
thf(fact_2979_prod_Onat__ivl__Suc_H,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M: nat,N: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M @ ( suc @ N ) )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ ( suc @ N ) ) )
= ( times_times @ A @ ( G @ ( suc @ N ) ) @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ) ) ).
% prod.nat_ivl_Suc'
thf(fact_2980_prod_OatLeast__Suc__atMost,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M: nat,N: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( times_times @ A @ ( G @ M ) @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ N ) ) ) ) ) ) ).
% prod.atLeast_Suc_atMost
thf(fact_2981_atLeastLessThan__subseteq__atLeastAtMost__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ A2 @ B2 ) @ ( set_or1337092689740270186AtMost @ A @ C2 @ D2 ) )
= ( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ A2 )
& ( ord_less_eq @ A @ B2 @ D2 ) ) ) ) ) ).
% atLeastLessThan_subseteq_atLeastAtMost_iff
thf(fact_2982_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) @ ( set_or7035219750837199246ssThan @ A @ C2 @ D2 ) )
= ( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ A2 )
& ( ord_less @ A @ B2 @ D2 ) ) ) ) ) ).
% atLeastAtMost_subseteq_atLeastLessThan_iff
thf(fact_2983_sum__shift__lb__Suc0__0__upt,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [F: nat > A,K: nat] :
( ( ( F @ ( zero_zero @ nat ) )
= ( zero_zero @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ F @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ ( zero_zero @ nat ) ) @ K ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ F @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K ) ) ) ) ) ).
% sum_shift_lb_Suc0_0_upt
thf(fact_2984_sum_OatLeast0__lessThan__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) @ ( G @ N ) ) ) ) ).
% sum.atLeast0_lessThan_Suc
thf(fact_2985_sum_OatLeast__Suc__lessThan,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M: nat,N: nat,G: nat > A] :
( ( ord_less @ nat @ M @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) )
= ( plus_plus @ A @ ( G @ M ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M ) @ N ) ) ) ) ) ) ).
% sum.atLeast_Suc_lessThan
thf(fact_2986_sum_OatLeastLessThan__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A2: nat,B2: nat,G: nat > A] :
( ( ord_less_eq @ nat @ A2 @ B2 )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ A2 @ ( suc @ B2 ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ A2 @ B2 ) ) @ ( G @ B2 ) ) ) ) ) ).
% sum.atLeastLessThan_Suc
thf(fact_2987_sum_Olast__plus,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M: nat,N: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( plus_plus @ A @ ( G @ N ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ) ) ).
% sum.last_plus
thf(fact_2988_bset_I9_J,axiom,
! [D2: int,D3: int,B5: set @ int,T2: int] :
( ( dvd_dvd @ int @ D2 @ D3 )
=> ! [X4: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D3 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ B5 )
=> ( X4
!= ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( dvd_dvd @ int @ D2 @ ( plus_plus @ int @ X4 @ T2 ) )
=> ( dvd_dvd @ int @ D2 @ ( plus_plus @ int @ ( minus_minus @ int @ X4 @ D3 ) @ T2 ) ) ) ) ) ).
% bset(9)
thf(fact_2989_bset_I10_J,axiom,
! [D2: int,D3: int,B5: set @ int,T2: int] :
( ( dvd_dvd @ int @ D2 @ D3 )
=> ! [X4: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D3 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ B5 )
=> ( X4
!= ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ~ ( dvd_dvd @ int @ D2 @ ( plus_plus @ int @ X4 @ T2 ) )
=> ~ ( dvd_dvd @ int @ D2 @ ( plus_plus @ int @ ( minus_minus @ int @ X4 @ D3 ) @ T2 ) ) ) ) ) ).
% bset(10)
thf(fact_2990_aset_I9_J,axiom,
! [D2: int,D3: int,A5: set @ int,T2: int] :
( ( dvd_dvd @ int @ D2 @ D3 )
=> ! [X4: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D3 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ A5 )
=> ( X4
!= ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( dvd_dvd @ int @ D2 @ ( plus_plus @ int @ X4 @ T2 ) )
=> ( dvd_dvd @ int @ D2 @ ( plus_plus @ int @ ( plus_plus @ int @ X4 @ D3 ) @ T2 ) ) ) ) ) ).
% aset(9)
thf(fact_2991_aset_I10_J,axiom,
! [D2: int,D3: int,A5: set @ int,T2: int] :
( ( dvd_dvd @ int @ D2 @ D3 )
=> ! [X4: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D3 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ A5 )
=> ( X4
!= ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ~ ( dvd_dvd @ int @ D2 @ ( plus_plus @ int @ X4 @ T2 ) )
=> ~ ( dvd_dvd @ int @ D2 @ ( plus_plus @ int @ ( plus_plus @ int @ X4 @ D3 ) @ T2 ) ) ) ) ) ).
% aset(10)
thf(fact_2992_prod_OSuc__reindex__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M: nat,N: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) )
= ( times_times @ A @ ( G @ M )
@ ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ) ) ).
% prod.Suc_reindex_ivl
thf(fact_2993_prod_OatMost__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_atMost @ nat @ ( suc @ N ) ) )
= ( times_times @ A @ ( G @ ( zero_zero @ nat ) )
@ ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ) ).
% prod.atMost_Suc_shift
thf(fact_2994_fact__prod,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiring_char_0_fact @ A )
= ( ^ [N4: nat] :
( semiring_1_of_nat @ A
@ ( groups7121269368397514597t_prod @ nat @ nat
@ ^ [X: nat] : X
@ ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ N4 ) ) ) ) ) ) ).
% fact_prod
thf(fact_2995_sum__Suc__diff_H,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [M: nat,N: nat,F: nat > A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( minus_minus @ A @ ( F @ ( suc @ I2 ) ) @ ( F @ I2 ) )
@ ( set_or7035219750837199246ssThan @ nat @ M @ N ) )
= ( minus_minus @ A @ ( F @ N ) @ ( F @ M ) ) ) ) ) ).
% sum_Suc_diff'
thf(fact_2996_sum_OatLeastLessThan__rev,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,N: nat,M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ N @ M ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( G @ ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N ) @ ( suc @ I2 ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ N @ M ) ) ) ) ).
% sum.atLeastLessThan_rev
thf(fact_2997_sum_Onested__swap,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A2: nat > nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ ( A2 @ I2 ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ I2 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J3: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( A2 @ I2 @ J3 )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J3 ) @ N ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% sum.nested_swap
thf(fact_2998_prod__atLeastAtMost__code,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [F: nat > A,A2: nat,B2: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ F @ ( set_or1337092689740270186AtMost @ nat @ A2 @ B2 ) )
= ( set_fo6178422350223883121st_nat @ A
@ ^ [A3: nat] : ( times_times @ A @ ( F @ A3 ) )
@ A2
@ B2
@ ( one_one @ A ) ) ) ) ).
% prod_atLeastAtMost_code
thf(fact_2999_prod_Oub__add__nat,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M: nat,N: nat,G: nat > A,P3: nat] :
( ( ord_less_eq @ nat @ M @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ ( plus_plus @ nat @ N @ P3 ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) @ ( plus_plus @ nat @ N @ P3 ) ) ) ) ) ) ) ).
% prod.ub_add_nat
thf(fact_3000_sum_Ohead__if,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [N: nat,M: nat,G: nat > A] :
( ( ( ord_less @ nat @ N @ M )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( zero_zero @ A ) ) )
& ( ~ ( ord_less @ nat @ N @ M )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) @ ( G @ N ) ) ) ) ) ) ).
% sum.head_if
thf(fact_3001_periodic__finite__ex,axiom,
! [D2: int,P: int > $o] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D2 )
=> ( ! [X3: int,K4: int] :
( ( P @ X3 )
= ( P @ ( minus_minus @ int @ X3 @ ( times_times @ int @ K4 @ D2 ) ) ) )
=> ( ( ? [X5: int] : ( P @ X5 ) )
= ( ? [X: int] :
( ( member @ int @ X @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D2 ) )
& ( P @ X ) ) ) ) ) ) ).
% periodic_finite_ex
thf(fact_3002_bset_I3_J,axiom,
! [D3: int,T2: int,B5: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D3 )
=> ( ( member @ int @ ( minus_minus @ int @ T2 @ ( one_one @ int ) ) @ B5 )
=> ! [X4: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D3 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ B5 )
=> ( X4
!= ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( X4 = T2 )
=> ( ( minus_minus @ int @ X4 @ D3 )
= T2 ) ) ) ) ) ).
% bset(3)
thf(fact_3003_bset_I4_J,axiom,
! [D3: int,T2: int,B5: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D3 )
=> ( ( member @ int @ T2 @ B5 )
=> ! [X4: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D3 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ B5 )
=> ( X4
!= ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( X4 != T2 )
=> ( ( minus_minus @ int @ X4 @ D3 )
!= T2 ) ) ) ) ) ).
% bset(4)
thf(fact_3004_bset_I5_J,axiom,
! [D3: int,B5: set @ int,T2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D3 )
=> ! [X4: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D3 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ B5 )
=> ( X4
!= ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less @ int @ X4 @ T2 )
=> ( ord_less @ int @ ( minus_minus @ int @ X4 @ D3 ) @ T2 ) ) ) ) ).
% bset(5)
thf(fact_3005_bset_I7_J,axiom,
! [D3: int,T2: int,B5: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D3 )
=> ( ( member @ int @ T2 @ B5 )
=> ! [X4: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D3 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ B5 )
=> ( X4
!= ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less @ int @ T2 @ X4 )
=> ( ord_less @ int @ T2 @ ( minus_minus @ int @ X4 @ D3 ) ) ) ) ) ) ).
% bset(7)
thf(fact_3006_aset_I3_J,axiom,
! [D3: int,T2: int,A5: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D3 )
=> ( ( member @ int @ ( plus_plus @ int @ T2 @ ( one_one @ int ) ) @ A5 )
=> ! [X4: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D3 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ A5 )
=> ( X4
!= ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( X4 = T2 )
=> ( ( plus_plus @ int @ X4 @ D3 )
= T2 ) ) ) ) ) ).
% aset(3)
thf(fact_3007_aset_I4_J,axiom,
! [D3: int,T2: int,A5: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D3 )
=> ( ( member @ int @ T2 @ A5 )
=> ! [X4: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D3 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ A5 )
=> ( X4
!= ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( X4 != T2 )
=> ( ( plus_plus @ int @ X4 @ D3 )
!= T2 ) ) ) ) ) ).
% aset(4)
thf(fact_3008_aset_I5_J,axiom,
! [D3: int,T2: int,A5: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D3 )
=> ( ( member @ int @ T2 @ A5 )
=> ! [X4: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D3 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ A5 )
=> ( X4
!= ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less @ int @ X4 @ T2 )
=> ( ord_less @ int @ ( plus_plus @ int @ X4 @ D3 ) @ T2 ) ) ) ) ) ).
% aset(5)
thf(fact_3009_aset_I7_J,axiom,
! [D3: int,A5: set @ int,T2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D3 )
=> ! [X4: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D3 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ A5 )
=> ( X4
!= ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less @ int @ T2 @ X4 )
=> ( ord_less @ int @ T2 @ ( plus_plus @ int @ X4 @ D3 ) ) ) ) ) ).
% aset(7)
thf(fact_3010_fact__eq__fact__times,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq @ nat @ N @ M )
=> ( ( semiring_char_0_fact @ nat @ M )
= ( times_times @ nat @ ( semiring_char_0_fact @ nat @ N )
@ ( groups7121269368397514597t_prod @ nat @ nat
@ ^ [X: nat] : X
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ N ) @ M ) ) ) ) ) ).
% fact_eq_fact_times
thf(fact_3011_sum_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,N: nat,M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ N @ M ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( G @ ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N ) @ I2 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ N ) @ M ) ) ) ) ).
% sum.atLeastLessThan_rev_at_least_Suc_atMost
thf(fact_3012_bset_I6_J,axiom,
! [D3: int,B5: set @ int,T2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D3 )
=> ! [X4: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D3 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ B5 )
=> ( X4
!= ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less_eq @ int @ X4 @ T2 )
=> ( ord_less_eq @ int @ ( minus_minus @ int @ X4 @ D3 ) @ T2 ) ) ) ) ).
% bset(6)
thf(fact_3013_bset_I8_J,axiom,
! [D3: int,T2: int,B5: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D3 )
=> ( ( member @ int @ ( minus_minus @ int @ T2 @ ( one_one @ int ) ) @ B5 )
=> ! [X4: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D3 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ B5 )
=> ( X4
!= ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less_eq @ int @ T2 @ X4 )
=> ( ord_less_eq @ int @ T2 @ ( minus_minus @ int @ X4 @ D3 ) ) ) ) ) ) ).
% bset(8)
thf(fact_3014_aset_I6_J,axiom,
! [D3: int,T2: int,A5: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D3 )
=> ( ( member @ int @ ( plus_plus @ int @ T2 @ ( one_one @ int ) ) @ A5 )
=> ! [X4: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D3 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ A5 )
=> ( X4
!= ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less_eq @ int @ X4 @ T2 )
=> ( ord_less_eq @ int @ ( plus_plus @ int @ X4 @ D3 ) @ T2 ) ) ) ) ) ).
% aset(6)
thf(fact_3015_aset_I8_J,axiom,
! [D3: int,A5: set @ int,T2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D3 )
=> ! [X4: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D3 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ A5 )
=> ( X4
!= ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less_eq @ int @ T2 @ X4 )
=> ( ord_less_eq @ int @ T2 @ ( plus_plus @ int @ X4 @ D3 ) ) ) ) ) ).
% aset(8)
thf(fact_3016_cppi,axiom,
! [D3: int,P: int > $o,P2: int > $o,A5: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D3 )
=> ( ? [Z5: int] :
! [X3: int] :
( ( ord_less @ int @ Z5 @ X3 )
=> ( ( P @ X3 )
= ( P2 @ X3 ) ) )
=> ( ! [X3: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D3 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ A5 )
=> ( X3
!= ( minus_minus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ( P @ X3 )
=> ( P @ ( plus_plus @ int @ X3 @ D3 ) ) ) )
=> ( ! [X3: int,K4: int] :
( ( P2 @ X3 )
= ( P2 @ ( minus_minus @ int @ X3 @ ( times_times @ int @ K4 @ D3 ) ) ) )
=> ( ( ? [X5: int] : ( P @ X5 ) )
= ( ? [X: int] :
( ( member @ int @ X @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D3 ) )
& ( P2 @ X ) )
| ? [X: int] :
( ( member @ int @ X @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D3 ) )
& ? [Y4: int] :
( ( member @ int @ Y4 @ A5 )
& ( P @ ( minus_minus @ int @ Y4 @ X ) ) ) ) ) ) ) ) ) ) ).
% cppi
thf(fact_3017_cpmi,axiom,
! [D3: int,P: int > $o,P2: int > $o,B5: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D3 )
=> ( ? [Z5: int] :
! [X3: int] :
( ( ord_less @ int @ X3 @ Z5 )
=> ( ( P @ X3 )
= ( P2 @ X3 ) ) )
=> ( ! [X3: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D3 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ B5 )
=> ( X3
!= ( plus_plus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ( P @ X3 )
=> ( P @ ( minus_minus @ int @ X3 @ D3 ) ) ) )
=> ( ! [X3: int,K4: int] :
( ( P2 @ X3 )
= ( P2 @ ( minus_minus @ int @ X3 @ ( times_times @ int @ K4 @ D3 ) ) ) )
=> ( ( ? [X5: int] : ( P @ X5 ) )
= ( ? [X: int] :
( ( member @ int @ X @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D3 ) )
& ( P2 @ X ) )
| ? [X: int] :
( ( member @ int @ X @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D3 ) )
& ? [Y4: int] :
( ( member @ int @ Y4 @ B5 )
& ( P @ ( plus_plus @ int @ Y4 @ X ) ) ) ) ) ) ) ) ) ) ).
% cpmi
thf(fact_3018_pochhammer__Suc__prod,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A2: A,N: nat] :
( ( comm_s3205402744901411588hammer @ A @ A2 @ ( suc @ N ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I2: nat] : ( plus_plus @ A @ A2 @ ( semiring_1_of_nat @ A @ I2 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% pochhammer_Suc_prod
thf(fact_3019_pochhammer__prod__rev,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ( ( comm_s3205402744901411588hammer @ A )
= ( ^ [A3: A,N4: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I2: nat] : ( plus_plus @ A @ A3 @ ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ N4 @ I2 ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ N4 ) ) ) ) ) ).
% pochhammer_prod_rev
thf(fact_3020_fact__div__fact,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq @ nat @ N @ M )
=> ( ( divide_divide @ nat @ ( semiring_char_0_fact @ nat @ M ) @ ( semiring_char_0_fact @ nat @ N ) )
= ( groups7121269368397514597t_prod @ nat @ nat
@ ^ [X: nat] : X
@ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) @ M ) ) ) ) ).
% fact_div_fact
thf(fact_3021_take__bit__sum,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ( ( bit_se2584673776208193580ke_bit @ A )
= ( ^ [N4: nat,A3: A] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K2: nat] : ( bit_se4730199178511100633sh_bit @ A @ K2 @ ( zero_neq_one_of_bool @ A @ ( bit_se5641148757651400278ts_bit @ A @ A3 @ K2 ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N4 ) ) ) ) ) ).
% take_bit_sum
thf(fact_3022_prod_Oin__pairs,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,M: nat,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I2: nat] : ( times_times @ A @ ( G @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I2 ) ) @ ( G @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I2 ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% prod.in_pairs
thf(fact_3023_prod_Oin__pairs__0,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_atMost @ nat @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I2: nat] : ( times_times @ A @ ( G @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I2 ) ) @ ( G @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I2 ) ) ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% prod.in_pairs_0
thf(fact_3024_pochhammer__Suc__prod__rev,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A2: A,N: nat] :
( ( comm_s3205402744901411588hammer @ A @ A2 @ ( suc @ N ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I2: nat] : ( plus_plus @ A @ A2 @ ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ N @ I2 ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% pochhammer_Suc_prod_rev
thf(fact_3025_horner__sum__eq__sum,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_1 @ A )
=> ( ( groups4207007520872428315er_sum @ B @ A )
= ( ^ [F2: B > A,A3: A,Xs3: list @ B] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N4: nat] : ( times_times @ A @ ( F2 @ ( nth @ B @ Xs3 @ N4 ) ) @ ( power_power @ A @ A3 @ N4 ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ B ) @ Xs3 ) ) ) ) ) ) ).
% horner_sum_eq_sum
thf(fact_3026_prod_Ozero__middle,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [P3: nat,K: nat,G: 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 ) @ ( G @ 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 ) @ ( G @ J3 ) @ ( H @ J3 ) )
@ ( set_ord_atMost @ nat @ ( minus_minus @ nat @ P3 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) ) ).
% prod.zero_middle
thf(fact_3027_case__nat__add__eq__if,axiom,
! [A: $tType,A2: A,F: nat > A,V2: num,N: nat] :
( ( case_nat @ A @ A2 @ F @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ V2 ) @ N ) )
= ( F @ ( plus_plus @ nat @ ( pred_numeral @ V2 ) @ N ) ) ) ).
% case_nat_add_eq_if
thf(fact_3028_integer__of__int__code,axiom,
( code_integer_of_int
= ( ^ [K2: int] :
( if @ code_integer @ ( ord_less @ int @ K2 @ ( zero_zero @ int ) ) @ ( uminus_uminus @ code_integer @ ( code_integer_of_int @ ( uminus_uminus @ int @ K2 ) ) )
@ ( if @ code_integer
@ ( K2
= ( zero_zero @ int ) )
@ ( zero_zero @ code_integer )
@ ( if @ code_integer
@ ( ( modulo_modulo @ int @ K2 @ ( 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 @ K2 @ ( 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 @ K2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) @ ( one_one @ code_integer ) ) ) ) ) ) ) ).
% integer_of_int_code
thf(fact_3029_sum__zero__power_H,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A5: set @ nat,C2: nat > A,D2: nat > A] :
( ( ( ( finite_finite2 @ nat @ A5 )
& ( member @ nat @ ( zero_zero @ nat ) @ A5 ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( divide_divide @ A @ ( times_times @ A @ ( C2 @ I2 ) @ ( power_power @ A @ ( zero_zero @ A ) @ I2 ) ) @ ( D2 @ I2 ) )
@ A5 )
= ( divide_divide @ A @ ( C2 @ ( zero_zero @ nat ) ) @ ( D2 @ ( zero_zero @ nat ) ) ) ) )
& ( ~ ( ( finite_finite2 @ nat @ A5 )
& ( member @ nat @ ( zero_zero @ nat ) @ A5 ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( divide_divide @ A @ ( times_times @ A @ ( C2 @ I2 ) @ ( power_power @ A @ ( zero_zero @ A ) @ I2 ) ) @ ( D2 @ I2 ) )
@ A5 )
= ( zero_zero @ A ) ) ) ) ) ).
% sum_zero_power'
thf(fact_3030_or__int__rec,axiom,
( ( bit_se1065995026697491101ons_or @ int )
= ( ^ [K2: int,L2: int] :
( plus_plus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K2 )
| ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L2 ) ) )
@ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ int @ ( divide_divide @ int @ K2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% or_int_rec
thf(fact_3031_or__numerals_I4_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X2: num,Y: num] :
( ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ ( bit0 @ X2 ) ) @ ( 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 @ X2 ) @ ( numeral_numeral @ A @ Y ) ) ) ) ) ) ).
% or_numerals(4)
thf(fact_3032_integer__of__int__eq__of__int,axiom,
( code_integer_of_int
= ( ring_1_of_int @ code_integer ) ) ).
% integer_of_int_eq_of_int
thf(fact_3033_or_Oright__neutral,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A] :
( ( bit_se1065995026697491101ons_or @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% or.right_neutral
thf(fact_3034_or_Oleft__neutral,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A] :
( ( bit_se1065995026697491101ons_or @ A @ ( zero_zero @ A ) @ A2 )
= A2 ) ) ).
% or.left_neutral
thf(fact_3035_finite__atLeastAtMost,axiom,
! [L: nat,U: nat] : ( finite_finite2 @ nat @ ( set_or1337092689740270186AtMost @ nat @ L @ U ) ) ).
% finite_atLeastAtMost
thf(fact_3036_finite__atLeastLessThan,axiom,
! [L: nat,U: nat] : ( finite_finite2 @ nat @ ( set_or7035219750837199246ssThan @ nat @ L @ U ) ) ).
% finite_atLeastLessThan
thf(fact_3037_finite__atMost,axiom,
! [K: nat] : ( finite_finite2 @ nat @ ( set_ord_atMost @ nat @ K ) ) ).
% finite_atMost
thf(fact_3038_sum_Oinfinite,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A5: set @ B,G: B > A] :
( ~ ( finite_finite2 @ B @ A5 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ A5 )
= ( zero_zero @ A ) ) ) ) ).
% sum.infinite
thf(fact_3039_sum__eq__0__iff,axiom,
! [A: $tType,B: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [F4: set @ B,F: B > A] :
( ( finite_finite2 @ B @ F4 )
=> ( ( ( groups7311177749621191930dd_sum @ B @ A @ F @ F4 )
= ( zero_zero @ A ) )
= ( ! [X: B] :
( ( member @ B @ X @ F4 )
=> ( ( F @ X )
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% sum_eq_0_iff
thf(fact_3040_prod__zero__iff,axiom,
! [A: $tType,B: $tType] :
( ( semidom @ A )
=> ! [A5: set @ B,F: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( ( groups7121269368397514597t_prod @ B @ A @ F @ A5 )
= ( zero_zero @ A ) )
= ( ? [X: B] :
( ( member @ B @ X @ A5 )
& ( ( F @ X )
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% prod_zero_iff
thf(fact_3041_infinite__Icc__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A] :
( ( ~ ( finite_finite2 @ A @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) ) )
= ( ord_less @ A @ A2 @ B2 ) ) ) ).
% infinite_Icc_iff
thf(fact_3042_infinite__Ico__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A] :
( ( ~ ( finite_finite2 @ A @ ( set_or7035219750837199246ssThan @ A @ A2 @ B2 ) ) )
= ( ord_less @ A @ A2 @ B2 ) ) ) ).
% infinite_Ico_iff
thf(fact_3043_prod_Oinfinite,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A5: set @ B,G: B > A] :
( ~ ( finite_finite2 @ B @ A5 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ A5 )
= ( one_one @ A ) ) ) ) ).
% prod.infinite
thf(fact_3044_bit_Odisj__one__left,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X2: A] :
( ( bit_se1065995026697491101ons_or @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ X2 )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% bit.disj_one_left
thf(fact_3045_bit_Odisj__one__right,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X2: A] :
( ( bit_se1065995026697491101ons_or @ A @ X2 @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% bit.disj_one_right
thf(fact_3046_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_3047_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_3048_sum_Odelta,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ! [S2: set @ B,A2: B,B2: B > A] :
( ( finite_finite2 @ B @ S2 )
=> ( ( ( member @ B @ A2 @ S2 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [K2: B] : ( if @ A @ ( K2 = A2 ) @ ( B2 @ K2 ) @ ( zero_zero @ A ) )
@ S2 )
= ( B2 @ A2 ) ) )
& ( ~ ( member @ B @ A2 @ S2 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [K2: B] : ( if @ A @ ( K2 = A2 ) @ ( B2 @ K2 ) @ ( zero_zero @ A ) )
@ S2 )
= ( zero_zero @ A ) ) ) ) ) ) ).
% sum.delta
thf(fact_3049_sum_Odelta_H,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ! [S2: set @ B,A2: B,B2: B > A] :
( ( finite_finite2 @ B @ S2 )
=> ( ( ( member @ B @ A2 @ S2 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [K2: B] : ( if @ A @ ( A2 = K2 ) @ ( B2 @ K2 ) @ ( zero_zero @ A ) )
@ S2 )
= ( B2 @ A2 ) ) )
& ( ~ ( member @ B @ A2 @ S2 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [K2: B] : ( if @ A @ ( A2 = K2 ) @ ( B2 @ K2 ) @ ( zero_zero @ A ) )
@ S2 )
= ( zero_zero @ A ) ) ) ) ) ) ).
% sum.delta'
thf(fact_3050_prod__eq__1__iff,axiom,
! [A: $tType,A5: set @ A,F: A > nat] :
( ( finite_finite2 @ A @ A5 )
=> ( ( ( groups7121269368397514597t_prod @ A @ nat @ F @ A5 )
= ( one_one @ nat ) )
= ( ! [X: A] :
( ( member @ A @ X @ A5 )
=> ( ( F @ X )
= ( one_one @ nat ) ) ) ) ) ) ).
% prod_eq_1_iff
thf(fact_3051_prod_Odelta_H,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S2: set @ B,A2: B,B2: B > A] :
( ( finite_finite2 @ B @ S2 )
=> ( ( ( member @ B @ A2 @ S2 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K2: B] : ( if @ A @ ( A2 = K2 ) @ ( B2 @ K2 ) @ ( one_one @ A ) )
@ S2 )
= ( B2 @ A2 ) ) )
& ( ~ ( member @ B @ A2 @ S2 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K2: B] : ( if @ A @ ( A2 = K2 ) @ ( B2 @ K2 ) @ ( one_one @ A ) )
@ S2 )
= ( one_one @ A ) ) ) ) ) ) ).
% prod.delta'
thf(fact_3052_prod_Odelta,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S2: set @ B,A2: B,B2: B > A] :
( ( finite_finite2 @ B @ S2 )
=> ( ( ( member @ B @ A2 @ S2 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K2: B] : ( if @ A @ ( K2 = A2 ) @ ( B2 @ K2 ) @ ( one_one @ A ) )
@ S2 )
= ( B2 @ A2 ) ) )
& ( ~ ( member @ B @ A2 @ S2 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K2: B] : ( if @ A @ ( K2 = A2 ) @ ( B2 @ K2 ) @ ( one_one @ A ) )
@ S2 )
= ( one_one @ A ) ) ) ) ) ) ).
% prod.delta
thf(fact_3053_case__nat__numeral,axiom,
! [A: $tType,A2: A,F: nat > A,V2: num] :
( ( case_nat @ A @ A2 @ F @ ( numeral_numeral @ nat @ V2 ) )
= ( F @ ( pred_numeral @ V2 ) ) ) ).
% case_nat_numeral
thf(fact_3054_or__nat__numerals_I4_J,axiom,
! [X2: num] :
( ( bit_se1065995026697491101ons_or @ nat @ ( numeral_numeral @ nat @ ( bit1 @ X2 ) ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( numeral_numeral @ nat @ ( bit1 @ X2 ) ) ) ).
% or_nat_numerals(4)
thf(fact_3055_or__nat__numerals_I2_J,axiom,
! [Y: num] :
( ( bit_se1065995026697491101ons_or @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( numeral_numeral @ nat @ ( bit1 @ Y ) ) )
= ( numeral_numeral @ nat @ ( bit1 @ Y ) ) ) ).
% or_nat_numerals(2)
thf(fact_3056_or__numerals_I2_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [Y: num] :
( ( bit_se1065995026697491101ons_or @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit1 @ Y ) ) )
= ( numeral_numeral @ A @ ( bit1 @ Y ) ) ) ) ).
% or_numerals(2)
thf(fact_3057_or__numerals_I8_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X2: num] :
( ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ ( bit1 @ X2 ) ) @ ( one_one @ A ) )
= ( numeral_numeral @ A @ ( bit1 @ X2 ) ) ) ) ).
% or_numerals(8)
thf(fact_3058_bit_Odisj__cancel__left,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X2: A] :
( ( bit_se1065995026697491101ons_or @ A @ ( bit_ri4277139882892585799ns_not @ A @ X2 ) @ X2 )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% bit.disj_cancel_left
thf(fact_3059_bit_Odisj__cancel__right,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X2: A] :
( ( bit_se1065995026697491101ons_or @ A @ X2 @ ( bit_ri4277139882892585799ns_not @ A @ X2 ) )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% bit.disj_cancel_right
thf(fact_3060_prod__pos__nat__iff,axiom,
! [A: $tType,A5: set @ A,F: A > nat] :
( ( finite_finite2 @ A @ A5 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( groups7121269368397514597t_prod @ A @ nat @ F @ A5 ) )
= ( ! [X: A] :
( ( member @ A @ X @ A5 )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F @ X ) ) ) ) ) ) ).
% prod_pos_nat_iff
thf(fact_3061_or__nat__numerals_I1_J,axiom,
! [Y: num] :
( ( bit_se1065995026697491101ons_or @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( numeral_numeral @ nat @ ( bit0 @ Y ) ) )
= ( numeral_numeral @ nat @ ( bit1 @ Y ) ) ) ).
% or_nat_numerals(1)
thf(fact_3062_or__nat__numerals_I3_J,axiom,
! [X2: num] :
( ( bit_se1065995026697491101ons_or @ nat @ ( numeral_numeral @ nat @ ( bit0 @ X2 ) ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( numeral_numeral @ nat @ ( bit1 @ X2 ) ) ) ).
% or_nat_numerals(3)
thf(fact_3063_or__numerals_I1_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [Y: num] :
( ( bit_se1065995026697491101ons_or @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ Y ) ) )
= ( numeral_numeral @ A @ ( bit1 @ Y ) ) ) ) ).
% or_numerals(1)
thf(fact_3064_or__numerals_I5_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X2: num] :
( ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ ( bit0 @ X2 ) ) @ ( one_one @ A ) )
= ( numeral_numeral @ A @ ( bit1 @ X2 ) ) ) ) ).
% or_numerals(5)
thf(fact_3065_or__minus__numerals_I2_J,axiom,
! [N: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( one_one @ int ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) )
= ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) ) ).
% or_minus_numerals(2)
thf(fact_3066_or__minus__numerals_I6_J,axiom,
! [N: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) @ ( one_one @ int ) )
= ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) ) ).
% or_minus_numerals(6)
thf(fact_3067_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
@ ^ [I2: nat] : ( times_times @ A @ ( C2 @ I2 ) @ ( power_power @ A @ ( zero_zero @ A ) @ I2 ) )
@ A5 )
= ( C2 @ ( zero_zero @ nat ) ) ) )
& ( ~ ( ( finite_finite2 @ nat @ A5 )
& ( member @ nat @ ( zero_zero @ nat ) @ A5 ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( times_times @ A @ ( C2 @ I2 ) @ ( power_power @ A @ ( zero_zero @ A ) @ I2 ) )
@ A5 )
= ( zero_zero @ A ) ) ) ) ) ).
% sum_zero_power
thf(fact_3068_and__minus__minus__numerals,axiom,
! [M: num,N: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
= ( bit_ri4277139882892585799ns_not @ int @ ( bit_se1065995026697491101ons_or @ int @ ( minus_minus @ int @ ( numeral_numeral @ int @ M ) @ ( one_one @ int ) ) @ ( minus_minus @ int @ ( numeral_numeral @ int @ N ) @ ( one_one @ int ) ) ) ) ) ).
% and_minus_minus_numerals
thf(fact_3069_or__minus__minus__numerals,axiom,
! [M: num,N: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
= ( bit_ri4277139882892585799ns_not @ int @ ( bit_se5824344872417868541ns_and @ int @ ( minus_minus @ int @ ( numeral_numeral @ int @ M ) @ ( one_one @ int ) ) @ ( minus_minus @ int @ ( numeral_numeral @ int @ N ) @ ( one_one @ int ) ) ) ) ) ).
% or_minus_minus_numerals
thf(fact_3070_or__numerals_I7_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X2: num,Y: num] :
( ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ ( bit1 @ X2 ) ) @ ( 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 @ X2 ) @ ( numeral_numeral @ A @ Y ) ) ) ) ) ) ).
% or_numerals(7)
thf(fact_3071_or__numerals_I6_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X2: num,Y: num] :
( ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ ( bit1 @ X2 ) ) @ ( 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 @ X2 ) @ ( numeral_numeral @ A @ Y ) ) ) ) ) ) ).
% or_numerals(6)
thf(fact_3072_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_3073_of__nat__or__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,N: nat] :
( ( semiring_1_of_nat @ A @ ( bit_se1065995026697491101ons_or @ nat @ M @ N ) )
= ( bit_se1065995026697491101ons_or @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% of_nat_or_eq
thf(fact_3074_int__prod,axiom,
! [B: $tType,F: B > nat,A5: set @ B] :
( ( semiring_1_of_nat @ int @ ( groups7121269368397514597t_prod @ B @ nat @ F @ A5 ) )
= ( groups7121269368397514597t_prod @ B @ int
@ ^ [X: B] : ( semiring_1_of_nat @ int @ ( F @ X ) )
@ A5 ) ) ).
% int_prod
thf(fact_3075_or__nat__def,axiom,
( ( bit_se1065995026697491101ons_or @ nat )
= ( ^ [M2: nat,N4: nat] : ( nat2 @ ( bit_se1065995026697491101ons_or @ int @ ( semiring_1_of_nat @ int @ M2 ) @ ( semiring_1_of_nat @ int @ N4 ) ) ) ) ) ).
% or_nat_def
thf(fact_3076_less__eq__integer_Oabs__eq,axiom,
! [Xa: int,X2: int] :
( ( ord_less_eq @ code_integer @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X2 ) )
= ( ord_less_eq @ int @ Xa @ X2 ) ) ).
% less_eq_integer.abs_eq
thf(fact_3077_less__integer_Oabs__eq,axiom,
! [Xa: int,X2: int] :
( ( ord_less @ code_integer @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X2 ) )
= ( ord_less @ int @ Xa @ X2 ) ) ).
% less_integer.abs_eq
thf(fact_3078_one__integer__def,axiom,
( ( one_one @ code_integer )
= ( code_integer_of_int @ ( one_one @ int ) ) ) ).
% one_integer_def
thf(fact_3079_times__integer_Oabs__eq,axiom,
! [Xa: int,X2: int] :
( ( times_times @ code_integer @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X2 ) )
= ( code_integer_of_int @ ( times_times @ int @ Xa @ X2 ) ) ) ).
% times_integer.abs_eq
thf(fact_3080_uminus__integer_Oabs__eq,axiom,
! [X2: int] :
( ( uminus_uminus @ code_integer @ ( code_integer_of_int @ X2 ) )
= ( code_integer_of_int @ ( uminus_uminus @ int @ X2 ) ) ) ).
% uminus_integer.abs_eq
thf(fact_3081_or__eq__0__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A,B2: A] :
( ( ( bit_se1065995026697491101ons_or @ A @ A2 @ B2 )
= ( zero_zero @ A ) )
= ( ( A2
= ( zero_zero @ A ) )
& ( B2
= ( zero_zero @ A ) ) ) ) ) ).
% or_eq_0_iff
thf(fact_3082_bit_Odisj__zero__right,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X2: A] :
( ( bit_se1065995026697491101ons_or @ A @ X2 @ ( zero_zero @ A ) )
= X2 ) ) ).
% bit.disj_zero_right
thf(fact_3083_zero__integer__def,axiom,
( ( zero_zero @ code_integer )
= ( code_integer_of_int @ ( zero_zero @ int ) ) ) ).
% zero_integer_def
thf(fact_3084_sgn__integer__code,axiom,
( ( sgn_sgn @ code_integer )
= ( ^ [K2: code_integer] :
( if @ code_integer
@ ( K2
= ( zero_zero @ code_integer ) )
@ ( zero_zero @ code_integer )
@ ( if @ code_integer @ ( ord_less @ code_integer @ K2 @ ( zero_zero @ code_integer ) ) @ ( uminus_uminus @ code_integer @ ( one_one @ code_integer ) ) @ ( one_one @ code_integer ) ) ) ) ) ).
% sgn_integer_code
thf(fact_3085_abs__integer__code,axiom,
( ( abs_abs @ code_integer )
= ( ^ [K2: code_integer] : ( if @ code_integer @ ( ord_less @ code_integer @ K2 @ ( zero_zero @ code_integer ) ) @ ( uminus_uminus @ code_integer @ K2 ) @ K2 ) ) ) ).
% abs_integer_code
thf(fact_3086_less__integer__code_I1_J,axiom,
~ ( ord_less @ code_integer @ ( zero_zero @ code_integer ) @ ( zero_zero @ code_integer ) ) ).
% less_integer_code(1)
thf(fact_3087_minus__integer__code_I2_J,axiom,
! [L: code_integer] :
( ( minus_minus @ code_integer @ ( zero_zero @ code_integer ) @ L )
= ( uminus_uminus @ code_integer @ L ) ) ).
% minus_integer_code(2)
thf(fact_3088_minus__integer__code_I1_J,axiom,
! [K: code_integer] :
( ( minus_minus @ code_integer @ K @ ( zero_zero @ code_integer ) )
= K ) ).
% minus_integer_code(1)
thf(fact_3089_finite__if__eq__beyond__finite,axiom,
! [A: $tType,S2: set @ A,S5: set @ A] :
( ( finite_finite2 @ A @ S2 )
=> ( finite_finite2 @ ( set @ A )
@ ( collect @ ( set @ A )
@ ^ [S6: set @ A] :
( ( minus_minus @ ( set @ A ) @ S6 @ S2 )
= ( minus_minus @ ( set @ A ) @ S5 @ S2 ) ) ) ) ) ).
% finite_if_eq_beyond_finite
thf(fact_3090_uminus__integer__code_I1_J,axiom,
( ( uminus_uminus @ code_integer @ ( zero_zero @ code_integer ) )
= ( zero_zero @ code_integer ) ) ).
% uminus_integer_code(1)
thf(fact_3091_or__integer_Oabs__eq,axiom,
! [Xa: int,X2: int] :
( ( bit_se1065995026697491101ons_or @ code_integer @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X2 ) )
= ( code_integer_of_int @ ( bit_se1065995026697491101ons_or @ int @ Xa @ X2 ) ) ) ).
% or_integer.abs_eq
thf(fact_3092_plus__integer__code_I2_J,axiom,
! [L: code_integer] :
( ( plus_plus @ code_integer @ ( zero_zero @ code_integer ) @ L )
= L ) ).
% plus_integer_code(2)
thf(fact_3093_plus__integer__code_I1_J,axiom,
! [K: code_integer] :
( ( plus_plus @ code_integer @ K @ ( zero_zero @ code_integer ) )
= K ) ).
% plus_integer_code(1)
thf(fact_3094_times__integer__code_I1_J,axiom,
! [K: code_integer] :
( ( times_times @ code_integer @ K @ ( zero_zero @ code_integer ) )
= ( zero_zero @ code_integer ) ) ).
% times_integer_code(1)
thf(fact_3095_times__integer__code_I2_J,axiom,
! [L: code_integer] :
( ( times_times @ code_integer @ ( zero_zero @ code_integer ) @ L )
= ( zero_zero @ code_integer ) ) ).
% times_integer_code(2)
thf(fact_3096_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 )
@ ^ [X: nat] : ( H @ ( F22 @ X ) )
@ Nat ) ) ).
% nat.case_distrib
thf(fact_3097_plus__integer_Oabs__eq,axiom,
! [Xa: int,X2: int] :
( ( plus_plus @ code_integer @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X2 ) )
= ( code_integer_of_int @ ( plus_plus @ int @ Xa @ X2 ) ) ) ).
% plus_integer.abs_eq
thf(fact_3098_minus__integer_Oabs__eq,axiom,
! [Xa: int,X2: int] :
( ( minus_minus @ code_integer @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X2 ) )
= ( code_integer_of_int @ ( minus_minus @ int @ Xa @ X2 ) ) ) ).
% minus_integer.abs_eq
thf(fact_3099_abs__integer_Oabs__eq,axiom,
! [X2: int] :
( ( abs_abs @ code_integer @ ( code_integer_of_int @ X2 ) )
= ( code_integer_of_int @ ( abs_abs @ int @ X2 ) ) ) ).
% abs_integer.abs_eq
thf(fact_3100_divide__integer_Oabs__eq,axiom,
! [Xa: int,X2: int] :
( ( divide_divide @ code_integer @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X2 ) )
= ( code_integer_of_int @ ( divide_divide @ int @ Xa @ X2 ) ) ) ).
% divide_integer.abs_eq
thf(fact_3101_modulo__integer_Oabs__eq,axiom,
! [Xa: int,X2: int] :
( ( modulo_modulo @ code_integer @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X2 ) )
= ( code_integer_of_int @ ( modulo_modulo @ int @ Xa @ X2 ) ) ) ).
% modulo_integer.abs_eq
thf(fact_3102_take__bit__integer_Oabs__eq,axiom,
! [Xa: nat,X2: int] :
( ( bit_se2584673776208193580ke_bit @ code_integer @ Xa @ ( code_integer_of_int @ X2 ) )
= ( code_integer_of_int @ ( bit_se2584673776208193580ke_bit @ int @ Xa @ X2 ) ) ) ).
% take_bit_integer.abs_eq
thf(fact_3103_not__integer_Oabs__eq,axiom,
! [X2: int] :
( ( bit_ri4277139882892585799ns_not @ code_integer @ ( code_integer_of_int @ X2 ) )
= ( code_integer_of_int @ ( bit_ri4277139882892585799ns_not @ int @ X2 ) ) ) ).
% not_integer.abs_eq
thf(fact_3104_and__integer_Oabs__eq,axiom,
! [Xa: int,X2: int] :
( ( bit_se5824344872417868541ns_and @ code_integer @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X2 ) )
= ( code_integer_of_int @ ( bit_se5824344872417868541ns_and @ int @ Xa @ X2 ) ) ) ).
% and_integer.abs_eq
thf(fact_3105_bounded__nat__set__is__finite,axiom,
! [N5: set @ nat,N: nat] :
( ! [X3: nat] :
( ( member @ nat @ X3 @ N5 )
=> ( ord_less @ nat @ X3 @ N ) )
=> ( finite_finite2 @ nat @ N5 ) ) ).
% bounded_nat_set_is_finite
thf(fact_3106_finite__nat__set__iff__bounded,axiom,
( ( finite_finite2 @ nat )
= ( ^ [N6: set @ nat] :
? [M2: nat] :
! [X: nat] :
( ( member @ nat @ X @ N6 )
=> ( ord_less @ nat @ X @ M2 ) ) ) ) ).
% finite_nat_set_iff_bounded
thf(fact_3107_sgn__integer_Oabs__eq,axiom,
! [X2: int] :
( ( sgn_sgn @ code_integer @ ( code_integer_of_int @ X2 ) )
= ( code_integer_of_int @ ( sgn_sgn @ int @ X2 ) ) ) ).
% sgn_integer.abs_eq
thf(fact_3108_finite__nat__set__iff__bounded__le,axiom,
( ( finite_finite2 @ nat )
= ( ^ [N6: set @ nat] :
? [M2: nat] :
! [X: nat] :
( ( member @ nat @ X @ N6 )
=> ( ord_less_eq @ nat @ X @ M2 ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_3109_bit__integer_Oabs__eq,axiom,
! [X2: int] :
( ( bit_se5641148757651400278ts_bit @ code_integer @ ( code_integer_of_int @ X2 ) )
= ( bit_se5641148757651400278ts_bit @ int @ X2 ) ) ).
% bit_integer.abs_eq
thf(fact_3110_infinite__Iic,axiom,
! [A: $tType] :
( ( ( linorder @ A )
& ( no_bot @ A ) )
=> ! [A2: A] :
~ ( finite_finite2 @ A @ ( set_ord_atMost @ A @ A2 ) ) ) ).
% infinite_Iic
thf(fact_3111_xor__integer_Oabs__eq,axiom,
! [Xa: int,X2: int] :
( ( bit_se5824344971392196577ns_xor @ code_integer @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X2 ) )
= ( code_integer_of_int @ ( bit_se5824344971392196577ns_xor @ int @ Xa @ X2 ) ) ) ).
% xor_integer.abs_eq
thf(fact_3112_push__bit__integer_Oabs__eq,axiom,
! [Xa: nat,X2: int] :
( ( bit_se4730199178511100633sh_bit @ code_integer @ Xa @ ( code_integer_of_int @ X2 ) )
= ( code_integer_of_int @ ( bit_se4730199178511100633sh_bit @ int @ Xa @ X2 ) ) ) ).
% push_bit_integer.abs_eq
thf(fact_3113_mask__integer_Oabs__eq,axiom,
( ( bit_se2239418461657761734s_mask @ code_integer )
= ( ^ [X: nat] : ( code_integer_of_int @ ( bit_se2239418461657761734s_mask @ int @ X ) ) ) ) ).
% mask_integer.abs_eq
thf(fact_3114_unset__bit__integer_Oabs__eq,axiom,
! [Xa: nat,X2: int] :
( ( bit_se2638667681897837118et_bit @ code_integer @ Xa @ ( code_integer_of_int @ X2 ) )
= ( code_integer_of_int @ ( bit_se2638667681897837118et_bit @ int @ Xa @ X2 ) ) ) ).
% unset_bit_integer.abs_eq
thf(fact_3115_set__bit__integer_Oabs__eq,axiom,
! [Xa: nat,X2: int] :
( ( bit_se5668285175392031749et_bit @ code_integer @ Xa @ ( code_integer_of_int @ X2 ) )
= ( code_integer_of_int @ ( bit_se5668285175392031749et_bit @ int @ Xa @ X2 ) ) ) ).
% set_bit_integer.abs_eq
thf(fact_3116_flip__bit__integer_Oabs__eq,axiom,
! [Xa: nat,X2: int] :
( ( bit_se8732182000553998342ip_bit @ code_integer @ Xa @ ( code_integer_of_int @ X2 ) )
= ( code_integer_of_int @ ( bit_se8732182000553998342ip_bit @ int @ Xa @ X2 ) ) ) ).
% flip_bit_integer.abs_eq
thf(fact_3117_finite__M__bounded__by__nat,axiom,
! [P: nat > $o,I: nat] :
( finite_finite2 @ nat
@ ( collect @ nat
@ ^ [K2: nat] :
( ( P @ K2 )
& ( ord_less @ nat @ K2 @ I ) ) ) ) ).
% finite_M_bounded_by_nat
thf(fact_3118_finite__less__ub,axiom,
! [F: nat > nat,U: nat] :
( ! [N2: nat] : ( ord_less_eq @ nat @ N2 @ ( F @ N2 ) )
=> ( finite_finite2 @ nat
@ ( collect @ nat
@ ^ [N4: nat] : ( ord_less_eq @ nat @ ( F @ N4 ) @ U ) ) ) ) ).
% finite_less_ub
thf(fact_3119_sum_Oswap__restrict,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( comm_monoid_add @ A )
=> ! [A5: set @ B,B5: set @ C,G: B > C > A,R3: B > C > $o] :
( ( finite_finite2 @ B @ A5 )
=> ( ( finite_finite2 @ C @ B5 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X: B] :
( groups7311177749621191930dd_sum @ C @ A @ ( G @ X )
@ ( collect @ C
@ ^ [Y4: C] :
( ( member @ C @ Y4 @ B5 )
& ( R3 @ X @ Y4 ) ) ) )
@ A5 )
= ( groups7311177749621191930dd_sum @ C @ A
@ ^ [Y4: C] :
( groups7311177749621191930dd_sum @ B @ A
@ ^ [X: B] : ( G @ X @ Y4 )
@ ( collect @ B
@ ^ [X: B] :
( ( member @ B @ X @ A5 )
& ( R3 @ X @ Y4 ) ) ) )
@ B5 ) ) ) ) ) ).
% sum.swap_restrict
thf(fact_3120_prod_Oswap__restrict,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A5: set @ B,B5: set @ C,G: B > C > A,R3: B > C > $o] :
( ( finite_finite2 @ B @ A5 )
=> ( ( finite_finite2 @ C @ B5 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [X: B] :
( groups7121269368397514597t_prod @ C @ A @ ( G @ X )
@ ( collect @ C
@ ^ [Y4: C] :
( ( member @ C @ Y4 @ B5 )
& ( R3 @ X @ Y4 ) ) ) )
@ A5 )
= ( groups7121269368397514597t_prod @ C @ A
@ ^ [Y4: C] :
( groups7121269368397514597t_prod @ B @ A
@ ^ [X: B] : ( G @ X @ Y4 )
@ ( collect @ B
@ ^ [X: B] :
( ( member @ B @ X @ A5 )
& ( R3 @ X @ Y4 ) ) ) )
@ B5 ) ) ) ) ) ).
% prod.swap_restrict
thf(fact_3121_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_3122_old_Onat_Osimps_I5_J,axiom,
! [A: $tType,F1: A,F22: nat > A,X22: nat] :
( ( case_nat @ A @ F1 @ F22 @ ( suc @ X22 ) )
= ( F22 @ X22 ) ) ).
% old.nat.simps(5)
thf(fact_3123_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_3124_OR__lower,axiom,
! [X2: int,Y: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X2 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y )
=> ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se1065995026697491101ons_or @ int @ X2 @ Y ) ) ) ) ).
% OR_lower
thf(fact_3125_disjunctive__add,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A,B2: A] :
( ! [N2: nat] :
( ~ ( bit_se5641148757651400278ts_bit @ A @ A2 @ N2 )
| ~ ( bit_se5641148757651400278ts_bit @ A @ B2 @ N2 ) )
=> ( ( plus_plus @ A @ A2 @ B2 )
= ( bit_se1065995026697491101ons_or @ A @ A2 @ B2 ) ) ) ) ).
% disjunctive_add
thf(fact_3126_plus__and__or,axiom,
! [X2: int,Y: int] :
( ( plus_plus @ int @ ( bit_se5824344872417868541ns_and @ int @ X2 @ Y ) @ ( bit_se1065995026697491101ons_or @ int @ X2 @ Y ) )
= ( plus_plus @ int @ X2 @ Y ) ) ).
% plus_and_or
thf(fact_3127_sum__mono__inv,axiom,
! [A: $tType,I6: $tType] :
( ( ordere8940638589300402666id_add @ A )
=> ! [F: I6 > A,I5: set @ I6,G: I6 > A,I: I6] :
( ( ( groups7311177749621191930dd_sum @ I6 @ A @ F @ I5 )
= ( groups7311177749621191930dd_sum @ I6 @ A @ G @ I5 ) )
=> ( ! [I3: I6] :
( ( member @ I6 @ I3 @ I5 )
=> ( ord_less_eq @ A @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ( member @ I6 @ I @ I5 )
=> ( ( finite_finite2 @ I6 @ I5 )
=> ( ( F @ I )
= ( G @ I ) ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_3128_sum_Oreindex__bij__witness__not__neutral,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comm_monoid_add @ A )
=> ! [S7: set @ B,T5: set @ C,S2: set @ B,I: C > B,J: B > C,T6: set @ C,G: B > A,H: C > A] :
( ( finite_finite2 @ B @ S7 )
=> ( ( finite_finite2 @ C @ T5 )
=> ( ! [A4: B] :
( ( member @ B @ A4 @ ( minus_minus @ ( set @ B ) @ S2 @ S7 ) )
=> ( ( I @ ( J @ A4 ) )
= A4 ) )
=> ( ! [A4: B] :
( ( member @ B @ A4 @ ( minus_minus @ ( set @ B ) @ S2 @ S7 ) )
=> ( member @ C @ ( J @ A4 ) @ ( minus_minus @ ( set @ C ) @ T6 @ T5 ) ) )
=> ( ! [B4: C] :
( ( member @ C @ B4 @ ( minus_minus @ ( set @ C ) @ T6 @ T5 ) )
=> ( ( J @ ( I @ B4 ) )
= B4 ) )
=> ( ! [B4: C] :
( ( member @ C @ B4 @ ( minus_minus @ ( set @ C ) @ T6 @ T5 ) )
=> ( member @ B @ ( I @ B4 ) @ ( minus_minus @ ( set @ B ) @ S2 @ S7 ) ) )
=> ( ! [A4: B] :
( ( member @ B @ A4 @ S7 )
=> ( ( G @ A4 )
= ( zero_zero @ A ) ) )
=> ( ! [B4: C] :
( ( member @ C @ B4 @ T5 )
=> ( ( H @ B4 )
= ( zero_zero @ A ) ) )
=> ( ! [A4: B] :
( ( member @ B @ A4 @ S2 )
=> ( ( H @ ( J @ A4 ) )
= ( G @ A4 ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ S2 )
= ( groups7311177749621191930dd_sum @ C @ A @ H @ T6 ) ) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
thf(fact_3129_prod__zero,axiom,
! [B: $tType,A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A5: set @ B,F: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ? [X4: B] :
( ( member @ B @ X4 @ A5 )
& ( ( F @ X4 )
= ( zero_zero @ A ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ F @ A5 )
= ( zero_zero @ A ) ) ) ) ) ).
% prod_zero
thf(fact_3130_infinite__Icc,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ~ ( finite_finite2 @ A @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) ) ) ) ).
% infinite_Icc
thf(fact_3131_infinite__Ico,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ~ ( finite_finite2 @ A @ ( set_or7035219750837199246ssThan @ A @ A2 @ B2 ) ) ) ) ).
% infinite_Ico
thf(fact_3132_nat_Odisc__eq__case_I1_J,axiom,
! [Nat: nat] :
( ( Nat
= ( zero_zero @ nat ) )
= ( case_nat @ $o @ $true
@ ^ [Uu: nat] : $false
@ Nat ) ) ).
% nat.disc_eq_case(1)
thf(fact_3133_nat_Odisc__eq__case_I2_J,axiom,
! [Nat: nat] :
( ( Nat
!= ( zero_zero @ nat ) )
= ( case_nat @ $o @ $false
@ ^ [Uu: nat] : $true
@ Nat ) ) ).
% nat.disc_eq_case(2)
thf(fact_3134_prod_Oreindex__bij__witness__not__neutral,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S7: set @ B,T5: set @ C,S2: set @ B,I: C > B,J: B > C,T6: set @ C,G: B > A,H: C > A] :
( ( finite_finite2 @ B @ S7 )
=> ( ( finite_finite2 @ C @ T5 )
=> ( ! [A4: B] :
( ( member @ B @ A4 @ ( minus_minus @ ( set @ B ) @ S2 @ S7 ) )
=> ( ( I @ ( J @ A4 ) )
= A4 ) )
=> ( ! [A4: B] :
( ( member @ B @ A4 @ ( minus_minus @ ( set @ B ) @ S2 @ S7 ) )
=> ( member @ C @ ( J @ A4 ) @ ( minus_minus @ ( set @ C ) @ T6 @ T5 ) ) )
=> ( ! [B4: C] :
( ( member @ C @ B4 @ ( minus_minus @ ( set @ C ) @ T6 @ T5 ) )
=> ( ( J @ ( I @ B4 ) )
= B4 ) )
=> ( ! [B4: C] :
( ( member @ C @ B4 @ ( minus_minus @ ( set @ C ) @ T6 @ T5 ) )
=> ( member @ B @ ( I @ B4 ) @ ( minus_minus @ ( set @ B ) @ S2 @ S7 ) ) )
=> ( ! [A4: B] :
( ( member @ B @ A4 @ S7 )
=> ( ( G @ A4 )
= ( one_one @ A ) ) )
=> ( ! [B4: C] :
( ( member @ C @ B4 @ T5 )
=> ( ( H @ B4 )
= ( one_one @ A ) ) )
=> ( ! [A4: B] :
( ( member @ B @ A4 @ S2 )
=> ( ( H @ ( J @ A4 ) )
= ( G @ A4 ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ S2 )
= ( groups7121269368397514597t_prod @ C @ A @ H @ T6 ) ) ) ) ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_witness_not_neutral
thf(fact_3135_set__bit__nat__def,axiom,
( ( bit_se5668285175392031749et_bit @ nat )
= ( ^ [M2: nat,N4: nat] : ( bit_se1065995026697491101ons_or @ nat @ N4 @ ( bit_se4730199178511100633sh_bit @ nat @ M2 @ ( one_one @ nat ) ) ) ) ) ).
% set_bit_nat_def
thf(fact_3136_sum_Ofinite__Collect__op,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [I5: set @ B,X2: B > A,Y: B > A] :
( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [I2: B] :
( ( member @ B @ I2 @ I5 )
& ( ( X2 @ I2 )
!= ( zero_zero @ A ) ) ) ) )
=> ( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [I2: B] :
( ( member @ B @ I2 @ I5 )
& ( ( Y @ I2 )
!= ( zero_zero @ A ) ) ) ) )
=> ( finite_finite2 @ B
@ ( collect @ B
@ ^ [I2: B] :
( ( member @ B @ I2 @ I5 )
& ( ( plus_plus @ A @ ( X2 @ I2 ) @ ( Y @ I2 ) )
!= ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_3137_prod_Ofinite__Collect__op,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [I5: set @ B,X2: B > A,Y: B > A] :
( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [I2: B] :
( ( member @ B @ I2 @ I5 )
& ( ( X2 @ I2 )
!= ( one_one @ A ) ) ) ) )
=> ( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [I2: B] :
( ( member @ B @ I2 @ I5 )
& ( ( Y @ I2 )
!= ( one_one @ A ) ) ) ) )
=> ( finite_finite2 @ B
@ ( collect @ B
@ ^ [I2: B] :
( ( member @ B @ I2 @ I5 )
& ( ( times_times @ A @ ( X2 @ I2 ) @ ( Y @ I2 ) )
!= ( one_one @ A ) ) ) ) ) ) ) ) ).
% prod.finite_Collect_op
thf(fact_3138_sum_Osetdiff__irrelevant,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A5: set @ B,G: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G
@ ( minus_minus @ ( set @ B ) @ A5
@ ( collect @ B
@ ^ [X: B] :
( ( G @ X )
= ( zero_zero @ A ) ) ) ) )
= ( groups7311177749621191930dd_sum @ B @ A @ G @ A5 ) ) ) ) ).
% sum.setdiff_irrelevant
thf(fact_3139_sum_Ointer__filter,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A5: set @ B,G: B > A,P: B > $o] :
( ( finite_finite2 @ B @ A5 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G
@ ( collect @ B
@ ^ [X: B] :
( ( member @ B @ X @ A5 )
& ( P @ X ) ) ) )
= ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X: B] : ( if @ A @ ( P @ X ) @ ( G @ X ) @ ( zero_zero @ A ) )
@ A5 ) ) ) ) ).
% sum.inter_filter
thf(fact_3140_filter__preserves__multiset,axiom,
! [A: $tType,M3: A > nat,P: A > $o] :
( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( M3 @ X ) ) ) )
=> ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( if @ nat @ ( P @ X ) @ ( M3 @ X ) @ ( zero_zero @ nat ) ) ) ) ) ) ).
% filter_preserves_multiset
thf(fact_3141_prod_Ointer__filter,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A5: set @ B,G: B > A,P: B > $o] :
( ( finite_finite2 @ B @ A5 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G
@ ( collect @ B
@ ^ [X: B] :
( ( member @ B @ X @ A5 )
& ( P @ X ) ) ) )
= ( groups7121269368397514597t_prod @ B @ A
@ ^ [X: B] : ( if @ A @ ( P @ X ) @ ( G @ X ) @ ( one_one @ A ) )
@ A5 ) ) ) ) ).
% prod.inter_filter
thf(fact_3142_prod_Osetdiff__irrelevant,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A5: set @ B,G: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G
@ ( minus_minus @ ( set @ B ) @ A5
@ ( collect @ B
@ ^ [X: B] :
( ( G @ X )
= ( one_one @ A ) ) ) ) )
= ( groups7121269368397514597t_prod @ B @ A @ G @ A5 ) ) ) ) ).
% prod.setdiff_irrelevant
thf(fact_3143_finite__int__segment,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [A2: A,B2: A] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X: A] :
( ( member @ A @ X @ ( ring_1_Ints @ A ) )
& ( ord_less_eq @ A @ A2 @ X )
& ( ord_less_eq @ A @ X @ B2 ) ) ) ) ) ).
% finite_int_segment
thf(fact_3144_or__not__numerals_I1_J,axiom,
( ( bit_se1065995026697491101ons_or @ int @ ( one_one @ int ) @ ( bit_ri4277139882892585799ns_not @ int @ ( one_one @ int ) ) )
= ( bit_ri4277139882892585799ns_not @ int @ ( zero_zero @ int ) ) ) ).
% or_not_numerals(1)
thf(fact_3145_atLeastLessThanPlusOne__atLeastAtMost__int,axiom,
! [L: int,U: int] :
( ( set_or7035219750837199246ssThan @ int @ L @ ( plus_plus @ int @ U @ ( one_one @ int ) ) )
= ( set_or1337092689740270186AtMost @ int @ L @ U ) ) ).
% atLeastLessThanPlusOne_atLeastAtMost_int
thf(fact_3146_sum__nonneg__eq__0__iff,axiom,
! [A: $tType,B: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A5: set @ B,F: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ A5 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F @ X3 ) ) )
=> ( ( ( groups7311177749621191930dd_sum @ B @ A @ F @ A5 )
= ( zero_zero @ A ) )
= ( ! [X: B] :
( ( member @ B @ X @ A5 )
=> ( ( F @ X )
= ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_3147_sum__le__included,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [S3: set @ B,T2: set @ C,G: C > A,I: C > B,F: B > A] :
( ( finite_finite2 @ B @ S3 )
=> ( ( finite_finite2 @ C @ T2 )
=> ( ! [X3: C] :
( ( member @ C @ X3 @ T2 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( G @ X3 ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ S3 )
=> ? [Xa2: C] :
( ( member @ C @ Xa2 @ T2 )
& ( ( I @ Xa2 )
= X3 )
& ( ord_less_eq @ A @ ( F @ X3 ) @ ( G @ Xa2 ) ) ) )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F @ S3 ) @ ( groups7311177749621191930dd_sum @ C @ A @ G @ T2 ) ) ) ) ) ) ) ).
% sum_le_included
thf(fact_3148_sum__strict__mono__ex1,axiom,
! [A: $tType,I6: $tType] :
( ( ordere8940638589300402666id_add @ A )
=> ! [A5: set @ I6,F: I6 > A,G: I6 > A] :
( ( finite_finite2 @ I6 @ A5 )
=> ( ! [X3: I6] :
( ( member @ I6 @ X3 @ A5 )
=> ( ord_less_eq @ A @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ? [X4: I6] :
( ( member @ I6 @ X4 @ A5 )
& ( ord_less @ A @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less @ A @ ( groups7311177749621191930dd_sum @ I6 @ A @ F @ A5 ) @ ( groups7311177749621191930dd_sum @ I6 @ A @ G @ A5 ) ) ) ) ) ) ).
% sum_strict_mono_ex1
thf(fact_3149_sum_Orelated,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [R3: A > A > $o,S2: set @ B,H: B > A,G: B > A] :
( ( R3 @ ( zero_zero @ A ) @ ( zero_zero @ A ) )
=> ( ! [X1: A,Y1: A,X23: A,Y23: A] :
( ( ( R3 @ X1 @ X23 )
& ( R3 @ Y1 @ Y23 ) )
=> ( R3 @ ( plus_plus @ A @ X1 @ Y1 ) @ ( plus_plus @ A @ X23 @ Y23 ) ) )
=> ( ( finite_finite2 @ B @ S2 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ S2 )
=> ( R3 @ ( H @ X3 ) @ ( G @ X3 ) ) )
=> ( R3 @ ( groups7311177749621191930dd_sum @ B @ A @ H @ S2 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ S2 ) ) ) ) ) ) ) ).
% sum.related
thf(fact_3150_sum_Osame__carrier,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [C6: set @ B,A5: set @ B,B5: set @ B,G: B > A,H: B > A] :
( ( finite_finite2 @ B @ C6 )
=> ( ( ord_less_eq @ ( set @ B ) @ A5 @ C6 )
=> ( ( ord_less_eq @ ( set @ B ) @ B5 @ C6 )
=> ( ! [A4: B] :
( ( member @ B @ A4 @ ( minus_minus @ ( set @ B ) @ C6 @ A5 ) )
=> ( ( G @ A4 )
= ( zero_zero @ A ) ) )
=> ( ! [B4: B] :
( ( member @ B @ B4 @ ( minus_minus @ ( set @ B ) @ C6 @ B5 ) )
=> ( ( H @ B4 )
= ( zero_zero @ A ) ) )
=> ( ( ( groups7311177749621191930dd_sum @ B @ A @ G @ A5 )
= ( groups7311177749621191930dd_sum @ B @ A @ H @ B5 ) )
= ( ( groups7311177749621191930dd_sum @ B @ A @ G @ C6 )
= ( groups7311177749621191930dd_sum @ B @ A @ H @ C6 ) ) ) ) ) ) ) ) ) ).
% sum.same_carrier
thf(fact_3151_sum_Osame__carrierI,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [C6: set @ B,A5: set @ B,B5: set @ B,G: B > A,H: B > A] :
( ( finite_finite2 @ B @ C6 )
=> ( ( ord_less_eq @ ( set @ B ) @ A5 @ C6 )
=> ( ( ord_less_eq @ ( set @ B ) @ B5 @ C6 )
=> ( ! [A4: B] :
( ( member @ B @ A4 @ ( minus_minus @ ( set @ B ) @ C6 @ A5 ) )
=> ( ( G @ A4 )
= ( zero_zero @ A ) ) )
=> ( ! [B4: B] :
( ( member @ B @ B4 @ ( minus_minus @ ( set @ B ) @ C6 @ B5 ) )
=> ( ( H @ B4 )
= ( zero_zero @ A ) ) )
=> ( ( ( groups7311177749621191930dd_sum @ B @ A @ G @ C6 )
= ( groups7311177749621191930dd_sum @ B @ A @ H @ C6 ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ A5 )
= ( groups7311177749621191930dd_sum @ B @ A @ H @ B5 ) ) ) ) ) ) ) ) ) ).
% sum.same_carrierI
thf(fact_3152_sum_Omono__neutral__left,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [T6: set @ B,S2: set @ B,G: B > A] :
( ( finite_finite2 @ B @ T6 )
=> ( ( ord_less_eq @ ( set @ B ) @ S2 @ T6 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T6 @ S2 ) )
=> ( ( G @ X3 )
= ( zero_zero @ A ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ S2 )
= ( groups7311177749621191930dd_sum @ B @ A @ G @ T6 ) ) ) ) ) ) ).
% sum.mono_neutral_left
thf(fact_3153_sum_Omono__neutral__right,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [T6: set @ B,S2: set @ B,G: B > A] :
( ( finite_finite2 @ B @ T6 )
=> ( ( ord_less_eq @ ( set @ B ) @ S2 @ T6 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T6 @ S2 ) )
=> ( ( G @ X3 )
= ( zero_zero @ A ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ T6 )
= ( groups7311177749621191930dd_sum @ B @ A @ G @ S2 ) ) ) ) ) ) ).
% sum.mono_neutral_right
thf(fact_3154_sum_Omono__neutral__cong__left,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [T6: set @ B,S2: set @ B,H: B > A,G: B > A] :
( ( finite_finite2 @ B @ T6 )
=> ( ( ord_less_eq @ ( set @ B ) @ S2 @ T6 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T6 @ S2 ) )
=> ( ( H @ X3 )
= ( zero_zero @ A ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ S2 )
=> ( ( G @ X3 )
= ( H @ X3 ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ S2 )
= ( groups7311177749621191930dd_sum @ B @ A @ H @ T6 ) ) ) ) ) ) ) ).
% sum.mono_neutral_cong_left
thf(fact_3155_sum_Omono__neutral__cong__right,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [T6: set @ B,S2: set @ B,G: B > A,H: B > A] :
( ( finite_finite2 @ B @ T6 )
=> ( ( ord_less_eq @ ( set @ B ) @ S2 @ T6 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T6 @ S2 ) )
=> ( ( G @ X3 )
= ( zero_zero @ A ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ S2 )
=> ( ( G @ X3 )
= ( H @ X3 ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ T6 )
= ( groups7311177749621191930dd_sum @ B @ A @ H @ S2 ) ) ) ) ) ) ) ).
% sum.mono_neutral_cong_right
thf(fact_3156_prod_Orelated,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [R3: A > A > $o,S2: set @ B,H: B > A,G: B > A] :
( ( R3 @ ( one_one @ A ) @ ( one_one @ A ) )
=> ( ! [X1: A,Y1: A,X23: A,Y23: A] :
( ( ( R3 @ X1 @ X23 )
& ( R3 @ Y1 @ Y23 ) )
=> ( R3 @ ( times_times @ A @ X1 @ Y1 ) @ ( times_times @ A @ X23 @ Y23 ) ) )
=> ( ( finite_finite2 @ B @ S2 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ S2 )
=> ( R3 @ ( H @ X3 ) @ ( G @ X3 ) ) )
=> ( R3 @ ( groups7121269368397514597t_prod @ B @ A @ H @ S2 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ S2 ) ) ) ) ) ) ) ).
% prod.related
thf(fact_3157_sum_Osubset__diff,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [B5: set @ B,A5: set @ B,G: B > A] :
( ( ord_less_eq @ ( set @ B ) @ B5 @ A5 )
=> ( ( finite_finite2 @ B @ A5 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ A5 )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A5 @ B5 ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ B5 ) ) ) ) ) ) ).
% sum.subset_diff
thf(fact_3158_prod_Omono__neutral__cong__right,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [T6: set @ B,S2: set @ B,G: B > A,H: B > A] :
( ( finite_finite2 @ B @ T6 )
=> ( ( ord_less_eq @ ( set @ B ) @ S2 @ T6 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T6 @ S2 ) )
=> ( ( G @ X3 )
= ( one_one @ A ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ S2 )
=> ( ( G @ X3 )
= ( H @ X3 ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ T6 )
= ( groups7121269368397514597t_prod @ B @ A @ H @ S2 ) ) ) ) ) ) ) ).
% prod.mono_neutral_cong_right
thf(fact_3159_prod_Omono__neutral__cong__left,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [T6: set @ B,S2: set @ B,H: B > A,G: B > A] :
( ( finite_finite2 @ B @ T6 )
=> ( ( ord_less_eq @ ( set @ B ) @ S2 @ T6 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T6 @ S2 ) )
=> ( ( H @ X3 )
= ( one_one @ A ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ S2 )
=> ( ( G @ X3 )
= ( H @ X3 ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ S2 )
= ( groups7121269368397514597t_prod @ B @ A @ H @ T6 ) ) ) ) ) ) ) ).
% prod.mono_neutral_cong_left
thf(fact_3160_prod_Omono__neutral__right,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [T6: set @ B,S2: set @ B,G: B > A] :
( ( finite_finite2 @ B @ T6 )
=> ( ( ord_less_eq @ ( set @ B ) @ S2 @ T6 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T6 @ S2 ) )
=> ( ( G @ X3 )
= ( one_one @ A ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ T6 )
= ( groups7121269368397514597t_prod @ B @ A @ G @ S2 ) ) ) ) ) ) ).
% prod.mono_neutral_right
thf(fact_3161_prod_Omono__neutral__left,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [T6: set @ B,S2: set @ B,G: B > A] :
( ( finite_finite2 @ B @ T6 )
=> ( ( ord_less_eq @ ( set @ B ) @ S2 @ T6 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T6 @ S2 ) )
=> ( ( G @ X3 )
= ( one_one @ A ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ S2 )
= ( groups7121269368397514597t_prod @ B @ A @ G @ T6 ) ) ) ) ) ) ).
% prod.mono_neutral_left
thf(fact_3162_prod_Osame__carrierI,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [C6: set @ B,A5: set @ B,B5: set @ B,G: B > A,H: B > A] :
( ( finite_finite2 @ B @ C6 )
=> ( ( ord_less_eq @ ( set @ B ) @ A5 @ C6 )
=> ( ( ord_less_eq @ ( set @ B ) @ B5 @ C6 )
=> ( ! [A4: B] :
( ( member @ B @ A4 @ ( minus_minus @ ( set @ B ) @ C6 @ A5 ) )
=> ( ( G @ A4 )
= ( one_one @ A ) ) )
=> ( ! [B4: B] :
( ( member @ B @ B4 @ ( minus_minus @ ( set @ B ) @ C6 @ B5 ) )
=> ( ( H @ B4 )
= ( one_one @ A ) ) )
=> ( ( ( groups7121269368397514597t_prod @ B @ A @ G @ C6 )
= ( groups7121269368397514597t_prod @ B @ A @ H @ C6 ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ A5 )
= ( groups7121269368397514597t_prod @ B @ A @ H @ B5 ) ) ) ) ) ) ) ) ) ).
% prod.same_carrierI
thf(fact_3163_prod_Osame__carrier,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [C6: set @ B,A5: set @ B,B5: set @ B,G: B > A,H: B > A] :
( ( finite_finite2 @ B @ C6 )
=> ( ( ord_less_eq @ ( set @ B ) @ A5 @ C6 )
=> ( ( ord_less_eq @ ( set @ B ) @ B5 @ C6 )
=> ( ! [A4: B] :
( ( member @ B @ A4 @ ( minus_minus @ ( set @ B ) @ C6 @ A5 ) )
=> ( ( G @ A4 )
= ( one_one @ A ) ) )
=> ( ! [B4: B] :
( ( member @ B @ B4 @ ( minus_minus @ ( set @ B ) @ C6 @ B5 ) )
=> ( ( H @ B4 )
= ( one_one @ A ) ) )
=> ( ( ( groups7121269368397514597t_prod @ B @ A @ G @ A5 )
= ( groups7121269368397514597t_prod @ B @ A @ H @ B5 ) )
= ( ( groups7121269368397514597t_prod @ B @ A @ G @ C6 )
= ( groups7121269368397514597t_prod @ B @ A @ H @ C6 ) ) ) ) ) ) ) ) ) ).
% prod.same_carrier
thf(fact_3164_set__bit__eq__or,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se5668285175392031749et_bit @ A )
= ( ^ [N4: nat,A3: A] : ( bit_se1065995026697491101ons_or @ A @ A3 @ ( bit_se4730199178511100633sh_bit @ A @ N4 @ ( one_one @ A ) ) ) ) ) ) ).
% set_bit_eq_or
thf(fact_3165_sum__eq__Suc0__iff,axiom,
! [A: $tType,A5: set @ A,F: A > nat] :
( ( finite_finite2 @ A @ A5 )
=> ( ( ( groups7311177749621191930dd_sum @ A @ nat @ F @ A5 )
= ( suc @ ( zero_zero @ nat ) ) )
= ( ? [X: A] :
( ( member @ A @ X @ A5 )
& ( ( F @ X )
= ( suc @ ( zero_zero @ nat ) ) )
& ! [Y4: A] :
( ( member @ A @ Y4 @ A5 )
=> ( ( X != Y4 )
=> ( ( F @ Y4 )
= ( zero_zero @ nat ) ) ) ) ) ) ) ) ).
% sum_eq_Suc0_iff
thf(fact_3166_sum__eq__1__iff,axiom,
! [A: $tType,A5: set @ A,F: A > nat] :
( ( finite_finite2 @ A @ A5 )
=> ( ( ( groups7311177749621191930dd_sum @ A @ nat @ F @ A5 )
= ( one_one @ nat ) )
= ( ? [X: A] :
( ( member @ A @ X @ A5 )
& ( ( F @ X )
= ( one_one @ nat ) )
& ! [Y4: A] :
( ( member @ A @ Y4 @ A5 )
=> ( ( X != Y4 )
=> ( ( F @ Y4 )
= ( zero_zero @ nat ) ) ) ) ) ) ) ) ).
% sum_eq_1_iff
thf(fact_3167_prod__int__eq,axiom,
! [I: nat,J: nat] :
( ( groups7121269368397514597t_prod @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( set_or1337092689740270186AtMost @ nat @ I @ J ) )
= ( groups7121269368397514597t_prod @ int @ int
@ ^ [X: int] : X
@ ( set_or1337092689740270186AtMost @ int @ ( semiring_1_of_nat @ int @ I ) @ ( semiring_1_of_nat @ int @ J ) ) ) ) ).
% prod_int_eq
thf(fact_3168_sum__nonneg__0,axiom,
! [B: $tType,A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [S3: set @ B,F: B > A,I: B] :
( ( finite_finite2 @ B @ S3 )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ S3 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F @ I3 ) ) )
=> ( ( ( groups7311177749621191930dd_sum @ B @ A @ F @ S3 )
= ( zero_zero @ A ) )
=> ( ( member @ B @ I @ S3 )
=> ( ( F @ I )
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_3169_sum__nonneg__leq__bound,axiom,
! [B: $tType,A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [S3: set @ B,F: B > A,B5: A,I: B] :
( ( finite_finite2 @ B @ S3 )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ S3 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F @ I3 ) ) )
=> ( ( ( groups7311177749621191930dd_sum @ B @ A @ F @ S3 )
= B5 )
=> ( ( member @ B @ I @ S3 )
=> ( ord_less_eq @ A @ ( F @ I ) @ B5 ) ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_3170_set__bit__int__def,axiom,
( ( bit_se5668285175392031749et_bit @ int )
= ( ^ [N4: nat,K2: int] : ( bit_se1065995026697491101ons_or @ int @ K2 @ ( bit_se4730199178511100633sh_bit @ int @ N4 @ ( one_one @ int ) ) ) ) ) ).
% set_bit_int_def
thf(fact_3171_add__mset__in__multiset,axiom,
! [A: $tType,M3: A > nat,A2: A] :
( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( M3 @ X ) ) ) )
=> ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( if @ nat @ ( X = A2 ) @ ( suc @ ( M3 @ X ) ) @ ( M3 @ X ) ) ) ) ) ) ).
% add_mset_in_multiset
thf(fact_3172_less__eq__nat_Osimps_I2_J,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ ( suc @ M ) @ N )
= ( case_nat @ $o @ $false @ ( ord_less_eq @ nat @ M ) @ N ) ) ).
% less_eq_nat.simps(2)
thf(fact_3173_diff__preserves__multiset,axiom,
! [A: $tType,M3: A > nat,N5: A > nat] :
( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( M3 @ X ) ) ) )
=> ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ ( M3 @ X ) @ ( N5 @ X ) ) ) ) ) ) ).
% diff_preserves_multiset
thf(fact_3174_finite__divisors__nat,axiom,
! [M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( finite_finite2 @ nat
@ ( collect @ nat
@ ^ [D5: nat] : ( dvd_dvd @ nat @ D5 @ M ) ) ) ) ).
% finite_divisors_nat
thf(fact_3175_subset__eq__atLeast0__atMost__finite,axiom,
! [N5: set @ nat,N: nat] :
( ( ord_less_eq @ ( set @ nat ) @ N5 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
=> ( finite_finite2 @ nat @ N5 ) ) ).
% subset_eq_atLeast0_atMost_finite
thf(fact_3176_subset__eq__atLeast0__lessThan__finite,axiom,
! [N5: set @ nat,N: nat] :
( ( ord_less_eq @ ( set @ nat ) @ N5 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) )
=> ( finite_finite2 @ nat @ N5 ) ) ).
% subset_eq_atLeast0_lessThan_finite
thf(fact_3177_finite__abs__int__segment,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [A2: A] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [K2: A] :
( ( member @ A @ K2 @ ( ring_1_Ints @ A ) )
& ( ord_less_eq @ A @ ( abs_abs @ A @ K2 ) @ A2 ) ) ) ) ) ).
% finite_abs_int_segment
thf(fact_3178_bit_Ocomplement__unique,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A2: A,X2: A,Y: A] :
( ( ( bit_se5824344872417868541ns_and @ A @ A2 @ X2 )
= ( zero_zero @ A ) )
=> ( ( ( bit_se1065995026697491101ons_or @ A @ A2 @ X2 )
= ( uminus_uminus @ A @ ( one_one @ A ) ) )
=> ( ( ( bit_se5824344872417868541ns_and @ A @ A2 @ Y )
= ( zero_zero @ A ) )
=> ( ( ( bit_se1065995026697491101ons_or @ A @ A2 @ Y )
= ( uminus_uminus @ A @ ( one_one @ A ) ) )
=> ( X2 = Y ) ) ) ) ) ) ).
% bit.complement_unique
thf(fact_3179_or__not__numerals_I2_J,axiom,
! [N: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( one_one @ int ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) )
= ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) ) ).
% or_not_numerals(2)
thf(fact_3180_or__not__numerals_I4_J,axiom,
! [M: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ ( bit0 @ M ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( one_one @ int ) ) )
= ( bit_ri4277139882892585799ns_not @ int @ ( one_one @ int ) ) ) ).
% or_not_numerals(4)
thf(fact_3181_sum__pos2,axiom,
! [A: $tType,B: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [I5: set @ B,I: B,F: B > A] :
( ( finite_finite2 @ B @ I5 )
=> ( ( member @ B @ I @ I5 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( F @ I ) )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ I5 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F @ I3 ) ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( groups7311177749621191930dd_sum @ B @ A @ F @ I5 ) ) ) ) ) ) ) ).
% sum_pos2
thf(fact_3182_nth__Cons,axiom,
! [A: $tType,X2: A,Xs: list @ A,N: nat] :
( ( nth @ A @ ( cons @ A @ X2 @ Xs ) @ N )
= ( case_nat @ A @ X2 @ ( nth @ A @ Xs ) @ N ) ) ).
% nth_Cons
thf(fact_3183_sum__mono2,axiom,
! [A: $tType,B: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [B5: set @ B,A5: set @ B,F: B > A] :
( ( finite_finite2 @ B @ B5 )
=> ( ( ord_less_eq @ ( set @ B ) @ A5 @ B5 )
=> ( ! [B4: B] :
( ( member @ B @ B4 @ ( minus_minus @ ( set @ B ) @ B5 @ A5 ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F @ B4 ) ) )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F @ A5 ) @ ( groups7311177749621191930dd_sum @ B @ A @ F @ B5 ) ) ) ) ) ) ).
% sum_mono2
thf(fact_3184_less__1__prod2,axiom,
! [B: $tType,A: $tType] :
( ( linordered_idom @ B )
=> ! [I5: set @ A,I: A,F: A > B] :
( ( finite_finite2 @ A @ I5 )
=> ( ( member @ A @ I @ I5 )
=> ( ( ord_less @ B @ ( one_one @ B ) @ ( F @ I ) )
=> ( ! [I3: A] :
( ( member @ A @ I3 @ I5 )
=> ( ord_less_eq @ B @ ( one_one @ B ) @ ( F @ I3 ) ) )
=> ( ord_less @ B @ ( one_one @ B ) @ ( groups7121269368397514597t_prod @ A @ B @ F @ I5 ) ) ) ) ) ) ) ).
% less_1_prod2
thf(fact_3185_prod__int__plus__eq,axiom,
! [I: nat,J: nat] :
( ( groups7121269368397514597t_prod @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( set_or1337092689740270186AtMost @ nat @ I @ ( plus_plus @ nat @ I @ J ) ) )
= ( groups7121269368397514597t_prod @ int @ int
@ ^ [X: int] : X
@ ( set_or1337092689740270186AtMost @ int @ ( semiring_1_of_nat @ int @ I ) @ ( semiring_1_of_nat @ int @ ( plus_plus @ nat @ I @ J ) ) ) ) ) ).
% prod_int_plus_eq
thf(fact_3186_diff__Suc,axiom,
! [M: nat,N: nat] :
( ( minus_minus @ nat @ M @ ( suc @ N ) )
= ( case_nat @ nat @ ( zero_zero @ nat )
@ ^ [K2: nat] : K2
@ ( minus_minus @ nat @ M @ N ) ) ) ).
% diff_Suc
thf(fact_3187_bit_Ocompl__unique,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X2: A,Y: A] :
( ( ( bit_se5824344872417868541ns_and @ A @ X2 @ Y )
= ( zero_zero @ A ) )
=> ( ( ( bit_se1065995026697491101ons_or @ A @ X2 @ Y )
= ( uminus_uminus @ A @ ( one_one @ A ) ) )
=> ( ( bit_ri4277139882892585799ns_not @ A @ X2 )
= Y ) ) ) ) ).
% bit.compl_unique
thf(fact_3188_or__not__numerals_I3_J,axiom,
! [N: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( one_one @ int ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) )
= ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) ) ).
% or_not_numerals(3)
thf(fact_3189_or__not__numerals_I7_J,axiom,
! [M: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ ( bit1 @ M ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( one_one @ int ) ) )
= ( bit_ri4277139882892585799ns_not @ int @ ( zero_zero @ int ) ) ) ).
% or_not_numerals(7)
thf(fact_3190_bit__numeral__rec_I1_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [W2: num,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ ( bit0 @ W2 ) ) @ N )
= ( case_nat @ $o @ $false @ ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ W2 ) ) @ N ) ) ) ).
% bit_numeral_rec(1)
thf(fact_3191_bit__numeral__rec_I2_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [W2: num,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ ( bit1 @ W2 ) ) @ N )
= ( case_nat @ $o @ $true @ ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ W2 ) ) @ N ) ) ) ).
% bit_numeral_rec(2)
thf(fact_3192_Nitpick_Ocase__nat__unfold,axiom,
! [A: $tType] :
( ( case_nat @ A )
= ( ^ [X: A,F2: nat > A,N4: nat] :
( if @ A
@ ( N4
= ( zero_zero @ nat ) )
@ X
@ ( F2 @ ( minus_minus @ nat @ N4 @ ( one_one @ nat ) ) ) ) ) ) ).
% Nitpick.case_nat_unfold
thf(fact_3193_sum__strict__mono2,axiom,
! [B: $tType,A: $tType] :
( ( ordere8940638589300402666id_add @ B )
=> ! [B5: set @ A,A5: set @ A,B2: A,F: A > B] :
( ( finite_finite2 @ A @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
=> ( ( member @ A @ B2 @ ( minus_minus @ ( set @ A ) @ B5 @ A5 ) )
=> ( ( ord_less @ B @ ( zero_zero @ B ) @ ( F @ B2 ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ B5 )
=> ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( F @ X3 ) ) )
=> ( ord_less @ B @ ( groups7311177749621191930dd_sum @ A @ B @ F @ A5 ) @ ( groups7311177749621191930dd_sum @ A @ B @ F @ B5 ) ) ) ) ) ) ) ) ).
% sum_strict_mono2
thf(fact_3194_prod__mono2,axiom,
! [B: $tType,A: $tType] :
( ( linordered_idom @ B )
=> ! [B5: set @ A,A5: set @ A,F: A > B] :
( ( finite_finite2 @ A @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
=> ( ! [B4: A] :
( ( member @ A @ B4 @ ( minus_minus @ ( set @ A ) @ B5 @ A5 ) )
=> ( ord_less_eq @ B @ ( one_one @ B ) @ ( F @ B4 ) ) )
=> ( ! [A4: A] :
( ( member @ A @ A4 @ A5 )
=> ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( F @ A4 ) ) )
=> ( ord_less_eq @ B @ ( groups7121269368397514597t_prod @ A @ B @ F @ A5 ) @ ( groups7121269368397514597t_prod @ A @ B @ F @ B5 ) ) ) ) ) ) ) ).
% prod_mono2
thf(fact_3195_one__or__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A] :
( ( bit_se1065995026697491101ons_or @ A @ ( one_one @ A ) @ A2 )
= ( plus_plus @ A @ A2 @ ( zero_neq_one_of_bool @ A @ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) ) ) ) ).
% one_or_eq
thf(fact_3196_or__one__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A] :
( ( bit_se1065995026697491101ons_or @ A @ A2 @ ( one_one @ A ) )
= ( plus_plus @ A @ A2 @ ( zero_neq_one_of_bool @ A @ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) ) ) ) ).
% or_one_eq
thf(fact_3197_mask__Suc__double,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat] :
( ( bit_se2239418461657761734s_mask @ A @ ( suc @ N ) )
= ( bit_se1065995026697491101ons_or @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se2239418461657761734s_mask @ A @ N ) ) ) ) ) ).
% mask_Suc_double
thf(fact_3198_OR__upper,axiom,
! [X2: int,N: nat,Y: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X2 )
=> ( ( ord_less @ int @ X2 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( ord_less @ int @ Y @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) )
=> ( ord_less @ int @ ( bit_se1065995026697491101ons_or @ int @ X2 @ Y ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ).
% OR_upper
thf(fact_3199_or__not__numerals_I5_J,axiom,
! [M: num,N: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ ( bit0 @ M ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) )
= ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ M ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N ) ) ) ) ) ) ).
% or_not_numerals(5)
thf(fact_3200_or__Suc__0__eq,axiom,
! [N: nat] :
( ( bit_se1065995026697491101ons_or @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) )
= ( plus_plus @ nat @ N @ ( zero_neq_one_of_bool @ nat @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% or_Suc_0_eq
thf(fact_3201_Suc__0__or__eq,axiom,
! [N: nat] :
( ( bit_se1065995026697491101ons_or @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N )
= ( plus_plus @ nat @ N @ ( zero_neq_one_of_bool @ nat @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% Suc_0_or_eq
thf(fact_3202_or__not__numerals_I9_J,axiom,
! [M: num,N: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ ( bit1 @ M ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) )
= ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ M ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N ) ) ) ) ) ) ).
% or_not_numerals(9)
thf(fact_3203_or__not__numerals_I8_J,axiom,
! [M: num,N: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ ( bit1 @ M ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) )
= ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ M ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N ) ) ) ) ) ) ).
% or_not_numerals(8)
thf(fact_3204_or__nat__rec,axiom,
( ( bit_se1065995026697491101ons_or @ nat )
= ( ^ [M2: nat,N4: nat] :
( plus_plus @ nat
@ ( zero_neq_one_of_bool @ nat
@ ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M2 )
| ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N4 ) ) )
@ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ nat @ ( divide_divide @ nat @ M2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% or_nat_rec
thf(fact_3205_finite__Collect__le__nat,axiom,
! [K: nat] :
( finite_finite2 @ nat
@ ( collect @ nat
@ ^ [N4: nat] : ( ord_less_eq @ nat @ N4 @ K ) ) ) ).
% finite_Collect_le_nat
thf(fact_3206_finite__Collect__less__nat,axiom,
! [K: nat] :
( finite_finite2 @ nat
@ ( collect @ nat
@ ^ [N4: nat] : ( ord_less @ nat @ N4 @ K ) ) ) ).
% finite_Collect_less_nat
thf(fact_3207_finite__Collect__subsets,axiom,
! [A: $tType,A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( finite_finite2 @ ( set @ A )
@ ( collect @ ( set @ A )
@ ^ [B8: set @ A] : ( ord_less_eq @ ( set @ A ) @ B8 @ A5 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_3208_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
@ ^ [X: A] :
( ( P @ X )
& ( Q @ X ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_3209_finite__Collect__disjI,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X: A] :
( ( P @ X )
| ( Q @ X ) ) ) )
= ( ( finite_finite2 @ A @ ( collect @ A @ P ) )
& ( finite_finite2 @ A @ ( collect @ A @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_3210_finite__atLeastAtMost__int,axiom,
! [L: int,U: int] : ( finite_finite2 @ int @ ( set_or1337092689740270186AtMost @ int @ L @ U ) ) ).
% finite_atLeastAtMost_int
thf(fact_3211_finite__atLeastLessThan__int,axiom,
! [L: int,U: int] : ( finite_finite2 @ int @ ( set_or7035219750837199246ssThan @ int @ L @ U ) ) ).
% finite_atLeastLessThan_int
thf(fact_3212_finite__interval__int1,axiom,
! [A2: int,B2: int] :
( finite_finite2 @ int
@ ( collect @ int
@ ^ [I2: int] :
( ( ord_less_eq @ int @ A2 @ I2 )
& ( ord_less_eq @ int @ I2 @ B2 ) ) ) ) ).
% finite_interval_int1
thf(fact_3213_finite__interval__int4,axiom,
! [A2: int,B2: int] :
( finite_finite2 @ int
@ ( collect @ int
@ ^ [I2: int] :
( ( ord_less @ int @ A2 @ I2 )
& ( ord_less @ int @ I2 @ B2 ) ) ) ) ).
% finite_interval_int4
thf(fact_3214_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_3215_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_3216_finite__interval__int3,axiom,
! [A2: int,B2: int] :
( finite_finite2 @ int
@ ( collect @ int
@ ^ [I2: int] :
( ( ord_less @ int @ A2 @ I2 )
& ( ord_less_eq @ int @ I2 @ B2 ) ) ) ) ).
% finite_interval_int3
thf(fact_3217_finite__interval__int2,axiom,
! [A2: int,B2: int] :
( finite_finite2 @ int
@ ( collect @ int
@ ^ [I2: int] :
( ( ord_less_eq @ int @ A2 @ I2 )
& ( ord_less @ int @ I2 @ B2 ) ) ) ) ).
% finite_interval_int2
thf(fact_3218_finite__atLeastZeroLessThan__int,axiom,
! [U: int] : ( finite_finite2 @ int @ ( set_or7035219750837199246ssThan @ int @ ( zero_zero @ int ) @ U ) ) ).
% finite_atLeastZeroLessThan_int
thf(fact_3219_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_3220_finite__maxlen,axiom,
! [A: $tType,M3: set @ ( list @ A )] :
( ( finite_finite2 @ ( list @ A ) @ M3 )
=> ? [N2: nat] :
! [X4: list @ A] :
( ( member @ ( list @ A ) @ X4 @ M3 )
=> ( ord_less @ nat @ ( size_size @ ( list @ A ) @ X4 ) @ N2 ) ) ) ).
% finite_maxlen
thf(fact_3221_finite__divisors__int,axiom,
! [I: int] :
( ( I
!= ( zero_zero @ int ) )
=> ( finite_finite2 @ int
@ ( collect @ int
@ ^ [D5: int] : ( dvd_dvd @ int @ D5 @ I ) ) ) ) ).
% finite_divisors_int
thf(fact_3222_pigeonhole__infinite__rel,axiom,
! [B: $tType,A: $tType,A5: set @ A,B5: set @ B,R3: A > B > $o] :
( ~ ( finite_finite2 @ A @ A5 )
=> ( ( finite_finite2 @ B @ B5 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ A5 )
=> ? [Xa2: B] :
( ( member @ B @ Xa2 @ B5 )
& ( R3 @ X3 @ Xa2 ) ) )
=> ? [X3: B] :
( ( member @ B @ X3 @ B5 )
& ~ ( finite_finite2 @ A
@ ( collect @ A
@ ^ [A3: A] :
( ( member @ A @ A3 @ A5 )
& ( R3 @ A3 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_3223_not__finite__existsD,axiom,
! [A: $tType,P: A > $o] :
( ~ ( finite_finite2 @ A @ ( collect @ A @ P ) )
=> ? [X_12: A] : ( P @ X_12 ) ) ).
% not_finite_existsD
thf(fact_3224_finite__has__maximal2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A5: set @ A,A2: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( member @ A @ A2 @ A5 )
=> ? [X3: A] :
( ( member @ A @ X3 @ A5 )
& ( ord_less_eq @ A @ A2 @ X3 )
& ! [Xa2: A] :
( ( member @ A @ Xa2 @ A5 )
=> ( ( ord_less_eq @ A @ X3 @ Xa2 )
=> ( X3 = Xa2 ) ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_3225_finite__has__minimal2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A5: set @ A,A2: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( member @ A @ A2 @ A5 )
=> ? [X3: A] :
( ( member @ A @ X3 @ A5 )
& ( ord_less_eq @ A @ X3 @ A2 )
& ! [Xa2: A] :
( ( member @ A @ Xa2 @ A5 )
=> ( ( ord_less_eq @ A @ Xa2 @ X3 )
=> ( X3 = Xa2 ) ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_3226_or__int__unfold,axiom,
( ( bit_se1065995026697491101ons_or @ int )
= ( ^ [K2: int,L2: int] :
( if @ int
@ ( ( K2
= ( uminus_uminus @ int @ ( one_one @ int ) ) )
| ( L2
= ( uminus_uminus @ int @ ( one_one @ int ) ) ) )
@ ( uminus_uminus @ int @ ( one_one @ int ) )
@ ( if @ int
@ ( K2
= ( zero_zero @ int ) )
@ L2
@ ( if @ int
@ ( L2
= ( zero_zero @ int ) )
@ K2
@ ( plus_plus @ int @ ( ord_max @ int @ ( modulo_modulo @ int @ K2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ int @ L2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ int @ ( divide_divide @ int @ K2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ).
% or_int_unfold
thf(fact_3227_bit_Oabstract__boolean__algebra__sym__diff__axioms,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( boolea3799213064322606851m_diff @ A @ ( bit_se5824344872417868541ns_and @ A ) @ ( bit_se1065995026697491101ons_or @ A ) @ ( bit_ri4277139882892585799ns_not @ A ) @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( bit_se5824344971392196577ns_xor @ A ) ) ) ).
% bit.abstract_boolean_algebra_sym_diff_axioms
thf(fact_3228_even__set__encode__iff,axiom,
! [A5: set @ nat] :
( ( finite_finite2 @ nat @ A5 )
=> ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( nat_set_encode @ A5 ) )
= ( ~ ( member @ nat @ ( zero_zero @ nat ) @ A5 ) ) ) ) ).
% even_set_encode_iff
thf(fact_3229_Code__Numeral_Opositive__def,axiom,
( code_positive
= ( numeral_numeral @ code_integer ) ) ).
% Code_Numeral.positive_def
thf(fact_3230_or__nat__unfold,axiom,
( ( bit_se1065995026697491101ons_or @ nat )
= ( ^ [M2: nat,N4: nat] :
( if @ nat
@ ( M2
= ( zero_zero @ nat ) )
@ N4
@ ( if @ nat
@ ( N4
= ( zero_zero @ nat ) )
@ M2
@ ( plus_plus @ nat @ ( ord_max @ nat @ ( modulo_modulo @ nat @ M2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ nat @ N4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ nat @ ( divide_divide @ nat @ M2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% or_nat_unfold
thf(fact_3231_max__0R,axiom,
! [N: nat] :
( ( ord_max @ nat @ N @ ( zero_zero @ nat ) )
= N ) ).
% max_0R
thf(fact_3232_max__0L,axiom,
! [N: nat] :
( ( ord_max @ nat @ ( zero_zero @ nat ) @ N )
= N ) ).
% max_0L
thf(fact_3233_max__nat_Oright__neutral,axiom,
! [A2: nat] :
( ( ord_max @ nat @ A2 @ ( zero_zero @ nat ) )
= A2 ) ).
% max_nat.right_neutral
thf(fact_3234_max__nat_Oneutr__eq__iff,axiom,
! [A2: nat,B2: nat] :
( ( ( zero_zero @ nat )
= ( ord_max @ nat @ A2 @ B2 ) )
= ( ( A2
= ( zero_zero @ nat ) )
& ( B2
= ( zero_zero @ nat ) ) ) ) ).
% max_nat.neutr_eq_iff
thf(fact_3235_max__nat_Oleft__neutral,axiom,
! [A2: nat] :
( ( ord_max @ nat @ ( zero_zero @ nat ) @ A2 )
= A2 ) ).
% max_nat.left_neutral
thf(fact_3236_max__nat_Oeq__neutr__iff,axiom,
! [A2: nat,B2: nat] :
( ( ( ord_max @ nat @ A2 @ B2 )
= ( zero_zero @ nat ) )
= ( ( A2
= ( zero_zero @ nat ) )
& ( B2
= ( zero_zero @ nat ) ) ) ) ).
% max_nat.eq_neutr_iff
thf(fact_3237_max__Suc__Suc,axiom,
! [M: nat,N: nat] :
( ( ord_max @ nat @ ( suc @ M ) @ ( suc @ N ) )
= ( suc @ ( ord_max @ nat @ M @ N ) ) ) ).
% max_Suc_Suc
thf(fact_3238_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_3239_max__0__1_I4_J,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [X2: num] :
( ( ord_max @ A @ ( numeral_numeral @ A @ X2 ) @ ( zero_zero @ A ) )
= ( numeral_numeral @ A @ X2 ) ) ) ).
% max_0_1(4)
thf(fact_3240_max__0__1_I3_J,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [X2: num] :
( ( ord_max @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ X2 ) )
= ( numeral_numeral @ A @ X2 ) ) ) ).
% max_0_1(3)
thf(fact_3241_max__0__1_I1_J,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ( ( ord_max @ A @ ( zero_zero @ A ) @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% max_0_1(1)
thf(fact_3242_max__0__1_I2_J,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ( ( ord_max @ A @ ( one_one @ A ) @ ( zero_zero @ A ) )
= ( one_one @ A ) ) ) ).
% max_0_1(2)
thf(fact_3243_max__0__1_I6_J,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [X2: num] :
( ( ord_max @ A @ ( numeral_numeral @ A @ X2 ) @ ( one_one @ A ) )
= ( numeral_numeral @ A @ X2 ) ) ) ).
% max_0_1(6)
thf(fact_3244_max__0__1_I5_J,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [X2: num] :
( ( ord_max @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ X2 ) )
= ( numeral_numeral @ A @ X2 ) ) ) ).
% max_0_1(5)
thf(fact_3245_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_3246_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_3247_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_3248_max__numeral__Suc,axiom,
! [K: num,N: nat] :
( ( ord_max @ nat @ ( numeral_numeral @ nat @ K ) @ ( suc @ N ) )
= ( suc @ ( ord_max @ nat @ ( pred_numeral @ K ) @ N ) ) ) ).
% max_numeral_Suc
thf(fact_3249_max__Suc__numeral,axiom,
! [N: nat,K: num] :
( ( ord_max @ nat @ ( suc @ N ) @ ( numeral_numeral @ nat @ K ) )
= ( suc @ ( ord_max @ nat @ N @ ( pred_numeral @ K ) ) ) ) ).
% max_Suc_numeral
thf(fact_3250_max__diff__distrib__left,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [X2: A,Y: A,Z2: A] :
( ( minus_minus @ A @ ( ord_max @ A @ X2 @ Y ) @ Z2 )
= ( ord_max @ A @ ( minus_minus @ A @ X2 @ Z2 ) @ ( minus_minus @ A @ Y @ Z2 ) ) ) ) ).
% max_diff_distrib_left
thf(fact_3251_max__absorb2,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X2: A,Y: A] :
( ( ord_less_eq @ A @ X2 @ Y )
=> ( ( ord_max @ A @ X2 @ Y )
= Y ) ) ) ).
% max_absorb2
thf(fact_3252_max__absorb1,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y: A,X2: A] :
( ( ord_less_eq @ A @ Y @ X2 )
=> ( ( ord_max @ A @ X2 @ Y )
= X2 ) ) ) ).
% max_absorb1
thf(fact_3253_max__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_max @ A )
= ( ^ [A3: A,B3: A] : ( if @ A @ ( ord_less_eq @ A @ A3 @ B3 ) @ B3 @ A3 ) ) ) ) ).
% max_def
thf(fact_3254_of__int__max,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X2: int,Y: int] :
( ( ring_1_of_int @ A @ ( ord_max @ int @ X2 @ Y ) )
= ( ord_max @ A @ ( ring_1_of_int @ A @ X2 ) @ ( ring_1_of_int @ A @ Y ) ) ) ) ).
% of_int_max
thf(fact_3255_of__nat__max,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [X2: nat,Y: nat] :
( ( semiring_1_of_nat @ A @ ( ord_max @ nat @ X2 @ Y ) )
= ( ord_max @ A @ ( semiring_1_of_nat @ A @ X2 ) @ ( semiring_1_of_nat @ A @ Y ) ) ) ) ).
% of_nat_max
thf(fact_3256_nat__mult__max__right,axiom,
! [M: nat,N: nat,Q2: nat] :
( ( times_times @ nat @ M @ ( ord_max @ nat @ N @ Q2 ) )
= ( ord_max @ nat @ ( times_times @ nat @ M @ N ) @ ( times_times @ nat @ M @ Q2 ) ) ) ).
% nat_mult_max_right
thf(fact_3257_nat__mult__max__left,axiom,
! [M: nat,N: nat,Q2: nat] :
( ( times_times @ nat @ ( ord_max @ nat @ M @ N ) @ Q2 )
= ( ord_max @ nat @ ( times_times @ nat @ M @ Q2 ) @ ( times_times @ nat @ N @ Q2 ) ) ) ).
% nat_mult_max_left
thf(fact_3258_nat__add__max__left,axiom,
! [M: nat,N: nat,Q2: nat] :
( ( plus_plus @ nat @ ( ord_max @ nat @ M @ N ) @ Q2 )
= ( ord_max @ nat @ ( plus_plus @ nat @ M @ Q2 ) @ ( plus_plus @ nat @ N @ Q2 ) ) ) ).
% nat_add_max_left
thf(fact_3259_nat__add__max__right,axiom,
! [M: nat,N: nat,Q2: nat] :
( ( plus_plus @ nat @ M @ ( ord_max @ nat @ N @ Q2 ) )
= ( ord_max @ nat @ ( plus_plus @ nat @ M @ N ) @ ( plus_plus @ nat @ M @ Q2 ) ) ) ).
% nat_add_max_right
thf(fact_3260_max__add__distrib__right,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [X2: A,Y: A,Z2: A] :
( ( plus_plus @ A @ X2 @ ( ord_max @ A @ Y @ Z2 ) )
= ( ord_max @ A @ ( plus_plus @ A @ X2 @ Y ) @ ( plus_plus @ A @ X2 @ Z2 ) ) ) ) ).
% max_add_distrib_right
thf(fact_3261_max__add__distrib__left,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [X2: A,Y: A,Z2: A] :
( ( plus_plus @ A @ ( ord_max @ A @ X2 @ Y ) @ Z2 )
= ( ord_max @ A @ ( plus_plus @ A @ X2 @ Z2 ) @ ( plus_plus @ A @ Y @ Z2 ) ) ) ) ).
% max_add_distrib_left
thf(fact_3262_max__def__raw,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_max @ A )
= ( ^ [A3: A,B3: A] : ( if @ A @ ( ord_less_eq @ A @ A3 @ B3 ) @ B3 @ A3 ) ) ) ) ).
% max_def_raw
thf(fact_3263_nat__minus__add__max,axiom,
! [N: nat,M: nat] :
( ( plus_plus @ nat @ ( minus_minus @ nat @ N @ M ) @ M )
= ( ord_max @ nat @ N @ M ) ) ).
% nat_minus_add_max
thf(fact_3264_max__Suc1,axiom,
! [N: nat,M: nat] :
( ( ord_max @ nat @ ( suc @ N ) @ M )
= ( case_nat @ nat @ ( suc @ N )
@ ^ [M8: nat] : ( suc @ ( ord_max @ nat @ N @ M8 ) )
@ M ) ) ).
% max_Suc1
thf(fact_3265_max__Suc2,axiom,
! [M: nat,N: nat] :
( ( ord_max @ nat @ M @ ( suc @ N ) )
= ( case_nat @ nat @ ( suc @ N )
@ ^ [M8: nat] : ( suc @ ( ord_max @ nat @ M8 @ N ) )
@ M ) ) ).
% max_Suc2
thf(fact_3266_set__encode__inf,axiom,
! [A5: set @ nat] :
( ~ ( finite_finite2 @ nat @ A5 )
=> ( ( nat_set_encode @ A5 )
= ( zero_zero @ nat ) ) ) ).
% set_encode_inf
thf(fact_3267_max_Oabsorb1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_max @ A @ A2 @ B2 )
= A2 ) ) ) ).
% max.absorb1
thf(fact_3268_max_Oabsorb2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_max @ A @ A2 @ B2 )
= B2 ) ) ) ).
% max.absorb2
thf(fact_3269_max_Obounded__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,C2: A,A2: A] :
( ( ord_less_eq @ A @ ( ord_max @ A @ B2 @ C2 ) @ A2 )
= ( ( ord_less_eq @ A @ B2 @ A2 )
& ( ord_less_eq @ A @ C2 @ A2 ) ) ) ) ).
% max.bounded_iff
thf(fact_3270_infinite__int__iff__unbounded__le,axiom,
! [S2: set @ int] :
( ( ~ ( finite_finite2 @ int @ S2 ) )
= ( ! [M2: int] :
? [N4: int] :
( ( ord_less_eq @ int @ M2 @ ( abs_abs @ int @ N4 ) )
& ( member @ int @ N4 @ S2 ) ) ) ) ).
% infinite_int_iff_unbounded_le
thf(fact_3271_set__encode__insert,axiom,
! [A5: set @ nat,N: nat] :
( ( finite_finite2 @ nat @ A5 )
=> ( ~ ( member @ nat @ N @ A5 )
=> ( ( nat_set_encode @ ( insert @ nat @ N @ A5 ) )
= ( plus_plus @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ ( nat_set_encode @ A5 ) ) ) ) ) ).
% set_encode_insert
thf(fact_3272_sum_Oinsert,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A5: set @ B,X2: B,G: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ~ ( member @ B @ X2 @ A5 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( insert @ B @ X2 @ A5 ) )
= ( plus_plus @ A @ ( G @ X2 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ A5 ) ) ) ) ) ) ).
% sum.insert
thf(fact_3273_atMost__Suc,axiom,
! [K: nat] :
( ( set_ord_atMost @ nat @ ( suc @ K ) )
= ( insert @ nat @ ( suc @ K ) @ ( set_ord_atMost @ nat @ K ) ) ) ).
% atMost_Suc
thf(fact_3274_sum_Oinsert__if,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A5: set @ B,X2: B,G: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( ( member @ B @ X2 @ A5 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( insert @ B @ X2 @ A5 ) )
= ( groups7311177749621191930dd_sum @ B @ A @ G @ A5 ) ) )
& ( ~ ( member @ B @ X2 @ A5 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( insert @ B @ X2 @ A5 ) )
= ( plus_plus @ A @ ( G @ X2 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ A5 ) ) ) ) ) ) ) ).
% sum.insert_if
thf(fact_3275_atLeast0__atMost__Suc,axiom,
! [N: nat] :
( ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) )
= ( insert @ nat @ ( suc @ N ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% atLeast0_atMost_Suc
thf(fact_3276_atLeast0__lessThan__Suc,axiom,
! [N: nat] :
( ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) )
= ( insert @ nat @ N @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% atLeast0_lessThan_Suc
thf(fact_3277_Icc__eq__insert__lb__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( set_or1337092689740270186AtMost @ nat @ M @ N )
= ( insert @ nat @ M @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ N ) ) ) ) ).
% Icc_eq_insert_lb_nat
thf(fact_3278_atLeastAtMostSuc__conv,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ ( suc @ N ) )
=> ( ( set_or1337092689740270186AtMost @ nat @ M @ ( suc @ N ) )
= ( insert @ nat @ ( suc @ N ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% atLeastAtMostSuc_conv
thf(fact_3279_atLeastAtMost__insertL,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( insert @ nat @ M @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ N ) )
= ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ).
% atLeastAtMost_insertL
thf(fact_3280_atMost__nat__numeral,axiom,
! [K: num] :
( ( set_ord_atMost @ nat @ ( numeral_numeral @ nat @ K ) )
= ( insert @ nat @ ( numeral_numeral @ nat @ K ) @ ( set_ord_atMost @ nat @ ( pred_numeral @ K ) ) ) ) ).
% atMost_nat_numeral
thf(fact_3281_set__decode__plus__power__2,axiom,
! [N: nat,Z2: nat] :
( ~ ( member @ nat @ N @ ( nat_set_decode @ Z2 ) )
=> ( ( nat_set_decode @ ( plus_plus @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ Z2 ) )
= ( insert @ nat @ N @ ( nat_set_decode @ Z2 ) ) ) ) ).
% set_decode_plus_power_2
thf(fact_3282_max_OcoboundedI2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C2: A,B2: A,A2: A] :
( ( ord_less_eq @ A @ C2 @ B2 )
=> ( ord_less_eq @ A @ C2 @ ( ord_max @ A @ A2 @ B2 ) ) ) ) ).
% max.coboundedI2
thf(fact_3283_max_OcoboundedI1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less_eq @ A @ C2 @ A2 )
=> ( ord_less_eq @ A @ C2 @ ( ord_max @ A @ A2 @ B2 ) ) ) ) ).
% max.coboundedI1
thf(fact_3284_max_Oabsorb__iff2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A3: A,B3: A] :
( ( ord_max @ A @ A3 @ B3 )
= B3 ) ) ) ) ).
% max.absorb_iff2
thf(fact_3285_max_Oabsorb__iff1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B3: A,A3: A] :
( ( ord_max @ A @ A3 @ B3 )
= A3 ) ) ) ) ).
% max.absorb_iff1
thf(fact_3286_le__max__iff__disj,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Z2: A,X2: A,Y: A] :
( ( ord_less_eq @ A @ Z2 @ ( ord_max @ A @ X2 @ Y ) )
= ( ( ord_less_eq @ A @ Z2 @ X2 )
| ( ord_less_eq @ A @ Z2 @ Y ) ) ) ) ).
% le_max_iff_disj
thf(fact_3287_max_Ocobounded2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,A2: A] : ( ord_less_eq @ A @ B2 @ ( ord_max @ A @ A2 @ B2 ) ) ) ).
% max.cobounded2
thf(fact_3288_max_Ocobounded1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A] : ( ord_less_eq @ A @ A2 @ ( ord_max @ A @ A2 @ B2 ) ) ) ).
% max.cobounded1
thf(fact_3289_max_Oorder__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B3: A,A3: A] :
( A3
= ( ord_max @ A @ A3 @ B3 ) ) ) ) ) ).
% max.order_iff
thf(fact_3290_max_OboundedI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ C2 @ A2 )
=> ( ord_less_eq @ A @ ( ord_max @ A @ B2 @ C2 ) @ A2 ) ) ) ) ).
% max.boundedI
thf(fact_3291_max_OboundedE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,C2: A,A2: A] :
( ( ord_less_eq @ A @ ( ord_max @ A @ B2 @ C2 ) @ A2 )
=> ~ ( ( ord_less_eq @ A @ B2 @ A2 )
=> ~ ( ord_less_eq @ A @ C2 @ A2 ) ) ) ) ).
% max.boundedE
thf(fact_3292_max_OorderI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A] :
( ( A2
= ( ord_max @ A @ A2 @ B2 ) )
=> ( ord_less_eq @ A @ B2 @ A2 ) ) ) ).
% max.orderI
thf(fact_3293_max_OorderE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( A2
= ( ord_max @ A @ A2 @ B2 ) ) ) ) ).
% max.orderE
thf(fact_3294_max_Omono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C2: A,A2: A,D2: A,B2: A] :
( ( ord_less_eq @ A @ C2 @ A2 )
=> ( ( ord_less_eq @ A @ D2 @ B2 )
=> ( ord_less_eq @ A @ ( ord_max @ A @ C2 @ D2 ) @ ( ord_max @ A @ A2 @ B2 ) ) ) ) ) ).
% max.mono
thf(fact_3295_infinite__nat__iff__unbounded__le,axiom,
! [S2: set @ nat] :
( ( ~ ( finite_finite2 @ nat @ S2 ) )
= ( ! [M2: nat] :
? [N4: nat] :
( ( ord_less_eq @ nat @ M2 @ N4 )
& ( member @ nat @ N4 @ S2 ) ) ) ) ).
% infinite_nat_iff_unbounded_le
thf(fact_3296_integer__of__num_I3_J,axiom,
! [N: num] :
( ( code_integer_of_num @ ( bit1 @ N ) )
= ( plus_plus @ code_integer @ ( plus_plus @ code_integer @ ( code_integer_of_num @ N ) @ ( code_integer_of_num @ N ) ) @ ( one_one @ code_integer ) ) ) ).
% integer_of_num(3)
thf(fact_3297_even__sum__iff,axiom,
! [A: $tType,B: $tType] :
( ( semiring_parity @ A )
=> ! [A5: set @ B,F: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ F @ A5 ) )
= ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) )
@ ( finite_card @ B
@ ( collect @ B
@ ^ [A3: B] :
( ( member @ B @ A3 @ A5 )
& ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( F @ A3 ) ) ) ) ) ) ) ) ) ).
% even_sum_iff
thf(fact_3298_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_3299_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_3300_drop__bit__rec,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se4197421643247451524op_bit @ A )
= ( ^ [N4: nat,A3: A] :
( if @ A
@ ( N4
= ( zero_zero @ nat ) )
@ A3
@ ( bit_se4197421643247451524op_bit @ A @ ( minus_minus @ nat @ N4 @ ( one_one @ nat ) ) @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% drop_bit_rec
thf(fact_3301_card__Collect__less__nat,axiom,
! [N: nat] :
( ( finite_card @ nat
@ ( collect @ nat
@ ^ [I2: nat] : ( ord_less @ nat @ I2 @ N ) ) )
= N ) ).
% card_Collect_less_nat
thf(fact_3302_drop__bit__nonnegative__int__iff,axiom,
! [N: nat,K: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se4197421643247451524op_bit @ int @ N @ K ) )
= ( ord_less_eq @ int @ ( zero_zero @ int ) @ K ) ) ).
% drop_bit_nonnegative_int_iff
thf(fact_3303_drop__bit__negative__int__iff,axiom,
! [N: nat,K: int] :
( ( ord_less @ int @ ( bit_se4197421643247451524op_bit @ int @ N @ K ) @ ( zero_zero @ int ) )
= ( ord_less @ int @ K @ ( zero_zero @ int ) ) ) ).
% drop_bit_negative_int_iff
thf(fact_3304_drop__bit__of__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat] :
( ( bit_se4197421643247451524op_bit @ A @ N @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% drop_bit_of_0
thf(fact_3305_drop__bit__minus__one,axiom,
! [N: nat] :
( ( bit_se4197421643247451524op_bit @ int @ N @ ( uminus_uminus @ int @ ( one_one @ int ) ) )
= ( uminus_uminus @ int @ ( one_one @ int ) ) ) ).
% drop_bit_minus_one
thf(fact_3306_card__atLeastLessThan,axiom,
! [L: nat,U: nat] :
( ( finite_card @ nat @ ( set_or7035219750837199246ssThan @ nat @ L @ U ) )
= ( minus_minus @ nat @ U @ L ) ) ).
% card_atLeastLessThan
thf(fact_3307_card__atMost,axiom,
! [U: nat] :
( ( finite_card @ nat @ ( set_ord_atMost @ nat @ U ) )
= ( suc @ U ) ) ).
% card_atMost
thf(fact_3308_drop__bit__drop__bit,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,N: nat,A2: A] :
( ( bit_se4197421643247451524op_bit @ A @ M @ ( bit_se4197421643247451524op_bit @ A @ N @ A2 ) )
= ( bit_se4197421643247451524op_bit @ A @ ( plus_plus @ nat @ M @ N ) @ A2 ) ) ) ).
% drop_bit_drop_bit
thf(fact_3309_card__Collect__le__nat,axiom,
! [N: nat] :
( ( finite_card @ nat
@ ( collect @ nat
@ ^ [I2: nat] : ( ord_less_eq @ nat @ I2 @ N ) ) )
= ( suc @ N ) ) ).
% card_Collect_le_nat
thf(fact_3310_card_Oinfinite,axiom,
! [A: $tType,A5: set @ A] :
( ~ ( finite_finite2 @ A @ A5 )
=> ( ( finite_card @ A @ A5 )
= ( zero_zero @ nat ) ) ) ).
% card.infinite
thf(fact_3311_card__atLeastAtMost,axiom,
! [L: nat,U: nat] :
( ( finite_card @ nat @ ( set_or1337092689740270186AtMost @ nat @ L @ U ) )
= ( minus_minus @ nat @ ( suc @ U ) @ L ) ) ).
% card_atLeastAtMost
thf(fact_3312_drop__bit__of__bool,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,B2: $o] :
( ( bit_se4197421643247451524op_bit @ A @ N @ ( zero_neq_one_of_bool @ A @ B2 ) )
= ( zero_neq_one_of_bool @ A
@ ( ( N
= ( zero_zero @ nat ) )
& B2 ) ) ) ) ).
% drop_bit_of_bool
thf(fact_3313_drop__bit__of__Suc__0,axiom,
! [N: nat] :
( ( bit_se4197421643247451524op_bit @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) )
= ( zero_neq_one_of_bool @ nat
@ ( N
= ( zero_zero @ nat ) ) ) ) ).
% drop_bit_of_Suc_0
thf(fact_3314_prod__constant,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [Y: A,A5: set @ B] :
( ( groups7121269368397514597t_prod @ B @ A
@ ^ [X: B] : Y
@ A5 )
= ( power_power @ A @ Y @ ( finite_card @ B @ A5 ) ) ) ) ).
% prod_constant
thf(fact_3315_card__atLeastLessThan__int,axiom,
! [L: int,U: int] :
( ( finite_card @ int @ ( set_or7035219750837199246ssThan @ int @ L @ U ) )
= ( nat2 @ ( minus_minus @ int @ U @ L ) ) ) ).
% card_atLeastLessThan_int
thf(fact_3316_drop__bit__of__1,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat] :
( ( bit_se4197421643247451524op_bit @ A @ N @ ( one_one @ A ) )
= ( zero_neq_one_of_bool @ A
@ ( N
= ( zero_zero @ nat ) ) ) ) ) ).
% drop_bit_of_1
thf(fact_3317_sum__constant,axiom,
! [B: $tType,A: $tType] :
( ( semiring_1 @ A )
=> ! [Y: A,A5: set @ B] :
( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X: B] : Y
@ A5 )
= ( times_times @ A @ ( semiring_1_of_nat @ A @ ( finite_card @ B @ A5 ) ) @ Y ) ) ) ).
% sum_constant
thf(fact_3318_card__Diff__insert,axiom,
! [A: $tType,A2: A,A5: set @ A,B5: set @ A] :
( ( member @ A @ A2 @ A5 )
=> ( ~ ( member @ A @ A2 @ B5 )
=> ( ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ A2 @ B5 ) ) )
= ( minus_minus @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A5 @ B5 ) ) @ ( one_one @ nat ) ) ) ) ) ).
% card_Diff_insert
thf(fact_3319_card__atLeastAtMost__int,axiom,
! [L: int,U: int] :
( ( finite_card @ int @ ( set_or1337092689740270186AtMost @ int @ L @ U ) )
= ( nat2 @ ( plus_plus @ int @ ( minus_minus @ int @ U @ L ) @ ( one_one @ int ) ) ) ) ).
% card_atLeastAtMost_int
thf(fact_3320_drop__bit__of__nat,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N: nat,M: nat] :
( ( bit_se4197421643247451524op_bit @ A @ N @ ( semiring_1_of_nat @ A @ M ) )
= ( semiring_1_of_nat @ A @ ( bit_se4197421643247451524op_bit @ nat @ N @ M ) ) ) ) ).
% drop_bit_of_nat
thf(fact_3321_of__nat__drop__bit,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [M: nat,N: nat] :
( ( semiring_1_of_nat @ A @ ( bit_se4197421643247451524op_bit @ nat @ M @ N ) )
= ( bit_se4197421643247451524op_bit @ A @ M @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% of_nat_drop_bit
thf(fact_3322_drop__bit__integer_Oabs__eq,axiom,
! [Xa: nat,X2: int] :
( ( bit_se4197421643247451524op_bit @ code_integer @ Xa @ ( code_integer_of_int @ X2 ) )
= ( code_integer_of_int @ ( bit_se4197421643247451524op_bit @ int @ Xa @ X2 ) ) ) ).
% drop_bit_integer.abs_eq
thf(fact_3323_n__subsets,axiom,
! [A: $tType,A5: set @ A,K: nat] :
( ( finite_finite2 @ A @ A5 )
=> ( ( finite_card @ ( set @ A )
@ ( collect @ ( set @ A )
@ ^ [B8: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B8 @ A5 )
& ( ( finite_card @ A @ B8 )
= K ) ) ) )
= ( binomial @ ( finite_card @ A @ A5 ) @ K ) ) ) ).
% n_subsets
thf(fact_3324_card__le__if__inj__on__rel,axiom,
! [B: $tType,A: $tType,B5: set @ A,A5: set @ B,R: B > A > $o] :
( ( finite_finite2 @ A @ B5 )
=> ( ! [A4: B] :
( ( member @ B @ A4 @ A5 )
=> ? [B9: A] :
( ( member @ A @ B9 @ B5 )
& ( R @ A4 @ B9 ) ) )
=> ( ! [A1: B,A22: B,B4: A] :
( ( member @ B @ A1 @ A5 )
=> ( ( member @ B @ A22 @ A5 )
=> ( ( member @ A @ B4 @ B5 )
=> ( ( R @ A1 @ B4 )
=> ( ( R @ A22 @ B4 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq @ nat @ ( finite_card @ B @ A5 ) @ ( finite_card @ A @ B5 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_3325_card__insert__le,axiom,
! [A: $tType,A5: set @ A,X2: A] : ( ord_less_eq @ nat @ ( finite_card @ A @ A5 ) @ ( finite_card @ A @ ( insert @ A @ X2 @ A5 ) ) ) ).
% card_insert_le
thf(fact_3326_take__bit__eq__self__iff__drop__bit__eq__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A2: A] :
( ( ( bit_se2584673776208193580ke_bit @ A @ N @ A2 )
= A2 )
= ( ( bit_se4197421643247451524op_bit @ A @ N @ A2 )
= ( zero_zero @ A ) ) ) ) ).
% take_bit_eq_self_iff_drop_bit_eq_0
thf(fact_3327_take__bit__drop__bit,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,N: nat,A2: A] :
( ( bit_se2584673776208193580ke_bit @ A @ M @ ( bit_se4197421643247451524op_bit @ A @ N @ A2 ) )
= ( bit_se4197421643247451524op_bit @ A @ N @ ( bit_se2584673776208193580ke_bit @ A @ ( plus_plus @ nat @ M @ N ) @ A2 ) ) ) ) ).
% take_bit_drop_bit
thf(fact_3328_sum__multicount__gen,axiom,
! [A: $tType,B: $tType,S3: set @ A,T2: set @ B,R3: A > B > $o,K: B > nat] :
( ( finite_finite2 @ A @ S3 )
=> ( ( finite_finite2 @ B @ T2 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ T2 )
=> ( ( finite_card @ A
@ ( collect @ A
@ ^ [I2: A] :
( ( member @ A @ I2 @ S3 )
& ( R3 @ I2 @ X3 ) ) ) )
= ( K @ X3 ) ) )
=> ( ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [I2: A] :
( finite_card @ B
@ ( collect @ B
@ ^ [J3: B] :
( ( member @ B @ J3 @ T2 )
& ( R3 @ I2 @ J3 ) ) ) )
@ S3 )
= ( groups7311177749621191930dd_sum @ B @ nat @ K @ T2 ) ) ) ) ) ).
% sum_multicount_gen
thf(fact_3329_card__eq__sum,axiom,
! [A: $tType] :
( ( finite_card @ A )
= ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [X: A] : ( one_one @ nat ) ) ) ).
% card_eq_sum
thf(fact_3330_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_3331_integer__of__num__def,axiom,
( code_integer_of_num
= ( numeral_numeral @ code_integer ) ) ).
% integer_of_num_def
thf(fact_3332_card__2__iff_H,axiom,
! [A: $tType,S2: set @ A] :
( ( ( finite_card @ A @ S2 )
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( ? [X: A] :
( ( member @ A @ X @ S2 )
& ? [Y4: A] :
( ( member @ A @ Y4 @ S2 )
& ( X != Y4 )
& ! [Z3: A] :
( ( member @ A @ Z3 @ S2 )
=> ( ( Z3 = X )
| ( Z3 = Y4 ) ) ) ) ) ) ) ).
% card_2_iff'
thf(fact_3333_obtain__subset__with__card__n,axiom,
! [A: $tType,N: nat,S2: set @ A] :
( ( ord_less_eq @ nat @ N @ ( finite_card @ A @ S2 ) )
=> ~ ! [T7: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ T7 @ S2 )
=> ( ( ( finite_card @ A @ T7 )
= N )
=> ~ ( finite_finite2 @ A @ T7 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_3334_card__mono,axiom,
! [A: $tType,B5: set @ A,A5: set @ A] :
( ( finite_finite2 @ A @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
=> ( ord_less_eq @ nat @ ( finite_card @ A @ A5 ) @ ( finite_card @ A @ B5 ) ) ) ) ).
% card_mono
thf(fact_3335_card__seteq,axiom,
! [A: $tType,B5: set @ A,A5: set @ A] :
( ( finite_finite2 @ A @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
=> ( ( ord_less_eq @ nat @ ( finite_card @ A @ B5 ) @ ( finite_card @ A @ A5 ) )
=> ( A5 = B5 ) ) ) ) ).
% card_seteq
thf(fact_3336_finite__if__finite__subsets__card__bdd,axiom,
! [A: $tType,F4: set @ A,C6: nat] :
( ! [G3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ G3 @ F4 )
=> ( ( finite_finite2 @ A @ G3 )
=> ( ord_less_eq @ nat @ ( finite_card @ A @ G3 ) @ C6 ) ) )
=> ( ( finite_finite2 @ A @ F4 )
& ( ord_less_eq @ nat @ ( finite_card @ A @ F4 ) @ C6 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_3337_card__le__sym__Diff,axiom,
! [A: $tType,A5: set @ A,B5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( finite_finite2 @ A @ B5 )
=> ( ( ord_less_eq @ nat @ ( finite_card @ A @ A5 ) @ ( finite_card @ A @ B5 ) )
=> ( ord_less_eq @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A5 @ B5 ) ) @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ B5 @ A5 ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_3338_card__less__Suc2,axiom,
! [M3: set @ nat,I: nat] :
( ~ ( member @ nat @ ( zero_zero @ nat ) @ M3 )
=> ( ( finite_card @ nat
@ ( collect @ nat
@ ^ [K2: nat] :
( ( member @ nat @ ( suc @ K2 ) @ M3 )
& ( ord_less @ nat @ K2 @ I ) ) ) )
= ( finite_card @ nat
@ ( collect @ nat
@ ^ [K2: nat] :
( ( member @ nat @ K2 @ M3 )
& ( ord_less @ nat @ K2 @ ( suc @ I ) ) ) ) ) ) ) ).
% card_less_Suc2
thf(fact_3339_card__less__Suc,axiom,
! [M3: set @ nat,I: nat] :
( ( member @ nat @ ( zero_zero @ nat ) @ M3 )
=> ( ( suc
@ ( finite_card @ nat
@ ( collect @ nat
@ ^ [K2: nat] :
( ( member @ nat @ ( suc @ K2 ) @ M3 )
& ( ord_less @ nat @ K2 @ I ) ) ) ) )
= ( finite_card @ nat
@ ( collect @ nat
@ ^ [K2: nat] :
( ( member @ nat @ K2 @ M3 )
& ( ord_less @ nat @ K2 @ ( suc @ I ) ) ) ) ) ) ) ).
% card_less_Suc
thf(fact_3340_card__less,axiom,
! [M3: set @ nat,I: nat] :
( ( member @ nat @ ( zero_zero @ nat ) @ M3 )
=> ( ( finite_card @ nat
@ ( collect @ nat
@ ^ [K2: nat] :
( ( member @ nat @ K2 @ M3 )
& ( ord_less @ nat @ K2 @ ( suc @ I ) ) ) ) )
!= ( zero_zero @ nat ) ) ) ).
% card_less
thf(fact_3341_card__atLeastZeroLessThan__int,axiom,
! [U: int] :
( ( finite_card @ int @ ( set_or7035219750837199246ssThan @ int @ ( zero_zero @ int ) @ U ) )
= ( nat2 @ U ) ) ).
% card_atLeastZeroLessThan_int
thf(fact_3342_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_3343_sum__Suc,axiom,
! [A: $tType,F: A > nat,A5: set @ A] :
( ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [X: A] : ( suc @ ( F @ X ) )
@ A5 )
= ( plus_plus @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ F @ A5 ) @ ( finite_card @ A @ A5 ) ) ) ).
% sum_Suc
thf(fact_3344_div__push__bit__of__1__eq__drop__bit,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A,N: nat] :
( ( divide_divide @ A @ A2 @ ( bit_se4730199178511100633sh_bit @ A @ N @ ( one_one @ A ) ) )
= ( bit_se4197421643247451524op_bit @ A @ N @ A2 ) ) ) ).
% div_push_bit_of_1_eq_drop_bit
thf(fact_3345_sum__multicount,axiom,
! [A: $tType,B: $tType,S2: set @ A,T6: set @ B,R3: A > B > $o,K: nat] :
( ( finite_finite2 @ A @ S2 )
=> ( ( finite_finite2 @ B @ T6 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ T6 )
=> ( ( finite_card @ A
@ ( collect @ A
@ ^ [I2: A] :
( ( member @ A @ I2 @ S2 )
& ( R3 @ I2 @ X3 ) ) ) )
= K ) )
=> ( ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [I2: A] :
( finite_card @ B
@ ( collect @ B
@ ^ [J3: B] :
( ( member @ B @ J3 @ T6 )
& ( R3 @ I2 @ J3 ) ) ) )
@ S2 )
= ( times_times @ nat @ K @ ( finite_card @ B @ T6 ) ) ) ) ) ) ).
% sum_multicount
thf(fact_3346_bit__iff__and__drop__bit__eq__1,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se5641148757651400278ts_bit @ A )
= ( ^ [A3: A,N4: nat] :
( ( bit_se5824344872417868541ns_and @ A @ ( bit_se4197421643247451524op_bit @ A @ N4 @ A3 ) @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ) ) ).
% bit_iff_and_drop_bit_eq_1
thf(fact_3347_bits__ident,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A2: A] :
( ( plus_plus @ A @ ( bit_se4730199178511100633sh_bit @ A @ N @ ( bit_se4197421643247451524op_bit @ A @ N @ A2 ) ) @ ( bit_se2584673776208193580ke_bit @ A @ N @ A2 ) )
= A2 ) ) ).
% bits_ident
thf(fact_3348_sum__bounded__below,axiom,
! [A: $tType,B: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( semiring_1 @ A ) )
=> ! [A5: set @ B,K3: A,F: B > A] :
( ! [I3: B] :
( ( member @ B @ I3 @ A5 )
=> ( ord_less_eq @ A @ K3 @ ( F @ I3 ) ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( finite_card @ B @ A5 ) ) @ K3 ) @ ( groups7311177749621191930dd_sum @ B @ A @ F @ A5 ) ) ) ) ).
% sum_bounded_below
thf(fact_3349_sum__bounded__above,axiom,
! [B: $tType,A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( semiring_1 @ A ) )
=> ! [A5: set @ B,F: B > A,K3: A] :
( ! [I3: B] :
( ( member @ B @ I3 @ A5 )
=> ( ord_less_eq @ A @ ( F @ I3 ) @ K3 ) )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F @ A5 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( finite_card @ B @ A5 ) ) @ K3 ) ) ) ) ).
% sum_bounded_above
thf(fact_3350_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 ) ) )
= ( ! [X: A] :
( ( member @ A @ X @ A5 )
=> ! [Y4: A] :
( ( member @ A @ Y4 @ A5 )
=> ( X = Y4 ) ) ) ) ) ) ).
% card_le_Suc0_iff_eq
thf(fact_3351_card__le__Suc__iff,axiom,
! [A: $tType,N: nat,A5: set @ A] :
( ( ord_less_eq @ nat @ ( suc @ N ) @ ( finite_card @ A @ A5 ) )
= ( ? [A3: A,B8: set @ A] :
( ( A5
= ( insert @ A @ A3 @ B8 ) )
& ~ ( member @ A @ A3 @ B8 )
& ( ord_less_eq @ nat @ N @ ( finite_card @ A @ B8 ) )
& ( finite_finite2 @ A @ B8 ) ) ) ) ).
% card_le_Suc_iff
thf(fact_3352_diff__card__le__card__Diff,axiom,
! [A: $tType,B5: set @ A,A5: set @ A] :
( ( finite_finite2 @ A @ B5 )
=> ( ord_less_eq @ nat @ ( minus_minus @ nat @ ( finite_card @ A @ A5 ) @ ( finite_card @ A @ B5 ) ) @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A5 @ B5 ) ) ) ) ).
% diff_card_le_card_Diff
thf(fact_3353_atLeastAtMostPlus1__int__conv,axiom,
! [M: int,N: int] :
( ( ord_less_eq @ int @ M @ ( plus_plus @ int @ ( one_one @ int ) @ N ) )
=> ( ( set_or1337092689740270186AtMost @ int @ M @ ( plus_plus @ int @ ( one_one @ int ) @ N ) )
= ( insert @ int @ ( plus_plus @ int @ ( one_one @ int ) @ N ) @ ( set_or1337092689740270186AtMost @ int @ M @ N ) ) ) ) ).
% atLeastAtMostPlus1_int_conv
thf(fact_3354_subset__eq__atLeast0__lessThan__card,axiom,
! [N5: set @ nat,N: nat] :
( ( ord_less_eq @ ( set @ nat ) @ N5 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) )
=> ( ord_less_eq @ nat @ ( finite_card @ nat @ N5 ) @ N ) ) ).
% subset_eq_atLeast0_lessThan_card
thf(fact_3355_card__sum__le__nat__sum,axiom,
! [S2: set @ nat] :
( ord_less_eq @ nat
@ ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [X: nat] : X
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( finite_card @ nat @ S2 ) ) )
@ ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [X: nat] : X
@ S2 ) ) ).
% card_sum_le_nat_sum
thf(fact_3356_card__insert__disjoint_H,axiom,
! [A: $tType,A5: set @ A,X2: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ~ ( member @ A @ X2 @ A5 )
=> ( ( minus_minus @ nat @ ( finite_card @ A @ ( insert @ A @ X2 @ A5 ) ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( finite_card @ A @ A5 ) ) ) ) ).
% card_insert_disjoint'
thf(fact_3357_integer__of__num__triv_I1_J,axiom,
( ( code_integer_of_num @ one2 )
= ( one_one @ code_integer ) ) ).
% integer_of_num_triv(1)
thf(fact_3358_integer__of__num_I2_J,axiom,
! [N: num] :
( ( code_integer_of_num @ ( bit0 @ N ) )
= ( plus_plus @ code_integer @ ( code_integer_of_num @ N ) @ ( code_integer_of_num @ N ) ) ) ).
% integer_of_num(2)
thf(fact_3359_prod__le__power,axiom,
! [B: $tType,A: $tType] :
( ( linordered_semidom @ A )
=> ! [A5: set @ B,F: B > A,N: A,K: nat] :
( ! [I3: B] :
( ( member @ B @ I3 @ A5 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F @ I3 ) )
& ( ord_less_eq @ A @ ( F @ I3 ) @ N ) ) )
=> ( ( ord_less_eq @ nat @ ( finite_card @ B @ A5 ) @ K )
=> ( ( ord_less_eq @ A @ ( one_one @ A ) @ N )
=> ( ord_less_eq @ A @ ( groups7121269368397514597t_prod @ B @ A @ F @ A5 ) @ ( power_power @ A @ N @ K ) ) ) ) ) ) ).
% prod_le_power
thf(fact_3360_sum__bounded__above__strict,axiom,
! [B: $tType,A: $tType] :
( ( ( ordere8940638589300402666id_add @ A )
& ( semiring_1 @ A ) )
=> ! [A5: set @ B,F: B > A,K3: A] :
( ! [I3: B] :
( ( member @ B @ I3 @ A5 )
=> ( ord_less @ A @ ( F @ I3 ) @ K3 ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( finite_card @ B @ A5 ) )
=> ( ord_less @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F @ A5 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( finite_card @ B @ A5 ) ) @ K3 ) ) ) ) ) ).
% sum_bounded_above_strict
thf(fact_3361_card__insert__le__m1,axiom,
! [A: $tType,N: nat,Y: set @ A,X2: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ nat @ ( finite_card @ A @ Y ) @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) )
=> ( ord_less_eq @ nat @ ( finite_card @ A @ ( insert @ A @ X2 @ Y ) ) @ N ) ) ) ).
% card_insert_le_m1
thf(fact_3362_prod__gen__delta,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S2: set @ B,A2: B,B2: B > A,C2: A] :
( ( finite_finite2 @ B @ S2 )
=> ( ( ( member @ B @ A2 @ S2 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K2: B] : ( if @ A @ ( K2 = A2 ) @ ( B2 @ K2 ) @ C2 )
@ S2 )
= ( times_times @ A @ ( B2 @ A2 ) @ ( power_power @ A @ C2 @ ( minus_minus @ nat @ ( finite_card @ B @ S2 ) @ ( one_one @ nat ) ) ) ) ) )
& ( ~ ( member @ B @ A2 @ S2 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K2: B] : ( if @ A @ ( K2 = A2 ) @ ( B2 @ K2 ) @ C2 )
@ S2 )
= ( power_power @ A @ C2 @ ( finite_card @ B @ S2 ) ) ) ) ) ) ) ).
% prod_gen_delta
thf(fact_3363_integer__of__num__triv_I2_J,axiom,
( ( code_integer_of_num @ ( bit0 @ one2 ) )
= ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) ).
% integer_of_num_triv(2)
thf(fact_3364_slice__eq__mask,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: nat,M: nat,A2: A] :
( ( bit_se4730199178511100633sh_bit @ A @ N @ ( bit_se2584673776208193580ke_bit @ A @ M @ ( bit_se4197421643247451524op_bit @ A @ N @ A2 ) ) )
= ( bit_se5824344872417868541ns_and @ A @ A2 @ ( bit_se5824344872417868541ns_and @ A @ ( bit_se2239418461657761734s_mask @ A @ ( plus_plus @ nat @ M @ N ) ) @ ( bit_ri4277139882892585799ns_not @ A @ ( bit_se2239418461657761734s_mask @ A @ N ) ) ) ) ) ) ).
% slice_eq_mask
thf(fact_3365_pred__def,axiom,
( pred
= ( case_nat @ nat @ ( zero_zero @ nat )
@ ^ [X24: nat] : X24 ) ) ).
% pred_def
thf(fact_3366_and__int_Osimps,axiom,
( ( bit_se5824344872417868541ns_and @ int )
= ( ^ [K2: int,L2: int] :
( if @ int
@ ( ( member @ int @ K2 @ ( 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 ) ) @ K2 )
& ~ ( 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 ) ) @ K2 )
& ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L2 ) ) )
@ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( divide_divide @ int @ K2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ).
% and_int.simps
thf(fact_3367_and__int_Oelims,axiom,
! [X2: int,Xa: int,Y: int] :
( ( ( bit_se5824344872417868541ns_and @ int @ X2 @ Xa )
= Y )
=> ( ( ( ( member @ int @ X2 @ ( 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 ) ) @ X2 )
& ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Xa ) ) ) ) ) )
& ( ~ ( ( member @ int @ X2 @ ( 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 ) ) @ X2 )
& ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Xa ) ) )
@ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( divide_divide @ int @ X2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ Xa @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% and_int.elims
thf(fact_3368_card__lists__length__le,axiom,
! [A: $tType,A5: set @ A,N: nat] :
( ( finite_finite2 @ A @ A5 )
=> ( ( finite_card @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Xs3: list @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs3 ) @ A5 )
& ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs3 ) @ N ) ) ) )
= ( groups7311177749621191930dd_sum @ nat @ nat @ ( power_power @ nat @ ( finite_card @ A @ A5 ) ) @ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% card_lists_length_le
thf(fact_3369_horner__sum__eq__sum__funpow,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_0 @ A )
=> ( ( groups4207007520872428315er_sum @ B @ A )
= ( ^ [F2: B > A,A3: A,Xs3: list @ B] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N4: nat] : ( compow @ ( A > A ) @ N4 @ ( times_times @ A @ A3 ) @ ( F2 @ ( nth @ B @ Xs3 @ N4 ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ B ) @ Xs3 ) ) ) ) ) ) ).
% horner_sum_eq_sum_funpow
thf(fact_3370_bot__apply,axiom,
! [C: $tType,D: $tType] :
( ( bot @ C )
=> ( ( bot_bot @ ( D > C ) )
= ( ^ [X: D] : ( bot_bot @ C ) ) ) ) ).
% bot_apply
thf(fact_3371_Suc__funpow,axiom,
! [N: nat] :
( ( compow @ ( nat > nat ) @ N @ suc )
= ( plus_plus @ nat @ N ) ) ).
% Suc_funpow
thf(fact_3372_max__bot,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [X2: A] :
( ( ord_max @ A @ ( bot_bot @ A ) @ X2 )
= X2 ) ) ).
% max_bot
thf(fact_3373_max__bot2,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [X2: A] :
( ( ord_max @ A @ X2 @ ( bot_bot @ A ) )
= X2 ) ) ).
% max_bot2
thf(fact_3374_funpow__0,axiom,
! [A: $tType,F: A > A,X2: A] :
( ( compow @ ( A > A ) @ ( zero_zero @ nat ) @ F @ X2 )
= X2 ) ).
% funpow_0
thf(fact_3375_sum_Oempty,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: B > A] :
( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( bot_bot @ ( set @ B ) ) )
= ( zero_zero @ A ) ) ) ).
% sum.empty
thf(fact_3376_atLeastatMost__empty__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A,B2: A] :
( ( ( set_or1337092689740270186AtMost @ A @ A2 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
= ( ~ ( ord_less_eq @ A @ A2 @ B2 ) ) ) ) ).
% atLeastatMost_empty_iff
thf(fact_3377_atLeastatMost__empty__iff2,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A,B2: A] :
( ( ( bot_bot @ ( set @ A ) )
= ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( ~ ( ord_less_eq @ A @ A2 @ B2 ) ) ) ) ).
% atLeastatMost_empty_iff2
thf(fact_3378_atLeastatMost__empty,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B2: A,A2: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ( ( set_or1337092689740270186AtMost @ A @ A2 @ B2 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% atLeastatMost_empty
thf(fact_3379_atLeastLessThan__empty,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( set_or7035219750837199246ssThan @ A @ A2 @ B2 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% atLeastLessThan_empty
thf(fact_3380_atLeastLessThan__empty__iff2,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A,B2: A] :
( ( ( bot_bot @ ( set @ A ) )
= ( set_or7035219750837199246ssThan @ A @ A2 @ B2 ) )
= ( ~ ( ord_less @ A @ A2 @ B2 ) ) ) ) ).
% atLeastLessThan_empty_iff2
thf(fact_3381_atLeastLessThan__empty__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A,B2: A] :
( ( ( set_or7035219750837199246ssThan @ A @ A2 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
= ( ~ ( ord_less @ A @ A2 @ B2 ) ) ) ) ).
% atLeastLessThan_empty_iff
thf(fact_3382_prod_Oempty,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: B > A] :
( ( groups7121269368397514597t_prod @ B @ A @ G @ ( bot_bot @ ( set @ B ) ) )
= ( one_one @ A ) ) ) ).
% prod.empty
thf(fact_3383_card_Oempty,axiom,
! [A: $tType] :
( ( finite_card @ A @ ( bot_bot @ ( set @ A ) ) )
= ( zero_zero @ nat ) ) ).
% card.empty
thf(fact_3384_atLeastAtMost__singleton__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ( set_or1337092689740270186AtMost @ A @ A2 @ B2 )
= ( insert @ A @ C2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( ( A2 = B2 )
& ( B2 = C2 ) ) ) ) ).
% atLeastAtMost_singleton_iff
thf(fact_3385_atLeastAtMost__singleton,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A] :
( ( set_or1337092689740270186AtMost @ A @ A2 @ A2 )
= ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% atLeastAtMost_singleton
thf(fact_3386_some__insert__self,axiom,
! [A: $tType,S2: set @ A] :
( ( S2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( insert @ A
@ ( fChoice @ A
@ ^ [X: A] : ( member @ A @ X @ S2 ) )
@ S2 )
= S2 ) ) ).
% some_insert_self
thf(fact_3387_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_3388_card__doubleton__eq__2__iff,axiom,
! [A: $tType,A2: A,B2: A] :
( ( ( finite_card @ A @ ( insert @ A @ A2 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( A2 != B2 ) ) ).
% card_doubleton_eq_2_iff
thf(fact_3389_funpow__mult,axiom,
! [A: $tType,N: nat,M: nat,F: A > A] :
( ( compow @ ( A > A ) @ N @ ( compow @ ( A > A ) @ M @ F ) )
= ( compow @ ( A > A ) @ ( times_times @ nat @ M @ N ) @ F ) ) ).
% funpow_mult
thf(fact_3390_funpow__swap1,axiom,
! [A: $tType,F: A > A,N: nat,X2: A] :
( ( F @ ( compow @ ( A > A ) @ N @ F @ X2 ) )
= ( compow @ ( A > A ) @ N @ F @ ( F @ X2 ) ) ) ).
% funpow_swap1
thf(fact_3391_bot__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( bot @ B )
=> ( ( bot_bot @ ( A > B ) )
= ( ^ [X: A] : ( bot_bot @ B ) ) ) ) ).
% bot_fun_def
thf(fact_3392_set__notEmptyE,axiom,
! [A: $tType,S2: set @ A] :
( ( S2
!= ( bot_bot @ ( set @ A ) ) )
=> ~ ! [X3: A] :
~ ( member @ A @ X3 @ S2 ) ) ).
% set_notEmptyE
thf(fact_3393_memb__imp__not__empty,axiom,
! [A: $tType,X2: A,S2: set @ A] :
( ( member @ A @ X2 @ S2 )
=> ( S2
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% memb_imp_not_empty
thf(fact_3394_bot_Oextremum__strict,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A2: A] :
~ ( ord_less @ A @ A2 @ ( bot_bot @ A ) ) ) ).
% bot.extremum_strict
thf(fact_3395_bot_Onot__eq__extremum,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A2: A] :
( ( A2
!= ( bot_bot @ A ) )
= ( ord_less @ A @ ( bot_bot @ A ) @ A2 ) ) ) ).
% bot.not_eq_extremum
thf(fact_3396_bot_Oextremum,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A2: A] : ( ord_less_eq @ A @ ( bot_bot @ A ) @ A2 ) ) ).
% bot.extremum
thf(fact_3397_bot_Oextremum__unique,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ A2 @ ( bot_bot @ A ) )
= ( A2
= ( bot_bot @ A ) ) ) ) ).
% bot.extremum_unique
thf(fact_3398_bot_Oextremum__uniqueI,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ A2 @ ( bot_bot @ A ) )
=> ( A2
= ( bot_bot @ A ) ) ) ) ).
% bot.extremum_uniqueI
thf(fact_3399_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_3400_some__elem,axiom,
! [A: $tType,S2: set @ A] :
( ( S2
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ A
@ ( fChoice @ A
@ ^ [X: A] : ( member @ A @ X @ S2 ) )
@ S2 ) ) ).
% some_elem
thf(fact_3401_some__in__eq,axiom,
! [A: $tType,A5: set @ A] :
( ( member @ A
@ ( fChoice @ A
@ ^ [X: A] : ( member @ A @ X @ A5 ) )
@ A5 )
= ( A5
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% some_in_eq
thf(fact_3402_diff__shunt__var,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X2: A,Y: A] :
( ( ( minus_minus @ A @ X2 @ Y )
= ( bot_bot @ A ) )
= ( ord_less_eq @ A @ X2 @ Y ) ) ) ).
% diff_shunt_var
thf(fact_3403_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_3404_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_3405_ex__min__if__finite,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [S2: set @ A] :
( ( finite_finite2 @ A @ S2 )
=> ( ( S2
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X3: A] :
( ( member @ A @ X3 @ S2 )
& ~ ? [Xa2: A] :
( ( member @ A @ Xa2 @ S2 )
& ( ord_less @ A @ Xa2 @ X3 ) ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_3406_infinite__growing,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X8: set @ A] :
( ( X8
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ X8 )
=> ? [Xa2: A] :
( ( member @ A @ Xa2 @ X8 )
& ( ord_less @ A @ X3 @ Xa2 ) ) )
=> ~ ( finite_finite2 @ A @ X8 ) ) ) ) ).
% infinite_growing
thf(fact_3407_subset__minus__empty,axiom,
! [A: $tType,A5: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
=> ( ( minus_minus @ ( set @ A ) @ A5 @ B5 )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_minus_empty
thf(fact_3408_insert__minus__eq,axiom,
! [A: $tType,X2: A,Y: A,A5: set @ A] :
( ( X2 != Y )
=> ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X2 @ A5 ) @ ( insert @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) )
= ( insert @ A @ X2 @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% insert_minus_eq
thf(fact_3409_set__minus__singleton__eq,axiom,
! [A: $tType,X2: A,X8: set @ A] :
( ~ ( member @ A @ X2 @ X8 )
=> ( ( minus_minus @ ( set @ A ) @ X8 @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) )
= X8 ) ) ).
% set_minus_singleton_eq
thf(fact_3410_atLeastAtMost__singleton_H,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B2: A] :
( ( A2 = B2 )
=> ( ( set_or1337092689740270186AtMost @ A @ A2 @ B2 )
= ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% atLeastAtMost_singleton'
thf(fact_3411_arg__min__if__finite_I1_J,axiom,
! [B: $tType,A: $tType] :
( ( order @ B )
=> ! [S2: set @ A,F: A > B] :
( ( finite_finite2 @ A @ S2 )
=> ( ( S2
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ A @ ( lattic7623131987881927897min_on @ A @ B @ F @ S2 ) @ S2 ) ) ) ) ).
% arg_min_if_finite(1)
thf(fact_3412_length__pos__if__in__set,axiom,
! [A: $tType,X2: A,Xs: list @ A] :
( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_3413_sorted__simps_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Ys: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( cons @ A @ X2 @ Ys ) )
= ( ! [X: A] :
( ( member @ A @ X @ ( set2 @ A @ Ys ) )
=> ( ord_less_eq @ A @ X2 @ X ) )
& ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Ys ) ) ) ) ).
% sorted_simps(2)
thf(fact_3414_all__set__conv__all__nth,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( ( ! [X: A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( P @ X ) ) )
= ( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( P @ ( nth @ A @ Xs @ I2 ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_3415_all__nth__imp__all__set,axiom,
! [A: $tType,Xs: list @ A,P: A > $o,X2: A] :
( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( P @ ( nth @ A @ Xs @ I3 ) ) )
=> ( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
=> ( P @ X2 ) ) ) ).
% all_nth_imp_all_set
thf(fact_3416_in__set__conv__nth,axiom,
! [A: $tType,X2: A,Xs: list @ A] :
( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
= ( ? [I2: nat] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( ( nth @ A @ Xs @ I2 )
= X2 ) ) ) ) ).
% in_set_conv_nth
thf(fact_3417_list__ball__nth,axiom,
! [A: $tType,N: nat,Xs: list @ A,P: A > $o] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( P @ X3 ) )
=> ( P @ ( nth @ A @ Xs @ N ) ) ) ) ).
% list_ball_nth
thf(fact_3418_nth__mem,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( member @ A @ ( nth @ A @ Xs @ N ) @ ( set2 @ A @ Xs ) ) ) ).
% nth_mem
thf(fact_3419_all__set__conv__nth,axiom,
! [A: $tType,L: list @ A,P: A > $o] :
( ( ! [X: A] :
( ( member @ A @ X @ ( set2 @ A @ L ) )
=> ( P @ X ) ) )
= ( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ L ) )
=> ( P @ ( nth @ A @ L @ I2 ) ) ) ) ) ).
% all_set_conv_nth
thf(fact_3420_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_3421_ex__is__arg__min__if__finite,axiom,
! [B: $tType,A: $tType] :
( ( order @ B )
=> ! [S2: set @ A,F: A > B] :
( ( finite_finite2 @ A @ S2 )
=> ( ( S2
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X_12: A] :
( lattic501386751177426532rg_min @ A @ B @ F
@ ^ [X: A] : ( member @ A @ X @ S2 )
@ X_12 ) ) ) ) ).
% ex_is_arg_min_if_finite
thf(fact_3422_finite__ranking__induct,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [S2: set @ B,P: ( set @ B ) > $o,F: B > A] :
( ( finite_finite2 @ B @ S2 )
=> ( ( P @ ( bot_bot @ ( set @ B ) ) )
=> ( ! [X3: B,S8: set @ B] :
( ( finite_finite2 @ B @ S8 )
=> ( ! [Y5: B] :
( ( member @ B @ Y5 @ S8 )
=> ( ord_less_eq @ A @ ( F @ Y5 ) @ ( F @ X3 ) ) )
=> ( ( P @ S8 )
=> ( P @ ( insert @ B @ X3 @ S8 ) ) ) ) )
=> ( P @ S2 ) ) ) ) ) ).
% finite_ranking_induct
thf(fact_3423_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 ) ) )
=> ( ! [B4: A,A7: set @ A] :
( ( finite_finite2 @ A @ A7 )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ A7 )
=> ( ord_less @ A @ B4 @ X4 ) )
=> ( ( P @ A7 )
=> ( P @ ( insert @ A @ B4 @ A7 ) ) ) ) )
=> ( P @ A5 ) ) ) ) ) ).
% finite_linorder_min_induct
thf(fact_3424_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 ) ) )
=> ( ! [B4: A,A7: set @ A] :
( ( finite_finite2 @ A @ A7 )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ A7 )
=> ( ord_less @ A @ X4 @ B4 ) )
=> ( ( P @ A7 )
=> ( P @ ( insert @ A @ B4 @ A7 ) ) ) ) )
=> ( P @ A5 ) ) ) ) ) ).
% finite_linorder_max_induct
thf(fact_3425_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_3426_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_3427_remove__subset,axiom,
! [A: $tType,X2: A,S2: set @ A] :
( ( member @ A @ X2 @ S2 )
=> ( ord_less @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ S2 @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) @ S2 ) ) ).
% remove_subset
thf(fact_3428_bot_Oordering__top__axioms,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ( ordering_top @ A
@ ^ [X: A,Y4: A] : ( ord_less_eq @ A @ Y4 @ X )
@ ^ [X: A,Y4: A] : ( ord_less @ A @ Y4 @ X )
@ ( bot_bot @ A ) ) ) ).
% bot.ordering_top_axioms
thf(fact_3429_of__nat__def,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( semiring_1_of_nat @ A )
= ( ^ [N4: nat] : ( compow @ ( A > A ) @ N4 @ ( plus_plus @ A @ ( one_one @ A ) ) @ ( zero_zero @ A ) ) ) ) ) ).
% of_nat_def
thf(fact_3430_numeral__add__unfold__funpow,axiom,
! [A: $tType] :
( ( semiring_numeral @ A )
=> ! [K: num,A2: A] :
( ( plus_plus @ A @ ( numeral_numeral @ A @ K ) @ A2 )
= ( compow @ ( A > A ) @ ( numeral_numeral @ nat @ K ) @ ( plus_plus @ A @ ( one_one @ A ) ) @ A2 ) ) ) ).
% numeral_add_unfold_funpow
thf(fact_3431_arg__min__least,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [S2: set @ A,Y: A,F: A > B] :
( ( finite_finite2 @ A @ S2 )
=> ( ( S2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( member @ A @ Y @ S2 )
=> ( ord_less_eq @ B @ ( F @ ( lattic7623131987881927897min_on @ A @ B @ F @ S2 ) ) @ ( F @ Y ) ) ) ) ) ) ).
% arg_min_least
thf(fact_3432_arg__min__if__finite_I2_J,axiom,
! [B: $tType,A: $tType] :
( ( order @ B )
=> ! [S2: set @ A,F: A > B] :
( ( finite_finite2 @ A @ S2 )
=> ( ( S2
!= ( bot_bot @ ( set @ A ) ) )
=> ~ ? [X4: A] :
( ( member @ A @ X4 @ S2 )
& ( ord_less @ B @ ( F @ X4 ) @ ( F @ ( lattic7623131987881927897min_on @ A @ B @ F @ S2 ) ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_3433_sum__pos,axiom,
! [A: $tType,B: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [I5: set @ B,F: B > A] :
( ( finite_finite2 @ B @ I5 )
=> ( ( I5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ I5 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( F @ I3 ) ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( groups7311177749621191930dd_sum @ B @ A @ F @ I5 ) ) ) ) ) ) ).
% sum_pos
thf(fact_3434_less__1__prod,axiom,
! [B: $tType,A: $tType] :
( ( linordered_idom @ B )
=> ! [I5: set @ A,F: A > B] :
( ( finite_finite2 @ A @ I5 )
=> ( ( I5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [I3: A] :
( ( member @ A @ I3 @ I5 )
=> ( ord_less @ B @ ( one_one @ B ) @ ( F @ I3 ) ) )
=> ( ord_less @ B @ ( one_one @ B ) @ ( groups7121269368397514597t_prod @ A @ B @ F @ I5 ) ) ) ) ) ) ).
% less_1_prod
thf(fact_3435_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_3436_card__Suc__eq,axiom,
! [A: $tType,A5: set @ A,K: nat] :
( ( ( finite_card @ A @ A5 )
= ( suc @ K ) )
= ( ? [B3: A,B8: set @ A] :
( ( A5
= ( insert @ A @ B3 @ B8 ) )
& ~ ( member @ A @ B3 @ B8 )
& ( ( finite_card @ A @ B8 )
= K )
& ( ( K
= ( zero_zero @ nat ) )
=> ( B8
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% card_Suc_eq
thf(fact_3437_card__eq__SucD,axiom,
! [A: $tType,A5: set @ A,K: nat] :
( ( ( finite_card @ A @ A5 )
= ( suc @ K ) )
=> ? [B4: A,B10: set @ A] :
( ( A5
= ( insert @ A @ B4 @ B10 ) )
& ~ ( member @ A @ B4 @ B10 )
& ( ( finite_card @ A @ B10 )
= K )
& ( ( K
= ( zero_zero @ nat ) )
=> ( B10
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% card_eq_SucD
thf(fact_3438_card__1__singleton__iff,axiom,
! [A: $tType,A5: set @ A] :
( ( ( finite_card @ A @ A5 )
= ( suc @ ( zero_zero @ nat ) ) )
= ( ? [X: A] :
( A5
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% card_1_singleton_iff
thf(fact_3439_card__1__singletonI,axiom,
! [A: $tType,S2: set @ A,X2: A] :
( ( finite_finite2 @ A @ S2 )
=> ( ( ( finite_card @ A @ S2 )
= ( one_one @ nat ) )
=> ( ( member @ A @ X2 @ S2 )
=> ( S2
= ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% card_1_singletonI
thf(fact_3440_card__Diff1__le,axiom,
! [A: $tType,A5: set @ A,X2: A] : ( ord_less_eq @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) @ ( finite_card @ A @ A5 ) ) ).
% card_Diff1_le
thf(fact_3441_atLeastLessThan__eq__atLeastAtMost__diff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( set_or7035219750837199246ssThan @ A )
= ( ^ [A3: A,B3: A] : ( minus_minus @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A3 @ B3 ) @ ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% atLeastLessThan_eq_atLeastAtMost_diff
thf(fact_3442_atLeastLessThan__upto,axiom,
( ( set_or7035219750837199246ssThan @ int )
= ( ^ [I2: int,J3: int] : ( set2 @ int @ ( upto @ I2 @ ( minus_minus @ int @ J3 @ ( one_one @ int ) ) ) ) ) ) ).
% atLeastLessThan_upto
thf(fact_3443_numeral__unfold__funpow,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( numeral_numeral @ A )
= ( ^ [K2: num] : ( compow @ ( A > A ) @ ( numeral_numeral @ nat @ K2 ) @ ( plus_plus @ A @ ( one_one @ A ) ) @ ( zero_zero @ A ) ) ) ) ) ).
% numeral_unfold_funpow
thf(fact_3444_simp__from__to,axiom,
( ( set_or1337092689740270186AtMost @ int )
= ( ^ [I2: int,J3: int] : ( if @ ( set @ int ) @ ( ord_less @ int @ J3 @ I2 ) @ ( bot_bot @ ( set @ int ) ) @ ( insert @ int @ I2 @ ( set_or1337092689740270186AtMost @ int @ ( plus_plus @ int @ I2 @ ( one_one @ int ) ) @ J3 ) ) ) ) ) ).
% simp_from_to
thf(fact_3445_finite__lists__length__eq,axiom,
! [A: $tType,A5: set @ A,N: nat] :
( ( finite_finite2 @ A @ A5 )
=> ( finite_finite2 @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Xs3: list @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs3 ) @ A5 )
& ( ( size_size @ ( list @ A ) @ Xs3 )
= N ) ) ) ) ) ).
% finite_lists_length_eq
thf(fact_3446_nth__equal__first__eq,axiom,
! [A: $tType,X2: A,Xs: list @ A,N: nat] :
( ~ ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
=> ( ( ord_less_eq @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ( nth @ A @ ( cons @ A @ X2 @ Xs ) @ N )
= X2 )
= ( N
= ( zero_zero @ nat ) ) ) ) ) ).
% nth_equal_first_eq
thf(fact_3447_sorted__list__of__set_Ofinite__set__strict__sorted,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ~ ! [L3: list @ A] :
( ( sorted_wrt @ A @ ( ord_less @ A ) @ L3 )
=> ( ( ( set2 @ A @ L3 )
= A5 )
=> ( ( size_size @ ( list @ A ) @ L3 )
!= ( finite_card @ A @ A5 ) ) ) ) ) ) ).
% sorted_list_of_set.finite_set_strict_sorted
thf(fact_3448_prod__mono__strict,axiom,
! [A: $tType,B: $tType] :
( ( linordered_semidom @ A )
=> ! [A5: set @ B,F: B > A,G: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ A5 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F @ I3 ) )
& ( ord_less @ A @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
=> ( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ord_less @ A @ ( groups7121269368397514597t_prod @ B @ A @ F @ A5 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ A5 ) ) ) ) ) ) ).
% prod_mono_strict
thf(fact_3449_sum_Oremove,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A5: set @ B,X2: B,G: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( member @ B @ X2 @ A5 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ A5 )
= ( plus_plus @ A @ ( G @ X2 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A5 @ ( insert @ B @ X2 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ).
% sum.remove
thf(fact_3450_sum_Oinsert__remove,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A5: set @ B,G: B > A,X2: B] :
( ( finite_finite2 @ B @ A5 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( insert @ B @ X2 @ A5 ) )
= ( plus_plus @ A @ ( G @ X2 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A5 @ ( insert @ B @ X2 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ).
% sum.insert_remove
thf(fact_3451_card__2__iff,axiom,
! [A: $tType,S2: set @ A] :
( ( ( finite_card @ A @ S2 )
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( ? [X: A,Y4: A] :
( ( S2
= ( insert @ A @ X @ ( insert @ A @ Y4 @ ( bot_bot @ ( set @ A ) ) ) ) )
& ( X != Y4 ) ) ) ) ).
% card_2_iff
thf(fact_3452_card__3__iff,axiom,
! [A: $tType,S2: set @ A] :
( ( ( finite_card @ A @ S2 )
= ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) )
= ( ? [X: A,Y4: A,Z3: A] :
( ( S2
= ( insert @ A @ X @ ( insert @ A @ Y4 @ ( insert @ A @ Z3 @ ( bot_bot @ ( set @ A ) ) ) ) ) )
& ( X != Y4 )
& ( Y4 != Z3 )
& ( X != Z3 ) ) ) ) ).
% card_3_iff
thf(fact_3453_card__lists__length__eq,axiom,
! [A: $tType,A5: set @ A,N: nat] :
( ( finite_finite2 @ A @ A5 )
=> ( ( finite_card @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Xs3: list @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs3 ) @ A5 )
& ( ( size_size @ ( list @ A ) @ Xs3 )
= N ) ) ) )
= ( power_power @ nat @ ( finite_card @ A @ A5 ) @ N ) ) ) ).
% card_lists_length_eq
thf(fact_3454_card__Diff__singleton__if,axiom,
! [A: $tType,X2: A,A5: set @ A] :
( ( ( member @ A @ X2 @ A5 )
=> ( ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( minus_minus @ nat @ ( finite_card @ A @ A5 ) @ ( one_one @ nat ) ) ) )
& ( ~ ( member @ A @ X2 @ A5 )
=> ( ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( finite_card @ A @ A5 ) ) ) ) ).
% card_Diff_singleton_if
thf(fact_3455_card__Diff__singleton,axiom,
! [A: $tType,X2: A,A5: set @ A] :
( ( member @ A @ X2 @ A5 )
=> ( ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( minus_minus @ nat @ ( finite_card @ A @ A5 ) @ ( one_one @ nat ) ) ) ) ).
% card_Diff_singleton
thf(fact_3456_finite__lists__length__le,axiom,
! [A: $tType,A5: set @ A,N: nat] :
( ( finite_finite2 @ A @ A5 )
=> ( finite_finite2 @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Xs3: list @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs3 ) @ A5 )
& ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs3 ) @ N ) ) ) ) ) ).
% finite_lists_length_le
thf(fact_3457_sum_Odelta__remove,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S2: set @ B,A2: B,B2: B > A,C2: B > A] :
( ( finite_finite2 @ B @ S2 )
=> ( ( ( member @ B @ A2 @ S2 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [K2: B] : ( if @ A @ ( K2 = A2 ) @ ( B2 @ K2 ) @ ( C2 @ K2 ) )
@ S2 )
= ( plus_plus @ A @ ( B2 @ A2 ) @ ( groups7311177749621191930dd_sum @ B @ A @ C2 @ ( minus_minus @ ( set @ B ) @ S2 @ ( insert @ B @ A2 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) )
& ( ~ ( member @ B @ A2 @ S2 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [K2: B] : ( if @ A @ ( K2 = A2 ) @ ( B2 @ K2 ) @ ( C2 @ K2 ) )
@ S2 )
= ( groups7311177749621191930dd_sum @ B @ A @ C2 @ ( minus_minus @ ( set @ B ) @ S2 @ ( insert @ B @ A2 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ).
% sum.delta_remove
thf(fact_3458_prod_Odelta__remove,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S2: set @ B,A2: B,B2: B > A,C2: B > A] :
( ( finite_finite2 @ B @ S2 )
=> ( ( ( member @ B @ A2 @ S2 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K2: B] : ( if @ A @ ( K2 = A2 ) @ ( B2 @ K2 ) @ ( C2 @ K2 ) )
@ S2 )
= ( times_times @ A @ ( B2 @ A2 ) @ ( groups7121269368397514597t_prod @ B @ A @ C2 @ ( minus_minus @ ( set @ B ) @ S2 @ ( insert @ B @ A2 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) )
& ( ~ ( member @ B @ A2 @ S2 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K2: B] : ( if @ A @ ( K2 = A2 ) @ ( B2 @ K2 ) @ ( C2 @ K2 ) )
@ S2 )
= ( groups7121269368397514597t_prod @ B @ A @ C2 @ ( minus_minus @ ( set @ B ) @ S2 @ ( insert @ B @ A2 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ).
% prod.delta_remove
thf(fact_3459_member__le__sum,axiom,
! [B: $tType,C: $tType] :
( ( ( ordere6911136660526730532id_add @ B )
& ( semiring_1 @ B ) )
=> ! [I: C,A5: set @ C,F: C > B] :
( ( member @ C @ I @ A5 )
=> ( ! [X3: C] :
( ( member @ C @ X3 @ ( minus_minus @ ( set @ C ) @ A5 @ ( insert @ C @ I @ ( bot_bot @ ( set @ C ) ) ) ) )
=> ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( F @ X3 ) ) )
=> ( ( finite_finite2 @ C @ A5 )
=> ( ord_less_eq @ B @ ( F @ I ) @ ( groups7311177749621191930dd_sum @ C @ B @ F @ A5 ) ) ) ) ) ) ).
% member_le_sum
thf(fact_3460_prod__diff1,axiom,
! [A: $tType,B: $tType] :
( ( semidom_divide @ A )
=> ! [A5: set @ B,F: B > A,A2: B] :
( ( finite_finite2 @ B @ A5 )
=> ( ( ( F @ A2 )
!= ( zero_zero @ A ) )
=> ( ( ( member @ B @ A2 @ A5 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ F @ ( minus_minus @ ( set @ B ) @ A5 @ ( insert @ B @ A2 @ ( bot_bot @ ( set @ B ) ) ) ) )
= ( divide_divide @ A @ ( groups7121269368397514597t_prod @ B @ A @ F @ A5 ) @ ( F @ A2 ) ) ) )
& ( ~ ( member @ B @ A2 @ A5 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ F @ ( minus_minus @ ( set @ B ) @ A5 @ ( insert @ B @ A2 @ ( bot_bot @ ( set @ B ) ) ) ) )
= ( groups7121269368397514597t_prod @ B @ A @ F @ A5 ) ) ) ) ) ) ) ).
% prod_diff1
thf(fact_3461_sum__bounded__above__divide,axiom,
! [B: $tType,A: $tType] :
( ( linordered_field @ A )
=> ! [A5: set @ B,F: B > A,K3: A] :
( ! [I3: B] :
( ( member @ B @ I3 @ A5 )
=> ( ord_less_eq @ A @ ( F @ I3 ) @ ( divide_divide @ A @ K3 @ ( semiring_1_of_nat @ A @ ( finite_card @ B @ A5 ) ) ) ) )
=> ( ( finite_finite2 @ B @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F @ A5 ) @ K3 ) ) ) ) ) ).
% sum_bounded_above_divide
thf(fact_3462_singleton__conv2,axiom,
! [A: $tType,A2: A] :
( ( collect @ A
@ ( ^ [Y3: A,Z: A] : Y3 = Z
@ A2 ) )
= ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% singleton_conv2
thf(fact_3463_singleton__conv,axiom,
! [A: $tType,A2: A] :
( ( collect @ A
@ ^ [X: A] : X = A2 )
= ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% singleton_conv
thf(fact_3464_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
@ ^ [X: nat] : X
@ ( 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_3465_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
@ ^ [X: nat] : X
@ ( 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_3466_relpowp__fun__conv,axiom,
! [A: $tType] :
( ( compow @ ( A > A > $o ) )
= ( ^ [N4: nat,P4: A > A > $o,X: A,Y4: A] :
? [F2: nat > A] :
( ( ( F2 @ ( zero_zero @ nat ) )
= X )
& ( ( F2 @ N4 )
= Y4 )
& ! [I2: nat] :
( ( ord_less @ nat @ I2 @ N4 )
=> ( P4 @ ( F2 @ I2 ) @ ( F2 @ ( suc @ I2 ) ) ) ) ) ) ) ).
% relpowp_fun_conv
thf(fact_3467_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_3468_relpowp__1,axiom,
! [A: $tType,P: A > A > $o] :
( ( compow @ ( A > A > $o ) @ ( one_one @ nat ) @ P )
= P ) ).
% relpowp_1
thf(fact_3469_set__decode__zero,axiom,
( ( nat_set_decode @ ( zero_zero @ nat ) )
= ( bot_bot @ ( set @ nat ) ) ) ).
% set_decode_zero
thf(fact_3470_set__encode__empty,axiom,
( ( nat_set_encode @ ( bot_bot @ ( set @ nat ) ) )
= ( zero_zero @ nat ) ) ).
% set_encode_empty
thf(fact_3471_atLeastLessThan__singleton,axiom,
! [M: nat] :
( ( set_or7035219750837199246ssThan @ nat @ M @ ( suc @ M ) )
= ( insert @ nat @ M @ ( bot_bot @ ( set @ nat ) ) ) ) ).
% atLeastLessThan_singleton
thf(fact_3472_atMost__0,axiom,
( ( set_ord_atMost @ nat @ ( zero_zero @ nat ) )
= ( insert @ nat @ ( zero_zero @ nat ) @ ( bot_bot @ ( set @ nat ) ) ) ) ).
% atMost_0
thf(fact_3473_finite__lists__distinct__length__eq,axiom,
! [A: $tType,A5: set @ A,N: nat] :
( ( finite_finite2 @ A @ A5 )
=> ( finite_finite2 @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Xs3: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs3 )
= N )
& ( distinct @ A @ Xs3 )
& ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs3 ) @ A5 ) ) ) ) ) ).
% finite_lists_distinct_length_eq
thf(fact_3474_bot__nat__def,axiom,
( ( bot_bot @ nat )
= ( zero_zero @ nat ) ) ).
% bot_nat_def
thf(fact_3475_relpowp__bot,axiom,
! [A: $tType,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( compow @ ( A > A > $o ) @ N @ ( bot_bot @ ( A > A > $o ) ) )
= ( bot_bot @ ( A > A > $o ) ) ) ) ).
% relpowp_bot
thf(fact_3476_atLeastLessThan0,axiom,
! [M: nat] :
( ( set_or7035219750837199246ssThan @ nat @ M @ ( zero_zero @ nat ) )
= ( bot_bot @ ( set @ nat ) ) ) ).
% atLeastLessThan0
thf(fact_3477_distinct__finite__set,axiom,
! [A: $tType,X2: set @ A] :
( finite_finite2 @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Ys3: list @ A] :
( ( ( set2 @ A @ Ys3 )
= X2 )
& ( distinct @ A @ Ys3 ) ) ) ) ).
% distinct_finite_set
thf(fact_3478_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_3479_distinct__length__le,axiom,
! [A: $tType,Ys: list @ A,Xs: list @ A] :
( ( distinct @ A @ Ys )
=> ( ( ( set2 @ A @ Ys )
= ( set2 @ A @ Xs ) )
=> ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Ys ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ).
% distinct_length_le
thf(fact_3480_sorted__distinct__set__unique,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,Ys: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( ( distinct @ A @ Xs )
=> ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Ys )
=> ( ( distinct @ A @ Ys )
=> ( ( ( set2 @ A @ Xs )
= ( set2 @ A @ Ys ) )
=> ( Xs = Ys ) ) ) ) ) ) ) ).
% sorted_distinct_set_unique
thf(fact_3481_distinct__conv__nth,axiom,
! [A: $tType] :
( ( distinct @ A )
= ( ^ [Xs3: list @ A] :
! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs3 ) )
=> ! [J3: nat] :
( ( ord_less @ nat @ J3 @ ( size_size @ ( list @ A ) @ Xs3 ) )
=> ( ( I2 != J3 )
=> ( ( nth @ A @ Xs3 @ I2 )
!= ( nth @ A @ Xs3 @ J3 ) ) ) ) ) ) ) ).
% distinct_conv_nth
thf(fact_3482_nth__eq__iff__index__eq,axiom,
! [A: $tType,Xs: list @ A,I: nat,J: nat] :
( ( distinct @ A @ Xs )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ( nth @ A @ Xs @ I )
= ( nth @ A @ Xs @ J ) )
= ( I = J ) ) ) ) ) ).
% nth_eq_iff_index_eq
thf(fact_3483_card__distinct,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( finite_card @ A @ ( set2 @ A @ Xs ) )
= ( size_size @ ( list @ A ) @ Xs ) )
=> ( distinct @ A @ Xs ) ) ).
% card_distinct
thf(fact_3484_distinct__card,axiom,
! [A: $tType,Xs: list @ A] :
( ( distinct @ A @ Xs )
=> ( ( finite_card @ A @ ( set2 @ A @ Xs ) )
= ( size_size @ ( list @ A ) @ Xs ) ) ) ).
% distinct_card
thf(fact_3485_relpowp_Osimps_I1_J,axiom,
! [A: $tType,R3: A > A > $o] :
( ( compow @ ( A > A > $o ) @ ( zero_zero @ nat ) @ R3 )
= ( ^ [Y3: A,Z: A] : Y3 = Z ) ) ).
% relpowp.simps(1)
thf(fact_3486_relpowp__0__E,axiom,
! [A: $tType,P: A > A > $o,X2: A,Y: A] :
( ( compow @ ( A > A > $o ) @ ( zero_zero @ nat ) @ P @ X2 @ Y )
=> ( X2 = Y ) ) ).
% relpowp_0_E
thf(fact_3487_relpowp__0__I,axiom,
! [A: $tType,P: A > A > $o,X2: A] : ( compow @ ( A > A > $o ) @ ( zero_zero @ nat ) @ P @ X2 @ X2 ) ).
% relpowp_0_I
thf(fact_3488_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 )
& ! [Y5: list @ A] :
( ( ( ( set2 @ A @ Y5 )
= A5 )
& ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Y5 )
& ( distinct @ A @ Y5 ) )
=> ( Y5 = X3 ) ) ) ) ) ).
% finite_sorted_distinct_unique
thf(fact_3489_distinct__Ex1,axiom,
! [A: $tType,Xs: list @ A,X2: A] :
( ( distinct @ A @ Xs )
=> ( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
=> ? [X3: nat] :
( ( ord_less @ nat @ X3 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( ( nth @ A @ Xs @ X3 )
= X2 )
& ! [Y5: nat] :
( ( ( ord_less @ nat @ Y5 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( ( nth @ A @ Xs @ Y5 )
= X2 ) )
=> ( Y5 = X3 ) ) ) ) ) ).
% distinct_Ex1
thf(fact_3490_distinct__finite__subset,axiom,
! [A: $tType,X2: set @ A] :
( ( finite_finite2 @ A @ X2 )
=> ( finite_finite2 @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Ys3: list @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Ys3 ) @ X2 )
& ( distinct @ A @ Ys3 ) ) ) ) ) ).
% distinct_finite_subset
thf(fact_3491_atLeastLessThanSuc,axiom,
! [M: nat,N: nat] :
( ( ( ord_less_eq @ nat @ M @ N )
=> ( ( set_or7035219750837199246ssThan @ nat @ M @ ( suc @ N ) )
= ( insert @ nat @ N @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) )
& ( ~ ( ord_less_eq @ nat @ M @ N )
=> ( ( set_or7035219750837199246ssThan @ nat @ M @ ( suc @ N ) )
= ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% atLeastLessThanSuc
thf(fact_3492_Set_Oempty__def,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
= ( collect @ A
@ ^ [X: A] : $false ) ) ).
% Set.empty_def
thf(fact_3493_insert__compr,axiom,
! [A: $tType] :
( ( insert @ A )
= ( ^ [A3: A,B8: set @ A] :
( collect @ A
@ ^ [X: A] :
( ( X = A3 )
| ( member @ A @ X @ B8 ) ) ) ) ) ).
% insert_compr
thf(fact_3494_insert__Collect,axiom,
! [A: $tType,A2: A,P: A > $o] :
( ( insert @ A @ A2 @ ( collect @ A @ P ) )
= ( collect @ A
@ ^ [U2: A] :
( ( U2 != A2 )
=> ( P @ U2 ) ) ) ) ).
% insert_Collect
thf(fact_3495_set__diff__eq,axiom,
! [A: $tType] :
( ( minus_minus @ ( set @ A ) )
= ( ^ [A8: set @ A,B8: set @ A] :
( collect @ A
@ ^ [X: A] :
( ( member @ A @ X @ A8 )
& ~ ( member @ A @ X @ B8 ) ) ) ) ) ).
% set_diff_eq
thf(fact_3496_minus__set__def,axiom,
! [A: $tType] :
( ( minus_minus @ ( set @ A ) )
= ( ^ [A8: set @ A,B8: set @ A] :
( collect @ A
@ ( minus_minus @ ( A > $o )
@ ^ [X: A] : ( member @ A @ X @ A8 )
@ ^ [X: A] : ( member @ A @ X @ B8 ) ) ) ) ) ).
% minus_set_def
thf(fact_3497_less__set__def,axiom,
! [A: $tType] :
( ( ord_less @ ( set @ A ) )
= ( ^ [A8: set @ A,B8: set @ A] :
( ord_less @ ( A > $o )
@ ^ [X: A] : ( member @ A @ X @ A8 )
@ ^ [X: A] : ( member @ A @ X @ B8 ) ) ) ) ).
% less_set_def
thf(fact_3498_distinct__sorted__strict__mono__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: list @ A,I: nat,J: nat] :
( ( distinct @ A @ L )
=> ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ L )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L ) )
=> ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ L ) )
=> ( ( ord_less @ A @ ( nth @ A @ L @ I ) @ ( nth @ A @ L @ J ) )
= ( ord_less @ nat @ I @ J ) ) ) ) ) ) ) ).
% distinct_sorted_strict_mono_iff
thf(fact_3499_distinct__sorted__mono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: list @ A,I: nat,J: nat] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ L )
=> ( ( distinct @ A @ L )
=> ( ( ord_less @ nat @ I @ J )
=> ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ L ) )
=> ( ord_less @ A @ ( nth @ A @ L @ I ) @ ( nth @ A @ L @ J ) ) ) ) ) ) ) ).
% distinct_sorted_mono
thf(fact_3500_distinct__sorted__mono__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: list @ A,I: nat,J: nat] :
( ( distinct @ A @ L )
=> ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ L )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L ) )
=> ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ L ) )
=> ( ( ord_less_eq @ A @ ( nth @ A @ L @ I ) @ ( nth @ A @ L @ J ) )
= ( ord_less_eq @ nat @ I @ J ) ) ) ) ) ) ) ).
% distinct_sorted_mono_iff
thf(fact_3501_atLeast1__atMost__eq__remove0,axiom,
! [N: nat] :
( ( set_or1337092689740270186AtMost @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N )
= ( minus_minus @ ( set @ nat ) @ ( set_ord_atMost @ nat @ N ) @ ( insert @ nat @ ( zero_zero @ nat ) @ ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% atLeast1_atMost_eq_remove0
thf(fact_3502_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_3503_relpowp__E2,axiom,
! [A: $tType,N: nat,P: A > A > $o,X2: A,Z2: A] :
( ( compow @ ( A > A > $o ) @ N @ P @ X2 @ Z2 )
=> ( ( ( N
= ( zero_zero @ nat ) )
=> ( X2 != Z2 ) )
=> ~ ! [Y2: A,M4: nat] :
( ( N
= ( suc @ M4 ) )
=> ( ( P @ X2 @ Y2 )
=> ~ ( compow @ ( A > A > $o ) @ M4 @ P @ Y2 @ Z2 ) ) ) ) ) ).
% relpowp_E2
thf(fact_3504_relpowp__E,axiom,
! [A: $tType,N: nat,P: A > A > $o,X2: A,Z2: A] :
( ( compow @ ( A > A > $o ) @ N @ P @ X2 @ Z2 )
=> ( ( ( N
= ( zero_zero @ nat ) )
=> ( X2 != Z2 ) )
=> ~ ! [Y2: A,M4: nat] :
( ( N
= ( suc @ M4 ) )
=> ( ( compow @ ( A > A > $o ) @ M4 @ P @ X2 @ Y2 )
=> ~ ( P @ Y2 @ Z2 ) ) ) ) ) ).
% relpowp_E
thf(fact_3505_Collect__conv__if,axiom,
! [A: $tType,P: A > $o,A2: A] :
( ( ( P @ A2 )
=> ( ( collect @ A
@ ^ [X: A] :
( ( X = A2 )
& ( P @ X ) ) )
= ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
& ( ~ ( P @ A2 )
=> ( ( collect @ A
@ ^ [X: A] :
( ( X = A2 )
& ( P @ X ) ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Collect_conv_if
thf(fact_3506_Collect__conv__if2,axiom,
! [A: $tType,P: A > $o,A2: A] :
( ( ( P @ A2 )
=> ( ( collect @ A
@ ^ [X: A] :
( ( A2 = X )
& ( P @ X ) ) )
= ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
& ( ~ ( P @ A2 )
=> ( ( collect @ A
@ ^ [X: A] :
( ( A2 = X )
& ( P @ X ) ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Collect_conv_if2
thf(fact_3507_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_3508_sum__count__set,axiom,
! [A: $tType,Xs: list @ A,X8: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ X8 )
=> ( ( finite_finite2 @ A @ X8 )
=> ( ( groups7311177749621191930dd_sum @ A @ nat @ ( count_list @ A @ Xs ) @ X8 )
= ( size_size @ ( list @ A ) @ Xs ) ) ) ) ).
% sum_count_set
thf(fact_3509_sum__diff1_H__aux,axiom,
! [B: $tType,A: $tType] :
( ( ab_group_add @ B )
=> ! [F4: set @ A,I5: set @ A,F: A > B,I: A] :
( ( finite_finite2 @ A @ F4 )
=> ( ( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [I2: A] :
( ( member @ A @ I2 @ I5 )
& ( ( F @ I2 )
!= ( zero_zero @ B ) ) ) )
@ F4 )
=> ( ( ( member @ A @ I @ I5 )
=> ( ( groups1027152243600224163dd_sum @ A @ B @ F @ ( minus_minus @ ( set @ A ) @ I5 @ ( insert @ A @ I @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( minus_minus @ B @ ( groups1027152243600224163dd_sum @ A @ B @ F @ I5 ) @ ( F @ I ) ) ) )
& ( ~ ( member @ A @ I @ I5 )
=> ( ( groups1027152243600224163dd_sum @ A @ B @ F @ ( minus_minus @ ( set @ A ) @ I5 @ ( insert @ A @ I @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( groups1027152243600224163dd_sum @ A @ B @ F @ I5 ) ) ) ) ) ) ) ).
% sum_diff1'_aux
thf(fact_3510_Nat_Ofunpow__code__def,axiom,
! [A: $tType] :
( ( funpow @ A )
= ( compow @ ( A > A ) ) ) ).
% Nat.funpow_code_def
thf(fact_3511_card__greaterThanLessThan__int,axiom,
! [L: int,U: int] :
( ( finite_card @ int @ ( set_or5935395276787703475ssThan @ int @ L @ U ) )
= ( nat2 @ ( minus_minus @ int @ U @ ( plus_plus @ int @ L @ ( one_one @ int ) ) ) ) ) ).
% card_greaterThanLessThan_int
thf(fact_3512_greaterThanLessThan__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [I: A,L: A,U: A] :
( ( member @ A @ I @ ( set_or5935395276787703475ssThan @ A @ L @ U ) )
= ( ( ord_less @ A @ L @ I )
& ( ord_less @ A @ I @ U ) ) ) ) ).
% greaterThanLessThan_iff
thf(fact_3513_finite__greaterThanLessThan__int,axiom,
! [L: int,U: int] : ( finite_finite2 @ int @ ( set_or5935395276787703475ssThan @ int @ L @ U ) ) ).
% finite_greaterThanLessThan_int
thf(fact_3514_greaterThanLessThan__empty__iff2,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A] :
( ( ( bot_bot @ ( set @ A ) )
= ( set_or5935395276787703475ssThan @ A @ A2 @ B2 ) )
= ( ord_less_eq @ A @ B2 @ A2 ) ) ) ).
% greaterThanLessThan_empty_iff2
thf(fact_3515_greaterThanLessThan__empty__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A] :
( ( ( set_or5935395276787703475ssThan @ A @ A2 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
= ( ord_less_eq @ A @ B2 @ A2 ) ) ) ).
% greaterThanLessThan_empty_iff
thf(fact_3516_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_3517_infinite__Ioo__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A] :
( ( ~ ( finite_finite2 @ A @ ( set_or5935395276787703475ssThan @ A @ A2 @ B2 ) ) )
= ( ord_less @ A @ A2 @ B2 ) ) ) ).
% infinite_Ioo_iff
thf(fact_3518_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_3519_count__notin,axiom,
! [A: $tType,X2: A,Xs: list @ A] :
( ~ ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
=> ( ( count_list @ A @ Xs @ X2 )
= ( zero_zero @ nat ) ) ) ).
% count_notin
thf(fact_3520_sum_Oinsert_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [I5: set @ B,P3: B > A,I: B] :
( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X: B] :
( ( member @ B @ X @ I5 )
& ( ( P3 @ X )
!= ( zero_zero @ A ) ) ) ) )
=> ( ( ( member @ B @ I @ I5 )
=> ( ( groups1027152243600224163dd_sum @ B @ A @ P3 @ ( insert @ B @ I @ I5 ) )
= ( groups1027152243600224163dd_sum @ B @ A @ P3 @ I5 ) ) )
& ( ~ ( member @ B @ I @ I5 )
=> ( ( groups1027152243600224163dd_sum @ B @ A @ P3 @ ( insert @ B @ I @ I5 ) )
= ( plus_plus @ A @ ( P3 @ I ) @ ( groups1027152243600224163dd_sum @ B @ A @ P3 @ I5 ) ) ) ) ) ) ) ).
% sum.insert'
thf(fact_3521_bot2E,axiom,
! [A: $tType,B: $tType,X2: A,Y: B] :
~ ( bot_bot @ ( A > B > $o ) @ X2 @ Y ) ).
% bot2E
thf(fact_3522_sum_Onon__neutral_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: B > A,I5: set @ B] :
( ( groups1027152243600224163dd_sum @ B @ A @ G
@ ( collect @ B
@ ^ [X: B] :
( ( member @ B @ X @ I5 )
& ( ( G @ X )
!= ( zero_zero @ A ) ) ) ) )
= ( groups1027152243600224163dd_sum @ B @ A @ G @ I5 ) ) ) ).
% sum.non_neutral'
thf(fact_3523_infinite__Ioo,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ~ ( finite_finite2 @ A @ ( set_or5935395276787703475ssThan @ A @ A2 @ B2 ) ) ) ) ).
% infinite_Ioo
thf(fact_3524_sum_Odistrib__triv_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [I5: set @ B,G: B > A,H: B > A] :
( ( finite_finite2 @ B @ I5 )
=> ( ( groups1027152243600224163dd_sum @ B @ A
@ ^ [I2: B] : ( plus_plus @ A @ ( G @ I2 ) @ ( H @ I2 ) )
@ I5 )
= ( plus_plus @ A @ ( groups1027152243600224163dd_sum @ B @ A @ G @ I5 ) @ ( groups1027152243600224163dd_sum @ B @ A @ H @ I5 ) ) ) ) ) ).
% sum.distrib_triv'
thf(fact_3525_greaterThanLessThan__subseteq__greaterThanLessThan,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ A2 @ B2 ) @ ( set_or5935395276787703475ssThan @ A @ C2 @ D2 ) )
= ( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ A2 )
& ( ord_less_eq @ A @ B2 @ D2 ) ) ) ) ) ).
% greaterThanLessThan_subseteq_greaterThanLessThan
thf(fact_3526_sum_Omono__neutral__left_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S2: set @ B,T6: set @ B,G: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S2 @ T6 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T6 @ S2 ) )
=> ( ( G @ X3 )
= ( zero_zero @ A ) ) )
=> ( ( groups1027152243600224163dd_sum @ B @ A @ G @ S2 )
= ( groups1027152243600224163dd_sum @ B @ A @ G @ T6 ) ) ) ) ) ).
% sum.mono_neutral_left'
thf(fact_3527_sum_Omono__neutral__right_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S2: set @ B,T6: set @ B,G: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S2 @ T6 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T6 @ S2 ) )
=> ( ( G @ X3 )
= ( zero_zero @ A ) ) )
=> ( ( groups1027152243600224163dd_sum @ B @ A @ G @ T6 )
= ( groups1027152243600224163dd_sum @ B @ A @ G @ S2 ) ) ) ) ) ).
% sum.mono_neutral_right'
thf(fact_3528_sum_Omono__neutral__cong__left_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S2: set @ B,T6: set @ B,H: B > A,G: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S2 @ T6 )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ ( minus_minus @ ( set @ B ) @ T6 @ S2 ) )
=> ( ( H @ I3 )
= ( zero_zero @ A ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ S2 )
=> ( ( G @ X3 )
= ( H @ X3 ) ) )
=> ( ( groups1027152243600224163dd_sum @ B @ A @ G @ S2 )
= ( groups1027152243600224163dd_sum @ B @ A @ H @ T6 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_left'
thf(fact_3529_sum_Omono__neutral__cong__right_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S2: set @ B,T6: set @ B,G: B > A,H: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S2 @ T6 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T6 @ S2 ) )
=> ( ( G @ X3 )
= ( zero_zero @ A ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ S2 )
=> ( ( G @ X3 )
= ( H @ X3 ) ) )
=> ( ( groups1027152243600224163dd_sum @ B @ A @ G @ T6 )
= ( groups1027152243600224163dd_sum @ B @ A @ H @ S2 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_right'
thf(fact_3530_atLeastPlusOneLessThan__greaterThanLessThan__int,axiom,
! [L: int,U: int] :
( ( set_or7035219750837199246ssThan @ int @ ( plus_plus @ int @ L @ ( one_one @ int ) ) @ U )
= ( set_or5935395276787703475ssThan @ int @ L @ U ) ) ).
% atLeastPlusOneLessThan_greaterThanLessThan_int
thf(fact_3531_sum_Odistrib_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [I5: set @ B,G: B > A,H: B > A] :
( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X: B] :
( ( member @ B @ X @ I5 )
& ( ( G @ X )
!= ( zero_zero @ A ) ) ) ) )
=> ( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X: B] :
( ( member @ B @ X @ I5 )
& ( ( H @ X )
!= ( zero_zero @ A ) ) ) ) )
=> ( ( groups1027152243600224163dd_sum @ B @ A
@ ^ [I2: B] : ( plus_plus @ A @ ( G @ I2 ) @ ( H @ I2 ) )
@ I5 )
= ( plus_plus @ A @ ( groups1027152243600224163dd_sum @ B @ A @ G @ I5 ) @ ( groups1027152243600224163dd_sum @ B @ A @ H @ I5 ) ) ) ) ) ) ).
% sum.distrib'
thf(fact_3532_sum_OG__def,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ( ( groups1027152243600224163dd_sum @ B @ A )
= ( ^ [P7: B > A,I7: set @ B] :
( if @ A
@ ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X: B] :
( ( member @ B @ X @ I7 )
& ( ( P7 @ X )
!= ( zero_zero @ A ) ) ) ) )
@ ( groups7311177749621191930dd_sum @ B @ A @ P7
@ ( collect @ B
@ ^ [X: B] :
( ( member @ B @ X @ I7 )
& ( ( P7 @ X )
!= ( zero_zero @ A ) ) ) ) )
@ ( zero_zero @ A ) ) ) ) ) ).
% sum.G_def
thf(fact_3533_count__list_Osimps_I1_J,axiom,
! [A: $tType,Y: A] :
( ( count_list @ A @ ( nil @ A ) @ Y )
= ( zero_zero @ nat ) ) ).
% count_list.simps(1)
thf(fact_3534_count__le__length,axiom,
! [A: $tType,Xs: list @ A,X2: A] : ( ord_less_eq @ nat @ ( count_list @ A @ Xs @ X2 ) @ ( size_size @ ( list @ A ) @ Xs ) ) ).
% count_le_length
thf(fact_3535_greaterThanLessThan__subseteq__atLeastAtMost__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ A2 @ B2 ) @ ( set_or1337092689740270186AtMost @ A @ C2 @ D2 ) )
= ( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ A2 )
& ( ord_less_eq @ A @ B2 @ D2 ) ) ) ) ) ).
% greaterThanLessThan_subseteq_atLeastAtMost_iff
thf(fact_3536_greaterThanLessThan__subseteq__atLeastLessThan__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ A2 @ B2 ) @ ( set_or7035219750837199246ssThan @ A @ C2 @ D2 ) )
= ( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ A2 )
& ( ord_less_eq @ A @ B2 @ D2 ) ) ) ) ) ).
% greaterThanLessThan_subseteq_atLeastLessThan_iff
thf(fact_3537_atLeastAtMost__diff__ends,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A] :
( ( minus_minus @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) @ ( insert @ A @ A2 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( set_or5935395276787703475ssThan @ A @ A2 @ B2 ) ) ) ).
% atLeastAtMost_diff_ends
thf(fact_3538_count__list_Osimps_I2_J,axiom,
! [A: $tType,X2: A,Y: A,Xs: list @ A] :
( ( ( X2 = Y )
=> ( ( count_list @ A @ ( cons @ A @ X2 @ Xs ) @ Y )
= ( plus_plus @ nat @ ( count_list @ A @ Xs @ Y ) @ ( one_one @ nat ) ) ) )
& ( ( X2 != Y )
=> ( ( count_list @ A @ ( cons @ A @ X2 @ Xs ) @ Y )
= ( count_list @ A @ Xs @ Y ) ) ) ) ).
% count_list.simps(2)
thf(fact_3539_greaterThanLessThan__upto,axiom,
( ( set_or5935395276787703475ssThan @ int )
= ( ^ [I2: int,J3: int] : ( set2 @ int @ ( upto @ ( plus_plus @ int @ I2 @ ( one_one @ int ) ) @ ( minus_minus @ int @ J3 @ ( one_one @ int ) ) ) ) ) ) ).
% greaterThanLessThan_upto
thf(fact_3540_sum__diff1_H,axiom,
! [B: $tType,A: $tType] :
( ( ab_group_add @ B )
=> ! [I5: set @ A,F: A > B,I: A] :
( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [I2: A] :
( ( member @ A @ I2 @ I5 )
& ( ( F @ I2 )
!= ( zero_zero @ B ) ) ) ) )
=> ( ( ( member @ A @ I @ I5 )
=> ( ( groups1027152243600224163dd_sum @ A @ B @ F @ ( minus_minus @ ( set @ A ) @ I5 @ ( insert @ A @ I @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( minus_minus @ B @ ( groups1027152243600224163dd_sum @ A @ B @ F @ I5 ) @ ( F @ I ) ) ) )
& ( ~ ( member @ A @ I @ I5 )
=> ( ( groups1027152243600224163dd_sum @ A @ B @ F @ ( minus_minus @ ( set @ A ) @ I5 @ ( insert @ A @ I @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( groups1027152243600224163dd_sum @ A @ B @ F @ I5 ) ) ) ) ) ) ).
% sum_diff1'
thf(fact_3541_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_3542_set__update__distinct,axiom,
! [A: $tType,Xs: list @ A,N: nat,X2: A] :
( ( distinct @ A @ Xs )
=> ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( set2 @ A @ ( list_update @ A @ Xs @ N @ X2 ) )
= ( insert @ A @ X2 @ ( minus_minus @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( insert @ A @ ( nth @ A @ Xs @ N ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% set_update_distinct
thf(fact_3543_max__nat_Osemilattice__neutr__order__axioms,axiom,
( semila1105856199041335345_order @ nat @ ( ord_max @ nat ) @ ( zero_zero @ nat )
@ ^ [X: nat,Y4: nat] : ( ord_less_eq @ nat @ Y4 @ X )
@ ^ [X: nat,Y4: nat] : ( ord_less @ nat @ Y4 @ X ) ) ).
% max_nat.semilattice_neutr_order_axioms
thf(fact_3544_bit_Oabstract__boolean__algebra__axioms,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( boolea2506097494486148201lgebra @ A @ ( bit_se5824344872417868541ns_and @ A ) @ ( bit_se1065995026697491101ons_or @ A ) @ ( bit_ri4277139882892585799ns_not @ A ) @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% bit.abstract_boolean_algebra_axioms
thf(fact_3545_sub__num__simps_I3_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [L: num] :
( ( neg_numeral_sub @ A @ one2 @ ( bit1 @ L ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( bit0 @ L ) ) ) ) ) ).
% sub_num_simps(3)
thf(fact_3546_finite__greaterThanLessThan,axiom,
! [L: nat,U: nat] : ( finite_finite2 @ nat @ ( set_or5935395276787703475ssThan @ nat @ L @ U ) ) ).
% finite_greaterThanLessThan
thf(fact_3547_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_3548_length__list__update,axiom,
! [A: $tType,Xs: list @ A,I: nat,X2: A] :
( ( size_size @ ( list @ A ) @ ( list_update @ A @ Xs @ I @ X2 ) )
= ( size_size @ ( list @ A ) @ Xs ) ) ).
% length_list_update
thf(fact_3549_sub__num__simps_I1_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( neg_numeral_sub @ A @ one2 @ one2 )
= ( zero_zero @ A ) ) ) ).
% sub_num_simps(1)
thf(fact_3550_diff__numeral__simps_I1_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M: num,N: num] :
( ( minus_minus @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N ) )
= ( neg_numeral_sub @ A @ M @ N ) ) ) ).
% diff_numeral_simps(1)
thf(fact_3551_list__update__beyond,axiom,
! [A: $tType,Xs: list @ A,I: nat,X2: A] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ I )
=> ( ( list_update @ A @ Xs @ I @ X2 )
= Xs ) ) ).
% list_update_beyond
thf(fact_3552_sorted__list__of__set_Olength__sorted__key__list__of__set,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A] :
( ( size_size @ ( list @ A ) @ ( linord4507533701916653071of_set @ A @ A5 ) )
= ( finite_card @ A @ A5 ) ) ) ).
% sorted_list_of_set.length_sorted_key_list_of_set
thf(fact_3553_card__greaterThanLessThan,axiom,
! [L: nat,U: nat] :
( ( finite_card @ nat @ ( set_or5935395276787703475ssThan @ nat @ L @ U ) )
= ( minus_minus @ nat @ U @ ( suc @ L ) ) ) ).
% card_greaterThanLessThan
thf(fact_3554_sub__num__simps_I6_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [K: num,L: num] :
( ( neg_numeral_sub @ A @ ( bit0 @ K ) @ ( bit0 @ L ) )
= ( neg_numeral_dbl @ A @ ( neg_numeral_sub @ A @ K @ L ) ) ) ) ).
% sub_num_simps(6)
thf(fact_3555_sub__num__simps_I9_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [K: num,L: num] :
( ( neg_numeral_sub @ A @ ( bit1 @ K ) @ ( bit1 @ L ) )
= ( neg_numeral_dbl @ A @ ( neg_numeral_sub @ A @ K @ L ) ) ) ) ).
% sub_num_simps(9)
thf(fact_3556_add__neg__numeral__simps_I2_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M: num,N: num] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( numeral_numeral @ A @ N ) )
= ( neg_numeral_sub @ A @ N @ M ) ) ) ).
% add_neg_numeral_simps(2)
thf(fact_3557_add__neg__numeral__simps_I1_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M: num,N: num] :
( ( plus_plus @ A @ ( numeral_numeral @ A @ M ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( neg_numeral_sub @ A @ M @ N ) ) ) ).
% add_neg_numeral_simps(1)
thf(fact_3558_semiring__norm_I166_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [V2: num,W2: num,Y: A] :
( ( plus_plus @ A @ ( numeral_numeral @ A @ V2 ) @ ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ Y ) )
= ( plus_plus @ A @ ( neg_numeral_sub @ A @ V2 @ W2 ) @ Y ) ) ) ).
% semiring_norm(166)
thf(fact_3559_semiring__norm_I167_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [V2: num,W2: num,Y: A] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) @ ( plus_plus @ A @ ( numeral_numeral @ A @ W2 ) @ Y ) )
= ( plus_plus @ A @ ( neg_numeral_sub @ A @ W2 @ V2 ) @ Y ) ) ) ).
% semiring_norm(167)
thf(fact_3560_diff__numeral__simps_I4_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M: num,N: num] :
( ( minus_minus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( neg_numeral_sub @ A @ N @ M ) ) ) ).
% diff_numeral_simps(4)
thf(fact_3561_nth__update__invalid,axiom,
! [A: $tType,I: nat,L: list @ A,J: nat,X2: A] :
( ~ ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L ) )
=> ( ( nth @ A @ ( list_update @ A @ L @ J @ X2 ) @ I )
= ( nth @ A @ L @ I ) ) ) ).
% nth_update_invalid
thf(fact_3562_nth__list__update__eq,axiom,
! [A: $tType,I: nat,Xs: list @ A,X2: A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ ( list_update @ A @ Xs @ I @ X2 ) @ I )
= X2 ) ) ).
% nth_list_update_eq
thf(fact_3563_sub__num__simps_I7_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [K: num,L: num] :
( ( neg_numeral_sub @ A @ ( bit0 @ K ) @ ( bit1 @ L ) )
= ( neg_numeral_dbl_dec @ A @ ( neg_numeral_sub @ A @ K @ L ) ) ) ) ).
% sub_num_simps(7)
thf(fact_3564_sub__num__simps_I8_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [K: num,L: num] :
( ( neg_numeral_sub @ A @ ( bit1 @ K ) @ ( bit0 @ L ) )
= ( neg_numeral_dbl_inc @ A @ ( neg_numeral_sub @ A @ K @ L ) ) ) ) ).
% sub_num_simps(8)
thf(fact_3565_diff__numeral__special_I2_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M: num] :
( ( minus_minus @ A @ ( numeral_numeral @ A @ M ) @ ( one_one @ A ) )
= ( neg_numeral_sub @ A @ M @ one2 ) ) ) ).
% diff_numeral_special(2)
thf(fact_3566_diff__numeral__special_I1_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [N: num] :
( ( minus_minus @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ N ) )
= ( neg_numeral_sub @ A @ one2 @ N ) ) ) ).
% diff_numeral_special(1)
thf(fact_3567_sub__num__simps_I5_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [K: num] :
( ( neg_numeral_sub @ A @ ( bit1 @ K ) @ one2 )
= ( numeral_numeral @ A @ ( bit0 @ K ) ) ) ) ).
% sub_num_simps(5)
thf(fact_3568_set__swap,axiom,
! [A: $tType,I: nat,Xs: list @ A,J: nat] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( set2 @ A @ ( list_update @ A @ ( list_update @ A @ Xs @ I @ ( nth @ A @ Xs @ J ) ) @ J @ ( nth @ A @ Xs @ I ) ) )
= ( set2 @ A @ Xs ) ) ) ) ).
% set_swap
thf(fact_3569_distinct__swap,axiom,
! [A: $tType,I: nat,Xs: list @ A,J: nat] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( distinct @ A @ ( list_update @ A @ ( list_update @ A @ Xs @ I @ ( nth @ A @ Xs @ J ) ) @ J @ ( nth @ A @ Xs @ I ) ) )
= ( distinct @ A @ Xs ) ) ) ) ).
% distinct_swap
thf(fact_3570_add__neg__numeral__special_I4_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [N: num] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ N ) )
= ( neg_numeral_sub @ A @ N @ one2 ) ) ) ).
% add_neg_numeral_special(4)
thf(fact_3571_add__neg__numeral__special_I3_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M: num] :
( ( plus_plus @ A @ ( numeral_numeral @ A @ M ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( neg_numeral_sub @ A @ M @ one2 ) ) ) ).
% add_neg_numeral_special(3)
thf(fact_3572_add__neg__numeral__special_I2_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M: num] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( one_one @ A ) )
= ( neg_numeral_sub @ A @ one2 @ M ) ) ) ).
% add_neg_numeral_special(2)
thf(fact_3573_add__neg__numeral__special_I1_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M: num] :
( ( plus_plus @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) )
= ( neg_numeral_sub @ A @ one2 @ M ) ) ) ).
% add_neg_numeral_special(1)
thf(fact_3574_minus__sub__one__diff__one,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [M: num] :
( ( minus_minus @ A @ ( uminus_uminus @ A @ ( neg_numeral_sub @ A @ M @ one2 ) ) @ ( one_one @ A ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) ) ) ).
% minus_sub_one_diff_one
thf(fact_3575_diff__numeral__special_I7_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [N: num] :
( ( minus_minus @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( neg_numeral_sub @ A @ N @ one2 ) ) ) ).
% diff_numeral_special(7)
thf(fact_3576_diff__numeral__special_I8_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M: num] :
( ( minus_minus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( neg_numeral_sub @ A @ one2 @ M ) ) ) ).
% diff_numeral_special(8)
thf(fact_3577_list__update_Osimps_I2_J,axiom,
! [A: $tType,X2: A,Xs: list @ A,I: nat,V2: A] :
( ( list_update @ A @ ( cons @ A @ X2 @ Xs ) @ I @ V2 )
= ( case_nat @ ( list @ A ) @ ( cons @ A @ V2 @ Xs )
@ ^ [J3: nat] : ( cons @ A @ X2 @ ( list_update @ A @ Xs @ J3 @ V2 ) )
@ I ) ) ).
% list_update.simps(2)
thf(fact_3578_list__update__code_I2_J,axiom,
! [A: $tType,X2: A,Xs: list @ A,Y: A] :
( ( list_update @ A @ ( cons @ A @ X2 @ Xs ) @ ( zero_zero @ nat ) @ Y )
= ( cons @ A @ Y @ Xs ) ) ).
% list_update_code(2)
thf(fact_3579_atLeastSucLessThan__greaterThanLessThan,axiom,
! [L: nat,U: nat] :
( ( set_or7035219750837199246ssThan @ nat @ ( suc @ L ) @ U )
= ( set_or5935395276787703475ssThan @ nat @ L @ U ) ) ).
% atLeastSucLessThan_greaterThanLessThan
thf(fact_3580_neg__numeral__class_Osub__def,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( neg_numeral_sub @ A )
= ( ^ [K2: num,L2: num] : ( minus_minus @ A @ ( numeral_numeral @ A @ K2 ) @ ( numeral_numeral @ A @ L2 ) ) ) ) ) ).
% neg_numeral_class.sub_def
thf(fact_3581_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_3582_in__set__upd__eq,axiom,
! [A: $tType,I: nat,L: list @ A,X2: A,Y: A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L ) )
=> ( ( member @ A @ X2 @ ( set2 @ A @ ( list_update @ A @ L @ I @ Y ) ) )
= ( ( X2 = Y )
| ( ( member @ A @ X2 @ ( set2 @ A @ L ) )
& ! [Y4: A] : ( member @ A @ X2 @ ( set2 @ A @ ( list_update @ A @ L @ I @ Y4 ) ) ) ) ) ) ) ).
% in_set_upd_eq
thf(fact_3583_in__set__upd__cases,axiom,
! [A: $tType,X2: A,L: list @ A,I: nat,Y: A] :
( ( member @ A @ X2 @ ( set2 @ A @ ( list_update @ A @ L @ I @ Y ) ) )
=> ( ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L ) )
=> ( X2 != Y ) )
=> ( member @ A @ X2 @ ( set2 @ A @ L ) ) ) ) ).
% in_set_upd_cases
thf(fact_3584_in__set__upd__eq__aux,axiom,
! [A: $tType,I: nat,L: list @ A,X2: A,Y: A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L ) )
=> ( ( member @ A @ X2 @ ( set2 @ A @ ( list_update @ A @ L @ I @ Y ) ) )
= ( ( X2 = Y )
| ! [Y4: A] : ( member @ A @ X2 @ ( set2 @ A @ ( list_update @ A @ L @ I @ Y4 ) ) ) ) ) ) ).
% in_set_upd_eq_aux
thf(fact_3585_set__update__memI,axiom,
! [A: $tType,N: nat,Xs: list @ A,X2: A] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( member @ A @ X2 @ ( set2 @ A @ ( list_update @ A @ Xs @ N @ X2 ) ) ) ) ).
% set_update_memI
thf(fact_3586_nth__list__update_H,axiom,
! [A: $tType,I: nat,J: nat,L: list @ A,X2: A] :
( ( ( ( I = J )
& ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L ) ) )
=> ( ( nth @ A @ ( list_update @ A @ L @ I @ X2 ) @ J )
= X2 ) )
& ( ~ ( ( I = J )
& ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L ) ) )
=> ( ( nth @ A @ ( list_update @ A @ L @ I @ X2 ) @ J )
= ( nth @ A @ L @ J ) ) ) ) ).
% nth_list_update'
thf(fact_3587_nth__list__update,axiom,
! [A: $tType,I: nat,Xs: list @ A,J: nat,X2: A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ( I = J )
=> ( ( nth @ A @ ( list_update @ A @ Xs @ I @ X2 ) @ J )
= X2 ) )
& ( ( I != J )
=> ( ( nth @ A @ ( list_update @ A @ Xs @ I @ X2 ) @ J )
= ( nth @ A @ Xs @ J ) ) ) ) ) ).
% nth_list_update
thf(fact_3588_list__update__same__conv,axiom,
! [A: $tType,I: nat,Xs: list @ A,X2: A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ( list_update @ A @ Xs @ I @ X2 )
= Xs )
= ( ( nth @ A @ Xs @ I )
= X2 ) ) ) ).
% list_update_same_conv
thf(fact_3589_nth__sorted__list__of__set__greaterThanLessThan,axiom,
! [N: nat,J: nat,I: nat] :
( ( ord_less @ nat @ N @ ( minus_minus @ nat @ J @ ( suc @ I ) ) )
=> ( ( nth @ nat @ ( linord4507533701916653071of_set @ nat @ ( set_or5935395276787703475ssThan @ nat @ I @ J ) ) @ N )
= ( suc @ ( plus_plus @ nat @ I @ N ) ) ) ) ).
% nth_sorted_list_of_set_greaterThanLessThan
thf(fact_3590_sub__non__positive,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: num,M: num] :
( ( ord_less_eq @ A @ ( neg_numeral_sub @ A @ N @ M ) @ ( zero_zero @ A ) )
= ( ord_less_eq @ num @ N @ M ) ) ) ).
% sub_non_positive
thf(fact_3591_sub__non__negative,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: num,M: num] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( neg_numeral_sub @ A @ N @ M ) )
= ( ord_less_eq @ num @ M @ N ) ) ) ).
% sub_non_negative
thf(fact_3592_sub__negative,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: num,M: num] :
( ( ord_less @ A @ ( neg_numeral_sub @ A @ N @ M ) @ ( zero_zero @ A ) )
= ( ord_less @ num @ N @ M ) ) ) ).
% sub_negative
thf(fact_3593_sub__positive,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: num,M: num] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( neg_numeral_sub @ A @ N @ M ) )
= ( ord_less @ num @ M @ N ) ) ) ).
% sub_positive
thf(fact_3594_sub__inc__One__eq,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [N: num] :
( ( neg_numeral_sub @ A @ ( inc @ N ) @ one2 )
= ( numeral_numeral @ A @ N ) ) ) ).
% sub_inc_One_eq
thf(fact_3595_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_3596_insert__swap__set__eq,axiom,
! [A: $tType,I: nat,L: list @ A,X2: A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L ) )
=> ( ( insert @ A @ ( nth @ A @ L @ I ) @ ( set2 @ A @ ( list_update @ A @ L @ I @ X2 ) ) )
= ( insert @ A @ X2 @ ( set2 @ A @ L ) ) ) ) ).
% insert_swap_set_eq
thf(fact_3597_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_3598_sub__num__simps_I2_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [L: num] :
( ( neg_numeral_sub @ A @ one2 @ ( bit0 @ L ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( bitM @ L ) ) ) ) ) ).
% sub_num_simps(2)
thf(fact_3599_subset__mset_Osum__list__update,axiom,
! [A: $tType,K: nat,Xs: list @ ( multiset @ A ),X2: 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 @ X2 ) )
= ( minus_minus @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ ( groups4543113879258116180m_list @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ Xs ) @ X2 ) @ ( nth @ ( multiset @ A ) @ Xs @ K ) ) ) ) ).
% subset_mset.sum_list_update
thf(fact_3600_num__of__nat_Osimps_I2_J,axiom,
! [N: nat] :
( ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( num_of_nat @ ( suc @ N ) )
= ( inc @ ( num_of_nat @ N ) ) ) )
& ( ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( num_of_nat @ ( suc @ N ) )
= one2 ) ) ) ).
% num_of_nat.simps(2)
thf(fact_3601_sorted__list__of__set_Osorted__key__list__of__set__insert__remove,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X2: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( linord4507533701916653071of_set @ A @ ( insert @ A @ X2 @ A5 ) )
= ( linorder_insort_key @ A @ A
@ ^ [X: A] : X
@ X2
@ ( linord4507533701916653071of_set @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_insert_remove
thf(fact_3602_eq__numeral__iff__iszero_I7_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X2: num] :
( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ X2 ) )
= ( one_one @ A ) )
= ( ring_1_iszero @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ X2 @ one2 ) ) ) ) ) ).
% eq_numeral_iff_iszero(7)
thf(fact_3603_num__of__nat__numeral__eq,axiom,
! [Q2: num] :
( ( num_of_nat @ ( numeral_numeral @ nat @ Q2 ) )
= Q2 ) ).
% num_of_nat_numeral_eq
thf(fact_3604_length__insort,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F: B > A,X2: B,Xs: list @ B] :
( ( size_size @ ( list @ B ) @ ( linorder_insort_key @ B @ A @ F @ X2 @ Xs ) )
= ( suc @ ( size_size @ ( list @ B ) @ Xs ) ) ) ) ).
% length_insort
thf(fact_3605_iszero__neg__numeral,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [W2: num] :
( ( ring_1_iszero @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) )
= ( ring_1_iszero @ A @ ( numeral_numeral @ A @ W2 ) ) ) ) ).
% iszero_neg_numeral
thf(fact_3606_subset__mset_Osum__list__eq__0__iff,axiom,
! [A: $tType,Ns: list @ ( multiset @ A )] :
( ( ( groups4543113879258116180m_list @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ Ns )
= ( zero_zero @ ( multiset @ A ) ) )
= ( ! [X: multiset @ A] :
( ( member @ ( multiset @ A ) @ X @ ( set2 @ ( multiset @ A ) @ Ns ) )
=> ( X
= ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ).
% subset_mset.sum_list_eq_0_iff
thf(fact_3607_dbl__dec__simps_I5_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [K: num] :
( ( neg_numeral_dbl_dec @ A @ ( numeral_numeral @ A @ K ) )
= ( numeral_numeral @ A @ ( bitM @ K ) ) ) ) ).
% dbl_dec_simps(5)
thf(fact_3608_pred__numeral__simps_I2_J,axiom,
! [K: num] :
( ( pred_numeral @ ( bit0 @ K ) )
= ( numeral_numeral @ nat @ ( bitM @ K ) ) ) ).
% pred_numeral_simps(2)
thf(fact_3609_sorted__list__of__set_Osorted__key__list__of__set__insert,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X2: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ~ ( member @ A @ X2 @ A5 )
=> ( ( linord4507533701916653071of_set @ A @ ( insert @ A @ X2 @ A5 ) )
= ( linorder_insort_key @ A @ A
@ ^ [X: A] : X
@ X2
@ ( linord4507533701916653071of_set @ A @ A5 ) ) ) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_insert
thf(fact_3610_sub__num__simps_I4_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [K: num] :
( ( neg_numeral_sub @ A @ ( bit0 @ K ) @ one2 )
= ( numeral_numeral @ A @ ( bitM @ K ) ) ) ) ).
% sub_num_simps(4)
thf(fact_3611_insort__left__comm,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y: A,Xs: list @ A] :
( ( linorder_insort_key @ A @ A
@ ^ [X: A] : X
@ X2
@ ( linorder_insort_key @ A @ A
@ ^ [X: A] : X
@ Y
@ Xs ) )
= ( linorder_insort_key @ A @ A
@ ^ [X: A] : X
@ Y
@ ( linorder_insort_key @ A @ A
@ ^ [X: A] : X
@ X2
@ Xs ) ) ) ) ).
% insort_left_comm
thf(fact_3612_iszero__def,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( ( ring_1_iszero @ A )
= ( ^ [Z3: A] :
( Z3
= ( zero_zero @ A ) ) ) ) ) ).
% iszero_def
thf(fact_3613_iszero__0,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( ring_1_iszero @ A @ ( zero_zero @ A ) ) ) ).
% iszero_0
thf(fact_3614_not__iszero__numeral,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [W2: num] :
~ ( ring_1_iszero @ A @ ( numeral_numeral @ A @ W2 ) ) ) ).
% not_iszero_numeral
thf(fact_3615_not__iszero__1,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ~ ( ring_1_iszero @ A @ ( one_one @ A ) ) ) ).
% not_iszero_1
thf(fact_3616_eq__iff__iszero__diff,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( ( ^ [Y3: A,Z: A] : Y3 = Z )
= ( ^ [X: A,Y4: A] : ( ring_1_iszero @ A @ ( minus_minus @ A @ X @ Y4 ) ) ) ) ) ).
% eq_iff_iszero_diff
thf(fact_3617_insort__key_Osimps_I2_J,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F: B > A,X2: B,Y: B,Ys: list @ B] :
( ( ( ord_less_eq @ A @ ( F @ X2 ) @ ( F @ Y ) )
=> ( ( linorder_insort_key @ B @ A @ F @ X2 @ ( cons @ B @ Y @ Ys ) )
= ( cons @ B @ X2 @ ( cons @ B @ Y @ Ys ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( F @ X2 ) @ ( F @ Y ) )
=> ( ( linorder_insort_key @ B @ A @ F @ X2 @ ( cons @ B @ Y @ Ys ) )
= ( cons @ B @ Y @ ( linorder_insort_key @ B @ A @ F @ X2 @ Ys ) ) ) ) ) ) ).
% insort_key.simps(2)
thf(fact_3618_sorted__insort,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Xs: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A )
@ ( linorder_insort_key @ A @ A
@ ^ [X: A] : X
@ X2
@ Xs ) )
= ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs ) ) ) ).
% sorted_insort
thf(fact_3619_inc__BitM__eq,axiom,
! [N: num] :
( ( inc @ ( bitM @ N ) )
= ( bit0 @ N ) ) ).
% inc_BitM_eq
thf(fact_3620_BitM__inc__eq,axiom,
! [N: num] :
( ( bitM @ ( inc @ N ) )
= ( bit1 @ N ) ) ).
% BitM_inc_eq
thf(fact_3621_eq__numeral__iff__iszero_I9_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X2: num] :
( ( ( numeral_numeral @ A @ X2 )
= ( zero_zero @ A ) )
= ( ring_1_iszero @ A @ ( numeral_numeral @ A @ X2 ) ) ) ) ).
% eq_numeral_iff_iszero(9)
thf(fact_3622_eq__numeral__iff__iszero_I10_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [Y: num] :
( ( ( zero_zero @ A )
= ( numeral_numeral @ A @ Y ) )
= ( ring_1_iszero @ A @ ( numeral_numeral @ A @ Y ) ) ) ) ).
% eq_numeral_iff_iszero(10)
thf(fact_3623_num__of__nat_Osimps_I1_J,axiom,
( ( num_of_nat @ ( zero_zero @ nat ) )
= one2 ) ).
% num_of_nat.simps(1)
thf(fact_3624_not__iszero__Numeral1,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ~ ( ring_1_iszero @ A @ ( numeral_numeral @ A @ one2 ) ) ) ).
% not_iszero_Numeral1
thf(fact_3625_not__iszero__neg__1,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ~ ( ring_1_iszero @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% not_iszero_neg_1
thf(fact_3626_eq__numeral__iff__iszero_I1_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X2: num,Y: num] :
( ( ( numeral_numeral @ A @ X2 )
= ( numeral_numeral @ A @ Y ) )
= ( ring_1_iszero @ A @ ( neg_numeral_sub @ A @ X2 @ Y ) ) ) ) ).
% eq_numeral_iff_iszero(1)
thf(fact_3627_insort__is__Cons,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ B,F: B > A,A2: B] :
( ! [X3: B] :
( ( member @ B @ X3 @ ( set2 @ B @ Xs ) )
=> ( ord_less_eq @ A @ ( F @ A2 ) @ ( F @ X3 ) ) )
=> ( ( linorder_insort_key @ B @ A @ F @ A2 @ Xs )
= ( cons @ B @ A2 @ Xs ) ) ) ) ).
% insort_is_Cons
thf(fact_3628_eval__nat__numeral_I2_J,axiom,
! [N: num] :
( ( numeral_numeral @ nat @ ( bit0 @ N ) )
= ( suc @ ( numeral_numeral @ nat @ ( bitM @ N ) ) ) ) ).
% eval_nat_numeral(2)
thf(fact_3629_BitM__plus__one,axiom,
! [N: num] :
( ( plus_plus @ num @ ( bitM @ N ) @ one2 )
= ( bit0 @ N ) ) ).
% BitM_plus_one
thf(fact_3630_one__plus__BitM,axiom,
! [N: num] :
( ( plus_plus @ num @ one2 @ ( bitM @ N ) )
= ( bit0 @ N ) ) ).
% one_plus_BitM
thf(fact_3631_eq__numeral__iff__iszero_I11_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X2: num] :
( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ X2 ) )
= ( zero_zero @ A ) )
= ( ring_1_iszero @ A @ ( numeral_numeral @ A @ X2 ) ) ) ) ).
% eq_numeral_iff_iszero(11)
thf(fact_3632_eq__numeral__iff__iszero_I12_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [Y: num] :
( ( ( zero_zero @ A )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ Y ) ) )
= ( ring_1_iszero @ A @ ( numeral_numeral @ A @ Y ) ) ) ) ).
% eq_numeral_iff_iszero(12)
thf(fact_3633_not__iszero__neg__Numeral1,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ~ ( ring_1_iszero @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ one2 ) ) ) ) ).
% not_iszero_neg_Numeral1
thf(fact_3634_numeral__num__of__nat,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( numeral_numeral @ nat @ ( num_of_nat @ N ) )
= N ) ) ).
% numeral_num_of_nat
thf(fact_3635_num__of__nat__One,axiom,
! [N: nat] :
( ( ord_less_eq @ nat @ N @ ( one_one @ nat ) )
=> ( ( num_of_nat @ N )
= one2 ) ) ).
% num_of_nat_One
thf(fact_3636_eq__numeral__iff__iszero_I3_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X2: num,Y: num] :
( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ X2 ) )
= ( numeral_numeral @ A @ Y ) )
= ( ring_1_iszero @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ X2 @ Y ) ) ) ) ) ).
% eq_numeral_iff_iszero(3)
thf(fact_3637_eq__numeral__iff__iszero_I2_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X2: num,Y: num] :
( ( ( numeral_numeral @ A @ X2 )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ Y ) ) )
= ( ring_1_iszero @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ X2 @ Y ) ) ) ) ) ).
% eq_numeral_iff_iszero(2)
thf(fact_3638_eq__numeral__iff__iszero_I4_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X2: num,Y: num] :
( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ X2 ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ Y ) ) )
= ( ring_1_iszero @ A @ ( neg_numeral_sub @ A @ Y @ X2 ) ) ) ) ).
% eq_numeral_iff_iszero(4)
thf(fact_3639_numeral__BitM,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [N: num] :
( ( numeral_numeral @ A @ ( bitM @ N ) )
= ( minus_minus @ A @ ( numeral_numeral @ A @ ( bit0 @ N ) ) @ ( one_one @ A ) ) ) ) ).
% numeral_BitM
thf(fact_3640_sub__BitM__One__eq,axiom,
! [N: num] :
( ( neg_numeral_sub @ int @ ( bitM @ N ) @ one2 )
= ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( neg_numeral_sub @ int @ N @ one2 ) ) ) ).
% sub_BitM_One_eq
thf(fact_3641_numeral__num__of__nat__unfold,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [N: nat] :
( ( ( N
= ( zero_zero @ nat ) )
=> ( ( numeral_numeral @ A @ ( num_of_nat @ N ) )
= ( one_one @ A ) ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ( ( numeral_numeral @ A @ ( num_of_nat @ N ) )
= ( semiring_1_of_nat @ A @ N ) ) ) ) ) ).
% numeral_num_of_nat_unfold
thf(fact_3642_num__of__nat__double,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( num_of_nat @ ( plus_plus @ nat @ N @ N ) )
= ( bit0 @ ( num_of_nat @ N ) ) ) ) ).
% num_of_nat_double
thf(fact_3643_num__of__nat__plus__distrib,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( num_of_nat @ ( plus_plus @ nat @ M @ N ) )
= ( plus_plus @ num @ ( num_of_nat @ M ) @ ( num_of_nat @ N ) ) ) ) ) ).
% num_of_nat_plus_distrib
thf(fact_3644_eq__numeral__iff__iszero_I6_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [Y: num] :
( ( ( one_one @ A )
= ( numeral_numeral @ A @ Y ) )
= ( ring_1_iszero @ A @ ( neg_numeral_sub @ A @ one2 @ Y ) ) ) ) ).
% eq_numeral_iff_iszero(6)
thf(fact_3645_eq__numeral__iff__iszero_I5_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X2: num] :
( ( ( numeral_numeral @ A @ X2 )
= ( one_one @ A ) )
= ( ring_1_iszero @ A @ ( neg_numeral_sub @ A @ X2 @ one2 ) ) ) ) ).
% eq_numeral_iff_iszero(5)
thf(fact_3646_sorted__list__of__set_Ofold__insort__key_Oremove,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X2: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( member @ A @ X2 @ A5 )
=> ( ( linord4507533701916653071of_set @ A @ A5 )
= ( linorder_insort_key @ A @ A
@ ^ [X: A] : X
@ X2
@ ( linord4507533701916653071of_set @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ).
% sorted_list_of_set.fold_insort_key.remove
thf(fact_3647_eq__numeral__iff__iszero_I8_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [Y: num] :
( ( ( one_one @ A )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ Y ) ) )
= ( ring_1_iszero @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ one2 @ Y ) ) ) ) ) ).
% eq_numeral_iff_iszero(8)
thf(fact_3648_or__minus__numerals_I5_J,axiom,
! [N: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) @ ( one_one @ int ) )
= ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit_or_not_num_neg @ one2 @ ( bitM @ N ) ) ) ) ) ).
% or_minus_numerals(5)
thf(fact_3649_or__minus__numerals_I1_J,axiom,
! [N: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( one_one @ int ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) )
= ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit_or_not_num_neg @ one2 @ ( bitM @ N ) ) ) ) ) ).
% or_minus_numerals(1)
thf(fact_3650_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_3651_num__of__integer_Oabs__eq,axiom,
! [X2: int] :
( ( code_num_of_integer @ ( code_integer_of_int @ X2 ) )
= ( num_of_nat @ ( nat2 @ X2 ) ) ) ).
% num_of_integer.abs_eq
thf(fact_3652_subset__mset_Oadd__le__cancel__left,axiom,
! [A: $tType,C2: multiset @ A,A2: multiset @ A,B2: multiset @ A] :
( ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ C2 @ A2 ) @ ( plus_plus @ ( multiset @ A ) @ C2 @ B2 ) )
= ( subseteq_mset @ A @ A2 @ B2 ) ) ).
% subset_mset.add_le_cancel_left
thf(fact_3653_subset__mset_Oadd__le__cancel__right,axiom,
! [A: $tType,A2: multiset @ A,C2: multiset @ A,B2: multiset @ A] :
( ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A2 @ C2 ) @ ( plus_plus @ ( multiset @ A ) @ B2 @ C2 ) )
= ( subseteq_mset @ A @ A2 @ B2 ) ) ).
% subset_mset.add_le_cancel_right
thf(fact_3654_mset__subset__eq__mono__add__left__cancel,axiom,
! [A: $tType,C6: multiset @ A,A5: multiset @ A,B5: multiset @ A] :
( ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ C6 @ A5 ) @ ( plus_plus @ ( multiset @ A ) @ C6 @ B5 ) )
= ( subseteq_mset @ A @ A5 @ B5 ) ) ).
% mset_subset_eq_mono_add_left_cancel
thf(fact_3655_mset__subset__eq__mono__add__right__cancel,axiom,
! [A: $tType,A5: multiset @ A,C6: multiset @ A,B5: multiset @ A] :
( ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A5 @ C6 ) @ ( plus_plus @ ( multiset @ A ) @ B5 @ C6 ) )
= ( subseteq_mset @ A @ A5 @ B5 ) ) ).
% mset_subset_eq_mono_add_right_cancel
thf(fact_3656_subset__mset_Oadd__le__same__cancel1,axiom,
! [A: $tType,B2: multiset @ A,A2: multiset @ A] :
( ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ B2 @ A2 ) @ B2 )
= ( subseteq_mset @ A @ A2 @ ( zero_zero @ ( multiset @ A ) ) ) ) ).
% subset_mset.add_le_same_cancel1
thf(fact_3657_subset__mset_Oadd__le__same__cancel2,axiom,
! [A: $tType,A2: multiset @ A,B2: multiset @ A] :
( ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A2 @ B2 ) @ B2 )
= ( subseteq_mset @ A @ A2 @ ( zero_zero @ ( multiset @ A ) ) ) ) ).
% subset_mset.add_le_same_cancel2
thf(fact_3658_subset__mset_Ole__add__same__cancel1,axiom,
! [A: $tType,A2: multiset @ A,B2: multiset @ A] :
( ( subseteq_mset @ A @ A2 @ ( plus_plus @ ( multiset @ A ) @ A2 @ B2 ) )
= ( subseteq_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ B2 ) ) ).
% subset_mset.le_add_same_cancel1
thf(fact_3659_subset__mset_Ole__add__same__cancel2,axiom,
! [A: $tType,A2: multiset @ A,B2: multiset @ A] :
( ( subseteq_mset @ A @ A2 @ ( plus_plus @ ( multiset @ A ) @ B2 @ A2 ) )
= ( subseteq_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ B2 ) ) ).
% subset_mset.le_add_same_cancel2
thf(fact_3660_subset__mset_Odiff__add,axiom,
! [A: $tType,A2: multiset @ A,B2: multiset @ A] :
( ( subseteq_mset @ A @ A2 @ B2 )
=> ( ( plus_plus @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ B2 @ A2 ) @ A2 )
= B2 ) ) ).
% subset_mset.diff_add
thf(fact_3661_subset__mset_Oadd__diff__assoc,axiom,
! [A: $tType,A2: multiset @ A,B2: multiset @ A,C2: multiset @ A] :
( ( subseteq_mset @ A @ A2 @ B2 )
=> ( ( plus_plus @ ( multiset @ A ) @ C2 @ ( minus_minus @ ( multiset @ A ) @ B2 @ A2 ) )
= ( minus_minus @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ C2 @ B2 ) @ A2 ) ) ) ).
% subset_mset.add_diff_assoc
thf(fact_3662_subset__mset_Oadd__diff__assoc2,axiom,
! [A: $tType,A2: multiset @ A,B2: multiset @ A,C2: multiset @ A] :
( ( subseteq_mset @ A @ A2 @ B2 )
=> ( ( plus_plus @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ B2 @ A2 ) @ C2 )
= ( minus_minus @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ B2 @ C2 ) @ A2 ) ) ) ).
% subset_mset.add_diff_assoc2
thf(fact_3663_mset__subset__eq__multiset__union__diff__commute,axiom,
! [A: $tType,B5: multiset @ A,A5: multiset @ A,C6: multiset @ A] :
( ( subseteq_mset @ A @ B5 @ A5 )
=> ( ( plus_plus @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ A5 @ B5 ) @ C6 )
= ( minus_minus @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ A5 @ C6 ) @ B5 ) ) ) ).
% mset_subset_eq_multiset_union_diff_commute
thf(fact_3664_mset__le__subtract,axiom,
! [A: $tType,A5: multiset @ A,B5: multiset @ A,C6: multiset @ A] :
( ( subseteq_mset @ A @ A5 @ B5 )
=> ( subseteq_mset @ A @ ( minus_minus @ ( multiset @ A ) @ A5 @ C6 ) @ ( minus_minus @ ( multiset @ A ) @ B5 @ C6 ) ) ) ).
% mset_le_subtract
thf(fact_3665_mset__subset__eq__exists__conv,axiom,
! [A: $tType] :
( ( subseteq_mset @ A )
= ( ^ [A8: multiset @ A,B8: multiset @ A] :
? [C8: multiset @ A] :
( B8
= ( plus_plus @ ( multiset @ A ) @ A8 @ C8 ) ) ) ) ).
% mset_subset_eq_exists_conv
thf(fact_3666_mset__subset__eq__add__right,axiom,
! [A: $tType,B5: multiset @ A,A5: multiset @ A] : ( subseteq_mset @ A @ B5 @ ( plus_plus @ ( multiset @ A ) @ A5 @ B5 ) ) ).
% mset_subset_eq_add_right
thf(fact_3667_mset__subset__eq__mono__add,axiom,
! [A: $tType,A5: multiset @ A,B5: multiset @ A,C6: multiset @ A,D3: multiset @ A] :
( ( subseteq_mset @ A @ A5 @ B5 )
=> ( ( subseteq_mset @ A @ C6 @ D3 )
=> ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A5 @ C6 ) @ ( plus_plus @ ( multiset @ A ) @ B5 @ D3 ) ) ) ) ).
% mset_subset_eq_mono_add
thf(fact_3668_mset__subset__eq__add__left,axiom,
! [A: $tType,A5: multiset @ A,B5: multiset @ A] : ( subseteq_mset @ A @ A5 @ ( plus_plus @ ( multiset @ A ) @ A5 @ B5 ) ) ).
% mset_subset_eq_add_left
thf(fact_3669_subset__mset_Oadd__le__imp__le__right,axiom,
! [A: $tType,A2: multiset @ A,C2: multiset @ A,B2: multiset @ A] :
( ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A2 @ C2 ) @ ( plus_plus @ ( multiset @ A ) @ B2 @ C2 ) )
=> ( subseteq_mset @ A @ A2 @ B2 ) ) ).
% subset_mset.add_le_imp_le_right
thf(fact_3670_subset__mset_Oadd__le__imp__le__left,axiom,
! [A: $tType,C2: multiset @ A,A2: multiset @ A,B2: multiset @ A] :
( ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ C2 @ A2 ) @ ( plus_plus @ ( multiset @ A ) @ C2 @ B2 ) )
=> ( subseteq_mset @ A @ A2 @ B2 ) ) ).
% subset_mset.add_le_imp_le_left
thf(fact_3671_subset__mset_Oadd__right__mono,axiom,
! [A: $tType,A2: multiset @ A,B2: multiset @ A,C2: multiset @ A] :
( ( subseteq_mset @ A @ A2 @ B2 )
=> ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A2 @ C2 ) @ ( plus_plus @ ( multiset @ A ) @ B2 @ C2 ) ) ) ).
% subset_mset.add_right_mono
thf(fact_3672_subset__mset_Oadd__left__mono,axiom,
! [A: $tType,A2: multiset @ A,B2: multiset @ A,C2: multiset @ A] :
( ( subseteq_mset @ A @ A2 @ B2 )
=> ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ C2 @ A2 ) @ ( plus_plus @ ( multiset @ A ) @ C2 @ B2 ) ) ) ).
% subset_mset.add_left_mono
thf(fact_3673_subset__mset_Ole__iff__add,axiom,
! [A: $tType] :
( ( subseteq_mset @ A )
= ( ^ [A3: multiset @ A,B3: multiset @ A] :
? [C5: multiset @ A] :
( B3
= ( plus_plus @ ( multiset @ A ) @ A3 @ C5 ) ) ) ) ).
% subset_mset.le_iff_add
thf(fact_3674_subset__mset_Oless__eqE,axiom,
! [A: $tType,A2: multiset @ A,B2: multiset @ A] :
( ( subseteq_mset @ A @ A2 @ B2 )
=> ~ ! [C4: multiset @ A] :
( B2
!= ( plus_plus @ ( multiset @ A ) @ A2 @ C4 ) ) ) ).
% subset_mset.less_eqE
thf(fact_3675_subset__mset_Oadd__mono,axiom,
! [A: $tType,A2: multiset @ A,B2: multiset @ A,C2: multiset @ A,D2: multiset @ A] :
( ( subseteq_mset @ A @ A2 @ B2 )
=> ( ( subseteq_mset @ A @ C2 @ D2 )
=> ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A2 @ C2 ) @ ( plus_plus @ ( multiset @ A ) @ B2 @ D2 ) ) ) ) ).
% subset_mset.add_mono
thf(fact_3676_mset__le__addE,axiom,
! [A: $tType,Xs: multiset @ A,Ys: multiset @ A] :
( ( subseteq_mset @ A @ Xs @ Ys )
=> ~ ! [Zs2: multiset @ A] :
( Ys
!= ( plus_plus @ ( multiset @ A ) @ Xs @ Zs2 ) ) ) ).
% mset_le_addE
thf(fact_3677_mset__le__distrib,axiom,
! [A: $tType,X8: multiset @ A,A5: multiset @ A,B5: multiset @ A] :
( ( subseteq_mset @ A @ X8 @ ( plus_plus @ ( multiset @ A ) @ A5 @ B5 ) )
=> ~ ! [Xa4: multiset @ A,Xb4: multiset @ A] :
( ( X8
= ( plus_plus @ ( multiset @ A ) @ Xa4 @ Xb4 ) )
=> ( ( subseteq_mset @ A @ Xa4 @ A5 )
=> ~ ( subseteq_mset @ A @ Xb4 @ B5 ) ) ) ) ).
% mset_le_distrib
thf(fact_3678_mset__union__subset,axiom,
! [A: $tType,A5: multiset @ A,B5: multiset @ A,C6: multiset @ A] :
( ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A5 @ B5 ) @ C6 )
=> ( ( subseteq_mset @ A @ A5 @ C6 )
& ( subseteq_mset @ A @ B5 @ C6 ) ) ) ).
% mset_union_subset
thf(fact_3679_mset__le__decr__left1,axiom,
! [A: $tType,A2: multiset @ A,C2: multiset @ A,B2: multiset @ A] :
( ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A2 @ C2 ) @ B2 )
=> ( subseteq_mset @ A @ A2 @ B2 ) ) ).
% mset_le_decr_left1
thf(fact_3680_mset__le__decr__left2,axiom,
! [A: $tType,C2: multiset @ A,A2: multiset @ A,B2: multiset @ A] :
( ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ C2 @ A2 ) @ B2 )
=> ( subseteq_mset @ A @ A2 @ B2 ) ) ).
% mset_le_decr_left2
thf(fact_3681_mset__le__incr__right1,axiom,
! [A: $tType,A2: multiset @ A,B2: multiset @ A,C2: multiset @ A] :
( ( subseteq_mset @ A @ A2 @ B2 )
=> ( subseteq_mset @ A @ A2 @ ( plus_plus @ ( multiset @ A ) @ B2 @ C2 ) ) ) ).
% mset_le_incr_right1
thf(fact_3682_mset__le__incr__right2,axiom,
! [A: $tType,A2: multiset @ A,B2: multiset @ A,C2: multiset @ A] :
( ( subseteq_mset @ A @ A2 @ B2 )
=> ( subseteq_mset @ A @ A2 @ ( plus_plus @ ( multiset @ A ) @ C2 @ B2 ) ) ) ).
% mset_le_incr_right2
thf(fact_3683_subset__mset_Olift__Suc__antimono__le,axiom,
! [A: $tType,F: nat > ( multiset @ A ),N: nat,N3: nat] :
( ! [N2: nat] : ( subseteq_mset @ A @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
=> ( ( ord_less_eq @ nat @ N @ N3 )
=> ( subseteq_mset @ A @ ( F @ N3 ) @ ( F @ N ) ) ) ) ).
% subset_mset.lift_Suc_antimono_le
thf(fact_3684_subset__mset_Olift__Suc__mono__le,axiom,
! [A: $tType,F: nat > ( multiset @ A ),N: nat,N3: nat] :
( ! [N2: nat] : ( subseteq_mset @ A @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq @ nat @ N @ N3 )
=> ( subseteq_mset @ A @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% subset_mset.lift_Suc_mono_le
thf(fact_3685_subset__mset_Oadd__decreasing,axiom,
! [A: $tType,A2: multiset @ A,C2: multiset @ A,B2: multiset @ A] :
( ( subseteq_mset @ A @ A2 @ ( zero_zero @ ( multiset @ A ) ) )
=> ( ( subseteq_mset @ A @ C2 @ B2 )
=> ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A2 @ C2 ) @ B2 ) ) ) ).
% subset_mset.add_decreasing
thf(fact_3686_subset__mset_Oadd__increasing,axiom,
! [A: $tType,A2: multiset @ A,B2: multiset @ A,C2: multiset @ A] :
( ( subseteq_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ A2 )
=> ( ( subseteq_mset @ A @ B2 @ C2 )
=> ( subseteq_mset @ A @ B2 @ ( plus_plus @ ( multiset @ A ) @ A2 @ C2 ) ) ) ) ).
% subset_mset.add_increasing
thf(fact_3687_subset__mset_Oadd__decreasing2,axiom,
! [A: $tType,C2: multiset @ A,A2: multiset @ A,B2: multiset @ A] :
( ( subseteq_mset @ A @ C2 @ ( zero_zero @ ( multiset @ A ) ) )
=> ( ( subseteq_mset @ A @ A2 @ B2 )
=> ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A2 @ C2 ) @ B2 ) ) ) ).
% subset_mset.add_decreasing2
thf(fact_3688_subset__mset_Oadd__increasing2,axiom,
! [A: $tType,C2: multiset @ A,B2: multiset @ A,A2: multiset @ A] :
( ( subseteq_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ C2 )
=> ( ( subseteq_mset @ A @ B2 @ A2 )
=> ( subseteq_mset @ A @ B2 @ ( plus_plus @ ( multiset @ A ) @ A2 @ C2 ) ) ) ) ).
% subset_mset.add_increasing2
thf(fact_3689_subset__mset_Oadd__nonneg__nonneg,axiom,
! [A: $tType,A2: multiset @ A,B2: multiset @ A] :
( ( subseteq_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ A2 )
=> ( ( subseteq_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ B2 )
=> ( subseteq_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ ( plus_plus @ ( multiset @ A ) @ A2 @ B2 ) ) ) ) ).
% subset_mset.add_nonneg_nonneg
thf(fact_3690_subset__mset_Oadd__nonpos__nonpos,axiom,
! [A: $tType,A2: multiset @ A,B2: multiset @ A] :
( ( subseteq_mset @ A @ A2 @ ( zero_zero @ ( multiset @ A ) ) )
=> ( ( subseteq_mset @ A @ B2 @ ( zero_zero @ ( multiset @ A ) ) )
=> ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A2 @ B2 ) @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ).
% subset_mset.add_nonpos_nonpos
thf(fact_3691_subset__mset_Oadd__nonneg__eq__0__iff,axiom,
! [A: $tType,X2: multiset @ A,Y: multiset @ A] :
( ( subseteq_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ X2 )
=> ( ( subseteq_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ Y )
=> ( ( ( plus_plus @ ( multiset @ A ) @ X2 @ Y )
= ( zero_zero @ ( multiset @ A ) ) )
= ( ( X2
= ( zero_zero @ ( multiset @ A ) ) )
& ( Y
= ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ) ).
% subset_mset.add_nonneg_eq_0_iff
thf(fact_3692_subset__mset_Oadd__nonpos__eq__0__iff,axiom,
! [A: $tType,X2: multiset @ A,Y: multiset @ A] :
( ( subseteq_mset @ A @ X2 @ ( zero_zero @ ( multiset @ A ) ) )
=> ( ( subseteq_mset @ A @ Y @ ( zero_zero @ ( multiset @ A ) ) )
=> ( ( ( plus_plus @ ( multiset @ A ) @ X2 @ Y )
= ( zero_zero @ ( multiset @ A ) ) )
= ( ( X2
= ( zero_zero @ ( multiset @ A ) ) )
& ( Y
= ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ) ).
% subset_mset.add_nonpos_eq_0_iff
thf(fact_3693_multiset__diff__union__assoc,axiom,
! [A: $tType,C6: multiset @ A,B5: multiset @ A,A5: multiset @ A] :
( ( subseteq_mset @ A @ C6 @ B5 )
=> ( ( minus_minus @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ A5 @ B5 ) @ C6 )
= ( plus_plus @ ( multiset @ A ) @ A5 @ ( minus_minus @ ( multiset @ A ) @ B5 @ C6 ) ) ) ) ).
% multiset_diff_union_assoc
thf(fact_3694_subset__eq__diff__conv,axiom,
! [A: $tType,A5: multiset @ A,C6: multiset @ A,B5: multiset @ A] :
( ( subseteq_mset @ A @ ( minus_minus @ ( multiset @ A ) @ A5 @ C6 ) @ B5 )
= ( subseteq_mset @ A @ A5 @ ( plus_plus @ ( multiset @ A ) @ B5 @ C6 ) ) ) ).
% subset_eq_diff_conv
thf(fact_3695_subset__mset_Ole__imp__diff__is__add,axiom,
! [A: $tType,A2: multiset @ A,B2: multiset @ A,C2: multiset @ A] :
( ( subseteq_mset @ A @ A2 @ B2 )
=> ( ( subseteq_mset @ A @ A2 @ B2 )
=> ( ( ( minus_minus @ ( multiset @ A ) @ B2 @ A2 )
= C2 )
= ( B2
= ( plus_plus @ ( multiset @ A ) @ C2 @ A2 ) ) ) ) ) ).
% subset_mset.le_imp_diff_is_add
thf(fact_3696_subset__mset_Oadd__diff__inverse,axiom,
! [A: $tType,A2: multiset @ A,B2: multiset @ A] :
( ( subseteq_mset @ A @ A2 @ B2 )
=> ( ( plus_plus @ ( multiset @ A ) @ A2 @ ( minus_minus @ ( multiset @ A ) @ B2 @ A2 ) )
= B2 ) ) ).
% subset_mset.add_diff_inverse
thf(fact_3697_subset__mset_Odiff__diff__right,axiom,
! [A: $tType,A2: multiset @ A,B2: multiset @ A,C2: multiset @ A] :
( ( subseteq_mset @ A @ A2 @ B2 )
=> ( ( minus_minus @ ( multiset @ A ) @ C2 @ ( minus_minus @ ( multiset @ A ) @ B2 @ A2 ) )
= ( minus_minus @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ C2 @ A2 ) @ B2 ) ) ) ).
% subset_mset.diff_diff_right
thf(fact_3698_subset__mset_Odiff__add__assoc2,axiom,
! [A: $tType,A2: multiset @ A,B2: multiset @ A,C2: multiset @ A] :
( ( subseteq_mset @ A @ A2 @ B2 )
=> ( ( minus_minus @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ B2 @ C2 ) @ A2 )
= ( plus_plus @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ B2 @ A2 ) @ C2 ) ) ) ).
% subset_mset.diff_add_assoc2
thf(fact_3699_subset__mset_Odiff__add__assoc,axiom,
! [A: $tType,A2: multiset @ A,B2: multiset @ A,C2: multiset @ A] :
( ( subseteq_mset @ A @ A2 @ B2 )
=> ( ( minus_minus @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ C2 @ B2 ) @ A2 )
= ( plus_plus @ ( multiset @ A ) @ C2 @ ( minus_minus @ ( multiset @ A ) @ B2 @ A2 ) ) ) ) ).
% subset_mset.diff_add_assoc
thf(fact_3700_subset__mset_Ole__diff__conv2,axiom,
! [A: $tType,A2: multiset @ A,B2: multiset @ A,C2: multiset @ A] :
( ( subseteq_mset @ A @ A2 @ B2 )
=> ( ( subseteq_mset @ A @ C2 @ ( minus_minus @ ( multiset @ A ) @ B2 @ A2 ) )
= ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ C2 @ A2 ) @ B2 ) ) ) ).
% subset_mset.le_diff_conv2
thf(fact_3701_subset__mset_Ole__add__diff,axiom,
! [A: $tType,A2: multiset @ A,B2: multiset @ A,C2: multiset @ A] :
( ( subseteq_mset @ A @ A2 @ B2 )
=> ( subseteq_mset @ A @ C2 @ ( minus_minus @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ B2 @ C2 ) @ A2 ) ) ) ).
% subset_mset.le_add_diff
thf(fact_3702_mset__le__subtract__left,axiom,
! [A: $tType,A5: multiset @ A,B5: multiset @ A,X8: multiset @ A] :
( ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A5 @ B5 ) @ X8 )
=> ( ( subseteq_mset @ A @ B5 @ ( minus_minus @ ( multiset @ A ) @ X8 @ A5 ) )
& ( subseteq_mset @ A @ A5 @ X8 ) ) ) ).
% mset_le_subtract_left
thf(fact_3703_mset__le__subtract__right,axiom,
! [A: $tType,A5: multiset @ A,B5: multiset @ A,X8: multiset @ A] :
( ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A5 @ B5 ) @ X8 )
=> ( ( subseteq_mset @ A @ A5 @ ( minus_minus @ ( multiset @ A ) @ X8 @ B5 ) )
& ( subseteq_mset @ A @ B5 @ X8 ) ) ) ).
% mset_le_subtract_right
thf(fact_3704_size__mset__mono,axiom,
! [A: $tType,A5: multiset @ A,B5: multiset @ A] :
( ( subseteq_mset @ A @ A5 @ B5 )
=> ( ord_less_eq @ nat @ ( size_size @ ( multiset @ A ) @ A5 ) @ ( size_size @ ( multiset @ A ) @ B5 ) ) ) ).
% size_mset_mono
thf(fact_3705_size__Diff__submset,axiom,
! [A: $tType,M3: multiset @ A,M9: multiset @ A] :
( ( subseteq_mset @ A @ M3 @ M9 )
=> ( ( size_size @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ M9 @ M3 ) )
= ( minus_minus @ nat @ ( size_size @ ( multiset @ A ) @ M9 ) @ ( size_size @ ( multiset @ A ) @ M3 ) ) ) ) ).
% size_Diff_submset
thf(fact_3706_subset__mset_Osum__list__nonneg__eq__0__iff,axiom,
! [A: $tType,Xs: list @ ( multiset @ A )] :
( ! [X3: multiset @ A] :
( ( member @ ( multiset @ A ) @ X3 @ ( set2 @ ( multiset @ A ) @ Xs ) )
=> ( subseteq_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ X3 ) )
=> ( ( ( groups4543113879258116180m_list @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ Xs )
= ( zero_zero @ ( multiset @ A ) ) )
= ( ! [X: multiset @ A] :
( ( member @ ( multiset @ A ) @ X @ ( set2 @ ( multiset @ A ) @ Xs ) )
=> ( X
= ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ) ).
% subset_mset.sum_list_nonneg_eq_0_iff
thf(fact_3707_subset__mset_Omember__le__sum__list,axiom,
! [A: $tType,X2: multiset @ A,Xs: list @ ( multiset @ A )] :
( ( member @ ( multiset @ A ) @ X2 @ ( set2 @ ( multiset @ A ) @ Xs ) )
=> ( subseteq_mset @ A @ X2 @ ( groups4543113879258116180m_list @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ Xs ) ) ) ).
% subset_mset.member_le_sum_list
thf(fact_3708_subset__mset_Osum__list__nonpos,axiom,
! [A: $tType,Xs: list @ ( multiset @ A )] :
( ! [X3: multiset @ A] :
( ( member @ ( multiset @ A ) @ X3 @ ( set2 @ ( multiset @ A ) @ Xs ) )
=> ( subseteq_mset @ A @ X3 @ ( zero_zero @ ( multiset @ A ) ) ) )
=> ( subseteq_mset @ A @ ( groups4543113879258116180m_list @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ Xs ) @ ( zero_zero @ ( multiset @ A ) ) ) ) ).
% subset_mset.sum_list_nonpos
thf(fact_3709_subset__mset_Osum__list__nonneg,axiom,
! [A: $tType,Xs: list @ ( multiset @ A )] :
( ! [X3: multiset @ A] :
( ( member @ ( multiset @ A ) @ X3 @ ( set2 @ ( multiset @ A ) @ Xs ) )
=> ( subseteq_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ X3 ) )
=> ( subseteq_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ ( groups4543113879258116180m_list @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ Xs ) ) ) ).
% subset_mset.sum_list_nonneg
thf(fact_3710_subset__mset_Osum__mono2,axiom,
! [A: $tType,B: $tType,B5: set @ B,A5: set @ B,F: B > ( multiset @ A )] :
( ( finite_finite2 @ B @ B5 )
=> ( ( ord_less_eq @ ( set @ B ) @ A5 @ B5 )
=> ( ! [B4: B] :
( ( member @ B @ B4 @ ( minus_minus @ ( set @ B ) @ B5 @ A5 ) )
=> ( subseteq_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ ( F @ B4 ) ) )
=> ( subseteq_mset @ A @ ( groups3894954378712506084id_sum @ ( multiset @ A ) @ B @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ F @ A5 ) @ ( groups3894954378712506084id_sum @ ( multiset @ A ) @ B @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ F @ B5 ) ) ) ) ) ).
% subset_mset.sum_mono2
thf(fact_3711_mset__size2elem,axiom,
! [A: $tType,P: multiset @ A,Q2: A,Q5: A] :
( ( ord_less_eq @ nat @ ( size_size @ ( multiset @ A ) @ P ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
=> ( ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ Q2 @ ( zero_zero @ ( multiset @ A ) ) ) @ ( add_mset @ A @ Q5 @ ( zero_zero @ ( multiset @ A ) ) ) ) @ P )
=> ( P
= ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ Q2 @ ( zero_zero @ ( multiset @ A ) ) ) @ ( add_mset @ A @ Q5 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ) ).
% mset_size2elem
thf(fact_3712_nth__sorted__list__of__set__greaterThanAtMost,axiom,
! [N: nat,J: nat,I: nat] :
( ( ord_less @ nat @ N @ ( minus_minus @ nat @ J @ I ) )
=> ( ( nth @ nat @ ( linord4507533701916653071of_set @ nat @ ( set_or3652927894154168847AtMost @ nat @ I @ J ) ) @ N )
= ( suc @ ( plus_plus @ nat @ I @ N ) ) ) ) ).
% nth_sorted_list_of_set_greaterThanAtMost
thf(fact_3713_take__bit__numeral__minus__numeral__int,axiom,
! [M: num,N: num] :
( ( bit_se2584673776208193580ke_bit @ int @ ( numeral_numeral @ nat @ M ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
= ( case_option @ int @ num @ ( zero_zero @ int )
@ ^ [Q6: 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 @ Q6 ) ) )
@ ( bit_take_bit_num @ ( numeral_numeral @ nat @ M ) @ N ) ) ) ).
% take_bit_numeral_minus_numeral_int
thf(fact_3714_union__mset__add__mset__right,axiom,
! [A: $tType,A5: multiset @ A,A2: A,B5: multiset @ A] :
( ( plus_plus @ ( multiset @ A ) @ A5 @ ( add_mset @ A @ A2 @ B5 ) )
= ( add_mset @ A @ A2 @ ( plus_plus @ ( multiset @ A ) @ A5 @ B5 ) ) ) ).
% union_mset_add_mset_right
thf(fact_3715_union__mset__add__mset__left,axiom,
! [A: $tType,A2: A,A5: multiset @ A,B5: multiset @ A] :
( ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ A2 @ A5 ) @ B5 )
= ( add_mset @ A @ A2 @ ( plus_plus @ ( multiset @ A ) @ A5 @ B5 ) ) ) ).
% union_mset_add_mset_left
thf(fact_3716_finite__greaterThanAtMost,axiom,
! [L: nat,U: nat] : ( finite_finite2 @ nat @ ( set_or3652927894154168847AtMost @ nat @ L @ U ) ) ).
% finite_greaterThanAtMost
thf(fact_3717_greaterThanAtMost__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [I: A,L: A,U: A] :
( ( member @ A @ I @ ( set_or3652927894154168847AtMost @ A @ L @ U ) )
= ( ( ord_less @ A @ L @ I )
& ( ord_less_eq @ A @ I @ U ) ) ) ) ).
% greaterThanAtMost_iff
thf(fact_3718_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_3719_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_3720_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_3721_infinite__Ioc__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A] :
( ( ~ ( finite_finite2 @ A @ ( set_or3652927894154168847AtMost @ A @ A2 @ B2 ) ) )
= ( ord_less @ A @ A2 @ B2 ) ) ) ).
% infinite_Ioc_iff
thf(fact_3722_size__add__mset,axiom,
! [A: $tType,A2: A,A5: multiset @ A] :
( ( size_size @ ( multiset @ A ) @ ( add_mset @ A @ A2 @ A5 ) )
= ( suc @ ( size_size @ ( multiset @ A ) @ A5 ) ) ) ).
% size_add_mset
thf(fact_3723_card__greaterThanAtMost,axiom,
! [L: nat,U: nat] :
( ( finite_card @ nat @ ( set_or3652927894154168847AtMost @ nat @ L @ U ) )
= ( minus_minus @ nat @ U @ L ) ) ).
% card_greaterThanAtMost
thf(fact_3724_subset__mset_Osum__eq__0__iff,axiom,
! [A: $tType,B: $tType,F4: set @ B,F: B > ( multiset @ A )] :
( ( finite_finite2 @ B @ F4 )
=> ( ( ( groups3894954378712506084id_sum @ ( multiset @ A ) @ B @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ F @ F4 )
= ( zero_zero @ ( multiset @ A ) ) )
= ( ! [X: B] :
( ( member @ B @ X @ F4 )
=> ( ( F @ X )
= ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ) ).
% subset_mset.sum_eq_0_iff
thf(fact_3725_size__multiset__add__mset,axiom,
! [A: $tType,F: A > nat,A2: A,M3: multiset @ A] :
( ( size_multiset @ A @ F @ ( add_mset @ A @ A2 @ M3 ) )
= ( plus_plus @ nat @ ( suc @ ( F @ A2 ) ) @ ( size_multiset @ A @ F @ M3 ) ) ) ).
% size_multiset_add_mset
thf(fact_3726_take__bit__numeral__numeral,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: num,N: num] :
( ( bit_se2584673776208193580ke_bit @ A @ ( numeral_numeral @ nat @ M ) @ ( numeral_numeral @ A @ N ) )
= ( case_option @ A @ num @ ( zero_zero @ A ) @ ( numeral_numeral @ A ) @ ( bit_take_bit_num @ ( numeral_numeral @ nat @ M ) @ N ) ) ) ) ).
% take_bit_numeral_numeral
thf(fact_3727_mset__le__add__mset__decr__left1,axiom,
! [A: $tType,C2: A,A2: multiset @ A,B2: multiset @ A] :
( ( subseteq_mset @ A @ ( add_mset @ A @ C2 @ A2 ) @ B2 )
=> ( subseteq_mset @ A @ A2 @ B2 ) ) ).
% mset_le_add_mset_decr_left1
thf(fact_3728_mset__union__2__elem,axiom,
! [A: $tType,A2: A,B2: A,C2: A,M3: multiset @ A] :
( ( ( add_mset @ A @ A2 @ ( add_mset @ A @ B2 @ ( zero_zero @ ( multiset @ A ) ) ) )
= ( add_mset @ A @ C2 @ M3 ) )
=> ( ( ( ( add_mset @ A @ A2 @ ( zero_zero @ ( multiset @ A ) ) )
= M3 )
& ( B2 = C2 ) )
| ( ( A2 = C2 )
& ( ( add_mset @ A @ B2 @ ( zero_zero @ ( multiset @ A ) ) )
= M3 ) ) ) ) ).
% mset_union_2_elem
thf(fact_3729_Ioc__inj,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( ( set_or3652927894154168847AtMost @ A @ A2 @ B2 )
= ( set_or3652927894154168847AtMost @ A @ C2 @ D2 ) )
= ( ( ( ord_less_eq @ A @ B2 @ A2 )
& ( ord_less_eq @ A @ D2 @ C2 ) )
| ( ( A2 = C2 )
& ( B2 = D2 ) ) ) ) ) ).
% Ioc_inj
thf(fact_3730_mset__le__add__mset,axiom,
! [A: $tType,X2: A,B5: multiset @ A,C6: multiset @ A] :
( ( subseteq_mset @ A @ ( add_mset @ A @ X2 @ B5 ) @ C6 )
=> ( ( subseteq_mset @ A @ ( add_mset @ A @ X2 @ ( zero_zero @ ( multiset @ A ) ) ) @ C6 )
& ( subseteq_mset @ A @ B5 @ C6 ) ) ) ).
% mset_le_add_mset
thf(fact_3731_mset__le__single__cases,axiom,
! [A: $tType,M3: multiset @ A,A2: A] :
( ( subseteq_mset @ A @ M3 @ ( add_mset @ A @ A2 @ ( zero_zero @ ( multiset @ A ) ) ) )
=> ( ( M3
!= ( zero_zero @ ( multiset @ A ) ) )
=> ( M3
= ( add_mset @ A @ A2 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ).
% mset_le_single_cases
thf(fact_3732_mset__le__add__mset__decr__left2,axiom,
! [A: $tType,C2: A,A2: multiset @ A,B2: multiset @ A] :
( ( subseteq_mset @ A @ ( add_mset @ A @ C2 @ A2 ) @ B2 )
=> ( subseteq_mset @ A @ ( add_mset @ A @ C2 @ ( zero_zero @ ( multiset @ A ) ) ) @ B2 ) ) ).
% mset_le_add_mset_decr_left2
thf(fact_3733_add__mset__add__single,axiom,
! [A: $tType] :
( ( add_mset @ A )
= ( ^ [A3: A,A8: multiset @ A] : ( plus_plus @ ( multiset @ A ) @ A8 @ ( add_mset @ A @ A3 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ).
% add_mset_add_single
thf(fact_3734_union__is__single,axiom,
! [A: $tType,M3: multiset @ A,N5: multiset @ A,A2: A] :
( ( ( plus_plus @ ( multiset @ A ) @ M3 @ N5 )
= ( add_mset @ A @ A2 @ ( zero_zero @ ( multiset @ A ) ) ) )
= ( ( ( M3
= ( add_mset @ A @ A2 @ ( zero_zero @ ( multiset @ A ) ) ) )
& ( N5
= ( zero_zero @ ( multiset @ A ) ) ) )
| ( ( M3
= ( zero_zero @ ( multiset @ A ) ) )
& ( N5
= ( add_mset @ A @ A2 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ) ).
% union_is_single
thf(fact_3735_single__is__union,axiom,
! [A: $tType,A2: A,M3: multiset @ A,N5: multiset @ A] :
( ( ( add_mset @ A @ A2 @ ( zero_zero @ ( multiset @ A ) ) )
= ( plus_plus @ ( multiset @ A ) @ M3 @ N5 ) )
= ( ( ( ( add_mset @ A @ A2 @ ( zero_zero @ ( multiset @ A ) ) )
= M3 )
& ( N5
= ( zero_zero @ ( multiset @ A ) ) ) )
| ( ( M3
= ( zero_zero @ ( multiset @ A ) ) )
& ( ( add_mset @ A @ A2 @ ( zero_zero @ ( multiset @ A ) ) )
= N5 ) ) ) ) ).
% single_is_union
thf(fact_3736_add__mset__replicate__mset__safe,axiom,
! [A: $tType,B: $tType,M3: multiset @ B,A2: B] :
( ( nO_MATCH @ ( multiset @ A ) @ ( multiset @ B ) @ ( zero_zero @ ( multiset @ A ) ) @ M3 )
=> ( ( add_mset @ B @ A2 @ M3 )
= ( plus_plus @ ( multiset @ B ) @ ( add_mset @ B @ A2 @ ( zero_zero @ ( multiset @ B ) ) ) @ M3 ) ) ) ).
% add_mset_replicate_mset_safe
thf(fact_3737_size__eq__Suc__imp__eq__union,axiom,
! [A: $tType,M3: multiset @ A,N: nat] :
( ( ( size_size @ ( multiset @ A ) @ M3 )
= ( suc @ N ) )
=> ? [A4: A,N10: multiset @ A] :
( M3
= ( add_mset @ A @ A4 @ N10 ) ) ) ).
% size_eq_Suc_imp_eq_union
thf(fact_3738_infinite__Ioc,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ~ ( finite_finite2 @ A @ ( set_or3652927894154168847AtMost @ A @ A2 @ B2 ) ) ) ) ).
% infinite_Ioc
thf(fact_3739_multiset__induct2__size,axiom,
! [D: $tType,C: $tType,A: $tType,B: $tType,M3: multiset @ A,N5: multiset @ B,P: ( multiset @ A ) > ( multiset @ B ) > $o] :
( ( ( size_size @ ( multiset @ A ) @ M3 )
= ( size_size @ ( multiset @ B ) @ N5 ) )
=> ( ( P @ ( zero_zero @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ B ) ) )
=> ( ! [A4: C,B4: D,M10: multiset @ A,N10: multiset @ B,Aa2: A,Ba: B] :
( ( P @ M10 @ N10 )
=> ( P @ ( add_mset @ A @ Aa2 @ M10 ) @ ( add_mset @ B @ Ba @ N10 ) ) )
=> ( P @ M3 @ N5 ) ) ) ) ).
% multiset_induct2_size
thf(fact_3740_Ioc__subset__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ A2 @ B2 ) @ ( set_or3652927894154168847AtMost @ A @ C2 @ D2 ) )
= ( ( ord_less_eq @ A @ B2 @ A2 )
| ( ( ord_less_eq @ A @ C2 @ A2 )
& ( ord_less_eq @ A @ B2 @ D2 ) ) ) ) ) ).
