TPTP Problem File: ITP207^4.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : ITP207^4 : TPTP v8.2.0. Released v8.0.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer problem Heap_Time_Monad 00386_012441
% 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 : 0018_Heap_Time_Monad_00386_012441 [Des22]
% Status : Theorem
% Rating : 0.67 v8.1.0
% Syntax : Number of formulae : 8823 (3051 unt; 905 typ; 0 def)
% Number of atoms : 22972 (8614 equ; 5 cnn)
% Maximal formula atoms : 23 ( 2 avg)
% Number of connectives : 166339 (1762 ~; 275 |;1533 &;153416 @)
% ( 0 <=>;9353 =>; 0 <=; 0 <~>)
% Maximal formula depth : 29 ( 8 avg)
% Number of types : 13 ( 12 usr)
% Number of type conns : 7778 (7778 >; 0 *; 0 +; 0 <<)
% Number of symbols : 897 ( 893 usr; 19 con; 0-10 aty)
% Number of variables : 30735 (4437 ^;24769 !; 517 ?;30735 :)
% (1012 !>; 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 13:57:16.283
%------------------------------------------------------------------------------
% Could-be-implicit typings (25)
thf(ty_t_Record_Otuple__isomorphism,type,
tuple_isomorphism: $tType > $tType > $tType > $tType ).
thf(ty_t_Heap__Time__Monad_OHeap,type,
heap_Time_Heap: $tType > $tType ).
thf(ty_t_Code__Numeral_Onatural,type,
code_natural: $tType ).
thf(ty_t_Code__Numeral_Ointeger,type,
code_integer: $tType ).
thf(ty_t_Code__Evaluation_Oterm,type,
code_term: $tType ).
thf(ty_t_Heap_Oheap_Oheap__ext,type,
heap_ext: $tType > $tType ).
thf(ty_t_Product__Type_Ounit,type,
product_unit: $tType ).
thf(ty_t_Product__Type_Oprod,type,
product_prod: $tType > $tType > $tType ).
thf(ty_t_Old__Datatype_Onode,type,
old_node: $tType > $tType > $tType ).
thf(ty_t_Typerep_Otyperep,type,
typerep: $tType ).
thf(ty_t_String_Oliteral,type,
literal: $tType ).
thf(ty_t_Predicate_Opred,type,
pred: $tType > $tType ).
thf(ty_t_Sum__Type_Osum,type,
sum_sum: $tType > $tType > $tType ).
thf(ty_t_Predicate_Oseq,type,
seq: $tType > $tType ).
thf(ty_t_Option_Ooption,type,
option: $tType > $tType ).
thf(ty_t_Filter_Ofilter,type,
filter: $tType > $tType ).
thf(ty_t_String_Ochar,type,
char: $tType ).
thf(ty_t_List_Olist,type,
list: $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 (880)
thf(sy_cl_Enum_Oenum,type,
enum:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Obounded__lattice,type,
bounded_lattice:
!>[A: $tType] : $o ).
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : $o ).
thf(sy_cl_Nat_Osize,type,
size:
!>[A: $tType] : $o ).
thf(sy_cl_HOL_Oequal,type,
cl_HOL_Oequal:
!>[A: $tType] : $o ).
thf(sy_cl_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_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_Oord,type,
ord:
!>[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_Rings_Oidom__modulo,type,
idom_modulo:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Omonoid__mult,type,
monoid_mult:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Opreorder,type,
preorder:
!>[A: $tType] : $o ).
thf(sy_cl_Parity_Oring__parity,type,
ring_parity:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oidom__abs__sgn,type,
idom_abs_sgn:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oordered__ring,type,
ordered_ring:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Ozero__neq__one,type,
zero_neq_one:
!>[A: $tType] : $o ).
thf(sy_cl_Countable_Ocountable,type,
countable:
!>[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_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_Odistrib__lattice,type,
distrib_lattice:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Osemilattice__inf,type,
semilattice_inf:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Osemilattice__sup,type,
semilattice_sup:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Odense__linorder,type,
dense_linorder:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__semidom,type,
linordered_semidom:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oordered__semiring__0,type,
ordered_semiring_0:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__semiring,type,
linordered_semiring:
!>[A: $tType] : $o ).
thf(sy_cl_Complete__Partial__Order_Occpo,type,
comple9053668089753744459l_ccpo:
!>[A: $tType] : $o ).
thf(sy_cl_Enum_Ofinite__distrib__lattice,type,
finite8700451911770168679attice:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocancel__semigroup__add,type,
cancel_semigroup_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__group__add,type,
ordered_ab_group_add:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__semiring__1,type,
linord6961819062388156250ring_1:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Onormalization__semidom,type,
normal8620421768224518004emidom:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oordered__comm__semiring,type,
ordere2520102378445227354miring:
!>[A: $tType] : $o ).
thf(sy_cl_Bit__Operations_Osemiring__bits,type,
bit_semiring_bits:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Ocomm__semiring__1__cancel,type,
comm_s4317794764714335236cancel:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__ring__strict,type,
linord4710134922213307826strict:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocancel__comm__monoid__add,type,
cancel1802427076303600483id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oring__1__no__zero__divisors,type,
ring_15535105094025558882visors:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocancel__ab__semigroup__add,type,
cancel2418104881723323429up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Olinordered__ab__group__add,type,
linord5086331880401160121up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__comm__monoid__add,type,
ordere6911136660526730532id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Archimedean__Field_Ofloor__ceiling,type,
archim2362893244070406136eiling:
!>[A: $tType] : $o ).
thf(sy_cl_GCD_Osemiring__gcd__mult__normalize,type,
semiri6843258321239162965malize:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__group__add__abs,type,
ordere166539214618696060dd_abs:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__semigroup__add,type,
ordere6658533253407199908up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemiring__no__zero__divisors,type,
semiri3467727345109120633visors:
!>[A: $tType] : $o ).
thf(sy_cl_Boolean__Algebras_Oboolean__algebra,type,
boolea8198339166811842893lgebra:
!>[A: $tType] : $o ).
thf(sy_cl_Quickcheck__Exhaustive_Oexhaustive,type,
quickc658316121487927005ustive:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__semiring__strict,type,
linord8928482502909563296strict:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemidom__divide__unit__factor,type,
semido2269285787275462019factor:
!>[A: $tType] : $o ).
thf(sy_cl_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__sup__bot,type,
bounde4967611905675639751up_bot:
!>[A: $tType] : $o ).
thf(sy_cl_Euclidean__Division_Oeuclidean__semiring,type,
euclid3725896446679973847miring:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
canoni5634975068530333245id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__cancel__comm__monoid__add,type,
ordere8940638589300402666id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ostrict__ordered__comm__monoid__add,type,
strict7427464778891057005id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Quickcheck__Exhaustive_Ofull__exhaustive,type,
quickc3360725361186068524ustive:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__comm__semiring__strict,type,
linord2810124833399127020strict:
!>[A: $tType] : $o ).
thf(sy_cl_Bit__Operations_Osemiring__bit__operations,type,
bit_se359711467146920520ations:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__semigroup__add__imp__le,type,
ordere2412721322843649153imp_le:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__cancel__ab__semigroup__add,type,
ordere580206878836729694up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__cancel__comm__monoid__diff,type,
ordere1170586879665033532d_diff:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ostrict__ordered__ab__semigroup__add,type,
strict9044650504122735259up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemiring__no__zero__divisors__cancel,type,
semiri6575147826004484403cancel:
!>[A: $tType] : $o ).
thf(sy_cl_Euclidean__Division_Oeuclidean__ring__cancel,type,
euclid8851590272496341667cancel:
!>[A: $tType] : $o ).
thf(sy_cl_Divides_Ounique__euclidean__semiring__numeral,type,
unique1627219031080169319umeral:
!>[A: $tType] : $o ).
thf(sy_cl_Complete__Lattices_Ocomplete__boolean__algebra,type,
comple489889107523837845lgebra:
!>[A: $tType] : $o ).
thf(sy_cl_Complete__Lattices_Ocomplete__distrib__lattice,type,
comple592849572758109894attice:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Onormalization__semidom__multiplicative,type,
normal6328177297339901930cative:
!>[A: $tType] : $o ).
thf(sy_cl_Euclidean__Division_Oeuclidean__semiring__cancel,type,
euclid4440199948858584721cancel:
!>[A: $tType] : $o ).
thf(sy_cl_Euclidean__Division_Ounique__euclidean__semiring,type,
euclid3128863361964157862miring:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__semigroup__monoid__add__imp__le,type,
ordere1937475149494474687imp_le:
!>[A: $tType] : $o ).
thf(sy_cl_Euclidean__Division_Ounique__euclidean__ring__with__nat,type,
euclid8789492081693882211th_nat:
!>[A: $tType] : $o ).
thf(sy_cl_Euclidean__Division_Ounique__euclidean__semiring__with__nat,type,
euclid5411537665997757685th_nat:
!>[A: $tType] : $o ).
thf(sy_cl_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,type,
semiri1453513574482234551roduct:
!>[A: $tType] : $o ).
thf(sy_cl_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations,type,
bit_un5681908812861735899ations:
!>[A: $tType] : $o ).
thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__lattice,type,
condit1219197933456340205attice:
!>[A: $tType] : $o ).
thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__linorder,type,
condit6923001295902523014norder:
!>[A: $tType] : $o ).
thf(sy_c_Archimedean__Field_Oceiling,type,
archimedean_ceiling:
!>[A: $tType] : ( A > int ) ).
thf(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor,type,
archim6421214686448440834_floor:
!>[A: $tType] : ( A > int ) ).
thf(sy_c_Archimedean__Field_Ofrac,type,
archimedean_frac:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Archimedean__Field_Oround,type,
archimedean_round:
!>[A: $tType] : ( A > int ) ).
thf(sy_c_BNF__Cardinal__Arithmetic_OCsum,type,
bNF_Cardinal_Csum:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( A > ( set @ ( product_prod @ B @ B ) ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) ) ).
thf(sy_c_BNF__Cardinal__Arithmetic_Ocexp,type,
bNF_Cardinal_cexp:
!>[B: $tType,A: $tType] : ( ( set @ ( product_prod @ B @ B ) ) > ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( A > B ) @ ( A > B ) ) ) ) ).
thf(sy_c_BNF__Cardinal__Arithmetic_Ocfinite,type,
bNF_Cardinal_cfinite:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_BNF__Cardinal__Arithmetic_Ocinfinite,type,
bNF_Ca4139267488887388095finite:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_BNF__Cardinal__Arithmetic_Ocone,type,
bNF_Cardinal_cone: set @ ( product_prod @ product_unit @ product_unit ) ).
thf(sy_c_BNF__Cardinal__Arithmetic_Ocprod,type,
bNF_Cardinal_cprod:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ B @ B ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) ) ).
thf(sy_c_BNF__Cardinal__Arithmetic_Ocsum,type,
bNF_Cardinal_csum:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ B @ B ) ) > ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) ) ).
thf(sy_c_BNF__Cardinal__Arithmetic_Octwo,type,
bNF_Cardinal_ctwo: set @ ( product_prod @ $o @ $o ) ).
thf(sy_c_BNF__Cardinal__Arithmetic_Oczero,type,
bNF_Cardinal_czero:
!>[A: $tType] : ( set @ ( product_prod @ A @ A ) ) ).
thf(sy_c_BNF__Cardinal__Order__Relation_OcardSuc,type,
bNF_Ca8387033319878233205ardSuc:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) ).
thf(sy_c_BNF__Cardinal__Order__Relation_Ocard__of,type,
bNF_Ca6860139660246222851ard_of:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_BNF__Cardinal__Order__Relation_Ocard__order__on,type,
bNF_Ca8970107618336181345der_on:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_BNF__Cardinal__Order__Relation_Ocofinal,type,
bNF_Ca7293521722713021262ofinal:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_BNF__Cardinal__Order__Relation_OisCardSuc,type,
bNF_Ca6246979054910435723ardSuc:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) > $o ) ).
thf(sy_c_BNF__Cardinal__Order__Relation_OnatLeq,type,
bNF_Ca8665028551170535155natLeq: set @ ( product_prod @ nat @ nat ) ).
thf(sy_c_BNF__Cardinal__Order__Relation_OnatLess,type,
bNF_Ca8459412986667044542atLess: set @ ( product_prod @ nat @ nat ) ).
thf(sy_c_BNF__Cardinal__Order__Relation_OregularCard,type,
bNF_Ca7133664381575040944arCard:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain,type,
bNF_Ca3754400796208372196lChain:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( A > B ) > $o ) ).
thf(sy_c_BNF__Composition_Oid__bnf,type,
bNF_id_bnf:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_BNF__Def_OGr,type,
bNF_Gr:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B ) > ( set @ ( product_prod @ A @ B ) ) ) ).
thf(sy_c_BNF__Def_OGrp,type,
bNF_Grp:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B ) > A > B > $o ) ).
thf(sy_c_BNF__Def_Ocollect,type,
bNF_collect:
!>[B: $tType,A: $tType] : ( ( set @ ( B > ( set @ A ) ) ) > B > ( set @ A ) ) ).
thf(sy_c_BNF__Def_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_OfstOp,type,
bNF_fstOp:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > B > $o ) > ( B > C > $o ) > ( product_prod @ A @ C ) > ( product_prod @ A @ B ) ) ).
thf(sy_c_BNF__Def_Opick__middlep,type,
bNF_pick_middlep:
!>[B: $tType,A: $tType,C: $tType] : ( ( B > A > $o ) > ( A > C > $o ) > B > C > A ) ).
thf(sy_c_BNF__Def_Orel__fun,type,
bNF_rel_fun:
!>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( A > C > $o ) > ( B > D > $o ) > ( A > B ) > ( C > D ) > $o ) ).
thf(sy_c_BNF__Def_Orel__set,type,
bNF_rel_set:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > ( set @ A ) > ( set @ B ) > $o ) ).
thf(sy_c_BNF__Def_Orel__sum,type,
bNF_rel_sum:
!>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( A > C > $o ) > ( B > D > $o ) > ( sum_sum @ A @ B ) > ( sum_sum @ C @ D ) > $o ) ).
thf(sy_c_BNF__Def_OsndOp,type,
bNF_sndOp:
!>[C: $tType,A: $tType,B: $tType] : ( ( C > A > $o ) > ( A > B > $o ) > ( product_prod @ C @ B ) > ( product_prod @ A @ B ) ) ).
thf(sy_c_BNF__Def_Ovimage2p,type,
bNF_vimage2p:
!>[A: $tType,D: $tType,B: $tType,E: $tType,C: $tType] : ( ( A > D ) > ( B > E ) > ( D > E > C ) > A > B > C ) ).
thf(sy_c_BNF__Greatest__Fixpoint_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_OfromCard,type,
bNF_Gr5436034075474128252omCard:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ B @ B ) ) > B > A ) ).
thf(sy_c_BNF__Greatest__Fixpoint_Oimage2,type,
bNF_Greatest_image2:
!>[C: $tType,A: $tType,B: $tType] : ( ( set @ C ) > ( C > A ) > ( C > B ) > ( set @ ( product_prod @ A @ B ) ) ) ).
thf(sy_c_BNF__Greatest__Fixpoint_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_OtoCard,type,
bNF_Greatest_toCard:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ B @ B ) ) > A > B ) ).
thf(sy_c_BNF__Greatest__Fixpoint_OtoCard__pred,type,
bNF_Gr1419584066657907630d_pred:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ B @ B ) ) > ( A > B ) > $o ) ).
thf(sy_c_BNF__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_Obsqr,type,
bNF_Wellorder_bsqr:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( product_prod @ A @ A ) @ ( product_prod @ A @ A ) ) ) ) ).
thf(sy_c_BNF__Wellorder__Constructions_Ocurr,type,
bNF_Wellorder_curr:
!>[A: $tType,B: $tType,C: $tType] : ( ( set @ A ) > ( ( product_prod @ A @ B ) > C ) > A > B > C ) ).
thf(sy_c_BNF__Wellorder__Constructions_Odir__image,type,
bNF_We2720479622203943262_image:
!>[A: $tType,A2: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( A > A2 ) > ( set @ ( product_prod @ A2 @ A2 ) ) ) ).
thf(sy_c_BNF__Wellorder__Constructions_OofilterIncl,type,
bNF_We413866401316099525erIncl:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) ).
thf(sy_c_BNF__Wellorder__Constructions_OordIso,type,
bNF_Wellorder_ordIso:
!>[A: $tType,A2: $tType] : ( set @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A2 @ A2 ) ) ) ) ).
thf(sy_c_BNF__Wellorder__Constructions_OordLeq,type,
bNF_Wellorder_ordLeq:
!>[A: $tType,A2: $tType] : ( set @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A2 @ A2 ) ) ) ) ).
thf(sy_c_BNF__Wellorder__Constructions_OordLess,type,
bNF_We4044943003108391690rdLess:
!>[A: $tType,A2: $tType] : ( set @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A2 @ A2 ) ) ) ) ).
thf(sy_c_BNF__Wellorder__Constructions_Oord__to__filter,type,
bNF_We8469521843155493636filter:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) ) ).
thf(sy_c_BNF__Wellorder__Embedding_Ocompat,type,
bNF_Wellorder_compat:
!>[A: $tType,A2: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ A2 @ A2 ) ) > ( A > A2 ) > $o ) ).
thf(sy_c_BNF__Wellorder__Embedding_Oembed,type,
bNF_Wellorder_embed:
!>[A: $tType,A2: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ A2 @ A2 ) ) > ( A > A2 ) > $o ) ).
thf(sy_c_BNF__Wellorder__Embedding_OembedS,type,
bNF_Wellorder_embedS:
!>[A: $tType,A2: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ A2 @ A2 ) ) > ( A > A2 ) > $o ) ).
thf(sy_c_BNF__Wellorder__Embedding_Oiso,type,
bNF_Wellorder_iso:
!>[A: $tType,A2: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ A2 @ A2 ) ) > ( A > A2 ) > $o ) ).
thf(sy_c_BNF__Wellorder__Relation_Owo__rel,type,
bNF_Wellorder_wo_rel:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_BNF__Wellorder__Relation_Owo__rel_OisMinim,type,
bNF_We4791949203932849705sMinim:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) > A > $o ) ).
thf(sy_c_BNF__Wellorder__Relation_Owo__rel_Omax2,type,
bNF_We1388413361240627857o_max2:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > A > A > A ) ).
thf(sy_c_BNF__Wellorder__Relation_Owo__rel_Ominim,type,
bNF_We6954850376910717587_minim:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) > A ) ).
thf(sy_c_BNF__Wellorder__Relation_Owo__rel_Osuc,type,
bNF_Wellorder_wo_suc:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) > A ) ).
thf(sy_c_Basic__BNF__LFPs_Octor__rec,type,
basic_BNF_ctor_rec:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Basic__BNF__LFPs_Oprod_Osize__prod,type,
basic_BNF_size_prod:
!>[A: $tType,B: $tType] : ( ( A > nat ) > ( B > nat ) > ( product_prod @ A @ B ) > nat ) ).
thf(sy_c_Basic__BNF__LFPs_Osum_Osize__sum,type,
basic_BNF_size_sum:
!>[A: $tType,B: $tType] : ( ( A > nat ) > ( B > nat ) > ( sum_sum @ A @ B ) > nat ) ).
thf(sy_c_Basic__BNF__LFPs_Oxtor,type,
basic_BNF_xtor:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Basic__BNFs_Ofsts,type,
basic_fsts:
!>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > ( set @ A ) ) ).
thf(sy_c_Basic__BNFs_Ofstsp,type,
basic_fstsp:
!>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > A > $o ) ).
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_Opred__sum,type,
basic_pred_sum:
!>[A: $tType,B: $tType] : ( ( A > $o ) > ( B > $o ) > ( sum_sum @ A @ B ) > $o ) ).
thf(sy_c_Basic__BNFs_Orel__prod,type,
basic_rel_prod:
!>[A: $tType,B: $tType,C: $tType,D: $tType] : ( ( A > B > $o ) > ( C > D > $o ) > ( product_prod @ A @ C ) > ( product_prod @ B @ D ) > $o ) ).
thf(sy_c_Basic__BNFs_Osetl,type,
basic_setl:
!>[A: $tType,B: $tType] : ( ( sum_sum @ A @ B ) > ( set @ A ) ) ).
thf(sy_c_Basic__BNFs_Osetlp,type,
basic_setlp:
!>[A: $tType,B: $tType] : ( ( sum_sum @ A @ B ) > A > $o ) ).
thf(sy_c_Basic__BNFs_Osetr,type,
basic_setr:
!>[A: $tType,B: $tType] : ( ( sum_sum @ A @ B ) > ( set @ B ) ) ).
thf(sy_c_Basic__BNFs_Osetrp,type,
basic_setrp:
!>[A: $tType,B: $tType] : ( ( sum_sum @ A @ B ) > B > $o ) ).
thf(sy_c_Basic__BNFs_Osnds,type,
basic_snds:
!>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > ( set @ B ) ) ).
thf(sy_c_Basic__BNFs_Osndsp,type,
basic_sndsp:
!>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > B > $o ) ).
thf(sy_c_Binomial_Obinomial,type,
binomial: nat > nat > nat ).
thf(sy_c_Binomial_Ogbinomial,type,
gbinomial:
!>[A: $tType] : ( A > nat > A ) ).
thf(sy_c_Bit__Operations_Oand__int__rel,type,
bit_and_int_rel: ( product_prod @ int @ int ) > ( product_prod @ int @ int ) > $o ).
thf(sy_c_Bit__Operations_Oand__not__num,type,
bit_and_not_num: num > num > ( option @ 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_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_Obit__cut__integer,type,
code_bit_cut_integer: code_integer > ( product_prod @ code_integer @ $o ) ).
thf(sy_c_Code__Numeral_Ocr__integer,type,
code_cr_integer: int > code_integer > $o ).
thf(sy_c_Code__Numeral_Ocr__natural,type,
code_cr_natural: nat > code_natural > $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__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_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_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__class_Obdd__above,type,
condit941137186595557371_above:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Conditionally__Complete__Lattices_Opreorder__class_Obdd__below,type,
condit1013018076250108175_below:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Conditionally__Complete__Lattices_Opreordering__bdd,type,
condit622319405099724424ng_bdd:
!>[A: $tType] : ( ( A > A > $o ) > ( A > A > $o ) > $o ) ).
thf(sy_c_Conditionally__Complete__Lattices_Opreordering__bdd_Obdd,type,
condit16957441358409770ng_bdd:
!>[A: $tType] : ( ( A > A > $o ) > ( set @ A ) > $o ) ).
thf(sy_c_Countable_Ofrom__nat,type,
from_nat:
!>[A: $tType] : ( nat > A ) ).
thf(sy_c_Countable_Onat__to__rat__surj,type,
nat_to_rat_surj: nat > rat ).
thf(sy_c_Countable_Onth__item,type,
nth_item:
!>[A: $tType] : ( nat > ( set @ ( old_node @ A @ product_unit ) ) ) ).
thf(sy_c_Countable_Onth__item__rel,type,
nth_item_rel: nat > nat > $o ).
thf(sy_c_Countable_Oto__nat,type,
to_nat:
!>[A: $tType] : ( A > nat ) ).
thf(sy_c_Divides_Oadjust__div,type,
adjust_div: ( product_prod @ int @ int ) > int ).
thf(sy_c_Divides_Odivmod__nat,type,
divmod_nat: nat > nat > ( product_prod @ nat @ nat ) ).
thf(sy_c_Divides_Oeucl__rel__int,type,
eucl_rel_int: int > int > ( product_prod @ int @ int ) > $o ).
thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivides__aux,type,
unique5940410009612947441es_aux:
!>[A: $tType] : ( ( product_prod @ A @ A ) > $o ) ).
thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod,type,
unique8689654367752047608divmod:
!>[A: $tType] : ( num > num > ( product_prod @ A @ A ) ) ).
thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod__step,type,
unique1321980374590559556d_step:
!>[A: $tType] : ( num > ( product_prod @ A @ A ) > ( product_prod @ A @ A ) ) ).
thf(sy_c_Equiv__Relations_Ocongruent,type,
equiv_congruent:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( A > B ) > $o ) ).
thf(sy_c_Equiv__Relations_Ocongruent2,type,
equiv_congruent2:
!>[A: $tType,B: $tType,C: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ B @ B ) ) > ( A > B > C ) > $o ) ).
thf(sy_c_Equiv__Relations_Oequiv,type,
equiv_equiv:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Equiv__Relations_Oequivp,type,
equiv_equivp:
!>[A: $tType] : ( ( A > A > $o ) > $o ) ).
thf(sy_c_Equiv__Relations_Opart__equivp,type,
equiv_part_equivp:
!>[A: $tType] : ( ( A > A > $o ) > $o ) ).
thf(sy_c_Equiv__Relations_Oproj,type,
equiv_proj:
!>[B: $tType,A: $tType] : ( ( set @ ( product_prod @ B @ A ) ) > B > ( set @ A ) ) ).
thf(sy_c_Equiv__Relations_Oquotient,type,
equiv_quotient:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > ( set @ ( set @ A ) ) ) ).
thf(sy_c_Euclidean__Division_Oeuclidean__semiring__class_Oeuclidean__size,type,
euclid6346220572633701492n_size:
!>[A: $tType] : ( A > nat ) ).
thf(sy_c_Euclidean__Division_Ounique__euclidean__semiring__class_Odivision__segment,type,
euclid7384307370059645450egment:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer,type,
comm_s3205402744901411588hammer:
!>[A: $tType] : ( A > nat > A ) ).
thf(sy_c_Factorial_Osemiring__char__0__class_Ofact,type,
semiring_char_0_fact:
!>[A: $tType] : ( nat > A ) ).
thf(sy_c_Fields_Oinverse__class_Oinverse,type,
inverse_inverse:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Filter_Oat__bot,type,
at_bot:
!>[A: $tType] : ( filter @ A ) ).
thf(sy_c_Filter_Oat__top,type,
at_top:
!>[A: $tType] : ( filter @ A ) ).
thf(sy_c_Filter_Ocofinite,type,
cofinite:
!>[A: $tType] : ( filter @ A ) ).
thf(sy_c_Filter_Oeventually,type,
eventually:
!>[A: $tType] : ( ( A > $o ) > ( filter @ A ) > $o ) ).
thf(sy_c_Filter_Ofilter_OAbs__filter,type,
abs_filter:
!>[A: $tType] : ( ( ( A > $o ) > $o ) > ( filter @ A ) ) ).
thf(sy_c_Filter_Ofiltercomap,type,
filtercomap:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( filter @ B ) > ( filter @ A ) ) ).
thf(sy_c_Filter_Ofilterlim,type,
filterlim:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( filter @ B ) > ( filter @ A ) > $o ) ).
thf(sy_c_Filter_Ofiltermap,type,
filtermap:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( filter @ A ) > ( filter @ B ) ) ).
thf(sy_c_Filter_Ofinite__subsets__at__top,type,
finite5375528669736107172at_top:
!>[A: $tType] : ( ( set @ A ) > ( filter @ ( set @ A ) ) ) ).
thf(sy_c_Filter_Ofrequently,type,
frequently:
!>[A: $tType] : ( ( A > $o ) > ( filter @ A ) > $o ) ).
thf(sy_c_Filter_Omap__filter__on,type,
map_filter_on:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B ) > ( filter @ A ) > ( filter @ B ) ) ).
thf(sy_c_Filter_Oprincipal,type,
principal:
!>[A: $tType] : ( ( set @ A ) > ( filter @ A ) ) ).
thf(sy_c_Filter_Oprod__filter,type,
prod_filter:
!>[A: $tType,B: $tType] : ( ( filter @ A ) > ( filter @ B ) > ( filter @ ( product_prod @ A @ B ) ) ) ).
thf(sy_c_Filter_Orel__filter,type,
rel_filter:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > ( filter @ A ) > ( filter @ B ) > $o ) ).
thf(sy_c_Finite__Set_OFpow,type,
finite_Fpow:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( set @ A ) ) ) ).
thf(sy_c_Finite__Set_Ocard,type,
finite_card:
!>[B: $tType] : ( ( set @ B ) > nat ) ).
thf(sy_c_Finite__Set_Ocomp__fun__commute,type,
finite6289374366891150609ommute:
!>[A: $tType,B: $tType] : ( ( A > B > B ) > $o ) ).
thf(sy_c_Finite__Set_Ocomp__fun__commute__on,type,
finite4664212375090638736ute_on:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B > B ) > $o ) ).
thf(sy_c_Finite__Set_Ocomp__fun__idem__on,type,
finite673082921795544331dem_on:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B > B ) > $o ) ).
thf(sy_c_Finite__Set_Ofinite,type,
finite_finite2:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Finite__Set_Ofold,type,
finite_fold:
!>[A: $tType,B: $tType] : ( ( A > B > B ) > B > ( set @ A ) > B ) ).
thf(sy_c_Finite__Set_Ofold__graph,type,
finite_fold_graph:
!>[A: $tType,B: $tType] : ( ( A > B > B ) > B > ( set @ A ) > B > $o ) ).
thf(sy_c_Finite__Set_Ofolding__on,type,
finite_folding_on:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B > B ) > $o ) ).
thf(sy_c_Finite__Set_Ofolding__on_OF,type,
finite_folding_F:
!>[A: $tType,B: $tType] : ( ( A > B > B ) > B > ( set @ A ) > B ) ).
thf(sy_c_Fun_Obij__betw,type,
bij_betw:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) > $o ) ).
thf(sy_c_Fun_Ocomp,type,
comp:
!>[B: $tType,C: $tType,A: $tType] : ( ( B > C ) > ( A > B ) > A > C ) ).
thf(sy_c_Fun_Ofun__upd,type,
fun_upd:
!>[A: $tType,B: $tType] : ( ( A > B ) > A > B > A > B ) ).
thf(sy_c_Fun_Oid,type,
id:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Fun_Oinj__on,type,
inj_on:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > $o ) ).
thf(sy_c_Fun_Omap__fun,type,
map_fun:
!>[C: $tType,A: $tType,B: $tType,D: $tType] : ( ( C > A ) > ( B > D ) > ( A > B ) > C > D ) ).
thf(sy_c_Fun_Ooverride__on,type,
override_on:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( A > B ) > ( set @ A ) > A > B ) ).
thf(sy_c_Fun_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_Omax__strict,type,
fun_max_strict: set @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) ).
thf(sy_c_Fun__Def_Omax__weak,type,
fun_max_weak: set @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) ).
thf(sy_c_Fun__Def_Omin__strict,type,
fun_min_strict: set @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) ).
thf(sy_c_Fun__Def_Omin__weak,type,
fun_min_weak: set @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) ).
thf(sy_c_Fun__Def_Opair__leq,type,
fun_pair_leq: set @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) ).
thf(sy_c_Fun__Def_Opair__less,type,
fun_pair_less: set @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) ).
thf(sy_c_Fun__Def_Oreduction__pair,type,
fun_reduction_pair:
!>[A: $tType] : ( ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) > $o ) ).
thf(sy_c_Fun__Def_Orp__inv__image,type,
fun_rp_inv_image:
!>[A: $tType,B: $tType] : ( ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) > ( B > A ) > ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) ) ) ).
thf(sy_c_GCD_OGcd__class_OGcd,type,
gcd_Gcd:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_GCD_OGcd__class_OLcm,type,
gcd_Lcm:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_GCD_Obezw,type,
bezw: nat > nat > ( product_prod @ int @ int ) ).
thf(sy_c_GCD_Obezw__rel,type,
bezw_rel: ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) > $o ).
thf(sy_c_GCD_Obounded__quasi__semilattice,type,
bounde8507323023520639062attice:
!>[A: $tType] : ( ( A > A > A ) > A > A > ( A > A ) > $o ) ).
thf(sy_c_GCD_Obounded__quasi__semilattice__axioms,type,
bounde5338082073960204281axioms:
!>[A: $tType] : ( ( A > A > A ) > A > A > ( A > A ) > $o ) ).
thf(sy_c_GCD_Obounded__quasi__semilattice__set,type,
bounde6485984586167503788ce_set:
!>[A: $tType] : ( ( A > A > A ) > A > A > ( A > A ) > $o ) ).
thf(sy_c_GCD_Obounded__quasi__semilattice__set_OF,type,
bounde2362111253966948842tice_F:
!>[A: $tType] : ( ( A > A > A ) > A > A > ( set @ A ) > A ) ).
thf(sy_c_GCD_Ogcd__class_Ogcd,type,
gcd_gcd:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_GCD_Ogcd__class_Olcm,type,
gcd_lcm:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_GCD_Ogcd__nat__rel,type,
gcd_nat_rel: ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) > $o ).
thf(sy_c_GCD_Osemiring__1__class_Osemiring__char,type,
semiri4206861660011772517g_char:
!>[A: $tType] : ( ( itself @ A ) > nat ) ).
thf(sy_c_GCD_Osemiring__gcd__class_OGcd__fin,type,
semiring_gcd_Gcd_fin:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_GCD_Osemiring__gcd__class_OLcm__fin,type,
semiring_gcd_Lcm_fin:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_Groups_Oabel__semigroup,type,
abel_semigroup:
!>[A: $tType] : ( ( A > A > A ) > $o ) ).
thf(sy_c_Groups_Oabel__semigroup__axioms,type,
abel_s757365448890700780axioms:
!>[A: $tType] : ( ( A > A > A ) > $o ) ).
thf(sy_c_Groups_Oabs__class_Oabs,type,
abs_abs:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Groups_Ocomm__monoid,type,
comm_monoid:
!>[A: $tType] : ( ( A > A > A ) > A > $o ) ).
thf(sy_c_Groups_Ocomm__monoid__axioms,type,
comm_monoid_axioms:
!>[A: $tType] : ( ( A > A > A ) > A > $o ) ).
thf(sy_c_Groups_Ogroup,type,
group:
!>[A: $tType] : ( ( A > A > A ) > A > ( A > A ) > $o ) ).
thf(sy_c_Groups_Ogroup__axioms,type,
group_axioms:
!>[A: $tType] : ( ( A > A > A ) > A > ( A > A ) > $o ) ).
thf(sy_c_Groups_Ominus__class_Ominus,type,
minus_minus:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Groups_Omonoid,type,
monoid:
!>[A: $tType] : ( ( A > A > A ) > A > $o ) ).
thf(sy_c_Groups_Omonoid__axioms,type,
monoid_axioms:
!>[A: $tType] : ( ( A > A > A ) > A > $o ) ).
thf(sy_c_Groups_Oone__class_Oone,type,
one_one:
!>[A: $tType] : A ).
thf(sy_c_Groups_Oplus__class_Oplus,type,
plus_plus:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Groups_Osemigroup,type,
semigroup:
!>[A: $tType] : ( ( A > A > A ) > $o ) ).
thf(sy_c_Groups_Osgn__class_Osgn,type,
sgn_sgn:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Groups_Otimes__class_Otimes,type,
times_times:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Groups_Ouminus__class_Ouminus,type,
uminus_uminus:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Groups_Ozero__class_Ozero,type,
zero_zero:
!>[A: $tType] : A ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum,type,
groups7311177749621191930dd_sum:
!>[B: $tType,A: $tType] : ( ( B > A ) > ( set @ B ) > A ) ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_H,type,
groups1027152243600224163dd_sum:
!>[C: $tType,A: $tType] : ( ( C > A ) > ( set @ C ) > A ) ).
thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod,type,
groups7121269368397514597t_prod:
!>[B: $tType,A: $tType] : ( ( B > A ) > ( set @ B ) > A ) ).
thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_H,type,
groups1962203154675924110t_prod:
!>[C: $tType,A: $tType] : ( ( C > A ) > ( set @ C ) > A ) ).
thf(sy_c_Groups__Big_Ocomm__monoid__set,type,
groups778175481326437816id_set:
!>[A: $tType] : ( ( A > A > A ) > A > $o ) ).
thf(sy_c_Groups__Big_Ocomm__monoid__set_OF,type,
groups_comm_monoid_F:
!>[A: $tType,B: $tType] : ( ( A > A > A ) > A > ( B > A ) > ( set @ B ) > A ) ).
thf(sy_c_Groups__Big_Ocomm__monoid__set_OG,type,
groups_comm_monoid_G:
!>[A: $tType,B: $tType] : ( ( A > A > A ) > A > ( B > A ) > ( set @ B ) > A ) ).
thf(sy_c_Groups__List_Ocomm__monoid__list,type,
groups1828464146339083142d_list:
!>[A: $tType] : ( ( A > A > A ) > A > $o ) ).
thf(sy_c_Groups__List_Ocomm__monoid__list__set,type,
groups4802862169904069756st_set:
!>[A: $tType] : ( ( A > A > A ) > A > $o ) ).
thf(sy_c_Groups__List_Ocomm__semiring__0__class_Ohorner__sum,type,
groups4207007520872428315er_sum:
!>[B: $tType,A: $tType] : ( ( B > A ) > A > ( list @ B ) > A ) ).
thf(sy_c_Groups__List_Omonoid__add__class_Osum__list,type,
groups8242544230860333062m_list:
!>[A: $tType] : ( ( list @ A ) > A ) ).
thf(sy_c_Groups__List_Omonoid__list,type,
groups_monoid_list:
!>[A: $tType] : ( ( A > A > A ) > A > $o ) ).
thf(sy_c_Groups__List_Omonoid__list_OF,type,
groups_monoid_F:
!>[A: $tType] : ( ( A > A > A ) > A > ( list @ A ) > A ) ).
thf(sy_c_Groups__List_Omonoid__mult__class_Oprod__list,type,
groups5270119922927024881d_list:
!>[A: $tType] : ( ( list @ A ) > A ) ).
thf(sy_c_HOL_OEx1,type,
ex1:
!>[A: $tType] : ( ( A > $o ) > $o ) ).
thf(sy_c_HOL_ONO__MATCH,type,
nO_MATCH:
!>[A: $tType,B: $tType] : ( A > B > $o ) ).
thf(sy_c_HOL_OThe,type,
the:
!>[A: $tType] : ( ( A > $o ) > A ) ).
thf(sy_c_HOL_OUniq,type,
uniq:
!>[A: $tType] : ( ( A > $o ) > $o ) ).
thf(sy_c_HOL_Odefault__class_Odefault,type,
default_default:
!>[A: $tType] : A ).
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_Ocase__Heap,type,
heap_Time_case_Heap:
!>[A: $tType,B: $tType] : ( ( ( ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) > B ) > ( heap_Time_Heap @ A ) > B ) ).
thf(sy_c_Heap__Time__Monad_OHeap_Omap__Heap,type,
heap_Time_map_Heap:
!>[A: $tType,Aa: $tType] : ( ( A > Aa ) > ( heap_Time_Heap @ A ) > ( heap_Time_Heap @ Aa ) ) ).
thf(sy_c_Heap__Time__Monad_OHeap_Opred__Heap,type,
heap_Time_pred_Heap:
!>[A: $tType] : ( ( A > $o ) > ( heap_Time_Heap @ A ) > $o ) ).
thf(sy_c_Heap__Time__Monad_OHeap_Orec__Heap,type,
heap_Time_rec_Heap:
!>[A: $tType,C: $tType] : ( ( ( ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) > C ) > ( heap_Time_Heap @ A ) > C ) ).
thf(sy_c_Heap__Time__Monad_OHeap_Orel__Heap,type,
heap_Time_rel_Heap:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > ( heap_Time_Heap @ A ) > ( heap_Time_Heap @ B ) > $o ) ).
thf(sy_c_Heap__Time__Monad_OHeap_Oset__Heap,type,
heap_Time_set_Heap:
!>[A: $tType] : ( ( heap_Time_Heap @ A ) > ( set @ 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_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_Hilbert__Choice_Obijection,type,
hilbert_bijection:
!>[A: $tType] : ( ( A > A ) > $o ) ).
thf(sy_c_Hilbert__Choice_Oinv__into,type,
hilbert_inv_into:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B ) > B > A ) ).
thf(sy_c_If,type,
if:
!>[A: $tType] : ( $o > A > A > A ) ).
thf(sy_c_Inductive_Ocomplete__lattice__class_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_Int_OAbs__Integ,type,
abs_Integ: ( product_prod @ nat @ nat ) > int ).
thf(sy_c_Int_ONeg,type,
neg: num > int ).
thf(sy_c_Int_OPos,type,
pos: num > int ).
thf(sy_c_Int_ORep__Integ,type,
rep_Integ: int > ( product_prod @ nat @ nat ) ).
thf(sy_c_Int_Ocr__int,type,
cr_int: ( product_prod @ nat @ nat ) > int > $o ).
thf(sy_c_Int_Odup,type,
dup: int > int ).
thf(sy_c_Int_Oint_OAbs__int,type,
abs_int: ( set @ ( product_prod @ nat @ nat ) ) > int ).
thf(sy_c_Int_Oint_ORep__int,type,
rep_int: int > ( set @ ( product_prod @ nat @ nat ) ) ).
thf(sy_c_Int_Oint__ge__less__than,type,
int_ge_less_than: int > ( set @ ( product_prod @ int @ int ) ) ).
thf(sy_c_Int_Oint__ge__less__than2,type,
int_ge_less_than2: int > ( set @ ( product_prod @ int @ int ) ) ).
thf(sy_c_Int_Ointrel,type,
intrel: ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) > $o ).
thf(sy_c_Int_Onat,type,
nat2: int > nat ).
thf(sy_c_Int_Opcr__int,type,
pcr_int: ( product_prod @ nat @ nat ) > int > $o ).
thf(sy_c_Int_Opower__int,type,
power_int:
!>[A: $tType] : ( A > int > A ) ).
thf(sy_c_Int_Oring__1__class_OInts,type,
ring_1_Ints:
!>[A: $tType] : ( set @ A ) ).
thf(sy_c_Int_Oring__1__class_Oof__int,type,
ring_1_of_int:
!>[A: $tType] : ( int > A ) ).
thf(sy_c_Int_Osub,type,
sub: num > num > int ).
thf(sy_c_Lattices_Oinf__class_Oinf,type,
inf_inf:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Lattices_Osemilattice__neutr,type,
semilattice_neutr:
!>[A: $tType] : ( ( A > A > A ) > A > $o ) ).
thf(sy_c_Lattices_Osemilattice__neutr__order,type,
semila1105856199041335345_order:
!>[A: $tType] : ( ( A > A > A ) > A > ( A > A > $o ) > ( A > A > $o ) > $o ) ).
thf(sy_c_Lattices_Osemilattice__order,type,
semilattice_order:
!>[A: $tType] : ( ( A > A > A ) > ( A > A > $o ) > ( A > A > $o ) > $o ) ).
thf(sy_c_Lattices_Osup__class_Osup,type,
sup_sup:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Lattices__Big_Olinorder_OMax,type,
lattices_Max:
!>[A: $tType] : ( ( A > A > $o ) > ( set @ A ) > A ) ).
thf(sy_c_Lattices__Big_Olinorder_OMin,type,
lattices_Min:
!>[A: $tType] : ( ( A > A > $o ) > ( set @ A ) > A ) ).
thf(sy_c_Lattices__Big_Olinorder__class_OMax,type,
lattic643756798349783984er_Max:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_Lattices__Big_Olinorder__class_OMin,type,
lattic643756798350308766er_Min:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min,type,
lattices_ord_arg_min:
!>[B: $tType,A: $tType] : ( ( B > A ) > ( B > $o ) > B ) ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on,type,
lattic7623131987881927897min_on:
!>[B: $tType,A: $tType] : ( ( B > A ) > ( set @ B ) > B ) ).
thf(sy_c_Lattices__Big_Oord__class_Ois__arg__min,type,
lattic501386751177426532rg_min:
!>[B: $tType,A: $tType] : ( ( B > A ) > ( B > $o ) > B > $o ) ).
thf(sy_c_Lattices__Big_Osemilattice__inf__class_OInf__fin,type,
lattic7752659483105999362nf_fin:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_Lattices__Big_Osemilattice__neutr__set_OF,type,
lattic5214292709420241887eutr_F:
!>[A: $tType] : ( ( A > A > A ) > A > ( set @ A ) > A ) ).
thf(sy_c_Lattices__Big_Osemilattice__order__set,type,
lattic4895041142388067077er_set:
!>[A: $tType] : ( ( A > A > A ) > ( A > A > $o ) > ( A > A > $o ) > $o ) ).
thf(sy_c_Lattices__Big_Osemilattice__set,type,
lattic149705377957585745ce_set:
!>[A: $tType] : ( ( A > A > A ) > $o ) ).
thf(sy_c_Lattices__Big_Osemilattice__set_OF,type,
lattic1715443433743089157tice_F:
!>[A: $tType] : ( ( A > A > A ) > ( set @ A ) > A ) ).
thf(sy_c_Lattices__Big_Osemilattice__sup__class_OSup__fin,type,
lattic5882676163264333800up_fin:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_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_Oinsort__key,type,
insort_key:
!>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( B > A ) > B > ( list @ B ) > ( list @ B ) ) ).
thf(sy_c_List_Olinorder__class_Oinsort__insert__key,type,
linord329482645794927042rt_key:
!>[B: $tType,A: $tType] : ( ( B > A ) > B > ( list @ B ) > ( list @ B ) ) ).
thf(sy_c_List_Olinorder__class_Oinsort__key,type,
linorder_insort_key:
!>[B: $tType,A: $tType] : ( ( B > A ) > B > ( list @ B ) > ( list @ B ) ) ).
thf(sy_c_List_Olinorder__class_Osort__key,type,
linorder_sort_key:
!>[B: $tType,A: $tType] : ( ( B > A ) > ( list @ B ) > ( list @ B ) ) ).
thf(sy_c_List_Olinorder__class_Osorted__key__list__of__set,type,
linord144544945434240204of_set:
!>[B: $tType,A: $tType] : ( ( B > A ) > ( set @ B ) > ( list @ B ) ) ).
thf(sy_c_List_Olinorder__class_Osorted__list__of__set,type,
linord4507533701916653071of_set:
!>[A: $tType] : ( ( set @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Olinorder__class_Ostable__sort__key,type,
linord3483353639454293061rt_key:
!>[B: $tType,A: $tType] : ( ( ( B > A ) > ( list @ B ) > ( list @ B ) ) > $o ) ).
thf(sy_c_List_Olist_OCons,type,
cons:
!>[A: $tType] : ( A > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Olist_ONil,type,
nil:
!>[A: $tType] : ( list @ A ) ).
thf(sy_c_List_Olist_Ocase__list,type,
case_list:
!>[B: $tType,A: $tType] : ( B > ( A > ( list @ A ) > B ) > ( list @ A ) > B ) ).
thf(sy_c_List_Olist_Ohd,type,
hd:
!>[A: $tType] : ( ( list @ A ) > A ) ).
thf(sy_c_List_Olist_Olist__all,type,
list_all:
!>[A: $tType] : ( ( A > $o ) > ( list @ A ) > $o ) ).
thf(sy_c_List_Olist_Olist__all2,type,
list_all2:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > ( list @ A ) > ( list @ B ) > $o ) ).
thf(sy_c_List_Olist_Omap,type,
map:
!>[A: $tType,Aa: $tType] : ( ( A > Aa ) > ( list @ A ) > ( list @ Aa ) ) ).
thf(sy_c_List_Olist_Orec__list,type,
rec_list:
!>[C: $tType,A: $tType] : ( C > ( A > ( list @ A ) > C > C ) > ( list @ A ) > C ) ).
thf(sy_c_List_Olist_Oset,type,
set2:
!>[A: $tType] : ( ( list @ A ) > ( set @ A ) ) ).
thf(sy_c_List_Olist_Osize__list,type,
size_list:
!>[A: $tType] : ( ( A > nat ) > ( list @ A ) > nat ) ).
thf(sy_c_List_Olist_Otl,type,
tl:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Olist__ex1,type,
list_ex1:
!>[A: $tType] : ( ( A > $o ) > ( list @ A ) > $o ) ).
thf(sy_c_List_Olist__update,type,
list_update:
!>[A: $tType] : ( ( list @ A ) > nat > A > ( list @ A ) ) ).
thf(sy_c_List_Olistrel,type,
listrel:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) ) ).
thf(sy_c_List_Olistrel1,type,
listrel1:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ).
thf(sy_c_List_Olistrel1p,type,
listrel1p:
!>[A: $tType] : ( ( A > A > $o ) > ( list @ A ) > ( list @ A ) > $o ) ).
thf(sy_c_List_Olistrelp,type,
listrelp:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > ( list @ A ) > ( list @ B ) > $o ) ).
thf(sy_c_List_Olists,type,
lists:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( list @ A ) ) ) ).
thf(sy_c_List_Omap__filter,type,
map_filter:
!>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( list @ A ) > ( list @ B ) ) ).
thf(sy_c_List_Omap__project,type,
map_project:
!>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( set @ A ) > ( set @ B ) ) ).
thf(sy_c_List_Omap__tailrec__rev,type,
map_tailrec_rev:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( list @ A ) > ( list @ B ) > ( list @ B ) ) ).
thf(sy_c_List_Omap__tailrec__rev__rel,type,
map_tailrec_rev_rel:
!>[A: $tType,B: $tType] : ( ( product_prod @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) > ( product_prod @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) > $o ) ).
thf(sy_c_List_Omeasures,type,
measures:
!>[A: $tType] : ( ( list @ ( A > nat ) ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_List_Omin__list,type,
min_list:
!>[A: $tType] : ( ( list @ A ) > A ) ).
thf(sy_c_List_Omin__list__rel,type,
min_list_rel:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > $o ) ).
thf(sy_c_List_On__lists,type,
n_lists:
!>[A: $tType] : ( nat > ( list @ A ) > ( list @ ( list @ A ) ) ) ).
thf(sy_c_List_Onth,type,
nth:
!>[A: $tType] : ( ( list @ A ) > nat > A ) ).
thf(sy_c_List_Onths,type,
nths:
!>[A: $tType] : ( ( list @ A ) > ( set @ nat ) > ( list @ A ) ) ).
thf(sy_c_List_Oord_Olexordp,type,
lexordp2:
!>[A: $tType] : ( ( A > A > $o ) > ( list @ A ) > ( list @ A ) > $o ) ).
thf(sy_c_List_Oord_Olexordp__eq,type,
lexordp_eq:
!>[A: $tType] : ( ( A > A > $o ) > ( list @ A ) > ( list @ A ) > $o ) ).
thf(sy_c_List_Oord__class_Olexordp,type,
ord_lexordp:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > $o ) ).
thf(sy_c_List_Opartition,type,
partition:
!>[A: $tType] : ( ( A > $o ) > ( list @ A ) > ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ).
thf(sy_c_List_Oproduct,type,
product:
!>[A: $tType,B: $tType] : ( ( list @ A ) > ( list @ B ) > ( list @ ( product_prod @ A @ B ) ) ) ).
thf(sy_c_List_Oproduct__lists,type,
product_lists:
!>[A: $tType] : ( ( list @ ( list @ A ) ) > ( list @ ( list @ A ) ) ) ).
thf(sy_c_List_Oremdups,type,
remdups:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Oremdups__adj,type,
remdups_adj:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Oremove1,type,
remove1:
!>[A: $tType] : ( A > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Oreplicate,type,
replicate:
!>[A: $tType] : ( nat > A > ( list @ A ) ) ).
thf(sy_c_List_Orev,type,
rev:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Orotate,type,
rotate:
!>[A: $tType] : ( nat > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Orotate1,type,
rotate1:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Oset__Cons,type,
set_Cons:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( list @ A ) ) > ( set @ ( list @ A ) ) ) ).
thf(sy_c_List_Oshuffles,type,
shuffles:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( set @ ( list @ A ) ) ) ).
thf(sy_c_List_Oshuffles__rel,type,
shuffles_rel:
!>[A: $tType] : ( ( product_prod @ ( list @ A ) @ ( list @ A ) ) > ( product_prod @ ( list @ A ) @ ( list @ A ) ) > $o ) ).
thf(sy_c_List_Osorted__wrt,type,
sorted_wrt:
!>[A: $tType] : ( ( A > A > $o ) > ( list @ A ) > $o ) ).
thf(sy_c_List_Osorted__wrt__rel,type,
sorted_wrt_rel:
!>[A: $tType] : ( ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) > ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) > $o ) ).
thf(sy_c_List_Osplice,type,
splice:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Osplice__rel,type,
splice_rel:
!>[A: $tType] : ( ( product_prod @ ( list @ A ) @ ( list @ A ) ) > ( product_prod @ ( list @ A ) @ ( list @ A ) ) > $o ) ).
thf(sy_c_List_Osubseqs,type,
subseqs:
!>[A: $tType] : ( ( list @ A ) > ( list @ ( list @ A ) ) ) ).
thf(sy_c_List_Otake,type,
take:
!>[A: $tType] : ( nat > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_OtakeWhile,type,
takeWhile:
!>[A: $tType] : ( ( A > $o ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Othose,type,
those:
!>[A: $tType] : ( ( list @ ( option @ A ) ) > ( option @ ( list @ A ) ) ) ).
thf(sy_c_List_Otranspose,type,
transpose:
!>[A: $tType] : ( ( list @ ( list @ A ) ) > ( list @ ( list @ A ) ) ) ).
thf(sy_c_List_Otranspose__rel,type,
transpose_rel:
!>[A: $tType] : ( ( list @ ( list @ A ) ) > ( list @ ( list @ A ) ) > $o ) ).
thf(sy_c_List_Oupt,type,
upt: nat > nat > ( list @ nat ) ).
thf(sy_c_List_Oupto,type,
upto: int > 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_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,
pred2: 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_Oint__decode,type,
nat_int_decode: nat > int ).
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_Osum__decode,type,
nat_sum_decode: nat > ( sum_sum @ nat @ nat ) ).
thf(sy_c_Nat__Bijection_Osum__encode,type,
nat_sum_encode: ( sum_sum @ nat @ nat ) > nat ).
thf(sy_c_Nat__Bijection_Otriangle,type,
nat_triangle: nat > nat ).
thf(sy_c_Num_OBitM,type,
bitM: num > num ).
thf(sy_c_Num_Oinc,type,
inc: num > num ).
thf(sy_c_Num_Onat__of__num,type,
nat_of_num: num > nat ).
thf(sy_c_Num_Oneg__numeral__class_Odbl,type,
neg_numeral_dbl:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__dec,type,
neg_numeral_dbl_dec:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__inc,type,
neg_numeral_dbl_inc:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Num_Oneg__numeral__class_Ois__num,type,
neg_numeral_is_num:
!>[A: $tType] : ( A > $o ) ).
thf(sy_c_Num_Oneg__numeral__class_Osub,type,
neg_numeral_sub:
!>[A: $tType] : ( num > num > A ) ).
thf(sy_c_Num_Onum_OBit0,type,
bit0: num > num ).
thf(sy_c_Num_Onum_OBit1,type,
bit1: num > num ).
thf(sy_c_Num_Onum_OOne,type,
one2: num ).
thf(sy_c_Num_Onum_Ocase__num,type,
case_num:
!>[A: $tType] : ( A > ( num > A ) > ( num > 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_Old__Datatype_OAtom,type,
old_Atom:
!>[A: $tType,B: $tType] : ( ( sum_sum @ A @ nat ) > ( set @ ( old_node @ A @ B ) ) ) ).
thf(sy_c_Old__Datatype_OCase,type,
old_Case:
!>[A: $tType,B: $tType,C: $tType] : ( ( ( set @ ( old_node @ A @ B ) ) > C ) > ( ( set @ ( old_node @ A @ B ) ) > C ) > ( set @ ( old_node @ A @ B ) ) > C ) ).
thf(sy_c_Old__Datatype_OIn0,type,
old_In0:
!>[A: $tType,B: $tType] : ( ( set @ ( old_node @ A @ B ) ) > ( set @ ( old_node @ A @ B ) ) ) ).
thf(sy_c_Old__Datatype_OIn1,type,
old_In1:
!>[A: $tType,B: $tType] : ( ( set @ ( old_node @ A @ B ) ) > ( set @ ( old_node @ A @ B ) ) ) ).
thf(sy_c_Old__Datatype_OLeaf,type,
old_Leaf:
!>[A: $tType,B: $tType] : ( A > ( set @ ( old_node @ A @ B ) ) ) ).
thf(sy_c_Old__Datatype_OLim,type,
old_Lim:
!>[B: $tType,A: $tType] : ( ( B > ( set @ ( old_node @ A @ B ) ) ) > ( set @ ( old_node @ A @ B ) ) ) ).
thf(sy_c_Old__Datatype_ONode,type,
old_Node:
!>[B: $tType,A: $tType] : ( set @ ( product_prod @ ( nat > ( sum_sum @ B @ nat ) ) @ ( sum_sum @ A @ nat ) ) ) ).
thf(sy_c_Old__Datatype_ONumb,type,
old_Numb:
!>[A: $tType,B: $tType] : ( nat > ( set @ ( old_node @ A @ B ) ) ) ).
thf(sy_c_Old__Datatype_OPush,type,
old_Push:
!>[B: $tType] : ( ( sum_sum @ B @ nat ) > ( nat > ( sum_sum @ B @ nat ) ) > nat > ( sum_sum @ B @ nat ) ) ).
thf(sy_c_Old__Datatype_OPush__Node,type,
old_Push_Node:
!>[B: $tType,A: $tType] : ( ( sum_sum @ B @ nat ) > ( old_node @ A @ B ) > ( old_node @ A @ B ) ) ).
thf(sy_c_Old__Datatype_OScons,type,
old_Scons:
!>[A: $tType,B: $tType] : ( ( set @ ( old_node @ A @ B ) ) > ( set @ ( old_node @ A @ B ) ) > ( set @ ( old_node @ A @ B ) ) ) ).
thf(sy_c_Old__Datatype_OSplit,type,
old_Split:
!>[A: $tType,B: $tType,C: $tType] : ( ( ( set @ ( old_node @ A @ B ) ) > ( set @ ( old_node @ A @ B ) ) > C ) > ( set @ ( old_node @ A @ B ) ) > C ) ).
thf(sy_c_Old__Datatype_Odprod,type,
old_dprod:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) ) > ( set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) ) > ( set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) ) ) ).
thf(sy_c_Old__Datatype_Odsum,type,
old_dsum:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) ) > ( set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) ) > ( set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) ) ) ).
thf(sy_c_Old__Datatype_Ondepth,type,
old_ndepth:
!>[A: $tType,B: $tType] : ( ( old_node @ A @ B ) > nat ) ).
thf(sy_c_Old__Datatype_Onode_OAbs__Node,type,
old_Abs_Node:
!>[B: $tType,A: $tType] : ( ( product_prod @ ( nat > ( sum_sum @ B @ nat ) ) @ ( sum_sum @ A @ nat ) ) > ( old_node @ A @ B ) ) ).
thf(sy_c_Old__Datatype_Onode_ORep__Node,type,
old_Rep_Node:
!>[A: $tType,B: $tType] : ( ( old_node @ A @ B ) > ( product_prod @ ( nat > ( sum_sum @ B @ nat ) ) @ ( sum_sum @ A @ nat ) ) ) ).
thf(sy_c_Old__Datatype_Ontrunc,type,
old_ntrunc:
!>[A: $tType,B: $tType] : ( nat > ( set @ ( old_node @ A @ B ) ) > ( set @ ( old_node @ A @ B ) ) ) ).
thf(sy_c_Old__Datatype_Ouprod,type,
old_uprod:
!>[A: $tType,B: $tType] : ( ( set @ ( set @ ( old_node @ A @ B ) ) ) > ( set @ ( set @ ( old_node @ A @ B ) ) ) > ( set @ ( set @ ( old_node @ A @ B ) ) ) ) ).
thf(sy_c_Old__Datatype_Ousum,type,
old_usum:
!>[A: $tType,B: $tType] : ( ( set @ ( set @ ( old_node @ A @ B ) ) ) > ( set @ ( set @ ( old_node @ A @ B ) ) ) > ( set @ ( set @ ( old_node @ A @ B ) ) ) ) ).
thf(sy_c_Option_Obind,type,
bind2:
!>[A: $tType,B: $tType] : ( ( option @ A ) > ( A > ( option @ B ) ) > ( option @ B ) ) ).
thf(sy_c_Option_Ocombine__options,type,
combine_options:
!>[A: $tType] : ( ( A > A > A ) > ( option @ A ) > ( option @ A ) > ( option @ A ) ) ).
thf(sy_c_Option_Ois__none,type,
is_none:
!>[A: $tType] : ( ( option @ A ) > $o ) ).
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_Opred__option,type,
pred_option:
!>[A: $tType] : ( ( A > $o ) > ( option @ A ) > $o ) ).
thf(sy_c_Option_Ooption_Orec__option,type,
rec_option:
!>[C: $tType,A: $tType] : ( C > ( A > C ) > ( option @ A ) > C ) ).
thf(sy_c_Option_Ooption_Orel__option,type,
rel_option:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > ( option @ A ) > ( option @ B ) > $o ) ).
thf(sy_c_Option_Ooption_Oset__option,type,
set_option:
!>[A: $tType] : ( ( option @ A ) > ( set @ A ) ) ).
thf(sy_c_Option_Ooption_Osize__option,type,
size_option:
!>[A: $tType] : ( ( A > nat ) > ( option @ A ) > nat ) ).
thf(sy_c_Option_Ooption_Othe,type,
the2:
!>[A: $tType] : ( ( option @ A ) > A ) ).
thf(sy_c_Option_Othese,type,
these:
!>[A: $tType] : ( ( set @ ( option @ A ) ) > ( set @ A ) ) ).
thf(sy_c_Order__Relation_OAbove,type,
order_Above:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) > ( set @ A ) ) ).
thf(sy_c_Order__Relation_OAboveS,type,
order_AboveS:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) > ( set @ A ) ) ).
thf(sy_c_Order__Relation_OUnder,type,
order_Under:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) > ( set @ A ) ) ).
thf(sy_c_Order__Relation_OUnderS,type,
order_UnderS:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) > ( set @ A ) ) ).
thf(sy_c_Order__Relation_Oabove,type,
order_above:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > A > ( set @ A ) ) ).
thf(sy_c_Order__Relation_OaboveS,type,
order_aboveS:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > A > ( set @ A ) ) ).
thf(sy_c_Order__Relation_Olinear__order__on,type,
order_679001287576687338der_on:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Order__Relation_Oofilter,type,
order_ofilter:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) > $o ) ).
thf(sy_c_Order__Relation_Opartial__order__on,type,
order_7125193373082350890der_on:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Order__Relation_Opreorder__on,type,
order_preorder_on:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Order__Relation_Orelation__of,type,
order_relation_of:
!>[A: $tType] : ( ( A > A > $o ) > ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_Order__Relation_Ounder,type,
order_under:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > A > ( set @ A ) ) ).
thf(sy_c_Order__Relation_OunderS,type,
order_underS:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > A > ( set @ A ) ) ).
thf(sy_c_Order__Relation_Owell__order__on,type,
order_well_order_on:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Orderings_Obot__class_Obot,type,
bot_bot:
!>[A: $tType] : A ).
thf(sy_c_Orderings_Oord_OLeast,type,
least:
!>[A: $tType] : ( ( A > A > $o ) > ( A > $o ) > A ) ).
thf(sy_c_Orderings_Oord_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__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_Oorder__class_Ostrict__mono,type,
order_strict_mono:
!>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
thf(sy_c_Orderings_Oordering,type,
ordering:
!>[A: $tType] : ( ( A > A > $o ) > ( A > A > $o ) > $o ) ).
thf(sy_c_Orderings_Oordering__top,type,
ordering_top:
!>[A: $tType] : ( ( A > A > $o ) > ( A > A > $o ) > A > $o ) ).
thf(sy_c_Orderings_Opreordering,type,
preordering:
!>[A: $tType] : ( ( A > A > $o ) > ( A > A > $o ) > $o ) ).
thf(sy_c_Orderings_Otop__class_Otop,type,
top_top:
!>[A: $tType] : A ).
thf(sy_c_Partial__Function_Oflat__lub,type,
partial_flat_lub:
!>[A: $tType] : ( A > ( set @ A ) > A ) ).
thf(sy_c_Partial__Function_Oimg__ord,type,
partial_img_ord:
!>[A: $tType,C: $tType,B: $tType] : ( ( A > C ) > ( C > C > B ) > A > A > B ) ).
thf(sy_c_Partial__Function_Omk__less,type,
partial_mk_less:
!>[A: $tType] : ( ( A > A > $o ) > A > A > $o ) ).
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_OSeq,type,
seq2:
!>[A: $tType] : ( ( product_unit > ( seq @ A ) ) > ( pred @ A ) ) ).
thf(sy_c_Predicate_Oadjunct,type,
adjunct:
!>[A: $tType] : ( ( pred @ A ) > ( seq @ A ) > ( seq @ A ) ) ).
thf(sy_c_Predicate_Oapply,type,
apply:
!>[A: $tType,B: $tType] : ( ( A > ( pred @ B ) ) > ( seq @ A ) > ( seq @ B ) ) ).
thf(sy_c_Predicate_Obind,type,
bind3:
!>[A: $tType,B: $tType] : ( ( pred @ A ) > ( A > ( pred @ B ) ) > ( pred @ B ) ) ).
thf(sy_c_Predicate_Ocontained,type,
contained:
!>[A: $tType] : ( ( seq @ A ) > ( pred @ A ) > $o ) ).
thf(sy_c_Predicate_Oiterate__upto,type,
iterate_upto:
!>[A: $tType] : ( ( code_natural > A ) > code_natural > code_natural > ( pred @ A ) ) ).
thf(sy_c_Predicate_Oiterate__upto__rel,type,
iterate_upto_rel:
!>[A: $tType] : ( ( product_prod @ ( code_natural > A ) @ ( product_prod @ code_natural @ code_natural ) ) > ( product_prod @ ( code_natural > A ) @ ( product_prod @ code_natural @ code_natural ) ) > $o ) ).
thf(sy_c_Predicate_Omap,type,
map2:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( pred @ A ) > ( pred @ B ) ) ).
thf(sy_c_Predicate_Opred_Ocase__pred,type,
case_pred:
!>[A: $tType,B: $tType] : ( ( ( A > $o ) > B ) > ( pred @ A ) > B ) ).
thf(sy_c_Predicate_Opred_Oeval,type,
eval:
!>[A: $tType] : ( ( pred @ A ) > A > $o ) ).
thf(sy_c_Predicate_Opred__of__seq,type,
pred_of_seq:
!>[A: $tType] : ( ( seq @ A ) > ( pred @ A ) ) ).
thf(sy_c_Predicate_Oseq_OEmpty,type,
empty:
!>[A: $tType] : ( seq @ A ) ).
thf(sy_c_Predicate_Oseq_OInsert,type,
insert:
!>[A: $tType] : ( A > ( pred @ A ) > ( seq @ A ) ) ).
thf(sy_c_Predicate_Oseq_OJoin,type,
join:
!>[A: $tType] : ( ( pred @ A ) > ( seq @ A ) > ( seq @ A ) ) ).
thf(sy_c_Predicate_Oseq_Ocase__seq,type,
case_seq:
!>[B: $tType,A: $tType] : ( B > ( A > ( pred @ A ) > B ) > ( ( pred @ A ) > ( seq @ A ) > B ) > ( seq @ A ) > B ) ).
thf(sy_c_Predicate_Oset__of__pred,type,
set_of_pred:
!>[A: $tType] : ( ( pred @ A ) > ( set @ A ) ) ).
thf(sy_c_Predicate_Oset__of__seq,type,
set_of_seq:
!>[A: $tType] : ( ( seq @ A ) > ( set @ A ) ) ).
thf(sy_c_Predicate_Osingle,type,
single:
!>[A: $tType] : ( A > ( pred @ A ) ) ).
thf(sy_c_Predicate_Othe,type,
the3:
!>[A: $tType] : ( ( pred @ A ) > A ) ).
thf(sy_c_Product__Type_OPair,type,
product_Pair:
!>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
thf(sy_c_Product__Type_OSigma,type,
product_Sigma:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > ( set @ B ) ) > ( set @ ( product_prod @ A @ B ) ) ) ).
thf(sy_c_Product__Type_OUnity,type,
product_Unity: product_unit ).
thf(sy_c_Product__Type_Oapfst,type,
product_apfst:
!>[A: $tType,C: $tType,B: $tType] : ( ( A > C ) > ( product_prod @ A @ B ) > ( product_prod @ C @ B ) ) ).
thf(sy_c_Product__Type_Oapsnd,type,
product_apsnd:
!>[B: $tType,C: $tType,A: $tType] : ( ( B > C ) > ( product_prod @ A @ B ) > ( product_prod @ A @ C ) ) ).
thf(sy_c_Product__Type_Ocurry,type,
product_curry:
!>[A: $tType,B: $tType,C: $tType] : ( ( ( product_prod @ A @ B ) > C ) > A > B > C ) ).
thf(sy_c_Product__Type_Ointernal__case__prod,type,
produc5280177257484947105e_prod:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
thf(sy_c_Product__Type_Omap__prod,type,
product_map_prod:
!>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( A > C ) > ( B > D ) > ( product_prod @ A @ B ) > ( product_prod @ C @ D ) ) ).
thf(sy_c_Product__Type_Oold_Obool_Orec__bool,type,
product_rec_bool:
!>[T: $tType] : ( T > T > $o > T ) ).
thf(sy_c_Product__Type_Oold_Obool_Orec__set__bool,type,
product_rec_set_bool:
!>[T: $tType] : ( T > T > $o > T > $o ) ).
thf(sy_c_Product__Type_Oold_Oprod_Orec__prod,type,
product_rec_prod:
!>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T ) ).
thf(sy_c_Product__Type_Oold_Oprod_Orec__set__prod,type,
product_rec_set_prod:
!>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T > $o ) ).
thf(sy_c_Product__Type_Oold_Ounit_Orec__set__unit,type,
product_rec_set_unit:
!>[T: $tType] : ( T > product_unit > T > $o ) ).
thf(sy_c_Product__Type_Oold_Ounit_Orec__unit,type,
product_rec_unit:
!>[T: $tType] : ( T > product_unit > T ) ).
thf(sy_c_Product__Type_Oprod_Ocase__prod,type,
product_case_prod:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
thf(sy_c_Product__Type_Oprod_Ofst,type,
product_fst:
!>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > A ) ).
thf(sy_c_Product__Type_Oprod_Osnd,type,
product_snd:
!>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > B ) ).
thf(sy_c_Product__Type_Oprod_Oswap,type,
product_swap:
!>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > ( product_prod @ B @ A ) ) ).
thf(sy_c_Product__Type_Oproduct,type,
product_product:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( set @ B ) > ( set @ ( product_prod @ A @ B ) ) ) ).
thf(sy_c_Product__Type_Oscomp,type,
product_scomp:
!>[A: $tType,B: $tType,C: $tType,D: $tType] : ( ( A > ( product_prod @ B @ C ) ) > ( B > C > D ) > A > D ) ).
thf(sy_c_Product__Type_Ounit_Ocase__unit,type,
product_case_unit:
!>[A: $tType] : ( A > product_unit > A ) ).
thf(sy_c_Pure_Otype,type,
type2:
!>[A: $tType] : ( itself @ A ) ).
thf(sy_c_Quotient_OQuotient3,type,
quotient3:
!>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( A > B ) > ( B > A ) > $o ) ).
thf(sy_c_Quotient_Oquot__type,type,
quot_type:
!>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( ( set @ A ) > B ) > ( B > ( set @ A ) ) > $o ) ).
thf(sy_c_Quotient_Oquot__type_Oabs,type,
quot_abs:
!>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( ( set @ A ) > B ) > A > B ) ).
thf(sy_c_Quotient_Oquot__type_Orep,type,
quot_rep:
!>[B: $tType,A: $tType] : ( ( B > ( set @ A ) ) > B > A ) ).
thf(sy_c_Random_Oiterate,type,
iterate:
!>[B: $tType,A: $tType] : ( code_natural > ( B > A > ( product_prod @ B @ A ) ) > B > A > ( product_prod @ B @ A ) ) ).
thf(sy_c_Random_Oiterate__rel,type,
iterate_rel:
!>[B: $tType,A: $tType] : ( ( product_prod @ code_natural @ ( product_prod @ ( B > A > ( product_prod @ B @ A ) ) @ B ) ) > ( product_prod @ code_natural @ ( product_prod @ ( B > A > ( product_prod @ B @ A ) ) @ B ) ) > $o ) ).
thf(sy_c_Random_Olog,type,
log: code_natural > code_natural > code_natural ).
thf(sy_c_Random_Olog__rel,type,
log_rel: ( product_prod @ code_natural @ code_natural ) > ( product_prod @ code_natural @ code_natural ) > $o ).
thf(sy_c_Random_Opick,type,
pick:
!>[A: $tType] : ( ( list @ ( product_prod @ code_natural @ A ) ) > code_natural > A ) ).
thf(sy_c_Random_Orange,type,
range: code_natural > ( product_prod @ code_natural @ code_natural ) > ( product_prod @ code_natural @ ( product_prod @ code_natural @ code_natural ) ) ).
thf(sy_c_Random_Oselect,type,
select:
!>[A: $tType] : ( ( list @ A ) > ( product_prod @ code_natural @ code_natural ) > ( product_prod @ A @ ( product_prod @ code_natural @ code_natural ) ) ) ).
thf(sy_c_Random_Oselect__weight,type,
select_weight:
!>[A: $tType] : ( ( list @ ( product_prod @ code_natural @ A ) ) > ( product_prod @ code_natural @ code_natural ) > ( product_prod @ A @ ( product_prod @ code_natural @ code_natural ) ) ) ).
thf(sy_c_Random_Osplit__seed,type,
split_seed: ( product_prod @ code_natural @ code_natural ) > ( product_prod @ ( product_prod @ code_natural @ code_natural ) @ ( product_prod @ code_natural @ code_natural ) ) ).
thf(sy_c_Random__Pred_ORandom,type,
random_Random:
!>[A: $tType] : ( ( ( product_prod @ code_natural @ code_natural ) > ( product_prod @ ( product_prod @ A @ ( product_unit > code_term ) ) @ ( product_prod @ code_natural @ code_natural ) ) ) > ( product_prod @ code_natural @ code_natural ) > ( product_prod @ ( pred @ A ) @ ( product_prod @ code_natural @ code_natural ) ) ) ).
thf(sy_c_Random__Pred_Obind,type,
random_bind:
!>[A: $tType,B: $tType] : ( ( ( product_prod @ code_natural @ code_natural ) > ( product_prod @ ( pred @ A ) @ ( product_prod @ code_natural @ code_natural ) ) ) > ( A > ( product_prod @ code_natural @ code_natural ) > ( product_prod @ ( pred @ B ) @ ( product_prod @ code_natural @ code_natural ) ) ) > ( product_prod @ code_natural @ code_natural ) > ( product_prod @ ( pred @ B ) @ ( product_prod @ code_natural @ code_natural ) ) ) ).
thf(sy_c_Random__Pred_Oempty,type,
random_empty:
!>[A: $tType] : ( ( product_prod @ code_natural @ code_natural ) > ( product_prod @ ( pred @ A ) @ ( product_prod @ code_natural @ code_natural ) ) ) ).
thf(sy_c_Random__Pred_Oiterate__upto,type,
random_iterate_upto:
!>[A: $tType] : ( ( code_natural > A ) > code_natural > code_natural > ( product_prod @ code_natural @ code_natural ) > ( product_prod @ ( pred @ A ) @ ( product_prod @ code_natural @ code_natural ) ) ) ).
thf(sy_c_Random__Pred_Omap,type,
random_map:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( ( product_prod @ code_natural @ code_natural ) > ( product_prod @ ( pred @ A ) @ ( product_prod @ code_natural @ code_natural ) ) ) > ( product_prod @ code_natural @ code_natural ) > ( product_prod @ ( pred @ B ) @ ( product_prod @ code_natural @ code_natural ) ) ) ).
thf(sy_c_Random__Pred_Onot__randompred,type,
random6974930770145893639ompred: ( ( product_prod @ code_natural @ code_natural ) > ( product_prod @ ( pred @ product_unit ) @ ( product_prod @ code_natural @ code_natural ) ) ) > ( product_prod @ code_natural @ code_natural ) > ( product_prod @ ( pred @ product_unit ) @ ( product_prod @ code_natural @ code_natural ) ) ).
thf(sy_c_Random__Pred_Osingle,type,
random_single:
!>[A: $tType] : ( A > ( product_prod @ code_natural @ code_natural ) > ( product_prod @ ( pred @ A ) @ ( product_prod @ code_natural @ code_natural ) ) ) ).
thf(sy_c_Random__Pred_Ounion,type,
random_union:
!>[A: $tType] : ( ( ( product_prod @ code_natural @ code_natural ) > ( product_prod @ ( pred @ A ) @ ( product_prod @ code_natural @ code_natural ) ) ) > ( ( product_prod @ code_natural @ code_natural ) > ( product_prod @ ( pred @ A ) @ ( product_prod @ code_natural @ code_natural ) ) ) > ( product_prod @ code_natural @ code_natural ) > ( product_prod @ ( pred @ A ) @ ( product_prod @ code_natural @ code_natural ) ) ) ).
thf(sy_c_Rat_OAbs__Rat,type,
abs_Rat: ( product_prod @ int @ int ) > rat ).
thf(sy_c_Rat_OFract,type,
fract: int > int > rat ).
thf(sy_c_Rat_OFrct,type,
frct: ( product_prod @ int @ int ) > rat ).
thf(sy_c_Rat_ORep__Rat,type,
rep_Rat: rat > ( product_prod @ int @ int ) ).
thf(sy_c_Rat_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_Record_Otuple__isomorphism_OTuple__Isomorphism,type,
tuple_1188178415141063261rphism:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > ( product_prod @ B @ C ) ) > ( ( product_prod @ B @ C ) > A ) > ( tuple_isomorphism @ A @ B @ C ) ) ).
thf(sy_c_Record_Otuple__isomorphism_Osize__tuple__isomorphism,type,
tuple_9181185373184732606rphism:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > nat ) > ( B > nat ) > ( C > nat ) > ( tuple_isomorphism @ A @ B @ C ) > nat ) ).
thf(sy_c_Relation_ODomain,type,
domain:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ A ) ) ).
thf(sy_c_Relation_ODomainp,type,
domainp:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > A > $o ) ).
thf(sy_c_Relation_OField,type,
field2:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) ) ).
thf(sy_c_Relation_OId,type,
id2:
!>[A: $tType] : ( set @ ( product_prod @ A @ A ) ) ).
thf(sy_c_Relation_OId__on,type,
id_on:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_Relation_OImage,type,
image:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ A ) > ( set @ B ) ) ).
thf(sy_c_Relation_OPowp,type,
powp:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) > $o ) ).
thf(sy_c_Relation_ORange,type,
range2:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ B ) ) ).
thf(sy_c_Relation_ORangep,type,
rangep:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > B > $o ) ).
thf(sy_c_Relation_Oantisym,type,
antisym:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Relation_Oantisymp,type,
antisymp:
!>[A: $tType] : ( ( A > A > $o ) > $o ) ).
thf(sy_c_Relation_Oasym,type,
asym:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Relation_Oasymp,type,
asymp:
!>[A: $tType] : ( ( A > A > $o ) > $o ) ).
thf(sy_c_Relation_Oconverse,type,
converse:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ ( product_prod @ B @ A ) ) ) ).
thf(sy_c_Relation_Oconversep,type,
conversep:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > B > A > $o ) ).
thf(sy_c_Relation_Oinv__image,type,
inv_image:
!>[B: $tType,A: $tType] : ( ( set @ ( product_prod @ B @ B ) ) > ( A > B ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_Relation_Oirrefl,type,
irrefl:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Relation_Oirreflp,type,
irreflp:
!>[A: $tType] : ( ( A > A > $o ) > $o ) ).
thf(sy_c_Relation_Orefl__on,type,
refl_on:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Relation_Oreflp,type,
reflp:
!>[A: $tType] : ( ( A > A > $o ) > $o ) ).
thf(sy_c_Relation_Orelcomp,type,
relcomp:
!>[A: $tType,B: $tType,C: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ ( product_prod @ B @ C ) ) > ( set @ ( product_prod @ A @ C ) ) ) ).
thf(sy_c_Relation_Orelcompp,type,
relcompp:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > B > $o ) > ( B > C > $o ) > A > C > $o ) ).
thf(sy_c_Relation_Osingle__valued,type,
single_valued:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > $o ) ).
thf(sy_c_Relation_Osingle__valuedp,type,
single_valuedp:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > $o ) ).
thf(sy_c_Relation_Ototal__on,type,
total_on:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Relation_Otrans,type,
trans:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Rings_Oalgebraic__semidom__class_Ocoprime,type,
algebr8660921524188924756oprime:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Rings_Odivide__class_Odivide,type,
divide_divide:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Rings_Odvd__class_Odvd,type,
dvd_dvd:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Rings_Omodulo__class_Omodulo,type,
modulo_modulo:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Rings_Onormalization__semidom__class_Onormalize,type,
normal6383669964737779283malize:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Rings_Ounit__factor__class_Ounit__factor,type,
unit_f5069060285200089521factor:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool,type,
zero_neq_one_of_bool:
!>[A: $tType] : ( $o > A ) ).
thf(sy_c_Set_OBall,type,
ball:
!>[A: $tType] : ( ( set @ A ) > ( A > $o ) > $o ) ).
thf(sy_c_Set_OBex,type,
bex:
!>[A: $tType] : ( ( set @ A ) > ( A > $o ) > $o ) ).
thf(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_Set_OPow,type,
pow2:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( set @ A ) ) ) ).
thf(sy_c_Set_Obind,type,
bind4:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > ( set @ B ) ) > ( set @ B ) ) ).
thf(sy_c_Set_Odisjnt,type,
disjnt:
!>[A: $tType] : ( ( set @ A ) > ( set @ A ) > $o ) ).
thf(sy_c_Set_Ofilter,type,
filter3:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) > ( set @ A ) ) ).
thf(sy_c_Set_Oimage,type,
image2:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) ) ).
thf(sy_c_Set_Oinsert,type,
insert2:
!>[A: $tType] : ( A > ( set @ A ) > ( set @ A ) ) ).
thf(sy_c_Set_Ois__singleton,type,
is_singleton:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Set_Opairwise,type,
pairwise:
!>[A: $tType] : ( ( A > A > $o ) > ( set @ A ) > $o ) ).
thf(sy_c_Set_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_OLiteral,type,
literal2: $o > $o > $o > $o > $o > $o > $o > literal > literal ).
thf(sy_c_String_Oascii__of,type,
ascii_of: char > char ).
thf(sy_c_String_Oasciis__of__literal,type,
asciis_of_literal: literal > ( list @ code_integer ) ).
thf(sy_c_String_Ochar_OChar,type,
char2: $o > $o > $o > $o > $o > $o > $o > $o > char ).
thf(sy_c_String_Ochar_Ocase__char,type,
case_char:
!>[A: $tType] : ( ( $o > $o > $o > $o > $o > $o > $o > $o > A ) > char > A ) ).
thf(sy_c_String_Ochar_Odigit0,type,
digit0: char > $o ).
thf(sy_c_String_Ochar_Odigit1,type,
digit1: char > $o ).
thf(sy_c_String_Ochar_Odigit2,type,
digit2: char > $o ).
thf(sy_c_String_Ochar_Odigit3,type,
digit3: char > $o ).
thf(sy_c_String_Ochar_Odigit4,type,
digit4: char > $o ).
thf(sy_c_String_Ochar_Odigit5,type,
digit5: char > $o ).
thf(sy_c_String_Ochar_Odigit6,type,
digit6: char > $o ).
thf(sy_c_String_Ochar_Odigit7,type,
digit7: char > $o ).
thf(sy_c_String_Ochar_Osize__char,type,
size_char: char > nat ).
thf(sy_c_String_Ocomm__semiring__1__class_Oof__char,type,
comm_s6883823935334413003f_char:
!>[A: $tType] : ( char > A ) ).
thf(sy_c_String_Ocr__literal,type,
cr_literal: ( list @ char ) > literal > $o ).
thf(sy_c_String_Oliteral_OAbs__literal,type,
abs_literal: ( list @ char ) > literal ).
thf(sy_c_String_Oliteral_Oexplode,type,
explode: literal > ( list @ char ) ).
thf(sy_c_String_Oliteral__of__asciis,type,
literal_of_asciis: ( list @ code_integer ) > literal ).
thf(sy_c_String_Opcr__literal,type,
pcr_literal: ( list @ char ) > literal > $o ).
thf(sy_c_String_Ounique__euclidean__semiring__with__bit__operations__class_Ochar__of,type,
unique5772411509450598832har_of:
!>[A: $tType] : ( A > char ) ).
thf(sy_c_Sum__Type_OInl,type,
sum_Inl:
!>[A: $tType,B: $tType] : ( A > ( sum_sum @ A @ B ) ) ).
thf(sy_c_Sum__Type_OInr,type,
sum_Inr:
!>[B: $tType,A: $tType] : ( B > ( sum_sum @ A @ B ) ) ).
thf(sy_c_Sum__Type_OPlus,type,
sum_Plus:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( set @ B ) > ( set @ ( sum_sum @ A @ B ) ) ) ).
thf(sy_c_Sum__Type_Omap__sum,type,
sum_map_sum:
!>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( A > C ) > ( B > D ) > ( sum_sum @ A @ B ) > ( sum_sum @ C @ D ) ) ).
thf(sy_c_Sum__Type_Oold_Osum_Orec__set__sum,type,
sum_rec_set_sum:
!>[A: $tType,T: $tType,B: $tType] : ( ( A > T ) > ( B > T ) > ( sum_sum @ A @ B ) > T > $o ) ).
thf(sy_c_Sum__Type_Oold_Osum_Orec__sum,type,
sum_rec_sum:
!>[A: $tType,T: $tType,B: $tType] : ( ( A > T ) > ( B > T ) > ( sum_sum @ A @ B ) > T ) ).
thf(sy_c_Sum__Type_Osum_Ocase__sum,type,
sum_case_sum:
!>[A: $tType,C: $tType,B: $tType] : ( ( A > C ) > ( B > C ) > ( sum_sum @ A @ B ) > C ) ).
thf(sy_c_Sum__Type_Osum_Oprojl,type,
sum_projl:
!>[A: $tType,B: $tType] : ( ( sum_sum @ A @ B ) > A ) ).
thf(sy_c_Sum__Type_Osum_Oprojr,type,
sum_projr:
!>[A: $tType,B: $tType] : ( ( sum_sum @ A @ B ) > B ) ).
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_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_Oacyclic,type,
transitive_acyclic:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Transitive__Closure_Ontrancl,type,
transitive_ntrancl:
!>[A: $tType] : ( nat > ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_Transitive__Closure_Ortrancl,type,
transitive_rtrancl:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_Transitive__Closure_Ortranclp,type,
transitive_rtranclp:
!>[A: $tType] : ( ( A > A > $o ) > A > A > $o ) ).
thf(sy_c_Transitive__Closure_Otrancl,type,
transitive_trancl:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_Transitive__Closure_Otranclp,type,
transitive_tranclp:
!>[A: $tType] : ( ( A > A > $o ) > A > A > $o ) ).
thf(sy_c_Typedef_Otype__definition,type,
type_definition:
!>[B: $tType,A: $tType] : ( ( B > A ) > ( A > B ) > ( set @ A ) > $o ) ).
thf(sy_c_Typerep_Otyperep_OTyperep,type,
typerep2: literal > ( list @ typerep ) > typerep ).
thf(sy_c_Typerep_Otyperep_Orec__typerep,type,
rec_typerep:
!>[A: $tType] : ( ( literal > ( list @ ( product_prod @ typerep @ A ) ) > A ) > typerep > A ) ).
thf(sy_c_Typerep_Otyperep_Osize__typerep,type,
size_typerep: typerep > nat ).
thf(sy_c_Wellfounded_Oacc,type,
acc:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) ) ).
thf(sy_c_Wellfounded_Oaccp,type,
accp:
!>[A: $tType] : ( ( A > A > $o ) > A > $o ) ).
thf(sy_c_Wellfounded_Ofinite__psubset,type,
finite_psubset:
!>[A: $tType] : ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ).
thf(sy_c_Wellfounded_Oless__than,type,
less_than: set @ ( product_prod @ nat @ nat ) ).
thf(sy_c_Wellfounded_Olex__prod,type,
lex_prod:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ B @ B ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) ) ).
thf(sy_c_Wellfounded_Omax__ext,type,
max_ext:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) ).
thf(sy_c_Wellfounded_Omax__extp,type,
max_extp:
!>[A: $tType] : ( ( A > A > $o ) > ( set @ A ) > ( set @ A ) > $o ) ).
thf(sy_c_Wellfounded_Omeasure,type,
measure:
!>[A: $tType] : ( ( A > nat ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_Wellfounded_Omin__ext,type,
min_ext:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) ).
thf(sy_c_Wellfounded_Omlex__prod,type,
mlex_prod:
!>[A: $tType] : ( ( A > nat ) > ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_Wellfounded_Opred__nat,type,
pred_nat: set @ ( product_prod @ nat @ nat ) ).
thf(sy_c_Wellfounded_Owf,type,
wf:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Wellfounded_OwfP,type,
wfP:
!>[A: $tType] : ( ( A > A > $o ) > $o ) ).
thf(sy_c_Wfrec_Oadm__wf,type,
adm_wf:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( ( A > B ) > A > B ) > $o ) ).
thf(sy_c_Wfrec_Ocut,type,
cut:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ ( product_prod @ A @ A ) ) > A > A > B ) ).
thf(sy_c_Wfrec_Osame__fst,type,
same_fst:
!>[A: $tType,B: $tType] : ( ( A > $o ) > ( A > ( set @ ( product_prod @ B @ B ) ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) ) ).
thf(sy_c_Zorn_OChains,type,
chains:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( set @ A ) ) ) ).
thf(sy_c_Zorn_Ochain__subset,type,
chain_subset:
!>[A: $tType] : ( ( set @ ( set @ A ) ) > $o ) ).
thf(sy_c_Zorn_Ochains,type,
chains2:
!>[A: $tType] : ( ( set @ ( set @ A ) ) > ( set @ ( set @ ( set @ A ) ) ) ) ).
thf(sy_c_Zorn_Oinit__seg__of,type,
init_seg_of:
!>[A: $tType] : ( set @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
thf(sy_c_Zorn_Opred__on_Ochain,type,
pred_chain:
!>[A: $tType] : ( ( set @ A ) > ( A > A > $o ) > ( set @ A ) > $o ) ).
thf(sy_c_Zorn_Opred__on_Omaxchain,type,
pred_maxchain:
!>[A: $tType] : ( ( set @ A ) > ( A > A > $o ) > ( set @ A ) > $o ) ).
thf(sy_c_Zorn_Opred__on_Osuc,type,
pred_suc:
!>[A: $tType] : ( ( set @ A ) > ( A > A > $o ) > ( set @ A ) > ( set @ A ) ) ).
thf(sy_c_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: a > $o ).
thf(sy_v_h,type,
h: heap_ext @ product_unit ).
thf(sy_v_x,type,
x: a ).
% Relevant facts (7223)
thf(fact_0_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_1_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_2_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_3_success__ureturnI,axiom,
! [A: $tType,X: A,H: heap_ext @ product_unit] : ( heap_Time_success @ A @ ( heap_Time_ureturn @ A @ X ) @ H ) ).
% success_ureturnI
thf(fact_4_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_5_success__returnI,axiom,
! [A: $tType,X: A,H: heap_ext @ product_unit] : ( heap_Time_success @ A @ ( heap_Time_return @ A @ X ) @ H ) ).
% success_returnI
thf(fact_6_effect__success,axiom,
! [A: $tType,C2: heap_Time_Heap @ A,H: heap_ext @ product_unit,H2: heap_ext @ product_unit,R: A,N: nat] :
( ( heap_Time_effect @ A @ C2 @ H @ H2 @ R @ N )
=> ( heap_Time_success @ A @ C2 @ H ) ) ).
% effect_success
thf(fact_7_success__effectE,axiom,
! [A: $tType,C2: heap_Time_Heap @ A,H: heap_ext @ product_unit] :
( ( heap_Time_success @ A @ C2 @ H )
=> ~ ! [R2: A,H3: heap_ext @ product_unit,N2: nat] :
~ ( heap_Time_effect @ A @ C2 @ H @ H3 @ R2 @ N2 ) ) ).
% success_effectE
thf(fact_8_execute__assert_I2_J,axiom,
! [A: $tType,P: A > $o,X: A,H: heap_ext @ product_unit] :
( ~ ( P @ X )
=> ( ( heap_Time_execute @ A @ ( heap_Time_assert @ A @ P @ X ) @ H )
= ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ).
% execute_assert(2)
thf(fact_9_success__bind__effectI,axiom,
! [A: $tType,B: $tType,F: heap_Time_Heap @ A,H: heap_ext @ product_unit,H2: heap_ext @ product_unit,X: A,N: nat,G: A > ( heap_Time_Heap @ B )] :
( ( heap_Time_effect @ A @ F @ H @ H2 @ X @ N )
=> ( ( heap_Time_success @ B @ ( G @ X ) @ H2 )
=> ( heap_Time_success @ B @ ( heap_Time_bind @ A @ B @ F @ G ) @ H ) ) ) ).
% success_bind_effectI
thf(fact_10_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_11_success__def,axiom,
! [A: $tType] :
( ( heap_Time_success @ A )
= ( ^ [F2: heap_Time_Heap @ A,H4: heap_ext @ product_unit] :
( ( heap_Time_execute @ A @ F2 @ H4 )
!= ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ) ).
% success_def
thf(fact_12_success__LetI,axiom,
! [A: $tType,B: $tType,X: A,T2: A,F: A > ( heap_Time_Heap @ B ),H: heap_ext @ product_unit] :
( ( X = T2 )
=> ( ( heap_Time_success @ B @ ( F @ X ) @ H )
=> ( heap_Time_success @ B @ ( F @ T2 ) @ H ) ) ) ).
% success_LetI
thf(fact_13_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
@ ^ [X2: A] : ( heap_Time_bind @ C @ B @ ( G @ X2 ) @ K ) ) ) ).
% Heap_Time_Monad.bind_bind
thf(fact_14_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_15_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_16_effect__deterministic_I3_J,axiom,
! [A: $tType,F: heap_Time_Heap @ A,H: heap_ext @ product_unit,H2: heap_ext @ product_unit,A3: A,N: nat,H5: heap_ext @ product_unit,B2: A,N3: nat] :
( ( heap_Time_effect @ A @ F @ H @ H2 @ A3 @ N )
=> ( ( heap_Time_effect @ A @ F @ H @ H5 @ B2 @ N3 )
=> ( N = N3 ) ) ) ).
% effect_deterministic(3)
thf(fact_17_effect__deterministic_I2_J,axiom,
! [A: $tType,F: heap_Time_Heap @ A,H: heap_ext @ product_unit,H2: heap_ext @ product_unit,A3: A,N: nat,H5: heap_ext @ product_unit,B2: A,N3: nat] :
( ( heap_Time_effect @ A @ F @ H @ H2 @ A3 @ N )
=> ( ( heap_Time_effect @ A @ F @ H @ H5 @ B2 @ N3 )
=> ( H2 = H5 ) ) ) ).
% effect_deterministic(2)
thf(fact_18_effect__deterministic_I1_J,axiom,
! [A: $tType,F: heap_Time_Heap @ A,H: heap_ext @ product_unit,H2: heap_ext @ product_unit,A3: A,N: nat,H5: heap_ext @ product_unit,B2: A,N3: nat] :
( ( heap_Time_effect @ A @ F @ H @ H2 @ A3 @ N )
=> ( ( heap_Time_effect @ A @ F @ H @ H5 @ B2 @ N3 )
=> ( A3 = B2 ) ) ) ).
% effect_deterministic(1)
thf(fact_19_Heap__eqI,axiom,
! [A: $tType,F: heap_Time_Heap @ A,G: heap_Time_Heap @ A] :
( ! [H6: heap_ext @ product_unit] :
( ( heap_Time_execute @ A @ F @ H6 )
= ( heap_Time_execute @ A @ G @ H6 ) )
=> ( F = G ) ) ).
% Heap_eqI
thf(fact_20_effect__ifE,axiom,
! [A: $tType,C2: $o,T2: heap_Time_Heap @ A,E2: heap_Time_Heap @ A,H: heap_ext @ product_unit,H2: heap_ext @ product_unit,R: A,N: nat] :
( ( heap_Time_effect @ A @ ( if @ ( heap_Time_Heap @ A ) @ C2 @ T2 @ E2 ) @ H @ H2 @ R @ N )
=> ( ( C2
=> ~ ( heap_Time_effect @ A @ T2 @ H @ H2 @ R @ N ) )
=> ~ ( ~ C2
=> ~ ( heap_Time_effect @ A @ E2 @ H @ H2 @ R @ N ) ) ) ) ).
% effect_ifE
thf(fact_21_effect__ifI,axiom,
! [A: $tType,C2: $o,T2: heap_Time_Heap @ A,H: heap_ext @ product_unit,H2: heap_ext @ product_unit,R: A,N: nat,E2: heap_Time_Heap @ A] :
( ( C2
=> ( heap_Time_effect @ A @ T2 @ H @ H2 @ R @ N ) )
=> ( ( ~ C2
=> ( heap_Time_effect @ A @ E2 @ H @ H2 @ R @ N ) )
=> ( heap_Time_effect @ A @ ( if @ ( heap_Time_Heap @ A ) @ C2 @ T2 @ E2 ) @ H @ H2 @ R @ N ) ) ) ).
% effect_ifI
thf(fact_22_effect__LetI,axiom,
! [B: $tType,A: $tType,X: A,T2: A,F: A > ( heap_Time_Heap @ B ),H: heap_ext @ product_unit,H2: heap_ext @ product_unit,R: B,N: nat] :
( ( X = T2 )
=> ( ( heap_Time_effect @ B @ ( F @ X ) @ H @ H2 @ R @ N )
=> ( heap_Time_effect @ B @ ( F @ T2 ) @ H @ H2 @ R @ N ) ) ) ).
% effect_LetI
thf(fact_23_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_24_ureturn__bind,axiom,
! [A: $tType,B: $tType,X: B,F: B > ( heap_Time_Heap @ A )] :
( ( heap_Time_bind @ B @ A @ ( heap_Time_ureturn @ B @ X ) @ F )
= ( F @ X ) ) ).
% ureturn_bind
thf(fact_25_execute__raise,axiom,
! [A: $tType,S: list @ char] :
( ( heap_Time_execute @ A @ ( heap_Time_raise @ A @ S ) )
= ( ^ [Uu: heap_ext @ product_unit] : ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ).
% execute_raise
thf(fact_26_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_27_effect__tapE,axiom,
! [A: $tType,F: ( heap_ext @ product_unit ) > A,H: heap_ext @ product_unit,H2: heap_ext @ product_unit,R: A,N: nat] :
( ( heap_Time_effect @ A @ ( heap_Time_tap @ A @ F ) @ H @ H2 @ R @ N )
=> ~ ( ( H2 = H )
=> ( ( R
= ( F @ H ) )
=> ( N
!= ( one_one @ nat ) ) ) ) ) ).
% effect_tapE
thf(fact_28_effect__tapI,axiom,
! [A: $tType,H2: heap_ext @ product_unit,H: heap_ext @ product_unit,R: A,F: ( heap_ext @ product_unit ) > A] :
( ( H2 = H )
=> ( ( R
= ( F @ H ) )
=> ( heap_Time_effect @ A @ ( heap_Time_tap @ A @ F ) @ H @ H2 @ R @ ( one_one @ nat ) ) ) ) ).
% effect_tapI
thf(fact_29_effect__raiseE,axiom,
! [A: $tType,X: list @ char,H: heap_ext @ product_unit,H2: heap_ext @ product_unit,R: A,N: nat] :
~ ( heap_Time_effect @ A @ ( heap_Time_raise @ A @ X ) @ H @ H2 @ R @ N ) ).
% effect_raiseE
thf(fact_30_effect__ureturnE,axiom,
! [A: $tType,X: A,H: heap_ext @ product_unit,H2: heap_ext @ product_unit,R: A,N: nat] :
( ( heap_Time_effect @ A @ ( heap_Time_ureturn @ A @ X ) @ H @ H2 @ R @ N )
=> ~ ( ( R = X )
=> ( ( H2 = H )
=> ( N
!= ( zero_zero @ nat ) ) ) ) ) ).
% effect_ureturnE
thf(fact_31_effect__ureturnI,axiom,
! [A: $tType,H: heap_ext @ product_unit,H2: heap_ext @ product_unit,X: A] :
( ( H = H2 )
=> ( heap_Time_effect @ A @ ( heap_Time_ureturn @ A @ X ) @ H @ H2 @ X @ ( zero_zero @ nat ) ) ) ).
% effect_ureturnI
thf(fact_32_effect__returnE,axiom,
! [A: $tType,X: A,H: heap_ext @ product_unit,H2: heap_ext @ product_unit,R: A,N: nat] :
( ( heap_Time_effect @ A @ ( heap_Time_return @ A @ X ) @ H @ H2 @ R @ N )
=> ~ ( ( R = X )
=> ( ( H2 = H )
=> ( N
!= ( one_one @ nat ) ) ) ) ) ).
% effect_returnE
thf(fact_33_effect__returnI,axiom,
! [A: $tType,H: heap_ext @ product_unit,H2: heap_ext @ product_unit,X: A] :
( ( H = H2 )
=> ( heap_Time_effect @ A @ ( heap_Time_return @ A @ X ) @ H @ H2 @ X @ ( one_one @ nat ) ) ) ).
% effect_returnI
thf(fact_34_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_35_effect__bindE,axiom,
! [B: $tType,A: $tType,F: heap_Time_Heap @ B,G: B > ( heap_Time_Heap @ A ),H: heap_ext @ product_unit,H5: heap_ext @ product_unit,R3: A,N: nat] :
( ( heap_Time_effect @ A @ ( heap_Time_bind @ B @ A @ F @ G ) @ H @ H5 @ R3 @ N )
=> ~ ! [H3: heap_ext @ product_unit,R2: B,N1: nat] :
( ( heap_Time_effect @ B @ F @ H @ H3 @ R2 @ N1 )
=> ! [N22: nat] :
( ( heap_Time_effect @ A @ ( G @ R2 ) @ H3 @ H5 @ R3 @ N22 )
=> ( N
!= ( plus_plus @ nat @ N1 @ N22 ) ) ) ) ) ).
% effect_bindE
thf(fact_36_option_Oinject,axiom,
! [A: $tType,X22: A,Y2: A] :
( ( ( some @ A @ X22 )
= ( some @ A @ Y2 ) )
= ( X22 = Y2 ) ) ).
% option.inject
thf(fact_37_not__Some__eq,axiom,
! [A: $tType,X: option @ A] :
( ( ! [Y: A] :
( X
!= ( some @ A @ Y ) ) )
= ( X
= ( none @ A ) ) ) ).
% not_Some_eq
thf(fact_38_not__None__eq,axiom,
! [A: $tType,X: option @ A] :
( ( X
!= ( none @ A ) )
= ( ? [Y: A] :
( X
= ( some @ A @ Y ) ) ) ) ).
% not_None_eq
thf(fact_39_combine__options__cases,axiom,
! [A: $tType,B: $tType,X: option @ A,P: ( option @ A ) > ( option @ B ) > $o,Y3: option @ B] :
( ( ( X
= ( none @ A ) )
=> ( P @ X @ Y3 ) )
=> ( ( ( Y3
= ( none @ B ) )
=> ( P @ X @ Y3 ) )
=> ( ! [A4: A,B3: B] :
( ( X
= ( some @ A @ A4 ) )
=> ( ( Y3
= ( some @ B @ B3 ) )
=> ( P @ X @ Y3 ) ) )
=> ( P @ X @ Y3 ) ) ) ) ).
% combine_options_cases
thf(fact_40_split__option__all,axiom,
! [A: $tType] :
( ( ^ [P2: ( option @ A ) > $o] :
! [X3: option @ A] : ( P2 @ X3 ) )
= ( ^ [P3: ( option @ A ) > $o] :
( ( P3 @ ( none @ A ) )
& ! [X2: A] : ( P3 @ ( some @ A @ X2 ) ) ) ) ) ).
% split_option_all
thf(fact_41_split__option__ex,axiom,
! [A: $tType] :
( ( ^ [P2: ( option @ A ) > $o] :
? [X3: option @ A] : ( P2 @ X3 ) )
= ( ^ [P3: ( option @ A ) > $o] :
( ( P3 @ ( none @ A ) )
| ? [X2: A] : ( P3 @ ( some @ A @ X2 ) ) ) ) ) ).
% split_option_ex
thf(fact_42_mem__Collect__eq,axiom,
! [A: $tType,A3: A,P: A > $o] :
( ( member @ A @ A3 @ ( collect @ A @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_43_Collect__mem__eq,axiom,
! [A: $tType,A5: set @ A] :
( ( collect @ A
@ ^ [X2: A] : ( member @ A @ X2 @ A5 ) )
= A5 ) ).
% Collect_mem_eq
thf(fact_44_Collect__cong,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X4: A] :
( ( P @ X4 )
= ( Q @ X4 ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_45_ext,axiom,
! [B: $tType,A: $tType,F: A > B,G: A > B] :
( ! [X4: A] :
( ( F @ X4 )
= ( G @ X4 ) )
=> ( F = G ) ) ).
% ext
thf(fact_46_option_Oexhaust,axiom,
! [A: $tType,Y3: option @ A] :
( ( Y3
!= ( none @ A ) )
=> ~ ! [X23: A] :
( Y3
!= ( some @ A @ X23 ) ) ) ).
% option.exhaust
thf(fact_47_option_OdiscI,axiom,
! [A: $tType,Option: option @ A,X22: A] :
( ( Option
= ( some @ A @ X22 ) )
=> ( Option
!= ( none @ A ) ) ) ).
% option.discI
thf(fact_48_option_Odistinct_I1_J,axiom,
! [A: $tType,X22: A] :
( ( none @ A )
!= ( some @ A @ X22 ) ) ).
% option.distinct(1)
thf(fact_49_effect__bindI,axiom,
! [B: $tType,A: $tType,F: heap_Time_Heap @ A,H: heap_ext @ product_unit,H2: heap_ext @ product_unit,R: A,N: nat,G: A > ( heap_Time_Heap @ B ),H5: heap_ext @ product_unit,R3: B,N3: nat] :
( ( heap_Time_effect @ A @ F @ H @ H2 @ R @ N )
=> ( ( heap_Time_effect @ B @ ( G @ R ) @ H2 @ H5 @ R3 @ N3 )
=> ( heap_Time_effect @ B @ ( heap_Time_bind @ A @ B @ F @ G ) @ H @ H5 @ R3 @ ( plus_plus @ nat @ N @ N3 ) ) ) ) ).
% effect_bindI
thf(fact_50_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus @ nat @ M @ ( zero_zero @ nat ) )
= M ) ).
% Nat.add_0_right
thf(fact_51_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_52_double__eq__0__iff,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A3: A] :
( ( ( plus_plus @ A @ A3 @ A3 )
= ( zero_zero @ A ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% double_eq_0_iff
thf(fact_53_add__0,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ! [A3: A] :
( ( plus_plus @ A @ ( zero_zero @ A ) @ A3 )
= A3 ) ) ).
% add_0
thf(fact_54_zero__eq__add__iff__both__eq__0,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [X: A,Y3: A] :
( ( ( zero_zero @ A )
= ( plus_plus @ A @ X @ Y3 ) )
= ( ( X
= ( zero_zero @ A ) )
& ( Y3
= ( zero_zero @ A ) ) ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_55_add__eq__0__iff__both__eq__0,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [X: A,Y3: A] :
( ( ( plus_plus @ A @ X @ Y3 )
= ( zero_zero @ A ) )
= ( ( X
= ( zero_zero @ A ) )
& ( Y3
= ( zero_zero @ A ) ) ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_56_add__cancel__right__right,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [A3: A,B2: A] :
( ( A3
= ( plus_plus @ A @ A3 @ B2 ) )
= ( B2
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_right_right
thf(fact_57_add__cancel__right__left,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [A3: A,B2: A] :
( ( A3
= ( plus_plus @ A @ B2 @ A3 ) )
= ( B2
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_right_left
thf(fact_58_add__cancel__left__right,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [A3: A,B2: A] :
( ( ( plus_plus @ A @ A3 @ B2 )
= A3 )
= ( B2
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_left_right
thf(fact_59_add__cancel__left__left,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [B2: A,A3: A] :
( ( ( plus_plus @ A @ B2 @ A3 )
= A3 )
= ( B2
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_left_left
thf(fact_60_double__zero__sym,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A3: A] :
( ( ( zero_zero @ A )
= ( plus_plus @ A @ A3 @ A3 ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% double_zero_sym
thf(fact_61_add_Oright__neutral,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ! [A3: A] :
( ( plus_plus @ A @ A3 @ ( zero_zero @ A ) )
= A3 ) ) ).
% add.right_neutral
thf(fact_62_add__left__cancel,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ( plus_plus @ A @ A3 @ B2 )
= ( plus_plus @ A @ A3 @ C2 ) )
= ( B2 = C2 ) ) ) ).
% add_left_cancel
thf(fact_63_add__right__cancel,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( ( plus_plus @ A @ B2 @ A3 )
= ( plus_plus @ A @ C2 @ A3 ) )
= ( B2 = C2 ) ) ) ).
% add_right_cancel
thf(fact_64_zero__reorient,axiom,
! [A: $tType] :
( ( zero @ A )
=> ! [X: A] :
( ( ( zero_zero @ A )
= X )
= ( X
= ( zero_zero @ A ) ) ) ) ).
% zero_reorient
thf(fact_65_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 )
= ( plus_plus @ A @ A3 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_66_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_67_group__cancel_Oadd1,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A5: A,K: A,A3: A,B2: A] :
( ( A5
= ( plus_plus @ A @ K @ A3 ) )
=> ( ( plus_plus @ A @ A5 @ B2 )
= ( plus_plus @ A @ K @ ( plus_plus @ A @ A3 @ B2 ) ) ) ) ) ).
% group_cancel.add1
thf(fact_68_group__cancel_Oadd2,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [B4: A,K: A,B2: A,A3: A] :
( ( B4
= ( plus_plus @ A @ K @ B2 ) )
=> ( ( plus_plus @ A @ A3 @ B4 )
= ( plus_plus @ A @ K @ ( plus_plus @ A @ A3 @ B2 ) ) ) ) ) ).
% group_cancel.add2
thf(fact_69_add_Oassoc,axiom,
! [A: $tType] :
( ( semigroup_add @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 )
= ( plus_plus @ A @ A3 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% add.assoc
thf(fact_70_add_Oleft__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ( plus_plus @ A @ A3 @ B2 )
= ( plus_plus @ A @ A3 @ C2 ) )
= ( B2 = C2 ) ) ) ).
% add.left_cancel
thf(fact_71_add_Oright__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( ( plus_plus @ A @ B2 @ A3 )
= ( plus_plus @ A @ C2 @ A3 ) )
= ( B2 = C2 ) ) ) ).
% add.right_cancel
thf(fact_72_add_Ocommute,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A )
=> ( ( plus_plus @ A )
= ( ^ [A6: A,B5: A] : ( plus_plus @ A @ B5 @ A6 ) ) ) ) ).
% add.commute
thf(fact_73_add_Oleft__commute,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( plus_plus @ A @ B2 @ ( plus_plus @ A @ A3 @ C2 ) )
= ( plus_plus @ A @ A3 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% add.left_commute
thf(fact_74_add__left__imp__eq,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ( plus_plus @ A @ A3 @ B2 )
= ( plus_plus @ A @ A3 @ C2 ) )
=> ( B2 = C2 ) ) ) ).
% add_left_imp_eq
thf(fact_75_add__right__imp__eq,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( ( plus_plus @ A @ B2 @ A3 )
= ( plus_plus @ A @ C2 @ A3 ) )
=> ( B2 = C2 ) ) ) ).
% add_right_imp_eq
thf(fact_76_one__reorient,axiom,
! [A: $tType] :
( ( one @ A )
=> ! [X: A] :
( ( ( one_one @ A )
= X )
= ( X
= ( one_one @ A ) ) ) ) ).
% one_reorient
thf(fact_77_comm__monoid__add__class_Oadd__0,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: A] :
( ( plus_plus @ A @ ( zero_zero @ A ) @ A3 )
= A3 ) ) ).
% comm_monoid_add_class.add_0
thf(fact_78_add_Ocomm__neutral,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: A] :
( ( plus_plus @ A @ A3 @ ( zero_zero @ A ) )
= A3 ) ) ).
% add.comm_neutral
thf(fact_79_add_Ogroup__left__neutral,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A] :
( ( plus_plus @ A @ ( zero_zero @ A ) @ A3 )
= A3 ) ) ).
% add.group_left_neutral
thf(fact_80_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus @ nat @ ( zero_zero @ nat ) @ N )
= N ) ).
% plus_nat.add_0
thf(fact_81_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_82_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_83_return__bind,axiom,
! [B: $tType,A: $tType,X: B,F: B > ( heap_Time_Heap @ A )] :
( ( heap_Time_bind @ B @ A @ ( heap_Time_return @ B @ X ) @ F )
= ( heap_Time_bind @ product_unit @ A @ ( heap_Time_wait @ ( one_one @ nat ) )
@ ^ [Uu: product_unit] : ( F @ X ) ) ) ).
% return_bind
thf(fact_84_Euclid__induct,axiom,
! [P: nat > nat > $o,A3: nat,B2: nat] :
( ! [A4: nat,B3: nat] :
( ( P @ A4 @ B3 )
= ( P @ B3 @ A4 ) )
=> ( ! [A4: nat] : ( P @ A4 @ ( zero_zero @ nat ) )
=> ( ! [A4: nat,B3: nat] :
( ( P @ A4 @ B3 )
=> ( P @ A4 @ ( plus_plus @ nat @ A4 @ B3 ) ) )
=> ( P @ A3 @ B2 ) ) ) ) ).
% Euclid_induct
thf(fact_85_zero__neq__one,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ( ( zero_zero @ A )
!= ( one_one @ A ) ) ) ).
% zero_neq_one
thf(fact_86_add__0__iff,axiom,
! [A: $tType] :
( ( semiri1453513574482234551roduct @ A )
=> ! [B2: A,A3: A] :
( ( B2
= ( plus_plus @ A @ B2 @ A3 ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% add_0_iff
thf(fact_87_verit__sum__simplify,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [A3: A] :
( ( plus_plus @ A @ A3 @ ( zero_zero @ A ) )
= A3 ) ) ).
% verit_sum_simplify
thf(fact_88_guard__def,axiom,
! [A: $tType] :
( ( heap_Time_guard @ A )
= ( ^ [P3: ( heap_ext @ product_unit ) > $o,F2: ( heap_ext @ product_unit ) > ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )] :
( heap_Time_Heap2 @ A
@ ^ [H4: heap_ext @ product_unit] : ( if @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ ( P3 @ H4 ) @ ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( F2 @ H4 ) ) @ ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ) ) ).
% guard_def
thf(fact_89_fun__cong__unused__0,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( zero @ B )
=> ! [F: ( A > B ) > C,G: C] :
( ( F
= ( ^ [X2: A > B] : G ) )
=> ( ( F
@ ^ [X2: A] : ( zero_zero @ B ) )
= G ) ) ) ).
% fun_cong_unused_0
thf(fact_90_raise__def,axiom,
! [A: $tType] :
( ( heap_Time_raise @ A )
= ( ^ [S2: 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_91_successE,axiom,
! [A: $tType,F: heap_Time_Heap @ A,H: heap_ext @ product_unit] :
( ( heap_Time_success @ A @ F @ H )
=> ~ ! [R2: A,H3: 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 ) @ R2 @ H3 ) ) ) ) ).
% successE
thf(fact_92_Heap__cases,axiom,
! [A: $tType,F: heap_Time_Heap @ A,H: heap_ext @ product_unit] :
( ! [X4: A,H3: 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 ) @ X4 @ H3 ) ) )
=> ( ( heap_Time_execute @ A @ F @ H )
= ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ).
% Heap_cases
thf(fact_93_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_94_Heap_Oinject,axiom,
! [A: $tType,X: ( 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 @ X )
= ( heap_Time_Heap2 @ A @ Ya ) )
= ( X = Ya ) ) ).
% Heap.inject
thf(fact_95_Heap__execute,axiom,
! [A: $tType,F: heap_Time_Heap @ A] :
( ( heap_Time_Heap2 @ A @ ( heap_Time_execute @ A @ F ) )
= F ) ).
% Heap_execute
thf(fact_96_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_97_rewriteL__comp__comp2,axiom,
! [A: $tType,C: $tType,B: $tType,D: $tType,E: $tType,F: C > B,G: A > C,L1: D > B,L2: A > D,H: E > A,R: E > D] :
( ( ( comp @ C @ B @ A @ F @ G )
= ( comp @ D @ B @ A @ L1 @ L2 ) )
=> ( ( ( comp @ A @ D @ E @ L2 @ H )
= R )
=> ( ( comp @ C @ B @ E @ F @ ( comp @ A @ C @ E @ G @ H ) )
= ( comp @ D @ B @ E @ L1 @ R ) ) ) ) ).
% rewriteL_comp_comp2
thf(fact_98_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_99_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_100_Heap_Oexhaust,axiom,
! [A: $tType,Y3: heap_Time_Heap @ A] :
~ ! [X4: ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )] :
( Y3
!= ( heap_Time_Heap2 @ A @ X4 ) ) ).
% Heap.exhaust
thf(fact_101_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_102_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_103_comp__apply,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( comp @ B @ A @ C )
= ( ^ [F2: B > A,G2: C > B,X2: C] : ( F2 @ ( G2 @ X2 ) ) ) ) ).
% comp_apply
thf(fact_104_old_Oprod_Oinject,axiom,
! [A: $tType,B: $tType,A3: A,B2: B,A7: A,B6: B] :
( ( ( product_Pair @ A @ B @ A3 @ B2 )
= ( product_Pair @ A @ B @ A7 @ B6 ) )
= ( ( A3 = A7 )
& ( B2 = B6 ) ) ) ).
% old.prod.inject
thf(fact_105_prod_Oinject,axiom,
! [A: $tType,B: $tType,X1: A,X22: B,Y1: A,Y2: B] :
( ( ( product_Pair @ A @ B @ X1 @ X22 )
= ( product_Pair @ A @ B @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X22 = Y2 ) ) ) ).
% prod.inject
thf(fact_106_execute__return,axiom,
! [A: $tType,X: A] :
( ( heap_Time_execute @ A @ ( heap_Time_return @ A @ X ) )
= ( comp @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ ( heap_ext @ product_unit ) @ ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )
@ ^ [H4: heap_ext @ product_unit] : ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ X @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H4 @ ( one_one @ nat ) ) ) ) ) ).
% execute_return
thf(fact_107_execute__ureturn,axiom,
! [A: $tType,X: A] :
( ( heap_Time_execute @ A @ ( heap_Time_ureturn @ A @ X ) )
= ( comp @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ ( heap_ext @ product_unit ) @ ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )
@ ^ [H4: heap_ext @ product_unit] : ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ X @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H4 @ ( zero_zero @ nat ) ) ) ) ) ).
% execute_ureturn
thf(fact_108_prod__induct7,axiom,
! [G3: $tType,F3: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) ) )] :
( ! [A4: A,B3: B,C3: C,D2: D,E3: E,F4: F3,G4: G3] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) ) @ B3 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F3 @ G3 ) @ E3 @ ( product_Pair @ F3 @ G3 @ F4 @ G4 ) ) ) ) ) ) )
=> ( P @ X ) ) ).
% prod_induct7
thf(fact_109_prod__induct6,axiom,
! [F3: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) )] :
( ! [A4: A,B3: B,C3: C,D2: D,E3: E,F4: F3] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) @ B3 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ F3 ) @ D2 @ ( product_Pair @ E @ F3 @ E3 @ F4 ) ) ) ) ) )
=> ( P @ X ) ) ).
% prod_induct6
thf(fact_110_prod__induct5,axiom,
! [E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
( ! [A4: A,B3: B,C3: C,D2: D,E3: E] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B3 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C3 @ ( product_Pair @ D @ E @ D2 @ E3 ) ) ) ) )
=> ( P @ X ) ) ).
% prod_induct5
thf(fact_111_prod__induct4,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
( ! [A4: A,B3: B,C3: C,D2: D] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B3 @ ( product_Pair @ C @ D @ C3 @ D2 ) ) ) )
=> ( P @ X ) ) ).
% prod_induct4
thf(fact_112_prod__induct3,axiom,
! [C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ C )] :
( ! [A4: A,B3: B,C3: C] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A4 @ ( product_Pair @ B @ C @ B3 @ C3 ) ) )
=> ( P @ X ) ) ).
% prod_induct3
thf(fact_113_prod__cases7,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F3: $tType,G3: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) ) )] :
~ ! [A4: A,B3: B,C3: C,D2: D,E3: E,F4: F3,G4: G3] :
( Y3
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) ) @ B3 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F3 @ G3 ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F3 @ G3 ) @ E3 @ ( product_Pair @ F3 @ G3 @ F4 @ G4 ) ) ) ) ) ) ) ).
% prod_cases7
thf(fact_114_prod__cases6,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F3: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) )] :
~ ! [A4: A,B3: B,C3: C,D2: D,E3: E,F4: F3] :
( Y3
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) ) @ B3 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F3 ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ F3 ) @ D2 @ ( product_Pair @ E @ F3 @ E3 @ F4 ) ) ) ) ) ) ).
% prod_cases6
thf(fact_115_timeFrame_Ocases,axiom,
! [A: $tType,X: product_prod @ nat @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )] :
( ! [N2: nat,R2: A,H6: heap_ext @ product_unit,N4: nat] :
( X
!= ( 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 ) @ R2 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H6 @ N4 ) ) ) ) )
=> ~ ! [N2: nat] :
( X
!= ( 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_116_old_Oprod_Oexhaust,axiom,
! [A: $tType,B: $tType,Y3: product_prod @ A @ B] :
~ ! [A4: A,B3: B] :
( Y3
!= ( product_Pair @ A @ B @ A4 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_117_surj__pair,axiom,
! [A: $tType,B: $tType,P4: product_prod @ A @ B] :
? [X4: A,Y4: B] :
( P4
= ( product_Pair @ A @ B @ X4 @ Y4 ) ) ).
% surj_pair
thf(fact_118_prod__cases,axiom,
! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,P4: product_prod @ A @ B] :
( ! [A4: A,B3: B] : ( P @ ( product_Pair @ A @ B @ A4 @ B3 ) )
=> ( P @ P4 ) ) ).
% prod_cases
thf(fact_119_Pair__inject,axiom,
! [A: $tType,B: $tType,A3: A,B2: B,A7: A,B6: B] :
( ( ( product_Pair @ A @ B @ A3 @ B2 )
= ( product_Pair @ A @ B @ A7 @ B6 ) )
=> ~ ( ( A3 = A7 )
=> ( B2 != B6 ) ) ) ).
% Pair_inject
thf(fact_120_comp__def,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comp @ B @ C @ A )
= ( ^ [F2: B > C,G2: A > B,X2: A] : ( F2 @ ( G2 @ X2 ) ) ) ) ).
% comp_def
thf(fact_121_comp__assoc,axiom,
! [B: $tType,D: $tType,C: $tType,A: $tType,F: D > B,G: C > D,H: A > C] :
( ( comp @ C @ B @ A @ ( comp @ D @ B @ C @ F @ G ) @ H )
= ( comp @ D @ B @ A @ F @ ( comp @ C @ D @ A @ G @ H ) ) ) ).
% comp_assoc
thf(fact_122_comp__eq__dest,axiom,
! [C: $tType,B: $tType,D: $tType,A: $tType,A3: C > B,B2: A > C,C2: D > B,D3: A > D,V: A] :
( ( ( comp @ C @ B @ A @ A3 @ B2 )
= ( comp @ D @ B @ A @ C2 @ D3 ) )
=> ( ( A3 @ ( B2 @ V ) )
= ( C2 @ ( D3 @ V ) ) ) ) ).
% comp_eq_dest
thf(fact_123_comp__eq__elim,axiom,
! [C: $tType,B: $tType,D: $tType,A: $tType,A3: C > B,B2: A > C,C2: D > B,D3: A > D] :
( ( ( comp @ C @ B @ A @ A3 @ B2 )
= ( comp @ D @ B @ A @ C2 @ D3 ) )
=> ! [V2: A] :
( ( A3 @ ( B2 @ V2 ) )
= ( C2 @ ( D3 @ V2 ) ) ) ) ).
% comp_eq_elim
thf(fact_124_comp__eq__dest__lhs,axiom,
! [C: $tType,B: $tType,A: $tType,A3: C > B,B2: A > C,C2: A > B,V: A] :
( ( ( comp @ C @ B @ A @ A3 @ B2 )
= C2 )
=> ( ( A3 @ ( B2 @ V ) )
= ( C2 @ V ) ) ) ).
% comp_eq_dest_lhs
thf(fact_125_effectI,axiom,
! [A: $tType,C2: heap_Time_Heap @ A,H: heap_ext @ product_unit,R: A,H2: 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 @ H2 @ N ) ) ) )
=> ( heap_Time_effect @ A @ C2 @ H @ H2 @ R @ N ) ) ).
% effectI
thf(fact_126_effect__def,axiom,
! [A: $tType] :
( ( heap_Time_effect @ A )
= ( ^ [C4: heap_Time_Heap @ A,H4: heap_ext @ product_unit,H7: heap_ext @ product_unit,R4: A,N5: nat] :
( ( heap_Time_execute @ A @ C4 @ H4 )
= ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R4 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H7 @ N5 ) ) ) ) ) ) ).
% effect_def
thf(fact_127_ureturn__def,axiom,
! [A: $tType] :
( ( heap_Time_ureturn @ A )
= ( ^ [X2: A] :
( heap_Time_heap @ A
@ ^ [H4: heap_ext @ product_unit] : ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ X2 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H4 @ ( zero_zero @ nat ) ) ) ) ) ) ).
% ureturn_def
thf(fact_128_return__def,axiom,
! [A: $tType] :
( ( heap_Time_return @ A )
= ( ^ [X2: A] :
( heap_Time_heap @ A
@ ^ [H4: heap_ext @ product_unit] : ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ X2 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H4 @ ( one_one @ nat ) ) ) ) ) ) ).
% return_def
thf(fact_129_execute__bind__eq__SomeI,axiom,
! [A: $tType,B: $tType,F: heap_Time_Heap @ A,H: heap_ext @ product_unit,X: A,H2: heap_ext @ product_unit,N: nat,G: A > ( heap_Time_Heap @ B ),Y3: B,H5: 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 ) @ X @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H2 @ N ) ) ) )
=> ( ( ( heap_Time_execute @ B @ ( G @ X ) @ H2 )
= ( some @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ Y3 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H5 @ 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 ) @ Y3 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H5 @ ( plus_plus @ nat @ N @ N3 ) ) ) ) ) ) ) ).
% execute_bind_eq_SomeI
thf(fact_130_success__bind__executeI,axiom,
! [A: $tType,B: $tType,F: heap_Time_Heap @ A,H: heap_ext @ product_unit,X: A,H2: 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 ) @ X @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H2 @ N ) ) ) )
=> ( ( heap_Time_success @ B @ ( G @ X ) @ H2 )
=> ( heap_Time_success @ B @ ( heap_Time_bind @ A @ B @ F @ G ) @ H ) ) ) ).
% success_bind_executeI
thf(fact_131_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_132_execute__assert_I1_J,axiom,
! [A: $tType,P: A > $o,X: A,H: heap_ext @ product_unit] :
( ( P @ X )
=> ( ( heap_Time_execute @ A @ ( heap_Time_assert @ A @ P @ X ) @ H )
= ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ X @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H @ ( one_one @ nat ) ) ) ) ) ) ).
% execute_assert(1)
thf(fact_133_tap__def,axiom,
! [A: $tType] :
( ( heap_Time_tap @ A )
= ( ^ [F2: ( heap_ext @ product_unit ) > A] :
( heap_Time_Heap2 @ A
@ ^ [H4: 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 @ H4 ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H4 @ ( one_one @ nat ) ) ) ) ) ) ) ).
% tap_def
thf(fact_134_prod__cases3,axiom,
! [A: $tType,B: $tType,C: $tType,Y3: product_prod @ A @ ( product_prod @ B @ C )] :
~ ! [A4: A,B3: B,C3: C] :
( Y3
!= ( product_Pair @ A @ ( product_prod @ B @ C ) @ A4 @ ( product_Pair @ B @ C @ B3 @ C3 ) ) ) ).
% prod_cases3
thf(fact_135_prod__cases4,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
~ ! [A4: A,B3: B,C3: C,D2: D] :
( Y3
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B3 @ ( product_Pair @ C @ D @ C3 @ D2 ) ) ) ) ).
% prod_cases4
thf(fact_136_prod__cases5,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
~ ! [A4: A,B3: B,C3: C,D2: D,E3: E] :
( Y3
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B3 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C3 @ ( product_Pair @ D @ E @ D2 @ E3 ) ) ) ) ) ).
% prod_cases5
thf(fact_137_old_Oprod_Orec,axiom,
! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A3: A,B2: B] :
( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A3 @ B2 ) )
= ( F1 @ A3 @ B2 ) ) ).
% old.prod.rec
thf(fact_138_execute__bind_I1_J,axiom,
! [A: $tType,B: $tType,F: heap_Time_Heap @ A,H: heap_ext @ product_unit,X: A,H2: 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 ) @ X @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H2 @ N ) ) ) )
=> ( ( heap_Time_execute @ B @ ( heap_Time_bind @ A @ B @ F @ G ) @ H )
= ( heap_Time_timeFrame @ B @ N @ ( heap_Time_execute @ B @ ( G @ X ) @ H2 ) ) ) ) ).
% execute_bind(1)
thf(fact_139_timeFrame_Oelims,axiom,
! [A: $tType,X: nat,Xa: option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ),Y3: option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )] :
( ( ( heap_Time_timeFrame @ A @ X @ Xa )
= Y3 )
=> ( ! [R2: A,H6: heap_ext @ product_unit,N4: nat] :
( ( Xa
= ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R2 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H6 @ N4 ) ) ) )
=> ( Y3
!= ( 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 @ H6 @ ( plus_plus @ nat @ X @ N4 ) ) ) ) ) )
=> ~ ( ( Xa
= ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) )
=> ( Y3
!= ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ) ) ).
% timeFrame.elims
thf(fact_140_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_141_K__record__comp,axiom,
! [C: $tType,B: $tType,A: $tType,C2: B,F: A > C] :
( ( comp @ C @ B @ A
@ ^ [X2: C] : C2
@ F )
= ( ^ [X2: A] : C2 ) ) ).
% K_record_comp
thf(fact_142_fun_Omap__ident,axiom,
! [A: $tType,D: $tType,T2: D > A] :
( ( comp @ A @ A @ D
@ ^ [X2: A] : X2
@ T2 )
= T2 ) ).
% fun.map_ident
thf(fact_143_pred__equals__eq2,axiom,
! [B: $tType,A: $tType,R5: set @ ( product_prod @ A @ B ),S3: set @ ( product_prod @ A @ B )] :
( ( ( ^ [X2: A,Y: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y ) @ R5 ) )
= ( ^ [X2: A,Y: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y ) @ S3 ) ) )
= ( R5 = S3 ) ) ).
% pred_equals_eq2
thf(fact_144_internal__case__prod__conv,axiom,
! [B: $tType,A: $tType,C: $tType,C2: B > C > A,A3: B,B2: C] :
( ( produc5280177257484947105e_prod @ B @ C @ A @ C2 @ ( product_Pair @ B @ C @ A3 @ B2 ) )
= ( C2 @ A3 @ B2 ) ) ).
% internal_case_prod_conv
thf(fact_145_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_146_wait__def,axiom,
( heap_Time_wait
= ( ^ [N5: nat] :
( heap_Time_Heap2 @ product_unit
@ ^ [H4: 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 @ H4 @ N5 ) ) ) ) ) ) ).
% wait_def
thf(fact_147_one__natural_Orsp,axiom,
( ( one_one @ nat )
= ( one_one @ nat ) ) ).
% one_natural.rsp
thf(fact_148_fun_Omap__comp,axiom,
! [B: $tType,C: $tType,A: $tType,D: $tType,G: B > C,F: A > B,V: D > A] :
( ( comp @ B @ C @ D @ G @ ( comp @ A @ B @ D @ F @ V ) )
= ( comp @ A @ C @ D @ ( comp @ B @ C @ A @ G @ F ) @ V ) ) ).
% fun.map_comp
thf(fact_149_unit__abs__eta__conv,axiom,
! [A: $tType,F: product_unit > A] :
( ( ^ [U: product_unit] : ( F @ product_Unity ) )
= F ) ).
% unit_abs_eta_conv
thf(fact_150_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_151_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_152_old_Ounit_Oexhaust,axiom,
! [Y3: product_unit] : Y3 = product_Unity ).
% old.unit.exhaust
thf(fact_153_unit__all__impI,axiom,
! [P: product_unit > $o,Q: product_unit > $o] :
( ( ( P @ product_Unity )
=> ( Q @ product_Unity ) )
=> ! [X5: product_unit] :
( ( P @ X5 )
=> ( Q @ X5 ) ) ) ).
% unit_all_impI
thf(fact_154_eq__sym__Unity__conv,axiom,
! [X: $o] :
( ( X
= ( product_Unity = product_Unity ) )
= X ) ).
% eq_sym_Unity_conv
thf(fact_155_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_156_zero__natural_Orsp,axiom,
( ( zero_zero @ nat )
= ( zero_zero @ nat ) ) ).
% zero_natural.rsp
thf(fact_157_old_Ounit_Orec,axiom,
! [T: $tType,F1: T] :
( ( product_rec_unit @ T @ F1 @ product_Unity )
= F1 ) ).
% old.unit.rec
thf(fact_158_default__unit__def,axiom,
( ( default_default @ product_unit )
= product_Unity ) ).
% default_unit_def
thf(fact_159_timeFrame_Opelims,axiom,
! [A: $tType,X: nat,Xa: option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ),Y3: option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )] :
( ( ( heap_Time_timeFrame @ A @ X @ Xa )
= Y3 )
=> ( ( accp @ ( product_prod @ nat @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) @ ( heap_T5500966940807335491me_rel @ A ) @ ( product_Pair @ nat @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ X @ Xa ) )
=> ( ! [R2: A,H6: heap_ext @ product_unit,N4: nat] :
( ( Xa
= ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R2 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H6 @ N4 ) ) ) )
=> ( ( Y3
= ( 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 @ H6 @ ( plus_plus @ nat @ X @ N4 ) ) ) ) )
=> ~ ( accp @ ( product_prod @ nat @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) @ ( heap_T5500966940807335491me_rel @ A ) @ ( product_Pair @ nat @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ X @ ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R2 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H6 @ N4 ) ) ) ) ) ) )
=> ~ ( ( Xa
= ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) )
=> ( ( Y3
= ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) )
=> ~ ( accp @ ( product_prod @ nat @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) @ ( heap_T5500966940807335491me_rel @ A ) @ ( product_Pair @ nat @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ X @ ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ) ) ) ) ) ).
% timeFrame.pelims
thf(fact_160_fold__atLeastAtMost__nat_Ocases,axiom,
! [A: $tType,X: product_prod @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) )] :
~ ! [F4: nat > A > A,A4: nat,B3: nat,Acc: A] :
( X
!= ( product_Pair @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) @ F4 @ ( product_Pair @ nat @ ( product_prod @ nat @ A ) @ A4 @ ( product_Pair @ nat @ A @ B3 @ Acc ) ) ) ) ).
% fold_atLeastAtMost_nat.cases
thf(fact_161_type__copy__map__cong0,axiom,
! [B: $tType,D: $tType,E: $tType,A: $tType,C: $tType,M2: B > A,G: C > B,X: C,N6: D > A,H: C > D,F: A > E] :
( ( ( M2 @ ( G @ X ) )
= ( N6 @ ( H @ X ) ) )
=> ( ( comp @ B @ E @ C @ ( comp @ A @ E @ B @ F @ M2 ) @ G @ X )
= ( comp @ D @ E @ C @ ( comp @ A @ E @ D @ F @ N6 ) @ H @ X ) ) ) ).
% type_copy_map_cong0
thf(fact_162_comp__apply__eq,axiom,
! [B: $tType,D: $tType,A: $tType,C: $tType,F: B > A,G: C > B,X: C,H: D > A,K: C > D] :
( ( ( F @ ( G @ X ) )
= ( H @ ( K @ X ) ) )
=> ( ( comp @ B @ A @ C @ F @ G @ X )
= ( comp @ D @ A @ C @ H @ K @ X ) ) ) ).
% comp_apply_eq
thf(fact_163_comp__cong,axiom,
! [C: $tType,B: $tType,D: $tType,A: $tType,E: $tType,F: B > A,G: C > B,X: C,F5: D > A,G5: E > D,X6: E] :
( ( ( F @ ( G @ X ) )
= ( F5 @ ( G5 @ X6 ) ) )
=> ( ( comp @ B @ A @ C @ F @ G @ X )
= ( comp @ D @ A @ E @ F5 @ G5 @ X6 ) ) ) ).
% comp_cong
thf(fact_164_Ints__odd__nonzero,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [A3: A] :
( ( member @ A @ A3 @ ( ring_1_Ints @ A ) )
=> ( ( plus_plus @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A3 ) @ A3 )
!= ( zero_zero @ A ) ) ) ) ).
% Ints_odd_nonzero
thf(fact_165_is__num__normalize_I1_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 )
= ( plus_plus @ A @ A3 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% is_num_normalize(1)
thf(fact_166_old_Ounit_Ocase,axiom,
! [A: $tType,F: A] :
( ( product_case_unit @ A @ F @ product_Unity )
= F ) ).
% old.unit.case
thf(fact_167_ssubst__Pair__rhs,axiom,
! [B: $tType,A: $tType,R: A,S: B,R5: set @ ( product_prod @ A @ B ),S4: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R @ S ) @ R5 )
=> ( ( S4 = S )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R @ S4 ) @ R5 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_168_unit_Ocase__distrib,axiom,
! [A: $tType,B: $tType,H: A > B,F: A,Unit: product_unit] :
( ( H @ ( product_case_unit @ A @ F @ Unit ) )
= ( product_case_unit @ B @ ( H @ F ) @ Unit ) ) ).
% unit.case_distrib
thf(fact_169_case__unit__Unity,axiom,
! [A: $tType] :
( ( product_case_unit @ A )
= ( ^ [F2: A,U: product_unit] : F2 ) ) ).
% case_unit_Unity
thf(fact_170_Ints__0,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( member @ A @ ( zero_zero @ A ) @ ( ring_1_Ints @ A ) ) ) ).
% Ints_0
thf(fact_171_Ints__add,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [A3: A,B2: A] :
( ( member @ A @ A3 @ ( ring_1_Ints @ A ) )
=> ( ( member @ A @ B2 @ ( ring_1_Ints @ A ) )
=> ( member @ A @ ( plus_plus @ A @ A3 @ B2 ) @ ( ring_1_Ints @ A ) ) ) ) ) ).
% Ints_add
thf(fact_172_Ints__1,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( member @ A @ ( one_one @ A ) @ ( ring_1_Ints @ A ) ) ) ).
% Ints_1
thf(fact_173_Ints__double__eq__0__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [A3: A] :
( ( member @ A @ A3 @ ( ring_1_Ints @ A ) )
=> ( ( ( plus_plus @ A @ A3 @ A3 )
= ( zero_zero @ A ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ) ).
% Ints_double_eq_0_iff
thf(fact_174_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_175_Ints__odd__less__0,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] :
( ( member @ A @ A3 @ ( ring_1_Ints @ A ) )
=> ( ( ord_less @ A @ ( plus_plus @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A3 ) @ A3 ) @ ( zero_zero @ A ) )
= ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ) ).
% Ints_odd_less_0
thf(fact_176_dbl__inc__def,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( neg_numeral_dbl_inc @ A )
= ( ^ [X2: A] : ( plus_plus @ A @ ( plus_plus @ A @ X2 @ X2 ) @ ( one_one @ A ) ) ) ) ) ).
% dbl_inc_def
thf(fact_177_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_178_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_179_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_180_curry__conv,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( product_curry @ B @ C @ A )
= ( ^ [F2: ( product_prod @ B @ C ) > A,A6: B,B5: C] : ( F2 @ ( product_Pair @ B @ C @ A6 @ B5 ) ) ) ) ).
% curry_conv
thf(fact_181_case__optionE,axiom,
! [A: $tType,P: $o,Q: A > $o,X: option @ A] :
( ( case_option @ $o @ A @ P @ Q @ X )
=> ( ( ( X
= ( none @ A ) )
=> ~ P )
=> ~ ! [Y4: A] :
( ( X
= ( some @ A @ Y4 ) )
=> ~ ( Q @ Y4 ) ) ) ) ).
% case_optionE
thf(fact_182_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_183_option_Ocollapse,axiom,
! [A: $tType,Option: option @ A] :
( ( Option
!= ( none @ A ) )
=> ( ( some @ A @ ( the2 @ A @ Option ) )
= Option ) ) ).
% option.collapse
thf(fact_184_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_185_curryI,axiom,
! [A: $tType,B: $tType,F: ( product_prod @ A @ B ) > $o,A3: A,B2: B] :
( ( F @ ( product_Pair @ A @ B @ A3 @ B2 ) )
=> ( product_curry @ A @ B @ $o @ F @ A3 @ B2 ) ) ).
% curryI
thf(fact_186_add_Oinverse__inverse,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A] :
( ( uminus_uminus @ A @ ( uminus_uminus @ A @ A3 ) )
= A3 ) ) ).
% add.inverse_inverse
thf(fact_187_neg__equal__iff__equal,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B2: A] :
( ( ( uminus_uminus @ A @ A3 )
= ( uminus_uminus @ A @ B2 ) )
= ( A3 = B2 ) ) ) ).
% neg_equal_iff_equal
thf(fact_188_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_189_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% less_nat_zero_code
thf(fact_190_neq0__conv,axiom,
! [N: nat] :
( ( N
!= ( zero_zero @ nat ) )
= ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% neq0_conv
thf(fact_191_bot__nat__0_Onot__eq__extremum,axiom,
! [A3: nat] :
( ( A3
!= ( zero_zero @ nat ) )
= ( ord_less @ nat @ ( zero_zero @ nat ) @ A3 ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_192_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_193_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_194_add__less__cancel__right,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A3 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) )
= ( ord_less @ A @ A3 @ B2 ) ) ) ).
% add_less_cancel_right
thf(fact_195_add__less__cancel__left,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ C2 @ A3 ) @ ( plus_plus @ A @ C2 @ B2 ) )
= ( ord_less @ A @ A3 @ B2 ) ) ) ).
% add_less_cancel_left
thf(fact_196_neg__equal__zero,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A3: A] :
( ( ( uminus_uminus @ A @ A3 )
= A3 )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% neg_equal_zero
thf(fact_197_equal__neg__zero,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A3: A] :
( ( A3
= ( uminus_uminus @ A @ A3 ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% equal_neg_zero
thf(fact_198_neg__equal__0__iff__equal,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A] :
( ( ( uminus_uminus @ A @ A3 )
= ( zero_zero @ A ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% neg_equal_0_iff_equal
thf(fact_199_neg__0__equal__iff__equal,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A] :
( ( ( zero_zero @ A )
= ( uminus_uminus @ A @ A3 ) )
= ( ( zero_zero @ A )
= A3 ) ) ) ).
% neg_0_equal_iff_equal
thf(fact_200_add_Oinverse__neutral,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ( ( uminus_uminus @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% add.inverse_neutral
thf(fact_201_neg__less__iff__less,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [B2: A,A3: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A3 ) )
= ( ord_less @ A @ A3 @ B2 ) ) ) ).
% neg_less_iff_less
thf(fact_202_minus__add__distrib,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A3: A,B2: A] :
( ( uminus_uminus @ A @ ( plus_plus @ A @ A3 @ B2 ) )
= ( plus_plus @ A @ ( uminus_uminus @ A @ A3 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ).
% minus_add_distrib
thf(fact_203_minus__add__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B2: A] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ A3 ) @ ( plus_plus @ A @ A3 @ B2 ) )
= B2 ) ) ).
% minus_add_cancel
thf(fact_204_add__minus__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B2: A] :
( ( plus_plus @ A @ A3 @ ( plus_plus @ A @ ( uminus_uminus @ A @ A3 ) @ B2 ) )
= B2 ) ) ).
% add_minus_cancel
thf(fact_205_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_206_less__one,axiom,
! [N: nat] :
( ( ord_less @ nat @ N @ ( one_one @ nat ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% less_one
thf(fact_207_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A3 @ A3 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_208_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A3 @ A3 ) @ ( zero_zero @ A ) )
= ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_209_less__add__same__cancel2,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ ( plus_plus @ A @ B2 @ A3 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ).
% less_add_same_cancel2
thf(fact_210_less__add__same__cancel1,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ ( plus_plus @ A @ A3 @ B2 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ).
% less_add_same_cancel1
thf(fact_211_add__less__same__cancel2,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A3 @ B2 ) @ B2 )
= ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% add_less_same_cancel2
thf(fact_212_add__less__same__cancel1,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [B2: A,A3: A] :
( ( ord_less @ A @ ( plus_plus @ A @ B2 @ A3 ) @ B2 )
= ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% add_less_same_cancel1
thf(fact_213_less__neg__neg,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A3: A] :
( ( ord_less @ A @ A3 @ ( uminus_uminus @ A @ A3 ) )
= ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% less_neg_neg
thf(fact_214_neg__less__pos,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ A3 ) @ A3 )
= ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% neg_less_pos
thf(fact_215_neg__0__less__iff__less,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ A3 ) )
= ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% neg_0_less_iff_less
thf(fact_216_neg__less__0__iff__less,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ A3 ) @ ( zero_zero @ A ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% neg_less_0_iff_less
thf(fact_217_ab__left__minus,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ A3 ) @ A3 )
= ( zero_zero @ A ) ) ) ).
% ab_left_minus
thf(fact_218_add_Oright__inverse,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A] :
( ( plus_plus @ A @ A3 @ ( uminus_uminus @ A @ A3 ) )
= ( zero_zero @ A ) ) ) ).
% add.right_inverse
thf(fact_219_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_220_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_221_equation__minus__iff,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B2: A] :
( ( A3
= ( uminus_uminus @ A @ B2 ) )
= ( B2
= ( uminus_uminus @ A @ A3 ) ) ) ) ).
% equation_minus_iff
thf(fact_222_minus__equation__iff,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B2: A] :
( ( ( uminus_uminus @ A @ A3 )
= B2 )
= ( ( uminus_uminus @ A @ B2 )
= A3 ) ) ) ).
% minus_equation_iff
thf(fact_223_less__minus__iff,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ ( uminus_uminus @ A @ B2 ) )
= ( ord_less @ A @ B2 @ ( uminus_uminus @ A @ A3 ) ) ) ) ).
% less_minus_iff
thf(fact_224_minus__less__iff,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ A3 ) @ B2 )
= ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ A3 ) ) ) ).
% minus_less_iff
thf(fact_225_measure__induct,axiom,
! [B: $tType,A: $tType] :
( ( wellorder @ B )
=> ! [F: A > B,P: A > $o,A3: A] :
( ! [X4: A] :
( ! [Y5: A] :
( ( ord_less @ B @ ( F @ Y5 ) @ ( F @ X4 ) )
=> ( P @ Y5 ) )
=> ( P @ X4 ) )
=> ( P @ A3 ) ) ) ).
% measure_induct
thf(fact_226_measure__induct__rule,axiom,
! [B: $tType,A: $tType] :
( ( wellorder @ B )
=> ! [F: A > B,P: A > $o,A3: A] :
( ! [X4: A] :
( ! [Y5: A] :
( ( ord_less @ B @ ( F @ Y5 ) @ ( F @ X4 ) )
=> ( P @ Y5 ) )
=> ( P @ X4 ) )
=> ( P @ A3 ) ) ) ).
% measure_induct_rule
thf(fact_227_verit__comp__simplify1_I1_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A] :
~ ( ord_less @ A @ A3 @ A3 ) ) ).
% verit_comp_simplify1(1)
thf(fact_228_verit__negate__coefficient_I3_J,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B2: A] :
( ( A3 = B2 )
=> ( ( uminus_uminus @ A @ A3 )
= ( uminus_uminus @ A @ B2 ) ) ) ) ).
% verit_negate_coefficient(3)
thf(fact_229_verit__negate__coefficient_I2_J,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A3 ) ) ) ) ).
% verit_negate_coefficient(2)
thf(fact_230_linorder__neqE__linordered__idom,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y3: A] :
( ( X != Y3 )
=> ( ~ ( ord_less @ A @ X @ Y3 )
=> ( ord_less @ A @ Y3 @ X ) ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_231_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_232_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_233_option_Ocase__eq__if,axiom,
! [A: $tType,B: $tType] :
( ( case_option @ B @ A )
= ( ^ [F12: B,F22: A > B,Option2: option @ A] :
( if @ B
@ ( Option2
= ( none @ A ) )
@ F12
@ ( F22 @ ( the2 @ A @ Option2 ) ) ) ) ) ).
% option.case_eq_if
thf(fact_234_option_Ocase__distrib,axiom,
! [C: $tType,B: $tType,A: $tType,H: B > C,F1: B,F23: A > B,Option: option @ A] :
( ( H @ ( case_option @ B @ A @ F1 @ F23 @ Option ) )
= ( case_option @ C @ A @ ( H @ F1 )
@ ^ [X2: A] : ( H @ ( F23 @ X2 ) )
@ Option ) ) ).
% option.case_distrib
thf(fact_235_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_236_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_237_add_Oinverse__distrib__swap,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B2: A] :
( ( uminus_uminus @ A @ ( plus_plus @ A @ A3 @ B2 ) )
= ( plus_plus @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A3 ) ) ) ) ).
% add.inverse_distrib_swap
thf(fact_238_group__cancel_Oneg1,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A5: A,K: A,A3: A] :
( ( A5
= ( plus_plus @ A @ K @ A3 ) )
=> ( ( uminus_uminus @ A @ A5 )
= ( plus_plus @ A @ ( uminus_uminus @ A @ K ) @ ( uminus_uminus @ A @ A3 ) ) ) ) ) ).
% group_cancel.neg1
thf(fact_239_is__num__normalize_I8_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [A3: A,B2: A] :
( ( uminus_uminus @ A @ ( plus_plus @ A @ A3 @ B2 ) )
= ( plus_plus @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A3 ) ) ) ) ).
% is_num_normalize(8)
thf(fact_240_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_241_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_242_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_243_not__less__zero,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [N: A] :
~ ( ord_less @ A @ N @ ( zero_zero @ A ) ) ) ).
% not_less_zero
thf(fact_244_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_245_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_246_add__less__imp__less__right,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A3 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) )
=> ( ord_less @ A @ A3 @ B2 ) ) ) ).
% add_less_imp_less_right
thf(fact_247_add__less__imp__less__left,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ C2 @ A3 ) @ ( plus_plus @ A @ C2 @ B2 ) )
=> ( ord_less @ A @ A3 @ B2 ) ) ) ).
% add_less_imp_less_left
thf(fact_248_add__strict__right__mono,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ord_less @ A @ ( plus_plus @ A @ A3 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% add_strict_right_mono
thf(fact_249_add__strict__left__mono,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ord_less @ A @ ( plus_plus @ A @ C2 @ A3 ) @ ( plus_plus @ A @ C2 @ B2 ) ) ) ) ).
% add_strict_left_mono
thf(fact_250_add__strict__mono,axiom,
! [A: $tType] :
( ( strict9044650504122735259up_add @ A )
=> ! [A3: A,B2: A,C2: A,D3: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ C2 @ D3 )
=> ( ord_less @ A @ ( plus_plus @ A @ A3 @ C2 ) @ ( plus_plus @ A @ B2 @ D3 ) ) ) ) ) ).
% add_strict_mono
thf(fact_251_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_252_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_253_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_254_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_255_curry__K,axiom,
! [B: $tType,C: $tType,A: $tType,C2: C] :
( ( product_curry @ A @ B @ C
@ ^ [X2: product_prod @ A @ B] : C2 )
= ( ^ [X2: A,Y: B] : C2 ) ) ).
% curry_K
thf(fact_256_infinite__descent0__measure,axiom,
! [A: $tType,V3: A > nat,P: A > $o,X: A] :
( ! [X4: A] :
( ( ( V3 @ X4 )
= ( zero_zero @ nat ) )
=> ( P @ X4 ) )
=> ( ! [X4: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( V3 @ X4 ) )
=> ( ~ ( P @ X4 )
=> ? [Y5: A] :
( ( ord_less @ nat @ ( V3 @ Y5 ) @ ( V3 @ X4 ) )
& ~ ( P @ Y5 ) ) ) )
=> ( P @ X ) ) ) ).
% infinite_descent0_measure
thf(fact_257_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ ( zero_zero @ nat ) )
=> ( ! [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ~ ( P @ N2 )
=> ? [M3: nat] :
( ( ord_less @ nat @ M3 @ N2 )
& ~ ( P @ M3 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_258_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ( N
!= ( zero_zero @ nat ) ) ) ).
% gr_implies_not0
thf(fact_259_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% less_zeroE
thf(fact_260_not__less0,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% not_less0
thf(fact_261_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% not_gr0
thf(fact_262_gr0I,axiom,
! [N: nat] :
( ( N
!= ( zero_zero @ nat ) )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% gr0I
thf(fact_263_bot__nat__0_Oextremum__strict,axiom,
! [A3: nat] :
~ ( ord_less @ nat @ A3 @ ( zero_zero @ nat ) ) ).
% bot_nat_0.extremum_strict
thf(fact_264_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_265_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_266_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_267_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_268_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less @ nat @ ( plus_plus @ nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_269_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less @ nat @ ( plus_plus @ nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_270_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_271_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_272_uminus__unit__def,axiom,
( ( uminus_uminus @ product_unit )
= ( ^ [Uu2: product_unit] : product_Unity ) ) ).
% uminus_unit_def
thf(fact_273_option_Osplit__sel,axiom,
! [B: $tType,A: $tType,P: B > $o,F1: B,F23: A > B,Option: option @ A] :
( ( P @ ( case_option @ B @ A @ F1 @ F23 @ Option ) )
= ( ( ( Option
= ( none @ A ) )
=> ( P @ F1 ) )
& ( ( Option
= ( some @ A @ ( the2 @ A @ Option ) ) )
=> ( P @ ( F23 @ ( the2 @ A @ Option ) ) ) ) ) ) ).
% option.split_sel
thf(fact_274_option_Osplit__sel__asm,axiom,
! [B: $tType,A: $tType,P: B > $o,F1: B,F23: A > B,Option: option @ A] :
( ( P @ ( case_option @ B @ A @ F1 @ F23 @ Option ) )
= ( ~ ( ( ( Option
= ( none @ A ) )
& ~ ( P @ F1 ) )
| ( ( Option
= ( some @ A @ ( the2 @ A @ Option ) ) )
& ~ ( P @ ( F23 @ ( the2 @ A @ Option ) ) ) ) ) ) ) ).
% option.split_sel_asm
thf(fact_275_Ints__minus,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [A3: A] :
( ( member @ A @ A3 @ ( ring_1_Ints @ A ) )
=> ( member @ A @ ( uminus_uminus @ A @ A3 ) @ ( ring_1_Ints @ A ) ) ) ) ).
% Ints_minus
thf(fact_276_minus__in__Ints__iff,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X: A] :
( ( member @ A @ ( uminus_uminus @ A @ X ) @ ( ring_1_Ints @ A ) )
= ( member @ A @ X @ ( ring_1_Ints @ A ) ) ) ) ).
% minus_in_Ints_iff
thf(fact_277_option_Oexpand,axiom,
! [A: $tType,Option: option @ A,Option3: option @ A] :
( ( ( Option
= ( none @ A ) )
= ( Option3
= ( none @ A ) ) )
=> ( ( ( Option
!= ( none @ A ) )
=> ( ( Option3
!= ( none @ A ) )
=> ( ( the2 @ A @ Option )
= ( the2 @ A @ Option3 ) ) ) )
=> ( Option = Option3 ) ) ) ).
% option.expand
thf(fact_278_option_Osel,axiom,
! [A: $tType,X22: A] :
( ( the2 @ A @ ( some @ A @ X22 ) )
= X22 ) ).
% option.sel
thf(fact_279_option_Osimps_I4_J,axiom,
! [A: $tType,B: $tType,F1: B,F23: A > B] :
( ( case_option @ B @ A @ F1 @ F23 @ ( none @ A ) )
= F1 ) ).
% option.simps(4)
thf(fact_280_option_Osimps_I5_J,axiom,
! [B: $tType,A: $tType,F1: B,F23: A > B,X22: A] :
( ( case_option @ B @ A @ F1 @ F23 @ ( some @ A @ X22 ) )
= ( F23 @ X22 ) ) ).
% option.simps(5)
thf(fact_281_curryE,axiom,
! [A: $tType,B: $tType,F: ( product_prod @ A @ B ) > $o,A3: A,B2: B] :
( ( product_curry @ A @ B @ $o @ F @ A3 @ B2 )
=> ( F @ ( product_Pair @ A @ B @ A3 @ B2 ) ) ) ).
% curryE
thf(fact_282_curryD,axiom,
! [A: $tType,B: $tType,F: ( product_prod @ A @ B ) > $o,A3: A,B2: B] :
( ( product_curry @ A @ B @ $o @ F @ A3 @ B2 )
=> ( F @ ( product_Pair @ A @ B @ A3 @ B2 ) ) ) ).
% curryD
thf(fact_283_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_284_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_285_neg__eq__iff__add__eq__0,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B2: A] :
( ( ( uminus_uminus @ A @ A3 )
= B2 )
= ( ( plus_plus @ A @ A3 @ B2 )
= ( zero_zero @ A ) ) ) ) ).
% neg_eq_iff_add_eq_0
thf(fact_286_eq__neg__iff__add__eq__0,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B2: A] :
( ( A3
= ( uminus_uminus @ A @ B2 ) )
= ( ( plus_plus @ A @ A3 @ B2 )
= ( zero_zero @ A ) ) ) ) ).
% eq_neg_iff_add_eq_0
thf(fact_287_add_Oinverse__unique,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B2: A] :
( ( ( plus_plus @ A @ A3 @ B2 )
= ( zero_zero @ A ) )
=> ( ( uminus_uminus @ A @ A3 )
= B2 ) ) ) ).
% add.inverse_unique
thf(fact_288_ab__group__add__class_Oab__left__minus,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A3: A] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ A3 ) @ A3 )
= ( zero_zero @ A ) ) ) ).
% ab_group_add_class.ab_left_minus
thf(fact_289_add__eq__0__iff,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B2: A] :
( ( ( plus_plus @ A @ A3 @ B2 )
= ( zero_zero @ A ) )
= ( B2
= ( uminus_uminus @ A @ A3 ) ) ) ) ).
% add_eq_0_iff
thf(fact_290_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_291_add__less__zeroD,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y3: A] :
( ( ord_less @ A @ ( plus_plus @ A @ X @ Y3 ) @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ X @ ( zero_zero @ A ) )
| ( ord_less @ A @ Y3 @ ( zero_zero @ A ) ) ) ) ) ).
% add_less_zeroD
thf(fact_292_pos__add__strict,axiom,
! [A: $tType] :
( ( strict7427464778891057005id_add @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ B2 @ C2 )
=> ( ord_less @ A @ B2 @ ( plus_plus @ A @ A3 @ C2 ) ) ) ) ) ).
% pos_add_strict
thf(fact_293_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ~ ! [C3: A] :
( ( B2
= ( plus_plus @ A @ A3 @ C3 ) )
=> ( C3
= ( zero_zero @ A ) ) ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_294_add__pos__pos,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A3 @ B2 ) ) ) ) ) ).
% add_pos_pos
thf(fact_295_add__neg__neg,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( plus_plus @ A @ A3 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% add_neg_neg
thf(fact_296_not__one__less__zero,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ~ ( ord_less @ A @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ).
% not_one_less_zero
thf(fact_297_zero__less__one,axiom,
! [A: $tType] :
( ( zero_less_one @ A )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).
% zero_less_one
thf(fact_298_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_299_add__mono1,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ord_less @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ ( plus_plus @ A @ B2 @ ( one_one @ A ) ) ) ) ) ).
% add_mono1
thf(fact_300_less__add__one,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A] : ( ord_less @ A @ A3 @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) ) ) ).
% less_add_one
thf(fact_301_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less @ nat @ I @ J )
=> ? [K2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K2 )
& ( ( plus_plus @ nat @ I @ K2 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_302_dbl__def,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( neg_numeral_dbl @ A )
= ( ^ [X2: A] : ( plus_plus @ A @ X2 @ X2 ) ) ) ) ).
% dbl_def
thf(fact_303_option_Oexhaust__sel,axiom,
! [A: $tType,Option: option @ A] :
( ( Option
!= ( none @ A ) )
=> ( Option
= ( some @ A @ ( the2 @ A @ Option ) ) ) ) ).
% option.exhaust_sel
thf(fact_304_curry__def,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( product_curry @ A @ B @ C )
= ( ^ [C4: ( product_prod @ A @ B ) > C,X2: A,Y: B] : ( C4 @ ( product_Pair @ A @ B @ X2 @ Y ) ) ) ) ).
% curry_def
thf(fact_305_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_306_disjE__realizer2,axiom,
! [B: $tType,A: $tType,P: $o,Q: A > $o,X: option @ A,R5: B > $o,F: B,G: A > B] :
( ( case_option @ $o @ A @ P @ Q @ X )
=> ( ( P
=> ( R5 @ F ) )
=> ( ! [Q3: A] :
( ( Q @ Q3 )
=> ( R5 @ ( G @ Q3 ) ) )
=> ( R5 @ ( case_option @ B @ A @ F @ G @ X ) ) ) ) ) ).
% disjE_realizer2
thf(fact_307_sgn__neg,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] :
( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( sgn_sgn @ A @ A3 )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ).
% sgn_neg
thf(fact_308_of__nat__code,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( semiring_1_of_nat @ A )
= ( ^ [N5: nat] :
( semiri8178284476397505188at_aux @ A
@ ^ [I2: A] : ( plus_plus @ A @ I2 @ ( one_one @ A ) )
@ N5
@ ( zero_zero @ A ) ) ) ) ) ).
% of_nat_code
thf(fact_309_add_Ogroup__axioms,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ( group @ A @ ( plus_plus @ A ) @ ( zero_zero @ A ) @ ( uminus_uminus @ A ) ) ) ).
% add.group_axioms
thf(fact_310_fold__atLeastAtMost__nat_Opinduct,axiom,
! [A: $tType,A0: nat > A > A,A1: nat,A22: nat,A32: A,P: ( nat > A > A ) > nat > nat > A > $o] :
( ( accp @ ( product_prod @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) ) @ ( set_fo1817059534552279752at_rel @ A ) @ ( product_Pair @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) @ A0 @ ( product_Pair @ nat @ ( product_prod @ nat @ A ) @ A1 @ ( product_Pair @ nat @ A @ A22 @ A32 ) ) ) )
=> ( ! [F4: nat > A > A,A4: nat,B3: nat,Acc: A] :
( ( accp @ ( product_prod @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) ) @ ( set_fo1817059534552279752at_rel @ A ) @ ( product_Pair @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) @ F4 @ ( product_Pair @ nat @ ( product_prod @ nat @ A ) @ A4 @ ( product_Pair @ nat @ A @ B3 @ Acc ) ) ) )
=> ( ( ~ ( ord_less @ nat @ B3 @ A4 )
=> ( P @ F4 @ ( plus_plus @ nat @ A4 @ ( one_one @ nat ) ) @ B3 @ ( F4 @ A4 @ Acc ) ) )
=> ( P @ F4 @ A4 @ B3 @ Acc ) ) )
=> ( P @ A0 @ A1 @ A22 @ A32 ) ) ) ).
% fold_atLeastAtMost_nat.pinduct
thf(fact_311_sgn__1__neg,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] :
( ( ( sgn_sgn @ A @ A3 )
= ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% sgn_1_neg
thf(fact_312_sgn__if,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( ( sgn_sgn @ A )
= ( ^ [X2: A] :
( if @ A
@ ( X2
= ( zero_zero @ A ) )
@ ( zero_zero @ A )
@ ( if @ A @ ( ord_less @ A @ ( zero_zero @ A ) @ X2 ) @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ) ) ).
% sgn_if
thf(fact_313_is__num_Osimps,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( neg_numeral_is_num @ A )
= ( ^ [A6: A] :
( ( A6
= ( one_one @ A ) )
| ? [X2: A] :
( ( A6
= ( uminus_uminus @ A @ X2 ) )
& ( neg_numeral_is_num @ A @ X2 ) )
| ? [X2: A,Y: A] :
( ( A6
= ( plus_plus @ A @ X2 @ Y ) )
& ( neg_numeral_is_num @ A @ X2 )
& ( neg_numeral_is_num @ A @ Y ) ) ) ) ) ) ).
% is_num.simps
thf(fact_314_is__num_Ocases,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [A3: A] :
( ( neg_numeral_is_num @ A @ A3 )
=> ( ( A3
!= ( one_one @ A ) )
=> ( ! [X4: A] :
( ( A3
= ( uminus_uminus @ A @ X4 ) )
=> ~ ( neg_numeral_is_num @ A @ X4 ) )
=> ~ ! [X4: A,Y4: A] :
( ( A3
= ( plus_plus @ A @ X4 @ Y4 ) )
=> ( ( neg_numeral_is_num @ A @ X4 )
=> ~ ( neg_numeral_is_num @ A @ Y4 ) ) ) ) ) ) ) ).
% is_num.cases
thf(fact_315_Ints__nonzero__abs__less1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A] :
( ( member @ A @ X @ ( ring_1_Ints @ A ) )
=> ( ( ord_less @ A @ ( abs_abs @ A @ X ) @ ( one_one @ A ) )
=> ( X
= ( zero_zero @ A ) ) ) ) ) ).
% Ints_nonzero_abs_less1
thf(fact_316_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_317_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_318_abs__abs,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A3: A] :
( ( abs_abs @ A @ ( abs_abs @ A @ A3 ) )
= ( abs_abs @ A @ A3 ) ) ) ).
% abs_abs
thf(fact_319_abs__idempotent,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] :
( ( abs_abs @ A @ ( abs_abs @ A @ A3 ) )
= ( abs_abs @ A @ A3 ) ) ) ).
% abs_idempotent
thf(fact_320_sgn__sgn,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A3: A] :
( ( sgn_sgn @ A @ ( sgn_sgn @ A @ A3 ) )
= ( sgn_sgn @ A @ A3 ) ) ) ).
% sgn_sgn
thf(fact_321_abs__0__eq,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] :
( ( ( zero_zero @ A )
= ( abs_abs @ A @ A3 ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% abs_0_eq
thf(fact_322_abs__eq__0,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] :
( ( ( abs_abs @ A @ A3 )
= ( zero_zero @ A ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% abs_eq_0
thf(fact_323_abs__zero,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ( ( abs_abs @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% abs_zero
thf(fact_324_abs__0,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ( ( abs_abs @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% abs_0
thf(fact_325_abs__add__abs,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A,B2: A] :
( ( abs_abs @ A @ ( plus_plus @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B2 ) ) )
= ( plus_plus @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B2 ) ) ) ) ).
% abs_add_abs
thf(fact_326_abs__1,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ( ( abs_abs @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% abs_1
thf(fact_327_abs__minus,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A3: A] :
( ( abs_abs @ A @ ( uminus_uminus @ A @ A3 ) )
= ( abs_abs @ A @ A3 ) ) ) ).
% abs_minus
thf(fact_328_abs__minus__cancel,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] :
( ( abs_abs @ A @ ( uminus_uminus @ A @ A3 ) )
= ( abs_abs @ A @ A3 ) ) ) ).
% abs_minus_cancel
thf(fact_329_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_330_sgn__0,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ( ( sgn_sgn @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% sgn_0
thf(fact_331_sgn__1,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ( ( sgn_sgn @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% sgn_1
thf(fact_332_sgn__minus,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A3: A] :
( ( sgn_sgn @ A @ ( uminus_uminus @ A @ A3 ) )
= ( uminus_uminus @ A @ ( sgn_sgn @ A @ A3 ) ) ) ) ).
% sgn_minus
thf(fact_333_zero__less__abs__iff,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( abs_abs @ A @ A3 ) )
= ( A3
!= ( zero_zero @ A ) ) ) ) ).
% zero_less_abs_iff
thf(fact_334_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_335_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_336_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_337_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_338_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_339_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_340_sgn__less,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( sgn_sgn @ A @ A3 ) @ ( zero_zero @ A ) )
= ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% sgn_less
thf(fact_341_sgn__greater,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( sgn_sgn @ A @ A3 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% sgn_greater
thf(fact_342_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_343_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_344_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_345_sgn__pos,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( sgn_sgn @ A @ A3 )
= ( one_one @ A ) ) ) ) ).
% sgn_pos
thf(fact_346_abs__sgn__eq__1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( abs_abs @ A @ ( sgn_sgn @ A @ A3 ) )
= ( one_one @ A ) ) ) ) ).
% abs_sgn_eq_1
thf(fact_347_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_348_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_349_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ N ) ).
% less_not_refl
thf(fact_350_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_351_less__not__refl3,axiom,
! [S: nat,T2: nat] :
( ( ord_less @ nat @ S @ T2 )
=> ( S != T2 ) ) ).
% less_not_refl3
thf(fact_352_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_353_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M3: nat] :
( ( ord_less @ nat @ M3 @ N2 )
=> ( P @ M3 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_354_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ~ ( P @ N2 )
=> ? [M3: nat] :
( ( ord_less @ nat @ M3 @ N2 )
& ~ ( P @ M3 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_355_linorder__neqE__nat,axiom,
! [X: nat,Y3: nat] :
( ( X != Y3 )
=> ( ~ ( ord_less @ nat @ X @ Y3 )
=> ( ord_less @ nat @ Y3 @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_356_infinite__descent__measure,axiom,
! [A: $tType,P: A > $o,V3: A > nat,X: A] :
( ! [X4: A] :
( ~ ( P @ X4 )
=> ? [Y5: A] :
( ( ord_less @ nat @ ( V3 @ Y5 ) @ ( V3 @ X4 ) )
& ~ ( P @ Y5 ) ) )
=> ( P @ X ) ) ).
% infinite_descent_measure
thf(fact_357_nat__int__comparison_I2_J,axiom,
( ( ord_less @ nat )
= ( ^ [A6: nat,B5: nat] : ( ord_less @ int @ ( semiring_1_of_nat @ int @ A6 ) @ ( semiring_1_of_nat @ int @ B5 ) ) ) ) ).
% nat_int_comparison(2)
thf(fact_358_group_Oinverse__distrib__swap,axiom,
! [A: $tType,F: A > A > A,Z: A,Inverse: A > A,A3: A,B2: A] :
( ( group @ A @ F @ Z @ Inverse )
=> ( ( Inverse @ ( F @ A3 @ B2 ) )
= ( F @ ( Inverse @ B2 ) @ ( Inverse @ A3 ) ) ) ) ).
% group.inverse_distrib_swap
thf(fact_359_group_Ogroup__left__neutral,axiom,
! [A: $tType,F: A > A > A,Z: A,Inverse: A > A,A3: A] :
( ( group @ A @ F @ Z @ Inverse )
=> ( ( F @ Z @ A3 )
= A3 ) ) ).
% group.group_left_neutral
thf(fact_360_group_Oinverse__neutral,axiom,
! [A: $tType,F: A > A > A,Z: A,Inverse: A > A] :
( ( group @ A @ F @ Z @ Inverse )
=> ( ( Inverse @ Z )
= Z ) ) ).
% group.inverse_neutral
thf(fact_361_group_Oinverse__inverse,axiom,
! [A: $tType,F: A > A > A,Z: A,Inverse: A > A,A3: A] :
( ( group @ A @ F @ Z @ Inverse )
=> ( ( Inverse @ ( Inverse @ A3 ) )
= A3 ) ) ).
% group.inverse_inverse
thf(fact_362_group_Oinverse__unique,axiom,
! [A: $tType,F: A > A > A,Z: A,Inverse: A > A,A3: A,B2: A] :
( ( group @ A @ F @ Z @ Inverse )
=> ( ( ( F @ A3 @ B2 )
= Z )
=> ( ( Inverse @ A3 )
= B2 ) ) ) ).
% group.inverse_unique
thf(fact_363_group_Oright__inverse,axiom,
! [A: $tType,F: A > A > A,Z: A,Inverse: A > A,A3: A] :
( ( group @ A @ F @ Z @ Inverse )
=> ( ( F @ A3 @ ( Inverse @ A3 ) )
= Z ) ) ).
% group.right_inverse
thf(fact_364_group_Oright__cancel,axiom,
! [A: $tType,F: A > A > A,Z: A,Inverse: A > A,B2: A,A3: A,C2: A] :
( ( group @ A @ F @ Z @ Inverse )
=> ( ( ( F @ B2 @ A3 )
= ( F @ C2 @ A3 ) )
= ( B2 = C2 ) ) ) ).
% group.right_cancel
thf(fact_365_group_Oleft__inverse,axiom,
! [A: $tType,F: A > A > A,Z: A,Inverse: A > A,A3: A] :
( ( group @ A @ F @ Z @ Inverse )
=> ( ( F @ ( Inverse @ A3 ) @ A3 )
= Z ) ) ).
% group.left_inverse
thf(fact_366_group_Oleft__cancel,axiom,
! [A: $tType,F: A > A > A,Z: A,Inverse: A > A,A3: A,B2: A,C2: A] :
( ( group @ A @ F @ Z @ Inverse )
=> ( ( ( F @ A3 @ B2 )
= ( F @ A3 @ C2 ) )
= ( B2 = C2 ) ) ) ).
% group.left_cancel
thf(fact_367_same__sgn__abs__add,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [B2: A,A3: A] :
( ( ( sgn_sgn @ A @ B2 )
= ( sgn_sgn @ A @ A3 ) )
=> ( ( abs_abs @ A @ ( plus_plus @ A @ A3 @ B2 ) )
= ( plus_plus @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B2 ) ) ) ) ) ).
% same_sgn_abs_add
thf(fact_368_nat__less__as__int,axiom,
( ( ord_less @ nat )
= ( ^ [A6: nat,B5: nat] : ( ord_less @ int @ ( semiring_1_of_nat @ int @ A6 ) @ ( semiring_1_of_nat @ int @ B5 ) ) ) ) ).
% nat_less_as_int
thf(fact_369_abs__sgn__eq,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] :
( ( ( A3
= ( zero_zero @ A ) )
=> ( ( abs_abs @ A @ ( sgn_sgn @ A @ A3 ) )
= ( zero_zero @ A ) ) )
& ( ( A3
!= ( zero_zero @ A ) )
=> ( ( abs_abs @ A @ ( sgn_sgn @ A @ A3 ) )
= ( one_one @ A ) ) ) ) ) ).
% abs_sgn_eq
thf(fact_370_abs__eq__0__iff,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A3: A] :
( ( ( abs_abs @ A @ A3 )
= ( zero_zero @ A ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% abs_eq_0_iff
thf(fact_371_abs__one,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( ( abs_abs @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% abs_one
thf(fact_372_abs__eq__iff,axiom,
! [A: $tType] :
( ( linordered_ring @ A )
=> ! [X: A,Y3: A] :
( ( ( abs_abs @ A @ X )
= ( abs_abs @ A @ Y3 ) )
= ( ( X = Y3 )
| ( X
= ( uminus_uminus @ A @ Y3 ) ) ) ) ) ).
% abs_eq_iff
thf(fact_373_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_374_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_375_int__ops_I1_J,axiom,
( ( semiring_1_of_nat @ int @ ( zero_zero @ nat ) )
= ( zero_zero @ int ) ) ).
% int_ops(1)
thf(fact_376_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_377_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_378_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_379_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_380_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_381_zadd__int__left,axiom,
! [M: nat,N: nat,Z: int] :
( ( plus_plus @ int @ ( semiring_1_of_nat @ int @ M ) @ ( plus_plus @ int @ ( semiring_1_of_nat @ int @ N ) @ Z ) )
= ( plus_plus @ int @ ( semiring_1_of_nat @ int @ ( plus_plus @ nat @ M @ N ) ) @ Z ) ) ).
% zadd_int_left
thf(fact_382_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_383_int__ops_I5_J,axiom,
! [A3: nat,B2: nat] :
( ( semiring_1_of_nat @ int @ ( plus_plus @ nat @ A3 @ B2 ) )
= ( plus_plus @ int @ ( semiring_1_of_nat @ int @ A3 ) @ ( semiring_1_of_nat @ int @ B2 ) ) ) ).
% int_ops(5)
thf(fact_384_sgn__0__0,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] :
( ( ( sgn_sgn @ A @ A3 )
= ( zero_zero @ A ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% sgn_0_0
thf(fact_385_sgn__eq__0__iff,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A3: A] :
( ( ( sgn_sgn @ A @ A3 )
= ( zero_zero @ A ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% sgn_eq_0_iff
thf(fact_386_same__sgn__sgn__add,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [B2: A,A3: A] :
( ( ( sgn_sgn @ A @ B2 )
= ( sgn_sgn @ A @ A3 ) )
=> ( ( sgn_sgn @ A @ ( plus_plus @ A @ A3 @ B2 ) )
= ( sgn_sgn @ A @ A3 ) ) ) ) ).
% same_sgn_sgn_add
thf(fact_387_int__ops_I2_J,axiom,
( ( semiring_1_of_nat @ int @ ( one_one @ nat ) )
= ( one_one @ int ) ) ).
% int_ops(2)
thf(fact_388_Ints__abs,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] :
( ( member @ A @ A3 @ ( ring_1_Ints @ A ) )
=> ( member @ A @ ( abs_abs @ A @ A3 ) @ ( ring_1_Ints @ A ) ) ) ) ).
% Ints_abs
thf(fact_389_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_390_is__num__normalize_I6_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [X: A,Y3: A] :
( ( neg_numeral_is_num @ A @ X )
=> ( ( neg_numeral_is_num @ A @ Y3 )
=> ( neg_numeral_is_num @ A @ ( plus_plus @ A @ X @ Y3 ) ) ) ) ) ).
% is_num_normalize(6)
thf(fact_391_is__num__add__commute,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [X: A,Y3: A] :
( ( neg_numeral_is_num @ A @ X )
=> ( ( neg_numeral_is_num @ A @ Y3 )
=> ( ( plus_plus @ A @ X @ Y3 )
= ( plus_plus @ A @ Y3 @ X ) ) ) ) ) ).
% is_num_add_commute
thf(fact_392_is__num__add__left__commute,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [X: A,Y3: A,Z: A] :
( ( neg_numeral_is_num @ A @ X )
=> ( ( neg_numeral_is_num @ A @ Y3 )
=> ( ( plus_plus @ A @ X @ ( plus_plus @ A @ Y3 @ Z ) )
= ( plus_plus @ A @ Y3 @ ( plus_plus @ A @ X @ Z ) ) ) ) ) ) ).
% is_num_add_left_commute
thf(fact_393_is__num__normalize_I4_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( neg_numeral_is_num @ A @ ( one_one @ A ) ) ) ).
% is_num_normalize(4)
thf(fact_394_abs__not__less__zero,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] :
~ ( ord_less @ A @ ( abs_abs @ A @ A3 ) @ ( zero_zero @ A ) ) ) ).
% abs_not_less_zero
thf(fact_395_abs__of__pos,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( abs_abs @ A @ A3 )
= A3 ) ) ) ).
% abs_of_pos
thf(fact_396_abs__less__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( abs_abs @ A @ A3 ) @ B2 )
= ( ( ord_less @ A @ A3 @ B2 )
& ( ord_less @ A @ ( uminus_uminus @ A @ A3 ) @ B2 ) ) ) ) ).
% abs_less_iff
thf(fact_397_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_398_sgn__not__eq__imp,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [B2: A,A3: A] :
( ( ( sgn_sgn @ A @ B2 )
!= ( sgn_sgn @ A @ A3 ) )
=> ( ( ( sgn_sgn @ A @ A3 )
!= ( zero_zero @ A ) )
=> ( ( ( sgn_sgn @ A @ B2 )
!= ( zero_zero @ A ) )
=> ( ( sgn_sgn @ A @ A3 )
= ( uminus_uminus @ A @ ( sgn_sgn @ A @ B2 ) ) ) ) ) ) ) ).
% sgn_not_eq_imp
thf(fact_399_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_400_abs__if__raw,axiom,
! [A: $tType] :
( ( abs_if @ A )
=> ( ( abs_abs @ A )
= ( ^ [A6: A] : ( if @ A @ ( ord_less @ A @ A6 @ ( zero_zero @ A ) ) @ ( uminus_uminus @ A @ A6 ) @ A6 ) ) ) ) ).
% abs_if_raw
thf(fact_401_abs__if,axiom,
! [A: $tType] :
( ( abs_if @ A )
=> ( ( abs_abs @ A )
= ( ^ [A6: A] : ( if @ A @ ( ord_less @ A @ A6 @ ( zero_zero @ A ) ) @ ( uminus_uminus @ A @ A6 ) @ A6 ) ) ) ) ).
% abs_if
thf(fact_402_abs__of__neg,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] :
( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( abs_abs @ A @ A3 )
= ( uminus_uminus @ A @ A3 ) ) ) ) ).
% abs_of_neg
thf(fact_403_sgn__1__pos,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] :
( ( ( sgn_sgn @ A @ A3 )
= ( one_one @ A ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% sgn_1_pos
thf(fact_404_abs__add__one__gt__zero,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A] : ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( one_one @ A ) @ ( abs_abs @ A @ X ) ) ) ) ).
% abs_add_one_gt_zero
thf(fact_405_fold__atLeastAtMost__nat_Opsimps,axiom,
! [A: $tType,F: nat > A > A,A3: nat,B2: nat,Acc2: 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 ) @ A3 @ ( product_Pair @ nat @ A @ B2 @ Acc2 ) ) ) )
=> ( ( ( ord_less @ nat @ B2 @ A3 )
=> ( ( set_fo6178422350223883121st_nat @ A @ F @ A3 @ B2 @ Acc2 )
= Acc2 ) )
& ( ~ ( ord_less @ nat @ B2 @ A3 )
=> ( ( set_fo6178422350223883121st_nat @ A @ F @ A3 @ B2 @ Acc2 )
= ( set_fo6178422350223883121st_nat @ A @ F @ ( plus_plus @ nat @ A3 @ ( one_one @ nat ) ) @ B2 @ ( F @ A3 @ Acc2 ) ) ) ) ) ) ).
% fold_atLeastAtMost_nat.psimps
thf(fact_406_fold__atLeastAtMost__nat_Opelims,axiom,
! [A: $tType,X: nat > A > A,Xa: nat,Xb: nat,Xc: A,Y3: A] :
( ( ( set_fo6178422350223883121st_nat @ A @ X @ Xa @ Xb @ Xc )
= Y3 )
=> ( ( accp @ ( product_prod @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) ) @ ( set_fo1817059534552279752at_rel @ A ) @ ( product_Pair @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) @ X @ ( product_Pair @ nat @ ( product_prod @ nat @ A ) @ Xa @ ( product_Pair @ nat @ A @ Xb @ Xc ) ) ) )
=> ~ ( ( ( ( ord_less @ nat @ Xb @ Xa )
=> ( Y3 = Xc ) )
& ( ~ ( ord_less @ nat @ Xb @ Xa )
=> ( Y3
= ( set_fo6178422350223883121st_nat @ A @ X @ ( plus_plus @ nat @ Xa @ ( one_one @ nat ) ) @ Xb @ ( X @ Xa @ Xc ) ) ) ) )
=> ~ ( accp @ ( product_prod @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) ) @ ( set_fo1817059534552279752at_rel @ A ) @ ( product_Pair @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) @ X @ ( product_Pair @ nat @ ( product_prod @ nat @ A ) @ Xa @ ( product_Pair @ nat @ A @ Xb @ Xc ) ) ) ) ) ) ) ).
% fold_atLeastAtMost_nat.pelims
thf(fact_407_fold__atLeastAtMost__nat_Oelims,axiom,
! [A: $tType,X: nat > A > A,Xa: nat,Xb: nat,Xc: A,Y3: A] :
( ( ( set_fo6178422350223883121st_nat @ A @ X @ Xa @ Xb @ Xc )
= Y3 )
=> ( ( ( ord_less @ nat @ Xb @ Xa )
=> ( Y3 = Xc ) )
& ( ~ ( ord_less @ nat @ Xb @ Xa )
=> ( Y3
= ( set_fo6178422350223883121st_nat @ A @ X @ ( plus_plus @ nat @ Xa @ ( one_one @ nat ) ) @ Xb @ ( X @ Xa @ Xc ) ) ) ) ) ) ).
% fold_atLeastAtMost_nat.elims
thf(fact_408_fold__atLeastAtMost__nat_Osimps,axiom,
! [A: $tType] :
( ( set_fo6178422350223883121st_nat @ A )
= ( ^ [F2: nat > A > A,A6: nat,B5: nat,Acc3: A] : ( if @ A @ ( ord_less @ nat @ B5 @ A6 ) @ Acc3 @ ( set_fo6178422350223883121st_nat @ A @ F2 @ ( plus_plus @ nat @ A6 @ ( one_one @ nat ) ) @ B5 @ ( F2 @ A6 @ Acc3 ) ) ) ) ) ).
% fold_atLeastAtMost_nat.simps
thf(fact_409_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_410_Ints__nonzero__abs__ge1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A] :
( ( member @ A @ X @ ( ring_1_Ints @ A ) )
=> ( ( X
!= ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ ( abs_abs @ A @ X ) ) ) ) ) ).
% Ints_nonzero_abs_ge1
thf(fact_411_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_412_Ints__eq__abs__less1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y3: A] :
( ( member @ A @ X @ ( ring_1_Ints @ A ) )
=> ( ( member @ A @ Y3 @ ( ring_1_Ints @ A ) )
=> ( ( X = Y3 )
= ( ord_less @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X @ Y3 ) ) @ ( one_one @ A ) ) ) ) ) ) ).
% Ints_eq_abs_less1
thf(fact_413_of__nat__zero__less__power__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X: nat,N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ ( semiring_1_of_nat @ A @ X ) @ N ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ X )
| ( N
= ( zero_zero @ nat ) ) ) ) ) ).
% of_nat_zero_less_power_iff
thf(fact_414_in__measure,axiom,
! [A: $tType,X: A,Y3: A,F: A > nat] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ ( measure @ A @ F ) )
= ( ord_less @ nat @ ( F @ X ) @ ( F @ Y3 ) ) ) ).
% in_measure
thf(fact_415_abs__of__nonpos,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( abs_abs @ A @ A3 )
= ( uminus_uminus @ A @ A3 ) ) ) ) ).
% abs_of_nonpos
thf(fact_416_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_417_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_418_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus @ nat @ M @ M )
= ( zero_zero @ nat ) ) ).
% diff_self_eq_0
thf(fact_419_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus @ nat @ ( zero_zero @ nat ) @ N )
= ( zero_zero @ nat ) ) ).
% diff_0_eq_0
thf(fact_420_le0,axiom,
! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% le0
thf(fact_421_bot__nat__0_Oextremum,axiom,
! [A3: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ A3 ) ).
% bot_nat_0.extremum
thf(fact_422_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_423_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_424_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_425_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_426_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_427_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_428_add__le__cancel__right,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) )
= ( ord_less_eq @ A @ A3 @ B2 ) ) ) ).
% add_le_cancel_right
thf(fact_429_add__le__cancel__left,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ C2 @ A3 ) @ ( plus_plus @ A @ C2 @ B2 ) )
= ( ord_less_eq @ A @ A3 @ B2 ) ) ) ).
% add_le_cancel_left
thf(fact_430_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [A3: A] :
( ( minus_minus @ A @ A3 @ A3 )
= ( zero_zero @ A ) ) ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_431_diff__zero,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [A3: A] :
( ( minus_minus @ A @ A3 @ ( zero_zero @ A ) )
= A3 ) ) ).
% diff_zero
thf(fact_432_zero__diff,axiom,
! [A: $tType] :
( ( comm_monoid_diff @ A )
=> ! [A3: A] :
( ( minus_minus @ A @ ( zero_zero @ A ) @ A3 )
= ( zero_zero @ A ) ) ) ).
% zero_diff
thf(fact_433_diff__0__right,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A] :
( ( minus_minus @ A @ A3 @ ( zero_zero @ A ) )
= A3 ) ) ).
% diff_0_right
thf(fact_434_diff__self,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A] :
( ( minus_minus @ A @ A3 @ A3 )
= ( zero_zero @ A ) ) ) ).
% diff_self
thf(fact_435_neg__le__iff__le,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [B2: A,A3: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A3 ) )
= ( ord_less_eq @ A @ A3 @ B2 ) ) ) ).
% neg_le_iff_le
thf(fact_436_add__diff__cancel__right_H,axiom,
! [A: $tType] :
( ( cancel2418104881723323429up_add @ A )
=> ! [A3: A,B2: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ A3 @ B2 ) @ B2 )
= A3 ) ) ).
% add_diff_cancel_right'
thf(fact_437_add__diff__cancel__right,axiom,
! [A: $tType] :
( ( cancel2418104881723323429up_add @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ A3 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) )
= ( minus_minus @ A @ A3 @ B2 ) ) ) ).
% add_diff_cancel_right
thf(fact_438_add__diff__cancel__left_H,axiom,
! [A: $tType] :
( ( cancel2418104881723323429up_add @ A )
=> ! [A3: A,B2: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ A3 @ B2 ) @ A3 )
= B2 ) ) ).
% add_diff_cancel_left'
thf(fact_439_add__diff__cancel__left,axiom,
! [A: $tType] :
( ( cancel2418104881723323429up_add @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ C2 @ A3 ) @ ( plus_plus @ A @ C2 @ B2 ) )
= ( minus_minus @ A @ A3 @ B2 ) ) ) ).
% add_diff_cancel_left
thf(fact_440_diff__add__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B2: A] :
( ( plus_plus @ A @ ( minus_minus @ A @ A3 @ B2 ) @ B2 )
= A3 ) ) ).
% diff_add_cancel
thf(fact_441_add__diff__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B2: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ A3 @ B2 ) @ B2 )
= A3 ) ) ).
% add_diff_cancel
thf(fact_442_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_443_minus__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B2: A] :
( ( uminus_uminus @ A @ ( minus_minus @ A @ A3 @ B2 ) )
= ( minus_minus @ A @ B2 @ A3 ) ) ) ).
% minus_diff_eq
thf(fact_444_power__one,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [N: nat] :
( ( power_power @ A @ ( one_one @ A ) @ N )
= ( one_one @ A ) ) ) ).
% power_one
thf(fact_445_nat__zero__less__power__iff,axiom,
! [X: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( power_power @ nat @ X @ N ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ X )
| ( N
= ( zero_zero @ nat ) ) ) ) ).
% nat_zero_less_power_iff
thf(fact_446_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_447_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_448_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_449_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_450_of__bool__less__iff,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [P: $o,Q: $o] :
( ( ord_less @ A @ ( zero_neq_one_of_bool @ A @ P ) @ ( zero_neq_one_of_bool @ A @ Q ) )
= ( ~ P
& Q ) ) ) ).
% of_bool_less_iff
thf(fact_451_power__one__right,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A3: A] :
( ( power_power @ A @ A3 @ ( one_one @ nat ) )
= A3 ) ) ).
% power_one_right
thf(fact_452_zle__diff1__eq,axiom,
! [W: int,Z: int] :
( ( ord_less_eq @ int @ W @ ( minus_minus @ int @ Z @ ( one_one @ int ) ) )
= ( ord_less @ int @ W @ Z ) ) ).
% zle_diff1_eq
thf(fact_453_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_454_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_455_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_456_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_457_abs__bool__eq,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [P: $o] :
( ( abs_abs @ A @ ( zero_neq_one_of_bool @ A @ P ) )
= ( zero_neq_one_of_bool @ A @ P ) ) ) ).
% abs_bool_eq
thf(fact_458_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A3 @ A3 ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_459_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ A3 ) @ ( zero_zero @ A ) )
= ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_460_le__add__same__cancel2,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ ( plus_plus @ A @ B2 @ A3 ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) ) ).
% le_add_same_cancel2
thf(fact_461_le__add__same__cancel1,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ ( plus_plus @ A @ A3 @ B2 ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) ) ).
% le_add_same_cancel1
thf(fact_462_add__le__same__cancel2,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ B2 ) @ B2 )
= ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% add_le_same_cancel2
thf(fact_463_add__le__same__cancel1,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [B2: A,A3: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ B2 @ A3 ) @ B2 )
= ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% add_le_same_cancel1
thf(fact_464_diff__ge__0__iff__ge,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A3 @ B2 ) )
= ( ord_less_eq @ A @ B2 @ A3 ) ) ) ).
% diff_ge_0_iff_ge
thf(fact_465_neg__0__le__iff__le,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ A3 ) )
= ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% neg_0_le_iff_le
thf(fact_466_neg__le__0__iff__le,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A3 ) @ ( zero_zero @ A ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% neg_le_0_iff_le
thf(fact_467_less__eq__neg__nonpos,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ A3 @ ( uminus_uminus @ A @ A3 ) )
= ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% less_eq_neg_nonpos
thf(fact_468_neg__less__eq__nonneg,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A3 ) @ A3 )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% neg_less_eq_nonneg
thf(fact_469_diff__gt__0__iff__gt,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A3 @ B2 ) )
= ( ord_less @ A @ B2 @ A3 ) ) ) ).
% diff_gt_0_iff_gt
thf(fact_470_le__add__diff__inverse2,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B2: A,A3: A] :
( ( ord_less_eq @ A @ B2 @ A3 )
=> ( ( plus_plus @ A @ ( minus_minus @ A @ A3 @ B2 ) @ B2 )
= A3 ) ) ) ).
% le_add_diff_inverse2
thf(fact_471_le__add__diff__inverse,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B2: A,A3: A] :
( ( ord_less_eq @ A @ B2 @ A3 )
=> ( ( plus_plus @ A @ B2 @ ( minus_minus @ A @ A3 @ B2 ) )
= A3 ) ) ) ).
% le_add_diff_inverse
thf(fact_472_diff__add__zero,axiom,
! [A: $tType] :
( ( comm_monoid_diff @ A )
=> ! [A3: A,B2: A] :
( ( minus_minus @ A @ A3 @ ( plus_plus @ A @ A3 @ B2 ) )
= ( zero_zero @ A ) ) ) ).