% Ioc_subset_iff
thf(fact_3741_atLeastSucAtMost__greaterThanAtMost,axiom,
! [L: nat,U: nat] :
( ( set_or1337092689740270186AtMost @ nat @ ( suc @ L ) @ U )
= ( set_or3652927894154168847AtMost @ nat @ L @ U ) ) ).
% atLeastSucAtMost_greaterThanAtMost
thf(fact_3742_mset__le__subtract__add__mset__left,axiom,
! [A: $tType,X2: A,B5: multiset @ A,X8: multiset @ A] :
( ( subseteq_mset @ A @ ( add_mset @ A @ X2 @ B5 ) @ X8 )
=> ( ( subseteq_mset @ A @ B5 @ ( minus_minus @ ( multiset @ A ) @ X8 @ ( add_mset @ A @ X2 @ ( zero_zero @ ( multiset @ A ) ) ) ) )
& ( subseteq_mset @ A @ ( add_mset @ A @ X2 @ ( zero_zero @ ( multiset @ A ) ) ) @ X8 ) ) ) ).
% mset_le_subtract_add_mset_left
thf(fact_3743_mset__le__subtract__add__mset__right,axiom,
! [A: $tType,X2: A,B5: multiset @ A,X8: multiset @ A] :
( ( subseteq_mset @ A @ ( add_mset @ A @ X2 @ B5 ) @ X8 )
=> ( ( subseteq_mset @ A @ ( add_mset @ A @ X2 @ ( zero_zero @ ( multiset @ A ) ) ) @ ( minus_minus @ ( multiset @ A ) @ X8 @ B5 ) )
& ( subseteq_mset @ A @ B5 @ X8 ) ) ) ).
% mset_le_subtract_add_mset_right
thf(fact_3744_mset__unplusm__dist__cases2,axiom,
! [A: $tType,B5: multiset @ A,C6: multiset @ A,S3: A,A5: multiset @ A] :
( ( ( plus_plus @ ( multiset @ A ) @ B5 @ C6 )
= ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ S3 @ ( zero_zero @ ( multiset @ A ) ) ) @ A5 ) )
=> ( ( ( B5
= ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ S3 @ ( zero_zero @ ( multiset @ A ) ) ) @ ( minus_minus @ ( multiset @ A ) @ B5 @ ( add_mset @ A @ S3 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) )
=> ( A5
!= ( plus_plus @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ B5 @ ( add_mset @ A @ S3 @ ( zero_zero @ ( multiset @ A ) ) ) ) @ C6 ) ) )
=> ~ ( ( C6
= ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ S3 @ ( zero_zero @ ( multiset @ A ) ) ) @ ( minus_minus @ ( multiset @ A ) @ C6 @ ( add_mset @ A @ S3 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) )
=> ( A5
!= ( plus_plus @ ( multiset @ A ) @ B5 @ ( minus_minus @ ( multiset @ A ) @ C6 @ ( add_mset @ A @ S3 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ) ) ) ).
% mset_unplusm_dist_cases2
thf(fact_3745_mset__unplusm__dist__cases,axiom,
! [A: $tType,S3: A,A5: multiset @ A,B5: multiset @ A,C6: multiset @ A] :
( ( ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ S3 @ ( zero_zero @ ( multiset @ A ) ) ) @ A5 )
= ( plus_plus @ ( multiset @ A ) @ B5 @ C6 ) )
=> ( ( ( B5
= ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ S3 @ ( zero_zero @ ( multiset @ A ) ) ) @ ( minus_minus @ ( multiset @ A ) @ B5 @ ( add_mset @ A @ S3 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) )
=> ( A5
!= ( plus_plus @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ B5 @ ( add_mset @ A @ S3 @ ( zero_zero @ ( multiset @ A ) ) ) ) @ C6 ) ) )
=> ~ ( ( C6
= ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ S3 @ ( zero_zero @ ( multiset @ A ) ) ) @ ( minus_minus @ ( multiset @ A ) @ C6 @ ( add_mset @ A @ S3 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) )
=> ( A5
!= ( plus_plus @ ( multiset @ A ) @ B5 @ ( minus_minus @ ( multiset @ A ) @ C6 @ ( add_mset @ A @ S3 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ) ) ) ).
% mset_unplusm_dist_cases
thf(fact_3746_mset__single__cases2_H,axiom,
! [A: $tType,S3: A,C2: multiset @ A,R4: A,C3: multiset @ A] :
( ( ( add_mset @ A @ S3 @ C2 )
= ( add_mset @ A @ R4 @ C3 ) )
=> ( ( ( S3 = R4 )
=> ( C2 != C3 ) )
=> ~ ! [Cc: multiset @ A] :
( ( C3
= ( plus_plus @ ( multiset @ A ) @ Cc @ ( add_mset @ A @ S3 @ ( zero_zero @ ( multiset @ A ) ) ) ) )
=> ( ( C2
= ( plus_plus @ ( multiset @ A ) @ Cc @ ( add_mset @ A @ R4 @ ( zero_zero @ ( multiset @ A ) ) ) ) )
=> ( ( ( minus_minus @ ( multiset @ A ) @ C3 @ ( add_mset @ A @ S3 @ ( zero_zero @ ( multiset @ A ) ) ) )
= Cc )
=> ( ( minus_minus @ ( multiset @ A ) @ C2 @ ( add_mset @ A @ R4 @ ( zero_zero @ ( multiset @ A ) ) ) )
!= Cc ) ) ) ) ) ) ).
% mset_single_cases2'
thf(fact_3747_mset__single__cases2,axiom,
! [A: $tType,S3: A,C2: multiset @ A,R4: A,C3: multiset @ A] :
( ( ( add_mset @ A @ S3 @ C2 )
= ( add_mset @ A @ R4 @ C3 ) )
=> ( ( ( S3 = R4 )
=> ( C2 != C3 ) )
=> ~ ( ( C3
= ( plus_plus @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ C3 @ ( add_mset @ A @ S3 @ ( zero_zero @ ( multiset @ A ) ) ) ) @ ( add_mset @ A @ S3 @ ( zero_zero @ ( multiset @ A ) ) ) ) )
=> ( ( C2
= ( plus_plus @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ C2 @ ( add_mset @ A @ R4 @ ( zero_zero @ ( multiset @ A ) ) ) ) @ ( add_mset @ A @ R4 @ ( zero_zero @ ( multiset @ A ) ) ) ) )
=> ( ( minus_minus @ ( multiset @ A ) @ C2 @ ( add_mset @ A @ R4 @ ( zero_zero @ ( multiset @ A ) ) ) )
!= ( minus_minus @ ( multiset @ A ) @ C3 @ ( add_mset @ A @ S3 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ) ) ) ).
% mset_single_cases2
thf(fact_3748_mset__single__cases_H,axiom,
! [A: $tType,S3: A,C2: multiset @ A,R4: A,C3: multiset @ A] :
( ( ( add_mset @ A @ S3 @ C2 )
= ( add_mset @ A @ R4 @ C3 ) )
=> ( ( ( S3 = R4 )
=> ( C2 != C3 ) )
=> ~ ! [Cc: multiset @ A] :
( ( C3
= ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ S3 @ ( zero_zero @ ( multiset @ A ) ) ) @ Cc ) )
=> ( ( C2
= ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ R4 @ ( zero_zero @ ( multiset @ A ) ) ) @ Cc ) )
=> ( ( ( minus_minus @ ( multiset @ A ) @ C3 @ ( add_mset @ A @ S3 @ ( zero_zero @ ( multiset @ A ) ) ) )
= Cc )
=> ( ( minus_minus @ ( multiset @ A ) @ C2 @ ( add_mset @ A @ R4 @ ( zero_zero @ ( multiset @ A ) ) ) )
!= Cc ) ) ) ) ) ) ).
% mset_single_cases'
thf(fact_3749_mset__single__cases,axiom,
! [A: $tType,S3: A,C2: multiset @ A,R4: A,C3: multiset @ A] :
( ( ( add_mset @ A @ S3 @ C2 )
= ( add_mset @ A @ R4 @ C3 ) )
=> ( ( ( S3 = R4 )
=> ( C2 != C3 ) )
=> ~ ( ( C3
= ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ S3 @ ( zero_zero @ ( multiset @ A ) ) ) @ ( minus_minus @ ( multiset @ A ) @ C3 @ ( add_mset @ A @ S3 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) )
=> ( ( C2
= ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ R4 @ ( zero_zero @ ( multiset @ A ) ) ) @ ( minus_minus @ ( multiset @ A ) @ C2 @ ( add_mset @ A @ R4 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) )
=> ( ( minus_minus @ ( multiset @ A ) @ C2 @ ( add_mset @ A @ R4 @ ( zero_zero @ ( multiset @ A ) ) ) )
!= ( minus_minus @ ( multiset @ A ) @ C3 @ ( add_mset @ A @ S3 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ) ) ) ).
% mset_single_cases
thf(fact_3750_size__1__singleton__mset,axiom,
! [A: $tType,M3: multiset @ A] :
( ( ( size_size @ ( multiset @ A ) @ M3 )
= ( one_one @ nat ) )
=> ? [A4: A] :
( M3
= ( add_mset @ A @ A4 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ).
% size_1_singleton_mset
thf(fact_3751_size__single,axiom,
! [A: $tType,B2: A] :
( ( size_size @ ( multiset @ A ) @ ( add_mset @ A @ B2 @ ( zero_zero @ ( multiset @ A ) ) ) )
= ( one_one @ nat ) ) ).
% size_single
thf(fact_3752_sorted__list__of__set__greaterThanAtMost,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq @ nat @ ( suc @ I ) @ J )
=> ( ( linord4507533701916653071of_set @ nat @ ( set_or3652927894154168847AtMost @ nat @ I @ J ) )
= ( cons @ nat @ ( suc @ I ) @ ( linord4507533701916653071of_set @ nat @ ( set_or3652927894154168847AtMost @ nat @ ( suc @ I ) @ J ) ) ) ) ) ).
% sorted_list_of_set_greaterThanAtMost
thf(fact_3753_size__mset__SucE,axiom,
! [A: $tType,A5: multiset @ A,N: nat] :
( ( ( size_size @ ( multiset @ A ) @ A5 )
= ( suc @ N ) )
=> ~ ! [A4: A,B10: multiset @ A] :
( ( A5
= ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ A4 @ ( zero_zero @ ( multiset @ A ) ) ) @ B10 ) )
=> ( ( size_size @ ( multiset @ A ) @ B10 )
!= N ) ) ) ).
% size_mset_SucE
thf(fact_3754_size__Diff1__le,axiom,
! [A: $tType,M3: multiset @ A,X2: A] : ( ord_less_eq @ nat @ ( size_size @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ M3 @ ( add_mset @ A @ X2 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) @ ( size_size @ ( multiset @ A ) @ M3 ) ) ).
% size_Diff1_le
thf(fact_3755_sum_Ohead,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M: nat,N: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( plus_plus @ A @ ( G @ M ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or3652927894154168847AtMost @ nat @ M @ N ) ) ) ) ) ) ).
% sum.head
thf(fact_3756_prod_Ohead,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M: nat,N: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( times_times @ A @ ( G @ M ) @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or3652927894154168847AtMost @ nat @ M @ N ) ) ) ) ) ) ).
% prod.head
thf(fact_3757_greaterThanAtMost__subseteq__atLeastAtMost__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ A2 @ B2 ) @ ( set_or1337092689740270186AtMost @ A @ C2 @ D2 ) )
= ( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ A2 )
& ( ord_less_eq @ A @ B2 @ D2 ) ) ) ) ) ).
% greaterThanAtMost_subseteq_atLeastAtMost_iff
thf(fact_3758_greaterThanAtMost__subseteq__atLeastLessThan__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ A2 @ B2 ) @ ( set_or7035219750837199246ssThan @ A @ C2 @ D2 ) )
= ( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ A2 )
& ( ord_less @ A @ B2 @ D2 ) ) ) ) ) ).
% greaterThanAtMost_subseteq_atLeastLessThan_iff
thf(fact_3759_subset__mset_Osum__nonneg,axiom,
! [A: $tType,B: $tType,A5: set @ B,F: B > ( multiset @ A )] :
( ! [X3: B] :
( ( member @ B @ X3 @ A5 )
=> ( subseteq_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ ( F @ X3 ) ) )
=> ( subseteq_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ ( groups3894954378712506084id_sum @ ( multiset @ A ) @ B @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ F @ A5 ) ) ) ).
% subset_mset.sum_nonneg
thf(fact_3760_subset__mset_Osum__nonpos,axiom,
! [B: $tType,A: $tType,A5: set @ B,F: B > ( multiset @ A )] :
( ! [X3: B] :
( ( member @ B @ X3 @ A5 )
=> ( subseteq_mset @ A @ ( F @ X3 ) @ ( zero_zero @ ( multiset @ A ) ) ) )
=> ( subseteq_mset @ A @ ( groups3894954378712506084id_sum @ ( multiset @ A ) @ B @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ F @ A5 ) @ ( zero_zero @ ( multiset @ A ) ) ) ) ).
% subset_mset.sum_nonpos
thf(fact_3761_greaterThanAtMost__eq__atLeastAtMost__diff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( set_or3652927894154168847AtMost @ A )
= ( ^ [A3: A,B3: A] : ( minus_minus @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A3 @ B3 ) @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% greaterThanAtMost_eq_atLeastAtMost_diff
thf(fact_3762_greaterThanLessThan__subseteq__greaterThanAtMost__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ A2 @ B2 ) @ ( set_or3652927894154168847AtMost @ A @ C2 @ D2 ) )
= ( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ A2 )
& ( ord_less_eq @ A @ B2 @ D2 ) ) ) ) ) ).
% greaterThanLessThan_subseteq_greaterThanAtMost_iff
thf(fact_3763_subset__mset_Osum__mono,axiom,
! [A: $tType,B: $tType,K3: set @ B,F: B > ( multiset @ A ),G: B > ( multiset @ A )] :
( ! [I3: B] :
( ( member @ B @ I3 @ K3 )
=> ( subseteq_mset @ A @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( subseteq_mset @ A @ ( groups3894954378712506084id_sum @ ( multiset @ A ) @ B @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ F @ K3 ) @ ( groups3894954378712506084id_sum @ ( multiset @ A ) @ B @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ G @ K3 ) ) ) ).
% subset_mset.sum_mono
thf(fact_3764_mset__size__le1__cases,axiom,
! [A: $tType,M3: multiset @ A] :
( ( ord_less_eq @ nat @ ( size_size @ ( multiset @ A ) @ M3 ) @ ( suc @ ( zero_zero @ nat ) ) )
=> ( ( M3
!= ( zero_zero @ ( multiset @ A ) ) )
=> ~ ! [M4: A] :
( M3
!= ( add_mset @ A @ M4 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ).
% mset_size_le1_cases
thf(fact_3765_subset__mset_Osum__nonneg__eq__0__iff,axiom,
! [A: $tType,B: $tType,A5: set @ B,F: B > ( multiset @ A )] :
( ( finite_finite2 @ B @ A5 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ A5 )
=> ( subseteq_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ ( F @ X3 ) ) )
=> ( ( ( groups3894954378712506084id_sum @ ( multiset @ A ) @ B @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ F @ A5 )
= ( zero_zero @ ( multiset @ A ) ) )
= ( ! [X: B] :
( ( member @ B @ X @ A5 )
=> ( ( F @ X )
= ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ) ) ).
% subset_mset.sum_nonneg_eq_0_iff
thf(fact_3766_subset__mset_Osum__le__included,axiom,
! [B: $tType,A: $tType,C: $tType,S3: set @ B,T2: set @ C,G: C > ( multiset @ A ),I: C > B,F: B > ( multiset @ A )] :
( ( finite_finite2 @ B @ S3 )
=> ( ( finite_finite2 @ C @ T2 )
=> ( ! [X3: C] :
( ( member @ C @ X3 @ T2 )
=> ( subseteq_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ ( G @ X3 ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ S3 )
=> ? [Xa2: C] :
( ( member @ C @ Xa2 @ T2 )
& ( ( I @ Xa2 )
= X3 )
& ( subseteq_mset @ A @ ( F @ X3 ) @ ( G @ Xa2 ) ) ) )
=> ( subseteq_mset @ A @ ( groups3894954378712506084id_sum @ ( multiset @ A ) @ B @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ F @ S3 ) @ ( groups3894954378712506084id_sum @ ( multiset @ A ) @ C @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ G @ T2 ) ) ) ) ) ) ).
% subset_mset.sum_le_included
thf(fact_3767_subset__mset_Osum__nonneg__0,axiom,
! [B: $tType,A: $tType,S3: set @ B,F: B > ( multiset @ A ),I: B] :
( ( finite_finite2 @ B @ S3 )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ S3 )
=> ( subseteq_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ ( F @ I3 ) ) )
=> ( ( ( groups3894954378712506084id_sum @ ( multiset @ A ) @ B @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ F @ S3 )
= ( zero_zero @ ( multiset @ A ) ) )
=> ( ( member @ B @ I @ S3 )
=> ( ( F @ I )
= ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ) ).
% subset_mset.sum_nonneg_0
thf(fact_3768_subset__mset_Osum__nonneg__leq__bound,axiom,
! [B: $tType,A: $tType,S3: set @ B,F: B > ( multiset @ A ),B5: multiset @ A,I: B] :
( ( finite_finite2 @ B @ S3 )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ S3 )
=> ( subseteq_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ ( F @ I3 ) ) )
=> ( ( ( groups3894954378712506084id_sum @ ( multiset @ A ) @ B @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ F @ S3 )
= B5 )
=> ( ( member @ B @ I @ S3 )
=> ( subseteq_mset @ A @ ( F @ I ) @ B5 ) ) ) ) ) ).
% subset_mset.sum_nonneg_leq_bound
thf(fact_3769_and__minus__numerals_I7_J,axiom,
! [N: num,M: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) @ ( numeral_numeral @ int @ M ) )
= ( case_option @ int @ num @ ( zero_zero @ int ) @ ( numeral_numeral @ int ) @ ( bit_and_not_num @ M @ ( bitM @ N ) ) ) ) ).
% and_minus_numerals(7)
thf(fact_3770_and__minus__numerals_I3_J,axiom,
! [M: num,N: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ M ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) )
= ( case_option @ int @ num @ ( zero_zero @ int ) @ ( numeral_numeral @ int ) @ ( bit_and_not_num @ M @ ( bitM @ N ) ) ) ) ).
% and_minus_numerals(3)
thf(fact_3771_and__minus__numerals_I8_J,axiom,
! [N: num,M: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) @ ( numeral_numeral @ int @ M ) )
= ( case_option @ int @ num @ ( zero_zero @ int ) @ ( numeral_numeral @ int ) @ ( bit_and_not_num @ M @ ( bit0 @ N ) ) ) ) ).
% and_minus_numerals(8)
thf(fact_3772_and__minus__numerals_I4_J,axiom,
! [M: num,N: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ M ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) )
= ( case_option @ int @ num @ ( zero_zero @ int ) @ ( numeral_numeral @ int ) @ ( bit_and_not_num @ M @ ( bit0 @ N ) ) ) ) ).
% and_minus_numerals(4)
thf(fact_3773_finite__greaterThanAtMost__int,axiom,
! [L: int,U: int] : ( finite_finite2 @ int @ ( set_or3652927894154168847AtMost @ int @ L @ U ) ) ).
% finite_greaterThanAtMost_int
thf(fact_3774_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_3775_card__greaterThanAtMost__int,axiom,
! [L: int,U: int] :
( ( finite_card @ int @ ( set_or3652927894154168847AtMost @ int @ L @ U ) )
= ( nat2 @ ( minus_minus @ int @ U @ L ) ) ) ).
% card_greaterThanAtMost_int
thf(fact_3776_atLeastPlusOneAtMost__greaterThanAtMost__int,axiom,
! [L: int,U: int] :
( ( set_or1337092689740270186AtMost @ int @ ( plus_plus @ int @ L @ ( one_one @ int ) ) @ U )
= ( set_or3652927894154168847AtMost @ int @ L @ U ) ) ).
% atLeastPlusOneAtMost_greaterThanAtMost_int
thf(fact_3777_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_3778_greaterThanAtMost__upto,axiom,
( ( set_or3652927894154168847AtMost @ int )
= ( ^ [I2: int,J3: int] : ( set2 @ int @ ( upto @ ( plus_plus @ int @ I2 @ ( one_one @ int ) ) @ J3 ) ) ) ) ).
% greaterThanAtMost_upto
thf(fact_3779_int__numeral__not__and__num,axiom,
! [M: num,N: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ M ) ) @ ( numeral_numeral @ int @ N ) )
= ( case_option @ int @ num @ ( zero_zero @ int ) @ ( numeral_numeral @ int ) @ ( bit_and_not_num @ N @ M ) ) ) ).
% int_numeral_not_and_num
thf(fact_3780_int__numeral__and__not__num,axiom,
! [M: num,N: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ M ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N ) ) )
= ( case_option @ int @ num @ ( zero_zero @ int ) @ ( numeral_numeral @ int ) @ ( bit_and_not_num @ M @ N ) ) ) ).
% int_numeral_and_not_num
thf(fact_3781_sorted__list__of__multiset__insert,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,M3: multiset @ A] :
( ( linord6283353356039996273ltiset @ A @ ( add_mset @ A @ X2 @ M3 ) )
= ( linorder_insort_key @ A @ A
@ ^ [X: A] : X
@ X2
@ ( linord6283353356039996273ltiset @ A @ M3 ) ) ) ) ).
% sorted_list_of_multiset_insert
thf(fact_3782_wcount__add__mset,axiom,
! [A: $tType,F: A > nat,X2: A,M3: multiset @ A,A2: A] :
( ( wcount @ A @ F @ ( add_mset @ A @ X2 @ M3 ) @ A2 )
= ( plus_plus @ nat @ ( if @ nat @ ( X2 = A2 ) @ ( suc @ ( F @ A2 ) ) @ ( zero_zero @ nat ) ) @ ( wcount @ A @ F @ M3 @ A2 ) ) ) ).
% wcount_add_mset
thf(fact_3783_nat__of__integer__less__iff,axiom,
! [X2: code_integer,Y: code_integer] :
( ( ord_less_eq @ code_integer @ ( zero_zero @ code_integer ) @ X2 )
=> ( ( ord_less_eq @ code_integer @ ( zero_zero @ code_integer ) @ Y )
=> ( ( ord_less @ nat @ ( code_nat_of_integer @ X2 ) @ ( code_nat_of_integer @ Y ) )
= ( ord_less @ code_integer @ X2 @ Y ) ) ) ) ).
% nat_of_integer_less_iff
thf(fact_3784_distinct__foldl__invar,axiom,
! [B: $tType,A: $tType,S2: list @ A,I5: ( set @ A ) > B > $o,Sigma_0: B,F: B > A > B] :
( ( distinct @ A @ S2 )
=> ( ( I5 @ ( set2 @ A @ S2 ) @ Sigma_0 )
=> ( ! [X3: A,It: set @ A,Sigma: B] :
( ( member @ A @ X3 @ It )
=> ( ( ord_less_eq @ ( set @ A ) @ It @ ( set2 @ A @ S2 ) )
=> ( ( I5 @ It @ Sigma )
=> ( I5 @ ( minus_minus @ ( set @ A ) @ It @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( F @ Sigma @ X3 ) ) ) ) )
=> ( I5 @ ( bot_bot @ ( set @ A ) ) @ ( foldl @ B @ A @ F @ Sigma_0 @ S2 ) ) ) ) ) ).
% distinct_foldl_invar
thf(fact_3785_nat__of__integer__of__nat,axiom,
! [N: nat] :
( ( code_nat_of_integer @ ( semiring_1_of_nat @ code_integer @ N ) )
= N ) ).
% nat_of_integer_of_nat
thf(fact_3786_of__nat__of__integer,axiom,
! [K: code_integer] :
( ( semiring_1_of_nat @ code_integer @ ( code_nat_of_integer @ K ) )
= ( ord_max @ code_integer @ ( zero_zero @ code_integer ) @ K ) ) ).
% of_nat_of_integer
thf(fact_3787_foldl__length,axiom,
! [A: $tType,L: list @ A] :
( ( foldl @ nat @ A
@ ^ [I2: nat,X: A] : ( suc @ I2 )
@ ( zero_zero @ nat )
@ L )
= ( size_size @ ( list @ A ) @ L ) ) ).
% foldl_length
thf(fact_3788_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_3789_foldl__A1__eq,axiom,
! [A: $tType,F: A > A > A,N: A,I: A,Ww: list @ A] :
( ! [E4: A] :
( ( F @ N @ E4 )
= E4 )
=> ( ! [E4: A] :
( ( F @ E4 @ N )
= E4 )
=> ( ! [A4: A,B4: A,C4: A] :
( ( F @ A4 @ ( F @ B4 @ C4 ) )
= ( F @ ( F @ A4 @ B4 ) @ C4 ) )
=> ( ( foldl @ A @ A @ F @ I @ Ww )
= ( F @ I @ ( foldl @ A @ A @ F @ N @ Ww ) ) ) ) ) ) ).
% foldl_A1_eq
thf(fact_3790_foldl__absorb1,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [X2: A,Zs: list @ A] :
( ( times_times @ A @ X2 @ ( foldl @ A @ A @ ( times_times @ A ) @ ( one_one @ A ) @ Zs ) )
= ( foldl @ A @ A @ ( times_times @ A ) @ X2 @ Zs ) ) ) ).
% foldl_absorb1
thf(fact_3791_nat__of__integer_Oabs__eq,axiom,
! [X2: int] :
( ( code_nat_of_integer @ ( code_integer_of_int @ X2 ) )
= ( nat2 @ X2 ) ) ).
% nat_of_integer.abs_eq
thf(fact_3792_foldl__length__aux,axiom,
! [A: $tType,A2: nat,L: list @ A] :
( ( foldl @ nat @ A
@ ^ [I2: nat,X: A] : ( suc @ I2 )
@ A2
@ L )
= ( plus_plus @ nat @ A2 @ ( size_size @ ( list @ A ) @ L ) ) ) ).
% foldl_length_aux
thf(fact_3793_wcount__union,axiom,
! [A: $tType,F: A > nat,M3: multiset @ A,N5: multiset @ A,A2: A] :
( ( wcount @ A @ F @ ( plus_plus @ ( multiset @ A ) @ M3 @ N5 ) @ A2 )
= ( plus_plus @ nat @ ( wcount @ A @ F @ M3 @ A2 ) @ ( wcount @ A @ F @ N5 @ A2 ) ) ) ).
% wcount_union
thf(fact_3794_length__n__lists,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( size_size @ ( list @ ( list @ A ) ) @ ( n_lists @ A @ N @ Xs ) )
= ( power_power @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N ) ) ).
% length_n_lists
thf(fact_3795_upto__rec2,axiom,
! [I: int,J: int] :
( ( ord_less_eq @ int @ I @ J )
=> ( ( upto @ I @ J )
= ( append @ int @ ( upto @ I @ ( minus_minus @ int @ J @ ( one_one @ int ) ) ) @ ( cons @ int @ J @ ( nil @ int ) ) ) ) ) ).
% upto_rec2
thf(fact_3796_prod_Oinsert_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [I5: set @ B,P3: B > A,I: B] :
( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X: B] :
( ( member @ B @ X @ I5 )
& ( ( P3 @ X )
!= ( one_one @ A ) ) ) ) )
=> ( ( ( member @ B @ I @ I5 )
=> ( ( groups1962203154675924110t_prod @ B @ A @ P3 @ ( insert @ B @ I @ I5 ) )
= ( groups1962203154675924110t_prod @ B @ A @ P3 @ I5 ) ) )
& ( ~ ( member @ B @ I @ I5 )
=> ( ( groups1962203154675924110t_prod @ B @ A @ P3 @ ( insert @ B @ I @ I5 ) )
= ( times_times @ A @ ( P3 @ I ) @ ( groups1962203154675924110t_prod @ B @ A @ P3 @ I5 ) ) ) ) ) ) ) ).
% prod.insert'
thf(fact_3797_set__n__lists,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( set2 @ ( list @ A ) @ ( n_lists @ A @ N @ Xs ) )
= ( collect @ ( list @ A )
@ ^ [Ys3: list @ A] :
( ( ( size_size @ ( list @ A ) @ Ys3 )
= N )
& ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Ys3 ) @ ( set2 @ A @ Xs ) ) ) ) ) ).
% set_n_lists
thf(fact_3798_empty__append__eq__id,axiom,
! [A: $tType] :
( ( append @ A @ ( nil @ A ) )
= ( ^ [X: list @ A] : X ) ) ).
% empty_append_eq_id
thf(fact_3799_append__eq__append__conv,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Us: list @ A,Vs: list @ A] :
( ( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ A ) @ Ys ) )
| ( ( size_size @ ( list @ A ) @ Us )
= ( size_size @ ( list @ A ) @ Vs ) ) )
=> ( ( ( append @ A @ Xs @ Us )
= ( append @ A @ Ys @ Vs ) )
= ( ( Xs = Ys )
& ( Us = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_3800_list__e__eq__lel_I2_J,axiom,
! [A: $tType,L1: list @ A,E3: A,L22: list @ A,E2: A] :
( ( ( append @ A @ L1 @ ( cons @ A @ E3 @ L22 ) )
= ( cons @ A @ E2 @ ( nil @ A ) ) )
= ( ( L1
= ( nil @ A ) )
& ( E3 = E2 )
& ( L22
= ( nil @ A ) ) ) ) ).
% list_e_eq_lel(2)
thf(fact_3801_list__e__eq__lel_I1_J,axiom,
! [A: $tType,E2: A,L1: list @ A,E3: A,L22: list @ A] :
( ( ( cons @ A @ E2 @ ( nil @ A ) )
= ( append @ A @ L1 @ ( cons @ A @ E3 @ L22 ) ) )
= ( ( L1
= ( nil @ A ) )
& ( E3 = E2 )
& ( L22
= ( nil @ A ) ) ) ) ).
% list_e_eq_lel(1)
thf(fact_3802_list__se__match_I4_J,axiom,
! [A: $tType,L22: list @ A,A2: A,L1: list @ A] :
( ( L22
!= ( nil @ A ) )
=> ( ( ( cons @ A @ A2 @ ( nil @ A ) )
= ( append @ A @ L1 @ L22 ) )
= ( ( L1
= ( nil @ A ) )
& ( L22
= ( cons @ A @ A2 @ ( nil @ A ) ) ) ) ) ) ).
% list_se_match(4)
thf(fact_3803_list__se__match_I3_J,axiom,
! [A: $tType,L1: list @ A,A2: A,L22: list @ A] :
( ( L1
!= ( nil @ A ) )
=> ( ( ( cons @ A @ A2 @ ( nil @ A ) )
= ( append @ A @ L1 @ L22 ) )
= ( ( L1
= ( cons @ A @ A2 @ ( nil @ A ) ) )
& ( L22
= ( nil @ A ) ) ) ) ) ).
% list_se_match(3)
thf(fact_3804_list__se__match_I2_J,axiom,
! [A: $tType,L22: list @ A,L1: list @ A,A2: A] :
( ( L22
!= ( nil @ A ) )
=> ( ( ( append @ A @ L1 @ L22 )
= ( cons @ A @ A2 @ ( nil @ A ) ) )
= ( ( L1
= ( nil @ A ) )
& ( L22
= ( cons @ A @ A2 @ ( nil @ A ) ) ) ) ) ) ).
% list_se_match(2)
thf(fact_3805_list__se__match_I1_J,axiom,
! [A: $tType,L1: list @ A,L22: list @ A,A2: A] :
( ( L1
!= ( nil @ A ) )
=> ( ( ( append @ A @ L1 @ L22 )
= ( cons @ A @ A2 @ ( nil @ A ) ) )
= ( ( L1
= ( cons @ A @ A2 @ ( nil @ A ) ) )
& ( L22
= ( nil @ A ) ) ) ) ) ).
% list_se_match(1)
thf(fact_3806_list__ee__eq__leel_I2_J,axiom,
! [A: $tType,L1: list @ A,E1: A,E22: A,L22: list @ A,E12: A,E23: A] :
( ( ( append @ A @ L1 @ ( cons @ A @ E1 @ ( cons @ A @ E22 @ L22 ) ) )
= ( cons @ A @ E12 @ ( cons @ A @ E23 @ ( nil @ A ) ) ) )
= ( ( L1
= ( nil @ A ) )
& ( E12 = E1 )
& ( E23 = E22 )
& ( L22
= ( nil @ A ) ) ) ) ).
% list_ee_eq_leel(2)
thf(fact_3807_list__ee__eq__leel_I1_J,axiom,
! [A: $tType,E12: A,E23: A,L1: list @ A,E1: A,E22: A,L22: list @ A] :
( ( ( cons @ A @ E12 @ ( cons @ A @ E23 @ ( nil @ A ) ) )
= ( append @ A @ L1 @ ( cons @ A @ E1 @ ( cons @ A @ E22 @ L22 ) ) ) )
= ( ( L1
= ( nil @ A ) )
& ( E12 = E1 )
& ( E23 = E22 )
& ( L22
= ( nil @ A ) ) ) ) ).
% list_ee_eq_leel(1)
thf(fact_3808_length__append,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( size_size @ ( list @ A ) @ ( append @ A @ Xs @ Ys ) )
= ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ A ) @ Ys ) ) ) ).
% length_append
thf(fact_3809_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_3810_n__lists__Nil,axiom,
! [A: $tType,N: nat] :
( ( ( N
= ( zero_zero @ nat ) )
=> ( ( n_lists @ A @ N @ ( nil @ A ) )
= ( cons @ ( list @ A ) @ ( nil @ A ) @ ( nil @ ( list @ A ) ) ) ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ( ( n_lists @ A @ N @ ( nil @ A ) )
= ( nil @ ( list @ A ) ) ) ) ) ).
% n_lists_Nil
thf(fact_3811_nth__append__length,axiom,
! [A: $tType,Xs: list @ A,X2: A,Ys: list @ A] :
( ( nth @ A @ ( append @ A @ Xs @ ( cons @ A @ X2 @ Ys ) ) @ ( size_size @ ( list @ A ) @ Xs ) )
= X2 ) ).
% nth_append_length
thf(fact_3812_nth__append__first,axiom,
! [A: $tType,I: nat,L: list @ A,L4: list @ A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L ) )
=> ( ( nth @ A @ ( append @ A @ L @ L4 ) @ I )
= ( nth @ A @ L @ I ) ) ) ).
% nth_append_first
thf(fact_3813_nth__append__length__plus,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,N: nat] :
( ( nth @ A @ ( append @ A @ Xs @ Ys ) @ ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N ) )
= ( nth @ A @ Ys @ N ) ) ).
% nth_append_length_plus
thf(fact_3814_list__update__length,axiom,
! [A: $tType,Xs: list @ A,X2: A,Ys: list @ A,Y: A] :
( ( list_update @ A @ ( append @ A @ Xs @ ( cons @ A @ X2 @ Ys ) ) @ ( size_size @ ( list @ A ) @ Xs ) @ Y )
= ( append @ A @ Xs @ ( cons @ A @ Y @ Ys ) ) ) ).
% list_update_length
thf(fact_3815_foldl__conc__empty__eq,axiom,
! [A: $tType,I: list @ A,Ww: list @ ( list @ A )] :
( ( foldl @ ( list @ A ) @ ( list @ A ) @ ( append @ A ) @ I @ Ww )
= ( append @ A @ I @ ( foldl @ ( list @ A ) @ ( list @ A ) @ ( append @ A ) @ ( nil @ A ) @ Ww ) ) ) ).
% foldl_conc_empty_eq
thf(fact_3816_list__match__lel__lel,axiom,
! [A: $tType,C1: list @ A,Qs: A,C22: list @ A,C12: list @ A,Qs2: A,C23: list @ A] :
( ( ( append @ A @ C1 @ ( cons @ A @ Qs @ C22 ) )
= ( append @ A @ C12 @ ( cons @ A @ Qs2 @ C23 ) ) )
=> ( ! [C21: list @ A] :
( ( C1
= ( append @ A @ C12 @ ( cons @ A @ Qs2 @ C21 ) ) )
=> ( C23
!= ( append @ A @ C21 @ ( cons @ A @ Qs @ C22 ) ) ) )
=> ( ( ( C12 = C1 )
=> ( ( Qs2 = Qs )
=> ( C23 != C22 ) ) )
=> ~ ! [C212: list @ A] :
( ( C12
= ( append @ A @ C1 @ ( cons @ A @ Qs @ C212 ) ) )
=> ( C22
!= ( append @ A @ C212 @ ( cons @ A @ Qs2 @ C23 ) ) ) ) ) ) ) ).
% list_match_lel_lel
thf(fact_3817_prod_Onon__neutral_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: B > A,I5: set @ B] :
( ( groups1962203154675924110t_prod @ B @ A @ G
@ ( collect @ B
@ ^ [X: B] :
( ( member @ B @ X @ I5 )
& ( ( G @ X )
!= ( one_one @ A ) ) ) ) )
= ( groups1962203154675924110t_prod @ B @ A @ G @ I5 ) ) ) ).
% prod.non_neutral'
thf(fact_3818_neq__Nil__revE,axiom,
! [A: $tType,L: list @ A] :
( ( L
!= ( nil @ A ) )
=> ~ ! [Ll2: list @ A,E4: A] :
( L
!= ( append @ A @ Ll2 @ ( cons @ A @ E4 @ ( nil @ A ) ) ) ) ) ).
% neq_Nil_revE
thf(fact_3819_rev__induct2_H,axiom,
! [A: $tType,B: $tType,P: ( list @ A ) > ( list @ B ) > $o,Xs: list @ A,Ys: list @ B] :
( ( P @ ( nil @ A ) @ ( nil @ B ) )
=> ( ! [X3: A,Xs2: list @ A] : ( P @ ( append @ A @ Xs2 @ ( cons @ A @ X3 @ ( nil @ A ) ) ) @ ( nil @ B ) )
=> ( ! [Y2: B,Ys4: list @ B] : ( P @ ( nil @ A ) @ ( append @ B @ Ys4 @ ( cons @ B @ Y2 @ ( nil @ B ) ) ) )
=> ( ! [X3: A,Xs2: list @ A,Y2: B,Ys4: list @ B] :
( ( P @ Xs2 @ Ys4 )
=> ( P @ ( append @ A @ Xs2 @ ( cons @ A @ X3 @ ( nil @ A ) ) ) @ ( append @ B @ Ys4 @ ( cons @ B @ Y2 @ ( nil @ B ) ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% rev_induct2'
thf(fact_3820_neq__Nil__rev__conv,axiom,
! [A: $tType,L: list @ A] :
( ( L
!= ( nil @ A ) )
= ( ? [Xs3: list @ A,X: A] :
( L
= ( append @ A @ Xs3 @ ( cons @ A @ X @ ( nil @ A ) ) ) ) ) ) ).
% neq_Nil_rev_conv
thf(fact_3821_rev__nonempty__induct2_H,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B,P: ( list @ A ) > ( list @ B ) > $o] :
( ( Xs
!= ( nil @ A ) )
=> ( ( Ys
!= ( nil @ B ) )
=> ( ! [X3: A,Y2: B] : ( P @ ( cons @ A @ X3 @ ( nil @ A ) ) @ ( cons @ B @ Y2 @ ( nil @ B ) ) )
=> ( ! [X3: A,Xs2: list @ A,Y2: B] :
( ( Xs2
!= ( nil @ A ) )
=> ( P @ ( append @ A @ Xs2 @ ( cons @ A @ X3 @ ( nil @ A ) ) ) @ ( cons @ B @ Y2 @ ( nil @ B ) ) ) )
=> ( ! [X3: A,Y2: B,Ys4: list @ B] :
( ( Ys4
!= ( nil @ B ) )
=> ( P @ ( cons @ A @ X3 @ ( nil @ A ) ) @ ( append @ B @ Ys4 @ ( cons @ B @ Y2 @ ( nil @ B ) ) ) ) )
=> ( ! [X3: A,Xs2: list @ A,Y2: 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 @ Y2 @ ( nil @ B ) ) ) ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ) ) ).
% rev_nonempty_induct2'
thf(fact_3822_list__Cons__eq__append__cases,axiom,
! [A: $tType,X2: A,Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ( ( cons @ A @ X2 @ Xs )
= ( append @ A @ Ys @ Zs ) )
=> ( ( ( Ys
= ( nil @ A ) )
=> ( Zs
!= ( cons @ A @ X2 @ Xs ) ) )
=> ~ ! [Ys5: list @ A] :
( ( Ys
= ( cons @ A @ X2 @ Ys5 ) )
=> ( ( append @ A @ Ys5 @ Zs )
!= Xs ) ) ) ) ).
% list_Cons_eq_append_cases
thf(fact_3823_list__append__eq__Cons__cases,axiom,
! [A: $tType,Ys: list @ A,Zs: list @ A,X2: A,Xs: list @ A] :
( ( ( append @ A @ Ys @ Zs )
= ( cons @ A @ X2 @ Xs ) )
=> ( ( ( Ys
= ( nil @ A ) )
=> ( Zs
!= ( cons @ A @ X2 @ Xs ) ) )
=> ~ ! [Ys5: list @ A] :
( ( Ys
= ( cons @ A @ X2 @ Ys5 ) )
=> ( ( append @ A @ Ys5 @ Zs )
!= Xs ) ) ) ) ).
% list_append_eq_Cons_cases
thf(fact_3824_in__set__list__format,axiom,
! [A: $tType,E2: A,L: list @ A] :
( ( member @ A @ E2 @ ( set2 @ A @ L ) )
=> ~ ! [L12: list @ A,L23: list @ A] :
( L
!= ( append @ A @ L12 @ ( cons @ A @ E2 @ L23 ) ) ) ) ).
% in_set_list_format
thf(fact_3825_xy__in__set__cases,axiom,
! [A: $tType,X2: A,L: list @ A,Y: A] :
( ( member @ A @ X2 @ ( set2 @ A @ L ) )
=> ( ( member @ A @ Y @ ( set2 @ A @ L ) )
=> ( ( ( X2 = Y )
=> ! [L12: list @ A,L23: list @ A] :
( L
!= ( append @ A @ L12 @ ( cons @ A @ Y @ L23 ) ) ) )
=> ( ( ( X2 != Y )
=> ! [L12: list @ A,L23: list @ A,L32: list @ A] :
( L
!= ( append @ A @ L12 @ ( cons @ A @ X2 @ ( append @ A @ L23 @ ( cons @ A @ Y @ L32 ) ) ) ) ) )
=> ~ ( ( X2 != Y )
=> ! [L12: list @ A,L23: list @ A,L32: list @ A] :
( L
!= ( append @ A @ L12 @ ( cons @ A @ Y @ ( append @ A @ L23 @ ( cons @ A @ X2 @ L32 ) ) ) ) ) ) ) ) ) ) ).
% xy_in_set_cases
thf(fact_3826_list__rest__coinc,axiom,
! [A: $tType,S22: list @ A,S1: list @ A,R1: list @ A,R22: list @ A] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ S22 ) @ ( size_size @ ( list @ A ) @ S1 ) )
=> ( ( ( append @ A @ S1 @ R1 )
= ( append @ A @ S22 @ R22 ) )
=> ? [R1p: list @ A] :
( R22
= ( append @ A @ R1p @ R1 ) ) ) ) ).
% list_rest_coinc
thf(fact_3827_distinct__match,axiom,
! [A: $tType,Al: list @ A,E2: A,Bl: list @ A,Al2: list @ A,Bl2: list @ A] :
( ( distinct @ A @ ( append @ A @ Al @ ( cons @ A @ E2 @ Bl ) ) )
=> ( ( ( append @ A @ Al @ ( cons @ A @ E2 @ Bl ) )
= ( append @ A @ Al2 @ ( cons @ A @ E2 @ Bl2 ) ) )
= ( ( Al = Al2 )
& ( Bl = Bl2 ) ) ) ) ).
% distinct_match
thf(fact_3828_rotate__append,axiom,
! [A: $tType,L: list @ A,Q2: list @ A] :
( ( rotate @ A @ ( size_size @ ( list @ A ) @ L ) @ ( append @ A @ L @ Q2 ) )
= ( append @ A @ Q2 @ L ) ) ).
% rotate_append
thf(fact_3829_length__n__lists__elem,axiom,
! [A: $tType,Ys: list @ A,N: nat,Xs: list @ A] :
( ( member @ ( list @ A ) @ Ys @ ( set2 @ ( list @ A ) @ ( n_lists @ A @ N @ Xs ) ) )
=> ( ( size_size @ ( list @ A ) @ Ys )
= N ) ) ).
% length_n_lists_elem
thf(fact_3830_same__length__different,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( Xs != Ys )
=> ( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ A ) @ Ys ) )
=> ? [Pre: list @ A,X3: A,Xs4: list @ A,Y2: A,Ys5: list @ A] :
( ( X3 != Y2 )
& ( Xs
= ( append @ A @ Pre @ ( append @ A @ ( cons @ A @ X3 @ ( nil @ A ) ) @ Xs4 ) ) )
& ( Ys
= ( append @ A @ Pre @ ( append @ A @ ( cons @ A @ Y2 @ ( nil @ A ) ) @ Ys5 ) ) ) ) ) ) ).
% same_length_different
thf(fact_3831_prod_Odistrib__triv_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [I5: set @ B,G: B > A,H: B > A] :
( ( finite_finite2 @ B @ I5 )
=> ( ( groups1962203154675924110t_prod @ B @ A
@ ^ [I2: B] : ( times_times @ A @ ( G @ I2 ) @ ( H @ I2 ) )
@ I5 )
= ( times_times @ A @ ( groups1962203154675924110t_prod @ B @ A @ G @ I5 ) @ ( groups1962203154675924110t_prod @ B @ A @ H @ I5 ) ) ) ) ) ).
% prod.distrib_triv'
thf(fact_3832_sorted__append,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,Ys: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( append @ A @ Xs @ Ys ) )
= ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
& ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Ys )
& ! [X: A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ! [Y4: A] :
( ( member @ A @ Y4 @ ( set2 @ A @ Ys ) )
=> ( ord_less_eq @ A @ X @ Y4 ) ) ) ) ) ) ).
% sorted_append
thf(fact_3833_list__update__append1,axiom,
! [A: $tType,I: nat,Xs: list @ A,Ys: list @ A,X2: A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( list_update @ A @ ( append @ A @ Xs @ Ys ) @ I @ X2 )
= ( append @ A @ ( list_update @ A @ Xs @ I @ X2 ) @ Ys ) ) ) ).
% list_update_append1
thf(fact_3834_foldl__rule,axiom,
! [Sigma2: $tType,A: $tType,I5: Sigma2 > ( list @ A ) > ( list @ A ) > $o,Sigma_0: Sigma2,L0: list @ A,F: Sigma2 > A > Sigma2] :
( ( I5 @ Sigma_0 @ ( nil @ A ) @ L0 )
=> ( ! [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 @ ( F @ Sigma @ X3 ) @ ( append @ A @ L12 @ ( cons @ A @ X3 @ ( nil @ A ) ) ) @ L23 ) ) )
=> ( I5 @ ( foldl @ Sigma2 @ A @ F @ Sigma_0 @ L0 ) @ L0 @ ( nil @ A ) ) ) ) ).
% foldl_rule
thf(fact_3835_foldl__rule__P,axiom,
! [Sigma2: $tType,A: $tType,I5: Sigma2 > ( list @ A ) > ( list @ A ) > $o,Sigma_0: Sigma2,L0: list @ A,F: Sigma2 > A > Sigma2,P: Sigma2 > $o] :
( ( I5 @ Sigma_0 @ ( nil @ A ) @ L0 )
=> ( ! [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 @ ( F @ Sigma @ X3 ) @ ( append @ A @ L12 @ ( cons @ A @ X3 @ ( nil @ A ) ) ) @ L23 ) ) )
=> ( ! [Sigma: Sigma2] :
( ( I5 @ Sigma @ L0 @ ( nil @ A ) )
=> ( P @ Sigma ) )
=> ( P @ ( foldl @ Sigma2 @ A @ F @ Sigma_0 @ L0 ) ) ) ) ) ).
% foldl_rule_P
thf(fact_3836_foldl__rule__aux,axiom,
! [Sigma2: $tType,A: $tType,I5: Sigma2 > ( list @ A ) > $o,Sigma_0: Sigma2,L0: list @ A,F: Sigma2 > A > Sigma2] :
( ( I5 @ Sigma_0 @ L0 )
=> ( ! [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 @ ( F @ Sigma @ X3 ) @ L23 ) ) )
=> ( I5 @ ( foldl @ Sigma2 @ A @ F @ Sigma_0 @ L0 ) @ ( nil @ A ) ) ) ) ).
% foldl_rule_aux
thf(fact_3837_foldl__rule__aux__P,axiom,
! [Sigma2: $tType,A: $tType,I5: Sigma2 > ( list @ A ) > $o,Sigma_0: Sigma2,L0: list @ A,F: Sigma2 > A > Sigma2,P: Sigma2 > $o] :
( ( I5 @ Sigma_0 @ L0 )
=> ( ! [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 @ ( F @ Sigma @ X3 ) @ L23 ) ) )
=> ( ! [Sigma: Sigma2] :
( ( I5 @ Sigma @ ( nil @ A ) )
=> ( P @ Sigma ) )
=> ( P @ ( foldl @ Sigma2 @ A @ F @ Sigma_0 @ L0 ) ) ) ) ) ).
% foldl_rule_aux_P
thf(fact_3838_prod_Omono__neutral__cong__right_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S2: set @ B,T6: set @ B,G: B > A,H: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S2 @ T6 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T6 @ S2 ) )
=> ( ( G @ X3 )
= ( one_one @ A ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ S2 )
=> ( ( G @ X3 )
= ( H @ X3 ) ) )
=> ( ( groups1962203154675924110t_prod @ B @ A @ G @ T6 )
= ( groups1962203154675924110t_prod @ B @ A @ H @ S2 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_right'
thf(fact_3839_prod_Omono__neutral__cong__left_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S2: set @ B,T6: set @ B,H: B > A,G: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S2 @ T6 )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ ( minus_minus @ ( set @ B ) @ T6 @ S2 ) )
=> ( ( H @ I3 )
= ( one_one @ A ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ S2 )
=> ( ( G @ X3 )
= ( H @ X3 ) ) )
=> ( ( groups1962203154675924110t_prod @ B @ A @ G @ S2 )
= ( groups1962203154675924110t_prod @ B @ A @ H @ T6 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_left'
thf(fact_3840_prod_Omono__neutral__right_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S2: set @ B,T6: set @ B,G: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S2 @ T6 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T6 @ S2 ) )
=> ( ( G @ X3 )
= ( one_one @ A ) ) )
=> ( ( groups1962203154675924110t_prod @ B @ A @ G @ T6 )
= ( groups1962203154675924110t_prod @ B @ A @ G @ S2 ) ) ) ) ) ).
% prod.mono_neutral_right'
thf(fact_3841_prod_Omono__neutral__left_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S2: set @ B,T6: set @ B,G: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S2 @ T6 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T6 @ S2 ) )
=> ( ( G @ X3 )
= ( one_one @ A ) ) )
=> ( ( groups1962203154675924110t_prod @ B @ A @ G @ S2 )
= ( groups1962203154675924110t_prod @ B @ A @ G @ T6 ) ) ) ) ) ).
% prod.mono_neutral_left'
thf(fact_3842_length__Suc__conv__rev,axiom,
! [A: $tType,Xs: list @ A,N: nat] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( suc @ N ) )
= ( ? [Y4: A,Ys3: list @ A] :
( ( Xs
= ( append @ A @ Ys3 @ ( cons @ A @ Y4 @ ( nil @ A ) ) ) )
& ( ( size_size @ ( list @ A ) @ Ys3 )
= N ) ) ) ) ).
% length_Suc_conv_rev
thf(fact_3843_length__Suc__rev__conv,axiom,
! [A: $tType,Xs: list @ A,N: nat] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( suc @ N ) )
= ( ? [Ys3: list @ A,Y4: A] :
( ( Xs
= ( append @ A @ Ys3 @ ( cons @ A @ Y4 @ ( nil @ A ) ) ) )
& ( ( size_size @ ( list @ A ) @ Ys3 )
= N ) ) ) ) ).
% length_Suc_rev_conv
thf(fact_3844_length__append__singleton,axiom,
! [A: $tType,Xs: list @ A,X2: A] :
( ( size_size @ ( list @ A ) @ ( append @ A @ Xs @ ( cons @ A @ X2 @ ( nil @ A ) ) ) )
= ( suc @ ( size_size @ ( list @ A ) @ Xs ) ) ) ).
% length_append_singleton
thf(fact_3845_length__compl__rev__induct,axiom,
! [A: $tType,P: ( list @ A ) > $o,L: list @ A] :
( ( P @ ( nil @ A ) )
=> ( ! [L3: list @ A,E4: A] :
( ! [Ll: list @ A] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Ll ) @ ( size_size @ ( list @ A ) @ L3 ) )
=> ( P @ Ll ) )
=> ( P @ ( append @ A @ L3 @ ( cons @ A @ E4 @ ( nil @ A ) ) ) ) )
=> ( P @ L ) ) ) ).
% length_compl_rev_induct
thf(fact_3846_not__distinct__split__distinct,axiom,
! [A: $tType,Xs: list @ A] :
( ~ ( distinct @ A @ Xs )
=> ~ ! [Y2: A,Ys4: list @ A] :
( ( distinct @ A @ Ys4 )
=> ( ( member @ A @ Y2 @ ( set2 @ A @ Ys4 ) )
=> ! [Zs2: list @ A] :
( Xs
!= ( append @ A @ Ys4 @ ( append @ A @ ( cons @ A @ Y2 @ ( nil @ A ) ) @ Zs2 ) ) ) ) ) ) ).
% not_distinct_split_distinct
thf(fact_3847_nth__append,axiom,
! [A: $tType,N: nat,Xs: list @ A,Ys: list @ A] :
( ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ ( append @ A @ Xs @ Ys ) @ N )
= ( nth @ A @ Xs @ N ) ) )
& ( ~ ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ ( append @ A @ Xs @ Ys ) @ N )
= ( nth @ A @ Ys @ ( minus_minus @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) ) ).
% nth_append
thf(fact_3848_list__update__append,axiom,
! [A: $tType,N: nat,Xs: list @ A,Ys: list @ A,X2: A] :
( ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( list_update @ A @ ( append @ A @ Xs @ Ys ) @ N @ X2 )
= ( append @ A @ ( list_update @ A @ Xs @ N @ X2 ) @ Ys ) ) )
& ( ~ ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( list_update @ A @ ( append @ A @ Xs @ Ys ) @ N @ X2 )
= ( append @ A @ Xs @ ( list_update @ A @ Ys @ ( minus_minus @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) ) @ X2 ) ) ) ) ) ).
% list_update_append
thf(fact_3849_n__lists_Osimps_I1_J,axiom,
! [A: $tType,Xs: list @ A] :
( ( n_lists @ A @ ( zero_zero @ nat ) @ Xs )
= ( cons @ ( list @ A ) @ ( nil @ A ) @ ( nil @ ( list @ A ) ) ) ) ).
% n_lists.simps(1)
thf(fact_3850_prod_Odistrib_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [I5: set @ B,G: B > A,H: B > A] :
( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X: B] :
( ( member @ B @ X @ I5 )
& ( ( G @ X )
!= ( one_one @ A ) ) ) ) )
=> ( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X: B] :
( ( member @ B @ X @ I5 )
& ( ( H @ X )
!= ( one_one @ A ) ) ) ) )
=> ( ( groups1962203154675924110t_prod @ B @ A
@ ^ [I2: B] : ( times_times @ A @ ( G @ I2 ) @ ( H @ I2 ) )
@ I5 )
= ( times_times @ A @ ( groups1962203154675924110t_prod @ B @ A @ G @ I5 ) @ ( groups1962203154675924110t_prod @ B @ A @ H @ I5 ) ) ) ) ) ) ).
% prod.distrib'
thf(fact_3851_upto__split2,axiom,
! [I: int,J: int,K: int] :
( ( ord_less_eq @ int @ I @ J )
=> ( ( ord_less_eq @ int @ J @ K )
=> ( ( upto @ I @ K )
= ( append @ int @ ( upto @ I @ J ) @ ( upto @ ( plus_plus @ int @ J @ ( one_one @ int ) ) @ K ) ) ) ) ) ).
% upto_split2
thf(fact_3852_upto__split1,axiom,
! [I: int,J: int,K: int] :
( ( ord_less_eq @ int @ I @ J )
=> ( ( ord_less_eq @ int @ J @ K )
=> ( ( upto @ I @ K )
= ( append @ int @ ( upto @ I @ ( minus_minus @ int @ J @ ( one_one @ int ) ) ) @ ( upto @ J @ K ) ) ) ) ) ).
% upto_split1
thf(fact_3853_prod_OG__def,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ( ( groups1962203154675924110t_prod @ B @ A )
= ( ^ [P7: B > A,I7: set @ B] :
( if @ A
@ ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X: B] :
( ( member @ B @ X @ I7 )
& ( ( P7 @ X )
!= ( one_one @ A ) ) ) ) )
@ ( groups7121269368397514597t_prod @ B @ A @ P7
@ ( collect @ B
@ ^ [X: B] :
( ( member @ B @ X @ I7 )
& ( ( P7 @ X )
!= ( one_one @ A ) ) ) ) )
@ ( one_one @ A ) ) ) ) ) ).
% prod.G_def
thf(fact_3854_slice__prepend,axiom,
! [A: $tType,I: nat,K: nat,Xs: list @ A,Ys: list @ A] :
( ( ord_less_eq @ nat @ I @ K )
=> ( ( ord_less_eq @ nat @ K @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( slice @ A @ I @ K @ Xs )
= ( slice @ A @ ( plus_plus @ nat @ I @ ( size_size @ ( list @ A ) @ Ys ) ) @ ( plus_plus @ nat @ K @ ( size_size @ ( list @ A ) @ Ys ) ) @ ( append @ A @ Ys @ Xs ) ) ) ) ) ).
% slice_prepend
thf(fact_3855_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_3856_horner__sum__append,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_1 @ A )
=> ! [F: B > A,A2: A,Xs: list @ B,Ys: list @ B] :
( ( groups4207007520872428315er_sum @ B @ A @ F @ A2 @ ( append @ B @ Xs @ Ys ) )
= ( plus_plus @ A @ ( groups4207007520872428315er_sum @ B @ A @ F @ A2 @ Xs ) @ ( times_times @ A @ ( power_power @ A @ A2 @ ( size_size @ ( list @ B ) @ Xs ) ) @ ( groups4207007520872428315er_sum @ B @ A @ F @ A2 @ Ys ) ) ) ) ) ).
% horner_sum_append
thf(fact_3857_sorted__insort__is__snoc,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,A2: A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( ord_less_eq @ A @ X3 @ A2 ) )
=> ( ( linorder_insort_key @ A @ A
@ ^ [X: A] : X
@ A2
@ Xs )
= ( append @ A @ Xs @ ( cons @ A @ A2 @ ( nil @ A ) ) ) ) ) ) ) ).
% sorted_insort_is_snoc
thf(fact_3858_upto__split3,axiom,
! [I: int,J: int,K: int] :
( ( ord_less_eq @ int @ I @ J )
=> ( ( ord_less_eq @ int @ J @ K )
=> ( ( upto @ I @ K )
= ( append @ int @ ( upto @ I @ ( minus_minus @ int @ J @ ( one_one @ int ) ) ) @ ( cons @ int @ J @ ( upto @ ( plus_plus @ int @ J @ ( one_one @ int ) ) @ K ) ) ) ) ) ) ).
% upto_split3
thf(fact_3859_Succ__def,axiom,
! [A: $tType] :
( ( bNF_Greatest_Succ @ A )
= ( ^ [Kl2: set @ ( list @ A ),Kl: list @ A] :
( collect @ A
@ ^ [K2: A] : ( member @ ( list @ A ) @ ( append @ A @ Kl @ ( cons @ A @ K2 @ ( nil @ A ) ) ) @ Kl2 ) ) ) ) ).
% Succ_def
thf(fact_3860_SuccI,axiom,
! [A: $tType,Kl3: list @ A,K: A,Kl4: set @ ( list @ A )] :
( ( member @ ( list @ A ) @ ( append @ A @ Kl3 @ ( cons @ A @ K @ ( nil @ A ) ) ) @ Kl4 )
=> ( member @ A @ K @ ( bNF_Greatest_Succ @ A @ Kl4 @ Kl3 ) ) ) ).
% SuccI
thf(fact_3861_SuccD,axiom,
! [A: $tType,K: A,Kl4: set @ ( list @ A ),Kl3: list @ A] :
( ( member @ A @ K @ ( bNF_Greatest_Succ @ A @ Kl4 @ Kl3 ) )
=> ( member @ ( list @ A ) @ ( append @ A @ Kl3 @ ( cons @ A @ K @ ( nil @ A ) ) ) @ Kl4 ) ) ).
% SuccD
thf(fact_3862_Gcd__remove0__nat,axiom,
! [M3: set @ nat] :
( ( finite_finite2 @ nat @ M3 )
=> ( ( gcd_Gcd @ nat @ M3 )
= ( gcd_Gcd @ nat @ ( minus_minus @ ( set @ nat ) @ M3 @ ( insert @ nat @ ( zero_zero @ nat ) @ ( bot_bot @ ( set @ nat ) ) ) ) ) ) ) ).
% Gcd_remove0_nat
thf(fact_3863_Gcd__empty,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ( ( gcd_Gcd @ A @ ( bot_bot @ ( set @ A ) ) )
= ( zero_zero @ A ) ) ) ).
% Gcd_empty
thf(fact_3864_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_3865_Gcd__nat__eq__one,axiom,
! [N5: set @ nat] :
( ( member @ nat @ ( one_one @ nat ) @ N5 )
=> ( ( gcd_Gcd @ nat @ N5 )
= ( one_one @ nat ) ) ) ).
% Gcd_nat_eq_one
thf(fact_3866_Gcd__dvd__nat,axiom,
! [A2: nat,A5: set @ nat] :
( ( member @ nat @ A2 @ A5 )
=> ( dvd_dvd @ nat @ ( gcd_Gcd @ nat @ A5 ) @ A2 ) ) ).
% Gcd_dvd_nat
thf(fact_3867_Gcd__greatest__nat,axiom,
! [A5: set @ nat,A2: nat] :
( ! [B4: nat] :
( ( member @ nat @ B4 @ A5 )
=> ( dvd_dvd @ nat @ A2 @ B4 ) )
=> ( dvd_dvd @ nat @ A2 @ ( gcd_Gcd @ nat @ A5 ) ) ) ).
% Gcd_greatest_nat
thf(fact_3868_Gcd__1,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A5: set @ A] :
( ( member @ A @ ( one_one @ A ) @ A5 )
=> ( ( gcd_Gcd @ A @ A5 )
= ( one_one @ A ) ) ) ) ).
% Gcd_1
thf(fact_3869_Gcd__dvd,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A2: A,A5: set @ A] :
( ( member @ A @ A2 @ A5 )
=> ( dvd_dvd @ A @ ( gcd_Gcd @ A @ A5 ) @ A2 ) ) ) ).
% Gcd_dvd
thf(fact_3870_dvd__GcdD,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [X2: A,A5: set @ A,Y: A] :
( ( dvd_dvd @ A @ X2 @ ( gcd_Gcd @ A @ A5 ) )
=> ( ( member @ A @ Y @ A5 )
=> ( dvd_dvd @ A @ X2 @ Y ) ) ) ) ).
% dvd_GcdD
thf(fact_3871_dvd__Gcd__iff,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [X2: A,A5: set @ A] :
( ( dvd_dvd @ A @ X2 @ ( gcd_Gcd @ A @ A5 ) )
= ( ! [X: A] :
( ( member @ A @ X @ A5 )
=> ( dvd_dvd @ A @ X2 @ X ) ) ) ) ) ).
% dvd_Gcd_iff
thf(fact_3872_Gcd__greatest,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A5: set @ A,A2: A] :
( ! [B4: A] :
( ( member @ A @ B4 @ A5 )
=> ( dvd_dvd @ A @ A2 @ B4 ) )
=> ( dvd_dvd @ A @ A2 @ ( gcd_Gcd @ A @ A5 ) ) ) ) ).
% Gcd_greatest
thf(fact_3873_Gcd__eq__1__I,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A2: A,A5: set @ A] :
( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
=> ( ( member @ A @ A2 @ A5 )
=> ( ( gcd_Gcd @ A @ A5 )
= ( one_one @ A ) ) ) ) ) ).
% Gcd_eq_1_I
thf(fact_3874_semiring__char__def,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( semiri4206861660011772517g_char @ A )
= ( ^ [Uu2: itself @ A] :
( gcd_Gcd @ nat
@ ( collect @ nat
@ ^ [N4: nat] :
( ( semiring_1_of_nat @ A @ N4 )
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% semiring_char_def
thf(fact_3875_empty__Shift,axiom,
! [A: $tType,Kl4: set @ ( list @ A ),K: A] :
( ( member @ ( list @ A ) @ ( nil @ A ) @ Kl4 )
=> ( ( member @ A @ K @ ( bNF_Greatest_Succ @ A @ Kl4 @ ( nil @ A ) ) )
=> ( member @ ( list @ A ) @ ( nil @ A ) @ ( bNF_Greatest_Shift @ A @ Kl4 @ K ) ) ) ) ).
% empty_Shift
thf(fact_3876_Succ__Shift,axiom,
! [A: $tType,Kl4: set @ ( list @ A ),K: A,Kl3: list @ A] :
( ( bNF_Greatest_Succ @ A @ ( bNF_Greatest_Shift @ A @ Kl4 @ K ) @ Kl3 )
= ( bNF_Greatest_Succ @ A @ Kl4 @ ( cons @ A @ K @ Kl3 ) ) ) ).
% Succ_Shift
thf(fact_3877_one__diff__power__eq_H,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X2: A,N: nat] :
( ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X2 @ N ) )
= ( times_times @ A @ ( minus_minus @ A @ ( one_one @ A ) @ X2 )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( power_power @ A @ X2 @ ( minus_minus @ nat @ N @ ( suc @ I2 ) ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% one_diff_power_eq'
thf(fact_3878_abs__Gcd__eq,axiom,
! [K3: set @ int] :
( ( abs_abs @ int @ ( gcd_Gcd @ int @ K3 ) )
= ( gcd_Gcd @ int @ K3 ) ) ).
% abs_Gcd_eq
thf(fact_3879_lessThan__eq__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y: A] :
( ( ( set_ord_lessThan @ A @ X2 )
= ( set_ord_lessThan @ A @ Y ) )
= ( X2 = Y ) ) ) ).
% lessThan_eq_iff
thf(fact_3880_lessThan__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [I: A,K: A] :
( ( member @ A @ I @ ( set_ord_lessThan @ A @ K ) )
= ( ord_less @ A @ I @ K ) ) ) ).
% lessThan_iff
thf(fact_3881_finite__lessThan,axiom,
! [K: nat] : ( finite_finite2 @ nat @ ( set_ord_lessThan @ nat @ K ) ) ).
% finite_lessThan
thf(fact_3882_card__lessThan,axiom,
! [U: nat] :
( ( finite_card @ nat @ ( set_ord_lessThan @ nat @ U ) )
= U ) ).
% card_lessThan
thf(fact_3883_lessThan__subset__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_ord_lessThan @ A @ X2 ) @ ( set_ord_lessThan @ A @ Y ) )
= ( ord_less_eq @ A @ X2 @ Y ) ) ) ).
% lessThan_subset_iff
thf(fact_3884_lessThan__minus__lessThan,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [N: A,M: A] :
( ( minus_minus @ ( set @ A ) @ ( set_ord_lessThan @ A @ N ) @ ( set_ord_lessThan @ A @ M ) )
= ( set_or7035219750837199246ssThan @ A @ M @ N ) ) ) ).
% lessThan_minus_lessThan
thf(fact_3885_lessThan__0,axiom,
( ( set_ord_lessThan @ nat @ ( zero_zero @ nat ) )
= ( bot_bot @ ( set @ nat ) ) ) ).
% lessThan_0
thf(fact_3886_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_3887_sum_OlessThan__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_lessThan @ nat @ ( suc @ N ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_lessThan @ nat @ N ) ) @ ( G @ N ) ) ) ) ).
% sum.lessThan_Suc
thf(fact_3888_prod_OlessThan__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_lessThan @ nat @ ( suc @ N ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_lessThan @ nat @ N ) ) @ ( G @ N ) ) ) ) ).
% prod.lessThan_Suc
thf(fact_3889_infinite__Iio,axiom,
! [A: $tType] :
( ( ( linorder @ A )
& ( no_bot @ A ) )
=> ! [A2: A] :
~ ( finite_finite2 @ A @ ( set_ord_lessThan @ A @ A2 ) ) ) ).
% infinite_Iio
thf(fact_3890_lessThan__non__empty,axiom,
! [A: $tType] :
( ( no_bot @ A )
=> ! [X2: A] :
( ( set_ord_lessThan @ A @ X2 )
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% lessThan_non_empty
thf(fact_3891_Gcd__dvd__int,axiom,
! [A2: int,A5: set @ int] :
( ( member @ int @ A2 @ A5 )
=> ( dvd_dvd @ int @ ( gcd_Gcd @ int @ A5 ) @ A2 ) ) ).
% Gcd_dvd_int
thf(fact_3892_Gcd__greatest__int,axiom,
! [A5: set @ int,A2: int] :
( ! [B4: int] :
( ( member @ int @ B4 @ A5 )
=> ( dvd_dvd @ int @ A2 @ B4 ) )
=> ( dvd_dvd @ int @ A2 @ ( gcd_Gcd @ int @ A5 ) ) ) ).
% Gcd_greatest_int
thf(fact_3893_lessThan__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( set_ord_lessThan @ A )
= ( ^ [U2: A] :
( collect @ A
@ ^ [X: A] : ( ord_less @ A @ X @ U2 ) ) ) ) ) ).
% lessThan_def
thf(fact_3894_lessThan__atLeast0,axiom,
( ( set_ord_lessThan @ nat )
= ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) ) ) ).
% lessThan_atLeast0
thf(fact_3895_lessThan__Suc__atMost,axiom,
! [K: nat] :
( ( set_ord_lessThan @ nat @ ( suc @ K ) )
= ( set_ord_atMost @ nat @ K ) ) ).
% lessThan_Suc_atMost
thf(fact_3896_lessThan__empty__iff,axiom,
! [N: nat] :
( ( ( set_ord_lessThan @ nat @ N )
= ( bot_bot @ ( set @ nat ) ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% lessThan_empty_iff
thf(fact_3897_lessThan__Suc,axiom,
! [K: nat] :
( ( set_ord_lessThan @ nat @ ( suc @ K ) )
= ( insert @ nat @ K @ ( set_ord_lessThan @ nat @ K ) ) ) ).
% lessThan_Suc
thf(fact_3898_Iio__eq__empty__iff,axiom,
! [A: $tType] :
( ( ( linorder @ A )
& ( order_bot @ A ) )
=> ! [N: A] :
( ( ( set_ord_lessThan @ A @ N )
= ( bot_bot @ ( set @ A ) ) )
= ( N
= ( bot_bot @ A ) ) ) ) ).
% Iio_eq_empty_iff
thf(fact_3899_lessThan__strict__subset__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [M: A,N: A] :
( ( ord_less @ ( set @ A ) @ ( set_ord_lessThan @ A @ M ) @ ( set_ord_lessThan @ A @ N ) )
= ( ord_less @ A @ M @ N ) ) ) ).
% lessThan_strict_subset_iff
thf(fact_3900_Gcd__int__greater__eq__0,axiom,
! [K3: set @ int] : ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( gcd_Gcd @ int @ K3 ) ) ).
% Gcd_int_greater_eq_0
thf(fact_3901_lessThan__nat__numeral,axiom,
! [K: num] :
( ( set_ord_lessThan @ nat @ ( numeral_numeral @ nat @ K ) )
= ( insert @ nat @ ( pred_numeral @ K ) @ ( set_ord_lessThan @ nat @ ( pred_numeral @ K ) ) ) ) ).
% lessThan_nat_numeral
thf(fact_3902_sum_Onat__diff__reindex,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( G @ ( minus_minus @ nat @ N @ ( suc @ I2 ) ) )
@ ( set_ord_lessThan @ nat @ N ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% sum.nat_diff_reindex
thf(fact_3903_ShiftD,axiom,
! [A: $tType,Kl3: list @ A,Kl4: set @ ( list @ A ),K: A] :
( ( member @ ( list @ A ) @ Kl3 @ ( bNF_Greatest_Shift @ A @ Kl4 @ K ) )
=> ( member @ ( list @ A ) @ ( cons @ A @ K @ Kl3 ) @ Kl4 ) ) ).
% ShiftD
thf(fact_3904_prod_Onat__diff__reindex,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I2: nat] : ( G @ ( minus_minus @ nat @ N @ ( suc @ I2 ) ) )
@ ( set_ord_lessThan @ nat @ N ) )
= ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% prod.nat_diff_reindex
thf(fact_3905_Iic__subset__Iio__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_ord_atMost @ A @ A2 ) @ ( set_ord_lessThan @ A @ B2 ) )
= ( ord_less @ A @ A2 @ B2 ) ) ) ).
% Iic_subset_Iio_iff
thf(fact_3906_sum__diff__distrib,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [Q: A > nat,P: A > nat,N: A] :
( ! [X3: A] : ( ord_less_eq @ nat @ ( Q @ X3 ) @ ( P @ X3 ) )
=> ( ( minus_minus @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ P @ ( set_ord_lessThan @ A @ N ) ) @ ( groups7311177749621191930dd_sum @ A @ nat @ Q @ ( set_ord_lessThan @ A @ N ) ) )
= ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [X: A] : ( minus_minus @ nat @ ( P @ X ) @ ( Q @ X ) )
@ ( set_ord_lessThan @ A @ N ) ) ) ) ) ).
% sum_diff_distrib
thf(fact_3907_sum_OlessThan__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_lessThan @ nat @ ( suc @ N ) ) )
= ( plus_plus @ A @ ( G @ ( zero_zero @ nat ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% sum.lessThan_Suc_shift
thf(fact_3908_sum__lessThan__telescope,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [F: nat > A,M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N4: nat] : ( minus_minus @ A @ ( F @ ( suc @ N4 ) ) @ ( F @ N4 ) )
@ ( set_ord_lessThan @ nat @ M ) )
= ( minus_minus @ A @ ( F @ M ) @ ( F @ ( zero_zero @ nat ) ) ) ) ) ).
% sum_lessThan_telescope
thf(fact_3909_sum__lessThan__telescope_H,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [F: nat > A,M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N4: nat] : ( minus_minus @ A @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
@ ( set_ord_lessThan @ nat @ M ) )
= ( minus_minus @ A @ ( F @ ( zero_zero @ nat ) ) @ ( F @ M ) ) ) ) ).
% sum_lessThan_telescope'
thf(fact_3910_prod_OlessThan__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_lessThan @ nat @ ( suc @ N ) ) )
= ( times_times @ A @ ( G @ ( zero_zero @ nat ) )
@ ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% prod.lessThan_Suc_shift
thf(fact_3911_sum_OatLeast1__atMost__eq,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K2: nat] : ( G @ ( suc @ K2 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% sum.atLeast1_atMost_eq
thf(fact_3912_prod_OatLeast1__atMost__eq,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [K2: nat] : ( G @ ( suc @ K2 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% prod.atLeast1_atMost_eq
thf(fact_3913_sum__bounds__lt__plus1,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [F: nat > A,Mm: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K2: nat] : ( F @ ( suc @ K2 ) )
@ ( set_ord_lessThan @ nat @ Mm ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ F @ ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ Mm ) ) ) ) ).
% sum_bounds_lt_plus1
thf(fact_3914_sum_Onat__group,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,K: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [M2: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( times_times @ nat @ M2 @ K ) @ ( plus_plus @ nat @ ( times_times @ nat @ M2 @ K ) @ K ) ) )
@ ( set_ord_lessThan @ nat @ N ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_lessThan @ nat @ ( times_times @ nat @ N @ K ) ) ) ) ) ).
% sum.nat_group
thf(fact_3915_Shift__def,axiom,
! [A: $tType] :
( ( bNF_Greatest_Shift @ A )
= ( ^ [Kl2: set @ ( list @ A ),K2: A] :
( collect @ ( list @ A )
@ ^ [Kl: list @ A] : ( member @ ( list @ A ) @ ( cons @ A @ K2 @ Kl ) @ Kl2 ) ) ) ) ).
% Shift_def
thf(fact_3916_prod_Onat__group,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,K: nat,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A
@ ^ [M2: nat] : ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( times_times @ nat @ M2 @ K ) @ ( plus_plus @ nat @ ( times_times @ nat @ M2 @ K ) @ K ) ) )
@ ( set_ord_lessThan @ nat @ N ) )
= ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_lessThan @ nat @ ( times_times @ nat @ N @ K ) ) ) ) ) ).
% prod.nat_group
thf(fact_3917_sum_Onested__swap_H,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A2: nat > nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ ( A2 @ I2 ) @ ( set_ord_lessThan @ nat @ I2 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J3: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( A2 @ I2 @ J3 )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J3 ) @ N ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% sum.nested_swap'
thf(fact_3918_prod_Onested__swap_H,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: nat > nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I2: nat] : ( groups7121269368397514597t_prod @ nat @ A @ ( A2 @ I2 ) @ ( set_ord_lessThan @ nat @ I2 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [J3: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I2: nat] : ( A2 @ I2 @ J3 )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J3 ) @ N ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% prod.nested_swap'
thf(fact_3919_one__diff__power__eq,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X2: A,N: nat] :
( ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X2 @ N ) )
= ( times_times @ A @ ( minus_minus @ A @ ( one_one @ A ) @ X2 ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X2 ) @ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% one_diff_power_eq
thf(fact_3920_power__diff__1__eq,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X2: A,N: nat] :
( ( minus_minus @ A @ ( power_power @ A @ X2 @ N ) @ ( one_one @ A ) )
= ( times_times @ A @ ( minus_minus @ A @ X2 @ ( one_one @ A ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X2 ) @ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% power_diff_1_eq
thf(fact_3921_geometric__sum,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [X2: A,N: nat] :
( ( X2
!= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X2 ) @ ( set_ord_lessThan @ nat @ N ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ X2 @ N ) @ ( one_one @ A ) ) @ ( minus_minus @ A @ X2 @ ( one_one @ A ) ) ) ) ) ) ).
% geometric_sum
thf(fact_3922_sum_OatMost__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_atMost @ nat @ N ) )
= ( plus_plus @ A @ ( G @ ( zero_zero @ nat ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% sum.atMost_shift
thf(fact_3923_prod_OatMost__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_atMost @ nat @ N ) )
= ( times_times @ A @ ( G @ ( zero_zero @ nat ) )
@ ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% prod.atMost_shift
thf(fact_3924_power__diff__sumr2,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X2: A,N: nat,Y: A] :
( ( minus_minus @ A @ ( power_power @ A @ X2 @ N ) @ ( power_power @ A @ Y @ N ) )
= ( times_times @ A @ ( minus_minus @ A @ X2 @ Y )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( times_times @ A @ ( power_power @ A @ Y @ ( minus_minus @ nat @ N @ ( suc @ I2 ) ) ) @ ( power_power @ A @ X2 @ I2 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% power_diff_sumr2
thf(fact_3925_diff__power__eq__sum,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X2: A,N: nat,Y: A] :
( ( minus_minus @ A @ ( power_power @ A @ X2 @ ( suc @ N ) ) @ ( power_power @ A @ Y @ ( suc @ N ) ) )
= ( times_times @ A @ ( minus_minus @ A @ X2 @ Y )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [P7: nat] : ( times_times @ A @ ( power_power @ A @ X2 @ P7 ) @ ( power_power @ A @ Y @ ( minus_minus @ nat @ N @ P7 ) ) )
@ ( set_ord_lessThan @ nat @ ( suc @ N ) ) ) ) ) ) ).
% diff_power_eq_sum
thf(fact_3926_sum__gp__strict,axiom,
! [A: $tType] :
( ( ( division_ring @ A )
& ( comm_ring @ A ) )
=> ! [X2: A,N: nat] :
( ( ( X2
= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X2 ) @ ( set_ord_lessThan @ nat @ N ) )
= ( semiring_1_of_nat @ A @ N ) ) )
& ( ( X2
!= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X2 ) @ ( set_ord_lessThan @ nat @ N ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X2 @ N ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ X2 ) ) ) ) ) ) ).
% sum_gp_strict
thf(fact_3927_atLeast1__lessThan__eq__remove0,axiom,
! [N: nat] :
( ( set_or7035219750837199246ssThan @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N )
= ( minus_minus @ ( set @ nat ) @ ( set_ord_lessThan @ nat @ N ) @ ( insert @ nat @ ( zero_zero @ nat ) @ ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% atLeast1_lessThan_eq_remove0
thf(fact_3928_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_3929_take__bit__num__simps_I7_J,axiom,
! [R: num,M: num] :
( ( bit_take_bit_num @ ( numeral_numeral @ nat @ R ) @ ( bit1 @ M ) )
= ( some @ num @ ( case_option @ num @ num @ one2 @ bit1 @ ( bit_take_bit_num @ ( pred_numeral @ R ) @ M ) ) ) ) ).
% take_bit_num_simps(7)
thf(fact_3930_sorted__list__of__set__def,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( linord4507533701916653071of_set @ A )
= ( linord144544945434240204of_set @ A @ A
@ ^ [X: A] : X ) ) ) ).
% sorted_list_of_set_def
thf(fact_3931_in__set__product__lists__length,axiom,
! [A: $tType,Xs: list @ A,Xss: list @ ( list @ A )] :
( ( member @ ( list @ A ) @ Xs @ ( set2 @ ( list @ A ) @ ( product_lists @ A @ Xss ) ) )
=> ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ ( list @ A ) ) @ Xss ) ) ) ).
% in_set_product_lists_length
thf(fact_3932_image__ident,axiom,
! [A: $tType,Y7: set @ A] :
( ( image2 @ A @ A
@ ^ [X: A] : X
@ Y7 )
= Y7 ) ).
% image_ident
thf(fact_3933_image__add__0,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [S2: set @ A] :
( ( image2 @ A @ A @ ( plus_plus @ A @ ( zero_zero @ A ) ) @ S2 )
= S2 ) ) ).
% image_add_0
thf(fact_3934_image__add__atLeastAtMost,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [K: A,I: A,J: A] :
( ( image2 @ A @ A @ ( plus_plus @ A @ K ) @ ( set_or1337092689740270186AtMost @ A @ I @ J ) )
= ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ K ) ) ) ) ).
% image_add_atLeastAtMost
thf(fact_3935_image__diff__atLeastAtMost,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [D2: A,A2: A,B2: A] :
( ( image2 @ A @ A @ ( minus_minus @ A @ D2 ) @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ D2 @ B2 ) @ ( minus_minus @ A @ D2 @ A2 ) ) ) ) ).
% image_diff_atLeastAtMost
thf(fact_3936_image__add__atLeastLessThan,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [K: A,I: A,J: A] :
( ( image2 @ A @ A @ ( plus_plus @ A @ K ) @ ( set_or7035219750837199246ssThan @ A @ I @ J ) )
= ( set_or7035219750837199246ssThan @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ K ) ) ) ) ).
% image_add_atLeastLessThan
thf(fact_3937_image__uminus__atLeastAtMost,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [X2: A,Y: A] :
( ( image2 @ A @ A @ ( uminus_uminus @ A ) @ ( set_or1337092689740270186AtMost @ A @ X2 @ Y ) )
= ( set_or1337092689740270186AtMost @ A @ ( uminus_uminus @ A @ Y ) @ ( uminus_uminus @ A @ X2 ) ) ) ) ).
% image_uminus_atLeastAtMost
thf(fact_3938_image__add__atMost,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [C2: A,A2: A] :
( ( image2 @ A @ A @ ( plus_plus @ A @ C2 ) @ ( set_ord_atMost @ A @ A2 ) )
= ( set_ord_atMost @ A @ ( plus_plus @ A @ C2 @ A2 ) ) ) ) ).
% image_add_atMost
thf(fact_3939_image__add__greaterThanAtMost,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( image2 @ A @ A @ ( plus_plus @ A @ C2 ) @ ( set_or3652927894154168847AtMost @ A @ A2 @ B2 ) )
= ( set_or3652927894154168847AtMost @ A @ ( plus_plus @ A @ C2 @ A2 ) @ ( plus_plus @ A @ C2 @ B2 ) ) ) ) ).
% image_add_greaterThanAtMost
thf(fact_3940_image__uminus__greaterThanLessThan,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [X2: A,Y: A] :
( ( image2 @ A @ A @ ( uminus_uminus @ A ) @ ( set_or5935395276787703475ssThan @ A @ X2 @ Y ) )
= ( set_or5935395276787703475ssThan @ A @ ( uminus_uminus @ A @ Y ) @ ( uminus_uminus @ A @ X2 ) ) ) ) ).
% image_uminus_greaterThanLessThan
thf(fact_3941_image__add__atLeastAtMost_H,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [K: A,I: A,J: A] :
( ( image2 @ A @ A
@ ^ [N4: A] : ( plus_plus @ A @ N4 @ K )
@ ( set_or1337092689740270186AtMost @ A @ I @ J ) )
= ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ K ) ) ) ) ).
% image_add_atLeastAtMost'
thf(fact_3942_image__minus__const__atLeastAtMost_H,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [D2: A,A2: A,B2: A] :
( ( image2 @ A @ A
@ ^ [T3: A] : ( minus_minus @ A @ T3 @ D2 )
@ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ A2 @ D2 ) @ ( minus_minus @ A @ B2 @ D2 ) ) ) ) ).
% image_minus_const_atLeastAtMost'
thf(fact_3943_image__add__atLeastLessThan_H,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [K: A,I: A,J: A] :
( ( image2 @ A @ A
@ ^ [N4: A] : ( plus_plus @ A @ N4 @ K )
@ ( set_or7035219750837199246ssThan @ A @ I @ J ) )
= ( set_or7035219750837199246ssThan @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ K ) ) ) ) ).
% image_add_atLeastLessThan'
thf(fact_3944_image__minus__const__greaterThanAtMost,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( image2 @ A @ A @ ( minus_minus @ A @ C2 ) @ ( set_or3652927894154168847AtMost @ A @ A2 @ B2 ) )
= ( set_or7035219750837199246ssThan @ A @ ( minus_minus @ A @ C2 @ B2 ) @ ( minus_minus @ A @ C2 @ A2 ) ) ) ) ).
% image_minus_const_greaterThanAtMost
thf(fact_3945_image__diff__atLeastLessThan,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( image2 @ A @ A @ ( minus_minus @ A @ C2 ) @ ( set_or7035219750837199246ssThan @ A @ A2 @ B2 ) )
= ( set_or3652927894154168847AtMost @ A @ ( minus_minus @ A @ C2 @ B2 ) @ ( minus_minus @ A @ C2 @ A2 ) ) ) ) ).
% image_diff_atLeastLessThan
thf(fact_3946_image__uminus__atLeastLessThan,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [X2: A,Y: A] :
( ( image2 @ A @ A @ ( uminus_uminus @ A ) @ ( set_or7035219750837199246ssThan @ A @ X2 @ Y ) )
= ( set_or3652927894154168847AtMost @ A @ ( uminus_uminus @ A @ Y ) @ ( uminus_uminus @ A @ X2 ) ) ) ) ).
% image_uminus_atLeastLessThan
thf(fact_3947_image__uminus__greaterThanAtMost,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [X2: A,Y: A] :
( ( image2 @ A @ A @ ( uminus_uminus @ A ) @ ( set_or3652927894154168847AtMost @ A @ X2 @ Y ) )
= ( set_or7035219750837199246ssThan @ A @ ( uminus_uminus @ A @ Y ) @ ( uminus_uminus @ A @ X2 ) ) ) ) ).
% image_uminus_greaterThanAtMost
thf(fact_3948_image__mult__atLeastAtMost,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [D2: A,A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ D2 )
=> ( ( image2 @ A @ A @ ( times_times @ A @ D2 ) @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ D2 @ A2 ) @ ( times_times @ A @ D2 @ B2 ) ) ) ) ) ).
% image_mult_atLeastAtMost
thf(fact_3949_image__divide__atLeastAtMost,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [D2: A,A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ D2 )
=> ( ( image2 @ A @ A
@ ^ [C5: A] : ( divide_divide @ A @ C5 @ D2 )
@ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( divide_divide @ A @ A2 @ D2 ) @ ( divide_divide @ A @ B2 @ D2 ) ) ) ) ) ).
% image_divide_atLeastAtMost
thf(fact_3950_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_3951_take__bit__num__simps_I4_J,axiom,
! [N: nat,M: num] :
( ( bit_take_bit_num @ ( suc @ N ) @ ( bit1 @ M ) )
= ( some @ num @ ( case_option @ num @ num @ one2 @ bit1 @ ( bit_take_bit_num @ N @ M ) ) ) ) ).
% take_bit_num_simps(4)
thf(fact_3952_imageE,axiom,
! [A: $tType,B: $tType,B2: A,F: B > A,A5: set @ B] :
( ( member @ A @ B2 @ ( image2 @ B @ A @ F @ A5 ) )
=> ~ ! [X3: B] :
( ( B2
= ( F @ X3 ) )
=> ~ ( member @ B @ X3 @ A5 ) ) ) ).
% imageE
thf(fact_3953_image__image,axiom,
! [A: $tType,B: $tType,C: $tType,F: B > A,G: C > B,A5: set @ C] :
( ( image2 @ B @ A @ F @ ( image2 @ C @ B @ G @ A5 ) )
= ( image2 @ C @ A
@ ^ [X: C] : ( F @ ( G @ X ) )
@ A5 ) ) ).
% image_image
thf(fact_3954_Compr__image__eq,axiom,
! [A: $tType,B: $tType,F: B > A,A5: set @ B,P: A > $o] :
( ( collect @ A
@ ^ [X: A] :
( ( member @ A @ X @ ( image2 @ B @ A @ F @ A5 ) )
& ( P @ X ) ) )
= ( image2 @ B @ A @ F
@ ( collect @ B
@ ^ [X: B] :
( ( member @ B @ X @ A5 )
& ( P @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_3955_le__some__optE,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [M: A,X2: option @ A] :
( ( ord_less_eq @ ( option @ A ) @ ( some @ A @ M ) @ X2 )
=> ~ ! [M11: A] :
( ( X2
= ( some @ A @ M11 ) )
=> ~ ( ord_less_eq @ A @ M @ M11 ) ) ) ) ).
% le_some_optE
thf(fact_3956_pigeonhole__infinite,axiom,
! [B: $tType,A: $tType,A5: set @ A,F: A > B] :
( ~ ( finite_finite2 @ A @ A5 )
=> ( ( finite_finite2 @ B @ ( image2 @ A @ B @ F @ A5 ) )
=> ? [X3: A] :
( ( member @ A @ X3 @ A5 )
& ~ ( finite_finite2 @ A
@ ( collect @ A
@ ^ [A3: A] :
( ( member @ A @ A3 @ A5 )
& ( ( F @ A3 )
= ( F @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_3957_image__constant__conv,axiom,
! [B: $tType,A: $tType,A5: set @ B,C2: A] :
( ( ( A5
= ( bot_bot @ ( set @ B ) ) )
=> ( ( image2 @ B @ A
@ ^ [X: B] : C2
@ A5 )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( image2 @ B @ A
@ ^ [X: B] : C2
@ A5 )
= ( insert @ A @ C2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% image_constant_conv
thf(fact_3958_image__constant,axiom,
! [A: $tType,B: $tType,X2: A,A5: set @ A,C2: B] :
( ( member @ A @ X2 @ A5 )
=> ( ( image2 @ A @ B
@ ^ [X: A] : C2
@ A5 )
= ( insert @ B @ C2 @ ( bot_bot @ ( set @ B ) ) ) ) ) ).
% image_constant
thf(fact_3959_finite__conv__nat__seg__image,axiom,
! [A: $tType] :
( ( finite_finite2 @ A )
= ( ^ [A8: set @ A] :
? [N4: nat,F2: nat > A] :
( A8
= ( image2 @ nat @ A @ F2
@ ( collect @ nat
@ ^ [I2: nat] : ( ord_less @ nat @ I2 @ N4 ) ) ) ) ) ) ).
% finite_conv_nat_seg_image
thf(fact_3960_nat__seg__image__imp__finite,axiom,
! [A: $tType,A5: set @ A,F: nat > A,N: nat] :
( ( A5
= ( image2 @ nat @ A @ F
@ ( collect @ nat
@ ^ [I2: nat] : ( ord_less @ nat @ I2 @ N ) ) ) )
=> ( finite_finite2 @ A @ A5 ) ) ).
% nat_seg_image_imp_finite
thf(fact_3961_sum_Oimage__gen,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S2: set @ B,H: B > A,G: B > C] :
( ( finite_finite2 @ B @ S2 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ H @ S2 )
= ( groups7311177749621191930dd_sum @ C @ A
@ ^ [Y4: C] :
( groups7311177749621191930dd_sum @ B @ A @ H
@ ( collect @ B
@ ^ [X: B] :
( ( member @ B @ X @ S2 )
& ( ( G @ X )
= Y4 ) ) ) )
@ ( image2 @ B @ C @ G @ S2 ) ) ) ) ) ).
% sum.image_gen
thf(fact_3962_prod_Oimage__gen,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S2: set @ B,H: B > A,G: B > C] :
( ( finite_finite2 @ B @ S2 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ H @ S2 )
= ( groups7121269368397514597t_prod @ C @ A
@ ^ [Y4: C] :
( groups7121269368397514597t_prod @ B @ A @ H
@ ( collect @ B
@ ^ [X: B] :
( ( member @ B @ X @ S2 )
& ( ( G @ X )
= Y4 ) ) ) )
@ ( image2 @ B @ C @ G @ S2 ) ) ) ) ) ).
% prod.image_gen
thf(fact_3963_Gcd__mono,axiom,
! [A: $tType,B: $tType] :
( ( semiring_Gcd @ A )
=> ! [A5: set @ B,F: B > A,G: B > A] :
( ! [X3: B] :
( ( member @ B @ X3 @ A5 )
=> ( dvd_dvd @ A @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( dvd_dvd @ A @ ( gcd_Gcd @ A @ ( image2 @ B @ A @ F @ A5 ) ) @ ( gcd_Gcd @ A @ ( image2 @ B @ A @ G @ A5 ) ) ) ) ) ).
% Gcd_mono
thf(fact_3964_card__image__le,axiom,
! [B: $tType,A: $tType,A5: set @ A,F: A > B] :
( ( finite_finite2 @ A @ A5 )
=> ( ord_less_eq @ nat @ ( finite_card @ B @ ( image2 @ A @ B @ F @ A5 ) ) @ ( finite_card @ A @ A5 ) ) ) ).
% card_image_le
thf(fact_3965_sum_Ogroup,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S2: set @ B,T6: set @ C,G: B > C,H: B > A] :
( ( finite_finite2 @ B @ S2 )
=> ( ( finite_finite2 @ C @ T6 )
=> ( ( ord_less_eq @ ( set @ C ) @ ( image2 @ B @ C @ G @ S2 ) @ T6 )
=> ( ( groups7311177749621191930dd_sum @ C @ A
@ ^ [Y4: C] :
( groups7311177749621191930dd_sum @ B @ A @ H
@ ( collect @ B
@ ^ [X: B] :
( ( member @ B @ X @ S2 )
& ( ( G @ X )
= Y4 ) ) ) )
@ T6 )
= ( groups7311177749621191930dd_sum @ B @ A @ H @ S2 ) ) ) ) ) ) ).
% sum.group
thf(fact_3966_prod_Ogroup,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S2: set @ B,T6: set @ C,G: B > C,H: B > A] :
( ( finite_finite2 @ B @ S2 )
=> ( ( finite_finite2 @ C @ T6 )
=> ( ( ord_less_eq @ ( set @ C ) @ ( image2 @ B @ C @ G @ S2 ) @ T6 )
=> ( ( groups7121269368397514597t_prod @ C @ A
@ ^ [Y4: C] :
( groups7121269368397514597t_prod @ B @ A @ H
@ ( collect @ B
@ ^ [X: B] :
( ( member @ B @ X @ S2 )
& ( ( G @ X )
= Y4 ) ) ) )
@ T6 )
= ( groups7121269368397514597t_prod @ B @ A @ H @ S2 ) ) ) ) ) ) ).
% prod.group
thf(fact_3967_surj__card__le,axiom,
! [B: $tType,A: $tType,A5: set @ A,B5: set @ B,F: A > B] :
( ( finite_finite2 @ A @ A5 )
=> ( ( ord_less_eq @ ( set @ B ) @ B5 @ ( image2 @ A @ B @ F @ A5 ) )
=> ( ord_less_eq @ nat @ ( finite_card @ B @ B5 ) @ ( finite_card @ A @ A5 ) ) ) ) ).
% surj_card_le
thf(fact_3968_set__image__eq__pointwiseI,axiom,
! [B: $tType,A: $tType,L: list @ A,L4: list @ A,F: A > B] :
( ( ( size_size @ ( list @ A ) @ L )
= ( size_size @ ( list @ A ) @ L4 ) )
=> ( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ L ) )
=> ( ( F @ ( nth @ A @ L @ I3 ) )
= ( F @ ( nth @ A @ L4 @ I3 ) ) ) )
=> ( ( image2 @ A @ B @ F @ ( set2 @ A @ L ) )
= ( image2 @ A @ B @ F @ ( set2 @ A @ L4 ) ) ) ) ) ).
% set_image_eq_pointwiseI
thf(fact_3969_in__set__image__conv__nth,axiom,
! [B: $tType,A: $tType,F: B > A,X2: B,L: list @ B] :
( ( member @ A @ ( F @ X2 ) @ ( image2 @ B @ A @ F @ ( set2 @ B @ L ) ) )
= ( ? [I2: nat] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ B ) @ L ) )
& ( ( F @ ( nth @ B @ L @ I2 ) )
= ( F @ X2 ) ) ) ) ) ).
% in_set_image_conv_nth
thf(fact_3970_and__not__num_Osimps_I8_J,axiom,
! [M: num,N: num] :
( ( bit_and_not_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
= ( case_option @ ( option @ num ) @ num @ ( some @ num @ one2 )
@ ^ [N11: num] : ( some @ num @ ( bit1 @ N11 ) )
@ ( bit_and_not_num @ M @ N ) ) ) ).
% and_not_num.simps(8)
thf(fact_3971_image__mult__atLeastAtMost__if,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,X2: A,Y: A] :
( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( image2 @ A @ A @ ( times_times @ A @ C2 ) @ ( set_or1337092689740270186AtMost @ A @ X2 @ Y ) )
= ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ C2 @ X2 ) @ ( times_times @ A @ C2 @ Y ) ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less_eq @ A @ X2 @ Y )
=> ( ( image2 @ A @ A @ ( times_times @ A @ C2 ) @ ( set_or1337092689740270186AtMost @ A @ X2 @ Y ) )
= ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ C2 @ Y ) @ ( times_times @ A @ C2 @ X2 ) ) ) )
& ( ~ ( ord_less_eq @ A @ X2 @ Y )
=> ( ( image2 @ A @ A @ ( times_times @ A @ C2 ) @ ( set_or1337092689740270186AtMost @ A @ X2 @ Y ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% image_mult_atLeastAtMost_if
thf(fact_3972_image__mult__atLeastAtMost__if_H,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X2: A,Y: A,C2: A] :
( ( ( ord_less_eq @ A @ X2 @ Y )
=> ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( image2 @ A @ A
@ ^ [X: A] : ( times_times @ A @ X @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ X2 @ Y ) )
= ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ X2 @ C2 ) @ ( times_times @ A @ Y @ C2 ) ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( image2 @ A @ A
@ ^ [X: A] : ( times_times @ A @ X @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ X2 @ Y ) )
= ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ Y @ C2 ) @ ( times_times @ A @ X2 @ C2 ) ) ) ) ) )
& ( ~ ( ord_less_eq @ A @ X2 @ Y )
=> ( ( image2 @ A @ A
@ ^ [X: A] : ( times_times @ A @ X @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ X2 @ Y ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% image_mult_atLeastAtMost_if'
thf(fact_3973_image__affinity__atLeastAtMost,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,M: A,C2: A] :
( ( ( ( set_or1337092689740270186AtMost @ A @ A2 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( image2 @ A @ A
@ ^ [X: A] : ( plus_plus @ A @ ( times_times @ A @ M @ X ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( ( set_or1337092689740270186AtMost @ A @ A2 @ B2 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
=> ( ( image2 @ A @ A
@ ^ [X: A] : ( plus_plus @ A @ ( times_times @ A @ M @ X ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ ( times_times @ A @ M @ A2 ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ M @ B2 ) @ C2 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
=> ( ( image2 @ A @ A
@ ^ [X: A] : ( plus_plus @ A @ ( times_times @ A @ M @ X ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ ( times_times @ A @ M @ B2 ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ M @ A2 ) @ C2 ) ) ) ) ) ) ) ) ).
% image_affinity_atLeastAtMost
thf(fact_3974_image__affinity__atLeastAtMost__diff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,M: A,C2: A] :
( ( ( ( set_or1337092689740270186AtMost @ A @ A2 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( image2 @ A @ A
@ ^ [X: A] : ( minus_minus @ A @ ( times_times @ A @ M @ X ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( ( set_or1337092689740270186AtMost @ A @ A2 @ B2 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
=> ( ( image2 @ A @ A
@ ^ [X: A] : ( minus_minus @ A @ ( times_times @ A @ M @ X ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ ( times_times @ A @ M @ A2 ) @ C2 ) @ ( minus_minus @ A @ ( times_times @ A @ M @ B2 ) @ C2 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
=> ( ( image2 @ A @ A
@ ^ [X: A] : ( minus_minus @ A @ ( times_times @ A @ M @ X ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ ( times_times @ A @ M @ B2 ) @ C2 ) @ ( minus_minus @ A @ ( times_times @ A @ M @ A2 ) @ C2 ) ) ) ) ) ) ) ) ).
% image_affinity_atLeastAtMost_diff
thf(fact_3975_image__affinity__atLeastAtMost__div,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,M: A,C2: A] :
( ( ( ( set_or1337092689740270186AtMost @ A @ A2 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( image2 @ A @ A
@ ^ [X: A] : ( plus_plus @ A @ ( divide_divide @ A @ X @ M ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( ( set_or1337092689740270186AtMost @ A @ A2 @ B2 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
=> ( ( image2 @ A @ A
@ ^ [X: A] : ( plus_plus @ A @ ( divide_divide @ A @ X @ M ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ ( divide_divide @ A @ A2 @ M ) @ C2 ) @ ( plus_plus @ A @ ( divide_divide @ A @ B2 @ M ) @ C2 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
=> ( ( image2 @ A @ A
@ ^ [X: A] : ( plus_plus @ A @ ( divide_divide @ A @ X @ M ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ ( divide_divide @ A @ B2 @ M ) @ C2 ) @ ( plus_plus @ A @ ( divide_divide @ A @ A2 @ M ) @ C2 ) ) ) ) ) ) ) ) ).
% image_affinity_atLeastAtMost_div
thf(fact_3976_image__affinity__atLeastAtMost__div__diff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,M: A,C2: A] :
( ( ( ( set_or1337092689740270186AtMost @ A @ A2 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( image2 @ A @ A
@ ^ [X: A] : ( minus_minus @ A @ ( divide_divide @ A @ X @ M ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( ( set_or1337092689740270186AtMost @ A @ A2 @ B2 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
=> ( ( image2 @ A @ A
@ ^ [X: A] : ( minus_minus @ A @ ( divide_divide @ A @ X @ M ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ ( divide_divide @ A @ A2 @ M ) @ C2 ) @ ( minus_minus @ A @ ( divide_divide @ A @ B2 @ M ) @ C2 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
=> ( ( image2 @ A @ A
@ ^ [X: A] : ( minus_minus @ A @ ( divide_divide @ A @ X @ M ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ ( divide_divide @ A @ B2 @ M ) @ C2 ) @ ( minus_minus @ A @ ( divide_divide @ A @ A2 @ M ) @ C2 ) ) ) ) ) ) ) ) ).
% image_affinity_atLeastAtMost_div_diff
thf(fact_3977_sum__fun__comp,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( semiring_1 @ C )
=> ! [S2: set @ A,R3: set @ B,G: A > B,F: B > C] :
( ( finite_finite2 @ A @ S2 )
=> ( ( finite_finite2 @ B @ R3 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ G @ S2 ) @ R3 )
=> ( ( groups7311177749621191930dd_sum @ A @ C
@ ^ [X: A] : ( F @ ( G @ X ) )
@ S2 )
= ( groups7311177749621191930dd_sum @ B @ C
@ ^ [Y4: B] :
( times_times @ C
@ ( semiring_1_of_nat @ C
@ ( finite_card @ A
@ ( collect @ A
@ ^ [X: A] :
( ( member @ A @ X @ S2 )
& ( ( G @ X )
= Y4 ) ) ) ) )
@ ( F @ Y4 ) )
@ R3 ) ) ) ) ) ) ).
% sum_fun_comp
thf(fact_3978_less__eq__option__Some,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A,Y: A] :
( ( ord_less_eq @ ( option @ A ) @ ( some @ A @ X2 ) @ ( some @ A @ Y ) )
= ( ord_less_eq @ A @ X2 @ Y ) ) ) ).
% less_eq_option_Some
thf(fact_3979_translation__subtract__diff,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A2: A,S3: set @ A,T2: set @ A] :
( ( image2 @ A @ A
@ ^ [X: A] : ( minus_minus @ A @ X @ A2 )
@ ( minus_minus @ ( set @ A ) @ S3 @ T2 ) )
= ( minus_minus @ ( set @ A )
@ ( image2 @ A @ A
@ ^ [X: A] : ( minus_minus @ A @ X @ A2 )
@ S3 )
@ ( image2 @ A @ A
@ ^ [X: A] : ( minus_minus @ A @ X @ A2 )
@ T2 ) ) ) ) ).
% translation_subtract_diff
thf(fact_3980_option_Osize_I4_J,axiom,
! [A: $tType,X22: A] :
( ( size_size @ ( option @ A ) @ ( some @ A @ X22 ) )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% option.size(4)
thf(fact_3981_translation__diff,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A2: A,S3: set @ A,T2: set @ A] :
( ( image2 @ A @ A @ ( plus_plus @ A @ A2 ) @ ( minus_minus @ ( set @ A ) @ S3 @ T2 ) )
= ( minus_minus @ ( set @ A ) @ ( image2 @ A @ A @ ( plus_plus @ A @ A2 ) @ S3 ) @ ( image2 @ A @ A @ ( plus_plus @ A @ A2 ) @ T2 ) ) ) ) ).
% translation_diff
thf(fact_3982_op__conc__empty__img__id,axiom,
! [A: $tType,L5: set @ ( list @ A )] :
( ( image2 @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ ( nil @ A ) ) @ L5 )
= L5 ) ).
% op_conc_empty_img_id
thf(fact_3983_image__Suc__atLeastAtMost,axiom,
! [I: nat,J: nat] :
( ( image2 @ nat @ nat @ suc @ ( set_or1337092689740270186AtMost @ nat @ I @ J ) )
= ( set_or1337092689740270186AtMost @ nat @ ( suc @ I ) @ ( suc @ J ) ) ) ).
% image_Suc_atLeastAtMost
thf(fact_3984_image__Suc__atLeastLessThan,axiom,
! [I: nat,J: nat] :
( ( image2 @ nat @ nat @ suc @ ( set_or7035219750837199246ssThan @ nat @ I @ J ) )
= ( set_or7035219750837199246ssThan @ nat @ ( suc @ I ) @ ( suc @ J ) ) ) ).
% image_Suc_atLeastLessThan
thf(fact_3985_Gcd__abs__eq,axiom,
! [K3: set @ int] :
( ( gcd_Gcd @ int @ ( image2 @ int @ int @ ( abs_abs @ int ) @ K3 ) )
= ( gcd_Gcd @ int @ K3 ) ) ).
% Gcd_abs_eq
thf(fact_3986_Gcd__int__eq,axiom,
! [N5: set @ nat] :
( ( gcd_Gcd @ int @ ( image2 @ nat @ int @ ( semiring_1_of_nat @ int ) @ N5 ) )
= ( semiring_1_of_nat @ int @ ( gcd_Gcd @ nat @ N5 ) ) ) ).
% Gcd_int_eq
thf(fact_3987_Gcd__nat__abs__eq,axiom,
! [K3: set @ int] :
( ( gcd_Gcd @ nat
@ ( image2 @ int @ nat
@ ^ [K2: int] : ( nat2 @ ( abs_abs @ int @ K2 ) )
@ K3 ) )
= ( nat2 @ ( gcd_Gcd @ int @ K3 ) ) ) ).
% Gcd_nat_abs_eq
thf(fact_3988_zero__notin__Suc__image,axiom,
! [A5: set @ nat] :
~ ( member @ nat @ ( zero_zero @ nat ) @ ( image2 @ nat @ nat @ suc @ A5 ) ) ).
% zero_notin_Suc_image
thf(fact_3989_image__int__atLeastAtMost,axiom,
! [A2: nat,B2: nat] :
( ( image2 @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( set_or1337092689740270186AtMost @ nat @ A2 @ B2 ) )
= ( set_or1337092689740270186AtMost @ int @ ( semiring_1_of_nat @ int @ A2 ) @ ( semiring_1_of_nat @ int @ B2 ) ) ) ).
% image_int_atLeastAtMost
thf(fact_3990_option_Osize__neq,axiom,
! [A: $tType,X2: option @ A] :
( ( size_size @ ( option @ A ) @ X2 )
!= ( zero_zero @ nat ) ) ).
% option.size_neq
thf(fact_3991_image__int__atLeastLessThan,axiom,
! [A2: nat,B2: nat] :
( ( image2 @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( set_or7035219750837199246ssThan @ nat @ A2 @ B2 ) )
= ( set_or7035219750837199246ssThan @ int @ ( semiring_1_of_nat @ int @ A2 ) @ ( semiring_1_of_nat @ int @ B2 ) ) ) ).
% image_int_atLeastLessThan
thf(fact_3992_image__Suc__atMost,axiom,
! [N: nat] :
( ( image2 @ nat @ nat @ suc @ ( set_ord_atMost @ nat @ N ) )
= ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ ( suc @ N ) ) ) ).
% image_Suc_atMost
thf(fact_3993_image__Suc__lessThan,axiom,
! [N: nat] :
( ( image2 @ nat @ nat @ suc @ ( set_ord_lessThan @ nat @ N ) )
= ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ N ) ) ).
% image_Suc_lessThan
thf(fact_3994_atLeast0__atMost__Suc__eq__insert__0,axiom,
! [N: nat] :
( ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) )
= ( insert @ nat @ ( zero_zero @ nat ) @ ( image2 @ nat @ nat @ suc @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% atLeast0_atMost_Suc_eq_insert_0
thf(fact_3995_atLeast0__lessThan__Suc__eq__insert__0,axiom,
! [N: nat] :
( ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) )
= ( insert @ nat @ ( zero_zero @ nat ) @ ( image2 @ nat @ nat @ suc @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% atLeast0_lessThan_Suc_eq_insert_0
thf(fact_3996_atMost__Suc__eq__insert__0,axiom,
! [N: nat] :
( ( set_ord_atMost @ nat @ ( suc @ N ) )
= ( insert @ nat @ ( zero_zero @ nat ) @ ( image2 @ nat @ nat @ suc @ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% atMost_Suc_eq_insert_0
thf(fact_3997_lessThan__Suc__eq__insert__0,axiom,
! [N: nat] :
( ( set_ord_lessThan @ nat @ ( suc @ N ) )
= ( insert @ nat @ ( zero_zero @ nat ) @ ( image2 @ nat @ nat @ suc @ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% lessThan_Suc_eq_insert_0
thf(fact_3998_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_3999_image__add__int__atLeastLessThan,axiom,
! [L: int,U: int] :
( ( image2 @ int @ int
@ ^ [X: int] : ( plus_plus @ int @ X @ L )
@ ( set_or7035219750837199246ssThan @ int @ ( zero_zero @ int ) @ ( minus_minus @ int @ U @ L ) ) )
= ( set_or7035219750837199246ssThan @ int @ L @ U ) ) ).
% image_add_int_atLeastLessThan
thf(fact_4000_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 )
@ ^ [X: A] : ( H @ ( F22 @ X ) )
@ Option ) ) ).
% option.case_distrib
thf(fact_4001_image__add__integer__atLeastLessThan,axiom,
! [L: code_integer,U: code_integer] :
( ( image2 @ code_integer @ code_integer
@ ^ [X: code_integer] : ( plus_plus @ code_integer @ X @ 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_4002_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_4003_image__minus__const__atLeastLessThan__nat,axiom,
! [C2: nat,Y: nat,X2: nat] :
( ( ( ord_less @ nat @ C2 @ Y )
=> ( ( image2 @ nat @ nat
@ ^ [I2: nat] : ( minus_minus @ nat @ I2 @ C2 )
@ ( set_or7035219750837199246ssThan @ nat @ X2 @ Y ) )
= ( set_or7035219750837199246ssThan @ nat @ ( minus_minus @ nat @ X2 @ C2 ) @ ( minus_minus @ nat @ Y @ C2 ) ) ) )
& ( ~ ( ord_less @ nat @ C2 @ Y )
=> ( ( ( ord_less @ nat @ X2 @ Y )
=> ( ( image2 @ nat @ nat
@ ^ [I2: nat] : ( minus_minus @ nat @ I2 @ C2 )
@ ( set_or7035219750837199246ssThan @ nat @ X2 @ Y ) )
= ( insert @ nat @ ( zero_zero @ nat ) @ ( bot_bot @ ( set @ nat ) ) ) ) )
& ( ~ ( ord_less @ nat @ X2 @ Y )
=> ( ( image2 @ nat @ nat
@ ^ [I2: nat] : ( minus_minus @ nat @ I2 @ C2 )
@ ( set_or7035219750837199246ssThan @ nat @ X2 @ Y ) )
= ( bot_bot @ ( set @ nat ) ) ) ) ) ) ) ).
% image_minus_const_atLeastLessThan_nat
thf(fact_4004_translation__Compl,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A2: A,T2: set @ A] :
( ( image2 @ A @ A @ ( plus_plus @ A @ A2 ) @ ( uminus_uminus @ ( set @ A ) @ T2 ) )
= ( uminus_uminus @ ( set @ A ) @ ( image2 @ A @ A @ ( plus_plus @ A @ A2 ) @ T2 ) ) ) ) ).
% translation_Compl
thf(fact_4005_translation__subtract__Compl,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A2: A,T2: set @ A] :
( ( image2 @ A @ A
@ ^ [X: A] : ( minus_minus @ A @ X @ A2 )
@ ( uminus_uminus @ ( set @ A ) @ T2 ) )
= ( uminus_uminus @ ( set @ A )
@ ( image2 @ A @ A
@ ^ [X: A] : ( minus_minus @ A @ X @ A2 )
@ T2 ) ) ) ) ).
% translation_subtract_Compl
thf(fact_4006_option_Osize__gen_I2_J,axiom,
! [A: $tType,X2: A > nat,X22: A] :
( ( size_option @ A @ X2 @ ( some @ A @ X22 ) )
= ( plus_plus @ nat @ ( X2 @ X22 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% option.size_gen(2)
thf(fact_4007_take__bit__num__simps_I6_J,axiom,
! [R: num,M: num] :
( ( bit_take_bit_num @ ( numeral_numeral @ nat @ R ) @ ( bit0 @ M ) )
= ( case_option @ ( option @ num ) @ num @ ( none @ num )
@ ^ [Q6: num] : ( some @ num @ ( bit0 @ Q6 ) )
@ ( bit_take_bit_num @ ( pred_numeral @ R ) @ M ) ) ) ).
% take_bit_num_simps(6)
thf(fact_4008_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 )
@ ^ [X: A] : ( cons @ A @ X @ ( nil @ A ) )
@ A5 ) ) ).
% set_Cons_sing_Nil
thf(fact_4009_image__Collect__subsetI,axiom,
! [A: $tType,B: $tType,P: A > $o,F: A > B,B5: set @ B] :
( ! [X3: A] :
( ( P @ X3 )
=> ( member @ B @ ( F @ X3 ) @ B5 ) )
=> ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F @ ( collect @ A @ P ) ) @ B5 ) ) ).
% image_Collect_subsetI
thf(fact_4010_take__bit__num__simps_I1_J,axiom,
! [M: num] :
( ( bit_take_bit_num @ ( zero_zero @ nat ) @ M )
= ( none @ num ) ) ).
% take_bit_num_simps(1)
thf(fact_4011_take__bit__num__simps_I3_J,axiom,
! [N: nat,M: num] :
( ( bit_take_bit_num @ ( suc @ N ) @ ( bit0 @ M ) )
= ( case_option @ ( option @ num ) @ num @ ( none @ num )
@ ^ [Q6: num] : ( some @ num @ ( bit0 @ Q6 ) )
@ ( bit_take_bit_num @ N @ M ) ) ) ).
% take_bit_num_simps(3)
thf(fact_4012_case__optionE,axiom,
! [A: $tType,P: $o,Q: A > $o,X2: option @ A] :
( ( case_option @ $o @ A @ P @ Q @ X2 )
=> ( ( ( X2
= ( none @ A ) )
=> ~ P )
=> ~ ! [Y2: A] :
( ( X2
= ( some @ A @ Y2 ) )
=> ~ ( Q @ Y2 ) ) ) ) ).
% case_optionE
thf(fact_4013_option_Odisc__eq__case_I1_J,axiom,
! [A: $tType,Option: option @ A] :
( ( Option
= ( none @ A ) )
= ( case_option @ $o @ A @ $true
@ ^ [Uu: A] : $false
@ Option ) ) ).
% option.disc_eq_case(1)
thf(fact_4014_option_Odisc__eq__case_I2_J,axiom,
! [A: $tType,Option: option @ A] :
( ( Option
!= ( none @ A ) )
= ( case_option @ $o @ A @ $false
@ ^ [Uu: A] : $true
@ Option ) ) ).
% option.disc_eq_case(2)
thf(fact_4015_uminus__set__def,axiom,
! [A: $tType] :
( ( uminus_uminus @ ( set @ A ) )
= ( ^ [A8: set @ A] :
( collect @ A
@ ( uminus_uminus @ ( A > $o )
@ ^ [X: A] : ( member @ A @ X @ A8 ) ) ) ) ) ).
% uminus_set_def
thf(fact_4016_Collect__neg__eq,axiom,
! [A: $tType,P: A > $o] :
( ( collect @ A
@ ^ [X: A] :
~ ( P @ X ) )
= ( uminus_uminus @ ( set @ A ) @ ( collect @ A @ P ) ) ) ).
% Collect_neg_eq
thf(fact_4017_Compl__eq,axiom,
! [A: $tType] :
( ( uminus_uminus @ ( set @ A ) )
= ( ^ [A8: set @ A] :
( collect @ A
@ ^ [X: A] :
~ ( member @ A @ X @ A8 ) ) ) ) ).
% Compl_eq
thf(fact_4018_option_Osize__gen_I1_J,axiom,
! [A: $tType,X2: A > nat] :
( ( size_option @ A @ X2 @ ( none @ A ) )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% option.size_gen(1)
thf(fact_4019_less__option__def,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( ord_less @ ( option @ A ) )
= ( ^ [X: option @ A] :
( case_option @ $o @ A @ $false
@ ^ [Y4: A] :
( case_option @ $o @ A @ $true
@ ^ [Z3: A] : ( ord_less @ A @ Z3 @ Y4 )
@ X ) ) ) ) ) ).
% less_option_def
thf(fact_4020_refines__option,axiom,
! [B: $tType,A: $tType,A2: option @ A,A6: option @ A,M1: heap_Time_Heap @ B,M12: heap_Time_Heap @ B,M22: A > ( heap_Time_Heap @ B ),M23: A > ( heap_Time_Heap @ B )] :
( ( A2 = A6 )
=> ( ( refine_Imp_refines @ B @ M1 @ M12 )
=> ( ! [X3: A] : ( refine_Imp_refines @ B @ ( M22 @ X3 ) @ ( M23 @ X3 ) )
=> ( refine_Imp_refines @ B @ ( case_option @ ( heap_Time_Heap @ B ) @ A @ M1 @ M22 @ A2 ) @ ( case_option @ ( heap_Time_Heap @ B ) @ A @ M12 @ M23 @ A6 ) ) ) ) ) ).
% refines_option
thf(fact_4021_less__eq__option__def,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( ord_less_eq @ ( option @ A ) )
= ( ^ [X: option @ A,Y4: option @ A] :
( case_option @ $o @ A @ $true
@ ^ [Z3: A] : ( case_option @ $o @ A @ $false @ ( ord_less_eq @ A @ Z3 ) @ Y4 )
@ X ) ) ) ) ).
% less_eq_option_def
thf(fact_4022_option_Osize_I3_J,axiom,
! [A: $tType] :
( ( size_size @ ( option @ A ) @ ( none @ A ) )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% option.size(3)
thf(fact_4023_take__bit__num__eq__None__imp,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,N: num] :
( ( ( bit_take_bit_num @ M @ N )
= ( none @ num ) )
=> ( ( bit_se2584673776208193580ke_bit @ A @ M @ ( numeral_numeral @ A @ N ) )
= ( zero_zero @ A ) ) ) ) ).
% take_bit_num_eq_None_imp
thf(fact_4024_and__not__num__eq__None__iff,axiom,
! [M: num,N: num] :
( ( ( bit_and_not_num @ M @ N )
= ( none @ num ) )
= ( ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ M ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N ) ) )
= ( zero_zero @ int ) ) ) ).
% and_not_num_eq_None_iff
thf(fact_4025_take__bit__num__def,axiom,
( bit_take_bit_num
= ( ^ [N4: nat,M2: num] :
( if @ ( option @ num )
@ ( ( bit_se2584673776208193580ke_bit @ nat @ N4 @ ( numeral_numeral @ nat @ M2 ) )
= ( zero_zero @ nat ) )
@ ( none @ num )
@ ( some @ num @ ( num_of_nat @ ( bit_se2584673776208193580ke_bit @ nat @ N4 @ ( numeral_numeral @ nat @ M2 ) ) ) ) ) ) ) ).
% take_bit_num_def
thf(fact_4026_rel__of__empty,axiom,
! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o] :
( ( rel_of @ A @ B
@ ^ [X: A] : ( none @ B )
@ P )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% rel_of_empty
thf(fact_4027_ran__nth__set__encoding__conv,axiom,
! [A: $tType,L: list @ A] :
( ( ran @ nat @ A
@ ^ [I2: nat] : ( if @ ( option @ A ) @ ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ L ) ) @ ( some @ A @ ( nth @ A @ L @ I2 ) ) @ ( none @ A ) ) )
= ( set2 @ A @ L ) ) ).
% ran_nth_set_encoding_conv
thf(fact_4028_and__not__num_Oelims,axiom,
! [X2: num,Xa: num,Y: option @ num] :
( ( ( bit_and_not_num @ X2 @ Xa )
= Y )
=> ( ( ( X2 = one2 )
=> ( ( Xa = one2 )
=> ( Y
!= ( none @ num ) ) ) )
=> ( ( ( X2 = one2 )
=> ( ? [N2: num] :
( Xa
= ( bit0 @ N2 ) )
=> ( Y
!= ( some @ num @ one2 ) ) ) )
=> ( ( ( X2 = one2 )
=> ( ? [N2: num] :
( Xa
= ( bit1 @ N2 ) )
=> ( Y
!= ( none @ num ) ) ) )
=> ( ! [M4: num] :
( ( X2
= ( bit0 @ M4 ) )
=> ( ( Xa = one2 )
=> ( Y
!= ( some @ num @ ( bit0 @ M4 ) ) ) ) )
=> ( ! [M4: num] :
( ( X2
= ( bit0 @ M4 ) )
=> ! [N2: num] :
( ( Xa
= ( bit0 @ N2 ) )
=> ( Y
!= ( map_option @ num @ num @ bit0 @ ( bit_and_not_num @ M4 @ N2 ) ) ) ) )
=> ( ! [M4: num] :
( ( X2
= ( bit0 @ M4 ) )
=> ! [N2: num] :
( ( Xa
= ( bit1 @ N2 ) )
=> ( Y
!= ( map_option @ num @ num @ bit0 @ ( bit_and_not_num @ M4 @ N2 ) ) ) ) )
=> ( ! [M4: num] :
( ( X2
= ( bit1 @ M4 ) )
=> ( ( Xa = one2 )
=> ( Y
!= ( some @ num @ ( bit0 @ M4 ) ) ) ) )
=> ( ! [M4: num] :
( ( X2
= ( bit1 @ M4 ) )
=> ! [N2: num] :
( ( Xa
= ( bit0 @ N2 ) )
=> ( Y
!= ( case_option @ ( option @ num ) @ num @ ( some @ num @ one2 )
@ ^ [N11: num] : ( some @ num @ ( bit1 @ N11 ) )
@ ( bit_and_not_num @ M4 @ N2 ) ) ) ) )
=> ~ ! [M4: num] :
( ( X2
= ( bit1 @ M4 ) )
=> ! [N2: num] :
( ( Xa
= ( bit1 @ N2 ) )
=> ( Y
!= ( map_option @ num @ num @ bit0 @ ( bit_and_not_num @ M4 @ N2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% and_not_num.elims
thf(fact_4029_Gcd__int__def,axiom,
( ( gcd_Gcd @ int )
= ( ^ [K6: set @ int] : ( semiring_1_of_nat @ int @ ( gcd_Gcd @ nat @ ( image2 @ int @ nat @ ( comp @ int @ nat @ int @ nat2 @ ( abs_abs @ int ) ) @ K6 ) ) ) ) ) ).
% Gcd_int_def
thf(fact_4030_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
@ ^ [X: A] : ( if @ ( option @ B ) @ ( X = K ) @ ( some @ B @ V2 ) @ ( M @ X ) ) )
= ( insert @ B @ V2 @ ( ran @ A @ B @ M ) ) ) ) ).
% map_update_eta_repair(2)
thf(fact_4031_fun__comp__eq__conv,axiom,
! [B: $tType,C: $tType,A: $tType,F: C > B,G: A > C,Fg: A > B] :
( ( ( comp @ C @ B @ A @ F @ G )
= Fg )
= ( ! [X: A] :
( ( F @ ( G @ X ) )
= ( Fg @ X ) ) ) ) ).
% fun_comp_eq_conv
thf(fact_4032_comp__cong__right,axiom,
! [C: $tType,B: $tType,A: $tType,X2: A > B,Y: A > B,F: B > C] :
( ( X2 = Y )
=> ( ( comp @ B @ C @ A @ F @ X2 )
= ( comp @ B @ C @ A @ F @ Y ) ) ) ).
% comp_cong_right
thf(fact_4033_comp__cong__left,axiom,
! [B: $tType,A: $tType,C: $tType,X2: A > B,Y: A > B,F: C > A] :
( ( X2 = Y )
=> ( ( comp @ A @ B @ C @ X2 @ F )
= ( comp @ A @ B @ C @ Y @ F ) ) ) ).
% comp_cong_left
thf(fact_4034_rewriteR__comp__comp2,axiom,
! [C: $tType,B: $tType,E: $tType,D: $tType,A: $tType,G: C > B,H: A > C,R1: D > B,R22: A > D,F: B > E,L: D > E] :
( ( ( comp @ C @ B @ A @ G @ H )
= ( comp @ D @ B @ A @ R1 @ R22 ) )
=> ( ( ( comp @ B @ E @ D @ F @ R1 )
= L )
=> ( ( comp @ C @ E @ A @ ( comp @ B @ E @ C @ F @ G ) @ H )
= ( comp @ D @ E @ A @ L @ R22 ) ) ) ) ).
% rewriteR_comp_comp2
thf(fact_4035_rewriteL__comp__comp2,axiom,
! [A: $tType,C: $tType,B: $tType,D: $tType,E: $tType,F: C > B,G: A > C,L1: D > B,L22: A > D,H: E > A,R: E > D] :
( ( ( comp @ C @ B @ A @ F @ G )
= ( comp @ D @ B @ A @ L1 @ L22 ) )
=> ( ( ( comp @ A @ D @ E @ L22 @ H )
= R )
=> ( ( comp @ C @ B @ E @ F @ ( comp @ A @ C @ E @ G @ H ) )
= ( comp @ D @ B @ E @ L1 @ R ) ) ) ) ).
% rewriteL_comp_comp2
thf(fact_4036_rewriteR__comp__comp,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,G: C > B,H: A > C,R: A > B,F: B > D] :
( ( ( comp @ C @ B @ A @ G @ H )
= R )
=> ( ( comp @ C @ D @ A @ ( comp @ B @ D @ C @ F @ G ) @ H )
= ( comp @ B @ D @ A @ F @ R ) ) ) ).
% rewriteR_comp_comp
thf(fact_4037_rewriteL__comp__comp,axiom,
! [C: $tType,B: $tType,A: $tType,D: $tType,F: C > B,G: A > C,L: A > B,H: D > A] :
( ( ( comp @ C @ B @ A @ F @ G )
= L )
=> ( ( comp @ C @ B @ D @ F @ ( comp @ A @ C @ D @ G @ H ) )
= ( comp @ A @ B @ D @ L @ H ) ) ) ).
% rewriteL_comp_comp
thf(fact_4038_comp__funpow,axiom,
! [B: $tType,A: $tType,N: nat,F: A > A] :
( ( compow @ ( ( B > A ) > B > A ) @ N @ ( comp @ A @ A @ B @ F ) )
= ( comp @ A @ A @ B @ ( compow @ ( A > A ) @ N @ F ) ) ) ).
% comp_funpow
thf(fact_4039_option_Omap__ident,axiom,
! [A: $tType,T2: option @ A] :
( ( map_option @ A @ A
@ ^ [X: A] : X
@ T2 )
= T2 ) ).
% option.map_ident
thf(fact_4040_timeFrame_Osimps_I2_J,axiom,
! [A: $tType,N: nat] :
( ( heap_Time_timeFrame @ A @ N @ ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) )
= ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ).
% timeFrame.simps(2)
thf(fact_4041_funpow_Osimps_I2_J,axiom,
! [A: $tType,N: nat,F: A > A] :
( ( compow @ ( A > A ) @ ( suc @ N ) @ F )
= ( comp @ A @ A @ A @ F @ ( compow @ ( A > A ) @ N @ F ) ) ) ).
% funpow.simps(2)
thf(fact_4042_funpow__Suc__right,axiom,
! [A: $tType,N: nat,F: A > A] :
( ( compow @ ( A > A ) @ ( suc @ N ) @ F )
= ( comp @ A @ A @ A @ ( compow @ ( A > A ) @ N @ F ) @ F ) ) ).
% funpow_Suc_right
thf(fact_4043_funpow__add,axiom,
! [A: $tType,M: nat,N: nat,F: A > A] :
( ( compow @ ( A > A ) @ ( plus_plus @ nat @ M @ N ) @ F )
= ( comp @ A @ A @ A @ ( compow @ ( A > A ) @ M @ F ) @ ( compow @ ( A > A ) @ N @ F ) ) ) ).
% funpow_add
thf(fact_4044_sum__comp__morphism,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( ( comm_monoid_add @ B )
& ( comm_monoid_add @ A ) )
=> ! [H: B > A,G: C > B,A5: set @ C] :
( ( ( H @ ( zero_zero @ B ) )
= ( zero_zero @ A ) )
=> ( ! [X3: B,Y2: B] :
( ( H @ ( plus_plus @ B @ X3 @ Y2 ) )
= ( plus_plus @ A @ ( H @ X3 ) @ ( H @ Y2 ) ) )
=> ( ( groups7311177749621191930dd_sum @ C @ A @ ( comp @ B @ A @ C @ H @ G ) @ A5 )
= ( H @ ( groups7311177749621191930dd_sum @ C @ B @ G @ A5 ) ) ) ) ) ) ).
% sum_comp_morphism
thf(fact_4045_map__option__case,axiom,
! [A: $tType,B: $tType] :
( ( map_option @ B @ A )
= ( ^ [F2: B > A] :
( case_option @ ( option @ A ) @ B @ ( none @ A )
@ ^ [X: B] : ( some @ A @ ( F2 @ X ) ) ) ) ) ).
% map_option_case
thf(fact_4046_sum_Oreindex__nontrivial,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A5: set @ B,H: B > C,G: C > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ! [X3: B,Y2: B] :
( ( member @ B @ X3 @ A5 )
=> ( ( member @ B @ Y2 @ A5 )
=> ( ( X3 != Y2 )
=> ( ( ( H @ X3 )
= ( H @ Y2 ) )
=> ( ( G @ ( H @ X3 ) )
= ( zero_zero @ A ) ) ) ) ) )
=> ( ( groups7311177749621191930dd_sum @ C @ A @ G @ ( image2 @ B @ C @ H @ A5 ) )
= ( groups7311177749621191930dd_sum @ B @ A @ ( comp @ C @ A @ B @ G @ H ) @ A5 ) ) ) ) ) ).
% sum.reindex_nontrivial
thf(fact_4047_prod_Oreindex__nontrivial,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A5: set @ B,H: B > C,G: C > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ! [X3: B,Y2: B] :
( ( member @ B @ X3 @ A5 )
=> ( ( member @ B @ Y2 @ A5 )
=> ( ( X3 != Y2 )
=> ( ( ( H @ X3 )
= ( H @ Y2 ) )
=> ( ( G @ ( H @ X3 ) )
= ( one_one @ A ) ) ) ) ) )
=> ( ( groups7121269368397514597t_prod @ C @ A @ G @ ( image2 @ B @ C @ H @ A5 ) )
= ( groups7121269368397514597t_prod @ B @ A @ ( comp @ C @ A @ B @ G @ H ) @ A5 ) ) ) ) ) ).
% prod.reindex_nontrivial
thf(fact_4048_sum__image__le,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( ordere6911136660526730532id_add @ B )
=> ! [I5: set @ C,G: A > B,F: C > A] :
( ( finite_finite2 @ C @ I5 )
=> ( ! [I3: C] :
( ( member @ C @ I3 @ I5 )
=> ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( G @ ( F @ I3 ) ) ) )
=> ( ord_less_eq @ B @ ( groups7311177749621191930dd_sum @ A @ B @ G @ ( image2 @ C @ A @ F @ I5 ) ) @ ( groups7311177749621191930dd_sum @ C @ B @ ( comp @ A @ B @ C @ G @ F ) @ I5 ) ) ) ) ) ).
% sum_image_le
thf(fact_4049_ran__empty,axiom,
! [B: $tType,A: $tType] :
( ( ran @ B @ A
@ ^ [X: B] : ( none @ A ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% ran_empty
thf(fact_4050_ran__map__option,axiom,
! [A: $tType,C: $tType,B: $tType,F: C > A,M: B > ( option @ C )] :
( ( ran @ B @ A
@ ^ [X: B] : ( map_option @ C @ A @ F @ ( M @ X ) ) )
= ( image2 @ C @ A @ F @ ( ran @ B @ C @ M ) ) ) ).
% ran_map_option
thf(fact_4051_and__num_Oelims,axiom,
! [X2: num,Xa: num,Y: option @ num] :
( ( ( bit_un7362597486090784418nd_num @ X2 @ Xa )
= Y )
=> ( ( ( X2 = one2 )
=> ( ( Xa = one2 )
=> ( Y
!= ( some @ num @ one2 ) ) ) )
=> ( ( ( X2 = one2 )
=> ( ? [N2: num] :
( Xa
= ( bit0 @ N2 ) )
=> ( Y
!= ( none @ num ) ) ) )
=> ( ( ( X2 = one2 )
=> ( ? [N2: num] :
( Xa
= ( bit1 @ N2 ) )
=> ( Y
!= ( some @ num @ one2 ) ) ) )
=> ( ( ? [M4: num] :
( X2
= ( bit0 @ M4 ) )
=> ( ( Xa = one2 )
=> ( Y
!= ( none @ num ) ) ) )
=> ( ! [M4: num] :
( ( X2
= ( bit0 @ M4 ) )
=> ! [N2: num] :
( ( Xa
= ( bit0 @ N2 ) )
=> ( Y
!= ( map_option @ num @ num @ bit0 @ ( bit_un7362597486090784418nd_num @ M4 @ N2 ) ) ) ) )
=> ( ! [M4: num] :
( ( X2
= ( bit0 @ M4 ) )
=> ! [N2: num] :
( ( Xa
= ( bit1 @ N2 ) )
=> ( Y
!= ( map_option @ num @ num @ bit0 @ ( bit_un7362597486090784418nd_num @ M4 @ N2 ) ) ) ) )
=> ( ( ? [M4: num] :
( X2
= ( bit1 @ M4 ) )
=> ( ( Xa = one2 )
=> ( Y
!= ( some @ num @ one2 ) ) ) )
=> ( ! [M4: num] :
( ( X2
= ( bit1 @ M4 ) )
=> ! [N2: num] :
( ( Xa
= ( bit0 @ N2 ) )
=> ( Y
!= ( map_option @ num @ num @ bit0 @ ( bit_un7362597486090784418nd_num @ M4 @ N2 ) ) ) ) )
=> ~ ! [M4: num] :
( ( X2
= ( bit1 @ M4 ) )
=> ! [N2: num] :
( ( Xa
= ( bit1 @ N2 ) )
=> ( Y
!= ( case_option @ ( option @ num ) @ num @ ( some @ num @ one2 )
@ ^ [N11: num] : ( some @ num @ ( bit1 @ N11 ) )
@ ( bit_un7362597486090784418nd_num @ M4 @ N2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% and_num.elims
thf(fact_4052_disjE__realizer2,axiom,
! [B: $tType,A: $tType,P: $o,Q: A > $o,X2: option @ A,R3: B > $o,F: B,G: A > B] :
( ( case_option @ $o @ A @ P @ Q @ X2 )
=> ( ( P
=> ( R3 @ F ) )
=> ( ! [Q4: A] :
( ( Q @ Q4 )
=> ( R3 @ ( G @ Q4 ) ) )
=> ( R3 @ ( case_option @ B @ A @ F @ G @ X2 ) ) ) ) ) ).
% disjE_realizer2
thf(fact_4053_map__option__o__empty,axiom,
! [C: $tType,B: $tType,A: $tType,F: C > B] :
( ( comp @ ( option @ C ) @ ( option @ B ) @ A @ ( map_option @ C @ B @ F )
@ ^ [X: A] : ( none @ C ) )
= ( ^ [X: A] : ( none @ B ) ) ) ).
% map_option_o_empty
thf(fact_4054_sorted__list__of__set_Ofold__insort__key_Ocomp__fun__commute__on,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Y: A,X2: A] :
( ( comp @ ( list @ A ) @ ( list @ A ) @ ( list @ A )
@ ( linorder_insort_key @ A @ A
@ ^ [X: A] : X
@ Y )
@ ( linorder_insort_key @ A @ A
@ ^ [X: A] : X
@ X2 ) )
= ( comp @ ( list @ A ) @ ( list @ A ) @ ( list @ A )
@ ( linorder_insort_key @ A @ A
@ ^ [X: A] : X
@ X2 )
@ ( linorder_insort_key @ A @ A
@ ^ [X: A] : X
@ Y ) ) ) ) ).
% sorted_list_of_set.fold_insort_key.comp_fun_commute_on
thf(fact_4055_rotate__add,axiom,
! [A: $tType,M: nat,N: nat] :
( ( rotate @ A @ ( plus_plus @ nat @ M @ N ) )
= ( comp @ ( list @ A ) @ ( list @ A ) @ ( list @ A ) @ ( rotate @ A @ M ) @ ( rotate @ A @ N ) ) ) ).
% rotate_add
thf(fact_4056_sum_OatLeast__Suc__atMost__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,M: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ ( suc @ N ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% sum.atLeast_Suc_atMost_Suc_shift
thf(fact_4057_sum_OatLeast__Suc__lessThan__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,M: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M ) @ ( suc @ N ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ).
% sum.atLeast_Suc_lessThan_Suc_shift
thf(fact_4058_sum_OatLeastAtMost__shift__bounds,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,M: nat,K: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ M @ K ) @ ( plus_plus @ nat @ N @ K ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ K ) ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% sum.atLeastAtMost_shift_bounds
thf(fact_4059_sum_OatLeastLessThan__shift__bounds,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,M: nat,K: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( plus_plus @ nat @ M @ K ) @ ( plus_plus @ nat @ N @ K ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ K ) ) @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ).
% sum.atLeastLessThan_shift_bounds
thf(fact_4060_prod_OatLeast__Suc__atMost__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,M: nat,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ ( suc @ N ) ) )
= ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% prod.atLeast_Suc_atMost_Suc_shift
thf(fact_4061_prod_OatLeast__Suc__lessThan__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,M: nat,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M ) @ ( suc @ N ) ) )
= ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ).
% prod.atLeast_Suc_lessThan_Suc_shift
thf(fact_4062_lift__def,axiom,
! [A: $tType,B: $tType] :
( ( heap_Time_lift @ A @ B )
= ( comp @ B @ ( heap_Time_Heap @ B ) @ A @ ( heap_Time_return @ B ) ) ) ).
% lift_def
thf(fact_4063_prod_OatLeastAtMost__shift__bounds,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,M: nat,K: nat,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ M @ K ) @ ( plus_plus @ nat @ N @ K ) ) )
= ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ K ) ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% prod.atLeastAtMost_shift_bounds
thf(fact_4064_prod_OatLeastLessThan__shift__bounds,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,M: nat,K: nat,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( plus_plus @ nat @ M @ K ) @ ( plus_plus @ nat @ N @ K ) ) )
= ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ K ) ) @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ).
% prod.atLeastLessThan_shift_bounds
thf(fact_4065_bit__drop__bit__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A2: A] :
( ( bit_se5641148757651400278ts_bit @ A @ ( bit_se4197421643247451524op_bit @ A @ N @ A2 ) )
= ( comp @ nat @ $o @ nat @ ( bit_se5641148757651400278ts_bit @ A @ A2 ) @ ( plus_plus @ nat @ N ) ) ) ) ).
% bit_drop_bit_eq
thf(fact_4066_sum_OatLeast0__atMost__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
= ( plus_plus @ A @ ( G @ ( zero_zero @ nat ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ) ).
% sum.atLeast0_atMost_Suc_shift
thf(fact_4067_sum_OatLeast0__lessThan__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
= ( plus_plus @ A @ ( G @ ( zero_zero @ nat ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ) ).
% sum.atLeast0_lessThan_Suc_shift
thf(fact_4068_prod_OatLeast0__atMost__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
= ( times_times @ A @ ( G @ ( zero_zero @ nat ) ) @ ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ) ).
% prod.atLeast0_atMost_Suc_shift
thf(fact_4069_prod_OatLeast0__lessThan__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
= ( times_times @ A @ ( G @ ( zero_zero @ nat ) ) @ ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ) ).
% prod.atLeast0_lessThan_Suc_shift
thf(fact_4070_sum_OatLeastLessThan__shift__0,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,M: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ M ) ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N @ M ) ) ) ) ) ).
% sum.atLeastLessThan_shift_0
thf(fact_4071_prod_OatLeastLessThan__shift__0,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,M: nat,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) )
= ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ M ) ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N @ M ) ) ) ) ) ).
% prod.atLeastLessThan_shift_0
thf(fact_4072_sum_OatLeast__atMost__pred__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,M: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ( comp @ nat @ A @ nat @ G
@ ^ [N4: nat] : ( minus_minus @ nat @ N4 @ ( suc @ ( zero_zero @ nat ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ ( suc @ N ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% sum.atLeast_atMost_pred_shift
thf(fact_4073_sum_OatLeast__lessThan__pred__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,M: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ( comp @ nat @ A @ nat @ G
@ ^ [N4: nat] : ( minus_minus @ nat @ N4 @ ( suc @ ( zero_zero @ nat ) ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M ) @ ( suc @ N ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ).
% sum.atLeast_lessThan_pred_shift
thf(fact_4074_prod_OatLeast__atMost__pred__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,M: nat,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A
@ ( comp @ nat @ A @ nat @ G
@ ^ [N4: nat] : ( minus_minus @ nat @ N4 @ ( suc @ ( zero_zero @ nat ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ ( suc @ N ) ) )
= ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% prod.atLeast_atMost_pred_shift
thf(fact_4075_prod_OatLeast__lessThan__pred__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,M: nat,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A
@ ( comp @ nat @ A @ nat @ G
@ ^ [N4: nat] : ( minus_minus @ nat @ N4 @ ( suc @ ( zero_zero @ nat ) ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M ) @ ( suc @ N ) ) )
= ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ).
% prod.atLeast_lessThan_pred_shift
thf(fact_4076_and__num__eq__None__iff,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [M: num,N: num] :
( ( ( bit_un7362597486090784418nd_num @ M @ N )
= ( none @ num ) )
= ( ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N ) )
= ( zero_zero @ A ) ) ) ) ).
% and_num_eq_None_iff
thf(fact_4077_sum_OatLeast__int__atMost__int__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: int > A,M: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ int @ A @ G @ ( set_or1337092689740270186AtMost @ int @ ( semiring_1_of_nat @ int @ M ) @ ( semiring_1_of_nat @ int @ N ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ int @ A @ nat @ G @ ( semiring_1_of_nat @ int ) ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% sum.atLeast_int_atMost_int_shift
thf(fact_4078_prod_OatLeast__int__atMost__int__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: int > A,M: nat,N: nat] :
( ( groups7121269368397514597t_prod @ int @ A @ G @ ( set_or1337092689740270186AtMost @ int @ ( semiring_1_of_nat @ int @ M ) @ ( semiring_1_of_nat @ int @ N ) ) )
= ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ int @ A @ nat @ G @ ( semiring_1_of_nat @ int ) ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% prod.atLeast_int_atMost_int_shift
thf(fact_4079_sum_OatLeast__int__lessThan__int__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: int > A,M: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ int @ A @ G @ ( set_or7035219750837199246ssThan @ int @ ( semiring_1_of_nat @ int @ M ) @ ( semiring_1_of_nat @ int @ N ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ int @ A @ nat @ G @ ( semiring_1_of_nat @ int ) ) @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ).
% sum.atLeast_int_lessThan_int_shift
thf(fact_4080_sum_OatLeastAtMost__shift__0,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M: nat,N: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ M ) ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N @ M ) ) ) ) ) ) ).
% sum.atLeastAtMost_shift_0
thf(fact_4081_prod_OatLeastAtMost__shift__0,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M: nat,N: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ M ) ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N @ M ) ) ) ) ) ) ).
% prod.atLeastAtMost_shift_0
thf(fact_4082_prod_OatLeast__int__lessThan__int__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: int > A,M: nat,N: nat] :
( ( groups7121269368397514597t_prod @ int @ A @ G @ ( set_or7035219750837199246ssThan @ int @ ( semiring_1_of_nat @ int @ M ) @ ( semiring_1_of_nat @ int @ N ) ) )
= ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ int @ A @ nat @ G @ ( semiring_1_of_nat @ int ) ) @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ).
% prod.atLeast_int_lessThan_int_shift
thf(fact_4083_numeral__and__num,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [M: num,N: num] :
( ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N ) )
= ( case_option @ A @ num @ ( zero_zero @ A ) @ ( numeral_numeral @ A ) @ ( bit_un7362597486090784418nd_num @ M @ N ) ) ) ) ).
% numeral_and_num
thf(fact_4084_and__num_Osimps_I9_J,axiom,
! [M: num,N: num] :
( ( bit_un7362597486090784418nd_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
= ( case_option @ ( option @ num ) @ num @ ( some @ num @ one2 )
@ ^ [N11: num] : ( some @ num @ ( bit1 @ N11 ) )
@ ( bit_un7362597486090784418nd_num @ M @ N ) ) ) ).
% and_num.simps(9)
thf(fact_4085_make_H__def,axiom,
! [B: $tType,A: $tType] :
( ( ( heap @ A )
& ( semiring_1 @ B ) )
=> ( ( array_make2 @ B @ A )
= ( ^ [I2: code_integer,F2: B > A] : ( array_make @ A @ ( code_nat_of_integer @ I2 ) @ ( comp @ B @ A @ nat @ F2 @ ( semiring_1_of_nat @ B ) ) ) ) ) ) ).
% make'_def
thf(fact_4086_Code__Numeral_Onegative__def,axiom,
( code_negative
= ( comp @ code_integer @ code_integer @ num @ ( uminus_uminus @ code_integer ) @ ( numeral_numeral @ code_integer ) ) ) ).
% Code_Numeral.negative_def
thf(fact_4087_empty__natural,axiom,
! [C: $tType,B: $tType,D: $tType,A: $tType,F: A > C,G: D > B] :
( ( comp @ C @ ( set @ B ) @ A
@ ^ [Uu: C] : ( bot_bot @ ( set @ B ) )
@ F )
= ( comp @ ( set @ D ) @ ( set @ B ) @ A @ ( image2 @ D @ B @ G )
@ ^ [Uu: A] : ( bot_bot @ ( set @ D ) ) ) ) ).
% empty_natural
thf(fact_4088_measure__function__int,axiom,
fun_is_measure @ int @ ( comp @ int @ nat @ int @ nat2 @ ( abs_abs @ int ) ) ).
% measure_function_int
thf(fact_4089_conj__comp__iff,axiom,
! [B: $tType,A: $tType,P: B > $o,Q: B > $o,G: A > B] :
( ( comp @ B @ $o @ A
@ ^ [X: B] :
( ( P @ X )
& ( Q @ X ) )
@ G )
= ( ^ [X: A] :
( ( comp @ B @ $o @ A @ P @ G @ X )
& ( comp @ B @ $o @ A @ Q @ G @ X ) ) ) ) ).
% conj_comp_iff
thf(fact_4090_measure__size,axiom,
! [A: $tType] :
( ( size @ A )
=> ( fun_is_measure @ A @ ( size_size @ A ) ) ) ).
% measure_size
thf(fact_4091_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
@ ^ [X: B] : ( zero_neq_one_of_bool @ A @ ( P @ X ) )
@ A5 )
= ( semiring_1_of_nat @ A @ ( finite_card @ B @ ( inf_inf @ ( set @ B ) @ A5 @ ( collect @ B @ P ) ) ) ) ) ) ) ) ).
% sum_of_bool_eq
thf(fact_4092_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_4093_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_4094_mset__update,axiom,
! [A: $tType,I: nat,Ls: list @ A,V2: A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Ls ) )
=> ( ( mset @ A @ ( list_update @ A @ Ls @ I @ V2 ) )
= ( add_mset @ A @ V2 @ ( minus_minus @ ( multiset @ A ) @ ( mset @ A @ Ls ) @ ( add_mset @ A @ ( nth @ A @ Ls @ I ) @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ) ).
% mset_update
thf(fact_4095_inf_Obounded__iff,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ ( inf_inf @ A @ B2 @ C2 ) )
= ( ( ord_less_eq @ A @ A2 @ B2 )
& ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% inf.bounded_iff
thf(fact_4096_le__inf__iff,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X2: A,Y: A,Z2: A] :
( ( ord_less_eq @ A @ X2 @ ( inf_inf @ A @ Y @ Z2 ) )
= ( ( ord_less_eq @ A @ X2 @ Y )
& ( ord_less_eq @ A @ X2 @ Z2 ) ) ) ) ).
% le_inf_iff
thf(fact_4097_length__rev,axiom,
! [A: $tType,Xs: list @ A] :
( ( size_size @ ( list @ A ) @ ( rev @ A @ Xs ) )
= ( size_size @ ( list @ A ) @ Xs ) ) ).
% length_rev
thf(fact_4098_take__update,axiom,
! [A: $tType,N: nat,L: list @ A,I: nat,X2: A] :
( ( take @ A @ N @ ( list_update @ A @ L @ I @ X2 ) )
= ( list_update @ A @ ( take @ A @ N @ L ) @ I @ X2 ) ) ).
% take_update
thf(fact_4099_take0,axiom,
! [A: $tType] :
( ( take @ A @ ( zero_zero @ nat ) )
= ( ^ [Xs3: list @ A] : ( nil @ A ) ) ) ).
% take0
thf(fact_4100_take__eq__Nil,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( ( take @ A @ N @ Xs )
= ( nil @ A ) )
= ( ( N
= ( zero_zero @ nat ) )
| ( Xs
= ( nil @ A ) ) ) ) ).
% take_eq_Nil
thf(fact_4101_take__eq__Nil2,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( ( nil @ A )
= ( take @ A @ N @ Xs ) )
= ( ( N
= ( zero_zero @ nat ) )
| ( Xs
= ( nil @ A ) ) ) ) ).
% take_eq_Nil2
thf(fact_4102_take__all,axiom,
! [A: $tType,Xs: list @ A,N: nat] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N )
=> ( ( take @ A @ N @ Xs )
= Xs ) ) ).
% take_all
thf(fact_4103_take__all__iff,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( ( take @ A @ N @ Xs )
= Xs )
= ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N ) ) ).
% take_all_iff
thf(fact_4104_mset__append,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( mset @ A @ ( append @ A @ Xs @ Ys ) )
= ( plus_plus @ ( multiset @ A ) @ ( mset @ A @ Xs ) @ ( mset @ A @ Ys ) ) ) ).
% mset_append
thf(fact_4105_inter__compl__diff__conv,axiom,
! [A: $tType,A5: set @ A,B5: set @ A] :
( ( inf_inf @ ( set @ A ) @ A5 @ ( uminus_uminus @ ( set @ A ) @ B5 ) )
= ( minus_minus @ ( set @ A ) @ A5 @ B5 ) ) ).
% inter_compl_diff_conv
thf(fact_4106_take__update__cancel,axiom,
! [A: $tType,N: nat,M: nat,Xs: list @ A,Y: A] :
( ( ord_less_eq @ nat @ N @ M )
=> ( ( take @ A @ N @ ( list_update @ A @ Xs @ M @ Y ) )
= ( take @ A @ N @ Xs ) ) ) ).
% take_update_cancel
thf(fact_4107_size__mset,axiom,
! [A: $tType,Xs: list @ A] :
( ( size_size @ ( multiset @ A ) @ ( mset @ A @ Xs ) )
= ( size_size @ ( list @ A ) @ Xs ) ) ).
% size_mset
thf(fact_4108_sorted__wrt__rev__linord,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: list @ A] :
( ( sorted_wrt @ A
@ ^ [X: A,Y4: A] : ( ord_less_eq @ A @ Y4 @ X )
@ L )
= ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( rev @ A @ L ) ) ) ) ).
% sorted_wrt_rev_linord
thf(fact_4109_Multiset_Omset__insort,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Xs: list @ A] :
( ( mset @ A
@ ( linorder_insort_key @ A @ A
@ ^ [X: A] : X
@ X2
@ Xs ) )
= ( add_mset @ A @ X2 @ ( mset @ A @ Xs ) ) ) ) ).
% Multiset.mset_insort
thf(fact_4110_take__append,axiom,
! [A: $tType,N: nat,Xs: list @ A,Ys: list @ A] :
( ( take @ A @ N @ ( append @ A @ Xs @ Ys ) )
= ( append @ A @ ( take @ A @ N @ Xs ) @ ( take @ A @ ( minus_minus @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) ) @ Ys ) ) ) ).
% take_append
thf(fact_4111_take__Cons__numeral,axiom,
! [A: $tType,V2: num,X2: A,Xs: list @ A] :
( ( take @ A @ ( numeral_numeral @ nat @ V2 ) @ ( cons @ A @ X2 @ Xs ) )
= ( cons @ A @ X2 @ ( take @ A @ ( minus_minus @ nat @ ( numeral_numeral @ nat @ V2 ) @ ( one_one @ nat ) ) @ Xs ) ) ) ).
% take_Cons_numeral
thf(fact_4112_sum__mult__of__bool__eq,axiom,
! [A: $tType,B: $tType] :
( ( semiring_1 @ A )
=> ! [A5: set @ B,F: B > A,P: B > $o] :
( ( finite_finite2 @ B @ A5 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X: B] : ( times_times @ A @ ( F @ X ) @ ( zero_neq_one_of_bool @ A @ ( P @ X ) ) )
@ A5 )
= ( groups7311177749621191930dd_sum @ B @ A @ F @ ( inf_inf @ ( set @ B ) @ A5 @ ( collect @ B @ P ) ) ) ) ) ) ).
% sum_mult_of_bool_eq
thf(fact_4113_sum__of__bool__mult__eq,axiom,
! [A: $tType,B: $tType] :
( ( semiring_1 @ A )
=> ! [A5: set @ B,P: B > $o,F: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X: B] : ( times_times @ A @ ( zero_neq_one_of_bool @ A @ ( P @ X ) ) @ ( F @ X ) )
@ A5 )
= ( groups7311177749621191930dd_sum @ B @ A @ F @ ( inf_inf @ ( set @ B ) @ A5 @ ( collect @ B @ P ) ) ) ) ) ) ).
% sum_of_bool_mult_eq
thf(fact_4114_disjointI,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ! [X3: A] :
( ( member @ A @ X3 @ A2 )
=> ~ ( member @ A @ X3 @ B2 ) )
=> ( ( inf_inf @ ( set @ A ) @ A2 @ B2 )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% disjointI
thf(fact_4115_mset__eq__length,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( mset @ A @ Xs )
= ( mset @ A @ Ys ) )
=> ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ A ) @ Ys ) ) ) ).
% mset_eq_length
thf(fact_4116_inf_OcoboundedI2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [B2: A,C2: A,A2: A] :
( ( ord_less_eq @ A @ B2 @ C2 )
=> ( ord_less_eq @ A @ ( inf_inf @ A @ A2 @ B2 ) @ C2 ) ) ) ).
% inf.coboundedI2
thf(fact_4117_inf_OcoboundedI1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ C2 )
=> ( ord_less_eq @ A @ ( inf_inf @ A @ A2 @ B2 ) @ C2 ) ) ) ).
% inf.coboundedI1
thf(fact_4118_inf_Oabsorb__iff2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B3: A,A3: A] :
( ( inf_inf @ A @ A3 @ B3 )
= B3 ) ) ) ) ).
% inf.absorb_iff2
thf(fact_4119_inf_Oabsorb__iff1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A3: A,B3: A] :
( ( inf_inf @ A @ A3 @ B3 )
= A3 ) ) ) ) ).
% inf.absorb_iff1
thf(fact_4120_inf_Ocobounded2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A2: A,B2: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ A2 @ B2 ) @ B2 ) ) ).
% inf.cobounded2
thf(fact_4121_inf_Ocobounded1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A2: A,B2: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ A2 @ B2 ) @ A2 ) ) ).
% inf.cobounded1
thf(fact_4122_inf_Oorder__iff,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A3: A,B3: A] :
( A3
= ( inf_inf @ A @ A3 @ B3 ) ) ) ) ) ).
% inf.order_iff
thf(fact_4123_inf__greatest,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X2: A,Y: A,Z2: A] :
( ( ord_less_eq @ A @ X2 @ Y )
=> ( ( ord_less_eq @ A @ X2 @ Z2 )
=> ( ord_less_eq @ A @ X2 @ ( inf_inf @ A @ Y @ Z2 ) ) ) ) ) ).
% inf_greatest
thf(fact_4124_inf_OboundedI,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ A2 @ C2 )
=> ( ord_less_eq @ A @ A2 @ ( inf_inf @ A @ B2 @ C2 ) ) ) ) ) ).
% inf.boundedI
thf(fact_4125_inf_OboundedE,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ ( inf_inf @ A @ B2 @ C2 ) )
=> ~ ( ( ord_less_eq @ A @ A2 @ B2 )
=> ~ ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% inf.boundedE
thf(fact_4126_inf__absorb2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [Y: A,X2: A] :
( ( ord_less_eq @ A @ Y @ X2 )
=> ( ( inf_inf @ A @ X2 @ Y )
= Y ) ) ) ).
% inf_absorb2
thf(fact_4127_inf__absorb1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X2: A,Y: A] :
( ( ord_less_eq @ A @ X2 @ Y )
=> ( ( inf_inf @ A @ X2 @ Y )
= X2 ) ) ) ).
% inf_absorb1
thf(fact_4128_inf_Oabsorb2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( inf_inf @ A @ A2 @ B2 )
= B2 ) ) ) ).
% inf.absorb2
thf(fact_4129_inf_Oabsorb1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( inf_inf @ A @ A2 @ B2 )
= A2 ) ) ) ).
% inf.absorb1
thf(fact_4130_le__iff__inf,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [X: A,Y4: A] :
( ( inf_inf @ A @ X @ Y4 )
= X ) ) ) ) ).
% le_iff_inf
thf(fact_4131_inf__unique,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [F: A > A > A,X2: A,Y: A] :
( ! [X3: A,Y2: A] : ( ord_less_eq @ A @ ( F @ X3 @ Y2 ) @ X3 )
=> ( ! [X3: A,Y2: A] : ( ord_less_eq @ A @ ( F @ X3 @ Y2 ) @ Y2 )
=> ( ! [X3: A,Y2: A,Z4: A] :
( ( ord_less_eq @ A @ X3 @ Y2 )
=> ( ( ord_less_eq @ A @ X3 @ Z4 )
=> ( ord_less_eq @ A @ X3 @ ( F @ Y2 @ Z4 ) ) ) )
=> ( ( inf_inf @ A @ X2 @ Y )
= ( F @ X2 @ Y ) ) ) ) ) ) ).
% inf_unique
thf(fact_4132_inf_OorderI,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A2: A,B2: A] :
( ( A2
= ( inf_inf @ A @ A2 @ B2 ) )
=> ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% inf.orderI
thf(fact_4133_inf_OorderE,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( A2
= ( inf_inf @ A @ A2 @ B2 ) ) ) ) ).
% inf.orderE
thf(fact_4134_le__infI2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [B2: A,X2: A,A2: A] :
( ( ord_less_eq @ A @ B2 @ X2 )
=> ( ord_less_eq @ A @ ( inf_inf @ A @ A2 @ B2 ) @ X2 ) ) ) ).
% le_infI2
thf(fact_4135_le__infI1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A2: A,X2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ X2 )
=> ( ord_less_eq @ A @ ( inf_inf @ A @ A2 @ B2 ) @ X2 ) ) ) ).
% le_infI1
thf(fact_4136_inf__mono,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A2: A,C2: A,B2: A,D2: A] :
( ( ord_less_eq @ A @ A2 @ C2 )
=> ( ( ord_less_eq @ A @ B2 @ D2 )
=> ( ord_less_eq @ A @ ( inf_inf @ A @ A2 @ B2 ) @ ( inf_inf @ A @ C2 @ D2 ) ) ) ) ) ).
% inf_mono
thf(fact_4137_le__infI,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X2: A,A2: A,B2: A] :
( ( ord_less_eq @ A @ X2 @ A2 )
=> ( ( ord_less_eq @ A @ X2 @ B2 )
=> ( ord_less_eq @ A @ X2 @ ( inf_inf @ A @ A2 @ B2 ) ) ) ) ) ).
% le_infI
thf(fact_4138_le__infE,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X2: A,A2: A,B2: A] :
( ( ord_less_eq @ A @ X2 @ ( inf_inf @ A @ A2 @ B2 ) )
=> ~ ( ( ord_less_eq @ A @ X2 @ A2 )
=> ~ ( ord_less_eq @ A @ X2 @ B2 ) ) ) ) ).
% le_infE
thf(fact_4139_inf__le2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X2: A,Y: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ X2 @ Y ) @ Y ) ) ).
% inf_le2
thf(fact_4140_inf__le1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X2: A,Y: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ X2 @ Y ) @ X2 ) ) ).
% inf_le1
thf(fact_4141_inf__sup__ord_I1_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X2: A,Y: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ X2 @ Y ) @ X2 ) ) ).
% inf_sup_ord(1)
thf(fact_4142_inf__sup__ord_I2_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X2: A,Y: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ X2 @ Y ) @ Y ) ) ).
% inf_sup_ord(2)
thf(fact_4143_inter__eq__subsetI,axiom,
! [A: $tType,S2: set @ A,S7: set @ A,A5: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ S2 @ S7 )
=> ( ( ( inf_inf @ ( set @ A ) @ A5 @ S7 )
= ( inf_inf @ ( set @ A ) @ B5 @ S7 ) )
=> ( ( inf_inf @ ( set @ A ) @ A5 @ S2 )
= ( inf_inf @ ( set @ A ) @ B5 @ S2 ) ) ) ) ).
% inter_eq_subsetI
thf(fact_4144_Int__def,axiom,
! [A: $tType] :
( ( inf_inf @ ( set @ A ) )
= ( ^ [A8: set @ A,B8: set @ A] :
( collect @ A
@ ^ [X: A] :
( ( member @ A @ X @ A8 )
& ( member @ A @ X @ B8 ) ) ) ) ) ).
% Int_def
thf(fact_4145_Int__Collect,axiom,
! [A: $tType,X2: A,A5: set @ A,P: A > $o] :
( ( member @ A @ X2 @ ( inf_inf @ ( set @ A ) @ A5 @ ( collect @ A @ P ) ) )
= ( ( member @ A @ X2 @ A5 )
& ( P @ X2 ) ) ) ).
% Int_Collect
thf(fact_4146_Collect__conj__eq,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( collect @ A
@ ^ [X: A] :
( ( P @ X )
& ( Q @ X ) ) )
= ( inf_inf @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_4147_sorted__wrt__rev,axiom,
! [A: $tType,P: A > A > $o,Xs: list @ A] :
( ( sorted_wrt @ A @ P @ ( rev @ A @ Xs ) )
= ( sorted_wrt @ A
@ ^ [X: A,Y4: A] : ( P @ Y4 @ X )
@ Xs ) ) ).
% sorted_wrt_rev
thf(fact_4148_mset__eq__finite,axiom,
! [A: $tType,Xs: list @ A] :
( finite_finite2 @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Ys3: list @ A] :
( ( mset @ A @ Ys3 )
= ( mset @ A @ Xs ) ) ) ) ).
% mset_eq_finite
thf(fact_4149_inf__option__def,axiom,
! [A: $tType] :
( ( inf @ A )
=> ( ( inf_inf @ ( option @ A ) )
= ( ^ [X: option @ A,Y4: option @ A] :
( case_option @ ( option @ A ) @ A @ ( none @ A )
@ ^ [Z3: A] :
( case_option @ ( option @ A ) @ A @ ( none @ A )
@ ^ [Aa3: A] : ( some @ A @ ( inf_inf @ A @ Z3 @ Aa3 ) )
@ Y4 )
@ X ) ) ) ) ).
% inf_option_def
thf(fact_4150_take__0,axiom,
! [A: $tType,Xs: list @ A] :
( ( take @ A @ ( zero_zero @ nat ) @ Xs )
= ( nil @ A ) ) ).
% take_0
thf(fact_4151_sorted__take,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,N: nat] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( take @ A @ N @ Xs ) ) ) ) ).
% sorted_take
thf(fact_4152_translation__Int,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A2: A,S3: set @ A,T2: set @ A] :
( ( image2 @ A @ A @ ( plus_plus @ A @ A2 ) @ ( inf_inf @ ( set @ A ) @ S3 @ T2 ) )
= ( inf_inf @ ( set @ A ) @ ( image2 @ A @ A @ ( plus_plus @ A @ A2 ) @ S3 ) @ ( image2 @ A @ A @ ( plus_plus @ A @ A2 ) @ T2 ) ) ) ) ).
% translation_Int
thf(fact_4153_disjoint__mono,axiom,
! [A: $tType,A2: set @ A,A6: set @ A,B2: set @ A,B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ A6 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ B6 )
=> ( ( ( inf_inf @ ( set @ A ) @ A6 @ B6 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( inf_inf @ ( set @ A ) @ A2 @ B2 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% disjoint_mono
thf(fact_4154_disjoint__alt__simp1,axiom,
! [A: $tType,A5: set @ A,B5: set @ A] :
( ( ( minus_minus @ ( set @ A ) @ A5 @ B5 )
= A5 )
= ( ( inf_inf @ ( set @ A ) @ A5 @ B5 )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% disjoint_alt_simp1
thf(fact_4155_disjoint__alt__simp2,axiom,
! [A: $tType,A5: set @ A,B5: set @ A] :
( ( ( minus_minus @ ( set @ A ) @ A5 @ B5 )
!= A5 )
= ( ( inf_inf @ ( set @ A ) @ A5 @ B5 )
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% disjoint_alt_simp2
thf(fact_4156_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_4157_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_4158_translation__subtract__Int,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A2: A,S3: set @ A,T2: set @ A] :
( ( image2 @ A @ A
@ ^ [X: A] : ( minus_minus @ A @ X @ A2 )
@ ( inf_inf @ ( set @ A ) @ S3 @ T2 ) )
= ( inf_inf @ ( set @ A )
@ ( image2 @ A @ A
@ ^ [X: A] : ( minus_minus @ A @ X @ A2 )
@ S3 )
@ ( image2 @ A @ A
@ ^ [X: A] : ( minus_minus @ A @ X @ A2 )
@ T2 ) ) ) ) ).
% translation_subtract_Int
thf(fact_4159_foldr__conv__foldl,axiom,
! [A: $tType,B: $tType] :
( ( foldr @ B @ A )
= ( ^ [F2: B > A > A,Xs3: list @ B,A3: A] :
( foldl @ A @ B
@ ^ [X: A,Y4: B] : ( F2 @ Y4 @ X )
@ A3
@ ( rev @ B @ Xs3 ) ) ) ) ).
% foldr_conv_foldl
thf(fact_4160_foldl__conv__foldr,axiom,
! [B: $tType,A: $tType] :
( ( foldl @ A @ B )
= ( ^ [F2: A > B > A,A3: A,Xs3: list @ B] :
( foldr @ B @ A
@ ^ [X: B,Y4: A] : ( F2 @ Y4 @ X )
@ ( rev @ B @ Xs3 )
@ A3 ) ) ) ).
% foldl_conv_foldr
thf(fact_4161_set__take__subset__set__take,axiom,
! [A: $tType,M: nat,N: nat,Xs: list @ A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( take @ A @ M @ Xs ) ) @ ( set2 @ A @ ( take @ A @ N @ Xs ) ) ) ) ).
% set_take_subset_set_take
thf(fact_4162_inf__shunt,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X2: A,Y: A] :
( ( ( inf_inf @ A @ X2 @ Y )
= ( bot_bot @ A ) )
= ( ord_less_eq @ A @ X2 @ ( uminus_uminus @ A @ Y ) ) ) ) ).
% inf_shunt
thf(fact_4163_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_4164_Ioc__disjoint,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( ( inf_inf @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ A2 @ B2 ) @ ( set_or3652927894154168847AtMost @ A @ C2 @ D2 ) )
= ( bot_bot @ ( set @ A ) ) )
= ( ( ord_less_eq @ A @ B2 @ A2 )
| ( ord_less_eq @ A @ D2 @ C2 )
| ( ord_less_eq @ A @ B2 @ C2 )
| ( ord_less_eq @ A @ D2 @ A2 ) ) ) ) ).
% Ioc_disjoint
thf(fact_4165_disjoint__alt__simp3,axiom,
! [A: $tType,A5: set @ A,B5: set @ A] :
( ( ord_less @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A5 @ B5 ) @ A5 )
= ( ( inf_inf @ ( set @ A ) @ A5 @ B5 )
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% disjoint_alt_simp3
thf(fact_4166_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_4167_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_4168_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_4169_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_4170_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_4171_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_4172_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_4173_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_4174_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_4175_sum_Ointer__restrict,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A5: set @ B,G: B > A,B5: set @ B] :
( ( finite_finite2 @ B @ A5 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( inf_inf @ ( set @ B ) @ A5 @ B5 ) )
= ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X: B] : ( if @ A @ ( member @ B @ X @ B5 ) @ ( G @ X ) @ ( zero_zero @ A ) )
@ A5 ) ) ) ) ).
% sum.inter_restrict
thf(fact_4176_prod_Ointer__restrict,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A5: set @ B,G: B > A,B5: set @ B] :
( ( finite_finite2 @ B @ A5 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( inf_inf @ ( set @ B ) @ A5 @ B5 ) )
= ( groups7121269368397514597t_prod @ B @ A
@ ^ [X: B] : ( if @ A @ ( member @ B @ X @ B5 ) @ ( G @ X ) @ ( one_one @ A ) )
@ A5 ) ) ) ) ).
% prod.inter_restrict
thf(fact_4177_nth__take__lemma,axiom,
! [A: $tType,K: nat,Xs: list @ A,Ys: list @ A] :
( ( ord_less_eq @ nat @ K @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ord_less_eq @ nat @ K @ ( size_size @ ( list @ A ) @ Ys ) )
=> ( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ K )
=> ( ( nth @ A @ Xs @ I3 )
= ( nth @ A @ Ys @ I3 ) ) )
=> ( ( take @ A @ K @ Xs )
= ( take @ A @ K @ Ys ) ) ) ) ) ).
% nth_take_lemma
thf(fact_4178_mset__swap,axiom,
! [A: $tType,I: nat,Ls: list @ A,J: nat] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Ls ) )
=> ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Ls ) )
=> ( ( mset @ A @ ( list_update @ A @ ( list_update @ A @ Ls @ J @ ( nth @ A @ Ls @ I ) ) @ I @ ( nth @ A @ Ls @ J ) ) )
= ( mset @ A @ Ls ) ) ) ) ).
% mset_swap
thf(fact_4179_take__Cons,axiom,
! [A: $tType,N: nat,X2: A,Xs: list @ A] :
( ( take @ A @ N @ ( cons @ A @ X2 @ Xs ) )
= ( case_nat @ ( list @ A ) @ ( nil @ A )
@ ^ [M2: nat] : ( cons @ A @ X2 @ ( take @ A @ M2 @ Xs ) )
@ N ) ) ).
% take_Cons
thf(fact_4180_rotate__rev,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( rotate @ A @ N @ ( rev @ A @ Xs ) )
= ( rev @ A @ ( rotate @ A @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( modulo_modulo @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) ) ) @ Xs ) ) ) ).
% rotate_rev
thf(fact_4181_sum_Omono__neutral__cong,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [T6: set @ B,S2: set @ B,H: B > A,G: B > A] :
( ( finite_finite2 @ B @ T6 )
=> ( ( finite_finite2 @ B @ S2 )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ ( minus_minus @ ( set @ B ) @ T6 @ S2 ) )
=> ( ( H @ I3 )
= ( zero_zero @ A ) ) )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ ( minus_minus @ ( set @ B ) @ S2 @ T6 ) )
=> ( ( G @ I3 )
= ( zero_zero @ A ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( inf_inf @ ( set @ B ) @ S2 @ T6 ) )
=> ( ( G @ X3 )
= ( H @ X3 ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ S2 )
= ( groups7311177749621191930dd_sum @ B @ A @ H @ T6 ) ) ) ) ) ) ) ) ).
% sum.mono_neutral_cong
thf(fact_4182_sum_OInt__Diff,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A5: set @ B,G: B > A,B5: set @ B] :
( ( finite_finite2 @ B @ A5 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ A5 )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( inf_inf @ ( set @ B ) @ A5 @ B5 ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A5 @ B5 ) ) ) ) ) ) ).
% sum.Int_Diff
thf(fact_4183_prod_Omono__neutral__cong,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [T6: set @ B,S2: set @ B,H: B > A,G: B > A] :
( ( finite_finite2 @ B @ T6 )
=> ( ( finite_finite2 @ B @ S2 )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ ( minus_minus @ ( set @ B ) @ T6 @ S2 ) )
=> ( ( H @ I3 )
= ( one_one @ A ) ) )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ ( minus_minus @ ( set @ B ) @ S2 @ T6 ) )
=> ( ( G @ I3 )
= ( one_one @ A ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( inf_inf @ ( set @ B ) @ S2 @ T6 ) )
=> ( ( G @ X3 )
= ( H @ X3 ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ S2 )
= ( groups7121269368397514597t_prod @ B @ A @ H @ T6 ) ) ) ) ) ) ) ) ).
% prod.mono_neutral_cong
thf(fact_4184_Iio__Int__singleton,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X2: A,K: A] :
( ( ( ord_less @ A @ X2 @ K )
=> ( ( inf_inf @ ( set @ A ) @ ( set_ord_lessThan @ A @ K ) @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) )
& ( ~ ( ord_less @ A @ X2 @ K )
=> ( ( inf_inf @ ( set @ A ) @ ( set_ord_lessThan @ A @ K ) @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% Iio_Int_singleton
thf(fact_4185_sum_OIf__cases,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A5: set @ B,P: B > $o,H: B > A,G: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X: B] : ( if @ A @ ( P @ X ) @ ( H @ X ) @ ( G @ X ) )
@ A5 )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ H @ ( inf_inf @ ( set @ B ) @ A5 @ ( collect @ B @ P ) ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( inf_inf @ ( set @ B ) @ A5 @ ( uminus_uminus @ ( set @ B ) @ ( collect @ B @ P ) ) ) ) ) ) ) ) ).
% sum.If_cases
thf(fact_4186_prod_OIf__cases,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A5: set @ B,P: B > $o,H: B > A,G: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [X: B] : ( if @ A @ ( P @ X ) @ ( H @ X ) @ ( G @ X ) )
@ A5 )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ H @ ( inf_inf @ ( set @ B ) @ A5 @ ( collect @ B @ P ) ) ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( inf_inf @ ( set @ B ) @ A5 @ ( uminus_uminus @ ( set @ B ) @ ( collect @ B @ P ) ) ) ) ) ) ) ) ).
% prod.If_cases
thf(fact_4187_take__Cons_H,axiom,
! [A: $tType,N: nat,X2: A,Xs: list @ A] :
( ( ( N
= ( zero_zero @ nat ) )
=> ( ( take @ A @ N @ ( cons @ A @ X2 @ Xs ) )
= ( nil @ A ) ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ( ( take @ A @ N @ ( cons @ A @ X2 @ Xs ) )
= ( cons @ A @ X2 @ ( take @ A @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ Xs ) ) ) ) ) ).
% take_Cons'
thf(fact_4188_rev__nth,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ ( rev @ A @ Xs ) @ N )
= ( nth @ A @ Xs @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( suc @ N ) ) ) ) ) ).
% rev_nth
thf(fact_4189_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_4190_sum__div__partition,axiom,
! [B: $tType,A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A5: set @ B,F: B > A,B2: A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( divide_divide @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F @ A5 ) @ B2 )
= ( plus_plus @ A
@ ( groups7311177749621191930dd_sum @ B @ A
@ ^ [A3: B] : ( divide_divide @ A @ ( F @ A3 ) @ B2 )
@ ( inf_inf @ ( set @ B ) @ A5
@ ( collect @ B
@ ^ [A3: B] : ( dvd_dvd @ A @ B2 @ ( F @ A3 ) ) ) ) )
@ ( divide_divide @ A
@ ( groups7311177749621191930dd_sum @ B @ A @ F
@ ( inf_inf @ ( set @ B ) @ A5
@ ( collect @ B
@ ^ [A3: B] :
~ ( dvd_dvd @ A @ B2 @ ( F @ A3 ) ) ) ) )
@ B2 ) ) ) ) ) ).
% sum_div_partition
thf(fact_4191_sorted__rev__iff__nth__Suc,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( rev @ A @ Xs ) )
= ( ! [I2: nat] :
( ( ord_less @ nat @ ( suc @ I2 ) @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ord_less_eq @ A @ ( nth @ A @ Xs @ ( suc @ I2 ) ) @ ( nth @ A @ Xs @ I2 ) ) ) ) ) ) ).
% sorted_rev_iff_nth_Suc
thf(fact_4192_sorted__rev__iff__nth__mono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( rev @ A @ Xs ) )
= ( ! [I2: nat,J3: nat] :
( ( ord_less_eq @ nat @ I2 @ J3 )
=> ( ( ord_less @ nat @ J3 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ord_less_eq @ A @ ( nth @ A @ Xs @ J3 ) @ ( nth @ A @ Xs @ I2 ) ) ) ) ) ) ) ).
% sorted_rev_iff_nth_mono
thf(fact_4193_sorted__rev__nth__mono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,I: nat,J: nat] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( rev @ A @ Xs ) )
=> ( ( ord_less_eq @ nat @ I @ J )
=> ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ord_less_eq @ A @ ( nth @ A @ Xs @ J ) @ ( nth @ A @ Xs @ I ) ) ) ) ) ) ).
% sorted_rev_nth_mono
thf(fact_4194_take__Suc__conv__app__nth,axiom,
! [A: $tType,I: nat,Xs: list @ A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( take @ A @ ( suc @ I ) @ Xs )
= ( append @ A @ ( take @ A @ I @ Xs ) @ ( cons @ A @ ( nth @ A @ Xs @ I ) @ ( nil @ A ) ) ) ) ) ).
% take_Suc_conv_app_nth
thf(fact_4195_take__update__last,axiom,
! [A: $tType,N: nat,List: list @ A,X2: A] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ List ) )
=> ( ( list_update @ A @ ( take @ A @ ( suc @ N ) @ List ) @ N @ X2 )
= ( append @ A @ ( take @ A @ N @ List ) @ ( cons @ A @ X2 @ ( nil @ A ) ) ) ) ) ).
% take_update_last
thf(fact_4196_butlast__upd__last__eq,axiom,
! [A: $tType,L: list @ A,X2: 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 ) ) ) @ X2 )
= ( append @ A @ ( take @ A @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ L ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ L ) @ ( cons @ A @ X2 @ ( nil @ A ) ) ) ) ) ).
% butlast_upd_last_eq
thf(fact_4197_card__disjoint__shuffles,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_card @ ( list @ A ) @ ( shuffles @ A @ Xs @ Ys ) )
= ( binomial @ ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ A ) @ Ys ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ).
% card_disjoint_shuffles
thf(fact_4198_last__take__nth__conv,axiom,
! [A: $tType,N: nat,L: list @ A] :
( ( ord_less_eq @ nat @ N @ ( size_size @ ( list @ A ) @ L ) )
=> ( ( N
!= ( zero_zero @ nat ) )
=> ( ( last @ A @ ( take @ A @ N @ L ) )
= ( nth @ A @ L @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ) ).
% last_take_nth_conv
thf(fact_4199_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_4200_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_4201_length__butlast,axiom,
! [A: $tType,Xs: list @ A] :
( ( size_size @ ( list @ A ) @ ( butlast @ A @ Xs ) )
= ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) ) ) ).
% length_butlast
thf(fact_4202_inf__set__def,axiom,
! [A: $tType] :
( ( inf_inf @ ( set @ A ) )
= ( ^ [A8: set @ A,B8: set @ A] :
( collect @ A
@ ( inf_inf @ ( A > $o )
@ ^ [X: A] : ( member @ A @ X @ A8 )
@ ^ [X: A] : ( member @ A @ X @ B8 ) ) ) ) ) ).
% inf_set_def
thf(fact_4203_butlast__update_H,axiom,
! [A: $tType,L: list @ A,I: nat,X2: A] :
( ( list_update @ A @ ( butlast @ A @ L ) @ I @ X2 )
= ( butlast @ A @ ( list_update @ A @ L @ I @ X2 ) ) ) ).
% butlast_update'
thf(fact_4204_distinct__butlast__swap,axiom,
! [A: $tType,Pq: list @ A,I: nat] :
( ( distinct @ A @ Pq )
=> ( distinct @ A @ ( butlast @ A @ ( list_update @ A @ Pq @ I @ ( last @ A @ Pq ) ) ) ) ) ).
% distinct_butlast_swap
thf(fact_4205_snoc__eq__iff__butlast_H,axiom,
! [A: $tType,Ys: list @ A,Xs: list @ A,X2: A] :
( ( Ys
= ( append @ A @ Xs @ ( cons @ A @ X2 @ ( nil @ A ) ) ) )
= ( ( Ys
!= ( nil @ A ) )
& ( ( butlast @ A @ Ys )
= Xs )
& ( ( last @ A @ Ys )
= X2 ) ) ) ).
% snoc_eq_iff_butlast'
thf(fact_4206_length__shuffles,axiom,
! [A: $tType,Zs: list @ A,Xs: list @ A,Ys: list @ A] :
( ( member @ ( list @ A ) @ Zs @ ( shuffles @ A @ Xs @ Ys ) )
=> ( ( size_size @ ( list @ A ) @ Zs )
= ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ A ) @ Ys ) ) ) ) ).
% length_shuffles
thf(fact_4207_mset__shuffles,axiom,
! [A: $tType,Zs: list @ A,Xs: list @ A,Ys: list @ A] :
( ( member @ ( list @ A ) @ Zs @ ( shuffles @ A @ Xs @ Ys ) )
=> ( ( mset @ A @ Zs )
= ( plus_plus @ ( multiset @ A ) @ ( mset @ A @ Xs ) @ ( mset @ A @ Ys ) ) ) ) ).
% mset_shuffles
thf(fact_4208_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_4209_butlast__eq__consE,axiom,
! [A: $tType,L: list @ A,X2: A,Xs: list @ A] :
( ( ( butlast @ A @ L )
= ( cons @ A @ X2 @ Xs ) )
=> ~ ! [Xl: A] :
( L
!= ( cons @ A @ X2 @ ( append @ A @ Xs @ ( cons @ A @ Xl @ ( nil @ A ) ) ) ) ) ) ).
% butlast_eq_consE
thf(fact_4210_butlast__eq__cons__conv,axiom,
! [A: $tType,L: list @ A,X2: A,Xs: list @ A] :
( ( ( butlast @ A @ L )
= ( cons @ A @ X2 @ Xs ) )
= ( ? [Xl2: A] :
( L
= ( cons @ A @ X2 @ ( append @ A @ Xs @ ( cons @ A @ Xl2 @ ( nil @ A ) ) ) ) ) ) ) ).
% butlast_eq_cons_conv
thf(fact_4211_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_4212_nth__butlast,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ ( butlast @ A @ Xs ) ) )
=> ( ( nth @ A @ ( butlast @ A @ Xs ) @ N )
= ( nth @ A @ Xs @ N ) ) ) ).
% nth_butlast
thf(fact_4213_take__butlast,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( take @ A @ N @ ( butlast @ A @ Xs ) )
= ( take @ A @ N @ Xs ) ) ) ).
% take_butlast
thf(fact_4214_butlast__power,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( compow @ ( ( list @ A ) > ( list @ A ) ) @ N @ ( butlast @ A ) @ Xs )
= ( take @ A @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N ) @ Xs ) ) ).
% butlast_power
thf(fact_4215_butlast__conv__take,axiom,
! [A: $tType] :
( ( butlast @ A )
= ( ^ [Xs3: list @ A] : ( take @ A @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs3 ) @ ( one_one @ nat ) ) @ Xs3 ) ) ) ).
% butlast_conv_take
thf(fact_4216_butlast__list__update,axiom,
! [A: $tType,K: nat,Xs: list @ A,X2: A] :
( ( ( K
= ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) ) )
=> ( ( butlast @ A @ ( list_update @ A @ Xs @ K @ X2 ) )
= ( butlast @ A @ Xs ) ) )
& ( ( K
!= ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) ) )
=> ( ( butlast @ A @ ( list_update @ A @ Xs @ K @ X2 ) )
= ( list_update @ A @ ( butlast @ A @ Xs ) @ K @ X2 ) ) ) ) ).
% butlast_list_update
thf(fact_4217_last__conv__nth,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( last @ A @ Xs )
= ( nth @ A @ Xs @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) ) ) ) ) ).
% last_conv_nth
thf(fact_4218_last__list__update,axiom,
! [A: $tType,Xs: list @ A,K: nat,X2: A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( ( K
= ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) ) )
=> ( ( last @ A @ ( list_update @ A @ Xs @ K @ X2 ) )
= X2 ) )
& ( ( K
!= ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) ) )
=> ( ( last @ A @ ( list_update @ A @ Xs @ K @ X2 ) )
= ( last @ A @ Xs ) ) ) ) ) ).
% last_list_update
thf(fact_4219_butlast__take,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( ord_less_eq @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( butlast @ A @ ( take @ A @ N @ Xs ) )
= ( take @ A @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ Xs ) ) ) ).
% butlast_take
thf(fact_4220_take__minus__one__conv__butlast,axiom,
! [A: $tType,N: nat,L: list @ A] :
( ( ord_less_eq @ nat @ N @ ( size_size @ ( list @ A ) @ L ) )
=> ( ( take @ A @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) @ L )
= ( butlast @ A @ ( take @ A @ N @ L ) ) ) ) ).
% take_minus_one_conv_butlast
thf(fact_4221_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_4222_upd__conv__take__nth__drop,axiom,
! [A: $tType,I: nat,Xs: list @ A,A2: A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( list_update @ A @ Xs @ I @ A2 )
= ( append @ A @ ( take @ A @ I @ Xs ) @ ( cons @ A @ A2 @ ( drop @ A @ ( suc @ I ) @ Xs ) ) ) ) ) ).
% upd_conv_take_nth_drop
thf(fact_4223_nat__of__integer__code,axiom,
( code_nat_of_integer
= ( ^ [K2: code_integer] :
( if @ nat @ ( ord_less_eq @ code_integer @ K2 @ ( 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 @ K2 @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% nat_of_integer_code
thf(fact_4224_revg__fun,axiom,
! [A: $tType] :
( ( revg @ A )
= ( ^ [A3: list @ A] : ( append @ A @ ( rev @ A @ A3 ) ) ) ) ).
% revg_fun
thf(fact_4225_drop0,axiom,
! [A: $tType] :
( ( drop @ A @ ( zero_zero @ nat ) )
= ( ^ [X: list @ A] : X ) ) ).
% drop0
thf(fact_4226_drop__drop,axiom,
! [A: $tType,N: nat,M: nat,Xs: list @ A] :
( ( drop @ A @ N @ ( drop @ A @ M @ Xs ) )
= ( drop @ A @ ( plus_plus @ nat @ N @ M ) @ Xs ) ) ).
% drop_drop
thf(fact_4227_length__drop,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( size_size @ ( list @ A ) @ ( drop @ A @ N @ Xs ) )
= ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N ) ) ).
% length_drop
thf(fact_4228_drop__upd__irrelevant,axiom,
! [A: $tType,M: nat,N: nat,L: list @ A,X2: A] :
( ( ord_less @ nat @ M @ N )
=> ( ( drop @ A @ N @ ( list_update @ A @ L @ M @ X2 ) )
= ( drop @ A @ N @ L ) ) ) ).
% drop_upd_irrelevant
thf(fact_4229_drop__all,axiom,
! [A: $tType,Xs: list @ A,N: nat] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N )
=> ( ( drop @ A @ N @ Xs )
= ( nil @ A ) ) ) ).
% drop_all
thf(fact_4230_drop__eq__Nil,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( ( drop @ A @ N @ Xs )
= ( nil @ A ) )
= ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N ) ) ).
% drop_eq_Nil
thf(fact_4231_drop__eq__Nil2,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( ( nil @ A )
= ( drop @ A @ N @ Xs ) )
= ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N ) ) ).
% drop_eq_Nil2
thf(fact_4232_drop__append,axiom,
! [A: $tType,N: nat,Xs: list @ A,Ys: list @ A] :
( ( drop @ A @ N @ ( append @ A @ Xs @ Ys ) )
= ( append @ A @ ( drop @ A @ N @ Xs ) @ ( drop @ A @ ( minus_minus @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) ) @ Ys ) ) ) ).
% drop_append
thf(fact_4233_last__drop,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( last @ A @ ( drop @ A @ N @ Xs ) )
= ( last @ A @ Xs ) ) ) ).
% last_drop
thf(fact_4234_drop__Cons__numeral,axiom,
! [A: $tType,V2: num,X2: A,Xs: list @ A] :
( ( drop @ A @ ( numeral_numeral @ nat @ V2 ) @ ( cons @ A @ X2 @ Xs ) )
= ( drop @ A @ ( minus_minus @ nat @ ( numeral_numeral @ nat @ V2 ) @ ( one_one @ nat ) ) @ Xs ) ) ).
% drop_Cons_numeral
thf(fact_4235_nth__drop,axiom,
! [A: $tType,N: nat,Xs: list @ A,I: nat] :
( ( ord_less_eq @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ ( drop @ A @ N @ Xs ) @ I )
= ( nth @ A @ Xs @ ( plus_plus @ nat @ N @ I ) ) ) ) ).
% nth_drop
thf(fact_4236_drop__0,axiom,
! [A: $tType,Xs: list @ A] :
( ( drop @ A @ ( zero_zero @ nat ) @ Xs )
= Xs ) ).
% drop_0
thf(fact_4237_case__prod__app,axiom,
! [A: $tType,D: $tType,C: $tType,B: $tType] :
( ( product_case_prod @ B @ C @ ( D > A ) )
= ( ^ [F2: B > C > D > A,X: product_prod @ B @ C,Y4: D] :
( product_case_prod @ B @ C @ A
@ ^ [L2: B,R2: C] : ( F2 @ L2 @ R2 @ Y4 )
@ X ) ) ) ).
% case_prod_app
thf(fact_4238_nested__case__prod__simp,axiom,
! [A: $tType,D: $tType,C: $tType,B: $tType] :
( ( product_case_prod @ B @ C @ ( D > A ) )
= ( ^ [F2: B > C > D > A,X: product_prod @ B @ C,Y4: D] :
( product_case_prod @ B @ C @ A
@ ^ [A3: B,B3: C] : ( F2 @ A3 @ B3 @ Y4 )
@ X ) ) ) ).
% nested_case_prod_simp
thf(fact_4239_drop__eq__ConsD,axiom,
! [A: $tType,N: nat,Xs: list @ A,X2: A,Xs5: list @ A] :
( ( ( drop @ A @ N @ Xs )
= ( cons @ A @ X2 @ Xs5 ) )
=> ( ( drop @ A @ ( suc @ N ) @ Xs )
= Xs5 ) ) ).
% drop_eq_ConsD
thf(fact_4240_sorted__drop,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,N: nat] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( drop @ A @ N @ Xs ) ) ) ) ).
% sorted_drop
thf(fact_4241_take__drop,axiom,
! [A: $tType,N: nat,M: nat,Xs: list @ A] :
( ( take @ A @ N @ ( drop @ A @ M @ Xs ) )
= ( drop @ A @ M @ ( take @ A @ ( plus_plus @ nat @ N @ M ) @ Xs ) ) ) ).
% take_drop
thf(fact_4242_revg_Osimps_I2_J,axiom,
! [A: $tType,A2: A,As: list @ A,B2: list @ A] :
( ( revg @ A @ ( cons @ A @ A2 @ As ) @ B2 )
= ( revg @ A @ As @ ( cons @ A @ A2 @ B2 ) ) ) ).
% revg.simps(2)
thf(fact_4243_revg_Osimps_I1_J,axiom,
! [A: $tType,B2: list @ A] :
( ( revg @ A @ ( nil @ A ) @ B2 )
= B2 ) ).
% revg.simps(1)
thf(fact_4244_set__drop__subset__set__drop,axiom,
! [A: $tType,N: nat,M: nat,Xs: list @ A] :
( ( ord_less_eq @ nat @ N @ M )
=> ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( drop @ A @ M @ Xs ) ) @ ( set2 @ A @ ( drop @ A @ N @ Xs ) ) ) ) ).
% set_drop_subset_set_drop
thf(fact_4245_append__eq__conv__conj,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ( ( append @ A @ Xs @ Ys )
= Zs )
= ( ( Xs
= ( take @ A @ ( size_size @ ( list @ A ) @ Xs ) @ Zs ) )
& ( Ys
= ( drop @ A @ ( size_size @ ( list @ A ) @ Xs ) @ Zs ) ) ) ) ).
% append_eq_conv_conj
thf(fact_4246_drop__take__drop__unsplit,axiom,
! [A: $tType,I: nat,J: nat,L: list @ A] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( append @ A @ ( drop @ A @ I @ ( take @ A @ J @ L ) ) @ ( drop @ A @ J @ L ) )
= ( drop @ A @ I @ L ) ) ) ).
% drop_take_drop_unsplit
thf(fact_4247_take__add,axiom,
! [A: $tType,I: nat,J: nat,Xs: list @ A] :
( ( take @ A @ ( plus_plus @ nat @ I @ J ) @ Xs )
= ( append @ A @ ( take @ A @ I @ Xs ) @ ( take @ A @ J @ ( drop @ A @ I @ Xs ) ) ) ) ).
% take_add
thf(fact_4248_drop__update__swap,axiom,
! [A: $tType,M: nat,N: nat,Xs: list @ A,X2: A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( drop @ A @ M @ ( list_update @ A @ Xs @ N @ X2 ) )
= ( list_update @ A @ ( drop @ A @ M @ Xs ) @ ( minus_minus @ nat @ N @ M ) @ X2 ) ) ) ).
% drop_update_swap
thf(fact_4249_drop__Cons,axiom,
! [A: $tType,N: nat,X2: A,Xs: list @ A] :
( ( drop @ A @ N @ ( cons @ A @ X2 @ Xs ) )
= ( case_nat @ ( list @ A ) @ ( cons @ A @ X2 @ Xs )
@ ^ [M2: nat] : ( drop @ A @ M2 @ Xs )
@ N ) ) ).
% drop_Cons
thf(fact_4250_slice__def,axiom,
! [A: $tType] :
( ( slice @ A )
= ( ^ [From2: nat,To2: nat,List2: list @ A] : ( take @ A @ ( minus_minus @ nat @ To2 @ From2 ) @ ( drop @ A @ From2 @ List2 ) ) ) ) ).
% slice_def
thf(fact_4251_drop__Cons_H,axiom,
! [A: $tType,N: nat,X2: A,Xs: list @ A] :
( ( ( N
= ( zero_zero @ nat ) )
=> ( ( drop @ A @ N @ ( cons @ A @ X2 @ Xs ) )
= ( cons @ A @ X2 @ Xs ) ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ( ( drop @ A @ N @ ( cons @ A @ X2 @ Xs ) )
= ( drop @ A @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ Xs ) ) ) ) ).
% drop_Cons'
thf(fact_4252_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_4253_take__rev,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( take @ A @ N @ ( rev @ A @ Xs ) )
= ( rev @ A @ ( drop @ A @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N ) @ Xs ) ) ) ).
% take_rev
thf(fact_4254_rev__take,axiom,
! [A: $tType,I: nat,Xs: list @ A] :
( ( rev @ A @ ( take @ A @ I @ Xs ) )
= ( drop @ A @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ I ) @ ( rev @ A @ Xs ) ) ) ).
% rev_take
thf(fact_4255_rev__drop,axiom,
! [A: $tType,I: nat,Xs: list @ A] :
( ( rev @ A @ ( drop @ A @ I @ Xs ) )
= ( take @ A @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ I ) @ ( rev @ A @ Xs ) ) ) ).
% rev_drop
thf(fact_4256_drop__rev,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( drop @ A @ N @ ( rev @ A @ Xs ) )
= ( rev @ A @ ( take @ A @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N ) @ Xs ) ) ) ).
% drop_rev
thf(fact_4257_revg_Oelims,axiom,
! [A: $tType,X2: list @ A,Xa: list @ A,Y: list @ A] :
( ( ( revg @ A @ X2 @ Xa )
= Y )
=> ( ( ( X2
= ( nil @ A ) )
=> ( Y != Xa ) )
=> ~ ! [A4: A,As2: list @ A] :
( ( X2
= ( cons @ A @ A4 @ As2 ) )
=> ( Y
!= ( revg @ A @ As2 @ ( cons @ A @ A4 @ Xa ) ) ) ) ) ) ).
% revg.elims
thf(fact_4258_Cons__nth__drop__Suc,axiom,
! [A: $tType,I: nat,Xs: list @ A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( cons @ A @ ( nth @ A @ Xs @ I ) @ ( drop @ A @ ( suc @ I ) @ Xs ) )
= ( drop @ A @ I @ Xs ) ) ) ).
% Cons_nth_drop_Suc
thf(fact_4259_in__set__drop__conv__nth,axiom,
! [A: $tType,X2: A,N: nat,L: list @ A] :
( ( member @ A @ X2 @ ( set2 @ A @ ( drop @ A @ N @ L ) ) )
= ( ? [I2: nat] :
( ( ord_less_eq @ nat @ N @ I2 )
& ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ L ) )
& ( X2
= ( nth @ A @ L @ I2 ) ) ) ) ) ).
% in_set_drop_conv_nth
thf(fact_4260_rotate__drop__take,axiom,
! [A: $tType] :
( ( rotate @ A )
= ( ^ [N4: nat,Xs3: list @ A] : ( append @ A @ ( drop @ A @ ( modulo_modulo @ nat @ N4 @ ( size_size @ ( list @ A ) @ Xs3 ) ) @ Xs3 ) @ ( take @ A @ ( modulo_modulo @ nat @ N4 @ ( size_size @ ( list @ A ) @ Xs3 ) ) @ Xs3 ) ) ) ) ).
% rotate_drop_take
thf(fact_4261_set__take__disj__set__drop__if__distinct,axiom,
! [A: $tType,Vs: list @ A,I: nat,J: nat] :
( ( distinct @ A @ Vs )
=> ( ( ord_less_eq @ nat @ I @ J )
=> ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ ( take @ A @ I @ Vs ) ) @ ( set2 @ A @ ( drop @ A @ J @ Vs ) ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% set_take_disj_set_drop_if_distinct
thf(fact_4262_foldl__list__update,axiom,
! [B: $tType,A: $tType,N: nat,Xs: list @ A,F: B > A > B,A2: B,X2: A] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( foldl @ B @ A @ F @ A2 @ ( list_update @ A @ Xs @ N @ X2 ) )
= ( foldl @ B @ A @ F @ ( F @ ( foldl @ B @ A @ F @ A2 @ ( take @ A @ N @ Xs ) ) @ X2 ) @ ( drop @ A @ ( suc @ N ) @ Xs ) ) ) ) ).
% foldl_list_update
thf(fact_4263_id__take__nth__drop,axiom,
! [A: $tType,I: nat,Xs: list @ A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( Xs
= ( append @ A @ ( take @ A @ I @ Xs ) @ ( cons @ A @ ( nth @ A @ Xs @ I ) @ ( drop @ A @ ( suc @ I ) @ Xs ) ) ) ) ) ).
% id_take_nth_drop
thf(fact_4264_num__of__integer__code,axiom,
( code_num_of_integer
= ( ^ [K2: code_integer] :
( if @ num @ ( ord_less_eq @ code_integer @ K2 @ ( 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 @ K2 @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% num_of_integer_code
thf(fact_4265_inf__Int__eq,axiom,
! [A: $tType,R3: set @ A,S2: set @ A] :
( ( inf_inf @ ( A > $o )
@ ^ [X: A] : ( member @ A @ X @ R3 )
@ ^ [X: A] : ( member @ A @ X @ S2 ) )
= ( ^ [X: A] : ( member @ A @ X @ ( inf_inf @ ( set @ A ) @ R3 @ S2 ) ) ) ) ).
% inf_Int_eq
thf(fact_4266_take__hd__drop,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( append @ A @ ( take @ A @ N @ Xs ) @ ( cons @ A @ ( hd @ A @ ( drop @ A @ N @ Xs ) ) @ ( nil @ A ) ) )
= ( take @ A @ ( suc @ N ) @ Xs ) ) ) ).
% take_hd_drop
thf(fact_4267_int__of__integer__code,axiom,
( code_int_of_integer
= ( ^ [K2: code_integer] :
( if @ int @ ( ord_less @ code_integer @ K2 @ ( zero_zero @ code_integer ) ) @ ( uminus_uminus @ int @ ( code_int_of_integer @ ( uminus_uminus @ code_integer @ K2 ) ) )
@ ( if @ int
@ ( K2
= ( 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 @ K2 @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% int_of_integer_code
thf(fact_4268_int__of__integer__inverse,axiom,
! [X2: code_integer] :
( ( code_integer_of_int @ ( code_int_of_integer @ X2 ) )
= X2 ) ).
% int_of_integer_inverse
thf(fact_4269_int__of__integer__integer__of__int,axiom,
! [K: int] :
( ( code_int_of_integer @ ( code_integer_of_int @ K ) )
= K ) ).
% int_of_integer_integer_of_int
thf(fact_4270_integer__of__int__int__of__integer,axiom,
! [K: code_integer] :
( ( code_integer_of_int @ ( code_int_of_integer @ K ) )
= K ) ).
% integer_of_int_int_of_integer
thf(fact_4271_int__of__integer__sub,axiom,
! [K: num,L: num] :
( ( code_int_of_integer @ ( neg_numeral_sub @ code_integer @ K @ L ) )
= ( neg_numeral_sub @ int @ K @ L ) ) ).
% int_of_integer_sub
thf(fact_4272_of__int__integer__of,axiom,
! [K: code_integer] :
( ( ring_1_of_int @ code_integer @ ( code_int_of_integer @ K ) )
= K ) ).
% of_int_integer_of
thf(fact_4273_int__of__integer__of__int,axiom,
! [K: int] :
( ( code_int_of_integer @ ( ring_1_of_int @ code_integer @ K ) )
= K ) ).
% int_of_integer_of_int
thf(fact_4274_zero__integer_Orep__eq,axiom,
( ( code_int_of_integer @ ( zero_zero @ code_integer ) )
= ( zero_zero @ int ) ) ).
% zero_integer.rep_eq
thf(fact_4275_int__of__integer__numeral,axiom,
! [K: num] :
( ( code_int_of_integer @ ( numeral_numeral @ code_integer @ K ) )
= ( numeral_numeral @ int @ K ) ) ).
% int_of_integer_numeral
thf(fact_4276_plus__integer_Orep__eq,axiom,
! [X2: code_integer,Xa: code_integer] :
( ( code_int_of_integer @ ( plus_plus @ code_integer @ X2 @ Xa ) )
= ( plus_plus @ int @ ( code_int_of_integer @ X2 ) @ ( code_int_of_integer @ Xa ) ) ) ).
% plus_integer.rep_eq
thf(fact_4277_one__integer_Orep__eq,axiom,
( ( code_int_of_integer @ ( one_one @ code_integer ) )
= ( one_one @ int ) ) ).
% one_integer.rep_eq
thf(fact_4278_uminus__integer_Orep__eq,axiom,
! [X2: code_integer] :
( ( code_int_of_integer @ ( uminus_uminus @ code_integer @ X2 ) )
= ( uminus_uminus @ int @ ( code_int_of_integer @ X2 ) ) ) ).
% uminus_integer.rep_eq
thf(fact_4279_times__integer_Orep__eq,axiom,
! [X2: code_integer,Xa: code_integer] :
( ( code_int_of_integer @ ( times_times @ code_integer @ X2 @ Xa ) )
= ( times_times @ int @ ( code_int_of_integer @ X2 ) @ ( code_int_of_integer @ Xa ) ) ) ).
% times_integer.rep_eq
thf(fact_4280_minus__integer_Orep__eq,axiom,
! [X2: code_integer,Xa: code_integer] :
( ( code_int_of_integer @ ( minus_minus @ code_integer @ X2 @ Xa ) )
= ( minus_minus @ int @ ( code_int_of_integer @ X2 ) @ ( code_int_of_integer @ Xa ) ) ) ).
% minus_integer.rep_eq
thf(fact_4281_abs__integer_Orep__eq,axiom,
! [X2: code_integer] :
( ( code_int_of_integer @ ( abs_abs @ code_integer @ X2 ) )
= ( abs_abs @ int @ ( code_int_of_integer @ X2 ) ) ) ).
% abs_integer.rep_eq
thf(fact_4282_divide__integer_Orep__eq,axiom,
! [X2: code_integer,Xa: code_integer] :
( ( code_int_of_integer @ ( divide_divide @ code_integer @ X2 @ Xa ) )
= ( divide_divide @ int @ ( code_int_of_integer @ X2 ) @ ( code_int_of_integer @ Xa ) ) ) ).
% divide_integer.rep_eq
thf(fact_4283_modulo__integer_Orep__eq,axiom,
! [X2: code_integer,Xa: code_integer] :
( ( code_int_of_integer @ ( modulo_modulo @ code_integer @ X2 @ Xa ) )
= ( modulo_modulo @ int @ ( code_int_of_integer @ X2 ) @ ( code_int_of_integer @ Xa ) ) ) ).
% modulo_integer.rep_eq
thf(fact_4284_int__of__integer__of__nat,axiom,
! [N: nat] :
( ( code_int_of_integer @ ( semiring_1_of_nat @ code_integer @ N ) )
= ( semiring_1_of_nat @ int @ N ) ) ).
% int_of_integer_of_nat
thf(fact_4285_sgn__integer_Orep__eq,axiom,
! [X2: code_integer] :
( ( code_int_of_integer @ ( sgn_sgn @ code_integer @ X2 ) )
= ( sgn_sgn @ int @ ( code_int_of_integer @ X2 ) ) ) ).
% sgn_integer.rep_eq
thf(fact_4286_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_4287_int__of__integer__max,axiom,
! [K: code_integer,L: code_integer] :
( ( code_int_of_integer @ ( ord_max @ code_integer @ K @ L ) )
= ( ord_max @ int @ ( code_int_of_integer @ K ) @ ( code_int_of_integer @ L ) ) ) ).
% int_of_integer_max
thf(fact_4288_integer__eq__iff,axiom,
( ( ^ [Y3: code_integer,Z: code_integer] : Y3 = Z )
= ( ^ [K2: code_integer,L2: code_integer] :
( ( code_int_of_integer @ K2 )
= ( code_int_of_integer @ L2 ) ) ) ) ).
% integer_eq_iff
thf(fact_4289_integer__eqI,axiom,
! [K: code_integer,L: code_integer] :
( ( ( code_int_of_integer @ K )
= ( code_int_of_integer @ L ) )
=> ( K = L ) ) ).
% integer_eqI
thf(fact_4290_int__of__integer__inject,axiom,
! [X2: code_integer,Y: code_integer] :
( ( ( code_int_of_integer @ X2 )
= ( code_int_of_integer @ Y ) )
= ( X2 = Y ) ) ).
% int_of_integer_inject
thf(fact_4291_prod__case__refines,axiom,
! [C: $tType,B: $tType,A: $tType,P3: product_prod @ A @ B,P8: product_prod @ A @ B,F: A > B > ( heap_Time_Heap @ C ),F3: A > B > ( heap_Time_Heap @ C )] :
( ( P3 = P8 )
=> ( ! [A4: A,B4: B] : ( refine_Imp_refines @ C @ ( F @ A4 @ B4 ) @ ( F3 @ A4 @ B4 ) )
=> ( refine_Imp_refines @ C @ ( product_case_prod @ A @ B @ ( heap_Time_Heap @ C ) @ F @ P3 ) @ ( product_case_prod @ A @ B @ ( heap_Time_Heap @ C ) @ F3 @ P8 ) ) ) ) ).
% prod_case_refines
thf(fact_4292_refines__case__prod__right,axiom,
! [C: $tType,B: $tType,A: $tType,M: heap_Time_Heap @ C,M5: A > B > ( heap_Time_Heap @ C ),T2: product_prod @ A @ B] :
( ! [A4: A,B4: B] : ( refine_Imp_refines @ C @ M @ ( M5 @ A4 @ B4 ) )
=> ( refine_Imp_refines @ C @ M @ ( product_case_prod @ A @ B @ ( heap_Time_Heap @ C ) @ M5 @ T2 ) ) ) ).
% refines_case_prod_right
thf(fact_4293_less__integer_Orep__eq,axiom,
( ( ord_less @ code_integer )
= ( ^ [X: code_integer,Xa5: code_integer] : ( ord_less @ int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa5 ) ) ) ) ).
% less_integer.rep_eq
thf(fact_4294_integer__less__iff,axiom,
( ( ord_less @ code_integer )
= ( ^ [K2: code_integer,L2: code_integer] : ( ord_less @ int @ ( code_int_of_integer @ K2 ) @ ( code_int_of_integer @ L2 ) ) ) ) ).
% integer_less_iff
thf(fact_4295_int__of__integer__less__iff,axiom,
! [X2: code_integer,Y: code_integer] :
( ( ord_less @ int @ ( code_int_of_integer @ X2 ) @ ( code_int_of_integer @ Y ) )
= ( ord_less @ code_integer @ X2 @ Y ) ) ).
% int_of_integer_less_iff
thf(fact_4296_less__eq__integer_Orep__eq,axiom,
( ( ord_less_eq @ code_integer )
= ( ^ [X: code_integer,Xa5: code_integer] : ( ord_less_eq @ int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa5 ) ) ) ) ).
% less_eq_integer.rep_eq
thf(fact_4297_integer__less__eq__iff,axiom,
( ( ord_less_eq @ code_integer )
= ( ^ [K2: code_integer,L2: code_integer] : ( ord_less_eq @ int @ ( code_int_of_integer @ K2 ) @ ( code_int_of_integer @ L2 ) ) ) ) ).
% integer_less_eq_iff
thf(fact_4298_nat__of__integer_Orep__eq,axiom,
( code_nat_of_integer
= ( ^ [X: code_integer] : ( nat2 @ ( code_int_of_integer @ X ) ) ) ) ).
% nat_of_integer.rep_eq
thf(fact_4299_sum_Otriangle__reindex__eq,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ ( product_prod @ nat @ nat ) @ A @ ( product_case_prod @ nat @ nat @ A @ G )
@ ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [I2: nat,J3: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ I2 @ J3 ) @ N ) ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K2: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( G @ I2 @ ( minus_minus @ nat @ K2 @ I2 ) )
@ ( set_ord_atMost @ nat @ K2 ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% sum.triangle_reindex_eq
thf(fact_4300_prod_Otriangle__reindex__eq,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ ( product_prod @ nat @ nat ) @ A @ ( product_case_prod @ nat @ nat @ A @ G )
@ ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [I2: nat,J3: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ I2 @ J3 ) @ N ) ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [K2: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I2: nat] : ( G @ I2 @ ( minus_minus @ nat @ K2 @ I2 ) )
@ ( set_ord_atMost @ nat @ K2 ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% prod.triangle_reindex_eq
thf(fact_4301_take__bit__integer_Orep__eq,axiom,
! [X2: nat,Xa: code_integer] :
( ( code_int_of_integer @ ( bit_se2584673776208193580ke_bit @ code_integer @ X2 @ Xa ) )
= ( bit_se2584673776208193580ke_bit @ int @ X2 @ ( code_int_of_integer @ Xa ) ) ) ).
% take_bit_integer.rep_eq
thf(fact_4302_not__integer_Orep__eq,axiom,
! [X2: code_integer] :
( ( code_int_of_integer @ ( bit_ri4277139882892585799ns_not @ code_integer @ X2 ) )
= ( bit_ri4277139882892585799ns_not @ int @ ( code_int_of_integer @ X2 ) ) ) ).
% not_integer.rep_eq
thf(fact_4303_and__integer_Orep__eq,axiom,
! [X2: code_integer,Xa: code_integer] :
( ( code_int_of_integer @ ( bit_se5824344872417868541ns_and @ code_integer @ X2 @ Xa ) )
= ( bit_se5824344872417868541ns_and @ int @ ( code_int_of_integer @ X2 ) @ ( code_int_of_integer @ Xa ) ) ) ).
% and_integer.rep_eq
thf(fact_4304_or__integer_Orep__eq,axiom,
! [X2: code_integer,Xa: code_integer] :
( ( code_int_of_integer @ ( bit_se1065995026697491101ons_or @ code_integer @ X2 @ Xa ) )
= ( bit_se1065995026697491101ons_or @ int @ ( code_int_of_integer @ X2 ) @ ( code_int_of_integer @ Xa ) ) ) ).
% or_integer.rep_eq
thf(fact_4305_bit__integer_Orep__eq,axiom,
( ( bit_se5641148757651400278ts_bit @ code_integer )
= ( ^ [X: code_integer] : ( bit_se5641148757651400278ts_bit @ int @ ( code_int_of_integer @ X ) ) ) ) ).
% bit_integer.rep_eq
thf(fact_4306_xor__integer_Orep__eq,axiom,
! [X2: code_integer,Xa: code_integer] :
( ( code_int_of_integer @ ( bit_se5824344971392196577ns_xor @ code_integer @ X2 @ Xa ) )
= ( bit_se5824344971392196577ns_xor @ int @ ( code_int_of_integer @ X2 ) @ ( code_int_of_integer @ Xa ) ) ) ).
% xor_integer.rep_eq
thf(fact_4307_push__bit__integer_Orep__eq,axiom,
! [X2: nat,Xa: code_integer] :
( ( code_int_of_integer @ ( bit_se4730199178511100633sh_bit @ code_integer @ X2 @ Xa ) )
= ( bit_se4730199178511100633sh_bit @ int @ X2 @ ( code_int_of_integer @ Xa ) ) ) ).
% push_bit_integer.rep_eq
thf(fact_4308_drop__bit__integer_Orep__eq,axiom,
! [X2: nat,Xa: code_integer] :
( ( code_int_of_integer @ ( bit_se4197421643247451524op_bit @ code_integer @ X2 @ Xa ) )
= ( bit_se4197421643247451524op_bit @ int @ X2 @ ( code_int_of_integer @ Xa ) ) ) ).
% drop_bit_integer.rep_eq
thf(fact_4309_mask__integer_Orep__eq,axiom,
! [X2: nat] :
( ( code_int_of_integer @ ( bit_se2239418461657761734s_mask @ code_integer @ X2 ) )
= ( bit_se2239418461657761734s_mask @ int @ X2 ) ) ).
% mask_integer.rep_eq
thf(fact_4310_unset__bit__integer_Orep__eq,axiom,
! [X2: nat,Xa: code_integer] :
( ( code_int_of_integer @ ( bit_se2638667681897837118et_bit @ code_integer @ X2 @ Xa ) )
= ( bit_se2638667681897837118et_bit @ int @ X2 @ ( code_int_of_integer @ Xa ) ) ) ).
% unset_bit_integer.rep_eq
thf(fact_4311_set__bit__integer_Orep__eq,axiom,
! [X2: nat,Xa: code_integer] :
( ( code_int_of_integer @ ( bit_se5668285175392031749et_bit @ code_integer @ X2 @ Xa ) )
= ( bit_se5668285175392031749et_bit @ int @ X2 @ ( code_int_of_integer @ Xa ) ) ) ).
% set_bit_integer.rep_eq
thf(fact_4312_sum_Otriangle__reindex,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ ( product_prod @ nat @ nat ) @ A @ ( product_case_prod @ nat @ nat @ A @ G )
@ ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [I2: nat,J3: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ I2 @ J3 ) @ N ) ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K2: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( G @ I2 @ ( minus_minus @ nat @ K2 @ I2 ) )
@ ( set_ord_atMost @ nat @ K2 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% sum.triangle_reindex
thf(fact_4313_flip__bit__integer_Orep__eq,axiom,
! [X2: nat,Xa: code_integer] :
( ( code_int_of_integer @ ( bit_se8732182000553998342ip_bit @ code_integer @ X2 @ Xa ) )
= ( bit_se8732182000553998342ip_bit @ int @ X2 @ ( code_int_of_integer @ Xa ) ) ) ).
% flip_bit_integer.rep_eq
thf(fact_4314_prod_Otriangle__reindex,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ ( product_prod @ nat @ nat ) @ A @ ( product_case_prod @ nat @ nat @ A @ G )
@ ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [I2: nat,J3: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ I2 @ J3 ) @ N ) ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [K2: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I2: nat] : ( G @ I2 @ ( minus_minus @ nat @ K2 @ I2 ) )
@ ( set_ord_atMost @ nat @ K2 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% prod.triangle_reindex
thf(fact_4315_hd__conv__nth,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( hd @ A @ Xs )
= ( nth @ A @ Xs @ ( zero_zero @ nat ) ) ) ) ).
% hd_conv_nth
thf(fact_4316_num__of__integer_Orep__eq,axiom,
( code_num_of_integer
= ( ^ [X: code_integer] : ( num_of_nat @ ( nat2 @ ( code_int_of_integer @ X ) ) ) ) ) ).
% num_of_integer.rep_eq
thf(fact_4317_sorted__hd__min,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
=> ( ord_less_eq @ A @ ( hd @ A @ Xs ) @ X4 ) ) ) ) ) ).
% sorted_hd_min
thf(fact_4318_hd__drop__conv__nth,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( hd @ A @ ( drop @ A @ N @ Xs ) )
= ( nth @ A @ Xs @ N ) ) ) ).
% hd_drop_conv_nth
thf(fact_4319_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_4320_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_4321_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_4322_hd__rotate__conv__nth,axiom,
! [A: $tType,Xs: list @ A,N: nat] :
( ( Xs
!= ( nil @ A ) )
=> ( ( hd @ A @ ( rotate @ A @ N @ Xs ) )
= ( nth @ A @ Xs @ ( modulo_modulo @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) ).
% hd_rotate_conv_nth
thf(fact_4323_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_4324_int__ge__less__than__def,axiom,
( int_ge_less_than
= ( ^ [D5: int] :
( collect @ ( product_prod @ int @ int )
@ ( product_case_prod @ int @ int @ $o
@ ^ [Z7: int,Z3: int] :
( ( ord_less_eq @ int @ D5 @ Z7 )
& ( ord_less @ int @ Z7 @ Z3 ) ) ) ) ) ) ).
% int_ge_less_than_def
thf(fact_4325_int__ge__less__than2__def,axiom,
( int_ge_less_than2
= ( ^ [D5: int] :
( collect @ ( product_prod @ int @ int )
@ ( product_case_prod @ int @ int @ $o
@ ^ [Z7: int,Z3: int] :
( ( ord_less_eq @ int @ D5 @ Z3 )
& ( ord_less @ int @ Z7 @ Z3 ) ) ) ) ) ) ).
% int_ge_less_than2_def
thf(fact_4326_Divides_Oadjust__div__def,axiom,
( adjust_div
= ( product_case_prod @ int @ int @ int
@ ^ [Q6: int,R2: int] :
( plus_plus @ int @ Q6
@ ( zero_neq_one_of_bool @ int
@ ( R2
!= ( zero_zero @ int ) ) ) ) ) ) ).
% Divides.adjust_div_def
thf(fact_4327_divmod__step__integer__def,axiom,
( ( unique1321980374590559556d_step @ code_integer )
= ( ^ [L2: num] :
( product_case_prod @ code_integer @ code_integer @ ( product_prod @ code_integer @ code_integer )
@ ^ [Q6: code_integer,R2: code_integer] : ( if @ ( product_prod @ code_integer @ code_integer ) @ ( ord_less_eq @ code_integer @ ( numeral_numeral @ code_integer @ L2 ) @ R2 ) @ ( product_Pair @ code_integer @ code_integer @ ( plus_plus @ code_integer @ ( times_times @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) @ Q6 ) @ ( one_one @ code_integer ) ) @ ( minus_minus @ code_integer @ R2 @ ( numeral_numeral @ code_integer @ L2 ) ) ) @ ( product_Pair @ code_integer @ code_integer @ ( times_times @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) @ Q6 ) @ R2 ) ) ) ) ) ).
% divmod_step_integer_def
thf(fact_4328_not__Some__eq2,axiom,
! [B: $tType,A: $tType,V2: option @ ( product_prod @ A @ B )] :
( ( ! [X: A,Y4: B] :
( V2
!= ( some @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y4 ) ) ) )
= ( V2
= ( none @ ( product_prod @ A @ B ) ) ) ) ).
% not_Some_eq2
thf(fact_4329_Eps__case__prod__eq,axiom,
! [A: $tType,B: $tType,X2: A,Y: B] :
( ( fChoice @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [X9: A,Y8: B] :
( ( X2 = X9 )
& ( Y = Y8 ) ) ) )
= ( product_Pair @ A @ B @ X2 @ Y ) ) ).
% Eps_case_prod_eq
thf(fact_4330_divmod__step__eq,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [L: num,R: A,Q2: A] :
( ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ L ) @ R )
=> ( ( unique1321980374590559556d_step @ A @ L @ ( product_Pair @ A @ A @ Q2 @ R ) )
= ( product_Pair @ A @ A @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Q2 ) @ ( one_one @ A ) ) @ ( minus_minus @ A @ R @ ( numeral_numeral @ A @ L ) ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ L ) @ R )
=> ( ( unique1321980374590559556d_step @ A @ L @ ( product_Pair @ A @ A @ Q2 @ R ) )
= ( product_Pair @ A @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Q2 ) @ R ) ) ) ) ) ).
% divmod_step_eq
thf(fact_4331_inf__Int__eq2,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ B ),S2: set @ ( product_prod @ A @ B )] :
( ( inf_inf @ ( A > B > $o )
@ ^ [X: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y4 ) @ R3 )
@ ^ [X: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y4 ) @ S2 ) )
= ( ^ [X: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y4 ) @ ( inf_inf @ ( set @ ( product_prod @ A @ B ) ) @ R3 @ S2 ) ) ) ) ).
% inf_Int_eq2
thf(fact_4332_split__paired__Eps,axiom,
! [B: $tType,A: $tType] :
( ( fChoice @ ( product_prod @ A @ B ) )
= ( ^ [P4: ( product_prod @ A @ B ) > $o] :
( fChoice @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [A3: A,B3: B] : ( P4 @ ( product_Pair @ A @ B @ A3 @ B3 ) ) ) ) ) ) ).
% split_paired_Eps
thf(fact_4333_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_4334_pred__equals__eq2,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ B ),S2: set @ ( product_prod @ A @ B )] :
( ( ( ^ [X: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y4 ) @ R3 ) )
= ( ^ [X: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y4 ) @ S2 ) ) )
= ( R3 = S2 ) ) ).
% pred_equals_eq2
thf(fact_4335_pred__subset__eq2,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ B ),S2: set @ ( product_prod @ A @ B )] :
( ( ord_less_eq @ ( A > B > $o )
@ ^ [X: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y4 ) @ R3 )
@ ^ [X: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y4 ) @ S2 ) )
= ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R3 @ S2 ) ) ).
% pred_subset_eq2
thf(fact_4336_pairself_Ocases,axiom,
! [B: $tType,A: $tType,X2: product_prod @ ( A > B ) @ ( product_prod @ A @ A )] :
~ ! [F5: A > B,A4: A,B4: A] :
( X2
!= ( product_Pair @ ( A > B ) @ ( product_prod @ A @ A ) @ F5 @ ( product_Pair @ A @ A @ A4 @ B4 ) ) ) ).
% pairself.cases
thf(fact_4337_bex2I,axiom,
! [A: $tType,B: $tType,A2: A,B2: B,S2: set @ ( product_prod @ A @ B ),P: A > B > $o] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ S2 )
=> ( ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ S2 )
=> ( P @ A2 @ B2 ) )
=> ? [A4: A,B4: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B4 ) @ S2 )
& ( P @ A4 @ B4 ) ) ) ) ).
% bex2I
thf(fact_4338_bot__empty__eq2,axiom,
! [B: $tType,A: $tType] :
( ( bot_bot @ ( A > B > $o ) )
= ( ^ [X: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y4 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% bot_empty_eq2
thf(fact_4339_rel__of__def,axiom,
! [B: $tType,A: $tType] :
( ( rel_of @ A @ B )
= ( ^ [M2: A > ( option @ B ),P4: ( product_prod @ A @ B ) > $o] :
( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [K2: A,V4: B] :
( ( ( M2 @ K2 )
= ( some @ B @ V4 ) )
& ( P4 @ ( product_Pair @ A @ B @ K2 @ V4 ) ) ) ) ) ) ) ).
% rel_of_def
thf(fact_4340_divmod__integer__def,axiom,
( code_divmod_integer
= ( ^ [K2: code_integer,L2: code_integer] : ( product_Pair @ code_integer @ code_integer @ ( divide_divide @ code_integer @ K2 @ L2 ) @ ( modulo_modulo @ code_integer @ K2 @ L2 ) ) ) ) ).
% divmod_integer_def
thf(fact_4341_divmod__step__def,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ( ( unique1321980374590559556d_step @ A )
= ( ^ [L2: num] :
( product_case_prod @ A @ A @ ( product_prod @ A @ A )
@ ^ [Q6: A,R2: A] : ( if @ ( product_prod @ A @ A ) @ ( ord_less_eq @ A @ ( numeral_numeral @ A @ L2 ) @ R2 ) @ ( product_Pair @ A @ A @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Q6 ) @ ( one_one @ A ) ) @ ( minus_minus @ A @ R2 @ ( numeral_numeral @ A @ L2 ) ) ) @ ( product_Pair @ A @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Q6 ) @ R2 ) ) ) ) ) ) ).
% divmod_step_def
thf(fact_4342_case__prodI2,axiom,
! [B: $tType,A: $tType,P3: product_prod @ A @ B,C2: A > B > $o] :
( ! [A4: A,B4: B] :
( ( P3
= ( product_Pair @ A @ B @ A4 @ B4 ) )
=> ( C2 @ A4 @ B4 ) )
=> ( product_case_prod @ A @ B @ $o @ C2 @ P3 ) ) ).
% case_prodI2
thf(fact_4343_case__prodI,axiom,
! [A: $tType,B: $tType,F: A > B > $o,A2: A,B2: B] :
( ( F @ A2 @ B2 )
=> ( product_case_prod @ A @ B @ $o @ F @ ( product_Pair @ A @ B @ A2 @ B2 ) ) ) ).
% case_prodI
thf(fact_4344_case__prod__conv,axiom,
! [B: $tType,A: $tType,C: $tType,F: B > C > A,A2: B,B2: C] :
( ( product_case_prod @ B @ C @ A @ F @ ( product_Pair @ B @ C @ A2 @ B2 ) )
= ( F @ A2 @ B2 ) ) ).
% case_prod_conv
thf(fact_4345_pair__imageI,axiom,
! [C: $tType,B: $tType,A: $tType,A2: A,B2: B,A5: set @ ( product_prod @ A @ B ),F: A > B > C] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ A5 )
=> ( member @ C @ ( F @ A2 @ B2 ) @ ( image2 @ ( product_prod @ A @ B ) @ C @ ( product_case_prod @ A @ B @ C @ F ) @ A5 ) ) ) ).
% pair_imageI
thf(fact_4346_case__prodI2_H,axiom,
! [A: $tType,B: $tType,C: $tType,P3: product_prod @ A @ B,C2: A > B > C > $o,X2: C] :
( ! [A4: A,B4: B] :
( ( ( product_Pair @ A @ B @ A4 @ B4 )
= P3 )
=> ( C2 @ A4 @ B4 @ X2 ) )
=> ( product_case_prod @ A @ B @ ( C > $o ) @ C2 @ P3 @ X2 ) ) ).
% case_prodI2'
thf(fact_4347_mem__case__prodI,axiom,
! [A: $tType,B: $tType,C: $tType,Z2: A,C2: B > C > ( set @ A ),A2: B,B2: C] :
( ( member @ A @ Z2 @ ( C2 @ A2 @ B2 ) )
=> ( member @ A @ Z2 @ ( product_case_prod @ B @ C @ ( set @ A ) @ C2 @ ( product_Pair @ B @ C @ A2 @ B2 ) ) ) ) ).
% mem_case_prodI
thf(fact_4348_mem__case__prodI2,axiom,
! [C: $tType,B: $tType,A: $tType,P3: product_prod @ A @ B,Z2: C,C2: A > B > ( set @ C )] :
( ! [A4: A,B4: B] :
( ( P3
= ( product_Pair @ A @ B @ A4 @ B4 ) )
=> ( member @ C @ Z2 @ ( C2 @ A4 @ B4 ) ) )
=> ( member @ C @ Z2 @ ( product_case_prod @ A @ B @ ( set @ C ) @ C2 @ P3 ) ) ) ).
% mem_case_prodI2
thf(fact_4349_split__part,axiom,
! [B: $tType,A: $tType,P: $o,Q: A > B > $o] :
( ( product_case_prod @ A @ B @ $o
@ ^ [A3: A,B3: B] :
( P
& ( Q @ A3 @ B3 ) ) )
= ( ^ [Ab: product_prod @ A @ B] :
( P
& ( product_case_prod @ A @ B @ $o @ Q @ Ab ) ) ) ) ).
% split_part
thf(fact_4350_Divides_Oadjust__div__eq,axiom,
! [Q2: int,R: int] :
( ( adjust_div @ ( product_Pair @ int @ int @ Q2 @ R ) )
= ( plus_plus @ int @ Q2
@ ( zero_neq_one_of_bool @ int
@ ( R
!= ( zero_zero @ int ) ) ) ) ) ).
% Divides.adjust_div_eq
thf(fact_4351_timeFrame_Osimps_I1_J,axiom,
! [A: $tType,N: nat,R: A,H: heap_ext @ product_unit,N3: nat] :
( ( heap_Time_timeFrame @ A @ N @ ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H @ N3 ) ) ) )
= ( 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 @ H @ ( plus_plus @ nat @ N @ N3 ) ) ) ) ) ).
% timeFrame.simps(1)
thf(fact_4352_mergesort__by__rel__split_Ocases,axiom,
! [A: $tType,X2: product_prod @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A )] :
( ! [Xs1: list @ A,Xs22: list @ A] :
( X2
!= ( 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] :
( X2
!= ( 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,X1: A,X23: A,Xs2: list @ A] :
( X2
!= ( product_Pair @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs1 @ Xs22 ) @ ( cons @ A @ X1 @ ( cons @ A @ X23 @ Xs2 ) ) ) ) ) ) ).
% mergesort_by_rel_split.cases
thf(fact_4353_partition__rev_Ocases,axiom,
! [A: $tType,X2: product_prod @ ( A > $o ) @ ( product_prod @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) )] :
( ! [P9: A > $o,Yes: list @ A,No: list @ A] :
( X2
!= ( product_Pair @ ( A > $o ) @ ( product_prod @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) ) @ P9 @ ( product_Pair @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes @ No ) @ ( nil @ A ) ) ) )
=> ~ ! [P9: A > $o,Yes: list @ A,No: list @ A,X3: A,Xs2: list @ A] :
( X2
!= ( product_Pair @ ( A > $o ) @ ( product_prod @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( list @ A ) ) @ P9 @ ( 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_4354_list__all__zip_Ocases,axiom,
! [A: $tType,B: $tType,X2: product_prod @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) )] :
( ! [P9: A > B > $o] :
( X2
!= ( product_Pair @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ P9 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ ( nil @ B ) ) ) )
=> ( ! [P9: A > B > $o,A4: A,As2: list @ A,B4: B,Bs2: list @ B] :
( X2
!= ( product_Pair @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ P9 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ A4 @ As2 ) @ ( cons @ B @ B4 @ Bs2 ) ) ) )
=> ( ! [P9: A > B > $o,V5: A,Va: list @ A] :
( X2
!= ( product_Pair @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ P9 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ V5 @ Va ) @ ( nil @ B ) ) ) )
=> ~ ! [P9: A > B > $o,V5: B,Va: list @ B] :
( X2
!= ( product_Pair @ ( A > B > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ P9 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ ( cons @ B @ V5 @ Va ) ) ) ) ) ) ) ).
% list_all_zip.cases
thf(fact_4355_quicksort__by__rel_Ocases,axiom,
! [A: $tType,X2: product_prod @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) )] :
( ! [R5: A > A > $o,Sl: list @ A] :
( X2
!= ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ R5 @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Sl @ ( nil @ A ) ) ) )
=> ~ ! [R5: A > A > $o,Sl: list @ A,X3: A,Xs2: list @ A] :
( X2
!= ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ R5 @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Sl @ ( cons @ A @ X3 @ Xs2 ) ) ) ) ) ).
% quicksort_by_rel.cases
thf(fact_4356_mergesort__by__rel__merge_Ocases,axiom,
! [A: $tType,X2: product_prod @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) )] :
( ! [R5: A > A > $o,X3: A,Xs2: list @ A,Y2: A,Ys4: list @ A] :
( X2
!= ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ R5 @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X3 @ Xs2 ) @ ( cons @ A @ Y2 @ Ys4 ) ) ) )
=> ( ! [R5: A > A > $o,Xs2: list @ A] :
( X2
!= ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ R5 @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ ( nil @ A ) ) ) )
=> ~ ! [R5: A > A > $o,V5: A,Va: list @ A] :
( X2
!= ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ R5 @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( cons @ A @ V5 @ Va ) ) ) ) ) ) ).
% mergesort_by_rel_merge.cases
thf(fact_4357_zipf_Ocases,axiom,
! [C: $tType,A: $tType,B: $tType,X2: product_prod @ ( A > B > C ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) )] :
( ! [F5: A > B > C] :
( X2
!= ( product_Pair @ ( A > B > C ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ F5 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ ( nil @ B ) ) ) )
=> ( ! [F5: A > B > C,A4: A,As2: list @ A,B4: B,Bs2: list @ B] :
( X2
!= ( product_Pair @ ( A > B > C ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ F5 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ A4 @ As2 ) @ ( cons @ B @ B4 @ Bs2 ) ) ) )
=> ( ! [A4: A > B > C,V5: A,Va: list @ A] :
( X2
!= ( product_Pair @ ( A > B > C ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ A4 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ V5 @ Va ) @ ( nil @ B ) ) ) )
=> ~ ! [A4: A > B > C,V5: B,Va: list @ B] :
( X2
!= ( product_Pair @ ( A > B > C ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ A4 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ ( cons @ B @ V5 @ Va ) ) ) ) ) ) ) ).
% zipf.cases
thf(fact_4358_timeFrame_Ocases,axiom,
! [A: $tType,X2: product_prod @ nat @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )] :
( ! [N2: nat,R6: A,H2: heap_ext @ product_unit,N8: nat] :
( X2
!= ( product_Pair @ nat @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ N2 @ ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R6 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H2 @ N8 ) ) ) ) )
=> ~ ! [N2: nat] :
( X2
!= ( product_Pair @ nat @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ N2 @ ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ) ).
% timeFrame.cases
thf(fact_4359_timeFrame_Oelims,axiom,
! [A: $tType,X2: 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 @ X2 @ Xa )
= Y )
=> ( ! [R6: A,H2: heap_ext @ product_unit,N8: nat] :
( ( Xa
= ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R6 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H2 @ N8 ) ) ) )
=> ( Y
!= ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R6 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H2 @ ( plus_plus @ nat @ X2 @ N8 ) ) ) ) ) )
=> ~ ( ( Xa
= ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) )
=> ( Y
!= ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ) ) ).
% timeFrame.elims
thf(fact_4360_merge_Ocases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: product_prod @ ( list @ A ) @ ( list @ A )] :
( ! [L23: list @ A] :
( X2
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ L23 ) )
=> ( ! [V5: A,Va: list @ A] :
( X2
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ V5 @ Va ) @ ( nil @ A ) ) )
=> ~ ! [X1: A,L12: list @ A,X23: A,L23: list @ A] :
( X2
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X1 @ L12 ) @ ( cons @ A @ X23 @ L23 ) ) ) ) ) ) ).
% merge.cases
thf(fact_4361_merge__list_Ocases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: product_prod @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) )] :
( ( X2
!= ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ ( nil @ ( list @ A ) ) @ ( nil @ ( list @ A ) ) ) )
=> ( ! [L3: list @ A] :
( X2
!= ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ ( nil @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ L3 @ ( nil @ ( list @ A ) ) ) ) )
=> ( ! [La: list @ A,Acc2: list @ ( list @ A )] :
( X2
!= ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ La @ Acc2 ) @ ( nil @ ( list @ A ) ) ) )
=> ( ! [La: list @ A,Acc2: list @ ( list @ A ),L3: list @ A] :
( X2
!= ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ La @ Acc2 ) @ ( cons @ ( list @ A ) @ L3 @ ( nil @ ( list @ A ) ) ) ) )
=> ~ ! [Acc2: list @ ( list @ A ),L12: list @ A,L23: list @ A,Ls2: list @ ( list @ A )] :
( X2
!= ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ Acc2 @ ( cons @ ( list @ A ) @ L12 @ ( cons @ ( list @ A ) @ L23 @ Ls2 ) ) ) ) ) ) ) ) ) ).
% merge_list.cases
thf(fact_4362_case__prodD_H,axiom,
! [B: $tType,A: $tType,C: $tType,R3: A > B > C > $o,A2: A,B2: B,C2: C] :
( ( product_case_prod @ A @ B @ ( C > $o ) @ R3 @ ( product_Pair @ A @ B @ A2 @ B2 ) @ C2 )
=> ( R3 @ A2 @ B2 @ C2 ) ) ).
% case_prodD'
thf(fact_4363_case__prodE_H,axiom,
! [B: $tType,A: $tType,C: $tType,C2: A > B > C > $o,P3: product_prod @ A @ B,Z2: C] :
( ( product_case_prod @ A @ B @ ( C > $o ) @ C2 @ P3 @ Z2 )
=> ~ ! [X3: A,Y2: B] :
( ( P3
= ( product_Pair @ A @ B @ X3 @ Y2 ) )
=> ~ ( C2 @ X3 @ Y2 @ Z2 ) ) ) ).
% case_prodE'
thf(fact_4364_prod_Ocase__distrib,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,H: C > D,F: A > B > C,Prod: product_prod @ A @ B] :
( ( H @ ( product_case_prod @ A @ B @ C @ F @ Prod ) )
= ( product_case_prod @ A @ B @ D
@ ^ [X12: A,X24: B] : ( H @ ( F @ X12 @ X24 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_4365_prod_Odisc__eq__case,axiom,
! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
( product_case_prod @ A @ B @ $o
@ ^ [Uu: A,Uv: B] : $true
@ Prod ) ).
% prod.disc_eq_case
thf(fact_4366_divmod__step__nat__def,axiom,
( ( unique1321980374590559556d_step @ nat )
= ( ^ [L2: num] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [Q6: nat,R2: nat] : ( if @ ( product_prod @ nat @ nat ) @ ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ L2 ) @ R2 ) @ ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Q6 ) @ ( one_one @ nat ) ) @ ( minus_minus @ nat @ R2 @ ( numeral_numeral @ nat @ L2 ) ) ) @ ( product_Pair @ nat @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Q6 ) @ R2 ) ) ) ) ) ).
% divmod_step_nat_def
thf(fact_4367_divmod__step__int__def,axiom,
( ( unique1321980374590559556d_step @ int )
= ( ^ [L2: num] :
( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [Q6: int,R2: int] : ( if @ ( product_prod @ int @ int ) @ ( ord_less_eq @ int @ ( numeral_numeral @ int @ L2 ) @ R2 ) @ ( product_Pair @ int @ int @ ( plus_plus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Q6 ) @ ( one_one @ int ) ) @ ( minus_minus @ int @ R2 @ ( numeral_numeral @ int @ L2 ) ) ) @ ( product_Pair @ int @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Q6 ) @ R2 ) ) ) ) ) ).
% divmod_step_int_def
thf(fact_4368_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,Y2: C] :
( ( Z2
= ( product_Pair @ B @ C @ X3 @ Y2 ) )
=> ~ ( Q @ ( P @ X3 @ Y2 ) ) ) ) ).
% case_prodE2
thf(fact_4369_case__prod__eta,axiom,
! [C: $tType,B: $tType,A: $tType,F: ( product_prod @ A @ B ) > C] :
( ( product_case_prod @ A @ B @ C
@ ^ [X: A,Y4: B] : ( F @ ( product_Pair @ A @ B @ X @ Y4 ) ) )
= F ) ).
% case_prod_eta
thf(fact_4370_cond__case__prod__eta,axiom,
! [C: $tType,B: $tType,A: $tType,F: A > B > C,G: ( product_prod @ A @ B ) > C] :
( ! [X3: A,Y2: B] :
( ( F @ X3 @ Y2 )
= ( G @ ( product_Pair @ A @ B @ X3 @ Y2 ) ) )
=> ( ( product_case_prod @ A @ B @ C @ F )
= G ) ) ).
% cond_case_prod_eta
thf(fact_4371_case__prodD,axiom,
! [A: $tType,B: $tType,F: A > B > $o,A2: A,B2: B] :
( ( product_case_prod @ A @ B @ $o @ F @ ( product_Pair @ A @ B @ A2 @ B2 ) )
=> ( F @ A2 @ B2 ) ) ).
% case_prodD
thf(fact_4372_case__prodE,axiom,
! [A: $tType,B: $tType,C2: A > B > $o,P3: product_prod @ A @ B] :
( ( product_case_prod @ A @ B @ $o @ C2 @ P3 )
=> ~ ! [X3: A,Y2: B] :
( ( P3
= ( product_Pair @ A @ B @ X3 @ Y2 ) )
=> ~ ( C2 @ X3 @ Y2 ) ) ) ).
% case_prodE
thf(fact_4373_take__bit__num__code,axiom,
( bit_take_bit_num
= ( ^ [N4: nat,M2: num] :
( product_case_prod @ nat @ num @ ( option @ num )
@ ^ [A3: nat,X: num] :
( case_nat @ ( option @ num ) @ ( none @ num )
@ ^ [O: nat] :
( case_num @ ( option @ num ) @ ( some @ num @ one2 )
@ ^ [P7: num] :
( case_option @ ( option @ num ) @ num @ ( none @ num )
@ ^ [Q6: num] : ( some @ num @ ( bit0 @ Q6 ) )
@ ( bit_take_bit_num @ O @ P7 ) )
@ ^ [P7: num] : ( some @ num @ ( case_option @ num @ num @ one2 @ bit1 @ ( bit_take_bit_num @ O @ P7 ) ) )
@ X )
@ A3 )
@ ( product_Pair @ nat @ num @ N4 @ M2 ) ) ) ) ).
% take_bit_num_code
thf(fact_4374_divmod__algorithm__code_I6_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M: num,N: num] :
( ( unique8689654367752047608divmod @ A @ ( bit1 @ M ) @ ( bit0 @ N ) )
= ( product_case_prod @ A @ A @ ( product_prod @ A @ A )
@ ^ [Q6: A,R2: A] : ( product_Pair @ A @ A @ Q6 @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ R2 ) @ ( one_one @ A ) ) )
@ ( unique8689654367752047608divmod @ A @ M @ N ) ) ) ) ).
% divmod_algorithm_code(6)
thf(fact_4375_divmod__algorithm__code_I7_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M: num,N: num] :
( ( ( ord_less_eq @ num @ M @ N )
=> ( ( unique8689654367752047608divmod @ A @ ( bit0 @ M ) @ ( bit1 @ N ) )
= ( product_Pair @ A @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ ( bit0 @ M ) ) ) ) )
& ( ~ ( ord_less_eq @ num @ M @ N )
=> ( ( unique8689654367752047608divmod @ A @ ( bit0 @ M ) @ ( bit1 @ N ) )
= ( unique1321980374590559556d_step @ A @ ( bit1 @ N ) @ ( unique8689654367752047608divmod @ A @ ( bit0 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ) ).
% divmod_algorithm_code(7)
thf(fact_4376_divmod__algorithm__code_I8_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M: num,N: num] :
( ( ( ord_less @ num @ M @ N )
=> ( ( unique8689654367752047608divmod @ A @ ( bit1 @ M ) @ ( bit1 @ N ) )
= ( product_Pair @ A @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ ( bit1 @ M ) ) ) ) )
& ( ~ ( ord_less @ num @ M @ N )
=> ( ( unique8689654367752047608divmod @ A @ ( bit1 @ M ) @ ( bit1 @ N ) )
= ( unique1321980374590559556d_step @ A @ ( bit1 @ N ) @ ( unique8689654367752047608divmod @ A @ ( bit1 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ) ).
% divmod_algorithm_code(8)
thf(fact_4377_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_4378_divmod__algorithm__code_I3_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [N: num] :
( ( unique8689654367752047608divmod @ A @ one2 @ ( bit0 @ N ) )
= ( product_Pair @ A @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ one2 ) ) ) ) ).
% divmod_algorithm_code(3)
thf(fact_4379_divmod__algorithm__code_I4_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [N: num] :
( ( unique8689654367752047608divmod @ A @ one2 @ ( bit1 @ N ) )
= ( product_Pair @ A @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ one2 ) ) ) ) ).
% divmod_algorithm_code(4)
thf(fact_4380_minus__one__div__numeral,axiom,
! [N: num] :
( ( divide_divide @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( numeral_numeral @ int @ N ) )
= ( uminus_uminus @ int @ ( adjust_div @ ( unique8689654367752047608divmod @ int @ one2 @ N ) ) ) ) ).
% minus_one_div_numeral
thf(fact_4381_one__div__minus__numeral,axiom,
! [N: num] :
( ( divide_divide @ int @ ( one_one @ int ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
= ( uminus_uminus @ int @ ( adjust_div @ ( unique8689654367752047608divmod @ int @ one2 @ N ) ) ) ) ).
% one_div_minus_numeral
thf(fact_4382_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_4383_divmod__algorithm__code_I5_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M: num,N: num] :
( ( unique8689654367752047608divmod @ A @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( product_case_prod @ A @ A @ ( product_prod @ A @ A )
@ ^ [Q6: A,R2: A] : ( product_Pair @ A @ A @ Q6 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ R2 ) )
@ ( unique8689654367752047608divmod @ A @ M @ N ) ) ) ) ).
% divmod_algorithm_code(5)
thf(fact_4384_fold__atLeastAtMost__nat_Ocases,axiom,
! [A: $tType,X2: product_prod @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) )] :
~ ! [F5: nat > A > A,A4: nat,B4: nat,Acc3: A] :
( X2
!= ( product_Pair @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) @ F5 @ ( product_Pair @ nat @ ( product_prod @ nat @ A ) @ A4 @ ( product_Pair @ nat @ A @ B4 @ Acc3 ) ) ) ) ).
% fold_atLeastAtMost_nat.cases
thf(fact_4385_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 )
@ ^ [X: num] : ( H @ ( F22 @ X ) )
@ ^ [X: num] : ( H @ ( F32 @ X ) )
@ Num ) ) ).
% num.case_distrib
thf(fact_4386_verit__eq__simplify_I17_J,axiom,
! [A: $tType,F1: A,F22: num > A,F32: num > A,X22: num] :
( ( case_num @ A @ F1 @ F22 @ F32 @ ( bit0 @ X22 ) )
= ( F22 @ X22 ) ) ).
% verit_eq_simplify(17)
thf(fact_4387_verit__eq__simplify_I16_J,axiom,
! [A: $tType,F1: A,F22: num > A,F32: num > A] :
( ( case_num @ A @ F1 @ F22 @ F32 @ one2 )
= F1 ) ).
% verit_eq_simplify(16)
thf(fact_4388_verit__eq__simplify_I18_J,axiom,
! [A: $tType,F1: A,F22: num > A,F32: num > A,X32: num] :
( ( case_num @ A @ F1 @ F22 @ F32 @ ( bit1 @ X32 ) )
= ( F32 @ X32 ) ) ).
% verit_eq_simplify(18)
thf(fact_4389_divmod__integer_H__def,axiom,
( ( unique8689654367752047608divmod @ code_integer )
= ( ^ [M2: num,N4: num] : ( product_Pair @ code_integer @ code_integer @ ( divide_divide @ code_integer @ ( numeral_numeral @ code_integer @ M2 ) @ ( numeral_numeral @ code_integer @ N4 ) ) @ ( modulo_modulo @ code_integer @ ( numeral_numeral @ code_integer @ M2 ) @ ( numeral_numeral @ code_integer @ N4 ) ) ) ) ) ).
% divmod_integer'_def
thf(fact_4390_divmod__divmod__step,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ( ( unique8689654367752047608divmod @ A )
= ( ^ [M2: num,N4: num] : ( if @ ( product_prod @ A @ A ) @ ( ord_less @ num @ M2 @ N4 ) @ ( product_Pair @ A @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ M2 ) ) @ ( unique1321980374590559556d_step @ A @ N4 @ ( unique8689654367752047608divmod @ A @ M2 @ ( bit0 @ N4 ) ) ) ) ) ) ) ).
% divmod_divmod_step
thf(fact_4391_divmod__nat__if,axiom,
( divmod_nat
= ( ^ [M2: nat,N4: nat] :
( if @ ( product_prod @ nat @ nat )
@ ( ( N4
= ( zero_zero @ nat ) )
| ( ord_less @ nat @ M2 @ N4 ) )
@ ( product_Pair @ nat @ nat @ ( zero_zero @ nat ) @ M2 )
@ ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [Q6: nat] : ( product_Pair @ nat @ nat @ ( suc @ Q6 ) )
@ ( divmod_nat @ ( minus_minus @ nat @ M2 @ N4 ) @ N4 ) ) ) ) ) ).
% divmod_nat_if
thf(fact_4392_neg__eucl__rel__int__mult__2,axiom,
! [B2: int,A2: int,Q2: int,R: int] :
( ( ord_less_eq @ int @ B2 @ ( zero_zero @ int ) )
=> ( ( eucl_rel_int @ ( plus_plus @ int @ A2 @ ( one_one @ int ) ) @ B2 @ ( product_Pair @ int @ int @ Q2 @ R ) )
=> ( eucl_rel_int @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ A2 ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ B2 ) @ ( product_Pair @ int @ int @ Q2 @ ( minus_minus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ R ) @ ( one_one @ int ) ) ) ) ) ) ).
% neg_eucl_rel_int_mult_2
thf(fact_4393_divides__aux__eq,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [Q2: A,R: A] :
( ( unique5940410009612947441es_aux @ A @ ( product_Pair @ A @ A @ Q2 @ R ) )
= ( R
= ( zero_zero @ A ) ) ) ) ).
% divides_aux_eq
thf(fact_4394_product__nth,axiom,
! [A: $tType,B: $tType,N: nat,Xs: list @ A,Ys: list @ B] :
( ( ord_less @ nat @ N @ ( times_times @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ B ) @ Ys ) ) )
=> ( ( nth @ ( product_prod @ A @ B ) @ ( product @ A @ B @ Xs @ Ys ) @ N )
= ( product_Pair @ A @ B @ ( nth @ A @ Xs @ ( divide_divide @ nat @ N @ ( size_size @ ( list @ B ) @ Ys ) ) ) @ ( nth @ B @ Ys @ ( modulo_modulo @ nat @ N @ ( size_size @ ( list @ B ) @ Ys ) ) ) ) ) ) ).
% product_nth
thf(fact_4395_length__product,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B] :
( ( size_size @ ( list @ ( product_prod @ A @ B ) ) @ ( product @ A @ B @ Xs @ Ys ) )
= ( times_times @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ B ) @ Ys ) ) ) ).
% length_product
thf(fact_4396_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_4397_eucl__rel__int__dividesI,axiom,
! [L: int,K: int,Q2: int] :
( ( L
!= ( zero_zero @ int ) )
=> ( ( K
= ( times_times @ int @ Q2 @ L ) )
=> ( eucl_rel_int @ K @ L @ ( product_Pair @ int @ int @ Q2 @ ( zero_zero @ int ) ) ) ) ) ).
% eucl_rel_int_dividesI
thf(fact_4398_zminus1__lemma,axiom,
! [A2: int,B2: int,Q2: int,R: int] :
( ( eucl_rel_int @ A2 @ B2 @ ( product_Pair @ int @ int @ Q2 @ R ) )
=> ( ( B2
!= ( zero_zero @ int ) )
=> ( eucl_rel_int @ ( uminus_uminus @ int @ A2 ) @ B2
@ ( product_Pair @ int @ int
@ ( if @ int
@ ( R
= ( zero_zero @ int ) )
@ ( uminus_uminus @ int @ Q2 )
@ ( minus_minus @ int @ ( uminus_uminus @ int @ Q2 ) @ ( one_one @ int ) ) )
@ ( if @ int
@ ( R
= ( zero_zero @ int ) )
@ ( zero_zero @ int )
@ ( minus_minus @ int @ B2 @ R ) ) ) ) ) ) ).
% zminus1_lemma
thf(fact_4399_eucl__rel__int__iff,axiom,
! [K: int,L: int,Q2: int,R: int] :
( ( eucl_rel_int @ K @ L @ ( product_Pair @ int @ int @ Q2 @ R ) )
= ( ( K
= ( plus_plus @ int @ ( times_times @ int @ L @ Q2 ) @ R ) )
& ( ( ord_less @ int @ ( zero_zero @ int ) @ L )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R )
& ( ord_less @ int @ R @ L ) ) )
& ( ~ ( ord_less @ int @ ( zero_zero @ int ) @ L )
=> ( ( ( ord_less @ int @ L @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ L @ R )
& ( ord_less_eq @ int @ R @ ( zero_zero @ int ) ) ) )
& ( ~ ( ord_less @ int @ L @ ( zero_zero @ int ) )
=> ( Q2
= ( zero_zero @ int ) ) ) ) ) ) ) ).
% eucl_rel_int_iff
thf(fact_4400_eucl__rel__int__remainderI,axiom,
! [R: int,L: int,K: int,Q2: int] :
( ( ( sgn_sgn @ int @ R )
= ( sgn_sgn @ int @ L ) )
=> ( ( ord_less @ int @ ( abs_abs @ int @ R ) @ ( abs_abs @ int @ L ) )
=> ( ( K
= ( plus_plus @ int @ ( times_times @ int @ Q2 @ L ) @ R ) )
=> ( eucl_rel_int @ K @ L @ ( product_Pair @ int @ int @ Q2 @ R ) ) ) ) ) ).
% eucl_rel_int_remainderI
thf(fact_4401_eucl__rel__int_Osimps,axiom,
( eucl_rel_int
= ( ^ [A12: int,A23: int,A32: product_prod @ int @ int] :
( ? [K2: int] :
( ( A12 = K2 )
& ( A23
= ( zero_zero @ int ) )
& ( A32
= ( product_Pair @ int @ int @ ( zero_zero @ int ) @ K2 ) ) )
| ? [L2: int,K2: int,Q6: int] :
( ( A12 = K2 )
& ( A23 = L2 )
& ( A32
= ( product_Pair @ int @ int @ Q6 @ ( zero_zero @ int ) ) )
& ( L2
!= ( zero_zero @ int ) )
& ( K2
= ( times_times @ int @ Q6 @ L2 ) ) )
| ? [R2: int,L2: int,K2: int,Q6: int] :
( ( A12 = K2 )
& ( A23 = L2 )
& ( A32
= ( product_Pair @ int @ int @ Q6 @ R2 ) )
& ( ( sgn_sgn @ int @ R2 )
= ( sgn_sgn @ int @ L2 ) )
& ( ord_less @ int @ ( abs_abs @ int @ R2 ) @ ( abs_abs @ int @ L2 ) )
& ( K2
= ( plus_plus @ int @ ( times_times @ int @ Q6 @ L2 ) @ R2 ) ) ) ) ) ) ).
% eucl_rel_int.simps
thf(fact_4402_eucl__rel__int_Ocases,axiom,
! [A13: int,A24: int,A33: product_prod @ int @ int] :
( ( eucl_rel_int @ A13 @ A24 @ A33 )
=> ( ( ( A24
= ( zero_zero @ int ) )
=> ( A33
!= ( product_Pair @ int @ int @ ( zero_zero @ int ) @ A13 ) ) )
=> ( ! [Q4: int] :
( ( A33
= ( product_Pair @ int @ int @ Q4 @ ( zero_zero @ int ) ) )
=> ( ( A24
!= ( zero_zero @ int ) )
=> ( A13
!= ( times_times @ int @ Q4 @ A24 ) ) ) )
=> ~ ! [R6: int,Q4: int] :
( ( A33
= ( product_Pair @ int @ int @ Q4 @ R6 ) )
=> ( ( ( sgn_sgn @ int @ R6 )
= ( sgn_sgn @ int @ A24 ) )
=> ( ( ord_less @ int @ ( abs_abs @ int @ R6 ) @ ( abs_abs @ int @ A24 ) )
=> ( A13
!= ( plus_plus @ int @ ( times_times @ int @ Q4 @ A24 ) @ R6 ) ) ) ) ) ) ) ) ).
% eucl_rel_int.cases
thf(fact_4403_pos__eucl__rel__int__mult__2,axiom,
! [B2: int,A2: int,Q2: int,R: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ B2 )
=> ( ( eucl_rel_int @ A2 @ B2 @ ( product_Pair @ int @ int @ Q2 @ R ) )
=> ( eucl_rel_int @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ A2 ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ B2 ) @ ( product_Pair @ int @ int @ Q2 @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ R ) ) ) ) ) ) ).
% pos_eucl_rel_int_mult_2
thf(fact_4404_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_4405_and__int_Opelims,axiom,
! [X2: int,Xa: int,Y: int] :
( ( ( bit_se5824344872417868541ns_and @ int @ X2 @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ int @ int ) @ bit_and_int_rel @ ( product_Pair @ int @ int @ X2 @ Xa ) )
=> ~ ( ( ( ( ( member @ int @ X2 @ ( 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 ) ) @ X2 )
& ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Xa ) ) ) ) ) )
& ( ~ ( ( member @ int @ X2 @ ( 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 ) ) @ X2 )
& ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Xa ) ) )
@ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( divide_divide @ int @ X2 @ ( 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 @ X2 @ Xa ) ) ) ) ) ).
% and_int.pelims
thf(fact_4406_fun__of__rel__def,axiom,
! [A: $tType,B: $tType] :
( ( fun_of_rel @ B @ A )
= ( ^ [R7: set @ ( product_prod @ B @ A ),X: B] :
( fChoice @ A
@ ^ [Y4: A] : ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X @ Y4 ) @ R7 ) ) ) ) ).
% fun_of_rel_def
thf(fact_4407_ureturn__def,axiom,
! [A: $tType] :
( ( heap_Time_ureturn @ A )
= ( ^ [X: A] :
( heap_Time_heap @ A
@ ^ [H3: heap_ext @ product_unit] : ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ X @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H3 @ ( zero_zero @ nat ) ) ) ) ) ) ).
% ureturn_def
thf(fact_4408_and__int_Opinduct,axiom,
! [A0: int,A13: int,P: int > int > $o] :
( ( accp @ ( product_prod @ int @ int ) @ bit_and_int_rel @ ( product_Pair @ int @ int @ A0 @ A13 ) )
=> ( ! [K4: int,L3: int] :
( ( accp @ ( product_prod @ int @ int ) @ bit_and_int_rel @ ( product_Pair @ int @ int @ K4 @ L3 ) )
=> ( ( ~ ( ( member @ int @ K4 @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
& ( member @ int @ L3 @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
=> ( P @ ( divide_divide @ int @ K4 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) )
=> ( P @ K4 @ L3 ) ) )
=> ( P @ A0 @ A13 ) ) ) ).
% and_int.pinduct
thf(fact_4409_return__def,axiom,
! [A: $tType] :
( ( heap_Time_return @ A )
= ( ^ [X: A] :
( heap_Time_heap @ A
@ ^ [H3: heap_ext @ product_unit] : ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ X @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H3 @ ( one_one @ nat ) ) ) ) ) ) ).
% return_def
thf(fact_4410_upto_Opelims,axiom,
! [X2: int,Xa: int,Y: list @ int] :
( ( ( upto @ X2 @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ int @ int ) @ upto_rel @ ( product_Pair @ int @ int @ X2 @ Xa ) )
=> ~ ( ( ( ( ord_less_eq @ int @ X2 @ Xa )
=> ( Y
= ( cons @ int @ X2 @ ( upto @ ( plus_plus @ int @ X2 @ ( one_one @ int ) ) @ Xa ) ) ) )
& ( ~ ( ord_less_eq @ int @ X2 @ Xa )
=> ( Y
= ( nil @ int ) ) ) )
=> ~ ( accp @ ( product_prod @ int @ int ) @ upto_rel @ ( product_Pair @ int @ int @ X2 @ Xa ) ) ) ) ) ).
% upto.pelims
thf(fact_4411_upto_Opsimps,axiom,
! [I: int,J: int] :
( ( accp @ ( product_prod @ int @ int ) @ upto_rel @ ( product_Pair @ int @ int @ I @ J ) )
=> ( ( ( ord_less_eq @ int @ I @ J )
=> ( ( upto @ I @ J )
= ( cons @ int @ I @ ( upto @ ( plus_plus @ int @ I @ ( one_one @ int ) ) @ J ) ) ) )
& ( ~ ( ord_less_eq @ int @ I @ J )
=> ( ( upto @ I @ J )
= ( nil @ int ) ) ) ) ) ).
% upto.psimps
thf(fact_4412_upto_Opinduct,axiom,
! [A0: int,A13: int,P: int > int > $o] :
( ( accp @ ( product_prod @ int @ int ) @ upto_rel @ ( product_Pair @ int @ int @ A0 @ A13 ) )
=> ( ! [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 @ A13 ) ) ) ).
% upto.pinduct
thf(fact_4413_divmod__integer__eq__cases,axiom,
( code_divmod_integer
= ( ^ [K2: code_integer,L2: code_integer] :
( if @ ( product_prod @ code_integer @ code_integer )
@ ( K2
= ( 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 ) @ K2 )
@ ( 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 @ K2 )
= ( sgn_sgn @ code_integer @ L2 ) )
@ ( code_divmod_abs @ K2 @ L2 )
@ ( product_case_prod @ code_integer @ code_integer @ ( product_prod @ code_integer @ code_integer )
@ ^ [R2: 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 @ R2 ) @ ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( minus_minus @ code_integer @ ( uminus_uminus @ code_integer @ R2 ) @ ( one_one @ code_integer ) ) @ ( minus_minus @ code_integer @ ( abs_abs @ code_integer @ L2 ) @ S6 ) ) )
@ ( code_divmod_abs @ K2 @ L2 ) ) ) ) ) ) ) ) ).
% divmod_integer_eq_cases
thf(fact_4414_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_4415_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_4416_divmod__integer__code,axiom,
( code_divmod_integer
= ( ^ [K2: code_integer,L2: code_integer] :
( if @ ( product_prod @ code_integer @ code_integer )
@ ( K2
= ( 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 ) @ K2 ) @ ( code_divmod_abs @ K2 @ L2 )
@ ( product_case_prod @ code_integer @ code_integer @ ( product_prod @ code_integer @ code_integer )
@ ^ [R2: 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 @ R2 ) @ ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( minus_minus @ code_integer @ ( uminus_uminus @ code_integer @ R2 ) @ ( one_one @ code_integer ) ) @ ( minus_minus @ code_integer @ L2 @ S6 ) ) )
@ ( code_divmod_abs @ K2 @ L2 ) ) )
@ ( if @ ( product_prod @ code_integer @ code_integer )
@ ( L2
= ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( zero_zero @ code_integer ) @ K2 )
@ ( product_apsnd @ code_integer @ code_integer @ code_integer @ ( uminus_uminus @ code_integer )
@ ( if @ ( product_prod @ code_integer @ code_integer ) @ ( ord_less @ code_integer @ K2 @ ( zero_zero @ code_integer ) ) @ ( code_divmod_abs @ K2 @ L2 )
@ ( product_case_prod @ code_integer @ code_integer @ ( product_prod @ code_integer @ code_integer )
@ ^ [R2: 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 @ R2 ) @ ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( minus_minus @ code_integer @ ( uminus_uminus @ code_integer @ R2 ) @ ( one_one @ code_integer ) ) @ ( minus_minus @ code_integer @ ( uminus_uminus @ code_integer @ L2 ) @ S6 ) ) )
@ ( code_divmod_abs @ K2 @ L2 ) ) ) ) ) ) ) ) ) ).
% divmod_integer_code
thf(fact_4417_divmod__abs__def,axiom,
( code_divmod_abs
= ( ^ [K2: code_integer,L2: code_integer] : ( product_Pair @ code_integer @ code_integer @ ( divide_divide @ code_integer @ ( abs_abs @ code_integer @ K2 ) @ ( abs_abs @ code_integer @ L2 ) ) @ ( modulo_modulo @ code_integer @ ( abs_abs @ code_integer @ K2 ) @ ( abs_abs @ code_integer @ L2 ) ) ) ) ) ).
% divmod_abs_def
thf(fact_4418_bit__cut__integer__code,axiom,
( code_bit_cut_integer
= ( ^ [K2: code_integer] :
( if @ ( product_prod @ code_integer @ $o )
@ ( K2
= ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ $o @ ( zero_zero @ code_integer ) @ $false )
@ ( product_case_prod @ code_integer @ code_integer @ ( product_prod @ code_integer @ $o )
@ ^ [R2: code_integer,S6: code_integer] :
( product_Pair @ code_integer @ $o @ ( if @ code_integer @ ( ord_less @ code_integer @ ( zero_zero @ code_integer ) @ K2 ) @ R2 @ ( minus_minus @ code_integer @ ( uminus_uminus @ code_integer @ R2 ) @ S6 ) )
@ ( S6
= ( one_one @ code_integer ) ) )
@ ( code_divmod_abs @ K2 @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% bit_cut_integer_code
thf(fact_4419_normalize__negative,axiom,
! [Q2: int,P3: int] :
( ( ord_less @ int @ Q2 @ ( zero_zero @ int ) )
=> ( ( normalize @ ( product_Pair @ int @ int @ P3 @ Q2 ) )
= ( normalize @ ( product_Pair @ int @ int @ ( uminus_uminus @ int @ P3 ) @ ( uminus_uminus @ int @ Q2 ) ) ) ) ) ).
% normalize_negative
thf(fact_4420_bit__cut__integer__def,axiom,
( code_bit_cut_integer
= ( ^ [K2: code_integer] :
( product_Pair @ code_integer @ $o @ ( divide_divide @ code_integer @ K2 @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) )
@ ~ ( dvd_dvd @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) @ K2 ) ) ) ) ).
% bit_cut_integer_def
thf(fact_4421_lex__take__index,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( lex @ A @ R ) )
=> ~ ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Ys ) )
=> ( ( ( take @ A @ I3 @ Xs )
= ( take @ A @ I3 @ Ys ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( nth @ A @ Xs @ I3 ) @ ( nth @ A @ Ys @ I3 ) ) @ R ) ) ) ) ) ).
% lex_take_index
thf(fact_4422_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_4423_Cons__in__lex,axiom,
! [A: $tType,X2: A,Xs: list @ A,Y: A,Ys: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X2 @ Xs ) @ ( cons @ A @ Y @ Ys ) ) @ ( lex @ A @ R ) )
= ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y ) @ R )
& ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ A ) @ Ys ) ) )
| ( ( X2 = Y )
& ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( lex @ A @ R ) ) ) ) ) ).
% Cons_in_lex
thf(fact_4424_lex__append__rightI,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,R: set @ ( product_prod @ A @ A ),Vs: list @ A,Us: list @ A] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( lex @ A @ R ) )
=> ( ( ( size_size @ ( list @ A ) @ Vs )
= ( size_size @ ( list @ A ) @ Us ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs @ Us ) @ ( append @ A @ Ys @ Vs ) ) @ ( lex @ A @ R ) ) ) ) ).
% lex_append_rightI
thf(fact_4425_normalize__denom__pos,axiom,
! [R: product_prod @ int @ int,P3: int,Q2: int] :
( ( ( normalize @ R )
= ( product_Pair @ int @ int @ P3 @ Q2 ) )
=> ( ord_less @ int @ ( zero_zero @ int ) @ Q2 ) ) ).
% normalize_denom_pos
thf(fact_4426_normalize__crossproduct,axiom,
! [Q2: int,S3: int,P3: int,R: int] :
( ( Q2
!= ( zero_zero @ int ) )
=> ( ( S3
!= ( zero_zero @ int ) )
=> ( ( ( normalize @ ( product_Pair @ int @ int @ P3 @ Q2 ) )
= ( normalize @ ( product_Pair @ int @ int @ R @ S3 ) ) )
=> ( ( times_times @ int @ P3 @ S3 )
= ( times_times @ int @ R @ Q2 ) ) ) ) ) ).
% normalize_crossproduct
thf(fact_4427_part__code_I2_J,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F: B > A,Pivot: A,X2: B,Xs: list @ B] :
( ( linorder_part @ B @ A @ F @ Pivot @ ( cons @ B @ X2 @ Xs ) )
= ( product_case_prod @ ( list @ B ) @ ( product_prod @ ( list @ B ) @ ( list @ B ) ) @ ( product_prod @ ( list @ B ) @ ( product_prod @ ( list @ B ) @ ( list @ B ) ) )
@ ^ [Lts: list @ B] :
( product_case_prod @ ( list @ B ) @ ( list @ B ) @ ( product_prod @ ( list @ B ) @ ( product_prod @ ( list @ B ) @ ( list @ B ) ) )
@ ^ [Eqs: list @ B,Gts: list @ B] : ( if @ ( product_prod @ ( list @ B ) @ ( product_prod @ ( list @ B ) @ ( list @ B ) ) ) @ ( ord_less @ A @ ( F @ X2 ) @ Pivot ) @ ( product_Pair @ ( list @ B ) @ ( product_prod @ ( list @ B ) @ ( list @ B ) ) @ ( cons @ B @ X2 @ Lts ) @ ( product_Pair @ ( list @ B ) @ ( list @ B ) @ Eqs @ Gts ) ) @ ( if @ ( product_prod @ ( list @ B ) @ ( product_prod @ ( list @ B ) @ ( list @ B ) ) ) @ ( ord_less @ A @ Pivot @ ( F @ X2 ) ) @ ( product_Pair @ ( list @ B ) @ ( product_prod @ ( list @ B ) @ ( list @ B ) ) @ Lts @ ( product_Pair @ ( list @ B ) @ ( list @ B ) @ Eqs @ ( cons @ B @ X2 @ Gts ) ) ) @ ( product_Pair @ ( list @ B ) @ ( product_prod @ ( list @ B ) @ ( list @ B ) ) @ Lts @ ( product_Pair @ ( list @ B ) @ ( list @ B ) @ ( cons @ B @ X2 @ Eqs ) @ Gts ) ) ) ) )
@ ( linorder_part @ B @ A @ F @ Pivot @ Xs ) ) ) ) ).
% part_code(2)
thf(fact_4428_prod__decode__aux_Oelims,axiom,
! [X2: nat,Xa: nat,Y: product_prod @ nat @ nat] :
( ( ( nat_prod_decode_aux @ X2 @ Xa )
= Y )
=> ( ( ( ord_less_eq @ nat @ Xa @ X2 )
=> ( Y
= ( product_Pair @ nat @ nat @ Xa @ ( minus_minus @ nat @ X2 @ Xa ) ) ) )
& ( ~ ( ord_less_eq @ nat @ Xa @ X2 )
=> ( Y
= ( nat_prod_decode_aux @ ( suc @ X2 ) @ ( minus_minus @ nat @ Xa @ ( suc @ X2 ) ) ) ) ) ) ) ).
% prod_decode_aux.elims
thf(fact_4429_prod__decode__aux_Osimps,axiom,
( nat_prod_decode_aux
= ( ^ [K2: nat,M2: nat] : ( if @ ( product_prod @ nat @ nat ) @ ( ord_less_eq @ nat @ M2 @ K2 ) @ ( product_Pair @ nat @ nat @ M2 @ ( minus_minus @ nat @ K2 @ M2 ) ) @ ( nat_prod_decode_aux @ ( suc @ K2 ) @ ( minus_minus @ nat @ M2 @ ( suc @ K2 ) ) ) ) ) ) ).
% prod_decode_aux.simps
thf(fact_4430_bezw__0,axiom,
! [X2: nat] :
( ( bezw @ X2 @ ( zero_zero @ nat ) )
= ( product_Pair @ int @ int @ ( one_one @ int ) @ ( zero_zero @ int ) ) ) ).
% bezw_0
thf(fact_4431_extract__Cons__code,axiom,
! [A: $tType,P: A > $o,X2: A,Xs: list @ A] :
( ( ( P @ X2 )
=> ( ( extract @ A @ P @ ( cons @ A @ X2 @ 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 ) @ X2 @ Xs ) ) ) ) )
& ( ~ ( P @ X2 )
=> ( ( extract @ A @ P @ ( cons @ A @ X2 @ Xs ) )
= ( case_option @ ( option @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) ) @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) @ ( none @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) )
@ ( product_case_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) @ ( option @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) )
@ ^ [Ys3: list @ A] :
( product_case_prod @ A @ ( list @ A ) @ ( option @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) )
@ ^ [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 @ X2 @ Ys3 ) @ ( product_Pair @ A @ ( list @ A ) @ Y4 @ Zs3 ) ) ) ) )
@ ( extract @ A @ P @ Xs ) ) ) ) ) ).
% extract_Cons_code
thf(fact_4432_of__list__def,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( ( array_of_list @ A )
= ( ^ [Xs3: list @ A] :
( heap_Time_heap @ ( array @ A )
@ ^ [H3: heap_ext @ product_unit] :
( product_case_prod @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( product_prod @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )
@ ^ [R2: array @ A,H5: heap_ext @ product_unit] : ( product_Pair @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R2 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H5 @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( size_size @ ( list @ A ) @ Xs3 ) ) ) )
@ ( array_alloc @ A @ Xs3 @ H3 ) ) ) ) ) ) ).
% of_list_def
thf(fact_4433_nth__enumerate__eq,axiom,
! [A: $tType,M: nat,Xs: list @ A,N: nat] :
( ( ord_less @ nat @ M @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ ( product_prod @ nat @ A ) @ ( enumerate @ A @ N @ Xs ) @ M )
= ( product_Pair @ nat @ A @ ( plus_plus @ nat @ N @ M ) @ ( nth @ A @ Xs @ M ) ) ) ) ).
% nth_enumerate_eq
thf(fact_4434_timeFrame_Opelims,axiom,
! [A: $tType,X2: 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 @ X2 @ 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 ) ) ) @ X2 @ Xa ) )
=> ( ! [R6: A,H2: heap_ext @ product_unit,N8: nat] :
( ( Xa
= ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R6 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H2 @ N8 ) ) ) )
=> ( ( Y
= ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R6 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H2 @ ( plus_plus @ nat @ X2 @ N8 ) ) ) ) )
=> ~ ( accp @ ( product_prod @ nat @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) @ ( heap_T5500966940807335491me_rel @ A ) @ ( product_Pair @ nat @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ X2 @ ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R6 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H2 @ N8 ) ) ) ) ) ) )
=> ~ ( ( Xa
= ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) )
=> ( ( Y
= ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) )
=> ~ ( accp @ ( product_prod @ nat @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) @ ( heap_T5500966940807335491me_rel @ A ) @ ( product_Pair @ nat @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ X2 @ ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ) ) ) ) ) ).
% timeFrame.pelims
thf(fact_4435_length__enumerate,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( size_size @ ( list @ ( product_prod @ nat @ A ) ) @ ( enumerate @ A @ N @ Xs ) )
= ( size_size @ ( list @ A ) @ Xs ) ) ).
% length_enumerate
thf(fact_4436_enumerate__append__eq,axiom,
! [A: $tType,N: nat,Xs: list @ A,Ys: list @ A] :
( ( enumerate @ A @ N @ ( append @ A @ Xs @ Ys ) )
= ( append @ ( product_prod @ nat @ A ) @ ( enumerate @ A @ N @ Xs ) @ ( enumerate @ A @ ( plus_plus @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) ) @ Ys ) ) ) ).
% enumerate_append_eq
thf(fact_4437_new__def,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( ( array_new @ A )
= ( ^ [N4: nat,X: A] :
( heap_Time_heap @ ( array @ A )
@ ^ [H3: heap_ext @ product_unit] :
( product_case_prod @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( product_prod @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )
@ ^ [R2: array @ A,H5: heap_ext @ product_unit] : ( product_Pair @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R2 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H5 @ ( plus_plus @ nat @ N4 @ ( one_one @ nat ) ) ) )
@ ( array_alloc @ A @ ( replicate @ A @ N4 @ X ) @ H3 ) ) ) ) ) ) ).
% new_def
thf(fact_4438_execute__of__list,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [Xs: list @ A,H: heap_ext @ product_unit] :
( ( heap_Time_execute @ ( array @ A ) @ ( array_of_list @ A @ Xs ) @ H )
= ( some @ ( product_prod @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )
@ ( product_case_prod @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( product_prod @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )
@ ^ [R2: array @ A,H5: heap_ext @ product_unit] : ( product_Pair @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R2 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H5 @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) )
@ ( array_alloc @ A @ Xs @ H ) ) ) ) ) ).
% execute_of_list
thf(fact_4439_subset__eq__mset__impl_Oelims,axiom,
! [A: $tType,X2: list @ A,Xa: list @ A,Y: option @ $o] :
( ( ( subset_eq_mset_impl @ A @ X2 @ Xa )
= Y )
=> ( ( ( X2
= ( nil @ A ) )
=> ( Y
!= ( some @ $o
@ ( Xa
!= ( nil @ A ) ) ) ) )
=> ~ ! [X3: A,Xs2: list @ A] :
( ( X2
= ( 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
@ ( ^ [Y3: A,Z: A] : Y3 = Z
@ X3 )
@ Xa ) ) ) ) ) ) ).
% subset_eq_mset_impl.elims
thf(fact_4440_Cons__lenlex__iff,axiom,
! [A: $tType,M: A,Ms: list @ A,N: A,Ns: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ M @ Ms ) @ ( cons @ A @ N @ Ns ) ) @ ( lenlex @ A @ R ) )
= ( ( 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 @ N ) @ R ) )
| ( ( M = N )
& ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ms @ Ns ) @ ( lenlex @ A @ R ) ) ) ) ) ).
% Cons_lenlex_iff
thf(fact_4441_replicate__eq__replicate,axiom,
! [A: $tType,M: nat,X2: A,N: nat,Y: A] :
( ( ( replicate @ A @ M @ X2 )
= ( replicate @ A @ N @ Y ) )
= ( ( M = N )
& ( ( M
!= ( zero_zero @ nat ) )
=> ( X2 = Y ) ) ) ) ).
% replicate_eq_replicate
thf(fact_4442_length__replicate,axiom,
! [A: $tType,N: nat,X2: A] :
( ( size_size @ ( list @ A ) @ ( replicate @ A @ N @ X2 ) )
= N ) ).
% length_replicate
thf(fact_4443_replicate__empty,axiom,
! [A: $tType,N: nat,X2: A] :
( ( ( replicate @ A @ N @ X2 )
= ( nil @ A ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% replicate_empty
thf(fact_4444_empty__replicate,axiom,
! [A: $tType,N: nat,X2: A] :
( ( ( nil @ A )
= ( replicate @ A @ N @ X2 ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% empty_replicate
thf(fact_4445_in__set__replicate,axiom,
! [A: $tType,X2: A,N: nat,Y: A] :
( ( member @ A @ X2 @ ( set2 @ A @ ( replicate @ A @ N @ Y ) ) )
= ( ( X2 = Y )
& ( N
!= ( zero_zero @ nat ) ) ) ) ).
% in_set_replicate
thf(fact_4446_Bex__set__replicate,axiom,
! [A: $tType,N: nat,A2: A,P: A > $o] :
( ( ? [X: A] :
( ( member @ A @ X @ ( set2 @ A @ ( replicate @ A @ N @ A2 ) ) )
& ( P @ X ) ) )
= ( ( P @ A2 )
& ( N
!= ( zero_zero @ nat ) ) ) ) ).
% Bex_set_replicate
thf(fact_4447_Ball__set__replicate,axiom,
! [A: $tType,N: nat,A2: A,P: A > $o] :
( ( ! [X: A] :
( ( member @ A @ X @ ( set2 @ A @ ( replicate @ A @ N @ A2 ) ) )
=> ( P @ X ) ) )
= ( ( P @ A2 )
| ( N
= ( zero_zero @ nat ) ) ) ) ).
% Ball_set_replicate
thf(fact_4448_hd__replicate,axiom,
! [A: $tType,N: nat,X2: A] :
( ( N
!= ( zero_zero @ nat ) )
=> ( ( hd @ A @ ( replicate @ A @ N @ X2 ) )
= X2 ) ) ).
% hd_replicate
thf(fact_4449_last__replicate,axiom,
! [A: $tType,N: nat,X2: A] :
( ( N
!= ( zero_zero @ nat ) )
=> ( ( last @ A @ ( replicate @ A @ N @ X2 ) )
= X2 ) ) ).
% last_replicate
thf(fact_4450_set__replicate,axiom,
! [A: $tType,N: nat,X2: A] :
( ( N
!= ( zero_zero @ nat ) )
=> ( ( set2 @ A @ ( replicate @ A @ N @ X2 ) )
= ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% set_replicate
thf(fact_4451_Heap__eqI,axiom,
! [A: $tType,F: heap_Time_Heap @ A,G: heap_Time_Heap @ A] :
( ! [H2: heap_ext @ product_unit] :
( ( heap_Time_execute @ A @ F @ H2 )
= ( heap_Time_execute @ A @ G @ H2 ) )
=> ( F = G ) ) ).
% Heap_eqI
thf(fact_4452_replicate__0,axiom,
! [A: $tType,X2: A] :
( ( replicate @ A @ ( zero_zero @ nat ) @ X2 )
= ( nil @ A ) ) ).
% replicate_0
thf(fact_4453_replicate__length__same,axiom,
! [A: $tType,Xs: list @ A,X2: A] :
( ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( X3 = X2 ) )
=> ( ( replicate @ A @ ( size_size @ ( list @ A ) @ Xs ) @ X2 )
= Xs ) ) ).
% replicate_length_same
thf(fact_4454_replicate__eqI,axiom,
! [A: $tType,Xs: list @ A,N: nat,X2: A] :
( ( ( size_size @ ( list @ A ) @ Xs )
= N )
=> ( ! [Y2: A] :
( ( member @ A @ Y2 @ ( set2 @ A @ Xs ) )
=> ( Y2 = X2 ) )
=> ( Xs
= ( replicate @ A @ N @ X2 ) ) ) ) ).
% replicate_eqI
thf(fact_4455_sorted__replicate,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [N: nat,X2: A] : ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( replicate @ A @ N @ X2 ) ) ) ).
% sorted_replicate
thf(fact_4456_replicate__add,axiom,
! [A: $tType,N: nat,M: nat,X2: A] :
( ( replicate @ A @ ( plus_plus @ nat @ N @ M ) @ X2 )
= ( append @ A @ ( replicate @ A @ N @ X2 ) @ ( replicate @ A @ M @ X2 ) ) ) ).
% replicate_add
thf(fact_4457_execute__bind_I2_J,axiom,
! [A: $tType,B: $tType,F: heap_Time_Heap @ A,H: heap_ext @ product_unit,G: A > ( heap_Time_Heap @ B )] :
( ( ( heap_Time_execute @ A @ F @ H )
= ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) )
=> ( ( heap_Time_execute @ B @ ( heap_Time_bind @ A @ B @ F @ G ) @ H )
= ( none @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ).
% execute_bind(2)
thf(fact_4458_fails__def,axiom,
! [A: $tType] :
( ( time_fails @ A )
= ( ^ [M2: heap_Time_Heap @ A,H3: heap_ext @ product_unit] :
( ( heap_Time_execute @ A @ M2 @ H3 )
= ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ) ).
% fails_def
thf(fact_4459_execute__heap,axiom,
! [A: $tType,F: ( heap_ext @ product_unit ) > ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )] :
( ( heap_Time_execute @ A @ ( heap_Time_heap @ A @ F ) )
= ( 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 ) ) ) @ F ) ) ).
% execute_heap
thf(fact_4460_time__def,axiom,
! [A: $tType] :
( ( time_time @ A )
= ( ^ [C5: heap_Time_Heap @ A,H3: heap_ext @ product_unit] :
( case_option @ nat @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( zero_zero @ nat )
@ ( product_case_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ nat
@ ^ [X: A] :
( product_case_prod @ ( heap_ext @ product_unit ) @ nat @ nat
@ ^ [Xa5: heap_ext @ product_unit,T3: nat] : T3 ) )
@ ( heap_Time_execute @ A @ C5 @ H3 ) ) ) ) ).
% time_def
thf(fact_4461_execute__raise,axiom,
! [A: $tType,S3: list @ char] :
( ( heap_Time_execute @ A @ ( heap_Time_raise @ A @ S3 ) )
= ( ^ [Uu: heap_ext @ product_unit] : ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ).
% execute_raise
thf(fact_4462_Heap__cases,axiom,
! [A: $tType,F: heap_Time_Heap @ A,H: heap_ext @ product_unit] :
( ! [X3: A,H6: product_prod @ ( heap_ext @ product_unit ) @ nat] :
( ( heap_Time_execute @ A @ F @ H )
!= ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ X3 @ H6 ) ) )
=> ( ( heap_Time_execute @ A @ F @ H )
= ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ).
% Heap_cases
thf(fact_4463_set__replicate__conv__if,axiom,
! [A: $tType,N: nat,X2: A] :
( ( ( N
= ( zero_zero @ nat ) )
=> ( ( set2 @ A @ ( replicate @ A @ N @ X2 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ( ( set2 @ A @ ( replicate @ A @ N @ X2 ) )
= ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% set_replicate_conv_if
thf(fact_4464_replicate__Suc__conv__snoc,axiom,
! [A: $tType,N: nat,X2: A] :
( ( replicate @ A @ ( suc @ N ) @ X2 )
= ( append @ A @ ( replicate @ A @ N @ X2 ) @ ( cons @ A @ X2 @ ( nil @ A ) ) ) ) ).
% replicate_Suc_conv_snoc
thf(fact_4465_execute__bind__case,axiom,
! [A: $tType,B: $tType,F: heap_Time_Heap @ B,G: B > ( heap_Time_Heap @ A ),H: heap_ext @ product_unit] :
( ( heap_Time_execute @ A @ ( heap_Time_bind @ B @ A @ F @ G ) @ 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 ) ) )
@ ^ [X: B] :
( product_case_prod @ ( heap_ext @ product_unit ) @ nat @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )
@ ^ [H5: heap_ext @ product_unit,N4: nat] : ( heap_Time_timeFrame @ A @ N4 @ ( heap_Time_execute @ A @ ( G @ X ) @ H5 ) ) ) )
@ ( heap_Time_execute @ B @ F @ H ) ) ) ).
% execute_bind_case
thf(fact_4466_execute__new,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [N: nat,X2: A,H: heap_ext @ product_unit] :
( ( heap_Time_execute @ ( array @ A ) @ ( array_new @ A @ N @ X2 ) @ H )
= ( some @ ( product_prod @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )
@ ( product_case_prod @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( product_prod @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )
@ ^ [R2: array @ A,H5: heap_ext @ product_unit] : ( product_Pair @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R2 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H5 @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) )
@ ( array_alloc @ A @ ( replicate @ A @ N @ X2 ) @ H ) ) ) ) ) ).
% execute_new
thf(fact_4467_lenlex__length,axiom,
! [A: $tType,Ms: list @ A,Ns: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ms @ Ns ) @ ( lenlex @ A @ R ) )
=> ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Ms ) @ ( size_size @ ( list @ A ) @ Ns ) ) ) ).
% lenlex_length
thf(fact_4468_lenlex__append1,axiom,
! [A: $tType,Us: list @ A,Xs: list @ A,R3: set @ ( product_prod @ A @ A ),Vs: list @ A,Ys: list @ A] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Us @ Xs ) @ ( lenlex @ A @ R3 ) )
=> ( ( ( size_size @ ( list @ A ) @ Vs )
= ( size_size @ ( list @ A ) @ Ys ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Us @ Vs ) @ ( append @ A @ Xs @ Ys ) ) @ ( lenlex @ A @ R3 ) ) ) ) ).
% lenlex_append1
thf(fact_4469_Cons__replicate__eq,axiom,
! [A: $tType,X2: A,Xs: list @ A,N: nat,Y: A] :
( ( ( cons @ A @ X2 @ Xs )
= ( replicate @ A @ N @ Y ) )
= ( ( X2 = Y )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
& ( Xs
= ( replicate @ A @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ X2 ) ) ) ) ).
% Cons_replicate_eq
thf(fact_4470_refines__def,axiom,
! [A: $tType] :
( ( refine_Imp_refines @ A )
= ( ^ [P7: heap_Time_Heap @ A,Q6: heap_Time_Heap @ A] :
! [H3: heap_ext @ product_unit] :
( case_option @ $o @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ $true
@ ( product_case_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ $o
@ ^ [R2: A] :
( product_case_prod @ ( heap_ext @ product_unit ) @ nat @ $o
@ ^ [H5: heap_ext @ product_unit,T3: nat] :
( ( heap_Time_execute @ A @ P7 @ H3 )
= ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R2 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H5 @ T3 ) ) ) ) ) )
@ ( heap_Time_execute @ A @ Q6 @ H3 ) ) ) ) ).
% refines_def
thf(fact_4471_execute__bind__eq__SomeI,axiom,
! [A: $tType,B: $tType,F: heap_Time_Heap @ A,H: heap_ext @ product_unit,X2: A,H4: heap_ext @ product_unit,N: nat,G: A > ( heap_Time_Heap @ B ),Y: B,H7: heap_ext @ product_unit,N3: nat] :
( ( ( heap_Time_execute @ A @ F @ H )
= ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ X2 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H4 @ N ) ) ) )
=> ( ( ( heap_Time_execute @ B @ ( G @ X2 ) @ H4 )
= ( some @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ Y @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H7 @ N3 ) ) ) )
=> ( ( heap_Time_execute @ B @ ( heap_Time_bind @ A @ B @ F @ G ) @ 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 @ H7 @ ( plus_plus @ nat @ N @ N3 ) ) ) ) ) ) ) ).
% execute_bind_eq_SomeI
thf(fact_4472_execute__bind_I1_J,axiom,
! [A: $tType,B: $tType,F: heap_Time_Heap @ A,H: heap_ext @ product_unit,X2: A,H4: heap_ext @ product_unit,N: nat,G: A > ( heap_Time_Heap @ B )] :
( ( ( heap_Time_execute @ A @ F @ H )
= ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ X2 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H4 @ N ) ) ) )
=> ( ( heap_Time_execute @ B @ ( heap_Time_bind @ A @ B @ F @ G ) @ H )
= ( heap_Time_timeFrame @ B @ N @ ( heap_Time_execute @ B @ ( G @ X2 ) @ H4 ) ) ) ) ).
% execute_bind(1)
thf(fact_4473_execute__return,axiom,
! [A: $tType,X2: A] :
( ( heap_Time_execute @ A @ ( heap_Time_return @ A @ X2 ) )
= ( comp @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ ( heap_ext @ product_unit ) @ ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )
@ ^ [H3: heap_ext @ product_unit] : ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ X2 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H3 @ ( one_one @ nat ) ) ) ) ) ).
% execute_return
thf(fact_4474_execute__ureturn,axiom,
! [A: $tType,X2: A] :
( ( heap_Time_execute @ A @ ( heap_Time_ureturn @ A @ X2 ) )
= ( comp @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ ( heap_ext @ product_unit ) @ ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )
@ ^ [H3: heap_ext @ product_unit] : ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ X2 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H3 @ ( zero_zero @ nat ) ) ) ) ) ).
% execute_ureturn
thf(fact_4475_lenlex__conv,axiom,
! [A: $tType] :
( ( lenlex @ A )
= ( ^ [R2: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ ( list @ A ) @ ( list @ A ) )
@ ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
@ ^ [Xs3: list @ A,Ys3: list @ A] :
( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Xs3 ) @ ( size_size @ ( list @ A ) @ Ys3 ) )
| ( ( ( size_size @ ( list @ A ) @ Xs3 )
= ( size_size @ ( list @ A ) @ Ys3 ) )
& ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs3 @ Ys3 ) @ ( lex @ A @ R2 ) ) ) ) ) ) ) ) ).
% lenlex_conv
thf(fact_4476_subset__eq__mset__impl_Osimps_I2_J,axiom,
! [A: $tType,X2: A,Xs: list @ A,Ys: list @ A] :
( ( subset_eq_mset_impl @ A @ ( cons @ A @ X2 @ Xs ) @ Ys )
= ( case_option @ ( option @ $o ) @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) @ ( none @ $o )
@ ( product_case_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) @ ( option @ $o )
@ ^ [Ys1: list @ A] :
( product_case_prod @ A @ ( list @ A ) @ ( option @ $o )
@ ^ [X: A,Ys22: list @ A] : ( subset_eq_mset_impl @ A @ Xs @ ( append @ A @ Ys1 @ Ys22 ) ) ) )
@ ( extract @ A
@ ( ^ [Y3: A,Z: A] : Y3 = Z
@ X2 )
@ Ys ) ) ) ).
% subset_eq_mset_impl.simps(2)
thf(fact_4477_execute__tap,axiom,
! [A: $tType,F: ( heap_ext @ product_unit ) > A,H: heap_ext @ product_unit] :
( ( heap_Time_execute @ A @ ( heap_Time_tap @ A @ F ) @ H )
= ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( F @ H ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H @ ( one_one @ nat ) ) ) ) ) ).
% execute_tap
thf(fact_4478_subset__eq__mset__impl_Opelims,axiom,
! [A: $tType,X2: list @ A,Xa: list @ A,Y: option @ $o] :
( ( ( subset_eq_mset_impl @ A @ X2 @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( subset751672762298770561pl_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X2 @ Xa ) )
=> ( ( ( X2
= ( 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] :
( ( X2
= ( 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
@ ( ^ [Y3: A,Z: A] : Y3 = Z
@ 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_4479_execute__assert_I1_J,axiom,
! [A: $tType,P: A > $o,X2: A,H: heap_ext @ product_unit] :
( ( P @ X2 )
=> ( ( heap_Time_execute @ A @ ( heap_Time_assert @ A @ P @ X2 ) @ H )
= ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ X2 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H @ ( one_one @ nat ) ) ) ) ) ) ).
% execute_assert(1)
thf(fact_4480_execute__lookup,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [R: ref @ A,H: heap_ext @ product_unit] :
( ( heap_Time_execute @ A @ ( ref_lookup @ A @ R ) @ 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 @ R ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H @ ( one_one @ nat ) ) ) ) ) ) ).
% execute_lookup
thf(fact_4481_lookup__def,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( ( ref_lookup @ A )
= ( ^ [R2: ref @ A] :
( heap_Time_tap @ A
@ ^ [H3: heap_ext @ product_unit] : ( ref_get @ A @ H3 @ R2 ) ) ) ) ) ).
% lookup_def
thf(fact_4482_assert__cong,axiom,
! [B: $tType,A: $tType,P: A > $o,P2: A > $o,F: A > ( heap_Time_Heap @ B ),F3: A > ( heap_Time_Heap @ B ),X2: A] :
( ( P = P2 )
=> ( ! [X3: A] :
( ( P2 @ X3 )
=> ( ( F @ X3 )
= ( F3 @ X3 ) ) )
=> ( ( heap_Time_bind @ A @ B @ ( heap_Time_assert @ A @ P @ X2 ) @ F )
= ( heap_Time_bind @ A @ B @ ( heap_Time_assert @ A @ P2 @ X2 ) @ F3 ) ) ) ) ).
% assert_cong
thf(fact_4483_execute__assert_I2_J,axiom,
! [A: $tType,P: A > $o,X2: A,H: heap_ext @ product_unit] :
( ~ ( P @ X2 )
=> ( ( heap_Time_execute @ A @ ( heap_Time_assert @ A @ P @ X2 ) @ H )
= ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ).
% execute_assert(2)
thf(fact_4484_revg_Opelims,axiom,
! [A: $tType,X2: list @ A,Xa: list @ A,Y: list @ A] :
( ( ( revg @ A @ X2 @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( revg_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X2 @ Xa ) )
=> ( ( ( X2
= ( nil @ A ) )
=> ( ( Y = Xa )
=> ~ ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( revg_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Xa ) ) ) )
=> ~ ! [A4: A,As2: list @ A] :
( ( X2
= ( cons @ A @ A4 @ As2 ) )
=> ( ( Y
= ( revg @ A @ As2 @ ( cons @ A @ A4 @ Xa ) ) )
=> ~ ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( revg_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ A4 @ As2 ) @ Xa ) ) ) ) ) ) ) ).
% revg.pelims
thf(fact_4485_Heap__Time__Monad_Obind__def,axiom,
! [B: $tType,A: $tType] :
( ( heap_Time_bind @ A @ B )
= ( ^ [F2: heap_Time_Heap @ A,G2: A > ( heap_Time_Heap @ B )] :
( heap_Time_Heap2 @ B
@ ^ [H3: 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 ) ) )
@ ^ [R2: A] :
( product_case_prod @ ( heap_ext @ product_unit ) @ nat @ ( option @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )
@ ^ [H5: heap_ext @ product_unit,N4: nat] : ( heap_Time_timeFrame @ B @ N4 @ ( heap_Time_execute @ B @ ( G2 @ R2 ) @ H5 ) ) ) )
@ ( heap_Time_execute @ A @ F2 @ H3 ) ) ) ) ) ).
% Heap_Time_Monad.bind_def
thf(fact_4486_the__res__def,axiom,
! [A: $tType] :
( ( time_the_res @ A )
= ( ^ [M2: heap_Time_Heap @ A,H3: heap_ext @ product_unit] :
( case_option @ A @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( undefined @ A )
@ ( product_case_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ A
@ ^ [R2: A] :
( product_case_prod @ ( heap_ext @ product_unit ) @ nat @ A
@ ^ [X: heap_ext @ product_unit,Xa5: nat] : R2 ) )
@ ( heap_Time_execute @ A @ M2 @ H3 ) ) ) ) ).
% the_res_def
thf(fact_4487_success__bind__executeI,axiom,
! [A: $tType,B: $tType,F: heap_Time_Heap @ A,H: heap_ext @ product_unit,X2: A,H4: heap_ext @ product_unit,N: nat,G: A > ( heap_Time_Heap @ B )] :
( ( ( heap_Time_execute @ A @ F @ H )
= ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ X2 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H4 @ N ) ) ) )
=> ( ( heap_Time_success @ B @ ( G @ X2 ) @ H4 )
=> ( heap_Time_success @ B @ ( heap_Time_bind @ A @ B @ F @ G ) @ H ) ) ) ).
% success_bind_executeI
thf(fact_4488_Heap_Oinject,axiom,
! [A: $tType,X2: ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ),Ya: ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )] :
( ( ( heap_Time_Heap2 @ A @ X2 )
= ( heap_Time_Heap2 @ A @ Ya ) )
= ( X2 = Ya ) ) ).
% Heap.inject
thf(fact_4489_Heap__execute,axiom,
! [A: $tType,F: heap_Time_Heap @ A] :
( ( heap_Time_Heap2 @ A @ ( heap_Time_execute @ A @ F ) )
= F ) ).
% Heap_execute
thf(fact_4490_execute_Osimps,axiom,
! [A: $tType,F: ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )] :
( ( heap_Time_execute @ A @ ( heap_Time_Heap2 @ A @ F ) )
= F ) ).
% execute.simps
thf(fact_4491_success__ifI,axiom,
! [A: $tType,C2: $o,T2: heap_Time_Heap @ A,H: heap_ext @ product_unit,E2: heap_Time_Heap @ A] :
( ( C2
=> ( heap_Time_success @ A @ T2 @ H ) )
=> ( ( ~ C2
=> ( heap_Time_success @ A @ E2 @ H ) )
=> ( heap_Time_success @ A @ ( if @ ( heap_Time_Heap @ A ) @ C2 @ T2 @ E2 ) @ H ) ) ) ).
% success_ifI
thf(fact_4492_Heap_Oexhaust,axiom,
! [A: $tType,Y: heap_Time_Heap @ A] :
~ ! [X3: ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )] :
( Y
!= ( heap_Time_Heap2 @ A @ X3 ) ) ).
% Heap.exhaust
thf(fact_4493_success__LetI,axiom,
! [A: $tType,B: $tType,X2: A,T2: A,F: A > ( heap_Time_Heap @ B ),H: heap_ext @ product_unit] :
( ( X2 = T2 )
=> ( ( heap_Time_success @ B @ ( F @ X2 ) @ H )
=> ( heap_Time_success @ B @ ( F @ T2 ) @ H ) ) ) ).
% success_LetI
thf(fact_4494_success__returnI,axiom,
! [A: $tType,X2: A,H: heap_ext @ product_unit] : ( heap_Time_success @ A @ ( heap_Time_return @ A @ X2 ) @ H ) ).
% success_returnI
thf(fact_4495_success__heapI,axiom,
! [A: $tType,F: ( heap_ext @ product_unit ) > ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ),H: heap_ext @ product_unit] : ( heap_Time_success @ A @ ( heap_Time_heap @ A @ F ) @ H ) ).
% success_heapI
thf(fact_4496_success__ureturnI,axiom,
! [A: $tType,X2: A,H: heap_ext @ product_unit] : ( heap_Time_success @ A @ ( heap_Time_ureturn @ A @ X2 ) @ H ) ).
% success_ureturnI
thf(fact_4497_success__assertI,axiom,
! [A: $tType,P: A > $o,X2: A,H: heap_ext @ product_unit] :
( ( P @ X2 )
=> ( heap_Time_success @ A @ ( heap_Time_assert @ A @ P @ X2 ) @ H ) ) ).
% success_assertI
thf(fact_4498_success__tapI,axiom,
! [A: $tType,F: ( heap_ext @ product_unit ) > A,H: heap_ext @ product_unit] : ( heap_Time_success @ A @ ( heap_Time_tap @ A @ F ) @ H ) ).
% success_tapI
thf(fact_4499_successI,axiom,
! [A: $tType,F: heap_Time_Heap @ A,H: heap_ext @ product_unit] :
( ( ( heap_Time_execute @ A @ F @ H )
!= ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) )
=> ( heap_Time_success @ A @ F @ H ) ) ).
% successI
thf(fact_4500_success__def,axiom,
! [A: $tType] :
( ( heap_Time_success @ A )
= ( ^ [F2: heap_Time_Heap @ A,H3: heap_ext @ product_unit] :
( ( heap_Time_execute @ A @ F2 @ H3 )
!= ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ) ).
% success_def
thf(fact_4501_refines__empty,axiom,
! [A: $tType,M: heap_Time_Heap @ A] :
( refine_Imp_refines @ A @ M
@ ( heap_Time_Heap2 @ A
@ ^ [X: heap_ext @ product_unit] : ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ).
% refines_empty
thf(fact_4502_TBOUND__empty,axiom,
! [A: $tType,T2: nat] :
( time_TBOUND @ A
@ ( heap_Time_Heap2 @ A
@ ^ [X: heap_ext @ product_unit] : ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) )
@ T2 ) ).
% TBOUND_empty
thf(fact_4503_raise__def,axiom,
! [A: $tType] :
( ( heap_Time_raise @ A )
= ( ^ [S6: list @ char] :
( heap_Time_Heap2 @ A
@ ^ [Uu: heap_ext @ product_unit] : ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ) ).
% raise_def
thf(fact_4504_the__heap__def,axiom,
! [A: $tType] :
( ( time_the_heap @ A )
= ( ^ [M2: heap_Time_Heap @ A,H3: heap_ext @ product_unit] :
( case_option @ ( heap_ext @ product_unit ) @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( undefined @ ( heap_ext @ product_unit ) )
@ ( product_case_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( heap_ext @ product_unit )
@ ^ [X: A] :
( product_case_prod @ ( heap_ext @ product_unit ) @ nat @ ( heap_ext @ product_unit )
@ ^ [H5: heap_ext @ product_unit,Xa5: nat] : H5 ) )
@ ( heap_Time_execute @ A @ M2 @ H3 ) ) ) ) ).
% the_heap_def
thf(fact_4505_successE,axiom,
! [A: $tType,F: heap_Time_Heap @ A,H: heap_ext @ product_unit] :
( ( heap_Time_success @ A @ F @ H )
=> ~ ! [R6: A,H6: product_prod @ ( heap_ext @ product_unit ) @ nat] :
( ( heap_Time_execute @ A @ F @ H )
!= ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R6 @ H6 ) ) ) ) ).
% successE
thf(fact_4506_Heap_Osize_I2_J,axiom,
! [A: $tType,X2: ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )] :
( ( size_size @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap2 @ A @ X2 ) )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% Heap.size(2)
thf(fact_4507_heap__def,axiom,
! [A: $tType] :
( ( heap_Time_heap @ A )
= ( ^ [F2: ( 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 ) ) ) @ F2 ) ) ) ) ).
% heap_def
thf(fact_4508_tap__def,axiom,
! [A: $tType] :
( ( heap_Time_tap @ A )
= ( ^ [F2: ( heap_ext @ product_unit ) > A] :
( heap_Time_Heap2 @ A
@ ^ [H3: heap_ext @ product_unit] : ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( F2 @ H3 ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H3 @ ( one_one @ nat ) ) ) ) ) ) ) ).
% tap_def
thf(fact_4509_Heap_Osize__gen,axiom,
! [A: $tType,Xa: A > nat,X2: ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )] :
( ( heap_Time_size_Heap @ A @ Xa @ ( heap_Time_Heap2 @ A @ X2 ) )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% Heap.size_gen
thf(fact_4510_arg__min__list_Oelims,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [X2: A > B,Xa: list @ A,Y: A] :
( ( ( arg_min_list @ A @ B @ X2 @ Xa )
= Y )
=> ( ! [X3: A] :
( ( Xa
= ( cons @ A @ X3 @ ( nil @ A ) ) )
=> ( Y != X3 ) )
=> ( ! [X3: A,Y2: A,Zs2: list @ A] :
( ( Xa
= ( cons @ A @ X3 @ ( cons @ A @ Y2 @ Zs2 ) ) )
=> ( Y
!= ( if @ A @ ( ord_less_eq @ B @ ( X2 @ X3 ) @ ( X2 @ ( arg_min_list @ A @ B @ X2 @ ( cons @ A @ Y2 @ Zs2 ) ) ) ) @ X3 @ ( arg_min_list @ A @ B @ X2 @ ( cons @ A @ Y2 @ Zs2 ) ) ) ) )
=> ~ ( ( Xa
= ( nil @ A ) )
=> ( Y
!= ( undefined @ A ) ) ) ) ) ) ) ).
% arg_min_list.elims
thf(fact_4511_guard__def,axiom,
! [A: $tType] :
( ( heap_Time_guard @ A )
= ( ^ [P4: ( heap_ext @ product_unit ) > $o,F2: ( heap_ext @ product_unit ) > ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )] :
( heap_Time_Heap2 @ A
@ ^ [H3: heap_ext @ product_unit] : ( if @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ ( P4 @ H3 ) @ ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( F2 @ H3 ) ) @ ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ) ) ).
% guard_def
thf(fact_4512_times__int_Oabs__eq,axiom,
! [Xa: product_prod @ nat @ nat,X2: product_prod @ nat @ nat] :
( ( times_times @ int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X2 ) )
= ( abs_Integ
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X: nat,Y4: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V4: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ X @ U2 ) @ ( times_times @ nat @ Y4 @ V4 ) ) @ ( plus_plus @ nat @ ( times_times @ nat @ X @ V4 ) @ ( times_times @ nat @ Y4 @ U2 ) ) ) )
@ Xa
@ X2 ) ) ) ).
% times_int.abs_eq
thf(fact_4513_int_Oabs__induct,axiom,
! [P: int > $o,X2: int] :
( ! [Y2: product_prod @ nat @ nat] : ( P @ ( abs_Integ @ Y2 ) )
=> ( P @ X2 ) ) ).
% int.abs_induct
thf(fact_4514_eq__Abs__Integ,axiom,
! [Z2: int] :
~ ! [X3: nat,Y2: nat] :
( Z2
!= ( abs_Integ @ ( product_Pair @ nat @ nat @ X3 @ Y2 ) ) ) ).
% eq_Abs_Integ
thf(fact_4515_success__guardI,axiom,
! [A: $tType,P: ( heap_ext @ product_unit ) > $o,H: heap_ext @ product_unit,F: ( heap_ext @ product_unit ) > ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )] :
( ( P @ H )
=> ( heap_Time_success @ A @ ( heap_Time_guard @ A @ P @ F ) @ H ) ) ).
% success_guardI
thf(fact_4516_nat_Oabs__eq,axiom,
! [X2: product_prod @ nat @ nat] :
( ( nat2 @ ( abs_Integ @ X2 ) )
= ( product_case_prod @ nat @ nat @ nat @ ( minus_minus @ nat ) @ X2 ) ) ).
% nat.abs_eq
thf(fact_4517_zero__int__def,axiom,
( ( zero_zero @ int )
= ( abs_Integ @ ( product_Pair @ nat @ nat @ ( zero_zero @ nat ) @ ( zero_zero @ nat ) ) ) ) ).
% zero_int_def
thf(fact_4518_int__def,axiom,
( ( semiring_1_of_nat @ int )
= ( ^ [N4: nat] : ( abs_Integ @ ( product_Pair @ nat @ nat @ N4 @ ( zero_zero @ nat ) ) ) ) ) ).
% int_def
thf(fact_4519_arg__min__list_Osimps_I2_J,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [F: A > B,X2: A,Y: A,Zs: list @ A] :
( ( arg_min_list @ A @ B @ F @ ( cons @ A @ X2 @ ( cons @ A @ Y @ Zs ) ) )
= ( if @ A @ ( ord_less_eq @ B @ ( F @ X2 ) @ ( F @ ( arg_min_list @ A @ B @ F @ ( cons @ A @ Y @ Zs ) ) ) ) @ X2 @ ( arg_min_list @ A @ B @ F @ ( cons @ A @ Y @ Zs ) ) ) ) ) ).
% arg_min_list.simps(2)
thf(fact_4520_execute__guard_I2_J,axiom,
! [A: $tType,P: ( heap_ext @ product_unit ) > $o,H: heap_ext @ product_unit,F: ( 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 @ F ) @ H )
= ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( F @ H ) ) ) ) ).
% execute_guard(2)
thf(fact_4521_execute__guard_I1_J,axiom,
! [A: $tType,P: ( heap_ext @ product_unit ) > $o,H: heap_ext @ product_unit,F: ( 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 @ F ) @ H )
= ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ).
% execute_guard(1)
thf(fact_4522_uminus__int_Oabs__eq,axiom,
! [X2: product_prod @ nat @ nat] :
( ( uminus_uminus @ int @ ( abs_Integ @ X2 ) )
= ( abs_Integ
@ ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [X: nat,Y4: nat] : ( product_Pair @ nat @ nat @ Y4 @ X )
@ X2 ) ) ) ).
% uminus_int.abs_eq
thf(fact_4523_one__int__def,axiom,
( ( one_one @ int )
= ( abs_Integ @ ( product_Pair @ nat @ nat @ ( one_one @ nat ) @ ( zero_zero @ nat ) ) ) ) ).
% one_int_def
thf(fact_4524_of__int_Oabs__eq,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X2: product_prod @ nat @ nat] :
( ( ring_1_of_int @ A @ ( abs_Integ @ X2 ) )
= ( product_case_prod @ nat @ nat @ A
@ ^ [I2: nat,J3: nat] : ( minus_minus @ A @ ( semiring_1_of_nat @ A @ I2 ) @ ( semiring_1_of_nat @ A @ J3 ) )
@ X2 ) ) ) ).
% of_int.abs_eq
thf(fact_4525_less__int_Oabs__eq,axiom,
! [Xa: product_prod @ nat @ nat,X2: product_prod @ nat @ nat] :
( ( ord_less @ int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X2 ) )
= ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X: nat,Y4: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V4: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ X @ V4 ) @ ( plus_plus @ nat @ U2 @ Y4 ) ) )
@ Xa
@ X2 ) ) ).
% less_int.abs_eq
thf(fact_4526_less__eq__int_Oabs__eq,axiom,
! [Xa: product_prod @ nat @ nat,X2: product_prod @ nat @ nat] :
( ( ord_less_eq @ int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X2 ) )
= ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X: nat,Y4: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V4: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ X @ V4 ) @ ( plus_plus @ nat @ U2 @ Y4 ) ) )
@ Xa
@ X2 ) ) ).
% less_eq_int.abs_eq
thf(fact_4527_plus__int_Oabs__eq,axiom,
! [Xa: product_prod @ nat @ nat,X2: product_prod @ nat @ nat] :
( ( plus_plus @ int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X2 ) )
= ( abs_Integ
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X: nat,Y4: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V4: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X @ U2 ) @ ( plus_plus @ nat @ Y4 @ V4 ) ) )
@ Xa
@ X2 ) ) ) ).
% plus_int.abs_eq
thf(fact_4528_minus__int_Oabs__eq,axiom,
! [Xa: product_prod @ nat @ nat,X2: product_prod @ nat @ nat] :
( ( minus_minus @ int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X2 ) )
= ( abs_Integ
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X: nat,Y4: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V4: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X @ V4 ) @ ( plus_plus @ nat @ Y4 @ U2 ) ) )
@ Xa
@ X2 ) ) ) ).
% minus_int.abs_eq
thf(fact_4529_rat__minus__code,axiom,
! [P3: rat,Q2: rat] :
( ( quotient_of @ ( minus_minus @ rat @ P3 @ Q2 ) )
= ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [A3: int,C5: int] :
( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [B3: int,D5: int] : ( normalize @ ( product_Pair @ int @ int @ ( minus_minus @ int @ ( times_times @ int @ A3 @ D5 ) @ ( times_times @ int @ B3 @ C5 ) ) @ ( times_times @ int @ C5 @ D5 ) ) )
@ ( quotient_of @ Q2 ) )
@ ( quotient_of @ P3 ) ) ) ).
% rat_minus_code
thf(fact_4530_rat__plus__code,axiom,
! [P3: rat,Q2: rat] :
( ( quotient_of @ ( plus_plus @ rat @ P3 @ Q2 ) )
= ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [A3: int,C5: int] :
( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [B3: int,D5: int] : ( normalize @ ( product_Pair @ int @ int @ ( plus_plus @ int @ ( times_times @ int @ A3 @ D5 ) @ ( times_times @ int @ B3 @ C5 ) ) @ ( times_times @ int @ C5 @ D5 ) ) )
@ ( quotient_of @ Q2 ) )
@ ( quotient_of @ P3 ) ) ) ).
% rat_plus_code
thf(fact_4531_listrel1__iff__update,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( listrel1 @ A @ R ) )
= ( ? [Y4: A,N4: nat] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( nth @ A @ Xs @ N4 ) @ Y4 ) @ R )
& ( ord_less @ nat @ N4 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( Ys
= ( list_update @ A @ Xs @ N4 @ Y4 ) ) ) ) ) ).
% listrel1_iff_update
thf(fact_4532_sort__key__by__quicksort__code,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ( ( linorder_sort_key @ B @ A )
= ( ^ [F2: B > A,Xs3: list @ B] :
( case_list @ ( list @ B ) @ B @ ( nil @ B )
@ ^ [X: B] :
( case_list @ ( list @ B ) @ B @ Xs3
@ ^ [Y4: B] :
( case_list @ ( list @ B ) @ B @ ( if @ ( list @ B ) @ ( ord_less_eq @ A @ ( F2 @ X ) @ ( F2 @ Y4 ) ) @ Xs3 @ ( cons @ B @ Y4 @ ( cons @ B @ X @ ( nil @ B ) ) ) )
@ ^ [Ab: B,List2: list @ B] :
( product_case_prod @ ( list @ B ) @ ( product_prod @ ( list @ B ) @ ( list @ B ) ) @ ( list @ B )
@ ^ [Lts: list @ B] :
( product_case_prod @ ( list @ B ) @ ( list @ B ) @ ( list @ B )
@ ^ [Eqs: list @ B,Gts: list @ B] : ( append @ B @ ( linorder_sort_key @ B @ A @ F2 @ Lts ) @ ( append @ B @ Eqs @ ( linorder_sort_key @ B @ A @ F2 @ Gts ) ) ) )
@ ( linorder_part @ B @ A @ F2 @ ( F2 @ ( 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_4533_sort__upto,axiom,
! [I: int,J: int] :
( ( linorder_sort_key @ int @ int
@ ^ [X: int] : X
@ ( upto @ I @ J ) )
= ( upto @ I @ J ) ) ).
% sort_upto
thf(fact_4534_length__sort,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F: B > A,Xs: list @ B] :
( ( size_size @ ( list @ B ) @ ( linorder_sort_key @ B @ A @ F @ Xs ) )
= ( size_size @ ( list @ B ) @ Xs ) ) ) ).
% length_sort
thf(fact_4535_rat__one__code,axiom,
( ( quotient_of @ ( one_one @ rat ) )
= ( product_Pair @ int @ int @ ( one_one @ int ) @ ( one_one @ int ) ) ) ).
% rat_one_code
thf(fact_4536_sorted__list__of__multiset__mset,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( linord6283353356039996273ltiset @ A @ ( mset @ A @ Xs ) )
= ( linorder_sort_key @ A @ A
@ ^ [X: A] : X
@ Xs ) ) ) ).
% sorted_list_of_multiset_mset
thf(fact_4537_rat__zero__code,axiom,
( ( quotient_of @ ( zero_zero @ rat ) )
= ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) ) ) ).
% rat_zero_code
thf(fact_4538_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_4539_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_4540_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_4541_list_Ocase__distrib,axiom,
! [B: $tType,C: $tType,A: $tType,H: B > C,F1: B,F22: A > ( list @ A ) > B,List: list @ A] :
( ( H @ ( case_list @ B @ A @ F1 @ F22 @ List ) )
= ( case_list @ C @ A @ ( H @ F1 )
@ ^ [X12: A,X24: list @ A] : ( H @ ( F22 @ X12 @ X24 ) )
@ List ) ) ).
% list.case_distrib
thf(fact_4542_sort__key__const,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [C2: B,Xs: list @ A] :
( ( linorder_sort_key @ A @ B
@ ^ [X: A] : C2
@ Xs )
= Xs ) ) ).
% sort_key_const
thf(fact_4543_diff__rat__def,axiom,
( ( minus_minus @ rat )
= ( ^ [Q6: rat,R2: rat] : ( plus_plus @ rat @ Q6 @ ( uminus_uminus @ rat @ R2 ) ) ) ) ).
% diff_rat_def
thf(fact_4544_listrel1__eq__len,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( listrel1 @ A @ R ) )
=> ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ A ) @ Ys ) ) ) ).
% listrel1_eq_len
thf(fact_4545_sorted__sort__id,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( ( linorder_sort_key @ A @ A
@ ^ [X: A] : X
@ Xs )
= Xs ) ) ) ).
% sorted_sort_id
thf(fact_4546_sorted__sort,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( sorted_wrt @ A @ ( ord_less_eq @ A )
@ ( linorder_sort_key @ A @ A
@ ^ [X: A] : X
@ Xs ) ) ) ).
% sorted_sort
thf(fact_4547_quotient__of__denom__pos,axiom,
! [R: rat,P3: int,Q2: int] :
( ( ( quotient_of @ R )
= ( product_Pair @ int @ int @ P3 @ Q2 ) )
=> ( ord_less @ int @ ( zero_zero @ int ) @ Q2 ) ) ).
% quotient_of_denom_pos
thf(fact_4548_properties__for__sort,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Ys: list @ A,Xs: list @ A] :
( ( ( mset @ A @ Ys )
= ( mset @ A @ Xs ) )
=> ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Ys )
=> ( ( linorder_sort_key @ A @ A
@ ^ [X: A] : X
@ Xs )
= Ys ) ) ) ) ).
% properties_for_sort
thf(fact_4549_hd__def,axiom,
! [A: $tType] :
( ( hd @ A )
= ( case_list @ A @ A @ ( undefined @ A )
@ ^ [X212: A,X223: list @ A] : X212 ) ) ).
% hd_def
thf(fact_4550_rat__floor__code,axiom,
( ( archim6421214686448440834_floor @ rat )
= ( ^ [P7: rat] : ( product_case_prod @ int @ int @ int @ ( divide_divide @ int ) @ ( quotient_of @ P7 ) ) ) ) ).
% rat_floor_code
thf(fact_4551_rat__uminus__code,axiom,
! [P3: rat] :
( ( quotient_of @ ( uminus_uminus @ rat @ P3 ) )
= ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [A3: int] : ( product_Pair @ int @ int @ ( uminus_uminus @ int @ A3 ) )
@ ( quotient_of @ P3 ) ) ) ).
% rat_uminus_code
thf(fact_4552_rat__abs__code,axiom,
! [P3: rat] :
( ( quotient_of @ ( abs_abs @ rat @ P3 ) )
= ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [A3: int] : ( product_Pair @ int @ int @ ( abs_abs @ int @ A3 ) )
@ ( quotient_of @ P3 ) ) ) ).
% rat_abs_code
thf(fact_4553_rat__less__eq__code,axiom,
( ( ord_less_eq @ rat )
= ( ^ [P7: rat,Q6: rat] :
( product_case_prod @ int @ int @ $o
@ ^ [A3: int,C5: int] :
( product_case_prod @ int @ int @ $o
@ ^ [B3: int,D5: int] : ( ord_less_eq @ int @ ( times_times @ int @ A3 @ D5 ) @ ( times_times @ int @ C5 @ B3 ) )
@ ( quotient_of @ Q6 ) )
@ ( quotient_of @ P7 ) ) ) ) ).
% rat_less_eq_code
thf(fact_4554_rat__less__code,axiom,
( ( ord_less @ rat )
= ( ^ [P7: rat,Q6: rat] :
( product_case_prod @ int @ int @ $o
@ ^ [A3: int,C5: int] :
( product_case_prod @ int @ int @ $o
@ ^ [B3: int,D5: int] : ( ord_less @ int @ ( times_times @ int @ A3 @ D5 ) @ ( times_times @ int @ C5 @ B3 ) )
@ ( quotient_of @ Q6 ) )
@ ( quotient_of @ P7 ) ) ) ) ).
% rat_less_code
thf(fact_4555_rat__divide__code,axiom,
! [P3: rat,Q2: rat] :
( ( quotient_of @ ( divide_divide @ rat @ P3 @ Q2 ) )
= ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [A3: int,C5: int] :
( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [B3: int,D5: int] : ( normalize @ ( product_Pair @ int @ int @ ( times_times @ int @ A3 @ D5 ) @ ( times_times @ int @ C5 @ B3 ) ) )
@ ( quotient_of @ Q2 ) )
@ ( quotient_of @ P3 ) ) ) ).
% rat_divide_code
thf(fact_4556_rat__times__code,axiom,
! [P3: rat,Q2: rat] :
( ( quotient_of @ ( times_times @ rat @ P3 @ Q2 ) )
= ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [A3: int,C5: int] :
( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [B3: int,D5: int] : ( normalize @ ( product_Pair @ int @ int @ ( times_times @ int @ A3 @ B3 ) @ ( times_times @ int @ C5 @ D5 ) ) )
@ ( quotient_of @ Q2 ) )
@ ( quotient_of @ P3 ) ) ) ).
% rat_times_code
thf(fact_4557_listrel1p__def,axiom,
! [A: $tType] :
( ( listrel1p @ A )
= ( ^ [R2: A > A > $o,Xs3: list @ A,Ys3: list @ A] : ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs3 @ Ys3 ) @ ( listrel1 @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ R2 ) ) ) ) ) ) ).
% listrel1p_def
thf(fact_4558_quotient__of__int,axiom,
! [A2: int] :
( ( quotient_of @ ( of_int @ A2 ) )
= ( product_Pair @ int @ int @ A2 @ ( one_one @ int ) ) ) ).
% quotient_of_int
thf(fact_4559_rat__inverse__code,axiom,
! [P3: rat] :
( ( quotient_of @ ( inverse_inverse @ rat @ P3 ) )
= ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [A3: int,B3: int] :
( if @ ( product_prod @ int @ int )
@ ( A3
= ( zero_zero @ int ) )
@ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
@ ( product_Pair @ int @ int @ ( times_times @ int @ ( sgn_sgn @ int @ A3 ) @ B3 ) @ ( abs_abs @ int @ A3 ) ) )
@ ( quotient_of @ P3 ) ) ) ).
% rat_inverse_code
thf(fact_4560_sort__key__by__quicksort,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ( ( linorder_sort_key @ B @ A )
= ( ^ [F2: B > A,Xs3: list @ B] :
( append @ B
@ ( linorder_sort_key @ B @ A @ F2
@ ( filter2 @ B
@ ^ [X: B] : ( ord_less @ A @ ( F2 @ X ) @ ( F2 @ ( nth @ B @ Xs3 @ ( divide_divide @ nat @ ( size_size @ ( list @ B ) @ Xs3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ Xs3 ) )
@ ( append @ B
@ ( filter2 @ B
@ ^ [X: B] :
( ( F2 @ X )
= ( F2 @ ( nth @ B @ Xs3 @ ( divide_divide @ nat @ ( size_size @ ( list @ B ) @ Xs3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ Xs3 )
@ ( linorder_sort_key @ B @ A @ F2
@ ( filter2 @ B
@ ^ [X: B] : ( ord_less @ A @ ( F2 @ ( nth @ B @ Xs3 @ ( divide_divide @ nat @ ( size_size @ ( list @ B ) @ Xs3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( F2 @ X ) )
@ Xs3 ) ) ) ) ) ) ) ).
% sort_key_by_quicksort
thf(fact_4561_filter__filter,axiom,
! [A: $tType,P: A > $o,Q: A > $o,Xs: list @ A] :
( ( filter2 @ A @ P @ ( filter2 @ A @ Q @ Xs ) )
= ( filter2 @ A
@ ^ [X: A] :
( ( Q @ X )
& ( P @ X ) )
@ Xs ) ) ).
% filter_filter
thf(fact_4562_inverse__zero,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ( ( inverse_inverse @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% inverse_zero
thf(fact_4563_inverse__nonzero__iff__nonzero,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A] :
( ( ( inverse_inverse @ A @ A2 )
= ( zero_zero @ A ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% inverse_nonzero_iff_nonzero
thf(fact_4564_inverse__eq__1__iff,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [X2: A] :
( ( ( inverse_inverse @ A @ X2 )
= ( one_one @ A ) )
= ( X2
= ( one_one @ A ) ) ) ) ).
% inverse_eq_1_iff
thf(fact_4565_inverse__1,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ( ( inverse_inverse @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% inverse_1
thf(fact_4566_set__filter,axiom,
! [A: $tType,P: A > $o,Xs: list @ A] :
( ( set2 @ A @ ( filter2 @ A @ P @ Xs ) )
= ( collect @ A
@ ^ [X: A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
& ( P @ X ) ) ) ) ).
% set_filter
thf(fact_4567_inverse__nonnegative__iff__nonnegative,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( inverse_inverse @ A @ A2 ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% inverse_nonnegative_iff_nonnegative
thf(fact_4568_inverse__nonpositive__iff__nonpositive,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A2 ) @ ( zero_zero @ A ) )
= ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% inverse_nonpositive_iff_nonpositive
thf(fact_4569_inverse__positive__iff__positive,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( inverse_inverse @ A @ A2 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% inverse_positive_iff_positive
thf(fact_4570_inverse__negative__iff__negative,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A] :
( ( ord_less @ A @ ( inverse_inverse @ A @ A2 ) @ ( zero_zero @ A ) )
= ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% inverse_negative_iff_negative
thf(fact_4571_inverse__less__iff__less__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) )
= ( ord_less @ A @ B2 @ A2 ) ) ) ) ) ).
% inverse_less_iff_less_neg
thf(fact_4572_inverse__less__iff__less,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) )
= ( ord_less @ A @ B2 @ A2 ) ) ) ) ) ).
% inverse_less_iff_less
thf(fact_4573_inverse__le__iff__le,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) )
= ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ) ).
% inverse_le_iff_le
thf(fact_4574_inverse__le__iff__le__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) )
= ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ) ).
% inverse_le_iff_le_neg
thf(fact_4575_left__inverse,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ ( inverse_inverse @ A @ A2 ) @ A2 )
= ( one_one @ A ) ) ) ) ).
% left_inverse
thf(fact_4576_right__inverse,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ A2 @ ( inverse_inverse @ A @ A2 ) )
= ( one_one @ A ) ) ) ) ).
% right_inverse
thf(fact_4577_inverse__eq__divide__numeral,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [W2: num] :
( ( inverse_inverse @ A @ ( numeral_numeral @ A @ W2 ) )
= ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ W2 ) ) ) ) ).
% inverse_eq_divide_numeral
thf(fact_4578_power__int__minus1__right,axiom,
! [A: $tType] :
( ( ( inverse @ A )
& ( monoid_mult @ A ) )
=> ! [Y: A] :
( ( power_int @ A @ Y @ ( uminus_uminus @ int @ ( one_one @ int ) ) )
= ( inverse_inverse @ A @ Y ) ) ) ).
% power_int_minus1_right
thf(fact_4579_inverse__eq__divide__neg__numeral,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [W2: num] :
( ( inverse_inverse @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) )
= ( divide_divide @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ).
% inverse_eq_divide_neg_numeral
thf(fact_4580_sort__key__stable,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [F: A > B,K: B,Xs: list @ A] :
( ( filter2 @ A
@ ^ [Y4: A] :
( ( F @ Y4 )
= K )
@ ( linorder_sort_key @ A @ B @ F @ Xs ) )
= ( filter2 @ A
@ ^ [Y4: A] :
( ( F @ Y4 )
= K )
@ Xs ) ) ) ).
% sort_key_stable
thf(fact_4581_partition__in__shuffles,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( member @ ( list @ A ) @ Xs
@ ( shuffles @ A @ ( filter2 @ A @ P @ Xs )
@ ( filter2 @ A
@ ^ [X: A] :
~ ( P @ X )
@ Xs ) ) ) ).
% partition_in_shuffles
thf(fact_4582_sgn__rat__def,axiom,
( ( sgn_sgn @ rat )
= ( ^ [A3: rat] :
( if @ rat
@ ( A3
= ( zero_zero @ rat ) )
@ ( zero_zero @ rat )
@ ( if @ rat @ ( ord_less @ rat @ ( zero_zero @ rat ) @ A3 ) @ ( one_one @ rat ) @ ( uminus_uminus @ rat @ ( one_one @ rat ) ) ) ) ) ) ).
% sgn_rat_def
thf(fact_4583_nonzero__imp__inverse__nonzero,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( inverse_inverse @ A @ A2 )
!= ( zero_zero @ A ) ) ) ) ).
% nonzero_imp_inverse_nonzero
thf(fact_4584_nonzero__inverse__inverse__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( inverse_inverse @ A @ ( inverse_inverse @ A @ A2 ) )
= A2 ) ) ) ).
% nonzero_inverse_inverse_eq
thf(fact_4585_nonzero__inverse__eq__imp__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A,B2: A] :
( ( ( inverse_inverse @ A @ A2 )
= ( inverse_inverse @ A @ B2 ) )
=> ( ( A2
!= ( zero_zero @ A ) )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( A2 = B2 ) ) ) ) ) ).
% nonzero_inverse_eq_imp_eq
thf(fact_4586_inverse__zero__imp__zero,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A] :
( ( ( inverse_inverse @ A @ A2 )
= ( zero_zero @ A ) )
=> ( A2
= ( zero_zero @ A ) ) ) ) ).
% inverse_zero_imp_zero
thf(fact_4587_field__class_Ofield__inverse__zero,axiom,
! [A: $tType] :
( ( field @ A )
=> ( ( inverse_inverse @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% field_class.field_inverse_zero
thf(fact_4588_power__int__inverse,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X2: A,N: int] :
( ( power_int @ A @ ( inverse_inverse @ A @ X2 ) @ N )
= ( inverse_inverse @ A @ ( power_int @ A @ X2 @ N ) ) ) ) ).
% power_int_inverse
thf(fact_4589_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_4590_positive__imp__inverse__positive,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( inverse_inverse @ A @ A2 ) ) ) ) ).
% positive_imp_inverse_positive
thf(fact_4591_negative__imp__inverse__negative,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A] :
( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( inverse_inverse @ A @ A2 ) @ ( zero_zero @ A ) ) ) ) ).
% negative_imp_inverse_negative
thf(fact_4592_inverse__positive__imp__positive,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( inverse_inverse @ A @ A2 ) )
=> ( ( A2
!= ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ) ).
% inverse_positive_imp_positive
thf(fact_4593_inverse__negative__imp__negative,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A] :
( ( ord_less @ A @ ( inverse_inverse @ A @ A2 ) @ ( zero_zero @ A ) )
=> ( ( A2
!= ( zero_zero @ A ) )
=> ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ) ).
% inverse_negative_imp_negative
thf(fact_4594_less__imp__inverse__less__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( inverse_inverse @ A @ B2 ) @ ( inverse_inverse @ A @ A2 ) ) ) ) ) ).
% less_imp_inverse_less_neg
thf(fact_4595_inverse__less__imp__less__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) )
=> ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B2 @ A2 ) ) ) ) ).
% inverse_less_imp_less_neg
thf(fact_4596_less__imp__inverse__less,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ord_less @ A @ ( inverse_inverse @ A @ B2 ) @ ( inverse_inverse @ A @ A2 ) ) ) ) ) ).
% less_imp_inverse_less
thf(fact_4597_inverse__less__imp__less,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ord_less @ A @ B2 @ A2 ) ) ) ) ).
% inverse_less_imp_less
thf(fact_4598_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_4599_nonzero__inverse__mult__distrib,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A,B2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( ( inverse_inverse @ A @ ( times_times @ A @ A2 @ B2 ) )
= ( times_times @ A @ ( inverse_inverse @ A @ B2 ) @ ( inverse_inverse @ A @ A2 ) ) ) ) ) ) ).
% nonzero_inverse_mult_distrib
thf(fact_4600_nonzero__inverse__minus__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( inverse_inverse @ A @ ( uminus_uminus @ A @ A2 ) )
= ( uminus_uminus @ A @ ( inverse_inverse @ A @ A2 ) ) ) ) ) ).
% nonzero_inverse_minus_eq
thf(fact_4601_inverse__unique,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A,B2: A] :
( ( ( times_times @ A @ A2 @ B2 )
= ( one_one @ A ) )
=> ( ( inverse_inverse @ A @ A2 )
= B2 ) ) ) ).
% inverse_unique
thf(fact_4602_inverse__numeral__1,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ( ( inverse_inverse @ A @ ( numeral_numeral @ A @ one2 ) )
= ( numeral_numeral @ A @ one2 ) ) ) ).
% inverse_numeral_1
thf(fact_4603_inverse__eq__divide,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ( ( inverse_inverse @ A )
= ( divide_divide @ A @ ( one_one @ A ) ) ) ) ).
% inverse_eq_divide
thf(fact_4604_power__mult__inverse__distrib,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X2: A,M: nat] :
( ( times_times @ A @ ( power_power @ A @ X2 @ M ) @ ( inverse_inverse @ A @ X2 ) )
= ( times_times @ A @ ( inverse_inverse @ A @ X2 ) @ ( power_power @ A @ X2 @ M ) ) ) ) ).
% power_mult_inverse_distrib
thf(fact_4605_power__mult__power__inverse__commute,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X2: A,M: nat,N: nat] :
( ( times_times @ A @ ( power_power @ A @ X2 @ M ) @ ( power_power @ A @ ( inverse_inverse @ A @ X2 ) @ N ) )
= ( times_times @ A @ ( power_power @ A @ ( inverse_inverse @ A @ X2 ) @ N ) @ ( power_power @ A @ X2 @ M ) ) ) ) ).
% power_mult_power_inverse_commute
thf(fact_4606_mult__inverse__of__nat__commute,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Xa: nat,X2: A] :
( ( times_times @ A @ ( inverse_inverse @ A @ ( semiring_1_of_nat @ A @ Xa ) ) @ X2 )
= ( times_times @ A @ X2 @ ( inverse_inverse @ A @ ( semiring_1_of_nat @ A @ Xa ) ) ) ) ) ).
% mult_inverse_of_nat_commute
thf(fact_4607_nonzero__abs__inverse,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( abs_abs @ A @ ( inverse_inverse @ A @ A2 ) )
= ( inverse_inverse @ A @ ( abs_abs @ A @ A2 ) ) ) ) ) ).
% nonzero_abs_inverse
thf(fact_4608_mult__inverse__of__int__commute,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Xa: int,X2: A] :
( ( times_times @ A @ ( inverse_inverse @ A @ ( ring_1_of_int @ A @ Xa ) ) @ X2 )
= ( times_times @ A @ X2 @ ( inverse_inverse @ A @ ( ring_1_of_int @ A @ Xa ) ) ) ) ) ).
% mult_inverse_of_int_commute
thf(fact_4609_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_4610_power__int__minus,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X2: A,N: int] :
( ( power_int @ A @ X2 @ ( uminus_uminus @ int @ N ) )
= ( inverse_inverse @ A @ ( power_int @ A @ X2 @ N ) ) ) ) ).
% power_int_minus
thf(fact_4611_list_Odisc__eq__case_I2_J,axiom,
! [A: $tType,List: list @ A] :
( ( List
!= ( nil @ A ) )
= ( case_list @ $o @ A @ $false
@ ^ [Uu: A,Uv: list @ A] : $true
@ List ) ) ).
% list.disc_eq_case(2)
thf(fact_4612_list_Odisc__eq__case_I1_J,axiom,
! [A: $tType,List: list @ A] :
( ( List
= ( nil @ A ) )
= ( case_list @ $o @ A @ $true
@ ^ [Uu: A,Uv: list @ A] : $false
@ List ) ) ).
% list.disc_eq_case(1)
thf(fact_4613_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
@ ^ [X: A] :
~ ( P @ X )
@ Xs ) ) )
= ( size_size @ ( list @ A ) @ Xs ) ) ).
% sum_length_filter_compl
thf(fact_4614_inter__set__filter,axiom,
! [A: $tType,A5: set @ A,Xs: list @ A] :
( ( inf_inf @ ( set @ A ) @ A5 @ ( set2 @ A @ Xs ) )
= ( set2 @ A
@ ( filter2 @ A
@ ^ [X: A] : ( member @ A @ X @ A5 )
@ Xs ) ) ) ).
% inter_set_filter
thf(fact_4615_sorted__same,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [G: ( list @ A ) > A,Xs: list @ A] :
( sorted_wrt @ A @ ( ord_less_eq @ A )
@ ( filter2 @ A
@ ^ [X: A] :
( X
= ( G @ Xs ) )
@ Xs ) ) ) ).
% sorted_same
thf(fact_4616_mset__eq__length__filter,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Z2: A] :
( ( ( mset @ A @ Xs )
= ( mset @ A @ Ys ) )
=> ( ( size_size @ ( list @ A )
@ ( filter2 @ A
@ ( ^ [Y3: A,Z: A] : Y3 = Z
@ Z2 )
@ Xs ) )
= ( size_size @ ( list @ A )
@ ( filter2 @ A
@ ( ^ [Y3: A,Z: A] : Y3 = Z
@ Z2 )
@ Ys ) ) ) ) ).
% mset_eq_length_filter
thf(fact_4617_replicate__length__filter,axiom,
! [A: $tType,X2: A,Xs: list @ A] :
( ( replicate @ A
@ ( size_size @ ( list @ A )
@ ( filter2 @ A
@ ( ^ [Y3: A,Z: A] : Y3 = Z
@ X2 )
@ Xs ) )
@ X2 )
= ( filter2 @ A
@ ( ^ [Y3: A,Z: A] : Y3 = Z
@ X2 )
@ Xs ) ) ).
% replicate_length_filter
thf(fact_4618_length__filter__less,axiom,
! [A: $tType,X2: A,Xs: list @ A,P: A > $o] :
( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
=> ( ~ ( P @ X2 )
=> ( ord_less @ nat @ ( size_size @ ( list @ A ) @ ( filter2 @ A @ P @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ).
% length_filter_less
thf(fact_4619_inverse__le__imp__le,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ).
% inverse_le_imp_le
thf(fact_4620_le__imp__inverse__le,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ord_less_eq @ A @ ( inverse_inverse @ A @ B2 ) @ ( inverse_inverse @ A @ A2 ) ) ) ) ) ).
% le_imp_inverse_le
thf(fact_4621_inverse__le__imp__le__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) )
=> ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ).
% inverse_le_imp_le_neg
thf(fact_4622_le__imp__inverse__le__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( inverse_inverse @ A @ B2 ) @ ( inverse_inverse @ A @ A2 ) ) ) ) ) ).
% le_imp_inverse_le_neg
thf(fact_4623_inverse__le__1__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X2: A] :
( ( ord_less_eq @ A @ ( inverse_inverse @ A @ X2 ) @ ( one_one @ A ) )
= ( ( ord_less_eq @ A @ X2 @ ( zero_zero @ A ) )
| ( ord_less_eq @ A @ ( one_one @ A ) @ X2 ) ) ) ) ).
% inverse_le_1_iff
thf(fact_4624_one__less__inverse,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ A2 @ ( one_one @ A ) )
=> ( ord_less @ A @ ( one_one @ A ) @ ( inverse_inverse @ A @ A2 ) ) ) ) ) ).
% one_less_inverse
thf(fact_4625_one__less__inverse__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X2: A] :
( ( ord_less @ A @ ( one_one @ A ) @ ( inverse_inverse @ A @ X2 ) )
= ( ( ord_less @ A @ ( zero_zero @ A ) @ X2 )
& ( ord_less @ A @ X2 @ ( one_one @ A ) ) ) ) ) ).
% one_less_inverse_iff
thf(fact_4626_division__ring__inverse__add,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A,B2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) )
= ( times_times @ A @ ( times_times @ A @ ( inverse_inverse @ A @ A2 ) @ ( plus_plus @ A @ A2 @ B2 ) ) @ ( inverse_inverse @ A @ B2 ) ) ) ) ) ) ).
% division_ring_inverse_add
thf(fact_4627_inverse__add,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A2: A,B2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) )
= ( times_times @ A @ ( times_times @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( inverse_inverse @ A @ A2 ) ) @ ( inverse_inverse @ A @ B2 ) ) ) ) ) ) ).
% inverse_add
thf(fact_4628_field__class_Ofield__inverse,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ ( inverse_inverse @ A @ A2 ) @ A2 )
= ( one_one @ A ) ) ) ) ).
% field_class.field_inverse
thf(fact_4629_division__ring__inverse__diff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A,B2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) )
= ( times_times @ A @ ( times_times @ A @ ( inverse_inverse @ A @ A2 ) @ ( minus_minus @ A @ B2 @ A2 ) ) @ ( inverse_inverse @ A @ B2 ) ) ) ) ) ) ).
% division_ring_inverse_diff
thf(fact_4630_nonzero__inverse__eq__divide,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( inverse_inverse @ A @ A2 )
= ( divide_divide @ A @ ( one_one @ A ) @ A2 ) ) ) ) ).
% nonzero_inverse_eq_divide
thf(fact_4631_filter__eq__snocD,axiom,
! [A: $tType,P: A > $o,L: list @ A,L4: list @ A,X2: A] :
( ( ( filter2 @ A @ P @ L )
= ( append @ A @ L4 @ ( cons @ A @ X2 @ ( nil @ A ) ) ) )
=> ( ( member @ A @ X2 @ ( set2 @ A @ L ) )
& ( P @ X2 ) ) ) ).
% filter_eq_snocD
thf(fact_4632_inverse__le__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) )
=> ( ord_less_eq @ A @ B2 @ A2 ) )
& ( ( ord_less_eq @ A @ ( times_times @ A @ A2 @ B2 ) @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ A2 @ B2 ) ) ) ) ) ).
% inverse_le_iff
thf(fact_4633_inverse__less__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) )
=> ( ord_less @ A @ B2 @ A2 ) )
& ( ( ord_less_eq @ A @ ( times_times @ A @ A2 @ B2 ) @ ( zero_zero @ A ) )
=> ( ord_less @ A @ A2 @ B2 ) ) ) ) ) ).
% inverse_less_iff
thf(fact_4634_one__le__inverse__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X2: A] :
( ( ord_less_eq @ A @ ( one_one @ A ) @ ( inverse_inverse @ A @ X2 ) )
= ( ( ord_less @ A @ ( zero_zero @ A ) @ X2 )
& ( ord_less_eq @ A @ X2 @ ( one_one @ A ) ) ) ) ) ).
% one_le_inverse_iff
thf(fact_4635_inverse__less__1__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X2: A] :
( ( ord_less @ A @ ( inverse_inverse @ A @ X2 ) @ ( one_one @ A ) )
= ( ( ord_less_eq @ A @ X2 @ ( zero_zero @ A ) )
| ( ord_less @ A @ ( one_one @ A ) @ X2 ) ) ) ) ).
% inverse_less_1_iff
thf(fact_4636_one__le__inverse,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ A2 @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ ( inverse_inverse @ A @ A2 ) ) ) ) ) ).
% one_le_inverse
thf(fact_4637_filter__conv__foldr,axiom,
! [A: $tType] :
( ( filter2 @ A )
= ( ^ [P4: A > $o,Xs3: list @ A] :
( foldr @ A @ ( list @ A )
@ ^ [X: A,Xt: list @ A] : ( if @ ( list @ A ) @ ( P4 @ X ) @ ( cons @ A @ X @ Xt ) @ Xt )
@ Xs3
@ ( nil @ A ) ) ) ) ).
% filter_conv_foldr
thf(fact_4638_reals__Archimedean,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X2 )
=> ? [N2: nat] : ( ord_less @ A @ ( inverse_inverse @ A @ ( semiring_1_of_nat @ A @ ( suc @ N2 ) ) ) @ X2 ) ) ) ).
% reals_Archimedean
thf(fact_4639_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
@ ^ [X: A] : X != Y
@ Xs ) ) ) ).
% set_minus_filter_out
thf(fact_4640_filter__shuffles__disjoint2_I1_J,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( member @ ( list @ A ) @ Zs @ ( shuffles @ A @ Xs @ Ys ) )
=> ( ( filter2 @ A
@ ^ [X: A] : ( member @ A @ X @ ( set2 @ A @ Ys ) )
@ Zs )
= Ys ) ) ) ).
% filter_shuffles_disjoint2(1)
thf(fact_4641_filter__shuffles__disjoint2_I2_J,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( member @ ( list @ A ) @ Zs @ ( shuffles @ A @ Xs @ Ys ) )
=> ( ( filter2 @ A
@ ^ [X: A] :
~ ( member @ A @ X @ ( set2 @ A @ Ys ) )
@ Zs )
= Xs ) ) ) ).
% filter_shuffles_disjoint2(2)
thf(fact_4642_filter__shuffles__disjoint1_I1_J,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( member @ ( list @ A ) @ Zs @ ( shuffles @ A @ Xs @ Ys ) )
=> ( ( filter2 @ A
@ ^ [X: A] : ( member @ A @ X @ ( set2 @ A @ Xs ) )
@ Zs )
= Xs ) ) ) ).
% filter_shuffles_disjoint1(1)
thf(fact_4643_filter__shuffles__disjoint1_I2_J,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( member @ ( list @ A ) @ Zs @ ( shuffles @ A @ Xs @ Ys ) )
=> ( ( filter2 @ A
@ ^ [X: A] :
~ ( member @ A @ X @ ( set2 @ A @ Xs ) )
@ Zs )
= Ys ) ) ) ).
% filter_shuffles_disjoint1(2)
thf(fact_4644_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
@ ^ [I2: nat] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( P3 @ ( nth @ A @ Xs @ I2 ) ) ) ) ) ) ).
% length_filter_conv_card
thf(fact_4645_filter__nth__ex__nth,axiom,
! [A: $tType,N: nat,P: A > $o,Xs: list @ A] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ ( filter2 @ A @ P @ Xs ) ) )
=> ? [M4: nat] :
( ( ord_less_eq @ nat @ N @ M4 )
& ( ord_less @ nat @ M4 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( ( nth @ A @ ( filter2 @ A @ P @ Xs ) @ N )
= ( nth @ A @ Xs @ M4 ) )
& ( ( filter2 @ A @ P @ ( take @ A @ M4 @ Xs ) )
= ( take @ A @ N @ ( filter2 @ A @ P @ Xs ) ) ) ) ) ).
% filter_nth_ex_nth
thf(fact_4646_ex__inverse__of__nat__less,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X2 )
=> ? [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
& ( ord_less @ A @ ( inverse_inverse @ A @ ( semiring_1_of_nat @ A @ N2 ) ) @ X2 ) ) ) ) ).
% ex_inverse_of_nat_less
thf(fact_4647_power__diff__conv__inverse,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X2: A,M: nat,N: nat] :
( ( X2
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ nat @ M @ N )
=> ( ( power_power @ A @ X2 @ ( minus_minus @ nat @ N @ M ) )
= ( times_times @ A @ ( power_power @ A @ X2 @ N ) @ ( power_power @ A @ ( inverse_inverse @ A @ X2 ) @ M ) ) ) ) ) ) ).
% power_diff_conv_inverse
thf(fact_4648_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_4649_part__def,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ( ( linorder_part @ B @ A )
= ( ^ [F2: B > A,Pivot2: A,Xs3: list @ B] :
( product_Pair @ ( list @ B ) @ ( product_prod @ ( list @ B ) @ ( list @ B ) )
@ ( filter2 @ B
@ ^ [X: B] : ( ord_less @ A @ ( F2 @ X ) @ Pivot2 )
@ Xs3 )
@ ( product_Pair @ ( list @ B ) @ ( list @ B )
@ ( filter2 @ B
@ ^ [X: B] :
( ( F2 @ X )
= Pivot2 )
@ Xs3 )
@ ( filter2 @ B
@ ^ [X: B] : ( ord_less @ A @ Pivot2 @ ( F2 @ X ) )
@ Xs3 ) ) ) ) ) ) ).
% part_def
thf(fact_4650_power__int__def,axiom,
! [A: $tType] :
( ( ( inverse @ A )
& ( power @ A ) )
=> ( ( power_int @ A )
= ( ^ [X: A,N4: int] : ( if @ A @ ( ord_less_eq @ int @ ( zero_zero @ int ) @ N4 ) @ ( power_power @ A @ X @ ( nat2 @ N4 ) ) @ ( power_power @ A @ ( inverse_inverse @ A @ X ) @ ( nat2 @ ( uminus_uminus @ int @ N4 ) ) ) ) ) ) ) ).
% power_int_def
thf(fact_4651_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
@ ^ [X: list @ B] : ( ord_max @ nat @ ( minus_minus @ nat @ ( size_size @ ( list @ B ) @ X ) @ ( suc @ ( zero_zero @ nat ) ) ) )
@ ( filter2 @ ( list @ B )
@ ^ [Ys3: list @ B] :
( Ys3
!= ( nil @ B ) )
@ Xss )
@ ( zero_zero @ nat ) ) ) ) ) ).
% transpose_aux_max
thf(fact_4652_sort__by__quicksort,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( linorder_sort_key @ A @ A
@ ^ [X: A] : X
@ Xs )
= ( append @ A
@ ( linorder_sort_key @ A @ A
@ ^ [X: A] : X
@ ( filter2 @ A
@ ^ [X: A] : ( ord_less @ A @ X @ ( nth @ A @ Xs @ ( divide_divide @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
@ Xs ) )
@ ( append @ A
@ ( filter2 @ A
@ ^ [X: A] :
( X
= ( nth @ A @ Xs @ ( divide_divide @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
@ Xs )
@ ( linorder_sort_key @ A @ A
@ ^ [X: A] : X
@ ( 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_4653_power__int__numeral__neg__numeral,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [M: num,N: num] :
( ( power_int @ A @ ( numeral_numeral @ A @ M ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
= ( inverse_inverse @ A @ ( numeral_numeral @ A @ ( pow @ M @ N ) ) ) ) ) ).
% power_int_numeral_neg_numeral
thf(fact_4654_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_4655_remove__rev__alt__def,axiom,
! [A: $tType] :
( ( remove_rev @ A )
= ( ^ [X: A,Xs3: list @ A] :
( filter2 @ A
@ ^ [Y4: A] : Y4 != X
@ ( rev @ A @ Xs3 ) ) ) ) ).
% remove_rev_alt_def
thf(fact_4656_pow_Osimps_I1_J,axiom,
! [X2: num] :
( ( pow @ X2 @ one2 )
= X2 ) ).
% pow.simps(1)
thf(fact_4657_Frct__code__post_I1_J,axiom,
! [A2: int] :
( ( frct @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ A2 ) )
= ( zero_zero @ rat ) ) ).
% Frct_code_post(1)
thf(fact_4658_Frct__code__post_I2_J,axiom,
! [A2: int] :
( ( frct @ ( product_Pair @ int @ int @ A2 @ ( zero_zero @ int ) ) )
= ( zero_zero @ rat ) ) ).
% Frct_code_post(2)
thf(fact_4659_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_4660_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_4661_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 )
@ ^ [X: list @ A] :
( X
!= ( nil @ A ) )
@ Xs ) ) ) ).
% transpose_max_length
thf(fact_4662_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 )
@ ^ [X: A,Xs3: list @ A] : X
@ ( filter2 @ A @ P
@ ( linorder_sort_key @ A @ A
@ ^ [X: A] : X
@ Xs ) ) ) ) ) ).
% Bleast_code
thf(fact_4663_quicksort_Oelims,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: list @ A,Y: list @ A] :
( ( ( linorder_quicksort @ A @ X2 )
= Y )
=> ( ( ( X2
= ( nil @ A ) )
=> ( Y
!= ( nil @ A ) ) )
=> ~ ! [X3: A,Xs2: list @ A] :
( ( X2
= ( 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_4664_sort__quicksort,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( linorder_sort_key @ A @ A
@ ^ [X: A] : X )
= ( linorder_quicksort @ A ) ) ) ).
% sort_quicksort
thf(fact_4665_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_4666_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_4667_quicksort_Osimps_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Xs: list @ A] :
( ( linorder_quicksort @ A @ ( cons @ A @ X2 @ Xs ) )
= ( append @ A
@ ( linorder_quicksort @ A
@ ( filter2 @ A
@ ^ [Y4: A] :
~ ( ord_less_eq @ A @ X2 @ Y4 )
@ Xs ) )
@ ( append @ A @ ( cons @ A @ X2 @ ( nil @ A ) ) @ ( linorder_quicksort @ A @ ( filter2 @ A @ ( ord_less_eq @ A @ X2 ) @ Xs ) ) ) ) ) ) ).
% quicksort.simps(2)
thf(fact_4668_nth__nth__transpose__sorted,axiom,
! [A: $tType,Xs: list @ ( list @ A ),I: nat,J: nat] :
( ( sorted_wrt @ nat @ ( ord_less_eq @ nat ) @ ( rev @ nat @ ( map @ ( list @ A ) @ nat @ ( size_size @ ( list @ A ) ) @ Xs ) ) )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ ( list @ A ) ) @ ( transpose @ A @ Xs ) ) )
=> ( ( ord_less @ nat @ J
@ ( size_size @ ( list @ ( list @ A ) )
@ ( filter2 @ ( list @ A )
@ ^ [Ys3: list @ A] : ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Ys3 ) )
@ Xs ) ) )
=> ( ( nth @ A @ ( nth @ ( list @ A ) @ ( transpose @ A @ Xs ) @ I ) @ J )
= ( nth @ A @ ( nth @ ( list @ A ) @ Xs @ J ) @ I ) ) ) ) ) ).
% nth_nth_transpose_sorted
thf(fact_4669_distinct__concat_H,axiom,
! [A: $tType,Xs: list @ ( list @ A )] :
( ( distinct @ ( list @ A )
@ ( filter2 @ ( list @ A )
@ ^ [Ys3: list @ A] :
( Ys3
!= ( 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_4670_less__eq__int_Orep__eq,axiom,
( ( ord_less_eq @ int )
= ( ^ [X: int,Xa5: int] :
( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [Y4: nat,Z3: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V4: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ Y4 @ V4 ) @ ( plus_plus @ nat @ U2 @ Z3 ) ) )
@ ( rep_Integ @ X )
@ ( rep_Integ @ Xa5 ) ) ) ) ).
% less_eq_int.rep_eq
thf(fact_4671_map__ident,axiom,
! [A: $tType] :
( ( map @ A @ A
@ ^ [X: A] : X )
= ( ^ [Xs3: list @ A] : Xs3 ) ) ).
% map_ident
thf(fact_4672_length__map,axiom,
! [A: $tType,B: $tType,F: B > A,Xs: list @ B] :
( ( size_size @ ( list @ A ) @ ( map @ B @ A @ F @ Xs ) )
= ( size_size @ ( list @ B ) @ Xs ) ) ).
% length_map
thf(fact_4673_length__concat__rev,axiom,
! [A: $tType,Xs: list @ ( list @ A )] :
( ( size_size @ ( list @ A ) @ ( concat @ A @ ( rev @ ( list @ A ) @ Xs ) ) )
= ( size_size @ ( list @ A ) @ ( concat @ A @ Xs ) ) ) ).
% length_concat_rev
thf(fact_4674_nth__map,axiom,
! [B: $tType,A: $tType,N: nat,Xs: list @ A,F: A > B] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ B @ ( map @ A @ B @ F @ Xs ) @ N )
= ( F @ ( nth @ A @ Xs @ N ) ) ) ) ).
% nth_map
thf(fact_4675_length__filter__map,axiom,
! [A: $tType,B: $tType,P: A > $o,F: B > A,Xs: list @ B] :
( ( size_size @ ( list @ A ) @ ( filter2 @ A @ P @ ( map @ B @ A @ F @ Xs ) ) )
= ( size_size @ ( list @ B ) @ ( filter2 @ B @ ( comp @ A @ $o @ B @ P @ F ) @ Xs ) ) ) ).
% length_filter_map
thf(fact_4676_concat__map__singleton,axiom,
! [A: $tType,B: $tType,F: B > A,Xs: list @ B] :
( ( concat @ A
@ ( map @ B @ ( list @ A )
@ ^ [X: B] : ( cons @ A @ ( F @ X ) @ ( nil @ A ) )
@ Xs ) )
= ( map @ B @ A @ F @ Xs ) ) ).
% concat_map_singleton
thf(fact_4677_map__eq__nth__eq,axiom,
! [A: $tType,B: $tType,F: B > A,L: list @ B,L4: list @ B,I: nat] :
( ( ( map @ B @ A @ F @ L )
= ( map @ B @ A @ F @ L4 ) )
=> ( ( F @ ( nth @ B @ L @ I ) )
= ( F @ ( nth @ B @ L4 @ I ) ) ) ) ).
% map_eq_nth_eq
thf(fact_4678_sorted__wrt__map,axiom,
! [A: $tType,B: $tType,R3: A > A > $o,F: B > A,Xs: list @ B] :
( ( sorted_wrt @ A @ R3 @ ( map @ B @ A @ F @ Xs ) )
= ( sorted_wrt @ B
@ ^ [X: B,Y4: B] : ( R3 @ ( F @ X ) @ ( F @ Y4 ) )
@ Xs ) ) ).
% sorted_wrt_map
thf(fact_4679_map__consI_I1_J,axiom,
! [A: $tType,B: $tType,W2: list @ A,F: B > A,Ww: list @ B,A2: B] :
( ( W2
= ( map @ B @ A @ F @ Ww ) )
=> ( ( cons @ A @ ( F @ A2 ) @ W2 )
= ( map @ B @ A @ F @ ( cons @ B @ A2 @ Ww ) ) ) ) ).
% map_consI(1)
thf(fact_4680_map__eq__consE,axiom,
! [B: $tType,A: $tType,F: B > A,Ls: list @ B,Fa: A,Fl: list @ A] :
( ( ( map @ B @ A @ F @ Ls )
= ( cons @ A @ Fa @ Fl ) )
=> ~ ! [A4: B,L3: list @ B] :
( ( Ls
= ( cons @ B @ A4 @ L3 ) )
=> ( ( ( F @ A4 )
= Fa )
=> ( ( map @ B @ A @ F @ L3 )
!= Fl ) ) ) ) ).
% map_eq_consE
thf(fact_4681_list_Omap__ident,axiom,
! [A: $tType,T2: list @ A] :
( ( map @ A @ A
@ ^ [X: A] : X
@ T2 )
= T2 ) ).
% list.map_ident
thf(fact_4682_distinct__mapI,axiom,
! [A: $tType,B: $tType,F: B > A,L: list @ B] :
( ( distinct @ A @ ( map @ B @ A @ F @ L ) )
=> ( distinct @ B @ L ) ) ).
% distinct_mapI
thf(fact_4683_foldl__map,axiom,
! [A: $tType,B: $tType,C: $tType,G: A > B > A,A2: A,F: C > B,Xs: list @ C] :
( ( foldl @ A @ B @ G @ A2 @ ( map @ C @ B @ F @ Xs ) )
= ( foldl @ A @ C
@ ^ [A3: A,X: C] : ( G @ A3 @ ( F @ X ) )
@ A2
@ Xs ) ) ).
% foldl_map
thf(fact_4684_append__eq__mapE,axiom,
! [B: $tType,A: $tType,Fl: list @ A,Fl2: list @ A,F: B > A,Ls: list @ B] :
( ( ( append @ A @ Fl @ Fl2 )
= ( map @ B @ A @ F @ Ls ) )
=> ~ ! [L3: list @ B,L6: list @ B] :
( ( Ls
= ( append @ B @ L3 @ L6 ) )
=> ( ( ( map @ B @ A @ F @ L3 )
= Fl )
=> ( ( map @ B @ A @ F @ L6 )
!= Fl2 ) ) ) ) ).
% append_eq_mapE
thf(fact_4685_map__eq__appendE,axiom,
! [B: $tType,A: $tType,F: B > A,Ls: list @ B,Fl: list @ A,Fl2: list @ A] :
( ( ( map @ B @ A @ F @ Ls )
= ( append @ A @ Fl @ Fl2 ) )
=> ~ ! [L3: list @ B,L6: list @ B] :
( ( Ls
= ( append @ B @ L3 @ L6 ) )
=> ( ( ( map @ B @ A @ F @ L3 )
= Fl )
=> ( ( map @ B @ A @ F @ L6 )
!= Fl2 ) ) ) ) ).
% map_eq_appendE
thf(fact_4686_Misc_Oappend__eq__map__conv,axiom,
! [A: $tType,B: $tType,Fl: list @ A,Fl2: list @ A,F: B > A,Ls: list @ B] :
( ( ( append @ A @ Fl @ Fl2 )
= ( map @ B @ A @ F @ Ls ) )
= ( ? [L2: list @ B,L7: list @ B] :
( ( Ls
= ( append @ B @ L2 @ L7 ) )
& ( ( map @ B @ A @ F @ L2 )
= Fl )
& ( ( map @ B @ A @ F @ L7 )
= Fl2 ) ) ) ) ).
% Misc.append_eq_map_conv
thf(fact_4687_Misc_Omap__eq__append__conv,axiom,
! [A: $tType,B: $tType,F: B > A,Ls: list @ B,Fl: list @ A,Fl2: list @ A] :
( ( ( map @ B @ A @ F @ Ls )
= ( append @ A @ Fl @ Fl2 ) )
= ( ? [L2: list @ B,L7: list @ B] :
( ( Ls
= ( append @ B @ L2 @ L7 ) )
& ( ( map @ B @ A @ F @ L2 )
= Fl )
& ( ( map @ B @ A @ F @ L7 )
= Fl2 ) ) ) ) ).
% Misc.map_eq_append_conv
thf(fact_4688_map__eq__imp__length__eq,axiom,
! [A: $tType,B: $tType,C: $tType,F: B > A,Xs: list @ B,G: C > A,Ys: list @ C] :
( ( ( map @ B @ A @ F @ Xs )
= ( map @ C @ A @ G @ Ys ) )
=> ( ( size_size @ ( list @ B ) @ Xs )
= ( size_size @ ( list @ C ) @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_4689_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 )
@ ^ [H3: A,T3: list @ A] : ( cons @ A @ H3 @ ( nil @ A ) ) )
@ Xss ) )
= ( map @ ( list @ A ) @ A @ ( hd @ A )
@ ( filter2 @ ( list @ A )
@ ^ [Ys3: list @ A] :
( Ys3
!= ( nil @ A ) )
@ Xss ) ) ) ).
% transpose_aux_filter_head
thf(fact_4690_set__oo__map__alt,axiom,
! [B: $tType,A: $tType,F: A > B] :
( ( comp @ ( list @ B ) @ ( set @ B ) @ ( list @ A ) @ ( set2 @ B ) @ ( map @ A @ B @ F ) )
= ( ^ [L2: list @ A] : ( image2 @ A @ B @ F @ ( set2 @ A @ L2 ) ) ) ) ).
% set_oo_map_alt
thf(fact_4691_map__consI_I2_J,axiom,
! [B: $tType,A: $tType,W2: list @ A,L: list @ A,F: B > A,Ww: list @ B,A2: B] :
( ( ( append @ A @ W2 @ L )
= ( append @ A @ ( map @ B @ A @ F @ Ww ) @ L ) )
=> ( ( cons @ A @ ( F @ A2 ) @ ( append @ A @ W2 @ L ) )
= ( append @ A @ ( map @ B @ A @ F @ ( cons @ B @ A2 @ Ww ) ) @ L ) ) ) ).
% map_consI(2)
thf(fact_4692_distinct__map__eq,axiom,
! [A: $tType,B: $tType,F: B > A,L: list @ B,X2: B,Y: B] :
( ( distinct @ A @ ( map @ B @ A @ F @ L ) )
=> ( ( ( F @ X2 )
= ( F @ Y ) )
=> ( ( member @ B @ X2 @ ( set2 @ B @ L ) )
=> ( ( member @ B @ Y @ ( set2 @ B @ L ) )
=> ( X2 = Y ) ) ) ) ) ).
% distinct_map_eq
thf(fact_4693_sorted__map,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F: B > A,Xs: list @ B] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F @ Xs ) )
= ( sorted_wrt @ B
@ ^ [X: B,Y4: B] : ( ord_less_eq @ A @ ( F @ X ) @ ( F @ Y4 ) )
@ Xs ) ) ) ).
% sorted_map
thf(fact_4694_map__replicate__const,axiom,
! [B: $tType,A: $tType,K: A,Lst: list @ B] :
( ( map @ B @ A
@ ^ [X: B] : K
@ Lst )
= ( replicate @ A @ ( size_size @ ( list @ B ) @ Lst ) @ K ) ) ).
% map_replicate_const
thf(fact_4695_sorted__filter,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F: B > A,Xs: list @ B,P: B > $o] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F @ Xs ) )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F @ ( filter2 @ B @ P @ Xs ) ) ) ) ) ).
% sorted_filter
thf(fact_4696_sorted__insort__key,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F: B > A,X2: B,Xs: list @ B] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F @ ( linorder_insort_key @ B @ A @ F @ X2 @ Xs ) ) )
= ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F @ Xs ) ) ) ) ).
% sorted_insort_key
thf(fact_4697_sorted__sort__key,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F: B > A,Xs: list @ B] : ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F @ ( linorder_sort_key @ B @ A @ F @ Xs ) ) ) ) ).
% sorted_sort_key
thf(fact_4698_concat__filter__neq__Nil,axiom,
! [A: $tType,Xs: list @ ( list @ A )] :
( ( concat @ A
@ ( filter2 @ ( list @ A )
@ ^ [Ys3: list @ A] :
( Ys3
!= ( nil @ A ) )
@ Xs ) )
= ( concat @ A @ Xs ) ) ).
% concat_filter_neq_Nil
thf(fact_4699_sorted__map__same,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F: B > A,G: ( list @ B ) > A,Xs: list @ B] :
( sorted_wrt @ A @ ( ord_less_eq @ A )
@ ( map @ B @ A @ F
@ ( filter2 @ B
@ ^ [X: B] :
( ( F @ X )
= ( G @ Xs ) )
@ Xs ) ) ) ) ).
% sorted_map_same
thf(fact_4700_foldl__foldl__conv__concat,axiom,
! [A: $tType,B: $tType,F: A > B > A,A2: A,Xs: list @ ( list @ B )] :
( ( foldl @ A @ ( list @ B ) @ ( foldl @ A @ B @ F ) @ A2 @ Xs )
= ( foldl @ A @ B @ F @ A2 @ ( concat @ B @ Xs ) ) ) ).
% foldl_foldl_conv_concat
thf(fact_4701_distinct__idx,axiom,
! [B: $tType,A: $tType,F: B > A,L: list @ B,I: nat,J: nat] :
( ( distinct @ A @ ( map @ B @ A @ F @ L ) )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ B ) @ L ) )
=> ( ( ord_less @ nat @ J @ ( size_size @ ( list @ B ) @ L ) )
=> ( ( ( F @ ( nth @ B @ L @ I ) )
= ( F @ ( nth @ B @ L @ J ) ) )
=> ( I = J ) ) ) ) ) ).
% distinct_idx
thf(fact_4702_map__upd__eq,axiom,
! [B: $tType,A: $tType,I: nat,L: list @ A,F: A > B,X2: A] :
( ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L ) )
=> ( ( F @ ( nth @ A @ L @ I ) )
= ( F @ X2 ) ) )
=> ( ( map @ A @ B @ F @ ( list_update @ A @ L @ I @ X2 ) )
= ( map @ A @ B @ F @ L ) ) ) ).
% map_upd_eq
thf(fact_4703_filter__insort,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F: B > A,Xs: list @ B,P: B > $o,X2: B] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F @ Xs ) )
=> ( ( P @ X2 )
=> ( ( filter2 @ B @ P @ ( linorder_insort_key @ B @ A @ F @ X2 @ Xs ) )
= ( linorder_insort_key @ B @ A @ F @ X2 @ ( filter2 @ B @ P @ Xs ) ) ) ) ) ) ).
% filter_insort
thf(fact_4704_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_4705_comm__append__is__replicate,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( Ys
!= ( nil @ A ) )
=> ( ( ( append @ A @ Xs @ Ys )
= ( append @ A @ Ys @ Xs ) )
=> ? [N2: nat,Zs2: list @ A] :
( ( ord_less @ nat @ ( one_one @ nat ) @ N2 )
& ( ( concat @ A @ ( replicate @ ( list @ A ) @ N2 @ Zs2 ) )
= ( append @ A @ Xs @ Ys ) ) ) ) ) ) ).
% comm_append_is_replicate
thf(fact_4706_transpose__column,axiom,
! [A: $tType,Xs: list @ ( list @ A ),I: nat] :
( ( sorted_wrt @ nat @ ( ord_less_eq @ nat ) @ ( rev @ nat @ ( map @ ( list @ A ) @ nat @ ( size_size @ ( list @ A ) ) @ Xs ) ) )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ ( list @ A ) ) @ Xs ) )
=> ( ( map @ ( list @ A ) @ A
@ ^ [Ys3: list @ A] : ( nth @ A @ Ys3 @ I )
@ ( filter2 @ ( list @ A )
@ ^ [Ys3: list @ A] : ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Ys3 ) )
@ ( transpose @ A @ Xs ) ) )
= ( nth @ ( list @ A ) @ Xs @ I ) ) ) ) ).
% transpose_column
thf(fact_4707_map__by__foldl,axiom,
! [B: $tType,A: $tType,F: A > B,L: list @ A] :
( ( foldl @ ( list @ B ) @ A
@ ^ [L2: list @ B,X: A] : ( append @ B @ L2 @ ( cons @ B @ ( F @ X ) @ ( nil @ B ) ) )
@ ( nil @ B )
@ L )
= ( map @ A @ B @ F @ L ) ) ).
% map_by_foldl
thf(fact_4708_nat_Orep__eq,axiom,
( nat2
= ( ^ [X: int] : ( product_case_prod @ nat @ nat @ nat @ ( minus_minus @ nat ) @ ( rep_Integ @ X ) ) ) ) ).
% nat.rep_eq
thf(fact_4709_length__product__lists,axiom,
! [B: $tType,Xss: list @ ( list @ B )] :
( ( size_size @ ( list @ ( list @ B ) ) @ ( product_lists @ B @ Xss ) )
= ( foldr @ nat @ nat @ ( times_times @ nat ) @ ( map @ ( list @ B ) @ nat @ ( size_size @ ( list @ B ) ) @ Xss ) @ ( one_one @ nat ) ) ) ).
% length_product_lists
thf(fact_4710_properties__for__sort__key,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [Ys: list @ B,Xs: list @ B,F: B > A] :
( ( ( mset @ B @ Ys )
= ( mset @ B @ Xs ) )
=> ( ! [K4: B] :
( ( member @ B @ K4 @ ( set2 @ B @ Ys ) )
=> ( ( filter2 @ B
@ ^ [X: B] :
( ( F @ K4 )
= ( F @ X ) )
@ Ys )
= ( filter2 @ B
@ ^ [X: B] :
( ( F @ K4 )
= ( F @ X ) )
@ Xs ) ) )
=> ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F @ Ys ) )
=> ( ( linorder_sort_key @ B @ A @ F @ Xs )
= Ys ) ) ) ) ) ).
% properties_for_sort_key
thf(fact_4711_of__int_Orep__eq,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( ( ring_1_of_int @ A )
= ( ^ [X: int] :
( product_case_prod @ nat @ nat @ A
@ ^ [I2: nat,J3: nat] : ( minus_minus @ A @ ( semiring_1_of_nat @ A @ I2 ) @ ( semiring_1_of_nat @ A @ J3 ) )
@ ( rep_Integ @ X ) ) ) ) ) ).
% of_int.rep_eq
thf(fact_4712_nth__transpose,axiom,
! [A: $tType,I: nat,Xs: list @ ( list @ A )] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ ( list @ A ) ) @ ( transpose @ A @ Xs ) ) )
=> ( ( nth @ ( list @ A ) @ ( transpose @ A @ Xs ) @ I )
= ( map @ ( list @ A ) @ A
@ ^ [Xs3: list @ A] : ( nth @ A @ Xs3 @ I )
@ ( filter2 @ ( list @ A )
@ ^ [Ys3: list @ A] : ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Ys3 ) )
@ Xs ) ) ) ) ).
% nth_transpose
thf(fact_4713_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_4714_transpose__column__length,axiom,
! [A: $tType,Xs: list @ ( list @ A ),I: nat] :
( ( sorted_wrt @ nat @ ( ord_less_eq @ nat ) @ ( rev @ nat @ ( map @ ( list @ A ) @ nat @ ( size_size @ ( list @ A ) ) @ Xs ) ) )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ ( list @ A ) ) @ Xs ) )
=> ( ( size_size @ ( list @ ( list @ A ) )
@ ( filter2 @ ( list @ A )
@ ^ [Ys3: list @ A] : ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Ys3 ) )
@ ( transpose @ A @ Xs ) ) )
= ( size_size @ ( list @ A ) @ ( nth @ ( list @ A ) @ Xs @ I ) ) ) ) ) ).
% transpose_column_length
thf(fact_4715_less__int_Orep__eq,axiom,
( ( ord_less @ int )
= ( ^ [X: int,Xa5: int] :
( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [Y4: nat,Z3: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V4: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ Y4 @ V4 ) @ ( plus_plus @ nat @ U2 @ Z3 ) ) )
@ ( rep_Integ @ X )
@ ( rep_Integ @ Xa5 ) ) ) ) ).
% less_int.rep_eq
thf(fact_4716_folding__insort__key_Ofinite__set__strict__sorted,axiom,
! [A: $tType,B: $tType,Less_eq: A > A > $o,Less: A > A > $o,S2: set @ B,F: B > A,A5: set @ B] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S2 @ F )
=> ( ( ord_less_eq @ ( set @ B ) @ A5 @ S2 )
=> ( ( finite_finite2 @ B @ A5 )
=> ~ ! [L3: list @ B] :
( ( sorted_wrt @ A @ Less @ ( map @ B @ A @ F @ L3 ) )
=> ( ( ( set2 @ B @ L3 )
= A5 )
=> ( ( size_size @ ( list @ B ) @ L3 )
!= ( finite_card @ B @ A5 ) ) ) ) ) ) ) ).
% folding_insort_key.finite_set_strict_sorted
thf(fact_4717_map__filter__map__filter,axiom,
! [A: $tType,B: $tType,F: B > A,P: B > $o,Xs: list @ B] :
( ( map @ B @ A @ F @ ( filter2 @ B @ P @ Xs ) )
= ( map_filter @ B @ A
@ ^ [X: B] : ( if @ ( option @ A ) @ ( P @ X ) @ ( some @ A @ ( F @ X ) ) @ ( none @ A ) )
@ Xs ) ) ).
% map_filter_map_filter
thf(fact_4718_insort__key__remove1,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [A2: B,Xs: list @ B,F: B > A] :
( ( member @ B @ A2 @ ( set2 @ B @ Xs ) )
=> ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F @ Xs ) )
=> ( ( ( hd @ B
@ ( filter2 @ B
@ ^ [X: B] :
( ( F @ A2 )
= ( F @ X ) )
@ Xs ) )
= A2 )
=> ( ( linorder_insort_key @ B @ A @ F @ A2 @ ( remove1 @ B @ A2 @ Xs ) )
= Xs ) ) ) ) ) ).
% insort_key_remove1
thf(fact_4719_product__concat__map,axiom,
! [B: $tType,A: $tType] :
( ( product @ A @ B )
= ( ^ [Xs3: list @ A,Ys3: list @ B] :
( concat @ ( product_prod @ A @ B )
@ ( map @ A @ ( list @ ( product_prod @ A @ B ) )
@ ^ [X: A] : ( map @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X ) @ Ys3 )
@ Xs3 ) ) ) ) ).
% product_concat_map
thf(fact_4720_n__lists_Osimps_I2_J,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( n_lists @ A @ ( suc @ N ) @ Xs )
= ( concat @ ( list @ A )
@ ( map @ ( list @ A ) @ ( list @ ( list @ A ) )
@ ^ [Ys3: list @ A] :
( map @ A @ ( list @ A )
@ ^ [Y4: A] : ( cons @ A @ Y4 @ Ys3 )
@ Xs )
@ ( n_lists @ A @ N @ Xs ) ) ) ) ).
% n_lists.simps(2)
thf(fact_4721_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 ) )
@ ^ [X: A] : ( map @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X ) @ ( product_lists @ A @ Xss ) )
@ Xs ) ) ) ).
% product_lists.simps(2)
thf(fact_4722_sorted__remove1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,A2: A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( remove1 @ A @ A2 @ Xs ) ) ) ) ).
% sorted_remove1
thf(fact_4723_subseqs_Osimps_I2_J,axiom,
! [A: $tType,X2: A,Xs: list @ A] :
( ( subseqs @ A @ ( cons @ A @ X2 @ Xs ) )
= ( append @ ( list @ A ) @ ( map @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X2 ) @ ( subseqs @ A @ Xs ) ) @ ( subseqs @ A @ Xs ) ) ) ).
% subseqs.simps(2)
thf(fact_4724_map__filter__simps_I1_J,axiom,
! [A: $tType,B: $tType,F: B > ( option @ A ),X2: B,Xs: list @ B] :
( ( map_filter @ B @ A @ F @ ( cons @ B @ X2 @ Xs ) )
= ( case_option @ ( list @ A ) @ A @ ( map_filter @ B @ A @ F @ Xs )
@ ^ [Y4: A] : ( cons @ A @ Y4 @ ( map_filter @ B @ A @ F @ Xs ) )
@ ( F @ X2 ) ) ) ).
% map_filter_simps(1)
thf(fact_4725_transpose_Oelims,axiom,
! [A: $tType,X2: list @ ( list @ A ),Y: list @ ( list @ A )] :
( ( ( transpose @ A @ X2 )
= Y )
=> ( ( ( X2
= ( nil @ ( list @ A ) ) )
=> ( Y
!= ( nil @ ( list @ A ) ) ) )
=> ( ! [Xss2: list @ ( list @ A )] :
( ( X2
= ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss2 ) )
=> ( Y
!= ( transpose @ A @ Xss2 ) ) )
=> ~ ! [X3: A,Xs2: list @ A,Xss2: list @ ( list @ A )] :
( ( X2
= ( 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 )
@ ^ [H3: A,T3: list @ A] : ( cons @ A @ H3 @ ( 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 ) )
@ ^ [H3: A,T3: list @ A] : ( cons @ ( list @ A ) @ T3 @ ( nil @ ( list @ A ) ) ) )
@ Xss2 ) ) ) ) ) ) ) ) ) ) ).
% transpose.elims
thf(fact_4726_transpose_Osimps_I3_J,axiom,
! [A: $tType,X2: A,Xs: list @ A,Xss: list @ ( list @ A )] :
( ( transpose @ A @ ( cons @ ( list @ A ) @ ( cons @ A @ X2 @ Xs ) @ Xss ) )
= ( cons @ ( list @ A )
@ ( cons @ A @ X2
@ ( concat @ A
@ ( map @ ( list @ A ) @ ( list @ A )
@ ( case_list @ ( list @ A ) @ A @ ( nil @ A )
@ ^ [H3: A,T3: list @ A] : ( cons @ A @ H3 @ ( nil @ A ) ) )
@ Xss ) ) )
@ ( transpose @ A
@ ( cons @ ( list @ A ) @ Xs
@ ( concat @ ( list @ A )
@ ( map @ ( list @ A ) @ ( list @ ( list @ A ) )
@ ( case_list @ ( list @ ( list @ A ) ) @ A @ ( nil @ ( list @ A ) )
@ ^ [H3: A,T3: list @ A] : ( cons @ ( list @ A ) @ T3 @ ( nil @ ( list @ A ) ) ) )
@ Xss ) ) ) ) ) ) ).
% transpose.simps(3)
thf(fact_4727_sorted__map__remove1,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F: B > A,Xs: list @ B,X2: B] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F @ Xs ) )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F @ ( remove1 @ B @ X2 @ Xs ) ) ) ) ) ).
% sorted_map_remove1
thf(fact_4728_length__remove1,axiom,
! [A: $tType,X2: A,Xs: list @ A] :
( ( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
=> ( ( size_size @ ( list @ A ) @ ( remove1 @ A @ X2 @ Xs ) )
= ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) ) ) )
& ( ~ ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
=> ( ( size_size @ ( list @ A ) @ ( remove1 @ A @ X2 @ Xs ) )
= ( size_size @ ( list @ A ) @ Xs ) ) ) ) ).
% length_remove1
thf(fact_4729_insort__remove1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,Xs: list @ A] :
( ( member @ A @ A2 @ ( set2 @ A @ Xs ) )
=> ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( ( linorder_insort_key @ A @ A
@ ^ [X: A] : X
@ A2
@ ( remove1 @ A @ A2 @ Xs ) )
= Xs ) ) ) ) ).
% insort_remove1
thf(fact_4730_execute__make,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [N: nat,F: nat > A,H: heap_ext @ product_unit] :
( ( heap_Time_execute @ ( array @ A ) @ ( array_make @ A @ N @ F ) @ H )
= ( some @ ( product_prod @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )
@ ( product_case_prod @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( product_prod @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )
@ ^ [R2: array @ A,H5: heap_ext @ product_unit] : ( product_Pair @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R2 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H5 @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) )
@ ( array_alloc @ A @ ( map @ nat @ A @ F @ ( upt @ ( zero_zero @ nat ) @ N ) ) @ H ) ) ) ) ) ).
% execute_make
thf(fact_4731_folding__insort__key_Osorted__key__list__of__set__unique,axiom,
! [A: $tType,B: $tType,Less_eq: A > A > $o,Less: A > A > $o,S2: set @ B,F: B > A,A5: set @ B,L: list @ B] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S2 @ F )
=> ( ( ord_less_eq @ ( set @ B ) @ A5 @ S2 )
=> ( ( finite_finite2 @ B @ A5 )
=> ( ( ( sorted_wrt @ A @ Less @ ( map @ B @ A @ F @ L ) )
& ( ( set2 @ B @ L )
= A5 )
& ( ( size_size @ ( list @ B ) @ L )
= ( finite_card @ B @ A5 ) ) )
= ( ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F @ A5 )
= L ) ) ) ) ) ).
% folding_insort_key.sorted_key_list_of_set_unique
thf(fact_4732_Array__Time_Omake__def,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( ( array_make @ A )
= ( ^ [N4: nat,F2: nat > A] :
( heap_Time_heap @ ( array @ A )
@ ^ [H3: heap_ext @ product_unit] :
( product_case_prod @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( product_prod @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )
@ ^ [R2: array @ A,H5: heap_ext @ product_unit] : ( product_Pair @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R2 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H5 @ ( plus_plus @ nat @ N4 @ ( one_one @ nat ) ) ) )
@ ( array_alloc @ A @ ( map @ nat @ A @ F2 @ ( upt @ ( zero_zero @ nat ) @ N4 ) ) @ H3 ) ) ) ) ) ) ).
% Array_Time.make_def
thf(fact_4733_sort__upt,axiom,
! [M: nat,N: nat] :
( ( linorder_sort_key @ nat @ nat
@ ^ [X: nat] : X
@ ( upt @ M @ N ) )
= ( upt @ M @ N ) ) ).
% sort_upt
thf(fact_4734_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_4735_upt__merge,axiom,
! [I: nat,J: nat,K: nat] :
( ( ( ord_less_eq @ nat @ I @ J )
& ( ord_less_eq @ nat @ J @ K ) )
=> ( ( append @ nat @ ( upt @ I @ J ) @ ( upt @ J @ K ) )
= ( upt @ I @ K ) ) ) ).
% upt_merge
thf(fact_4736_upt__conv__Nil,axiom,
! [J: nat,I: nat] :
( ( ord_less_eq @ nat @ J @ I )
=> ( ( upt @ I @ J )
= ( nil @ nat ) ) ) ).
% upt_conv_Nil
thf(fact_4737_drop__upt,axiom,
! [M: nat,I: nat,J: nat] :
( ( drop @ nat @ M @ ( upt @ I @ J ) )
= ( upt @ ( plus_plus @ nat @ I @ M ) @ J ) ) ).
% drop_upt
thf(fact_4738_upt__eq__Nil__conv,axiom,
! [I: nat,J: nat] :
( ( ( upt @ I @ J )
= ( nil @ nat ) )
= ( ( J
= ( zero_zero @ nat ) )
| ( ord_less_eq @ nat @ J @ I ) ) ) ).
% upt_eq_Nil_conv
thf(fact_4739_take__upt,axiom,
! [I: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ I @ M ) @ N )
=> ( ( take @ nat @ M @ ( upt @ I @ N ) )
= ( upt @ I @ ( plus_plus @ nat @ I @ M ) ) ) ) ).
% take_upt
thf(fact_4740_length__upt,axiom,
! [I: nat,J: nat] :
( ( size_size @ ( list @ nat ) @ ( upt @ I @ J ) )
= ( minus_minus @ nat @ J @ I ) ) ).
% length_upt
thf(fact_4741_last__upt,axiom,
! [I: nat,J: nat] :
( ( ord_less @ nat @ I @ J )
=> ( ( last @ nat @ ( upt @ I @ J ) )
= ( minus_minus @ nat @ J @ ( one_one @ nat ) ) ) ) ).
% last_upt
thf(fact_4742_nth__upt,axiom,
! [I: nat,K: nat,J: nat] :
( ( ord_less @ nat @ ( plus_plus @ nat @ I @ K ) @ J )
=> ( ( nth @ nat @ ( upt @ I @ J ) @ K )
= ( plus_plus @ nat @ I @ K ) ) ) ).
% nth_upt
thf(fact_4743_map__add__upt_H,axiom,
! [Ofs: nat,A2: nat,B2: nat] :
( ( map @ nat @ nat
@ ^ [I2: nat] : ( plus_plus @ nat @ I2 @ Ofs )
@ ( upt @ A2 @ B2 ) )
= ( upt @ ( plus_plus @ nat @ A2 @ Ofs ) @ ( plus_plus @ nat @ B2 @ Ofs ) ) ) ).
% map_add_upt'
thf(fact_4744_upt__eq__append__conv,axiom,
! [I: nat,J: nat,Xs: list @ nat,Ys: list @ nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( ( upt @ I @ J )
= ( append @ nat @ Xs @ Ys ) )
= ( ? [K2: nat] :
( ( ord_less_eq @ nat @ I @ K2 )
& ( ord_less_eq @ nat @ K2 @ J )
& ( ( upt @ I @ K2 )
= Xs )
& ( ( upt @ K2 @ J )
= Ys ) ) ) ) ) ).
% upt_eq_append_conv
thf(fact_4745_upt__0,axiom,
! [I: nat] :
( ( upt @ I @ ( zero_zero @ nat ) )
= ( nil @ nat ) ) ).
% upt_0
thf(fact_4746_upt__append,axiom,
! [I: nat,J: nat] :
( ( ord_less @ nat @ I @ J )
=> ( ( append @ nat @ ( upt @ ( zero_zero @ nat ) @ I ) @ ( upt @ I @ J ) )
= ( upt @ ( zero_zero @ nat ) @ J ) ) ) ).
% upt_append
thf(fact_4747_upt__add__eq__append,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( upt @ I @ ( plus_plus @ nat @ J @ K ) )
= ( append @ nat @ ( upt @ I @ J ) @ ( upt @ J @ ( plus_plus @ nat @ J @ K ) ) ) ) ) ).
% upt_add_eq_append
thf(fact_4748_butlast__upt,axiom,
! [M: nat,N: nat] :
( ( butlast @ nat @ ( upt @ M @ N ) )
= ( upt @ M @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ).
% butlast_upt
thf(fact_4749_atLeast__upt,axiom,
( ( set_ord_lessThan @ nat )
= ( ^ [N4: nat] : ( set2 @ nat @ ( upt @ ( zero_zero @ nat ) @ N4 ) ) ) ) ).
% atLeast_upt
thf(fact_4750_map__add__upt,axiom,
! [N: nat,M: nat] :
( ( map @ nat @ nat
@ ^ [I2: nat] : ( plus_plus @ nat @ I2 @ N )
@ ( upt @ ( zero_zero @ nat ) @ M ) )
= ( upt @ N @ ( plus_plus @ nat @ M @ N ) ) ) ).
% map_add_upt
thf(fact_4751_sorted__upt,axiom,
! [M: nat,N: nat] : ( sorted_wrt @ nat @ ( ord_less_eq @ nat ) @ ( upt @ M @ N ) ) ).
% sorted_upt
thf(fact_4752_map__replicate__trivial,axiom,
! [A: $tType,X2: A,I: nat] :
( ( map @ nat @ A
@ ^ [I2: nat] : X2
@ ( upt @ ( zero_zero @ nat ) @ I ) )
= ( replicate @ A @ I @ X2 ) ) ).
% map_replicate_trivial
thf(fact_4753_filter__upt__take__conv,axiom,
! [A: $tType,P: A > $o,M: nat,L: list @ A,N: nat] :
( ( filter2 @ nat
@ ^ [I2: nat] : ( P @ ( nth @ A @ ( take @ A @ M @ L ) @ I2 ) )
@ ( upt @ N @ M ) )
= ( filter2 @ nat
@ ^ [I2: nat] : ( P @ ( nth @ A @ L @ I2 ) )
@ ( upt @ N @ M ) ) ) ).
% filter_upt_take_conv
thf(fact_4754_enumerate__map__upt,axiom,
! [A: $tType,N: nat,F: nat > A,M: nat] :
( ( enumerate @ A @ N @ ( map @ nat @ A @ F @ ( upt @ N @ M ) ) )
= ( map @ nat @ ( product_prod @ nat @ A )
@ ^ [K2: nat] : ( product_Pair @ nat @ A @ K2 @ ( F @ K2 ) )
@ ( upt @ N @ M ) ) ) ).
% enumerate_map_upt
thf(fact_4755_atMost__upto,axiom,
( ( set_ord_atMost @ nat )
= ( ^ [N4: nat] : ( set2 @ nat @ ( upt @ ( zero_zero @ nat ) @ ( suc @ N4 ) ) ) ) ) ).
% atMost_upto
thf(fact_4756_upt__Suc__append,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( upt @ I @ ( suc @ J ) )
= ( append @ nat @ ( upt @ I @ J ) @ ( cons @ nat @ J @ ( nil @ nat ) ) ) ) ) ).
% upt_Suc_append
thf(fact_4757_upt__Suc,axiom,
! [I: nat,J: nat] :
( ( ( ord_less_eq @ nat @ I @ J )
=> ( ( upt @ I @ ( suc @ J ) )
= ( append @ nat @ ( upt @ I @ J ) @ ( cons @ nat @ J @ ( nil @ nat ) ) ) ) )
& ( ~ ( ord_less_eq @ nat @ I @ J )
=> ( ( upt @ I @ ( suc @ J ) )
= ( nil @ nat ) ) ) ) ).
% upt_Suc
thf(fact_4758_map__upt__Suc,axiom,
! [A: $tType,F: nat > A,N: nat] :
( ( map @ nat @ A @ F @ ( upt @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
= ( cons @ A @ ( F @ ( zero_zero @ nat ) )
@ ( map @ nat @ A
@ ^ [I2: nat] : ( F @ ( suc @ I2 ) )
@ ( upt @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% map_upt_Suc
thf(fact_4759_upt__filter__extend,axiom,
! [U: nat,U3: nat,P: nat > $o] :
( ( ord_less_eq @ nat @ U @ U3 )
=> ( ! [I3: nat] :
( ( ( ord_less_eq @ nat @ U @ I3 )
& ( ord_less @ nat @ I3 @ U3 ) )
=> ~ ( P @ I3 ) )
=> ( ( filter2 @ nat @ P @ ( upt @ ( zero_zero @ nat ) @ U ) )
= ( filter2 @ nat @ P @ ( upt @ ( zero_zero @ nat ) @ U3 ) ) ) ) ) ).
% upt_filter_extend
thf(fact_4760_map__decr__upt,axiom,
! [M: nat,N: nat] :
( ( map @ nat @ nat
@ ^ [N4: nat] : ( minus_minus @ nat @ N4 @ ( suc @ ( zero_zero @ nat ) ) )
@ ( upt @ ( suc @ M ) @ ( suc @ N ) ) )
= ( upt @ M @ N ) ) ).
% map_decr_upt
thf(fact_4761_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_4762_nth__map__upt,axiom,
! [A: $tType,I: nat,N: nat,M: nat,F: nat > A] :
( ( ord_less @ nat @ I @ ( minus_minus @ nat @ N @ M ) )
=> ( ( nth @ A @ ( map @ nat @ A @ F @ ( upt @ M @ N ) ) @ I )
= ( F @ ( plus_plus @ nat @ M @ I ) ) ) ) ).
% nth_map_upt
thf(fact_4763_upt__eq__lel__conv,axiom,
! [L: nat,H: nat,Is1: list @ nat,I: nat,Is2: list @ nat] :
( ( ( upt @ L @ H )
= ( append @ nat @ Is1 @ ( cons @ nat @ I @ Is2 ) ) )
= ( ( Is1
= ( upt @ L @ I ) )
& ( Is2
= ( upt @ ( suc @ I ) @ H ) )
& ( ord_less_eq @ nat @ L @ I )
& ( ord_less @ nat @ I @ H ) ) ) ).
% upt_eq_lel_conv
thf(fact_4764_upt__eq__Cons__conv,axiom,
! [I: nat,J: nat,X2: nat,Xs: list @ nat] :
( ( ( upt @ I @ J )
= ( cons @ nat @ X2 @ Xs ) )
= ( ( ord_less @ nat @ I @ J )
& ( I = X2 )
& ( ( upt @ ( plus_plus @ nat @ I @ ( one_one @ nat ) ) @ J )
= Xs ) ) ) ).
% upt_eq_Cons_conv
thf(fact_4765_enumerate__replicate__eq,axiom,
! [A: $tType,N: nat,M: nat,A2: A] :
( ( enumerate @ A @ N @ ( replicate @ A @ M @ A2 ) )
= ( map @ nat @ ( product_prod @ nat @ A )
@ ^ [Q6: nat] : ( product_Pair @ nat @ A @ Q6 @ A2 )
@ ( upt @ N @ ( plus_plus @ nat @ N @ M ) ) ) ) ).
% enumerate_replicate_eq
thf(fact_4766_map__upt__eqI,axiom,
! [A: $tType,Xs: list @ A,N: nat,M: nat,F: nat > A] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( minus_minus @ nat @ N @ M ) )
=> ( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ Xs @ I3 )
= ( F @ ( plus_plus @ nat @ M @ I3 ) ) ) )
=> ( ( map @ nat @ A @ F @ ( upt @ M @ N ) )
= Xs ) ) ) ).
% map_upt_eqI
thf(fact_4767_map__nth__upt__drop__take__conv,axiom,
! [A: $tType,N5: nat,L: list @ A,M3: nat] :
( ( ord_less_eq @ nat @ N5 @ ( size_size @ ( list @ A ) @ L ) )
=> ( ( map @ nat @ A @ ( nth @ A @ L ) @ ( upt @ M3 @ N5 ) )
= ( drop @ A @ M3 @ ( take @ A @ N5 @ L ) ) ) ) ).
% map_nth_upt_drop_take_conv
thf(fact_4768_filter__upt__last,axiom,
! [A: $tType,P: A > $o,L: list @ A,Js2: list @ nat,J: nat,I: nat] :
( ( ( filter2 @ nat
@ ^ [K2: nat] : ( P @ ( nth @ A @ L @ K2 ) )
@ ( upt @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ L ) ) )
= ( append @ nat @ Js2 @ ( cons @ nat @ J @ ( nil @ nat ) ) ) )
=> ( ( ord_less @ nat @ J @ I )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L ) )
=> ~ ( P @ ( nth @ A @ L @ I ) ) ) ) ) ).
% filter_upt_last
thf(fact_4769_folding__insort__key_Olength__sorted__key__list__of__set,axiom,
! [A: $tType,B: $tType,Less_eq: A > A > $o,Less: A > A > $o,S2: set @ B,F: B > A,A5: set @ B] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S2 @ F )
=> ( ( ord_less_eq @ ( set @ B ) @ A5 @ S2 )
=> ( ( size_size @ ( list @ B ) @ ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F @ A5 ) )
= ( finite_card @ B @ A5 ) ) ) ) ).
% folding_insort_key.length_sorted_key_list_of_set
thf(fact_4770_horner__sum__bit__eq__take__bit,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A,N: nat] :
( ( groups4207007520872428315er_sum @ $o @ A @ ( zero_neq_one_of_bool @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( map @ nat @ $o @ ( bit_se5641148757651400278ts_bit @ A @ A2 ) @ ( upt @ ( zero_zero @ nat ) @ N ) ) )
= ( bit_se2584673776208193580ke_bit @ A @ N @ A2 ) ) ) ).
% horner_sum_bit_eq_take_bit
thf(fact_4771_transpose__rectangle,axiom,
! [A: $tType,Xs: list @ ( list @ A ),N: nat] :
( ( ( Xs
= ( nil @ ( list @ A ) ) )
=> ( N
= ( zero_zero @ nat ) ) )
=> ( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ ( list @ A ) ) @ Xs ) )
=> ( ( size_size @ ( list @ A ) @ ( nth @ ( list @ A ) @ Xs @ I3 ) )
= N ) )
=> ( ( transpose @ A @ Xs )
= ( map @ nat @ ( list @ A )
@ ^ [I2: nat] :
( map @ nat @ A
@ ^ [J3: nat] : ( nth @ A @ ( nth @ ( list @ A ) @ Xs @ J3 ) @ I2 )
@ ( upt @ ( zero_zero @ nat ) @ ( size_size @ ( list @ ( list @ A ) ) @ Xs ) ) )
@ ( upt @ ( zero_zero @ nat ) @ N ) ) ) ) ) ).
% transpose_rectangle
thf(fact_4772_product__code,axiom,
! [B: $tType,A: $tType,Xs: list @ A,Ys: list @ B] :
( ( product_product @ A @ B @ ( set2 @ A @ Xs ) @ ( set2 @ B @ Ys ) )
= ( set2 @ ( product_prod @ A @ B )
@ ( concat @ ( product_prod @ A @ B )
@ ( map @ A @ ( list @ ( product_prod @ A @ B ) )
@ ^ [X: A] : ( map @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X ) @ Ys )
@ Xs ) ) ) ) ).
% product_code
thf(fact_4773_sorted__wrt__less__sum__mono__lowerbound,axiom,
! [B: $tType] :
( ( ordere6911136660526730532id_add @ B )
=> ! [F: nat > B,Ns: list @ nat] :
( ! [X3: nat,Y2: nat] :
( ( ord_less_eq @ nat @ X3 @ Y2 )
=> ( ord_less_eq @ B @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ( sorted_wrt @ nat @ ( ord_less @ nat ) @ Ns )
=> ( ord_less_eq @ B @ ( groups7311177749621191930dd_sum @ nat @ B @ F @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ nat ) @ Ns ) ) ) @ ( groups8242544230860333062m_list @ B @ ( map @ nat @ B @ F @ Ns ) ) ) ) ) ) ).
% sorted_wrt_less_sum_mono_lowerbound
thf(fact_4774_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 ) )
@ ^ [H3: A,T3: list @ A] : ( cons @ ( list @ A ) @ T3 @ ( nil @ ( list @ A ) ) ) )
@ Xss ) )
= ( map @ ( list @ A ) @ ( list @ A ) @ ( tl @ A )
@ ( filter2 @ ( list @ A )
@ ^ [Ys3: list @ A] :
( Ys3
!= ( nil @ A ) )
@ Xss ) ) ) ).
% transpose_aux_filter_tail
thf(fact_4775_sum__list_ONil,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ( ( groups8242544230860333062m_list @ A @ ( nil @ A ) )
= ( zero_zero @ A ) ) ) ).
% sum_list.Nil
thf(fact_4776_sum__list_OCons,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ! [X2: A,Xs: list @ A] :
( ( groups8242544230860333062m_list @ A @ ( cons @ A @ X2 @ Xs ) )
= ( plus_plus @ A @ X2 @ ( groups8242544230860333062m_list @ A @ Xs ) ) ) ) ).
% sum_list.Cons
thf(fact_4777_sum__list__eq__0__iff,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [Ns: list @ A] :
( ( ( groups8242544230860333062m_list @ A @ Ns )
= ( zero_zero @ A ) )
= ( ! [X: A] :
( ( member @ A @ X @ ( set2 @ A @ Ns ) )
=> ( X
= ( zero_zero @ A ) ) ) ) ) ) ).
% sum_list_eq_0_iff
thf(fact_4778_sum__list__append,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ! [Xs: list @ A,Ys: list @ A] :
( ( groups8242544230860333062m_list @ A @ ( append @ A @ Xs @ Ys ) )
= ( plus_plus @ A @ ( groups8242544230860333062m_list @ A @ Xs ) @ ( groups8242544230860333062m_list @ A @ Ys ) ) ) ) ).
% sum_list_append
thf(fact_4779_sum__list__0,axiom,
! [B: $tType,A: $tType] :
( ( monoid_add @ A )
=> ! [Xs: list @ B] :
( ( groups8242544230860333062m_list @ A
@ ( map @ B @ A
@ ^ [X: B] : ( zero_zero @ A )
@ Xs ) )
= ( zero_zero @ A ) ) ) ).
% sum_list_0
thf(fact_4780_length__tl,axiom,
! [A: $tType,Xs: list @ A] :
( ( size_size @ ( list @ A ) @ ( tl @ A @ Xs ) )
= ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) ) ) ).
% length_tl
thf(fact_4781_in__hd__or__tl__conv,axiom,
! [A: $tType,L: list @ A,X2: A] :
( ( L
!= ( nil @ A ) )
=> ( ( ( X2
= ( hd @ A @ L ) )
| ( member @ A @ X2 @ ( set2 @ A @ ( tl @ A @ L ) ) ) )
= ( member @ A @ X2 @ ( set2 @ A @ L ) ) ) ) ).
% in_hd_or_tl_conv
thf(fact_4782_tl__replicate,axiom,
! [A: $tType,N: nat,X2: A] :
( ( tl @ A @ ( replicate @ A @ N @ X2 ) )
= ( replicate @ A @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ X2 ) ) ).
% tl_replicate
thf(fact_4783_sum__list__upt,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( groups8242544230860333062m_list @ nat @ ( upt @ M @ N ) )
= ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [X: nat] : X
@ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ).
% sum_list_upt
thf(fact_4784_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_4785_in__set__tlD,axiom,
! [A: $tType,X2: A,Xs: list @ A] :
( ( member @ A @ X2 @ ( set2 @ A @ ( tl @ A @ Xs ) ) )
=> ( member @ A @ X2 @ ( set2 @ A @ Xs ) ) ) ).
% in_set_tlD
thf(fact_4786_tl__obtain__elem,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( ( tl @ A @ Xs )
= ( nil @ A ) )
=> ~ ! [E4: A] :
( Xs
!= ( cons @ A @ E4 @ ( nil @ A ) ) ) ) ) ).
% tl_obtain_elem
thf(fact_4787_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_4788_member__le__sum__list,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [X2: A,Xs: list @ A] :
( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
=> ( ord_less_eq @ A @ X2 @ ( groups8242544230860333062m_list @ A @ Xs ) ) ) ) ).
% member_le_sum_list
thf(fact_4789_not__hd__in__tl,axiom,
! [A: $tType,X2: A,Xs: list @ A] :
( ( X2
!= ( hd @ A @ Xs ) )
=> ( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
=> ( member @ A @ X2 @ ( set2 @ A @ ( tl @ A @ Xs ) ) ) ) ) ).
% not_hd_in_tl
thf(fact_4790_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_4791_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_4792_sum__list__addf,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [F: B > A,G: B > A,Xs: list @ B] :
( ( groups8242544230860333062m_list @ A
@ ( map @ B @ A
@ ^ [X: B] : ( plus_plus @ A @ ( F @ X ) @ ( G @ X ) )
@ Xs ) )
= ( plus_plus @ A @ ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F @ Xs ) ) @ ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ G @ Xs ) ) ) ) ) ).
% sum_list_addf
thf(fact_4793_sum__list__mult__const,axiom,
! [B: $tType,A: $tType] :
( ( semiring_0 @ A )
=> ! [F: B > A,C2: A,Xs: list @ B] :
( ( groups8242544230860333062m_list @ A
@ ( map @ B @ A
@ ^ [X: B] : ( times_times @ A @ ( F @ X ) @ C2 )
@ Xs ) )
= ( times_times @ A @ ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F @ Xs ) ) @ C2 ) ) ) ).
% sum_list_mult_const
thf(fact_4794_sum__list__const__mult,axiom,
! [A: $tType,B: $tType] :
( ( semiring_0 @ A )
=> ! [C2: A,F: B > A,Xs: list @ B] :
( ( groups8242544230860333062m_list @ A
@ ( map @ B @ A
@ ^ [X: B] : ( times_times @ A @ C2 @ ( F @ X ) )
@ Xs ) )
= ( times_times @ A @ C2 @ ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F @ Xs ) ) ) ) ) ).
% sum_list_const_mult
thf(fact_4795_sum__list__subtractf,axiom,
! [A: $tType,B: $tType] :
( ( ab_group_add @ A )
=> ! [F: B > A,G: B > A,Xs: list @ B] :
( ( groups8242544230860333062m_list @ A
@ ( map @ B @ A
@ ^ [X: B] : ( minus_minus @ A @ ( F @ X ) @ ( G @ X ) )
@ Xs ) )
= ( minus_minus @ A @ ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F @ Xs ) ) @ ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ G @ Xs ) ) ) ) ) ).
% sum_list_subtractf
thf(fact_4796_tl__def,axiom,
! [A: $tType] :
( ( tl @ A )
= ( case_list @ ( list @ A ) @ A @ ( nil @ A )
@ ^ [X212: A,X223: list @ A] : X223 ) ) ).
% tl_def
thf(fact_4797_tl__append,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( tl @ A @ ( append @ A @ Xs @ Ys ) )
= ( case_list @ ( list @ A ) @ A @ ( tl @ A @ Ys )
@ ^ [Z3: A,Zs3: list @ A] : ( append @ A @ Zs3 @ Ys )
@ Xs ) ) ).
% tl_append
thf(fact_4798_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_4799_Misc_Onth__tl,axiom,
! [A: $tType,Xs: list @ A,N: nat] :
( ( Xs
!= ( nil @ A ) )
=> ( ( nth @ A @ ( tl @ A @ Xs ) @ N )
= ( nth @ A @ Xs @ ( suc @ N ) ) ) ) ).
% Misc.nth_tl
thf(fact_4800_list__take__induct__tl2,axiom,
! [B: $tType,A: $tType,Xs: list @ A,Ys: list @ B,P: B > A > $o] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ! [N2: nat] :
( ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( P @ ( nth @ B @ Ys @ N2 ) @ ( nth @ A @ Xs @ N2 ) ) )
=> ! [N9: nat] :
( ( ord_less @ nat @ N9 @ ( size_size @ ( list @ A ) @ ( tl @ A @ Xs ) ) )
=> ( P @ ( nth @ B @ ( tl @ B @ Ys ) @ N9 ) @ ( nth @ A @ ( tl @ A @ Xs ) @ N9 ) ) ) ) ) ).
% list_take_induct_tl2
thf(fact_4801_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_4802_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 ) )
= ( ! [X: A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( X
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% sum_list_nonneg_eq_0_iff
thf(fact_4803_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_4804_tl__take,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( tl @ A @ ( take @ A @ N @ Xs ) )
= ( take @ A @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ ( tl @ A @ Xs ) ) ) ).
% tl_take
thf(fact_4805_distinct__hd__tl,axiom,
! [A: $tType,Xs: list @ A,X2: A] :
( ( distinct @ A @ Xs )
=> ( ( X2
= ( hd @ A @ Xs ) )
=> ~ ( member @ A @ X2 @ ( set2 @ A @ ( tl @ A @ Xs ) ) ) ) ) ).
% distinct_hd_tl
thf(fact_4806_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_4807_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_4808_sum__list_Oeq__foldr,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ( ( groups8242544230860333062m_list @ A )
= ( ^ [Xs3: list @ A] : ( foldr @ A @ A @ ( plus_plus @ A ) @ Xs3 @ ( zero_zero @ A ) ) ) ) ) ).
% sum_list.eq_foldr
thf(fact_4809_sum__list__replicate,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [N: nat,C2: A] :
( ( groups8242544230860333062m_list @ A @ ( replicate @ A @ N @ C2 ) )
= ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ C2 ) ) ) ).
% sum_list_replicate
thf(fact_4810_remove1__tl,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( remove1 @ A @ ( hd @ A @ Xs ) @ Xs )
= ( tl @ A @ Xs ) ) ) ).
% remove1_tl
thf(fact_4811_sum__list__filter__le__nat,axiom,
! [A: $tType,F: A > nat,P: A > $o,Xs: list @ A] : ( ord_less_eq @ nat @ ( groups8242544230860333062m_list @ nat @ ( map @ A @ nat @ F @ ( filter2 @ A @ P @ Xs ) ) ) @ ( groups8242544230860333062m_list @ nat @ ( map @ A @ nat @ F @ Xs ) ) ) ).
% sum_list_filter_le_nat
thf(fact_4812_sum__list__mono,axiom,
! [B: $tType,A: $tType] :
( ( ( monoid_add @ B )
& ( ordere6658533253407199908up_add @ B ) )
=> ! [Xs: list @ A,F: A > B,G: A > B] :
( ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( ord_less_eq @ B @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_eq @ B @ ( groups8242544230860333062m_list @ B @ ( map @ A @ B @ F @ Xs ) ) @ ( groups8242544230860333062m_list @ B @ ( map @ A @ B @ G @ Xs ) ) ) ) ) ).
% sum_list_mono
thf(fact_4813_length__concat,axiom,
! [B: $tType,Xss: list @ ( list @ B )] :
( ( size_size @ ( list @ B ) @ ( concat @ B @ Xss ) )
= ( groups8242544230860333062m_list @ nat @ ( map @ ( list @ B ) @ nat @ ( size_size @ ( list @ B ) ) @ Xss ) ) ) ).
% length_concat
thf(fact_4814_sum__list__map__filter_H,axiom,
! [A: $tType,B: $tType] :
( ( monoid_add @ A )
=> ! [F: B > A,P: B > $o,Xs: list @ B] :
( ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F @ ( filter2 @ B @ P @ Xs ) ) )
= ( groups8242544230860333062m_list @ A
@ ( map @ B @ A
@ ^ [X: B] : ( if @ A @ ( P @ X ) @ ( F @ X ) @ ( zero_zero @ A ) )
@ Xs ) ) ) ) ).
% sum_list_map_filter'
thf(fact_4815_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
@ ^ [X: A] : X
@ ( set2 @ A @ Xs ) ) ) ) ) ).
% distinct_sum_list_conv_Sum
thf(fact_4816_Nitpick_Osize__list__simp_I2_J,axiom,
! [A: $tType] :
( ( size_size @ ( list @ A ) )
= ( ^ [Xs3: list @ A] :
( if @ nat
@ ( Xs3
= ( nil @ A ) )
@ ( zero_zero @ nat )
@ ( suc @ ( size_size @ ( list @ A ) @ ( tl @ A @ Xs3 ) ) ) ) ) ) ).
% Nitpick.size_list_simp(2)
thf(fact_4817_List_Onth__tl,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ ( tl @ A @ Xs ) ) )
=> ( ( nth @ A @ ( tl @ A @ Xs ) @ N )
= ( nth @ A @ Xs @ ( suc @ N ) ) ) ) ).
% List.nth_tl
thf(fact_4818_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_4819_sum__list__map__filter,axiom,
! [A: $tType,B: $tType] :
( ( monoid_add @ A )
=> ! [Xs: list @ B,P: B > $o,F: B > A] :
( ! [X3: B] :
( ( member @ B @ X3 @ ( set2 @ B @ Xs ) )
=> ( ~ ( P @ X3 )
=> ( ( F @ X3 )
= ( zero_zero @ A ) ) ) )
=> ( ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F @ ( filter2 @ B @ P @ Xs ) ) )
= ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F @ Xs ) ) ) ) ) ).
% sum_list_map_filter
thf(fact_4820_sum__list__map__remove1,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [X2: B,Xs: list @ B,F: B > A] :
( ( member @ B @ X2 @ ( set2 @ B @ Xs ) )
=> ( ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F @ Xs ) )
= ( plus_plus @ A @ ( F @ X2 ) @ ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F @ ( remove1 @ B @ X2 @ Xs ) ) ) ) ) ) ) ).
% sum_list_map_remove1
thf(fact_4821_sum__list__triv,axiom,
! [C: $tType,B: $tType] :
( ( semiring_1 @ B )
=> ! [R: B,Xs: list @ C] :
( ( groups8242544230860333062m_list @ B
@ ( map @ C @ B
@ ^ [X: C] : R
@ Xs ) )
= ( times_times @ B @ ( semiring_1_of_nat @ B @ ( size_size @ ( list @ C ) @ Xs ) ) @ R ) ) ) ).
% sum_list_triv
thf(fact_4822_sum__list__Suc,axiom,
! [A: $tType,F: A > nat,Xs: list @ A] :
( ( groups8242544230860333062m_list @ nat
@ ( map @ A @ nat
@ ^ [X: A] : ( suc @ ( F @ X ) )
@ Xs ) )
= ( plus_plus @ nat @ ( groups8242544230860333062m_list @ nat @ ( map @ A @ nat @ F @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ).
% sum_list_Suc
thf(fact_4823_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_4824_card__length__sum__list__rec,axiom,
! [M: nat,N5: 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 )
= N5 ) ) ) )
= ( 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 )
= N5 ) ) ) )
@ ( 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 ) )
= N5 ) ) ) ) ) ) ) ).
% card_length_sum_list_rec
thf(fact_4825_card__length__sum__list,axiom,
! [M: nat,N5: nat] :
( ( finite_card @ ( list @ nat )
@ ( collect @ ( list @ nat )
@ ^ [L2: list @ nat] :
( ( ( size_size @ ( list @ nat ) @ L2 )
= M )
& ( ( groups8242544230860333062m_list @ nat @ L2 )
= N5 ) ) ) )
= ( binomial @ ( minus_minus @ nat @ ( plus_plus @ nat @ N5 @ M ) @ ( one_one @ nat ) ) @ N5 ) ) ).
% card_length_sum_list
thf(fact_4826_sum__list__map__eq__sum__count,axiom,
! [A: $tType,F: A > nat,Xs: list @ A] :
( ( groups8242544230860333062m_list @ nat @ ( map @ A @ nat @ F @ Xs ) )
= ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [X: A] : ( times_times @ nat @ ( count_list @ A @ Xs @ X ) @ ( F @ X ) )
@ ( set2 @ A @ Xs ) ) ) ).
% sum_list_map_eq_sum_count
thf(fact_4827_sum__list__update,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [K: nat,Xs: list @ A,X2: A] :
( ( ord_less @ nat @ K @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( groups8242544230860333062m_list @ A @ ( list_update @ A @ Xs @ K @ X2 ) )
= ( minus_minus @ A @ ( plus_plus @ A @ ( groups8242544230860333062m_list @ A @ Xs ) @ X2 ) @ ( nth @ A @ Xs @ K ) ) ) ) ) ).
% sum_list_update
thf(fact_4828_sum__list__map__eq__sum__count2,axiom,
! [A: $tType,Xs: list @ A,X8: set @ A,F: A > nat] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ X8 )
=> ( ( finite_finite2 @ A @ X8 )
=> ( ( groups8242544230860333062m_list @ nat @ ( map @ A @ nat @ F @ Xs ) )
= ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [X: A] : ( times_times @ nat @ ( count_list @ A @ Xs @ X ) @ ( F @ X ) )
@ X8 ) ) ) ) ).
% sum_list_map_eq_sum_count2
thf(fact_4829_merge__list__correct,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Ls: list @ ( list @ A ),As: list @ ( list @ A )] :
( ! [L3: list @ A] :
( ( member @ ( list @ A ) @ L3 @ ( set2 @ ( list @ A ) @ Ls ) )
=> ( ( distinct @ A @ L3 )
& ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ L3 ) ) )
=> ( ! [L3: list @ A] :
( ( member @ ( list @ A ) @ L3 @ ( set2 @ ( list @ A ) @ As ) )
=> ( ( distinct @ A @ L3 )
& ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ L3 ) ) )
=> ( ( distinct @ A @ ( merge_list @ A @ As @ Ls ) )
& ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( merge_list @ A @ As @ Ls ) )
& ( ( set2 @ A @ ( merge_list @ A @ As @ Ls ) )
= ( set2 @ A @ ( concat @ A @ ( append @ ( list @ A ) @ As @ Ls ) ) ) ) ) ) ) ) ).
% merge_list_correct
thf(fact_4830_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 )
@ ^ [X: list @ A] :
( X
!= ( nil @ A ) )
@ Xs ) ) ) ).
% transpose_transpose
thf(fact_4831_Nitpick_Osize__list__simp_I1_J,axiom,
! [A: $tType] :
( ( size_list @ A )
= ( ^ [F2: A > nat,Xs3: list @ A] :
( if @ nat
@ ( Xs3
= ( nil @ A ) )
@ ( zero_zero @ nat )
@ ( suc @ ( plus_plus @ nat @ ( F2 @ ( hd @ A @ Xs3 ) ) @ ( size_list @ A @ F2 @ ( tl @ A @ Xs3 ) ) ) ) ) ) ) ).
% Nitpick.size_list_simp(1)
thf(fact_4832_size__list__append,axiom,
! [A: $tType,F: A > nat,Xs: list @ A,Ys: list @ A] :
( ( size_list @ A @ F @ ( append @ A @ Xs @ Ys ) )
= ( plus_plus @ nat @ ( size_list @ A @ F @ Xs ) @ ( size_list @ A @ F @ Ys ) ) ) ).
% size_list_append
thf(fact_4833_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_4834_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_4835_takeWhile__eq__take,axiom,
! [A: $tType] :
( ( takeWhile @ A )
= ( ^ [P4: A > $o,Xs3: list @ A] : ( take @ A @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P4 @ Xs3 ) ) @ Xs3 ) ) ) ).
% takeWhile_eq_take
thf(fact_4836_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_4837_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_4838_merge__list_Osimps_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [La2: list @ A,Acc22: list @ ( list @ A )] :
( ( merge_list @ A @ ( cons @ ( list @ A ) @ La2 @ Acc22 ) @ ( nil @ ( list @ A ) ) )
= ( merge_list @ A @ ( nil @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ La2 @ Acc22 ) ) ) ) ).
% merge_list.simps(3)
thf(fact_4839_merge__list_Osimps_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [La2: list @ A,Acc22: list @ ( list @ A ),L: list @ A] :
( ( merge_list @ A @ ( cons @ ( list @ A ) @ La2 @ Acc22 ) @ ( cons @ ( list @ A ) @ L @ ( nil @ ( list @ A ) ) ) )
= ( merge_list @ A @ ( nil @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ L @ ( cons @ ( list @ A ) @ La2 @ Acc22 ) ) ) ) ) ).
% merge_list.simps(4)
thf(fact_4840_list_Osize__gen_I1_J,axiom,
! [A: $tType,X2: A > nat] :
( ( size_list @ A @ X2 @ ( nil @ A ) )
= ( zero_zero @ nat ) ) ).
% list.size_gen(1)
thf(fact_4841_size__list__pointwise,axiom,
! [A: $tType,Xs: list @ A,F: A > nat,G: A > nat] :
( ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( ord_less_eq @ nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_eq @ nat @ ( size_list @ A @ F @ Xs ) @ ( size_list @ A @ G @ Xs ) ) ) ).
% size_list_pointwise
thf(fact_4842_size__list__estimation_H,axiom,
! [A: $tType,X2: A,Xs: list @ A,Y: nat,F: A > nat] :
( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
=> ( ( ord_less_eq @ nat @ Y @ ( F @ X2 ) )
=> ( ord_less_eq @ nat @ Y @ ( size_list @ A @ F @ Xs ) ) ) ) ).
% size_list_estimation'
thf(fact_4843_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_4844_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_4845_drop__takeWhile,axiom,
! [A: $tType,I: nat,P: A > $o,L: list @ A] :
( ( ord_less_eq @ nat @ I @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P @ L ) ) )
=> ( ( drop @ A @ I @ ( takeWhile @ A @ P @ L ) )
= ( takeWhile @ A @ P @ ( drop @ A @ I @ L ) ) ) ) ).
% drop_takeWhile
thf(fact_4846_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_4847_less__length__takeWhile__conv,axiom,
! [A: $tType,I: nat,P: A > $o,L: list @ A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P @ L ) ) )
= ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L ) )
& ! [J3: nat] :
( ( ord_less_eq @ nat @ J3 @ I )
=> ( P @ ( nth @ A @ L @ J3 ) ) ) ) ) ).
% less_length_takeWhile_conv
thf(fact_4848_eq__len__takeWhile__conv,axiom,
! [A: $tType,I: nat,P: A > $o,L: list @ A] :
( ( I
= ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P @ L ) ) )
= ( ( ord_less_eq @ nat @ I @ ( size_size @ ( list @ A ) @ L ) )
& ! [J3: nat] :
( ( ord_less @ nat @ J3 @ I )
=> ( P @ ( nth @ A @ L @ J3 ) ) )
& ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L ) )
=> ~ ( P @ ( nth @ A @ L @ I ) ) ) ) ) ).
% eq_len_takeWhile_conv
thf(fact_4849_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_4850_takeWhile__eq__take__P__nth,axiom,
! [A: $tType,N: nat,Xs: list @ A,P: A > $o] :
( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ N )
=> ( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( P @ ( nth @ A @ Xs @ I3 ) ) ) )
=> ( ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ~ ( P @ ( nth @ A @ Xs @ N ) ) )
=> ( ( takeWhile @ A @ P @ Xs )
= ( take @ A @ N @ Xs ) ) ) ) ).
% takeWhile_eq_take_P_nth
thf(fact_4851_size__list__conv__sum__list,axiom,
! [B: $tType] :
( ( size_list @ B )
= ( ^ [F2: B > nat,Xs3: list @ B] : ( plus_plus @ nat @ ( groups8242544230860333062m_list @ nat @ ( map @ B @ nat @ F2 @ Xs3 ) ) @ ( size_size @ ( list @ B ) @ Xs3 ) ) ) ) ).
% size_list_conv_sum_list
thf(fact_4852_list_Osize__gen_I2_J,axiom,
! [A: $tType,X2: A > nat,X21: A,X222: list @ A] :
( ( size_list @ A @ X2 @ ( cons @ A @ X21 @ X222 ) )
= ( plus_plus @ nat @ ( plus_plus @ nat @ ( X2 @ X21 ) @ ( size_list @ A @ X2 @ X222 ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% list.size_gen(2)
thf(fact_4853_filter__equals__takeWhile__sorted__rev,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F: B > A,Xs: list @ B,T2: A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( rev @ A @ ( map @ B @ A @ F @ Xs ) ) )
=> ( ( filter2 @ B
@ ^ [X: B] : ( ord_less @ A @ T2 @ ( F @ X ) )
@ Xs )
= ( takeWhile @ B
@ ^ [X: B] : ( ord_less @ A @ T2 @ ( F @ X ) )
@ Xs ) ) ) ) ).
% filter_equals_takeWhile_sorted_rev
thf(fact_4854_mergesort__remdups__def,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( mergesort_remdups @ A )
= ( ^ [Xs3: list @ A] :
( merge_list @ A @ ( nil @ ( list @ A ) )
@ ( map @ A @ ( list @ A )
@ ^ [X: A] : ( cons @ A @ X @ ( nil @ A ) )
@ Xs3 ) ) ) ) ) ).
% mergesort_remdups_def
thf(fact_4855_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,Ys3: list @ A] : ( some @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) @ ( product_Pair @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) @ ( takeWhile @ A @ ( comp @ $o @ $o @ A @ (~) @ P4 ) @ Xs3 ) @ ( product_Pair @ A @ ( list @ A ) @ Y4 @ Ys3 ) ) )
@ ( dropWhile @ A @ ( comp @ $o @ $o @ A @ (~) @ P4 ) @ Xs3 ) ) ) ) ).
% extract_def
thf(fact_4856_transpose_Opelims,axiom,
! [A: $tType,X2: list @ ( list @ A ),Y: list @ ( list @ A )] :
( ( ( transpose @ A @ X2 )
= Y )
=> ( ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ X2 )
=> ( ( ( X2
= ( nil @ ( list @ A ) ) )
=> ( ( Y
= ( nil @ ( list @ A ) ) )
=> ~ ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ ( nil @ ( list @ A ) ) ) ) )
=> ( ! [Xss2: list @ ( list @ A )] :
( ( X2
= ( 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 )] :
( ( X2
= ( 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 )
@ ^ [H3: A,T3: list @ A] : ( cons @ A @ H3 @ ( 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 ) )
@ ^ [H3: A,T3: list @ A] : ( cons @ ( list @ A ) @ T3 @ ( nil @ ( list @ A ) ) ) )
@ Xss2 ) ) ) ) ) )
=> ~ ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ ( cons @ ( list @ A ) @ ( cons @ A @ X3 @ Xs2 ) @ Xss2 ) ) ) ) ) ) ) ) ).
% transpose.pelims
thf(fact_4857_transpose_Opsimps_I3_J,axiom,
! [A: $tType,X2: A,Xs: list @ A,Xss: list @ ( list @ A )] :
( ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ ( cons @ ( list @ A ) @ ( cons @ A @ X2 @ Xs ) @ Xss ) )
=> ( ( transpose @ A @ ( cons @ ( list @ A ) @ ( cons @ A @ X2 @ Xs ) @ Xss ) )
= ( cons @ ( list @ A )
@ ( cons @ A @ X2
@ ( concat @ A
@ ( map @ ( list @ A ) @ ( list @ A )
@ ( case_list @ ( list @ A ) @ A @ ( nil @ A )
@ ^ [H3: A,T3: list @ A] : ( cons @ A @ H3 @ ( nil @ A ) ) )
@ Xss ) ) )
@ ( transpose @ A
@ ( cons @ ( list @ A ) @ Xs
@ ( concat @ ( list @ A )
@ ( map @ ( list @ A ) @ ( list @ ( list @ A ) )
@ ( case_list @ ( list @ ( list @ A ) ) @ A @ ( nil @ ( list @ A ) )
@ ^ [H3: A,T3: list @ A] : ( cons @ ( list @ A ) @ T3 @ ( nil @ ( list @ A ) ) ) )
@ Xss ) ) ) ) ) ) ) ).
% transpose.psimps(3)
thf(fact_4858_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_4859_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_4860_dropWhile__eq__drop,axiom,
! [A: $tType] :
( ( dropWhile @ A )
= ( ^ [P4: A > $o,Xs3: list @ A] : ( drop @ A @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P4 @ Xs3 ) ) @ Xs3 ) ) ) ).
% dropWhile_eq_drop
thf(fact_4861_length__dropWhile__takeWhile,axiom,
! [A: $tType,X2: nat,P: A > $o,Xs: list @ A] :
( ( ord_less @ nat @ X2 @ ( size_size @ ( list @ A ) @ ( dropWhile @ A @ P @ Xs ) ) )
=> ( ord_less @ nat @ ( plus_plus @ nat @ X2 @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P @ Xs ) ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ).
% length_dropWhile_takeWhile
thf(fact_4862_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_4863_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 ) )
@ ^ [H3: A,T3: list @ A] : ( cons @ ( list @ A ) @ T3 @ ( nil @ ( list @ A ) ) ) )
@ Xss2 ) ) ) )
=> ( P @ ( cons @ ( list @ A ) @ ( cons @ A @ X3 @ Xs2 ) @ Xss2 ) ) ) )
=> ( P @ A0 ) ) ) ) ) ).
% transpose.pinduct
thf(fact_4864_dropWhile__neq__rev,axiom,
! [A: $tType,Xs: list @ A,X2: A] :
( ( distinct @ A @ Xs )
=> ( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
=> ( ( dropWhile @ A
@ ^ [Y4: A] : Y4 != X2
@ ( rev @ A @ Xs ) )
= ( cons @ A @ X2
@ ( rev @ A
@ ( takeWhile @ A
@ ^ [Y4: A] : Y4 != X2
@ Xs ) ) ) ) ) ) ).
% dropWhile_neq_rev
thf(fact_4865_takeWhile__neq__rev,axiom,
! [A: $tType,Xs: list @ A,X2: A] :
( ( distinct @ A @ Xs )
=> ( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
=> ( ( takeWhile @ A
@ ^ [Y4: A] : Y4 != X2
@ ( rev @ A @ Xs ) )
= ( rev @ A
@ ( tl @ A
@ ( dropWhile @ A
@ ^ [Y4: A] : Y4 != X2
@ Xs ) ) ) ) ) ) ).
% takeWhile_neq_rev
thf(fact_4866_quicksort_Opelims,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: list @ A,Y: list @ A] :
( ( ( linorder_quicksort @ A @ X2 )
= Y )
=> ( ( accp @ ( list @ A ) @ ( linord6200660962353139674rt_rel @ A ) @ X2 )
=> ( ( ( X2
= ( nil @ A ) )
=> ( ( Y
= ( nil @ A ) )
=> ~ ( accp @ ( list @ A ) @ ( linord6200660962353139674rt_rel @ A ) @ ( nil @ A ) ) ) )
=> ~ ! [X3: A,Xs2: list @ A] :
( ( X2
= ( 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_4867_remove__rev__def,axiom,
! [A: $tType] :
( ( remove_rev @ A )
= ( ^ [X: A] :
( filter_rev @ A
@ ( comp @ $o @ $o @ A @ (~)
@ ( ^ [Y3: A,Z: A] : Y3 = Z
@ X ) ) ) ) ) ).
% remove_rev_def
thf(fact_4868_find__dropWhile,axiom,
! [A: $tType] :
( ( find @ A )
= ( ^ [P4: A > $o,Xs3: list @ A] :
( case_list @ ( option @ A ) @ A @ ( none @ A )
@ ^ [X: A,Xa5: list @ A] : ( some @ A @ X )
@ ( dropWhile @ A @ ( comp @ $o @ $o @ A @ (~) @ P4 ) @ Xs3 ) ) ) ) ).
% find_dropWhile
thf(fact_4869_find__SomeD_I1_J,axiom,
! [A: $tType,P: A > $o,Xs: list @ A,X2: A] :
( ( ( find @ A @ P @ Xs )
= ( some @ A @ X2 ) )
=> ( P @ X2 ) ) ).
% find_SomeD(1)
thf(fact_4870_find__SomeD_I2_J,axiom,
! [A: $tType,P: A > $o,Xs: list @ A,X2: A] :
( ( ( find @ A @ P @ Xs )
= ( some @ A @ X2 ) )
=> ( member @ A @ X2 @ ( set2 @ A @ Xs ) ) ) ).
% find_SomeD(2)
thf(fact_4871_find__Some__iff,axiom,
! [A: $tType,P: A > $o,Xs: list @ A,X2: A] :
( ( ( find @ A @ P @ Xs )
= ( some @ A @ X2 ) )
= ( ? [I2: nat] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( P @ ( nth @ A @ Xs @ I2 ) )
& ( X2
= ( nth @ A @ Xs @ I2 ) )
& ! [J3: nat] :
( ( ord_less @ nat @ J3 @ I2 )
=> ~ ( P @ ( nth @ A @ Xs @ J3 ) ) ) ) ) ) ).
% find_Some_iff
thf(fact_4872_find__Some__iff2,axiom,
! [A: $tType,X2: A,P: A > $o,Xs: list @ A] :
( ( ( some @ A @ X2 )
= ( find @ A @ P @ Xs ) )
= ( ? [I2: nat] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( P @ ( nth @ A @ Xs @ I2 ) )
& ( X2
= ( nth @ A @ Xs @ I2 ) )
& ! [J3: nat] :
( ( ord_less @ nat @ J3 @ I2 )
=> ~ ( P @ ( nth @ A @ Xs @ J3 ) ) ) ) ) ) ).
% find_Some_iff2
thf(fact_4873_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_4874_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_4875_effect__makeI,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [A2: array @ A,H4: heap_ext @ product_unit,F: nat > A,N: nat,H: heap_ext @ product_unit] :
( ( ( product_Pair @ ( array @ A ) @ ( heap_ext @ product_unit ) @ A2 @ H4 )
= ( array_alloc @ A @ ( map @ nat @ A @ F @ ( upt @ ( zero_zero @ nat ) @ N ) ) @ H ) )
=> ( heap_Time_effect @ ( array @ A ) @ ( array_make @ A @ N @ F ) @ H @ H4 @ A2 @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ).
% effect_makeI
thf(fact_4876_map__distinct__upd__conv,axiom,
! [B: $tType,A: $tType,I: nat,L: list @ A,F: A > B,X2: B] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ L ) )
=> ( ( distinct @ A @ L )
=> ( ( list_update @ B @ ( map @ A @ B @ F @ L ) @ I @ X2 )
= ( map @ A @ B @ ( fun_upd @ A @ B @ F @ ( nth @ A @ L @ I ) @ X2 ) @ L ) ) ) ) ).
% map_distinct_upd_conv
thf(fact_4877_empty__upd__none,axiom,
! [B: $tType,A: $tType,X2: A] :
( ( fun_upd @ A @ ( option @ B )
@ ^ [X: A] : ( none @ B )
@ X2
@ ( none @ B ) )
= ( ^ [X: A] : ( none @ B ) ) ) ).
% empty_upd_none
thf(fact_4878_image__update,axiom,
! [B: $tType,A: $tType,X2: A,A5: set @ A,F: A > B,N: B] :
( ~ ( member @ A @ X2 @ A5 )
=> ( ( image2 @ A @ B @ ( fun_upd @ A @ B @ F @ X2 @ N ) @ A5 )
= ( image2 @ A @ B @ F @ A5 ) ) ) ).
% image_update
thf(fact_4879_success__effectE,axiom,
! [A: $tType,C2: heap_Time_Heap @ A,H: heap_ext @ product_unit] :
( ( heap_Time_success @ A @ C2 @ H )
=> ~ ! [R6: A,H6: heap_ext @ product_unit,N2: nat] :
~ ( heap_Time_effect @ A @ C2 @ H @ H6 @ R6 @ N2 ) ) ).
% success_effectE
thf(fact_4880_effect__success,axiom,
! [A: $tType,C2: heap_Time_Heap @ A,H: heap_ext @ product_unit,H4: heap_ext @ product_unit,R: A,N: nat] :
( ( heap_Time_effect @ A @ C2 @ H @ H4 @ R @ N )
=> ( heap_Time_success @ A @ C2 @ H ) ) ).
% effect_success
thf(fact_4881_effect__deterministic_I3_J,axiom,
! [A: $tType,F: heap_Time_Heap @ A,H: heap_ext @ product_unit,H4: heap_ext @ product_unit,A2: A,N: nat,H7: heap_ext @ product_unit,B2: A,N3: nat] :
( ( heap_Time_effect @ A @ F @ H @ H4 @ A2 @ N )
=> ( ( heap_Time_effect @ A @ F @ H @ H7 @ B2 @ N3 )
=> ( N = N3 ) ) ) ).
% effect_deterministic(3)
thf(fact_4882_effect__deterministic_I2_J,axiom,
! [A: $tType,F: heap_Time_Heap @ A,H: heap_ext @ product_unit,H4: heap_ext @ product_unit,A2: A,N: nat,H7: heap_ext @ product_unit,B2: A,N3: nat] :
( ( heap_Time_effect @ A @ F @ H @ H4 @ A2 @ N )
=> ( ( heap_Time_effect @ A @ F @ H @ H7 @ B2 @ N3 )
=> ( H4 = H7 ) ) ) ).
% effect_deterministic(2)
thf(fact_4883_effect__deterministic_I1_J,axiom,
! [A: $tType,F: heap_Time_Heap @ A,H: heap_ext @ product_unit,H4: heap_ext @ product_unit,A2: A,N: nat,H7: heap_ext @ product_unit,B2: A,N3: nat] :
( ( heap_Time_effect @ A @ F @ H @ H4 @ A2 @ N )
=> ( ( heap_Time_effect @ A @ F @ H @ H7 @ B2 @ N3 )
=> ( A2 = B2 ) ) ) ).
% effect_deterministic(1)
thf(fact_4884_effect__ifE,axiom,
! [A: $tType,C2: $o,T2: heap_Time_Heap @ A,E2: heap_Time_Heap @ A,H: heap_ext @ product_unit,H4: heap_ext @ product_unit,R: A,N: nat] :
( ( heap_Time_effect @ A @ ( if @ ( heap_Time_Heap @ A ) @ C2 @ T2 @ E2 ) @ H @ H4 @ R @ N )
=> ( ( C2
=> ~ ( heap_Time_effect @ A @ T2 @ H @ H4 @ R @ N ) )
=> ~ ( ~ C2
=> ~ ( heap_Time_effect @ A @ E2 @ H @ H4 @ R @ N ) ) ) ) ).
% effect_ifE
thf(fact_4885_effect__ifI,axiom,
! [A: $tType,C2: $o,T2: heap_Time_Heap @ A,H: heap_ext @ product_unit,H4: heap_ext @ product_unit,R: A,N: nat,E2: heap_Time_Heap @ A] :
( ( C2
=> ( heap_Time_effect @ A @ T2 @ H @ H4 @ R @ N ) )
=> ( ( ~ C2
=> ( heap_Time_effect @ A @ E2 @ H @ H4 @ R @ N ) )
=> ( heap_Time_effect @ A @ ( if @ ( heap_Time_Heap @ A ) @ C2 @ T2 @ E2 ) @ H @ H4 @ R @ N ) ) ) ).
% effect_ifI
thf(fact_4886_effect__LetI,axiom,
! [B: $tType,A: $tType,X2: A,T2: A,F: A > ( heap_Time_Heap @ B ),H: heap_ext @ product_unit,H4: heap_ext @ product_unit,R: B,N: nat] :
( ( X2 = T2 )
=> ( ( heap_Time_effect @ B @ ( F @ X2 ) @ H @ H4 @ R @ N )
=> ( heap_Time_effect @ B @ ( F @ T2 ) @ H @ H4 @ R @ N ) ) ) ).
% effect_LetI
thf(fact_4887_effect__raiseE,axiom,
! [A: $tType,X2: list @ char,H: heap_ext @ product_unit,H4: heap_ext @ product_unit,R: A,N: nat] :
~ ( heap_Time_effect @ A @ ( heap_Time_raise @ A @ X2 ) @ H @ H4 @ R @ N ) ).
% effect_raiseE
thf(fact_4888_finite__update__induct,axiom,
! [B: $tType,A: $tType,F: A > B,C2: B,P: ( A > B ) > $o] :
( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [A3: A] :
( ( F @ A3 )
!= C2 ) ) )
=> ( ( P
@ ^ [A3: A] : C2 )
=> ( ! [A4: A,B4: B,F5: A > B] :
( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [C5: A] :
( ( F5 @ C5 )
!= C2 ) ) )
=> ( ( ( F5 @ A4 )
= C2 )
=> ( ( B4 != C2 )
=> ( ( P @ F5 )
=> ( P @ ( fun_upd @ A @ B @ F5 @ A4 @ B4 ) ) ) ) ) )
=> ( P @ F ) ) ) ) ).
% finite_update_induct
thf(fact_4889_effect__bindI,axiom,
! [B: $tType,A: $tType,F: heap_Time_Heap @ A,H: heap_ext @ product_unit,H4: heap_ext @ product_unit,R: A,N: nat,G: A > ( heap_Time_Heap @ B ),H7: heap_ext @ product_unit,R4: B,N3: nat] :
( ( heap_Time_effect @ A @ F @ H @ H4 @ R @ N )
=> ( ( heap_Time_effect @ B @ ( G @ R ) @ H4 @ H7 @ R4 @ N3 )
=> ( heap_Time_effect @ B @ ( heap_Time_bind @ A @ B @ F @ G ) @ H @ H7 @ R4 @ ( plus_plus @ nat @ N @ N3 ) ) ) ) ).
% effect_bindI
thf(fact_4890_effect__bindE,axiom,
! [B: $tType,A: $tType,F: heap_Time_Heap @ B,G: B > ( heap_Time_Heap @ A ),H: heap_ext @ product_unit,H7: heap_ext @ product_unit,R4: A,N: nat] :
( ( heap_Time_effect @ A @ ( heap_Time_bind @ B @ A @ F @ G ) @ H @ H7 @ R4 @ N )
=> ~ ! [H6: heap_ext @ product_unit,R6: B,N12: nat] :
( ( heap_Time_effect @ B @ F @ H @ H6 @ R6 @ N12 )
=> ! [N23: nat] :
( ( heap_Time_effect @ A @ ( G @ R6 ) @ H6 @ H7 @ R4 @ N23 )
=> ( N
!= ( plus_plus @ nat @ N12 @ N23 ) ) ) ) ) ).
% effect_bindE
thf(fact_4891_effect__returnI,axiom,
! [A: $tType,H: heap_ext @ product_unit,H4: heap_ext @ product_unit,X2: A] :
( ( H = H4 )
=> ( heap_Time_effect @ A @ ( heap_Time_return @ A @ X2 ) @ H @ H4 @ X2 @ ( one_one @ nat ) ) ) ).
% effect_returnI
thf(fact_4892_effect__returnE,axiom,
! [A: $tType,X2: A,H: heap_ext @ product_unit,H4: heap_ext @ product_unit,R: A,N: nat] :
( ( heap_Time_effect @ A @ ( heap_Time_return @ A @ X2 ) @ H @ H4 @ R @ N )
=> ~ ( ( R = X2 )
=> ( ( H4 = H )
=> ( N
!= ( one_one @ nat ) ) ) ) ) ).
% effect_returnE
thf(fact_4893_effect__ureturnE,axiom,
! [A: $tType,X2: A,H: heap_ext @ product_unit,H4: heap_ext @ product_unit,R: A,N: nat] :
( ( heap_Time_effect @ A @ ( heap_Time_ureturn @ A @ X2 ) @ H @ H4 @ R @ N )
=> ~ ( ( R = X2 )
=> ( ( H4 = H )
=> ( N
!= ( zero_zero @ nat ) ) ) ) ) ).
% effect_ureturnE
thf(fact_4894_effect__ureturnI,axiom,
! [A: $tType,H: heap_ext @ product_unit,H4: heap_ext @ product_unit,X2: A] :
( ( H = H4 )
=> ( heap_Time_effect @ A @ ( heap_Time_ureturn @ A @ X2 ) @ H @ H4 @ X2 @ ( zero_zero @ nat ) ) ) ).
% effect_ureturnI
thf(fact_4895_success__bind__effectI,axiom,
! [A: $tType,B: $tType,F: heap_Time_Heap @ A,H: heap_ext @ product_unit,H4: heap_ext @ product_unit,X2: A,N: nat,G: A > ( heap_Time_Heap @ B )] :
( ( heap_Time_effect @ A @ F @ H @ H4 @ X2 @ N )
=> ( ( heap_Time_success @ B @ ( G @ X2 ) @ H4 )
=> ( heap_Time_success @ B @ ( heap_Time_bind @ A @ B @ F @ G ) @ H ) ) ) ).
% success_bind_effectI
thf(fact_4896_effect__assertI,axiom,
! [A: $tType,P: A > $o,X2: A,H4: heap_ext @ product_unit,H: heap_ext @ product_unit,R: A,N: nat] :
( ( P @ X2 )
=> ( ( H4 = H )
=> ( ( R = X2 )
=> ( ( N
= ( one_one @ nat ) )
=> ( heap_Time_effect @ A @ ( heap_Time_assert @ A @ P @ X2 ) @ H @ H4 @ R @ N ) ) ) ) ) ).
% effect_assertI
thf(fact_4897_effect__assertE,axiom,
! [A: $tType,P: A > $o,X2: A,H: heap_ext @ product_unit,H4: heap_ext @ product_unit,R: A,N: nat] :
( ( heap_Time_effect @ A @ ( heap_Time_assert @ A @ P @ X2 ) @ H @ H4 @ R @ N )
=> ~ ( ( P @ X2 )
=> ( ( R = X2 )
=> ( ( H4 = H )
=> ( N
!= ( one_one @ nat ) ) ) ) ) ) ).
% effect_assertE
thf(fact_4898_effect__tapI,axiom,
! [A: $tType,H4: heap_ext @ product_unit,H: heap_ext @ product_unit,R: A,F: ( heap_ext @ product_unit ) > A] :
( ( H4 = H )
=> ( ( R
= ( F @ H ) )
=> ( heap_Time_effect @ A @ ( heap_Time_tap @ A @ F ) @ H @ H4 @ R @ ( one_one @ nat ) ) ) ) ).
% effect_tapI
thf(fact_4899_effect__tapE,axiom,
! [A: $tType,F: ( heap_ext @ product_unit ) > A,H: heap_ext @ product_unit,H4: heap_ext @ product_unit,R: A,N: nat] :
( ( heap_Time_effect @ A @ ( heap_Time_tap @ A @ F ) @ H @ H4 @ R @ N )
=> ~ ( ( H4 = H )
=> ( ( R
= ( F @ H ) )
=> ( N
!= ( one_one @ nat ) ) ) ) ) ).
% effect_tapE
thf(fact_4900_partition__rev_Osimps_I2_J,axiom,
! [A: $tType,P: A > $o,Yes2: list @ A,No2: list @ A,X2: A,Xs: list @ A] :
( ( partition_rev @ A @ P @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes2 @ No2 ) @ ( cons @ A @ X2 @ Xs ) )
= ( partition_rev @ A @ P @ ( if @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( P @ X2 ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X2 @ Yes2 ) @ No2 ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes2 @ ( cons @ A @ X2 @ No2 ) ) ) @ Xs ) ) ).
% partition_rev.simps(2)
thf(fact_4901_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_4902_effect__lookupE,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [R: ref @ A,H: heap_ext @ product_unit,H4: heap_ext @ product_unit,X2: A,N: nat] :
( ( heap_Time_effect @ A @ ( ref_lookup @ A @ R ) @ H @ H4 @ X2 @ N )
=> ~ ( ( H4 = H )
=> ( ( X2
= ( ref_get @ A @ H @ R ) )
=> ( N
!= ( one_one @ nat ) ) ) ) ) ) ).
% effect_lookupE
thf(fact_4903_effect__lookupI,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [H4: heap_ext @ product_unit,H: heap_ext @ product_unit,X2: A,R: ref @ A,N: nat] :
( ( H4 = H )
=> ( ( X2
= ( ref_get @ A @ H @ R ) )
=> ( ( N
= ( one_one @ nat ) )
=> ( heap_Time_effect @ A @ ( ref_lookup @ A @ R ) @ H @ H4 @ X2 @ N ) ) ) ) ) ).
% effect_lookupI
thf(fact_4904_partition__rev_Oelims,axiom,
! [A: $tType,X2: A > $o,Xa: product_prod @ ( list @ A ) @ ( list @ A ),Xb: list @ A,Y: product_prod @ ( list @ A ) @ ( list @ A )] :
( ( ( partition_rev @ A @ X2 @ 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 @ X2 @ ( if @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( X2 @ 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_4905_effectI,axiom,
! [A: $tType,C2: heap_Time_Heap @ A,H: heap_ext @ product_unit,R: A,H4: heap_ext @ product_unit,N: 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 @ H4 @ N ) ) ) )
=> ( heap_Time_effect @ A @ C2 @ H @ H4 @ R @ N ) ) ).
% effectI
thf(fact_4906_effect__def,axiom,
! [A: $tType] :
( ( heap_Time_effect @ A )
= ( ^ [C5: heap_Time_Heap @ A,H3: heap_ext @ product_unit,H5: heap_ext @ product_unit,R2: A,N4: nat] :
( ( heap_Time_execute @ A @ C5 @ H3 )
= ( 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 @ H5 @ N4 ) ) ) ) ) ) ).
% effect_def
thf(fact_4907_effect__of__listI,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [A2: array @ A,H4: heap_ext @ product_unit,Xs: list @ A,H: heap_ext @ product_unit] :
( ( ( product_Pair @ ( array @ A ) @ ( heap_ext @ product_unit ) @ A2 @ H4 )
= ( array_alloc @ A @ Xs @ H ) )
=> ( heap_Time_effect @ ( array @ A ) @ ( array_of_list @ A @ Xs ) @ H @ H4 @ A2 @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) ).
% effect_of_listI
thf(fact_4908_effect__newI,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [A2: array @ A,H4: heap_ext @ product_unit,N: nat,X2: A,H: heap_ext @ product_unit] :
( ( ( product_Pair @ ( array @ A ) @ ( heap_ext @ product_unit ) @ A2 @ H4 )
= ( array_alloc @ A @ ( replicate @ A @ N @ X2 ) @ H ) )
=> ( heap_Time_effect @ ( array @ A ) @ ( array_new @ A @ N @ X2 ) @ H @ H4 @ A2 @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ).
% effect_newI
thf(fact_4909_map__upds__append1,axiom,
! [B: $tType,A: $tType,Xs: list @ A,Ys: list @ B,M: A > ( option @ B ),X2: A] :
( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( map_upds @ A @ B @ M @ ( append @ A @ Xs @ ( cons @ A @ X2 @ ( nil @ A ) ) ) @ Ys )
= ( fun_upd @ A @ ( option @ B ) @ ( map_upds @ A @ B @ M @ Xs @ Ys ) @ X2 @ ( some @ B @ ( nth @ B @ Ys @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) ) ).
% map_upds_append1
thf(fact_4910_quicksort__by__rel_Oelims,axiom,
! [A: $tType,X2: A > A > $o,Xa: list @ A,Xb: list @ A,Y: list @ A] :
( ( ( quicksort_by_rel @ A @ X2 @ 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 @ X2 @ ( cons @ A @ X3 @ ( quicksort_by_rel @ A @ X2 @ Xa @ Xs_b ) ) @ Xs_s )
@ ( partition_rev @ A
@ ^ [Y4: A] : ( X2 @ Y4 @ X3 )
@ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) )
@ Xs2 ) ) ) ) ) ) ).
% quicksort_by_rel.elims
thf(fact_4911_quicksort__by__rel_Osimps_I2_J,axiom,
! [A: $tType,R3: A > A > $o,Sl2: list @ A,X2: A,Xs: list @ A] :
( ( quicksort_by_rel @ A @ R3 @ Sl2 @ ( cons @ A @ X2 @ Xs ) )
= ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ ( list @ A )
@ ^ [Xs_s: list @ A,Xs_b: list @ A] : ( quicksort_by_rel @ A @ R3 @ ( cons @ A @ X2 @ ( quicksort_by_rel @ A @ R3 @ Sl2 @ Xs_b ) ) @ Xs_s )
@ ( partition_rev @ A
@ ^ [Y4: A] : ( R3 @ Y4 @ X2 )
@ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) )
@ Xs ) ) ) ).
% quicksort_by_rel.simps(2)
thf(fact_4912_fun__upds__append2__drop,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B,M: A > ( option @ B ),Zs: list @ B] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( map_upds @ A @ B @ M @ Xs @ ( append @ B @ Ys @ Zs ) )
= ( map_upds @ A @ B @ M @ Xs @ Ys ) ) ) ).
% fun_upds_append2_drop
thf(fact_4913_fun__upds__append__drop,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B,M: A > ( option @ B ),Zs: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( map_upds @ A @ B @ M @ ( append @ A @ Xs @ Zs ) @ Ys )
= ( map_upds @ A @ B @ M @ Xs @ Ys ) ) ) ).
% fun_upds_append_drop
thf(fact_4914_set__quicksort__by__rel,axiom,
! [A: $tType,R3: A > A > $o,Sl2: list @ A,Xs: list @ A] :
( ( set2 @ A @ ( quicksort_by_rel @ A @ R3 @ Sl2 @ Xs ) )
= ( set2 @ A @ ( append @ A @ Xs @ Sl2 ) ) ) ).
% set_quicksort_by_rel
thf(fact_4915_quicksort__by__rel__permutes,axiom,
! [A: $tType,R3: A > A > $o,Sl2: list @ A,Xs: list @ A] :
( ( mset @ A @ ( quicksort_by_rel @ A @ R3 @ Sl2 @ Xs ) )
= ( mset @ A @ ( append @ A @ Xs @ Sl2 ) ) ) ).
% quicksort_by_rel_permutes
thf(fact_4916_map__upds__list__update2__drop,axiom,
! [A: $tType,B: $tType,Xs: list @ A,I: nat,M: A > ( option @ B ),Ys: list @ B,Y: B] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ I )
=> ( ( map_upds @ A @ B @ M @ Xs @ ( list_update @ B @ Ys @ I @ Y ) )
= ( map_upds @ A @ B @ M @ Xs @ Ys ) ) ) ).
% map_upds_list_update2_drop
thf(fact_4917_quicksort__by__rel_Osimps_I1_J,axiom,
! [A: $tType,R3: A > A > $o,Sl2: list @ A] :
( ( quicksort_by_rel @ A @ R3 @ Sl2 @ ( nil @ A ) )
= Sl2 ) ).
% quicksort_by_rel.simps(1)
thf(fact_4918_quicksort__by__rel__remove__acc__guared,axiom,
! [A: $tType,Sl2: list @ A,R3: A > A > $o,Xs: list @ A] :
( ( Sl2
!= ( nil @ A ) )
=> ( ( quicksort_by_rel @ A @ R3 @ Sl2 @ Xs )
= ( append @ A @ ( quicksort_by_rel @ A @ R3 @ ( nil @ A ) @ Xs ) @ Sl2 ) ) ) ).
% quicksort_by_rel_remove_acc_guared
thf(fact_4919_quicksort__by__rel__remove__acc,axiom,
! [A: $tType] :
( ( quicksort_by_rel @ A )
= ( ^ [R7: A > A > $o,Sl3: list @ A,Xs3: list @ A] : ( append @ A @ ( quicksort_by_rel @ A @ R7 @ ( nil @ A ) @ Xs3 ) @ Sl3 ) ) ) ).
% quicksort_by_rel_remove_acc
thf(fact_4920_sorted__wrt__quicksort__by__rel,axiom,
! [X10: $tType,R3: X10 > X10 > $o,Xs: list @ X10] :
( ! [X3: X10,Y2: X10] :
( ( R3 @ X3 @ Y2 )
| ( R3 @ Y2 @ X3 ) )
=> ( ! [X3: X10,Y2: X10,Z4: X10] :
( ( R3 @ X3 @ Y2 )
=> ( ( R3 @ Y2 @ Z4 )
=> ( R3 @ X3 @ Z4 ) ) )
=> ( sorted_wrt @ X10 @ R3 @ ( quicksort_by_rel @ X10 @ R3 @ ( nil @ X10 ) @ Xs ) ) ) ) ).
% sorted_wrt_quicksort_by_rel
thf(fact_4921_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_4922_sort__quicksort__by__rel,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( linorder_sort_key @ A @ A
@ ^ [X: A] : X )
= ( quicksort_by_rel @ A @ ( ord_less_eq @ A ) @ ( nil @ A ) ) ) ) ).
% sort_quicksort_by_rel
thf(fact_4923_map__upd__upds__conv__if,axiom,
! [A: $tType,B: $tType,X2: A,Ys: list @ B,Xs: list @ A,F: A > ( option @ B ),Y: B] :
( ( ( member @ A @ X2 @ ( set2 @ A @ ( take @ A @ ( size_size @ ( list @ B ) @ Ys ) @ Xs ) ) )
=> ( ( map_upds @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ F @ X2 @ ( some @ B @ Y ) ) @ Xs @ Ys )
= ( map_upds @ A @ B @ F @ Xs @ Ys ) ) )
& ( ~ ( member @ A @ X2 @ ( set2 @ A @ ( take @ A @ ( size_size @ ( list @ B ) @ Ys ) @ Xs ) ) )
=> ( ( map_upds @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ F @ X2 @ ( some @ B @ Y ) ) @ Xs @ Ys )
= ( fun_upd @ A @ ( option @ B ) @ ( map_upds @ A @ B @ F @ Xs @ Ys ) @ X2 @ ( some @ B @ Y ) ) ) ) ) ).
% map_upd_upds_conv_if
thf(fact_4924_floor__rat__def,axiom,
( ( archim6421214686448440834_floor @ rat )
= ( ^ [X: rat] :
( the @ int
@ ^ [Z3: int] :
( ( ord_less_eq @ rat @ ( ring_1_of_int @ rat @ Z3 ) @ X )
& ( ord_less @ rat @ X @ ( ring_1_of_int @ rat @ ( plus_plus @ int @ Z3 @ ( one_one @ int ) ) ) ) ) ) ) ) ).
% floor_rat_def
thf(fact_4925_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 )
@ ^ [X: nat,Y4: nat] : ( product_Pair @ nat @ nat @ Y4 @ X ) ) ) ) ).
% uminus_int_def
thf(fact_4926_map__filter__def,axiom,
! [B: $tType,A: $tType] :
( ( map_filter @ A @ B )
= ( ^ [F2: A > ( option @ B ),Xs3: list @ A] :
( map @ A @ B @ ( comp @ ( option @ B ) @ B @ A @ ( the2 @ B ) @ F2 )
@ ( filter2 @ A
@ ^ [X: A] :
( ( F2 @ X )
!= ( none @ B ) )
@ Xs3 ) ) ) ) ).
% map_filter_def
thf(fact_4927_option_Othe__def,axiom,
! [A: $tType] :
( ( the2 @ A )
= ( case_option @ A @ A @ ( undefined @ A )
@ ^ [X24: A] : X24 ) ) ).
% option.the_def
thf(fact_4928_old_Orec__nat__def,axiom,
! [T: $tType] :
( ( rec_nat @ T )
= ( ^ [F12: T,F23: nat > T > T,X: nat] : ( the @ T @ ( rec_set_nat @ T @ F12 @ F23 @ X ) ) ) ) ).
% old.rec_nat_def
thf(fact_4929_the__sym__eq__trivial,axiom,
! [A: $tType,X2: A] :
( ( the @ A
@ ( ^ [Y3: A,Z: A] : Y3 = Z
@ X2 ) )
= X2 ) ).
% the_sym_eq_trivial
thf(fact_4930_the__eq__trivial,axiom,
! [A: $tType,A2: A] :
( ( the @ A
@ ^ [X: A] : X = A2 )
= A2 ) ).
% the_eq_trivial
thf(fact_4931_the__equality,axiom,
! [A: $tType,P: A > $o,A2: A] :
( ( P @ A2 )
=> ( ! [X3: A] :
( ( P @ X3 )
=> ( X3 = A2 ) )
=> ( ( the @ A @ P )
= A2 ) ) ) ).
% the_equality
thf(fact_4932_The__split__eq,axiom,
! [A: $tType,B: $tType,X2: A,Y: B] :
( ( the @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [X9: A,Y8: B] :
( ( X2 = X9 )
& ( Y = Y8 ) ) ) )
= ( product_Pair @ A @ B @ X2 @ Y ) ) ).
% The_split_eq
thf(fact_4933_theI,axiom,
! [A: $tType,P: A > $o,A2: A] :
( ( P @ A2 )
=> ( ! [X3: A] :
( ( P @ X3 )
=> ( X3 = A2 ) )
=> ( P @ ( the @ A @ P ) ) ) ) ).
% theI
thf(fact_4934_theI_H,axiom,
! [A: $tType,P: A > $o] :
( ? [X4: A] :
( ( P @ X4 )
& ! [Y2: A] :
( ( P @ Y2 )
=> ( Y2 = X4 ) ) )
=> ( P @ ( the @ A @ P ) ) ) ).
% theI'
thf(fact_4935_theI2,axiom,
! [A: $tType,P: A > $o,A2: A,Q: A > $o] :
( ( P @ A2 )
=> ( ! [X3: A] :
( ( P @ X3 )
=> ( X3 = A2 ) )
=> ( ! [X3: A] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( Q @ ( the @ A @ P ) ) ) ) ) ).
% theI2
thf(fact_4936_If__def,axiom,
! [A: $tType] :
( ( if @ A )
= ( ^ [P4: $o,X: A,Y4: A] :
( the @ A
@ ^ [Z3: A] :
( ( P4
=> ( Z3 = X ) )
& ( ~ P4
=> ( Z3 = Y4 ) ) ) ) ) ) ).
% If_def
thf(fact_4937_the1I2,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ? [X4: A] :
( ( P @ X4 )
& ! [Y2: A] :
( ( P @ Y2 )
=> ( Y2 = X4 ) ) )
=> ( ! [X3: A] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( Q @ ( the @ A @ P ) ) ) ) ).
% the1I2
thf(fact_4938_the1__equality,axiom,
! [A: $tType,P: A > $o,A2: A] :
( ? [X4: A] :
( ( P @ X4 )
& ! [Y2: A] :
( ( P @ Y2 )
=> ( Y2 = X4 ) ) )
=> ( ( P @ A2 )
=> ( ( the @ A @ P )
= A2 ) ) ) ).
% the1_equality
thf(fact_4939_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 ) )
@ ^ [X: nat,Y4: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V4: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ X @ U2 ) @ ( times_times @ nat @ Y4 @ V4 ) ) @ ( plus_plus @ nat @ ( times_times @ nat @ X @ V4 ) @ ( times_times @ nat @ Y4 @ U2 ) ) ) ) ) ) ) ).
% times_int_def
thf(fact_4940_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 ) )
@ ^ [X: nat,Y4: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V4: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X @ V4 ) @ ( plus_plus @ nat @ Y4 @ U2 ) ) ) ) ) ) ).
% minus_int_def
thf(fact_4941_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 ) )
@ ^ [X: nat,Y4: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V4: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X @ U2 ) @ ( plus_plus @ nat @ Y4 @ V4 ) ) ) ) ) ) ).
% plus_int_def
thf(fact_4942_the__elem__def,axiom,
! [A: $tType] :
( ( the_elem @ A )
= ( ^ [X5: set @ A] :
( the @ A
@ ^ [X: A] :
( X5
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% the_elem_def
thf(fact_4943_sort__mergesort,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( linorder_sort_key @ A @ A
@ ^ [X: A] : X )
= ( mergesort @ A ) ) ) ).
% sort_mergesort
thf(fact_4944_prod__encode__def,axiom,
( nat_prod_encode
= ( product_case_prod @ nat @ nat @ nat
@ ^ [M2: nat,N4: nat] : ( plus_plus @ nat @ ( nat_triangle @ ( plus_plus @ nat @ M2 @ N4 ) ) @ M2 ) ) ) ).
% prod_encode_def
thf(fact_4945_le__prod__encode__1,axiom,
! [A2: nat,B2: nat] : ( ord_less_eq @ nat @ A2 @ ( nat_prod_encode @ ( product_Pair @ nat @ nat @ A2 @ B2 ) ) ) ).
% le_prod_encode_1
thf(fact_4946_le__prod__encode__2,axiom,
! [B2: nat,A2: nat] : ( ord_less_eq @ nat @ B2 @ ( nat_prod_encode @ ( product_Pair @ nat @ nat @ A2 @ B2 ) ) ) ).
% le_prod_encode_2
thf(fact_4947_prod__encode__prod__decode__aux,axiom,
! [K: nat,M: nat] :
( ( nat_prod_encode @ ( nat_prod_decode_aux @ K @ M ) )
= ( plus_plus @ nat @ ( nat_triangle @ K ) @ M ) ) ).
% prod_encode_prod_decode_aux
thf(fact_4948_list__encode_Oelims,axiom,
! [X2: list @ nat,Y: nat] :
( ( ( nat_list_encode @ X2 )
= Y )
=> ( ( ( X2
= ( nil @ nat ) )
=> ( Y
!= ( zero_zero @ nat ) ) )
=> ~ ! [X3: nat,Xs2: list @ nat] :
( ( X2
= ( cons @ nat @ X3 @ Xs2 ) )
=> ( Y
!= ( suc @ ( nat_prod_encode @ ( product_Pair @ nat @ nat @ X3 @ ( nat_list_encode @ Xs2 ) ) ) ) ) ) ) ) ).
% list_encode.elims
thf(fact_4949_list__encode_Opelims,axiom,
! [X2: list @ nat,Y: nat] :
( ( ( nat_list_encode @ X2 )
= Y )
=> ( ( accp @ ( list @ nat ) @ nat_list_encode_rel @ X2 )
=> ( ( ( X2
= ( nil @ nat ) )
=> ( ( Y
= ( zero_zero @ nat ) )
=> ~ ( accp @ ( list @ nat ) @ nat_list_encode_rel @ ( nil @ nat ) ) ) )
=> ~ ! [X3: nat,Xs2: list @ nat] :
( ( X2
= ( cons @ nat @ X3 @ Xs2 ) )
=> ( ( Y
= ( suc @ ( nat_prod_encode @ ( product_Pair @ nat @ nat @ X3 @ ( nat_list_encode @ Xs2 ) ) ) ) )
=> ~ ( accp @ ( list @ nat ) @ nat_list_encode_rel @ ( cons @ nat @ X3 @ Xs2 ) ) ) ) ) ) ) ).
% list_encode.pelims
thf(fact_4950_merge__list_Opsimps_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [La2: list @ A,Acc22: list @ ( list @ A ),L: list @ A] :
( ( accp @ ( product_prod @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) ) @ ( merge_list_rel @ A ) @ ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ La2 @ Acc22 ) @ ( cons @ ( list @ A ) @ L @ ( nil @ ( list @ A ) ) ) ) )
=> ( ( merge_list @ A @ ( cons @ ( list @ A ) @ La2 @ Acc22 ) @ ( cons @ ( list @ A ) @ L @ ( nil @ ( list @ A ) ) ) )
= ( merge_list @ A @ ( nil @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ L @ ( cons @ ( list @ A ) @ La2 @ Acc22 ) ) ) ) ) ) ).
% merge_list.psimps(4)
thf(fact_4951_list__encode_Osimps_I1_J,axiom,
( ( nat_list_encode @ ( nil @ nat ) )
= ( zero_zero @ nat ) ) ).
% list_encode.simps(1)
thf(fact_4952_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_4953_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_4954_merge__list_Opsimps_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [La2: list @ A,Acc22: list @ ( list @ A )] :
( ( accp @ ( product_prod @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) ) @ ( merge_list_rel @ A ) @ ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ La2 @ Acc22 ) @ ( nil @ ( list @ A ) ) ) )
=> ( ( merge_list @ A @ ( cons @ ( list @ A ) @ La2 @ Acc22 ) @ ( nil @ ( list @ A ) ) )
= ( merge_list @ A @ ( nil @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ La2 @ Acc22 ) ) ) ) ) ).
% merge_list.psimps(3)
thf(fact_4955_merge__list_Opelims,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: list @ ( list @ A ),Xa: list @ ( list @ A ),Y: list @ A] :
( ( ( merge_list @ A @ X2 @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) ) @ ( merge_list_rel @ A ) @ ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ X2 @ Xa ) )
=> ( ( ( X2
= ( 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 ) ) ) ) ) ) )
=> ( ( ( X2
= ( nil @ ( list @ A ) ) )
=> ! [L3: list @ A] :
( ( Xa
= ( cons @ ( list @ A ) @ L3 @ ( nil @ ( list @ A ) ) ) )
=> ( ( Y = L3 )
=> ~ ( accp @ ( product_prod @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) ) @ ( merge_list_rel @ A ) @ ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ ( nil @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ L3 @ ( nil @ ( list @ A ) ) ) ) ) ) ) )
=> ( ! [La: list @ A,Acc2: list @ ( list @ A )] :
( ( X2
= ( cons @ ( list @ A ) @ La @ Acc2 ) )
=> ( ( Xa
= ( nil @ ( list @ A ) ) )
=> ( ( Y
= ( merge_list @ A @ ( nil @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ La @ Acc2 ) ) )
=> ~ ( accp @ ( product_prod @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) ) @ ( merge_list_rel @ A ) @ ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ La @ Acc2 ) @ ( nil @ ( list @ A ) ) ) ) ) ) )
=> ( ! [La: list @ A,Acc2: list @ ( list @ A )] :
( ( X2
= ( cons @ ( list @ A ) @ La @ Acc2 ) )
=> ! [L3: list @ A] :
( ( Xa
= ( cons @ ( list @ A ) @ L3 @ ( nil @ ( list @ A ) ) ) )
=> ( ( Y
= ( merge_list @ A @ ( nil @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ L3 @ ( cons @ ( list @ A ) @ La @ Acc2 ) ) ) )
=> ~ ( accp @ ( product_prod @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) ) @ ( merge_list_rel @ A ) @ ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ La @ Acc2 ) @ ( cons @ ( list @ A ) @ L3 @ ( nil @ ( list @ A ) ) ) ) ) ) ) )
=> ~ ! [L12: list @ A,L23: list @ A,Ls2: list @ ( list @ A )] :
( ( Xa
= ( cons @ ( list @ A ) @ L12 @ ( cons @ ( list @ A ) @ L23 @ Ls2 ) ) )
=> ( ( Y
= ( merge_list @ A @ ( cons @ ( list @ A ) @ ( merge @ A @ L12 @ L23 ) @ X2 ) @ Ls2 ) )
=> ~ ( accp @ ( product_prod @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) ) @ ( merge_list_rel @ A ) @ ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ X2 @ ( cons @ ( list @ A ) @ L12 @ ( cons @ ( list @ A ) @ L23 @ Ls2 ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% merge_list.pelims
thf(fact_4956_merge__list_Opsimps_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Acc22: list @ ( list @ A ),L1: list @ A,L22: list @ A,Ls: list @ ( list @ A )] :
( ( accp @ ( product_prod @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) ) @ ( merge_list_rel @ A ) @ ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ Acc22 @ ( cons @ ( list @ A ) @ L1 @ ( cons @ ( list @ A ) @ L22 @ Ls ) ) ) )
=> ( ( merge_list @ A @ Acc22 @ ( cons @ ( list @ A ) @ L1 @ ( cons @ ( list @ A ) @ L22 @ Ls ) ) )
= ( merge_list @ A @ ( cons @ ( list @ A ) @ ( merge @ A @ L1 @ L22 ) @ Acc22 ) @ Ls ) ) ) ) ).
% merge_list.psimps(5)
thf(fact_4957_merge__list_Opinduct,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A0: list @ ( list @ A ),A13: 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 @ A13 ) )
=> ( ( ( accp @ ( product_prod @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) ) @ ( merge_list_rel @ A ) @ ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ ( nil @ ( list @ A ) ) @ ( nil @ ( list @ A ) ) ) )
=> ( P @ ( nil @ ( list @ A ) ) @ ( nil @ ( list @ A ) ) ) )
=> ( ! [L3: list @ A] :
( ( accp @ ( product_prod @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) ) @ ( merge_list_rel @ A ) @ ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ ( nil @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ L3 @ ( nil @ ( list @ A ) ) ) ) )
=> ( P @ ( nil @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ L3 @ ( nil @ ( list @ A ) ) ) ) )
=> ( ! [La: list @ A,Acc2: list @ ( list @ A )] :
( ( accp @ ( product_prod @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) ) @ ( merge_list_rel @ A ) @ ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ La @ Acc2 ) @ ( nil @ ( list @ A ) ) ) )
=> ( ( P @ ( nil @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ La @ Acc2 ) )
=> ( P @ ( cons @ ( list @ A ) @ La @ Acc2 ) @ ( nil @ ( list @ A ) ) ) ) )
=> ( ! [La: list @ A,Acc2: list @ ( list @ A ),L3: list @ A] :
( ( accp @ ( product_prod @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) ) @ ( merge_list_rel @ A ) @ ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ La @ Acc2 ) @ ( cons @ ( list @ A ) @ L3 @ ( nil @ ( list @ A ) ) ) ) )
=> ( ( P @ ( nil @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ L3 @ ( cons @ ( list @ A ) @ La @ Acc2 ) ) )
=> ( P @ ( cons @ ( list @ A ) @ La @ Acc2 ) @ ( cons @ ( list @ A ) @ L3 @ ( nil @ ( list @ A ) ) ) ) ) )
=> ( ! [Acc2: list @ ( list @ A ),L12: list @ A,L23: list @ A,Ls2: list @ ( list @ A )] :
( ( accp @ ( product_prod @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) ) @ ( merge_list_rel @ A ) @ ( product_Pair @ ( list @ ( list @ A ) ) @ ( list @ ( list @ A ) ) @ Acc2 @ ( cons @ ( list @ A ) @ L12 @ ( cons @ ( list @ A ) @ L23 @ Ls2 ) ) ) )
=> ( ( P @ ( cons @ ( list @ A ) @ ( merge @ A @ L12 @ L23 ) @ Acc2 ) @ Ls2 )
=> ( P @ Acc2 @ ( cons @ ( list @ A ) @ L12 @ ( cons @ ( list @ A ) @ L23 @ Ls2 ) ) ) ) )
=> ( P @ A0 @ A13 ) ) ) ) ) ) ) ) ).
% merge_list.pinduct
thf(fact_4958_merge_Osimps_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L22: list @ A] :
( ( merge @ A @ ( nil @ A ) @ L22 )
= L22 ) ) ).
% merge.simps(1)
thf(fact_4959_merge_Osimps_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X13: A,X22: A,L1: list @ A,L22: list @ A] :
( ( ( ord_less @ A @ X13 @ X22 )
=> ( ( merge @ A @ ( cons @ A @ X13 @ L1 ) @ ( cons @ A @ X22 @ L22 ) )
= ( cons @ A @ X13 @ ( merge @ A @ L1 @ ( cons @ A @ X22 @ L22 ) ) ) ) )
& ( ~ ( ord_less @ A @ X13 @ X22 )
=> ( ( ( X13 = X22 )
=> ( ( merge @ A @ ( cons @ A @ X13 @ L1 ) @ ( cons @ A @ X22 @ L22 ) )
= ( cons @ A @ X13 @ ( merge @ A @ L1 @ L22 ) ) ) )
& ( ( X13 != X22 )
=> ( ( merge @ A @ ( cons @ A @ X13 @ L1 ) @ ( cons @ A @ X22 @ L22 ) )
= ( cons @ A @ X22 @ ( merge @ A @ ( cons @ A @ X13 @ L1 ) @ L22 ) ) ) ) ) ) ) ) ).
% merge.simps(3)
thf(fact_4960_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_4961_merge__list_Osimps_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Acc22: list @ ( list @ A ),L1: list @ A,L22: list @ A,Ls: list @ ( list @ A )] :
( ( merge_list @ A @ Acc22 @ ( cons @ ( list @ A ) @ L1 @ ( cons @ ( list @ A ) @ L22 @ Ls ) ) )
= ( merge_list @ A @ ( cons @ ( list @ A ) @ ( merge @ A @ L1 @ L22 ) @ Acc22 ) @ Ls ) ) ) ).
% merge_list.simps(5)
thf(fact_4962_merge_Oelims,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: list @ A,Xa: list @ A,Y: list @ A] :
( ( ( merge @ A @ X2 @ Xa )
= Y )
=> ( ( ( X2
= ( nil @ A ) )
=> ( Y != Xa ) )
=> ( ! [V5: A,Va: list @ A] :
( ( X2
= ( cons @ A @ V5 @ Va ) )
=> ( ( Xa
= ( nil @ A ) )
=> ( Y
!= ( cons @ A @ V5 @ Va ) ) ) )
=> ~ ! [X1: A,L12: list @ A] :
( ( X2
= ( cons @ A @ X1 @ L12 ) )
=> ! [X23: A,L23: list @ A] :
( ( Xa
= ( cons @ A @ X23 @ L23 ) )
=> ~ ( ( ( ord_less @ A @ X1 @ X23 )
=> ( Y
= ( cons @ A @ X1 @ ( merge @ A @ L12 @ ( cons @ A @ X23 @ L23 ) ) ) ) )
& ( ~ ( ord_less @ A @ X1 @ X23 )
=> ( ( ( X1 = X23 )
=> ( Y
= ( cons @ A @ X1 @ ( merge @ A @ L12 @ L23 ) ) ) )
& ( ( X1 != X23 )
=> ( Y
= ( cons @ A @ X23 @ ( merge @ A @ ( cons @ A @ X1 @ L12 ) @ L23 ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% merge.elims
thf(fact_4963_merge__list_Oelims,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: list @ ( list @ A ),Xa: list @ ( list @ A ),Y: list @ A] :
( ( ( merge_list @ A @ X2 @ Xa )
= Y )
=> ( ( ( X2
= ( nil @ ( list @ A ) ) )
=> ( ( Xa
= ( nil @ ( list @ A ) ) )
=> ( Y
!= ( nil @ A ) ) ) )
=> ( ( ( X2
= ( nil @ ( list @ A ) ) )
=> ! [L3: list @ A] :
( ( Xa
= ( cons @ ( list @ A ) @ L3 @ ( nil @ ( list @ A ) ) ) )
=> ( Y != L3 ) ) )
=> ( ! [La: list @ A,Acc2: list @ ( list @ A )] :
( ( X2
= ( cons @ ( list @ A ) @ La @ Acc2 ) )
=> ( ( Xa
= ( nil @ ( list @ A ) ) )
=> ( Y
!= ( merge_list @ A @ ( nil @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ La @ Acc2 ) ) ) ) )
=> ( ! [La: list @ A,Acc2: list @ ( list @ A )] :
( ( X2
= ( cons @ ( list @ A ) @ La @ Acc2 ) )
=> ! [L3: list @ A] :
( ( Xa
= ( cons @ ( list @ A ) @ L3 @ ( nil @ ( list @ A ) ) ) )
=> ( Y
!= ( merge_list @ A @ ( nil @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ L3 @ ( cons @ ( list @ A ) @ La @ Acc2 ) ) ) ) ) )
=> ~ ! [L12: list @ A,L23: list @ A,Ls2: list @ ( list @ A )] :
( ( Xa
= ( cons @ ( list @ A ) @ L12 @ ( cons @ ( list @ A ) @ L23 @ Ls2 ) ) )
=> ( Y
!= ( merge_list @ A @ ( cons @ ( list @ A ) @ ( merge @ A @ L12 @ L23 ) @ X2 ) @ Ls2 ) ) ) ) ) ) ) ) ) ).
% merge_list.elims
thf(fact_4964_merge_Opelims,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: list @ A,Xa: list @ A,Y: list @ A] :
( ( ( merge @ A @ X2 @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( merge_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X2 @ Xa ) )
=> ( ( ( X2
= ( nil @ A ) )
=> ( ( Y = Xa )
=> ~ ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( merge_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Xa ) ) ) )
=> ( ! [V5: A,Va: list @ A] :
( ( X2
= ( cons @ A @ V5 @ Va ) )
=> ( ( Xa
= ( nil @ A ) )
=> ( ( Y
= ( cons @ A @ V5 @ Va ) )
=> ~ ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( merge_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ V5 @ Va ) @ ( nil @ A ) ) ) ) ) )
=> ~ ! [X1: A,L12: list @ A] :
( ( X2
= ( cons @ A @ X1 @ L12 ) )
=> ! [X23: A,L23: list @ A] :
( ( Xa
= ( cons @ A @ X23 @ L23 ) )
=> ( ( ( ( ord_less @ A @ X1 @ X23 )
=> ( Y
= ( cons @ A @ X1 @ ( merge @ A @ L12 @ ( cons @ A @ X23 @ L23 ) ) ) ) )
& ( ~ ( ord_less @ A @ X1 @ X23 )
=> ( ( ( X1 = X23 )
=> ( Y
= ( cons @ A @ X1 @ ( merge @ A @ L12 @ L23 ) ) ) )
& ( ( X1 != X23 )
=> ( Y
= ( cons @ A @ X23 @ ( merge @ A @ ( cons @ A @ X1 @ L12 ) @ L23 ) ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( merge_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X1 @ L12 ) @ ( cons @ A @ X23 @ L23 ) ) ) ) ) ) ) ) ) ) ) ).
% merge.pelims
thf(fact_4965_filter__rev__aux__alt,axiom,
! [A: $tType] :
( ( filter_rev_aux @ A )
= ( ^ [A3: list @ A,P4: A > $o,L2: list @ A] : ( append @ A @ ( filter2 @ A @ P4 @ ( rev @ A @ L2 ) ) @ A3 ) ) ) ).
% filter_rev_aux_alt
thf(fact_4966_rat__floor__lemma,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq @ rat @ ( ring_1_of_int @ rat @ ( divide_divide @ int @ A2 @ B2 ) ) @ ( fract @ A2 @ B2 ) )
& ( ord_less @ rat @ ( fract @ A2 @ B2 ) @ ( ring_1_of_int @ rat @ ( plus_plus @ int @ ( divide_divide @ int @ A2 @ B2 ) @ ( one_one @ int ) ) ) ) ) ).
% rat_floor_lemma
thf(fact_4967_less__rat,axiom,
! [B2: int,D2: int,A2: int,C2: int] :
( ( B2
!= ( zero_zero @ int ) )
=> ( ( D2
!= ( zero_zero @ int ) )
=> ( ( ord_less @ rat @ ( fract @ A2 @ B2 ) @ ( fract @ C2 @ D2 ) )
= ( ord_less @ int @ ( times_times @ int @ ( times_times @ int @ A2 @ D2 ) @ ( times_times @ int @ B2 @ D2 ) ) @ ( times_times @ int @ ( times_times @ int @ C2 @ B2 ) @ ( times_times @ int @ B2 @ D2 ) ) ) ) ) ) ).
% less_rat
thf(fact_4968_le__rat,axiom,
! [B2: int,D2: int,A2: int,C2: int] :
( ( B2
!= ( zero_zero @ int ) )
=> ( ( D2
!= ( zero_zero @ int ) )
=> ( ( ord_less_eq @ rat @ ( fract @ A2 @ B2 ) @ ( fract @ C2 @ D2 ) )
= ( ord_less_eq @ int @ ( times_times @ int @ ( times_times @ int @ A2 @ D2 ) @ ( times_times @ int @ B2 @ D2 ) ) @ ( times_times @ int @ ( times_times @ int @ C2 @ B2 ) @ ( times_times @ int @ B2 @ D2 ) ) ) ) ) ) ).
% le_rat
thf(fact_4969_add__rat,axiom,
! [B2: int,D2: int,A2: int,C2: int] :
( ( B2
!= ( zero_zero @ int ) )
=> ( ( D2
!= ( zero_zero @ int ) )
=> ( ( plus_plus @ rat @ ( fract @ A2 @ B2 ) @ ( fract @ C2 @ D2 ) )
= ( fract @ ( plus_plus @ int @ ( times_times @ int @ A2 @ D2 ) @ ( times_times @ int @ C2 @ B2 ) ) @ ( times_times @ int @ B2 @ D2 ) ) ) ) ) ).
% add_rat
thf(fact_4970_diff__rat,axiom,
! [B2: int,D2: int,A2: int,C2: int] :
( ( B2
!= ( zero_zero @ int ) )
=> ( ( D2
!= ( zero_zero @ int ) )
=> ( ( minus_minus @ rat @ ( fract @ A2 @ B2 ) @ ( fract @ C2 @ D2 ) )
= ( fract @ ( minus_minus @ int @ ( times_times @ int @ A2 @ D2 ) @ ( times_times @ int @ C2 @ B2 ) ) @ ( times_times @ int @ B2 @ D2 ) ) ) ) ) ).
% diff_rat
thf(fact_4971_eq__rat_I3_J,axiom,
! [A2: int,C2: int] :
( ( fract @ ( zero_zero @ int ) @ A2 )
= ( fract @ ( zero_zero @ int ) @ C2 ) ) ).
% eq_rat(3)
thf(fact_4972_Rat__induct__pos,axiom,
! [P: rat > $o,Q2: rat] :
( ! [A4: int,B4: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B4 )
=> ( P @ ( fract @ A4 @ B4 ) ) )
=> ( P @ Q2 ) ) ).
% Rat_induct_pos
thf(fact_4973_eq__rat_I2_J,axiom,
! [A2: int] :
( ( fract @ A2 @ ( zero_zero @ int ) )
= ( fract @ ( zero_zero @ int ) @ ( one_one @ int ) ) ) ).
% eq_rat(2)
thf(fact_4974_mult__rat__cancel,axiom,
! [C2: int,A2: int,B2: int] :
( ( C2
!= ( zero_zero @ int ) )
=> ( ( fract @ ( times_times @ int @ C2 @ A2 ) @ ( times_times @ int @ C2 @ B2 ) )
= ( fract @ A2 @ B2 ) ) ) ).
% mult_rat_cancel
thf(fact_4975_eq__rat_I1_J,axiom,
! [B2: int,D2: int,A2: int,C2: int] :
( ( B2
!= ( zero_zero @ int ) )
=> ( ( D2
!= ( zero_zero @ int ) )
=> ( ( ( fract @ A2 @ B2 )
= ( fract @ C2 @ D2 ) )
= ( ( times_times @ int @ A2 @ D2 )
= ( times_times @ int @ C2 @ B2 ) ) ) ) ) ).
% eq_rat(1)
thf(fact_4976_Fract__of__nat__eq,axiom,
! [K: nat] :
( ( fract @ ( semiring_1_of_nat @ int @ K ) @ ( one_one @ int ) )
= ( semiring_1_of_nat @ rat @ K ) ) ).
% Fract_of_nat_eq
thf(fact_4977_rat__number__collapse_I1_J,axiom,
! [K: int] :
( ( fract @ ( zero_zero @ int ) @ K )
= ( zero_zero @ rat ) ) ).
% rat_number_collapse(1)
thf(fact_4978_rat__number__collapse_I6_J,axiom,
! [K: int] :
( ( fract @ K @ ( zero_zero @ int ) )
= ( zero_zero @ rat ) ) ).
% rat_number_collapse(6)
thf(fact_4979_One__rat__def,axiom,
( ( one_one @ rat )
= ( fract @ ( one_one @ int ) @ ( one_one @ int ) ) ) ).
% One_rat_def
thf(fact_4980_Fract__of__int__eq,axiom,
! [K: int] :
( ( fract @ K @ ( one_one @ int ) )
= ( ring_1_of_int @ rat @ K ) ) ).
% Fract_of_int_eq
thf(fact_4981_filter__rev__aux_Osimps_I2_J,axiom,
! [A: $tType,P: A > $o,X2: A,A2: list @ A,Xs: list @ A] :
( ( ( P @ X2 )
=> ( ( filter_rev_aux @ A @ A2 @ P @ ( cons @ A @ X2 @ Xs ) )
= ( filter_rev_aux @ A @ ( cons @ A @ X2 @ A2 ) @ P @ Xs ) ) )
& ( ~ ( P @ X2 )
=> ( ( filter_rev_aux @ A @ A2 @ P @ ( cons @ A @ X2 @ Xs ) )
= ( filter_rev_aux @ A @ A2 @ P @ Xs ) ) ) ) ).
% filter_rev_aux.simps(2)
thf(fact_4982_filter__rev__aux_Osimps_I1_J,axiom,
! [A: $tType,A2: list @ A,P: A > $o] :
( ( filter_rev_aux @ A @ A2 @ P @ ( nil @ A ) )
= A2 ) ).
% filter_rev_aux.simps(1)
thf(fact_4983_Zero__rat__def,axiom,
( ( zero_zero @ rat )
= ( fract @ ( zero_zero @ int ) @ ( one_one @ int ) ) ) ).
% Zero_rat_def
thf(fact_4984_rat__number__expand_I3_J,axiom,
( ( numeral_numeral @ rat )
= ( ^ [K2: num] : ( fract @ ( numeral_numeral @ int @ K2 ) @ ( one_one @ int ) ) ) ) ).
% rat_number_expand(3)
thf(fact_4985_rat__number__collapse_I3_J,axiom,
! [W2: num] :
( ( fract @ ( numeral_numeral @ int @ W2 ) @ ( one_one @ int ) )
= ( numeral_numeral @ rat @ W2 ) ) ).
% rat_number_collapse(3)
thf(fact_4986_zero__less__Fract__iff,axiom,
! [B2: int,A2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ( ord_less @ rat @ ( zero_zero @ rat ) @ ( fract @ A2 @ B2 ) )
= ( ord_less @ int @ ( zero_zero @ int ) @ A2 ) ) ) ).
% zero_less_Fract_iff
thf(fact_4987_Fract__less__zero__iff,axiom,
! [B2: int,A2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ( ord_less @ rat @ ( fract @ A2 @ B2 ) @ ( zero_zero @ rat ) )
= ( ord_less @ int @ A2 @ ( zero_zero @ int ) ) ) ) ).
% Fract_less_zero_iff
thf(fact_4988_Fract__less__one__iff,axiom,
! [B2: int,A2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ( ord_less @ rat @ ( fract @ A2 @ B2 ) @ ( one_one @ rat ) )
= ( ord_less @ int @ A2 @ B2 ) ) ) ).
% Fract_less_one_iff
thf(fact_4989_one__less__Fract__iff,axiom,
! [B2: int,A2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ( ord_less @ rat @ ( one_one @ rat ) @ ( fract @ A2 @ B2 ) )
= ( ord_less @ int @ B2 @ A2 ) ) ) ).
% one_less_Fract_iff
thf(fact_4990_rat__number__collapse_I5_J,axiom,
( ( fract @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( one_one @ int ) )
= ( uminus_uminus @ rat @ ( one_one @ rat ) ) ) ).
% rat_number_collapse(5)
thf(fact_4991_Fract__add__one,axiom,
! [N: int,M: int] :
( ( N
!= ( zero_zero @ int ) )
=> ( ( fract @ ( plus_plus @ int @ M @ N ) @ N )
= ( plus_plus @ rat @ ( fract @ M @ N ) @ ( one_one @ rat ) ) ) ) ).
% Fract_add_one
thf(fact_4992_Fract__le__zero__iff,axiom,
! [B2: int,A2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ( ord_less_eq @ rat @ ( fract @ A2 @ B2 ) @ ( zero_zero @ rat ) )
= ( ord_less_eq @ int @ A2 @ ( zero_zero @ int ) ) ) ) ).
% Fract_le_zero_iff
thf(fact_4993_zero__le__Fract__iff,axiom,
! [B2: int,A2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ( ord_less_eq @ rat @ ( zero_zero @ rat ) @ ( fract @ A2 @ B2 ) )
= ( ord_less_eq @ int @ ( zero_zero @ int ) @ A2 ) ) ) ).
% zero_le_Fract_iff
thf(fact_4994_Fract__le__one__iff,axiom,
! [B2: int,A2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ( ord_less_eq @ rat @ ( fract @ A2 @ B2 ) @ ( one_one @ rat ) )
= ( ord_less_eq @ int @ A2 @ B2 ) ) ) ).
% Fract_le_one_iff
thf(fact_4995_one__le__Fract__iff,axiom,
! [B2: int,A2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ( ord_less_eq @ rat @ ( one_one @ rat ) @ ( fract @ A2 @ B2 ) )
= ( ord_less_eq @ int @ B2 @ A2 ) ) ) ).
% one_le_Fract_iff
thf(fact_4996_rat__number__expand_I5_J,axiom,
! [K: num] :
( ( uminus_uminus @ rat @ ( numeral_numeral @ rat @ K ) )
= ( fract @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) @ ( one_one @ int ) ) ) ).
% rat_number_expand(5)
thf(fact_4997_rat__number__collapse_I4_J,axiom,
! [W2: num] :
( ( fract @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ W2 ) ) @ ( one_one @ int ) )
= ( uminus_uminus @ rat @ ( numeral_numeral @ rat @ W2 ) ) ) ).
% rat_number_collapse(4)
thf(fact_4998_filter__rev__def,axiom,
! [A: $tType] :
( ( filter_rev @ A )
= ( filter_rev_aux @ A @ ( nil @ A ) ) ) ).
% filter_rev_def
thf(fact_4999_pow_Osimps_I3_J,axiom,
! [X2: num,Y: num] :
( ( pow @ X2 @ ( bit1 @ Y ) )
= ( times_times @ num @ ( sqr @ ( pow @ X2 @ Y ) ) @ X2 ) ) ).
% pow.simps(3)
thf(fact_5000_restrict__map__upds,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B,D3: set @ A,M: A > ( option @ B )] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ D3 )
=> ( ( restrict_map @ A @ B @ ( map_upds @ A @ B @ M @ Xs @ Ys ) @ D3 )
= ( map_upds @ A @ B @ ( restrict_map @ A @ B @ M @ ( minus_minus @ ( set @ A ) @ D3 @ ( set2 @ A @ Xs ) ) ) @ Xs @ Ys ) ) ) ) ).
% restrict_map_upds
thf(fact_5001_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
@ ^ [I2: nat,J3: nat] : ( minus_minus @ A @ ( semiring_1_of_nat @ A @ I2 ) @ ( semiring_1_of_nat @ A @ J3 ) ) ) ) ) ) ).
% ring_1_class.of_int_def
thf(fact_5002_of__nat__eq__id,axiom,
( ( semiring_1_of_nat @ nat )
= ( id @ nat ) ) ).
% of_nat_eq_id
thf(fact_5003_id__funpow,axiom,
! [A: $tType,N: nat] :
( ( compow @ ( A > A ) @ N @ ( id @ A ) )
= ( id @ A ) ) ).
% id_funpow
thf(fact_5004_rotate0,axiom,
! [A: $tType] :
( ( rotate @ A @ ( zero_zero @ nat ) )
= ( id @ ( list @ A ) ) ) ).
% rotate0
thf(fact_5005_restrict__map__empty,axiom,
! [B: $tType,A: $tType,D3: set @ A] :
( ( restrict_map @ A @ B
@ ^ [X: A] : ( none @ B )
@ D3 )
= ( ^ [X: A] : ( none @ B ) ) ) ).
% restrict_map_empty
thf(fact_5006_push__bit__0__id,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se4730199178511100633sh_bit @ A @ ( zero_zero @ nat ) )
= ( id @ A ) ) ) ).
% push_bit_0_id
thf(fact_5007_drop__bit__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se4197421643247451524op_bit @ A @ ( zero_zero @ nat ) )
= ( id @ A ) ) ) ).
% drop_bit_0
thf(fact_5008_map__fun_Oidentity,axiom,
! [B: $tType,A: $tType] :
( ( map_fun @ A @ A @ B @ B
@ ^ [X: A] : X
@ ^ [X: B] : X )
= ( id @ ( A > B ) ) ) ).
% map_fun.identity
thf(fact_5009_pointfree__idE,axiom,
! [B: $tType,A: $tType,F: B > A,G: A > B,X2: A] :
( ( ( comp @ B @ A @ A @ F @ G )
= ( id @ A ) )
=> ( ( F @ ( G @ X2 ) )
= X2 ) ) ).
% pointfree_idE
thf(fact_5010_restrict__map__subset__eq,axiom,
! [B: $tType,A: $tType,M: A > ( option @ B ),R3: set @ A,M5: A > ( option @ B ),R8: set @ A] :
( ( ( restrict_map @ A @ B @ M @ R3 )
= M5 )
=> ( ( ord_less_eq @ ( set @ A ) @ R8 @ R3 )
=> ( ( restrict_map @ A @ B @ M @ R8 )
= ( restrict_map @ A @ B @ M5 @ R8 ) ) ) ) ).
% restrict_map_subset_eq
thf(fact_5011_le__map__restrict,axiom,
! [B: $tType,A: $tType] :
( ( order @ B )
=> ! [M: A > ( option @ B ),X8: set @ A] : ( ord_less_eq @ ( A > ( option @ B ) ) @ ( restrict_map @ A @ B @ M @ X8 ) @ M ) ) ).
% le_map_restrict
thf(fact_5012_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_5013_List_Omap_Oidentity,axiom,
! [A: $tType] :
( ( map @ A @ A
@ ^ [X: A] : X )
= ( id @ ( list @ A ) ) ) ).
% List.map.identity
thf(fact_5014_map__option_Oidentity,axiom,
! [A: $tType] :
( ( map_option @ A @ A
@ ^ [X: A] : X )
= ( id @ ( option @ A ) ) ) ).
% map_option.identity
thf(fact_5015_funpow__simps__right_I1_J,axiom,
! [A: $tType,F: A > A] :
( ( compow @ ( A > A ) @ ( zero_zero @ nat ) @ F )
= ( id @ A ) ) ).
% funpow_simps_right(1)
thf(fact_5016_sqr_Osimps_I2_J,axiom,
! [N: num] :
( ( sqr @ ( bit0 @ N ) )
= ( bit0 @ ( bit0 @ ( sqr @ N ) ) ) ) ).
% sqr.simps(2)
thf(fact_5017_sqr_Osimps_I1_J,axiom,
( ( sqr @ one2 )
= one2 ) ).
% sqr.simps(1)
thf(fact_5018_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 )
@ ^ [X: nat,Y4: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V4: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ X @ V4 ) @ ( plus_plus @ nat @ U2 @ Y4 ) ) ) ) ) ) ).
% less_int_def
thf(fact_5019_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 )
@ ^ [X: nat,Y4: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V4: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ X @ V4 ) @ ( plus_plus @ nat @ U2 @ Y4 ) ) ) ) ) ) ).
% less_eq_int_def
thf(fact_5020_sqr__conv__mult,axiom,
( sqr
= ( ^ [X: num] : ( times_times @ num @ X @ X ) ) ) ).
% sqr_conv_mult
thf(fact_5021_foldr__filter,axiom,
! [A: $tType,B: $tType,F: B > A > A,P: B > $o,Xs: list @ B] :
( ( foldr @ B @ A @ F @ ( filter2 @ B @ P @ Xs ) )
= ( foldr @ B @ A
@ ^ [X: B] : ( if @ ( A > A ) @ ( P @ X ) @ ( F @ X ) @ ( id @ A ) )
@ Xs ) ) ).
% foldr_filter
thf(fact_5022_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_5023_map__restrict__insert__none__simp,axiom,
! [A: $tType,B: $tType,M: B > ( option @ A ),X2: B,S3: set @ B] :
( ( ( M @ X2 )
= ( none @ A ) )
=> ( ( restrict_map @ B @ A @ M @ ( uminus_uminus @ ( set @ B ) @ ( insert @ B @ X2 @ S3 ) ) )
= ( restrict_map @ B @ A @ M @ ( uminus_uminus @ ( set @ B ) @ S3 ) ) ) ) ).
% map_restrict_insert_none_simp
thf(fact_5024_restrict__map__upd,axiom,
! [B: $tType,A: $tType,F: A > ( option @ B ),S2: set @ A,K: A,V2: B] :
( ( fun_upd @ A @ ( option @ B ) @ ( restrict_map @ A @ B @ F @ S2 ) @ K @ ( some @ B @ V2 ) )
= ( restrict_map @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ F @ K @ ( some @ B @ V2 ) ) @ ( insert @ A @ K @ S2 ) ) ) ).
% restrict_map_upd
thf(fact_5025_pow_Osimps_I2_J,axiom,
! [X2: num,Y: num] :
( ( pow @ X2 @ ( bit0 @ Y ) )
= ( sqr @ ( pow @ X2 @ Y ) ) ) ).
% pow.simps(2)
thf(fact_5026_map__upd__eq__restrict,axiom,
! [B: $tType,A: $tType,M: A > ( option @ B ),X2: A] :
( ( fun_upd @ A @ ( option @ B ) @ M @ X2 @ ( none @ B ) )
= ( restrict_map @ A @ B @ M @ ( uminus_uminus @ ( set @ A ) @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% map_upd_eq_restrict
thf(fact_5027_sqr_Osimps_I3_J,axiom,
! [N: num] :
( ( sqr @ ( bit1 @ N ) )
= ( bit1 @ ( bit0 @ ( plus_plus @ num @ ( sqr @ N ) @ N ) ) ) ) ).
% sqr.simps(3)
thf(fact_5028_map__of__map__restrict,axiom,
! [B: $tType,A: $tType,F: A > B,Ks: list @ A] :
( ( map_of @ A @ B
@ ( map @ A @ ( product_prod @ A @ B )
@ ^ [K2: A] : ( product_Pair @ A @ B @ K2 @ ( F @ K2 ) )
@ Ks ) )
= ( restrict_map @ A @ B @ ( comp @ B @ ( option @ B ) @ A @ ( some @ B ) @ F ) @ ( set2 @ A @ Ks ) ) ) ).
% map_of_map_restrict
thf(fact_5029_nths__Cons,axiom,
! [A: $tType,X2: A,L: list @ A,A5: set @ nat] :
( ( nths @ A @ ( cons @ A @ X2 @ L ) @ A5 )
= ( append @ A @ ( if @ ( list @ A ) @ ( member @ nat @ ( zero_zero @ nat ) @ A5 ) @ ( cons @ A @ X2 @ ( nil @ A ) ) @ ( nil @ A ) )
@ ( nths @ A @ L
@ ( collect @ nat
@ ^ [J3: nat] : ( member @ nat @ ( suc @ J3 ) @ A5 ) ) ) ) ) ).
% nths_Cons
thf(fact_5030_positive__rat,axiom,
! [A2: int,B2: int] :
( ( positive @ ( fract @ A2 @ B2 ) )
= ( ord_less @ int @ ( zero_zero @ int ) @ ( times_times @ int @ A2 @ B2 ) ) ) ).
% positive_rat
thf(fact_5031_empty__eq__map__of__iff,axiom,
! [B: $tType,A: $tType,Xys: list @ ( product_prod @ A @ B )] :
( ( ( ^ [X: A] : ( none @ B ) )
= ( map_of @ A @ B @ Xys ) )
= ( Xys
= ( nil @ ( product_prod @ A @ B ) ) ) ) ).
% empty_eq_map_of_iff
thf(fact_5032_nths__singleton,axiom,
! [A: $tType,A5: set @ nat,X2: A] :
( ( ( member @ nat @ ( zero_zero @ nat ) @ A5 )
=> ( ( nths @ A @ ( cons @ A @ X2 @ ( nil @ A ) ) @ A5 )
= ( cons @ A @ X2 @ ( nil @ A ) ) ) )
& ( ~ ( member @ nat @ ( zero_zero @ nat ) @ A5 )
=> ( ( nths @ A @ ( cons @ A @ X2 @ ( nil @ A ) ) @ A5 )
= ( nil @ A ) ) ) ) ).
% nths_singleton
thf(fact_5033_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_5034_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_5035_map__of__None__filterD,axiom,
! [B: $tType,A: $tType,Xs: list @ ( product_prod @ B @ A ),X2: B,P: ( product_prod @ B @ A ) > $o] :
( ( ( map_of @ B @ A @ Xs @ X2 )
= ( none @ A ) )
=> ( ( map_of @ B @ A @ ( filter2 @ ( product_prod @ B @ A ) @ P @ Xs ) @ X2 )
= ( none @ A ) ) ) ).
% map_of_None_filterD
thf(fact_5036_plus__integer__def,axiom,
( ( plus_plus @ code_integer )
= ( map_fun @ code_integer @ int @ ( int > int ) @ ( code_integer > code_integer ) @ code_int_of_integer @ ( map_fun @ code_integer @ int @ int @ code_integer @ code_int_of_integer @ code_integer_of_int ) @ ( plus_plus @ int ) ) ) ).
% plus_integer_def
thf(fact_5037_times__integer__def,axiom,
( ( times_times @ code_integer )
= ( map_fun @ code_integer @ int @ ( int > int ) @ ( code_integer > code_integer ) @ code_int_of_integer @ ( map_fun @ code_integer @ int @ int @ code_integer @ code_int_of_integer @ code_integer_of_int ) @ ( times_times @ int ) ) ) ).
% times_integer_def
thf(fact_5038_minus__integer__def,axiom,
( ( minus_minus @ code_integer )
= ( map_fun @ code_integer @ int @ ( int > int ) @ ( code_integer > code_integer ) @ code_int_of_integer @ ( map_fun @ code_integer @ int @ int @ code_integer @ code_int_of_integer @ code_integer_of_int ) @ ( minus_minus @ int ) ) ) ).
% minus_integer_def
thf(fact_5039_divide__integer__def,axiom,
( ( divide_divide @ code_integer )
= ( map_fun @ code_integer @ int @ ( int > int ) @ ( code_integer > code_integer ) @ code_int_of_integer @ ( map_fun @ code_integer @ int @ int @ code_integer @ code_int_of_integer @ code_integer_of_int ) @ ( divide_divide @ int ) ) ) ).
% divide_integer_def
thf(fact_5040_modulo__integer__def,axiom,
( ( modulo_modulo @ code_integer )
= ( map_fun @ code_integer @ int @ ( int > int ) @ ( code_integer > code_integer ) @ code_int_of_integer @ ( map_fun @ code_integer @ int @ int @ code_integer @ code_int_of_integer @ code_integer_of_int ) @ ( modulo_modulo @ int ) ) ) ).
% modulo_integer_def
thf(fact_5041_positive__add,axiom,
! [X2: rat,Y: rat] :
( ( positive @ X2 )
=> ( ( positive @ Y )
=> ( positive @ ( plus_plus @ rat @ X2 @ Y ) ) ) ) ).
% positive_add
thf(fact_5042_and__integer__def,axiom,
( ( bit_se5824344872417868541ns_and @ code_integer )
= ( map_fun @ code_integer @ int @ ( int > int ) @ ( code_integer > code_integer ) @ code_int_of_integer @ ( map_fun @ code_integer @ int @ int @ code_integer @ code_int_of_integer @ code_integer_of_int ) @ ( bit_se5824344872417868541ns_and @ int ) ) ) ).
% and_integer_def
thf(fact_5043_or__integer__def,axiom,
( ( bit_se1065995026697491101ons_or @ code_integer )
= ( map_fun @ code_integer @ int @ ( int > int ) @ ( code_integer > code_integer ) @ code_int_of_integer @ ( map_fun @ code_integer @ int @ int @ code_integer @ code_int_of_integer @ code_integer_of_int ) @ ( bit_se1065995026697491101ons_or @ int ) ) ) ).
% or_integer_def
thf(fact_5044_take__bit__integer__def,axiom,
( ( bit_se2584673776208193580ke_bit @ code_integer )
= ( map_fun @ nat @ nat @ ( int > int ) @ ( code_integer > code_integer ) @ ( id @ nat ) @ ( map_fun @ code_integer @ int @ int @ code_integer @ code_int_of_integer @ code_integer_of_int ) @ ( bit_se2584673776208193580ke_bit @ int ) ) ) ).
% take_bit_integer_def
thf(fact_5045_xor__integer__def,axiom,
( ( bit_se5824344971392196577ns_xor @ code_integer )
= ( map_fun @ code_integer @ int @ ( int > int ) @ ( code_integer > code_integer ) @ code_int_of_integer @ ( map_fun @ code_integer @ int @ int @ code_integer @ code_int_of_integer @ code_integer_of_int ) @ ( bit_se5824344971392196577ns_xor @ int ) ) ) ).
% xor_integer_def
thf(fact_5046_drop__bit__integer__def,axiom,
( ( bit_se4197421643247451524op_bit @ code_integer )
= ( map_fun @ nat @ nat @ ( int > int ) @ ( code_integer > code_integer ) @ ( id @ nat ) @ ( map_fun @ code_integer @ int @ int @ code_integer @ code_int_of_integer @ code_integer_of_int ) @ ( bit_se4197421643247451524op_bit @ int ) ) ) ).
% drop_bit_integer_def
thf(fact_5047_push__bit__integer__def,axiom,
( ( bit_se4730199178511100633sh_bit @ code_integer )
= ( map_fun @ nat @ nat @ ( int > int ) @ ( code_integer > code_integer ) @ ( id @ nat ) @ ( map_fun @ code_integer @ int @ int @ code_integer @ code_int_of_integer @ code_integer_of_int ) @ ( bit_se4730199178511100633sh_bit @ int ) ) ) ).
% push_bit_integer_def
thf(fact_5048_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_5049_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_5050_unset__bit__integer__def,axiom,
( ( bit_se2638667681897837118et_bit @ code_integer )
= ( map_fun @ nat @ nat @ ( int > int ) @ ( code_integer > code_integer ) @ ( id @ nat ) @ ( map_fun @ code_integer @ int @ int @ code_integer @ code_int_of_integer @ code_integer_of_int ) @ ( bit_se2638667681897837118et_bit @ int ) ) ) ).
% unset_bit_integer_def
thf(fact_5051_set__bit__integer__def,axiom,
( ( bit_se5668285175392031749et_bit @ code_integer )
= ( map_fun @ nat @ nat @ ( int > int ) @ ( code_integer > code_integer ) @ ( id @ nat ) @ ( map_fun @ code_integer @ int @ int @ code_integer @ code_int_of_integer @ code_integer_of_int ) @ ( bit_se5668285175392031749et_bit @ int ) ) ) ).
% set_bit_integer_def
thf(fact_5052_flip__bit__integer__def,axiom,
( ( bit_se8732182000553998342ip_bit @ code_integer )
= ( map_fun @ nat @ nat @ ( int > int ) @ ( code_integer > code_integer ) @ ( id @ nat ) @ ( map_fun @ code_integer @ int @ int @ code_integer @ code_int_of_integer @ code_integer_of_int ) @ ( bit_se8732182000553998342ip_bit @ int ) ) ) ).
% flip_bit_integer_def
thf(fact_5053_drop__eq__nths,axiom,
! [A: $tType] :
( ( drop @ A )
= ( ^ [N4: nat,Xs3: list @ A] : ( nths @ A @ Xs3 @ ( collect @ nat @ ( ord_less_eq @ nat @ N4 ) ) ) ) ) ).
% drop_eq_nths
thf(fact_5054_nat__of__integer__def,axiom,
( code_nat_of_integer
= ( map_fun @ code_integer @ int @ nat @ nat @ code_int_of_integer @ ( id @ nat ) @ nat2 ) ) ).
% nat_of_integer_def
thf(fact_5055_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_5056_nths__append,axiom,
! [A: $tType,L: list @ A,L4: list @ A,A5: set @ nat] :
( ( nths @ A @ ( append @ A @ L @ L4 ) @ A5 )
= ( append @ A @ ( nths @ A @ L @ A5 )
@ ( nths @ A @ L4
@ ( collect @ nat
@ ^ [J3: nat] : ( member @ nat @ ( plus_plus @ nat @ J3 @ ( size_size @ ( list @ A ) @ L ) ) @ A5 ) ) ) ) ) ).
% nths_append
thf(fact_5057_nths__drop,axiom,
! [A: $tType,N: nat,Xs: list @ A,I5: set @ nat] :
( ( nths @ A @ ( drop @ A @ N @ Xs ) @ I5 )
= ( nths @ A @ Xs @ ( image2 @ nat @ nat @ ( plus_plus @ nat @ N ) @ I5 ) ) ) ).
% nths_drop
thf(fact_5058_filter__in__nths,axiom,
! [A: $tType,Xs: list @ A,S3: set @ nat] :
( ( distinct @ A @ Xs )
=> ( ( filter2 @ A
@ ^ [X: A] : ( member @ A @ X @ ( set2 @ A @ ( nths @ A @ Xs @ S3 ) ) )
@ Xs )
= ( nths @ A @ Xs @ S3 ) ) ) ).
% filter_in_nths
thf(fact_5059_bit__integer__def,axiom,
( ( bit_se5641148757651400278ts_bit @ code_integer )
= ( map_fun @ code_integer @ int @ ( nat > $o ) @ ( nat > $o ) @ code_int_of_integer @ ( id @ ( nat > $o ) ) @ ( bit_se5641148757651400278ts_bit @ int ) ) ) ).
% bit_integer_def
thf(fact_5060_length__nths,axiom,
! [A: $tType,Xs: list @ A,I5: set @ nat] :
( ( size_size @ ( list @ A ) @ ( nths @ A @ Xs @ I5 ) )
= ( finite_card @ nat
@ ( collect @ nat
@ ^ [I2: nat] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( member @ nat @ I2 @ I5 ) ) ) ) ) ).
% length_nths
thf(fact_5061_uminus__integer__def,axiom,
( ( uminus_uminus @ code_integer )
= ( map_fun @ code_integer @ int @ int @ code_integer @ code_int_of_integer @ code_integer_of_int @ ( uminus_uminus @ int ) ) ) ).
% uminus_integer_def
thf(fact_5062_abs__integer__def,axiom,
( ( abs_abs @ code_integer )
= ( map_fun @ code_integer @ int @ int @ code_integer @ code_int_of_integer @ code_integer_of_int @ ( abs_abs @ int ) ) ) ).
% abs_integer_def
thf(fact_5063_map__of__map,axiom,
! [B: $tType,C: $tType,A: $tType,F: 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 )
@ ^ [K2: A,V4: C] : ( product_Pair @ A @ B @ K2 @ ( F @ V4 ) ) )
@ Xs ) )
= ( comp @ ( option @ C ) @ ( option @ B ) @ A @ ( map_option @ C @ B @ F ) @ ( map_of @ A @ C @ Xs ) ) ) ).
% map_of_map
thf(fact_5064_mask__integer__def,axiom,
( ( bit_se2239418461657761734s_mask @ code_integer )
= ( map_fun @ nat @ nat @ int @ code_integer @ ( id @ nat ) @ code_integer_of_int @ ( bit_se2239418461657761734s_mask @ int ) ) ) ).
% mask_integer_def
thf(fact_5065_sgn__integer__def,axiom,
( ( sgn_sgn @ code_integer )
= ( map_fun @ code_integer @ int @ int @ code_integer @ code_int_of_integer @ code_integer_of_int @ ( sgn_sgn @ int ) ) ) ).
% sgn_integer_def
thf(fact_5066_not__integer__def,axiom,
( ( bit_ri4277139882892585799ns_not @ code_integer )
= ( map_fun @ code_integer @ int @ int @ code_integer @ code_int_of_integer @ code_integer_of_int @ ( bit_ri4277139882892585799ns_not @ int ) ) ) ).
% not_integer_def
thf(fact_5067_filter__eq__nths,axiom,
! [A: $tType] :
( ( filter2 @ A )
= ( ^ [P4: A > $o,Xs3: list @ A] :
( nths @ A @ Xs3
@ ( collect @ nat
@ ^ [I2: nat] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs3 ) )
& ( P4 @ ( nth @ A @ Xs3 @ I2 ) ) ) ) ) ) ) ).
% filter_eq_nths
thf(fact_5068_num__of__integer__def,axiom,
( code_num_of_integer
= ( map_fun @ code_integer @ int @ num @ num @ code_int_of_integer @ ( id @ num ) @ ( comp @ nat @ num @ int @ num_of_nat @ nat2 ) ) ) ).
% num_of_integer_def
thf(fact_5069_mset__zip__take__Cons__drop__twice,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B,J: nat,X2: A,Y: B] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( ord_less_eq @ nat @ J @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( mset @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ ( append @ A @ ( take @ A @ J @ Xs ) @ ( cons @ A @ X2 @ ( drop @ A @ J @ Xs ) ) ) @ ( append @ B @ ( take @ B @ J @ Ys ) @ ( cons @ B @ Y @ ( drop @ B @ J @ Ys ) ) ) ) )
= ( add_mset @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y ) @ ( mset @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) ) ) ) ) ) ).
% mset_zip_take_Cons_drop_twice
thf(fact_5070_dual__min,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( min @ A
@ ^ [X: A,Y4: A] : ( ord_less_eq @ A @ Y4 @ X ) )
= ( ord_max @ A ) ) ) ).
% dual_min
thf(fact_5071_mask__mod__exp,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [N: nat,M: nat] :
( ( modulo_modulo @ A @ ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ A ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) )
= ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( ord_min @ nat @ M @ N ) ) @ ( one_one @ A ) ) ) ) ).
% mask_mod_exp
thf(fact_5072_subset__mset_Omin__arg__le_I1_J,axiom,
! [A: $tType,N: multiset @ A,M: multiset @ A] :
( ( subseteq_mset @ A @ N @ ( min @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ M @ N ) )
= ( ( min @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ M @ N )
= N ) ) ).
% subset_mset.min_arg_le(1)
thf(fact_5073_subset__mset_Omin__arg__le_I2_J,axiom,
! [A: $tType,M: multiset @ A,N: multiset @ A] :
( ( subseteq_mset @ A @ M @ ( min @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ M @ N ) )
= ( ( min @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ M @ N )
= M ) ) ).
% subset_mset.min_arg_le(2)
thf(fact_5074_min_Obounded__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ ( ord_min @ A @ B2 @ C2 ) )
= ( ( ord_less_eq @ A @ A2 @ B2 )
& ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% min.bounded_iff
thf(fact_5075_min_Oabsorb2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_min @ A @ A2 @ B2 )
= B2 ) ) ) ).
% min.absorb2
thf(fact_5076_min_Oabsorb1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_min @ A @ A2 @ B2 )
= A2 ) ) ) ).
% min.absorb1
thf(fact_5077_min__arg__le_I2_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [M: A,N: A] :
( ( ord_less_eq @ A @ M @ ( ord_min @ A @ M @ N ) )
= ( ( ord_min @ A @ M @ N )
= M ) ) ) ).
% min_arg_le(2)
thf(fact_5078_min__arg__le_I1_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [N: A,M: A] :
( ( ord_less_eq @ A @ N @ ( ord_min @ A @ M @ N ) )
= ( ( ord_min @ A @ M @ N )
= N ) ) ) ).
% min_arg_le(1)
thf(fact_5079_min__eq__arg_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [M: A,N: A] :
( ( ( ord_min @ A @ M @ N )
= N )
= ( ord_less_eq @ A @ N @ M ) ) ) ).
% min_eq_arg(2)
thf(fact_5080_min__eq__arg_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [M: A,N: A] :
( ( ( ord_min @ A @ M @ N )
= M )
= ( ord_less_eq @ A @ M @ N ) ) ) ).
% min_eq_arg(1)
thf(fact_5081_min__arg__not__ge_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [M: A,N: A] :
( ( ~ ( ord_less @ A @ ( ord_min @ A @ M @ N ) @ M ) )
= ( ( ord_min @ A @ M @ N )
= M ) ) ) ).
% min_arg_not_ge(1)
thf(fact_5082_min__arg__not__ge_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [M: A,N: A] :
( ( ~ ( ord_less @ A @ ( ord_min @ A @ M @ N ) @ N ) )
= ( ( ord_min @ A @ M @ N )
= N ) ) ) ).
% min_arg_not_ge(2)
thf(fact_5083_min__less__self__conv_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( ord_min @ A @ A2 @ B2 ) @ A2 )
= ( ord_less @ A @ B2 @ A2 ) ) ) ).
% min_less_self_conv(1)
thf(fact_5084_min__less__self__conv_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( ord_min @ A @ A2 @ B2 ) @ B2 )
= ( ord_less @ A @ A2 @ B2 ) ) ) ).
% min_less_self_conv(2)
thf(fact_5085_min__simps_I1_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_min @ A @ A2 @ B2 )
= A2 ) ) ) ).
% min_simps(1)
thf(fact_5086_min__simps_I2_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B2: A,A2: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ( ( ord_min @ A @ A2 @ B2 )
= B2 ) ) ) ).
% min_simps(2)
thf(fact_5087_min__bot,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [X2: A] :
( ( ord_min @ A @ ( bot_bot @ A ) @ X2 )
= ( bot_bot @ A ) ) ) ).
% min_bot
thf(fact_5088_min__bot2,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [X2: A] :
( ( ord_min @ A @ X2 @ ( bot_bot @ A ) )
= ( bot_bot @ A ) ) ) ).
% min_bot2
thf(fact_5089_zip__eq__zip__same__len,axiom,
! [A: $tType,B: $tType,A2: list @ A,B2: list @ B,A6: list @ A,B6: list @ B] :
( ( ( size_size @ ( list @ A ) @ A2 )
= ( size_size @ ( list @ B ) @ B2 ) )
=> ( ( ( size_size @ ( list @ A ) @ A6 )
= ( size_size @ ( list @ B ) @ B6 ) )
=> ( ( ( zip @ A @ B @ A2 @ B2 )
= ( zip @ A @ B @ A6 @ B6 ) )
= ( ( A2 = A6 )
& ( B2 = B6 ) ) ) ) ) ).
% zip_eq_zip_same_len
thf(fact_5090_max__min__same_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y: A] :
( ( ord_max @ A @ X2 @ ( ord_min @ A @ X2 @ Y ) )
= X2 ) ) ).
% max_min_same(1)
thf(fact_5091_max__min__same_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y: A] :
( ( ord_max @ A @ ( ord_min @ A @ X2 @ Y ) @ X2 )
= X2 ) ) ).
% max_min_same(2)
thf(fact_5092_max__min__same_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y: A] :
( ( ord_max @ A @ ( ord_min @ A @ X2 @ Y ) @ Y )
= Y ) ) ).
% max_min_same(3)
thf(fact_5093_max__min__same_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Y: A,X2: A] :
( ( ord_max @ A @ Y @ ( ord_min @ A @ X2 @ Y ) )
= Y ) ) ).
% max_min_same(4)
thf(fact_5094_min__0R,axiom,
! [N: nat] :
( ( ord_min @ nat @ N @ ( zero_zero @ nat ) )
= ( zero_zero @ nat ) ) ).
% min_0R
thf(fact_5095_min__0L,axiom,
! [N: nat] :
( ( ord_min @ nat @ ( zero_zero @ nat ) @ N )
= ( zero_zero @ nat ) ) ).
% min_0L
thf(fact_5096_min__Suc__Suc,axiom,
! [M: nat,N: nat] :
( ( ord_min @ nat @ ( suc @ M ) @ ( suc @ N ) )
= ( suc @ ( ord_min @ nat @ M @ N ) ) ) ).
% min_Suc_Suc
thf(fact_5097_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_5098_min__0__1_I4_J,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [X2: num] :
( ( ord_min @ A @ ( numeral_numeral @ A @ X2 ) @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% min_0_1(4)
thf(fact_5099_min__0__1_I3_J,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [X2: num] :
( ( ord_min @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ X2 ) )
= ( zero_zero @ A ) ) ) ).
% min_0_1(3)
thf(fact_5100_min__0__1_I1_J,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ( ( ord_min @ A @ ( zero_zero @ A ) @ ( one_one @ A ) )
= ( zero_zero @ A ) ) ) ).
% min_0_1(1)
thf(fact_5101_min__0__1_I2_J,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ( ( ord_min @ A @ ( one_one @ A ) @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% min_0_1(2)
thf(fact_5102_min__0__1_I6_J,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [X2: num] :
( ( ord_min @ A @ ( numeral_numeral @ A @ X2 ) @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% min_0_1(6)
thf(fact_5103_min__0__1_I5_J,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [X2: num] :
( ( ord_min @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ X2 ) )
= ( one_one @ A ) ) ) ).
% min_0_1(5)
thf(fact_5104_Int__atMost,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A] :
( ( inf_inf @ ( set @ A ) @ ( set_ord_atMost @ A @ A2 ) @ ( set_ord_atMost @ A @ B2 ) )
= ( set_ord_atMost @ A @ ( ord_min @ A @ A2 @ B2 ) ) ) ) ).
% Int_atMost
thf(fact_5105_min__Suc__gt_I2_J,axiom,
! [A2: nat,B2: nat] :
( ( ord_less @ nat @ A2 @ B2 )
=> ( ( ord_min @ nat @ B2 @ ( suc @ A2 ) )
= ( suc @ A2 ) ) ) ).
% min_Suc_gt(2)
thf(fact_5106_min__Suc__gt_I1_J,axiom,
! [A2: nat,B2: nat] :
( ( ord_less @ nat @ A2 @ B2 )
=> ( ( ord_min @ nat @ ( suc @ A2 ) @ B2 )
= ( suc @ A2 ) ) ) ).
% min_Suc_gt(1)
thf(fact_5107_length__take,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( size_size @ ( list @ A ) @ ( take @ A @ N @ Xs ) )
= ( ord_min @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N ) ) ).
% length_take
thf(fact_5108_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_5109_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_5110_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_5111_Int__atLeastAtMost,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) @ ( set_or1337092689740270186AtMost @ A @ C2 @ D2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( ord_max @ A @ A2 @ C2 ) @ ( ord_min @ A @ B2 @ D2 ) ) ) ) ).
% Int_atLeastAtMost
thf(fact_5112_Int__atLeastLessThan,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ A2 @ B2 ) @ ( set_or7035219750837199246ssThan @ A @ C2 @ D2 ) )
= ( set_or7035219750837199246ssThan @ A @ ( ord_max @ A @ A2 @ C2 ) @ ( ord_min @ A @ B2 @ D2 ) ) ) ) ).
% Int_atLeastLessThan
thf(fact_5113_Int__atLeastAtMostR1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,C2: A,D2: A] :
( ( inf_inf @ ( set @ A ) @ ( set_ord_atMost @ A @ B2 ) @ ( set_or1337092689740270186AtMost @ A @ C2 @ D2 ) )
= ( set_or1337092689740270186AtMost @ A @ C2 @ ( ord_min @ A @ B2 @ D2 ) ) ) ) ).
% Int_atLeastAtMostR1
thf(fact_5114_Int__atLeastAtMostL1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A,D2: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) @ ( set_ord_atMost @ A @ D2 ) )
= ( set_or1337092689740270186AtMost @ A @ A2 @ ( ord_min @ A @ B2 @ D2 ) ) ) ) ).
% Int_atLeastAtMostL1
thf(fact_5115_Int__greaterThanAtMost,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ A2 @ B2 ) @ ( set_or3652927894154168847AtMost @ A @ C2 @ D2 ) )
= ( set_or3652927894154168847AtMost @ A @ ( ord_max @ A @ A2 @ C2 ) @ ( ord_min @ A @ B2 @ D2 ) ) ) ) ).
% Int_greaterThanAtMost
thf(fact_5116_Int__greaterThanLessThan,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ A2 @ B2 ) @ ( set_or5935395276787703475ssThan @ A @ C2 @ D2 ) )
= ( set_or5935395276787703475ssThan @ A @ ( ord_max @ A @ A2 @ C2 ) @ ( ord_min @ A @ B2 @ D2 ) ) ) ) ).
% Int_greaterThanLessThan
thf(fact_5117_min__numeral__Suc,axiom,
! [K: num,N: nat] :
( ( ord_min @ nat @ ( numeral_numeral @ nat @ K ) @ ( suc @ N ) )
= ( suc @ ( ord_min @ nat @ ( pred_numeral @ K ) @ N ) ) ) ).
% min_numeral_Suc
thf(fact_5118_min__Suc__numeral,axiom,
! [N: nat,K: num] :
( ( ord_min @ nat @ ( suc @ N ) @ ( numeral_numeral @ nat @ K ) )
= ( suc @ ( ord_min @ nat @ N @ ( pred_numeral @ K ) ) ) ) ).
% min_Suc_numeral
thf(fact_5119_zip__append,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Us: list @ B,Ys: list @ A,Vs: list @ B] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Us ) )
=> ( ( zip @ A @ B @ ( append @ A @ Xs @ Ys ) @ ( append @ B @ Us @ Vs ) )
= ( append @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Us ) @ ( zip @ A @ B @ Ys @ Vs ) ) ) ) ).
% zip_append
thf(fact_5120_length__zip,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B] :
( ( size_size @ ( list @ ( product_prod @ A @ B ) ) @ ( zip @ A @ B @ Xs @ Ys ) )
= ( ord_min @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ B ) @ Ys ) ) ) ).
% length_zip
thf(fact_5121_map__of__zip__is__None,axiom,
! [B: $tType,A: $tType,Xs: list @ A,Ys: list @ B,X2: A] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( ( map_of @ A @ B @ ( zip @ A @ B @ Xs @ Ys ) @ X2 )
= ( none @ B ) )
= ( ~ ( member @ A @ X2 @ ( set2 @ A @ Xs ) ) ) ) ) ).
% map_of_zip_is_None
thf(fact_5122_nth__zip,axiom,
! [A: $tType,B: $tType,I: nat,Xs: list @ A,Ys: list @ B] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( nth @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) @ I )
= ( product_Pair @ A @ B @ ( nth @ A @ Xs @ I ) @ ( nth @ B @ Ys @ I ) ) ) ) ) ).
% nth_zip
thf(fact_5123_min__def__raw,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_min @ A )
= ( ^ [A3: A,B3: A] : ( if @ A @ ( ord_less_eq @ A @ A3 @ B3 ) @ A3 @ B3 ) ) ) ) ).
% min_def_raw
thf(fact_5124_min__le__iff__disj,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y: A,Z2: A] :
( ( ord_less_eq @ A @ ( ord_min @ A @ X2 @ Y ) @ Z2 )
= ( ( ord_less_eq @ A @ X2 @ Z2 )
| ( ord_less_eq @ A @ Y @ Z2 ) ) ) ) ).
% min_le_iff_disj
thf(fact_5125_min_OcoboundedI2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,C2: A,A2: A] :
( ( ord_less_eq @ A @ B2 @ C2 )
=> ( ord_less_eq @ A @ ( ord_min @ A @ A2 @ B2 ) @ C2 ) ) ) ).
% min.coboundedI2
thf(fact_5126_min_OcoboundedI1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ C2 )
=> ( ord_less_eq @ A @ ( ord_min @ A @ A2 @ B2 ) @ C2 ) ) ) ).
% min.coboundedI1
thf(fact_5127_min_Oabsorb__iff2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B3: A,A3: A] :
( ( ord_min @ A @ A3 @ B3 )
= B3 ) ) ) ) ).
% min.absorb_iff2
thf(fact_5128_min_Oabsorb__iff1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A3: A,B3: A] :
( ( ord_min @ A @ A3 @ B3 )
= A3 ) ) ) ) ).
% min.absorb_iff1
thf(fact_5129_min_Ocobounded2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A] : ( ord_less_eq @ A @ ( ord_min @ A @ A2 @ B2 ) @ B2 ) ) ).
% min.cobounded2
thf(fact_5130_min_Ocobounded1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A] : ( ord_less_eq @ A @ ( ord_min @ A @ A2 @ B2 ) @ A2 ) ) ).
% min.cobounded1
thf(fact_5131_min_Oorder__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A3: A,B3: A] :
( A3
= ( ord_min @ A @ A3 @ B3 ) ) ) ) ) ).
% min.order_iff
thf(fact_5132_min_OboundedI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ A2 @ C2 )
=> ( ord_less_eq @ A @ A2 @ ( ord_min @ A @ B2 @ C2 ) ) ) ) ) ).
% min.boundedI
thf(fact_5133_min_OboundedE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ ( ord_min @ A @ B2 @ C2 ) )
=> ~ ( ( ord_less_eq @ A @ A2 @ B2 )
=> ~ ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% min.boundedE
thf(fact_5134_min_OorderI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A] :
( ( A2
= ( ord_min @ A @ A2 @ B2 ) )
=> ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% min.orderI
thf(fact_5135_min_OorderE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( A2
= ( ord_min @ A @ A2 @ B2 ) ) ) ) ).
% min.orderE
thf(fact_5136_min_Omono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,C2: A,B2: A,D2: A] :
( ( ord_less_eq @ A @ A2 @ C2 )
=> ( ( ord_less_eq @ A @ B2 @ D2 )
=> ( ord_less_eq @ A @ ( ord_min @ A @ A2 @ B2 ) @ ( ord_min @ A @ C2 @ D2 ) ) ) ) ) ).
% min.mono
thf(fact_5137_min__absorb2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y: A,X2: A] :
( ( ord_less_eq @ A @ Y @ X2 )
=> ( ( ord_min @ A @ X2 @ Y )
= Y ) ) ) ).
% min_absorb2
thf(fact_5138_min__absorb1,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X2: A,Y: A] :
( ( ord_less_eq @ A @ X2 @ Y )
=> ( ( ord_min @ A @ X2 @ Y )
= X2 ) ) ) ).
% min_absorb1
thf(fact_5139_min__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_min @ A )
= ( ^ [A3: A,B3: A] : ( if @ A @ ( ord_less_eq @ A @ A3 @ B3 ) @ A3 @ B3 ) ) ) ) ).
% min_def
thf(fact_5140_of__int__min,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X2: int,Y: int] :
( ( ring_1_of_int @ A @ ( ord_min @ int @ X2 @ Y ) )
= ( ord_min @ A @ ( ring_1_of_int @ A @ X2 ) @ ( ring_1_of_int @ A @ Y ) ) ) ) ).
% of_int_min
thf(fact_5141_zip__obtain__same__length,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B,P: ( list @ ( product_prod @ A @ B ) ) > $o] :
( ! [Zs2: list @ A,Ws2: list @ B,N2: nat] :
( ( ( size_size @ ( list @ A ) @ Zs2 )
= ( size_size @ ( list @ B ) @ Ws2 ) )
=> ( ( N2
= ( ord_min @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ B ) @ Ys ) ) )
=> ( ( Zs2
= ( take @ A @ N2 @ Xs ) )
=> ( ( Ws2
= ( take @ B @ N2 @ Ys ) )
=> ( P @ ( zip @ A @ B @ Zs2 @ Ws2 ) ) ) ) ) )
=> ( P @ ( zip @ A @ B @ Xs @ Ys ) ) ) ).
% zip_obtain_same_length
thf(fact_5142_nat__mult__min__left,axiom,
! [M: nat,N: nat,Q2: nat] :
( ( times_times @ nat @ ( ord_min @ nat @ M @ N ) @ Q2 )
= ( ord_min @ nat @ ( times_times @ nat @ M @ Q2 ) @ ( times_times @ nat @ N @ Q2 ) ) ) ).
% nat_mult_min_left
thf(fact_5143_nat__mult__min__right,axiom,
! [M: nat,N: nat,Q2: nat] :
( ( times_times @ nat @ M @ ( ord_min @ nat @ N @ Q2 ) )
= ( ord_min @ nat @ ( times_times @ nat @ M @ N ) @ ( times_times @ nat @ M @ Q2 ) ) ) ).
% nat_mult_min_right
thf(fact_5144_ord_Omin_Ocong,axiom,
! [A: $tType] :
( ( min @ A )
= ( min @ A ) ) ).
% ord.min.cong
thf(fact_5145_ord_Omin__def,axiom,
! [A: $tType] :
( ( min @ A )
= ( ^ [Less_eq2: A > A > $o,A3: A,B3: A] : ( if @ A @ ( Less_eq2 @ A3 @ B3 ) @ A3 @ B3 ) ) ) ).
% ord.min_def
thf(fact_5146_inf__nat__def,axiom,
( ( inf_inf @ nat )
= ( ord_min @ nat ) ) ).
% inf_nat_def
thf(fact_5147_zip__inj,axiom,
! [A: $tType,B: $tType,A2: list @ A,B2: list @ B,A6: list @ A,B6: list @ B] :
( ( ( size_size @ ( list @ A ) @ A2 )
= ( size_size @ ( list @ B ) @ B2 ) )
=> ( ( ( size_size @ ( list @ A ) @ A6 )
= ( size_size @ ( list @ B ) @ B6 ) )
=> ( ( ( zip @ A @ B @ A2 @ B2 )
= ( zip @ A @ B @ A6 @ B6 ) )
=> ( ( A2 = A6 )
& ( B2 = B6 ) ) ) ) ) ).
% zip_inj
thf(fact_5148_min__add__distrib__left,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [X2: A,Y: A,Z2: A] :
( ( plus_plus @ A @ ( ord_min @ A @ X2 @ Y ) @ Z2 )
= ( ord_min @ A @ ( plus_plus @ A @ X2 @ Z2 ) @ ( plus_plus @ A @ Y @ Z2 ) ) ) ) ).
% min_add_distrib_left
thf(fact_5149_min__add__distrib__right,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [X2: A,Y: A,Z2: A] :
( ( plus_plus @ A @ X2 @ ( ord_min @ A @ Y @ Z2 ) )
= ( ord_min @ A @ ( plus_plus @ A @ X2 @ Y ) @ ( plus_plus @ A @ X2 @ Z2 ) ) ) ) ).
% min_add_distrib_right
thf(fact_5150_min__diff__distrib__left,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [X2: A,Y: A,Z2: A] :
( ( minus_minus @ A @ ( ord_min @ A @ X2 @ Y ) @ Z2 )
= ( ord_min @ A @ ( minus_minus @ A @ X2 @ Z2 ) @ ( minus_minus @ A @ Y @ Z2 ) ) ) ) ).
% min_diff_distrib_left
thf(fact_5151_min__diff,axiom,
! [M: nat,I: nat,N: nat] :
( ( ord_min @ nat @ ( minus_minus @ nat @ M @ I ) @ ( minus_minus @ nat @ N @ I ) )
= ( minus_minus @ nat @ ( ord_min @ nat @ M @ N ) @ I ) ) ).
% min_diff
thf(fact_5152_zip__assoc,axiom,
! [B: $tType,A: $tType,C: $tType,Xs: list @ A,Ys: list @ B,Zs: list @ C] :
( ( zip @ A @ ( product_prod @ B @ C ) @ Xs @ ( zip @ B @ C @ Ys @ Zs ) )
= ( map @ ( product_prod @ ( product_prod @ A @ B ) @ C ) @ ( product_prod @ A @ ( product_prod @ B @ C ) )
@ ( product_case_prod @ ( product_prod @ A @ B ) @ C @ ( product_prod @ A @ ( product_prod @ B @ C ) )
@ ( product_case_prod @ A @ B @ ( C > ( product_prod @ A @ ( product_prod @ B @ C ) ) )
@ ^ [X: A,Y4: B,Z3: C] : ( product_Pair @ A @ ( product_prod @ B @ C ) @ X @ ( product_Pair @ B @ C @ Y4 @ Z3 ) ) ) )
@ ( zip @ ( product_prod @ A @ B ) @ C @ ( zip @ A @ B @ Xs @ Ys ) @ Zs ) ) ) ).
% zip_assoc
thf(fact_5153_of__nat__min,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [X2: nat,Y: nat] :
( ( semiring_1_of_nat @ A @ ( ord_min @ nat @ X2 @ Y ) )
= ( ord_min @ A @ ( semiring_1_of_nat @ A @ X2 ) @ ( semiring_1_of_nat @ A @ Y ) ) ) ) ).
% of_nat_min
thf(fact_5154_minus__max__eq__min,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [X2: A,Y: A] :
( ( uminus_uminus @ A @ ( ord_max @ A @ X2 @ Y ) )
= ( ord_min @ A @ ( uminus_uminus @ A @ X2 ) @ ( uminus_uminus @ A @ Y ) ) ) ) ).
% minus_max_eq_min
thf(fact_5155_minus__min__eq__max,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [X2: A,Y: A] :
( ( uminus_uminus @ A @ ( ord_min @ A @ X2 @ Y ) )
= ( ord_max @ A @ ( uminus_uminus @ A @ X2 ) @ ( uminus_uminus @ A @ Y ) ) ) ) ).
% minus_min_eq_max
thf(fact_5156_ex__mset__zip__left,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B,Xs5: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( ( mset @ A @ Xs5 )
= ( mset @ A @ Xs ) )
=> ? [Ys5: list @ B] :
( ( ( size_size @ ( list @ B ) @ Ys5 )
= ( size_size @ ( list @ A ) @ Xs5 ) )
& ( ( mset @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs5 @ Ys5 ) )
= ( mset @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) ) ) ) ) ) ).
% ex_mset_zip_left
thf(fact_5157_zip__rev,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( zip @ A @ B @ ( rev @ A @ Xs ) @ ( rev @ B @ Ys ) )
= ( rev @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) ) ) ) ).
% zip_rev
thf(fact_5158_greaterThan__Int__greaterThan,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A] :
( ( inf_inf @ ( set @ A ) @ ( set_ord_lessThan @ A @ A2 ) @ ( set_ord_lessThan @ A @ B2 ) )
= ( set_ord_lessThan @ A @ ( ord_min @ A @ A2 @ B2 ) ) ) ) ).
% greaterThan_Int_greaterThan
thf(fact_5159_zip__Cons1,axiom,
! [A: $tType,B: $tType,X2: A,Xs: list @ A,Ys: list @ B] :
( ( zip @ A @ B @ ( cons @ A @ X2 @ Xs ) @ Ys )
= ( case_list @ ( list @ ( product_prod @ A @ B ) ) @ B @ ( nil @ ( product_prod @ A @ B ) )
@ ^ [Y4: B,Ys3: list @ B] : ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y4 ) @ ( zip @ A @ B @ Xs @ Ys3 ) )
@ Ys ) ) ).
% zip_Cons1
thf(fact_5160_zip__Cons,axiom,
! [B: $tType,A: $tType,Xs: list @ A,Y: B,Ys: list @ B] :
( ( zip @ A @ B @ Xs @ ( cons @ B @ Y @ Ys ) )
= ( case_list @ ( list @ ( product_prod @ A @ B ) ) @ A @ ( nil @ ( product_prod @ A @ B ) )
@ ^ [Z3: A,Zs3: list @ A] : ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Z3 @ Y ) @ ( zip @ A @ B @ Zs3 @ Ys ) )
@ Xs ) ) ).