% diff_add_zero
thf(fact_473_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_474_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_475_diff__0,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A] :
( ( minus_minus @ A @ ( zero_zero @ A ) @ A3 )
= ( uminus_uminus @ A @ A3 ) ) ) ).
% diff_0
thf(fact_476_diff__minus__eq__add,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B2: A] :
( ( minus_minus @ A @ A3 @ ( uminus_uminus @ A @ B2 ) )
= ( plus_plus @ A @ A3 @ B2 ) ) ) ).
% diff_minus_eq_add
thf(fact_477_uminus__add__conv__diff,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A3: A,B2: A] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ A3 ) @ B2 )
= ( minus_minus @ A @ B2 @ A3 ) ) ) ).
% uminus_add_conv_diff
thf(fact_478_abs__of__nonneg,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( abs_abs @ A @ A3 )
= A3 ) ) ) ).
% abs_of_nonneg
thf(fact_479_abs__le__self__iff,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ A3 ) @ A3 )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% abs_le_self_iff
thf(fact_480_abs__le__zero__iff,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ A3 ) @ ( zero_zero @ A ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% abs_le_zero_iff
thf(fact_481_power__inject__exp,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,M: nat,N: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A3 )
=> ( ( ( power_power @ A @ A3 @ M )
= ( power_power @ A @ A3 @ N ) )
= ( M = N ) ) ) ) ).
% power_inject_exp
thf(fact_482_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_483_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_484_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_485_zabs__less__one__iff,axiom,
! [Z: int] :
( ( ord_less @ int @ ( abs_abs @ int @ Z ) @ ( one_one @ int ) )
= ( Z
= ( zero_zero @ int ) ) ) ).
% zabs_less_one_iff
thf(fact_486_zle__add1__eq__le,axiom,
! [W: int,Z: int] :
( ( ord_less @ int @ W @ ( plus_plus @ int @ Z @ ( one_one @ int ) ) )
= ( ord_less_eq @ int @ W @ Z ) ) ).
% zle_add1_eq_le
thf(fact_487_power__increasing__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B2: A,X: nat,Y3: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ B2 )
=> ( ( ord_less_eq @ A @ ( power_power @ A @ B2 @ X ) @ ( power_power @ A @ B2 @ Y3 ) )
= ( ord_less_eq @ nat @ X @ Y3 ) ) ) ) ).
% power_increasing_iff
thf(fact_488_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_489_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_490_power__strict__increasing__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B2: A,X: nat,Y3: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ B2 )
=> ( ( ord_less @ A @ ( power_power @ A @ B2 @ X ) @ ( power_power @ A @ B2 @ Y3 ) )
= ( ord_less @ nat @ X @ Y3 ) ) ) ) ).
% power_strict_increasing_iff
thf(fact_491_power__eq__0__iff,axiom,
! [A: $tType] :
( ( semiri2026040879449505780visors @ A )
=> ! [A3: A,N: nat] :
( ( ( power_power @ A @ A3 @ N )
= ( zero_zero @ A ) )
= ( ( A3
= ( zero_zero @ A ) )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% power_eq_0_iff
thf(fact_492_idom__abs__sgn__class_Oabs__sgn,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A3: A] :
( ( sgn_sgn @ A @ ( abs_abs @ A @ A3 ) )
= ( zero_neq_one_of_bool @ A
@ ( A3
!= ( zero_zero @ A ) ) ) ) ) ).
% idom_abs_sgn_class.abs_sgn
thf(fact_493_sgn__abs,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A3: A] :
( ( abs_abs @ A @ ( sgn_sgn @ A @ A3 ) )
= ( zero_neq_one_of_bool @ A
@ ( A3
!= ( zero_zero @ A ) ) ) ) ) ).
% sgn_abs
thf(fact_494_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_495_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_496_power__mono__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,B2: A,N: nat] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ A @ ( power_power @ A @ A3 @ N ) @ ( power_power @ A @ B2 @ N ) )
= ( ord_less_eq @ A @ A3 @ B2 ) ) ) ) ) ) ).
% power_mono_iff
thf(fact_497_zero__less__power__abs__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ ( abs_abs @ A @ A3 ) @ N ) )
= ( ( A3
!= ( zero_zero @ A ) )
| ( N
= ( zero_zero @ nat ) ) ) ) ) ).
% zero_less_power_abs_iff
thf(fact_498_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_499_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_500_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_501_uminus__int__code_I1_J,axiom,
( ( uminus_uminus @ int @ ( zero_zero @ int ) )
= ( zero_zero @ int ) ) ).
% uminus_int_code(1)
thf(fact_502_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_503_less__int__code_I1_J,axiom,
~ ( ord_less @ int @ ( zero_zero @ int ) @ ( zero_zero @ int ) ) ).
% less_int_code(1)
thf(fact_504_int__cases2,axiom,
! [Z: int] :
( ! [N2: nat] :
( Z
!= ( semiring_1_of_nat @ int @ N2 ) )
=> ~ ! [N2: nat] :
( Z
!= ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N2 ) ) ) ) ).
% int_cases2
thf(fact_505_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_506_power__le__imp__le__exp,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,M: nat,N: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( power_power @ A @ A3 @ M ) @ ( power_power @ A @ A3 @ N ) )
=> ( ord_less_eq @ nat @ M @ N ) ) ) ) ).
% power_le_imp_le_exp
thf(fact_507_int__ops_I6_J,axiom,
! [A3: nat,B2: nat] :
( ( ( ord_less @ int @ ( semiring_1_of_nat @ int @ A3 ) @ ( semiring_1_of_nat @ int @ B2 ) )
=> ( ( semiring_1_of_nat @ int @ ( minus_minus @ nat @ A3 @ B2 ) )
= ( zero_zero @ int ) ) )
& ( ~ ( ord_less @ int @ ( semiring_1_of_nat @ int @ A3 ) @ ( semiring_1_of_nat @ int @ B2 ) )
=> ( ( semiring_1_of_nat @ int @ ( minus_minus @ nat @ A3 @ B2 ) )
= ( minus_minus @ int @ ( semiring_1_of_nat @ int @ A3 ) @ ( semiring_1_of_nat @ int @ B2 ) ) ) ) ) ).
% int_ops(6)
thf(fact_508_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_509_add1__zle__eq,axiom,
! [W: int,Z: int] :
( ( ord_less_eq @ int @ ( plus_plus @ int @ W @ ( one_one @ int ) ) @ Z )
= ( ord_less @ int @ W @ Z ) ) ).
% add1_zle_eq
thf(fact_510_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_511_le__imp__0__less,axiom,
! [Z: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
=> ( ord_less @ int @ ( zero_zero @ int ) @ ( plus_plus @ int @ ( one_one @ int ) @ Z ) ) ) ).
% le_imp_0_less
thf(fact_512_zless__add1__eq,axiom,
! [W: int,Z: int] :
( ( ord_less @ int @ W @ ( plus_plus @ int @ Z @ ( one_one @ int ) ) )
= ( ( ord_less @ int @ W @ Z )
| ( W = Z ) ) ) ).
% zless_add1_eq
thf(fact_513_odd__less__0__iff,axiom,
! [Z: int] :
( ( ord_less @ int @ ( plus_plus @ int @ ( plus_plus @ int @ ( one_one @ int ) @ Z ) @ Z ) @ ( zero_zero @ int ) )
= ( ord_less @ int @ Z @ ( zero_zero @ int ) ) ) ).
% odd_less_0_iff
thf(fact_514_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_515_zless__imp__add1__zle,axiom,
! [W: int,Z: int] :
( ( ord_less @ int @ W @ Z )
=> ( ord_less_eq @ int @ ( plus_plus @ int @ W @ ( one_one @ int ) ) @ Z ) ) ).
% zless_imp_add1_zle
thf(fact_516_int__one__le__iff__zero__less,axiom,
! [Z: int] :
( ( ord_less_eq @ int @ ( one_one @ int ) @ Z )
= ( ord_less @ int @ ( zero_zero @ int ) @ Z ) ) ).
% int_one_le_iff_zero_less
thf(fact_517_nat__leq__as__int,axiom,
( ( ord_less_eq @ nat )
= ( ^ [A6: nat,B5: nat] : ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ A6 ) @ ( semiring_1_of_nat @ int @ B5 ) ) ) ) ).
% nat_leq_as_int
thf(fact_518_int__if,axiom,
! [P: $o,A3: nat,B2: nat] :
( ( P
=> ( ( semiring_1_of_nat @ int @ ( if @ nat @ P @ A3 @ B2 ) )
= ( semiring_1_of_nat @ int @ A3 ) ) )
& ( ~ P
=> ( ( semiring_1_of_nat @ int @ ( if @ nat @ P @ A3 @ B2 ) )
= ( semiring_1_of_nat @ int @ B2 ) ) ) ) ).
% int_if
thf(fact_519_nat__int__comparison_I1_J,axiom,
( ( ^ [Y6: nat,Z2: nat] : Y6 = Z2 )
= ( ^ [A6: nat,B5: nat] :
( ( semiring_1_of_nat @ int @ A6 )
= ( semiring_1_of_nat @ int @ B5 ) ) ) ) ).
% nat_int_comparison(1)
thf(fact_520_nat__int__comparison_I3_J,axiom,
( ( ord_less_eq @ nat )
= ( ^ [A6: nat,B5: nat] : ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ A6 ) @ ( semiring_1_of_nat @ int @ B5 ) ) ) ) ).
% nat_int_comparison(3)
thf(fact_521_int__diff__cases,axiom,
! [Z: int] :
~ ! [M4: nat,N2: nat] :
( Z
!= ( minus_minus @ int @ ( semiring_1_of_nat @ int @ M4 ) @ ( semiring_1_of_nat @ int @ N2 ) ) ) ).
% int_diff_cases
thf(fact_522_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_523_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_524_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_525_zle__iff__zadd,axiom,
( ( ord_less_eq @ int )
= ( ^ [W2: int,Z3: int] :
? [N5: nat] :
( Z3
= ( plus_plus @ int @ W2 @ ( semiring_1_of_nat @ int @ N5 ) ) ) ) ) ).
% zle_iff_zadd
thf(fact_526_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_527_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_528_zero__le__power,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N: nat] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A3 @ N ) ) ) ) ).
% zero_le_power
thf(fact_529_power__mono,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,B2: A,N: nat] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ord_less_eq @ A @ ( power_power @ A @ A3 @ N ) @ ( power_power @ A @ B2 @ N ) ) ) ) ) ).
% power_mono
thf(fact_530_less__eq__int__code_I1_J,axiom,
ord_less_eq @ int @ ( zero_zero @ int ) @ ( zero_zero @ int ) ).
% less_eq_int_code(1)
thf(fact_531_minus__int__code_I1_J,axiom,
! [K: int] :
( ( minus_minus @ int @ K @ ( zero_zero @ int ) )
= K ) ).
% minus_int_code(1)
thf(fact_532_diff__eq__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B2: A,C2: A,D3: A] :
( ( ( minus_minus @ A @ A3 @ B2 )
= ( minus_minus @ A @ C2 @ D3 ) )
=> ( ( A3 = B2 )
= ( C2 = D3 ) ) ) ) ).
% diff_eq_diff_eq
thf(fact_533_diff__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B2: A,D3: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ D3 @ C2 )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ A3 @ C2 ) @ ( minus_minus @ A @ B2 @ D3 ) ) ) ) ) ).
% diff_mono
thf(fact_534_diff__left__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( ord_less_eq @ A @ B2 @ A3 )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ C2 @ A3 ) @ ( minus_minus @ A @ C2 @ B2 ) ) ) ) ).
% diff_left_mono
thf(fact_535_diff__right__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ A3 @ C2 ) @ ( minus_minus @ A @ B2 @ C2 ) ) ) ) ).
% diff_right_mono
thf(fact_536_diff__eq__diff__less__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B2: A,C2: A,D3: A] :
( ( ( minus_minus @ A @ A3 @ B2 )
= ( minus_minus @ A @ C2 @ D3 ) )
=> ( ( ord_less_eq @ A @ A3 @ B2 )
= ( ord_less_eq @ A @ C2 @ D3 ) ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_537_diff__right__commute,axiom,
! [A: $tType] :
( ( cancel2418104881723323429up_add @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( minus_minus @ A @ ( minus_minus @ A @ A3 @ C2 ) @ B2 )
= ( minus_minus @ A @ ( minus_minus @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% diff_right_commute
thf(fact_538_power__increasing,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N: nat,N6: nat,A3: A] :
( ( ord_less_eq @ nat @ N @ N6 )
=> ( ( ord_less_eq @ A @ ( one_one @ A ) @ A3 )
=> ( ord_less_eq @ A @ ( power_power @ A @ A3 @ N ) @ ( power_power @ A @ A3 @ N6 ) ) ) ) ) ).
% power_increasing
thf(fact_539_one__le__power,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N: nat] :
( ( ord_less_eq @ A @ ( one_one @ A ) @ A3 )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ ( power_power @ A @ A3 @ N ) ) ) ) ).
% one_le_power
thf(fact_540_verit__comp__simplify1_I2_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A] : ( ord_less_eq @ A @ A3 @ A3 ) ) ).
% verit_comp_simplify1(2)
thf(fact_541_of__bool__eq__iff,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ! [P4: $o,Q2: $o] :
( ( ( zero_neq_one_of_bool @ A @ P4 )
= ( zero_neq_one_of_bool @ A @ Q2 ) )
= ( P4 = Q2 ) ) ) ).
% of_bool_eq_iff
thf(fact_542_verit__la__disequality,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A] :
( ( A3 = B2 )
| ~ ( ord_less_eq @ A @ A3 @ B2 )
| ~ ( ord_less_eq @ A @ B2 @ A3 ) ) ) ).
% verit_la_disequality
thf(fact_543_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_544_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_545_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_546_power__decreasing,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N: nat,N6: nat,A3: A] :
( ( ord_less_eq @ nat @ N @ N6 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ A3 @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( power_power @ A @ A3 @ N6 ) @ ( power_power @ A @ A3 @ N ) ) ) ) ) ) ).
% power_decreasing
thf(fact_547_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_548_le__iff__diff__le__0,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A6: A,B5: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ A6 @ B5 ) @ ( zero_zero @ A ) ) ) ) ) ).
% le_iff_diff_le_0
thf(fact_549_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_550_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_551_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_552_diff__le__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ ( minus_minus @ A @ A3 @ B2 ) @ C2 )
= ( ord_less_eq @ A @ A3 @ ( plus_plus @ A @ C2 @ B2 ) ) ) ) ).
% diff_le_eq
thf(fact_553_le__diff__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ ( minus_minus @ A @ C2 @ B2 ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% le_diff_eq
thf(fact_554_diff__add,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( plus_plus @ A @ ( minus_minus @ A @ B2 @ A3 ) @ A3 )
= B2 ) ) ) ).
% diff_add
thf(fact_555_le__add__diff,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ord_less_eq @ A @ C2 @ ( minus_minus @ A @ ( plus_plus @ A @ B2 @ C2 ) @ A3 ) ) ) ) ).
% le_add_diff
thf(fact_556_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ ( minus_minus @ A @ B2 @ A3 ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ C2 @ A3 ) @ B2 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_557_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( plus_plus @ A @ C2 @ ( minus_minus @ A @ B2 @ A3 ) )
= ( minus_minus @ A @ ( plus_plus @ A @ C2 @ B2 ) @ A3 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_558_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( minus_minus @ A @ ( plus_plus @ A @ C2 @ B2 ) @ A3 )
= ( plus_plus @ A @ C2 @ ( minus_minus @ A @ B2 @ A3 ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_559_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( plus_plus @ A @ ( minus_minus @ A @ B2 @ A3 ) @ C2 )
= ( minus_minus @ A @ ( plus_plus @ A @ B2 @ C2 ) @ A3 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_560_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( minus_minus @ A @ ( plus_plus @ A @ B2 @ C2 ) @ A3 )
= ( plus_plus @ A @ ( minus_minus @ A @ B2 @ A3 ) @ C2 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_561_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( minus_minus @ A @ C2 @ ( minus_minus @ A @ B2 @ A3 ) )
= ( minus_minus @ A @ ( plus_plus @ A @ C2 @ A3 ) @ B2 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_562_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( plus_plus @ A @ A3 @ ( minus_minus @ A @ B2 @ A3 ) )
= B2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_563_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ( minus_minus @ A @ B2 @ A3 )
= C2 )
= ( B2
= ( plus_plus @ A @ C2 @ A3 ) ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_564_diff__less__mono,axiom,
! [A3: nat,B2: nat,C2: nat] :
( ( ord_less @ nat @ A3 @ B2 )
=> ( ( ord_less_eq @ nat @ C2 @ A3 )
=> ( ord_less @ nat @ ( minus_minus @ nat @ A3 @ C2 ) @ ( minus_minus @ nat @ B2 @ C2 ) ) ) ) ).
% diff_less_mono
thf(fact_565_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_566_abs__triangle__ineq2,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A,B2: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B2 ) ) @ ( abs_abs @ A @ ( minus_minus @ A @ A3 @ B2 ) ) ) ) ).
% abs_triangle_ineq2
thf(fact_567_abs__triangle__ineq3,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A,B2: A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B2 ) ) ) @ ( abs_abs @ A @ ( minus_minus @ A @ A3 @ B2 ) ) ) ) ).
% abs_triangle_ineq3
thf(fact_568_abs__triangle__ineq2__sym,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A,B2: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B2 ) ) @ ( abs_abs @ A @ ( minus_minus @ A @ B2 @ A3 ) ) ) ) ).
% abs_triangle_ineq2_sym
thf(fact_569_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_570_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_571_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_572_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_573_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_574_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_575_power__not__zero,axiom,
! [A: $tType] :
( ( semiri2026040879449505780visors @ A )
=> ! [A3: A,N: nat] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( power_power @ A @ A3 @ N )
!= ( zero_zero @ A ) ) ) ) ).
% power_not_zero
thf(fact_576_power__less__imp__less__base,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N: nat,B2: A] :
( ( ord_less @ A @ ( power_power @ A @ A3 @ N ) @ ( power_power @ A @ B2 @ N ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less @ A @ A3 @ B2 ) ) ) ) ).
% power_less_imp_less_base
thf(fact_577_power__le__one,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N: nat] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ A3 @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( power_power @ A @ A3 @ N ) @ ( one_one @ A ) ) ) ) ) ).
% power_le_one
thf(fact_578_pred__subset__eq2,axiom,
! [B: $tType,A: $tType,R5: set @ ( product_prod @ A @ B ),S3: set @ ( product_prod @ A @ B )] :
( ( ord_less_eq @ ( A > B > $o )
@ ^ [X2: A,Y: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y ) @ R5 )
@ ^ [X2: A,Y: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y ) @ S3 ) )
= ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R5 @ S3 ) ) ).
% pred_subset_eq2
thf(fact_579_zero__le__power__abs,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,N: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ ( abs_abs @ A @ A3 ) @ N ) ) ) ).
% zero_le_power_abs
thf(fact_580_abs__diff__triangle__ineq,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A,B2: A,C2: A,D3: A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ ( plus_plus @ A @ A3 @ B2 ) @ ( plus_plus @ A @ C2 @ D3 ) ) ) @ ( plus_plus @ A @ ( abs_abs @ A @ ( minus_minus @ A @ A3 @ C2 ) ) @ ( abs_abs @ A @ ( minus_minus @ A @ B2 @ D3 ) ) ) ) ) ).
% abs_diff_triangle_ineq
thf(fact_581_abs__triangle__ineq4,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A,B2: A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ A3 @ B2 ) ) @ ( plus_plus @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B2 ) ) ) ) ).
% abs_triangle_ineq4
thf(fact_582_abs__diff__le__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,A3: A,R: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X @ A3 ) ) @ R )
= ( ( ord_less_eq @ A @ ( minus_minus @ A @ A3 @ R ) @ X )
& ( ord_less_eq @ A @ X @ ( plus_plus @ A @ A3 @ R ) ) ) ) ) ).
% abs_diff_le_iff
thf(fact_583_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_584_power__eq__iff__eq__base,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N: nat,A3: A,B2: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ( power_power @ A @ A3 @ N )
= ( power_power @ A @ B2 @ N ) )
= ( A3 = B2 ) ) ) ) ) ) ).
% power_eq_iff_eq_base
thf(fact_585_power__eq__imp__eq__base,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N: nat,B2: A] :
( ( ( power_power @ A @ A3 @ N )
= ( power_power @ A @ B2 @ N ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( A3 = B2 ) ) ) ) ) ) ).
% power_eq_imp_eq_base
thf(fact_586_self__le__power,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N: nat] :
( ( ord_less_eq @ A @ ( one_one @ A ) @ A3 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less_eq @ A @ A3 @ ( power_power @ A @ A3 @ N ) ) ) ) ) ).
% self_le_power
thf(fact_587_eq__iff__diff__eq__0,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ( ( ^ [Y6: A,Z2: A] : Y6 = Z2 )
= ( ^ [A6: A,B5: A] :
( ( minus_minus @ A @ A6 @ B5 )
= ( zero_zero @ A ) ) ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_588_diff__strict__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B2: A,D3: A,C2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ D3 @ C2 )
=> ( ord_less @ A @ ( minus_minus @ A @ A3 @ C2 ) @ ( minus_minus @ A @ B2 @ D3 ) ) ) ) ) ).
% diff_strict_mono
thf(fact_589_diff__eq__diff__less,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B2: A,C2: A,D3: A] :
( ( ( minus_minus @ A @ A3 @ B2 )
= ( minus_minus @ A @ C2 @ D3 ) )
=> ( ( ord_less @ A @ A3 @ B2 )
= ( ord_less @ A @ C2 @ D3 ) ) ) ) ).
% diff_eq_diff_less
thf(fact_590_diff__strict__left__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( ord_less @ A @ B2 @ A3 )
=> ( ord_less @ A @ ( minus_minus @ A @ C2 @ A3 ) @ ( minus_minus @ A @ C2 @ B2 ) ) ) ) ).
% diff_strict_left_mono
thf(fact_591_diff__strict__right__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ord_less @ A @ ( minus_minus @ A @ A3 @ C2 ) @ ( minus_minus @ A @ B2 @ C2 ) ) ) ) ).
% diff_strict_right_mono
thf(fact_592_diff__diff__eq,axiom,
! [A: $tType] :
( ( cancel2418104881723323429up_add @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( minus_minus @ A @ ( minus_minus @ A @ A3 @ B2 ) @ C2 )
= ( minus_minus @ A @ A3 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% diff_diff_eq
thf(fact_593_add__implies__diff,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [C2: A,B2: A,A3: A] :
( ( ( plus_plus @ A @ C2 @ B2 )
= A3 )
=> ( C2
= ( minus_minus @ A @ A3 @ B2 ) ) ) ) ).
% add_implies_diff
thf(fact_594_diff__add__eq__diff__diff__swap,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( minus_minus @ A @ A3 @ ( plus_plus @ A @ B2 @ C2 ) )
= ( minus_minus @ A @ ( minus_minus @ A @ A3 @ C2 ) @ B2 ) ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_595_diff__add__eq,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( plus_plus @ A @ ( minus_minus @ A @ A3 @ B2 ) @ C2 )
= ( minus_minus @ A @ ( plus_plus @ A @ A3 @ C2 ) @ B2 ) ) ) ).
% diff_add_eq
thf(fact_596_diff__diff__eq2,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( minus_minus @ A @ A3 @ ( minus_minus @ A @ B2 @ C2 ) )
= ( minus_minus @ A @ ( plus_plus @ A @ A3 @ C2 ) @ B2 ) ) ) ).
% diff_diff_eq2
thf(fact_597_add__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( plus_plus @ A @ A3 @ ( minus_minus @ A @ B2 @ C2 ) )
= ( minus_minus @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% add_diff_eq
thf(fact_598_eq__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( A3
= ( minus_minus @ A @ C2 @ B2 ) )
= ( ( plus_plus @ A @ A3 @ B2 )
= C2 ) ) ) ).
% eq_diff_eq
thf(fact_599_diff__eq__eq,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ( minus_minus @ A @ A3 @ B2 )
= C2 )
= ( A3
= ( plus_plus @ A @ C2 @ B2 ) ) ) ) ).
% diff_eq_eq
thf(fact_600_group__cancel_Osub1,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A5: A,K: A,A3: A,B2: A] :
( ( A5
= ( plus_plus @ A @ K @ A3 ) )
=> ( ( minus_minus @ A @ A5 @ B2 )
= ( plus_plus @ A @ K @ ( minus_minus @ A @ A3 @ B2 ) ) ) ) ) ).
% group_cancel.sub1
thf(fact_601_minus__diff__commute,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [B2: A,A3: A] :
( ( minus_minus @ A @ ( uminus_uminus @ A @ B2 ) @ A3 )
= ( minus_minus @ A @ ( uminus_uminus @ A @ A3 ) @ B2 ) ) ) ).
% minus_diff_commute
thf(fact_602_zero__le,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [X: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X ) ) ).
% zero_le
thf(fact_603_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_604_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_605_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus @ nat @ M @ ( zero_zero @ nat ) )
= M ) ).
% minus_nat.diff_0
thf(fact_606_verit__comp__simplify1_I3_J,axiom,
! [B: $tType] :
( ( linorder @ B )
=> ! [B6: B,A7: B] :
( ( ~ ( ord_less_eq @ B @ B6 @ A7 ) )
= ( ord_less @ B @ A7 @ B6 ) ) ) ).
% verit_comp_simplify1(3)
thf(fact_607_abs__minus__commute,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A,B2: A] :
( ( abs_abs @ A @ ( minus_minus @ A @ A3 @ B2 ) )
= ( abs_abs @ A @ ( minus_minus @ A @ B2 @ A3 ) ) ) ) ).
% abs_minus_commute
thf(fact_608_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_609_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_610_add__le__imp__le__right,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) )
=> ( ord_less_eq @ A @ A3 @ B2 ) ) ) ).
% add_le_imp_le_right
thf(fact_611_add__le__imp__le__left,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ C2 @ A3 ) @ ( plus_plus @ A @ C2 @ B2 ) )
=> ( ord_less_eq @ A @ A3 @ B2 ) ) ) ).
% add_le_imp_le_left
thf(fact_612_le__iff__add,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A6: A,B5: A] :
? [C4: A] :
( B5
= ( plus_plus @ A @ A6 @ C4 ) ) ) ) ) ).
% le_iff_add
thf(fact_613_add__right__mono,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% add_right_mono
thf(fact_614_less__eqE,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ~ ! [C3: A] :
( B2
!= ( plus_plus @ A @ A3 @ C3 ) ) ) ) ).
% less_eqE
thf(fact_615_add__left__mono,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ C2 @ A3 ) @ ( plus_plus @ A @ C2 @ B2 ) ) ) ) ).
% add_left_mono
thf(fact_616_add__mono,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [A3: A,B2: A,C2: A,D3: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ D3 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ C2 ) @ ( plus_plus @ A @ B2 @ D3 ) ) ) ) ) ).
% add_mono
thf(fact_617_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_618_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_619_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_620_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_621_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N ) @ N )
= M ) ).
% diff_add_inverse2
thf(fact_622_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus @ nat @ ( plus_plus @ nat @ N @ M ) @ N )
= M ) ).
% diff_add_inverse
thf(fact_623_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_624_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_625_subrelI,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
( ! [X4: A,Y4: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y4 ) @ R )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y4 ) @ S ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R @ S ) ) ).
% subrelI
thf(fact_626_le__minus__iff,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ ( uminus_uminus @ A @ B2 ) )
= ( ord_less_eq @ A @ B2 @ ( uminus_uminus @ A @ A3 ) ) ) ) ).
% le_minus_iff
thf(fact_627_minus__le__iff,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A3 ) @ B2 )
= ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ A3 ) ) ) ).
% minus_le_iff
thf(fact_628_le__imp__neg__le,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A3 ) ) ) ) ).
% le_imp_neg_le
thf(fact_629_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_630_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq @ nat @ N @ ( zero_zero @ nat ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% le_0_eq
thf(fact_631_bot__nat__0_Oextremum__uniqueI,axiom,
! [A3: nat] :
( ( ord_less_eq @ nat @ A3 @ ( zero_zero @ nat ) )
=> ( A3
= ( zero_zero @ nat ) ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_632_bot__nat__0_Oextremum__unique,axiom,
! [A3: nat] :
( ( ord_less_eq @ nat @ A3 @ ( zero_zero @ nat ) )
= ( A3
= ( zero_zero @ nat ) ) ) ).
% bot_nat_0.extremum_unique
thf(fact_633_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% less_eq_nat.simps(1)
thf(fact_634_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_635_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_636_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_637_le__eq__less__or__eq,axiom,
( ( ord_less_eq @ nat )
= ( ^ [M5: nat,N5: nat] :
( ( ord_less @ nat @ M5 @ N5 )
| ( M5 = N5 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_638_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_639_nat__less__le,axiom,
( ( ord_less @ nat )
= ( ^ [M5: nat,N5: nat] :
( ( ord_less_eq @ nat @ M5 @ N5 )
& ( M5 != N5 ) ) ) ) ).
% nat_less_le
thf(fact_640_nat__le__iff__add,axiom,
( ( ord_less_eq @ nat )
= ( ^ [M5: nat,N5: nat] :
? [K3: nat] :
( N5
= ( plus_plus @ nat @ M5 @ K3 ) ) ) ) ).
% nat_le_iff_add
thf(fact_641_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_642_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_643_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_644_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_645_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_646_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_647_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_648_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq @ nat @ N @ ( plus_plus @ nat @ M @ N ) ) ).
% le_add2
thf(fact_649_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq @ nat @ N @ ( plus_plus @ nat @ N @ M ) ) ).
% le_add1
thf(fact_650_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_651_abs__le__D1,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ A3 ) @ B2 )
=> ( ord_less_eq @ A @ A3 @ B2 ) ) ) ).
% abs_le_D1
thf(fact_652_abs__ge__self,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] : ( ord_less_eq @ A @ A3 @ ( abs_abs @ A @ A3 ) ) ) ).
% abs_ge_self
thf(fact_653_zero__less__power,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A3 @ N ) ) ) ) ).
% zero_less_power
thf(fact_654_Ints__diff,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [A3: A,B2: A] :
( ( member @ A @ A3 @ ( ring_1_Ints @ A ) )
=> ( ( member @ A @ B2 @ ( ring_1_Ints @ A ) )
=> ( member @ A @ ( minus_minus @ A @ A3 @ B2 ) @ ( ring_1_Ints @ A ) ) ) ) ) ).
% Ints_diff
thf(fact_655_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_656_Ints__power,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [A3: A,N: nat] :
( ( member @ A @ A3 @ ( ring_1_Ints @ A ) )
=> ( member @ A @ ( power_power @ A @ A3 @ N ) @ ( ring_1_Ints @ A ) ) ) ) ).
% Ints_power
thf(fact_657_power__0,axiom,
! [A: $tType] :
( ( power @ A )
=> ! [A3: A] :
( ( power_power @ A @ A3 @ ( zero_zero @ nat ) )
= ( one_one @ A ) ) ) ).
% power_0
thf(fact_658_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_659_power__strict__mono,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,B2: A,N: nat] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less @ A @ ( power_power @ A @ A3 @ N ) @ ( power_power @ A @ B2 @ N ) ) ) ) ) ) ).
% power_strict_mono
thf(fact_660_split__of__bool__asm,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ! [P: A > $o,P4: $o] :
( ( P @ ( zero_neq_one_of_bool @ A @ P4 ) )
= ( ~ ( ( P4
& ~ ( P @ ( one_one @ A ) ) )
| ( ~ P4
& ~ ( P @ ( zero_zero @ A ) ) ) ) ) ) ) ).
% split_of_bool_asm
thf(fact_661_split__of__bool,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ! [P: A > $o,P4: $o] :
( ( P @ ( zero_neq_one_of_bool @ A @ P4 ) )
= ( ( P4
=> ( P @ ( one_one @ A ) ) )
& ( ~ P4
=> ( P @ ( zero_zero @ A ) ) ) ) ) ) ).
% split_of_bool
thf(fact_662_of__bool__def,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ( ( zero_neq_one_of_bool @ A )
= ( ^ [P5: $o] : ( if @ A @ P5 @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ) ) ).
% of_bool_def
thf(fact_663_less__iff__diff__less__0,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ( ( ord_less @ A )
= ( ^ [A6: A,B5: A] : ( ord_less @ A @ ( minus_minus @ A @ A6 @ B5 ) @ ( zero_zero @ A ) ) ) ) ) ).
% less_iff_diff_less_0
thf(fact_664_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_665_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,B2: A] :
( ~ ( ord_less @ A @ A3 @ B2 )
=> ( ( plus_plus @ A @ B2 @ ( minus_minus @ A @ A3 @ B2 ) )
= A3 ) ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_666_diff__less__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less @ A @ ( minus_minus @ A @ A3 @ B2 ) @ C2 )
= ( ord_less @ A @ A3 @ ( plus_plus @ A @ C2 @ B2 ) ) ) ) ).
% diff_less_eq
thf(fact_667_less__diff__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( ord_less @ A @ A3 @ ( minus_minus @ A @ C2 @ B2 ) )
= ( ord_less @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% less_diff_eq
thf(fact_668_power__strict__increasing,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N: nat,N6: nat,A3: A] :
( ( ord_less @ nat @ N @ N6 )
=> ( ( ord_less @ A @ ( one_one @ A ) @ A3 )
=> ( ord_less @ A @ ( power_power @ A @ A3 @ N ) @ ( power_power @ A @ A3 @ N6 ) ) ) ) ) ).
% power_strict_increasing
thf(fact_669_power__less__imp__less__exp,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,M: nat,N: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A3 )
=> ( ( ord_less @ A @ ( power_power @ A @ A3 @ M ) @ ( power_power @ A @ A3 @ N ) )
=> ( ord_less @ nat @ M @ N ) ) ) ) ).
% power_less_imp_less_exp
thf(fact_670_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_671_group__cancel_Osub2,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [B4: A,K: A,B2: A,A3: A] :
( ( B4
= ( plus_plus @ A @ K @ B2 ) )
=> ( ( minus_minus @ A @ A3 @ B4 )
= ( plus_plus @ A @ ( uminus_uminus @ A @ K ) @ ( minus_minus @ A @ A3 @ B2 ) ) ) ) ) ).
% group_cancel.sub2
thf(fact_672_diff__conv__add__uminus,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ( ( minus_minus @ A )
= ( ^ [A6: A,B5: A] : ( plus_plus @ A @ A6 @ ( uminus_uminus @ A @ B5 ) ) ) ) ) ).
% diff_conv_add_uminus
thf(fact_673_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ( ( minus_minus @ A )
= ( ^ [A6: A,B5: A] : ( plus_plus @ A @ A6 @ ( uminus_uminus @ A @ B5 ) ) ) ) ) ).
% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_674_add__nonpos__eq__0__iff,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [X: A,Y3: A] :
( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ Y3 @ ( zero_zero @ A ) )
=> ( ( ( plus_plus @ A @ X @ Y3 )
= ( zero_zero @ A ) )
= ( ( X
= ( zero_zero @ A ) )
& ( Y3
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_675_add__nonneg__eq__0__iff,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [X: A,Y3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y3 )
=> ( ( ( plus_plus @ A @ X @ Y3 )
= ( zero_zero @ A ) )
= ( ( X
= ( zero_zero @ A ) )
& ( Y3
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_676_add__nonpos__nonpos,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% add_nonpos_nonpos
thf(fact_677_add__nonneg__nonneg,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A3 @ B2 ) ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_678_add__increasing2,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [C2: A,B2: A,A3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less_eq @ A @ B2 @ A3 )
=> ( ord_less_eq @ A @ B2 @ ( plus_plus @ A @ A3 @ C2 ) ) ) ) ) ).
% add_increasing2
thf(fact_679_add__decreasing2,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ C2 ) @ B2 ) ) ) ) ).
% add_decreasing2
thf(fact_680_add__increasing,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ B2 @ C2 )
=> ( ord_less_eq @ A @ B2 @ ( plus_plus @ A @ A3 @ C2 ) ) ) ) ) ).
% add_increasing
thf(fact_681_add__decreasing,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ C2 @ B2 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ C2 ) @ B2 ) ) ) ) ).
% add_decreasing
thf(fact_682_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_683_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_684_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_685_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_686_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_687_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_688_add__le__less__mono,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [A3: A,B2: A,C2: A,D3: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ C2 @ D3 )
=> ( ord_less @ A @ ( plus_plus @ A @ A3 @ C2 ) @ ( plus_plus @ A @ B2 @ D3 ) ) ) ) ) ).
% add_le_less_mono
thf(fact_689_add__less__le__mono,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [A3: A,B2: A,C2: A,D3: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ D3 )
=> ( ord_less @ A @ ( plus_plus @ A @ A3 @ C2 ) @ ( plus_plus @ A @ B2 @ D3 ) ) ) ) ) ).
% add_less_le_mono
thf(fact_690_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_691_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_692_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_693_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_694_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_695_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_696_abs__ge__zero,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( abs_abs @ A @ A3 ) ) ) ).
% abs_ge_zero
thf(fact_697_abs__triangle__ineq,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A,B2: A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( plus_plus @ A @ A3 @ B2 ) ) @ ( plus_plus @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B2 ) ) ) ) ).
% abs_triangle_ineq
thf(fact_698_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ ( zero_zero @ nat ) )
=> ? [K2: nat] :
( ( ord_less_eq @ nat @ K2 @ N )
& ! [I4: nat] :
( ( ord_less @ nat @ I4 @ K2 )
=> ~ ( P @ I4 ) )
& ( P @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_699_abs__leI,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A3 ) @ B2 )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ A3 ) @ B2 ) ) ) ) ).
% abs_leI
thf(fact_700_abs__le__D2,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ A3 ) @ B2 )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ A3 ) @ B2 ) ) ) ).
% abs_le_D2
thf(fact_701_abs__le__iff,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ A3 ) @ B2 )
= ( ( ord_less_eq @ A @ A3 @ B2 )
& ( ord_less_eq @ A @ ( uminus_uminus @ A @ A3 ) @ B2 ) ) ) ) ).
% abs_le_iff
thf(fact_702_abs__ge__minus__self,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ A3 ) @ ( abs_abs @ A @ A3 ) ) ) ).
% abs_ge_minus_self
thf(fact_703_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_704_power__strict__decreasing,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N: nat,N6: nat,A3: A] :
( ( ord_less @ nat @ N @ N6 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ A3 @ ( one_one @ A ) )
=> ( ord_less @ A @ ( power_power @ A @ A3 @ N6 ) @ ( power_power @ A @ A3 @ N ) ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_705_one__less__power,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A3 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less @ A @ ( one_one @ A ) @ ( power_power @ A @ A3 @ N ) ) ) ) ) ).
% one_less_power
thf(fact_706_abs__diff__less__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,A3: A,R: A] :
( ( ord_less @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X @ A3 ) ) @ R )
= ( ( ord_less @ A @ ( minus_minus @ A @ A3 @ R ) @ X )
& ( ord_less @ A @ X @ ( plus_plus @ A @ A3 @ R ) ) ) ) ) ).
% abs_diff_less_iff
thf(fact_707_add__strict__increasing2,axiom,
! [A: $tType] :
( ( ordere8940638589300402666id_add @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ B2 @ C2 )
=> ( ord_less @ A @ B2 @ ( plus_plus @ A @ A3 @ C2 ) ) ) ) ) ).
% add_strict_increasing2
thf(fact_708_add__strict__increasing,axiom,
! [A: $tType] :
( ( ordere8940638589300402666id_add @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ B2 @ C2 )
=> ( ord_less @ A @ B2 @ ( plus_plus @ A @ A3 @ C2 ) ) ) ) ) ).
% add_strict_increasing
thf(fact_709_add__pos__nonneg,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A3 @ B2 ) ) ) ) ) ).
% add_pos_nonneg
thf(fact_710_add__nonpos__neg,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( plus_plus @ A @ A3 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% add_nonpos_neg
thf(fact_711_add__nonneg__pos,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A3 @ B2 ) ) ) ) ) ).
% add_nonneg_pos
thf(fact_712_add__neg__nonpos,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( plus_plus @ A @ A3 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% add_neg_nonpos
thf(fact_713_discrete,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ( ( ord_less @ A )
= ( ^ [A6: A] : ( ord_less_eq @ A @ ( plus_plus @ A @ A6 @ ( one_one @ A ) ) ) ) ) ) ).
% discrete
thf(fact_714_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_715_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_716_nat__diff__split__asm,axiom,
! [P: nat > $o,A3: nat,B2: nat] :
( ( P @ ( minus_minus @ nat @ A3 @ B2 ) )
= ( ~ ( ( ( ord_less @ nat @ A3 @ B2 )
& ~ ( P @ ( zero_zero @ nat ) ) )
| ? [D4: nat] :
( ( A3
= ( plus_plus @ nat @ B2 @ D4 ) )
& ~ ( P @ D4 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_717_nat__diff__split,axiom,
! [P: nat > $o,A3: nat,B2: nat] :
( ( P @ ( minus_minus @ nat @ A3 @ B2 ) )
= ( ( ( ord_less @ nat @ A3 @ B2 )
=> ( P @ ( zero_zero @ nat ) ) )
& ! [D4: nat] :
( ( A3
= ( plus_plus @ nat @ B2 @ D4 ) )
=> ( P @ D4 ) ) ) ) ).
% nat_diff_split
thf(fact_718_dense__eq0__I,axiom,
! [A: $tType] :
( ( ( ordere166539214618696060dd_abs @ A )
& ( dense_linorder @ A ) )
=> ! [X: A] :
( ! [E3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ E3 )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ X ) @ E3 ) )
=> ( X
= ( zero_zero @ A ) ) ) ) ).
% dense_eq0_I
thf(fact_719_abs__minus__le__zero,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( abs_abs @ A @ A3 ) ) @ ( zero_zero @ A ) ) ) ).
% abs_minus_le_zero
thf(fact_720_abs__eq__iff_H,axiom,
! [A: $tType] :
( ( linordered_ring @ A )
=> ! [A3: A,B2: A] :
( ( ( abs_abs @ A @ A3 )
= B2 )
= ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
& ( ( A3 = B2 )
| ( A3
= ( uminus_uminus @ A @ B2 ) ) ) ) ) ) ).
% abs_eq_iff'
thf(fact_721_eq__abs__iff_H,axiom,
! [A: $tType] :
( ( linordered_ring @ A )
=> ! [A3: A,B2: A] :
( ( A3
= ( abs_abs @ A @ B2 ) )
= ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
& ( ( B2 = A3 )
| ( B2
= ( uminus_uminus @ A @ A3 ) ) ) ) ) ) ).
% eq_abs_iff'
thf(fact_722_dbl__dec__def,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( neg_numeral_dbl_dec @ A )
= ( ^ [X2: A] : ( minus_minus @ A @ ( plus_plus @ A @ X2 @ X2 ) @ ( one_one @ A ) ) ) ) ) ).
% dbl_dec_def
thf(fact_723_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_724_field__le__epsilon,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y3: A] :
( ! [E3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ E3 )
=> ( ord_less_eq @ A @ X @ ( plus_plus @ A @ Y3 @ E3 ) ) )
=> ( ord_less_eq @ A @ X @ Y3 ) ) ) ).
% field_le_epsilon
thf(fact_725_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_726_ex__has__greatest__nat__lemma,axiom,
! [A: $tType,P: A > $o,K: A,F: A > nat,N: nat] :
( ( P @ K )
=> ( ! [X4: A] :
( ( P @ X4 )
=> ? [Y5: A] :
( ( P @ Y5 )
& ~ ( ord_less_eq @ nat @ ( F @ Y5 ) @ ( F @ X4 ) ) ) )
=> ? [Y4: A] :
( ( P @ Y4 )
& ~ ( ord_less @ nat @ ( F @ Y4 ) @ ( plus_plus @ nat @ ( F @ K ) @ N ) ) ) ) ) ).
% ex_has_greatest_nat_lemma
thf(fact_727_bezw__0,axiom,
! [X: nat] :
( ( bezw @ X @ ( zero_zero @ nat ) )
= ( product_Pair @ int @ int @ ( one_one @ int ) @ ( zero_zero @ int ) ) ) ).
% bezw_0
thf(fact_728_frac__unique__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,A3: A] :
( ( ( archimedean_frac @ A @ X )
= A3 )
= ( ( member @ A @ ( minus_minus @ A @ X @ A3 ) @ ( ring_1_Ints @ A ) )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less @ A @ A3 @ ( one_one @ A ) ) ) ) ) ).
% frac_unique_iff
thf(fact_729_Nats__altdef2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( ( semiring_1_Nats @ A )
= ( collect @ A
@ ^ [N5: A] :
( ( member @ A @ N5 @ ( ring_1_Ints @ A ) )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ N5 ) ) ) ) ) ).
% Nats_altdef2
thf(fact_730_convex__bound__lt,axiom,
! [A: $tType] :
( ( linord715952674999750819strict @ A )
=> ! [X: A,A3: A,Y3: A,U2: A,V: A] :
( ( ord_less @ A @ X @ A3 )
=> ( ( ord_less @ A @ Y3 @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ U2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ V )
=> ( ( ( plus_plus @ A @ U2 @ V )
= ( one_one @ A ) )
=> ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ U2 @ X ) @ ( times_times @ A @ V @ Y3 ) ) @ A3 ) ) ) ) ) ) ) ).
% convex_bound_lt
thf(fact_731_zero__less__nat__eq,axiom,
! [Z: int] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( nat2 @ Z ) )
= ( ord_less @ int @ ( zero_zero @ int ) @ Z ) ) ).
% zero_less_nat_eq
thf(fact_732_of__int__1__less__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z: int] :
( ( ord_less @ A @ ( one_one @ A ) @ ( ring_1_of_int @ A @ Z ) )
= ( ord_less @ int @ ( one_one @ int ) @ Z ) ) ) ).
% of_int_1_less_iff
thf(fact_733_of__int__less__1__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z: int] :
( ( ord_less @ A @ ( ring_1_of_int @ A @ Z ) @ ( one_one @ A ) )
= ( ord_less @ int @ Z @ ( one_one @ int ) ) ) ) ).
% of_int_less_1_iff
thf(fact_734_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_735_mult__0__right,axiom,
! [M: nat] :
( ( times_times @ nat @ M @ ( zero_zero @ nat ) )
= ( zero_zero @ nat ) ) ).
% mult_0_right
thf(fact_736_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_737_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_738_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_739_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_740_of__int__eq__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [W: int,Z: int] :
( ( ( ring_1_of_int @ A @ W )
= ( ring_1_of_int @ A @ Z ) )
= ( W = Z ) ) ) ).
% of_int_eq_iff
thf(fact_741_mult__zero__left,axiom,
! [A: $tType] :
( ( mult_zero @ A )
=> ! [A3: A] :
( ( times_times @ A @ ( zero_zero @ A ) @ A3 )
= ( zero_zero @ A ) ) ) ).
% mult_zero_left
thf(fact_742_mult__zero__right,axiom,
! [A: $tType] :
( ( mult_zero @ A )
=> ! [A3: A] :
( ( times_times @ A @ A3 @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% mult_zero_right
thf(fact_743_mult__eq__0__iff,axiom,
! [A: $tType] :
( ( semiri3467727345109120633visors @ A )
=> ! [A3: A,B2: A] :
( ( ( times_times @ A @ A3 @ B2 )
= ( zero_zero @ A ) )
= ( ( A3
= ( zero_zero @ A ) )
| ( B2
= ( zero_zero @ A ) ) ) ) ) ).
% mult_eq_0_iff
thf(fact_744_mult__cancel__left,axiom,
! [A: $tType] :
( ( semiri6575147826004484403cancel @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( ( times_times @ A @ C2 @ A3 )
= ( times_times @ A @ C2 @ B2 ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( A3 = B2 ) ) ) ) ).
% mult_cancel_left
thf(fact_745_mult__cancel__right,axiom,
! [A: $tType] :
( ( semiri6575147826004484403cancel @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( ( times_times @ A @ A3 @ C2 )
= ( times_times @ A @ B2 @ C2 ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( A3 = B2 ) ) ) ) ).
% mult_cancel_right
thf(fact_746_mult_Oright__neutral,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A3: A] :
( ( times_times @ A @ A3 @ ( one_one @ A ) )
= A3 ) ) ).
% mult.right_neutral
thf(fact_747_mult__1,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A3: A] :
( ( times_times @ A @ ( one_one @ A ) @ A3 )
= A3 ) ) ).
% mult_1
thf(fact_748_mult__minus__right,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A3: A,B2: A] :
( ( times_times @ A @ A3 @ ( uminus_uminus @ A @ B2 ) )
= ( uminus_uminus @ A @ ( times_times @ A @ A3 @ B2 ) ) ) ) ).
% mult_minus_right
thf(fact_749_minus__mult__minus,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A3: A,B2: A] :
( ( times_times @ A @ ( uminus_uminus @ A @ A3 ) @ ( uminus_uminus @ A @ B2 ) )
= ( times_times @ A @ A3 @ B2 ) ) ) ).
% minus_mult_minus
thf(fact_750_mult__minus__left,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A3: A,B2: A] :
( ( times_times @ A @ ( uminus_uminus @ A @ A3 ) @ B2 )
= ( uminus_uminus @ A @ ( times_times @ A @ A3 @ B2 ) ) ) ) ).
% mult_minus_left
thf(fact_751_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_752_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_753_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_754_abs__mult__self__eq,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] :
( ( times_times @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ A3 ) )
= ( times_times @ A @ A3 @ A3 ) ) ) ).
% abs_mult_self_eq
thf(fact_755_of__int__mult,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [W: int,Z: int] :
( ( ring_1_of_int @ A @ ( times_times @ int @ W @ Z ) )
= ( times_times @ A @ ( ring_1_of_int @ A @ W ) @ ( ring_1_of_int @ A @ Z ) ) ) ) ).
% of_int_mult
thf(fact_756_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_757_of__int__power__eq__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [X: int,B2: int,W: nat] :
( ( ( ring_1_of_int @ A @ X )
= ( power_power @ A @ ( ring_1_of_int @ A @ B2 ) @ W ) )
= ( X
= ( power_power @ int @ B2 @ W ) ) ) ) ).
% of_int_power_eq_of_int_cancel_iff
thf(fact_758_of__int__eq__of__int__power__cancel__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [B2: int,W: nat,X: int] :
( ( ( power_power @ A @ ( ring_1_of_int @ A @ B2 ) @ W )
= ( ring_1_of_int @ A @ X ) )
= ( ( power_power @ int @ B2 @ W )
= X ) ) ) ).
% of_int_eq_of_int_power_cancel_iff
thf(fact_759_of__int__power,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [Z: int,N: nat] :
( ( ring_1_of_int @ A @ ( power_power @ int @ Z @ N ) )
= ( power_power @ A @ ( ring_1_of_int @ A @ Z ) @ N ) ) ) ).
% of_int_power
thf(fact_760_nat__int,axiom,
! [N: nat] :
( ( nat2 @ ( semiring_1_of_nat @ int @ N ) )
= N ) ).
% nat_int
thf(fact_761_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_762_mult__cancel__left2,axiom,
! [A: $tType] :
( ( ring_15535105094025558882visors @ A )
=> ! [C2: A,A3: A] :
( ( ( times_times @ A @ C2 @ A3 )
= C2 )
= ( ( C2
= ( zero_zero @ A ) )
| ( A3
= ( one_one @ A ) ) ) ) ) ).
% mult_cancel_left2
thf(fact_763_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_764_mult__cancel__right2,axiom,
! [A: $tType] :
( ( ring_15535105094025558882visors @ A )
=> ! [A3: A,C2: A] :
( ( ( times_times @ A @ A3 @ C2 )
= C2 )
= ( ( C2
= ( zero_zero @ A ) )
| ( A3
= ( one_one @ A ) ) ) ) ) ).
% mult_cancel_right2
thf(fact_765_mult__minus1,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [Z: A] :
( ( times_times @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ Z )
= ( uminus_uminus @ A @ Z ) ) ) ).
% mult_minus1
thf(fact_766_mult__minus1__right,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [Z: A] :
( ( times_times @ A @ Z @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( uminus_uminus @ A @ Z ) ) ) ).
% mult_minus1_right
thf(fact_767_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_768_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_769_of__int__0__eq__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [Z: int] :
( ( ( zero_zero @ A )
= ( ring_1_of_int @ A @ Z ) )
= ( Z
= ( zero_zero @ int ) ) ) ) ).
% of_int_0_eq_iff
thf(fact_770_of__int__eq__0__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [Z: int] :
( ( ( ring_1_of_int @ A @ Z )
= ( zero_zero @ A ) )
= ( Z
= ( zero_zero @ int ) ) ) ) ).
% of_int_eq_0_iff
thf(fact_771_of__int__le__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [W: int,Z: int] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ W ) @ ( ring_1_of_int @ A @ Z ) )
= ( ord_less_eq @ int @ W @ Z ) ) ) ).
% of_int_le_iff
thf(fact_772_of__int__less__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [W: int,Z: int] :
( ( ord_less @ A @ ( ring_1_of_int @ A @ W ) @ ( ring_1_of_int @ A @ Z ) )
= ( ord_less @ int @ W @ Z ) ) ) ).
% of_int_less_iff
thf(fact_773_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_774_of__int__eq__1__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [Z: int] :
( ( ( ring_1_of_int @ A @ Z )
= ( one_one @ A ) )
= ( Z
= ( one_one @ int ) ) ) ) ).
% of_int_eq_1_iff
thf(fact_775_of__int__add,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [W: int,Z: int] :
( ( ring_1_of_int @ A @ ( plus_plus @ int @ W @ Z ) )
= ( plus_plus @ A @ ( ring_1_of_int @ A @ W ) @ ( ring_1_of_int @ A @ Z ) ) ) ) ).
% of_int_add
thf(fact_776_of__int__minus,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [Z: int] :
( ( ring_1_of_int @ A @ ( uminus_uminus @ int @ Z ) )
= ( uminus_uminus @ A @ ( ring_1_of_int @ A @ Z ) ) ) ) ).
% of_int_minus
thf(fact_777_of__int__diff,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [W: int,Z: int] :
( ( ring_1_of_int @ A @ ( minus_minus @ int @ W @ Z ) )
= ( minus_minus @ A @ ( ring_1_of_int @ A @ W ) @ ( ring_1_of_int @ A @ Z ) ) ) ) ).
% of_int_diff
thf(fact_778_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_779_of__int__abs,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: int] :
( ( ring_1_of_int @ A @ ( abs_abs @ int @ X ) )
= ( abs_abs @ A @ ( ring_1_of_int @ A @ X ) ) ) ) ).
% of_int_abs
thf(fact_780_frac__of__int,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z: int] :
( ( archimedean_frac @ A @ ( ring_1_of_int @ A @ Z ) )
= ( zero_zero @ A ) ) ) ).
% frac_of_int
thf(fact_781_frac__eq__0__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ( archimedean_frac @ A @ X )
= ( zero_zero @ A ) )
= ( member @ A @ X @ ( ring_1_Ints @ A ) ) ) ) ).
% frac_eq_0_iff
thf(fact_782_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_783_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_784_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_785_left__minus__one__mult__self,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [N: nat,A3: A] :
( ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N ) @ ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N ) @ A3 ) )
= A3 ) ) ).
% left_minus_one_mult_self
thf(fact_786_of__int__power__le__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: int,B2: int,W: nat] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ X ) @ ( power_power @ A @ ( ring_1_of_int @ A @ B2 ) @ W ) )
= ( ord_less_eq @ int @ X @ ( power_power @ int @ B2 @ W ) ) ) ) ).
% of_int_power_le_of_int_cancel_iff
thf(fact_787_of__int__le__of__int__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [B2: int,W: nat,X: int] :
( ( ord_less_eq @ A @ ( power_power @ A @ ( ring_1_of_int @ A @ B2 ) @ W ) @ ( ring_1_of_int @ A @ X ) )
= ( ord_less_eq @ int @ ( power_power @ int @ B2 @ W ) @ X ) ) ) ).
% of_int_le_of_int_power_cancel_iff
thf(fact_788_of__int__power__less__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: int,B2: int,W: nat] :
( ( ord_less @ A @ ( ring_1_of_int @ A @ X ) @ ( power_power @ A @ ( ring_1_of_int @ A @ B2 ) @ W ) )
= ( ord_less @ int @ X @ ( power_power @ int @ B2 @ W ) ) ) ) ).
% of_int_power_less_of_int_cancel_iff
thf(fact_789_of__int__less__of__int__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [B2: int,W: nat,X: int] :
( ( ord_less @ A @ ( power_power @ A @ ( ring_1_of_int @ A @ B2 ) @ W ) @ ( ring_1_of_int @ A @ X ) )
= ( ord_less @ int @ ( power_power @ int @ B2 @ W ) @ X ) ) ) ).
% of_int_less_of_int_power_cancel_iff
thf(fact_790_nat__0__iff,axiom,
! [I: int] :
( ( ( nat2 @ I )
= ( zero_zero @ nat ) )
= ( ord_less_eq @ int @ I @ ( zero_zero @ int ) ) ) ).
% nat_0_iff
thf(fact_791_nat__le__0,axiom,
! [Z: int] :
( ( ord_less_eq @ int @ Z @ ( zero_zero @ int ) )
=> ( ( nat2 @ Z )
= ( zero_zero @ nat ) ) ) ).
% nat_le_0
thf(fact_792_zless__nat__conj,axiom,
! [W: int,Z: int] :
( ( ord_less @ nat @ ( nat2 @ W ) @ ( nat2 @ Z ) )
= ( ( ord_less @ int @ ( zero_zero @ int ) @ Z )
& ( ord_less @ int @ W @ Z ) ) ) ).
% zless_nat_conj
thf(fact_793_sgn__mult__self__eq,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] :
( ( times_times @ A @ ( sgn_sgn @ A @ A3 ) @ ( sgn_sgn @ A @ A3 ) )
= ( zero_neq_one_of_bool @ A
@ ( A3
!= ( zero_zero @ A ) ) ) ) ) ).
% sgn_mult_self_eq
thf(fact_794_int__nat__eq,axiom,
! [Z: int] :
( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
=> ( ( semiring_1_of_nat @ int @ ( nat2 @ Z ) )
= Z ) )
& ( ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
=> ( ( semiring_1_of_nat @ int @ ( nat2 @ Z ) )
= ( zero_zero @ int ) ) ) ) ).
% int_nat_eq
thf(fact_795_nat__zminus__int,axiom,
! [N: nat] :
( ( nat2 @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N ) ) )
= ( zero_zero @ nat ) ) ).
% nat_zminus_int
thf(fact_796_frac__gt__0__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( archimedean_frac @ A @ X ) )
= ( ~ ( member @ A @ X @ ( ring_1_Ints @ A ) ) ) ) ) ).
% frac_gt_0_iff
thf(fact_797_of__int__0__le__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z: int] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( ring_1_of_int @ A @ Z ) )
= ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z ) ) ) ).
% of_int_0_le_iff
thf(fact_798_of__int__le__0__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z: int] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z ) @ ( zero_zero @ A ) )
= ( ord_less_eq @ int @ Z @ ( zero_zero @ int ) ) ) ) ).
% of_int_le_0_iff
thf(fact_799_of__int__0__less__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z: int] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( ring_1_of_int @ A @ Z ) )
= ( ord_less @ int @ ( zero_zero @ int ) @ Z ) ) ) ).
% of_int_0_less_iff
thf(fact_800_of__int__less__0__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z: int] :
( ( ord_less @ A @ ( ring_1_of_int @ A @ Z ) @ ( zero_zero @ A ) )
= ( ord_less @ int @ Z @ ( zero_zero @ int ) ) ) ) ).
% of_int_less_0_iff
thf(fact_801_of__int__le__1__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z: int] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z ) @ ( one_one @ A ) )
= ( ord_less_eq @ int @ Z @ ( one_one @ int ) ) ) ) ).
% of_int_le_1_iff
thf(fact_802_of__int__1__le__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z: int] :
( ( ord_less_eq @ A @ ( one_one @ A ) @ ( ring_1_of_int @ A @ Z ) )
= ( ord_less_eq @ int @ ( one_one @ int ) @ Z ) ) ) ).
% of_int_1_le_iff
thf(fact_803_of__nat__nat,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [Z: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
=> ( ( semiring_1_of_nat @ A @ ( nat2 @ Z ) )
= ( ring_1_of_int @ A @ Z ) ) ) ) ).
% of_nat_nat
thf(fact_804_nat__times__as__int,axiom,
( ( times_times @ nat )
= ( ^ [A6: nat,B5: nat] : ( nat2 @ ( times_times @ int @ ( semiring_1_of_nat @ int @ A6 ) @ ( semiring_1_of_nat @ int @ B5 ) ) ) ) ) ).
% nat_times_as_int
thf(fact_805_nat__mult__distrib,axiom,
! [Z: int,Z4: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
=> ( ( nat2 @ ( times_times @ int @ Z @ Z4 ) )
= ( times_times @ nat @ ( nat2 @ Z ) @ ( nat2 @ Z4 ) ) ) ) ).
% nat_mult_distrib
thf(fact_806_int__distrib_I4_J,axiom,
! [W: int,Z1: int,Z22: int] :
( ( times_times @ int @ W @ ( minus_minus @ int @ Z1 @ Z22 ) )
= ( minus_minus @ int @ ( times_times @ int @ W @ Z1 ) @ ( times_times @ int @ W @ Z22 ) ) ) ).
% int_distrib(4)
thf(fact_807_int__distrib_I3_J,axiom,
! [Z1: int,Z22: int,W: int] :
( ( times_times @ int @ ( minus_minus @ int @ Z1 @ Z22 ) @ W )
= ( minus_minus @ int @ ( times_times @ int @ Z1 @ W ) @ ( times_times @ int @ Z22 @ W ) ) ) ).
% int_distrib(3)
thf(fact_808_le__cube,axiom,
! [M: nat] : ( ord_less_eq @ nat @ M @ ( times_times @ nat @ M @ ( times_times @ nat @ M @ M ) ) ) ).
% le_cube
thf(fact_809_le__refl,axiom,
! [N: nat] : ( ord_less_eq @ nat @ N @ N ) ).
% le_refl
thf(fact_810_nat__mono,axiom,
! [X: int,Y3: int] :
( ( ord_less_eq @ int @ X @ Y3 )
=> ( ord_less_eq @ nat @ ( nat2 @ X ) @ ( nat2 @ Y3 ) ) ) ).
% nat_mono
thf(fact_811_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_812_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq @ nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_813_le__square,axiom,
! [M: nat] : ( ord_less_eq @ nat @ M @ ( times_times @ nat @ M @ M ) ) ).
% le_square
thf(fact_814_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_815_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_816_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_817_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_818_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_819_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_820_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_821_le__diff__iff_H,axiom,
! [A3: nat,C2: nat,B2: nat] :
( ( ord_less_eq @ nat @ A3 @ C2 )
=> ( ( ord_less_eq @ nat @ B2 @ C2 )
=> ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ C2 @ A3 ) @ ( minus_minus @ nat @ C2 @ B2 ) )
= ( ord_less_eq @ nat @ B2 @ A3 ) ) ) ) ).
% le_diff_iff'
thf(fact_822_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_823_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_824_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_825_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_826_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_827_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_828_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_829_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B2: nat] :
( ( P @ K )
=> ( ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq @ nat @ Y4 @ B2 ) )
=> ? [X4: nat] :
( ( P @ X4 )
& ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq @ nat @ Y5 @ X4 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_830_verit__la__generic,axiom,
! [A3: int,X: int] :
( ( ord_less_eq @ int @ A3 @ X )
| ( A3 = X )
| ( ord_less_eq @ int @ X @ A3 ) ) ).
% verit_la_generic
thf(fact_831_predicate2D__conj,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Q: A > B > $o,R5: $o,X: A,Y3: B] :
( ( ( ord_less_eq @ ( A > B > $o ) @ P @ Q )
& R5 )
=> ( R5
& ( ( P @ X @ Y3 )
=> ( Q @ X @ Y3 ) ) ) ) ).
% predicate2D_conj
thf(fact_832_pred__subset__eq,axiom,
! [A: $tType,R5: set @ A,S3: set @ A] :
( ( ord_less_eq @ ( A > $o )
@ ^ [X2: A] : ( member @ A @ X2 @ R5 )
@ ^ [X2: A] : ( member @ A @ X2 @ S3 ) )
= ( ord_less_eq @ ( set @ A ) @ R5 @ S3 ) ) ).
% pred_subset_eq
thf(fact_833_eq__subset,axiom,
! [A: $tType,P: A > A > $o] :
( ord_less_eq @ ( A > A > $o )
@ ^ [Y6: A,Z2: A] : Y6 = Z2
@ ^ [A6: A,B5: A] :
( ( P @ A6 @ B5 )
| ( A6 = B5 ) ) ) ).
% eq_subset
thf(fact_834_conj__subset__def,axiom,
! [A: $tType,A5: set @ A,P: A > $o,Q: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ A5
@ ( collect @ A
@ ^ [X2: A] :
( ( P @ X2 )
& ( Q @ X2 ) ) ) )
= ( ( ord_less_eq @ ( set @ A ) @ A5 @ ( collect @ A @ P ) )
& ( ord_less_eq @ ( set @ A ) @ A5 @ ( collect @ A @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_835_prop__restrict,axiom,
! [A: $tType,X: A,Z5: set @ A,X7: set @ A,P: A > $o] :
( ( member @ A @ X @ Z5 )
=> ( ( ord_less_eq @ ( set @ A ) @ Z5
@ ( collect @ A
@ ^ [X2: A] :
( ( member @ A @ X2 @ X7 )
& ( P @ X2 ) ) ) )
=> ( P @ X ) ) ) ).
% prop_restrict
thf(fact_836_Collect__restrict,axiom,
! [A: $tType,X7: set @ A,P: A > $o] :
( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [X2: A] :
( ( member @ A @ X2 @ X7 )
& ( P @ X2 ) ) )
@ X7 ) ).
% Collect_restrict
thf(fact_837_mult_Oleft__commute,axiom,
! [A: $tType] :
( ( ab_semigroup_mult @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( times_times @ A @ B2 @ ( times_times @ A @ A3 @ C2 ) )
= ( times_times @ A @ A3 @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% mult.left_commute
thf(fact_838_mult_Ocommute,axiom,
! [A: $tType] :
( ( ab_semigroup_mult @ A )
=> ( ( times_times @ A )
= ( ^ [A6: A,B5: A] : ( times_times @ A @ B5 @ A6 ) ) ) ) ).
% mult.commute
thf(fact_839_mult_Oassoc,axiom,
! [A: $tType] :
( ( semigroup_mult @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( times_times @ A @ ( times_times @ A @ A3 @ B2 ) @ C2 )
= ( times_times @ A @ A3 @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% mult.assoc
thf(fact_840_mult__of__int__commute,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X: int,Y3: A] :
( ( times_times @ A @ ( ring_1_of_int @ A @ X ) @ Y3 )
= ( times_times @ A @ Y3 @ ( ring_1_of_int @ A @ X ) ) ) ) ).
% mult_of_int_commute
thf(fact_841_Nats__mult,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [A3: A,B2: A] :
( ( member @ A @ A3 @ ( semiring_1_Nats @ A ) )
=> ( ( member @ A @ B2 @ ( semiring_1_Nats @ A ) )
=> ( member @ A @ ( times_times @ A @ A3 @ B2 ) @ ( semiring_1_Nats @ A ) ) ) ) ) ).
% Nats_mult
thf(fact_842_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: $tType] :
( ( ab_semigroup_mult @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( times_times @ A @ ( times_times @ A @ A3 @ B2 ) @ C2 )
= ( times_times @ A @ A3 @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_843_nat__abs__mult__distrib,axiom,
! [W: int,Z: int] :
( ( nat2 @ ( abs_abs @ int @ ( times_times @ int @ W @ Z ) ) )
= ( times_times @ nat @ ( nat2 @ ( abs_abs @ int @ W ) ) @ ( nat2 @ ( abs_abs @ int @ Z ) ) ) ) ).
% nat_abs_mult_distrib
thf(fact_844_int__ops_I7_J,axiom,
! [A3: nat,B2: nat] :
( ( semiring_1_of_nat @ int @ ( times_times @ nat @ A3 @ B2 ) )
= ( times_times @ int @ ( semiring_1_of_nat @ int @ A3 ) @ ( semiring_1_of_nat @ int @ B2 ) ) ) ).
% int_ops(7)
thf(fact_845_nat__mult__distrib__neg,axiom,
! [Z: int,Z4: int] :
( ( ord_less_eq @ int @ Z @ ( zero_zero @ int ) )
=> ( ( nat2 @ ( times_times @ int @ Z @ Z4 ) )
= ( times_times @ nat @ ( nat2 @ ( uminus_uminus @ int @ Z ) ) @ ( nat2 @ ( uminus_uminus @ int @ Z4 ) ) ) ) ) ).
% nat_mult_distrib_neg
thf(fact_846_ex__nat,axiom,
( ( ^ [P2: nat > $o] :
? [X3: nat] : ( P2 @ X3 ) )
= ( ^ [P3: nat > $o] :
? [X2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X2 )
& ( P3 @ ( nat2 @ X2 ) ) ) ) ) ).
% ex_nat
thf(fact_847_all__nat,axiom,
( ( ^ [P2: nat > $o] :
! [X3: nat] : ( P2 @ X3 ) )
= ( ^ [P3: nat > $o] :
! [X2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X2 )
=> ( P3 @ ( nat2 @ X2 ) ) ) ) ) ).
% all_nat
thf(fact_848_nat__power__eq,axiom,
! [Z: int,N: nat] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
=> ( ( nat2 @ ( power_power @ int @ Z @ N ) )
= ( power_power @ nat @ ( nat2 @ Z ) @ N ) ) ) ).
% nat_power_eq
thf(fact_849_eq__nat__nat__iff,axiom,
! [Z: int,Z4: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z4 )
=> ( ( ( nat2 @ Z )
= ( nat2 @ Z4 ) )
= ( Z = Z4 ) ) ) ) ).
% eq_nat_nat_iff
thf(fact_850_nat__diff__distrib,axiom,
! [Z4: int,Z: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z4 )
=> ( ( ord_less_eq @ int @ Z4 @ Z )
=> ( ( nat2 @ ( minus_minus @ int @ Z @ Z4 ) )
= ( minus_minus @ nat @ ( nat2 @ Z ) @ ( nat2 @ Z4 ) ) ) ) ) ).
% nat_diff_distrib
thf(fact_851_nat__diff__distrib_H,axiom,
! [X: int,Y3: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y3 )
=> ( ( nat2 @ ( minus_minus @ int @ X @ Y3 ) )
= ( minus_minus @ nat @ ( nat2 @ X ) @ ( nat2 @ Y3 ) ) ) ) ) ).
% nat_diff_distrib'
thf(fact_852_nat__le__iff,axiom,
! [X: int,N: nat] :
( ( ord_less_eq @ nat @ ( nat2 @ X ) @ N )
= ( ord_less_eq @ int @ X @ ( semiring_1_of_nat @ int @ N ) ) ) ).
% nat_le_iff
thf(fact_853_nat__abs__int__diff,axiom,
! [A3: nat,B2: nat] :
( ( ( ord_less_eq @ nat @ A3 @ B2 )
=> ( ( nat2 @ ( abs_abs @ int @ ( minus_minus @ int @ ( semiring_1_of_nat @ int @ A3 ) @ ( semiring_1_of_nat @ int @ B2 ) ) ) )
= ( minus_minus @ nat @ B2 @ A3 ) ) )
& ( ~ ( ord_less_eq @ nat @ A3 @ B2 )
=> ( ( nat2 @ ( abs_abs @ int @ ( minus_minus @ int @ ( semiring_1_of_nat @ int @ A3 ) @ ( semiring_1_of_nat @ int @ B2 ) ) ) )
= ( minus_minus @ nat @ A3 @ B2 ) ) ) ) ).
% nat_abs_int_diff
thf(fact_854_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_855_mult__not__zero,axiom,
! [A: $tType] :
( ( mult_zero @ A )
=> ! [A3: A,B2: A] :
( ( ( times_times @ A @ A3 @ B2 )
!= ( zero_zero @ A ) )
=> ( ( A3
!= ( zero_zero @ A ) )
& ( B2
!= ( zero_zero @ A ) ) ) ) ) ).
% mult_not_zero
thf(fact_856_divisors__zero,axiom,
! [A: $tType] :
( ( semiri3467727345109120633visors @ A )
=> ! [A3: A,B2: A] :
( ( ( times_times @ A @ A3 @ B2 )
= ( zero_zero @ A ) )
=> ( ( A3
= ( zero_zero @ A ) )
| ( B2
= ( zero_zero @ A ) ) ) ) ) ).
% divisors_zero
thf(fact_857_no__zero__divisors,axiom,
! [A: $tType] :
( ( semiri3467727345109120633visors @ A )
=> ! [A3: A,B2: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ A3 @ B2 )
!= ( zero_zero @ A ) ) ) ) ) ).
% no_zero_divisors
thf(fact_858_mult__left__cancel,axiom,
! [A: $tType] :
( ( semiri6575147826004484403cancel @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( ( times_times @ A @ C2 @ A3 )
= ( times_times @ A @ C2 @ B2 ) )
= ( A3 = B2 ) ) ) ) ).
% mult_left_cancel
thf(fact_859_mult__right__cancel,axiom,
! [A: $tType] :
( ( semiri6575147826004484403cancel @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( ( times_times @ A @ A3 @ C2 )
= ( times_times @ A @ B2 @ C2 ) )
= ( A3 = B2 ) ) ) ) ).
% mult_right_cancel
thf(fact_860_ring__class_Oring__distribs_I2_J,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( times_times @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 )
= ( plus_plus @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_861_ring__class_Oring__distribs_I1_J,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( times_times @ A @ A3 @ ( plus_plus @ A @ B2 @ C2 ) )
= ( plus_plus @ A @ ( times_times @ A @ A3 @ B2 ) @ ( times_times @ A @ A3 @ C2 ) ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_862_comm__semiring__class_Odistrib,axiom,
! [A: $tType] :
( ( comm_semiring @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( times_times @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 )
= ( plus_plus @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% comm_semiring_class.distrib
thf(fact_863_distrib__left,axiom,
! [A: $tType] :
( ( semiring @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( times_times @ A @ A3 @ ( plus_plus @ A @ B2 @ C2 ) )
= ( plus_plus @ A @ ( times_times @ A @ A3 @ B2 ) @ ( times_times @ A @ A3 @ C2 ) ) ) ) ).
% distrib_left
thf(fact_864_distrib__right,axiom,
! [A: $tType] :
( ( semiring @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( times_times @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 )
= ( plus_plus @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% distrib_right
thf(fact_865_combine__common__factor,axiom,
! [A: $tType] :
( ( semiring @ A )
=> ! [A3: A,E2: A,B2: A,C2: A] :
( ( plus_plus @ A @ ( times_times @ A @ A3 @ E2 ) @ ( plus_plus @ A @ ( times_times @ A @ B2 @ E2 ) @ C2 ) )
= ( plus_plus @ A @ ( times_times @ A @ ( plus_plus @ A @ A3 @ B2 ) @ E2 ) @ C2 ) ) ) ).
% combine_common_factor
thf(fact_866_crossproduct__eq,axiom,
! [A: $tType] :
( ( semiri1453513574482234551roduct @ A )
=> ! [W: A,Y3: A,X: A,Z: A] :
( ( ( plus_plus @ A @ ( times_times @ A @ W @ Y3 ) @ ( times_times @ A @ X @ Z ) )
= ( plus_plus @ A @ ( times_times @ A @ W @ Z ) @ ( times_times @ A @ X @ Y3 ) ) )
= ( ( W = X )
| ( Y3 = Z ) ) ) ) ).
% crossproduct_eq
thf(fact_867_crossproduct__noteq,axiom,
! [A: $tType] :
( ( semiri1453513574482234551roduct @ A )
=> ! [A3: A,B2: A,C2: A,D3: A] :
( ( ( A3 != B2 )
& ( C2 != D3 ) )
= ( ( plus_plus @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ D3 ) )
!= ( plus_plus @ A @ ( times_times @ A @ A3 @ D3 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ) ).
% crossproduct_noteq
thf(fact_868_comm__monoid__mult__class_Omult__1,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: A] :
( ( times_times @ A @ ( one_one @ A ) @ A3 )
= A3 ) ) ).
% comm_monoid_mult_class.mult_1
thf(fact_869_mult_Ocomm__neutral,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: A] :
( ( times_times @ A @ A3 @ ( one_one @ A ) )
= A3 ) ) ).
% mult.comm_neutral
thf(fact_870_right__diff__distrib_H,axiom,
! [A: $tType] :
( ( comm_s4317794764714335236cancel @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( times_times @ A @ A3 @ ( minus_minus @ A @ B2 @ C2 ) )
= ( minus_minus @ A @ ( times_times @ A @ A3 @ B2 ) @ ( times_times @ A @ A3 @ C2 ) ) ) ) ).
% right_diff_distrib'
thf(fact_871_left__diff__distrib_H,axiom,
! [A: $tType] :
( ( comm_s4317794764714335236cancel @ A )
=> ! [B2: A,C2: A,A3: A] :
( ( times_times @ A @ ( minus_minus @ A @ B2 @ C2 ) @ A3 )
= ( minus_minus @ A @ ( times_times @ A @ B2 @ A3 ) @ ( times_times @ A @ C2 @ A3 ) ) ) ) ).
% left_diff_distrib'
thf(fact_872_right__diff__distrib,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( times_times @ A @ A3 @ ( minus_minus @ A @ B2 @ C2 ) )
= ( minus_minus @ A @ ( times_times @ A @ A3 @ B2 ) @ ( times_times @ A @ A3 @ C2 ) ) ) ) ).
% right_diff_distrib
thf(fact_873_left__diff__distrib,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( times_times @ A @ ( minus_minus @ A @ A3 @ B2 ) @ C2 )
= ( minus_minus @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% left_diff_distrib
thf(fact_874_minus__mult__commute,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A3: A,B2: A] :
( ( times_times @ A @ ( uminus_uminus @ A @ A3 ) @ B2 )
= ( times_times @ A @ A3 @ ( uminus_uminus @ A @ B2 ) ) ) ) ).
% minus_mult_commute
thf(fact_875_square__eq__iff,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [A3: A,B2: A] :
( ( ( times_times @ A @ A3 @ A3 )
= ( times_times @ A @ B2 @ B2 ) )
= ( ( A3 = B2 )
| ( A3
= ( uminus_uminus @ A @ B2 ) ) ) ) ) ).
% square_eq_iff
thf(fact_876_mult__0,axiom,
! [N: nat] :
( ( times_times @ nat @ ( zero_zero @ nat ) @ N )
= ( zero_zero @ nat ) ) ).
% mult_0
thf(fact_877_mult__of__nat__commute,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [X: nat,Y3: A] :
( ( times_times @ A @ ( semiring_1_of_nat @ A @ X ) @ Y3 )
= ( times_times @ A @ Y3 @ ( semiring_1_of_nat @ A @ X ) ) ) ) ).
% mult_of_nat_commute
thf(fact_878_times__int__code_I2_J,axiom,
! [L: int] :
( ( times_times @ int @ ( zero_zero @ int ) @ L )
= ( zero_zero @ int ) ) ).
% times_int_code(2)
thf(fact_879_times__int__code_I1_J,axiom,
! [K: int] :
( ( times_times @ int @ K @ ( zero_zero @ int ) )
= ( zero_zero @ int ) ) ).
% times_int_code(1)
thf(fact_880_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_881_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_882_abs__mult,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A3: A,B2: A] :
( ( abs_abs @ A @ ( times_times @ A @ A3 @ B2 ) )
= ( times_times @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B2 ) ) ) ) ).
% abs_mult
thf(fact_883_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_884_Ints__of__int,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [Z: int] : ( member @ A @ ( ring_1_of_int @ A @ Z ) @ ( ring_1_Ints @ A ) ) ) ).
% Ints_of_int
thf(fact_885_Ints__induct,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [Q2: A,P: A > $o] :
( ( member @ A @ Q2 @ ( ring_1_Ints @ A ) )
=> ( ! [Z6: int] : ( P @ ( ring_1_of_int @ A @ Z6 ) )
=> ( P @ Q2 ) ) ) ) ).
% Ints_induct
thf(fact_886_Ints__cases,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [Q2: A] :
( ( member @ A @ Q2 @ ( ring_1_Ints @ A ) )
=> ~ ! [Z6: int] :
( Q2
!= ( ring_1_of_int @ A @ Z6 ) ) ) ) ).
% Ints_cases
thf(fact_887_int__distrib_I2_J,axiom,
! [W: int,Z1: int,Z22: int] :
( ( times_times @ int @ W @ ( plus_plus @ int @ Z1 @ Z22 ) )
= ( plus_plus @ int @ ( times_times @ int @ W @ Z1 ) @ ( times_times @ int @ W @ Z22 ) ) ) ).
% int_distrib(2)
thf(fact_888_int__distrib_I1_J,axiom,
! [Z1: int,Z22: int,W: int] :
( ( times_times @ int @ ( plus_plus @ int @ Z1 @ Z22 ) @ W )
= ( plus_plus @ int @ ( times_times @ int @ Z1 @ W ) @ ( times_times @ int @ Z22 @ W ) ) ) ).
% int_distrib(1)
thf(fact_889_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_890_nat__mult__1,axiom,
! [N: nat] :
( ( times_times @ nat @ ( one_one @ nat ) @ N )
= N ) ).
% nat_mult_1
thf(fact_891_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times @ nat @ N @ ( one_one @ nat ) )
= N ) ).
% nat_mult_1_right
thf(fact_892_sgn__mult,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A3: A,B2: A] :
( ( sgn_sgn @ A @ ( times_times @ A @ A3 @ B2 ) )
= ( times_times @ A @ ( sgn_sgn @ A @ A3 ) @ ( sgn_sgn @ A @ B2 ) ) ) ) ).
% sgn_mult
thf(fact_893_of__bool__conj,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [P: $o,Q: $o] :
( ( zero_neq_one_of_bool @ A
@ ( P
& Q ) )
= ( times_times @ A @ ( zero_neq_one_of_bool @ A @ P ) @ ( zero_neq_one_of_bool @ A @ Q ) ) ) ) ).
% of_bool_conj
thf(fact_894_Ints__mult,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [A3: A,B2: A] :
( ( member @ A @ A3 @ ( ring_1_Ints @ A ) )
=> ( ( member @ A @ B2 @ ( ring_1_Ints @ A ) )
=> ( member @ A @ ( times_times @ A @ A3 @ B2 ) @ ( ring_1_Ints @ A ) ) ) ) ) ).
% Ints_mult
thf(fact_895_nat__minus__as__int,axiom,
( ( minus_minus @ nat )
= ( ^ [A6: nat,B5: nat] : ( nat2 @ ( minus_minus @ int @ ( semiring_1_of_nat @ int @ A6 ) @ ( semiring_1_of_nat @ int @ B5 ) ) ) ) ) ).
% nat_minus_as_int
thf(fact_896_lambda__zero,axiom,
! [A: $tType] :
( ( mult_zero @ A )
=> ( ( ^ [H4: A] : ( zero_zero @ A ) )
= ( times_times @ A @ ( zero_zero @ A ) ) ) ) ).
% lambda_zero
thf(fact_897_lambda__one,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ( ( ^ [X2: A] : X2 )
= ( times_times @ A @ ( one_one @ A ) ) ) ) ).
% lambda_one
thf(fact_898_frac__ge__0,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( archimedean_frac @ A @ X ) ) ) ).
% frac_ge_0
thf(fact_899_frac__lt__1,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] : ( ord_less @ A @ ( archimedean_frac @ A @ X ) @ ( one_one @ A ) ) ) ).
% frac_lt_1
thf(fact_900_frac__1__eq,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( archimedean_frac @ A @ ( plus_plus @ A @ X @ ( one_one @ A ) ) )
= ( archimedean_frac @ A @ X ) ) ) ).
% frac_1_eq
thf(fact_901_Nats__0,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( member @ A @ ( zero_zero @ A ) @ ( semiring_1_Nats @ A ) ) ) ).
% Nats_0
thf(fact_902_Nats__add,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [A3: A,B2: A] :
( ( member @ A @ A3 @ ( semiring_1_Nats @ A ) )
=> ( ( member @ A @ B2 @ ( semiring_1_Nats @ A ) )
=> ( member @ A @ ( plus_plus @ A @ A3 @ B2 ) @ ( semiring_1_Nats @ A ) ) ) ) ) ).
% Nats_add
thf(fact_903_Nats__1,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( member @ A @ ( one_one @ A ) @ ( semiring_1_Nats @ A ) ) ) ).
% Nats_1
thf(fact_904_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_905_Nats__induct,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [X: A,P: A > $o] :
( ( member @ A @ X @ ( semiring_1_Nats @ A ) )
=> ( ! [N2: nat] : ( P @ ( semiring_1_of_nat @ A @ N2 ) )
=> ( P @ X ) ) ) ) ).
% Nats_induct
thf(fact_906_Nats__cases,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [X: A] :
( ( member @ A @ X @ ( semiring_1_Nats @ A ) )
=> ~ ! [N2: nat] :
( X
!= ( semiring_1_of_nat @ A @ N2 ) ) ) ) ).
% Nats_cases
thf(fact_907_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_908_nat__zero__as__int,axiom,
( ( zero_zero @ nat )
= ( nat2 @ ( zero_zero @ int ) ) ) ).
% nat_zero_as_int
thf(fact_909_nat__le__eq__zle,axiom,
! [W: int,Z: int] :
( ( ( ord_less @ int @ ( zero_zero @ int ) @ W )
| ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z ) )
=> ( ( ord_less_eq @ nat @ ( nat2 @ W ) @ ( nat2 @ Z ) )
= ( ord_less_eq @ int @ W @ Z ) ) ) ).
% nat_le_eq_zle
thf(fact_910_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_911_int__eq__iff,axiom,
! [M: nat,Z: int] :
( ( ( semiring_1_of_nat @ int @ M )
= Z )
= ( ( M
= ( nat2 @ Z ) )
& ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z ) ) ) ).
% int_eq_iff
thf(fact_912_nat__0__le,axiom,
! [Z: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
=> ( ( semiring_1_of_nat @ int @ ( nat2 @ Z ) )
= Z ) ) ).
% nat_0_le
thf(fact_913_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_914_nat__one__as__int,axiom,
( ( one_one @ nat )
= ( nat2 @ ( one_one @ int ) ) ) ).
% nat_one_as_int
thf(fact_915_ex__less__of__nat__mult,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X: A,Y3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X )
=> ? [N2: nat] : ( ord_less @ A @ Y3 @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N2 ) @ X ) ) ) ) ).
% ex_less_of_nat_mult
thf(fact_916_mult__mono,axiom,
! [A: $tType] :
( ( ordered_semiring @ A )
=> ! [A3: A,B2: A,C2: A,D3: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ D3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ D3 ) ) ) ) ) ) ) ).
% mult_mono
thf(fact_917_mult__mono_H,axiom,
! [A: $tType] :
( ( ordered_semiring @ A )
=> ! [A3: A,B2: A,C2: A,D3: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ D3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ D3 ) ) ) ) ) ) ) ).
% mult_mono'
thf(fact_918_zero__le__square,axiom,
! [A: $tType] :
( ( linordered_ring @ A )
=> ! [A3: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ A3 ) ) ) ).
% zero_le_square
thf(fact_919_split__mult__pos__le,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [A3: A,B2: A] :
( ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) )
| ( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B2 ) ) ) ) ).
% split_mult_pos_le
thf(fact_920_mult__left__mono__neg,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( ord_less_eq @ A @ B2 @ A3 )
=> ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) ) ) ) ) ).
% mult_left_mono_neg
thf(fact_921_mult__nonpos__nonpos,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B2 ) ) ) ) ) ).
% mult_nonpos_nonpos
thf(fact_922_mult__left__mono,axiom,
! [A: $tType] :
( ( ordered_semiring @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) ) ) ) ) ).
% mult_left_mono
thf(fact_923_mult__right__mono__neg,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( ord_less_eq @ A @ B2 @ A3 )
=> ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ) ).
% mult_right_mono_neg
thf(fact_924_mult__right__mono,axiom,
! [A: $tType] :
( ( ordered_semiring @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ) ).
% mult_right_mono
thf(fact_925_mult__le__0__iff,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ A3 @ B2 ) @ ( zero_zero @ A ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) )
| ( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) ) ) ) ).
% mult_le_0_iff
thf(fact_926_split__mult__neg__le,axiom,
! [A: $tType] :
( ( ordered_semiring_0 @ A )
=> ! [A3: A,B2: A] :
( ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) )
| ( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ A3 @ B2 ) @ ( zero_zero @ A ) ) ) ) ).
% split_mult_neg_le
thf(fact_927_mult__nonneg__nonneg,axiom,
! [A: $tType] :
( ( ordered_semiring_0 @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B2 ) ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_928_mult__nonneg__nonpos,axiom,
! [A: $tType] :
( ( ordered_semiring_0 @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ A3 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% mult_nonneg_nonpos
thf(fact_929_mult__nonpos__nonneg,axiom,
! [A: $tType] :
( ( ordered_semiring_0 @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A3 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% mult_nonpos_nonneg
thf(fact_930_mult__nonneg__nonpos2,axiom,
! [A: $tType] :
( ( ordered_semiring_0 @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ B2 @ A3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_931_zero__le__mult__iff,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B2 ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) )
| ( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) ) ) ) ) ).
% zero_le_mult_iff
thf(fact_932_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: $tType] :
( ( ordere2520102378445227354miring @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_933_mult__neg__neg,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B2 ) ) ) ) ) ).
% mult_neg_neg
thf(fact_934_not__square__less__zero,axiom,
! [A: $tType] :
( ( linordered_ring @ A )
=> ! [A3: A] :
~ ( ord_less @ A @ ( times_times @ A @ A3 @ A3 ) @ ( zero_zero @ A ) ) ) ).
% not_square_less_zero
thf(fact_935_mult__less__0__iff,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( times_times @ A @ A3 @ B2 ) @ ( zero_zero @ A ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less @ A @ B2 @ ( zero_zero @ A ) ) )
| ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
& ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ) ) ).
% mult_less_0_iff
thf(fact_936_mult__neg__pos,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less @ A @ ( times_times @ A @ A3 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% mult_neg_pos
thf(fact_937_mult__pos__neg,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ A3 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% mult_pos_neg
thf(fact_938_mult__pos__pos,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B2 ) ) ) ) ) ).
% mult_pos_pos
thf(fact_939_mult__pos__neg2,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ B2 @ A3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% mult_pos_neg2
thf(fact_940_zero__less__mult__iff,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) )
| ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
& ( ord_less @ A @ B2 @ ( zero_zero @ A ) ) ) ) ) ) ).
% zero_less_mult_iff
thf(fact_941_zero__less__mult__pos,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B2 ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ) ).
% zero_less_mult_pos
thf(fact_942_zero__less__mult__pos2,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [B2: A,A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ B2 @ A3 ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ) ).
% zero_less_mult_pos2
thf(fact_943_mult__less__cancel__left__neg,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( ord_less @ A @ B2 @ A3 ) ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_944_mult__less__cancel__left__pos,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( ord_less @ A @ A3 @ B2 ) ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_945_mult__strict__left__mono__neg,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( ord_less @ A @ B2 @ A3 )
=> ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_946_mult__strict__left__mono,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) ) ) ) ) ).
% mult_strict_left_mono
thf(fact_947_mult__less__cancel__left__disj,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( ord_less @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
& ( ord_less @ A @ A3 @ B2 ) )
| ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
& ( ord_less @ A @ B2 @ A3 ) ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_948_mult__strict__right__mono__neg,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( ord_less @ A @ B2 @ A3 )
=> ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_949_mult__strict__right__mono,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ) ).
% mult_strict_right_mono
thf(fact_950_mult__less__cancel__right__disj,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( ord_less @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
& ( ord_less @ A @ A3 @ B2 ) )
| ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
& ( ord_less @ A @ B2 @ A3 ) ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_951_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: $tType] :
( ( linord2810124833399127020strict @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_952_add__scale__eq__noteq,axiom,
! [A: $tType] :
( ( semiri1453513574482234551roduct @ A )
=> ! [R: A,A3: A,B2: A,C2: A,D3: A] :
( ( R
!= ( zero_zero @ A ) )
=> ( ( ( A3 = B2 )
& ( C2 != D3 ) )
=> ( ( plus_plus @ A @ A3 @ ( times_times @ A @ R @ C2 ) )
!= ( plus_plus @ A @ B2 @ ( times_times @ A @ R @ D3 ) ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_953_sum__squares__eq__zero__iff,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [X: A,Y3: A] :
( ( ( plus_plus @ A @ ( times_times @ A @ X @ X ) @ ( times_times @ A @ Y3 @ Y3 ) )
= ( zero_zero @ A ) )
= ( ( X
= ( zero_zero @ A ) )
& ( Y3
= ( zero_zero @ A ) ) ) ) ) ).
% sum_squares_eq_zero_iff
thf(fact_954_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_955_eq__add__iff1,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A3: A,E2: A,C2: A,B2: A,D3: A] :
( ( ( plus_plus @ A @ ( times_times @ A @ A3 @ E2 ) @ C2 )
= ( plus_plus @ A @ ( times_times @ A @ B2 @ E2 ) @ D3 ) )
= ( ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ A3 @ B2 ) @ E2 ) @ C2 )
= D3 ) ) ) ).
% eq_add_iff1
thf(fact_956_eq__add__iff2,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A3: A,E2: A,C2: A,B2: A,D3: A] :
( ( ( plus_plus @ A @ ( times_times @ A @ A3 @ E2 ) @ C2 )
= ( plus_plus @ A @ ( times_times @ A @ B2 @ E2 ) @ D3 ) )
= ( C2
= ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ B2 @ A3 ) @ E2 ) @ D3 ) ) ) ) ).
% eq_add_iff2
thf(fact_957_square__diff__square__factored,axiom,
! [A: $tType] :
( ( comm_ring @ A )
=> ! [X: A,Y3: A] :
( ( minus_minus @ A @ ( times_times @ A @ X @ X ) @ ( times_times @ A @ Y3 @ Y3 ) )
= ( times_times @ A @ ( plus_plus @ A @ X @ Y3 ) @ ( minus_minus @ A @ X @ Y3 ) ) ) ) ).
% square_diff_square_factored
thf(fact_958_square__eq__1__iff,axiom,
! [A: $tType] :
( ( ring_15535105094025558882visors @ A )
=> ! [X: A] :
( ( ( times_times @ A @ X @ X )
= ( one_one @ A ) )
= ( ( X
= ( one_one @ A ) )
| ( X
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ) ).
% square_eq_1_iff
thf(fact_959_left__right__inverse__power,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [X: A,Y3: A,N: nat] :
( ( ( times_times @ A @ X @ Y3 )
= ( one_one @ A ) )
=> ( ( times_times @ A @ ( power_power @ A @ X @ N ) @ ( power_power @ A @ Y3 @ N ) )
= ( one_one @ A ) ) ) ) ).
% left_right_inverse_power
thf(fact_960_abs__mult__less,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,C2: A,B2: A,D3: A] :
( ( ord_less @ A @ ( abs_abs @ A @ A3 ) @ C2 )
=> ( ( ord_less @ A @ ( abs_abs @ A @ B2 ) @ D3 )
=> ( ord_less @ A @ ( times_times @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B2 ) ) @ ( times_times @ A @ C2 @ D3 ) ) ) ) ) ).
% abs_mult_less
thf(fact_961_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_962_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_963_power__add,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A3: A,M: nat,N: nat] :
( ( power_power @ A @ A3 @ ( plus_plus @ nat @ M @ N ) )
= ( times_times @ A @ ( power_power @ A @ A3 @ M ) @ ( power_power @ A @ A3 @ N ) ) ) ) ).
% power_add
thf(fact_964_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_965_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_966_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_967_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_968_mult__sgn__abs,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A] :
( ( times_times @ A @ ( sgn_sgn @ A @ X ) @ ( abs_abs @ A @ X ) )
= X ) ) ).
% mult_sgn_abs
thf(fact_969_sgn__mult__abs,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A3: A] :
( ( times_times @ A @ ( sgn_sgn @ A @ A3 ) @ ( abs_abs @ A @ A3 ) )
= A3 ) ) ).
% sgn_mult_abs
thf(fact_970_abs__mult__sgn,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A3: A] :
( ( times_times @ A @ ( abs_abs @ A @ A3 ) @ ( sgn_sgn @ A @ A3 ) )
= A3 ) ) ).
% abs_mult_sgn
thf(fact_971_linordered__idom__class_Oabs__sgn,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( ( abs_abs @ A )
= ( ^ [K3: A] : ( times_times @ A @ K3 @ ( sgn_sgn @ A @ K3 ) ) ) ) ) ).
% linordered_idom_class.abs_sgn
thf(fact_972_of__int__of__nat,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( ( ring_1_of_int @ A )
= ( ^ [K3: int] : ( if @ A @ ( ord_less @ int @ K3 @ ( zero_zero @ int ) ) @ ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ ( nat2 @ ( uminus_uminus @ int @ K3 ) ) ) ) @ ( semiring_1_of_nat @ A @ ( nat2 @ K3 ) ) ) ) ) ) ).