% zip_Cons
thf(fact_5161_zip__left__commute,axiom,
! [B: $tType,A: $tType,C: $tType,Xs: list @ A,Ys: list @ B,Zs: list @ C] :
( ( zip @ A @ ( product_prod @ B @ C ) @ Xs @ ( zip @ B @ C @ Ys @ Zs ) )
= ( map @ ( product_prod @ B @ ( product_prod @ A @ C ) ) @ ( product_prod @ A @ ( product_prod @ B @ C ) )
@ ( product_case_prod @ B @ ( product_prod @ A @ C ) @ ( product_prod @ A @ ( product_prod @ B @ C ) )
@ ^ [Y4: B] :
( product_case_prod @ A @ C @ ( product_prod @ A @ ( product_prod @ B @ C ) )
@ ^ [X: A,Z3: C] : ( product_Pair @ A @ ( product_prod @ B @ C ) @ X @ ( product_Pair @ B @ C @ Y4 @ Z3 ) ) ) )
@ ( zip @ B @ ( product_prod @ A @ C ) @ Ys @ ( zip @ A @ C @ Xs @ Zs ) ) ) ) ).
% zip_left_commute
thf(fact_5162_map2__map__map,axiom,
! [B: $tType,A: $tType,C: $tType,D: $tType,H: B > C > A,F: D > B,Xs: list @ D,G: D > C] :
( ( map @ ( product_prod @ B @ C ) @ A @ ( product_case_prod @ B @ C @ A @ H ) @ ( zip @ B @ C @ ( map @ D @ B @ F @ Xs ) @ ( map @ D @ C @ G @ Xs ) ) )
= ( map @ D @ A
@ ^ [X: D] : ( H @ ( F @ X ) @ ( G @ X ) )
@ Xs ) ) ).
% map2_map_map
thf(fact_5163_zip__commute,axiom,
! [B: $tType,A: $tType] :
( ( zip @ A @ B )
= ( ^ [Xs3: list @ A,Ys3: list @ B] :
( map @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B )
@ ( product_case_prod @ B @ A @ ( product_prod @ A @ B )
@ ^ [X: B,Y4: A] : ( product_Pair @ A @ B @ Y4 @ X ) )
@ ( zip @ B @ A @ Ys3 @ Xs3 ) ) ) ) ).
% zip_commute
thf(fact_5164_zip__same__conv__map,axiom,
! [A: $tType,Xs: list @ A] :
( ( zip @ A @ A @ Xs @ Xs )
= ( map @ A @ ( product_prod @ A @ A )
@ ^ [X: A] : ( product_Pair @ A @ A @ X @ X )
@ Xs ) ) ).
% zip_same_conv_map
thf(fact_5165_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_5166_map__of__zip__inject,axiom,
! [B: $tType,A: $tType,Ys: list @ A,Xs: list @ B,Zs: list @ A] :
( ( ( size_size @ ( list @ A ) @ Ys )
= ( size_size @ ( list @ B ) @ Xs ) )
=> ( ( ( size_size @ ( list @ A ) @ Zs )
= ( size_size @ ( list @ B ) @ Xs ) )
=> ( ( distinct @ B @ Xs )
=> ( ( ( map_of @ B @ A @ ( zip @ B @ A @ Xs @ Ys ) )
= ( map_of @ B @ A @ ( zip @ B @ A @ Xs @ Zs ) ) )
=> ( Ys = Zs ) ) ) ) ) ).
% map_of_zip_inject
thf(fact_5167_map__zip__map2,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,F: ( product_prod @ B @ C ) > A,Xs: list @ B,G: D > C,Ys: list @ D] :
( ( map @ ( product_prod @ B @ C ) @ A @ F @ ( zip @ B @ C @ Xs @ ( map @ D @ C @ G @ Ys ) ) )
= ( map @ ( product_prod @ B @ D ) @ A
@ ( product_case_prod @ B @ D @ A
@ ^ [X: B,Y4: D] : ( F @ ( product_Pair @ B @ C @ X @ ( G @ Y4 ) ) ) )
@ ( zip @ B @ D @ Xs @ Ys ) ) ) ).
% map_zip_map2
thf(fact_5168_map__zip__map,axiom,
! [B: $tType,A: $tType,D: $tType,C: $tType,F: ( product_prod @ B @ C ) > A,G: D > B,Xs: list @ D,Ys: list @ C] :
( ( map @ ( product_prod @ B @ C ) @ A @ F @ ( zip @ B @ C @ ( map @ D @ B @ G @ Xs ) @ Ys ) )
= ( map @ ( product_prod @ D @ C ) @ A
@ ( product_case_prod @ D @ C @ A
@ ^ [X: D,Y4: C] : ( F @ ( product_Pair @ B @ C @ ( G @ X ) @ Y4 ) ) )
@ ( zip @ D @ C @ Xs @ Ys ) ) ) ).
% map_zip_map
thf(fact_5169_subset__mset_Omin__add__distrib__right,axiom,
! [A: $tType,X2: multiset @ A,Y: multiset @ A,Z2: multiset @ A] :
( ( plus_plus @ ( multiset @ A ) @ X2 @ ( min @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ Y @ Z2 ) )
= ( min @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( plus_plus @ ( multiset @ A ) @ X2 @ Y ) @ ( plus_plus @ ( multiset @ A ) @ X2 @ Z2 ) ) ) ).
% subset_mset.min_add_distrib_right
thf(fact_5170_subset__mset_Omin__add__distrib__left,axiom,
! [A: $tType,X2: multiset @ A,Y: multiset @ A,Z2: multiset @ A] :
( ( plus_plus @ ( multiset @ A ) @ ( min @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ X2 @ Y ) @ Z2 )
= ( min @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( plus_plus @ ( multiset @ A ) @ X2 @ Z2 ) @ ( plus_plus @ ( multiset @ A ) @ Y @ Z2 ) ) ) ).
% subset_mset.min_add_distrib_left
thf(fact_5171_map__prod__fun__zip,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,F: C > A,G: D > B,Xs: list @ C,Ys: list @ D] :
( ( map @ ( product_prod @ C @ D ) @ ( product_prod @ A @ B )
@ ( product_case_prod @ C @ D @ ( product_prod @ A @ B )
@ ^ [X: C,Y4: D] : ( product_Pair @ A @ B @ ( F @ X ) @ ( G @ Y4 ) ) )
@ ( zip @ C @ D @ Xs @ Ys ) )
= ( zip @ A @ B @ ( map @ C @ A @ F @ Xs ) @ ( map @ D @ B @ G @ Ys ) ) ) ).
% map_prod_fun_zip
thf(fact_5172_zip__map2,axiom,
! [B: $tType,A: $tType,C: $tType,Xs: list @ A,F: C > B,Ys: list @ C] :
( ( zip @ A @ B @ Xs @ ( map @ C @ B @ F @ Ys ) )
= ( map @ ( product_prod @ A @ C ) @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ C @ ( product_prod @ A @ B )
@ ^ [X: A,Y4: C] : ( product_Pair @ A @ B @ X @ ( F @ Y4 ) ) )
@ ( zip @ A @ C @ Xs @ Ys ) ) ) ).
% zip_map2
thf(fact_5173_zip__map1,axiom,
! [A: $tType,C: $tType,B: $tType,F: C > A,Xs: list @ C,Ys: list @ B] :
( ( zip @ A @ B @ ( map @ C @ A @ F @ Xs ) @ Ys )
= ( map @ ( product_prod @ C @ B ) @ ( product_prod @ A @ B )
@ ( product_case_prod @ C @ B @ ( product_prod @ A @ B )
@ ^ [X: C] : ( product_Pair @ A @ B @ ( F @ X ) ) )
@ ( zip @ C @ B @ Xs @ Ys ) ) ) ).
% zip_map1
thf(fact_5174_max__mult__distrib__left,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [P3: A,X2: A,Y: A] :
( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P3 )
=> ( ( times_times @ A @ P3 @ ( ord_max @ A @ X2 @ Y ) )
= ( ord_max @ A @ ( times_times @ A @ P3 @ X2 ) @ ( times_times @ A @ P3 @ Y ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P3 )
=> ( ( times_times @ A @ P3 @ ( ord_max @ A @ X2 @ Y ) )
= ( ord_min @ A @ ( times_times @ A @ P3 @ X2 ) @ ( times_times @ A @ P3 @ Y ) ) ) ) ) ) ).
% max_mult_distrib_left
thf(fact_5175_min__mult__distrib__left,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [P3: A,X2: A,Y: A] :
( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P3 )
=> ( ( times_times @ A @ P3 @ ( ord_min @ A @ X2 @ Y ) )
= ( ord_min @ A @ ( times_times @ A @ P3 @ X2 ) @ ( times_times @ A @ P3 @ Y ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P3 )
=> ( ( times_times @ A @ P3 @ ( ord_min @ A @ X2 @ Y ) )
= ( ord_max @ A @ ( times_times @ A @ P3 @ X2 ) @ ( times_times @ A @ P3 @ Y ) ) ) ) ) ) ).
% min_mult_distrib_left
thf(fact_5176_max__mult__distrib__right,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [P3: A,X2: A,Y: A] :
( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P3 )
=> ( ( times_times @ A @ ( ord_max @ A @ X2 @ Y ) @ P3 )
= ( ord_max @ A @ ( times_times @ A @ X2 @ P3 ) @ ( times_times @ A @ Y @ P3 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P3 )
=> ( ( times_times @ A @ ( ord_max @ A @ X2 @ Y ) @ P3 )
= ( ord_min @ A @ ( times_times @ A @ X2 @ P3 ) @ ( times_times @ A @ Y @ P3 ) ) ) ) ) ) ).
% max_mult_distrib_right
thf(fact_5177_min__mult__distrib__right,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [P3: A,X2: A,Y: A] :
( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P3 )
=> ( ( times_times @ A @ ( ord_min @ A @ X2 @ Y ) @ P3 )
= ( ord_min @ A @ ( times_times @ A @ X2 @ P3 ) @ ( times_times @ A @ Y @ P3 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P3 )
=> ( ( times_times @ A @ ( ord_min @ A @ X2 @ Y ) @ P3 )
= ( ord_max @ A @ ( times_times @ A @ X2 @ P3 ) @ ( times_times @ A @ Y @ P3 ) ) ) ) ) ) ).
% min_mult_distrib_right
thf(fact_5178_max__divide__distrib__right,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [P3: A,X2: A,Y: A] :
( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P3 )
=> ( ( divide_divide @ A @ ( ord_max @ A @ X2 @ Y ) @ P3 )
= ( ord_max @ A @ ( divide_divide @ A @ X2 @ P3 ) @ ( divide_divide @ A @ Y @ P3 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P3 )
=> ( ( divide_divide @ A @ ( ord_max @ A @ X2 @ Y ) @ P3 )
= ( ord_min @ A @ ( divide_divide @ A @ X2 @ P3 ) @ ( divide_divide @ A @ Y @ P3 ) ) ) ) ) ) ).
% max_divide_distrib_right
thf(fact_5179_min__divide__distrib__right,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [P3: A,X2: A,Y: A] :
( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P3 )
=> ( ( divide_divide @ A @ ( ord_min @ A @ X2 @ Y ) @ P3 )
= ( ord_min @ A @ ( divide_divide @ A @ X2 @ P3 ) @ ( divide_divide @ A @ Y @ P3 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P3 )
=> ( ( divide_divide @ A @ ( ord_min @ A @ X2 @ Y ) @ P3 )
= ( ord_max @ A @ ( divide_divide @ A @ X2 @ P3 ) @ ( divide_divide @ A @ Y @ P3 ) ) ) ) ) ) ).
% min_divide_distrib_right
thf(fact_5180_min__Suc1,axiom,
! [N: nat,M: nat] :
( ( ord_min @ nat @ ( suc @ N ) @ M )
= ( case_nat @ nat @ ( zero_zero @ nat )
@ ^ [M8: nat] : ( suc @ ( ord_min @ nat @ N @ M8 ) )
@ M ) ) ).
% min_Suc1
thf(fact_5181_min__Suc2,axiom,
! [M: nat,N: nat] :
( ( ord_min @ nat @ M @ ( suc @ N ) )
= ( case_nat @ nat @ ( zero_zero @ nat )
@ ^ [M8: nat] : ( suc @ ( ord_min @ nat @ M8 @ N ) )
@ M ) ) ).
% min_Suc2
thf(fact_5182_last__zip,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B] :
( ( Xs
!= ( nil @ A ) )
=> ( ( Ys
!= ( nil @ B ) )
=> ( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( last @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) )
= ( product_Pair @ A @ B @ ( last @ A @ Xs ) @ ( last @ B @ Ys ) ) ) ) ) ) ).
% last_zip
thf(fact_5183_in__set__impl__in__set__zip2,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B,Y: B] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( member @ B @ Y @ ( set2 @ B @ Ys ) )
=> ~ ! [X3: A] :
~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y ) @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) ) ) ) ) ).
% in_set_impl_in_set_zip2
thf(fact_5184_in__set__impl__in__set__zip1,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B,X2: A] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
=> ~ ! [Y2: B] :
~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y2 ) @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) ) ) ) ) ).
% in_set_impl_in_set_zip1
thf(fact_5185_map__of__zip__is__Some,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B,X2: A] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
= ( ? [Y4: B] :
( ( map_of @ A @ B @ ( zip @ A @ B @ Xs @ Ys ) @ X2 )
= ( some @ B @ Y4 ) ) ) ) ) ).
% map_of_zip_is_Some
thf(fact_5186_map__upds__fold__map__upd,axiom,
! [B: $tType,A: $tType] :
( ( map_upds @ A @ B )
= ( ^ [M2: A > ( option @ B ),Ks2: list @ A,Vs2: list @ B] :
( foldl @ ( A > ( option @ B ) ) @ ( product_prod @ A @ B )
@ ^ [N4: A > ( option @ B )] :
( product_case_prod @ A @ B @ ( A > ( option @ B ) )
@ ^ [K2: A,V4: B] : ( fun_upd @ A @ ( option @ B ) @ N4 @ K2 @ ( some @ B @ V4 ) ) )
@ M2
@ ( zip @ A @ B @ Ks2 @ Vs2 ) ) ) ) ).
% map_upds_fold_map_upd
thf(fact_5187_enumerate__eq__zip,axiom,
! [A: $tType] :
( ( enumerate @ A )
= ( ^ [N4: nat,Xs3: list @ A] : ( zip @ nat @ A @ ( upt @ N4 @ ( plus_plus @ nat @ N4 @ ( size_size @ ( list @ A ) @ Xs3 ) ) ) @ Xs3 ) ) ) ).
% enumerate_eq_zip
thf(fact_5188_foldl__snd__zip,axiom,
! [B: $tType,C: $tType,A: $tType,Ys: list @ A,Xs: list @ B,F: C > A > C,B2: C] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Ys ) @ ( size_size @ ( list @ B ) @ Xs ) )
=> ( ( foldl @ C @ ( product_prod @ B @ A )
@ ^ [B3: C] :
( product_case_prod @ B @ A @ C
@ ^ [X: B] : ( F @ B3 ) )
@ B2
@ ( zip @ B @ A @ Xs @ Ys ) )
= ( foldl @ C @ A @ F @ B2 @ Ys ) ) ) ).
% foldl_snd_zip
thf(fact_5189_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_5190_map__zip1,axiom,
! [A: $tType,B: $tType,K: B,L: list @ A] :
( ( map @ A @ ( product_prod @ A @ B )
@ ^ [X: A] : ( product_Pair @ A @ B @ X @ K )
@ L )
= ( zip @ A @ B @ L @ ( replicate @ B @ ( size_size @ ( list @ A ) @ L ) @ K ) ) ) ).
% map_zip1
thf(fact_5191_foldr__snd__zip,axiom,
! [B: $tType,A: $tType,C: $tType,Ys: list @ A,Xs: list @ B,F: A > C > C,B2: C] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Ys ) @ ( size_size @ ( list @ B ) @ Xs ) )
=> ( ( foldr @ ( product_prod @ B @ A ) @ C
@ ( product_case_prod @ B @ A @ ( C > C )
@ ^ [X: B] : F )
@ ( zip @ B @ A @ Xs @ Ys )
@ B2 )
= ( foldr @ A @ C @ F @ Ys @ B2 ) ) ) ).
% foldr_snd_zip
thf(fact_5192_zip__replicate2,axiom,
! [B: $tType,A: $tType,Xs: list @ A,N: nat,Y: B] :
( ( zip @ A @ B @ Xs @ ( replicate @ B @ N @ Y ) )
= ( map @ A @ ( product_prod @ A @ B )
@ ^ [X: A] : ( product_Pair @ A @ B @ X @ Y )
@ ( take @ A @ N @ Xs ) ) ) ).
% zip_replicate2
thf(fact_5193_map__of__zip__upd,axiom,
! [A: $tType,B: $tType,Ys: list @ B,Xs: list @ A,Zs: list @ B,X2: A,Y: B,Z2: B] :
( ( ( size_size @ ( list @ B ) @ Ys )
= ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ( size_size @ ( list @ B ) @ Zs )
= ( size_size @ ( list @ A ) @ Xs ) )
=> ( ~ ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
=> ( ( ( fun_upd @ A @ ( option @ B ) @ ( map_of @ A @ B @ ( zip @ A @ B @ Xs @ Ys ) ) @ X2 @ ( some @ B @ Y ) )
= ( fun_upd @ A @ ( option @ B ) @ ( map_of @ A @ B @ ( zip @ A @ B @ Xs @ Zs ) ) @ X2 @ ( some @ B @ Z2 ) ) )
=> ( ( map_of @ A @ B @ ( zip @ A @ B @ Xs @ Ys ) )
= ( map_of @ A @ B @ ( zip @ A @ B @ Xs @ Zs ) ) ) ) ) ) ) ).
% map_of_zip_upd
thf(fact_5194_ran__map__of__zip,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( distinct @ A @ Xs )
=> ( ( ran @ A @ B @ ( map_of @ A @ B @ ( zip @ A @ B @ Xs @ Ys ) ) )
= ( set2 @ B @ Ys ) ) ) ) ).
% ran_map_of_zip
thf(fact_5195_zip__append1,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ A,Zs: list @ B] :
( ( zip @ A @ B @ ( append @ A @ Xs @ Ys ) @ Zs )
= ( append @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ ( take @ B @ ( size_size @ ( list @ A ) @ Xs ) @ Zs ) ) @ ( zip @ A @ B @ Ys @ ( drop @ B @ ( size_size @ ( list @ A ) @ Xs ) @ Zs ) ) ) ) ).
% zip_append1
thf(fact_5196_zip__append2,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B,Zs: list @ B] :
( ( zip @ A @ B @ Xs @ ( append @ B @ Ys @ Zs ) )
= ( append @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ ( take @ A @ ( size_size @ ( list @ B ) @ Ys ) @ Xs ) @ Ys ) @ ( zip @ A @ B @ ( drop @ A @ ( size_size @ ( list @ B ) @ Ys ) @ Xs ) @ Zs ) ) ) ).
% zip_append2
thf(fact_5197_map__of__zip__nth,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B,I: nat] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( distinct @ A @ Xs )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( map_of @ A @ B @ ( zip @ A @ B @ Xs @ Ys ) @ ( nth @ A @ Xs @ I ) )
= ( some @ B @ ( nth @ B @ Ys @ I ) ) ) ) ) ) ).
% map_of_zip_nth
thf(fact_5198_min__list_Oelims,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X2: list @ A,Y: A] :
( ( ( min_list @ A @ X2 )
= Y )
=> ( ! [X3: A,Xs2: list @ A] :
( ( X2
= ( cons @ A @ X3 @ Xs2 ) )
=> ( Y
!= ( case_list @ A @ A @ X3
@ ^ [A3: A,List2: list @ A] : ( ord_min @ A @ X3 @ ( min_list @ A @ Xs2 ) )
@ Xs2 ) ) )
=> ~ ( ( X2
= ( nil @ A ) )
=> ( Y
!= ( undefined @ A ) ) ) ) ) ) ).
% min_list.elims
thf(fact_5199_integer__of__nat__def,axiom,
( code_integer_of_nat
= ( map_fun @ nat @ nat @ int @ code_integer @ ( id @ nat ) @ code_integer_of_int @ ( semiring_1_of_nat @ int ) ) ) ).
% integer_of_nat_def
thf(fact_5200_map__of__distinct__upd4,axiom,
! [A: $tType,B: $tType,X2: A,Xs: list @ ( product_prod @ A @ B ),Ys: list @ ( product_prod @ A @ B ),Y: B] :
( ~ ( member @ A @ X2 @ ( set2 @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xs ) ) )
=> ( ~ ( member @ A @ X2 @ ( set2 @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Ys ) ) )
=> ( ( map_of @ A @ B @ ( append @ ( product_prod @ A @ B ) @ Xs @ Ys ) )
= ( fun_upd @ A @ ( option @ B ) @ ( map_of @ A @ B @ ( append @ ( product_prod @ A @ B ) @ Xs @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y ) @ Ys ) ) ) @ X2 @ ( none @ B ) ) ) ) ) ).
% map_of_distinct_upd4
thf(fact_5201_nat__of__integer__integer__of__nat,axiom,
! [N: nat] :
( ( code_nat_of_integer @ ( code_integer_of_nat @ N ) )
= N ) ).
% nat_of_integer_integer_of_nat
thf(fact_5202_fst__divmod__integer,axiom,
! [K: code_integer,L: code_integer] :
( ( product_fst @ code_integer @ code_integer @ ( code_divmod_integer @ K @ L ) )
= ( divide_divide @ code_integer @ K @ L ) ) ).
% fst_divmod_integer
thf(fact_5203_int__of__integer__min,axiom,
! [K: code_integer,L: code_integer] :
( ( code_int_of_integer @ ( ord_min @ code_integer @ K @ L ) )
= ( ord_min @ int @ ( code_int_of_integer @ K ) @ ( code_int_of_integer @ L ) ) ) ).
% int_of_integer_min
thf(fact_5204_img__fst,axiom,
! [B: $tType,A: $tType,A2: A,B2: B,S2: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ S2 )
=> ( member @ A @ A2 @ ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ S2 ) ) ) ).
% img_fst
thf(fact_5205_int__of__integer__integer__of__nat,axiom,
! [N: nat] :
( ( code_int_of_integer @ ( code_integer_of_nat @ N ) )
= ( semiring_1_of_nat @ int @ N ) ) ).
% int_of_integer_integer_of_nat
thf(fact_5206_integer__of__nat_Orep__eq,axiom,
! [X2: nat] :
( ( code_int_of_integer @ ( code_integer_of_nat @ X2 ) )
= ( semiring_1_of_nat @ int @ X2 ) ) ).
% integer_of_nat.rep_eq
thf(fact_5207_fst__divmod__abs,axiom,
! [K: code_integer,L: code_integer] :
( ( product_fst @ code_integer @ code_integer @ ( code_divmod_abs @ K @ L ) )
= ( divide_divide @ code_integer @ ( abs_abs @ code_integer @ K ) @ ( abs_abs @ code_integer @ L ) ) ) ).
% fst_divmod_abs
thf(fact_5208_map__fst__zip,axiom,
! [B: $tType,A: $tType,Xs: list @ A,Ys: list @ B] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) )
= Xs ) ) ).
% map_fst_zip
thf(fact_5209_map__fst__enumerate,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( map @ ( product_prod @ nat @ A ) @ nat @ ( product_fst @ nat @ A ) @ ( enumerate @ A @ N @ Xs ) )
= ( upt @ N @ ( plus_plus @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ).
% map_fst_enumerate
thf(fact_5210_sorted__wrt__map__linord,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ A )
=> ! [L: list @ ( product_prod @ A @ B )] :
( ( sorted_wrt @ ( product_prod @ A @ B )
@ ^ [X: product_prod @ A @ B,Y4: product_prod @ A @ B] : ( ord_less_eq @ A @ ( product_fst @ A @ B @ X ) @ ( 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_5211_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 )
@ ^ [X: A] : ( product_Pair @ A @ B @ X @ K )
@ L ) )
= L ) ).
% map_fst_mk_snd
thf(fact_5212_Suc__0__div__numeral,axiom,
! [K: num] :
( ( divide_divide @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( numeral_numeral @ nat @ K ) )
= ( product_fst @ nat @ nat @ ( unique8689654367752047608divmod @ nat @ one2 @ K ) ) ) ).
% Suc_0_div_numeral
thf(fact_5213_sorted__wrt__map__rev__linord,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ A )
=> ! [L: list @ ( product_prod @ A @ B )] :
( ( sorted_wrt @ ( product_prod @ A @ B )
@ ^ [X: product_prod @ A @ B,Y4: product_prod @ A @ B] : ( ord_less_eq @ A @ ( product_fst @ A @ B @ Y4 ) @ ( product_fst @ A @ B @ X ) )
@ 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_5214_one__div__numeral,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [N: num] :
( ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ N ) )
= ( product_fst @ A @ A @ ( unique8689654367752047608divmod @ A @ one2 @ N ) ) ) ) ).
% one_div_numeral
thf(fact_5215_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_5216_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_5217_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 )
@ ^ [X: A] : ( product_Pair @ A @ A @ X @ X )
@ ( product_fst @ A @ B ) ) )
= ( product_fst @ A @ B ) ) ).
% fst_diag_fst
thf(fact_5218_inf__int__def,axiom,
( ( inf_inf @ int )
= ( ord_min @ int ) ) ).
% inf_int_def
thf(fact_5219_measure__fst,axiom,
! [B: $tType,A: $tType,F: A > nat] :
( ( fun_is_measure @ A @ F )
=> ( fun_is_measure @ ( product_prod @ A @ B )
@ ^ [P7: product_prod @ A @ B] : ( F @ ( product_fst @ A @ B @ P7 ) ) ) ) ).
% measure_fst
thf(fact_5220_fst__def,axiom,
! [B: $tType,A: $tType] :
( ( product_fst @ A @ B )
= ( product_case_prod @ A @ B @ A
@ ^ [X12: A,X24: B] : X12 ) ) ).
% fst_def
thf(fact_5221_fn__fst__conv,axiom,
! [B: $tType,C: $tType,A: $tType,F: A > C] :
( ( ^ [X: product_prod @ A @ B] : ( F @ ( product_fst @ A @ B @ X ) ) )
= ( product_case_prod @ A @ B @ C
@ ^ [A3: A,Uu: B] : ( F @ A3 ) ) ) ).
% fn_fst_conv
thf(fact_5222_fstE,axiom,
! [B: $tType,A: $tType,X2: product_prod @ A @ B,A2: A,B2: B,P: A > $o] :
( ( X2
= ( product_Pair @ A @ B @ A2 @ B2 ) )
=> ( ( P @ ( product_fst @ A @ B @ X2 ) )
=> ( P @ A2 ) ) ) ).
% fstE
thf(fact_5223_in__fst__imageE,axiom,
! [B: $tType,A: $tType,X2: A,S2: set @ ( product_prod @ A @ B )] :
( ( member @ A @ X2 @ ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ S2 ) )
=> ~ ! [Y2: B] :
~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y2 ) @ S2 ) ) ).
% in_fst_imageE
thf(fact_5224_fst__image__mp,axiom,
! [B: $tType,A: $tType,A5: set @ ( product_prod @ A @ B ),B5: set @ A,X2: A,Y: B] :
( ( ord_less_eq @ ( set @ A ) @ ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ A5 ) @ B5 )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y ) @ A5 )
=> ( member @ A @ X2 @ B5 ) ) ) ).
% fst_image_mp
thf(fact_5225_integer__of__nat__eq__of__nat,axiom,
( code_integer_of_nat
= ( semiring_1_of_nat @ code_integer ) ) ).
% integer_of_nat_eq_of_nat
thf(fact_5226_fst__diag__id,axiom,
! [A: $tType,Z2: A] :
( ( comp @ ( product_prod @ A @ A ) @ A @ A @ ( product_fst @ A @ A )
@ ^ [X: A] : ( product_Pair @ A @ A @ X @ X )
@ Z2 )
= ( id @ A @ Z2 ) ) ).
% fst_diag_id
thf(fact_5227_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_5228_sorted__enumerate,axiom,
! [A: $tType,N: nat,Xs: list @ A] : ( sorted_wrt @ nat @ ( ord_less_eq @ nat ) @ ( map @ ( product_prod @ nat @ A ) @ nat @ ( product_fst @ nat @ A ) @ ( enumerate @ A @ N @ Xs ) ) ) ).
% sorted_enumerate
thf(fact_5229_fst__foldl,axiom,
! [B: $tType,A: $tType,C: $tType,F: A > C > A,G: A > B > C > B,A2: 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 ) )
@ ^ [A3: A,B3: B,X: C] : ( product_Pair @ A @ B @ ( F @ A3 @ X ) @ ( G @ A3 @ B3 @ X ) ) )
@ ( product_Pair @ A @ B @ A2 @ B2 )
@ Xs ) )
= ( foldl @ A @ C @ F @ A2 @ Xs ) ) ).
% fst_foldl
thf(fact_5230_integer__of__nat__0,axiom,
( ( code_integer_of_nat @ ( zero_zero @ nat ) )
= ( zero_zero @ code_integer ) ) ).
% integer_of_nat_0
thf(fact_5231_integer__of__nat_Oabs__eq,axiom,
( code_integer_of_nat
= ( ^ [X: nat] : ( code_integer_of_int @ ( semiring_1_of_nat @ int @ X ) ) ) ) ).
% integer_of_nat.abs_eq
thf(fact_5232_distinct__map__fstD,axiom,
! [A: $tType,B: $tType,Xs: list @ ( product_prod @ A @ B ),X2: 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 @ X2 @ Y ) @ ( set2 @ ( product_prod @ A @ B ) @ Xs ) )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Z2 ) @ ( set2 @ ( product_prod @ A @ B ) @ Xs ) )
=> ( Y = Z2 ) ) ) ) ).
% distinct_map_fstD
thf(fact_5233_integer__of__nat__numeral,axiom,
! [N: num] :
( ( code_integer_of_nat @ ( numeral_numeral @ nat @ N ) )
= ( numeral_numeral @ code_integer @ N ) ) ).
% integer_of_nat_numeral
thf(fact_5234_integer__of__nat__1,axiom,
( ( code_integer_of_nat @ ( one_one @ nat ) )
= ( one_one @ code_integer ) ) ).
% integer_of_nat_1
thf(fact_5235_map__fst__zip__take,axiom,
! [B: $tType,A: $tType,Xs: list @ A,Ys: list @ B] :
( ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) )
= ( take @ A @ ( ord_min @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ B ) @ Ys ) ) @ Xs ) ) ).
% map_fst_zip_take
thf(fact_5236_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_5237_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
@ ^ [K2: A,V4: B] :
( ( map_of @ A @ B @ Xs @ K2 )
= ( some @ B @ V4 ) ) ) ) ) ) ).
% set_map_of_compr
thf(fact_5238_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_5239_map__of__distinct__lookup,axiom,
! [A: $tType,B: $tType,X2: A,Xs: list @ ( product_prod @ A @ B ),Ys: list @ ( product_prod @ A @ B ),Y: B] :
( ~ ( member @ A @ X2 @ ( set2 @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xs ) ) )
=> ( ~ ( member @ A @ X2 @ ( set2 @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Ys ) ) )
=> ( ( map_of @ A @ B @ ( append @ ( product_prod @ A @ B ) @ Xs @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y ) @ Ys ) ) @ X2 )
= ( some @ B @ Y ) ) ) ) ).
% map_of_distinct_lookup
thf(fact_5240_min__list_Osimps,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X2: A,Xs: list @ A] :
( ( min_list @ A @ ( cons @ A @ X2 @ Xs ) )
= ( case_list @ A @ A @ X2
@ ^ [A3: A,List2: list @ A] : ( ord_min @ A @ X2 @ ( min_list @ A @ Xs ) )
@ Xs ) ) ) ).
% min_list.simps
thf(fact_5241_map__of__distinct__upd3,axiom,
! [A: $tType,B: $tType,X2: A,Xs: list @ ( product_prod @ A @ B ),Ys: list @ ( product_prod @ A @ B ),Y: B,Y9: B] :
( ~ ( member @ A @ X2 @ ( set2 @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xs ) ) )
=> ( ~ ( member @ A @ X2 @ ( set2 @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Ys ) ) )
=> ( ( map_of @ A @ B @ ( append @ ( product_prod @ A @ B ) @ Xs @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y ) @ Ys ) ) )
= ( fun_upd @ A @ ( option @ B ) @ ( map_of @ A @ B @ ( append @ ( product_prod @ A @ B ) @ Xs @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y9 ) @ Ys ) ) ) @ X2 @ ( some @ B @ Y ) ) ) ) ) ).
% map_of_distinct_upd3
thf(fact_5242_map__of__distinct__upd2,axiom,
! [A: $tType,B: $tType,X2: A,Xs: list @ ( product_prod @ A @ B ),Ys: list @ ( product_prod @ A @ B ),Y: B] :
( ~ ( member @ A @ X2 @ ( set2 @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xs ) ) )
=> ( ~ ( member @ A @ X2 @ ( set2 @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Ys ) ) )
=> ( ( map_of @ A @ B @ ( append @ ( product_prod @ A @ B ) @ Xs @ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y ) @ Ys ) ) )
= ( fun_upd @ A @ ( option @ B ) @ ( map_of @ A @ B @ ( append @ ( product_prod @ A @ B ) @ Xs @ Ys ) ) @ X2 @ ( some @ B @ Y ) ) ) ) ) ).
% map_of_distinct_upd2
thf(fact_5243_min__list_Opelims,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X2: list @ A,Y: A] :
( ( ( min_list @ A @ X2 )
= Y )
=> ( ( accp @ ( list @ A ) @ ( min_list_rel @ A ) @ X2 )
=> ( ! [X3: A,Xs2: list @ A] :
( ( X2
= ( cons @ A @ X3 @ Xs2 ) )
=> ( ( Y
= ( case_list @ A @ A @ X3
@ ^ [A3: A,List2: list @ A] : ( ord_min @ A @ X3 @ ( min_list @ A @ Xs2 ) )
@ Xs2 ) )
=> ~ ( accp @ ( list @ A ) @ ( min_list_rel @ A ) @ ( cons @ A @ X3 @ Xs2 ) ) ) )
=> ~ ( ( X2
= ( nil @ A ) )
=> ( ( Y
= ( undefined @ A ) )
=> ~ ( accp @ ( list @ A ) @ ( min_list_rel @ A ) @ ( nil @ A ) ) ) ) ) ) ) ) ).
% min_list.pelims
thf(fact_5244_bezw_Oelims,axiom,
! [X2: nat,Xa: nat,Y: product_prod @ int @ int] :
( ( ( bezw @ X2 @ 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 @ X2 @ Xa ) ) ) @ ( minus_minus @ int @ ( product_fst @ int @ int @ ( bezw @ Xa @ ( modulo_modulo @ nat @ X2 @ Xa ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ ( bezw @ Xa @ ( modulo_modulo @ nat @ X2 @ Xa ) ) ) @ ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ X2 @ Xa ) ) ) ) ) ) ) ) ) ).
% bezw.elims
thf(fact_5245_bezw_Osimps,axiom,
( bezw
= ( ^ [X: 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 @ X @ Y4 ) ) ) @ ( minus_minus @ int @ ( product_fst @ int @ int @ ( bezw @ Y4 @ ( modulo_modulo @ nat @ X @ Y4 ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ ( bezw @ Y4 @ ( modulo_modulo @ nat @ X @ Y4 ) ) ) @ ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ X @ Y4 ) ) ) ) ) ) ) ) ).
% bezw.simps
thf(fact_5246_img__snd,axiom,
! [B: $tType,A: $tType,A2: A,B2: B,S2: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ S2 )
=> ( member @ B @ B2 @ ( image2 @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ S2 ) ) ) ).
% img_snd
thf(fact_5247_map__snd__zip,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( map @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) )
= Ys ) ) ).
% map_snd_zip
thf(fact_5248_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_5249_one__mod__numeral,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [N: num] :
( ( modulo_modulo @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ N ) )
= ( product_snd @ A @ A @ ( unique8689654367752047608divmod @ A @ one2 @ N ) ) ) ) ).
% one_mod_numeral
thf(fact_5250_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 )
@ ^ [X: B] : ( product_Pair @ B @ A @ X @ K )
@ L ) )
= ( replicate @ A @ ( size_size @ ( list @ B ) @ L ) @ K ) ) ).
% map_snd_mk_snd
thf(fact_5251_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 )
@ ^ [X: B] : ( product_Pair @ B @ B @ X @ X )
@ ( product_snd @ A @ B ) ) )
= ( product_snd @ A @ B ) ) ).
% snd_diag_snd
thf(fact_5252_snd__def,axiom,
! [B: $tType,A: $tType] :
( ( product_snd @ A @ B )
= ( product_case_prod @ A @ B @ B
@ ^ [X12: A,X24: B] : X24 ) ) ).
% snd_def
thf(fact_5253_fn__snd__conv,axiom,
! [B: $tType,C: $tType,A: $tType,F: B > C] :
( ( ^ [X: product_prod @ A @ B] : ( F @ ( product_snd @ A @ B @ X ) ) )
= ( product_case_prod @ A @ B @ C
@ ^ [Uu: A] : F ) ) ).
% fn_snd_conv
thf(fact_5254_Ex__prod__contract,axiom,
! [B: $tType,A: $tType,P: A > B > $o] :
( ( ? [A3: A,X5: B] : ( P @ A3 @ X5 ) )
= ( ? [Z3: product_prod @ A @ B] : ( P @ ( product_fst @ A @ B @ Z3 ) @ ( product_snd @ A @ B @ Z3 ) ) ) ) ).
% Ex_prod_contract
thf(fact_5255_All__prod__contract,axiom,
! [B: $tType,A: $tType,P: A > B > $o] :
( ( ! [A3: A,X5: B] : ( P @ A3 @ X5 ) )
= ( ! [Z3: product_prod @ A @ B] : ( P @ ( product_fst @ A @ B @ Z3 ) @ ( product_snd @ A @ B @ Z3 ) ) ) ) ).
% All_prod_contract
thf(fact_5256_sndE,axiom,
! [A: $tType,B: $tType,X2: product_prod @ A @ B,A2: A,B2: B,P: B > $o] :
( ( X2
= ( product_Pair @ A @ B @ A2 @ B2 ) )
=> ( ( P @ ( product_snd @ A @ B @ X2 ) )
=> ( P @ B2 ) ) ) ).
% sndE
thf(fact_5257_measure__snd,axiom,
! [A: $tType,B: $tType,F: A > nat] :
( ( fun_is_measure @ A @ F )
=> ( fun_is_measure @ ( product_prod @ B @ A )
@ ^ [P7: product_prod @ B @ A] : ( F @ ( product_snd @ B @ A @ P7 ) ) ) ) ).
% measure_snd
thf(fact_5258_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [B: $tType,A: $tType,P: A > B > $o,X2: A,Y: B,A2: product_prod @ A @ B] :
( ( P @ X2 @ Y )
=> ( ( A2
= ( product_Pair @ A @ B @ X2 @ Y ) )
=> ( P @ ( product_fst @ A @ B @ A2 ) @ ( product_snd @ A @ B @ A2 ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_5259_in__snd__imageE,axiom,
! [A: $tType,B: $tType,Y: A,S2: set @ ( product_prod @ B @ A )] :
( ( member @ A @ Y @ ( image2 @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) @ S2 ) )
=> ~ ! [X3: B] :
~ ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X3 @ Y ) @ S2 ) ) ).
% in_snd_imageE
thf(fact_5260_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 )
@ ^ [X: A] : ( product_Pair @ A @ A @ X @ X )
@ ( product_fst @ A @ B ) ) )
= ( product_fst @ A @ B ) ) ).
% snd_diag_fst
thf(fact_5261_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 )
@ ^ [X: B] : ( product_Pair @ B @ B @ X @ X )
@ ( product_snd @ A @ B ) ) )
= ( product_snd @ A @ B ) ) ).
% fst_diag_snd
thf(fact_5262_exE__realizer,axiom,
! [C: $tType,A: $tType,B: $tType,P: A > B > $o,P3: product_prod @ B @ A,Q: C > $o,F: B > A > C] :
( ( P @ ( product_snd @ B @ A @ P3 ) @ ( product_fst @ B @ A @ P3 ) )
=> ( ! [X3: B,Y2: A] :
( ( P @ Y2 @ X3 )
=> ( Q @ ( F @ X3 @ Y2 ) ) )
=> ( Q @ ( product_case_prod @ B @ A @ C @ F @ P3 ) ) ) ) ).
% exE_realizer
thf(fact_5263_split__comp__eq,axiom,
! [A: $tType,C: $tType,B: $tType,D: $tType,F: A > B > C,G: D > A] :
( ( ^ [U2: product_prod @ D @ B] : ( F @ ( G @ ( product_fst @ D @ B @ U2 ) ) @ ( product_snd @ D @ B @ U2 ) ) )
= ( product_case_prod @ D @ B @ C
@ ^ [X: D] : ( F @ ( G @ X ) ) ) ) ).
% split_comp_eq
thf(fact_5264_case__prod__beta_H,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( product_case_prod @ A @ B @ C )
= ( ^ [F2: A > B > C,X: product_prod @ A @ B] : ( F2 @ ( product_fst @ A @ B @ X ) @ ( product_snd @ A @ B @ X ) ) ) ) ).
% case_prod_beta'
thf(fact_5265_case__prod__unfold,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( product_case_prod @ A @ B @ C )
= ( ^ [C5: A > B > C,P7: product_prod @ A @ B] : ( C5 @ ( product_fst @ A @ B @ P7 ) @ ( product_snd @ A @ B @ P7 ) ) ) ) ).
% case_prod_unfold
thf(fact_5266_divides__aux__def,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ( ( unique5940410009612947441es_aux @ A )
= ( ^ [Qr: product_prod @ A @ A] :
( ( product_snd @ A @ A @ Qr )
= ( zero_zero @ A ) ) ) ) ) ).
% divides_aux_def
thf(fact_5267_snd__image__mp,axiom,
! [B: $tType,A: $tType,A5: set @ ( product_prod @ B @ A ),B5: set @ A,X2: B,Y: A] :
( ( ord_less_eq @ ( set @ A ) @ ( image2 @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) @ A5 ) @ B5 )
=> ( ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X2 @ Y ) @ A5 )
=> ( member @ A @ Y @ B5 ) ) ) ).
% snd_image_mp
thf(fact_5268_quotient__of__denom__pos_H,axiom,
! [R: rat] : ( ord_less @ int @ ( zero_zero @ int ) @ ( product_snd @ int @ int @ ( quotient_of @ R ) ) ) ).
% quotient_of_denom_pos'
thf(fact_5269_snd__diag__id,axiom,
! [A: $tType,Z2: A] :
( ( comp @ ( product_prod @ A @ A ) @ A @ A @ ( product_snd @ A @ A )
@ ^ [X: A] : ( product_Pair @ A @ A @ X @ X )
@ Z2 )
= ( id @ A @ Z2 ) ) ).
% snd_diag_id
thf(fact_5270_case__prod__comp,axiom,
! [D: $tType,A: $tType,C: $tType,B: $tType,F: D > C > A,G: B > D,X2: product_prod @ B @ C] :
( ( product_case_prod @ B @ C @ A @ ( comp @ D @ ( C > A ) @ B @ F @ G ) @ X2 )
= ( F @ ( G @ ( product_fst @ B @ C @ X2 ) ) @ ( product_snd @ B @ C @ X2 ) ) ) ).
% case_prod_comp
thf(fact_5271_Eps__case__prod,axiom,
! [B: $tType,A: $tType,P: A > B > $o] :
( ( fChoice @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ P ) )
= ( fChoice @ ( product_prod @ A @ B )
@ ^ [Xy: product_prod @ A @ B] : ( P @ ( product_fst @ A @ B @ Xy ) @ ( product_snd @ A @ B @ Xy ) ) ) ) ).
% Eps_case_prod
thf(fact_5272_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_5273_zip__eq__conv,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B,Zs: list @ ( product_prod @ A @ B )] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( ( zip @ A @ B @ Xs @ Ys )
= Zs )
= ( ( ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Zs )
= Xs )
& ( ( map @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ Zs )
= Ys ) ) ) ) ).
% zip_eq_conv
thf(fact_5274_map__snd__zip__take,axiom,
! [B: $tType,A: $tType,Xs: list @ B,Ys: list @ A] :
( ( map @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) @ ( zip @ B @ A @ Xs @ Ys ) )
= ( take @ A @ ( ord_min @ nat @ ( size_size @ ( list @ B ) @ Xs ) @ ( size_size @ ( list @ A ) @ Ys ) ) @ Ys ) ) ).
% map_snd_zip_take
thf(fact_5275_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_5276_in__set__zip,axiom,
! [B: $tType,A: $tType,P3: product_prod @ A @ B,Xs: list @ A,Ys: list @ B] :
( ( member @ ( product_prod @ A @ B ) @ P3 @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) ) )
= ( ? [N4: nat] :
( ( ( nth @ A @ Xs @ N4 )
= ( product_fst @ A @ B @ P3 ) )
& ( ( nth @ B @ Ys @ N4 )
= ( product_snd @ A @ B @ P3 ) )
& ( ord_less @ nat @ N4 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( ord_less @ nat @ N4 @ ( size_size @ ( list @ B ) @ Ys ) ) ) ) ) ).
% in_set_zip
thf(fact_5277_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_5278_bezw__non__0,axiom,
! [Y: nat,X2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ Y )
=> ( ( bezw @ X2 @ Y )
= ( product_Pair @ int @ int @ ( product_snd @ int @ int @ ( bezw @ Y @ ( modulo_modulo @ nat @ X2 @ Y ) ) ) @ ( minus_minus @ int @ ( product_fst @ int @ int @ ( bezw @ Y @ ( modulo_modulo @ nat @ X2 @ Y ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ ( bezw @ Y @ ( modulo_modulo @ nat @ X2 @ Y ) ) ) @ ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ X2 @ Y ) ) ) ) ) ) ) ).
% bezw_non_0
thf(fact_5279_minus__one__mod__numeral,axiom,
! [N: num] :
( ( modulo_modulo @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( numeral_numeral @ int @ N ) )
= ( adjust_mod @ ( numeral_numeral @ int @ N ) @ ( product_snd @ int @ int @ ( unique8689654367752047608divmod @ int @ one2 @ N ) ) ) ) ).
% minus_one_mod_numeral
thf(fact_5280_one__mod__minus__numeral,axiom,
! [N: num] :
( ( modulo_modulo @ int @ ( one_one @ int ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
= ( uminus_uminus @ int @ ( adjust_mod @ ( numeral_numeral @ int @ N ) @ ( product_snd @ int @ int @ ( unique8689654367752047608divmod @ int @ one2 @ N ) ) ) ) ) ).
% one_mod_minus_numeral
thf(fact_5281_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 )
@ ^ [X: A,Y4: B] : ( product_Pair @ B @ A @ Y4 @ X ) ) ) ) ).
% fst_snd_flip
thf(fact_5282_snd__divmod__integer,axiom,
! [K: code_integer,L: code_integer] :
( ( product_snd @ code_integer @ code_integer @ ( code_divmod_integer @ K @ L ) )
= ( modulo_modulo @ code_integer @ K @ L ) ) ).
% snd_divmod_integer
thf(fact_5283_snd__divmod__abs,axiom,
! [K: code_integer,L: code_integer] :
( ( product_snd @ code_integer @ code_integer @ ( code_divmod_abs @ K @ L ) )
= ( modulo_modulo @ code_integer @ ( abs_abs @ code_integer @ K ) @ ( abs_abs @ code_integer @ L ) ) ) ).
% snd_divmod_abs
thf(fact_5284_Suc__0__mod__numeral,axiom,
! [K: num] :
( ( modulo_modulo @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( numeral_numeral @ nat @ K ) )
= ( product_snd @ nat @ nat @ ( unique8689654367752047608divmod @ nat @ one2 @ K ) ) ) ).
% Suc_0_mod_numeral
thf(fact_5285_effect__heapE,axiom,
! [A: $tType,F: ( heap_ext @ product_unit ) > ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ),H: heap_ext @ product_unit,H4: heap_ext @ product_unit,R: A,N: nat] :
( ( heap_Time_effect @ A @ ( heap_Time_heap @ A @ F ) @ H @ H4 @ R @ N )
=> ~ ( ( H4
= ( product_fst @ ( heap_ext @ product_unit ) @ nat @ ( product_snd @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( F @ H ) ) ) )
=> ( ( N
= ( product_snd @ ( heap_ext @ product_unit ) @ nat @ ( product_snd @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( F @ H ) ) ) )
=> ( R
!= ( product_fst @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( F @ H ) ) ) ) ) ) ).
% effect_heapE
thf(fact_5286_effect__heapI,axiom,
! [A: $tType,N: nat,F: ( heap_ext @ product_unit ) > ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ),H: heap_ext @ product_unit,H4: heap_ext @ product_unit,R: A] :
( ( N
= ( product_snd @ ( heap_ext @ product_unit ) @ nat @ ( product_snd @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( F @ H ) ) ) )
=> ( ( H4
= ( product_fst @ ( heap_ext @ product_unit ) @ nat @ ( product_snd @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( F @ H ) ) ) )
=> ( ( R
= ( product_fst @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( F @ H ) ) )
=> ( heap_Time_effect @ A @ ( heap_Time_heap @ A @ F ) @ H @ H4 @ R @ N ) ) ) ) ).
% effect_heapI
thf(fact_5287_effect__guardI,axiom,
! [A: $tType,P: ( heap_ext @ product_unit ) > $o,H: heap_ext @ product_unit,H4: heap_ext @ product_unit,F: ( heap_ext @ product_unit ) > ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ),N: nat,R: A] :
( ( P @ H )
=> ( ( H4
= ( product_fst @ ( heap_ext @ product_unit ) @ nat @ ( product_snd @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( F @ H ) ) ) )
=> ( ( N
= ( product_snd @ ( heap_ext @ product_unit ) @ nat @ ( product_snd @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( F @ H ) ) ) )
=> ( ( R
= ( product_fst @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( F @ H ) ) )
=> ( heap_Time_effect @ A @ ( heap_Time_guard @ A @ P @ F ) @ H @ H4 @ R @ N ) ) ) ) ) ).
% effect_guardI
thf(fact_5288_effect__guardE,axiom,
! [A: $tType,P: ( heap_ext @ product_unit ) > $o,F: ( heap_ext @ product_unit ) > ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ),H: heap_ext @ product_unit,H4: heap_ext @ product_unit,R: A,N: nat] :
( ( heap_Time_effect @ A @ ( heap_Time_guard @ A @ P @ F ) @ H @ H4 @ R @ N )
=> ~ ( ( H4
= ( product_fst @ ( heap_ext @ product_unit ) @ nat @ ( product_snd @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( F @ H ) ) ) )
=> ( ( N
= ( product_snd @ ( heap_ext @ product_unit ) @ nat @ ( product_snd @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( F @ H ) ) ) )
=> ( ( R
= ( product_fst @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( F @ H ) ) )
=> ~ ( P @ H ) ) ) ) ) ).
% effect_guardE
thf(fact_5289_effectE,axiom,
! [A: $tType,C2: heap_Time_Heap @ A,H: heap_ext @ product_unit,H4: heap_ext @ product_unit,R: A,N: nat] :
( ( heap_Time_effect @ A @ C2 @ H @ H4 @ R @ N )
=> ~ ( ( R
= ( 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 ) ) ) )
=> ( ( H4
= ( 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 ) ) ) ) )
=> ( ( N
= ( 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_5290_execute__bind__success,axiom,
! [B: $tType,A: $tType,F: heap_Time_Heap @ A,H: heap_ext @ product_unit,G: A > ( heap_Time_Heap @ B )] :
( ( heap_Time_success @ A @ F @ H )
=> ( ( heap_Time_execute @ B @ ( heap_Time_bind @ A @ B @ F @ G ) @ 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 @ F @ H ) ) ) ) @ ( heap_Time_execute @ B @ ( G @ ( 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 @ F @ 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 @ F @ H ) ) ) ) ) ) ) ) ).
% execute_bind_success
thf(fact_5291_nths__shift__lemma__Suc,axiom,
! [A: $tType,P: nat > $o,Xs: list @ A,Is: list @ nat] :
( ( map @ ( product_prod @ A @ nat ) @ A @ ( product_fst @ A @ nat )
@ ( filter2 @ ( product_prod @ A @ nat )
@ ^ [P7: product_prod @ A @ nat] : ( P @ ( suc @ ( product_snd @ A @ nat @ P7 ) ) )
@ ( zip @ A @ nat @ Xs @ Is ) ) )
= ( map @ ( product_prod @ A @ nat ) @ A @ ( product_fst @ A @ nat )
@ ( filter2 @ ( product_prod @ A @ nat )
@ ^ [P7: product_prod @ A @ nat] : ( P @ ( product_snd @ A @ nat @ P7 ) )
@ ( zip @ A @ nat @ Xs @ ( map @ nat @ nat @ suc @ Is ) ) ) ) ) ).
% nths_shift_lemma_Suc
thf(fact_5292_Divides_Oadjust__mod__def,axiom,
( adjust_mod
= ( ^ [L2: int,R2: int] :
( if @ int
@ ( R2
= ( zero_zero @ int ) )
@ ( zero_zero @ int )
@ ( minus_minus @ int @ L2 @ R2 ) ) ) ) ).
% Divides.adjust_mod_def
thf(fact_5293_nths__shift__lemma,axiom,
! [A: $tType,A5: set @ nat,Xs: list @ A,I: nat] :
( ( map @ ( product_prod @ A @ nat ) @ A @ ( product_fst @ A @ nat )
@ ( filter2 @ ( product_prod @ A @ nat )
@ ^ [P7: product_prod @ A @ nat] : ( member @ nat @ ( product_snd @ A @ nat @ P7 ) @ A5 )
@ ( zip @ A @ nat @ Xs @ ( upt @ I @ ( plus_plus @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) )
= ( map @ ( product_prod @ A @ nat ) @ A @ ( product_fst @ A @ nat )
@ ( filter2 @ ( product_prod @ A @ nat )
@ ^ [P7: product_prod @ A @ nat] : ( member @ nat @ ( plus_plus @ nat @ ( product_snd @ A @ nat @ P7 ) @ I ) @ A5 )
@ ( zip @ A @ nat @ Xs @ ( upt @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) ) ).
% nths_shift_lemma
thf(fact_5294_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 )
@ ^ [P7: product_prod @ A @ nat] : ( member @ nat @ ( product_snd @ A @ nat @ P7 ) @ A8 )
@ ( zip @ A @ nat @ Xs3 @ ( upt @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs3 ) ) ) ) ) ) ) ).
% nths_def
thf(fact_5295_in__set__enumerate__eq,axiom,
! [A: $tType,P3: product_prod @ nat @ A,N: nat,Xs: list @ A] :
( ( member @ ( product_prod @ nat @ A ) @ P3 @ ( set2 @ ( product_prod @ nat @ A ) @ ( enumerate @ A @ N @ Xs ) ) )
= ( ( ord_less_eq @ nat @ N @ ( product_fst @ nat @ A @ P3 ) )
& ( ord_less @ nat @ ( product_fst @ nat @ A @ P3 ) @ ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N ) )
& ( ( nth @ A @ Xs @ ( minus_minus @ nat @ ( product_fst @ nat @ A @ P3 ) @ N ) )
= ( product_snd @ nat @ A @ P3 ) ) ) ) ).
% in_set_enumerate_eq
thf(fact_5296_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 )
@ ^ [X: B,Y4: A] : ( product_Pair @ A @ B @ Y4 @ X ) ) ) ) ).
% snd_fst_flip
thf(fact_5297_size__prod__simp,axiom,
! [B: $tType,A: $tType] :
( ( basic_BNF_size_prod @ A @ B )
= ( ^ [F2: A > nat,G2: B > nat,P7: product_prod @ A @ B] : ( plus_plus @ nat @ ( plus_plus @ nat @ ( F2 @ ( product_fst @ A @ B @ P7 ) ) @ ( G2 @ ( product_snd @ A @ B @ P7 ) ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% size_prod_simp
thf(fact_5298_positive__def,axiom,
( positive
= ( map_fun @ rat @ ( product_prod @ int @ int ) @ $o @ $o @ rep_Rat @ ( id @ $o )
@ ^ [X: product_prod @ int @ int] : ( ord_less @ int @ ( zero_zero @ int ) @ ( times_times @ int @ ( product_fst @ int @ int @ X ) @ ( product_snd @ int @ int @ X ) ) ) ) ) ).
% positive_def
thf(fact_5299_bezw_Opelims,axiom,
! [X2: nat,Xa: nat,Y: product_prod @ int @ int] :
( ( ( bezw @ X2 @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ nat @ nat ) @ bezw_rel @ ( product_Pair @ nat @ nat @ X2 @ 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 @ X2 @ Xa ) ) ) @ ( minus_minus @ int @ ( product_fst @ int @ int @ ( bezw @ Xa @ ( modulo_modulo @ nat @ X2 @ Xa ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ ( bezw @ Xa @ ( modulo_modulo @ nat @ X2 @ Xa ) ) ) @ ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ X2 @ Xa ) ) ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ nat @ nat ) @ bezw_rel @ ( product_Pair @ nat @ nat @ X2 @ Xa ) ) ) ) ) ).
% bezw.pelims
thf(fact_5300_positive_Orep__eq,axiom,
( positive
= ( ^ [X: rat] : ( ord_less @ int @ ( zero_zero @ int ) @ ( times_times @ int @ ( product_fst @ int @ int @ ( rep_Rat @ X ) ) @ ( product_snd @ int @ int @ ( rep_Rat @ X ) ) ) ) ) ) ).
% positive.rep_eq
thf(fact_5301_prod__decode__aux_Opelims,axiom,
! [X2: nat,Xa: nat,Y: product_prod @ nat @ nat] :
( ( ( nat_prod_decode_aux @ X2 @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ nat @ nat ) @ nat_pr5047031295181774490ux_rel @ ( product_Pair @ nat @ nat @ X2 @ Xa ) )
=> ~ ( ( ( ( ord_less_eq @ nat @ Xa @ X2 )
=> ( Y
= ( product_Pair @ nat @ nat @ Xa @ ( minus_minus @ nat @ X2 @ Xa ) ) ) )
& ( ~ ( ord_less_eq @ nat @ Xa @ X2 )
=> ( Y
= ( nat_prod_decode_aux @ ( suc @ X2 ) @ ( minus_minus @ nat @ Xa @ ( suc @ X2 ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ nat @ nat ) @ nat_pr5047031295181774490ux_rel @ ( product_Pair @ nat @ nat @ X2 @ Xa ) ) ) ) ) ).
% prod_decode_aux.pelims
thf(fact_5302_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 )
@ ^ [X: product_prod @ int @ int,Y4: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( plus_plus @ int @ ( times_times @ int @ ( product_fst @ int @ int @ X ) @ ( product_snd @ int @ int @ Y4 ) ) @ ( times_times @ int @ ( product_fst @ int @ int @ Y4 ) @ ( product_snd @ int @ int @ X ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ X ) @ ( product_snd @ int @ int @ Y4 ) ) ) ) ) ).
% plus_rat_def
thf(fact_5303_inverse__rat__def,axiom,
( ( inverse_inverse @ rat )
= ( map_fun @ rat @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ rat @ rep_Rat @ abs_Rat
@ ^ [X: product_prod @ int @ int] :
( 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_def
thf(fact_5304_one__rat__def,axiom,
( ( one_one @ rat )
= ( abs_Rat @ ( product_Pair @ int @ int @ ( one_one @ int ) @ ( one_one @ int ) ) ) ) ).
% one_rat_def
thf(fact_5305_Fract_Oabs__eq,axiom,
( fract
= ( ^ [Xa5: int,X: int] :
( abs_Rat
@ ( if @ ( product_prod @ int @ int )
@ ( X
= ( zero_zero @ int ) )
@ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
@ ( product_Pair @ int @ int @ Xa5 @ X ) ) ) ) ) ).
% Fract.abs_eq
thf(fact_5306_zero__rat__def,axiom,
( ( zero_zero @ rat )
= ( abs_Rat @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) ) ) ) ).
% zero_rat_def
thf(fact_5307_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 )
@ ^ [A3: int,B3: int] :
( if @ ( product_prod @ int @ int )
@ ( B3
= ( zero_zero @ int ) )
@ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
@ ( product_Pair @ int @ int @ A3 @ B3 ) ) ) ) ).
% Fract_def
thf(fact_5308_uminus__rat__def,axiom,
( ( uminus_uminus @ rat )
= ( map_fun @ rat @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ rat @ rep_Rat @ abs_Rat
@ ^ [X: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( uminus_uminus @ int @ ( product_fst @ int @ int @ X ) ) @ ( product_snd @ int @ int @ X ) ) ) ) ).
% uminus_rat_def
thf(fact_5309_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 )
@ ^ [X: product_prod @ int @ int,Y4: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( times_times @ int @ ( product_fst @ int @ int @ X ) @ ( product_fst @ int @ int @ Y4 ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ X ) @ ( product_snd @ int @ int @ Y4 ) ) ) ) ) ).
% times_rat_def
thf(fact_5310_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_5311_normalize__def,axiom,
( normalize
= ( ^ [P7: product_prod @ int @ int] :
( if @ ( product_prod @ int @ int ) @ ( ord_less @ int @ ( zero_zero @ int ) @ ( product_snd @ int @ int @ P7 ) ) @ ( product_Pair @ int @ int @ ( divide_divide @ int @ ( product_fst @ int @ int @ P7 ) @ ( gcd_gcd @ int @ ( product_fst @ int @ int @ P7 ) @ ( product_snd @ int @ int @ P7 ) ) ) @ ( divide_divide @ int @ ( product_snd @ int @ int @ P7 ) @ ( gcd_gcd @ int @ ( product_fst @ int @ int @ P7 ) @ ( product_snd @ int @ int @ P7 ) ) ) )
@ ( if @ ( product_prod @ int @ int )
@ ( ( product_snd @ int @ int @ P7 )
= ( zero_zero @ int ) )
@ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
@ ( product_Pair @ int @ int @ ( divide_divide @ int @ ( product_fst @ int @ int @ P7 ) @ ( uminus_uminus @ int @ ( gcd_gcd @ int @ ( product_fst @ int @ int @ P7 ) @ ( product_snd @ int @ int @ P7 ) ) ) ) @ ( divide_divide @ int @ ( product_snd @ int @ int @ P7 ) @ ( uminus_uminus @ int @ ( gcd_gcd @ int @ ( product_fst @ int @ int @ P7 ) @ ( product_snd @ int @ int @ P7 ) ) ) ) ) ) ) ) ) ).
% normalize_def
thf(fact_5312_set__to__map__insert,axiom,
! [B: $tType,A: $tType,Kv: product_prod @ A @ B,S2: set @ ( product_prod @ A @ B )] :
( ~ ( member @ A @ ( product_fst @ A @ B @ Kv ) @ ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ S2 ) )
=> ( ( set_to_map @ A @ B @ ( insert @ ( product_prod @ A @ B ) @ Kv @ S2 ) )
= ( fun_upd @ A @ ( option @ B ) @ ( set_to_map @ A @ B @ S2 ) @ ( product_fst @ A @ B @ Kv ) @ ( some @ B @ ( product_snd @ A @ B @ Kv ) ) ) ) ) ).
% set_to_map_insert
thf(fact_5313_gcd__right__idem,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A2: A,B2: A] :
( ( gcd_gcd @ A @ ( gcd_gcd @ A @ A2 @ B2 ) @ B2 )
= ( gcd_gcd @ A @ A2 @ B2 ) ) ) ).
% gcd_right_idem
thf(fact_5314_gcd__left__idem,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A2: A,B2: A] :
( ( gcd_gcd @ A @ A2 @ ( gcd_gcd @ A @ A2 @ B2 ) )
= ( gcd_gcd @ A @ A2 @ B2 ) ) ) ).
% gcd_left_idem
thf(fact_5315_of__int__eq__id,axiom,
( ( ring_1_of_int @ int )
= ( id @ int ) ) ).
% of_int_eq_id
thf(fact_5316_gcd__eq__0__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A2: A,B2: A] :
( ( ( gcd_gcd @ A @ A2 @ B2 )
= ( zero_zero @ A ) )
= ( ( A2
= ( zero_zero @ A ) )
& ( B2
= ( zero_zero @ A ) ) ) ) ) ).
% gcd_eq_0_iff
thf(fact_5317_gcd__add1,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [M: A,N: A] :
( ( gcd_gcd @ A @ ( plus_plus @ A @ M @ N ) @ N )
= ( gcd_gcd @ A @ M @ N ) ) ) ).
% gcd_add1
thf(fact_5318_gcd__add2,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [M: A,N: A] :
( ( gcd_gcd @ A @ M @ ( plus_plus @ A @ M @ N ) )
= ( gcd_gcd @ A @ M @ N ) ) ) ).
% gcd_add2
thf(fact_5319_gcd_Obottom__left__bottom,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A2: A] :
( ( gcd_gcd @ A @ ( one_one @ A ) @ A2 )
= ( one_one @ A ) ) ) ).
% gcd.bottom_left_bottom
thf(fact_5320_gcd_Obottom__right__bottom,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A2: A] :
( ( gcd_gcd @ A @ A2 @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% gcd.bottom_right_bottom
thf(fact_5321_gcd__neg2,axiom,
! [A: $tType] :
( ( ring_gcd @ A )
=> ! [A2: A,B2: A] :
( ( gcd_gcd @ A @ A2 @ ( uminus_uminus @ A @ B2 ) )
= ( gcd_gcd @ A @ A2 @ B2 ) ) ) ).
% gcd_neg2
thf(fact_5322_gcd__neg1,axiom,
! [A: $tType] :
( ( ring_gcd @ A )
=> ! [A2: A,B2: A] :
( ( gcd_gcd @ A @ ( uminus_uminus @ A @ A2 ) @ B2 )
= ( gcd_gcd @ A @ A2 @ B2 ) ) ) ).
% gcd_neg1
thf(fact_5323_gcd__greatest__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A2 @ ( gcd_gcd @ A @ B2 @ C2 ) )
= ( ( dvd_dvd @ A @ A2 @ B2 )
& ( dvd_dvd @ A @ A2 @ C2 ) ) ) ) ).
% gcd_greatest_iff
thf(fact_5324_gcd__dvd2,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A2: A,B2: A] : ( dvd_dvd @ A @ ( gcd_gcd @ A @ A2 @ B2 ) @ B2 ) ) ).
% gcd_dvd2
thf(fact_5325_gcd__dvd1,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A2: A,B2: A] : ( dvd_dvd @ A @ ( gcd_gcd @ A @ A2 @ B2 ) @ A2 ) ) ).
% gcd_dvd1
thf(fact_5326_gcd__exp,axiom,
! [A: $tType] :
( ( semiri6843258321239162965malize @ A )
=> ! [A2: A,N: nat,B2: A] :
( ( gcd_gcd @ A @ ( power_power @ A @ A2 @ N ) @ ( power_power @ A @ B2 @ N ) )
= ( power_power @ A @ ( gcd_gcd @ A @ A2 @ B2 ) @ N ) ) ) ).
% gcd_exp
thf(fact_5327_gcd__1__int,axiom,
! [M: int] :
( ( gcd_gcd @ int @ M @ ( one_one @ int ) )
= ( one_one @ int ) ) ).
% gcd_1_int
thf(fact_5328_gcd__neg1__int,axiom,
! [X2: int,Y: int] :
( ( gcd_gcd @ int @ ( uminus_uminus @ int @ X2 ) @ Y )
= ( gcd_gcd @ int @ X2 @ Y ) ) ).
% gcd_neg1_int
thf(fact_5329_gcd__neg2__int,axiom,
! [X2: int,Y: int] :
( ( gcd_gcd @ int @ X2 @ ( uminus_uminus @ int @ Y ) )
= ( gcd_gcd @ int @ X2 @ Y ) ) ).
% gcd_neg2_int
thf(fact_5330_gcd__abs2__int,axiom,
! [X2: int,Y: int] :
( ( gcd_gcd @ int @ X2 @ ( abs_abs @ int @ Y ) )
= ( gcd_gcd @ int @ X2 @ Y ) ) ).
% gcd_abs2_int
thf(fact_5331_gcd__abs1__int,axiom,
! [X2: int,Y: int] :
( ( gcd_gcd @ int @ ( abs_abs @ int @ X2 ) @ Y )
= ( gcd_gcd @ int @ X2 @ Y ) ) ).
% gcd_abs1_int
thf(fact_5332_abs__gcd__int,axiom,
! [X2: int,Y: int] :
( ( abs_abs @ int @ ( gcd_gcd @ int @ X2 @ Y ) )
= ( gcd_gcd @ int @ X2 @ Y ) ) ).
% abs_gcd_int
thf(fact_5333_gcd__neg__numeral__1,axiom,
! [A: $tType] :
( ( ring_gcd @ A )
=> ! [N: num,A2: A] :
( ( gcd_gcd @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) @ A2 )
= ( gcd_gcd @ A @ ( numeral_numeral @ A @ N ) @ A2 ) ) ) ).
% gcd_neg_numeral_1
thf(fact_5334_gcd__neg__numeral__2,axiom,
! [A: $tType] :
( ( ring_gcd @ A )
=> ! [A2: A,N: num] :
( ( gcd_gcd @ A @ A2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( gcd_gcd @ A @ A2 @ ( numeral_numeral @ A @ N ) ) ) ) ).
% gcd_neg_numeral_2
thf(fact_5335_is__unit__gcd__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A2: A,B2: A] :
( ( dvd_dvd @ A @ ( gcd_gcd @ A @ A2 @ B2 ) @ ( one_one @ A ) )
= ( ( gcd_gcd @ A @ A2 @ B2 )
= ( one_one @ A ) ) ) ) ).
% is_unit_gcd_iff
thf(fact_5336_gcd__pos__int,axiom,
! [M: int,N: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ ( gcd_gcd @ int @ M @ N ) )
= ( ( M
!= ( zero_zero @ int ) )
| ( N
!= ( zero_zero @ int ) ) ) ) ).
% gcd_pos_int
thf(fact_5337_Gcd__insert,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A2: A,A5: set @ A] :
( ( gcd_Gcd @ A @ ( insert @ A @ A2 @ A5 ) )
= ( gcd_gcd @ A @ A2 @ ( gcd_Gcd @ A @ A5 ) ) ) ) ).
% Gcd_insert
thf(fact_5338_gcd__neg__numeral__2__int,axiom,
! [X2: int,N: num] :
( ( gcd_gcd @ int @ X2 @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
= ( gcd_gcd @ int @ X2 @ ( numeral_numeral @ int @ N ) ) ) ).
% gcd_neg_numeral_2_int
thf(fact_5339_gcd__neg__numeral__1__int,axiom,
! [N: num,X2: int] :
( ( gcd_gcd @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) @ X2 )
= ( gcd_gcd @ int @ ( numeral_numeral @ int @ N ) @ X2 ) ) ).
% gcd_neg_numeral_1_int
thf(fact_5340_gcd__0__int,axiom,
! [X2: int] :
( ( gcd_gcd @ int @ X2 @ ( zero_zero @ int ) )
= ( abs_abs @ int @ X2 ) ) ).
% gcd_0_int
thf(fact_5341_gcd__0__left__int,axiom,
! [X2: int] :
( ( gcd_gcd @ int @ ( zero_zero @ int ) @ X2 )
= ( abs_abs @ int @ X2 ) ) ).
% gcd_0_left_int
thf(fact_5342_gcd__proj2__if__dvd__int,axiom,
! [Y: int,X2: int] :
( ( dvd_dvd @ int @ Y @ X2 )
=> ( ( gcd_gcd @ int @ X2 @ Y )
= ( abs_abs @ int @ Y ) ) ) ).
% gcd_proj2_if_dvd_int
thf(fact_5343_gcd__proj1__if__dvd__int,axiom,
! [X2: int,Y: int] :
( ( dvd_dvd @ int @ X2 @ Y )
=> ( ( gcd_gcd @ int @ X2 @ Y )
= ( abs_abs @ int @ X2 ) ) ) ).
% gcd_proj1_if_dvd_int
thf(fact_5344_set__to__map__empty,axiom,
! [B: $tType,A: $tType] :
( ( set_to_map @ A @ B @ ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) )
= ( ^ [X: A] : ( none @ B ) ) ) ).
% set_to_map_empty
thf(fact_5345_Gcd__2,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A2: A,B2: A] :
( ( gcd_Gcd @ A @ ( insert @ A @ A2 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( gcd_gcd @ A @ A2 @ B2 ) ) ) ).
% Gcd_2
thf(fact_5346_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_5347_gcd__add__mult,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [M: A,K: A,N: A] :
( ( gcd_gcd @ A @ M @ ( plus_plus @ A @ ( times_times @ A @ K @ M ) @ N ) )
= ( gcd_gcd @ A @ M @ N ) ) ) ).
% gcd_add_mult
thf(fact_5348_gcd__idem__int,axiom,
! [X2: int] :
( ( gcd_gcd @ int @ X2 @ X2 )
= ( abs_abs @ int @ X2 ) ) ).
% gcd_idem_int
thf(fact_5349_gcd__dvd__antisym,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( dvd_dvd @ A @ ( gcd_gcd @ A @ A2 @ B2 ) @ ( gcd_gcd @ A @ C2 @ D2 ) )
=> ( ( dvd_dvd @ A @ ( gcd_gcd @ A @ C2 @ D2 ) @ ( gcd_gcd @ A @ A2 @ B2 ) )
=> ( ( gcd_gcd @ A @ A2 @ B2 )
= ( gcd_gcd @ A @ C2 @ D2 ) ) ) ) ) ).
% gcd_dvd_antisym
thf(fact_5350_gcd__greatest,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( dvd_dvd @ A @ C2 @ A2 )
=> ( ( dvd_dvd @ A @ C2 @ B2 )
=> ( dvd_dvd @ A @ C2 @ ( gcd_gcd @ A @ A2 @ B2 ) ) ) ) ) ).
% gcd_greatest
thf(fact_5351_gcd__dvdI2,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [B2: A,C2: A,A2: A] :
( ( dvd_dvd @ A @ B2 @ C2 )
=> ( dvd_dvd @ A @ ( gcd_gcd @ A @ A2 @ B2 ) @ C2 ) ) ) ).
% gcd_dvdI2
thf(fact_5352_gcd__dvdI1,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( dvd_dvd @ A @ A2 @ C2 )
=> ( dvd_dvd @ A @ ( gcd_gcd @ A @ A2 @ B2 ) @ C2 ) ) ) ).
% gcd_dvdI1
thf(fact_5353_dvd__gcdD2,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A2 @ ( gcd_gcd @ A @ B2 @ C2 ) )
=> ( dvd_dvd @ A @ A2 @ C2 ) ) ) ).
% dvd_gcdD2
thf(fact_5354_dvd__gcdD1,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A2 @ ( gcd_gcd @ A @ B2 @ C2 ) )
=> ( dvd_dvd @ A @ A2 @ B2 ) ) ) ).
% dvd_gcdD1
thf(fact_5355_gcd__mono,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A2: A,C2: A,B2: A,D2: A] :
( ( dvd_dvd @ A @ A2 @ C2 )
=> ( ( dvd_dvd @ A @ B2 @ D2 )
=> ( dvd_dvd @ A @ ( gcd_gcd @ A @ A2 @ B2 ) @ ( gcd_gcd @ A @ C2 @ D2 ) ) ) ) ) ).
% gcd_mono
thf(fact_5356_gcd__red__int,axiom,
( ( gcd_gcd @ int )
= ( ^ [X: int,Y4: int] : ( gcd_gcd @ int @ Y4 @ ( modulo_modulo @ int @ X @ Y4 ) ) ) ) ).
% gcd_red_int
thf(fact_5357_gcd_Oleft__commute,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( gcd_gcd @ A @ B2 @ ( gcd_gcd @ A @ A2 @ C2 ) )
= ( gcd_gcd @ A @ A2 @ ( gcd_gcd @ A @ B2 @ C2 ) ) ) ) ).
% gcd.left_commute
thf(fact_5358_gcd_Ocommute,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ( ( gcd_gcd @ A )
= ( ^ [A3: A,B3: A] : ( gcd_gcd @ A @ B3 @ A3 ) ) ) ) ).
% gcd.commute
thf(fact_5359_gcd_Oassoc,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( gcd_gcd @ A @ ( gcd_gcd @ A @ A2 @ B2 ) @ C2 )
= ( gcd_gcd @ A @ A2 @ ( gcd_gcd @ A @ B2 @ C2 ) ) ) ) ).
% gcd.assoc
thf(fact_5360_gcd__diff1,axiom,
! [A: $tType] :
( ( ring_gcd @ A )
=> ! [M: A,N: A] :
( ( gcd_gcd @ A @ ( minus_minus @ A @ M @ N ) @ N )
= ( gcd_gcd @ A @ M @ N ) ) ) ).
% gcd_diff1
thf(fact_5361_gcd__diff2,axiom,
! [A: $tType] :
( ( ring_gcd @ A )
=> ! [N: A,M: A] :
( ( gcd_gcd @ A @ ( minus_minus @ A @ N @ M ) @ N )
= ( gcd_gcd @ A @ M @ N ) ) ) ).
% gcd_diff2
thf(fact_5362_gcd__dvd__prod,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A2: A,B2: A,K: A] : ( dvd_dvd @ A @ ( gcd_gcd @ A @ A2 @ B2 ) @ ( times_times @ A @ K @ B2 ) ) ) ).
% gcd_dvd_prod
thf(fact_5363_gcd__ge__0__int,axiom,
! [X2: int,Y: int] : ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( gcd_gcd @ int @ X2 @ Y ) ) ).
% gcd_ge_0_int
thf(fact_5364_bezout__int,axiom,
! [X2: int,Y: int] :
? [U4: int,V5: int] :
( ( plus_plus @ int @ ( times_times @ int @ U4 @ X2 ) @ ( times_times @ int @ V5 @ Y ) )
= ( gcd_gcd @ int @ X2 @ Y ) ) ).
% bezout_int
thf(fact_5365_gcd__mult__distrib__int,axiom,
! [K: int,M: int,N: int] :
( ( times_times @ int @ ( abs_abs @ int @ K ) @ ( gcd_gcd @ int @ M @ N ) )
= ( gcd_gcd @ int @ ( times_times @ int @ K @ M ) @ ( times_times @ int @ K @ N ) ) ) ).
% gcd_mult_distrib_int
thf(fact_5366_gcd__mult__unit1,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
=> ( ( gcd_gcd @ A @ ( times_times @ A @ B2 @ A2 ) @ C2 )
= ( gcd_gcd @ A @ B2 @ C2 ) ) ) ) ).
% gcd_mult_unit1
thf(fact_5367_gcd__mult__unit2,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
=> ( ( gcd_gcd @ A @ B2 @ ( times_times @ A @ C2 @ A2 ) )
= ( gcd_gcd @ A @ B2 @ C2 ) ) ) ) ).
% gcd_mult_unit2
thf(fact_5368_mergesort__by__rel__split_Osimps_I3_J,axiom,
! [A: $tType,Xs12: list @ A,Xs23: list @ A,X13: A,X22: A,Xs: list @ A] :
( ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs12 @ Xs23 ) @ ( cons @ A @ X13 @ ( cons @ A @ X22 @ Xs ) ) )
= ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X13 @ Xs12 ) @ ( cons @ A @ X22 @ Xs23 ) ) @ Xs ) ) ).
% mergesort_by_rel_split.simps(3)
thf(fact_5369_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_5370_gcd__div__unit1,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
=> ( ( gcd_gcd @ A @ ( divide_divide @ A @ B2 @ A2 ) @ C2 )
= ( gcd_gcd @ A @ B2 @ C2 ) ) ) ) ).
% gcd_div_unit1
thf(fact_5371_gcd__div__unit2,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
=> ( ( gcd_gcd @ A @ B2 @ ( divide_divide @ A @ C2 @ A2 ) )
= ( gcd_gcd @ A @ B2 @ C2 ) ) ) ) ).
% gcd_div_unit2
thf(fact_5372_gcd__le2__int,axiom,
! [B2: int,A2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ord_less_eq @ int @ ( gcd_gcd @ int @ A2 @ B2 ) @ B2 ) ) ).
% gcd_le2_int
thf(fact_5373_gcd__le1__int,axiom,
! [A2: int,B2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ A2 )
=> ( ord_less_eq @ int @ ( gcd_gcd @ int @ A2 @ B2 ) @ A2 ) ) ).
% gcd_le1_int
thf(fact_5374_gcd__cases__int,axiom,
! [X2: int,Y: int,P: int > $o] :
( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X2 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y )
=> ( P @ ( gcd_gcd @ int @ X2 @ Y ) ) ) )
=> ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X2 )
=> ( ( ord_less_eq @ int @ Y @ ( zero_zero @ int ) )
=> ( P @ ( gcd_gcd @ int @ X2 @ ( uminus_uminus @ int @ Y ) ) ) ) )
=> ( ( ( ord_less_eq @ int @ X2 @ ( zero_zero @ int ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y )
=> ( P @ ( gcd_gcd @ int @ ( uminus_uminus @ int @ X2 ) @ Y ) ) ) )
=> ( ( ( ord_less_eq @ int @ X2 @ ( zero_zero @ int ) )
=> ( ( ord_less_eq @ int @ Y @ ( zero_zero @ int ) )
=> ( P @ ( gcd_gcd @ int @ ( uminus_uminus @ int @ X2 ) @ ( uminus_uminus @ int @ Y ) ) ) ) )
=> ( P @ ( gcd_gcd @ int @ X2 @ Y ) ) ) ) ) ) ).
% gcd_cases_int
thf(fact_5375_gcd__unique__int,axiom,
! [D2: int,A2: int,B2: int] :
( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ D2 )
& ( dvd_dvd @ int @ D2 @ A2 )
& ( dvd_dvd @ int @ D2 @ B2 )
& ! [E5: int] :
( ( ( dvd_dvd @ int @ E5 @ A2 )
& ( dvd_dvd @ int @ E5 @ B2 ) )
=> ( dvd_dvd @ int @ E5 @ D2 ) ) )
= ( D2
= ( gcd_gcd @ int @ A2 @ B2 ) ) ) ).
% gcd_unique_int
thf(fact_5376_gcd__non__0__int,axiom,
! [Y: int,X2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ Y )
=> ( ( gcd_gcd @ int @ X2 @ Y )
= ( gcd_gcd @ int @ Y @ ( modulo_modulo @ int @ X2 @ Y ) ) ) ) ).
% gcd_non_0_int
thf(fact_5377_gcd__code__int,axiom,
( ( gcd_gcd @ int )
= ( ^ [K2: int,L2: int] :
( abs_abs @ int
@ ( if @ int
@ ( L2
= ( zero_zero @ int ) )
@ K2
@ ( gcd_gcd @ int @ L2 @ ( modulo_modulo @ int @ ( abs_abs @ int @ K2 ) @ ( abs_abs @ int @ L2 ) ) ) ) ) ) ) ).
% gcd_code_int
thf(fact_5378_set__to__map__empty__iff_I1_J,axiom,
! [B: $tType,A: $tType,S2: set @ ( product_prod @ A @ B )] :
( ( ( set_to_map @ A @ B @ S2 )
= ( ^ [X: A] : ( none @ B ) ) )
= ( S2
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ).
% set_to_map_empty_iff(1)
thf(fact_5379_mergesort__by__rel__split_Osimps_I2_J,axiom,
! [A: $tType,Xs12: list @ A,Xs23: list @ A,X2: A] :
( ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs12 @ Xs23 ) @ ( cons @ A @ X2 @ ( nil @ A ) ) )
= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X2 @ Xs12 ) @ Xs23 ) ) ).
% mergesort_by_rel_split.simps(2)
thf(fact_5380_mergesort__by__rel__split_Oelims,axiom,
! [A: $tType,X2: product_prod @ ( list @ A ) @ ( list @ A ),Xa: list @ A,Y: product_prod @ ( list @ A ) @ ( list @ A )] :
( ( ( merges295452479951948502_split @ A @ X2 @ Xa )
= Y )
=> ( ! [Xs1: list @ A,Xs22: list @ A] :
( ( X2
= ( 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] :
( ( X2
= ( 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] :
( ( X2
= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs1 @ Xs22 ) )
=> ! [X1: A,X23: A,Xs2: list @ A] :
( ( Xa
= ( cons @ A @ X1 @ ( cons @ A @ X23 @ Xs2 ) ) )
=> ( Y
!= ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X1 @ Xs1 ) @ ( cons @ A @ X23 @ Xs22 ) ) @ Xs2 ) ) ) ) ) ) ) ).
% mergesort_by_rel_split.elims
thf(fact_5381_set__to__map__empty__iff_I2_J,axiom,
! [B: $tType,A: $tType,S2: set @ ( product_prod @ A @ B )] :
( ( ( ^ [X: A] : ( none @ B ) )
= ( set_to_map @ A @ B @ S2 ) )
= ( S2
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ).
% set_to_map_empty_iff(2)
thf(fact_5382_set__to__map__ran,axiom,
! [A: $tType,B: $tType,S2: set @ ( product_prod @ B @ A )] : ( ord_less_eq @ ( set @ A ) @ ( ran @ B @ A @ ( set_to_map @ B @ A @ S2 ) ) @ ( image2 @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) @ S2 ) ) ).
% set_to_map_ran
thf(fact_5383_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 )
@ ^ [X: product_prod @ int @ int] : ( divide_divide @ A @ ( ring_1_of_int @ A @ ( product_fst @ int @ int @ X ) ) @ ( ring_1_of_int @ A @ ( product_snd @ int @ int @ X ) ) ) ) ) ) ).
% of_rat_def
thf(fact_5384_Code__Numeral_Odup__def,axiom,
( code_dup
= ( map_fun @ code_integer @ int @ int @ code_integer @ code_int_of_integer @ code_integer_of_int
@ ^ [K2: int] : ( plus_plus @ int @ K2 @ K2 ) ) ) ).
% Code_Numeral.dup_def
thf(fact_5385_lexord__take__index__conv,axiom,
! [A: $tType,X2: list @ A,Y: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X2 @ Y ) @ ( lexord @ A @ R ) )
= ( ( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ X2 ) @ ( size_size @ ( list @ A ) @ Y ) )
& ( ( take @ A @ ( size_size @ ( list @ A ) @ X2 ) @ Y )
= X2 ) )
| ? [I2: nat] :
( ( ord_less @ nat @ I2 @ ( ord_min @ nat @ ( size_size @ ( list @ A ) @ X2 ) @ ( size_size @ ( list @ A ) @ Y ) ) )
& ( ( take @ A @ I2 @ X2 )
= ( take @ A @ I2 @ Y ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( nth @ A @ X2 @ I2 ) @ ( nth @ A @ Y @ I2 ) ) @ R ) ) ) ) ).
% lexord_take_index_conv
thf(fact_5386_gcd__nat_Oeq__neutr__iff,axiom,
! [A2: nat,B2: nat] :
( ( ( gcd_gcd @ nat @ A2 @ B2 )
= ( zero_zero @ nat ) )
= ( ( A2
= ( zero_zero @ nat ) )
& ( B2
= ( zero_zero @ nat ) ) ) ) ).
% gcd_nat.eq_neutr_iff
thf(fact_5387_gcd__nat_Oleft__neutral,axiom,
! [A2: nat] :
( ( gcd_gcd @ nat @ ( zero_zero @ nat ) @ A2 )
= A2 ) ).
% gcd_nat.left_neutral
thf(fact_5388_gcd__nat_Oneutr__eq__iff,axiom,
! [A2: nat,B2: nat] :
( ( ( zero_zero @ nat )
= ( gcd_gcd @ nat @ A2 @ B2 ) )
= ( ( A2
= ( zero_zero @ nat ) )
& ( B2
= ( zero_zero @ nat ) ) ) ) ).
% gcd_nat.neutr_eq_iff
thf(fact_5389_gcd__nat_Oright__neutral,axiom,
! [A2: nat] :
( ( gcd_gcd @ nat @ A2 @ ( zero_zero @ nat ) )
= A2 ) ).
% gcd_nat.right_neutral
thf(fact_5390_gcd__0__nat,axiom,
! [X2: nat] :
( ( gcd_gcd @ nat @ X2 @ ( zero_zero @ nat ) )
= X2 ) ).
% gcd_0_nat
thf(fact_5391_gcd__0__left__nat,axiom,
! [X2: nat] :
( ( gcd_gcd @ nat @ ( zero_zero @ nat ) @ X2 )
= X2 ) ).
% gcd_0_left_nat
thf(fact_5392_gcd__1__nat,axiom,
! [M: nat] :
( ( gcd_gcd @ nat @ M @ ( one_one @ nat ) )
= ( one_one @ nat ) ) ).
% gcd_1_nat
thf(fact_5393_gcd__nat_Oabsorb1,axiom,
! [A2: nat,B2: nat] :
( ( dvd_dvd @ nat @ A2 @ B2 )
=> ( ( gcd_gcd @ nat @ A2 @ B2 )
= A2 ) ) ).
% gcd_nat.absorb1
thf(fact_5394_gcd__nat_Oabsorb2,axiom,
! [B2: nat,A2: nat] :
( ( dvd_dvd @ nat @ B2 @ A2 )
=> ( ( gcd_gcd @ nat @ A2 @ B2 )
= B2 ) ) ).
% gcd_nat.absorb2
thf(fact_5395_gcd__nat_Obounded__iff,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( dvd_dvd @ nat @ A2 @ ( gcd_gcd @ nat @ B2 @ C2 ) )
= ( ( dvd_dvd @ nat @ A2 @ B2 )
& ( dvd_dvd @ nat @ A2 @ C2 ) ) ) ).
% gcd_nat.bounded_iff
thf(fact_5396_gcd__proj1__if__dvd__nat,axiom,
! [X2: nat,Y: nat] :
( ( dvd_dvd @ nat @ X2 @ Y )
=> ( ( gcd_gcd @ nat @ X2 @ Y )
= X2 ) ) ).
% gcd_proj1_if_dvd_nat
thf(fact_5397_gcd__proj2__if__dvd__nat,axiom,
! [Y: nat,X2: nat] :
( ( dvd_dvd @ nat @ Y @ X2 )
=> ( ( gcd_gcd @ nat @ X2 @ Y )
= Y ) ) ).
% gcd_proj2_if_dvd_nat
thf(fact_5398_gcd__Suc__0,axiom,
! [M: nat] :
( ( gcd_gcd @ nat @ M @ ( suc @ ( zero_zero @ nat ) ) )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% gcd_Suc_0
thf(fact_5399_gcd__pos__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( gcd_gcd @ nat @ M @ N ) )
= ( ( M
!= ( zero_zero @ nat ) )
| ( N
!= ( zero_zero @ nat ) ) ) ) ).
% gcd_pos_nat
thf(fact_5400_gcd__int__int__eq,axiom,
! [M: nat,N: nat] :
( ( gcd_gcd @ int @ ( semiring_1_of_nat @ int @ M ) @ ( semiring_1_of_nat @ int @ N ) )
= ( semiring_1_of_nat @ int @ ( gcd_gcd @ nat @ M @ N ) ) ) ).
% gcd_int_int_eq
thf(fact_5401_of__rat__0,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( ( field_char_0_of_rat @ A @ ( zero_zero @ rat ) )
= ( zero_zero @ A ) ) ) ).
% of_rat_0
thf(fact_5402_of__rat__eq__0__iff,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A2: rat] :
( ( ( field_char_0_of_rat @ A @ A2 )
= ( zero_zero @ A ) )
= ( A2
= ( zero_zero @ rat ) ) ) ) ).
% of_rat_eq_0_iff
thf(fact_5403_zero__eq__of__rat__iff,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A2: rat] :
( ( ( zero_zero @ A )
= ( field_char_0_of_rat @ A @ A2 ) )
= ( ( zero_zero @ rat )
= A2 ) ) ) ).
% zero_eq_of_rat_iff
thf(fact_5404_of__rat__1,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( ( field_char_0_of_rat @ A @ ( one_one @ rat ) )
= ( one_one @ A ) ) ) ).
% of_rat_1
thf(fact_5405_of__rat__eq__1__iff,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A2: rat] :
( ( ( field_char_0_of_rat @ A @ A2 )
= ( one_one @ A ) )
= ( A2
= ( one_one @ rat ) ) ) ) ).
% of_rat_eq_1_iff
thf(fact_5406_one__eq__of__rat__iff,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A2: rat] :
( ( ( one_one @ A )
= ( field_char_0_of_rat @ A @ A2 ) )
= ( ( one_one @ rat )
= A2 ) ) ) ).
% one_eq_of_rat_iff
thf(fact_5407_of__rat__of__nat__eq,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [N: nat] :
( ( field_char_0_of_rat @ A @ ( semiring_1_of_nat @ rat @ N ) )
= ( semiring_1_of_nat @ A @ N ) ) ) ).
% of_rat_of_nat_eq
thf(fact_5408_gcd__nat__abs__left__eq,axiom,
! [K: int,N: nat] :
( ( gcd_gcd @ nat @ ( nat2 @ ( abs_abs @ int @ K ) ) @ N )
= ( nat2 @ ( gcd_gcd @ int @ K @ ( semiring_1_of_nat @ int @ N ) ) ) ) ).
% gcd_nat_abs_left_eq
thf(fact_5409_gcd__nat__abs__right__eq,axiom,
! [N: nat,K: int] :
( ( gcd_gcd @ nat @ N @ ( nat2 @ ( abs_abs @ int @ K ) ) )
= ( nat2 @ ( gcd_gcd @ int @ ( semiring_1_of_nat @ int @ N ) @ K ) ) ) ).
% gcd_nat_abs_right_eq
thf(fact_5410_zero__less__of__rat__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [R: rat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( field_char_0_of_rat @ A @ R ) )
= ( ord_less @ rat @ ( zero_zero @ rat ) @ R ) ) ) ).
% zero_less_of_rat_iff
thf(fact_5411_of__rat__less__0__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [R: rat] :
( ( ord_less @ A @ ( field_char_0_of_rat @ A @ R ) @ ( zero_zero @ A ) )
= ( ord_less @ rat @ R @ ( zero_zero @ rat ) ) ) ) ).
% of_rat_less_0_iff
thf(fact_5412_of__rat__le__0__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [R: rat] :
( ( ord_less_eq @ A @ ( field_char_0_of_rat @ A @ R ) @ ( zero_zero @ A ) )
= ( ord_less_eq @ rat @ R @ ( zero_zero @ rat ) ) ) ) ).
% of_rat_le_0_iff
thf(fact_5413_zero__le__of__rat__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [R: rat] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( field_char_0_of_rat @ A @ R ) )
= ( ord_less_eq @ rat @ ( zero_zero @ rat ) @ R ) ) ) ).
% zero_le_of_rat_iff
thf(fact_5414_of__rat__neg__one,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( ( field_char_0_of_rat @ A @ ( uminus_uminus @ rat @ ( one_one @ rat ) ) )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% of_rat_neg_one
thf(fact_5415_one__less__of__rat__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [R: rat] :
( ( ord_less @ A @ ( one_one @ A ) @ ( field_char_0_of_rat @ A @ R ) )
= ( ord_less @ rat @ ( one_one @ rat ) @ R ) ) ) ).
% one_less_of_rat_iff
thf(fact_5416_of__rat__less__1__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [R: rat] :
( ( ord_less @ A @ ( field_char_0_of_rat @ A @ R ) @ ( one_one @ A ) )
= ( ord_less @ rat @ R @ ( one_one @ rat ) ) ) ) ).
% of_rat_less_1_iff
thf(fact_5417_of__rat__le__1__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [R: rat] :
( ( ord_less_eq @ A @ ( field_char_0_of_rat @ A @ R ) @ ( one_one @ A ) )
= ( ord_less_eq @ rat @ R @ ( one_one @ rat ) ) ) ) ).
% of_rat_le_1_iff
thf(fact_5418_one__le__of__rat__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [R: rat] :
( ( ord_less_eq @ A @ ( one_one @ A ) @ ( field_char_0_of_rat @ A @ R ) )
= ( ord_less_eq @ rat @ ( one_one @ rat ) @ R ) ) ) ).
% one_le_of_rat_iff
thf(fact_5419_gcd__mult__distrib__nat,axiom,
! [K: nat,M: nat,N: nat] :
( ( times_times @ nat @ K @ ( gcd_gcd @ nat @ M @ N ) )
= ( gcd_gcd @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N ) ) ) ).
% gcd_mult_distrib_nat
thf(fact_5420_gcd__nat_Omono,axiom,
! [A2: nat,C2: nat,B2: nat,D2: nat] :
( ( dvd_dvd @ nat @ A2 @ C2 )
=> ( ( dvd_dvd @ nat @ B2 @ D2 )
=> ( dvd_dvd @ nat @ ( gcd_gcd @ nat @ A2 @ B2 ) @ ( gcd_gcd @ nat @ C2 @ D2 ) ) ) ) ).
% gcd_nat.mono
thf(fact_5421_gcd__nat_OorderE,axiom,
! [A2: nat,B2: nat] :
( ( dvd_dvd @ nat @ A2 @ B2 )
=> ( A2
= ( gcd_gcd @ nat @ A2 @ B2 ) ) ) ).
% gcd_nat.orderE
thf(fact_5422_gcd__nat_OorderI,axiom,
! [A2: nat,B2: nat] :
( ( A2
= ( gcd_gcd @ nat @ A2 @ B2 ) )
=> ( dvd_dvd @ nat @ A2 @ B2 ) ) ).
% gcd_nat.orderI
thf(fact_5423_gcd__nat_Oabsorb3,axiom,
! [A2: nat,B2: nat] :
( ( ( dvd_dvd @ nat @ A2 @ B2 )
& ( A2 != B2 ) )
=> ( ( gcd_gcd @ nat @ A2 @ B2 )
= A2 ) ) ).
% gcd_nat.absorb3
thf(fact_5424_gcd__nat_Oabsorb4,axiom,
! [B2: nat,A2: nat] :
( ( ( dvd_dvd @ nat @ B2 @ A2 )
& ( B2 != A2 ) )
=> ( ( gcd_gcd @ nat @ A2 @ B2 )
= B2 ) ) ).
% gcd_nat.absorb4
thf(fact_5425_gcd__nat_OboundedE,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( dvd_dvd @ nat @ A2 @ ( gcd_gcd @ nat @ B2 @ C2 ) )
=> ~ ( ( dvd_dvd @ nat @ A2 @ B2 )
=> ~ ( dvd_dvd @ nat @ A2 @ C2 ) ) ) ).
% gcd_nat.boundedE
thf(fact_5426_gcd__nat_OboundedI,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( dvd_dvd @ nat @ A2 @ B2 )
=> ( ( dvd_dvd @ nat @ A2 @ C2 )
=> ( dvd_dvd @ nat @ A2 @ ( gcd_gcd @ nat @ B2 @ C2 ) ) ) ) ).
% gcd_nat.boundedI
thf(fact_5427_gcd__nat_Oorder__iff,axiom,
( ( dvd_dvd @ nat )
= ( ^ [A3: nat,B3: nat] :
( A3
= ( gcd_gcd @ nat @ A3 @ B3 ) ) ) ) ).
% gcd_nat.order_iff
thf(fact_5428_gcd__nat_Ocobounded1,axiom,
! [A2: nat,B2: nat] : ( dvd_dvd @ nat @ ( gcd_gcd @ nat @ A2 @ B2 ) @ A2 ) ).
% gcd_nat.cobounded1
thf(fact_5429_gcd__nat_Ocobounded2,axiom,
! [A2: nat,B2: nat] : ( dvd_dvd @ nat @ ( gcd_gcd @ nat @ A2 @ B2 ) @ B2 ) ).
% gcd_nat.cobounded2
thf(fact_5430_gcd__nat_Oabsorb__iff1,axiom,
( ( dvd_dvd @ nat )
= ( ^ [A3: nat,B3: nat] :
( ( gcd_gcd @ nat @ A3 @ B3 )
= A3 ) ) ) ).
% gcd_nat.absorb_iff1
thf(fact_5431_gcd__nat_Oabsorb__iff2,axiom,
( ( dvd_dvd @ nat )
= ( ^ [B3: nat,A3: nat] :
( ( gcd_gcd @ nat @ A3 @ B3 )
= B3 ) ) ) ).
% gcd_nat.absorb_iff2
thf(fact_5432_gcd__nat_OcoboundedI1,axiom,
! [A2: nat,C2: nat,B2: nat] :
( ( dvd_dvd @ nat @ A2 @ C2 )
=> ( dvd_dvd @ nat @ ( gcd_gcd @ nat @ A2 @ B2 ) @ C2 ) ) ).
% gcd_nat.coboundedI1
thf(fact_5433_gcd__nat_OcoboundedI2,axiom,
! [B2: nat,C2: nat,A2: nat] :
( ( dvd_dvd @ nat @ B2 @ C2 )
=> ( dvd_dvd @ nat @ ( gcd_gcd @ nat @ A2 @ B2 ) @ C2 ) ) ).
% gcd_nat.coboundedI2
thf(fact_5434_gcd__nat_Ostrict__boundedE,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( ( dvd_dvd @ nat @ A2 @ ( gcd_gcd @ nat @ B2 @ C2 ) )
& ( A2
!= ( gcd_gcd @ nat @ B2 @ C2 ) ) )
=> ~ ( ( ( dvd_dvd @ nat @ A2 @ B2 )
& ( A2 != B2 ) )
=> ~ ( ( dvd_dvd @ nat @ A2 @ C2 )
& ( A2 != C2 ) ) ) ) ).
% gcd_nat.strict_boundedE
thf(fact_5435_gcd__nat_Ostrict__order__iff,axiom,
! [A2: nat,B2: nat] :
( ( ( dvd_dvd @ nat @ A2 @ B2 )
& ( A2 != B2 ) )
= ( ( A2
= ( gcd_gcd @ nat @ A2 @ B2 ) )
& ( A2 != B2 ) ) ) ).
% gcd_nat.strict_order_iff
thf(fact_5436_gcd__nat_Ostrict__coboundedI1,axiom,
! [A2: nat,C2: nat,B2: nat] :
( ( ( dvd_dvd @ nat @ A2 @ C2 )
& ( A2 != C2 ) )
=> ( ( dvd_dvd @ nat @ ( gcd_gcd @ nat @ A2 @ B2 ) @ C2 )
& ( ( gcd_gcd @ nat @ A2 @ B2 )
!= C2 ) ) ) ).
% gcd_nat.strict_coboundedI1
thf(fact_5437_gcd__nat_Ostrict__coboundedI2,axiom,
! [B2: nat,C2: nat,A2: nat] :
( ( ( dvd_dvd @ nat @ B2 @ C2 )
& ( B2 != C2 ) )
=> ( ( dvd_dvd @ nat @ ( gcd_gcd @ nat @ A2 @ B2 ) @ C2 )
& ( ( gcd_gcd @ nat @ A2 @ B2 )
!= C2 ) ) ) ).
% gcd_nat.strict_coboundedI2
thf(fact_5438_gcd__unique__nat,axiom,
! [D2: nat,A2: nat,B2: nat] :
( ( ( dvd_dvd @ nat @ D2 @ A2 )
& ( dvd_dvd @ nat @ D2 @ B2 )
& ! [E5: nat] :
( ( ( dvd_dvd @ nat @ E5 @ A2 )
& ( dvd_dvd @ nat @ E5 @ B2 ) )
=> ( dvd_dvd @ nat @ E5 @ D2 ) ) )
= ( D2
= ( gcd_gcd @ nat @ A2 @ B2 ) ) ) ).
% gcd_unique_nat
thf(fact_5439_gcd__red__nat,axiom,
( ( gcd_gcd @ nat )
= ( ^ [X: nat,Y4: nat] : ( gcd_gcd @ nat @ Y4 @ ( modulo_modulo @ nat @ X @ Y4 ) ) ) ) ).
% gcd_red_nat
thf(fact_5440_of__rat__sum,axiom,
! [A: $tType,B: $tType] :
( ( field_char_0 @ A )
=> ! [F: B > rat,A5: set @ B] :
( ( field_char_0_of_rat @ A @ ( groups7311177749621191930dd_sum @ B @ rat @ F @ A5 ) )
= ( groups7311177749621191930dd_sum @ B @ A
@ ^ [A3: B] : ( field_char_0_of_rat @ A @ ( F @ A3 ) )
@ A5 ) ) ) ).
% of_rat_sum
thf(fact_5441_of__rat__prod,axiom,
! [A: $tType,B: $tType] :
( ( field_char_0 @ A )
=> ! [F: B > rat,A5: set @ B] :
( ( field_char_0_of_rat @ A @ ( groups7121269368397514597t_prod @ B @ rat @ F @ A5 ) )
= ( groups7121269368397514597t_prod @ B @ A
@ ^ [A3: B] : ( field_char_0_of_rat @ A @ ( F @ A3 ) )
@ A5 ) ) ) ).
% of_rat_prod
thf(fact_5442_gcd__le1__nat,axiom,
! [A2: nat,B2: nat] :
( ( A2
!= ( zero_zero @ nat ) )
=> ( ord_less_eq @ nat @ ( gcd_gcd @ nat @ A2 @ B2 ) @ A2 ) ) ).
% gcd_le1_nat
thf(fact_5443_gcd__le2__nat,axiom,
! [B2: nat,A2: nat] :
( ( B2
!= ( zero_zero @ nat ) )
=> ( ord_less_eq @ nat @ ( gcd_gcd @ nat @ A2 @ B2 ) @ B2 ) ) ).
% gcd_le2_nat
thf(fact_5444_gcd__diff2__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( gcd_gcd @ nat @ ( minus_minus @ nat @ N @ M ) @ N )
= ( gcd_gcd @ nat @ M @ N ) ) ) ).
% gcd_diff2_nat
thf(fact_5445_gcd__diff1__nat,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq @ nat @ N @ M )
=> ( ( gcd_gcd @ nat @ ( minus_minus @ nat @ M @ N ) @ N )
= ( gcd_gcd @ nat @ M @ N ) ) ) ).
% gcd_diff1_nat
thf(fact_5446_gcd__non__0__nat,axiom,
! [Y: nat,X2: nat] :
( ( Y
!= ( zero_zero @ nat ) )
=> ( ( gcd_gcd @ nat @ X2 @ Y )
= ( gcd_gcd @ nat @ Y @ ( modulo_modulo @ nat @ X2 @ Y ) ) ) ) ).
% gcd_non_0_nat
thf(fact_5447_gcd__nat_Osimps,axiom,
( ( gcd_gcd @ nat )
= ( ^ [X: nat,Y4: nat] :
( if @ nat
@ ( Y4
= ( zero_zero @ nat ) )
@ X
@ ( gcd_gcd @ nat @ Y4 @ ( modulo_modulo @ nat @ X @ Y4 ) ) ) ) ) ).
% gcd_nat.simps
thf(fact_5448_gcd__nat_Oelims,axiom,
! [X2: nat,Xa: nat,Y: nat] :
( ( ( gcd_gcd @ nat @ X2 @ Xa )
= Y )
=> ( ( ( Xa
= ( zero_zero @ nat ) )
=> ( Y = X2 ) )
& ( ( Xa
!= ( zero_zero @ nat ) )
=> ( Y
= ( gcd_gcd @ nat @ Xa @ ( modulo_modulo @ nat @ X2 @ Xa ) ) ) ) ) ) ).
% gcd_nat.elims
thf(fact_5449_gcd__integer_Orep__eq,axiom,
! [X2: code_integer,Xa: code_integer] :
( ( code_int_of_integer @ ( gcd_gcd @ code_integer @ X2 @ Xa ) )
= ( gcd_gcd @ int @ ( code_int_of_integer @ X2 ) @ ( code_int_of_integer @ Xa ) ) ) ).
% gcd_integer.rep_eq
thf(fact_5450_gcd__integer_Oabs__eq,axiom,
! [Xa: int,X2: int] :
( ( gcd_gcd @ code_integer @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X2 ) )
= ( code_integer_of_int @ ( gcd_gcd @ int @ Xa @ X2 ) ) ) ).
% gcd_integer.abs_eq
thf(fact_5451_Code__Numeral_Odup__code_I1_J,axiom,
( ( code_dup @ ( zero_zero @ code_integer ) )
= ( zero_zero @ code_integer ) ) ).
% Code_Numeral.dup_code(1)
thf(fact_5452_of__rat__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [R: rat,S3: rat] :
( ( ord_less_eq @ A @ ( field_char_0_of_rat @ A @ R ) @ ( field_char_0_of_rat @ A @ S3 ) )
= ( ord_less_eq @ rat @ R @ S3 ) ) ) ).
% of_rat_less_eq
thf(fact_5453_Gcd__in,axiom,
! [A5: set @ nat] :
( ! [A4: nat,B4: nat] :
( ( member @ nat @ A4 @ A5 )
=> ( ( member @ nat @ B4 @ A5 )
=> ( member @ nat @ ( gcd_gcd @ nat @ A4 @ B4 ) @ A5 ) ) )
=> ( ( A5
!= ( bot_bot @ ( set @ nat ) ) )
=> ( member @ nat @ ( gcd_Gcd @ nat @ A5 ) @ A5 ) ) ) ).
% Gcd_in
thf(fact_5454_of__rat__add,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A2: rat,B2: rat] :
( ( field_char_0_of_rat @ A @ ( plus_plus @ rat @ A2 @ B2 ) )
= ( plus_plus @ A @ ( field_char_0_of_rat @ A @ A2 ) @ ( field_char_0_of_rat @ A @ B2 ) ) ) ) ).
% of_rat_add
thf(fact_5455_bezout__nat,axiom,
! [A2: nat,B2: nat] :
( ( A2
!= ( zero_zero @ nat ) )
=> ? [X3: nat,Y2: nat] :
( ( times_times @ nat @ A2 @ X3 )
= ( plus_plus @ nat @ ( times_times @ nat @ B2 @ Y2 ) @ ( gcd_gcd @ nat @ A2 @ B2 ) ) ) ) ).
% bezout_nat
thf(fact_5456_bezout__gcd__nat_H,axiom,
! [B2: nat,A2: nat] :
? [X3: nat,Y2: nat] :
( ( ( ord_less_eq @ nat @ ( times_times @ nat @ B2 @ Y2 ) @ ( times_times @ nat @ A2 @ X3 ) )
& ( ( minus_minus @ nat @ ( times_times @ nat @ A2 @ X3 ) @ ( times_times @ nat @ B2 @ Y2 ) )
= ( gcd_gcd @ nat @ A2 @ B2 ) ) )
| ( ( ord_less_eq @ nat @ ( times_times @ nat @ A2 @ Y2 ) @ ( times_times @ nat @ B2 @ X3 ) )
& ( ( minus_minus @ nat @ ( times_times @ nat @ B2 @ X3 ) @ ( times_times @ nat @ A2 @ Y2 ) )
= ( gcd_gcd @ nat @ A2 @ B2 ) ) ) ) ).
% bezout_gcd_nat'
thf(fact_5457_gcd__code__integer,axiom,
( ( gcd_gcd @ code_integer )
= ( ^ [K2: code_integer,L2: code_integer] :
( abs_abs @ code_integer
@ ( if @ code_integer
@ ( L2
= ( zero_zero @ code_integer ) )
@ K2
@ ( gcd_gcd @ code_integer @ L2 @ ( modulo_modulo @ code_integer @ ( abs_abs @ code_integer @ K2 ) @ ( abs_abs @ code_integer @ L2 ) ) ) ) ) ) ) ).
% gcd_code_integer
thf(fact_5458_lexord__sufE,axiom,
! [A: $tType,Xs: list @ A,Zs: list @ A,Ys: list @ A,Qs: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs @ Zs ) @ ( append @ A @ Ys @ Qs ) ) @ ( lexord @ A @ R ) )
=> ( ( Xs != Ys )
=> ( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ A ) @ Ys ) )
=> ( ( ( size_size @ ( list @ A ) @ Zs )
= ( size_size @ ( list @ A ) @ Qs ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( lexord @ A @ R ) ) ) ) ) ) ).
% lexord_sufE
thf(fact_5459_lexord__lex,axiom,
! [A: $tType,X2: list @ A,Y: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X2 @ Y ) @ ( lex @ A @ R ) )
= ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X2 @ Y ) @ ( lexord @ A @ R ) )
& ( ( size_size @ ( list @ A ) @ X2 )
= ( size_size @ ( list @ A ) @ Y ) ) ) ) ).
% lexord_lex
thf(fact_5460_dup_Orep__eq,axiom,
! [X2: code_integer] :
( ( code_int_of_integer @ ( code_dup @ X2 ) )
= ( plus_plus @ int @ ( code_int_of_integer @ X2 ) @ ( code_int_of_integer @ X2 ) ) ) ).
% dup.rep_eq
thf(fact_5461_dup_Oabs__eq,axiom,
! [X2: int] :
( ( code_dup @ ( code_integer_of_int @ X2 ) )
= ( code_integer_of_int @ ( plus_plus @ int @ X2 @ X2 ) ) ) ).
% dup.abs_eq
thf(fact_5462_gcd__int__def,axiom,
( ( gcd_gcd @ int )
= ( ^ [X: int,Y4: int] : ( semiring_1_of_nat @ int @ ( gcd_gcd @ nat @ ( nat2 @ ( abs_abs @ int @ X ) ) @ ( nat2 @ ( abs_abs @ int @ Y4 ) ) ) ) ) ) ).
% gcd_int_def
thf(fact_5463_of__rat__rat,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [B2: int,A2: int] :
( ( B2
!= ( zero_zero @ int ) )
=> ( ( field_char_0_of_rat @ A @ ( fract @ A2 @ B2 ) )
= ( divide_divide @ A @ ( ring_1_of_int @ A @ A2 ) @ ( ring_1_of_int @ A @ B2 ) ) ) ) ) ).
% of_rat_rat
thf(fact_5464_gcd__nat_Osemilattice__neutr__order__axioms,axiom,
( semila1105856199041335345_order @ nat @ ( gcd_gcd @ nat ) @ ( zero_zero @ nat ) @ ( dvd_dvd @ nat )
@ ^ [M2: nat,N4: nat] :
( ( dvd_dvd @ nat @ M2 @ N4 )
& ( M2 != N4 ) ) ) ).
% gcd_nat.semilattice_neutr_order_axioms
thf(fact_5465_lexord__sufI,axiom,
! [A: $tType,U: list @ A,W2: list @ A,R: 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 @ R ) )
=> ( ( 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 @ R ) ) ) ) ).
% lexord_sufI
thf(fact_5466_gcd__integer__def,axiom,
( ( gcd_gcd @ code_integer )
= ( map_fun @ code_integer @ int @ ( int > int ) @ ( code_integer > code_integer ) @ code_int_of_integer @ ( map_fun @ code_integer @ int @ int @ code_integer @ code_int_of_integer @ code_integer_of_int ) @ ( gcd_gcd @ int ) ) ) ).
% gcd_integer_def
thf(fact_5467_bezw__aux,axiom,
! [X2: nat,Y: nat] :
( ( semiring_1_of_nat @ int @ ( gcd_gcd @ nat @ X2 @ Y ) )
= ( plus_plus @ int @ ( times_times @ int @ ( product_fst @ int @ int @ ( bezw @ X2 @ Y ) ) @ ( semiring_1_of_nat @ int @ X2 ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ ( bezw @ X2 @ Y ) ) @ ( semiring_1_of_nat @ int @ Y ) ) ) ) ).
% bezw_aux
thf(fact_5468_List_Olexordp__def,axiom,
! [A: $tType] :
( ( lexordp @ A )
= ( ^ [R2: A > A > $o,Xs3: list @ A,Ys3: list @ A] : ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs3 @ Ys3 ) @ ( lexord @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ R2 ) ) ) ) ) ) ).
% List.lexordp_def
thf(fact_5469_gcd__nat_Opelims,axiom,
! [X2: nat,Xa: nat,Y: nat] :
( ( ( gcd_gcd @ nat @ X2 @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ nat @ nat ) @ gcd_nat_rel @ ( product_Pair @ nat @ nat @ X2 @ Xa ) )
=> ~ ( ( ( ( Xa
= ( zero_zero @ nat ) )
=> ( Y = X2 ) )
& ( ( Xa
!= ( zero_zero @ nat ) )
=> ( Y
= ( gcd_gcd @ nat @ Xa @ ( modulo_modulo @ nat @ X2 @ Xa ) ) ) ) )
=> ~ ( accp @ ( product_prod @ nat @ nat ) @ gcd_nat_rel @ ( product_Pair @ nat @ nat @ X2 @ Xa ) ) ) ) ) ).
% gcd_nat.pelims
thf(fact_5470_plus__rat_Oabs__eq,axiom,
! [Xa: product_prod @ int @ int,X2: product_prod @ int @ int] :
( ( ratrel @ Xa @ Xa )
=> ( ( ratrel @ X2 @ X2 )
=> ( ( plus_plus @ rat @ ( abs_Rat @ Xa ) @ ( abs_Rat @ X2 ) )
= ( abs_Rat @ ( product_Pair @ int @ int @ ( plus_plus @ int @ ( times_times @ int @ ( product_fst @ int @ int @ Xa ) @ ( product_snd @ int @ int @ X2 ) ) @ ( times_times @ int @ ( product_fst @ int @ int @ X2 ) @ ( product_snd @ int @ int @ Xa ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ Xa ) @ ( product_snd @ int @ int @ X2 ) ) ) ) ) ) ) ).
% plus_rat.abs_eq
thf(fact_5471_gcd__idem__nat,axiom,
! [X2: nat] :
( ( gcd_gcd @ nat @ X2 @ X2 )
= X2 ) ).
% gcd_idem_nat
thf(fact_5472_gcd__nat_Oright__idem,axiom,
! [A2: nat,B2: nat] :
( ( gcd_gcd @ nat @ ( gcd_gcd @ nat @ A2 @ B2 ) @ B2 )
= ( gcd_gcd @ nat @ A2 @ B2 ) ) ).
% gcd_nat.right_idem
thf(fact_5473_gcd__nat_Oleft__idem,axiom,
! [A2: nat,B2: nat] :
( ( gcd_gcd @ nat @ A2 @ ( gcd_gcd @ nat @ A2 @ B2 ) )
= ( gcd_gcd @ nat @ A2 @ B2 ) ) ).
% gcd_nat.left_idem
thf(fact_5474_gcd__nat_Oidem,axiom,
! [A2: nat] :
( ( gcd_gcd @ nat @ A2 @ A2 )
= A2 ) ).
% gcd_nat.idem
thf(fact_5475_ratrel__iff,axiom,
( ratrel
= ( ^ [X: product_prod @ int @ int,Y4: product_prod @ int @ int] :
( ( ( product_snd @ int @ int @ X )
!= ( zero_zero @ int ) )
& ( ( product_snd @ int @ int @ Y4 )
!= ( zero_zero @ int ) )
& ( ( times_times @ int @ ( product_fst @ int @ int @ X ) @ ( product_snd @ int @ int @ Y4 ) )
= ( times_times @ int @ ( product_fst @ int @ int @ Y4 ) @ ( product_snd @ int @ int @ X ) ) ) ) ) ) ).
% ratrel_iff
thf(fact_5476_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_5477_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_5478_ratrel__def,axiom,
( ratrel
= ( ^ [X: product_prod @ int @ int,Y4: product_prod @ int @ int] :
( ( ( product_snd @ int @ int @ X )
!= ( zero_zero @ int ) )
& ( ( product_snd @ int @ int @ Y4 )
!= ( zero_zero @ int ) )
& ( ( times_times @ int @ ( product_fst @ int @ int @ X ) @ ( product_snd @ int @ int @ Y4 ) )
= ( times_times @ int @ ( product_fst @ int @ int @ Y4 ) @ ( product_snd @ int @ int @ X ) ) ) ) ) ) ).
% ratrel_def
thf(fact_5479_positive_Oabs__eq,axiom,
! [X2: product_prod @ int @ int] :
( ( ratrel @ X2 @ X2 )
=> ( ( positive @ ( abs_Rat @ X2 ) )
= ( ord_less @ int @ ( zero_zero @ int ) @ ( times_times @ int @ ( product_fst @ int @ int @ X2 ) @ ( product_snd @ int @ int @ X2 ) ) ) ) ) ).
% positive.abs_eq
thf(fact_5480_inverse__rat_Oabs__eq,axiom,
! [X2: product_prod @ int @ int] :
( ( ratrel @ X2 @ X2 )
=> ( ( inverse_inverse @ rat @ ( abs_Rat @ X2 ) )
= ( abs_Rat
@ ( if @ ( product_prod @ int @ int )
@ ( ( product_fst @ int @ int @ X2 )
= ( zero_zero @ int ) )
@ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
@ ( product_Pair @ int @ int @ ( product_snd @ int @ int @ X2 ) @ ( product_fst @ int @ int @ X2 ) ) ) ) ) ) ).
% inverse_rat.abs_eq
thf(fact_5481_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_5482_Code__Numeral_Osub__code_I9_J,axiom,
! [M: num,N: num] :
( ( code_sub @ ( bit0 @ M ) @ ( bit1 @ N ) )
= ( minus_minus @ code_integer @ ( code_dup @ ( code_sub @ M @ N ) ) @ ( one_one @ code_integer ) ) ) ).
% Code_Numeral.sub_code(9)
thf(fact_5483_Code__Numeral_Osub__code_I8_J,axiom,
! [M: num,N: num] :
( ( code_sub @ ( bit1 @ M ) @ ( bit0 @ N ) )
= ( plus_plus @ code_integer @ ( code_dup @ ( code_sub @ M @ N ) ) @ ( one_one @ code_integer ) ) ) ).
% Code_Numeral.sub_code(8)
thf(fact_5484_Code__Numeral_Osub__code_I1_J,axiom,
( ( code_sub @ one2 @ one2 )
= ( zero_zero @ code_integer ) ) ).
% Code_Numeral.sub_code(1)
thf(fact_5485_Code__Numeral_Osub__code_I6_J,axiom,
! [M: num,N: num] :
( ( code_sub @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( code_dup @ ( code_sub @ M @ N ) ) ) ).
% Code_Numeral.sub_code(6)
thf(fact_5486_Code__Numeral_Osub__code_I7_J,axiom,
! [M: num,N: num] :
( ( code_sub @ ( bit1 @ M ) @ ( bit1 @ N ) )
= ( code_dup @ ( code_sub @ M @ N ) ) ) ).
% Code_Numeral.sub_code(7)
thf(fact_5487_sub_Orep__eq,axiom,
! [X2: num,Xa: num] :
( ( code_int_of_integer @ ( code_sub @ X2 @ Xa ) )
= ( minus_minus @ int @ ( numeral_numeral @ int @ X2 ) @ ( numeral_numeral @ int @ Xa ) ) ) ).
% sub.rep_eq
thf(fact_5488_sub_Oabs__eq,axiom,
( code_sub
= ( ^ [Xa5: num,X: num] : ( code_integer_of_int @ ( minus_minus @ int @ ( numeral_numeral @ int @ Xa5 ) @ ( numeral_numeral @ int @ X ) ) ) ) ) ).
% sub.abs_eq
thf(fact_5489_Code__Numeral_Osub__def,axiom,
( code_sub
= ( map_fun @ num @ num @ ( num > int ) @ ( num > code_integer ) @ ( id @ num ) @ ( map_fun @ num @ num @ int @ code_integer @ ( id @ num ) @ code_integer_of_int )
@ ^ [M2: num,N4: num] : ( minus_minus @ int @ ( numeral_numeral @ int @ M2 ) @ ( numeral_numeral @ int @ N4 ) ) ) ) ).
% Code_Numeral.sub_def
thf(fact_5490_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_5491_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_5492_map__of__distinct__upd,axiom,
! [A: $tType,B: $tType,X2: A,Xs: list @ ( product_prod @ A @ B ),Y: B] :
( ~ ( member @ A @ X2 @ ( set2 @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xs ) ) )
=> ( ( map_add @ A @ B
@ ( fun_upd @ A @ ( option @ B )
@ ^ [X: A] : ( none @ B )
@ X2
@ ( some @ B @ Y ) )
@ ( map_of @ A @ B @ Xs ) )
= ( fun_upd @ A @ ( option @ B ) @ ( map_of @ A @ B @ Xs ) @ X2 @ ( some @ B @ Y ) ) ) ) ).
% map_of_distinct_upd
thf(fact_5493_empty__eq__map__add__iff,axiom,
! [B: $tType,A: $tType,F: A > ( option @ B ),G: A > ( option @ B )] :
( ( ( ^ [X: A] : ( none @ B ) )
= ( map_add @ A @ B @ F @ G ) )
= ( ( F
= ( ^ [X: A] : ( none @ B ) ) )
& ( G
= ( ^ [X: A] : ( none @ B ) ) ) ) ) ).
% empty_eq_map_add_iff
thf(fact_5494_map__add__empty,axiom,
! [B: $tType,A: $tType,M: A > ( option @ B )] :
( ( map_add @ A @ B @ M
@ ^ [X: A] : ( none @ B ) )
= M ) ).
% map_add_empty
thf(fact_5495_empty__map__add,axiom,
! [B: $tType,A: $tType,M: A > ( option @ B )] :
( ( map_add @ A @ B
@ ^ [X: A] : ( none @ B )
@ M )
= M ) ).
% empty_map_add
thf(fact_5496_map__to__set__empty,axiom,
! [B: $tType,A: $tType] :
( ( map_to_set @ A @ B
@ ^ [X: A] : ( none @ B ) )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% map_to_set_empty
thf(fact_5497_map__add__first__le,axiom,
! [B: $tType,A: $tType] :
( ( order @ B )
=> ! [M: A > ( option @ B ),M5: A > ( option @ B ),N: A > ( option @ B )] :
( ( ord_less_eq @ ( A > ( option @ B ) ) @ M @ M5 )
=> ( ord_less_eq @ ( A > ( option @ B ) ) @ ( map_add @ A @ B @ M @ N ) @ ( map_add @ A @ B @ M5 @ N ) ) ) ) ).
% map_add_first_le
thf(fact_5498_map__add__left__None,axiom,
! [A: $tType,B: $tType,F: B > ( option @ A ),K: B,G: B > ( option @ A )] :
( ( ( F @ K )
= ( none @ A ) )
=> ( ( map_add @ B @ A @ F @ G @ K )
= ( G @ K ) ) ) ).
% map_add_left_None
thf(fact_5499_map__add__find__left,axiom,
! [A: $tType,B: $tType,G: B > ( option @ A ),K: B,F: B > ( option @ A )] :
( ( ( G @ K )
= ( none @ A ) )
=> ( ( map_add @ B @ A @ F @ G @ K )
= ( F @ K ) ) ) ).
% map_add_find_left
thf(fact_5500_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_5501_map__add__def,axiom,
! [B: $tType,A: $tType] :
( ( map_add @ A @ B )
= ( ^ [M13: A > ( option @ B ),M24: A > ( option @ B ),X: A] : ( case_option @ ( option @ B ) @ B @ ( M13 @ X ) @ ( some @ B ) @ ( M24 @ X ) ) ) ) ).
% map_add_def
thf(fact_5502_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
= ( ^ [X: A] : ( none @ B ) ) ) ) ).
% map_to_set_empty_iff(2)
thf(fact_5503_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
= ( ^ [X: A] : ( none @ B ) ) ) ) ).
% map_to_set_empty_iff(1)
thf(fact_5504_map__to__set__def,axiom,
! [B: $tType,A: $tType] :
( ( map_to_set @ A @ B )
= ( ^ [M2: A > ( option @ B )] :
( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [K2: A,V4: B] :
( ( M2 @ K2 )
= ( some @ B @ V4 ) ) ) ) ) ) ).
% map_to_set_def
thf(fact_5505_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 ) )
@ ^ [K2: A,V4: B,M2: A > ( option @ B )] : ( fun_upd @ A @ ( option @ B ) @ M2 @ K2 @ ( some @ B @ V4 ) ) )
@ Ps
@ M ) ) ).
% map_add_map_of_foldr
thf(fact_5506_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 ),F2: A > ( option @ B )] : ( map_add @ A @ B @ F2 @ ( map_of @ A @ B @ Xs3 ) )
@ Xss
@ ^ [X: A] : ( none @ B ) ) ) ).
% map_of_concat
thf(fact_5507_map__to__set__ran,axiom,
! [A: $tType,B: $tType] :
( ( ran @ B @ A )
= ( ^ [M2: B > ( option @ A )] : ( image2 @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) @ ( map_to_set @ B @ A @ M2 ) ) ) ) ).
% map_to_set_ran
thf(fact_5508_cr__rat__def,axiom,
( cr_rat
= ( ^ [X: product_prod @ int @ int,Y4: rat] :
( ( ratrel @ X @ X )
& ( ( abs_Rat @ X )
= Y4 ) ) ) ) ).
% cr_rat_def
thf(fact_5509_set__zip,axiom,
! [B: $tType,A: $tType,Xs: list @ A,Ys: list @ B] :
( ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) )
= ( collect @ ( product_prod @ A @ B )
@ ^ [Uu: product_prod @ A @ B] :
? [I2: nat] :
( ( Uu
= ( product_Pair @ A @ B @ ( nth @ A @ Xs @ I2 ) @ ( nth @ B @ Ys @ I2 ) ) )
& ( ord_less @ nat @ I2 @ ( ord_min @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ B ) @ Ys ) ) ) ) ) ) ).
% set_zip
thf(fact_5510_mergesort__by__rel_Opinduct,axiom,
! [A: $tType,A0: A > A > $o,A13: 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 @ A13 ) )
=> ( ! [R5: A > A > $o,Xs2: list @ A] :
( ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( mergesort_by_rel_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ R5 @ Xs2 ) )
=> ( ( ~ ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
=> ( P @ R5 @ ( product_fst @ ( list @ A ) @ ( list @ A ) @ ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) ) @ Xs2 ) ) ) )
=> ( ( ~ ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
=> ( P @ R5 @ ( product_snd @ ( list @ A ) @ ( list @ A ) @ ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) ) @ Xs2 ) ) ) )
=> ( P @ R5 @ Xs2 ) ) ) )
=> ( P @ A0 @ A13 ) ) ) ).
% mergesort_by_rel.pinduct
thf(fact_5511_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
@ ^ [X: B] :
? [Y4: A] :
( ( P @ Y4 )
& ( Q @ X @ Y4 ) ) ) )
= ( ! [Y4: A] :
( ( P @ Y4 )
=> ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X: B] : ( Q @ X @ Y4 ) ) ) ) ) ) ) ).
% finite_Collect_bounded_ex
thf(fact_5512_eq__or__mem__image__simp,axiom,
! [B: $tType,A: $tType,F: B > A,A2: B,B5: set @ B] :
( ( collect @ A
@ ^ [Uu: A] :
? [L2: B] :
( ( Uu
= ( F @ L2 ) )
& ( ( L2 = A2 )
| ( member @ B @ L2 @ B5 ) ) ) )
= ( insert @ A @ ( F @ A2 )
@ ( collect @ A
@ ^ [Uu: A] :
? [L2: B] :
( ( Uu
= ( F @ L2 ) )
& ( member @ B @ L2 @ B5 ) ) ) ) ) ).
% eq_or_mem_image_simp
thf(fact_5513_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 )
@ ^ [Uu: product_prod @ A @ B] :
? [V6: B] :
( Uu
= ( product_Pair @ A @ B @ K @ V6 ) ) ) ) ) ) ).
% map_to_set_upd
thf(fact_5514_fs__contract,axiom,
! [B: $tType,C: $tType,A: $tType,F: A > B > C,S2: set @ C] :
( ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B )
@ ( collect @ ( product_prod @ A @ B )
@ ^ [Uu: product_prod @ A @ B] :
? [P7: product_prod @ A @ B] :
( ( Uu = P7 )
& ( member @ C @ ( F @ ( product_fst @ A @ B @ P7 ) @ ( product_snd @ A @ B @ P7 ) ) @ S2 ) ) ) )
= ( collect @ A
@ ^ [A3: A] :
? [B3: B] : ( member @ C @ ( F @ A3 @ B3 ) @ S2 ) ) ) ).
% fs_contract
thf(fact_5515_set__Cons__def,axiom,
! [A: $tType] :
( ( set_Cons @ A )
= ( ^ [A8: set @ A,XS: set @ ( list @ A )] :
( collect @ ( list @ A )
@ ^ [Z3: list @ A] :
? [X: A,Xs3: list @ A] :
( ( Z3
= ( cons @ A @ X @ Xs3 ) )
& ( member @ A @ X @ A8 )
& ( member @ ( list @ A ) @ Xs3 @ XS ) ) ) ) ) ).
% set_Cons_def
thf(fact_5516_finite__image__set,axiom,
! [A: $tType,B: $tType,P: A > $o,F: A > B] :
( ( finite_finite2 @ A @ ( collect @ A @ P ) )
=> ( finite_finite2 @ B
@ ( collect @ B
@ ^ [Uu: B] :
? [X: A] :
( ( Uu
= ( F @ X ) )
& ( P @ X ) ) ) ) ) ).
% finite_image_set
thf(fact_5517_finite__image__set2,axiom,
! [A: $tType,B: $tType,C: $tType,P: A > $o,Q: B > $o,F: A > B > C] :
( ( finite_finite2 @ A @ ( collect @ A @ P ) )
=> ( ( finite_finite2 @ B @ ( collect @ B @ Q ) )
=> ( finite_finite2 @ C
@ ( collect @ C
@ ^ [Uu: C] :
? [X: A,Y4: B] :
( ( Uu
= ( F @ X @ Y4 ) )
& ( P @ X )
& ( Q @ Y4 ) ) ) ) ) ) ).
% finite_image_set2
thf(fact_5518_setcompr__eq__image,axiom,
! [A: $tType,B: $tType,F: B > A,P: B > $o] :
( ( collect @ A
@ ^ [Uu: A] :
? [X: B] :
( ( Uu
= ( F @ X ) )
& ( P @ X ) ) )
= ( image2 @ B @ A @ F @ ( collect @ B @ P ) ) ) ).
% setcompr_eq_image
thf(fact_5519_Setcompr__eq__image,axiom,
! [A: $tType,B: $tType,F: B > A,A5: set @ B] :
( ( collect @ A
@ ^ [Uu: A] :
? [X: B] :
( ( Uu
= ( F @ X ) )
& ( member @ B @ X @ A5 ) ) )
= ( image2 @ B @ A @ F @ A5 ) ) ).
% Setcompr_eq_image
thf(fact_5520_exE__some,axiom,
! [A: $tType,P: A > $o,C2: A] :
( ? [X_1: A] : ( P @ X_1 )
=> ( ( C2
= ( fChoice @ A @ P ) )
=> ( P @ C2 ) ) ) ).
% exE_some
thf(fact_5521_mergesort__by__rel_Ocases,axiom,
! [A: $tType,X2: product_prod @ ( A > A > $o ) @ ( list @ A )] :
~ ! [R5: A > A > $o,Xs2: list @ A] :
( X2
!= ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ R5 @ Xs2 ) ) ).
% mergesort_by_rel.cases
thf(fact_5522_ran__def,axiom,
! [B: $tType,A: $tType] :
( ( ran @ A @ B )
= ( ^ [M2: A > ( option @ B )] :
( collect @ B
@ ^ [B3: B] :
? [A3: A] :
( ( M2 @ A3 )
= ( some @ B @ B3 ) ) ) ) ) ).
% ran_def
thf(fact_5523_some__theI,axiom,
! [A: $tType,B: $tType,P: A > B > $o] :
( ? [A9: A,X_1: B] : ( P @ A9 @ X_1 )
=> ( ! [B1: B,B22: B] :
( ? [A4: A] : ( P @ A4 @ B1 )
=> ( ? [A4: A] : ( P @ A4 @ B22 )
=> ( B1 = B22 ) ) )
=> ( P
@ ( fChoice @ A
@ ^ [A3: A] :
? [X5: B] : ( P @ A3 @ X5 ) )
@ ( the @ B
@ ^ [B3: B] :
? [A3: A] : ( P @ A3 @ B3 ) ) ) ) ) ).
% some_theI
thf(fact_5524_listrel1__def,axiom,
! [A: $tType] :
( ( listrel1 @ A )
= ( ^ [R2: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ ( list @ A ) @ ( list @ A ) )
@ ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
@ ^ [Xs3: list @ A,Ys3: list @ A] :
? [Us2: list @ A,Z3: A,Z7: A,Vs2: list @ A] :
( ( Xs3
= ( append @ A @ Us2 @ ( cons @ A @ Z3 @ Vs2 ) ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z3 @ Z7 ) @ R2 )
& ( Ys3
= ( append @ A @ Us2 @ ( cons @ A @ Z7 @ Vs2 ) ) ) ) ) ) ) ) ).
% listrel1_def
thf(fact_5525_lexord__def,axiom,
! [A: $tType] :
( ( lexord @ A )
= ( ^ [R2: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ ( list @ A ) @ ( list @ A ) )
@ ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
@ ^ [X: list @ A,Y4: list @ A] :
? [A3: A,V4: list @ A] :
( ( Y4
= ( append @ A @ X @ ( cons @ A @ A3 @ V4 ) ) )
| ? [U2: list @ A,B3: A,C5: A,W: list @ A,Z3: list @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B3 @ C5 ) @ R2 )
& ( X
= ( append @ A @ U2 @ ( cons @ A @ B3 @ W ) ) )
& ( Y4
= ( append @ A @ U2 @ ( cons @ A @ C5 @ Z3 ) ) ) ) ) ) ) ) ) ).
% lexord_def
thf(fact_5526_lex__conv,axiom,
! [A: $tType] :
( ( lex @ A )
= ( ^ [R2: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ ( list @ A ) @ ( list @ A ) )
@ ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
@ ^ [Xs3: list @ A,Ys3: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs3 )
= ( size_size @ ( list @ A ) @ Ys3 ) )
& ? [Xys2: list @ A,X: A,Y4: A,Xs6: list @ A,Ys6: list @ A] :
( ( Xs3
= ( append @ A @ Xys2 @ ( cons @ A @ X @ Xs6 ) ) )
& ( Ys3
= ( append @ A @ Xys2 @ ( cons @ A @ Y4 @ Ys6 ) ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y4 ) @ R2 ) ) ) ) ) ) ) ).
% lex_conv
thf(fact_5527_relcomp__unfold,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( relcomp @ A @ C @ B )
= ( ^ [R2: set @ ( product_prod @ A @ C ),S6: set @ ( product_prod @ C @ B )] :
( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [X: A,Z3: B] :
? [Y4: C] :
( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ X @ Y4 ) @ R2 )
& ( member @ ( product_prod @ C @ B ) @ ( product_Pair @ C @ B @ Y4 @ Z3 ) @ S6 ) ) ) ) ) ) ).
% relcomp_unfold
thf(fact_5528_set__map__filter,axiom,
! [B: $tType,A: $tType,G: B > ( option @ A ),Xs: list @ B] :
( ( set2 @ A @ ( map_filter @ B @ A @ G @ Xs ) )
= ( collect @ A
@ ^ [Y4: A] :
? [X: B] :
( ( member @ B @ X @ ( set2 @ B @ Xs ) )
& ( ( G @ X )
= ( some @ A @ Y4 ) ) ) ) ) ).
% set_map_filter
thf(fact_5529_set__conv__nth,axiom,
! [A: $tType] :
( ( set2 @ A )
= ( ^ [Xs3: list @ A] :
( collect @ A
@ ^ [Uu: A] :
? [I2: nat] :
( ( Uu
= ( nth @ A @ Xs3 @ I2 ) )
& ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs3 ) ) ) ) ) ) ).
% set_conv_nth
thf(fact_5530_set__nths,axiom,
! [A: $tType,Xs: list @ A,I5: set @ nat] :
( ( set2 @ A @ ( nths @ A @ Xs @ I5 ) )
= ( collect @ A
@ ^ [Uu: A] :
? [I2: nat] :
( ( Uu
= ( nth @ A @ Xs @ I2 ) )
& ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( member @ nat @ I2 @ I5 ) ) ) ) ).
% set_nths
thf(fact_5531_set__drop__conv,axiom,
! [A: $tType,N: nat,L: list @ A] :
( ( set2 @ A @ ( drop @ A @ N @ L ) )
= ( collect @ A
@ ^ [Uu: A] :
? [I2: nat] :
( ( Uu
= ( nth @ A @ L @ I2 ) )
& ( ord_less_eq @ nat @ N @ I2 )
& ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ L ) ) ) ) ) ).
% set_drop_conv
thf(fact_5532_arg__min__list_Opelims,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [X2: A > B,Xa: list @ A,Y: A] :
( ( ( arg_min_list @ A @ B @ X2 @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ ( A > B ) @ ( list @ A ) ) @ ( arg_min_list_rel @ A @ B ) @ ( product_Pair @ ( A > B ) @ ( list @ A ) @ X2 @ 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 ) @ X2 @ ( cons @ A @ X3 @ ( nil @ A ) ) ) ) ) )
=> ( ! [X3: A,Y2: A,Zs2: list @ A] :
( ( Xa
= ( cons @ A @ X3 @ ( cons @ A @ Y2 @ Zs2 ) ) )
=> ( ( Y
= ( if @ A @ ( ord_less_eq @ B @ ( X2 @ X3 ) @ ( X2 @ ( arg_min_list @ A @ B @ X2 @ ( cons @ A @ Y2 @ Zs2 ) ) ) ) @ X3 @ ( arg_min_list @ A @ B @ X2 @ ( cons @ A @ Y2 @ Zs2 ) ) ) )
=> ~ ( accp @ ( product_prod @ ( A > B ) @ ( list @ A ) ) @ ( arg_min_list_rel @ A @ B ) @ ( product_Pair @ ( A > B ) @ ( list @ A ) @ X2 @ ( cons @ A @ X3 @ ( cons @ A @ Y2 @ Zs2 ) ) ) ) ) )
=> ~ ( ( Xa
= ( nil @ A ) )
=> ( ( Y
= ( undefined @ A ) )
=> ~ ( accp @ ( product_prod @ ( A > B ) @ ( list @ A ) ) @ ( arg_min_list_rel @ A @ B ) @ ( product_Pair @ ( A > B ) @ ( list @ A ) @ X2 @ ( nil @ A ) ) ) ) ) ) ) ) ) ) ).
% arg_min_list.pelims
thf(fact_5533_lexn__conv,axiom,
! [A: $tType] :
( ( lexn @ A )
= ( ^ [R2: set @ ( product_prod @ A @ A ),N4: nat] :
( collect @ ( product_prod @ ( list @ A ) @ ( list @ A ) )
@ ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
@ ^ [Xs3: list @ A,Ys3: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs3 )
= N4 )
& ( ( size_size @ ( list @ A ) @ Ys3 )
= N4 )
& ? [Xys2: list @ A,X: A,Y4: A,Xs6: list @ A,Ys6: list @ A] :
( ( Xs3
= ( append @ A @ Xys2 @ ( cons @ A @ X @ Xs6 ) ) )
& ( Ys3
= ( append @ A @ Xys2 @ ( cons @ A @ Y4 @ Ys6 ) ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y4 ) @ R2 ) ) ) ) ) ) ) ).
% lexn_conv
thf(fact_5534_pairself__image__eq,axiom,
! [B: $tType,A: $tType,F: B > A,P: B > B > $o] :
( ( image2 @ ( product_prod @ B @ B ) @ ( product_prod @ A @ A ) @ ( pairself @ B @ A @ F ) @ ( collect @ ( product_prod @ B @ B ) @ ( product_case_prod @ B @ B @ $o @ P ) ) )
= ( collect @ ( product_prod @ A @ A )
@ ^ [Uu: product_prod @ A @ A] :
? [A3: B,B3: B] :
( ( Uu
= ( product_Pair @ A @ A @ ( F @ A3 ) @ ( F @ B3 ) ) )
& ( P @ A3 @ B3 ) ) ) ) ).
% pairself_image_eq
thf(fact_5535_lexn_Osimps_I1_J,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( lexn @ A @ R @ ( zero_zero @ nat ) )
= ( bot_bot @ ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ) ).
% lexn.simps(1)
thf(fact_5536_pairself_Osimps,axiom,
! [B: $tType,A: $tType,F: A > B,A2: A,B2: A] :
( ( pairself @ A @ B @ F @ ( product_Pair @ A @ A @ A2 @ B2 ) )
= ( product_Pair @ B @ B @ ( F @ A2 ) @ ( F @ B2 ) ) ) ).
% pairself.simps
thf(fact_5537_pairself_Oelims,axiom,
! [B: $tType,A: $tType,X2: A > B,Xa: product_prod @ A @ A,Y: product_prod @ B @ B] :
( ( ( pairself @ A @ B @ X2 @ Xa )
= Y )
=> ~ ! [A4: A,B4: A] :
( ( Xa
= ( product_Pair @ A @ A @ A4 @ B4 ) )
=> ( Y
!= ( product_Pair @ B @ B @ ( X2 @ A4 ) @ ( X2 @ B4 ) ) ) ) ) ).
% pairself.elims
thf(fact_5538_lexn__length,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,R: set @ ( product_prod @ A @ A ),N: nat] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( lexn @ A @ R @ N ) )
=> ( ( ( size_size @ ( list @ A ) @ Xs )
= N )
& ( ( size_size @ ( list @ A ) @ Ys )
= N ) ) ) ).
% lexn_length
thf(fact_5539_mergesort__by__rel_Opelims,axiom,
! [A: $tType,X2: A > A > $o,Xa: list @ A,Y: list @ A] :
( ( ( mergesort_by_rel @ A @ X2 @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( mergesort_by_rel_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X2 @ 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 @ X2 @ ( mergesort_by_rel @ A @ X2 @ ( product_fst @ ( list @ A ) @ ( list @ A ) @ ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) ) @ Xa ) ) ) @ ( mergesort_by_rel @ A @ X2 @ ( 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 ) @ X2 @ Xa ) ) ) ) ) ).
% mergesort_by_rel.pelims
thf(fact_5540_mergesort__by__rel_Opsimps,axiom,
! [A: $tType,R3: 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 ) @ R3 @ Xs ) )
=> ( ( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
=> ( ( mergesort_by_rel @ A @ R3 @ Xs )
= Xs ) )
& ( ~ ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
=> ( ( mergesort_by_rel @ A @ R3 @ Xs )
= ( merges9089515139780605204_merge @ A @ R3 @ ( mergesort_by_rel @ A @ R3 @ ( product_fst @ ( list @ A ) @ ( list @ A ) @ ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) ) @ Xs ) ) ) @ ( mergesort_by_rel @ A @ R3 @ ( 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_5541_Eps__Opt__def,axiom,
! [A: $tType] :
( ( eps_Opt @ A )
= ( ^ [P4: A > $o] :
( if @ ( option @ A )
@ ? [X5: A] : ( P4 @ X5 )
@ ( some @ A @ ( fChoice @ A @ P4 ) )
@ ( none @ A ) ) ) ) ).
% Eps_Opt_def
thf(fact_5542_mergesort__by__rel__simps_I1_J,axiom,
! [A: $tType,R3: A > A > $o] :
( ( mergesort_by_rel @ A @ R3 @ ( nil @ A ) )
= ( nil @ A ) ) ).
% mergesort_by_rel_simps(1)
thf(fact_5543_set__mergesort__by__rel,axiom,
! [A: $tType,R3: A > A > $o,Xs: list @ A] :
( ( set2 @ A @ ( mergesort_by_rel @ A @ R3 @ Xs ) )
= ( set2 @ A @ Xs ) ) ).
% set_mergesort_by_rel
thf(fact_5544_mergesort__by__rel__merge__simps_I3_J,axiom,
! [A: $tType,R3: A > A > $o,Ys: list @ A] :
( ( merges9089515139780605204_merge @ A @ R3 @ ( nil @ A ) @ Ys )
= Ys ) ).
% mergesort_by_rel_merge_simps(3)
thf(fact_5545_sorted__wrt__mergesort__by__rel__merge,axiom,
! [A: $tType,R3: A > A > $o,Xs: list @ A,Ys: list @ A] :
( ! [X3: A,Y2: A] :
( ( R3 @ X3 @ Y2 )
| ( R3 @ Y2 @ X3 ) )
=> ( ! [X3: A,Y2: A,Z4: A] :
( ( R3 @ X3 @ Y2 )
=> ( ( R3 @ Y2 @ Z4 )
=> ( R3 @ X3 @ Z4 ) ) )
=> ( ( sorted_wrt @ A @ R3 @ ( merges9089515139780605204_merge @ A @ R3 @ Xs @ Ys ) )
= ( ( sorted_wrt @ A @ R3 @ Xs )
& ( sorted_wrt @ A @ R3 @ Ys ) ) ) ) ) ).
% sorted_wrt_mergesort_by_rel_merge
thf(fact_5546_mergesort__by__rel__permutes,axiom,
! [A: $tType,R3: A > A > $o,Xs: list @ A] :
( ( mset @ A @ ( mergesort_by_rel @ A @ R3 @ Xs ) )
= ( mset @ A @ Xs ) ) ).
% mergesort_by_rel_permutes
thf(fact_5547_some__opt__sym__eq__trivial,axiom,
! [A: $tType,X2: A] :
( ( eps_Opt @ A
@ ( ^ [Y3: A,Z: A] : Y3 = Z
@ X2 ) )
= ( some @ A @ X2 ) ) ).
% some_opt_sym_eq_trivial
thf(fact_5548_some__opt__eq__trivial,axiom,
! [A: $tType,X2: A] :
( ( eps_Opt @ A
@ ^ [Y4: A] : Y4 = X2 )
= ( some @ A @ X2 ) ) ).
% some_opt_eq_trivial
thf(fact_5549_some__opt__false__trivial,axiom,
! [A: $tType] :
( ( eps_Opt @ A
@ ^ [Uu: A] : $false )
= ( none @ A ) ) ).
% some_opt_false_trivial
thf(fact_5550_mergesort__by__rel__simps_I2_J,axiom,
! [A: $tType,R3: A > A > $o,X2: A] :
( ( mergesort_by_rel @ A @ R3 @ ( cons @ A @ X2 @ ( nil @ A ) ) )
= ( cons @ A @ X2 @ ( nil @ A ) ) ) ).
% mergesort_by_rel_simps(2)
thf(fact_5551_mset__mergesort__by__rel__merge,axiom,
! [A: $tType,R3: A > A > $o,Xs: list @ A,Ys: list @ A] :
( ( mset @ A @ ( merges9089515139780605204_merge @ A @ R3 @ Xs @ Ys ) )
= ( plus_plus @ ( multiset @ A ) @ ( mset @ A @ Xs ) @ ( mset @ A @ Ys ) ) ) ).
% mset_mergesort_by_rel_merge
thf(fact_5552_Eps__Opt__eq__None,axiom,
! [A: $tType,P: A > $o] :
( ( ( eps_Opt @ A @ P )
= ( none @ A ) )
= ( ~ ? [X5: A] : ( P @ X5 ) ) ) ).
% Eps_Opt_eq_None
thf(fact_5553_mergesort__by__rel__simps_I3_J,axiom,
! [A: $tType,R3: A > A > $o,X13: A,X22: A,Xs: list @ A] :
( ( mergesort_by_rel @ A @ R3 @ ( cons @ A @ X13 @ ( cons @ A @ X22 @ Xs ) ) )
= ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ ( list @ A )
@ ^ [Xs13: list @ A,Xs24: list @ A] : ( merges9089515139780605204_merge @ A @ R3 @ ( mergesort_by_rel @ A @ R3 @ Xs13 ) @ ( mergesort_by_rel @ A @ R3 @ Xs24 ) )
@ ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X13 @ ( nil @ A ) ) @ ( cons @ A @ X22 @ ( nil @ A ) ) ) @ Xs ) ) ) ).
% mergesort_by_rel_simps(3)
thf(fact_5554_sorted__wrt__mergesort__by__rel,axiom,
! [X10: $tType,R3: X10 > X10 > $o,Xs: list @ X10] :
( ! [X3: X10,Y2: X10] :
( ( R3 @ X3 @ Y2 )
| ( R3 @ Y2 @ X3 ) )
=> ( ! [X3: X10,Y2: X10,Z4: X10] :
( ( R3 @ X3 @ Y2 )
=> ( ( R3 @ Y2 @ Z4 )
=> ( R3 @ X3 @ Z4 ) ) )
=> ( sorted_wrt @ X10 @ R3 @ ( mergesort_by_rel @ X10 @ R3 @ Xs ) ) ) ) ).
% sorted_wrt_mergesort_by_rel
thf(fact_5555_mergesort__by__rel__merge__simps_I2_J,axiom,
! [A: $tType,R3: A > A > $o,Xs: list @ A] :
( ( merges9089515139780605204_merge @ A @ R3 @ Xs @ ( nil @ A ) )
= Xs ) ).
% mergesort_by_rel_merge_simps(2)
thf(fact_5556_mergesort__by__rel__merge__simps_I1_J,axiom,
! [A: $tType,R3: A > A > $o,X2: A,Y: A,Xs: list @ A,Ys: list @ A] :
( ( ( R3 @ X2 @ Y )
=> ( ( merges9089515139780605204_merge @ A @ R3 @ ( cons @ A @ X2 @ Xs ) @ ( cons @ A @ Y @ Ys ) )
= ( cons @ A @ X2 @ ( merges9089515139780605204_merge @ A @ R3 @ Xs @ ( cons @ A @ Y @ Ys ) ) ) ) )
& ( ~ ( R3 @ X2 @ Y )
=> ( ( merges9089515139780605204_merge @ A @ R3 @ ( cons @ A @ X2 @ Xs ) @ ( cons @ A @ Y @ Ys ) )
= ( cons @ A @ Y @ ( merges9089515139780605204_merge @ A @ R3 @ ( cons @ A @ X2 @ Xs ) @ Ys ) ) ) ) ) ).
% mergesort_by_rel_merge_simps(1)
thf(fact_5557_Eps__Opt__eq__Some,axiom,
! [A: $tType,P: A > $o,X2: A] :
( ! [X11: A] :
( ( P @ X2 )
=> ( ( P @ X11 )
=> ( X11 = X2 ) ) )
=> ( ( ( eps_Opt @ A @ P )
= ( some @ A @ X2 ) )
= ( P @ X2 ) ) ) ).
% Eps_Opt_eq_Some
thf(fact_5558_Eps__Opt__eq__Some__implies,axiom,
! [A: $tType,P: A > $o,X2: A] :
( ( ( eps_Opt @ A @ P )
= ( some @ A @ X2 ) )
=> ( P @ X2 ) ) ).
% Eps_Opt_eq_Some_implies
thf(fact_5559_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_5560_mergesort__by__rel__merge_Oelims,axiom,
! [A: $tType,X2: A > A > $o,Xa: list @ A,Xb: list @ A,Y: list @ A] :
( ( ( merges9089515139780605204_merge @ A @ X2 @ Xa @ Xb )
= Y )
=> ( ! [X3: A,Xs2: list @ A] :
( ( Xa
= ( cons @ A @ X3 @ Xs2 ) )
=> ! [Y2: A,Ys4: list @ A] :
( ( Xb
= ( cons @ A @ Y2 @ Ys4 ) )
=> ~ ( ( ( X2 @ X3 @ Y2 )
=> ( Y
= ( cons @ A @ X3 @ ( merges9089515139780605204_merge @ A @ X2 @ Xs2 @ ( cons @ A @ Y2 @ Ys4 ) ) ) ) )
& ( ~ ( X2 @ X3 @ Y2 )
=> ( Y
= ( cons @ A @ Y2 @ ( merges9089515139780605204_merge @ A @ X2 @ ( cons @ A @ X3 @ Xs2 ) @ Ys4 ) ) ) ) ) ) )
=> ( ( ( Xb
= ( nil @ A ) )
=> ( Y != Xa ) )
=> ~ ( ( Xa
= ( nil @ A ) )
=> ! [V5: A,Va: list @ A] :
( ( Xb
= ( cons @ A @ V5 @ Va ) )
=> ( Y
!= ( cons @ A @ V5 @ Va ) ) ) ) ) ) ) ).
% mergesort_by_rel_merge.elims
thf(fact_5561_mergesort__by__rel__merge_Osimps_I3_J,axiom,
! [A: $tType,R3: A > A > $o,V2: A,Va2: list @ A] :
( ( merges9089515139780605204_merge @ A @ R3 @ ( nil @ A ) @ ( cons @ A @ V2 @ Va2 ) )
= ( cons @ A @ V2 @ Va2 ) ) ).
% mergesort_by_rel_merge.simps(3)
thf(fact_5562_sort__mergesort__by__rel,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( linorder_sort_key @ A @ A
@ ^ [X: A] : X )
= ( mergesort_by_rel @ A @ ( ord_less_eq @ A ) ) ) ) ).
% sort_mergesort_by_rel
thf(fact_5563_mergesort__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( mergesort @ A )
= ( mergesort_by_rel @ A @ ( ord_less_eq @ A ) ) ) ) ).
% mergesort_def
thf(fact_5564_set__to__map__def,axiom,
! [A: $tType,B: $tType] :
( ( set_to_map @ B @ A )
= ( ^ [S: set @ ( product_prod @ B @ A ),K2: B] :
( eps_Opt @ A
@ ^ [V4: A] : ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ K2 @ V4 ) @ S ) ) ) ) ).
% set_to_map_def
thf(fact_5565_mergesort__by__rel_Oelims,axiom,
! [A: $tType,X2: A > A > $o,Xa: list @ A,Y: list @ A] :
( ( ( mergesort_by_rel @ A @ X2 @ 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 @ X2 @ ( mergesort_by_rel @ A @ X2 @ ( product_fst @ ( list @ A ) @ ( list @ A ) @ ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) ) @ Xa ) ) ) @ ( mergesort_by_rel @ A @ X2 @ ( 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_5566_mergesort__by__rel_Osimps,axiom,
! [A: $tType] :
( ( mergesort_by_rel @ A )
= ( ^ [R7: 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 @ R7 @ ( mergesort_by_rel @ A @ R7 @ ( product_fst @ ( list @ A ) @ ( list @ A ) @ ( merges295452479951948502_split @ A @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) ) @ Xs3 ) ) ) @ ( mergesort_by_rel @ A @ R7 @ ( 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_5567_image2__def,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( bNF_Greatest_image2 @ C @ A @ B )
= ( ^ [A8: set @ C,F2: C > A,G2: C > B] :
( collect @ ( product_prod @ A @ B )
@ ^ [Uu: product_prod @ A @ B] :
? [A3: C] :
( ( Uu
= ( product_Pair @ A @ B @ ( F2 @ A3 ) @ ( G2 @ A3 ) ) )
& ( member @ C @ A3 @ A8 ) ) ) ) ) ).
% image2_def
thf(fact_5568_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
@ ^ [I2: A] : ( the2 @ B @ ( map_of @ A @ B @ Xs @ I2 ) )
@ ( 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_5569_finite__range__prod,axiom,
! [A: $tType,C: $tType,B: $tType,F: B > ( product_prod @ A @ C )] :
( ( finite_finite2 @ A @ ( image2 @ B @ A @ ( comp @ ( product_prod @ A @ C ) @ A @ B @ ( product_fst @ A @ C ) @ F ) @ ( top_top @ ( set @ B ) ) ) )
=> ( ( finite_finite2 @ C @ ( image2 @ B @ C @ ( comp @ ( product_prod @ A @ C ) @ C @ B @ ( product_snd @ A @ C ) @ F ) @ ( top_top @ ( set @ B ) ) ) )
=> ( finite_finite2 @ ( product_prod @ A @ C ) @ ( image2 @ B @ ( product_prod @ A @ C ) @ F @ ( top_top @ ( set @ B ) ) ) ) ) ) ).
% finite_range_prod
thf(fact_5570_top__apply,axiom,
! [C: $tType,D: $tType] :
( ( top @ C )
=> ( ( top_top @ ( D > C ) )
= ( ^ [X: D] : ( top_top @ C ) ) ) ) ).
% top_apply
thf(fact_5571_atMost__UNIV__triv,axiom,
! [A: $tType] :
( ( set_ord_atMost @ ( set @ A ) @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ ( set @ A ) ) ) ) ).
% atMost_UNIV_triv
thf(fact_5572_card__UNIV__unit,axiom,
( ( finite_card @ product_unit @ ( top_top @ ( set @ product_unit ) ) )
= ( one_one @ nat ) ) ).
% card_UNIV_unit
thf(fact_5573_image__mset_Oidentity,axiom,
! [A: $tType] :
( ( image_mset @ A @ A
@ ^ [X: A] : X )
= ( id @ ( multiset @ A ) ) ) ).
% image_mset.identity
thf(fact_5574_card__eq__UNIV2,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [S2: set @ A] :
( ( ( finite_card @ A @ ( top_top @ ( set @ A ) ) )
= ( finite_card @ A @ S2 ) )
= ( S2
= ( top_top @ ( set @ A ) ) ) ) ) ).
% card_eq_UNIV2
thf(fact_5575_card__eq__UNIV,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [S2: set @ A] :
( ( ( finite_card @ A @ S2 )
= ( finite_card @ A @ ( top_top @ ( set @ A ) ) ) )
= ( S2
= ( top_top @ ( set @ A ) ) ) ) ) ).
% card_eq_UNIV
thf(fact_5576_min__top,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [X2: A] :
( ( ord_min @ A @ ( top_top @ A ) @ X2 )
= X2 ) ) ).
% min_top
thf(fact_5577_min__top2,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [X2: A] :
( ( ord_min @ A @ X2 @ ( top_top @ A ) )
= X2 ) ) ).
% min_top2
thf(fact_5578_max__top,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [X2: A] :
( ( ord_max @ A @ ( top_top @ A ) @ X2 )
= ( top_top @ A ) ) ) ).
% max_top
thf(fact_5579_max__top2,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [X2: A] :
( ( ord_max @ A @ X2 @ ( top_top @ A ) )
= ( top_top @ A ) ) ) ).
% max_top2
thf(fact_5580_image__mset__union,axiom,
! [A: $tType,B: $tType,F: B > A,M3: multiset @ B,N5: multiset @ B] :
( ( image_mset @ B @ A @ F @ ( plus_plus @ ( multiset @ B ) @ M3 @ N5 ) )
= ( plus_plus @ ( multiset @ A ) @ ( image_mset @ B @ A @ F @ M3 ) @ ( image_mset @ B @ A @ F @ N5 ) ) ) ).
% image_mset_union
thf(fact_5581_restrict__map__UNIV,axiom,
! [B: $tType,A: $tType,F: A > ( option @ B )] :
( ( restrict_map @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
= F ) ).
% restrict_map_UNIV
thf(fact_5582_size__image__mset,axiom,
! [A: $tType,B: $tType,F: B > A,M3: multiset @ B] :
( ( size_size @ ( multiset @ A ) @ ( image_mset @ B @ A @ F @ M3 ) )
= ( size_size @ ( multiset @ B ) @ M3 ) ) ).
% size_image_mset
thf(fact_5583_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_5584_finite__Collect__not,axiom,
! [A: $tType,P: A > $o] :
( ( finite_finite2 @ A @ ( collect @ A @ P ) )
=> ( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X: A] :
~ ( P @ X ) ) )
= ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ) ).
% finite_Collect_not
thf(fact_5585_Collect__const__case__prod,axiom,
! [B: $tType,A: $tType,P: $o] :
( ( P
=> ( ( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [A3: A,B3: B] : P ) )
= ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) )
& ( ~ P
=> ( ( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [A3: A,B3: B] : P ) )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% Collect_const_case_prod
thf(fact_5586_surj__plus,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A2: A] :
( ( image2 @ A @ A @ ( plus_plus @ A @ A2 ) @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ) ).
% surj_plus
thf(fact_5587_card__ge__UNIV,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [S2: set @ A] :
( ( ord_less_eq @ nat @ ( finite_card @ A @ ( top_top @ ( set @ A ) ) ) @ ( finite_card @ A @ S2 ) )
= ( S2
= ( top_top @ ( set @ A ) ) ) ) ) ).
% card_ge_UNIV
thf(fact_5588_surj__fn,axiom,
! [A: $tType,F: A > A,N: nat] :
( ( ( image2 @ A @ A @ F @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) )
=> ( ( image2 @ A @ A @ ( compow @ ( A > A ) @ N @ F ) @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ) ).
% surj_fn
thf(fact_5589_Gcd__UNIV,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ( ( gcd_Gcd @ A @ ( top_top @ ( set @ A ) ) )
= ( one_one @ A ) ) ) ).
% Gcd_UNIV
thf(fact_5590_surj__diff__right,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A2: A] :
( ( image2 @ A @ A
@ ^ [X: A] : ( minus_minus @ A @ X @ A2 )
@ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ) ).
% surj_diff_right
thf(fact_5591_range__constant,axiom,
! [B: $tType,A: $tType,X2: A] :
( ( image2 @ B @ A
@ ^ [Uu: B] : X2
@ ( top_top @ ( set @ B ) ) )
= ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% range_constant
thf(fact_5592_atLeastAtMost__eq__UNIV__iff,axiom,
! [A: $tType] :
( ( bounded_lattice @ A )
=> ! [X2: A,Y: A] :
( ( ( set_or1337092689740270186AtMost @ A @ X2 @ Y )
= ( top_top @ ( set @ A ) ) )
= ( ( X2
= ( bot_bot @ A ) )
& ( Y
= ( top_top @ A ) ) ) ) ) ).
% atLeastAtMost_eq_UNIV_iff
thf(fact_5593_finite__fun__UNIVD1,axiom,
! [B: $tType,A: $tType] :
( ( finite_finite2 @ ( A > B ) @ ( top_top @ ( set @ ( A > B ) ) ) )
=> ( ( ( finite_card @ B @ ( top_top @ ( set @ B ) ) )
!= ( suc @ ( zero_zero @ nat ) ) )
=> ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ) ).
% finite_fun_UNIVD1
thf(fact_5594_mset__map__split__orig__le,axiom,
! [B: $tType,A: $tType,F: B > A,P: multiset @ B,M14: multiset @ A,M25: multiset @ A] :
( ( subseteq_mset @ A @ ( image_mset @ B @ A @ F @ P ) @ ( plus_plus @ ( multiset @ A ) @ M14 @ M25 ) )
=> ~ ! [P12: multiset @ B,P22: multiset @ B] :
( ( P
= ( plus_plus @ ( multiset @ B ) @ P12 @ P22 ) )
=> ( ( subseteq_mset @ A @ ( image_mset @ B @ A @ F @ P12 ) @ M14 )
=> ~ ( subseteq_mset @ A @ ( image_mset @ B @ A @ F @ P22 ) @ M25 ) ) ) ) ).
% mset_map_split_orig_le
thf(fact_5595_top_Oextremum__uniqueI,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( top_top @ A ) @ A2 )
=> ( A2
= ( top_top @ A ) ) ) ) ).
% top.extremum_uniqueI
thf(fact_5596_top_Oextremum__unique,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( top_top @ A ) @ A2 )
= ( A2
= ( top_top @ A ) ) ) ) ).
% top.extremum_unique
thf(fact_5597_top__greatest,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A2: A] : ( ord_less_eq @ A @ A2 @ ( top_top @ A ) ) ) ).
% top_greatest
thf(fact_5598_top_Oextremum__strict,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A2: A] :
~ ( ord_less @ A @ ( top_top @ A ) @ A2 ) ) ).
% top.extremum_strict
thf(fact_5599_top_Onot__eq__extremum,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A2: A] :
( ( A2
!= ( top_top @ A ) )
= ( ord_less @ A @ A2 @ ( top_top @ A ) ) ) ) ).
% top.not_eq_extremum
thf(fact_5600_infinite__UNIV__int,axiom,
~ ( finite_finite2 @ int @ ( top_top @ ( set @ int ) ) ) ).
% infinite_UNIV_int
thf(fact_5601_not__UNIV__eq__Iic,axiom,
! [A: $tType] :
( ( no_top @ A )
=> ! [H4: A] :
( ( top_top @ ( set @ A ) )
!= ( set_ord_atMost @ A @ H4 ) ) ) ).
% not_UNIV_eq_Iic
thf(fact_5602_atMost__eq__UNIV__iff,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [X2: A] :
( ( ( set_ord_atMost @ A @ X2 )
= ( top_top @ ( set @ A ) ) )
= ( X2
= ( top_top @ A ) ) ) ) ).
% atMost_eq_UNIV_iff
thf(fact_5603_mset__map__id,axiom,
! [B: $tType,A: $tType,F: B > A,G: A > B,X8: multiset @ A] :
( ! [X3: A] :
( ( F @ ( G @ X3 ) )
= X3 )
=> ( ( image_mset @ B @ A @ F @ ( image_mset @ A @ B @ G @ X8 ) )
= X8 ) ) ).
% mset_map_id
thf(fact_5604_not__UNIV__eq__Icc,axiom,
! [A: $tType] :
( ( no_top @ A )
=> ! [L4: A,H4: A] :
( ( top_top @ ( set @ A ) )
!= ( set_or1337092689740270186AtMost @ A @ L4 @ H4 ) ) ) ).
% not_UNIV_eq_Icc
thf(fact_5605_integer__of__int__cases,axiom,
! [X2: code_integer] :
~ ! [Y2: int] :
( ( X2
= ( code_integer_of_int @ Y2 ) )
=> ~ ( member @ int @ Y2 @ ( top_top @ ( set @ int ) ) ) ) ).
% integer_of_int_cases
thf(fact_5606_integer__of__int__induct,axiom,
! [P: code_integer > $o,X2: code_integer] :
( ! [Y2: int] :
( ( member @ int @ Y2 @ ( top_top @ ( set @ int ) ) )
=> ( P @ ( code_integer_of_int @ Y2 ) ) )
=> ( P @ X2 ) ) ).
% integer_of_int_induct
thf(fact_5607_integer__of__int__inject,axiom,
! [X2: int,Y: int] :
( ( member @ int @ X2 @ ( top_top @ ( set @ int ) ) )
=> ( ( member @ int @ Y @ ( top_top @ ( set @ int ) ) )
=> ( ( ( code_integer_of_int @ X2 )
= ( code_integer_of_int @ Y ) )
= ( X2 = Y ) ) ) ) ).
% integer_of_int_inject
thf(fact_5608_int__of__integer,axiom,
! [X2: code_integer] : ( member @ int @ ( code_int_of_integer @ X2 ) @ ( top_top @ ( set @ int ) ) ) ).
% int_of_integer
thf(fact_5609_int__of__integer__cases,axiom,
! [Y: int] :
( ( member @ int @ Y @ ( top_top @ ( set @ int ) ) )
=> ~ ! [X3: code_integer] :
( Y
!= ( code_int_of_integer @ X3 ) ) ) ).
% int_of_integer_cases
thf(fact_5610_int__of__integer__induct,axiom,
! [Y: int,P: int > $o] :
( ( member @ int @ Y @ ( top_top @ ( set @ int ) ) )
=> ( ! [X3: code_integer] : ( P @ ( code_int_of_integer @ X3 ) )
=> ( P @ Y ) ) ) ).
% int_of_integer_induct
thf(fact_5611_mset__map__split__orig,axiom,
! [B: $tType,A: $tType,F: B > A,P: multiset @ B,M14: multiset @ A,M25: multiset @ A] :
( ( ( image_mset @ B @ A @ F @ P )
= ( plus_plus @ ( multiset @ A ) @ M14 @ M25 ) )
=> ~ ! [P12: multiset @ B,P22: multiset @ B] :
( ( P
= ( plus_plus @ ( multiset @ B ) @ P12 @ P22 ) )
=> ( ( ( image_mset @ B @ A @ F @ P12 )
= M14 )
=> ( ( image_mset @ B @ A @ F @ P22 )
!= M25 ) ) ) ) ).
% mset_map_split_orig
thf(fact_5612_UNIV__def,axiom,
! [A: $tType] :
( ( top_top @ ( set @ A ) )
= ( collect @ A
@ ^ [X: A] : $true ) ) ).
% UNIV_def
thf(fact_5613_multiset_Omap__ident,axiom,
! [A: $tType,T2: multiset @ A] :
( ( image_mset @ A @ A
@ ^ [X: A] : X
@ T2 )
= T2 ) ).
% multiset.map_ident
thf(fact_5614_rangeE,axiom,
! [A: $tType,B: $tType,B2: A,F: B > A] :
( ( member @ A @ B2 @ ( image2 @ B @ A @ F @ ( top_top @ ( set @ B ) ) ) )
=> ~ ! [X3: B] :
( B2
!= ( F @ X3 ) ) ) ).
% rangeE
thf(fact_5615_range__composition,axiom,
! [A: $tType,C: $tType,B: $tType,F: C > A,G: B > C] :
( ( image2 @ B @ A
@ ^ [X: B] : ( F @ ( G @ X ) )
@ ( top_top @ ( set @ B ) ) )
= ( image2 @ C @ A @ F @ ( image2 @ B @ C @ G @ ( top_top @ ( set @ B ) ) ) ) ) ).
% range_composition
thf(fact_5616_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_5617_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_5618_integer__of__int__inverse,axiom,
! [Y: int] :
( ( member @ int @ Y @ ( top_top @ ( set @ int ) ) )
=> ( ( code_int_of_integer @ ( code_integer_of_int @ Y ) )
= Y ) ) ).
% integer_of_int_inverse
thf(fact_5619_finite__range__imageI,axiom,
! [C: $tType,A: $tType,B: $tType,G: B > A,F: A > C] :
( ( finite_finite2 @ A @ ( image2 @ B @ A @ G @ ( top_top @ ( set @ B ) ) ) )
=> ( finite_finite2 @ C
@ ( image2 @ B @ C
@ ^ [X: B] : ( F @ ( G @ X ) )
@ ( top_top @ ( set @ B ) ) ) ) ) ).
% finite_range_imageI
thf(fact_5620_full__SetCompr__eq,axiom,
! [A: $tType,B: $tType,F: B > A] :
( ( collect @ A
@ ^ [U2: A] :
? [X: B] :
( U2
= ( F @ X ) ) )
= ( image2 @ B @ A @ F @ ( top_top @ ( set @ B ) ) ) ) ).
% full_SetCompr_eq
thf(fact_5621_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_5622_image2__eqI,axiom,
! [A: $tType,C: $tType,B: $tType,B2: A,F: B > A,X2: B,C2: C,G: B > C,A5: set @ B] :
( ( B2
= ( F @ X2 ) )
=> ( ( C2
= ( G @ X2 ) )
=> ( ( member @ B @ X2 @ A5 )
=> ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ B2 @ C2 ) @ ( bNF_Greatest_image2 @ B @ A @ C @ A5 @ F @ G ) ) ) ) ) ).
% image2_eqI
thf(fact_5623_Ints__def,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( ( ring_1_Ints @ A )
= ( image2 @ int @ A @ ( ring_1_of_int @ A ) @ ( top_top @ ( set @ int ) ) ) ) ) ).
% Ints_def
thf(fact_5624_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 ) )
@ ^ [X: A] : X ) ) ).
% sorted_list_of_set.folding_insort_key_axioms
thf(fact_5625_int__in__range__abs,axiom,
! [N: nat] : ( member @ int @ ( semiring_1_of_nat @ int @ N ) @ ( image2 @ int @ int @ ( abs_abs @ int ) @ ( top_top @ ( set @ int ) ) ) ) ).
% int_in_range_abs
thf(fact_5626_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_5627_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_5628_card__range__greater__zero,axiom,
! [A: $tType,B: $tType,F: B > A] :
( ( finite_finite2 @ A @ ( image2 @ B @ A @ F @ ( top_top @ ( set @ B ) ) ) )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ ( finite_card @ A @ ( image2 @ B @ A @ F @ ( top_top @ ( set @ B ) ) ) ) ) ) ).
% card_range_greater_zero
thf(fact_5629_range__mod,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( image2 @ nat @ nat
@ ^ [M2: nat] : ( modulo_modulo @ nat @ M2 @ N )
@ ( top_top @ ( set @ nat ) ) )
= ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% range_mod
thf(fact_5630_range__prod,axiom,
! [C: $tType,B: $tType,A: $tType,F: C > ( product_prod @ A @ B )] :
( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( image2 @ C @ ( product_prod @ A @ B ) @ F @ ( top_top @ ( set @ C ) ) )
@ ( product_Sigma @ A @ B @ ( image2 @ C @ A @ ( comp @ ( product_prod @ A @ B ) @ A @ C @ ( product_fst @ A @ B ) @ F ) @ ( top_top @ ( set @ C ) ) )
@ ^ [Uu: A] : ( image2 @ C @ B @ ( comp @ ( product_prod @ A @ B ) @ B @ C @ ( product_snd @ A @ B ) @ F ) @ ( top_top @ ( set @ C ) ) ) ) ) ).
% range_prod
thf(fact_5631_dual__max,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( max @ A
@ ^ [X: A,Y4: A] : ( ord_less_eq @ A @ Y4 @ X ) )
= ( ord_min @ A ) ) ) ).
% dual_max
thf(fact_5632_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
@ ^ [I2: A] : ( finite_card @ B @ ( A5 @ I2 ) )
@ I5 ) ) ) ) ) ).
% card_UN_disjoint
thf(fact_5633_Sup__atMost,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [Y: A] :
( ( complete_Sup_Sup @ A @ ( set_ord_atMost @ A @ Y ) )
= Y ) ) ).
% Sup_atMost
thf(fact_5634_Sup__lessThan,axiom,
! [A: $tType] :
( ( ( comple6319245703460814977attice @ A )
& ( dense_linorder @ A ) )
=> ! [Y: A] :
( ( complete_Sup_Sup @ A @ ( set_ord_lessThan @ A @ Y ) )
= Y ) ) ).
% Sup_lessThan
thf(fact_5635_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
@ ^ [A3: A,B3: B] :
( ( P @ A3 )
& ( Q @ B3 ) ) ) )
= ( product_Sigma @ A @ B @ ( collect @ A @ P )
@ ^ [Uu: A] : ( collect @ B @ Q ) ) ) ).
% Collect_case_prod
thf(fact_5636_Sup__atLeastAtMost,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X2: A,Y: A] :
( ( ord_less_eq @ A @ X2 @ Y )
=> ( ( complete_Sup_Sup @ A @ ( set_or1337092689740270186AtMost @ A @ X2 @ Y ) )
= Y ) ) ) ).
% Sup_atLeastAtMost
thf(fact_5637_cSup__atLeastAtMost,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [Y: A,X2: A] :
( ( ord_less_eq @ A @ Y @ X2 )
=> ( ( complete_Sup_Sup @ A @ ( set_or1337092689740270186AtMost @ A @ Y @ X2 ) )
= X2 ) ) ) ).
% cSup_atLeastAtMost
thf(fact_5638_Sup__atLeastLessThan,axiom,
! [A: $tType] :
( ( ( comple6319245703460814977attice @ A )
& ( dense_linorder @ A ) )
=> ! [X2: A,Y: A] :
( ( ord_less @ A @ X2 @ Y )
=> ( ( complete_Sup_Sup @ A @ ( set_or7035219750837199246ssThan @ A @ X2 @ Y ) )
= Y ) ) ) ).
% Sup_atLeastLessThan
thf(fact_5639_Sup__greaterThanAtMost,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X2: A,Y: A] :
( ( ord_less @ A @ X2 @ Y )
=> ( ( complete_Sup_Sup @ A @ ( set_or3652927894154168847AtMost @ A @ X2 @ Y ) )
= Y ) ) ) ).
% Sup_greaterThanAtMost
thf(fact_5640_Sup__greaterThanLessThan,axiom,
! [A: $tType] :
( ( ( comple6319245703460814977attice @ A )
& ( dense_linorder @ A ) )
=> ! [X2: A,Y: A] :
( ( ord_less @ A @ X2 @ Y )
=> ( ( complete_Sup_Sup @ A @ ( set_or5935395276787703475ssThan @ A @ X2 @ Y ) )
= Y ) ) ) ).
% Sup_greaterThanLessThan
thf(fact_5641_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
@ ^ [X: B] : C2
@ A5 ) )
= C2 ) ) ) ).
% cSUP_const
thf(fact_5642_finite__UN__I,axiom,
! [B: $tType,A: $tType,A5: set @ A,B5: A > ( set @ B )] :
( ( finite_finite2 @ A @ A5 )
=> ( ! [A4: A] :
( ( member @ A @ A4 @ A5 )
=> ( finite_finite2 @ B @ ( B5 @ A4 ) ) )
=> ( finite_finite2 @ B @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ B5 @ A5 ) ) ) ) ) ).
% finite_UN_I
thf(fact_5643_Sigma__empty2,axiom,
! [B: $tType,A: $tType,A5: set @ A] :
( ( product_Sigma @ A @ B @ A5
@ ^ [Uu: A] : ( bot_bot @ ( set @ B ) ) )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% Sigma_empty2
thf(fact_5644_Times__empty,axiom,
! [A: $tType,B: $tType,A5: set @ A,B5: set @ B] :
( ( ( product_Sigma @ A @ B @ A5
@ ^ [Uu: A] : B5 )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) )
= ( ( A5
= ( bot_bot @ ( set @ A ) ) )
| ( B5
= ( bot_bot @ ( set @ B ) ) ) ) ) ).
% Times_empty
thf(fact_5645_Compl__Times__UNIV2,axiom,
! [B: $tType,A: $tType,A5: set @ A] :
( ( uminus_uminus @ ( set @ ( product_prod @ A @ B ) )
@ ( product_Sigma @ A @ B @ A5
@ ^ [Uu: A] : ( top_top @ ( set @ B ) ) ) )
= ( product_Sigma @ A @ B @ ( uminus_uminus @ ( set @ A ) @ A5 )
@ ^ [Uu: A] : ( top_top @ ( set @ B ) ) ) ) ).
% Compl_Times_UNIV2
thf(fact_5646_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 ) )
@ ^ [Uu: A] : A5 ) )
= ( product_Sigma @ A @ B @ ( top_top @ ( set @ A ) )
@ ^ [Uu: A] : ( uminus_uminus @ ( set @ B ) @ A5 ) ) ) ).
% Compl_Times_UNIV1
thf(fact_5647_finite__SigmaI,axiom,
! [B: $tType,A: $tType,A5: set @ A,B5: A > ( set @ B )] :
( ( finite_finite2 @ A @ A5 )
=> ( ! [A4: A] :
( ( member @ A @ A4 @ A5 )
=> ( finite_finite2 @ B @ ( B5 @ A4 ) ) )
=> ( finite_finite2 @ ( product_prod @ A @ B ) @ ( product_Sigma @ A @ B @ A5 @ B5 ) ) ) ) ).
% finite_SigmaI
thf(fact_5648_UNIV__Times__UNIV,axiom,
! [B: $tType,A: $tType] :
( ( product_Sigma @ A @ B @ ( top_top @ ( set @ A ) )
@ ^ [Uu: A] : ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% UNIV_Times_UNIV
thf(fact_5649_fst__image__times,axiom,
! [B: $tType,A: $tType,B5: set @ B,A5: set @ A] :
( ( ( B5
= ( bot_bot @ ( set @ B ) ) )
=> ( ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B )
@ ( product_Sigma @ A @ B @ A5
@ ^ [Uu: A] : B5 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( B5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B )
@ ( product_Sigma @ A @ B @ A5
@ ^ [Uu: A] : B5 ) )
= A5 ) ) ) ).
% fst_image_times
thf(fact_5650_snd__image__times,axiom,
! [B: $tType,A: $tType,A5: set @ B,B5: set @ A] :
( ( ( A5
= ( bot_bot @ ( set @ B ) ) )
=> ( ( image2 @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A )
@ ( product_Sigma @ B @ A @ A5
@ ^ [Uu: B] : B5 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( image2 @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A )
@ ( product_Sigma @ B @ A @ A5
@ ^ [Uu: B] : B5 ) )
= B5 ) ) ) ).
% snd_image_times
thf(fact_5651_card__SigmaI,axiom,
! [B: $tType,A: $tType,A5: set @ A,B5: A > ( set @ B )] :
( ( finite_finite2 @ A @ A5 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ A5 )
=> ( finite_finite2 @ B @ ( B5 @ X3 ) ) )
=> ( ( finite_card @ ( product_prod @ A @ B ) @ ( product_Sigma @ A @ B @ A5 @ B5 ) )
= ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [A3: A] : ( finite_card @ B @ ( B5 @ A3 ) )
@ A5 ) ) ) ) ).
% card_SigmaI
thf(fact_5652_Sigma__UNIV__cancel,axiom,
! [B: $tType,A: $tType,A5: set @ A,X8: set @ B] :
( ( minus_minus @ ( set @ ( product_prod @ A @ B ) )
@ ( product_Sigma @ A @ B @ A5
@ ^ [Uu: A] : X8 )
@ ( product_Sigma @ A @ B @ A5
@ ^ [Uu: A] : ( top_top @ ( set @ B ) ) ) )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% Sigma_UNIV_cancel
thf(fact_5653_set__product,axiom,
! [B: $tType,A: $tType,Xs: list @ A,Ys: list @ B] :
( ( set2 @ ( product_prod @ A @ B ) @ ( product @ A @ B @ Xs @ Ys ) )
= ( product_Sigma @ A @ B @ ( set2 @ A @ Xs )
@ ^ [Uu: A] : ( set2 @ B @ Ys ) ) ) ).
% set_product
thf(fact_5654_pairself__image__cart,axiom,
! [B: $tType,A: $tType,F: B > A,A5: set @ B,B5: set @ B] :
( ( image2 @ ( product_prod @ B @ B ) @ ( product_prod @ A @ A ) @ ( pairself @ B @ A @ F )
@ ( product_Sigma @ B @ B @ A5
@ ^ [Uu: B] : B5 ) )
= ( product_Sigma @ A @ A @ ( image2 @ B @ A @ F @ A5 )
@ ^ [Uu: A] : ( image2 @ B @ A @ F @ B5 ) ) ) ).
% pairself_image_cart
thf(fact_5655_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_5656_Sigma__def,axiom,
! [B: $tType,A: $tType] :
( ( product_Sigma @ A @ B )
= ( ^ [A8: set @ A,B8: A > ( set @ B )] :
( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) )
@ ( image2 @ A @ ( set @ ( product_prod @ A @ B ) )
@ ^ [X: 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 @ X @ Y4 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) )
@ ( B8 @ X ) ) )
@ A8 ) ) ) ) ).
% Sigma_def
thf(fact_5657_Some__SUP,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ B )
=> ! [A5: set @ A,F: A > B] :
( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( some @ B @ ( complete_Sup_Sup @ B @ ( image2 @ A @ B @ F @ A5 ) ) )
= ( complete_Sup_Sup @ ( option @ B )
@ ( image2 @ A @ ( option @ B )
@ ^ [X: A] : ( some @ B @ ( F @ X ) )
@ A5 ) ) ) ) ) ).
% Some_SUP
thf(fact_5658_le__cSup__finite,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X8: set @ A,X2: A] :
( ( finite_finite2 @ A @ X8 )
=> ( ( member @ A @ X2 @ X8 )
=> ( ord_less_eq @ A @ X2 @ ( complete_Sup_Sup @ A @ X8 ) ) ) ) ) ).
% le_cSup_finite
thf(fact_5659_inf__Sup,axiom,
! [A: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [A2: A,B5: set @ A] :
( ( inf_inf @ A @ A2 @ ( complete_Sup_Sup @ A @ B5 ) )
= ( complete_Sup_Sup @ A @ ( image2 @ A @ A @ ( inf_inf @ A @ A2 ) @ B5 ) ) ) ) ).
% inf_Sup
thf(fact_5660_snd__image__Sigma,axiom,
! [A: $tType,B: $tType,A5: set @ B,B5: B > ( set @ A )] :
( ( image2 @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) @ ( product_Sigma @ B @ A @ A5 @ B5 ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B5 @ A5 ) ) ) ).
% snd_image_Sigma
thf(fact_5661_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_5662_Sigma__Union,axiom,
! [B: $tType,A: $tType,X8: set @ ( set @ A ),B5: A > ( set @ B )] :
( ( product_Sigma @ A @ B @ ( complete_Sup_Sup @ ( set @ A ) @ X8 ) @ B5 )
= ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) )
@ ( image2 @ ( set @ A ) @ ( set @ ( product_prod @ A @ B ) )
@ ^ [A8: set @ A] : ( product_Sigma @ A @ B @ A8 @ B5 )
@ X8 ) ) ) ).
% Sigma_Union
thf(fact_5663_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_5664_Union__natural,axiom,
! [B: $tType,A: $tType,F: A > B] :
( ( comp @ ( set @ ( set @ B ) ) @ ( set @ B ) @ ( set @ ( set @ A ) ) @ ( complete_Sup_Sup @ ( set @ B ) ) @ ( image2 @ ( set @ A ) @ ( set @ B ) @ ( image2 @ A @ B @ F ) ) )
= ( comp @ ( set @ A ) @ ( set @ B ) @ ( set @ ( set @ A ) ) @ ( image2 @ A @ B @ F ) @ ( complete_Sup_Sup @ ( set @ A ) ) ) ) ).
% Union_natural
thf(fact_5665_relcomp__UNION__distrib,axiom,
! [A: $tType,C: $tType,B: $tType,D: $tType,S3: set @ ( product_prod @ A @ C ),R: D > ( set @ ( product_prod @ C @ B ) ),I5: set @ D] :
( ( relcomp @ A @ C @ B @ S3 @ ( complete_Sup_Sup @ ( set @ ( product_prod @ C @ B ) ) @ ( image2 @ D @ ( set @ ( product_prod @ C @ B ) ) @ R @ I5 ) ) )
= ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) )
@ ( image2 @ D @ ( set @ ( product_prod @ A @ B ) )
@ ^ [I2: D] : ( relcomp @ A @ C @ B @ S3 @ ( R @ I2 ) )
@ I5 ) ) ) ).
% relcomp_UNION_distrib
thf(fact_5666_relcomp__UNION__distrib2,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,R: D > ( set @ ( product_prod @ A @ C ) ),I5: set @ D,S3: set @ ( product_prod @ C @ B )] :
( ( relcomp @ A @ C @ B @ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ C ) ) @ ( image2 @ D @ ( set @ ( product_prod @ A @ C ) ) @ R @ I5 ) ) @ S3 )
= ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) )
@ ( image2 @ D @ ( set @ ( product_prod @ A @ B ) )
@ ^ [I2: D] : ( relcomp @ A @ C @ B @ ( R @ I2 ) @ S3 )
@ I5 ) ) ) ).
% relcomp_UNION_distrib2
thf(fact_5667_top__empty__eq2,axiom,
! [B: $tType,A: $tType] :
( ( top_top @ ( A > B > $o ) )
= ( ^ [X: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y4 ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% top_empty_eq2
thf(fact_5668_UN__Times__distrib,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,E6: C > ( set @ A ),F4: D > ( set @ B ),A5: set @ C,B5: set @ D] :
( ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) )
@ ( image2 @ ( product_prod @ C @ D ) @ ( set @ ( product_prod @ A @ B ) )
@ ( product_case_prod @ C @ D @ ( set @ ( product_prod @ A @ B ) )
@ ^ [A3: C,B3: D] :
( product_Sigma @ A @ B @ ( E6 @ A3 )
@ ^ [Uu: A] : ( F4 @ B3 ) ) )
@ ( product_Sigma @ C @ D @ A5
@ ^ [Uu: C] : B5 ) ) )
= ( product_Sigma @ A @ B @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ C @ ( set @ A ) @ E6 @ A5 ) )
@ ^ [Uu: A] : ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ D @ ( set @ B ) @ F4 @ B5 ) ) ) ) ).
% UN_Times_distrib
thf(fact_5669_Times__eq__cancel2,axiom,
! [A: $tType,B: $tType,X2: A,C6: set @ A,A5: set @ B,B5: set @ B] :
( ( member @ A @ X2 @ C6 )
=> ( ( ( product_Sigma @ B @ A @ A5
@ ^ [Uu: B] : C6 )
= ( product_Sigma @ B @ A @ B5
@ ^ [Uu: B] : C6 ) )
= ( A5 = B5 ) ) ) ).
% Times_eq_cancel2
thf(fact_5670_Sigma__cong,axiom,
! [B: $tType,A: $tType,A5: set @ A,B5: set @ A,C6: A > ( set @ B ),D3: A > ( set @ B )] :
( ( A5 = B5 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ B5 )
=> ( ( C6 @ X3 )
= ( D3 @ X3 ) ) )
=> ( ( product_Sigma @ A @ B @ A5 @ C6 )
= ( product_Sigma @ A @ B @ B5 @ D3 ) ) ) ) ).
% Sigma_cong
thf(fact_5671_ord_Omax__def,axiom,
! [A: $tType] :
( ( max @ A )
= ( ^ [Less_eq2: A > A > $o,A3: A,B3: A] : ( if @ A @ ( Less_eq2 @ A3 @ B3 ) @ B3 @ A3 ) ) ) ).
% ord.max_def
thf(fact_5672_ord_Omax_Ocong,axiom,
! [A: $tType] :
( ( max @ A )
= ( max @ A ) ) ).
% ord.max.cong
thf(fact_5673_cSup__eq__maximum,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [Z2: A,X8: set @ A] :
( ( member @ A @ Z2 @ X8 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ X8 )
=> ( ord_less_eq @ A @ X3 @ Z2 ) )
=> ( ( complete_Sup_Sup @ A @ X8 )
= Z2 ) ) ) ) ).
% cSup_eq_maximum
thf(fact_5674_cSup__eq,axiom,
! [A: $tType] :
( ( ( condit1219197933456340205attice @ A )
& ( no_bot @ A ) )
=> ! [X8: set @ A,A2: A] :
( ! [X3: A] :
( ( member @ A @ X3 @ X8 )
=> ( ord_less_eq @ A @ X3 @ A2 ) )
=> ( ! [Y2: A] :
( ! [X4: A] :
( ( member @ A @ X4 @ X8 )
=> ( ord_less_eq @ A @ X4 @ Y2 ) )
=> ( ord_less_eq @ A @ A2 @ Y2 ) )
=> ( ( complete_Sup_Sup @ A @ X8 )
= A2 ) ) ) ) ).
% cSup_eq
thf(fact_5675_in__Union__o__assoc,axiom,
! [B: $tType,A: $tType,C: $tType,X2: A,Gset: B > ( set @ ( set @ A ) ),Gmap: C > B,A5: C] :
( ( member @ A @ X2 @ ( comp @ B @ ( set @ A ) @ C @ ( comp @ ( set @ ( set @ A ) ) @ ( set @ A ) @ B @ ( complete_Sup_Sup @ ( set @ A ) ) @ Gset ) @ Gmap @ A5 ) )
=> ( member @ A @ X2 @ ( 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_5676_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
@ ^ [X: A,Y4: B] :
( ( P @ X )
& ( Q @ X @ Y4 ) ) ) )
= ( product_Sigma @ A @ B @ ( collect @ A @ P )
@ ^ [X: A] : ( collect @ B @ ( Q @ X ) ) ) ) ).
% Collect_case_prod_Sigma
thf(fact_5677_Sup__inf__eq__bot__iff,axiom,
! [A: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [B5: set @ A,A2: A] :
( ( ( inf_inf @ A @ ( complete_Sup_Sup @ A @ B5 ) @ A2 )
= ( bot_bot @ A ) )
= ( ! [X: A] :
( ( member @ A @ X @ B5 )
=> ( ( inf_inf @ A @ X @ A2 )
= ( bot_bot @ A ) ) ) ) ) ) ).
% Sup_inf_eq_bot_iff
thf(fact_5678_times__eq__iff,axiom,
! [A: $tType,B: $tType,A5: set @ A,B5: set @ B,C6: set @ A,D3: set @ B] :
( ( ( product_Sigma @ A @ B @ A5
@ ^ [Uu: A] : B5 )
= ( product_Sigma @ A @ B @ C6
@ ^ [Uu: A] : D3 ) )
= ( ( ( A5 = C6 )
& ( B5 = D3 ) )
| ( ( ( A5
= ( bot_bot @ ( set @ A ) ) )
| ( B5
= ( bot_bot @ ( set @ B ) ) ) )
& ( ( C6
= ( bot_bot @ ( set @ A ) ) )
| ( D3
= ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ).
% times_eq_iff
thf(fact_5679_cSup__eq__non__empty,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X8: set @ A,A2: A] :
( ( X8
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ X8 )
=> ( ord_less_eq @ A @ X3 @ A2 ) )
=> ( ! [Y2: A] :
( ! [X4: A] :
( ( member @ A @ X4 @ X8 )
=> ( ord_less_eq @ A @ X4 @ Y2 ) )
=> ( ord_less_eq @ A @ A2 @ Y2 ) )
=> ( ( complete_Sup_Sup @ A @ X8 )
= A2 ) ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_5680_cSup__least,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X8: set @ A,Z2: A] :
( ( X8
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ X8 )
=> ( ord_less_eq @ A @ X3 @ Z2 ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ X8 ) @ Z2 ) ) ) ) ).
% cSup_least
thf(fact_5681_finite__UNION__then__finite,axiom,
! [A: $tType,B: $tType,B5: B > ( set @ A ),A5: set @ B,A2: B] :
( ( finite_finite2 @ A @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B5 @ A5 ) ) )
=> ( ( member @ B @ A2 @ A5 )
=> ( finite_finite2 @ A @ ( B5 @ A2 ) ) ) ) ).
% finite_UNION_then_finite
thf(fact_5682_card__Union__le__sum__card,axiom,
! [A: $tType,U5: set @ ( set @ A )] : ( ord_less_eq @ nat @ ( finite_card @ A @ ( complete_Sup_Sup @ ( set @ A ) @ U5 ) ) @ ( groups7311177749621191930dd_sum @ ( set @ A ) @ nat @ ( finite_card @ A ) @ U5 ) ) ).
% card_Union_le_sum_card
thf(fact_5683_Product__Type_Oproduct__def,axiom,
! [B: $tType,A: $tType] :
( ( product_product @ A @ B )
= ( ^ [A8: set @ A,B8: set @ B] :
( product_Sigma @ A @ B @ A8
@ ^ [Uu: A] : B8 ) ) ) ).
% Product_Type.product_def
thf(fact_5684_member__product,axiom,
! [B: $tType,A: $tType,X2: product_prod @ A @ B,A5: set @ A,B5: set @ B] :
( ( member @ ( product_prod @ A @ B ) @ X2 @ ( product_product @ A @ B @ A5 @ B5 ) )
= ( member @ ( product_prod @ A @ B ) @ X2
@ ( product_Sigma @ A @ B @ A5
@ ^ [Uu: A] : B5 ) ) ) ).
% member_product
thf(fact_5685_Times__subset__cancel2,axiom,
! [A: $tType,B: $tType,X2: A,C6: set @ A,A5: set @ B,B5: set @ B] :
( ( member @ A @ X2 @ C6 )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ B @ A ) )
@ ( product_Sigma @ B @ A @ A5
@ ^ [Uu: B] : C6 )
@ ( product_Sigma @ B @ A @ B5
@ ^ [Uu: B] : C6 ) )
= ( ord_less_eq @ ( set @ B ) @ A5 @ B5 ) ) ) ).
% Times_subset_cancel2
thf(fact_5686_mem__Times__iff,axiom,
! [A: $tType,B: $tType,X2: product_prod @ A @ B,A5: set @ A,B5: set @ B] :
( ( member @ ( product_prod @ A @ B ) @ X2
@ ( product_Sigma @ A @ B @ A5
@ ^ [Uu: A] : B5 ) )
= ( ( member @ A @ ( product_fst @ A @ B @ X2 ) @ A5 )
& ( member @ B @ ( product_snd @ A @ B @ X2 ) @ B5 ) ) ) ).
% mem_Times_iff
thf(fact_5687_in__prod__fst__sndI,axiom,
! [B: $tType,A: $tType,X2: product_prod @ A @ B,A5: set @ A,B5: set @ B] :
( ( member @ A @ ( product_fst @ A @ B @ X2 ) @ A5 )
=> ( ( member @ B @ ( product_snd @ A @ B @ X2 ) @ B5 )
=> ( member @ ( product_prod @ A @ B ) @ X2
@ ( product_Sigma @ A @ B @ A5
@ ^ [Uu: A] : B5 ) ) ) ) ).
% in_prod_fst_sndI
thf(fact_5688_card__cartesian__product,axiom,
! [A: $tType,B: $tType,A5: set @ A,B5: set @ B] :
( ( finite_card @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A5
@ ^ [Uu: A] : B5 ) )
= ( times_times @ nat @ ( finite_card @ A @ A5 ) @ ( finite_card @ B @ B5 ) ) ) ).
% card_cartesian_product
thf(fact_5689_Sigma__empty__iff,axiom,
! [B: $tType,A: $tType,I5: set @ A,X8: A > ( set @ B )] :
( ( ( product_Sigma @ A @ B @ I5 @ X8 )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) )
= ( ! [X: A] :
( ( member @ A @ X @ I5 )
=> ( ( X8 @ X )
= ( bot_bot @ ( set @ B ) ) ) ) ) ) ).
% Sigma_empty_iff
thf(fact_5690_SUP__inf__distrib2,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [F: B > A,A5: set @ B,G: C > A,B5: set @ C] :
( ( inf_inf @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F @ A5 ) ) @ ( complete_Sup_Sup @ A @ ( image2 @ C @ A @ G @ B5 ) ) )
= ( complete_Sup_Sup @ A
@ ( image2 @ B @ A
@ ^ [A3: B] :
( complete_Sup_Sup @ A
@ ( image2 @ C @ A
@ ^ [B3: C] : ( inf_inf @ A @ ( F @ A3 ) @ ( G @ B3 ) )
@ B5 ) )
@ A5 ) ) ) ) ).
% SUP_inf_distrib2
thf(fact_5691_inf__SUP,axiom,
! [A: $tType,B: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [A2: A,F: B > A,B5: set @ B] :
( ( inf_inf @ A @ A2 @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F @ B5 ) ) )
= ( complete_Sup_Sup @ A
@ ( image2 @ B @ A
@ ^ [B3: B] : ( inf_inf @ A @ A2 @ ( F @ B3 ) )
@ B5 ) ) ) ) ).
% inf_SUP
thf(fact_5692_Sup__inf,axiom,
! [A: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [B5: set @ A,A2: A] :
( ( inf_inf @ A @ ( complete_Sup_Sup @ A @ B5 ) @ A2 )
= ( complete_Sup_Sup @ A
@ ( image2 @ A @ A
@ ^ [B3: A] : ( inf_inf @ A @ B3 @ A2 )
@ B5 ) ) ) ) ).
% Sup_inf
thf(fact_5693_SUP__inf,axiom,
! [A: $tType,B: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [F: B > A,B5: set @ B,A2: A] :
( ( inf_inf @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F @ B5 ) ) @ A2 )
= ( complete_Sup_Sup @ A
@ ( image2 @ B @ A
@ ^ [B3: B] : ( inf_inf @ A @ ( F @ B3 ) @ A2 )
@ B5 ) ) ) ) ).
% SUP_inf
thf(fact_5694_Times__Int__Times,axiom,
! [B: $tType,A: $tType,A5: set @ A,B5: set @ B,C6: set @ A,D3: set @ B] :
( ( inf_inf @ ( set @ ( product_prod @ A @ B ) )
@ ( product_Sigma @ A @ B @ A5
@ ^ [Uu: A] : B5 )
@ ( product_Sigma @ A @ B @ C6
@ ^ [Uu: A] : D3 ) )
= ( product_Sigma @ A @ B @ ( inf_inf @ ( set @ A ) @ A5 @ C6 )
@ ^ [Uu: A] : ( inf_inf @ ( set @ B ) @ B5 @ D3 ) ) ) ).
% Times_Int_Times
thf(fact_5695_Sigma__Int__distrib2,axiom,
! [B: $tType,A: $tType,I5: set @ A,A5: A > ( set @ B ),B5: A > ( set @ B )] :
( ( product_Sigma @ A @ B @ I5
@ ^ [I2: A] : ( inf_inf @ ( set @ B ) @ ( A5 @ I2 ) @ ( B5 @ I2 ) ) )
= ( inf_inf @ ( set @ ( product_prod @ A @ B ) ) @ ( product_Sigma @ A @ B @ I5 @ A5 ) @ ( product_Sigma @ A @ B @ I5 @ B5 ) ) ) ).
% Sigma_Int_distrib2
thf(fact_5696_Times__Int__distrib1,axiom,
! [B: $tType,A: $tType,A5: set @ A,B5: set @ A,C6: set @ B] :
( ( product_Sigma @ A @ B @ ( inf_inf @ ( set @ A ) @ A5 @ B5 )
@ ^ [Uu: A] : C6 )
= ( inf_inf @ ( set @ ( product_prod @ A @ B ) )
@ ( product_Sigma @ A @ B @ A5
@ ^ [Uu: A] : C6 )
@ ( product_Sigma @ A @ B @ B5
@ ^ [Uu: A] : C6 ) ) ) ).
% Times_Int_distrib1
thf(fact_5697_finite__cartesian__product,axiom,
! [B: $tType,A: $tType,A5: set @ A,B5: set @ B] :
( ( finite_finite2 @ A @ A5 )
=> ( ( finite_finite2 @ B @ B5 )
=> ( finite_finite2 @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A5
@ ^ [Uu: A] : B5 ) ) ) ) ).
% finite_cartesian_product
thf(fact_5698_relcomp__subset__Sigma,axiom,
! [B: $tType,C: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),A5: set @ A,B5: set @ B,S3: set @ ( product_prod @ B @ C ),C6: set @ C] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R
@ ( product_Sigma @ A @ B @ A5
@ ^ [Uu: A] : B5 ) )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ B @ C ) ) @ S3
@ ( product_Sigma @ B @ C @ B5
@ ^ [Uu: B] : C6 ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ C ) ) @ ( relcomp @ A @ B @ C @ R @ S3 )
@ ( product_Sigma @ A @ C @ A5
@ ^ [Uu: A] : C6 ) ) ) ) ).
% relcomp_subset_Sigma
thf(fact_5699_Times__Diff__distrib1,axiom,
! [B: $tType,A: $tType,A5: set @ A,B5: set @ A,C6: set @ B] :
( ( product_Sigma @ A @ B @ ( minus_minus @ ( set @ A ) @ A5 @ B5 )
@ ^ [Uu: A] : C6 )
= ( minus_minus @ ( set @ ( product_prod @ A @ B ) )
@ ( product_Sigma @ A @ B @ A5
@ ^ [Uu: A] : C6 )
@ ( product_Sigma @ A @ B @ B5
@ ^ [Uu: A] : C6 ) ) ) ).
% Times_Diff_distrib1
thf(fact_5700_Sigma__Diff__distrib2,axiom,
! [B: $tType,A: $tType,I5: set @ A,A5: A > ( set @ B ),B5: A > ( set @ B )] :
( ( product_Sigma @ A @ B @ I5
@ ^ [I2: A] : ( minus_minus @ ( set @ B ) @ ( A5 @ I2 ) @ ( B5 @ I2 ) ) )
= ( minus_minus @ ( set @ ( product_prod @ A @ B ) ) @ ( product_Sigma @ A @ B @ I5 @ A5 ) @ ( product_Sigma @ A @ B @ I5 @ B5 ) ) ) ).
% Sigma_Diff_distrib2
thf(fact_5701_UN__image__subset,axiom,
! [C: $tType,A: $tType,B: $tType,F: B > ( set @ A ),G: C > ( set @ B ),X2: C,X8: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ F @ ( G @ X2 ) ) ) @ X8 )
= ( ord_less_eq @ ( set @ B ) @ ( G @ X2 )
@ ( collect @ B
@ ^ [X: B] : ( ord_less_eq @ ( set @ A ) @ ( F @ X ) @ X8 ) ) ) ) ).
% UN_image_subset
thf(fact_5702_UN__UN__finite__eq,axiom,
! [A: $tType,A5: nat > ( set @ A )] :
( ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ nat @ ( set @ A )
@ ^ [N4: nat] : ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ A5 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N4 ) ) )
@ ( top_top @ ( set @ nat ) ) ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ A5 @ ( top_top @ ( set @ nat ) ) ) ) ) ).
% UN_UN_finite_eq
thf(fact_5703_cSUP__least,axiom,
! [B: $tType,A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A5: set @ B,F: B > A,M3: A] :
( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ A5 )
=> ( ord_less_eq @ A @ ( F @ X3 ) @ M3 ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F @ A5 ) ) @ M3 ) ) ) ) ).
% cSUP_least
thf(fact_5704_cSup__abs__le,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( linordered_idom @ A ) )
=> ! [S2: set @ A,A2: A] :
( ( S2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ S2 )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ X3 ) @ A2 ) )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ ( complete_Sup_Sup @ A @ S2 ) ) @ A2 ) ) ) ) ).
% cSup_abs_le
thf(fact_5705_sum_OUnion__comp,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [B5: set @ ( set @ B ),G: B > A] :
( ! [X3: set @ B] :
( ( member @ ( set @ B ) @ X3 @ B5 )
=> ( finite_finite2 @ B @ X3 ) )
=> ( ! [A14: set @ B] :
( ( member @ ( set @ B ) @ A14 @ B5 )
=> ! [A25: set @ B] :
( ( member @ ( set @ B ) @ A25 @ B5 )
=> ( ( A14 != A25 )
=> ! [X3: B] :
( ( member @ B @ X3 @ A14 )
=> ( ( member @ B @ X3 @ A25 )
=> ( ( G @ X3 )
= ( zero_zero @ A ) ) ) ) ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( complete_Sup_Sup @ ( set @ B ) @ B5 ) )
= ( comp @ ( ( set @ B ) > A ) @ ( ( set @ ( set @ B ) ) > A ) @ ( B > A ) @ ( groups7311177749621191930dd_sum @ ( set @ B ) @ A ) @ ( groups7311177749621191930dd_sum @ B @ A ) @ G @ B5 ) ) ) ) ) ).
% sum.Union_comp
thf(fact_5706_prod_OUnion__comp,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [B5: set @ ( set @ B ),G: B > A] :
( ! [X3: set @ B] :
( ( member @ ( set @ B ) @ X3 @ B5 )
=> ( finite_finite2 @ B @ X3 ) )
=> ( ! [A14: set @ B] :
( ( member @ ( set @ B ) @ A14 @ B5 )
=> ! [A25: set @ B] :
( ( member @ ( set @ B ) @ A25 @ B5 )
=> ( ( A14 != A25 )
=> ! [X3: B] :
( ( member @ B @ X3 @ A14 )
=> ( ( member @ B @ X3 @ A25 )
=> ( ( G @ X3 )
= ( one_one @ A ) ) ) ) ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( complete_Sup_Sup @ ( set @ B ) @ B5 ) )
= ( comp @ ( ( set @ B ) > A ) @ ( ( set @ ( set @ B ) ) > A ) @ ( B > A ) @ ( groups7121269368397514597t_prod @ ( set @ B ) @ A ) @ ( groups7121269368397514597t_prod @ B @ A ) @ G @ B5 ) ) ) ) ) ).
% prod.Union_comp
thf(fact_5707_card__Union__le__sum__card__weak,axiom,
! [A: $tType,U5: set @ ( set @ A )] :
( ! [X3: set @ A] :
( ( member @ ( set @ A ) @ X3 @ U5 )
=> ( finite_finite2 @ A @ X3 ) )
=> ( ord_less_eq @ nat @ ( finite_card @ A @ ( complete_Sup_Sup @ ( set @ A ) @ U5 ) ) @ ( groups7311177749621191930dd_sum @ ( set @ A ) @ nat @ ( finite_card @ A ) @ U5 ) ) ) ).
% card_Union_le_sum_card_weak
thf(fact_5708_lex__def,axiom,
! [A: $tType] :
( ( lex @ A )
= ( ^ [R2: 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 @ R2 ) @ ( top_top @ ( set @ nat ) ) ) ) ) ) ).
% lex_def
thf(fact_5709_card__cartesian__product__singleton,axiom,
! [A: $tType,B: $tType,X2: A,A5: set @ B] :
( ( finite_card @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) )
@ ^ [Uu: A] : A5 ) )
= ( finite_card @ B @ A5 ) ) ).
% card_cartesian_product_singleton
thf(fact_5710_times__subset__iff,axiom,
! [A: $tType,B: $tType,A5: set @ A,C6: set @ B,B5: set @ A,D3: set @ B] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) )
@ ( product_Sigma @ A @ B @ A5
@ ^ [Uu: A] : C6 )
@ ( product_Sigma @ A @ B @ B5
@ ^ [Uu: A] : D3 ) )
= ( ( A5
= ( bot_bot @ ( set @ A ) ) )
| ( C6
= ( bot_bot @ ( set @ B ) ) )
| ( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
& ( ord_less_eq @ ( set @ B ) @ C6 @ D3 ) ) ) ) ).
% times_subset_iff
thf(fact_5711_finite__SigmaI2,axiom,
! [B: $tType,A: $tType,A5: set @ A,B5: A > ( set @ B )] :
( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X: A] :
( ( member @ A @ X @ A5 )
& ( ( B5 @ X )
!= ( bot_bot @ ( set @ B ) ) ) ) ) )
=> ( ! [A4: A] :
( ( member @ A @ A4 @ A5 )
=> ( finite_finite2 @ B @ ( B5 @ A4 ) ) )
=> ( finite_finite2 @ ( product_prod @ A @ B ) @ ( product_Sigma @ A @ B @ A5 @ B5 ) ) ) ) ).
% finite_SigmaI2
thf(fact_5712_finite__cartesian__productD1,axiom,
! [B: $tType,A: $tType,A5: set @ A,B5: set @ B] :
( ( finite_finite2 @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A5
@ ^ [Uu: A] : B5 ) )
=> ( ( B5
!= ( bot_bot @ ( set @ B ) ) )
=> ( finite_finite2 @ A @ A5 ) ) ) ).
% finite_cartesian_productD1
thf(fact_5713_finite__cartesian__productD2,axiom,
! [A: $tType,B: $tType,A5: set @ A,B5: set @ B] :
( ( finite_finite2 @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A5
@ ^ [Uu: A] : B5 ) )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( finite_finite2 @ B @ B5 ) ) ) ).
% finite_cartesian_productD2
thf(fact_5714_finite__cartesian__product__iff,axiom,
! [A: $tType,B: $tType,A5: set @ A,B5: set @ B] :
( ( finite_finite2 @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A5
@ ^ [Uu: A] : B5 ) )
= ( ( A5
= ( bot_bot @ ( set @ A ) ) )
| ( B5
= ( bot_bot @ ( set @ B ) ) )
| ( ( finite_finite2 @ A @ A5 )
& ( finite_finite2 @ B @ B5 ) ) ) ) ).
% finite_cartesian_product_iff
thf(fact_5715_fst__image__Sigma,axiom,
! [B: $tType,A: $tType,A5: set @ A,B5: A > ( set @ B )] :
( ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ ( product_Sigma @ A @ B @ A5 @ B5 ) )
= ( collect @ A
@ ^ [X: A] :
( ( member @ A @ X @ A5 )
& ( ( B5 @ X )
!= ( bot_bot @ ( set @ B ) ) ) ) ) ) ).
% fst_image_Sigma
thf(fact_5716_finite__subset__Union,axiom,
! [A: $tType,A5: set @ A,B11: set @ ( set @ A )] :
( ( finite_finite2 @ A @ A5 )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ ( complete_Sup_Sup @ ( set @ A ) @ B11 ) )
=> ~ ! [F6: set @ ( set @ A )] :
( ( finite_finite2 @ ( set @ A ) @ F6 )
=> ( ( ord_less_eq @ ( set @ ( set @ A ) ) @ F6 @ B11 )
=> ~ ( ord_less_eq @ ( set @ A ) @ A5 @ ( complete_Sup_Sup @ ( set @ A ) @ F6 ) ) ) ) ) ) ).
% finite_subset_Union
thf(fact_5717_set__zip__cart,axiom,
! [B: $tType,A: $tType,X2: product_prod @ A @ B,L: list @ A,L4: list @ B] :
( ( member @ ( product_prod @ A @ B ) @ X2 @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ L @ L4 ) ) )
=> ( member @ ( product_prod @ A @ B ) @ X2
@ ( product_Sigma @ A @ B @ ( set2 @ A @ L )
@ ^ [Uu: A] : ( set2 @ B @ L4 ) ) ) ) ).
% set_zip_cart
thf(fact_5718_UN__finite__subset,axiom,
! [A: $tType,A5: nat > ( set @ A ),C6: set @ A] :
( ! [N2: nat] : ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ A5 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) @ C6 )
=> ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ A5 @ ( top_top @ ( set @ nat ) ) ) ) @ C6 ) ) ).
% UN_finite_subset
thf(fact_5719_UN__finite2__eq,axiom,
! [A: $tType,A5: nat > ( set @ A ),B5: nat > ( set @ A ),K: nat] :
( ! [N2: nat] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ A5 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ B5 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( plus_plus @ nat @ N2 @ K ) ) ) ) )
=> ( ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ A5 @ ( top_top @ ( set @ nat ) ) ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ B5 @ ( top_top @ ( set @ nat ) ) ) ) ) ) ).
% UN_finite2_eq
thf(fact_5720_swap__product,axiom,
! [B: $tType,A: $tType,A5: set @ B,B5: set @ A] :
( ( image2 @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B )
@ ( product_case_prod @ B @ A @ ( product_prod @ A @ B )
@ ^ [I2: B,J3: A] : ( product_Pair @ A @ B @ J3 @ I2 ) )
@ ( product_Sigma @ B @ A @ A5
@ ^ [Uu: B] : B5 ) )
= ( product_Sigma @ A @ B @ B5
@ ^ [Uu: A] : A5 ) ) ).
% swap_product
thf(fact_5721_subset__mset_Omax__add__distrib__left,axiom,
! [A: $tType,X2: multiset @ A,Y: multiset @ A,Z2: multiset @ A] :
( ( plus_plus @ ( multiset @ A ) @ ( max @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ X2 @ Y ) @ Z2 )
= ( max @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( plus_plus @ ( multiset @ A ) @ X2 @ Z2 ) @ ( plus_plus @ ( multiset @ A ) @ Y @ Z2 ) ) ) ).
% subset_mset.max_add_distrib_left
thf(fact_5722_subset__mset_Omax__add__distrib__right,axiom,
! [A: $tType,X2: multiset @ A,Y: multiset @ A,Z2: multiset @ A] :
( ( plus_plus @ ( multiset @ A ) @ X2 @ ( max @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ Y @ Z2 ) )
= ( max @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( plus_plus @ ( multiset @ A ) @ X2 @ Y ) @ ( plus_plus @ ( multiset @ A ) @ X2 @ Z2 ) ) ) ).
% subset_mset.max_add_distrib_right
thf(fact_5723_UN__le__add__shift,axiom,
! [A: $tType,M3: nat > ( set @ A ),K: nat,N: nat] :
( ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ nat @ ( set @ A )
@ ^ [I2: nat] : ( M3 @ ( plus_plus @ nat @ I2 @ K ) )
@ ( set_ord_atMost @ nat @ N ) ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ M3 @ ( set_or1337092689740270186AtMost @ nat @ K @ ( plus_plus @ nat @ N @ K ) ) ) ) ) ).
% UN_le_add_shift
thf(fact_5724_UN__le__add__shift__strict,axiom,
! [A: $tType,M3: nat > ( set @ A ),K: nat,N: nat] :
( ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ nat @ ( set @ A )
@ ^ [I2: nat] : ( M3 @ ( plus_plus @ nat @ I2 @ K ) )
@ ( set_ord_lessThan @ nat @ N ) ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ M3 @ ( set_or7035219750837199246ssThan @ nat @ K @ ( plus_plus @ nat @ N @ K ) ) ) ) ) ).
% UN_le_add_shift_strict
thf(fact_5725_cSup__asclose,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( linordered_idom @ A ) )
=> ! [S2: set @ A,L: A,E2: A] :
( ( S2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ S2 )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X3 @ L ) ) @ E2 ) )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ ( complete_Sup_Sup @ A @ S2 ) @ L ) ) @ E2 ) ) ) ) ).
% cSup_asclose
thf(fact_5726_sum_Ocartesian__product,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: B > C > A,B5: set @ C,A5: set @ B] :
( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X: B] : ( groups7311177749621191930dd_sum @ C @ A @ ( G @ X ) @ B5 )
@ A5 )
= ( groups7311177749621191930dd_sum @ ( product_prod @ B @ C ) @ A @ ( product_case_prod @ B @ C @ A @ G )
@ ( product_Sigma @ B @ C @ A5
@ ^ [Uu: B] : B5 ) ) ) ) ).
% sum.cartesian_product
thf(fact_5727_prod_Ocartesian__product,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: B > C > A,B5: set @ C,A5: set @ B] :
( ( groups7121269368397514597t_prod @ B @ A
@ ^ [X: B] : ( groups7121269368397514597t_prod @ C @ A @ ( G @ X ) @ B5 )
@ A5 )
= ( groups7121269368397514597t_prod @ ( product_prod @ B @ C ) @ A @ ( product_case_prod @ B @ C @ A @ G )
@ ( product_Sigma @ B @ C @ A5
@ ^ [Uu: B] : B5 ) ) ) ) ).
% prod.cartesian_product
thf(fact_5728_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 )
@ ^ [Uu: A] : ( image2 @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ A5 ) ) ) ).
% subset_fst_snd
thf(fact_5729_image__paired__Times,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,F: C > A,G: D > B,A5: set @ C,B5: set @ D] :
( ( image2 @ ( product_prod @ C @ D ) @ ( product_prod @ A @ B )
@ ( product_case_prod @ C @ D @ ( product_prod @ A @ B )
@ ^ [X: C,Y4: D] : ( product_Pair @ A @ B @ ( F @ X ) @ ( G @ Y4 ) ) )
@ ( product_Sigma @ C @ D @ A5
@ ^ [Uu: C] : B5 ) )
= ( product_Sigma @ A @ B @ ( image2 @ C @ A @ F @ A5 )
@ ^ [Uu: A] : ( image2 @ D @ B @ G @ B5 ) ) ) ).
% image_paired_Times
thf(fact_5730_UN__finite2__subset,axiom,
! [A: $tType,A5: nat > ( set @ A ),B5: nat > ( set @ A ),K: nat] :
( ! [N2: nat] : ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ A5 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ B5 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( plus_plus @ nat @ N2 @ 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 ) @ B5 @ ( top_top @ ( set @ nat ) ) ) ) ) ) ).
% UN_finite2_subset
thf(fact_5731_sum_OSigma,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A5: set @ B,B5: B > ( set @ C ),G: B > C > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ A5 )
=> ( finite_finite2 @ C @ ( B5 @ X3 ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X: B] : ( groups7311177749621191930dd_sum @ C @ A @ ( G @ X ) @ ( B5 @ X ) )
@ A5 )
= ( groups7311177749621191930dd_sum @ ( product_prod @ B @ C ) @ A @ ( product_case_prod @ B @ C @ A @ G ) @ ( product_Sigma @ B @ C @ A5 @ B5 ) ) ) ) ) ) ).
% sum.Sigma
thf(fact_5732_prod_OSigma,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A5: set @ B,B5: B > ( set @ C ),G: B > C > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ A5 )
=> ( finite_finite2 @ C @ ( B5 @ X3 ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [X: B] : ( groups7121269368397514597t_prod @ C @ A @ ( G @ X ) @ ( B5 @ X ) )
@ A5 )
= ( groups7121269368397514597t_prod @ ( product_prod @ B @ C ) @ A @ ( product_case_prod @ B @ C @ A @ G ) @ ( product_Sigma @ B @ C @ A5 @ B5 ) ) ) ) ) ) ).
% prod.Sigma
thf(fact_5733_sum_OUNION__disjoint,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [I5: set @ B,A5: B > ( set @ C ),G: C > A] :
( ( finite_finite2 @ B @ I5 )
=> ( ! [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 @ G @ ( complete_Sup_Sup @ ( set @ C ) @ ( image2 @ B @ ( set @ C ) @ A5 @ I5 ) ) )
= ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X: B] : ( groups7311177749621191930dd_sum @ C @ A @ G @ ( A5 @ X ) )
@ I5 ) ) ) ) ) ) ).
% sum.UNION_disjoint
thf(fact_5734_prod_OUNION__disjoint,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [I5: set @ B,A5: B > ( set @ C ),G: C > A] :
( ( finite_finite2 @ B @ I5 )
=> ( ! [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 @ G @ ( complete_Sup_Sup @ ( set @ C ) @ ( image2 @ B @ ( set @ C ) @ A5 @ I5 ) ) )
= ( groups7121269368397514597t_prod @ B @ A
@ ^ [X: B] : ( groups7121269368397514597t_prod @ C @ A @ G @ ( A5 @ X ) )
@ I5 ) ) ) ) ) ) ).
% prod.UNION_disjoint
thf(fact_5735_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
@ ^ [I2: A] : ( finite_card @ B @ ( A5 @ I2 ) )
@ I5 ) ) ) ).
% card_UN_le
thf(fact_5736_lists__length__Suc__eq,axiom,
! [A: $tType,A5: set @ A,N: nat] :
( ( collect @ ( list @ A )
@ ^ [Xs3: list @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs3 ) @ A5 )
& ( ( size_size @ ( list @ A ) @ Xs3 )
= ( suc @ N ) ) ) )
= ( image2 @ ( product_prod @ ( list @ A ) @ A ) @ ( list @ A )
@ ( product_case_prod @ ( list @ A ) @ A @ ( list @ A )
@ ^ [Xs3: list @ A,N4: A] : ( cons @ A @ N4 @ Xs3 ) )
@ ( product_Sigma @ ( list @ A ) @ A
@ ( collect @ ( list @ A )
@ ^ [Xs3: list @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs3 ) @ A5 )
& ( ( size_size @ ( list @ A ) @ Xs3 )
= N ) ) )
@ ^ [Uu: list @ A] : A5 ) ) ) ).
% lists_length_Suc_eq
thf(fact_5737_finite__mono__strict__prefix__implies__finite__fixpoint,axiom,
! [A: $tType,F: nat > ( set @ A ),S2: set @ A] :
( ! [I3: nat] : ( ord_less_eq @ ( set @ A ) @ ( F @ I3 ) @ S2 )
=> ( ( finite_finite2 @ A @ S2 )
=> ( ? [N13: nat] :
( ! [N2: nat] :
( ( ord_less_eq @ nat @ N2 @ N13 )
=> ! [M4: nat] :
( ( ord_less_eq @ nat @ M4 @ N13 )
=> ( ( ord_less @ nat @ M4 @ N2 )
=> ( ord_less @ ( set @ A ) @ ( F @ M4 ) @ ( F @ N2 ) ) ) ) )
& ! [N2: nat] :
( ( ord_less_eq @ nat @ N13 @ N2 )
=> ( ( F @ N13 )
= ( F @ N2 ) ) ) )
=> ( ( F @ ( finite_card @ A @ S2 ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ F @ ( top_top @ ( set @ nat ) ) ) ) ) ) ) ) ).
% finite_mono_strict_prefix_implies_finite_fixpoint
thf(fact_5738_UN__singleton,axiom,
! [A: $tType,A5: set @ A] :
( ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ A @ ( set @ A )
@ ^ [X: A] : ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) )
@ A5 ) )
= A5 ) ).
% UN_singleton
thf(fact_5739_UN__simps_I1_J,axiom,
! [A: $tType,B: $tType,C6: set @ B,A2: A,B5: B > ( set @ A )] :
( ( ( C6
= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [X: B] : ( insert @ A @ A2 @ ( B5 @ X ) )
@ C6 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( C6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [X: B] : ( insert @ A @ A2 @ ( B5 @ X ) )
@ C6 ) )
= ( insert @ A @ A2 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B5 @ C6 ) ) ) ) ) ) ).
% UN_simps(1)
thf(fact_5740_SUP__eq__top__iff,axiom,
! [B: $tType,A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [F: B > A,A5: set @ B] :
( ( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F @ A5 ) )
= ( top_top @ A ) )
= ( ! [X: A] :
( ( ord_less @ A @ X @ ( top_top @ A ) )
=> ? [Y4: B] :
( ( member @ B @ Y4 @ A5 )
& ( ord_less @ A @ X @ ( F @ Y4 ) ) ) ) ) ) ) ).
% SUP_eq_top_iff
thf(fact_5741_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 )
@ ^ [X: B] : C2
@ A5 ) )
= C2 ) ) ).
% subset_mset.cSUP_const
thf(fact_5742_Sup__apply,axiom,
! [B: $tType,A: $tType] :
( ( complete_Sup @ B )
=> ( ( complete_Sup_Sup @ ( A > B ) )
= ( ^ [A8: set @ ( A > B ),X: A] :
( complete_Sup_Sup @ B
@ ( image2 @ ( A > B ) @ B
@ ^ [F2: A > B] : ( F2 @ X )
@ A8 ) ) ) ) ) ).
% Sup_apply
thf(fact_5743_Sup__nat__empty,axiom,
( ( complete_Sup_Sup @ nat @ ( bot_bot @ ( set @ nat ) ) )
= ( zero_zero @ nat ) ) ).
% Sup_nat_empty
thf(fact_5744_top2I,axiom,
! [A: $tType,B: $tType,X2: A,Y: B] : ( top_top @ ( A > B > $o ) @ X2 @ Y ) ).
% top2I
thf(fact_5745_SUP__apply,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( complete_Sup @ A )
=> ! [F: C > B > A,A5: set @ C,X2: B] :
( ( complete_Sup_Sup @ ( B > A ) @ ( image2 @ C @ ( B > A ) @ F @ A5 ) @ X2 )
= ( complete_Sup_Sup @ A
@ ( image2 @ C @ A
@ ^ [Y4: C] : ( F @ Y4 @ X2 )
@ A5 ) ) ) ) ).
% SUP_apply
thf(fact_5746_SUP__identity__eq,axiom,
! [A: $tType] :
( ( complete_Sup @ A )
=> ! [A5: set @ A] :
( ( complete_Sup_Sup @ A
@ ( image2 @ A @ A
@ ^ [X: A] : X
@ A5 ) )
= ( complete_Sup_Sup @ A @ A5 ) ) ) ).
% SUP_identity_eq
thf(fact_5747_UN__iff,axiom,
! [A: $tType,B: $tType,B2: A,B5: B > ( set @ A ),A5: set @ B] :
( ( member @ A @ B2 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B5 @ A5 ) ) )
= ( ? [X: B] :
( ( member @ B @ X @ A5 )
& ( member @ A @ B2 @ ( B5 @ X ) ) ) ) ) ).
% UN_iff
thf(fact_5748_UN__I,axiom,
! [B: $tType,A: $tType,A2: A,A5: set @ A,B2: B,B5: A > ( set @ B )] :
( ( member @ A @ A2 @ A5 )
=> ( ( member @ B @ B2 @ ( B5 @ A2 ) )
=> ( member @ B @ B2 @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ B5 @ A5 ) ) ) ) ) ).
% UN_I
thf(fact_5749_SUP__bot__conv_I2_J,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [B5: B > A,A5: set @ B] :
( ( ( bot_bot @ A )
= ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ B5 @ A5 ) ) )
= ( ! [X: B] :
( ( member @ B @ X @ A5 )
=> ( ( B5 @ X )
= ( bot_bot @ A ) ) ) ) ) ) ).
% SUP_bot_conv(2)
thf(fact_5750_SUP__bot__conv_I1_J,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [B5: B > A,A5: set @ B] :
( ( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ B5 @ A5 ) )
= ( bot_bot @ A ) )
= ( ! [X: B] :
( ( member @ B @ X @ A5 )
=> ( ( B5 @ X )
= ( bot_bot @ A ) ) ) ) ) ) ).
% SUP_bot_conv(1)
thf(fact_5751_SUP__bot,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ B] :
( ( complete_Sup_Sup @ A
@ ( image2 @ B @ A
@ ^ [X: B] : ( bot_bot @ A )
@ A5 ) )
= ( bot_bot @ A ) ) ) ).
% SUP_bot
thf(fact_5752_SUP__const,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ B,F: A] :
( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ A
@ ( image2 @ B @ A
@ ^ [I2: B] : F
@ A5 ) )
= F ) ) ) ).
% SUP_const
thf(fact_5753_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_5754_SUP__Sup__eq,axiom,
! [A: $tType,S2: set @ ( set @ A )] :
( ( complete_Sup_Sup @ ( A > $o )
@ ( image2 @ ( set @ A ) @ ( A > $o )
@ ^ [I2: set @ A,X: A] : ( member @ A @ X @ I2 )
@ S2 ) )
= ( ^ [X: A] : ( member @ A @ X @ ( complete_Sup_Sup @ ( set @ A ) @ S2 ) ) ) ) ).
% SUP_Sup_eq
thf(fact_5755_Sup__set__def,axiom,
! [A: $tType] :
( ( complete_Sup_Sup @ ( set @ A ) )
= ( ^ [A8: set @ ( set @ A )] :
( collect @ A
@ ^ [X: A] : ( complete_Sup_Sup @ $o @ ( image2 @ ( set @ A ) @ $o @ ( member @ A @ X ) @ A8 ) ) ) ) ) ).
% Sup_set_def
thf(fact_5756_SUP__UN__eq2,axiom,
! [B: $tType,C: $tType,A: $tType,R: C > ( set @ ( product_prod @ A @ B ) ),S2: set @ C] :
( ( complete_Sup_Sup @ ( A > B > $o )
@ ( image2 @ C @ ( A > B > $o )
@ ^ [I2: C,X: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y4 ) @ ( R @ I2 ) )
@ S2 ) )
= ( ^ [X: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y4 ) @ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) ) @ ( image2 @ C @ ( set @ ( product_prod @ A @ B ) ) @ R @ S2 ) ) ) ) ) ).
% SUP_UN_eq2
thf(fact_5757_SUP__Sup__eq2,axiom,
! [B: $tType,A: $tType,S2: set @ ( set @ ( product_prod @ A @ B ) )] :
( ( complete_Sup_Sup @ ( A > B > $o )
@ ( image2 @ ( set @ ( product_prod @ A @ B ) ) @ ( A > B > $o )
@ ^ [I2: set @ ( product_prod @ A @ B ),X: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y4 ) @ I2 )
@ S2 ) )
= ( ^ [X: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y4 ) @ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) ) @ S2 ) ) ) ) ).
% SUP_Sup_eq2
thf(fact_5758_Sup__SUP__eq2,axiom,
! [B: $tType,A: $tType] :
( ( complete_Sup_Sup @ ( A > B > $o ) )
= ( ^ [S: set @ ( A > B > $o ),X: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ 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 ) @ S ) ) ) ) ) ) ).
% Sup_SUP_eq2
thf(fact_5759_SUP__UN__eq,axiom,
! [B: $tType,A: $tType,R: B > ( set @ A ),S2: set @ B] :
( ( complete_Sup_Sup @ ( A > $o )
@ ( image2 @ B @ ( A > $o )
@ ^ [I2: B,X: A] : ( member @ A @ X @ ( R @ I2 ) )
@ S2 ) )
= ( ^ [X: A] : ( member @ A @ X @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ R @ S2 ) ) ) ) ) ).
% SUP_UN_eq
thf(fact_5760_Inf_OINF__identity__eq,axiom,
! [A: $tType,Inf: ( set @ A ) > A,A5: set @ A] :
( ( Inf
@ ( image2 @ A @ A
@ ^ [X: A] : X
@ A5 ) )
= ( Inf @ A5 ) ) ).
% Inf.INF_identity_eq
thf(fact_5761_Sup_OSUP__identity__eq,axiom,
! [A: $tType,Sup: ( set @ A ) > A,A5: set @ A] :
( ( Sup
@ ( image2 @ A @ A
@ ^ [X: A] : X
@ A5 ) )
= ( Sup @ A5 ) ) ).
% Sup.SUP_identity_eq
thf(fact_5762_Sup__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( complete_Sup @ B )
=> ( ( complete_Sup_Sup @ ( A > B ) )
= ( ^ [A8: set @ ( A > B ),X: A] :
( complete_Sup_Sup @ B
@ ( image2 @ ( A > B ) @ B
@ ^ [F2: A > B] : ( F2 @ X )
@ A8 ) ) ) ) ) ).
% Sup_fun_def
thf(fact_5763_Sup__eqI,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ A,X2: A] :
( ! [Y2: A] :
( ( member @ A @ Y2 @ A5 )
=> ( ord_less_eq @ A @ Y2 @ X2 ) )
=> ( ! [Y2: A] :
( ! [Z5: A] :
( ( member @ A @ Z5 @ A5 )
=> ( ord_less_eq @ A @ Z5 @ Y2 ) )
=> ( ord_less_eq @ A @ X2 @ Y2 ) )
=> ( ( complete_Sup_Sup @ A @ A5 )
= X2 ) ) ) ) ).
% Sup_eqI
thf(fact_5764_Sup__mono,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ A,B5: set @ A] :
( ! [A4: A] :
( ( member @ A @ A4 @ A5 )
=> ? [X4: A] :
( ( member @ A @ X4 @ B5 )
& ( ord_less_eq @ A @ A4 @ X4 ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A5 ) @ ( complete_Sup_Sup @ A @ B5 ) ) ) ) ).
% Sup_mono
thf(fact_5765_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_5766_Sup__upper,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X2: A,A5: set @ A] :
( ( member @ A @ X2 @ A5 )
=> ( ord_less_eq @ A @ X2 @ ( complete_Sup_Sup @ A @ A5 ) ) ) ) ).
% Sup_upper
thf(fact_5767_Sup__le__iff,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ A,B2: A] :
( ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A5 ) @ B2 )
= ( ! [X: A] :
( ( member @ A @ X @ A5 )
=> ( ord_less_eq @ A @ X @ B2 ) ) ) ) ) ).
% Sup_le_iff
thf(fact_5768_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_5769_SUP__commute,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F: B > C > A,B5: set @ C,A5: set @ B] :
( ( complete_Sup_Sup @ A
@ ( image2 @ B @ A
@ ^ [I2: B] : ( complete_Sup_Sup @ A @ ( image2 @ C @ A @ ( F @ I2 ) @ B5 ) )
@ A5 ) )
= ( complete_Sup_Sup @ A
@ ( image2 @ C @ A
@ ^ [J3: C] :
( complete_Sup_Sup @ A
@ ( image2 @ B @ A
@ ^ [I2: B] : ( F @ I2 @ J3 )
@ A5 ) )
@ B5 ) ) ) ) ).
% SUP_commute
thf(fact_5770_image__Union,axiom,
! [A: $tType,B: $tType,F: B > A,S2: set @ ( set @ B )] :
( ( image2 @ B @ A @ F @ ( complete_Sup_Sup @ ( set @ B ) @ S2 ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ ( set @ B ) @ ( set @ A ) @ ( image2 @ B @ A @ F ) @ S2 ) ) ) ).
% image_Union
thf(fact_5771_Union__SetCompr__eq,axiom,
! [B: $tType,A: $tType,F: B > ( set @ A ),P: B > $o] :
( ( complete_Sup_Sup @ ( set @ A )
@ ( collect @ ( set @ A )
@ ^ [Uu: set @ A] :
? [X: B] :
( ( Uu
= ( F @ X ) )
& ( P @ X ) ) ) )
= ( collect @ A
@ ^ [A3: A] :
? [X: B] :
( ( P @ X )
& ( member @ A @ A3 @ ( F @ X ) ) ) ) ) ).
% Union_SetCompr_eq
thf(fact_5772_Int__Union,axiom,
! [A: $tType,A5: set @ A,B5: set @ ( set @ A )] :
( ( inf_inf @ ( set @ A ) @ A5 @ ( complete_Sup_Sup @ ( set @ A ) @ B5 ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ ( set @ A ) @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A5 ) @ B5 ) ) ) ).
% Int_Union
thf(fact_5773_Int__Union2,axiom,
! [A: $tType,B5: set @ ( set @ A ),A5: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ B5 ) @ A5 )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ ( set @ A ) @ ( set @ A )
@ ^ [C8: set @ A] : ( inf_inf @ ( set @ A ) @ C8 @ A5 )
@ B5 ) ) ) ).
% Int_Union2
thf(fact_5774_UN__UN__flatten,axiom,
! [A: $tType,B: $tType,C: $tType,C6: B > ( set @ A ),B5: C > ( set @ B ),A5: set @ C] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ C6 @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ C @ ( set @ B ) @ B5 @ A5 ) ) ) )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ C @ ( set @ A )
@ ^ [Y4: C] : ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ C6 @ ( B5 @ Y4 ) ) )
@ A5 ) ) ) ).
% UN_UN_flatten
thf(fact_5775_UN__E,axiom,
! [A: $tType,B: $tType,B2: A,B5: B > ( set @ A ),A5: set @ B] :
( ( member @ A @ B2 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B5 @ A5 ) ) )
=> ~ ! [X3: B] :
( ( member @ B @ X3 @ A5 )
=> ~ ( member @ A @ B2 @ ( B5 @ X3 ) ) ) ) ).
% UN_E
thf(fact_5776_UN__extend__simps_I8_J,axiom,
! [P10: $tType,O2: $tType,B5: O2 > ( set @ P10 ),A5: set @ ( set @ O2 )] :
( ( complete_Sup_Sup @ ( set @ P10 )
@ ( image2 @ ( set @ O2 ) @ ( set @ P10 )
@ ^ [Y4: set @ O2] : ( complete_Sup_Sup @ ( set @ P10 ) @ ( image2 @ O2 @ ( set @ P10 ) @ B5 @ Y4 ) )
@ A5 ) )
= ( complete_Sup_Sup @ ( set @ P10 ) @ ( image2 @ O2 @ ( set @ P10 ) @ B5 @ ( complete_Sup_Sup @ ( set @ O2 ) @ A5 ) ) ) ) ).
% UN_extend_simps(8)
thf(fact_5777_UN__extend__simps_I9_J,axiom,
! [S9: $tType,R9: $tType,Q7: $tType,C6: R9 > ( set @ S9 ),B5: Q7 > ( set @ R9 ),A5: set @ Q7] :
( ( complete_Sup_Sup @ ( set @ S9 )
@ ( image2 @ Q7 @ ( set @ S9 )
@ ^ [X: Q7] : ( complete_Sup_Sup @ ( set @ S9 ) @ ( image2 @ R9 @ ( set @ S9 ) @ C6 @ ( B5 @ X ) ) )
@ A5 ) )
= ( complete_Sup_Sup @ ( set @ S9 ) @ ( image2 @ R9 @ ( set @ S9 ) @ C6 @ ( complete_Sup_Sup @ ( set @ R9 ) @ ( image2 @ Q7 @ ( set @ R9 ) @ B5 @ A5 ) ) ) ) ) ).
% UN_extend_simps(9)
thf(fact_5778_le__Sup__iff,axiom,
! [A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [X2: A,A5: set @ A] :
( ( ord_less_eq @ A @ X2 @ ( complete_Sup_Sup @ A @ A5 ) )
= ( ! [Y4: A] :
( ( ord_less @ A @ Y4 @ X2 )
=> ? [X: A] :
( ( member @ A @ X @ A5 )
& ( ord_less @ A @ Y4 @ X ) ) ) ) ) ) ).
% le_Sup_iff
thf(fact_5779_SUP__eq,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ B,B5: set @ C,F: B > A,G: C > A] :
( ! [I3: B] :
( ( member @ B @ I3 @ A5 )
=> ? [X4: C] :
( ( member @ C @ X4 @ B5 )
& ( ord_less_eq @ A @ ( F @ I3 ) @ ( G @ X4 ) ) ) )
=> ( ! [J2: C] :
( ( member @ C @ J2 @ B5 )
=> ? [X4: B] :
( ( member @ B @ X4 @ A5 )
& ( ord_less_eq @ A @ ( G @ J2 ) @ ( F @ X4 ) ) ) )
=> ( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F @ A5 ) )
= ( complete_Sup_Sup @ A @ ( image2 @ C @ A @ G @ B5 ) ) ) ) ) ) ).
% SUP_eq
thf(fact_5780_less__eq__Sup,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ A,U: A] :
( ! [V5: A] :
( ( member @ A @ V5 @ A5 )
=> ( ord_less_eq @ A @ U @ V5 ) )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ A @ U @ ( complete_Sup_Sup @ A @ A5 ) ) ) ) ) ).
% less_eq_Sup
thf(fact_5781_Sup__subset__mono,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A5 ) @ ( complete_Sup_Sup @ A @ B5 ) ) ) ) ).
% Sup_subset_mono
thf(fact_5782_SUP__eqI,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ B,F: B > A,X2: A] :
( ! [I3: B] :
( ( member @ B @ I3 @ A5 )
=> ( ord_less_eq @ A @ ( F @ I3 ) @ X2 ) )
=> ( ! [Y2: A] :
( ! [I4: B] :
( ( member @ B @ I4 @ A5 )
=> ( ord_less_eq @ A @ ( F @ I4 ) @ Y2 ) )
=> ( ord_less_eq @ A @ X2 @ Y2 ) )
=> ( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F @ A5 ) )
= X2 ) ) ) ) ).
% SUP_eqI
thf(fact_5783_SUP__mono,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ B,B5: set @ C,F: B > A,G: C > A] :
( ! [N2: B] :
( ( member @ B @ N2 @ A5 )
=> ? [X4: C] :
( ( member @ C @ X4 @ B5 )
& ( ord_less_eq @ A @ ( F @ N2 ) @ ( G @ X4 ) ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F @ A5 ) ) @ ( complete_Sup_Sup @ A @ ( image2 @ C @ A @ G @ B5 ) ) ) ) ) ).
% SUP_mono
thf(fact_5784_SUP__least,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ B,F: B > A,U: A] :
( ! [I3: B] :
( ( member @ B @ I3 @ A5 )
=> ( ord_less_eq @ A @ ( F @ I3 ) @ U ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F @ A5 ) ) @ U ) ) ) ).
% SUP_least
thf(fact_5785_SUP__mono_H,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F: B > A,G: B > A,A5: set @ B] :
( ! [X3: B] : ( ord_less_eq @ A @ ( F @ X3 ) @ ( G @ X3 ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F @ A5 ) ) @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ G @ A5 ) ) ) ) ) ).
% SUP_mono'
thf(fact_5786_SUP__upper,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I: B,A5: set @ B,F: B > A] :
( ( member @ B @ I @ A5 )
=> ( ord_less_eq @ A @ ( F @ I ) @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F @ A5 ) ) ) ) ) ).
% SUP_upper
thf(fact_5787_SUP__le__iff,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F: B > A,A5: set @ B,U: A] :
( ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F @ A5 ) ) @ U )
= ( ! [X: B] :
( ( member @ B @ X @ A5 )
=> ( ord_less_eq @ A @ ( F @ X ) @ U ) ) ) ) ) ).
% SUP_le_iff
thf(fact_5788_SUP__upper2,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I: B,A5: set @ B,U: A,F: B > A] :
( ( member @ B @ I @ A5 )
=> ( ( ord_less_eq @ A @ U @ ( F @ I ) )
=> ( ord_less_eq @ A @ U @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F @ A5 ) ) ) ) ) ) ).
% SUP_upper2
thf(fact_5789_SUP__lessD,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F: B > A,A5: set @ B,Y: A,I: B] :
( ( ord_less @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F @ A5 ) ) @ Y )
=> ( ( member @ B @ I @ A5 )
=> ( ord_less @ A @ ( F @ I ) @ Y ) ) ) ) ).
% SUP_lessD
thf(fact_5790_less__SUP__iff,axiom,
! [A: $tType,B: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [A2: A,F: B > A,A5: set @ B] :
( ( ord_less @ A @ A2 @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F @ A5 ) ) )
= ( ? [X: B] :
( ( member @ B @ X @ A5 )
& ( ord_less @ A @ A2 @ ( F @ X ) ) ) ) ) ) ).
% less_SUP_iff
thf(fact_5791_UN__extend__simps_I10_J,axiom,
! [V7: $tType,U6: $tType,T: $tType,B5: U6 > ( set @ V7 ),F: T > U6,A5: set @ T] :
( ( complete_Sup_Sup @ ( set @ V7 )
@ ( image2 @ T @ ( set @ V7 )
@ ^ [A3: T] : ( B5 @ ( F @ A3 ) )
@ A5 ) )
= ( complete_Sup_Sup @ ( set @ V7 ) @ ( image2 @ U6 @ ( set @ V7 ) @ B5 @ ( image2 @ T @ U6 @ F @ A5 ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_5792_image__UN,axiom,
! [A: $tType,B: $tType,C: $tType,F: B > A,B5: C > ( set @ B ),A5: set @ C] :
( ( image2 @ B @ A @ F @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ C @ ( set @ B ) @ B5 @ A5 ) ) )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ C @ ( set @ A )
@ ^ [X: C] : ( image2 @ B @ A @ F @ ( B5 @ X ) )
@ A5 ) ) ) ).
% image_UN
thf(fact_5793_UNION__empty__conv_I2_J,axiom,
! [A: $tType,B: $tType,B5: B > ( set @ A ),A5: set @ B] :
( ( ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B5 @ A5 ) )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X: B] :
( ( member @ B @ X @ A5 )
=> ( ( B5 @ X )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% UNION_empty_conv(2)
thf(fact_5794_UNION__empty__conv_I1_J,axiom,
! [A: $tType,B: $tType,B5: B > ( set @ A ),A5: set @ B] :
( ( ( bot_bot @ ( set @ A ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B5 @ A5 ) ) )
= ( ! [X: B] :
( ( member @ B @ X @ A5 )
=> ( ( B5 @ X )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% UNION_empty_conv(1)
thf(fact_5795_UN__empty,axiom,
! [B: $tType,A: $tType,B5: B > ( set @ A )] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B5 @ ( bot_bot @ ( set @ B ) ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% UN_empty
thf(fact_5796_UN__empty2,axiom,
! [B: $tType,A: $tType,A5: set @ B] :
( ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [X: B] : ( bot_bot @ ( set @ A ) )
@ A5 ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% UN_empty2
thf(fact_5797_UN__subset__iff,axiom,
! [A: $tType,B: $tType,A5: B > ( set @ A ),I5: set @ B,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ A5 @ I5 ) ) @ B5 )
= ( ! [X: B] :
( ( member @ B @ X @ I5 )
=> ( ord_less_eq @ ( set @ A ) @ ( A5 @ X ) @ B5 ) ) ) ) ).
% UN_subset_iff
thf(fact_5798_UN__upper,axiom,
! [B: $tType,A: $tType,A2: A,A5: set @ A,B5: A > ( set @ B )] :
( ( member @ A @ A2 @ A5 )
=> ( ord_less_eq @ ( set @ B ) @ ( B5 @ A2 ) @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ B5 @ A5 ) ) ) ) ).
% UN_upper
thf(fact_5799_UN__least,axiom,
! [A: $tType,B: $tType,A5: set @ A,B5: A > ( set @ B ),C6: set @ B] :
( ! [X3: A] :
( ( member @ A @ X3 @ A5 )
=> ( ord_less_eq @ ( set @ B ) @ ( B5 @ X3 ) @ C6 ) )
=> ( ord_less_eq @ ( set @ B ) @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ B5 @ A5 ) ) @ C6 ) ) ).
% UN_least
thf(fact_5800_UN__mono,axiom,
! [B: $tType,A: $tType,A5: set @ A,B5: set @ A,F: A > ( set @ B ),G: A > ( set @ B )] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ A5 )
=> ( ord_less_eq @ ( set @ B ) @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_eq @ ( set @ B ) @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ F @ A5 ) ) @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ G @ B5 ) ) ) ) ) ).
% UN_mono
thf(fact_5801_UN__insert__distrib,axiom,
! [B: $tType,A: $tType,U: A,A5: set @ A,A2: B,B5: A > ( set @ B )] :
( ( member @ A @ U @ A5 )
=> ( ( complete_Sup_Sup @ ( set @ B )
@ ( image2 @ A @ ( set @ B )
@ ^ [X: A] : ( insert @ B @ A2 @ ( B5 @ X ) )
@ A5 ) )
= ( insert @ B @ A2 @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ B5 @ A5 ) ) ) ) ) ).
% UN_insert_distrib
thf(fact_5802_Int__UN__distrib2,axiom,
! [A: $tType,C: $tType,B: $tType,A5: B > ( set @ A ),I5: set @ B,B5: C > ( set @ A ),J4: set @ C] :
( ( inf_inf @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ A5 @ I5 ) ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ C @ ( set @ A ) @ B5 @ J4 ) ) )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [I2: B] :
( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ C @ ( set @ A )
@ ^ [J3: C] : ( inf_inf @ ( set @ A ) @ ( A5 @ I2 ) @ ( B5 @ J3 ) )
@ J4 ) )
@ I5 ) ) ) ).
% Int_UN_distrib2
thf(fact_5803_Int__UN__distrib,axiom,
! [A: $tType,B: $tType,B5: set @ A,A5: B > ( set @ A ),I5: set @ B] :
( ( inf_inf @ ( set @ A ) @ B5 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ A5 @ I5 ) ) )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [I2: B] : ( inf_inf @ ( set @ A ) @ B5 @ ( A5 @ I2 ) )
@ I5 ) ) ) ).
% Int_UN_distrib
thf(fact_5804_UN__extend__simps_I4_J,axiom,
! [H8: $tType,G4: $tType,A5: G4 > ( set @ H8 ),C6: set @ G4,B5: set @ H8] :
( ( inf_inf @ ( set @ H8 ) @ ( complete_Sup_Sup @ ( set @ H8 ) @ ( image2 @ G4 @ ( set @ H8 ) @ A5 @ C6 ) ) @ B5 )
= ( complete_Sup_Sup @ ( set @ H8 )
@ ( image2 @ G4 @ ( set @ H8 )
@ ^ [X: G4] : ( inf_inf @ ( set @ H8 ) @ ( A5 @ X ) @ B5 )
@ C6 ) ) ) ).
% UN_extend_simps(4)
thf(fact_5805_UN__extend__simps_I5_J,axiom,
! [I6: $tType,J5: $tType,A5: set @ I6,B5: J5 > ( set @ I6 ),C6: set @ J5] :
( ( inf_inf @ ( set @ I6 ) @ A5 @ ( complete_Sup_Sup @ ( set @ I6 ) @ ( image2 @ J5 @ ( set @ I6 ) @ B5 @ C6 ) ) )
= ( complete_Sup_Sup @ ( set @ I6 )
@ ( image2 @ J5 @ ( set @ I6 )
@ ^ [X: J5] : ( inf_inf @ ( set @ I6 ) @ A5 @ ( B5 @ X ) )
@ C6 ) ) ) ).
% UN_extend_simps(5)
thf(fact_5806_UN__extend__simps_I6_J,axiom,
! [L8: $tType,K7: $tType,A5: K7 > ( set @ L8 ),C6: set @ K7,B5: set @ L8] :
( ( minus_minus @ ( set @ L8 ) @ ( complete_Sup_Sup @ ( set @ L8 ) @ ( image2 @ K7 @ ( set @ L8 ) @ A5 @ C6 ) ) @ B5 )
= ( complete_Sup_Sup @ ( set @ L8 )
@ ( image2 @ K7 @ ( set @ L8 )
@ ^ [X: K7] : ( minus_minus @ ( set @ L8 ) @ ( A5 @ X ) @ B5 )
@ C6 ) ) ) ).
% UN_extend_simps(6)
thf(fact_5807_le__SUP__iff,axiom,
! [B: $tType,A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [X2: A,F: B > A,A5: set @ B] :
( ( ord_less_eq @ A @ X2 @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F @ A5 ) ) )
= ( ! [Y4: A] :
( ( ord_less @ A @ Y4 @ X2 )
=> ? [X: B] :
( ( member @ B @ X @ A5 )
& ( ord_less @ A @ Y4 @ ( F @ X ) ) ) ) ) ) ) ).
% le_SUP_iff
thf(fact_5808_SUP__eq__iff,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I5: set @ B,C2: A,F: B > A] :
( ( I5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ I5 )
=> ( ord_less_eq @ A @ C2 @ ( F @ I3 ) ) )
=> ( ( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F @ I5 ) )
= C2 )
= ( ! [X: B] :
( ( member @ B @ X @ I5 )
=> ( ( F @ X )
= C2 ) ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_5809_Sup__inter__less__eq,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ A,B5: set @ A] : ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( inf_inf @ ( set @ A ) @ A5 @ B5 ) ) @ ( inf_inf @ A @ ( complete_Sup_Sup @ A @ A5 ) @ ( complete_Sup_Sup @ A @ B5 ) ) ) ) ).
% Sup_inter_less_eq
thf(fact_5810_SUP__subset__mono,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ B,B5: set @ B,F: B > A,G: B > A] :
( ( ord_less_eq @ ( set @ B ) @ A5 @ B5 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ A5 )
=> ( ord_less_eq @ A @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F @ A5 ) ) @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ G @ B5 ) ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_5811_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_5812_SUP__empty,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F: B > A] :
( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F @ ( bot_bot @ ( set @ B ) ) ) )
= ( bot_bot @ A ) ) ) ).
% SUP_empty
thf(fact_5813_UN__extend__simps_I1_J,axiom,
! [A: $tType,B: $tType,C6: set @ B,A2: A,B5: B > ( set @ A )] :
( ( ( C6
= ( bot_bot @ ( set @ B ) ) )
=> ( ( insert @ A @ A2 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B5 @ C6 ) ) )
= ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
& ( ( C6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( insert @ A @ A2 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B5 @ C6 ) ) )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [X: B] : ( insert @ A @ A2 @ ( B5 @ X ) )
@ C6 ) ) ) ) ) ).
% UN_extend_simps(1)
thf(fact_5814_SUP__UNION,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F: B > A,G: C > ( set @ B ),A5: set @ C] :
( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ C @ ( set @ B ) @ G @ A5 ) ) ) )
= ( complete_Sup_Sup @ A
@ ( image2 @ C @ A
@ ^ [Y4: C] : ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F @ ( G @ Y4 ) ) )
@ A5 ) ) ) ) ).
% SUP_UNION
thf(fact_5815_Collect__ex__eq,axiom,
! [A: $tType,B: $tType,P: A > B > $o] :
( ( collect @ A
@ ^ [X: A] :
? [X5: B] : ( P @ X @ X5 ) )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [Y4: B] :
( collect @ A
@ ^ [X: A] : ( P @ X @ Y4 ) )
@ ( top_top @ ( set @ B ) ) ) ) ) ).
% Collect_ex_eq
thf(fact_5816_UNION__singleton__eq__range,axiom,
! [A: $tType,B: $tType,F: B > A,A5: set @ B] :
( ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [X: B] : ( insert @ A @ ( F @ X ) @ ( bot_bot @ ( set @ A ) ) )
@ A5 ) )
= ( image2 @ B @ A @ F @ A5 ) ) ).
% UNION_singleton_eq_range
thf(fact_5817_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_5818_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 ),X8: set @ ( product_prod @ B @ C )] :
( ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ ( product_prod @ B @ C ) @ ( set @ A )
@ ( product_case_prod @ B @ C @ ( set @ A )
@ ^ [X12: B,X24: C] : ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ ( product_prod @ D @ E ) @ ( set @ A ) @ ( product_case_prod @ D @ E @ ( set @ A ) @ ( A5 @ X12 @ X24 ) ) @ Y7 ) ) )
@ X8 ) )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ ( product_prod @ B @ C ) @ ( set @ A )
@ ^ [X: 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 )
@ ^ [X12: B,X24: C] : ( product_case_prod @ D @ E @ ( set @ A ) @ ( A5 @ X12 @ X24 ) @ Y4 )
@ X )
@ Y7 ) )
@ X8 ) ) ) ).
% UN_UN_split_split_eq
thf(fact_5819_infinite__cartesian__product,axiom,
! [B: $tType,A: $tType,A5: set @ A,B5: set @ B] :
( ~ ( finite_finite2 @ A @ A5 )
=> ( ~ ( finite_finite2 @ B @ B5 )
=> ~ ( finite_finite2 @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A5
@ ^ [Uu: A] : B5 ) ) ) ) ).
% infinite_cartesian_product
thf(fact_5820_INF__identity__eq,axiom,
! [A: $tType] :
( ( complete_Inf @ A )
=> ! [A5: set @ A] :
( ( complete_Inf_Inf @ A
@ ( image2 @ A @ A
@ ^ [X: A] : X
@ A5 ) )
= ( complete_Inf_Inf @ A @ A5 ) ) ) ).
% INF_identity_eq
thf(fact_5821_INT__I,axiom,
! [B: $tType,A: $tType,A5: set @ A,B2: B,B5: A > ( set @ B )] :
( ! [X3: A] :
( ( member @ A @ X3 @ A5 )
=> ( member @ B @ B2 @ ( B5 @ X3 ) ) )
=> ( member @ B @ B2 @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ B5 @ A5 ) ) ) ) ).
% INT_I
thf(fact_5822_INT__iff,axiom,
! [A: $tType,B: $tType,B2: A,B5: B > ( set @ A ),A5: set @ B] :
( ( member @ A @ B2 @ ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B5 @ A5 ) ) )
= ( ! [X: B] :
( ( member @ B @ X @ A5 )
=> ( member @ A @ B2 @ ( B5 @ X ) ) ) ) ) ).
% INT_iff
thf(fact_5823_Inf__apply,axiom,
! [B: $tType,A: $tType] :
( ( complete_Inf @ B )
=> ( ( complete_Inf_Inf @ ( A > B ) )
= ( ^ [A8: set @ ( A > B ),X: A] :
( complete_Inf_Inf @ B
@ ( image2 @ ( A > B ) @ B
@ ^ [F2: A > B] : ( F2 @ X )
@ A8 ) ) ) ) ) ).
% Inf_apply
thf(fact_5824_SUP2__I,axiom,
! [B: $tType,A: $tType,C: $tType,A2: A,A5: set @ A,B5: A > B > C > $o,B2: B,C2: C] :
( ( member @ A @ A2 @ A5 )
=> ( ( B5 @ A2 @ B2 @ C2 )
=> ( complete_Sup_Sup @ ( B > C > $o ) @ ( image2 @ A @ ( B > C > $o ) @ B5 @ A5 ) @ B2 @ C2 ) ) ) ).
% SUP2_I
thf(fact_5825_Inf__atLeastAtMost,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X2: A,Y: A] :
( ( ord_less_eq @ A @ X2 @ Y )
=> ( ( complete_Inf_Inf @ A @ ( set_or1337092689740270186AtMost @ A @ X2 @ Y ) )
= X2 ) ) ) ).
% Inf_atLeastAtMost
thf(fact_5826_cInf__atLeastAtMost,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [Y: A,X2: A] :
( ( ord_less_eq @ A @ Y @ X2 )
=> ( ( complete_Inf_Inf @ A @ ( set_or1337092689740270186AtMost @ A @ Y @ X2 ) )
= Y ) ) ) ).
% cInf_atLeastAtMost
thf(fact_5827_Inf__atLeastLessThan,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X2: A,Y: A] :
( ( ord_less @ A @ X2 @ Y )
=> ( ( complete_Inf_Inf @ A @ ( set_or7035219750837199246ssThan @ A @ X2 @ Y ) )
= X2 ) ) ) ).
% Inf_atLeastLessThan
thf(fact_5828_SUP1__I,axiom,
! [A: $tType,B: $tType,A2: A,A5: set @ A,B5: A > B > $o,B2: B] :
( ( member @ A @ A2 @ A5 )
=> ( ( B5 @ A2 @ B2 )
=> ( complete_Sup_Sup @ ( B > $o ) @ ( image2 @ A @ ( B > $o ) @ B5 @ A5 ) @ B2 ) ) ) ).
% SUP1_I
thf(fact_5829_Inf__atMost,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X2: A] :
( ( complete_Inf_Inf @ A @ ( set_ord_atMost @ A @ X2 ) )
= ( bot_bot @ A ) ) ) ).
% Inf_atMost
thf(fact_5830_Inf__greaterThanAtMost,axiom,
! [A: $tType] :
( ( ( comple6319245703460814977attice @ A )
& ( dense_linorder @ A ) )
=> ! [X2: A,Y: A] :
( ( ord_less @ A @ X2 @ Y )
=> ( ( complete_Inf_Inf @ A @ ( set_or3652927894154168847AtMost @ A @ X2 @ Y ) )
= X2 ) ) ) ).
% Inf_greaterThanAtMost
thf(fact_5831_Inf__greaterThanLessThan,axiom,
! [A: $tType] :
( ( ( comple6319245703460814977attice @ A )
& ( dense_linorder @ A ) )
=> ! [X2: A,Y: A] :
( ( ord_less @ A @ X2 @ Y )
=> ( ( complete_Inf_Inf @ A @ ( set_or5935395276787703475ssThan @ A @ X2 @ Y ) )
= X2 ) ) ) ).
% Inf_greaterThanLessThan
thf(fact_5832_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
@ ^ [X: B] : C2
@ A5 ) )
= C2 ) ) ) ).
% cINF_const
thf(fact_5833_INF__const,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ B,F: A] :
( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [I2: B] : F
@ A5 ) )
= F ) ) ) ).
% INF_const
thf(fact_5834_INF__top,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ B] :
( ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [X: B] : ( top_top @ A )
@ A5 ) )
= ( top_top @ A ) ) ) ).
% INF_top
thf(fact_5835_INF__top__conv_I1_J,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [B5: B > A,A5: set @ B] :
( ( ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ B5 @ A5 ) )
= ( top_top @ A ) )
= ( ! [X: B] :
( ( member @ B @ X @ A5 )
=> ( ( B5 @ X )
= ( top_top @ A ) ) ) ) ) ) ).
% INF_top_conv(1)
thf(fact_5836_INF__top__conv_I2_J,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [B5: B > A,A5: set @ B] :
( ( ( top_top @ A )
= ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ B5 @ A5 ) ) )
= ( ! [X: B] :
( ( member @ B @ X @ A5 )
=> ( ( B5 @ X )
= ( top_top @ A ) ) ) ) ) ) ).
% INF_top_conv(2)
thf(fact_5837_finite__INT,axiom,
! [B: $tType,A: $tType,I5: set @ A,A5: A > ( set @ B )] :
( ? [X4: A] :
( ( member @ A @ X4 @ I5 )
& ( finite_finite2 @ B @ ( A5 @ X4 ) ) )
=> ( finite_finite2 @ B @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ A5 @ I5 ) ) ) ) ).
% finite_INT
thf(fact_5838_INF__apply,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( complete_Inf @ A )
=> ! [F: C > B > A,A5: set @ C,X2: B] :
( ( complete_Inf_Inf @ ( B > A ) @ ( image2 @ C @ ( B > A ) @ F @ A5 ) @ X2 )
= ( complete_Inf_Inf @ A
@ ( image2 @ C @ A
@ ^ [Y4: C] : ( F @ Y4 @ X2 )
@ A5 ) ) ) ) ).
% INF_apply
thf(fact_5839_INF__eq__bot__iff,axiom,
! [B: $tType,A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [F: B > A,A5: set @ B] :
( ( ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F @ A5 ) )
= ( bot_bot @ A ) )
= ( ! [X: A] :
( ( ord_less @ A @ ( bot_bot @ A ) @ X )
=> ? [Y4: B] :
( ( member @ B @ Y4 @ A5 )
& ( ord_less @ A @ ( F @ Y4 ) @ X ) ) ) ) ) ) ).
% INF_eq_bot_iff
thf(fact_5840_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_5841_INT__insert,axiom,
! [A: $tType,B: $tType,B5: B > ( set @ A ),A2: B,A5: set @ B] :
( ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B5 @ ( insert @ B @ A2 @ A5 ) ) )
= ( inf_inf @ ( set @ A ) @ ( B5 @ A2 ) @ ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B5 @ A5 ) ) ) ) ).
% INT_insert
thf(fact_5842_Compl__INT,axiom,
! [A: $tType,B: $tType,B5: B > ( set @ A ),A5: set @ B] :
( ( uminus_uminus @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B5 @ A5 ) ) )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [X: B] : ( uminus_uminus @ ( set @ A ) @ ( B5 @ X ) )
@ A5 ) ) ) ).
% Compl_INT
thf(fact_5843_Compl__UN,axiom,
! [A: $tType,B: $tType,B5: B > ( set @ A ),A5: set @ B] :
( ( uminus_uminus @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B5 @ A5 ) ) )
= ( complete_Inf_Inf @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [X: B] : ( uminus_uminus @ ( set @ A ) @ ( B5 @ X ) )
@ A5 ) ) ) ).
% Compl_UN
thf(fact_5844_Inf__atMostLessThan,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X2: A] :
( ( ord_less @ A @ ( top_top @ A ) @ X2 )
=> ( ( complete_Inf_Inf @ A @ ( set_ord_lessThan @ A @ X2 ) )
= ( bot_bot @ A ) ) ) ) ).
% Inf_atMostLessThan
thf(fact_5845_INT__simps_I1_J,axiom,
! [A: $tType,B: $tType,C6: set @ A,A5: A > ( set @ B ),B5: set @ B] :
( ( ( C6
= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Inf_Inf @ ( set @ B )
@ ( image2 @ A @ ( set @ B )
@ ^ [X: A] : ( inf_inf @ ( set @ B ) @ ( A5 @ X ) @ B5 )
@ C6 ) )
= ( top_top @ ( set @ B ) ) ) )
& ( ( C6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Inf_Inf @ ( set @ B )
@ ( image2 @ A @ ( set @ B )
@ ^ [X: A] : ( inf_inf @ ( set @ B ) @ ( A5 @ X ) @ B5 )
@ C6 ) )
= ( inf_inf @ ( set @ B ) @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ A5 @ C6 ) ) @ B5 ) ) ) ) ).
% INT_simps(1)
thf(fact_5846_INT__simps_I2_J,axiom,
! [C: $tType,D: $tType,C6: set @ D,A5: set @ C,B5: D > ( set @ C )] :
( ( ( C6
= ( bot_bot @ ( set @ D ) ) )
=> ( ( complete_Inf_Inf @ ( set @ C )
@ ( image2 @ D @ ( set @ C )
@ ^ [X: D] : ( inf_inf @ ( set @ C ) @ A5 @ ( B5 @ X ) )
@ C6 ) )
= ( top_top @ ( set @ C ) ) ) )
& ( ( C6
!= ( bot_bot @ ( set @ D ) ) )
=> ( ( complete_Inf_Inf @ ( set @ C )
@ ( image2 @ D @ ( set @ C )
@ ^ [X: D] : ( inf_inf @ ( set @ C ) @ A5 @ ( B5 @ X ) )
@ C6 ) )
= ( inf_inf @ ( set @ C ) @ A5 @ ( complete_Inf_Inf @ ( set @ C ) @ ( image2 @ D @ ( set @ C ) @ B5 @ C6 ) ) ) ) ) ) ).
% INT_simps(2)
thf(fact_5847_INT__simps_I3_J,axiom,
! [E: $tType,F7: $tType,C6: set @ E,A5: E > ( set @ F7 ),B5: set @ F7] :
( ( ( C6
= ( bot_bot @ ( set @ E ) ) )
=> ( ( complete_Inf_Inf @ ( set @ F7 )
@ ( image2 @ E @ ( set @ F7 )
@ ^ [X: E] : ( minus_minus @ ( set @ F7 ) @ ( A5 @ X ) @ B5 )
@ C6 ) )
= ( top_top @ ( set @ F7 ) ) ) )
& ( ( C6
!= ( bot_bot @ ( set @ E ) ) )
=> ( ( complete_Inf_Inf @ ( set @ F7 )
@ ( image2 @ E @ ( set @ F7 )
@ ^ [X: E] : ( minus_minus @ ( set @ F7 ) @ ( A5 @ X ) @ B5 )
@ C6 ) )
= ( minus_minus @ ( set @ F7 ) @ ( complete_Inf_Inf @ ( set @ F7 ) @ ( image2 @ E @ ( set @ F7 ) @ A5 @ C6 ) ) @ B5 ) ) ) ) ).
% INT_simps(3)
thf(fact_5848_INT__simps_I4_J,axiom,
! [G4: $tType,H8: $tType,C6: set @ H8,A5: set @ G4,B5: H8 > ( set @ G4 )] :
( ( ( C6
= ( bot_bot @ ( set @ H8 ) ) )
=> ( ( complete_Inf_Inf @ ( set @ G4 )
@ ( image2 @ H8 @ ( set @ G4 )
@ ^ [X: H8] : ( minus_minus @ ( set @ G4 ) @ A5 @ ( B5 @ X ) )
@ C6 ) )
= ( top_top @ ( set @ G4 ) ) ) )
& ( ( C6
!= ( bot_bot @ ( set @ H8 ) ) )
=> ( ( complete_Inf_Inf @ ( set @ G4 )
@ ( image2 @ H8 @ ( set @ G4 )
@ ^ [X: H8] : ( minus_minus @ ( set @ G4 ) @ A5 @ ( B5 @ X ) )
@ C6 ) )
= ( minus_minus @ ( set @ G4 ) @ A5 @ ( complete_Sup_Sup @ ( set @ G4 ) @ ( image2 @ H8 @ ( set @ G4 ) @ B5 @ C6 ) ) ) ) ) ) ).
% INT_simps(4)
thf(fact_5849_cInf__le__finite,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X8: set @ A,X2: A] :
( ( finite_finite2 @ A @ X8 )
=> ( ( member @ A @ X2 @ X8 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ X8 ) @ X2 ) ) ) ) ).