% of_int_of_nat
thf(fact_973_field__le__mult__one__interval,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y3: A] :
( ! [Z6: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Z6 )
=> ( ( ord_less @ A @ Z6 @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ Z6 @ X ) @ Y3 ) ) )
=> ( ord_less_eq @ A @ X @ Y3 ) ) ) ).
% field_le_mult_one_interval
thf(fact_974_of__int__nonneg,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( ring_1_of_int @ A @ Z ) ) ) ) ).
% of_int_nonneg
thf(fact_975_nat__mono__iff,axiom,
! [Z: int,W: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ Z )
=> ( ( ord_less @ nat @ ( nat2 @ W ) @ ( nat2 @ Z ) )
= ( ord_less @ int @ W @ Z ) ) ) ).
% nat_mono_iff
thf(fact_976_nat__less__eq__zless,axiom,
! [W: int,Z: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ W )
=> ( ( ord_less @ nat @ ( nat2 @ W ) @ ( nat2 @ Z ) )
= ( ord_less @ int @ W @ Z ) ) ) ).
% nat_less_eq_zless
thf(fact_977_nat__eq__iff,axiom,
! [W: int,M: nat] :
( ( ( nat2 @ W )
= M )
= ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ W )
=> ( W
= ( semiring_1_of_nat @ int @ M ) ) )
& ( ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ W )
=> ( M
= ( zero_zero @ nat ) ) ) ) ) ).
% nat_eq_iff
thf(fact_978_nat__eq__iff2,axiom,
! [M: nat,W: int] :
( ( M
= ( nat2 @ W ) )
= ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ W )
=> ( W
= ( semiring_1_of_nat @ int @ M ) ) )
& ( ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ W )
=> ( M
= ( zero_zero @ nat ) ) ) ) ) ).
% nat_eq_iff2
thf(fact_979_nat__add__distrib,axiom,
! [Z: int,Z4: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z4 )
=> ( ( nat2 @ ( plus_plus @ int @ Z @ Z4 ) )
= ( plus_plus @ nat @ ( nat2 @ Z ) @ ( nat2 @ Z4 ) ) ) ) ) ).
% nat_add_distrib
thf(fact_980_zless__nat__eq__int__zless,axiom,
! [M: nat,Z: int] :
( ( ord_less @ nat @ M @ ( nat2 @ Z ) )
= ( ord_less @ int @ ( semiring_1_of_nat @ int @ M ) @ Z ) ) ).
% zless_nat_eq_int_zless
thf(fact_981_nat__int__add,axiom,
! [A3: nat,B2: nat] :
( ( nat2 @ ( plus_plus @ int @ ( semiring_1_of_nat @ int @ A3 ) @ ( semiring_1_of_nat @ int @ B2 ) ) )
= ( plus_plus @ nat @ A3 @ B2 ) ) ).
% nat_int_add
thf(fact_982_Nats__diff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,B2: A] :
( ( member @ A @ A3 @ ( semiring_1_Nats @ A ) )
=> ( ( member @ A @ B2 @ ( semiring_1_Nats @ A ) )
=> ( ( ord_less_eq @ A @ B2 @ A3 )
=> ( member @ A @ ( minus_minus @ A @ A3 @ B2 ) @ ( semiring_1_Nats @ A ) ) ) ) ) ) ).
% Nats_diff
thf(fact_983_mult__less__le__imp__less,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A3: A,B2: A,C2: A,D3: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ D3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ D3 ) ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_984_mult__le__less__imp__less,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A3: A,B2: A,C2: A,D3: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ C2 @ D3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ D3 ) ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_985_mult__right__le__imp__le,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ A3 @ B2 ) ) ) ) ).
% mult_right_le_imp_le
thf(fact_986_mult__left__le__imp__le,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ A3 @ B2 ) ) ) ) ).
% mult_left_le_imp_le
thf(fact_987_mult__le__cancel__left__pos,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( ord_less_eq @ A @ A3 @ B2 ) ) ) ) ).
% mult_le_cancel_left_pos
thf(fact_988_mult__le__cancel__left__neg,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( ord_less_eq @ A @ B2 @ A3 ) ) ) ) ).
% mult_le_cancel_left_neg
thf(fact_989_mult__less__cancel__right,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( ord_less @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ A3 @ B2 ) )
& ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B2 @ A3 ) ) ) ) ) ).
% mult_less_cancel_right
thf(fact_990_mult__strict__mono_H,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A3: A,B2: A,C2: A,D3: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ C2 @ D3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ D3 ) ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_991_mult__right__less__imp__less,axiom,
! [A: $tType] :
( ( linordered_semiring @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( ord_less @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ A3 @ B2 ) ) ) ) ).
% mult_right_less_imp_less
thf(fact_992_mult__less__cancel__left,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( ord_less @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ A3 @ B2 ) )
& ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B2 @ A3 ) ) ) ) ) ).
% mult_less_cancel_left
thf(fact_993_mult__strict__mono,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A3: A,B2: A,C2: A,D3: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ C2 @ D3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ D3 ) ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_994_mult__left__less__imp__less,axiom,
! [A: $tType] :
( ( linordered_semiring @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( ord_less @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ A3 @ B2 ) ) ) ) ).
% mult_left_less_imp_less
thf(fact_995_mult__le__cancel__right,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ A3 @ B2 ) )
& ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B2 @ A3 ) ) ) ) ) ).
% mult_le_cancel_right
thf(fact_996_mult__le__cancel__left,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ A3 @ B2 ) )
& ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B2 @ A3 ) ) ) ) ) ).
% mult_le_cancel_left
thf(fact_997_sum__squares__ge__zero,axiom,
! [A: $tType] :
( ( linordered_ring @ A )
=> ! [X: A,Y3: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( times_times @ A @ X @ X ) @ ( times_times @ A @ Y3 @ Y3 ) ) ) ) ).
% sum_squares_ge_zero
thf(fact_998_sum__squares__le__zero__iff,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [X: A,Y3: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ X @ X ) @ ( times_times @ A @ Y3 @ Y3 ) ) @ ( zero_zero @ A ) )
= ( ( X
= ( zero_zero @ A ) )
& ( Y3
= ( zero_zero @ A ) ) ) ) ) ).
% sum_squares_le_zero_iff
thf(fact_999_mult__left__le__one__le,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y3 )
=> ( ( ord_less_eq @ A @ Y3 @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ Y3 @ X ) @ X ) ) ) ) ) ).
% mult_left_le_one_le
thf(fact_1000_mult__right__le__one__le,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y3 )
=> ( ( ord_less_eq @ A @ Y3 @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ X @ Y3 ) @ X ) ) ) ) ) ).
% mult_right_le_one_le
thf(fact_1001_mult__le__one,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ ( one_one @ A ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ A3 @ B2 ) @ ( one_one @ A ) ) ) ) ) ) ).
% mult_le_one
thf(fact_1002_mult__left__le,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [C2: A,A3: A] :
( ( ord_less_eq @ A @ C2 @ ( one_one @ A ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C2 ) @ A3 ) ) ) ) ).
% mult_left_le
thf(fact_1003_not__sum__squares__lt__zero,axiom,
! [A: $tType] :
( ( linordered_ring @ A )
=> ! [X: A,Y3: A] :
~ ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ X @ X ) @ ( times_times @ A @ Y3 @ Y3 ) ) @ ( zero_zero @ A ) ) ) ).
% not_sum_squares_lt_zero
thf(fact_1004_sum__squares__gt__zero__iff,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [X: A,Y3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( times_times @ A @ X @ X ) @ ( times_times @ A @ Y3 @ Y3 ) ) )
= ( ( X
!= ( zero_zero @ A ) )
| ( Y3
!= ( zero_zero @ A ) ) ) ) ) ).
% sum_squares_gt_zero_iff
thf(fact_1005_le__add__iff2,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [A3: A,E2: A,C2: A,B2: A,D3: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ A3 @ E2 ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ B2 @ E2 ) @ D3 ) )
= ( ord_less_eq @ A @ C2 @ ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ B2 @ A3 ) @ E2 ) @ D3 ) ) ) ) ).
% le_add_iff2
thf(fact_1006_le__add__iff1,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [A3: A,E2: A,C2: A,B2: A,D3: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ A3 @ E2 ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ B2 @ E2 ) @ D3 ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ A3 @ B2 ) @ E2 ) @ C2 ) @ D3 ) ) ) ).
% le_add_iff1
thf(fact_1007_less__add__iff2,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [A3: A,E2: A,C2: A,B2: A,D3: A] :
( ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ A3 @ E2 ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ B2 @ E2 ) @ D3 ) )
= ( ord_less @ A @ C2 @ ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ B2 @ A3 ) @ E2 ) @ D3 ) ) ) ) ).
% less_add_iff2
thf(fact_1008_less__add__iff1,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [A3: A,E2: A,C2: A,B2: A,D3: A] :
( ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ A3 @ E2 ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ B2 @ E2 ) @ D3 ) )
= ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ A3 @ B2 ) @ E2 ) @ C2 ) @ D3 ) ) ) ).
% less_add_iff1
thf(fact_1009_square__diff__one__factored,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X: A] :
( ( minus_minus @ A @ ( times_times @ A @ X @ X ) @ ( one_one @ A ) )
= ( times_times @ A @ ( plus_plus @ A @ X @ ( one_one @ A ) ) @ ( minus_minus @ A @ X @ ( one_one @ A ) ) ) ) ) ).
% square_diff_one_factored
thf(fact_1010_abs__eq__mult,axiom,
! [A: $tType] :
( ( ordered_ring_abs @ A )
=> ! [A3: A,B2: A] :
( ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
| ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) )
& ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
| ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) ) )
=> ( ( abs_abs @ A @ ( times_times @ A @ A3 @ B2 ) )
= ( times_times @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B2 ) ) ) ) ) ).
% abs_eq_mult
thf(fact_1011_abs__mult__pos,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( times_times @ A @ ( abs_abs @ A @ Y3 ) @ X )
= ( abs_abs @ A @ ( times_times @ A @ Y3 @ X ) ) ) ) ) ).
% abs_mult_pos
thf(fact_1012_power__gt1__lemma,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A3 )
=> ( ord_less @ A @ ( one_one @ A ) @ ( times_times @ A @ A3 @ ( power_power @ A @ A3 @ N ) ) ) ) ) ).
% power_gt1_lemma
thf(fact_1013_power__less__power__Suc,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A3 )
=> ( ord_less @ A @ ( power_power @ A @ A3 @ N ) @ ( times_times @ A @ A3 @ ( power_power @ A @ A3 @ N ) ) ) ) ) ).
% power_less_power_Suc
thf(fact_1014_power__minus,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [A3: A,N: nat] :
( ( power_power @ A @ ( uminus_uminus @ A @ A3 ) @ N )
= ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N ) @ ( power_power @ A @ A3 @ N ) ) ) ) ).
% power_minus
thf(fact_1015_mult__eq__if,axiom,
( ( times_times @ nat )
= ( ^ [M5: nat,N5: nat] :
( if @ nat
@ ( M5
= ( zero_zero @ nat ) )
@ ( zero_zero @ nat )
@ ( plus_plus @ nat @ N5 @ ( times_times @ nat @ ( minus_minus @ nat @ M5 @ ( one_one @ nat ) ) @ N5 ) ) ) ) ) ).
% mult_eq_if
thf(fact_1016_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_1017_diff__nat__eq__if,axiom,
! [Z4: int,Z: int] :
( ( ( ord_less @ int @ Z4 @ ( zero_zero @ int ) )
=> ( ( minus_minus @ nat @ ( nat2 @ Z ) @ ( nat2 @ Z4 ) )
= ( nat2 @ Z ) ) )
& ( ~ ( ord_less @ int @ Z4 @ ( zero_zero @ int ) )
=> ( ( minus_minus @ nat @ ( nat2 @ Z ) @ ( nat2 @ Z4 ) )
= ( if @ nat @ ( ord_less @ int @ ( minus_minus @ int @ Z @ Z4 ) @ ( zero_zero @ int ) ) @ ( zero_zero @ nat ) @ ( nat2 @ ( minus_minus @ int @ Z @ Z4 ) ) ) ) ) ) ).
% diff_nat_eq_if
thf(fact_1018_frac__eq,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ( archimedean_frac @ A @ X )
= X )
= ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
& ( ord_less @ A @ X @ ( one_one @ A ) ) ) ) ) ).
% frac_eq
thf(fact_1019_nat__plus__as__int,axiom,
( ( plus_plus @ nat )
= ( ^ [A6: nat,B5: nat] : ( nat2 @ ( plus_plus @ int @ ( semiring_1_of_nat @ int @ A6 ) @ ( semiring_1_of_nat @ int @ B5 ) ) ) ) ) ).
% nat_plus_as_int
thf(fact_1020_frac__add,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Y3: A] :
( ( ( ord_less @ A @ ( plus_plus @ A @ ( archimedean_frac @ A @ X ) @ ( archimedean_frac @ A @ Y3 ) ) @ ( one_one @ A ) )
=> ( ( archimedean_frac @ A @ ( plus_plus @ A @ X @ Y3 ) )
= ( plus_plus @ A @ ( archimedean_frac @ A @ X ) @ ( archimedean_frac @ A @ Y3 ) ) ) )
& ( ~ ( ord_less @ A @ ( plus_plus @ A @ ( archimedean_frac @ A @ X ) @ ( archimedean_frac @ A @ Y3 ) ) @ ( one_one @ A ) )
=> ( ( archimedean_frac @ A @ ( plus_plus @ A @ X @ Y3 ) )
= ( minus_minus @ A @ ( plus_plus @ A @ ( archimedean_frac @ A @ X ) @ ( archimedean_frac @ A @ Y3 ) ) @ ( one_one @ A ) ) ) ) ) ) ).
% frac_add
thf(fact_1021_of__int__pos,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ Z )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( ring_1_of_int @ A @ Z ) ) ) ) ).
% of_int_pos
thf(fact_1022_of__int__leD,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: int,X: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ ( ring_1_of_int @ A @ N ) ) @ X )
=> ( ( N
= ( zero_zero @ int ) )
| ( ord_less_eq @ A @ ( one_one @ A ) @ X ) ) ) ) ).
% of_int_leD
thf(fact_1023_of__int__lessD,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: int,X: A] :
( ( ord_less @ A @ ( abs_abs @ A @ ( ring_1_of_int @ A @ N ) ) @ X )
=> ( ( N
= ( zero_zero @ int ) )
| ( ord_less @ A @ ( one_one @ A ) @ X ) ) ) ) ).
% of_int_lessD
thf(fact_1024_of__nat__less__of__int__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: nat,X: int] :
( ( ord_less @ A @ ( semiring_1_of_nat @ A @ N ) @ ( ring_1_of_int @ A @ X ) )
= ( ord_less @ int @ ( semiring_1_of_nat @ int @ N ) @ X ) ) ) ).
% of_nat_less_of_int_iff
thf(fact_1025_split__nat,axiom,
! [P: nat > $o,I: int] :
( ( P @ ( nat2 @ I ) )
= ( ! [N5: nat] :
( ( I
= ( semiring_1_of_nat @ int @ N5 ) )
=> ( P @ N5 ) )
& ( ( ord_less @ int @ I @ ( zero_zero @ int ) )
=> ( P @ ( zero_zero @ nat ) ) ) ) ) ).
% split_nat
thf(fact_1026_nat__less__iff,axiom,
! [W: int,M: nat] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ W )
=> ( ( ord_less @ nat @ ( nat2 @ W ) @ M )
= ( ord_less @ int @ W @ ( semiring_1_of_nat @ int @ M ) ) ) ) ).
% nat_less_iff
thf(fact_1027_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_1028_mult__le__cancel__left2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C2: A,A3: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A3 ) @ C2 )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ A3 @ ( one_one @ A ) ) )
& ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ A3 ) ) ) ) ) ).
% mult_le_cancel_left2
thf(fact_1029_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_1030_mult__le__cancel__right2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,C2: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C2 ) @ C2 )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ A3 @ ( one_one @ A ) ) )
& ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ A3 ) ) ) ) ) ).
% mult_le_cancel_right2
thf(fact_1031_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_1032_mult__less__cancel__left2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C2: A,A3: A] :
( ( ord_less @ A @ ( times_times @ A @ C2 @ A3 ) @ C2 )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ A3 @ ( one_one @ A ) ) )
& ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( one_one @ A ) @ A3 ) ) ) ) ) ).
% mult_less_cancel_left2
thf(fact_1033_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_1034_mult__less__cancel__right2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,C2: A] :
( ( ord_less @ A @ ( times_times @ A @ A3 @ C2 ) @ C2 )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ A3 @ ( one_one @ A ) ) )
& ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( one_one @ A ) @ A3 ) ) ) ) ) ).
% mult_less_cancel_right2
thf(fact_1035_convex__bound__le,axiom,
! [A: $tType] :
( ( linord6961819062388156250ring_1 @ A )
=> ! [X: A,A3: A,Y3: A,U2: A,V: A] :
( ( ord_less_eq @ A @ X @ A3 )
=> ( ( ord_less_eq @ A @ Y3 @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ U2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ V )
=> ( ( ( plus_plus @ A @ U2 @ V )
= ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ U2 @ X ) @ ( times_times @ A @ V @ Y3 ) ) @ A3 ) ) ) ) ) ) ) ).
% convex_bound_le
thf(fact_1036_power__Suc__less,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ A3 @ ( one_one @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ A3 @ ( power_power @ A @ A3 @ N ) ) @ ( power_power @ A @ A3 @ N ) ) ) ) ) ).
% power_Suc_less
thf(fact_1037_power__eq__if,axiom,
! [A: $tType] :
( ( power @ A )
=> ( ( power_power @ A )
= ( ^ [P5: A,M5: nat] :
( if @ A
@ ( M5
= ( zero_zero @ nat ) )
@ ( one_one @ A )
@ ( times_times @ A @ P5 @ ( power_power @ A @ P5 @ ( minus_minus @ nat @ M5 @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% power_eq_if
thf(fact_1038_power__minus__mult,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [N: nat,A3: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( times_times @ A @ ( power_power @ A @ A3 @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) @ A3 )
= ( power_power @ A @ A3 @ N ) ) ) ) ).
% power_minus_mult
thf(fact_1039_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_1040_frac__neg,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ( member @ A @ X @ ( ring_1_Ints @ A ) )
=> ( ( archimedean_frac @ A @ ( uminus_uminus @ A @ X ) )
= ( zero_zero @ A ) ) )
& ( ~ ( member @ A @ X @ ( ring_1_Ints @ A ) )
=> ( ( archimedean_frac @ A @ ( uminus_uminus @ A @ X ) )
= ( minus_minus @ A @ ( one_one @ A ) @ ( archimedean_frac @ A @ X ) ) ) ) ) ) ).
% frac_neg
thf(fact_1041_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_1042_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_1043_nat__less__add__iff1,axiom,
! [J: nat,I: nat,U2: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ J @ I )
=> ( ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I @ U2 ) @ M ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U2 ) @ N ) )
= ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ I @ J ) @ U2 ) @ M ) @ N ) ) ) ).
% nat_less_add_iff1
thf(fact_1044_nat__less__add__iff2,axiom,
! [I: nat,J: nat,U2: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I @ U2 ) @ M ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U2 ) @ N ) )
= ( ord_less @ nat @ M @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ J @ I ) @ U2 ) @ N ) ) ) ) ).
% nat_less_add_iff2
thf(fact_1045_mult__le__cancel__iff1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z: A,X: A,Y3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Z )
=> ( ( ord_less_eq @ A @ ( times_times @ A @ X @ Z ) @ ( times_times @ A @ Y3 @ Z ) )
= ( ord_less_eq @ A @ X @ Y3 ) ) ) ) ).
% mult_le_cancel_iff1
thf(fact_1046_mult__le__cancel__iff2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z: A,X: A,Y3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Z )
=> ( ( ord_less_eq @ A @ ( times_times @ A @ Z @ X ) @ ( times_times @ A @ Z @ Y3 ) )
= ( ord_less_eq @ A @ X @ Y3 ) ) ) ) ).
% mult_le_cancel_iff2
thf(fact_1047_nat__eq__add__iff1,axiom,
! [J: nat,I: nat,U2: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ J @ I )
=> ( ( ( plus_plus @ nat @ ( times_times @ nat @ I @ U2 ) @ M )
= ( plus_plus @ nat @ ( times_times @ nat @ J @ U2 ) @ N ) )
= ( ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ I @ J ) @ U2 ) @ M )
= N ) ) ) ).
% nat_eq_add_iff1
thf(fact_1048_nat__eq__add__iff2,axiom,
! [I: nat,J: nat,U2: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( ( plus_plus @ nat @ ( times_times @ nat @ I @ U2 ) @ M )
= ( plus_plus @ nat @ ( times_times @ nat @ J @ U2 ) @ N ) )
= ( M
= ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ J @ I ) @ U2 ) @ N ) ) ) ) ).
% nat_eq_add_iff2
thf(fact_1049_nat__le__add__iff1,axiom,
! [J: nat,I: nat,U2: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ J @ I )
=> ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I @ U2 ) @ M ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U2 ) @ N ) )
= ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ I @ J ) @ U2 ) @ M ) @ N ) ) ) ).
% nat_le_add_iff1
thf(fact_1050_nat__le__add__iff2,axiom,
! [I: nat,J: nat,U2: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I @ U2 ) @ M ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U2 ) @ N ) )
= ( ord_less_eq @ nat @ M @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ J @ I ) @ U2 ) @ N ) ) ) ) ).
% nat_le_add_iff2
thf(fact_1051_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_1052_left__add__mult__distrib,axiom,
! [I: nat,U2: nat,J: nat,K: nat] :
( ( plus_plus @ nat @ ( times_times @ nat @ I @ U2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U2 ) @ K ) )
= ( plus_plus @ nat @ ( times_times @ nat @ ( plus_plus @ nat @ I @ J ) @ U2 ) @ K ) ) ).
% left_add_mult_distrib
thf(fact_1053_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_1054_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_1055_mult__less__iff1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z: A,X: A,Y3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Z )
=> ( ( ord_less @ A @ ( times_times @ A @ X @ Z ) @ ( times_times @ A @ Y3 @ Z ) )
= ( ord_less @ A @ X @ Y3 ) ) ) ) ).
% mult_less_iff1
thf(fact_1056_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_1057_nat__diff__add__eq2,axiom,
! [I: nat,J: nat,U2: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( minus_minus @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I @ U2 ) @ M ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U2 ) @ N ) )
= ( minus_minus @ nat @ M @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ J @ I ) @ U2 ) @ N ) ) ) ) ).
% nat_diff_add_eq2
thf(fact_1058_nat__diff__add__eq1,axiom,
! [J: nat,I: nat,U2: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ J @ I )
=> ( ( minus_minus @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I @ U2 ) @ M ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U2 ) @ N ) )
= ( minus_minus @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ I @ J ) @ U2 ) @ M ) @ N ) ) ) ).
% nat_diff_add_eq1
thf(fact_1059_eucl__rel__int_Osimps,axiom,
( eucl_rel_int
= ( ^ [A12: int,A23: int,A33: product_prod @ int @ int] :
( ? [K3: int] :
( ( A12 = K3 )
& ( A23
= ( zero_zero @ int ) )
& ( A33
= ( product_Pair @ int @ int @ ( zero_zero @ int ) @ K3 ) ) )
| ? [L4: int,K3: int,Q4: int] :
( ( A12 = K3 )
& ( A23 = L4 )
& ( A33
= ( product_Pair @ int @ int @ Q4 @ ( zero_zero @ int ) ) )
& ( L4
!= ( zero_zero @ int ) )
& ( K3
= ( times_times @ int @ Q4 @ L4 ) ) )
| ? [R4: int,L4: int,K3: int,Q4: int] :
( ( A12 = K3 )
& ( A23 = L4 )
& ( A33
= ( product_Pair @ int @ int @ Q4 @ R4 ) )
& ( ( sgn_sgn @ int @ R4 )
= ( sgn_sgn @ int @ L4 ) )
& ( ord_less @ int @ ( abs_abs @ int @ R4 ) @ ( abs_abs @ int @ L4 ) )
& ( K3
= ( plus_plus @ int @ ( times_times @ int @ Q4 @ L4 ) @ R4 ) ) ) ) ) ) ).
% eucl_rel_int.simps
thf(fact_1060_eucl__rel__int_Ocases,axiom,
! [A1: int,A22: int,A32: product_prod @ int @ int] :
( ( eucl_rel_int @ A1 @ A22 @ A32 )
=> ( ( ( A22
= ( zero_zero @ int ) )
=> ( A32
!= ( product_Pair @ int @ int @ ( zero_zero @ int ) @ A1 ) ) )
=> ( ! [Q3: int] :
( ( A32
= ( product_Pair @ int @ int @ Q3 @ ( zero_zero @ int ) ) )
=> ( ( A22
!= ( zero_zero @ int ) )
=> ( A1
!= ( times_times @ int @ Q3 @ A22 ) ) ) )
=> ~ ! [R2: int,Q3: int] :
( ( A32
= ( product_Pair @ int @ int @ Q3 @ R2 ) )
=> ( ( ( sgn_sgn @ int @ R2 )
= ( sgn_sgn @ int @ A22 ) )
=> ( ( ord_less @ int @ ( abs_abs @ int @ R2 ) @ ( abs_abs @ int @ A22 ) )
=> ( A1
!= ( plus_plus @ int @ ( times_times @ int @ Q3 @ A22 ) @ R2 ) ) ) ) ) ) ) ) ).
% eucl_rel_int.cases
thf(fact_1061_pochhammer__code,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ( ( comm_s3205402744901411588hammer @ A )
= ( ^ [A6: A,N5: nat] :
( if @ A
@ ( N5
= ( zero_zero @ nat ) )
@ ( one_one @ A )
@ ( set_fo6178422350223883121st_nat @ A
@ ^ [O: nat] : ( times_times @ A @ ( plus_plus @ A @ A6 @ ( semiring_1_of_nat @ A @ O ) ) )
@ ( zero_zero @ nat )
@ ( minus_minus @ nat @ N5 @ ( one_one @ nat ) )
@ ( one_one @ A ) ) ) ) ) ) ).
% pochhammer_code
thf(fact_1062_less__eq__set__def,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A8: set @ A,B7: set @ A] :
( ord_less_eq @ ( A > $o )
@ ^ [X2: A] : ( member @ A @ X2 @ A8 )
@ ^ [X2: A] : ( member @ A @ X2 @ B7 ) ) ) ) ).
% less_eq_set_def
thf(fact_1063_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_1064_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_1065_zminus1__lemma,axiom,
! [A3: int,B2: int,Q2: int,R: int] :
( ( eucl_rel_int @ A3 @ B2 @ ( product_Pair @ int @ int @ Q2 @ R ) )
=> ( ( B2
!= ( zero_zero @ int ) )
=> ( eucl_rel_int @ ( uminus_uminus @ int @ A3 ) @ 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_1066_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_1067_pochhammer__1,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A3: A] :
( ( comm_s3205402744901411588hammer @ A @ A3 @ ( one_one @ nat ) )
= A3 ) ) ).
% pochhammer_1
thf(fact_1068_pochhammer__0,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A3: A] :
( ( comm_s3205402744901411588hammer @ A @ A3 @ ( zero_zero @ nat ) )
= ( one_one @ A ) ) ) ).
% pochhammer_0
thf(fact_1069_unique__quotient,axiom,
! [A3: int,B2: int,Q2: int,R: int,Q5: int,R3: int] :
( ( eucl_rel_int @ A3 @ B2 @ ( product_Pair @ int @ int @ Q2 @ R ) )
=> ( ( eucl_rel_int @ A3 @ B2 @ ( product_Pair @ int @ int @ Q5 @ R3 ) )
=> ( Q2 = Q5 ) ) ) ).
% unique_quotient
thf(fact_1070_unique__remainder,axiom,
! [A3: int,B2: int,Q2: int,R: int,Q5: int,R3: int] :
( ( eucl_rel_int @ A3 @ B2 @ ( product_Pair @ int @ int @ Q2 @ R ) )
=> ( ( eucl_rel_int @ A3 @ B2 @ ( product_Pair @ int @ int @ Q5 @ R3 ) )
=> ( R = R3 ) ) ) ).
% unique_remainder
thf(fact_1071_pochhammer__pos,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X: A,N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( comm_s3205402744901411588hammer @ A @ X @ N ) ) ) ) ).
% pochhammer_pos
thf(fact_1072_pochhammer__eq__0__mono,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,N: nat,M: nat] :
( ( ( comm_s3205402744901411588hammer @ A @ A3 @ N )
= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ nat @ N @ M )
=> ( ( comm_s3205402744901411588hammer @ A @ A3 @ M )
= ( zero_zero @ A ) ) ) ) ) ).
% pochhammer_eq_0_mono
thf(fact_1073_pochhammer__neq__0__mono,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,M: nat,N: nat] :
( ( ( comm_s3205402744901411588hammer @ A @ A3 @ M )
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ nat @ N @ M )
=> ( ( comm_s3205402744901411588hammer @ A @ A3 @ N )
!= ( zero_zero @ A ) ) ) ) ) ).
% pochhammer_neq_0_mono
thf(fact_1074_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_1075_pochhammer__nonneg,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X: A,N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( comm_s3205402744901411588hammer @ A @ X @ N ) ) ) ) ).
% pochhammer_nonneg
thf(fact_1076_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_1077_pochhammer__eq__0__iff,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,N: nat] :
( ( ( comm_s3205402744901411588hammer @ A @ A3 @ N )
= ( zero_zero @ A ) )
= ( ? [K3: nat] :
( ( ord_less @ nat @ K3 @ N )
& ( A3
= ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ K3 ) ) ) ) ) ) ) ).
% pochhammer_eq_0_iff
thf(fact_1078_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_1079_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_1080_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_1081_pochhammer__product_H,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [Z: A,N: nat,M: nat] :
( ( comm_s3205402744901411588hammer @ A @ Z @ ( plus_plus @ nat @ N @ M ) )
= ( times_times @ A @ ( comm_s3205402744901411588hammer @ A @ Z @ N ) @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ Z @ ( semiring_1_of_nat @ A @ N ) ) @ M ) ) ) ) ).
% pochhammer_product'
thf(fact_1082_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_1083_Collect__subset,axiom,
! [A: $tType,A5: set @ A,P: A > $o] :
( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [X2: A] :
( ( member @ A @ X2 @ A5 )
& ( P @ X2 ) ) )
@ A5 ) ).
% Collect_subset
thf(fact_1084_pochhammer__product,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [M: nat,N: nat,Z: A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( comm_s3205402744901411588hammer @ A @ Z @ N )
= ( times_times @ A @ ( comm_s3205402744901411588hammer @ A @ Z @ M ) @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ Z @ ( semiring_1_of_nat @ A @ M ) ) @ ( minus_minus @ nat @ N @ M ) ) ) ) ) ) ).
% pochhammer_product
thf(fact_1085_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_1086_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_1087_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_1088_power__minus_H,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X: A,N: nat] :
( ( nO_MATCH @ A @ A @ ( one_one @ A ) @ X )
=> ( ( power_power @ A @ ( uminus_uminus @ A @ X ) @ N )
= ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N ) @ ( power_power @ A @ X @ N ) ) ) ) ) ).
% power_minus'
thf(fact_1089_one__less__nat__eq,axiom,
! [Z: int] :
( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( nat2 @ Z ) )
= ( ord_less @ int @ ( one_one @ int ) @ Z ) ) ).
% one_less_nat_eq
thf(fact_1090_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_1091_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_1092_power_Opower__eq__if,axiom,
! [A: $tType] :
( ( power2 @ A )
= ( ^ [One: A,Times: A > A > A,P5: A,M5: nat] :
( if @ A
@ ( M5
= ( zero_zero @ nat ) )
@ One
@ ( Times @ P5 @ ( power2 @ A @ One @ Times @ P5 @ ( minus_minus @ nat @ M5 @ ( one_one @ nat ) ) ) ) ) ) ) ).
% power.power_eq_if
thf(fact_1093_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_1094_fact__num__eq__if,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiring_char_0_fact @ A )
= ( ^ [M5: nat] :
( if @ A
@ ( M5
= ( zero_zero @ nat ) )
@ ( one_one @ A )
@ ( times_times @ A @ ( semiring_1_of_nat @ A @ M5 ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ M5 @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% fact_num_eq_if
thf(fact_1095_mult__ceiling__le__Ints,axiom,
! [A: $tType,B: $tType] :
( ( ( archim2362893244070406136eiling @ B )
& ( linordered_idom @ A ) )
=> ! [A3: B,B2: B] :
( ( ord_less_eq @ B @ ( zero_zero @ B ) @ A3 )
=> ( ( member @ B @ A3 @ ( ring_1_Ints @ B ) )
=> ( ord_less_eq @ A @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ B @ ( times_times @ B @ A3 @ B2 ) ) ) @ ( ring_1_of_int @ A @ ( times_times @ int @ ( archimedean_ceiling @ B @ A3 ) @ ( archimedean_ceiling @ B @ B2 ) ) ) ) ) ) ) ).
% mult_ceiling_le_Ints
thf(fact_1096_nat_Oinject,axiom,
! [X22: nat,Y2: nat] :
( ( ( suc @ X22 )
= ( suc @ Y2 ) )
= ( X22 = Y2 ) ) ).
% nat.inject
thf(fact_1097_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_1098_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_1099_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ( ord_less @ nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_1100_lessI,axiom,
! [N: nat] : ( ord_less @ nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_1101_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_1102_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_1103_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_1104_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_1105_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_1106_less__Suc0,axiom,
! [N: nat] :
( ( ord_less @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% less_Suc0
thf(fact_1107_zero__less__Suc,axiom,
! [N: nat] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_1108_power__Suc0__right,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A3: A] :
( ( power_power @ A @ A3 @ ( suc @ ( zero_zero @ nat ) ) )
= A3 ) ) ).
% power_Suc0_right
thf(fact_1109_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_1110_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_1111_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_1112_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus @ nat @ ( suc @ N ) @ ( one_one @ nat ) )
= N ) ).
% diff_Suc_1
thf(fact_1113_fact__0,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiring_char_0_fact @ A @ ( zero_zero @ nat ) )
= ( one_one @ A ) ) ) ).
% fact_0
thf(fact_1114_ceiling__zero,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archimedean_ceiling @ A @ ( zero_zero @ A ) )
= ( zero_zero @ int ) ) ) ).
% ceiling_zero
thf(fact_1115_nat__power__eq__Suc__0__iff,axiom,
! [X: nat,M: nat] :
( ( ( power_power @ nat @ X @ M )
= ( suc @ ( zero_zero @ nat ) ) )
= ( ( M
= ( zero_zero @ nat ) )
| ( X
= ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% nat_power_eq_Suc_0_iff
thf(fact_1116_power__Suc__0,axiom,
! [N: nat] :
( ( power_power @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% power_Suc_0
thf(fact_1117_pochhammer__Suc0,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A3: A] :
( ( comm_s3205402744901411588hammer @ A @ A3 @ ( suc @ ( zero_zero @ nat ) ) )
= A3 ) ) ).
% pochhammer_Suc0
thf(fact_1118_ceiling__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archimedean_ceiling @ A @ ( one_one @ A ) )
= ( one_one @ int ) ) ) ).
% ceiling_one
thf(fact_1119_fact__1,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiring_char_0_fact @ A @ ( one_one @ nat ) )
= ( one_one @ A ) ) ) ).
% fact_1
thf(fact_1120_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_1121_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_1122_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_1123_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_1124_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_1125_nat__1,axiom,
( ( nat2 @ ( one_one @ int ) )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% nat_1
thf(fact_1126_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_1127_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_1128_ceiling__add__of__int,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Z: int] :
( ( archimedean_ceiling @ A @ ( plus_plus @ A @ X @ ( ring_1_of_int @ A @ Z ) ) )
= ( plus_plus @ int @ ( archimedean_ceiling @ A @ X ) @ Z ) ) ) ).
% ceiling_add_of_int
thf(fact_1129_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_1130_ceiling__le__zero,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X ) @ ( zero_zero @ int ) )
= ( ord_less_eq @ A @ X @ ( zero_zero @ A ) ) ) ) ).
% ceiling_le_zero
thf(fact_1131_zero__less__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less @ int @ ( zero_zero @ int ) @ ( archimedean_ceiling @ A @ X ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ X ) ) ) ).
% zero_less_ceiling
thf(fact_1132_ceiling__less__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less @ int @ ( archimedean_ceiling @ A @ X ) @ ( one_one @ int ) )
= ( ord_less_eq @ A @ X @ ( zero_zero @ A ) ) ) ) ).
% ceiling_less_one
thf(fact_1133_one__le__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less_eq @ int @ ( one_one @ int ) @ ( archimedean_ceiling @ A @ X ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ X ) ) ) ).
% one_le_ceiling
thf(fact_1134_ceiling__le__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X ) @ ( one_one @ int ) )
= ( ord_less_eq @ A @ X @ ( one_one @ A ) ) ) ) ).
% ceiling_le_one
thf(fact_1135_one__less__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less @ int @ ( one_one @ int ) @ ( archimedean_ceiling @ A @ X ) )
= ( ord_less @ A @ ( one_one @ A ) @ X ) ) ) ).
% one_less_ceiling
thf(fact_1136_ceiling__add__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( archimedean_ceiling @ A @ ( plus_plus @ A @ X @ ( one_one @ A ) ) )
= ( plus_plus @ int @ ( archimedean_ceiling @ A @ X ) @ ( one_one @ int ) ) ) ) ).
% ceiling_add_one
thf(fact_1137_ceiling__diff__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( archimedean_ceiling @ A @ ( minus_minus @ A @ X @ ( one_one @ A ) ) )
= ( minus_minus @ int @ ( archimedean_ceiling @ A @ X ) @ ( one_one @ int ) ) ) ) ).
% ceiling_diff_one
thf(fact_1138_ceiling__less__zero,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less @ int @ ( archimedean_ceiling @ A @ X ) @ ( zero_zero @ int ) )
= ( ord_less_eq @ A @ X @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ).
% ceiling_less_zero
thf(fact_1139_zero__le__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( archimedean_ceiling @ A @ X ) )
= ( ord_less @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ X ) ) ) ).
% zero_le_ceiling
thf(fact_1140_Suc__inject,axiom,
! [X: nat,Y3: nat] :
( ( ( suc @ X )
= ( suc @ Y3 ) )
=> ( X = Y3 ) ) ).
% Suc_inject
thf(fact_1141_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_1142_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_1143_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_1144_not0__implies__Suc,axiom,
! [N: nat] :
( ( N
!= ( zero_zero @ nat ) )
=> ? [M4: nat] :
( N
= ( suc @ M4 ) ) ) ).
% not0_implies_Suc
thf(fact_1145_Zero__not__Suc,axiom,
! [M: nat] :
( ( zero_zero @ nat )
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_1146_Zero__neq__Suc,axiom,
! [M: nat] :
( ( zero_zero @ nat )
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_1147_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= ( zero_zero @ nat ) ) ).
% Suc_neq_Zero
thf(fact_1148_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_1149_diff__induct,axiom,
! [P: nat > nat > $o,M: nat,N: nat] :
( ! [X4: nat] : ( P @ X4 @ ( zero_zero @ nat ) )
=> ( ! [Y4: nat] : ( P @ ( zero_zero @ nat ) @ ( suc @ Y4 ) )
=> ( ! [X4: nat,Y4: nat] :
( ( P @ X4 @ Y4 )
=> ( P @ ( suc @ X4 ) @ ( suc @ Y4 ) ) )
=> ( P @ M @ N ) ) ) ) ).
% diff_induct
thf(fact_1150_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_1151_old_Onat_Oexhaust,axiom,
! [Y3: nat] :
( ( Y3
!= ( zero_zero @ nat ) )
=> ~ ! [Nat3: nat] :
( Y3
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_1152_nat_OdiscI,axiom,
! [Nat: nat,X22: nat] :
( ( Nat
= ( suc @ X22 ) )
=> ( Nat
!= ( zero_zero @ nat ) ) ) ).
% nat.discI
thf(fact_1153_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( ( zero_zero @ nat )
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_1154_old_Onat_Odistinct_I2_J,axiom,
! [Nat4: nat] :
( ( suc @ Nat4 )
!= ( zero_zero @ nat ) ) ).
% old.nat.distinct(2)
thf(fact_1155_nat_Odistinct_I1_J,axiom,
! [X22: nat] :
( ( zero_zero @ nat )
!= ( suc @ X22 ) ) ).
% nat.distinct(1)
thf(fact_1156_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_1157_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_1158_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_1159_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,K2: nat] :
( ( ord_less @ nat @ I3 @ J2 )
=> ( ( ord_less @ nat @ J2 @ K2 )
=> ( ( P @ I3 @ J2 )
=> ( ( P @ J2 @ K2 )
=> ( P @ I3 @ K2 ) ) ) ) )
=> ( P @ I @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_1160_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_1161_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_1162_less__antisym,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less @ nat @ N @ M )
=> ( ( ord_less @ nat @ N @ ( suc @ M ) )
=> ( M = N ) ) ) ).
% less_antisym
thf(fact_1163_Suc__less__eq2,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ ( suc @ N ) @ M )
= ( ? [M6: nat] :
( ( M
= ( suc @ M6 ) )
& ( ord_less @ nat @ N @ M6 ) ) ) ) ).
% Suc_less_eq2
thf(fact_1164_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_1165_not__less__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less @ nat @ M @ N ) )
= ( ord_less @ nat @ N @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_1166_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_1167_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_1168_less__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ( ord_less @ nat @ M @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_1169_less__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less @ nat @ M @ N )
=> ( M = N ) ) ) ).
% less_SucE
thf(fact_1170_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_1171_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_1172_Suc__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ ( suc @ M ) @ N )
=> ( ord_less @ nat @ M @ N ) ) ).
% Suc_lessD
thf(fact_1173_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_1174_transitive__stepwise__le,axiom,
! [M: nat,N: nat,R5: nat > nat > $o] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ! [X4: nat] : ( R5 @ X4 @ X4 )
=> ( ! [X4: nat,Y4: nat,Z6: nat] :
( ( R5 @ X4 @ Y4 )
=> ( ( R5 @ Y4 @ Z6 )
=> ( R5 @ X4 @ Z6 ) ) )
=> ( ! [N2: nat] : ( R5 @ N2 @ ( suc @ N2 ) )
=> ( R5 @ M @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_1175_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_1176_full__nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M3: nat] :
( ( ord_less_eq @ nat @ ( suc @ M3 ) @ N2 )
=> ( P @ M3 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% full_nat_induct
thf(fact_1177_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_1178_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq @ nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_1179_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_1180_Suc__le__D,axiom,
! [N: nat,M7: nat] :
( ( ord_less_eq @ nat @ ( suc @ N ) @ M7 )
=> ? [M4: nat] :
( M7
= ( suc @ M4 ) ) ) ).
% Suc_le_D
thf(fact_1181_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ord_less_eq @ nat @ M @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_1182_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_1183_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ ( suc @ M ) @ N )
=> ( ord_less_eq @ nat @ M @ N ) ) ).
% Suc_leD
thf(fact_1184_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_1185_add__Suc,axiom,
! [M: nat,N: nat] :
( ( plus_plus @ nat @ ( suc @ M ) @ N )
= ( suc @ ( plus_plus @ nat @ M @ N ) ) ) ).
% add_Suc
thf(fact_1186_nat__arith_Osuc1,axiom,
! [A5: nat,K: nat,A3: nat] :
( ( A5
= ( plus_plus @ nat @ K @ A3 ) )
=> ( ( suc @ A5 )
= ( plus_plus @ nat @ K @ ( suc @ A3 ) ) ) ) ).
% nat_arith.suc1
thf(fact_1187_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_1188_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_1189_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_1190_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_1191_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_1192_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_1193_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_1194_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_1195_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_1196_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_1197_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_1198_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_1199_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_1200_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_1201_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
= ( ? [M5: nat] :
( N
= ( suc @ M5 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_1202_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_1203_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ? [M4: nat] :
( N
= ( suc @ M4 ) ) ) ).
% gr0_implies_Suc
thf(fact_1204_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_1205_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_1206_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_1207_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_1208_less__eq__Suc__le,axiom,
( ( ord_less @ nat )
= ( ^ [N5: nat] : ( ord_less_eq @ nat @ ( suc @ N5 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_1209_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_1210_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_1211_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_1212_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_1213_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_1214_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_1215_Suc__leI,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ( ord_less_eq @ nat @ ( suc @ M ) @ N ) ) ).
% Suc_leI
thf(fact_1216_less__natE,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ~ ! [Q3: nat] :
( N
!= ( suc @ ( plus_plus @ nat @ M @ Q3 ) ) ) ) ).
% less_natE
thf(fact_1217_less__add__Suc1,axiom,
! [I: nat,M: nat] : ( ord_less @ nat @ I @ ( suc @ ( plus_plus @ nat @ I @ M ) ) ) ).
% less_add_Suc1
thf(fact_1218_less__add__Suc2,axiom,
! [I: nat,M: nat] : ( ord_less @ nat @ I @ ( suc @ ( plus_plus @ nat @ M @ I ) ) ) ).
% less_add_Suc2
thf(fact_1219_less__iff__Suc__add,axiom,
( ( ord_less @ nat )
= ( ^ [M5: nat,N5: nat] :
? [K3: nat] :
( N5
= ( suc @ ( plus_plus @ nat @ M5 @ K3 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_1220_less__imp__Suc__add,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ? [K2: nat] :
( N
= ( suc @ ( plus_plus @ nat @ M @ K2 ) ) ) ) ).
% less_imp_Suc_add
thf(fact_1221_diff__less__Suc,axiom,
! [M: nat,N: nat] : ( ord_less @ nat @ ( minus_minus @ nat @ M @ N ) @ ( suc @ M ) ) ).
% diff_less_Suc
thf(fact_1222_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_1223_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_1224_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_1225_One__nat__def,axiom,
( ( one_one @ nat )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% One_nat_def
thf(fact_1226_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_1227_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_1228_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus @ nat @ ( one_one @ nat ) ) ) ).
% Suc_eq_plus1_left
thf(fact_1229_plus__1__eq__Suc,axiom,
( ( plus_plus @ nat @ ( one_one @ nat ) )
= suc ) ).
% plus_1_eq_Suc
thf(fact_1230_Suc__eq__plus1,axiom,
( suc
= ( ^ [N5: nat] : ( plus_plus @ nat @ N5 @ ( one_one @ nat ) ) ) ) ).
% Suc_eq_plus1
thf(fact_1231_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_1232_int__cases,axiom,
! [Z: int] :
( ! [N2: nat] :
( Z
!= ( semiring_1_of_nat @ int @ N2 ) )
=> ~ ! [N2: nat] :
( Z
!= ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ ( suc @ N2 ) ) ) ) ) ).
% int_cases
thf(fact_1233_int__of__nat__induct,axiom,
! [P: int > $o,Z: int] :
( ! [N2: nat] : ( P @ ( semiring_1_of_nat @ int @ N2 ) )
=> ( ! [N2: nat] : ( P @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ ( suc @ N2 ) ) ) )
=> ( P @ Z ) ) ) ).
% int_of_nat_induct
thf(fact_1234_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_1235_power_Opower_Opower__0,axiom,
! [A: $tType,One2: A,Times2: A > A > A,A3: A] :
( ( power2 @ A @ One2 @ Times2 @ A3 @ ( zero_zero @ nat ) )
= One2 ) ).
% power.power.power_0
thf(fact_1236_power__le__imp__le__base,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N: nat,B2: A] :
( ( ord_less_eq @ A @ ( power_power @ A @ A3 @ ( suc @ N ) ) @ ( power_power @ A @ B2 @ ( suc @ N ) ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less_eq @ A @ A3 @ B2 ) ) ) ) ).
% power_le_imp_le_base
thf(fact_1237_power__inject__base,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N: nat,B2: A] :
( ( ( power_power @ A @ A3 @ ( suc @ N ) )
= ( power_power @ A @ B2 @ ( suc @ N ) ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( A3 = B2 ) ) ) ) ) ).
% power_inject_base
thf(fact_1238_power__gt1,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A3 )
=> ( ord_less @ A @ ( one_one @ A ) @ ( power_power @ A @ A3 @ ( suc @ N ) ) ) ) ) ).
% power_gt1
thf(fact_1239_ceiling__add__le,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Y3: A] : ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ ( plus_plus @ A @ X @ Y3 ) ) @ ( plus_plus @ int @ ( archimedean_ceiling @ A @ X ) @ ( archimedean_ceiling @ A @ Y3 ) ) ) ) ).
% ceiling_add_le
thf(fact_1240_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_1241_ex__least__nat__less,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ ( zero_zero @ nat ) )
=> ? [K2: nat] :
( ( ord_less @ nat @ K2 @ N )
& ! [I4: nat] :
( ( ord_less_eq @ nat @ I4 @ K2 )
=> ~ ( P @ I4 ) )
& ( P @ ( suc @ K2 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_1242_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_1243_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_1244_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_1245_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_1246_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_1247_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_1248_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_1249_zless__iff__Suc__zadd,axiom,
( ( ord_less @ int )
= ( ^ [W2: int,Z3: int] :
? [N5: nat] :
( Z3
= ( plus_plus @ int @ W2 @ ( semiring_1_of_nat @ int @ ( suc @ N5 ) ) ) ) ) ) ).
% zless_iff_Suc_zadd
thf(fact_1250_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_1251_int__ops_I4_J,axiom,
! [A3: nat] :
( ( semiring_1_of_nat @ int @ ( suc @ A3 ) )
= ( plus_plus @ int @ ( semiring_1_of_nat @ int @ A3 ) @ ( one_one @ int ) ) ) ).
% int_ops(4)
thf(fact_1252_power__Suc__le__self,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N: nat] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ A3 @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( power_power @ A @ A3 @ ( suc @ N ) ) @ A3 ) ) ) ) ).
% power_Suc_le_self
thf(fact_1253_power__Suc__less__one,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ A3 @ ( one_one @ A ) )
=> ( ord_less @ A @ ( power_power @ A @ A3 @ ( suc @ N ) ) @ ( one_one @ A ) ) ) ) ) ).
% power_Suc_less_one
thf(fact_1254_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_1255_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_1256_pochhammer__rec,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A3: A,N: nat] :
( ( comm_s3205402744901411588hammer @ A @ A3 @ ( suc @ N ) )
= ( times_times @ A @ A3 @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ N ) ) ) ) ).
% pochhammer_rec
thf(fact_1257_add__eq__if,axiom,
( ( plus_plus @ nat )
= ( ^ [M5: nat,N5: nat] :
( if @ nat
@ ( M5
= ( zero_zero @ nat ) )
@ N5
@ ( suc @ ( plus_plus @ nat @ ( minus_minus @ nat @ M5 @ ( one_one @ nat ) ) @ N5 ) ) ) ) ) ).
% add_eq_if
thf(fact_1258_pochhammer__rec_H,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [Z: A,N: nat] :
( ( comm_s3205402744901411588hammer @ A @ Z @ ( suc @ N ) )
= ( times_times @ A @ ( plus_plus @ A @ Z @ ( semiring_1_of_nat @ A @ N ) ) @ ( comm_s3205402744901411588hammer @ A @ Z @ N ) ) ) ) ).
% pochhammer_rec'
thf(fact_1259_pochhammer__Suc,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A3: A,N: nat] :
( ( comm_s3205402744901411588hammer @ A @ A3 @ ( suc @ N ) )
= ( times_times @ A @ ( comm_s3205402744901411588hammer @ A @ A3 @ N ) @ ( plus_plus @ A @ A3 @ ( semiring_1_of_nat @ A @ N ) ) ) ) ) ).
% pochhammer_Suc
thf(fact_1260_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_1261_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_1262_negD,axiom,
! [X: int] :
( ( ord_less @ int @ X @ ( zero_zero @ int ) )
=> ? [N2: nat] :
( X
= ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ ( suc @ N2 ) ) ) ) ) ).
% negD
thf(fact_1263_Suc__as__int,axiom,
( suc
= ( ^ [A6: nat] : ( nat2 @ ( plus_plus @ int @ ( semiring_1_of_nat @ int @ A6 ) @ ( one_one @ int ) ) ) ) ) ).
% Suc_as_int
thf(fact_1264_ceiling__correct,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ X ) ) @ ( one_one @ A ) ) @ X )
& ( ord_less_eq @ A @ X @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ X ) ) ) ) ) ).
% ceiling_correct
thf(fact_1265_ceiling__unique,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z: int,X: A] :
( ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ Z ) @ ( one_one @ A ) ) @ X )
=> ( ( ord_less_eq @ A @ X @ ( ring_1_of_int @ A @ Z ) )
=> ( ( archimedean_ceiling @ A @ X )
= Z ) ) ) ) ).
% ceiling_unique
thf(fact_1266_ceiling__eq__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,A3: int] :
( ( ( archimedean_ceiling @ A @ X )
= A3 )
= ( ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ A3 ) @ ( one_one @ A ) ) @ X )
& ( ord_less_eq @ A @ X @ ( ring_1_of_int @ A @ A3 ) ) ) ) ) ).
% ceiling_eq_iff
thf(fact_1267_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_1268_mult__ceiling__le,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ ( times_times @ A @ A3 @ B2 ) ) @ ( times_times @ int @ ( archimedean_ceiling @ A @ A3 ) @ ( archimedean_ceiling @ A @ B2 ) ) ) ) ) ) ).
% mult_ceiling_le
thf(fact_1269_ceiling__less__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Z: int] :
( ( ord_less @ int @ ( archimedean_ceiling @ A @ X ) @ Z )
= ( ord_less_eq @ A @ X @ ( minus_minus @ A @ ( ring_1_of_int @ A @ Z ) @ ( one_one @ A ) ) ) ) ) ).
% ceiling_less_iff
thf(fact_1270_le__ceiling__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z: int,X: A] :
( ( ord_less_eq @ int @ Z @ ( archimedean_ceiling @ A @ X ) )
= ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ Z ) @ ( one_one @ A ) ) @ X ) ) ) ).
% le_ceiling_iff
thf(fact_1271_Suc__nat__eq__nat__zadd1,axiom,
! [Z: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
=> ( ( suc @ ( nat2 @ Z ) )
= ( nat2 @ ( plus_plus @ int @ ( one_one @ int ) @ Z ) ) ) ) ).
% Suc_nat_eq_nat_zadd1
thf(fact_1272_option_Osize__gen_I2_J,axiom,
! [A: $tType,X: A > nat,X22: A] :
( ( size_option @ A @ X @ ( some @ A @ X22 ) )
= ( plus_plus @ nat @ ( X @ X22 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% option.size_gen(2)
thf(fact_1273_Heap_Osize__gen,axiom,
! [A: $tType,Xa: A > nat,X: ( 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 @ X ) )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% Heap.size_gen
thf(fact_1274_option_Osize__gen_I1_J,axiom,
! [A: $tType,X: A > nat] :
( ( size_option @ A @ X @ ( none @ A ) )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% option.size_gen(1)
thf(fact_1275_Heap_Osize_I2_J,axiom,
! [A: $tType,X: ( 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 @ X ) )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% Heap.size(2)
thf(fact_1276_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_1277_ceiling__divide__lower,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Q2: A,P4: 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 @ P4 @ Q2 ) ) ) @ ( one_one @ A ) ) @ Q2 ) @ P4 ) ) ) ).
% ceiling_divide_lower
thf(fact_1278_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_1279_option_Osize_I3_J,axiom,
! [A: $tType] :
( ( size_size @ ( option @ A ) @ ( none @ A ) )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% option.size(3)
thf(fact_1280_list__decode_Ocases,axiom,
! [X: nat] :
( ( X
!= ( zero_zero @ nat ) )
=> ~ ! [N2: nat] :
( X
!= ( suc @ N2 ) ) ) ).
% list_decode.cases
thf(fact_1281_divide__eq__0__iff,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: A,B2: A] :
( ( ( divide_divide @ A @ A3 @ B2 )
= ( zero_zero @ A ) )
= ( ( A3
= ( zero_zero @ A ) )
| ( B2
= ( zero_zero @ A ) ) ) ) ) ).
% divide_eq_0_iff
thf(fact_1282_divide__cancel__left,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( ( divide_divide @ A @ C2 @ A3 )
= ( divide_divide @ A @ C2 @ B2 ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( A3 = B2 ) ) ) ) ).
% divide_cancel_left
thf(fact_1283_divide__cancel__right,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( ( divide_divide @ A @ A3 @ C2 )
= ( divide_divide @ A @ B2 @ C2 ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( A3 = B2 ) ) ) ) ).
% divide_cancel_right
thf(fact_1284_division__ring__divide__zero,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A] :
( ( divide_divide @ A @ A3 @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% division_ring_divide_zero
thf(fact_1285_div__0,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A3: A] :
( ( divide_divide @ A @ ( zero_zero @ A ) @ A3 )
= ( zero_zero @ A ) ) ) ).
% div_0
thf(fact_1286_div__by__0,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A3: A] :
( ( divide_divide @ A @ A3 @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% div_by_0
thf(fact_1287_div__by__1,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A3: A] :
( ( divide_divide @ A @ A3 @ ( one_one @ A ) )
= A3 ) ) ).
% div_by_1
thf(fact_1288_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_1289_nonzero__mult__divide__mult__cancel__right2,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( divide_divide @ A @ A3 @ B2 ) ) ) ) ).
% nonzero_mult_divide_mult_cancel_right2
thf(fact_1290_nonzero__mult__divide__mult__cancel__right,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) )
= ( divide_divide @ A @ A3 @ B2 ) ) ) ) ).
% nonzero_mult_divide_mult_cancel_right
thf(fact_1291_nonzero__mult__divide__mult__cancel__left2,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ B2 @ C2 ) )
= ( divide_divide @ A @ A3 @ B2 ) ) ) ) ).
% nonzero_mult_divide_mult_cancel_left2
thf(fact_1292_nonzero__mult__divide__mult__cancel__left,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( divide_divide @ A @ A3 @ B2 ) ) ) ) ).
% nonzero_mult_divide_mult_cancel_left
thf(fact_1293_mult__divide__mult__cancel__left__if,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( ( C2
= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( zero_zero @ A ) ) )
& ( ( C2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( divide_divide @ A @ A3 @ B2 ) ) ) ) ) ).
% mult_divide_mult_cancel_left_if
thf(fact_1294_nonzero__mult__div__cancel__left,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A3: A,B2: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ A3 @ B2 ) @ A3 )
= B2 ) ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_1295_nonzero__mult__div__cancel__right,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [B2: A,A3: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ A3 @ B2 ) @ B2 )
= A3 ) ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_1296_divide__eq__1__iff,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: A,B2: A] :
( ( ( divide_divide @ A @ A3 @ B2 )
= ( one_one @ A ) )
= ( ( B2
!= ( zero_zero @ A ) )
& ( A3 = B2 ) ) ) ) ).
% divide_eq_1_iff
thf(fact_1297_one__eq__divide__iff,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: A,B2: A] :
( ( ( one_one @ A )
= ( divide_divide @ A @ A3 @ B2 ) )
= ( ( B2
!= ( zero_zero @ A ) )
& ( A3 = B2 ) ) ) ) ).
% one_eq_divide_iff
thf(fact_1298_divide__self,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ A3 @ A3 )
= ( one_one @ A ) ) ) ) ).
% divide_self
thf(fact_1299_divide__self__if,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A] :
( ( ( A3
= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ A3 @ A3 )
= ( zero_zero @ A ) ) )
& ( ( A3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ A3 @ A3 )
= ( one_one @ A ) ) ) ) ) ).
% divide_self_if
thf(fact_1300_divide__eq__eq__1,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,A3: A] :
( ( ( divide_divide @ A @ B2 @ A3 )
= ( one_one @ A ) )
= ( ( A3
!= ( zero_zero @ A ) )
& ( A3 = B2 ) ) ) ) ).
% divide_eq_eq_1
thf(fact_1301_eq__divide__eq__1,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,A3: A] :
( ( ( one_one @ A )
= ( divide_divide @ A @ B2 @ A3 ) )
= ( ( A3
!= ( zero_zero @ A ) )
& ( A3 = B2 ) ) ) ) ).
% eq_divide_eq_1
thf(fact_1302_one__divide__eq__0__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ( divide_divide @ A @ ( one_one @ A ) @ A3 )
= ( zero_zero @ A ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% one_divide_eq_0_iff
thf(fact_1303_zero__eq__1__divide__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ( zero_zero @ A )
= ( divide_divide @ A @ ( one_one @ A ) @ A3 ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% zero_eq_1_divide_iff
thf(fact_1304_div__self,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ A3 @ A3 )
= ( one_one @ A ) ) ) ) ).
% div_self
thf(fact_1305_divide__minus1,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [X: A] :
( ( divide_divide @ A @ X @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( uminus_uminus @ A @ X ) ) ) ).
% divide_minus1
thf(fact_1306_divide__le__0__1__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ ( one_one @ A ) @ A3 ) @ ( zero_zero @ A ) )
= ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% divide_le_0_1_iff
thf(fact_1307_zero__le__divide__1__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ ( one_one @ A ) @ A3 ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% zero_le_divide_1_iff
thf(fact_1308_divide__less__0__1__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( divide_divide @ A @ ( one_one @ A ) @ A3 ) @ ( zero_zero @ A ) )
= ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% divide_less_0_1_iff
thf(fact_1309_divide__less__eq__1__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( divide_divide @ A @ B2 @ A3 ) @ ( one_one @ A ) )
= ( ord_less @ A @ A3 @ B2 ) ) ) ) ).
% divide_less_eq_1_neg
thf(fact_1310_divide__less__eq__1__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ ( divide_divide @ A @ B2 @ A3 ) @ ( one_one @ A ) )
= ( ord_less @ A @ B2 @ A3 ) ) ) ) ).
% divide_less_eq_1_pos
thf(fact_1311_less__divide__eq__1__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B2 @ A3 ) )
= ( ord_less @ A @ B2 @ A3 ) ) ) ) ).
% less_divide_eq_1_neg
thf(fact_1312_less__divide__eq__1__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B2 @ A3 ) )
= ( ord_less @ A @ A3 @ B2 ) ) ) ) ).
% less_divide_eq_1_pos
thf(fact_1313_zero__less__divide__1__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ ( one_one @ A ) @ A3 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% zero_less_divide_1_iff
thf(fact_1314_nonzero__divide__mult__cancel__left,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: A,B2: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ A3 @ ( times_times @ A @ A3 @ B2 ) )
= ( divide_divide @ A @ ( one_one @ A ) @ B2 ) ) ) ) ).
% nonzero_divide_mult_cancel_left
thf(fact_1315_nonzero__divide__mult__cancel__right,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [B2: A,A3: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ B2 @ ( times_times @ A @ A3 @ B2 ) )
= ( divide_divide @ A @ ( one_one @ A ) @ A3 ) ) ) ) ).
% nonzero_divide_mult_cancel_right
thf(fact_1316_divide__le__0__abs__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ A3 @ ( abs_abs @ A @ B2 ) ) @ ( zero_zero @ A ) )
= ( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
| ( B2
= ( zero_zero @ A ) ) ) ) ) ).
% divide_le_0_abs_iff
thf(fact_1317_zero__le__divide__abs__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A3 @ ( abs_abs @ A @ B2 ) ) )
= ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
| ( B2
= ( zero_zero @ A ) ) ) ) ) ).
% zero_le_divide_abs_iff
thf(fact_1318_divide__le__eq__1__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ A3 ) @ ( one_one @ A ) )
= ( ord_less_eq @ A @ A3 @ B2 ) ) ) ) ).
% divide_le_eq_1_neg
thf(fact_1319_divide__le__eq__1__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ A3 ) @ ( one_one @ A ) )
= ( ord_less_eq @ A @ B2 @ A3 ) ) ) ) ).
% divide_le_eq_1_pos
thf(fact_1320_le__divide__eq__1__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B2 @ A3 ) )
= ( ord_less_eq @ A @ B2 @ A3 ) ) ) ) ).
% le_divide_eq_1_neg
thf(fact_1321_le__divide__eq__1__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B2 @ A3 ) )
= ( ord_less_eq @ A @ A3 @ B2 ) ) ) ) ).
% le_divide_eq_1_pos
thf(fact_1322_size__neq__size__imp__neq,axiom,
! [A: $tType] :
( ( size @ A )
=> ! [X: A,Y3: A] :
( ( ( size_size @ A @ X )
!= ( size_size @ A @ Y3 ) )
=> ( X != Y3 ) ) ) ).
% size_neq_size_imp_neq
thf(fact_1323_add__divide__distrib,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( divide_divide @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 )
= ( plus_plus @ A @ ( divide_divide @ A @ A3 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ).
% add_divide_distrib
thf(fact_1324_nat__div__as__int,axiom,
( ( divide_divide @ nat )
= ( ^ [A6: nat,B5: nat] : ( nat2 @ ( divide_divide @ int @ ( semiring_1_of_nat @ int @ A6 ) @ ( semiring_1_of_nat @ int @ B5 ) ) ) ) ) ).
% nat_div_as_int
thf(fact_1325_divide__le__0__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ A3 @ B2 ) @ ( zero_zero @ A ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) )
| ( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) ) ) ) ).
% divide_le_0_iff
thf(fact_1326_divide__right__mono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ A3 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ) ).
% divide_right_mono
thf(fact_1327_zero__le__divide__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A3 @ B2 ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) )
| ( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) ) ) ) ) ).
% zero_le_divide_iff
thf(fact_1328_divide__nonneg__nonneg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y3 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X @ Y3 ) ) ) ) ) ).
% divide_nonneg_nonneg
thf(fact_1329_divide__nonneg__nonpos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less_eq @ A @ Y3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Y3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% divide_nonneg_nonpos
thf(fact_1330_divide__nonpos__nonneg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y3: A] :
( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y3 )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Y3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% divide_nonpos_nonneg
thf(fact_1331_divide__nonpos__nonpos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y3: A] :
( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ Y3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X @ Y3 ) ) ) ) ) ).
% divide_nonpos_nonpos
thf(fact_1332_divide__right__mono__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ C2 ) @ ( divide_divide @ A @ A3 @ C2 ) ) ) ) ) ).
% divide_right_mono_neg
thf(fact_1333_divide__neg__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y3: A] :
( ( ord_less @ A @ X @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ Y3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X @ Y3 ) ) ) ) ) ).
% divide_neg_neg
thf(fact_1334_divide__neg__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y3: A] :
( ( ord_less @ A @ X @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ Y3 )
=> ( ord_less @ A @ ( divide_divide @ A @ X @ Y3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% divide_neg_pos
thf(fact_1335_divide__pos__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less @ A @ Y3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( divide_divide @ A @ X @ Y3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% divide_pos_neg
thf(fact_1336_divide__pos__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ Y3 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X @ Y3 ) ) ) ) ) ).
% divide_pos_pos
thf(fact_1337_divide__less__0__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( divide_divide @ A @ A3 @ B2 ) @ ( zero_zero @ A ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less @ A @ B2 @ ( zero_zero @ A ) ) )
| ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
& ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ) ) ).
% divide_less_0_iff
thf(fact_1338_divide__less__cancel,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( ord_less @ A @ ( divide_divide @ A @ A3 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ A3 @ B2 ) )
& ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B2 @ A3 ) )
& ( C2
!= ( zero_zero @ A ) ) ) ) ) ).
% divide_less_cancel
thf(fact_1339_zero__less__divide__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A3 @ B2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) )
| ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
& ( ord_less @ A @ B2 @ ( zero_zero @ A ) ) ) ) ) ) ).
% zero_less_divide_iff
thf(fact_1340_divide__strict__right__mono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( divide_divide @ A @ A3 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ) ).
% divide_strict_right_mono
thf(fact_1341_divide__strict__right__mono__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( ord_less @ A @ B2 @ A3 )
=> ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( divide_divide @ A @ A3 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ) ).
% divide_strict_right_mono_neg
thf(fact_1342_nonzero__eq__divide__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( A3
= ( divide_divide @ A @ B2 @ C2 ) )
= ( ( times_times @ A @ A3 @ C2 )
= B2 ) ) ) ) ).
% nonzero_eq_divide_eq
thf(fact_1343_nonzero__divide__eq__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [C2: A,B2: A,A3: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( ( divide_divide @ A @ B2 @ C2 )
= A3 )
= ( B2
= ( times_times @ A @ A3 @ C2 ) ) ) ) ) ).
% nonzero_divide_eq_eq
thf(fact_1344_eq__divide__imp,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( ( times_times @ A @ A3 @ C2 )
= B2 )
=> ( A3
= ( divide_divide @ A @ B2 @ C2 ) ) ) ) ) ).
% eq_divide_imp
thf(fact_1345_divide__eq__imp,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [C2: A,B2: A,A3: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( B2
= ( times_times @ A @ A3 @ C2 ) )
=> ( ( divide_divide @ A @ B2 @ C2 )
= A3 ) ) ) ) ).
% divide_eq_imp
thf(fact_1346_eq__divide__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( A3
= ( divide_divide @ A @ B2 @ C2 ) )
= ( ( ( C2
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ A3 @ C2 )
= B2 ) )
& ( ( C2
= ( zero_zero @ A ) )
=> ( A3
= ( zero_zero @ A ) ) ) ) ) ) ).
% eq_divide_eq
thf(fact_1347_divide__eq__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,C2: A,A3: A] :
( ( ( divide_divide @ A @ B2 @ C2 )
= A3 )
= ( ( ( C2
!= ( zero_zero @ A ) )
=> ( B2
= ( times_times @ A @ A3 @ C2 ) ) )
& ( ( C2
= ( zero_zero @ A ) )
=> ( A3
= ( zero_zero @ A ) ) ) ) ) ) ).
% divide_eq_eq
thf(fact_1348_frac__eq__eq,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [Y3: A,Z: A,X: A,W: A] :
( ( Y3
!= ( zero_zero @ A ) )
=> ( ( Z
!= ( zero_zero @ A ) )
=> ( ( ( divide_divide @ A @ X @ Y3 )
= ( divide_divide @ A @ W @ Z ) )
= ( ( times_times @ A @ X @ Z )
= ( times_times @ A @ W @ Y3 ) ) ) ) ) ) ).
% frac_eq_eq
thf(fact_1349_right__inverse__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,A3: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( ( divide_divide @ A @ A3 @ B2 )
= ( one_one @ A ) )
= ( A3 = B2 ) ) ) ) ).
% right_inverse_eq
thf(fact_1350_nonzero__minus__divide__right,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,A3: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( uminus_uminus @ A @ ( divide_divide @ A @ A3 @ B2 ) )
= ( divide_divide @ A @ A3 @ ( uminus_uminus @ A @ B2 ) ) ) ) ) ).
% nonzero_minus_divide_right
thf(fact_1351_nonzero__minus__divide__divide,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,A3: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( uminus_uminus @ A @ A3 ) @ ( uminus_uminus @ A @ B2 ) )
= ( divide_divide @ A @ A3 @ B2 ) ) ) ) ).
% nonzero_minus_divide_divide
thf(fact_1352_power__one__over,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,N: nat] :
( ( power_power @ A @ ( divide_divide @ A @ ( one_one @ A ) @ A3 ) @ N )
= ( divide_divide @ A @ ( one_one @ A ) @ ( power_power @ A @ A3 @ N ) ) ) ) ).
% power_one_over
thf(fact_1353_nonzero__abs__divide,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,A3: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( abs_abs @ A @ ( divide_divide @ A @ A3 @ B2 ) )
= ( divide_divide @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B2 ) ) ) ) ) ).
% nonzero_abs_divide
thf(fact_1354_div__int__unique,axiom,
! [K: int,L: int,Q2: int,R: int] :
( ( eucl_rel_int @ K @ L @ ( product_Pair @ int @ int @ Q2 @ R ) )
=> ( ( divide_divide @ int @ K @ L )
= Q2 ) ) ).
% div_int_unique
thf(fact_1355_option_Osize__neq,axiom,
! [A: $tType,X: option @ A] :
( ( size_size @ ( option @ A ) @ X )
!= ( zero_zero @ nat ) ) ).
% option.size_neq
thf(fact_1356_Heap_Osize__neq,axiom,
! [A: $tType,X: heap_Time_Heap @ A] :
( ( size_size @ ( heap_Time_Heap @ A ) @ X )
!= ( zero_zero @ nat ) ) ).
% Heap.size_neq
thf(fact_1357_frac__le,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y3: A,X: A,W: A,Z: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y3 )
=> ( ( ord_less_eq @ A @ X @ Y3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ W )
=> ( ( ord_less_eq @ A @ W @ Z )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Z ) @ ( divide_divide @ A @ Y3 @ W ) ) ) ) ) ) ) ).
% frac_le
thf(fact_1358_frac__less,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y3: A,W: A,Z: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less @ A @ X @ Y3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ W )
=> ( ( ord_less_eq @ A @ W @ Z )
=> ( ord_less @ A @ ( divide_divide @ A @ X @ Z ) @ ( divide_divide @ A @ Y3 @ W ) ) ) ) ) ) ) ).
% frac_less
thf(fact_1359_frac__less2,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y3: A,W: A,Z: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less_eq @ A @ X @ Y3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ W )
=> ( ( ord_less @ A @ W @ Z )
=> ( ord_less @ A @ ( divide_divide @ A @ X @ Z ) @ ( divide_divide @ A @ Y3 @ W ) ) ) ) ) ) ) ).
% frac_less2
thf(fact_1360_divide__le__cancel,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ A3 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ A3 @ B2 ) )
& ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B2 @ A3 ) ) ) ) ) ).
% divide_le_cancel
thf(fact_1361_divide__nonneg__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less @ A @ Y3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Y3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% divide_nonneg_neg
thf(fact_1362_divide__nonneg__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ Y3 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X @ Y3 ) ) ) ) ) ).
% divide_nonneg_pos
thf(fact_1363_divide__nonpos__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y3: A] :
( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ Y3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X @ Y3 ) ) ) ) ) ).
% divide_nonpos_neg
thf(fact_1364_divide__nonpos__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y3: A] :
( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ Y3 )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Y3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% divide_nonpos_pos
thf(fact_1365_div__positive,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [B2: A,A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ A3 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A3 @ B2 ) ) ) ) ) ).
% div_positive
thf(fact_1366_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ A3 @ B2 )
=> ( ( divide_divide @ A @ A3 @ B2 )
= ( zero_zero @ A ) ) ) ) ) ).
% unique_euclidean_semiring_numeral_class.div_less
thf(fact_1367_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( divide_divide @ A @ A3 @ ( times_times @ A @ B2 @ C2 ) )
= ( divide_divide @ A @ ( divide_divide @ A @ A3 @ B2 ) @ C2 ) ) ) ) ).
% unique_euclidean_semiring_numeral_class.div_mult2_eq
thf(fact_1368_divide__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,C2: A,A3: A] :
( ( ord_less @ A @ ( divide_divide @ A @ B2 @ C2 ) @ A3 )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ B2 @ ( times_times @ A @ A3 @ C2 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ A3 @ C2 ) @ B2 ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ) ) ) ) ).
% divide_less_eq
thf(fact_1369_less__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less @ A @ A3 @ ( divide_divide @ A @ B2 @ C2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ A3 @ C2 ) @ B2 ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B2 @ ( times_times @ A @ A3 @ C2 ) ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% less_divide_eq
thf(fact_1370_neg__divide__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,B2: A,A3: A] :
( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( divide_divide @ A @ B2 @ C2 ) @ A3 )
= ( ord_less @ A @ ( times_times @ A @ A3 @ C2 ) @ B2 ) ) ) ) ).
% neg_divide_less_eq
thf(fact_1371_neg__less__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ A3 @ ( divide_divide @ A @ B2 @ C2 ) )
= ( ord_less @ A @ B2 @ ( times_times @ A @ A3 @ C2 ) ) ) ) ) ).
% neg_less_divide_eq
thf(fact_1372_pos__divide__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,B2: A,A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less @ A @ ( divide_divide @ A @ B2 @ C2 ) @ A3 )
= ( ord_less @ A @ B2 @ ( times_times @ A @ A3 @ C2 ) ) ) ) ) ).
% pos_divide_less_eq
thf(fact_1373_pos__less__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less @ A @ A3 @ ( divide_divide @ A @ B2 @ C2 ) )
= ( ord_less @ A @ ( times_times @ A @ A3 @ C2 ) @ B2 ) ) ) ) ).
% pos_less_divide_eq
thf(fact_1374_mult__imp__div__pos__less,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y3: A,X: A,Z: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Y3 )
=> ( ( ord_less @ A @ X @ ( times_times @ A @ Z @ Y3 ) )
=> ( ord_less @ A @ ( divide_divide @ A @ X @ Y3 ) @ Z ) ) ) ) ).
% mult_imp_div_pos_less
thf(fact_1375_mult__imp__less__div__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y3: A,Z: A,X: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Y3 )
=> ( ( ord_less @ A @ ( times_times @ A @ Z @ Y3 ) @ X )
=> ( ord_less @ A @ Z @ ( divide_divide @ A @ X @ Y3 ) ) ) ) ) ).
% mult_imp_less_div_pos
thf(fact_1376_divide__strict__left__mono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( ord_less @ A @ B2 @ A3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B2 ) )
=> ( ord_less @ A @ ( divide_divide @ A @ C2 @ A3 ) @ ( divide_divide @ A @ C2 @ B2 ) ) ) ) ) ) ).
% divide_strict_left_mono
thf(fact_1377_divide__strict__left__mono__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B2 ) )
=> ( ord_less @ A @ ( divide_divide @ A @ C2 @ A3 ) @ ( divide_divide @ A @ C2 @ B2 ) ) ) ) ) ) ).
% divide_strict_left_mono_neg
thf(fact_1378_divide__less__eq__1,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,A3: A] :
( ( ord_less @ A @ ( divide_divide @ A @ B2 @ A3 ) @ ( one_one @ A ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less @ A @ B2 @ A3 ) )
| ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
& ( ord_less @ A @ A3 @ B2 ) )
| ( A3
= ( zero_zero @ A ) ) ) ) ) ).
% divide_less_eq_1
thf(fact_1379_less__divide__eq__1,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,A3: A] :
( ( ord_less @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B2 @ A3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less @ A @ A3 @ B2 ) )
| ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
& ( ord_less @ A @ B2 @ A3 ) ) ) ) ) ).
% less_divide_eq_1
thf(fact_1380_add__divide__eq__if__simps_I2_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z: A,A3: A,B2: A] :
( ( ( Z
= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( divide_divide @ A @ A3 @ Z ) @ B2 )
= B2 ) )
& ( ( Z
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( divide_divide @ A @ A3 @ Z ) @ B2 )
= ( divide_divide @ A @ ( plus_plus @ A @ A3 @ ( times_times @ A @ B2 @ Z ) ) @ Z ) ) ) ) ) ).
% add_divide_eq_if_simps(2)
thf(fact_1381_add__divide__eq__if__simps_I1_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z: A,A3: A,B2: A] :
( ( ( Z
= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ A3 @ ( divide_divide @ A @ B2 @ Z ) )
= A3 ) )
& ( ( Z
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ A3 @ ( divide_divide @ A @ B2 @ Z ) )
= ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ A3 @ Z ) @ B2 ) @ Z ) ) ) ) ) ).
% add_divide_eq_if_simps(1)
thf(fact_1382_add__frac__eq,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [Y3: A,Z: A,X: A,W: A] :
( ( Y3
!= ( zero_zero @ A ) )
=> ( ( Z
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( divide_divide @ A @ X @ Y3 ) @ ( divide_divide @ A @ W @ Z ) )
= ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ X @ Z ) @ ( times_times @ A @ W @ Y3 ) ) @ ( times_times @ A @ Y3 @ Z ) ) ) ) ) ) ).
% add_frac_eq
thf(fact_1383_add__frac__num,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [Y3: A,X: A,Z: A] :
( ( Y3
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( divide_divide @ A @ X @ Y3 ) @ Z )
= ( divide_divide @ A @ ( plus_plus @ A @ X @ ( times_times @ A @ Z @ Y3 ) ) @ Y3 ) ) ) ) ).
% add_frac_num
thf(fact_1384_add__num__frac,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [Y3: A,Z: A,X: A] :
( ( Y3
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ Z @ ( divide_divide @ A @ X @ Y3 ) )
= ( divide_divide @ A @ ( plus_plus @ A @ X @ ( times_times @ A @ Z @ Y3 ) ) @ Y3 ) ) ) ) ).
% add_num_frac
thf(fact_1385_add__divide__eq__iff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z: A,X: A,Y3: A] :
( ( Z
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ X @ ( divide_divide @ A @ Y3 @ Z ) )
= ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ X @ Z ) @ Y3 ) @ Z ) ) ) ) ).
% add_divide_eq_iff
thf(fact_1386_divide__add__eq__iff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z: A,X: A,Y3: A] :
( ( Z
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( divide_divide @ A @ X @ Z ) @ Y3 )
= ( divide_divide @ A @ ( plus_plus @ A @ X @ ( times_times @ A @ Y3 @ Z ) ) @ Z ) ) ) ) ).
% divide_add_eq_iff
thf(fact_1387_add__divide__eq__if__simps_I4_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z: A,A3: A,B2: A] :
( ( ( Z
= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ A3 @ ( divide_divide @ A @ B2 @ Z ) )
= A3 ) )
& ( ( Z
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ A3 @ ( divide_divide @ A @ B2 @ Z ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ A3 @ Z ) @ B2 ) @ Z ) ) ) ) ) ).
% add_divide_eq_if_simps(4)
thf(fact_1388_diff__frac__eq,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [Y3: A,Z: A,X: A,W: A] :
( ( Y3
!= ( zero_zero @ A ) )
=> ( ( Z
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( divide_divide @ A @ X @ Y3 ) @ ( divide_divide @ A @ W @ Z ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ X @ Z ) @ ( times_times @ A @ W @ Y3 ) ) @ ( times_times @ A @ Y3 @ Z ) ) ) ) ) ) ).
% diff_frac_eq
thf(fact_1389_diff__divide__eq__iff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z: A,X: A,Y3: A] :
( ( Z
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ X @ ( divide_divide @ A @ Y3 @ Z ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ X @ Z ) @ Y3 ) @ Z ) ) ) ) ).
% diff_divide_eq_iff
thf(fact_1390_divide__diff__eq__iff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z: A,X: A,Y3: A] :
( ( Z
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( divide_divide @ A @ X @ Z ) @ Y3 )
= ( divide_divide @ A @ ( minus_minus @ A @ X @ ( times_times @ A @ Y3 @ Z ) ) @ Z ) ) ) ) ).
% divide_diff_eq_iff
thf(fact_1391_gt__half__sum,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ord_less @ A @ ( divide_divide @ A @ ( plus_plus @ A @ A3 @ B2 ) @ ( plus_plus @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) @ B2 ) ) ) ).
% gt_half_sum
thf(fact_1392_less__half__sum,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ord_less @ A @ A3 @ ( divide_divide @ A @ ( plus_plus @ A @ A3 @ B2 ) @ ( plus_plus @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) ) ) ) ).
% less_half_sum
thf(fact_1393_eq__minus__divide__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( A3
= ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) )
= ( ( ( C2
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ A3 @ C2 )
= ( uminus_uminus @ A @ B2 ) ) )
& ( ( C2
= ( zero_zero @ A ) )
=> ( A3
= ( zero_zero @ A ) ) ) ) ) ) ).
% eq_minus_divide_eq
thf(fact_1394_minus__divide__eq__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,C2: A,A3: A] :
( ( ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) )
= A3 )
= ( ( ( C2
!= ( zero_zero @ A ) )
=> ( ( uminus_uminus @ A @ B2 )
= ( times_times @ A @ A3 @ C2 ) ) )
& ( ( C2
= ( zero_zero @ A ) )
=> ( A3
= ( zero_zero @ A ) ) ) ) ) ) ).
% minus_divide_eq_eq
thf(fact_1395_nonzero__neg__divide__eq__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( ( uminus_uminus @ A @ ( divide_divide @ A @ A3 @ B2 ) )
= C2 )
= ( ( uminus_uminus @ A @ A3 )
= ( times_times @ A @ C2 @ B2 ) ) ) ) ) ).
% nonzero_neg_divide_eq_eq
thf(fact_1396_nonzero__neg__divide__eq__eq2,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,C2: A,A3: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( C2
= ( uminus_uminus @ A @ ( divide_divide @ A @ A3 @ B2 ) ) )
= ( ( times_times @ A @ C2 @ B2 )
= ( uminus_uminus @ A @ A3 ) ) ) ) ) ).
% nonzero_neg_divide_eq_eq2
thf(fact_1397_divide__eq__minus__1__iff,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: A,B2: A] :
( ( ( divide_divide @ A @ A3 @ B2 )
= ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( ( B2
!= ( zero_zero @ A ) )
& ( A3
= ( uminus_uminus @ A @ B2 ) ) ) ) ) ).
% divide_eq_minus_1_iff
thf(fact_1398_abs__div__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y3: A,X: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Y3 )
=> ( ( divide_divide @ A @ ( abs_abs @ A @ X ) @ Y3 )
= ( abs_abs @ A @ ( divide_divide @ A @ X @ Y3 ) ) ) ) ) ).
% abs_div_pos
thf(fact_1399_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_1400_distrib__left__NO__MATCH,axiom,
! [B: $tType,A: $tType] :
( ( semiring @ A )
=> ! [X: B,Y3: B,A3: A,B2: A,C2: A] :
( ( nO_MATCH @ B @ A @ ( divide_divide @ B @ X @ Y3 ) @ A3 )
=> ( ( times_times @ A @ A3 @ ( plus_plus @ A @ B2 @ C2 ) )
= ( plus_plus @ A @ ( times_times @ A @ A3 @ B2 ) @ ( times_times @ A @ A3 @ C2 ) ) ) ) ) ).
% distrib_left_NO_MATCH
thf(fact_1401_distrib__right__NO__MATCH,axiom,
! [B: $tType,A: $tType] :
( ( semiring @ A )
=> ! [X: B,Y3: B,C2: A,A3: A,B2: A] :
( ( nO_MATCH @ B @ A @ ( divide_divide @ B @ X @ Y3 ) @ C2 )
=> ( ( times_times @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 )
= ( plus_plus @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ) ).
% distrib_right_NO_MATCH
thf(fact_1402_left__diff__distrib__NO__MATCH,axiom,
! [B: $tType,A: $tType] :
( ( ring @ A )
=> ! [X: B,Y3: B,C2: A,A3: A,B2: A] :
( ( nO_MATCH @ B @ A @ ( divide_divide @ B @ X @ Y3 ) @ C2 )
=> ( ( times_times @ A @ ( minus_minus @ A @ A3 @ B2 ) @ C2 )
= ( minus_minus @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ) ).
% left_diff_distrib_NO_MATCH
thf(fact_1403_right__diff__distrib__NO__MATCH,axiom,
! [B: $tType,A: $tType] :
( ( ring @ A )
=> ! [X: B,Y3: B,A3: A,B2: A,C2: A] :
( ( nO_MATCH @ B @ A @ ( divide_divide @ B @ X @ Y3 ) @ A3 )
=> ( ( times_times @ A @ A3 @ ( minus_minus @ A @ B2 @ C2 ) )
= ( minus_minus @ A @ ( times_times @ A @ A3 @ B2 ) @ ( times_times @ A @ A3 @ C2 ) ) ) ) ) ).
% right_diff_distrib_NO_MATCH
thf(fact_1404_divide__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,C2: A,A3: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ C2 ) @ A3 )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ B2 @ ( times_times @ A @ A3 @ C2 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C2 ) @ B2 ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ) ) ) ) ).
% divide_le_eq
thf(fact_1405_le__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ ( divide_divide @ A @ B2 @ C2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C2 ) @ B2 ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B2 @ ( times_times @ A @ A3 @ C2 ) ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% le_divide_eq
thf(fact_1406_divide__left__mono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( ord_less_eq @ A @ B2 @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B2 ) )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ C2 @ A3 ) @ ( divide_divide @ A @ C2 @ B2 ) ) ) ) ) ) ).
% divide_left_mono
thf(fact_1407_neg__divide__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,B2: A,A3: A] :
( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ C2 ) @ A3 )
= ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C2 ) @ B2 ) ) ) ) ).
% neg_divide_le_eq
thf(fact_1408_neg__le__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ A3 @ ( divide_divide @ A @ B2 @ C2 ) )
= ( ord_less_eq @ A @ B2 @ ( times_times @ A @ A3 @ C2 ) ) ) ) ) ).
% neg_le_divide_eq
thf(fact_1409_pos__divide__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,B2: A,A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ C2 ) @ A3 )
= ( ord_less_eq @ A @ B2 @ ( times_times @ A @ A3 @ C2 ) ) ) ) ) ).
% pos_divide_le_eq
thf(fact_1410_pos__le__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less_eq @ A @ A3 @ ( divide_divide @ A @ B2 @ C2 ) )
= ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C2 ) @ B2 ) ) ) ) ).
% pos_le_divide_eq
thf(fact_1411_mult__imp__div__pos__le,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y3: A,X: A,Z: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Y3 )
=> ( ( ord_less_eq @ A @ X @ ( times_times @ A @ Z @ Y3 ) )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Y3 ) @ Z ) ) ) ) ).
% mult_imp_div_pos_le
thf(fact_1412_mult__imp__le__div__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y3: A,Z: A,X: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Y3 )
=> ( ( ord_less_eq @ A @ ( times_times @ A @ Z @ Y3 ) @ X )
=> ( ord_less_eq @ A @ Z @ ( divide_divide @ A @ X @ Y3 ) ) ) ) ) ).
% mult_imp_le_div_pos
thf(fact_1413_divide__left__mono__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B2 ) )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ C2 @ A3 ) @ ( divide_divide @ A @ C2 @ B2 ) ) ) ) ) ) ).
% divide_left_mono_neg
thf(fact_1414_le__divide__eq__1,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,A3: A] :
( ( ord_less_eq @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B2 @ A3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less_eq @ A @ A3 @ B2 ) )
| ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ B2 @ A3 ) ) ) ) ) ).
% le_divide_eq_1
thf(fact_1415_divide__le__eq__1,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,A3: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ A3 ) @ ( one_one @ A ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less_eq @ A @ B2 @ A3 ) )
| ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ A3 @ B2 ) )
| ( A3
= ( zero_zero @ A ) ) ) ) ) ).
% divide_le_eq_1
thf(fact_1416_frac__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y3: A,Z: A,X: A,W: A] :
( ( Y3
!= ( zero_zero @ A ) )
=> ( ( Z
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Y3 ) @ ( divide_divide @ A @ W @ Z ) )
= ( ord_less_eq @ A @ ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ X @ Z ) @ ( times_times @ A @ W @ Y3 ) ) @ ( times_times @ A @ Y3 @ Z ) ) @ ( zero_zero @ A ) ) ) ) ) ) ).
% frac_le_eq
thf(fact_1417_frac__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y3: A,Z: A,X: A,W: A] :
( ( Y3
!= ( zero_zero @ A ) )
=> ( ( Z
!= ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( divide_divide @ A @ X @ Y3 ) @ ( divide_divide @ A @ W @ Z ) )
= ( ord_less @ A @ ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ X @ Z ) @ ( times_times @ A @ W @ Y3 ) ) @ ( times_times @ A @ Y3 @ Z ) ) @ ( zero_zero @ A ) ) ) ) ) ) ).
% frac_less_eq
thf(fact_1418_less__minus__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less @ A @ A3 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ A3 @ C2 ) @ ( uminus_uminus @ A @ B2 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A3 @ C2 ) ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% less_minus_divide_eq
thf(fact_1419_minus__divide__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,C2: A,A3: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) @ A3 )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A3 @ C2 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ A3 @ C2 ) @ ( uminus_uminus @ A @ B2 ) ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ) ) ) ) ).
% minus_divide_less_eq
thf(fact_1420_neg__less__minus__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ A3 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) )
= ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A3 @ C2 ) ) ) ) ) ).
% neg_less_minus_divide_eq
thf(fact_1421_neg__minus__divide__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,B2: A,A3: A] :
( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) @ A3 )
= ( ord_less @ A @ ( times_times @ A @ A3 @ C2 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ) ).
% neg_minus_divide_less_eq
thf(fact_1422_pos__less__minus__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less @ A @ A3 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) )
= ( ord_less @ A @ ( times_times @ A @ A3 @ C2 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ) ).
% pos_less_minus_divide_eq
thf(fact_1423_pos__minus__divide__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,B2: A,A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) @ A3 )
= ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A3 @ C2 ) ) ) ) ) ).
% pos_minus_divide_less_eq
thf(fact_1424_minus__divide__add__eq__iff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z: A,X: A,Y3: A] :
( ( Z
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ X @ Z ) ) @ Y3 )
= ( divide_divide @ A @ ( plus_plus @ A @ ( uminus_uminus @ A @ X ) @ ( times_times @ A @ Y3 @ Z ) ) @ Z ) ) ) ) ).
% minus_divide_add_eq_iff
thf(fact_1425_add__divide__eq__if__simps_I3_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z: A,A3: A,B2: A] :
( ( ( Z
= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ A3 @ Z ) ) @ B2 )
= B2 ) )
& ( ( Z
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ A3 @ Z ) ) @ B2 )
= ( divide_divide @ A @ ( plus_plus @ A @ ( uminus_uminus @ A @ A3 ) @ ( times_times @ A @ B2 @ Z ) ) @ Z ) ) ) ) ) ).
% add_divide_eq_if_simps(3)
thf(fact_1426_add__divide__eq__if__simps_I6_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z: A,A3: A,B2: A] :
( ( ( Z
= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ A3 @ Z ) ) @ B2 )
= ( uminus_uminus @ A @ B2 ) ) )
& ( ( Z
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ A3 @ Z ) ) @ B2 )
= ( divide_divide @ A @ ( minus_minus @ A @ ( uminus_uminus @ A @ A3 ) @ ( times_times @ A @ B2 @ Z ) ) @ Z ) ) ) ) ) ).
% add_divide_eq_if_simps(6)
thf(fact_1427_add__divide__eq__if__simps_I5_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z: A,A3: A,B2: A] :
( ( ( Z
= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( divide_divide @ A @ A3 @ Z ) @ B2 )
= ( uminus_uminus @ A @ B2 ) ) )
& ( ( Z
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( divide_divide @ A @ A3 @ Z ) @ B2 )
= ( divide_divide @ A @ ( minus_minus @ A @ A3 @ ( times_times @ A @ B2 @ Z ) ) @ Z ) ) ) ) ) ).
% add_divide_eq_if_simps(5)
thf(fact_1428_minus__divide__diff__eq__iff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z: A,X: A,Y3: A] :
( ( Z
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ X @ Z ) ) @ Y3 )
= ( divide_divide @ A @ ( minus_minus @ A @ ( uminus_uminus @ A @ X ) @ ( times_times @ A @ Y3 @ Z ) ) @ Z ) ) ) ) ).
% minus_divide_diff_eq_iff
thf(fact_1429_power__diff,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A3: A,N: nat,M: nat] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ nat @ N @ M )
=> ( ( power_power @ A @ A3 @ ( minus_minus @ nat @ M @ N ) )
= ( divide_divide @ A @ ( power_power @ A @ A3 @ M ) @ ( power_power @ A @ A3 @ N ) ) ) ) ) ) ).
% power_diff
thf(fact_1430_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_1431_verit__less__mono__div__int2,axiom,
! [A5: int,B4: int,N: int] :
( ( ord_less_eq @ int @ A5 @ B4 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ ( uminus_uminus @ int @ N ) )
=> ( ord_less_eq @ int @ ( divide_divide @ int @ B4 @ N ) @ ( divide_divide @ int @ A5 @ N ) ) ) ) ).
% verit_less_mono_div_int2
thf(fact_1432_le__minus__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C2 ) @ ( uminus_uminus @ A @ B2 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A3 @ C2 ) ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% le_minus_divide_eq
thf(fact_1433_minus__divide__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,C2: A,A3: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) @ A3 )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A3 @ C2 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C2 ) @ ( uminus_uminus @ A @ B2 ) ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ) ) ) ) ).
% minus_divide_le_eq
thf(fact_1434_neg__le__minus__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ A3 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) )
= ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A3 @ C2 ) ) ) ) ) ).
% neg_le_minus_divide_eq
thf(fact_1435_neg__minus__divide__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,B2: A,A3: A] :
( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) @ A3 )
= ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C2 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ) ).
% neg_minus_divide_le_eq
thf(fact_1436_pos__le__minus__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less_eq @ A @ A3 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) )
= ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C2 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ) ).
% pos_le_minus_divide_eq
thf(fact_1437_pos__minus__divide__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,B2: A,A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) @ A3 )
= ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A3 @ C2 ) ) ) ) ) ).
% pos_minus_divide_le_eq
thf(fact_1438_scaling__mono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [U2: A,V: A,R: A,S: A] :
( ( ord_less_eq @ A @ U2 @ V )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ R )
=> ( ( ord_less_eq @ A @ R @ S )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ U2 @ ( divide_divide @ A @ ( times_times @ A @ R @ ( minus_minus @ A @ V @ U2 ) ) @ S ) ) @ V ) ) ) ) ) ).
% scaling_mono
thf(fact_1439_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_1440_ceiling__divide__upper,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Q2: A,P4: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Q2 )
=> ( ord_less_eq @ A @ P4 @ ( times_times @ A @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ ( divide_divide @ A @ P4 @ Q2 ) ) ) @ Q2 ) ) ) ) ).
% ceiling_divide_upper
thf(fact_1441_div__add__self1__no__field,axiom,
! [B: $tType,A: $tType] :
( ( ( euclid4440199948858584721cancel @ A )
& ( field @ B ) )
=> ! [X: B,B2: A,A3: A] :
( ( nO_MATCH @ B @ A @ X @ B2 )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ B2 @ A3 ) @ B2 )
= ( plus_plus @ A @ ( divide_divide @ A @ A3 @ B2 ) @ ( one_one @ A ) ) ) ) ) ) ).
% div_add_self1_no_field
thf(fact_1442_div__add__self2__no__field,axiom,
! [B: $tType,A: $tType] :
( ( ( euclid4440199948858584721cancel @ A )
& ( field @ B ) )
=> ! [X: B,B2: A,A3: A] :
( ( nO_MATCH @ B @ A @ X @ B2 )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A3 @ B2 ) @ B2 )
= ( plus_plus @ A @ ( divide_divide @ A @ A3 @ B2 ) @ ( one_one @ A ) ) ) ) ) ) ).
% div_add_self2_no_field
thf(fact_1443_div__mult__self1,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A3 @ ( times_times @ A @ C2 @ B2 ) ) @ B2 )
= ( plus_plus @ A @ C2 @ ( divide_divide @ A @ A3 @ B2 ) ) ) ) ) ).
% div_mult_self1
thf(fact_1444_div__mult__self2,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A3 @ ( times_times @ A @ B2 @ C2 ) ) @ B2 )
= ( plus_plus @ A @ C2 @ ( divide_divide @ A @ A3 @ B2 ) ) ) ) ) ).
% div_mult_self2
thf(fact_1445_div__mult__self3,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B2: A,C2: A,A3: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ C2 @ B2 ) @ A3 ) @ B2 )
= ( plus_plus @ A @ C2 @ ( divide_divide @ A @ A3 @ B2 ) ) ) ) ) ).
% div_mult_self3
thf(fact_1446_div__mult__self4,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B2: A,C2: A,A3: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ B2 @ C2 ) @ A3 ) @ B2 )
= ( plus_plus @ A @ C2 @ ( divide_divide @ A @ A3 @ B2 ) ) ) ) ) ).
% div_mult_self4
thf(fact_1447_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_1448_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_1449_div__mult__mult1,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( divide_divide @ A @ A3 @ B2 ) ) ) ) ).
% div_mult_mult1
thf(fact_1450_div__mult__mult2,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) )
= ( divide_divide @ A @ A3 @ B2 ) ) ) ) ).
% div_mult_mult2
thf(fact_1451_div__mult__mult1__if,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( ( C2
= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( zero_zero @ A ) ) )
& ( ( C2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( divide_divide @ A @ A3 @ B2 ) ) ) ) ) ).
% div_mult_mult1_if
thf(fact_1452_div__minus1__right,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A3: A] :
( ( divide_divide @ A @ A3 @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( uminus_uminus @ A @ A3 ) ) ) ).
% div_minus1_right
thf(fact_1453_div__by__Suc__0,axiom,
! [M: nat] :
( ( divide_divide @ nat @ M @ ( suc @ ( zero_zero @ nat ) ) )
= M ) ).
% div_by_Suc_0
thf(fact_1454_div__less,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ( ( divide_divide @ nat @ M @ N )
= ( zero_zero @ nat ) ) ) ).
% div_less
thf(fact_1455_tuple__isomorphism_Osize__neq,axiom,
! [A: $tType,B: $tType,C: $tType,X: tuple_isomorphism @ A @ B @ C] :
( ( size_size @ ( tuple_isomorphism @ A @ B @ C ) @ X )
!= ( zero_zero @ nat ) ) ).
% tuple_isomorphism.size_neq
thf(fact_1456_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_1457_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_1458_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_1459_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_1460_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_1461_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_1462_div__add__self2,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B2: A,A3: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A3 @ B2 ) @ B2 )
= ( plus_plus @ A @ ( divide_divide @ A @ A3 @ B2 ) @ ( one_one @ A ) ) ) ) ) ).
% div_add_self2
thf(fact_1463_div__add__self1,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B2: A,A3: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ B2 @ A3 ) @ B2 )
= ( plus_plus @ A @ ( divide_divide @ A @ A3 @ B2 ) @ ( one_one @ A ) ) ) ) ) ).
% div_add_self1
thf(fact_1464_div__if,axiom,
( ( divide_divide @ nat )
= ( ^ [M5: nat,N5: nat] :
( if @ nat
@ ( ( ord_less @ nat @ M5 @ N5 )
| ( N5
= ( zero_zero @ nat ) ) )
@ ( zero_zero @ nat )
@ ( suc @ ( divide_divide @ nat @ ( minus_minus @ nat @ M5 @ N5 ) @ N5 ) ) ) ) ) ).
% div_if
thf(fact_1465_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_1466_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_1467_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_1468_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_1469_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_1470_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 ) ) )
| ? [Q4: nat] :
( ( ord_less_eq @ nat @ ( times_times @ nat @ N @ Q4 ) @ M )
& ( ord_less @ nat @ M @ ( times_times @ nat @ N @ ( suc @ Q4 ) ) )
& ( P @ Q4 ) ) ) ) ).
% split_div'
thf(fact_1471_power__diff__power__eq,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A3: A,N: nat,M: nat] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( ( ord_less_eq @ nat @ N @ M )
=> ( ( divide_divide @ A @ ( power_power @ A @ A3 @ M ) @ ( power_power @ A @ A3 @ N ) )
= ( power_power @ A @ A3 @ ( minus_minus @ nat @ M @ N ) ) ) )
& ( ~ ( ord_less_eq @ nat @ N @ M )
=> ( ( divide_divide @ A @ ( power_power @ A @ A3 @ M ) @ ( power_power @ A @ A3 @ N ) )
= ( divide_divide @ A @ ( one_one @ A ) @ ( power_power @ A @ A3 @ ( minus_minus @ nat @ N @ M ) ) ) ) ) ) ) ) ).
% power_diff_power_eq
thf(fact_1472_bits__div__by__1,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A] :
( ( divide_divide @ A @ A3 @ ( one_one @ A ) )
= A3 ) ) ).
% bits_div_by_1
thf(fact_1473_bits__div__by__0,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A] :
( ( divide_divide @ A @ A3 @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% bits_div_by_0
thf(fact_1474_bits__div__0,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A] :
( ( divide_divide @ A @ ( zero_zero @ A ) @ A3 )
= ( zero_zero @ A ) ) ) ).
% bits_div_0
thf(fact_1475_divide__int__def,axiom,
( ( divide_divide @ int )
= ( ^ [K3: int,L4: int] :
( if @ int
@ ( L4
= ( zero_zero @ int ) )
@ ( zero_zero @ int )
@ ( if @ int
@ ( ( sgn_sgn @ int @ K3 )
= ( sgn_sgn @ int @ L4 ) )
@ ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ ( nat2 @ ( abs_abs @ int @ K3 ) ) @ ( nat2 @ ( abs_abs @ int @ L4 ) ) ) )
@ ( uminus_uminus @ int
@ ( semiring_1_of_nat @ int
@ ( plus_plus @ nat @ ( divide_divide @ nat @ ( nat2 @ ( abs_abs @ int @ K3 ) ) @ ( nat2 @ ( abs_abs @ int @ L4 ) ) )
@ ( zero_neq_one_of_bool @ nat
@ ~ ( dvd_dvd @ int @ L4 @ K3 ) ) ) ) ) ) ) ) ) ).
% divide_int_def
thf(fact_1476_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_1477_normalize__negative,axiom,
! [Q2: int,P4: int] :
( ( ord_less @ int @ Q2 @ ( zero_zero @ int ) )
=> ( ( normalize @ ( product_Pair @ int @ int @ P4 @ Q2 ) )
= ( normalize @ ( product_Pair @ int @ int @ ( uminus_uminus @ int @ P4 ) @ ( uminus_uminus @ int @ Q2 ) ) ) ) ) ).
% normalize_negative
thf(fact_1478_gbinomial__code,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( ( gbinomial @ A )
= ( ^ [A6: A,K3: nat] :
( if @ A
@ ( K3
= ( zero_zero @ nat ) )
@ ( one_one @ A )
@ ( divide_divide @ A
@ ( set_fo6178422350223883121st_nat @ A
@ ^ [L4: nat] : ( times_times @ A @ ( minus_minus @ A @ A6 @ ( semiring_1_of_nat @ A @ L4 ) ) )
@ ( zero_zero @ nat )
@ ( minus_minus @ nat @ K3 @ ( one_one @ nat ) )
@ ( one_one @ A ) )
@ ( semiring_char_0_fact @ A @ K3 ) ) ) ) ) ) ).
% gbinomial_code
thf(fact_1479_dependent__nat__choice,axiom,
! [A: $tType,P: nat > A > $o,Q: nat > A > A > $o] :
( ? [X_1: A] : ( P @ ( zero_zero @ nat ) @ X_1 )
=> ( ! [X4: A,N2: nat] :
( ( P @ N2 @ X4 )
=> ? [Y5: A] :
( ( P @ ( suc @ N2 ) @ Y5 )
& ( Q @ N2 @ X4 @ Y5 ) ) )
=> ? [F4: nat > A] :
! [N7: nat] :
( ( P @ N7 @ ( F4 @ N7 ) )
& ( Q @ N7 @ ( F4 @ N7 ) @ ( F4 @ ( suc @ N7 ) ) ) ) ) ) ).
% dependent_nat_choice
thf(fact_1480_dvd__0__right,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A3: A] : ( dvd_dvd @ A @ A3 @ ( zero_zero @ A ) ) ) ).
% dvd_0_right
thf(fact_1481_dvd__0__left__iff,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A3: A] :
( ( dvd_dvd @ A @ ( zero_zero @ A ) @ A3 )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% dvd_0_left_iff
thf(fact_1482_dvd__add__triv__left__iff,axiom,
! [A: $tType] :
( ( comm_s4317794764714335236cancel @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ ( plus_plus @ A @ A3 @ B2 ) )
= ( dvd_dvd @ A @ A3 @ B2 ) ) ) ).
% dvd_add_triv_left_iff
thf(fact_1483_dvd__add__triv__right__iff,axiom,
! [A: $tType] :
( ( comm_s4317794764714335236cancel @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ ( plus_plus @ A @ B2 @ A3 ) )
= ( dvd_dvd @ A @ A3 @ B2 ) ) ) ).
% dvd_add_triv_right_iff
thf(fact_1484_minus__dvd__iff,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [X: A,Y3: A] :
( ( dvd_dvd @ A @ ( uminus_uminus @ A @ X ) @ Y3 )
= ( dvd_dvd @ A @ X @ Y3 ) ) ) ).
% minus_dvd_iff
thf(fact_1485_dvd__minus__iff,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [X: A,Y3: A] :
( ( dvd_dvd @ A @ X @ ( uminus_uminus @ A @ Y3 ) )
= ( dvd_dvd @ A @ X @ Y3 ) ) ) ).
% dvd_minus_iff
thf(fact_1486_div__dvd__div,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A3 @ B2 )
=> ( ( dvd_dvd @ A @ A3 @ C2 )
=> ( ( dvd_dvd @ A @ ( divide_divide @ A @ B2 @ A3 ) @ ( divide_divide @ A @ C2 @ A3 ) )
= ( dvd_dvd @ A @ B2 @ C2 ) ) ) ) ) ).
% div_dvd_div
thf(fact_1487_dvd__abs__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M: A,K: A] :
( ( dvd_dvd @ A @ M @ ( abs_abs @ A @ K ) )
= ( dvd_dvd @ A @ M @ K ) ) ) ).
% dvd_abs_iff
thf(fact_1488_abs__dvd__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M: A,K: A] :
( ( dvd_dvd @ A @ ( abs_abs @ A @ M ) @ K )
= ( dvd_dvd @ A @ M @ K ) ) ) ).
% abs_dvd_iff
thf(fact_1489_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_1490_gbinomial__1,axiom,
! [A: $tType] :
( ( ( semiring_char_0 @ A )
& ( semidom_divide @ A ) )
=> ! [A3: A] :
( ( gbinomial @ A @ A3 @ ( one_one @ nat ) )
= A3 ) ) ).
% gbinomial_1
thf(fact_1491_dvd__mult__cancel__left,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( dvd_dvd @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( dvd_dvd @ A @ A3 @ B2 ) ) ) ) ).
% dvd_mult_cancel_left
thf(fact_1492_dvd__mult__cancel__right,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( dvd_dvd @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( dvd_dvd @ A @ A3 @ B2 ) ) ) ) ).
% dvd_mult_cancel_right
thf(fact_1493_dvd__times__left__cancel__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ ( times_times @ A @ A3 @ B2 ) @ ( times_times @ A @ A3 @ C2 ) )
= ( dvd_dvd @ A @ B2 @ C2 ) ) ) ) ).
% dvd_times_left_cancel_iff
thf(fact_1494_dvd__times__right__cancel__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ ( times_times @ A @ B2 @ A3 ) @ ( times_times @ A @ C2 @ A3 ) )
= ( dvd_dvd @ A @ B2 @ C2 ) ) ) ) ).
% dvd_times_right_cancel_iff
thf(fact_1495_dvd__add__times__triv__left__iff,axiom,
! [A: $tType] :
( ( comm_s4317794764714335236cancel @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ ( plus_plus @ A @ ( times_times @ A @ C2 @ A3 ) @ B2 ) )
= ( dvd_dvd @ A @ A3 @ B2 ) ) ) ).
% dvd_add_times_triv_left_iff
thf(fact_1496_dvd__add__times__triv__right__iff,axiom,
! [A: $tType] :
( ( comm_s4317794764714335236cancel @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A3 @ ( plus_plus @ A @ B2 @ ( times_times @ A @ C2 @ A3 ) ) )
= ( dvd_dvd @ A @ A3 @ B2 ) ) ) ).
% dvd_add_times_triv_right_iff
thf(fact_1497_unit__prod,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( dvd_dvd @ A @ ( times_times @ A @ A3 @ B2 ) @ ( one_one @ A ) ) ) ) ) ).
% unit_prod
thf(fact_1498_dvd__div__mult__self,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ B2 )
=> ( ( times_times @ A @ ( divide_divide @ A @ B2 @ A3 ) @ A3 )
= B2 ) ) ) ).
% dvd_div_mult_self
thf(fact_1499_dvd__mult__div__cancel,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ B2 )
=> ( ( times_times @ A @ A3 @ ( divide_divide @ A @ B2 @ A3 ) )
= B2 ) ) ) ).
% dvd_mult_div_cancel
thf(fact_1500_div__add,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( dvd_dvd @ A @ C2 @ A3 )
=> ( ( dvd_dvd @ A @ C2 @ B2 )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 )
= ( plus_plus @ A @ ( divide_divide @ A @ A3 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ) ) ).
% div_add
thf(fact_1501_unit__div,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( dvd_dvd @ A @ ( divide_divide @ A @ A3 @ B2 ) @ ( one_one @ A ) ) ) ) ) ).
% unit_div
thf(fact_1502_unit__div__1__unit,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( dvd_dvd @ A @ ( divide_divide @ A @ ( one_one @ A ) @ A3 ) @ ( one_one @ A ) ) ) ) ).
% unit_div_1_unit
thf(fact_1503_unit__div__1__div__1,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( divide_divide @ A @ ( one_one @ A ) @ ( divide_divide @ A @ ( one_one @ A ) @ A3 ) )
= A3 ) ) ) ).
% unit_div_1_div_1
thf(fact_1504_div__diff,axiom,
! [A: $tType] :
( ( idom_modulo @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( dvd_dvd @ A @ C2 @ A3 )
=> ( ( dvd_dvd @ A @ C2 @ B2 )
=> ( ( divide_divide @ A @ ( minus_minus @ A @ A3 @ B2 ) @ C2 )
= ( minus_minus @ A @ ( divide_divide @ A @ A3 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ) ) ).
% div_diff
thf(fact_1505_dvd__1__left,axiom,
! [K: nat] : ( dvd_dvd @ nat @ ( suc @ ( zero_zero @ nat ) ) @ K ) ).
% dvd_1_left
thf(fact_1506_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_1507_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_1508_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_1509_gbinomial__0_I1_J,axiom,
! [A: $tType] :
( ( ( semiring_char_0 @ A )
& ( semidom_divide @ A ) )
=> ! [A3: A] :
( ( gbinomial @ A @ A3 @ ( zero_zero @ nat ) )
= ( one_one @ A ) ) ) ).
% gbinomial_0(1)
thf(fact_1510_gbinomial__Suc0,axiom,
! [A: $tType] :
( ( ( semiring_char_0 @ A )
& ( semidom_divide @ A ) )
=> ! [A3: A] :
( ( gbinomial @ A @ A3 @ ( suc @ ( zero_zero @ nat ) ) )
= A3 ) ) ).
% gbinomial_Suc0
thf(fact_1511_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_1512_zdvd1__eq,axiom,
! [X: int] :
( ( dvd_dvd @ int @ X @ ( one_one @ int ) )
= ( ( abs_abs @ int @ X )
= ( one_one @ int ) ) ) ).
% zdvd1_eq
thf(fact_1513_unit__div__mult__self,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( times_times @ A @ ( divide_divide @ A @ B2 @ A3 ) @ A3 )
= B2 ) ) ) ).
% unit_div_mult_self
thf(fact_1514_unit__mult__div__div,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( times_times @ A @ B2 @ ( divide_divide @ A @ ( one_one @ A ) @ A3 ) )
= ( divide_divide @ A @ B2 @ A3 ) ) ) ) ).
% unit_mult_div_div
thf(fact_1515_pow__divides__pow__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [N: nat,A3: A,B2: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( dvd_dvd @ A @ ( power_power @ A @ A3 @ N ) @ ( power_power @ A @ B2 @ N ) )
= ( dvd_dvd @ A @ A3 @ B2 ) ) ) ) ).
% pow_divides_pow_iff
thf(fact_1516_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_1517_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_1518_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_1519_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_1520_normalize__denom__zero,axiom,
! [P4: int] :
( ( normalize @ ( product_Pair @ int @ int @ P4 @ ( zero_zero @ int ) ) )
= ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) ) ) ).
% normalize_denom_zero
thf(fact_1521_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_1522_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_1523_strict__subset__divisors__dvd,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ ( set @ A )
@ ( collect @ A
@ ^ [C4: A] : ( dvd_dvd @ A @ C4 @ A3 ) )
@ ( collect @ A
@ ^ [C4: A] : ( dvd_dvd @ A @ C4 @ B2 ) ) )
= ( ( dvd_dvd @ A @ A3 @ B2 )
& ~ ( dvd_dvd @ A @ B2 @ A3 ) ) ) ) ).
% strict_subset_divisors_dvd
thf(fact_1524_gcd__nat_Onot__eq__order__implies__strict,axiom,
! [A3: nat,B2: nat] :
( ( A3 != B2 )
=> ( ( dvd_dvd @ nat @ A3 @ B2 )
=> ( ( dvd_dvd @ nat @ A3 @ B2 )
& ( A3 != B2 ) ) ) ) ).
% gcd_nat.not_eq_order_implies_strict
thf(fact_1525_dvd__trans,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A3 @ B2 )
=> ( ( dvd_dvd @ A @ B2 @ C2 )
=> ( dvd_dvd @ A @ A3 @ C2 ) ) ) ) ).
% dvd_trans
thf(fact_1526_dvd__refl,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: A] : ( dvd_dvd @ A @ A3 @ A3 ) ) ).
% dvd_refl
thf(fact_1527_gcd__nat_Ostrict__implies__not__eq,axiom,
! [A3: nat,B2: nat] :
( ( ( dvd_dvd @ nat @ A3 @ B2 )
& ( A3 != B2 ) )
=> ( A3 != B2 ) ) ).
% gcd_nat.strict_implies_not_eq
thf(fact_1528_gcd__nat_Ostrict__implies__order,axiom,
! [A3: nat,B2: nat] :
( ( ( dvd_dvd @ nat @ A3 @ B2 )
& ( A3 != B2 ) )
=> ( dvd_dvd @ nat @ A3 @ B2 ) ) ).
% gcd_nat.strict_implies_order
thf(fact_1529_gcd__nat_Ostrict__iff__order,axiom,
! [A3: nat,B2: nat] :
( ( ( dvd_dvd @ nat @ A3 @ B2 )
& ( A3 != B2 ) )
= ( ( dvd_dvd @ nat @ A3 @ B2 )
& ( A3 != B2 ) ) ) ).
% gcd_nat.strict_iff_order
thf(fact_1530_gcd__nat_Oorder__iff__strict,axiom,
( ( dvd_dvd @ nat )
= ( ^ [A6: nat,B5: nat] :
( ( ( dvd_dvd @ nat @ A6 @ B5 )
& ( A6 != B5 ) )
| ( A6 = B5 ) ) ) ) ).
% gcd_nat.order_iff_strict
thf(fact_1531_gcd__nat_Ostrict__iff__not,axiom,
! [A3: nat,B2: nat] :
( ( ( dvd_dvd @ nat @ A3 @ B2 )
& ( A3 != B2 ) )
= ( ( dvd_dvd @ nat @ A3 @ B2 )
& ~ ( dvd_dvd @ nat @ B2 @ A3 ) ) ) ).
% gcd_nat.strict_iff_not
thf(fact_1532_gcd__nat_Ostrict__trans2,axiom,
! [A3: nat,B2: nat,C2: nat] :
( ( ( dvd_dvd @ nat @ A3 @ B2 )
& ( A3 != B2 ) )
=> ( ( dvd_dvd @ nat @ B2 @ C2 )
=> ( ( dvd_dvd @ nat @ A3 @ C2 )
& ( A3 != C2 ) ) ) ) ).
% gcd_nat.strict_trans2
thf(fact_1533_gcd__nat_Ostrict__trans1,axiom,
! [A3: nat,B2: nat,C2: nat] :
( ( dvd_dvd @ nat @ A3 @ B2 )
=> ( ( ( dvd_dvd @ nat @ B2 @ C2 )
& ( B2 != C2 ) )
=> ( ( dvd_dvd @ nat @ A3 @ C2 )
& ( A3 != C2 ) ) ) ) ).
% gcd_nat.strict_trans1
thf(fact_1534_gcd__nat_Ostrict__trans,axiom,
! [A3: nat,B2: nat,C2: nat] :
( ( ( dvd_dvd @ nat @ A3 @ B2 )
& ( A3 != B2 ) )
=> ( ( ( dvd_dvd @ nat @ B2 @ C2 )
& ( B2 != C2 ) )
=> ( ( dvd_dvd @ nat @ A3 @ C2 )
& ( A3 != C2 ) ) ) ) ).
% gcd_nat.strict_trans
thf(fact_1535_gcd__nat_Oantisym,axiom,
! [A3: nat,B2: nat] :
( ( dvd_dvd @ nat @ A3 @ B2 )
=> ( ( dvd_dvd @ nat @ B2 @ A3 )
=> ( A3 = B2 ) ) ) ).
% gcd_nat.antisym
thf(fact_1536_gcd__nat_Oirrefl,axiom,
! [A3: nat] :
~ ( ( dvd_dvd @ nat @ A3 @ A3 )
& ( A3 != A3 ) ) ).
% gcd_nat.irrefl
thf(fact_1537_gcd__nat_Oeq__iff,axiom,
( ( ^ [Y6: nat,Z2: nat] : Y6 = Z2 )
= ( ^ [A6: nat,B5: nat] :
( ( dvd_dvd @ nat @ A6 @ B5 )
& ( dvd_dvd @ nat @ B5 @ A6 ) ) ) ) ).
% gcd_nat.eq_iff
thf(fact_1538_gcd__nat_Otrans,axiom,
! [A3: nat,B2: nat,C2: nat] :
( ( dvd_dvd @ nat @ A3 @ B2 )
=> ( ( dvd_dvd @ nat @ B2 @ C2 )
=> ( dvd_dvd @ nat @ A3 @ C2 ) ) ) ).
% gcd_nat.trans
thf(fact_1539_gcd__nat_Orefl,axiom,
! [A3: nat] : ( dvd_dvd @ nat @ A3 @ A3 ) ).
% gcd_nat.refl
thf(fact_1540_gcd__nat_Oasym,axiom,
! [A3: nat,B2: nat] :
( ( ( dvd_dvd @ nat @ A3 @ B2 )
& ( A3 != B2 ) )
=> ~ ( ( dvd_dvd @ nat @ B2 @ A3 )
& ( B2 != A3 ) ) ) ).
% gcd_nat.asym
thf(fact_1541_dvd__antisym,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd @ nat @ M @ N )
=> ( ( dvd_dvd @ nat @ N @ M )
=> ( M = N ) ) ) ).
% dvd_antisym
thf(fact_1542_dvd__field__iff,axiom,
! [A: $tType] :
( ( field @ A )
=> ( ( dvd_dvd @ A )
= ( ^ [A6: A,B5: A] :
( ( A6
= ( zero_zero @ A ) )
=> ( B5
= ( zero_zero @ A ) ) ) ) ) ) ).
% dvd_field_iff
thf(fact_1543_dvd__0__left,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A3: A] :
( ( dvd_dvd @ A @ ( zero_zero @ A ) @ A3 )
=> ( A3
= ( zero_zero @ A ) ) ) ) ).
% dvd_0_left
thf(fact_1544_dvd__triv__right,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: A,B2: A] : ( dvd_dvd @ A @ A3 @ ( times_times @ A @ B2 @ A3 ) ) ) ).
% dvd_triv_right
thf(fact_1545_dvd__mult__right,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ ( times_times @ A @ A3 @ B2 ) @ C2 )
=> ( dvd_dvd @ A @ B2 @ C2 ) ) ) ).
% dvd_mult_right
thf(fact_1546_mult__dvd__mono,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: A,B2: A,C2: A,D3: A] :
( ( dvd_dvd @ A @ A3 @ B2 )
=> ( ( dvd_dvd @ A @ C2 @ D3 )
=> ( dvd_dvd @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ D3 ) ) ) ) ) ).
% mult_dvd_mono
thf(fact_1547_dvd__triv__left,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: A,B2: A] : ( dvd_dvd @ A @ A3 @ ( times_times @ A @ A3 @ B2 ) ) ) ).
% dvd_triv_left
thf(fact_1548_dvd__mult__left,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ ( times_times @ A @ A3 @ B2 ) @ C2 )
=> ( dvd_dvd @ A @ A3 @ C2 ) ) ) ).
% dvd_mult_left
thf(fact_1549_dvd__mult2,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A3 @ B2 )
=> ( dvd_dvd @ A @ A3 @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% dvd_mult2
thf(fact_1550_division__decomp,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A3 @ ( times_times @ A @ B2 @ C2 ) )
=> ? [B8: A,C5: A] :
( ( A3
= ( times_times @ A @ B8 @ C5 ) )
& ( dvd_dvd @ A @ B8 @ B2 )
& ( dvd_dvd @ A @ C5 @ C2 ) ) ) ) ).
% division_decomp
thf(fact_1551_dvd__mult,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ C2 )
=> ( dvd_dvd @ A @ A3 @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% dvd_mult
thf(fact_1552_dvd__productE,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [P4: A,A3: A,B2: A] :
( ( dvd_dvd @ A @ P4 @ ( times_times @ A @ A3 @ B2 ) )
=> ~ ! [X4: A,Y4: A] :
( ( P4
= ( times_times @ A @ X4 @ Y4 ) )
=> ( ( dvd_dvd @ A @ X4 @ A3 )
=> ~ ( dvd_dvd @ A @ Y4 @ B2 ) ) ) ) ) ).
% dvd_productE
thf(fact_1553_dvd__def,axiom,
! [A: $tType] :
( ( dvd @ A )
=> ( ( dvd_dvd @ A )
= ( ^ [B5: A,A6: A] :
? [K3: A] :
( A6
= ( times_times @ A @ B5 @ K3 ) ) ) ) ) ).
% dvd_def
thf(fact_1554_dvdI,axiom,
! [A: $tType] :
( ( dvd @ A )
=> ! [A3: A,B2: A,K: A] :
( ( A3
= ( times_times @ A @ B2 @ K ) )
=> ( dvd_dvd @ A @ B2 @ A3 ) ) ) ).
% dvdI
thf(fact_1555_dvdE,axiom,
! [A: $tType] :
( ( dvd @ A )
=> ! [B2: A,A3: A] :
( ( dvd_dvd @ A @ B2 @ A3 )
=> ~ ! [K2: A] :
( A3
!= ( times_times @ A @ B2 @ K2 ) ) ) ) ).
% dvdE
thf(fact_1556_dvd__add,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A3 @ B2 )
=> ( ( dvd_dvd @ A @ A3 @ C2 )
=> ( dvd_dvd @ A @ A3 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ) ).
% dvd_add
thf(fact_1557_dvd__add__left__iff,axiom,
! [A: $tType] :
( ( comm_s4317794764714335236cancel @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ C2 )
=> ( ( dvd_dvd @ A @ A3 @ ( plus_plus @ A @ B2 @ C2 ) )
= ( dvd_dvd @ A @ A3 @ B2 ) ) ) ) ).
% dvd_add_left_iff
thf(fact_1558_dvd__add__right__iff,axiom,
! [A: $tType] :
( ( comm_s4317794764714335236cancel @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A3 @ B2 )
=> ( ( dvd_dvd @ A @ A3 @ ( plus_plus @ A @ B2 @ C2 ) )
= ( dvd_dvd @ A @ A3 @ C2 ) ) ) ) ).
% dvd_add_right_iff
thf(fact_1559_one__dvd,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: A] : ( dvd_dvd @ A @ ( one_one @ A ) @ A3 ) ) ).
% one_dvd
thf(fact_1560_unit__imp__dvd,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A3: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( dvd_dvd @ A @ B2 @ A3 ) ) ) ).
% unit_imp_dvd
thf(fact_1561_dvd__unit__imp__unit,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ B2 )
=> ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( dvd_dvd @ A @ A3 @ ( one_one @ A ) ) ) ) ) ).
% dvd_unit_imp_unit
thf(fact_1562_dvd__diff,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [X: A,Y3: A,Z: A] :
( ( dvd_dvd @ A @ X @ Y3 )
=> ( ( dvd_dvd @ A @ X @ Z )
=> ( dvd_dvd @ A @ X @ ( minus_minus @ A @ Y3 @ Z ) ) ) ) ) ).
% dvd_diff
thf(fact_1563_div__div__div__same,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [D3: A,B2: A,A3: A] :
( ( dvd_dvd @ A @ D3 @ B2 )
=> ( ( dvd_dvd @ A @ B2 @ A3 )
=> ( ( divide_divide @ A @ ( divide_divide @ A @ A3 @ D3 ) @ ( divide_divide @ A @ B2 @ D3 ) )
= ( divide_divide @ A @ A3 @ B2 ) ) ) ) ) ).
% div_div_div_same
thf(fact_1564_dvd__div__eq__cancel,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( ( divide_divide @ A @ A3 @ C2 )
= ( divide_divide @ A @ B2 @ C2 ) )
=> ( ( dvd_dvd @ A @ C2 @ A3 )
=> ( ( dvd_dvd @ A @ C2 @ B2 )
=> ( A3 = B2 ) ) ) ) ) ).
% dvd_div_eq_cancel
thf(fact_1565_dvd__div__eq__iff,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( dvd_dvd @ A @ C2 @ A3 )
=> ( ( dvd_dvd @ A @ C2 @ B2 )
=> ( ( ( divide_divide @ A @ A3 @ C2 )
= ( divide_divide @ A @ B2 @ C2 ) )
= ( A3 = B2 ) ) ) ) ) ).
% dvd_div_eq_iff
thf(fact_1566_gcd__nat_Oextremum,axiom,
! [A3: nat] : ( dvd_dvd @ nat @ A3 @ ( zero_zero @ nat ) ) ).
% gcd_nat.extremum
thf(fact_1567_gcd__nat_Oextremum__strict,axiom,
! [A3: nat] :
~ ( ( dvd_dvd @ nat @ ( zero_zero @ nat ) @ A3 )
& ( ( zero_zero @ nat )
!= A3 ) ) ).
% gcd_nat.extremum_strict
thf(fact_1568_gcd__nat_Oextremum__unique,axiom,
! [A3: nat] :
( ( dvd_dvd @ nat @ ( zero_zero @ nat ) @ A3 )
= ( A3
= ( zero_zero @ nat ) ) ) ).
% gcd_nat.extremum_unique
thf(fact_1569_gcd__nat_Onot__eq__extremum,axiom,
! [A3: nat] :
( ( A3
!= ( zero_zero @ nat ) )
= ( ( dvd_dvd @ nat @ A3 @ ( zero_zero @ nat ) )
& ( A3
!= ( zero_zero @ nat ) ) ) ) ).
% gcd_nat.not_eq_extremum
thf(fact_1570_gcd__nat_Oextremum__uniqueI,axiom,
! [A3: nat] :
( ( dvd_dvd @ nat @ ( zero_zero @ nat ) @ A3 )
=> ( A3
= ( zero_zero @ nat ) ) ) ).
% gcd_nat.extremum_uniqueI
thf(fact_1571_dvd__if__abs__eq,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [L: A,K: A] :
( ( ( abs_abs @ A @ L )
= ( abs_abs @ A @ K ) )
=> ( dvd_dvd @ A @ L @ K ) ) ) ).
% dvd_if_abs_eq
thf(fact_1572_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_1573_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_1574_zdvd__antisym__abs,axiom,
! [A3: int,B2: int] :
( ( dvd_dvd @ int @ A3 @ B2 )
=> ( ( dvd_dvd @ int @ B2 @ A3 )
=> ( ( abs_abs @ int @ A3 )
= ( abs_abs @ int @ B2 ) ) ) ) ).
% zdvd_antisym_abs
thf(fact_1575_subset__divisors__dvd,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [C4: A] : ( dvd_dvd @ A @ C4 @ A3 ) )
@ ( collect @ A
@ ^ [C4: A] : ( dvd_dvd @ A @ C4 @ B2 ) ) )
= ( dvd_dvd @ A @ A3 @ B2 ) ) ) ).
% subset_divisors_dvd
thf(fact_1576_gbinomial__Suc__Suc,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,K: nat] :
( ( gbinomial @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ ( suc @ K ) )
= ( plus_plus @ A @ ( gbinomial @ A @ A3 @ K ) @ ( gbinomial @ A @ A3 @ ( suc @ K ) ) ) ) ) ).
% gbinomial_Suc_Suc
thf(fact_1577_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_1578_unit__mult__right__cancel,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( ( times_times @ A @ B2 @ A3 )
= ( times_times @ A @ C2 @ A3 ) )
= ( B2 = C2 ) ) ) ) ).
% unit_mult_right_cancel
thf(fact_1579_unit__mult__left__cancel,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( ( times_times @ A @ A3 @ B2 )
= ( times_times @ A @ A3 @ C2 ) )
= ( B2 = C2 ) ) ) ) ).
% unit_mult_left_cancel
thf(fact_1580_mult__unit__dvd__iff_H,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ ( times_times @ A @ A3 @ B2 ) @ C2 )
= ( dvd_dvd @ A @ B2 @ C2 ) ) ) ) ).
% mult_unit_dvd_iff'
thf(fact_1581_dvd__mult__unit__iff_H,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ A3 @ ( times_times @ A @ B2 @ C2 ) )
= ( dvd_dvd @ A @ A3 @ C2 ) ) ) ) ).
% dvd_mult_unit_iff'
thf(fact_1582_mult__unit__dvd__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ ( times_times @ A @ A3 @ B2 ) @ C2 )
= ( dvd_dvd @ A @ A3 @ C2 ) ) ) ) ).
% mult_unit_dvd_iff
thf(fact_1583_dvd__mult__unit__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ A3 @ ( times_times @ A @ C2 @ B2 ) )
= ( dvd_dvd @ A @ A3 @ C2 ) ) ) ) ).
% dvd_mult_unit_iff
thf(fact_1584_is__unit__mult__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ ( times_times @ A @ A3 @ B2 ) @ ( one_one @ A ) )
= ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
& ( dvd_dvd @ A @ B2 @ ( one_one @ A ) ) ) ) ) ).
% is_unit_mult_iff
thf(fact_1585_dvd__div__eq__0__iff,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [B2: A,A3: A] :
( ( dvd_dvd @ A @ B2 @ A3 )
=> ( ( ( divide_divide @ A @ A3 @ B2 )
= ( zero_zero @ A ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ) ).
% dvd_div_eq_0_iff
thf(fact_1586_div__mult__div__if__dvd,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A3: A,D3: A,C2: A] :
( ( dvd_dvd @ A @ B2 @ A3 )
=> ( ( dvd_dvd @ A @ D3 @ C2 )
=> ( ( times_times @ A @ ( divide_divide @ A @ A3 @ B2 ) @ ( divide_divide @ A @ C2 @ D3 ) )
= ( divide_divide @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ D3 ) ) ) ) ) ) ).
% div_mult_div_if_dvd
thf(fact_1587_dvd__mult__imp__div,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( dvd_dvd @ A @ ( times_times @ A @ A3 @ C2 ) @ B2 )
=> ( dvd_dvd @ A @ A3 @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ).
% dvd_mult_imp_div
thf(fact_1588_dvd__div__mult2__eq,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,C2: A,A3: A] :
( ( dvd_dvd @ A @ ( times_times @ A @ B2 @ C2 ) @ A3 )
=> ( ( divide_divide @ A @ A3 @ ( times_times @ A @ B2 @ C2 ) )
= ( divide_divide @ A @ ( divide_divide @ A @ A3 @ B2 ) @ C2 ) ) ) ) ).
% dvd_div_mult2_eq
thf(fact_1589_div__div__eq__right,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C2: A,B2: A,A3: A] :
( ( dvd_dvd @ A @ C2 @ B2 )
=> ( ( dvd_dvd @ A @ B2 @ A3 )
=> ( ( divide_divide @ A @ A3 @ ( divide_divide @ A @ B2 @ C2 ) )
= ( times_times @ A @ ( divide_divide @ A @ A3 @ B2 ) @ C2 ) ) ) ) ) ).
% div_div_eq_right
thf(fact_1590_div__mult__swap,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C2: A,B2: A,A3: A] :
( ( dvd_dvd @ A @ C2 @ B2 )
=> ( ( times_times @ A @ A3 @ ( divide_divide @ A @ B2 @ C2 ) )
= ( divide_divide @ A @ ( times_times @ A @ A3 @ B2 ) @ C2 ) ) ) ) ).
% div_mult_swap
thf(fact_1591_dvd__div__mult,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C2: A,B2: A,A3: A] :
( ( dvd_dvd @ A @ C2 @ B2 )
=> ( ( times_times @ A @ ( divide_divide @ A @ B2 @ C2 ) @ A3 )
= ( divide_divide @ A @ ( times_times @ A @ B2 @ A3 ) @ C2 ) ) ) ) ).
% dvd_div_mult
thf(fact_1592_div__plus__div__distrib__dvd__right,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [C2: A,B2: A,A3: A] :
( ( dvd_dvd @ A @ C2 @ B2 )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 )
= ( plus_plus @ A @ ( divide_divide @ A @ A3 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ) ).
% div_plus_div_distrib_dvd_right
thf(fact_1593_div__plus__div__distrib__dvd__left,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( dvd_dvd @ A @ C2 @ A3 )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 )
= ( plus_plus @ A @ ( divide_divide @ A @ A3 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ) ).
% div_plus_div_distrib_dvd_left
thf(fact_1594_unit__div__cancel,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( ( divide_divide @ A @ B2 @ A3 )
= ( divide_divide @ A @ C2 @ A3 ) )
= ( B2 = C2 ) ) ) ) ).
% unit_div_cancel
thf(fact_1595_div__unit__dvd__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ ( divide_divide @ A @ A3 @ B2 ) @ C2 )
= ( dvd_dvd @ A @ A3 @ C2 ) ) ) ) ).
% div_unit_dvd_iff
thf(fact_1596_dvd__div__unit__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ A3 @ ( divide_divide @ A @ C2 @ B2 ) )
= ( dvd_dvd @ A @ A3 @ C2 ) ) ) ) ).
% dvd_div_unit_iff
thf(fact_1597_dvd__div__neg,axiom,
! [A: $tType] :
( ( idom_divide @ A )
=> ! [B2: A,A3: A] :
( ( dvd_dvd @ A @ B2 @ A3 )
=> ( ( divide_divide @ A @ A3 @ ( uminus_uminus @ A @ B2 ) )
= ( uminus_uminus @ A @ ( divide_divide @ A @ A3 @ B2 ) ) ) ) ) ).
% dvd_div_neg
thf(fact_1598_dvd__neg__div,axiom,
! [A: $tType] :
( ( idom_divide @ A )
=> ! [B2: A,A3: A] :
( ( dvd_dvd @ A @ B2 @ A3 )
=> ( ( divide_divide @ A @ ( uminus_uminus @ A @ A3 ) @ B2 )
= ( uminus_uminus @ A @ ( divide_divide @ A @ A3 @ B2 ) ) ) ) ) ).
% dvd_neg_div
thf(fact_1599_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_1600_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_1601_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_1602_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_1603_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_1604_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_1605_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_1606_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_1607_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_1608_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_1609_bezout__add__nat,axiom,
! [A3: nat,B2: nat] :
? [D2: nat,X4: nat,Y4: nat] :
( ( dvd_dvd @ nat @ D2 @ A3 )
& ( dvd_dvd @ nat @ D2 @ B2 )
& ( ( ( times_times @ nat @ A3 @ X4 )
= ( plus_plus @ nat @ ( times_times @ nat @ B2 @ Y4 ) @ D2 ) )
| ( ( times_times @ nat @ B2 @ X4 )
= ( plus_plus @ nat @ ( times_times @ nat @ A3 @ Y4 ) @ D2 ) ) ) ) ).
% bezout_add_nat
thf(fact_1610_bezout__lemma__nat,axiom,
! [D3: nat,A3: nat,B2: nat,X: nat,Y3: nat] :
( ( dvd_dvd @ nat @ D3 @ A3 )
=> ( ( dvd_dvd @ nat @ D3 @ B2 )
=> ( ( ( ( times_times @ nat @ A3 @ X )
= ( plus_plus @ nat @ ( times_times @ nat @ B2 @ Y3 ) @ D3 ) )
| ( ( times_times @ nat @ B2 @ X )
= ( plus_plus @ nat @ ( times_times @ nat @ A3 @ Y3 ) @ D3 ) ) )
=> ? [X4: nat,Y4: nat] :
( ( dvd_dvd @ nat @ D3 @ A3 )
& ( dvd_dvd @ nat @ D3 @ ( plus_plus @ nat @ A3 @ B2 ) )
& ( ( ( times_times @ nat @ A3 @ X4 )
= ( plus_plus @ nat @ ( times_times @ nat @ ( plus_plus @ nat @ A3 @ B2 ) @ Y4 ) @ D3 ) )
| ( ( times_times @ nat @ ( plus_plus @ nat @ A3 @ B2 ) @ X4 )
= ( plus_plus @ nat @ ( times_times @ nat @ A3 @ Y4 ) @ D3 ) ) ) ) ) ) ) ).
% bezout_lemma_nat
thf(fact_1611_bezout1__nat,axiom,
! [A3: nat,B2: nat] :
? [D2: nat,X4: nat,Y4: nat] :
( ( dvd_dvd @ nat @ D2 @ A3 )
& ( dvd_dvd @ nat @ D2 @ B2 )
& ( ( ( minus_minus @ nat @ ( times_times @ nat @ A3 @ X4 ) @ ( times_times @ nat @ B2 @ Y4 ) )
= D2 )
| ( ( minus_minus @ nat @ ( times_times @ nat @ B2 @ X4 ) @ ( times_times @ nat @ A3 @ Y4 ) )
= D2 ) ) ) ).
% bezout1_nat
thf(fact_1612_zdvd__period,axiom,
! [A3: int,D3: int,X: int,T2: int,C2: int] :
( ( dvd_dvd @ int @ A3 @ D3 )
=> ( ( dvd_dvd @ int @ A3 @ ( plus_plus @ int @ X @ T2 ) )
= ( dvd_dvd @ int @ A3 @ ( plus_plus @ int @ ( plus_plus @ int @ X @ ( times_times @ int @ C2 @ D3 ) ) @ T2 ) ) ) ) ).
% zdvd_period
thf(fact_1613_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_1614_nat__dvd__iff,axiom,
! [Z: int,M: nat] :
( ( dvd_dvd @ nat @ ( nat2 @ Z ) @ M )
= ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
=> ( dvd_dvd @ int @ Z @ ( semiring_1_of_nat @ int @ M ) ) )
& ( ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
=> ( M
= ( zero_zero @ nat ) ) ) ) ) ).
% nat_dvd_iff
thf(fact_1615_gbinomial__addition__formula,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,K: nat] :
( ( gbinomial @ A @ A3 @ ( suc @ K ) )
= ( plus_plus @ A @ ( gbinomial @ A @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) @ ( suc @ K ) ) @ ( gbinomial @ A @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) @ K ) ) ) ) ).
% gbinomial_addition_formula
thf(fact_1616_gbinomial__absorb__comp,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,K: nat] :
( ( times_times @ A @ ( minus_minus @ A @ A3 @ ( semiring_1_of_nat @ A @ K ) ) @ ( gbinomial @ A @ A3 @ K ) )
= ( times_times @ A @ A3 @ ( gbinomial @ A @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) @ K ) ) ) ) ).
% gbinomial_absorb_comp
thf(fact_1617_gbinomial__mult__1,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,K: nat] :
( ( times_times @ A @ A3 @ ( gbinomial @ A @ A3 @ K ) )
= ( plus_plus @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ K ) @ ( gbinomial @ A @ A3 @ K ) ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ K ) ) @ ( gbinomial @ A @ A3 @ ( suc @ K ) ) ) ) ) ) ).
% gbinomial_mult_1
thf(fact_1618_gbinomial__mult__1_H,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,K: nat] :
( ( times_times @ A @ ( gbinomial @ A @ A3 @ K ) @ A3 )
= ( plus_plus @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ K ) @ ( gbinomial @ A @ A3 @ K ) ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ K ) ) @ ( gbinomial @ A @ A3 @ ( suc @ K ) ) ) ) ) ) ).
% gbinomial_mult_1'
thf(fact_1619_unit__dvdE,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ~ ( ( A3
!= ( zero_zero @ A ) )
=> ! [C3: A] :
( B2
!= ( times_times @ A @ A3 @ C3 ) ) ) ) ) ).
% unit_dvdE
thf(fact_1620_dvd__div__eq__mult,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ A3 @ B2 )
=> ( ( ( divide_divide @ A @ B2 @ A3 )
= C2 )
= ( B2
= ( times_times @ A @ C2 @ A3 ) ) ) ) ) ) ).
% dvd_div_eq_mult
thf(fact_1621_div__dvd__iff__mult,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ B2 @ A3 )
=> ( ( dvd_dvd @ A @ ( divide_divide @ A @ A3 @ B2 ) @ C2 )
= ( dvd_dvd @ A @ A3 @ ( times_times @ A @ C2 @ B2 ) ) ) ) ) ) ).
% div_dvd_iff_mult
thf(fact_1622_dvd__div__iff__mult,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C2: A,B2: A,A3: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ C2 @ B2 )
=> ( ( dvd_dvd @ A @ A3 @ ( divide_divide @ A @ B2 @ C2 ) )
= ( dvd_dvd @ A @ ( times_times @ A @ A3 @ C2 ) @ B2 ) ) ) ) ) ).
% dvd_div_iff_mult
thf(fact_1623_dvd__div__div__eq__mult,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,C2: A,B2: A,D3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( C2
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ A3 @ B2 )
=> ( ( dvd_dvd @ A @ C2 @ D3 )
=> ( ( ( divide_divide @ A @ B2 @ A3 )
= ( divide_divide @ A @ D3 @ C2 ) )
= ( ( times_times @ A @ B2 @ C2 )
= ( times_times @ A @ A3 @ D3 ) ) ) ) ) ) ) ) ).
% dvd_div_div_eq_mult
thf(fact_1624_unit__div__eq__0__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A3: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( ( divide_divide @ A @ A3 @ B2 )
= ( zero_zero @ A ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ) ).
% unit_div_eq_0_iff
thf(fact_1625_is__unit__div__mult2__eq,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,C2: A,A3: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ C2 @ ( one_one @ A ) )
=> ( ( divide_divide @ A @ A3 @ ( times_times @ A @ B2 @ C2 ) )
= ( divide_divide @ A @ ( divide_divide @ A @ A3 @ B2 ) @ C2 ) ) ) ) ) ).
% is_unit_div_mult2_eq
thf(fact_1626_unit__div__mult__swap,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( dvd_dvd @ A @ C2 @ ( one_one @ A ) )
=> ( ( times_times @ A @ A3 @ ( divide_divide @ A @ B2 @ C2 ) )
= ( divide_divide @ A @ ( times_times @ A @ A3 @ B2 ) @ C2 ) ) ) ) ).
% unit_div_mult_swap
thf(fact_1627_unit__div__commute,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( times_times @ A @ ( divide_divide @ A @ A3 @ B2 ) @ C2 )
= ( divide_divide @ A @ ( times_times @ A @ A3 @ C2 ) @ B2 ) ) ) ) ).
% unit_div_commute
thf(fact_1628_div__mult__unit2,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C2: A,B2: A,A3: A] :
( ( dvd_dvd @ A @ C2 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ B2 @ A3 )
=> ( ( divide_divide @ A @ A3 @ ( times_times @ A @ B2 @ C2 ) )
= ( divide_divide @ A @ ( divide_divide @ A @ A3 @ B2 ) @ C2 ) ) ) ) ) ).
% div_mult_unit2
thf(fact_1629_unit__eq__div2,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( A3
= ( divide_divide @ A @ C2 @ B2 ) )
= ( ( times_times @ A @ A3 @ B2 )
= C2 ) ) ) ) ).
% unit_eq_div2
thf(fact_1630_unit__eq__div1,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( ( divide_divide @ A @ A3 @ B2 )
= C2 )
= ( A3
= ( times_times @ A @ C2 @ B2 ) ) ) ) ) ).
% unit_eq_div1
thf(fact_1631_is__unit__power__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,N: nat] :
( ( dvd_dvd @ A @ ( power_power @ A @ A3 @ N ) @ ( one_one @ A ) )
= ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
| ( N
= ( zero_zero @ nat ) ) ) ) ) ).
% is_unit_power_iff
thf(fact_1632_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_1633_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_1634_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_1635_bezout__add__strong__nat,axiom,
! [A3: nat,B2: nat] :
( ( A3
!= ( zero_zero @ nat ) )
=> ? [D2: nat,X4: nat,Y4: nat] :
( ( dvd_dvd @ nat @ D2 @ A3 )
& ( dvd_dvd @ nat @ D2 @ B2 )
& ( ( times_times @ nat @ A3 @ X4 )
= ( plus_plus @ nat @ ( times_times @ nat @ B2 @ Y4 ) @ D2 ) ) ) ) ).
% bezout_add_strong_nat
thf(fact_1636_zdvd__imp__le,axiom,
! [Z: int,N: int] :
( ( dvd_dvd @ int @ Z @ N )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ N )
=> ( ord_less_eq @ int @ Z @ N ) ) ) ).
% zdvd_imp_le
thf(fact_1637_dvd__imp__le__int,axiom,
! [I: int,D3: int] :
( ( I
!= ( zero_zero @ int ) )
=> ( ( dvd_dvd @ int @ D3 @ I )
=> ( ord_less_eq @ int @ ( abs_abs @ int @ D3 ) @ ( abs_abs @ int @ I ) ) ) ) ).
% dvd_imp_le_int
thf(fact_1638_of__int__divide__in__Ints,axiom,
! [A: $tType] :
( ( idom_divide @ A )
=> ! [B2: int,A3: int] :
( ( dvd_dvd @ int @ B2 @ A3 )
=> ( member @ A @ ( divide_divide @ A @ ( ring_1_of_int @ A @ A3 ) @ ( ring_1_of_int @ A @ B2 ) ) @ ( ring_1_Ints @ A ) ) ) ) ).
% of_int_divide_in_Ints
thf(fact_1639_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_1640_Suc__times__gbinomial,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,A3: A] :
( ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ K ) ) @ ( gbinomial @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ ( suc @ K ) ) )
= ( times_times @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ ( gbinomial @ A @ A3 @ K ) ) ) ) ).
% Suc_times_gbinomial
thf(fact_1641_gbinomial__absorption,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,A3: A] :
( ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ K ) ) @ ( gbinomial @ A @ A3 @ ( suc @ K ) ) )
= ( times_times @ A @ A3 @ ( gbinomial @ A @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) @ K ) ) ) ) ).
% gbinomial_absorption
thf(fact_1642_is__unitE,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,C2: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ~ ( ( A3
!= ( zero_zero @ A ) )
=> ! [B3: A] :
( ( B3
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ B3 @ ( one_one @ A ) )
=> ( ( ( divide_divide @ A @ ( one_one @ A ) @ A3 )
= B3 )
=> ( ( ( divide_divide @ A @ ( one_one @ A ) @ B3 )
= A3 )
=> ( ( ( times_times @ A @ A3 @ B3 )
= ( one_one @ A ) )
=> ( ( divide_divide @ A @ C2 @ A3 )
!= ( times_times @ A @ C2 @ B3 ) ) ) ) ) ) ) ) ) ) ).
% is_unitE
thf(fact_1643_is__unit__div__mult__cancel__left,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( divide_divide @ A @ A3 @ ( times_times @ A @ A3 @ B2 ) )
= ( divide_divide @ A @ ( one_one @ A ) @ B2 ) ) ) ) ) ).
% is_unit_div_mult_cancel_left
thf(fact_1644_is__unit__div__mult__cancel__right,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( divide_divide @ A @ A3 @ ( times_times @ A @ B2 @ A3 ) )
= ( divide_divide @ A @ ( one_one @ A ) @ B2 ) ) ) ) ) ).
% is_unit_div_mult_cancel_right
thf(fact_1645_dvd__power__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [X: A,M: nat,N: nat] :
( ( X
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ ( power_power @ A @ X @ M ) @ ( power_power @ A @ X @ N ) )
= ( ( dvd_dvd @ A @ X @ ( one_one @ A ) )
| ( ord_less_eq @ nat @ M @ N ) ) ) ) ) ).
% dvd_power_iff
thf(fact_1646_dvd__power,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [N: nat,X: A] :
( ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
| ( X
= ( one_one @ A ) ) )
=> ( dvd_dvd @ A @ X @ ( power_power @ A @ X @ N ) ) ) ) ).
% dvd_power
thf(fact_1647_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_1648_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_1649_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_1650_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_1651_gbinomial__factors,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,K: nat] :
( ( gbinomial @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ ( suc @ K ) )
= ( times_times @ A @ ( divide_divide @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ ( semiring_1_of_nat @ A @ ( suc @ K ) ) ) @ ( gbinomial @ A @ A3 @ K ) ) ) ) ).
% gbinomial_factors
thf(fact_1652_gbinomial__rec,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,K: nat] :
( ( gbinomial @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ ( suc @ K ) )
= ( times_times @ A @ ( gbinomial @ A @ A3 @ K ) @ ( divide_divide @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ ( semiring_1_of_nat @ A @ ( suc @ K ) ) ) ) ) ) ).
% gbinomial_rec
thf(fact_1653_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_1654_gbinomial__negated__upper,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( ( gbinomial @ A )
= ( ^ [A6: A,K3: nat] : ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K3 ) @ ( gbinomial @ A @ ( minus_minus @ A @ ( minus_minus @ A @ ( semiring_1_of_nat @ A @ K3 ) @ A6 ) @ ( one_one @ A ) ) @ K3 ) ) ) ) ) ).
% gbinomial_negated_upper
thf(fact_1655_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_1656_normalize__denom__pos,axiom,
! [R: product_prod @ int @ int,P4: int,Q2: int] :
( ( ( normalize @ R )
= ( product_Pair @ int @ int @ P4 @ Q2 ) )
=> ( ord_less @ int @ ( zero_zero @ int ) @ Q2 ) ) ).
% normalize_denom_pos
thf(fact_1657_normalize__crossproduct,axiom,
! [Q2: int,S: int,P4: int,R: int] :
( ( Q2
!= ( zero_zero @ int ) )
=> ( ( S
!= ( zero_zero @ int ) )
=> ( ( ( normalize @ ( product_Pair @ int @ int @ P4 @ Q2 ) )
= ( normalize @ ( product_Pair @ int @ int @ R @ S ) ) )
=> ( ( times_times @ int @ P4 @ S )
= ( times_times @ int @ R @ Q2 ) ) ) ) ) ).
% normalize_crossproduct
thf(fact_1658_gbinomial__minus,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,K: nat] :
( ( gbinomial @ A @ ( uminus_uminus @ A @ A3 ) @ K )
= ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K ) @ ( gbinomial @ A @ ( minus_minus @ A @ ( plus_plus @ A @ A3 @ ( semiring_1_of_nat @ A @ K ) ) @ ( one_one @ A ) ) @ K ) ) ) ) ).
% gbinomial_minus
thf(fact_1659_gbinomial__reduce__nat,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,A3: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ( gbinomial @ A @ A3 @ K )
= ( plus_plus @ A @ ( gbinomial @ A @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) @ ( minus_minus @ nat @ K @ ( one_one @ nat ) ) ) @ ( gbinomial @ A @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) @ K ) ) ) ) ) ).
% gbinomial_reduce_nat
thf(fact_1660_gbinomial__pochhammer,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( ( gbinomial @ A )
= ( ^ [A6: A,K3: nat] : ( divide_divide @ A @ ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K3 ) @ ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ A6 ) @ K3 ) ) @ ( semiring_char_0_fact @ A @ K3 ) ) ) ) ) ).
% gbinomial_pochhammer
thf(fact_1661_gbinomial__pochhammer_H,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( ( gbinomial @ A )
= ( ^ [A6: A,K3: nat] : ( divide_divide @ A @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ ( minus_minus @ A @ A6 @ ( semiring_1_of_nat @ A @ K3 ) ) @ ( one_one @ A ) ) @ K3 ) @ ( semiring_char_0_fact @ A @ K3 ) ) ) ) ) ).
% gbinomial_pochhammer'
thf(fact_1662_gbinomial__absorption_H,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,A3: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ( gbinomial @ A @ A3 @ K )
= ( times_times @ A @ ( divide_divide @ A @ A3 @ ( semiring_1_of_nat @ A @ K ) ) @ ( gbinomial @ A @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) @ ( minus_minus @ nat @ K @ ( one_one @ nat ) ) ) ) ) ) ) ).
% gbinomial_absorption'
thf(fact_1663_inf__period_I4_J,axiom,
! [A: $tType] :
( ( ( comm_ring @ A )
& ( dvd @ A ) )
=> ! [D3: A,D5: A,T2: A] :
( ( dvd_dvd @ A @ D3 @ D5 )
=> ! [X5: A,K4: A] :
( ( ~ ( dvd_dvd @ A @ D3 @ ( plus_plus @ A @ X5 @ T2 ) ) )
= ( ~ ( dvd_dvd @ A @ D3 @ ( plus_plus @ A @ ( minus_minus @ A @ X5 @ ( times_times @ A @ K4 @ D5 ) ) @ T2 ) ) ) ) ) ) ).
% inf_period(4)
thf(fact_1664_inf__period_I3_J,axiom,
! [A: $tType] :
( ( ( comm_ring @ A )
& ( dvd @ A ) )
=> ! [D3: A,D5: A,T2: A] :
( ( dvd_dvd @ A @ D3 @ D5 )
=> ! [X5: A,K4: A] :
( ( dvd_dvd @ A @ D3 @ ( plus_plus @ A @ X5 @ T2 ) )
= ( dvd_dvd @ A @ D3 @ ( plus_plus @ A @ ( minus_minus @ A @ X5 @ ( times_times @ A @ K4 @ D5 ) ) @ T2 ) ) ) ) ) ).
% inf_period(3)
thf(fact_1665_unity__coeff__ex,axiom,
! [A: $tType] :
( ( ( dvd @ A )
& ( semiring_0 @ A ) )
=> ! [P: A > $o,L: A] :
( ( ? [X2: A] : ( P @ ( times_times @ A @ L @ X2 ) ) )
= ( ? [X2: A] :
( ( dvd_dvd @ A @ L @ ( plus_plus @ A @ X2 @ ( zero_zero @ A ) ) )
& ( P @ X2 ) ) ) ) ) ).
% unity_coeff_ex
thf(fact_1666_minf_I10_J,axiom,
! [B: $tType] :
( ( ( plus @ B )
& ( linorder @ B )
& ( dvd @ B ) )
=> ! [D3: B,S: B] :
? [Z6: B] :
! [X5: B] :
( ( ord_less @ B @ X5 @ Z6 )
=> ( ( ~ ( dvd_dvd @ B @ D3 @ ( plus_plus @ B @ X5 @ S ) ) )
= ( ~ ( dvd_dvd @ B @ D3 @ ( plus_plus @ B @ X5 @ S ) ) ) ) ) ) ).
% minf(10)
thf(fact_1667_minf_I9_J,axiom,
! [B: $tType] :
( ( ( plus @ B )
& ( linorder @ B )
& ( dvd @ B ) )
=> ! [D3: B,S: B] :
? [Z6: B] :
! [X5: B] :
( ( ord_less @ B @ X5 @ Z6 )
=> ( ( dvd_dvd @ B @ D3 @ ( plus_plus @ B @ X5 @ S ) )
= ( dvd_dvd @ B @ D3 @ ( plus_plus @ B @ X5 @ S ) ) ) ) ) ).
% minf(9)
thf(fact_1668_pinf_I10_J,axiom,
! [B: $tType] :
( ( ( plus @ B )
& ( linorder @ B )
& ( dvd @ B ) )
=> ! [D3: B,S: B] :
? [Z6: B] :
! [X5: B] :
( ( ord_less @ B @ Z6 @ X5 )
=> ( ( ~ ( dvd_dvd @ B @ D3 @ ( plus_plus @ B @ X5 @ S ) ) )
= ( ~ ( dvd_dvd @ B @ D3 @ ( plus_plus @ B @ X5 @ S ) ) ) ) ) ) ).
% pinf(10)
thf(fact_1669_pinf_I9_J,axiom,
! [B: $tType] :
( ( ( plus @ B )
& ( linorder @ B )
& ( dvd @ B ) )
=> ! [D3: B,S: B] :
? [Z6: B] :
! [X5: B] :
( ( ord_less @ B @ Z6 @ X5 )
=> ( ( dvd_dvd @ B @ D3 @ ( plus_plus @ B @ X5 @ S ) )
= ( dvd_dvd @ B @ D3 @ ( plus_plus @ B @ X5 @ S ) ) ) ) ) ).
% pinf(9)
thf(fact_1670_tuple__isomorphism_Osize_I2_J,axiom,
! [B: $tType,C: $tType,A: $tType,X1: A > ( product_prod @ B @ C ),X22: ( product_prod @ B @ C ) > A] :
( ( size_size @ ( tuple_isomorphism @ A @ B @ C ) @ ( tuple_1188178415141063261rphism @ A @ B @ C @ X1 @ X22 ) )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% tuple_isomorphism.size(2)
thf(fact_1671_less__set__def,axiom,
! [A: $tType] :
( ( ord_less @ ( set @ A ) )
= ( ^ [A8: set @ A,B7: set @ A] :
( ord_less @ ( A > $o )
@ ^ [X2: A] : ( member @ A @ X2 @ A8 )
@ ^ [X2: A] : ( member @ A @ X2 @ B7 ) ) ) ) ).
% less_set_def
thf(fact_1672_tuple__isomorphism_Osize__gen,axiom,
! [B: $tType,C: $tType,A: $tType,Xb: A > nat,Xa: B > nat,X: C > nat,X1: A > ( product_prod @ B @ C ),X22: ( product_prod @ B @ C ) > A] :
( ( tuple_9181185373184732606rphism @ A @ B @ C @ Xb @ Xa @ X @ ( tuple_1188178415141063261rphism @ A @ B @ C @ X1 @ X22 ) )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% tuple_isomorphism.size_gen
thf(fact_1673_triangle__Suc,axiom,
! [N: nat] :
( ( nat_triangle @ ( suc @ N ) )
= ( plus_plus @ nat @ ( nat_triangle @ N ) @ ( suc @ N ) ) ) ).
% triangle_Suc
thf(fact_1674_floor__divide__upper,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Q2: A,P4: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Q2 )
=> ( ord_less @ A @ P4 @ ( times_times @ A @ ( plus_plus @ A @ ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ ( divide_divide @ A @ P4 @ Q2 ) ) ) @ ( one_one @ A ) ) @ Q2 ) ) ) ) ).
% floor_divide_upper
thf(fact_1675_verit__le__mono__div__int,axiom,
! [A5: int,B4: int,N: int] :
( ( ord_less @ int @ A5 @ B4 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ N )
=> ( ord_less_eq @ int
@ ( plus_plus @ int @ ( divide_divide @ int @ A5 @ N )
@ ( if @ int
@ ( ( modulo_modulo @ int @ B4 @ N )
= ( zero_zero @ int ) )
@ ( one_one @ int )
@ ( zero_zero @ int ) ) )
@ ( divide_divide @ int @ B4 @ N ) ) ) ) ).
% verit_le_mono_div_int
thf(fact_1676_verit__le__mono__div,axiom,
! [A5: nat,B4: nat,N: nat] :
( ( ord_less @ nat @ A5 @ B4 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less_eq @ nat
@ ( plus_plus @ nat @ ( divide_divide @ nat @ A5 @ N )
@ ( if @ nat
@ ( ( modulo_modulo @ nat @ B4 @ N )
= ( zero_zero @ nat ) )
@ ( one_one @ nat )
@ ( zero_zero @ nat ) ) )
@ ( divide_divide @ nat @ B4 @ N ) ) ) ) ).
% verit_le_mono_div
thf(fact_1677_divide__le__eq__numeral_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,C2: A,W: num] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ C2 ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ B2 @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ C2 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ C2 ) @ B2 ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) ) ) ) ) ) ) ).
% divide_le_eq_numeral(2)
thf(fact_1678_le__divide__eq__numeral_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [W: num,B2: A,C2: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ ( 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 @ W ) ) @ 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 @ W ) ) @ C2 ) ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% le_divide_eq_numeral(2)
thf(fact_1679_floor__add,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Y3: A] :
( ( ( ord_less @ A @ ( plus_plus @ A @ ( archimedean_frac @ A @ X ) @ ( archimedean_frac @ A @ Y3 ) ) @ ( one_one @ A ) )
=> ( ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X @ Y3 ) )
= ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( archim6421214686448440834_floor @ A @ Y3 ) ) ) )
& ( ~ ( ord_less @ A @ ( plus_plus @ A @ ( archimedean_frac @ A @ X ) @ ( archimedean_frac @ A @ Y3 ) ) @ ( one_one @ A ) )
=> ( ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X @ Y3 ) )
= ( plus_plus @ int @ ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( archim6421214686448440834_floor @ A @ Y3 ) ) @ ( one_one @ int ) ) ) ) ) ) ).
% floor_add
thf(fact_1680_int__eq__iff__numeral,axiom,
! [M: nat,V: num] :
( ( ( semiring_1_of_nat @ int @ M )
= ( numeral_numeral @ int @ V ) )
= ( M
= ( numeral_numeral @ nat @ V ) ) ) ).
% int_eq_iff_numeral
thf(fact_1681_nat__numeral,axiom,
! [K: num] :
( ( nat2 @ ( numeral_numeral @ int @ K ) )
= ( numeral_numeral @ nat @ K ) ) ).
% nat_numeral
thf(fact_1682_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_1683_bits__mod__0,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A] :
( ( modulo_modulo @ A @ ( zero_zero @ A ) @ A3 )
= ( zero_zero @ A ) ) ) ).
% bits_mod_0
thf(fact_1684_mod__0,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [A3: A] :
( ( modulo_modulo @ A @ ( zero_zero @ A ) @ A3 )
= ( zero_zero @ A ) ) ) ).
% mod_0
thf(fact_1685_mod__by__0,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [A3: A] :
( ( modulo_modulo @ A @ A3 @ ( zero_zero @ A ) )
= A3 ) ) ).
% mod_by_0
thf(fact_1686_mod__self,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [A3: A] :
( ( modulo_modulo @ A @ A3 @ A3 )
= ( zero_zero @ A ) ) ) ).
% mod_self
thf(fact_1687_mod__add__self2,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A3: A,B2: A] :
( ( modulo_modulo @ A @ ( plus_plus @ A @ A3 @ B2 ) @ B2 )
= ( modulo_modulo @ A @ A3 @ B2 ) ) ) ).
% mod_add_self2
thf(fact_1688_mod__add__self1,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B2: A,A3: A] :
( ( modulo_modulo @ A @ ( plus_plus @ A @ B2 @ A3 ) @ B2 )
= ( modulo_modulo @ A @ A3 @ B2 ) ) ) ).
% mod_add_self1
thf(fact_1689_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_1690_of__int__eq__numeral__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [Z: int,N: num] :
( ( ( ring_1_of_int @ A @ Z )
= ( numeral_numeral @ A @ N ) )
= ( Z
= ( numeral_numeral @ int @ N ) ) ) ) ).
% of_int_eq_numeral_iff
thf(fact_1691_triangle__0,axiom,
( ( nat_triangle @ ( zero_zero @ nat ) )
= ( zero_zero @ nat ) ) ).
% triangle_0
thf(fact_1692_distrib__left__numeral,axiom,
! [A: $tType] :
( ( ( numeral @ A )
& ( semiring @ A ) )
=> ! [V: num,B2: A,C2: A] :
( ( times_times @ A @ ( numeral_numeral @ A @ V ) @ ( plus_plus @ A @ B2 @ C2 ) )
= ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ V ) @ B2 ) @ ( times_times @ A @ ( numeral_numeral @ A @ V ) @ C2 ) ) ) ) ).
% distrib_left_numeral
thf(fact_1693_distrib__right__numeral,axiom,
! [A: $tType] :
( ( ( numeral @ A )
& ( semiring @ A ) )
=> ! [A3: A,B2: A,V: num] :
( ( times_times @ A @ ( plus_plus @ A @ A3 @ B2 ) @ ( numeral_numeral @ A @ V ) )
= ( plus_plus @ A @ ( times_times @ A @ A3 @ ( numeral_numeral @ A @ V ) ) @ ( times_times @ A @ B2 @ ( numeral_numeral @ A @ V ) ) ) ) ) ).
% distrib_right_numeral
thf(fact_1694_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_1695_mod__mult__self2__is__0,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A3: A,B2: A] :
( ( modulo_modulo @ A @ ( times_times @ A @ A3 @ B2 ) @ B2 )
= ( zero_zero @ A ) ) ) ).
% mod_mult_self2_is_0
thf(fact_1696_mod__mult__self1__is__0,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B2: A,A3: A] :
( ( modulo_modulo @ A @ ( times_times @ A @ B2 @ A3 ) @ B2 )
= ( zero_zero @ A ) ) ) ).
% mod_mult_self1_is_0
thf(fact_1697_bits__mod__by__1,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A] :
( ( modulo_modulo @ A @ A3 @ ( one_one @ A ) )
= ( zero_zero @ A ) ) ) ).
% bits_mod_by_1
thf(fact_1698_mod__by__1,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [A3: A] :
( ( modulo_modulo @ A @ A3 @ ( one_one @ A ) )
= ( zero_zero @ A ) ) ) ).
% mod_by_1
thf(fact_1699_mod__mult__self4,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B2: A,C2: A,A3: A] :
( ( modulo_modulo @ A @ ( plus_plus @ A @ ( times_times @ A @ B2 @ C2 ) @ A3 ) @ B2 )
= ( modulo_modulo @ A @ A3 @ B2 ) ) ) ).
% mod_mult_self4
thf(fact_1700_mod__mult__self3,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [C2: A,B2: A,A3: A] :
( ( modulo_modulo @ A @ ( plus_plus @ A @ ( times_times @ A @ C2 @ B2 ) @ A3 ) @ B2 )
= ( modulo_modulo @ A @ A3 @ B2 ) ) ) ).
% mod_mult_self3
thf(fact_1701_mod__mult__self2,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( modulo_modulo @ A @ ( plus_plus @ A @ A3 @ ( times_times @ A @ B2 @ C2 ) ) @ B2 )
= ( modulo_modulo @ A @ A3 @ B2 ) ) ) ).
% mod_mult_self2
thf(fact_1702_mod__mult__self1,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( modulo_modulo @ A @ ( plus_plus @ A @ A3 @ ( times_times @ A @ C2 @ B2 ) ) @ B2 )
= ( modulo_modulo @ A @ A3 @ B2 ) ) ) ).
% mod_mult_self1
thf(fact_1703_dvd__imp__mod__0,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ B2 )
=> ( ( modulo_modulo @ A @ B2 @ A3 )
= ( zero_zero @ A ) ) ) ) ).
% dvd_imp_mod_0
thf(fact_1704_bits__mod__div__trivial,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A,B2: A] :
( ( divide_divide @ A @ ( modulo_modulo @ A @ A3 @ B2 ) @ B2 )
= ( zero_zero @ A ) ) ) ).
% bits_mod_div_trivial
thf(fact_1705_mod__div__trivial,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A3: A,B2: A] :
( ( divide_divide @ A @ ( modulo_modulo @ A @ A3 @ B2 ) @ B2 )
= ( zero_zero @ A ) ) ) ).
% mod_div_trivial
thf(fact_1706_mod__by__Suc__0,axiom,
! [M: nat] :
( ( modulo_modulo @ nat @ M @ ( suc @ ( zero_zero @ nat ) ) )
= ( zero_zero @ nat ) ) ).
% mod_by_Suc_0
thf(fact_1707_floor__zero,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archim6421214686448440834_floor @ A @ ( zero_zero @ A ) )
= ( zero_zero @ int ) ) ) ).
% floor_zero
thf(fact_1708_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_1709_add__numeral__left,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ! [V: num,W: num,Z: A] :
( ( plus_plus @ A @ ( numeral_numeral @ A @ V ) @ ( plus_plus @ A @ ( numeral_numeral @ A @ W ) @ Z ) )
= ( plus_plus @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ V @ W ) ) @ Z ) ) ) ).
% add_numeral_left
thf(fact_1710_nat__neg__numeral,axiom,
! [K: num] :
( ( nat2 @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) )
= ( zero_zero @ nat ) ) ).
% nat_neg_numeral
thf(fact_1711_floor__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archim6421214686448440834_floor @ A @ ( one_one @ A ) )
= ( one_one @ int ) ) ) ).
% floor_one
thf(fact_1712_diff__nat__numeral,axiom,
! [V: num,V4: num] :
( ( minus_minus @ nat @ ( numeral_numeral @ nat @ V ) @ ( numeral_numeral @ nat @ V4 ) )
= ( nat2 @ ( minus_minus @ int @ ( numeral_numeral @ int @ V ) @ ( numeral_numeral @ int @ V4 ) ) ) ) ).
% diff_nat_numeral
thf(fact_1713_nat__eq__numeral__power__cancel__iff,axiom,
! [Y3: int,X: num,N: nat] :
( ( ( nat2 @ Y3 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ X ) @ N ) )
= ( Y3
= ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N ) ) ) ).
% nat_eq_numeral_power_cancel_iff
thf(fact_1714_numeral__power__eq__nat__cancel__iff,axiom,
! [X: num,N: nat,Y3: int] :
( ( ( power_power @ nat @ ( numeral_numeral @ nat @ X ) @ N )
= ( nat2 @ Y3 ) )
= ( ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N )
= Y3 ) ) ).
% numeral_power_eq_nat_cancel_iff
thf(fact_1715_divide__eq__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,W: num,A3: A] :
( ( ( divide_divide @ A @ B2 @ ( numeral_numeral @ A @ W ) )
= A3 )
= ( ( ( ( numeral_numeral @ A @ W )
!= ( zero_zero @ A ) )
=> ( B2
= ( times_times @ A @ A3 @ ( numeral_numeral @ A @ W ) ) ) )
& ( ( ( numeral_numeral @ A @ W )
= ( zero_zero @ A ) )
=> ( A3
= ( zero_zero @ A ) ) ) ) ) ) ).
% divide_eq_eq_numeral1(1)
thf(fact_1716_eq__divide__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,B2: A,W: num] :
( ( A3
= ( divide_divide @ A @ B2 @ ( numeral_numeral @ A @ W ) ) )
= ( ( ( ( numeral_numeral @ A @ W )
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ A3 @ ( numeral_numeral @ A @ W ) )
= B2 ) )
& ( ( ( numeral_numeral @ A @ W )
= ( zero_zero @ A ) )
=> ( A3
= ( zero_zero @ A ) ) ) ) ) ) ).
% eq_divide_eq_numeral1(1)
thf(fact_1717_mod__minus1__right,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A3: A] :
( ( modulo_modulo @ A @ A3 @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( zero_zero @ A ) ) ) ).
% mod_minus1_right
thf(fact_1718_of__int__le__numeral__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z: int,N: num] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z ) @ ( numeral_numeral @ A @ N ) )
= ( ord_less_eq @ int @ Z @ ( numeral_numeral @ int @ N ) ) ) ) ).
% of_int_le_numeral_iff
thf(fact_1719_of__int__numeral__le__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: num,Z: int] :
( ( ord_less_eq @ A @ ( numeral_numeral @ A @ N ) @ ( ring_1_of_int @ A @ Z ) )
= ( ord_less_eq @ int @ ( numeral_numeral @ int @ N ) @ Z ) ) ) ).
% of_int_numeral_le_iff
thf(fact_1720_of__int__numeral__less__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: num,Z: int] :
( ( ord_less @ A @ ( numeral_numeral @ A @ N ) @ ( ring_1_of_int @ A @ Z ) )
= ( ord_less @ int @ ( numeral_numeral @ int @ N ) @ Z ) ) ) ).
% of_int_numeral_less_iff
thf(fact_1721_of__int__less__numeral__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z: int,N: num] :
( ( ord_less @ A @ ( ring_1_of_int @ A @ Z ) @ ( numeral_numeral @ A @ N ) )
= ( ord_less @ int @ Z @ ( numeral_numeral @ int @ N ) ) ) ) ).
% of_int_less_numeral_iff
thf(fact_1722_semiring__norm_I168_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [V: num,W: num,Y3: A] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) @ ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ Y3 ) )
= ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ V @ W ) ) ) @ Y3 ) ) ) ).
% semiring_norm(168)
thf(fact_1723_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_1724_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_1725_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_1726_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_1727_ceiling__add__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,V: num] :
( ( archimedean_ceiling @ A @ ( plus_plus @ A @ X @ ( numeral_numeral @ A @ V ) ) )
= ( plus_plus @ int @ ( archimedean_ceiling @ A @ X ) @ ( numeral_numeral @ int @ V ) ) ) ) ).
% ceiling_add_numeral
thf(fact_1728_numeral__power__eq__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [X: num,N: nat,Y3: int] :
( ( ( power_power @ A @ ( numeral_numeral @ A @ X ) @ N )
= ( ring_1_of_int @ A @ Y3 ) )
= ( ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N )
= Y3 ) ) ) ).
% numeral_power_eq_of_int_cancel_iff
thf(fact_1729_of__int__eq__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [Y3: int,X: num,N: nat] :
( ( ( ring_1_of_int @ A @ Y3 )
= ( power_power @ A @ ( numeral_numeral @ A @ X ) @ N ) )
= ( Y3
= ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N ) ) ) ) ).
% of_int_eq_numeral_power_cancel_iff
thf(fact_1730_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_1731_floor__add2,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Y3: A] :
( ( ( member @ A @ X @ ( ring_1_Ints @ A ) )
| ( member @ A @ Y3 @ ( ring_1_Ints @ A ) ) )
=> ( ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X @ Y3 ) )
= ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( archim6421214686448440834_floor @ A @ Y3 ) ) ) ) ) ).
% floor_add2
thf(fact_1732_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_1733_eq__divide__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,B2: A,W: num] :
( ( A3
= ( divide_divide @ A @ B2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) )
= ( ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) )
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ A3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) )
= B2 ) )
& ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) )
= ( zero_zero @ A ) )
=> ( A3
= ( zero_zero @ A ) ) ) ) ) ) ).
% eq_divide_eq_numeral1(2)
thf(fact_1734_divide__eq__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,W: num,A3: A] :
( ( ( divide_divide @ A @ B2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) )
= A3 )
= ( ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) )
!= ( zero_zero @ A ) )
=> ( B2
= ( times_times @ A @ A3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) ) )
& ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) )
= ( zero_zero @ A ) )
=> ( A3
= ( zero_zero @ A ) ) ) ) ) ) ).
% divide_eq_eq_numeral1(2)
thf(fact_1735_zero__le__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( archim6421214686448440834_floor @ A @ X ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ X ) ) ) ).
% zero_le_floor
thf(fact_1736_floor__less__zero,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( zero_zero @ int ) )
= ( ord_less @ A @ X @ ( zero_zero @ A ) ) ) ) ).
% floor_less_zero
thf(fact_1737_zero__less__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less @ int @ ( zero_zero @ int ) @ ( archim6421214686448440834_floor @ A @ X ) )
= ( ord_less_eq @ A @ ( one_one @ A ) @ X ) ) ) ).
% zero_less_floor
thf(fact_1738_floor__le__zero,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( zero_zero @ int ) )
= ( ord_less @ A @ X @ ( one_one @ A ) ) ) ) ).
% floor_le_zero
thf(fact_1739_one__le__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less_eq @ int @ ( one_one @ int ) @ ( archim6421214686448440834_floor @ A @ X ) )
= ( ord_less_eq @ A @ ( one_one @ A ) @ X ) ) ) ).
% one_le_floor
thf(fact_1740_floor__less__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( one_one @ int ) )
= ( ord_less @ A @ X @ ( one_one @ A ) ) ) ) ).
% floor_less_one
thf(fact_1741_floor__diff__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( archim6421214686448440834_floor @ A @ ( minus_minus @ A @ X @ ( one_one @ A ) ) )
= ( minus_minus @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( one_one @ int ) ) ) ) ).
% floor_diff_one
thf(fact_1742_nat__numeral__diff__1,axiom,
! [V: num] :
( ( minus_minus @ nat @ ( numeral_numeral @ nat @ V ) @ ( one_one @ nat ) )
= ( nat2 @ ( minus_minus @ int @ ( numeral_numeral @ int @ V ) @ ( one_one @ int ) ) ) ) ).
% nat_numeral_diff_1
thf(fact_1743_numeral__power__less__nat__cancel__iff,axiom,
! [X: num,N: nat,A3: int] :
( ( ord_less @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ X ) @ N ) @ ( nat2 @ A3 ) )
= ( ord_less @ int @ ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N ) @ A3 ) ) ).
% numeral_power_less_nat_cancel_iff
thf(fact_1744_nat__less__numeral__power__cancel__iff,axiom,
! [A3: int,X: num,N: nat] :
( ( ord_less @ nat @ ( nat2 @ A3 ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ X ) @ N ) )
= ( ord_less @ int @ A3 @ ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N ) ) ) ).
% nat_less_numeral_power_cancel_iff
thf(fact_1745_nat__le__numeral__power__cancel__iff,axiom,
! [A3: int,X: num,N: nat] :
( ( ord_less_eq @ nat @ ( nat2 @ A3 ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ X ) @ N ) )
= ( ord_less_eq @ int @ A3 @ ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N ) ) ) ).
% nat_le_numeral_power_cancel_iff
thf(fact_1746_numeral__power__le__nat__cancel__iff,axiom,
! [X: num,N: nat,A3: int] :
( ( ord_less_eq @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ X ) @ N ) @ ( nat2 @ A3 ) )
= ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N ) @ A3 ) ) ).
% numeral_power_le_nat_cancel_iff
thf(fact_1747_numeral__less__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V: num,X: A] :
( ( ord_less @ int @ ( numeral_numeral @ int @ V ) @ ( archim6421214686448440834_floor @ A @ X ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ ( numeral_numeral @ A @ V ) @ ( one_one @ A ) ) @ X ) ) ) ).
% numeral_less_floor
thf(fact_1748_floor__le__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,V: num] :
( ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( numeral_numeral @ int @ V ) )
= ( ord_less @ A @ X @ ( plus_plus @ A @ ( numeral_numeral @ A @ V ) @ ( one_one @ A ) ) ) ) ) ).
% floor_le_numeral
thf(fact_1749_ceiling__less__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,V: num] :
( ( ord_less @ int @ ( archimedean_ceiling @ A @ X ) @ ( numeral_numeral @ int @ V ) )
= ( ord_less_eq @ A @ X @ ( minus_minus @ A @ ( numeral_numeral @ A @ V ) @ ( one_one @ A ) ) ) ) ) ).
% ceiling_less_numeral
thf(fact_1750_numeral__le__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V: num,X: A] :
( ( ord_less_eq @ int @ ( numeral_numeral @ int @ V ) @ ( archimedean_ceiling @ A @ X ) )
= ( ord_less @ A @ ( minus_minus @ A @ ( numeral_numeral @ A @ V ) @ ( one_one @ A ) ) @ X ) ) ) ).
% numeral_le_ceiling
thf(fact_1751_of__int__le__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: int,X: num,N: nat] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ A3 ) @ ( power_power @ A @ ( numeral_numeral @ A @ X ) @ N ) )
= ( ord_less_eq @ int @ A3 @ ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N ) ) ) ) ).
% of_int_le_numeral_power_cancel_iff
thf(fact_1752_numeral__power__le__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: num,N: nat,A3: int] :
( ( ord_less_eq @ A @ ( power_power @ A @ ( numeral_numeral @ A @ X ) @ N ) @ ( ring_1_of_int @ A @ A3 ) )
= ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N ) @ A3 ) ) ) ).
% numeral_power_le_of_int_cancel_iff
thf(fact_1753_of__int__less__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: int,X: num,N: nat] :
( ( ord_less @ A @ ( ring_1_of_int @ A @ A3 ) @ ( power_power @ A @ ( numeral_numeral @ A @ X ) @ N ) )
= ( ord_less @ int @ A3 @ ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N ) ) ) ) ).
% of_int_less_numeral_power_cancel_iff
thf(fact_1754_numeral__power__less__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: num,N: nat,A3: int] :
( ( ord_less @ A @ ( power_power @ A @ ( numeral_numeral @ A @ X ) @ N ) @ ( ring_1_of_int @ A @ A3 ) )
= ( ord_less @ int @ ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N ) @ A3 ) ) ) ).
% numeral_power_less_of_int_cancel_iff
thf(fact_1755_of__int__eq__neg__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [Y3: int,X: num,N: nat] :
( ( ( ring_1_of_int @ A @ Y3 )
= ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X ) ) @ N ) )
= ( Y3
= ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X ) ) @ N ) ) ) ) ).
% of_int_eq_neg_numeral_power_cancel_iff
thf(fact_1756_neg__numeral__power__eq__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [X: num,N: nat,Y3: int] :
( ( ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X ) ) @ N )
= ( ring_1_of_int @ A @ Y3 ) )
= ( ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X ) ) @ N )
= Y3 ) ) ) ).
% neg_numeral_power_eq_of_int_cancel_iff
thf(fact_1757_neg__numeral__less__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V: num,X: A] :
( ( ord_less @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V ) ) @ ( archim6421214686448440834_floor @ A @ X ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) @ ( one_one @ A ) ) @ X ) ) ) ).
% neg_numeral_less_floor
thf(fact_1758_floor__le__neg__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,V: num] :
( ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V ) ) )
= ( ord_less @ A @ X @ ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) @ ( one_one @ A ) ) ) ) ) ).
% floor_le_neg_numeral
thf(fact_1759_ceiling__less__neg__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,V: num] :
( ( ord_less @ int @ ( archimedean_ceiling @ A @ X ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V ) ) )
= ( ord_less_eq @ A @ X @ ( minus_minus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) @ ( one_one @ A ) ) ) ) ) ).
% ceiling_less_neg_numeral
thf(fact_1760_neg__numeral__le__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V: num,X: A] :
( ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V ) ) @ ( archimedean_ceiling @ A @ X ) )
= ( ord_less @ A @ ( minus_minus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) @ ( one_one @ A ) ) @ X ) ) ) ).
% neg_numeral_le_ceiling
thf(fact_1761_of__int__le__neg__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: int,X: num,N: nat] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ A3 ) @ ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X ) ) @ N ) )
= ( ord_less_eq @ int @ A3 @ ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X ) ) @ N ) ) ) ) ).
% of_int_le_neg_numeral_power_cancel_iff
thf(fact_1762_neg__numeral__power__le__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: num,N: nat,A3: int] :
( ( ord_less_eq @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X ) ) @ N ) @ ( ring_1_of_int @ A @ A3 ) )
= ( ord_less_eq @ int @ ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X ) ) @ N ) @ A3 ) ) ) ).
% neg_numeral_power_le_of_int_cancel_iff
thf(fact_1763_of__int__less__neg__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: int,X: num,N: nat] :
( ( ord_less @ A @ ( ring_1_of_int @ A @ A3 ) @ ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X ) ) @ N ) )
= ( ord_less @ int @ A3 @ ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X ) ) @ N ) ) ) ) ).
% of_int_less_neg_numeral_power_cancel_iff
thf(fact_1764_neg__numeral__power__less__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: num,N: nat,A3: int] :
( ( ord_less @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X ) ) @ N ) @ ( ring_1_of_int @ A @ A3 ) )
= ( ord_less @ int @ ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X ) ) @ N ) @ A3 ) ) ) ).
% neg_numeral_power_less_of_int_cancel_iff
thf(fact_1765_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_1766_nat__numeral__as__int,axiom,
( ( numeral_numeral @ nat )
= ( ^ [I2: num] : ( nat2 @ ( numeral_numeral @ int @ I2 ) ) ) ) ).
% nat_numeral_as_int
thf(fact_1767_mod__add__right__eq,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( modulo_modulo @ A @ ( plus_plus @ A @ A3 @ ( modulo_modulo @ A @ B2 @ C2 ) ) @ C2 )
= ( modulo_modulo @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% mod_add_right_eq
thf(fact_1768_mod__add__left__eq,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( modulo_modulo @ A @ ( plus_plus @ A @ ( modulo_modulo @ A @ A3 @ C2 ) @ B2 ) @ C2 )
= ( modulo_modulo @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% mod_add_left_eq
thf(fact_1769_mod__add__cong,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A3: A,C2: A,A7: A,B2: A,B6: A] :
( ( ( modulo_modulo @ A @ A3 @ C2 )
= ( modulo_modulo @ A @ A7 @ C2 ) )
=> ( ( ( modulo_modulo @ A @ B2 @ C2 )
= ( modulo_modulo @ A @ B6 @ C2 ) )
=> ( ( modulo_modulo @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 )
= ( modulo_modulo @ A @ ( plus_plus @ A @ A7 @ B6 ) @ C2 ) ) ) ) ) ).
% mod_add_cong
thf(fact_1770_mod__add__eq,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( modulo_modulo @ A @ ( plus_plus @ A @ ( modulo_modulo @ A @ A3 @ C2 ) @ ( modulo_modulo @ A @ B2 @ C2 ) ) @ C2 )
= ( modulo_modulo @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% mod_add_eq
thf(fact_1771_dvd__mod__imp__dvd,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( dvd_dvd @ A @ C2 @ ( modulo_modulo @ A @ A3 @ B2 ) )
=> ( ( dvd_dvd @ A @ C2 @ B2 )
=> ( dvd_dvd @ A @ C2 @ A3 ) ) ) ) ).
% dvd_mod_imp_dvd
thf(fact_1772_dvd__mod__iff,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [C2: A,B2: A,A3: A] :
( ( dvd_dvd @ A @ C2 @ B2 )
=> ( ( dvd_dvd @ A @ C2 @ ( modulo_modulo @ A @ A3 @ B2 ) )
= ( dvd_dvd @ A @ C2 @ A3 ) ) ) ) ).
% dvd_mod_iff
thf(fact_1773_zero__neq__numeral,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [N: num] :
( ( zero_zero @ A )
!= ( numeral_numeral @ A @ N ) ) ) ).
% zero_neq_numeral
thf(fact_1774_nat__mod__as__int,axiom,
( ( modulo_modulo @ nat )
= ( ^ [A6: nat,B5: nat] : ( nat2 @ ( modulo_modulo @ int @ ( semiring_1_of_nat @ int @ A6 ) @ ( semiring_1_of_nat @ int @ B5 ) ) ) ) ) ).
% nat_mod_as_int
thf(fact_1775_Ints__numeral,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [N: num] : ( member @ A @ ( numeral_numeral @ A @ N ) @ ( ring_1_Ints @ A ) ) ) ).
% Ints_numeral
thf(fact_1776_Nats__numeral,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [W: num] : ( member @ A @ ( numeral_numeral @ A @ W ) @ ( semiring_1_Nats @ A ) ) ) ).
% Nats_numeral
thf(fact_1777_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_1778_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ord_less_eq @ A @ ( modulo_modulo @ A @ A3 @ B2 ) @ A3 ) ) ) ).
% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
thf(fact_1779_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [B2: A,A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less @ A @ ( modulo_modulo @ A @ A3 @ B2 ) @ B2 ) ) ) ).
% unique_euclidean_semiring_numeral_class.pos_mod_bound
thf(fact_1780_mod__eqE,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( ( modulo_modulo @ A @ A3 @ C2 )
= ( modulo_modulo @ A @ B2 @ C2 ) )
=> ~ ! [D2: A] :
( B2
!= ( plus_plus @ A @ A3 @ ( times_times @ A @ C2 @ D2 ) ) ) ) ) ).
% mod_eqE
thf(fact_1781_mod__eq__0__iff__dvd,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [A3: A,B2: A] :
( ( ( modulo_modulo @ A @ A3 @ B2 )
= ( zero_zero @ A ) )
= ( dvd_dvd @ A @ B2 @ A3 ) ) ) ).
% mod_eq_0_iff_dvd
thf(fact_1782_dvd__eq__mod__eq__0,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ( ( dvd_dvd @ A )
= ( ^ [A6: A,B5: A] :
( ( modulo_modulo @ A @ B5 @ A6 )
= ( zero_zero @ A ) ) ) ) ) ).
% dvd_eq_mod_eq_0
thf(fact_1783_mod__0__imp__dvd,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [A3: A,B2: A] :
( ( ( modulo_modulo @ A @ A3 @ B2 )
= ( zero_zero @ A ) )
=> ( dvd_dvd @ A @ B2 @ A3 ) ) ) ).
% mod_0_imp_dvd
thf(fact_1784_mod__eq__self__iff__div__eq__0,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [A3: A,B2: A] :
( ( ( modulo_modulo @ A @ A3 @ B2 )
= A3 )
= ( ( divide_divide @ A @ A3 @ B2 )
= ( zero_zero @ A ) ) ) ) ).
% mod_eq_self_iff_div_eq_0
thf(fact_1785_div__add1__eq,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( divide_divide @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 )
= ( plus_plus @ A @ ( plus_plus @ A @ ( divide_divide @ A @ A3 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) @ ( divide_divide @ A @ ( plus_plus @ A @ ( modulo_modulo @ A @ A3 @ C2 ) @ ( modulo_modulo @ A @ B2 @ C2 ) ) @ C2 ) ) ) ) ).
% div_add1_eq
thf(fact_1786_dvd__minus__mod,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [B2: A,A3: A] : ( dvd_dvd @ A @ B2 @ ( minus_minus @ A @ A3 @ ( modulo_modulo @ A @ A3 @ B2 ) ) ) ) ).
% dvd_minus_mod
thf(fact_1787_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_1788_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_1789_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_1790_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_1791_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_1792_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_1793_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_1794_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_1795_mod__eq__0D,axiom,
! [M: nat,D3: nat] :
( ( ( modulo_modulo @ nat @ M @ D3 )
= ( zero_zero @ nat ) )
=> ? [Q3: nat] :
( M
= ( times_times @ nat @ D3 @ Q3 ) ) ) ).
% mod_eq_0D
thf(fact_1796_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_1797_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_1798_one__plus__numeral__commute,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ! [X: num] :
( ( plus_plus @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ X ) )
= ( plus_plus @ A @ ( numeral_numeral @ A @ X ) @ ( one_one @ A ) ) ) ) ).
% one_plus_numeral_commute
thf(fact_1799_nat__mod__eq__iff,axiom,
! [X: nat,N: nat,Y3: nat] :
( ( ( modulo_modulo @ nat @ X @ N )
= ( modulo_modulo @ nat @ Y3 @ N ) )
= ( ? [Q1: nat,Q22: nat] :
( ( plus_plus @ nat @ X @ ( times_times @ nat @ N @ Q1 ) )
= ( plus_plus @ nat @ Y3 @ ( times_times @ nat @ N @ Q22 ) ) ) ) ) ).
% nat_mod_eq_iff
thf(fact_1800_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_1801_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_1802_mod__int__unique,axiom,
! [K: int,L: int,Q2: int,R: int] :
( ( eucl_rel_int @ K @ L @ ( product_Pair @ int @ int @ Q2 @ R ) )
=> ( ( modulo_modulo @ int @ K @ L )
= R ) ) ).
% mod_int_unique
thf(fact_1803_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [B2: A,A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( modulo_modulo @ A @ A3 @ B2 ) ) ) ) ).
% unique_euclidean_semiring_numeral_class.pos_mod_sign
thf(fact_1804_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ A3 @ B2 )
=> ( ( modulo_modulo @ A @ A3 @ B2 )
= A3 ) ) ) ) ).