% cInf_le_finite
thf(fact_5850_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_5851_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_5852_INF__commute,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F: B > C > A,B5: set @ C,A5: set @ B] :
( ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [I2: B] : ( complete_Inf_Inf @ A @ ( image2 @ C @ A @ ( F @ I2 ) @ B5 ) )
@ A5 ) )
= ( complete_Inf_Inf @ A
@ ( image2 @ C @ A
@ ^ [J3: C] :
( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [I2: B] : ( F @ I2 @ J3 )
@ A5 ) )
@ B5 ) ) ) ) ).
% INF_commute
thf(fact_5853_INT__D,axiom,
! [A: $tType,B: $tType,B2: A,B5: B > ( set @ A ),A5: set @ B,A2: B] :
( ( member @ A @ B2 @ ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B5 @ A5 ) ) )
=> ( ( member @ B @ A2 @ A5 )
=> ( member @ A @ B2 @ ( B5 @ A2 ) ) ) ) ).
% INT_D
thf(fact_5854_INT__E,axiom,
! [A: $tType,B: $tType,B2: A,B5: B > ( set @ A ),A5: set @ B,A2: B] :
( ( member @ A @ B2 @ ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B5 @ A5 ) ) )
=> ( ~ ( member @ A @ B2 @ ( B5 @ A2 ) )
=> ~ ( member @ B @ A2 @ A5 ) ) ) ).
% INT_E
thf(fact_5855_SUP1__E,axiom,
! [B: $tType,A: $tType,B5: B > A > $o,A5: set @ B,B2: A] :
( ( complete_Sup_Sup @ ( A > $o ) @ ( image2 @ B @ ( A > $o ) @ B5 @ A5 ) @ B2 )
=> ~ ! [X3: B] :
( ( member @ B @ X3 @ A5 )
=> ~ ( B5 @ X3 @ B2 ) ) ) ).
% SUP1_E
thf(fact_5856_SUP2__E,axiom,
! [A: $tType,C: $tType,B: $tType,B5: C > A > B > $o,A5: set @ C,B2: A,C2: B] :
( ( complete_Sup_Sup @ ( A > B > $o ) @ ( image2 @ C @ ( A > B > $o ) @ B5 @ A5 ) @ B2 @ C2 )
=> ~ ! [X3: C] :
( ( member @ C @ X3 @ A5 )
=> ~ ( B5 @ X3 @ B2 @ C2 ) ) ) ).
% SUP2_E
thf(fact_5857_Inf__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( complete_Inf @ B )
=> ( ( complete_Inf_Inf @ ( A > B ) )
= ( ^ [A8: set @ ( A > B ),X: A] :
( complete_Inf_Inf @ B
@ ( image2 @ ( A > B ) @ B
@ ^ [F2: A > B] : ( F2 @ X )
@ A8 ) ) ) ) ) ).
% Inf_fun_def
thf(fact_5858_Some__INF,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F: B > A,A5: set @ B] :
( ( some @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F @ A5 ) ) )
= ( complete_Inf_Inf @ ( option @ A )
@ ( image2 @ B @ ( option @ A )
@ ^ [X: B] : ( some @ A @ ( F @ X ) )
@ A5 ) ) ) ) ).
% Some_INF
thf(fact_5859_INF__eq,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ B,B5: set @ C,G: C > A,F: B > A] :
( ! [I3: B] :
( ( member @ B @ I3 @ A5 )
=> ? [X4: C] :
( ( member @ C @ X4 @ B5 )
& ( ord_less_eq @ A @ ( G @ X4 ) @ ( F @ I3 ) ) ) )
=> ( ! [J2: C] :
( ( member @ C @ J2 @ B5 )
=> ? [X4: B] :
( ( member @ B @ X4 @ A5 )
& ( ord_less_eq @ A @ ( F @ X4 ) @ ( G @ J2 ) ) ) )
=> ( ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F @ A5 ) )
= ( complete_Inf_Inf @ A @ ( image2 @ C @ A @ G @ B5 ) ) ) ) ) ) ).
% INF_eq
thf(fact_5860_cInf__greatest,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X8: set @ A,Z2: A] :
( ( X8
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ X8 )
=> ( ord_less_eq @ A @ Z2 @ X3 ) )
=> ( ord_less_eq @ A @ Z2 @ ( complete_Inf_Inf @ A @ X8 ) ) ) ) ) ).
% cInf_greatest
thf(fact_5861_cInf__eq__non__empty,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X8: set @ A,A2: A] :
( ( X8
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ X8 )
=> ( ord_less_eq @ A @ A2 @ X3 ) )
=> ( ! [Y2: A] :
( ! [X4: A] :
( ( member @ A @ X4 @ X8 )
=> ( ord_less_eq @ A @ Y2 @ X4 ) )
=> ( ord_less_eq @ A @ Y2 @ A2 ) )
=> ( ( complete_Inf_Inf @ A @ X8 )
= A2 ) ) ) ) ) ).
% cInf_eq_non_empty
thf(fact_5862_Inf__less__eq,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ A,U: A] :
( ! [V5: A] :
( ( member @ A @ V5 @ A5 )
=> ( ord_less_eq @ A @ V5 @ U ) )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A5 ) @ U ) ) ) ) ).
% Inf_less_eq
thf(fact_5863_Inf__le__iff,axiom,
! [A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [A5: set @ A,X2: A] :
( ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A5 ) @ X2 )
= ( ! [Y4: A] :
( ( ord_less @ A @ X2 @ Y4 )
=> ? [X: A] :
( ( member @ A @ X @ A5 )
& ( ord_less @ A @ X @ Y4 ) ) ) ) ) ) ).
% Inf_le_iff
thf(fact_5864_Inf__eqI,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ A,X2: A] :
( ! [I3: A] :
( ( member @ A @ I3 @ A5 )
=> ( ord_less_eq @ A @ X2 @ I3 ) )
=> ( ! [Y2: A] :
( ! [I4: A] :
( ( member @ A @ I4 @ A5 )
=> ( ord_less_eq @ A @ Y2 @ I4 ) )
=> ( ord_less_eq @ A @ Y2 @ X2 ) )
=> ( ( complete_Inf_Inf @ A @ A5 )
= X2 ) ) ) ) ).
% Inf_eqI
thf(fact_5865_Inf__mono,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [B5: set @ A,A5: set @ A] :
( ! [B4: A] :
( ( member @ A @ B4 @ B5 )
=> ? [X4: A] :
( ( member @ A @ X4 @ A5 )
& ( ord_less_eq @ A @ X4 @ B4 ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A5 ) @ ( complete_Inf_Inf @ A @ B5 ) ) ) ) ).
% Inf_mono
thf(fact_5866_Inf__lower,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X2: A,A5: set @ A] :
( ( member @ A @ X2 @ A5 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A5 ) @ X2 ) ) ) ).
% Inf_lower
thf(fact_5867_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_5868_le__Inf__iff,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [B2: A,A5: set @ A] :
( ( ord_less_eq @ A @ B2 @ ( complete_Inf_Inf @ A @ A5 ) )
= ( ! [X: A] :
( ( member @ A @ X @ A5 )
=> ( ord_less_eq @ A @ B2 @ X ) ) ) ) ) ).
% le_Inf_iff
thf(fact_5869_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_5870_cInf__eq,axiom,
! [A: $tType] :
( ( ( condit1219197933456340205attice @ A )
& ( no_top @ A ) )
=> ! [X8: set @ A,A2: A] :
( ! [X3: A] :
( ( member @ A @ X3 @ X8 )
=> ( ord_less_eq @ A @ A2 @ X3 ) )
=> ( ! [Y2: A] :
( ! [X4: A] :
( ( member @ A @ X4 @ X8 )
=> ( ord_less_eq @ A @ Y2 @ X4 ) )
=> ( ord_less_eq @ A @ Y2 @ A2 ) )
=> ( ( complete_Inf_Inf @ A @ X8 )
= A2 ) ) ) ) ).
% cInf_eq
thf(fact_5871_cInf__eq__minimum,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [Z2: A,X8: set @ A] :
( ( member @ A @ Z2 @ X8 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ X8 )
=> ( ord_less_eq @ A @ Z2 @ X3 ) )
=> ( ( complete_Inf_Inf @ A @ X8 )
= Z2 ) ) ) ) ).
% cInf_eq_minimum
thf(fact_5872_Inf__superset__mono,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [B5: set @ A,A5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B5 @ A5 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A5 ) @ ( complete_Inf_Inf @ A @ B5 ) ) ) ) ).
% Inf_superset_mono
thf(fact_5873_INF__eqI,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ B,X2: A,F: B > A] :
( ! [I3: B] :
( ( member @ B @ I3 @ A5 )
=> ( ord_less_eq @ A @ X2 @ ( F @ I3 ) ) )
=> ( ! [Y2: A] :
( ! [I4: B] :
( ( member @ B @ I4 @ A5 )
=> ( ord_less_eq @ A @ Y2 @ ( F @ I4 ) ) )
=> ( ord_less_eq @ A @ Y2 @ X2 ) )
=> ( ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F @ A5 ) )
= X2 ) ) ) ) ).
% INF_eqI
thf(fact_5874_INF__mono,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [B5: set @ B,A5: set @ C,F: C > A,G: B > A] :
( ! [M4: B] :
( ( member @ B @ M4 @ B5 )
=> ? [X4: C] :
( ( member @ C @ X4 @ A5 )
& ( ord_less_eq @ A @ ( F @ X4 ) @ ( G @ M4 ) ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ C @ A @ F @ A5 ) ) @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ G @ B5 ) ) ) ) ) ).
% INF_mono
thf(fact_5875_INF__lower,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I: B,A5: set @ B,F: B > A] :
( ( member @ B @ I @ A5 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F @ A5 ) ) @ ( F @ I ) ) ) ) ).
% INF_lower
thf(fact_5876_INF__mono_H,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F: B > A,G: B > A,A5: set @ B] :
( ! [X3: B] : ( ord_less_eq @ A @ ( F @ X3 ) @ ( G @ X3 ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F @ A5 ) ) @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ G @ A5 ) ) ) ) ) ).
% INF_mono'
thf(fact_5877_INF__lower2,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I: B,A5: set @ B,F: B > A,U: A] :
( ( member @ B @ I @ A5 )
=> ( ( ord_less_eq @ A @ ( F @ I ) @ U )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F @ A5 ) ) @ U ) ) ) ) ).
% INF_lower2
thf(fact_5878_le__INF__iff,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [U: A,F: B > A,A5: set @ B] :
( ( ord_less_eq @ A @ U @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F @ A5 ) ) )
= ( ! [X: B] :
( ( member @ B @ X @ A5 )
=> ( ord_less_eq @ A @ U @ ( F @ X ) ) ) ) ) ) ).
% le_INF_iff
thf(fact_5879_INF__greatest,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ B,U: A,F: B > A] :
( ! [I3: B] :
( ( member @ B @ I3 @ A5 )
=> ( ord_less_eq @ A @ U @ ( F @ I3 ) ) )
=> ( ord_less_eq @ A @ U @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F @ A5 ) ) ) ) ) ).
% INF_greatest
thf(fact_5880_less__INF__D,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [Y: A,F: B > A,A5: set @ B,I: B] :
( ( ord_less @ A @ Y @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F @ A5 ) ) )
=> ( ( member @ B @ I @ A5 )
=> ( ord_less @ A @ Y @ ( F @ I ) ) ) ) ) ).
% less_INF_D
thf(fact_5881_INF__less__iff,axiom,
! [A: $tType,B: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [F: B > A,A5: set @ B,A2: A] :
( ( ord_less @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F @ A5 ) ) @ A2 )
= ( ? [X: B] :
( ( member @ B @ X @ A5 )
& ( ord_less @ A @ ( F @ X ) @ A2 ) ) ) ) ) ).
% INF_less_iff
thf(fact_5882_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_5883_INF__inf__distrib,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F: B > A,A5: set @ B,G: B > A] :
( ( inf_inf @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F @ A5 ) ) @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ G @ A5 ) ) )
= ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [A3: B] : ( inf_inf @ A @ ( F @ A3 ) @ ( G @ A3 ) )
@ A5 ) ) ) ) ).
% INF_inf_distrib
thf(fact_5884_INT__extend__simps_I10_J,axiom,
! [V7: $tType,U6: $tType,T: $tType,B5: U6 > ( set @ V7 ),F: T > U6,A5: set @ T] :
( ( complete_Inf_Inf @ ( set @ V7 )
@ ( image2 @ T @ ( set @ V7 )
@ ^ [A3: T] : ( B5 @ ( F @ A3 ) )
@ A5 ) )
= ( complete_Inf_Inf @ ( set @ V7 ) @ ( image2 @ U6 @ ( set @ V7 ) @ B5 @ ( image2 @ T @ U6 @ F @ A5 ) ) ) ) ).
% INT_extend_simps(10)
thf(fact_5885_INT__subset__iff,axiom,
! [A: $tType,B: $tType,B5: set @ A,A5: B > ( set @ A ),I5: set @ B] :
( ( ord_less_eq @ ( set @ A ) @ B5 @ ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ A5 @ I5 ) ) )
= ( ! [X: B] :
( ( member @ B @ X @ I5 )
=> ( ord_less_eq @ ( set @ A ) @ B5 @ ( A5 @ X ) ) ) ) ) ).
% INT_subset_iff
thf(fact_5886_INT__anti__mono,axiom,
! [B: $tType,A: $tType,A5: set @ A,B5: set @ A,F: A > ( set @ B ),G: A > ( set @ B )] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ A5 )
=> ( ord_less_eq @ ( set @ B ) @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_eq @ ( set @ B ) @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ F @ B5 ) ) @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ G @ A5 ) ) ) ) ) ).
% INT_anti_mono
thf(fact_5887_INT__greatest,axiom,
! [B: $tType,A: $tType,A5: set @ A,C6: set @ B,B5: A > ( set @ B )] :
( ! [X3: A] :
( ( member @ A @ X3 @ A5 )
=> ( ord_less_eq @ ( set @ B ) @ C6 @ ( B5 @ X3 ) ) )
=> ( ord_less_eq @ ( set @ B ) @ C6 @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ B5 @ A5 ) ) ) ) ).
% INT_greatest
thf(fact_5888_INT__lower,axiom,
! [B: $tType,A: $tType,A2: A,A5: set @ A,B5: A > ( set @ B )] :
( ( member @ A @ A2 @ A5 )
=> ( ord_less_eq @ ( set @ B ) @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ B5 @ A5 ) ) @ ( B5 @ A2 ) ) ) ).
% INT_lower
thf(fact_5889_INT__extend__simps_I5_J,axiom,
! [I6: $tType,J5: $tType,A2: I6,B5: J5 > ( set @ I6 ),C6: set @ J5] :
( ( insert @ I6 @ A2 @ ( complete_Inf_Inf @ ( set @ I6 ) @ ( image2 @ J5 @ ( set @ I6 ) @ B5 @ C6 ) ) )
= ( complete_Inf_Inf @ ( set @ I6 )
@ ( image2 @ J5 @ ( set @ I6 )
@ ^ [X: J5] : ( insert @ I6 @ A2 @ ( B5 @ X ) )
@ C6 ) ) ) ).
% INT_extend_simps(5)
thf(fact_5890_INT__insert__distrib,axiom,
! [B: $tType,A: $tType,U: A,A5: set @ A,A2: B,B5: A > ( set @ B )] :
( ( member @ A @ U @ A5 )
=> ( ( complete_Inf_Inf @ ( set @ B )
@ ( image2 @ A @ ( set @ B )
@ ^ [X: A] : ( insert @ B @ A2 @ ( B5 @ X ) )
@ A5 ) )
= ( insert @ B @ A2 @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ B5 @ A5 ) ) ) ) ) ).
% INT_insert_distrib
thf(fact_5891_INTER__UNIV__conv_I1_J,axiom,
! [A: $tType,B: $tType,B5: B > ( set @ A ),A5: set @ B] :
( ( ( top_top @ ( set @ A ) )
= ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B5 @ A5 ) ) )
= ( ! [X: B] :
( ( member @ B @ X @ A5 )
=> ( ( B5 @ X )
= ( top_top @ ( set @ A ) ) ) ) ) ) ).
% INTER_UNIV_conv(1)
thf(fact_5892_INTER__UNIV__conv_I2_J,axiom,
! [A: $tType,B: $tType,B5: B > ( set @ A ),A5: set @ B] :
( ( ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B5 @ A5 ) )
= ( top_top @ ( set @ A ) ) )
= ( ! [X: B] :
( ( member @ B @ X @ A5 )
=> ( ( B5 @ X )
= ( top_top @ ( set @ A ) ) ) ) ) ) ).
% INTER_UNIV_conv(2)
thf(fact_5893_Int__Inter__image,axiom,
! [A: $tType,B: $tType,A5: B > ( set @ A ),B5: B > ( set @ A ),C6: set @ B] :
( ( complete_Inf_Inf @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [X: B] : ( inf_inf @ ( set @ A ) @ ( A5 @ X ) @ ( B5 @ X ) )
@ C6 ) )
= ( inf_inf @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ A5 @ C6 ) ) @ ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B5 @ C6 ) ) ) ) ).
% Int_Inter_image
thf(fact_5894_INT__Int__distrib,axiom,
! [A: $tType,B: $tType,A5: B > ( set @ A ),B5: B > ( set @ A ),I5: set @ B] :
( ( complete_Inf_Inf @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [I2: B] : ( inf_inf @ ( set @ A ) @ ( A5 @ I2 ) @ ( B5 @ I2 ) )
@ I5 ) )
= ( inf_inf @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ A5 @ I5 ) ) @ ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B5 @ I5 ) ) ) ) ).
% INT_Int_distrib
thf(fact_5895_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_5896_INT__extend__simps_I9_J,axiom,
! [S9: $tType,R9: $tType,Q7: $tType,C6: R9 > ( set @ S9 ),B5: Q7 > ( set @ R9 ),A5: set @ Q7] :
( ( complete_Inf_Inf @ ( set @ S9 )
@ ( image2 @ Q7 @ ( set @ S9 )
@ ^ [X: Q7] : ( complete_Inf_Inf @ ( set @ S9 ) @ ( image2 @ R9 @ ( set @ S9 ) @ C6 @ ( B5 @ X ) ) )
@ A5 ) )
= ( complete_Inf_Inf @ ( set @ S9 ) @ ( image2 @ R9 @ ( set @ S9 ) @ C6 @ ( complete_Sup_Sup @ ( set @ R9 ) @ ( image2 @ Q7 @ ( set @ R9 ) @ B5 @ A5 ) ) ) ) ) ).
% INT_extend_simps(9)
thf(fact_5897_Int__Inter__eq_I1_J,axiom,
! [A: $tType,B11: set @ ( set @ A ),A5: set @ A] :
( ( ( B11
= ( bot_bot @ ( set @ ( set @ A ) ) ) )
=> ( ( inf_inf @ ( set @ A ) @ A5 @ ( complete_Inf_Inf @ ( set @ A ) @ B11 ) )
= A5 ) )
& ( ( B11
!= ( bot_bot @ ( set @ ( set @ A ) ) ) )
=> ( ( inf_inf @ ( set @ A ) @ A5 @ ( complete_Inf_Inf @ ( set @ A ) @ B11 ) )
= ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ ( set @ A ) @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A5 ) @ B11 ) ) ) ) ) ).
% Int_Inter_eq(1)
thf(fact_5898_Int__Inter__eq_I2_J,axiom,
! [A: $tType,B11: set @ ( set @ A ),A5: set @ A] :
( ( ( B11
= ( bot_bot @ ( set @ ( set @ A ) ) ) )
=> ( ( inf_inf @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ B11 ) @ A5 )
= A5 ) )
& ( ( B11
!= ( bot_bot @ ( set @ ( set @ A ) ) ) )
=> ( ( inf_inf @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ B11 ) @ A5 )
= ( complete_Inf_Inf @ ( set @ A )
@ ( image2 @ ( set @ A ) @ ( set @ A )
@ ^ [B8: set @ A] : ( inf_inf @ ( set @ A ) @ B8 @ A5 )
@ B11 ) ) ) ) ) ).
% Int_Inter_eq(2)
thf(fact_5899_INF__le__iff,axiom,
! [B: $tType,A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [F: B > A,A5: set @ B,X2: A] :
( ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F @ A5 ) ) @ X2 )
= ( ! [Y4: A] :
( ( ord_less @ A @ X2 @ Y4 )
=> ? [X: B] :
( ( member @ B @ X @ A5 )
& ( ord_less @ A @ ( F @ X ) @ Y4 ) ) ) ) ) ) ).
% INF_le_iff
thf(fact_5900_cINF__greatest,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A5: set @ B,M: A,F: B > A] :
( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ A5 )
=> ( ord_less_eq @ A @ M @ ( F @ X3 ) ) )
=> ( ord_less_eq @ A @ M @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F @ A5 ) ) ) ) ) ) ).
% cINF_greatest
thf(fact_5901_INF__eq__iff,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I5: set @ B,F: B > A,C2: A] :
( ( I5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ I5 )
=> ( ord_less_eq @ A @ ( F @ I3 ) @ C2 ) )
=> ( ( ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F @ I5 ) )
= C2 )
= ( ! [X: B] :
( ( member @ B @ X @ I5 )
=> ( ( F @ X )
= C2 ) ) ) ) ) ) ) ).
% INF_eq_iff
thf(fact_5902_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_5903_cInf__abs__ge,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( linordered_idom @ A ) )
=> ! [S2: set @ A,A2: A] :
( ( S2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ S2 )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ X3 ) @ A2 ) )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ ( complete_Inf_Inf @ A @ S2 ) ) @ A2 ) ) ) ) ).
% cInf_abs_ge
thf(fact_5904_INF__superset__mono,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [B5: set @ B,A5: set @ B,F: B > A,G: B > A] :
( ( ord_less_eq @ ( set @ B ) @ B5 @ A5 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ B5 )
=> ( ord_less_eq @ A @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F @ A5 ) ) @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ G @ B5 ) ) ) ) ) ) ).
% INF_superset_mono
thf(fact_5905_INF__empty,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F: B > A] :
( ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F @ ( bot_bot @ ( set @ B ) ) ) )
= ( top_top @ A ) ) ) ).
% INF_empty
thf(fact_5906_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_5907_INF__inf__const1,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I5: set @ B,X2: A,F: B > A] :
( ( I5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [I2: B] : ( inf_inf @ A @ X2 @ ( F @ I2 ) )
@ I5 ) )
= ( inf_inf @ A @ X2 @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F @ I5 ) ) ) ) ) ) ).
% INF_inf_const1
thf(fact_5908_INF__inf__const2,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I5: set @ B,F: B > A,X2: A] :
( ( I5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [I2: B] : ( inf_inf @ A @ ( F @ I2 ) @ X2 )
@ I5 ) )
= ( inf_inf @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F @ I5 ) ) @ X2 ) ) ) ) ).
% INF_inf_const2
thf(fact_5909_uminus__SUP,axiom,
! [A: $tType,B: $tType] :
( ( comple489889107523837845lgebra @ A )
=> ! [B5: B > A,A5: set @ B] :
( ( uminus_uminus @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ B5 @ A5 ) ) )
= ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [X: B] : ( uminus_uminus @ A @ ( B5 @ X ) )
@ A5 ) ) ) ) ).
% uminus_SUP
thf(fact_5910_uminus__INF,axiom,
! [A: $tType,B: $tType] :
( ( comple489889107523837845lgebra @ A )
=> ! [B5: B > A,A5: set @ B] :
( ( uminus_uminus @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ B5 @ A5 ) ) )
= ( complete_Sup_Sup @ A
@ ( image2 @ B @ A
@ ^ [X: B] : ( uminus_uminus @ A @ ( B5 @ X ) )
@ A5 ) ) ) ) ).
% uminus_INF
thf(fact_5911_INF__insert,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F: B > A,A2: B,A5: set @ B] :
( ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F @ ( insert @ B @ A2 @ A5 ) ) )
= ( inf_inf @ A @ ( F @ A2 ) @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F @ A5 ) ) ) ) ) ).
% INF_insert
thf(fact_5912_INT__empty,axiom,
! [B: $tType,A: $tType,B5: B > ( set @ A )] :
( ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B5 @ ( bot_bot @ ( set @ B ) ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% INT_empty
thf(fact_5913_INT__extend__simps_I2_J,axiom,
! [C: $tType,D: $tType,C6: set @ D,A5: set @ C,B5: D > ( set @ C )] :
( ( ( C6
= ( bot_bot @ ( set @ D ) ) )
=> ( ( inf_inf @ ( set @ C ) @ A5 @ ( complete_Inf_Inf @ ( set @ C ) @ ( image2 @ D @ ( set @ C ) @ B5 @ C6 ) ) )
= A5 ) )
& ( ( C6
!= ( bot_bot @ ( set @ D ) ) )
=> ( ( inf_inf @ ( set @ C ) @ A5 @ ( complete_Inf_Inf @ ( set @ C ) @ ( image2 @ D @ ( set @ C ) @ B5 @ C6 ) ) )
= ( complete_Inf_Inf @ ( set @ C )
@ ( image2 @ D @ ( set @ C )
@ ^ [X: D] : ( inf_inf @ ( set @ C ) @ A5 @ ( B5 @ X ) )
@ C6 ) ) ) ) ) ).
% INT_extend_simps(2)
thf(fact_5914_INT__extend__simps_I1_J,axiom,
! [B: $tType,A: $tType,C6: set @ A,A5: A > ( set @ B ),B5: set @ B] :
( ( ( C6
= ( bot_bot @ ( set @ A ) ) )
=> ( ( inf_inf @ ( set @ B ) @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ A5 @ C6 ) ) @ B5 )
= B5 ) )
& ( ( C6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( inf_inf @ ( set @ B ) @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ A5 @ C6 ) ) @ B5 )
= ( complete_Inf_Inf @ ( set @ B )
@ ( image2 @ A @ ( set @ B )
@ ^ [X: A] : ( inf_inf @ ( set @ B ) @ ( A5 @ X ) @ B5 )
@ C6 ) ) ) ) ) ).
% INT_extend_simps(1)
thf(fact_5915_UN__extend__simps_I7_J,axiom,
! [M15: $tType,N14: $tType,A5: set @ M15,B5: N14 > ( set @ M15 ),C6: set @ N14] :
( ( minus_minus @ ( set @ M15 ) @ A5 @ ( complete_Inf_Inf @ ( set @ M15 ) @ ( image2 @ N14 @ ( set @ M15 ) @ B5 @ C6 ) ) )
= ( complete_Sup_Sup @ ( set @ M15 )
@ ( image2 @ N14 @ ( set @ M15 )
@ ^ [X: N14] : ( minus_minus @ ( set @ M15 ) @ A5 @ ( B5 @ X ) )
@ C6 ) ) ) ).
% UN_extend_simps(7)
thf(fact_5916_INT__extend__simps_I8_J,axiom,
! [P10: $tType,O2: $tType,B5: O2 > ( set @ P10 ),A5: set @ ( set @ O2 )] :
( ( complete_Inf_Inf @ ( set @ P10 )
@ ( image2 @ ( set @ O2 ) @ ( set @ P10 )
@ ^ [Y4: set @ O2] : ( complete_Inf_Inf @ ( set @ P10 ) @ ( image2 @ O2 @ ( set @ P10 ) @ B5 @ Y4 ) )
@ A5 ) )
= ( complete_Inf_Inf @ ( set @ P10 ) @ ( image2 @ O2 @ ( set @ P10 ) @ B5 @ ( complete_Sup_Sup @ ( set @ O2 ) @ A5 ) ) ) ) ).
% INT_extend_simps(8)
thf(fact_5917_INF__le__SUP,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ B,F: B > A] :
( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F @ A5 ) ) @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F @ A5 ) ) ) ) ) ).
% INF_le_SUP
thf(fact_5918_cInf__asclose,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( linordered_idom @ A ) )
=> ! [S2: set @ A,L: A,E2: A] :
( ( S2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ S2 )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X3 @ L ) ) @ E2 ) )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ ( complete_Inf_Inf @ A @ S2 ) @ L ) ) @ E2 ) ) ) ) ).
% cInf_asclose
thf(fact_5919_INT__extend__simps_I3_J,axiom,
! [F7: $tType,E: $tType,C6: set @ E,A5: E > ( set @ F7 ),B5: set @ F7] :
( ( ( C6
= ( bot_bot @ ( set @ E ) ) )
=> ( ( minus_minus @ ( set @ F7 ) @ ( complete_Inf_Inf @ ( set @ F7 ) @ ( image2 @ E @ ( set @ F7 ) @ A5 @ C6 ) ) @ B5 )
= ( minus_minus @ ( set @ F7 ) @ ( top_top @ ( set @ F7 ) ) @ B5 ) ) )
& ( ( C6
!= ( bot_bot @ ( set @ E ) ) )
=> ( ( minus_minus @ ( set @ F7 ) @ ( complete_Inf_Inf @ ( set @ F7 ) @ ( image2 @ E @ ( set @ F7 ) @ A5 @ C6 ) ) @ B5 )
= ( complete_Inf_Inf @ ( set @ F7 )
@ ( image2 @ E @ ( set @ F7 )
@ ^ [X: E] : ( minus_minus @ ( set @ F7 ) @ ( A5 @ X ) @ B5 )
@ C6 ) ) ) ) ) ).
% INT_extend_simps(3)
thf(fact_5920_INT__extend__simps_I4_J,axiom,
! [G4: $tType,H8: $tType,C6: set @ H8,A5: set @ G4,B5: H8 > ( set @ G4 )] :
( ( ( C6
= ( bot_bot @ ( set @ H8 ) ) )
=> ( ( minus_minus @ ( set @ G4 ) @ A5 @ ( complete_Sup_Sup @ ( set @ G4 ) @ ( image2 @ H8 @ ( set @ G4 ) @ B5 @ C6 ) ) )
= A5 ) )
& ( ( C6
!= ( bot_bot @ ( set @ H8 ) ) )
=> ( ( minus_minus @ ( set @ G4 ) @ A5 @ ( complete_Sup_Sup @ ( set @ G4 ) @ ( image2 @ H8 @ ( set @ G4 ) @ B5 @ C6 ) ) )
= ( complete_Inf_Inf @ ( set @ G4 )
@ ( image2 @ H8 @ ( set @ G4 )
@ ^ [X: H8] : ( minus_minus @ ( set @ G4 ) @ A5 @ ( B5 @ X ) )
@ C6 ) ) ) ) ) ).
% INT_extend_simps(4)
thf(fact_5921_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
@ ^ [X: C] : ( P @ X @ Y4 )
@ ( top_top @ ( set @ C ) ) ) )
@ ( top_top @ ( set @ B ) ) ) )
= ( complete_Inf_Inf @ A
@ ( image2 @ ( B > C ) @ A
@ ^ [X: B > C] :
( complete_Sup_Sup @ A
@ ( image2 @ B @ A
@ ^ [Y4: B] : ( P @ ( X @ Y4 ) @ Y4 )
@ ( top_top @ ( set @ B ) ) ) )
@ ( top_top @ ( set @ ( B > C ) ) ) ) ) ) ) ).
% SUP_INF
thf(fact_5922_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
@ ^ [X: C] : ( P @ X @ Y4 )
@ ( top_top @ ( set @ C ) ) ) )
@ ( top_top @ ( set @ B ) ) ) )
= ( complete_Sup_Sup @ A
@ ( image2 @ ( B > C ) @ A
@ ^ [F2: B > C] :
( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [X: B] : ( P @ ( F2 @ X ) @ X )
@ ( top_top @ ( set @ B ) ) ) )
@ ( top_top @ ( set @ ( B > C ) ) ) ) ) ) ) ).
% INF_SUP
thf(fact_5923_UN__constant__eq,axiom,
! [A: $tType,B: $tType,A2: A,A5: set @ A,F: A > ( set @ B ),C2: set @ B] :
( ( member @ A @ A2 @ A5 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ A5 )
=> ( ( F @ X3 )
= C2 ) )
=> ( ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ F @ A5 ) )
= C2 ) ) ) ).
% UN_constant_eq
thf(fact_5924_relChain__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ( ( bNF_Ca3754400796208372196lChain @ A @ B )
= ( ^ [R2: set @ ( product_prod @ A @ A ),As3: A > B] :
! [I2: A,J3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ I2 @ J3 ) @ R2 )
=> ( ord_less_eq @ B @ ( As3 @ I2 ) @ ( As3 @ J3 ) ) ) ) ) ) ).
% relChain_def
thf(fact_5925_Restr__subset,axiom,
! [A: $tType,A5: set @ A,B5: set @ A,R: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
=> ( ( inf_inf @ ( set @ ( product_prod @ A @ A ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R
@ ( product_Sigma @ A @ A @ B5
@ ^ [Uu: A] : B5 ) )
@ ( product_Sigma @ A @ A @ A5
@ ^ [Uu: A] : A5 ) )
= ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R
@ ( product_Sigma @ A @ A @ A5
@ ^ [Uu: A] : A5 ) ) ) ) ).
% Restr_subset
thf(fact_5926_finite__inf__Sup,axiom,
! [A: $tType] :
( ( finite8700451911770168679attice @ A )
=> ! [A2: A,A5: set @ A] :
( ( inf_inf @ A @ A2 @ ( complete_Sup_Sup @ A @ A5 ) )
= ( complete_Sup_Sup @ A
@ ( collect @ A
@ ^ [Uu: A] :
? [B3: A] :
( ( Uu
= ( inf_inf @ A @ A2 @ B3 ) )
& ( member @ A @ B3 @ A5 ) ) ) ) ) ) ).
% finite_inf_Sup
thf(fact_5927_INF2__I,axiom,
! [B: $tType,A: $tType,C: $tType,A5: set @ A,B5: A > B > C > $o,B2: B,C2: C] :
( ! [X3: A] :
( ( member @ A @ X3 @ A5 )
=> ( B5 @ X3 @ B2 @ C2 ) )
=> ( complete_Inf_Inf @ ( B > C > $o ) @ ( image2 @ A @ ( B > C > $o ) @ B5 @ A5 ) @ B2 @ C2 ) ) ).
% INF2_I
thf(fact_5928_INF1__I,axiom,
! [A: $tType,B: $tType,A5: set @ A,B5: A > B > $o,B2: B] :
( ! [X3: A] :
( ( member @ A @ X3 @ A5 )
=> ( B5 @ X3 @ B2 ) )
=> ( complete_Inf_Inf @ ( B > $o ) @ ( image2 @ A @ ( B > $o ) @ B5 @ A5 ) @ B2 ) ) ).
% INF1_I
thf(fact_5929_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 )
@ ^ [X: B] : C2
@ A5 ) )
= C2 ) ) ).
% subset_mset.cINF_const
thf(fact_5930_INF1__D,axiom,
! [B: $tType,A: $tType,B5: B > A > $o,A5: set @ B,B2: A,A2: B] :
( ( complete_Inf_Inf @ ( A > $o ) @ ( image2 @ B @ ( A > $o ) @ B5 @ A5 ) @ B2 )
=> ( ( member @ B @ A2 @ A5 )
=> ( B5 @ A2 @ B2 ) ) ) ).
% INF1_D
thf(fact_5931_INF1__E,axiom,
! [A: $tType,B: $tType,B5: B > A > $o,A5: set @ B,B2: A,A2: B] :
( ( complete_Inf_Inf @ ( A > $o ) @ ( image2 @ B @ ( A > $o ) @ B5 @ A5 ) @ B2 )
=> ( ~ ( B5 @ A2 @ B2 )
=> ~ ( member @ B @ A2 @ A5 ) ) ) ).
% INF1_E
thf(fact_5932_INF2__D,axiom,
! [A: $tType,C: $tType,B: $tType,B5: C > A > B > $o,A5: set @ C,B2: A,C2: B,A2: C] :
( ( complete_Inf_Inf @ ( A > B > $o ) @ ( image2 @ C @ ( A > B > $o ) @ B5 @ A5 ) @ B2 @ C2 )
=> ( ( member @ C @ A2 @ A5 )
=> ( B5 @ A2 @ B2 @ C2 ) ) ) ).
% INF2_D
thf(fact_5933_INF2__E,axiom,
! [B: $tType,A: $tType,C: $tType,B5: C > A > B > $o,A5: set @ C,B2: A,C2: B,A2: C] :
( ( complete_Inf_Inf @ ( A > B > $o ) @ ( image2 @ C @ ( A > B > $o ) @ B5 @ A5 ) @ B2 @ C2 )
=> ( ~ ( B5 @ A2 @ B2 @ C2 )
=> ~ ( member @ C @ A2 @ A5 ) ) ) ).
% INF2_E
thf(fact_5934_INF__Int__eq2,axiom,
! [B: $tType,A: $tType,S2: set @ ( set @ ( product_prod @ A @ B ) )] :
( ( complete_Inf_Inf @ ( A > B > $o )
@ ( image2 @ ( set @ ( product_prod @ A @ B ) ) @ ( A > B > $o )
@ ^ [I2: set @ ( product_prod @ A @ B ),X: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y4 ) @ I2 )
@ S2 ) )
= ( ^ [X: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y4 ) @ ( complete_Inf_Inf @ ( set @ ( product_prod @ A @ B ) ) @ S2 ) ) ) ) ).
% INF_Int_eq2
thf(fact_5935_INF__Int__eq,axiom,
! [A: $tType,S2: set @ ( set @ A )] :
( ( complete_Inf_Inf @ ( A > $o )
@ ( image2 @ ( set @ A ) @ ( A > $o )
@ ^ [I2: set @ A,X: A] : ( member @ A @ X @ I2 )
@ S2 ) )
= ( ^ [X: A] : ( member @ A @ X @ ( complete_Inf_Inf @ ( set @ A ) @ S2 ) ) ) ) ).
% INF_Int_eq
thf(fact_5936_INF__INT__eq,axiom,
! [B: $tType,A: $tType,R: B > ( set @ A ),S2: set @ B] :
( ( complete_Inf_Inf @ ( A > $o )
@ ( image2 @ B @ ( A > $o )
@ ^ [I2: B,X: A] : ( member @ A @ X @ ( R @ I2 ) )
@ S2 ) )
= ( ^ [X: A] : ( member @ A @ X @ ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ R @ S2 ) ) ) ) ) ).
% INF_INT_eq
thf(fact_5937_INF__INT__eq2,axiom,
! [B: $tType,C: $tType,A: $tType,R: C > ( set @ ( product_prod @ A @ B ) ),S2: set @ C] :
( ( complete_Inf_Inf @ ( A > B > $o )
@ ( image2 @ C @ ( A > B > $o )
@ ^ [I2: C,X: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y4 ) @ ( R @ I2 ) )
@ S2 ) )
= ( ^ [X: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y4 ) @ ( complete_Inf_Inf @ ( set @ ( product_prod @ A @ B ) ) @ ( image2 @ C @ ( set @ ( product_prod @ A @ B ) ) @ R @ S2 ) ) ) ) ) ).
% INF_INT_eq2
thf(fact_5938_Inf__set__def,axiom,
! [A: $tType] :
( ( complete_Inf_Inf @ ( set @ A ) )
= ( ^ [A8: set @ ( set @ A )] :
( collect @ A
@ ^ [X: A] : ( complete_Inf_Inf @ $o @ ( image2 @ ( set @ A ) @ $o @ ( member @ A @ X ) @ A8 ) ) ) ) ) ).
% Inf_set_def
thf(fact_5939_Inf__INT__eq2,axiom,
! [B: $tType,A: $tType] :
( ( complete_Inf_Inf @ ( A > B > $o ) )
= ( ^ [S: set @ ( A > B > $o ),X: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ 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 ) @ S ) ) ) ) ) ) ).
% Inf_INT_eq2
thf(fact_5940_natLess__def,axiom,
( bNF_Ca8459412986667044542atLess
= ( collect @ ( product_prod @ nat @ nat ) @ ( product_case_prod @ nat @ nat @ $o @ ( ord_less @ nat ) ) ) ) ).
% natLess_def
thf(fact_5941_relpow__finite__bounded1,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),K: nat] :
( ( finite_finite2 @ ( product_prod @ A @ A ) @ R3 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ K @ R3 )
@ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ A ) )
@ ( image2 @ nat @ ( set @ ( product_prod @ A @ A ) )
@ ^ [N4: nat] : ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N4 @ R3 )
@ ( collect @ nat
@ ^ [N4: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N4 )
& ( ord_less_eq @ nat @ N4 @ ( finite_card @ ( product_prod @ A @ A ) @ R3 ) ) ) ) ) ) ) ) ) ).
% relpow_finite_bounded1
thf(fact_5942_mlex__eq,axiom,
! [A: $tType] :
( ( mlex_prod @ A )
= ( ^ [F2: A > nat,R7: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ^ [X: A,Y4: A] :
( ( ord_less @ nat @ ( F2 @ X ) @ ( F2 @ Y4 ) )
| ( ( ord_less_eq @ nat @ ( F2 @ X ) @ ( F2 @ Y4 ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y4 ) @ R7 ) ) ) ) ) ) ) ).
% mlex_eq
thf(fact_5943_relpow__1,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( one_one @ nat ) @ R3 )
= R3 ) ).
% relpow_1
thf(fact_5944_finite__relpow,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),N: nat] :
( ( finite_finite2 @ ( product_prod @ A @ A ) @ R3 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( finite_finite2 @ ( product_prod @ A @ A ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R3 ) ) ) ) ).
% finite_relpow
thf(fact_5945_relpow__0__E,axiom,
! [A: $tType,X2: A,Y: A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( zero_zero @ nat ) @ R3 ) )
=> ( X2 = Y ) ) ).
% relpow_0_E
thf(fact_5946_relpow__0__I,axiom,
! [A: $tType,X2: A,R3: set @ ( product_prod @ A @ A )] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ X2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( zero_zero @ nat ) @ R3 ) ) ).
% relpow_0_I
thf(fact_5947_relpow__add,axiom,
! [A: $tType,M: nat,N: nat,R3: set @ ( product_prod @ A @ A )] :
( ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( plus_plus @ nat @ M @ N ) @ R3 )
= ( relcomp @ A @ A @ A @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ M @ R3 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R3 ) ) ) ).
% relpow_add
thf(fact_5948_relpowp__relpow__eq,axiom,
! [A: $tType,N: nat,R3: set @ ( product_prod @ A @ A )] :
( ( compow @ ( A > A > $o ) @ N
@ ^ [X: A,Y4: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y4 ) @ R3 ) )
= ( ^ [X: A,Y4: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y4 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R3 ) ) ) ) ).
% relpowp_relpow_eq
thf(fact_5949_relpow__E2,axiom,
! [A: $tType,X2: A,Z2: A,N: nat,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Z2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R3 ) )
=> ( ( ( N
= ( zero_zero @ nat ) )
=> ( X2 != Z2 ) )
=> ~ ! [Y2: A,M4: nat] :
( ( N
= ( suc @ M4 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y2 ) @ R3 )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ Z2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ M4 @ R3 ) ) ) ) ) ) ).
% relpow_E2
thf(fact_5950_relpow__E,axiom,
! [A: $tType,X2: A,Z2: A,N: nat,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Z2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R3 ) )
=> ( ( ( N
= ( zero_zero @ nat ) )
=> ( X2 != Z2 ) )
=> ~ ! [Y2: A,M4: nat] :
( ( N
= ( suc @ M4 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ M4 @ R3 ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ Z2 ) @ R3 ) ) ) ) ) ).
% relpow_E
thf(fact_5951_relpow__empty,axiom,
! [A: $tType,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ).
% relpow_empty
thf(fact_5952_relpow__fun__conv,axiom,
! [A: $tType,A2: A,B2: A,N: nat,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R3 ) )
= ( ? [F2: nat > A] :
( ( ( F2 @ ( zero_zero @ nat ) )
= A2 )
& ( ( F2 @ N )
= B2 )
& ! [I2: nat] :
( ( ord_less @ nat @ I2 @ N )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( F2 @ I2 ) @ ( F2 @ ( suc @ I2 ) ) ) @ R3 ) ) ) ) ) ).
% relpow_fun_conv
thf(fact_5953_relpow__finite__bounded,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),K: nat] :
( ( finite_finite2 @ ( product_prod @ A @ A ) @ R3 )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ K @ R3 )
@ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ A ) )
@ ( image2 @ nat @ ( set @ ( product_prod @ A @ A ) )
@ ^ [N4: nat] : ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N4 @ R3 )
@ ( collect @ nat
@ ^ [N4: nat] : ( ord_less_eq @ nat @ N4 @ ( finite_card @ ( product_prod @ A @ A ) @ R3 ) ) ) ) ) ) ) ).
% relpow_finite_bounded
thf(fact_5954_ntrancl__def,axiom,
! [A: $tType] :
( ( transitive_ntrancl @ A )
= ( ^ [N4: nat,R7: set @ ( product_prod @ A @ A )] :
( complete_Sup_Sup @ ( set @ ( product_prod @ A @ A ) )
@ ( image2 @ nat @ ( set @ ( product_prod @ A @ A ) )
@ ^ [I2: nat] : ( compow @ ( set @ ( product_prod @ A @ A ) ) @ I2 @ R7 )
@ ( collect @ nat
@ ^ [I2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ I2 )
& ( ord_less_eq @ nat @ I2 @ ( suc @ N4 ) ) ) ) ) ) ) ) ).
% ntrancl_def
thf(fact_5955_mlex__leq,axiom,
! [A: $tType,F: A > nat,X2: A,Y: A,R3: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ nat @ ( F @ X2 ) @ ( F @ Y ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y ) @ R3 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y ) @ ( mlex_prod @ A @ F @ R3 ) ) ) ) ).
% mlex_leq
thf(fact_5956_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_5957_length__remdups__eq,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( size_size @ ( list @ A ) @ ( remdups @ A @ Xs ) )
= ( size_size @ ( list @ A ) @ Xs ) )
= ( ( remdups @ A @ Xs )
= Xs ) ) ).
% length_remdups_eq
thf(fact_5958_ntrancl__Zero,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( transitive_ntrancl @ A @ ( zero_zero @ nat ) @ R3 )
= R3 ) ).
% ntrancl_Zero
thf(fact_5959_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_5960_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_5961_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_5962_length__remdups__card__conv,axiom,
! [A: $tType,Xs: list @ A] :
( ( size_size @ ( list @ A ) @ ( remdups @ A @ Xs ) )
= ( finite_card @ A @ ( set2 @ A @ Xs ) ) ) ).
% length_remdups_card_conv
thf(fact_5963_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
@ ^ [X: A] : X
@ ( remdups @ A @ Xs ) ) ) ) ).
% sorted_list_of_set_sort_remdups
thf(fact_5964_remdup__sort__mergesort__remdups,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( comp @ ( list @ A ) @ ( list @ A ) @ ( list @ A ) @ ( remdups @ A )
@ ( linorder_sort_key @ A @ A
@ ^ [X: A] : X ) )
= ( mergesort_remdups @ A ) ) ) ).
% remdup_sort_mergesort_remdups
thf(fact_5965_trancl__finite__eq__relpow,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( finite_finite2 @ ( product_prod @ A @ A ) @ R3 )
=> ( ( transitive_trancl @ A @ R3 )
= ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ A ) )
@ ( image2 @ nat @ ( set @ ( product_prod @ A @ A ) )
@ ^ [N4: nat] : ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N4 @ R3 )
@ ( collect @ nat
@ ^ [N4: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N4 )
& ( ord_less_eq @ nat @ N4 @ ( finite_card @ ( product_prod @ A @ A ) @ R3 ) ) ) ) ) ) ) ) ).
% trancl_finite_eq_relpow
thf(fact_5966_INF__filter__not__bot,axiom,
! [I6: $tType,A: $tType,B5: set @ I6,F4: I6 > ( filter @ A )] :
( ! [X14: set @ I6] :
( ( ord_less_eq @ ( set @ I6 ) @ X14 @ B5 )
=> ( ( finite_finite2 @ I6 @ X14 )
=> ( ( complete_Inf_Inf @ ( filter @ A ) @ ( image2 @ I6 @ ( filter @ A ) @ F4 @ X14 ) )
!= ( bot_bot @ ( filter @ A ) ) ) ) )
=> ( ( complete_Inf_Inf @ ( filter @ A ) @ ( image2 @ I6 @ ( filter @ A ) @ F4 @ B5 ) )
!= ( bot_bot @ ( filter @ A ) ) ) ) ).
% INF_filter_not_bot
thf(fact_5967_trancl__single,axiom,
! [A: $tType,A2: A,B2: A] :
( ( transitive_trancl @ A @ ( insert @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) )
= ( insert @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ).
% trancl_single
thf(fact_5968_trancl__mono__mp,axiom,
! [A: $tType,U5: set @ ( product_prod @ A @ A ),V: set @ ( product_prod @ A @ A ),X2: product_prod @ A @ A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ U5 @ V )
=> ( ( member @ ( product_prod @ A @ A ) @ X2 @ ( transitive_trancl @ A @ U5 ) )
=> ( member @ ( product_prod @ A @ A ) @ X2 @ ( transitive_trancl @ A @ V ) ) ) ) ).
% trancl_mono_mp
thf(fact_5969_trancl__sub,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ ( transitive_trancl @ A @ R3 ) ) ).
% trancl_sub
thf(fact_5970_trancl__sub__insert__trancl,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),X2: product_prod @ A @ A] : ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( transitive_trancl @ A @ R3 ) @ ( transitive_trancl @ A @ ( insert @ ( product_prod @ A @ A ) @ X2 @ R3 ) ) ) ).
% trancl_sub_insert_trancl
thf(fact_5971_INF__filter__bot__base,axiom,
! [B: $tType,A: $tType,I5: set @ A,F4: A > ( filter @ B )] :
( ! [I3: A] :
( ( member @ A @ I3 @ I5 )
=> ! [J2: A] :
( ( member @ A @ J2 @ I5 )
=> ? [X4: A] :
( ( member @ A @ X4 @ I5 )
& ( ord_less_eq @ ( filter @ B ) @ ( F4 @ X4 ) @ ( inf_inf @ ( filter @ B ) @ ( F4 @ I3 ) @ ( F4 @ J2 ) ) ) ) ) )
=> ( ( ( complete_Inf_Inf @ ( filter @ B ) @ ( image2 @ A @ ( filter @ B ) @ F4 @ I5 ) )
= ( bot_bot @ ( filter @ B ) ) )
= ( ? [X: A] :
( ( member @ A @ X @ I5 )
& ( ( F4 @ X )
= ( bot_bot @ ( filter @ B ) ) ) ) ) ) ) ).
% INF_filter_bot_base
thf(fact_5972_trancl__subset__Sigma,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A5: set @ A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R
@ ( product_Sigma @ A @ A @ A5
@ ^ [Uu: A] : A5 ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( transitive_trancl @ A @ R )
@ ( product_Sigma @ A @ A @ A5
@ ^ [Uu: A] : A5 ) ) ) ).
% trancl_subset_Sigma
thf(fact_5973_trancl__power,axiom,
! [A: $tType,P3: product_prod @ A @ A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ P3 @ ( transitive_trancl @ A @ R3 ) )
= ( ? [N4: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N4 )
& ( member @ ( product_prod @ A @ A ) @ P3 @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N4 @ R3 ) ) ) ) ) ).
% trancl_power
thf(fact_5974_Restr__trancl__mono,axiom,
! [A: $tType,V2: A,W2: A,E6: set @ ( product_prod @ A @ A ),U5: 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 @ U5
@ ^ [Uu: A] : U5 ) ) ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V2 @ W2 ) @ ( transitive_trancl @ A @ E6 ) ) ) ).
% Restr_trancl_mono
thf(fact_5975_trancl__set__ntrancl,axiom,
! [A: $tType,Xs: list @ ( product_prod @ A @ A )] :
( ( transitive_trancl @ A @ ( set2 @ ( product_prod @ A @ A ) @ Xs ) )
= ( transitive_ntrancl @ A @ ( minus_minus @ nat @ ( finite_card @ ( product_prod @ A @ A ) @ ( set2 @ ( product_prod @ A @ A ) @ Xs ) ) @ ( one_one @ nat ) ) @ ( set2 @ ( product_prod @ A @ A ) @ Xs ) ) ) ).
% trancl_set_ntrancl
thf(fact_5976_finite__trancl__ntranl,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( finite_finite2 @ ( product_prod @ A @ A ) @ R3 )
=> ( ( transitive_trancl @ A @ R3 )
= ( transitive_ntrancl @ A @ ( minus_minus @ nat @ ( finite_card @ ( product_prod @ A @ A ) @ R3 ) @ ( one_one @ nat ) ) @ R3 ) ) ) ).
% finite_trancl_ntranl
thf(fact_5977_nth__step__trancl,axiom,
! [A: $tType,Xs: list @ A,R3: set @ ( product_prod @ A @ A ),N: nat,M: nat] :
( ! [N2: nat] :
( ( ord_less @ nat @ N2 @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( nth @ A @ Xs @ ( suc @ N2 ) ) @ ( nth @ A @ Xs @ N2 ) ) @ R3 ) )
=> ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ord_less @ nat @ M @ N )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( nth @ A @ Xs @ N ) @ ( nth @ A @ Xs @ M ) ) @ ( transitive_trancl @ A @ R3 ) ) ) ) ) ).
% nth_step_trancl
thf(fact_5978_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 ) )
@ ^ [X: A] : ( insert @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ X ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
@ A8 ) ) ) ) ).
% Id_on_def
thf(fact_5979_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 )
@ ^ [A3: A,B3: B] :
( product_case_prod @ A @ B @ $o
@ ^ [A10: A,B12: B] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ A10 ) @ Ra )
| ( ( A3 = A10 )
& ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ B3 @ B12 ) @ Rb ) ) ) ) ) ) ) ) ) ).
% lex_prod_def
thf(fact_5980_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
@ ^ [X: A,Y4: A] :
( ( X = Y4 )
& ( A5 @ X ) ) ) ) ) ).
% Id_on_def'
thf(fact_5981_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
@ ^ [Uu: A] : A5 ) ) ).
% Id_on_subset_Times
thf(fact_5982_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 )
@ ^ [X: A] : ( product_Pair @ A @ A @ X @ X )
@ Xs ) ) ) ).
% Id_on_set
thf(fact_5983_list__collect__set__alt,axiom,
! [A: $tType,B: $tType] :
( ( list_collect_set @ B @ A )
= ( ^ [F2: B > ( set @ A ),L2: list @ B] :
( complete_Sup_Sup @ ( set @ A )
@ ( collect @ ( set @ A )
@ ^ [Uu: set @ A] :
? [I2: nat] :
( ( Uu
= ( F2 @ ( nth @ B @ L2 @ I2 ) ) )
& ( ord_less @ nat @ I2 @ ( size_size @ ( list @ B ) @ L2 ) ) ) ) ) ) ) ).
% list_collect_set_alt
thf(fact_5984_Gcd__eq__Max,axiom,
! [M3: set @ nat] :
( ( finite_finite2 @ nat @ M3 )
=> ( ( M3
!= ( bot_bot @ ( set @ nat ) ) )
=> ( ~ ( member @ nat @ ( zero_zero @ nat ) @ M3 )
=> ( ( gcd_Gcd @ nat @ M3 )
= ( lattic643756798349783984er_Max @ nat
@ ( complete_Inf_Inf @ ( set @ nat )
@ ( image2 @ nat @ ( set @ nat )
@ ^ [M2: nat] :
( collect @ nat
@ ^ [D5: nat] : ( dvd_dvd @ nat @ D5 @ M2 ) )
@ M3 ) ) ) ) ) ) ) ).
% Gcd_eq_Max
thf(fact_5985_Max__singleton,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A] :
( ( lattic643756798349783984er_Max @ A @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) )
= X2 ) ) ).
% Max_singleton
thf(fact_5986_list__collect__set__simps_I2_J,axiom,
! [A: $tType,B: $tType,F: B > ( set @ A ),A2: B] :
( ( list_collect_set @ B @ A @ F @ ( cons @ B @ A2 @ ( nil @ B ) ) )
= ( F @ A2 ) ) ).
% list_collect_set_simps(2)
thf(fact_5987_list__collect__set__simps_I1_J,axiom,
! [B: $tType,A: $tType,F: B > ( set @ A )] :
( ( list_collect_set @ B @ A @ F @ ( nil @ B ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% list_collect_set_simps(1)
thf(fact_5988_Max__divisors__self__nat,axiom,
! [N: nat] :
( ( N
!= ( zero_zero @ nat ) )
=> ( ( lattic643756798349783984er_Max @ nat
@ ( collect @ nat
@ ^ [D5: nat] : ( dvd_dvd @ nat @ D5 @ N ) ) )
= N ) ) ).
% Max_divisors_self_nat
thf(fact_5989_Max_Obounded__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X2: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ ( lattic643756798349783984er_Max @ A @ A5 ) @ X2 )
= ( ! [X: A] :
( ( member @ A @ X @ A5 )
=> ( ord_less_eq @ A @ X @ X2 ) ) ) ) ) ) ) ).
% Max.bounded_iff
thf(fact_5990_Max__less__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X2: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ ( lattic643756798349783984er_Max @ A @ A5 ) @ X2 )
= ( ! [X: A] :
( ( member @ A @ X @ A5 )
=> ( ord_less @ A @ X @ X2 ) ) ) ) ) ) ) ).
% Max_less_iff
thf(fact_5991_list__collect__set__map__simps_I2_J,axiom,
! [A: $tType,B: $tType,C: $tType,F: B > ( set @ A ),X2: C > B,A2: C] :
( ( list_collect_set @ B @ A @ F @ ( map @ C @ B @ X2 @ ( cons @ C @ A2 @ ( nil @ C ) ) ) )
= ( F @ ( X2 @ A2 ) ) ) ).
% list_collect_set_map_simps(2)
thf(fact_5992_list__collect__set__map__simps_I1_J,axiom,
! [C: $tType,B: $tType,A: $tType,F: B > ( set @ A ),X2: C > B] :
( ( list_collect_set @ B @ A @ F @ ( map @ C @ B @ X2 @ ( nil @ C ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% list_collect_set_map_simps(1)
thf(fact_5993_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
@ ^ [Uu: B] : C2
@ A5 ) )
= C2 ) ) ) ) ).
% Max_const
thf(fact_5994_Max__insert,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X2: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ ( insert @ A @ X2 @ A5 ) )
= ( ord_max @ A @ X2 @ ( lattic643756798349783984er_Max @ A @ A5 ) ) ) ) ) ) ).
% Max_insert
thf(fact_5995_Max__ge,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X2: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( member @ A @ X2 @ A5 )
=> ( ord_less_eq @ A @ X2 @ ( lattic643756798349783984er_Max @ A @ A5 ) ) ) ) ) ).
% Max_ge
thf(fact_5996_Max__eqI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X2: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ! [Y2: A] :
( ( member @ A @ Y2 @ A5 )
=> ( ord_less_eq @ A @ Y2 @ X2 ) )
=> ( ( member @ A @ X2 @ A5 )
=> ( ( lattic643756798349783984er_Max @ A @ A5 )
= X2 ) ) ) ) ) ).
% Max_eqI
thf(fact_5997_Max__eq__if,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,B5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( finite_finite2 @ A @ B5 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ A5 )
=> ? [Xa2: A] :
( ( member @ A @ Xa2 @ B5 )
& ( ord_less_eq @ A @ X3 @ Xa2 ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ B5 )
=> ? [Xa2: A] :
( ( member @ A @ Xa2 @ A5 )
& ( ord_less_eq @ A @ X3 @ Xa2 ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ A5 )
= ( lattic643756798349783984er_Max @ A @ B5 ) ) ) ) ) ) ) ).
% Max_eq_if
thf(fact_5998_Max_OcoboundedI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,A2: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( member @ A @ A2 @ A5 )
=> ( ord_less_eq @ A @ A2 @ ( lattic643756798349783984er_Max @ A @ A5 ) ) ) ) ) ).
% Max.coboundedI
thf(fact_5999_list__collect__set__as__map,axiom,
! [A: $tType,B: $tType] :
( ( list_collect_set @ B @ A )
= ( ^ [F2: B > ( set @ A ),L2: list @ B] : ( complete_Sup_Sup @ ( set @ A ) @ ( set2 @ ( set @ A ) @ ( map @ B @ ( set @ A ) @ F2 @ L2 ) ) ) ) ) ).
% list_collect_set_as_map
thf(fact_6000_Max_Oin__idem,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X2: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( member @ A @ X2 @ A5 )
=> ( ( ord_max @ A @ X2 @ ( lattic643756798349783984er_Max @ A @ A5 ) )
= ( lattic643756798349783984er_Max @ A @ A5 ) ) ) ) ) ).
% Max.in_idem
thf(fact_6001_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_6002_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 )
& ! [X: A] :
( ( member @ A @ X @ A5 )
=> ( ord_less_eq @ A @ X @ M ) ) ) ) ) ) ) ).
% Max_eq_iff
thf(fact_6003_Max__ge__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X2: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ X2 @ ( lattic643756798349783984er_Max @ A @ A5 ) )
= ( ? [X: A] :
( ( member @ A @ X @ A5 )
& ( ord_less_eq @ A @ X2 @ X ) ) ) ) ) ) ) ).
% Max_ge_iff
thf(fact_6004_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 )
& ! [X: A] :
( ( member @ A @ X @ A5 )
=> ( ord_less_eq @ A @ X @ M ) ) ) ) ) ) ) ).
% eq_Max_iff
thf(fact_6005_Max_OboundedE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X2: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ ( lattic643756798349783984er_Max @ A @ A5 ) @ X2 )
=> ! [A9: A] :
( ( member @ A @ A9 @ A5 )
=> ( ord_less_eq @ A @ A9 @ X2 ) ) ) ) ) ) ).
% Max.boundedE
thf(fact_6006_Max_OboundedI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X2: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [A4: A] :
( ( member @ A @ A4 @ A5 )
=> ( ord_less_eq @ A @ A4 @ X2 ) )
=> ( ord_less_eq @ A @ ( lattic643756798349783984er_Max @ A @ A5 ) @ X2 ) ) ) ) ) ).
% Max.boundedI
thf(fact_6007_Max__gr__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X2: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ X2 @ ( lattic643756798349783984er_Max @ A @ A5 ) )
= ( ? [X: A] :
( ( member @ A @ X @ A5 )
& ( ord_less @ A @ X2 @ X ) ) ) ) ) ) ) ).
% Max_gr_iff
thf(fact_6008_Max__insert2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,A2: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ! [B4: A] :
( ( member @ A @ B4 @ A5 )
=> ( ord_less_eq @ A @ B4 @ A2 ) )
=> ( ( lattic643756798349783984er_Max @ A @ ( insert @ A @ A2 @ A5 ) )
= A2 ) ) ) ) ).
% Max_insert2
thf(fact_6009_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_6010_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_6011_Sup__nat__def,axiom,
( ( complete_Sup_Sup @ nat )
= ( ^ [X5: set @ nat] :
( if @ nat
@ ( X5
= ( bot_bot @ ( set @ nat ) ) )
@ ( zero_zero @ nat )
@ ( lattic643756798349783984er_Max @ nat @ X5 ) ) ) ) ).
% Sup_nat_def
thf(fact_6012_Max_Osubset__imp,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B5 )
=> ( ord_less_eq @ A @ ( lattic643756798349783984er_Max @ A @ A5 ) @ ( lattic643756798349783984er_Max @ A @ B5 ) ) ) ) ) ) ).
% Max.subset_imp
thf(fact_6013_Max__mono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [M3: set @ A,N5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ M3 @ N5 )
=> ( ( M3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ N5 )
=> ( ord_less_eq @ A @ ( lattic643756798349783984er_Max @ A @ M3 ) @ ( lattic643756798349783984er_Max @ A @ N5 ) ) ) ) ) ) ).
% Max_mono
thf(fact_6014_hom__Max__commute,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [H: A > A,N5: set @ A] :
( ! [X3: A,Y2: A] :
( ( H @ ( ord_max @ A @ X3 @ Y2 ) )
= ( ord_max @ A @ ( H @ X3 ) @ ( H @ Y2 ) ) )
=> ( ( finite_finite2 @ A @ N5 )
=> ( ( N5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( H @ ( lattic643756798349783984er_Max @ A @ N5 ) )
= ( lattic643756798349783984er_Max @ A @ ( image2 @ A @ A @ H @ N5 ) ) ) ) ) ) ) ).
% hom_Max_commute
thf(fact_6015_Max_Osubset,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,B5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( B5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ A5 )
=> ( ( ord_max @ A @ ( lattic643756798349783984er_Max @ A @ B5 ) @ ( lattic643756798349783984er_Max @ A @ A5 ) )
= ( lattic643756798349783984er_Max @ A @ A5 ) ) ) ) ) ) ).
% Max.subset
thf(fact_6016_Max_Oinsert__not__elem,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X2: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ~ ( member @ A @ X2 @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ ( insert @ A @ X2 @ A5 ) )
= ( ord_max @ A @ X2 @ ( lattic643756798349783984er_Max @ A @ A5 ) ) ) ) ) ) ) ).
% Max.insert_not_elem
thf(fact_6017_Max_Oclosed,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A,Y2: A] : ( member @ A @ ( ord_max @ A @ X3 @ Y2 ) @ ( insert @ A @ X3 @ ( insert @ A @ Y2 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( member @ A @ ( lattic643756798349783984er_Max @ A @ A5 ) @ A5 ) ) ) ) ) ).
% Max.closed
thf(fact_6018_card__le__Suc__Max,axiom,
! [S2: set @ nat] :
( ( finite_finite2 @ nat @ S2 )
=> ( ord_less_eq @ nat @ ( finite_card @ nat @ S2 ) @ ( suc @ ( lattic643756798349783984er_Max @ nat @ S2 ) ) ) ) ).
% card_le_Suc_Max
thf(fact_6019_Max__add__commute,axiom,
! [B: $tType,A: $tType] :
( ( linord4140545234300271783up_add @ A )
=> ! [S2: set @ B,F: B > A,K: A] :
( ( finite_finite2 @ B @ S2 )
=> ( ( S2
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( lattic643756798349783984er_Max @ A
@ ( image2 @ B @ A
@ ^ [X: B] : ( plus_plus @ A @ ( F @ X ) @ K )
@ S2 ) )
= ( plus_plus @ A @ ( lattic643756798349783984er_Max @ A @ ( image2 @ B @ A @ F @ S2 ) ) @ K ) ) ) ) ) ).
% Max_add_commute
thf(fact_6020_divide__nat__def,axiom,
( ( divide_divide @ nat )
= ( ^ [M2: nat,N4: nat] :
( if @ nat
@ ( N4
= ( zero_zero @ nat ) )
@ ( zero_zero @ nat )
@ ( lattic643756798349783984er_Max @ nat
@ ( collect @ nat
@ ^ [K2: nat] : ( ord_less_eq @ nat @ ( times_times @ nat @ K2 @ N4 ) @ M2 ) ) ) ) ) ) ).
% divide_nat_def
thf(fact_6021_gcd__is__Max__divisors__nat,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( gcd_gcd @ nat @ M @ N )
= ( lattic643756798349783984er_Max @ nat
@ ( collect @ nat
@ ^ [D5: nat] :
( ( dvd_dvd @ nat @ D5 @ M )
& ( dvd_dvd @ nat @ D5 @ N ) ) ) ) ) ) ).
% gcd_is_Max_divisors_nat
thf(fact_6022_list__collect__set__def,axiom,
! [A: $tType,B: $tType] :
( ( list_collect_set @ B @ A )
= ( ^ [F2: B > ( set @ A ),L2: list @ B] :
( complete_Sup_Sup @ ( set @ A )
@ ( collect @ ( set @ A )
@ ^ [Uu: set @ A] :
? [A3: B] :
( ( Uu
= ( F2 @ A3 ) )
& ( member @ B @ A3 @ ( set2 @ B @ L2 ) ) ) ) ) ) ) ).
% list_collect_set_def
thf(fact_6023_Max_Oinsert__remove,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X2: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ ( insert @ A @ X2 @ A5 ) )
= X2 ) )
& ( ( ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ ( insert @ A @ X2 @ A5 ) )
= ( ord_max @ A @ X2 @ ( lattic643756798349783984er_Max @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% Max.insert_remove
thf(fact_6024_Max_Oremove,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X2: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( member @ A @ X2 @ A5 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ A5 )
= X2 ) )
& ( ( ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ A5 )
= ( ord_max @ A @ X2 @ ( lattic643756798349783984er_Max @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ) ).
% Max.remove
thf(fact_6025_sum__le__card__Max,axiom,
! [A: $tType,A5: set @ A,F: A > nat] :
( ( finite_finite2 @ A @ A5 )
=> ( ord_less_eq @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ F @ A5 ) @ ( times_times @ nat @ ( finite_card @ A @ A5 ) @ ( lattic643756798349783984er_Max @ nat @ ( image2 @ A @ nat @ F @ A5 ) ) ) ) ) ).
% sum_le_card_Max
thf(fact_6026_dual__Min,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( lattices_Min @ A
@ ^ [X: A,Y4: A] : ( ord_less_eq @ A @ Y4 @ X ) )
= ( lattic643756798349783984er_Max @ A ) ) ) ).
% dual_Min
thf(fact_6027_lexn_Osimps_I2_J,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),N: nat] :
( ( lexn @ A @ R @ ( suc @ N ) )
= ( inf_inf @ ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( image2 @ ( product_prod @ ( product_prod @ A @ ( list @ A ) ) @ ( product_prod @ A @ ( list @ A ) ) ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_map_prod @ ( product_prod @ A @ ( list @ A ) ) @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) @ ( list @ A ) @ ( product_case_prod @ A @ ( list @ A ) @ ( list @ A ) @ ( cons @ A ) ) @ ( product_case_prod @ A @ ( list @ A ) @ ( list @ A ) @ ( cons @ A ) ) ) @ ( lex_prod @ A @ ( list @ A ) @ R @ ( lexn @ A @ R @ N ) ) )
@ ( collect @ ( product_prod @ ( list @ A ) @ ( list @ A ) )
@ ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
@ ^ [Xs3: list @ A,Ys3: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs3 )
= ( suc @ N ) )
& ( ( size_size @ ( list @ A ) @ Ys3 )
= ( suc @ N ) ) ) ) ) ) ) ).
% lexn.simps(2)
thf(fact_6028_map__prod__ident,axiom,
! [B: $tType,A: $tType] :
( ( product_map_prod @ A @ A @ B @ B
@ ^ [X: A] : X
@ ^ [Y4: B] : Y4 )
= ( ^ [Z3: product_prod @ A @ B] : Z3 ) ) ).
% map_prod_ident
thf(fact_6029_Max__divisors__self__int,axiom,
! [N: int] :
( ( N
!= ( zero_zero @ int ) )
=> ( ( lattic643756798349783984er_Max @ int
@ ( collect @ int
@ ^ [D5: int] : ( dvd_dvd @ int @ D5 @ N ) ) )
= ( abs_abs @ int @ N ) ) ) ).
% Max_divisors_self_int
thf(fact_6030_map__prod_Oidentity,axiom,
! [B: $tType,A: $tType] :
( ( product_map_prod @ A @ A @ B @ B
@ ^ [X: A] : X
@ ^ [X: B] : X )
= ( id @ ( product_prod @ A @ B ) ) ) ).
% map_prod.identity
thf(fact_6031_linorder_OMin_Ocong,axiom,
! [A: $tType] :
( ( lattices_Min @ A )
= ( lattices_Min @ A ) ) ).
% linorder.Min.cong
thf(fact_6032_case__prod__map__prod,axiom,
! [C: $tType,A: $tType,B: $tType,E: $tType,D: $tType,H: B > C > A,F: D > B,G: E > C,X2: product_prod @ D @ E] :
( ( product_case_prod @ B @ C @ A @ H @ ( product_map_prod @ D @ B @ E @ C @ F @ G @ X2 ) )
= ( product_case_prod @ D @ E @ A
@ ^ [L2: D,R2: E] : ( H @ ( F @ L2 ) @ ( G @ R2 ) )
@ X2 ) ) ).
% case_prod_map_prod
thf(fact_6033_map__prod__def,axiom,
! [B: $tType,D: $tType,C: $tType,A: $tType] :
( ( product_map_prod @ A @ C @ B @ D )
= ( ^ [F2: A > C,G2: B > D] :
( product_case_prod @ A @ B @ ( product_prod @ C @ D )
@ ^ [X: A,Y4: B] : ( product_Pair @ C @ D @ ( F2 @ X ) @ ( G2 @ Y4 ) ) ) ) ) ).
% map_prod_def
thf(fact_6034_case__prod__o__map__prod,axiom,
! [B: $tType,D: $tType,C: $tType,E: $tType,A: $tType,F: 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 @ F ) @ ( product_map_prod @ A @ D @ B @ E @ G1 @ G22 ) )
= ( product_case_prod @ A @ B @ C
@ ^ [L2: A,R2: B] : ( F @ ( G1 @ L2 ) @ ( G22 @ R2 ) ) ) ) ).
% case_prod_o_map_prod
thf(fact_6035_map__prod__surj__on,axiom,
! [B: $tType,D: $tType,C: $tType,A: $tType,F: B > A,A5: set @ B,A11: set @ A,G: D > C,B5: set @ D,B13: set @ C] :
( ( ( image2 @ B @ A @ F @ A5 )
= A11 )
=> ( ( ( image2 @ D @ C @ G @ B5 )
= B13 )
=> ( ( image2 @ ( product_prod @ B @ D ) @ ( product_prod @ A @ C ) @ ( product_map_prod @ B @ A @ D @ C @ F @ G )
@ ( product_Sigma @ B @ D @ A5
@ ^ [Uu: B] : B5 ) )
= ( product_Sigma @ A @ C @ A11
@ ^ [Uu: A] : B13 ) ) ) ) ).
% map_prod_surj_on
thf(fact_6036_gcd__is__Max__divisors__int,axiom,
! [N: int,M: int] :
( ( N
!= ( zero_zero @ int ) )
=> ( ( gcd_gcd @ int @ M @ N )
= ( lattic643756798349783984er_Max @ int
@ ( collect @ int
@ ^ [D5: int] :
( ( dvd_dvd @ int @ D5 @ M )
& ( dvd_dvd @ int @ D5 @ N ) ) ) ) ) ) ).
% gcd_is_Max_divisors_int
thf(fact_6037_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_6038_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_6039_greaterThan__eq__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y: A] :
( ( ( set_ord_greaterThan @ A @ X2 )
= ( set_ord_greaterThan @ A @ Y ) )
= ( X2 = Y ) ) ) ).
% greaterThan_eq_iff
thf(fact_6040_le__sup__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X2: A,Y: A,Z2: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ X2 @ Y ) @ Z2 )
= ( ( ord_less_eq @ A @ X2 @ Z2 )
& ( ord_less_eq @ A @ Y @ Z2 ) ) ) ) ).
% le_sup_iff
thf(fact_6041_sup_Obounded__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B2: A,C2: A,A2: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ B2 @ C2 ) @ A2 )
= ( ( ord_less_eq @ A @ B2 @ A2 )
& ( ord_less_eq @ A @ C2 @ A2 ) ) ) ) ).
% sup.bounded_iff
thf(fact_6042_greaterThan__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [I: A,K: A] :
( ( member @ A @ I @ ( set_ord_greaterThan @ A @ K ) )
= ( ord_less @ A @ K @ I ) ) ) ).
% greaterThan_iff
thf(fact_6043_Inf__greaterThan,axiom,
! [A: $tType] :
( ( ( comple6319245703460814977attice @ A )
& ( dense_linorder @ A ) )
=> ! [X2: A] :
( ( complete_Inf_Inf @ A @ ( set_ord_greaterThan @ A @ X2 ) )
= X2 ) ) ).
% Inf_greaterThan
thf(fact_6044_greaterThan__subset__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_ord_greaterThan @ A @ X2 ) @ ( set_ord_greaterThan @ A @ Y ) )
= ( ord_less_eq @ A @ Y @ X2 ) ) ) ).
% greaterThan_subset_iff
thf(fact_6045_set__mergesort__by__rel__merge,axiom,
! [A: $tType,R3: A > A > $o,Xs: list @ A,Ys: list @ A] :
( ( set2 @ A @ ( merges9089515139780605204_merge @ A @ R3 @ Xs @ Ys ) )
= ( sup_sup @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys ) ) ) ).
% set_mergesort_by_rel_merge
thf(fact_6046_list__collect__set__simps_I3_J,axiom,
! [A: $tType,B: $tType,F: B > ( set @ A ),A2: B,L: list @ B] :
( ( list_collect_set @ B @ A @ F @ ( cons @ B @ A2 @ L ) )
= ( sup_sup @ ( set @ A ) @ ( F @ A2 ) @ ( list_collect_set @ B @ A @ F @ L ) ) ) ).
% list_collect_set_simps(3)
thf(fact_6047_list__collect__set__simps_I4_J,axiom,
! [A: $tType,B: $tType,F: B > ( set @ A ),L: list @ B,L4: list @ B] :
( ( list_collect_set @ B @ A @ F @ ( append @ B @ L @ L4 ) )
= ( sup_sup @ ( set @ A ) @ ( list_collect_set @ B @ A @ F @ L ) @ ( list_collect_set @ B @ A @ F @ L4 ) ) ) ).
% list_collect_set_simps(4)
thf(fact_6048_Compl__atMost,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [K: A] :
( ( uminus_uminus @ ( set @ A ) @ ( set_ord_atMost @ A @ K ) )
= ( set_ord_greaterThan @ A @ K ) ) ) ).
% Compl_atMost
thf(fact_6049_Compl__greaterThan,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [K: A] :
( ( uminus_uminus @ ( set @ A ) @ ( set_ord_greaterThan @ A @ K ) )
= ( set_ord_atMost @ A @ K ) ) ) ).
% Compl_greaterThan
thf(fact_6050_if__image__distrib,axiom,
! [A: $tType,B: $tType,P: B > $o,F: B > A,G: B > A,S2: set @ B] :
( ( image2 @ B @ A
@ ^ [X: B] : ( if @ A @ ( P @ X ) @ ( F @ X ) @ ( G @ X ) )
@ S2 )
= ( sup_sup @ ( set @ A ) @ ( image2 @ B @ A @ F @ ( inf_inf @ ( set @ B ) @ S2 @ ( collect @ B @ P ) ) )
@ ( image2 @ B @ A @ G
@ ( inf_inf @ ( set @ B ) @ S2
@ ( collect @ B
@ ^ [X: B] :
~ ( P @ X ) ) ) ) ) ) ).
% if_image_distrib
thf(fact_6051_UN__Un,axiom,
! [A: $tType,B: $tType,M3: B > ( set @ A ),A5: set @ B,B5: set @ B] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ M3 @ ( sup_sup @ ( set @ B ) @ A5 @ B5 ) ) )
= ( sup_sup @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ M3 @ A5 ) ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ M3 @ B5 ) ) ) ) ).
% UN_Un
thf(fact_6052_Sup__greaterThanAtLeast,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X2: A] :
( ( ord_less @ A @ X2 @ ( top_top @ A ) )
=> ( ( complete_Sup_Sup @ A @ ( set_ord_greaterThan @ A @ X2 ) )
= ( top_top @ A ) ) ) ) ).
% Sup_greaterThanAtLeast
thf(fact_6053_image__uminus__lessThan,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [X2: A] :
( ( image2 @ A @ A @ ( uminus_uminus @ A ) @ ( set_ord_lessThan @ A @ X2 ) )
= ( set_ord_greaterThan @ A @ ( uminus_uminus @ A @ X2 ) ) ) ) ).
% image_uminus_lessThan
thf(fact_6054_image__uminus__greaterThan,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [X2: A] :
( ( image2 @ A @ A @ ( uminus_uminus @ A ) @ ( set_ord_greaterThan @ A @ X2 ) )
= ( set_ord_lessThan @ A @ ( uminus_uminus @ A @ X2 ) ) ) ) ).
% image_uminus_greaterThan
thf(fact_6055_list__collect__set__map__simps_I3_J,axiom,
! [A: $tType,B: $tType,C: $tType,F: B > ( set @ A ),X2: C > B,A2: C,L: list @ C] :
( ( list_collect_set @ B @ A @ F @ ( map @ C @ B @ X2 @ ( cons @ C @ A2 @ L ) ) )
= ( sup_sup @ ( set @ A ) @ ( F @ ( X2 @ A2 ) ) @ ( list_collect_set @ B @ A @ F @ ( map @ C @ B @ X2 @ L ) ) ) ) ).
% list_collect_set_map_simps(3)
thf(fact_6056_list__collect__set__map__simps_I4_J,axiom,
! [A: $tType,B: $tType,C: $tType,F: B > ( set @ A ),X2: C > B,L: list @ C,L4: list @ C] :
( ( list_collect_set @ B @ A @ F @ ( map @ C @ B @ X2 @ ( append @ C @ L @ L4 ) ) )
= ( sup_sup @ ( set @ A ) @ ( list_collect_set @ B @ A @ F @ ( map @ C @ B @ X2 @ L ) ) @ ( list_collect_set @ B @ A @ F @ ( map @ C @ B @ X2 @ L4 ) ) ) ) ).
% list_collect_set_map_simps(4)
thf(fact_6057_UN__simps_I2_J,axiom,
! [C: $tType,D: $tType,C6: set @ C,A5: C > ( set @ D ),B5: set @ D] :
( ( ( C6
= ( bot_bot @ ( set @ C ) ) )
=> ( ( complete_Sup_Sup @ ( set @ D )
@ ( image2 @ C @ ( set @ D )
@ ^ [X: C] : ( sup_sup @ ( set @ D ) @ ( A5 @ X ) @ B5 )
@ C6 ) )
= ( bot_bot @ ( set @ D ) ) ) )
& ( ( C6
!= ( bot_bot @ ( set @ C ) ) )
=> ( ( complete_Sup_Sup @ ( set @ D )
@ ( image2 @ C @ ( set @ D )
@ ^ [X: C] : ( sup_sup @ ( set @ D ) @ ( A5 @ X ) @ B5 )
@ C6 ) )
= ( sup_sup @ ( set @ D ) @ ( complete_Sup_Sup @ ( set @ D ) @ ( image2 @ C @ ( set @ D ) @ A5 @ C6 ) ) @ B5 ) ) ) ) ).
% UN_simps(2)
thf(fact_6058_UN__simps_I3_J,axiom,
! [E: $tType,F7: $tType,C6: set @ F7,A5: set @ E,B5: F7 > ( set @ E )] :
( ( ( C6
= ( bot_bot @ ( set @ F7 ) ) )
=> ( ( complete_Sup_Sup @ ( set @ E )
@ ( image2 @ F7 @ ( set @ E )
@ ^ [X: F7] : ( sup_sup @ ( set @ E ) @ A5 @ ( B5 @ X ) )
@ C6 ) )
= ( bot_bot @ ( set @ E ) ) ) )
& ( ( C6
!= ( bot_bot @ ( set @ F7 ) ) )
=> ( ( complete_Sup_Sup @ ( set @ E )
@ ( image2 @ F7 @ ( set @ E )
@ ^ [X: F7] : ( sup_sup @ ( set @ E ) @ A5 @ ( B5 @ X ) )
@ C6 ) )
= ( sup_sup @ ( set @ E ) @ A5 @ ( complete_Sup_Sup @ ( set @ E ) @ ( image2 @ F7 @ ( set @ E ) @ B5 @ C6 ) ) ) ) ) ) ).
% UN_simps(3)
thf(fact_6059_UN__insert,axiom,
! [A: $tType,B: $tType,B5: B > ( set @ A ),A2: B,A5: set @ B] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B5 @ ( insert @ B @ A2 @ A5 ) ) )
= ( sup_sup @ ( set @ A ) @ ( B5 @ A2 ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B5 @ A5 ) ) ) ) ).
% UN_insert
thf(fact_6060_insert__Times__insert,axiom,
! [B: $tType,A: $tType,A2: A,A5: set @ A,B2: B,B5: set @ B] :
( ( product_Sigma @ A @ B @ ( insert @ A @ A2 @ A5 )
@ ^ [Uu: A] : ( insert @ B @ B2 @ B5 ) )
= ( insert @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 )
@ ( sup_sup @ ( set @ ( product_prod @ A @ B ) )
@ ( product_Sigma @ A @ B @ A5
@ ^ [Uu: A] : ( insert @ B @ B2 @ B5 ) )
@ ( product_Sigma @ A @ B @ ( insert @ A @ A2 @ A5 )
@ ^ [Uu: A] : B5 ) ) ) ) ).
% insert_Times_insert
thf(fact_6061_sup__Inf,axiom,
! [A: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [A2: A,B5: set @ A] :
( ( sup_sup @ A @ A2 @ ( complete_Inf_Inf @ A @ B5 ) )
= ( complete_Inf_Inf @ A @ ( image2 @ A @ A @ ( sup_sup @ A @ A2 ) @ B5 ) ) ) ) ).
% sup_Inf
thf(fact_6062_SUP__union,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [M3: B > A,A5: set @ B,B5: set @ B] :
( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ M3 @ ( sup_sup @ ( set @ B ) @ A5 @ B5 ) ) )
= ( sup_sup @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ M3 @ A5 ) ) @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ M3 @ B5 ) ) ) ) ) ).
% SUP_union
thf(fact_6063_set__union__code,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( sup_sup @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys ) )
= ( set2 @ A @ ( append @ A @ Xs @ Ys ) ) ) ).
% set_union_code
thf(fact_6064_distrib__inf__le,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X2: A,Y: A,Z2: A] : ( ord_less_eq @ A @ ( sup_sup @ A @ ( inf_inf @ A @ X2 @ Y ) @ ( inf_inf @ A @ X2 @ Z2 ) ) @ ( inf_inf @ A @ X2 @ ( sup_sup @ A @ Y @ Z2 ) ) ) ) ).
% distrib_inf_le
thf(fact_6065_distrib__sup__le,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X2: A,Y: A,Z2: A] : ( ord_less_eq @ A @ ( sup_sup @ A @ X2 @ ( inf_inf @ A @ Y @ Z2 ) ) @ ( inf_inf @ A @ ( sup_sup @ A @ X2 @ Y ) @ ( sup_sup @ A @ X2 @ Z2 ) ) ) ) ).
% distrib_sup_le
thf(fact_6066_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_6067_inf__sup__ord_I4_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [Y: A,X2: A] : ( ord_less_eq @ A @ Y @ ( sup_sup @ A @ X2 @ Y ) ) ) ).
% inf_sup_ord(4)
thf(fact_6068_inf__sup__ord_I3_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X2: A,Y: A] : ( ord_less_eq @ A @ X2 @ ( sup_sup @ A @ X2 @ Y ) ) ) ).
% inf_sup_ord(3)
thf(fact_6069_le__supE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A2: A,B2: A,X2: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ A2 @ B2 ) @ X2 )
=> ~ ( ( ord_less_eq @ A @ A2 @ X2 )
=> ~ ( ord_less_eq @ A @ B2 @ X2 ) ) ) ) ).
% le_supE
thf(fact_6070_le__supI,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A2: A,X2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ X2 )
=> ( ( ord_less_eq @ A @ B2 @ X2 )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ A2 @ B2 ) @ X2 ) ) ) ) ).
% le_supI
thf(fact_6071_sup__ge1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X2: A,Y: A] : ( ord_less_eq @ A @ X2 @ ( sup_sup @ A @ X2 @ Y ) ) ) ).
% sup_ge1
thf(fact_6072_sup__ge2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [Y: A,X2: A] : ( ord_less_eq @ A @ Y @ ( sup_sup @ A @ X2 @ Y ) ) ) ).
% sup_ge2
thf(fact_6073_le__supI1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X2: A,A2: A,B2: A] :
( ( ord_less_eq @ A @ X2 @ A2 )
=> ( ord_less_eq @ A @ X2 @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ).
% le_supI1
thf(fact_6074_le__supI2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X2: A,B2: A,A2: A] :
( ( ord_less_eq @ A @ X2 @ B2 )
=> ( ord_less_eq @ A @ X2 @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ).
% le_supI2
thf(fact_6075_sup_Omono,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [C2: A,A2: A,D2: A,B2: A] :
( ( ord_less_eq @ A @ C2 @ A2 )
=> ( ( ord_less_eq @ A @ D2 @ B2 )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ C2 @ D2 ) @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ) ).
% sup.mono
thf(fact_6076_sup__mono,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A2: A,C2: A,B2: A,D2: A] :
( ( ord_less_eq @ A @ A2 @ C2 )
=> ( ( ord_less_eq @ A @ B2 @ D2 )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ A2 @ B2 ) @ ( sup_sup @ A @ C2 @ D2 ) ) ) ) ) ).
% sup_mono
thf(fact_6077_sup__least,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [Y: A,X2: A,Z2: A] :
( ( ord_less_eq @ A @ Y @ X2 )
=> ( ( ord_less_eq @ A @ Z2 @ X2 )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ Y @ Z2 ) @ X2 ) ) ) ) ).
% sup_least
thf(fact_6078_le__iff__sup,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [X: A,Y4: A] :
( ( sup_sup @ A @ X @ Y4 )
= Y4 ) ) ) ) ).
% le_iff_sup
thf(fact_6079_sup_OorderE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( A2
= ( sup_sup @ A @ A2 @ B2 ) ) ) ) ).
% sup.orderE
thf(fact_6080_sup_OorderI,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A2: A,B2: A] :
( ( A2
= ( sup_sup @ A @ A2 @ B2 ) )
=> ( ord_less_eq @ A @ B2 @ A2 ) ) ) ).
% sup.orderI
thf(fact_6081_sup__unique,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [F: A > A > A,X2: A,Y: A] :
( ! [X3: A,Y2: A] : ( ord_less_eq @ A @ X3 @ ( F @ X3 @ Y2 ) )
=> ( ! [X3: A,Y2: A] : ( ord_less_eq @ A @ Y2 @ ( F @ X3 @ Y2 ) )
=> ( ! [X3: A,Y2: A,Z4: A] :
( ( ord_less_eq @ A @ Y2 @ X3 )
=> ( ( ord_less_eq @ A @ Z4 @ X3 )
=> ( ord_less_eq @ A @ ( F @ Y2 @ Z4 ) @ X3 ) ) )
=> ( ( sup_sup @ A @ X2 @ Y )
= ( F @ X2 @ Y ) ) ) ) ) ) ).
% sup_unique
thf(fact_6082_sup_Oabsorb1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( sup_sup @ A @ A2 @ B2 )
= A2 ) ) ) ).
% sup.absorb1
thf(fact_6083_sup_Oabsorb2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( sup_sup @ A @ A2 @ B2 )
= B2 ) ) ) ).
% sup.absorb2
thf(fact_6084_sup__absorb1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [Y: A,X2: A] :
( ( ord_less_eq @ A @ Y @ X2 )
=> ( ( sup_sup @ A @ X2 @ Y )
= X2 ) ) ) ).
% sup_absorb1
thf(fact_6085_sup__absorb2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X2: A,Y: A] :
( ( ord_less_eq @ A @ X2 @ Y )
=> ( ( sup_sup @ A @ X2 @ Y )
= Y ) ) ) ).
% sup_absorb2
thf(fact_6086_sup_OboundedE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B2: A,C2: A,A2: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ B2 @ C2 ) @ A2 )
=> ~ ( ( ord_less_eq @ A @ B2 @ A2 )
=> ~ ( ord_less_eq @ A @ C2 @ A2 ) ) ) ) ).
% sup.boundedE
thf(fact_6087_sup_OboundedI,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ C2 @ A2 )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ B2 @ C2 ) @ A2 ) ) ) ) ).
% sup.boundedI
thf(fact_6088_sup_Oorder__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B3: A,A3: A] :
( A3
= ( sup_sup @ A @ A3 @ B3 ) ) ) ) ) ).
% sup.order_iff
thf(fact_6089_sup_Ocobounded1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A2: A,B2: A] : ( ord_less_eq @ A @ A2 @ ( sup_sup @ A @ A2 @ B2 ) ) ) ).
% sup.cobounded1
thf(fact_6090_sup_Ocobounded2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B2: A,A2: A] : ( ord_less_eq @ A @ B2 @ ( sup_sup @ A @ A2 @ B2 ) ) ) ).
% sup.cobounded2
thf(fact_6091_sup_Oabsorb__iff1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B3: A,A3: A] :
( ( sup_sup @ A @ A3 @ B3 )
= A3 ) ) ) ) ).
% sup.absorb_iff1
thf(fact_6092_sup_Oabsorb__iff2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A3: A,B3: A] :
( ( sup_sup @ A @ A3 @ B3 )
= B3 ) ) ) ) ).
% sup.absorb_iff2
thf(fact_6093_sup_OcoboundedI1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less_eq @ A @ C2 @ A2 )
=> ( ord_less_eq @ A @ C2 @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ).
% sup.coboundedI1
thf(fact_6094_sup_OcoboundedI2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [C2: A,B2: A,A2: A] :
( ( ord_less_eq @ A @ C2 @ B2 )
=> ( ord_less_eq @ A @ C2 @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ).
% sup.coboundedI2
thf(fact_6095_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_6096_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_6097_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_6098_Times__Un__distrib1,axiom,
! [B: $tType,A: $tType,A5: set @ A,B5: set @ A,C6: set @ B] :
( ( product_Sigma @ A @ B @ ( sup_sup @ ( set @ A ) @ A5 @ B5 )
@ ^ [Uu: A] : C6 )
= ( sup_sup @ ( set @ ( product_prod @ A @ B ) )
@ ( product_Sigma @ A @ B @ A5
@ ^ [Uu: A] : C6 )
@ ( product_Sigma @ A @ B @ B5
@ ^ [Uu: A] : C6 ) ) ) ).
% Times_Un_distrib1
thf(fact_6099_Sigma__Un__distrib2,axiom,
! [B: $tType,A: $tType,I5: set @ A,A5: A > ( set @ B ),B5: A > ( set @ B )] :
( ( product_Sigma @ A @ B @ I5
@ ^ [I2: A] : ( sup_sup @ ( set @ B ) @ ( A5 @ I2 ) @ ( B5 @ I2 ) ) )
= ( sup_sup @ ( set @ ( product_prod @ A @ B ) ) @ ( product_Sigma @ A @ B @ I5 @ A5 ) @ ( product_Sigma @ A @ B @ I5 @ B5 ) ) ) ).
% Sigma_Un_distrib2
thf(fact_6100_Collect__imp__eq,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( collect @ A
@ ^ [X: A] :
( ( P @ X )
=> ( Q @ X ) ) )
= ( sup_sup @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ ( collect @ A @ P ) ) @ ( collect @ A @ Q ) ) ) ).
% Collect_imp_eq
thf(fact_6101_set__diff__diff__left,axiom,
! [A: $tType,A5: set @ A,B5: set @ A,C6: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A5 @ B5 ) @ C6 )
= ( minus_minus @ ( set @ A ) @ A5 @ ( sup_sup @ ( set @ A ) @ B5 @ C6 ) ) ) ).
% set_diff_diff_left
thf(fact_6102_insert__def,axiom,
! [A: $tType] :
( ( insert @ A )
= ( ^ [A3: A] :
( sup_sup @ ( set @ A )
@ ( collect @ A
@ ^ [X: A] : X = A3 ) ) ) ) ).
% insert_def
thf(fact_6103_greaterThan__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( set_ord_greaterThan @ A )
= ( ^ [L2: A] : ( collect @ A @ ( ord_less @ A @ L2 ) ) ) ) ) ).
% greaterThan_def
thf(fact_6104_Collect__disj__eq,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( collect @ A
@ ^ [X: A] :
( ( P @ X )
| ( Q @ X ) ) )
= ( sup_sup @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% Collect_disj_eq
thf(fact_6105_Un__def,axiom,
! [A: $tType] :
( ( sup_sup @ ( set @ A ) )
= ( ^ [A8: set @ A,B8: set @ A] :
( collect @ A
@ ^ [X: A] :
( ( member @ A @ X @ A8 )
| ( member @ A @ X @ B8 ) ) ) ) ) ).
% Un_def
thf(fact_6106_sup__option__def,axiom,
! [A: $tType] :
( ( sup @ A )
=> ( ( sup_sup @ ( option @ A ) )
= ( ^ [X: option @ A,Y4: option @ A] :
( case_option @ ( option @ A ) @ A @ Y4
@ ^ [X9: A] :
( case_option @ ( option @ A ) @ A @ X
@ ^ [Z3: A] : ( some @ A @ ( sup_sup @ A @ X9 @ Z3 ) )
@ Y4 )
@ X ) ) ) ) ).
% sup_option_def
thf(fact_6107_infinite__Ioi,axiom,
! [A: $tType] :
( ( ( linorder @ A )
& ( no_top @ A ) )
=> ! [A2: A] :
~ ( finite_finite2 @ A @ ( set_ord_greaterThan @ A @ A2 ) ) ) ).
% infinite_Ioi
thf(fact_6108_greaterThan__non__empty,axiom,
! [A: $tType] :
( ( no_top @ A )
=> ! [X2: A] :
( ( set_ord_greaterThan @ A @ X2 )
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% greaterThan_non_empty
thf(fact_6109_foldl__un__empty__eq,axiom,
! [A: $tType,I: set @ A,Ww: list @ ( set @ A )] :
( ( foldl @ ( set @ A ) @ ( set @ A ) @ ( sup_sup @ ( set @ A ) ) @ I @ Ww )
= ( sup_sup @ ( set @ A ) @ I @ ( foldl @ ( set @ A ) @ ( set @ A ) @ ( sup_sup @ ( set @ A ) ) @ ( bot_bot @ ( set @ A ) ) @ Ww ) ) ) ).
% foldl_un_empty_eq
thf(fact_6110_Inf__sup__eq__top__iff,axiom,
! [A: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [B5: set @ A,A2: A] :
( ( ( sup_sup @ A @ ( complete_Inf_Inf @ A @ B5 ) @ A2 )
= ( top_top @ A ) )
= ( ! [X: A] :
( ( member @ A @ X @ B5 )
=> ( ( sup_sup @ A @ X @ A2 )
= ( top_top @ A ) ) ) ) ) ) ).
% Inf_sup_eq_top_iff
thf(fact_6111_complete__lattice__class_OSUP__sup__distrib,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F: B > A,A5: set @ B,G: B > A] :
( ( sup_sup @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F @ A5 ) ) @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ G @ A5 ) ) )
= ( complete_Sup_Sup @ A
@ ( image2 @ B @ A
@ ^ [A3: B] : ( sup_sup @ A @ ( F @ A3 ) @ ( G @ A3 ) )
@ A5 ) ) ) ) ).
% complete_lattice_class.SUP_sup_distrib
thf(fact_6112_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_6113_INF__sup__distrib2,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [F: B > A,A5: set @ B,G: C > A,B5: set @ C] :
( ( sup_sup @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F @ A5 ) ) @ ( complete_Inf_Inf @ A @ ( image2 @ C @ A @ G @ B5 ) ) )
= ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [A3: B] :
( complete_Inf_Inf @ A
@ ( image2 @ C @ A
@ ^ [B3: C] : ( sup_sup @ A @ ( F @ A3 ) @ ( G @ B3 ) )
@ B5 ) )
@ A5 ) ) ) ) ).
% INF_sup_distrib2
thf(fact_6114_sup__INF,axiom,
! [A: $tType,B: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [A2: A,F: B > A,B5: set @ B] :
( ( sup_sup @ A @ A2 @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F @ B5 ) ) )
= ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [B3: B] : ( sup_sup @ A @ A2 @ ( F @ B3 ) )
@ B5 ) ) ) ) ).
% sup_INF
thf(fact_6115_Inf__sup,axiom,
! [A: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [B5: set @ A,A2: A] :
( ( sup_sup @ A @ ( complete_Inf_Inf @ A @ B5 ) @ A2 )
= ( complete_Inf_Inf @ A
@ ( image2 @ A @ A
@ ^ [B3: A] : ( sup_sup @ A @ B3 @ A2 )
@ B5 ) ) ) ) ).
% Inf_sup
thf(fact_6116_INF__sup,axiom,
! [A: $tType,B: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [F: B > A,B5: set @ B,A2: A] :
( ( sup_sup @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F @ B5 ) ) @ A2 )
= ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [B3: B] : ( sup_sup @ A @ ( F @ B3 ) @ A2 )
@ B5 ) ) ) ) ).
% INF_sup
thf(fact_6117_Un__Union__image,axiom,
! [A: $tType,B: $tType,A5: B > ( set @ A ),B5: B > ( set @ A ),C6: set @ B] :
( ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [X: B] : ( sup_sup @ ( set @ A ) @ ( A5 @ X ) @ ( B5 @ X ) )
@ C6 ) )
= ( sup_sup @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ A5 @ C6 ) ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B5 @ C6 ) ) ) ) ).
% Un_Union_image
thf(fact_6118_UN__Un__distrib,axiom,
! [A: $tType,B: $tType,A5: B > ( set @ A ),B5: B > ( set @ A ),I5: set @ B] :
( ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [I2: B] : ( sup_sup @ ( set @ A ) @ ( A5 @ I2 ) @ ( B5 @ I2 ) )
@ 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 ) @ B5 @ I5 ) ) ) ) ).
% UN_Un_distrib
thf(fact_6119_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_6120_INT__extend__simps_I7_J,axiom,
! [M15: $tType,N14: $tType,A5: set @ M15,B5: N14 > ( set @ M15 ),C6: set @ N14] :
( ( sup_sup @ ( set @ M15 ) @ A5 @ ( complete_Inf_Inf @ ( set @ M15 ) @ ( image2 @ N14 @ ( set @ M15 ) @ B5 @ C6 ) ) )
= ( complete_Inf_Inf @ ( set @ M15 )
@ ( image2 @ N14 @ ( set @ M15 )
@ ^ [X: N14] : ( sup_sup @ ( set @ M15 ) @ A5 @ ( B5 @ X ) )
@ C6 ) ) ) ).
% INT_extend_simps(7)
thf(fact_6121_INT__extend__simps_I6_J,axiom,
! [L8: $tType,K7: $tType,A5: K7 > ( set @ L8 ),C6: set @ K7,B5: set @ L8] :
( ( sup_sup @ ( set @ L8 ) @ ( complete_Inf_Inf @ ( set @ L8 ) @ ( image2 @ K7 @ ( set @ L8 ) @ A5 @ C6 ) ) @ B5 )
= ( complete_Inf_Inf @ ( set @ L8 )
@ ( image2 @ K7 @ ( set @ L8 )
@ ^ [X: K7] : ( sup_sup @ ( set @ L8 ) @ ( A5 @ X ) @ B5 )
@ C6 ) ) ) ).
% INT_extend_simps(6)
thf(fact_6122_Un__INT__distrib,axiom,
! [A: $tType,B: $tType,B5: set @ A,A5: B > ( set @ A ),I5: set @ B] :
( ( sup_sup @ ( set @ A ) @ B5 @ ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ A5 @ I5 ) ) )
= ( complete_Inf_Inf @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [I2: B] : ( sup_sup @ ( set @ A ) @ B5 @ ( A5 @ I2 ) )
@ I5 ) ) ) ).
% Un_INT_distrib
thf(fact_6123_Un__INT__distrib2,axiom,
! [A: $tType,C: $tType,B: $tType,A5: B > ( set @ A ),I5: set @ B,B5: C > ( set @ A ),J4: 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 ) @ B5 @ J4 ) ) )
= ( complete_Inf_Inf @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [I2: B] :
( complete_Inf_Inf @ ( set @ A )
@ ( image2 @ C @ ( set @ A )
@ ^ [J3: C] : ( sup_sup @ ( set @ A ) @ ( A5 @ I2 ) @ ( B5 @ J3 ) )
@ J4 ) )
@ I5 ) ) ) ).
% Un_INT_distrib2
thf(fact_6124_Union__take__drop__id,axiom,
! [A: $tType,N: nat,L: list @ ( set @ A )] :
( ( sup_sup @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( set2 @ ( set @ A ) @ ( drop @ ( set @ A ) @ N @ L ) ) ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( set2 @ ( set @ A ) @ ( take @ ( set @ A ) @ N @ L ) ) ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( set2 @ ( set @ A ) @ L ) ) ) ).
% Union_take_drop_id
thf(fact_6125_lessThan__Int__lessThan,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A] :
( ( inf_inf @ ( set @ A ) @ ( set_ord_greaterThan @ A @ A2 ) @ ( set_ord_greaterThan @ A @ B2 ) )
= ( set_ord_greaterThan @ A @ ( ord_max @ A @ A2 @ B2 ) ) ) ) ).
% lessThan_Int_lessThan
thf(fact_6126_Un__Inter,axiom,
! [A: $tType,A5: set @ A,B5: set @ ( set @ A )] :
( ( sup_sup @ ( set @ A ) @ A5 @ ( complete_Inf_Inf @ ( set @ A ) @ B5 ) )
= ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ ( set @ A ) @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A5 ) @ B5 ) ) ) ).
% Un_Inter
thf(fact_6127_sup__shunt,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X2: A,Y: A] :
( ( ( sup_sup @ A @ X2 @ Y )
= ( top_top @ A ) )
= ( ord_less_eq @ A @ ( uminus_uminus @ A @ X2 ) @ Y ) ) ) ).
% sup_shunt
thf(fact_6128_sup__neg__inf,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [P3: A,Q2: A,R: A] :
( ( ord_less_eq @ A @ P3 @ ( sup_sup @ A @ Q2 @ R ) )
= ( ord_less_eq @ A @ ( inf_inf @ A @ P3 @ ( uminus_uminus @ A @ Q2 ) ) @ R ) ) ) ).
% sup_neg_inf
thf(fact_6129_shunt2,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X2: A,Y: A,Z2: A] :
( ( ord_less_eq @ A @ ( inf_inf @ A @ X2 @ ( uminus_uminus @ A @ Y ) ) @ Z2 )
= ( ord_less_eq @ A @ X2 @ ( sup_sup @ A @ Y @ Z2 ) ) ) ) ).
% shunt2
thf(fact_6130_shunt1,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X2: A,Y: A,Z2: A] :
( ( ord_less_eq @ A @ ( inf_inf @ A @ X2 @ Y ) @ Z2 )
= ( ord_less_eq @ A @ X2 @ ( sup_sup @ A @ ( uminus_uminus @ A @ Y ) @ Z2 ) ) ) ) ).
% shunt1
thf(fact_6131_less__eq__Inf__inter,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ A,B5: set @ A] : ( ord_less_eq @ A @ ( sup_sup @ A @ ( complete_Inf_Inf @ A @ A5 ) @ ( complete_Inf_Inf @ A @ B5 ) ) @ ( complete_Inf_Inf @ A @ ( inf_inf @ ( set @ A ) @ A5 @ B5 ) ) ) ) ).
% less_eq_Inf_inter
thf(fact_6132_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_6133_card__Un__le,axiom,
! [A: $tType,A5: set @ A,B5: set @ A] : ( ord_less_eq @ nat @ ( finite_card @ A @ ( sup_sup @ ( set @ A ) @ A5 @ B5 ) ) @ ( plus_plus @ nat @ ( finite_card @ A @ A5 ) @ ( finite_card @ A @ B5 ) ) ) ).
% card_Un_le
thf(fact_6134_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_6135_atLeastLessThan__add__Un,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( set_or7035219750837199246ssThan @ nat @ I @ ( plus_plus @ nat @ J @ K ) )
= ( sup_sup @ ( set @ nat ) @ ( set_or7035219750837199246ssThan @ nat @ I @ J ) @ ( set_or7035219750837199246ssThan @ nat @ J @ ( plus_plus @ nat @ J @ K ) ) ) ) ) ).
% atLeastLessThan_add_Un
thf(fact_6136_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_6137_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_6138_SUP__insert,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F: B > A,A2: B,A5: set @ B] :
( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F @ ( insert @ B @ A2 @ A5 ) ) )
= ( sup_sup @ A @ ( F @ A2 ) @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F @ A5 ) ) ) ) ) ).
% SUP_insert
thf(fact_6139_Un__interval,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [B14: A,B23: A,B32: A,F: A > B] :
( ( ord_less_eq @ A @ B14 @ B23 )
=> ( ( ord_less_eq @ A @ B23 @ B32 )
=> ( ( sup_sup @ ( set @ B )
@ ( collect @ B
@ ^ [Uu: B] :
? [I2: A] :
( ( Uu
= ( F @ I2 ) )
& ( ord_less_eq @ A @ B14 @ I2 )
& ( ord_less @ A @ I2 @ B23 ) ) )
@ ( collect @ B
@ ^ [Uu: B] :
? [I2: A] :
( ( Uu
= ( F @ I2 ) )
& ( ord_less_eq @ A @ B23 @ I2 )
& ( ord_less @ A @ I2 @ B32 ) ) ) )
= ( collect @ B
@ ^ [Uu: B] :
? [I2: A] :
( ( Uu
= ( F @ I2 ) )
& ( ord_less_eq @ A @ B14 @ I2 )
& ( ord_less @ A @ I2 @ B32 ) ) ) ) ) ) ) ).
% Un_interval
thf(fact_6140_INF__union,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [M3: B > A,A5: set @ B,B5: set @ B] :
( ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ M3 @ ( sup_sup @ ( set @ B ) @ A5 @ B5 ) ) )
= ( inf_inf @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ M3 @ A5 ) ) @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ M3 @ B5 ) ) ) ) ) ).
% INF_union
thf(fact_6141_UN__extend__simps_I2_J,axiom,
! [D: $tType,C: $tType,C6: set @ C,A5: C > ( set @ D ),B5: set @ D] :
( ( ( C6
= ( bot_bot @ ( set @ C ) ) )
=> ( ( sup_sup @ ( set @ D ) @ ( complete_Sup_Sup @ ( set @ D ) @ ( image2 @ C @ ( set @ D ) @ A5 @ C6 ) ) @ B5 )
= B5 ) )
& ( ( C6
!= ( bot_bot @ ( set @ C ) ) )
=> ( ( sup_sup @ ( set @ D ) @ ( complete_Sup_Sup @ ( set @ D ) @ ( image2 @ C @ ( set @ D ) @ A5 @ C6 ) ) @ B5 )
= ( complete_Sup_Sup @ ( set @ D )
@ ( image2 @ C @ ( set @ D )
@ ^ [X: C] : ( sup_sup @ ( set @ D ) @ ( A5 @ X ) @ B5 )
@ C6 ) ) ) ) ) ).
% UN_extend_simps(2)
thf(fact_6142_UN__extend__simps_I3_J,axiom,
! [E: $tType,F7: $tType,C6: set @ F7,A5: set @ E,B5: F7 > ( set @ E )] :
( ( ( C6
= ( bot_bot @ ( set @ F7 ) ) )
=> ( ( sup_sup @ ( set @ E ) @ A5 @ ( complete_Sup_Sup @ ( set @ E ) @ ( image2 @ F7 @ ( set @ E ) @ B5 @ C6 ) ) )
= A5 ) )
& ( ( C6
!= ( bot_bot @ ( set @ F7 ) ) )
=> ( ( sup_sup @ ( set @ E ) @ A5 @ ( complete_Sup_Sup @ ( set @ E ) @ ( image2 @ F7 @ ( set @ E ) @ B5 @ C6 ) ) )
= ( complete_Sup_Sup @ ( set @ E )
@ ( image2 @ F7 @ ( set @ E )
@ ^ [X: F7] : ( sup_sup @ ( set @ E ) @ A5 @ ( B5 @ X ) )
@ C6 ) ) ) ) ) ).
% UN_extend_simps(3)
thf(fact_6143_INT__Un,axiom,
! [A: $tType,B: $tType,M3: B > ( set @ A ),A5: set @ B,B5: set @ B] :
( ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ M3 @ ( sup_sup @ ( set @ B ) @ A5 @ B5 ) ) )
= ( inf_inf @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ M3 @ A5 ) ) @ ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ M3 @ B5 ) ) ) ) ).
% INT_Un
thf(fact_6144_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_6145_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_6146_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 )
@ ^ [X: set @ A] : ( member @ ( set @ A ) @ X @ ( set2 @ ( set @ A ) @ L ) ) ) ) ) ).
% foldl_set
thf(fact_6147_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_6148_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_6149_sum_Ounion__inter,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A5: set @ B,B5: set @ B,G: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( finite_finite2 @ B @ B5 )
=> ( ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( sup_sup @ ( set @ B ) @ A5 @ B5 ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( inf_inf @ ( set @ B ) @ A5 @ B5 ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G @ A5 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ B5 ) ) ) ) ) ) ).
% sum.union_inter
thf(fact_6150_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_6151_greaterThanLessThan__eq,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( set_or5935395276787703475ssThan @ A )
= ( ^ [A3: A,B3: A] : ( inf_inf @ ( set @ A ) @ ( set_ord_greaterThan @ A @ A3 ) @ ( set_ord_lessThan @ A @ B3 ) ) ) ) ) ).
% greaterThanLessThan_eq
thf(fact_6152_card__Un__Int,axiom,
! [A: $tType,A5: set @ A,B5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( finite_finite2 @ A @ B5 )
=> ( ( plus_plus @ nat @ ( finite_card @ A @ A5 ) @ ( finite_card @ A @ B5 ) )
= ( plus_plus @ nat @ ( finite_card @ A @ ( sup_sup @ ( set @ A ) @ A5 @ B5 ) ) @ ( finite_card @ A @ ( inf_inf @ ( set @ A ) @ A5 @ B5 ) ) ) ) ) ) ).
% card_Un_Int
thf(fact_6153_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_6154_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_6155_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_6156_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_6157_Max_Ounion,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,B5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B5 )
=> ( ( B5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ ( sup_sup @ ( set @ A ) @ A5 @ B5 ) )
= ( ord_max @ A @ ( lattic643756798349783984er_Max @ A @ A5 ) @ ( lattic643756798349783984er_Max @ A @ B5 ) ) ) ) ) ) ) ) ).
% Max.union
thf(fact_6158_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_6159_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_6160_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_6161_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_6162_sup__bot_Osemilattice__neutr__order__axioms,axiom,
! [A: $tType] :
( ( bounde4967611905675639751up_bot @ A )
=> ( semila1105856199041335345_order @ A @ ( sup_sup @ A ) @ ( bot_bot @ A )
@ ^ [X: A,Y4: A] : ( ord_less_eq @ A @ Y4 @ X )
@ ^ [X: A,Y4: A] : ( ord_less @ A @ Y4 @ X ) ) ) ).
% sup_bot.semilattice_neutr_order_axioms
thf(fact_6163_trancl__insert2,axiom,
! [A: $tType,A2: A,B2: A,R: set @ ( product_prod @ A @ A )] :
( ( transitive_trancl @ A @ ( insert @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ R ) )
= ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ ( transitive_trancl @ A @ R )
@ ( collect @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ^ [X: A,Y4: A] :
( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ A2 ) @ ( transitive_trancl @ A @ R ) )
| ( X = A2 ) )
& ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ Y4 ) @ ( transitive_trancl @ A @ R ) )
| ( Y4 = B2 ) ) ) ) ) ) ) ).
% trancl_insert2
thf(fact_6164_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_6165_Un__eq__UN,axiom,
! [A: $tType] :
( ( sup_sup @ ( set @ A ) )
= ( ^ [A8: set @ A,B8: set @ A] :
( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ $o @ ( set @ A )
@ ^ [B3: $o] : ( if @ ( set @ A ) @ B3 @ A8 @ B8 )
@ ( top_top @ ( set @ $o ) ) ) ) ) ) ).
% Un_eq_UN
thf(fact_6166_sum_Ounion__inter__neutral,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A5: set @ B,B5: set @ B,G: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( finite_finite2 @ B @ B5 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( inf_inf @ ( set @ B ) @ A5 @ B5 ) )
=> ( ( G @ X3 )
= ( zero_zero @ A ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( sup_sup @ ( set @ B ) @ A5 @ B5 ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G @ A5 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ B5 ) ) ) ) ) ) ) ).
% sum.union_inter_neutral
thf(fact_6167_sum__Un,axiom,
! [A: $tType,B: $tType] :
( ( ab_group_add @ A )
=> ! [A5: set @ B,B5: set @ B,F: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( finite_finite2 @ B @ B5 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ F @ ( sup_sup @ ( set @ B ) @ A5 @ B5 ) )
= ( minus_minus @ A @ ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F @ A5 ) @ ( groups7311177749621191930dd_sum @ B @ A @ F @ B5 ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ F @ ( inf_inf @ ( set @ B ) @ A5 @ B5 ) ) ) ) ) ) ) ).
% sum_Un
thf(fact_6168_sum_Ounion__disjoint,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A5: set @ B,B5: set @ B,G: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( finite_finite2 @ B @ B5 )
=> ( ( ( inf_inf @ ( set @ B ) @ A5 @ B5 )
= ( bot_bot @ ( set @ B ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( sup_sup @ ( set @ B ) @ A5 @ B5 ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G @ A5 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ B5 ) ) ) ) ) ) ) ).
% sum.union_disjoint
thf(fact_6169_prod_Ounion__inter__neutral,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A5: set @ B,B5: set @ B,G: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( finite_finite2 @ B @ B5 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( inf_inf @ ( set @ B ) @ A5 @ B5 ) )
=> ( ( G @ X3 )
= ( one_one @ A ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( sup_sup @ ( set @ B ) @ A5 @ B5 ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G @ A5 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ B5 ) ) ) ) ) ) ) ).
% prod.union_inter_neutral
thf(fact_6170_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_6171_sum_Ounion__diff2,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A5: set @ B,B5: set @ B,G: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( finite_finite2 @ B @ B5 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( sup_sup @ ( set @ B ) @ A5 @ B5 ) )
= ( plus_plus @ A @ ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A5 @ B5 ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ B5 @ A5 ) ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( inf_inf @ ( set @ B ) @ A5 @ B5 ) ) ) ) ) ) ) ).
% sum.union_diff2
thf(fact_6172_sum__Un2,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ B )
=> ! [A5: set @ A,B5: set @ A,F: A > B] :
( ( finite_finite2 @ A @ ( sup_sup @ ( set @ A ) @ A5 @ B5 ) )
=> ( ( groups7311177749621191930dd_sum @ A @ B @ F @ ( sup_sup @ ( set @ A ) @ A5 @ B5 ) )
= ( plus_plus @ B @ ( plus_plus @ B @ ( groups7311177749621191930dd_sum @ A @ B @ F @ ( minus_minus @ ( set @ A ) @ A5 @ B5 ) ) @ ( groups7311177749621191930dd_sum @ A @ B @ F @ ( minus_minus @ ( set @ A ) @ B5 @ A5 ) ) ) @ ( groups7311177749621191930dd_sum @ A @ B @ F @ ( inf_inf @ ( set @ A ) @ A5 @ B5 ) ) ) ) ) ) ).
% sum_Un2
thf(fact_6173_card__Un__disjoint,axiom,
! [A: $tType,A5: set @ A,B5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( finite_finite2 @ A @ B5 )
=> ( ( ( inf_inf @ ( set @ A ) @ A5 @ B5 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_card @ A @ ( sup_sup @ ( set @ A ) @ A5 @ B5 ) )
= ( plus_plus @ nat @ ( finite_card @ A @ A5 ) @ ( finite_card @ A @ B5 ) ) ) ) ) ) ).
% card_Un_disjoint
thf(fact_6174_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_6175_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_6176_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_6177_sum__Un__nat,axiom,
! [A: $tType,A5: set @ A,B5: set @ A,F: A > nat] :
( ( finite_finite2 @ A @ A5 )
=> ( ( finite_finite2 @ A @ B5 )
=> ( ( groups7311177749621191930dd_sum @ A @ nat @ F @ ( sup_sup @ ( set @ A ) @ A5 @ B5 ) )
= ( minus_minus @ nat @ ( plus_plus @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ F @ A5 ) @ ( groups7311177749621191930dd_sum @ A @ nat @ F @ B5 ) ) @ ( groups7311177749621191930dd_sum @ A @ nat @ F @ ( inf_inf @ ( set @ A ) @ A5 @ B5 ) ) ) ) ) ) ).
% sum_Un_nat
thf(fact_6178_greaterThan__0,axiom,
( ( set_ord_greaterThan @ nat @ ( zero_zero @ nat ) )
= ( image2 @ nat @ nat @ suc @ ( top_top @ ( set @ nat ) ) ) ) ).
% greaterThan_0
thf(fact_6179_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_6180_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_6181_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_6182_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_6183_prod__Un,axiom,
! [A: $tType,B: $tType] :
( ( field @ A )
=> ! [A5: set @ B,B5: set @ B,F: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( finite_finite2 @ B @ B5 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( inf_inf @ ( set @ B ) @ A5 @ B5 ) )
=> ( ( F @ X3 )
!= ( zero_zero @ A ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ F @ ( sup_sup @ ( set @ B ) @ A5 @ B5 ) )
= ( divide_divide @ A @ ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ F @ A5 ) @ ( groups7121269368397514597t_prod @ B @ A @ F @ B5 ) ) @ ( groups7121269368397514597t_prod @ B @ A @ F @ ( inf_inf @ ( set @ B ) @ A5 @ B5 ) ) ) ) ) ) ) ) ).
% prod_Un
thf(fact_6184_UN__le__eq__Un0,axiom,
! [A: $tType,M3: nat > ( set @ A ),N: nat] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ M3 @ ( set_ord_atMost @ nat @ N ) ) )
= ( sup_sup @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ M3 @ ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ N ) ) ) @ ( M3 @ ( zero_zero @ nat ) ) ) ) ).
% UN_le_eq_Un0
thf(fact_6185_interval__cases,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [S2: set @ A] :
( ! [A4: A,B4: A,X3: A] :
( ( member @ A @ A4 @ S2 )
=> ( ( member @ A @ B4 @ S2 )
=> ( ( ord_less_eq @ A @ A4 @ X3 )
=> ( ( ord_less_eq @ A @ X3 @ B4 )
=> ( member @ A @ X3 @ S2 ) ) ) ) )
=> ? [A4: A,B4: A] :
( ( S2
= ( bot_bot @ ( set @ A ) ) )
| ( S2
= ( top_top @ ( set @ A ) ) )
| ( S2
= ( set_ord_lessThan @ A @ B4 ) )
| ( S2
= ( set_ord_atMost @ A @ B4 ) )
| ( S2
= ( set_ord_greaterThan @ A @ A4 ) )
| ( S2
= ( set_ord_atLeast @ A @ A4 ) )
| ( S2
= ( set_or5935395276787703475ssThan @ A @ A4 @ B4 ) )
| ( S2
= ( set_or3652927894154168847AtMost @ A @ A4 @ B4 ) )
| ( S2
= ( set_or7035219750837199246ssThan @ A @ A4 @ B4 ) )
| ( S2
= ( set_or1337092689740270186AtMost @ A @ A4 @ B4 ) ) ) ) ) ).
% interval_cases
thf(fact_6186_brk__rel__def,axiom,
! [B: $tType,A: $tType] :
( ( brk_rel @ A @ B )
= ( ^ [R7: set @ ( product_prod @ A @ B )] :
( sup_sup @ ( set @ ( product_prod @ ( product_prod @ $o @ A ) @ ( product_prod @ $o @ B ) ) )
@ ( collect @ ( product_prod @ ( product_prod @ $o @ A ) @ ( product_prod @ $o @ B ) )
@ ^ [Uu: product_prod @ ( product_prod @ $o @ A ) @ ( product_prod @ $o @ B )] :
? [X: A,Y4: B] :
( ( Uu
= ( product_Pair @ ( product_prod @ $o @ A ) @ ( product_prod @ $o @ B ) @ ( product_Pair @ $o @ A @ $false @ X ) @ ( product_Pair @ $o @ B @ $false @ Y4 ) ) )
& ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y4 ) @ R7 ) ) )
@ ( collect @ ( product_prod @ ( product_prod @ $o @ A ) @ ( product_prod @ $o @ B ) )
@ ^ [Uu: product_prod @ ( product_prod @ $o @ A ) @ ( product_prod @ $o @ B )] :
? [X: A,Y4: B] :
( Uu
= ( product_Pair @ ( product_prod @ $o @ A ) @ ( product_prod @ $o @ B ) @ ( product_Pair @ $o @ A @ $true @ X ) @ ( product_Pair @ $o @ B @ $false @ Y4 ) ) ) ) ) ) ) ).
% brk_rel_def
thf(fact_6187_atLeast__eq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X2: A,Y: A] :
( ( ( set_ord_atLeast @ A @ X2 )
= ( set_ord_atLeast @ A @ Y ) )
= ( X2 = Y ) ) ) ).
% atLeast_eq_iff
thf(fact_6188_atLeast__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [I: A,K: A] :
( ( member @ A @ I @ ( set_ord_atLeast @ A @ K ) )
= ( ord_less_eq @ A @ K @ I ) ) ) ).
% atLeast_iff
thf(fact_6189_atLeast__0,axiom,
( ( set_ord_atLeast @ nat @ ( zero_zero @ nat ) )
= ( top_top @ ( set @ nat ) ) ) ).
% atLeast_0
thf(fact_6190_Inf__atLeast,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X2: A] :
( ( complete_Inf_Inf @ A @ ( set_ord_atLeast @ A @ X2 ) )
= X2 ) ) ).
% Inf_atLeast
thf(fact_6191_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_6192_atLeast__subset__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A,Y: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_ord_atLeast @ A @ X2 ) @ ( set_ord_atLeast @ A @ Y ) )
= ( ord_less_eq @ A @ Y @ X2 ) ) ) ).
% atLeast_subset_iff
thf(fact_6193_image__add__atLeast,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [K: A,I: A] :
( ( image2 @ A @ A @ ( plus_plus @ A @ K ) @ ( set_ord_atLeast @ A @ I ) )
= ( set_ord_atLeast @ A @ ( plus_plus @ A @ K @ I ) ) ) ) ).
% image_add_atLeast
thf(fact_6194_Sup__atLeast,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X2: A] :
( ( complete_Sup_Sup @ A @ ( set_ord_atLeast @ A @ X2 ) )
= ( top_top @ A ) ) ) ).
% Sup_atLeast
thf(fact_6195_Compl__atLeast,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [K: A] :
( ( uminus_uminus @ ( set @ A ) @ ( set_ord_atLeast @ A @ K ) )
= ( set_ord_lessThan @ A @ K ) ) ) ).
% Compl_atLeast
thf(fact_6196_Compl__lessThan,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [K: A] :
( ( uminus_uminus @ ( set @ A ) @ ( set_ord_lessThan @ A @ K ) )
= ( set_ord_atLeast @ A @ K ) ) ) ).
% Compl_lessThan
thf(fact_6197_Icc__subset__Ici__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [L: A,H: A,L4: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ H ) @ ( set_ord_atLeast @ A @ L4 ) )
= ( ~ ( ord_less_eq @ A @ L @ H )
| ( ord_less_eq @ A @ L4 @ L ) ) ) ) ).
% Icc_subset_Ici_iff
thf(fact_6198_image__minus__const__atLeast,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C2: A,A2: A] :
( ( image2 @ A @ A @ ( minus_minus @ A @ C2 ) @ ( set_ord_atLeast @ A @ A2 ) )
= ( set_ord_atMost @ A @ ( minus_minus @ A @ C2 @ A2 ) ) ) ) ).
% image_minus_const_atLeast
thf(fact_6199_image__minus__const__AtMost,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C2: A,B2: A] :
( ( image2 @ A @ A @ ( minus_minus @ A @ C2 ) @ ( set_ord_atMost @ A @ B2 ) )
= ( set_ord_atLeast @ A @ ( minus_minus @ A @ C2 @ B2 ) ) ) ) ).
% image_minus_const_AtMost
thf(fact_6200_image__uminus__atMost,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [X2: A] :
( ( image2 @ A @ A @ ( uminus_uminus @ A ) @ ( set_ord_atMost @ A @ X2 ) )
= ( set_ord_atLeast @ A @ ( uminus_uminus @ A @ X2 ) ) ) ) ).
% image_uminus_atMost
thf(fact_6201_image__uminus__atLeast,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [X2: A] :
( ( image2 @ A @ A @ ( uminus_uminus @ A ) @ ( set_ord_atLeast @ A @ X2 ) )
= ( set_ord_atMost @ A @ ( uminus_uminus @ A @ X2 ) ) ) ) ).
% image_uminus_atLeast
thf(fact_6202_Int__atLeastAtMostR2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,C2: A,D2: A] :
( ( inf_inf @ ( set @ A ) @ ( set_ord_atLeast @ A @ A2 ) @ ( set_or1337092689740270186AtMost @ A @ C2 @ D2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( ord_max @ A @ A2 @ C2 ) @ D2 ) ) ) ).
% Int_atLeastAtMostR2
thf(fact_6203_Int__atLeastAtMostL2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) @ ( set_ord_atLeast @ A @ C2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( ord_max @ A @ A2 @ C2 ) @ B2 ) ) ) ).
% Int_atLeastAtMostL2
thf(fact_6204_sup__Un__eq2,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ B ),S2: set @ ( product_prod @ A @ B )] :
( ( sup_sup @ ( A > B > $o )
@ ^ [X: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y4 ) @ R3 )
@ ^ [X: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y4 ) @ S2 ) )
= ( ^ [X: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y4 ) @ ( sup_sup @ ( set @ ( product_prod @ A @ B ) ) @ R3 @ S2 ) ) ) ) ).
% sup_Un_eq2
thf(fact_6205_sup__set__def,axiom,
! [A: $tType] :
( ( sup_sup @ ( set @ A ) )
= ( ^ [A8: set @ A,B8: set @ A] :
( collect @ A
@ ( sup_sup @ ( A > $o )
@ ^ [X: A] : ( member @ A @ X @ A8 )
@ ^ [X: A] : ( member @ A @ X @ B8 ) ) ) ) ) ).
% sup_set_def
thf(fact_6206_sup__Un__eq,axiom,
! [A: $tType,R3: set @ A,S2: set @ A] :
( ( sup_sup @ ( A > $o )
@ ^ [X: A] : ( member @ A @ X @ R3 )
@ ^ [X: A] : ( member @ A @ X @ S2 ) )
= ( ^ [X: A] : ( member @ A @ X @ ( sup_sup @ ( set @ A ) @ R3 @ S2 ) ) ) ) ).
% sup_Un_eq
thf(fact_6207_sup__nat__def,axiom,
( ( sup_sup @ nat )
= ( ord_max @ nat ) ) ).
% sup_nat_def
thf(fact_6208_sup__int__def,axiom,
( ( sup_sup @ int )
= ( ord_max @ int ) ) ).
% sup_int_def
thf(fact_6209_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_6210_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_6211_infinite__Ici,axiom,
! [A: $tType] :
( ( ( linorder @ A )
& ( no_top @ A ) )
=> ! [A2: A] :
~ ( finite_finite2 @ A @ ( set_ord_atLeast @ A @ A2 ) ) ) ).
% infinite_Ici
thf(fact_6212_not__Iic__eq__Ici,axiom,
! [A: $tType] :
( ( no_top @ A )
=> ! [H: A,L4: A] :
( ( set_ord_atMost @ A @ H )
!= ( set_ord_atLeast @ A @ L4 ) ) ) ).
% not_Iic_eq_Ici
thf(fact_6213_not__Ici__eq__Icc,axiom,
! [A: $tType] :
( ( no_top @ A )
=> ! [L4: A,L: A,H: A] :
( ( set_ord_atLeast @ A @ L4 )
!= ( set_or1337092689740270186AtMost @ A @ L @ H ) ) ) ).
% not_Ici_eq_Icc
thf(fact_6214_not__UNIV__eq__Ici,axiom,
! [A: $tType] :
( ( no_bot @ A )
=> ! [L4: A] :
( ( top_top @ ( set @ A ) )
!= ( set_ord_atLeast @ A @ L4 ) ) ) ).
% not_UNIV_eq_Ici
thf(fact_6215_atLeast__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( set_ord_atLeast @ A )
= ( ^ [L2: A] : ( collect @ A @ ( ord_less_eq @ A @ L2 ) ) ) ) ) ).
% atLeast_def
thf(fact_6216_atLeast__eq__UNIV__iff,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [X2: A] :
( ( ( set_ord_atLeast @ A @ X2 )
= ( top_top @ ( set @ A ) ) )
= ( X2
= ( bot_bot @ A ) ) ) ) ).
% atLeast_eq_UNIV_iff
thf(fact_6217_not__Ici__le__Icc,axiom,
! [A: $tType] :
( ( no_top @ A )
=> ! [L: A,L4: A,H4: A] :
~ ( ord_less_eq @ ( set @ A ) @ ( set_ord_atLeast @ A @ L ) @ ( set_or1337092689740270186AtMost @ A @ L4 @ H4 ) ) ) ).
% not_Ici_le_Icc
thf(fact_6218_not__Iic__le__Ici,axiom,
! [A: $tType] :
( ( no_bot @ A )
=> ! [H: A,L4: A] :
~ ( ord_less_eq @ ( set @ A ) @ ( set_ord_atMost @ A @ H ) @ ( set_ord_atLeast @ A @ L4 ) ) ) ).
% not_Iic_le_Ici
thf(fact_6219_not__Ici__le__Iic,axiom,
! [A: $tType] :
( ( no_top @ A )
=> ! [L: A,H4: A] :
~ ( ord_less_eq @ ( set @ A ) @ ( set_ord_atLeast @ A @ L ) @ ( set_ord_atMost @ A @ H4 ) ) ) ).
% not_Ici_le_Iic
thf(fact_6220_Ioi__le__Ico,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A] : ( ord_less_eq @ ( set @ A ) @ ( set_ord_greaterThan @ A @ A2 ) @ ( set_ord_atLeast @ A @ A2 ) ) ) ).
% Ioi_le_Ico
thf(fact_6221_atLeast__Suc__greaterThan,axiom,
! [K: nat] :
( ( set_ord_atLeast @ nat @ ( suc @ K ) )
= ( set_ord_greaterThan @ nat @ K ) ) ).
% atLeast_Suc_greaterThan
thf(fact_6222_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_6223_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_6224_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_6225_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_6226_Ici__subset__Ioi__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_ord_atLeast @ A @ A2 ) @ ( set_ord_greaterThan @ A @ B2 ) )
= ( ord_less @ A @ B2 @ A2 ) ) ) ).
% Ici_subset_Ioi_iff
thf(fact_6227_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_6228_atMost__Int__atLeast,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [N: A] :
( ( inf_inf @ ( set @ A ) @ ( set_ord_atMost @ A @ N ) @ ( set_ord_atLeast @ A @ N ) )
= ( insert @ A @ N @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% atMost_Int_atLeast
thf(fact_6229_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_6230_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_6231_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_6232_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_6233_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_6234_insert__relcomp__union__fold,axiom,
! [C: $tType,B: $tType,A: $tType,S2: set @ ( product_prod @ A @ B ),X2: product_prod @ C @ A,X8: set @ ( product_prod @ C @ B )] :
( ( finite_finite2 @ ( product_prod @ A @ B ) @ S2 )
=> ( ( sup_sup @ ( set @ ( product_prod @ C @ B ) ) @ ( relcomp @ C @ A @ B @ ( insert @ ( product_prod @ C @ A ) @ X2 @ ( bot_bot @ ( set @ ( product_prod @ C @ A ) ) ) ) @ S2 ) @ X8 )
= ( 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 ) ) )
@ ^ [W: A,Z3: B,A15: set @ ( product_prod @ C @ B )] :
( if @ ( set @ ( product_prod @ C @ B ) )
@ ( ( product_snd @ C @ A @ X2 )
= W )
@ ( insert @ ( product_prod @ C @ B ) @ ( product_Pair @ C @ B @ ( product_fst @ C @ A @ X2 ) @ Z3 ) @ A15 )
@ A15 ) )
@ X8
@ S2 ) ) ) ).
% insert_relcomp_union_fold
thf(fact_6235_fold__union__pair,axiom,
! [B: $tType,A: $tType,B5: set @ A,X2: B,A5: set @ ( product_prod @ B @ A )] :
( ( finite_finite2 @ A @ B5 )
=> ( ( sup_sup @ ( set @ ( product_prod @ B @ A ) )
@ ( complete_Sup_Sup @ ( set @ ( product_prod @ B @ A ) )
@ ( image2 @ A @ ( set @ ( product_prod @ B @ A ) )
@ ^ [Y4: A] : ( insert @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X2 @ Y4 ) @ ( bot_bot @ ( set @ ( product_prod @ B @ A ) ) ) )
@ B5 ) )
@ A5 )
= ( finite_fold @ A @ ( set @ ( product_prod @ B @ A ) )
@ ^ [Y4: A] : ( insert @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X2 @ Y4 ) )
@ A5
@ B5 ) ) ) ).
% fold_union_pair
thf(fact_6236_sum_Oeq__fold,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ( ( groups7311177749621191930dd_sum @ B @ A )
= ( ^ [G2: B > A] : ( finite_fold @ B @ A @ ( comp @ A @ ( A > A ) @ B @ ( plus_plus @ A ) @ G2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% sum.eq_fold
thf(fact_6237_prod_Oeq__fold,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ( ( groups7121269368397514597t_prod @ B @ A )
= ( ^ [G2: B > A] : ( finite_fold @ B @ A @ ( comp @ A @ ( A > A ) @ B @ ( times_times @ A ) @ G2 ) @ ( one_one @ A ) ) ) ) ) ).
% prod.eq_fold
thf(fact_6238_Max_Oeq__fold,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X2: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( lattic643756798349783984er_Max @ A @ ( insert @ A @ X2 @ A5 ) )
= ( finite_fold @ A @ A @ ( ord_max @ A ) @ X2 @ A5 ) ) ) ) ).
% Max.eq_fold
thf(fact_6239_image__fold__insert,axiom,
! [B: $tType,A: $tType,A5: set @ A,F: A > B] :
( ( finite_finite2 @ A @ A5 )
=> ( ( image2 @ A @ B @ F @ A5 )
= ( finite_fold @ A @ ( set @ B )
@ ^ [K2: A] : ( insert @ B @ ( F @ K2 ) )
@ ( bot_bot @ ( set @ B ) )
@ A5 ) ) ) ).
% image_fold_insert
thf(fact_6240_product__fold,axiom,
! [B: $tType,A: $tType,A5: set @ A,B5: set @ B] :
( ( finite_finite2 @ A @ A5 )
=> ( ( finite_finite2 @ B @ B5 )
=> ( ( product_Sigma @ A @ B @ A5
@ ^ [Uu: A] : B5 )
= ( finite_fold @ A @ ( set @ ( product_prod @ A @ B ) )
@ ^ [X: A,Z3: set @ ( product_prod @ A @ B )] :
( finite_fold @ B @ ( set @ ( product_prod @ A @ B ) )
@ ^ [Y4: B] : ( insert @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y4 ) )
@ Z3
@ B5 )
@ ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) )
@ A5 ) ) ) ) ).
% product_fold
thf(fact_6241_relcomp__fold,axiom,
! [C: $tType,B: $tType,A: $tType,R3: set @ ( product_prod @ A @ B ),S2: set @ ( product_prod @ B @ C )] :
( ( finite_finite2 @ ( product_prod @ A @ B ) @ R3 )
=> ( ( finite_finite2 @ ( product_prod @ B @ C ) @ S2 )
=> ( ( relcomp @ A @ B @ C @ R3 @ S2 )
= ( 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 ) ) )
@ ^ [X: 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 ) ) )
@ ^ [W: B,Z3: C,A15: set @ ( product_prod @ A @ C )] : ( if @ ( set @ ( product_prod @ A @ C ) ) @ ( Y4 = W ) @ ( insert @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ X @ Z3 ) @ A15 ) @ A15 ) )
@ A8
@ S2 ) )
@ ( bot_bot @ ( set @ ( product_prod @ A @ C ) ) )
@ R3 ) ) ) ) ).
% relcomp_fold
thf(fact_6242_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 ) )
@ ^ [X: A] : ( insert @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ X ) )
@ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) )
@ A5 ) ) ) ).
% Id_on_fold
thf(fact_6243_insert__relcomp__fold,axiom,
! [C: $tType,B: $tType,A: $tType,S2: set @ ( product_prod @ A @ B ),X2: product_prod @ C @ A,R3: set @ ( product_prod @ C @ A )] :
( ( finite_finite2 @ ( product_prod @ A @ B ) @ S2 )
=> ( ( relcomp @ C @ A @ B @ ( insert @ ( product_prod @ C @ A ) @ X2 @ R3 ) @ S2 )
= ( 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 ) ) )
@ ^ [W: A,Z3: B,A15: set @ ( product_prod @ C @ B )] :
( if @ ( set @ ( product_prod @ C @ B ) )
@ ( ( product_snd @ C @ A @ X2 )
= W )
@ ( insert @ ( product_prod @ C @ B ) @ ( product_Pair @ C @ B @ ( product_fst @ C @ A @ X2 ) @ Z3 ) @ A15 )
@ A15 ) )
@ ( relcomp @ C @ A @ B @ R3 @ S2 )
@ S2 ) ) ) ).
% insert_relcomp_fold
thf(fact_6244_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 )
@ ^ [X: A,A15: set @ A] : ( if @ ( set @ A ) @ ( P @ X ) @ ( insert @ A @ X @ A15 ) @ A15 )
@ ( bot_bot @ ( set @ A ) )
@ A5 ) ) ) ).
% Set_filter_fold
thf(fact_6245_comp__fun__commute__relcomp__fold,axiom,
! [A: $tType,B: $tType,C: $tType,S2: set @ ( product_prod @ A @ B )] :
( ( finite_finite2 @ ( product_prod @ A @ B ) @ S2 )
=> ( 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 ) ) )
@ ^ [X: 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 ) ) )
@ ^ [W: A,Z3: B,A15: set @ ( product_prod @ C @ B )] : ( if @ ( set @ ( product_prod @ C @ B ) ) @ ( Y4 = W ) @ ( insert @ ( product_prod @ C @ B ) @ ( product_Pair @ C @ B @ X @ Z3 ) @ A15 ) @ A15 ) )
@ A8
@ S2 ) ) ) ) ).
% comp_fun_commute_relcomp_fold
thf(fact_6246_comp__fun__commute_Ocomp__fun__commute__funpow,axiom,
! [B: $tType,A: $tType,F: A > B > B,G: A > nat] :
( ( finite6289374366891150609ommute @ A @ B @ F )
=> ( finite6289374366891150609ommute @ A @ B
@ ^ [X: A] : ( compow @ ( B > B ) @ ( G @ X ) @ ( F @ X ) ) ) ) ).
% comp_fun_commute.comp_fun_commute_funpow
thf(fact_6247_Set_Ofilter__def,axiom,
! [A: $tType] :
( ( filter3 @ A )
= ( ^ [P4: A > $o,A8: set @ A] :
( collect @ A
@ ^ [A3: A] :
( ( member @ A @ A3 @ A8 )
& ( P4 @ A3 ) ) ) ) ) ).
% Set.filter_def
thf(fact_6248_comp__fun__commute__const,axiom,
! [B: $tType,A: $tType,F: B > B] :
( finite6289374366891150609ommute @ A @ B
@ ^ [Uu: A] : F ) ).
% comp_fun_commute_const
thf(fact_6249_comp__fun__commute__filter__fold,axiom,
! [A: $tType,P: A > $o] :
( finite6289374366891150609ommute @ A @ ( set @ A )
@ ^ [X: A,A15: set @ A] : ( if @ ( set @ A ) @ ( P @ X ) @ ( insert @ A @ X @ A15 ) @ A15 ) ) ).
% comp_fun_commute_filter_fold
thf(fact_6250_comp__fun__commute_Ofoldl__f__commute,axiom,
! [B: $tType,A: $tType,F: A > B > B,A2: A,B2: B,Xs: list @ A] :
( ( finite6289374366891150609ommute @ A @ B @ F )
=> ( ( F @ A2
@ ( foldl @ B @ A
@ ^ [A3: B,B3: A] : ( F @ B3 @ A3 )
@ B2
@ Xs ) )
= ( foldl @ B @ A
@ ^ [A3: B,B3: A] : ( F @ B3 @ A3 )
@ ( F @ A2 @ B2 )
@ Xs ) ) ) ).
% comp_fun_commute.foldl_f_commute
thf(fact_6251_card_Oeq__fold,axiom,
! [A: $tType] :
( ( finite_card @ A )
= ( finite_fold @ A @ nat
@ ^ [Uu: A] : suc
@ ( zero_zero @ nat ) ) ) ).
% card.eq_fold
thf(fact_6252_comp__fun__commute_Ofoldr__conv__foldl,axiom,
! [B: $tType,A: $tType,F: A > B > B,Xs: list @ A,A2: B] :
( ( finite6289374366891150609ommute @ A @ B @ F )
=> ( ( foldr @ A @ B @ F @ Xs @ A2 )
= ( foldl @ B @ A
@ ^ [A3: B,B3: A] : ( F @ B3 @ A3 )
@ A2
@ Xs ) ) ) ).
% comp_fun_commute.foldr_conv_foldl
thf(fact_6253_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
@ ^ [X: A] : X )
@ ( nil @ A ) ) ) ) ).
% sorted_list_of_set.fold_insort_key.eq_fold
thf(fact_6254_comp__fun__commute__Image__fold,axiom,
! [B: $tType,A: $tType,S2: set @ A] :
( finite6289374366891150609ommute @ ( product_prod @ A @ B ) @ ( set @ B )
@ ( product_case_prod @ A @ B @ ( ( set @ B ) > ( set @ B ) )
@ ^ [X: A,Y4: B,A8: set @ B] : ( if @ ( set @ B ) @ ( member @ A @ X @ S2 ) @ ( insert @ B @ Y4 @ A8 ) @ A8 ) ) ) ).
% comp_fun_commute_Image_fold
thf(fact_6255_inter__Set__filter,axiom,
! [A: $tType,B5: set @ A,A5: set @ A] :
( ( finite_finite2 @ A @ B5 )
=> ( ( inf_inf @ ( set @ A ) @ A5 @ B5 )
= ( filter3 @ A
@ ^ [X: A] : ( member @ A @ X @ A5 )
@ B5 ) ) ) ).
% inter_Set_filter
thf(fact_6256_comp__fun__commute__product__fold,axiom,
! [A: $tType,B: $tType,B5: set @ A] :
( ( finite_finite2 @ A @ B5 )
=> ( finite6289374366891150609ommute @ B @ ( set @ ( product_prod @ B @ A ) )
@ ^ [X: B,Z3: set @ ( product_prod @ B @ A )] :
( finite_fold @ A @ ( set @ ( product_prod @ B @ A ) )
@ ^ [Y4: A] : ( insert @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X @ Y4 ) )
@ Z3
@ B5 ) ) ) ).
% comp_fun_commute_product_fold
thf(fact_6257_Max_Oeq__fold_H,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( lattic643756798349783984er_Max @ A )
= ( ^ [A8: set @ A] :
( the2 @ A
@ ( finite_fold @ A @ ( option @ A )
@ ^ [X: A,Y4: option @ A] : ( some @ A @ ( case_option @ A @ A @ X @ ( ord_max @ A @ X ) @ Y4 ) )
@ ( none @ A )
@ A8 ) ) ) ) ) ).
% Max.eq_fold'
thf(fact_6258_range__abs__Nats,axiom,
( ( image2 @ int @ int @ ( abs_abs @ int ) @ ( top_top @ ( set @ int ) ) )
= ( semiring_1_Nats @ int ) ) ).
% range_abs_Nats
thf(fact_6259_same__fst__def,axiom,
! [B: $tType,A: $tType] :
( ( same_fst @ A @ B )
= ( ^ [P4: A > $o,R7: A > ( set @ ( product_prod @ B @ B ) )] :
( collect @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) )
@ ( product_case_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ $o
@ ( product_case_prod @ A @ B @ ( ( product_prod @ A @ B ) > $o )
@ ^ [X9: A,Y8: B] :
( product_case_prod @ A @ B @ $o
@ ^ [X: A,Y4: B] :
( ( X9 = X )
& ( P4 @ X )
& ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y8 @ Y4 ) @ ( R7 @ X ) ) ) ) ) ) ) ) ) ).
% same_fst_def
thf(fact_6260_same__fst__trancl,axiom,
! [B: $tType,A: $tType,P: A > $o,R3: A > ( set @ ( product_prod @ B @ B ) )] :
( ( transitive_trancl @ ( product_prod @ A @ B ) @ ( same_fst @ A @ B @ P @ R3 ) )
= ( same_fst @ A @ B @ P
@ ^ [X: A] : ( transitive_trancl @ B @ ( R3 @ X ) ) ) ) ).
% same_fst_trancl
thf(fact_6261_comp__fun__commute__plus__mset,axiom,
! [A: $tType] : ( finite6289374366891150609ommute @ ( multiset @ A ) @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) ) ) ).
% comp_fun_commute_plus_mset
thf(fact_6262_comp__fun__commute__insort,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( finite6289374366891150609ommute @ A @ ( list @ A )
@ ( linorder_insort_key @ A @ A
@ ^ [X: A] : X ) ) ) ).
% comp_fun_commute_insort
thf(fact_6263_Nats__cases,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [X2: A] :
( ( member @ A @ X2 @ ( semiring_1_Nats @ A ) )
=> ~ ! [N2: nat] :
( X2
!= ( semiring_1_of_nat @ A @ N2 ) ) ) ) ).
% Nats_cases
thf(fact_6264_Nats__induct,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [X2: A,P: A > $o] :
( ( member @ A @ X2 @ ( semiring_1_Nats @ A ) )
=> ( ! [N2: nat] : ( P @ ( semiring_1_of_nat @ A @ N2 ) )
=> ( P @ X2 ) ) ) ) ).
% Nats_induct
thf(fact_6265_of__nat__in__Nats,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [N: nat] : ( member @ A @ ( semiring_1_of_nat @ A @ N ) @ ( semiring_1_Nats @ A ) ) ) ).
% of_nat_in_Nats
thf(fact_6266_Nats__0,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( member @ A @ ( zero_zero @ A ) @ ( semiring_1_Nats @ A ) ) ) ).
% Nats_0
thf(fact_6267_Nats__numeral,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [W2: num] : ( member @ A @ ( numeral_numeral @ A @ W2 ) @ ( semiring_1_Nats @ A ) ) ) ).
% Nats_numeral
thf(fact_6268_Nats__add,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [A2: A,B2: A] :
( ( member @ A @ A2 @ ( semiring_1_Nats @ A ) )
=> ( ( member @ A @ B2 @ ( semiring_1_Nats @ A ) )
=> ( member @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( semiring_1_Nats @ A ) ) ) ) ) ).
% Nats_add
thf(fact_6269_Nats__mult,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [A2: A,B2: A] :
( ( member @ A @ A2 @ ( semiring_1_Nats @ A ) )
=> ( ( member @ A @ B2 @ ( semiring_1_Nats @ A ) )
=> ( member @ A @ ( times_times @ A @ A2 @ B2 ) @ ( semiring_1_Nats @ A ) ) ) ) ) ).
% Nats_mult
thf(fact_6270_Nats__1,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( member @ A @ ( one_one @ A ) @ ( semiring_1_Nats @ A ) ) ) ).
% Nats_1
thf(fact_6271_Nats__diff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,B2: A] :
( ( member @ A @ A2 @ ( semiring_1_Nats @ A ) )
=> ( ( member @ A @ B2 @ ( semiring_1_Nats @ A ) )
=> ( ( ord_less_eq @ A @ B2 @ A2 )
=> ( member @ A @ ( minus_minus @ A @ A2 @ B2 ) @ ( semiring_1_Nats @ A ) ) ) ) ) ) ).
% Nats_diff
thf(fact_6272_fold__atLeastAtMost__nat,axiom,
! [A: $tType,F: nat > A > A,A2: nat,B2: nat,Acc4: A] :
( ( finite6289374366891150609ommute @ nat @ A @ F )
=> ( ( set_fo6178422350223883121st_nat @ A @ F @ A2 @ B2 @ Acc4 )
= ( finite_fold @ nat @ A @ F @ Acc4 @ ( set_or1337092689740270186AtMost @ nat @ A2 @ B2 ) ) ) ) ).
% fold_atLeastAtMost_nat
thf(fact_6273_Nats__subset__Ints,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( ord_less_eq @ ( set @ A ) @ ( semiring_1_Nats @ A ) @ ( ring_1_Ints @ A ) ) ) ).
% Nats_subset_Ints
thf(fact_6274_comp__fun__commute__Pow__fold,axiom,
! [A: $tType] :
( finite6289374366891150609ommute @ A @ ( set @ ( set @ A ) )
@ ^ [X: A,A8: set @ ( set @ A )] : ( sup_sup @ ( set @ ( set @ A ) ) @ A8 @ ( image2 @ ( set @ A ) @ ( set @ A ) @ ( insert @ A @ X ) @ A8 ) ) ) ).
% comp_fun_commute_Pow_fold
thf(fact_6275_Nats__def,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( semiring_1_Nats @ A )
= ( image2 @ nat @ A @ ( semiring_1_of_nat @ A ) @ ( top_top @ ( set @ nat ) ) ) ) ) ).
% Nats_def
thf(fact_6276_Nats__altdef2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( ( semiring_1_Nats @ A )
= ( collect @ A
@ ^ [N4: A] :
( ( member @ A @ N4 @ ( ring_1_Ints @ A ) )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ N4 ) ) ) ) ) ).
% Nats_altdef2
thf(fact_6277_Nats__altdef1,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( ( semiring_1_Nats @ A )
= ( collect @ A
@ ^ [Uu: A] :
? [N4: int] :
( ( Uu
= ( ring_1_of_int @ A @ N4 ) )
& ( ord_less_eq @ int @ ( zero_zero @ int ) @ N4 ) ) ) ) ) ).
% Nats_altdef1
thf(fact_6278_Sup__fin_Oeq__fold_H,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( lattic5882676163264333800up_fin @ A )
= ( ^ [A8: set @ A] :
( the2 @ A
@ ( finite_fold @ A @ ( option @ A )
@ ^ [X: A,Y4: option @ A] : ( some @ A @ ( case_option @ A @ A @ X @ ( sup_sup @ A @ X ) @ Y4 ) )
@ ( none @ A )
@ A8 ) ) ) ) ) ).
% Sup_fin.eq_fold'
thf(fact_6279_Inf__fin_Oeq__fold_H,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( lattic7752659483105999362nf_fin @ A )
= ( ^ [A8: set @ A] :
( the2 @ A
@ ( finite_fold @ A @ ( option @ A )
@ ^ [X: A,Y4: option @ A] : ( some @ A @ ( case_option @ A @ A @ X @ ( inf_inf @ A @ X ) @ Y4 ) )
@ ( none @ A )
@ A8 ) ) ) ) ) ).
% Inf_fin.eq_fold'
thf(fact_6280_Sup__fin_Osingleton,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X2: A] :
( ( lattic5882676163264333800up_fin @ A @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) )
= X2 ) ) ).
% Sup_fin.singleton
thf(fact_6281_Inf__fin_Osingleton,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X2: A] :
( ( lattic7752659483105999362nf_fin @ A @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) )
= X2 ) ) ).
% Inf_fin.singleton
thf(fact_6282_inf__Sup__absorb,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [A5: set @ A,A2: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( member @ A @ A2 @ A5 )
=> ( ( inf_inf @ A @ A2 @ ( lattic5882676163264333800up_fin @ A @ A5 ) )
= A2 ) ) ) ) ).
% inf_Sup_absorb
thf(fact_6283_sup__Inf__absorb,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [A5: set @ A,A2: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( member @ A @ A2 @ A5 )
=> ( ( sup_sup @ A @ ( lattic7752659483105999362nf_fin @ A @ A5 ) @ A2 )
= A2 ) ) ) ) ).
% sup_Inf_absorb
thf(fact_6284_Inf__fin_Oinsert,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A5: set @ A,X2: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic7752659483105999362nf_fin @ A @ ( insert @ A @ X2 @ A5 ) )
= ( inf_inf @ A @ X2 @ ( lattic7752659483105999362nf_fin @ A @ A5 ) ) ) ) ) ) ).
% Inf_fin.insert
thf(fact_6285_Sup__fin_Oinsert,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A5: set @ A,X2: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic5882676163264333800up_fin @ A @ ( insert @ A @ X2 @ A5 ) )
= ( sup_sup @ A @ X2 @ ( lattic5882676163264333800up_fin @ A @ A5 ) ) ) ) ) ) ).
% Sup_fin.insert
thf(fact_6286_Inf__fin_Oin__idem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A5: set @ A,X2: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( member @ A @ X2 @ A5 )
=> ( ( inf_inf @ A @ X2 @ ( lattic7752659483105999362nf_fin @ A @ A5 ) )
= ( lattic7752659483105999362nf_fin @ A @ A5 ) ) ) ) ) ).
% Inf_fin.in_idem
thf(fact_6287_Sup__fin_Oin__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A5: set @ A,X2: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( member @ A @ X2 @ A5 )
=> ( ( sup_sup @ A @ X2 @ ( lattic5882676163264333800up_fin @ A @ A5 ) )
= ( lattic5882676163264333800up_fin @ A @ A5 ) ) ) ) ) ).
% Sup_fin.in_idem
thf(fact_6288_Sup__fin__Max,axiom,
! [A: $tType] :
( ( ( semilattice_sup @ A )
& ( linorder @ A ) )
=> ( ( lattic5882676163264333800up_fin @ A )
= ( lattic643756798349783984er_Max @ A ) ) ) ).
% Sup_fin_Max
thf(fact_6289_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_6290_Inf__fin_OcoboundedI,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A5: set @ A,A2: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( member @ A @ A2 @ A5 )
=> ( ord_less_eq @ A @ ( lattic7752659483105999362nf_fin @ A @ A5 ) @ A2 ) ) ) ) ).
% Inf_fin.coboundedI
thf(fact_6291_Sup__fin_OcoboundedI,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A5: set @ A,A2: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( member @ A @ A2 @ A5 )
=> ( ord_less_eq @ A @ A2 @ ( lattic5882676163264333800up_fin @ A @ A5 ) ) ) ) ) ).
% Sup_fin.coboundedI
thf(fact_6292_Sup__fin_Obounded__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A5: set @ A,X2: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ ( lattic5882676163264333800up_fin @ A @ A5 ) @ X2 )
= ( ! [X: A] :
( ( member @ A @ X @ A5 )
=> ( ord_less_eq @ A @ X @ X2 ) ) ) ) ) ) ) ).
% Sup_fin.bounded_iff
thf(fact_6293_Inf__fin_Obounded__iff,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A5: set @ A,X2: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ X2 @ ( lattic7752659483105999362nf_fin @ A @ A5 ) )
= ( ! [X: A] :
( ( member @ A @ X @ A5 )
=> ( ord_less_eq @ A @ X2 @ X ) ) ) ) ) ) ) ).
% Inf_fin.bounded_iff
thf(fact_6294_Sup__fin_OboundedI,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A5: set @ A,X2: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [A4: A] :
( ( member @ A @ A4 @ A5 )
=> ( ord_less_eq @ A @ A4 @ X2 ) )
=> ( ord_less_eq @ A @ ( lattic5882676163264333800up_fin @ A @ A5 ) @ X2 ) ) ) ) ) ).
% Sup_fin.boundedI
thf(fact_6295_Sup__fin_OboundedE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A5: set @ A,X2: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ ( lattic5882676163264333800up_fin @ A @ A5 ) @ X2 )
=> ! [A9: A] :
( ( member @ A @ A9 @ A5 )
=> ( ord_less_eq @ A @ A9 @ X2 ) ) ) ) ) ) ).
% Sup_fin.boundedE
thf(fact_6296_Inf__fin_OboundedI,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A5: set @ A,X2: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [A4: A] :
( ( member @ A @ A4 @ A5 )
=> ( ord_less_eq @ A @ X2 @ A4 ) )
=> ( ord_less_eq @ A @ X2 @ ( lattic7752659483105999362nf_fin @ A @ A5 ) ) ) ) ) ) ).
% Inf_fin.boundedI
thf(fact_6297_Inf__fin_OboundedE,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A5: set @ A,X2: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ X2 @ ( lattic7752659483105999362nf_fin @ A @ A5 ) )
=> ! [A9: A] :
( ( member @ A @ A9 @ A5 )
=> ( ord_less_eq @ A @ X2 @ A9 ) ) ) ) ) ) ).
% Inf_fin.boundedE
thf(fact_6298_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_6299_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_6300_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_6301_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_6302_Inf__fin_Osubset__imp,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A5: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B5 )
=> ( ord_less_eq @ A @ ( lattic7752659483105999362nf_fin @ A @ B5 ) @ ( lattic7752659483105999362nf_fin @ A @ A5 ) ) ) ) ) ) ).
% Inf_fin.subset_imp
thf(fact_6303_Sup__fin_Osubset__imp,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A5: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B5 )
=> ( ord_less_eq @ A @ ( lattic5882676163264333800up_fin @ A @ A5 ) @ ( lattic5882676163264333800up_fin @ A @ B5 ) ) ) ) ) ) ).
% Sup_fin.subset_imp
thf(fact_6304_Inf__fin_Ohom__commute,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [H: A > A,N5: set @ A] :
( ! [X3: A,Y2: A] :
( ( H @ ( inf_inf @ A @ X3 @ Y2 ) )
= ( inf_inf @ A @ ( H @ X3 ) @ ( H @ Y2 ) ) )
=> ( ( finite_finite2 @ A @ N5 )
=> ( ( N5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( H @ ( lattic7752659483105999362nf_fin @ A @ N5 ) )
= ( lattic7752659483105999362nf_fin @ A @ ( image2 @ A @ A @ H @ N5 ) ) ) ) ) ) ) ).
% Inf_fin.hom_commute
thf(fact_6305_Sup__fin_Ohom__commute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [H: A > A,N5: set @ A] :
( ! [X3: A,Y2: A] :
( ( H @ ( sup_sup @ A @ X3 @ Y2 ) )
= ( sup_sup @ A @ ( H @ X3 ) @ ( H @ Y2 ) ) )
=> ( ( finite_finite2 @ A @ N5 )
=> ( ( N5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( H @ ( lattic5882676163264333800up_fin @ A @ N5 ) )
= ( lattic5882676163264333800up_fin @ A @ ( image2 @ A @ A @ H @ N5 ) ) ) ) ) ) ) ).
% Sup_fin.hom_commute
thf(fact_6306_Inf__fin_Osubset,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A5: set @ A,B5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( B5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ A5 )
=> ( ( inf_inf @ A @ ( lattic7752659483105999362nf_fin @ A @ B5 ) @ ( lattic7752659483105999362nf_fin @ A @ A5 ) )
= ( lattic7752659483105999362nf_fin @ A @ A5 ) ) ) ) ) ) ).
% Inf_fin.subset
thf(fact_6307_Sup__fin_Osubset,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A5: set @ A,B5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( B5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ A5 )
=> ( ( sup_sup @ A @ ( lattic5882676163264333800up_fin @ A @ B5 ) @ ( lattic5882676163264333800up_fin @ A @ A5 ) )
= ( lattic5882676163264333800up_fin @ A @ A5 ) ) ) ) ) ) ).
% Sup_fin.subset
thf(fact_6308_Inf__fin_Oclosed,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A,Y2: A] : ( member @ A @ ( inf_inf @ A @ X3 @ Y2 ) @ ( insert @ A @ X3 @ ( insert @ A @ Y2 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( member @ A @ ( lattic7752659483105999362nf_fin @ A @ A5 ) @ A5 ) ) ) ) ) ).
% Inf_fin.closed
thf(fact_6309_Inf__fin_Oinsert__not__elem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A5: set @ A,X2: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ~ ( member @ A @ X2 @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic7752659483105999362nf_fin @ A @ ( insert @ A @ X2 @ A5 ) )
= ( inf_inf @ A @ X2 @ ( lattic7752659483105999362nf_fin @ A @ A5 ) ) ) ) ) ) ) ).
% Inf_fin.insert_not_elem
thf(fact_6310_Sup__fin_Oinsert__not__elem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A5: set @ A,X2: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ~ ( member @ A @ X2 @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic5882676163264333800up_fin @ A @ ( insert @ A @ X2 @ A5 ) )
= ( sup_sup @ A @ X2 @ ( lattic5882676163264333800up_fin @ A @ A5 ) ) ) ) ) ) ) ).
% Sup_fin.insert_not_elem
thf(fact_6311_Sup__fin_Oclosed,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A,Y2: A] : ( member @ A @ ( sup_sup @ A @ X3 @ Y2 ) @ ( insert @ A @ X3 @ ( insert @ A @ Y2 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( member @ A @ ( lattic5882676163264333800up_fin @ A @ A5 ) @ A5 ) ) ) ) ) ).
% Sup_fin.closed
thf(fact_6312_Inf__fin_Ounion,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A5: set @ A,B5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B5 )
=> ( ( B5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic7752659483105999362nf_fin @ A @ ( sup_sup @ ( set @ A ) @ A5 @ B5 ) )
= ( inf_inf @ A @ ( lattic7752659483105999362nf_fin @ A @ A5 ) @ ( lattic7752659483105999362nf_fin @ A @ B5 ) ) ) ) ) ) ) ) ).
% Inf_fin.union
thf(fact_6313_Sup__fin_Ounion,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A5: set @ A,B5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B5 )
=> ( ( B5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic5882676163264333800up_fin @ A @ ( sup_sup @ ( set @ A ) @ A5 @ B5 ) )
= ( sup_sup @ A @ ( lattic5882676163264333800up_fin @ A @ A5 ) @ ( lattic5882676163264333800up_fin @ A @ B5 ) ) ) ) ) ) ) ) ).
% Sup_fin.union
thf(fact_6314_Inf__fin_Oeq__fold,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A5: set @ A,X2: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( lattic7752659483105999362nf_fin @ A @ ( insert @ A @ X2 @ A5 ) )
= ( finite_fold @ A @ A @ ( inf_inf @ A ) @ X2 @ A5 ) ) ) ) ).
% Inf_fin.eq_fold
thf(fact_6315_Sup__fin_Oeq__fold,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A5: set @ A,X2: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( lattic5882676163264333800up_fin @ A @ ( insert @ A @ X2 @ A5 ) )
= ( finite_fold @ A @ A @ ( sup_sup @ A ) @ X2 @ A5 ) ) ) ) ).
% Sup_fin.eq_fold
thf(fact_6316_inf__Sup2__distrib,axiom,
! [A: $tType] :
( ( distrib_lattice @ A )
=> ! [A5: set @ A,B5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B5 )
=> ( ( B5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( inf_inf @ A @ ( lattic5882676163264333800up_fin @ A @ A5 ) @ ( lattic5882676163264333800up_fin @ A @ B5 ) )
= ( lattic5882676163264333800up_fin @ A
@ ( collect @ A
@ ^ [Uu: A] :
? [A3: A,B3: A] :
( ( Uu
= ( inf_inf @ A @ A3 @ B3 ) )
& ( member @ A @ A3 @ A5 )
& ( member @ A @ B3 @ B5 ) ) ) ) ) ) ) ) ) ) ).
% inf_Sup2_distrib
thf(fact_6317_inf__Sup1__distrib,axiom,
! [A: $tType] :
( ( distrib_lattice @ A )
=> ! [A5: set @ A,X2: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( inf_inf @ A @ X2 @ ( lattic5882676163264333800up_fin @ A @ A5 ) )
= ( lattic5882676163264333800up_fin @ A
@ ( collect @ A
@ ^ [Uu: A] :
? [A3: A] :
( ( Uu
= ( inf_inf @ A @ X2 @ A3 ) )
& ( member @ A @ A3 @ A5 ) ) ) ) ) ) ) ) ).
% inf_Sup1_distrib
thf(fact_6318_sup__Inf2__distrib,axiom,
! [A: $tType] :
( ( distrib_lattice @ A )
=> ! [A5: set @ A,B5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B5 )
=> ( ( B5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( sup_sup @ A @ ( lattic7752659483105999362nf_fin @ A @ A5 ) @ ( lattic7752659483105999362nf_fin @ A @ B5 ) )
= ( lattic7752659483105999362nf_fin @ A
@ ( collect @ A
@ ^ [Uu: A] :
? [A3: A,B3: A] :
( ( Uu
= ( sup_sup @ A @ A3 @ B3 ) )
& ( member @ A @ A3 @ A5 )
& ( member @ A @ B3 @ B5 ) ) ) ) ) ) ) ) ) ) ).
% sup_Inf2_distrib
thf(fact_6319_sup__Inf1__distrib,axiom,
! [A: $tType] :
( ( distrib_lattice @ A )
=> ! [A5: set @ A,X2: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( sup_sup @ A @ X2 @ ( lattic7752659483105999362nf_fin @ A @ A5 ) )
= ( lattic7752659483105999362nf_fin @ A
@ ( collect @ A
@ ^ [Uu: A] :
? [A3: A] :
( ( Uu
= ( sup_sup @ A @ X2 @ A3 ) )
& ( member @ A @ A3 @ A5 ) ) ) ) ) ) ) ) ).
% sup_Inf1_distrib
thf(fact_6320_Inf__fin_Oremove,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A5: set @ A,X2: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( member @ A @ X2 @ A5 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic7752659483105999362nf_fin @ A @ A5 )
= X2 ) )
& ( ( ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic7752659483105999362nf_fin @ A @ A5 )
= ( inf_inf @ A @ X2 @ ( lattic7752659483105999362nf_fin @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ) ).
% Inf_fin.remove
thf(fact_6321_Inf__fin_Oinsert__remove,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A5: set @ A,X2: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic7752659483105999362nf_fin @ A @ ( insert @ A @ X2 @ A5 ) )
= X2 ) )
& ( ( ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic7752659483105999362nf_fin @ A @ ( insert @ A @ X2 @ A5 ) )
= ( inf_inf @ A @ X2 @ ( lattic7752659483105999362nf_fin @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% Inf_fin.insert_remove
thf(fact_6322_Sup__fin_Oinsert__remove,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A5: set @ A,X2: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic5882676163264333800up_fin @ A @ ( insert @ A @ X2 @ A5 ) )
= X2 ) )
& ( ( ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic5882676163264333800up_fin @ A @ ( insert @ A @ X2 @ A5 ) )
= ( sup_sup @ A @ X2 @ ( lattic5882676163264333800up_fin @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% Sup_fin.insert_remove
thf(fact_6323_Sup__fin_Oremove,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A5: set @ A,X2: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( member @ A @ X2 @ A5 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic5882676163264333800up_fin @ A @ A5 )
= X2 ) )
& ( ( ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic5882676163264333800up_fin @ A @ A5 )
= ( sup_sup @ A @ X2 @ ( lattic5882676163264333800up_fin @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ) ).
% Sup_fin.remove
thf(fact_6324_Pow__fold,axiom,
! [A: $tType,A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( pow2 @ A @ A5 )
= ( finite_fold @ A @ ( set @ ( set @ A ) )
@ ^ [X: A,A8: set @ ( set @ A )] : ( sup_sup @ ( set @ ( set @ A ) ) @ A8 @ ( image2 @ ( set @ A ) @ ( set @ A ) @ ( insert @ A @ X ) @ A8 ) )
@ ( insert @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ ( set @ A ) ) ) )
@ A5 ) ) ) ).
% Pow_fold
thf(fact_6325_Min_Oeq__fold_H,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( lattic643756798350308766er_Min @ A )
= ( ^ [A8: set @ A] :
( the2 @ A
@ ( finite_fold @ A @ ( option @ A )
@ ^ [X: A,Y4: option @ A] : ( some @ A @ ( case_option @ A @ A @ X @ ( ord_min @ A @ X ) @ Y4 ) )
@ ( none @ A )
@ A8 ) ) ) ) ) ).
% Min.eq_fold'
thf(fact_6326_Min__singleton,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A] :
( ( lattic643756798350308766er_Min @ A @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) )
= X2 ) ) ).
% Min_singleton
thf(fact_6327_Min_Obounded__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X2: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ X2 @ ( lattic643756798350308766er_Min @ A @ A5 ) )
= ( ! [X: A] :
( ( member @ A @ X @ A5 )
=> ( ord_less_eq @ A @ X2 @ X ) ) ) ) ) ) ) ).
% Min.bounded_iff
thf(fact_6328_Min__gr__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X2: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ X2 @ ( lattic643756798350308766er_Min @ A @ A5 ) )
= ( ! [X: A] :
( ( member @ A @ X @ A5 )
=> ( ord_less @ A @ X2 @ X ) ) ) ) ) ) ) ).
% Min_gr_iff
thf(fact_6329_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
@ ^ [Uu: B] : C2
@ A5 ) )
= C2 ) ) ) ) ).
% Min_const
thf(fact_6330_Min__insert,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X2: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798350308766er_Min @ A @ ( insert @ A @ X2 @ A5 ) )
= ( ord_min @ A @ X2 @ ( lattic643756798350308766er_Min @ A @ A5 ) ) ) ) ) ) ).
% Min_insert
thf(fact_6331_minus__Min__eq__Max,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [S2: set @ A] :
( ( finite_finite2 @ A @ S2 )
=> ( ( S2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( uminus_uminus @ A @ ( lattic643756798350308766er_Min @ A @ S2 ) )
= ( lattic643756798349783984er_Max @ A @ ( image2 @ A @ A @ ( uminus_uminus @ A ) @ S2 ) ) ) ) ) ) ).
% minus_Min_eq_Max
thf(fact_6332_minus__Max__eq__Min,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [S2: set @ A] :
( ( finite_finite2 @ A @ S2 )
=> ( ( S2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( uminus_uminus @ A @ ( lattic643756798349783984er_Max @ A @ S2 ) )
= ( lattic643756798350308766er_Min @ A @ ( image2 @ A @ A @ ( uminus_uminus @ A ) @ S2 ) ) ) ) ) ) ).
% minus_Max_eq_Min
thf(fact_6333_Inf__fin__Min,axiom,
! [A: $tType] :
( ( ( semilattice_inf @ A )
& ( linorder @ A ) )
=> ( ( lattic7752659483105999362nf_fin @ A )
= ( lattic643756798350308766er_Min @ A ) ) ) ).
% Inf_fin_Min
thf(fact_6334_Pow__Compl,axiom,
! [A: $tType,A5: set @ A] :
( ( pow2 @ A @ ( uminus_uminus @ ( set @ A ) @ A5 ) )
= ( collect @ ( set @ A )
@ ^ [Uu: set @ A] :
? [B8: set @ A] :
( ( Uu
= ( uminus_uminus @ ( set @ A ) @ B8 ) )
& ( member @ ( set @ A ) @ A5 @ ( pow2 @ A @ B8 ) ) ) ) ) ).
% Pow_Compl
thf(fact_6335_Pow__def,axiom,
! [A: $tType] :
( ( pow2 @ A )
= ( ^ [A8: set @ A] :
( collect @ ( set @ A )
@ ^ [B8: set @ A] : ( ord_less_eq @ ( set @ A ) @ B8 @ A8 ) ) ) ) ).
% Pow_def
thf(fact_6336_Min_OcoboundedI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,A2: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( member @ A @ A2 @ A5 )
=> ( ord_less_eq @ A @ ( lattic643756798350308766er_Min @ A @ A5 ) @ A2 ) ) ) ) ).
% Min.coboundedI
thf(fact_6337_Min__eqI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X2: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ! [Y2: A] :
( ( member @ A @ Y2 @ A5 )
=> ( ord_less_eq @ A @ X2 @ Y2 ) )
=> ( ( member @ A @ X2 @ A5 )
=> ( ( lattic643756798350308766er_Min @ A @ A5 )
= X2 ) ) ) ) ) ).
% Min_eqI
thf(fact_6338_Min__le,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X2: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( member @ A @ X2 @ A5 )
=> ( ord_less_eq @ A @ ( lattic643756798350308766er_Min @ A @ A5 ) @ X2 ) ) ) ) ).
% Min_le
thf(fact_6339_Min_Oin__idem,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X2: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( member @ A @ X2 @ A5 )
=> ( ( ord_min @ A @ X2 @ ( lattic643756798350308766er_Min @ A @ A5 ) )
= ( lattic643756798350308766er_Min @ A @ A5 ) ) ) ) ) ).
% Min.in_idem
thf(fact_6340_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_6341_Pow__INT__eq,axiom,
! [A: $tType,B: $tType,B5: B > ( set @ A ),A5: set @ B] :
( ( pow2 @ A @ ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B5 @ A5 ) ) )
= ( complete_Inf_Inf @ ( set @ ( set @ A ) )
@ ( image2 @ B @ ( set @ ( set @ A ) )
@ ^ [X: B] : ( pow2 @ A @ ( B5 @ X ) )
@ A5 ) ) ) ).
% Pow_INT_eq
thf(fact_6342_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 )
& ! [X: A] :
( ( member @ A @ X @ A5 )
=> ( ord_less_eq @ A @ M @ X ) ) ) ) ) ) ) ).
% Min_eq_iff
thf(fact_6343_Min__le__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X2: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ ( lattic643756798350308766er_Min @ A @ A5 ) @ X2 )
= ( ? [X: A] :
( ( member @ A @ X @ A5 )
& ( ord_less_eq @ A @ X @ X2 ) ) ) ) ) ) ) ).
% Min_le_iff
thf(fact_6344_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 )
& ! [X: A] :
( ( member @ A @ X @ A5 )
=> ( ord_less_eq @ A @ M @ X ) ) ) ) ) ) ) ).
% eq_Min_iff
thf(fact_6345_Min_OboundedE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X2: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ X2 @ ( lattic643756798350308766er_Min @ A @ A5 ) )
=> ! [A9: A] :
( ( member @ A @ A9 @ A5 )
=> ( ord_less_eq @ A @ X2 @ A9 ) ) ) ) ) ) ).
% Min.boundedE
thf(fact_6346_Min_OboundedI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X2: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [A4: A] :
( ( member @ A @ A4 @ A5 )
=> ( ord_less_eq @ A @ X2 @ A4 ) )
=> ( ord_less_eq @ A @ X2 @ ( lattic643756798350308766er_Min @ A @ A5 ) ) ) ) ) ) ).
% Min.boundedI
thf(fact_6347_Min__less__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X2: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ ( lattic643756798350308766er_Min @ A @ A5 ) @ X2 )
= ( ? [X: A] :
( ( member @ A @ X @ A5 )
& ( ord_less @ A @ X @ X2 ) ) ) ) ) ) ) ).
% Min_less_iff
thf(fact_6348_Min__insert2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,A2: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ! [B4: A] :
( ( member @ A @ B4 @ A5 )
=> ( ord_less_eq @ A @ A2 @ B4 ) )
=> ( ( lattic643756798350308766er_Min @ A @ ( insert @ A @ A2 @ A5 ) )
= A2 ) ) ) ) ).
% Min_insert2
thf(fact_6349_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_6350_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_6351_UN__Pow__subset,axiom,
! [A: $tType,B: $tType,B5: B > ( set @ A ),A5: set @ B] :
( ord_less_eq @ ( set @ ( set @ A ) )
@ ( complete_Sup_Sup @ ( set @ ( set @ A ) )
@ ( image2 @ B @ ( set @ ( set @ A ) )
@ ^ [X: B] : ( pow2 @ A @ ( B5 @ X ) )
@ A5 ) )
@ ( pow2 @ A @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B5 @ A5 ) ) ) ) ).
% UN_Pow_subset
thf(fact_6352_Min_Osubset__imp,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B5 )
=> ( ord_less_eq @ A @ ( lattic643756798350308766er_Min @ A @ B5 ) @ ( lattic643756798350308766er_Min @ A @ A5 ) ) ) ) ) ) ).
% Min.subset_imp
thf(fact_6353_Min__antimono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [M3: set @ A,N5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ M3 @ N5 )
=> ( ( M3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ N5 )
=> ( ord_less_eq @ A @ ( lattic643756798350308766er_Min @ A @ N5 ) @ ( lattic643756798350308766er_Min @ A @ M3 ) ) ) ) ) ) ).
% Min_antimono
thf(fact_6354_hom__Min__commute,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [H: A > A,N5: set @ A] :
( ! [X3: A,Y2: A] :
( ( H @ ( ord_min @ A @ X3 @ Y2 ) )
= ( ord_min @ A @ ( H @ X3 ) @ ( H @ Y2 ) ) )
=> ( ( finite_finite2 @ A @ N5 )
=> ( ( N5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( H @ ( lattic643756798350308766er_Min @ A @ N5 ) )
= ( lattic643756798350308766er_Min @ A @ ( image2 @ A @ A @ H @ N5 ) ) ) ) ) ) ) ).
% hom_Min_commute
thf(fact_6355_Min_Osubset,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,B5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( B5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ A5 )
=> ( ( ord_min @ A @ ( lattic643756798350308766er_Min @ A @ B5 ) @ ( lattic643756798350308766er_Min @ A @ A5 ) )
= ( lattic643756798350308766er_Min @ A @ A5 ) ) ) ) ) ) ).
% Min.subset
thf(fact_6356_Min_Oclosed,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A,Y2: A] : ( member @ A @ ( ord_min @ A @ X3 @ Y2 ) @ ( insert @ A @ X3 @ ( insert @ A @ Y2 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( member @ A @ ( lattic643756798350308766er_Min @ A @ A5 ) @ A5 ) ) ) ) ) ).
% Min.closed
thf(fact_6357_Min_Oinsert__not__elem,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X2: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ~ ( member @ A @ X2 @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798350308766er_Min @ A @ ( insert @ A @ X2 @ A5 ) )
= ( ord_min @ A @ X2 @ ( lattic643756798350308766er_Min @ A @ A5 ) ) ) ) ) ) ) ).
% Min.insert_not_elem
thf(fact_6358_Min_Ounion,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,B5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B5 )
=> ( ( B5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798350308766er_Min @ A @ ( sup_sup @ ( set @ A ) @ A5 @ B5 ) )
= ( ord_min @ A @ ( lattic643756798350308766er_Min @ A @ A5 ) @ ( lattic643756798350308766er_Min @ A @ B5 ) ) ) ) ) ) ) ) ).
% Min.union
thf(fact_6359_Min_Oeq__fold,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X2: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( lattic643756798350308766er_Min @ A @ ( insert @ A @ X2 @ A5 ) )
= ( finite_fold @ A @ A @ ( ord_min @ A ) @ X2 @ A5 ) ) ) ) ).
% Min.eq_fold
thf(fact_6360_Min__add__commute,axiom,
! [B: $tType,A: $tType] :
( ( linord4140545234300271783up_add @ A )
=> ! [S2: set @ B,F: B > A,K: A] :
( ( finite_finite2 @ B @ S2 )
=> ( ( S2
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( lattic643756798350308766er_Min @ A
@ ( image2 @ B @ A
@ ^ [X: B] : ( plus_plus @ A @ ( F @ X ) @ K )
@ S2 ) )
= ( plus_plus @ A @ ( lattic643756798350308766er_Min @ A @ ( image2 @ B @ A @ F @ S2 ) ) @ K ) ) ) ) ) ).
% Min_add_commute
thf(fact_6361_binomial__def,axiom,
( binomial
= ( ^ [N4: nat,K2: nat] :
( finite_card @ ( set @ nat )
@ ( collect @ ( set @ nat )
@ ^ [K6: set @ nat] :
( ( member @ ( set @ nat ) @ K6 @ ( pow2 @ nat @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N4 ) ) )
& ( ( finite_card @ nat @ K6 )
= K2 ) ) ) ) ) ) ).
% binomial_def
thf(fact_6362_Min_Oinsert__remove,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X2: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798350308766er_Min @ A @ ( insert @ A @ X2 @ A5 ) )
= X2 ) )
& ( ( ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798350308766er_Min @ A @ ( insert @ A @ X2 @ A5 ) )
= ( ord_min @ A @ X2 @ ( lattic643756798350308766er_Min @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% Min.insert_remove
thf(fact_6363_Min_Oremove,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X2: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( member @ A @ X2 @ A5 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798350308766er_Min @ A @ A5 )
= X2 ) )
& ( ( ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798350308766er_Min @ A @ A5 )
= ( ord_min @ A @ X2 @ ( lattic643756798350308766er_Min @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ) ).
% Min.remove
thf(fact_6364_arg__min__SOME__Min,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [S2: set @ A,F: A > B] :
( ( finite_finite2 @ A @ S2 )
=> ( ( lattic7623131987881927897min_on @ A @ B @ F @ S2 )
= ( fChoice @ A
@ ^ [Y4: A] :
( ( member @ A @ Y4 @ S2 )
& ( ( F @ Y4 )
= ( lattic643756798350308766er_Min @ B @ ( image2 @ A @ B @ F @ S2 ) ) ) ) ) ) ) ) ).
% arg_min_SOME_Min
thf(fact_6365_sorted__find__Min,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,P: A > $o] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( ? [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
& ( P @ X4 ) )
=> ( ( find @ A @ P @ Xs )
= ( some @ A
@ ( lattic643756798350308766er_Min @ A
@ ( collect @ A
@ ^ [X: A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
& ( P @ X ) ) ) ) ) ) ) ) ) ).
% sorted_find_Min
thf(fact_6366_Pow__set_I2_J,axiom,
! [B: $tType,X2: B,Xs: list @ B] :
( ( pow2 @ B @ ( set2 @ B @ ( cons @ B @ X2 @ Xs ) ) )
= ( sup_sup @ ( set @ ( set @ B ) ) @ ( pow2 @ B @ ( set2 @ B @ Xs ) ) @ ( image2 @ ( set @ B ) @ ( set @ B ) @ ( insert @ B @ X2 ) @ ( pow2 @ B @ ( set2 @ B @ Xs ) ) ) ) ) ).
% Pow_set(2)
thf(fact_6367_card__Min__le__sum,axiom,
! [A: $tType,A5: set @ A,F: A > nat] :
( ( finite_finite2 @ A @ A5 )
=> ( ord_less_eq @ nat @ ( times_times @ nat @ ( finite_card @ A @ A5 ) @ ( lattic643756798350308766er_Min @ nat @ ( image2 @ A @ nat @ F @ A5 ) ) ) @ ( groups7311177749621191930dd_sum @ A @ nat @ F @ A5 ) ) ) ).
% card_Min_le_sum
thf(fact_6368_dual__Max,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( lattices_Max @ A
@ ^ [X: A,Y4: A] : ( ord_less_eq @ A @ Y4 @ X ) )
= ( lattic643756798350308766er_Min @ A ) ) ) ).
% dual_Max
thf(fact_6369_rtrancl__finite__eq__relpow,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( finite_finite2 @ ( product_prod @ A @ A ) @ R3 )
=> ( ( transitive_rtrancl @ A @ R3 )
= ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ A ) )
@ ( image2 @ nat @ ( set @ ( product_prod @ A @ A ) )
@ ^ [N4: nat] : ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N4 @ R3 )
@ ( collect @ nat
@ ^ [N4: nat] : ( ord_less_eq @ nat @ N4 @ ( finite_card @ ( product_prod @ A @ A ) @ R3 ) ) ) ) ) ) ) ).
% rtrancl_finite_eq_relpow
thf(fact_6370_rtrancl__sub__insert__rtrancl,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),X2: product_prod @ A @ A] : ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( transitive_rtrancl @ A @ R3 ) @ ( transitive_rtrancl @ A @ ( insert @ ( product_prod @ A @ A ) @ X2 @ R3 ) ) ) ).
% rtrancl_sub_insert_rtrancl
thf(fact_6371_in__rtrancl__insert,axiom,
! [A: $tType,X2: product_prod @ A @ A,R3: set @ ( product_prod @ A @ A ),R: product_prod @ A @ A] :
( ( member @ ( product_prod @ A @ A ) @ X2 @ ( transitive_rtrancl @ A @ R3 ) )
=> ( member @ ( product_prod @ A @ A ) @ X2 @ ( transitive_rtrancl @ A @ ( insert @ ( product_prod @ A @ A ) @ R @ R3 ) ) ) ) ).
% in_rtrancl_insert
thf(fact_6372_linorder_OMax_Ocong,axiom,
! [A: $tType] :
( ( lattices_Max @ A )
= ( lattices_Max @ A ) ) ).
% linorder.Max.cong
thf(fact_6373_rtrancl__mono__rightI,axiom,
! [A: $tType,S2: set @ ( product_prod @ A @ A ),S7: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ S2 @ S7 )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ S2 @ ( transitive_rtrancl @ A @ S7 ) ) ) ).
% rtrancl_mono_rightI
thf(fact_6374_rtrancl__mono__mp,axiom,
! [A: $tType,U5: set @ ( product_prod @ A @ A ),V: set @ ( product_prod @ A @ A ),X2: product_prod @ A @ A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ U5 @ V )
=> ( ( member @ ( product_prod @ A @ A ) @ X2 @ ( transitive_rtrancl @ A @ U5 ) )
=> ( member @ ( product_prod @ A @ A ) @ X2 @ ( transitive_rtrancl @ A @ V ) ) ) ) ).
% rtrancl_mono_mp
thf(fact_6375_r__le__rtrancl,axiom,
! [A: $tType,S2: set @ ( product_prod @ A @ A )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ S2 @ ( transitive_rtrancl @ A @ S2 ) ) ).
% r_le_rtrancl
thf(fact_6376_converse__rtranclE_H,axiom,
! [A: $tType,U: A,V2: A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ V2 ) @ ( transitive_rtrancl @ A @ R3 ) )
=> ( ( U != V2 )
=> ~ ! [Vh: A] :
( ( U != Vh )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ Vh ) @ R3 )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Vh @ V2 ) @ ( transitive_rtrancl @ A @ R3 ) ) ) ) ) ) ).
% converse_rtranclE'
thf(fact_6377_trancl__union__outside,axiom,
! [A: $tType,V2: A,W2: A,E6: set @ ( product_prod @ A @ A ),U5: 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 @ U5 ) ) )
=> ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V2 @ W2 ) @ ( transitive_trancl @ A @ E6 ) )
=> ? [X3: A,Y2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V2 @ X3 ) @ ( transitive_rtrancl @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ E6 @ U5 ) ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y2 ) @ U5 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ W2 ) @ ( transitive_rtrancl @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ E6 @ U5 ) ) ) ) ) ) ).
% trancl_union_outside
thf(fact_6378_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_6379_rtrancl__listrel1__eq__len,axiom,
! [A: $tType,X2: list @ A,Y: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X2 @ Y ) @ ( transitive_rtrancl @ ( list @ A ) @ ( listrel1 @ A @ R ) ) )
=> ( ( size_size @ ( list @ A ) @ X2 )
= ( size_size @ ( list @ A ) @ Y ) ) ) ).
% rtrancl_listrel1_eq_len
thf(fact_6380_trancl__subset__Sigma__aux,axiom,
! [A: $tType,A2: A,B2: A,R: set @ ( product_prod @ A @ A ),A5: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ ( transitive_rtrancl @ A @ R ) )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R
@ ( product_Sigma @ A @ A @ A5
@ ^ [Uu: A] : A5 ) )
=> ( ( A2 = B2 )
| ( member @ A @ A2 @ A5 ) ) ) ) ).
% trancl_subset_Sigma_aux
thf(fact_6381_Restr__rtrancl__mono,axiom,
! [A: $tType,V2: A,W2: A,E6: set @ ( product_prod @ A @ A ),U5: 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 @ U5
@ ^ [Uu: A] : U5 ) ) ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V2 @ W2 ) @ ( transitive_rtrancl @ A @ E6 ) ) ) ).
% Restr_rtrancl_mono
thf(fact_6382_rtrancl__mapI,axiom,
! [B: $tType,A: $tType,A2: A,B2: A,E6: set @ ( product_prod @ A @ A ),F: A > B] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ ( transitive_rtrancl @ A @ E6 ) )
=> ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F @ A2 ) @ ( F @ B2 ) ) @ ( transitive_rtrancl @ B @ ( image2 @ ( product_prod @ A @ A ) @ ( product_prod @ B @ B ) @ ( pairself @ A @ B @ F ) @ E6 ) ) ) ) ).
% rtrancl_mapI
thf(fact_6383_rtrancl__last__touch,axiom,
! [A: $tType,Q2: A,Q5: A,R3: set @ ( product_prod @ A @ A ),S2: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Q2 @ Q5 ) @ ( transitive_rtrancl @ A @ R3 ) )
=> ( ( member @ A @ Q2 @ S2 )
=> ~ ! [Qt: A] :
( ( member @ A @ Qt @ S2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Q2 @ Qt ) @ ( transitive_rtrancl @ A @ R3 ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Qt @ Q5 )
@ ( transitive_rtrancl @ A
@ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ ( top_top @ ( set @ A ) )
@ ^ [Uu: A] : S2 ) ) ) ) ) ) ) ) ).
% rtrancl_last_touch
thf(fact_6384_rtrancl__last__visit_H,axiom,
! [A: $tType,Q2: A,Q5: A,R3: set @ ( product_prod @ A @ A ),S2: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Q2 @ Q5 ) @ ( transitive_rtrancl @ A @ R3 ) )
=> ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Q2 @ Q5 )
@ ( transitive_rtrancl @ A
@ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ ( top_top @ ( set @ A ) )
@ ^ [Uu: A] : S2 ) ) ) )
=> ~ ! [Qt: A] :
( ( member @ A @ Qt @ S2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Q2 @ Qt ) @ ( transitive_rtrancl @ A @ R3 ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Qt @ Q5 )
@ ( transitive_rtrancl @ A
@ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ ( top_top @ ( set @ A ) )
@ ^ [Uu: A] : S2 ) ) ) ) ) ) ) ) ).
% rtrancl_last_visit'
thf(fact_6385_rtrancl__insert,axiom,
! [A: $tType,A2: A,B2: A,R: set @ ( product_prod @ A @ A )] :
( ( transitive_rtrancl @ A @ ( insert @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ R ) )
= ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ ( transitive_rtrancl @ A @ R )
@ ( collect @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ^ [X: A,Y4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ A2 ) @ ( transitive_rtrancl @ A @ R ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ Y4 ) @ ( transitive_rtrancl @ A @ R ) ) ) ) ) ) ) ).
% rtrancl_insert
thf(fact_6386_rtrancl__imp__UN__relpow,axiom,
! [A: $tType,P3: product_prod @ A @ A,R3: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ P3 @ ( transitive_rtrancl @ A @ R3 ) )
=> ( member @ ( product_prod @ A @ A ) @ P3
@ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ A ) )
@ ( image2 @ nat @ ( set @ ( product_prod @ A @ A ) )
@ ^ [N4: nat] : ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N4 @ R3 )
@ ( top_top @ ( set @ nat ) ) ) ) ) ) ).
% rtrancl_imp_UN_relpow
thf(fact_6387_rtrancl__is__UN__relpow,axiom,
! [A: $tType] :
( ( transitive_rtrancl @ A )
= ( ^ [R7: set @ ( product_prod @ A @ A )] :
( complete_Sup_Sup @ ( set @ ( product_prod @ A @ A ) )
@ ( image2 @ nat @ ( set @ ( product_prod @ A @ A ) )
@ ^ [N4: nat] : ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N4 @ R7 )
@ ( top_top @ ( set @ nat ) ) ) ) ) ) ).
% rtrancl_is_UN_relpow
thf(fact_6388_rtrancl__last__visit,axiom,
! [A: $tType,Q2: A,Q5: A,R3: set @ ( product_prod @ A @ A ),S2: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Q2 @ Q5 ) @ ( transitive_rtrancl @ A @ R3 ) )
=> ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Q2 @ Q5 )
@ ( transitive_rtrancl @ A
@ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ ( top_top @ ( set @ A ) )
@ ^ [Uu: A] : S2 ) ) ) )
=> ~ ! [Qt: A] :
( ( member @ A @ Qt @ S2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Q2 @ Qt ) @ ( transitive_trancl @ A @ R3 ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Qt @ Q5 )
@ ( transitive_rtrancl @ A
@ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ ( top_top @ ( set @ A ) )
@ ^ [Uu: A] : S2 ) ) ) ) ) ) ) ) ).
% rtrancl_last_visit
thf(fact_6389_trancl__insert,axiom,
! [A: $tType,Y: A,X2: A,R: set @ ( product_prod @ A @ A )] :
( ( transitive_trancl @ A @ ( insert @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X2 ) @ R ) )
= ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ ( transitive_trancl @ A @ R )
@ ( collect @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ^ [A3: A,B3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ Y ) @ ( transitive_rtrancl @ A @ R ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ B3 ) @ ( transitive_rtrancl @ A @ R ) ) ) ) ) ) ) ).
% trancl_insert
thf(fact_6390_trancl__multi__insert2,axiom,
! [A: $tType,A2: A,B2: A,R: set @ ( product_prod @ A @ A ),M: A,X8: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 )
@ ( transitive_trancl @ A
@ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ R
@ ( product_Sigma @ A @ A @ ( insert @ A @ M @ ( bot_bot @ ( set @ A ) ) )
@ ^ [Uu: A] : X8 ) ) ) )
=> ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ ( transitive_trancl @ A @ R ) )
=> ~ ! [X3: A] :
( ( member @ A @ X3 @ X8 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ M ) @ ( transitive_rtrancl @ A @ R ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ B2 ) @ ( transitive_rtrancl @ A @ R ) ) ) ) ) ) ).
% trancl_multi_insert2
thf(fact_6391_trancl__multi__insert,axiom,
! [A: $tType,A2: A,B2: A,R: set @ ( product_prod @ A @ A ),X8: set @ A,M: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 )
@ ( transitive_trancl @ A
@ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ R
@ ( product_Sigma @ A @ A @ X8
@ ^ [Uu: A] : ( insert @ A @ M @ ( bot_bot @ ( set @ A ) ) ) ) ) ) )
=> ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ ( transitive_trancl @ A @ R ) )
=> ~ ! [X3: A] :
( ( member @ A @ X3 @ X8 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ X3 ) @ ( transitive_rtrancl @ A @ R ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ M @ B2 ) @ ( transitive_rtrancl @ A @ R ) ) ) ) ) ) ).
% trancl_multi_insert
thf(fact_6392_rtrancl__last__visit__node,axiom,
! [A: $tType,S3: A,S5: A,R3: set @ ( product_prod @ A @ A ),Sh: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ S3 @ S5 ) @ ( transitive_rtrancl @ A @ R3 ) )
=> ( ( ( S3 != Sh )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ S3 @ S5 )
@ ( transitive_rtrancl @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ ( top_top @ ( set @ A ) )
@ ^ [Uu: A] : ( uminus_uminus @ ( set @ A ) @ ( insert @ A @ Sh @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) )
| ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ S3 @ Sh ) @ ( transitive_rtrancl @ A @ R3 ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Sh @ S5 )
@ ( transitive_rtrancl @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ ( top_top @ ( set @ A ) )
@ ^ [Uu: A] : ( uminus_uminus @ ( set @ A ) @ ( insert @ A @ Sh @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% rtrancl_last_visit_node
thf(fact_6393_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,B8: set @ A] :
( ( ord_less @ ( set @ A ) @ A8 @ B8 )
& ( finite_finite2 @ A @ B8 ) ) ) ) ) ).
% finite_psubset_def
thf(fact_6394_Gr__incl,axiom,
! [A: $tType,B: $tType,A5: set @ A,F: A > B,B5: set @ B] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( bNF_Gr @ A @ B @ A5 @ F )
@ ( product_Sigma @ A @ B @ A5
@ ^ [Uu: A] : B5 ) )
= ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F @ A5 ) @ B5 ) ) ).
% Gr_incl
thf(fact_6395_GrD2,axiom,
! [A: $tType,B: $tType,X2: A,Fx: B,A5: set @ A,F: A > B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Fx ) @ ( bNF_Gr @ A @ B @ A5 @ F ) )
=> ( ( F @ X2 )
= Fx ) ) ).
% GrD2
thf(fact_6396_GrD1,axiom,
! [B: $tType,A: $tType,X2: A,Fx: B,A5: set @ A,F: A > B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Fx ) @ ( bNF_Gr @ A @ B @ A5 @ F ) )
=> ( member @ A @ X2 @ A5 ) ) ).
% GrD1
thf(fact_6397_Id__on__Gr,axiom,
! [A: $tType] :
( ( id_on @ A )
= ( ^ [A8: set @ A] : ( bNF_Gr @ A @ A @ A8 @ ( id @ A ) ) ) ) ).
% Id_on_Gr
thf(fact_6398_Gr__def,axiom,
! [B: $tType,A: $tType] :
( ( bNF_Gr @ A @ B )
= ( ^ [A8: set @ A,F2: A > B] :
( collect @ ( product_prod @ A @ B )
@ ^ [Uu: product_prod @ A @ B] :
? [A3: A] :
( ( Uu
= ( product_Pair @ A @ B @ A3 @ ( F2 @ A3 ) ) )
& ( member @ A @ A3 @ A8 ) ) ) ) ) ).
% Gr_def
thf(fact_6399_set__list__bind,axiom,
! [A: $tType,B: $tType,Xs: list @ B,F: B > ( list @ A )] :
( ( set2 @ A @ ( bind @ B @ A @ Xs @ F ) )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [X: B] : ( set2 @ A @ ( F @ X ) )
@ ( set2 @ B @ Xs ) ) ) ) ).
% set_list_bind
thf(fact_6400_pred__nat__def,axiom,
( pred_nat
= ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [M2: nat,N4: nat] :
( N4
= ( suc @ M2 ) ) ) ) ) ).
% pred_nat_def
thf(fact_6401_pred__nat__trancl__eq__le,axiom,
! [M: nat,N: nat] :
( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ M @ N ) @ ( transitive_rtrancl @ nat @ pred_nat ) )
= ( ord_less_eq @ nat @ M @ N ) ) ).
% pred_nat_trancl_eq_le
thf(fact_6402_rtrancl__restrictI,axiom,
! [A: $tType,U: A,V2: A,E6: set @ ( product_prod @ A @ A ),R3: 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 ) )
@ ^ [Uu: A] : R3 ) ) ) )
=> ( ~ ( member @ A @ U @ R3 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ V2 ) @ ( transitive_rtrancl @ A @ ( rel_restrict @ A @ E6 @ R3 ) ) ) ) ) ).
% rtrancl_restrictI
thf(fact_6403_image__split__eq__Sigma,axiom,
! [C: $tType,B: $tType,A: $tType,F: C > A,G: C > B,A5: set @ C] :
( ( image2 @ C @ ( product_prod @ A @ B )
@ ^ [X: C] : ( product_Pair @ A @ B @ ( F @ X ) @ ( G @ X ) )
@ A5 )
= ( product_Sigma @ A @ B @ ( image2 @ C @ A @ F @ A5 )
@ ^ [X: A] : ( image2 @ C @ B @ G @ ( inf_inf @ ( set @ C ) @ ( vimage @ C @ A @ F @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ A5 ) ) ) ) ).
% image_split_eq_Sigma
thf(fact_6404_vimage__Collect__eq,axiom,
! [B: $tType,A: $tType,F: A > B,P: B > $o] :
( ( vimage @ A @ B @ F @ ( collect @ B @ P ) )
= ( collect @ A
@ ^ [Y4: A] : ( P @ ( F @ Y4 ) ) ) ) ).
% vimage_Collect_eq
thf(fact_6405_vimage__ident,axiom,
! [A: $tType,Y7: set @ A] :
( ( vimage @ A @ A
@ ^ [X: A] : X
@ Y7 )
= Y7 ) ).
% vimage_ident
thf(fact_6406_rel__restrict__empty,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( rel_restrict @ A @ R3 @ ( bot_bot @ ( set @ A ) ) )
= R3 ) ).
% rel_restrict_empty
thf(fact_6407_vimage__const,axiom,
! [B: $tType,A: $tType,C2: B,A5: set @ B] :
( ( ( member @ B @ C2 @ A5 )
=> ( ( vimage @ A @ B
@ ^ [X: A] : C2
@ A5 )
= ( top_top @ ( set @ A ) ) ) )
& ( ~ ( member @ B @ C2 @ A5 )
=> ( ( vimage @ A @ B
@ ^ [X: A] : C2
@ A5 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% vimage_const
thf(fact_6408_vimage__if,axiom,
! [B: $tType,A: $tType,C2: B,A5: set @ B,D2: B,B5: set @ A] :
( ( ( member @ B @ C2 @ A5 )
=> ( ( ( member @ B @ D2 @ A5 )
=> ( ( vimage @ A @ B
@ ^ [X: A] : ( if @ B @ ( member @ A @ X @ B5 ) @ C2 @ D2 )
@ A5 )
= ( top_top @ ( set @ A ) ) ) )
& ( ~ ( member @ B @ D2 @ A5 )
=> ( ( vimage @ A @ B
@ ^ [X: A] : ( if @ B @ ( member @ A @ X @ B5 ) @ C2 @ D2 )
@ A5 )
= B5 ) ) ) )
& ( ~ ( member @ B @ C2 @ A5 )
=> ( ( ( member @ B @ D2 @ A5 )
=> ( ( vimage @ A @ B
@ ^ [X: A] : ( if @ B @ ( member @ A @ X @ B5 ) @ C2 @ D2 )
@ A5 )
= ( uminus_uminus @ ( set @ A ) @ B5 ) ) )
& ( ~ ( member @ B @ D2 @ A5 )
=> ( ( vimage @ A @ B
@ ^ [X: A] : ( if @ B @ ( member @ A @ X @ B5 ) @ C2 @ D2 )
@ A5 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% vimage_if
thf(fact_6409_vimage__Union,axiom,
! [A: $tType,B: $tType,F: A > B,A5: set @ ( set @ B )] :
( ( vimage @ A @ B @ F @ ( complete_Sup_Sup @ ( set @ B ) @ A5 ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ ( set @ B ) @ ( set @ A ) @ ( vimage @ A @ B @ F ) @ A5 ) ) ) ).
% vimage_Union
thf(fact_6410_vimage__INT,axiom,
! [A: $tType,B: $tType,C: $tType,F: A > B,B5: C > ( set @ B ),A5: set @ C] :
( ( vimage @ A @ B @ F @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ C @ ( set @ B ) @ B5 @ A5 ) ) )
= ( complete_Inf_Inf @ ( set @ A )
@ ( image2 @ C @ ( set @ A )
@ ^ [X: C] : ( vimage @ A @ B @ F @ ( B5 @ X ) )
@ A5 ) ) ) ).
% vimage_INT
thf(fact_6411_vimage__UN,axiom,
! [A: $tType,B: $tType,C: $tType,F: A > B,B5: C > ( set @ B ),A5: set @ C] :
( ( vimage @ A @ B @ F @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ C @ ( set @ B ) @ B5 @ A5 ) ) )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ C @ ( set @ A )
@ ^ [X: C] : ( vimage @ A @ B @ F @ ( B5 @ X ) )
@ A5 ) ) ) ).
% vimage_UN
thf(fact_6412_rel__restrict__union,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A5: set @ A,B5: set @ A] :
( ( rel_restrict @ A @ R3 @ ( sup_sup @ ( set @ A ) @ A5 @ B5 ) )
= ( rel_restrict @ A @ ( rel_restrict @ A @ R3 @ A5 ) @ B5 ) ) ).
% rel_restrict_union
thf(fact_6413_rel__restrict__notR_I2_J,axiom,
! [A: $tType,X2: A,Y: A,A5: set @ ( product_prod @ A @ A ),R3: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y ) @ ( rel_restrict @ A @ A5 @ R3 ) )
=> ~ ( member @ A @ Y @ R3 ) ) ).
% rel_restrict_notR(2)
thf(fact_6414_rel__restrict__notR_I1_J,axiom,
! [A: $tType,X2: A,Y: A,A5: set @ ( product_prod @ A @ A ),R3: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y ) @ ( rel_restrict @ A @ A5 @ R3 ) )
=> ~ ( member @ A @ X2 @ R3 ) ) ).
% rel_restrict_notR(1)
thf(fact_6415_rel__restrictI,axiom,
! [A: $tType,X2: A,R3: set @ A,Y: A,E6: set @ ( product_prod @ A @ A )] :
( ~ ( member @ A @ X2 @ R3 )
=> ( ~ ( member @ A @ Y @ R3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y ) @ E6 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y ) @ ( rel_restrict @ A @ E6 @ R3 ) ) ) ) ) ).
% rel_restrictI
thf(fact_6416_rel__restrict__lift,axiom,
! [A: $tType,X2: A,Y: A,E6: set @ ( product_prod @ A @ A ),R3: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y ) @ ( rel_restrict @ A @ E6 @ R3 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y ) @ E6 ) ) ).
% rel_restrict_lift
thf(fact_6417_rel__restrict__sub,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A5: set @ A] : ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( rel_restrict @ A @ R3 @ A5 ) @ R3 ) ).
% rel_restrict_sub
thf(fact_6418_rel__restrict__mono,axiom,
! [A: $tType,A5: set @ ( product_prod @ A @ A ),B5: set @ ( product_prod @ A @ A ),R3: set @ A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ A5 @ B5 )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( rel_restrict @ A @ A5 @ R3 ) @ ( rel_restrict @ A @ B5 @ R3 ) ) ) ).
% rel_restrict_mono
thf(fact_6419_vimage__def,axiom,
! [B: $tType,A: $tType] :
( ( vimage @ A @ B )
= ( ^ [F2: A > B,B8: set @ B] :
( collect @ A
@ ^ [X: A] : ( member @ B @ ( F2 @ X ) @ B8 ) ) ) ) ).
% vimage_def
thf(fact_6420_finite__rel__restrict,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A5: set @ A] :
( ( finite_finite2 @ ( product_prod @ A @ A ) @ R3 )
=> ( finite_finite2 @ ( product_prod @ A @ A ) @ ( rel_restrict @ A @ R3 @ A5 ) ) ) ).
% finite_rel_restrict
thf(fact_6421_set_Oidentity,axiom,
! [A: $tType] :
( ( vimage @ A @ A
@ ^ [X: A] : X )
= ( id @ ( set @ A ) ) ) ).
% set.identity
thf(fact_6422_vimage__Times,axiom,
! [A: $tType,B: $tType,C: $tType,F: A > ( product_prod @ B @ C ),A5: set @ B,B5: set @ C] :
( ( vimage @ A @ ( product_prod @ B @ C ) @ F
@ ( product_Sigma @ B @ C @ A5
@ ^ [Uu: B] : B5 ) )
= ( inf_inf @ ( set @ A ) @ ( vimage @ A @ B @ ( comp @ ( product_prod @ B @ C ) @ B @ A @ ( product_fst @ B @ C ) @ F ) @ A5 ) @ ( vimage @ A @ C @ ( comp @ ( product_prod @ B @ C ) @ C @ A @ ( product_snd @ B @ C ) @ F ) @ B5 ) ) ) ).
% vimage_Times
thf(fact_6423_rel__restrict__trancl__mem,axiom,
! [A: $tType,A2: A,B2: A,A5: set @ ( product_prod @ A @ A ),R3: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ ( transitive_trancl @ A @ ( rel_restrict @ A @ A5 @ R3 ) ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ ( rel_restrict @ A @ ( transitive_trancl @ A @ A5 ) @ R3 ) ) ) ).
% rel_restrict_trancl_mem
thf(fact_6424_rel__restrict__trancl__notR_I1_J,axiom,
! [A: $tType,V2: A,W2: A,E6: set @ ( product_prod @ A @ A ),R3: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V2 @ W2 ) @ ( transitive_trancl @ A @ ( rel_restrict @ A @ E6 @ R3 ) ) )
=> ~ ( member @ A @ V2 @ R3 ) ) ).
% rel_restrict_trancl_notR(1)
thf(fact_6425_rel__restrict__trancl__notR_I2_J,axiom,
! [A: $tType,V2: A,W2: A,E6: set @ ( product_prod @ A @ A ),R3: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V2 @ W2 ) @ ( transitive_trancl @ A @ ( rel_restrict @ A @ E6 @ R3 ) ) )
=> ~ ( member @ A @ W2 @ R3 ) ) ).
% rel_restrict_trancl_notR(2)
thf(fact_6426_rel__restrict__mono2,axiom,
! [A: $tType,R3: set @ A,S2: set @ A,A5: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ A ) @ R3 @ S2 )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( rel_restrict @ A @ A5 @ S2 ) @ ( rel_restrict @ A @ A5 @ R3 ) ) ) ).
% rel_restrict_mono2
thf(fact_6427_rel__restrict__trancl__sub,axiom,
! [A: $tType,A5: set @ ( product_prod @ A @ A ),R3: set @ A] : ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( transitive_trancl @ A @ ( rel_restrict @ A @ A5 @ R3 ) ) @ ( rel_restrict @ A @ ( transitive_trancl @ A @ A5 ) @ R3 ) ) ).
% rel_restrict_trancl_sub
thf(fact_6428_vimage__Suc__insert__0,axiom,
! [A5: set @ nat] :
( ( vimage @ nat @ nat @ suc @ ( insert @ nat @ ( zero_zero @ nat ) @ A5 ) )
= ( vimage @ nat @ nat @ suc @ A5 ) ) ).
% vimage_Suc_insert_0
thf(fact_6429_rel__restrict__def,axiom,
! [A: $tType] :
( ( rel_restrict @ A )
= ( ^ [R7: set @ ( product_prod @ A @ A ),A8: set @ A] :
( collect @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ^ [V4: A,W: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ V4 @ W ) @ R7 )
& ~ ( member @ A @ V4 @ A8 )
& ~ ( member @ A @ W @ A8 ) ) ) ) ) ) ).
% rel_restrict_def
thf(fact_6430_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
@ ^ [Uu: A] : ( top_top @ ( set @ B ) ) ) ) ).
% vimage_fst
thf(fact_6431_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 ) )
@ ^ [Uu: A] : A5 ) ) ).
% vimage_snd
thf(fact_6432_vimage__eq__UN,axiom,
! [B: $tType,A: $tType] :
( ( vimage @ A @ B )
= ( ^ [F2: A > B,B8: set @ B] :
( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [Y4: B] : ( vimage @ A @ B @ F2 @ ( insert @ B @ Y4 @ ( bot_bot @ ( set @ B ) ) ) )
@ B8 ) ) ) ) ).
% vimage_eq_UN
thf(fact_6433_rel__restrict__compl,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),A5: set @ A] :
( ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ ( rel_restrict @ A @ R3 @ A5 ) @ ( rel_restrict @ A @ R3 @ ( uminus_uminus @ ( set @ A ) @ A5 ) ) )
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% rel_restrict_compl
thf(fact_6434_homo__rel__restrict__mono,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),B5: set @ A,A5: set @ A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ B5
@ ^ [Uu: A] : B5 ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( rel_restrict @ A @ R3 @ A5 )
@ ( product_Sigma @ A @ A @ ( minus_minus @ ( set @ A ) @ B5 @ A5 )
@ ^ [Uu: A] : ( minus_minus @ ( set @ A ) @ B5 @ A5 ) ) ) ) ).
% homo_rel_restrict_mono
thf(fact_6435_rel__restrict__alt__def,axiom,
! [A: $tType] :
( ( rel_restrict @ A )
= ( ^ [R7: set @ ( product_prod @ A @ A ),A8: set @ A] :
( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R7
@ ( product_Sigma @ A @ A @ ( uminus_uminus @ ( set @ A ) @ A8 )
@ ^ [Uu: A] : ( uminus_uminus @ ( set @ A ) @ A8 ) ) ) ) ) ).
% rel_restrict_alt_def
thf(fact_6436_rel__restrict__Sigma__sub,axiom,
! [A: $tType,A5: set @ A,R3: set @ A] :
( ord_less_eq @ ( set @ ( product_prod @ A @ A ) )
@ ( rel_restrict @ A
@ ( transitive_trancl @ A
@ ( product_Sigma @ A @ A @ A5
@ ^ [Uu: A] : A5 ) )
@ R3 )
@ ( transitive_trancl @ A
@ ( product_Sigma @ A @ A @ ( minus_minus @ ( set @ A ) @ A5 @ R3 )
@ ^ [Uu: A] : ( minus_minus @ ( set @ A ) @ A5 @ R3 ) ) ) ) ).
% rel_restrict_Sigma_sub
thf(fact_6437_finite__mono__remains__stable__implies__strict__prefix,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F: nat > A] :
( ( finite_finite2 @ A @ ( image2 @ nat @ A @ F @ ( top_top @ ( set @ nat ) ) ) )
=> ( ( order_mono @ nat @ A @ F )
=> ( ! [N2: nat] :
( ( ( F @ N2 )
= ( F @ ( suc @ N2 ) ) )
=> ( ( F @ ( suc @ N2 ) )
= ( F @ ( suc @ ( suc @ N2 ) ) ) ) )
=> ? [N10: nat] :
( ! [N9: nat] :
( ( ord_less_eq @ nat @ N9 @ N10 )
=> ! [M7: nat] :
( ( ord_less_eq @ nat @ M7 @ N10 )
=> ( ( ord_less @ nat @ M7 @ N9 )
=> ( ord_less @ A @ ( F @ M7 ) @ ( F @ N9 ) ) ) ) )
& ! [N9: nat] :
( ( ord_less_eq @ nat @ N10 @ N9 )
=> ( ( F @ N10 )
= ( F @ N9 ) ) ) ) ) ) ) ) ).
% finite_mono_remains_stable_implies_strict_prefix
thf(fact_6438_finite__def,axiom,
! [A: $tType] :
( ( finite_finite2 @ A )
= ( complete_lattice_lfp @ ( ( set @ A ) > $o )
@ ^ [P7: ( set @ A ) > $o,X: set @ A] :
( ( X
= ( bot_bot @ ( set @ A ) ) )
| ? [A8: set @ A,A3: A] :
( ( X
= ( insert @ A @ A3 @ A8 ) )
& ( P7 @ A8 ) ) ) ) ) ).
% finite_def
thf(fact_6439_lfp__Kleene__iter,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F: A > A,K: nat] :
( ( order_mono @ A @ A @ F )
=> ( ( ( compow @ ( A > A ) @ ( suc @ K ) @ F @ ( bot_bot @ A ) )
= ( compow @ ( A > A ) @ K @ F @ ( bot_bot @ A ) ) )
=> ( ( complete_lattice_lfp @ A @ F )
= ( compow @ ( A > A ) @ K @ F @ ( bot_bot @ A ) ) ) ) ) ) ).
% lfp_Kleene_iter
thf(fact_6440_mono__funpow,axiom,
! [A: $tType] :
( ( ( lattice @ A )
& ( order_bot @ A ) )
=> ! [Q: A > A] :
( ( order_mono @ A @ A @ Q )
=> ( order_mono @ nat @ A
@ ^ [I2: nat] : ( compow @ ( A > A ) @ I2 @ Q @ ( bot_bot @ A ) ) ) ) ) ).
% mono_funpow
thf(fact_6441_lfp__funpow,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F: A > A,N: nat] :
( ( order_mono @ A @ A @ F )
=> ( ( complete_lattice_lfp @ A @ ( compow @ ( A > A ) @ ( suc @ N ) @ F ) )
= ( complete_lattice_lfp @ A @ F ) ) ) ) ).
% lfp_funpow
thf(fact_6442_mono__pow,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F: A > A,N: nat] :
( ( order_mono @ A @ A @ F )
=> ( order_mono @ A @ A @ ( compow @ ( A > A ) @ N @ F ) ) ) ) ).
% mono_pow
thf(fact_6443_mono__strict__invE,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( order @ B ) )
=> ! [F: A > B,X2: A,Y: A] :
( ( order_mono @ A @ B @ F )
=> ( ( ord_less @ B @ ( F @ X2 ) @ ( F @ Y ) )
=> ( ord_less @ A @ X2 @ Y ) ) ) ) ).
% mono_strict_invE
thf(fact_6444_mono__Suc,axiom,
order_mono @ nat @ nat @ suc ).
% mono_Suc
thf(fact_6445_def__lfp__unfold,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [H: A,F: A > A] :
( ( H
= ( complete_lattice_lfp @ A @ F ) )
=> ( ( order_mono @ A @ A @ F )
=> ( H
= ( F @ H ) ) ) ) ) ).
% def_lfp_unfold
thf(fact_6446_lfp__fixpoint,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F: A > A] :
( ( order_mono @ A @ A @ F )
=> ( ( F @ ( complete_lattice_lfp @ A @ F ) )
= ( complete_lattice_lfp @ A @ F ) ) ) ) ).
% lfp_fixpoint
thf(fact_6447_lfp__unfold,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F: A > A] :
( ( order_mono @ A @ A @ F )
=> ( ( complete_lattice_lfp @ A @ F )
= ( F @ ( complete_lattice_lfp @ A @ F ) ) ) ) ) ).
% lfp_unfold
thf(fact_6448_mono__add,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [A2: A] : ( order_mono @ A @ A @ ( plus_plus @ A @ A2 ) ) ) ).
% mono_add
thf(fact_6449_lfp__const,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [T2: A] :
( ( complete_lattice_lfp @ A
@ ^ [X: A] : T2 )
= T2 ) ) ).
% lfp_const
thf(fact_6450_lfp__rolling,axiom,
! [A: $tType,B: $tType] :
( ( ( comple6319245703460814977attice @ B )
& ( comple6319245703460814977attice @ A ) )
=> ! [G: A > B,F: B > A] :
( ( order_mono @ A @ B @ G )
=> ( ( order_mono @ B @ A @ F )
=> ( ( G
@ ( complete_lattice_lfp @ A
@ ^ [X: A] : ( F @ ( G @ X ) ) ) )
= ( complete_lattice_lfp @ B
@ ^ [X: B] : ( G @ ( F @ X ) ) ) ) ) ) ) ).
% lfp_rolling
thf(fact_6451_mono__invE,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( order @ B ) )
=> ! [F: A > B,X2: A,Y: A] :
( ( order_mono @ A @ B @ F )
=> ( ( ord_less @ B @ ( F @ X2 ) @ ( F @ Y ) )
=> ( ord_less_eq @ A @ X2 @ Y ) ) ) ) ).
% mono_invE
thf(fact_6452_mono__iff__le__Suc,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( order_mono @ nat @ A )
= ( ^ [F2: nat > A] :
! [N4: nat] : ( ord_less_eq @ A @ ( F2 @ N4 ) @ ( F2 @ ( suc @ N4 ) ) ) ) ) ) ).
% mono_iff_le_Suc
thf(fact_6453_lfp__lfp,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F: A > A > A] :
( ! [X3: A,Y2: A,W3: A,Z4: A] :
( ( ord_less_eq @ A @ X3 @ Y2 )
=> ( ( ord_less_eq @ A @ W3 @ Z4 )
=> ( ord_less_eq @ A @ ( F @ X3 @ W3 ) @ ( F @ Y2 @ Z4 ) ) ) )
=> ( ( complete_lattice_lfp @ A
@ ^ [X: A] : ( complete_lattice_lfp @ A @ ( F @ X ) ) )
= ( complete_lattice_lfp @ A
@ ^ [X: A] : ( F @ X @ X ) ) ) ) ) ).
% lfp_lfp
thf(fact_6454_mono__def,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ( ( order_mono @ A @ B )
= ( ^ [F2: A > B] :
! [X: A,Y4: A] :
( ( ord_less_eq @ A @ X @ Y4 )
=> ( ord_less_eq @ B @ ( F2 @ X ) @ ( F2 @ Y4 ) ) ) ) ) ) ).
% mono_def
thf(fact_6455_monoI,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F: A > B] :
( ! [X3: A,Y2: A] :
( ( ord_less_eq @ A @ X3 @ Y2 )
=> ( ord_less_eq @ B @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( order_mono @ A @ B @ F ) ) ) ).
% monoI
thf(fact_6456_monoE,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F: A > B,X2: A,Y: A] :
( ( order_mono @ A @ B @ F )
=> ( ( ord_less_eq @ A @ X2 @ Y )
=> ( ord_less_eq @ B @ ( F @ X2 ) @ ( F @ Y ) ) ) ) ) ).
% monoE
thf(fact_6457_monoD,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F: A > B,X2: A,Y: A] :
( ( order_mono @ A @ B @ F )
=> ( ( ord_less_eq @ A @ X2 @ Y )
=> ( ord_less_eq @ B @ ( F @ X2 ) @ ( F @ Y ) ) ) ) ) ).
% monoD
thf(fact_6458_lfp__eqI,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F4: A > A,X2: A] :
( ( order_mono @ A @ A @ F4 )
=> ( ( ( F4 @ X2 )
= X2 )
=> ( ! [Z4: A] :
( ( ( F4 @ Z4 )
= Z4 )
=> ( ord_less_eq @ A @ X2 @ Z4 ) )
=> ( ( complete_lattice_lfp @ A @ F4 )
= X2 ) ) ) ) ) ).
% lfp_eqI
thf(fact_6459_lfp__greatest,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F: A > A,A5: A] :
( ! [U4: A] :
( ( ord_less_eq @ A @ ( F @ U4 ) @ U4 )
=> ( ord_less_eq @ A @ A5 @ U4 ) )
=> ( ord_less_eq @ A @ A5 @ ( complete_lattice_lfp @ A @ F ) ) ) ) ).
% lfp_greatest
thf(fact_6460_lfp__lowerbound,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F: A > A,A5: A] :
( ( ord_less_eq @ A @ ( F @ A5 ) @ A5 )
=> ( ord_less_eq @ A @ ( complete_lattice_lfp @ A @ F ) @ A5 ) ) ) ).
% lfp_lowerbound
thf(fact_6461_lfp__mono,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F: A > A,G: A > A] :
( ! [Z8: A] : ( ord_less_eq @ A @ ( F @ Z8 ) @ ( G @ Z8 ) )
=> ( ord_less_eq @ A @ ( complete_lattice_lfp @ A @ F ) @ ( complete_lattice_lfp @ A @ G ) ) ) ) ).
% lfp_mono
thf(fact_6462_funpow__mono,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F: A > A,A5: A,B5: A,N: nat] :
( ( order_mono @ A @ A @ F )
=> ( ( ord_less_eq @ A @ A5 @ B5 )
=> ( ord_less_eq @ A @ ( compow @ ( A > A ) @ N @ F @ A5 ) @ ( compow @ ( A > A ) @ N @ F @ B5 ) ) ) ) ) ).
% funpow_mono
thf(fact_6463_def__lfp__induct,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: A,F: A > A,P: A] :
( ( A5
= ( complete_lattice_lfp @ A @ F ) )
=> ( ( order_mono @ A @ A @ F )
=> ( ( ord_less_eq @ A @ ( F @ ( inf_inf @ A @ A5 @ P ) ) @ P )
=> ( ord_less_eq @ A @ A5 @ P ) ) ) ) ) ).
% def_lfp_induct
thf(fact_6464_lfp__induct,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F: A > A,P: A] :
( ( order_mono @ A @ A @ F )
=> ( ( ord_less_eq @ A @ ( F @ ( inf_inf @ A @ ( complete_lattice_lfp @ A @ F ) @ P ) ) @ P )
=> ( ord_less_eq @ A @ ( complete_lattice_lfp @ A @ F ) @ P ) ) ) ) ).
% lfp_induct
thf(fact_6465_mono__inf,axiom,
! [B: $tType,A: $tType] :
( ( ( semilattice_inf @ A )
& ( semilattice_inf @ B ) )
=> ! [F: A > B,A5: A,B5: A] :
( ( order_mono @ A @ B @ F )
=> ( ord_less_eq @ B @ ( F @ ( inf_inf @ A @ A5 @ B5 ) ) @ ( inf_inf @ B @ ( F @ A5 ) @ ( F @ B5 ) ) ) ) ) ).
% mono_inf
thf(fact_6466_mono__sup,axiom,
! [B: $tType,A: $tType] :
( ( ( semilattice_sup @ A )
& ( semilattice_sup @ B ) )
=> ! [F: A > B,A5: A,B5: A] :
( ( order_mono @ A @ B @ F )
=> ( ord_less_eq @ B @ ( sup_sup @ B @ ( F @ A5 ) @ ( F @ B5 ) ) @ ( F @ ( sup_sup @ A @ A5 @ B5 ) ) ) ) ) ).
% mono_sup
thf(fact_6467_lfp__ordinal__induct,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F: A > A,P: A > $o] :
( ( order_mono @ A @ A @ F )
=> ( ! [S8: A] :
( ( P @ S8 )
=> ( ( ord_less_eq @ A @ S8 @ ( complete_lattice_lfp @ A @ F ) )
=> ( P @ ( F @ S8 ) ) ) )
=> ( ! [M10: set @ A] :
( ! [X4: A] :
( ( member @ A @ X4 @ M10 )
=> ( P @ X4 ) )
=> ( P @ ( complete_Sup_Sup @ A @ M10 ) ) )
=> ( P @ ( complete_lattice_lfp @ A @ F ) ) ) ) ) ) ).
% lfp_ordinal_induct
thf(fact_6468_Rings_Omono__mult,axiom,
! [A: $tType] :
( ( ordered_semiring @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( order_mono @ A @ A @ ( times_times @ A @ A2 ) ) ) ) ).
% Rings.mono_mult
thf(fact_6469_mono__times__nat,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( order_mono @ nat @ nat @ ( times_times @ nat @ N ) ) ) ).
% mono_times_nat
thf(fact_6470_mono__image__least,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [F: A > B,M: A,N: A,M5: B,N3: B] :
( ( order_mono @ A @ B @ F )
=> ( ( ( image2 @ A @ B @ F @ ( set_or7035219750837199246ssThan @ A @ M @ N ) )
= ( set_or7035219750837199246ssThan @ B @ M5 @ N3 ) )
=> ( ( ord_less @ A @ M @ N )
=> ( ( F @ M )
= M5 ) ) ) ) ) ).
% mono_image_least
thf(fact_6471_Kleene__iter__lpfp,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [F: A > A,P3: A,K: nat] :
( ( order_mono @ A @ A @ F )
=> ( ( ord_less_eq @ A @ ( F @ P3 ) @ P3 )
=> ( ord_less_eq @ A @ ( compow @ ( A > A ) @ K @ F @ ( bot_bot @ A ) ) @ P3 ) ) ) ) ).
% Kleene_iter_lpfp
thf(fact_6472_Kleene__iter__gpfp,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [F: A > A,P3: A,K: nat] :
( ( order_mono @ A @ A @ F )
=> ( ( ord_less_eq @ A @ P3 @ ( F @ P3 ) )
=> ( ord_less_eq @ A @ P3 @ ( compow @ ( A > A ) @ K @ F @ ( top_top @ A ) ) ) ) ) ) ).
% Kleene_iter_gpfp
thf(fact_6473_funpow__mono2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F: A > A,I: nat,J: nat,X2: A,Y: A] :
( ( order_mono @ A @ A @ F )
=> ( ( ord_less_eq @ nat @ I @ J )
=> ( ( ord_less_eq @ A @ X2 @ Y )
=> ( ( ord_less_eq @ A @ X2 @ ( F @ X2 ) )
=> ( ord_less_eq @ A @ ( compow @ ( A > A ) @ I @ F @ X2 ) @ ( compow @ ( A > A ) @ J @ F @ Y ) ) ) ) ) ) ) ).
% funpow_mono2
thf(fact_6474_lfp__def,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ( ( complete_lattice_lfp @ A )
= ( ^ [F2: A > A] :
( complete_Inf_Inf @ A
@ ( collect @ A
@ ^ [U2: A] : ( ord_less_eq @ A @ ( F2 @ U2 ) @ U2 ) ) ) ) ) ) ).
% lfp_def
thf(fact_6475_mono__SUP,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ( comple6319245703460814977attice @ A )
& ( comple6319245703460814977attice @ B ) )
=> ! [F: A > B,A5: C > A,I5: set @ C] :
( ( order_mono @ A @ B @ F )
=> ( ord_less_eq @ B
@ ( complete_Sup_Sup @ B
@ ( image2 @ C @ B
@ ^ [X: C] : ( F @ ( A5 @ X ) )
@ I5 ) )
@ ( F @ ( complete_Sup_Sup @ A @ ( image2 @ C @ A @ A5 @ I5 ) ) ) ) ) ) ).
% mono_SUP
thf(fact_6476_mono__Sup,axiom,
! [B: $tType,A: $tType] :
( ( ( comple6319245703460814977attice @ A )
& ( comple6319245703460814977attice @ B ) )
=> ! [F: A > B,A5: set @ A] :
( ( order_mono @ A @ B @ F )
=> ( ord_less_eq @ B @ ( complete_Sup_Sup @ B @ ( image2 @ A @ B @ F @ A5 ) ) @ ( F @ ( complete_Sup_Sup @ A @ A5 ) ) ) ) ) ).
% mono_Sup
thf(fact_6477_mono__INF,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ( comple6319245703460814977attice @ A )
& ( comple6319245703460814977attice @ B ) )
=> ! [F: A > B,A5: C > A,I5: set @ C] :
( ( order_mono @ A @ B @ F )
=> ( ord_less_eq @ B @ ( F @ ( complete_Inf_Inf @ A @ ( image2 @ C @ A @ A5 @ I5 ) ) )
@ ( complete_Inf_Inf @ B
@ ( image2 @ C @ B
@ ^ [X: C] : ( F @ ( A5 @ X ) )
@ I5 ) ) ) ) ) ).
% mono_INF
thf(fact_6478_mono__Inf,axiom,
! [B: $tType,A: $tType] :
( ( ( comple6319245703460814977attice @ A )
& ( comple6319245703460814977attice @ B ) )
=> ! [F: A > B,A5: set @ A] :
( ( order_mono @ A @ B @ F )
=> ( ord_less_eq @ B @ ( F @ ( complete_Inf_Inf @ A @ A5 ) ) @ ( complete_Inf_Inf @ B @ ( image2 @ A @ B @ F @ A5 ) ) ) ) ) ).
% mono_Inf
thf(fact_6479_funpow__decreasing,axiom,
! [A: $tType] :
( ( ( lattice @ A )
& ( order_bot @ A ) )
=> ! [M: nat,N: nat,F: A > A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( order_mono @ A @ A @ F )
=> ( ord_less_eq @ A @ ( compow @ ( A > A ) @ M @ F @ ( bot_bot @ A ) ) @ ( compow @ ( A > A ) @ N @ F @ ( bot_bot @ A ) ) ) ) ) ) ).
% funpow_decreasing
thf(fact_6480_funpow__increasing,axiom,
! [A: $tType] :
( ( ( lattice @ A )
& ( order_top @ A ) )
=> ! [M: nat,N: nat,F: A > A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( order_mono @ A @ A @ F )
=> ( ord_less_eq @ A @ ( compow @ ( A > A ) @ N @ F @ ( top_top @ A ) ) @ ( compow @ ( A > A ) @ M @ F @ ( top_top @ A ) ) ) ) ) ) ).
% funpow_increasing
thf(fact_6481_mono__Max__commute,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( linorder @ B ) )
=> ! [F: A > B,A5: set @ A] :
( ( order_mono @ A @ B @ F )
=> ( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( F @ ( lattic643756798349783984er_Max @ A @ A5 ) )
= ( lattic643756798349783984er_Max @ B @ ( image2 @ A @ B @ F @ A5 ) ) ) ) ) ) ) ).
% mono_Max_commute
thf(fact_6482_mono__Min__commute,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( linorder @ B ) )
=> ! [F: A > B,A5: set @ A] :
( ( order_mono @ A @ B @ F )
=> ( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( F @ ( lattic643756798350308766er_Min @ A @ A5 ) )
= ( lattic643756798350308766er_Min @ B @ ( image2 @ A @ B @ F @ A5 ) ) ) ) ) ) ) ).
% mono_Min_commute
thf(fact_6483_mono__ge2__power__minus__self,axiom,
! [K: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K )
=> ( order_mono @ nat @ nat
@ ^ [M2: nat] : ( minus_minus @ nat @ ( power_power @ nat @ K @ M2 ) @ M2 ) ) ) ).
% mono_ge2_power_minus_self
thf(fact_6484_remdups__adj__altdef,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( remdups_adj @ A @ Xs )
= Ys )
= ( ? [F2: nat > nat] :
( ( order_mono @ nat @ nat @ F2 )
& ( ( image2 @ nat @ nat @ F2 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) ) )
= ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Ys ) ) )
& ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ Xs @ I2 )
= ( nth @ A @ Ys @ ( F2 @ I2 ) ) ) )
& ! [I2: nat] :
( ( ord_less @ nat @ ( plus_plus @ nat @ I2 @ ( one_one @ nat ) ) @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ( nth @ A @ Xs @ I2 )
= ( nth @ A @ Xs @ ( plus_plus @ nat @ I2 @ ( one_one @ nat ) ) ) )
= ( ( F2 @ I2 )
= ( F2 @ ( plus_plus @ nat @ I2 @ ( one_one @ nat ) ) ) ) ) ) ) ) ) ).
% remdups_adj_altdef
thf(fact_6485_INF__principal__finite,axiom,
! [B: $tType,A: $tType,X8: set @ A,F: A > ( set @ B )] :
( ( finite_finite2 @ A @ X8 )
=> ( ( complete_Inf_Inf @ ( filter @ B )
@ ( image2 @ A @ ( filter @ B )
@ ^ [X: A] : ( principal @ B @ ( F @ X ) )
@ X8 ) )
= ( principal @ B @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ F @ X8 ) ) ) ) ) ).
% INF_principal_finite
thf(fact_6486_SUP__principal,axiom,
! [A: $tType,B: $tType,A5: B > ( set @ A ),I5: set @ B] :
( ( complete_Sup_Sup @ ( filter @ A )
@ ( image2 @ B @ ( filter @ A )
@ ^ [I2: B] : ( principal @ A @ ( A5 @ I2 ) )
@ I5 ) )
= ( principal @ A @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ A5 @ I5 ) ) ) ) ).
% SUP_principal
thf(fact_6487_lfp__induct2,axiom,
! [A: $tType,B: $tType,A2: A,B2: B,F: ( set @ ( product_prod @ A @ B ) ) > ( set @ ( product_prod @ A @ B ) ),P: A > B > $o] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ ( complete_lattice_lfp @ ( set @ ( product_prod @ A @ B ) ) @ F ) )
=> ( ( order_mono @ ( set @ ( product_prod @ A @ B ) ) @ ( set @ ( product_prod @ A @ B ) ) @ F )
=> ( ! [A4: A,B4: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B4 ) @ ( F @ ( inf_inf @ ( set @ ( product_prod @ A @ B ) ) @ ( complete_lattice_lfp @ ( set @ ( product_prod @ A @ B ) ) @ F ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ P ) ) ) ) )
=> ( P @ A4 @ B4 ) )
=> ( P @ A2 @ B2 ) ) ) ) ).
% lfp_induct2
thf(fact_6488_lfp__induct__set,axiom,
! [A: $tType,A2: A,F: ( set @ A ) > ( set @ A ),P: A > $o] :
( ( member @ A @ A2 @ ( complete_lattice_lfp @ ( set @ A ) @ F ) )
=> ( ( order_mono @ ( set @ A ) @ ( set @ A ) @ F )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ ( F @ ( inf_inf @ ( set @ A ) @ ( complete_lattice_lfp @ ( set @ A ) @ F ) @ ( collect @ A @ P ) ) ) )
=> ( P @ X3 ) )
=> ( P @ A2 ) ) ) ) ).
% lfp_induct_set
thf(fact_6489_def__lfp__induct__set,axiom,
! [A: $tType,A5: set @ A,F: ( set @ A ) > ( set @ A ),A2: A,P: A > $o] :
( ( A5
= ( complete_lattice_lfp @ ( set @ A ) @ F ) )
=> ( ( order_mono @ ( set @ A ) @ ( set @ A ) @ F )
=> ( ( member @ A @ A2 @ A5 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ ( F @ ( inf_inf @ ( set @ A ) @ A5 @ ( collect @ A @ P ) ) ) )
=> ( P @ X3 ) )
=> ( P @ A2 ) ) ) ) ) ).
% def_lfp_induct_set
thf(fact_6490_coinduct3__mono__lemma,axiom,
! [B: $tType,A: $tType] :
( ( order @ A )
=> ! [F: A > ( set @ B ),X8: set @ B,B5: set @ B] :
( ( order_mono @ A @ ( set @ B ) @ F )
=> ( order_mono @ A @ ( set @ B )
@ ^ [X: A] : ( sup_sup @ ( set @ B ) @ ( sup_sup @ ( set @ B ) @ ( F @ X ) @ X8 ) @ B5 ) ) ) ) ).
% coinduct3_mono_lemma
thf(fact_6491_tranclp_Omono,axiom,
! [A: $tType,R: A > A > $o] :
( order_mono @ ( A > A > $o ) @ ( A > A > $o )
@ ^ [P7: A > A > $o,X12: A,X24: A] :
( ? [A3: A,B3: A] :
( ( X12 = A3 )
& ( X24 = B3 )
& ( R @ A3 @ B3 ) )
| ? [A3: A,B3: A,C5: A] :
( ( X12 = A3 )
& ( X24 = C5 )
& ( P7 @ A3 @ B3 )
& ( R @ B3 @ C5 ) ) ) ) ).
% tranclp.mono
thf(fact_6492_rtranclp_Omono,axiom,
! [A: $tType,R: A > A > $o] :
( order_mono @ ( A > A > $o ) @ ( A > A > $o )
@ ^ [P7: A > A > $o,X12: A,X24: A] :
( ? [A3: A] :
( ( X12 = A3 )
& ( X24 = A3 ) )
| ? [A3: A,B3: A,C5: A] :
( ( X12 = A3 )
& ( X24 = C5 )
& ( P7 @ A3 @ B3 )
& ( R @ B3 @ C5 ) ) ) ) ).
% rtranclp.mono
thf(fact_6493_lfp__ordinal__induct__set,axiom,
! [A: $tType,F: ( set @ A ) > ( set @ A ),P: ( set @ A ) > $o] :
( ( order_mono @ ( set @ A ) @ ( set @ A ) @ F )
=> ( ! [S8: set @ A] :
( ( P @ S8 )
=> ( P @ ( F @ S8 ) ) )
=> ( ! [M10: set @ ( set @ A )] :
( ! [X4: set @ A] :
( ( member @ ( set @ A ) @ X4 @ M10 )
=> ( P @ X4 ) )
=> ( P @ ( complete_Sup_Sup @ ( set @ A ) @ M10 ) ) )
=> ( P @ ( complete_lattice_lfp @ ( set @ A ) @ F ) ) ) ) ) ).
% lfp_ordinal_induct_set
thf(fact_6494_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_6495_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_6496_remdups__adj__Cons_H,axiom,
! [A: $tType,X2: A,Xs: list @ A] :
( ( remdups_adj @ A @ ( cons @ A @ X2 @ Xs ) )
= ( cons @ A @ X2
@ ( remdups_adj @ A
@ ( dropWhile @ A
@ ^ [Y4: A] : Y4 = X2
@ Xs ) ) ) ) ).
% remdups_adj_Cons'
thf(fact_6497_remdups__adj__Cons,axiom,
! [A: $tType,X2: A,Xs: list @ A] :
( ( remdups_adj @ A @ ( cons @ A @ X2 @ Xs ) )
= ( case_list @ ( list @ A ) @ A @ ( cons @ A @ X2 @ ( nil @ A ) )
@ ^ [Y4: A,Xs3: list @ A] : ( if @ ( list @ A ) @ ( X2 = Y4 ) @ ( cons @ A @ Y4 @ Xs3 ) @ ( cons @ A @ X2 @ ( cons @ A @ Y4 @ Xs3 ) ) )
@ ( remdups_adj @ A @ Xs ) ) ) ).
% remdups_adj_Cons
thf(fact_6498_remdups__adj__adjacent,axiom,
! [A: $tType,I: nat,Xs: list @ A] :
( ( ord_less @ nat @ ( suc @ I ) @ ( size_size @ ( list @ A ) @ ( remdups_adj @ A @ Xs ) ) )
=> ( ( nth @ A @ ( remdups_adj @ A @ Xs ) @ I )
!= ( nth @ A @ ( remdups_adj @ A @ Xs ) @ ( suc @ I ) ) ) ) ).
% remdups_adj_adjacent
thf(fact_6499_ord_Olexordp_Omono,axiom,
! [A: $tType,Less: A > A > $o] :
( order_mono @ ( ( list @ A ) > ( list @ A ) > $o ) @ ( ( list @ A ) > ( list @ A ) > $o )
@ ^ [P7: ( list @ A ) > ( list @ A ) > $o,X12: list @ A,X24: list @ A] :
( ? [Y4: A,Ys3: list @ A] :
( ( X12
= ( nil @ A ) )
& ( X24
= ( cons @ A @ Y4 @ Ys3 ) ) )
| ? [X: A,Y4: A,Xs3: list @ A,Ys3: list @ A] :
( ( X12
= ( cons @ A @ X @ Xs3 ) )
& ( X24
= ( cons @ A @ Y4 @ Ys3 ) )
& ( Less @ X @ Y4 ) )
| ? [X: A,Y4: A,Xs3: list @ A,Ys3: list @ A] :
( ( X12
= ( cons @ A @ X @ Xs3 ) )
& ( X24
= ( cons @ A @ Y4 @ Ys3 ) )
& ~ ( Less @ X @ Y4 )
& ~ ( Less @ Y4 @ X )
& ( P7 @ Xs3 @ Ys3 ) ) ) ) ).
% ord.lexordp.mono
thf(fact_6500_remdups__adj__replicate,axiom,
! [A: $tType,N: nat,X2: A] :
( ( ( N
= ( zero_zero @ nat ) )
=> ( ( remdups_adj @ A @ ( replicate @ A @ N @ X2 ) )
= ( nil @ A ) ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ( ( remdups_adj @ A @ ( replicate @ A @ N @ X2 ) )
= ( cons @ A @ X2 @ ( nil @ A ) ) ) ) ) ).
% remdups_adj_replicate
thf(fact_6501_remdups__adj__singleton,axiom,
! [A: $tType,Xs: list @ A,X2: A] :
( ( ( remdups_adj @ A @ Xs )
= ( cons @ A @ X2 @ ( nil @ A ) ) )
=> ( Xs
= ( replicate @ A @ ( size_size @ ( list @ A ) @ Xs ) @ X2 ) ) ) ).
% remdups_adj_singleton
thf(fact_6502_finite_Omono,axiom,
! [A: $tType] :
( order_mono @ ( ( set @ A ) > $o ) @ ( ( set @ A ) > $o )
@ ^ [P7: ( set @ A ) > $o,X: set @ A] :
( ( X
= ( bot_bot @ ( set @ A ) ) )
| ? [A8: set @ A,A3: A] :
( ( X
= ( insert @ A @ A3 @ A8 ) )
& ( P7 @ A8 ) ) ) ) ).
% finite.mono
thf(fact_6503_remdups__adj__append__dropWhile,axiom,
! [A: $tType,Xs: list @ A,Y: A,Ys: list @ A] :
( ( remdups_adj @ A @ ( append @ A @ Xs @ ( cons @ A @ Y @ Ys ) ) )
= ( append @ A @ ( remdups_adj @ A @ ( append @ A @ Xs @ ( cons @ A @ Y @ ( nil @ A ) ) ) )
@ ( remdups_adj @ A
@ ( dropWhile @ A
@ ^ [X: A] : X = Y
@ Ys ) ) ) ) ).
% remdups_adj_append_dropWhile
thf(fact_6504_remdups__adj__append_H_H,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( remdups_adj @ A @ ( append @ A @ Xs @ Ys ) )
= ( append @ A @ ( remdups_adj @ A @ Xs )
@ ( remdups_adj @ A
@ ( dropWhile @ A
@ ^ [Y4: A] :
( Y4
= ( last @ A @ Xs ) )
@ Ys ) ) ) ) ) ).
% remdups_adj_append''
thf(fact_6505_tl__remdups__adj,axiom,
! [A: $tType,Ys: list @ A] :
( ( Ys
!= ( nil @ A ) )
=> ( ( tl @ A @ ( remdups_adj @ A @ Ys ) )
= ( remdups_adj @ A
@ ( dropWhile @ A
@ ^ [X: A] :
( X
= ( hd @ A @ Ys ) )
@ ( tl @ A @ Ys ) ) ) ) ) ).
% tl_remdups_adj
thf(fact_6506_lexordp_Omono,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( order_mono @ ( ( list @ A ) > ( list @ A ) > $o ) @ ( ( list @ A ) > ( list @ A ) > $o )
@ ^ [P7: ( list @ A ) > ( list @ A ) > $o,X12: list @ A,X24: list @ A] :
( ? [Y4: A,Ys3: list @ A] :
( ( X12
= ( nil @ A ) )
& ( X24
= ( cons @ A @ Y4 @ Ys3 ) ) )
| ? [X: A,Y4: A,Xs3: list @ A,Ys3: list @ A] :
( ( X12
= ( cons @ A @ X @ Xs3 ) )
& ( X24
= ( cons @ A @ Y4 @ Ys3 ) )
& ( ord_less @ A @ X @ Y4 ) )
| ? [X: A,Y4: A,Xs3: list @ A,Ys3: list @ A] :
( ( X12
= ( cons @ A @ X @ Xs3 ) )
& ( X24
= ( cons @ A @ Y4 @ Ys3 ) )
& ~ ( ord_less @ A @ X @ Y4 )
& ~ ( ord_less @ A @ Y4 @ X )
& ( P7 @ Xs3 @ Ys3 ) ) ) ) ) ).
% lexordp.mono
thf(fact_6507_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_6508_remdups__adj__singleton__iff,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( size_size @ ( list @ A ) @ ( remdups_adj @ A @ Xs ) )
= ( suc @ ( zero_zero @ nat ) ) )
= ( ( Xs
!= ( nil @ A ) )
& ( Xs
= ( replicate @ A @ ( size_size @ ( list @ A ) @ Xs ) @ ( hd @ A @ Xs ) ) ) ) ) ).
% remdups_adj_singleton_iff
thf(fact_6509_at__bot__def,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( at_bot @ A )
= ( complete_Inf_Inf @ ( filter @ A )
@ ( image2 @ A @ ( filter @ A )
@ ^ [K2: A] : ( principal @ A @ ( set_ord_atMost @ A @ K2 ) )
@ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% at_bot_def
thf(fact_6510_at__top__def,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( at_top @ A )
= ( complete_Inf_Inf @ ( filter @ A )
@ ( image2 @ A @ ( filter @ A )
@ ^ [K2: A] : ( principal @ A @ ( set_ord_atLeast @ A @ K2 ) )
@ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% at_top_def
thf(fact_6511_mono__compose,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ C ) )
=> ! [Q: A > B > C,F: D > B] :
( ( order_mono @ A @ ( B > C ) @ Q )
=> ( order_mono @ A @ ( D > C )
@ ^ [I2: A,X: D] : ( Q @ I2 @ ( F @ X ) ) ) ) ) ).
% mono_compose
thf(fact_6512_at__top__sub,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C2: A] :
( ( at_top @ A )
= ( complete_Inf_Inf @ ( filter @ A )
@ ( image2 @ A @ ( filter @ A )
@ ^ [K2: A] : ( principal @ A @ ( set_ord_atLeast @ A @ K2 ) )
@ ( set_ord_atLeast @ A @ C2 ) ) ) ) ) ).
% at_top_sub
thf(fact_6513_at__bot__sub,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C2: A] :
( ( at_bot @ A )
= ( complete_Inf_Inf @ ( filter @ A )
@ ( image2 @ A @ ( filter @ A )
@ ^ [K2: A] : ( principal @ A @ ( set_ord_atMost @ A @ K2 ) )
@ ( set_ord_atMost @ A @ C2 ) ) ) ) ) ).
% at_bot_sub
thf(fact_6514_ord__class_Olexordp__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_lexordp @ A )
= ( complete_lattice_lfp @ ( ( list @ A ) > ( list @ A ) > $o )
@ ^ [P7: ( list @ A ) > ( list @ A ) > $o,X12: list @ A,X24: list @ A] :
( ? [Y4: A,Ys3: list @ A] :
( ( X12
= ( nil @ A ) )
& ( X24
= ( cons @ A @ Y4 @ Ys3 ) ) )
| ? [X: A,Y4: A,Xs3: list @ A,Ys3: list @ A] :
( ( X12
= ( cons @ A @ X @ Xs3 ) )
& ( X24
= ( cons @ A @ Y4 @ Ys3 ) )
& ( ord_less @ A @ X @ Y4 ) )
| ? [X: A,Y4: A,Xs3: list @ A,Ys3: list @ A] :
( ( X12
= ( cons @ A @ X @ Xs3 ) )
& ( X24
= ( cons @ A @ Y4 @ Ys3 ) )
& ~ ( ord_less @ A @ X @ Y4 )
& ~ ( ord_less @ A @ Y4 @ X )
& ( P7 @ Xs3 @ Ys3 ) ) ) ) ) ) ).
% ord_class.lexordp_def
thf(fact_6515_image__o__collect,axiom,
! [B: $tType,C: $tType,A: $tType,G: C > B,F4: set @ ( A > ( set @ C ) )] :
( ( bNF_collect @ A @ B @ ( image2 @ ( A > ( set @ C ) ) @ ( A > ( set @ B ) ) @ ( comp @ ( set @ C ) @ ( set @ B ) @ A @ ( image2 @ C @ B @ G ) ) @ F4 ) )
= ( comp @ ( set @ C ) @ ( set @ B ) @ A @ ( image2 @ C @ B @ G ) @ ( bNF_collect @ A @ C @ F4 ) ) ) ).
% image_o_collect
thf(fact_6516_collect__def,axiom,
! [A: $tType,B: $tType] :
( ( bNF_collect @ B @ A )
= ( ^ [F8: set @ ( B > ( set @ A ) ),X: B] :
( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ ( B > ( set @ A ) ) @ ( set @ A )
@ ^ [F2: B > ( set @ A )] : ( F2 @ X )
@ F8 ) ) ) ) ).
% collect_def
thf(fact_6517_collect__comp,axiom,
! [A: $tType,B: $tType,C: $tType,F4: set @ ( C > ( set @ B ) ),G: A > C] :
( ( comp @ C @ ( set @ B ) @ A @ ( bNF_collect @ C @ B @ F4 ) @ G )
= ( bNF_collect @ A @ B
@ ( image2 @ ( C > ( set @ B ) ) @ ( A > ( set @ B ) )
@ ^ [F2: C > ( set @ B )] : ( comp @ C @ ( set @ B ) @ A @ F2 @ G )
@ F4 ) ) ) ).
% collect_comp
thf(fact_6518_lexordp__conv__lexord,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_lexordp @ A )
= ( ^ [Xs3: list @ A,Ys3: list @ A] : ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs3 @ Ys3 ) @ ( lexord @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ ( ord_less @ A ) ) ) ) ) ) ) ) ).
% lexordp_conv_lexord
thf(fact_6519_ord_Olexordp__def,axiom,
! [A: $tType] :
( ( lexordp2 @ A )
= ( ^ [Less2: A > A > $o] :
( complete_lattice_lfp @ ( ( list @ A ) > ( list @ A ) > $o )
@ ^ [P7: ( list @ A ) > ( list @ A ) > $o,X12: list @ A,X24: list @ A] :
( ? [Y4: A,Ys3: list @ A] :
( ( X12
= ( nil @ A ) )
& ( X24
= ( cons @ A @ Y4 @ Ys3 ) ) )
| ? [X: A,Y4: A,Xs3: list @ A,Ys3: list @ A] :
( ( X12
= ( cons @ A @ X @ Xs3 ) )
& ( X24
= ( cons @ A @ Y4 @ Ys3 ) )
& ( Less2 @ X @ Y4 ) )
| ? [X: A,Y4: A,Xs3: list @ A,Ys3: list @ A] :
( ( X12
= ( cons @ A @ X @ Xs3 ) )
& ( X24
= ( cons @ A @ Y4 @ Ys3 ) )
& ~ ( Less2 @ X @ Y4 )
& ~ ( Less2 @ Y4 @ X )
& ( P7 @ Xs3 @ Ys3 ) ) ) ) ) ) ).
% ord.lexordp_def
thf(fact_6520_bijective__Empty,axiom,
! [B: $tType,A: $tType] : ( bijective @ A @ B @ ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% bijective_Empty
thf(fact_6521_bijective__def,axiom,
! [B: $tType,A: $tType] :
( ( bijective @ A @ B )
= ( ^ [R7: set @ ( product_prod @ A @ B )] :
( ! [X: A,Y4: B,Z3: B] :
( ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y4 ) @ R7 )
& ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Z3 ) @ R7 ) )
=> ( Y4 = Z3 ) )
& ! [X: A,Y4: A,Z3: B] :
( ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Z3 ) @ R7 )
& ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Y4 @ Z3 ) @ R7 ) )
=> ( X = Y4 ) ) ) ) ) ).
% bijective_def
thf(fact_6522_map__of__mapk__SomeI,axiom,
! [A: $tType,B: $tType,C: $tType,F: A > B,T2: list @ ( product_prod @ A @ C ),K: A,X2: C] :
( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
=> ( ( ( map_of @ A @ C @ T2 @ K )
= ( some @ C @ X2 ) )
=> ( ( map_of @ B @ C
@ ( map @ ( product_prod @ A @ C ) @ ( product_prod @ B @ C )
@ ( product_case_prod @ A @ C @ ( product_prod @ B @ C )
@ ^ [K2: A] : ( product_Pair @ B @ C @ ( F @ K2 ) ) )
@ T2 )
@ ( F @ K ) )
= ( some @ C @ X2 ) ) ) ) ).
% map_of_mapk_SomeI
thf(fact_6523_quotient__of__def,axiom,
( quotient_of
= ( ^ [X: rat] :
( the @ ( product_prod @ int @ int )
@ ^ [Pair: product_prod @ int @ int] :
( ( X
= ( 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_6524_coprime__mult__left__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( algebr8660921524188924756oprime @ A @ ( times_times @ A @ A2 @ B2 ) @ C2 )
= ( ( algebr8660921524188924756oprime @ A @ A2 @ C2 )
& ( algebr8660921524188924756oprime @ A @ B2 @ C2 ) ) ) ) ).
% coprime_mult_left_iff
thf(fact_6525_coprime__mult__right__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( algebr8660921524188924756oprime @ A @ C2 @ ( times_times @ A @ A2 @ B2 ) )
= ( ( algebr8660921524188924756oprime @ A @ C2 @ A2 )
& ( algebr8660921524188924756oprime @ A @ C2 @ B2 ) ) ) ) ).
% coprime_mult_right_iff
thf(fact_6526_coprime__minus__right__iff,axiom,
! [A: $tType] :
( ( ring_gcd @ A )
=> ! [A2: A,B2: A] :
( ( algebr8660921524188924756oprime @ A @ A2 @ ( uminus_uminus @ A @ B2 ) )
= ( algebr8660921524188924756oprime @ A @ A2 @ B2 ) ) ) ).
% coprime_minus_right_iff
thf(fact_6527_coprime__minus__left__iff,axiom,
! [A: $tType] :
( ( ring_gcd @ A )
=> ! [A2: A,B2: A] :
( ( algebr8660921524188924756oprime @ A @ ( uminus_uminus @ A @ A2 ) @ B2 )
= ( algebr8660921524188924756oprime @ A @ A2 @ B2 ) ) ) ).
% coprime_minus_left_iff
thf(fact_6528_coprime__self,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A] :
( ( algebr8660921524188924756oprime @ A @ A2 @ A2 )
= ( dvd_dvd @ A @ A2 @ ( one_one @ A ) ) ) ) ).
% coprime_self
thf(fact_6529_coprime__mod__right__iff,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [A2: A,B2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( algebr8660921524188924756oprime @ A @ A2 @ ( modulo_modulo @ A @ B2 @ A2 ) )
= ( algebr8660921524188924756oprime @ A @ A2 @ B2 ) ) ) ) ).
% coprime_mod_right_iff
thf(fact_6530_coprime__mod__left__iff,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [B2: A,A2: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( algebr8660921524188924756oprime @ A @ ( modulo_modulo @ A @ A2 @ B2 ) @ B2 )
= ( algebr8660921524188924756oprime @ A @ A2 @ B2 ) ) ) ) ).
% coprime_mod_left_iff
thf(fact_6531_coprime__power__left__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A2: A,N: nat,B2: A] :
( ( algebr8660921524188924756oprime @ A @ ( power_power @ A @ A2 @ N ) @ B2 )
= ( ( algebr8660921524188924756oprime @ A @ A2 @ B2 )
| ( N
= ( zero_zero @ nat ) ) ) ) ) ).
% coprime_power_left_iff
thf(fact_6532_coprime__power__right__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A2: A,B2: A,N: nat] :
( ( algebr8660921524188924756oprime @ A @ A2 @ ( power_power @ A @ B2 @ N ) )
= ( ( algebr8660921524188924756oprime @ A @ A2 @ B2 )
| ( N
= ( zero_zero @ nat ) ) ) ) ) ).
% coprime_power_right_iff
thf(fact_6533_coprime__imp__gcd__eq__1,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A2: A,B2: A] :
( ( algebr8660921524188924756oprime @ A @ A2 @ B2 )
=> ( ( gcd_gcd @ A @ A2 @ B2 )
= ( one_one @ A ) ) ) ) ).
% coprime_imp_gcd_eq_1
thf(fact_6534_inj__mult__left,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [A2: A] :
( ( inj_on @ A @ A @ ( times_times @ A @ A2 ) @ ( top_top @ ( set @ A ) ) )
= ( A2
!= ( zero_zero @ A ) ) ) ) ).
% inj_mult_left
thf(fact_6535_coprime__0__left__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A] :
( ( algebr8660921524188924756oprime @ A @ ( zero_zero @ A ) @ A2 )
= ( dvd_dvd @ A @ A2 @ ( one_one @ A ) ) ) ) ).
% coprime_0_left_iff
thf(fact_6536_coprime__0__right__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A] :
( ( algebr8660921524188924756oprime @ A @ A2 @ ( zero_zero @ A ) )
= ( dvd_dvd @ A @ A2 @ ( one_one @ A ) ) ) ) ).
% coprime_0_right_iff
thf(fact_6537_coprime__mult__self__right__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( algebr8660921524188924756oprime @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) )
= ( ( dvd_dvd @ A @ C2 @ ( one_one @ A ) )
& ( algebr8660921524188924756oprime @ A @ A2 @ B2 ) ) ) ) ).
% coprime_mult_self_right_iff
thf(fact_6538_coprime__mult__self__left__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( algebr8660921524188924756oprime @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( ( dvd_dvd @ A @ C2 @ ( one_one @ A ) )
& ( algebr8660921524188924756oprime @ A @ A2 @ B2 ) ) ) ) ).
% coprime_mult_self_left_iff
thf(fact_6539_is__unit__gcd,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A2: A,B2: A] :
( ( dvd_dvd @ A @ ( gcd_gcd @ A @ A2 @ B2 ) @ ( one_one @ A ) )
= ( algebr8660921524188924756oprime @ A @ A2 @ B2 ) ) ) ).
% is_unit_gcd
thf(fact_6540_inj__divide__right,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A2: A] :
( ( inj_on @ A @ A
@ ^ [B3: A] : ( divide_divide @ A @ B3 @ A2 )
@ ( top_top @ ( set @ A ) ) )
= ( A2
!= ( zero_zero @ A ) ) ) ) ).
% inj_divide_right
thf(fact_6541_inj__on__apsnd,axiom,
! [A: $tType,C: $tType,B: $tType,F: B > C,A5: set @ B] :
( ( inj_on @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) @ ( product_apsnd @ B @ C @ A @ F )
@ ( product_Sigma @ A @ B @ ( top_top @ ( set @ A ) )
@ ^ [Uu: A] : A5 ) )
= ( inj_on @ B @ C @ F @ A5 ) ) ).
% inj_on_apsnd
thf(fact_6542_normalize__stable,axiom,
! [Q2: int,P3: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ Q2 )
=> ( ( algebr8660921524188924756oprime @ int @ P3 @ Q2 )
=> ( ( normalize @ ( product_Pair @ int @ int @ P3 @ Q2 ) )
= ( product_Pair @ int @ int @ P3 @ Q2 ) ) ) ) ).
% normalize_stable
thf(fact_6543_subset__image__inj,axiom,
! [A: $tType,B: $tType,S2: set @ A,F: B > A,T6: set @ B] :
( ( ord_less_eq @ ( set @ A ) @ S2 @ ( image2 @ B @ A @ F @ T6 ) )
= ( ? [U7: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ U7 @ T6 )
& ( inj_on @ B @ A @ F @ U7 )
& ( S2
= ( image2 @ B @ A @ F @ U7 ) ) ) ) ) ).
% subset_image_inj
thf(fact_6544_prod_Oimage__eq,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: B > A,A5: set @ B] :
( ( inj_on @ B @ A @ G @ A5 )
=> ( ( groups7121269368397514597t_prod @ A @ A
@ ^ [X: A] : X
@ ( image2 @ B @ A @ G @ A5 ) )
= ( groups7121269368397514597t_prod @ B @ A @ G @ A5 ) ) ) ) ).
% prod.image_eq
thf(fact_6545_sum_Oimage__eq,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: B > A,A5: set @ B] :
( ( inj_on @ B @ A @ G @ A5 )
=> ( ( groups7311177749621191930dd_sum @ A @ A
@ ^ [X: A] : X
@ ( image2 @ B @ A @ G @ A5 ) )
= ( groups7311177749621191930dd_sum @ B @ A @ G @ A5 ) ) ) ) ).
% sum.image_eq
thf(fact_6546_inj__Pair_I1_J,axiom,
! [B: $tType,A: $tType,C2: A > B,S2: set @ A] :
( inj_on @ A @ ( product_prod @ A @ B )
@ ^ [X: A] : ( product_Pair @ A @ B @ X @ ( C2 @ X ) )
@ S2 ) ).
% inj_Pair(1)
thf(fact_6547_inj__Pair_I2_J,axiom,
! [B: $tType,A: $tType,C2: A > B,S2: set @ A] :
( inj_on @ A @ ( product_prod @ B @ A )
@ ^ [X: A] : ( product_Pair @ B @ A @ ( C2 @ X ) @ X )
@ S2 ) ).
% inj_Pair(2)
thf(fact_6548_inj__on__convol__ident,axiom,
! [B: $tType,A: $tType,F: A > B,X8: set @ A] :
( inj_on @ A @ ( product_prod @ A @ B )
@ ^ [X: A] : ( product_Pair @ A @ B @ X @ ( F @ X ) )
@ X8 ) ).
% inj_on_convol_ident
thf(fact_6549_inj__on__diff__nat,axiom,
! [N5: set @ nat,K: nat] :
( ! [N2: nat] :
( ( member @ nat @ N2 @ N5 )
=> ( ord_less_eq @ nat @ K @ N2 ) )
=> ( inj_on @ nat @ nat
@ ^ [N4: nat] : ( minus_minus @ nat @ N4 @ K )
@ N5 ) ) ).
% inj_on_diff_nat
thf(fact_6550_linorder__inj__onI,axiom,
! [B: $tType,A: $tType] :
( ( order @ A )
=> ! [A5: set @ A,F: A > B] :
( ! [X3: A,Y2: A] :
( ( ord_less @ A @ X3 @ Y2 )
=> ( ( member @ A @ X3 @ A5 )
=> ( ( member @ A @ Y2 @ A5 )
=> ( ( F @ X3 )
!= ( F @ Y2 ) ) ) ) )
=> ( ! [X3: A,Y2: A] :
( ( member @ A @ X3 @ A5 )
=> ( ( member @ A @ Y2 @ A5 )
=> ( ( ord_less_eq @ A @ X3 @ Y2 )
| ( ord_less_eq @ A @ Y2 @ X3 ) ) ) )
=> ( inj_on @ A @ B @ F @ A5 ) ) ) ) ).
% linorder_inj_onI
thf(fact_6551_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_6552_sorted__list__of__set_Oinj__on,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( inj_on @ A @ A
@ ^ [X: A] : X
@ ( top_top @ ( set @ A ) ) ) ) ).
% sorted_list_of_set.inj_on
thf(fact_6553_inj__fun,axiom,
! [B: $tType,C: $tType,A: $tType,F: A > B] :
( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
=> ( inj_on @ A @ ( C > B )
@ ^ [X: A,Y4: C] : ( F @ X )
@ ( top_top @ ( set @ A ) ) ) ) ).
% inj_fun
thf(fact_6554_inj__add__left,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [A2: A] : ( inj_on @ A @ A @ ( plus_plus @ A @ A2 ) @ ( top_top @ ( set @ A ) ) ) ) ).
% inj_add_left
thf(fact_6555_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_6556_inj__diff__right,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A2: A] :
( inj_on @ A @ A
@ ^ [B3: A] : ( minus_minus @ A @ B3 @ A2 )
@ ( top_top @ ( set @ A ) ) ) ) ).
% inj_diff_right
thf(fact_6557_finite__Collect,axiom,
! [A: $tType,B: $tType,S2: set @ A,F: B > A] :
( ( finite_finite2 @ A @ S2 )
=> ( ( inj_on @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
=> ( finite_finite2 @ B
@ ( collect @ B
@ ^ [A3: B] : ( member @ A @ ( F @ A3 ) @ S2 ) ) ) ) ) ).
% finite_Collect
thf(fact_6558_finite__inverse__image,axiom,
! [A: $tType,B: $tType,A5: set @ A,F: B > A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( inj_on @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
=> ( finite_finite2 @ B
@ ( collect @ B
@ ^ [J3: B] : ( member @ A @ ( F @ J3 ) @ A5 ) ) ) ) ) ).
% finite_inverse_image
thf(fact_6559_prod__coprime__right,axiom,
! [A: $tType,B: $tType] :
( ( semiring_gcd @ A )
=> ! [A5: set @ B,A2: A,F: B > A] :
( ! [I3: B] :
( ( member @ B @ I3 @ A5 )
=> ( algebr8660921524188924756oprime @ A @ A2 @ ( F @ I3 ) ) )
=> ( algebr8660921524188924756oprime @ A @ A2 @ ( groups7121269368397514597t_prod @ B @ A @ F @ A5 ) ) ) ) ).
% prod_coprime_right
thf(fact_6560_prod__coprime__left,axiom,
! [B: $tType,A: $tType] :
( ( semiring_gcd @ A )
=> ! [A5: set @ B,F: B > A,A2: A] :
( ! [I3: B] :
( ( member @ B @ I3 @ A5 )
=> ( algebr8660921524188924756oprime @ A @ ( F @ I3 ) @ A2 ) )
=> ( algebr8660921524188924756oprime @ A @ ( groups7121269368397514597t_prod @ B @ A @ F @ A5 ) @ A2 ) ) ) ).
% prod_coprime_left
thf(fact_6561_coprime__diff__one__left,axiom,
! [A: $tType] :
( ( ring_gcd @ A )
=> ! [A2: A] : ( algebr8660921524188924756oprime @ A @ ( minus_minus @ A @ A2 @ ( one_one @ A ) ) @ A2 ) ) ).
% coprime_diff_one_left
thf(fact_6562_coprime__doff__one__right,axiom,
! [A: $tType] :
( ( ring_gcd @ A )
=> ! [A2: A] : ( algebr8660921524188924756oprime @ A @ A2 @ ( minus_minus @ A @ A2 @ ( one_one @ A ) ) ) ) ).
% coprime_doff_one_right
thf(fact_6563_gcd__mult__left__left__cancel,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [B2: A,C2: A,A2: A] :
( ( algebr8660921524188924756oprime @ A @ B2 @ C2 )
=> ( ( gcd_gcd @ A @ ( times_times @ A @ C2 @ A2 ) @ B2 )
= ( gcd_gcd @ A @ A2 @ B2 ) ) ) ) ).
% gcd_mult_left_left_cancel
thf(fact_6564_gcd__mult__left__right__cancel,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [B2: A,C2: A,A2: A] :
( ( algebr8660921524188924756oprime @ A @ B2 @ C2 )
=> ( ( gcd_gcd @ A @ ( times_times @ A @ A2 @ C2 ) @ B2 )
= ( gcd_gcd @ A @ A2 @ B2 ) ) ) ) ).
% gcd_mult_left_right_cancel
thf(fact_6565_gcd__mult__right__left__cancel,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( algebr8660921524188924756oprime @ A @ A2 @ C2 )
=> ( ( gcd_gcd @ A @ A2 @ ( times_times @ A @ C2 @ B2 ) )
= ( gcd_gcd @ A @ A2 @ B2 ) ) ) ) ).
% gcd_mult_right_left_cancel
thf(fact_6566_gcd__mult__right__right__cancel,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( algebr8660921524188924756oprime @ A @ A2 @ C2 )
=> ( ( gcd_gcd @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) )
= ( gcd_gcd @ A @ A2 @ B2 ) ) ) ) ).
% gcd_mult_right_right_cancel
thf(fact_6567_gcd__eq__1__imp__coprime,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A2: A,B2: A] :
( ( ( gcd_gcd @ A @ A2 @ B2 )
= ( one_one @ A ) )
=> ( algebr8660921524188924756oprime @ A @ A2 @ B2 ) ) ) ).
% gcd_eq_1_imp_coprime
thf(fact_6568_coprime__iff__gcd__eq__1,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ( ( algebr8660921524188924756oprime @ A )
= ( ^ [A3: A,B3: A] :
( ( gcd_gcd @ A @ A3 @ B3 )
= ( one_one @ A ) ) ) ) ) ).