% unique_euclidean_semiring_numeral_class.mod_less
thf(fact_1805_unit__imp__mod__eq__0,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [B2: A,A3: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( modulo_modulo @ A @ A3 @ B2 )
= ( zero_zero @ A ) ) ) ) ).
% unit_imp_mod_eq_0
thf(fact_1806_cancel__div__mod__rules_I2_J,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ ( times_times @ A @ B2 @ ( divide_divide @ A @ A3 @ B2 ) ) @ ( modulo_modulo @ A @ A3 @ B2 ) ) @ C2 )
= ( plus_plus @ A @ A3 @ C2 ) ) ) ).
% cancel_div_mod_rules(2)
thf(fact_1807_cancel__div__mod__rules_I1_J,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ ( times_times @ A @ ( divide_divide @ A @ A3 @ B2 ) @ B2 ) @ ( modulo_modulo @ A @ A3 @ B2 ) ) @ C2 )
= ( plus_plus @ A @ A3 @ C2 ) ) ) ).
% cancel_div_mod_rules(1)
thf(fact_1808_mod__div__decomp,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [A3: A,B2: A] :
( A3
= ( plus_plus @ A @ ( times_times @ A @ ( divide_divide @ A @ A3 @ B2 ) @ B2 ) @ ( modulo_modulo @ A @ A3 @ B2 ) ) ) ) ).
% mod_div_decomp
thf(fact_1809_div__mult__mod__eq,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [A3: A,B2: A] :
( ( plus_plus @ A @ ( times_times @ A @ ( divide_divide @ A @ A3 @ B2 ) @ B2 ) @ ( modulo_modulo @ A @ A3 @ B2 ) )
= A3 ) ) ).
% div_mult_mod_eq
thf(fact_1810_mod__div__mult__eq,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [A3: A,B2: A] :
( ( plus_plus @ A @ ( modulo_modulo @ A @ A3 @ B2 ) @ ( times_times @ A @ ( divide_divide @ A @ A3 @ B2 ) @ B2 ) )
= A3 ) ) ).
% mod_div_mult_eq
thf(fact_1811_mod__mult__div__eq,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [A3: A,B2: A] :
( ( plus_plus @ A @ ( modulo_modulo @ A @ A3 @ B2 ) @ ( times_times @ A @ B2 @ ( divide_divide @ A @ A3 @ B2 ) ) )
= A3 ) ) ).
% mod_mult_div_eq
thf(fact_1812_mult__div__mod__eq,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [B2: A,A3: A] :
( ( plus_plus @ A @ ( times_times @ A @ B2 @ ( divide_divide @ A @ A3 @ B2 ) ) @ ( modulo_modulo @ A @ A3 @ B2 ) )
= A3 ) ) ).
% mult_div_mod_eq
thf(fact_1813_div__mult1__eq,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( divide_divide @ A @ ( times_times @ A @ A3 @ B2 ) @ C2 )
= ( plus_plus @ A @ ( times_times @ A @ A3 @ ( divide_divide @ A @ B2 @ C2 ) ) @ ( divide_divide @ A @ ( times_times @ A @ A3 @ ( modulo_modulo @ A @ B2 @ C2 ) ) @ C2 ) ) ) ) ).
% div_mult1_eq
thf(fact_1814_minus__div__mult__eq__mod,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [A3: A,B2: A] :
( ( minus_minus @ A @ A3 @ ( times_times @ A @ ( divide_divide @ A @ A3 @ B2 ) @ B2 ) )
= ( modulo_modulo @ A @ A3 @ B2 ) ) ) ).
% minus_div_mult_eq_mod
thf(fact_1815_minus__mod__eq__div__mult,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [A3: A,B2: A] :
( ( minus_minus @ A @ A3 @ ( modulo_modulo @ A @ A3 @ B2 ) )
= ( times_times @ A @ ( divide_divide @ A @ A3 @ B2 ) @ B2 ) ) ) ).
% minus_mod_eq_div_mult
thf(fact_1816_minus__mod__eq__mult__div,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [A3: A,B2: A] :
( ( minus_minus @ A @ A3 @ ( modulo_modulo @ A @ A3 @ B2 ) )
= ( times_times @ A @ B2 @ ( divide_divide @ A @ A3 @ B2 ) ) ) ) ).
% minus_mod_eq_mult_div
thf(fact_1817_minus__mult__div__eq__mod,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [A3: A,B2: A] :
( ( minus_minus @ A @ A3 @ ( times_times @ A @ B2 @ ( divide_divide @ A @ A3 @ B2 ) ) )
= ( modulo_modulo @ A @ A3 @ B2 ) ) ) ).
% minus_mult_div_eq_mod
thf(fact_1818_le__floor__add,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Y3: A] : ( ord_less_eq @ int @ ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( archim6421214686448440834_floor @ A @ Y3 ) ) @ ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X @ Y3 ) ) ) ) ).
% le_floor_add
thf(fact_1819_floor__add__int,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Z: int] :
( ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X ) @ Z )
= ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X @ ( ring_1_of_int @ A @ Z ) ) ) ) ) ).
% floor_add_int
thf(fact_1820_int__add__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z: int,X: A] :
( ( plus_plus @ int @ Z @ ( archim6421214686448440834_floor @ A @ X ) )
= ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ ( ring_1_of_int @ A @ Z ) @ X ) ) ) ) ).
% int_add_floor
thf(fact_1821_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_1822_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_1823_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_1824_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_1825_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_1826_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_1827_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_1828_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_1829_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_1830_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_1831_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_1832_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_1833_divide__eq__eq__numeral_I1_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,C2: A,W: num] :
( ( ( divide_divide @ A @ B2 @ C2 )
= ( numeral_numeral @ A @ W ) )
= ( ( ( C2
!= ( zero_zero @ A ) )
=> ( B2
= ( times_times @ A @ ( numeral_numeral @ A @ W ) @ C2 ) ) )
& ( ( C2
= ( zero_zero @ A ) )
=> ( ( numeral_numeral @ A @ W )
= ( zero_zero @ A ) ) ) ) ) ) ).
% divide_eq_eq_numeral(1)
thf(fact_1834_eq__divide__eq__numeral_I1_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [W: num,B2: A,C2: A] :
( ( ( numeral_numeral @ A @ W )
= ( divide_divide @ A @ B2 @ C2 ) )
= ( ( ( C2
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ ( numeral_numeral @ A @ W ) @ C2 )
= B2 ) )
& ( ( C2
= ( zero_zero @ A ) )
=> ( ( numeral_numeral @ A @ W )
= ( zero_zero @ A ) ) ) ) ) ) ).
% eq_divide_eq_numeral(1)
thf(fact_1835_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_1836_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_1837_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_1838_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_1839_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_1840_nat__mod__eq__lemma,axiom,
! [X: nat,N: nat,Y3: nat] :
( ( ( modulo_modulo @ nat @ X @ N )
= ( modulo_modulo @ nat @ Y3 @ N ) )
=> ( ( ord_less_eq @ nat @ Y3 @ X )
=> ? [Q3: nat] :
( X
= ( plus_plus @ nat @ Y3 @ ( times_times @ nat @ N @ Q3 ) ) ) ) ) ).
% nat_mod_eq_lemma
thf(fact_1841_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 )
=> ~ ! [S5: nat] :
( N
!= ( plus_plus @ nat @ M @ ( times_times @ nat @ Q2 @ S5 ) ) ) ) ) ).
% mod_eq_nat2E
thf(fact_1842_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 )
=> ~ ! [S5: nat] :
( M
!= ( plus_plus @ nat @ N @ ( times_times @ nat @ Q2 @ S5 ) ) ) ) ) ).
% mod_eq_nat1E
thf(fact_1843_div__less__mono,axiom,
! [A5: nat,B4: nat,N: nat] :
( ( ord_less @ nat @ A5 @ B4 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ( modulo_modulo @ nat @ A5 @ N )
= ( zero_zero @ nat ) )
=> ( ( ( modulo_modulo @ nat @ B4 @ N )
= ( zero_zero @ nat ) )
=> ( ord_less @ nat @ ( divide_divide @ nat @ A5 @ N ) @ ( divide_divide @ nat @ B4 @ N ) ) ) ) ) ) ).
% div_less_mono
thf(fact_1844_zdiv__mono__strict,axiom,
! [A5: int,B4: int,N: int] :
( ( ord_less @ int @ A5 @ B4 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ N )
=> ( ( ( modulo_modulo @ int @ A5 @ N )
= ( zero_zero @ int ) )
=> ( ( ( modulo_modulo @ int @ B4 @ N )
= ( zero_zero @ int ) )
=> ( ord_less @ int @ ( divide_divide @ int @ A5 @ N ) @ ( divide_divide @ int @ B4 @ N ) ) ) ) ) ) ).
% zdiv_mono_strict
thf(fact_1845_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_1846_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_1847_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_1848_eucl__rel__int,axiom,
! [K: int,L: int] : ( eucl_rel_int @ K @ L @ ( product_Pair @ int @ int @ ( divide_divide @ int @ K @ L ) @ ( modulo_modulo @ int @ K @ L ) ) ) ).
% eucl_rel_int
thf(fact_1849_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_1850_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_1851_mod__mult2__eq_H,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A3: A,M: nat,N: nat] :
( ( modulo_modulo @ A @ A3 @ ( 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 @ A3 @ ( semiring_1_of_nat @ A @ M ) ) @ ( semiring_1_of_nat @ A @ N ) ) ) @ ( modulo_modulo @ A @ A3 @ ( semiring_1_of_nat @ A @ M ) ) ) ) ) ).
% mod_mult2_eq'
thf(fact_1852_one__add__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( one_one @ int ) )
= ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X @ ( one_one @ A ) ) ) ) ) ).
% one_add_floor
thf(fact_1853_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_1854_less__divide__eq__numeral_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [W: num,B2: A,C2: A] :
( ( ord_less @ A @ ( numeral_numeral @ A @ W ) @ ( divide_divide @ A @ B2 @ C2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ ( numeral_numeral @ A @ W ) @ 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 @ W ) @ C2 ) ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( numeral_numeral @ A @ W ) @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% less_divide_eq_numeral(1)
thf(fact_1855_divide__less__eq__numeral_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,C2: A,W: num] :
( ( ord_less @ A @ ( divide_divide @ A @ B2 @ C2 ) @ ( numeral_numeral @ A @ W ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ B2 @ ( times_times @ A @ ( numeral_numeral @ A @ W ) @ C2 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ ( numeral_numeral @ A @ W ) @ C2 ) @ B2 ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ W ) ) ) ) ) ) ) ) ).
% divide_less_eq_numeral(1)
thf(fact_1856_eq__divide__eq__numeral_I2_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [W: num,B2: A,C2: A] :
( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) )
= ( divide_divide @ A @ B2 @ C2 ) )
= ( ( ( C2
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ C2 )
= B2 ) )
& ( ( C2
= ( zero_zero @ A ) )
=> ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) )
= ( zero_zero @ A ) ) ) ) ) ) ).
% eq_divide_eq_numeral(2)
thf(fact_1857_divide__eq__eq__numeral_I2_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,C2: A,W: num] :
( ( ( divide_divide @ A @ B2 @ C2 )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) )
= ( ( ( C2
!= ( zero_zero @ A ) )
=> ( B2
= ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ C2 ) ) )
& ( ( C2
= ( zero_zero @ A ) )
=> ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) )
= ( zero_zero @ A ) ) ) ) ) ) ).
% divide_eq_eq_numeral(2)
thf(fact_1858_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_1859_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_1860_floor__unique,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z: int,X: A] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z ) @ X )
=> ( ( ord_less @ A @ X @ ( plus_plus @ A @ ( ring_1_of_int @ A @ Z ) @ ( one_one @ A ) ) )
=> ( ( archim6421214686448440834_floor @ A @ X )
= Z ) ) ) ) ).
% floor_unique
thf(fact_1861_floor__eq__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,A3: int] :
( ( ( archim6421214686448440834_floor @ A @ X )
= A3 )
= ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ A3 ) @ X )
& ( ord_less @ A @ X @ ( plus_plus @ A @ ( ring_1_of_int @ A @ A3 ) @ ( one_one @ A ) ) ) ) ) ) ).
% floor_eq_iff
thf(fact_1862_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_1863_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( modulo_modulo @ A @ A3 @ ( times_times @ A @ B2 @ C2 ) )
= ( plus_plus @ A @ ( times_times @ A @ B2 @ ( modulo_modulo @ A @ ( divide_divide @ A @ A3 @ B2 ) @ C2 ) ) @ ( modulo_modulo @ A @ A3 @ B2 ) ) ) ) ) ).
% unique_euclidean_semiring_numeral_class.mod_mult2_eq
thf(fact_1864_le__mult__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less_eq @ int @ ( times_times @ int @ ( archim6421214686448440834_floor @ A @ A3 ) @ ( archim6421214686448440834_floor @ A @ B2 ) ) @ ( archim6421214686448440834_floor @ A @ ( times_times @ A @ A3 @ B2 ) ) ) ) ) ) ).
% le_mult_floor
thf(fact_1865_less__floor__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z: int,X: A] :
( ( ord_less @ int @ Z @ ( archim6421214686448440834_floor @ A @ X ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ ( ring_1_of_int @ A @ Z ) @ ( one_one @ A ) ) @ X ) ) ) ).
% less_floor_iff
thf(fact_1866_floor__le__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Z: int] :
( ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X ) @ Z )
= ( ord_less @ A @ X @ ( plus_plus @ A @ ( ring_1_of_int @ A @ Z ) @ ( one_one @ A ) ) ) ) ) ).
% floor_le_iff
thf(fact_1867_le__divide__eq__numeral_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [W: num,B2: A,C2: A] :
( ( ord_less_eq @ A @ ( numeral_numeral @ A @ W ) @ ( divide_divide @ A @ B2 @ C2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ ( numeral_numeral @ A @ W ) @ 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 @ W ) @ C2 ) ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( numeral_numeral @ A @ W ) @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% le_divide_eq_numeral(1)
thf(fact_1868_divide__le__eq__numeral_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,C2: A,W: num] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ C2 ) @ ( numeral_numeral @ A @ W ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ B2 @ ( times_times @ A @ ( numeral_numeral @ A @ W ) @ C2 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ ( numeral_numeral @ A @ W ) @ C2 ) @ B2 ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ W ) ) ) ) ) ) ) ) ).
% divide_le_eq_numeral(1)
thf(fact_1869_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_1870_less__divide__eq__numeral_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [W: num,B2: A,C2: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ ( divide_divide @ A @ B2 @ C2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ 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 @ W ) ) @ C2 ) ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% less_divide_eq_numeral(2)
thf(fact_1871_divide__less__eq__numeral_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,C2: A,W: num] :
( ( ord_less @ A @ ( divide_divide @ A @ B2 @ C2 ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ B2 @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ C2 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ C2 ) @ B2 ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) ) ) ) ) ) ) ).
% divide_less_eq_numeral(2)
thf(fact_1872_floor__divide__lower,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Q2: A,P4: 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 @ P4 @ Q2 ) ) ) @ Q2 ) @ P4 ) ) ) ).
% floor_divide_lower
thf(fact_1873_le__mult__floor__Ints,axiom,
! [A: $tType,B: $tType] :
( ( ( archim2362893244070406136eiling @ B )
& ( linordered_idom @ A ) )
=> ! [A3: B,B2: B] :
( ( ord_less_eq @ B @ ( zero_zero @ B ) @ A3 )
=> ( ( member @ B @ A3 @ ( ring_1_Ints @ B ) )
=> ( ord_less_eq @ A @ ( ring_1_of_int @ A @ ( times_times @ int @ ( archim6421214686448440834_floor @ B @ A3 ) @ ( archim6421214686448440834_floor @ B @ B2 ) ) ) @ ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ B @ ( times_times @ B @ A3 @ B2 ) ) ) ) ) ) ) ).
% le_mult_floor_Ints
thf(fact_1874_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_1875_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_1876_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_1877_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_1878_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_1879_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_1880_prod__decode__aux_Osimps,axiom,
( nat_prod_decode_aux
= ( ^ [K3: nat,M5: nat] : ( if @ ( product_prod @ nat @ nat ) @ ( ord_less_eq @ nat @ M5 @ K3 ) @ ( product_Pair @ nat @ nat @ M5 @ ( minus_minus @ nat @ K3 @ M5 ) ) @ ( nat_prod_decode_aux @ ( suc @ K3 ) @ ( minus_minus @ nat @ M5 @ ( suc @ K3 ) ) ) ) ) ) ).
% prod_decode_aux.simps
thf(fact_1881_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_1882_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_1883_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_1884_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_1885_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_1886_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_1887_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_1888_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_1889_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_1890_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_1891_prod__decode__aux_Ocases,axiom,
! [X: product_prod @ nat @ nat] :
~ ! [K2: nat,M4: nat] :
( X
!= ( product_Pair @ nat @ nat @ K2 @ M4 ) ) ).
% prod_decode_aux.cases
thf(fact_1892_num_Osize_I4_J,axiom,
( ( size_size @ num @ one2 )
= ( zero_zero @ nat ) ) ).
% num.size(4)
thf(fact_1893_numerals_I1_J,axiom,
( ( numeral_numeral @ nat @ one2 )
= ( one_one @ nat ) ) ).
% numerals(1)
thf(fact_1894_numeral__One,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ( ( numeral_numeral @ A @ one2 )
= ( one_one @ A ) ) ) ).
% numeral_One
thf(fact_1895_numeral__1__eq__Suc__0,axiom,
( ( numeral_numeral @ nat @ one2 )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% numeral_1_eq_Suc_0
thf(fact_1896_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_1897_num__of__nat_Osimps_I1_J,axiom,
( ( num_of_nat @ ( zero_zero @ nat ) )
= one2 ) ).
% num_of_nat.simps(1)
thf(fact_1898_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_1899_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_1900_Suc__nat__number__of__add,axiom,
! [V: num,N: nat] :
( ( suc @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ V ) @ N ) )
= ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( plus_plus @ num @ V @ one2 ) ) @ N ) ) ).
% Suc_nat_number_of_add
thf(fact_1901_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_1902_numeral__inc,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ! [X: num] :
( ( numeral_numeral @ A @ ( inc @ X ) )
= ( plus_plus @ A @ ( numeral_numeral @ A @ X ) @ ( one_one @ A ) ) ) ) ).
% numeral_inc
thf(fact_1903_prod__decode__aux_Oelims,axiom,
! [X: nat,Xa: nat,Y3: product_prod @ nat @ nat] :
( ( ( nat_prod_decode_aux @ X @ Xa )
= Y3 )
=> ( ( ( ord_less_eq @ nat @ Xa @ X )
=> ( Y3
= ( product_Pair @ nat @ nat @ Xa @ ( minus_minus @ nat @ X @ Xa ) ) ) )
& ( ~ ( ord_less_eq @ nat @ Xa @ X )
=> ( Y3
= ( nat_prod_decode_aux @ ( suc @ X ) @ ( minus_minus @ nat @ Xa @ ( suc @ X ) ) ) ) ) ) ) ).
% prod_decode_aux.elims
thf(fact_1904_pochhammer__double,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [Z: A,N: nat] :
( ( comm_s3205402744901411588hammer @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Z ) @ ( 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 @ Z @ N ) ) @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ Z @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ N ) ) ) ) ).
% pochhammer_double
thf(fact_1905_even__succ__mod__exp,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A,N: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( modulo_modulo @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A3 ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
= ( plus_plus @ A @ ( one_one @ A ) @ ( modulo_modulo @ A @ A3 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ) ) ).
% even_succ_mod_exp
thf(fact_1906_even__succ__div__exp,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A,N: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A3 ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
= ( divide_divide @ A @ A3 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ) ).
% even_succ_div_exp
thf(fact_1907_neg__eucl__rel__int__mult__2,axiom,
! [B2: int,A3: int,Q2: int,R: int] :
( ( ord_less_eq @ int @ B2 @ ( zero_zero @ int ) )
=> ( ( eucl_rel_int @ ( plus_plus @ int @ A3 @ ( 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 ) ) @ A3 ) ) @ ( 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_1908_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_1909_even__mult__exp__div__exp__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A,M: nat,N: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ ( times_times @ A @ A3 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ 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 @ A3 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N @ M ) ) ) ) ) ) ) ) ).
% even_mult_exp_div_exp_iff
thf(fact_1910_divmod__digit__1_I1_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ ( modulo_modulo @ A @ A3 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) )
=> ( ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A3 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) ) @ ( one_one @ A ) )
= ( divide_divide @ A @ A3 @ B2 ) ) ) ) ) ) ).
% divmod_digit_1(1)
thf(fact_1911_verit__eq__simplify_I8_J,axiom,
! [X22: num,Y2: num] :
( ( ( bit0 @ X22 )
= ( bit0 @ Y2 ) )
= ( X22 = Y2 ) ) ).
% verit_eq_simplify(8)
thf(fact_1912_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_1913_odd__add,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A3: A,B2: A] :
( ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ A @ A3 @ B2 ) ) )
= ( ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) )
!= ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) ) ) ) ).
% odd_add
thf(fact_1914_even__add,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ A @ A3 @ B2 ) )
= ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
= ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) ) ) ).
% even_add
thf(fact_1915_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_1916_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_1917_zero__eq__power2,axiom,
! [A: $tType] :
( ( semiri2026040879449505780visors @ A )
=> ! [A3: A] :
( ( ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( zero_zero @ A ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% zero_eq_power2
thf(fact_1918_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_1919_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_1920_Suc__1,axiom,
( ( suc @ ( one_one @ nat ) )
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ).
% Suc_1
thf(fact_1921_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_1922_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_1923_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_1924_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_1925_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_1926_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_1927_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_1928_even__plus__one__iff,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A3: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) )
= ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) ) ) ).
% even_plus_one_iff
thf(fact_1929_even__diff,axiom,
! [A: $tType] :
( ( ring_parity @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( minus_minus @ A @ A3 @ B2 ) )
= ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ A @ A3 @ B2 ) ) ) ) ).
% even_diff
thf(fact_1930_not__mod__2__eq__1__eq__0,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A3: A] :
( ( ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
!= ( one_one @ A ) )
= ( ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( zero_zero @ A ) ) ) ) ).
% not_mod_2_eq_1_eq_0
thf(fact_1931_not__mod__2__eq__0__eq__1,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A3: A] :
( ( ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
!= ( zero_zero @ A ) )
= ( ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( one_one @ A ) ) ) ) ).
% not_mod_2_eq_0_eq_1
thf(fact_1932_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_1933_power2__less__eq__zero__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( zero_zero @ A ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% power2_less_eq_zero_iff
thf(fact_1934_power2__eq__iff__nonneg,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X: A,Y3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y3 )
=> ( ( ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( power_power @ A @ Y3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( X = Y3 ) ) ) ) ) ).
% power2_eq_iff_nonneg
thf(fact_1935_zero__less__power2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( A3
!= ( zero_zero @ A ) ) ) ) ).
% zero_less_power2
thf(fact_1936_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_1937_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_1938_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_1939_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_1940_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_1941_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_1942_even__succ__div__2,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A3 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% even_succ_div_2
thf(fact_1943_odd__succ__div__two,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A3: A] :
( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( plus_plus @ A @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) ) ) ) ) ).
% odd_succ_div_two
thf(fact_1944_even__succ__div__two,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A3: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% even_succ_div_two
thf(fact_1945_even__power,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A3: A,N: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( power_power @ A @ A3 @ N ) )
= ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% even_power
thf(fact_1946_zero__le__power__eq__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,W: num] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ W ) ) )
= ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W ) )
| ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W ) )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ) ) ).
% zero_le_power_eq_numeral
thf(fact_1947_power__less__zero__eq__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,W: num] :
( ( ord_less @ A @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ W ) ) @ ( zero_zero @ A ) )
= ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W ) )
& ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ) ).
% power_less_zero_eq_numeral
thf(fact_1948_power__less__zero__eq,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,N: nat] :
( ( ord_less @ A @ ( power_power @ A @ A3 @ N ) @ ( zero_zero @ A ) )
= ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
& ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ) ).
% power_less_zero_eq
thf(fact_1949_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_1950_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_1951_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_1952_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_1953_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_1954_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_1955_odd__two__times__div__two__succ,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A3: A] :
( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( one_one @ A ) )
= A3 ) ) ) ).
% odd_two_times_div_two_succ
thf(fact_1956_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_1957_zero__less__power__eq__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,W: num] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ W ) ) )
= ( ( ( numeral_numeral @ nat @ W )
= ( zero_zero @ nat ) )
| ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W ) )
& ( A3
!= ( zero_zero @ A ) ) )
| ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W ) )
& ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ) ) ).
% zero_less_power_eq_numeral
thf(fact_1958_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_1959_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_1960_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_1961_power__le__zero__eq__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,W: num] :
( ( ord_less_eq @ A @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ W ) ) @ ( zero_zero @ A ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( numeral_numeral @ nat @ W ) )
& ( ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W ) )
& ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) )
| ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W ) )
& ( A3
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% power_le_zero_eq_numeral
thf(fact_1962_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_1963_verit__eq__simplify_I10_J,axiom,
! [X22: num] :
( one2
!= ( bit0 @ X22 ) ) ).
% verit_eq_simplify(10)
thf(fact_1964_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_1965_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_1966_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_1967_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_1968_power__numeral__even,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [Z: A,W: num] :
( ( power_power @ A @ Z @ ( numeral_numeral @ nat @ ( bit0 @ W ) ) )
= ( times_times @ A @ ( power_power @ A @ Z @ ( numeral_numeral @ nat @ W ) ) @ ( power_power @ A @ Z @ ( numeral_numeral @ nat @ W ) ) ) ) ) ).
% power_numeral_even
thf(fact_1969_left__add__twice,axiom,
! [A: $tType] :
( ( semiring_numeral @ A )
=> ! [A3: A,B2: A] :
( ( plus_plus @ A @ A3 @ ( plus_plus @ A @ A3 @ B2 ) )
= ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) @ B2 ) ) ) ).
% left_add_twice
thf(fact_1970_mult__2__right,axiom,
! [A: $tType] :
( ( semiring_numeral @ A )
=> ! [Z: A] :
( ( times_times @ A @ Z @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( plus_plus @ A @ Z @ Z ) ) ) ).
% mult_2_right
thf(fact_1971_mult__2,axiom,
! [A: $tType] :
( ( semiring_numeral @ A )
=> ! [Z: A] :
( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Z )
= ( plus_plus @ A @ Z @ Z ) ) ) ).
% mult_2
thf(fact_1972_even__zero,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( zero_zero @ A ) ) ) ).
% even_zero
thf(fact_1973_odd__even__add,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A3: A,B2: A] :
( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 )
=> ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ A @ A3 @ B2 ) ) ) ) ) ).
% odd_even_add
thf(fact_1974_odd__one,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( one_one @ A ) ) ) ).
% odd_one
thf(fact_1975_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_1976_numeral__2__eq__2,axiom,
( ( numeral_numeral @ nat @ ( bit0 @ one2 ) )
= ( suc @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% numeral_2_eq_2
thf(fact_1977_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_1978_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_1979_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_1980_half__gt__zero,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% half_gt_zero
thf(fact_1981_half__gt__zero__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% half_gt_zero_iff
thf(fact_1982_zero__le__power2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% zero_le_power2
thf(fact_1983_power2__eq__imp__eq,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X: A,Y3: A] :
( ( ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( power_power @ A @ Y3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y3 )
=> ( X = Y3 ) ) ) ) ) ).
% power2_eq_imp_eq
thf(fact_1984_power2__le__imp__le,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X: A,Y3: A] :
( ( ord_less_eq @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y3 )
=> ( ord_less_eq @ A @ X @ Y3 ) ) ) ) ).
% power2_le_imp_le
thf(fact_1985_even__iff__mod__2__eq__zero,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A3: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
= ( ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( zero_zero @ A ) ) ) ) ).
% even_iff_mod_2_eq_zero
thf(fact_1986_power2__less__0,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] :
~ ( ord_less @ A @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( zero_zero @ A ) ) ) ).
% power2_less_0
thf(fact_1987_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_1988_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_1989_odd__iff__mod__2__eq__one,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A3: A] :
( ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) )
= ( ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( one_one @ A ) ) ) ) ).
% odd_iff_mod_2_eq_one
thf(fact_1990_sum__power2__eq__zero__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y3: A] :
( ( ( plus_plus @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( zero_zero @ A ) )
= ( ( X
= ( zero_zero @ A ) )
& ( Y3
= ( zero_zero @ A ) ) ) ) ) ).
% sum_power2_eq_zero_iff
thf(fact_1991_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_1992_div__exp__eq,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A,M: nat,N: nat] :
( ( divide_divide @ A @ ( divide_divide @ A @ A3 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
= ( divide_divide @ A @ A3 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ M @ N ) ) ) ) ) ).
% div_exp_eq
thf(fact_1993_power2__eq__1__iff,axiom,
! [A: $tType] :
( ( ring_15535105094025558882visors @ A )
=> ! [A3: A] :
( ( ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( one_one @ A ) )
= ( ( A3
= ( one_one @ A ) )
| ( A3
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ) ).
% power2_eq_1_iff
thf(fact_1994_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_1995_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_1996_abs__square__eq__1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A] :
( ( ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( one_one @ A ) )
= ( ( abs_abs @ A @ X )
= ( one_one @ A ) ) ) ) ).
% abs_square_eq_1
thf(fact_1997_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_1998_nat__2,axiom,
( ( nat2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) )
= ( suc @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% nat_2
thf(fact_1999_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_2000_divmod__digit__0_I2_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [B2: A,A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less @ A @ ( modulo_modulo @ A @ A3 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) @ B2 )
=> ( ( modulo_modulo @ A @ A3 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) )
= ( modulo_modulo @ A @ A3 @ B2 ) ) ) ) ) ).
% divmod_digit_0(2)
thf(fact_2001_oddE,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A3: A] :
( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ~ ! [B3: A] :
( A3
!= ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B3 ) @ ( one_one @ A ) ) ) ) ) ).
% oddE
thf(fact_2002_power2__less__imp__less,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X: A,Y3: A] :
( ( ord_less @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y3 )
=> ( ord_less @ A @ X @ Y3 ) ) ) ) ).
% power2_less_imp_less
thf(fact_2003_sum__power2__ge__zero,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y3: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% sum_power2_ge_zero
thf(fact_2004_sum__power2__le__zero__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y3: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( zero_zero @ A ) )
= ( ( X
= ( zero_zero @ A ) )
& ( Y3
= ( zero_zero @ A ) ) ) ) ) ).
% sum_power2_le_zero_iff
thf(fact_2005_parity__cases,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A3: A] :
( ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
!= ( zero_zero @ A ) ) )
=> ~ ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
!= ( one_one @ A ) ) ) ) ) ).
% parity_cases
thf(fact_2006_mod2__eq__if,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A3: A] :
( ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( zero_zero @ A ) ) )
& ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( one_one @ A ) ) ) ) ) ).
% mod2_eq_if
thf(fact_2007_bits__stable__imp__add__self,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A] :
( ( ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= A3 )
=> ( ( plus_plus @ A @ A3 @ ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
= ( zero_zero @ A ) ) ) ) ).
% bits_stable_imp_add_self
thf(fact_2008_not__sum__power2__lt__zero,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y3: A] :
~ ( ord_less @ A @ ( plus_plus @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( zero_zero @ A ) ) ) ).
% not_sum_power2_lt_zero
thf(fact_2009_sum__power2__gt__zero__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
= ( ( X
!= ( zero_zero @ A ) )
| ( Y3
!= ( zero_zero @ A ) ) ) ) ) ).
% sum_power2_gt_zero_iff
thf(fact_2010_power2__sum,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [X: A,Y3: A] :
( ( power_power @ A @ ( plus_plus @ A @ X @ Y3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( plus_plus @ A @ ( plus_plus @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( times_times @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X ) @ Y3 ) ) ) ) ).
% power2_sum
thf(fact_2011_square__le__1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ X )
=> ( ( ord_less_eq @ A @ X @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) ) ) ) ) ).
% square_le_1
thf(fact_2012_zero__le__even__power_H,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,N: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A3 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% zero_le_even_power'
thf(fact_2013_power2__le__iff__abs__le,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Y3: A,X: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y3 )
=> ( ( ord_less_eq @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( ord_less_eq @ A @ ( abs_abs @ A @ X ) @ Y3 ) ) ) ) ).
% power2_le_iff_abs_le
thf(fact_2014_zero__le__even__power,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: nat,A3: A] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A3 @ N ) ) ) ) ).
% zero_le_even_power
thf(fact_2015_zero__le__odd__power,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: nat,A3: A] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A3 @ N ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ) ).
% zero_le_odd_power
thf(fact_2016_zero__le__power__eq,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,N: nat] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A3 @ 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 ) @ A3 ) ) ) ) ) ).
% zero_le_power_eq
thf(fact_2017_abs__square__le__1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A] :
( ( ord_less_eq @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) )
= ( ord_less_eq @ A @ ( abs_abs @ A @ X ) @ ( one_one @ A ) ) ) ) ).
% abs_square_le_1
thf(fact_2018_abs__square__less__1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A] :
( ( ord_less @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) )
= ( ord_less @ A @ ( abs_abs @ A @ X ) @ ( one_one @ A ) ) ) ) ).
% abs_square_less_1
thf(fact_2019_bits__induct,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [P: A > $o,A3: A] :
( ! [A4: A] :
( ( ( divide_divide @ A @ A4 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= A4 )
=> ( P @ A4 ) )
=> ( ! [A4: A,B3: $o] :
( ( P @ A4 )
=> ( ( ( divide_divide @ A @ ( plus_plus @ A @ ( zero_neq_one_of_bool @ A @ B3 ) @ ( 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 @ B3 ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A4 ) ) ) ) )
=> ( P @ A3 ) ) ) ) ).
% bits_induct
thf(fact_2020_div__exp__mod__exp__eq,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A,N: nat,M: nat] :
( ( modulo_modulo @ A @ ( divide_divide @ A @ A3 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) )
= ( divide_divide @ A @ ( modulo_modulo @ A @ A3 @ ( 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_2021_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_2022_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_2023_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_2024_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_2025_fact__code,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiring_char_0_fact @ A )
= ( ^ [N5: nat] : ( semiring_1_of_nat @ A @ ( set_fo6178422350223883121st_nat @ nat @ ( times_times @ nat ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N5 @ ( one_one @ nat ) ) ) ) ) ) ).
% fact_code
thf(fact_2026_divmod__digit__0_I1_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [B2: A,A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less @ A @ ( modulo_modulo @ A @ A3 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) @ B2 )
=> ( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A3 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) )
= ( divide_divide @ A @ A3 @ B2 ) ) ) ) ) ).
% divmod_digit_0(1)
thf(fact_2027_power2__diff,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [X: A,Y3: A] :
( ( power_power @ A @ ( minus_minus @ A @ X @ Y3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( minus_minus @ A @ ( plus_plus @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( times_times @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X ) @ Y3 ) ) ) ) ).
% power2_diff
thf(fact_2028_odd__0__le__power__imp__0__le,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,N: nat] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A3 @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% odd_0_le_power_imp_0_le
thf(fact_2029_odd__power__less__zero,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,N: nat] :
( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( power_power @ A @ A3 @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) @ ( zero_zero @ A ) ) ) ) ).
% odd_power_less_zero
thf(fact_2030_zero__less__power__eq,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A3 @ N ) )
= ( ( N
= ( zero_zero @ nat ) )
| ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
& ( A3
!= ( zero_zero @ A ) ) )
| ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
& ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ) ) ).
% zero_less_power_eq
thf(fact_2031_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_2032_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_2033_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_2034_mod__double__modulus,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M: A,X: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ M )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ( modulo_modulo @ A @ X @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) )
= ( modulo_modulo @ A @ X @ M ) )
| ( ( modulo_modulo @ A @ X @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) )
= ( plus_plus @ A @ ( modulo_modulo @ A @ X @ M ) @ M ) ) ) ) ) ) ).
% mod_double_modulus
thf(fact_2035_divmod__digit__1_I2_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ ( modulo_modulo @ A @ A3 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) )
=> ( ( minus_minus @ A @ ( modulo_modulo @ A @ A3 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) @ B2 )
= ( modulo_modulo @ A @ A3 @ B2 ) ) ) ) ) ) ).
% divmod_digit_1(2)
thf(fact_2036_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_2037_power__le__zero__eq,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,N: nat] :
( ( ord_less_eq @ A @ ( power_power @ A @ A3 @ N ) @ ( zero_zero @ A ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
& ( ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
& ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) )
| ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
& ( A3
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% power_le_zero_eq
thf(fact_2038_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_2039_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_2040_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_2041_pos__eucl__rel__int__mult__2,axiom,
! [B2: int,A3: int,Q2: int,R: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ B2 )
=> ( ( eucl_rel_int @ A3 @ B2 @ ( product_Pair @ int @ int @ Q2 @ R ) )
=> ( eucl_rel_int @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ A3 ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ B2 ) @ ( product_Pair @ int @ int @ Q2 @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ R ) ) ) ) ) ) ).
% pos_eucl_rel_int_mult_2
thf(fact_2042_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_2043_flip__bit__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A] :
( ( bit_se8732182000553998342ip_bit @ A @ ( zero_zero @ nat ) @ A3 )
= ( plus_plus @ A @ ( zero_neq_one_of_bool @ A @ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% flip_bit_0
thf(fact_2044_set__bit__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A] :
( ( bit_se5668285175392031749et_bit @ A @ ( zero_zero @ nat ) @ A3 )
= ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% set_bit_0
thf(fact_2045_unset__bit__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A] :
( ( bit_se2638667681897837118et_bit @ A @ ( zero_zero @ nat ) @ A3 )
= ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% unset_bit_0
thf(fact_2046_set__bit__Suc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A3: A] :
( ( bit_se5668285175392031749et_bit @ A @ ( suc @ N ) @ A3 )
= ( plus_plus @ A @ ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5668285175392031749et_bit @ A @ N @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% set_bit_Suc
thf(fact_2047_flip__bit__Suc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A3: A] :
( ( bit_se8732182000553998342ip_bit @ A @ ( suc @ N ) @ A3 )
= ( plus_plus @ A @ ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se8732182000553998342ip_bit @ A @ N @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% flip_bit_Suc
thf(fact_2048_unset__bit__Suc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A3: A] :
( ( bit_se2638667681897837118et_bit @ A @ ( suc @ N ) @ A3 )
= ( plus_plus @ A @ ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se2638667681897837118et_bit @ A @ N @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% unset_bit_Suc
thf(fact_2049_even__set__bit__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,A3: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5668285175392031749et_bit @ A @ M @ A3 ) )
= ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
& ( M
!= ( zero_zero @ nat ) ) ) ) ) ).
% even_set_bit_iff
thf(fact_2050_even__flip__bit__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,A3: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se8732182000553998342ip_bit @ A @ M @ A3 ) )
= ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
!= ( M
= ( zero_zero @ nat ) ) ) ) ) ).
% even_flip_bit_iff
thf(fact_2051_even__unset__bit__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,A3: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se2638667681897837118et_bit @ A @ M @ A3 ) )
= ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
| ( M
= ( zero_zero @ nat ) ) ) ) ) ).
% even_unset_bit_iff
thf(fact_2052_signed__take__bit__rec,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( bit_ri4674362597316999326ke_bit @ A )
= ( ^ [N5: nat,A6: A] :
( if @ A
@ ( N5
= ( zero_zero @ nat ) )
@ ( uminus_uminus @ A @ ( modulo_modulo @ A @ A6 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
@ ( plus_plus @ A @ ( modulo_modulo @ A @ A6 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_ri4674362597316999326ke_bit @ A @ ( minus_minus @ nat @ N5 @ ( one_one @ nat ) ) @ ( divide_divide @ A @ A6 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% signed_take_bit_rec
thf(fact_2053_round__unique,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Y3: int] :
( ( ord_less @ A @ ( minus_minus @ A @ X @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( ring_1_of_int @ A @ Y3 ) )
=> ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Y3 ) @ ( plus_plus @ A @ X @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) )
=> ( ( archimedean_round @ A @ X )
= Y3 ) ) ) ) ).
% round_unique
thf(fact_2054_round__altdef,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archimedean_round @ A )
= ( ^ [X2: A] : ( if @ int @ ( ord_less_eq @ A @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( archimedean_frac @ A @ X2 ) ) @ ( archimedean_ceiling @ A @ X2 ) @ ( archim6421214686448440834_floor @ A @ X2 ) ) ) ) ) ).
% round_altdef
thf(fact_2055_binomial__code,axiom,
( binomial
= ( ^ [N5: nat,K3: nat] : ( if @ nat @ ( ord_less @ nat @ N5 @ K3 ) @ ( zero_zero @ nat ) @ ( if @ nat @ ( ord_less @ nat @ N5 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K3 ) ) @ ( binomial @ N5 @ ( minus_minus @ nat @ N5 @ K3 ) ) @ ( divide_divide @ nat @ ( set_fo6178422350223883121st_nat @ nat @ ( times_times @ nat ) @ ( plus_plus @ nat @ ( minus_minus @ nat @ N5 @ K3 ) @ ( one_one @ nat ) ) @ N5 @ ( one_one @ nat ) ) @ ( semiring_char_0_fact @ nat @ K3 ) ) ) ) ) ) ).
% binomial_code
thf(fact_2056_round__unique_H,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,N: int] :
( ( ord_less @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X @ ( ring_1_of_int @ A @ N ) ) ) @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
=> ( ( archimedean_round @ A @ X )
= N ) ) ) ).
% round_unique'
thf(fact_2057_of__int__round__abs__le,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ ( archimedean_round @ A @ X ) ) @ X ) ) @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% of_int_round_abs_le
thf(fact_2058_of__int__round__gt,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] : ( ord_less @ A @ ( minus_minus @ A @ X @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( ring_1_of_int @ A @ ( archimedean_round @ A @ X ) ) ) ) ).
% of_int_round_gt
thf(fact_2059_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_2060_binomial__n__n,axiom,
! [N: nat] :
( ( binomial @ N @ N )
= ( one_one @ nat ) ) ).
% binomial_n_n
thf(fact_2061_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_2062_binomial__0__Suc,axiom,
! [K: nat] :
( ( binomial @ ( zero_zero @ nat ) @ ( suc @ K ) )
= ( zero_zero @ nat ) ) ).
% binomial_0_Suc
thf(fact_2063_binomial__1,axiom,
! [N: nat] :
( ( binomial @ N @ ( suc @ ( zero_zero @ nat ) ) )
= N ) ).
% binomial_1
thf(fact_2064_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_2065_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_2066_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_2067_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_2068_binomial__n__0,axiom,
! [N: nat] :
( ( binomial @ N @ ( zero_zero @ nat ) )
= ( one_one @ nat ) ) ).
% binomial_n_0
thf(fact_2069_round__0,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archimedean_round @ A @ ( zero_zero @ A ) )
= ( zero_zero @ int ) ) ) ).
% round_0
thf(fact_2070_round__1,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archimedean_round @ A @ ( one_one @ A ) )
= ( one_one @ int ) ) ) ).
% round_1
thf(fact_2071_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_2072_signed__take__bit__0,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A3: A] :
( ( bit_ri4674362597316999326ke_bit @ A @ ( zero_zero @ nat ) @ A3 )
= ( uminus_uminus @ A @ ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% signed_take_bit_0
thf(fact_2073_choose__one,axiom,
! [N: nat] :
( ( binomial @ N @ ( one_one @ nat ) )
= N ) ).
% choose_one
thf(fact_2074_binomial__eq__0,axiom,
! [N: nat,K: nat] :
( ( ord_less @ nat @ N @ K )
=> ( ( binomial @ N @ K )
= ( zero_zero @ nat ) ) ) ).
% binomial_eq_0
thf(fact_2075_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_2076_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_2077_Suc__times__binomial__add,axiom,
! [A3: nat,B2: nat] :
( ( times_times @ nat @ ( suc @ A3 ) @ ( binomial @ ( suc @ ( plus_plus @ nat @ A3 @ B2 ) ) @ ( suc @ A3 ) ) )
= ( times_times @ nat @ ( suc @ B2 ) @ ( binomial @ ( suc @ ( plus_plus @ nat @ A3 @ B2 ) ) @ A3 ) ) ) ).
% Suc_times_binomial_add
thf(fact_2078_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_2079_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_2080_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_2081_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_2082_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_2083_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_2084_round__def,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archimedean_round @ A )
= ( ^ [X2: A] : ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X2 @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% round_def
thf(fact_2085_signed__take__bit__Suc,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: nat,A3: A] :
( ( bit_ri4674362597316999326ke_bit @ A @ ( suc @ N ) @ A3 )
= ( plus_plus @ A @ ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_ri4674362597316999326ke_bit @ A @ N @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% signed_take_bit_Suc
thf(fact_2086_of__int__round__le,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] : ( ord_less_eq @ A @ ( ring_1_of_int @ A @ ( archimedean_round @ A @ X ) ) @ ( plus_plus @ A @ X @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% of_int_round_le
thf(fact_2087_of__int__round__ge,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ X @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( ring_1_of_int @ A @ ( archimedean_round @ A @ X ) ) ) ) ).
% of_int_round_ge
thf(fact_2088_set__decode__0,axiom,
! [X: nat] :
( ( member @ nat @ ( zero_zero @ nat ) @ ( nat_set_decode @ X ) )
= ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ X ) ) ) ).
% set_decode_0
thf(fact_2089_set__decode__def,axiom,
( nat_set_decode
= ( ^ [X2: nat] :
( collect @ nat
@ ^ [N5: nat] :
~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ X2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N5 ) ) ) ) ) ) ).
% set_decode_def
thf(fact_2090_divmod__step__def,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ( ( unique1321980374590559556d_step @ A )
= ( ^ [L4: num] :
( product_case_prod @ A @ A @ ( product_prod @ A @ A )
@ ^ [Q4: A,R4: A] : ( if @ ( product_prod @ A @ A ) @ ( ord_less_eq @ A @ ( numeral_numeral @ A @ L4 ) @ R4 ) @ ( product_Pair @ A @ A @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Q4 ) @ ( one_one @ A ) ) @ ( minus_minus @ A @ R4 @ ( numeral_numeral @ A @ L4 ) ) ) @ ( product_Pair @ A @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Q4 ) @ R4 ) ) ) ) ) ) ).
% divmod_step_def
thf(fact_2091_take__bit__rec,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se2584673776208193580ke_bit @ A )
= ( ^ [N5: nat,A6: A] :
( if @ A
@ ( N5
= ( zero_zero @ nat ) )
@ ( zero_zero @ A )
@ ( plus_plus @ A @ ( times_times @ A @ ( bit_se2584673776208193580ke_bit @ A @ ( minus_minus @ nat @ N5 @ ( one_one @ nat ) ) @ ( divide_divide @ A @ A6 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ A @ A6 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% take_bit_rec
thf(fact_2092_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_2093_divmod__divmod__step,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ( ( unique8689654367752047608divmod @ A )
= ( ^ [M5: num,N5: num] : ( if @ ( product_prod @ A @ A ) @ ( ord_less @ num @ M5 @ N5 ) @ ( product_Pair @ A @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ M5 ) ) @ ( unique1321980374590559556d_step @ A @ N5 @ ( unique8689654367752047608divmod @ A @ M5 @ ( bit0 @ N5 ) ) ) ) ) ) ) ).
% divmod_divmod_step
thf(fact_2094_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_2095_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_2096_bit_Oxor__self,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X: A] :
( ( bit_se5824344971392196577ns_xor @ A @ X @ X )
= ( zero_zero @ A ) ) ) ).
% bit.xor_self
thf(fact_2097_xor__self__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A] :
( ( bit_se5824344971392196577ns_xor @ A @ A3 @ A3 )
= ( zero_zero @ A ) ) ) ).
% xor_self_eq
thf(fact_2098_xor_Oleft__neutral,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A] :
( ( bit_se5824344971392196577ns_xor @ A @ ( zero_zero @ A ) @ A3 )
= A3 ) ) ).
% xor.left_neutral
thf(fact_2099_xor_Oright__neutral,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A] :
( ( bit_se5824344971392196577ns_xor @ A @ A3 @ ( zero_zero @ A ) )
= A3 ) ) ).
% xor.right_neutral
thf(fact_2100_case__prod__conv,axiom,
! [B: $tType,A: $tType,C: $tType,F: B > C > A,A3: B,B2: C] :
( ( product_case_prod @ B @ C @ A @ F @ ( product_Pair @ B @ C @ A3 @ B2 ) )
= ( F @ A3 @ B2 ) ) ).
% case_prod_conv
thf(fact_2101_case__prod__curry,axiom,
! [C: $tType,B: $tType,A: $tType,F: ( product_prod @ A @ B ) > C] :
( ( product_case_prod @ A @ B @ C @ ( product_curry @ A @ B @ C @ F ) )
= F ) ).
% case_prod_curry
thf(fact_2102_curry__case__prod,axiom,
! [C: $tType,B: $tType,A: $tType,F: A > B > C] :
( ( product_curry @ A @ B @ C @ ( product_case_prod @ A @ B @ C @ F ) )
= F ) ).
% curry_case_prod
thf(fact_2103_take__bit__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A] :
( ( bit_se2584673776208193580ke_bit @ A @ ( zero_zero @ nat ) @ A3 )
= ( zero_zero @ A ) ) ) ).
% take_bit_0
thf(fact_2104_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_2105_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_2106_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_2107_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_2108_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_2109_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_2110_even__take__bit__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A3: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se2584673776208193580ke_bit @ A @ N @ A3 ) )
= ( ( N
= ( zero_zero @ nat ) )
| ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) ) ) ).
% even_take_bit_eq
thf(fact_2111_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_2112_take__bit__Suc__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A] :
( ( bit_se2584673776208193580ke_bit @ A @ ( suc @ ( zero_zero @ nat ) ) @ A3 )
= ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% take_bit_Suc_0
thf(fact_2113_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 )
@ ^ [Q4: A,R4: A] : ( product_Pair @ A @ A @ Q4 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ R4 ) )
@ ( unique8689654367752047608divmod @ A @ M @ N ) ) ) ) ).
% divmod_algorithm_code(5)
thf(fact_2114_case__prod__app,axiom,
! [A: $tType,D: $tType,C: $tType,B: $tType] :
( ( product_case_prod @ B @ C @ ( D > A ) )
= ( ^ [F2: B > C > D > A,X2: product_prod @ B @ C,Y: D] :
( product_case_prod @ B @ C @ A
@ ^ [L4: B,R4: C] : ( F2 @ L4 @ R4 @ Y )
@ X2 ) ) ) ).
% case_prod_app
thf(fact_2115_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_2116_take__bit__add,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N: nat,A3: A,B2: A] :
( ( bit_se2584673776208193580ke_bit @ A @ N @ ( plus_plus @ A @ ( bit_se2584673776208193580ke_bit @ A @ N @ A3 ) @ ( bit_se2584673776208193580ke_bit @ A @ N @ B2 ) ) )
= ( bit_se2584673776208193580ke_bit @ A @ N @ ( plus_plus @ A @ A3 @ B2 ) ) ) ) ).
% take_bit_add
thf(fact_2117_old_Oprod_Ocase,axiom,
! [A: $tType,C: $tType,B: $tType,F: A > B > C,X1: A,X22: B] :
( ( product_case_prod @ A @ B @ C @ F @ ( product_Pair @ A @ B @ X1 @ X22 ) )
= ( F @ X1 @ X22 ) ) ).
% old.prod.case
thf(fact_2118_split__cong,axiom,
! [C: $tType,B: $tType,A: $tType,Q2: product_prod @ A @ B,F: A > B > C,G: A > B > C,P4: product_prod @ A @ B] :
( ! [X4: A,Y4: B] :
( ( ( product_Pair @ A @ B @ X4 @ Y4 )
= Q2 )
=> ( ( F @ X4 @ Y4 )
= ( G @ X4 @ Y4 ) ) )
=> ( ( P4 = Q2 )
=> ( ( product_case_prod @ A @ B @ C @ F @ P4 )
= ( product_case_prod @ A @ B @ C @ G @ Q2 ) ) ) ) ).
% split_cong
thf(fact_2119_case__prod__Pair__iden,axiom,
! [B: $tType,A: $tType,P4: product_prod @ A @ B] :
( ( product_case_prod @ A @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B ) @ P4 )
= P4 ) ).
% case_prod_Pair_iden
thf(fact_2120_case__prodE2,axiom,
! [B: $tType,A: $tType,C: $tType,Q: A > $o,P: B > C > A,Z: product_prod @ B @ C] :
( ( Q @ ( product_case_prod @ B @ C @ A @ P @ Z ) )
=> ~ ! [X4: B,Y4: C] :
( ( Z
= ( product_Pair @ B @ C @ X4 @ Y4 ) )
=> ~ ( Q @ ( P @ X4 @ Y4 ) ) ) ) ).
% case_prodE2
thf(fact_2121_case__prod__eta,axiom,
! [C: $tType,B: $tType,A: $tType,F: ( product_prod @ A @ B ) > C] :
( ( product_case_prod @ A @ B @ C
@ ^ [X2: A,Y: B] : ( F @ ( product_Pair @ A @ B @ X2 @ Y ) ) )
= F ) ).
% case_prod_eta
thf(fact_2122_cond__case__prod__eta,axiom,
! [C: $tType,B: $tType,A: $tType,F: A > B > C,G: ( product_prod @ A @ B ) > C] :
( ! [X4: A,Y4: B] :
( ( F @ X4 @ Y4 )
= ( G @ ( product_Pair @ A @ B @ X4 @ Y4 ) ) )
=> ( ( product_case_prod @ A @ B @ C @ F )
= G ) ) ).
% cond_case_prod_eta
thf(fact_2123_divmod__step__integer__def,axiom,
( ( unique1321980374590559556d_step @ code_integer )
= ( ^ [L4: num] :
( product_case_prod @ code_integer @ code_integer @ ( product_prod @ code_integer @ code_integer )
@ ^ [Q4: code_integer,R4: code_integer] : ( if @ ( product_prod @ code_integer @ code_integer ) @ ( ord_less_eq @ code_integer @ ( numeral_numeral @ code_integer @ L4 ) @ R4 ) @ ( product_Pair @ code_integer @ code_integer @ ( plus_plus @ code_integer @ ( times_times @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) @ Q4 ) @ ( one_one @ code_integer ) ) @ ( minus_minus @ code_integer @ R4 @ ( numeral_numeral @ code_integer @ L4 ) ) ) @ ( product_Pair @ code_integer @ code_integer @ ( times_times @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) @ Q4 ) @ R4 ) ) ) ) ) ).
% divmod_step_integer_def
thf(fact_2124_internal__case__prod__def,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( produc5280177257484947105e_prod @ A @ B @ C )
= ( product_case_prod @ A @ B @ C ) ) ).
% internal_case_prod_def
thf(fact_2125_num_Osize__gen_I1_J,axiom,
( ( size_num @ one2 )
= ( zero_zero @ nat ) ) ).
% num.size_gen(1)
thf(fact_2126_divmod__int__def,axiom,
( ( unique8689654367752047608divmod @ int )
= ( ^ [M5: num,N5: num] : ( product_Pair @ int @ int @ ( divide_divide @ int @ ( numeral_numeral @ int @ M5 ) @ ( numeral_numeral @ int @ N5 ) ) @ ( modulo_modulo @ int @ ( numeral_numeral @ int @ M5 ) @ ( numeral_numeral @ int @ N5 ) ) ) ) ) ).
% divmod_int_def
thf(fact_2127_take__bit__eq__0__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A3: A] :
( ( ( bit_se2584673776208193580ke_bit @ A @ N @ A3 )
= ( zero_zero @ A ) )
= ( dvd_dvd @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) @ A3 ) ) ) ).
% take_bit_eq_0_iff
thf(fact_2128_divmod__def,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ( ( unique8689654367752047608divmod @ A )
= ( ^ [M5: num,N5: num] : ( product_Pair @ A @ A @ ( divide_divide @ A @ ( numeral_numeral @ A @ M5 ) @ ( numeral_numeral @ A @ N5 ) ) @ ( modulo_modulo @ A @ ( numeral_numeral @ A @ M5 ) @ ( numeral_numeral @ A @ N5 ) ) ) ) ) ) ).
% divmod_def
thf(fact_2129_divmod_H__nat__def,axiom,
( ( unique8689654367752047608divmod @ nat )
= ( ^ [M5: num,N5: num] : ( product_Pair @ nat @ nat @ ( divide_divide @ nat @ ( numeral_numeral @ nat @ M5 ) @ ( numeral_numeral @ nat @ N5 ) ) @ ( modulo_modulo @ nat @ ( numeral_numeral @ nat @ M5 ) @ ( numeral_numeral @ nat @ N5 ) ) ) ) ) ).
% divmod'_nat_def
thf(fact_2130_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_2131_take__bit__Suc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A3: A] :
( ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N ) @ A3 )
= ( plus_plus @ A @ ( times_times @ A @ ( bit_se2584673776208193580ke_bit @ A @ N @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% take_bit_Suc
thf(fact_2132_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_2133_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_2134_xor__nat__unfold,axiom,
( ( bit_se5824344971392196577ns_xor @ nat )
= ( ^ [M5: nat,N5: nat] :
( if @ nat
@ ( M5
= ( zero_zero @ nat ) )
@ N5
@ ( if @ nat
@ ( N5
= ( zero_zero @ nat ) )
@ M5
@ ( plus_plus @ nat @ ( modulo_modulo @ nat @ ( plus_plus @ nat @ ( modulo_modulo @ nat @ M5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ nat @ N5 @ ( 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 @ M5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% xor_nat_unfold
thf(fact_2135_xor__nat__rec,axiom,
( ( bit_se5824344971392196577ns_xor @ nat )
= ( ^ [M5: nat,N5: nat] :
( plus_plus @ nat
@ ( zero_neq_one_of_bool @ nat
@ ( ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M5 ) )
!= ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N5 ) ) ) )
@ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ nat @ ( divide_divide @ nat @ M5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% xor_nat_rec
thf(fact_2136_xor__one__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A] :
( ( bit_se5824344971392196577ns_xor @ A @ A3 @ ( one_one @ A ) )
= ( minus_minus @ A @ ( plus_plus @ A @ A3 @ ( zero_neq_one_of_bool @ A @ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) )
@ ( zero_neq_one_of_bool @ A
@ ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) ) ) ) ).
% xor_one_eq
thf(fact_2137_one__xor__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A] :
( ( bit_se5824344971392196577ns_xor @ A @ ( one_one @ A ) @ A3 )
= ( minus_minus @ A @ ( plus_plus @ A @ A3 @ ( zero_neq_one_of_bool @ A @ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) )
@ ( zero_neq_one_of_bool @ A
@ ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) ) ) ) ).
% one_xor_eq
thf(fact_2138_stable__imp__take__bit__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A,N: nat] :
( ( ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= A3 )
=> ( ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ( bit_se2584673776208193580ke_bit @ A @ N @ A3 )
= ( zero_zero @ A ) ) )
& ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ( bit_se2584673776208193580ke_bit @ A @ N @ A3 )
= ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ A ) ) ) ) ) ) ) ).
% stable_imp_take_bit_eq
thf(fact_2139_divmod__step__nat__def,axiom,
( ( unique1321980374590559556d_step @ nat )
= ( ^ [L4: num] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [Q4: nat,R4: nat] : ( if @ ( product_prod @ nat @ nat ) @ ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ L4 ) @ R4 ) @ ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Q4 ) @ ( one_one @ nat ) ) @ ( minus_minus @ nat @ R4 @ ( numeral_numeral @ nat @ L4 ) ) ) @ ( product_Pair @ nat @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Q4 ) @ R4 ) ) ) ) ) ).
% divmod_step_nat_def
thf(fact_2140_divmod__step__int__def,axiom,
( ( unique1321980374590559556d_step @ int )
= ( ^ [L4: num] :
( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [Q4: int,R4: int] : ( if @ ( product_prod @ int @ int ) @ ( ord_less_eq @ int @ ( numeral_numeral @ int @ L4 ) @ R4 ) @ ( product_Pair @ int @ int @ ( plus_plus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Q4 ) @ ( one_one @ int ) ) @ ( minus_minus @ int @ R4 @ ( numeral_numeral @ int @ L4 ) ) ) @ ( product_Pair @ int @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Q4 ) @ R4 ) ) ) ) ) ).
% divmod_step_int_def
thf(fact_2141_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_2142_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 )
@ ^ [Q4: 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 @ Q4 ) ) )
@ ( bit_take_bit_num @ ( numeral_numeral @ nat @ M ) @ N ) ) ) ).
% take_bit_numeral_minus_numeral_int
thf(fact_2143_divmod__nat__if,axiom,
( divmod_nat
= ( ^ [M5: nat,N5: nat] :
( if @ ( product_prod @ nat @ nat )
@ ( ( N5
= ( zero_zero @ nat ) )
| ( ord_less @ nat @ M5 @ N5 ) )
@ ( product_Pair @ nat @ nat @ ( zero_zero @ nat ) @ M5 )
@ ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [Q4: nat] : ( product_Pair @ nat @ nat @ ( suc @ Q4 ) )
@ ( divmod_nat @ ( minus_minus @ nat @ M5 @ N5 ) @ N5 ) ) ) ) ) ).
% divmod_nat_if
thf(fact_2144_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 )
@ ^ [Q4: A,R4: A] : ( product_Pair @ A @ A @ Q4 @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ R4 ) @ ( one_one @ A ) ) )
@ ( unique8689654367752047608divmod @ A @ M @ N ) ) ) ) ).
% divmod_algorithm_code(6)
thf(fact_2145_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_2146_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_2147_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_2148_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_2149_verit__eq__simplify_I9_J,axiom,
! [X32: num,Y32: num] :
( ( ( bit1 @ X32 )
= ( bit1 @ Y32 ) )
= ( X32 = Y32 ) ) ).
% verit_eq_simplify(9)
thf(fact_2150_case__prodI2,axiom,
! [B: $tType,A: $tType,P4: product_prod @ A @ B,C2: A > B > $o] :
( ! [A4: A,B3: B] :
( ( P4
= ( product_Pair @ A @ B @ A4 @ B3 ) )
=> ( C2 @ A4 @ B3 ) )
=> ( product_case_prod @ A @ B @ $o @ C2 @ P4 ) ) ).
% case_prodI2
thf(fact_2151_case__prodI,axiom,
! [A: $tType,B: $tType,F: A > B > $o,A3: A,B2: B] :
( ( F @ A3 @ B2 )
=> ( product_case_prod @ A @ B @ $o @ F @ ( product_Pair @ A @ B @ A3 @ B2 ) ) ) ).
% case_prodI
thf(fact_2152_mem__case__prodI2,axiom,
! [C: $tType,B: $tType,A: $tType,P4: product_prod @ A @ B,Z: C,C2: A > B > ( set @ C )] :
( ! [A4: A,B3: B] :
( ( P4
= ( product_Pair @ A @ B @ A4 @ B3 ) )
=> ( member @ C @ Z @ ( C2 @ A4 @ B3 ) ) )
=> ( member @ C @ Z @ ( product_case_prod @ A @ B @ ( set @ C ) @ C2 @ P4 ) ) ) ).
% mem_case_prodI2
thf(fact_2153_mem__case__prodI,axiom,
! [A: $tType,B: $tType,C: $tType,Z: A,C2: B > C > ( set @ A ),A3: B,B2: C] :
( ( member @ A @ Z @ ( C2 @ A3 @ B2 ) )
=> ( member @ A @ Z @ ( product_case_prod @ B @ C @ ( set @ A ) @ C2 @ ( product_Pair @ B @ C @ A3 @ B2 ) ) ) ) ).
% mem_case_prodI
thf(fact_2154_case__prodI2_H,axiom,
! [A: $tType,B: $tType,C: $tType,P4: product_prod @ A @ B,C2: A > B > C > $o,X: C] :
( ! [A4: A,B3: B] :
( ( ( product_Pair @ A @ B @ A4 @ B3 )
= P4 )
=> ( C2 @ A4 @ B3 @ X ) )
=> ( product_case_prod @ A @ B @ ( C > $o ) @ C2 @ P4 @ X ) ) ).
% case_prodI2'
thf(fact_2155_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_2156_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 )
@ ^ [Q4: num] : ( some @ num @ ( bit0 @ Q4 ) )
@ ( bit_take_bit_num @ N @ M ) ) ) ).
% take_bit_num_simps(3)
thf(fact_2157_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_2158_xor__numerals_I8_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num] :
( ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ ( bit1 @ X ) ) @ ( one_one @ A ) )
= ( numeral_numeral @ A @ ( bit0 @ X ) ) ) ) ).
% xor_numerals(8)
thf(fact_2159_xor__numerals_I5_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num] :
( ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ ( bit0 @ X ) ) @ ( one_one @ A ) )
= ( numeral_numeral @ A @ ( bit1 @ X ) ) ) ) ).
% xor_numerals(5)
thf(fact_2160_xor__numerals_I2_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [Y3: num] :
( ( bit_se5824344971392196577ns_xor @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit1 @ Y3 ) ) )
= ( numeral_numeral @ A @ ( bit0 @ Y3 ) ) ) ) ).
% xor_numerals(2)
thf(fact_2161_xor__numerals_I1_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [Y3: num] :
( ( bit_se5824344971392196577ns_xor @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ Y3 ) ) )
= ( numeral_numeral @ A @ ( bit1 @ Y3 ) ) ) ) ).
% xor_numerals(1)
thf(fact_2162_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_2163_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_2164_Suc__mod__eq__add3__mod__numeral,axiom,
! [M: nat,V: num] :
( ( modulo_modulo @ nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ ( numeral_numeral @ nat @ V ) )
= ( modulo_modulo @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) @ M ) @ ( numeral_numeral @ nat @ V ) ) ) ).
% Suc_mod_eq_add3_mod_numeral
thf(fact_2165_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_2166_Suc__div__eq__add3__div__numeral,axiom,
! [M: nat,V: num] :
( ( divide_divide @ nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ ( numeral_numeral @ nat @ V ) )
= ( divide_divide @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) @ M ) @ ( numeral_numeral @ nat @ V ) ) ) ).
% Suc_div_eq_add3_div_numeral
thf(fact_2167_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_2168_xor__nat__numerals_I4_J,axiom,
! [X: num] :
( ( bit_se5824344971392196577ns_xor @ nat @ ( numeral_numeral @ nat @ ( bit1 @ X ) ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( numeral_numeral @ nat @ ( bit0 @ X ) ) ) ).
% xor_nat_numerals(4)
thf(fact_2169_xor__nat__numerals_I3_J,axiom,
! [X: num] :
( ( bit_se5824344971392196577ns_xor @ nat @ ( numeral_numeral @ nat @ ( bit0 @ X ) ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( numeral_numeral @ nat @ ( bit1 @ X ) ) ) ).
% xor_nat_numerals(3)
thf(fact_2170_xor__nat__numerals_I2_J,axiom,
! [Y3: num] :
( ( bit_se5824344971392196577ns_xor @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( numeral_numeral @ nat @ ( bit1 @ Y3 ) ) )
= ( numeral_numeral @ nat @ ( bit0 @ Y3 ) ) ) ).
% xor_nat_numerals(2)
thf(fact_2171_xor__nat__numerals_I1_J,axiom,
! [Y3: num] :
( ( bit_se5824344971392196577ns_xor @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( numeral_numeral @ nat @ ( bit0 @ Y3 ) ) )
= ( numeral_numeral @ nat @ ( bit1 @ Y3 ) ) ) ).
% xor_nat_numerals(1)
thf(fact_2172_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_2173_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_2174_xor__numerals_I4_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num,Y3: num] :
( ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ ( bit0 @ X ) ) @ ( numeral_numeral @ A @ ( bit1 @ Y3 ) ) )
= ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ X ) @ ( numeral_numeral @ A @ Y3 ) ) ) ) ) ) ).
% xor_numerals(4)
thf(fact_2175_xor__numerals_I6_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num,Y3: num] :
( ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ ( bit1 @ X ) ) @ ( numeral_numeral @ A @ ( bit0 @ Y3 ) ) )
= ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ X ) @ ( numeral_numeral @ A @ Y3 ) ) ) ) ) ) ).
% xor_numerals(6)
thf(fact_2176_divmod__integer_H__def,axiom,
( ( unique8689654367752047608divmod @ code_integer )
= ( ^ [M5: num,N5: num] : ( product_Pair @ code_integer @ code_integer @ ( divide_divide @ code_integer @ ( numeral_numeral @ code_integer @ M5 ) @ ( numeral_numeral @ code_integer @ N5 ) ) @ ( modulo_modulo @ code_integer @ ( numeral_numeral @ code_integer @ M5 ) @ ( numeral_numeral @ code_integer @ N5 ) ) ) ) ) ).
% divmod_integer'_def
thf(fact_2177_verit__eq__simplify_I14_J,axiom,
! [X22: num,X32: num] :
( ( bit0 @ X22 )
!= ( bit1 @ X32 ) ) ).
% verit_eq_simplify(14)
thf(fact_2178_verit__eq__simplify_I12_J,axiom,
! [X32: num] :
( one2
!= ( bit1 @ X32 ) ) ).
% verit_eq_simplify(12)
thf(fact_2179_mem__case__prodE,axiom,
! [B: $tType,A: $tType,C: $tType,Z: A,C2: B > C > ( set @ A ),P4: product_prod @ B @ C] :
( ( member @ A @ Z @ ( product_case_prod @ B @ C @ ( set @ A ) @ C2 @ P4 ) )
=> ~ ! [X4: B,Y4: C] :
( ( P4
= ( product_Pair @ B @ C @ X4 @ Y4 ) )
=> ~ ( member @ A @ Z @ ( C2 @ X4 @ Y4 ) ) ) ) ).
% mem_case_prodE
thf(fact_2180_case__prodE,axiom,
! [A: $tType,B: $tType,C2: A > B > $o,P4: product_prod @ A @ B] :
( ( product_case_prod @ A @ B @ $o @ C2 @ P4 )
=> ~ ! [X4: A,Y4: B] :
( ( P4
= ( product_Pair @ A @ B @ X4 @ Y4 ) )
=> ~ ( C2 @ X4 @ Y4 ) ) ) ).
% case_prodE
thf(fact_2181_case__prodD,axiom,
! [A: $tType,B: $tType,F: A > B > $o,A3: A,B2: B] :
( ( product_case_prod @ A @ B @ $o @ F @ ( product_Pair @ A @ B @ A3 @ B2 ) )
=> ( F @ A3 @ B2 ) ) ).
% case_prodD
thf(fact_2182_case__prodE_H,axiom,
! [B: $tType,A: $tType,C: $tType,C2: A > B > C > $o,P4: product_prod @ A @ B,Z: C] :
( ( product_case_prod @ A @ B @ ( C > $o ) @ C2 @ P4 @ Z )
=> ~ ! [X4: A,Y4: B] :
( ( P4
= ( product_Pair @ A @ B @ X4 @ Y4 ) )
=> ~ ( C2 @ X4 @ Y4 @ Z ) ) ) ).
% case_prodE'
thf(fact_2183_case__prodD_H,axiom,
! [B: $tType,A: $tType,C: $tType,R5: A > B > C > $o,A3: A,B2: B,C2: C] :
( ( product_case_prod @ A @ B @ ( C > $o ) @ R5 @ ( product_Pair @ A @ B @ A3 @ B2 ) @ C2 )
=> ( R5 @ A3 @ B2 @ C2 ) ) ).
% case_prodD'
thf(fact_2184_xor__num_Ocases,axiom,
! [X: product_prod @ num @ num] :
( ( X
!= ( product_Pair @ num @ num @ one2 @ one2 ) )
=> ( ! [N2: num] :
( X
!= ( product_Pair @ num @ num @ one2 @ ( bit0 @ N2 ) ) )
=> ( ! [N2: num] :
( X
!= ( product_Pair @ num @ num @ one2 @ ( bit1 @ N2 ) ) )
=> ( ! [M4: num] :
( X
!= ( product_Pair @ num @ num @ ( bit0 @ M4 ) @ one2 ) )
=> ( ! [M4: num,N2: num] :
( X
!= ( product_Pair @ num @ num @ ( bit0 @ M4 ) @ ( bit0 @ N2 ) ) )
=> ( ! [M4: num,N2: num] :
( X
!= ( product_Pair @ num @ num @ ( bit0 @ M4 ) @ ( bit1 @ N2 ) ) )
=> ( ! [M4: num] :
( X
!= ( product_Pair @ num @ num @ ( bit1 @ M4 ) @ one2 ) )
=> ( ! [M4: num,N2: num] :
( X
!= ( product_Pair @ num @ num @ ( bit1 @ M4 ) @ ( bit0 @ N2 ) ) )
=> ~ ! [M4: num,N2: num] :
( X
!= ( product_Pair @ num @ num @ ( bit1 @ M4 ) @ ( bit1 @ N2 ) ) ) ) ) ) ) ) ) ) ) ).
% xor_num.cases
thf(fact_2185_Collect__case__prod__mono,axiom,
! [B: $tType,A: $tType,A5: A > B > $o,B4: A > B > $o] :
( ( ord_less_eq @ ( A > B > $o ) @ A5 @ B4 )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ A5 ) ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ B4 ) ) ) ) ).
% Collect_case_prod_mono
thf(fact_2186_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 ) ) )
@ ^ [X2: B] :
( product_case_prod @ ( heap_ext @ product_unit ) @ nat @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )
@ ^ [H7: heap_ext @ product_unit,N5: nat] : ( heap_Time_timeFrame @ A @ N5 @ ( heap_Time_execute @ A @ ( G @ X2 ) @ H7 ) ) ) )
@ ( heap_Time_execute @ B @ F @ H ) ) ) ).
% execute_bind_case
thf(fact_2187_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_2188_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_2189_power__numeral__odd,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [Z: A,W: num] :
( ( power_power @ A @ Z @ ( numeral_numeral @ nat @ ( bit1 @ W ) ) )
= ( times_times @ A @ ( times_times @ A @ Z @ ( power_power @ A @ Z @ ( numeral_numeral @ nat @ W ) ) ) @ ( power_power @ A @ Z @ ( numeral_numeral @ nat @ W ) ) ) ) ) ).
% power_numeral_odd
thf(fact_2190_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
@ ^ [H4: heap_ext @ product_unit] :
( case_option @ ( option @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( none @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )
@ ( product_case_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( option @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )
@ ^ [R4: A] :
( product_case_prod @ ( heap_ext @ product_unit ) @ nat @ ( option @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )
@ ^ [H7: heap_ext @ product_unit,N5: nat] : ( heap_Time_timeFrame @ B @ N5 @ ( heap_Time_execute @ B @ ( G2 @ R4 ) @ H7 ) ) ) )
@ ( heap_Time_execute @ A @ F2 @ H4 ) ) ) ) ) ).
% Heap_Time_Monad.bind_def
thf(fact_2191_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_2192_numeral__3__eq__3,axiom,
( ( numeral_numeral @ nat @ ( bit1 @ one2 ) )
= ( suc @ ( suc @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% numeral_3_eq_3
thf(fact_2193_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_2194_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_2195_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_2196_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_2197_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_2198_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_2199_take__bit__num__def,axiom,
( bit_take_bit_num
= ( ^ [N5: nat,M5: num] :
( if @ ( option @ num )
@ ( ( bit_se2584673776208193580ke_bit @ nat @ N5 @ ( numeral_numeral @ nat @ M5 ) )
= ( zero_zero @ nat ) )
@ ( none @ num )
@ ( some @ num @ ( num_of_nat @ ( bit_se2584673776208193580ke_bit @ nat @ N5 @ ( numeral_numeral @ nat @ M5 ) ) ) ) ) ) ) ).
% take_bit_num_def
thf(fact_2200_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_2201_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_2202_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_2203_Divides_Oadjust__div__def,axiom,
( adjust_div
= ( product_case_prod @ int @ int @ int
@ ^ [Q4: int,R4: int] :
( plus_plus @ int @ Q4
@ ( zero_neq_one_of_bool @ int
@ ( R4
!= ( zero_zero @ int ) ) ) ) ) ) ).
% Divides.adjust_div_def
thf(fact_2204_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_2205_divmod__nat__def,axiom,
( divmod_nat
= ( ^ [M5: nat,N5: nat] : ( product_Pair @ nat @ nat @ ( divide_divide @ nat @ M5 @ N5 ) @ ( modulo_modulo @ nat @ M5 @ N5 ) ) ) ) ).
% divmod_nat_def
thf(fact_2206_bit__cut__integer__def,axiom,
( code_bit_cut_integer
= ( ^ [K3: code_integer] :
( product_Pair @ code_integer @ $o @ ( divide_divide @ code_integer @ K3 @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) )
@ ~ ( dvd_dvd @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) @ K3 ) ) ) ) ).
% bit_cut_integer_def
thf(fact_2207_integer__of__int__code,axiom,
( code_integer_of_int
= ( ^ [K3: int] :
( if @ code_integer @ ( ord_less @ int @ K3 @ ( zero_zero @ int ) ) @ ( uminus_uminus @ code_integer @ ( code_integer_of_int @ ( uminus_uminus @ int @ K3 ) ) )
@ ( if @ code_integer
@ ( K3
= ( zero_zero @ int ) )
@ ( zero_zero @ code_integer )
@ ( if @ code_integer
@ ( ( modulo_modulo @ int @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) )
= ( zero_zero @ int ) )
@ ( times_times @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) @ ( code_integer_of_int @ ( divide_divide @ int @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) )
@ ( plus_plus @ code_integer @ ( times_times @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) @ ( code_integer_of_int @ ( divide_divide @ int @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) @ ( one_one @ code_integer ) ) ) ) ) ) ) ).
% integer_of_int_code
thf(fact_2208_int__ge__less__than__def,axiom,
( int_ge_less_than
= ( ^ [D4: int] :
( collect @ ( product_prod @ int @ int )
@ ( product_case_prod @ int @ int @ $o
@ ^ [Z7: int,Z3: int] :
( ( ord_less_eq @ int @ D4 @ Z7 )
& ( ord_less @ int @ Z7 @ Z3 ) ) ) ) ) ) ).
% int_ge_less_than_def
thf(fact_2209_int__ge__less__than2__def,axiom,
( int_ge_less_than2
= ( ^ [D4: int] :
( collect @ ( product_prod @ int @ int )
@ ( product_case_prod @ int @ int @ $o
@ ^ [Z7: int,Z3: int] :
( ( ord_less_eq @ int @ D4 @ Z3 )
& ( ord_less @ int @ Z7 @ Z3 ) ) ) ) ) ) ).
% int_ge_less_than2_def
thf(fact_2210_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_2211_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_2212_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_2213_split__part,axiom,
! [B: $tType,A: $tType,P: $o,Q: A > B > $o] :
( ( product_case_prod @ A @ B @ $o
@ ^ [A6: A,B5: B] :
( P
& ( Q @ A6 @ B5 ) ) )
= ( ^ [Ab: product_prod @ A @ B] :
( P
& ( product_case_prod @ A @ B @ $o @ Q @ Ab ) ) ) ) ).
% split_part
thf(fact_2214_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_2215_push__bit__eq__0__iff,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N: nat,A3: A] :
( ( ( bit_se4730199178511100633sh_bit @ A @ N @ A3 )
= ( zero_zero @ A ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% push_bit_eq_0_iff
thf(fact_2216_push__bit__push__bit,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,N: nat,A3: A] :
( ( bit_se4730199178511100633sh_bit @ A @ M @ ( bit_se4730199178511100633sh_bit @ A @ N @ A3 ) )
= ( bit_se4730199178511100633sh_bit @ A @ ( plus_plus @ nat @ M @ N ) @ A3 ) ) ) ).
% push_bit_push_bit
thf(fact_2217_pred__numeral__simps_I1_J,axiom,
( ( pred_numeral @ one2 )
= ( zero_zero @ nat ) ) ).
% pred_numeral_simps(1)
thf(fact_2218_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_2219_even__push__bit__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A3: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se4730199178511100633sh_bit @ A @ N @ A3 ) )
= ( ( N
!= ( zero_zero @ nat ) )
| ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) ) ) ).
% even_push_bit_iff
thf(fact_2220_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_2221_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_2222_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 )
@ ^ [Q4: num] : ( some @ num @ ( bit0 @ Q4 ) )
@ ( bit_take_bit_num @ ( pred_numeral @ R ) @ M ) ) ) ).
% take_bit_num_simps(6)
thf(fact_2223_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_2224_push__bit__add,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A3: A,B2: A] :
( ( bit_se4730199178511100633sh_bit @ A @ N @ ( plus_plus @ A @ A3 @ B2 ) )
= ( plus_plus @ A @ ( bit_se4730199178511100633sh_bit @ A @ N @ A3 ) @ ( bit_se4730199178511100633sh_bit @ A @ N @ B2 ) ) ) ) ).
% push_bit_add
thf(fact_2225_push__bit__take__bit,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,N: nat,A3: A] :
( ( bit_se4730199178511100633sh_bit @ A @ M @ ( bit_se2584673776208193580ke_bit @ A @ N @ A3 ) )
= ( bit_se2584673776208193580ke_bit @ A @ ( plus_plus @ nat @ M @ N ) @ ( bit_se4730199178511100633sh_bit @ A @ M @ A3 ) ) ) ) ).
% push_bit_take_bit
thf(fact_2226_flip__bit__nat__def,axiom,
( ( bit_se8732182000553998342ip_bit @ nat )
= ( ^ [M5: nat,N5: nat] : ( bit_se5824344971392196577ns_xor @ nat @ N5 @ ( bit_se4730199178511100633sh_bit @ nat @ M5 @ ( one_one @ nat ) ) ) ) ) ).
% flip_bit_nat_def
thf(fact_2227_pred__numeral__def,axiom,
( pred_numeral
= ( ^ [K3: num] : ( minus_minus @ nat @ ( numeral_numeral @ nat @ K3 ) @ ( one_one @ nat ) ) ) ) ).
% pred_numeral_def
thf(fact_2228_flip__bit__eq__xor,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se8732182000553998342ip_bit @ A )
= ( ^ [N5: nat,A6: A] : ( bit_se5824344971392196577ns_xor @ A @ A6 @ ( bit_se4730199178511100633sh_bit @ A @ N5 @ ( one_one @ A ) ) ) ) ) ) ).
% flip_bit_eq_xor
thf(fact_2229_bit__cut__integer__code,axiom,
( code_bit_cut_integer
= ( ^ [K3: code_integer] :
( if @ ( product_prod @ code_integer @ $o )
@ ( K3
= ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ $o @ ( zero_zero @ code_integer ) @ $false )
@ ( product_case_prod @ code_integer @ code_integer @ ( product_prod @ code_integer @ $o )
@ ^ [R4: code_integer,S2: code_integer] :
( product_Pair @ code_integer @ $o @ ( if @ code_integer @ ( ord_less @ code_integer @ ( zero_zero @ code_integer ) @ K3 ) @ R4 @ ( minus_minus @ code_integer @ ( uminus_uminus @ code_integer @ R4 ) @ S2 ) )
@ ( S2
= ( one_one @ code_integer ) ) )
@ ( code_divmod_abs @ K3 @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% bit_cut_integer_code
thf(fact_2230_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_2231_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_2232_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 )
@ ^ [N8: num] : ( some @ num @ ( bit1 @ N8 ) )
@ ( bit_and_not_num @ M @ N ) ) ) ).
% and_not_num.simps(8)
thf(fact_2233_take__bit__num__code,axiom,
( bit_take_bit_num
= ( ^ [N5: nat,M5: num] :
( product_case_prod @ nat @ num @ ( option @ num )
@ ^ [A6: nat,X2: num] :
( case_nat @ ( option @ num ) @ ( none @ num )
@ ^ [O: nat] :
( case_num @ ( option @ num ) @ ( some @ num @ one2 )
@ ^ [P5: num] :
( case_option @ ( option @ num ) @ num @ ( none @ num )
@ ^ [Q4: num] : ( some @ num @ ( bit0 @ Q4 ) )
@ ( bit_take_bit_num @ O @ P5 ) )
@ ^ [P5: num] : ( some @ num @ ( case_option @ num @ num @ one2 @ bit1 @ ( bit_take_bit_num @ O @ P5 ) ) )
@ X2 )
@ A6 )
@ ( product_Pair @ nat @ num @ N5 @ M5 ) ) ) ) ).
% take_bit_num_code
thf(fact_2234_num__of__integer__code,axiom,
( code_num_of_integer
= ( ^ [K3: code_integer] :
( if @ num @ ( ord_less_eq @ code_integer @ K3 @ ( one_one @ code_integer ) ) @ one2
@ ( product_case_prod @ code_integer @ code_integer @ num
@ ^ [L4: code_integer,J3: code_integer] :
( if @ num
@ ( J3
= ( zero_zero @ code_integer ) )
@ ( plus_plus @ num @ ( code_num_of_integer @ L4 ) @ ( code_num_of_integer @ L4 ) )
@ ( plus_plus @ num @ ( plus_plus @ num @ ( code_num_of_integer @ L4 ) @ ( code_num_of_integer @ L4 ) ) @ one2 ) )
@ ( code_divmod_integer @ K3 @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% num_of_integer_code
thf(fact_2235_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_2236_mask__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se2239418461657761734s_mask @ A @ ( zero_zero @ nat ) )
= ( zero_zero @ A ) ) ) ).
% mask_0
thf(fact_2237_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_2238_case__nat__numeral,axiom,
! [A: $tType,A3: A,F: nat > A,V: num] :
( ( case_nat @ A @ A3 @ F @ ( numeral_numeral @ nat @ V ) )
= ( F @ ( pred_numeral @ V ) ) ) ).
% case_nat_numeral
thf(fact_2239_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_2240_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_2241_case__nat__add__eq__if,axiom,
! [A: $tType,A3: A,F: nat > A,V: num,N: nat] :
( ( case_nat @ A @ A3 @ F @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ V ) @ N ) )
= ( F @ ( plus_plus @ nat @ ( pred_numeral @ V ) @ N ) ) ) ).
% case_nat_add_eq_if
thf(fact_2242_nat_Ocase__distrib,axiom,
! [B: $tType,A: $tType,H: A > B,F1: A,F23: nat > A,Nat: nat] :
( ( H @ ( case_nat @ A @ F1 @ F23 @ Nat ) )
= ( case_nat @ B @ ( H @ F1 )
@ ^ [X2: nat] : ( H @ ( F23 @ X2 ) )
@ Nat ) ) ).
% nat.case_distrib
thf(fact_2243_num_Ocase__distrib,axiom,
! [B: $tType,A: $tType,H: A > B,F1: A,F23: num > A,F32: num > A,Num: num] :
( ( H @ ( case_num @ A @ F1 @ F23 @ F32 @ Num ) )
= ( case_num @ B @ ( H @ F1 )
@ ^ [X2: num] : ( H @ ( F23 @ X2 ) )
@ ^ [X2: num] : ( H @ ( F32 @ X2 ) )
@ Num ) ) ).
% num.case_distrib
thf(fact_2244_old_Onat_Osimps_I5_J,axiom,
! [A: $tType,F1: A,F23: nat > A,X22: nat] :
( ( case_nat @ A @ F1 @ F23 @ ( suc @ X22 ) )
= ( F23 @ X22 ) ) ).
% old.nat.simps(5)
thf(fact_2245_old_Onat_Osimps_I4_J,axiom,
! [A: $tType,F1: A,F23: nat > A] :
( ( case_nat @ A @ F1 @ F23 @ ( zero_zero @ nat ) )
= F1 ) ).
% old.nat.simps(4)
thf(fact_2246_verit__eq__simplify_I17_J,axiom,
! [A: $tType,F1: A,F23: num > A,F32: num > A,X22: num] :
( ( case_num @ A @ F1 @ F23 @ F32 @ ( bit0 @ X22 ) )
= ( F23 @ X22 ) ) ).
% verit_eq_simplify(17)
thf(fact_2247_verit__eq__simplify_I16_J,axiom,
! [A: $tType,F1: A,F23: num > A,F32: num > A] :
( ( case_num @ A @ F1 @ F23 @ F32 @ one2 )
= F1 ) ).
% verit_eq_simplify(16)
thf(fact_2248_verit__eq__simplify_I18_J,axiom,
! [A: $tType,F1: A,F23: num > A,F32: num > A,X32: num] :
( ( case_num @ A @ F1 @ F23 @ F32 @ ( bit1 @ X32 ) )
= ( F32 @ X32 ) ) ).
% verit_eq_simplify(18)
thf(fact_2249_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_2250_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_2251_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_2252_Nitpick_Ocase__nat__unfold,axiom,
! [A: $tType] :
( ( case_nat @ A )
= ( ^ [X2: A,F2: nat > A,N5: nat] :
( if @ A
@ ( N5
= ( zero_zero @ nat ) )
@ X2
@ ( F2 @ ( minus_minus @ nat @ N5 @ ( one_one @ nat ) ) ) ) ) ) ).
% Nitpick.case_nat_unfold
thf(fact_2253_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_2254_divmod__abs__def,axiom,
( code_divmod_abs
= ( ^ [K3: code_integer,L4: code_integer] : ( product_Pair @ code_integer @ code_integer @ ( divide_divide @ code_integer @ ( abs_abs @ code_integer @ K3 ) @ ( abs_abs @ code_integer @ L4 ) ) @ ( modulo_modulo @ code_integer @ ( abs_abs @ code_integer @ K3 ) @ ( abs_abs @ code_integer @ L4 ) ) ) ) ) ).
% divmod_abs_def
thf(fact_2255_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_2256_mask__nat__def,axiom,
( ( bit_se2239418461657761734s_mask @ nat )
= ( ^ [N5: nat] : ( minus_minus @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N5 ) @ ( one_one @ nat ) ) ) ) ).
% mask_nat_def
thf(fact_2257_mask__eq__exp__minus__1,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se2239418461657761734s_mask @ A )
= ( ^ [N5: nat] : ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N5 ) @ ( one_one @ A ) ) ) ) ) ).
% mask_eq_exp_minus_1
thf(fact_2258_divmod__integer__def,axiom,
( code_divmod_integer
= ( ^ [K3: code_integer,L4: code_integer] : ( product_Pair @ code_integer @ code_integer @ ( divide_divide @ code_integer @ K3 @ L4 ) @ ( modulo_modulo @ code_integer @ K3 @ L4 ) ) ) ) ).
% divmod_integer_def
thf(fact_2259_nat__of__integer__code,axiom,
( code_nat_of_integer
= ( ^ [K3: code_integer] :
( if @ nat @ ( ord_less_eq @ code_integer @ K3 @ ( zero_zero @ code_integer ) ) @ ( zero_zero @ nat )
@ ( product_case_prod @ code_integer @ code_integer @ nat
@ ^ [L4: code_integer,J3: code_integer] :
( if @ nat
@ ( J3
= ( zero_zero @ code_integer ) )
@ ( plus_plus @ nat @ ( code_nat_of_integer @ L4 ) @ ( code_nat_of_integer @ L4 ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( code_nat_of_integer @ L4 ) @ ( code_nat_of_integer @ L4 ) ) @ ( one_one @ nat ) ) )
@ ( code_divmod_integer @ K3 @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% nat_of_integer_code
thf(fact_2260_divmod__integer__code,axiom,
( code_divmod_integer
= ( ^ [K3: code_integer,L4: code_integer] :
( if @ ( product_prod @ code_integer @ code_integer )
@ ( K3
= ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( zero_zero @ code_integer ) @ ( zero_zero @ code_integer ) )
@ ( if @ ( product_prod @ code_integer @ code_integer ) @ ( ord_less @ code_integer @ ( zero_zero @ code_integer ) @ L4 )
@ ( if @ ( product_prod @ code_integer @ code_integer ) @ ( ord_less @ code_integer @ ( zero_zero @ code_integer ) @ K3 ) @ ( code_divmod_abs @ K3 @ L4 )
@ ( product_case_prod @ code_integer @ code_integer @ ( product_prod @ code_integer @ code_integer )
@ ^ [R4: code_integer,S2: code_integer] :
( if @ ( product_prod @ code_integer @ code_integer )
@ ( S2
= ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( uminus_uminus @ code_integer @ R4 ) @ ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( minus_minus @ code_integer @ ( uminus_uminus @ code_integer @ R4 ) @ ( one_one @ code_integer ) ) @ ( minus_minus @ code_integer @ L4 @ S2 ) ) )
@ ( code_divmod_abs @ K3 @ L4 ) ) )
@ ( if @ ( product_prod @ code_integer @ code_integer )
@ ( L4
= ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( zero_zero @ code_integer ) @ K3 )
@ ( product_apsnd @ code_integer @ code_integer @ code_integer @ ( uminus_uminus @ code_integer )
@ ( if @ ( product_prod @ code_integer @ code_integer ) @ ( ord_less @ code_integer @ K3 @ ( zero_zero @ code_integer ) ) @ ( code_divmod_abs @ K3 @ L4 )
@ ( product_case_prod @ code_integer @ code_integer @ ( product_prod @ code_integer @ code_integer )
@ ^ [R4: code_integer,S2: code_integer] :
( if @ ( product_prod @ code_integer @ code_integer )
@ ( S2
= ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( uminus_uminus @ code_integer @ R4 ) @ ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( minus_minus @ code_integer @ ( uminus_uminus @ code_integer @ R4 ) @ ( one_one @ code_integer ) ) @ ( minus_minus @ code_integer @ ( uminus_uminus @ code_integer @ L4 ) @ S2 ) ) )
@ ( code_divmod_abs @ K3 @ L4 ) ) ) ) ) ) ) ) ) ).
% divmod_integer_code
thf(fact_2261_divmod__integer__eq__cases,axiom,
( code_divmod_integer
= ( ^ [K3: code_integer,L4: code_integer] :
( if @ ( product_prod @ code_integer @ code_integer )
@ ( K3
= ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( zero_zero @ code_integer ) @ ( zero_zero @ code_integer ) )
@ ( if @ ( product_prod @ code_integer @ code_integer )
@ ( L4
= ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( zero_zero @ code_integer ) @ K3 )
@ ( comp @ code_integer @ ( ( product_prod @ code_integer @ code_integer ) > ( product_prod @ code_integer @ code_integer ) ) @ code_integer @ ( comp @ ( code_integer > code_integer ) @ ( ( product_prod @ code_integer @ code_integer ) > ( product_prod @ code_integer @ code_integer ) ) @ code_integer @ ( product_apsnd @ code_integer @ code_integer @ code_integer ) @ ( times_times @ code_integer ) ) @ ( sgn_sgn @ code_integer ) @ L4
@ ( if @ ( product_prod @ code_integer @ code_integer )
@ ( ( sgn_sgn @ code_integer @ K3 )
= ( sgn_sgn @ code_integer @ L4 ) )
@ ( code_divmod_abs @ K3 @ L4 )
@ ( product_case_prod @ code_integer @ code_integer @ ( product_prod @ code_integer @ code_integer )
@ ^ [R4: code_integer,S2: code_integer] :
( if @ ( product_prod @ code_integer @ code_integer )
@ ( S2
= ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( uminus_uminus @ code_integer @ R4 ) @ ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( minus_minus @ code_integer @ ( uminus_uminus @ code_integer @ R4 ) @ ( one_one @ code_integer ) ) @ ( minus_minus @ code_integer @ ( abs_abs @ code_integer @ L4 ) @ S2 ) ) )
@ ( code_divmod_abs @ K3 @ L4 ) ) ) ) ) ) ) ) ).
% divmod_integer_eq_cases
thf(fact_2262_int__of__integer__code,axiom,
( code_int_of_integer
= ( ^ [K3: code_integer] :
( if @ int @ ( ord_less @ code_integer @ K3 @ ( zero_zero @ code_integer ) ) @ ( uminus_uminus @ int @ ( code_int_of_integer @ ( uminus_uminus @ code_integer @ K3 ) ) )
@ ( if @ int
@ ( K3
= ( zero_zero @ code_integer ) )
@ ( zero_zero @ int )
@ ( product_case_prod @ code_integer @ code_integer @ int
@ ^ [L4: code_integer,J3: code_integer] :
( if @ int
@ ( J3
= ( zero_zero @ code_integer ) )
@ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( code_int_of_integer @ L4 ) )
@ ( plus_plus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( code_int_of_integer @ L4 ) ) @ ( one_one @ int ) ) )
@ ( code_divmod_integer @ K3 @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% int_of_integer_code
thf(fact_2263_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_2264_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_2265_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_2266_and__zero__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A] :
( ( bit_se5824344872417868541ns_and @ A @ A3 @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% and_zero_eq
thf(fact_2267_zero__and__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A] :
( ( bit_se5824344872417868541ns_and @ A @ ( zero_zero @ A ) @ A3 )
= ( zero_zero @ A ) ) ) ).
% zero_and_eq
thf(fact_2268_bit_Oconj__zero__left,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X: A] :
( ( bit_se5824344872417868541ns_and @ A @ ( zero_zero @ A ) @ X )
= ( zero_zero @ A ) ) ) ).
% bit.conj_zero_left
thf(fact_2269_bit_Oconj__zero__right,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X: A] :
( ( bit_se5824344872417868541ns_and @ A @ X @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% bit.conj_zero_right
thf(fact_2270_apsnd__conv,axiom,
! [A: $tType,B: $tType,C: $tType,F: C > B,X: A,Y3: C] :
( ( product_apsnd @ C @ B @ A @ F @ ( product_Pair @ A @ C @ X @ Y3 ) )
= ( product_Pair @ A @ B @ X @ ( F @ Y3 ) ) ) ).
% apsnd_conv
thf(fact_2271_bit_Oconj__one__right,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X: A] :
( ( bit_se5824344872417868541ns_and @ A @ X @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= X ) ) ).
% bit.conj_one_right
thf(fact_2272_and_Oright__neutral,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A3: A] :
( ( bit_se5824344872417868541ns_and @ A @ A3 @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= A3 ) ) ).
% and.right_neutral
thf(fact_2273_and_Oleft__neutral,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A3: A] :
( ( bit_se5824344872417868541ns_and @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ A3 )
= A3 ) ) ).
% and.left_neutral
thf(fact_2274_and__numerals_I2_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [Y3: num] :
( ( bit_se5824344872417868541ns_and @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit1 @ Y3 ) ) )
= ( one_one @ A ) ) ) ).
% and_numerals(2)
thf(fact_2275_and__numerals_I8_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num] :
( ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ ( bit1 @ X ) ) @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% and_numerals(8)
thf(fact_2276_and__numerals_I5_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num] :
( ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ ( bit0 @ X ) ) @ ( one_one @ A ) )
= ( zero_zero @ A ) ) ) ).
% and_numerals(5)
thf(fact_2277_and__numerals_I1_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [Y3: num] :
( ( bit_se5824344872417868541ns_and @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ Y3 ) ) )
= ( zero_zero @ A ) ) ) ).
% and_numerals(1)
thf(fact_2278_and__numerals_I7_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num,Y3: num] :
( ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ ( bit1 @ X ) ) @ ( numeral_numeral @ A @ ( bit1 @ Y3 ) ) )
= ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ X ) @ ( numeral_numeral @ A @ Y3 ) ) ) ) ) ) ).
% and_numerals(7)
thf(fact_2279_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_2280_apsnd__compose,axiom,
! [C: $tType,B: $tType,D: $tType,A: $tType,F: C > B,G: D > C,X: product_prod @ A @ D] :
( ( product_apsnd @ C @ B @ A @ F @ ( product_apsnd @ D @ C @ A @ G @ X ) )
= ( product_apsnd @ D @ B @ A @ ( comp @ C @ B @ D @ F @ G ) @ X ) ) ).
% apsnd_compose
thf(fact_2281_and__eq__minus__1__iff,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A3: A,B2: A] :
( ( ( bit_se5824344872417868541ns_and @ A @ A3 @ B2 )
= ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( ( A3
= ( uminus_uminus @ A @ ( one_one @ A ) ) )
& ( B2
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ) ).
% and_eq_minus_1_iff
thf(fact_2282_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_2283_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_2284_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_2285_one__and__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A] :
( ( bit_se5824344872417868541ns_and @ A @ ( one_one @ A ) @ A3 )
= ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% one_and_eq
thf(fact_2286_and__one__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A] :
( ( bit_se5824344872417868541ns_and @ A @ A3 @ ( one_one @ A ) )
= ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% and_one_eq
thf(fact_2287_diff__Suc,axiom,
! [M: nat,N: nat] :
( ( minus_minus @ nat @ M @ ( suc @ N ) )
= ( case_nat @ nat @ ( zero_zero @ nat )
@ ^ [K3: nat] : K3
@ ( minus_minus @ nat @ M @ N ) ) ) ).
% diff_Suc
thf(fact_2288_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_2289_nat_Osplit__sels_I1_J,axiom,
! [A: $tType,P: A > $o,F1: A,F23: nat > A,Nat: nat] :
( ( P @ ( case_nat @ A @ F1 @ F23 @ Nat ) )
= ( ( ( Nat
= ( zero_zero @ nat ) )
=> ( P @ F1 ) )
& ( ( Nat
= ( suc @ ( pred2 @ Nat ) ) )
=> ( P @ ( F23 @ ( pred2 @ Nat ) ) ) ) ) ) ).
% nat.split_sels(1)
thf(fact_2290_nat_Osplit__sels_I2_J,axiom,
! [A: $tType,P: A > $o,F1: A,F23: nat > A,Nat: nat] :
( ( P @ ( case_nat @ A @ F1 @ F23 @ Nat ) )
= ( ~ ( ( ( Nat
= ( zero_zero @ nat ) )
& ~ ( P @ F1 ) )
| ( ( Nat
= ( suc @ ( pred2 @ Nat ) ) )
& ~ ( P @ ( F23 @ ( pred2 @ Nat ) ) ) ) ) ) ) ).
% nat.split_sels(2)
thf(fact_2291_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
@ ^ [Q4: nat] : ( ord_less @ nat @ Q4 @ N ) ) ) ) ).
% mask_eq_sum_exp_nat
thf(fact_2292_rec__nat__add__eq__if,axiom,
! [A: $tType,A3: A,F: nat > A > A,V: num,N: nat] :
( ( rec_nat @ A @ A3 @ F @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ V ) @ N ) )
= ( F @ ( plus_plus @ nat @ ( pred_numeral @ V ) @ N ) @ ( rec_nat @ A @ A3 @ F @ ( plus_plus @ nat @ ( pred_numeral @ V ) @ N ) ) ) ) ).
% rec_nat_add_eq_if
thf(fact_2293_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_2294_Ints__sum,axiom,
! [A: $tType,B: $tType] :
( ( ring_1 @ B )
=> ! [A5: set @ A,F: A > B] :
( ! [X4: A] :
( ( member @ A @ X4 @ A5 )
=> ( member @ B @ ( F @ X4 ) @ ( ring_1_Ints @ B ) ) )
=> ( member @ B @ ( groups7311177749621191930dd_sum @ A @ B @ F @ A5 ) @ ( ring_1_Ints @ B ) ) ) ) ).
% Ints_sum
thf(fact_2295_fst__comp__apsnd,axiom,
! [C: $tType,B: $tType,A: $tType,F: B > C] :
( ( comp @ ( product_prod @ A @ C ) @ A @ ( product_prod @ A @ B ) @ ( product_fst @ A @ C ) @ ( product_apsnd @ B @ C @ A @ F ) )
= ( product_fst @ A @ B ) ) ).
% fst_comp_apsnd
thf(fact_2296_fst__apsnd,axiom,
! [B: $tType,C: $tType,A: $tType,F: C > B,X: product_prod @ A @ C] :
( ( product_fst @ A @ B @ ( product_apsnd @ C @ B @ A @ F @ X ) )
= ( product_fst @ A @ C @ X ) ) ).
% fst_apsnd
thf(fact_2297_old_Onat_Osimps_I7_J,axiom,
! [T: $tType,F1: T,F23: nat > T > T,Nat: nat] :
( ( rec_nat @ T @ F1 @ F23 @ ( suc @ Nat ) )
= ( F23 @ Nat @ ( rec_nat @ T @ F1 @ F23 @ Nat ) ) ) ).
% old.nat.simps(7)
thf(fact_2298_old_Onat_Osimps_I6_J,axiom,
! [T: $tType,F1: T,F23: nat > T > T] :
( ( rec_nat @ T @ F1 @ F23 @ ( zero_zero @ nat ) )
= F1 ) ).
% old.nat.simps(6)
thf(fact_2299_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
@ ^ [X2: B] : ( ring_1_of_int @ A @ ( F @ X2 ) )
@ A5 ) ) ) ).
% of_int_sum
thf(fact_2300_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
@ ^ [X2: B] : ( semiring_1_of_nat @ A @ ( F @ X2 ) )
@ A5 ) ) ) ).
% of_nat_sum
thf(fact_2301_rec__nat__numeral,axiom,
! [A: $tType,A3: A,F: nat > A > A,V: num] :
( ( rec_nat @ A @ A3 @ F @ ( numeral_numeral @ nat @ V ) )
= ( F @ ( pred_numeral @ V ) @ ( rec_nat @ A @ A3 @ F @ ( pred_numeral @ V ) ) ) ) ).
% rec_nat_numeral
thf(fact_2302_and__nat__numerals_I1_J,axiom,
! [Y3: num] :
( ( bit_se5824344872417868541ns_and @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( numeral_numeral @ nat @ ( bit0 @ Y3 ) ) )
= ( zero_zero @ nat ) ) ).
% and_nat_numerals(1)
thf(fact_2303_and__nat__numerals_I3_J,axiom,
! [X: num] :
( ( bit_se5824344872417868541ns_and @ nat @ ( numeral_numeral @ nat @ ( bit0 @ X ) ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( zero_zero @ nat ) ) ).
% and_nat_numerals(3)
thf(fact_2304_and__nat__numerals_I4_J,axiom,
! [X: num] :
( ( bit_se5824344872417868541ns_and @ nat @ ( numeral_numeral @ nat @ ( bit1 @ X ) ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( one_one @ nat ) ) ).
% and_nat_numerals(4)
thf(fact_2305_and__nat__numerals_I2_J,axiom,
! [Y3: num] :
( ( bit_se5824344872417868541ns_and @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( numeral_numeral @ nat @ ( bit1 @ Y3 ) ) )
= ( one_one @ nat ) ) ).
% and_nat_numerals(2)
thf(fact_2306_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_2307_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_2308_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_2309_fstI,axiom,
! [B: $tType,A: $tType,X: product_prod @ A @ B,Y3: A,Z: B] :
( ( X
= ( product_Pair @ A @ B @ Y3 @ Z ) )
=> ( ( product_fst @ A @ B @ X )
= Y3 ) ) ).
% fstI
thf(fact_2310_fst__eqD,axiom,
! [B: $tType,A: $tType,X: A,Y3: B,A3: A] :
( ( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X @ Y3 ) )
= A3 )
=> ( X = A3 ) ) ).
% fst_eqD
thf(fact_2311_fst__conv,axiom,
! [B: $tType,A: $tType,X1: A,X22: B] :
( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X1 @ X22 ) )
= X1 ) ).
% fst_conv
thf(fact_2312_mod__sum__eq,axiom,
! [B: $tType,A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [F: B > A,A3: A,A5: set @ B] :
( ( modulo_modulo @ A
@ ( groups7311177749621191930dd_sum @ B @ A
@ ^ [I2: B] : ( modulo_modulo @ A @ ( F @ I2 ) @ A3 )
@ A5 )
@ A3 )
= ( modulo_modulo @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F @ A5 ) @ A3 ) ) ) ).
% mod_sum_eq
thf(fact_2313_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_2314_sum__power__add,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X: A,M: nat,I5: set @ nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( power_power @ A @ X @ ( plus_plus @ nat @ M @ I2 ) )
@ I5 )
= ( times_times @ A @ ( power_power @ A @ X @ M ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ I5 ) ) ) ) ).
% sum_power_add
thf(fact_2315_pred__def,axiom,
( pred2
= ( case_nat @ nat @ ( zero_zero @ nat )
@ ^ [X24: nat] : X24 ) ) ).
% pred_def
thf(fact_2316_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_2317_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
@ ^ [Q4: nat] : ( ord_less @ nat @ Q4 @ N ) ) ) ) ) ).
% mask_eq_sum_exp
thf(fact_2318_and__nat__unfold,axiom,
( ( bit_se5824344872417868541ns_and @ nat )
= ( ^ [M5: nat,N5: nat] :
( if @ nat
@ ( ( M5
= ( zero_zero @ nat ) )
| ( N5
= ( zero_zero @ nat ) ) )
@ ( zero_zero @ nat )
@ ( plus_plus @ nat @ ( times_times @ nat @ ( modulo_modulo @ nat @ M5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ nat @ N5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ nat @ ( divide_divide @ nat @ M5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ).
% and_nat_unfold
thf(fact_2319_and__nat__rec,axiom,
( ( bit_se5824344872417868541ns_and @ nat )
= ( ^ [M5: nat,N5: nat] :
( plus_plus @ nat
@ ( zero_neq_one_of_bool @ nat
@ ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M5 )
& ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N5 ) ) )
@ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ nat @ ( divide_divide @ nat @ M5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% and_nat_rec
thf(fact_2320_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_2321_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_2322_convex__sum__bound__le,axiom,
! [A: $tType,B: $tType] :
( ( linordered_idom @ B )
=> ! [I5: set @ A,X: A > B,A3: A > B,B2: B,Delta: B] :
( ! [I3: A] :
( ( member @ A @ I3 @ I5 )
=> ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( X @ I3 ) ) )
=> ( ( ( groups7311177749621191930dd_sum @ A @ B @ X @ I5 )
= ( one_one @ B ) )
=> ( ! [I3: A] :
( ( member @ A @ I3 @ I5 )
=> ( ord_less_eq @ B @ ( abs_abs @ B @ ( minus_minus @ B @ ( A3 @ I3 ) @ B2 ) ) @ Delta ) )
=> ( ord_less_eq @ B
@ ( abs_abs @ B
@ ( minus_minus @ B
@ ( groups7311177749621191930dd_sum @ A @ B
@ ^ [I2: A] : ( times_times @ B @ ( A3 @ I2 ) @ ( X @ I2 ) )
@ I5 )
@ B2 ) )
@ Delta ) ) ) ) ) ).
% convex_sum_bound_le
thf(fact_2323_abs__sum__abs,axiom,
! [B: $tType,A: $tType] :
( ( ordere166539214618696060dd_abs @ B )
=> ! [F: A > B,A5: set @ A] :
( ( abs_abs @ B
@ ( groups7311177749621191930dd_sum @ A @ B
@ ^ [A6: A] : ( abs_abs @ B @ ( F @ A6 ) )
@ A5 ) )
= ( groups7311177749621191930dd_sum @ A @ B
@ ^ [A6: A] : ( abs_abs @ B @ ( F @ A6 ) )
@ A5 ) ) ) ).
% abs_sum_abs
thf(fact_2324_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_2325_sum__SucD,axiom,
! [A: $tType,F: A > nat,A5: set @ A,N: nat] :
( ( ( groups7311177749621191930dd_sum @ A @ nat @ F @ A5 )
= ( suc @ N ) )
=> ? [X4: A] :
( ( member @ A @ X4 @ A5 )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F @ X4 ) ) ) ) ).
% sum_SucD
thf(fact_2326_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
@ ^ [X2: B] : ( semiring_1_of_nat @ int @ ( F @ X2 ) )
@ A5 ) ) ).
% int_sum
thf(fact_2327_sum_Oswap,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: B > C > A,B4: set @ C,A5: set @ B] :
( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [I2: B] : ( groups7311177749621191930dd_sum @ C @ A @ ( G @ I2 ) @ B4 )
@ A5 )
= ( groups7311177749621191930dd_sum @ C @ A
@ ^ [J3: C] :
( groups7311177749621191930dd_sum @ B @ A
@ ^ [I2: B] : ( G @ I2 @ J3 )
@ A5 )
@ B4 ) ) ) ).
% sum.swap
thf(fact_2328_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_2329_sum_Oneutral,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A5: set @ B,G: B > A] :
( ! [X4: B] :
( ( member @ B @ X4 @ A5 )
=> ( ( G @ X4 )
= ( zero_zero @ A ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ A5 )
= ( zero_zero @ A ) ) ) ) ).
% sum.neutral
thf(fact_2330_sum__mono,axiom,
! [A: $tType,B: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [K5: set @ B,F: B > A,G: B > A] :
( ! [I3: B] :
( ( member @ B @ I3 @ K5 )
=> ( ord_less_eq @ A @ ( F @ I3 ) @ ( G @ I3 ) ) )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F @ K5 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ K5 ) ) ) ) ).
% sum_mono
thf(fact_2331_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
@ ^ [N5: B] : ( times_times @ A @ R @ ( F @ N5 ) )
@ A5 ) ) ) ).
% sum_distrib_left
thf(fact_2332_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
@ ^ [N5: B] : ( times_times @ A @ ( F @ N5 ) @ R )
@ A5 ) ) ) ).
% sum_distrib_right
thf(fact_2333_sum__product,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( semiring_0 @ B )
=> ! [F: A > B,A5: set @ A,G: C > B,B4: set @ C] :
( ( times_times @ B @ ( groups7311177749621191930dd_sum @ A @ B @ F @ A5 ) @ ( groups7311177749621191930dd_sum @ C @ B @ G @ B4 ) )
= ( groups7311177749621191930dd_sum @ A @ B
@ ^ [I2: A] :
( groups7311177749621191930dd_sum @ C @ B
@ ^ [J3: C] : ( times_times @ B @ ( F @ I2 ) @ ( G @ J3 ) )
@ B4 )
@ A5 ) ) ) ).
% sum_product
thf(fact_2334_sum_Odistrib,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: B > A,H: B > A,A5: set @ B] :
( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X2: B] : ( plus_plus @ A @ ( G @ X2 ) @ ( H @ X2 ) )
@ A5 )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G @ A5 ) @ ( groups7311177749621191930dd_sum @ B @ A @ H @ A5 ) ) ) ) ).
% sum.distrib
thf(fact_2335_sum__subtractf,axiom,
! [A: $tType,B: $tType] :
( ( ab_group_add @ A )
=> ! [F: B > A,G: B > A,A5: set @ B] :
( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X2: B] : ( minus_minus @ A @ ( F @ X2 ) @ ( G @ X2 ) )
@ A5 )
= ( minus_minus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F @ A5 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ A5 ) ) ) ) ).
% sum_subtractf
thf(fact_2336_sum__negf,axiom,
! [A: $tType,B: $tType] :
( ( ab_group_add @ A )
=> ! [F: B > A,A5: set @ B] :
( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X2: B] : ( uminus_uminus @ A @ ( F @ X2 ) )
@ A5 )
= ( uminus_uminus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F @ A5 ) ) ) ) ).
% sum_negf
thf(fact_2337_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
@ ^ [N5: B] : ( divide_divide @ A @ ( F @ N5 ) @ R )
@ A5 ) ) ) ).
% sum_divide_distrib
thf(fact_2338_sum__nonpos,axiom,
! [B: $tType,A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A5: set @ B,F: B > A] :
( ! [X4: B] :
( ( member @ B @ X4 @ A5 )
=> ( ord_less_eq @ A @ ( F @ X4 ) @ ( zero_zero @ A ) ) )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F @ A5 ) @ ( zero_zero @ A ) ) ) ) ).
% sum_nonpos
thf(fact_2339_sum__nonneg,axiom,
! [A: $tType,B: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A5: set @ B,F: B > A] :
( ! [X4: B] :
( ( member @ B @ X4 @ A5 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F @ X4 ) ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( groups7311177749621191930dd_sum @ B @ A @ F @ A5 ) ) ) ) ).
% sum_nonneg
thf(fact_2340_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 )
=> ( ! [X4: nat] :
( ( member @ nat @ ( suc @ X4 ) @ A5 )
=> ( ( F @ ( suc @ X4 ) )
= ( G @ ( suc @ X4 ) ) ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ F @ A5 )
= ( groups7311177749621191930dd_sum @ nat @ A @ G @ A5 ) ) ) ) ) ).
% sum_cong_Suc
thf(fact_2341_sum__subtractf__nat,axiom,
! [A: $tType,A5: set @ A,G: A > nat,F: A > nat] :
( ! [X4: A] :
( ( member @ A @ X4 @ A5 )
=> ( ord_less_eq @ nat @ ( G @ X4 ) @ ( F @ X4 ) ) )
=> ( ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [X2: A] : ( minus_minus @ nat @ ( F @ X2 ) @ ( G @ X2 ) )
@ A5 )
= ( minus_minus @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ F @ A5 ) @ ( groups7311177749621191930dd_sum @ A @ nat @ G @ A5 ) ) ) ) ).
% sum_subtractf_nat
thf(fact_2342_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 ) )
=> ( ! [X4: B,Y4: B] :
( ( H @ ( plus_plus @ B @ X4 @ Y4 ) )
= ( plus_plus @ A @ ( H @ X4 ) @ ( H @ Y4 ) ) )
=> ( ( groups7311177749621191930dd_sum @ C @ A @ ( comp @ B @ A @ C @ H @ G ) @ A5 )
= ( H @ ( groups7311177749621191930dd_sum @ C @ B @ G @ A5 ) ) ) ) ) ) ).
% sum_comp_morphism
thf(fact_2343_fst__diag__fst,axiom,
! [B: $tType,A: $tType] :
( ( comp @ ( product_prod @ A @ A ) @ A @ ( product_prod @ A @ B ) @ ( product_fst @ A @ A )
@ ( comp @ A @ ( product_prod @ A @ A ) @ ( product_prod @ A @ B )
@ ^ [X2: A] : ( product_Pair @ A @ A @ X2 @ X2 )
@ ( product_fst @ A @ B ) ) )
= ( product_fst @ A @ B ) ) ).
% fst_diag_fst
thf(fact_2344_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_2345_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_2346_gbinomial__partial__row__sum,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( times_times @ A @ ( gbinomial @ A @ A3 @ K3 ) @ ( minus_minus @ A @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( semiring_1_of_nat @ A @ K3 ) ) )
@ ( set_ord_atMost @ nat @ M ) )
= ( times_times @ A @ ( divide_divide @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ M ) @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( gbinomial @ A @ A3 @ ( plus_plus @ nat @ M @ ( one_one @ nat ) ) ) ) ) ) ).
% gbinomial_partial_row_sum
thf(fact_2347_sum__gp,axiom,
! [A: $tType] :
( ( ( division_ring @ A )
& ( comm_ring @ A ) )
=> ! [N: nat,M: nat,X: A] :
( ( ( ord_less @ nat @ N @ M )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( zero_zero @ A ) ) )
& ( ~ ( ord_less @ nat @ N @ M )
=> ( ( ( X
= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) @ M ) ) ) )
& ( ( X
!= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ X @ M ) @ ( power_power @ A @ X @ ( suc @ N ) ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ X ) ) ) ) ) ) ) ) ).
% sum_gp
thf(fact_2348_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_2349_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_2350_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_2351_atMost__atLeast0,axiom,
( ( set_ord_atMost @ nat )
= ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) ) ) ).
% atMost_atLeast0
thf(fact_2352_atMost__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( set_ord_atMost @ A )
= ( ^ [U: A] :
( collect @ A
@ ^ [X2: A] : ( ord_less_eq @ A @ X2 @ U ) ) ) ) ) ).
% atMost_def
thf(fact_2353_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_2354_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_2355_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_2356_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_2357_sum__power__shift,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [M: nat,N: nat,X: A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( times_times @ A @ ( power_power @ A @ X @ M ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_atMost @ nat @ ( minus_minus @ nat @ N @ M ) ) ) ) ) ) ) ).
% sum_power_shift
thf(fact_2358_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_2359_sum__choose__upper,axiom,
! [M: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [K3: nat] : ( binomial @ K3 @ M )
@ ( set_ord_atMost @ nat @ N ) )
= ( binomial @ ( suc @ N ) @ ( suc @ M ) ) ) ).
% sum_choose_upper
thf(fact_2360_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_2361_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_2362_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_2363_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_2364_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_2365_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_2366_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_2367_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_2368_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_2369_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
@ ^ [N5: nat] : ( minus_minus @ nat @ N5 @ ( 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_2370_sum__choose__lower,axiom,
! [R: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [K3: nat] : ( binomial @ ( plus_plus @ nat @ R @ K3 ) @ K3 )
@ ( set_ord_atMost @ nat @ N ) )
= ( binomial @ ( suc @ ( plus_plus @ nat @ R @ N ) ) @ N ) ) ).
% sum_choose_lower
thf(fact_2371_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_2372_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_2373_sum__atLeastAtMost__code,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [F: nat > A,A3: nat,B2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ F @ ( set_or1337092689740270186AtMost @ nat @ A3 @ B2 ) )
= ( set_fo6178422350223883121st_nat @ A
@ ^ [A6: nat] : ( plus_plus @ A @ ( F @ A6 ) )
@ A3
@ B2
@ ( zero_zero @ A ) ) ) ) ).
% sum_atLeastAtMost_code
thf(fact_2374_subset__CollectI,axiom,
! [A: $tType,B4: set @ A,A5: set @ A,Q: A > $o,P: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ B4 @ A5 )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ B4 )
=> ( ( Q @ X4 )
=> ( P @ X4 ) ) )
=> ( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [X2: A] :
( ( member @ A @ X2 @ B4 )
& ( Q @ X2 ) ) )
@ ( collect @ A
@ ^ [X2: A] :
( ( member @ A @ X2 @ A5 )
& ( P @ X2 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_2375_subset__Collect__iff,axiom,
! [A: $tType,B4: set @ A,A5: set @ A,P: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ B4 @ A5 )
=> ( ( ord_less_eq @ ( set @ A ) @ B4
@ ( collect @ A
@ ^ [X2: A] :
( ( member @ A @ X2 @ A5 )
& ( P @ X2 ) ) ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ B4 )
=> ( P @ X2 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_2376_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_2377_sum_Oub__add__nat,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M: nat,N: nat,G: nat > A,P4: 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 @ P4 ) ) )
= ( 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 @ P4 ) ) ) ) ) ) ) ).
% sum.ub_add_nat
thf(fact_2378_gbinomial__parallel__sum,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( gbinomial @ A @ ( plus_plus @ A @ A3 @ ( semiring_1_of_nat @ A @ K3 ) ) @ K3 )
@ ( set_ord_atMost @ nat @ N ) )
= ( gbinomial @ A @ ( plus_plus @ A @ ( plus_plus @ A @ A3 @ ( semiring_1_of_nat @ A @ N ) ) @ ( one_one @ A ) ) @ N ) ) ) ).
% gbinomial_parallel_sum
thf(fact_2379_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
@ ^ [K3: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( G @ I2 @ ( minus_minus @ nat @ K3 @ I2 ) )
@ ( set_ord_atMost @ nat @ K3 ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% sum.triangle_reindex_eq
thf(fact_2380_sum__choose__diagonal,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [K3: nat] : ( binomial @ ( minus_minus @ nat @ N @ K3 ) @ ( minus_minus @ nat @ M @ K3 ) )
@ ( set_ord_atMost @ nat @ M ) )
= ( binomial @ ( suc @ N ) @ M ) ) ) ).
% sum_choose_diagonal
thf(fact_2381_vandermonde,axiom,
! [M: nat,N: nat,R: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [K3: nat] : ( times_times @ nat @ ( binomial @ M @ K3 ) @ ( binomial @ N @ ( minus_minus @ nat @ R @ K3 ) ) )
@ ( set_ord_atMost @ nat @ R ) )
= ( binomial @ ( plus_plus @ nat @ M @ N ) @ R ) ) ).
% vandermonde
thf(fact_2382_sum__gp__basic,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X: A,N: nat] :
( ( times_times @ A @ ( minus_minus @ A @ ( one_one @ A ) @ X ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_atMost @ nat @ N ) ) )
= ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X @ ( suc @ N ) ) ) ) ) ).
% sum_gp_basic
thf(fact_2383_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
@ ^ [K3: nat] : ( minus_minus @ A @ ( F @ K3 ) @ ( F @ ( plus_plus @ nat @ K3 @ ( 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
@ ^ [K3: nat] : ( minus_minus @ A @ ( F @ K3 ) @ ( F @ ( plus_plus @ nat @ K3 @ ( one_one @ nat ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( zero_zero @ A ) ) ) ) ) ).
% sum_natinterval_diff
thf(fact_2384_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
@ ^ [K3: nat] : ( minus_minus @ A @ ( F @ K3 ) @ ( F @ ( minus_minus @ nat @ K3 @ ( one_one @ nat ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ N ) )
= ( minus_minus @ A @ ( F @ N ) @ ( F @ M ) ) ) ) ) ).
% sum_telescope''
thf(fact_2385_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_2386_binomial,axiom,
! [A3: nat,B2: nat,N: nat] :
( ( power_power @ nat @ ( plus_plus @ nat @ A3 @ B2 ) @ N )
= ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [K3: nat] : ( times_times @ nat @ ( times_times @ nat @ ( semiring_1_of_nat @ nat @ ( binomial @ N @ K3 ) ) @ ( power_power @ nat @ A3 @ K3 ) ) @ ( power_power @ nat @ B2 @ ( minus_minus @ nat @ N @ K3 ) ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ).
% binomial
thf(fact_2387_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_2388_gbinomial__sum__lower__neg,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( times_times @ A @ ( gbinomial @ A @ A3 @ K3 ) @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K3 ) )
@ ( set_ord_atMost @ nat @ M ) )
= ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ M ) @ ( gbinomial @ A @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) @ M ) ) ) ) ).
% gbinomial_sum_lower_neg
thf(fact_2389_binomial__ring,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A3: A,B2: A,N: nat] :
( ( power_power @ A @ ( plus_plus @ A @ A3 @ B2 ) @ N )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( times_times @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( binomial @ N @ K3 ) ) @ ( power_power @ A @ A3 @ K3 ) ) @ ( power_power @ A @ B2 @ ( minus_minus @ nat @ N @ K3 ) ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% binomial_ring
thf(fact_2390_pochhammer__binomial__sum,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [A3: A,B2: A,N: nat] :
( ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ A3 @ B2 ) @ N )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( times_times @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( binomial @ N @ K3 ) ) @ ( comm_s3205402744901411588hammer @ A @ A3 @ K3 ) ) @ ( comm_s3205402744901411588hammer @ A @ B2 @ ( minus_minus @ nat @ N @ K3 ) ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% pochhammer_binomial_sum
thf(fact_2391_sum__gp__multiplied,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [M: nat,N: nat,X: A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( times_times @ A @ ( minus_minus @ A @ ( one_one @ A ) @ X ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) )
= ( minus_minus @ A @ ( power_power @ A @ X @ M ) @ ( power_power @ A @ X @ ( suc @ N ) ) ) ) ) ) ).
% sum_gp_multiplied
thf(fact_2392_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_2393_choose__square__sum,axiom,
! [N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [K3: nat] : ( power_power @ nat @ ( binomial @ N @ K3 ) @ ( 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_2394_sum_Ozero__middle,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [P4: nat,K: nat,G: nat > A,H: nat > A] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ P4 )
=> ( ( ord_less_eq @ nat @ K @ P4 )
=> ( ( 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 @ P4 ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J3: nat] : ( if @ A @ ( ord_less @ nat @ J3 @ K ) @ ( G @ J3 ) @ ( H @ J3 ) )
@ ( set_ord_atMost @ nat @ ( minus_minus @ nat @ P4 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) ) ).
% sum.zero_middle
thf(fact_2395_gbinomial__partial__sum__poly,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [M: nat,A3: A,X: A,Y3: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( times_times @ A @ ( times_times @ A @ ( gbinomial @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ M ) @ A3 ) @ K3 ) @ ( power_power @ A @ X @ K3 ) ) @ ( power_power @ A @ Y3 @ ( minus_minus @ nat @ M @ K3 ) ) )
@ ( set_ord_atMost @ nat @ M ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( times_times @ A @ ( times_times @ A @ ( gbinomial @ A @ ( uminus_uminus @ A @ A3 ) @ K3 ) @ ( power_power @ A @ ( uminus_uminus @ A @ X ) @ K3 ) ) @ ( power_power @ A @ ( plus_plus @ A @ X @ Y3 ) @ ( minus_minus @ nat @ M @ K3 ) ) )
@ ( set_ord_atMost @ nat @ M ) ) ) ) ).
% gbinomial_partial_sum_poly
thf(fact_2396_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_2397_gauss__sum__nat,axiom,
! [N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [X2: nat] : X2
@ ( 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_2398_sum__gp0,axiom,
! [A: $tType] :
( ( ( division_ring @ A )
& ( comm_ring @ A ) )
=> ! [X: A,N: nat] :
( ( ( X
= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_atMost @ nat @ N ) )
= ( semiring_1_of_nat @ A @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) ) )
& ( ( X
!= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_atMost @ nat @ N ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X @ ( suc @ N ) ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ X ) ) ) ) ) ) ).
% sum_gp0
thf(fact_2399_gbinomial__partial__sum__poly__xpos,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [M: nat,A3: A,X: A,Y3: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( times_times @ A @ ( times_times @ A @ ( gbinomial @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ M ) @ A3 ) @ K3 ) @ ( power_power @ A @ X @ K3 ) ) @ ( power_power @ A @ Y3 @ ( minus_minus @ nat @ M @ K3 ) ) )
@ ( set_ord_atMost @ nat @ M ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( times_times @ A @ ( times_times @ A @ ( gbinomial @ A @ ( minus_minus @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ K3 ) @ A3 ) @ ( one_one @ A ) ) @ K3 ) @ ( power_power @ A @ X @ K3 ) ) @ ( power_power @ A @ ( plus_plus @ A @ X @ Y3 ) @ ( minus_minus @ nat @ M @ K3 ) ) )
@ ( set_ord_atMost @ nat @ M ) ) ) ) ).
% gbinomial_partial_sum_poly_xpos
thf(fact_2400_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_2401_gbinomial__sum__nat__pow2,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( divide_divide @ A @ ( gbinomial @ A @ ( semiring_1_of_nat @ A @ ( plus_plus @ nat @ M @ K3 ) ) @ K3 ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ K3 ) )
@ ( set_ord_atMost @ nat @ M ) )
= ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) ) ) ).
% gbinomial_sum_nat_pow2
thf(fact_2402_double__arith__series,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A3: A,D3: A,N: nat] :
( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( plus_plus @ A @ A3 @ ( times_times @ A @ ( semiring_1_of_nat @ A @ I2 ) @ D3 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) )
= ( times_times @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N ) @ ( one_one @ A ) ) @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ D3 ) ) ) ) ) ).
% double_arith_series
thf(fact_2403_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_2404_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_2405_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_2406_arith__series__nat,axiom,
! [A3: nat,D3: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [I2: nat] : ( plus_plus @ nat @ A3 @ ( times_times @ nat @ I2 @ D3 ) )
@ ( 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 ) ) @ A3 ) @ ( times_times @ nat @ N @ D3 ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% arith_series_nat
thf(fact_2407_Sum__Icc__nat,axiom,
! [M: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [X2: nat] : X2
@ ( 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_2408_rec__nat__Suc__imp,axiom,
! [A: $tType,F: nat > A,F1: A,F23: nat > A > A,N: nat] :
( ( F
= ( rec_nat @ A @ F1 @ F23 ) )
=> ( ( F @ ( suc @ N ) )
= ( F23 @ N @ ( F @ N ) ) ) ) ).
% rec_nat_Suc_imp
thf(fact_2409_rec__nat__0__imp,axiom,
! [A: $tType,F: nat > A,F1: A,F23: nat > A > A] :
( ( F
= ( rec_nat @ A @ F1 @ F23 ) )
=> ( ( F @ ( zero_zero @ nat ) )
= F1 ) ) ).
% rec_nat_0_imp
thf(fact_2410_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_2411_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_2412_arith__series,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A3: A,D3: A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( plus_plus @ A @ A3 @ ( times_times @ A @ ( semiring_1_of_nat @ A @ I2 ) @ D3 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
= ( divide_divide @ A @ ( times_times @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N ) @ ( one_one @ A ) ) @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ D3 ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% arith_series
thf(fact_2413_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_2414_sum__gp__offset,axiom,
! [A: $tType] :
( ( ( division_ring @ A )
& ( comm_ring @ A ) )
=> ! [X: A,M: nat,N: nat] :
( ( ( X
= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M @ ( plus_plus @ nat @ M @ N ) ) )
= ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N ) @ ( one_one @ A ) ) ) )
& ( ( X
!= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M @ ( plus_plus @ nat @ M @ N ) ) )
= ( divide_divide @ A @ ( times_times @ A @ ( power_power @ A @ X @ M ) @ ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X @ ( suc @ N ) ) ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ X ) ) ) ) ) ) ).
% sum_gp_offset
thf(fact_2415_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_2416_gchoose__row__sum__weighted,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [R: A,M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( times_times @ A @ ( gbinomial @ A @ R @ K3 ) @ ( minus_minus @ A @ ( divide_divide @ A @ R @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( semiring_1_of_nat @ A @ K3 ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ M ) )
= ( times_times @ A @ ( divide_divide @ A @ ( semiring_1_of_nat @ A @ ( suc @ M ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( gbinomial @ A @ R @ ( suc @ M ) ) ) ) ) ).
% gchoose_row_sum_weighted
thf(fact_2417_pochhammer__times__pochhammer__half,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [Z: A,N: nat] :
( ( times_times @ A @ ( comm_s3205402744901411588hammer @ A @ Z @ ( suc @ N ) ) @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ Z @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( suc @ N ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [K3: nat] : ( plus_plus @ A @ Z @ ( divide_divide @ A @ ( semiring_1_of_nat @ A @ K3 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ nat ) ) ) ) ) ) ).
% pochhammer_times_pochhammer_half
thf(fact_2418_ex__nat__less,axiom,
! [N: nat,P: nat > $o] :
( ( ? [M5: nat] :
( ( ord_less_eq @ nat @ M5 @ N )
& ( P @ M5 ) ) )
= ( ? [X2: nat] :
( ( member @ nat @ X2 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
& ( P @ X2 ) ) ) ) ).
% ex_nat_less
thf(fact_2419_all__nat__less,axiom,
! [N: nat,P: nat > $o] :
( ( ! [M5: nat] :
( ( ord_less_eq @ nat @ M5 @ N )
=> ( P @ M5 ) ) )
= ( ! [X2: nat] :
( ( member @ nat @ X2 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
=> ( P @ X2 ) ) ) ) ).
% all_nat_less
thf(fact_2420_bezw_Osimps,axiom,
( bezw
= ( ^ [X2: nat,Y: nat] :
( if @ ( product_prod @ int @ int )
@ ( Y
= ( zero_zero @ nat ) )
@ ( product_Pair @ int @ int @ ( one_one @ int ) @ ( zero_zero @ int ) )
@ ( 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.simps
thf(fact_2421_bezw_Oelims,axiom,
! [X: nat,Xa: nat,Y3: product_prod @ int @ int] :
( ( ( bezw @ X @ Xa )
= Y3 )
=> ( ( ( Xa
= ( zero_zero @ nat ) )
=> ( Y3
= ( product_Pair @ int @ int @ ( one_one @ int ) @ ( zero_zero @ int ) ) ) )
& ( ( Xa
!= ( zero_zero @ nat ) )
=> ( Y3
= ( product_Pair @ int @ int @ ( product_snd @ int @ int @ ( bezw @ Xa @ ( modulo_modulo @ nat @ X @ Xa ) ) ) @ ( minus_minus @ int @ ( product_fst @ int @ int @ ( bezw @ Xa @ ( modulo_modulo @ nat @ X @ Xa ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ ( bezw @ Xa @ ( modulo_modulo @ nat @ X @ Xa ) ) ) @ ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ X @ Xa ) ) ) ) ) ) ) ) ) ).
% bezw.elims
thf(fact_2422_bezw__non__0,axiom,
! [Y3: nat,X: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ Y3 )
=> ( ( bezw @ X @ Y3 )
= ( product_Pair @ int @ int @ ( product_snd @ int @ int @ ( bezw @ Y3 @ ( modulo_modulo @ nat @ X @ Y3 ) ) ) @ ( minus_minus @ int @ ( product_fst @ int @ int @ ( bezw @ Y3 @ ( modulo_modulo @ nat @ X @ Y3 ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ ( bezw @ Y3 @ ( modulo_modulo @ nat @ X @ Y3 ) ) ) @ ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ X @ Y3 ) ) ) ) ) ) ) ).
% bezw_non_0
thf(fact_2423_of__nat__id,axiom,
( ( semiring_1_of_nat @ nat )
= ( ^ [N5: nat] : N5 ) ) ).
% of_nat_id
thf(fact_2424_Ints__prod,axiom,
! [A: $tType,B: $tType] :
( ( ( comm_monoid_mult @ B )
& ( ring_1 @ B ) )
=> ! [A5: set @ A,F: A > B] :
( ! [X4: A] :
( ( member @ A @ X4 @ A5 )
=> ( member @ B @ ( F @ X4 ) @ ( ring_1_Ints @ B ) ) )
=> ( member @ B @ ( groups7121269368397514597t_prod @ A @ B @ F @ A5 ) @ ( ring_1_Ints @ B ) ) ) ) ).
% Ints_prod
thf(fact_2425_snd__apsnd,axiom,
! [A: $tType,C: $tType,B: $tType,F: C > A,X: product_prod @ B @ C] :
( ( product_snd @ B @ A @ ( product_apsnd @ C @ A @ B @ F @ X ) )
= ( F @ ( product_snd @ B @ C @ X ) ) ) ).
% snd_apsnd
thf(fact_2426_apsnd__eq__conv,axiom,
! [B: $tType,C: $tType,A: $tType,F: C > B,X: product_prod @ A @ C,G: C > B] :
( ( ( product_apsnd @ C @ B @ A @ F @ X )
= ( product_apsnd @ C @ B @ A @ G @ X ) )
= ( ( F @ ( product_snd @ A @ C @ X ) )
= ( G @ ( product_snd @ A @ C @ X ) ) ) ) ).
% apsnd_eq_conv
thf(fact_2427_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_2428_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
@ ^ [X2: B] : ( semiring_1_of_nat @ A @ ( F @ X2 ) )
@ A5 ) ) ) ).
% of_nat_prod
thf(fact_2429_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
@ ^ [X2: B] : ( ring_1_of_int @ A @ ( F @ X2 ) )
@ A5 ) ) ) ).
% of_int_prod
thf(fact_2430_prod_Ocollapse,axiom,
! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
( ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_2431_snd__comp__apsnd,axiom,
! [C: $tType,B: $tType,A: $tType,F: B > C] :
( ( comp @ ( product_prod @ A @ C ) @ C @ ( product_prod @ A @ B ) @ ( product_snd @ A @ C ) @ ( product_apsnd @ B @ C @ A @ F ) )
= ( comp @ B @ C @ ( product_prod @ A @ B ) @ F @ ( product_snd @ A @ B ) ) ) ).
% snd_comp_apsnd
thf(fact_2432_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_2433_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_2434_prod_Oswap,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: B > C > A,B4: set @ C,A5: set @ B] :
( ( groups7121269368397514597t_prod @ B @ A
@ ^ [I2: B] : ( groups7121269368397514597t_prod @ C @ A @ ( G @ I2 ) @ B4 )
@ A5 )
= ( groups7121269368397514597t_prod @ C @ A
@ ^ [J3: C] :
( groups7121269368397514597t_prod @ B @ A
@ ^ [I2: B] : ( G @ I2 @ J3 )
@ A5 )
@ B4 ) ) ) ).
% prod.swap
thf(fact_2435_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_2436_prod_Oneutral,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A5: set @ B,G: B > A] :
( ! [X4: B] :
( ( member @ B @ X4 @ A5 )
=> ( ( G @ X4 )
= ( one_one @ A ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ A5 )
= ( one_one @ A ) ) ) ) ).
% prod.neutral
thf(fact_2437_snd__conv,axiom,
! [Aa: $tType,A: $tType,X1: Aa,X22: A] :
( ( product_snd @ Aa @ A @ ( product_Pair @ Aa @ A @ X1 @ X22 ) )
= X22 ) ).
% snd_conv
thf(fact_2438_snd__eqD,axiom,
! [B: $tType,A: $tType,X: B,Y3: A,A3: A] :
( ( ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X @ Y3 ) )
= A3 )
=> ( Y3 = A3 ) ) ).
% snd_eqD
thf(fact_2439_sndI,axiom,
! [A: $tType,B: $tType,X: product_prod @ A @ B,Y3: A,Z: B] :
( ( X
= ( product_Pair @ A @ B @ Y3 @ Z ) )
=> ( ( product_snd @ A @ B @ X )
= Z ) ) ).
% sndI
thf(fact_2440_prod_Oexpand,axiom,
! [B: $tType,A: $tType,Prod: product_prod @ A @ B,Prod2: product_prod @ A @ B] :
( ( ( ( product_fst @ A @ B @ Prod )
= ( product_fst @ A @ B @ Prod2 ) )
& ( ( product_snd @ A @ B @ Prod )
= ( product_snd @ A @ B @ Prod2 ) ) )
=> ( Prod = Prod2 ) ) ).
% prod.expand
thf(fact_2441_prod__eqI,axiom,
! [B: $tType,A: $tType,P4: product_prod @ A @ B,Q2: product_prod @ A @ B] :
( ( ( product_fst @ A @ B @ P4 )
= ( product_fst @ A @ B @ Q2 ) )
=> ( ( ( product_snd @ A @ B @ P4 )
= ( product_snd @ A @ B @ Q2 ) )
=> ( P4 = Q2 ) ) ) ).
% prod_eqI
thf(fact_2442_prod__eq__iff,axiom,
! [B: $tType,A: $tType] :
( ( ^ [Y6: product_prod @ A @ B,Z2: product_prod @ A @ B] : Y6 = Z2 )
= ( ^ [S2: product_prod @ A @ B,T3: product_prod @ A @ B] :
( ( ( product_fst @ A @ B @ S2 )
= ( product_fst @ A @ B @ T3 ) )
& ( ( product_snd @ A @ B @ S2 )
= ( product_snd @ A @ B @ T3 ) ) ) ) ) ).
% prod_eq_iff
thf(fact_2443_prod_Odistrib,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: B > A,H: B > A,A5: set @ B] :
( ( groups7121269368397514597t_prod @ B @ A
@ ^ [X2: B] : ( times_times @ A @ ( G @ X2 ) @ ( H @ X2 ) )
@ A5 )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G @ A5 ) @ ( groups7121269368397514597t_prod @ B @ A @ H @ A5 ) ) ) ) ).
% prod.distrib
thf(fact_2444_prod__dividef,axiom,
! [A: $tType,B: $tType] :
( ( field @ A )
=> ! [F: B > A,G: B > A,A5: set @ B] :
( ( groups7121269368397514597t_prod @ B @ A
@ ^ [X2: B] : ( divide_divide @ A @ ( F @ X2 ) @ ( G @ X2 ) )
@ A5 )
= ( divide_divide @ A @ ( groups7121269368397514597t_prod @ B @ A @ F @ A5 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ A5 ) ) ) ) ).
% prod_dividef
thf(fact_2445_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
@ ^ [X2: A] : ( power_power @ B @ ( F @ X2 ) @ N )
@ A5 ) ) ) ).
% prod_power_distrib
thf(fact_2446_mod__prod__eq,axiom,
! [B: $tType,A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [F: B > A,A3: A,A5: set @ B] :
( ( modulo_modulo @ A
@ ( groups7121269368397514597t_prod @ B @ A
@ ^ [I2: B] : ( modulo_modulo @ A @ ( F @ I2 ) @ A3 )
@ A5 )
@ A3 )
= ( modulo_modulo @ A @ ( groups7121269368397514597t_prod @ B @ A @ F @ A5 ) @ A3 ) ) ) ).
% mod_prod_eq
thf(fact_2447_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
@ ^ [X2: B] : ( abs_abs @ A @ ( F @ X2 ) )
@ A5 ) ) ) ).
% abs_prod
thf(fact_2448_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_2449_prod__nonneg,axiom,
! [A: $tType,B: $tType] :
( ( linordered_semidom @ A )
=> ! [A5: set @ B,F: B > A] :
( ! [X4: B] :
( ( member @ B @ X4 @ A5 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F @ X4 ) ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( groups7121269368397514597t_prod @ B @ A @ F @ A5 ) ) ) ) ).
% prod_nonneg
thf(fact_2450_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_2451_prod__pos,axiom,
! [A: $tType,B: $tType] :
( ( linordered_semidom @ A )
=> ! [A5: set @ B,F: B > A] :
( ! [X4: B] :
( ( member @ B @ X4 @ A5 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( F @ X4 ) ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( groups7121269368397514597t_prod @ B @ A @ F @ A5 ) ) ) ) ).
% prod_pos
thf(fact_2452_prod__ge__1,axiom,
! [A: $tType,B: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [A5: set @ B,F: B > A] :
( ! [X4: B] :
( ( member @ B @ X4 @ A5 )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ ( F @ X4 ) ) )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ ( groups7121269368397514597t_prod @ B @ A @ F @ A5 ) ) ) ) ).
% prod_ge_1
thf(fact_2453_prod_Oexhaust__sel,axiom,
! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
( Prod
= ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_2454_surjective__pairing,axiom,
! [B: $tType,A: $tType,T2: product_prod @ A @ B] :
( T2
= ( product_Pair @ A @ B @ ( product_fst @ A @ B @ T2 ) @ ( product_snd @ A @ B @ T2 ) ) ) ).
% surjective_pairing
thf(fact_2455_conjI__realizer,axiom,
! [A: $tType,B: $tType,P: A > $o,P4: A,Q: B > $o,Q2: B] :
( ( P @ P4 )
=> ( ( Q @ Q2 )
=> ( ( P @ ( product_fst @ A @ B @ ( product_Pair @ A @ B @ P4 @ Q2 ) ) )
& ( Q @ ( product_snd @ A @ B @ ( product_Pair @ A @ B @ P4 @ Q2 ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_2456_exI__realizer,axiom,
! [B: $tType,A: $tType,P: A > B > $o,Y3: A,X: B] :
( ( P @ Y3 @ X )
=> ( P @ ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X @ Y3 ) ) @ ( product_fst @ B @ A @ ( product_Pair @ B @ A @ X @ Y3 ) ) ) ) ).
% exI_realizer
thf(fact_2457_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [B: $tType,A: $tType,P: A > B > $o,X: A,Y3: B,A3: product_prod @ A @ B] :
( ( P @ X @ Y3 )
=> ( ( A3
= ( product_Pair @ A @ B @ X @ Y3 ) )
=> ( P @ ( product_fst @ A @ B @ A3 ) @ ( product_snd @ A @ B @ A3 ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_2458_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_2459_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_2460_case__prod__beta,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( product_case_prod @ B @ C @ A )
= ( ^ [F2: B > C > A,P5: product_prod @ B @ C] : ( F2 @ ( product_fst @ B @ C @ P5 ) @ ( product_snd @ B @ C @ P5 ) ) ) ) ).
% case_prod_beta
thf(fact_2461_split__beta,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( product_case_prod @ A @ B @ C )
= ( ^ [F2: A > B > C,Prod3: product_prod @ A @ B] : ( F2 @ ( product_fst @ A @ B @ Prod3 ) @ ( product_snd @ A @ B @ Prod3 ) ) ) ) ).
% split_beta
thf(fact_2462_Product__Type_OCollect__case__prodD,axiom,
! [B: $tType,A: $tType,X: product_prod @ A @ B,A5: A > B > $o] :
( ( member @ ( product_prod @ A @ B ) @ X @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ A5 ) ) )
=> ( A5 @ ( product_fst @ A @ B @ X ) @ ( product_snd @ A @ B @ X ) ) ) ).
% Product_Type.Collect_case_prodD
thf(fact_2463_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_2464_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_2465_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
@ ^ [A6: B] : ( power_power @ A @ C2 @ ( F @ A6 ) )
@ A5 ) ) ) ).
% power_sum
thf(fact_2466_split__comp__eq,axiom,
! [A: $tType,C: $tType,B: $tType,D: $tType,F: A > B > C,G: D > A] :
( ( ^ [U: product_prod @ D @ B] : ( F @ ( G @ ( product_fst @ D @ B @ U ) ) @ ( product_snd @ D @ B @ U ) ) )
= ( product_case_prod @ D @ B @ C
@ ^ [X2: D] : ( F @ ( G @ X2 ) ) ) ) ).
% split_comp_eq
thf(fact_2467_case__prod__beta_H,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( product_case_prod @ A @ B @ C )
= ( ^ [F2: A > B > C,X2: product_prod @ A @ B] : ( F2 @ ( product_fst @ A @ B @ X2 ) @ ( product_snd @ A @ B @ X2 ) ) ) ) ).
% case_prod_beta'
thf(fact_2468_case__prod__unfold,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( product_case_prod @ A @ B @ C )
= ( ^ [C4: A > B > C,P5: product_prod @ A @ B] : ( C4 @ ( product_fst @ A @ B @ P5 ) @ ( product_snd @ A @ B @ P5 ) ) ) ) ).
% case_prod_unfold
thf(fact_2469_exE__realizer,axiom,
! [C: $tType,A: $tType,B: $tType,P: A > B > $o,P4: product_prod @ B @ A,Q: C > $o,F: B > A > C] :
( ( P @ ( product_snd @ B @ A @ P4 ) @ ( product_fst @ B @ A @ P4 ) )
=> ( ! [X4: B,Y4: A] :
( ( P @ Y4 @ X4 )
=> ( Q @ ( F @ X4 @ Y4 ) ) )
=> ( Q @ ( product_case_prod @ B @ A @ C @ F @ P4 ) ) ) ) ).
% exE_realizer
thf(fact_2470_prod__le__1,axiom,
! [B: $tType,A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [A5: set @ B,F: B > A] :
( ! [X4: B] :
( ( member @ B @ X4 @ A5 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F @ X4 ) )
& ( ord_less_eq @ A @ ( F @ X4 ) @ ( one_one @ A ) ) ) )
=> ( ord_less_eq @ A @ ( groups7121269368397514597t_prod @ B @ A @ F @ A5 ) @ ( one_one @ A ) ) ) ) ).
% prod_le_1
thf(fact_2471_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_2472_prod_Osplit__sel__asm,axiom,
! [C: $tType,B: $tType,A: $tType,P: C > $o,F: A > B > C,Prod: product_prod @ A @ B] :
( ( P @ ( product_case_prod @ A @ B @ C @ F @ Prod ) )
= ( ~ ( ( Prod
= ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) )
& ~ ( P @ ( F @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ) ) ) ).
% prod.split_sel_asm
thf(fact_2473_prod_Osplit__sel,axiom,
! [C: $tType,B: $tType,A: $tType,P: C > $o,F: A > B > C,Prod: product_prod @ A @ B] :
( ( P @ ( product_case_prod @ A @ B @ C @ F @ Prod ) )
= ( ( Prod
= ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) )
=> ( P @ ( F @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ) ) ).
% prod.split_sel
thf(fact_2474_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_2475_case__prod__comp,axiom,
! [D: $tType,A: $tType,C: $tType,B: $tType,F: D > C > A,G: B > D,X: product_prod @ B @ C] :
( ( product_case_prod @ B @ C @ A @ ( comp @ D @ ( C > A ) @ B @ F @ G ) @ X )
= ( F @ ( G @ ( product_fst @ B @ C @ X ) ) @ ( product_snd @ B @ C @ X ) ) ) ).
% case_prod_comp
thf(fact_2476_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_2477_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_2478_snd__diag__snd,axiom,
! [B: $tType,A: $tType] :
( ( comp @ ( product_prod @ B @ B ) @ B @ ( product_prod @ A @ B ) @ ( product_snd @ B @ B )
@ ( comp @ B @ ( product_prod @ B @ B ) @ ( product_prod @ A @ B )
@ ^ [X2: B] : ( product_Pair @ B @ B @ X2 @ X2 )
@ ( product_snd @ A @ B ) ) )
= ( product_snd @ A @ B ) ) ).
% snd_diag_snd
thf(fact_2479_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_2480_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_2481_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
@ ^ [N5: nat] : ( minus_minus @ nat @ N5 @ ( 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_2482_fact__prod,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiring_char_0_fact @ A )
= ( ^ [N5: nat] :
( semiring_1_of_nat @ A
@ ( groups7121269368397514597t_prod @ nat @ nat
@ ^ [X2: nat] : X2
@ ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ N5 ) ) ) ) ) ) ).
% fact_prod
thf(fact_2483_prod__atLeastAtMost__code,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [F: nat > A,A3: nat,B2: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ F @ ( set_or1337092689740270186AtMost @ nat @ A3 @ B2 ) )
= ( set_fo6178422350223883121st_nat @ A
@ ^ [A6: nat] : ( times_times @ A @ ( F @ A6 ) )
@ A3
@ B2
@ ( one_one @ A ) ) ) ) ).
% prod_atLeastAtMost_code
thf(fact_2484_fst__snd__flip,axiom,
! [B: $tType,A: $tType] :
( ( product_fst @ A @ B )
= ( comp @ ( product_prod @ B @ A ) @ A @ ( product_prod @ A @ B ) @ ( product_snd @ B @ A )
@ ( product_case_prod @ A @ B @ ( product_prod @ B @ A )
@ ^ [X2: A,Y: B] : ( product_Pair @ B @ A @ Y @ X2 ) ) ) ) ).
% fst_snd_flip
thf(fact_2485_snd__fst__flip,axiom,
! [A: $tType,B: $tType] :
( ( product_snd @ B @ A )
= ( comp @ ( product_prod @ A @ B ) @ A @ ( product_prod @ B @ A ) @ ( product_fst @ A @ B )
@ ( product_case_prod @ B @ A @ ( product_prod @ A @ B )
@ ^ [X2: B,Y: A] : ( product_Pair @ A @ B @ Y @ X2 ) ) ) ) ).
% snd_fst_flip
thf(fact_2486_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_2487_prod_Oub__add__nat,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M: nat,N: nat,G: nat > A,P4: 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 @ P4 ) ) )
= ( 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 @ P4 ) ) ) ) ) ) ) ).
% prod.ub_add_nat
thf(fact_2488_fst__diag__snd,axiom,
! [B: $tType,A: $tType] :
( ( comp @ ( product_prod @ B @ B ) @ B @ ( product_prod @ A @ B ) @ ( product_fst @ B @ B )
@ ( comp @ B @ ( product_prod @ B @ B ) @ ( product_prod @ A @ B )
@ ^ [X2: B] : ( product_Pair @ B @ B @ X2 @ X2 )
@ ( product_snd @ A @ B ) ) )
= ( product_snd @ A @ B ) ) ).
% fst_diag_snd
thf(fact_2489_snd__diag__fst,axiom,
! [B: $tType,A: $tType] :
( ( comp @ ( product_prod @ A @ A ) @ A @ ( product_prod @ A @ B ) @ ( product_snd @ A @ A )
@ ( comp @ A @ ( product_prod @ A @ A ) @ ( product_prod @ A @ B )
@ ^ [X2: A] : ( product_Pair @ A @ A @ X2 @ X2 )
@ ( product_fst @ A @ B ) ) )
= ( product_fst @ A @ B ) ) ).
% snd_diag_fst
thf(fact_2490_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
@ ^ [X2: nat] : X2
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ N ) @ M ) ) ) ) ) ).
% fact_eq_fact_times
thf(fact_2491_pochhammer__Suc__prod,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A3: A,N: nat] :
( ( comm_s3205402744901411588hammer @ A @ A3 @ ( suc @ N ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I2: nat] : ( plus_plus @ A @ A3 @ ( semiring_1_of_nat @ A @ I2 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% pochhammer_Suc_prod
thf(fact_2492_pochhammer__prod__rev,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ( ( comm_s3205402744901411588hammer @ A )
= ( ^ [A6: A,N5: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I2: nat] : ( plus_plus @ A @ A6 @ ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ N5 @ I2 ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ N5 ) ) ) ) ) ).
% pochhammer_prod_rev
thf(fact_2493_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
@ ^ [X2: nat] : X2
@ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) @ M ) ) ) ) ).
% fact_div_fact
thf(fact_2494_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_2495_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_2496_pochhammer__Suc__prod__rev,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A3: A,N: nat] :
( ( comm_s3205402744901411588hammer @ A @ A3 @ ( suc @ N ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I2: nat] : ( plus_plus @ A @ A3 @ ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ N @ I2 ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% pochhammer_Suc_prod_rev
thf(fact_2497_prod_Ozero__middle,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [P4: nat,K: nat,G: nat > A,H: nat > A] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ P4 )
=> ( ( ord_less_eq @ nat @ K @ P4 )
=> ( ( groups7121269368397514597t_prod @ nat @ A
@ ^ [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 @ P4 ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [J3: nat] : ( if @ A @ ( ord_less @ nat @ J3 @ K ) @ ( G @ J3 ) @ ( H @ J3 ) )
@ ( set_ord_atMost @ nat @ ( minus_minus @ nat @ P4 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) ) ).
% prod.zero_middle
thf(fact_2498_gbinomial__Suc,axiom,
! [A: $tType] :
( ( ( semiring_char_0 @ A )
& ( semidom_divide @ A ) )
=> ! [A3: A,K: nat] :
( ( gbinomial @ A @ A3 @ ( suc @ K ) )
= ( divide_divide @ A
@ ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I2: nat] : ( minus_minus @ A @ A3 @ ( semiring_1_of_nat @ A @ I2 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ K ) )
@ ( semiring_char_0_fact @ A @ ( suc @ K ) ) ) ) ) ).
% gbinomial_Suc
thf(fact_2499_size__prod__simp,axiom,
! [B: $tType,A: $tType] :
( ( basic_BNF_size_prod @ A @ B )
= ( ^ [F2: A > nat,G2: B > nat,P5: product_prod @ A @ B] : ( plus_plus @ nat @ ( plus_plus @ nat @ ( F2 @ ( product_fst @ A @ B @ P5 ) ) @ ( G2 @ ( product_snd @ A @ B @ P5 ) ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% size_prod_simp
thf(fact_2500_bezw_Opelims,axiom,
! [X: nat,Xa: nat,Y3: product_prod @ int @ int] :
( ( ( bezw @ X @ Xa )
= Y3 )
=> ( ( accp @ ( product_prod @ nat @ nat ) @ bezw_rel @ ( product_Pair @ nat @ nat @ X @ Xa ) )
=> ~ ( ( ( ( Xa
= ( zero_zero @ nat ) )
=> ( Y3
= ( product_Pair @ int @ int @ ( one_one @ int ) @ ( zero_zero @ int ) ) ) )
& ( ( Xa
!= ( zero_zero @ nat ) )
=> ( Y3
= ( product_Pair @ int @ int @ ( product_snd @ int @ int @ ( bezw @ Xa @ ( modulo_modulo @ nat @ X @ Xa ) ) ) @ ( minus_minus @ int @ ( product_fst @ int @ int @ ( bezw @ Xa @ ( modulo_modulo @ nat @ X @ Xa ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ ( bezw @ Xa @ ( modulo_modulo @ nat @ X @ Xa ) ) ) @ ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ X @ Xa ) ) ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ nat @ nat ) @ bezw_rel @ ( product_Pair @ nat @ nat @ X @ Xa ) ) ) ) ) ).
% bezw.pelims
thf(fact_2501_normalize__def,axiom,
( normalize
= ( ^ [P5: product_prod @ int @ int] :
( if @ ( product_prod @ int @ int ) @ ( ord_less @ int @ ( zero_zero @ int ) @ ( product_snd @ int @ int @ P5 ) ) @ ( product_Pair @ int @ int @ ( divide_divide @ int @ ( product_fst @ int @ int @ P5 ) @ ( gcd_gcd @ int @ ( product_fst @ int @ int @ P5 ) @ ( product_snd @ int @ int @ P5 ) ) ) @ ( divide_divide @ int @ ( product_snd @ int @ int @ P5 ) @ ( gcd_gcd @ int @ ( product_fst @ int @ int @ P5 ) @ ( product_snd @ int @ int @ P5 ) ) ) )
@ ( if @ ( product_prod @ int @ int )
@ ( ( product_snd @ int @ int @ P5 )
= ( zero_zero @ int ) )
@ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
@ ( product_Pair @ int @ int @ ( divide_divide @ int @ ( product_fst @ int @ int @ P5 ) @ ( uminus_uminus @ int @ ( gcd_gcd @ int @ ( product_fst @ int @ int @ P5 ) @ ( product_snd @ int @ int @ P5 ) ) ) ) @ ( divide_divide @ int @ ( product_snd @ int @ int @ P5 ) @ ( uminus_uminus @ int @ ( gcd_gcd @ int @ ( product_fst @ int @ int @ P5 ) @ ( product_snd @ int @ int @ P5 ) ) ) ) ) ) ) ) ) ).
% normalize_def
thf(fact_2502_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
@ ^ [K3: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( G @ I2 @ ( minus_minus @ nat @ K3 @ I2 ) )
@ ( set_ord_atMost @ nat @ K3 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% sum.triangle_reindex
thf(fact_2503_Sum__Icc__int,axiom,
! [M: int,N: int] :
( ( ord_less_eq @ int @ M @ N )
=> ( ( groups7311177749621191930dd_sum @ int @ int
@ ^ [X2: int] : X2
@ ( 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_2504_Sum__Ico__nat,axiom,
! [M: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [X2: nat] : X2
@ ( 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_2505_gcd__right__idem,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A] :
( ( gcd_gcd @ A @ ( gcd_gcd @ A @ A3 @ B2 ) @ B2 )
= ( gcd_gcd @ A @ A3 @ B2 ) ) ) ).
% gcd_right_idem
thf(fact_2506_gcd__left__idem,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A] :
( ( gcd_gcd @ A @ A3 @ ( gcd_gcd @ A @ A3 @ B2 ) )
= ( gcd_gcd @ A @ A3 @ B2 ) ) ) ).
% gcd_left_idem
thf(fact_2507_gcd__eq__0__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A] :
( ( ( gcd_gcd @ A @ A3 @ B2 )
= ( zero_zero @ A ) )
= ( ( A3
= ( zero_zero @ A ) )
& ( B2
= ( zero_zero @ A ) ) ) ) ) ).
% gcd_eq_0_iff
thf(fact_2508_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_2509_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_2510_gcd_Obottom__left__bottom,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A] :
( ( gcd_gcd @ A @ ( one_one @ A ) @ A3 )
= ( one_one @ A ) ) ) ).
% gcd.bottom_left_bottom
thf(fact_2511_gcd_Obottom__right__bottom,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A] :
( ( gcd_gcd @ A @ A3 @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% gcd.bottom_right_bottom
thf(fact_2512_gcd__neg2,axiom,
! [A: $tType] :
( ( ring_gcd @ A )
=> ! [A3: A,B2: A] :
( ( gcd_gcd @ A @ A3 @ ( uminus_uminus @ A @ B2 ) )
= ( gcd_gcd @ A @ A3 @ B2 ) ) ) ).
% gcd_neg2
thf(fact_2513_gcd__neg1,axiom,
! [A: $tType] :
( ( ring_gcd @ A )
=> ! [A3: A,B2: A] :
( ( gcd_gcd @ A @ ( uminus_uminus @ A @ A3 ) @ B2 )
= ( gcd_gcd @ A @ A3 @ B2 ) ) ) ).
% gcd_neg1
thf(fact_2514_gcd__greatest__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A3 @ ( gcd_gcd @ A @ B2 @ C2 ) )
= ( ( dvd_dvd @ A @ A3 @ B2 )
& ( dvd_dvd @ A @ A3 @ C2 ) ) ) ) ).
% gcd_greatest_iff
thf(fact_2515_gcd__dvd2,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A] : ( dvd_dvd @ A @ ( gcd_gcd @ A @ A3 @ B2 ) @ B2 ) ) ).
% gcd_dvd2
thf(fact_2516_gcd__dvd1,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A] : ( dvd_dvd @ A @ ( gcd_gcd @ A @ A3 @ B2 ) @ A3 ) ) ).
% gcd_dvd1
thf(fact_2517_gcd__exp,axiom,
! [A: $tType] :
( ( semiri6843258321239162965malize @ A )
=> ! [A3: A,N: nat,B2: A] :
( ( gcd_gcd @ A @ ( power_power @ A @ A3 @ N ) @ ( power_power @ A @ B2 @ N ) )
= ( power_power @ A @ ( gcd_gcd @ A @ A3 @ B2 ) @ N ) ) ) ).
% gcd_exp
thf(fact_2518_gcd__1__int,axiom,
! [M: int] :
( ( gcd_gcd @ int @ M @ ( one_one @ int ) )
= ( one_one @ int ) ) ).
% gcd_1_int
thf(fact_2519_gcd__neg2__int,axiom,
! [X: int,Y3: int] :
( ( gcd_gcd @ int @ X @ ( uminus_uminus @ int @ Y3 ) )
= ( gcd_gcd @ int @ X @ Y3 ) ) ).
% gcd_neg2_int
thf(fact_2520_gcd__neg1__int,axiom,
! [X: int,Y3: int] :
( ( gcd_gcd @ int @ ( uminus_uminus @ int @ X ) @ Y3 )
= ( gcd_gcd @ int @ X @ Y3 ) ) ).
% gcd_neg1_int
thf(fact_2521_abs__gcd__int,axiom,
! [X: int,Y3: int] :
( ( abs_abs @ int @ ( gcd_gcd @ int @ X @ Y3 ) )
= ( gcd_gcd @ int @ X @ Y3 ) ) ).
% abs_gcd_int
thf(fact_2522_gcd__abs1__int,axiom,
! [X: int,Y3: int] :
( ( gcd_gcd @ int @ ( abs_abs @ int @ X ) @ Y3 )
= ( gcd_gcd @ int @ X @ Y3 ) ) ).
% gcd_abs1_int
thf(fact_2523_gcd__abs2__int,axiom,
! [X: int,Y3: int] :
( ( gcd_gcd @ int @ X @ ( abs_abs @ int @ Y3 ) )
= ( gcd_gcd @ int @ X @ Y3 ) ) ).
% gcd_abs2_int
thf(fact_2524_gcd__neg__numeral__1,axiom,
! [A: $tType] :
( ( ring_gcd @ A )
=> ! [N: num,A3: A] :
( ( gcd_gcd @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) @ A3 )
= ( gcd_gcd @ A @ ( numeral_numeral @ A @ N ) @ A3 ) ) ) ).
% gcd_neg_numeral_1
thf(fact_2525_gcd__neg__numeral__2,axiom,
! [A: $tType] :
( ( ring_gcd @ A )
=> ! [A3: A,N: num] :
( ( gcd_gcd @ A @ A3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( gcd_gcd @ A @ A3 @ ( numeral_numeral @ A @ N ) ) ) ) ).
% gcd_neg_numeral_2
thf(fact_2526_is__unit__gcd__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ ( gcd_gcd @ A @ A3 @ B2 ) @ ( one_one @ A ) )
= ( ( gcd_gcd @ A @ A3 @ B2 )
= ( one_one @ A ) ) ) ) ).
% is_unit_gcd_iff
thf(fact_2527_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_2528_gcd__neg__numeral__1__int,axiom,
! [N: num,X: int] :
( ( gcd_gcd @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) @ X )
= ( gcd_gcd @ int @ ( numeral_numeral @ int @ N ) @ X ) ) ).
% gcd_neg_numeral_1_int
thf(fact_2529_gcd__neg__numeral__2__int,axiom,
! [X: int,N: num] :
( ( gcd_gcd @ int @ X @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
= ( gcd_gcd @ int @ X @ ( numeral_numeral @ int @ N ) ) ) ).
% gcd_neg_numeral_2_int
thf(fact_2530_gcd__0__int,axiom,
! [X: int] :
( ( gcd_gcd @ int @ X @ ( zero_zero @ int ) )
= ( abs_abs @ int @ X ) ) ).
% gcd_0_int
thf(fact_2531_gcd__0__left__int,axiom,
! [X: int] :
( ( gcd_gcd @ int @ ( zero_zero @ int ) @ X )
= ( abs_abs @ int @ X ) ) ).
% gcd_0_left_int
thf(fact_2532_gcd__proj2__if__dvd__int,axiom,
! [Y3: int,X: int] :
( ( dvd_dvd @ int @ Y3 @ X )
=> ( ( gcd_gcd @ int @ X @ Y3 )
= ( abs_abs @ int @ Y3 ) ) ) ).
% gcd_proj2_if_dvd_int
thf(fact_2533_gcd__proj1__if__dvd__int,axiom,
! [X: int,Y3: int] :
( ( dvd_dvd @ int @ X @ Y3 )
=> ( ( gcd_gcd @ int @ X @ Y3 )
= ( abs_abs @ int @ X ) ) ) ).
% gcd_proj1_if_dvd_int
thf(fact_2534_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_2535_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_2536_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_2537_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_2538_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
@ ^ [X2: B] : ( semiring_1_of_nat @ int @ ( F @ X2 ) )
@ A5 ) ) ).
% int_prod
thf(fact_2539_lessThan__atLeast0,axiom,
( ( set_ord_lessThan @ nat )
= ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) ) ) ).
% lessThan_atLeast0
thf(fact_2540_gcd_Oleft__commute,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( gcd_gcd @ A @ B2 @ ( gcd_gcd @ A @ A3 @ C2 ) )
= ( gcd_gcd @ A @ A3 @ ( gcd_gcd @ A @ B2 @ C2 ) ) ) ) ).
% gcd.left_commute
thf(fact_2541_gcd_Ocommute,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ( ( gcd_gcd @ A )
= ( ^ [A6: A,B5: A] : ( gcd_gcd @ A @ B5 @ A6 ) ) ) ) ).
% gcd.commute
thf(fact_2542_gcd_Oassoc,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( gcd_gcd @ A @ ( gcd_gcd @ A @ A3 @ B2 ) @ C2 )
= ( gcd_gcd @ A @ A3 @ ( gcd_gcd @ A @ B2 @ C2 ) ) ) ) ).
% gcd.assoc
thf(fact_2543_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_2544_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_2545_gcd__dvd__antisym,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A,C2: A,D3: A] :
( ( dvd_dvd @ A @ ( gcd_gcd @ A @ A3 @ B2 ) @ ( gcd_gcd @ A @ C2 @ D3 ) )
=> ( ( dvd_dvd @ A @ ( gcd_gcd @ A @ C2 @ D3 ) @ ( gcd_gcd @ A @ A3 @ B2 ) )
=> ( ( gcd_gcd @ A @ A3 @ B2 )
= ( gcd_gcd @ A @ C2 @ D3 ) ) ) ) ) ).
% gcd_dvd_antisym
thf(fact_2546_gcd__greatest,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( dvd_dvd @ A @ C2 @ A3 )
=> ( ( dvd_dvd @ A @ C2 @ B2 )
=> ( dvd_dvd @ A @ C2 @ ( gcd_gcd @ A @ A3 @ B2 ) ) ) ) ) ).
% gcd_greatest
thf(fact_2547_gcd__dvdI2,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [B2: A,C2: A,A3: A] :
( ( dvd_dvd @ A @ B2 @ C2 )
=> ( dvd_dvd @ A @ ( gcd_gcd @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% gcd_dvdI2
thf(fact_2548_gcd__dvdI1,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ C2 )
=> ( dvd_dvd @ A @ ( gcd_gcd @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% gcd_dvdI1
thf(fact_2549_dvd__gcdD2,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A3 @ ( gcd_gcd @ A @ B2 @ C2 ) )
=> ( dvd_dvd @ A @ A3 @ C2 ) ) ) ).
% dvd_gcdD2
thf(fact_2550_dvd__gcdD1,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A3 @ ( gcd_gcd @ A @ B2 @ C2 ) )
=> ( dvd_dvd @ A @ A3 @ B2 ) ) ) ).
% dvd_gcdD1
thf(fact_2551_gcd__mono,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,C2: A,B2: A,D3: A] :
( ( dvd_dvd @ A @ A3 @ C2 )
=> ( ( dvd_dvd @ A @ B2 @ D3 )
=> ( dvd_dvd @ A @ ( gcd_gcd @ A @ A3 @ B2 ) @ ( gcd_gcd @ A @ C2 @ D3 ) ) ) ) ) ).
% gcd_mono
thf(fact_2552_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
@ ^ [X2: int] : X2
@ ( set_or1337092689740270186AtMost @ int @ ( semiring_1_of_nat @ int @ I ) @ ( semiring_1_of_nat @ int @ J ) ) ) ) ).
% prod_int_eq
thf(fact_2553_gcd__idem__int,axiom,
! [X: int] :
( ( gcd_gcd @ int @ X @ X )
= ( abs_abs @ int @ X ) ) ).
% gcd_idem_int
thf(fact_2554_gcd__red__int,axiom,
( ( gcd_gcd @ int )
= ( ^ [X2: int,Y: int] : ( gcd_gcd @ int @ Y @ ( modulo_modulo @ int @ X2 @ Y ) ) ) ) ).
% gcd_red_int
thf(fact_2555_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,H2: heap_ext @ product_unit,R: A,N: nat] :
( ( heap_Time_effect @ A @ ( heap_Time_heap @ A @ F ) @ H @ H2 @ R @ N )
=> ~ ( ( H2
= ( 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_2556_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,H2: 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 ) ) ) )
=> ( ( H2
= ( 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 @ H2 @ R @ N ) ) ) ) ).
% effect_heapI
thf(fact_2557_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,H2: heap_ext @ product_unit,R: A,N: nat] :
( ( heap_Time_effect @ A @ ( heap_Time_guard @ A @ P @ F ) @ H @ H2 @ R @ N )
=> ~ ( ( H2
= ( 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_2558_effect__guardI,axiom,
! [A: $tType,P: ( heap_ext @ product_unit ) > $o,H: heap_ext @ product_unit,H2: 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 )
=> ( ( H2
= ( 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 @ H2 @ R @ N ) ) ) ) ) ).
% effect_guardI
thf(fact_2559_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_2560_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
@ ^ [X2: int] : X2
@ ( 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_2561_sum_Onat__group,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,K: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [M5: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( times_times @ nat @ M5 @ K ) @ ( plus_plus @ nat @ ( times_times @ nat @ M5 @ 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_2562_prod_Onat__group,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,K: nat,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A
@ ^ [M5: nat] : ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( times_times @ nat @ M5 @ K ) @ ( plus_plus @ nat @ ( times_times @ nat @ M5 @ 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_2563_effectE,axiom,
! [A: $tType,C2: heap_Time_Heap @ A,H: heap_ext @ product_unit,H2: heap_ext @ product_unit,R: A,N: nat] :
( ( heap_Time_effect @ A @ C2 @ H @ H2 @ 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 ) ) ) )
=> ( ( H2
= ( 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_2564_lessThan__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( set_ord_lessThan @ A )
= ( ^ [U: A] :
( collect @ A
@ ^ [X2: A] : ( ord_less @ A @ X2 @ U ) ) ) ) ) ).
% lessThan_def
thf(fact_2565_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_2566_gcd__dvd__prod,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A,K: A] : ( dvd_dvd @ A @ ( gcd_gcd @ A @ A3 @ B2 ) @ ( times_times @ A @ K @ B2 ) ) ) ).
% gcd_dvd_prod
thf(fact_2567_ex__nat__less__eq,axiom,
! [N: nat,P: nat > $o] :
( ( ? [M5: nat] :
( ( ord_less @ nat @ M5 @ N )
& ( P @ M5 ) ) )
= ( ? [X2: nat] :
( ( member @ nat @ X2 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) )
& ( P @ X2 ) ) ) ) ).
% ex_nat_less_eq
thf(fact_2568_all__nat__less__eq,axiom,
! [N: nat,P: nat > $o] :
( ( ! [M5: nat] :
( ( ord_less @ nat @ M5 @ N )
=> ( P @ M5 ) ) )
= ( ! [X2: nat] :
( ( member @ nat @ X2 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) )
=> ( P @ X2 ) ) ) ) ).
% all_nat_less_eq
thf(fact_2569_gcd__ge__0__int,axiom,
! [X: int,Y3: int] : ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( gcd_gcd @ int @ X @ Y3 ) ) ).
% gcd_ge_0_int
thf(fact_2570_bezout__int,axiom,
! [X: int,Y3: int] :
? [U3: int,V5: int] :
( ( plus_plus @ int @ ( times_times @ int @ U3 @ X ) @ ( times_times @ int @ V5 @ Y3 ) )
= ( gcd_gcd @ int @ X @ Y3 ) ) ).
% bezout_int
thf(fact_2571_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_2572_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_2573_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_2574_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_2575_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_2576_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_2577_gcd__mult__unit2,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( gcd_gcd @ A @ B2 @ ( times_times @ A @ C2 @ A3 ) )
= ( gcd_gcd @ A @ B2 @ C2 ) ) ) ) ).
% gcd_mult_unit2
thf(fact_2578_gcd__mult__unit1,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( gcd_gcd @ A @ ( times_times @ A @ B2 @ A3 ) @ C2 )
= ( gcd_gcd @ A @ B2 @ C2 ) ) ) ) ).
% gcd_mult_unit1
thf(fact_2579_gcd__div__unit2,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( gcd_gcd @ A @ B2 @ ( divide_divide @ A @ C2 @ A3 ) )
= ( gcd_gcd @ A @ B2 @ C2 ) ) ) ) ).
% gcd_div_unit2
thf(fact_2580_gcd__div__unit1,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( gcd_gcd @ A @ ( divide_divide @ A @ B2 @ A3 ) @ C2 )
= ( gcd_gcd @ A @ B2 @ C2 ) ) ) ) ).
% gcd_div_unit1
thf(fact_2581_gcd__le2__int,axiom,
! [B2: int,A3: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ord_less_eq @ int @ ( gcd_gcd @ int @ A3 @ B2 ) @ B2 ) ) ).
% gcd_le2_int
thf(fact_2582_gcd__le1__int,axiom,
! [A3: int,B2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ A3 )
=> ( ord_less_eq @ int @ ( gcd_gcd @ int @ A3 @ B2 ) @ A3 ) ) ).
% gcd_le1_int
thf(fact_2583_gcd__cases__int,axiom,
! [X: int,Y3: int,P: int > $o] :
( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y3 )
=> ( P @ ( gcd_gcd @ int @ X @ Y3 ) ) ) )
=> ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
=> ( ( ord_less_eq @ int @ Y3 @ ( zero_zero @ int ) )
=> ( P @ ( gcd_gcd @ int @ X @ ( uminus_uminus @ int @ Y3 ) ) ) ) )
=> ( ( ( ord_less_eq @ int @ X @ ( zero_zero @ int ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y3 )
=> ( P @ ( gcd_gcd @ int @ ( uminus_uminus @ int @ X ) @ Y3 ) ) ) )
=> ( ( ( ord_less_eq @ int @ X @ ( zero_zero @ int ) )
=> ( ( ord_less_eq @ int @ Y3 @ ( zero_zero @ int ) )
=> ( P @ ( gcd_gcd @ int @ ( uminus_uminus @ int @ X ) @ ( uminus_uminus @ int @ Y3 ) ) ) ) )
=> ( P @ ( gcd_gcd @ int @ X @ Y3 ) ) ) ) ) ) ).
% gcd_cases_int
thf(fact_2584_sum_OatLeastLessThan__concat,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M: nat,N: nat,P4: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( ord_less_eq @ nat @ N @ P4 )
=> ( ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ N @ P4 ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ P4 ) ) ) ) ) ) ).
% sum.atLeastLessThan_concat
thf(fact_2585_gcd__unique__int,axiom,
! [D3: int,A3: int,B2: int] :
( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ D3 )
& ( dvd_dvd @ int @ D3 @ A3 )
& ( dvd_dvd @ int @ D3 @ B2 )
& ! [E4: int] :
( ( ( dvd_dvd @ int @ E4 @ A3 )
& ( dvd_dvd @ int @ E4 @ B2 ) )
=> ( dvd_dvd @ int @ E4 @ D3 ) ) )
= ( D3
= ( gcd_gcd @ int @ A3 @ B2 ) ) ) ).
% gcd_unique_int
thf(fact_2586_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_2587_gcd__non__0__int,axiom,
! [Y3: int,X: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ Y3 )
=> ( ( gcd_gcd @ int @ X @ Y3 )
= ( gcd_gcd @ int @ Y3 @ ( modulo_modulo @ int @ X @ Y3 ) ) ) ) ).
% gcd_non_0_int
thf(fact_2588_gcd__code__int,axiom,
( ( gcd_gcd @ int )
= ( ^ [K3: int,L4: int] :
( abs_abs @ int
@ ( if @ int
@ ( L4
= ( zero_zero @ int ) )
@ K3
@ ( gcd_gcd @ int @ L4 @ ( modulo_modulo @ int @ ( abs_abs @ int @ K3 ) @ ( abs_abs @ int @ L4 ) ) ) ) ) ) ) ).
% gcd_code_int
thf(fact_2589_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_2590_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_2591_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_2592_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_2593_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_2594_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_2595_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_2596_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_2597_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
@ ^ [K3: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I2: nat] : ( G @ I2 @ ( minus_minus @ nat @ K3 @ I2 ) )
@ ( set_ord_atMost @ nat @ K3 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% prod.triangle_reindex
thf(fact_2598_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_2599_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_2600_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_2601_sum_OatLeastLessThan__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: nat,B2: nat,G: nat > A] :
( ( ord_less_eq @ nat @ A3 @ B2 )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ A3 @ ( suc @ B2 ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ A3 @ B2 ) ) @ ( G @ B2 ) ) ) ) ) ).
% sum.atLeastLessThan_Suc
thf(fact_2602_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_2603_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_2604_fact__prod__Suc,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiring_char_0_fact @ A )
= ( ^ [N5: nat] : ( semiring_1_of_nat @ A @ ( groups7121269368397514597t_prod @ nat @ nat @ suc @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N5 ) ) ) ) ) ) ).
% fact_prod_Suc
thf(fact_2605_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_2606_sum__lessThan__telescope,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [F: nat > A,M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N5: nat] : ( minus_minus @ A @ ( F @ ( suc @ N5 ) ) @ ( F @ N5 ) )
@ ( set_ord_lessThan @ nat @ M ) )
= ( minus_minus @ A @ ( F @ M ) @ ( F @ ( zero_zero @ nat ) ) ) ) ) ).
% sum_lessThan_telescope
thf(fact_2607_sum__lessThan__telescope_H,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [F: nat > A,M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N5: nat] : ( minus_minus @ A @ ( F @ N5 ) @ ( F @ ( suc @ N5 ) ) )
@ ( set_ord_lessThan @ nat @ M ) )
= ( minus_minus @ A @ ( F @ ( zero_zero @ nat ) ) @ ( F @ M ) ) ) ) ).
% sum_lessThan_telescope'
thf(fact_2608_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_2609_sum__diff__distrib,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [Q: A > nat,P: A > nat,N: A] :
( ! [X4: A] : ( ord_less_eq @ nat @ ( Q @ X4 ) @ ( P @ X4 ) )
=> ( ( 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
@ ^ [X2: A] : ( minus_minus @ nat @ ( P @ X2 ) @ ( Q @ X2 ) )
@ ( set_ord_lessThan @ A @ N ) ) ) ) ) ).
% sum_diff_distrib
thf(fact_2610_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
@ ^ [K3: nat] : ( G @ ( suc @ K3 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% sum.atLeast1_atMost_eq
thf(fact_2611_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
@ ^ [K3: nat] : ( G @ ( suc @ K3 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% prod.atLeast1_atMost_eq
thf(fact_2612_sum__bounds__lt__plus1,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [F: nat > A,Mm: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( F @ ( suc @ K3 ) )
@ ( set_ord_lessThan @ nat @ Mm ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ F @ ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ Mm ) ) ) ) ).
% sum_bounds_lt_plus1
thf(fact_2613_sum_Onested__swap_H,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: nat > nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ ( A3 @ I2 ) @ ( set_ord_lessThan @ nat @ I2 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J3: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( A3 @ I2 @ J3 )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J3 ) @ N ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% sum.nested_swap'
thf(fact_2614_prod_Onested__swap_H,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: nat > nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I2: nat] : ( groups7121269368397514597t_prod @ nat @ A @ ( A3 @ I2 ) @ ( set_ord_lessThan @ nat @ I2 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [J3: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I2: nat] : ( A3 @ I2 @ J3 )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J3 ) @ N ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% prod.nested_swap'
thf(fact_2615_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_2616_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_2617_sum_Onested__swap,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: nat > nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ ( A3 @ 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] : ( A3 @ I2 @ J3 )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J3 ) @ N ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% sum.nested_swap
thf(fact_2618_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_2619_prod_Onested__swap,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: nat > nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I2: nat] : ( groups7121269368397514597t_prod @ nat @ A @ ( A3 @ 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] : ( A3 @ I2 @ J3 )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J3 ) @ N ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% prod.nested_swap
thf(fact_2620_fact__prod__rev,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiring_char_0_fact @ A )
= ( ^ [N5: nat] : ( semiring_1_of_nat @ A @ ( groups7121269368397514597t_prod @ nat @ nat @ ( minus_minus @ nat @ N5 ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N5 ) ) ) ) ) ) ).
% fact_prod_rev
thf(fact_2621_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_2622_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_2623_power__diff__1__eq,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X: A,N: nat] :
( ( minus_minus @ A @ ( power_power @ A @ X @ N ) @ ( one_one @ A ) )
= ( times_times @ A @ ( minus_minus @ A @ X @ ( one_one @ A ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% power_diff_1_eq
thf(fact_2624_one__diff__power__eq,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X: A,N: nat] :
( ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X @ N ) )
= ( times_times @ A @ ( minus_minus @ A @ ( one_one @ A ) @ X ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% one_diff_power_eq
thf(fact_2625_geometric__sum,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [X: A,N: nat] :
( ( X
!= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_lessThan @ nat @ N ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ X @ N ) @ ( one_one @ A ) ) @ ( minus_minus @ A @ X @ ( one_one @ A ) ) ) ) ) ) ).
% geometric_sum
thf(fact_2626_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_2627_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_2628_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_2629_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_2630_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_2631_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_2632_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_2633_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_2634_pochhammer__prod,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ( ( comm_s3205402744901411588hammer @ A )
= ( ^ [A6: A,N5: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I2: nat] : ( plus_plus @ A @ A6 @ ( semiring_1_of_nat @ A @ I2 ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N5 ) ) ) ) ) ).
% pochhammer_prod
thf(fact_2635_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
@ ^ [N5: nat] : ( minus_minus @ nat @ N5 @ ( 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_2636_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
@ ^ [N5: nat] : ( minus_minus @ nat @ N5 @ ( 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_2637_sum__gp__strict,axiom,
! [A: $tType] :
( ( ( division_ring @ A )
& ( comm_ring @ A ) )
=> ! [X: A,N: nat] :
( ( ( X
= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_lessThan @ nat @ N ) )
= ( semiring_1_of_nat @ A @ N ) ) )
& ( ( X
!= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_lessThan @ nat @ N ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X @ N ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ X ) ) ) ) ) ) ).
% sum_gp_strict
thf(fact_2638_power__diff__sumr2,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X: A,N: nat,Y3: A] :
( ( minus_minus @ A @ ( power_power @ A @ X @ N ) @ ( power_power @ A @ Y3 @ N ) )
= ( times_times @ A @ ( minus_minus @ A @ X @ Y3 )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( times_times @ A @ ( power_power @ A @ Y3 @ ( minus_minus @ nat @ N @ ( suc @ I2 ) ) ) @ ( power_power @ A @ X @ I2 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% power_diff_sumr2
thf(fact_2639_diff__power__eq__sum,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X: A,N: nat,Y3: A] :
( ( minus_minus @ A @ ( power_power @ A @ X @ ( suc @ N ) ) @ ( power_power @ A @ Y3 @ ( suc @ N ) ) )
= ( times_times @ A @ ( minus_minus @ A @ X @ Y3 )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [P5: nat] : ( times_times @ A @ ( power_power @ A @ X @ P5 ) @ ( power_power @ A @ Y3 @ ( minus_minus @ nat @ N @ P5 ) ) )
@ ( set_ord_lessThan @ nat @ ( suc @ N ) ) ) ) ) ) ).
% diff_power_eq_sum
thf(fact_2640_one__diff__power__eq_H,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X: A,N: nat] :
( ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X @ N ) )
= ( times_times @ A @ ( minus_minus @ A @ ( one_one @ A ) @ X )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( power_power @ A @ X @ ( minus_minus @ nat @ N @ ( suc @ I2 ) ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% one_diff_power_eq'
thf(fact_2641_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
@ ^ [K3: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I2: nat] : ( G @ I2 @ ( minus_minus @ nat @ K3 @ I2 ) )
@ ( set_ord_atMost @ nat @ K3 ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% prod.triangle_reindex_eq
thf(fact_2642_gbinomial__altdef__of__nat,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( ( gbinomial @ A )
= ( ^ [A6: A,K3: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I2: nat] : ( divide_divide @ A @ ( minus_minus @ A @ A6 @ ( semiring_1_of_nat @ A @ I2 ) ) @ ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ K3 @ I2 ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K3 ) ) ) ) ) ).
% gbinomial_altdef_of_nat
thf(fact_2643_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_2644_gbinomial__mult__fact_H,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,K: nat] :
( ( times_times @ A @ ( gbinomial @ A @ A3 @ K ) @ ( semiring_char_0_fact @ A @ K ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I2: nat] : ( minus_minus @ A @ A3 @ ( semiring_1_of_nat @ A @ I2 ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K ) ) ) ) ).
% gbinomial_mult_fact'
thf(fact_2645_gbinomial__mult__fact,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,A3: A] :
( ( times_times @ A @ ( semiring_char_0_fact @ A @ K ) @ ( gbinomial @ A @ A3 @ K ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I2: nat] : ( minus_minus @ A @ A3 @ ( semiring_1_of_nat @ A @ I2 ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K ) ) ) ) ).
% gbinomial_mult_fact
thf(fact_2646_gbinomial__prod__rev,axiom,
! [A: $tType] :
( ( ( semiring_char_0 @ A )
& ( semidom_divide @ A ) )
=> ( ( gbinomial @ A )
= ( ^ [A6: A,K3: nat] :
( divide_divide @ A
@ ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I2: nat] : ( minus_minus @ A @ A6 @ ( semiring_1_of_nat @ A @ I2 ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K3 ) )
@ ( semiring_char_0_fact @ A @ K3 ) ) ) ) ) ).
% gbinomial_prod_rev
thf(fact_2647_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_2648_prod__decode__aux_Opelims,axiom,
! [X: nat,Xa: nat,Y3: product_prod @ nat @ nat] :
( ( ( nat_prod_decode_aux @ X @ Xa )
= Y3 )
=> ( ( accp @ ( product_prod @ nat @ nat ) @ nat_pr5047031295181774490ux_rel @ ( product_Pair @ nat @ nat @ X @ Xa ) )
=> ~ ( ( ( ( ord_less_eq @ nat @ Xa @ X )
=> ( Y3
= ( product_Pair @ nat @ nat @ Xa @ ( minus_minus @ nat @ X @ Xa ) ) ) )
& ( ~ ( ord_less_eq @ nat @ Xa @ X )
=> ( Y3
= ( nat_prod_decode_aux @ ( suc @ X ) @ ( minus_minus @ nat @ Xa @ ( suc @ X ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ nat @ nat ) @ nat_pr5047031295181774490ux_rel @ ( product_Pair @ nat @ nat @ X @ Xa ) ) ) ) ) ).
% prod_decode_aux.pelims
thf(fact_2649_sum__zero__power_H,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A5: set @ nat,C2: nat > A,D3: nat > A] :
( ( ( ( finite_finite2 @ nat @ A5 )
& ( member @ nat @ ( zero_zero @ nat ) @ A5 ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( divide_divide @ A @ ( times_times @ A @ ( C2 @ I2 ) @ ( power_power @ A @ ( zero_zero @ A ) @ I2 ) ) @ ( D3 @ I2 ) )
@ A5 )
= ( divide_divide @ A @ ( C2 @ ( zero_zero @ nat ) ) @ ( D3 @ ( zero_zero @ nat ) ) ) ) )
& ( ~ ( ( finite_finite2 @ nat @ A5 )
& ( member @ nat @ ( zero_zero @ nat ) @ A5 ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( divide_divide @ A @ ( times_times @ A @ ( C2 @ I2 ) @ ( power_power @ A @ ( zero_zero @ A ) @ I2 ) ) @ ( D3 @ I2 ) )
@ A5 )
= ( zero_zero @ A ) ) ) ) ) ).
% sum_zero_power'
thf(fact_2650_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 )
@ ^ [N8: num] : ( some @ num @ ( bit1 @ N8 ) )
@ ( bit_un7362597486090784418nd_num @ M @ N ) ) ) ).
% and_num.simps(9)
thf(fact_2651_or__numerals_I4_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num,Y3: num] :
( ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ ( bit0 @ X ) ) @ ( numeral_numeral @ A @ ( bit1 @ Y3 ) ) )
= ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ X ) @ ( numeral_numeral @ A @ Y3 ) ) ) ) ) ) ).
% or_numerals(4)
thf(fact_2652_or__numerals_I6_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num,Y3: num] :
( ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ ( bit1 @ X ) ) @ ( numeral_numeral @ A @ ( bit0 @ Y3 ) ) )
= ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ X ) @ ( numeral_numeral @ A @ Y3 ) ) ) ) ) ) ).
% or_numerals(6)
thf(fact_2653_or__numerals_I7_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num,Y3: num] :
( ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ ( bit1 @ X ) ) @ ( numeral_numeral @ A @ ( bit1 @ Y3 ) ) )
= ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ X ) @ ( numeral_numeral @ A @ Y3 ) ) ) ) ) ) ).
% or_numerals(7)
thf(fact_2654_gcd__0__left__nat,axiom,
! [X: nat] :
( ( gcd_gcd @ nat @ ( zero_zero @ nat ) @ X )
= X ) ).
% gcd_0_left_nat
thf(fact_2655_gcd__0__nat,axiom,
! [X: nat] :
( ( gcd_gcd @ nat @ X @ ( zero_zero @ nat ) )
= X ) ).
% gcd_0_nat
thf(fact_2656_gcd__nat_Oright__neutral,axiom,
! [A3: nat] :
( ( gcd_gcd @ nat @ A3 @ ( zero_zero @ nat ) )
= A3 ) ).
% gcd_nat.right_neutral
thf(fact_2657_gcd__nat_Oneutr__eq__iff,axiom,
! [A3: nat,B2: nat] :
( ( ( zero_zero @ nat )
= ( gcd_gcd @ nat @ A3 @ B2 ) )
= ( ( A3
= ( zero_zero @ nat ) )
& ( B2
= ( zero_zero @ nat ) ) ) ) ).
% gcd_nat.neutr_eq_iff
thf(fact_2658_gcd__nat_Oleft__neutral,axiom,
! [A3: nat] :
( ( gcd_gcd @ nat @ ( zero_zero @ nat ) @ A3 )
= A3 ) ).