% coprime_iff_gcd_eq_1
thf(fact_6569_coprime__crossproduct__int,axiom,
! [A2: int,D2: int,B2: int,C2: int] :
( ( algebr8660921524188924756oprime @ int @ A2 @ D2 )
=> ( ( algebr8660921524188924756oprime @ int @ B2 @ C2 )
=> ( ( ( times_times @ int @ ( abs_abs @ int @ A2 ) @ ( abs_abs @ int @ C2 ) )
= ( times_times @ int @ ( abs_abs @ int @ B2 ) @ ( abs_abs @ int @ D2 ) ) )
= ( ( ( abs_abs @ int @ A2 )
= ( abs_abs @ int @ B2 ) )
& ( ( abs_abs @ int @ C2 )
= ( abs_abs @ int @ D2 ) ) ) ) ) ) ).
% coprime_crossproduct_int
thf(fact_6570_coprimeI,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,B2: A] :
( ! [C4: A] :
( ( dvd_dvd @ A @ C4 @ A2 )
=> ( ( dvd_dvd @ A @ C4 @ B2 )
=> ( dvd_dvd @ A @ C4 @ ( one_one @ A ) ) ) )
=> ( algebr8660921524188924756oprime @ A @ A2 @ B2 ) ) ) ).
% coprimeI
thf(fact_6571_coprime__def,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ( ( algebr8660921524188924756oprime @ A )
= ( ^ [A3: A,B3: A] :
! [C5: A] :
( ( dvd_dvd @ A @ C5 @ A3 )
=> ( ( dvd_dvd @ A @ C5 @ B3 )
=> ( dvd_dvd @ A @ C5 @ ( one_one @ A ) ) ) ) ) ) ) ).
% coprime_def
thf(fact_6572_not__coprimeE,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,B2: A] :
( ~ ( algebr8660921524188924756oprime @ A @ A2 @ B2 )
=> ~ ! [C4: A] :
( ( dvd_dvd @ A @ C4 @ A2 )
=> ( ( dvd_dvd @ A @ C4 @ B2 )
=> ( dvd_dvd @ A @ C4 @ ( one_one @ A ) ) ) ) ) ) ).
% not_coprimeE
thf(fact_6573_not__coprimeI,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( dvd_dvd @ A @ C2 @ A2 )
=> ( ( dvd_dvd @ A @ C2 @ B2 )
=> ( ~ ( dvd_dvd @ A @ C2 @ ( one_one @ A ) )
=> ~ ( algebr8660921524188924756oprime @ A @ A2 @ B2 ) ) ) ) ) ).
% not_coprimeI
thf(fact_6574_coprime__absorb__left,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [X2: A,Y: A] :
( ( dvd_dvd @ A @ X2 @ Y )
=> ( ( algebr8660921524188924756oprime @ A @ X2 @ Y )
= ( dvd_dvd @ A @ X2 @ ( one_one @ A ) ) ) ) ) ).
% coprime_absorb_left
thf(fact_6575_coprime__imp__coprime,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C2: A,D2: A,A2: A,B2: A] :
( ( algebr8660921524188924756oprime @ A @ C2 @ D2 )
=> ( ! [E4: A] :
( ~ ( dvd_dvd @ A @ E4 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ E4 @ A2 )
=> ( ( dvd_dvd @ A @ E4 @ B2 )
=> ( dvd_dvd @ A @ E4 @ C2 ) ) ) )
=> ( ! [E4: A] :
( ~ ( dvd_dvd @ A @ E4 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ E4 @ A2 )
=> ( ( dvd_dvd @ A @ E4 @ B2 )
=> ( dvd_dvd @ A @ E4 @ D2 ) ) ) )
=> ( algebr8660921524188924756oprime @ A @ A2 @ B2 ) ) ) ) ) ).
% coprime_imp_coprime
thf(fact_6576_coprime__absorb__right,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [Y: A,X2: A] :
( ( dvd_dvd @ A @ Y @ X2 )
=> ( ( algebr8660921524188924756oprime @ A @ X2 @ Y )
= ( dvd_dvd @ A @ Y @ ( one_one @ A ) ) ) ) ) ).
% coprime_absorb_right
thf(fact_6577_coprime__common__divisor,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( algebr8660921524188924756oprime @ A @ A2 @ B2 )
=> ( ( dvd_dvd @ A @ C2 @ A2 )
=> ( ( dvd_dvd @ A @ C2 @ B2 )
=> ( dvd_dvd @ A @ C2 @ ( one_one @ A ) ) ) ) ) ) ).
% coprime_common_divisor
thf(fact_6578_is__unit__left__imp__coprime,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,B2: A] :
( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
=> ( algebr8660921524188924756oprime @ A @ A2 @ B2 ) ) ) ).
% is_unit_left_imp_coprime
thf(fact_6579_is__unit__right__imp__coprime,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A2: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( algebr8660921524188924756oprime @ A @ A2 @ B2 ) ) ) ).
% is_unit_right_imp_coprime
thf(fact_6580_divides__mult,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( dvd_dvd @ A @ A2 @ C2 )
=> ( ( dvd_dvd @ A @ B2 @ C2 )
=> ( ( algebr8660921524188924756oprime @ A @ A2 @ B2 )
=> ( dvd_dvd @ A @ ( times_times @ A @ A2 @ B2 ) @ C2 ) ) ) ) ) ).
% divides_mult
thf(fact_6581_coprime__dvd__mult__left__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( algebr8660921524188924756oprime @ A @ A2 @ C2 )
=> ( ( dvd_dvd @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) )
= ( dvd_dvd @ A @ A2 @ B2 ) ) ) ) ).
% coprime_dvd_mult_left_iff
thf(fact_6582_coprime__dvd__mult__right__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( algebr8660921524188924756oprime @ A @ A2 @ C2 )
=> ( ( dvd_dvd @ A @ A2 @ ( times_times @ A @ C2 @ B2 ) )
= ( dvd_dvd @ A @ A2 @ B2 ) ) ) ) ).
% coprime_dvd_mult_right_iff
thf(fact_6583_inj__on__mult,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A2: A,A5: set @ A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( inj_on @ A @ A @ ( times_times @ A @ A2 ) @ A5 ) ) ) ).
% inj_on_mult
thf(fact_6584_inj__on__of__nat,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [N5: set @ nat] : ( inj_on @ nat @ A @ ( semiring_1_of_nat @ A ) @ N5 ) ) ).
% inj_on_of_nat
thf(fact_6585_inj__on__id2,axiom,
! [A: $tType,A5: set @ A] :
( inj_on @ A @ A
@ ^ [X: A] : X
@ A5 ) ).
% inj_on_id2
thf(fact_6586_inj__on__add_H,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [A2: A,A5: set @ A] :
( inj_on @ A @ A
@ ^ [B3: A] : ( plus_plus @ A @ B3 @ A2 )
@ A5 ) ) ).
% inj_on_add'
thf(fact_6587_inj__on__add,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [A2: A,A5: set @ A] : ( inj_on @ A @ A @ ( plus_plus @ A @ A2 ) @ A5 ) ) ).
% inj_on_add
thf(fact_6588_inj__Suc,axiom,
! [N5: set @ nat] : ( inj_on @ nat @ nat @ suc @ N5 ) ).
% inj_Suc
thf(fact_6589_coprime__1__left,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A] : ( algebr8660921524188924756oprime @ A @ ( one_one @ A ) @ A2 ) ) ).
% coprime_1_left
thf(fact_6590_coprime__1__right,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A] : ( algebr8660921524188924756oprime @ A @ A2 @ ( one_one @ A ) ) ) ).
% coprime_1_right
thf(fact_6591_coprime__add__one__right,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A2: A] : ( algebr8660921524188924756oprime @ A @ A2 @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) ) ) ).
% coprime_add_one_right
thf(fact_6592_coprime__add__one__left,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A2: A] : ( algebr8660921524188924756oprime @ A @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) @ A2 ) ) ).
% coprime_add_one_left
thf(fact_6593_mult__mod__cancel__right,axiom,
! [A: $tType] :
( ( ( euclid8851590272496341667cancel @ A )
& ( semiring_gcd @ A ) )
=> ! [A2: A,N: A,M: A,B2: A] :
( ( ( modulo_modulo @ A @ ( times_times @ A @ A2 @ N ) @ M )
= ( modulo_modulo @ A @ ( times_times @ A @ B2 @ N ) @ M ) )
=> ( ( algebr8660921524188924756oprime @ A @ M @ N )
=> ( ( modulo_modulo @ A @ A2 @ M )
= ( modulo_modulo @ A @ B2 @ M ) ) ) ) ) ).
% mult_mod_cancel_right
thf(fact_6594_mult__mod__cancel__left,axiom,
! [A: $tType] :
( ( ( euclid8851590272496341667cancel @ A )
& ( semiring_gcd @ A ) )
=> ! [N: A,A2: A,M: A,B2: A] :
( ( ( modulo_modulo @ A @ ( times_times @ A @ N @ A2 ) @ M )
= ( modulo_modulo @ A @ ( times_times @ A @ N @ B2 ) @ M ) )
=> ( ( algebr8660921524188924756oprime @ A @ M @ N )
=> ( ( modulo_modulo @ A @ A2 @ M )
= ( modulo_modulo @ A @ B2 @ M ) ) ) ) ) ).
% mult_mod_cancel_left
thf(fact_6595_finite__inverse__image__gen,axiom,
! [A: $tType,B: $tType,A5: set @ A,F: B > A,D3: set @ B] :
( ( finite_finite2 @ A @ A5 )
=> ( ( inj_on @ B @ A @ F @ D3 )
=> ( finite_finite2 @ B
@ ( collect @ B
@ ^ [J3: B] :
( ( member @ B @ J3 @ D3 )
& ( member @ A @ ( F @ J3 ) @ A5 ) ) ) ) ) ) ).
% finite_inverse_image_gen
thf(fact_6596_inj__fn,axiom,
! [A: $tType,F: A > A,N: nat] :
( ( inj_on @ A @ A @ F @ ( top_top @ ( set @ A ) ) )
=> ( inj_on @ A @ A @ ( compow @ ( A > A ) @ N @ F ) @ ( top_top @ ( set @ A ) ) ) ) ).
% inj_fn
thf(fact_6597_map__prod__inj__on,axiom,
! [D: $tType,B: $tType,C: $tType,A: $tType,F: A > B,A5: set @ A,G: C > D,B5: set @ C] :
( ( inj_on @ A @ B @ F @ A5 )
=> ( ( inj_on @ C @ D @ G @ B5 )
=> ( inj_on @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) @ ( product_map_prod @ A @ B @ C @ D @ F @ G )
@ ( product_Sigma @ A @ C @ A5
@ ^ [Uu: A] : B5 ) ) ) ) ).
% map_prod_inj_on
thf(fact_6598_inj__singleton,axiom,
! [A: $tType,A5: set @ A] :
( inj_on @ A @ ( set @ A )
@ ^ [X: A] : ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) )
@ A5 ) ).
% inj_singleton
thf(fact_6599_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 )
@ ^ [I2: A,J3: B] : ( product_Pair @ B @ A @ J3 @ I2 ) )
@ A5 ) ).
% swap_inj_on
thf(fact_6600_arg__min__inj__eq,axiom,
! [B: $tType,A: $tType] :
( ( order @ B )
=> ! [F: A > B,P: A > $o,A2: A] :
( ( inj_on @ A @ B @ F @ ( collect @ A @ P ) )
=> ( ( P @ A2 )
=> ( ! [Y2: A] :
( ( P @ Y2 )
=> ( ord_less_eq @ B @ ( F @ A2 ) @ ( F @ Y2 ) ) )
=> ( ( lattices_ord_arg_min @ A @ B @ F @ P )
= A2 ) ) ) ) ) ).
% arg_min_inj_eq
thf(fact_6601_inj__split__Cons,axiom,
! [A: $tType,X8: 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,N4: A] : ( cons @ A @ N4 @ Xs3 ) )
@ X8 ) ).
% inj_split_Cons
thf(fact_6602_inj__on__iff__surj,axiom,
! [A: $tType,B: $tType,A5: set @ A,A11: set @ B] :
( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ? [F2: A > B] :
( ( inj_on @ A @ B @ F2 @ A5 )
& ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F2 @ A5 ) @ A11 ) ) )
= ( ? [G2: B > A] :
( ( image2 @ B @ A @ G2 @ A11 )
= A5 ) ) ) ) ).
% inj_on_iff_surj
thf(fact_6603_invertible__coprime,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ( modulo_modulo @ A @ ( times_times @ A @ A2 @ B2 ) @ C2 )
= ( one_one @ A ) )
=> ( algebr8660921524188924756oprime @ A @ A2 @ C2 ) ) ) ).
% invertible_coprime
thf(fact_6604_gcd__coprime__exists,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A2: A,B2: A] :
( ( ( gcd_gcd @ A @ A2 @ B2 )
!= ( zero_zero @ A ) )
=> ? [A16: A,B7: A] :
( ( A2
= ( times_times @ A @ A16 @ ( gcd_gcd @ A @ A2 @ B2 ) ) )
& ( B2
= ( times_times @ A @ B7 @ ( gcd_gcd @ A @ A2 @ B2 ) ) )
& ( algebr8660921524188924756oprime @ A @ A16 @ B7 ) ) ) ) ).
% gcd_coprime_exists
thf(fact_6605_gcd__coprime,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A2: A,B2: A,A6: A,B6: A] :
( ( ( gcd_gcd @ A @ A2 @ B2 )
!= ( zero_zero @ A ) )
=> ( ( A2
= ( times_times @ A @ A6 @ ( gcd_gcd @ A @ A2 @ B2 ) ) )
=> ( ( B2
= ( times_times @ A @ B6 @ ( gcd_gcd @ A @ A2 @ B2 ) ) )
=> ( algebr8660921524188924756oprime @ A @ A6 @ B6 ) ) ) ) ) ).
% gcd_coprime
thf(fact_6606_div__gcd__coprime,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A2: A,B2: A] :
( ( ( A2
!= ( zero_zero @ A ) )
| ( B2
!= ( zero_zero @ A ) ) )
=> ( algebr8660921524188924756oprime @ A @ ( divide_divide @ A @ A2 @ ( gcd_gcd @ A @ A2 @ B2 ) ) @ ( divide_divide @ A @ B2 @ ( gcd_gcd @ A @ A2 @ B2 ) ) ) ) ) ).
% div_gcd_coprime
thf(fact_6607_Rat__cases,axiom,
! [Q2: rat] :
~ ! [A4: int,B4: int] :
( ( Q2
= ( fract @ A4 @ B4 ) )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B4 )
=> ~ ( algebr8660921524188924756oprime @ int @ A4 @ B4 ) ) ) ).
% Rat_cases
thf(fact_6608_Rat__induct,axiom,
! [P: rat > $o,Q2: rat] :
( ! [A4: int,B4: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B4 )
=> ( ( algebr8660921524188924756oprime @ int @ A4 @ B4 )
=> ( P @ ( fract @ A4 @ B4 ) ) ) )
=> ( P @ Q2 ) ) ).
% Rat_induct
thf(fact_6609_coprime__common__divisor__int,axiom,
! [A2: int,B2: int,X2: int] :
( ( algebr8660921524188924756oprime @ int @ A2 @ B2 )
=> ( ( dvd_dvd @ int @ X2 @ A2 )
=> ( ( dvd_dvd @ int @ X2 @ B2 )
=> ( ( abs_abs @ int @ X2 )
= ( one_one @ int ) ) ) ) ) ).
% coprime_common_divisor_int
thf(fact_6610_inj__graph,axiom,
! [B: $tType,A: $tType] :
( inj_on @ ( A > B ) @ ( set @ ( product_prod @ A @ B ) )
@ ^ [F2: A > B] :
( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [X: A,Y4: B] :
( Y4
= ( F2 @ X ) ) ) )
@ ( top_top @ ( set @ ( A > B ) ) ) ) ).
% inj_graph
thf(fact_6611_inj__on__UNION__chain,axiom,
! [C: $tType,B: $tType,A: $tType,I5: set @ A,A5: A > ( set @ B ),F: 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 @ F @ ( A5 @ I3 ) ) )
=> ( inj_on @ B @ C @ F @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ A5 @ I5 ) ) ) ) ) ).
% inj_on_UNION_chain
thf(fact_6612_inj__on__filter__key__eq,axiom,
! [B: $tType,A: $tType,F: A > B,Y: A,Xs: list @ A] :
( ( inj_on @ A @ B @ F @ ( insert @ A @ Y @ ( set2 @ A @ Xs ) ) )
=> ( ( filter2 @ A
@ ^ [X: A] :
( ( F @ Y )
= ( F @ X ) )
@ Xs )
= ( filter2 @ A
@ ( ^ [Y3: A,Z: A] : Y3 = Z
@ Y )
@ Xs ) ) ) ).
% inj_on_filter_key_eq
thf(fact_6613_inj__on__INTER,axiom,
! [C: $tType,B: $tType,A: $tType,I5: set @ A,F: B > C,A5: A > ( set @ B )] :
( ( I5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [I3: A] :
( ( member @ A @ I3 @ I5 )
=> ( inj_on @ B @ C @ F @ ( A5 @ I3 ) ) )
=> ( inj_on @ B @ C @ F @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ A5 @ I5 ) ) ) ) ) ).
% inj_on_INTER
thf(fact_6614_finite__imp__nat__seg__image__inj__on,axiom,
! [A: $tType,A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ? [N2: nat,F5: nat > A] :
( ( A5
= ( image2 @ nat @ A @ F5
@ ( collect @ nat
@ ^ [I2: nat] : ( ord_less @ nat @ I2 @ N2 ) ) ) )
& ( inj_on @ nat @ A @ F5
@ ( collect @ nat
@ ^ [I2: nat] : ( ord_less @ nat @ I2 @ N2 ) ) ) ) ) ).
% finite_imp_nat_seg_image_inj_on
thf(fact_6615_finite__imp__inj__to__nat__seg_H,axiom,
! [A: $tType,A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ~ ! [F5: A > nat] :
( ? [N2: nat] :
( ( image2 @ A @ nat @ F5 @ A5 )
= ( collect @ nat
@ ^ [I2: nat] : ( ord_less @ nat @ I2 @ N2 ) ) )
=> ~ ( inj_on @ A @ nat @ F5 @ A5 ) ) ) ).
% finite_imp_inj_to_nat_seg'
thf(fact_6616_finite__imp__inj__to__nat__seg,axiom,
! [A: $tType,A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ? [F5: A > nat,N2: nat] :
( ( ( image2 @ A @ nat @ F5 @ A5 )
= ( collect @ nat
@ ^ [I2: nat] : ( ord_less @ nat @ I2 @ N2 ) ) )
& ( inj_on @ A @ nat @ F5 @ A5 ) ) ) ).
% finite_imp_inj_to_nat_seg
thf(fact_6617_set__to__map__inverse,axiom,
! [B: $tType,A: $tType,S2: set @ ( product_prod @ A @ B )] :
( ( inj_on @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ S2 )
=> ( ( map_to_set @ A @ B @ ( set_to_map @ A @ B @ S2 ) )
= S2 ) ) ).
% set_to_map_inverse
thf(fact_6618_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_6619_infinite__countable__subset,axiom,
! [A: $tType,S2: set @ A] :
( ~ ( finite_finite2 @ A @ S2 )
=> ? [F5: nat > A] :
( ( inj_on @ nat @ A @ F5 @ ( top_top @ ( set @ nat ) ) )
& ( ord_less_eq @ ( set @ A ) @ ( image2 @ nat @ A @ F5 @ ( top_top @ ( set @ nat ) ) ) @ S2 ) ) ) ).
% infinite_countable_subset
thf(fact_6620_infinite__iff__countable__subset,axiom,
! [A: $tType,S2: set @ A] :
( ( ~ ( finite_finite2 @ A @ S2 ) )
= ( ? [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 ) ) ) @ S2 ) ) ) ) ).
% infinite_iff_countable_subset
thf(fact_6621_Rat__cases__nonzero,axiom,
! [Q2: rat] :
( ! [A4: int,B4: int] :
( ( Q2
= ( fract @ A4 @ B4 ) )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B4 )
=> ( ( A4
!= ( zero_zero @ int ) )
=> ~ ( algebr8660921524188924756oprime @ int @ A4 @ B4 ) ) ) )
=> ( Q2
= ( zero_zero @ rat ) ) ) ).
% Rat_cases_nonzero
thf(fact_6622_inj__on__disjoint__Un,axiom,
! [B: $tType,A: $tType,F: A > B,A5: set @ A,G: A > B,B5: set @ A] :
( ( inj_on @ A @ B @ F @ A5 )
=> ( ( inj_on @ A @ B @ G @ B5 )
=> ( ( ( inf_inf @ ( set @ B ) @ ( image2 @ A @ B @ F @ A5 ) @ ( image2 @ A @ B @ G @ B5 ) )
= ( bot_bot @ ( set @ B ) ) )
=> ( inj_on @ A @ B
@ ^ [X: A] : ( if @ B @ ( member @ A @ X @ A5 ) @ ( F @ X ) @ ( G @ X ) )
@ ( sup_sup @ ( set @ A ) @ A5 @ B5 ) ) ) ) ) ).
% inj_on_disjoint_Un
thf(fact_6623_set__to__map__simp,axiom,
! [B: $tType,A: $tType,S2: set @ ( product_prod @ A @ B ),K: A,V2: B] :
( ( inj_on @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ S2 )
=> ( ( ( set_to_map @ A @ B @ S2 @ K )
= ( some @ B @ V2 ) )
= ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ V2 ) @ S2 ) ) ) ).
% set_to_map_simp
thf(fact_6624_image__INT,axiom,
! [B: $tType,A: $tType,C: $tType,F: A > B,C6: set @ A,A5: set @ C,B5: C > ( set @ A ),J: C] :
( ( inj_on @ A @ B @ F @ C6 )
=> ( ! [X3: C] :
( ( member @ C @ X3 @ A5 )
=> ( ord_less_eq @ ( set @ A ) @ ( B5 @ X3 ) @ C6 ) )
=> ( ( member @ C @ J @ A5 )
=> ( ( image2 @ A @ B @ F @ ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ C @ ( set @ A ) @ B5 @ A5 ) ) )
= ( complete_Inf_Inf @ ( set @ B )
@ ( image2 @ C @ ( set @ B )
@ ^ [X: C] : ( image2 @ A @ B @ F @ ( B5 @ X ) )
@ A5 ) ) ) ) ) ) ).
% image_INT
thf(fact_6625_inj__on__funpow__least,axiom,
! [A: $tType,N: nat,F: A > A,S3: A] :
( ( ( compow @ ( A > A ) @ N @ F @ S3 )
= S3 )
=> ( ! [M4: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M4 )
=> ( ( ord_less @ nat @ M4 @ N )
=> ( ( compow @ ( A > A ) @ M4 @ F @ S3 )
!= S3 ) ) )
=> ( inj_on @ nat @ A
@ ^ [K2: nat] : ( compow @ ( A > A ) @ K2 @ F @ S3 )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% inj_on_funpow_least
thf(fact_6626_card__le__inj,axiom,
! [B: $tType,A: $tType,A5: set @ A,B5: set @ B] :
( ( finite_finite2 @ A @ A5 )
=> ( ( finite_finite2 @ B @ B5 )
=> ( ( ord_less_eq @ nat @ ( finite_card @ A @ A5 ) @ ( finite_card @ B @ B5 ) )
=> ? [F5: A > B] :
( ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F5 @ A5 ) @ B5 )
& ( inj_on @ A @ B @ F5 @ A5 ) ) ) ) ) ).
% card_le_inj
thf(fact_6627_card__inj__on__le,axiom,
! [A: $tType,B: $tType,F: A > B,A5: set @ A,B5: set @ B] :
( ( inj_on @ A @ B @ F @ A5 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F @ A5 ) @ B5 )
=> ( ( finite_finite2 @ B @ B5 )
=> ( ord_less_eq @ nat @ ( finite_card @ A @ A5 ) @ ( finite_card @ B @ B5 ) ) ) ) ) ).
% card_inj_on_le
thf(fact_6628_inj__on__iff__card__le,axiom,
! [A: $tType,B: $tType,A5: set @ A,B5: set @ B] :
( ( finite_finite2 @ A @ A5 )
=> ( ( finite_finite2 @ B @ B5 )
=> ( ( ? [F2: A > B] :
( ( inj_on @ A @ B @ F2 @ A5 )
& ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F2 @ A5 ) @ B5 ) ) )
= ( ord_less_eq @ nat @ ( finite_card @ A @ A5 ) @ ( finite_card @ B @ B5 ) ) ) ) ) ).
% inj_on_iff_card_le
thf(fact_6629_card__vimage__inj__on__le,axiom,
! [A: $tType,B: $tType,F: A > B,D3: set @ A,A5: set @ B] :
( ( inj_on @ A @ B @ F @ D3 )
=> ( ( finite_finite2 @ B @ A5 )
=> ( ord_less_eq @ nat @ ( finite_card @ A @ ( inf_inf @ ( set @ A ) @ ( vimage @ A @ B @ F @ A5 ) @ D3 ) ) @ ( finite_card @ B @ A5 ) ) ) ) ).
% card_vimage_inj_on_le
thf(fact_6630_quotient__of__unique,axiom,
! [R: rat] :
? [X3: product_prod @ int @ int] :
( ( R
= ( 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 ) )
& ! [Y5: product_prod @ int @ int] :
( ( ( R
= ( fract @ ( product_fst @ int @ int @ Y5 ) @ ( product_snd @ int @ int @ Y5 ) ) )
& ( ord_less @ int @ ( zero_zero @ int ) @ ( product_snd @ int @ int @ Y5 ) )
& ( algebr8660921524188924756oprime @ int @ ( product_fst @ int @ int @ Y5 ) @ ( product_snd @ int @ int @ Y5 ) ) )
=> ( Y5 = X3 ) ) ) ).
% quotient_of_unique
thf(fact_6631_Ex__inj__on__UNION__Sigma,axiom,
! [A: $tType,B: $tType,A5: B > ( set @ A ),I5: set @ B] :
? [F5: A > ( product_prod @ B @ A )] :
( ( inj_on @ A @ ( product_prod @ B @ A ) @ F5 @ ( 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 ) @ F5 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ A5 @ I5 ) ) ) @ ( product_Sigma @ B @ A @ I5 @ A5 ) ) ) ).
% Ex_inj_on_UNION_Sigma
thf(fact_6632_map__sorted__distinct__set__unique,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F: B > A,Xs: list @ B,Ys: list @ B] :
( ( inj_on @ B @ A @ F @ ( sup_sup @ ( set @ B ) @ ( set2 @ B @ Xs ) @ ( set2 @ B @ Ys ) ) )
=> ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F @ Xs ) )
=> ( ( distinct @ A @ ( map @ B @ A @ F @ Xs ) )
=> ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F @ Ys ) )
=> ( ( distinct @ A @ ( map @ B @ A @ F @ Ys ) )
=> ( ( ( set2 @ B @ Xs )
= ( set2 @ B @ Ys ) )
=> ( Xs = Ys ) ) ) ) ) ) ) ) ).
% map_sorted_distinct_set_unique
thf(fact_6633_sort__key__inj__key__eq,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ B,Ys: list @ B,F: B > A] :
( ( ( mset @ B @ Xs )
= ( mset @ B @ Ys ) )
=> ( ( inj_on @ B @ A @ F @ ( set2 @ B @ Xs ) )
=> ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F @ Ys ) )
=> ( ( linorder_sort_key @ B @ A @ F @ Xs )
= Ys ) ) ) ) ) ).
% sort_key_inj_key_eq
thf(fact_6634_sum__mult__sum__if__inj,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( semiring_0 @ B )
=> ! [F: A > B,G: C > B,A5: set @ A,B5: set @ C] :
( ( inj_on @ ( product_prod @ A @ C ) @ B
@ ( product_case_prod @ A @ C @ B
@ ^ [A3: A,B3: C] : ( times_times @ B @ ( F @ A3 ) @ ( G @ B3 ) ) )
@ ( product_Sigma @ A @ C @ A5
@ ^ [Uu: A] : B5 ) )
=> ( ( times_times @ B @ ( groups7311177749621191930dd_sum @ A @ B @ F @ A5 ) @ ( groups7311177749621191930dd_sum @ C @ B @ G @ B5 ) )
= ( groups7311177749621191930dd_sum @ B @ B @ ( id @ B )
@ ( collect @ B
@ ^ [Uu: B] :
? [A3: A,B3: C] :
( ( Uu
= ( times_times @ B @ ( F @ A3 ) @ ( G @ B3 ) ) )
& ( member @ A @ A3 @ A5 )
& ( member @ C @ B3 @ B5 ) ) ) ) ) ) ) ).
% sum_mult_sum_if_inj
thf(fact_6635_funpow__inj__finite,axiom,
! [A: $tType,P3: A > A,X2: A] :
( ( inj_on @ A @ A @ P3 @ ( top_top @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [Y4: A] :
? [N4: nat] :
( Y4
= ( compow @ ( A > A ) @ N4 @ P3 @ X2 ) ) ) )
=> ~ ! [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( compow @ ( A > A ) @ N2 @ P3 @ X2 )
!= X2 ) ) ) ) ).
% funpow_inj_finite
thf(fact_6636_inj__vimage__singleton,axiom,
! [B: $tType,A: $tType,F: A > B,A2: B] :
( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
=> ( ord_less_eq @ ( set @ A ) @ ( vimage @ A @ B @ F @ ( insert @ B @ A2 @ ( bot_bot @ ( set @ B ) ) ) )
@ ( insert @ A
@ ( the @ A
@ ^ [X: A] :
( ( F @ X )
= A2 ) )
@ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% inj_vimage_singleton
thf(fact_6637_inj__on__vimage__singleton,axiom,
! [B: $tType,A: $tType,F: A > B,A5: set @ A,A2: B] :
( ( inj_on @ A @ B @ F @ A5 )
=> ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ ( vimage @ A @ B @ F @ ( insert @ B @ A2 @ ( bot_bot @ ( set @ B ) ) ) ) @ A5 )
@ ( insert @ A
@ ( the @ A
@ ^ [X: A] :
( ( member @ A @ X @ A5 )
& ( ( F @ X )
= A2 ) ) )
@ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% inj_on_vimage_singleton
thf(fact_6638_inj__on__map__inv__f,axiom,
! [B: $tType,A: $tType,L: list @ A,A5: set @ A,F: A > B] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ L ) @ A5 )
=> ( ( inj_on @ A @ B @ F @ A5 )
=> ( ( map @ B @ A @ ( inv_on @ A @ B @ F @ A5 ) @ ( map @ A @ B @ F @ L ) )
= L ) ) ) ).
% inj_on_map_inv_f
thf(fact_6639_Rats__cases_H,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [X2: A] :
( ( member @ A @ X2 @ ( field_char_0_Rats @ A ) )
=> ~ ! [A4: int,B4: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B4 )
=> ( ( algebr8660921524188924756oprime @ int @ A4 @ B4 )
=> ( X2
!= ( divide_divide @ A @ ( ring_1_of_int @ A @ A4 ) @ ( ring_1_of_int @ A @ B4 ) ) ) ) ) ) ) ).
% Rats_cases'
thf(fact_6640_coprime__int__iff,axiom,
! [M: nat,N: nat] :
( ( algebr8660921524188924756oprime @ int @ ( semiring_1_of_nat @ int @ M ) @ ( semiring_1_of_nat @ int @ N ) )
= ( algebr8660921524188924756oprime @ nat @ M @ N ) ) ).
% coprime_int_iff
thf(fact_6641_inv__on__f__f,axiom,
! [B: $tType,A: $tType,F: A > B,A5: set @ A,X2: A] :
( ( inj_on @ A @ B @ F @ A5 )
=> ( ( member @ A @ X2 @ A5 )
=> ( ( inv_on @ A @ B @ F @ A5 @ ( F @ X2 ) )
= X2 ) ) ) ).
% inv_on_f_f
thf(fact_6642_coprime__nat__abs__left__iff,axiom,
! [K: int,N: nat] :
( ( algebr8660921524188924756oprime @ nat @ ( nat2 @ ( abs_abs @ int @ K ) ) @ N )
= ( algebr8660921524188924756oprime @ int @ K @ ( semiring_1_of_nat @ int @ N ) ) ) ).
% coprime_nat_abs_left_iff
thf(fact_6643_coprime__nat__abs__right__iff,axiom,
! [N: nat,K: int] :
( ( algebr8660921524188924756oprime @ nat @ N @ ( nat2 @ ( abs_abs @ int @ K ) ) )
= ( algebr8660921524188924756oprime @ int @ ( semiring_1_of_nat @ int @ N ) @ K ) ) ).
% coprime_nat_abs_right_iff
thf(fact_6644_coprime__common__divisor__nat,axiom,
! [A2: nat,B2: nat,X2: nat] :
( ( algebr8660921524188924756oprime @ nat @ A2 @ B2 )
=> ( ( dvd_dvd @ nat @ X2 @ A2 )
=> ( ( dvd_dvd @ nat @ X2 @ B2 )
=> ( X2
= ( one_one @ nat ) ) ) ) ) ).
% coprime_common_divisor_nat
thf(fact_6645_coprime__Suc__right__nat,axiom,
! [N: nat] : ( algebr8660921524188924756oprime @ nat @ N @ ( suc @ N ) ) ).
% coprime_Suc_right_nat
thf(fact_6646_coprime__Suc__left__nat,axiom,
! [N: nat] : ( algebr8660921524188924756oprime @ nat @ ( suc @ N ) @ N ) ).
% coprime_Suc_left_nat
thf(fact_6647_coprime__crossproduct__nat,axiom,
! [A2: nat,D2: nat,B2: nat,C2: nat] :
( ( algebr8660921524188924756oprime @ nat @ A2 @ D2 )
=> ( ( algebr8660921524188924756oprime @ nat @ B2 @ C2 )
=> ( ( ( times_times @ nat @ A2 @ C2 )
= ( times_times @ nat @ B2 @ D2 ) )
= ( ( A2 = B2 )
& ( C2 = D2 ) ) ) ) ) ).
% coprime_crossproduct_nat
thf(fact_6648_coprime__Suc__0__left,axiom,
! [N: nat] : ( algebr8660921524188924756oprime @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N ) ).
% coprime_Suc_0_left
thf(fact_6649_coprime__Suc__0__right,axiom,
! [N: nat] : ( algebr8660921524188924756oprime @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ).
% coprime_Suc_0_right
thf(fact_6650_Rats__0,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( member @ A @ ( zero_zero @ A ) @ ( field_char_0_Rats @ A ) ) ) ).
% Rats_0
thf(fact_6651_Rats__1,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( member @ A @ ( one_one @ A ) @ ( field_char_0_Rats @ A ) ) ) ).
% Rats_1
thf(fact_6652_Rats__add,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A2: A,B2: A] :
( ( member @ A @ A2 @ ( field_char_0_Rats @ A ) )
=> ( ( member @ A @ B2 @ ( field_char_0_Rats @ A ) )
=> ( member @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( field_char_0_Rats @ A ) ) ) ) ) ).
% Rats_add
thf(fact_6653_Rats__of__nat,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [N: nat] : ( member @ A @ ( semiring_1_of_nat @ A @ N ) @ ( field_char_0_Rats @ A ) ) ) ).
% Rats_of_nat
thf(fact_6654_inv__on__f__range,axiom,
! [A: $tType,B: $tType,Y: A,F: B > A,A5: set @ B] :
( ( member @ A @ Y @ ( image2 @ B @ A @ F @ A5 ) )
=> ( member @ B @ ( inv_on @ B @ A @ F @ A5 @ Y ) @ A5 ) ) ).
% inv_on_f_range
thf(fact_6655_f__inv__on__f,axiom,
! [B: $tType,A: $tType,Y: A,F: B > A,A5: set @ B] :
( ( member @ A @ Y @ ( image2 @ B @ A @ F @ A5 ) )
=> ( ( F @ ( inv_on @ B @ A @ F @ A5 @ Y ) )
= Y ) ) ).
% f_inv_on_f
thf(fact_6656_mult__inj__if__coprime__nat,axiom,
! [B: $tType,A: $tType,F: A > nat,A5: set @ A,G: B > nat,B5: set @ B] :
( ( inj_on @ A @ nat @ F @ A5 )
=> ( ( inj_on @ B @ nat @ G @ B5 )
=> ( ! [A4: A,B4: B] :
( ( member @ A @ A4 @ A5 )
=> ( ( member @ B @ B4 @ B5 )
=> ( algebr8660921524188924756oprime @ nat @ ( F @ A4 ) @ ( G @ B4 ) ) ) )
=> ( inj_on @ ( product_prod @ A @ B ) @ nat
@ ( product_case_prod @ A @ B @ nat
@ ^ [A3: A,B3: B] : ( times_times @ nat @ ( F @ A3 ) @ ( G @ B3 ) ) )
@ ( product_Sigma @ A @ B @ A5
@ ^ [Uu: A] : B5 ) ) ) ) ) ).
% mult_inj_if_coprime_nat
thf(fact_6657_coprime__diff__one__left__nat,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( algebr8660921524188924756oprime @ nat @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ N ) ) ).
% coprime_diff_one_left_nat
thf(fact_6658_coprime__diff__one__right__nat,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( algebr8660921524188924756oprime @ nat @ N @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ).
% coprime_diff_one_right_nat
thf(fact_6659_inv__on__def,axiom,
! [B: $tType,A: $tType] :
( ( inv_on @ A @ B )
= ( ^ [F2: A > B,A8: set @ A,X: B] :
( fChoice @ A
@ ^ [Y4: A] :
( ( member @ A @ Y4 @ A8 )
& ( ( F2 @ Y4 )
= X ) ) ) ) ) ).
% inv_on_def
thf(fact_6660_inj__on__apfst,axiom,
! [B: $tType,C: $tType,A: $tType,F: A > C,A5: set @ A] :
( ( inj_on @ ( product_prod @ A @ B ) @ ( product_prod @ C @ B ) @ ( product_apfst @ A @ C @ B @ F )
@ ( product_Sigma @ A @ B @ A5
@ ^ [Uu: A] : ( top_top @ ( set @ B ) ) ) )
= ( inj_on @ A @ C @ F @ A5 ) ) ).
% inj_on_apfst
thf(fact_6661_If__the__inv__into__f__f,axiom,
! [B: $tType,A: $tType,I: A,C6: set @ A,G: A > B,X2: A] :
( ( member @ A @ I @ C6 )
=> ( ( inj_on @ A @ B @ G @ C6 )
=> ( ( comp @ B @ A @ A
@ ^ [I2: B] : ( if @ A @ ( member @ B @ I2 @ ( image2 @ A @ B @ G @ C6 ) ) @ ( the_inv_into @ A @ B @ C6 @ G @ I2 ) @ X2 )
@ G
@ I )
= ( id @ A @ I ) ) ) ) ).
% If_the_inv_into_f_f
thf(fact_6662_the__inv__into__def,axiom,
! [B: $tType,A: $tType] :
( ( the_inv_into @ A @ B )
= ( ^ [A8: set @ A,F2: A > B,X: B] :
( the @ A
@ ^ [Y4: A] :
( ( member @ A @ Y4 @ A8 )
& ( ( F2 @ Y4 )
= X ) ) ) ) ) ).
% the_inv_into_def
thf(fact_6663_If__the__inv__into__in__Func,axiom,
! [B: $tType,A: $tType,G: A > B,C6: set @ A,B5: set @ A,X2: A] :
( ( inj_on @ A @ B @ G @ C6 )
=> ( ( ord_less_eq @ ( set @ A ) @ C6 @ ( sup_sup @ ( set @ A ) @ B5 @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( member @ ( B > A )
@ ^ [I2: B] : ( if @ A @ ( member @ B @ I2 @ ( image2 @ A @ B @ G @ C6 ) ) @ ( the_inv_into @ A @ B @ C6 @ G @ I2 ) @ X2 )
@ ( bNF_Wellorder_Func @ B @ A @ ( top_top @ ( set @ B ) ) @ ( sup_sup @ ( set @ A ) @ B5 @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% If_the_inv_into_in_Func
thf(fact_6664_card__quotient__disjoint,axiom,
! [A: $tType,A5: set @ A,R: set @ ( product_prod @ A @ A )] :
( ( finite_finite2 @ A @ A5 )
=> ( ( inj_on @ A @ ( set @ ( set @ A ) )
@ ^ [X: A] : ( equiv_quotient @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) @ R )
@ A5 )
=> ( ( finite_card @ ( set @ A ) @ ( equiv_quotient @ A @ A5 @ R ) )
= ( finite_card @ A @ A5 ) ) ) ) ).
% card_quotient_disjoint
thf(fact_6665_quotient__diff1,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A5: set @ A,A2: A] :
( ( inj_on @ A @ ( set @ ( set @ A ) )
@ ^ [A3: A] : ( equiv_quotient @ A @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) @ R )
@ A5 )
=> ( ( member @ A @ A2 @ A5 )
=> ( ( equiv_quotient @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) @ R )
= ( minus_minus @ ( set @ ( set @ A ) ) @ ( equiv_quotient @ A @ A5 @ R ) @ ( equiv_quotient @ A @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) @ R ) ) ) ) ) ).
% quotient_diff1
thf(fact_6666_Func__empty,axiom,
! [B: $tType,A: $tType,B5: set @ B] :
( ( bNF_Wellorder_Func @ A @ B @ ( bot_bot @ ( set @ A ) ) @ B5 )
= ( insert @ ( A > B )
@ ^ [X: A] : ( undefined @ B )
@ ( bot_bot @ ( set @ ( A > B ) ) ) ) ) ).
% Func_empty
thf(fact_6667_finite__equiv__class,axiom,
! [A: $tType,A5: set @ A,R: set @ ( product_prod @ A @ A ),X8: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R
@ ( product_Sigma @ A @ A @ A5
@ ^ [Uu: A] : A5 ) )
=> ( ( member @ ( set @ A ) @ X8 @ ( equiv_quotient @ A @ A5 @ R ) )
=> ( finite_finite2 @ A @ X8 ) ) ) ) ).
% finite_equiv_class
thf(fact_6668_finite__quotient,axiom,
! [A: $tType,A5: set @ A,R: set @ ( product_prod @ A @ A )] :
( ( finite_finite2 @ A @ A5 )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R
@ ( product_Sigma @ A @ A @ A5
@ ^ [Uu: A] : A5 ) )
=> ( finite_finite2 @ ( set @ A ) @ ( equiv_quotient @ A @ A5 @ R ) ) ) ) ).
% finite_quotient
thf(fact_6669_quotient__def,axiom,
! [A: $tType] :
( ( equiv_quotient @ A )
= ( ^ [A8: set @ A,R2: set @ ( product_prod @ A @ A )] :
( complete_Sup_Sup @ ( set @ ( set @ A ) )
@ ( image2 @ A @ ( set @ ( set @ A ) )
@ ^ [X: A] : ( insert @ ( set @ A ) @ ( image @ A @ A @ R2 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ ( bot_bot @ ( set @ ( set @ A ) ) ) )
@ A8 ) ) ) ) ).
% quotient_def
thf(fact_6670_positive_Orsp,axiom,
( bNF_rel_fun @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ $o @ $o @ ratrel
@ ^ [Y3: $o,Z: $o] : Y3 = Z
@ ^ [X: product_prod @ int @ int] : ( ord_less @ int @ ( zero_zero @ int ) @ ( times_times @ int @ ( product_fst @ int @ int @ X ) @ ( product_snd @ int @ int @ X ) ) )
@ ^ [X: product_prod @ int @ int] : ( ord_less @ int @ ( zero_zero @ int ) @ ( times_times @ int @ ( product_fst @ int @ int @ X ) @ ( product_snd @ int @ int @ X ) ) ) ) ).
% positive.rsp
thf(fact_6671_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_6672_Image__UN,axiom,
! [A: $tType,B: $tType,C: $tType,R: set @ ( product_prod @ B @ A ),B5: C > ( set @ B ),A5: set @ C] :
( ( image @ B @ A @ R @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ C @ ( set @ B ) @ B5 @ A5 ) ) )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ C @ ( set @ A )
@ ^ [X: C] : ( image @ B @ A @ R @ ( B5 @ X ) )
@ A5 ) ) ) ).
% Image_UN
thf(fact_6673_id__transfer,axiom,
! [A: $tType,B: $tType,A5: A > B > $o] : ( bNF_rel_fun @ A @ B @ A @ B @ A5 @ A5 @ ( id @ A ) @ ( id @ B ) ) ).
% id_transfer
thf(fact_6674_finite__Image__subset,axiom,
! [A: $tType,B: $tType,A5: set @ ( product_prod @ B @ A ),B5: set @ B,C6: set @ ( product_prod @ B @ A )] :
( ( finite_finite2 @ A @ ( image @ B @ A @ A5 @ B5 ) )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ B @ A ) ) @ C6 @ A5 )
=> ( finite_finite2 @ A @ ( image @ B @ A @ C6 @ B5 ) ) ) ) ).
% finite_Image_subset
thf(fact_6675_transfer__rule__of__int,axiom,
! [A: $tType,B: $tType] :
( ( ( ring_1 @ B )
& ( ring_1 @ A ) )
=> ! [R3: A > B > $o] :
( ( R3 @ ( zero_zero @ A ) @ ( zero_zero @ B ) )
=> ( ( R3 @ ( one_one @ A ) @ ( one_one @ B ) )
=> ( ( bNF_rel_fun @ A @ B @ ( A > A ) @ ( B > B ) @ R3 @ ( bNF_rel_fun @ A @ B @ A @ B @ R3 @ R3 ) @ ( plus_plus @ A ) @ ( plus_plus @ B ) )
=> ( ( bNF_rel_fun @ A @ B @ A @ B @ R3 @ R3 @ ( uminus_uminus @ A ) @ ( uminus_uminus @ B ) )
=> ( bNF_rel_fun @ int @ int @ A @ B
@ ^ [Y3: int,Z: int] : Y3 = Z
@ R3
@ ( ring_1_of_int @ A )
@ ( ring_1_of_int @ B ) ) ) ) ) ) ) ).
% transfer_rule_of_int
thf(fact_6676_comp__transfer,axiom,
! [A: $tType,B: $tType,E: $tType,F7: $tType,D: $tType,C: $tType,B5: A > C > $o,C6: B > D > $o,A5: E > F7 > $o] : ( bNF_rel_fun @ ( A > B ) @ ( C > D ) @ ( ( E > A ) > E > B ) @ ( ( F7 > C ) > F7 > D ) @ ( bNF_rel_fun @ A @ C @ B @ D @ B5 @ C6 ) @ ( bNF_rel_fun @ ( E > A ) @ ( F7 > C ) @ ( E > B ) @ ( F7 > D ) @ ( bNF_rel_fun @ E @ F7 @ A @ C @ A5 @ B5 ) @ ( bNF_rel_fun @ E @ F7 @ B @ D @ A5 @ C6 ) ) @ ( comp @ A @ B @ E ) @ ( comp @ C @ D @ F7 ) ) ).
% comp_transfer
thf(fact_6677_If__transfer,axiom,
! [A: $tType,B: $tType,A5: A > B > $o] :
( bNF_rel_fun @ $o @ $o @ ( A > A > A ) @ ( B > B > B )
@ ^ [Y3: $o,Z: $o] : Y3 = Z
@ ( bNF_rel_fun @ A @ B @ ( A > A ) @ ( B > B ) @ A5 @ ( bNF_rel_fun @ A @ B @ A @ B @ A5 @ A5 ) )
@ ( if @ A )
@ ( if @ B ) ) ).
% If_transfer
thf(fact_6678_rel__fun__def__butlast,axiom,
! [B: $tType,D: $tType,C: $tType,E: $tType,F7: $tType,A: $tType,R3: A > B > $o,S2: C > E > $o,T6: D > F7 > $o,F: A > C > D,G: B > E > F7] :
( ( bNF_rel_fun @ A @ B @ ( C > D ) @ ( E > F7 ) @ R3 @ ( bNF_rel_fun @ C @ E @ D @ F7 @ S2 @ T6 ) @ F @ G )
= ( ! [X: A,Y4: B] :
( ( R3 @ X @ Y4 )
=> ( bNF_rel_fun @ C @ E @ D @ F7 @ S2 @ T6 @ ( F @ X ) @ ( G @ Y4 ) ) ) ) ) ).
% rel_fun_def_butlast
thf(fact_6679_power__transfer,axiom,
! [A: $tType,B: $tType] :
( ( ( power @ B )
& ( power @ A ) )
=> ! [R3: A > B > $o] :
( ( R3 @ ( one_one @ A ) @ ( one_one @ B ) )
=> ( ( bNF_rel_fun @ A @ B @ ( A > A ) @ ( B > B ) @ R3 @ ( bNF_rel_fun @ A @ B @ A @ B @ R3 @ R3 ) @ ( times_times @ A ) @ ( times_times @ B ) )
=> ( bNF_rel_fun @ A @ B @ ( nat > A ) @ ( nat > B ) @ R3
@ ( bNF_rel_fun @ nat @ nat @ A @ B
@ ^ [Y3: nat,Z: nat] : Y3 = Z
@ R3 )
@ ( power_power @ A )
@ ( power_power @ B ) ) ) ) ) ).
% power_transfer
thf(fact_6680_transfer__rule__numeral,axiom,
! [A: $tType,B: $tType] :
( ( ( monoid_add @ B )
& ( semiring_numeral @ B )
& ( monoid_add @ A )
& ( semiring_numeral @ A ) )
=> ! [R3: A > B > $o] :
( ( R3 @ ( zero_zero @ A ) @ ( zero_zero @ B ) )
=> ( ( R3 @ ( one_one @ A ) @ ( one_one @ B ) )
=> ( ( bNF_rel_fun @ A @ B @ ( A > A ) @ ( B > B ) @ R3 @ ( bNF_rel_fun @ A @ B @ A @ B @ R3 @ R3 ) @ ( plus_plus @ A ) @ ( plus_plus @ B ) )
=> ( bNF_rel_fun @ num @ num @ A @ B
@ ^ [Y3: num,Z: num] : Y3 = Z
@ R3
@ ( numeral_numeral @ A )
@ ( numeral_numeral @ B ) ) ) ) ) ) ).
% transfer_rule_numeral
thf(fact_6681_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_6682_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_6683_rtrancl__image__advance__rtrancl,axiom,
! [A: $tType,Q2: A,R3: set @ ( product_prod @ A @ A ),Q0: set @ A,X2: A] :
( ( member @ A @ Q2 @ ( image @ A @ A @ ( transitive_rtrancl @ A @ R3 ) @ Q0 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Q2 @ X2 ) @ ( transitive_rtrancl @ A @ R3 ) )
=> ( member @ A @ X2 @ ( image @ A @ A @ ( transitive_rtrancl @ A @ R3 ) @ Q0 ) ) ) ) ).
% rtrancl_image_advance_rtrancl
thf(fact_6684_rtrancl__image__advance,axiom,
! [A: $tType,Q2: A,R3: set @ ( product_prod @ A @ A ),Q0: set @ A,X2: A] :
( ( member @ A @ Q2 @ ( image @ A @ A @ ( transitive_rtrancl @ A @ R3 ) @ Q0 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Q2 @ X2 ) @ R3 )
=> ( member @ A @ X2 @ ( image @ A @ A @ ( transitive_rtrancl @ A @ R3 ) @ Q0 ) ) ) ) ).
% rtrancl_image_advance
thf(fact_6685_trancl__Image__unfold__right,axiom,
! [A: $tType,E6: set @ ( product_prod @ A @ A ),S2: set @ A] :
( ( image @ A @ A @ ( transitive_trancl @ A @ E6 ) @ S2 )
= ( image @ A @ A @ E6 @ ( image @ A @ A @ ( transitive_rtrancl @ A @ E6 ) @ S2 ) ) ) ).
% trancl_Image_unfold_right
thf(fact_6686_trancl__Image__unfold__left,axiom,
! [A: $tType,E6: set @ ( product_prod @ A @ A ),S2: set @ A] :
( ( image @ A @ A @ ( transitive_trancl @ A @ E6 ) @ S2 )
= ( image @ A @ A @ ( transitive_rtrancl @ A @ E6 ) @ ( image @ A @ A @ E6 @ S2 ) ) ) ).
% trancl_Image_unfold_left
thf(fact_6687_Image__empty__rtrancl__Image__id,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),V2: A] :
( ( ( image @ A @ A @ R3 @ ( insert @ A @ V2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( image @ A @ A @ ( transitive_rtrancl @ A @ R3 ) @ ( insert @ A @ V2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( insert @ A @ V2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Image_empty_rtrancl_Image_id
thf(fact_6688_Image__empty__trancl__Image__empty,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),V2: A] :
( ( ( image @ A @ A @ R3 @ ( insert @ A @ V2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( image @ A @ A @ ( transitive_trancl @ A @ R3 ) @ ( insert @ A @ V2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% Image_empty_trancl_Image_empty
thf(fact_6689_reachable__mono,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),R8: set @ ( product_prod @ A @ A ),X8: set @ A,X15: set @ A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ R8 )
=> ( ( ord_less_eq @ ( set @ A ) @ X8 @ X15 )
=> ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ ( transitive_rtrancl @ A @ R3 ) @ X8 ) @ ( image @ A @ A @ ( transitive_rtrancl @ A @ R8 ) @ X15 ) ) ) ) ).
% reachable_mono
thf(fact_6690_Image__subset__snd__image,axiom,
! [A: $tType,B: $tType,A5: set @ ( product_prod @ B @ A ),B5: set @ B] : ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ A5 @ B5 ) @ ( image2 @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) @ A5 ) ) ).
% Image_subset_snd_image
thf(fact_6691_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_6692_Image__singleton,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ B @ A ),A2: B] :
( ( image @ B @ A @ R @ ( insert @ B @ A2 @ ( bot_bot @ ( set @ B ) ) ) )
= ( collect @ A
@ ^ [B3: A] : ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ A2 @ B3 ) @ R ) ) ) ).
% Image_singleton
thf(fact_6693_Image__subset,axiom,
! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ B ),A5: set @ A,B5: set @ B,C6: set @ A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R
@ ( product_Sigma @ A @ B @ A5
@ ^ [Uu: A] : B5 ) )
=> ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ R @ C6 ) @ B5 ) ) ).
% Image_subset
thf(fact_6694_Image__INT__subset,axiom,
! [A: $tType,B: $tType,C: $tType,R: set @ ( product_prod @ B @ A ),B5: C > ( set @ B ),A5: set @ C] :
( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ R @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ C @ ( set @ B ) @ B5 @ A5 ) ) )
@ ( complete_Inf_Inf @ ( set @ A )
@ ( image2 @ C @ ( set @ A )
@ ^ [X: C] : ( image @ B @ A @ R @ ( B5 @ X ) )
@ A5 ) ) ) ).
% Image_INT_subset
thf(fact_6695_rtrancl__apply__insert,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),X2: A,S2: set @ A] :
( ( image @ A @ A @ ( transitive_rtrancl @ A @ R3 ) @ ( insert @ A @ X2 @ S2 ) )
= ( insert @ A @ X2 @ ( image @ A @ A @ ( transitive_rtrancl @ A @ R3 ) @ ( sup_sup @ ( set @ A ) @ S2 @ ( image @ A @ A @ R3 @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% rtrancl_apply_insert
thf(fact_6696_E__closed__restr__reach__cases,axiom,
! [A: $tType,U: A,V2: A,E6: set @ ( product_prod @ A @ A ),R3: 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 @ R3 ) @ R3 )
=> ( ~ ( member @ A @ V2 @ R3 )
=> ~ ( ~ ( member @ A @ U @ R3 )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ U @ V2 ) @ ( transitive_rtrancl @ A @ ( rel_restrict @ A @ E6 @ R3 ) ) ) ) ) ) ) ).
% E_closed_restr_reach_cases
thf(fact_6697_rel__restrict__tranclI,axiom,
! [A: $tType,X2: A,Y: A,E6: set @ ( product_prod @ A @ A ),R3: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y ) @ ( transitive_trancl @ A @ E6 ) )
=> ( ~ ( member @ A @ X2 @ R3 )
=> ( ~ ( member @ A @ Y @ R3 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ E6 @ R3 ) @ R3 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y ) @ ( transitive_trancl @ A @ ( rel_restrict @ A @ E6 @ R3 ) ) ) ) ) ) ) ).
% rel_restrict_tranclI
thf(fact_6698_Image__eq__UN,axiom,
! [A: $tType,B: $tType] :
( ( image @ B @ A )
= ( ^ [R2: set @ ( product_prod @ B @ A ),B8: set @ B] :
( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [Y4: B] : ( image @ B @ A @ R2 @ ( insert @ B @ Y4 @ ( bot_bot @ ( set @ B ) ) ) )
@ B8 ) ) ) ) ).
% Image_eq_UN
thf(fact_6699_Sigma__Image,axiom,
! [A: $tType,B: $tType,A5: set @ B,B5: B > ( set @ A ),X8: set @ B] :
( ( image @ B @ A @ ( product_Sigma @ B @ A @ A5 @ B5 ) @ X8 )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B5 @ ( inf_inf @ ( set @ B ) @ X8 @ A5 ) ) ) ) ).
% Sigma_Image
thf(fact_6700_UN__Image,axiom,
! [A: $tType,B: $tType,C: $tType,X8: C > ( set @ ( product_prod @ B @ A ) ),I5: set @ C,S2: set @ B] :
( ( image @ B @ A @ ( complete_Sup_Sup @ ( set @ ( product_prod @ B @ A ) ) @ ( image2 @ C @ ( set @ ( product_prod @ B @ A ) ) @ X8 @ I5 ) ) @ S2 )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ C @ ( set @ A )
@ ^ [I2: C] : ( image @ B @ A @ ( X8 @ I2 ) @ S2 )
@ I5 ) ) ) ).
% UN_Image
thf(fact_6701_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_6702_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_6703_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_6704_trancl__restrict__reachable,axiom,
! [A: $tType,U: A,V2: A,E6: set @ ( product_prod @ A @ A ),S2: 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 @ S2 ) @ S2 )
=> ( ( member @ A @ U @ S2 )
=> ( 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 @ S2
@ ^ [Uu: A] : S2 ) ) ) ) ) ) ) ).
% trancl_restrict_reachable
thf(fact_6705_of__rat_Orsp,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( bNF_rel_fun @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ A @ A @ ratrel
@ ^ [Y3: A,Z: A] : Y3 = Z
@ ^ [X: product_prod @ int @ int] : ( divide_divide @ A @ ( ring_1_of_int @ A @ ( product_fst @ int @ int @ X ) ) @ ( ring_1_of_int @ A @ ( product_snd @ int @ int @ X ) ) )
@ ^ [X: product_prod @ int @ int] : ( divide_divide @ A @ ( ring_1_of_int @ A @ ( product_fst @ int @ int @ X ) ) @ ( ring_1_of_int @ A @ ( product_snd @ int @ int @ X ) ) ) ) ) ).
% of_rat.rsp
thf(fact_6706_Image__fold,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ B ),S2: set @ A] :
( ( finite_finite2 @ ( product_prod @ A @ B ) @ R3 )
=> ( ( image @ A @ B @ R3 @ S2 )
= ( finite_fold @ ( product_prod @ A @ B ) @ ( set @ B )
@ ( product_case_prod @ A @ B @ ( ( set @ B ) > ( set @ B ) )
@ ^ [X: A,Y4: B,A8: set @ B] : ( if @ ( set @ B ) @ ( member @ A @ X @ S2 ) @ ( insert @ B @ Y4 @ A8 ) @ A8 ) )
@ ( bot_bot @ ( set @ B ) )
@ R3 ) ) ) ).
% Image_fold
thf(fact_6707_fun_Oin__rel,axiom,
! [B: $tType,A: $tType,D: $tType,R3: A > B > $o,A2: D > A,B2: D > B] :
( ( bNF_rel_fun @ D @ D @ A @ B
@ ^ [Y3: D,Z: D] : Y3 = Z
@ R3
@ A2
@ B2 )
= ( ? [Z3: D > ( product_prod @ A @ B )] :
( ( member @ ( D > ( product_prod @ A @ B ) ) @ Z3
@ ( collect @ ( D > ( product_prod @ A @ B ) )
@ ^ [X: D > ( product_prod @ A @ B )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( image2 @ D @ ( product_prod @ A @ B ) @ X @ ( top_top @ ( set @ D ) ) ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R3 ) ) ) ) )
& ( ( comp @ ( product_prod @ A @ B ) @ A @ D @ ( product_fst @ A @ B ) @ Z3 )
= A2 )
& ( ( comp @ ( product_prod @ A @ B ) @ B @ D @ ( product_snd @ A @ B ) @ Z3 )
= B2 ) ) ) ) ).
% fun.in_rel
thf(fact_6708_transfer__rule__of__nat,axiom,
! [A: $tType,B: $tType] :
( ( ( semiring_1 @ B )
& ( semiring_1 @ A ) )
=> ! [R3: A > B > $o] :
( ( R3 @ ( zero_zero @ A ) @ ( zero_zero @ B ) )
=> ( ( R3 @ ( one_one @ A ) @ ( one_one @ B ) )
=> ( ( bNF_rel_fun @ A @ B @ ( A > A ) @ ( B > B ) @ R3 @ ( bNF_rel_fun @ A @ B @ A @ B @ R3 @ R3 ) @ ( plus_plus @ A ) @ ( plus_plus @ B ) )
=> ( bNF_rel_fun @ nat @ nat @ A @ B
@ ^ [Y3: nat,Z: nat] : Y3 = Z
@ R3
@ ( semiring_1_of_nat @ A )
@ ( semiring_1_of_nat @ B ) ) ) ) ) ) ).
% transfer_rule_of_nat
thf(fact_6709_Fract_Orsp,axiom,
( bNF_rel_fun @ int @ int @ ( int > ( product_prod @ int @ int ) ) @ ( int > ( product_prod @ int @ int ) )
@ ^ [Y3: int,Z: int] : Y3 = Z
@ ( bNF_rel_fun @ int @ int @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int )
@ ^ [Y3: int,Z: int] : Y3 = Z
@ ratrel )
@ ^ [A3: int,B3: int] :
( if @ ( product_prod @ int @ int )
@ ( B3
= ( zero_zero @ int ) )
@ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
@ ( product_Pair @ int @ int @ A3 @ B3 ) )
@ ^ [A3: int,B3: int] :
( if @ ( product_prod @ int @ int )
@ ( B3
= ( zero_zero @ int ) )
@ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
@ ( product_Pair @ int @ int @ A3 @ B3 ) ) ) ).
% Fract.rsp
thf(fact_6710_less__eq__integer_Orsp,axiom,
( bNF_rel_fun @ int @ int @ ( int > $o ) @ ( int > $o )
@ ^ [Y3: int,Z: int] : Y3 = Z
@ ( bNF_rel_fun @ int @ int @ $o @ $o
@ ^ [Y3: int,Z: int] : Y3 = Z
@ ^ [Y3: $o,Z: $o] : Y3 = Z )
@ ( ord_less_eq @ int )
@ ( ord_less_eq @ int ) ) ).
% less_eq_integer.rsp
thf(fact_6711_less__eq__natural_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ ( nat > $o ) @ ( nat > $o )
@ ^ [Y3: nat,Z: nat] : Y3 = Z
@ ( bNF_rel_fun @ nat @ nat @ $o @ $o
@ ^ [Y3: nat,Z: nat] : Y3 = Z
@ ^ [Y3: $o,Z: $o] : Y3 = Z )
@ ( ord_less_eq @ nat )
@ ( ord_less_eq @ nat ) ) ).
% less_eq_natural.rsp
thf(fact_6712_uminus__integer_Orsp,axiom,
( bNF_rel_fun @ int @ int @ int @ int
@ ^ [Y3: int,Z: int] : Y3 = Z
@ ^ [Y3: int,Z: int] : Y3 = Z
@ ( uminus_uminus @ int )
@ ( uminus_uminus @ int ) ) ).
% uminus_integer.rsp
thf(fact_6713_minus__natural_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( nat > nat )
@ ^ [Y3: nat,Z: nat] : Y3 = Z
@ ( bNF_rel_fun @ nat @ nat @ nat @ nat
@ ^ [Y3: nat,Z: nat] : Y3 = Z
@ ^ [Y3: nat,Z: nat] : Y3 = Z )
@ ( minus_minus @ nat )
@ ( minus_minus @ nat ) ) ).
% minus_natural.rsp
thf(fact_6714_minus__integer_Orsp,axiom,
( bNF_rel_fun @ int @ int @ ( int > int ) @ ( int > int )
@ ^ [Y3: int,Z: int] : Y3 = Z
@ ( bNF_rel_fun @ int @ int @ int @ int
@ ^ [Y3: int,Z: int] : Y3 = Z
@ ^ [Y3: int,Z: int] : Y3 = Z )
@ ( minus_minus @ int )
@ ( minus_minus @ int ) ) ).
% minus_integer.rsp
thf(fact_6715_sub_Orsp,axiom,
( bNF_rel_fun @ num @ num @ ( num > int ) @ ( num > int )
@ ^ [Y3: num,Z: num] : Y3 = Z
@ ( bNF_rel_fun @ num @ num @ int @ int
@ ^ [Y3: num,Z: num] : Y3 = Z
@ ^ [Y3: int,Z: int] : Y3 = Z )
@ ^ [M2: num,N4: num] : ( minus_minus @ int @ ( numeral_numeral @ int @ M2 ) @ ( numeral_numeral @ int @ N4 ) )
@ ^ [M2: num,N4: num] : ( minus_minus @ int @ ( numeral_numeral @ int @ M2 ) @ ( numeral_numeral @ int @ N4 ) ) ) ).
% sub.rsp
thf(fact_6716_gcd__integer_Orsp,axiom,
( bNF_rel_fun @ int @ int @ ( int > int ) @ ( int > int )
@ ^ [Y3: int,Z: int] : Y3 = Z
@ ( bNF_rel_fun @ int @ int @ int @ int
@ ^ [Y3: int,Z: int] : Y3 = Z
@ ^ [Y3: int,Z: int] : Y3 = Z )
@ ( gcd_gcd @ int )
@ ( gcd_gcd @ int ) ) ).
% gcd_integer.rsp
thf(fact_6717_integer__of__natural_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ int @ int
@ ^ [Y3: nat,Z: nat] : Y3 = Z
@ ^ [Y3: int,Z: int] : Y3 = Z
@ ( semiring_1_of_nat @ int )
@ ( semiring_1_of_nat @ int ) ) ).
% integer_of_natural.rsp
thf(fact_6718_dup_Orsp,axiom,
( bNF_rel_fun @ int @ int @ int @ int
@ ^ [Y3: int,Z: int] : Y3 = Z
@ ^ [Y3: int,Z: int] : Y3 = Z
@ ^ [K2: int] : ( plus_plus @ int @ K2 @ K2 )
@ ^ [K2: int] : ( plus_plus @ int @ K2 @ K2 ) ) ).
% dup.rsp
thf(fact_6719_sgn__integer_Orsp,axiom,
( bNF_rel_fun @ int @ int @ int @ int
@ ^ [Y3: int,Z: int] : Y3 = Z
@ ^ [Y3: int,Z: int] : Y3 = Z
@ ( sgn_sgn @ int )
@ ( sgn_sgn @ int ) ) ).
% sgn_integer.rsp
thf(fact_6720_plus__natural_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( nat > nat )
@ ^ [Y3: nat,Z: nat] : Y3 = Z
@ ( bNF_rel_fun @ nat @ nat @ nat @ nat
@ ^ [Y3: nat,Z: nat] : Y3 = Z
@ ^ [Y3: nat,Z: nat] : Y3 = Z )
@ ( plus_plus @ nat )
@ ( plus_plus @ nat ) ) ).
% plus_natural.rsp
thf(fact_6721_plus__integer_Orsp,axiom,
( bNF_rel_fun @ int @ int @ ( int > int ) @ ( int > int )
@ ^ [Y3: int,Z: int] : Y3 = Z
@ ( bNF_rel_fun @ int @ int @ int @ int
@ ^ [Y3: int,Z: int] : Y3 = Z
@ ^ [Y3: int,Z: int] : Y3 = Z )
@ ( plus_plus @ int )
@ ( plus_plus @ int ) ) ).
% plus_integer.rsp
thf(fact_6722_num_Ocase__transfer,axiom,
! [A: $tType,B: $tType,S2: A > B > $o] :
( bNF_rel_fun @ A @ B @ ( ( num > A ) > ( num > A ) > num > A ) @ ( ( num > B ) > ( num > B ) > num > B ) @ S2
@ ( bNF_rel_fun @ ( num > A ) @ ( num > B ) @ ( ( num > A ) > num > A ) @ ( ( num > B ) > num > B )
@ ( bNF_rel_fun @ num @ num @ A @ B
@ ^ [Y3: num,Z: num] : Y3 = Z
@ S2 )
@ ( bNF_rel_fun @ ( num > A ) @ ( num > B ) @ ( num > A ) @ ( num > B )
@ ( bNF_rel_fun @ num @ num @ A @ B
@ ^ [Y3: num,Z: num] : Y3 = Z
@ S2 )
@ ( bNF_rel_fun @ num @ num @ A @ B
@ ^ [Y3: num,Z: num] : Y3 = Z
@ S2 ) ) )
@ ( case_num @ A )
@ ( case_num @ B ) ) ).
% num.case_transfer
thf(fact_6723_bit__natural_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ ( nat > $o ) @ ( nat > $o )
@ ^ [Y3: nat,Z: nat] : Y3 = Z
@ ^ [Y3: nat > $o,Z: nat > $o] : Y3 = Z
@ ( bit_se5641148757651400278ts_bit @ nat )
@ ( bit_se5641148757651400278ts_bit @ nat ) ) ).
% bit_natural.rsp
thf(fact_6724_Suc_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ nat @ nat
@ ^ [Y3: nat,Z: nat] : Y3 = Z
@ ^ [Y3: nat,Z: nat] : Y3 = Z
@ suc
@ suc ) ).
% Suc.rsp
thf(fact_6725_natural__of__integer_Orsp,axiom,
( bNF_rel_fun @ int @ int @ nat @ nat
@ ^ [Y3: int,Z: int] : Y3 = Z
@ ^ [Y3: nat,Z: nat] : Y3 = Z
@ nat2
@ nat2 ) ).
% natural_of_integer.rsp
thf(fact_6726_not__integer_Orsp,axiom,
( bNF_rel_fun @ int @ int @ int @ int
@ ^ [Y3: int,Z: int] : Y3 = Z
@ ^ [Y3: int,Z: int] : Y3 = Z
@ ( bit_ri4277139882892585799ns_not @ int )
@ ( bit_ri4277139882892585799ns_not @ int ) ) ).
% not_integer.rsp
thf(fact_6727_and__natural_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( nat > nat )
@ ^ [Y3: nat,Z: nat] : Y3 = Z
@ ( bNF_rel_fun @ nat @ nat @ nat @ nat
@ ^ [Y3: nat,Z: nat] : Y3 = Z
@ ^ [Y3: nat,Z: nat] : Y3 = Z )
@ ( bit_se5824344872417868541ns_and @ nat )
@ ( bit_se5824344872417868541ns_and @ nat ) ) ).
% and_natural.rsp
thf(fact_6728_and__integer_Orsp,axiom,
( bNF_rel_fun @ int @ int @ ( int > int ) @ ( int > int )
@ ^ [Y3: int,Z: int] : Y3 = Z
@ ( bNF_rel_fun @ int @ int @ int @ int
@ ^ [Y3: int,Z: int] : Y3 = Z
@ ^ [Y3: int,Z: int] : Y3 = Z )
@ ( bit_se5824344872417868541ns_and @ int )
@ ( bit_se5824344872417868541ns_and @ int ) ) ).
% and_integer.rsp
thf(fact_6729_take__bit__natural_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( nat > nat )
@ ^ [Y3: nat,Z: nat] : Y3 = Z
@ ( bNF_rel_fun @ nat @ nat @ nat @ nat
@ ^ [Y3: nat,Z: nat] : Y3 = Z
@ ^ [Y3: nat,Z: nat] : Y3 = Z )
@ ( bit_se2584673776208193580ke_bit @ nat )
@ ( bit_se2584673776208193580ke_bit @ nat ) ) ).
% take_bit_natural.rsp
thf(fact_6730_take__bit__integer_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ ( int > int ) @ ( int > int )
@ ^ [Y3: nat,Z: nat] : Y3 = Z
@ ( bNF_rel_fun @ int @ int @ int @ int
@ ^ [Y3: int,Z: int] : Y3 = Z
@ ^ [Y3: int,Z: int] : Y3 = Z )
@ ( bit_se2584673776208193580ke_bit @ int )
@ ( bit_se2584673776208193580ke_bit @ int ) ) ).
% take_bit_integer.rsp
thf(fact_6731_modulo__natural_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( nat > nat )
@ ^ [Y3: nat,Z: nat] : Y3 = Z
@ ( bNF_rel_fun @ nat @ nat @ nat @ nat
@ ^ [Y3: nat,Z: nat] : Y3 = Z
@ ^ [Y3: nat,Z: nat] : Y3 = Z )
@ ( modulo_modulo @ nat )
@ ( modulo_modulo @ nat ) ) ).
% modulo_natural.rsp
thf(fact_6732_modulo__integer_Orsp,axiom,
( bNF_rel_fun @ int @ int @ ( int > int ) @ ( int > int )
@ ^ [Y3: int,Z: int] : Y3 = Z
@ ( bNF_rel_fun @ int @ int @ int @ int
@ ^ [Y3: int,Z: int] : Y3 = Z
@ ^ [Y3: int,Z: int] : Y3 = Z )
@ ( modulo_modulo @ int )
@ ( modulo_modulo @ int ) ) ).
% modulo_integer.rsp
thf(fact_6733_unset__bit__natural_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( nat > nat )
@ ^ [Y3: nat,Z: nat] : Y3 = Z
@ ( bNF_rel_fun @ nat @ nat @ nat @ nat
@ ^ [Y3: nat,Z: nat] : Y3 = Z
@ ^ [Y3: nat,Z: nat] : Y3 = Z )
@ ( bit_se2638667681897837118et_bit @ nat )
@ ( bit_se2638667681897837118et_bit @ nat ) ) ).
% unset_bit_natural.rsp
thf(fact_6734_unset__bit__integer_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ ( int > int ) @ ( int > int )
@ ^ [Y3: nat,Z: nat] : Y3 = Z
@ ( bNF_rel_fun @ int @ int @ int @ int
@ ^ [Y3: int,Z: int] : Y3 = Z
@ ^ [Y3: int,Z: int] : Y3 = Z )
@ ( bit_se2638667681897837118et_bit @ int )
@ ( bit_se2638667681897837118et_bit @ int ) ) ).
% unset_bit_integer.rsp
thf(fact_6735_set__bit__natural_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( nat > nat )
@ ^ [Y3: nat,Z: nat] : Y3 = Z
@ ( bNF_rel_fun @ nat @ nat @ nat @ nat
@ ^ [Y3: nat,Z: nat] : Y3 = Z
@ ^ [Y3: nat,Z: nat] : Y3 = Z )
@ ( bit_se5668285175392031749et_bit @ nat )
@ ( bit_se5668285175392031749et_bit @ nat ) ) ).
% set_bit_natural.rsp
thf(fact_6736_set__bit__integer_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ ( int > int ) @ ( int > int )
@ ^ [Y3: nat,Z: nat] : Y3 = Z
@ ( bNF_rel_fun @ int @ int @ int @ int
@ ^ [Y3: int,Z: int] : Y3 = Z
@ ^ [Y3: int,Z: int] : Y3 = Z )
@ ( bit_se5668285175392031749et_bit @ int )
@ ( bit_se5668285175392031749et_bit @ int ) ) ).
% set_bit_integer.rsp
thf(fact_6737_flip__bit__integer_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ ( int > int ) @ ( int > int )
@ ^ [Y3: nat,Z: nat] : Y3 = Z
@ ( bNF_rel_fun @ int @ int @ int @ int
@ ^ [Y3: int,Z: int] : Y3 = Z
@ ^ [Y3: int,Z: int] : Y3 = Z )
@ ( bit_se8732182000553998342ip_bit @ int )
@ ( bit_se8732182000553998342ip_bit @ int ) ) ).
% flip_bit_integer.rsp
thf(fact_6738_flip__bit__natural_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( nat > nat )
@ ^ [Y3: nat,Z: nat] : Y3 = Z
@ ( bNF_rel_fun @ nat @ nat @ nat @ nat
@ ^ [Y3: nat,Z: nat] : Y3 = Z
@ ^ [Y3: nat,Z: nat] : Y3 = Z )
@ ( bit_se8732182000553998342ip_bit @ nat )
@ ( bit_se8732182000553998342ip_bit @ nat ) ) ).
% flip_bit_natural.rsp
thf(fact_6739_push__bit__natural_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( nat > nat )
@ ^ [Y3: nat,Z: nat] : Y3 = Z
@ ( bNF_rel_fun @ nat @ nat @ nat @ nat
@ ^ [Y3: nat,Z: nat] : Y3 = Z
@ ^ [Y3: nat,Z: nat] : Y3 = Z )
@ ( bit_se4730199178511100633sh_bit @ nat )
@ ( bit_se4730199178511100633sh_bit @ nat ) ) ).
% push_bit_natural.rsp
thf(fact_6740_push__bit__integer_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ ( int > int ) @ ( int > int )
@ ^ [Y3: nat,Z: nat] : Y3 = Z
@ ( bNF_rel_fun @ int @ int @ int @ int
@ ^ [Y3: int,Z: int] : Y3 = Z
@ ^ [Y3: int,Z: int] : Y3 = Z )
@ ( bit_se4730199178511100633sh_bit @ int )
@ ( bit_se4730199178511100633sh_bit @ int ) ) ).
% push_bit_integer.rsp
thf(fact_6741_xor__natural_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( nat > nat )
@ ^ [Y3: nat,Z: nat] : Y3 = Z
@ ( bNF_rel_fun @ nat @ nat @ nat @ nat
@ ^ [Y3: nat,Z: nat] : Y3 = Z
@ ^ [Y3: nat,Z: nat] : Y3 = Z )
@ ( bit_se5824344971392196577ns_xor @ nat )
@ ( bit_se5824344971392196577ns_xor @ nat ) ) ).
% xor_natural.rsp
thf(fact_6742_xor__integer_Orsp,axiom,
( bNF_rel_fun @ int @ int @ ( int > int ) @ ( int > int )
@ ^ [Y3: int,Z: int] : Y3 = Z
@ ( bNF_rel_fun @ int @ int @ int @ int
@ ^ [Y3: int,Z: int] : Y3 = Z
@ ^ [Y3: int,Z: int] : Y3 = Z )
@ ( bit_se5824344971392196577ns_xor @ int )
@ ( bit_se5824344971392196577ns_xor @ int ) ) ).
% xor_integer.rsp
thf(fact_6743_bit__integer_Orsp,axiom,
( bNF_rel_fun @ int @ int @ ( nat > $o ) @ ( nat > $o )
@ ^ [Y3: int,Z: int] : Y3 = Z
@ ^ [Y3: nat > $o,Z: nat > $o] : Y3 = Z
@ ( bit_se5641148757651400278ts_bit @ int )
@ ( bit_se5641148757651400278ts_bit @ int ) ) ).
% bit_integer.rsp
thf(fact_6744_or__integer_Orsp,axiom,
( bNF_rel_fun @ int @ int @ ( int > int ) @ ( int > int )
@ ^ [Y3: int,Z: int] : Y3 = Z
@ ( bNF_rel_fun @ int @ int @ int @ int
@ ^ [Y3: int,Z: int] : Y3 = Z
@ ^ [Y3: int,Z: int] : Y3 = Z )
@ ( bit_se1065995026697491101ons_or @ int )
@ ( bit_se1065995026697491101ons_or @ int ) ) ).
% or_integer.rsp
thf(fact_6745_or__natural_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( nat > nat )
@ ^ [Y3: nat,Z: nat] : Y3 = Z
@ ( bNF_rel_fun @ nat @ nat @ nat @ nat
@ ^ [Y3: nat,Z: nat] : Y3 = Z
@ ^ [Y3: nat,Z: nat] : Y3 = Z )
@ ( bit_se1065995026697491101ons_or @ nat )
@ ( bit_se1065995026697491101ons_or @ nat ) ) ).
% or_natural.rsp
thf(fact_6746_drop__bit__integer_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ ( int > int ) @ ( int > int )
@ ^ [Y3: nat,Z: nat] : Y3 = Z
@ ( bNF_rel_fun @ int @ int @ int @ int
@ ^ [Y3: int,Z: int] : Y3 = Z
@ ^ [Y3: int,Z: int] : Y3 = Z )
@ ( bit_se4197421643247451524op_bit @ int )
@ ( bit_se4197421643247451524op_bit @ int ) ) ).
% drop_bit_integer.rsp
thf(fact_6747_drop__bit__natural_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( nat > nat )
@ ^ [Y3: nat,Z: nat] : Y3 = Z
@ ( bNF_rel_fun @ nat @ nat @ nat @ nat
@ ^ [Y3: nat,Z: nat] : Y3 = Z
@ ^ [Y3: nat,Z: nat] : Y3 = Z )
@ ( bit_se4197421643247451524op_bit @ nat )
@ ( bit_se4197421643247451524op_bit @ nat ) ) ).
% drop_bit_natural.rsp
thf(fact_6748_mask__integer_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ int @ int
@ ^ [Y3: nat,Z: nat] : Y3 = Z
@ ^ [Y3: int,Z: int] : Y3 = Z
@ ( bit_se2239418461657761734s_mask @ int )
@ ( bit_se2239418461657761734s_mask @ int ) ) ).
% mask_integer.rsp
thf(fact_6749_mask__natural_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ nat @ nat
@ ^ [Y3: nat,Z: nat] : Y3 = Z
@ ^ [Y3: nat,Z: nat] : Y3 = Z
@ ( bit_se2239418461657761734s_mask @ nat )
@ ( bit_se2239418461657761734s_mask @ nat ) ) ).
% mask_natural.rsp
thf(fact_6750_abs__integer_Orsp,axiom,
( bNF_rel_fun @ int @ int @ int @ int
@ ^ [Y3: int,Z: int] : Y3 = Z
@ ^ [Y3: int,Z: int] : Y3 = Z
@ ( abs_abs @ int )
@ ( abs_abs @ int ) ) ).
% abs_integer.rsp
thf(fact_6751_divide__integer_Orsp,axiom,
( bNF_rel_fun @ int @ int @ ( int > int ) @ ( int > int )
@ ^ [Y3: int,Z: int] : Y3 = Z
@ ( bNF_rel_fun @ int @ int @ int @ int
@ ^ [Y3: int,Z: int] : Y3 = Z
@ ^ [Y3: int,Z: int] : Y3 = Z )
@ ( divide_divide @ int )
@ ( divide_divide @ int ) ) ).
% divide_integer.rsp
thf(fact_6752_divide__natural_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( nat > nat )
@ ^ [Y3: nat,Z: nat] : Y3 = Z
@ ( bNF_rel_fun @ nat @ nat @ nat @ nat
@ ^ [Y3: nat,Z: nat] : Y3 = Z
@ ^ [Y3: nat,Z: nat] : Y3 = Z )
@ ( divide_divide @ nat )
@ ( divide_divide @ nat ) ) ).
% divide_natural.rsp
thf(fact_6753_times__natural_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( nat > nat )
@ ^ [Y3: nat,Z: nat] : Y3 = Z
@ ( bNF_rel_fun @ nat @ nat @ nat @ nat
@ ^ [Y3: nat,Z: nat] : Y3 = Z
@ ^ [Y3: nat,Z: nat] : Y3 = Z )
@ ( times_times @ nat )
@ ( times_times @ nat ) ) ).
% times_natural.rsp
thf(fact_6754_times__integer_Orsp,axiom,
( bNF_rel_fun @ int @ int @ ( int > int ) @ ( int > int )
@ ^ [Y3: int,Z: int] : Y3 = Z
@ ( bNF_rel_fun @ int @ int @ int @ int
@ ^ [Y3: int,Z: int] : Y3 = Z
@ ^ [Y3: int,Z: int] : Y3 = Z )
@ ( times_times @ int )
@ ( times_times @ int ) ) ).
% times_integer.rsp
thf(fact_6755_less__integer_Orsp,axiom,
( bNF_rel_fun @ int @ int @ ( int > $o ) @ ( int > $o )
@ ^ [Y3: int,Z: int] : Y3 = Z
@ ( bNF_rel_fun @ int @ int @ $o @ $o
@ ^ [Y3: int,Z: int] : Y3 = Z
@ ^ [Y3: $o,Z: $o] : Y3 = Z )
@ ( ord_less @ int )
@ ( ord_less @ int ) ) ).
% less_integer.rsp
thf(fact_6756_less__natural_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ ( nat > $o ) @ ( nat > $o )
@ ^ [Y3: nat,Z: nat] : Y3 = Z
@ ( bNF_rel_fun @ nat @ nat @ $o @ $o
@ ^ [Y3: nat,Z: nat] : Y3 = Z
@ ^ [Y3: $o,Z: $o] : Y3 = Z )
@ ( ord_less @ nat )
@ ( ord_less @ nat ) ) ).
% less_natural.rsp
thf(fact_6757_transfer__rule__of__bool,axiom,
! [A: $tType,B: $tType] :
( ( ( zero_neq_one @ B )
& ( zero_neq_one @ A ) )
=> ! [R3: A > B > $o] :
( ( R3 @ ( zero_zero @ A ) @ ( zero_zero @ B ) )
=> ( ( R3 @ ( one_one @ A ) @ ( one_one @ B ) )
=> ( bNF_rel_fun @ $o @ $o @ A @ B
@ ^ [Y3: $o,Z: $o] : Y3 = Z
@ R3
@ ( zero_neq_one_of_bool @ A )
@ ( zero_neq_one_of_bool @ B ) ) ) ) ) ).
% transfer_rule_of_bool
thf(fact_6758_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 )
@ ^ [X: product_prod @ int @ int,Y4: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( times_times @ int @ ( product_fst @ int @ int @ X ) @ ( product_fst @ int @ int @ Y4 ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ X ) @ ( product_snd @ int @ int @ Y4 ) ) )
@ ^ [X: product_prod @ int @ int,Y4: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( times_times @ int @ ( product_fst @ int @ int @ X ) @ ( product_fst @ int @ int @ Y4 ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ X ) @ ( product_snd @ int @ int @ Y4 ) ) ) ) ).
% times_rat.rsp
thf(fact_6759_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 )
@ ^ [X: product_prod @ int @ int,Y4: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( plus_plus @ int @ ( times_times @ int @ ( product_fst @ int @ int @ X ) @ ( product_snd @ int @ int @ Y4 ) ) @ ( times_times @ int @ ( product_fst @ int @ int @ Y4 ) @ ( product_snd @ int @ int @ X ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ X ) @ ( product_snd @ int @ int @ Y4 ) ) )
@ ^ [X: product_prod @ int @ int,Y4: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( plus_plus @ int @ ( times_times @ int @ ( product_fst @ int @ int @ X ) @ ( product_snd @ int @ int @ Y4 ) ) @ ( times_times @ int @ ( product_fst @ int @ int @ Y4 ) @ ( product_snd @ int @ int @ X ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ X ) @ ( product_snd @ int @ int @ Y4 ) ) ) ) ).
% plus_rat.rsp
thf(fact_6760_num__of__integer_Orsp,axiom,
( bNF_rel_fun @ int @ int @ num @ num
@ ^ [Y3: int,Z: int] : Y3 = Z
@ ^ [Y3: num,Z: num] : Y3 = Z
@ ( comp @ nat @ num @ int @ num_of_nat @ nat2 )
@ ( comp @ nat @ num @ int @ num_of_nat @ nat2 ) ) ).
% num_of_integer.rsp
thf(fact_6761_fun_Omap__ident,axiom,
! [A: $tType,D: $tType,T2: D > A] :
( ( comp @ A @ A @ D
@ ^ [X: A] : X
@ T2 )
= T2 ) ).
% fun.map_ident
thf(fact_6762_predicate2__transferD,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,R12: A > B > $o,R23: C > D > $o,P: A > C > $o,Q: B > D > $o,A2: product_prod @ A @ B,A5: set @ ( product_prod @ A @ B ),B2: product_prod @ C @ D,B5: set @ ( product_prod @ C @ D )] :
( ( bNF_rel_fun @ A @ B @ ( C > $o ) @ ( D > $o ) @ R12
@ ( bNF_rel_fun @ C @ D @ $o @ $o @ R23
@ ^ [Y3: $o,Z: $o] : Y3 = Z )
@ P
@ Q )
=> ( ( member @ ( product_prod @ A @ B ) @ A2 @ A5 )
=> ( ( member @ ( product_prod @ C @ D ) @ B2 @ B5 )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ A5 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R12 ) ) )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ C @ D ) ) @ B5 @ ( collect @ ( product_prod @ C @ D ) @ ( product_case_prod @ C @ D @ $o @ R23 ) ) )
=> ( ( P @ ( product_fst @ A @ B @ A2 ) @ ( product_fst @ C @ D @ B2 ) )
= ( Q @ ( product_snd @ A @ B @ A2 ) @ ( product_snd @ C @ D @ B2 ) ) ) ) ) ) ) ) ).
% predicate2_transferD
thf(fact_6763_prod_Omap__ident,axiom,
! [B: $tType,A: $tType,T2: product_prod @ A @ B] :
( ( product_map_prod @ A @ A @ B @ B
@ ^ [X: A] : X
@ ^ [X: B] : X
@ T2 )
= T2 ) ).
% prod.map_ident
thf(fact_6764_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
@ ^ [X: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( uminus_uminus @ int @ ( product_fst @ int @ int @ X ) ) @ ( product_snd @ int @ int @ X ) )
@ ^ [X: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( uminus_uminus @ int @ ( product_fst @ int @ int @ X ) ) @ ( product_snd @ int @ int @ X ) ) ) ).
% uminus_rat.rsp
thf(fact_6765_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
@ ^ [X: product_prod @ int @ int] :
( 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 ) ) )
@ ^ [X: product_prod @ int @ int] :
( 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.rsp
thf(fact_6766_rec__nat__transfer,axiom,
! [A: $tType,B: $tType,A5: A > B > $o] :
( bNF_rel_fun @ A @ B @ ( ( nat > A > A ) > nat > A ) @ ( ( nat > B > B ) > nat > B ) @ A5
@ ( bNF_rel_fun @ ( nat > A > A ) @ ( nat > B > B ) @ ( nat > A ) @ ( nat > B )
@ ( bNF_rel_fun @ nat @ nat @ ( A > A ) @ ( B > B )
@ ^ [Y3: nat,Z: nat] : Y3 = Z
@ ( bNF_rel_fun @ A @ B @ A @ B @ A5 @ A5 ) )
@ ( bNF_rel_fun @ nat @ nat @ A @ B
@ ^ [Y3: nat,Z: nat] : Y3 = Z
@ A5 ) )
@ ( rec_nat @ A )
@ ( rec_nat @ B ) ) ).
% rec_nat_transfer
thf(fact_6767_fun_Orel__map_I1_J,axiom,
! [A: $tType,C: $tType,B: $tType,D: $tType,Sb: C > B > $o,I: A > C,X2: D > A,Y: D > B] :
( ( bNF_rel_fun @ D @ D @ C @ B
@ ^ [Y3: D,Z: D] : Y3 = Z
@ Sb
@ ( comp @ A @ C @ D @ I @ X2 )
@ Y )
= ( bNF_rel_fun @ D @ D @ A @ B
@ ^ [Y3: D,Z: D] : Y3 = Z
@ ^ [X: A] : ( Sb @ ( I @ X ) )
@ X2
@ Y ) ) ).
% fun.rel_map(1)
thf(fact_6768_fun_Orel__map_I2_J,axiom,
! [A: $tType,C: $tType,B: $tType,D: $tType,Sa: A > C > $o,X2: D > A,G: B > C,Y: D > B] :
( ( bNF_rel_fun @ D @ D @ A @ C
@ ^ [Y3: D,Z: D] : Y3 = Z
@ Sa
@ X2
@ ( comp @ B @ C @ D @ G @ Y ) )
= ( bNF_rel_fun @ D @ D @ A @ B
@ ^ [Y3: D,Z: D] : Y3 = Z
@ ^ [X: A,Y4: B] : ( Sa @ X @ ( G @ Y4 ) )
@ X2
@ Y ) ) ).
% fun.rel_map(2)
thf(fact_6769_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 )
@ ^ [X: product_prod @ int @ int,Y4: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( plus_plus @ int @ ( times_times @ int @ ( product_fst @ int @ int @ X ) @ ( product_snd @ int @ int @ Y4 ) ) @ ( times_times @ int @ ( product_fst @ int @ int @ Y4 ) @ ( product_snd @ int @ int @ X ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ X ) @ ( product_snd @ int @ int @ Y4 ) ) )
@ ( plus_plus @ rat ) ) ).
% plus_rat.transfer
thf(fact_6770_inverse__rat_Otransfer,axiom,
( bNF_rel_fun @ ( product_prod @ int @ int ) @ rat @ ( product_prod @ int @ int ) @ rat @ pcr_rat @ pcr_rat
@ ^ [X: product_prod @ int @ int] :
( 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_inverse @ rat ) ) ).
% inverse_rat.transfer
thf(fact_6771_one__rat_Otransfer,axiom,
pcr_rat @ ( product_Pair @ int @ int @ ( one_one @ int ) @ ( one_one @ int ) ) @ ( one_one @ rat ) ).
% one_rat.transfer
thf(fact_6772_zero__rat_Otransfer,axiom,
pcr_rat @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) ) @ ( zero_zero @ rat ) ).
% zero_rat.transfer
thf(fact_6773_Fract_Otransfer,axiom,
( bNF_rel_fun @ int @ int @ ( int > ( product_prod @ int @ int ) ) @ ( int > rat )
@ ^ [Y3: int,Z: int] : Y3 = Z
@ ( bNF_rel_fun @ int @ int @ ( product_prod @ int @ int ) @ rat
@ ^ [Y3: int,Z: int] : Y3 = Z
@ pcr_rat )
@ ^ [A3: int,B3: int] :
( if @ ( product_prod @ int @ int )
@ ( B3
= ( zero_zero @ int ) )
@ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
@ ( product_Pair @ int @ int @ A3 @ B3 ) )
@ fract ) ).
% Fract.transfer
thf(fact_6774_of__rat_Otransfer,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( bNF_rel_fun @ ( product_prod @ int @ int ) @ rat @ A @ A @ pcr_rat
@ ^ [Y3: A,Z: A] : Y3 = Z
@ ^ [X: product_prod @ int @ int] : ( divide_divide @ A @ ( ring_1_of_int @ A @ ( product_fst @ int @ int @ X ) ) @ ( ring_1_of_int @ A @ ( product_snd @ int @ int @ X ) ) )
@ ( field_char_0_of_rat @ A ) ) ) ).
% of_rat.transfer
thf(fact_6775_uminus__rat_Otransfer,axiom,
( bNF_rel_fun @ ( product_prod @ int @ int ) @ rat @ ( product_prod @ int @ int ) @ rat @ pcr_rat @ pcr_rat
@ ^ [X: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( uminus_uminus @ int @ ( product_fst @ int @ int @ X ) ) @ ( product_snd @ int @ int @ X ) )
@ ( uminus_uminus @ rat ) ) ).
% uminus_rat.transfer
thf(fact_6776_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 )
@ ^ [X: product_prod @ int @ int,Y4: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( times_times @ int @ ( product_fst @ int @ int @ X ) @ ( product_fst @ int @ int @ Y4 ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ X ) @ ( product_snd @ int @ int @ Y4 ) ) )
@ ( times_times @ rat ) ) ).
% times_rat.transfer
thf(fact_6777_positive_Otransfer,axiom,
( bNF_rel_fun @ ( product_prod @ int @ int ) @ rat @ $o @ $o @ pcr_rat
@ ^ [Y3: $o,Z: $o] : Y3 = Z
@ ^ [X: product_prod @ int @ int] : ( ord_less @ int @ ( zero_zero @ int ) @ ( times_times @ int @ ( product_fst @ int @ int @ X ) @ ( product_snd @ int @ int @ X ) ) )
@ positive ) ).
% positive.transfer
thf(fact_6778_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 ) )
@ ^ [X: nat,Y4: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V4: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ X @ U2 ) @ ( times_times @ nat @ Y4 @ V4 ) ) @ ( plus_plus @ nat @ ( times_times @ nat @ X @ V4 ) @ ( times_times @ nat @ Y4 @ U2 ) ) ) ) )
@ ( times_times @ int ) ) ).
% times_int.transfer
thf(fact_6779_num_Orec__transfer,axiom,
! [A: $tType,B: $tType,S2: A > B > $o] :
( bNF_rel_fun @ A @ B @ ( ( num > A > A ) > ( num > A > A ) > num > A ) @ ( ( num > B > B ) > ( num > B > B ) > num > B ) @ S2
@ ( bNF_rel_fun @ ( num > A > A ) @ ( num > B > B ) @ ( ( num > A > A ) > num > A ) @ ( ( num > B > B ) > num > B )
@ ( bNF_rel_fun @ num @ num @ ( A > A ) @ ( B > B )
@ ^ [Y3: num,Z: num] : Y3 = Z
@ ( bNF_rel_fun @ A @ B @ A @ B @ S2 @ S2 ) )
@ ( bNF_rel_fun @ ( num > A > A ) @ ( num > B > B ) @ ( num > A ) @ ( num > B )
@ ( bNF_rel_fun @ num @ num @ ( A > A ) @ ( B > B )
@ ^ [Y3: num,Z: num] : Y3 = Z
@ ( bNF_rel_fun @ A @ B @ A @ B @ S2 @ S2 ) )
@ ( bNF_rel_fun @ num @ num @ A @ B
@ ^ [Y3: num,Z: num] : Y3 = Z
@ S2 ) ) )
@ ( rec_num @ A )
@ ( rec_num @ B ) ) ).
% num.rec_transfer
thf(fact_6780_verit__eq__simplify_I21_J,axiom,
! [A: $tType,F1: A,F22: num > A > A,F32: num > A > A,X32: num] :
( ( rec_num @ A @ F1 @ F22 @ F32 @ ( bit1 @ X32 ) )
= ( F32 @ X32 @ ( rec_num @ A @ F1 @ F22 @ F32 @ X32 ) ) ) ).
% verit_eq_simplify(21)
thf(fact_6781_verit__eq__simplify_I19_J,axiom,
! [A: $tType,F1: A,F22: num > A > A,F32: num > A > A] :
( ( rec_num @ A @ F1 @ F22 @ F32 @ one2 )
= F1 ) ).
% verit_eq_simplify(19)
thf(fact_6782_verit__eq__simplify_I20_J,axiom,
! [A: $tType,F1: A,F22: num > A > A,F32: num > A > A,X22: num] :
( ( rec_num @ A @ F1 @ F22 @ F32 @ ( bit0 @ X22 ) )
= ( F22 @ X22 @ ( rec_num @ A @ F1 @ F22 @ F32 @ X22 ) ) ) ).
% verit_eq_simplify(20)
thf(fact_6783_zero__int_Otransfer,axiom,
pcr_int @ ( product_Pair @ nat @ nat @ ( zero_zero @ nat ) @ ( zero_zero @ nat ) ) @ ( zero_zero @ int ) ).
% zero_int.transfer
thf(fact_6784_int__transfer,axiom,
( bNF_rel_fun @ nat @ nat @ ( product_prod @ nat @ nat ) @ int
@ ^ [Y3: nat,Z: nat] : Y3 = Z
@ pcr_int
@ ^ [N4: nat] : ( product_Pair @ nat @ nat @ N4 @ ( zero_zero @ nat ) )
@ ( semiring_1_of_nat @ int ) ) ).
% int_transfer
thf(fact_6785_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 )
@ ^ [X: nat,Y4: nat] : ( product_Pair @ nat @ nat @ Y4 @ X ) )
@ ( uminus_uminus @ int ) ) ).
% uminus_int.transfer
thf(fact_6786_nat_Otransfer,axiom,
( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ nat @ nat @ pcr_int
@ ^ [Y3: nat,Z: nat] : Y3 = Z
@ ( product_case_prod @ nat @ nat @ nat @ ( minus_minus @ nat ) )
@ nat2 ) ).
% nat.transfer
thf(fact_6787_one__int_Otransfer,axiom,
pcr_int @ ( product_Pair @ nat @ nat @ ( one_one @ nat ) @ ( zero_zero @ nat ) ) @ ( one_one @ int ) ).
% one_int.transfer
thf(fact_6788_of__int_Otransfer,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ A @ A @ pcr_int
@ ^ [Y3: A,Z: A] : Y3 = Z
@ ( product_case_prod @ nat @ nat @ A
@ ^ [I2: nat,J3: nat] : ( minus_minus @ A @ ( semiring_1_of_nat @ A @ I2 ) @ ( semiring_1_of_nat @ A @ J3 ) ) )
@ ( ring_1_of_int @ A ) ) ) ).
% of_int.transfer
thf(fact_6789_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
@ ^ [Y3: $o,Z: $o] : Y3 = Z )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X: nat,Y4: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V4: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ X @ V4 ) @ ( plus_plus @ nat @ U2 @ Y4 ) ) ) )
@ ( ord_less @ int ) ) ).
% less_int.transfer
thf(fact_6790_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
@ ^ [Y3: $o,Z: $o] : Y3 = Z )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X: nat,Y4: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V4: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ X @ V4 ) @ ( plus_plus @ nat @ U2 @ Y4 ) ) ) )
@ ( ord_less_eq @ int ) ) ).
% less_eq_int.transfer
thf(fact_6791_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 ) )
@ ^ [X: nat,Y4: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V4: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X @ U2 ) @ ( plus_plus @ nat @ Y4 @ V4 ) ) ) )
@ ( plus_plus @ int ) ) ).
% plus_int.transfer
thf(fact_6792_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 ) )
@ ^ [X: nat,Y4: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V4: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X @ V4 ) @ ( plus_plus @ nat @ Y4 @ U2 ) ) ) )
@ ( minus_minus @ int ) ) ).
% minus_int.transfer
thf(fact_6793_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 ) )
@ ^ [X: nat,Y4: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V4: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ X @ U2 ) @ ( times_times @ nat @ Y4 @ V4 ) ) @ ( plus_plus @ nat @ ( times_times @ nat @ X @ V4 ) @ ( times_times @ nat @ Y4 @ U2 ) ) ) ) )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X: nat,Y4: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V4: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ X @ U2 ) @ ( times_times @ nat @ Y4 @ V4 ) ) @ ( plus_plus @ nat @ ( times_times @ nat @ X @ V4 ) @ ( times_times @ nat @ Y4 @ U2 ) ) ) ) ) ) ).
% times_int.rsp
thf(fact_6794_antimono__funpow,axiom,
! [A: $tType] :
( ( ( lattice @ A )
& ( order_top @ A ) )
=> ! [Q: A > A] :
( ( order_mono @ A @ A @ Q )
=> ( order_antimono @ nat @ A
@ ^ [I2: nat] : ( compow @ ( A > A ) @ I2 @ Q @ ( top_top @ A ) ) ) ) ) ).
% antimono_funpow
thf(fact_6795_intrel__iff,axiom,
! [X2: nat,Y: nat,U: nat,V2: nat] :
( ( intrel @ ( product_Pair @ nat @ nat @ X2 @ Y ) @ ( product_Pair @ nat @ nat @ U @ V2 ) )
= ( ( plus_plus @ nat @ X2 @ V2 )
= ( plus_plus @ nat @ U @ Y ) ) ) ).
% intrel_iff
thf(fact_6796_int_Orel__eq__transfer,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
@ ^ [Y3: $o,Z: $o] : Y3 = Z )
@ intrel
@ ^ [Y3: int,Z: int] : Y3 = Z ) ).
% int.rel_eq_transfer
thf(fact_6797_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_6798_antimonoD,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F: A > B,X2: A,Y: A] :
( ( order_antimono @ A @ B @ F )
=> ( ( ord_less_eq @ A @ X2 @ Y )
=> ( ord_less_eq @ B @ ( F @ Y ) @ ( F @ X2 ) ) ) ) ) ).
% antimonoD
thf(fact_6799_antimonoE,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F: A > B,X2: A,Y: A] :
( ( order_antimono @ A @ B @ F )
=> ( ( ord_less_eq @ A @ X2 @ Y )
=> ( ord_less_eq @ B @ ( F @ Y ) @ ( F @ X2 ) ) ) ) ) ).
% antimonoE
thf(fact_6800_antimonoI,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F: A > B] :
( ! [X3: A,Y2: A] :
( ( ord_less_eq @ A @ X3 @ Y2 )
=> ( ord_less_eq @ B @ ( F @ Y2 ) @ ( F @ X3 ) ) )
=> ( order_antimono @ A @ B @ F ) ) ) ).
% antimonoI
thf(fact_6801_antimono__def,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ( ( order_antimono @ A @ B )
= ( ^ [F2: A > B] :
! [X: A,Y4: A] :
( ( ord_less_eq @ A @ X @ Y4 )
=> ( ord_less_eq @ B @ ( F2 @ Y4 ) @ ( F2 @ X ) ) ) ) ) ) ).
% antimono_def
thf(fact_6802_int_Oabs__eq__iff,axiom,
! [X2: product_prod @ nat @ nat,Y: product_prod @ nat @ nat] :
( ( ( abs_Integ @ X2 )
= ( abs_Integ @ Y ) )
= ( intrel @ X2 @ Y ) ) ).
% int.abs_eq_iff
thf(fact_6803_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 )
@ ^ [X: nat,Y4: nat] : ( product_Pair @ nat @ nat @ Y4 @ X ) )
@ ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [X: nat,Y4: nat] : ( product_Pair @ nat @ nat @ Y4 @ X ) ) ) ).
% uminus_int.rsp
thf(fact_6804_nat_Orsp,axiom,
( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ nat @ nat @ intrel
@ ^ [Y3: nat,Z: nat] : Y3 = Z
@ ( product_case_prod @ nat @ nat @ nat @ ( minus_minus @ nat ) )
@ ( product_case_prod @ nat @ nat @ nat @ ( minus_minus @ nat ) ) ) ).
% nat.rsp
thf(fact_6805_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_6806_antimono__iff__le__Suc,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( order_antimono @ nat @ A )
= ( ^ [F2: nat > A] :
! [N4: nat] : ( ord_less_eq @ A @ ( F2 @ ( suc @ N4 ) ) @ ( F2 @ N4 ) ) ) ) ) ).
% antimono_iff_le_Suc
thf(fact_6807_of__int_Orsp,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ A @ A @ intrel
@ ^ [Y3: A,Z: A] : Y3 = Z
@ ( product_case_prod @ nat @ nat @ A
@ ^ [I2: nat,J3: nat] : ( minus_minus @ A @ ( semiring_1_of_nat @ A @ I2 ) @ ( semiring_1_of_nat @ A @ J3 ) ) )
@ ( product_case_prod @ nat @ nat @ A
@ ^ [I2: nat,J3: nat] : ( minus_minus @ A @ ( semiring_1_of_nat @ A @ I2 ) @ ( semiring_1_of_nat @ A @ J3 ) ) ) ) ) ).
% of_int.rsp
thf(fact_6808_intrel__def,axiom,
( intrel
= ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X: nat,Y4: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V4: nat] :
( ( plus_plus @ nat @ X @ V4 )
= ( plus_plus @ nat @ U2 @ Y4 ) ) ) ) ) ).
% intrel_def
thf(fact_6809_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
@ ^ [Y3: $o,Z: $o] : Y3 = Z )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X: nat,Y4: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V4: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ X @ V4 ) @ ( plus_plus @ nat @ U2 @ Y4 ) ) ) )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X: nat,Y4: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V4: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ X @ V4 ) @ ( plus_plus @ nat @ U2 @ Y4 ) ) ) ) ) ).
% less_int.rsp
thf(fact_6810_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
@ ^ [Y3: $o,Z: $o] : Y3 = Z )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X: nat,Y4: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V4: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ X @ V4 ) @ ( plus_plus @ nat @ U2 @ Y4 ) ) ) )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X: nat,Y4: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V4: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ X @ V4 ) @ ( plus_plus @ nat @ U2 @ Y4 ) ) ) ) ) ).
% less_eq_int.rsp
thf(fact_6811_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 ) )
@ ^ [X: nat,Y4: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V4: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X @ V4 ) @ ( plus_plus @ nat @ Y4 @ U2 ) ) ) )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X: nat,Y4: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V4: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X @ V4 ) @ ( plus_plus @ nat @ Y4 @ U2 ) ) ) ) ) ).
% minus_int.rsp
thf(fact_6812_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 ) )
@ ^ [X: nat,Y4: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V4: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X @ U2 ) @ ( plus_plus @ nat @ Y4 @ V4 ) ) ) )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X: nat,Y4: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V4: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X @ U2 ) @ ( plus_plus @ nat @ Y4 @ V4 ) ) ) ) ) ).
% plus_int.rsp
thf(fact_6813_sum__mset__constant,axiom,
! [A: $tType,B: $tType] :
( ( semiring_1 @ B )
=> ! [Y: B,A5: multiset @ A] :
( ( comm_m7189776963980413722m_mset @ B
@ ( image_mset @ A @ B
@ ^ [X: A] : Y
@ A5 ) )
= ( times_times @ B @ ( semiring_1_of_nat @ B @ ( size_size @ ( multiset @ A ) @ A5 ) ) @ Y ) ) ) ).
% sum_mset_constant
thf(fact_6814_finite__refines__card__le,axiom,
! [A: $tType,A5: set @ A,R3: set @ ( product_prod @ A @ A ),S2: set @ ( product_prod @ A @ A )] :
( ( finite_finite2 @ ( set @ A ) @ ( equiv_quotient @ A @ A5 @ R3 ) )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ S2 )
=> ( ( equiv_equiv @ A @ A5 @ R3 )
=> ( ( equiv_equiv @ A @ A5 @ S2 )
=> ( ord_less_eq @ nat @ ( finite_card @ ( set @ A ) @ ( equiv_quotient @ A @ A5 @ S2 ) ) @ ( finite_card @ ( set @ A ) @ ( equiv_quotient @ A @ A5 @ R3 ) ) ) ) ) ) ) ).
% finite_refines_card_le
thf(fact_6815_of__nat__sum__mset,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [A5: multiset @ nat] :
( ( semiring_1_of_nat @ A @ ( comm_m7189776963980413722m_mset @ nat @ A5 ) )
= ( comm_m7189776963980413722m_mset @ A @ ( image_mset @ nat @ A @ ( semiring_1_of_nat @ A ) @ A5 ) ) ) ) ).
% of_nat_sum_mset
thf(fact_6816_Union__image__single__mset,axiom,
! [A: $tType,M: multiset @ A] :
( ( comm_m7189776963980413722m_mset @ ( multiset @ A )
@ ( image_mset @ A @ ( multiset @ A )
@ ^ [X: A] : ( add_mset @ A @ X @ ( zero_zero @ ( multiset @ A ) ) )
@ M ) )
= M ) ).
% Union_image_single_mset
thf(fact_6817_sum__mset_Oempty,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ( ( comm_m7189776963980413722m_mset @ A @ ( zero_zero @ ( multiset @ A ) ) )
= ( zero_zero @ A ) ) ) ).
% sum_mset.empty
thf(fact_6818_sum__mset_Oadd__mset,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [X2: A,N5: multiset @ A] :
( ( comm_m7189776963980413722m_mset @ A @ ( add_mset @ A @ X2 @ N5 ) )
= ( plus_plus @ A @ X2 @ ( comm_m7189776963980413722m_mset @ A @ N5 ) ) ) ) ).
% sum_mset.add_mset
thf(fact_6819_sum__mset_Ounion,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M3: multiset @ A,N5: multiset @ A] :
( ( comm_m7189776963980413722m_mset @ A @ ( plus_plus @ ( multiset @ A ) @ M3 @ N5 ) )
= ( plus_plus @ A @ ( comm_m7189776963980413722m_mset @ A @ M3 ) @ ( comm_m7189776963980413722m_mset @ A @ N5 ) ) ) ) ).
% sum_mset.union
thf(fact_6820_sum__mset_Oneutral__const,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A5: multiset @ B] :
( ( comm_m7189776963980413722m_mset @ A
@ ( image_mset @ B @ A
@ ^ [Uu: B] : ( zero_zero @ A )
@ A5 ) )
= ( zero_zero @ A ) ) ) ).
% sum_mset.neutral_const
thf(fact_6821_sum__mset_Oinsert,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: B > A,X2: B,A5: multiset @ B] :
( ( comm_m7189776963980413722m_mset @ A @ ( image_mset @ B @ A @ G @ ( add_mset @ B @ X2 @ A5 ) ) )
= ( plus_plus @ A @ ( G @ X2 ) @ ( comm_m7189776963980413722m_mset @ A @ ( image_mset @ B @ A @ G @ A5 ) ) ) ) ) ).
% sum_mset.insert
thf(fact_6822_equiv__type,axiom,
! [A: $tType,A5: set @ A,R: set @ ( product_prod @ A @ A )] :
( ( equiv_equiv @ A @ A5 @ R )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R
@ ( product_Sigma @ A @ A @ A5
@ ^ [Uu: A] : A5 ) ) ) ).
% equiv_type
thf(fact_6823_sum__mset_Oswap,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: B > C > A,B5: multiset @ C,A5: multiset @ B] :
( ( comm_m7189776963980413722m_mset @ A
@ ( image_mset @ B @ A
@ ^ [I2: B] : ( comm_m7189776963980413722m_mset @ A @ ( image_mset @ C @ A @ ( G @ I2 ) @ B5 ) )
@ A5 ) )
= ( comm_m7189776963980413722m_mset @ A
@ ( image_mset @ C @ A
@ ^ [J3: C] :
( comm_m7189776963980413722m_mset @ A
@ ( image_mset @ B @ A
@ ^ [I2: B] : ( G @ I2 @ J3 )
@ A5 ) )
@ B5 ) ) ) ) ).
% sum_mset.swap
thf(fact_6824_sum__mset_Odistrib,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: B > A,H: B > A,A5: multiset @ B] :
( ( comm_m7189776963980413722m_mset @ A
@ ( image_mset @ B @ A
@ ^ [X: B] : ( plus_plus @ A @ ( G @ X ) @ ( H @ X ) )
@ A5 ) )
= ( plus_plus @ A @ ( comm_m7189776963980413722m_mset @ A @ ( image_mset @ B @ A @ G @ A5 ) ) @ ( comm_m7189776963980413722m_mset @ A @ ( image_mset @ B @ A @ H @ A5 ) ) ) ) ) ).
% sum_mset.distrib
thf(fact_6825_sum__mset__distrib__left,axiom,
! [A: $tType,B: $tType] :
( ( semiring_0 @ A )
=> ! [C2: A,F: B > A,M3: multiset @ B] :
( ( times_times @ A @ C2 @ ( comm_m7189776963980413722m_mset @ A @ ( image_mset @ B @ A @ F @ M3 ) ) )
= ( comm_m7189776963980413722m_mset @ A
@ ( image_mset @ B @ A
@ ^ [X: B] : ( times_times @ A @ C2 @ ( F @ X ) )
@ M3 ) ) ) ) ).
% sum_mset_distrib_left
thf(fact_6826_sum__mset__distrib__right,axiom,
! [A: $tType,B: $tType] :
( ( semiring_0 @ A )
=> ! [F: B > A,M3: multiset @ B,C2: A] :
( ( times_times @ A @ ( comm_m7189776963980413722m_mset @ A @ ( image_mset @ B @ A @ F @ M3 ) ) @ C2 )
= ( comm_m7189776963980413722m_mset @ A
@ ( image_mset @ B @ A
@ ^ [X: B] : ( times_times @ A @ ( F @ X ) @ C2 )
@ M3 ) ) ) ) ).
% sum_mset_distrib_right
thf(fact_6827_sum__mset__product,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( ( comm_monoid_add @ A )
& ( times @ A )
& ( semiring_0 @ B ) )
=> ! [F: A > B,A5: multiset @ A,G: C > B,B5: multiset @ C] :
( ( times_times @ B @ ( comm_m7189776963980413722m_mset @ B @ ( image_mset @ A @ B @ F @ A5 ) ) @ ( comm_m7189776963980413722m_mset @ B @ ( image_mset @ C @ B @ G @ B5 ) ) )
= ( comm_m7189776963980413722m_mset @ B
@ ( image_mset @ A @ B
@ ^ [I2: A] :
( comm_m7189776963980413722m_mset @ B
@ ( image_mset @ C @ B
@ ^ [J3: C] : ( times_times @ B @ ( F @ I2 ) @ ( G @ J3 ) )
@ B5 ) )
@ A5 ) ) ) ) ).
% sum_mset_product
thf(fact_6828_equiv__Eps__preserves,axiom,
! [A: $tType,A5: set @ A,R: set @ ( product_prod @ A @ A ),X8: set @ A] :
( ( equiv_equiv @ A @ A5 @ R )
=> ( ( member @ ( set @ A ) @ X8 @ ( equiv_quotient @ A @ A5 @ R ) )
=> ( member @ A
@ ( fChoice @ A
@ ^ [X: A] : ( member @ A @ X @ X8 ) )
@ A5 ) ) ) ).
% equiv_Eps_preserves
thf(fact_6829_equiv__Eps__in,axiom,
! [A: $tType,A5: set @ A,R: set @ ( product_prod @ A @ A ),X8: set @ A] :
( ( equiv_equiv @ A @ A5 @ R )
=> ( ( member @ ( set @ A ) @ X8 @ ( equiv_quotient @ A @ A5 @ R ) )
=> ( member @ A
@ ( fChoice @ A
@ ^ [X: A] : ( member @ A @ X @ X8 ) )
@ X8 ) ) ) ).
% equiv_Eps_in
thf(fact_6830_size__eq__sum__mset,axiom,
! [A: $tType] :
( ( size_size @ ( multiset @ A ) )
= ( ^ [M6: multiset @ A] :
( comm_m7189776963980413722m_mset @ nat
@ ( image_mset @ A @ nat
@ ^ [A3: A] : ( one_one @ nat )
@ M6 ) ) ) ) ).
% size_eq_sum_mset
thf(fact_6831_refines__equiv__image__eq,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),S2: set @ ( product_prod @ A @ A ),A5: set @ A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ S2 )
=> ( ( equiv_equiv @ A @ A5 @ R3 )
=> ( ( equiv_equiv @ A @ A5 @ S2 )
=> ( ( image2 @ ( set @ A ) @ ( set @ A ) @ ( image @ A @ A @ S2 ) @ ( equiv_quotient @ A @ A5 @ R3 ) )
= ( equiv_quotient @ A @ A5 @ S2 ) ) ) ) ) ).
% refines_equiv_image_eq
thf(fact_6832_UN__equiv__class2,axiom,
! [A: $tType,C: $tType,B: $tType,A17: set @ A,R1: set @ ( product_prod @ A @ A ),A26: set @ B,R22: set @ ( product_prod @ B @ B ),F: A > B > ( set @ C ),A13: A,A24: B] :
( ( equiv_equiv @ A @ A17 @ R1 )
=> ( ( equiv_equiv @ B @ A26 @ R22 )
=> ( ( equiv_congruent2 @ A @ B @ ( set @ C ) @ R1 @ R22 @ F )
=> ( ( member @ A @ A13 @ A17 )
=> ( ( member @ B @ A24 @ A26 )
=> ( ( complete_Sup_Sup @ ( set @ C )
@ ( image2 @ A @ ( set @ C )
@ ^ [X12: A] : ( complete_Sup_Sup @ ( set @ C ) @ ( image2 @ B @ ( set @ C ) @ ( F @ X12 ) @ ( image @ B @ B @ R22 @ ( insert @ B @ A24 @ ( bot_bot @ ( set @ B ) ) ) ) ) )
@ ( image @ A @ A @ R1 @ ( insert @ A @ A13 @ ( bot_bot @ ( set @ A ) ) ) ) ) )
= ( F @ A13 @ A24 ) ) ) ) ) ) ) ).
% UN_equiv_class2
thf(fact_6833_UN__equiv__class,axiom,
! [B: $tType,A: $tType,A5: set @ A,R: set @ ( product_prod @ A @ A ),F: A > ( set @ B ),A2: A] :
( ( equiv_equiv @ A @ A5 @ R )
=> ( ( equiv_congruent @ A @ ( set @ B ) @ R @ F )
=> ( ( member @ A @ A2 @ A5 )
=> ( ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ F @ ( image @ A @ A @ R @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) )
= ( F @ A2 ) ) ) ) ) ).
% UN_equiv_class
thf(fact_6834_congruent2__implies__congruent__UN,axiom,
! [B: $tType,C: $tType,A: $tType,A17: set @ A,R1: set @ ( product_prod @ A @ A ),A26: set @ B,R22: set @ ( product_prod @ B @ B ),F: A > B > ( set @ C ),A2: B] :
( ( equiv_equiv @ A @ A17 @ R1 )
=> ( ( equiv_equiv @ B @ A26 @ R22 )
=> ( ( equiv_congruent2 @ A @ B @ ( set @ C ) @ R1 @ R22 @ F )
=> ( ( member @ B @ A2 @ A26 )
=> ( equiv_congruent @ A @ ( set @ C ) @ R1
@ ^ [X12: A] : ( complete_Sup_Sup @ ( set @ C ) @ ( image2 @ B @ ( set @ C ) @ ( F @ X12 ) @ ( image @ B @ B @ R22 @ ( insert @ B @ A2 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ) ).
% congruent2_implies_congruent_UN
thf(fact_6835_UN__equiv__class__type,axiom,
! [A: $tType,B: $tType,A5: set @ A,R: set @ ( product_prod @ A @ A ),F: A > ( set @ B ),X8: set @ A,B5: set @ ( set @ B )] :
( ( equiv_equiv @ A @ A5 @ R )
=> ( ( equiv_congruent @ A @ ( set @ B ) @ R @ F )
=> ( ( member @ ( set @ A ) @ X8 @ ( equiv_quotient @ A @ A5 @ R ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ A5 )
=> ( member @ ( set @ B ) @ ( F @ X3 ) @ B5 ) )
=> ( member @ ( set @ B ) @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ F @ X8 ) ) @ B5 ) ) ) ) ) ).
% UN_equiv_class_type
thf(fact_6836_UN__equiv__class__type2,axiom,
! [A: $tType,B: $tType,C: $tType,A17: set @ A,R1: set @ ( product_prod @ A @ A ),A26: set @ B,R22: set @ ( product_prod @ B @ B ),F: A > B > ( set @ C ),X16: set @ A,X25: set @ B,B5: set @ ( set @ C )] :
( ( equiv_equiv @ A @ A17 @ R1 )
=> ( ( equiv_equiv @ B @ A26 @ R22 )
=> ( ( equiv_congruent2 @ A @ B @ ( set @ C ) @ R1 @ R22 @ F )
=> ( ( member @ ( set @ A ) @ X16 @ ( equiv_quotient @ A @ A17 @ R1 ) )
=> ( ( member @ ( set @ B ) @ X25 @ ( equiv_quotient @ B @ A26 @ R22 ) )
=> ( ! [X1: A,X23: B] :
( ( member @ A @ X1 @ A17 )
=> ( ( member @ B @ X23 @ A26 )
=> ( member @ ( set @ C ) @ ( F @ X1 @ X23 ) @ B5 ) ) )
=> ( member @ ( set @ C )
@ ( complete_Sup_Sup @ ( set @ C )
@ ( image2 @ A @ ( set @ C )
@ ^ [X12: A] : ( complete_Sup_Sup @ ( set @ C ) @ ( image2 @ B @ ( set @ C ) @ ( F @ X12 ) @ X25 ) )
@ X16 ) )
@ B5 ) ) ) ) ) ) ) ).
% UN_equiv_class_type2
thf(fact_6837_univ__preserves,axiom,
! [A: $tType,B: $tType,A5: set @ A,R: set @ ( product_prod @ A @ A ),F: A > B,B5: set @ B] :
( ( equiv_equiv @ A @ A5 @ R )
=> ( ( equiv_congruent @ A @ B @ R @ F )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ A5 )
=> ( member @ B @ ( F @ X3 ) @ B5 ) )
=> ! [X4: set @ A] :
( ( member @ ( set @ A ) @ X4 @ ( equiv_quotient @ A @ A5 @ R ) )
=> ( member @ B @ ( bNF_Greatest_univ @ A @ B @ F @ X4 ) @ B5 ) ) ) ) ) ).
% univ_preserves
thf(fact_6838_UN__equiv__class__inject,axiom,
! [B: $tType,A: $tType,A5: set @ A,R: set @ ( product_prod @ A @ A ),F: A > ( set @ B ),X8: set @ A,Y7: set @ A] :
( ( equiv_equiv @ A @ A5 @ R )
=> ( ( equiv_congruent @ A @ ( set @ B ) @ R @ F )
=> ( ( ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ F @ X8 ) )
= ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ F @ Y7 ) ) )
=> ( ( member @ ( set @ A ) @ X8 @ ( equiv_quotient @ A @ A5 @ R ) )
=> ( ( member @ ( set @ A ) @ Y7 @ ( equiv_quotient @ A @ A5 @ R ) )
=> ( ! [X3: A,Y2: A] :
( ( member @ A @ X3 @ A5 )
=> ( ( member @ A @ Y2 @ A5 )
=> ( ( ( F @ X3 )
= ( F @ Y2 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y2 ) @ R ) ) ) )
=> ( X8 = Y7 ) ) ) ) ) ) ) ).
% UN_equiv_class_inject
thf(fact_6839_univ__commute,axiom,
! [B: $tType,A: $tType,A5: set @ A,R: set @ ( product_prod @ A @ A ),F: A > B,X2: A] :
( ( equiv_equiv @ A @ A5 @ R )
=> ( ( equiv_congruent @ A @ B @ R @ F )
=> ( ( member @ A @ X2 @ A5 )
=> ( ( bNF_Greatest_univ @ A @ B @ F @ ( equiv_proj @ A @ A @ R @ X2 ) )
= ( F @ X2 ) ) ) ) ) ).
% univ_commute
thf(fact_6840_proj__Eps,axiom,
! [A: $tType,A5: set @ A,R: set @ ( product_prod @ A @ A ),X8: set @ A] :
( ( equiv_equiv @ A @ A5 @ R )
=> ( ( member @ ( set @ A ) @ X8 @ ( equiv_quotient @ A @ A5 @ R ) )
=> ( ( equiv_proj @ A @ A @ R
@ ( fChoice @ A
@ ^ [X: A] : ( member @ A @ X @ X8 ) ) )
= X8 ) ) ) ).
% proj_Eps
thf(fact_6841_equiv__proj,axiom,
! [A: $tType,A5: set @ A,R3: set @ ( product_prod @ A @ A ),Z2: product_prod @ A @ A] :
( ( equiv_equiv @ A @ A5 @ R3 )
=> ( ( member @ ( product_prod @ A @ A ) @ Z2 @ R3 )
=> ( ( comp @ A @ ( set @ A ) @ ( product_prod @ A @ A ) @ ( equiv_proj @ A @ A @ R3 ) @ ( product_fst @ A @ A ) @ Z2 )
= ( comp @ A @ ( set @ A ) @ ( product_prod @ A @ A ) @ ( equiv_proj @ A @ A @ R3 ) @ ( product_snd @ A @ A ) @ Z2 ) ) ) ) ).
% equiv_proj
thf(fact_6842_relImage__proj,axiom,
! [A: $tType,A5: set @ A,R3: set @ ( product_prod @ A @ A )] :
( ( equiv_equiv @ A @ A5 @ R3 )
=> ( ord_less_eq @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( bNF_Gr4221423524335903396lImage @ A @ ( set @ A ) @ R3 @ ( equiv_proj @ A @ A @ R3 ) ) @ ( id_on @ ( set @ A ) @ ( equiv_quotient @ A @ A5 @ R3 ) ) ) ) ).
% relImage_proj
thf(fact_6843_list_Oin__rel,axiom,
! [B: $tType,A: $tType] :
( ( list_all2 @ A @ B )
= ( ^ [R7: A > B > $o,A3: list @ A,B3: list @ B] :
? [Z3: list @ ( product_prod @ A @ B )] :
( ( member @ ( list @ ( product_prod @ A @ B ) ) @ Z3
@ ( collect @ ( list @ ( product_prod @ A @ B ) )
@ ^ [X: list @ ( product_prod @ A @ B )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( set2 @ ( product_prod @ A @ B ) @ X ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R7 ) ) ) ) )
& ( ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Z3 )
= A3 )
& ( ( map @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ Z3 )
= B3 ) ) ) ) ).
% list.in_rel
thf(fact_6844_list_Odisc__transfer_I2_J,axiom,
! [A: $tType,B: $tType,R3: A > B > $o] :
( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ $o @ $o @ ( list_all2 @ A @ B @ R3 )
@ ^ [Y3: $o,Z: $o] : Y3 = Z
@ ^ [List2: list @ A] :
( List2
!= ( nil @ A ) )
@ ^ [List2: list @ B] :
( List2
!= ( nil @ B ) ) ) ).
% list.disc_transfer(2)
thf(fact_6845_list_Odisc__transfer_I1_J,axiom,
! [A: $tType,B: $tType,R3: A > B > $o] :
( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ $o @ $o @ ( list_all2 @ A @ B @ R3 )
@ ^ [Y3: $o,Z: $o] : Y3 = Z
@ ^ [List2: list @ A] :
( List2
= ( nil @ A ) )
@ ^ [List2: list @ B] :
( List2
= ( nil @ B ) ) ) ).
% list.disc_transfer(1)
thf(fact_6846_length__transfer,axiom,
! [A: $tType,B: $tType,A5: A > B > $o] :
( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ nat @ nat @ ( list_all2 @ A @ B @ A5 )
@ ^ [Y3: nat,Z: nat] : Y3 = Z
@ ( size_size @ ( list @ A ) )
@ ( size_size @ ( list @ B ) ) ) ).
% length_transfer
thf(fact_6847_list_Orel__map_I2_J,axiom,
! [A: $tType,C: $tType,B: $tType,Sa: A > C > $o,X2: list @ A,G: B > C,Y: list @ B] :
( ( list_all2 @ A @ C @ Sa @ X2 @ ( map @ B @ C @ G @ Y ) )
= ( list_all2 @ A @ B
@ ^ [X: A,Y4: B] : ( Sa @ X @ ( G @ Y4 ) )
@ X2
@ Y ) ) ).
% list.rel_map(2)
thf(fact_6848_list_Orel__map_I1_J,axiom,
! [A: $tType,C: $tType,B: $tType,Sb: C > B > $o,I: A > C,X2: list @ A,Y: list @ B] :
( ( list_all2 @ C @ B @ Sb @ ( map @ A @ C @ I @ X2 ) @ Y )
= ( list_all2 @ A @ B
@ ^ [X: A] : ( Sb @ ( I @ X ) )
@ X2
@ Y ) ) ).
% list.rel_map(1)
thf(fact_6849_list__all2__map1,axiom,
! [C: $tType,A: $tType,B: $tType,P: A > B > $o,F: C > A,As: list @ C,Bs: list @ B] :
( ( list_all2 @ A @ B @ P @ ( map @ C @ A @ F @ As ) @ Bs )
= ( list_all2 @ C @ B
@ ^ [X: C] : ( P @ ( F @ X ) )
@ As
@ Bs ) ) ).
% list_all2_map1
thf(fact_6850_list__all2__map2,axiom,
! [A: $tType,B: $tType,C: $tType,P: A > B > $o,As: list @ A,F: C > B,Bs: list @ C] :
( ( list_all2 @ A @ B @ P @ As @ ( map @ C @ B @ F @ Bs ) )
= ( list_all2 @ A @ C
@ ^ [X: A,Y4: C] : ( P @ X @ ( F @ Y4 ) )
@ As
@ Bs ) ) ).
% list_all2_map2
thf(fact_6851_Misc_Olist__all2__induct,axiom,
! [A: $tType,B: $tType,P: A > B > $o,L: list @ A,L4: list @ B,Q: ( list @ A ) > ( list @ B ) > $o] :
( ( list_all2 @ A @ B @ P @ L @ L4 )
=> ( ( Q @ ( nil @ A ) @ ( nil @ B ) )
=> ( ! [X3: A,X11: B,Ls2: list @ A,Ls3: list @ B] :
( ( P @ X3 @ X11 )
=> ( ( list_all2 @ A @ B @ P @ Ls2 @ Ls3 )
=> ( ( Q @ Ls2 @ Ls3 )
=> ( Q @ ( cons @ A @ X3 @ Ls2 ) @ ( cons @ B @ X11 @ Ls3 ) ) ) ) )
=> ( Q @ L @ L4 ) ) ) ) ).
% Misc.list_all2_induct
thf(fact_6852_list__all2__append2,axiom,
! [B: $tType,A: $tType,P: A > B > $o,Xs: list @ A,Ys: list @ B,Zs: list @ B] :
( ( list_all2 @ A @ B @ P @ Xs @ ( append @ B @ Ys @ Zs ) )
= ( ? [Us2: list @ A,Vs2: list @ A] :
( ( Xs
= ( append @ A @ Us2 @ Vs2 ) )
& ( ( size_size @ ( list @ A ) @ Us2 )
= ( size_size @ ( list @ B ) @ Ys ) )
& ( ( size_size @ ( list @ A ) @ Vs2 )
= ( size_size @ ( list @ B ) @ Zs ) )
& ( list_all2 @ A @ B @ P @ Us2 @ Ys )
& ( list_all2 @ A @ B @ P @ Vs2 @ Zs ) ) ) ) ).
% list_all2_append2
thf(fact_6853_list__all2__append1,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Xs: list @ A,Ys: list @ A,Zs: list @ B] :
( ( list_all2 @ A @ B @ P @ ( append @ A @ Xs @ Ys ) @ Zs )
= ( ? [Us2: list @ B,Vs2: list @ B] :
( ( Zs
= ( append @ B @ Us2 @ Vs2 ) )
& ( ( size_size @ ( list @ B ) @ Us2 )
= ( size_size @ ( list @ A ) @ Xs ) )
& ( ( size_size @ ( list @ B ) @ Vs2 )
= ( size_size @ ( list @ A ) @ Ys ) )
& ( list_all2 @ A @ B @ P @ Xs @ Us2 )
& ( list_all2 @ A @ B @ P @ Ys @ Vs2 ) ) ) ) ).
% list_all2_append1
thf(fact_6854_list__all2__append,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B,P: A > B > $o,Us: list @ A,Vs: list @ B] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( list_all2 @ A @ B @ P @ ( append @ A @ Xs @ Us ) @ ( append @ B @ Ys @ Vs ) )
= ( ( list_all2 @ A @ B @ P @ Xs @ Ys )
& ( list_all2 @ A @ B @ P @ Us @ Vs ) ) ) ) ).
% list_all2_append
thf(fact_6855_list__all2__lengthD,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Xs: list @ A,Ys: list @ B] :
( ( list_all2 @ A @ B @ P @ Xs @ Ys )
=> ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) ) ) ).
% list_all2_lengthD
thf(fact_6856_relImage__mono,axiom,
! [B: $tType,A: $tType,R12: set @ ( product_prod @ A @ A ),R23: set @ ( product_prod @ A @ A ),F: A > B] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R12 @ R23 )
=> ( ord_less_eq @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Gr4221423524335903396lImage @ A @ B @ R12 @ F ) @ ( bNF_Gr4221423524335903396lImage @ A @ B @ R23 @ F ) ) ) ).
% relImage_mono
thf(fact_6857_list__all2__conv__all__nth,axiom,
! [B: $tType,A: $tType] :
( ( list_all2 @ A @ B )
= ( ^ [P4: A > B > $o,Xs3: list @ A,Ys3: list @ B] :
( ( ( size_size @ ( list @ A ) @ Xs3 )
= ( size_size @ ( list @ B ) @ Ys3 ) )
& ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs3 ) )
=> ( P4 @ ( nth @ A @ Xs3 @ I2 ) @ ( nth @ B @ Ys3 @ I2 ) ) ) ) ) ) ).
% list_all2_conv_all_nth
thf(fact_6858_list__all2__all__nthI,axiom,
! [A: $tType,B: $tType,A2: list @ A,B2: list @ B,P: A > B > $o] :
( ( ( size_size @ ( list @ A ) @ A2 )
= ( size_size @ ( list @ B ) @ B2 ) )
=> ( ! [N2: nat] :
( ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ A2 ) )
=> ( P @ ( nth @ A @ A2 @ N2 ) @ ( nth @ B @ B2 @ N2 ) ) )
=> ( list_all2 @ A @ B @ P @ A2 @ B2 ) ) ) ).
% list_all2_all_nthI
thf(fact_6859_list__all2__nthD2,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Xs: list @ A,Ys: list @ B,P3: nat] :
( ( list_all2 @ A @ B @ P @ Xs @ Ys )
=> ( ( ord_less @ nat @ P3 @ ( size_size @ ( list @ B ) @ Ys ) )
=> ( P @ ( nth @ A @ Xs @ P3 ) @ ( nth @ B @ Ys @ P3 ) ) ) ) ).
% list_all2_nthD2
thf(fact_6860_list__all2__nthD,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Xs: list @ A,Ys: list @ B,P3: nat] :
( ( list_all2 @ A @ B @ P @ Xs @ Ys )
=> ( ( ord_less @ nat @ P3 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( P @ ( nth @ A @ Xs @ P3 ) @ ( nth @ B @ Ys @ P3 ) ) ) ) ).
% list_all2_nthD
thf(fact_6861_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 )
@ ^ [X: A,Ys3: list @ A] : ( member @ A @ X @ ( set2 @ A @ Ys3 ) )
@ Xs3
@ Xss ) ) ) ).
% product_lists_set
thf(fact_6862_list__all2I,axiom,
! [A: $tType,B: $tType,A2: list @ A,B2: list @ B,P: A > B > $o] :
( ! [X3: product_prod @ A @ B] :
( ( member @ ( product_prod @ A @ B ) @ X3 @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ A2 @ B2 ) ) )
=> ( product_case_prod @ A @ B @ $o @ P @ X3 ) )
=> ( ( ( size_size @ ( list @ A ) @ A2 )
= ( size_size @ ( list @ B ) @ B2 ) )
=> ( list_all2 @ A @ B @ P @ A2 @ B2 ) ) ) ).
% list_all2I
thf(fact_6863_sum__list__transfer,axiom,
! [A: $tType,B: $tType] :
( ( ( monoid_add @ B )
& ( monoid_add @ A ) )
=> ! [A5: A > B > $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 @ ( list @ A ) @ ( list @ B ) @ A @ B @ ( list_all2 @ A @ B @ A5 ) @ A5 @ ( groups8242544230860333062m_list @ A ) @ ( groups8242544230860333062m_list @ B ) ) ) ) ) ).
% sum_list_transfer
thf(fact_6864_relImage__def,axiom,
! [A: $tType,B: $tType] :
( ( bNF_Gr4221423524335903396lImage @ B @ A )
= ( ^ [R7: set @ ( product_prod @ B @ B ),F2: B > A] :
( collect @ ( product_prod @ A @ A )
@ ^ [Uu: product_prod @ A @ A] :
? [A12: B,A23: B] :
( ( Uu
= ( product_Pair @ A @ A @ ( F2 @ A12 ) @ ( F2 @ A23 ) ) )
& ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ A12 @ A23 ) @ R7 ) ) ) ) ) ).
% relImage_def
thf(fact_6865_horner__sum__transfer,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType] :
( ( ( comm_semiring_0 @ B )
& ( comm_semiring_0 @ A ) )
=> ! [A5: A > B > $o,B5: 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 @ B5 @ 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 @ B5 ) @ A5 ) ) @ ( groups4207007520872428315er_sum @ C @ A ) @ ( groups4207007520872428315er_sum @ D @ B ) ) ) ) ) ) ).
% horner_sum_transfer
thf(fact_6866_relImage__relInvImage,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),F: B > A,A5: set @ B] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ ( image2 @ B @ A @ F @ A5 )
@ ^ [Uu: A] : ( image2 @ B @ A @ F @ A5 ) ) )
=> ( ( bNF_Gr4221423524335903396lImage @ B @ A @ ( bNF_Gr7122648621184425601vImage @ B @ A @ A5 @ R3 @ F ) @ F )
= R3 ) ) ).
% relImage_relInvImage
thf(fact_6867_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_6868_prod__list_ONil,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ( ( groups5270119922927024881d_list @ A @ ( nil @ A ) )
= ( one_one @ A ) ) ) ).
% prod_list.Nil
thf(fact_6869_relInvImage__mono,axiom,
! [A: $tType,B: $tType,R12: set @ ( product_prod @ A @ A ),R23: set @ ( product_prod @ A @ A ),A5: set @ B,F: B > A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R12 @ R23 )
=> ( ord_less_eq @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Gr7122648621184425601vImage @ B @ A @ A5 @ R12 @ F ) @ ( bNF_Gr7122648621184425601vImage @ B @ A @ A5 @ R23 @ F ) ) ) ).
% relInvImage_mono
thf(fact_6870_prod__list__zero__iff,axiom,
! [A: $tType] :
( ( ( semiring_1 @ A )
& ( semiri3467727345109120633visors @ A ) )
=> ! [Xs: list @ A] :
( ( ( groups5270119922927024881d_list @ A @ Xs )
= ( zero_zero @ A ) )
= ( member @ A @ ( zero_zero @ A ) @ ( set2 @ A @ Xs ) ) ) ) ).
% prod_list_zero_iff
thf(fact_6871_prod__list__coprime__left,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [Xs: list @ A,A2: A] :
( ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( algebr8660921524188924756oprime @ A @ X3 @ A2 ) )
=> ( algebr8660921524188924756oprime @ A @ ( groups5270119922927024881d_list @ A @ Xs ) @ A2 ) ) ) ).
% prod_list_coprime_left
thf(fact_6872_prod__list__coprime__right,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [Xs: list @ A,A2: A] :
( ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( algebr8660921524188924756oprime @ A @ A2 @ X3 ) )
=> ( algebr8660921524188924756oprime @ A @ A2 @ ( groups5270119922927024881d_list @ A @ Xs ) ) ) ) ).
% prod_list_coprime_right
thf(fact_6873_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_6874_relInvImage__def,axiom,
! [B: $tType,A: $tType] :
( ( bNF_Gr7122648621184425601vImage @ A @ B )
= ( ^ [A8: set @ A,R7: set @ ( product_prod @ B @ B ),F2: A > B] :
( collect @ ( product_prod @ A @ A )
@ ^ [Uu: product_prod @ A @ A] :
? [A12: A,A23: A] :
( ( Uu
= ( product_Pair @ A @ A @ A12 @ A23 ) )
& ( member @ A @ A12 @ A8 )
& ( member @ A @ A23 @ A8 )
& ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F2 @ A12 ) @ ( F2 @ A23 ) ) @ R7 ) ) ) ) ) ).
% relInvImage_def
thf(fact_6875_relInvImage__UNIV__relImage,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),F: A > B] : ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ ( bNF_Gr7122648621184425601vImage @ A @ B @ ( top_top @ ( set @ A ) ) @ ( bNF_Gr4221423524335903396lImage @ A @ B @ R3 @ F ) @ F ) ) ).
% relInvImage_UNIV_relImage
thf(fact_6876_Some__image__these__eq,axiom,
! [A: $tType,A5: set @ ( option @ A )] :
( ( image2 @ A @ ( option @ A ) @ ( some @ A ) @ ( these @ A @ A5 ) )
= ( collect @ ( option @ A )
@ ^ [X: option @ A] :
( ( member @ ( option @ A ) @ X @ A5 )
& ( X
!= ( none @ A ) ) ) ) ) ).
% Some_image_these_eq
thf(fact_6877_possible__bit__def,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ( ( bit_se6407376104438227557le_bit @ A )
= ( ^ [Tyrep: itself @ A,N4: nat] :
( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N4 )
!= ( zero_zero @ A ) ) ) ) ) ).
% possible_bit_def
thf(fact_6878_possible__bit__0,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [Ty: itself @ A] : ( bit_se6407376104438227557le_bit @ A @ Ty @ ( zero_zero @ nat ) ) ) ).
% possible_bit_0
thf(fact_6879_possible__bit__less__imp,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [Tyrep2: itself @ A,I: nat,J: nat] :
( ( bit_se6407376104438227557le_bit @ A @ Tyrep2 @ I )
=> ( ( ord_less_eq @ nat @ J @ I )
=> ( bit_se6407376104438227557le_bit @ A @ Tyrep2 @ J ) ) ) ) ).
% possible_bit_less_imp
thf(fact_6880_Option_Othese__def,axiom,
! [A: $tType] :
( ( these @ A )
= ( ^ [A8: set @ ( option @ A )] :
( image2 @ ( option @ A ) @ A @ ( the2 @ A )
@ ( collect @ ( option @ A )
@ ^ [X: option @ A] :
( ( member @ ( option @ A ) @ X @ A8 )
& ( X
!= ( none @ A ) ) ) ) ) ) ) ).
% Option.these_def
thf(fact_6881_drop__bit__exp__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,N: nat] :
( ( bit_se4197421643247451524op_bit @ A @ M @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
= ( times_times @ A
@ ( zero_neq_one_of_bool @ A
@ ( ( ord_less_eq @ nat @ M @ N )
& ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N ) ) )
@ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N @ M ) ) ) ) ) ).
% drop_bit_exp_eq
thf(fact_6882_bit__minus__2__iff,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ N )
= ( ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% bit_minus_2_iff
thf(fact_6883_CHAR__eq__0,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiri4206861660011772517g_char @ A @ ( type2 @ A ) )
= ( zero_zero @ nat ) ) ) ).
% CHAR_eq_0
thf(fact_6884_of__nat__CHAR,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( semiring_1_of_nat @ A @ ( semiri4206861660011772517g_char @ A @ ( type2 @ A ) ) )
= ( zero_zero @ A ) ) ) ).
% of_nat_CHAR
thf(fact_6885_bit__minus__1__iff,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N )
= ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N ) ) ) ).
% bit_minus_1_iff
thf(fact_6886_CHAR__eqI,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [C2: nat] :
( ( ( semiring_1_of_nat @ A @ C2 )
= ( zero_zero @ A ) )
=> ( ! [X3: nat] :
( ( ( semiring_1_of_nat @ A @ X3 )
= ( zero_zero @ A ) )
=> ( dvd_dvd @ nat @ C2 @ X3 ) )
=> ( ( semiri4206861660011772517g_char @ A @ ( type2 @ A ) )
= C2 ) ) ) ) ).
% CHAR_eqI
thf(fact_6887_of__nat__eq__0__iff__char__dvd,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [N: nat] :
( ( ( semiring_1_of_nat @ A @ N )
= ( zero_zero @ A ) )
= ( dvd_dvd @ nat @ ( semiri4206861660011772517g_char @ A @ ( type2 @ A ) ) @ N ) ) ) ).
% of_nat_eq_0_iff_char_dvd
thf(fact_6888_bit__of__nat__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [M: nat,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( semiring_1_of_nat @ A @ M ) @ N )
= ( ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N )
& ( bit_se5641148757651400278ts_bit @ nat @ M @ N ) ) ) ) ).
% bit_of_nat_iff
thf(fact_6889_bit__minus__iff,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A2: A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( uminus_uminus @ A @ A2 ) @ N )
= ( ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N )
& ~ ( bit_se5641148757651400278ts_bit @ A @ ( minus_minus @ A @ A2 @ ( one_one @ A ) ) @ N ) ) ) ) ).
% bit_minus_iff
thf(fact_6890_bit__push__bit__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,A2: A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( bit_se4730199178511100633sh_bit @ A @ M @ A2 ) @ N )
= ( ( ord_less_eq @ nat @ M @ N )
& ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N )
& ( bit_se5641148757651400278ts_bit @ A @ A2 @ ( minus_minus @ nat @ N @ M ) ) ) ) ) ).
% bit_push_bit_iff
thf(fact_6891_CHAR__pos__iff,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( semiri4206861660011772517g_char @ A @ ( type2 @ A ) ) )
= ( ? [N4: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N4 )
& ( ( semiring_1_of_nat @ A @ N4 )
= ( zero_zero @ A ) ) ) ) ) ) ).
% CHAR_pos_iff
thf(fact_6892_CHAR__eq__posI,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [C2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ C2 )
=> ( ( ( semiring_1_of_nat @ A @ C2 )
= ( zero_zero @ A ) )
=> ( ! [X3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ X3 )
=> ( ( ord_less @ nat @ X3 @ C2 )
=> ( ( semiring_1_of_nat @ A @ X3 )
!= ( zero_zero @ A ) ) ) )
=> ( ( semiri4206861660011772517g_char @ A @ ( type2 @ A ) )
= C2 ) ) ) ) ) ).
% CHAR_eq_posI
thf(fact_6893_CHAR__eq0__iff,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( ( semiri4206861660011772517g_char @ A @ ( type2 @ A ) )
= ( zero_zero @ nat ) )
= ( ! [N4: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N4 )
=> ( ( semiring_1_of_nat @ A @ N4 )
!= ( zero_zero @ A ) ) ) ) ) ) ).
% CHAR_eq0_iff
thf(fact_6894_fold__possible__bit,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [N: nat] :
( ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N )
= ( zero_zero @ A ) )
= ( ~ ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N ) ) ) ) ).
% fold_possible_bit
thf(fact_6895_bit__2__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N )
= ( ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ ( one_one @ nat ) )
& ( N
= ( one_one @ nat ) ) ) ) ) ).
% bit_2_iff
thf(fact_6896_bit__minus__exp__iff,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [M: nat,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( uminus_uminus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) ) @ N )
= ( ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N )
& ( ord_less_eq @ nat @ M @ N ) ) ) ) ).
% bit_minus_exp_iff
thf(fact_6897_bit__mask__sub__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [M: nat,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) @ ( one_one @ A ) ) @ N )
= ( ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N )
& ( ord_less @ nat @ N @ M ) ) ) ) ).
% bit_mask_sub_iff
thf(fact_6898_bit__double__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A2: A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) @ N )
= ( ( bit_se5641148757651400278ts_bit @ A @ A2 @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) )
& ( N
!= ( zero_zero @ nat ) )
& ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N ) ) ) ) ).
% bit_double_iff
thf(fact_6899_Gcd__fin_Oeq__fold,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ( ( semiring_gcd_Gcd_fin @ A )
= ( ^ [A8: set @ A] : ( if @ A @ ( finite_finite2 @ A @ A8 ) @ ( finite_fold @ A @ A @ ( gcd_gcd @ A ) @ ( zero_zero @ A ) @ A8 ) @ ( one_one @ A ) ) ) ) ) ).
% Gcd_fin.eq_fold
thf(fact_6900_fold__atLeastAtMost__nat_Opsimps,axiom,
! [A: $tType,F: nat > A > A,A2: nat,B2: nat,Acc4: A] :
( ( accp @ ( product_prod @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) ) @ ( set_fo1817059534552279752at_rel @ A ) @ ( product_Pair @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) @ F @ ( product_Pair @ nat @ ( product_prod @ nat @ A ) @ A2 @ ( product_Pair @ nat @ A @ B2 @ Acc4 ) ) ) )
=> ( ( ( ord_less @ nat @ B2 @ A2 )
=> ( ( set_fo6178422350223883121st_nat @ A @ F @ A2 @ B2 @ Acc4 )
= Acc4 ) )
& ( ~ ( ord_less @ nat @ B2 @ A2 )
=> ( ( set_fo6178422350223883121st_nat @ A @ F @ A2 @ B2 @ Acc4 )
= ( set_fo6178422350223883121st_nat @ A @ F @ ( plus_plus @ nat @ A2 @ ( one_one @ nat ) ) @ B2 @ ( F @ A2 @ Acc4 ) ) ) ) ) ) ).
% fold_atLeastAtMost_nat.psimps
thf(fact_6901_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_6902_Gcd__fin_Oinfinite,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A5: set @ A] :
( ~ ( finite_finite2 @ A @ A5 )
=> ( ( semiring_gcd_Gcd_fin @ A @ A5 )
= ( one_one @ A ) ) ) ) ).
% Gcd_fin.infinite
thf(fact_6903_is__unit__Gcd__fin__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A5: set @ A] :
( ( dvd_dvd @ A @ ( semiring_gcd_Gcd_fin @ A @ A5 ) @ ( one_one @ A ) )
= ( ( semiring_gcd_Gcd_fin @ A @ A5 )
= ( one_one @ A ) ) ) ) ).
% is_unit_Gcd_fin_iff
thf(fact_6904_Gcd__fin_Oinsert,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A2: A,A5: set @ A] :
( ( semiring_gcd_Gcd_fin @ A @ ( insert @ A @ A2 @ A5 ) )
= ( gcd_gcd @ A @ A2 @ ( semiring_gcd_Gcd_fin @ A @ A5 ) ) ) ) ).
% Gcd_fin.insert
thf(fact_6905_Gcd__fin__eq__Gcd,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( semiring_gcd_Gcd_fin @ A @ A5 )
= ( gcd_Gcd @ A @ A5 ) ) ) ) ).
% Gcd_fin_eq_Gcd
thf(fact_6906_Gcd__fin_Ounion,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A5: set @ A,B5: set @ A] :
( ( semiring_gcd_Gcd_fin @ A @ ( sup_sup @ ( set @ A ) @ A5 @ B5 ) )
= ( gcd_gcd @ A @ ( semiring_gcd_Gcd_fin @ A @ A5 ) @ ( semiring_gcd_Gcd_fin @ A @ B5 ) ) ) ) ).
% Gcd_fin.union
thf(fact_6907_Gcd__fin_Oin__idem,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A2: A,A5: set @ A] :
( ( member @ A @ A2 @ A5 )
=> ( ( gcd_gcd @ A @ A2 @ ( semiring_gcd_Gcd_fin @ A @ A5 ) )
= ( semiring_gcd_Gcd_fin @ A @ A5 ) ) ) ) ).
% Gcd_fin.in_idem
thf(fact_6908_Gcd__fin__dvd,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A2: A,A5: set @ A] :
( ( member @ A @ A2 @ A5 )
=> ( dvd_dvd @ A @ ( semiring_gcd_Gcd_fin @ A @ A5 ) @ A2 ) ) ) ).
% Gcd_fin_dvd
thf(fact_6909_gcd__list__greatest,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [Bs: list @ A,A2: A] :
( ! [B4: A] :
( ( member @ A @ B4 @ ( set2 @ A @ Bs ) )
=> ( dvd_dvd @ A @ A2 @ B4 ) )
=> ( dvd_dvd @ A @ A2 @ ( semiring_gcd_Gcd_fin @ A @ ( set2 @ A @ Bs ) ) ) ) ) ).
% gcd_list_greatest
thf(fact_6910_dvd__gcd__list__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [B2: A,Xs: list @ A] :
( ( dvd_dvd @ A @ B2 @ ( semiring_gcd_Gcd_fin @ A @ ( set2 @ A @ Xs ) ) )
= ( ! [X: A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( dvd_dvd @ A @ B2 @ X ) ) ) ) ) ).
% dvd_gcd_list_iff
thf(fact_6911_Gcd__fin__greatest,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A5: set @ A,A2: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ! [B4: A] :
( ( member @ A @ B4 @ A5 )
=> ( dvd_dvd @ A @ A2 @ B4 ) )
=> ( dvd_dvd @ A @ A2 @ ( semiring_gcd_Gcd_fin @ A @ A5 ) ) ) ) ) ).
% Gcd_fin_greatest
thf(fact_6912_dvd__Gcd__fin__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A5: set @ A,B2: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( dvd_dvd @ A @ B2 @ ( semiring_gcd_Gcd_fin @ A @ A5 ) )
= ( ! [X: A] :
( ( member @ A @ X @ A5 )
=> ( dvd_dvd @ A @ B2 @ X ) ) ) ) ) ) ).
% dvd_Gcd_fin_iff
thf(fact_6913_Gcd__fin_Osubset,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [B5: set @ A,A5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B5 @ A5 )
=> ( ( gcd_gcd @ A @ ( semiring_gcd_Gcd_fin @ A @ B5 ) @ ( semiring_gcd_Gcd_fin @ A @ A5 ) )
= ( semiring_gcd_Gcd_fin @ A @ A5 ) ) ) ) ).
% Gcd_fin.subset
thf(fact_6914_Gcd__fin_Oremove,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A2: A,A5: set @ A] :
( ( member @ A @ A2 @ A5 )
=> ( ( semiring_gcd_Gcd_fin @ A @ A5 )
= ( gcd_gcd @ A @ A2 @ ( semiring_gcd_Gcd_fin @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% Gcd_fin.remove
thf(fact_6915_Gcd__fin_Oinsert__remove,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A2: A,A5: set @ A] :
( ( semiring_gcd_Gcd_fin @ A @ ( insert @ A @ A2 @ A5 ) )
= ( gcd_gcd @ A @ A2 @ ( semiring_gcd_Gcd_fin @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% Gcd_fin.insert_remove
thf(fact_6916_fold__atLeastAtMost__nat_Opinduct,axiom,
! [A: $tType,A0: nat > A > A,A13: nat,A24: nat,A33: A,P: ( nat > A > A ) > nat > nat > A > $o] :
( ( accp @ ( product_prod @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) ) @ ( set_fo1817059534552279752at_rel @ A ) @ ( product_Pair @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) @ A0 @ ( product_Pair @ nat @ ( product_prod @ nat @ A ) @ A13 @ ( product_Pair @ nat @ A @ A24 @ A33 ) ) ) )
=> ( ! [F5: nat > A > A,A4: nat,B4: 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 ) ) @ F5 @ ( product_Pair @ nat @ ( product_prod @ nat @ A ) @ A4 @ ( product_Pair @ nat @ A @ B4 @ Acc3 ) ) ) )
=> ( ( ~ ( ord_less @ nat @ B4 @ A4 )
=> ( P @ F5 @ ( plus_plus @ nat @ A4 @ ( one_one @ nat ) ) @ B4 @ ( F5 @ A4 @ Acc3 ) ) )
=> ( P @ F5 @ A4 @ B4 @ Acc3 ) ) )
=> ( P @ A0 @ A13 @ A24 @ A33 ) ) ) ).
% fold_atLeastAtMost_nat.pinduct
thf(fact_6917_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_6918_fold__atLeastAtMost__nat_Opelims,axiom,
! [A: $tType,X2: nat > A > A,Xa: nat,Xb: nat,Xc: A,Y: A] :
( ( ( set_fo6178422350223883121st_nat @ A @ X2 @ 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 ) ) @ X2 @ ( 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 @ X2 @ ( plus_plus @ nat @ Xa @ ( one_one @ nat ) ) @ Xb @ ( X2 @ 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 ) ) @ X2 @ ( product_Pair @ nat @ ( product_prod @ nat @ A ) @ Xa @ ( product_Pair @ nat @ A @ Xb @ Xc ) ) ) ) ) ) ) ).
% fold_atLeastAtMost_nat.pelims
thf(fact_6919_pairself_Opelims,axiom,
! [B: $tType,A: $tType,X2: A > B,Xa: product_prod @ A @ A,Y: product_prod @ B @ B] :
( ( ( pairself @ A @ B @ X2 @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ ( A > B ) @ ( product_prod @ A @ A ) ) @ ( pairself_rel @ A @ B ) @ ( product_Pair @ ( A > B ) @ ( product_prod @ A @ A ) @ X2 @ Xa ) )
=> ~ ! [A4: A,B4: A] :
( ( Xa
= ( product_Pair @ A @ A @ A4 @ B4 ) )
=> ( ( Y
= ( product_Pair @ B @ B @ ( X2 @ A4 ) @ ( X2 @ B4 ) ) )
=> ~ ( accp @ ( product_prod @ ( A > B ) @ ( product_prod @ A @ A ) ) @ ( pairself_rel @ A @ B ) @ ( product_Pair @ ( A > B ) @ ( product_prod @ A @ A ) @ X2 @ ( product_Pair @ A @ A @ A4 @ B4 ) ) ) ) ) ) ) ).
% pairself.pelims
thf(fact_6920_finite__enum__subset,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [X8: set @ A,Y7: set @ A] :
( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( finite_card @ A @ X8 ) )
=> ( ( infini527867602293511546merate @ A @ X8 @ I3 )
= ( infini527867602293511546merate @ A @ Y7 @ I3 ) ) )
=> ( ( finite_finite2 @ A @ X8 )
=> ( ( finite_finite2 @ A @ Y7 )
=> ( ( ord_less_eq @ nat @ ( finite_card @ A @ X8 ) @ ( finite_card @ A @ Y7 ) )
=> ( ord_less_eq @ ( set @ A ) @ X8 @ Y7 ) ) ) ) ) ) ).
% finite_enum_subset
thf(fact_6921_le__enumerate,axiom,
! [S2: set @ nat,N: nat] :
( ~ ( finite_finite2 @ nat @ S2 )
=> ( ord_less_eq @ nat @ N @ ( infini527867602293511546merate @ nat @ S2 @ N ) ) ) ).
% le_enumerate
thf(fact_6922_finite__le__enumerate,axiom,
! [S2: set @ nat,N: nat] :
( ( finite_finite2 @ nat @ S2 )
=> ( ( ord_less @ nat @ N @ ( finite_card @ nat @ S2 ) )
=> ( ord_less_eq @ nat @ N @ ( infini527867602293511546merate @ nat @ S2 @ N ) ) ) ) ).
% finite_le_enumerate
thf(fact_6923_enumerate__Suc_H,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S2: set @ A,N: nat] :
( ( infini527867602293511546merate @ A @ S2 @ ( suc @ N ) )
= ( infini527867602293511546merate @ A @ ( minus_minus @ ( set @ A ) @ S2 @ ( insert @ A @ ( infini527867602293511546merate @ A @ S2 @ ( zero_zero @ nat ) ) @ ( bot_bot @ ( set @ A ) ) ) ) @ N ) ) ) ).
% enumerate_Suc'
thf(fact_6924_partition__rev_Opelims,axiom,
! [A: $tType,X2: A > $o,Xa: product_prod @ ( list @ A ) @ ( list @ A ),Xb: list @ A,Y: product_prod @ ( list @ A ) @ ( list @ A )] :
( ( ( partition_rev @ A @ X2 @ 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 ) ) @ X2 @ ( 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 ) ) @ X2 @ ( 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 @ X2 @ ( if @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( X2 @ 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 ) ) @ X2 @ ( 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_6925_quicksort__by__rel_Opsimps_I2_J,axiom,
! [A: $tType,R3: A > A > $o,Sl2: list @ A,X2: 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 ) ) @ R3 @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Sl2 @ ( cons @ A @ X2 @ Xs ) ) ) )
=> ( ( quicksort_by_rel @ A @ R3 @ Sl2 @ ( cons @ A @ X2 @ Xs ) )
= ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ ( list @ A )
@ ^ [Xs_s: list @ A,Xs_b: list @ A] : ( quicksort_by_rel @ A @ R3 @ ( cons @ A @ X2 @ ( quicksort_by_rel @ A @ R3 @ Sl2 @ Xs_b ) ) @ Xs_s )
@ ( partition_rev @ A
@ ^ [Y4: A] : ( R3 @ Y4 @ X2 )
@ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) )
@ Xs ) ) ) ) ).
% quicksort_by_rel.psimps(2)
thf(fact_6926_quicksort__by__rel_Opsimps_I1_J,axiom,
! [A: $tType,R3: 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 ) ) @ R3 @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Sl2 @ ( nil @ A ) ) ) )
=> ( ( quicksort_by_rel @ A @ R3 @ Sl2 @ ( nil @ A ) )
= Sl2 ) ) ).
% quicksort_by_rel.psimps(1)
thf(fact_6927_quicksort__by__rel_Opinduct,axiom,
! [A: $tType,A0: A > A > $o,A13: list @ A,A24: 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 ) @ A13 @ A24 ) ) )
=> ( ! [R5: A > A > $o,Sl: list @ A] :
( ( accp @ ( product_prod @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( quicksort_by_rel_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ R5 @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Sl @ ( nil @ A ) ) ) )
=> ( P @ R5 @ Sl @ ( nil @ A ) ) )
=> ( ! [R5: A > A > $o,Sl: list @ A,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 ) ) @ R5 @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Sl @ ( cons @ A @ X3 @ Xs2 ) ) ) )
=> ( ! [Xa2: product_prod @ ( list @ A ) @ ( list @ A ),Xb2: list @ A,Y5: list @ A] :
( ( Xa2
= ( partition_rev @ A
@ ^ [Z3: A] : ( R5 @ Z3 @ X3 )
@ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) )
@ Xs2 ) )
=> ( ( ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xb2 @ Y5 )
= Xa2 )
=> ( P @ R5 @ Sl @ Y5 ) ) )
=> ( ! [Xa2: product_prod @ ( list @ A ) @ ( list @ A ),Xb2: list @ A,Y5: list @ A] :
( ( Xa2
= ( partition_rev @ A
@ ^ [Z3: A] : ( R5 @ Z3 @ X3 )
@ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) )
@ Xs2 ) )
=> ( ( ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xb2 @ Y5 )
= Xa2 )
=> ( P @ R5 @ ( cons @ A @ X3 @ ( quicksort_by_rel @ A @ R5 @ Sl @ Y5 ) ) @ Xb2 ) ) )
=> ( P @ R5 @ Sl @ ( cons @ A @ X3 @ Xs2 ) ) ) ) )
=> ( P @ A0 @ A13 @ A24 ) ) ) ) ).
% quicksort_by_rel.pinduct
thf(fact_6928_quicksort__by__rel_Opelims,axiom,
! [A: $tType,X2: A > A > $o,Xa: list @ A,Xb: list @ A,Y: list @ A] :
( ( ( quicksort_by_rel @ A @ X2 @ 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 ) ) @ X2 @ ( 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 ) ) @ X2 @ ( 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 @ X2 @ ( cons @ A @ X3 @ ( quicksort_by_rel @ A @ X2 @ Xa @ Xs_b ) ) @ Xs_s )
@ ( partition_rev @ A
@ ^ [Y4: A] : ( X2 @ 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 ) ) @ X2 @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xa @ ( cons @ A @ X3 @ Xs2 ) ) ) ) ) ) ) ) ) ).
% quicksort_by_rel.pelims
thf(fact_6929_mergesort__by__rel__merge_Opelims,axiom,
! [A: $tType,X2: A > A > $o,Xa: list @ A,Xb: list @ A,Y: list @ A] :
( ( ( merges9089515139780605204_merge @ A @ X2 @ 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 ) ) @ X2 @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xa @ Xb ) ) )
=> ( ! [X3: A,Xs2: list @ A] :
( ( Xa
= ( cons @ A @ X3 @ Xs2 ) )
=> ! [Y2: A,Ys4: list @ A] :
( ( Xb
= ( cons @ A @ Y2 @ Ys4 ) )
=> ( ( ( ( X2 @ X3 @ Y2 )
=> ( Y
= ( cons @ A @ X3 @ ( merges9089515139780605204_merge @ A @ X2 @ Xs2 @ ( cons @ A @ Y2 @ Ys4 ) ) ) ) )
& ( ~ ( X2 @ X3 @ Y2 )
=> ( Y
= ( cons @ A @ Y2 @ ( merges9089515139780605204_merge @ A @ X2 @ ( 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 ) ) @ X2 @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X3 @ Xs2 ) @ ( cons @ A @ Y2 @ 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 ) ) @ X2 @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xa @ ( nil @ A ) ) ) ) ) )
=> ~ ( ( Xa
= ( nil @ A ) )
=> ! [V5: A,Va: list @ A] :
( ( Xb
= ( cons @ A @ V5 @ Va ) )
=> ( ( Y
= ( cons @ A @ V5 @ 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 ) ) @ X2 @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( cons @ A @ V5 @ Va ) ) ) ) ) ) ) ) ) ) ) ).
% mergesort_by_rel_merge.pelims
thf(fact_6930_finite__enumerate__Suc_H_H,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S2: set @ A,N: nat] :
( ( finite_finite2 @ A @ S2 )
=> ( ( ord_less @ nat @ ( suc @ N ) @ ( finite_card @ A @ S2 ) )
=> ( ( infini527867602293511546merate @ A @ S2 @ ( suc @ N ) )
= ( ord_Least @ A
@ ^ [S6: A] :
( ( member @ A @ S6 @ S2 )
& ( ord_less @ A @ ( infini527867602293511546merate @ A @ S2 @ N ) @ S6 ) ) ) ) ) ) ) ).
% finite_enumerate_Suc''
thf(fact_6931_Least__eq__0,axiom,
! [P: nat > $o] :
( ( P @ ( zero_zero @ nat ) )
=> ( ( ord_Least @ nat @ P )
= ( zero_zero @ nat ) ) ) ).
% Least_eq_0
thf(fact_6932_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_6933_Least1I,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P: A > $o] :
( ? [X4: A] :
( ( P @ X4 )
& ! [Y2: A] :
( ( P @ Y2 )
=> ( ord_less_eq @ A @ X4 @ Y2 ) )
& ! [Y2: A] :
( ( ( P @ Y2 )
& ! [Ya2: A] :
( ( P @ Ya2 )
=> ( ord_less_eq @ A @ Y2 @ Ya2 ) ) )
=> ( Y2 = X4 ) ) )
=> ( P @ ( ord_Least @ A @ P ) ) ) ) ).
% Least1I
thf(fact_6934_Least1__le,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P: A > $o,Z2: A] :
( ? [X4: A] :
( ( P @ X4 )
& ! [Y2: A] :
( ( P @ Y2 )
=> ( ord_less_eq @ A @ X4 @ Y2 ) )
& ! [Y2: A] :
( ( ( P @ Y2 )
& ! [Ya2: A] :
( ( P @ Ya2 )
=> ( ord_less_eq @ A @ Y2 @ Ya2 ) ) )
=> ( Y2 = X4 ) ) )
=> ( ( P @ Z2 )
=> ( ord_less_eq @ A @ ( ord_Least @ A @ P ) @ Z2 ) ) ) ) ).
% Least1_le
thf(fact_6935_LeastI2__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P: A > $o,X2: A,Q: A > $o] :
( ( P @ X2 )
=> ( ! [Y2: A] :
( ( P @ Y2 )
=> ( ord_less_eq @ A @ X2 @ Y2 ) )
=> ( ! [X3: A] :
( ( P @ X3 )
=> ( ! [Y5: A] :
( ( P @ Y5 )
=> ( ord_less_eq @ A @ X3 @ Y5 ) )
=> ( Q @ X3 ) ) )
=> ( Q @ ( ord_Least @ A @ P ) ) ) ) ) ) ).
% LeastI2_order
thf(fact_6936_Least__equality,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P: A > $o,X2: A] :
( ( P @ X2 )
=> ( ! [Y2: A] :
( ( P @ Y2 )
=> ( ord_less_eq @ A @ X2 @ Y2 ) )
=> ( ( ord_Least @ A @ P )
= X2 ) ) ) ) ).
% Least_equality
thf(fact_6937_LeastI2__wellorder,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P: A > $o,A2: A,Q: A > $o] :
( ( P @ A2 )
=> ( ! [A4: A] :
( ( P @ A4 )
=> ( ! [B9: A] :
( ( P @ B9 )
=> ( ord_less_eq @ A @ A4 @ B9 ) )
=> ( Q @ A4 ) ) )
=> ( Q @ ( ord_Least @ A @ P ) ) ) ) ) ).
% LeastI2_wellorder
thf(fact_6938_LeastI2__wellorder__ex,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P: A > $o,Q: A > $o] :
( ? [X_1: A] : ( P @ X_1 )
=> ( ! [A4: A] :
( ( P @ A4 )
=> ( ! [B9: A] :
( ( P @ B9 )
=> ( ord_less_eq @ A @ A4 @ B9 ) )
=> ( Q @ A4 ) ) )
=> ( Q @ ( ord_Least @ A @ P ) ) ) ) ) ).
% LeastI2_wellorder_ex
thf(fact_6939_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_6940_LeastI2__ex,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P: A > $o,Q: A > $o] :
( ? [X_1: A] : ( P @ X_1 )
=> ( ! [X3: A] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( Q @ ( ord_Least @ A @ P ) ) ) ) ) ).
% LeastI2_ex
thf(fact_6941_LeastI__ex,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P: A > $o] :
( ? [X_1: A] : ( P @ X_1 )
=> ( P @ ( ord_Least @ A @ P ) ) ) ) ).
% LeastI_ex
thf(fact_6942_LeastI2,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P: A > $o,A2: A,Q: A > $o] :
( ( P @ A2 )
=> ( ! [X3: A] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( Q @ ( ord_Least @ A @ P ) ) ) ) ) ).
% LeastI2
thf(fact_6943_LeastI,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P: A > $o,K: A] :
( ( P @ K )
=> ( P @ ( ord_Least @ A @ P ) ) ) ) ).
% LeastI
thf(fact_6944_Least__Suc2,axiom,
! [P: nat > $o,N: nat,Q: nat > $o,M: nat] :
( ( P @ N )
=> ( ( Q @ M )
=> ( ~ ( P @ ( zero_zero @ nat ) )
=> ( ! [K4: nat] :
( ( P @ ( suc @ K4 ) )
= ( Q @ K4 ) )
=> ( ( ord_Least @ nat @ P )
= ( suc @ ( ord_Least @ nat @ Q ) ) ) ) ) ) ) ).
% Least_Suc2
thf(fact_6945_Least__Suc,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ ( zero_zero @ nat ) )
=> ( ( ord_Least @ nat @ P )
= ( suc
@ ( ord_Least @ nat
@ ^ [M2: nat] : ( P @ ( suc @ M2 ) ) ) ) ) ) ) ).
% Least_Suc
thf(fact_6946_Least__Min,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o] :
( ( finite_finite2 @ A @ ( collect @ A @ P ) )
=> ( ? [X_1: A] : ( P @ X_1 )
=> ( ( ord_Least @ A @ P )
= ( lattic643756798350308766er_Min @ A @ ( collect @ A @ P ) ) ) ) ) ) ).
% Least_Min
thf(fact_6947_abort__Bleast__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( abort_Bleast @ A )
= ( ^ [S: set @ A,P4: A > $o] :
( ord_Least @ A
@ ^ [X: A] :
( ( member @ A @ X @ S )
& ( P4 @ X ) ) ) ) ) ) ).
% abort_Bleast_def
thf(fact_6948_Bleast__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( bleast @ A )
= ( ^ [S: set @ A,P4: A > $o] :
( ord_Least @ A
@ ^ [X: A] :
( ( member @ A @ X @ S )
& ( P4 @ X ) ) ) ) ) ) ).
% Bleast_def
thf(fact_6949_enumerate__0,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S2: set @ A] :
( ( infini527867602293511546merate @ A @ S2 @ ( zero_zero @ nat ) )
= ( ord_Least @ A
@ ^ [N4: A] : ( member @ A @ N4 @ S2 ) ) ) ) ).
% enumerate_0
thf(fact_6950_Least__mono,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F: A > B,S2: set @ A] :
( ( order_mono @ A @ B @ F )
=> ( ? [X4: A] :
( ( member @ A @ X4 @ S2 )
& ! [Xa3: A] :
( ( member @ A @ Xa3 @ S2 )
=> ( ord_less_eq @ A @ X4 @ Xa3 ) ) )
=> ( ( ord_Least @ B
@ ^ [Y4: B] : ( member @ B @ Y4 @ ( image2 @ A @ B @ F @ S2 ) ) )
= ( F
@ ( ord_Least @ A
@ ^ [X: A] : ( member @ A @ X @ S2 ) ) ) ) ) ) ) ).
% Least_mono
thf(fact_6951_enumerate__Suc_H_H,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S2: set @ A,N: nat] :
( ~ ( finite_finite2 @ A @ S2 )
=> ( ( infini527867602293511546merate @ A @ S2 @ ( suc @ N ) )
= ( ord_Least @ A
@ ^ [S6: A] :
( ( member @ A @ S6 @ S2 )
& ( ord_less @ A @ ( infini527867602293511546merate @ A @ S2 @ N ) @ S6 ) ) ) ) ) ) ).
% enumerate_Suc''
thf(fact_6952_enumerate__Suc,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S2: set @ A,N: nat] :
( ( infini527867602293511546merate @ A @ S2 @ ( suc @ N ) )
= ( infini527867602293511546merate @ A
@ ( minus_minus @ ( set @ A ) @ S2
@ ( insert @ A
@ ( ord_Least @ A
@ ^ [N4: A] : ( member @ A @ N4 @ S2 ) )
@ ( bot_bot @ ( set @ A ) ) ) )
@ N ) ) ) ).
% enumerate_Suc
thf(fact_6953_size__Diff__singleton,axiom,
! [A: $tType,X2: A,M3: multiset @ A] :
( ( member @ A @ X2 @ ( set_mset @ A @ M3 ) )
=> ( ( size_size @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ M3 @ ( add_mset @ A @ X2 @ ( zero_zero @ ( multiset @ A ) ) ) ) )
= ( minus_minus @ nat @ ( size_size @ ( multiset @ A ) @ M3 ) @ ( one_one @ nat ) ) ) ) ).
% size_Diff_singleton
thf(fact_6954_size__Diff__singleton__if,axiom,
! [A: $tType,X2: A,A5: multiset @ A] :
( ( ( member @ A @ X2 @ ( set_mset @ A @ A5 ) )
=> ( ( size_size @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ A5 @ ( add_mset @ A @ X2 @ ( zero_zero @ ( multiset @ A ) ) ) ) )
= ( minus_minus @ nat @ ( size_size @ ( multiset @ A ) @ A5 ) @ ( one_one @ nat ) ) ) )
& ( ~ ( member @ A @ X2 @ ( set_mset @ A @ A5 ) )
=> ( ( size_size @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ A5 @ ( add_mset @ A @ X2 @ ( zero_zero @ ( multiset @ A ) ) ) ) )
= ( size_size @ ( multiset @ A ) @ A5 ) ) ) ) ).
% size_Diff_singleton_if
thf(fact_6955_set__mset__union,axiom,
! [A: $tType,M3: multiset @ A,N5: multiset @ A] :
( ( set_mset @ A @ ( plus_plus @ ( multiset @ A ) @ M3 @ N5 ) )
= ( sup_sup @ ( set @ A ) @ ( set_mset @ A @ M3 ) @ ( set_mset @ A @ N5 ) ) ) ).
% set_mset_union
thf(fact_6956_sum__mset__0__iff,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [M3: multiset @ A] :
( ( ( comm_m7189776963980413722m_mset @ A @ M3 )
= ( zero_zero @ A ) )
= ( ! [X: A] :
( ( member @ A @ X @ ( set_mset @ A @ M3 ) )
=> ( X
= ( zero_zero @ A ) ) ) ) ) ) ).
% sum_mset_0_iff
thf(fact_6957_mset__diff__cancel1elem,axiom,
! [A: $tType,A2: A,B5: multiset @ A] :
( ~ ( member @ A @ A2 @ ( set_mset @ A @ B5 ) )
=> ( ( minus_minus @ ( multiset @ A ) @ ( add_mset @ A @ A2 @ ( zero_zero @ ( multiset @ A ) ) ) @ B5 )
= ( add_mset @ A @ A2 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ).
% mset_diff_cancel1elem
thf(fact_6958_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_6959_Inf__nat__def,axiom,
( ( complete_Inf_Inf @ nat )
= ( ^ [X5: set @ nat] :
( ord_Least @ nat
@ ^ [N4: nat] : ( member @ nat @ N4 @ X5 ) ) ) ) ).
% Inf_nat_def
thf(fact_6960_in__image__mset,axiom,
! [A: $tType,B: $tType,Y: A,F: B > A,M3: multiset @ B] :
( ( member @ A @ Y @ ( set_mset @ A @ ( image_mset @ B @ A @ F @ M3 ) ) )
= ( member @ A @ Y @ ( image2 @ B @ A @ F @ ( set_mset @ B @ M3 ) ) ) ) ).
% in_image_mset
thf(fact_6961_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_6962_union__iff,axiom,
! [A: $tType,A2: A,A5: multiset @ A,B5: multiset @ A] :
( ( member @ A @ A2 @ ( set_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A5 @ B5 ) ) )
= ( ( member @ A @ A2 @ ( set_mset @ A @ A5 ) )
| ( member @ A @ A2 @ ( set_mset @ A @ B5 ) ) ) ) ).
% union_iff
thf(fact_6963_mset__un__cases,axiom,
! [A: $tType,A2: A,A5: multiset @ A,B5: multiset @ A] :
( ( member @ A @ A2 @ ( set_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A5 @ B5 ) ) )
=> ( ~ ( member @ A @ A2 @ ( set_mset @ A @ A5 ) )
=> ( member @ A @ A2 @ ( set_mset @ A @ B5 ) ) ) ) ).
% mset_un_cases
thf(fact_6964_mset__left__cancel__union,axiom,
! [A: $tType,A2: A,A5: multiset @ A,B5: multiset @ A] :
( ( member @ A @ A2 @ ( set_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A5 @ B5 ) ) )
=> ( ~ ( member @ A @ A2 @ ( set_mset @ A @ A5 ) )
=> ( member @ A @ A2 @ ( set_mset @ A @ B5 ) ) ) ) ).
% mset_left_cancel_union
thf(fact_6965_mset__right__cancel__union,axiom,
! [A: $tType,A2: A,A5: multiset @ A,B5: multiset @ A] :
( ( member @ A @ A2 @ ( set_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A5 @ B5 ) ) )
=> ( ~ ( member @ A @ A2 @ ( set_mset @ A @ B5 ) )
=> ( member @ A @ A2 @ ( set_mset @ A @ A5 ) ) ) ) ).
% mset_right_cancel_union
thf(fact_6966_ex__Melem__conv,axiom,
! [A: $tType,A5: multiset @ A] :
( ( ? [X: A] : ( member @ A @ X @ ( set_mset @ A @ A5 ) ) )
= ( A5
!= ( zero_zero @ ( multiset @ A ) ) ) ) ).
% ex_Melem_conv
thf(fact_6967_size__eq__Suc__imp__elem,axiom,
! [A: $tType,M3: multiset @ A,N: nat] :
( ( ( size_size @ ( multiset @ A ) @ M3 )
= ( suc @ N ) )
=> ? [A4: A] : ( member @ A @ A4 @ ( set_mset @ A @ M3 ) ) ) ).
% size_eq_Suc_imp_elem
thf(fact_6968_image__mset__cong__pair,axiom,
! [C: $tType,B: $tType,A: $tType,M3: multiset @ ( product_prod @ A @ B ),F: A > B > C,G: A > B > C] :
( ! [X3: A,Y2: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y2 ) @ ( set_mset @ ( product_prod @ A @ B ) @ M3 ) )
=> ( ( F @ X3 @ Y2 )
= ( G @ X3 @ Y2 ) ) )
=> ( ( image_mset @ ( product_prod @ A @ B ) @ C @ ( product_case_prod @ A @ B @ C @ F ) @ M3 )
= ( image_mset @ ( product_prod @ A @ B ) @ C @ ( product_case_prod @ A @ B @ C @ G ) @ M3 ) ) ) ).
% image_mset_cong_pair
thf(fact_6969_image__mset__cong,axiom,
! [B: $tType,A: $tType,M3: multiset @ A,F: A > B,G: A > B] :
( ! [X3: A] :
( ( member @ A @ X3 @ ( set_mset @ A @ M3 ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_mset @ A @ B @ F @ M3 )
= ( image_mset @ A @ B @ G @ M3 ) ) ) ).
% image_mset_cong
thf(fact_6970_sum__mset_Oneutral,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A5: multiset @ A] :
( ! [X3: A] :
( ( member @ A @ X3 @ ( set_mset @ A @ A5 ) )
=> ( X3
= ( zero_zero @ A ) ) )
=> ( ( comm_m7189776963980413722m_mset @ A @ A5 )
= ( zero_zero @ A ) ) ) ) ).
% sum_mset.neutral
thf(fact_6971_multiset__induct__min,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: ( multiset @ A ) > $o,M3: multiset @ A] :
( ( P @ ( zero_zero @ ( multiset @ A ) ) )
=> ( ! [X3: A,M10: multiset @ A] :
( ( P @ M10 )
=> ( ! [Xa2: A] :
( ( member @ A @ Xa2 @ ( set_mset @ A @ M10 ) )
=> ( ord_less_eq @ A @ X3 @ Xa2 ) )
=> ( P @ ( add_mset @ A @ X3 @ M10 ) ) ) )
=> ( P @ M3 ) ) ) ) ).
% multiset_induct_min
thf(fact_6972_multiset__induct__max,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: ( multiset @ A ) > $o,M3: multiset @ A] :
( ( P @ ( zero_zero @ ( multiset @ A ) ) )
=> ( ! [X3: A,M10: multiset @ A] :
( ( P @ M10 )
=> ( ! [Xa2: A] :
( ( member @ A @ Xa2 @ ( set_mset @ A @ M10 ) )
=> ( ord_less_eq @ A @ Xa2 @ X3 ) )
=> ( P @ ( add_mset @ A @ X3 @ M10 ) ) ) )
=> ( P @ M3 ) ) ) ) ).
% multiset_induct_max
thf(fact_6973_mset__right__cancel__elem,axiom,
! [A: $tType,A2: A,A5: multiset @ A,B2: A] :
( ( member @ A @ A2 @ ( set_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A5 @ ( add_mset @ A @ B2 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) )
=> ( ( A2 != B2 )
=> ( member @ A @ A2 @ ( set_mset @ A @ A5 ) ) ) ) ).
% mset_right_cancel_elem
thf(fact_6974_multi__member__this,axiom,
! [A: $tType,X2: A,XS2: multiset @ A] : ( member @ A @ X2 @ ( set_mset @ A @ ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ X2 @ ( zero_zero @ ( multiset @ A ) ) ) @ XS2 ) ) ) ).
% multi_member_this
thf(fact_6975_multi__member__skip,axiom,
! [A: $tType,X2: A,XS2: multiset @ A,Y: A] :
( ( member @ A @ X2 @ ( set_mset @ A @ XS2 ) )
=> ( member @ A @ X2 @ ( set_mset @ A @ ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ Y @ ( zero_zero @ ( multiset @ A ) ) ) @ XS2 ) ) ) ) ).
% multi_member_skip
thf(fact_6976_mset__left__cancel__elem,axiom,
! [A: $tType,A2: A,B2: A,A5: multiset @ A] :
( ( member @ A @ A2 @ ( set_mset @ A @ ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ B2 @ ( zero_zero @ ( multiset @ A ) ) ) @ A5 ) ) )
=> ( ( A2 != B2 )
=> ( member @ A @ A2 @ ( set_mset @ A @ A5 ) ) ) ) ).
% mset_left_cancel_elem
thf(fact_6977_sum__mset__mono,axiom,
! [A: $tType,B: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [K3: multiset @ B,F: B > A,G: B > A] :
( ! [I3: B] :
( ( member @ B @ I3 @ ( set_mset @ B @ K3 ) )
=> ( ord_less_eq @ A @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ord_less_eq @ A @ ( comm_m7189776963980413722m_mset @ A @ ( image_mset @ B @ A @ F @ K3 ) ) @ ( comm_m7189776963980413722m_mset @ A @ ( image_mset @ B @ A @ G @ K3 ) ) ) ) ) ).
% sum_mset_mono
thf(fact_6978_mset__2dist2__cases,axiom,
! [A: $tType,A2: A,B2: A,A5: multiset @ A,B5: multiset @ A] :
( ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ A2 @ ( zero_zero @ ( multiset @ A ) ) ) @ ( add_mset @ A @ B2 @ ( zero_zero @ ( multiset @ A ) ) ) ) @ ( plus_plus @ ( multiset @ A ) @ A5 @ B5 ) )
=> ( ~ ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ A2 @ ( zero_zero @ ( multiset @ A ) ) ) @ ( add_mset @ A @ B2 @ ( zero_zero @ ( multiset @ A ) ) ) ) @ A5 )
=> ( ~ ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ A2 @ ( zero_zero @ ( multiset @ A ) ) ) @ ( add_mset @ A @ B2 @ ( zero_zero @ ( multiset @ A ) ) ) ) @ B5 )
=> ( ( ( member @ A @ A2 @ ( set_mset @ A @ A5 ) )
=> ~ ( member @ A @ B2 @ ( set_mset @ A @ B5 ) ) )
=> ~ ( ( member @ A @ A2 @ ( set_mset @ A @ B5 ) )
=> ~ ( member @ A @ B2 @ ( set_mset @ A @ A5 ) ) ) ) ) ) ) ).
% mset_2dist2_cases
thf(fact_6979_mset__union__subset__s,axiom,
! [A: $tType,A2: A,B5: multiset @ A,C6: multiset @ A] :
( ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ A2 @ ( zero_zero @ ( multiset @ A ) ) ) @ B5 ) @ C6 )
=> ( ( member @ A @ A2 @ ( set_mset @ A @ C6 ) )
& ( subseteq_mset @ A @ B5 @ C6 ) ) ) ).
% mset_union_subset_s
thf(fact_6980_mset__le__mono__add__single,axiom,
! [A: $tType,A2: A,Ys: multiset @ A,B2: A,Ws: multiset @ A] :
( ( member @ A @ A2 @ ( set_mset @ A @ Ys ) )
=> ( ( member @ A @ B2 @ ( set_mset @ A @ Ws ) )
=> ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ A2 @ ( zero_zero @ ( multiset @ A ) ) ) @ ( add_mset @ A @ B2 @ ( zero_zero @ ( multiset @ A ) ) ) ) @ ( plus_plus @ ( multiset @ A ) @ Ys @ Ws ) ) ) ) ).
% mset_le_mono_add_single
thf(fact_6981_nth__mem__mset,axiom,
! [A: $tType,I: nat,Ls: list @ A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Ls ) )
=> ( member @ A @ ( nth @ A @ Ls @ I ) @ ( set_mset @ A @ ( mset @ A @ Ls ) ) ) ) ).
% nth_mem_mset
thf(fact_6982_diff__union__single__conv,axiom,
! [A: $tType,A2: A,J4: multiset @ A,I5: multiset @ A] :
( ( member @ A @ A2 @ ( set_mset @ A @ J4 ) )
=> ( ( minus_minus @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ I5 @ J4 ) @ ( add_mset @ A @ A2 @ ( zero_zero @ ( multiset @ A ) ) ) )
= ( plus_plus @ ( multiset @ A ) @ I5 @ ( minus_minus @ ( multiset @ A ) @ J4 @ ( add_mset @ A @ A2 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ) ).
% diff_union_single_conv
thf(fact_6983_mset__un__single__un__cases,axiom,
! [A: $tType,A2: A,A5: multiset @ A,B5: multiset @ A,C6: multiset @ A] :
( ( ( add_mset @ A @ A2 @ A5 )
= ( plus_plus @ ( multiset @ A ) @ B5 @ C6 ) )
=> ( ( ( member @ A @ A2 @ ( set_mset @ A @ B5 ) )
=> ( A5
!= ( plus_plus @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ B5 @ ( add_mset @ A @ A2 @ ( zero_zero @ ( multiset @ A ) ) ) ) @ C6 ) ) )
=> ~ ( ( member @ A @ A2 @ ( set_mset @ A @ C6 ) )
=> ( A5
!= ( plus_plus @ ( multiset @ A ) @ B5 @ ( minus_minus @ ( multiset @ A ) @ C6 @ ( add_mset @ A @ A2 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ) ) ) ).
% mset_un_single_un_cases
thf(fact_6984_diff__union__single__conv2,axiom,
! [A: $tType,A2: A,J4: multiset @ A,I5: multiset @ A] :
( ( member @ A @ A2 @ ( set_mset @ A @ J4 ) )
=> ( ( minus_minus @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ J4 @ I5 ) @ ( add_mset @ A @ A2 @ ( zero_zero @ ( multiset @ A ) ) ) )
= ( plus_plus @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ J4 @ ( add_mset @ A @ A2 @ ( zero_zero @ ( multiset @ A ) ) ) ) @ I5 ) ) ) ).
% diff_union_single_conv2
thf(fact_6985_mset__union__diff__comm,axiom,
! [A: $tType,T2: A,S2: multiset @ A,T6: multiset @ A] :
( ( member @ A @ T2 @ ( set_mset @ A @ S2 ) )
=> ( ( plus_plus @ ( multiset @ A ) @ T6 @ ( minus_minus @ ( multiset @ A ) @ S2 @ ( add_mset @ A @ T2 @ ( zero_zero @ ( multiset @ A ) ) ) ) )
= ( minus_minus @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ T6 @ S2 ) @ ( add_mset @ A @ T2 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ).
% mset_union_diff_comm
thf(fact_6986_insert__DiffM2,axiom,
! [A: $tType,X2: A,M3: multiset @ A] :
( ( member @ A @ X2 @ ( set_mset @ A @ M3 ) )
=> ( ( plus_plus @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ M3 @ ( add_mset @ A @ X2 @ ( zero_zero @ ( multiset @ A ) ) ) ) @ ( add_mset @ A @ X2 @ ( zero_zero @ ( multiset @ A ) ) ) )
= M3 ) ) ).
% insert_DiffM2
thf(fact_6987_mset__contains__eq,axiom,
! [A: $tType,M: A,M3: multiset @ A] :
( ( member @ A @ M @ ( set_mset @ A @ M3 ) )
= ( ( plus_plus @ ( multiset @ A ) @ ( add_mset @ A @ M @ ( zero_zero @ ( multiset @ A ) ) ) @ ( minus_minus @ ( multiset @ A ) @ M3 @ ( add_mset @ A @ M @ ( zero_zero @ ( multiset @ A ) ) ) ) )
= M3 ) ) ).
% mset_contains_eq
thf(fact_6988_mset__size1elem,axiom,
! [A: $tType,P: multiset @ A,Q2: A] :
( ( ord_less_eq @ nat @ ( size_size @ ( multiset @ A ) @ P ) @ ( one_one @ nat ) )
=> ( ( member @ A @ Q2 @ ( set_mset @ A @ P ) )
=> ( P
= ( add_mset @ A @ Q2 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ).
% mset_size1elem
thf(fact_6989_size__Suc__Diff1,axiom,
! [A: $tType,X2: A,M3: multiset @ A] :
( ( member @ A @ X2 @ ( set_mset @ A @ M3 ) )
=> ( ( suc @ ( size_size @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ M3 @ ( add_mset @ A @ X2 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) )
= ( size_size @ ( multiset @ A ) @ M3 ) ) ) ).
% size_Suc_Diff1
thf(fact_6990_size__Diff1__less,axiom,
! [A: $tType,X2: A,M3: multiset @ A] :
( ( member @ A @ X2 @ ( set_mset @ A @ M3 ) )
=> ( ord_less @ nat @ ( size_size @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ M3 @ ( add_mset @ A @ X2 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) @ ( size_size @ ( multiset @ A ) @ M3 ) ) ) ).
% size_Diff1_less
thf(fact_6991_size__Diff2__less,axiom,
! [A: $tType,X2: A,M3: multiset @ A,Y: A] :
( ( member @ A @ X2 @ ( set_mset @ A @ M3 ) )
=> ( ( member @ A @ Y @ ( set_mset @ A @ M3 ) )
=> ( ord_less @ nat @ ( size_size @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ M3 @ ( add_mset @ A @ X2 @ ( zero_zero @ ( multiset @ A ) ) ) ) @ ( add_mset @ A @ Y @ ( zero_zero @ ( multiset @ A ) ) ) ) ) @ ( size_size @ ( multiset @ A ) @ M3 ) ) ) ) ).
% size_Diff2_less
thf(fact_6992_sum__mset_Oremove,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [X2: A,A5: multiset @ A] :
( ( member @ A @ X2 @ ( set_mset @ A @ A5 ) )
=> ( ( comm_m7189776963980413722m_mset @ A @ A5 )
= ( plus_plus @ A @ X2 @ ( comm_m7189776963980413722m_mset @ A @ ( minus_minus @ ( multiset @ A ) @ A5 @ ( add_mset @ A @ X2 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ) ) ) ).
% sum_mset.remove
thf(fact_6993_size__diff__se,axiom,
! [A: $tType,T2: A,S2: multiset @ A] :
( ( member @ A @ T2 @ ( set_mset @ A @ S2 ) )
=> ( ( size_size @ ( multiset @ A ) @ S2 )
= ( plus_plus @ nat @ ( size_size @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ S2 @ ( add_mset @ A @ T2 @ ( zero_zero @ ( multiset @ A ) ) ) ) ) @ ( one_one @ nat ) ) ) ) ).
% size_diff_se
thf(fact_6994_subset__mset_Osum__mset__mono,axiom,
! [A: $tType,B: $tType,K3: multiset @ B,F: B > ( multiset @ A ),G: B > ( multiset @ A )] :
( ! [I3: B] :
( ( member @ B @ I3 @ ( set_mset @ B @ K3 ) )
=> ( subseteq_mset @ A @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( subseteq_mset @ A @ ( comm_monoid_sum_mset @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ ( image_mset @ B @ ( multiset @ A ) @ F @ K3 ) ) @ ( comm_monoid_sum_mset @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ ( image_mset @ B @ ( multiset @ A ) @ G @ K3 ) ) ) ) ).
% subset_mset.sum_mset_mono
thf(fact_6995_subset__mset_Osum__mset__0__iff,axiom,
! [A: $tType,M3: multiset @ ( multiset @ A )] :
( ( ( comm_monoid_sum_mset @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ M3 )
= ( zero_zero @ ( multiset @ A ) ) )
= ( ! [X: multiset @ A] :
( ( member @ ( multiset @ A ) @ X @ ( set_mset @ ( multiset @ A ) @ M3 ) )
=> ( X
= ( zero_zero @ ( multiset @ A ) ) ) ) ) ) ).
% subset_mset.sum_mset_0_iff
thf(fact_6996_multiset_Oin__rel,axiom,
! [B: $tType,A: $tType] :
( ( rel_mset @ A @ B )
= ( ^ [R7: A > B > $o,A3: multiset @ A,B3: multiset @ B] :
? [Z3: multiset @ ( product_prod @ A @ B )] :
( ( member @ ( multiset @ ( product_prod @ A @ B ) ) @ Z3
@ ( collect @ ( multiset @ ( product_prod @ A @ B ) )
@ ^ [X: multiset @ ( product_prod @ A @ B )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( set_mset @ ( product_prod @ A @ B ) @ X ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R7 ) ) ) ) )
& ( ( image_mset @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Z3 )
= A3 )
& ( ( image_mset @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ Z3 )
= B3 ) ) ) ) ).
% multiset.in_rel
thf(fact_6997_count__image__mset,axiom,
! [A: $tType,B: $tType,F: B > A,A5: multiset @ B,X2: A] :
( ( count @ A @ ( image_mset @ B @ A @ F @ A5 ) @ X2 )
= ( groups7311177749621191930dd_sum @ B @ nat @ ( count @ B @ A5 ) @ ( inf_inf @ ( set @ B ) @ ( vimage @ B @ A @ F @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( set_mset @ B @ A5 ) ) ) ) ).
% count_image_mset
thf(fact_6998_mset__empty__count,axiom,
! [A: $tType,M3: multiset @ A] :
( ( ! [P7: A] :
( ( count @ A @ M3 @ P7 )
= ( zero_zero @ nat ) ) )
= ( M3
= ( zero_zero @ ( multiset @ A ) ) ) ) ).
% mset_empty_count
thf(fact_6999_count__empty,axiom,
! [A: $tType,A2: A] :
( ( count @ A @ ( zero_zero @ ( multiset @ A ) ) @ A2 )
= ( zero_zero @ nat ) ) ).
% count_empty
thf(fact_7000_count__union,axiom,
! [A: $tType,M3: multiset @ A,N5: multiset @ A,A2: A] :
( ( count @ A @ ( plus_plus @ ( multiset @ A ) @ M3 @ N5 ) @ A2 )
= ( plus_plus @ nat @ ( count @ A @ M3 @ A2 ) @ ( count @ A @ N5 @ A2 ) ) ) ).
% count_union
thf(fact_7001_count__greater__zero__iff,axiom,
! [A: $tType,M3: multiset @ A,X2: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( count @ A @ M3 @ X2 ) )
= ( member @ A @ X2 @ ( set_mset @ A @ M3 ) ) ) ).
% count_greater_zero_iff
thf(fact_7002_count__mset__0__iff,axiom,
! [A: $tType,Xs: list @ A,X2: A] :
( ( ( count @ A @ ( mset @ A @ Xs ) @ X2 )
= ( zero_zero @ nat ) )
= ( ~ ( member @ A @ X2 @ ( set2 @ A @ Xs ) ) ) ) ).
% count_mset_0_iff
thf(fact_7003_count__greater__eq__one__iff,axiom,
! [A: $tType,M3: multiset @ A,X2: A] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ ( count @ A @ M3 @ X2 ) )
= ( member @ A @ X2 @ ( set_mset @ A @ M3 ) ) ) ).
% count_greater_eq_one_iff
thf(fact_7004_count__greater__eq__Suc__zero__iff,axiom,
! [A: $tType,M3: multiset @ A,X2: A] :
( ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( count @ A @ M3 @ X2 ) )
= ( member @ A @ X2 @ ( set_mset @ A @ M3 ) ) ) ).
% count_greater_eq_Suc_zero_iff
thf(fact_7005_count__eq__zero__iff,axiom,
! [A: $tType,M3: multiset @ A,X2: A] :
( ( ( count @ A @ M3 @ X2 )
= ( zero_zero @ nat ) )
= ( ~ ( member @ A @ X2 @ ( set_mset @ A @ M3 ) ) ) ) ).
% count_eq_zero_iff
thf(fact_7006_count__inI,axiom,
! [A: $tType,M3: multiset @ A,X2: A] :
( ( ( count @ A @ M3 @ X2 )
!= ( zero_zero @ nat ) )
=> ( member @ A @ X2 @ ( set_mset @ A @ M3 ) ) ) ).
% count_inI
thf(fact_7007_multiset_Orel__map_I2_J,axiom,
! [A: $tType,C: $tType,B: $tType,Sa: A > C > $o,X2: multiset @ A,G: B > C,Y: multiset @ B] :
( ( rel_mset @ A @ C @ Sa @ X2 @ ( image_mset @ B @ C @ G @ Y ) )
= ( rel_mset @ A @ B
@ ^ [X: A,Y4: B] : ( Sa @ X @ ( G @ Y4 ) )
@ X2
@ Y ) ) ).
% multiset.rel_map(2)
thf(fact_7008_multiset_Orel__map_I1_J,axiom,
! [A: $tType,C: $tType,B: $tType,Sb: C > B > $o,I: A > C,X2: multiset @ A,Y: multiset @ B] :
( ( rel_mset @ C @ B @ Sb @ ( image_mset @ A @ C @ I @ X2 ) @ Y )
= ( rel_mset @ A @ B
@ ^ [X: A] : ( Sb @ ( I @ X ) )
@ X2
@ Y ) ) ).
% multiset.rel_map(1)
thf(fact_7009_zero__multiset_Orep__eq,axiom,
! [A: $tType] :
( ( count @ A @ ( zero_zero @ ( multiset @ A ) ) )
= ( ^ [A3: A] : ( zero_zero @ nat ) ) ) ).
% zero_multiset.rep_eq
thf(fact_7010_plus__multiset_Orep__eq,axiom,
! [A: $tType,X2: multiset @ A,Xa: multiset @ A] :
( ( count @ A @ ( plus_plus @ ( multiset @ A ) @ X2 @ Xa ) )
= ( ^ [A3: A] : ( plus_plus @ nat @ ( count @ A @ X2 @ A3 ) @ ( count @ A @ Xa @ A3 ) ) ) ) ).
% plus_multiset.rep_eq
thf(fact_7011_count__sum,axiom,
! [A: $tType,B: $tType,F: B > ( multiset @ A ),A5: set @ B,X2: A] :
( ( count @ A @ ( groups7311177749621191930dd_sum @ B @ ( multiset @ A ) @ F @ A5 ) @ X2 )
= ( groups7311177749621191930dd_sum @ B @ nat
@ ^ [A3: B] : ( count @ A @ ( F @ A3 ) @ X2 )
@ A5 ) ) ).
% count_sum
thf(fact_7012_rel__mset__size,axiom,
! [A: $tType,B: $tType,R3: A > B > $o,M3: multiset @ A,N5: multiset @ B] :
( ( rel_mset @ A @ B @ R3 @ M3 @ N5 )
=> ( ( size_size @ ( multiset @ A ) @ M3 )
= ( size_size @ ( multiset @ B ) @ N5 ) ) ) ).
% rel_mset_size
thf(fact_7013_mset__subset__eq__count,axiom,
! [A: $tType,A5: multiset @ A,B5: multiset @ A,A2: A] :
( ( subseteq_mset @ A @ A5 @ B5 )
=> ( ord_less_eq @ nat @ ( count @ A @ A5 @ A2 ) @ ( count @ A @ B5 @ A2 ) ) ) ).
% mset_subset_eq_count
thf(fact_7014_subseteq__mset__def,axiom,
! [A: $tType] :
( ( subseteq_mset @ A )
= ( ^ [A8: multiset @ A,B8: multiset @ A] :
! [A3: A] : ( ord_less_eq @ nat @ ( count @ A @ A8 @ A3 ) @ ( count @ A @ B8 @ A3 ) ) ) ) ).
% subseteq_mset_def
thf(fact_7015_mset__subset__eqI,axiom,
! [A: $tType,A5: multiset @ A,B5: multiset @ A] :
( ! [A4: A] : ( ord_less_eq @ nat @ ( count @ A @ A5 @ A4 ) @ ( count @ A @ B5 @ A4 ) )
=> ( subseteq_mset @ A @ A5 @ B5 ) ) ).
% mset_subset_eqI
thf(fact_7016_count__ne__remove,axiom,
! [A: $tType,X2: A,T2: A,S2: multiset @ A] :
( ( X2 != T2 )
=> ( ( count @ A @ S2 @ X2 )
= ( count @ A @ ( minus_minus @ ( multiset @ A ) @ S2 @ ( add_mset @ A @ T2 @ ( zero_zero @ ( multiset @ A ) ) ) ) @ X2 ) ) ) ).
% count_ne_remove
thf(fact_7017_count__in__diffI,axiom,
! [A: $tType,N5: multiset @ A,X2: A,M3: multiset @ A] :
( ! [N2: nat] :
( ( count @ A @ N5 @ X2 )
!= ( plus_plus @ nat @ N2 @ ( count @ A @ M3 @ X2 ) ) )
=> ( member @ A @ X2 @ ( set_mset @ A @ ( minus_minus @ ( multiset @ A ) @ M3 @ N5 ) ) ) ) ).
% count_in_diffI
thf(fact_7018_set__mset__def,axiom,
! [A: $tType] :
( ( set_mset @ A )
= ( ^ [M6: multiset @ A] :
( collect @ A
@ ^ [X: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( count @ A @ M6 @ X ) ) ) ) ) ).
% set_mset_def
thf(fact_7019_set__mset__diff,axiom,
! [A: $tType,M3: multiset @ A,N5: multiset @ A] :
( ( set_mset @ A @ ( minus_minus @ ( multiset @ A ) @ M3 @ N5 ) )
= ( collect @ A
@ ^ [A3: A] : ( ord_less @ nat @ ( count @ A @ N5 @ A3 ) @ ( count @ A @ M3 @ A3 ) ) ) ) ).
% set_mset_diff
thf(fact_7020_count__mset,axiom,
! [A: $tType,Xs: list @ A,X2: A] :
( ( count @ A @ ( mset @ A @ Xs ) @ X2 )
= ( size_size @ ( list @ A )
@ ( filter2 @ A
@ ( ^ [Y3: A,Z: A] : Y3 = Z
@ X2 )
@ Xs ) ) ) ).
% count_mset
thf(fact_7021_count__image__mset_H,axiom,
! [A: $tType,B: $tType,F: B > A,X8: multiset @ B,Y: A] :
( ( count @ A @ ( image_mset @ B @ A @ F @ X8 ) @ Y )
= ( groups7311177749621191930dd_sum @ B @ nat @ ( count @ B @ X8 )
@ ( collect @ B
@ ^ [X: B] :
( ( member @ B @ X @ ( set_mset @ B @ X8 ) )
& ( Y
= ( F @ X ) ) ) ) ) ) ).
% count_image_mset'
thf(fact_7022_size__multiset__overloaded__eq,axiom,
! [B: $tType] :
( ( size_size @ ( multiset @ B ) )
= ( ^ [X: multiset @ B] : ( groups7311177749621191930dd_sum @ B @ nat @ ( count @ B @ X ) @ ( set_mset @ B @ X ) ) ) ) ).
% size_multiset_overloaded_eq
thf(fact_7023_count,axiom,
! [A: $tType,X2: multiset @ A] :
( member @ ( A > nat ) @ ( count @ A @ X2 )
@ ( collect @ ( A > nat )
@ ^ [F2: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F2 @ X ) ) ) ) ) ) ).
% count
thf(fact_7024_count__cases,axiom,
! [A: $tType,Y: A > nat] :
( ( member @ ( A > nat ) @ Y
@ ( collect @ ( A > nat )
@ ^ [F2: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F2 @ X ) ) ) ) ) )
=> ~ ! [X3: multiset @ A] :
( Y
!= ( count @ A @ X3 ) ) ) ).
% count_cases
thf(fact_7025_count__induct,axiom,
! [A: $tType,Y: A > nat,P: ( A > nat ) > $o] :
( ( member @ ( A > nat ) @ Y
@ ( collect @ ( A > nat )
@ ^ [F2: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F2 @ X ) ) ) ) ) )
=> ( ! [X3: multiset @ A] : ( P @ ( count @ A @ X3 ) )
=> ( P @ Y ) ) ) ).
% count_induct
thf(fact_7026_Inf__multiset_Orep__eq,axiom,
! [A: $tType,X2: set @ ( multiset @ A )] :
( ( count @ A @ ( complete_Inf_Inf @ ( multiset @ A ) @ X2 ) )
= ( ^ [I2: A] :
( if @ nat
@ ( ( image2 @ ( multiset @ A ) @ ( A > nat ) @ ( count @ A ) @ X2 )
= ( bot_bot @ ( set @ ( A > nat ) ) ) )
@ ( zero_zero @ nat )
@ ( complete_Inf_Inf @ nat
@ ( image2 @ ( A > nat ) @ nat
@ ^ [F2: A > nat] : ( F2 @ I2 )
@ ( image2 @ ( multiset @ A ) @ ( A > nat ) @ ( count @ A ) @ X2 ) ) ) ) ) ) ).
% Inf_multiset.rep_eq
thf(fact_7027_count__Inf__multiset__nonempty,axiom,
! [A: $tType,A5: set @ ( multiset @ A ),X2: A] :
( ( A5
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( count @ A @ ( complete_Inf_Inf @ ( multiset @ A ) @ A5 ) @ X2 )
= ( complete_Inf_Inf @ nat
@ ( image2 @ ( multiset @ A ) @ nat
@ ^ [X5: multiset @ A] : ( count @ A @ X5 @ X2 )
@ A5 ) ) ) ) ).
% count_Inf_multiset_nonempty
thf(fact_7028_count__single,axiom,
! [A: $tType,B2: A,A2: A] :
( ( ( B2 = A2 )
=> ( ( count @ A @ ( add_mset @ A @ B2 @ ( zero_zero @ ( multiset @ A ) ) ) @ A2 )
= ( one_one @ nat ) ) )
& ( ( B2 != A2 )
=> ( ( count @ A @ ( add_mset @ A @ B2 @ ( zero_zero @ ( multiset @ A ) ) ) @ A2 )
= ( zero_zero @ nat ) ) ) ) ).
% count_single
thf(fact_7029_count__mset__gt__0,axiom,
! [A: $tType,X2: A,Xs: list @ A] :
( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ ( count @ A @ ( mset @ A @ Xs ) @ X2 ) ) ) ).
% count_mset_gt_0
thf(fact_7030_in__diff__countE,axiom,
! [A: $tType,X2: A,M3: multiset @ A,N5: multiset @ A] :
( ( member @ A @ X2 @ ( set_mset @ A @ ( minus_minus @ ( multiset @ A ) @ M3 @ N5 ) ) )
=> ~ ! [N2: nat] :
( ( count @ A @ M3 @ X2 )
!= ( plus_plus @ nat @ ( suc @ N2 ) @ ( count @ A @ N5 @ X2 ) ) ) ) ).
% in_diff_countE
thf(fact_7031_distinct__count__atmost__1,axiom,
! [A: $tType] :
( ( distinct @ A )
= ( ^ [X: list @ A] :
! [A3: A] :
( ( ( member @ A @ A3 @ ( set2 @ A @ X ) )
=> ( ( count @ A @ ( mset @ A @ X ) @ A3 )
= ( one_one @ nat ) ) )
& ( ~ ( member @ A @ A3 @ ( set2 @ A @ X ) )
=> ( ( count @ A @ ( mset @ A @ X ) @ A3 )
= ( zero_zero @ nat ) ) ) ) ) ) ).
% distinct_count_atmost_1
thf(fact_7032_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
@ ^ [X: B] : ( if @ A @ ( Y = X ) @ C2 @ ( zero_zero @ A ) )
@ A5 ) )
= ( times_times @ A @ C2 @ ( semiring_1_of_nat @ A @ ( count @ B @ A5 @ Y ) ) ) ) ) ).
% sum_mset_delta'
thf(fact_7033_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
@ ^ [X: B] : ( if @ A @ ( X = Y ) @ C2 @ ( zero_zero @ A ) )
@ A5 ) )
= ( times_times @ A @ C2 @ ( semiring_1_of_nat @ A @ ( count @ B @ A5 @ Y ) ) ) ) ) ).
% sum_mset_delta
thf(fact_7034_size__multiset__eq,axiom,
! [A: $tType] :
( ( size_multiset @ A )
= ( ^ [F2: A > nat,M6: multiset @ A] :
( groups7311177749621191930dd_sum @ A @ nat
@ ^ [X: A] : ( times_times @ nat @ ( count @ A @ M6 @ X ) @ ( suc @ ( F2 @ X ) ) )
@ ( set_mset @ A @ M6 ) ) ) ) ).
% size_multiset_eq
thf(fact_7035_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 )
@ ^ [X5: multiset @ A] :
( collect @ A
@ ^ [X: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( count @ A @ X5 @ X ) ) )
@ A5 ) ) ) ) ) ).
% bdd_above_multiset_imp_finite_support
thf(fact_7036_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
@ ^ [I2: A] :
( ord_less @ nat @ ( zero_zero @ nat )
@ ( complete_Sup_Sup @ nat
@ ( image2 @ ( multiset @ A ) @ nat
@ ^ [M6: multiset @ A] : ( count @ A @ M6 @ I2 )
@ A5 ) ) ) ) ) ) ) ).
% Sup_multiset_in_multiset
thf(fact_7037_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 ) ) )
= ( ! [X: B] :
( ( member @ B @ X @ I5 )
=> ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( A5 @ X ) ) ) ) ) ) ).
% subset_mset.bdd_above_UN
thf(fact_7038_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_7039_count__Sup__multiset__nonempty,axiom,
! [A: $tType,A5: set @ ( multiset @ A ),X2: A] :
( ( A5
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ A5 )
=> ( ( count @ A @ ( complete_Sup_Sup @ ( multiset @ A ) @ A5 ) @ X2 )
= ( complete_Sup_Sup @ nat
@ ( image2 @ ( multiset @ A ) @ nat
@ ^ [X5: multiset @ A] : ( count @ A @ X5 @ X2 )
@ A5 ) ) ) ) ) ).
% count_Sup_multiset_nonempty
thf(fact_7040_subset__mset_Omono__cSUP,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( condit1219197933456340205attice @ B )
=> ! [F: ( multiset @ A ) > B,A5: C > ( multiset @ A ),I5: set @ C] :
( ( mono @ ( multiset @ A ) @ B @ ( subseteq_mset @ A ) @ F )
=> ( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( image2 @ C @ ( multiset @ A ) @ A5 @ I5 ) )
=> ( ( I5
!= ( bot_bot @ ( set @ C ) ) )
=> ( ord_less_eq @ B
@ ( complete_Sup_Sup @ B
@ ( image2 @ C @ B
@ ^ [X: C] : ( F @ ( A5 @ X ) )
@ I5 ) )
@ ( F @ ( complete_Sup_Sup @ ( multiset @ A ) @ ( image2 @ C @ ( multiset @ A ) @ A5 @ I5 ) ) ) ) ) ) ) ) ).
% subset_mset.mono_cSUP
thf(fact_7041_subset__mset_Omono__cSup,axiom,
! [B: $tType,A: $tType] :
( ( condit1219197933456340205attice @ B )
=> ! [F: ( multiset @ A ) > B,A5: set @ ( multiset @ A )] :
( ( mono @ ( multiset @ A ) @ B @ ( subseteq_mset @ A ) @ F )
=> ( ( 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 @ F @ A5 ) ) @ ( F @ ( complete_Sup_Sup @ ( multiset @ A ) @ A5 ) ) ) ) ) ) ) ).
% subset_mset.mono_cSup
thf(fact_7042_order_Omono_Ocong,axiom,
! [B: $tType,A: $tType] :
( ( order @ B )
=> ( ( mono @ A @ B )
= ( mono @ A @ B ) ) ) ).
% order.mono.cong
thf(fact_7043_subset__mset_Omono__def,axiom,
! [B: $tType,A: $tType] :
( ( order @ B )
=> ! [F: ( multiset @ A ) > B] :
( ( mono @ ( multiset @ A ) @ B @ ( subseteq_mset @ A ) @ F )
= ( ! [X: multiset @ A,Y4: multiset @ A] :
( ( subseteq_mset @ A @ X @ Y4 )
=> ( ord_less_eq @ B @ ( F @ X ) @ ( F @ Y4 ) ) ) ) ) ) ).
% subset_mset.mono_def
thf(fact_7044_subset__mset_OmonoI,axiom,
! [B: $tType,A: $tType] :
( ( order @ B )
=> ! [F: ( multiset @ A ) > B] :
( ! [X3: multiset @ A,Y2: multiset @ A] :
( ( subseteq_mset @ A @ X3 @ Y2 )
=> ( ord_less_eq @ B @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( mono @ ( multiset @ A ) @ B @ ( subseteq_mset @ A ) @ F ) ) ) ).
% subset_mset.monoI
thf(fact_7045_subset__mset_OmonoE,axiom,
! [B: $tType,A: $tType] :
( ( order @ B )
=> ! [F: ( multiset @ A ) > B,X2: multiset @ A,Y: multiset @ A] :
( ( mono @ ( multiset @ A ) @ B @ ( subseteq_mset @ A ) @ F )
=> ( ( subseteq_mset @ A @ X2 @ Y )
=> ( ord_less_eq @ B @ ( F @ X2 ) @ ( F @ Y ) ) ) ) ) ).
% subset_mset.monoE
thf(fact_7046_subset__mset_OmonoD,axiom,
! [B: $tType,A: $tType] :
( ( order @ B )
=> ! [F: ( multiset @ A ) > B,X2: multiset @ A,Y: multiset @ A] :
( ( mono @ ( multiset @ A ) @ B @ ( subseteq_mset @ A ) @ F )
=> ( ( subseteq_mset @ A @ X2 @ Y )
=> ( ord_less_eq @ B @ ( F @ X2 ) @ ( F @ Y ) ) ) ) ) ).
% subset_mset.monoD
thf(fact_7047_Inf__multiset__def,axiom,
! [A: $tType] :
( ( complete_Inf_Inf @ ( multiset @ A ) )
= ( map_fun @ ( set @ ( multiset @ A ) ) @ ( set @ ( A > nat ) ) @ ( A > nat ) @ ( multiset @ A ) @ ( image2 @ ( multiset @ A ) @ ( A > nat ) @ ( count @ A ) ) @ ( abs_multiset @ A )
@ ^ [A8: set @ ( A > nat ),I2: A] :
( if @ nat
@ ( A8
= ( bot_bot @ ( set @ ( A > nat ) ) ) )
@ ( zero_zero @ nat )
@ ( complete_Inf_Inf @ nat
@ ( image2 @ ( A > nat ) @ nat
@ ^ [F2: A > nat] : ( F2 @ I2 )
@ A8 ) ) ) ) ) ).
% Inf_multiset_def
thf(fact_7048_Sup__multiset__def,axiom,
! [A: $tType] :
( ( complete_Sup_Sup @ ( multiset @ A ) )
= ( ^ [A8: set @ ( multiset @ A )] :
( if @ ( multiset @ A )
@ ( ( A8
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
& ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ A8 ) )
@ ( abs_multiset @ A
@ ^ [I2: A] :
( complete_Sup_Sup @ nat
@ ( image2 @ ( multiset @ A ) @ nat
@ ^ [X5: multiset @ A] : ( count @ A @ X5 @ I2 )
@ A8 ) ) )
@ ( zero_zero @ ( multiset @ A ) ) ) ) ) ).
% Sup_multiset_def
thf(fact_7049_count__Abs__multiset,axiom,
! [A: $tType,F: A > nat] :
( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F @ X ) ) ) )
=> ( ( count @ A @ ( abs_multiset @ A @ F ) )
= F ) ) ).
% count_Abs_multiset
thf(fact_7050_zero__multiset__def,axiom,
! [A: $tType] :
( ( zero_zero @ ( multiset @ A ) )
= ( abs_multiset @ A
@ ^ [A3: A] : ( zero_zero @ nat ) ) ) ).
% zero_multiset_def
thf(fact_7051_Abs__multiset__inject,axiom,
! [A: $tType,X2: A > nat,Y: A > nat] :
( ( member @ ( A > nat ) @ X2
@ ( collect @ ( A > nat )
@ ^ [F2: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F2 @ X ) ) ) ) ) )
=> ( ( member @ ( A > nat ) @ Y
@ ( collect @ ( A > nat )
@ ^ [F2: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F2 @ X ) ) ) ) ) )
=> ( ( ( abs_multiset @ A @ X2 )
= ( abs_multiset @ A @ Y ) )
= ( X2 = Y ) ) ) ) ).
% Abs_multiset_inject
thf(fact_7052_Abs__multiset__induct,axiom,
! [A: $tType,P: ( multiset @ A ) > $o,X2: multiset @ A] :
( ! [Y2: A > nat] :
( ( member @ ( A > nat ) @ Y2
@ ( collect @ ( A > nat )
@ ^ [F2: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F2 @ X ) ) ) ) ) )
=> ( P @ ( abs_multiset @ A @ Y2 ) ) )
=> ( P @ X2 ) ) ).
% Abs_multiset_induct
thf(fact_7053_Abs__multiset__cases,axiom,
! [A: $tType,X2: multiset @ A] :
~ ! [Y2: A > nat] :
( ( X2
= ( abs_multiset @ A @ Y2 ) )
=> ~ ( member @ ( A > nat ) @ Y2
@ ( collect @ ( A > nat )
@ ^ [F2: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F2 @ X ) ) ) ) ) ) ) ).
% Abs_multiset_cases
thf(fact_7054_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 ) )
@ ^ [M6: A > nat,N6: A > nat,A3: A] : ( plus_plus @ nat @ ( M6 @ A3 ) @ ( N6 @ A3 ) ) ) ) ).
% plus_multiset_def
thf(fact_7055_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 ) )
@ ^ [M6: A > nat,N6: A > nat,A3: A] : ( minus_minus @ nat @ ( M6 @ A3 ) @ ( N6 @ A3 ) ) ) ) ).
% minus_multiset_def
thf(fact_7056_Abs__multiset__inverse,axiom,
! [A: $tType,Y: A > nat] :
( ( member @ ( A > nat ) @ Y
@ ( collect @ ( A > nat )
@ ^ [F2: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F2 @ X ) ) ) ) ) )
=> ( ( count @ A @ ( abs_multiset @ A @ Y ) )
= Y ) ) ).
% Abs_multiset_inverse
thf(fact_7057_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 ) )
@ ^ [A3: A,M6: A > nat,B3: A] : ( if @ nat @ ( B3 = A3 ) @ ( suc @ ( M6 @ B3 ) ) @ ( M6 @ B3 ) ) ) ) ).
% add_mset_def
thf(fact_7058_subset__mset_Omono__cINF,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( condit1219197933456340205attice @ B )
=> ! [F: ( multiset @ A ) > B,A5: C > ( multiset @ A ),I5: set @ C] :
( ( mono @ ( multiset @ A ) @ B @ ( subseteq_mset @ A ) @ F )
=> ( ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( image2 @ C @ ( multiset @ A ) @ A5 @ I5 ) )
=> ( ( I5
!= ( bot_bot @ ( set @ C ) ) )
=> ( ord_less_eq @ B @ ( F @ ( complete_Inf_Inf @ ( multiset @ A ) @ ( image2 @ C @ ( multiset @ A ) @ A5 @ I5 ) ) )
@ ( complete_Inf_Inf @ B
@ ( image2 @ C @ B
@ ^ [X: C] : ( F @ ( A5 @ X ) )
@ I5 ) ) ) ) ) ) ) ).
% subset_mset.mono_cINF
thf(fact_7059_subset__mset_Omono__cInf,axiom,
! [B: $tType,A: $tType] :
( ( condit1219197933456340205attice @ B )
=> ! [F: ( multiset @ A ) > B,A5: set @ ( multiset @ A )] :
( ( mono @ ( multiset @ A ) @ B @ ( subseteq_mset @ A ) @ F )
=> ( ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ord_less_eq @ B @ ( F @ ( complete_Inf_Inf @ ( multiset @ A ) @ A5 ) ) @ ( complete_Inf_Inf @ B @ ( image2 @ ( multiset @ A ) @ B @ F @ A5 ) ) ) ) ) ) ) ).
% subset_mset.mono_cInf
thf(fact_7060_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 ) ) )
= ( ! [X: B] :
( ( member @ B @ X @ I5 )
=> ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( A5 @ X ) ) ) ) ) ) ).
% subset_mset.bdd_below_UN
thf(fact_7061_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,M6: A > nat,X: A] : ( if @ nat @ ( P4 @ X ) @ ( M6 @ X ) @ ( zero_zero @ nat ) ) ) ) ).
% filter_mset_def
thf(fact_7062_repeat__mset__def,axiom,
! [A: $tType] :
( ( repeat_mset @ A )
= ( map_fun @ nat @ nat @ ( ( A > nat ) > A > nat ) @ ( ( multiset @ A ) > ( multiset @ A ) ) @ ( id @ nat ) @ ( map_fun @ ( multiset @ A ) @ ( A > nat ) @ ( A > nat ) @ ( multiset @ A ) @ ( count @ A ) @ ( abs_multiset @ A ) )
@ ^ [N4: nat,M6: A > nat,A3: A] : ( times_times @ nat @ N4 @ ( M6 @ A3 ) ) ) ) ).
% repeat_mset_def
thf(fact_7063_filter__mset__True,axiom,
! [A: $tType,M3: multiset @ A] :
( ( filter_mset @ A
@ ^ [Y4: A] : $true
@ M3 )
= M3 ) ).
% filter_mset_True
thf(fact_7064_repeat__mset__distrib2,axiom,
! [A: $tType,N: nat,A5: multiset @ A,B5: multiset @ A] :
( ( repeat_mset @ A @ N @ ( plus_plus @ ( multiset @ A ) @ A5 @ B5 ) )
= ( plus_plus @ ( multiset @ A ) @ ( repeat_mset @ A @ N @ A5 ) @ ( repeat_mset @ A @ N @ B5 ) ) ) ).
% repeat_mset_distrib2
thf(fact_7065_filter__union__mset,axiom,
! [A: $tType,P: A > $o,M3: multiset @ A,N5: multiset @ A] :
( ( filter_mset @ A @ P @ ( plus_plus @ ( multiset @ A ) @ M3 @ N5 ) )
= ( plus_plus @ ( multiset @ A ) @ ( filter_mset @ A @ P @ M3 ) @ ( filter_mset @ A @ P @ N5 ) ) ) ).
% filter_union_mset
thf(fact_7066_union__filter__mset__complement,axiom,
! [A: $tType,P: A > $o,Q: A > $o,M3: multiset @ A] :
( ! [X3: A] :
( ( P @ X3 )
= ( ~ ( Q @ X3 ) ) )
=> ( ( plus_plus @ ( multiset @ A ) @ ( filter_mset @ A @ P @ M3 ) @ ( filter_mset @ A @ Q @ M3 ) )
= M3 ) ) ).
% union_filter_mset_complement
thf(fact_7067_filter__mset__False,axiom,
! [A: $tType,M3: multiset @ A] :
( ( filter_mset @ A
@ ^ [Y4: A] : $false
@ M3 )
= ( zero_zero @ ( multiset @ A ) ) ) ).
% filter_mset_False
thf(fact_7068_set__mset__filter,axiom,
! [A: $tType,P: A > $o,M3: multiset @ A] :
( ( set_mset @ A @ ( filter_mset @ A @ P @ M3 ) )
= ( collect @ A
@ ^ [A3: A] :
( ( member @ A @ A3 @ ( set_mset @ A @ M3 ) )
& ( P @ A3 ) ) ) ) ).
% set_mset_filter
thf(fact_7069_repeat__mset__0,axiom,
! [A: $tType,M3: multiset @ A] :
( ( repeat_mset @ A @ ( zero_zero @ nat ) @ M3 )
= ( zero_zero @ ( multiset @ A ) ) ) ).
% repeat_mset_0
thf(fact_7070_repeat__mset__Suc,axiom,
! [A: $tType,N: nat,M3: multiset @ A] :
( ( repeat_mset @ A @ ( suc @ N ) @ M3 )
= ( plus_plus @ ( multiset @ A ) @ M3 @ ( repeat_mset @ A @ N @ M3 ) ) ) ).
% repeat_mset_Suc
thf(fact_7071_count__filter__mset,axiom,
! [A: $tType,P: A > $o,A2: A,M3: multiset @ A] :
( ( ( P @ A2 )
=> ( ( count @ A @ ( filter_mset @ A @ P @ M3 ) @ A2 )
= ( count @ A @ M3 @ A2 ) ) )
& ( ~ ( P @ A2 )
=> ( ( count @ A @ ( filter_mset @ A @ P @ M3 ) @ A2 )
= ( zero_zero @ nat ) ) ) ) ).
% count_filter_mset
thf(fact_7072_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_7073_filter__mset_Orep__eq,axiom,
! [A: $tType,X2: A > $o,Xa: multiset @ A] :
( ( count @ A @ ( filter_mset @ A @ X2 @ Xa ) )
= ( ^ [X: A] : ( if @ nat @ ( X2 @ X ) @ ( count @ A @ Xa @ X ) @ ( zero_zero @ nat ) ) ) ) ).
% filter_mset.rep_eq
thf(fact_7074_size__filter__mset__lesseq,axiom,
! [A: $tType,F: A > $o,M3: multiset @ A] : ( ord_less_eq @ nat @ ( size_size @ ( multiset @ A ) @ ( filter_mset @ A @ F @ M3 ) ) @ ( size_size @ ( multiset @ A ) @ M3 ) ) ).
% size_filter_mset_lesseq
thf(fact_7075_repeat__mset__distrib,axiom,
! [A: $tType,M: nat,N: nat,A5: multiset @ A] :
( ( repeat_mset @ A @ ( plus_plus @ nat @ M @ N ) @ A5 )
= ( plus_plus @ ( multiset @ A ) @ ( repeat_mset @ A @ M @ A5 ) @ ( repeat_mset @ A @ N @ A5 ) ) ) ).
% repeat_mset_distrib
thf(fact_7076_left__add__mult__distrib__mset,axiom,
! [A: $tType,I: nat,U: multiset @ A,J: nat,K: multiset @ A] :
( ( plus_plus @ ( multiset @ A ) @ ( repeat_mset @ A @ I @ U ) @ ( plus_plus @ ( multiset @ A ) @ ( repeat_mset @ A @ J @ U ) @ K ) )
= ( plus_plus @ ( multiset @ A ) @ ( repeat_mset @ A @ ( plus_plus @ nat @ I @ J ) @ U ) @ K ) ) ).
% left_add_mult_distrib_mset
thf(fact_7077_repeat__mset__cancel1,axiom,
! [A: $tType,A2: nat,A5: multiset @ A,B5: multiset @ A] :
( ( ( repeat_mset @ A @ A2 @ A5 )
= ( repeat_mset @ A @ A2 @ B5 ) )
= ( ( A5 = B5 )
| ( A2
= ( zero_zero @ nat ) ) ) ) ).
% repeat_mset_cancel1
thf(fact_7078_multiset__partition,axiom,
! [A: $tType,M3: multiset @ A,P: A > $o] :
( M3
= ( plus_plus @ ( multiset @ A ) @ ( filter_mset @ A @ P @ M3 )
@ ( filter_mset @ A
@ ^ [X: A] :
~ ( P @ X )
@ M3 ) ) ) ).
% multiset_partition
thf(fact_7079_filter__filter__mset,axiom,
! [A: $tType,P: A > $o,Q: A > $o,M3: multiset @ A] :
( ( filter_mset @ A @ P @ ( filter_mset @ A @ Q @ M3 ) )
= ( filter_mset @ A
@ ^ [X: A] :
( ( Q @ X )
& ( P @ X ) )
@ M3 ) ) ).
% filter_filter_mset
thf(fact_7080_repeat__mset__eq__empty__iff,axiom,
! [A: $tType,N: nat,A5: multiset @ A] :
( ( ( repeat_mset @ A @ N @ A5 )
= ( zero_zero @ ( multiset @ A ) ) )
= ( ( N
= ( zero_zero @ nat ) )
| ( A5
= ( zero_zero @ ( multiset @ A ) ) ) ) ) ).
% repeat_mset_eq_empty_iff
thf(fact_7081_image__mset__If,axiom,
! [A: $tType,B: $tType,P: B > $o,F: B > A,G: B > A,A5: multiset @ B] :
( ( image_mset @ B @ A
@ ^ [X: B] : ( if @ A @ ( P @ X ) @ ( F @ X ) @ ( G @ X ) )
@ A5 )
= ( plus_plus @ ( multiset @ A ) @ ( image_mset @ B @ A @ F @ ( filter_mset @ B @ P @ A5 ) )
@ ( image_mset @ B @ A @ G
@ ( filter_mset @ B
@ ^ [X: B] :
~ ( P @ X )
@ A5 ) ) ) ) ).
% image_mset_If
thf(fact_7082_mset__subseteq__add__iff1,axiom,
! [A: $tType,J: nat,I: nat,U: multiset @ A,M: multiset @ A,N: multiset @ A] :
( ( ord_less_eq @ nat @ J @ I )
=> ( ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ ( repeat_mset @ A @ I @ U ) @ M ) @ ( plus_plus @ ( multiset @ A ) @ ( repeat_mset @ A @ J @ U ) @ N ) )
= ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ ( repeat_mset @ A @ ( minus_minus @ nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% mset_subseteq_add_iff1
thf(fact_7083_mset__subseteq__add__iff2,axiom,
! [A: $tType,I: nat,J: nat,U: multiset @ A,M: multiset @ A,N: multiset @ A] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( subseteq_mset @ A @ ( plus_plus @ ( multiset @ A ) @ ( repeat_mset @ A @ I @ U ) @ M ) @ ( plus_plus @ ( multiset @ A ) @ ( repeat_mset @ A @ J @ U ) @ N ) )
= ( subseteq_mset @ A @ M @ ( plus_plus @ ( multiset @ A ) @ ( repeat_mset @ A @ ( minus_minus @ nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% mset_subseteq_add_iff2
thf(fact_7084_fold__mset__def,axiom,
! [B: $tType,A: $tType] :
( ( fold_mset @ A @ B )
= ( ^ [F2: A > B > B,S6: B,M6: multiset @ A] :
( finite_fold @ A @ B
@ ^ [X: A] : ( compow @ ( B > B ) @ ( count @ A @ M6 @ X ) @ ( F2 @ X ) )
@ S6
@ ( set_mset @ A @ M6 ) ) ) ) ).
% fold_mset_def
thf(fact_7085_set__mset__replicate__mset__subset,axiom,
! [A: $tType,N: nat,X2: A] :
( ( ( N
= ( zero_zero @ nat ) )
=> ( ( set_mset @ A @ ( replicate_mset @ A @ N @ X2 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ( ( set_mset @ A @ ( replicate_mset @ A @ N @ X2 ) )
= ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% set_mset_replicate_mset_subset
thf(fact_7086_size__replicate__mset,axiom,
! [A: $tType,N: nat,M3: A] :
( ( size_size @ ( multiset @ A ) @ ( replicate_mset @ A @ N @ M3 ) )
= N ) ).
% size_replicate_mset
thf(fact_7087_replicate__mset__0,axiom,
! [A: $tType,X2: A] :
( ( replicate_mset @ A @ ( zero_zero @ nat ) @ X2 )
= ( zero_zero @ ( multiset @ A ) ) ) ).
% replicate_mset_0
thf(fact_7088_replicate__mset__eq__empty__iff,axiom,
! [A: $tType,N: nat,A2: A] :
( ( ( replicate_mset @ A @ N @ A2 )
= ( zero_zero @ ( multiset @ A ) ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% replicate_mset_eq_empty_iff
thf(fact_7089_count__replicate__mset,axiom,
! [A: $tType,Y: A,X2: A,N: nat] :
( ( ( Y = X2 )
=> ( ( count @ A @ ( replicate_mset @ A @ N @ X2 ) @ Y )
= N ) )
& ( ( Y != X2 )
=> ( ( count @ A @ ( replicate_mset @ A @ N @ X2 ) @ Y )
= ( zero_zero @ nat ) ) ) ) ).
% count_replicate_mset
thf(fact_7090_in__mset__fold__plus__iff,axiom,
! [A: $tType,X2: A,M3: multiset @ A,NN: multiset @ ( multiset @ A )] :
( ( member @ A @ X2 @ ( set_mset @ A @ ( fold_mset @ ( multiset @ A ) @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) ) @ M3 @ NN ) ) )
= ( ( member @ A @ X2 @ ( set_mset @ A @ M3 ) )
| ? [N6: multiset @ A] :
( ( member @ ( multiset @ A ) @ N6 @ ( set_mset @ ( multiset @ A ) @ NN ) )
& ( member @ A @ X2 @ ( set_mset @ A @ N6 ) ) ) ) ) ).
% in_mset_fold_plus_iff
thf(fact_7091_in__replicate__mset,axiom,
! [A: $tType,X2: A,N: nat,Y: A] :
( ( member @ A @ X2 @ ( set_mset @ A @ ( replicate_mset @ A @ N @ Y ) ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
& ( X2 = Y ) ) ) ).
% in_replicate_mset
thf(fact_7092_sum__mset__replicate__mset,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [N: nat,A2: A] :
( ( comm_m7189776963980413722m_mset @ A @ ( replicate_mset @ A @ N @ A2 ) )
= ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ A2 ) ) ) ).
% sum_mset_replicate_mset
thf(fact_7093_repeat__mset__distrib__add__mset,axiom,
! [A: $tType,N: nat,A2: A,A5: multiset @ A] :
( ( repeat_mset @ A @ N @ ( add_mset @ A @ A2 @ A5 ) )
= ( plus_plus @ ( multiset @ A ) @ ( replicate_mset @ A @ N @ A2 ) @ ( repeat_mset @ A @ N @ A5 ) ) ) ).
% repeat_mset_distrib_add_mset
thf(fact_7094_msubseteq__replicate__msetE,axiom,
! [A: $tType,A5: multiset @ A,N: nat,A2: A] :
( ( subseteq_mset @ A @ A5 @ ( replicate_mset @ A @ N @ A2 ) )
=> ~ ! [M4: nat] :
( ( ord_less_eq @ nat @ M4 @ N )
=> ( A5
!= ( replicate_mset @ A @ M4 @ A2 ) ) ) ) ).
% msubseteq_replicate_msetE
thf(fact_7095_replicate__mset__eq__iff,axiom,
! [A: $tType,M: nat,A2: A,N: nat,B2: A] :
( ( ( replicate_mset @ A @ M @ A2 )
= ( replicate_mset @ A @ N @ B2 ) )
= ( ( ( M
= ( zero_zero @ nat ) )
& ( N
= ( zero_zero @ nat ) ) )
| ( ( M = N )
& ( A2 = B2 ) ) ) ) ).
% replicate_mset_eq_iff
thf(fact_7096_union__fold__mset__add__mset,axiom,
! [A: $tType] :
( ( plus_plus @ ( multiset @ A ) )
= ( fold_mset @ A @ ( multiset @ A ) @ ( add_mset @ A ) ) ) ).
% union_fold_mset_add_mset
thf(fact_7097_comp__fun__commute_Ofold__mset__union,axiom,
! [B: $tType,A: $tType,F: A > B > B,S3: B,M3: multiset @ A,N5: multiset @ A] :
( ( finite6289374366891150609ommute @ A @ B @ F )
=> ( ( fold_mset @ A @ B @ F @ S3 @ ( plus_plus @ ( multiset @ A ) @ M3 @ N5 ) )
= ( fold_mset @ A @ B @ F @ ( fold_mset @ A @ B @ F @ S3 @ M3 ) @ N5 ) ) ) ).
% comp_fun_commute.fold_mset_union
thf(fact_7098_image__mset__const__eq,axiom,
! [B: $tType,A: $tType,C2: A,M3: multiset @ B] :
( ( image_mset @ B @ A
@ ^ [A3: B] : C2
@ M3 )
= ( replicate_mset @ A @ ( size_size @ ( multiset @ B ) @ M3 ) @ C2 ) ) ).
% image_mset_const_eq
thf(fact_7099_filter__eq__replicate__mset,axiom,
! [A: $tType,X2: A,D3: multiset @ A] :
( ( filter_mset @ A
@ ^ [Y4: A] : Y4 = X2
@ D3 )
= ( replicate_mset @ A @ ( count @ A @ D3 @ X2 ) @ X2 ) ) ).
% filter_eq_replicate_mset
thf(fact_7100_replicate__mset__msubseteq__iff,axiom,
! [A: $tType,M: nat,A2: A,N: nat,B2: A] :
( ( subseteq_mset @ A @ ( replicate_mset @ A @ M @ A2 ) @ ( replicate_mset @ A @ N @ B2 ) )
= ( ( M
= ( zero_zero @ nat ) )
| ( ( A2 = B2 )
& ( ord_less_eq @ nat @ M @ N ) ) ) ) ).
% replicate_mset_msubseteq_iff
thf(fact_7101_count__le__replicate__mset__subset__eq,axiom,
! [A: $tType,N: nat,M3: multiset @ A,X2: A] :
( ( ord_less_eq @ nat @ N @ ( count @ A @ M3 @ X2 ) )
= ( subseteq_mset @ A @ ( replicate_mset @ A @ N @ X2 ) @ M3 ) ) ).
% count_le_replicate_mset_subset_eq
thf(fact_7102_sum__mset_Oeq__fold,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ( ( comm_m7189776963980413722m_mset @ A )
= ( fold_mset @ A @ A @ ( plus_plus @ A ) @ ( zero_zero @ A ) ) ) ) ).
% sum_mset.eq_fold
thf(fact_7103_sorted__list__of__multiset__def,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( linord6283353356039996273ltiset @ A )
= ( fold_mset @ A @ ( list @ A )
@ ( linorder_insort_key @ A @ A
@ ^ [X: A] : X )
@ ( nil @ A ) ) ) ) ).
% sorted_list_of_multiset_def
thf(fact_7104_mult1E,axiom,
! [A: $tType,N5: multiset @ A,M3: multiset @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ N5 @ M3 ) @ ( mult1 @ A @ R ) )
=> ~ ! [A4: A,M0: multiset @ A] :
( ( M3
= ( add_mset @ A @ A4 @ M0 ) )
=> ! [K8: multiset @ A] :
( ( N5
= ( plus_plus @ ( multiset @ A ) @ M0 @ K8 ) )
=> ~ ! [B9: A] :
( ( member @ A @ B9 @ ( set_mset @ A @ K8 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B9 @ A4 ) @ R ) ) ) ) ) ).
% mult1E
thf(fact_7105_mult1I,axiom,
! [A: $tType,M3: multiset @ A,A2: A,M02: multiset @ A,N5: multiset @ A,K3: multiset @ A,R: set @ ( product_prod @ A @ A )] :
( ( M3
= ( add_mset @ A @ A2 @ M02 ) )
=> ( ( N5
= ( plus_plus @ ( multiset @ A ) @ M02 @ K3 ) )
=> ( ! [B4: A] :
( ( member @ A @ B4 @ ( set_mset @ A @ K3 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B4 @ A2 ) @ R ) )
=> ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ N5 @ M3 ) @ ( mult1 @ A @ R ) ) ) ) ) ).
% mult1I
thf(fact_7106_mult1__union,axiom,
! [A: $tType,B5: multiset @ A,D3: multiset @ A,R: set @ ( product_prod @ A @ A ),C6: multiset @ A] :
( ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ B5 @ D3 ) @ ( mult1 @ A @ R ) )
=> ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ C6 @ B5 ) @ ( plus_plus @ ( multiset @ A ) @ C6 @ D3 ) ) @ ( mult1 @ A @ R ) ) ) ).
% mult1_union
thf(fact_7107_less__add,axiom,
! [A: $tType,N5: multiset @ A,A2: A,M02: multiset @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ N5 @ ( add_mset @ A @ A2 @ M02 ) ) @ ( mult1 @ A @ R ) )
=> ( ? [M10: multiset @ A] :
( ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ M10 @ M02 ) @ ( mult1 @ A @ R ) )
& ( N5
= ( add_mset @ A @ A2 @ M10 ) ) )
| ? [K8: multiset @ A] :
( ! [B9: A] :
( ( member @ A @ B9 @ ( set_mset @ A @ K8 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B9 @ A2 ) @ R ) )
& ( N5
= ( plus_plus @ ( multiset @ A ) @ M02 @ K8 ) ) ) ) ) ).
% less_add
thf(fact_7108_mult1__lessE,axiom,
! [A: $tType,N5: multiset @ A,M3: multiset @ A,R: A > A > $o] :
( ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ N5 @ M3 ) @ ( mult1 @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ R ) ) ) )
=> ( ( asymp @ A @ R )
=> ~ ! [A4: A,M0: multiset @ A] :
( ( M3
= ( add_mset @ A @ A4 @ M0 ) )
=> ! [K8: multiset @ A] :
( ( N5
= ( plus_plus @ ( multiset @ A ) @ M0 @ K8 ) )
=> ( ~ ( member @ A @ A4 @ ( set_mset @ A @ K8 ) )
=> ~ ! [B9: A] :
( ( member @ A @ B9 @ ( set_mset @ A @ K8 ) )
=> ( R @ B9 @ A4 ) ) ) ) ) ) ) ).
% mult1_lessE
thf(fact_7109_one__step__implies__mult,axiom,
! [A: $tType,J4: multiset @ A,K3: multiset @ A,R: set @ ( product_prod @ A @ A ),I5: multiset @ A] :
( ( J4
!= ( zero_zero @ ( multiset @ A ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ ( set_mset @ A @ K3 ) )
=> ? [Xa2: A] :
( ( member @ A @ Xa2 @ ( set_mset @ A @ J4 ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Xa2 ) @ R ) ) )
=> ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ I5 @ K3 ) @ ( plus_plus @ ( multiset @ A ) @ I5 @ J4 ) ) @ ( mult @ A @ R ) ) ) ) ).
% one_step_implies_mult
thf(fact_7110_asymp__greater,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( asymp @ A
@ ^ [X: A,Y4: A] : ( ord_less @ A @ Y4 @ X ) ) ) ).
% asymp_greater
thf(fact_7111_prod__mset__minus,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A2: A,A5: multiset @ A] :
( ( member @ A @ A2 @ ( set_mset @ A @ A5 ) )
=> ( ( A2
!= ( zero_zero @ A ) )
=> ( ( comm_m9189036328036947845d_mset @ A @ ( minus_minus @ ( multiset @ A ) @ A5 @ ( add_mset @ A @ A2 @ ( zero_zero @ ( multiset @ A ) ) ) ) )
= ( divide_divide @ A @ ( comm_m9189036328036947845d_mset @ A @ A5 ) @ A2 ) ) ) ) ) ).
% prod_mset_minus
thf(fact_7112_mult__implies__one__step,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),M3: multiset @ A,N5: multiset @ A] :
( ( trans @ A @ R )
=> ( ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ M3 @ N5 ) @ ( mult @ A @ R ) )
=> ? [I8: multiset @ A,J6: multiset @ A] :
( ( N5
= ( plus_plus @ ( multiset @ A ) @ I8 @ J6 ) )
& ? [K8: multiset @ A] :
( ( M3
= ( plus_plus @ ( multiset @ A ) @ I8 @ K8 ) )
& ( J6
!= ( zero_zero @ ( multiset @ A ) ) )
& ! [X4: A] :
( ( member @ A @ X4 @ ( set_mset @ A @ K8 ) )
=> ? [Xa3: A] :
( ( member @ A @ Xa3 @ ( set_mset @ A @ J6 ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Xa3 ) @ R ) ) ) ) ) ) ) ).
% mult_implies_one_step
thf(fact_7113_prod__mset__empty,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ( ( comm_m9189036328036947845d_mset @ A @ ( zero_zero @ ( multiset @ A ) ) )
= ( one_one @ A ) ) ) ).
% prod_mset_empty
thf(fact_7114_prod__mset_Ounion,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M3: multiset @ A,N5: multiset @ A] :
( ( comm_m9189036328036947845d_mset @ A @ ( plus_plus @ ( multiset @ A ) @ M3 @ N5 ) )
= ( times_times @ A @ ( comm_m9189036328036947845d_mset @ A @ M3 ) @ ( comm_m9189036328036947845d_mset @ A @ N5 ) ) ) ) ).
% prod_mset.union
thf(fact_7115_prod__mset__Un,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A5: multiset @ A,B5: multiset @ A] :
( ( comm_m9189036328036947845d_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A5 @ B5 ) )
= ( times_times @ A @ ( comm_m9189036328036947845d_mset @ A @ A5 ) @ ( comm_m9189036328036947845d_mset @ A @ B5 ) ) ) ) ).
% prod_mset_Un
thf(fact_7116_prod__mset__zero__iff,axiom,
! [A: $tType] :
( ( semidom @ A )
=> ! [A5: multiset @ A] :
( ( ( comm_m9189036328036947845d_mset @ A @ A5 )
= ( zero_zero @ A ) )
= ( member @ A @ ( zero_zero @ A ) @ ( set_mset @ A @ A5 ) ) ) ) ).
% prod_mset_zero_iff
thf(fact_7117_prod__mset_Oneutral__const,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A5: multiset @ B] :
( ( comm_m9189036328036947845d_mset @ A
@ ( image_mset @ B @ A
@ ^ [Uu: B] : ( one_one @ A )
@ A5 ) )
= ( one_one @ A ) ) ) ).
% prod_mset.neutral_const
thf(fact_7118_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
@ ^ [Uu: B] : C2
@ A5 ) )
= ( power_power @ A @ C2 @ ( size_size @ ( multiset @ B ) @ A5 ) ) ) ) ).
% prod_mset_constant
thf(fact_7119_trans__Restr,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A5: set @ A] :
( ( trans @ A @ R )
=> ( trans @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R
@ ( product_Sigma @ A @ A @ A5
@ ^ [Uu: A] : A5 ) ) ) ) ).
% trans_Restr
thf(fact_7120_prod__mset_Oswap,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: B > C > A,B5: multiset @ C,A5: multiset @ B] :
( ( comm_m9189036328036947845d_mset @ A
@ ( image_mset @ B @ A
@ ^ [I2: B] : ( comm_m9189036328036947845d_mset @ A @ ( image_mset @ C @ A @ ( G @ I2 ) @ B5 ) )
@ A5 ) )
= ( comm_m9189036328036947845d_mset @ A
@ ( image_mset @ C @ A
@ ^ [J3: C] :
( comm_m9189036328036947845d_mset @ A
@ ( image_mset @ B @ A
@ ^ [I2: B] : ( G @ I2 @ J3 )
@ A5 ) )
@ B5 ) ) ) ) ).
% prod_mset.swap
thf(fact_7121_prod__mset_Odistrib,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: B > A,H: B > A,A5: multiset @ B] :
( ( comm_m9189036328036947845d_mset @ A
@ ( image_mset @ B @ A
@ ^ [X: B] : ( times_times @ A @ ( G @ X ) @ ( H @ X ) )
@ A5 ) )
= ( times_times @ A @ ( comm_m9189036328036947845d_mset @ A @ ( image_mset @ B @ A @ G @ A5 ) ) @ ( comm_m9189036328036947845d_mset @ A @ ( image_mset @ B @ A @ H @ A5 ) ) ) ) ) ).
% prod_mset.distrib
thf(fact_7122_prod__mset_Oneutral,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A5: multiset @ A] :
( ! [X3: A] :
( ( member @ A @ X3 @ ( set_mset @ A @ A5 ) )
=> ( X3
= ( one_one @ A ) ) )
=> ( ( comm_m9189036328036947845d_mset @ A @ A5 )
= ( one_one @ A ) ) ) ) ).
% prod_mset.neutral
thf(fact_7123_is__unit__prod__mset__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A5: multiset @ A] :
( ( dvd_dvd @ A @ ( comm_m9189036328036947845d_mset @ A @ A5 ) @ ( one_one @ A ) )
= ( ! [X: A] :
( ( member @ A @ X @ ( set_mset @ A @ A5 ) )
=> ( dvd_dvd @ A @ X @ ( one_one @ A ) ) ) ) ) ) ).
% is_unit_prod_mset_iff
thf(fact_7124_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_7125_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
@ ^ [X: B] : ( if @ A @ ( X = Y ) @ C2 @ ( one_one @ A ) )
@ A5 ) )
= ( power_power @ A @ C2 @ ( count @ B @ A5 @ Y ) ) ) ) ).
% prod_mset_delta
thf(fact_7126_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
@ ^ [X: B] : ( if @ A @ ( Y = X ) @ C2 @ ( one_one @ A ) )
@ A5 ) )
= ( power_power @ A @ C2 @ ( count @ B @ A5 @ Y ) ) ) ) ).
% prod_mset_delta'
thf(fact_7127_prod__mset__multiplicity,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ( ( comm_m9189036328036947845d_mset @ A )
= ( ^ [M6: multiset @ A] :
( groups7121269368397514597t_prod @ A @ A
@ ^ [X: A] : ( power_power @ A @ X @ ( count @ A @ M6 @ X ) )
@ ( set_mset @ A @ M6 ) ) ) ) ) ).
% prod_mset_multiplicity
thf(fact_7128_prod__mset__diff,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [B5: multiset @ A,A5: multiset @ A] :
( ( subseteq_mset @ A @ B5 @ A5 )
=> ( ~ ( member @ A @ ( zero_zero @ A ) @ ( set_mset @ A @ B5 ) )
=> ( ( comm_m9189036328036947845d_mset @ A @ ( minus_minus @ ( multiset @ A ) @ A5 @ B5 ) )
= ( divide_divide @ A @ ( comm_m9189036328036947845d_mset @ A @ A5 ) @ ( comm_m9189036328036947845d_mset @ A @ B5 ) ) ) ) ) ) ).
% prod_mset_diff
thf(fact_7129_mult1__def,axiom,
! [A: $tType] :
( ( mult1 @ A )
= ( ^ [R2: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) )
@ ( product_case_prod @ ( multiset @ A ) @ ( multiset @ A ) @ $o
@ ^ [N6: multiset @ A,M6: multiset @ A] :
? [A3: A,M03: multiset @ A,K6: multiset @ A] :
( ( M6
= ( add_mset @ A @ A3 @ M03 ) )
& ( N6
= ( plus_plus @ ( multiset @ A ) @ M03 @ K6 ) )
& ! [B3: A] :
( ( member @ A @ B3 @ ( set_mset @ A @ K6 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B3 @ A3 ) @ R2 ) ) ) ) ) ) ) ).
% mult1_def
thf(fact_7130_mult__cancel,axiom,
! [A: $tType,S3: set @ ( product_prod @ A @ A ),X8: multiset @ A,Z9: multiset @ A,Y7: multiset @ A] :
( ( trans @ A @ S3 )
=> ( ( irrefl @ A @ S3 )
=> ( ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ X8 @ Z9 ) @ ( plus_plus @ ( multiset @ A ) @ Y7 @ Z9 ) ) @ ( mult @ A @ S3 ) )
= ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ X8 @ Y7 ) @ ( mult @ A @ S3 ) ) ) ) ) ).
% mult_cancel
thf(fact_7131_rel__fun__def,axiom,
! [B: $tType,D: $tType,C: $tType,A: $tType] :
( ( bNF_rel_fun @ A @ C @ B @ D )
= ( ^ [A8: A > C > $o,B8: B > D > $o,F2: A > B,G2: C > D] :
! [X: A,Y4: C] :
( ( A8 @ X @ Y4 )
=> ( B8 @ ( F2 @ X ) @ ( G2 @ Y4 ) ) ) ) ) ).
% rel_fun_def
thf(fact_7132_rel__fun__eq__rel,axiom,
! [C: $tType,B: $tType,A: $tType,R3: B > C > $o] :
( ( bNF_rel_fun @ A @ A @ B @ C
@ ^ [Y3: A,Z: A] : Y3 = Z
@ R3 )
= ( ^ [F2: A > B,G2: A > C] :
! [X: A] : ( R3 @ ( F2 @ X ) @ ( G2 @ X ) ) ) ) ).
% rel_fun_eq_rel
thf(fact_7133_finite__set__of__finite__funs,axiom,
! [B: $tType,A: $tType,A5: set @ A,B5: set @ B,D2: B] :
( ( finite_finite2 @ A @ A5 )
=> ( ( finite_finite2 @ B @ B5 )
=> ( finite_finite2 @ ( A > B )
@ ( collect @ ( A > B )
@ ^ [F2: A > B] :
! [X: A] :
( ( ( member @ A @ X @ A5 )
=> ( member @ B @ ( F2 @ X ) @ B5 ) )
& ( ~ ( member @ A @ X @ A5 )
=> ( ( F2 @ X )
= D2 ) ) ) ) ) ) ) ).
% finite_set_of_finite_funs
thf(fact_7134_Collect__all__eq,axiom,
! [A: $tType,B: $tType,P: A > B > $o] :
( ( collect @ A
@ ^ [X: A] :
! [X5: B] : ( P @ X @ X5 ) )
= ( complete_Inf_Inf @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [Y4: B] :
( collect @ A
@ ^ [X: A] : ( P @ X @ Y4 ) )
@ ( top_top @ ( set @ B ) ) ) ) ) ).
% Collect_all_eq
thf(fact_7135_Least__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_Least @ A )
= ( ^ [P4: A > $o] :
( the @ A
@ ^ [X: A] :
( ( P4 @ X )
& ! [Y4: A] :
( ( P4 @ Y4 )
=> ( ord_less_eq @ A @ X @ Y4 ) ) ) ) ) ) ) ).
% Least_def
thf(fact_7136_Greatest__def,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( order_Greatest @ A )
= ( ^ [P4: A > $o] :
( the @ A
@ ^ [X: A] :
( ( P4 @ X )
& ! [Y4: A] :
( ( P4 @ Y4 )
=> ( ord_less_eq @ A @ Y4 @ X ) ) ) ) ) ) ) ).
% Greatest_def
thf(fact_7137_ord_OLeast__def,axiom,
! [A: $tType] :
( ( least @ A )
= ( ^ [Less_eq2: A > A > $o,P4: A > $o] :
( the @ A
@ ^ [X: A] :
( ( P4 @ X )
& ! [Y4: A] :
( ( P4 @ Y4 )
=> ( Less_eq2 @ X @ Y4 ) ) ) ) ) ) ).
% ord.Least_def
thf(fact_7138_transfer__bforall__def,axiom,
! [A: $tType] :
( ( transfer_bforall @ A )
= ( ^ [P4: A > $o,Q8: A > $o] :
! [X: A] :
( ( P4 @ X )
=> ( Q8 @ X ) ) ) ) ).
% transfer_bforall_def
thf(fact_7139_ord_OLeast_Ocong,axiom,
! [A: $tType] :
( ( least @ A )
= ( least @ A ) ) ).
% ord.Least.cong
thf(fact_7140_subset__mset_OLeast__def,axiom,
! [A: $tType,P: ( multiset @ A ) > $o] :
( ( least @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ P )
= ( the @ ( multiset @ A )
@ ^ [X: multiset @ A] :
( ( P @ X )
& ! [Y4: multiset @ A] :
( ( P @ Y4 )
=> ( subseteq_mset @ A @ X @ Y4 ) ) ) ) ) ).
% subset_mset.Least_def
thf(fact_7141_repeat__mset_Oabs__eq,axiom,
! [A: $tType,X2: A > nat,Xa: nat] :
( ( bNF_eq_onp @ ( A > nat )
@ ^ [F2: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F2 @ X ) ) ) )
@ X2
@ X2 )
=> ( ( repeat_mset @ A @ Xa @ ( abs_multiset @ A @ X2 ) )
= ( abs_multiset @ A
@ ^ [A3: A] : ( times_times @ nat @ Xa @ ( X2 @ A3 ) ) ) ) ) ).
% repeat_mset.abs_eq
thf(fact_7142_type__definition__multiset,axiom,
! [A: $tType] :
( type_definition @ ( multiset @ A ) @ ( A > nat ) @ ( count @ A ) @ ( abs_multiset @ A )
@ ( collect @ ( A > nat )
@ ^ [F2: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F2 @ X ) ) ) ) ) ) ).
% type_definition_multiset
thf(fact_7143_type__copy__obj__one__point__absE,axiom,
! [A: $tType,B: $tType,Rep: A > B,Abs: B > A,S3: A] :
( ( type_definition @ A @ B @ Rep @ Abs @ ( top_top @ ( set @ B ) ) )
=> ~ ! [X3: B] :
( S3
!= ( Abs @ X3 ) ) ) ).
% type_copy_obj_one_point_absE
thf(fact_7144_eq__onp__live__step,axiom,
! [A: $tType,X2: $o,Y: $o,P: A > $o,A2: A] :
( ( X2 = Y )
=> ( ( ( bNF_eq_onp @ A @ P @ A2 @ A2 )
& X2 )
= ( ( P @ A2 )
& Y ) ) ) ).
% eq_onp_live_step
thf(fact_7145_eq__onp__True,axiom,
! [A: $tType] :
( ( bNF_eq_onp @ A
@ ^ [Uu: A] : $true )
= ( ^ [Y3: A,Z: A] : Y3 = Z ) ) ).
% eq_onp_True
thf(fact_7146_eq__onp__def,axiom,
! [A: $tType] :
( ( bNF_eq_onp @ A )
= ( ^ [R7: A > $o,X: A,Y4: A] :
( ( R7 @ X )
& ( X = Y4 ) ) ) ) ).
% eq_onp_def
thf(fact_7147_type__copy__map__comp0__undo,axiom,
! [E: $tType,A: $tType,C: $tType,B: $tType,D: $tType,F7: $tType,Rep: A > B,Abs: B > A,Rep2: C > D,Abs2: D > C,Rep3: E > F7,Abs3: F7 > E,M3: F7 > D,M14: B > D,M25: F7 > B] :
( ( type_definition @ A @ B @ Rep @ Abs @ ( top_top @ ( set @ B ) ) )
=> ( ( type_definition @ C @ D @ Rep2 @ Abs2 @ ( top_top @ ( set @ D ) ) )
=> ( ( type_definition @ E @ F7 @ Rep3 @ Abs3 @ ( top_top @ ( set @ F7 ) ) )
=> ( ( ( comp @ F7 @ C @ E @ ( comp @ D @ C @ F7 @ Abs2 @ M3 ) @ Rep3 )
= ( comp @ A @ C @ E @ ( comp @ B @ C @ A @ ( comp @ D @ C @ B @ Abs2 @ M14 ) @ Rep ) @ ( comp @ F7 @ A @ E @ ( comp @ B @ A @ F7 @ Abs @ M25 ) @ Rep3 ) ) )
=> ( ( comp @ B @ D @ F7 @ M14 @ M25 )
= M3 ) ) ) ) ) ).
% type_copy_map_comp0_undo
thf(fact_7148_rel__fun__eq__eq__onp,axiom,
! [B: $tType,A: $tType,P: B > $o] :
( ( bNF_rel_fun @ A @ A @ B @ B
@ ^ [Y3: A,Z: A] : Y3 = Z
@ ( bNF_eq_onp @ B @ P ) )
= ( bNF_eq_onp @ ( A > B )
@ ^ [F2: A > B] :
! [X: A] : ( P @ ( F2 @ X ) ) ) ) ).
% rel_fun_eq_eq_onp
thf(fact_7149_rel__fun__eq__onp__rel,axiom,
! [C: $tType,B: $tType,A: $tType,R3: A > $o,S2: B > C > $o] :
( ( bNF_rel_fun @ A @ A @ B @ C @ ( bNF_eq_onp @ A @ R3 ) @ S2 )
= ( ^ [F2: A > B,G2: A > C] :
! [X: A] :
( ( R3 @ X )
=> ( S2 @ ( F2 @ X ) @ ( G2 @ X ) ) ) ) ) ).
% rel_fun_eq_onp_rel
thf(fact_7150_zero__multiset_Orsp,axiom,
! [A: $tType] :
( bNF_eq_onp @ ( A > nat )
@ ^ [F2: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F2 @ X ) ) ) )
@ ^ [A3: A] : ( zero_zero @ nat )
@ ^ [A3: A] : ( zero_zero @ nat ) ) ).
% zero_multiset.rsp
thf(fact_7151_filter__mset_Orsp,axiom,
! [A: $tType] :
( bNF_rel_fun @ ( A > $o ) @ ( A > $o ) @ ( ( A > nat ) > A > nat ) @ ( ( A > nat ) > A > nat )
@ ^ [Y3: A > $o,Z: A > $o] : Y3 = Z
@ ( bNF_rel_fun @ ( A > nat ) @ ( A > nat ) @ ( A > nat ) @ ( A > nat )
@ ( bNF_eq_onp @ ( A > nat )
@ ^ [F2: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F2 @ X ) ) ) ) )
@ ( bNF_eq_onp @ ( A > nat )
@ ^ [F2: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F2 @ X ) ) ) ) ) )
@ ^ [P4: A > $o,M6: A > nat,X: A] : ( if @ nat @ ( P4 @ X ) @ ( M6 @ X ) @ ( zero_zero @ nat ) )
@ ^ [P4: A > $o,M6: A > nat,X: A] : ( if @ nat @ ( P4 @ X ) @ ( M6 @ X ) @ ( zero_zero @ nat ) ) ) ).
% filter_mset.rsp
thf(fact_7152_type__copy__Rep__o__Abs,axiom,
! [A: $tType,B: $tType,Rep: A > B,Abs: B > A] :
( ( type_definition @ A @ B @ Rep @ Abs @ ( top_top @ ( set @ B ) ) )
=> ( ( comp @ A @ B @ B @ Rep @ Abs )
= ( id @ B ) ) ) ).
% type_copy_Rep_o_Abs
thf(fact_7153_type__copy__Abs__o__Rep,axiom,
! [B: $tType,A: $tType,Rep: A > B,Abs: B > A] :
( ( type_definition @ A @ B @ Rep @ Abs @ ( top_top @ ( set @ B ) ) )
=> ( ( comp @ B @ A @ A @ Abs @ Rep )
= ( id @ A ) ) ) ).
% type_copy_Abs_o_Rep
thf(fact_7154_add__mset_Orsp,axiom,
! [A: $tType] :
( bNF_rel_fun @ A @ A @ ( ( A > nat ) > A > nat ) @ ( ( A > nat ) > A > nat )
@ ^ [Y3: A,Z: A] : Y3 = Z
@ ( bNF_rel_fun @ ( A > nat ) @ ( A > nat ) @ ( A > nat ) @ ( A > nat )
@ ( bNF_eq_onp @ ( A > nat )
@ ^ [F2: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F2 @ X ) ) ) ) )
@ ( bNF_eq_onp @ ( A > nat )
@ ^ [F2: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F2 @ X ) ) ) ) ) )
@ ^ [A3: A,M6: A > nat,B3: A] : ( if @ nat @ ( B3 = A3 ) @ ( suc @ ( M6 @ B3 ) ) @ ( M6 @ B3 ) )
@ ^ [A3: A,M6: A > nat,B3: A] : ( if @ nat @ ( B3 = A3 ) @ ( suc @ ( M6 @ B3 ) ) @ ( M6 @ B3 ) ) ) ).
% add_mset.rsp
thf(fact_7155_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 )
@ ^ [F2: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F2 @ X ) ) ) ) )
@ ( bNF_rel_fun @ ( A > nat ) @ ( A > nat ) @ ( A > nat ) @ ( A > nat )
@ ( bNF_eq_onp @ ( A > nat )
@ ^ [F2: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F2 @ X ) ) ) ) )
@ ( bNF_eq_onp @ ( A > nat )
@ ^ [F2: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F2 @ X ) ) ) ) ) )
@ ^ [M6: A > nat,N6: A > nat,A3: A] : ( plus_plus @ nat @ ( M6 @ A3 ) @ ( N6 @ A3 ) )
@ ^ [M6: A > nat,N6: A > nat,A3: A] : ( plus_plus @ nat @ ( M6 @ A3 ) @ ( N6 @ A3 ) ) ) ).
% plus_multiset.rsp
thf(fact_7156_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 )
@ ^ [F2: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F2 @ X ) ) ) ) )
@ ( bNF_rel_fun @ ( A > nat ) @ ( A > nat ) @ ( A > nat ) @ ( A > nat )
@ ( bNF_eq_onp @ ( A > nat )
@ ^ [F2: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F2 @ X ) ) ) ) )
@ ( bNF_eq_onp @ ( A > nat )
@ ^ [F2: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F2 @ X ) ) ) ) ) )
@ ^ [M6: A > nat,N6: A > nat,A3: A] : ( minus_minus @ nat @ ( M6 @ A3 ) @ ( N6 @ A3 ) )
@ ^ [M6: A > nat,N6: A > nat,A3: A] : ( minus_minus @ nat @ ( M6 @ A3 ) @ ( N6 @ A3 ) ) ) ).
% minus_multiset.rsp
thf(fact_7157_repeat__mset_Orsp,axiom,
! [A: $tType] :
( bNF_rel_fun @ nat @ nat @ ( ( A > nat ) > A > nat ) @ ( ( A > nat ) > A > nat )
@ ^ [Y3: nat,Z: nat] : Y3 = Z
@ ( bNF_rel_fun @ ( A > nat ) @ ( A > nat ) @ ( A > nat ) @ ( A > nat )
@ ( bNF_eq_onp @ ( A > nat )
@ ^ [F2: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F2 @ X ) ) ) ) )
@ ( bNF_eq_onp @ ( A > nat )
@ ^ [F2: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F2 @ X ) ) ) ) ) )
@ ^ [N4: nat,M6: A > nat,A3: A] : ( times_times @ nat @ N4 @ ( M6 @ A3 ) )
@ ^ [N4: nat,M6: A > nat,A3: A] : ( times_times @ nat @ N4 @ ( M6 @ A3 ) ) ) ).
% repeat_mset.rsp
thf(fact_7158_filter__mset_Oabs__eq,axiom,
! [A: $tType,X2: A > nat,Xa: A > $o] :
( ( bNF_eq_onp @ ( A > nat )
@ ^ [F2: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F2 @ X ) ) ) )
@ X2
@ X2 )
=> ( ( filter_mset @ A @ Xa @ ( abs_multiset @ A @ X2 ) )
= ( abs_multiset @ A
@ ^ [X: A] : ( if @ nat @ ( Xa @ X ) @ ( X2 @ X ) @ ( zero_zero @ nat ) ) ) ) ) ).
% filter_mset.abs_eq
thf(fact_7159_add__mset_Oabs__eq,axiom,
! [A: $tType,X2: A > nat,Xa: A] :
( ( bNF_eq_onp @ ( A > nat )
@ ^ [F2: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F2 @ X ) ) ) )
@ X2
@ X2 )
=> ( ( add_mset @ A @ Xa @ ( abs_multiset @ A @ X2 ) )
= ( abs_multiset @ A
@ ^ [B3: A] : ( if @ nat @ ( B3 = Xa ) @ ( suc @ ( X2 @ B3 ) ) @ ( X2 @ B3 ) ) ) ) ) ).
% add_mset.abs_eq
thf(fact_7160_plus__multiset_Oabs__eq,axiom,
! [A: $tType,Xa: A > nat,X2: A > nat] :
( ( bNF_eq_onp @ ( A > nat )
@ ^ [F2: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F2 @ X ) ) ) )
@ Xa
@ Xa )
=> ( ( bNF_eq_onp @ ( A > nat )
@ ^ [F2: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F2 @ X ) ) ) )
@ X2
@ X2 )
=> ( ( plus_plus @ ( multiset @ A ) @ ( abs_multiset @ A @ Xa ) @ ( abs_multiset @ A @ X2 ) )
= ( abs_multiset @ A
@ ^ [A3: A] : ( plus_plus @ nat @ ( Xa @ A3 ) @ ( X2 @ A3 ) ) ) ) ) ) ).
% plus_multiset.abs_eq
thf(fact_7161_minus__multiset_Oabs__eq,axiom,
! [A: $tType,Xa: A > nat,X2: A > nat] :
( ( bNF_eq_onp @ ( A > nat )
@ ^ [F2: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F2 @ X ) ) ) )
@ Xa
@ Xa )
=> ( ( bNF_eq_onp @ ( A > nat )
@ ^ [F2: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F2 @ X ) ) ) )
@ X2
@ X2 )
=> ( ( minus_minus @ ( multiset @ A ) @ ( abs_multiset @ A @ Xa ) @ ( abs_multiset @ A @ X2 ) )
= ( abs_multiset @ A
@ ^ [A3: A] : ( minus_minus @ nat @ ( Xa @ A3 ) @ ( X2 @ A3 ) ) ) ) ) ) ).
% minus_multiset.abs_eq
thf(fact_7162_subset__mset_OGreatest__def,axiom,
! [A: $tType,P: ( multiset @ A ) > $o] :
( ( greatest @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ P )
= ( the @ ( multiset @ A )
@ ^ [X: multiset @ A] :
( ( P @ X )
& ! [Y4: multiset @ A] :
( ( P @ Y4 )
=> ( subseteq_mset @ A @ Y4 @ X ) ) ) ) ) ).
% subset_mset.Greatest_def
thf(fact_7163_Inf__multiset_Oabs__eq,axiom,
! [A: $tType,X2: set @ ( A > nat )] :
( ( bNF_rel_set @ ( A > nat ) @ ( A > nat )
@ ( bNF_eq_onp @ ( A > nat )
@ ^ [F2: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F2 @ X ) ) ) ) )
@ X2
@ X2 )
=> ( ( complete_Inf_Inf @ ( multiset @ A ) @ ( image2 @ ( A > nat ) @ ( multiset @ A ) @ ( abs_multiset @ A ) @ X2 ) )
= ( abs_multiset @ A
@ ^ [I2: A] :
( if @ nat
@ ( X2
= ( bot_bot @ ( set @ ( A > nat ) ) ) )
@ ( zero_zero @ nat )
@ ( complete_Inf_Inf @ nat
@ ( image2 @ ( A > nat ) @ nat
@ ^ [F2: A > nat] : ( F2 @ I2 )
@ X2 ) ) ) ) ) ) ).
% Inf_multiset.abs_eq
thf(fact_7164_order_OGreatest_Ocong,axiom,
! [A: $tType] :
( ( greatest @ A )
= ( greatest @ A ) ) ).
% order.Greatest.cong
thf(fact_7165_fun_Oset__transfer,axiom,
! [A: $tType,B: $tType,D: $tType,R3: A > B > $o] :
( bNF_rel_fun @ ( D > A ) @ ( D > B ) @ ( set @ A ) @ ( set @ B )
@ ( bNF_rel_fun @ D @ D @ A @ B
@ ^ [Y3: D,Z: D] : Y3 = Z
@ R3 )
@ ( bNF_rel_set @ A @ B @ R3 )
@ ^ [F2: D > A] : ( image2 @ D @ A @ F2 @ ( top_top @ ( set @ D ) ) )
@ ^ [F2: D > B] : ( image2 @ D @ B @ F2 @ ( top_top @ ( set @ D ) ) ) ) ).
% fun.set_transfer
thf(fact_7166_type__definition__integer,axiom,
type_definition @ code_integer @ int @ code_int_of_integer @ code_integer_of_int @ ( top_top @ ( set @ int ) ) ).
% type_definition_integer
thf(fact_7167_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 )
@ ^ [F2: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F2 @ X ) ) ) ) ) )
@ ( bNF_eq_onp @ ( A > nat )
@ ^ [F2: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F2 @ X ) ) ) ) )
@ ^ [A8: set @ ( A > nat ),I2: A] :
( if @ nat
@ ( A8
= ( bot_bot @ ( set @ ( A > nat ) ) ) )
@ ( zero_zero @ nat )
@ ( complete_Inf_Inf @ nat
@ ( image2 @ ( A > nat ) @ nat
@ ^ [F2: A > nat] : ( F2 @ I2 )
@ A8 ) ) )
@ ^ [A8: set @ ( A > nat ),I2: A] :
( if @ nat
@ ( A8
= ( bot_bot @ ( set @ ( A > nat ) ) ) )
@ ( zero_zero @ nat )
@ ( complete_Inf_Inf @ nat
@ ( image2 @ ( A > nat ) @ nat
@ ^ [F2: A > nat] : ( F2 @ I2 )
@ A8 ) ) ) ) ).
% Inf_multiset.rsp
thf(fact_7168_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
@ ^ [Y3: C,Z: C] : Y3 = Z )
@ ^ [Y3: C,Z: C] : Y3 = Z )
@ ^ [A8: set @ A,F2: A > C] : ( complete_Inf_Inf @ C @ ( image2 @ A @ C @ F2 @ A8 ) )
@ ^ [A8: set @ B,F2: B > C] : ( complete_Inf_Inf @ C @ ( image2 @ B @ C @ F2 @ A8 ) ) ) ) ).
% INF_parametric
thf(fact_7169_SUP__parametric,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( complete_Sup @ C )
=> ! [R3: A > B > $o] :
( bNF_rel_fun @ ( set @ A ) @ ( set @ B ) @ ( ( A > C ) > C ) @ ( ( B > C ) > C ) @ ( bNF_rel_set @ A @ B @ R3 )
@ ( bNF_rel_fun @ ( A > C ) @ ( B > C ) @ C @ C
@ ( bNF_rel_fun @ A @ B @ C @ C @ R3
@ ^ [Y3: C,Z: C] : Y3 = Z )
@ ^ [Y3: C,Z: C] : Y3 = Z )
@ ^ [A8: set @ A,F2: A > C] : ( complete_Sup_Sup @ C @ ( image2 @ A @ C @ F2 @ A8 ) )
@ ^ [A8: set @ B,F2: B > C] : ( complete_Sup_Sup @ C @ ( image2 @ B @ C @ F2 @ A8 ) ) ) ) ).
% SUP_parametric
thf(fact_7170_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_7171_UNION__transfer,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,A5: A > B > $o,B5: 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 @ B5 ) ) @ ( bNF_rel_set @ C @ D @ B5 ) )
@ ^ [A8: set @ A,F2: A > ( set @ C )] : ( complete_Sup_Sup @ ( set @ C ) @ ( image2 @ A @ ( set @ C ) @ F2 @ A8 ) )
@ ^ [A8: set @ B,F2: B > ( set @ D )] : ( complete_Sup_Sup @ ( set @ D ) @ ( image2 @ B @ ( set @ D ) @ F2 @ A8 ) ) ) ).
% UNION_transfer
thf(fact_7172_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
@ ^ [Y3: A,Z: A] : Y3 = Z ) )
@ ( pcr_multiset @ A @ A
@ ^ [Y3: A,Z: A] : Y3 = Z )
@ ^ [A8: set @ ( A > nat ),I2: A] :
( if @ nat
@ ( A8
= ( bot_bot @ ( set @ ( A > nat ) ) ) )
@ ( zero_zero @ nat )
@ ( complete_Inf_Inf @ nat
@ ( image2 @ ( A > nat ) @ nat
@ ^ [F2: A > nat] : ( F2 @ I2 )
@ A8 ) ) )
@ ( complete_Inf_Inf @ ( multiset @ A ) ) ) ).
% Inf_multiset.transfer
thf(fact_7173_typedef__rep__transfer,axiom,
! [A: $tType,B: $tType,Rep: B > A,Abs: A > B,A5: set @ A,T6: A > B > $o] :
( ( type_definition @ B @ A @ Rep @ Abs @ A5 )
=> ( ( T6
= ( ^ [X: A,Y4: B] :
( X
= ( Rep @ Y4 ) ) ) )
=> ( bNF_rel_fun @ A @ B @ A @ A @ T6
@ ^ [Y3: A,Z: A] : Y3 = Z
@ ^ [X: A] : X
@ Rep ) ) ) ).
% typedef_rep_transfer
thf(fact_7174_zero__multiset_Otransfer,axiom,
! [A: $tType] :
( pcr_multiset @ A @ A
@ ^ [Y3: A,Z: A] : Y3 = Z
@ ^ [A3: A] : ( zero_zero @ nat )
@ ( zero_zero @ ( multiset @ A ) ) ) ).
% zero_multiset.transfer
thf(fact_7175_multiset_Orep__transfer,axiom,
! [D: $tType,E: $tType,T6: D > E > $o] :
( bNF_rel_fun @ ( D > nat ) @ ( multiset @ E ) @ ( D > nat ) @ ( E > nat ) @ ( pcr_multiset @ D @ E @ T6 )
@ ( bNF_rel_fun @ D @ E @ nat @ nat @ T6
@ ^ [Y3: nat,Z: nat] : Y3 = Z )
@ ^ [X: D > nat] : X
@ ( count @ E ) ) ).
% multiset.rep_transfer
thf(fact_7176_add__mset_Otransfer,axiom,
! [A: $tType] :
( bNF_rel_fun @ A @ A @ ( ( A > nat ) > A > nat ) @ ( ( multiset @ A ) > ( multiset @ A ) )
@ ^ [Y3: A,Z: A] : Y3 = Z
@ ( bNF_rel_fun @ ( A > nat ) @ ( multiset @ A ) @ ( A > nat ) @ ( multiset @ A )
@ ( pcr_multiset @ A @ A
@ ^ [Y3: A,Z: A] : Y3 = Z )
@ ( pcr_multiset @ A @ A
@ ^ [Y3: A,Z: A] : Y3 = Z ) )
@ ^ [A3: A,M6: A > nat,B3: A] : ( if @ nat @ ( B3 = A3 ) @ ( suc @ ( M6 @ B3 ) ) @ ( M6 @ B3 ) )
@ ( add_mset @ A ) ) ).
% add_mset.transfer
thf(fact_7177_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
@ ^ [Y3: A,Z: A] : Y3 = Z )
@ ( bNF_rel_fun @ ( A > nat ) @ ( multiset @ A ) @ ( A > nat ) @ ( multiset @ A )
@ ( pcr_multiset @ A @ A
@ ^ [Y3: A,Z: A] : Y3 = Z )
@ ( pcr_multiset @ A @ A
@ ^ [Y3: A,Z: A] : Y3 = Z ) )
@ ^ [M6: A > nat,N6: A > nat,A3: A] : ( plus_plus @ nat @ ( M6 @ A3 ) @ ( N6 @ A3 ) )
@ ( plus_plus @ ( multiset @ A ) ) ) ).
% plus_multiset.transfer
thf(fact_7178_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
@ ^ [Y3: A,Z: A] : Y3 = Z )
@ ( bNF_rel_fun @ ( A > nat ) @ ( multiset @ A ) @ ( A > nat ) @ ( multiset @ A )
@ ( pcr_multiset @ A @ A
@ ^ [Y3: A,Z: A] : Y3 = Z )
@ ( pcr_multiset @ A @ A
@ ^ [Y3: A,Z: A] : Y3 = Z ) )
@ ^ [M6: A > nat,N6: A > nat,A3: A] : ( minus_minus @ nat @ ( M6 @ A3 ) @ ( N6 @ A3 ) )
@ ( minus_minus @ ( multiset @ A ) ) ) ).
% minus_multiset.transfer
thf(fact_7179_filter__mset_Otransfer,axiom,
! [A: $tType] :
( bNF_rel_fun @ ( A > $o ) @ ( A > $o ) @ ( ( A > nat ) > A > nat ) @ ( ( multiset @ A ) > ( multiset @ A ) )
@ ^ [Y3: A > $o,Z: A > $o] : Y3 = Z
@ ( bNF_rel_fun @ ( A > nat ) @ ( multiset @ A ) @ ( A > nat ) @ ( multiset @ A )
@ ( pcr_multiset @ A @ A
@ ^ [Y3: A,Z: A] : Y3 = Z )
@ ( pcr_multiset @ A @ A
@ ^ [Y3: A,Z: A] : Y3 = Z ) )
@ ^ [P4: A > $o,M6: A > nat,X: A] : ( if @ nat @ ( P4 @ X ) @ ( M6 @ X ) @ ( zero_zero @ nat ) )
@ ( filter_mset @ A ) ) ).
% filter_mset.transfer
thf(fact_7180_repeat__mset_Otransfer,axiom,
! [A: $tType] :
( bNF_rel_fun @ nat @ nat @ ( ( A > nat ) > A > nat ) @ ( ( multiset @ A ) > ( multiset @ A ) )
@ ^ [Y3: nat,Z: nat] : Y3 = Z
@ ( bNF_rel_fun @ ( A > nat ) @ ( multiset @ A ) @ ( A > nat ) @ ( multiset @ A )
@ ( pcr_multiset @ A @ A
@ ^ [Y3: A,Z: A] : Y3 = Z )
@ ( pcr_multiset @ A @ A
@ ^ [Y3: A,Z: A] : Y3 = Z ) )
@ ^ [N4: nat,M6: A > nat,A3: A] : ( times_times @ nat @ N4 @ ( M6 @ A3 ) )
@ ( repeat_mset @ A ) ) ).
% repeat_mset.transfer
thf(fact_7181_rel__pred__comp__def,axiom,
! [B: $tType,A: $tType] :
( ( rel_pred_comp @ A @ B )
= ( ^ [R7: A > B > $o,P4: B > $o,X: A] :
? [Y4: B] :
( ( R7 @ X @ Y4 )
& ( P4 @ Y4 ) ) ) ) ).
% rel_pred_comp_def
thf(fact_7182_sum__multiset__singleton,axiom,
! [A: $tType,A5: set @ A] :
( ( groups7311177749621191930dd_sum @ A @ ( multiset @ A )
@ ^ [N4: A] : ( add_mset @ A @ N4 @ ( zero_zero @ ( multiset @ A ) ) )
@ A5 )
= ( mset_set @ A @ A5 ) ) ).
% sum_multiset_singleton
thf(fact_7183_count__mset__set_I3_J,axiom,
! [A: $tType,X2: A,A5: set @ A] :
( ~ ( member @ A @ X2 @ A5 )
=> ( ( count @ A @ ( mset_set @ A @ A5 ) @ X2 )
= ( zero_zero @ nat ) ) ) ).
% count_mset_set(3)
thf(fact_7184_size__mset__set,axiom,
! [A: $tType,A5: set @ A] :
( ( size_size @ ( multiset @ A ) @ ( mset_set @ A @ A5 ) )
= ( finite_card @ A @ A5 ) ) ).
% size_mset_set
thf(fact_7185_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
@ ^ [X: A] :
( ( member @ A @ X @ A5 )
& ( P @ X ) ) ) ) ) ) ).
% filter_mset_mset_set
thf(fact_7186_count__mset__set_I2_J,axiom,
! [A: $tType,A5: set @ A,X2: A] :
( ~ ( finite_finite2 @ A @ A5 )
=> ( ( count @ A @ ( mset_set @ A @ A5 ) @ X2 )
= ( zero_zero @ nat ) ) ) ).
% count_mset_set(2)
thf(fact_7187_count__mset__set_I1_J,axiom,
! [A: $tType,A5: set @ A,X2: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( member @ A @ X2 @ A5 )
=> ( ( count @ A @ ( mset_set @ A @ A5 ) @ X2 )
= ( one_one @ nat ) ) ) ) ).
% count_mset_set(1)
thf(fact_7188_mset__set__upto__eq__mset__upto,axiom,
! [N: nat] :
( ( mset_set @ nat @ ( set_ord_lessThan @ nat @ N ) )
= ( mset @ nat @ ( upt @ ( zero_zero @ nat ) @ N ) ) ) ).
% mset_set_upto_eq_mset_upto
thf(fact_7189_count__mset__set_H,axiom,
! [A: $tType,A5: set @ A,X2: A] :
( ( ( ( finite_finite2 @ A @ A5 )
& ( member @ A @ X2 @ A5 ) )
=> ( ( count @ A @ ( mset_set @ A @ A5 ) @ X2 )
= ( one_one @ nat ) ) )
& ( ~ ( ( finite_finite2 @ A @ A5 )
& ( member @ A @ X2 @ A5 ) )
=> ( ( count @ A @ ( mset_set @ A @ A5 ) @ X2 )
= ( zero_zero @ nat ) ) ) ) ).
% count_mset_set'
thf(fact_7190_count__mset__set__finite__iff,axiom,
! [A: $tType,S2: set @ A,A2: A] :
( ( finite_finite2 @ A @ S2 )
=> ( ( ( member @ A @ A2 @ S2 )
=> ( ( count @ A @ ( mset_set @ A @ S2 ) @ A2 )
= ( one_one @ nat ) ) )
& ( ~ ( member @ A @ A2 @ S2 )
=> ( ( count @ A @ ( mset_set @ A @ S2 ) @ A2 )
= ( zero_zero @ nat ) ) ) ) ) ).
% count_mset_set_finite_iff
thf(fact_7191_mset__set__Union,axiom,
! [A: $tType,A5: set @ A,B5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( finite_finite2 @ A @ B5 )
=> ( ( ( inf_inf @ ( set @ A ) @ A5 @ B5 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( mset_set @ A @ ( sup_sup @ ( set @ A ) @ A5 @ B5 ) )
= ( plus_plus @ ( multiset @ A ) @ ( mset_set @ A @ A5 ) @ ( mset_set @ A @ B5 ) ) ) ) ) ) ).
% mset_set_Union
thf(fact_7192_relImage__Gr,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),A5: set @ A,F: A > B] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ A5
@ ^ [Uu: A] : A5 ) )
=> ( ( bNF_Gr4221423524335903396lImage @ A @ B @ R3 @ F )
= ( relcomp @ B @ A @ B @ ( converse @ A @ B @ ( bNF_Gr @ A @ B @ A5 @ F ) ) @ ( relcomp @ A @ A @ B @ R3 @ ( bNF_Gr @ A @ B @ A5 @ F ) ) ) ) ) ).
% relImage_Gr
thf(fact_7193_relInvImage__Gr,axiom,
! [A: $tType,B: $tType,R3: set @ ( product_prod @ A @ A ),B5: set @ A,A5: set @ B,F: B > A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( product_Sigma @ A @ A @ B5
@ ^ [Uu: A] : B5 ) )
=> ( ( bNF_Gr7122648621184425601vImage @ B @ A @ A5 @ R3 @ F )
= ( relcomp @ B @ A @ B @ ( bNF_Gr @ B @ A @ A5 @ F ) @ ( relcomp @ A @ A @ B @ R3 @ ( converse @ B @ A @ ( bNF_Gr @ B @ A @ A5 @ F ) ) ) ) ) ) ).
% relInvImage_Gr
thf(fact_7194_pair__set__inverse,axiom,
! [B: $tType,A: $tType,P: B > A > $o] :
( ( converse @ B @ A @ ( collect @ ( product_prod @ B @ A ) @ ( product_case_prod @ B @ A @ $o @ P ) ) )
= ( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [B3: A,A3: B] : ( P @ A3 @ B3 ) ) ) ) ).
% pair_set_inverse
thf(fact_7195_converse__unfold,axiom,
! [A: $tType,B: $tType] :
( ( converse @ B @ A )
= ( ^ [R2: set @ ( product_prod @ B @ A )] :
( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [Y4: A,X: B] : ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X @ Y4 ) @ R2 ) ) ) ) ) ).
% converse_unfold
thf(fact_7196_converse__Times,axiom,
! [B: $tType,A: $tType,A5: set @ B,B5: set @ A] :
( ( converse @ B @ A
@ ( product_Sigma @ B @ A @ A5
@ ^ [Uu: B] : B5 ) )
= ( product_Sigma @ A @ B @ B5
@ ^ [Uu: A] : A5 ) ) ).
% converse_Times
thf(fact_7197_converse__INTER,axiom,
! [B: $tType,A: $tType,C: $tType,R: C > ( set @ ( product_prod @ B @ A ) ),S2: set @ C] :
( ( converse @ B @ A @ ( complete_Inf_Inf @ ( set @ ( product_prod @ B @ A ) ) @ ( image2 @ C @ ( set @ ( product_prod @ B @ A ) ) @ R @ S2 ) ) )
= ( complete_Inf_Inf @ ( set @ ( product_prod @ A @ B ) )
@ ( image2 @ C @ ( set @ ( product_prod @ A @ B ) )
@ ^ [X: C] : ( converse @ B @ A @ ( R @ X ) )
@ S2 ) ) ) ).
% converse_INTER
thf(fact_7198_converse__UNION,axiom,
! [B: $tType,A: $tType,C: $tType,R: C > ( set @ ( product_prod @ B @ A ) ),S2: set @ C] :
( ( converse @ B @ A @ ( complete_Sup_Sup @ ( set @ ( product_prod @ B @ A ) ) @ ( image2 @ C @ ( set @ ( product_prod @ B @ A ) ) @ R @ S2 ) ) )
= ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) )
@ ( image2 @ C @ ( set @ ( product_prod @ A @ B ) )
@ ^ [X: C] : ( converse @ B @ A @ ( R @ X ) )
@ S2 ) ) ) ).
% converse_UNION
thf(fact_7199_image2__Gr,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( bNF_Greatest_image2 @ C @ A @ B )
= ( ^ [A8: set @ C,F2: C > A,G2: C > B] : ( relcomp @ A @ C @ B @ ( converse @ C @ A @ ( bNF_Gr @ C @ A @ A8 @ F2 ) ) @ ( bNF_Gr @ C @ B @ A8 @ G2 ) ) ) ) ).
% image2_Gr
thf(fact_7200_Image__INT__eq,axiom,
! [A: $tType,B: $tType,C: $tType,R: set @ ( product_prod @ B @ A ),A5: set @ C,B5: C > ( set @ B )] :
( ( single_valued @ A @ B @ ( converse @ B @ A @ R ) )
=> ( ( A5
!= ( bot_bot @ ( set @ C ) ) )
=> ( ( image @ B @ A @ R @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ C @ ( set @ B ) @ B5 @ A5 ) ) )
= ( complete_Inf_Inf @ ( set @ A )
@ ( image2 @ C @ ( set @ A )
@ ^ [X: C] : ( image @ B @ A @ R @ ( B5 @ X ) )
@ A5 ) ) ) ) ) ).
% Image_INT_eq
thf(fact_7201_stable__sort__key__def,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ( ( linord3483353639454293061rt_key @ B @ A )
= ( ^ [Sk: ( B > A ) > ( list @ B ) > ( list @ B )] :
! [F2: B > A,Xs3: list @ B,K2: A] :
( ( filter2 @ B
@ ^ [Y4: B] :
( ( F2 @ Y4 )
= K2 )
@ ( Sk @ F2 @ Xs3 ) )
= ( filter2 @ B
@ ^ [Y4: B] :
( ( F2 @ Y4 )
= K2 )
@ Xs3 ) ) ) ) ) ).
% stable_sort_key_def
thf(fact_7202_single__valued__inter1,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ B ),S2: set @ ( product_prod @ A @ B )] :
( ( single_valued @ A @ B @ R3 )
=> ( single_valued @ A @ B @ ( inf_inf @ ( set @ ( product_prod @ A @ B ) ) @ R3 @ S2 ) ) ) ).
% single_valued_inter1
thf(fact_7203_single__valued__inter2,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ B ),S2: set @ ( product_prod @ A @ B )] :
( ( single_valued @ A @ B @ R3 )
=> ( single_valued @ A @ B @ ( inf_inf @ ( set @ ( product_prod @ A @ B ) ) @ S2 @ R3 ) ) ) ).
% single_valued_inter2
thf(fact_7204_bijective__alt,axiom,
! [B: $tType,A: $tType] :
( ( bijective @ A @ B )
= ( ^ [R7: set @ ( product_prod @ A @ B )] :
( ( single_valued @ A @ B @ R7 )
& ( single_valued @ B @ A @ ( converse @ A @ B @ R7 ) ) ) ) ) ).
% bijective_alt
thf(fact_7205_numeral__xor__num,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [M: num,N: num] :
( ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N ) )
= ( case_option @ A @ num @ ( zero_zero @ A ) @ ( numeral_numeral @ A ) @ ( bit_un2480387367778600638or_num @ M @ N ) ) ) ) ).
% numeral_xor_num
thf(fact_7206_filterlim__base__iff,axiom,
! [A: $tType,C: $tType,B: $tType,D: $tType,I5: set @ A,F4: A > ( set @ B ),F: B > C,G5: D > ( set @ C ),J4: set @ D] :
( ( I5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [I3: A] :
( ( member @ A @ I3 @ I5 )
=> ! [J2: A] :
( ( member @ A @ J2 @ I5 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( F4 @ I3 ) @ ( F4 @ J2 ) )
| ( ord_less_eq @ ( set @ B ) @ ( F4 @ J2 ) @ ( F4 @ I3 ) ) ) ) )
=> ( ( filterlim @ B @ C @ F
@ ( complete_Inf_Inf @ ( filter @ C )
@ ( image2 @ D @ ( filter @ C )
@ ^ [J3: D] : ( principal @ C @ ( G5 @ J3 ) )
@ J4 ) )
@ ( complete_Inf_Inf @ ( filter @ B )
@ ( image2 @ A @ ( filter @ B )
@ ^ [I2: A] : ( principal @ B @ ( F4 @ I2 ) )
@ I5 ) ) )
= ( ! [X: D] :
( ( member @ D @ X @ J4 )
=> ? [Y4: A] :
( ( member @ A @ Y4 @ I5 )
& ! [Z3: B] :
( ( member @ B @ Z3 @ ( F4 @ Y4 ) )
=> ( member @ C @ ( F @ Z3 ) @ ( G5 @ X ) ) ) ) ) ) ) ) ) ).
% filterlim_base_iff
thf(fact_7207_filterlim__ident,axiom,
! [A: $tType,F4: filter @ A] :
( filterlim @ A @ A
@ ^ [X: A] : X
@ F4
@ F4 ) ).
% filterlim_ident
thf(fact_7208_filterlim__compose,axiom,
! [B: $tType,A: $tType,C: $tType,G: A > B,F33: filter @ B,F24: filter @ A,F: C > A,F13: filter @ C] :
( ( filterlim @ A @ B @ G @ F33 @ F24 )
=> ( ( filterlim @ C @ A @ F @ F24 @ F13 )
=> ( filterlim @ C @ B
@ ^ [X: C] : ( G @ ( F @ X ) )
@ F33
@ F13 ) ) ) ).
% filterlim_compose
thf(fact_7209_filterlim__inf,axiom,
! [B: $tType,A: $tType,F: A > B,F24: filter @ B,F33: filter @ B,F13: filter @ A] :
( ( filterlim @ A @ B @ F @ ( inf_inf @ ( filter @ B ) @ F24 @ F33 ) @ F13 )
= ( ( filterlim @ A @ B @ F @ F24 @ F13 )
& ( filterlim @ A @ B @ F @ F33 @ F13 ) ) ) ).
% filterlim_inf
thf(fact_7210_filterlim__sequentially__Suc,axiom,
! [A: $tType,F: nat > A,F4: filter @ A] :
( ( filterlim @ nat @ A
@ ^ [X: nat] : ( F @ ( suc @ X ) )
@ F4
@ ( at_top @ nat ) )
= ( filterlim @ nat @ A @ F @ F4 @ ( at_top @ nat ) ) ) ).
% filterlim_sequentially_Suc
thf(fact_7211_filterlim__INF,axiom,
! [A: $tType,B: $tType,C: $tType,F: A > B,G5: C > ( filter @ B ),B5: set @ C,F4: filter @ A] :
( ( filterlim @ A @ B @ F @ ( complete_Inf_Inf @ ( filter @ B ) @ ( image2 @ C @ ( filter @ B ) @ G5 @ B5 ) ) @ F4 )
= ( ! [X: C] :
( ( member @ C @ X @ B5 )
=> ( filterlim @ A @ B @ F @ ( G5 @ X ) @ F4 ) ) ) ) ).
% filterlim_INF
thf(fact_7212_filterlim__INF_H,axiom,
! [C: $tType,B: $tType,A: $tType,X2: A,A5: set @ A,F: B > C,F4: filter @ C,G5: A > ( filter @ B )] :
( ( member @ A @ X2 @ A5 )
=> ( ( filterlim @ B @ C @ F @ F4 @ ( G5 @ X2 ) )
=> ( filterlim @ B @ C @ F @ F4 @ ( complete_Inf_Inf @ ( filter @ B ) @ ( image2 @ A @ ( filter @ B ) @ G5 @ A5 ) ) ) ) ) ).
% filterlim_INF'
thf(fact_7213_filterlim__If,axiom,
! [B: $tType,A: $tType,F: A > B,G5: filter @ B,F4: filter @ A,P: A > $o,G: A > B] :
( ( filterlim @ A @ B @ F @ G5 @ ( inf_inf @ ( filter @ A ) @ F4 @ ( principal @ A @ ( collect @ A @ P ) ) ) )
=> ( ( filterlim @ A @ B @ G @ G5
@ ( inf_inf @ ( filter @ A ) @ F4
@ ( principal @ A
@ ( collect @ A
@ ^ [X: A] :
~ ( P @ X ) ) ) ) )
=> ( filterlim @ A @ B
@ ^ [X: A] : ( if @ B @ ( P @ X ) @ ( F @ X ) @ ( G @ X ) )
@ G5
@ F4 ) ) ) ).
% filterlim_If
thf(fact_7214_filterlim__base,axiom,
! [B: $tType,A: $tType,E: $tType,D: $tType,C: $tType,J4: set @ A,I: A > C,I5: set @ C,F4: C > ( set @ D ),F: D > E,G5: A > ( set @ E )] :
( ! [M4: A,X3: B] :
( ( member @ A @ M4 @ J4 )
=> ( member @ C @ ( I @ M4 ) @ I5 ) )
=> ( ! [M4: A,X3: D] :
( ( member @ A @ M4 @ J4 )
=> ( ( member @ D @ X3 @ ( F4 @ ( I @ M4 ) ) )
=> ( member @ E @ ( F @ X3 ) @ ( G5 @ M4 ) ) ) )
=> ( filterlim @ D @ E @ F
@ ( complete_Inf_Inf @ ( filter @ E )
@ ( image2 @ A @ ( filter @ E )
@ ^ [J3: A] : ( principal @ E @ ( G5 @ J3 ) )
@ J4 ) )
@ ( complete_Inf_Inf @ ( filter @ D )
@ ( image2 @ C @ ( filter @ D )
@ ^ [I2: C] : ( principal @ D @ ( F4 @ I2 ) )
@ I5 ) ) ) ) ) ).
% filterlim_base
thf(fact_7215_xor__num__eq__None__iff,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [M: num,N: num] :
( ( ( bit_un2480387367778600638or_num @ M @ N )
= ( none @ num ) )
= ( ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N ) )
= ( zero_zero @ A ) ) ) ) ).
% xor_num_eq_None_iff
thf(fact_7216_map__rec,axiom,
! [A: $tType,B: $tType] :
( ( map @ B @ A )
= ( ^ [F2: B > A] :
( rec_list @ ( list @ A ) @ B @ ( nil @ A )
@ ^ [X: B,Uu: list @ B] : ( cons @ A @ ( F2 @ X ) ) ) ) ) ).
% map_rec
thf(fact_7217_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_7218_rec__list__Cons__imp,axiom,
! [B: $tType,A: $tType,F: ( list @ A ) > B,F1: B,F22: A > ( list @ A ) > B > B,X2: A,Xs: list @ A] :
( ( F
= ( rec_list @ B @ A @ F1 @ F22 ) )
=> ( ( F @ ( cons @ A @ X2 @ Xs ) )
= ( F22 @ X2 @ Xs @ ( F @ Xs ) ) ) ) ).
% rec_list_Cons_imp
thf(fact_7219_zipf_Osimps_I2_J,axiom,
! [A: $tType,C: $tType,B: $tType,F: A > B > C,A2: A,As: list @ A,B2: B,Bs: list @ B] :
( ( zipf @ A @ B @ C @ F @ ( cons @ A @ A2 @ As ) @ ( cons @ B @ B2 @ Bs ) )
= ( cons @ C @ ( F @ A2 @ B2 ) @ ( zipf @ A @ B @ C @ F @ As @ Bs ) ) ) ).
% zipf.simps(2)
thf(fact_7220_zipf_Osimps_I1_J,axiom,
! [B: $tType,A: $tType,C: $tType,F: A > B > C] :
( ( zipf @ A @ B @ C @ F @ ( nil @ A ) @ ( nil @ B ) )
= ( nil @ C ) ) ).
% zipf.simps(1)
thf(fact_7221_rec__list__Nil__imp,axiom,
! [A: $tType,B: $tType,F: ( list @ A ) > B,F1: B,F22: A > ( list @ A ) > B > B] :
( ( F
= ( rec_list @ B @ A @ F1 @ F22 ) )
=> ( ( F @ ( nil @ A ) )
= F1 ) ) ).
% rec_list_Nil_imp
thf(fact_7222_zipf_Oelims,axiom,
! [B: $tType,A: $tType,C: $tType,X2: A > B > C,Xa: list @ A,Xb: list @ B,Y: list @ C] :
( ( ( zipf @ A @ B @ C @ X2 @ Xa @ Xb )
= Y )
=> ( ( ( Xa
= ( nil @ A ) )
=> ( ( Xb
= ( nil @ B ) )
=> ( Y
!= ( nil @ C ) ) ) )
=> ( ! [A4: A,As2: list @ A] :
( ( Xa
= ( cons @ A @ A4 @ As2 ) )
=> ! [B4: B,Bs2: list @ B] :
( ( Xb
= ( cons @ B @ B4 @ Bs2 ) )
=> ( Y
!= ( cons @ C @ ( X2 @ A4 @ B4 ) @ ( zipf @ A @ B @ C @ X2 @ As2 @ Bs2 ) ) ) ) )
=> ( ( ? [V5: A,Va: list @ A] :
( Xa
= ( cons @ A @ V5 @ Va ) )
=> ( ( Xb
= ( nil @ B ) )
=> ( Y
!= ( undefined @ ( list @ C ) ) ) ) )
=> ~ ( ( Xa
= ( nil @ A ) )
=> ( ? [V5: B,Va: list @ B] :
( Xb
= ( cons @ B @ V5 @ Va ) )
=> ( Y
!= ( undefined @ ( list @ C ) ) ) ) ) ) ) ) ) ).
% zipf.elims
thf(fact_7223_list_Orec__o__map,axiom,
! [C: $tType,B: $tType,A: $tType,G: C,Ga: B > ( list @ B ) > C > C,F: A > B] :
( ( comp @ ( list @ B ) @ C @ ( list @ A ) @ ( rec_list @ C @ B @ G @ Ga ) @ ( map @ A @ B @ F ) )
= ( rec_list @ C @ A @ G
@ ^ [X: A,Xa5: list @ A] : ( Ga @ ( F @ X ) @ ( map @ A @ B @ F @ Xa5 ) ) ) ) ).
% list.rec_o_map
thf(fact_7224_zipf_Opelims,axiom,
! [C: $tType,A: $tType,B: $tType,X2: A > B > C,Xa: list @ A,Xb: list @ B,Y: list @ C] :
( ( ( zipf @ A @ B @ C @ X2 @ 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 ) ) @ X2 @ ( 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 ) ) @ X2 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ ( nil @ B ) ) ) ) ) ) )
=> ( ! [A4: A,As2: list @ A] :
( ( Xa
= ( cons @ A @ A4 @ As2 ) )
=> ! [B4: B,Bs2: list @ B] :
( ( Xb
= ( cons @ B @ B4 @ Bs2 ) )
=> ( ( Y
= ( cons @ C @ ( X2 @ A4 @ B4 ) @ ( zipf @ A @ B @ C @ X2 @ As2 @ Bs2 ) ) )
=> ~ ( accp @ ( product_prod @ ( A > B > C ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) @ ( zipf_rel @ A @ B @ C ) @ ( product_Pair @ ( A > B > C ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ X2 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ A4 @ As2 ) @ ( cons @ B @ B4 @ Bs2 ) ) ) ) ) ) )
=> ( ! [V5: A,Va: list @ A] :
( ( Xa
= ( cons @ A @ V5 @ 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 ) ) @ X2 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ V5 @ Va ) @ ( nil @ B ) ) ) ) ) ) )
=> ~ ( ( Xa
= ( nil @ A ) )
=> ! [V5: B,Va: list @ B] :
( ( Xb
= ( cons @ B @ V5 @ 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 ) ) @ X2 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ ( cons @ B @ V5 @ Va ) ) ) ) ) ) ) ) ) ) ) ) ).
% zipf.pelims
thf(fact_7225_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_7226_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 ) )
@ ^ [X5: set @ A] :
( principal @ ( set @ A )
@ ( collect @ ( set @ A )
@ ^ [Y6: set @ A] :
( ( finite_finite2 @ A @ Y6 )
& ( ord_less_eq @ ( set @ A ) @ X5 @ Y6 )
& ( ord_less_eq @ ( set @ A ) @ Y6 @ A8 ) ) ) )
@ ( collect @ ( set @ A )
@ ^ [X5: set @ A] :
( ( finite_finite2 @ A @ X5 )
& ( ord_less_eq @ ( set @ A ) @ X5 @ A8 ) ) ) ) ) ) ) ).
% finite_subsets_at_top_def
thf(fact_7227_set__rec,axiom,
! [A: $tType] :
( ( set2 @ A )
= ( rec_list @ ( set @ A ) @ A @ ( bot_bot @ ( set @ A ) )
@ ^ [X: A,Uu: list @ A] : ( insert @ A @ X ) ) ) ).
% set_rec
thf(fact_7228_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_7229_the__dflt__None__empty,axiom,
! [A: $tType] :
( ( dflt_None_set @ A @ ( bot_bot @ ( set @ A ) ) )
= ( none @ ( set @ A ) ) ) ).
% the_dflt_None_empty
thf(fact_7230_the__dflt__None__nonempty,axiom,
! [A: $tType,S2: set @ A] :
( ( S2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( dflt_None_set @ A @ S2 )
= ( some @ ( set @ A ) @ S2 ) ) ) ).
% the_dflt_None_nonempty
thf(fact_7231_dflt__None__set__def,axiom,
! [A: $tType] :
( ( dflt_None_set @ A )
= ( ^ [S: set @ A] :
( if @ ( option @ ( set @ A ) )
@ ( S
= ( bot_bot @ ( set @ A ) ) )
@ ( none @ ( set @ A ) )
@ ( some @ ( set @ A ) @ S ) ) ) ) ).
% dflt_None_set_def
thf(fact_7232_the__dflt__None__set,axiom,
! [A: $tType,X2: set @ A] :
( ( the_default @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ ( dflt_None_set @ A @ X2 ) )
= X2 ) ).
% the_dflt_None_set
thf(fact_7233_curr__surj,axiom,
! [C: $tType,B: $tType,A: $tType,G: A > B > C,A5: set @ A,B5: set @ B,C6: set @ C] :
( ( member @ ( A > B > C ) @ G @ ( bNF_Wellorder_Func @ A @ ( B > C ) @ A5 @ ( bNF_Wellorder_Func @ B @ C @ B5 @ C6 ) ) )
=> ? [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
@ ^ [Uu: A] : B5 )
@ C6 ) )
& ( ( bNF_Wellorder_curr @ A @ B @ C @ A5 @ X3 )
= G ) ) ) ).
% curr_surj
thf(fact_7234_curr__def,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( bNF_Wellorder_curr @ A @ B @ C )
= ( ^ [A8: set @ A,F2: ( product_prod @ A @ B ) > C,A3: A] :
( if @ ( B > C ) @ ( member @ A @ A3 @ A8 )
@ ^ [B3: B] : ( F2 @ ( product_Pair @ A @ B @ A3 @ B3 ) )
@ ( undefined @ ( B > C ) ) ) ) ) ).
% curr_def
thf(fact_7235_curr__inj,axiom,
! [C: $tType,B: $tType,A: $tType,F1: ( product_prod @ A @ B ) > C,A5: set @ A,B5: set @ B,C6: 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
@ ^ [Uu: A] : B5 )
@ C6 ) )
=> ( ( member @ ( ( product_prod @ A @ B ) > C ) @ F22
@ ( bNF_Wellorder_Func @ ( product_prod @ A @ B ) @ C
@ ( product_Sigma @ A @ B @ A5
@ ^ [Uu: A] : B5 )
@ C6 ) )
=> ( ( ( bNF_Wellorder_curr @ A @ B @ C @ A5 @ F1 )
= ( bNF_Wellorder_curr @ A @ B @ C @ A5 @ F22 ) )
= ( F1 = F22 ) ) ) ) ).
% curr_inj
thf(fact_7236_the__default_Osimps_I1_J,axiom,
! [A: $tType,Uu3: A,X2: A] :
( ( the_default @ A @ Uu3 @ ( some @ A @ X2 ) )
= X2 ) ).
% the_default.simps(1)
thf(fact_7237_the__default_Osimps_I2_J,axiom,
! [A: $tType,X2: A] :
( ( the_default @ A @ X2 @ ( none @ A ) )
= X2 ) ).
% the_default.simps(2)
thf(fact_7238_curr__in,axiom,
! [A: $tType,B: $tType,C: $tType,F: ( product_prod @ A @ B ) > C,A5: set @ A,B5: set @ B,C6: set @ C] :
( ( member @ ( ( product_prod @ A @ B ) > C ) @ F
@ ( bNF_Wellorder_Func @ ( product_prod @ A @ B ) @ C
@ ( product_Sigma @ A @ B @ A5
@ ^ [Uu: A] : B5 )
@ C6 ) )
=> ( member @ ( A > B > C ) @ ( bNF_Wellorder_curr @ A @ B @ C @ A5 @ F ) @ ( bNF_Wellorder_Func @ A @ ( B > C ) @ A5 @ ( bNF_Wellorder_Func @ B @ C @ B5 @ C6 ) ) ) ) ).
% curr_in
thf(fact_7239_bij__betw__curr,axiom,
! [A: $tType,B: $tType,C: $tType,A5: set @ A,B5: set @ B,C6: 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
@ ^ [Uu: A] : B5 )
@ C6 )
@ ( bNF_Wellorder_Func @ A @ ( B > C ) @ A5 @ ( bNF_Wellorder_Func @ B @ C @ B5 @ C6 ) ) ) ).
% bij_betw_curr
thf(fact_7240_listrel__iff__nth,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B,R: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs @ Ys ) @ ( listrel @ A @ B @ R ) )
= ( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
& ! [N4: nat] :
( ( ord_less @ nat @ N4 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ ( nth @ A @ Xs @ N4 ) @ ( nth @ B @ Ys @ N4 ) ) @ R ) ) ) ) ).
% listrel_iff_nth
thf(fact_7241_bij__betw__add,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [A2: A,A5: set @ A,B5: set @ A] :
( ( bij_betw @ A @ A @ ( plus_plus @ A @ A2 ) @ A5 @ B5 )
= ( ( image2 @ A @ A @ ( plus_plus @ A @ A2 ) @ A5 )
= B5 ) ) ) ).
% bij_betw_add
thf(fact_7242_o__bij,axiom,
! [A: $tType,B: $tType,G: B > A,F: A > B] :
( ( ( comp @ B @ A @ A @ G @ F )
= ( id @ A ) )
=> ( ( ( comp @ A @ B @ B @ F @ G )
= ( id @ B ) )
=> ( bij_betw @ A @ B @ F @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) ) ) ) ).
% o_bij
thf(fact_7243_listrel__eq__len,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B,R: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs @ Ys ) @ ( listrel @ A @ B @ R ) )
=> ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) ) ) ).
% listrel_eq_len
thf(fact_7244_sum_Oreindex__bij__betw,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comm_monoid_add @ A )
=> ! [H: B > C,S2: set @ B,T6: set @ C,G: C > A] :
( ( bij_betw @ B @ C @ H @ S2 @ T6 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X: B] : ( G @ ( H @ X ) )
@ S2 )
= ( groups7311177749621191930dd_sum @ C @ A @ G @ T6 ) ) ) ) ).
% sum.reindex_bij_betw
thf(fact_7245_bij__betw__iff__card,axiom,
! [A: $tType,B: $tType,A5: set @ A,B5: set @ B] :
( ( finite_finite2 @ A @ A5 )
=> ( ( finite_finite2 @ B @ B5 )
=> ( ( ? [F2: A > B] : ( bij_betw @ A @ B @ F2 @ A5 @ B5 ) )
= ( ( finite_card @ A @ A5 )
= ( finite_card @ B @ B5 ) ) ) ) ) ).
% bij_betw_iff_card
thf(fact_7246_finite__same__card__bij,axiom,
! [A: $tType,B: $tType,A5: set @ A,B5: set @ B] :
( ( finite_finite2 @ A @ A5 )
=> ( ( finite_finite2 @ B @ B5 )
=> ( ( ( finite_card @ A @ A5 )
= ( finite_card @ B @ B5 ) )
=> ? [H2: A > B] : ( bij_betw @ A @ B @ H2 @ A5 @ B5 ) ) ) ) ).
% finite_same_card_bij
thf(fact_7247_prod_Oreindex__bij__betw,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comm_monoid_mult @ A )
=> ! [H: B > C,S2: set @ B,T6: set @ C,G: C > A] :
( ( bij_betw @ B @ C @ H @ S2 @ T6 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [X: B] : ( G @ ( H @ X ) )
@ S2 )
= ( groups7121269368397514597t_prod @ C @ A @ G @ T6 ) ) ) ) ).
% prod.reindex_bij_betw
thf(fact_7248_bij__betw__funpow,axiom,
! [A: $tType,F: A > A,S2: set @ A,N: nat] :
( ( bij_betw @ A @ A @ F @ S2 @ S2 )
=> ( bij_betw @ A @ A @ ( compow @ ( A > A ) @ N @ F ) @ S2 @ S2 ) ) ).
% bij_betw_funpow
thf(fact_7249_bij__fn,axiom,
! [A: $tType,F: A > A,N: nat] :
( ( bij_betw @ A @ A @ F @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ A ) ) )
=> ( bij_betw @ A @ A @ ( compow @ ( A > A ) @ N @ F ) @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ A ) ) ) ) ).
% bij_fn
thf(fact_7250_Schroeder__Bernstein,axiom,
! [A: $tType,B: $tType,F: A > B,A5: set @ A,B5: set @ B,G: B > A] :
( ( inj_on @ A @ B @ F @ A5 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F @ A5 ) @ B5 )
=> ( ( inj_on @ B @ A @ G @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image2 @ B @ A @ G @ B5 ) @ A5 )
=> ? [H2: A > B] : ( bij_betw @ A @ B @ H2 @ A5 @ B5 ) ) ) ) ) ).
% Schroeder_Bernstein
thf(fact_7251_bij__betw__disjoint__Un,axiom,
! [A: $tType,B: $tType,F: A > B,A5: set @ A,C6: set @ B,G: A > B,B5: set @ A,D3: set @ B] :
( ( bij_betw @ A @ B @ F @ A5 @ C6 )
=> ( ( bij_betw @ A @ B @ G @ B5 @ D3 )
=> ( ( ( inf_inf @ ( set @ A ) @ A5 @ B5 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( inf_inf @ ( set @ B ) @ C6 @ D3 )
= ( bot_bot @ ( set @ B ) ) )
=> ( bij_betw @ A @ B
@ ^ [X: A] : ( if @ B @ ( member @ A @ X @ A5 ) @ ( F @ X ) @ ( G @ X ) )
@ ( sup_sup @ ( set @ A ) @ A5 @ B5 )
@ ( sup_sup @ ( set @ B ) @ C6 @ D3 ) ) ) ) ) ) ).
% bij_betw_disjoint_Un
thf(fact_7252_bij__betw__UNION__chain,axiom,
! [B: $tType,C: $tType,A: $tType,I5: set @ A,A5: A > ( set @ B ),F: B > C,A11: 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 @ F @ ( A5 @ I3 ) @ ( A11 @ I3 ) ) )
=> ( bij_betw @ B @ C @ F @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ A5 @ I5 ) ) @ ( complete_Sup_Sup @ ( set @ C ) @ ( image2 @ A @ ( set @ C ) @ A11 @ I5 ) ) ) ) ) ).
% bij_betw_UNION_chain
thf(fact_7253_infinite__imp__bij__betw2,axiom,
! [A: $tType,A5: set @ A,A2: A] :
( ~ ( finite_finite2 @ A @ A5 )
=> ? [H2: A > A] : ( bij_betw @ A @ A @ H2 @ A5 @ ( sup_sup @ ( set @ A ) @ A5 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% infinite_imp_bij_betw2
thf(fact_7254_infinite__imp__bij__betw,axiom,
! [A: $tType,A5: set @ A,A2: A] :
( ~ ( finite_finite2 @ A @ A5 )
=> ? [H2: A > A] : ( bij_betw @ A @ A @ H2 @ A5 @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% infinite_imp_bij_betw
thf(fact_7255_mono__bij__Inf,axiom,
! [B: $tType,A: $tType] :
( ( ( comple5582772986160207858norder @ A )
& ( comple5582772986160207858norder @ B ) )
=> ! [F: A > B,A5: set @ A] :
( ( order_mono @ A @ B @ F )
=> ( ( bij_betw @ A @ B @ F @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
=> ( ( F @ ( complete_Inf_Inf @ A @ A5 ) )
= ( complete_Inf_Inf @ B @ ( image2 @ A @ B @ F @ A5 ) ) ) ) ) ) ).
% mono_bij_Inf
thf(fact_7256_sum_Oreindex__bij__betw__not__neutral,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comm_monoid_add @ A )
=> ! [S7: set @ B,T5: set @ C,H: B > C,S2: set @ B,T6: set @ C,G: C > A] :
( ( finite_finite2 @ B @ S7 )
=> ( ( finite_finite2 @ C @ T5 )
=> ( ( bij_betw @ B @ C @ H @ ( minus_minus @ ( set @ B ) @ S2 @ S7 ) @ ( minus_minus @ ( set @ C ) @ T6 @ T5 ) )
=> ( ! [A4: B] :
( ( member @ B @ A4 @ S7 )
=> ( ( G @ ( H @ A4 ) )
= ( zero_zero @ A ) ) )
=> ( ! [B4: C] :
( ( member @ C @ B4 @ T5 )
=> ( ( G @ B4 )
= ( zero_zero @ A ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X: B] : ( G @ ( H @ X ) )
@ S2 )
= ( groups7311177749621191930dd_sum @ C @ A @ G @ T6 ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_betw_not_neutral
thf(fact_7257_prod_Oreindex__bij__betw__not__neutral,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S7: set @ B,T5: set @ C,H: B > C,S2: set @ B,T6: set @ C,G: C > A] :
( ( finite_finite2 @ B @ S7 )
=> ( ( finite_finite2 @ C @ T5 )
=> ( ( bij_betw @ B @ C @ H @ ( minus_minus @ ( set @ B ) @ S2 @ S7 ) @ ( minus_minus @ ( set @ C ) @ T6 @ T5 ) )
=> ( ! [A4: B] :
( ( member @ B @ A4 @ S7 )
=> ( ( G @ ( H @ A4 ) )
= ( one_one @ A ) ) )
=> ( ! [B4: C] :
( ( member @ C @ B4 @ T5 )
=> ( ( G @ B4 )
= ( one_one @ A ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [X: B] : ( G @ ( H @ X ) )
@ S2 )
= ( groups7121269368397514597t_prod @ C @ A @ G @ T6 ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_betw_not_neutral
thf(fact_7258_bij__image__INT,axiom,
! [B: $tType,A: $tType,C: $tType,F: A > B,B5: C > ( set @ A ),A5: set @ C] :
( ( bij_betw @ A @ B @ F @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
=> ( ( image2 @ A @ B @ F @ ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ C @ ( set @ A ) @ B5 @ A5 ) ) )
= ( complete_Inf_Inf @ ( set @ B )
@ ( image2 @ C @ ( set @ B )
@ ^ [X: C] : ( image2 @ A @ B @ F @ ( B5 @ X ) )
@ A5 ) ) ) ) ).
% bij_image_INT
thf(fact_7259_listrel__iff__zip,axiom,
! [B: $tType,A: $tType,Xs: list @ A,Ys: list @ B,R: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs @ Ys ) @ ( listrel @ A @ B @ R ) )
= ( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
& ! [X: product_prod @ A @ B] :
( ( member @ ( product_prod @ A @ B ) @ X @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) ) )
=> ( product_case_prod @ A @ B @ $o
@ ^ [Y4: A,Z3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Y4 @ Z3 ) @ R )
@ X ) ) ) ) ).
% listrel_iff_zip
thf(fact_7260_listrel__def,axiom,
! [B: $tType,A: $tType] :
( ( listrel @ A @ B )
= ( ^ [R2: set @ ( product_prod @ A @ B )] :
( collect @ ( product_prod @ ( list @ A ) @ ( list @ B ) )
@ ( product_case_prod @ ( list @ A ) @ ( list @ B ) @ $o
@ ( listrelp @ A @ B
@ ^ [X: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y4 ) @ R2 ) ) ) ) ) ) ).
% listrel_def
thf(fact_7261_bij__betw__of__nat,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [N5: set @ nat,A5: set @ A] :
( ( bij_betw @ nat @ A @ ( semiring_1_of_nat @ A ) @ N5 @ A5 )
= ( ( image2 @ nat @ A @ ( semiring_1_of_nat @ A ) @ N5 )
= A5 ) ) ) ).
% bij_betw_of_nat
thf(fact_7262_bij__betw__Suc,axiom,
! [M3: set @ nat,N5: set @ nat] :
( ( bij_betw @ nat @ nat @ suc @ M3 @ N5 )
= ( ( image2 @ nat @ nat @ suc @ M3 )
= N5 ) ) ).
% bij_betw_Suc
thf(fact_7263_INF__bool__eq,axiom,
! [A: $tType] :
( ( ^ [A8: set @ A,F2: A > $o] : ( complete_Inf_Inf @ $o @ ( image2 @ A @ $o @ F2 @ A8 ) ) )
= ( ball @ A ) ) ).
% INF_bool_eq
thf(fact_7264_Collect__ball__eq,axiom,
! [A: $tType,B: $tType,A5: set @ B,P: A > B > $o] :
( ( collect @ A
@ ^ [X: A] :
! [Y4: B] :
( ( member @ B @ Y4 @ A5 )
=> ( P @ X @ Y4 ) ) )
= ( complete_Inf_Inf @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [Y4: B] :
( collect @ A
@ ^ [X: A] : ( P @ X @ Y4 ) )
@ A5 ) ) ) ).
% Collect_ball_eq
thf(fact_7265_INTER__eq,axiom,
! [B: $tType,A: $tType,B5: B > ( set @ A ),A5: set @ B] :
( ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B5 @ A5 ) )
= ( collect @ A
@ ^ [Y4: A] :
! [X: B] :
( ( member @ B @ X @ A5 )
=> ( member @ A @ Y4 @ ( B5 @ X ) ) ) ) ) ).
% INTER_eq
thf(fact_7266_Ball__def__raw,axiom,
! [A: $tType] :
( ( ball @ A )
= ( ^ [A8: set @ A,P4: A > $o] :
! [X: A] :
( ( member @ A @ X @ A8 )
=> ( P4 @ X ) ) ) ) ).
% Ball_def_raw
thf(fact_7267_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 ),F2: A > B > C] :
! [X: product_prod @ A @ A] :
( ( member @ ( product_prod @ A @ A ) @ X @ R13 )
=> ( product_case_prod @ A @ A @ $o
@ ^ [Y12: A,Z12: A] :
! [Y4: product_prod @ B @ B] :
( ( member @ ( product_prod @ B @ B ) @ Y4 @ R24 )
=> ( product_case_prod @ B @ B @ $o
@ ^ [Y24: B,Z23: B] :
( ( F2 @ Y12 @ Y24 )
= ( F2 @ Z12 @ Z23 ) )
@ Y4 ) )
@ X ) ) ) ) ).
% congruent2_def
thf(fact_7268_irrefl__distinct,axiom,
! [A: $tType] :
( ( irrefl @ A )
= ( ^ [R2: set @ ( product_prod @ A @ A )] :
! [X: product_prod @ A @ A] :
( ( member @ ( product_prod @ A @ A ) @ X @ R2 )
=> ( product_case_prod @ A @ A @ $o
@ ^ [A3: A,B3: A] : A3 != B3
@ X ) ) ) ) ).
% irrefl_distinct
thf(fact_7269_congruent__def,axiom,
! [B: $tType,A: $tType] :
( ( equiv_congruent @ A @ B )
= ( ^ [R2: set @ ( product_prod @ A @ A ),F2: A > B] :
! [X: product_prod @ A @ A] :
( ( member @ ( product_prod @ A @ A ) @ X @ R2 )
=> ( product_case_prod @ A @ A @ $o
@ ^ [Y4: A,Z3: A] :
( ( F2 @ Y4 )
= ( F2 @ Z3 ) )
@ X ) ) ) ) ).
% congruent_def
thf(fact_7270_Ball__fold,axiom,
! [A: $tType,A5: set @ A,P: A > $o] :
( ( finite_finite2 @ A @ A5 )
=> ( ( ! [X: A] :
( ( member @ A @ X @ A5 )
=> ( P @ X ) ) )
= ( finite_fold @ A @ $o
@ ^ [K2: A,S6: $o] :
( S6
& ( P @ K2 ) )
@ $true
@ A5 ) ) ) ).
% Ball_fold
thf(fact_7271_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] :
! [X: A] :
( ( member @ A @ X @ A8 )
=> ( P @ X ) ) ) ) ).
% set_relator_eq_onp
thf(fact_7272_Ball__comp__iff,axiom,
! [C: $tType,B: $tType,A: $tType,A5: B > ( set @ C ),F: C > $o,G: A > B] :
( ( comp @ B @ $o @ A
@ ^ [X: B] :
! [Y4: C] :
( ( member @ C @ Y4 @ ( A5 @ X ) )
=> ( F @ Y4 ) )
@ G )
= ( ^ [X: A] :
! [Y4: C] :
( ( member @ C @ Y4 @ ( comp @ B @ ( set @ C ) @ A @ A5 @ G @ X ) )
=> ( F @ Y4 ) ) ) ) ).
% Ball_comp_iff
thf(fact_7273_Inf__eq__Sup,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ( ( complete_Inf_Inf @ A )
= ( ^ [A8: set @ A] :
( complete_Sup_Sup @ A
@ ( collect @ A
@ ^ [B3: A] :
! [X: A] :
( ( member @ A @ X @ A8 )
=> ( ord_less_eq @ A @ B3 @ X ) ) ) ) ) ) ) ).
% Inf_eq_Sup
thf(fact_7274_Sup__eq__Inf,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ( ( complete_Sup_Sup @ A )
= ( ^ [A8: set @ A] :
( complete_Inf_Inf @ A
@ ( collect @ A
@ ^ [B3: A] :
! [X: A] :
( ( member @ A @ X @ A8 )
=> ( ord_less_eq @ A @ X @ B3 ) ) ) ) ) ) ) ).
% Sup_eq_Inf
thf(fact_7275_trans__join,axiom,
! [A: $tType] :
( ( trans @ A )
= ( ^ [R2: set @ ( product_prod @ A @ A )] :
! [X: product_prod @ A @ A] :
( ( member @ ( product_prod @ A @ A ) @ X @ R2 )
=> ( product_case_prod @ A @ A @ $o
@ ^ [Y4: A,Y12: A] :
! [Z3: product_prod @ A @ A] :
( ( member @ ( product_prod @ A @ A ) @ Z3 @ R2 )
=> ( product_case_prod @ A @ A @ $o
@ ^ [Y24: A,Aa3: A] :
( ( Y12 = Y24 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Aa3 ) @ R2 ) )
@ Z3 ) )
@ X ) ) ) ) ).
% trans_join
thf(fact_7276_takeWhile__append,axiom,
! [A: $tType,Xs: list @ A,P: A > $o,Ys: list @ A] :
( ( ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( P @ X3 ) )
=> ( ( takeWhile @ A @ P @ ( append @ A @ Xs @ Ys ) )
= ( append @ A @ Xs @ ( takeWhile @ A @ P @ Ys ) ) ) )
& ( ~ ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
=> ( P @ X4 ) )
=> ( ( takeWhile @ A @ P @ ( append @ A @ Xs @ Ys ) )
= ( takeWhile @ A @ P @ Xs ) ) ) ) ).
% takeWhile_append
thf(fact_7277_dropWhile__append,axiom,
! [A: $tType,Xs: list @ A,P: A > $o,Ys: list @ A] :
( ( ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( P @ X3 ) )
=> ( ( dropWhile @ A @ P @ ( append @ A @ Xs @ Ys ) )
= ( dropWhile @ A @ P @ Ys ) ) )
& ( ~ ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
=> ( P @ X4 ) )
=> ( ( dropWhile @ A @ P @ ( append @ A @ Xs @ Ys ) )
= ( append @ A @ ( dropWhile @ A @ P @ Xs ) @ Ys ) ) ) ) ).
% dropWhile_append
thf(fact_7278_ex__bij__betw__finite__nat,axiom,
! [A: $tType,M3: set @ A] :
( ( finite_finite2 @ A @ M3 )
=> ? [H2: A > nat] : ( bij_betw @ A @ nat @ H2 @ M3 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( finite_card @ A @ M3 ) ) ) ) ).
% ex_bij_betw_finite_nat
thf(fact_7279_ex__bij__betw__nat__finite,axiom,
! [A: $tType,M3: set @ A] :
( ( finite_finite2 @ A @ M3 )
=> ? [H2: nat > A] : ( bij_betw @ nat @ A @ H2 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( finite_card @ A @ M3 ) ) @ M3 ) ) ).
% ex_bij_betw_nat_finite
thf(fact_7280_ex__bij__betw__nat__finite__1,axiom,
! [A: $tType,M3: set @ A] :
( ( finite_finite2 @ A @ M3 )
=> ? [H2: nat > A] : ( bij_betw @ nat @ A @ H2 @ ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ ( finite_card @ A @ M3 ) ) @ M3 ) ) ).
% ex_bij_betw_nat_finite_1
thf(fact_7281_Func__def,axiom,
! [B: $tType,A: $tType] :
( ( bNF_Wellorder_Func @ A @ B )
= ( ^ [A8: set @ A,B8: set @ B] :
( collect @ ( A > B )
@ ^ [F2: A > B] :
( ! [X: A] :
( ( member @ A @ X @ A8 )
=> ( member @ B @ ( F2 @ X ) @ B8 ) )
& ! [A3: A] :
( ~ ( member @ A @ A3 @ A8 )
=> ( ( F2 @ A3 )
= ( undefined @ B ) ) ) ) ) ) ) ).
% Func_def
thf(fact_7282_list__eq__iff__zip__eq,axiom,
! [A: $tType] :
( ( ^ [Y3: list @ A,Z: list @ A] : Y3 = Z )
= ( ^ [Xs3: list @ A,Ys3: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs3 )
= ( size_size @ ( list @ A ) @ Ys3 ) )
& ! [X: product_prod @ A @ A] :
( ( member @ ( product_prod @ A @ A ) @ X @ ( set2 @ ( product_prod @ A @ A ) @ ( zip @ A @ A @ Xs3 @ Ys3 ) ) )
=> ( product_case_prod @ A @ A @ $o
@ ^ [Y3: A,Z: A] : Y3 = Z
@ X ) ) ) ) ) ).
% list_eq_iff_zip_eq
thf(fact_7283_bij__betw__nth,axiom,
! [A: $tType,Xs: list @ A,A5: set @ nat,B5: set @ A] :
( ( distinct @ A @ Xs )
=> ( ( A5
= ( set_ord_lessThan @ nat @ ( size_size @ ( list @ A ) @ Xs ) ) )
=> ( ( B5
= ( set2 @ A @ Xs ) )
=> ( bij_betw @ nat @ A @ ( nth @ A @ Xs ) @ A5 @ B5 ) ) ) ) ).
% bij_betw_nth
thf(fact_7284_sum_OatLeastAtMost__reindex,axiom,
! [B: $tType,A: $tType] :
( ( ( comm_monoid_add @ A )
& ( ord @ B ) )
=> ! [H: nat > B,M: nat,N: nat,G: B > A] :
( ( bij_betw @ nat @ B @ H @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) @ ( set_or1337092689740270186AtMost @ B @ ( H @ M ) @ ( H @ N ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( set_or1337092689740270186AtMost @ B @ ( H @ M ) @ ( H @ N ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ B @ A @ nat @ G @ H ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ) ).
% sum.atLeastAtMost_reindex
thf(fact_7285_sum_OatLeastLessThan__reindex,axiom,
! [B: $tType,A: $tType] :
( ( ( comm_monoid_add @ A )
& ( ord @ B ) )
=> ! [H: nat > B,M: nat,N: nat,G: B > A] :
( ( bij_betw @ nat @ B @ H @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) @ ( set_or7035219750837199246ssThan @ B @ ( H @ M ) @ ( H @ N ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( set_or7035219750837199246ssThan @ B @ ( H @ M ) @ ( H @ N ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ B @ A @ nat @ G @ H ) @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ) ).
% sum.atLeastLessThan_reindex
thf(fact_7286_prod_OatLeastAtMost__reindex,axiom,
! [B: $tType,A: $tType] :
( ( ( comm_monoid_mult @ A )
& ( ord @ B ) )
=> ! [H: nat > B,M: nat,N: nat,G: B > A] :
( ( bij_betw @ nat @ B @ H @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) @ ( set_or1337092689740270186AtMost @ B @ ( H @ M ) @ ( H @ N ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( set_or1337092689740270186AtMost @ B @ ( H @ M ) @ ( H @ N ) ) )
= ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ B @ A @ nat @ G @ H ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ) ).
% prod.atLeastAtMost_reindex
thf(fact_7287_prod_OatLeastLessThan__reindex,axiom,
! [B: $tType,A: $tType] :
( ( ( comm_monoid_mult @ A )
& ( ord @ B ) )
=> ! [H: nat > B,M: nat,N: nat,G: B > A] :
( ( bij_betw @ nat @ B @ H @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) @ ( set_or7035219750837199246ssThan @ B @ ( H @ M ) @ ( H @ N ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( set_or7035219750837199246ssThan @ B @ ( H @ M ) @ ( H @ N ) ) )
= ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ B @ A @ nat @ G @ H ) @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ) ).
% prod.atLeastLessThan_reindex
thf(fact_7288_concat__eq__concat__iff,axiom,
! [A: $tType,Xs: list @ ( list @ A ),Ys: 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 @ Ys ) ) )
=> ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
@ ^ [Y4: list @ A,Z3: list @ A] :
( ( size_size @ ( list @ A ) @ Y4 )
= ( size_size @ ( list @ A ) @ Z3 ) )
@ X3 ) )
=> ( ( ( size_size @ ( list @ ( list @ A ) ) @ Xs )
= ( size_size @ ( list @ ( list @ A ) ) @ Ys ) )
=> ( ( ( concat @ A @ Xs )
= ( concat @ A @ Ys ) )
= ( Xs = Ys ) ) ) ) ).
% concat_eq_concat_iff
thf(fact_7289_concat__injective,axiom,
! [A: $tType,Xs: list @ ( list @ A ),Ys: list @ ( list @ A )] :
( ( ( concat @ A @ Xs )
= ( concat @ A @ Ys ) )
=> ( ( ( size_size @ ( list @ ( list @ A ) ) @ Xs )
= ( size_size @ ( list @ ( list @ A ) ) @ Ys ) )
=> ( ! [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 @ Ys ) ) )
=> ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
@ ^ [Y4: list @ A,Z3: list @ A] :
( ( size_size @ ( list @ A ) @ Y4 )
= ( size_size @ ( list @ A ) @ Z3 ) )
@ X3 ) )
=> ( Xs = Ys ) ) ) ) ).
% concat_injective
thf(fact_7290_list__all2__iff,axiom,
! [B: $tType,A: $tType] :
( ( list_all2 @ A @ B )
= ( ^ [P4: A > B > $o,Xs3: list @ A,Ys3: list @ B] :
( ( ( size_size @ ( list @ A ) @ Xs3 )
= ( size_size @ ( list @ B ) @ Ys3 ) )
& ! [X: product_prod @ A @ B] :
( ( member @ ( product_prod @ A @ B ) @ X @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs3 @ Ys3 ) ) )
=> ( product_case_prod @ A @ B @ $o @ P4 @ X ) ) ) ) ) ).
% list_all2_iff
thf(fact_7291_listrelp__listrel__eq,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B )] :
( ( listrelp @ A @ B
@ ^ [X: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y4 ) @ R ) )
= ( ^ [X: list @ A,Y4: list @ B] : ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ X @ Y4 ) @ ( listrel @ A @ B @ R ) ) ) ) ).
% listrelp_listrel_eq
thf(fact_7292_iteratesp_Omono,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [F: A > A] :
( order_mono @ ( A > $o ) @ ( A > $o )
@ ^ [P7: A > $o,X: A] :
( ? [Y4: A] :
( ( X
= ( F @ Y4 ) )
& ( P7 @ Y4 ) )
| ? [M6: set @ A] :
( ( X
= ( complete_Sup_Sup @ A @ M6 ) )
& ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ M6 )
& ! [Y4: A] :
( ( member @ A @ Y4 @ M6 )
=> ( P7 @ Y4 ) ) ) ) ) ) ).
% iteratesp.mono
thf(fact_7293_effect__makeE,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [N: nat,F: nat > A,H: heap_ext @ product_unit,H4: heap_ext @ product_unit,R: array @ A,N3: nat] :
( ( heap_Time_effect @ ( array @ A ) @ ( array_make @ A @ N @ F ) @ H @ H4 @ R @ N3 )
=> ~ ( ( R
= ( product_fst @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( array_alloc @ A @ ( map @ nat @ A @ F @ ( upt @ ( zero_zero @ nat ) @ N ) ) @ H ) ) )
=> ( ( H4
= ( product_snd @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( array_alloc @ A @ ( map @ nat @ A @ F @ ( upt @ ( zero_zero @ nat ) @ N ) ) @ H ) ) )
=> ( ( ( array_get @ A @ H4 @ R )
= ( map @ nat @ A @ F @ ( upt @ ( zero_zero @ nat ) @ N ) ) )
=> ( ( array_present @ A @ H4 @ R )
=> ( ~ ( array_present @ A @ H @ R )
=> ( ( plus_plus @ nat @ N @ ( one_one @ nat ) )
!= N3 ) ) ) ) ) ) ) ) ).
% effect_makeE
thf(fact_7294_Inter__eq,axiom,
! [A: $tType] :
( ( complete_Inf_Inf @ ( set @ A ) )
= ( ^ [A8: set @ ( set @ A )] :
( collect @ A
@ ^ [X: A] :
! [Y4: set @ A] :
( ( member @ ( set @ A ) @ Y4 @ A8 )
=> ( member @ A @ X @ Y4 ) ) ) ) ) ).
% Inter_eq
thf(fact_7295_chain__singleton,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [X2: A] : ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% chain_singleton
thf(fact_7296_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
@ ^ [X: A] :
( ( member @ A @ X @ A5 )
& ( P @ X ) ) ) ) ) ).
% chain_compr
thf(fact_7297_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_7298_ccpo__Sup__upper,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [A5: set @ A,X2: A] :
( ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ A5 )
=> ( ( member @ A @ X2 @ A5 )
=> ( ord_less_eq @ A @ X2 @ ( complete_Sup_Sup @ A @ A5 ) ) ) ) ) ).
% ccpo_Sup_upper
thf(fact_7299_ccpo__Sup__mono,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [A5: set @ A,B5: set @ A] :
( ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ A5 )
=> ( ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ B5 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ A5 )
=> ? [Xa2: A] :
( ( member @ A @ Xa2 @ B5 )
& ( ord_less_eq @ A @ X3 @ Xa2 ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A5 ) @ ( complete_Sup_Sup @ A @ B5 ) ) ) ) ) ) ).
% ccpo_Sup_mono
thf(fact_7300_Sup__int__def,axiom,
( ( complete_Sup_Sup @ int )
= ( ^ [X5: set @ int] :
( the @ int
@ ^ [X: int] :
( ( member @ int @ X @ X5 )
& ! [Y4: int] :
( ( member @ int @ Y4 @ X5 )
=> ( ord_less_eq @ int @ Y4 @ X ) ) ) ) ) ) ).
% Sup_int_def
thf(fact_7301_Union__maximal__sets,axiom,
! [A: $tType,F9: set @ ( set @ A )] :
( ( finite_finite2 @ ( set @ A ) @ F9 )
=> ( ( complete_Sup_Sup @ ( set @ A )
@ ( collect @ ( set @ A )
@ ^ [T8: set @ A] :
( ( member @ ( set @ A ) @ T8 @ F9 )
& ! [X: set @ A] :
( ( member @ ( set @ A ) @ X @ F9 )
=> ~ ( ord_less @ ( set @ A ) @ T8 @ X ) ) ) ) )
= ( complete_Sup_Sup @ ( set @ A ) @ F9 ) ) ) ).
% Union_maximal_sets
thf(fact_7302_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 )
@ ^ [Uu: set @ A] :
? [F2: ( set @ A ) > A] :
( ( Uu
= ( image2 @ ( set @ A ) @ A @ F2 @ A5 ) )
& ! [X: set @ A] :
( ( member @ ( set @ A ) @ X @ A5 )
=> ( member @ A @ ( F2 @ X ) @ X ) ) ) ) ) ) ) ) ).
% Sup_Inf
thf(fact_7303_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 )
@ ^ [Uu: set @ A] :
? [F2: ( set @ A ) > A] :
( ( Uu
= ( image2 @ ( set @ A ) @ A @ F2 @ A5 ) )
& ! [X: set @ A] :
( ( member @ ( set @ A ) @ X @ A5 )
=> ( member @ A @ ( F2 @ X ) @ X ) ) ) ) ) ) ) ) ).
% Inf_Sup
thf(fact_7304_Inf__filter__def,axiom,
! [A: $tType] :
( ( complete_Inf_Inf @ ( filter @ A ) )
= ( ^ [S: set @ ( filter @ A )] :
( complete_Sup_Sup @ ( filter @ A )
@ ( collect @ ( filter @ A )
@ ^ [F8: filter @ A] :
! [X: filter @ A] :
( ( member @ ( filter @ A ) @ X @ S )
=> ( ord_less_eq @ ( filter @ A ) @ F8 @ X ) ) ) ) ) ) ).
% Inf_filter_def
thf(fact_7305_effect__of__listE,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [Xs: list @ A,H: heap_ext @ product_unit,H4: heap_ext @ product_unit,R: array @ A,N3: nat] :
( ( heap_Time_effect @ ( array @ A ) @ ( array_of_list @ A @ Xs ) @ H @ H4 @ R @ N3 )
=> ~ ( ( R
= ( product_fst @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( array_alloc @ A @ Xs @ H ) ) )
=> ( ( H4
= ( product_snd @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( array_alloc @ A @ Xs @ H ) ) )
=> ( ( ( array_get @ A @ H4 @ R )
= Xs )
=> ( ( array_present @ A @ H4 @ R )
=> ( ~ ( array_present @ A @ H @ R )
=> ( N3
!= ( plus_plus @ nat @ ( one_one @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) ) ) ) ) ) ).
% effect_of_listE
thf(fact_7306_effect__newE,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [N: nat,X2: A,H: heap_ext @ product_unit,H4: heap_ext @ product_unit,R: array @ A,N3: nat] :
( ( heap_Time_effect @ ( array @ A ) @ ( array_new @ A @ N @ X2 ) @ H @ H4 @ R @ N3 )
=> ~ ( ( R
= ( product_fst @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( array_alloc @ A @ ( replicate @ A @ N @ X2 ) @ H ) ) )
=> ( ( H4
= ( product_snd @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( array_alloc @ A @ ( replicate @ A @ N @ X2 ) @ H ) ) )
=> ( ( ( array_get @ A @ H4 @ R )
= ( replicate @ A @ N @ X2 ) )
=> ( ( array_present @ A @ H4 @ R )
=> ( ~ ( array_present @ A @ H @ R )
=> ( ( plus_plus @ nat @ N @ ( one_one @ nat ) )
!= N3 ) ) ) ) ) ) ) ) ).
% effect_newE
thf(fact_7307_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 )
@ ^ [Uu: set @ A] :
? [F2: ( set @ A ) > A] :
( ( Uu
= ( image2 @ ( set @ A ) @ A @ F2 @ A5 ) )
& ! [X: set @ A] :
( ( member @ ( set @ A ) @ X @ A5 )
=> ( member @ A @ ( F2 @ X ) @ X ) ) ) ) ) )
@ ( complete_Inf_Inf @ A @ ( image2 @ ( set @ A ) @ A @ ( complete_Sup_Sup @ A ) @ A5 ) ) ) ) ).
% Sup_Inf_le
thf(fact_7308_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 )
@ ^ [Uu: set @ A] :
? [F2: ( set @ A ) > A] :
( ( Uu
= ( image2 @ ( set @ A ) @ A @ F2 @ A5 ) )
& ! [X: set @ A] :
( ( member @ ( set @ A ) @ X @ A5 )
=> ( member @ A @ ( F2 @ X ) @ X ) ) ) ) ) ) ) ) ).
% Inf_Sup_le
thf(fact_7309_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 )
@ ^ [Uu: set @ A] :
? [F2: ( set @ A ) > A] :
( ( Uu
= ( image2 @ ( set @ A ) @ A @ F2 @ A5 ) )
& ! [X: set @ A] :
( ( member @ ( set @ A ) @ X @ A5 )
=> ( member @ A @ ( F2 @ X ) @ X ) ) ) ) ) ) ) ) ).
% finite_Inf_Sup
thf(fact_7310_INF__SUP__set,axiom,
! [A: $tType,B: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [G: B > A,A5: set @ ( set @ B )] :
( ( complete_Inf_Inf @ A
@ ( image2 @ ( set @ B ) @ A
@ ^ [B8: set @ B] : ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ G @ B8 ) )
@ A5 ) )
= ( complete_Sup_Sup @ A
@ ( image2 @ ( set @ B ) @ A
@ ^ [B8: set @ B] : ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ G @ B8 ) )
@ ( collect @ ( set @ B )
@ ^ [Uu: set @ B] :
? [F2: ( set @ B ) > B] :
( ( Uu
= ( image2 @ ( set @ B ) @ B @ F2 @ A5 ) )
& ! [X: set @ B] :
( ( member @ ( set @ B ) @ X @ A5 )
=> ( member @ B @ ( F2 @ X ) @ X ) ) ) ) ) ) ) ) ).
% INF_SUP_set
thf(fact_7311_SUP__INF__set,axiom,
! [A: $tType,B: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [G: B > A,A5: set @ ( set @ B )] :
( ( complete_Sup_Sup @ A
@ ( image2 @ ( set @ B ) @ A
@ ^ [X: set @ B] : ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ G @ X ) )
@ A5 ) )
= ( complete_Inf_Inf @ A
@ ( image2 @ ( set @ B ) @ A
@ ^ [X: set @ B] : ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ G @ X ) )
@ ( collect @ ( set @ B )
@ ^ [Uu: set @ B] :
? [F2: ( set @ B ) > B] :
( ( Uu
= ( image2 @ ( set @ B ) @ B @ F2 @ A5 ) )
& ! [X: set @ B] :
( ( member @ ( set @ B ) @ X @ A5 )
=> ( member @ B @ ( F2 @ X ) @ X ) ) ) ) ) ) ) ) ).
% SUP_INF_set
thf(fact_7312_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 ) )
@ ^ [Uu: set @ ( option @ A )] :
? [F2: ( set @ ( option @ A ) ) > ( option @ A )] :
( ( Uu
= ( image2 @ ( set @ ( option @ A ) ) @ ( option @ A ) @ F2 @ A5 ) )
& ! [X: set @ ( option @ A )] :
( ( member @ ( set @ ( option @ A ) ) @ X @ A5 )
=> ( member @ ( option @ A ) @ ( F2 @ X ) @ X ) ) ) ) ) ) ) ) ).
% option_Inf_Sup
thf(fact_7313_subset__mset_OcSup__cInf,axiom,
! [A: $tType,S2: set @ ( multiset @ A )] :
( ( S2
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ S2 )
=> ( ( complete_Sup_Sup @ ( multiset @ A ) @ S2 )
= ( complete_Inf_Inf @ ( multiset @ A )
@ ( collect @ ( multiset @ A )
@ ^ [X: multiset @ A] :
! [Y4: multiset @ A] :
( ( member @ ( multiset @ A ) @ Y4 @ S2 )
=> ( subseteq_mset @ A @ Y4 @ X ) ) ) ) ) ) ) ).
% subset_mset.cSup_cInf
thf(fact_7314_subset__mset_OcInf__cSup,axiom,
! [A: $tType,S2: set @ ( multiset @ A )] :
( ( S2
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ S2 )
=> ( ( complete_Inf_Inf @ ( multiset @ A ) @ S2 )
= ( complete_Sup_Sup @ ( multiset @ A )
@ ( collect @ ( multiset @ A )
@ ^ [X: multiset @ A] :
! [Y4: multiset @ A] :
( ( member @ ( multiset @ A ) @ Y4 @ S2 )
=> ( subseteq_mset @ A @ X @ Y4 ) ) ) ) ) ) ) ).
% subset_mset.cInf_cSup
thf(fact_7315_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_7316_iteratesp__def,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ( ( comple7512665784863727008ratesp @ A )
= ( ^ [F2: A > A] :
( complete_lattice_lfp @ ( A > $o )
@ ^ [P7: A > $o,X: A] :
( ? [Y4: A] :
( ( X
= ( F2 @ Y4 ) )
& ( P7 @ Y4 ) )
| ? [M6: set @ A] :
( ( X
= ( complete_Sup_Sup @ A @ M6 ) )
& ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ M6 )
& ! [Y4: A] :
( ( member @ A @ Y4 @ M6 )
=> ( P7 @ Y4 ) ) ) ) ) ) ) ) ).
% iteratesp_def
thf(fact_7317_iteratesp_Osimps,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ( ( comple7512665784863727008ratesp @ A )
= ( ^ [F2: A > A,A3: A] :
( ? [X: A] :
( ( A3
= ( F2 @ X ) )
& ( comple7512665784863727008ratesp @ A @ F2 @ X ) )
| ? [M6: set @ A] :
( ( A3
= ( complete_Sup_Sup @ A @ M6 ) )
& ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ M6 )
& ! [X: A] :
( ( member @ A @ X @ M6 )
=> ( comple7512665784863727008ratesp @ A @ F2 @ X ) ) ) ) ) ) ) ).
% iteratesp.simps
thf(fact_7318_iteratesp_Ocases,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [F: A > A,A2: A] :
( ( comple7512665784863727008ratesp @ A @ F @ A2 )
=> ( ! [X3: A] :
( ( A2
= ( F @ X3 ) )
=> ~ ( comple7512665784863727008ratesp @ A @ F @ X3 ) )
=> ~ ! [M10: set @ A] :
( ( A2
= ( complete_Sup_Sup @ A @ M10 ) )
=> ( ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ M10 )
=> ~ ! [X4: A] :
( ( member @ A @ X4 @ M10 )
=> ( comple7512665784863727008ratesp @ A @ F @ X4 ) ) ) ) ) ) ) ).
% iteratesp.cases
thf(fact_7319_iteratesp_OSup,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [M3: set @ A,F: A > A] :
( ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ M3 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ M3 )
=> ( comple7512665784863727008ratesp @ A @ F @ X3 ) )
=> ( comple7512665784863727008ratesp @ A @ F @ ( complete_Sup_Sup @ A @ M3 ) ) ) ) ) ).
% iteratesp.Sup
thf(fact_7320_execute__nth_I1_J,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [I: nat,H: heap_ext @ product_unit,A2: array @ A] :
( ( ord_less @ nat @ I @ ( array_length @ A @ H @ A2 ) )
=> ( ( heap_Time_execute @ A @ ( array_nth @ A @ A2 @ I ) @ H )
= ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( nth @ A @ ( array_get @ A @ H @ A2 ) @ I ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H @ ( one_one @ nat ) ) ) ) ) ) ) ).
% execute_nth(1)
thf(fact_7321_Array__Time_Onth__def,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( ( array_nth @ A )
= ( ^ [A3: array @ A,I2: nat] :
( heap_Time_guard @ A
@ ^ [H3: heap_ext @ product_unit] : ( ord_less @ nat @ I2 @ ( array_length @ A @ H3 @ A3 ) )
@ ^ [H3: heap_ext @ product_unit] : ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( nth @ A @ ( array_get @ A @ H3 @ A3 ) @ I2 ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H3 @ ( one_one @ nat ) ) ) ) ) ) ) ).
% Array_Time.nth_def
thf(fact_7322_length__def,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( ( array_length @ A )
= ( ^ [H3: heap_ext @ product_unit,A3: array @ A] : ( size_size @ ( list @ A ) @ ( array_get @ A @ H3 @ A3 ) ) ) ) ) ).
% length_def
thf(fact_7323_fails__array__nth,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [P3: array @ A,I: nat,H: heap_ext @ product_unit] :
( ( time_fails @ A @ ( array_nth @ A @ P3 @ I ) @ H )
= ( ~ ( ord_less @ nat @ I @ ( array_length @ A @ H @ P3 ) ) ) ) ) ).
% fails_array_nth
thf(fact_7324_fails__array__swap,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [I: nat,X2: A,P3: array @ A,H: heap_ext @ product_unit] :
( ( time_fails @ A @ ( array_swap @ A @ I @ X2 @ P3 ) @ H )
= ( ~ ( ord_less @ nat @ I @ ( array_length @ A @ H @ P3 ) ) ) ) ) ).
% fails_array_swap
thf(fact_7325_fails__array__upd,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [I: nat,X2: A,P3: array @ A,H: heap_ext @ product_unit] :
( ( time_fails @ ( array @ A ) @ ( array_upd @ A @ I @ X2 @ P3 ) @ H )
= ( ~ ( ord_less @ nat @ I @ ( array_length @ A @ H @ P3 ) ) ) ) ) ).
% fails_array_upd
thf(fact_7326_execute__upd_I2_J,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [H: heap_ext @ product_unit,A2: array @ A,I: nat,X2: A] :
( ( ord_less_eq @ nat @ ( array_length @ A @ H @ A2 ) @ I )
=> ( ( heap_Time_execute @ ( array @ A ) @ ( array_upd @ A @ I @ X2 @ A2 ) @ H )
= ( none @ ( product_prod @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ) ).
% execute_upd(2)
thf(fact_7327_fails__array__map__entry,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [I: nat,X2: A > A,P3: array @ A,H: heap_ext @ product_unit] :
( ( time_fails @ ( array @ A ) @ ( array_map_entry @ A @ I @ X2 @ P3 ) @ H )
= ( ~ ( ord_less @ nat @ I @ ( array_length @ A @ H @ P3 ) ) ) ) ) ).
% fails_array_map_entry
thf(fact_7328_execute__map__entry_I2_J,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [H: heap_ext @ product_unit,A2: array @ A,I: nat,F: A > A] :
( ( ord_less_eq @ nat @ ( array_length @ A @ H @ A2 ) @ I )
=> ( ( heap_Time_execute @ ( array @ A ) @ ( array_map_entry @ A @ I @ F @ A2 ) @ H )
= ( none @ ( product_prod @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ) ).
% execute_map_entry(2)
thf(fact_7329_execute__nth_I2_J,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [H: heap_ext @ product_unit,A2: array @ A,I: nat] :
( ( ord_less_eq @ nat @ ( array_length @ A @ H @ A2 ) @ I )
=> ( ( heap_Time_execute @ A @ ( array_nth @ A @ A2 @ I ) @ H )
= ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ) ).
% execute_nth(2)
thf(fact_7330_execute__swap_I2_J,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [H: heap_ext @ product_unit,A2: array @ A,I: nat,X2: A] :
( ( ord_less_eq @ nat @ ( array_length @ A @ H @ A2 ) @ I )
=> ( ( heap_Time_execute @ A @ ( array_swap @ A @ I @ X2 @ A2 ) @ H )
= ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ) ).
% execute_swap(2)
thf(fact_7331_Array__Time_Olength__alloc,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [Xs: list @ A,H: heap_ext @ product_unit,Ys: list @ A] :
( ( array_length @ A @ ( product_snd @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( array_alloc @ A @ Xs @ H ) ) @ ( product_fst @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( array_alloc @ A @ Ys @ H ) ) )
= ( size_size @ ( list @ A ) @ Xs ) ) ) ).
% Array_Time.length_alloc
thf(fact_7332_effect__nthI,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [I: nat,H: heap_ext @ product_unit,A2: array @ A,H4: heap_ext @ product_unit,R: A,N: nat] :
( ( ord_less @ nat @ I @ ( array_length @ A @ H @ A2 ) )
=> ( ( H4 = H )
=> ( ( R
= ( nth @ A @ ( array_get @ A @ H @ A2 ) @ I ) )
=> ( ( N
= ( one_one @ nat ) )
=> ( heap_Time_effect @ A @ ( array_nth @ A @ A2 @ I ) @ H @ H4 @ R @ N ) ) ) ) ) ) ).
% effect_nthI
thf(fact_7333_effect__nthE,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [A2: array @ A,I: nat,H: heap_ext @ product_unit,H4: heap_ext @ product_unit,R: A,N: nat] :
( ( heap_Time_effect @ A @ ( array_nth @ A @ A2 @ I ) @ H @ H4 @ R @ N )
=> ~ ( ( ord_less @ nat @ I @ ( array_length @ A @ H @ A2 ) )
=> ( ( R
= ( nth @ A @ ( array_get @ A @ H @ A2 ) @ I ) )
=> ( ( H4 = H )
=> ( N
!= ( one_one @ nat ) ) ) ) ) ) ) ).
% effect_nthE
thf(fact_7334_swap__def,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( ( array_swap @ A )
= ( ^ [I2: nat,X: A,A3: array @ A] :
( heap_Time_guard @ A
@ ^ [H3: heap_ext @ product_unit] : ( ord_less @ nat @ I2 @ ( array_length @ A @ H3 @ A3 ) )
@ ^ [H3: heap_ext @ product_unit] : ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( nth @ A @ ( array_get @ A @ H3 @ A3 ) @ I2 ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ ( array_update @ A @ A3 @ I2 @ X @ H3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% swap_def
thf(fact_7335_map__entry__def,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( ( array_map_entry @ A )
= ( ^ [I2: nat,F2: A > A,A3: array @ A] :
( heap_Time_guard @ ( array @ A )
@ ^ [H3: heap_ext @ product_unit] : ( ord_less @ nat @ I2 @ ( array_length @ A @ H3 @ A3 ) )
@ ^ [H3: heap_ext @ product_unit] : ( product_Pair @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ A3 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ ( array_update @ A @ A3 @ I2 @ ( F2 @ ( nth @ A @ ( array_get @ A @ H3 @ A3 ) @ I2 ) ) @ H3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% map_entry_def
thf(fact_7336_effect__updI,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [I: nat,H: heap_ext @ product_unit,A2: array @ A,H4: heap_ext @ product_unit,V2: A,N: nat] :
( ( ord_less @ nat @ I @ ( array_length @ A @ H @ A2 ) )
=> ( ( H4
= ( array_update @ A @ A2 @ I @ V2 @ H ) )
=> ( ( N
= ( one_one @ nat ) )
=> ( heap_Time_effect @ ( array @ A ) @ ( array_upd @ A @ I @ V2 @ A2 ) @ H @ H4 @ A2 @ N ) ) ) ) ) ).
% effect_updI
thf(fact_7337_effect__updE,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [I: nat,V2: A,A2: array @ A,H: heap_ext @ product_unit,H4: heap_ext @ product_unit,R: array @ A,N: nat] :
( ( heap_Time_effect @ ( array @ A ) @ ( array_upd @ A @ I @ V2 @ A2 ) @ H @ H4 @ R @ N )
=> ~ ( ( R = A2 )
=> ( ( H4
= ( array_update @ A @ A2 @ I @ V2 @ H ) )
=> ( ( ord_less @ nat @ I @ ( array_length @ A @ H @ A2 ) )
=> ( N
!= ( one_one @ nat ) ) ) ) ) ) ) ).
% effect_updE
thf(fact_7338_upd__def,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( ( array_upd @ A )
= ( ^ [I2: nat,X: A,A3: array @ A] :
( heap_Time_guard @ ( array @ A )
@ ^ [H3: heap_ext @ product_unit] : ( ord_less @ nat @ I2 @ ( array_length @ A @ H3 @ A3 ) )
@ ^ [H3: heap_ext @ product_unit] : ( product_Pair @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ A3 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ ( array_update @ A @ A3 @ I2 @ X @ H3 ) @ ( one_one @ nat ) ) ) ) ) ) ) ).
% upd_def
thf(fact_7339_execute__upd_I1_J,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [I: nat,H: heap_ext @ product_unit,A2: array @ A,X2: A] :
( ( ord_less @ nat @ I @ ( array_length @ A @ H @ A2 ) )
=> ( ( heap_Time_execute @ ( array @ A ) @ ( array_upd @ A @ I @ X2 @ A2 ) @ H )
= ( some @ ( product_prod @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ A2 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ ( array_update @ A @ A2 @ I @ X2 @ H ) @ ( one_one @ nat ) ) ) ) ) ) ) ).
% execute_upd(1)
thf(fact_7340_freeze__def,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( ( array_freeze @ A )
= ( ^ [A3: array @ A] :
( heap_Time_heap @ ( list @ A )
@ ^ [H3: heap_ext @ product_unit] : ( product_Pair @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( array_get @ A @ H3 @ A3 ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H3 @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( array_length @ A @ H3 @ A3 ) ) ) ) ) ) ) ) ).
% freeze_def
thf(fact_7341_execute__freeze,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [A2: array @ A,H: heap_ext @ product_unit] :
( ( heap_Time_execute @ ( list @ A ) @ ( array_freeze @ A @ A2 ) @ H )
= ( some @ ( product_prod @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( array_get @ A @ H @ A2 ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( array_length @ A @ H @ A2 ) ) ) ) ) ) ) ).
% execute_freeze
thf(fact_7342_time__array__freeze,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [P3: array @ A,H: heap_ext @ product_unit] :
( ( time_time @ ( list @ A ) @ ( array_freeze @ A @ P3 ) @ H )
= ( plus_plus @ nat @ ( array_length @ A @ H @ P3 ) @ ( one_one @ nat ) ) ) ) ).
% time_array_freeze
thf(fact_7343_effect__freezeE,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [A2: array @ A,H: heap_ext @ product_unit,H4: heap_ext @ product_unit,R: list @ A,N: nat] :
( ( heap_Time_effect @ ( list @ A ) @ ( array_freeze @ A @ A2 ) @ H @ H4 @ R @ N )
=> ~ ( ( H4 = H )
=> ( ( R
= ( array_get @ A @ H @ A2 ) )
=> ( N
!= ( plus_plus @ nat @ ( array_length @ A @ H @ A2 ) @ ( one_one @ nat ) ) ) ) ) ) ) ).
% effect_freezeE
thf(fact_7344_effect__freezeI,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [H4: heap_ext @ product_unit,H: heap_ext @ product_unit,R: list @ A,A2: array @ A,N: nat] :
( ( H4 = H )
=> ( ( R
= ( array_get @ A @ H @ A2 ) )
=> ( ( N
= ( array_length @ A @ H @ A2 ) )
=> ( heap_Time_effect @ ( list @ A ) @ ( array_freeze @ A @ A2 ) @ H @ H4 @ R @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ) ) ).
% effect_freezeI
thf(fact_7345_flat__lub__def,axiom,
! [A: $tType] :
( ( partial_flat_lub @ A )
= ( ^ [B3: A,A8: set @ A] :
( if @ A @ ( ord_less_eq @ ( set @ A ) @ A8 @ ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) @ B3
@ ( the @ A
@ ^ [X: A] : ( member @ A @ X @ ( minus_minus @ ( set @ A ) @ A8 @ ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% flat_lub_def
thf(fact_7346_execute__len,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [A2: array @ A,H: heap_ext @ product_unit] :
( ( heap_Time_execute @ nat @ ( array_len @ A @ A2 ) @ H )
= ( some @ ( product_prod @ nat @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ nat @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( array_length @ A @ H @ A2 ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H @ ( one_one @ nat ) ) ) ) ) ) ).
% execute_len
thf(fact_7347_fails__array__len,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [P3: array @ A,H: heap_ext @ product_unit] :
~ ( time_fails @ nat @ ( array_len @ A @ P3 ) @ H ) ) ).
% fails_array_len
thf(fact_7348_time__array__len,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [P3: array @ A,H: heap_ext @ product_unit] :
( ( time_time @ nat @ ( array_len @ A @ P3 ) @ H )
= ( one_one @ nat ) ) ) ).
% time_array_len
thf(fact_7349_len__def,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( ( array_len @ A )
= ( ^ [A3: array @ A] :
( heap_Time_tap @ nat
@ ^ [H3: heap_ext @ product_unit] : ( array_length @ A @ H3 @ A3 ) ) ) ) ) ).
% len_def
thf(fact_7350_effect__lengthI,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [H4: heap_ext @ product_unit,H: heap_ext @ product_unit,R: nat,A2: array @ A,N: nat] :
( ( H4 = H )
=> ( ( R
= ( array_length @ A @ H @ A2 ) )
=> ( ( N
= ( one_one @ nat ) )
=> ( heap_Time_effect @ nat @ ( array_len @ A @ A2 ) @ H @ H4 @ R @ N ) ) ) ) ) ).
% effect_lengthI
thf(fact_7351_effect__lengthE,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [A2: array @ A,H: heap_ext @ product_unit,H4: heap_ext @ product_unit,R: nat,N: nat] :
( ( heap_Time_effect @ nat @ ( array_len @ A @ A2 ) @ H @ H4 @ R @ N )
=> ~ ( ( R
= ( array_length @ A @ H4 @ A2 ) )
=> ( ( H4 = H )
=> ( N
!= ( one_one @ nat ) ) ) ) ) ) ).
% effect_lengthE
thf(fact_7352_len_H__def,axiom,
! [A: $tType,B: $tType] :
( ( ( heap @ B )
& ( semiring_1 @ A ) )
=> ( ( array_len2 @ B @ A )
= ( ^ [A3: array @ B] :
( heap_Time_bind @ nat @ A @ ( array_len @ B @ A3 )
@ ^ [N4: nat] : ( heap_Time_ureturn @ A @ ( semiring_1_of_nat @ A @ N4 ) ) ) ) ) ) ).
% len'_def
thf(fact_7353_minus__coset__filter,axiom,
! [A: $tType,A5: set @ A,Xs: list @ A] :
( ( minus_minus @ ( set @ A ) @ A5 @ ( coset @ A @ Xs ) )
= ( set2 @ A
@ ( filter2 @ A
@ ^ [X: A] : ( member @ A @ X @ A5 )
@ Xs ) ) ) ).
% minus_coset_filter
thf(fact_7354_union__coset__filter,axiom,
! [A: $tType,Xs: list @ A,A5: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( coset @ A @ Xs ) @ A5 )
= ( coset @ A
@ ( filter2 @ A
@ ^ [X: A] :
~ ( member @ A @ X @ A5 )
@ Xs ) ) ) ).
% union_coset_filter
thf(fact_7355_multiset_Odomain,axiom,
! [B: $tType,C: $tType,T6: C > B > $o] :
( ( domainp @ ( C > nat ) @ ( multiset @ B ) @ ( pcr_multiset @ C @ B @ T6 ) )
= ( ^ [X: C > nat] :
? [Y4: B > nat] :
( ( bNF_rel_fun @ C @ B @ nat @ nat @ T6
@ ^ [Y3: nat,Z: nat] : Y3 = Z
@ X
@ Y4 )
& ( finite_finite2 @ B
@ ( collect @ B
@ ^ [Z3: B] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( Y4 @ Z3 ) ) ) ) ) ) ) ).
% multiset.domain
thf(fact_7356_and__not__num_Opelims,axiom,
! [X2: num,Xa: num,Y: option @ num] :
( ( ( bit_and_not_num @ X2 @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ num @ num ) @ bit_and_not_num_rel @ ( product_Pair @ num @ num @ X2 @ Xa ) )
=> ( ( ( X2 = one2 )
=> ( ( Xa = one2 )
=> ( ( Y
= ( none @ num ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_and_not_num_rel @ ( product_Pair @ num @ num @ one2 @ one2 ) ) ) ) )
=> ( ( ( X2 = one2 )
=> ! [N2: num] :
( ( Xa
= ( bit0 @ N2 ) )
=> ( ( Y
= ( some @ num @ one2 ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_and_not_num_rel @ ( product_Pair @ num @ num @ one2 @ ( bit0 @ N2 ) ) ) ) ) )
=> ( ( ( X2 = one2 )
=> ! [N2: num] :
( ( Xa
= ( bit1 @ N2 ) )
=> ( ( Y
= ( none @ num ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_and_not_num_rel @ ( product_Pair @ num @ num @ one2 @ ( bit1 @ N2 ) ) ) ) ) )
=> ( ! [M4: num] :
( ( X2
= ( bit0 @ M4 ) )
=> ( ( Xa = one2 )
=> ( ( Y
= ( some @ num @ ( bit0 @ M4 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_and_not_num_rel @ ( product_Pair @ num @ num @ ( bit0 @ M4 ) @ one2 ) ) ) ) )
=> ( ! [M4: num] :
( ( X2
= ( bit0 @ M4 ) )
=> ! [N2: num] :
( ( Xa
= ( bit0 @ N2 ) )
=> ( ( Y
= ( map_option @ num @ num @ bit0 @ ( bit_and_not_num @ M4 @ N2 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_and_not_num_rel @ ( product_Pair @ num @ num @ ( bit0 @ M4 ) @ ( bit0 @ N2 ) ) ) ) ) )
=> ( ! [M4: num] :
( ( X2
= ( bit0 @ M4 ) )
=> ! [N2: num] :
( ( Xa
= ( bit1 @ N2 ) )
=> ( ( Y
= ( map_option @ num @ num @ bit0 @ ( bit_and_not_num @ M4 @ N2 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_and_not_num_rel @ ( product_Pair @ num @ num @ ( bit0 @ M4 ) @ ( bit1 @ N2 ) ) ) ) ) )
=> ( ! [M4: num] :
( ( X2
= ( bit1 @ M4 ) )
=> ( ( Xa = one2 )
=> ( ( Y
= ( some @ num @ ( bit0 @ M4 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_and_not_num_rel @ ( product_Pair @ num @ num @ ( bit1 @ M4 ) @ one2 ) ) ) ) )
=> ( ! [M4: num] :
( ( X2
= ( bit1 @ M4 ) )
=> ! [N2: num] :
( ( Xa
= ( bit0 @ N2 ) )
=> ( ( Y
= ( case_option @ ( option @ num ) @ num @ ( some @ num @ one2 )
@ ^ [N11: num] : ( some @ num @ ( bit1 @ N11 ) )
@ ( bit_and_not_num @ M4 @ N2 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_and_not_num_rel @ ( product_Pair @ num @ num @ ( bit1 @ M4 ) @ ( bit0 @ N2 ) ) ) ) ) )
=> ~ ! [M4: num] :
( ( X2
= ( bit1 @ M4 ) )
=> ! [N2: num] :
( ( Xa
= ( bit1 @ N2 ) )
=> ( ( Y
= ( map_option @ num @ num @ bit0 @ ( bit_and_not_num @ M4 @ N2 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_and_not_num_rel @ ( product_Pair @ num @ num @ ( bit1 @ M4 ) @ ( bit1 @ N2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% and_not_num.pelims
thf(fact_7357_multiset_Odomain__eq,axiom,
! [A: $tType] :
( ( domainp @ ( A > nat ) @ ( multiset @ A )
@ ( pcr_multiset @ A @ A
@ ^ [Y3: A,Z: A] : Y3 = Z ) )
= ( ^ [F2: A > nat] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X: A] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F2 @ X ) ) ) ) ) ) ).
% multiset.domain_eq
thf(fact_7358_and__num_Opelims,axiom,
! [X2: num,Xa: num,Y: option @ num] :
( ( ( bit_un7362597486090784418nd_num @ X2 @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ num @ num ) @ bit_un4731106466462545111um_rel @ ( product_Pair @ num @ num @ X2 @ Xa ) )
=> ( ( ( X2 = one2 )
=> ( ( Xa = one2 )
=> ( ( Y
= ( some @ num @ one2 ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4731106466462545111um_rel @ ( product_Pair @ num @ num @ one2 @ one2 ) ) ) ) )
=> ( ( ( X2 = one2 )
=> ! [N2: num] :
( ( Xa
= ( bit0 @ N2 ) )
=> ( ( Y
= ( none @ num ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4731106466462545111um_rel @ ( product_Pair @ num @ num @ one2 @ ( bit0 @ N2 ) ) ) ) ) )
=> ( ( ( X2 = one2 )
=> ! [N2: num] :
( ( Xa
= ( bit1 @ N2 ) )
=> ( ( Y
= ( some @ num @ one2 ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4731106466462545111um_rel @ ( product_Pair @ num @ num @ one2 @ ( bit1 @ N2 ) ) ) ) ) )
=> ( ! [M4: num] :
( ( X2
= ( bit0 @ M4 ) )
=> ( ( Xa = one2 )
=> ( ( Y
= ( none @ num ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4731106466462545111um_rel @ ( product_Pair @ num @ num @ ( bit0 @ M4 ) @ one2 ) ) ) ) )
=> ( ! [M4: num] :
( ( X2
= ( bit0 @ M4 ) )
=> ! [N2: num] :
( ( Xa
= ( bit0 @ N2 ) )
=> ( ( Y
= ( map_option @ num @ num @ bit0 @ ( bit_un7362597486090784418nd_num @ M4 @ N2 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4731106466462545111um_rel @ ( product_Pair @ num @ num @ ( bit0 @ M4 ) @ ( bit0 @ N2 ) ) ) ) ) )
=> ( ! [M4: num] :
( ( X2
= ( bit0 @ M4 ) )
=> ! [N2: num] :
( ( Xa
= ( bit1 @ N2 ) )
=> ( ( Y
= ( map_option @ num @ num @ bit0 @ ( bit_un7362597486090784418nd_num @ M4 @ N2 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4731106466462545111um_rel @ ( product_Pair @ num @ num @ ( bit0 @ M4 ) @ ( bit1 @ N2 ) ) ) ) ) )
=> ( ! [M4: num] :
( ( X2
= ( bit1 @ M4 ) )
=> ( ( Xa = one2 )
=> ( ( Y
= ( some @ num @ one2 ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4731106466462545111um_rel @ ( product_Pair @ num @ num @ ( bit1 @ M4 ) @ one2 ) ) ) ) )
=> ( ! [M4: num] :
( ( X2
= ( bit1 @ M4 ) )
=> ! [N2: num] :
( ( Xa
= ( bit0 @ N2 ) )
=> ( ( Y
= ( map_option @ num @ num @ bit0 @ ( bit_un7362597486090784418nd_num @ M4 @ N2 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4731106466462545111um_rel @ ( product_Pair @ num @ num @ ( bit1 @ M4 ) @ ( bit0 @ N2 ) ) ) ) ) )
=> ~ ! [M4: num] :
( ( X2
= ( bit1 @ M4 ) )
=> ! [N2: num] :
( ( Xa
= ( bit1 @ N2 ) )
=> ( ( Y
= ( case_option @ ( option @ num ) @ num @ ( some @ num @ one2 )
@ ^ [N11: num] : ( some @ num @ ( bit1 @ N11 ) )
@ ( bit_un7362597486090784418nd_num @ M4 @ N2 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4731106466462545111um_rel @ ( product_Pair @ num @ num @ ( bit1 @ M4 ) @ ( bit1 @ N2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% and_num.pelims
thf(fact_7359_multiset_Odomain__par__left__total,axiom,
! [B: $tType,C: $tType,T6: C > B > $o,P2: ( C > nat ) > $o] :
( ( left_total @ ( C > nat ) @ ( B > nat )
@ ( bNF_rel_fun @ C @ B @ nat @ nat @ T6
@ ^ [Y3: nat,Z: nat] : Y3 = Z ) )
=> ( ( bNF_rel_fun @ ( C > nat ) @ ( B > nat ) @ $o @ $o
@ ( bNF_rel_fun @ C @ B @ nat @ nat @ T6
@ ^ [Y3: nat,Z: nat] : Y3 = Z )
@ ^ [Y3: $o,Z: $o] : Y3 = Z
@ P2
@ ^ [F2: B > nat] :
( finite_finite2 @ B
@ ( collect @ B
@ ^ [X: B] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F2 @ X ) ) ) ) )
=> ( ( domainp @ ( C > nat ) @ ( multiset @ B ) @ ( pcr_multiset @ C @ B @ T6 ) )
= P2 ) ) ) ).
% multiset.domain_par_left_total
thf(fact_7360_multiset_Odomain__par,axiom,
! [B: $tType,C: $tType,T6: C > B > $o,DT: C > $o,DS: nat > $o,P23: ( C > nat ) > $o] :
( ( ( domainp @ C @ B @ T6 )
= DT )
=> ( ( ( domainp @ nat @ nat
@ ^ [Y3: nat,Z: nat] : Y3 = Z )
= DS )
=> ( ( left_unique @ C @ B @ T6 )
=> ( ( bNF_rel_fun @ ( C > nat ) @ ( B > nat ) @ $o @ $o
@ ( bNF_rel_fun @ C @ B @ nat @ nat @ T6
@ ^ [Y3: nat,Z: nat] : Y3 = Z )
@ ^ [Y3: $o,Z: $o] : Y3 = Z
@ P23
@ ^ [F2: B > nat] :
( finite_finite2 @ B
@ ( collect @ B
@ ^ [X: B] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F2 @ X ) ) ) ) )
=> ( ( domainp @ ( C > nat ) @ ( multiset @ B ) @ ( pcr_multiset @ C @ B @ T6 ) )
= ( inf_inf @ ( ( C > nat ) > $o ) @ ( basic_pred_fun @ C @ nat @ DT @ DS ) @ P23 ) ) ) ) ) ) ).
% multiset.domain_par
thf(fact_7361_mono__transfer,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ D )
& ( order @ C )
& ( order @ A ) )
=> ! [A5: A > B > $o,B5: 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
@ ^ [Y3: $o,Z: $o] : Y3 = Z )
@ ( ord_less_eq @ A )
@ ( ord_less_eq @ B ) )
=> ( ( bNF_rel_fun @ C @ D @ ( C > $o ) @ ( D > $o ) @ B5
@ ( bNF_rel_fun @ C @ D @ $o @ $o @ B5
@ ^ [Y3: $o,Z: $o] : Y3 = Z )
@ ( 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 @ B5 )
@ ^ [Y3: $o,Z: $o] : Y3 = Z
@ ( order_mono @ A @ C )
@ ( order_mono @ B @ D ) ) ) ) ) ) ).
% mono_transfer
thf(fact_7362_pred__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( basic_pred_fun @ A @ B )
= ( ^ [A8: A > $o,B8: B > $o,F2: A > B] :
! [X: A] :
( ( A8 @ X )
=> ( B8 @ ( F2 @ X ) ) ) ) ) ).
% pred_fun_def
thf(fact_7363_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
@ ^ [Uu: D] : $true
@ P )
@ ( basic_pred_fun @ D @ A
@ ^ [Uu: D] : $true
@ Pa ) ) ) ).
% fun.pred_mono
thf(fact_7364_fun_Opred__True,axiom,
! [A: $tType,D: $tType] :
( ( basic_pred_fun @ D @ A
@ ^ [Uu: D] : $true
@ ^ [Uu: A] : $true )
= ( ^ [Uu: D > A] : $true ) ) ).
% fun.pred_True
thf(fact_7365_int_Obi__total,axiom,
bi_total @ ( product_prod @ nat @ nat ) @ int @ pcr_int ).
% int.bi_total
thf(fact_7366_typedef__left__unique,axiom,
! [B: $tType,A: $tType,Rep: B > A,Abs: A > B,A5: set @ A,T6: A > B > $o] :
( ( type_definition @ B @ A @ Rep @ Abs @ A5 )
=> ( ( T6
= ( ^ [X: A,Y4: B] :
( X
= ( Rep @ Y4 ) ) ) )
=> ( left_unique @ A @ B @ T6 ) ) ) ).
% typedef_left_unique
thf(fact_7367_pred__fun__True__id,axiom,
! [A: $tType,B: $tType,C: $tType,P3: B > $o,F: C > B] :
( ( nO_MATCH @ ( A > A ) @ ( B > $o ) @ ( id @ A ) @ P3 )
=> ( ( basic_pred_fun @ C @ B
@ ^ [X: C] : $true
@ P3
@ F )
= ( basic_pred_fun @ C @ $o
@ ^ [X: C] : $true
@ ( id @ $o )
@ ( comp @ B @ $o @ C @ P3 @ F ) ) ) ) ).
% pred_fun_True_id
thf(fact_7368_fun_Omap__cong__pred,axiom,
! [B: $tType,A: $tType,D: $tType,X2: D > A,Ya: D > A,F: A > B,G: A > B] :
( ( X2 = Ya )
=> ( ( basic_pred_fun @ D @ A
@ ^ [Uu: D] : $true
@ ^ [Z3: A] :
( ( F @ Z3 )
= ( G @ Z3 ) )
@ Ya )
=> ( ( comp @ A @ B @ D @ F @ X2 )
= ( comp @ A @ B @ D @ G @ Ya ) ) ) ) ).
% fun.map_cong_pred
thf(fact_7369_fun_Opred__mono__strong,axiom,
! [A: $tType,D: $tType,P: A > $o,X2: D > A,Pa: A > $o] :
( ( basic_pred_fun @ D @ A
@ ^ [Uu: D] : $true
@ P
@ X2 )
=> ( ! [Z4: A] :
( ( member @ A @ Z4 @ ( image2 @ D @ A @ X2 @ ( top_top @ ( set @ D ) ) ) )
=> ( ( P @ Z4 )
=> ( Pa @ Z4 ) ) )
=> ( basic_pred_fun @ D @ A
@ ^ [Uu: D] : $true
@ Pa
@ X2 ) ) ) ).
% fun.pred_mono_strong
thf(fact_7370_fun_Opred__cong,axiom,
! [A: $tType,D: $tType,X2: D > A,Ya: D > A,P: A > $o,Pa: A > $o] :
( ( X2 = Ya )
=> ( ! [Z4: A] :
( ( member @ A @ Z4 @ ( image2 @ D @ A @ Ya @ ( top_top @ ( set @ D ) ) ) )
=> ( ( P @ Z4 )
= ( Pa @ Z4 ) ) )
=> ( ( basic_pred_fun @ D @ A
@ ^ [Uu: D] : $true
@ P
@ X2 )
= ( basic_pred_fun @ D @ A
@ ^ [Uu: D] : $true
@ Pa
@ Ya ) ) ) ) ).
% fun.pred_cong
thf(fact_7371_Domainp__cr__rat,axiom,
( ( domainp @ ( product_prod @ int @ int ) @ rat @ pcr_rat )
= ( ^ [X: product_prod @ int @ int] :
( ( product_snd @ int @ int @ X )
!= ( zero_zero @ int ) ) ) ) ).
% Domainp_cr_rat
thf(fact_7372_fun_Opred__rel,axiom,
! [A: $tType,D: $tType,P: A > $o,X2: D > A] :
( ( basic_pred_fun @ D @ A
@ ^ [Uu: D] : $true
@ P
@ X2 )
= ( bNF_rel_fun @ D @ D @ A @ A
@ ^ [Y3: D,Z: D] : Y3 = Z
@ ( bNF_eq_onp @ A @ P )
@ X2
@ X2 ) ) ).
% fun.pred_rel
thf(fact_7373_fun_ODomainp__rel,axiom,
! [C: $tType,B: $tType,A: $tType,R3: A > B > $o] :
( ( domainp @ ( C > A ) @ ( C > B )
@ ( bNF_rel_fun @ C @ C @ A @ B
@ ^ [Y3: C,Z: C] : Y3 = Z
@ R3 ) )
= ( basic_pred_fun @ C @ A
@ ^ [Uu: C] : $true
@ ( domainp @ A @ B @ R3 ) ) ) ).
% fun.Domainp_rel
thf(fact_7374_fun_Opred__map,axiom,
! [B: $tType,A: $tType,D: $tType,Q: B > $o,F: A > B,X2: D > A] :
( ( basic_pred_fun @ D @ B
@ ^ [Uu: D] : $true
@ Q
@ ( comp @ A @ B @ D @ F @ X2 ) )
= ( basic_pred_fun @ D @ A
@ ^ [Uu: D] : $true
@ ( comp @ B @ $o @ A @ Q @ F )
@ X2 ) ) ).
% fun.pred_map
thf(fact_7375_fun_Opred__set,axiom,
! [A: $tType,D: $tType,P: A > $o] :
( ( basic_pred_fun @ D @ A
@ ^ [Uu: D] : $true
@ P )
= ( ^ [X: D > A] :
! [Y4: A] :
( ( member @ A @ Y4 @ ( image2 @ D @ A @ X @ ( top_top @ ( set @ D ) ) ) )
=> ( P @ Y4 ) ) ) ) ).
% fun.pred_set
thf(fact_7376_fun_Opred__transfer,axiom,
! [A: $tType,B: $tType,D: $tType,R3: A > B > $o] :
( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ ( ( D > A ) > $o ) @ ( ( D > B ) > $o )
@ ( bNF_rel_fun @ A @ B @ $o @ $o @ R3
@ ^ [Y3: $o,Z: $o] : Y3 = Z )
@ ( bNF_rel_fun @ ( D > A ) @ ( D > B ) @ $o @ $o
@ ( bNF_rel_fun @ D @ D @ A @ B
@ ^ [Y3: D,Z: D] : Y3 = Z
@ R3 )
@ ^ [Y3: $o,Z: $o] : Y3 = Z )
@ ( basic_pred_fun @ D @ A
@ ^ [Uu: D] : $true )
@ ( basic_pred_fun @ D @ B
@ ^ [Uu: D] : $true ) ) ).
% fun.pred_transfer
thf(fact_7377_fun_Orel__eq__onp,axiom,
! [D: $tType,A: $tType,P: A > $o] :
( ( bNF_rel_fun @ D @ D @ A @ A
@ ^ [Y3: D,Z: D] : Y3 = Z
@ ( bNF_eq_onp @ A @ P ) )
= ( bNF_eq_onp @ ( D > A )
@ ( basic_pred_fun @ D @ A
@ ^ [Uu: D] : $true
@ P ) ) ) ).
% fun.rel_eq_onp
thf(fact_7378_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_7379_eventually__INF__base,axiom,
! [B: $tType,A: $tType,B5: set @ A,F4: A > ( filter @ B ),P: B > $o] :
( ( B5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [A4: A] :
( ( member @ A @ A4 @ B5 )
=> ! [B4: A] :
( ( member @ A @ B4 @ B5 )
=> ? [X4: A] :
( ( member @ A @ X4 @ B5 )
& ( ord_less_eq @ ( filter @ B ) @ ( F4 @ X4 ) @ ( inf_inf @ ( filter @ B ) @ ( F4 @ A4 ) @ ( F4 @ B4 ) ) ) ) ) )
=> ( ( eventually @ B @ P @ ( complete_Inf_Inf @ ( filter @ B ) @ ( image2 @ A @ ( filter @ B ) @ F4 @ B5 ) ) )
= ( ? [X: A] :
( ( member @ A @ X @ B5 )
& ( eventually @ B @ P @ ( F4 @ X ) ) ) ) ) ) ) ).
% eventually_INF_base
thf(fact_7380_eventually__const,axiom,
! [A: $tType,F4: filter @ A,P: $o] :
( ( F4
!= ( bot_bot @ ( filter @ A ) ) )
=> ( ( eventually @ A
@ ^ [X: A] : P
@ F4 )
= P ) ) ).
% eventually_const
thf(fact_7381_eventually__sequentially__Suc,axiom,
! [P: nat > $o] :
( ( eventually @ nat
@ ^ [I2: nat] : ( P @ ( suc @ I2 ) )
@ ( at_top @ nat ) )
= ( eventually @ nat @ P @ ( at_top @ nat ) ) ) ).
% eventually_sequentially_Suc
thf(fact_7382_eventually__sequentially__seg,axiom,
! [P: nat > $o,K: nat] :
( ( eventually @ nat
@ ^ [N4: nat] : ( P @ ( plus_plus @ nat @ N4 @ K ) )
@ ( at_top @ nat ) )
= ( eventually @ nat @ P @ ( at_top @ nat ) ) ) ).
% eventually_sequentially_seg
thf(fact_7383_eventually__inf__principal,axiom,
! [A: $tType,P: A > $o,F4: filter @ A,S3: set @ A] :
( ( eventually @ A @ P @ ( inf_inf @ ( filter @ A ) @ F4 @ ( principal @ A @ S3 ) ) )
= ( eventually @ A
@ ^ [X: A] :
( ( member @ A @ X @ S3 )
=> ( P @ X ) )
@ F4 ) ) ).
% eventually_inf_principal
thf(fact_7384_eventually__ex,axiom,
! [B: $tType,A: $tType,P: A > B > $o,F4: filter @ A] :
( ( eventually @ A
@ ^ [X: A] :
? [X5: B] : ( P @ X @ X5 )
@ F4 )
= ( ? [Y6: A > B] :
( eventually @ A
@ ^ [X: A] : ( P @ X @ ( Y6 @ X ) )
@ F4 ) ) ) ).
% eventually_ex
thf(fact_7385_filterlim__at__top__at__top,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( linorder @ B ) )
=> ! [Q: A > $o,F: A > B,P: B > $o,G: B > A] :
( ! [X3: A,Y2: A] :
( ( Q @ X3 )
=> ( ( Q @ Y2 )
=> ( ( ord_less_eq @ A @ X3 @ Y2 )
=> ( ord_less_eq @ B @ ( F @ X3 ) @ ( F @ Y2 ) ) ) ) )
=> ( ! [X3: B] :
( ( P @ X3 )
=> ( ( F @ ( G @ X3 ) )
= X3 ) )
=> ( ! [X3: B] :
( ( P @ X3 )
=> ( Q @ ( G @ X3 ) ) )
=> ( ( eventually @ A @ Q @ ( at_top @ A ) )
=> ( ( eventually @ B @ P @ ( at_top @ B ) )
=> ( filterlim @ A @ B @ F @ ( at_top @ B ) @ ( at_top @ A ) ) ) ) ) ) ) ) ).
% filterlim_at_top_at_top
thf(fact_7386_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
@ ^ [X: B] :
! [Y4: A] :
( ( member @ A @ Y4 @ A5 )
=> ( P @ X @ Y4 ) )
@ Net )
= ( ! [X: A] :
( ( member @ A @ X @ A5 )
=> ( eventually @ B
@ ^ [Y4: B] : ( P @ Y4 @ X )
@ Net ) ) ) ) ) ).
% eventually_ball_finite_distrib
thf(fact_7387_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
@ ^ [X: B] :
! [Y4: A] :
( ( member @ A @ Y4 @ A5 )
=> ( P @ X @ Y4 ) )
@ Net ) ) ) ).
% eventually_ball_finite
thf(fact_7388_eventually__INF1,axiom,
! [B: $tType,A: $tType,I: A,I5: set @ A,P: B > $o,F4: A > ( filter @ B )] :
( ( member @ A @ I @ I5 )
=> ( ( eventually @ B @ P @ ( F4 @ I ) )
=> ( eventually @ B @ P @ ( complete_Inf_Inf @ ( filter @ B ) @ ( image2 @ A @ ( filter @ B ) @ F4 @ I5 ) ) ) ) ) ).
% eventually_INF1
thf(fact_7389_filterlim__mono__eventually,axiom,
! [B: $tType,A: $tType,F: A > B,F4: filter @ B,G5: filter @ A,F10: filter @ B,G6: filter @ A,F3: A > B] :
( ( filterlim @ A @ B @ F @ F4 @ G5 )
=> ( ( ord_less_eq @ ( filter @ B ) @ F4 @ F10 )
=> ( ( ord_less_eq @ ( filter @ A ) @ G6 @ G5 )
=> ( ( eventually @ A
@ ^ [X: A] :
( ( F @ X )
= ( F3 @ X ) )
@ G6 )
=> ( filterlim @ A @ B @ F3 @ F10 @ G6 ) ) ) ) ) ).
% filterlim_mono_eventually
thf(fact_7390_filterlim__iff,axiom,
! [B: $tType,A: $tType] :
( ( filterlim @ A @ B )
= ( ^ [F2: A > B,F25: filter @ B,F14: filter @ A] :
! [P4: B > $o] :
( ( eventually @ B @ P4 @ F25 )
=> ( eventually @ A
@ ^ [X: A] : ( P4 @ ( F2 @ X ) )
@ F14 ) ) ) ) ).
% filterlim_iff
thf(fact_7391_filterlim__cong,axiom,
! [A: $tType,B: $tType,F13: filter @ A,F15: filter @ A,F24: filter @ B,F26: filter @ B,F: B > A,G: B > A] :
( ( F13 = F15 )
=> ( ( F24 = F26 )
=> ( ( eventually @ B
@ ^ [X: B] :
( ( F @ X )
= ( G @ X ) )
@ F24 )
=> ( ( filterlim @ B @ A @ F @ F13 @ F24 )
= ( filterlim @ B @ A @ G @ F15 @ F26 ) ) ) ) ) ).
% filterlim_cong
thf(fact_7392_eventually__compose__filterlim,axiom,
! [A: $tType,B: $tType,P: A > $o,F4: filter @ A,F: B > A,G5: filter @ B] :
( ( eventually @ A @ P @ F4 )
=> ( ( filterlim @ B @ A @ F @ F4 @ G5 )
=> ( eventually @ B
@ ^ [X: B] : ( P @ ( F @ X ) )
@ G5 ) ) ) ).
% eventually_compose_filterlim
thf(fact_7393_typedef__bi__unique,axiom,
! [B: $tType,A: $tType,Rep: B > A,Abs: A > B,A5: set @ A,T6: A > B > $o] :
( ( type_definition @ B @ A @ Rep @ Abs @ A5 )
=> ( ( T6
= ( ^ [X: A,Y4: B] :
( X
= ( Rep @ Y4 ) ) ) )
=> ( bi_unique @ A @ B @ T6 ) ) ) ).
% typedef_bi_unique
thf(fact_7394_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_7395_eventually__const__iff,axiom,
! [A: $tType,P: $o,F4: filter @ A] :
( ( eventually @ A
@ ^ [X: A] : P
@ F4 )
= ( P
| ( F4
= ( bot_bot @ ( filter @ A ) ) ) ) ) ).
% eventually_const_iff
thf(fact_7396_trivial__limit__def,axiom,
! [A: $tType,F4: filter @ A] :
( ( F4
= ( bot_bot @ ( filter @ A ) ) )
= ( eventually @ A
@ ^ [X: A] : $false
@ F4 ) ) ).
% trivial_limit_def
thf(fact_7397_eventually__gt__at__bot,axiom,
! [A: $tType] :
( ( unboun7993243217541854897norder @ A )
=> ! [C2: A] :
( eventually @ A
@ ^ [X: A] : ( ord_less @ A @ X @ C2 )
@ ( at_bot @ A ) ) ) ).
% eventually_gt_at_bot
thf(fact_7398_eventually__at__bot__not__equal,axiom,
! [A: $tType] :
( ( ( linorder @ A )
& ( no_bot @ A ) )
=> ! [C2: A] :
( eventually @ A
@ ^ [X: A] : X != C2
@ ( at_bot @ A ) ) ) ).
% eventually_at_bot_not_equal
thf(fact_7399_not__eventually__impI,axiom,
! [A: $tType,P: A > $o,F4: filter @ A,Q: A > $o] :
( ( eventually @ A @ P @ F4 )
=> ( ~ ( eventually @ A @ Q @ F4 )
=> ~ ( eventually @ A
@ ^ [X: A] :
( ( P @ X )
=> ( Q @ X ) )
@ F4 ) ) ) ).
% not_eventually_impI
thf(fact_7400_eventually__conj__iff,axiom,
! [A: $tType,P: A > $o,Q: A > $o,F4: filter @ A] :
( ( eventually @ A
@ ^ [X: A] :
( ( P @ X )
& ( Q @ X ) )
@ F4 )
= ( ( eventually @ A @ P @ F4 )
& ( eventually @ A @ Q @ F4 ) ) ) ).
% eventually_conj_iff
thf(fact_7401_eventually__rev__mp,axiom,
! [A: $tType,P: A > $o,F4: filter @ A,Q: A > $o] :
( ( eventually @ A @ P @ F4 )
=> ( ( eventually @ A
@ ^ [X: A] :
( ( P @ X )
=> ( Q @ X ) )
@ F4 )
=> ( eventually @ A @ Q @ F4 ) ) ) ).
% eventually_rev_mp
thf(fact_7402_eventually__subst,axiom,
! [A: $tType,P: A > $o,Q: A > $o,F4: filter @ A] :
( ( eventually @ A
@ ^ [N4: A] :
( ( P @ N4 )
= ( Q @ N4 ) )
@ F4 )
=> ( ( eventually @ A @ P @ F4 )
= ( eventually @ A @ Q @ F4 ) ) ) ).
% eventually_subst
thf(fact_7403_eventually__elim2,axiom,
! [A: $tType,P: A > $o,F4: filter @ A,Q: A > $o,R3: A > $o] :
( ( eventually @ A @ P @ F4 )
=> ( ( eventually @ A @ Q @ F4 )
=> ( ! [I3: A] :
( ( P @ I3 )
=> ( ( Q @ I3 )
=> ( R3 @ I3 ) ) )
=> ( eventually @ A @ R3 @ F4 ) ) ) ) ).
% eventually_elim2
thf(fact_7404_eventually__conj,axiom,
! [A: $tType,P: A > $o,F4: filter @ A,Q: A > $o] :
( ( eventually @ A @ P @ F4 )
=> ( ( eventually @ A @ Q @ F4 )
=> ( eventually @ A
@ ^ [X: A] :
( ( P @ X )
& ( Q @ X ) )
@ F4 ) ) ) ).
% eventually_conj
thf(fact_7405_eventually__True,axiom,
! [A: $tType,F4: filter @ A] :
( eventually @ A
@ ^ [X: A] : $true
@ F4 ) ).
% eventually_True
thf(fact_7406_eventually__mp,axiom,
! [A: $tType,P: A > $o,Q: A > $o,F4: filter @ A] :
( ( eventually @ A
@ ^ [X: A] :
( ( P @ X )
=> ( Q @ X ) )
@ F4 )
=> ( ( eventually @ A @ P @ F4 )
=> ( eventually @ A @ Q @ F4 ) ) ) ).
% eventually_mp
thf(fact_7407_eventually__frequently__const__simps_I3_J,axiom,
! [A: $tType,P: A > $o,C6: $o,F4: filter @ A] :
( ( eventually @ A
@ ^ [X: A] :
( ( P @ X )
| C6 )
@ F4 )
= ( ( eventually @ A @ P @ F4 )
| C6 ) ) ).
% eventually_frequently_const_simps(3)
thf(fact_7408_eventually__frequently__const__simps_I4_J,axiom,
! [A: $tType,C6: $o,P: A > $o,F4: filter @ A] :
( ( eventually @ A
@ ^ [X: A] :
( C6
| ( P @ X ) )
@ F4 )
= ( C6
| ( eventually @ A @ P @ F4 ) ) ) ).
% eventually_frequently_const_simps(4)
thf(fact_7409_eventually__frequently__const__simps_I6_J,axiom,
! [A: $tType,C6: $o,P: A > $o,F4: filter @ A] :
( ( eventually @ A
@ ^ [X: A] :
( C6
=> ( P @ X ) )
@ F4 )
= ( C6
=> ( eventually @ A @ P @ F4 ) ) ) ).
% eventually_frequently_const_simps(6)
thf(fact_7410_eventually__le__at__bot,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C2: A] :
( eventually @ A
@ ^ [X: A] : ( ord_less_eq @ A @ X @ C2 )
@ ( at_bot @ A ) ) ) ).
% eventually_le_at_bot
thf(fact_7411_eventually__at__bot__linorder,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o] :
( ( eventually @ A @ P @ ( at_bot @ A ) )
= ( ? [N6: A] :
! [N4: A] :
( ( ord_less_eq @ A @ N4 @ N6 )
=> ( P @ N4 ) ) ) ) ) ).
% eventually_at_bot_linorder
thf(fact_7412_eventually__all__finite,axiom,
! [B: $tType,A: $tType] :
( ( finite_finite @ B )
=> ! [P: A > B > $o,Net: filter @ A] :
( ! [Y2: B] :
( eventually @ A
@ ^ [X: A] : ( P @ X @ Y2 )
@ Net )
=> ( eventually @ A
@ ^ [X: A] :
! [X5: B] : ( P @ X @ X5 )
@ Net ) ) ) ).
% eventually_all_finite
thf(fact_7413_eventually__sequentially,axiom,
! [P: nat > $o] :
( ( eventually @ nat @ P @ ( at_top @ nat ) )
= ( ? [N6: nat] :
! [N4: nat] :
( ( ord_less_eq @ nat @ N6 @ N4 )
=> ( P @ N4 ) ) ) ) ).
% eventually_sequentially
thf(fact_7414_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_7415_eventually__at__top__linorder,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o] :
( ( eventually @ A @ P @ ( at_top @ A ) )
= ( ? [N6: A] :
! [N4: A] :
( ( ord_less_eq @ A @ N6 @ N4 )
=> ( P @ N4 ) ) ) ) ) ).
% eventually_at_top_linorder
thf(fact_7416_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_7417_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_7418_le__sequentially,axiom,
! [F4: filter @ nat] :
( ( ord_less_eq @ ( filter @ nat ) @ F4 @ ( at_top @ nat ) )
= ( ! [N6: nat] : ( eventually @ nat @ ( ord_less_eq @ nat @ N6 ) @ F4 ) ) ) ).
% le_sequentially
thf(fact_7419_eventually__False__sequentially,axiom,
~ ( eventually @ nat
@ ^ [N4: nat] : $false
@ ( at_top @ nat ) ) ).
% eventually_False_sequentially
thf(fact_7420_eventually__at__top__not__equal,axiom,
! [A: $tType] :
( ( ( linorder @ A )
& ( no_top @ A ) )
=> ! [C2: A] :
( eventually @ A
@ ^ [X: A] : X != C2
@ ( at_top @ A ) ) ) ).
% eventually_at_top_not_equal
thf(fact_7421_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_7422_le__principal,axiom,
! [A: $tType,F4: filter @ A,A5: set @ A] :
( ( ord_less_eq @ ( filter @ A ) @ F4 @ ( principal @ A @ A5 ) )
= ( eventually @ A
@ ^ [X: A] : ( member @ A @ X @ A5 )
@ F4 ) ) ).
% le_principal
thf(fact_7423_filterlim__principal,axiom,
! [B: $tType,A: $tType,F: A > B,S2: set @ B,F4: filter @ A] :
( ( filterlim @ A @ B @ F @ ( principal @ B @ S2 ) @ F4 )
= ( eventually @ A
@ ^ [X: A] : ( member @ B @ ( F @ X ) @ S2 )
@ F4 ) ) ).
% filterlim_principal
thf(fact_7424_eventually__all__ge__at__top,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o] :
( ( eventually @ A @ P @ ( at_top @ A ) )
=> ( eventually @ A
@ ^ [X: A] :
! [Y4: A] :
( ( ord_less_eq @ A @ X @ Y4 )
=> ( P @ Y4 ) )
@ ( at_top @ A ) ) ) ) ).
% eventually_all_ge_at_top
thf(fact_7425_filterlim__at__top,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ B )
=> ! [F: A > B,F4: filter @ A] :
( ( filterlim @ A @ B @ F @ ( at_top @ B ) @ F4 )
= ( ! [Z10: B] :
( eventually @ A
@ ^ [X: A] : ( ord_less_eq @ B @ Z10 @ ( F @ X ) )
@ F4 ) ) ) ) ).
% filterlim_at_top
thf(fact_7426_filterlim__at__top__ge,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ B )
=> ! [F: A > B,F4: filter @ A,C2: B] :
( ( filterlim @ A @ B @ F @ ( at_top @ B ) @ F4 )
= ( ! [Z10: B] :
( ( ord_less_eq @ B @ C2 @ Z10 )
=> ( eventually @ A
@ ^ [X: A] : ( ord_less_eq @ B @ Z10 @ ( F @ X ) )
@ F4 ) ) ) ) ) ).
% filterlim_at_top_ge
thf(fact_7427_filterlim__at__top__mono,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F: B > A,F4: filter @ B,G: B > A] :
( ( filterlim @ B @ A @ F @ ( at_top @ A ) @ F4 )
=> ( ( eventually @ B
@ ^ [X: B] : ( ord_less_eq @ A @ ( F @ X ) @ ( G @ X ) )
@ F4 )
=> ( filterlim @ B @ A @ G @ ( at_top @ A ) @ F4 ) ) ) ) ).
% filterlim_at_top_mono
thf(fact_7428_filterlim__at__top__dense,axiom,
! [A: $tType,B: $tType] :
( ( unboun7993243217541854897norder @ B )
=> ! [F: A > B,F4: filter @ A] :
( ( filterlim @ A @ B @ F @ ( at_top @ B ) @ F4 )
= ( ! [Z10: B] :
( eventually @ A
@ ^ [X: A] : ( ord_less @ B @ Z10 @ ( F @ X ) )
@ F4 ) ) ) ) ).
% filterlim_at_top_dense
thf(fact_7429_eventually__INF__finite,axiom,
! [B: $tType,A: $tType,A5: set @ A,P: B > $o,F4: A > ( filter @ B )] :
( ( finite_finite2 @ A @ A5 )
=> ( ( eventually @ B @ P @ ( complete_Inf_Inf @ ( filter @ B ) @ ( image2 @ A @ ( filter @ B ) @ F4 @ A5 ) ) )
= ( ? [Q8: A > B > $o] :
( ! [X: A] :
( ( member @ A @ X @ A5 )
=> ( eventually @ B @ ( Q8 @ X ) @ ( F4 @ X ) ) )
& ! [Y4: B] :
( ! [X: A] :
( ( member @ A @ X @ A5 )
=> ( Q8 @ X @ Y4 ) )
=> ( P @ Y4 ) ) ) ) ) ) ).
% eventually_INF_finite
thf(fact_7430_filterlim__at__bot,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ B )
=> ! [F: A > B,F4: filter @ A] :
( ( filterlim @ A @ B @ F @ ( at_bot @ B ) @ F4 )
= ( ! [Z10: B] :
( eventually @ A
@ ^ [X: A] : ( ord_less_eq @ B @ ( F @ X ) @ Z10 )
@ F4 ) ) ) ) ).
% filterlim_at_bot
thf(fact_7431_filterlim__at__bot__le,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ B )
=> ! [F: A > B,F4: filter @ A,C2: B] :
( ( filterlim @ A @ B @ F @ ( at_bot @ B ) @ F4 )
= ( ! [Z10: B] :
( ( ord_less_eq @ B @ Z10 @ C2 )
=> ( eventually @ A
@ ^ [X: A] : ( ord_less_eq @ B @ ( F @ X ) @ Z10 )
@ F4 ) ) ) ) ) ).
% filterlim_at_bot_le
thf(fact_7432_filterlim__at__bot__dense,axiom,
! [A: $tType,B: $tType] :
( ( ( dense_linorder @ B )
& ( no_bot @ B ) )
=> ! [F: A > B,F4: filter @ A] :
( ( filterlim @ A @ B @ F @ ( at_bot @ B ) @ F4 )
= ( ! [Z10: B] :
( eventually @ A
@ ^ [X: A] : ( ord_less @ B @ ( F @ X ) @ Z10 )
@ F4 ) ) ) ) ).
% filterlim_at_bot_dense
thf(fact_7433_filterlim__at__top__gt,axiom,
! [A: $tType,B: $tType] :
( ( unboun7993243217541854897norder @ B )
=> ! [F: A > B,F4: filter @ A,C2: B] :
( ( filterlim @ A @ B @ F @ ( at_top @ B ) @ F4 )
= ( ! [Z10: B] :
( ( ord_less @ B @ C2 @ Z10 )
=> ( eventually @ A
@ ^ [X: A] : ( ord_less_eq @ B @ Z10 @ ( F @ X ) )
@ F4 ) ) ) ) ) ).
% filterlim_at_top_gt
thf(fact_7434_eventually__INF,axiom,
! [A: $tType,B: $tType,P: A > $o,F4: B > ( filter @ A ),B5: set @ B] :
( ( eventually @ A @ P @ ( complete_Inf_Inf @ ( filter @ A ) @ ( image2 @ B @ ( filter @ A ) @ F4 @ B5 ) ) )
= ( ? [X5: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ X5 @ B5 )
& ( finite_finite2 @ B @ X5 )
& ( eventually @ A @ P @ ( complete_Inf_Inf @ ( filter @ A ) @ ( image2 @ B @ ( filter @ A ) @ F4 @ X5 ) ) ) ) ) ) ).
% eventually_INF
thf(fact_7435_filterlim__at__bot__lt,axiom,
! [A: $tType,B: $tType] :
( ( unboun7993243217541854897norder @ B )
=> ! [F: A > B,F4: filter @ A,C2: B] :
( ( filterlim @ A @ B @ F @ ( at_bot @ B ) @ F4 )
= ( ! [Z10: B] :
( ( ord_less @ B @ Z10 @ C2 )
=> ( eventually @ A
@ ^ [X: A] : ( ord_less_eq @ B @ ( F @ X ) @ Z10 )
@ F4 ) ) ) ) ) ).
% filterlim_at_bot_lt
thf(fact_7436_filterlim__finite__subsets__at__top,axiom,
! [A: $tType,B: $tType,F: A > ( set @ B ),A5: set @ B,F4: filter @ A] :
( ( filterlim @ A @ ( set @ B ) @ F @ ( finite5375528669736107172at_top @ B @ A5 ) @ F4 )
= ( ! [X5: set @ B] :
( ( ( finite_finite2 @ B @ X5 )
& ( ord_less_eq @ ( set @ B ) @ X5 @ A5 ) )
=> ( eventually @ A
@ ^ [Y4: A] :
( ( finite_finite2 @ B @ ( F @ Y4 ) )
& ( ord_less_eq @ ( set @ B ) @ X5 @ ( F @ Y4 ) )
& ( ord_less_eq @ ( set @ B ) @ ( F @ Y4 ) @ A5 ) )
@ F4 ) ) ) ) ).
% filterlim_finite_subsets_at_top
thf(fact_7437_map__filter__on__comp,axiom,
! [A: $tType,C: $tType,B: $tType,G: B > A,Y7: set @ B,X8: set @ A,F4: filter @ B,F: A > C] :
( ( ord_less_eq @ ( set @ A ) @ ( image2 @ B @ A @ G @ Y7 ) @ X8 )
=> ( ( eventually @ B
@ ^ [X: B] : ( member @ B @ X @ Y7 )
@ F4 )
=> ( ( map_filter_on @ A @ C @ X8 @ F @ ( map_filter_on @ B @ A @ Y7 @ G @ F4 ) )
= ( map_filter_on @ B @ C @ Y7 @ ( comp @ A @ C @ B @ F @ G ) @ F4 ) ) ) ) ).
% map_filter_on_comp
thf(fact_7438_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 )
@ ^ [S: set @ ( set @ A )] : ( inf_inf @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ S ) @ ( collect @ A @ ( domainp @ A @ B @ A5 ) ) )
@ ( complete_Inf_Inf @ ( set @ B ) ) ) ) ) ).
% right_total_Inter_transfer
thf(fact_7439_int_Oright__total,axiom,
right_total @ ( product_prod @ nat @ nat ) @ int @ pcr_int ).
% int.right_total
thf(fact_7440_typedef__right__total,axiom,
! [B: $tType,A: $tType,Rep: B > A,Abs: A > B,A5: set @ A,T6: A > B > $o] :
( ( type_definition @ B @ A @ Rep @ Abs @ A5 )
=> ( ( T6
= ( ^ [X: A,Y4: B] :
( X
= ( Rep @ Y4 ) ) ) )
=> ( right_total @ A @ B @ T6 ) ) ) ).
% typedef_right_total
thf(fact_7441_eventually__map__filter__on,axiom,
! [B: $tType,A: $tType,X8: set @ A,F4: filter @ A,P: B > $o,F: A > B] :
( ( eventually @ A
@ ^ [X: A] : ( member @ A @ X @ X8 )
@ F4 )
=> ( ( eventually @ B @ P @ ( map_filter_on @ A @ B @ X8 @ F @ F4 ) )
= ( eventually @ A
@ ^ [X: A] :
( ( P @ ( F @ X ) )
& ( member @ A @ X @ X8 ) )
@ F4 ) ) ) ).
% eventually_map_filter_on
thf(fact_7442_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
@ ^ [Y3: $o,Z: $o] : Y3 = Z )
@ ( bNF_rel_set @ A @ B @ A5 )
@ ^ [P4: A > $o] :
( collect @ A
@ ^ [X: A] :
( ( P4 @ X )
& ( domainp @ A @ B @ A5 @ X ) ) )
@ ( collect @ B ) ) ) ).
% right_total_Collect_transfer
thf(fact_7443_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 )
@ ^ [S: set @ A] : ( inf_inf @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ S ) @ ( collect @ A @ ( domainp @ A @ B @ A5 ) ) )
@ ( uminus_uminus @ ( set @ B ) ) ) ) ) ).
% right_total_Compl_transfer
thf(fact_7444_right__total__fun__eq__transfer,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,A5: A > B > $o,B5: C > D > $o] :
( ( right_total @ A @ B @ A5 )
=> ( ( bi_unique @ C @ D @ B5 )
=> ( bNF_rel_fun @ ( A > C ) @ ( B > D ) @ ( ( A > C ) > $o ) @ ( ( B > D ) > $o ) @ ( bNF_rel_fun @ A @ B @ C @ D @ A5 @ B5 )
@ ( bNF_rel_fun @ ( A > C ) @ ( B > D ) @ $o @ $o @ ( bNF_rel_fun @ A @ B @ C @ D @ A5 @ B5 )
@ ^ [Y3: $o,Z: $o] : Y3 = Z )
@ ^ [F2: A > C,G2: A > C] :
! [X: A] :
( ( member @ A @ X @ ( collect @ A @ ( domainp @ A @ B @ A5 ) ) )
=> ( ( F2 @ X )
= ( G2 @ X ) ) )
@ ^ [Y3: B > D,Z: B > D] : Y3 = Z ) ) ) ).
% right_total_fun_eq_transfer
thf(fact_7445_vimage__right__total__transfer,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,B5: A > B > $o,A5: C > D > $o] :
( ( bi_unique @ A @ B @ B5 )
=> ( ( 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 @ B5 ) @ ( bNF_rel_fun @ ( set @ A ) @ ( set @ B ) @ ( set @ C ) @ ( set @ D ) @ ( bNF_rel_set @ A @ B @ B5 ) @ ( bNF_rel_set @ C @ D @ A5 ) )
@ ^ [F2: C > A,X5: set @ A] : ( inf_inf @ ( set @ C ) @ ( vimage @ C @ A @ F2 @ X5 ) @ ( collect @ C @ ( domainp @ C @ D @ A5 ) ) )
@ ( vimage @ D @ B ) ) ) ) ).
% vimage_right_total_transfer
thf(fact_7446_Sup__filter__def,axiom,
! [A: $tType] :
( ( complete_Sup_Sup @ ( filter @ A ) )
= ( ^ [S: set @ ( filter @ A )] :
( abs_filter @ A
@ ^ [P4: A > $o] :
! [X: filter @ A] :
( ( member @ ( filter @ A ) @ X @ S )
=> ( eventually @ A @ P4 @ X ) ) ) ) ) ).
% Sup_filter_def
thf(fact_7447_sorted__insort__insert__key,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F: B > A,Xs: list @ B,X2: B] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F @ Xs ) )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F @ ( linord329482645794927042rt_key @ B @ A @ F @ X2 @ Xs ) ) ) ) ) ).
% sorted_insort_insert_key
thf(fact_7448_sup__filter__def,axiom,
! [A: $tType] :
( ( sup_sup @ ( filter @ A ) )
= ( ^ [F8: filter @ A,F11: filter @ A] :
( abs_filter @ A
@ ^ [P4: A > $o] :
( ( eventually @ A @ P4 @ F8 )
& ( eventually @ A @ P4 @ F11 ) ) ) ) ) ).
% sup_filter_def
thf(fact_7449_bot__filter__def,axiom,
! [A: $tType] :
( ( bot_bot @ ( filter @ A ) )
= ( abs_filter @ A
@ ^ [P4: A > $o] : $true ) ) ).
% bot_filter_def
thf(fact_7450_top__filter__def,axiom,
! [A: $tType] :
( ( top_top @ ( filter @ A ) )
= ( abs_filter @ A
@ ^ [P5: A > $o] :
! [X6: A] : ( P5 @ X6 ) ) ) ).
% top_filter_def
thf(fact_7451_insort__insert__triv,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Xs: list @ A] :
( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
=> ( ( linord329482645794927042rt_key @ A @ A
@ ^ [X: A] : X
@ X2
@ Xs )
= Xs ) ) ) ).
% insort_insert_triv
thf(fact_7452_principal__def,axiom,
! [A: $tType] :
( ( principal @ A )
= ( ^ [S: set @ A] : ( abs_filter @ A @ ( ball @ A @ S ) ) ) ) ).
% principal_def
thf(fact_7453_map__filter__on__def,axiom,
! [B: $tType,A: $tType] :
( ( map_filter_on @ A @ B )
= ( ^ [X5: set @ A,F2: A > B,F8: filter @ A] :
( abs_filter @ B
@ ^ [P4: B > $o] :
( eventually @ A
@ ^ [X: A] :
( ( P4 @ ( F2 @ X ) )
& ( member @ A @ X @ X5 ) )
@ F8 ) ) ) ) ).
% map_filter_on_def
thf(fact_7454_set__insort__insert,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Xs: list @ A] :
( ( set2 @ A
@ ( linord329482645794927042rt_key @ A @ A
@ ^ [X: A] : X
@ X2
@ Xs ) )
= ( insert @ A @ X2 @ ( set2 @ A @ Xs ) ) ) ) ).
% set_insort_insert
thf(fact_7455_sorted__insort__insert,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,X2: A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A )
@ ( linord329482645794927042rt_key @ A @ A
@ ^ [X: A] : X
@ X2
@ Xs ) ) ) ) ).
% sorted_insort_insert
thf(fact_7456_insort__insert__insort,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Xs: list @ A] :
( ~ ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
=> ( ( linord329482645794927042rt_key @ A @ A
@ ^ [X: A] : X
@ X2
@ Xs )
= ( linorder_insort_key @ A @ A
@ ^ [X: A] : X
@ X2
@ Xs ) ) ) ) ).
% insort_insert_insort
thf(fact_7457_inf__filter__def,axiom,
! [A: $tType] :
( ( inf_inf @ ( filter @ A ) )
= ( ^ [F8: filter @ A,F11: filter @ A] :
( abs_filter @ A
@ ^ [P4: A > $o] :
? [Q8: A > $o,R7: A > $o] :
( ( eventually @ A @ Q8 @ F8 )
& ( eventually @ A @ R7 @ F11 )
& ! [X: A] :
( ( ( Q8 @ X )
& ( R7 @ X ) )
=> ( P4 @ X ) ) ) ) ) ) ).
% inf_filter_def
thf(fact_7458_subset__mset_OcINF__inf__distrib,axiom,
! [A: $tType,B: $tType,A5: set @ B,F: B > ( multiset @ A ),G: B > ( multiset @ A )] :
( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F @ A5 ) )
=> ( ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ G @ A5 ) )
=> ( ( inter_mset @ A @ ( complete_Inf_Inf @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F @ A5 ) ) @ ( complete_Inf_Inf @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ G @ A5 ) ) )
= ( complete_Inf_Inf @ ( multiset @ A )
@ ( image2 @ B @ ( multiset @ A )
@ ^ [A3: B] : ( inter_mset @ A @ ( F @ A3 ) @ ( G @ A3 ) )
@ A5 ) ) ) ) ) ) ).
% subset_mset.cINF_inf_distrib
thf(fact_7459_admissible__chfin,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [P: A > $o] :
( ! [S8: set @ A] :
( ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ S8 )
=> ( finite_finite2 @ A @ S8 ) )
=> ( comple1908693960933563346ssible @ A @ ( complete_Sup_Sup @ A ) @ ( ord_less_eq @ A ) @ P ) ) ) ).
% admissible_chfin
thf(fact_7460_subset__mset_Obdd__below__image__inf,axiom,
! [A: $tType,B: $tType,F: B > ( multiset @ A ),G: B > ( multiset @ A ),A5: set @ B] :
( ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A )
@ ( image2 @ B @ ( multiset @ A )
@ ^ [X: B] : ( inter_mset @ A @ ( F @ X ) @ ( G @ X ) )
@ A5 ) )
= ( ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F @ A5 ) )
& ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ G @ A5 ) ) ) ) ).
% subset_mset.bdd_below_image_inf
thf(fact_7461_admissible__True,axiom,
! [A: $tType,Lub: ( set @ A ) > A,Ord: A > A > $o] :
( comple1908693960933563346ssible @ A @ Lub @ Ord
@ ^ [X: A] : $true ) ).
% admissible_True
thf(fact_7462_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
@ ^ [X: A] :
( ( P @ X )
& ( Q @ X ) ) ) ) ) ).
% admissible_conj
thf(fact_7463_admissible__const,axiom,
! [A: $tType,Lub: ( set @ A ) > A,Ord: A > A > $o,T2: $o] :
( comple1908693960933563346ssible @ A @ Lub @ Ord
@ ^ [X: A] : T2 ) ).
% admissible_const
thf(fact_7464_admissible__all,axiom,
! [A: $tType,B: $tType,Lub: ( set @ B ) > B,Ord: B > B > $o,P: B > A > $o] :
( ! [Y2: A] :
( comple1908693960933563346ssible @ B @ Lub @ Ord
@ ^ [X: B] : ( P @ X @ Y2 ) )
=> ( comple1908693960933563346ssible @ B @ Lub @ Ord
@ ^ [X: B] :
! [X5: A] : ( P @ X @ X5 ) ) ) ).
% admissible_all
thf(fact_7465_inter__subset__eq__union,axiom,
! [A: $tType,A5: multiset @ A,B5: multiset @ A] : ( subseteq_mset @ A @ ( inter_mset @ A @ A5 @ B5 ) @ ( plus_plus @ ( multiset @ A ) @ A5 @ B5 ) ) ).
% inter_subset_eq_union
thf(fact_7466_inter__union__distrib__left,axiom,
! [A: $tType,A5: multiset @ A,B5: multiset @ A,C6: multiset @ A] :
( ( plus_plus @ ( multiset @ A ) @ ( inter_mset @ A @ A5 @ B5 ) @ C6 )
= ( inter_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A5 @ C6 ) @ ( plus_plus @ ( multiset @ A ) @ B5 @ C6 ) ) ) ).
% inter_union_distrib_left
thf(fact_7467_inter__union__distrib__right,axiom,
! [A: $tType,C6: multiset @ A,A5: multiset @ A,B5: multiset @ A] :
( ( plus_plus @ ( multiset @ A ) @ C6 @ ( inter_mset @ A @ A5 @ B5 ) )
= ( inter_mset @ A @ ( plus_plus @ ( multiset @ A ) @ C6 @ A5 ) @ ( plus_plus @ ( multiset @ A ) @ C6 @ B5 ) ) ) ).
% inter_union_distrib_right
thf(fact_7468_inter__mset__empty__distrib__right,axiom,
! [A: $tType,A5: multiset @ A,B5: multiset @ A,C6: multiset @ A] :
( ( ( inter_mset @ A @ A5 @ ( plus_plus @ ( multiset @ A ) @ B5 @ C6 ) )
= ( zero_zero @ ( multiset @ A ) ) )
= ( ( ( inter_mset @ A @ A5 @ B5 )
= ( zero_zero @ ( multiset @ A ) ) )
& ( ( inter_mset @ A @ A5 @ C6 )
= ( zero_zero @ ( multiset @ A ) ) ) ) ) ).
% inter_mset_empty_distrib_right
thf(fact_7469_inter__mset__empty__distrib__left,axiom,
! [A: $tType,A5: multiset @ A,B5: multiset @ A,C6: multiset @ A] :
( ( ( inter_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A5 @ B5 ) @ C6 )
= ( zero_zero @ ( multiset @ A ) ) )
= ( ( ( inter_mset @ A @ A5 @ C6 )
= ( zero_zero @ ( multiset @ A ) ) )
& ( ( inter_mset @ A @ B5 @ C6 )
= ( zero_zero @ ( multiset @ A ) ) ) ) ) ).
% inter_mset_empty_distrib_left
thf(fact_7470_admissible__ball,axiom,
! [A: $tType,B: $tType,A5: set @ A,Lub: ( set @ B ) > B,Ord: B > B > $o,P: B > A > $o] :
( ! [Y2: A] :
( ( member @ A @ Y2 @ A5 )
=> ( comple1908693960933563346ssible @ B @ Lub @ Ord
@ ^ [X: B] : ( P @ X @ Y2 ) ) )
=> ( comple1908693960933563346ssible @ B @ Lub @ Ord
@ ^ [X: B] :
! [Y4: A] :
( ( member @ A @ Y4 @ A5 )
=> ( P @ X @ Y4 ) ) ) ) ).
% admissible_ball
thf(fact_7471_multiset__union__diff__commute,axiom,
! [A: $tType,B5: multiset @ A,C6: multiset @ A,A5: multiset @ A] :
( ( ( inter_mset @ A @ B5 @ C6 )
= ( zero_zero @ ( multiset @ A ) ) )
=> ( ( minus_minus @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ A5 @ B5 ) @ C6 )
= ( plus_plus @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ A5 @ C6 ) @ B5 ) ) ) ).
% multiset_union_diff_commute
thf(fact_7472_size__Diff__subset__Int,axiom,
! [A: $tType,M3: multiset @ A,M9: multiset @ A] :
( ( size_size @ ( multiset @ A ) @ ( minus_minus @ ( multiset @ A ) @ M3 @ M9 ) )
= ( minus_minus @ nat @ ( size_size @ ( multiset @ A ) @ M3 ) @ ( size_size @ ( multiset @ A ) @ ( inter_mset @ A @ M3 @ M9 ) ) ) ) ).
% size_Diff_subset_Int
thf(fact_7473_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 )
@ ^ [X: A] :
( ( P @ X )
| ( Q @ X ) ) ) ) ) ) ).
% admissible_disj
thf(fact_7474_subset__mset_Omono__inf,axiom,
! [B: $tType,A: $tType] :
( ( semilattice_inf @ B )
=> ! [F: ( multiset @ A ) > B,A5: multiset @ A,B5: multiset @ A] :
( ( mono @ ( multiset @ A ) @ B @ ( subseteq_mset @ A ) @ F )
=> ( ord_less_eq @ B @ ( F @ ( inter_mset @ A @ A5 @ B5 ) ) @ ( inf_inf @ B @ ( F @ A5 ) @ ( F @ B5 ) ) ) ) ) ).
% subset_mset.mono_inf
thf(fact_7475_filtercomap__def,axiom,
! [B: $tType,A: $tType] :
( ( filtercomap @ A @ B )
= ( ^ [F2: A > B,F8: filter @ B] :
( abs_filter @ A
@ ^ [P4: A > $o] :
? [Q8: B > $o] :
( ( eventually @ B @ Q8 @ F8 )
& ! [X: A] :
( ( Q8 @ ( F2 @ X ) )
=> ( P4 @ X ) ) ) ) ) ) ).
% filtercomap_def
thf(fact_7476_multp__def,axiom,
! [A: $tType] :
( ( multp @ A )
= ( ^ [R2: A > A > $o,M6: multiset @ A,N6: multiset @ A] : ( member @ ( product_prod @ ( multiset @ A ) @ ( multiset @ A ) ) @ ( product_Pair @ ( multiset @ A ) @ ( multiset @ A ) @ M6 @ N6 ) @ ( mult @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ R2 ) ) ) ) ) ) ).
% multp_def
thf(fact_7477_eventually__filtercomapI,axiom,
! [B: $tType,A: $tType,P: A > $o,F4: filter @ A,F: B > A] :
( ( eventually @ A @ P @ F4 )
=> ( eventually @ B
@ ^ [X: B] : ( P @ ( F @ X ) )
@ ( filtercomap @ B @ A @ F @ F4 ) ) ) ).
% eventually_filtercomapI
thf(fact_7478_filtercomap__filtercomap,axiom,
! [A: $tType,B: $tType,C: $tType,F: A > B,G: B > C,F4: filter @ C] :
( ( filtercomap @ A @ B @ F @ ( filtercomap @ B @ C @ G @ F4 ) )
= ( filtercomap @ A @ C
@ ^ [X: A] : ( G @ ( F @ X ) )
@ F4 ) ) ).
% filtercomap_filtercomap
thf(fact_7479_filtercomap__ident,axiom,
! [A: $tType,F4: filter @ A] :
( ( filtercomap @ A @ A
@ ^ [X: A] : X
@ F4 )
= F4 ) ).
% filtercomap_ident
thf(fact_7480_filtercomap__INF,axiom,
! [A: $tType,B: $tType,C: $tType,F: A > B,F4: C > ( filter @ B ),B5: set @ C] :
( ( filtercomap @ A @ B @ F @ ( complete_Inf_Inf @ ( filter @ B ) @ ( image2 @ C @ ( filter @ B ) @ F4 @ B5 ) ) )
= ( complete_Inf_Inf @ ( filter @ A )
@ ( image2 @ C @ ( filter @ A )
@ ^ [B3: C] : ( filtercomap @ A @ B @ F @ ( F4 @ B3 ) )
@ B5 ) ) ) ).
% filtercomap_INF
thf(fact_7481_eventually__filtercomap__at__top__linorder,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ A )
=> ! [P: B > $o,F: B > A] :
( ( eventually @ B @ P @ ( filtercomap @ B @ A @ F @ ( at_top @ A ) ) )
= ( ? [N6: A] :
! [X: B] :
( ( ord_less_eq @ A @ N6 @ ( F @ X ) )
=> ( P @ X ) ) ) ) ) ).
% eventually_filtercomap_at_top_linorder
thf(fact_7482_eventually__filtercomap__at__bot__linorder,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ A )
=> ! [P: B > $o,F: B > A] :
( ( eventually @ B @ P @ ( filtercomap @ B @ A @ F @ ( at_bot @ A ) ) )
= ( ? [N6: A] :
! [X: B] :
( ( ord_less_eq @ A @ ( F @ X ) @ N6 )
=> ( P @ X ) ) ) ) ) ).
% eventually_filtercomap_at_bot_linorder
thf(fact_7483_one__step__implies__multp,axiom,
! [A: $tType,J4: multiset @ A,K3: multiset @ A,R: A > A > $o,I5: multiset @ A] :
( ( J4
!= ( zero_zero @ ( multiset @ A ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ ( set_mset @ A @ K3 ) )
=> ? [Xa2: A] :
( ( member @ A @ Xa2 @ ( set_mset @ A @ J4 ) )
& ( R @ X3 @ Xa2 ) ) )
=> ( multp @ A @ R @ ( plus_plus @ ( multiset @ A ) @ I5 @ K3 ) @ ( plus_plus @ ( multiset @ A ) @ I5 @ J4 ) ) ) ) ).
% one_step_implies_multp
thf(fact_7484_filtercomap__SUP,axiom,
! [A: $tType,C: $tType,B: $tType,F: A > C,F4: B > ( filter @ C ),B5: set @ B] :
( ord_less_eq @ ( filter @ A )
@ ( complete_Sup_Sup @ ( filter @ A )
@ ( image2 @ B @ ( filter @ A )
@ ^ [B3: B] : ( filtercomap @ A @ C @ F @ ( F4 @ B3 ) )
@ B5 ) )
@ ( filtercomap @ A @ C @ F @ ( complete_Sup_Sup @ ( filter @ C ) @ ( image2 @ B @ ( filter @ C ) @ F4 @ B5 ) ) ) ) ).
% filtercomap_SUP
thf(fact_7485_pair__lessI2,axiom,
! [A2: nat,B2: nat,S3: nat,T2: nat] :
( ( ord_less_eq @ nat @ A2 @ B2 )
=> ( ( ord_less @ nat @ S3 @ T2 )
=> ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ A2 @ S3 ) @ ( product_Pair @ nat @ nat @ B2 @ T2 ) ) @ fun_pair_less ) ) ) ).
% pair_lessI2
thf(fact_7486_subset__mset_OSUP__sup__distrib,axiom,
! [A: $tType,B: $tType,A5: set @ B,F: B > ( multiset @ A ),G: B > ( multiset @ A )] :
( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F @ A5 ) )
=> ( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ G @ A5 ) )
=> ( ( union_mset @ A @ ( complete_Sup_Sup @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F @ A5 ) ) @ ( complete_Sup_Sup @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ G @ A5 ) ) )
= ( complete_Sup_Sup @ ( multiset @ A )
@ ( image2 @ B @ ( multiset @ A )
@ ^ [A3: B] : ( union_mset @ A @ ( F @ A3 ) @ ( G @ A3 ) )
@ A5 ) ) ) ) ) ) ).
% subset_mset.SUP_sup_distrib
thf(fact_7487_subset__mset_Obdd__above__image__sup,axiom,
! [A: $tType,B: $tType,F: B > ( multiset @ A ),G: B > ( multiset @ A ),A5: set @ B] :
( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A )
@ ( image2 @ B @ ( multiset @ A )
@ ^ [X: B] : ( union_mset @ A @ ( F @ X ) @ ( G @ X ) )
@ A5 ) )
= ( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F @ A5 ) )
& ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ G @ A5 ) ) ) ) ).
% subset_mset.bdd_above_image_sup
thf(fact_7488_size__Un__Int,axiom,
! [A: $tType,A5: multiset @ A,B5: multiset @ A] :
( ( plus_plus @ nat @ ( size_size @ ( multiset @ A ) @ A5 ) @ ( size_size @ ( multiset @ A ) @ B5 ) )
= ( plus_plus @ nat @ ( size_size @ ( multiset @ A ) @ ( union_mset @ A @ A5 @ B5 ) ) @ ( size_size @ ( multiset @ A ) @ ( inter_mset @ A @ A5 @ B5 ) ) ) ) ).
% size_Un_Int
thf(fact_7489_union__mset__def,axiom,
! [A: $tType] :
( ( union_mset @ A )
= ( ^ [A8: multiset @ A,B8: multiset @ A] : ( plus_plus @ ( multiset @ A ) @ A8 @ ( minus_minus @ ( multiset @ A ) @ B8 @ A8 ) ) ) ) ).
% union_mset_def
thf(fact_7490_sup__union__distrib__left,axiom,
! [A: $tType,A5: multiset @ A,B5: multiset @ A,C6: multiset @ A] :
( ( plus_plus @ ( multiset @ A ) @ ( union_mset @ A @ A5 @ B5 ) @ C6 )
= ( union_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A5 @ C6 ) @ ( plus_plus @ ( multiset @ A ) @ B5 @ C6 ) ) ) ).
% sup_union_distrib_left
thf(fact_7491_union__sup__distrib__right,axiom,
! [A: $tType,C6: multiset @ A,A5: multiset @ A,B5: multiset @ A] :
( ( plus_plus @ ( multiset @ A ) @ C6 @ ( union_mset @ A @ A5 @ B5 ) )
= ( union_mset @ A @ ( plus_plus @ ( multiset @ A ) @ C6 @ A5 ) @ ( plus_plus @ ( multiset @ A ) @ C6 @ B5 ) ) ) ).
% union_sup_distrib_right
thf(fact_7492_union__diff__inter__eq__sup,axiom,
! [A: $tType,A5: multiset @ A,B5: multiset @ A] :
( ( minus_minus @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ A5 @ B5 ) @ ( inter_mset @ A @ A5 @ B5 ) )
= ( union_mset @ A @ A5 @ B5 ) ) ).
% union_diff_inter_eq_sup
thf(fact_7493_union__diff__sup__eq__inter,axiom,
! [A: $tType,A5: multiset @ A,B5: multiset @ A] :
( ( minus_minus @ ( multiset @ A ) @ ( plus_plus @ ( multiset @ A ) @ A5 @ B5 ) @ ( union_mset @ A @ A5 @ B5 ) )
= ( inter_mset @ A @ A5 @ B5 ) ) ).
% union_diff_sup_eq_inter
thf(fact_7494_size__Un__disjoint,axiom,
! [A: $tType,A5: multiset @ A,B5: multiset @ A] :
( ( ( inter_mset @ A @ A5 @ B5 )
= ( zero_zero @ ( multiset @ A ) ) )
=> ( ( size_size @ ( multiset @ A ) @ ( union_mset @ A @ A5 @ B5 ) )
= ( plus_plus @ nat @ ( size_size @ ( multiset @ A ) @ A5 ) @ ( size_size @ ( multiset @ A ) @ B5 ) ) ) ) ).
% size_Un_disjoint
thf(fact_7495_subset__mset_Omono__sup,axiom,
! [B: $tType,A: $tType] :
( ( semilattice_sup @ B )
=> ! [F: ( multiset @ A ) > B,A5: multiset @ A,B5: multiset @ A] :
( ( mono @ ( multiset @ A ) @ B @ ( subseteq_mset @ A ) @ F )
=> ( ord_less_eq @ B @ ( sup_sup @ B @ ( F @ A5 ) @ ( F @ B5 ) ) @ ( F @ ( union_mset @ A @ A5 @ B5 ) ) ) ) ) ).
% subset_mset.mono_sup
thf(fact_7496_sum__mset_Ounion__disjoint,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A5: multiset @ B,B5: multiset @ B,G: B > A] :
( ( ( inter_mset @ B @ A5 @ B5 )
= ( zero_zero @ ( multiset @ B ) ) )
=> ( ( comm_m7189776963980413722m_mset @ A @ ( image_mset @ B @ A @ G @ ( union_mset @ B @ A5 @ B5 ) ) )
= ( plus_plus @ A @ ( comm_m7189776963980413722m_mset @ A @ ( image_mset @ B @ A @ G @ A5 ) ) @ ( comm_m7189776963980413722m_mset @ A @ ( image_mset @ B @ A @ G @ B5 ) ) ) ) ) ) ).
% sum_mset.union_disjoint
thf(fact_7497_pair__leqI2,axiom,
! [A2: nat,B2: nat,S3: nat,T2: nat] :
( ( ord_less_eq @ nat @ A2 @ B2 )
=> ( ( ord_less_eq @ nat @ S3 @ T2 )
=> ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ A2 @ S3 ) @ ( product_Pair @ nat @ nat @ B2 @ T2 ) ) @ fun_pair_leq ) ) ) ).
% pair_leqI2
thf(fact_7498_euclidean__size__times__nonunit,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [A2: A,B2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( ~ ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
=> ( ord_less @ nat @ ( euclid6346220572633701492n_size @ A @ B2 ) @ ( euclid6346220572633701492n_size @ A @ ( times_times @ A @ A2 @ B2 ) ) ) ) ) ) ) ).
% euclidean_size_times_nonunit
thf(fact_7499_euclidean__size__integer_Orep__eq,axiom,
( ( euclid6346220572633701492n_size @ code_integer )
= ( ^ [X: code_integer] : ( euclid6346220572633701492n_size @ int @ ( code_int_of_integer @ X ) ) ) ) ).
% euclidean_size_integer.rep_eq
thf(fact_7500_euclidean__size__of__nat,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [N: nat] :
( ( euclid6346220572633701492n_size @ A @ ( semiring_1_of_nat @ A @ N ) )
= N ) ) ).
% euclidean_size_of_nat
thf(fact_7501_size__0,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ( ( euclid6346220572633701492n_size @ A @ ( zero_zero @ A ) )
= ( zero_zero @ nat ) ) ) ).
% size_0
thf(fact_7502_euclidean__size__eq__0__iff,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [B2: A] :
( ( ( euclid6346220572633701492n_size @ A @ B2 )
= ( zero_zero @ nat ) )
= ( B2
= ( zero_zero @ A ) ) ) ) ).
% euclidean_size_eq_0_iff
thf(fact_7503_euclidean__size__1,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ( ( euclid6346220572633701492n_size @ A @ ( one_one @ A ) )
= ( one_one @ nat ) ) ) ).
% euclidean_size_1
thf(fact_7504_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_7505_euclidean__size__integer__def,axiom,
( ( euclid6346220572633701492n_size @ code_integer )
= ( map_fun @ code_integer @ int @ nat @ nat @ code_int_of_integer @ ( id @ nat ) @ ( euclid6346220572633701492n_size @ int ) ) ) ).
% euclidean_size_integer_def
thf(fact_7506_dvd__euclidean__size__eq__imp__dvd,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [A2: A,B2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( ( euclid6346220572633701492n_size @ A @ A2 )
= ( euclid6346220572633701492n_size @ A @ B2 ) )
=> ( ( dvd_dvd @ A @ B2 @ A2 )
=> ( dvd_dvd @ A @ A2 @ B2 ) ) ) ) ) ).
% dvd_euclidean_size_eq_imp_dvd
thf(fact_7507_euclidean__size__unit,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [A2: A] :
( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
=> ( ( euclid6346220572633701492n_size @ A @ A2 )
= ( euclid6346220572633701492n_size @ A @ ( one_one @ A ) ) ) ) ) ).
% euclidean_size_unit
thf(fact_7508_euclidean__size__integer_Oabs__eq,axiom,
! [X2: int] :
( ( euclid6346220572633701492n_size @ code_integer @ ( code_integer_of_int @ X2 ) )
= ( euclid6346220572633701492n_size @ int @ X2 ) ) ).
% euclidean_size_integer.abs_eq
thf(fact_7509_euclidean__size__natural_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ nat @ nat
@ ^ [Y3: nat,Z: nat] : Y3 = Z
@ ^ [Y3: nat,Z: nat] : Y3 = Z
@ ( euclid6346220572633701492n_size @ nat )
@ ( euclid6346220572633701492n_size @ nat ) ) ).
% euclidean_size_natural.rsp
thf(fact_7510_euclidean__size__integer_Orsp,axiom,
( bNF_rel_fun @ int @ int @ nat @ nat
@ ^ [Y3: int,Z: int] : Y3 = Z
@ ^ [Y3: nat,Z: nat] : Y3 = Z
@ ( euclid6346220572633701492n_size @ int )
@ ( euclid6346220572633701492n_size @ int ) ) ).
% euclidean_size_integer.rsp
thf(fact_7511_unit__iff__euclidean__size,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [A2: A] :
( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
= ( ( ( euclid6346220572633701492n_size @ A @ A2 )
= ( euclid6346220572633701492n_size @ A @ ( one_one @ A ) ) )
& ( A2
!= ( zero_zero @ A ) ) ) ) ) ).
% unit_iff_euclidean_size
thf(fact_7512_size__mult__mono,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [B2: A,A2: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ord_less_eq @ nat @ ( euclid6346220572633701492n_size @ A @ A2 ) @ ( euclid6346220572633701492n_size @ A @ ( times_times @ A @ A2 @ B2 ) ) ) ) ) ).
% size_mult_mono
thf(fact_7513_size__mult__mono_H,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [B2: A,A2: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ord_less_eq @ nat @ ( euclid6346220572633701492n_size @ A @ A2 ) @ ( euclid6346220572633701492n_size @ A @ ( times_times @ A @ B2 @ A2 ) ) ) ) ) ).
% size_mult_mono'
thf(fact_7514_euclidean__size__times__unit,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [A2: A,B2: A] :
( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
=> ( ( euclid6346220572633701492n_size @ A @ ( times_times @ A @ A2 @ B2 ) )
= ( euclid6346220572633701492n_size @ A @ B2 ) ) ) ) ).
% euclidean_size_times_unit
thf(fact_7515_dvd__proper__imp__size__less,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [A2: A,B2: A] :
( ( dvd_dvd @ A @ A2 @ B2 )
=> ( ~ ( dvd_dvd @ A @ B2 @ A2 )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( ord_less @ nat @ ( euclid6346220572633701492n_size @ A @ A2 ) @ ( euclid6346220572633701492n_size @ A @ B2 ) ) ) ) ) ) ).
% dvd_proper_imp_size_less
thf(fact_7516_dvd__imp__size__le,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [A2: A,B2: A] :
( ( dvd_dvd @ A @ A2 @ B2 )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( ord_less_eq @ nat @ ( euclid6346220572633701492n_size @ A @ A2 ) @ ( euclid6346220572633701492n_size @ A @ B2 ) ) ) ) ) ).
% dvd_imp_size_le
thf(fact_7517_mod__size__less,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [B2: A,A2: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ord_less @ nat @ ( euclid6346220572633701492n_size @ A @ ( modulo_modulo @ A @ A2 @ B2 ) ) @ ( euclid6346220572633701492n_size @ A @ B2 ) ) ) ) ).
% mod_size_less
thf(fact_7518_divmod__cases,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [B2: A,A2: A] :
( ( ( B2
!= ( zero_zero @ A ) )
=> ( ( ( modulo_modulo @ A @ A2 @ B2 )
= ( zero_zero @ A ) )
=> ( A2
!= ( times_times @ A @ ( divide_divide @ A @ A2 @ B2 ) @ B2 ) ) ) )
=> ( ( ( B2
!= ( zero_zero @ A ) )
=> ! [Q4: A,R6: A] :
( ( ( euclid7384307370059645450egment @ A @ R6 )
= ( euclid7384307370059645450egment @ A @ B2 ) )
=> ( ( ord_less @ nat @ ( euclid6346220572633701492n_size @ A @ R6 ) @ ( euclid6346220572633701492n_size @ A @ B2 ) )
=> ( ( R6
!= ( zero_zero @ A ) )
=> ( ( ( divide_divide @ A @ A2 @ B2 )
= Q4 )
=> ( ( ( modulo_modulo @ A @ A2 @ B2 )
= R6 )
=> ( A2
!= ( plus_plus @ A @ ( times_times @ A @ Q4 @ B2 ) @ R6 ) ) ) ) ) ) ) )
=> ( B2
= ( zero_zero @ A ) ) ) ) ) ).
% divmod_cases
thf(fact_7519_mod__eqI,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [B2: A,R: A,Q2: A,A2: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( ( euclid7384307370059645450egment @ A @ R )
= ( euclid7384307370059645450egment @ A @ B2 ) )
=> ( ( ord_less @ nat @ ( euclid6346220572633701492n_size @ A @ R ) @ ( euclid6346220572633701492n_size @ A @ B2 ) )
=> ( ( ( plus_plus @ A @ ( times_times @ A @ Q2 @ B2 ) @ R )
= A2 )
=> ( ( modulo_modulo @ A @ A2 @ B2 )
= R ) ) ) ) ) ) ).
% mod_eqI
thf(fact_7520_division__segment__integer_Orep__eq,axiom,
! [X2: code_integer] :
( ( code_int_of_integer @ ( euclid7384307370059645450egment @ code_integer @ X2 ) )
= ( euclid7384307370059645450egment @ int @ ( code_int_of_integer @ X2 ) ) ) ).
% division_segment_integer.rep_eq
thf(fact_7521_abs__division__segment,axiom,
! [K: int] :
( ( abs_abs @ int @ ( euclid7384307370059645450egment @ int @ K ) )
= ( one_one @ int ) ) ).
% abs_division_segment
thf(fact_7522_division__segment__1,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ( ( euclid7384307370059645450egment @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% division_segment_1
thf(fact_7523_division__segment__numeral,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [K: num] :
( ( euclid7384307370059645450egment @ A @ ( numeral_numeral @ A @ K ) )
= ( one_one @ A ) ) ) ).
% division_segment_numeral
thf(fact_7524_division__segment__of__nat,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [N: nat] :
( ( euclid7384307370059645450egment @ A @ ( semiring_1_of_nat @ A @ N ) )
= ( one_one @ A ) ) ) ).
% division_segment_of_nat
thf(fact_7525_division__segment__euclidean__size,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A2: A] :
( ( times_times @ A @ ( euclid7384307370059645450egment @ A @ A2 ) @ ( semiring_1_of_nat @ A @ ( euclid6346220572633701492n_size @ A @ A2 ) ) )
= A2 ) ) ).
% division_segment_euclidean_size
thf(fact_7526_division__segment__integer_Orsp,axiom,
( bNF_rel_fun @ int @ int @ int @ int
@ ^ [Y3: int,Z: int] : Y3 = Z
@ ^ [Y3: int,Z: int] : Y3 = Z
@ ( euclid7384307370059645450egment @ int )
@ ( euclid7384307370059645450egment @ int ) ) ).
% division_segment_integer.rsp
thf(fact_7527_division__segment__natural_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ nat @ nat
@ ^ [Y3: nat,Z: nat] : Y3 = Z
@ ^ [Y3: nat,Z: nat] : Y3 = Z
@ ( euclid7384307370059645450egment @ nat )
@ ( euclid7384307370059645450egment @ nat ) ) ).
% division_segment_natural.rsp
thf(fact_7528_is__unit__division__segment,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [A2: A] : ( dvd_dvd @ A @ ( euclid7384307370059645450egment @ A @ A2 ) @ ( one_one @ A ) ) ) ).
% is_unit_division_segment
thf(fact_7529_division__segment__nat__def,axiom,
( ( euclid7384307370059645450egment @ nat )
= ( ^ [N4: nat] : ( one_one @ nat ) ) ) ).
% division_segment_nat_def
thf(fact_7530_division__segment__not__0,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [A2: A] :
( ( euclid7384307370059645450egment @ A @ A2 )
!= ( zero_zero @ A ) ) ) ).
% division_segment_not_0
thf(fact_7531_division__segment__integer_Oabs__eq,axiom,
! [X2: int] :
( ( euclid7384307370059645450egment @ code_integer @ ( code_integer_of_int @ X2 ) )
= ( code_integer_of_int @ ( euclid7384307370059645450egment @ int @ X2 ) ) ) ).
% division_segment_integer.abs_eq
thf(fact_7532_division__segment__mult,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [A2: A,B2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( ( euclid7384307370059645450egment @ A @ ( times_times @ A @ A2 @ B2 ) )
= ( times_times @ A @ ( euclid7384307370059645450egment @ A @ A2 ) @ ( euclid7384307370059645450egment @ A @ B2 ) ) ) ) ) ) ).
% division_segment_mult
thf(fact_7533_division__segment__eq__sgn,axiom,
! [K: int] :
( ( K
!= ( zero_zero @ int ) )
=> ( ( euclid7384307370059645450egment @ int @ K )
= ( sgn_sgn @ int @ K ) ) ) ).
% division_segment_eq_sgn
thf(fact_7534_division__segment__integer__def,axiom,
( ( euclid7384307370059645450egment @ code_integer )
= ( map_fun @ code_integer @ int @ int @ code_integer @ code_int_of_integer @ code_integer_of_int @ ( euclid7384307370059645450egment @ int ) ) ) ).
% division_segment_integer_def
thf(fact_7535_division__segment__mod,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [B2: A,A2: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ~ ( dvd_dvd @ A @ B2 @ A2 )
=> ( ( euclid7384307370059645450egment @ A @ ( modulo_modulo @ A @ A2 @ B2 ) )
= ( euclid7384307370059645450egment @ A @ B2 ) ) ) ) ) ).
% division_segment_mod
thf(fact_7536_of__nat__euclidean__size,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A2: A] :
( ( semiring_1_of_nat @ A @ ( euclid6346220572633701492n_size @ A @ A2 ) )
= ( divide_divide @ A @ A2 @ ( euclid7384307370059645450egment @ A @ A2 ) ) ) ) ).
% of_nat_euclidean_size
thf(fact_7537_division__segment__int__def,axiom,
( ( euclid7384307370059645450egment @ int )
= ( ^ [K2: int] : ( if @ int @ ( ord_less_eq @ int @ ( zero_zero @ int ) @ K2 ) @ ( one_one @ int ) @ ( uminus_uminus @ int @ ( one_one @ int ) ) ) ) ) ).
% division_segment_int_def
thf(fact_7538_unique__euclidean__semiring__class_Odiv__eq__0__iff,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [A2: A,B2: A] :
( ( ( euclid7384307370059645450egment @ A @ A2 )
= ( euclid7384307370059645450egment @ A @ B2 ) )
=> ( ( ( divide_divide @ A @ A2 @ B2 )
= ( zero_zero @ A ) )
= ( ( ord_less @ nat @ ( euclid6346220572633701492n_size @ A @ A2 ) @ ( euclid6346220572633701492n_size @ A @ B2 ) )
| ( B2
= ( zero_zero @ A ) ) ) ) ) ) ).
% unique_euclidean_semiring_class.div_eq_0_iff
thf(fact_7539_div__eqI,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [B2: A,R: A,Q2: A,A2: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( ( euclid7384307370059645450egment @ A @ R )
= ( euclid7384307370059645450egment @ A @ B2 ) )
=> ( ( ord_less @ nat @ ( euclid6346220572633701492n_size @ A @ R ) @ ( euclid6346220572633701492n_size @ A @ B2 ) )
=> ( ( ( plus_plus @ A @ ( times_times @ A @ Q2 @ B2 ) @ R )
= A2 )
=> ( ( divide_divide @ A @ A2 @ B2 )
= Q2 ) ) ) ) ) ) ).
% div_eqI
thf(fact_7540_div__bounded,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [B2: A,R: A,Q2: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( ( euclid7384307370059645450egment @ A @ R )
= ( euclid7384307370059645450egment @ A @ B2 ) )
=> ( ( ord_less @ nat @ ( euclid6346220572633701492n_size @ A @ R ) @ ( euclid6346220572633701492n_size @ A @ B2 ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ Q2 @ B2 ) @ R ) @ B2 )
= Q2 ) ) ) ) ) ).
% div_bounded
thf(fact_7541_ran__add,axiom,
! [B: $tType,A: $tType,F: A > ( option @ B ),G: A > ( option @ B )] :
( ( ( inf_inf @ ( set @ A ) @ ( dom @ A @ B @ F ) @ ( dom @ A @ B @ G ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( ran @ A @ B @ ( map_add @ A @ B @ F @ G ) )
= ( sup_sup @ ( set @ B ) @ ( ran @ A @ B @ F ) @ ( ran @ A @ B @ G ) ) ) ) ).
% ran_add
thf(fact_7542_coinduct3__lemma,axiom,
! [A: $tType,X8: set @ A,F: ( set @ A ) > ( set @ A )] :
( ( ord_less_eq @ ( set @ A ) @ X8
@ ( F
@ ( complete_lattice_lfp @ ( set @ A )
@ ^ [X: set @ A] : ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ ( F @ X ) @ X8 ) @ ( complete_lattice_gfp @ ( set @ A ) @ F ) ) ) ) )
=> ( ( order_mono @ ( set @ A ) @ ( set @ A ) @ F )
=> ( ord_less_eq @ ( set @ A )
@ ( complete_lattice_lfp @ ( set @ A )
@ ^ [X: set @ A] : ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ ( F @ X ) @ X8 ) @ ( complete_lattice_gfp @ ( set @ A ) @ F ) ) )
@ ( F
@ ( complete_lattice_lfp @ ( set @ A )
@ ^ [X: set @ A] : ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ ( F @ X ) @ X8 ) @ ( complete_lattice_gfp @ ( set @ A ) @ F ) ) ) ) ) ) ) ).
% coinduct3_lemma
thf(fact_7543_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_7544_dom__empty,axiom,
! [B: $tType,A: $tType] :
( ( dom @ A @ B
@ ^ [X: A] : ( none @ B ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% dom_empty
thf(fact_7545_map__update__eta__repair_I1_J,axiom,
! [B: $tType,A: $tType,K: A,V2: B,M: A > ( option @ B )] :
( ( dom @ A @ B
@ ^ [X: A] : ( if @ ( option @ B ) @ ( X = K ) @ ( some @ B @ V2 ) @ ( M @ X ) ) )
= ( insert @ A @ K @ ( dom @ A @ B @ M ) ) ) ).
% map_update_eta_repair(1)
thf(fact_7546_dom__const_H,axiom,
! [B: $tType,A: $tType,F: A > B] :
( ( dom @ A @ B
@ ^ [X: A] : ( some @ B @ ( F @ X ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% dom_const'
thf(fact_7547_restrict__map__inv,axiom,
! [B: $tType,A: $tType,F: A > ( option @ B )] :
( ( restrict_map @ A @ B @ F @ ( uminus_uminus @ ( set @ A ) @ ( dom @ A @ B @ F ) ) )
= ( ^ [X: A] : ( none @ B ) ) ) ).
% restrict_map_inv
thf(fact_7548_dom__map__of__zip,axiom,
! [B: $tType,A: $tType,Xs: list @ A,Ys: list @ B] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( dom @ A @ B @ ( map_of @ A @ B @ ( zip @ A @ B @ Xs @ Ys ) ) )
= ( set2 @ A @ Xs ) ) ) ).
% dom_map_of_zip
thf(fact_7549_dom__map__upds,axiom,
! [B: $tType,A: $tType,M: A > ( option @ B ),Xs: list @ A,Ys: list @ B] :
( ( dom @ A @ B @ ( map_upds @ A @ B @ M @ Xs @ Ys ) )
= ( sup_sup @ ( set @ A ) @ ( set2 @ A @ ( take @ A @ ( size_size @ ( list @ B ) @ Ys ) @ Xs ) ) @ ( dom @ A @ B @ M ) ) ) ).
% dom_map_upds
thf(fact_7550_set__to__map__dom,axiom,
! [B: $tType,A: $tType,S2: set @ ( product_prod @ A @ B )] :
( ( dom @ A @ B @ ( set_to_map @ A @ B @ S2 ) )
= ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ S2 ) ) ).
% set_to_map_dom
thf(fact_7551_def__Collect__coinduct,axiom,
! [A: $tType,A5: set @ A,P: ( set @ A ) > A > $o,A2: A,X8: set @ A] :
( ( A5
= ( complete_lattice_gfp @ ( set @ A )
@ ^ [W: set @ A] : ( collect @ A @ ( P @ W ) ) ) )
=> ( ( order_mono @ ( set @ A ) @ ( set @ A )
@ ^ [W: set @ A] : ( collect @ A @ ( P @ W ) ) )
=> ( ( member @ A @ A2 @ X8 )
=> ( ! [Z4: A] :
( ( member @ A @ Z4 @ X8 )
=> ( P @ ( sup_sup @ ( set @ A ) @ X8 @ A5 ) @ Z4 ) )
=> ( member @ A @ A2 @ A5 ) ) ) ) ) ).
% def_Collect_coinduct
thf(fact_7552_gfp__fun__UnI2,axiom,
! [A: $tType,F: ( set @ A ) > ( set @ A ),A2: A,X8: set @ A] :
( ( order_mono @ ( set @ A ) @ ( set @ A ) @ F )
=> ( ( member @ A @ A2 @ ( complete_lattice_gfp @ ( set @ A ) @ F ) )
=> ( member @ A @ A2 @ ( F @ ( sup_sup @ ( set @ A ) @ X8 @ ( complete_lattice_gfp @ ( set @ A ) @ F ) ) ) ) ) ) ).
% gfp_fun_UnI2
thf(fact_7553_dom__if,axiom,
! [B: $tType,A: $tType,P: A > $o,F: A > ( option @ B ),G: A > ( option @ B )] :
( ( dom @ A @ B
@ ^ [X: A] : ( if @ ( option @ B ) @ ( P @ X ) @ ( F @ X ) @ ( G @ X ) ) )
= ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ ( dom @ A @ B @ F ) @ ( collect @ A @ P ) )
@ ( inf_inf @ ( set @ A ) @ ( dom @ A @ B @ G )
@ ( collect @ A
@ ^ [X: A] :
~ ( P @ X ) ) ) ) ) ).
% dom_if
thf(fact_7554_weak__coinduct__image,axiom,
! [A: $tType,B: $tType,A2: A,X8: set @ A,G: A > B,F: ( set @ B ) > ( set @ B )] :
( ( member @ A @ A2 @ X8 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ G @ X8 ) @ ( F @ ( image2 @ A @ B @ G @ X8 ) ) )
=> ( member @ B @ ( G @ A2 ) @ ( complete_lattice_gfp @ ( set @ B ) @ F ) ) ) ) ).
% weak_coinduct_image
thf(fact_7555_gfp__def,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ( ( complete_lattice_gfp @ A )
= ( ^ [F2: A > A] :
( complete_Sup_Sup @ A
@ ( collect @ A
@ ^ [U2: A] : ( ord_less_eq @ A @ U2 @ ( F2 @ U2 ) ) ) ) ) ) ) ).
% gfp_def
thf(fact_7556_le__map__dom__mono,axiom,
! [B: $tType,A: $tType] :
( ( preorder @ B )
=> ! [M: A > ( option @ B ),M5: A > ( option @ B )] :
( ( ord_less_eq @ ( A > ( option @ B ) ) @ M @ M5 )
=> ( ord_less_eq @ ( set @ A ) @ ( dom @ A @ B @ M ) @ ( dom @ A @ B @ M5 ) ) ) ) ).
% le_map_dom_mono
thf(fact_7557_gfp__gfp,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F: A > A > A] :
( ! [X3: A,Y2: A,W3: A,Z4: A] :
( ( ord_less_eq @ A @ X3 @ Y2 )
=> ( ( ord_less_eq @ A @ W3 @ Z4 )
=> ( ord_less_eq @ A @ ( F @ X3 @ W3 ) @ ( F @ Y2 @ Z4 ) ) ) )
=> ( ( complete_lattice_gfp @ A
@ ^ [X: A] : ( complete_lattice_gfp @ A @ ( F @ X ) ) )
= ( complete_lattice_gfp @ A
@ ^ [X: A] : ( F @ X @ X ) ) ) ) ) ).
% gfp_gfp
thf(fact_7558_gfp__least,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F: A > A,X8: A] :
( ! [U4: A] :
( ( ord_less_eq @ A @ U4 @ ( F @ U4 ) )
=> ( ord_less_eq @ A @ U4 @ X8 ) )
=> ( ord_less_eq @ A @ ( complete_lattice_gfp @ A @ F ) @ X8 ) ) ) ).
% gfp_least
thf(fact_7559_gfp__upperbound,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X8: A,F: A > A] :
( ( ord_less_eq @ A @ X8 @ ( F @ X8 ) )
=> ( ord_less_eq @ A @ X8 @ ( complete_lattice_gfp @ A @ F ) ) ) ) ).
% gfp_upperbound
thf(fact_7560_gfp__mono,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F: A > A,G: A > A] :
( ! [Z8: A] : ( ord_less_eq @ A @ ( F @ Z8 ) @ ( G @ Z8 ) )
=> ( ord_less_eq @ A @ ( complete_lattice_gfp @ A @ F ) @ ( complete_lattice_gfp @ A @ G ) ) ) ) ).
% gfp_mono
thf(fact_7561_weak__coinduct,axiom,
! [A: $tType,A2: A,X8: set @ A,F: ( set @ A ) > ( set @ A )] :
( ( member @ A @ A2 @ X8 )
=> ( ( ord_less_eq @ ( set @ A ) @ X8 @ ( F @ X8 ) )
=> ( member @ A @ A2 @ ( complete_lattice_gfp @ ( set @ A ) @ F ) ) ) ) ).
% weak_coinduct
thf(fact_7562_gfp__eqI,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F4: A > A,X2: A] :
( ( order_mono @ A @ A @ F4 )
=> ( ( ( F4 @ X2 )
= X2 )
=> ( ! [Z4: A] :
( ( ( F4 @ Z4 )
= Z4 )
=> ( ord_less_eq @ A @ Z4 @ X2 ) )
=> ( ( complete_lattice_gfp @ A @ F4 )
= X2 ) ) ) ) ) ).
% gfp_eqI
thf(fact_7563_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_7564_map__to__set__dom,axiom,
! [B: $tType,A: $tType] :
( ( dom @ A @ B )
= ( ^ [M2: A > ( option @ B )] : ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ ( map_to_set @ A @ B @ M2 ) ) ) ) ).
% map_to_set_dom
thf(fact_7565_map__add__left__comm,axiom,
! [B: $tType,A: $tType,A5: A > ( option @ B ),B5: A > ( option @ B ),C6: A > ( option @ B )] :
( ( ( inf_inf @ ( set @ A ) @ ( dom @ A @ B @ A5 ) @ ( dom @ A @ B @ B5 ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( map_add @ A @ B @ A5 @ ( map_add @ A @ B @ B5 @ C6 ) )
= ( map_add @ A @ B @ B5 @ ( map_add @ A @ B @ A5 @ C6 ) ) ) ) ).
% map_add_left_comm
thf(fact_7566_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_7567_dom__def,axiom,
! [B: $tType,A: $tType] :
( ( dom @ A @ B )
= ( ^ [M2: A > ( option @ B )] :
( collect @ A
@ ^ [A3: A] :
( ( M2 @ A3 )
!= ( none @ B ) ) ) ) ) ).
% dom_def
thf(fact_7568_nempty__dom,axiom,
! [B: $tType,A: $tType,E2: A > ( option @ B )] :
( ( E2
!= ( ^ [X: A] : ( none @ B ) ) )
=> ~ ! [M4: A] :
~ ( member @ A @ M4 @ ( dom @ A @ B @ E2 ) ) ) ).
% nempty_dom
thf(fact_7569_dom__map__option,axiom,
! [B: $tType,C: $tType,A: $tType,F: A > C > B,M: A > ( option @ C )] :
( ( dom @ A @ B
@ ^ [K2: A] : ( map_option @ C @ B @ ( F @ K2 ) @ ( M @ K2 ) ) )
= ( dom @ A @ C @ M ) ) ).
% dom_map_option
thf(fact_7570_gfp__const,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [T2: A] :
( ( complete_lattice_gfp @ A
@ ^ [X: A] : T2 )
= T2 ) ) ).
% gfp_const
thf(fact_7571_map__dom__ran__finite,axiom,
! [B: $tType,A: $tType,M3: A > ( option @ B )] :
( ( finite_finite2 @ A @ ( dom @ A @ B @ M3 ) )
=> ( finite_finite2 @ B @ ( ran @ A @ B @ M3 ) ) ) ).
% map_dom_ran_finite
thf(fact_7572_map__card__eq__iff,axiom,
! [B: $tType,A: $tType,M3: A > ( option @ B ),X2: A,Y: A] :
( ( finite_finite2 @ A @ ( dom @ A @ B @ M3 ) )
=> ( ( ( finite_card @ A @ ( dom @ A @ B @ M3 ) )
= ( finite_card @ B @ ( ran @ A @ B @ M3 ) ) )
=> ( ( member @ A @ X2 @ ( dom @ A @ B @ M3 ) )
=> ( ( ( M3 @ X2 )
= ( M3 @ Y ) )
= ( X2 = Y ) ) ) ) ) ).
% map_card_eq_iff
thf(fact_7573_def__gfp__unfold,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: A,F: A > A] :
( ( A5
= ( complete_lattice_gfp @ A @ F ) )
=> ( ( order_mono @ A @ A @ F )
=> ( A5
= ( F @ A5 ) ) ) ) ) ).
% def_gfp_unfold
thf(fact_7574_gfp__fixpoint,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F: A > A] :
( ( order_mono @ A @ A @ F )
=> ( ( F @ ( complete_lattice_gfp @ A @ F ) )
= ( complete_lattice_gfp @ A @ F ) ) ) ) ).
% gfp_fixpoint
thf(fact_7575_gfp__unfold,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F: A > A] :
( ( order_mono @ A @ A @ F )
=> ( ( complete_lattice_gfp @ A @ F )
= ( F @ ( complete_lattice_gfp @ A @ F ) ) ) ) ) ).
% gfp_unfold
thf(fact_7576_gfp__rolling,axiom,
! [A: $tType,B: $tType] :
( ( ( comple6319245703460814977attice @ B )
& ( comple6319245703460814977attice @ A ) )
=> ! [G: A > B,F: B > A] :
( ( order_mono @ A @ B @ G )
=> ( ( order_mono @ B @ A @ F )
=> ( ( G
@ ( complete_lattice_gfp @ A
@ ^ [X: A] : ( F @ ( G @ X ) ) ) )
= ( complete_lattice_gfp @ B
@ ^ [X: B] : ( G @ ( F @ X ) ) ) ) ) ) ) ).
% gfp_rolling
thf(fact_7577_finite__set__of__finite__maps,axiom,
! [B: $tType,A: $tType,A5: set @ A,B5: set @ B] :
( ( finite_finite2 @ A @ A5 )
=> ( ( finite_finite2 @ B @ B5 )
=> ( finite_finite2 @ ( A > ( option @ B ) )
@ ( collect @ ( A > ( option @ B ) )
@ ^ [M2: A > ( option @ B )] :
( ( ( dom @ A @ B @ M2 )
= A5 )
& ( ord_less_eq @ ( set @ B ) @ ( ran @ A @ B @ M2 ) @ B5 ) ) ) ) ) ) ).
% finite_set_of_finite_maps
thf(fact_7578_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_7579_coinduct__set,axiom,
! [A: $tType,F: ( set @ A ) > ( set @ A ),A2: A,X8: set @ A] :
( ( order_mono @ ( set @ A ) @ ( set @ A ) @ F )
=> ( ( member @ A @ A2 @ X8 )
=> ( ( ord_less_eq @ ( set @ A ) @ X8 @ ( F @ ( sup_sup @ ( set @ A ) @ X8 @ ( complete_lattice_gfp @ ( set @ A ) @ F ) ) ) )
=> ( member @ A @ A2 @ ( complete_lattice_gfp @ ( set @ A ) @ F ) ) ) ) ) ).
% coinduct_set
thf(fact_7580_def__coinduct__set,axiom,
! [A: $tType,A5: set @ A,F: ( set @ A ) > ( set @ A ),A2: A,X8: set @ A] :
( ( A5
= ( complete_lattice_gfp @ ( set @ A ) @ F ) )
=> ( ( order_mono @ ( set @ A ) @ ( set @ A ) @ F )
=> ( ( member @ A @ A2 @ X8 )
=> ( ( ord_less_eq @ ( set @ A ) @ X8 @ ( F @ ( sup_sup @ ( set @ A ) @ X8 @ A5 ) ) )
=> ( member @ A @ A2 @ A5 ) ) ) ) ) ).
% def_coinduct_set
thf(fact_7581_coinduct,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F: A > A,X8: A] :
( ( order_mono @ A @ A @ F )
=> ( ( ord_less_eq @ A @ X8 @ ( F @ ( sup_sup @ A @ X8 @ ( complete_lattice_gfp @ A @ F ) ) ) )
=> ( ord_less_eq @ A @ X8 @ ( complete_lattice_gfp @ A @ F ) ) ) ) ) ).
% coinduct
thf(fact_7582_def__coinduct,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: A,F: A > A,X8: A] :
( ( A5
= ( complete_lattice_gfp @ A @ F ) )
=> ( ( order_mono @ A @ A @ F )
=> ( ( ord_less_eq @ A @ X8 @ ( F @ ( sup_sup @ A @ X8 @ A5 ) ) )
=> ( ord_less_eq @ A @ X8 @ A5 ) ) ) ) ) ).
% def_coinduct
thf(fact_7583_coinduct__lemma,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X8: A,F: A > A] :
( ( ord_less_eq @ A @ X8 @ ( F @ ( sup_sup @ A @ X8 @ ( complete_lattice_gfp @ A @ F ) ) ) )
=> ( ( order_mono @ A @ A @ F )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ X8 @ ( complete_lattice_gfp @ A @ F ) ) @ ( F @ ( sup_sup @ A @ X8 @ ( complete_lattice_gfp @ A @ F ) ) ) ) ) ) ) ).
% coinduct_lemma
thf(fact_7584_gfp__ordinal__induct,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F: A > A,P: A > $o] :
( ( order_mono @ A @ A @ F )
=> ( ! [S8: A] :
( ( P @ S8 )
=> ( ( ord_less_eq @ A @ ( complete_lattice_gfp @ A @ F ) @ S8 )
=> ( P @ ( F @ S8 ) ) ) )
=> ( ! [M10: set @ A] :
( ! [X4: A] :
( ( member @ A @ X4 @ M10 )
=> ( P @ X4 ) )
=> ( P @ ( complete_Inf_Inf @ A @ M10 ) ) )
=> ( P @ ( complete_lattice_gfp @ A @ F ) ) ) ) ) ) ).
% gfp_ordinal_induct
thf(fact_7585_gfp__funpow,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F: A > A,N: nat] :
( ( order_mono @ A @ A @ F )
=> ( ( complete_lattice_gfp @ A @ ( compow @ ( A > A ) @ ( suc @ N ) @ F ) )
= ( complete_lattice_gfp @ A @ F ) ) ) ) ).
% gfp_funpow
thf(fact_7586_inj__on__map__the,axiom,
! [B: $tType,A: $tType,D3: set @ A,M: A > ( option @ B )] :
( ( ord_less_eq @ ( set @ A ) @ D3 @ ( dom @ A @ B @ M ) )
=> ( ( inj_on @ A @ ( option @ B ) @ M @ D3 )
=> ( inj_on @ A @ B @ ( comp @ ( option @ B ) @ B @ A @ ( the2 @ B ) @ M ) @ D3 ) ) ) ).
% inj_on_map_the
thf(fact_7587_lfp__le__gfp,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F: A > A] :
( ( order_mono @ A @ A @ F )
=> ( ord_less_eq @ A @ ( complete_lattice_lfp @ A @ F ) @ ( complete_lattice_gfp @ A @ F ) ) ) ) ).
% lfp_le_gfp
thf(fact_7588_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
@ ^ [X: A] : ( none @ B ) )
=> ( ! [K4: A,V5: B,M4: A > ( option @ B )] :
( ( finite_finite2 @ A @ ( dom @ A @ B @ M4 ) )
=> ( ~ ( member @ A @ K4 @ ( dom @ A @ B @ M4 ) )
=> ( ( P @ M4 )
=> ( P @ ( fun_upd @ A @ ( option @ B ) @ M4 @ K4 @ ( some @ B @ V5 ) ) ) ) ) )
=> ( P @ M ) ) ) ) ).
% finite_Map_induct
thf(fact_7589_ran__is__image,axiom,
! [A: $tType,B: $tType] :
( ( ran @ B @ A )
= ( ^ [M6: B > ( option @ A )] : ( image2 @ B @ A @ ( comp @ ( option @ A ) @ A @ B @ ( the2 @ A ) @ M6 ) @ ( dom @ B @ A @ M6 ) ) ) ) ).
% ran_is_image
thf(fact_7590_map__add__distinct__le,axiom,
! [B: $tType,A: $tType] :
( ( preorder @ B )
=> ! [M: A > ( option @ B ),M5: A > ( option @ B ),N: A > ( option @ B ),N3: A > ( option @ B )] :
( ( ord_less_eq @ ( A > ( option @ B ) ) @ M @ M5 )
=> ( ( ord_less_eq @ ( A > ( option @ B ) ) @ N @ N3 )
=> ( ( ( inf_inf @ ( set @ A ) @ ( dom @ A @ B @ M5 ) @ ( dom @ A @ B @ N3 ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ ( A > ( option @ B ) ) @ ( map_add @ A @ B @ M @ N ) @ ( map_add @ A @ B @ M5 @ N3 ) ) ) ) ) ) ).
% map_add_distinct_le
thf(fact_7591_dom__eq__singleton__conv,axiom,
! [A: $tType,B: $tType,F: A > ( option @ B ),X2: A] :
( ( ( dom @ A @ B @ F )
= ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( ? [V4: B] :
( F
= ( fun_upd @ A @ ( option @ B )
@ ^ [X: A] : ( none @ B )
@ X2
@ ( some @ B @ V4 ) ) ) ) ) ).
% dom_eq_singleton_conv
thf(fact_7592_gfp__Kleene__iter,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F: A > A,K: nat] :
( ( order_mono @ A @ A @ F )
=> ( ( ( compow @ ( A > A ) @ ( suc @ K ) @ F @ ( top_top @ A ) )
= ( compow @ ( A > A ) @ K @ F @ ( top_top @ A ) ) )
=> ( ( complete_lattice_gfp @ A @ F )
= ( compow @ ( A > A ) @ K @ F @ ( top_top @ A ) ) ) ) ) ) ).
% gfp_Kleene_iter
thf(fact_7593_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 )
@ ^ [K2: A] : ( product_Pair @ A @ B @ K2 @ ( the2 @ B @ ( M @ K2 ) ) )
@ Xs ) )
= M ) ) ).
% map_of_map_keys
thf(fact_7594_coinduct3,axiom,
! [A: $tType,F: ( set @ A ) > ( set @ A ),A2: A,X8: set @ A] :
( ( order_mono @ ( set @ A ) @ ( set @ A ) @ F )
=> ( ( member @ A @ A2 @ X8 )
=> ( ( ord_less_eq @ ( set @ A ) @ X8
@ ( F
@ ( complete_lattice_lfp @ ( set @ A )
@ ^ [X: set @ A] : ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ ( F @ X ) @ X8 ) @ ( complete_lattice_gfp @ ( set @ A ) @ F ) ) ) ) )
=> ( member @ A @ A2 @ ( complete_lattice_gfp @ ( set @ A ) @ F ) ) ) ) ) ).
% coinduct3
thf(fact_7595_def__coinduct3,axiom,
! [A: $tType,A5: set @ A,F: ( set @ A ) > ( set @ A ),A2: A,X8: set @ A] :
( ( A5
= ( complete_lattice_gfp @ ( set @ A ) @ F ) )
=> ( ( order_mono @ ( set @ A ) @ ( set @ A ) @ F )
=> ( ( member @ A @ A2 @ X8 )
=> ( ( ord_less_eq @ ( set @ A ) @ X8
@ ( F
@ ( complete_lattice_lfp @ ( set @ A )
@ ^ [X: set @ A] : ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ ( F @ X ) @ X8 ) @ A5 ) ) ) )
=> ( member @ A @ A2 @ A5 ) ) ) ) ) ).
% def_coinduct3
thf(fact_7596_dom__override__on,axiom,
! [B: $tType,A: $tType,F: A > ( option @ B ),G: A > ( option @ B ),A5: set @ A] :
( ( dom @ A @ B @ ( override_on @ A @ ( option @ B ) @ F @ G @ A5 ) )
= ( sup_sup @ ( set @ A )
@ ( minus_minus @ ( set @ A ) @ ( dom @ A @ B @ F )
@ ( collect @ A
@ ^ [A3: A] : ( member @ A @ A3 @ ( minus_minus @ ( set @ A ) @ A5 @ ( dom @ A @ B @ G ) ) ) ) )
@ ( collect @ A
@ ^ [A3: A] : ( member @ A @ A3 @ ( inf_inf @ ( set @ A ) @ A5 @ ( dom @ A @ B @ G ) ) ) ) ) ) ).
% dom_override_on
thf(fact_7597_eq__f__restr__conv,axiom,
! [B: $tType,A: $tType,S3: set @ A,F: ( A > ( option @ B ) ) > A > ( option @ B ),A5: A > ( option @ B )] :
( ( ( ord_less_eq @ ( set @ A ) @ S3 @ ( dom @ A @ B @ ( F @ A5 ) ) )
& ( A5
= ( restrict_map @ A @ B @ ( F @ A5 ) @ ( uminus_uminus @ ( set @ A ) @ S3 ) ) ) )
= ( ( map_le @ A @ B @ A5 @ ( F @ A5 ) )
& ( S3
= ( minus_minus @ ( set @ A ) @ ( dom @ A @ B @ ( F @ A5 ) ) @ ( dom @ A @ B @ A5 ) ) ) ) ) ).
% eq_f_restr_conv
thf(fact_7598_map__le__empty,axiom,
! [B: $tType,A: $tType,G: A > ( option @ B )] :
( map_le @ A @ B
@ ^ [X: A] : ( none @ B )
@ G ) ).
% map_le_empty
thf(fact_7599_map__leD,axiom,
! [A: $tType,B: $tType,M1: A > ( option @ B ),M22: A > ( option @ B ),K: A,V2: B] :
( ( map_le @ A @ B @ M1 @ M22 )
=> ( ( ( M1 @ K )
= ( some @ B @ V2 ) )
=> ( ( M22 @ K )
= ( some @ B @ V2 ) ) ) ) ).
% map_leD
thf(fact_7600_map__leI,axiom,
! [B: $tType,A: $tType,M1: A > ( option @ B ),M22: A > ( option @ B )] :
( ! [X3: A,V5: B] :
( ( ( M1 @ X3 )
= ( some @ B @ V5 ) )
=> ( ( M22 @ X3 )
= ( some @ B @ V5 ) ) )
=> ( map_le @ A @ B @ M1 @ M22 ) ) ).
% map_leI
thf(fact_7601_eq__f__restr__ss__eq,axiom,
! [B: $tType,A: $tType,S3: set @ A,F: ( A > ( option @ B ) ) > A > ( option @ B ),A5: A > ( option @ B )] :
( ( ord_less_eq @ ( set @ A ) @ S3 @ ( dom @ A @ B @ ( F @ A5 ) ) )
=> ( ( A5
= ( restrict_map @ A @ B @ ( F @ A5 ) @ ( uminus_uminus @ ( set @ A ) @ S3 ) ) )
= ( ( map_le @ A @ B @ A5 @ ( F @ A5 ) )
& ( S3
= ( minus_minus @ ( set @ A ) @ ( dom @ A @ B @ ( F @ A5 ) ) @ ( dom @ A @ B @ A5 ) ) ) ) ) ) ).
% eq_f_restr_ss_eq
thf(fact_7602_le__map__mmupd__not__dom,axiom,
! [A: $tType,B: $tType,M: A > ( option @ B ),K3: set @ A,V2: B] : ( map_le @ A @ B @ M @ ( map_mmupd @ A @ B @ M @ ( minus_minus @ ( set @ A ) @ K3 @ ( dom @ A @ B @ M ) ) @ V2 ) ) ).
% le_map_mmupd_not_dom
thf(fact_7603_map__mmupd__update__less,axiom,
! [A: $tType,B: $tType,K3: set @ A,K9: set @ A,M: A > ( option @ B ),V2: B] :
( ( ord_less_eq @ ( set @ A ) @ K3 @ K9 )
=> ( map_le @ A @ B @ ( map_mmupd @ A @ B @ M @ ( minus_minus @ ( set @ A ) @ K3 @ ( dom @ A @ B @ M ) ) @ V2 ) @ ( map_mmupd @ A @ B @ M @ ( minus_minus @ ( set @ A ) @ K9 @ ( dom @ A @ B @ M ) ) @ V2 ) ) ) ).
% map_mmupd_update_less
thf(fact_7604_mmupd__notin__upd,axiom,
! [B: $tType,A: $tType,K: A,K3: set @ A,M: A > ( option @ B ),V2: B] :
( ~ ( member @ A @ K @ K3 )
=> ( ( map_mmupd @ A @ B @ M @ K3 @ V2 @ K )
= ( M @ K ) ) ) ).
% mmupd_notin_upd
thf(fact_7605_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_7606_mmupd__in__upd,axiom,
! [A: $tType,B: $tType,K: A,K3: set @ A,M: A > ( option @ B ),V2: B] :
( ( member @ A @ K @ K3 )
=> ( ( map_mmupd @ A @ B @ M @ K3 @ V2 @ K )
= ( some @ B @ V2 ) ) ) ).
% mmupd_in_upd
thf(fact_7607_dom__mmupd,axiom,
! [B: $tType,A: $tType,M: A > ( option @ B ),K3: set @ A,V2: B] :
( ( dom @ A @ B @ ( map_mmupd @ A @ B @ M @ K3 @ V2 ) )
= ( sup_sup @ ( set @ A ) @ ( dom @ A @ B @ M ) @ K3 ) ) ).
% dom_mmupd
thf(fact_7608_map__mmupdE,axiom,
! [B: $tType,A: $tType,M: B > ( option @ A ),K3: set @ B,V2: A,K: B,X2: A] :
( ( ( map_mmupd @ B @ A @ M @ K3 @ V2 @ K )
= ( some @ A @ X2 ) )
=> ( ( ~ ( member @ B @ K @ K3 )
=> ( ( M @ K )
!= ( some @ A @ X2 ) ) )
=> ~ ( ( member @ B @ K @ K3 )
=> ( X2 != V2 ) ) ) ) ).
% map_mmupdE
thf(fact_7609_map__mmupd__def,axiom,
! [A: $tType,B: $tType] :
( ( map_mmupd @ B @ A )
= ( ^ [M2: B > ( option @ A ),K6: set @ B,V4: A,K2: B] : ( if @ ( option @ A ) @ ( member @ B @ K2 @ K6 ) @ ( some @ A @ V4 ) @ ( M2 @ K2 ) ) ) ) ).
% map_mmupd_def
thf(fact_7610_graph__eq__to__snd__dom,axiom,
! [B: $tType,A: $tType] :
( ( graph @ A @ B )
= ( ^ [M2: A > ( option @ B )] :
( image2 @ A @ ( product_prod @ A @ B )
@ ^ [X: A] : ( product_Pair @ A @ B @ X @ ( the2 @ B @ ( M2 @ X ) ) )
@ ( dom @ A @ B @ M2 ) ) ) ) ).
% graph_eq_to_snd_dom
thf(fact_7611_cSUP__UNION,axiom,
! [B: $tType,D: $tType,C: $tType] :
( ( condit1219197933456340205attice @ B )
=> ! [A5: set @ C,B5: C > ( set @ D ),F: D > B] :
( ( A5
!= ( bot_bot @ ( set @ C ) ) )
=> ( ! [X3: C] :
( ( member @ C @ X3 @ A5 )
=> ( ( B5 @ X3 )
!= ( bot_bot @ ( set @ D ) ) ) )
=> ( ( condit941137186595557371_above @ B
@ ( complete_Sup_Sup @ ( set @ B )
@ ( image2 @ C @ ( set @ B )
@ ^ [X: C] : ( image2 @ D @ B @ F @ ( B5 @ X ) )
@ A5 ) ) )
=> ( ( complete_Sup_Sup @ B @ ( image2 @ D @ B @ F @ ( complete_Sup_Sup @ ( set @ D ) @ ( image2 @ C @ ( set @ D ) @ B5 @ A5 ) ) ) )
= ( complete_Sup_Sup @ B
@ ( image2 @ C @ B
@ ^ [X: C] : ( complete_Sup_Sup @ B @ ( image2 @ D @ B @ F @ ( B5 @ X ) ) )
@ A5 ) ) ) ) ) ) ) ).
% cSUP_UNION
thf(fact_7612_bdd__above_OI,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A5: set @ A,M3: A] :
( ! [X3: A] :
( ( member @ A @ X3 @ A5 )
=> ( ord_less_eq @ A @ X3 @ M3 ) )
=> ( condit941137186595557371_above @ A @ A5 ) ) ) ).
% bdd_above.I
thf(fact_7613_bdd__above__image__sup,axiom,
! [A: $tType,B: $tType] :
( ( lattice @ A )
=> ! [F: B > A,G: B > A,A5: set @ B] :
( ( condit941137186595557371_above @ A
@ ( image2 @ B @ A
@ ^ [X: B] : ( sup_sup @ A @ ( F @ X ) @ ( G @ X ) )
@ A5 ) )
= ( ( condit941137186595557371_above @ A @ ( image2 @ B @ A @ F @ A5 ) )
& ( condit941137186595557371_above @ A @ ( image2 @ B @ A @ G @ A5 ) ) ) ) ) ).
% bdd_above_image_sup
thf(fact_7614_graph__empty,axiom,
! [B: $tType,A: $tType] :
( ( graph @ A @ B
@ ^ [X: A] : ( none @ B ) )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% graph_empty
thf(fact_7615_bdd__above__UN,axiom,
! [A: $tType,B: $tType] :
( ( lattice @ A )
=> ! [I5: set @ B,A5: B > ( set @ A )] :
( ( finite_finite2 @ B @ I5 )
=> ( ( condit941137186595557371_above @ A @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ A5 @ I5 ) ) )
= ( ! [X: B] :
( ( member @ B @ X @ I5 )
=> ( condit941137186595557371_above @ A @ ( A5 @ X ) ) ) ) ) ) ) ).
% bdd_above_UN
thf(fact_7616_cSUP__upper2,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [F: B > A,A5: set @ B,X2: B,U: A] :
( ( condit941137186595557371_above @ A @ ( image2 @ B @ A @ F @ A5 ) )
=> ( ( member @ B @ X2 @ A5 )
=> ( ( ord_less_eq @ A @ U @ ( F @ X2 ) )
=> ( ord_less_eq @ A @ U @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F @ A5 ) ) ) ) ) ) ) ).
% cSUP_upper2
thf(fact_7617_cSUP__upper,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X2: B,A5: set @ B,F: B > A] :
( ( member @ B @ X2 @ A5 )
=> ( ( condit941137186595557371_above @ A @ ( image2 @ B @ A @ F @ A5 ) )
=> ( ord_less_eq @ A @ ( F @ X2 ) @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F @ A5 ) ) ) ) ) ) ).
% cSUP_upper
thf(fact_7618_cSup__le__iff,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [S2: set @ A,A2: A] :
( ( S2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit941137186595557371_above @ A @ S2 )
=> ( ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ S2 ) @ A2 )
= ( ! [X: A] :
( ( member @ A @ X @ S2 )
=> ( ord_less_eq @ A @ X @ A2 ) ) ) ) ) ) ) ).
% cSup_le_iff
thf(fact_7619_cSup__mono,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [B5: set @ A,A5: set @ A] :
( ( B5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit941137186595557371_above @ A @ A5 )
=> ( ! [B4: A] :
( ( member @ A @ B4 @ B5 )
=> ? [X4: A] :
( ( member @ A @ X4 @ A5 )
& ( ord_less_eq @ A @ B4 @ X4 ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ B5 ) @ ( complete_Sup_Sup @ A @ A5 ) ) ) ) ) ) ).
% cSup_mono
thf(fact_7620_bdd__above_OE,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A5: set @ A] :
( ( condit941137186595557371_above @ A @ A5 )
=> ~ ! [M10: A] :
~ ! [X4: A] :
( ( member @ A @ X4 @ A5 )
=> ( ord_less_eq @ A @ X4 @ M10 ) ) ) ) ).
% bdd_above.E
thf(fact_7621_bdd__above_Ounfold,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( condit941137186595557371_above @ A )
= ( ^ [A8: set @ A] :
? [M6: A] :
! [X: A] :
( ( member @ A @ X @ A8 )
=> ( ord_less_eq @ A @ X @ M6 ) ) ) ) ) ).
% bdd_above.unfold
thf(fact_7622_cSup__upper2,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X2: A,X8: set @ A,Y: A] :
( ( member @ A @ X2 @ X8 )
=> ( ( ord_less_eq @ A @ Y @ X2 )
=> ( ( condit941137186595557371_above @ A @ X8 )
=> ( ord_less_eq @ A @ Y @ ( complete_Sup_Sup @ A @ X8 ) ) ) ) ) ) ).
% cSup_upper2
thf(fact_7623_cSup__upper,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X2: A,X8: set @ A] :
( ( member @ A @ X2 @ X8 )
=> ( ( condit941137186595557371_above @ A @ X8 )
=> ( ord_less_eq @ A @ X2 @ ( complete_Sup_Sup @ A @ X8 ) ) ) ) ) ).
% cSup_upper
thf(fact_7624_bdd__above_OI2,axiom,
! [A: $tType,B: $tType] :
( ( preorder @ A )
=> ! [A5: set @ B,F: B > A,M3: A] :
( ! [X3: B] :
( ( member @ B @ X3 @ A5 )
=> ( ord_less_eq @ A @ ( F @ X3 ) @ M3 ) )
=> ( condit941137186595557371_above @ A @ ( image2 @ B @ A @ F @ A5 ) ) ) ) ).
% bdd_above.I2
thf(fact_7625_cSUP__lessD,axiom,
! [B: $tType,A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [F: B > A,A5: set @ B,Y: A,I: B] :
( ( condit941137186595557371_above @ A @ ( image2 @ B @ A @ F @ A5 ) )
=> ( ( ord_less @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F @ A5 ) ) @ Y )
=> ( ( member @ B @ I @ A5 )
=> ( ord_less @ A @ ( F @ I ) @ Y ) ) ) ) ) ).
% cSUP_lessD
thf(fact_7626_cSUP__le__iff,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A5: set @ B,F: B > A,U: A] :
( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit941137186595557371_above @ A @ ( image2 @ B @ A @ F @ A5 ) )
=> ( ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F @ A5 ) ) @ U )
= ( ! [X: B] :
( ( member @ B @ X @ A5 )
=> ( ord_less_eq @ A @ ( F @ X ) @ U ) ) ) ) ) ) ) ).
% cSUP_le_iff
thf(fact_7627_cSUP__mono,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A5: set @ B,G: C > A,B5: set @ C,F: B > A] :
( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit941137186595557371_above @ A @ ( image2 @ C @ A @ G @ B5 ) )
=> ( ! [N2: B] :
( ( member @ B @ N2 @ A5 )
=> ? [X4: C] :
( ( member @ C @ X4 @ B5 )
& ( ord_less_eq @ A @ ( F @ N2 ) @ ( G @ X4 ) ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F @ A5 ) ) @ ( complete_Sup_Sup @ A @ ( image2 @ C @ A @ G @ B5 ) ) ) ) ) ) ) ).
% cSUP_mono
thf(fact_7628_cSup__subset__mono,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A5: set @ A,B5: set @ A] :
( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit941137186595557371_above @ A @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A5 ) @ ( complete_Sup_Sup @ A @ B5 ) ) ) ) ) ) ).
% cSup_subset_mono
thf(fact_7629_graph__def,axiom,
! [B: $tType,A: $tType] :
( ( graph @ A @ B )
= ( ^ [M2: A > ( option @ B )] :
( collect @ ( product_prod @ A @ B )
@ ^ [Uu: product_prod @ A @ B] :
? [A3: A,B3: B] :
( ( Uu
= ( product_Pair @ A @ B @ A3 @ B3 ) )
& ( ( M2 @ A3 )
= ( some @ B @ B3 ) ) ) ) ) ) ).
% graph_def
thf(fact_7630_less__cSUP__iff,axiom,
! [A: $tType,B: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [A5: set @ B,F: B > A,A2: A] :
( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit941137186595557371_above @ A @ ( image2 @ B @ A @ F @ A5 ) )
=> ( ( ord_less @ A @ A2 @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F @ A5 ) ) )
= ( ? [X: B] :
( ( member @ B @ X @ A5 )
& ( ord_less @ A @ A2 @ ( F @ X ) ) ) ) ) ) ) ) ).
% less_cSUP_iff
thf(fact_7631_conditionally__complete__lattice__class_OSUP__sup__distrib,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A5: set @ B,F: B > A,G: B > A] :
( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit941137186595557371_above @ A @ ( image2 @ B @ A @ F @ A5 ) )
=> ( ( condit941137186595557371_above @ A @ ( image2 @ B @ A @ G @ A5 ) )
=> ( ( sup_sup @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F @ A5 ) ) @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ G @ A5 ) ) )
= ( complete_Sup_Sup @ A
@ ( image2 @ B @ A
@ ^ [A3: B] : ( sup_sup @ A @ ( F @ A3 ) @ ( G @ A3 ) )
@ A5 ) ) ) ) ) ) ) ).
% conditionally_complete_lattice_class.SUP_sup_distrib
thf(fact_7632_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_7633_cSUP__subset__mono,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A5: set @ B,G: B > A,B5: set @ B,F: B > A] :
( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit941137186595557371_above @ A @ ( image2 @ B @ A @ G @ B5 ) )
=> ( ( ord_less_eq @ ( set @ B ) @ A5 @ B5 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ A5 )
=> ( ord_less_eq @ A @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F @ A5 ) ) @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ G @ B5 ) ) ) ) ) ) ) ) ).
% cSUP_subset_mono
thf(fact_7634_cSup__inter__less__eq,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A5: set @ A,B5: set @ A] :
( ( condit941137186595557371_above @ A @ A5 )
=> ( ( condit941137186595557371_above @ A @ B5 )
=> ( ( ( inf_inf @ ( set @ A ) @ A5 @ B5 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( inf_inf @ ( set @ A ) @ A5 @ B5 ) ) @ ( sup_sup @ A @ ( complete_Sup_Sup @ A @ A5 ) @ ( complete_Sup_Sup @ A @ B5 ) ) ) ) ) ) ) ).
% cSup_inter_less_eq
thf(fact_7635_bdd__above__multiset__imp__bdd__above__count,axiom,
! [A: $tType,A5: set @ ( multiset @ A ),X2: A] :
( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ A5 )
=> ( condit941137186595557371_above @ nat
@ ( image2 @ ( multiset @ A ) @ nat
@ ^ [X5: multiset @ A] : ( count @ A @ X5 @ X2 )
@ A5 ) ) ) ).
% bdd_above_multiset_imp_bdd_above_count
thf(fact_7636_cSup__cInf,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [S2: set @ A] :
( ( S2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit941137186595557371_above @ A @ S2 )
=> ( ( complete_Sup_Sup @ A @ S2 )
= ( complete_Inf_Inf @ A
@ ( collect @ A
@ ^ [X: A] :
! [Y4: A] :
( ( member @ A @ Y4 @ S2 )
=> ( ord_less_eq @ A @ Y4 @ X ) ) ) ) ) ) ) ) ).
% cSup_cInf
thf(fact_7637_mono__cSup,axiom,
! [B: $tType,A: $tType] :
( ( ( condit1219197933456340205attice @ A )
& ( condit1219197933456340205attice @ B ) )
=> ! [F: A > B,A5: set @ A] :
( ( order_mono @ A @ B @ F )
=> ( ( condit941137186595557371_above @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ B @ ( complete_Sup_Sup @ B @ ( image2 @ A @ B @ F @ A5 ) ) @ ( F @ ( complete_Sup_Sup @ A @ A5 ) ) ) ) ) ) ) ).
% mono_cSup
thf(fact_7638_mono__cSUP,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ( condit1219197933456340205attice @ A )
& ( condit1219197933456340205attice @ B ) )
=> ! [F: A > B,A5: C > A,I5: set @ C] :
( ( order_mono @ A @ B @ F )
=> ( ( condit941137186595557371_above @ A @ ( image2 @ C @ A @ A5 @ I5 ) )
=> ( ( I5
!= ( bot_bot @ ( set @ C ) ) )
=> ( ord_less_eq @ B
@ ( complete_Sup_Sup @ B
@ ( image2 @ C @ B
@ ^ [X: C] : ( F @ ( A5 @ X ) )
@ I5 ) )
@ ( F @ ( complete_Sup_Sup @ A @ ( image2 @ C @ A @ A5 @ I5 ) ) ) ) ) ) ) ) ).
% mono_cSUP
thf(fact_7639_trans__wf__iff,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( trans @ A @ R )
=> ( ( wf @ A @ R )
= ( ! [A3: A] :
( wf @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R
@ ( product_Sigma @ A @ A @ ( image @ A @ A @ ( converse @ A @ A @ R ) @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
@ ^ [Uu: A] : ( image @ A @ A @ ( converse @ A @ A @ R ) @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ).
% trans_wf_iff
thf(fact_7640_cINF__UNION,axiom,
! [B: $tType,D: $tType,C: $tType] :
( ( condit1219197933456340205attice @ B )
=> ! [A5: set @ C,B5: C > ( set @ D ),F: D > B] :
( ( A5
!= ( bot_bot @ ( set @ C ) ) )
=> ( ! [X3: C] :
( ( member @ C @ X3 @ A5 )
=> ( ( B5 @ X3 )
!= ( bot_bot @ ( set @ D ) ) ) )
=> ( ( condit1013018076250108175_below @ B
@ ( complete_Sup_Sup @ ( set @ B )
@ ( image2 @ C @ ( set @ B )
@ ^ [X: C] : ( image2 @ D @ B @ F @ ( B5 @ X ) )
@ A5 ) ) )
=> ( ( complete_Inf_Inf @ B @ ( image2 @ D @ B @ F @ ( complete_Sup_Sup @ ( set @ D ) @ ( image2 @ C @ ( set @ D ) @ B5 @ A5 ) ) ) )
= ( complete_Inf_Inf @ B
@ ( image2 @ C @ B
@ ^ [X: C] : ( complete_Inf_Inf @ B @ ( image2 @ D @ B @ F @ ( B5 @ X ) ) )
@ A5 ) ) ) ) ) ) ) ).
% cINF_UNION
thf(fact_7641_bdd__belowI,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A5: set @ A,M: A] :
( ! [X3: A] :
( ( member @ A @ X3 @ A5 )
=> ( ord_less_eq @ A @ M @ X3 ) )
=> ( condit1013018076250108175_below @ A @ A5 ) ) ) ).
% bdd_belowI
thf(fact_7642_bdd__below_OI,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A5: set @ A,M3: A] :
( ! [X3: A] :
( ( member @ A @ X3 @ A5 )
=> ( ord_less_eq @ A @ M3 @ X3 ) )
=> ( condit1013018076250108175_below @ A @ A5 ) ) ) ).
% bdd_below.I
thf(fact_7643_brk__rel__wf,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( wf @ A @ R3 )
=> ( wf @ ( product_prod @ $o @ A ) @ ( brk_rel @ A @ A @ R3 ) ) ) ).
% brk_rel_wf
thf(fact_7644_bdd__below__image__inf,axiom,
! [A: $tType,B: $tType] :
( ( lattice @ A )
=> ! [F: B > A,G: B > A,A5: set @ B] :
( ( condit1013018076250108175_below @ A
@ ( image2 @ B @ A
@ ^ [X: B] : ( inf_inf @ A @ ( F @ X ) @ ( G @ X ) )
@ A5 ) )
= ( ( condit1013018076250108175_below @ A @ ( image2 @ B @ A @ F @ A5 ) )
& ( condit1013018076250108175_below @ A @ ( image2 @ B @ A @ G @ A5 ) ) ) ) ) ).
% bdd_below_image_inf
thf(fact_7645_bdd__below__UN,axiom,
! [A: $tType,B: $tType] :
( ( lattice @ A )
=> ! [I5: set @ B,A5: B > ( set @ A )] :
( ( finite_finite2 @ B @ I5 )
=> ( ( condit1013018076250108175_below @ A @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ A5 @ I5 ) ) )
= ( ! [X: B] :
( ( member @ B @ X @ I5 )
=> ( condit1013018076250108175_below @ A @ ( A5 @ X ) ) ) ) ) ) ) ).
% bdd_below_UN
thf(fact_7646_finite__wf__eq__wf__converse,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( finite_finite2 @ ( product_prod @ A @ A ) @ R3 )
=> ( ( wf @ A @ ( converse @ A @ A @ R3 ) )
= ( wf @ A @ R3 ) ) ) ).
% finite_wf_eq_wf_converse
thf(fact_7647_wf__finite__segments,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( irrefl @ A @ R )
=> ( ( trans @ A @ R )
=> ( ! [X3: A] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [Y4: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ X3 ) @ R ) ) )
=> ( wf @ A @ R ) ) ) ) ).
% wf_finite_segments
thf(fact_7648_wf__linord__ex__has__least,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),P: B > $o,K: B,M: B > A] :
( ( wf @ A @ R )
=> ( ! [X3: A,Y2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y2 ) @ ( transitive_trancl @ A @ R ) )
= ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ X3 ) @ ( transitive_rtrancl @ A @ R ) ) ) )
=> ( ( P @ K )
=> ? [X3: B] :
( ( P @ X3 )
& ! [Y5: B] :
( ( P @ Y5 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( M @ X3 ) @ ( M @ Y5 ) ) @ ( transitive_rtrancl @ A @ R ) ) ) ) ) ) ) ).
% wf_linord_ex_has_least
thf(fact_7649_wf__no__infinite__down__chainE,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),F: nat > A] :
( ( wf @ A @ R )
=> ~ ! [K4: nat] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( F @ ( suc @ K4 ) ) @ ( F @ K4 ) ) @ R ) ) ).
% wf_no_infinite_down_chainE
thf(fact_7650_wf__iff__no__infinite__down__chain,axiom,
! [A: $tType] :
( ( wf @ A )
= ( ^ [R2: set @ ( product_prod @ A @ A )] :
~ ? [F2: nat > A] :
! [I2: nat] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( F2 @ ( suc @ I2 ) ) @ ( F2 @ I2 ) ) @ R2 ) ) ) ).
% wf_iff_no_infinite_down_chain
thf(fact_7651_wf__bounded__measure,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),Ub: A > nat,F: A > nat] :
( ! [A4: A,B4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B4 @ A4 ) @ R )
=> ( ( ord_less_eq @ nat @ ( Ub @ B4 ) @ ( Ub @ A4 ) )
& ( ord_less_eq @ nat @ ( F @ B4 ) @ ( Ub @ A4 ) )
& ( ord_less @ nat @ ( F @ A4 ) @ ( F @ B4 ) ) ) )
=> ( wf @ A @ R ) ) ).
% wf_bounded_measure
thf(fact_7652_cINF__lower,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [F: B > A,A5: set @ B,X2: B] :
( ( condit1013018076250108175_below @ A @ ( image2 @ B @ A @ F @ A5 ) )
=> ( ( member @ B @ X2 @ A5 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F @ A5 ) ) @ ( F @ X2 ) ) ) ) ) ).
% cINF_lower
thf(fact_7653_cINF__lower2,axiom,
! [B: $tType,A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [F: B > A,A5: set @ B,X2: B,U: A] :
( ( condit1013018076250108175_below @ A @ ( image2 @ B @ A @ F @ A5 ) )
=> ( ( member @ B @ X2 @ A5 )
=> ( ( ord_less_eq @ A @ ( F @ X2 ) @ U )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F @ A5 ) ) @ U ) ) ) ) ) ).
% cINF_lower2
thf(fact_7654_le__cInf__iff,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [S2: set @ A,A2: A] :
( ( S2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit1013018076250108175_below @ A @ S2 )
=> ( ( ord_less_eq @ A @ A2 @ ( complete_Inf_Inf @ A @ S2 ) )
= ( ! [X: A] :
( ( member @ A @ X @ S2 )
=> ( ord_less_eq @ A @ A2 @ X ) ) ) ) ) ) ) ).
% le_cInf_iff
thf(fact_7655_cInf__mono,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [B5: set @ A,A5: set @ A] :
( ( B5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit1013018076250108175_below @ A @ A5 )
=> ( ! [B4: A] :
( ( member @ A @ B4 @ B5 )
=> ? [X4: A] :
( ( member @ A @ X4 @ A5 )
& ( ord_less_eq @ A @ X4 @ B4 ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A5 ) @ ( complete_Inf_Inf @ A @ B5 ) ) ) ) ) ) ).
% cInf_mono
thf(fact_7656_bdd__belowI2,axiom,
! [A: $tType,B: $tType] :
( ( preorder @ A )
=> ! [A5: set @ B,M: A,F: B > A] :
( ! [X3: B] :
( ( member @ B @ X3 @ A5 )
=> ( ord_less_eq @ A @ M @ ( F @ X3 ) ) )
=> ( condit1013018076250108175_below @ A @ ( image2 @ B @ A @ F @ A5 ) ) ) ) ).
% bdd_belowI2
thf(fact_7657_bdd__below_OI2,axiom,
! [A: $tType,B: $tType] :
( ( preorder @ A )
=> ! [A5: set @ B,M3: A,F: B > A] :
( ! [X3: B] :
( ( member @ B @ X3 @ A5 )
=> ( ord_less_eq @ A @ M3 @ ( F @ X3 ) ) )
=> ( condit1013018076250108175_below @ A @ ( image2 @ B @ A @ F @ A5 ) ) ) ) ).
% bdd_below.I2
thf(fact_7658_bdd__below_Ounfold,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( condit1013018076250108175_below @ A )
= ( ^ [A8: set @ A] :
? [M6: A] :
! [X: A] :
( ( member @ A @ X @ A8 )
=> ( ord_less_eq @ A @ M6 @ X ) ) ) ) ) ).
% bdd_below.unfold
thf(fact_7659_bdd__below_OE,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A5: set @ A] :
( ( condit1013018076250108175_below @ A @ A5 )
=> ~ ! [M10: A] :
~ ! [X4: A] :
( ( member @ A @ X4 @ A5 )
=> ( ord_less_eq @ A @ M10 @ X4 ) ) ) ) ).
% bdd_below.E
thf(fact_7660_cInf__lower,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X2: A,X8: set @ A] :
( ( member @ A @ X2 @ X8 )
=> ( ( condit1013018076250108175_below @ A @ X8 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ X8 ) @ X2 ) ) ) ) ).
% cInf_lower
thf(fact_7661_cInf__lower2,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X2: A,X8: set @ A,Y: A] :
( ( member @ A @ X2 @ X8 )
=> ( ( ord_less_eq @ A @ X2 @ Y )
=> ( ( condit1013018076250108175_below @ A @ X8 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ X8 ) @ Y ) ) ) ) ) ).
% cInf_lower2
thf(fact_7662_wf__int__ge__less__than2,axiom,
! [D2: int] : ( wf @ int @ ( int_ge_less_than2 @ D2 ) ) ).
% wf_int_ge_less_than2
thf(fact_7663_wf__int__ge__less__than,axiom,
! [D2: int] : ( wf @ int @ ( int_ge_less_than @ D2 ) ) ).
% wf_int_ge_less_than
thf(fact_7664_wf__less,axiom,
wf @ nat @ ( collect @ ( product_prod @ nat @ nat ) @ ( product_case_prod @ nat @ nat @ $o @ ( ord_less @ nat ) ) ) ).
% wf_less
thf(fact_7665_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_7666_wf__if__measure,axiom,
! [A: $tType,P: A > $o,F: A > nat,G: A > A] :
( ! [X3: A] :
( ( P @ X3 )
=> ( ord_less @ nat @ ( F @ ( G @ X3 ) ) @ ( F @ X3 ) ) )
=> ( wf @ A
@ ( collect @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ^ [Y4: A,X: A] :
( ( P @ X )
& ( Y4
= ( G @ X ) ) ) ) ) ) ) ).
% wf_if_measure
thf(fact_7667_wfI,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A5: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R
@ ( product_Sigma @ A @ A @ A5
@ ^ [Uu: A] : B5 ) )
=> ( ! [X3: A,P9: A > $o] :
( ! [Xa2: A] :
( ! [Y2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ Xa2 ) @ R )
=> ( P9 @ Y2 ) )
=> ( P9 @ Xa2 ) )
=> ( ( member @ A @ X3 @ A5 )
=> ( ( member @ A @ X3 @ B5 )
=> ( P9 @ X3 ) ) ) )
=> ( wf @ A @ R ) ) ) ).
% wfI
thf(fact_7668_less__cINF__D,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [F: B > A,A5: set @ B,Y: A,I: B] :
( ( condit1013018076250108175_below @ A @ ( image2 @ B @ A @ F @ A5 ) )
=> ( ( ord_less @ A @ Y @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F @ A5 ) ) )
=> ( ( member @ B @ I @ A5 )
=> ( ord_less @ A @ Y @ ( F @ I ) ) ) ) ) ) ).
% less_cINF_D
thf(fact_7669_cINF__mono,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [B5: set @ B,F: C > A,A5: set @ C,G: B > A] :
( ( B5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit1013018076250108175_below @ A @ ( image2 @ C @ A @ F @ A5 ) )
=> ( ! [M4: B] :
( ( member @ B @ M4 @ B5 )
=> ? [X4: C] :
( ( member @ C @ X4 @ A5 )
& ( ord_less_eq @ A @ ( F @ X4 ) @ ( G @ M4 ) ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ C @ A @ F @ A5 ) ) @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ G @ B5 ) ) ) ) ) ) ) ).
% cINF_mono
thf(fact_7670_le__cINF__iff,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A5: set @ B,F: B > A,U: A] :
( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit1013018076250108175_below @ A @ ( image2 @ B @ A @ F @ A5 ) )
=> ( ( ord_less_eq @ A @ U @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F @ A5 ) ) )
= ( ! [X: B] :
( ( member @ B @ X @ A5 )
=> ( ord_less_eq @ A @ U @ ( F @ X ) ) ) ) ) ) ) ) ).
% le_cINF_iff
thf(fact_7671_cInf__superset__mono,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A5: set @ A,B5: set @ A] :
( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit1013018076250108175_below @ A @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ B5 ) @ ( complete_Inf_Inf @ A @ A5 ) ) ) ) ) ) ).
% cInf_superset_mono
thf(fact_7672_cINF__less__iff,axiom,
! [A: $tType,B: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [A5: set @ B,F: B > A,A2: A] :
( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit1013018076250108175_below @ A @ ( image2 @ B @ A @ F @ A5 ) )
=> ( ( ord_less @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F @ A5 ) ) @ A2 )
= ( ? [X: B] :
( ( member @ B @ X @ A5 )
& ( ord_less @ A @ ( F @ X ) @ A2 ) ) ) ) ) ) ) ).
% cINF_less_iff
thf(fact_7673_cINF__inf__distrib,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A5: set @ B,F: B > A,G: B > A] :
( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit1013018076250108175_below @ A @ ( image2 @ B @ A @ F @ A5 ) )
=> ( ( condit1013018076250108175_below @ A @ ( image2 @ B @ A @ G @ A5 ) )
=> ( ( inf_inf @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F @ A5 ) ) @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ G @ A5 ) ) )
= ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [A3: B] : ( inf_inf @ A @ ( F @ A3 ) @ ( G @ A3 ) )
@ A5 ) ) ) ) ) ) ) ).
% cINF_inf_distrib
thf(fact_7674_wf__bounded__supset,axiom,
! [A: $tType,S2: set @ A] :
( ( finite_finite2 @ A @ S2 )
=> ( wf @ ( set @ A )
@ ( collect @ ( product_prod @ ( set @ A ) @ ( set @ A ) )
@ ( product_case_prod @ ( set @ A ) @ ( set @ A ) @ $o
@ ^ [Q9: set @ A,Q8: set @ A] :
( ( ord_less @ ( set @ A ) @ Q8 @ Q9 )
& ( ord_less_eq @ ( set @ A ) @ Q9 @ S2 ) ) ) ) ) ) ).
% wf_bounded_supset
thf(fact_7675_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
@ ^ [X5: set @ A,Y6: set @ A] :
( ( ord_less @ ( set @ A ) @ X5 @ Y6 )
& ( ord_less_eq @ ( set @ A ) @ Y6 @ A5 ) ) ) ) ) ) ).
% finite_subset_wf
thf(fact_7676_cSUP__eq__cINF__D,axiom,
! [B: $tType,C: $tType] :
( ( condit1219197933456340205attice @ B )
=> ! [F: C > B,A5: set @ C,A2: C] :
( ( ( complete_Sup_Sup @ B @ ( image2 @ C @ B @ F @ A5 ) )
= ( complete_Inf_Inf @ B @ ( image2 @ C @ B @ F @ A5 ) ) )
=> ( ( condit941137186595557371_above @ B @ ( image2 @ C @ B @ F @ A5 ) )
=> ( ( condit1013018076250108175_below @ B @ ( image2 @ C @ B @ F @ A5 ) )
=> ( ( member @ C @ A2 @ A5 )
=> ( ( F @ A2 )
= ( complete_Inf_Inf @ B @ ( image2 @ C @ B @ F @ A5 ) ) ) ) ) ) ) ) ).
% cSUP_eq_cINF_D
thf(fact_7677_cINF__superset__mono,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A5: set @ B,G: B > A,B5: set @ B,F: B > A] :
( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit1013018076250108175_below @ A @ ( image2 @ B @ A @ G @ B5 ) )
=> ( ( ord_less_eq @ ( set @ B ) @ A5 @ B5 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ B5 )
=> ( ord_less_eq @ A @ ( G @ X3 ) @ ( F @ X3 ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ G @ B5 ) ) @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F @ A5 ) ) ) ) ) ) ) ) ).
% cINF_superset_mono
thf(fact_7678_less__eq__cInf__inter,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A5: set @ A,B5: set @ A] :
( ( condit1013018076250108175_below @ A @ A5 )
=> ( ( condit1013018076250108175_below @ A @ B5 )
=> ( ( ( inf_inf @ ( set @ A ) @ A5 @ B5 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ A @ ( inf_inf @ A @ ( complete_Inf_Inf @ A @ A5 ) @ ( complete_Inf_Inf @ A @ B5 ) ) @ ( complete_Inf_Inf @ A @ ( inf_inf @ ( set @ A ) @ A5 @ B5 ) ) ) ) ) ) ) ).
% less_eq_cInf_inter
thf(fact_7679_cInf__le__cSup,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A5: set @ A] :
( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit941137186595557371_above @ A @ A5 )
=> ( ( condit1013018076250108175_below @ A @ A5 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A5 ) @ ( complete_Sup_Sup @ A @ A5 ) ) ) ) ) ) ).
% cInf_le_cSup
thf(fact_7680_cInf__cSup,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [S2: set @ A] :
( ( S2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit1013018076250108175_below @ A @ S2 )
=> ( ( complete_Inf_Inf @ A @ S2 )
= ( complete_Sup_Sup @ A
@ ( collect @ A
@ ^ [X: A] :
! [Y4: A] :
( ( member @ A @ Y4 @ S2 )
=> ( ord_less_eq @ A @ X @ Y4 ) ) ) ) ) ) ) ) ).
% cInf_cSup
thf(fact_7681_mono__cInf,axiom,
! [B: $tType,A: $tType] :
( ( ( condit1219197933456340205attice @ A )
& ( condit1219197933456340205attice @ B ) )
=> ! [F: A > B,A5: set @ A] :
( ( order_mono @ A @ B @ F )
=> ( ( condit1013018076250108175_below @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ B @ ( F @ ( complete_Inf_Inf @ A @ A5 ) ) @ ( complete_Inf_Inf @ B @ ( image2 @ A @ B @ F @ A5 ) ) ) ) ) ) ) ).
% mono_cInf
thf(fact_7682_mono__cINF,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ( condit1219197933456340205attice @ A )
& ( condit1219197933456340205attice @ B ) )
=> ! [F: A > B,A5: C > A,I5: set @ C] :
( ( order_mono @ A @ B @ F )
=> ( ( condit1013018076250108175_below @ A @ ( image2 @ C @ A @ A5 @ I5 ) )
=> ( ( I5
!= ( bot_bot @ ( set @ C ) ) )
=> ( ord_less_eq @ B @ ( F @ ( complete_Inf_Inf @ A @ ( image2 @ C @ A @ A5 @ I5 ) ) )
@ ( complete_Inf_Inf @ B
@ ( image2 @ C @ B
@ ^ [X: C] : ( F @ ( A5 @ X ) )
@ I5 ) ) ) ) ) ) ) ).
% mono_cINF
thf(fact_7683_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
@ ^ [A3: A,B3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B3 )
@ ( product_Sigma @ A @ A @ A8
@ ^ [Uu: A] : A8 ) )
& ( P4 @ A3 @ B3 ) ) ) ) ) ) ).
% relation_of_def
thf(fact_7684_filterlim__INF__INF,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,J4: set @ A,I5: set @ B,F: D > C,F4: B > ( filter @ D ),G5: A > ( filter @ C )] :
( ! [M4: A] :
( ( member @ A @ M4 @ J4 )
=> ? [X4: B] :
( ( member @ B @ X4 @ I5 )
& ( ord_less_eq @ ( filter @ C ) @ ( filtermap @ D @ C @ F @ ( F4 @ X4 ) ) @ ( G5 @ M4 ) ) ) )
=> ( filterlim @ D @ C @ F @ ( complete_Inf_Inf @ ( filter @ C ) @ ( image2 @ A @ ( filter @ C ) @ G5 @ J4 ) ) @ ( complete_Inf_Inf @ ( filter @ D ) @ ( image2 @ B @ ( filter @ D ) @ F4 @ I5 ) ) ) ) ).
% filterlim_INF_INF
thf(fact_7685_filtermap__id_H,axiom,
! [A: $tType] :
( ( filtermap @ A @ A
@ ^ [X: A] : X )
= ( ^ [F8: filter @ A] : F8 ) ) ).
% filtermap_id'
thf(fact_7686_filtermap__ident,axiom,
! [A: $tType,F4: filter @ A] :
( ( filtermap @ A @ A
@ ^ [X: A] : X
@ F4 )
= F4 ) ).
% filtermap_ident
thf(fact_7687_filtermap__filtermap,axiom,
! [A: $tType,B: $tType,C: $tType,F: B > A,G: C > B,F4: filter @ C] :
( ( filtermap @ B @ A @ F @ ( filtermap @ C @ B @ G @ F4 ) )
= ( filtermap @ C @ A
@ ^ [X: C] : ( F @ ( G @ X ) )
@ F4 ) ) ).
% filtermap_filtermap
thf(fact_7688_filterlim__filtermap,axiom,
! [B: $tType,A: $tType,C: $tType,F: A > B,F13: filter @ B,G: C > A,F24: filter @ C] :
( ( filterlim @ A @ B @ F @ F13 @ ( filtermap @ C @ A @ G @ F24 ) )
= ( filterlim @ C @ B
@ ^ [X: C] : ( F @ ( G @ X ) )
@ F13
@ F24 ) ) ).
% filterlim_filtermap
thf(fact_7689_eventually__filtermap,axiom,
! [A: $tType,B: $tType,P: A > $o,F: B > A,F4: filter @ B] :
( ( eventually @ A @ P @ ( filtermap @ B @ A @ F @ F4 ) )
= ( eventually @ B
@ ^ [X: B] : ( P @ ( F @ X ) )
@ F4 ) ) ).
% eventually_filtermap
thf(fact_7690_filtermap__fun__inverse,axiom,
! [B: $tType,A: $tType,G: A > B,F4: filter @ B,G5: filter @ A,F: B > A] :
( ( filterlim @ A @ B @ G @ F4 @ G5 )
=> ( ( filterlim @ B @ A @ F @ G5 @ F4 )
=> ( ( eventually @ A
@ ^ [X: A] :
( ( F @ ( G @ X ) )
= X )
@ G5 )
=> ( ( filtermap @ B @ A @ F @ F4 )
= G5 ) ) ) ) ).
% filtermap_fun_inverse
thf(fact_7691_filtermap__SUP,axiom,
! [A: $tType,B: $tType,C: $tType,F: B > A,F4: C > ( filter @ B ),B5: set @ C] :
( ( filtermap @ B @ A @ F @ ( complete_Sup_Sup @ ( filter @ B ) @ ( image2 @ C @ ( filter @ B ) @ F4 @ B5 ) ) )
= ( complete_Sup_Sup @ ( filter @ A )
@ ( image2 @ C @ ( filter @ A )
@ ^ [B3: C] : ( filtermap @ B @ A @ F @ ( F4 @ B3 ) )
@ B5 ) ) ) ).
% filtermap_SUP
thf(fact_7692_filtermap__def,axiom,
! [B: $tType,A: $tType] :
( ( filtermap @ A @ B )
= ( ^ [F2: A > B,F8: filter @ A] :
( abs_filter @ B
@ ^ [P4: B > $o] :
( eventually @ A
@ ^ [X: A] : ( P4 @ ( F2 @ X ) )
@ F8 ) ) ) ) ).
% filtermap_def
thf(fact_7693_filtermap__INF,axiom,
! [A: $tType,B: $tType,C: $tType,F: B > A,F4: C > ( filter @ B ),B5: set @ C] :
( ord_less_eq @ ( filter @ A ) @ ( filtermap @ B @ A @ F @ ( complete_Inf_Inf @ ( filter @ B ) @ ( image2 @ C @ ( filter @ B ) @ F4 @ B5 ) ) )
@ ( complete_Inf_Inf @ ( filter @ A )
@ ( image2 @ C @ ( filter @ A )
@ ^ [B3: C] : ( filtermap @ B @ A @ F @ ( F4 @ B3 ) )
@ B5 ) ) ) ).
% filtermap_INF
thf(fact_7694_prod__filter__principal__singleton2,axiom,
! [B: $tType,A: $tType,F4: filter @ A,X2: B] :
( ( prod_filter @ A @ B @ F4 @ ( principal @ B @ ( insert @ B @ X2 @ ( bot_bot @ ( set @ B ) ) ) ) )
= ( filtermap @ A @ ( product_prod @ A @ B )
@ ^ [A3: A] : ( product_Pair @ A @ B @ A3 @ X2 )
@ F4 ) ) ).
% prod_filter_principal_singleton2
thf(fact_7695_num__of__integer_Otransfer,axiom,
( bNF_rel_fun @ int @ code_integer @ num @ num @ code_pcr_integer
@ ^ [Y3: num,Z: num] : Y3 = Z
@ ( comp @ nat @ num @ int @ num_of_nat @ nat2 )
@ code_num_of_integer ) ).
% num_of_integer.transfer
thf(fact_7696_filtermap__Pair,axiom,
! [A: $tType,B: $tType,C: $tType,F: C > A,G: C > B,F4: filter @ C] :
( ord_less_eq @ ( filter @ ( product_prod @ A @ B ) )
@ ( filtermap @ C @ ( product_prod @ A @ B )
@ ^ [X: C] : ( product_Pair @ A @ B @ ( F @ X ) @ ( G @ X ) )
@ F4 )
@ ( prod_filter @ A @ B @ ( filtermap @ C @ A @ F @ F4 ) @ ( filtermap @ C @ B @ G @ F4 ) ) ) ).
% filtermap_Pair
thf(fact_7697_prod__filter__assoc,axiom,
! [A: $tType,B: $tType,C: $tType,F4: filter @ A,G5: filter @ B,H9: filter @ C] :
( ( prod_filter @ ( product_prod @ A @ B ) @ C @ ( prod_filter @ A @ B @ F4 @ G5 ) @ H9 )
= ( filtermap @ ( product_prod @ A @ ( product_prod @ B @ C ) ) @ ( product_prod @ ( product_prod @ A @ B ) @ C )
@ ( product_case_prod @ A @ ( product_prod @ B @ C ) @ ( product_prod @ ( product_prod @ A @ B ) @ C )
@ ^ [X: A] :
( product_case_prod @ B @ C @ ( product_prod @ ( product_prod @ A @ B ) @ C )
@ ^ [Y4: B] : ( product_Pair @ ( product_prod @ A @ B ) @ C @ ( product_Pair @ A @ B @ X @ Y4 ) ) ) )
@ ( prod_filter @ A @ ( product_prod @ B @ C ) @ F4 @ ( prod_filter @ B @ C @ G5 @ H9 ) ) ) ) ).
% prod_filter_assoc
thf(fact_7698_euclidean__size__integer_Otransfer,axiom,
( bNF_rel_fun @ int @ code_integer @ nat @ nat @ code_pcr_integer
@ ^ [Y3: nat,Z: nat] : Y3 = Z
@ ( euclid6346220572633701492n_size @ int )
@ ( euclid6346220572633701492n_size @ code_integer ) ) ).
% euclidean_size_integer.transfer
thf(fact_7699_filterlim__Pair,axiom,
! [C: $tType,B: $tType,A: $tType,F: A > B,G5: filter @ B,F4: filter @ A,G: A > C,H9: filter @ C] :
( ( filterlim @ A @ B @ F @ G5 @ F4 )
=> ( ( filterlim @ A @ C @ G @ H9 @ F4 )
=> ( filterlim @ A @ ( product_prod @ B @ C )
@ ^ [X: A] : ( product_Pair @ B @ C @ ( F @ X ) @ ( G @ X ) )
@ ( prod_filter @ B @ C @ G5 @ H9 )
@ F4 ) ) ) ).
% filterlim_Pair
thf(fact_7700_integer_Oleft__unique,axiom,
left_unique @ int @ code_integer @ code_pcr_integer ).
% integer.left_unique
thf(fact_7701_zero__integer_Otransfer,axiom,
code_pcr_integer @ ( zero_zero @ int ) @ ( zero_zero @ code_integer ) ).
% zero_integer.transfer
thf(fact_7702_one__integer_Otransfer,axiom,
code_pcr_integer @ ( one_one @ int ) @ ( one_one @ code_integer ) ).
% one_integer.transfer
thf(fact_7703_integer_Obi__unique,axiom,
bi_unique @ int @ code_integer @ code_pcr_integer ).
% integer.bi_unique
thf(fact_7704_integer_Odomain,axiom,
! [P: int > $o] :
( ( ( domainp @ int @ int
@ ^ [Y3: int,Z: int] : Y3 = Z )
= P )
=> ( ( domainp @ int @ code_integer @ code_pcr_integer )
= P ) ) ).
% integer.domain
thf(fact_7705_integer_Obi__total,axiom,
bi_total @ int @ code_integer @ code_pcr_integer ).
% integer.bi_total
thf(fact_7706_flip__bit__integer_Otransfer,axiom,
( bNF_rel_fun @ nat @ nat @ ( int > int ) @ ( code_integer > code_integer )
@ ^ [Y3: nat,Z: nat] : Y3 = Z
@ ( bNF_rel_fun @ int @ code_integer @ int @ code_integer @ code_pcr_integer @ code_pcr_integer )
@ ( bit_se8732182000553998342ip_bit @ int )
@ ( bit_se8732182000553998342ip_bit @ code_integer ) ) ).
% flip_bit_integer.transfer
thf(fact_7707_push__bit__integer_Otransfer,axiom,
( bNF_rel_fun @ nat @ nat @ ( int > int ) @ ( code_integer > code_integer )
@ ^ [Y3: nat,Z: nat] : Y3 = Z
@ ( bNF_rel_fun @ int @ code_integer @ int @ code_integer @ code_pcr_integer @ code_pcr_integer )
@ ( bit_se4730199178511100633sh_bit @ int )
@ ( bit_se4730199178511100633sh_bit @ code_integer ) ) ).
% push_bit_integer.transfer
thf(fact_7708_drop__bit__integer_Otransfer,axiom,
( bNF_rel_fun @ nat @ nat @ ( int > int ) @ ( code_integer > code_integer )
@ ^ [Y3: nat,Z: nat] : Y3 = Z
@ ( bNF_rel_fun @ int @ code_integer @ int @ code_integer @ code_pcr_integer @ code_pcr_integer )
@ ( bit_se4197421643247451524op_bit @ int )
@ ( bit_se4197421643247451524op_bit @ code_integer ) ) ).
% drop_bit_integer.transfer
thf(fact_7709_xor__integer_Otransfer,axiom,
bNF_rel_fun @ int @ code_integer @ ( int > int ) @ ( code_integer > code_integer ) @ code_pcr_integer @ ( bNF_rel_fun @ int @ code_integer @ int @ code_integer @ code_pcr_integer @ code_pcr_integer ) @ ( bit_se5824344971392196577ns_xor @ int ) @ ( bit_se5824344971392196577ns_xor @ code_integer ) ).
% xor_integer.transfer
thf(fact_7710_bit__integer_Otransfer,axiom,
( bNF_rel_fun @ int @ code_integer @ ( nat > $o ) @ ( nat > $o ) @ code_pcr_integer
@ ^ [Y3: nat > $o,Z: nat > $o] : Y3 = Z
@ ( bit_se5641148757651400278ts_bit @ int )
@ ( bit_se5641148757651400278ts_bit @ code_integer ) ) ).
% bit_integer.transfer
thf(fact_7711_or__integer_Otransfer,axiom,
bNF_rel_fun @ int @ code_integer @ ( int > int ) @ ( code_integer > code_integer ) @ code_pcr_integer @ ( bNF_rel_fun @ int @ code_integer @ int @ code_integer @ code_pcr_integer @ code_pcr_integer ) @ ( bit_se1065995026697491101ons_or @ int ) @ ( bit_se1065995026697491101ons_or @ code_integer ) ).
% or_integer.transfer
thf(fact_7712_take__bit__integer_Otransfer,axiom,
( bNF_rel_fun @ nat @ nat @ ( int > int ) @ ( code_integer > code_integer )
@ ^ [Y3: nat,Z: nat] : Y3 = Z
@ ( bNF_rel_fun @ int @ code_integer @ int @ code_integer @ code_pcr_integer @ code_pcr_integer )
@ ( bit_se2584673776208193580ke_bit @ int )
@ ( bit_se2584673776208193580ke_bit @ code_integer ) ) ).
% take_bit_integer.transfer
thf(fact_7713_not__integer_Otransfer,axiom,
bNF_rel_fun @ int @ code_integer @ int @ code_integer @ code_pcr_integer @ code_pcr_integer @ ( bit_ri4277139882892585799ns_not @ int ) @ ( bit_ri4277139882892585799ns_not @ code_integer ) ).
% not_integer.transfer
thf(fact_7714_and__integer_Otransfer,axiom,
bNF_rel_fun @ int @ code_integer @ ( int > int ) @ ( code_integer > code_integer ) @ code_pcr_integer @ ( bNF_rel_fun @ int @ code_integer @ int @ code_integer @ code_pcr_integer @ code_pcr_integer ) @ ( bit_se5824344872417868541ns_and @ int ) @ ( bit_se5824344872417868541ns_and @ code_integer ) ).
% and_integer.transfer
thf(fact_7715_dup_Otransfer,axiom,
( bNF_rel_fun @ int @ code_integer @ int @ code_integer @ code_pcr_integer @ code_pcr_integer
@ ^ [K2: int] : ( plus_plus @ int @ K2 @ K2 )
@ code_dup ) ).
% dup.transfer
thf(fact_7716_integer__of__nat_Otransfer,axiom,
( bNF_rel_fun @ nat @ nat @ int @ code_integer
@ ^ [Y3: nat,Z: nat] : Y3 = Z
@ code_pcr_integer
@ ( semiring_1_of_nat @ int )
@ code_integer_of_nat ) ).
% integer_of_nat.transfer
thf(fact_7717_less__integer_Otransfer,axiom,
( bNF_rel_fun @ int @ code_integer @ ( int > $o ) @ ( code_integer > $o ) @ code_pcr_integer
@ ( bNF_rel_fun @ int @ code_integer @ $o @ $o @ code_pcr_integer
@ ^ [Y3: $o,Z: $o] : Y3 = Z )
@ ( ord_less @ int )
@ ( ord_less @ code_integer ) ) ).
% less_integer.transfer
thf(fact_7718_times__integer_Otransfer,axiom,
bNF_rel_fun @ int @ code_integer @ ( int > int ) @ ( code_integer > code_integer ) @ code_pcr_integer @ ( bNF_rel_fun @ int @ code_integer @ int @ code_integer @ code_pcr_integer @ code_pcr_integer ) @ ( times_times @ int ) @ ( times_times @ code_integer ) ).
% times_integer.transfer
thf(fact_7719_divide__integer_Otransfer,axiom,
bNF_rel_fun @ int @ code_integer @ ( int > int ) @ ( code_integer > code_integer ) @ code_pcr_integer @ ( bNF_rel_fun @ int @ code_integer @ int @ code_integer @ code_pcr_integer @ code_pcr_integer ) @ ( divide_divide @ int ) @ ( divide_divide @ code_integer ) ).
% divide_integer.transfer
thf(fact_7720_abs__integer_Otransfer,axiom,
bNF_rel_fun @ int @ code_integer @ int @ code_integer @ code_pcr_integer @ code_pcr_integer @ ( abs_abs @ int ) @ ( abs_abs @ code_integer ) ).
% abs_integer.transfer
thf(fact_7721_nat__of__integer_Otransfer,axiom,
( bNF_rel_fun @ int @ code_integer @ nat @ nat @ code_pcr_integer
@ ^ [Y3: nat,Z: nat] : Y3 = Z
@ nat2
@ code_nat_of_integer ) ).
% nat_of_integer.transfer
thf(fact_7722_modulo__integer_Otransfer,axiom,
bNF_rel_fun @ int @ code_integer @ ( int > int ) @ ( code_integer > code_integer ) @ code_pcr_integer @ ( bNF_rel_fun @ int @ code_integer @ int @ code_integer @ code_pcr_integer @ code_pcr_integer ) @ ( modulo_modulo @ int ) @ ( modulo_modulo @ code_integer ) ).
% modulo_integer.transfer
thf(fact_7723_plus__integer_Otransfer,axiom,
bNF_rel_fun @ int @ code_integer @ ( int > int ) @ ( code_integer > code_integer ) @ code_pcr_integer @ ( bNF_rel_fun @ int @ code_integer @ int @ code_integer @ code_pcr_integer @ code_pcr_integer ) @ ( plus_plus @ int ) @ ( plus_plus @ code_integer ) ).
% plus_integer.transfer
thf(fact_7724_sgn__integer_Otransfer,axiom,
bNF_rel_fun @ int @ code_integer @ int @ code_integer @ code_pcr_integer @ code_pcr_integer @ ( sgn_sgn @ int ) @ ( sgn_sgn @ code_integer ) ).
% sgn_integer.transfer
thf(fact_7725_integer_Orep__transfer,axiom,
( bNF_rel_fun @ int @ code_integer @ int @ int @ code_pcr_integer
@ ^ [Y3: int,Z: int] : Y3 = Z
@ ^ [X: int] : X
@ code_int_of_integer ) ).
% integer.rep_transfer
thf(fact_7726_integer_Oid__abs__transfer,axiom,
( bNF_rel_fun @ int @ int @ int @ code_integer
@ ^ [Y3: int,Z: int] : Y3 = Z
@ code_pcr_integer
@ ^ [X: int] : X
@ code_integer_of_int ) ).
% integer.id_abs_transfer
thf(fact_7727_gcd__integer_Otransfer,axiom,
bNF_rel_fun @ int @ code_integer @ ( int > int ) @ ( code_integer > code_integer ) @ code_pcr_integer @ ( bNF_rel_fun @ int @ code_integer @ int @ code_integer @ code_pcr_integer @ code_pcr_integer ) @ ( gcd_gcd @ int ) @ ( gcd_gcd @ code_integer ) ).
% gcd_integer.transfer
thf(fact_7728_minus__integer_Otransfer,axiom,
bNF_rel_fun @ int @ code_integer @ ( int > int ) @ ( code_integer > code_integer ) @ code_pcr_integer @ ( bNF_rel_fun @ int @ code_integer @ int @ code_integer @ code_pcr_integer @ code_pcr_integer ) @ ( minus_minus @ int ) @ ( minus_minus @ code_integer ) ).
% minus_integer.transfer
thf(fact_7729_uminus__integer_Otransfer,axiom,
bNF_rel_fun @ int @ code_integer @ int @ code_integer @ code_pcr_integer @ code_pcr_integer @ ( uminus_uminus @ int ) @ ( uminus_uminus @ code_integer ) ).
% uminus_integer.transfer
thf(fact_7730_mask__integer_Otransfer,axiom,
( bNF_rel_fun @ nat @ nat @ int @ code_integer
@ ^ [Y3: nat,Z: nat] : Y3 = Z
@ code_pcr_integer
@ ( bit_se2239418461657761734s_mask @ int )
@ ( bit_se2239418461657761734s_mask @ code_integer ) ) ).
% mask_integer.transfer
thf(fact_7731_unset__bit__integer_Otransfer,axiom,
( bNF_rel_fun @ nat @ nat @ ( int > int ) @ ( code_integer > code_integer )
@ ^ [Y3: nat,Z: nat] : Y3 = Z
@ ( bNF_rel_fun @ int @ code_integer @ int @ code_integer @ code_pcr_integer @ code_pcr_integer )
@ ( bit_se2638667681897837118et_bit @ int )
@ ( bit_se2638667681897837118et_bit @ code_integer ) ) ).
% unset_bit_integer.transfer
thf(fact_7732_set__bit__integer_Otransfer,axiom,
( bNF_rel_fun @ nat @ nat @ ( int > int ) @ ( code_integer > code_integer )
@ ^ [Y3: nat,Z: nat] : Y3 = Z
@ ( bNF_rel_fun @ int @ code_integer @ int @ code_integer @ code_pcr_integer @ code_pcr_integer )
@ ( bit_se5668285175392031749et_bit @ int )
@ ( bit_se5668285175392031749et_bit @ code_integer ) ) ).
% set_bit_integer.transfer
thf(fact_7733_integer_Oright__total,axiom,
right_total @ int @ code_integer @ code_pcr_integer ).
% integer.right_total
thf(fact_7734_less__eq__integer_Otransfer,axiom,
( bNF_rel_fun @ int @ code_integer @ ( int > $o ) @ ( code_integer > $o ) @ code_pcr_integer
@ ( bNF_rel_fun @ int @ code_integer @ $o @ $o @ code_pcr_integer
@ ^ [Y3: $o,Z: $o] : Y3 = Z )
@ ( ord_less_eq @ int )
@ ( ord_less_eq @ code_integer ) ) ).
% less_eq_integer.transfer
thf(fact_7735_division__segment__integer_Otransfer,axiom,
bNF_rel_fun @ int @ code_integer @ int @ code_integer @ code_pcr_integer @ code_pcr_integer @ ( euclid7384307370059645450egment @ int ) @ ( euclid7384307370059645450egment @ code_integer ) ).
% division_segment_integer.transfer
thf(fact_7736_principal__prod__principal,axiom,
! [B: $tType,A: $tType,A5: set @ A,B5: set @ B] :
( ( prod_filter @ A @ B @ ( principal @ A @ A5 ) @ ( principal @ B @ B5 ) )
= ( principal @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A5
@ ^ [Uu: A] : B5 ) ) ) ).
% principal_prod_principal
thf(fact_7737_prod__filter__def,axiom,
! [B: $tType,A: $tType] :
( ( prod_filter @ A @ B )
= ( ^ [F8: filter @ A,G7: filter @ B] :
( complete_Inf_Inf @ ( filter @ ( product_prod @ A @ B ) )
@ ( image2 @ ( product_prod @ ( A > $o ) @ ( B > $o ) ) @ ( filter @ ( product_prod @ A @ B ) )
@ ( product_case_prod @ ( A > $o ) @ ( B > $o ) @ ( filter @ ( product_prod @ A @ B ) )
@ ^ [P4: A > $o,Q8: B > $o] :
( principal @ ( product_prod @ A @ B )
@ ( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [X: A,Y4: B] :
( ( P4 @ X )
& ( Q8 @ Y4 ) ) ) ) ) )
@ ( collect @ ( product_prod @ ( A > $o ) @ ( B > $o ) )
@ ( product_case_prod @ ( A > $o ) @ ( B > $o ) @ $o
@ ^ [P4: A > $o,Q8: B > $o] :
( ( eventually @ A @ P4 @ F8 )
& ( eventually @ B @ Q8 @ G7 ) ) ) ) ) ) ) ) ).
% prod_filter_def
thf(fact_7738_sub_Otransfer,axiom,
( bNF_rel_fun @ num @ num @ ( num > int ) @ ( num > code_integer )
@ ^ [Y3: num,Z: num] : Y3 = Z
@ ( bNF_rel_fun @ num @ num @ int @ code_integer
@ ^ [Y3: num,Z: num] : Y3 = Z
@ code_pcr_integer )
@ ^ [M2: num,N4: num] : ( minus_minus @ int @ ( numeral_numeral @ int @ M2 ) @ ( numeral_numeral @ int @ N4 ) )
@ code_sub ) ).
% sub.transfer
thf(fact_7739_eventually__prod__sequentially,axiom,
! [P: ( product_prod @ nat @ nat ) > $o] :
( ( eventually @ ( product_prod @ nat @ nat ) @ P @ ( prod_filter @ nat @ nat @ ( at_top @ nat ) @ ( at_top @ nat ) ) )
= ( ? [N6: nat] :
! [M2: nat] :
( ( ord_less_eq @ nat @ N6 @ M2 )
=> ! [N4: nat] :
( ( ord_less_eq @ nat @ N6 @ N4 )
=> ( P @ ( product_Pair @ nat @ nat @ N4 @ M2 ) ) ) ) ) ) ).
% eventually_prod_sequentially
thf(fact_7740_eventually__prodI,axiom,
! [A: $tType,B: $tType,P: A > $o,F4: filter @ A,Q: B > $o,G5: filter @ B] :
( ( eventually @ A @ P @ F4 )
=> ( ( eventually @ B @ Q @ G5 )
=> ( eventually @ ( product_prod @ A @ B )
@ ^ [X: product_prod @ A @ B] :
( ( P @ ( product_fst @ A @ B @ X ) )
& ( Q @ ( product_snd @ A @ B @ X ) ) )
@ ( prod_filter @ A @ B @ F4 @ G5 ) ) ) ) ).
% eventually_prodI
thf(fact_7741_eventually__prod2,axiom,
! [A: $tType,B: $tType,A5: filter @ A,P: B > $o,B5: filter @ B] :
( ( A5
!= ( bot_bot @ ( filter @ A ) ) )
=> ( ( eventually @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [X: A] : P )
@ ( prod_filter @ A @ B @ A5 @ B5 ) )
= ( eventually @ B @ P @ B5 ) ) ) ).
% eventually_prod2
thf(fact_7742_eventually__prod1,axiom,
! [A: $tType,B: $tType,B5: filter @ A,P: B > $o,A5: filter @ B] :
( ( B5
!= ( bot_bot @ ( filter @ A ) ) )
=> ( ( eventually @ ( product_prod @ B @ A )
@ ( product_case_prod @ B @ A @ $o
@ ^ [X: B,Y4: A] : ( P @ X ) )
@ ( prod_filter @ B @ A @ A5 @ B5 ) )
= ( eventually @ B @ P @ A5 ) ) ) ).
% eventually_prod1
thf(fact_7743_prod__filter__INF,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,I5: set @ A,J4: set @ B,A5: A > ( filter @ C ),B5: B > ( filter @ D )] :
( ( I5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( J4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( prod_filter @ C @ D @ ( complete_Inf_Inf @ ( filter @ C ) @ ( image2 @ A @ ( filter @ C ) @ A5 @ I5 ) ) @ ( complete_Inf_Inf @ ( filter @ D ) @ ( image2 @ B @ ( filter @ D ) @ B5 @ J4 ) ) )
= ( complete_Inf_Inf @ ( filter @ ( product_prod @ C @ D ) )
@ ( image2 @ A @ ( filter @ ( product_prod @ C @ D ) )
@ ^ [I2: A] :
( complete_Inf_Inf @ ( filter @ ( product_prod @ C @ D ) )
@ ( image2 @ B @ ( filter @ ( product_prod @ C @ D ) )
@ ^ [J3: B] : ( prod_filter @ C @ D @ ( A5 @ I2 ) @ ( B5 @ J3 ) )
@ J4 ) )
@ I5 ) ) ) ) ) ).
% prod_filter_INF
thf(fact_7744_prod__filter__INF1,axiom,
! [B: $tType,C: $tType,A: $tType,I5: set @ A,A5: A > ( filter @ B ),B5: filter @ C] :
( ( I5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( prod_filter @ B @ C @ ( complete_Inf_Inf @ ( filter @ B ) @ ( image2 @ A @ ( filter @ B ) @ A5 @ I5 ) ) @ B5 )
= ( complete_Inf_Inf @ ( filter @ ( product_prod @ B @ C ) )
@ ( image2 @ A @ ( filter @ ( product_prod @ B @ C ) )
@ ^ [I2: A] : ( prod_filter @ B @ C @ ( A5 @ I2 ) @ B5 )
@ I5 ) ) ) ) ).
% prod_filter_INF1
thf(fact_7745_prod__filter__INF2,axiom,
! [B: $tType,C: $tType,A: $tType,J4: set @ A,A5: filter @ B,B5: A > ( filter @ C )] :
( ( J4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( prod_filter @ B @ C @ A5 @ ( complete_Inf_Inf @ ( filter @ C ) @ ( image2 @ A @ ( filter @ C ) @ B5 @ J4 ) ) )
= ( complete_Inf_Inf @ ( filter @ ( product_prod @ B @ C ) )
@ ( image2 @ A @ ( filter @ ( product_prod @ B @ C ) )
@ ^ [I2: A] : ( prod_filter @ B @ C @ A5 @ ( B5 @ I2 ) )
@ J4 ) ) ) ) ).
% prod_filter_INF2
thf(fact_7746_char__of__take__bit__eq,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N: nat,M: A] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) @ N )
=> ( ( unique5772411509450598832har_of @ A @ ( bit_se2584673776208193580ke_bit @ A @ N @ M ) )
= ( unique5772411509450598832har_of @ A @ M ) ) ) ) ).
% char_of_take_bit_eq
thf(fact_7747_Max_Osemilattice__order__set__axioms,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( lattic4895041142388067077er_set @ A @ ( ord_max @ A )
@ ^ [X: A,Y4: A] : ( ord_less_eq @ A @ Y4 @ X )
@ ^ [X: A,Y4: A] : ( ord_less @ A @ Y4 @ X ) ) ) ).
% Max.semilattice_order_set_axioms
thf(fact_7748_char__of__nat,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N: nat] :
( ( unique5772411509450598832har_of @ A @ ( semiring_1_of_nat @ A @ N ) )
= ( unique5772411509450598832har_of @ nat @ N ) ) ) ).
% char_of_nat
thf(fact_7749_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_7750_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_7751_UNIV__char__of__nat,axiom,
( ( top_top @ ( set @ char ) )
= ( image2 @ nat @ char @ ( unique5772411509450598832har_of @ nat ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ).
% UNIV_char_of_nat
thf(fact_7752_inj__on__char__of__nat,axiom,
inj_on @ nat @ char @ ( unique5772411509450598832har_of @ nat ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% inj_on_char_of_nat
thf(fact_7753_Sup__fin_Osemilattice__order__set__axioms,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( lattic4895041142388067077er_set @ A @ ( sup_sup @ A )
@ ^ [X: A,Y4: A] : ( ord_less_eq @ A @ Y4 @ X )
@ ^ [X: A,Y4: A] : ( ord_less @ A @ Y4 @ X ) ) ) ).
% Sup_fin.semilattice_order_set_axioms
thf(fact_7754_char__of__def,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ( ( unique5772411509450598832har_of @ A )
= ( ^ [N4: A] :
( char2
@ ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N4 )
@ ( bit_se5641148757651400278ts_bit @ A @ N4 @ ( one_one @ nat ) )
@ ( bit_se5641148757651400278ts_bit @ A @ N4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
@ ( bit_se5641148757651400278ts_bit @ A @ N4 @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) )
@ ( bit_se5641148757651400278ts_bit @ A @ N4 @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) )
@ ( bit_se5641148757651400278ts_bit @ A @ N4 @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ one2 ) ) ) )
@ ( bit_se5641148757651400278ts_bit @ A @ N4 @ ( numeral_numeral @ nat @ ( bit0 @ ( bit1 @ one2 ) ) ) )
@ ( bit_se5641148757651400278ts_bit @ A @ N4 @ ( numeral_numeral @ nat @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ).
% char_of_def
thf(fact_7755_range__nat__of__char,axiom,
( ( image2 @ char @ nat @ ( comm_s6883823935334413003f_char @ nat ) @ ( top_top @ ( set @ char ) ) )
= ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ).
% range_nat_of_char
thf(fact_7756_char_Osize_I2_J,axiom,
! [X13: $o,X22: $o,X32: $o,X42: $o,X52: $o,X62: $o,X72: $o,X82: $o] :
( ( size_size @ char @ ( char2 @ X13 @ X22 @ X32 @ X42 @ X52 @ X62 @ X72 @ X82 ) )
= ( zero_zero @ nat ) ) ).
% char.size(2)
thf(fact_7757_assert__def,axiom,
! [A: $tType] :
( ( heap_Time_assert @ A )
= ( ^ [P4: A > $o,X: A] : ( if @ ( heap_Time_Heap @ A ) @ ( P4 @ X ) @ ( heap_Time_return @ A @ X ) @ ( heap_Time_raise @ A @ ( cons @ char @ ( char2 @ $true @ $false @ $false @ $false @ $false @ $true @ $true @ $false ) @ ( cons @ char @ ( char2 @ $true @ $true @ $false @ $false @ $true @ $true @ $true @ $false ) @ ( cons @ char @ ( char2 @ $true @ $true @ $false @ $false @ $true @ $true @ $true @ $false ) @ ( cons @ char @ ( char2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ $false ) @ ( cons @ char @ ( char2 @ $false @ $true @ $false @ $false @ $true @ $true @ $true @ $false ) @ ( cons @ char @ ( char2 @ $false @ $false @ $true @ $false @ $true @ $true @ $true @ $false ) @ ( nil @ char ) ) ) ) ) ) ) ) ) ) ) ).
% assert_def
thf(fact_7758_integer__of__char__code,axiom,
! [B0: $o,B14: $o,B23: $o,B32: $o,B42: $o,B52: $o,B62: $o,B72: $o] :
( ( integer_of_char @ ( char2 @ B0 @ B14 @ B23 @ B32 @ B42 @ B52 @ B62 @ B72 ) )
= ( plus_plus @ code_integer @ ( times_times @ code_integer @ ( plus_plus @ code_integer @ ( times_times @ code_integer @ ( plus_plus @ code_integer @ ( times_times @ code_integer @ ( plus_plus @ code_integer @ ( times_times @ code_integer @ ( plus_plus @ code_integer @ ( times_times @ code_integer @ ( plus_plus @ code_integer @ ( times_times @ code_integer @ ( plus_plus @ code_integer @ ( times_times @ code_integer @ ( zero_neq_one_of_bool @ code_integer @ B72 ) @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) @ ( zero_neq_one_of_bool @ code_integer @ B62 ) ) @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) @ ( zero_neq_one_of_bool @ code_integer @ B52 ) ) @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) @ ( zero_neq_one_of_bool @ code_integer @ B42 ) ) @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) @ ( zero_neq_one_of_bool @ code_integer @ B32 ) ) @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) @ ( zero_neq_one_of_bool @ code_integer @ B23 ) ) @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) @ ( zero_neq_one_of_bool @ code_integer @ B14 ) ) @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) @ ( zero_neq_one_of_bool @ code_integer @ B0 ) ) ) ).
% integer_of_char_code
thf(fact_7759_char_Osize__gen,axiom,
! [X13: $o,X22: $o,X32: $o,X42: $o,X52: $o,X62: $o,X72: $o,X82: $o] :
( ( size_char @ ( char2 @ X13 @ X22 @ X32 @ X42 @ X52 @ X62 @ X72 @ X82 ) )
= ( zero_zero @ nat ) ) ).
% char.size_gen
thf(fact_7760_char__of__integer__code,axiom,
( char_of_integer
= ( ^ [K2: code_integer] :
( product_case_prod @ code_integer @ $o @ char
@ ^ [Q02: code_integer,B02: $o] :
( product_case_prod @ code_integer @ $o @ char
@ ^ [Q1: code_integer,B15: $o] :
( product_case_prod @ code_integer @ $o @ char
@ ^ [Q22: code_integer,B24: $o] :
( product_case_prod @ code_integer @ $o @ char
@ ^ [Q32: code_integer,B33: $o] :
( product_case_prod @ code_integer @ $o @ char
@ ^ [Q42: code_integer,B43: $o] :
( product_case_prod @ code_integer @ $o @ char
@ ^ [Q52: code_integer,B53: $o] :
( product_case_prod @ code_integer @ $o @ char
@ ^ [Q62: code_integer,B63: $o] :
( product_case_prod @ code_integer @ $o @ char
@ ^ [Uu: code_integer] : ( char2 @ B02 @ B15 @ B24 @ B33 @ B43 @ B53 @ B63 )
@ ( code_bit_cut_integer @ Q62 ) )
@ ( code_bit_cut_integer @ Q52 ) )
@ ( code_bit_cut_integer @ Q42 ) )
@ ( code_bit_cut_integer @ Q32 ) )
@ ( code_bit_cut_integer @ Q22 ) )
@ ( code_bit_cut_integer @ Q1 ) )
@ ( code_bit_cut_integer @ Q02 ) )
@ ( code_bit_cut_integer @ K2 ) ) ) ) ).
% char_of_integer_code
thf(fact_7761_assert_H__def,axiom,
( refine_Imp_assert
= ( ^ [P4: $o] : ( if @ ( heap_Time_Heap @ product_unit ) @ P4 @ ( heap_Time_ureturn @ product_unit @ product_Unity ) @ ( heap_Time_raise @ product_unit @ ( cons @ char @ ( char2 @ $true @ $false @ $false @ $false @ $false @ $true @ $true @ $false ) @ ( cons @ char @ ( char2 @ $true @ $true @ $false @ $false @ $true @ $true @ $true @ $false ) @ ( cons @ char @ ( char2 @ $true @ $true @ $false @ $false @ $true @ $true @ $true @ $false ) @ ( cons @ char @ ( char2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ $false ) @ ( cons @ char @ ( char2 @ $false @ $true @ $false @ $false @ $true @ $true @ $true @ $false ) @ ( cons @ char @ ( char2 @ $false @ $false @ $true @ $false @ $true @ $true @ $true @ $false ) @ ( nil @ char ) ) ) ) ) ) ) ) ) ) ) ).
% assert'_def
thf(fact_7762_unit__abs__eta__conv,axiom,
! [A: $tType,F: product_unit > A] :
( ( ^ [U2: product_unit] : ( F @ product_Unity ) )
= F ) ).
% unit_abs_eta_conv
thf(fact_7763_wait__def,axiom,
( heap_Time_wait
= ( ^ [N4: nat] :
( heap_Time_Heap2 @ product_unit
@ ^ [H3: heap_ext @ product_unit] : ( some @ ( product_prod @ product_unit @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ product_unit @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ product_Unity @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H3 @ N4 ) ) ) ) ) ) ).
% wait_def
thf(fact_7764_unit__all__impI,axiom,
! [P: product_unit > $o,Q: product_unit > $o] :
( ( ( P @ product_Unity )
=> ( Q @ product_Unity ) )
=> ! [X4: product_unit] :
( ( P @ X4 )
=> ( Q @ X4 ) ) ) ).
% unit_all_impI
thf(fact_7765_eq__sym__Unity__conv,axiom,
! [X2: $o] :
( ( X2
= ( product_Unity = product_Unity ) )
= X2 ) ).
% eq_sym_Unity_conv
thf(fact_7766_refines__assert_H,axiom,
! [Phi: $o] : ( refine_Imp_refines @ product_unit @ ( heap_Time_ureturn @ product_unit @ product_Unity ) @ ( refine_Imp_assert @ Phi ) ) ).
% refines_assert'
thf(fact_7767_execute__assert_H,axiom,
! [P: $o,H: heap_ext @ product_unit] :
( ( P
=> ( ( heap_Time_execute @ product_unit @ ( refine_Imp_assert @ P ) @ 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 @ H @ ( zero_zero @ nat ) ) ) ) ) )
& ( ~ P
=> ( ( heap_Time_execute @ product_unit @ ( refine_Imp_assert @ P ) @ H )
= ( none @ ( product_prod @ product_unit @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ) ).
% execute_assert'
thf(fact_7768_upd_H__def,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( ( array_upd2 @ A )
= ( ^ [A3: array @ A,I2: code_integer,X: A] :
( heap_Time_bind @ ( array @ A ) @ product_unit @ ( array_upd @ A @ ( code_nat_of_integer @ I2 ) @ X @ A3 )
@ ^ [Uu: array @ A] : ( heap_Time_ureturn @ product_unit @ product_Unity ) ) ) ) ) ).
% upd'_def
thf(fact_7769_CODE__ABORT__def,axiom,
! [A: $tType] :
( ( cODE_ABORT @ A )
= ( ^ [F2: product_unit > A] : ( F2 @ product_Unity ) ) ) ).
% CODE_ABORT_def
thf(fact_7770_natural__zero__minus__one,axiom,
( ( minus_minus @ code_natural @ ( zero_zero @ code_natural ) @ ( one_one @ code_natural ) )
= ( zero_zero @ code_natural ) ) ).
% natural_zero_minus_one
thf(fact_7771_execute__update,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [R: ref @ A,V2: A,H: heap_ext @ product_unit] :
( ( heap_Time_execute @ product_unit @ ( ref_update @ A @ R @ 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 @ R @ V2 @ H ) @ ( one_one @ nat ) ) ) ) ) ) ).
% execute_update
thf(fact_7772_minus__shift__def,axiom,
( minus_shift
= ( ^ [R2: code_natural,K2: code_natural,L2: code_natural] : ( if @ code_natural @ ( ord_less @ code_natural @ K2 @ L2 ) @ ( minus_minus @ code_natural @ ( plus_plus @ code_natural @ R2 @ K2 ) @ L2 ) @ ( minus_minus @ code_natural @ K2 @ L2 ) ) ) ) ).
% minus_shift_def
thf(fact_7773_effect__updateE,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [R4: ref @ A,V2: A,H: heap_ext @ product_unit,H4: heap_ext @ product_unit,R: product_unit,N: nat] :
( ( heap_Time_effect @ product_unit @ ( ref_update @ A @ R4 @ V2 ) @ H @ H4 @ R @ N )
=> ~ ( ( H4
= ( ref_set @ A @ R4 @ V2 @ H ) )
=> ( N
!= ( one_one @ nat ) ) ) ) ) ).
% effect_updateE
thf(fact_7774_effect__updateI,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [H4: heap_ext @ product_unit,R: ref @ A,V2: A,H: heap_ext @ product_unit,N: nat,X2: product_unit] :
( ( H4
= ( ref_set @ A @ R @ V2 @ H ) )
=> ( ( N
= ( one_one @ nat ) )
=> ( heap_Time_effect @ product_unit @ ( ref_update @ A @ R @ V2 ) @ H @ H4 @ X2 @ N ) ) ) ) ).
% effect_updateI
thf(fact_7775_Ref__Time_Oupdate__def,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( ( ref_update @ A )
= ( ^ [R2: ref @ A,V4: A] :
( heap_Time_heap @ product_unit
@ ^ [H3: heap_ext @ product_unit] : ( product_Pair @ product_unit @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ product_Unity @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ ( ref_set @ A @ R2 @ V4 @ H3 ) @ ( one_one @ nat ) ) ) ) ) ) ) ).
% Ref_Time.update_def
thf(fact_7776_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_7777_log_Oelims,axiom,
! [X2: code_natural,Xa: code_natural,Y: code_natural] :
( ( ( log @ X2 @ Xa )
= Y )
=> ( ( ( ( ord_less_eq @ code_natural @ X2 @ ( one_one @ code_natural ) )
| ( ord_less @ code_natural @ Xa @ X2 ) )
=> ( Y
= ( one_one @ code_natural ) ) )
& ( ~ ( ( ord_less_eq @ code_natural @ X2 @ ( one_one @ code_natural ) )
| ( ord_less @ code_natural @ Xa @ X2 ) )
=> ( Y
= ( plus_plus @ code_natural @ ( one_one @ code_natural ) @ ( log @ X2 @ ( divide_divide @ code_natural @ Xa @ X2 ) ) ) ) ) ) ) ).
% log.elims
thf(fact_7778_log_Osimps,axiom,
( log
= ( ^ [B3: code_natural,I2: code_natural] :
( if @ code_natural
@ ( ( ord_less_eq @ code_natural @ B3 @ ( one_one @ code_natural ) )
| ( ord_less @ code_natural @ I2 @ B3 ) )
@ ( one_one @ code_natural )
@ ( plus_plus @ code_natural @ ( one_one @ code_natural ) @ ( log @ B3 @ ( divide_divide @ code_natural @ I2 @ B3 ) ) ) ) ) ) ).
% log.simps
thf(fact_7779_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 ) )
@ ^ [V4: code_natural,W: 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 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ V4 ) @ 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 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ W ) ) )
@ ( product_snd @ code_natural @ ( product_prod @ code_natural @ code_natural ) @ ( next @ S6 ) ) )
@ S6 ) ) ) ).
% split_seed_def
thf(fact_7780_Random_Orange__def,axiom,
( range
= ( ^ [K2: 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 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ K2 )
@ ^ [L2: code_natural] :
( product_scomp @ ( product_prod @ code_natural @ code_natural ) @ code_natural @ ( product_prod @ code_natural @ code_natural ) @ ( product_prod @ code_natural @ ( product_prod @ code_natural @ code_natural ) ) @ next
@ ^ [V4: code_natural] : ( product_Pair @ code_natural @ ( product_prod @ code_natural @ code_natural ) @ ( plus_plus @ code_natural @ V4 @ ( times_times @ code_natural @ L2 @ ( numeral_numeral @ code_natural @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
@ ( one_one @ code_natural ) )
@ ^ [V4: code_natural] : ( product_Pair @ code_natural @ ( product_prod @ code_natural @ code_natural ) @ ( modulo_modulo @ code_natural @ V4 @ K2 ) ) ) ) ) ).
% Random.range_def
thf(fact_7781_scomp__scomp,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,F7: $tType,E: $tType,F: A > ( product_prod @ E @ F7 ),G: E > F7 > ( product_prod @ C @ D ),H: C > D > B] :
( ( product_scomp @ A @ C @ D @ B @ ( product_scomp @ A @ E @ F7 @ ( product_prod @ C @ D ) @ F @ G ) @ H )
= ( product_scomp @ A @ E @ F7 @ B @ F
@ ^ [X: E] : ( product_scomp @ F7 @ C @ D @ B @ ( G @ X ) @ H ) ) ) ).
% scomp_scomp
thf(fact_7782_iterate_Oelims,axiom,
! [A: $tType,B: $tType,X2: code_natural,Xa: B > A > ( product_prod @ B @ A ),Xb: B,Y: A > ( product_prod @ B @ A )] :
( ( ( iterate @ B @ A @ X2 @ Xa @ Xb )
= Y )
=> ( ( ( X2
= ( zero_zero @ code_natural ) )
=> ( Y
= ( product_Pair @ B @ A @ Xb ) ) )
& ( ( X2
!= ( zero_zero @ code_natural ) )
=> ( Y
= ( product_scomp @ A @ B @ A @ ( product_prod @ B @ A ) @ ( Xa @ Xb ) @ ( iterate @ B @ A @ ( minus_minus @ code_natural @ X2 @ ( one_one @ code_natural ) ) @ Xa ) ) ) ) ) ) ).
% iterate.elims
thf(fact_7783_iterate_Osimps,axiom,
! [B: $tType,A: $tType] :
( ( iterate @ B @ A )
= ( ^ [K2: code_natural,F2: B > A > ( product_prod @ B @ A ),X: B] :
( if @ ( A > ( product_prod @ B @ A ) )
@ ( K2
= ( zero_zero @ code_natural ) )
@ ( product_Pair @ B @ A @ X )
@ ( product_scomp @ A @ B @ A @ ( product_prod @ B @ A ) @ ( F2 @ X ) @ ( iterate @ B @ A @ ( minus_minus @ code_natural @ K2 @ ( one_one @ code_natural ) ) @ F2 ) ) ) ) ) ).
% iterate.simps
thf(fact_7784_scomp__unfold,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType] :
( ( product_scomp @ A @ B @ C @ D )
= ( ^ [F2: A > ( product_prod @ B @ C ),G2: B > C > D,X: A] : ( G2 @ ( product_fst @ B @ C @ ( F2 @ X ) ) @ ( product_snd @ B @ C @ ( F2 @ X ) ) ) ) ) ).
% scomp_unfold
thf(fact_7785_inc__shift__def,axiom,
( inc_shift
= ( ^ [V4: code_natural,K2: code_natural] : ( if @ code_natural @ ( V4 = K2 ) @ ( one_one @ code_natural ) @ ( plus_plus @ code_natural @ K2 @ ( one_one @ code_natural ) ) ) ) ) ).
% inc_shift_def
thf(fact_7786_natural_Osimps_I4_J,axiom,
! [T: $tType,F1: T,F22: code_natural > T] :
( ( code_case_natural @ T @ F1 @ F22 @ ( zero_zero @ code_natural ) )
= F1 ) ).
% natural.simps(4)
thf(fact_7787_log_Opelims,axiom,
! [X2: code_natural,Xa: code_natural,Y: code_natural] :
( ( ( log @ X2 @ Xa )
= Y )
=> ( ( accp @ ( product_prod @ code_natural @ code_natural ) @ log_rel @ ( product_Pair @ code_natural @ code_natural @ X2 @ Xa ) )
=> ~ ( ( ( ( ( ord_less_eq @ code_natural @ X2 @ ( one_one @ code_natural ) )
| ( ord_less @ code_natural @ Xa @ X2 ) )
=> ( Y
= ( one_one @ code_natural ) ) )
& ( ~ ( ( ord_less_eq @ code_natural @ X2 @ ( one_one @ code_natural ) )
| ( ord_less @ code_natural @ Xa @ X2 ) )
=> ( Y
= ( plus_plus @ code_natural @ ( one_one @ code_natural ) @ ( log @ X2 @ ( divide_divide @ code_natural @ Xa @ X2 ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ code_natural @ code_natural ) @ log_rel @ ( product_Pair @ code_natural @ code_natural @ X2 @ Xa ) ) ) ) ) ).
% log.pelims
thf(fact_7788_Predicate_Oiterate__upto_Opinduct,axiom,
! [A: $tType,A0: code_natural > A,A13: code_natural,A24: 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 @ A13 @ A24 ) ) )
=> ( ! [F5: code_natural > A,N2: code_natural,M4: code_natural] :
( ( accp @ ( product_prod @ ( code_natural > A ) @ ( product_prod @ code_natural @ code_natural ) ) @ ( iterate_upto_rel @ A ) @ ( product_Pair @ ( code_natural > A ) @ ( product_prod @ code_natural @ code_natural ) @ F5 @ ( product_Pair @ code_natural @ code_natural @ N2 @ M4 ) ) )
=> ( ! [X4: product_unit] :
( ~ ( ord_less @ code_natural @ M4 @ N2 )
=> ( P @ F5 @ ( plus_plus @ code_natural @ N2 @ ( one_one @ code_natural ) ) @ M4 ) )
=> ( P @ F5 @ N2 @ M4 ) ) )
=> ( P @ A0 @ A13 @ A24 ) ) ) ).
% Predicate.iterate_upto.pinduct
thf(fact_7789_case__natural__def,axiom,
! [T: $tType] :
( ( code_case_natural @ T )
= ( ^ [F12: T,F23: code_natural > T] :
( code_rec_natural @ T @ F12
@ ^ [X12: code_natural,X24: T] : ( F23 @ X12 ) ) ) ) ).
% case_natural_def
thf(fact_7790_natural_Osimps_I6_J,axiom,
! [T: $tType,F1: T,F22: code_natural > T > T] :
( ( code_rec_natural @ T @ F1 @ F22 @ ( zero_zero @ code_natural ) )
= F1 ) ).
% natural.simps(6)
thf(fact_7791_rec__natural__def,axiom,
! [T: $tType] :
( ( code_rec_natural @ T )
= ( ^ [F12: T,F23: code_natural > T > T,X: code_natural] : ( the @ T @ ( code_rec_set_natural @ T @ F12 @ F23 @ X ) ) ) ) ).
% rec_natural_def
thf(fact_7792_iterate_Opelims,axiom,
! [A: $tType,B: $tType,X2: code_natural,Xa: B > A > ( product_prod @ B @ A ),Xb: B,Y: A > ( product_prod @ B @ A )] :
( ( ( iterate @ B @ A @ X2 @ 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 ) @ X2 @ ( product_Pair @ ( B > A > ( product_prod @ B @ A ) ) @ B @ Xa @ Xb ) ) )
=> ~ ( ( ( ( X2
= ( zero_zero @ code_natural ) )
=> ( Y
= ( product_Pair @ B @ A @ Xb ) ) )
& ( ( X2
!= ( zero_zero @ code_natural ) )
=> ( Y
= ( product_scomp @ A @ B @ A @ ( product_prod @ B @ A ) @ ( Xa @ Xb ) @ ( iterate @ B @ A @ ( minus_minus @ code_natural @ X2 @ ( 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 ) @ X2 @ ( product_Pair @ ( B > A > ( product_prod @ B @ A ) ) @ B @ Xa @ Xb ) ) ) ) ) ) ).
% iterate.pelims
thf(fact_7793_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
@ ^ [K2: code_natural,Uu: A] : ( ord_less @ code_natural @ ( zero_zero @ code_natural ) @ K2 ) )
@ Xs ) )
= ( pick @ A @ Xs ) ) ).
% pick_drop_zero
thf(fact_7794_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
@ ^ [K2: code_natural,Uu: A] : ( ord_less @ code_natural @ ( zero_zero @ code_natural ) @ K2 ) )
@ Xs ) )
= ( select_weight @ A @ Xs ) ) ).
% select_weight_drop_zero
thf(fact_7795_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 ) ) )
@ ^ [K2: code_natural] : ( product_Pair @ A @ ( product_prod @ code_natural @ code_natural ) @ ( pick @ A @ Xs3 @ K2 ) ) ) ) ) ).
% select_weight_def
thf(fact_7796_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_7797_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_7798_natural__of__nat__of__nat,axiom,
( code_natural_of_nat
= ( semiring_1_of_nat @ code_natural ) ) ).
% natural_of_nat_of_nat
thf(fact_7799_divide__natural_Oabs__eq,axiom,
! [Xa: nat,X2: nat] :
( ( divide_divide @ code_natural @ ( code_natural_of_nat @ Xa ) @ ( code_natural_of_nat @ X2 ) )
= ( code_natural_of_nat @ ( divide_divide @ nat @ Xa @ X2 ) ) ) ).
% divide_natural.abs_eq
thf(fact_7800_less__eq__natural_Oabs__eq,axiom,
! [Xa: nat,X2: nat] :
( ( ord_less_eq @ code_natural @ ( code_natural_of_nat @ Xa ) @ ( code_natural_of_nat @ X2 ) )
= ( ord_less_eq @ nat @ Xa @ X2 ) ) ).
% less_eq_natural.abs_eq
thf(fact_7801_minus__natural_Oabs__eq,axiom,
! [Xa: nat,X2: nat] :
( ( minus_minus @ code_natural @ ( code_natural_of_nat @ Xa ) @ ( code_natural_of_nat @ X2 ) )
= ( code_natural_of_nat @ ( minus_minus @ nat @ Xa @ X2 ) ) ) ).
% minus_natural.abs_eq
thf(fact_7802_zero__natural__def,axiom,
( ( zero_zero @ code_natural )
= ( code_natural_of_nat @ ( zero_zero @ nat ) ) ) ).
% zero_natural_def
thf(fact_7803_one__natural__def,axiom,
( ( one_one @ code_natural )
= ( code_natural_of_nat @ ( one_one @ nat ) ) ) ).
% one_natural_def
thf(fact_7804_euclidean__size__natural_Oabs__eq,axiom,
! [X2: nat] :
( ( euclid6346220572633701492n_size @ code_natural @ ( code_natural_of_nat @ X2 ) )
= ( euclid6346220572633701492n_size @ nat @ X2 ) ) ).
% euclidean_size_natural.abs_eq
thf(fact_7805_mask__natural__def,axiom,
( ( bit_se2239418461657761734s_mask @ code_natural )
= ( map_fun @ nat @ nat @ nat @ code_natural @ ( id @ nat ) @ code_natural_of_nat @ ( bit_se2239418461657761734s_mask @ nat ) ) ) ).
% mask_natural_def
thf(fact_7806_drop__bit__natural_Oabs__eq,axiom,
! [Xa: nat,X2: nat] :
( ( bit_se4197421643247451524op_bit @ code_natural @ Xa @ ( code_natural_of_nat @ X2 ) )
= ( code_natural_of_nat @ ( bit_se4197421643247451524op_bit @ nat @ Xa @ X2 ) ) ) ).
% drop_bit_natural.abs_eq
thf(fact_7807_and__natural_Oabs__eq,axiom,
! [Xa: nat,X2: nat] :
( ( bit_se5824344872417868541ns_and @ code_natural @ ( code_natural_of_nat @ Xa ) @ ( code_natural_of_nat @ X2 ) )
= ( code_natural_of_nat @ ( bit_se5824344872417868541ns_and @ nat @ Xa @ X2 ) ) ) ).
% and_natural.abs_eq
thf(fact_7808_bit__natural_Oabs__eq,axiom,
! [X2: nat] :
( ( bit_se5641148757651400278ts_bit @ code_natural @ ( code_natural_of_nat @ X2 ) )
= ( bit_se5641148757651400278ts_bit @ nat @ X2 ) ) ).
% bit_natural.abs_eq
thf(fact_7809_take__bit__natural_Oabs__eq,axiom,
! [Xa: nat,X2: nat] :
( ( bit_se2584673776208193580ke_bit @ code_natural @ Xa @ ( code_natural_of_nat @ X2 ) )
= ( code_natural_of_nat @ ( bit_se2584673776208193580ke_bit @ nat @ Xa @ X2 ) ) ) ).
% take_bit_natural.abs_eq
thf(fact_7810_xor__natural_Oabs__eq,axiom,
! [Xa: nat,X2: nat] :
( ( bit_se5824344971392196577ns_xor @ code_natural @ ( code_natural_of_nat @ Xa ) @ ( code_natural_of_nat @ X2 ) )
= ( code_natural_of_nat @ ( bit_se5824344971392196577ns_xor @ nat @ Xa @ X2 ) ) ) ).
% xor_natural.abs_eq
thf(fact_7811_or__natural_Oabs__eq,axiom,
! [Xa: nat,X2: nat] :
( ( bit_se1065995026697491101ons_or @ code_natural @ ( code_natural_of_nat @ Xa ) @ ( code_natural_of_nat @ X2 ) )
= ( code_natural_of_nat @ ( bit_se1065995026697491101ons_or @ nat @ Xa @ X2 ) ) ) ).
% or_natural.abs_eq
thf(fact_7812_push__bit__natural_Oabs__eq,axiom,
! [Xa: nat,X2: nat] :
( ( bit_se4730199178511100633sh_bit @ code_natural @ Xa @ ( code_natural_of_nat @ X2 ) )
= ( code_natural_of_nat @ ( bit_se4730199178511100633sh_bit @ nat @ Xa @ X2 ) ) ) ).
% push_bit_natural.abs_eq
thf(fact_7813_mask__natural_Oabs__eq,axiom,
( ( bit_se2239418461657761734s_mask @ code_natural )
= ( ^ [X: nat] : ( code_natural_of_nat @ ( bit_se2239418461657761734s_mask @ nat @ X ) ) ) ) ).
% mask_natural.abs_eq
thf(fact_7814_flip__bit__natural_Oabs__eq,axiom,
! [Xa: nat,X2: nat] :
( ( bit_se8732182000553998342ip_bit @ code_natural @ Xa @ ( code_natural_of_nat @ X2 ) )
= ( code_natural_of_nat @ ( bit_se8732182000553998342ip_bit @ nat @ Xa @ X2 ) ) ) ).
% flip_bit_natural.abs_eq
thf(fact_7815_unset__bit__natural_Oabs__eq,axiom,
! [Xa: nat,X2: nat] :
( ( bit_se2638667681897837118et_bit @ code_natural @ Xa @ ( code_natural_of_nat @ X2 ) )
= ( code_natural_of_nat @ ( bit_se2638667681897837118et_bit @ nat @ Xa @ X2 ) ) ) ).
% unset_bit_natural.abs_eq
thf(fact_7816_set__bit__natural_Oabs__eq,axiom,
! [Xa: nat,X2: nat] :
( ( bit_se5668285175392031749et_bit @ code_natural @ Xa @ ( code_natural_of_nat @ X2 ) )
= ( code_natural_of_nat @ ( bit_se5668285175392031749et_bit @ nat @ Xa @ X2 ) ) ) ).
% set_bit_natural.abs_eq
thf(fact_7817_division__segment__natural_Oabs__eq,axiom,
! [X2: nat] :
( ( euclid7384307370059645450egment @ code_natural @ ( code_natural_of_nat @ X2 ) )
= ( code_natural_of_nat @ ( euclid7384307370059645450egment @ nat @ X2 ) ) ) ).
% division_segment_natural.abs_eq
thf(fact_7818_natural__of__nat__cases,axiom,
! [X2: code_natural] :
~ ! [Y2: nat] :
( ( X2
= ( code_natural_of_nat @ Y2 ) )
=> ~ ( member @ nat @ Y2 @ ( top_top @ ( set @ nat ) ) ) ) ).
% natural_of_nat_cases
thf(fact_7819_natural__of__nat__induct,axiom,
! [P: code_natural > $o,X2: code_natural] :
( ! [Y2: nat] :
( ( member @ nat @ Y2 @ ( top_top @ ( set @ nat ) ) )
=> ( P @ ( code_natural_of_nat @ Y2 ) ) )
=> ( P @ X2 ) ) ).
% natural_of_nat_induct
thf(fact_7820_natural__of__nat__inject,axiom,
! [X2: nat,Y: nat] :
( ( member @ nat @ X2 @ ( top_top @ ( set @ nat ) ) )
=> ( ( member @ nat @ Y @ ( top_top @ ( set @ nat ) ) )
=> ( ( ( code_natural_of_nat @ X2 )
= ( code_natural_of_nat @ Y ) )
= ( X2 = Y ) ) ) ) ).
% natural_of_nat_inject
thf(fact_7821_less__natural_Oabs__eq,axiom,
! [Xa: nat,X2: nat] :
( ( ord_less @ code_natural @ ( code_natural_of_nat @ Xa ) @ ( code_natural_of_nat @ X2 ) )
= ( ord_less @ nat @ Xa @ X2 ) ) ).
% less_natural.abs_eq
thf(fact_7822_plus__natural_Oabs__eq,axiom,
! [Xa: nat,X2: nat] :
( ( plus_plus @ code_natural @ ( code_natural_of_nat @ Xa ) @ ( code_natural_of_nat @ X2 ) )
= ( code_natural_of_nat @ ( plus_plus @ nat @ Xa @ X2 ) ) ) ).
% plus_natural.abs_eq
thf(fact_7823_times__natural_Oabs__eq,axiom,
! [Xa: nat,X2: nat] :
( ( times_times @ code_natural @ ( code_natural_of_nat @ Xa ) @ ( code_natural_of_nat @ X2 ) )
= ( code_natural_of_nat @ ( times_times @ nat @ Xa @ X2 ) ) ) ).
% times_natural.abs_eq
thf(fact_7824_modulo__natural_Oabs__eq,axiom,
! [Xa: nat,X2: nat] :
( ( modulo_modulo @ code_natural @ ( code_natural_of_nat @ Xa ) @ ( code_natural_of_nat @ X2 ) )
= ( code_natural_of_nat @ ( modulo_modulo @ nat @ Xa @ X2 ) ) ) ).
% modulo_natural.abs_eq
thf(fact_7825_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 ) ) )
@ ^ [K2: code_natural] : ( product_Pair @ A @ ( product_prod @ code_natural @ code_natural ) @ ( nth @ A @ Xs3 @ ( code_nat_of_natural @ K2 ) ) ) ) ) ) ).
% select_def
thf(fact_7826_Nat_Oabs__eq,axiom,
! [X2: int] :
( ( code_Nat @ ( code_integer_of_int @ X2 ) )
= ( code_natural_of_nat @ ( nat2 @ X2 ) ) ) ).
% Nat.abs_eq
thf(fact_7827_nat__of__natural__inverse,axiom,
! [X2: code_natural] :
( ( code_natural_of_nat @ ( code_nat_of_natural @ X2 ) )
= X2 ) ).
% nat_of_natural_inverse
thf(fact_7828_nat__of__natural__of__nat__inverse,axiom,
! [N: nat] :
( ( code_nat_of_natural @ ( code_natural_of_nat @ N ) )
= N ) ).
% nat_of_natural_of_nat_inverse
thf(fact_7829_natural__of__nat__of__natural__inverse,axiom,
! [N: code_natural] :
( ( code_natural_of_nat @ ( code_nat_of_natural @ N ) )
= N ) ).
% natural_of_nat_of_natural_inverse
thf(fact_7830_nat__of__natural__min,axiom,
! [K: code_natural,L: code_natural] :
( ( code_nat_of_natural @ ( ord_min @ code_natural @ K @ L ) )
= ( ord_min @ nat @ ( code_nat_of_natural @ K ) @ ( code_nat_of_natural @ L ) ) ) ).
% nat_of_natural_min
thf(fact_7831_nat__of__natural__max,axiom,
! [K: code_natural,L: code_natural] :
( ( code_nat_of_natural @ ( ord_max @ code_natural @ K @ L ) )
= ( ord_max @ nat @ ( code_nat_of_natural @ K ) @ ( code_nat_of_natural @ L ) ) ) ).
% nat_of_natural_max
thf(fact_7832_nat__of__natural__of__nat,axiom,
! [N: nat] :
( ( code_nat_of_natural @ ( semiring_1_of_nat @ code_natural @ N ) )
= N ) ).
% nat_of_natural_of_nat
thf(fact_7833_of__nat__of__natural,axiom,
! [N: code_natural] :
( ( semiring_1_of_nat @ code_natural @ ( code_nat_of_natural @ N ) )
= N ) ).
% of_nat_of_natural
thf(fact_7834_zero__natural_Orep__eq,axiom,
( ( code_nat_of_natural @ ( zero_zero @ code_natural ) )
= ( zero_zero @ nat ) ) ).
% zero_natural.rep_eq
thf(fact_7835_one__natural_Orep__eq,axiom,
( ( code_nat_of_natural @ ( one_one @ code_natural ) )
= ( one_one @ nat ) ) ).
% one_natural.rep_eq
thf(fact_7836_plus__natural_Orep__eq,axiom,
! [X2: code_natural,Xa: code_natural] :
( ( code_nat_of_natural @ ( plus_plus @ code_natural @ X2 @ Xa ) )
= ( plus_plus @ nat @ ( code_nat_of_natural @ X2 ) @ ( code_nat_of_natural @ Xa ) ) ) ).
% plus_natural.rep_eq
thf(fact_7837_minus__natural_Orep__eq,axiom,
! [X2: code_natural,Xa: code_natural] :
( ( code_nat_of_natural @ ( minus_minus @ code_natural @ X2 @ Xa ) )
= ( minus_minus @ nat @ ( code_nat_of_natural @ X2 ) @ ( code_nat_of_natural @ Xa ) ) ) ).
% minus_natural.rep_eq
thf(fact_7838_nat__of__natural__numeral,axiom,
! [K: num] :
( ( code_nat_of_natural @ ( numeral_numeral @ code_natural @ K ) )
= ( numeral_numeral @ nat @ K ) ) ).
% nat_of_natural_numeral
thf(fact_7839_times__natural_Orep__eq,axiom,
! [X2: code_natural,Xa: code_natural] :
( ( code_nat_of_natural @ ( times_times @ code_natural @ X2 @ Xa ) )
= ( times_times @ nat @ ( code_nat_of_natural @ X2 ) @ ( code_nat_of_natural @ Xa ) ) ) ).
% times_natural.rep_eq
thf(fact_7840_divide__natural_Orep__eq,axiom,
! [X2: code_natural,Xa: code_natural] :
( ( code_nat_of_natural @ ( divide_divide @ code_natural @ X2 @ Xa ) )
= ( divide_divide @ nat @ ( code_nat_of_natural @ X2 ) @ ( code_nat_of_natural @ Xa ) ) ) ).
% divide_natural.rep_eq
thf(fact_7841_modulo__natural_Orep__eq,axiom,
! [X2: code_natural,Xa: code_natural] :
( ( code_nat_of_natural @ ( modulo_modulo @ code_natural @ X2 @ Xa ) )
= ( modulo_modulo @ nat @ ( code_nat_of_natural @ X2 ) @ ( code_nat_of_natural @ Xa ) ) ) ).
% modulo_natural.rep_eq
thf(fact_7842_division__segment__natural_Orep__eq,axiom,
! [X2: code_natural] :
( ( code_nat_of_natural @ ( euclid7384307370059645450egment @ code_natural @ X2 ) )
= ( euclid7384307370059645450egment @ nat @ ( code_nat_of_natural @ X2 ) ) ) ).
% division_segment_natural.rep_eq
thf(fact_7843_euclidean__size__natural_Orep__eq,axiom,
( ( euclid6346220572633701492n_size @ code_natural )
= ( ^ [X: code_natural] : ( euclid6346220572633701492n_size @ nat @ ( code_nat_of_natural @ X ) ) ) ) ).
% euclidean_size_natural.rep_eq
thf(fact_7844_minus__natural__def,axiom,
( ( minus_minus @ code_natural )
= ( map_fun @ code_natural @ nat @ ( nat > nat ) @ ( code_natural > code_natural ) @ code_nat_of_natural @ ( map_fun @ code_natural @ nat @ nat @ code_natural @ code_nat_of_natural @ code_natural_of_nat ) @ ( minus_minus @ nat ) ) ) ).
% minus_natural_def
thf(fact_7845_or__natural__def,axiom,
( ( bit_se1065995026697491101ons_or @ code_natural )
= ( map_fun @ code_natural @ nat @ ( nat > nat ) @ ( code_natural > code_natural ) @ code_nat_of_natural @ ( map_fun @ code_natural @ nat @ nat @ code_natural @ code_nat_of_natural @ code_natural_of_nat ) @ ( bit_se1065995026697491101ons_or @ nat ) ) ) ).
% or_natural_def
thf(fact_7846_xor__natural__def,axiom,
( ( bit_se5824344971392196577ns_xor @ code_natural )
= ( map_fun @ code_natural @ nat @ ( nat > nat ) @ ( code_natural > code_natural ) @ code_nat_of_natural @ ( map_fun @ code_natural @ nat @ nat @ code_natural @ code_nat_of_natural @ code_natural_of_nat ) @ ( bit_se5824344971392196577ns_xor @ nat ) ) ) ).
% xor_natural_def
thf(fact_7847_and__natural__def,axiom,
( ( bit_se5824344872417868541ns_and @ code_natural )
= ( map_fun @ code_natural @ nat @ ( nat > nat ) @ ( code_natural > code_natural ) @ code_nat_of_natural @ ( map_fun @ code_natural @ nat @ nat @ code_natural @ code_nat_of_natural @ code_natural_of_nat ) @ ( bit_se5824344872417868541ns_and @ nat ) ) ) ).
% and_natural_def
thf(fact_7848_division__segment__natural__def,axiom,
( ( euclid7384307370059645450egment @ code_natural )
= ( map_fun @ code_natural @ nat @ nat @ code_natural @ code_nat_of_natural @ code_natural_of_nat @ ( euclid7384307370059645450egment @ nat ) ) ) ).
% division_segment_natural_def
thf(fact_7849_natural__of__nat__inverse,axiom,
! [Y: nat] :
( ( member @ nat @ Y @ ( top_top @ ( set @ nat ) ) )
=> ( ( code_nat_of_natural @ ( code_natural_of_nat @ Y ) )
= Y ) ) ).
% natural_of_nat_inverse
thf(fact_7850_type__definition__natural,axiom,
type_definition @ code_natural @ nat @ code_nat_of_natural @ code_natural_of_nat @ ( top_top @ ( set @ nat ) ) ).
% type_definition_natural
thf(fact_7851_plus__natural__def,axiom,
( ( plus_plus @ code_natural )
= ( map_fun @ code_natural @ nat @ ( nat > nat ) @ ( code_natural > code_natural ) @ code_nat_of_natural @ ( map_fun @ code_natural @ nat @ nat @ code_natural @ code_nat_of_natural @ code_natural_of_nat ) @ ( plus_plus @ nat ) ) ) ).
% plus_natural_def
thf(fact_7852_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_7853_less__natural_Orep__eq,axiom,
( ( ord_less @ code_natural )
= ( ^ [X: code_natural,Xa5: code_natural] : ( ord_less @ nat @ ( code_nat_of_natural @ X ) @ ( code_nat_of_natural @ Xa5 ) ) ) ) ).
% less_natural.rep_eq
thf(fact_7854_nat__of__natural__induct,axiom,
! [Y: nat,P: nat > $o] :
( ( member @ nat @ Y @ ( top_top @ ( set @ nat ) ) )
=> ( ! [X3: code_natural] : ( P @ ( code_nat_of_natural @ X3 ) )
=> ( P @ Y ) ) ) ).
% nat_of_natural_induct
thf(fact_7855_nat__of__natural__cases,axiom,
! [Y: nat] :
( ( member @ nat @ Y @ ( top_top @ ( set @ nat ) ) )
=> ~ ! [X3: code_natural] :
( Y
!= ( code_nat_of_natural @ X3 ) ) ) ).
% nat_of_natural_cases
thf(fact_7856_nat__of__natural,axiom,
! [X2: code_natural] : ( member @ nat @ ( code_nat_of_natural @ X2 ) @ ( top_top @ ( set @ nat ) ) ) ).
% nat_of_natural
thf(fact_7857_drop__bit__natural_Orep__eq,axiom,
! [X2: nat,Xa: code_natural] :
( ( code_nat_of_natural @ ( bit_se4197421643247451524op_bit @ code_natural @ X2 @ Xa ) )
= ( bit_se4197421643247451524op_bit @ nat @ X2 @ ( code_nat_of_natural @ Xa ) ) ) ).
% drop_bit_natural.rep_eq
thf(fact_7858_bit__natural_Orep__eq,axiom,
( ( bit_se5641148757651400278ts_bit @ code_natural )
= ( ^ [X: code_natural] : ( bit_se5641148757651400278ts_bit @ nat @ ( code_nat_of_natural @ X ) ) ) ) ).
% bit_natural.rep_eq
thf(fact_7859_or__natural_Orep__eq,axiom,
! [X2: code_natural,Xa: code_natural] :
( ( code_nat_of_natural @ ( bit_se1065995026697491101ons_or @ code_natural @ X2 @ Xa ) )
= ( bit_se1065995026697491101ons_or @ nat @ ( code_nat_of_natural @ X2 ) @ ( code_nat_of_natural @ Xa ) ) ) ).
% or_natural.rep_eq
thf(fact_7860_xor__natural_Orep__eq,axiom,
! [X2: code_natural,Xa: code_natural] :
( ( code_nat_of_natural @ ( bit_se5824344971392196577ns_xor @ code_natural @ X2 @ Xa ) )
= ( bit_se5824344971392196577ns_xor @ nat @ ( code_nat_of_natural @ X2 ) @ ( code_nat_of_natural @ Xa ) ) ) ).
% xor_natural.rep_eq
thf(fact_7861_natural__eq__iff,axiom,
( ( ^ [Y3: code_natural,Z: code_natural] : Y3 = Z )
= ( ^ [M2: code_natural,N4: code_natural] :
( ( code_nat_of_natural @ M2 )
= ( code_nat_of_natural @ N4 ) ) ) ) ).
% natural_eq_iff
thf(fact_7862_natural__eqI,axiom,
! [M: code_natural,N: code_natural] :
( ( ( code_nat_of_natural @ M )
= ( code_nat_of_natural @ N ) )
=> ( M = N ) ) ).
% natural_eqI
thf(fact_7863_nat__of__natural__inject,axiom,
! [X2: code_natural,Y: code_natural] :
( ( ( code_nat_of_natural @ X2 )
= ( code_nat_of_natural @ Y ) )
= ( X2 = Y ) ) ).
% nat_of_natural_inject
thf(fact_7864_take__bit__natural_Orep__eq,axiom,
! [X2: nat,Xa: code_natural] :
( ( code_nat_of_natural @ ( bit_se2584673776208193580ke_bit @ code_natural @ X2 @ Xa ) )
= ( bit_se2584673776208193580ke_bit @ nat @ X2 @ ( code_nat_of_natural @ Xa ) ) ) ).
% take_bit_natural.rep_eq
thf(fact_7865_and__natural_Orep__eq,axiom,
! [X2: code_natural,Xa: code_natural] :
( ( code_nat_of_natural @ ( bit_se5824344872417868541ns_and @ code_natural @ X2 @ Xa ) )
= ( bit_se5824344872417868541ns_and @ nat @ ( code_nat_of_natural @ X2 ) @ ( code_nat_of_natural @ Xa ) ) ) ).
% and_natural.rep_eq
thf(fact_7866_mask__natural_Orep__eq,axiom,
! [X2: nat] :
( ( code_nat_of_natural @ ( bit_se2239418461657761734s_mask @ code_natural @ X2 ) )
= ( bit_se2239418461657761734s_mask @ nat @ X2 ) ) ).
% mask_natural.rep_eq
thf(fact_7867_push__bit__natural_Orep__eq,axiom,
! [X2: nat,Xa: code_natural] :
( ( code_nat_of_natural @ ( bit_se4730199178511100633sh_bit @ code_natural @ X2 @ Xa ) )
= ( bit_se4730199178511100633sh_bit @ nat @ X2 @ ( code_nat_of_natural @ Xa ) ) ) ).
% push_bit_natural.rep_eq
thf(fact_7868_set__bit__natural_Orep__eq,axiom,
! [X2: nat,Xa: code_natural] :
( ( code_nat_of_natural @ ( bit_se5668285175392031749et_bit @ code_natural @ X2 @ Xa ) )
= ( bit_se5668285175392031749et_bit @ nat @ X2 @ ( code_nat_of_natural @ Xa ) ) ) ).
% set_bit_natural.rep_eq
thf(fact_7869_unset__bit__natural_Orep__eq,axiom,
! [X2: nat,Xa: code_natural] :
( ( code_nat_of_natural @ ( bit_se2638667681897837118et_bit @ code_natural @ X2 @ Xa ) )
= ( bit_se2638667681897837118et_bit @ nat @ X2 @ ( code_nat_of_natural @ Xa ) ) ) ).
% unset_bit_natural.rep_eq
thf(fact_7870_flip__bit__natural_Orep__eq,axiom,
! [X2: nat,Xa: code_natural] :
( ( code_nat_of_natural @ ( bit_se8732182000553998342ip_bit @ code_natural @ X2 @ Xa ) )
= ( bit_se8732182000553998342ip_bit @ nat @ X2 @ ( code_nat_of_natural @ Xa ) ) ) ).
% flip_bit_natural.rep_eq
thf(fact_7871_Nat_Orep__eq,axiom,
! [X2: code_integer] :
( ( code_nat_of_natural @ ( code_Nat @ X2 ) )
= ( nat2 @ ( code_int_of_integer @ X2 ) ) ) ).
% Nat.rep_eq
thf(fact_7872_euclidean__size__natural__def,axiom,
( ( euclid6346220572633701492n_size @ code_natural )
= ( map_fun @ code_natural @ nat @ nat @ nat @ code_nat_of_natural @ ( id @ nat ) @ ( euclid6346220572633701492n_size @ nat ) ) ) ).
% euclidean_size_natural_def
thf(fact_7873_bit__natural__def,axiom,
( ( bit_se5641148757651400278ts_bit @ code_natural )
= ( map_fun @ code_natural @ nat @ ( nat > $o ) @ ( nat > $o ) @ code_nat_of_natural @ ( id @ ( nat > $o ) ) @ ( bit_se5641148757651400278ts_bit @ nat ) ) ) ).
% bit_natural_def
thf(fact_7874_less__eq__natural_Orep__eq,axiom,
( ( ord_less_eq @ code_natural )
= ( ^ [X: code_natural,Xa5: code_natural] : ( ord_less_eq @ nat @ ( code_nat_of_natural @ X ) @ ( code_nat_of_natural @ Xa5 ) ) ) ) ).
% less_eq_natural.rep_eq
thf(fact_7875_times__natural__def,axiom,
( ( times_times @ code_natural )
= ( map_fun @ code_natural @ nat @ ( nat > nat ) @ ( code_natural > code_natural ) @ code_nat_of_natural @ ( map_fun @ code_natural @ nat @ nat @ code_natural @ code_nat_of_natural @ code_natural_of_nat ) @ ( times_times @ nat ) ) ) ).
% times_natural_def
thf(fact_7876_divide__natural__def,axiom,
( ( divide_divide @ code_natural )
= ( map_fun @ code_natural @ nat @ ( nat > nat ) @ ( code_natural > code_natural ) @ code_nat_of_natural @ ( map_fun @ code_natural @ nat @ nat @ code_natural @ code_nat_of_natural @ code_natural_of_nat ) @ ( divide_divide @ nat ) ) ) ).
% divide_natural_def
thf(fact_7877_take__bit__natural__def,axiom,
( ( bit_se2584673776208193580ke_bit @ code_natural )
= ( map_fun @ nat @ nat @ ( nat > nat ) @ ( code_natural > code_natural ) @ ( id @ nat ) @ ( map_fun @ code_natural @ nat @ nat @ code_natural @ code_nat_of_natural @ code_natural_of_nat ) @ ( bit_se2584673776208193580ke_bit @ nat ) ) ) ).
% take_bit_natural_def
thf(fact_7878_push__bit__natural__def,axiom,
( ( bit_se4730199178511100633sh_bit @ code_natural )
= ( map_fun @ nat @ nat @ ( nat > nat ) @ ( code_natural > code_natural ) @ ( id @ nat ) @ ( map_fun @ code_natural @ nat @ nat @ code_natural @ code_nat_of_natural @ code_natural_of_nat ) @ ( bit_se4730199178511100633sh_bit @ nat ) ) ) ).
% push_bit_natural_def
thf(fact_7879_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_7880_modulo__natural__def,axiom,
( ( modulo_modulo @ code_natural )
= ( map_fun @ code_natural @ nat @ ( nat > nat ) @ ( code_natural > code_natural ) @ code_nat_of_natural @ ( map_fun @ code_natural @ nat @ nat @ code_natural @ code_nat_of_natural @ code_natural_of_nat ) @ ( modulo_modulo @ nat ) ) ) ).
% modulo_natural_def
thf(fact_7881_drop__bit__natural__def,axiom,
( ( bit_se4197421643247451524op_bit @ code_natural )
= ( map_fun @ nat @ nat @ ( nat > nat ) @ ( code_natural > code_natural ) @ ( id @ nat ) @ ( map_fun @ code_natural @ nat @ nat @ code_natural @ code_nat_of_natural @ code_natural_of_nat ) @ ( bit_se4197421643247451524op_bit @ nat ) ) ) ).
% drop_bit_natural_def
thf(fact_7882_flip__bit__natural__def,axiom,
( ( bit_se8732182000553998342ip_bit @ code_natural )
= ( map_fun @ nat @ nat @ ( nat > nat ) @ ( code_natural > code_natural ) @ ( id @ nat ) @ ( map_fun @ code_natural @ nat @ nat @ code_natural @ code_nat_of_natural @ code_natural_of_nat ) @ ( bit_se8732182000553998342ip_bit @ nat ) ) ) ).
% flip_bit_natural_def
thf(fact_7883_set__bit__natural__def,axiom,
( ( bit_se5668285175392031749et_bit @ code_natural )
= ( map_fun @ nat @ nat @ ( nat > nat ) @ ( code_natural > code_natural ) @ ( id @ nat ) @ ( map_fun @ code_natural @ nat @ nat @ code_natural @ code_nat_of_natural @ code_natural_of_nat ) @ ( bit_se5668285175392031749et_bit @ nat ) ) ) ).
% set_bit_natural_def
thf(fact_7884_unset__bit__natural__def,axiom,
( ( bit_se2638667681897837118et_bit @ code_natural )
= ( map_fun @ nat @ nat @ ( nat > nat ) @ ( code_natural > code_natural ) @ ( id @ nat ) @ ( map_fun @ code_natural @ nat @ nat @ code_natural @ code_nat_of_natural @ code_natural_of_nat ) @ ( bit_se2638667681897837118et_bit @ nat ) ) ) ).
% unset_bit_natural_def
thf(fact_7885_natural__decr,axiom,
! [N: code_natural] :
( ( N
!= ( zero_zero @ code_natural ) )
=> ( ord_less @ nat @ ( minus_minus @ nat @ ( code_nat_of_natural @ N ) @ ( suc @ ( zero_zero @ nat ) ) ) @ ( code_nat_of_natural @ N ) ) ) ).
% natural_decr
thf(fact_7886_Nat__def,axiom,
( code_Nat
= ( map_fun @ code_integer @ int @ nat @ code_natural @ code_int_of_integer @ code_natural_of_nat @ nat2 ) ) ).
% Nat_def
thf(fact_7887_integer__of__natural_Orep__eq,axiom,
! [X2: code_natural] :
( ( code_int_of_integer @ ( code_i5400310926305786745atural @ X2 ) )
= ( semiring_1_of_nat @ int @ ( code_nat_of_natural @ X2 ) ) ) ).
% integer_of_natural.rep_eq
thf(fact_7888_int__of__integer__of__natural,axiom,
! [N: code_natural] :
( ( code_int_of_integer @ ( code_i5400310926305786745atural @ N ) )
= ( semiring_1_of_nat @ int @ ( code_nat_of_natural @ N ) ) ) ).
% int_of_integer_of_natural
thf(fact_7889_integer__of__natural__of__nat,axiom,
! [N: nat] :
( ( code_i5400310926305786745atural @ ( semiring_1_of_nat @ code_natural @ N ) )
= ( semiring_1_of_nat @ code_integer @ N ) ) ).
% integer_of_natural_of_nat
thf(fact_7890_integer__of__natural__def,axiom,
( code_i5400310926305786745atural
= ( map_fun @ code_natural @ nat @ int @ code_integer @ code_nat_of_natural @ code_integer_of_int @ ( semiring_1_of_nat @ int ) ) ) ).
% integer_of_natural_def
thf(fact_7891_integer__of__natural_Oabs__eq,axiom,
! [X2: nat] :
( ( code_i5400310926305786745atural @ ( code_natural_of_nat @ X2 ) )
= ( code_integer_of_int @ ( semiring_1_of_nat @ int @ X2 ) ) ) ).
% integer_of_natural.abs_eq
thf(fact_7892_integer__of__natural__of__integer,axiom,
! [K: code_integer] :
( ( code_i5400310926305786745atural @ ( code_n4118661773612635043nteger @ K ) )
= ( ord_max @ code_integer @ ( zero_zero @ code_integer ) @ K ) ) ).
% integer_of_natural_of_integer
thf(fact_7893_natural__of__integer__def,axiom,
( code_n4118661773612635043nteger
= ( map_fun @ code_integer @ int @ nat @ code_natural @ code_int_of_integer @ code_natural_of_nat @ nat2 ) ) ).
% natural_of_integer_def
thf(fact_7894_natural__of__integer__of__natural,axiom,
! [N: code_natural] :
( ( code_n4118661773612635043nteger @ ( code_i5400310926305786745atural @ N ) )
= N ) ).
% natural_of_integer_of_natural
thf(fact_7895_natural__of__integer_Orep__eq,axiom,
! [X2: code_integer] :
( ( code_nat_of_natural @ ( code_n4118661773612635043nteger @ X2 ) )
= ( nat2 @ ( code_int_of_integer @ X2 ) ) ) ).
% natural_of_integer.rep_eq
thf(fact_7896_natural__of__integer_Oabs__eq,axiom,
! [X2: int] :
( ( code_n4118661773612635043nteger @ ( code_integer_of_int @ X2 ) )
= ( code_natural_of_nat @ ( nat2 @ X2 ) ) ) ).
% natural_of_integer.abs_eq
thf(fact_7897_Code__Numeral_Osize__nat__def,axiom,
code_size_nat = code_nat_of_natural ).
% Code_Numeral.size_nat_def
thf(fact_7898_Code__Numeral_OSuc__def,axiom,
( code_Suc
= ( map_fun @ code_natural @ nat @ nat @ code_natural @ code_nat_of_natural @ code_natural_of_nat @ suc ) ) ).
% Code_Numeral.Suc_def
thf(fact_7899_natural_Oinject,axiom,
! [Natural: code_natural,Natural2: code_natural] :
( ( ( code_Suc @ Natural )
= ( code_Suc @ Natural2 ) )
= ( Natural = Natural2 ) ) ).
% natural.inject
thf(fact_7900_natural_Osimps_I7_J,axiom,
! [T: $tType,F1: T,F22: code_natural > T > T,Natural: code_natural] :
( ( code_rec_natural @ T @ F1 @ F22 @ ( code_Suc @ Natural ) )
= ( F22 @ Natural @ ( code_rec_natural @ T @ F1 @ F22 @ Natural ) ) ) ).
% natural.simps(7)
thf(fact_7901_natural_Osimps_I5_J,axiom,
! [T: $tType,F1: T,F22: code_natural > T,Natural: code_natural] :
( ( code_case_natural @ T @ F1 @ F22 @ ( code_Suc @ Natural ) )
= ( F22 @ Natural ) ) ).
% natural.simps(5)
thf(fact_7902_Suc_Orep__eq,axiom,
! [X2: code_natural] :
( ( code_nat_of_natural @ ( code_Suc @ X2 ) )
= ( suc @ ( code_nat_of_natural @ X2 ) ) ) ).
% Suc.rep_eq
thf(fact_7903_natural_Odistinct_I2_J,axiom,
! [Natural3: code_natural] :
( ( code_Suc @ Natural3 )
!= ( zero_zero @ code_natural ) ) ).
% natural.distinct(2)
thf(fact_7904_natural_Odistinct_I1_J,axiom,
! [Natural2: code_natural] :
( ( zero_zero @ code_natural )
!= ( code_Suc @ Natural2 ) ) ).
% natural.distinct(1)
thf(fact_7905_natural_Oexhaust,axiom,
! [Y: code_natural] :
( ( Y
!= ( zero_zero @ code_natural ) )
=> ~ ! [Natural4: code_natural] :
( Y
!= ( code_Suc @ Natural4 ) ) ) ).
% natural.exhaust
thf(fact_7906_Suc_Oabs__eq,axiom,
! [X2: nat] :
( ( code_Suc @ ( code_natural_of_nat @ X2 ) )
= ( code_natural_of_nat @ ( suc @ X2 ) ) ) ).
% Suc.abs_eq
thf(fact_7907_Suc__natural__minus__one,axiom,
! [N: code_natural] :
( ( minus_minus @ code_natural @ ( code_Suc @ N ) @ ( one_one @ code_natural ) )
= N ) ).
% Suc_natural_minus_one
thf(fact_7908_subset__singleton__iff__Uniq,axiom,
! [A: $tType,A5: set @ A] :
( ( ? [A3: A] : ( ord_less_eq @ ( set @ A ) @ A5 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( uniq @ A
@ ^ [X: A] : ( member @ A @ X @ A5 ) ) ) ).
% subset_singleton_iff_Uniq
thf(fact_7909_prod__decode__triangle__add,axiom,
! [K: nat,M: nat] :
( ( nat_prod_decode @ ( plus_plus @ nat @ ( nat_triangle @ K ) @ M ) )
= ( nat_prod_decode_aux @ K @ M ) ) ).
% prod_decode_triangle_add
thf(fact_7910_prod__decode__def,axiom,
( nat_prod_decode
= ( nat_prod_decode_aux @ ( zero_zero @ nat ) ) ) ).
% prod_decode_def
thf(fact_7911_inj__on__iff__Uniq,axiom,
! [B: $tType,A: $tType] :
( ( inj_on @ A @ B )
= ( ^ [F2: A > B,A8: set @ A] :
! [X: A] :
( ( member @ A @ X @ A8 )
=> ( uniq @ A
@ ^ [Y4: A] :
( ( member @ A @ Y4 @ A8 )
& ( ( F2 @ X )
= ( F2 @ Y4 ) ) ) ) ) ) ) ).
% inj_on_iff_Uniq
thf(fact_7912_the1__equality_H,axiom,
! [A: $tType,P: A > $o,A2: A] :
( ( uniq @ A @ P )
=> ( ( P @ A2 )
=> ( ( the @ A @ P )
= A2 ) ) ) ).
% the1_equality'
thf(fact_7913_ex1__iff__ex__Uniq,axiom,
! [A: $tType] :
( ( ex1 @ A )
= ( ^ [P4: A > $o] :
( ? [X5: A] : ( P4 @ X5 )
& ( uniq @ A @ P4 ) ) ) ) ).
% ex1_iff_ex_Uniq
thf(fact_7914_alt__ex1E_H,axiom,
! [A: $tType,P: A > $o] :
( ? [X4: A] :
( ( P @ X4 )
& ! [Y2: A] :
( ( P @ Y2 )
=> ( Y2 = X4 ) ) )
=> ~ ( ? [X_12: A] : ( P @ X_12 )
=> ~ ( uniq @ A @ P ) ) ) ).
% alt_ex1E'
thf(fact_7915_left__unique__iff,axiom,
! [B: $tType,A: $tType] :
( ( left_unique @ A @ B )
= ( ^ [R7: A > B > $o] :
! [Z3: B] :
( uniq @ A
@ ^ [X: A] : ( R7 @ X @ Z3 ) ) ) ) ).
% left_unique_iff
thf(fact_7916_bi__unique__iff,axiom,
! [B: $tType,A: $tType] :
( ( bi_unique @ A @ B )
= ( ^ [R7: A > B > $o] :
( ! [Z3: B] :
( uniq @ A
@ ^ [X: A] : ( R7 @ X @ Z3 ) )
& ! [Z3: A] : ( uniq @ B @ ( R7 @ Z3 ) ) ) ) ) ).
% bi_unique_iff
thf(fact_7917_strict__sorted__equal__Uniq,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A] :
( uniq @ ( list @ A )
@ ^ [Xs3: list @ A] :
( ( sorted_wrt @ A @ ( ord_less @ A ) @ Xs3 )
& ( ( set2 @ A @ Xs3 )
= A5 ) ) ) ) ).
% strict_sorted_equal_Uniq
thf(fact_7918_list__decode_Oelims,axiom,
! [X2: nat,Y: list @ nat] :
( ( ( nat_list_decode @ X2 )
= Y )
=> ( ( ( X2
= ( zero_zero @ nat ) )
=> ( Y
!= ( nil @ nat ) ) )
=> ~ ! [N2: nat] :
( ( X2
= ( suc @ N2 ) )
=> ( Y
!= ( product_case_prod @ nat @ nat @ ( list @ nat )
@ ^ [X: nat,Y4: nat] : ( cons @ nat @ X @ ( nat_list_decode @ Y4 ) )
@ ( nat_prod_decode @ N2 ) ) ) ) ) ) ).
% list_decode.elims
thf(fact_7919_list__decode_Osimps_I2_J,axiom,
! [N: nat] :
( ( nat_list_decode @ ( suc @ N ) )
= ( product_case_prod @ nat @ nat @ ( list @ nat )
@ ^ [X: nat,Y4: nat] : ( cons @ nat @ X @ ( nat_list_decode @ Y4 ) )
@ ( nat_prod_decode @ N ) ) ) ).
% list_decode.simps(2)
thf(fact_7920_list__decode_Osimps_I1_J,axiom,
( ( nat_list_decode @ ( zero_zero @ nat ) )
= ( nil @ nat ) ) ).
% list_decode.simps(1)
thf(fact_7921_list__decode_Opelims,axiom,
! [X2: nat,Y: list @ nat] :
( ( ( nat_list_decode @ X2 )
= Y )
=> ( ( accp @ nat @ nat_list_decode_rel @ X2 )
=> ( ( ( X2
= ( zero_zero @ nat ) )
=> ( ( Y
= ( nil @ nat ) )
=> ~ ( accp @ nat @ nat_list_decode_rel @ ( zero_zero @ nat ) ) ) )
=> ~ ! [N2: nat] :
( ( X2
= ( suc @ N2 ) )
=> ( ( Y
= ( product_case_prod @ nat @ nat @ ( list @ nat )
@ ^ [X: nat,Y4: nat] : ( cons @ nat @ X @ ( nat_list_decode @ Y4 ) )
@ ( nat_prod_decode @ N2 ) ) )
=> ~ ( accp @ nat @ nat_list_decode_rel @ ( suc @ N2 ) ) ) ) ) ) ) ).
% list_decode.pelims
thf(fact_7922_Gcd__fin__def,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ( ( semiring_gcd_Gcd_fin @ A )
= ( bounde2362111253966948842tice_F @ A @ ( gcd_gcd @ A ) @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ) ).
% Gcd_fin_def
thf(fact_7923_list__decode_Opsimps_I1_J,axiom,
( ( accp @ nat @ nat_list_decode_rel @ ( zero_zero @ nat ) )
=> ( ( nat_list_decode @ ( zero_zero @ nat ) )
= ( nil @ nat ) ) ) ).
% list_decode.psimps(1)
thf(fact_7924_bounded__quasi__semilattice__set_OF_Ocong,axiom,
! [A: $tType] :
( ( bounde2362111253966948842tice_F @ A )
= ( bounde2362111253966948842tice_F @ A ) ) ).
% bounded_quasi_semilattice_set.F.cong
thf(fact_7925_list__decode_Opinduct,axiom,
! [A0: nat,P: nat > $o] :
( ( accp @ nat @ nat_list_decode_rel @ A0 )
=> ( ( ( accp @ nat @ nat_list_decode_rel @ ( zero_zero @ nat ) )
=> ( P @ ( zero_zero @ nat ) ) )
=> ( ! [N2: nat] :
( ( accp @ nat @ nat_list_decode_rel @ ( suc @ N2 ) )
=> ( ! [X4: nat,Y5: nat] :
( ( ( product_Pair @ nat @ nat @ X4 @ Y5 )
= ( nat_prod_decode @ N2 ) )
=> ( P @ Y5 ) )
=> ( P @ ( suc @ N2 ) ) ) )
=> ( P @ A0 ) ) ) ) ).
% list_decode.pinduct
thf(fact_7926_list__decode_Opsimps_I2_J,axiom,
! [N: nat] :
( ( accp @ nat @ nat_list_decode_rel @ ( suc @ N ) )
=> ( ( nat_list_decode @ ( suc @ N ) )
= ( product_case_prod @ nat @ nat @ ( list @ nat )
@ ^ [X: nat,Y4: nat] : ( cons @ nat @ X @ ( nat_list_decode @ Y4 ) )
@ ( nat_prod_decode @ N ) ) ) ) ).
% list_decode.psimps(2)
thf(fact_7927_bounded__quasi__semilattice__set_Oinsert__remove,axiom,
! [A: $tType,F: A > A > A,Top: A,Bot: A,Normalize: A > A,A2: A,A5: set @ A] :
( ( bounde6485984586167503788ce_set @ A @ F @ Top @ Bot @ Normalize )
=> ( ( bounde2362111253966948842tice_F @ A @ F @ Top @ Bot @ ( insert @ A @ A2 @ A5 ) )
= ( F @ A2 @ ( bounde2362111253966948842tice_F @ A @ F @ Top @ Bot @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% bounded_quasi_semilattice_set.insert_remove
thf(fact_7928_bounded__quasi__semilattice__set_Oremove,axiom,
! [A: $tType,F: A > A > A,Top: A,Bot: A,Normalize: A > A,A2: A,A5: set @ A] :
( ( bounde6485984586167503788ce_set @ A @ F @ Top @ Bot @ Normalize )
=> ( ( member @ A @ A2 @ A5 )
=> ( ( bounde2362111253966948842tice_F @ A @ F @ Top @ Bot @ A5 )
= ( F @ A2 @ ( bounde2362111253966948842tice_F @ A @ F @ Top @ Bot @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% bounded_quasi_semilattice_set.remove
thf(fact_7929_bounded__quasi__semilattice__set_Onormalize,axiom,
! [A: $tType,F: A > A > A,Top: A,Bot: A,Normalize: A > A,A5: set @ A] :
( ( bounde6485984586167503788ce_set @ A @ F @ Top @ Bot @ Normalize )
=> ( ( Normalize @ ( bounde2362111253966948842tice_F @ A @ F @ Top @ Bot @ A5 ) )
= ( bounde2362111253966948842tice_F @ A @ F @ Top @ Bot @ A5 ) ) ) ).
% bounded_quasi_semilattice_set.normalize
thf(fact_7930_bounded__quasi__semilattice__set_Oin__idem,axiom,
! [A: $tType,F: A > A > A,Top: A,Bot: A,Normalize: A > A,A2: A,A5: set @ A] :
( ( bounde6485984586167503788ce_set @ A @ F @ Top @ Bot @ Normalize )
=> ( ( member @ A @ A2 @ A5 )
=> ( ( F @ A2 @ ( bounde2362111253966948842tice_F @ A @ F @ Top @ Bot @ A5 ) )
= ( bounde2362111253966948842tice_F @ A @ F @ Top @ Bot @ A5 ) ) ) ) ).
% bounded_quasi_semilattice_set.in_idem
thf(fact_7931_bounded__quasi__semilattice__set_Oempty,axiom,
! [A: $tType,F: A > A > A,Top: A,Bot: A,Normalize: A > A] :
( ( bounde6485984586167503788ce_set @ A @ F @ Top @ Bot @ Normalize )
=> ( ( bounde2362111253966948842tice_F @ A @ F @ Top @ Bot @ ( bot_bot @ ( set @ A ) ) )
= Top ) ) ).
% bounded_quasi_semilattice_set.empty
thf(fact_7932_bounded__quasi__semilattice__set_Oinfinite,axiom,
! [A: $tType,F: A > A > A,Top: A,Bot: A,Normalize: A > A,A5: set @ A] :
( ( bounde6485984586167503788ce_set @ A @ F @ Top @ Bot @ Normalize )
=> ( ~ ( finite_finite2 @ A @ A5 )
=> ( ( bounde2362111253966948842tice_F @ A @ F @ Top @ Bot @ A5 )
= Bot ) ) ) ).
% bounded_quasi_semilattice_set.infinite
thf(fact_7933_bounded__quasi__semilattice__set_Osubset,axiom,
! [A: $tType,F: A > A > A,Top: A,Bot: A,Normalize: A > A,B5: set @ A,A5: set @ A] :
( ( bounde6485984586167503788ce_set @ A @ F @ Top @ Bot @ Normalize )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ A5 )
=> ( ( F @ ( bounde2362111253966948842tice_F @ A @ F @ Top @ Bot @ B5 ) @ ( bounde2362111253966948842tice_F @ A @ F @ Top @ Bot @ A5 ) )
= ( bounde2362111253966948842tice_F @ A @ F @ Top @ Bot @ A5 ) ) ) ) ).
% bounded_quasi_semilattice_set.subset
thf(fact_7934_bounded__quasi__semilattice__set_Oinsert,axiom,
! [A: $tType,F: A > A > A,Top: A,Bot: A,Normalize: A > A,A2: A,A5: set @ A] :
( ( bounde6485984586167503788ce_set @ A @ F @ Top @ Bot @ Normalize )
=> ( ( bounde2362111253966948842tice_F @ A @ F @ Top @ Bot @ ( insert @ A @ A2 @ A5 ) )
= ( F @ A2 @ ( bounde2362111253966948842tice_F @ A @ F @ Top @ Bot @ A5 ) ) ) ) ).
% bounded_quasi_semilattice_set.insert
thf(fact_7935_bounded__quasi__semilattice__set_Ounion,axiom,
! [A: $tType,F: A > A > A,Top: A,Bot: A,Normalize: A > A,A5: set @ A,B5: set @ A] :
( ( bounde6485984586167503788ce_set @ A @ F @ Top @ Bot @ Normalize )
=> ( ( bounde2362111253966948842tice_F @ A @ F @ Top @ Bot @ ( sup_sup @ ( set @ A ) @ A5 @ B5 ) )
= ( F @ ( bounde2362111253966948842tice_F @ A @ F @ Top @ Bot @ A5 ) @ ( bounde2362111253966948842tice_F @ A @ F @ Top @ Bot @ B5 ) ) ) ) ).
% bounded_quasi_semilattice_set.union
thf(fact_7936_bounded__quasi__semilattice__set_Oeq__fold,axiom,
! [A: $tType,F: A > A > A,Top: A,Bot: A,Normalize: A > A,A5: set @ A] :
( ( bounde6485984586167503788ce_set @ A @ F @ Top @ Bot @ Normalize )
=> ( ( ( finite_finite2 @ A @ A5 )
=> ( ( bounde2362111253966948842tice_F @ A @ F @ Top @ Bot @ A5 )
= ( finite_fold @ A @ A @ F @ Top @ A5 ) ) )
& ( ~ ( finite_finite2 @ A @ A5 )
=> ( ( bounde2362111253966948842tice_F @ A @ F @ Top @ Bot @ A5 )
= Bot ) ) ) ) ).
% bounded_quasi_semilattice_set.eq_fold
thf(fact_7937_uncurry__apply,axiom,
! [B: $tType,A: $tType,C: $tType,F: B > C > A,A2: B,B2: C] :
( ( uncurry @ B @ C @ A @ F @ ( product_Pair @ B @ C @ A2 @ B2 ) )
= ( F @ A2 @ B2 ) ) ).
% uncurry_apply
thf(fact_7938_subset__mset_Osum__pos2,axiom,
! [A: $tType,B: $tType,I5: set @ B,I: B,F: B > ( multiset @ A )] :
( ( finite_finite2 @ B @ I5 )
=> ( ( member @ B @ I @ I5 )
=> ( ( subset_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ ( F @ I ) )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ I5 )
=> ( subseteq_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ ( F @ I3 ) ) )
=> ( subset_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ ( groups3894954378712506084id_sum @ ( multiset @ A ) @ B @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ F @ I5 ) ) ) ) ) ) ).
% subset_mset.sum_pos2
thf(fact_7939_subset__mset_Oadd__less__cancel__right,axiom,
! [A: $tType,A2: multiset @ A,C2: multiset @ A,B2: multiset @ A] :
( ( subset_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A2 @ C2 ) @ ( plus_plus @ ( multiset @ A ) @ B2 @ C2 ) )
= ( subset_mset @ A @ A2 @ B2 ) ) ).
% subset_mset.add_less_cancel_right
thf(fact_7940_subset__mset_Oadd__less__cancel__left,axiom,
! [A: $tType,C2: multiset @ A,A2: multiset @ A,B2: multiset @ A] :
( ( subset_mset @ A @ ( plus_plus @ ( multiset @ A ) @ C2 @ A2 ) @ ( plus_plus @ ( multiset @ A ) @ C2 @ B2 ) )
= ( subset_mset @ A @ A2 @ B2 ) ) ).
% subset_mset.add_less_cancel_left
thf(fact_7941_subset__mset_Oless__add__same__cancel2,axiom,
! [A: $tType,A2: multiset @ A,B2: multiset @ A] :
( ( subset_mset @ A @ A2 @ ( plus_plus @ ( multiset @ A ) @ B2 @ A2 ) )
= ( subset_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ B2 ) ) ).
% subset_mset.less_add_same_cancel2
thf(fact_7942_subset__mset_Oless__add__same__cancel1,axiom,
! [A: $tType,A2: multiset @ A,B2: multiset @ A] :
( ( subset_mset @ A @ A2 @ ( plus_plus @ ( multiset @ A ) @ A2 @ B2 ) )
= ( subset_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ B2 ) ) ).
% subset_mset.less_add_same_cancel1
thf(fact_7943_subset__mset_Oadd__less__same__cancel2,axiom,
! [A: $tType,A2: multiset @ A,B2: multiset @ A] :
( ( subset_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A2 @ B2 ) @ B2 )
= ( subset_mset @ A @ A2 @ ( zero_zero @ ( multiset @ A ) ) ) ) ).
% subset_mset.add_less_same_cancel2
thf(fact_7944_subset__mset_Oadd__less__same__cancel1,axiom,
! [A: $tType,B2: multiset @ A,A2: multiset @ A] :
( ( subset_mset @ A @ ( plus_plus @ ( multiset @ A ) @ B2 @ A2 ) @ B2 )
= ( subset_mset @ A @ A2 @ ( zero_zero @ ( multiset @ A ) ) ) ) ).
% subset_mset.add_less_same_cancel1
thf(fact_7945_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_7946_subset__mset_Opos__add__strict,axiom,
! [A: $tType,A2: multiset @ A,B2: multiset @ A,C2: multiset @ A] :
( ( subset_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ A2 )
=> ( ( subset_mset @ A @ B2 @ C2 )
=> ( subset_mset @ A @ B2 @ ( plus_plus @ ( multiset @ A ) @ A2 @ C2 ) ) ) ) ).
% subset_mset.pos_add_strict
thf(fact_7947_subset__mset_Oadd__pos__pos,axiom,
! [A: $tType,A2: multiset @ A,B2: multiset @ A] :
( ( subset_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ A2 )
=> ( ( subset_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ B2 )
=> ( subset_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ ( plus_plus @ ( multiset @ A ) @ A2 @ B2 ) ) ) ) ).
% subset_mset.add_pos_pos
thf(fact_7948_subset__mset_Oadd__neg__neg,axiom,
! [A: $tType,A2: multiset @ A,B2: multiset @ A] :
( ( subset_mset @ A @ A2 @ ( zero_zero @ ( multiset @ A ) ) )
=> ( ( subset_mset @ A @ B2 @ ( zero_zero @ ( multiset @ A ) ) )
=> ( subset_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A2 @ B2 ) @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ).
% subset_mset.add_neg_neg
thf(fact_7949_subset__mset_OlessE,axiom,
! [A: $tType,A2: multiset @ A,B2: multiset @ A] :
( ( subset_mset @ A @ A2 @ B2 )
=> ~ ! [C4: multiset @ A] :
( ( B2
= ( plus_plus @ ( multiset @ A ) @ A2 @ C4 ) )
=> ( C4
= ( zero_zero @ ( multiset @ A ) ) ) ) ) ).
% subset_mset.lessE
thf(fact_7950_subset__mset_Oadd__strict__increasing2,axiom,
! [A: $tType,A2: multiset @ A,B2: multiset @ A,C2: multiset @ A] :
( ( subseteq_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ A2 )
=> ( ( subset_mset @ A @ B2 @ C2 )
=> ( subset_mset @ A @ B2 @ ( plus_plus @ ( multiset @ A ) @ A2 @ C2 ) ) ) ) ).
% subset_mset.add_strict_increasing2
thf(fact_7951_subset__mset_Oadd__strict__increasing,axiom,
! [A: $tType,A2: multiset @ A,B2: multiset @ A,C2: multiset @ A] :
( ( subset_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ A2 )
=> ( ( subseteq_mset @ A @ B2 @ C2 )
=> ( subset_mset @ A @ B2 @ ( plus_plus @ ( multiset @ A ) @ A2 @ C2 ) ) ) ) ).
% subset_mset.add_strict_increasing
thf(fact_7952_subset__mset_Oadd__pos__nonneg,axiom,
! [A: $tType,A2: multiset @ A,B2: multiset @ A] :
( ( subset_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ A2 )
=> ( ( subseteq_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ B2 )
=> ( subset_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ ( plus_plus @ ( multiset @ A ) @ A2 @ B2 ) ) ) ) ).
% subset_mset.add_pos_nonneg
thf(fact_7953_subset__mset_Oadd__nonpos__neg,axiom,
! [A: $tType,A2: multiset @ A,B2: multiset @ A] :
( ( subseteq_mset @ A @ A2 @ ( zero_zero @ ( multiset @ A ) ) )
=> ( ( subset_mset @ A @ B2 @ ( zero_zero @ ( multiset @ A ) ) )
=> ( subset_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A2 @ B2 ) @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ).
% subset_mset.add_nonpos_neg
thf(fact_7954_subset__mset_Oadd__nonneg__pos,axiom,
! [A: $tType,A2: multiset @ A,B2: multiset @ A] :
( ( subseteq_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ A2 )
=> ( ( subset_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ B2 )
=> ( subset_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ ( plus_plus @ ( multiset @ A ) @ A2 @ B2 ) ) ) ) ).
% subset_mset.add_nonneg_pos
thf(fact_7955_subset__mset_Oadd__neg__nonpos,axiom,
! [A: $tType,A2: multiset @ A,B2: multiset @ A] :
( ( subset_mset @ A @ A2 @ ( zero_zero @ ( multiset @ A ) ) )
=> ( ( subseteq_mset @ A @ B2 @ ( zero_zero @ ( multiset @ A ) ) )
=> ( subset_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A2 @ B2 ) @ ( zero_zero @ ( multiset @ A ) ) ) ) ) ).
% subset_mset.add_neg_nonpos
thf(fact_7956_mset__subset__size,axiom,
! [A: $tType,A5: multiset @ A,B5: multiset @ A] :
( ( subset_mset @ A @ A5 @ B5 )
=> ( ord_less @ nat @ ( size_size @ ( multiset @ A ) @ A5 ) @ ( size_size @ ( multiset @ A ) @ B5 ) ) ) ).
% mset_subset_size
thf(fact_7957_size__psubset,axiom,
! [A: $tType,M3: multiset @ A,M9: multiset @ A] :
( ( subseteq_mset @ A @ M3 @ M9 )
=> ( ( ord_less @ nat @ ( size_size @ ( multiset @ A ) @ M3 ) @ ( size_size @ ( multiset @ A ) @ M9 ) )
=> ( subset_mset @ A @ M3 @ M9 ) ) ) ).
% size_psubset
thf(fact_7958_subset__mset_Oadd__less__imp__less__right,axiom,
! [A: $tType,A2: multiset @ A,C2: multiset @ A,B2: multiset @ A] :
( ( subset_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A2 @ C2 ) @ ( plus_plus @ ( multiset @ A ) @ B2 @ C2 ) )
=> ( subset_mset @ A @ A2 @ B2 ) ) ).
% subset_mset.add_less_imp_less_right
thf(fact_7959_subset__mset_Oadd__less__imp__less__left,axiom,
! [A: $tType,C2: multiset @ A,A2: multiset @ A,B2: multiset @ A] :
( ( subset_mset @ A @ ( plus_plus @ ( multiset @ A ) @ C2 @ A2 ) @ ( plus_plus @ ( multiset @ A ) @ C2 @ B2 ) )
=> ( subset_mset @ A @ A2 @ B2 ) ) ).
% subset_mset.add_less_imp_less_left
thf(fact_7960_subset__mset_Oadd__strict__right__mono,axiom,
! [A: $tType,A2: multiset @ A,B2: multiset @ A,C2: multiset @ A] :
( ( subset_mset @ A @ A2 @ B2 )
=> ( subset_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A2 @ C2 ) @ ( plus_plus @ ( multiset @ A ) @ B2 @ C2 ) ) ) ).
% subset_mset.add_strict_right_mono
thf(fact_7961_subset__mset_Oadd__strict__left__mono,axiom,
! [A: $tType,A2: multiset @ A,B2: multiset @ A,C2: multiset @ A] :
( ( subset_mset @ A @ A2 @ B2 )
=> ( subset_mset @ A @ ( plus_plus @ ( multiset @ A ) @ C2 @ A2 ) @ ( plus_plus @ ( multiset @ A ) @ C2 @ B2 ) ) ) ).
% subset_mset.add_strict_left_mono
thf(fact_7962_subset__mset_Oadd__strict__mono,axiom,
! [A: $tType,A2: multiset @ A,B2: multiset @ A,C2: multiset @ A,D2: multiset @ A] :
( ( subset_mset @ A @ A2 @ B2 )
=> ( ( subset_mset @ A @ C2 @ D2 )
=> ( subset_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A2 @ C2 ) @ ( plus_plus @ ( multiset @ A ) @ B2 @ D2 ) ) ) ) ).
% subset_mset.add_strict_mono
thf(fact_7963_subset__mset_Oadd__less__le__mono,axiom,
! [A: $tType,A2: multiset @ A,B2: multiset @ A,C2: multiset @ A,D2: multiset @ A] :
( ( subset_mset @ A @ A2 @ B2 )
=> ( ( subseteq_mset @ A @ C2 @ D2 )
=> ( subset_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A2 @ C2 ) @ ( plus_plus @ ( multiset @ A ) @ B2 @ D2 ) ) ) ) ).
% subset_mset.add_less_le_mono
thf(fact_7964_subset__mset_Oadd__le__less__mono,axiom,
! [A: $tType,A2: multiset @ A,B2: multiset @ A,C2: multiset @ A,D2: multiset @ A] :
( ( subseteq_mset @ A @ A2 @ B2 )
=> ( ( subset_mset @ A @ C2 @ D2 )
=> ( subset_mset @ A @ ( plus_plus @ ( multiset @ A ) @ A2 @ C2 ) @ ( plus_plus @ ( multiset @ A ) @ B2 @ D2 ) ) ) ) ).
% subset_mset.add_le_less_mono
thf(fact_7965_subset__mset_Oasymp__greater,axiom,
! [A: $tType] :
( asymp @ ( multiset @ A )
@ ^ [X: multiset @ A,Y4: multiset @ A] : ( subset_mset @ A @ Y4 @ X ) ) ).
% subset_mset.asymp_greater
thf(fact_7966_subset__mset_OSup__fin_Osemilattice__order__set__axioms,axiom,
! [A: $tType] :
( lattic4895041142388067077er_set @ ( multiset @ A ) @ ( union_mset @ A )
@ ^ [X: multiset @ A,Y4: multiset @ A] : ( subseteq_mset @ A @ Y4 @ X )
@ ^ [X: multiset @ A,Y4: multiset @ A] : ( subset_mset @ A @ Y4 @ X ) ) ).
% subset_mset.Sup_fin.semilattice_order_set_axioms
thf(fact_7967_subset__mset_Olexordp__def,axiom,
! [A: $tType] :
( ( lexordp2 @ ( multiset @ A ) @ ( subset_mset @ A ) )
= ( complete_lattice_lfp @ ( ( list @ ( multiset @ A ) ) > ( list @ ( multiset @ A ) ) > $o )
@ ^ [P7: ( list @ ( multiset @ A ) ) > ( list @ ( multiset @ A ) ) > $o,X12: list @ ( multiset @ A ),X24: list @ ( multiset @ A )] :
( ? [Y4: multiset @ A,Ys3: list @ ( multiset @ A )] :
( ( X12
= ( nil @ ( multiset @ A ) ) )
& ( X24
= ( cons @ ( multiset @ A ) @ Y4 @ Ys3 ) ) )
| ? [X: multiset @ A,Y4: multiset @ A,Xs3: list @ ( multiset @ A ),Ys3: list @ ( multiset @ A )] :
( ( X12
= ( cons @ ( multiset @ A ) @ X @ Xs3 ) )
& ( X24
= ( cons @ ( multiset @ A ) @ Y4 @ Ys3 ) )
& ( subset_mset @ A @ X @ Y4 ) )
| ? [X: multiset @ A,Y4: multiset @ A,Xs3: list @ ( multiset @ A ),Ys3: list @ ( multiset @ A )] :
( ( X12
= ( cons @ ( multiset @ A ) @ X @ Xs3 ) )
& ( X24
= ( cons @ ( multiset @ A ) @ Y4 @ Ys3 ) )
& ~ ( subset_mset @ A @ X @ Y4 )
& ~ ( subset_mset @ A @ Y4 @ X )
& ( P7 @ Xs3 @ Ys3 ) ) ) ) ) ).
% subset_mset.lexordp_def
thf(fact_7968_subset__mset_Olexordp_Omono,axiom,
! [A: $tType] :
( order_mono @ ( ( list @ ( multiset @ A ) ) > ( list @ ( multiset @ A ) ) > $o ) @ ( ( list @ ( multiset @ A ) ) > ( list @ ( multiset @ A ) ) > $o )
@ ^ [P7: ( list @ ( multiset @ A ) ) > ( list @ ( multiset @ A ) ) > $o,X12: list @ ( multiset @ A ),X24: list @ ( multiset @ A )] :
( ? [Y4: multiset @ A,Ys3: list @ ( multiset @ A )] :
( ( X12
= ( nil @ ( multiset @ A ) ) )
& ( X24
= ( cons @ ( multiset @ A ) @ Y4 @ Ys3 ) ) )
| ? [X: multiset @ A,Y4: multiset @ A,Xs3: list @ ( multiset @ A ),Ys3: list @ ( multiset @ A )] :
( ( X12
= ( cons @ ( multiset @ A ) @ X @ Xs3 ) )
& ( X24
= ( cons @ ( multiset @ A ) @ Y4 @ Ys3 ) )
& ( subset_mset @ A @ X @ Y4 ) )
| ? [X: multiset @ A,Y4: multiset @ A,Xs3: list @ ( multiset @ A ),Ys3: list @ ( multiset @ A )] :
( ( X12
= ( cons @ ( multiset @ A ) @ X @ Xs3 ) )
& ( X24
= ( cons @ ( multiset @ A ) @ Y4 @ Ys3 ) )
& ~ ( subset_mset @ A @ X @ Y4 )
& ~ ( subset_mset @ A @ Y4 @ X )
& ( P7 @ Xs3 @ Ys3 ) ) ) ) ).
% subset_mset.lexordp.mono
thf(fact_7969_subset__mset_Oordering__top__axioms,axiom,
! [A: $tType] :
( ordering_top @ ( multiset @ A )
@ ^ [A8: multiset @ A,B8: multiset @ A] : ( subseteq_mset @ A @ B8 @ A8 )
@ ^ [A8: multiset @ A,B8: multiset @ A] : ( subset_mset @ A @ B8 @ A8 )
@ ( zero_zero @ ( multiset @ A ) ) ) ).
% subset_mset.ordering_top_axioms
thf(fact_7970_subset__mset_Osemilattice__neutr__order__axioms,axiom,
! [A: $tType] :
( semila1105856199041335345_order @ ( multiset @ A ) @ ( union_mset @ A ) @ ( zero_zero @ ( multiset @ A ) )
@ ^ [A8: multiset @ A,B8: multiset @ A] : ( subseteq_mset @ A @ B8 @ A8 )
@ ^ [A8: multiset @ A,B8: multiset @ A] : ( subset_mset @ A @ B8 @ A8 ) ) ).
% subset_mset.semilattice_neutr_order_axioms
thf(fact_7971_uncurry__def,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( uncurry @ A @ B @ C )
= ( product_case_prod @ A @ B @ C ) ) ).
% uncurry_def
thf(fact_7972_mset__subset__add__iff1,axiom,
! [A: $tType,J: nat,I: nat,U: multiset @ A,M: multiset @ A,N: multiset @ A] :
( ( ord_less_eq @ nat @ J @ I )
=> ( ( subset_mset @ A @ ( plus_plus @ ( multiset @ A ) @ ( repeat_mset @ A @ I @ U ) @ M ) @ ( plus_plus @ ( multiset @ A ) @ ( repeat_mset @ A @ J @ U ) @ N ) )
= ( subset_mset @ A @ ( plus_plus @ ( multiset @ A ) @ ( repeat_mset @ A @ ( minus_minus @ nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% mset_subset_add_iff1
thf(fact_7973_mset__subset__add__iff2,axiom,
! [A: $tType,I: nat,J: nat,U: multiset @ A,M: multiset @ A,N: multiset @ A] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( subset_mset @ A @ ( plus_plus @ ( multiset @ A ) @ ( repeat_mset @ A @ I @ U ) @ M ) @ ( plus_plus @ ( multiset @ A ) @ ( repeat_mset @ A @ J @ U ) @ N ) )
= ( subset_mset @ A @ M @ ( plus_plus @ ( multiset @ A ) @ ( repeat_mset @ A @ ( minus_minus @ nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% mset_subset_add_iff2
thf(fact_7974_subset__mset_OcSUP__lessD,axiom,
! [B: $tType,A: $tType,F: B > ( multiset @ A ),A5: set @ B,Y: multiset @ A,I: B] :
( ( condit8047198070973881523_above @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F @ A5 ) )
=> ( ( subset_mset @ A @ ( complete_Sup_Sup @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F @ A5 ) ) @ Y )
=> ( ( member @ B @ I @ A5 )
=> ( subset_mset @ A @ ( F @ I ) @ Y ) ) ) ) ).
% subset_mset.cSUP_lessD
thf(fact_7975_subset__mset_Oless__cINF__D,axiom,
! [A: $tType,B: $tType,F: B > ( multiset @ A ),A5: set @ B,Y: multiset @ A,I: B] :
( ( condit8119078960628432327_below @ ( multiset @ A ) @ ( subseteq_mset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F @ A5 ) )
=> ( ( subset_mset @ A @ Y @ ( complete_Inf_Inf @ ( multiset @ A ) @ ( image2 @ B @ ( multiset @ A ) @ F @ A5 ) ) )
=> ( ( member @ B @ I @ A5 )
=> ( subset_mset @ A @ Y @ ( F @ I ) ) ) ) ) ).
% subset_mset.less_cINF_D
thf(fact_7976_subset__mset_Osum__strict__mono,axiom,
! [A: $tType,B: $tType,A5: set @ B,F: B > ( multiset @ A ),G: B > ( multiset @ A )] :
( ( finite_finite2 @ B @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ A5 )
=> ( subset_mset @ A @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( subset_mset @ A @ ( groups3894954378712506084id_sum @ ( multiset @ A ) @ B @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ F @ A5 ) @ ( groups3894954378712506084id_sum @ ( multiset @ A ) @ B @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ G @ A5 ) ) ) ) ) ).
% subset_mset.sum_strict_mono
thf(fact_7977_subset__mset_Osum__pos,axiom,
! [A: $tType,B: $tType,I5: set @ B,F: 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 ) ) @ ( F @ I3 ) ) )
=> ( subset_mset @ A @ ( zero_zero @ ( multiset @ A ) ) @ ( groups3894954378712506084id_sum @ ( multiset @ A ) @ B @ ( plus_plus @ ( multiset @ A ) ) @ ( zero_zero @ ( multiset @ A ) ) @ F @ I5 ) ) ) ) ) ).
% subset_mset.sum_pos
thf(fact_7978_card__def,axiom,
! [B: $tType] :
( ( finite_card @ B )
= ( finite_folding_F @ B @ nat
@ ^ [Uu: B] : suc
@ ( zero_zero @ nat ) ) ) ).
% card_def
thf(fact_7979_rel__filter_Ocases,axiom,
! [A: $tType,B: $tType,R3: A > B > $o,F4: filter @ A,G5: filter @ B] :
( ( rel_filter @ A @ B @ R3 @ F4 @ G5 )
=> ~ ! [Z8: filter @ ( product_prod @ A @ B )] :
( ( eventually @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R3 ) @ Z8 )
=> ( ( ( map_filter_on @ ( product_prod @ A @ B ) @ A @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R3 ) ) @ ( product_fst @ A @ B ) @ Z8 )
= F4 )
=> ( ( map_filter_on @ ( product_prod @ A @ B ) @ B @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R3 ) ) @ ( product_snd @ A @ B ) @ Z8 )
!= G5 ) ) ) ) ).
% rel_filter.cases
thf(fact_7980_rel__filter_Osimps,axiom,
! [B: $tType,A: $tType] :
( ( rel_filter @ A @ B )
= ( ^ [R7: A > B > $o,F8: filter @ A,G7: filter @ B] :
? [Z10: filter @ ( product_prod @ A @ B )] :
( ( eventually @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R7 ) @ Z10 )
& ( ( map_filter_on @ ( product_prod @ A @ B ) @ A @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R7 ) ) @ ( product_fst @ A @ B ) @ Z10 )
= F8 )
& ( ( map_filter_on @ ( product_prod @ A @ B ) @ B @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R7 ) ) @ ( product_snd @ A @ B ) @ Z10 )
= G7 ) ) ) ) ).
% rel_filter.simps
thf(fact_7981_rel__filter_Ointros,axiom,
! [A: $tType,B: $tType,R3: A > B > $o,Z9: filter @ ( product_prod @ A @ B ),F4: filter @ A,G5: filter @ B] :
( ( eventually @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R3 ) @ Z9 )
=> ( ( ( map_filter_on @ ( product_prod @ A @ B ) @ A @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R3 ) ) @ ( product_fst @ A @ B ) @ Z9 )
= F4 )
=> ( ( ( map_filter_on @ ( product_prod @ A @ B ) @ B @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R3 ) ) @ ( product_snd @ A @ B ) @ Z9 )
= G5 )
=> ( rel_filter @ A @ B @ R3 @ F4 @ G5 ) ) ) ) ).
% rel_filter.intros
thf(fact_7982_map__comp__def,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( map_comp @ B @ C @ A )
= ( ^ [F2: B > ( option @ C ),G2: A > ( option @ B ),K2: A] : ( case_option @ ( option @ C ) @ B @ ( none @ C ) @ F2 @ ( G2 @ K2 ) ) ) ) ).
% map_comp_def
thf(fact_7983_rel__fun__iff__geq__image2p,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType] :
( ( bNF_rel_fun @ A @ B @ C @ D )
= ( ^ [R7: A > B > $o,S: C > D > $o,F2: A > C,G2: B > D] : ( ord_less_eq @ ( C > D > $o ) @ ( bNF_Greatest_image2p @ A @ C @ B @ D @ F2 @ G2 @ R7 ) @ S ) ) ) ).
% rel_fun_iff_geq_image2p
thf(fact_7984_map__comp__empty_I2_J,axiom,
! [B: $tType,D: $tType,C: $tType,M: C > ( option @ B )] :
( ( map_comp @ B @ D @ C
@ ^ [X: B] : ( none @ D )
@ M )
= ( ^ [X: C] : ( none @ D ) ) ) ).
% map_comp_empty(2)
thf(fact_7985_map__comp__empty_I1_J,axiom,
! [C: $tType,B: $tType,A: $tType,M: C > ( option @ B )] :
( ( map_comp @ C @ B @ A @ M
@ ^ [X: A] : ( none @ C ) )
= ( ^ [X: A] : ( none @ B ) ) ) ).
% map_comp_empty(1)
thf(fact_7986_image2pI,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,R3: A > B > $o,X2: A,Y: B,F: A > C,G: B > D] :
( ( R3 @ X2 @ Y )
=> ( bNF_Greatest_image2p @ A @ C @ B @ D @ F @ G @ R3 @ ( F @ X2 ) @ ( G @ Y ) ) ) ).
% image2pI
thf(fact_7987_image2pE,axiom,
! [D: $tType,B: $tType,A: $tType,C: $tType,F: A > B,G: C > D,R3: A > C > $o,Fx: B,Gy: D] :
( ( bNF_Greatest_image2p @ A @ B @ C @ D @ F @ G @ R3 @ Fx @ Gy )
=> ~ ! [X3: A] :
( ( Fx
= ( F @ X3 ) )
=> ! [Y2: C] :
( ( Gy
= ( G @ Y2 ) )
=> ~ ( R3 @ X3 @ Y2 ) ) ) ) ).
% image2pE
thf(fact_7988_rel__fun__image2p,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,R3: A > B > $o,F: A > C,G: B > D] : ( bNF_rel_fun @ A @ B @ C @ D @ R3 @ ( bNF_Greatest_image2p @ A @ C @ B @ D @ F @ G @ R3 ) @ F @ G ) ).
% rel_fun_image2p
thf(fact_7989_image2p__def,axiom,
! [D: $tType,B: $tType,A: $tType,C: $tType] :
( ( bNF_Greatest_image2p @ C @ A @ D @ B )
= ( ^ [F2: C > A,G2: D > B,R7: C > D > $o,X: A,Y4: B] :
? [X9: C,Y8: D] :
( ( R7 @ X9 @ Y8 )
& ( ( F2 @ X9 )
= X )
& ( ( G2 @ Y8 )
= Y4 ) ) ) ) ).
% image2p_def
thf(fact_7990_subset__mset_Oinf__Sup1__distrib,axiom,
! [A: $tType,A5: set @ ( multiset @ A ),X2: multiset @ A] :
( ( finite_finite2 @ ( multiset @ A ) @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( inter_mset @ A @ X2 @ ( lattic4630905495605216202up_fin @ ( multiset @ A ) @ ( union_mset @ A ) @ A5 ) )
= ( lattic4630905495605216202up_fin @ ( multiset @ A ) @ ( union_mset @ A )
@ ( collect @ ( multiset @ A )
@ ^ [Uu: multiset @ A] :
? [A3: multiset @ A] :
( ( Uu
= ( inter_mset @ A @ X2 @ A3 ) )
& ( member @ ( multiset @ A ) @ A3 @ A5 ) ) ) ) ) ) ) ).
% subset_mset.inf_Sup1_distrib
thf(fact_7991_subset__mset_Oinf__Sup2__distrib,axiom,
! [A: $tType,A5: set @ ( multiset @ A ),B5: set @ ( multiset @ A )] :
( ( finite_finite2 @ ( multiset @ A ) @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( finite_finite2 @ ( multiset @ A ) @ B5 )
=> ( ( B5
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( inter_mset @ A @ ( lattic4630905495605216202up_fin @ ( multiset @ A ) @ ( union_mset @ A ) @ A5 ) @ ( lattic4630905495605216202up_fin @ ( multiset @ A ) @ ( union_mset @ A ) @ B5 ) )
= ( lattic4630905495605216202up_fin @ ( multiset @ A ) @ ( union_mset @ A )
@ ( collect @ ( multiset @ A )
@ ^ [Uu: multiset @ A] :
? [A3: multiset @ A,B3: multiset @ A] :
( ( Uu
= ( inter_mset @ A @ A3 @ B3 ) )
& ( member @ ( multiset @ A ) @ A3 @ A5 )
& ( member @ ( multiset @ A ) @ B3 @ B5 ) ) ) ) ) ) ) ) ) ).
% subset_mset.inf_Sup2_distrib
thf(fact_7992_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 ) )
@ ^ [X: multiset @ A,Y4: option @ ( multiset @ A )] : ( some @ ( multiset @ A ) @ ( case_option @ ( multiset @ A ) @ ( multiset @ A ) @ X @ ( union_mset @ A @ X ) @ Y4 ) )
@ ( none @ ( multiset @ A ) )
@ A5 ) ) ) ).
% subset_mset.Sup_fin.eq_fold'
thf(fact_7993_semilattice__sup_OSup__fin_Ocong,axiom,
! [A: $tType] :
( ( lattic4630905495605216202up_fin @ A )
= ( lattic4630905495605216202up_fin @ A ) ) ).
% semilattice_sup.Sup_fin.cong
thf(fact_7994_subset__mset_Osup__Inf2__distrib,axiom,
! [A: $tType,A5: set @ ( multiset @ A ),B5: set @ ( multiset @ A )] :
( ( finite_finite2 @ ( multiset @ A ) @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( finite_finite2 @ ( multiset @ A ) @ B5 )
=> ( ( B5
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( union_mset @ A @ ( lattic8678736583308907530nf_fin @ ( multiset @ A ) @ ( inter_mset @ A ) @ A5 ) @ ( lattic8678736583308907530nf_fin @ ( multiset @ A ) @ ( inter_mset @ A ) @ B5 ) )
= ( lattic8678736583308907530nf_fin @ ( multiset @ A ) @ ( inter_mset @ A )
@ ( collect @ ( multiset @ A )
@ ^ [Uu: multiset @ A] :
? [A3: multiset @ A,B3: multiset @ A] :
( ( Uu
= ( union_mset @ A @ A3 @ B3 ) )
& ( member @ ( multiset @ A ) @ A3 @ A5 )
& ( member @ ( multiset @ A ) @ B3 @ B5 ) ) ) ) ) ) ) ) ) ).
% subset_mset.sup_Inf2_distrib
thf(fact_7995_subset__mset_Osup__Inf1__distrib,axiom,
! [A: $tType,A5: set @ ( multiset @ A ),X2: multiset @ A] :
( ( finite_finite2 @ ( multiset @ A ) @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ ( multiset @ A ) ) ) )
=> ( ( union_mset @ A @ X2 @ ( lattic8678736583308907530nf_fin @ ( multiset @ A ) @ ( inter_mset @ A ) @ A5 ) )
= ( lattic8678736583308907530nf_fin @ ( multiset @ A ) @ ( inter_mset @ A )
@ ( collect @ ( multiset @ A )
@ ^ [Uu: multiset @ A] :
? [A3: multiset @ A] :
( ( Uu
= ( union_mset @ A @ X2 @ A3 ) )
& ( member @ ( multiset @ A ) @ A3 @ A5 ) ) ) ) ) ) ) ).
% subset_mset.sup_Inf1_distrib
thf(fact_7996_semilattice__inf_OInf__fin_Ocong,axiom,
! [A: $tType] :
( ( lattic8678736583308907530nf_fin @ A )
= ( lattic8678736583308907530nf_fin @ A ) ) ).
% semilattice_inf.Inf_fin.cong
thf(fact_7997_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 ) )
@ ^ [X: multiset @ A,Y4: option @ ( multiset @ A )] : ( some @ ( multiset @ A ) @ ( case_option @ ( multiset @ A ) @ ( multiset @ A ) @ X @ ( inter_mset @ A @ X ) @ Y4 ) )
@ ( none @ ( multiset @ A ) )
@ A5 ) ) ) ).
% subset_mset.Inf_fin.eq_fold'
thf(fact_7998_Code__Numeral_Osub__code_I4_J,axiom,
! [N: num] :
( ( code_sub @ one2 @ ( bit0 @ N ) )
= ( code_Neg @ ( bitM @ N ) ) ) ).
% Code_Numeral.sub_code(4)
thf(fact_7999_flip__pred,axiom,
! [A: $tType,B: $tType,A5: set @ ( product_prod @ A @ B ),R3: 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 @ R3 ) ) ) )
=> ( 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 )
@ ^ [X: A,Y4: B] : ( product_Pair @ B @ A @ Y4 @ X ) )
@ A5 )
@ ( collect @ ( product_prod @ B @ A ) @ ( product_case_prod @ B @ A @ $o @ R3 ) ) ) ) ).
% flip_pred
thf(fact_8000_conversep__noteq,axiom,
! [A: $tType] :
( ( conversep @ A @ A
@ ^ [X: A,Y4: A] : X != Y4 )
= ( ^ [X: A,Y4: A] : X != Y4 ) ) ).
% conversep_noteq
thf(fact_8001_less__eq__integer__code_I9_J,axiom,
! [K: num,L: num] :
( ( ord_less_eq @ code_integer @ ( code_Neg @ K ) @ ( code_Neg @ L ) )
= ( ord_less_eq @ num @ L @ K ) ) ).
% less_eq_integer_code(9)
thf(fact_8002_less__eq__integer__code_I3_J,axiom,
! [L: num] :
~ ( ord_less_eq @ code_integer @ ( zero_zero @ code_integer ) @ ( code_Neg @ L ) ) ).
% less_eq_integer_code(3)
thf(fact_8003_less__eq__integer__code_I7_J,axiom,
! [K: num] : ( ord_less_eq @ code_integer @ ( code_Neg @ K ) @ ( zero_zero @ code_integer ) ) ).
% less_eq_integer_code(7)
thf(fact_8004_minus__integer__code_I6_J,axiom,
! [M: num,N: num] :
( ( minus_minus @ code_integer @ ( code_Neg @ M ) @ ( code_Neg @ N ) )
= ( code_sub @ N @ M ) ) ).
% minus_integer_code(6)
thf(fact_8005_Code__Numeral_Odup__code_I3_J,axiom,
! [N: num] :
( ( code_dup @ ( code_Neg @ N ) )
= ( code_Neg @ ( bit0 @ N ) ) ) ).
% Code_Numeral.dup_code(3)
thf(fact_8006_less__integer__code_I9_J,axiom,
! [K: num,L: num] :
( ( ord_less @ code_integer @ ( code_Neg @ K ) @ ( code_Neg @ L ) )
= ( ord_less @ num @ L @ K ) ) ).
% less_integer_code(9)
thf(fact_8007_plus__integer__code_I6_J,axiom,
! [M: num,N: num] :
( ( plus_plus @ code_integer @ ( code_Neg @ M ) @ ( code_Neg @ N ) )
= ( code_Neg @ ( plus_plus @ num @ M @ N ) ) ) ).
% plus_integer_code(6)
thf(fact_8008_less__integer__code_I7_J,axiom,
! [K: num] : ( ord_less @ code_integer @ ( code_Neg @ K ) @ ( zero_zero @ code_integer ) ) ).
% less_integer_code(7)
thf(fact_8009_less__integer__code_I3_J,axiom,
! [L: num] :
~ ( ord_less @ code_integer @ ( zero_zero @ code_integer ) @ ( code_Neg @ L ) ) ).
% less_integer_code(3)
thf(fact_8010_conversep__converse__eq,axiom,
! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ B )] :
( ( conversep @ A @ B
@ ^ [X: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y4 ) @ R ) )
= ( ^ [X: B,Y4: A] : ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X @ Y4 ) @ ( converse @ A @ B @ R ) ) ) ) ).
% conversep_converse_eq
thf(fact_8011_converse__def,axiom,
! [B: $tType,A: $tType] :
( ( converse @ A @ B )
= ( ^ [R2: set @ ( product_prod @ A @ B )] :
( collect @ ( product_prod @ B @ A )
@ ( product_case_prod @ B @ A @ $o
@ ( conversep @ A @ B
@ ^ [X: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y4 ) @ R2 ) ) ) ) ) ) ).
% converse_def
thf(fact_8012_divmod__abs__code_I2_J,axiom,
! [K: num,L: num] :
( ( code_divmod_abs @ ( code_Neg @ K ) @ ( code_Neg @ L ) )
= ( unique8689654367752047608divmod @ code_integer @ K @ L ) ) ).
% divmod_abs_code(2)
thf(fact_8013_Code__Numeral_Osub__code_I5_J,axiom,
! [N: num] :
( ( code_sub @ one2 @ ( bit1 @ N ) )
= ( code_Neg @ ( bit0 @ N ) ) ) ).
% Code_Numeral.sub_code(5)
thf(fact_8014_sorted__wrt__iff__nth__Suc__transp,axiom,
! [A: $tType,P: A > A > $o,Xs: list @ A] :
( ( transp @ A @ P )
=> ( ( sorted_wrt @ A @ P @ Xs )
= ( ! [I2: nat] :
( ( ord_less @ nat @ ( suc @ I2 ) @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( P @ ( nth @ A @ Xs @ I2 ) @ ( nth @ A @ Xs @ ( suc @ I2 ) ) ) ) ) ) ) ).
% sorted_wrt_iff_nth_Suc_transp
thf(fact_8015_prod__set__simps_I2_J,axiom,
! [A: $tType,B: $tType,X2: A,Y: B] :
( ( basic_snds @ A @ B @ ( product_Pair @ A @ B @ X2 @ Y ) )
= ( insert @ B @ Y @ ( bot_bot @ ( set @ B ) ) ) ) ).
% prod_set_simps(2)
thf(fact_8016_transp__trans,axiom,
! [A: $tType] :
( ( transp @ A )
= ( ^ [R2: A > A > $o] : ( trans @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ R2 ) ) ) ) ) ).
% transp_trans
thf(fact_8017_transp__trans__eq,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( transp @ A
@ ^ [X: A,Y4: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y4 ) @ R ) )
= ( trans @ A @ R ) ) ).
% transp_trans_eq
thf(fact_8018_transp__gr,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( transp @ A
@ ^ [X: A,Y4: A] : ( ord_less @ A @ Y4 @ X ) ) ) ).
% transp_gr
thf(fact_8019_transp__empty,axiom,
! [A: $tType] :
( transp @ A
@ ^ [X: A,Y4: A] : $false ) ).
% transp_empty
thf(fact_8020_transp__singleton,axiom,
! [A: $tType,A2: A] :
( transp @ A
@ ^ [X: A,Y4: A] :
( ( X = A2 )
& ( Y4 = A2 ) ) ) ).
% transp_singleton
thf(fact_8021_subset__mset_Otransp__ge,axiom,
! [A: $tType] :
( transp @ ( multiset @ A )
@ ^ [X: multiset @ A,Y4: multiset @ A] : ( subseteq_mset @ A @ Y4 @ X ) ) ).
% subset_mset.transp_ge
thf(fact_8022_transp__le,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( transp @ A @ ( ord_less_eq @ A ) ) ) ).
% transp_le
thf(fact_8023_transp__ge,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( transp @ A
@ ^ [X: A,Y4: A] : ( ord_less_eq @ A @ Y4 @ X ) ) ) ).
% transp_ge
thf(fact_8024_subset__mset_Otransp__gr,axiom,
! [A: $tType] :
( transp @ ( multiset @ A )
@ ^ [X: multiset @ A,Y4: multiset @ A] : ( subset_mset @ A @ Y4 @ X ) ) ).
% subset_mset.transp_gr
thf(fact_8025_multp__implies__one__step,axiom,
! [A: $tType,R: A > A > $o,M3: multiset @ A,N5: multiset @ A] :
( ( transp @ A @ R )
=> ( ( multp @ A @ R @ M3 @ N5 )
=> ? [I8: multiset @ A,J6: multiset @ A] :
( ( N5
= ( plus_plus @ ( multiset @ A ) @ I8 @ J6 ) )
& ? [K8: multiset @ A] :
( ( M3
= ( plus_plus @ ( multiset @ A ) @ I8 @ K8 ) )
& ( J6
!= ( zero_zero @ ( multiset @ A ) ) )
& ! [X4: A] :
( ( member @ A @ X4 @ ( set_mset @ A @ K8 ) )
=> ? [Xa3: A] :
( ( member @ A @ Xa3 @ ( set_mset @ A @ J6 ) )
& ( R @ X4 @ Xa3 ) ) ) ) ) ) ) ).
% multp_implies_one_step
thf(fact_8026_multp__cancel__max,axiom,
! [A: $tType,R: A > A > $o,X8: multiset @ A,Y7: multiset @ A] :
( ( transp @ A @ R )
=> ( ( irreflp @ A @ R )
=> ( ( multp @ A @ R @ X8 @ Y7 )
= ( multp @ A @ R @ ( minus_minus @ ( multiset @ A ) @ X8 @ Y7 ) @ ( minus_minus @ ( multiset @ A ) @ Y7 @ X8 ) ) ) ) ) ).
% multp_cancel_max
thf(fact_8027_multp__cancel,axiom,
! [A: $tType,R: A > A > $o,X8: multiset @ A,Z9: multiset @ A,Y7: multiset @ A] :
( ( transp @ A @ R )
=> ( ( irreflp @ A @ R )
=> ( ( multp @ A @ R @ ( plus_plus @ ( multiset @ A ) @ X8 @ Z9 ) @ ( plus_plus @ ( multiset @ A ) @ Y7 @ Z9 ) )
= ( multp @ A @ R @ X8 @ Y7 ) ) ) ) ).
% multp_cancel
thf(fact_8028_irreflp__multp,axiom,
! [A: $tType,R: A > A > $o] :
( ( transp @ A @ R )
=> ( ( irreflp @ A @ R )
=> ( irreflp @ ( multiset @ A ) @ ( multp @ A @ R ) ) ) ) ).
% irreflp_multp
thf(fact_8029_subset__mset_Oirreflp__greater,axiom,
! [A: $tType] :
( irreflp @ ( multiset @ A )
@ ^ [X: multiset @ A,Y4: multiset @ A] : ( subset_mset @ A @ Y4 @ X ) ) ).
% subset_mset.irreflp_greater
thf(fact_8030_irreflp__greater,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( irreflp @ A
@ ^ [X: A,Y4: A] : ( ord_less @ A @ Y4 @ X ) ) ) ).
% irreflp_greater
thf(fact_8031_irreflp__irrefl__eq,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( irreflp @ A
@ ^ [A3: A,B3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B3 ) @ R3 ) )
= ( irrefl @ A @ R3 ) ) ).
% irreflp_irrefl_eq
thf(fact_8032_multp__cancel__add__mset,axiom,
! [A: $tType,R: A > A > $o,Uu3: A,X8: multiset @ A,Y7: multiset @ A] :
( ( transp @ A @ R )
=> ( ( irreflp @ A @ R )
=> ( ( multp @ A @ R @ ( add_mset @ A @ Uu3 @ X8 ) @ ( add_mset @ A @ Uu3 @ Y7 ) )
= ( multp @ A @ R @ X8 @ Y7 ) ) ) ) ).
% multp_cancel_add_mset
thf(fact_8033_prod__set__simps_I1_J,axiom,
! [B: $tType,A: $tType,X2: A,Y: B] :
( ( basic_fsts @ A @ B @ ( product_Pair @ A @ B @ X2 @ Y ) )
= ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% prod_set_simps(1)
thf(fact_8034_single__valuedp__single__valued__eq,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B )] :
( ( single_valuedp @ A @ B
@ ^ [X: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y4 ) @ R ) )
= ( single_valued @ A @ B @ R ) ) ).
% single_valuedp_single_valued_eq
thf(fact_8035_single__valuedp__iff__Uniq,axiom,
! [B: $tType,A: $tType] :
( ( single_valuedp @ A @ B )
= ( ^ [R2: A > B > $o] :
! [X: A] : ( uniq @ B @ ( R2 @ X ) ) ) ) ).
% single_valuedp_iff_Uniq
thf(fact_8036_prod_Oin__rel,axiom,
! [B: $tType,D: $tType,C: $tType,A: $tType] :
( ( basic_rel_prod @ A @ C @ B @ D )
= ( ^ [R14: A > C > $o,R25: B > D > $o,A3: product_prod @ A @ B,B3: product_prod @ C @ D] :
? [Z3: product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D )] :
( ( member @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) ) @ Z3
@ ( collect @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) )
@ ^ [X: product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ C ) ) @ ( basic_fsts @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) @ X ) @ ( collect @ ( product_prod @ A @ C ) @ ( product_case_prod @ A @ C @ $o @ R14 ) ) )
& ( ord_less_eq @ ( set @ ( product_prod @ B @ D ) ) @ ( basic_snds @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) @ X ) @ ( collect @ ( product_prod @ B @ D ) @ ( product_case_prod @ B @ D @ $o @ R25 ) ) ) ) ) )
& ( ( product_map_prod @ ( product_prod @ A @ C ) @ A @ ( product_prod @ B @ D ) @ B @ ( product_fst @ A @ C ) @ ( product_fst @ B @ D ) @ Z3 )
= A3 )
& ( ( product_map_prod @ ( product_prod @ A @ C ) @ C @ ( product_prod @ B @ D ) @ D @ ( product_snd @ A @ C ) @ ( product_snd @ B @ D ) @ Z3 )
= B3 ) ) ) ) ).
% prod.in_rel
thf(fact_8037_Gcd__int__set__eq__fold,axiom,
! [Xs: list @ int] :
( ( gcd_Gcd @ int @ ( set2 @ int @ Xs ) )
= ( fold @ int @ int @ ( gcd_gcd @ int ) @ Xs @ ( zero_zero @ int ) ) ) ).
% Gcd_int_set_eq_fold
thf(fact_8038_fold__filter,axiom,
! [A: $tType,B: $tType,F: B > A > A,P: B > $o,Xs: list @ B] :
( ( fold @ B @ A @ F @ ( filter2 @ B @ P @ Xs ) )
= ( fold @ B @ A
@ ^ [X: B] : ( if @ ( A > A ) @ ( P @ X ) @ ( F @ X ) @ ( id @ A ) )
@ Xs ) ) ).
% fold_filter
thf(fact_8039_snd__transfer,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,A5: A > C > $o,B5: B > D > $o] : ( bNF_rel_fun @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) @ B @ D @ ( basic_rel_prod @ A @ C @ B @ D @ A5 @ B5 ) @ B5 @ ( product_snd @ A @ B ) @ ( product_snd @ C @ D ) ) ).
% snd_transfer
thf(fact_8040_int_Oid__abs__transfer,axiom,
( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ int
@ ( basic_rel_prod @ nat @ nat @ nat @ nat
@ ^ [Y3: nat,Z: nat] : Y3 = Z
@ ^ [Y3: nat,Z: nat] : Y3 = Z )
@ pcr_int
@ ^ [X: product_prod @ nat @ nat] : X
@ abs_Integ ) ).
% int.id_abs_transfer
thf(fact_8041_prod_Orel__map_I1_J,axiom,
! [A: $tType,B: $tType,E: $tType,F7: $tType,D: $tType,C: $tType,S1b: E > C > $o,S2b: F7 > D > $o,I1: A > E,I22: B > F7,X2: product_prod @ A @ B,Y: product_prod @ C @ D] :
( ( basic_rel_prod @ E @ C @ F7 @ D @ S1b @ S2b @ ( product_map_prod @ A @ E @ B @ F7 @ I1 @ I22 @ X2 ) @ Y )
= ( basic_rel_prod @ A @ C @ B @ D
@ ^ [X: A] : ( S1b @ ( I1 @ X ) )
@ ^ [X: B] : ( S2b @ ( I22 @ X ) )
@ X2
@ Y ) ) ).
% prod.rel_map(1)
thf(fact_8042_prod_Orel__map_I2_J,axiom,
! [A: $tType,B: $tType,E: $tType,F7: $tType,D: $tType,C: $tType,S1a: A > E > $o,S2a: B > F7 > $o,X2: product_prod @ A @ B,G1: C > E,G22: D > F7,Y: product_prod @ C @ D] :
( ( basic_rel_prod @ A @ E @ B @ F7 @ S1a @ S2a @ X2 @ ( product_map_prod @ C @ E @ D @ F7 @ G1 @ G22 @ Y ) )
= ( basic_rel_prod @ A @ C @ B @ D
@ ^ [X: A,Y4: C] : ( S1a @ X @ ( G1 @ Y4 ) )
@ ^ [X: B,Y4: D] : ( S2a @ X @ ( G22 @ Y4 ) )
@ X2
@ Y ) ) ).
% prod.rel_map(2)
thf(fact_8043_foldl__conv__fold,axiom,
! [B: $tType,A: $tType] :
( ( foldl @ A @ B )
= ( ^ [F2: A > B > A,S6: A,Xs3: list @ B] :
( fold @ B @ A
@ ^ [X: B,T3: A] : ( F2 @ T3 @ X )
@ Xs3
@ S6 ) ) ) ).
% foldl_conv_fold
thf(fact_8044_fold__plus__sum__list__rev,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ! [Xs: list @ A] :
( ( fold @ A @ A @ ( plus_plus @ A ) @ Xs )
= ( plus_plus @ A @ ( groups8242544230860333062m_list @ A @ ( rev @ A @ Xs ) ) ) ) ) ).
% fold_plus_sum_list_rev
thf(fact_8045_fst__transfer,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,A5: A > C > $o,B5: B > D > $o] : ( bNF_rel_fun @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) @ A @ C @ ( basic_rel_prod @ A @ C @ B @ D @ A5 @ B5 ) @ A5 @ ( product_fst @ A @ B ) @ ( product_fst @ C @ D ) ) ).
% fst_transfer
thf(fact_8046_rel__prod__conv,axiom,
! [B: $tType,D: $tType,C: $tType,A: $tType] :
( ( basic_rel_prod @ A @ C @ B @ D )
= ( ^ [R14: A > C > $o,R25: B > D > $o] :
( product_case_prod @ A @ B @ ( ( product_prod @ C @ D ) > $o )
@ ^ [A3: A,B3: B] :
( product_case_prod @ C @ D @ $o
@ ^ [C5: C,D5: D] :
( ( R14 @ A3 @ C5 )
& ( R25 @ B3 @ D5 ) ) ) ) ) ) ).
% rel_prod_conv
thf(fact_8047_Pair__transfer,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,A5: A > B > $o,B5: C > D > $o] : ( bNF_rel_fun @ A @ B @ ( C > ( product_prod @ A @ C ) ) @ ( D > ( product_prod @ B @ D ) ) @ A5 @ ( bNF_rel_fun @ C @ D @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) @ B5 @ ( basic_rel_prod @ A @ B @ C @ D @ A5 @ B5 ) ) @ ( product_Pair @ A @ C ) @ ( product_Pair @ B @ D ) ) ).
% Pair_transfer
thf(fact_8048_bounded__quasi__semilattice__set_Oset__eq__fold,axiom,
! [A: $tType,F: A > A > A,Top: A,Bot: A,Normalize: A > A,Xs: list @ A] :
( ( bounde6485984586167503788ce_set @ A @ F @ Top @ Bot @ Normalize )
=> ( ( bounde2362111253966948842tice_F @ A @ F @ Top @ Bot @ ( set2 @ A @ Xs ) )
= ( fold @ A @ A @ F @ Xs @ Top ) ) ) ).
% bounded_quasi_semilattice_set.set_eq_fold
thf(fact_8049_case__prod__transfer,axiom,
! [A: $tType,B: $tType,C: $tType,F7: $tType,E: $tType,D: $tType,A5: A > D > $o,B5: B > E > $o,C6: C > F7 > $o] : ( bNF_rel_fun @ ( A > B > C ) @ ( D > E > F7 ) @ ( ( product_prod @ A @ B ) > C ) @ ( ( product_prod @ D @ E ) > F7 ) @ ( bNF_rel_fun @ A @ D @ ( B > C ) @ ( E > F7 ) @ A5 @ ( bNF_rel_fun @ B @ E @ C @ F7 @ B5 @ C6 ) ) @ ( bNF_rel_fun @ ( product_prod @ A @ B ) @ ( product_prod @ D @ E ) @ C @ F7 @ ( basic_rel_prod @ A @ D @ B @ E @ A5 @ B5 ) @ C6 ) @ ( product_case_prod @ A @ B @ C ) @ ( product_case_prod @ D @ E @ F7 ) ) ).
% case_prod_transfer
thf(fact_8050_rat_Odomain,axiom,
( ( domainp @ ( product_prod @ int @ int ) @ rat @ pcr_rat )
= ( ^ [X: product_prod @ int @ int] :
? [Y4: product_prod @ int @ int] :
( ( basic_rel_prod @ int @ int @ int @ int
@ ^ [Y3: int,Z: int] : Y3 = Z
@ ^ [Y3: int,Z: int] : Y3 = Z
@ X
@ Y4 )
& ( ratrel @ Y4 @ Y4 ) ) ) ) ).
% rat.domain
thf(fact_8051_Gcd__set__eq__fold,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [Xs: list @ A] :
( ( gcd_Gcd @ A @ ( set2 @ A @ Xs ) )
= ( fold @ A @ A @ ( gcd_gcd @ A ) @ Xs @ ( zero_zero @ A ) ) ) ) ).
% Gcd_set_eq_fold
thf(fact_8052_Gcd__fin_Oset__eq__fold,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [Xs: list @ A] :
( ( semiring_gcd_Gcd_fin @ A @ ( set2 @ A @ Xs ) )
= ( fold @ A @ A @ ( gcd_gcd @ A ) @ Xs @ ( zero_zero @ A ) ) ) ) ).
% Gcd_fin.set_eq_fold
thf(fact_8053_rat_Odomain__par,axiom,
! [DR1: int > $o,DR2: int > $o,P23: ( product_prod @ int @ int ) > $o] :
( ( ( domainp @ int @ int
@ ^ [Y3: int,Z: int] : Y3 = Z )
= DR1 )
=> ( ( ( domainp @ int @ int
@ ^ [Y3: int,Z: int] : Y3 = Z )
= DR2 )
=> ( ( bNF_rel_fun @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ $o @ $o
@ ( basic_rel_prod @ int @ int @ int @ int
@ ^ [Y3: int,Z: int] : Y3 = Z
@ ^ [Y3: int,Z: int] : Y3 = Z )
@ ^ [Y3: $o,Z: $o] : Y3 = Z
@ P23
@ ^ [X: product_prod @ int @ int] : ( ratrel @ X @ X ) )
=> ( ( domainp @ ( product_prod @ int @ int ) @ rat @ pcr_rat )
= ( inf_inf @ ( ( product_prod @ int @ int ) > $o ) @ ( basic_pred_prod @ int @ int @ DR1 @ DR2 ) @ P23 ) ) ) ) ) ).
% rat.domain_par
thf(fact_8054_rat_Odomain__par__left__total,axiom,
! [P2: ( product_prod @ int @ int ) > $o] :
( ( left_total @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int )
@ ( basic_rel_prod @ int @ int @ int @ int
@ ^ [Y3: int,Z: int] : Y3 = Z
@ ^ [Y3: int,Z: int] : Y3 = Z ) )
=> ( ( bNF_rel_fun @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ $o @ $o
@ ( basic_rel_prod @ int @ int @ int @ int
@ ^ [Y3: int,Z: int] : Y3 = Z
@ ^ [Y3: int,Z: int] : Y3 = Z )
@ ^ [Y3: $o,Z: $o] : Y3 = Z
@ P2
@ ^ [X: product_prod @ int @ int] : ( ratrel @ X @ X ) )
=> ( ( domainp @ ( product_prod @ int @ int ) @ rat @ pcr_rat )
= P2 ) ) ) ).
% rat.domain_par_left_total
thf(fact_8055_prod_Omap__cong__pred,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,X2: product_prod @ A @ B,Ya: product_prod @ A @ B,F1: A > C,G1: A > C,F22: B > D,G22: B > D] :
( ( X2 = Ya )
=> ( ( basic_pred_prod @ A @ B
@ ^ [Z12: A] :
( ( F1 @ Z12 )
= ( G1 @ Z12 ) )
@ ^ [Z23: B] :
( ( F22 @ Z23 )
= ( G22 @ Z23 ) )
@ Ya )
=> ( ( product_map_prod @ A @ C @ B @ D @ F1 @ F22 @ X2 )
= ( product_map_prod @ A @ C @ B @ D @ G1 @ G22 @ Ya ) ) ) ) ).
% prod.map_cong_pred
thf(fact_8056_prod_Opred__True,axiom,
! [B: $tType,A: $tType] :
( ( basic_pred_prod @ A @ B
@ ^ [Uu: A] : $true
@ ^ [Uu: B] : $true )
= ( ^ [Uu: product_prod @ A @ B] : $true ) ) ).
% prod.pred_True
thf(fact_8057_sort__conv__fold,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( linorder_sort_key @ A @ A
@ ^ [X: A] : X
@ Xs )
= ( fold @ A @ ( list @ A )
@ ( linorder_insort_key @ A @ A
@ ^ [X: A] : X )
@ Xs
@ ( nil @ A ) ) ) ) ).
% sort_conv_fold
thf(fact_8058_Gcd__nat__set__eq__fold,axiom,
! [Xs: list @ nat] :
( ( gcd_Gcd @ nat @ ( set2 @ nat @ Xs ) )
= ( fold @ nat @ nat @ ( gcd_gcd @ nat ) @ Xs @ ( zero_zero @ nat ) ) ) ).
% Gcd_nat_set_eq_fold
thf(fact_8059_Abs__rat__cases,axiom,
! [X2: rat] :
~ ! [Y2: set @ ( product_prod @ int @ int )] :
( ( X2
= ( abs_rat @ Y2 ) )
=> ~ ( member @ ( set @ ( product_prod @ int @ int ) ) @ Y2
@ ( collect @ ( set @ ( product_prod @ int @ int ) )
@ ^ [C5: set @ ( product_prod @ int @ int )] :
? [X: product_prod @ int @ int] :
( ( ratrel @ X @ X )
& ( C5
= ( collect @ ( product_prod @ int @ int ) @ ( ratrel @ X ) ) ) ) ) ) ) ).
% Abs_rat_cases
thf(fact_8060_Abs__rat__induct,axiom,
! [P: rat > $o,X2: rat] :
( ! [Y2: set @ ( product_prod @ int @ int )] :
( ( member @ ( set @ ( product_prod @ int @ int ) ) @ Y2
@ ( collect @ ( set @ ( product_prod @ int @ int ) )
@ ^ [C5: set @ ( product_prod @ int @ int )] :
? [X: product_prod @ int @ int] :
( ( ratrel @ X @ X )
& ( C5
= ( collect @ ( product_prod @ int @ int ) @ ( ratrel @ X ) ) ) ) ) )
=> ( P @ ( abs_rat @ Y2 ) ) )
=> ( P @ X2 ) ) ).
% Abs_rat_induct
thf(fact_8061_int_Odomain,axiom,
! [DR1: nat > $o,DR2: nat > $o] :
( ( ( domainp @ nat @ nat
@ ^ [Y3: nat,Z: nat] : Y3 = Z )
= DR1 )
=> ( ( ( domainp @ nat @ nat
@ ^ [Y3: nat,Z: nat] : Y3 = Z )
= DR2 )
=> ( ( domainp @ ( product_prod @ nat @ nat ) @ int @ pcr_int )
= ( basic_pred_prod @ nat @ nat @ DR1 @ DR2 ) ) ) ) ).
% int.domain
thf(fact_8062_Abs__rat__inject,axiom,
! [X2: set @ ( product_prod @ int @ int ),Y: set @ ( product_prod @ int @ int )] :
( ( member @ ( set @ ( product_prod @ int @ int ) ) @ X2
@ ( collect @ ( set @ ( product_prod @ int @ int ) )
@ ^ [C5: set @ ( product_prod @ int @ int )] :
? [X: product_prod @ int @ int] :
( ( ratrel @ X @ X )
& ( C5
= ( collect @ ( product_prod @ int @ int ) @ ( ratrel @ X ) ) ) ) ) )
=> ( ( member @ ( set @ ( product_prod @ int @ int ) ) @ Y
@ ( collect @ ( set @ ( product_prod @ int @ int ) )
@ ^ [C5: set @ ( product_prod @ int @ int )] :
? [X: product_prod @ int @ int] :
( ( ratrel @ X @ X )
& ( C5
= ( collect @ ( product_prod @ int @ int ) @ ( ratrel @ X ) ) ) ) ) )
=> ( ( ( abs_rat @ X2 )
= ( abs_rat @ Y ) )
= ( X2 = Y ) ) ) ) ).
% Abs_rat_inject
thf(fact_8063_Abs__rat__inverse,axiom,
! [Y: set @ ( product_prod @ int @ int )] :
( ( member @ ( set @ ( product_prod @ int @ int ) ) @ Y
@ ( collect @ ( set @ ( product_prod @ int @ int ) )
@ ^ [C5: set @ ( product_prod @ int @ int )] :
? [X: product_prod @ int @ int] :
( ( ratrel @ X @ X )
& ( C5
= ( collect @ ( product_prod @ int @ int ) @ ( ratrel @ X ) ) ) ) ) )
=> ( ( rep_rat @ ( abs_rat @ Y ) )
= Y ) ) ).
% Abs_rat_inverse
thf(fact_8064_type__definition__rat,axiom,
( type_definition @ rat @ ( set @ ( product_prod @ int @ int ) ) @ rep_rat @ abs_rat
@ ( collect @ ( set @ ( product_prod @ int @ int ) )
@ ^ [C5: set @ ( product_prod @ int @ int )] :
? [X: product_prod @ int @ int] :
( ( ratrel @ X @ X )
& ( C5
= ( collect @ ( product_prod @ int @ int ) @ ( ratrel @ X ) ) ) ) ) ) ).
% type_definition_rat
thf(fact_8065_Rep__rat,axiom,
! [X2: rat] :
( member @ ( set @ ( product_prod @ int @ int ) ) @ ( rep_rat @ X2 )
@ ( collect @ ( set @ ( product_prod @ int @ int ) )
@ ^ [C5: set @ ( product_prod @ int @ int )] :
? [X: product_prod @ int @ int] :
( ( ratrel @ X @ X )
& ( C5
= ( collect @ ( product_prod @ int @ int ) @ ( ratrel @ X ) ) ) ) ) ) ).
% Rep_rat
thf(fact_8066_Rep__rat__cases,axiom,
! [Y: set @ ( product_prod @ int @ int )] :
( ( member @ ( set @ ( product_prod @ int @ int ) ) @ Y
@ ( collect @ ( set @ ( product_prod @ int @ int ) )
@ ^ [C5: set @ ( product_prod @ int @ int )] :
? [X: product_prod @ int @ int] :
( ( ratrel @ X @ X )
& ( C5
= ( collect @ ( product_prod @ int @ int ) @ ( ratrel @ X ) ) ) ) ) )
=> ~ ! [X3: rat] :
( Y
!= ( rep_rat @ X3 ) ) ) ).
% Rep_rat_cases
thf(fact_8067_Rep__rat__induct,axiom,
! [Y: set @ ( product_prod @ int @ int ),P: ( set @ ( product_prod @ int @ int ) ) > $o] :
( ( member @ ( set @ ( product_prod @ int @ int ) ) @ Y
@ ( collect @ ( set @ ( product_prod @ int @ int ) )
@ ^ [C5: set @ ( product_prod @ int @ int )] :
? [X: product_prod @ int @ int] :
( ( ratrel @ X @ X )
& ( C5
= ( collect @ ( product_prod @ int @ int ) @ ( ratrel @ X ) ) ) ) ) )
=> ( ! [X3: rat] : ( P @ ( rep_rat @ X3 ) )
=> ( P @ Y ) ) ) ).
% Rep_rat_induct
thf(fact_8068_prod_Orel__compp__Grp,axiom,
! [B: $tType,D: $tType,C: $tType,A: $tType] :
( ( basic_rel_prod @ A @ C @ B @ D )
= ( ^ [R14: A > C > $o,R25: B > D > $o] :
( relcompp @ ( product_prod @ A @ B ) @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) ) @ ( product_prod @ C @ D )
@ ( conversep @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) ) @ ( product_prod @ A @ B )
@ ( bNF_Grp @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) ) @ ( product_prod @ A @ B )
@ ( collect @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) )
@ ^ [X: product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ C ) ) @ ( basic_fsts @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) @ X ) @ ( collect @ ( product_prod @ A @ C ) @ ( product_case_prod @ A @ C @ $o @ R14 ) ) )
& ( ord_less_eq @ ( set @ ( product_prod @ B @ D ) ) @ ( basic_snds @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) @ X ) @ ( collect @ ( product_prod @ B @ D ) @ ( product_case_prod @ B @ D @ $o @ R25 ) ) ) ) )
@ ( product_map_prod @ ( product_prod @ A @ C ) @ A @ ( product_prod @ B @ D ) @ B @ ( product_fst @ A @ C ) @ ( product_fst @ B @ D ) ) ) )
@ ( bNF_Grp @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) ) @ ( product_prod @ C @ D )
@ ( collect @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) )
@ ^ [X: product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ C ) ) @ ( basic_fsts @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) @ X ) @ ( collect @ ( product_prod @ A @ C ) @ ( product_case_prod @ A @ C @ $o @ R14 ) ) )
& ( ord_less_eq @ ( set @ ( product_prod @ B @ D ) ) @ ( basic_snds @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) @ X ) @ ( collect @ ( product_prod @ B @ D ) @ ( product_case_prod @ B @ D @ $o @ R25 ) ) ) ) )
@ ( product_map_prod @ ( product_prod @ A @ C ) @ C @ ( product_prod @ B @ D ) @ D @ ( product_snd @ A @ C ) @ ( product_snd @ B @ D ) ) ) ) ) ) ).
% prod.rel_compp_Grp
thf(fact_8069_convol__transfer,axiom,
! [A: $tType,B: $tType,E: $tType,F7: $tType,D: $tType,C: $tType,R3: A > C > $o,S2: B > D > $o,T6: E > F7 > $o] : ( bNF_rel_fun @ ( A > B ) @ ( C > D ) @ ( ( A > E ) > A > ( product_prod @ B @ E ) ) @ ( ( C > F7 ) > C > ( product_prod @ D @ F7 ) ) @ ( bNF_rel_fun @ A @ C @ B @ D @ R3 @ S2 ) @ ( bNF_rel_fun @ ( A > E ) @ ( C > F7 ) @ ( A > ( product_prod @ B @ E ) ) @ ( C > ( product_prod @ D @ F7 ) ) @ ( bNF_rel_fun @ A @ C @ E @ F7 @ R3 @ T6 ) @ ( bNF_rel_fun @ A @ C @ ( product_prod @ B @ E ) @ ( product_prod @ D @ F7 ) @ R3 @ ( basic_rel_prod @ B @ D @ E @ F7 @ S2 @ T6 ) ) ) @ ( bNF_convol @ A @ B @ E ) @ ( bNF_convol @ C @ D @ F7 ) ) ).
% convol_transfer
thf(fact_8070_pcr__Domainp,axiom,
! [B: $tType,A: $tType,C: $tType,B5: A > B > $o,P: A > $o,A5: C > A > $o] :
( ( ( domainp @ A @ B @ B5 )
= P )
=> ( ( domainp @ C @ B @ ( relcompp @ C @ A @ B @ A5 @ B5 ) )
= ( ^ [X: C] :
? [Y4: A] :
( ( A5 @ X @ Y4 )
& ( P @ Y4 ) ) ) ) ) ).
% pcr_Domainp
thf(fact_8071_prod_Orel__Grp,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,A17: set @ A,F1: A > C,A26: set @ B,F22: B > D] :
( ( basic_rel_prod @ A @ C @ B @ D @ ( bNF_Grp @ A @ C @ A17 @ F1 ) @ ( bNF_Grp @ B @ D @ A26 @ F22 ) )
= ( bNF_Grp @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D )
@ ( collect @ ( product_prod @ A @ B )
@ ^ [X: product_prod @ A @ B] :
( ( ord_less_eq @ ( set @ A ) @ ( basic_fsts @ A @ B @ X ) @ A17 )
& ( ord_less_eq @ ( set @ B ) @ ( basic_snds @ A @ B @ X ) @ A26 ) ) )
@ ( product_map_prod @ A @ C @ B @ D @ F1 @ F22 ) ) ) ).
% prod.rel_Grp
thf(fact_8072_OO__Grp__alt,axiom,
! [B: $tType,C: $tType,A: $tType,A5: set @ C,F: C > A,G: C > B] :
( ( relcompp @ A @ C @ B @ ( conversep @ C @ A @ ( bNF_Grp @ C @ A @ A5 @ F ) ) @ ( bNF_Grp @ C @ B @ A5 @ G ) )
= ( ^ [X: A,Y4: B] :
? [Z3: C] :
( ( member @ C @ Z3 @ A5 )
& ( ( F @ Z3 )
= X )
& ( ( G @ Z3 )
= Y4 ) ) ) ) ).
% OO_Grp_alt
thf(fact_8073_OO__def,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( relcompp @ A @ C @ B )
= ( ^ [R7: A > C > $o,S: C > B > $o,X: A,Z3: B] :
? [Y4: C] :
( ( R7 @ X @ Y4 )
& ( S @ Y4 @ Z3 ) ) ) ) ).
% OO_def
thf(fact_8074_fun_Orel__Grp,axiom,
! [D: $tType,B: $tType,A: $tType,A5: set @ A,F: A > B] :
( ( bNF_rel_fun @ D @ D @ A @ B
@ ^ [Y3: D,Z: D] : Y3 = Z
@ ( bNF_Grp @ A @ B @ A5 @ F ) )
= ( bNF_Grp @ ( D > A ) @ ( D > B )
@ ( collect @ ( D > A )
@ ^ [X: D > A] : ( ord_less_eq @ ( set @ A ) @ ( image2 @ D @ A @ X @ ( top_top @ ( set @ D ) ) ) @ A5 ) )
@ ( comp @ A @ B @ D @ F ) ) ) ).
% fun.rel_Grp
thf(fact_8075_eq__le__Grp__id__iff,axiom,
! [A: $tType,R3: A > $o] :
( ( ord_less_eq @ ( A > A > $o )
@ ^ [Y3: A,Z: A] : Y3 = Z
@ ( bNF_Grp @ A @ A @ ( collect @ A @ R3 ) @ ( id @ A ) ) )
= ( ! [X5: A] : ( R3 @ X5 ) ) ) ).
% eq_le_Grp_id_iff
thf(fact_8076_snd__convol_H,axiom,
! [B: $tType,A: $tType,C: $tType,F: C > B,G: C > A,X2: C] :
( ( product_snd @ B @ A @ ( bNF_convol @ C @ B @ A @ F @ G @ X2 ) )
= ( G @ X2 ) ) ).
% snd_convol'
thf(fact_8077_multiset_Orel__Grp,axiom,
! [B: $tType,A: $tType,A5: set @ A,F: A > B] :
( ( rel_mset @ A @ B @ ( bNF_Grp @ A @ B @ A5 @ F ) )
= ( bNF_Grp @ ( multiset @ A ) @ ( multiset @ B )
@ ( collect @ ( multiset @ A )
@ ^ [X: multiset @ A] : ( ord_less_eq @ ( set @ A ) @ ( set_mset @ A @ X ) @ A5 ) )
@ ( image_mset @ A @ B @ F ) ) ) ).
% multiset.rel_Grp
thf(fact_8078_Grp__def,axiom,
! [B: $tType,A: $tType] :
( ( bNF_Grp @ A @ B )
= ( ^ [A8: set @ A,F2: A > B,A3: A,B3: B] :
( ( B3
= ( F2 @ A3 ) )
& ( member @ A @ A3 @ A8 ) ) ) ) ).
% Grp_def
thf(fact_8079_relcompp__SUP__distrib,axiom,
! [A: $tType,C: $tType,B: $tType,D: $tType,S3: A > C > $o,R: D > C > B > $o,I5: set @ D] :
( ( relcompp @ A @ C @ B @ S3 @ ( complete_Sup_Sup @ ( C > B > $o ) @ ( image2 @ D @ ( C > B > $o ) @ R @ I5 ) ) )
= ( complete_Sup_Sup @ ( A > B > $o )
@ ( image2 @ D @ ( A > B > $o )
@ ^ [I2: D] : ( relcompp @ A @ C @ B @ S3 @ ( R @ I2 ) )
@ I5 ) ) ) ).
% relcompp_SUP_distrib
thf(fact_8080_relcompp__SUP__distrib2,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,R: D > A > C > $o,I5: set @ D,S3: C > B > $o] :
( ( relcompp @ A @ C @ B @ ( complete_Sup_Sup @ ( A > C > $o ) @ ( image2 @ D @ ( A > C > $o ) @ R @ I5 ) ) @ S3 )
= ( complete_Sup_Sup @ ( A > B > $o )
@ ( image2 @ D @ ( A > B > $o )
@ ^ [I2: D] : ( relcompp @ A @ C @ B @ ( R @ I2 ) @ S3 )
@ I5 ) ) ) ).
% relcompp_SUP_distrib2
thf(fact_8081_relpowp__add,axiom,
! [A: $tType,M: nat,N: nat,P: A > A > $o] :
( ( compow @ ( A > A > $o ) @ ( plus_plus @ nat @ M @ N ) @ P )
= ( relcompp @ A @ A @ A @ ( compow @ ( A > A > $o ) @ M @ P ) @ ( compow @ ( A > A > $o ) @ N @ P ) ) ) ).
% relpowp_add
thf(fact_8082_relcompp__relcomp__eq,axiom,
! [C: $tType,B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S3: set @ ( product_prod @ B @ C )] :
( ( relcompp @ A @ B @ C
@ ^ [X: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y4 ) @ R )
@ ^ [X: B,Y4: C] : ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ X @ Y4 ) @ S3 ) )
= ( ^ [X: A,Y4: C] : ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ X @ Y4 ) @ ( relcomp @ A @ B @ C @ R @ S3 ) ) ) ) ).
% relcompp_relcomp_eq
thf(fact_8083_fst__convol_H,axiom,
! [B: $tType,A: $tType,C: $tType,F: C > A,G: C > B,X2: C] :
( ( product_fst @ A @ B @ ( bNF_convol @ C @ A @ B @ F @ G @ X2 ) )
= ( F @ X2 ) ) ).
% fst_convol'
thf(fact_8084_list_Orel__Grp,axiom,
! [B: $tType,A: $tType,A5: set @ A,F: A > B] :
( ( list_all2 @ A @ B @ ( bNF_Grp @ A @ B @ A5 @ F ) )
= ( bNF_Grp @ ( list @ A ) @ ( list @ B )
@ ( collect @ ( list @ A )
@ ^ [X: list @ A] : ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ X ) @ A5 ) )
@ ( map @ A @ B @ F ) ) ) ).
% list.rel_Grp
thf(fact_8085_relcomp__def,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( relcomp @ A @ B @ C )
= ( ^ [R2: 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
@ ^ [X: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y4 ) @ R2 )
@ ^ [X: B,Y4: C] : ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ X @ Y4 ) @ S6 ) ) ) ) ) ) ).
% relcomp_def
thf(fact_8086_map__prod__o__convol,axiom,
! [D: $tType,B: $tType,C: $tType,E: $tType,A: $tType,H1: D > B,H22: E > C,F: A > D,G: A > E] :
( ( comp @ ( product_prod @ D @ E ) @ ( product_prod @ B @ C ) @ A @ ( product_map_prod @ D @ B @ E @ C @ H1 @ H22 ) @ ( bNF_convol @ A @ D @ E @ F @ G ) )
= ( bNF_convol @ A @ B @ C @ ( comp @ D @ B @ A @ H1 @ F ) @ ( comp @ E @ C @ A @ H22 @ G ) ) ) ).
% map_prod_o_convol
thf(fact_8087_convol__o,axiom,
! [B: $tType,C: $tType,D: $tType,A: $tType,F: D > B,G: D > C,H: A > D] :
( ( comp @ D @ ( product_prod @ B @ C ) @ A @ ( bNF_convol @ D @ B @ C @ F @ G ) @ H )
= ( bNF_convol @ A @ B @ C @ ( comp @ D @ B @ A @ F @ H ) @ ( comp @ D @ C @ A @ G @ H ) ) ) ).
% convol_o
thf(fact_8088_map__prod__o__convol__id,axiom,
! [B: $tType,A: $tType,C: $tType,F: C > A,G: C > B,X2: C] :
( ( comp @ ( product_prod @ C @ B ) @ ( product_prod @ A @ B ) @ C @ ( product_map_prod @ C @ A @ B @ B @ F @ ( id @ B ) ) @ ( bNF_convol @ C @ C @ B @ ( id @ C ) @ G ) @ X2 )
= ( bNF_convol @ C @ A @ B @ ( comp @ A @ A @ C @ ( id @ A ) @ F ) @ G @ X2 ) ) ).
% map_prod_o_convol_id
thf(fact_8089_convol__expand__snd,axiom,
! [C: $tType,B: $tType,A: $tType,F: A > ( product_prod @ B @ C ),G: A > B] :
( ( ( comp @ ( product_prod @ B @ C ) @ B @ A @ ( product_fst @ B @ C ) @ F )
= G )
=> ( ( bNF_convol @ A @ B @ C @ G @ ( comp @ ( product_prod @ B @ C ) @ C @ A @ ( product_snd @ B @ C ) @ F ) )
= F ) ) ).
% convol_expand_snd
thf(fact_8090_convol__expand__snd_H,axiom,
! [C: $tType,B: $tType,A: $tType,F: A > ( product_prod @ B @ C ),G: A > B,H: A > C] :
( ( ( comp @ ( product_prod @ B @ C ) @ B @ A @ ( product_fst @ B @ C ) @ F )
= G )
=> ( ( H
= ( comp @ ( product_prod @ B @ C ) @ C @ A @ ( product_snd @ B @ C ) @ F ) )
= ( ( bNF_convol @ A @ B @ C @ G @ H )
= F ) ) ) ).
% convol_expand_snd'
thf(fact_8091_type__copy__vimage2p__Grp__Abs,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,Rep: A > B,Abs: B > A,G: D > C,P: C > $o,H: C > A] :
( ( type_definition @ A @ B @ Rep @ Abs @ ( top_top @ ( set @ B ) ) )
=> ( ( bNF_vimage2p @ D @ C @ B @ A @ $o @ G @ Abs @ ( bNF_Grp @ C @ A @ ( collect @ C @ P ) @ H ) )
= ( bNF_Grp @ D @ B
@ ( collect @ D
@ ^ [X: D] : ( P @ ( G @ X ) ) )
@ ( comp @ C @ B @ D @ ( comp @ A @ B @ C @ Rep @ H ) @ G ) ) ) ) ).
% type_copy_vimage2p_Grp_Abs
thf(fact_8092_type__copy__vimage2p__Grp__Rep,axiom,
! [B: $tType,A: $tType,D: $tType,C: $tType,Rep: A > B,Abs: B > A,F: C > D,P: D > $o,H: D > B] :
( ( type_definition @ A @ B @ Rep @ Abs @ ( top_top @ ( set @ B ) ) )
=> ( ( bNF_vimage2p @ C @ D @ A @ B @ $o @ F @ Rep @ ( bNF_Grp @ D @ B @ ( collect @ D @ P ) @ H ) )
= ( bNF_Grp @ C @ A
@ ( collect @ C
@ ^ [X: C] : ( P @ ( F @ X ) ) )
@ ( comp @ D @ A @ C @ ( comp @ B @ A @ D @ Abs @ H ) @ F ) ) ) ) ).
% type_copy_vimage2p_Grp_Rep
thf(fact_8093_vimage2p__rel__fun,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,F: A > C,G: B > D,R3: C > D > $o] : ( bNF_rel_fun @ A @ B @ C @ D @ ( bNF_vimage2p @ A @ C @ B @ D @ $o @ F @ G @ R3 ) @ R3 @ F @ G ) ).
% vimage2p_rel_fun
thf(fact_8094_vimage2p__def,axiom,
! [B: $tType,E: $tType,C: $tType,D: $tType,A: $tType] :
( ( bNF_vimage2p @ A @ D @ B @ E @ C )
= ( ^ [F2: A > D,G2: B > E,R7: D > E > C,X: A,Y4: B] : ( R7 @ ( F2 @ X ) @ ( G2 @ Y4 ) ) ) ) ).
% vimage2p_def
thf(fact_8095_vimage2p__refl,axiom,
! [A: $tType,B: $tType,R3: A > A > $o,F: B > A,X2: B] :
( ! [X3: A] : ( R3 @ X3 @ X3 )
=> ( bNF_vimage2p @ B @ A @ B @ A @ $o @ F @ F @ R3 @ X2 @ X2 ) ) ).
% vimage2p_refl
thf(fact_8096_vimage2p__id,axiom,
! [C: $tType,B: $tType,A: $tType,R3: A > B > C] :
( ( bNF_vimage2p @ A @ A @ B @ B @ C @ ( id @ A ) @ ( id @ B ) @ R3 )
= R3 ) ).
% vimage2p_id
thf(fact_8097_vimage2p__comp,axiom,
! [E: $tType,D: $tType,F7: $tType,A: $tType,C: $tType,B: $tType,G4: $tType,F1: F7 > A,F22: D > F7,G1: G4 > B,G22: E > G4] :
( ( bNF_vimage2p @ D @ A @ E @ B @ C @ ( comp @ F7 @ A @ D @ F1 @ F22 ) @ ( comp @ G4 @ B @ E @ G1 @ G22 ) )
= ( comp @ ( F7 > G4 > C ) @ ( D > E > C ) @ ( A > B > C ) @ ( bNF_vimage2p @ D @ F7 @ E @ G4 @ C @ F22 @ G22 ) @ ( bNF_vimage2p @ F7 @ A @ G4 @ B @ C @ F1 @ G1 ) ) ) ).
% vimage2p_comp
thf(fact_8098_vimage2p__mono,axiom,
! [A: $tType,D: $tType,B: $tType,C: $tType,F: A > B,G: C > D,R3: B > D > $o,X2: A,Y: C,S2: B > D > $o] :
( ( bNF_vimage2p @ A @ B @ C @ D @ $o @ F @ G @ R3 @ X2 @ Y )
=> ( ( ord_less_eq @ ( B > D > $o ) @ R3 @ S2 )
=> ( bNF_vimage2p @ A @ B @ C @ D @ $o @ F @ G @ S2 @ X2 @ Y ) ) ) ).
% vimage2p_mono
thf(fact_8099_list_Orel__compp__Grp,axiom,
! [B: $tType,A: $tType] :
( ( list_all2 @ A @ B )
= ( ^ [R7: 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 ) )
@ ^ [X: list @ ( product_prod @ A @ B )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( set2 @ ( product_prod @ A @ B ) @ X ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R7 ) ) ) )
@ ( 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 ) )
@ ^ [X: list @ ( product_prod @ A @ B )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( set2 @ ( product_prod @ A @ B ) @ X ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R7 ) ) ) )
@ ( map @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) ) ) ) ) ) ).
% list.rel_compp_Grp
thf(fact_8100_Abs__transfer,axiom,
! [B: $tType,A: $tType,C: $tType,D: $tType,Rep1: A > B,Abs1: B > A,Rep22: C > D,Abs22: D > C,R3: B > D > $o] :
( ( type_definition @ A @ B @ Rep1 @ Abs1 @ ( top_top @ ( set @ B ) ) )
=> ( ( type_definition @ C @ D @ Rep22 @ Abs22 @ ( top_top @ ( set @ D ) ) )
=> ( bNF_rel_fun @ B @ D @ A @ C @ R3 @ ( bNF_vimage2p @ A @ B @ C @ D @ $o @ Rep1 @ Rep22 @ R3 ) @ Abs1 @ Abs22 ) ) ) ).
% Abs_transfer
thf(fact_8101_fun_Orel__compp__Grp,axiom,
! [D: $tType,B: $tType,A: $tType,R3: A > B > $o] :
( ( bNF_rel_fun @ D @ D @ A @ B
@ ^ [Y3: D,Z: D] : Y3 = Z
@ R3 )
= ( 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 ) )
@ ^ [X: D > ( product_prod @ A @ B )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( image2 @ D @ ( product_prod @ A @ B ) @ X @ ( top_top @ ( set @ D ) ) ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R3 ) ) ) )
@ ( 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 ) )
@ ^ [X: D > ( product_prod @ A @ B )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( image2 @ D @ ( product_prod @ A @ B ) @ X @ ( top_top @ ( set @ D ) ) ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R3 ) ) ) )
@ ( comp @ ( product_prod @ A @ B ) @ B @ D @ ( product_snd @ A @ B ) ) ) ) ) ).
% fun.rel_compp_Grp
thf(fact_8102_multiset_Orel__compp__Grp,axiom,
! [B: $tType,A: $tType] :
( ( rel_mset @ A @ B )
= ( ^ [R7: 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 ) )
@ ^ [X: multiset @ ( product_prod @ A @ B )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( set_mset @ ( product_prod @ A @ B ) @ X ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R7 ) ) ) )
@ ( 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 ) )
@ ^ [X: multiset @ ( product_prod @ A @ B )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( set_mset @ ( product_prod @ A @ B ) @ X ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R7 ) ) ) )
@ ( image_mset @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) ) ) ) ) ) ).
% multiset.rel_compp_Grp
thf(fact_8103_pcr__int__def,axiom,
( pcr_int
= ( relcompp @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ int
@ ( basic_rel_prod @ nat @ nat @ nat @ nat
@ ^ [Y3: nat,Z: nat] : Y3 = Z
@ ^ [Y3: nat,Z: nat] : Y3 = Z )
@ cr_int ) ) ).
% pcr_int_def
thf(fact_8104_right__total__relcompp__transfer,axiom,
! [C: $tType,A: $tType,E: $tType,F7: $tType,B: $tType,D: $tType,B5: A > B > $o,A5: C > D > $o,C6: E > F7 > $o] :
( ( right_total @ A @ B @ B5 )
=> ( bNF_rel_fun @ ( C > A > $o ) @ ( D > B > $o ) @ ( ( A > E > $o ) > C > E > $o ) @ ( ( B > F7 > $o ) > D > F7 > $o )
@ ( bNF_rel_fun @ C @ D @ ( A > $o ) @ ( B > $o ) @ A5
@ ( bNF_rel_fun @ A @ B @ $o @ $o @ B5
@ ^ [Y3: $o,Z: $o] : Y3 = Z ) )
@ ( bNF_rel_fun @ ( A > E > $o ) @ ( B > F7 > $o ) @ ( C > E > $o ) @ ( D > F7 > $o )
@ ( bNF_rel_fun @ A @ B @ ( E > $o ) @ ( F7 > $o ) @ B5
@ ( bNF_rel_fun @ E @ F7 @ $o @ $o @ C6
@ ^ [Y3: $o,Z: $o] : Y3 = Z ) )
@ ( bNF_rel_fun @ C @ D @ ( E > $o ) @ ( F7 > $o ) @ A5
@ ( bNF_rel_fun @ E @ F7 @ $o @ $o @ C6
@ ^ [Y3: $o,Z: $o] : Y3 = Z ) ) )
@ ^ [R7: C > A > $o,S: A > E > $o,X: C,Z3: E] :
? [Y4: A] :
( ( member @ A @ Y4 @ ( collect @ A @ ( domainp @ A @ B @ B5 ) ) )
& ( R7 @ X @ Y4 )
& ( S @ Y4 @ Z3 ) )
@ ( relcompp @ D @ B @ F7 ) ) ) ).
% right_total_relcompp_transfer
thf(fact_8105_finite__Collect__bex,axiom,
! [B: $tType,A: $tType,A5: set @ A,Q: B > A > $o] :
( ( finite_finite2 @ A @ A5 )
=> ( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X: B] :
? [Y4: A] :
( ( member @ A @ Y4 @ A5 )
& ( Q @ X @ Y4 ) ) ) )
= ( ! [X: A] :
( ( member @ A @ X @ A5 )
=> ( finite_finite2 @ B
@ ( collect @ B
@ ^ [Y4: B] : ( Q @ Y4 @ X ) ) ) ) ) ) ) ).
% finite_Collect_bex
thf(fact_8106_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] :
? [X: B] :
( ( member @ B @ X @ A5 )
& ( P @ X @ Y4 ) ) ) ) ).
% Image_Collect_case_prod
thf(fact_8107_SUP__bool__eq,axiom,
! [A: $tType] :
( ( ^ [A8: set @ A,F2: A > $o] : ( complete_Sup_Sup @ $o @ ( image2 @ A @ $o @ F2 @ A8 ) ) )
= ( bex @ A ) ) ).
% SUP_bool_eq
thf(fact_8108_nths__nths,axiom,
! [A: $tType,Xs: list @ A,A5: set @ nat,B5: set @ nat] :
( ( nths @ A @ ( nths @ A @ Xs @ A5 ) @ B5 )
= ( nths @ A @ Xs
@ ( collect @ nat
@ ^ [I2: nat] :
( ( member @ nat @ I2 @ A5 )
& ( member @ nat
@ ( finite_card @ nat
@ ( collect @ nat
@ ^ [I9: nat] :
( ( member @ nat @ I9 @ A5 )
& ( ord_less @ nat @ I9 @ I2 ) ) ) )
@ B5 ) ) ) ) ) ).
% nths_nths
thf(fact_8109_Bex__def__raw,axiom,
! [A: $tType] :
( ( bex @ A )
= ( ^ [A8: set @ A,P4: A > $o] :
? [X: A] :
( ( member @ A @ X @ A8 )
& ( P4 @ X ) ) ) ) ).
% Bex_def_raw
thf(fact_8110_Bex__fold,axiom,
! [A: $tType,A5: set @ A,P: A > $o] :
( ( finite_finite2 @ A @ A5 )
=> ( ( ? [X: A] :
( ( member @ A @ X @ A5 )
& ( P @ X ) ) )
= ( finite_fold @ A @ $o
@ ^ [K2: A,S6: $o] :
( S6
| ( P @ K2 ) )
@ $false
@ A5 ) ) ) ).
% Bex_fold
thf(fact_8111_rel__set__def,axiom,
! [B: $tType,A: $tType] :
( ( bNF_rel_set @ A @ B )
= ( ^ [R7: A > B > $o,A8: set @ A,B8: set @ B] :
( ! [X: A] :
( ( member @ A @ X @ A8 )
=> ? [Y4: B] :
( ( member @ B @ Y4 @ B8 )
& ( R7 @ X @ Y4 ) ) )
& ! [X: B] :
( ( member @ B @ X @ B8 )
=> ? [Y4: A] :
( ( member @ A @ Y4 @ A8 )
& ( R7 @ Y4 @ X ) ) ) ) ) ) ).
% rel_set_def
thf(fact_8112_Union__eq,axiom,
! [A: $tType] :
( ( complete_Sup_Sup @ ( set @ A ) )
= ( ^ [A8: set @ ( set @ A )] :
( collect @ A
@ ^ [X: A] :
? [Y4: set @ A] :
( ( member @ ( set @ A ) @ Y4 @ A8 )
& ( member @ A @ X @ Y4 ) ) ) ) ) ).
% Union_eq
thf(fact_8113_UNION__eq,axiom,
! [B: $tType,A: $tType,B5: B > ( set @ A ),A5: set @ B] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B5 @ A5 ) )
= ( collect @ A
@ ^ [Y4: A] :
? [X: B] :
( ( member @ B @ X @ A5 )
& ( member @ A @ Y4 @ ( B5 @ X ) ) ) ) ) ).
% UNION_eq
thf(fact_8114_Collect__bex__eq,axiom,
! [A: $tType,B: $tType,A5: set @ B,P: A > B > $o] :
( ( collect @ A
@ ^ [X: A] :
? [Y4: B] :
( ( member @ B @ Y4 @ A5 )
& ( P @ X @ Y4 ) ) )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [Y4: B] :
( collect @ A
@ ^ [X: A] : ( P @ X @ Y4 ) )
@ A5 ) ) ) ).
% Collect_bex_eq
thf(fact_8115_vimage__image__eq,axiom,
! [B: $tType,A: $tType,F: A > B,A5: set @ A] :
( ( vimage @ A @ B @ F @ ( image2 @ A @ B @ F @ A5 ) )
= ( collect @ A
@ ^ [Y4: A] :
? [X: A] :
( ( member @ A @ X @ A5 )
& ( ( F @ X )
= ( F @ Y4 ) ) ) ) ) ).
% vimage_image_eq
thf(fact_8116_image__def,axiom,
! [B: $tType,A: $tType] :
( ( image2 @ A @ B )
= ( ^ [F2: A > B,A8: set @ A] :
( collect @ B
@ ^ [Y4: B] :
? [X: A] :
( ( member @ A @ X @ A8 )
& ( Y4
= ( F2 @ X ) ) ) ) ) ) ).
% image_def
thf(fact_8117_Image__def,axiom,
! [B: $tType,A: $tType] :
( ( image @ A @ B )
= ( ^ [R2: set @ ( product_prod @ A @ B ),S6: set @ A] :
( collect @ B
@ ^ [Y4: B] :
? [X: A] :
( ( member @ A @ X @ S6 )
& ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y4 ) @ R2 ) ) ) ) ) ).
% Image_def
thf(fact_8118_int_Opcr__cr__eq,axiom,
pcr_int = cr_int ).
% int.pcr_cr_eq
thf(fact_8119_right__total__Domainp__transfer,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,B5: A > B > $o,A5: C > D > $o] :
( ( right_total @ A @ B @ B5 )
=> ( 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 @ B5
@ ^ [Y3: $o,Z: $o] : Y3 = Z ) )
@ ( bNF_rel_fun @ C @ D @ $o @ $o @ A5
@ ^ [Y3: $o,Z: $o] : Y3 = Z )
@ ^ [T8: C > A > $o,X: C] :
? [Y4: A] :
( ( member @ A @ Y4 @ ( collect @ A @ ( domainp @ A @ B @ B5 ) ) )
& ( T8 @ X @ Y4 ) )
@ ( domainp @ D @ B ) ) ) ).
% right_total_Domainp_transfer
thf(fact_8120_cr__int__def,axiom,
( cr_int
= ( ^ [X: product_prod @ nat @ nat] :
( ^ [Y3: int,Z: int] : Y3 = Z
@ ( abs_Integ @ X ) ) ) ) ).
% cr_int_def
thf(fact_8121_multp__code__def,axiom,
! [A: $tType] :
( ( multp_code @ A )
= ( ^ [P4: A > A > $o,N6: multiset @ A,M6: multiset @ A] :
( ( ( minus_minus @ ( multiset @ A ) @ M6 @ ( inter_mset @ A @ M6 @ N6 ) )
!= ( zero_zero @ ( multiset @ A ) ) )
& ! [X: A] :
( ( member @ A @ X @ ( set_mset @ A @ ( minus_minus @ ( multiset @ A ) @ N6 @ ( inter_mset @ A @ M6 @ N6 ) ) ) )
=> ? [Y4: A] :
( ( member @ A @ Y4 @ ( set_mset @ A @ ( minus_minus @ ( multiset @ A ) @ M6 @ ( inter_mset @ A @ M6 @ N6 ) ) ) )
& ( P4 @ X @ Y4 ) ) ) ) ) ) ).
% multp_code_def
thf(fact_8122_multeqp__code__def,axiom,
! [A: $tType] :
( ( multeqp_code @ A )
= ( ^ [P4: A > A > $o,N6: multiset @ A,M6: multiset @ A] :
! [X: A] :
( ( member @ A @ X @ ( set_mset @ A @ ( minus_minus @ ( multiset @ A ) @ N6 @ ( inter_mset @ A @ M6 @ N6 ) ) ) )
=> ? [Y4: A] :
( ( member @ A @ Y4 @ ( set_mset @ A @ ( minus_minus @ ( multiset @ A ) @ M6 @ ( inter_mset @ A @ M6 @ N6 ) ) ) )
& ( P4 @ X @ Y4 ) ) ) ) ) ).
% multeqp_code_def
thf(fact_8123_map__project__def,axiom,
! [B: $tType,A: $tType] :
( ( map_project @ A @ B )
= ( ^ [F2: A > ( option @ B ),A8: set @ A] :
( collect @ B
@ ^ [B3: B] :
? [X: A] :
( ( member @ A @ X @ A8 )
& ( ( F2 @ X )
= ( some @ B @ B3 ) ) ) ) ) ) ).
% map_project_def
thf(fact_8124_min__ext__def,axiom,
! [A: $tType] :
( ( min_ext @ A )
= ( ^ [R2: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ ( set @ A ) @ ( set @ A ) )
@ ^ [Uu: product_prod @ ( set @ A ) @ ( set @ A )] :
? [X5: set @ A,Y6: set @ A] :
( ( Uu
= ( product_Pair @ ( set @ A ) @ ( set @ A ) @ X5 @ Y6 ) )
& ( X5
!= ( bot_bot @ ( set @ A ) ) )
& ! [X: A] :
( ( member @ A @ X @ Y6 )
=> ? [Y4: A] :
( ( member @ A @ Y4 @ X5 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ X ) @ R2 ) ) ) ) ) ) ) ).
% min_ext_def
thf(fact_8125_max__ext__eq,axiom,
! [A: $tType] :
( ( max_ext @ A )
= ( ^ [R7: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ ( set @ A ) @ ( set @ A ) )
@ ( product_case_prod @ ( set @ A ) @ ( set @ A ) @ $o
@ ^ [X5: set @ A,Y6: set @ A] :
( ( finite_finite2 @ A @ X5 )
& ( finite_finite2 @ A @ Y6 )
& ( Y6
!= ( bot_bot @ ( set @ A ) ) )
& ! [X: A] :
( ( member @ A @ X @ X5 )
=> ? [Y4: A] :
( ( member @ A @ Y4 @ Y6 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y4 ) @ R7 ) ) ) ) ) ) ) ) ).
% max_ext_eq
thf(fact_8126_max__extp_Omax__extI,axiom,
! [A: $tType,X8: set @ A,Y7: set @ A,R3: A > A > $o] :
( ( finite_finite2 @ A @ X8 )
=> ( ( finite_finite2 @ A @ Y7 )
=> ( ( Y7
!= ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ X8 )
=> ? [Xa2: A] :
( ( member @ A @ Xa2 @ Y7 )
& ( R3 @ X3 @ Xa2 ) ) )
=> ( max_extp @ A @ R3 @ X8 @ Y7 ) ) ) ) ) ).
% max_extp.max_extI
thf(fact_8127_max__extp__max__ext__eq,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( max_extp @ A
@ ^ [X: A,Y4: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y4 ) @ R3 ) )
= ( ^ [X: set @ A,Y4: set @ A] : ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ X @ Y4 ) @ ( max_ext @ A @ R3 ) ) ) ) ).
% max_extp_max_ext_eq
thf(fact_8128_max__ext__def,axiom,
! [A: $tType] :
( ( max_ext @ A )
= ( ^ [R7: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ ( set @ A ) @ ( set @ A ) )
@ ( product_case_prod @ ( set @ A ) @ ( set @ A ) @ $o
@ ( max_extp @ A
@ ^ [X: A,Y4: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y4 ) @ R7 ) ) ) ) ) ) ).
% max_ext_def
thf(fact_8129_max__extp__eq,axiom,
! [A: $tType] :
( ( max_extp @ A )
= ( ^ [R2: A > A > $o,X: set @ A,Y4: set @ A] : ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ X @ Y4 ) @ ( max_ext @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ R2 ) ) ) ) ) ) ).
% max_extp_eq
thf(fact_8130_max__extp_Ocases,axiom,
! [A: $tType,R3: A > A > $o,A13: set @ A,A24: set @ A] :
( ( max_extp @ A @ R3 @ A13 @ A24 )
=> ~ ( ( finite_finite2 @ A @ A13 )
=> ( ( finite_finite2 @ A @ A24 )
=> ( ( A24
!= ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) )
=> ~ ! [X4: A] :
( ( member @ A @ X4 @ A13 )
=> ? [Xa3: A] :
( ( member @ A @ Xa3 @ A24 )
& ( R3 @ X4 @ Xa3 ) ) ) ) ) ) ) ).
% max_extp.cases
thf(fact_8131_max__extp_Osimps,axiom,
! [A: $tType] :
( ( max_extp @ A )
= ( ^ [R7: A > A > $o,A12: set @ A,A23: set @ A] :
( ( finite_finite2 @ A @ A12 )
& ( finite_finite2 @ A @ A23 )
& ( A23
!= ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) )
& ! [X: A] :
( ( member @ A @ X @ A12 )
=> ? [Y4: A] :
( ( member @ A @ Y4 @ A23 )
& ( R7 @ X @ Y4 ) ) ) ) ) ) ).
% max_extp.simps
thf(fact_8132_Quotient__int,axiom,
quotient @ ( product_prod @ nat @ nat ) @ int @ intrel @ abs_Integ @ rep_Integ @ cr_int ).
% Quotient_int
thf(fact_8133_lcm__code__integer,axiom,
( ( gcd_lcm @ code_integer )
= ( ^ [A3: code_integer,B3: code_integer] : ( divide_divide @ code_integer @ ( times_times @ code_integer @ ( abs_abs @ code_integer @ A3 ) @ ( abs_abs @ code_integer @ B3 ) ) @ ( gcd_gcd @ code_integer @ A3 @ B3 ) ) ) ) ).
% lcm_code_integer
thf(fact_8134_lcm__left__idem,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A2: A,B2: A] :
( ( gcd_lcm @ A @ A2 @ ( gcd_lcm @ A @ A2 @ B2 ) )
= ( gcd_lcm @ A @ A2 @ B2 ) ) ) ).
% lcm_left_idem
thf(fact_8135_lcm__right__idem,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A2: A,B2: A] :
( ( gcd_lcm @ A @ ( gcd_lcm @ A @ A2 @ B2 ) @ B2 )
= ( gcd_lcm @ A @ A2 @ B2 ) ) ) ).
% lcm_right_idem
thf(fact_8136_lcm_Obottom__left__bottom,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A2: A] :
( ( gcd_lcm @ A @ ( zero_zero @ A ) @ A2 )
= ( zero_zero @ A ) ) ) ).
% lcm.bottom_left_bottom
thf(fact_8137_lcm_Obottom__right__bottom,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A2: A] :
( ( gcd_lcm @ A @ A2 @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% lcm.bottom_right_bottom
thf(fact_8138_lcm__neg2,axiom,
! [A: $tType] :
( ( ring_gcd @ A )
=> ! [A2: A,B2: A] :
( ( gcd_lcm @ A @ A2 @ ( uminus_uminus @ A @ B2 ) )
= ( gcd_lcm @ A @ A2 @ B2 ) ) ) ).
% lcm_neg2
thf(fact_8139_lcm__neg1,axiom,
! [A: $tType] :
( ( ring_gcd @ A )
=> ! [A2: A,B2: A] :
( ( gcd_lcm @ A @ ( uminus_uminus @ A @ A2 ) @ B2 )
= ( gcd_lcm @ A @ A2 @ B2 ) ) ) ).
% lcm_neg1
thf(fact_8140_dvd__lcm1,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A2: A,B2: A] : ( dvd_dvd @ A @ A2 @ ( gcd_lcm @ A @ A2 @ B2 ) ) ) ).
% dvd_lcm1
thf(fact_8141_dvd__lcm2,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [B2: A,A2: A] : ( dvd_dvd @ A @ B2 @ ( gcd_lcm @ A @ A2 @ B2 ) ) ) ).
% dvd_lcm2
thf(fact_8142_lcm__least__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ ( gcd_lcm @ A @ A2 @ B2 ) @ C2 )
= ( ( dvd_dvd @ A @ A2 @ C2 )
& ( dvd_dvd @ A @ B2 @ C2 ) ) ) ) ).
% lcm_least_iff
thf(fact_8143_lcm__neg__numeral__2,axiom,
! [A: $tType] :
( ( ring_gcd @ A )
=> ! [A2: A,N: num] :
( ( gcd_lcm @ A @ A2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( gcd_lcm @ A @ A2 @ ( numeral_numeral @ A @ N ) ) ) ) ).
% lcm_neg_numeral_2
thf(fact_8144_lcm__neg__numeral__1,axiom,
! [A: $tType] :
( ( ring_gcd @ A )
=> ! [N: num,A2: A] :
( ( gcd_lcm @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) @ A2 )
= ( gcd_lcm @ A @ ( numeral_numeral @ A @ N ) @ A2 ) ) ) ).
% lcm_neg_numeral_1
thf(fact_8145_lcm__eq__1__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A2: A,B2: A] :
( ( ( gcd_lcm @ A @ A2 @ B2 )
= ( one_one @ A ) )
= ( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
& ( dvd_dvd @ A @ B2 @ ( one_one @ A ) ) ) ) ) ).
% lcm_eq_1_iff
thf(fact_8146_open__typedef__to__Quotient,axiom,
! [A: $tType,B: $tType,Rep: A > B,Abs: B > A,P: B > $o,T6: B > A > $o] :
( ( type_definition @ A @ B @ Rep @ Abs @ ( collect @ B @ P ) )
=> ( ( T6
= ( ^ [X: B,Y4: A] :
( X
= ( Rep @ Y4 ) ) ) )
=> ( quotient @ B @ A @ ( bNF_eq_onp @ B @ P ) @ Abs @ Rep @ T6 ) ) ) ).
% open_typedef_to_Quotient
thf(fact_8147_typedef__to__Quotient,axiom,
! [A: $tType,B: $tType,Rep: A > B,Abs: B > A,S2: set @ B,T6: B > A > $o] :
( ( type_definition @ A @ B @ Rep @ Abs @ S2 )
=> ( ( T6
= ( ^ [X: B,Y4: A] :
( X
= ( Rep @ Y4 ) ) ) )
=> ( quotient @ B @ A
@ ( bNF_eq_onp @ B
@ ^ [X: B] : ( member @ B @ X @ S2 ) )
@ Abs
@ Rep
@ T6 ) ) ) ).
% typedef_to_Quotient
thf(fact_8148_UNIV__typedef__to__Quotient,axiom,
! [A: $tType,B: $tType,Rep: A > B,Abs: B > A,T6: B > A > $o] :
( ( type_definition @ A @ B @ Rep @ Abs @ ( top_top @ ( set @ B ) ) )
=> ( ( T6
= ( ^ [X: B,Y4: A] :
( X
= ( Rep @ Y4 ) ) ) )
=> ( quotient @ B @ A
@ ^ [Y3: B,Z: B] : Y3 = Z
@ Abs
@ Rep
@ T6 ) ) ) ).
% UNIV_typedef_to_Quotient
thf(fact_8149_lcm__div__unit1,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
=> ( ( gcd_lcm @ A @ ( divide_divide @ A @ B2 @ A2 ) @ C2 )
= ( gcd_lcm @ A @ B2 @ C2 ) ) ) ) ).
% lcm_div_unit1
thf(fact_8150_lcm__div__unit2,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
=> ( ( gcd_lcm @ A @ B2 @ ( divide_divide @ A @ C2 @ A2 ) )
= ( gcd_lcm @ A @ B2 @ C2 ) ) ) ) ).
% lcm_div_unit2
thf(fact_8151_zero__eq__lcm__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A2: A,B2: A] :
( ( ( zero_zero @ A )
= ( gcd_lcm @ A @ A2 @ B2 ) )
= ( ( A2
= ( zero_zero @ A ) )
| ( B2
= ( zero_zero @ A ) ) ) ) ) ).
% zero_eq_lcm_iff
thf(fact_8152_lcm__eq__0__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A2: A,B2: A] :
( ( ( gcd_lcm @ A @ A2 @ B2 )
= ( zero_zero @ A ) )
= ( ( A2
= ( zero_zero @ A ) )
| ( B2
= ( zero_zero @ A ) ) ) ) ) ).
% lcm_eq_0_iff
thf(fact_8153_Quotient__cr__rel,axiom,
! [B: $tType,A: $tType,R3: A > A > $o,Abs: A > B,Rep: B > A,T6: A > B > $o] :
( ( quotient @ A @ B @ R3 @ Abs @ Rep @ T6 )
=> ( T6
= ( ^ [X: A,Y4: B] :
( ( R3 @ X @ X )
& ( ( Abs @ X )
= Y4 ) ) ) ) ) ).
% Quotient_cr_rel
thf(fact_8154_Quotient__def,axiom,
! [B: $tType,A: $tType] :
( ( quotient @ A @ B )
= ( ^ [R7: A > A > $o,Abs4: A > B,Rep4: B > A,T8: A > B > $o] :
( ! [A3: B] :
( ( Abs4 @ ( Rep4 @ A3 ) )
= A3 )
& ! [A3: B] : ( R7 @ ( Rep4 @ A3 ) @ ( Rep4 @ A3 ) )
& ! [R2: A,S6: A] :
( ( R7 @ R2 @ S6 )
= ( ( R7 @ R2 @ R2 )
& ( R7 @ S6 @ S6 )
& ( ( Abs4 @ R2 )
= ( Abs4 @ S6 ) ) ) )
& ( T8
= ( ^ [X: A,Y4: B] :
( ( R7 @ X @ X )
& ( ( Abs4 @ X )
= Y4 ) ) ) ) ) ) ) ).
% Quotient_def
thf(fact_8155_QuotientI,axiom,
! [A: $tType,B: $tType,Abs: B > A,Rep: A > B,R3: B > B > $o,T6: B > A > $o] :
( ! [A4: A] :
( ( Abs @ ( Rep @ A4 ) )
= A4 )
=> ( ! [A4: A] : ( R3 @ ( Rep @ A4 ) @ ( Rep @ A4 ) )
=> ( ! [R6: B,S4: B] :
( ( R3 @ R6 @ S4 )
= ( ( R3 @ R6 @ R6 )
& ( R3 @ S4 @ S4 )
& ( ( Abs @ R6 )
= ( Abs @ S4 ) ) ) )
=> ( ( T6
= ( ^ [X: B,Y4: A] :
( ( R3 @ X @ X )
& ( ( Abs @ X )
= Y4 ) ) ) )
=> ( quotient @ B @ A @ R3 @ Abs @ Rep @ T6 ) ) ) ) ) ).
% QuotientI
thf(fact_8156_gcd__dvd__lcm,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A2: A,B2: A] : ( dvd_dvd @ A @ ( gcd_gcd @ A @ A2 @ B2 ) @ ( gcd_lcm @ A @ A2 @ B2 ) ) ) ).
% gcd_dvd_lcm
thf(fact_8157_lcm_Oassoc,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( gcd_lcm @ A @ ( gcd_lcm @ A @ A2 @ B2 ) @ C2 )
= ( gcd_lcm @ A @ A2 @ ( gcd_lcm @ A @ B2 @ C2 ) ) ) ) ).
% lcm.assoc
thf(fact_8158_lcm_Ocommute,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ( ( gcd_lcm @ A )
= ( ^ [A3: A,B3: A] : ( gcd_lcm @ A @ B3 @ A3 ) ) ) ) ).
% lcm.commute
thf(fact_8159_lcm_Oleft__commute,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( gcd_lcm @ A @ B2 @ ( gcd_lcm @ A @ A2 @ C2 ) )
= ( gcd_lcm @ A @ A2 @ ( gcd_lcm @ A @ B2 @ C2 ) ) ) ) ).
% lcm.left_commute
thf(fact_8160_lcm__mono,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A2: A,C2: A,B2: A,D2: A] :
( ( dvd_dvd @ A @ A2 @ C2 )
=> ( ( dvd_dvd @ A @ B2 @ D2 )
=> ( dvd_dvd @ A @ ( gcd_lcm @ A @ A2 @ B2 ) @ ( gcd_lcm @ A @ C2 @ D2 ) ) ) ) ) ).
% lcm_mono
thf(fact_8161_dvd__lcmI1,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A2 @ B2 )
=> ( dvd_dvd @ A @ A2 @ ( gcd_lcm @ A @ B2 @ C2 ) ) ) ) ).
% dvd_lcmI1
thf(fact_8162_dvd__lcmI2,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( dvd_dvd @ A @ A2 @ C2 )
=> ( dvd_dvd @ A @ A2 @ ( gcd_lcm @ A @ B2 @ C2 ) ) ) ) ).
% dvd_lcmI2
thf(fact_8163_lcm__dvdD1,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ ( gcd_lcm @ A @ A2 @ B2 ) @ C2 )
=> ( dvd_dvd @ A @ A2 @ C2 ) ) ) ).
% lcm_dvdD1
thf(fact_8164_lcm__dvdD2,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ ( gcd_lcm @ A @ A2 @ B2 ) @ C2 )
=> ( dvd_dvd @ A @ B2 @ C2 ) ) ) ).
% lcm_dvdD2
thf(fact_8165_lcm__least,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( dvd_dvd @ A @ A2 @ C2 )
=> ( ( dvd_dvd @ A @ B2 @ C2 )
=> ( dvd_dvd @ A @ ( gcd_lcm @ A @ A2 @ B2 ) @ C2 ) ) ) ) ).
% lcm_least
thf(fact_8166_lcm__mult__unit1,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
=> ( ( gcd_lcm @ A @ ( times_times @ A @ B2 @ A2 ) @ C2 )
= ( gcd_lcm @ A @ B2 @ C2 ) ) ) ) ).
% lcm_mult_unit1
thf(fact_8167_lcm__mult__unit2,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
=> ( ( gcd_lcm @ A @ B2 @ ( times_times @ A @ C2 @ A2 ) )
= ( gcd_lcm @ A @ B2 @ C2 ) ) ) ) ).
% lcm_mult_unit2
thf(fact_8168_Quotient__eq__onp__typedef,axiom,
! [B: $tType,A: $tType,P: A > $o,Abs: A > B,Rep: B > A,Cr: A > B > $o] :
( ( quotient @ A @ B @ ( bNF_eq_onp @ A @ P ) @ Abs @ Rep @ Cr )
=> ( type_definition @ B @ A @ Rep @ Abs @ ( collect @ A @ P ) ) ) ).
% Quotient_eq_onp_typedef
thf(fact_8169_Quotient__crel__typedef,axiom,
! [B: $tType,A: $tType,P: A > $o,Abs: A > B,Rep: B > A,T6: A > B > $o] :
( ( quotient @ A @ B @ ( bNF_eq_onp @ A @ P ) @ Abs @ Rep @ T6 )
=> ( T6
= ( ^ [X: A,Y4: B] :
( X
= ( Rep @ Y4 ) ) ) ) ) ).
% Quotient_crel_typedef
thf(fact_8170_lcm__0__iff__nat,axiom,
! [M: nat,N: nat] :
( ( ( gcd_lcm @ nat @ M @ N )
= ( zero_zero @ nat ) )
= ( ( M
= ( zero_zero @ nat ) )
| ( N
= ( zero_zero @ nat ) ) ) ) ).
% lcm_0_iff_nat
thf(fact_8171_lcm__0__iff__int,axiom,
! [M: int,N: int] :
( ( ( gcd_lcm @ int @ M @ N )
= ( zero_zero @ int ) )
= ( ( M
= ( zero_zero @ int ) )
| ( N
= ( zero_zero @ int ) ) ) ) ).
% lcm_0_iff_int
thf(fact_8172_lcm__proj2__if__dvd__nat,axiom,
! [X2: nat,Y: nat] :
( ( dvd_dvd @ nat @ X2 @ Y )
=> ( ( gcd_lcm @ nat @ X2 @ Y )
= Y ) ) ).
% lcm_proj2_if_dvd_nat
thf(fact_8173_lcm__proj1__if__dvd__nat,axiom,
! [X2: nat,Y: nat] :
( ( dvd_dvd @ nat @ X2 @ Y )
=> ( ( gcd_lcm @ nat @ Y @ X2 )
= Y ) ) ).
% lcm_proj1_if_dvd_nat
thf(fact_8174_lcm__proj2__iff__nat,axiom,
! [M: nat,N: nat] :
( ( ( gcd_lcm @ nat @ M @ N )
= N )
= ( dvd_dvd @ nat @ M @ N ) ) ).
% lcm_proj2_iff_nat
thf(fact_8175_lcm__proj1__iff__nat,axiom,
! [M: nat,N: nat] :
( ( ( gcd_lcm @ nat @ M @ N )
= M )
= ( dvd_dvd @ nat @ N @ M ) ) ).
% lcm_proj1_iff_nat
thf(fact_8176_lcm__int__int__eq,axiom,
! [M: nat,N: nat] :
( ( gcd_lcm @ int @ ( semiring_1_of_nat @ int @ M ) @ ( semiring_1_of_nat @ int @ N ) )
= ( semiring_1_of_nat @ int @ ( gcd_lcm @ nat @ M @ N ) ) ) ).
% lcm_int_int_eq
thf(fact_8177_abs__lcm__int,axiom,
! [I: int,J: int] :
( ( abs_abs @ int @ ( gcd_lcm @ int @ I @ J ) )
= ( gcd_lcm @ int @ I @ J ) ) ).
% abs_lcm_int
thf(fact_8178_lcm__abs1__int,axiom,
! [X2: int,Y: int] :
( ( gcd_lcm @ int @ ( abs_abs @ int @ X2 ) @ Y )
= ( gcd_lcm @ int @ X2 @ Y ) ) ).
% lcm_abs1_int
thf(fact_8179_lcm__abs2__int,axiom,
! [X2: int,Y: int] :
( ( gcd_lcm @ int @ X2 @ ( abs_abs @ int @ Y ) )
= ( gcd_lcm @ int @ X2 @ Y ) ) ).
% lcm_abs2_int
% Type constructors (677)
thf(tcon_Product__Type_Ounit___Lattices_Obounded__lattice__top,axiom,
bounded_lattice_top @ product_unit ).
thf(tcon_Option_Ooption___Lattices_Obounded__lattice__top_1,axiom,
! [A18: $tType] :
( ( bounded_lattice_top @ A18 )
=> ( bounded_lattice_top @ ( option @ A18 ) ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Obounded__lattice__top_2,axiom,
! [A18: $tType] : ( bounded_lattice_top @ ( filter @ A18 ) ) ).
thf(tcon_HOL_Obool___Lattices_Obounded__lattice__top_3,axiom,
bounded_lattice_top @ $o ).
thf(tcon_Set_Oset___Lattices_Obounded__lattice__top_4,axiom,
! [A18: $tType] : ( bounded_lattice_top @ ( set @ A18 ) ) ).
thf(tcon_fun___Lattices_Obounded__lattice__top_5,axiom,
! [A18: $tType,A19: $tType] :
( ( bounded_lattice @ A19 )
=> ( bounded_lattice_top @ ( A18 > A19 ) ) ) ).
thf(tcon_fun___Conditionally__Complete__Lattices_Oconditionally__complete__lattice,axiom,
! [A18: $tType,A19: $tType] :
( ( comple6319245703460814977attice @ A19 )
=> ( condit1219197933456340205attice @ ( A18 > A19 ) ) ) ).
thf(tcon_fun___Complete__Lattices_Ocomplete__distrib__lattice,axiom,
! [A18: $tType,A19: $tType] :
( ( comple592849572758109894attice @ A19 )
=> ( comple592849572758109894attice @ ( A18 > A19 ) ) ) ).
thf(tcon_fun___Complete__Lattices_Ocomplete__boolean__algebra,axiom,
! [A18: $tType,A19: $tType] :
( ( comple489889107523837845lgebra @ A19 )
=> ( comple489889107523837845lgebra @ ( A18 > A19 ) ) ) ).
thf(tcon_fun___Lattices_Obounded__semilattice__sup__bot,axiom,
! [A18: $tType,A19: $tType] :
( ( bounded_lattice @ A19 )
=> ( bounde4967611905675639751up_bot @ ( A18 > A19 ) ) ) ).
thf(tcon_fun___Lattices_Obounded__semilattice__inf__top,axiom,
! [A18: $tType,A19: $tType] :
( ( bounded_lattice @ A19 )
=> ( bounde4346867609351753570nf_top @ ( A18 > A19 ) ) ) ).
thf(tcon_fun___Complete__Lattices_Ocomplete__lattice,axiom,
! [A18: $tType,A19: $tType] :
( ( comple6319245703460814977attice @ A19 )
=> ( comple6319245703460814977attice @ ( A18 > A19 ) ) ) ).
thf(tcon_fun___Boolean__Algebras_Oboolean__algebra,axiom,
! [A18: $tType,A19: $tType] :
( ( boolea8198339166811842893lgebra @ A19 )
=> ( boolea8198339166811842893lgebra @ ( A18 > A19 ) ) ) ).
thf(tcon_fun___Complete__Partial__Order_Occpo,axiom,
! [A18: $tType,A19: $tType] :
( ( comple6319245703460814977attice @ A19 )
=> ( comple9053668089753744459l_ccpo @ ( A18 > A19 ) ) ) ).
thf(tcon_fun___Lattices_Osemilattice__sup,axiom,
! [A18: $tType,A19: $tType] :
( ( semilattice_sup @ A19 )
=> ( semilattice_sup @ ( A18 > A19 ) ) ) ).
thf(tcon_fun___Lattices_Osemilattice__inf,axiom,
! [A18: $tType,A19: $tType] :
( ( semilattice_inf @ A19 )
=> ( semilattice_inf @ ( A18 > A19 ) ) ) ).
thf(tcon_fun___Lattices_Odistrib__lattice,axiom,
! [A18: $tType,A19: $tType] :
( ( distrib_lattice @ A19 )
=> ( distrib_lattice @ ( A18 > A19 ) ) ) ).
thf(tcon_fun___Lattices_Obounded__lattice,axiom,
! [A18: $tType,A19: $tType] :
( ( bounded_lattice @ A19 )
=> ( bounded_lattice @ ( A18 > A19 ) ) ) ).
thf(tcon_fun___Complete__Lattices_OSup,axiom,
! [A18: $tType,A19: $tType] :
( ( complete_Sup @ A19 )
=> ( complete_Sup @ ( A18 > A19 ) ) ) ).
thf(tcon_fun___Complete__Lattices_OInf,axiom,
! [A18: $tType,A19: $tType] :
( ( complete_Inf @ A19 )
=> ( complete_Inf @ ( A18 > A19 ) ) ) ).
thf(tcon_fun___Orderings_Oorder__top,axiom,
! [A18: $tType,A19: $tType] :
( ( order_top @ A19 )
=> ( order_top @ ( A18 > A19 ) ) ) ).
thf(tcon_fun___Orderings_Oorder__bot,axiom,
! [A18: $tType,A19: $tType] :
( ( order_bot @ A19 )
=> ( order_bot @ ( A18 > A19 ) ) ) ).
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A18: $tType,A19: $tType] :
( ( preorder @ A19 )
=> ( preorder @ ( A18 > A19 ) ) ) ).
thf(tcon_fun___Finite__Set_Ofinite,axiom,
! [A18: $tType,A19: $tType] :
( ( ( finite_finite @ A18 )
& ( finite_finite @ A19 ) )
=> ( finite_finite @ ( A18 > A19 ) ) ) ).
thf(tcon_fun___Lattices_Olattice,axiom,
! [A18: $tType,A19: $tType] :
( ( lattice @ A19 )
=> ( lattice @ ( A18 > A19 ) ) ) ).
thf(tcon_fun___Orderings_Oorder,axiom,
! [A18: $tType,A19: $tType] :
( ( order @ A19 )
=> ( order @ ( A18 > A19 ) ) ) ).
thf(tcon_fun___Orderings_Otop,axiom,
! [A18: $tType,A19: $tType] :
( ( top @ A19 )
=> ( top @ ( A18 > A19 ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A18: $tType,A19: $tType] :
( ( ord @ A19 )
=> ( ord @ ( A18 > A19 ) ) ) ).
thf(tcon_fun___Orderings_Obot,axiom,
! [A18: $tType,A19: $tType] :
( ( bot @ A19 )
=> ( bot @ ( A18 > A19 ) ) ) ).
thf(tcon_fun___Groups_Ouminus,axiom,
! [A18: $tType,A19: $tType] :
( ( uminus @ A19 )
=> ( uminus @ ( A18 > A19 ) ) ) ).
thf(tcon_fun___Lattices_Osup,axiom,
! [A18: $tType,A19: $tType] :
( ( semilattice_sup @ A19 )
=> ( sup @ ( A18 > A19 ) ) ) ).
thf(tcon_fun___Lattices_Oinf,axiom,
! [A18: $tType,A19: $tType] :
( ( semilattice_inf @ A19 )
=> ( inf @ ( A18 > A19 ) ) ) ).
thf(tcon_Int_Oint___Conditionally__Complete__Lattices_Oconditionally__complete__linorder,axiom,
condit6923001295902523014norder @ int ).
thf(tcon_Int_Oint___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_6,axiom,
condit1219197933456340205attice @ int ).
thf(tcon_Int_Oint___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations,axiom,
bit_un5681908812861735899ations @ int ).
thf(tcon_Int_Oint___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,axiom,
semiri1453513574482234551roduct @ int ).
thf(tcon_Int_Oint___Euclidean__Division_Ounique__euclidean__semiring__with__nat,axiom,
euclid5411537665997757685th_nat @ int ).
thf(tcon_Int_Oint___Euclidean__Division_Ounique__euclidean__ring__with__nat,axiom,
euclid8789492081693882211th_nat @ int ).
thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__monoid__add__imp__le,axiom,
ordere1937475149494474687imp_le @ int ).
thf(tcon_Int_Oint___Euclidean__Division_Ounique__euclidean__semiring,axiom,
euclid3128863361964157862miring @ int ).
thf(tcon_Int_Oint___Euclidean__Division_Oeuclidean__semiring__cancel,axiom,
euclid4440199948858584721cancel @ int ).
thf(tcon_Int_Oint___Divides_Ounique__euclidean__semiring__numeral,axiom,
unique1627219031080169319umeral @ int ).
thf(tcon_Int_Oint___Euclidean__Division_Oeuclidean__ring__cancel,axiom,
euclid8851590272496341667cancel @ int ).
thf(tcon_Int_Oint___Rings_Osemiring__no__zero__divisors__cancel,axiom,
semiri6575147826004484403cancel @ int ).
thf(tcon_Int_Oint___Groups_Ostrict__ordered__ab__semigroup__add,axiom,
strict9044650504122735259up_add @ int ).
thf(tcon_Int_Oint___Groups_Oordered__cancel__ab__semigroup__add,axiom,
ordere580206878836729694up_add @ int ).
thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
ordere2412721322843649153imp_le @ int ).
thf(tcon_Int_Oint___Bit__Operations_Osemiring__bit__operations,axiom,
bit_se359711467146920520ations @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__comm__semiring__strict,axiom,
linord2810124833399127020strict @ int ).
thf(tcon_Int_Oint___Groups_Ostrict__ordered__comm__monoid__add,axiom,
strict7427464778891057005id_add @ int ).
thf(tcon_Int_Oint___Groups_Oordered__cancel__comm__monoid__add,axiom,
ordere8940638589300402666id_add @ int ).
thf(tcon_Int_Oint___Euclidean__Division_Oeuclidean__semiring,axiom,
euclid3725896446679973847miring @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__semiring__1__strict,axiom,
linord715952674999750819strict @ int ).
thf(tcon_Int_Oint___Groups_Olinordered__ab__semigroup__add,axiom,
linord4140545234300271783up_add @ int ).
thf(tcon_Int_Oint___Bit__Operations_Oring__bit__operations,axiom,
bit_ri3973907225187159222ations @ int ).
thf(tcon_Int_Oint___Rings_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___Rings_Osemiring__no__zero__divisors,axiom,
semiri3467727345109120633visors @ int ).
thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__add,axiom,
ordere6658533253407199908up_add @ int ).
thf(tcon_Int_Oint___Groups_Oordered__ab__group__add__abs,axiom,
ordere166539214618696060dd_abs @ int ).
thf(tcon_Int_Oint___GCD_Osemiring__gcd__mult__normalize,axiom,
semiri6843258321239162965malize @ int ).
thf(tcon_Int_Oint___Groups_Oordered__comm__monoid__add,axiom,
ordere6911136660526730532id_add @ int ).
thf(tcon_Int_Oint___Groups_Olinordered__ab__group__add,axiom,
linord5086331880401160121up_add @ int ).
thf(tcon_Int_Oint___Groups_Ocancel__ab__semigroup__add,axiom,
cancel2418104881723323429up_add @ int ).
thf(tcon_Int_Oint___Rings_Oring__1__no__zero__divisors,axiom,
ring_15535105094025558882visors @ int ).
thf(tcon_Int_Oint___Groups_Ocancel__comm__monoid__add,axiom,
cancel1802427076303600483id_add @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__ring__strict,axiom,
linord4710134922213307826strict @ int ).
thf(tcon_Int_Oint___Rings_Ocomm__semiring__1__cancel,axiom,
comm_s4317794764714335236cancel @ int ).
thf(tcon_Int_Oint___Bit__Operations_Osemiring__bits,axiom,
bit_semiring_bits @ int ).
thf(tcon_Int_Oint___Rings_Oordered__comm__semiring,axiom,
ordere2520102378445227354miring @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__semiring__1,axiom,
linord6961819062388156250ring_1 @ int ).
thf(tcon_Int_Oint___Groups_Oordered__ab__group__add,axiom,
ordered_ab_group_add @ int ).
thf(tcon_Int_Oint___Groups_Ocancel__semigroup__add,axiom,
cancel_semigroup_add @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__semiring,axiom,
linordered_semiring @ int ).
thf(tcon_Int_Oint___Rings_Oordered__semiring__0,axiom,
ordered_semiring_0 @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__semidom,axiom,
linordered_semidom @ int ).
thf(tcon_Int_Oint___Lattices_Osemilattice__sup_7,axiom,
semilattice_sup @ int ).
thf(tcon_Int_Oint___Lattices_Osemilattice__inf_8,axiom,
semilattice_inf @ int ).
thf(tcon_Int_Oint___Lattices_Odistrib__lattice_9,axiom,
distrib_lattice @ int ).
thf(tcon_Int_Oint___Groups_Oab__semigroup__mult,axiom,
ab_semigroup_mult @ int ).
thf(tcon_Int_Oint___Rings_Osemiring__1__cancel,axiom,
semiring_1_cancel @ int ).
thf(tcon_Int_Oint___Rings_Oalgebraic__semidom,axiom,
algebraic_semidom @ int ).
thf(tcon_Int_Oint___Groups_Ocomm__monoid__mult,axiom,
comm_monoid_mult @ int ).
thf(tcon_Int_Oint___Groups_Oab__semigroup__add,axiom,
ab_semigroup_add @ int ).
thf(tcon_Int_Oint___Rings_Oordered__semiring,axiom,
ordered_semiring @ int ).
thf(tcon_Int_Oint___Rings_Oordered__ring__abs,axiom,
ordered_ring_abs @ int ).
thf(tcon_Int_Oint___Parity_Osemiring__parity,axiom,
semiring_parity @ int ).
thf(tcon_Int_Oint___Groups_Ocomm__monoid__add,axiom,
comm_monoid_add @ int ).
thf(tcon_Int_Oint___Rings_Osemiring__modulo,axiom,
semiring_modulo @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__ring,axiom,
linordered_ring @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__idom,axiom,
linordered_idom @ int ).
thf(tcon_Int_Oint___Rings_Ocomm__semiring__1,axiom,
comm_semiring_1 @ int ).
thf(tcon_Int_Oint___Rings_Ocomm__semiring__0,axiom,
comm_semiring_0 @ int ).
thf(tcon_Int_Oint___Groups_Osemigroup__mult,axiom,
semigroup_mult @ int ).
thf(tcon_Int_Oint___Complete__Lattices_OSup_10,axiom,
complete_Sup @ int ).
thf(tcon_Int_Oint___Complete__Lattices_OInf_11,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___Groups_Osemigroup__add,axiom,
semigroup_add @ int ).
thf(tcon_Int_Oint___Rings_Ozero__less__one,axiom,
zero_less_one @ int ).
thf(tcon_Int_Oint___Rings_Ocomm__semiring,axiom,
comm_semiring @ int ).
thf(tcon_Int_Oint___Nat_Osemiring__char__0,axiom,
semiring_char_0 @ int ).
thf(tcon_Int_Oint___Groups_Oab__group__add,axiom,
ab_group_add @ int ).
thf(tcon_Int_Oint___Rings_Ozero__neq__one,axiom,
zero_neq_one @ int ).
thf(tcon_Int_Oint___Rings_Oordered__ring,axiom,
ordered_ring @ int ).
thf(tcon_Int_Oint___Rings_Oidom__abs__sgn,axiom,
idom_abs_sgn @ int ).
thf(tcon_Int_Oint___Parity_Oring__parity,axiom,
ring_parity @ int ).
thf(tcon_Int_Oint___Orderings_Opreorder_12,axiom,
preorder @ int ).
thf(tcon_Int_Oint___Orderings_Olinorder,axiom,
linorder @ int ).
thf(tcon_Int_Oint___Groups_Omonoid__mult,axiom,
monoid_mult @ int ).
thf(tcon_Int_Oint___Rings_Oidom__divide,axiom,
idom_divide @ int ).
thf(tcon_Int_Oint___Rings_Ocomm__ring__1,axiom,
comm_ring_1 @ int ).
thf(tcon_Int_Oint___Groups_Omonoid__add,axiom,
monoid_add @ int ).
thf(tcon_Int_Oint___Rings_Osemiring__1,axiom,
semiring_1 @ int ).
thf(tcon_Int_Oint___Rings_Osemiring__0,axiom,
semiring_0 @ int ).
thf(tcon_Int_Oint___Orderings_Ono__top,axiom,
no_top @ int ).
thf(tcon_Int_Oint___Orderings_Ono__bot,axiom,
no_bot @ int ).
thf(tcon_Int_Oint___Lattices_Olattice_13,axiom,
lattice @ int ).
thf(tcon_Int_Oint___Groups_Ogroup__add,axiom,
group_add @ int ).
thf(tcon_Int_Oint___GCD_Osemiring__gcd,axiom,
semiring_gcd @ int ).
thf(tcon_Int_Oint___GCD_Osemiring__Gcd,axiom,
semiring_Gcd @ int ).
thf(tcon_Int_Oint___Rings_Omult__zero,axiom,
mult_zero @ int ).
thf(tcon_Int_Oint___Rings_Ocomm__ring,axiom,
comm_ring @ int ).
thf(tcon_Int_Oint___Orderings_Oorder_14,axiom,
order @ int ).
thf(tcon_Int_Oint___Num_Oneg__numeral,axiom,
neg_numeral @ int ).
thf(tcon_Int_Oint___Nat_Oring__char__0,axiom,
ring_char_0 @ int ).
thf(tcon_Int_Oint___Rings_Osemiring,axiom,
semiring @ int ).
thf(tcon_Int_Oint___Rings_Osemidom,axiom,
semidom @ int ).
thf(tcon_Int_Oint___Orderings_Oord_15,axiom,
ord @ int ).
thf(tcon_Int_Oint___Groups_Ouminus_16,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_17,axiom,
sup @ int ).
thf(tcon_Int_Oint___Lattices_Oinf_18,axiom,
inf @ int ).
thf(tcon_Int_Oint___Groups_Otimes,axiom,
times @ int ).
thf(tcon_Int_Oint___GCD_Oring__gcd,axiom,
ring_gcd @ int ).
thf(tcon_Int_Oint___Power_Opower,axiom,
power @ int ).
thf(tcon_Int_Oint___Num_Onumeral,axiom,
numeral @ int ).
thf(tcon_Int_Oint___Groups_Ozero,axiom,
zero @ int ).
thf(tcon_Int_Oint___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_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_19,axiom,
condit6923001295902523014norder @ nat ).
thf(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_20,axiom,
condit1219197933456340205attice @ nat ).
thf(tcon_Nat_Onat___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_21,axiom,
bit_un5681908812861735899ations @ nat ).
thf(tcon_Nat_Onat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_22,axiom,
semiri1453513574482234551roduct @ nat ).
thf(tcon_Nat_Onat___Euclidean__Division_Ounique__euclidean__semiring__with__nat_23,axiom,
euclid5411537665997757685th_nat @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_24,axiom,
ordere1937475149494474687imp_le @ nat ).
thf(tcon_Nat_Onat___Euclidean__Division_Ounique__euclidean__semiring_25,axiom,
euclid3128863361964157862miring @ nat ).
thf(tcon_Nat_Onat___Euclidean__Division_Oeuclidean__semiring__cancel_26,axiom,
euclid4440199948858584721cancel @ nat ).
thf(tcon_Nat_Onat___Divides_Ounique__euclidean__semiring__numeral_27,axiom,
unique1627219031080169319umeral @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors__cancel_28,axiom,
semiri6575147826004484403cancel @ nat ).
thf(tcon_Nat_Onat___Groups_Ostrict__ordered__ab__semigroup__add_29,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_30,axiom,
ordere580206878836729694up_add @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le_31,axiom,
ordere2412721322843649153imp_le @ nat ).
thf(tcon_Nat_Onat___Bit__Operations_Osemiring__bit__operations_32,axiom,
bit_se359711467146920520ations @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__comm__semiring__strict_33,axiom,
linord2810124833399127020strict @ nat ).
thf(tcon_Nat_Onat___Groups_Ostrict__ordered__comm__monoid__add_34,axiom,
strict7427464778891057005id_add @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__cancel__comm__monoid__add_35,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_36,axiom,
euclid3725896446679973847miring @ nat ).
thf(tcon_Nat_Onat___Groups_Olinordered__ab__semigroup__add_37,axiom,
linord4140545234300271783up_add @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__1__no__zero__divisors_38,axiom,
semiri2026040879449505780visors @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring_39,axiom,
linord181362715937106298miring @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__semiring__strict_40,axiom,
linord8928482502909563296strict @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors_41,axiom,
semiri3467727345109120633visors @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add_42,axiom,
ordere6658533253407199908up_add @ nat ).
thf(tcon_Nat_Onat___GCD_Osemiring__gcd__mult__normalize_43,axiom,
semiri6843258321239162965malize @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__comm__monoid__add_44,axiom,
ordere6911136660526730532id_add @ nat ).
thf(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add_45,axiom,
cancel2418104881723323429up_add @ nat ).
thf(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add_46,axiom,
cancel1802427076303600483id_add @ nat ).
thf(tcon_Nat_Onat___Rings_Ocomm__semiring__1__cancel_47,axiom,
comm_s4317794764714335236cancel @ nat ).
thf(tcon_Nat_Onat___Bit__Operations_Osemiring__bits_48,axiom,
bit_semiring_bits @ nat ).
thf(tcon_Nat_Onat___Rings_Oordered__comm__semiring_49,axiom,
ordere2520102378445227354miring @ nat ).
thf(tcon_Nat_Onat___Groups_Ocancel__semigroup__add_50,axiom,
cancel_semigroup_add @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__semiring_51,axiom,
linordered_semiring @ nat ).
thf(tcon_Nat_Onat___Rings_Oordered__semiring__0_52,axiom,
ordered_semiring_0 @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__semidom_53,axiom,
linordered_semidom @ nat ).
thf(tcon_Nat_Onat___Lattices_Osemilattice__sup_54,axiom,
semilattice_sup @ nat ).
thf(tcon_Nat_Onat___Lattices_Osemilattice__inf_55,axiom,
semilattice_inf @ nat ).
thf(tcon_Nat_Onat___Lattices_Odistrib__lattice_56,axiom,
distrib_lattice @ nat ).
thf(tcon_Nat_Onat___Groups_Oab__semigroup__mult_57,axiom,
ab_semigroup_mult @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__1__cancel_58,axiom,
semiring_1_cancel @ nat ).
thf(tcon_Nat_Onat___Rings_Oalgebraic__semidom_59,axiom,
algebraic_semidom @ nat ).
thf(tcon_Nat_Onat___Groups_Ocomm__monoid__mult_60,axiom,
comm_monoid_mult @ nat ).
thf(tcon_Nat_Onat___Groups_Ocomm__monoid__diff,axiom,
comm_monoid_diff @ nat ).
thf(tcon_Nat_Onat___Groups_Oab__semigroup__add_61,axiom,
ab_semigroup_add @ nat ).
thf(tcon_Nat_Onat___Rings_Oordered__semiring_62,axiom,
ordered_semiring @ nat ).
thf(tcon_Nat_Onat___Parity_Osemiring__parity_63,axiom,
semiring_parity @ nat ).
thf(tcon_Nat_Onat___Groups_Ocomm__monoid__add_64,axiom,
comm_monoid_add @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__modulo_65,axiom,
semiring_modulo @ nat ).
thf(tcon_Nat_Onat___Rings_Ocomm__semiring__1_66,axiom,
comm_semiring_1 @ nat ).
thf(tcon_Nat_Onat___Rings_Ocomm__semiring__0_67,axiom,
comm_semiring_0 @ nat ).
thf(tcon_Nat_Onat___Groups_Osemigroup__mult_68,axiom,
semigroup_mult @ nat ).
thf(tcon_Nat_Onat___Complete__Lattices_OSup_69,axiom,
complete_Sup @ nat ).
thf(tcon_Nat_Onat___Complete__Lattices_OInf_70,axiom,
complete_Inf @ nat ).
thf(tcon_Nat_Onat___Rings_Osemidom__modulo_71,axiom,
semidom_modulo @ nat ).
thf(tcon_Nat_Onat___Rings_Osemidom__divide_72,axiom,
semidom_divide @ nat ).
thf(tcon_Nat_Onat___Num_Osemiring__numeral_73,axiom,
semiring_numeral @ nat ).
thf(tcon_Nat_Onat___Groups_Osemigroup__add_74,axiom,
semigroup_add @ nat ).
thf(tcon_Nat_Onat___Rings_Ozero__less__one_75,axiom,
zero_less_one @ nat ).
thf(tcon_Nat_Onat___Rings_Ocomm__semiring_76,axiom,
comm_semiring @ nat ).
thf(tcon_Nat_Onat___Orderings_Owellorder,axiom,
wellorder @ nat ).
thf(tcon_Nat_Onat___Orderings_Oorder__bot_77,axiom,
order_bot @ nat ).
thf(tcon_Nat_Onat___Nat_Osemiring__char__0_78,axiom,
semiring_char_0 @ nat ).
thf(tcon_Nat_Onat___Rings_Ozero__neq__one_79,axiom,
zero_neq_one @ nat ).
thf(tcon_Nat_Onat___Orderings_Opreorder_80,axiom,
preorder @ nat ).
thf(tcon_Nat_Onat___Orderings_Olinorder_81,axiom,
linorder @ nat ).
thf(tcon_Nat_Onat___Groups_Omonoid__mult_82,axiom,
monoid_mult @ nat ).
thf(tcon_Nat_Onat___Groups_Omonoid__add_83,axiom,
monoid_add @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__1_84,axiom,
semiring_1 @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__0_85,axiom,
semiring_0 @ nat ).
thf(tcon_Nat_Onat___Orderings_Ono__top_86,axiom,
no_top @ nat ).
thf(tcon_Nat_Onat___Lattices_Olattice_87,axiom,
lattice @ nat ).
thf(tcon_Nat_Onat___GCD_Osemiring__gcd_88,axiom,
semiring_gcd @ nat ).
thf(tcon_Nat_Onat___GCD_Osemiring__Gcd_89,axiom,
semiring_Gcd @ nat ).
thf(tcon_Nat_Onat___Rings_Omult__zero_90,axiom,
mult_zero @ nat ).
thf(tcon_Nat_Onat___Orderings_Oorder_91,axiom,
order @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring_92,axiom,
semiring @ nat ).
thf(tcon_Nat_Onat___Rings_Osemidom_93,axiom,
semidom @ nat ).
thf(tcon_Nat_Onat___Orderings_Oord_94,axiom,
ord @ nat ).
thf(tcon_Nat_Onat___Orderings_Obot_95,axiom,
bot @ nat ).
thf(tcon_Nat_Onat___Lattices_Osup_96,axiom,
sup @ nat ).
thf(tcon_Nat_Onat___Lattices_Oinf_97,axiom,
inf @ nat ).
thf(tcon_Nat_Onat___Groups_Otimes_98,axiom,
times @ nat ).
thf(tcon_Nat_Onat___Power_Opower_99,axiom,
power @ nat ).
thf(tcon_Nat_Onat___Num_Onumeral_100,axiom,
numeral @ nat ).
thf(tcon_Nat_Onat___Groups_Ozero_101,axiom,
zero @ nat ).
thf(tcon_Nat_Onat___Groups_Oplus_102,axiom,
plus @ nat ).
thf(tcon_Nat_Onat___Groups_Oone_103,axiom,
one @ nat ).
thf(tcon_Nat_Onat___Rings_Odvd_104,axiom,
dvd @ nat ).
thf(tcon_Nat_Onat___Heap_Oheap_105,axiom,
heap @ nat ).
thf(tcon_Nat_Onat___Nat_Osize,axiom,
size @ nat ).
thf(tcon_Num_Onum___Orderings_Opreorder_106,axiom,
preorder @ num ).
thf(tcon_Num_Onum___Orderings_Olinorder_107,axiom,
linorder @ num ).
thf(tcon_Num_Onum___Orderings_Oorder_108,axiom,
order @ num ).
thf(tcon_Num_Onum___Orderings_Oord_109,axiom,
ord @ num ).
thf(tcon_Num_Onum___Groups_Otimes_110,axiom,
times @ num ).
thf(tcon_Num_Onum___Groups_Oplus_111,axiom,
plus @ num ).
thf(tcon_Num_Onum___Nat_Osize_112,axiom,
size @ num ).
thf(tcon_Rat_Orat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_113,axiom,
semiri1453513574482234551roduct @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_114,axiom,
ordere1937475149494474687imp_le @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__no__zero__divisors__cancel_115,axiom,
semiri6575147826004484403cancel @ rat ).
thf(tcon_Rat_Orat___Groups_Ostrict__ordered__ab__semigroup__add_116,axiom,
strict9044650504122735259up_add @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__cancel__ab__semigroup__add_117,axiom,
ordere580206878836729694up_add @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__add__imp__le_118,axiom,
ordere2412721322843649153imp_le @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__comm__semiring__strict_119,axiom,
linord2810124833399127020strict @ rat ).
thf(tcon_Rat_Orat___Groups_Ostrict__ordered__comm__monoid__add_120,axiom,
strict7427464778891057005id_add @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__cancel__comm__monoid__add_121,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_122,axiom,
linord715952674999750819strict @ rat ).
thf(tcon_Rat_Orat___Orderings_Ounbounded__dense__linorder,axiom,
unboun7993243217541854897norder @ rat ).
thf(tcon_Rat_Orat___Groups_Olinordered__ab__semigroup__add_123,axiom,
linord4140545234300271783up_add @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__1__no__zero__divisors_124,axiom,
semiri2026040879449505780visors @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__nonzero__semiring_125,axiom,
linord181362715937106298miring @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__semiring__strict_126,axiom,
linord8928482502909563296strict @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__no__zero__divisors_127,axiom,
semiri3467727345109120633visors @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__add_128,axiom,
ordere6658533253407199908up_add @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__ab__group__add__abs_129,axiom,
ordere166539214618696060dd_abs @ rat ).
thf(tcon_Rat_Orat___Archimedean__Field_Ofloor__ceiling,axiom,
archim2362893244070406136eiling @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__comm__monoid__add_130,axiom,
ordere6911136660526730532id_add @ rat ).
thf(tcon_Rat_Orat___Groups_Olinordered__ab__group__add_131,axiom,
linord5086331880401160121up_add @ rat ).
thf(tcon_Rat_Orat___Groups_Ocancel__ab__semigroup__add_132,axiom,
cancel2418104881723323429up_add @ rat ).
thf(tcon_Rat_Orat___Rings_Oring__1__no__zero__divisors_133,axiom,
ring_15535105094025558882visors @ rat ).
thf(tcon_Rat_Orat___Groups_Ocancel__comm__monoid__add_134,axiom,
cancel1802427076303600483id_add @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__ring__strict_135,axiom,
linord4710134922213307826strict @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__semiring__1__cancel_136,axiom,
comm_s4317794764714335236cancel @ rat ).
thf(tcon_Rat_Orat___Rings_Oordered__comm__semiring_137,axiom,
ordere2520102378445227354miring @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__semiring__1_138,axiom,
linord6961819062388156250ring_1 @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__ab__group__add_139,axiom,
ordered_ab_group_add @ rat ).
thf(tcon_Rat_Orat___Groups_Ocancel__semigroup__add_140,axiom,
cancel_semigroup_add @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__semiring_141,axiom,
linordered_semiring @ rat ).
thf(tcon_Rat_Orat___Rings_Oordered__semiring__0_142,axiom,
ordered_semiring_0 @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__semidom_143,axiom,
linordered_semidom @ rat ).
thf(tcon_Rat_Orat___Orderings_Odense__linorder,axiom,
dense_linorder @ rat ).
thf(tcon_Rat_Orat___Lattices_Osemilattice__sup_144,axiom,
semilattice_sup @ rat ).
thf(tcon_Rat_Orat___Lattices_Osemilattice__inf_145,axiom,
semilattice_inf @ rat ).
thf(tcon_Rat_Orat___Lattices_Odistrib__lattice_146,axiom,
distrib_lattice @ rat ).
thf(tcon_Rat_Orat___Groups_Oab__semigroup__mult_147,axiom,
ab_semigroup_mult @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__1__cancel_148,axiom,
semiring_1_cancel @ rat ).
thf(tcon_Rat_Orat___Groups_Ocomm__monoid__mult_149,axiom,
comm_monoid_mult @ rat ).
thf(tcon_Rat_Orat___Groups_Oab__semigroup__add_150,axiom,
ab_semigroup_add @ rat ).
thf(tcon_Rat_Orat___Fields_Olinordered__field,axiom,
linordered_field @ rat ).
thf(tcon_Rat_Orat___Rings_Oordered__semiring_151,axiom,
ordered_semiring @ rat ).
thf(tcon_Rat_Orat___Rings_Oordered__ring__abs_152,axiom,
ordered_ring_abs @ rat ).
thf(tcon_Rat_Orat___Groups_Ocomm__monoid__add_153,axiom,
comm_monoid_add @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__ring_154,axiom,
linordered_ring @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__idom_155,axiom,
linordered_idom @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__semiring__1_156,axiom,
comm_semiring_1 @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__semiring__0_157,axiom,
comm_semiring_0 @ rat ).
thf(tcon_Rat_Orat___Orderings_Odense__order,axiom,
dense_order @ rat ).
thf(tcon_Rat_Orat___Groups_Osemigroup__mult_158,axiom,
semigroup_mult @ rat ).
thf(tcon_Rat_Orat___Rings_Osemidom__divide_159,axiom,
semidom_divide @ rat ).
thf(tcon_Rat_Orat___Num_Osemiring__numeral_160,axiom,
semiring_numeral @ rat ).
thf(tcon_Rat_Orat___Groups_Osemigroup__add_161,axiom,
semigroup_add @ rat ).
thf(tcon_Rat_Orat___Fields_Odivision__ring,axiom,
division_ring @ rat ).
thf(tcon_Rat_Orat___Rings_Ozero__less__one_162,axiom,
zero_less_one @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__semiring_163,axiom,
comm_semiring @ rat ).
thf(tcon_Rat_Orat___Nat_Osemiring__char__0_164,axiom,
semiring_char_0 @ rat ).
thf(tcon_Rat_Orat___Groups_Oab__group__add_165,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_166,axiom,
zero_neq_one @ rat ).
thf(tcon_Rat_Orat___Rings_Oordered__ring_167,axiom,
ordered_ring @ rat ).
thf(tcon_Rat_Orat___Rings_Oidom__abs__sgn_168,axiom,
idom_abs_sgn @ rat ).
thf(tcon_Rat_Orat___Orderings_Opreorder_169,axiom,
preorder @ rat ).
thf(tcon_Rat_Orat___Orderings_Olinorder_170,axiom,
linorder @ rat ).
thf(tcon_Rat_Orat___Groups_Omonoid__mult_171,axiom,
monoid_mult @ rat ).
thf(tcon_Rat_Orat___Rings_Oidom__divide_172,axiom,
idom_divide @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__ring__1_173,axiom,
comm_ring_1 @ rat ).
thf(tcon_Rat_Orat___Groups_Omonoid__add_174,axiom,
monoid_add @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__1_175,axiom,
semiring_1 @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__0_176,axiom,
semiring_0 @ rat ).
thf(tcon_Rat_Orat___Orderings_Ono__top_177,axiom,
no_top @ rat ).
thf(tcon_Rat_Orat___Orderings_Ono__bot_178,axiom,
no_bot @ rat ).
thf(tcon_Rat_Orat___Lattices_Olattice_179,axiom,
lattice @ rat ).
thf(tcon_Rat_Orat___Groups_Ogroup__add_180,axiom,
group_add @ rat ).
thf(tcon_Rat_Orat___Rings_Omult__zero_181,axiom,
mult_zero @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__ring_182,axiom,
comm_ring @ rat ).
thf(tcon_Rat_Orat___Orderings_Oorder_183,axiom,
order @ rat ).
thf(tcon_Rat_Orat___Num_Oneg__numeral_184,axiom,
neg_numeral @ rat ).
thf(tcon_Rat_Orat___Nat_Oring__char__0_185,axiom,
ring_char_0 @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring_186,axiom,
semiring @ rat ).
thf(tcon_Rat_Orat___Fields_Oinverse,axiom,
inverse @ rat ).
thf(tcon_Rat_Orat___Rings_Osemidom_187,axiom,
semidom @ rat ).
thf(tcon_Rat_Orat___Orderings_Oord_188,axiom,
ord @ rat ).
thf(tcon_Rat_Orat___Groups_Ouminus_189,axiom,
uminus @ rat ).
thf(tcon_Rat_Orat___Rings_Oring__1_190,axiom,
ring_1 @ rat ).
thf(tcon_Rat_Orat___Rings_Oabs__if_191,axiom,
abs_if @ rat ).
thf(tcon_Rat_Orat___Lattices_Osup_192,axiom,
sup @ rat ).
thf(tcon_Rat_Orat___Lattices_Oinf_193,axiom,
inf @ rat ).
thf(tcon_Rat_Orat___Groups_Otimes_194,axiom,
times @ rat ).
thf(tcon_Rat_Orat___Fields_Ofield,axiom,
field @ rat ).
thf(tcon_Rat_Orat___Power_Opower_195,axiom,
power @ rat ).
thf(tcon_Rat_Orat___Num_Onumeral_196,axiom,
numeral @ rat ).
thf(tcon_Rat_Orat___Groups_Ozero_197,axiom,
zero @ rat ).
thf(tcon_Rat_Orat___Groups_Oplus_198,axiom,
plus @ rat ).
thf(tcon_Rat_Orat___Rings_Oring_199,axiom,
ring @ rat ).
thf(tcon_Rat_Orat___Rings_Oidom_200,axiom,
idom @ rat ).
thf(tcon_Rat_Orat___Groups_Oone_201,axiom,
one @ rat ).
thf(tcon_Rat_Orat___Rings_Odvd_202,axiom,
dvd @ rat ).
thf(tcon_Set_Oset___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_203,axiom,
! [A18: $tType] : ( condit1219197933456340205attice @ ( set @ A18 ) ) ).
thf(tcon_Set_Oset___Complete__Lattices_Ocomplete__distrib__lattice_204,axiom,
! [A18: $tType] : ( comple592849572758109894attice @ ( set @ A18 ) ) ).
thf(tcon_Set_Oset___Complete__Lattices_Ocomplete__boolean__algebra_205,axiom,
! [A18: $tType] : ( comple489889107523837845lgebra @ ( set @ A18 ) ) ).
thf(tcon_Set_Oset___Lattices_Obounded__semilattice__sup__bot_206,axiom,
! [A18: $tType] : ( bounde4967611905675639751up_bot @ ( set @ A18 ) ) ).
thf(tcon_Set_Oset___Lattices_Obounded__semilattice__inf__top_207,axiom,
! [A18: $tType] : ( bounde4346867609351753570nf_top @ ( set @ A18 ) ) ).
thf(tcon_Set_Oset___Complete__Lattices_Ocomplete__lattice_208,axiom,
! [A18: $tType] : ( comple6319245703460814977attice @ ( set @ A18 ) ) ).
thf(tcon_Set_Oset___Boolean__Algebras_Oboolean__algebra_209,axiom,
! [A18: $tType] : ( boolea8198339166811842893lgebra @ ( set @ A18 ) ) ).
thf(tcon_Set_Oset___Complete__Partial__Order_Occpo_210,axiom,
! [A18: $tType] : ( comple9053668089753744459l_ccpo @ ( set @ A18 ) ) ).
thf(tcon_Set_Oset___Lattices_Osemilattice__sup_211,axiom,
! [A18: $tType] : ( semilattice_sup @ ( set @ A18 ) ) ).
thf(tcon_Set_Oset___Lattices_Osemilattice__inf_212,axiom,
! [A18: $tType] : ( semilattice_inf @ ( set @ A18 ) ) ).
thf(tcon_Set_Oset___Lattices_Odistrib__lattice_213,axiom,
! [A18: $tType] : ( distrib_lattice @ ( set @ A18 ) ) ).
thf(tcon_Set_Oset___Lattices_Obounded__lattice_214,axiom,
! [A18: $tType] : ( bounded_lattice @ ( set @ A18 ) ) ).
thf(tcon_Set_Oset___Complete__Lattices_OSup_215,axiom,
! [A18: $tType] : ( complete_Sup @ ( set @ A18 ) ) ).
thf(tcon_Set_Oset___Complete__Lattices_OInf_216,axiom,
! [A18: $tType] : ( complete_Inf @ ( set @ A18 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder__top_217,axiom,
! [A18: $tType] : ( order_top @ ( set @ A18 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder__bot_218,axiom,
! [A18: $tType] : ( order_bot @ ( set @ A18 ) ) ).
thf(tcon_Set_Oset___Orderings_Opreorder_219,axiom,
! [A18: $tType] : ( preorder @ ( set @ A18 ) ) ).
thf(tcon_Set_Oset___Finite__Set_Ofinite_220,axiom,
! [A18: $tType] :
( ( finite_finite @ A18 )
=> ( finite_finite @ ( set @ A18 ) ) ) ).
thf(tcon_Set_Oset___Lattices_Olattice_221,axiom,
! [A18: $tType] : ( lattice @ ( set @ A18 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_222,axiom,
! [A18: $tType] : ( order @ ( set @ A18 ) ) ).
thf(tcon_Set_Oset___Orderings_Otop_223,axiom,
! [A18: $tType] : ( top @ ( set @ A18 ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_224,axiom,
! [A18: $tType] : ( ord @ ( set @ A18 ) ) ).
thf(tcon_Set_Oset___Orderings_Obot_225,axiom,
! [A18: $tType] : ( bot @ ( set @ A18 ) ) ).
thf(tcon_Set_Oset___Groups_Ouminus_226,axiom,
! [A18: $tType] : ( uminus @ ( set @ A18 ) ) ).
thf(tcon_Set_Oset___Lattices_Osup_227,axiom,
! [A18: $tType] : ( sup @ ( set @ A18 ) ) ).
thf(tcon_Set_Oset___Lattices_Oinf_228,axiom,
! [A18: $tType] : ( inf @ ( set @ A18 ) ) ).
thf(tcon_HOL_Obool___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_229,axiom,
condit1219197933456340205attice @ $o ).
thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__distrib__lattice_230,axiom,
comple592849572758109894attice @ $o ).
thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__boolean__algebra_231,axiom,
comple489889107523837845lgebra @ $o ).
thf(tcon_HOL_Obool___Lattices_Obounded__semilattice__sup__bot_232,axiom,
bounde4967611905675639751up_bot @ $o ).
thf(tcon_HOL_Obool___Lattices_Obounded__semilattice__inf__top_233,axiom,
bounde4346867609351753570nf_top @ $o ).
thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__lattice_234,axiom,
comple6319245703460814977attice @ $o ).
thf(tcon_HOL_Obool___Boolean__Algebras_Oboolean__algebra_235,axiom,
boolea8198339166811842893lgebra @ $o ).
thf(tcon_HOL_Obool___Complete__Partial__Order_Occpo_236,axiom,
comple9053668089753744459l_ccpo @ $o ).
thf(tcon_HOL_Obool___Lattices_Osemilattice__sup_237,axiom,
semilattice_sup @ $o ).
thf(tcon_HOL_Obool___Lattices_Osemilattice__inf_238,axiom,
semilattice_inf @ $o ).
thf(tcon_HOL_Obool___Lattices_Odistrib__lattice_239,axiom,
distrib_lattice @ $o ).
thf(tcon_HOL_Obool___Lattices_Obounded__lattice_240,axiom,
bounded_lattice @ $o ).
thf(tcon_HOL_Obool___Complete__Lattices_OSup_241,axiom,
complete_Sup @ $o ).
thf(tcon_HOL_Obool___Complete__Lattices_OInf_242,axiom,
complete_Inf @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder__top_243,axiom,
order_top @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder__bot_244,axiom,
order_bot @ $o ).
thf(tcon_HOL_Obool___Orderings_Opreorder_245,axiom,
preorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Olinorder_246,axiom,
linorder @ $o ).
thf(tcon_HOL_Obool___Finite__Set_Ofinite_247,axiom,
finite_finite @ $o ).
thf(tcon_HOL_Obool___Lattices_Olattice_248,axiom,
lattice @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder_249,axiom,
order @ $o ).
thf(tcon_HOL_Obool___Orderings_Otop_250,axiom,
top @ $o ).
thf(tcon_HOL_Obool___Orderings_Oord_251,axiom,
ord @ $o ).
thf(tcon_HOL_Obool___Orderings_Obot_252,axiom,
bot @ $o ).
thf(tcon_HOL_Obool___Groups_Ouminus_253,axiom,
uminus @ $o ).
thf(tcon_HOL_Obool___Lattices_Osup_254,axiom,
sup @ $o ).
thf(tcon_HOL_Obool___Lattices_Oinf_255,axiom,
inf @ $o ).
thf(tcon_HOL_Obool___Heap_Oheap_256,axiom,
heap @ $o ).
thf(tcon_List_Olist___Heap_Oheap_257,axiom,
! [A18: $tType] :
( ( heap @ A18 )
=> ( heap @ ( list @ A18 ) ) ) ).
thf(tcon_List_Olist___Nat_Osize_258,axiom,
! [A18: $tType] : ( size @ ( list @ A18 ) ) ).
thf(tcon_Heap_Oarray___Heap_Oheap_259,axiom,
! [A18: $tType] : ( heap @ ( array @ A18 ) ) ).
thf(tcon_Heap_Oarray___Nat_Osize_260,axiom,
! [A18: $tType] : ( size @ ( array @ A18 ) ) ).
thf(tcon_String_Ochar___Finite__Set_Ofinite_261,axiom,
finite_finite @ char ).
thf(tcon_String_Ochar___Heap_Oheap_262,axiom,
heap @ char ).
thf(tcon_String_Ochar___Nat_Osize_263,axiom,
size @ char ).
thf(tcon_Filter_Ofilter___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_264,axiom,
! [A18: $tType] : ( condit1219197933456340205attice @ ( filter @ A18 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Obounded__semilattice__sup__bot_265,axiom,
! [A18: $tType] : ( bounde4967611905675639751up_bot @ ( filter @ A18 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Obounded__semilattice__inf__top_266,axiom,
! [A18: $tType] : ( bounde4346867609351753570nf_top @ ( filter @ A18 ) ) ).
thf(tcon_Filter_Ofilter___Complete__Lattices_Ocomplete__lattice_267,axiom,
! [A18: $tType] : ( comple6319245703460814977attice @ ( filter @ A18 ) ) ).
thf(tcon_Filter_Ofilter___Complete__Partial__Order_Occpo_268,axiom,
! [A18: $tType] : ( comple9053668089753744459l_ccpo @ ( filter @ A18 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Osemilattice__sup_269,axiom,
! [A18: $tType] : ( semilattice_sup @ ( filter @ A18 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Osemilattice__inf_270,axiom,
! [A18: $tType] : ( semilattice_inf @ ( filter @ A18 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Odistrib__lattice_271,axiom,
! [A18: $tType] : ( distrib_lattice @ ( filter @ A18 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Obounded__lattice_272,axiom,
! [A18: $tType] : ( bounded_lattice @ ( filter @ A18 ) ) ).
thf(tcon_Filter_Ofilter___Complete__Lattices_OSup_273,axiom,
! [A18: $tType] : ( complete_Sup @ ( filter @ A18 ) ) ).
thf(tcon_Filter_Ofilter___Complete__Lattices_OInf_274,axiom,
! [A18: $tType] : ( complete_Inf @ ( filter @ A18 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Oorder__top_275,axiom,
! [A18: $tType] : ( order_top @ ( filter @ A18 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Oorder__bot_276,axiom,
! [A18: $tType] : ( order_bot @ ( filter @ A18 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Opreorder_277,axiom,
! [A18: $tType] : ( preorder @ ( filter @ A18 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Olattice_278,axiom,
! [A18: $tType] : ( lattice @ ( filter @ A18 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Oorder_279,axiom,
! [A18: $tType] : ( order @ ( filter @ A18 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Otop_280,axiom,
! [A18: $tType] : ( top @ ( filter @ A18 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Oord_281,axiom,
! [A18: $tType] : ( ord @ ( filter @ A18 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Obot_282,axiom,
! [A18: $tType] : ( bot @ ( filter @ A18 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Osup_283,axiom,
! [A18: $tType] : ( sup @ ( filter @ A18 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Oinf_284,axiom,
! [A18: $tType] : ( inf @ ( filter @ A18 ) ) ).
thf(tcon_Option_Ooption___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_285,axiom,
! [A18: $tType] :
( ( comple5582772986160207858norder @ A18 )
=> ( condit6923001295902523014norder @ ( option @ A18 ) ) ) ).
thf(tcon_Option_Ooption___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_286,axiom,
! [A18: $tType] :
( ( comple6319245703460814977attice @ A18 )
=> ( condit1219197933456340205attice @ ( option @ A18 ) ) ) ).
thf(tcon_Option_Ooption___Complete__Lattices_Ocomplete__distrib__lattice_287,axiom,
! [A18: $tType] :
( ( comple592849572758109894attice @ A18 )
=> ( comple592849572758109894attice @ ( option @ A18 ) ) ) ).
thf(tcon_Option_Ooption___Lattices_Obounded__semilattice__sup__bot_288,axiom,
! [A18: $tType] :
( ( lattice @ A18 )
=> ( bounde4967611905675639751up_bot @ ( option @ A18 ) ) ) ).
thf(tcon_Option_Ooption___Lattices_Obounded__semilattice__inf__top_289,axiom,
! [A18: $tType] :
( ( bounded_lattice_top @ A18 )
=> ( bounde4346867609351753570nf_top @ ( option @ A18 ) ) ) ).
thf(tcon_Option_Ooption___Complete__Lattices_Ocomplete__linorder,axiom,
! [A18: $tType] :
( ( comple5582772986160207858norder @ A18 )
=> ( comple5582772986160207858norder @ ( option @ A18 ) ) ) ).
thf(tcon_Option_Ooption___Complete__Lattices_Ocomplete__lattice_290,axiom,
! [A18: $tType] :
( ( comple6319245703460814977attice @ A18 )
=> ( comple6319245703460814977attice @ ( option @ A18 ) ) ) ).
thf(tcon_Option_Ooption___Complete__Partial__Order_Occpo_291,axiom,
! [A18: $tType] :
( ( comple6319245703460814977attice @ A18 )
=> ( comple9053668089753744459l_ccpo @ ( option @ A18 ) ) ) ).
thf(tcon_Option_Ooption___Lattices_Osemilattice__sup_292,axiom,
! [A18: $tType] :
( ( semilattice_sup @ A18 )
=> ( semilattice_sup @ ( option @ A18 ) ) ) ).
thf(tcon_Option_Ooption___Lattices_Osemilattice__inf_293,axiom,
! [A18: $tType] :
( ( semilattice_inf @ A18 )
=> ( semilattice_inf @ ( option @ A18 ) ) ) ).
thf(tcon_Option_Ooption___Lattices_Odistrib__lattice_294,axiom,
! [A18: $tType] :
( ( distrib_lattice @ A18 )
=> ( distrib_lattice @ ( option @ A18 ) ) ) ).
thf(tcon_Option_Ooption___Lattices_Obounded__lattice_295,axiom,
! [A18: $tType] :
( ( bounded_lattice_top @ A18 )
=> ( bounded_lattice @ ( option @ A18 ) ) ) ).
thf(tcon_Option_Ooption___Complete__Lattices_OSup_296,axiom,
! [A18: $tType] :
( ( comple6319245703460814977attice @ A18 )
=> ( complete_Sup @ ( option @ A18 ) ) ) ).
thf(tcon_Option_Ooption___Complete__Lattices_OInf_297,axiom,
! [A18: $tType] :
( ( comple6319245703460814977attice @ A18 )
=> ( complete_Inf @ ( option @ A18 ) ) ) ).
thf(tcon_Option_Ooption___Orderings_Owellorder_298,axiom,
! [A18: $tType] :
( ( wellorder @ A18 )
=> ( wellorder @ ( option @ A18 ) ) ) ).
thf(tcon_Option_Ooption___Orderings_Oorder__top_299,axiom,
! [A18: $tType] :
( ( order_top @ A18 )
=> ( order_top @ ( option @ A18 ) ) ) ).
thf(tcon_Option_Ooption___Orderings_Oorder__bot_300,axiom,
! [A18: $tType] :
( ( order @ A18 )
=> ( order_bot @ ( option @ A18 ) ) ) ).
thf(tcon_Option_Ooption___Orderings_Opreorder_301,axiom,
! [A18: $tType] :
( ( preorder @ A18 )
=> ( preorder @ ( option @ A18 ) ) ) ).
thf(tcon_Option_Ooption___Orderings_Olinorder_302,axiom,
! [A18: $tType] :
( ( linorder @ A18 )
=> ( linorder @ ( option @ A18 ) ) ) ).
thf(tcon_Option_Ooption___Finite__Set_Ofinite_303,axiom,
! [A18: $tType] :
( ( finite_finite @ A18 )
=> ( finite_finite @ ( option @ A18 ) ) ) ).
thf(tcon_Option_Ooption___Lattices_Olattice_304,axiom,
! [A18: $tType] :
( ( lattice @ A18 )
=> ( lattice @ ( option @ A18 ) ) ) ).
thf(tcon_Option_Ooption___Orderings_Oorder_305,axiom,
! [A18: $tType] :
( ( order @ A18 )
=> ( order @ ( option @ A18 ) ) ) ).
thf(tcon_Option_Ooption___Orderings_Otop_306,axiom,
! [A18: $tType] :
( ( order_top @ A18 )
=> ( top @ ( option @ A18 ) ) ) ).
thf(tcon_Option_Ooption___Orderings_Oord_307,axiom,
! [A18: $tType] :
( ( preorder @ A18 )
=> ( ord @ ( option @ A18 ) ) ) ).
thf(tcon_Option_Ooption___Orderings_Obot_308,axiom,
! [A18: $tType] :
( ( order @ A18 )
=> ( bot @ ( option @ A18 ) ) ) ).
thf(tcon_Option_Ooption___Lattices_Osup_309,axiom,
! [A18: $tType] :
( ( sup @ A18 )
=> ( sup @ ( option @ A18 ) ) ) ).
thf(tcon_Option_Ooption___Lattices_Oinf_310,axiom,
! [A18: $tType] :
( ( inf @ A18 )
=> ( inf @ ( option @ A18 ) ) ) ).
thf(tcon_Option_Ooption___Heap_Oheap_311,axiom,
! [A18: $tType] :
( ( heap @ A18 )
=> ( heap @ ( option @ A18 ) ) ) ).
thf(tcon_Option_Ooption___Nat_Osize_312,axiom,
! [A18: $tType] : ( size @ ( option @ A18 ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Oordered__ab__semigroup__add_313,axiom,
! [A18: $tType] :
( ( preorder @ A18 )
=> ( ordere6658533253407199908up_add @ ( multiset @ A18 ) ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Ocancel__ab__semigroup__add_314,axiom,
! [A18: $tType] : ( cancel2418104881723323429up_add @ ( multiset @ A18 ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Ocancel__comm__monoid__add_315,axiom,
! [A18: $tType] : ( cancel1802427076303600483id_add @ ( multiset @ A18 ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Ocancel__semigroup__add_316,axiom,
! [A18: $tType] : ( cancel_semigroup_add @ ( multiset @ A18 ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Ocomm__monoid__diff_317,axiom,
! [A18: $tType] : ( comm_monoid_diff @ ( multiset @ A18 ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Oab__semigroup__add_318,axiom,
! [A18: $tType] : ( ab_semigroup_add @ ( multiset @ A18 ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Ocomm__monoid__add_319,axiom,
! [A18: $tType] : ( comm_monoid_add @ ( multiset @ A18 ) ) ).
thf(tcon_Multiset_Omultiset___Complete__Lattices_OSup_320,axiom,
! [A18: $tType] : ( complete_Sup @ ( multiset @ A18 ) ) ).
thf(tcon_Multiset_Omultiset___Complete__Lattices_OInf_321,axiom,
! [A18: $tType] : ( complete_Inf @ ( multiset @ A18 ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Osemigroup__add_322,axiom,
! [A18: $tType] : ( semigroup_add @ ( multiset @ A18 ) ) ).
thf(tcon_Multiset_Omultiset___Orderings_Opreorder_323,axiom,
! [A18: $tType] :
( ( preorder @ A18 )
=> ( preorder @ ( multiset @ A18 ) ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Omonoid__add_324,axiom,
! [A18: $tType] : ( monoid_add @ ( multiset @ A18 ) ) ).
thf(tcon_Multiset_Omultiset___Orderings_Oorder_325,axiom,
! [A18: $tType] :
( ( preorder @ A18 )
=> ( order @ ( multiset @ A18 ) ) ) ).
thf(tcon_Multiset_Omultiset___Orderings_Oord_326,axiom,
! [A18: $tType] :
( ( preorder @ A18 )
=> ( ord @ ( multiset @ A18 ) ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Ozero_327,axiom,
! [A18: $tType] : ( zero @ ( multiset @ A18 ) ) ).
thf(tcon_Multiset_Omultiset___Groups_Oplus_328,axiom,
! [A18: $tType] : ( plus @ ( multiset @ A18 ) ) ).
thf(tcon_Multiset_Omultiset___Nat_Osize_329,axiom,
! [A18: $tType] : ( size @ ( multiset @ A18 ) ) ).
thf(tcon_Product__Type_Oprod___Finite__Set_Ofinite_330,axiom,
! [A18: $tType,A19: $tType] :
( ( ( finite_finite @ A18 )
& ( finite_finite @ A19 ) )
=> ( finite_finite @ ( product_prod @ A18 @ A19 ) ) ) ).
thf(tcon_Product__Type_Oprod___Heap_Oheap_331,axiom,
! [A18: $tType,A19: $tType] :
( ( ( heap @ A18 )
& ( heap @ A19 ) )
=> ( heap @ ( product_prod @ A18 @ A19 ) ) ) ).
thf(tcon_Product__Type_Oprod___Nat_Osize_332,axiom,
! [A18: $tType,A19: $tType] : ( size @ ( product_prod @ A18 @ A19 ) ) ).
thf(tcon_Product__Type_Ounit___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_333,axiom,
condit6923001295902523014norder @ product_unit ).
thf(tcon_Product__Type_Ounit___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_334,axiom,
condit1219197933456340205attice @ product_unit ).
thf(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__distrib__lattice_335,axiom,
comple592849572758109894attice @ product_unit ).
thf(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__boolean__algebra_336,axiom,
comple489889107523837845lgebra @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Obounded__semilattice__sup__bot_337,axiom,
bounde4967611905675639751up_bot @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Obounded__semilattice__inf__top_338,axiom,
bounde4346867609351753570nf_top @ product_unit ).
thf(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__linorder_339,axiom,
comple5582772986160207858norder @ product_unit ).
thf(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__lattice_340,axiom,
comple6319245703460814977attice @ product_unit ).
thf(tcon_Product__Type_Ounit___Boolean__Algebras_Oboolean__algebra_341,axiom,
boolea8198339166811842893lgebra @ product_unit ).
thf(tcon_Product__Type_Ounit___Complete__Partial__Order_Occpo_342,axiom,
comple9053668089753744459l_ccpo @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Osemilattice__sup_343,axiom,
semilattice_sup @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Osemilattice__inf_344,axiom,
semilattice_inf @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Odistrib__lattice_345,axiom,
distrib_lattice @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Obounded__lattice_346,axiom,
bounded_lattice @ product_unit ).
thf(tcon_Product__Type_Ounit___Complete__Lattices_OSup_347,axiom,
complete_Sup @ product_unit ).
thf(tcon_Product__Type_Ounit___Complete__Lattices_OInf_348,axiom,
complete_Inf @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Owellorder_349,axiom,
wellorder @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oorder__top_350,axiom,
order_top @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oorder__bot_351,axiom,
order_bot @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Opreorder_352,axiom,
preorder @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Olinorder_353,axiom,
linorder @ product_unit ).
thf(tcon_Product__Type_Ounit___Finite__Set_Ofinite_354,axiom,
finite_finite @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Olattice_355,axiom,
lattice @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oorder_356,axiom,
order @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Otop_357,axiom,
top @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oord_358,axiom,
ord @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Obot_359,axiom,
bot @ product_unit ).
thf(tcon_Product__Type_Ounit___Groups_Ouminus_360,axiom,
uminus @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Osup_361,axiom,
sup @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Oinf_362,axiom,
inf @ product_unit ).
thf(tcon_Product__Type_Ounit___Heap_Oheap_363,axiom,
heap @ product_unit ).
thf(tcon_Code__Numeral_Ointeger___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_364,axiom,
bit_un5681908812861735899ations @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_365,axiom,
semiri1453513574482234551roduct @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__semiring__with__nat_366,axiom,
euclid5411537665997757685th_nat @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__ring__with__nat_367,axiom,
euclid8789492081693882211th_nat @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__monoid__add__imp__le_368,axiom,
ordere1937475149494474687imp_le @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__semiring_369,axiom,
euclid3128863361964157862miring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__semiring__cancel_370,axiom,
euclid4440199948858584721cancel @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Divides_Ounique__euclidean__semiring__numeral_371,axiom,
unique1627219031080169319umeral @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__ring__cancel_372,axiom,
euclid8851590272496341667cancel @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__no__zero__divisors__cancel_373,axiom,
semiri6575147826004484403cancel @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ostrict__ordered__ab__semigroup__add_374,axiom,
strict9044650504122735259up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__cancel__ab__semigroup__add_375,axiom,
ordere580206878836729694up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__add__imp__le_376,axiom,
ordere2412721322843649153imp_le @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Bit__Operations_Osemiring__bit__operations_377,axiom,
bit_se359711467146920520ations @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__comm__semiring__strict_378,axiom,
linord2810124833399127020strict @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ostrict__ordered__comm__monoid__add_379,axiom,
strict7427464778891057005id_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__cancel__comm__monoid__add_380,axiom,
ordere8940638589300402666id_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__semiring_381,axiom,
euclid3725896446679973847miring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__1__strict_382,axiom,
linord715952674999750819strict @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Olinordered__ab__semigroup__add_383,axiom,
linord4140545234300271783up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Bit__Operations_Oring__bit__operations_384,axiom,
bit_ri3973907225187159222ations @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1__no__zero__divisors_385,axiom,
semiri2026040879449505780visors @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__nonzero__semiring_386,axiom,
linord181362715937106298miring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__strict_387,axiom,
linord8928482502909563296strict @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__no__zero__divisors_388,axiom,
semiri3467727345109120633visors @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__add_389,axiom,
ordere6658533253407199908up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__group__add__abs_390,axiom,
ordere166539214618696060dd_abs @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__comm__monoid__add_391,axiom,
ordere6911136660526730532id_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Olinordered__ab__group__add_392,axiom,
linord5086331880401160121up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ocancel__ab__semigroup__add_393,axiom,
cancel2418104881723323429up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oring__1__no__zero__divisors_394,axiom,
ring_15535105094025558882visors @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ocancel__comm__monoid__add_395,axiom,
cancel1802427076303600483id_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__ring__strict_396,axiom,
linord4710134922213307826strict @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__1__cancel_397,axiom,
comm_s4317794764714335236cancel @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Bit__Operations_Osemiring__bits_398,axiom,
bit_semiring_bits @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__comm__semiring_399,axiom,
ordere2520102378445227354miring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__1_400,axiom,
linord6961819062388156250ring_1 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__group__add_401,axiom,
ordered_ab_group_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ocancel__semigroup__add_402,axiom,
cancel_semigroup_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring_403,axiom,
linordered_semiring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__semiring__0_404,axiom,
ordered_semiring_0 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semidom_405,axiom,
linordered_semidom @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oab__semigroup__mult_406,axiom,
ab_semigroup_mult @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1__cancel_407,axiom,
semiring_1_cancel @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oalgebraic__semidom_408,axiom,
algebraic_semidom @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ocomm__monoid__mult_409,axiom,
comm_monoid_mult @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oab__semigroup__add_410,axiom,
ab_semigroup_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__semiring_411,axiom,
ordered_semiring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__ring__abs_412,axiom,
ordered_ring_abs @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Parity_Osemiring__parity_413,axiom,
semiring_parity @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ocomm__monoid__add_414,axiom,
comm_monoid_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__modulo_415,axiom,
semiring_modulo @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__ring_416,axiom,
linordered_ring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__idom_417,axiom,
linordered_idom @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__1_418,axiom,
comm_semiring_1 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__0_419,axiom,
comm_semiring_0 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Osemigroup__mult_420,axiom,
semigroup_mult @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemidom__modulo_421,axiom,
semidom_modulo @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemidom__divide_422,axiom,
semidom_divide @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Num_Osemiring__numeral_423,axiom,
semiring_numeral @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Osemigroup__add_424,axiom,
semigroup_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ozero__less__one_425,axiom,
zero_less_one @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring_426,axiom,
comm_semiring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Nat_Osemiring__char__0_427,axiom,
semiring_char_0 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oab__group__add_428,axiom,
ab_group_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ozero__neq__one_429,axiom,
zero_neq_one @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__ring_430,axiom,
ordered_ring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oidom__abs__sgn_431,axiom,
idom_abs_sgn @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Parity_Oring__parity_432,axiom,
ring_parity @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Orderings_Opreorder_433,axiom,
preorder @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Orderings_Olinorder_434,axiom,
linorder @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Omonoid__mult_435,axiom,
monoid_mult @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oidom__divide_436,axiom,
idom_divide @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__ring__1_437,axiom,
comm_ring_1 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Omonoid__add_438,axiom,
monoid_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1_439,axiom,
semiring_1 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__0_440,axiom,
semiring_0 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ogroup__add_441,axiom,
group_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Omult__zero_442,axiom,
mult_zero @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__ring_443,axiom,
comm_ring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Orderings_Oorder_444,axiom,
order @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Num_Oneg__numeral_445,axiom,
neg_numeral @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Nat_Oring__char__0_446,axiom,
ring_char_0 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring_447,axiom,
semiring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemidom_448,axiom,
semidom @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Orderings_Oord_449,axiom,
ord @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ouminus_450,axiom,
uminus @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oring__1_451,axiom,
ring_1 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oabs__if_452,axiom,
abs_if @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Otimes_453,axiom,
times @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Power_Opower_454,axiom,
power @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Num_Onumeral_455,axiom,
numeral @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ozero_456,axiom,
zero @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oplus_457,axiom,
plus @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oring_458,axiom,
ring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oidom_459,axiom,
idom @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oone_460,axiom,
one @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Odvd_461,axiom,
dvd @ code_integer ).
thf(tcon_Code__Numeral_Onatural___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_462,axiom,
bit_un5681908812861735899ations @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Euclidean__Division_Ounique__euclidean__semiring__with__nat_463,axiom,
euclid5411537665997757685th_nat @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Oordered__ab__semigroup__monoid__add__imp__le_464,axiom,
ordere1937475149494474687imp_le @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Euclidean__Division_Ounique__euclidean__semiring_465,axiom,
euclid3128863361964157862miring @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Euclidean__Division_Oeuclidean__semiring__cancel_466,axiom,
euclid4440199948858584721cancel @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Osemiring__no__zero__divisors__cancel_467,axiom,
semiri6575147826004484403cancel @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Ostrict__ordered__ab__semigroup__add_468,axiom,
strict9044650504122735259up_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Oordered__cancel__ab__semigroup__add_469,axiom,
ordere580206878836729694up_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Oordered__ab__semigroup__add__imp__le_470,axiom,
ordere2412721322843649153imp_le @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Bit__Operations_Osemiring__bit__operations_471,axiom,
bit_se359711467146920520ations @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Olinordered__comm__semiring__strict_472,axiom,
linord2810124833399127020strict @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Ostrict__ordered__comm__monoid__add_473,axiom,
strict7427464778891057005id_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Oordered__cancel__comm__monoid__add_474,axiom,
ordere8940638589300402666id_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Euclidean__Division_Oeuclidean__semiring_475,axiom,
euclid3725896446679973847miring @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Olinordered__ab__semigroup__add_476,axiom,
linord4140545234300271783up_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Osemiring__1__no__zero__divisors_477,axiom,
semiri2026040879449505780visors @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Olinordered__nonzero__semiring_478,axiom,
linord181362715937106298miring @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Olinordered__semiring__strict_479,axiom,
linord8928482502909563296strict @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Osemiring__no__zero__divisors_480,axiom,
semiri3467727345109120633visors @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Oordered__ab__semigroup__add_481,axiom,
ordere6658533253407199908up_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Oordered__comm__monoid__add_482,axiom,
ordere6911136660526730532id_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Ocancel__ab__semigroup__add_483,axiom,
cancel2418104881723323429up_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Ocancel__comm__monoid__add_484,axiom,
cancel1802427076303600483id_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring__1__cancel_485,axiom,
comm_s4317794764714335236cancel @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Bit__Operations_Osemiring__bits_486,axiom,
bit_semiring_bits @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Oordered__comm__semiring_487,axiom,
ordere2520102378445227354miring @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Ocancel__semigroup__add_488,axiom,
cancel_semigroup_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Olinordered__semiring_489,axiom,
linordered_semiring @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Oordered__semiring__0_490,axiom,
ordered_semiring_0 @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Olinordered__semidom_491,axiom,
linordered_semidom @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Oab__semigroup__mult_492,axiom,
ab_semigroup_mult @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Osemiring__1__cancel_493,axiom,
semiring_1_cancel @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Oalgebraic__semidom_494,axiom,
algebraic_semidom @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Ocomm__monoid__mult_495,axiom,
comm_monoid_mult @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Ocomm__monoid__diff_496,axiom,
comm_monoid_diff @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Oab__semigroup__add_497,axiom,
ab_semigroup_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Oordered__semiring_498,axiom,
ordered_semiring @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Parity_Osemiring__parity_499,axiom,
semiring_parity @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Ocomm__monoid__add_500,axiom,
comm_monoid_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Osemiring__modulo_501,axiom,
semiring_modulo @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring__1_502,axiom,
comm_semiring_1 @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring__0_503,axiom,
comm_semiring_0 @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Osemigroup__mult_504,axiom,
semigroup_mult @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Osemidom__modulo_505,axiom,
semidom_modulo @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Osemidom__divide_506,axiom,
semidom_divide @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Num_Osemiring__numeral_507,axiom,
semiring_numeral @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Osemigroup__add_508,axiom,
semigroup_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Ozero__less__one_509,axiom,
zero_less_one @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring_510,axiom,
comm_semiring @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Nat_Osemiring__char__0_511,axiom,
semiring_char_0 @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Ozero__neq__one_512,axiom,
zero_neq_one @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Orderings_Opreorder_513,axiom,
preorder @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Orderings_Olinorder_514,axiom,
linorder @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Omonoid__mult_515,axiom,
monoid_mult @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Omonoid__add_516,axiom,
monoid_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Osemiring__1_517,axiom,
semiring_1 @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Osemiring__0_518,axiom,
semiring_0 @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Omult__zero_519,axiom,
mult_zero @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Orderings_Oorder_520,axiom,
order @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Osemiring_521,axiom,
semiring @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Osemidom_522,axiom,
semidom @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Orderings_Oord_523,axiom,
ord @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Otimes_524,axiom,
times @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Power_Opower_525,axiom,
power @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Num_Onumeral_526,axiom,
numeral @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Ozero_527,axiom,
zero @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Oplus_528,axiom,
plus @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Oone_529,axiom,
one @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Odvd_530,axiom,
dvd @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Nat_Osize_531,axiom,
size @ code_natural ).
thf(tcon_Heap__Time__Monad_OHeap___Nat_Osize_532,axiom,
! [A18: $tType] : ( size @ ( heap_Time_Heap @ A18 ) ) ).
% Helper facts (4)
thf(help_If_3_1_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_T,axiom,
! [A: $tType,X2: A,Y: A] :
( ( if @ A @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_T,axiom,
! [A: $tType,X2: A,Y: A] :
( ( if @ A @ $true @ X2 @ Y )
= X2 ) ).
thf(help_fChoice_1_1_T,axiom,
! [A: $tType,P: A > $o] :
( ( P @ ( fChoice @ A @ P ) )
= ( ? [X5: A] : ( P @ X5 ) ) ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ord_less_eq @ nat
@ ( time_time @ a
@ ( heap_Time_bind @ product_unit @ a @ ( refine_Imp_assert @ p )
@ ^ [Uu: product_unit] : m )
@ h )
@ t ) ).
%------------------------------------------------------------------------------