% gcd_nat.left_neutral
thf(fact_2659_gcd__nat_Oeq__neutr__iff,axiom,
! [A3: nat,B2: nat] :
( ( ( gcd_gcd @ nat @ A3 @ B2 )
= ( zero_zero @ nat ) )
= ( ( A3
= ( zero_zero @ nat ) )
& ( B2
= ( zero_zero @ nat ) ) ) ) ).
% gcd_nat.eq_neutr_iff
thf(fact_2660_gcd__nat_Oabsorb1,axiom,
! [A3: nat,B2: nat] :
( ( dvd_dvd @ nat @ A3 @ B2 )
=> ( ( gcd_gcd @ nat @ A3 @ B2 )
= A3 ) ) ).
% gcd_nat.absorb1
thf(fact_2661_gcd__nat_Oabsorb2,axiom,
! [B2: nat,A3: nat] :
( ( dvd_dvd @ nat @ B2 @ A3 )
=> ( ( gcd_gcd @ nat @ A3 @ B2 )
= B2 ) ) ).
% gcd_nat.absorb2
thf(fact_2662_gcd__nat_Obounded__iff,axiom,
! [A3: nat,B2: nat,C2: nat] :
( ( dvd_dvd @ nat @ A3 @ ( gcd_gcd @ nat @ B2 @ C2 ) )
= ( ( dvd_dvd @ nat @ A3 @ B2 )
& ( dvd_dvd @ nat @ A3 @ C2 ) ) ) ).
% gcd_nat.bounded_iff
thf(fact_2663_gcd__proj1__if__dvd__nat,axiom,
! [X: nat,Y3: nat] :
( ( dvd_dvd @ nat @ X @ Y3 )
=> ( ( gcd_gcd @ nat @ X @ Y3 )
= X ) ) ).
% gcd_proj1_if_dvd_nat
thf(fact_2664_gcd__proj2__if__dvd__nat,axiom,
! [Y3: nat,X: nat] :
( ( dvd_dvd @ nat @ Y3 @ X )
=> ( ( gcd_gcd @ nat @ X @ Y3 )
= Y3 ) ) ).
% gcd_proj2_if_dvd_nat
thf(fact_2665_gcd__1__nat,axiom,
! [M: nat] :
( ( gcd_gcd @ nat @ M @ ( one_one @ nat ) )
= ( one_one @ nat ) ) ).
% gcd_1_nat
thf(fact_2666_gcd__Suc__0,axiom,
! [M: nat] :
( ( gcd_gcd @ nat @ M @ ( suc @ ( zero_zero @ nat ) ) )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% gcd_Suc_0
thf(fact_2667_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_2668_or_Oleft__neutral,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A] :
( ( bit_se1065995026697491101ons_or @ A @ ( zero_zero @ A ) @ A3 )
= A3 ) ) ).
% or.left_neutral
thf(fact_2669_or_Oright__neutral,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A] :
( ( bit_se1065995026697491101ons_or @ A @ A3 @ ( zero_zero @ A ) )
= A3 ) ) ).
% or.right_neutral
thf(fact_2670_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_2671_sum__eq__0__iff,axiom,
! [A: $tType,B: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [F6: set @ B,F: B > A] :
( ( finite_finite2 @ B @ F6 )
=> ( ( ( groups7311177749621191930dd_sum @ B @ A @ F @ F6 )
= ( zero_zero @ A ) )
= ( ! [X2: B] :
( ( member @ B @ X2 @ F6 )
=> ( ( F @ X2 )
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% sum_eq_0_iff
thf(fact_2672_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_2673_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 ) )
= ( ? [X2: B] :
( ( member @ B @ X2 @ A5 )
& ( ( F @ X2 )
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% prod_zero_iff
thf(fact_2674_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_2675_bit_Odisj__one__right,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X: A] :
( ( bit_se1065995026697491101ons_or @ A @ X @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% bit.disj_one_right
thf(fact_2676_bit_Odisj__one__left,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X: A] :
( ( bit_se1065995026697491101ons_or @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ X )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% bit.disj_one_left
thf(fact_2677_sum_Odelta_H,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ! [S3: set @ B,A3: B,B2: B > A] :
( ( finite_finite2 @ B @ S3 )
=> ( ( ( member @ B @ A3 @ S3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [K3: B] : ( if @ A @ ( A3 = K3 ) @ ( B2 @ K3 ) @ ( zero_zero @ A ) )
@ S3 )
= ( B2 @ A3 ) ) )
& ( ~ ( member @ B @ A3 @ S3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [K3: B] : ( if @ A @ ( A3 = K3 ) @ ( B2 @ K3 ) @ ( zero_zero @ A ) )
@ S3 )
= ( zero_zero @ A ) ) ) ) ) ) ).
% sum.delta'
thf(fact_2678_sum_Odelta,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ! [S3: set @ B,A3: B,B2: B > A] :
( ( finite_finite2 @ B @ S3 )
=> ( ( ( member @ B @ A3 @ S3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [K3: B] : ( if @ A @ ( K3 = A3 ) @ ( B2 @ K3 ) @ ( zero_zero @ A ) )
@ S3 )
= ( B2 @ A3 ) ) )
& ( ~ ( member @ B @ A3 @ S3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [K3: B] : ( if @ A @ ( K3 = A3 ) @ ( B2 @ K3 ) @ ( zero_zero @ A ) )
@ S3 )
= ( zero_zero @ A ) ) ) ) ) ) ).
% sum.delta
thf(fact_2679_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 ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ A5 )
=> ( ( F @ X2 )
= ( one_one @ nat ) ) ) ) ) ) ).
% prod_eq_1_iff
thf(fact_2680_prod_Odelta,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S3: set @ B,A3: B,B2: B > A] :
( ( finite_finite2 @ B @ S3 )
=> ( ( ( member @ B @ A3 @ S3 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K3: B] : ( if @ A @ ( K3 = A3 ) @ ( B2 @ K3 ) @ ( one_one @ A ) )
@ S3 )
= ( B2 @ A3 ) ) )
& ( ~ ( member @ B @ A3 @ S3 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K3: B] : ( if @ A @ ( K3 = A3 ) @ ( B2 @ K3 ) @ ( one_one @ A ) )
@ S3 )
= ( one_one @ A ) ) ) ) ) ) ).
% prod.delta
thf(fact_2681_prod_Odelta_H,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S3: set @ B,A3: B,B2: B > A] :
( ( finite_finite2 @ B @ S3 )
=> ( ( ( member @ B @ A3 @ S3 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K3: B] : ( if @ A @ ( A3 = K3 ) @ ( B2 @ K3 ) @ ( one_one @ A ) )
@ S3 )
= ( B2 @ A3 ) ) )
& ( ~ ( member @ B @ A3 @ S3 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K3: B] : ( if @ A @ ( A3 = K3 ) @ ( B2 @ K3 ) @ ( one_one @ A ) )
@ S3 )
= ( one_one @ A ) ) ) ) ) ) ).
% prod.delta'
thf(fact_2682_or__nat__numerals_I4_J,axiom,
! [X: num] :
( ( bit_se1065995026697491101ons_or @ nat @ ( numeral_numeral @ nat @ ( bit1 @ X ) ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( numeral_numeral @ nat @ ( bit1 @ X ) ) ) ).
% or_nat_numerals(4)
thf(fact_2683_or__nat__numerals_I2_J,axiom,
! [Y3: num] :
( ( bit_se1065995026697491101ons_or @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( numeral_numeral @ nat @ ( bit1 @ Y3 ) ) )
= ( numeral_numeral @ nat @ ( bit1 @ Y3 ) ) ) ).
% or_nat_numerals(2)
thf(fact_2684_or__numerals_I2_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [Y3: num] :
( ( bit_se1065995026697491101ons_or @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit1 @ Y3 ) ) )
= ( numeral_numeral @ A @ ( bit1 @ Y3 ) ) ) ) ).
% or_numerals(2)
thf(fact_2685_or__numerals_I8_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num] :
( ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ ( bit1 @ X ) ) @ ( one_one @ A ) )
= ( numeral_numeral @ A @ ( bit1 @ X ) ) ) ) ).
% or_numerals(8)
thf(fact_2686_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 ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ A5 )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F @ X2 ) ) ) ) ) ) ).
% prod_pos_nat_iff
thf(fact_2687_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_2688_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_2689_or__nat__numerals_I1_J,axiom,
! [Y3: num] :
( ( bit_se1065995026697491101ons_or @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( numeral_numeral @ nat @ ( bit0 @ Y3 ) ) )
= ( numeral_numeral @ nat @ ( bit1 @ Y3 ) ) ) ).
% or_nat_numerals(1)
thf(fact_2690_or__nat__numerals_I3_J,axiom,
! [X: num] :
( ( bit_se1065995026697491101ons_or @ nat @ ( numeral_numeral @ nat @ ( bit0 @ X ) ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( numeral_numeral @ nat @ ( bit1 @ X ) ) ) ).
% or_nat_numerals(3)
thf(fact_2691_or__numerals_I1_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [Y3: num] :
( ( bit_se1065995026697491101ons_or @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ Y3 ) ) )
= ( numeral_numeral @ A @ ( bit1 @ Y3 ) ) ) ) ).
% or_numerals(1)
thf(fact_2692_or__numerals_I5_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num] :
( ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ ( bit0 @ X ) ) @ ( one_one @ A ) )
= ( numeral_numeral @ A @ ( bit1 @ X ) ) ) ) ).
% or_numerals(5)
thf(fact_2693_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_2694_or__eq__0__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A,B2: A] :
( ( ( bit_se1065995026697491101ons_or @ A @ A3 @ B2 )
= ( zero_zero @ A ) )
= ( ( A3
= ( zero_zero @ A ) )
& ( B2
= ( zero_zero @ A ) ) ) ) ) ).
% or_eq_0_iff
thf(fact_2695_bit_Odisj__zero__right,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X: A] :
( ( bit_se1065995026697491101ons_or @ A @ X @ ( zero_zero @ A ) )
= X ) ) ).
% bit.disj_zero_right
thf(fact_2696_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_2697_gcd__nat_Omono,axiom,
! [A3: nat,C2: nat,B2: nat,D3: nat] :
( ( dvd_dvd @ nat @ A3 @ C2 )
=> ( ( dvd_dvd @ nat @ B2 @ D3 )
=> ( dvd_dvd @ nat @ ( gcd_gcd @ nat @ A3 @ B2 ) @ ( gcd_gcd @ nat @ C2 @ D3 ) ) ) ) ).
% gcd_nat.mono
thf(fact_2698_gcd__nat_OorderE,axiom,
! [A3: nat,B2: nat] :
( ( dvd_dvd @ nat @ A3 @ B2 )
=> ( A3
= ( gcd_gcd @ nat @ A3 @ B2 ) ) ) ).
% gcd_nat.orderE
thf(fact_2699_gcd__nat_OorderI,axiom,
! [A3: nat,B2: nat] :
( ( A3
= ( gcd_gcd @ nat @ A3 @ B2 ) )
=> ( dvd_dvd @ nat @ A3 @ B2 ) ) ).
% gcd_nat.orderI
thf(fact_2700_gcd__nat_Oabsorb3,axiom,
! [A3: nat,B2: nat] :
( ( ( dvd_dvd @ nat @ A3 @ B2 )
& ( A3 != B2 ) )
=> ( ( gcd_gcd @ nat @ A3 @ B2 )
= A3 ) ) ).
% gcd_nat.absorb3
thf(fact_2701_gcd__nat_Oabsorb4,axiom,
! [B2: nat,A3: nat] :
( ( ( dvd_dvd @ nat @ B2 @ A3 )
& ( B2 != A3 ) )
=> ( ( gcd_gcd @ nat @ A3 @ B2 )
= B2 ) ) ).
% gcd_nat.absorb4
thf(fact_2702_gcd__nat_OboundedE,axiom,
! [A3: nat,B2: nat,C2: nat] :
( ( dvd_dvd @ nat @ A3 @ ( gcd_gcd @ nat @ B2 @ C2 ) )
=> ~ ( ( dvd_dvd @ nat @ A3 @ B2 )
=> ~ ( dvd_dvd @ nat @ A3 @ C2 ) ) ) ).
% gcd_nat.boundedE
thf(fact_2703_gcd__nat_OboundedI,axiom,
! [A3: nat,B2: nat,C2: nat] :
( ( dvd_dvd @ nat @ A3 @ B2 )
=> ( ( dvd_dvd @ nat @ A3 @ C2 )
=> ( dvd_dvd @ nat @ A3 @ ( gcd_gcd @ nat @ B2 @ C2 ) ) ) ) ).
% gcd_nat.boundedI
thf(fact_2704_gcd__nat_Oorder__iff,axiom,
( ( dvd_dvd @ nat )
= ( ^ [A6: nat,B5: nat] :
( A6
= ( gcd_gcd @ nat @ A6 @ B5 ) ) ) ) ).
% gcd_nat.order_iff
thf(fact_2705_gcd__nat_Ocobounded1,axiom,
! [A3: nat,B2: nat] : ( dvd_dvd @ nat @ ( gcd_gcd @ nat @ A3 @ B2 ) @ A3 ) ).
% gcd_nat.cobounded1
thf(fact_2706_gcd__nat_Ocobounded2,axiom,
! [A3: nat,B2: nat] : ( dvd_dvd @ nat @ ( gcd_gcd @ nat @ A3 @ B2 ) @ B2 ) ).
% gcd_nat.cobounded2
thf(fact_2707_gcd__nat_Oabsorb__iff1,axiom,
( ( dvd_dvd @ nat )
= ( ^ [A6: nat,B5: nat] :
( ( gcd_gcd @ nat @ A6 @ B5 )
= A6 ) ) ) ).
% gcd_nat.absorb_iff1
thf(fact_2708_gcd__nat_Oabsorb__iff2,axiom,
( ( dvd_dvd @ nat )
= ( ^ [B5: nat,A6: nat] :
( ( gcd_gcd @ nat @ A6 @ B5 )
= B5 ) ) ) ).
% gcd_nat.absorb_iff2
thf(fact_2709_gcd__nat_OcoboundedI1,axiom,
! [A3: nat,C2: nat,B2: nat] :
( ( dvd_dvd @ nat @ A3 @ C2 )
=> ( dvd_dvd @ nat @ ( gcd_gcd @ nat @ A3 @ B2 ) @ C2 ) ) ).
% gcd_nat.coboundedI1
thf(fact_2710_gcd__nat_OcoboundedI2,axiom,
! [B2: nat,C2: nat,A3: nat] :
( ( dvd_dvd @ nat @ B2 @ C2 )
=> ( dvd_dvd @ nat @ ( gcd_gcd @ nat @ A3 @ B2 ) @ C2 ) ) ).
% gcd_nat.coboundedI2
thf(fact_2711_gcd__nat_Ostrict__boundedE,axiom,
! [A3: nat,B2: nat,C2: nat] :
( ( ( dvd_dvd @ nat @ A3 @ ( gcd_gcd @ nat @ B2 @ C2 ) )
& ( A3
!= ( gcd_gcd @ nat @ B2 @ C2 ) ) )
=> ~ ( ( ( dvd_dvd @ nat @ A3 @ B2 )
& ( A3 != B2 ) )
=> ~ ( ( dvd_dvd @ nat @ A3 @ C2 )
& ( A3 != C2 ) ) ) ) ).
% gcd_nat.strict_boundedE
thf(fact_2712_gcd__nat_Ostrict__order__iff,axiom,
! [A3: nat,B2: nat] :
( ( ( dvd_dvd @ nat @ A3 @ B2 )
& ( A3 != B2 ) )
= ( ( A3
= ( gcd_gcd @ nat @ A3 @ B2 ) )
& ( A3 != B2 ) ) ) ).
% gcd_nat.strict_order_iff
thf(fact_2713_gcd__nat_Ostrict__coboundedI1,axiom,
! [A3: nat,C2: nat,B2: nat] :
( ( ( dvd_dvd @ nat @ A3 @ C2 )
& ( A3 != C2 ) )
=> ( ( dvd_dvd @ nat @ ( gcd_gcd @ nat @ A3 @ B2 ) @ C2 )
& ( ( gcd_gcd @ nat @ A3 @ B2 )
!= C2 ) ) ) ).
% gcd_nat.strict_coboundedI1
thf(fact_2714_gcd__nat_Ostrict__coboundedI2,axiom,
! [B2: nat,C2: nat,A3: nat] :
( ( ( dvd_dvd @ nat @ B2 @ C2 )
& ( B2 != C2 ) )
=> ( ( dvd_dvd @ nat @ ( gcd_gcd @ nat @ A3 @ B2 ) @ C2 )
& ( ( gcd_gcd @ nat @ A3 @ B2 )
!= C2 ) ) ) ).
% gcd_nat.strict_coboundedI2
thf(fact_2715_gcd__unique__nat,axiom,
! [D3: nat,A3: nat,B2: nat] :
( ( ( dvd_dvd @ nat @ D3 @ A3 )
& ( dvd_dvd @ nat @ D3 @ B2 )
& ! [E4: nat] :
( ( ( dvd_dvd @ nat @ E4 @ A3 )
& ( dvd_dvd @ nat @ E4 @ B2 ) )
=> ( dvd_dvd @ nat @ E4 @ D3 ) ) )
= ( D3
= ( gcd_gcd @ nat @ A3 @ B2 ) ) ) ).
% gcd_unique_nat
thf(fact_2716_gcd__red__nat,axiom,
( ( gcd_gcd @ nat )
= ( ^ [X2: nat,Y: nat] : ( gcd_gcd @ nat @ Y @ ( modulo_modulo @ nat @ X2 @ Y ) ) ) ) ).
% gcd_red_nat
thf(fact_2717_finite__M__bounded__by__nat,axiom,
! [P: nat > $o,I: nat] :
( finite_finite2 @ nat
@ ( collect @ nat
@ ^ [K3: nat] :
( ( P @ K3 )
& ( ord_less @ nat @ K3 @ I ) ) ) ) ).
% finite_M_bounded_by_nat
thf(fact_2718_finite__less__ub,axiom,
! [F: nat > nat,U2: nat] :
( ! [N2: nat] : ( ord_less_eq @ nat @ N2 @ ( F @ N2 ) )
=> ( finite_finite2 @ nat
@ ( collect @ nat
@ ^ [N5: nat] : ( ord_less_eq @ nat @ ( F @ N5 ) @ U2 ) ) ) ) ).
% finite_less_ub
thf(fact_2719_sum_Oswap__restrict,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( comm_monoid_add @ A )
=> ! [A5: set @ B,B4: set @ C,G: B > C > A,R5: B > C > $o] :
( ( finite_finite2 @ B @ A5 )
=> ( ( finite_finite2 @ C @ B4 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X2: B] :
( groups7311177749621191930dd_sum @ C @ A @ ( G @ X2 )
@ ( collect @ C
@ ^ [Y: C] :
( ( member @ C @ Y @ B4 )
& ( R5 @ X2 @ Y ) ) ) )
@ A5 )
= ( groups7311177749621191930dd_sum @ C @ A
@ ^ [Y: C] :
( groups7311177749621191930dd_sum @ B @ A
@ ^ [X2: B] : ( G @ X2 @ Y )
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ A5 )
& ( R5 @ X2 @ Y ) ) ) )
@ B4 ) ) ) ) ) ).
% sum.swap_restrict
thf(fact_2720_prod_Oswap__restrict,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A5: set @ B,B4: set @ C,G: B > C > A,R5: B > C > $o] :
( ( finite_finite2 @ B @ A5 )
=> ( ( finite_finite2 @ C @ B4 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [X2: B] :
( groups7121269368397514597t_prod @ C @ A @ ( G @ X2 )
@ ( collect @ C
@ ^ [Y: C] :
( ( member @ C @ Y @ B4 )
& ( R5 @ X2 @ Y ) ) ) )
@ A5 )
= ( groups7121269368397514597t_prod @ C @ A
@ ^ [Y: C] :
( groups7121269368397514597t_prod @ B @ A
@ ^ [X2: B] : ( G @ X2 @ Y )
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ A5 )
& ( R5 @ X2 @ Y ) ) ) )
@ B4 ) ) ) ) ) ).
% prod.swap_restrict
thf(fact_2721_gcd__le2__nat,axiom,
! [B2: nat,A3: nat] :
( ( B2
!= ( zero_zero @ nat ) )
=> ( ord_less_eq @ nat @ ( gcd_gcd @ nat @ A3 @ B2 ) @ B2 ) ) ).
% gcd_le2_nat
thf(fact_2722_gcd__le1__nat,axiom,
! [A3: nat,B2: nat] :
( ( A3
!= ( zero_zero @ nat ) )
=> ( ord_less_eq @ nat @ ( gcd_gcd @ nat @ A3 @ B2 ) @ A3 ) ) ).
% gcd_le1_nat
thf(fact_2723_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_2724_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_2725_gcd__nat_Oelims,axiom,
! [X: nat,Xa: nat,Y3: nat] :
( ( ( gcd_gcd @ nat @ X @ Xa )
= Y3 )
=> ( ( ( Xa
= ( zero_zero @ nat ) )
=> ( Y3 = X ) )
& ( ( Xa
!= ( zero_zero @ nat ) )
=> ( Y3
= ( gcd_gcd @ nat @ Xa @ ( modulo_modulo @ nat @ X @ Xa ) ) ) ) ) ) ).
% gcd_nat.elims
thf(fact_2726_gcd__nat_Osimps,axiom,
( ( gcd_gcd @ nat )
= ( ^ [X2: nat,Y: nat] :
( if @ nat
@ ( Y
= ( zero_zero @ nat ) )
@ X2
@ ( gcd_gcd @ nat @ Y @ ( modulo_modulo @ nat @ X2 @ Y ) ) ) ) ) ).
% gcd_nat.simps
thf(fact_2727_gcd__non__0__nat,axiom,
! [Y3: nat,X: nat] :
( ( Y3
!= ( zero_zero @ nat ) )
=> ( ( gcd_gcd @ nat @ X @ Y3 )
= ( gcd_gcd @ nat @ Y3 @ ( modulo_modulo @ nat @ X @ Y3 ) ) ) ) ).
% gcd_non_0_nat
thf(fact_2728_sum_Oreindex__bij__witness__not__neutral,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comm_monoid_add @ A )
=> ! [S6: set @ B,T4: set @ C,S3: set @ B,I: C > B,J: B > C,T5: set @ C,G: B > A,H: C > A] :
( ( finite_finite2 @ B @ S6 )
=> ( ( finite_finite2 @ C @ T4 )
=> ( ! [A4: B] :
( ( member @ B @ A4 @ ( minus_minus @ ( set @ B ) @ S3 @ S6 ) )
=> ( ( I @ ( J @ A4 ) )
= A4 ) )
=> ( ! [A4: B] :
( ( member @ B @ A4 @ ( minus_minus @ ( set @ B ) @ S3 @ S6 ) )
=> ( member @ C @ ( J @ A4 ) @ ( minus_minus @ ( set @ C ) @ T5 @ T4 ) ) )
=> ( ! [B3: C] :
( ( member @ C @ B3 @ ( minus_minus @ ( set @ C ) @ T5 @ T4 ) )
=> ( ( J @ ( I @ B3 ) )
= B3 ) )
=> ( ! [B3: C] :
( ( member @ C @ B3 @ ( minus_minus @ ( set @ C ) @ T5 @ T4 ) )
=> ( member @ B @ ( I @ B3 ) @ ( minus_minus @ ( set @ B ) @ S3 @ S6 ) ) )
=> ( ! [A4: B] :
( ( member @ B @ A4 @ S6 )
=> ( ( G @ A4 )
= ( zero_zero @ A ) ) )
=> ( ! [B3: C] :
( ( member @ C @ B3 @ T4 )
=> ( ( H @ B3 )
= ( zero_zero @ A ) ) )
=> ( ! [A4: B] :
( ( member @ B @ A4 @ S3 )
=> ( ( H @ ( J @ A4 ) )
= ( G @ A4 ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ S3 )
= ( groups7311177749621191930dd_sum @ C @ A @ H @ T5 ) ) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
thf(fact_2729_prod__zero,axiom,
! [B: $tType,A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A5: set @ B,F: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ? [X5: B] :
( ( member @ B @ X5 @ A5 )
& ( ( F @ X5 )
= ( zero_zero @ A ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ F @ A5 )
= ( zero_zero @ A ) ) ) ) ) ).
% prod_zero
thf(fact_2730_prod_Oreindex__bij__witness__not__neutral,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S6: set @ B,T4: set @ C,S3: set @ B,I: C > B,J: B > C,T5: set @ C,G: B > A,H: C > A] :
( ( finite_finite2 @ B @ S6 )
=> ( ( finite_finite2 @ C @ T4 )
=> ( ! [A4: B] :
( ( member @ B @ A4 @ ( minus_minus @ ( set @ B ) @ S3 @ S6 ) )
=> ( ( I @ ( J @ A4 ) )
= A4 ) )
=> ( ! [A4: B] :
( ( member @ B @ A4 @ ( minus_minus @ ( set @ B ) @ S3 @ S6 ) )
=> ( member @ C @ ( J @ A4 ) @ ( minus_minus @ ( set @ C ) @ T5 @ T4 ) ) )
=> ( ! [B3: C] :
( ( member @ C @ B3 @ ( minus_minus @ ( set @ C ) @ T5 @ T4 ) )
=> ( ( J @ ( I @ B3 ) )
= B3 ) )
=> ( ! [B3: C] :
( ( member @ C @ B3 @ ( minus_minus @ ( set @ C ) @ T5 @ T4 ) )
=> ( member @ B @ ( I @ B3 ) @ ( minus_minus @ ( set @ B ) @ S3 @ S6 ) ) )
=> ( ! [A4: B] :
( ( member @ B @ A4 @ S6 )
=> ( ( G @ A4 )
= ( one_one @ A ) ) )
=> ( ! [B3: C] :
( ( member @ C @ B3 @ T4 )
=> ( ( H @ B3 )
= ( one_one @ A ) ) )
=> ( ! [A4: B] :
( ( member @ B @ A4 @ S3 )
=> ( ( H @ ( J @ A4 ) )
= ( G @ A4 ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ S3 )
= ( groups7121269368397514597t_prod @ C @ A @ H @ T5 ) ) ) ) ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_witness_not_neutral
thf(fact_2731_gcd__integer_Orep__eq,axiom,
! [X: code_integer,Xa: code_integer] :
( ( code_int_of_integer @ ( gcd_gcd @ code_integer @ X @ Xa ) )
= ( gcd_gcd @ int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa ) ) ) ).
% gcd_integer.rep_eq
thf(fact_2732_gcd__integer_Oabs__eq,axiom,
! [Xa: int,X: int] :
( ( gcd_gcd @ code_integer @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X ) )
= ( code_integer_of_int @ ( gcd_gcd @ int @ Xa @ X ) ) ) ).
% gcd_integer.abs_eq
thf(fact_2733_subset__eq__atLeast0__atMost__finite,axiom,
! [N6: set @ nat,N: nat] :
( ( ord_less_eq @ ( set @ nat ) @ N6 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
=> ( finite_finite2 @ nat @ N6 ) ) ).
% subset_eq_atLeast0_atMost_finite
thf(fact_2734_subset__eq__atLeast0__lessThan__finite,axiom,
! [N6: set @ nat,N: nat] :
( ( ord_less_eq @ ( set @ nat ) @ N6 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) )
=> ( finite_finite2 @ nat @ N6 ) ) ).
% subset_eq_atLeast0_lessThan_finite
thf(fact_2735_set__bit__nat__def,axiom,
( ( bit_se5668285175392031749et_bit @ nat )
= ( ^ [M5: nat,N5: nat] : ( bit_se1065995026697491101ons_or @ nat @ N5 @ ( bit_se4730199178511100633sh_bit @ nat @ M5 @ ( one_one @ nat ) ) ) ) ) ).
% set_bit_nat_def
thf(fact_2736_sum_Ofinite__Collect__op,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [I5: set @ B,X: B > A,Y3: B > A] :
( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [I2: B] :
( ( member @ B @ I2 @ I5 )
& ( ( X @ I2 )
!= ( zero_zero @ A ) ) ) ) )
=> ( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [I2: B] :
( ( member @ B @ I2 @ I5 )
& ( ( Y3 @ I2 )
!= ( zero_zero @ A ) ) ) ) )
=> ( finite_finite2 @ B
@ ( collect @ B
@ ^ [I2: B] :
( ( member @ B @ I2 @ I5 )
& ( ( plus_plus @ A @ ( X @ I2 ) @ ( Y3 @ I2 ) )
!= ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_2737_prod_Ofinite__Collect__op,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [I5: set @ B,X: B > A,Y3: B > A] :
( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [I2: B] :
( ( member @ B @ I2 @ I5 )
& ( ( X @ I2 )
!= ( one_one @ A ) ) ) ) )
=> ( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [I2: B] :
( ( member @ B @ I2 @ I5 )
& ( ( Y3 @ I2 )
!= ( one_one @ A ) ) ) ) )
=> ( finite_finite2 @ B
@ ( collect @ B
@ ^ [I2: B] :
( ( member @ B @ I2 @ I5 )
& ( ( times_times @ A @ ( X @ I2 ) @ ( Y3 @ I2 ) )
!= ( one_one @ A ) ) ) ) ) ) ) ) ).
% prod.finite_Collect_op
thf(fact_2738_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
@ ^ [X2: B] :
( ( member @ B @ X2 @ A5 )
& ( P @ X2 ) ) ) )
= ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X2: B] : ( if @ A @ ( P @ X2 ) @ ( G @ X2 ) @ ( zero_zero @ A ) )
@ A5 ) ) ) ) ).
% sum.inter_filter
thf(fact_2739_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
@ ^ [X2: B] :
( ( G @ X2 )
= ( zero_zero @ A ) ) ) ) )
= ( groups7311177749621191930dd_sum @ B @ A @ G @ A5 ) ) ) ) ).
% sum.setdiff_irrelevant
thf(fact_2740_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
@ ^ [X2: B] :
( ( G @ X2 )
= ( one_one @ A ) ) ) ) )
= ( groups7121269368397514597t_prod @ B @ A @ G @ A5 ) ) ) ) ).
% prod.setdiff_irrelevant
thf(fact_2741_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
@ ^ [X2: B] :
( ( member @ B @ X2 @ A5 )
& ( P @ X2 ) ) ) )
= ( groups7121269368397514597t_prod @ B @ A
@ ^ [X2: B] : ( if @ A @ ( P @ X2 ) @ ( G @ X2 ) @ ( one_one @ A ) )
@ A5 ) ) ) ) ).
% prod.inter_filter
thf(fact_2742_finite__int__segment,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [A3: A,B2: A] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X2: A] :
( ( member @ A @ X2 @ ( ring_1_Ints @ A ) )
& ( ord_less_eq @ A @ A3 @ X2 )
& ( ord_less_eq @ A @ X2 @ B2 ) ) ) ) ) ).
% finite_int_segment
thf(fact_2743_bezout__nat,axiom,
! [A3: nat,B2: nat] :
( ( A3
!= ( zero_zero @ nat ) )
=> ? [X4: nat,Y4: nat] :
( ( times_times @ nat @ A3 @ X4 )
= ( plus_plus @ nat @ ( times_times @ nat @ B2 @ Y4 ) @ ( gcd_gcd @ nat @ A3 @ B2 ) ) ) ) ).
% bezout_nat
thf(fact_2744_bezout__gcd__nat_H,axiom,
! [B2: nat,A3: nat] :
? [X4: nat,Y4: nat] :
( ( ( ord_less_eq @ nat @ ( times_times @ nat @ B2 @ Y4 ) @ ( times_times @ nat @ A3 @ X4 ) )
& ( ( minus_minus @ nat @ ( times_times @ nat @ A3 @ X4 ) @ ( times_times @ nat @ B2 @ Y4 ) )
= ( gcd_gcd @ nat @ A3 @ B2 ) ) )
| ( ( ord_less_eq @ nat @ ( times_times @ nat @ A3 @ Y4 ) @ ( times_times @ nat @ B2 @ X4 ) )
& ( ( minus_minus @ nat @ ( times_times @ nat @ B2 @ X4 ) @ ( times_times @ nat @ A3 @ Y4 ) )
= ( gcd_gcd @ nat @ A3 @ B2 ) ) ) ) ).
% bezout_gcd_nat'
thf(fact_2745_sum__nonneg__eq__0__iff,axiom,
! [A: $tType,B: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A5: set @ B,F: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ A5 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F @ X4 ) ) )
=> ( ( ( groups7311177749621191930dd_sum @ B @ A @ F @ A5 )
= ( zero_zero @ A ) )
= ( ! [X2: B] :
( ( member @ B @ X2 @ A5 )
=> ( ( F @ X2 )
= ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_2746_sum__le__included,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [S: set @ B,T2: set @ C,G: C > A,I: C > B,F: B > A] :
( ( finite_finite2 @ B @ S )
=> ( ( finite_finite2 @ C @ T2 )
=> ( ! [X4: C] :
( ( member @ C @ X4 @ T2 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( G @ X4 ) ) )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ S )
=> ? [Xa2: C] :
( ( member @ C @ Xa2 @ T2 )
& ( ( I @ Xa2 )
= X4 )
& ( ord_less_eq @ A @ ( F @ X4 ) @ ( G @ Xa2 ) ) ) )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F @ S ) @ ( groups7311177749621191930dd_sum @ C @ A @ G @ T2 ) ) ) ) ) ) ) ).
% sum_le_included
thf(fact_2747_sum_Orelated,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [R5: A > A > $o,S3: set @ B,H: B > A,G: B > A] :
( ( R5 @ ( zero_zero @ A ) @ ( zero_zero @ A ) )
=> ( ! [X13: A,Y12: A,X23: A,Y22: A] :
( ( ( R5 @ X13 @ X23 )
& ( R5 @ Y12 @ Y22 ) )
=> ( R5 @ ( plus_plus @ A @ X13 @ Y12 ) @ ( plus_plus @ A @ X23 @ Y22 ) ) )
=> ( ( finite_finite2 @ B @ S3 )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ S3 )
=> ( R5 @ ( H @ X4 ) @ ( G @ X4 ) ) )
=> ( R5 @ ( groups7311177749621191930dd_sum @ B @ A @ H @ S3 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ S3 ) ) ) ) ) ) ) ).
% sum.related
thf(fact_2748_sum_Osame__carrier,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [C6: set @ B,A5: set @ B,B4: set @ B,G: B > A,H: B > A] :
( ( finite_finite2 @ B @ C6 )
=> ( ( ord_less_eq @ ( set @ B ) @ A5 @ C6 )
=> ( ( ord_less_eq @ ( set @ B ) @ B4 @ C6 )
=> ( ! [A4: B] :
( ( member @ B @ A4 @ ( minus_minus @ ( set @ B ) @ C6 @ A5 ) )
=> ( ( G @ A4 )
= ( zero_zero @ A ) ) )
=> ( ! [B3: B] :
( ( member @ B @ B3 @ ( minus_minus @ ( set @ B ) @ C6 @ B4 ) )
=> ( ( H @ B3 )
= ( zero_zero @ A ) ) )
=> ( ( ( groups7311177749621191930dd_sum @ B @ A @ G @ A5 )
= ( groups7311177749621191930dd_sum @ B @ A @ H @ B4 ) )
= ( ( groups7311177749621191930dd_sum @ B @ A @ G @ C6 )
= ( groups7311177749621191930dd_sum @ B @ A @ H @ C6 ) ) ) ) ) ) ) ) ) ).
% sum.same_carrier
thf(fact_2749_sum_Osame__carrierI,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [C6: set @ B,A5: set @ B,B4: set @ B,G: B > A,H: B > A] :
( ( finite_finite2 @ B @ C6 )
=> ( ( ord_less_eq @ ( set @ B ) @ A5 @ C6 )
=> ( ( ord_less_eq @ ( set @ B ) @ B4 @ C6 )
=> ( ! [A4: B] :
( ( member @ B @ A4 @ ( minus_minus @ ( set @ B ) @ C6 @ A5 ) )
=> ( ( G @ A4 )
= ( zero_zero @ A ) ) )
=> ( ! [B3: B] :
( ( member @ B @ B3 @ ( minus_minus @ ( set @ B ) @ C6 @ B4 ) )
=> ( ( H @ B3 )
= ( 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 @ B4 ) ) ) ) ) ) ) ) ) ).
% sum.same_carrierI
thf(fact_2750_sum_Omono__neutral__left,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [T5: set @ B,S3: set @ B,G: B > A] :
( ( finite_finite2 @ B @ T5 )
=> ( ( ord_less_eq @ ( set @ B ) @ S3 @ T5 )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ ( minus_minus @ ( set @ B ) @ T5 @ S3 ) )
=> ( ( G @ X4 )
= ( zero_zero @ A ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ S3 )
= ( groups7311177749621191930dd_sum @ B @ A @ G @ T5 ) ) ) ) ) ) ).
% sum.mono_neutral_left
thf(fact_2751_sum_Omono__neutral__right,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [T5: set @ B,S3: set @ B,G: B > A] :
( ( finite_finite2 @ B @ T5 )
=> ( ( ord_less_eq @ ( set @ B ) @ S3 @ T5 )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ ( minus_minus @ ( set @ B ) @ T5 @ S3 ) )
=> ( ( G @ X4 )
= ( zero_zero @ A ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ T5 )
= ( groups7311177749621191930dd_sum @ B @ A @ G @ S3 ) ) ) ) ) ) ).
% sum.mono_neutral_right
thf(fact_2752_sum_Omono__neutral__cong__left,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [T5: set @ B,S3: set @ B,H: B > A,G: B > A] :
( ( finite_finite2 @ B @ T5 )
=> ( ( ord_less_eq @ ( set @ B ) @ S3 @ T5 )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ ( minus_minus @ ( set @ B ) @ T5 @ S3 ) )
=> ( ( H @ X4 )
= ( zero_zero @ A ) ) )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ S3 )
=> ( ( G @ X4 )
= ( H @ X4 ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ S3 )
= ( groups7311177749621191930dd_sum @ B @ A @ H @ T5 ) ) ) ) ) ) ) ).
% sum.mono_neutral_cong_left
thf(fact_2753_sum_Omono__neutral__cong__right,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [T5: set @ B,S3: set @ B,G: B > A,H: B > A] :
( ( finite_finite2 @ B @ T5 )
=> ( ( ord_less_eq @ ( set @ B ) @ S3 @ T5 )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ ( minus_minus @ ( set @ B ) @ T5 @ S3 ) )
=> ( ( G @ X4 )
= ( zero_zero @ A ) ) )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ S3 )
=> ( ( G @ X4 )
= ( H @ X4 ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ T5 )
= ( groups7311177749621191930dd_sum @ B @ A @ H @ S3 ) ) ) ) ) ) ) ).
% sum.mono_neutral_cong_right
thf(fact_2754_prod_Orelated,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [R5: A > A > $o,S3: set @ B,H: B > A,G: B > A] :
( ( R5 @ ( one_one @ A ) @ ( one_one @ A ) )
=> ( ! [X13: A,Y12: A,X23: A,Y22: A] :
( ( ( R5 @ X13 @ X23 )
& ( R5 @ Y12 @ Y22 ) )
=> ( R5 @ ( times_times @ A @ X13 @ Y12 ) @ ( times_times @ A @ X23 @ Y22 ) ) )
=> ( ( finite_finite2 @ B @ S3 )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ S3 )
=> ( R5 @ ( H @ X4 ) @ ( G @ X4 ) ) )
=> ( R5 @ ( groups7121269368397514597t_prod @ B @ A @ H @ S3 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ S3 ) ) ) ) ) ) ) ).
% prod.related
thf(fact_2755_sum_Osubset__diff,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [B4: set @ B,A5: set @ B,G: B > A] :
( ( ord_less_eq @ ( set @ B ) @ B4 @ A5 )
=> ( ( finite_finite2 @ B @ A5 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ A5 )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A5 @ B4 ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ B4 ) ) ) ) ) ) ).
% sum.subset_diff
thf(fact_2756_prod_Osame__carrier,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [C6: set @ B,A5: set @ B,B4: set @ B,G: B > A,H: B > A] :
( ( finite_finite2 @ B @ C6 )
=> ( ( ord_less_eq @ ( set @ B ) @ A5 @ C6 )
=> ( ( ord_less_eq @ ( set @ B ) @ B4 @ C6 )
=> ( ! [A4: B] :
( ( member @ B @ A4 @ ( minus_minus @ ( set @ B ) @ C6 @ A5 ) )
=> ( ( G @ A4 )
= ( one_one @ A ) ) )
=> ( ! [B3: B] :
( ( member @ B @ B3 @ ( minus_minus @ ( set @ B ) @ C6 @ B4 ) )
=> ( ( H @ B3 )
= ( one_one @ A ) ) )
=> ( ( ( groups7121269368397514597t_prod @ B @ A @ G @ A5 )
= ( groups7121269368397514597t_prod @ B @ A @ H @ B4 ) )
= ( ( groups7121269368397514597t_prod @ B @ A @ G @ C6 )
= ( groups7121269368397514597t_prod @ B @ A @ H @ C6 ) ) ) ) ) ) ) ) ) ).
% prod.same_carrier
thf(fact_2757_prod_Osame__carrierI,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [C6: set @ B,A5: set @ B,B4: set @ B,G: B > A,H: B > A] :
( ( finite_finite2 @ B @ C6 )
=> ( ( ord_less_eq @ ( set @ B ) @ A5 @ C6 )
=> ( ( ord_less_eq @ ( set @ B ) @ B4 @ C6 )
=> ( ! [A4: B] :
( ( member @ B @ A4 @ ( minus_minus @ ( set @ B ) @ C6 @ A5 ) )
=> ( ( G @ A4 )
= ( one_one @ A ) ) )
=> ( ! [B3: B] :
( ( member @ B @ B3 @ ( minus_minus @ ( set @ B ) @ C6 @ B4 ) )
=> ( ( H @ B3 )
= ( 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 @ B4 ) ) ) ) ) ) ) ) ) ).
% prod.same_carrierI
thf(fact_2758_prod_Omono__neutral__left,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [T5: set @ B,S3: set @ B,G: B > A] :
( ( finite_finite2 @ B @ T5 )
=> ( ( ord_less_eq @ ( set @ B ) @ S3 @ T5 )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ ( minus_minus @ ( set @ B ) @ T5 @ S3 ) )
=> ( ( G @ X4 )
= ( one_one @ A ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ S3 )
= ( groups7121269368397514597t_prod @ B @ A @ G @ T5 ) ) ) ) ) ) ).
% prod.mono_neutral_left
thf(fact_2759_prod_Omono__neutral__right,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [T5: set @ B,S3: set @ B,G: B > A] :
( ( finite_finite2 @ B @ T5 )
=> ( ( ord_less_eq @ ( set @ B ) @ S3 @ T5 )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ ( minus_minus @ ( set @ B ) @ T5 @ S3 ) )
=> ( ( G @ X4 )
= ( one_one @ A ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ T5 )
= ( groups7121269368397514597t_prod @ B @ A @ G @ S3 ) ) ) ) ) ) ).
% prod.mono_neutral_right
thf(fact_2760_prod_Omono__neutral__cong__left,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [T5: set @ B,S3: set @ B,H: B > A,G: B > A] :
( ( finite_finite2 @ B @ T5 )
=> ( ( ord_less_eq @ ( set @ B ) @ S3 @ T5 )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ ( minus_minus @ ( set @ B ) @ T5 @ S3 ) )
=> ( ( H @ X4 )
= ( one_one @ A ) ) )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ S3 )
=> ( ( G @ X4 )
= ( H @ X4 ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ S3 )
= ( groups7121269368397514597t_prod @ B @ A @ H @ T5 ) ) ) ) ) ) ) ).
% prod.mono_neutral_cong_left
thf(fact_2761_prod_Omono__neutral__cong__right,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [T5: set @ B,S3: set @ B,G: B > A,H: B > A] :
( ( finite_finite2 @ B @ T5 )
=> ( ( ord_less_eq @ ( set @ B ) @ S3 @ T5 )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ ( minus_minus @ ( set @ B ) @ T5 @ S3 ) )
=> ( ( G @ X4 )
= ( one_one @ A ) ) )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ S3 )
=> ( ( G @ X4 )
= ( H @ X4 ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ T5 )
= ( groups7121269368397514597t_prod @ B @ A @ H @ S3 ) ) ) ) ) ) ) ).
% prod.mono_neutral_cong_right
thf(fact_2762_set__bit__eq__or,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se5668285175392031749et_bit @ A )
= ( ^ [N5: nat,A6: A] : ( bit_se1065995026697491101ons_or @ A @ A6 @ ( bit_se4730199178511100633sh_bit @ A @ N5 @ ( one_one @ A ) ) ) ) ) ) ).
% set_bit_eq_or
thf(fact_2763_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 ) ) )
= ( ? [X2: A] :
( ( member @ A @ X2 @ A5 )
& ( ( F @ X2 )
= ( suc @ ( zero_zero @ nat ) ) )
& ! [Y: A] :
( ( member @ A @ Y @ A5 )
=> ( ( X2 != Y )
=> ( ( F @ Y )
= ( zero_zero @ nat ) ) ) ) ) ) ) ) ).
% sum_eq_Suc0_iff
thf(fact_2764_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 ) )
= ( ? [X2: A] :
( ( member @ A @ X2 @ A5 )
& ( ( F @ X2 )
= ( one_one @ nat ) )
& ! [Y: A] :
( ( member @ A @ Y @ A5 )
=> ( ( X2 != Y )
=> ( ( F @ Y )
= ( zero_zero @ nat ) ) ) ) ) ) ) ) ).
% sum_eq_1_iff
thf(fact_2765_sum__nonneg__0,axiom,
! [B: $tType,A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [S: set @ B,F: B > A,I: B] :
( ( finite_finite2 @ B @ S )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ S )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F @ I3 ) ) )
=> ( ( ( groups7311177749621191930dd_sum @ B @ A @ F @ S )
= ( zero_zero @ A ) )
=> ( ( member @ B @ I @ S )
=> ( ( F @ I )
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_2766_sum__nonneg__leq__bound,axiom,
! [B: $tType,A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [S: set @ B,F: B > A,B4: A,I: B] :
( ( finite_finite2 @ B @ S )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ S )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F @ I3 ) ) )
=> ( ( ( groups7311177749621191930dd_sum @ B @ A @ F @ S )
= B4 )
=> ( ( member @ B @ I @ S )
=> ( ord_less_eq @ A @ ( F @ I ) @ B4 ) ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_2767_finite__divisors__nat,axiom,
! [M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( finite_finite2 @ nat
@ ( collect @ nat
@ ^ [D4: nat] : ( dvd_dvd @ nat @ D4 @ M ) ) ) ) ).
% finite_divisors_nat
thf(fact_2768_gcd__code__integer,axiom,
( ( gcd_gcd @ code_integer )
= ( ^ [K3: code_integer,L4: code_integer] :
( abs_abs @ code_integer
@ ( if @ code_integer
@ ( L4
= ( zero_zero @ code_integer ) )
@ K3
@ ( gcd_gcd @ code_integer @ L4 @ ( modulo_modulo @ code_integer @ ( abs_abs @ code_integer @ K3 ) @ ( abs_abs @ code_integer @ L4 ) ) ) ) ) ) ) ).
% gcd_code_integer
thf(fact_2769_finite__abs__int__segment,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [A3: A] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [K3: A] :
( ( member @ A @ K3 @ ( ring_1_Ints @ A ) )
& ( ord_less_eq @ A @ ( abs_abs @ A @ K3 ) @ A3 ) ) ) ) ) ).
% finite_abs_int_segment
thf(fact_2770_bit_Ocomplement__unique,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A3: A,X: A,Y3: A] :
( ( ( bit_se5824344872417868541ns_and @ A @ A3 @ X )
= ( zero_zero @ A ) )
=> ( ( ( bit_se1065995026697491101ons_or @ A @ A3 @ X )
= ( uminus_uminus @ A @ ( one_one @ A ) ) )
=> ( ( ( bit_se5824344872417868541ns_and @ A @ A3 @ Y3 )
= ( zero_zero @ A ) )
=> ( ( ( bit_se1065995026697491101ons_or @ A @ A3 @ Y3 )
= ( uminus_uminus @ A @ ( one_one @ A ) ) )
=> ( X = Y3 ) ) ) ) ) ) ).
% bit.complement_unique
thf(fact_2771_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_2772_sum__mono2,axiom,
! [A: $tType,B: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [B4: set @ B,A5: set @ B,F: B > A] :
( ( finite_finite2 @ B @ B4 )
=> ( ( ord_less_eq @ ( set @ B ) @ A5 @ B4 )
=> ( ! [B3: B] :
( ( member @ B @ B3 @ ( minus_minus @ ( set @ B ) @ B4 @ A5 ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F @ B3 ) ) )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F @ A5 ) @ ( groups7311177749621191930dd_sum @ B @ A @ F @ B4 ) ) ) ) ) ) ).
% sum_mono2
thf(fact_2773_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_2774_gcd__int__def,axiom,
( ( gcd_gcd @ int )
= ( ^ [X2: int,Y: int] : ( semiring_1_of_nat @ int @ ( gcd_gcd @ nat @ ( nat2 @ ( abs_abs @ int @ X2 ) ) @ ( nat2 @ ( abs_abs @ int @ Y ) ) ) ) ) ) ).
% gcd_int_def
thf(fact_2775_sum__strict__mono2,axiom,
! [B: $tType,A: $tType] :
( ( ordere8940638589300402666id_add @ B )
=> ! [B4: set @ A,A5: set @ A,B2: A,F: A > B] :
( ( finite_finite2 @ A @ B4 )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
=> ( ( member @ A @ B2 @ ( minus_minus @ ( set @ A ) @ B4 @ A5 ) )
=> ( ( ord_less @ B @ ( zero_zero @ B ) @ ( F @ B2 ) )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ B4 )
=> ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( F @ X4 ) ) )
=> ( ord_less @ B @ ( groups7311177749621191930dd_sum @ A @ B @ F @ A5 ) @ ( groups7311177749621191930dd_sum @ A @ B @ F @ B4 ) ) ) ) ) ) ) ) ).
% sum_strict_mono2
thf(fact_2776_prod__mono2,axiom,
! [B: $tType,A: $tType] :
( ( linordered_idom @ B )
=> ! [B4: set @ A,A5: set @ A,F: A > B] :
( ( finite_finite2 @ A @ B4 )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
=> ( ! [B3: A] :
( ( member @ A @ B3 @ ( minus_minus @ ( set @ A ) @ B4 @ A5 ) )
=> ( ord_less_eq @ B @ ( one_one @ B ) @ ( F @ B3 ) ) )
=> ( ! [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 @ B4 ) ) ) ) ) ) ) ).
% prod_mono2
thf(fact_2777_bezw__aux,axiom,
! [X: nat,Y3: nat] :
( ( semiring_1_of_nat @ int @ ( gcd_gcd @ nat @ X @ Y3 ) )
= ( plus_plus @ int @ ( times_times @ int @ ( product_fst @ int @ int @ ( bezw @ X @ Y3 ) ) @ ( semiring_1_of_nat @ int @ X ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ ( bezw @ X @ Y3 ) ) @ ( semiring_1_of_nat @ int @ Y3 ) ) ) ) ).
% bezw_aux
thf(fact_2778_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_2779_one__or__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A] :
( ( bit_se1065995026697491101ons_or @ A @ ( one_one @ A ) @ A3 )
= ( plus_plus @ A @ A3 @ ( zero_neq_one_of_bool @ A @ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) ) ) ) ).
% one_or_eq
thf(fact_2780_or__one__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A] :
( ( bit_se1065995026697491101ons_or @ A @ A3 @ ( one_one @ A ) )
= ( plus_plus @ A @ A3 @ ( zero_neq_one_of_bool @ A @ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) ) ) ) ).
% or_one_eq
thf(fact_2781_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_2782_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_2783_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_2784_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_2785_or__nat__rec,axiom,
( ( bit_se1065995026697491101ons_or @ nat )
= ( ^ [M5: nat,N5: nat] :
( plus_plus @ nat
@ ( zero_neq_one_of_bool @ nat
@ ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M5 )
| ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N5 ) ) )
@ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ nat @ ( divide_divide @ nat @ M5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% or_nat_rec
thf(fact_2786_finite__Collect__le__nat,axiom,
! [K: nat] :
( finite_finite2 @ nat
@ ( collect @ nat
@ ^ [N5: nat] : ( ord_less_eq @ nat @ N5 @ K ) ) ) ).
% finite_Collect_le_nat
thf(fact_2787_finite__Collect__less__nat,axiom,
! [K: nat] :
( finite_finite2 @ nat
@ ( collect @ nat
@ ^ [N5: nat] : ( ord_less @ nat @ N5 @ K ) ) ) ).
% finite_Collect_less_nat
thf(fact_2788_finite__Collect__subsets,axiom,
! [A: $tType,A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( finite_finite2 @ ( set @ A )
@ ( collect @ ( set @ A )
@ ^ [B7: set @ A] : ( ord_less_eq @ ( set @ A ) @ B7 @ A5 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_2789_gcd__nat_Opelims,axiom,
! [X: nat,Xa: nat,Y3: nat] :
( ( ( gcd_gcd @ nat @ X @ Xa )
= Y3 )
=> ( ( accp @ ( product_prod @ nat @ nat ) @ gcd_nat_rel @ ( product_Pair @ nat @ nat @ X @ Xa ) )
=> ~ ( ( ( ( Xa
= ( zero_zero @ nat ) )
=> ( Y3 = X ) )
& ( ( Xa
!= ( zero_zero @ nat ) )
=> ( Y3
= ( gcd_gcd @ nat @ Xa @ ( modulo_modulo @ nat @ X @ Xa ) ) ) ) )
=> ~ ( accp @ ( product_prod @ nat @ nat ) @ gcd_nat_rel @ ( product_Pair @ nat @ nat @ X @ Xa ) ) ) ) ) ).
% gcd_nat.pelims
thf(fact_2790_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
@ ^ [X2: A] :
( ( P @ X2 )
& ( Q @ X2 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_2791_finite__Collect__disjI,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X2: A] :
( ( P @ X2 )
| ( Q @ X2 ) ) ) )
= ( ( finite_finite2 @ A @ ( collect @ A @ P ) )
& ( finite_finite2 @ A @ ( collect @ A @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_2792_gcd__idem__nat,axiom,
! [X: nat] :
( ( gcd_gcd @ nat @ X @ X )
= X ) ).
% gcd_idem_nat
thf(fact_2793_gcd__nat_Oright__idem,axiom,
! [A3: nat,B2: nat] :
( ( gcd_gcd @ nat @ ( gcd_gcd @ nat @ A3 @ B2 ) @ B2 )
= ( gcd_gcd @ nat @ A3 @ B2 ) ) ).
% gcd_nat.right_idem
thf(fact_2794_gcd__nat_Oleft__idem,axiom,
! [A3: nat,B2: nat] :
( ( gcd_gcd @ nat @ A3 @ ( gcd_gcd @ nat @ A3 @ B2 ) )
= ( gcd_gcd @ nat @ A3 @ B2 ) ) ).
% gcd_nat.left_idem
thf(fact_2795_gcd__nat_Oidem,axiom,
! [A3: nat] :
( ( gcd_gcd @ nat @ A3 @ A3 )
= A3 ) ).
% gcd_nat.idem
thf(fact_2796_finite__interval__int1,axiom,
! [A3: int,B2: int] :
( finite_finite2 @ int
@ ( collect @ int
@ ^ [I2: int] :
( ( ord_less_eq @ int @ A3 @ I2 )
& ( ord_less_eq @ int @ I2 @ B2 ) ) ) ) ).
% finite_interval_int1
thf(fact_2797_finite__interval__int4,axiom,
! [A3: int,B2: int] :
( finite_finite2 @ int
@ ( collect @ int
@ ^ [I2: int] :
( ( ord_less @ int @ A3 @ I2 )
& ( ord_less @ int @ I2 @ B2 ) ) ) ) ).
% finite_interval_int4
thf(fact_2798_finite__interval__int2,axiom,
! [A3: int,B2: int] :
( finite_finite2 @ int
@ ( collect @ int
@ ^ [I2: int] :
( ( ord_less_eq @ int @ A3 @ I2 )
& ( ord_less @ int @ I2 @ B2 ) ) ) ) ).
% finite_interval_int2
thf(fact_2799_finite__interval__int3,axiom,
! [A3: int,B2: int] :
( finite_finite2 @ int
@ ( collect @ int
@ ^ [I2: int] :
( ( ord_less @ int @ A3 @ I2 )
& ( ord_less_eq @ int @ I2 @ B2 ) ) ) ) ).
% finite_interval_int3
thf(fact_2800_minus__set__def,axiom,
! [A: $tType] :
( ( minus_minus @ ( set @ A ) )
= ( ^ [A8: set @ A,B7: set @ A] :
( collect @ A
@ ( minus_minus @ ( A > $o )
@ ^ [X2: A] : ( member @ A @ X2 @ A8 )
@ ^ [X2: A] : ( member @ A @ X2 @ B7 ) ) ) ) ) ).
% minus_set_def
thf(fact_2801_set__diff__eq,axiom,
! [A: $tType] :
( ( minus_minus @ ( set @ A ) )
= ( ^ [A8: set @ A,B7: set @ A] :
( collect @ A
@ ^ [X2: A] :
( ( member @ A @ X2 @ A8 )
& ~ ( member @ A @ X2 @ B7 ) ) ) ) ) ).
% set_diff_eq
thf(fact_2802_finite__divisors__int,axiom,
! [I: int] :
( ( I
!= ( zero_zero @ int ) )
=> ( finite_finite2 @ int
@ ( collect @ int
@ ^ [D4: int] : ( dvd_dvd @ int @ D4 @ I ) ) ) ) ).
% finite_divisors_int
thf(fact_2803_card_Ocomp__fun__commute__on,axiom,
( ( comp @ nat @ nat @ nat @ suc @ suc )
= ( comp @ nat @ nat @ nat @ suc @ suc ) ) ).
% card.comp_fun_commute_on
thf(fact_2804_pigeonhole__infinite__rel,axiom,
! [B: $tType,A: $tType,A5: set @ A,B4: set @ B,R5: A > B > $o] :
( ~ ( finite_finite2 @ A @ A5 )
=> ( ( finite_finite2 @ B @ B4 )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ A5 )
=> ? [Xa2: B] :
( ( member @ B @ Xa2 @ B4 )
& ( R5 @ X4 @ Xa2 ) ) )
=> ? [X4: B] :
( ( member @ B @ X4 @ B4 )
& ~ ( finite_finite2 @ A
@ ( collect @ A
@ ^ [A6: A] :
( ( member @ A @ A6 @ A5 )
& ( R5 @ A6 @ X4 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_2805_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_2806_in__finite__psubset,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ A5 @ B4 ) @ ( finite_psubset @ A ) )
= ( ( ord_less @ ( set @ A ) @ A5 @ B4 )
& ( finite_finite2 @ A @ B4 ) ) ) ).
% in_finite_psubset
thf(fact_2807_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_2808_or__nat__unfold,axiom,
( ( bit_se1065995026697491101ons_or @ nat )
= ( ^ [M5: nat,N5: nat] :
( if @ nat
@ ( M5
= ( zero_zero @ nat ) )
@ N5
@ ( if @ nat
@ ( N5
= ( zero_zero @ nat ) )
@ M5
@ ( plus_plus @ nat @ ( ord_max @ nat @ ( modulo_modulo @ nat @ M5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ nat @ N5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ nat @ ( divide_divide @ nat @ M5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% or_nat_unfold
thf(fact_2809_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
@ ^ [A6: B] :
( ( member @ B @ A6 @ A5 )
& ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( F @ A6 ) ) ) ) ) ) ) ) ) ).
% even_sum_iff
thf(fact_2810_drop__bit__rec,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se4197421643247451524op_bit @ A )
= ( ^ [N5: nat,A6: A] :
( if @ A
@ ( N5
= ( zero_zero @ nat ) )
@ A6
@ ( bit_se4197421643247451524op_bit @ A @ ( minus_minus @ nat @ N5 @ ( one_one @ nat ) ) @ ( divide_divide @ A @ A6 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% drop_bit_rec
thf(fact_2811_finite__psubset__def,axiom,
! [A: $tType] :
( ( finite_psubset @ A )
= ( collect @ ( product_prod @ ( set @ A ) @ ( set @ A ) )
@ ( product_case_prod @ ( set @ A ) @ ( set @ A ) @ $o
@ ^ [A8: set @ A,B7: set @ A] :
( ( ord_less @ ( set @ A ) @ A8 @ B7 )
& ( finite_finite2 @ A @ B7 ) ) ) ) ) ).
% finite_psubset_def
thf(fact_2812_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_2813_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_2814_drop__bit__drop__bit,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,N: nat,A3: A] :
( ( bit_se4197421643247451524op_bit @ A @ M @ ( bit_se4197421643247451524op_bit @ A @ N @ A3 ) )
= ( bit_se4197421643247451524op_bit @ A @ ( plus_plus @ nat @ M @ N ) @ A3 ) ) ) ).
% drop_bit_drop_bit
thf(fact_2815_of__bool__or__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [P: $o,Q: $o] :
( ( zero_neq_one_of_bool @ A
@ ( P
| Q ) )
= ( ord_max @ A @ ( zero_neq_one_of_bool @ A @ P ) @ ( zero_neq_one_of_bool @ A @ Q ) ) ) ) ).
% of_bool_or_iff
thf(fact_2816_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_2817_max__0R,axiom,
! [N: nat] :
( ( ord_max @ nat @ N @ ( zero_zero @ nat ) )
= N ) ).
% max_0R
thf(fact_2818_max__0L,axiom,
! [N: nat] :
( ( ord_max @ nat @ ( zero_zero @ nat ) @ N )
= N ) ).
% max_0L
thf(fact_2819_max__nat_Oright__neutral,axiom,
! [A3: nat] :
( ( ord_max @ nat @ A3 @ ( zero_zero @ nat ) )
= A3 ) ).
% max_nat.right_neutral
thf(fact_2820_max__nat_Oneutr__eq__iff,axiom,
! [A3: nat,B2: nat] :
( ( ( zero_zero @ nat )
= ( ord_max @ nat @ A3 @ B2 ) )
= ( ( A3
= ( zero_zero @ nat ) )
& ( B2
= ( zero_zero @ nat ) ) ) ) ).
% max_nat.neutr_eq_iff
thf(fact_2821_max__nat_Oleft__neutral,axiom,
! [A3: nat] :
( ( ord_max @ nat @ ( zero_zero @ nat ) @ A3 )
= A3 ) ).
% max_nat.left_neutral
thf(fact_2822_max__nat_Oeq__neutr__iff,axiom,
! [A3: nat,B2: nat] :
( ( ( ord_max @ nat @ A3 @ B2 )
= ( zero_zero @ nat ) )
= ( ( A3
= ( zero_zero @ nat ) )
& ( B2
= ( zero_zero @ nat ) ) ) ) ).
% max_nat.eq_neutr_iff
thf(fact_2823_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_2824_max__0__1_I4_J,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [X: num] :
( ( ord_max @ A @ ( numeral_numeral @ A @ X ) @ ( zero_zero @ A ) )
= ( numeral_numeral @ A @ X ) ) ) ).
% max_0_1(4)
thf(fact_2825_max__0__1_I3_J,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [X: num] :
( ( ord_max @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ X ) )
= ( numeral_numeral @ A @ X ) ) ) ).
% max_0_1(3)
thf(fact_2826_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_2827_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_2828_max__0__1_I6_J,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [X: num] :
( ( ord_max @ A @ ( numeral_numeral @ A @ X ) @ ( one_one @ A ) )
= ( numeral_numeral @ A @ X ) ) ) ).
% max_0_1(6)
thf(fact_2829_max__0__1_I5_J,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [X: num] :
( ( ord_max @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ X ) )
= ( numeral_numeral @ A @ X ) ) ) ).
% max_0_1(5)
thf(fact_2830_card_Oinfinite,axiom,
! [A: $tType,A5: set @ A] :
( ~ ( finite_finite2 @ A @ A5 )
=> ( ( finite_card @ A @ A5 )
= ( zero_zero @ nat ) ) ) ).
% card.infinite
thf(fact_2831_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_2832_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_2833_prod__constant,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [Y3: A,A5: set @ B] :
( ( groups7121269368397514597t_prod @ B @ A
@ ^ [X2: B] : Y3
@ A5 )
= ( power_power @ A @ Y3 @ ( finite_card @ B @ A5 ) ) ) ) ).
% prod_constant
thf(fact_2834_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_2835_sum__constant,axiom,
! [B: $tType,A: $tType] :
( ( semiring_1 @ A )
=> ! [Y3: A,A5: set @ B] :
( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X2: B] : Y3
@ A5 )
= ( times_times @ A @ ( semiring_1_of_nat @ A @ ( finite_card @ B @ A5 ) ) @ Y3 ) ) ) ).
% sum_constant
thf(fact_2836_max__add__distrib__left,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [X: A,Y3: A,Z: A] :
( ( plus_plus @ A @ ( ord_max @ A @ X @ Y3 ) @ Z )
= ( ord_max @ A @ ( plus_plus @ A @ X @ Z ) @ ( plus_plus @ A @ Y3 @ Z ) ) ) ) ).
% max_add_distrib_left
thf(fact_2837_max__add__distrib__right,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [X: A,Y3: A,Z: A] :
( ( plus_plus @ A @ X @ ( ord_max @ A @ Y3 @ Z ) )
= ( ord_max @ A @ ( plus_plus @ A @ X @ Y3 ) @ ( plus_plus @ A @ X @ Z ) ) ) ) ).
% max_add_distrib_right
thf(fact_2838_max__diff__distrib__left,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [X: A,Y3: A,Z: A] :
( ( minus_minus @ A @ ( ord_max @ A @ X @ Y3 ) @ Z )
= ( ord_max @ A @ ( minus_minus @ A @ X @ Z ) @ ( minus_minus @ A @ Y3 @ Z ) ) ) ) ).
% max_diff_distrib_left
thf(fact_2839_of__int__max,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: int,Y3: int] :
( ( ring_1_of_int @ A @ ( ord_max @ int @ X @ Y3 ) )
= ( ord_max @ A @ ( ring_1_of_int @ A @ X ) @ ( ring_1_of_int @ A @ Y3 ) ) ) ) ).
% of_int_max
thf(fact_2840_of__nat__max,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [X: nat,Y3: nat] :
( ( semiring_1_of_nat @ A @ ( ord_max @ nat @ X @ Y3 ) )
= ( ord_max @ A @ ( semiring_1_of_nat @ A @ X ) @ ( semiring_1_of_nat @ A @ Y3 ) ) ) ) ).
% of_nat_max
thf(fact_2841_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_2842_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_2843_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_2844_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_2845_max__def__raw,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_max @ A )
= ( ^ [A6: A,B5: A] : ( if @ A @ ( ord_less_eq @ A @ A6 @ B5 ) @ B5 @ A6 ) ) ) ) ).
% max_def_raw
thf(fact_2846_n__subsets,axiom,
! [A: $tType,A5: set @ A,K: nat] :
( ( finite_finite2 @ A @ A5 )
=> ( ( finite_card @ ( set @ A )
@ ( collect @ ( set @ A )
@ ^ [B7: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B7 @ A5 )
& ( ( finite_card @ A @ B7 )
= K ) ) ) )
= ( binomial @ ( finite_card @ A @ A5 ) @ K ) ) ) ).
% n_subsets
thf(fact_2847_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_2848_take__bit__eq__self__iff__drop__bit__eq__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A3: A] :
( ( ( bit_se2584673776208193580ke_bit @ A @ N @ A3 )
= A3 )
= ( ( bit_se4197421643247451524op_bit @ A @ N @ A3 )
= ( zero_zero @ A ) ) ) ) ).
% take_bit_eq_self_iff_drop_bit_eq_0
thf(fact_2849_take__bit__drop__bit,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,N: nat,A3: A] :
( ( bit_se2584673776208193580ke_bit @ A @ M @ ( bit_se4197421643247451524op_bit @ A @ N @ A3 ) )
= ( bit_se4197421643247451524op_bit @ A @ N @ ( bit_se2584673776208193580ke_bit @ A @ ( plus_plus @ nat @ M @ N ) @ A3 ) ) ) ) ).
% take_bit_drop_bit
thf(fact_2850_sum__multicount__gen,axiom,
! [A: $tType,B: $tType,S: set @ A,T2: set @ B,R5: A > B > $o,K: B > nat] :
( ( finite_finite2 @ A @ S )
=> ( ( finite_finite2 @ B @ T2 )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ T2 )
=> ( ( finite_card @ A
@ ( collect @ A
@ ^ [I2: A] :
( ( member @ A @ I2 @ S )
& ( R5 @ I2 @ X4 ) ) ) )
= ( K @ X4 ) ) )
=> ( ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [I2: A] :
( finite_card @ B
@ ( collect @ B
@ ^ [J3: B] :
( ( member @ B @ J3 @ T2 )
& ( R5 @ I2 @ J3 ) ) ) )
@ S )
= ( groups7311177749621191930dd_sum @ B @ nat @ K @ T2 ) ) ) ) ) ).
% sum_multicount_gen
thf(fact_2851_card__eq__sum,axiom,
! [A: $tType] :
( ( finite_card @ A )
= ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [X2: A] : ( one_one @ nat ) ) ) ).
% card_eq_sum
thf(fact_2852_max__Suc2,axiom,
! [M: nat,N: nat] :
( ( ord_max @ nat @ M @ ( suc @ N ) )
= ( case_nat @ nat @ ( suc @ N )
@ ^ [M6: nat] : ( suc @ ( ord_max @ nat @ M6 @ N ) )
@ M ) ) ).
% max_Suc2
thf(fact_2853_max__Suc1,axiom,
! [N: nat,M: nat] :
( ( ord_max @ nat @ ( suc @ N ) @ M )
= ( case_nat @ nat @ ( suc @ N )
@ ^ [M6: nat] : ( suc @ ( ord_max @ nat @ N @ M6 ) )
@ M ) ) ).
% max_Suc1
thf(fact_2854_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_2855_card__less__Suc2,axiom,
! [M2: set @ nat,I: nat] :
( ~ ( member @ nat @ ( zero_zero @ nat ) @ M2 )
=> ( ( finite_card @ nat
@ ( collect @ nat
@ ^ [K3: nat] :
( ( member @ nat @ ( suc @ K3 ) @ M2 )
& ( ord_less @ nat @ K3 @ I ) ) ) )
= ( finite_card @ nat
@ ( collect @ nat
@ ^ [K3: nat] :
( ( member @ nat @ K3 @ M2 )
& ( ord_less @ nat @ K3 @ ( suc @ I ) ) ) ) ) ) ) ).
% card_less_Suc2
thf(fact_2856_card__less__Suc,axiom,
! [M2: set @ nat,I: nat] :
( ( member @ nat @ ( zero_zero @ nat ) @ M2 )
=> ( ( suc
@ ( finite_card @ nat
@ ( collect @ nat
@ ^ [K3: nat] :
( ( member @ nat @ ( suc @ K3 ) @ M2 )
& ( ord_less @ nat @ K3 @ I ) ) ) ) )
= ( finite_card @ nat
@ ( collect @ nat
@ ^ [K3: nat] :
( ( member @ nat @ K3 @ M2 )
& ( ord_less @ nat @ K3 @ ( suc @ I ) ) ) ) ) ) ) ).
% card_less_Suc
thf(fact_2857_card__less,axiom,
! [M2: set @ nat,I: nat] :
( ( member @ nat @ ( zero_zero @ nat ) @ M2 )
=> ( ( finite_card @ nat
@ ( collect @ nat
@ ^ [K3: nat] :
( ( member @ nat @ K3 @ M2 )
& ( ord_less @ nat @ K3 @ ( suc @ I ) ) ) ) )
!= ( zero_zero @ nat ) ) ) ).
% card_less
thf(fact_2858_sum__Suc,axiom,
! [A: $tType,F: A > nat,A5: set @ A] :
( ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [X2: A] : ( suc @ ( F @ X2 ) )
@ A5 )
= ( plus_plus @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ F @ A5 ) @ ( finite_card @ A @ A5 ) ) ) ).
% sum_Suc
thf(fact_2859_sum__multicount,axiom,
! [A: $tType,B: $tType,S3: set @ A,T5: set @ B,R5: A > B > $o,K: nat] :
( ( finite_finite2 @ A @ S3 )
=> ( ( finite_finite2 @ B @ T5 )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ T5 )
=> ( ( finite_card @ A
@ ( collect @ A
@ ^ [I2: A] :
( ( member @ A @ I2 @ S3 )
& ( R5 @ I2 @ X4 ) ) ) )
= K ) )
=> ( ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [I2: A] :
( finite_card @ B
@ ( collect @ B
@ ^ [J3: B] :
( ( member @ B @ J3 @ T5 )
& ( R5 @ I2 @ J3 ) ) ) )
@ S3 )
= ( times_times @ nat @ K @ ( finite_card @ B @ T5 ) ) ) ) ) ) ).
% sum_multicount
thf(fact_2860_div__push__bit__of__1__eq__drop__bit,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A,N: nat] :
( ( divide_divide @ A @ A3 @ ( bit_se4730199178511100633sh_bit @ A @ N @ ( one_one @ A ) ) )
= ( bit_se4197421643247451524op_bit @ A @ N @ A3 ) ) ) ).
% div_push_bit_of_1_eq_drop_bit
thf(fact_2861_bits__ident,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A3: A] :
( ( plus_plus @ A @ ( bit_se4730199178511100633sh_bit @ A @ N @ ( bit_se4197421643247451524op_bit @ A @ N @ A3 ) ) @ ( bit_se2584673776208193580ke_bit @ A @ N @ A3 ) )
= A3 ) ) ).
% bits_ident
thf(fact_2862_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_2863_set__encode__inf,axiom,
! [A5: set @ nat] :
( ~ ( finite_finite2 @ nat @ A5 )
=> ( ( nat_set_encode @ A5 )
= ( zero_zero @ nat ) ) ) ).
% set_encode_inf
thf(fact_2864_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 ) ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ A5 )
=> ! [Y: A] :
( ( member @ A @ Y @ A5 )
=> ( X2 = Y ) ) ) ) ) ) ).
% card_le_Suc0_iff_eq
thf(fact_2865_subset__eq__atLeast0__lessThan__card,axiom,
! [N6: set @ nat,N: nat] :
( ( ord_less_eq @ ( set @ nat ) @ N6 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) )
=> ( ord_less_eq @ nat @ ( finite_card @ nat @ N6 ) @ N ) ) ).
% subset_eq_atLeast0_lessThan_card
thf(fact_2866_card__sum__le__nat__sum,axiom,
! [S3: set @ nat] :
( ord_less_eq @ nat
@ ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [X2: nat] : X2
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( finite_card @ nat @ S3 ) ) )
@ ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [X2: nat] : X2
@ S3 ) ) ).
% card_sum_le_nat_sum
thf(fact_2867_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_2868_sum__bounded__above__strict,axiom,
! [B: $tType,A: $tType] :
( ( ( ordere8940638589300402666id_add @ A )
& ( semiring_1 @ A ) )
=> ! [A5: set @ B,F: B > A,K5: A] :
( ! [I3: B] :
( ( member @ B @ I3 @ A5 )
=> ( ord_less @ A @ ( F @ I3 ) @ K5 ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( finite_card @ B @ A5 ) )
=> ( ord_less @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F @ A5 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( finite_card @ B @ A5 ) ) @ K5 ) ) ) ) ) ).
% sum_bounded_above_strict
thf(fact_2869_prod__gen__delta,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S3: set @ B,A3: B,B2: B > A,C2: A] :
( ( finite_finite2 @ B @ S3 )
=> ( ( ( member @ B @ A3 @ S3 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K3: B] : ( if @ A @ ( K3 = A3 ) @ ( B2 @ K3 ) @ C2 )
@ S3 )
= ( times_times @ A @ ( B2 @ A3 ) @ ( power_power @ A @ C2 @ ( minus_minus @ nat @ ( finite_card @ B @ S3 ) @ ( one_one @ nat ) ) ) ) ) )
& ( ~ ( member @ B @ A3 @ S3 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K3: B] : ( if @ A @ ( K3 = A3 ) @ ( B2 @ K3 ) @ C2 )
@ S3 )
= ( power_power @ A @ C2 @ ( finite_card @ B @ S3 ) ) ) ) ) ) ) ).
% prod_gen_delta
thf(fact_2870_set__encode__insert,axiom,
! [A5: set @ nat,N: nat] :
( ( finite_finite2 @ nat @ A5 )
=> ( ~ ( member @ nat @ N @ A5 )
=> ( ( nat_set_encode @ ( insert2 @ nat @ N @ A5 ) )
= ( plus_plus @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ ( nat_set_encode @ A5 ) ) ) ) ) ).
% set_encode_insert
thf(fact_2871_card__lists__length__le,axiom,
! [A: $tType,A5: set @ A,N: nat] :
( ( finite_finite2 @ A @ A5 )
=> ( ( finite_card @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Xs: list @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ A5 )
& ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N ) ) ) )
= ( groups7311177749621191930dd_sum @ nat @ nat @ ( power_power @ nat @ ( finite_card @ A @ A5 ) ) @ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% card_lists_length_le
thf(fact_2872_bit__rec,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ( ( bit_se5641148757651400278ts_bit @ A )
= ( ^ [A6: A,N5: nat] :
( ( ( N5
= ( zero_zero @ nat ) )
=> ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A6 ) )
& ( ( N5
!= ( zero_zero @ nat ) )
=> ( bit_se5641148757651400278ts_bit @ A @ ( divide_divide @ A @ A6 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( minus_minus @ nat @ N5 @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% bit_rec
thf(fact_2873_bit__sum__mult__2__cases,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A,B2: A,N: nat] :
( ! [J2: nat] :
~ ( bit_se5641148757651400278ts_bit @ A @ A3 @ ( suc @ J2 ) )
=> ( ( bit_se5641148757651400278ts_bit @ A @ ( plus_plus @ A @ A3 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) @ N )
= ( ( ( N
= ( zero_zero @ nat ) )
=> ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ( bit_se5641148757651400278ts_bit @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) @ N ) ) ) ) ) ) ).
% bit_sum_mult_2_cases
thf(fact_2874_max__nat_Osemilattice__neutr__order__axioms,axiom,
( semila1105856199041335345_order @ nat @ ( ord_max @ nat ) @ ( zero_zero @ nat )
@ ^ [X2: nat,Y: nat] : ( ord_less_eq @ nat @ Y @ X2 )
@ ^ [X2: nat,Y: nat] : ( ord_less @ nat @ Y @ X2 ) ) ).
% max_nat.semilattice_neutr_order_axioms
thf(fact_2875_apsnd__apfst,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,F: C > B,G: D > A,X: product_prod @ D @ C] :
( ( product_apsnd @ C @ B @ A @ F @ ( product_apfst @ D @ A @ C @ G @ X ) )
= ( product_Pair @ A @ B @ ( G @ ( product_fst @ D @ C @ X ) ) @ ( F @ ( product_snd @ D @ C @ X ) ) ) ) ).
% apsnd_apfst
thf(fact_2876_apfst__conv,axiom,
! [C: $tType,A: $tType,B: $tType,F: C > A,X: C,Y3: B] :
( ( product_apfst @ C @ A @ B @ F @ ( product_Pair @ C @ B @ X @ Y3 ) )
= ( product_Pair @ A @ B @ ( F @ X ) @ Y3 ) ) ).
% apfst_conv
thf(fact_2877_apfst__eq__conv,axiom,
! [A: $tType,B: $tType,C: $tType,F: C > A,X: product_prod @ C @ B,G: C > A] :
( ( ( product_apfst @ C @ A @ B @ F @ X )
= ( product_apfst @ C @ A @ B @ G @ X ) )
= ( ( F @ ( product_fst @ C @ B @ X ) )
= ( G @ ( product_fst @ C @ B @ X ) ) ) ) ).
% apfst_eq_conv
thf(fact_2878_fst__apfst,axiom,
! [A: $tType,B: $tType,C: $tType,F: C > A,X: product_prod @ C @ B] :
( ( product_fst @ A @ B @ ( product_apfst @ C @ A @ B @ F @ X ) )
= ( F @ ( product_fst @ C @ B @ X ) ) ) ).
% fst_apfst
thf(fact_2879_snd__apfst,axiom,
! [B: $tType,A: $tType,C: $tType,F: C > B,X: product_prod @ C @ A] :
( ( product_snd @ B @ A @ ( product_apfst @ C @ B @ A @ F @ X ) )
= ( product_snd @ C @ A @ X ) ) ).
% snd_apfst
thf(fact_2880_snd__comp__apfst,axiom,
! [C: $tType,B: $tType,A: $tType,F: A > C] :
( ( comp @ ( product_prod @ C @ B ) @ B @ ( product_prod @ A @ B ) @ ( product_snd @ C @ B ) @ ( product_apfst @ A @ C @ B @ F ) )
= ( product_snd @ A @ B ) ) ).
% snd_comp_apfst
thf(fact_2881_sum_Oinsert,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A5: set @ B,X: B,G: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ~ ( member @ B @ X @ A5 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( insert2 @ B @ X @ A5 ) )
= ( plus_plus @ A @ ( G @ X ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ A5 ) ) ) ) ) ) ).
% sum.insert
thf(fact_2882_fst__comp__apfst,axiom,
! [C: $tType,B: $tType,A: $tType,F: A > C] :
( ( comp @ ( product_prod @ C @ B ) @ C @ ( product_prod @ A @ B ) @ ( product_fst @ C @ B ) @ ( product_apfst @ A @ C @ B @ F ) )
= ( comp @ A @ C @ ( product_prod @ A @ B ) @ F @ ( product_fst @ A @ B ) ) ) ).
% fst_comp_apfst
thf(fact_2883_card__Diff__insert,axiom,
! [A: $tType,A3: A,A5: set @ A,B4: set @ A] :
( ( member @ A @ A3 @ A5 )
=> ( ~ ( member @ A @ A3 @ B4 )
=> ( ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert2 @ A @ A3 @ B4 ) ) )
= ( minus_minus @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A5 @ B4 ) ) @ ( one_one @ nat ) ) ) ) ) ).
% card_Diff_insert
thf(fact_2884_apfst__apsnd,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,F: C > A,G: D > B,X: product_prod @ C @ D] :
( ( product_apfst @ C @ A @ B @ F @ ( product_apsnd @ D @ B @ C @ G @ X ) )
= ( product_Pair @ A @ B @ ( F @ ( product_fst @ C @ D @ X ) ) @ ( G @ ( product_snd @ C @ D @ X ) ) ) ) ).
% apfst_apsnd
thf(fact_2885_bit__0,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A] :
( ( bit_se5641148757651400278ts_bit @ A @ A3 @ ( zero_zero @ nat ) )
= ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) ) ) ).
% bit_0
thf(fact_2886_bit__mod__2__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ N )
= ( ( N
= ( zero_zero @ nat ) )
& ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) ) ) ).
% bit_mod_2_iff
thf(fact_2887_insert__compr,axiom,
! [A: $tType] :
( ( insert2 @ A )
= ( ^ [A6: A,B7: set @ A] :
( collect @ A
@ ^ [X2: A] :
( ( X2 = A6 )
| ( member @ A @ X2 @ B7 ) ) ) ) ) ).
% insert_compr
thf(fact_2888_insert__Collect,axiom,
! [A: $tType,A3: A,P: A > $o] :
( ( insert2 @ A @ A3 @ ( collect @ A @ P ) )
= ( collect @ A
@ ^ [U: A] :
( ( U != A3 )
=> ( P @ U ) ) ) ) ).
% insert_Collect
thf(fact_2889_bit__disjunctive__add__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A,B2: A,N: nat] :
( ! [N2: nat] :
( ~ ( bit_se5641148757651400278ts_bit @ A @ A3 @ N2 )
| ~ ( bit_se5641148757651400278ts_bit @ A @ B2 @ N2 ) )
=> ( ( bit_se5641148757651400278ts_bit @ A @ ( plus_plus @ A @ A3 @ B2 ) @ N )
= ( ( bit_se5641148757651400278ts_bit @ A @ A3 @ N )
| ( bit_se5641148757651400278ts_bit @ A @ B2 @ N ) ) ) ) ) ).
% bit_disjunctive_add_iff
thf(fact_2890_apfst__compose,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,F: C > A,G: D > C,X: product_prod @ D @ B] :
( ( product_apfst @ C @ A @ B @ F @ ( product_apfst @ D @ C @ B @ G @ X ) )
= ( product_apfst @ D @ A @ B @ ( comp @ C @ A @ D @ F @ G ) @ X ) ) ).
% apfst_compose
thf(fact_2891_apsnd__apfst__commute,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,F: C > B,G: D > A,P4: product_prod @ D @ C] :
( ( product_apsnd @ C @ B @ A @ F @ ( product_apfst @ D @ A @ C @ G @ P4 ) )
= ( product_apfst @ D @ A @ B @ G @ ( product_apsnd @ C @ B @ D @ F @ P4 ) ) ) ).
% apsnd_apfst_commute
thf(fact_2892_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_2893_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_2894_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_2895_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_2896_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_2897_disjunctive__add,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A,B2: A] :
( ! [N2: nat] :
( ~ ( bit_se5641148757651400278ts_bit @ A @ A3 @ N2 )
| ~ ( bit_se5641148757651400278ts_bit @ A @ B2 @ N2 ) )
=> ( ( plus_plus @ A @ A3 @ B2 )
= ( bit_se1065995026697491101ons_or @ A @ A3 @ B2 ) ) ) ) ).
% disjunctive_add
thf(fact_2898_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_2899_bit__drop__bit__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A3: A] :
( ( bit_se5641148757651400278ts_bit @ A @ ( bit_se4197421643247451524op_bit @ A @ N @ A3 ) )
= ( comp @ nat @ $o @ nat @ ( bit_se5641148757651400278ts_bit @ A @ A3 ) @ ( plus_plus @ nat @ N ) ) ) ) ).
% bit_drop_bit_eq
thf(fact_2900_length__pos__if__in__set,axiom,
! [A: $tType,X: A,Xs2: list @ A] :
( ( member @ A @ X @ ( set2 @ A @ Xs2 ) )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs2 ) ) ) ).
% length_pos_if_in_set
thf(fact_2901_finite__lists__length__eq,axiom,
! [A: $tType,A5: set @ A,N: nat] :
( ( finite_finite2 @ A @ A5 )
=> ( finite_finite2 @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Xs: list @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ A5 )
& ( ( size_size @ ( list @ A ) @ Xs )
= N ) ) ) ) ) ).
% finite_lists_length_eq
thf(fact_2902_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_2903_bit__iff__and__drop__bit__eq__1,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se5641148757651400278ts_bit @ A )
= ( ^ [A6: A,N5: nat] :
( ( bit_se5824344872417868541ns_and @ A @ ( bit_se4197421643247451524op_bit @ A @ N5 @ A6 ) @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ) ) ).
% bit_iff_and_drop_bit_eq_1
thf(fact_2904_sum_Oinsert__if,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A5: set @ B,X: B,G: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( ( member @ B @ X @ A5 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( insert2 @ B @ X @ A5 ) )
= ( groups7311177749621191930dd_sum @ B @ A @ G @ A5 ) ) )
& ( ~ ( member @ B @ X @ A5 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( insert2 @ B @ X @ A5 ) )
= ( plus_plus @ A @ ( G @ X ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ A5 ) ) ) ) ) ) ) ).
% sum.insert_if
thf(fact_2905_atLeast0__atMost__Suc,axiom,
! [N: nat] :
( ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) )
= ( insert2 @ nat @ ( suc @ N ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% atLeast0_atMost_Suc
thf(fact_2906_atLeast0__lessThan__Suc,axiom,
! [N: nat] :
( ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) )
= ( insert2 @ nat @ N @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% atLeast0_lessThan_Suc
thf(fact_2907_finite__lists__length__le,axiom,
! [A: $tType,A5: set @ A,N: nat] :
( ( finite_finite2 @ A @ A5 )
=> ( finite_finite2 @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Xs: list @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ A5 )
& ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N ) ) ) ) ) ).
% finite_lists_length_le
thf(fact_2908_bit__numeral__rec_I1_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [W: num,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ ( bit0 @ W ) ) @ N )
= ( case_nat @ $o @ $false @ ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ W ) ) @ N ) ) ) ).
% bit_numeral_rec(1)
thf(fact_2909_bit__numeral__rec_I2_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [W: num,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ ( bit1 @ W ) ) @ N )
= ( case_nat @ $o @ $true @ ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ W ) ) @ N ) ) ) ).
% bit_numeral_rec(2)
thf(fact_2910_bit__iff__and__push__bit__not__eq__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se5641148757651400278ts_bit @ A )
= ( ^ [A6: A,N5: nat] :
( ( bit_se5824344872417868541ns_and @ A @ A6 @ ( bit_se4730199178511100633sh_bit @ A @ N5 @ ( one_one @ A ) ) )
!= ( zero_zero @ A ) ) ) ) ) ).
% bit_iff_and_push_bit_not_eq_0
thf(fact_2911_card__lists__length__eq,axiom,
! [A: $tType,A5: set @ A,N: nat] :
( ( finite_finite2 @ A @ A5 )
=> ( ( finite_card @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Xs: list @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ A5 )
& ( ( size_size @ ( list @ A ) @ Xs )
= N ) ) ) )
= ( power_power @ nat @ ( finite_card @ A @ A5 ) @ N ) ) ) ).
% card_lists_length_eq
thf(fact_2912_exp__eq__0__imp__not__bit,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [N: nat,A3: A] :
( ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N )
= ( zero_zero @ A ) )
=> ~ ( bit_se5641148757651400278ts_bit @ A @ A3 @ N ) ) ) ).
% exp_eq_0_imp_not_bit
thf(fact_2913_int__bit__bound,axiom,
! [K: int] :
~ ! [N2: nat] :
( ! [M3: nat] :
( ( ord_less_eq @ nat @ N2 @ M3 )
=> ( ( bit_se5641148757651400278ts_bit @ int @ K @ M3 )
= ( 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_2914_and__exp__eq__0__iff__not__bit,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A,N: nat] :
( ( ( bit_se5824344872417868541ns_and @ A @ A3 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
= ( zero_zero @ A ) )
= ( ~ ( bit_se5641148757651400278ts_bit @ A @ A3 @ N ) ) ) ) ).
% and_exp_eq_0_iff_not_bit
thf(fact_2915_gcd__nat_Osemilattice__neutr__order__axioms,axiom,
( semila1105856199041335345_order @ nat @ ( gcd_gcd @ nat ) @ ( zero_zero @ nat ) @ ( dvd_dvd @ nat )
@ ^ [M5: nat,N5: nat] :
( ( dvd_dvd @ nat @ M5 @ N5 )
& ( M5 != N5 ) ) ) ).
% gcd_nat.semilattice_neutr_order_axioms
thf(fact_2916_card__insert__le__m1,axiom,
! [A: $tType,N: nat,Y3: set @ A,X: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ nat @ ( finite_card @ A @ Y3 ) @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) )
=> ( ord_less_eq @ nat @ ( finite_card @ A @ ( insert2 @ A @ X @ Y3 ) ) @ N ) ) ) ).
% card_insert_le_m1
thf(fact_2917_take__bit__sum,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ( ( bit_se2584673776208193580ke_bit @ A )
= ( ^ [N5: nat,A6: A] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( bit_se4730199178511100633sh_bit @ A @ K3 @ ( zero_neq_one_of_bool @ A @ ( bit_se5641148757651400278ts_bit @ A @ A6 @ K3 ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N5 ) ) ) ) ) ).
% take_bit_sum
thf(fact_2918_even__bit__succ__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A,N: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ( bit_se5641148757651400278ts_bit @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A3 ) @ N )
= ( ( bit_se5641148757651400278ts_bit @ A @ A3 @ N )
| ( N
= ( zero_zero @ nat ) ) ) ) ) ) ).
% even_bit_succ_iff
thf(fact_2919_odd__bit__iff__bit__pred,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A,N: nat] :
( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ( bit_se5641148757651400278ts_bit @ A @ A3 @ N )
= ( ( bit_se5641148757651400278ts_bit @ A @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) @ N )
| ( N
= ( zero_zero @ nat ) ) ) ) ) ) ).
% odd_bit_iff_bit_pred
thf(fact_2920_set__decode__plus__power__2,axiom,
! [N: nat,Z: nat] :
( ~ ( member @ nat @ N @ ( nat_set_decode @ Z ) )
=> ( ( nat_set_decode @ ( plus_plus @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ Z ) )
= ( insert2 @ nat @ N @ ( nat_set_decode @ Z ) ) ) ) ).
% set_decode_plus_power_2
thf(fact_2921_signed__take__bit__code,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( bit_ri4674362597316999326ke_bit @ A )
= ( ^ [N5: nat,A6: A] : ( if @ A @ ( bit_se5641148757651400278ts_bit @ A @ ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N5 ) @ A6 ) @ N5 ) @ ( plus_plus @ A @ ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N5 ) @ A6 ) @ ( bit_se4730199178511100633sh_bit @ A @ ( suc @ N5 ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) @ ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N5 ) @ A6 ) ) ) ) ) ).
% signed_take_bit_code
thf(fact_2922_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 )
@ ^ [Xs: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs )
= K )
& ( distinct @ A @ Xs )
& ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ A5 ) ) ) )
= ( groups7121269368397514597t_prod @ nat @ nat
@ ^ [X2: nat] : X2
@ ( 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_2923_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 )
@ ^ [Xs: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs )
= K )
& ( distinct @ A @ Xs )
& ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ A5 ) ) ) )
= ( groups7121269368397514597t_prod @ nat @ nat
@ ^ [X2: nat] : X2
@ ( 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_2924_apfst__convE,axiom,
! [C: $tType,A: $tType,B: $tType,Q2: product_prod @ A @ B,F: C > A,P4: product_prod @ C @ B] :
( ( Q2
= ( product_apfst @ C @ A @ B @ F @ P4 ) )
=> ~ ! [X4: C,Y4: B] :
( ( P4
= ( product_Pair @ C @ B @ X4 @ Y4 ) )
=> ( Q2
!= ( product_Pair @ A @ B @ ( F @ X4 ) @ Y4 ) ) ) ) ).
% apfst_convE
thf(fact_2925_times__int_Oabs__eq,axiom,
! [Xa: product_prod @ nat @ nat,X: product_prod @ nat @ nat] :
( ( times_times @ int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X ) )
= ( abs_Integ
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X2: nat,Y: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U: nat,V6: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ X2 @ U ) @ ( times_times @ nat @ Y @ V6 ) ) @ ( plus_plus @ nat @ ( times_times @ nat @ X2 @ V6 ) @ ( times_times @ nat @ Y @ U ) ) ) )
@ Xa
@ X ) ) ) ).
% times_int.abs_eq
thf(fact_2926_rat__minus__code,axiom,
! [P4: rat,Q2: rat] :
( ( quotient_of @ ( minus_minus @ rat @ P4 @ Q2 ) )
= ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [A6: int,C4: int] :
( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [B5: int,D4: int] : ( normalize @ ( product_Pair @ int @ int @ ( minus_minus @ int @ ( times_times @ int @ A6 @ D4 ) @ ( times_times @ int @ B5 @ C4 ) ) @ ( times_times @ int @ C4 @ D4 ) ) )
@ ( quotient_of @ Q2 ) )
@ ( quotient_of @ P4 ) ) ) ).
% rat_minus_code
thf(fact_2927_rat__one__code,axiom,
( ( quotient_of @ ( one_one @ rat ) )
= ( product_Pair @ int @ int @ ( one_one @ int ) @ ( one_one @ int ) ) ) ).
% rat_one_code
thf(fact_2928_rat__zero__code,axiom,
( ( quotient_of @ ( zero_zero @ rat ) )
= ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) ) ) ).
% rat_zero_code
thf(fact_2929_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_2930_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_2931_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_2932_finite__lists__distinct__length__eq,axiom,
! [A: $tType,A5: set @ A,N: nat] :
( ( finite_finite2 @ A @ A5 )
=> ( finite_finite2 @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Xs: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs )
= N )
& ( distinct @ A @ Xs )
& ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ A5 ) ) ) ) ) ).
% finite_lists_distinct_length_eq
thf(fact_2933_int_Oabs__induct,axiom,
! [P: int > $o,X: int] :
( ! [Y4: product_prod @ nat @ nat] : ( P @ ( abs_Integ @ Y4 ) )
=> ( P @ X ) ) ).
% int.abs_induct
thf(fact_2934_conj__comp__iff,axiom,
! [B: $tType,A: $tType,P: B > $o,Q: B > $o,G: A > B] :
( ( comp @ B @ $o @ A
@ ^ [X2: B] :
( ( P @ X2 )
& ( Q @ X2 ) )
@ G )
= ( ^ [X2: A] :
( ( comp @ B @ $o @ A @ P @ G @ X2 )
& ( comp @ B @ $o @ A @ Q @ G @ X2 ) ) ) ) ).
% conj_comp_iff
thf(fact_2935_quotient__of__div,axiom,
! [R: rat,N: int,D3: int] :
( ( ( quotient_of @ R )
= ( product_Pair @ int @ int @ N @ D3 ) )
=> ( R
= ( divide_divide @ rat @ ( ring_1_of_int @ rat @ N ) @ ( ring_1_of_int @ rat @ D3 ) ) ) ) ).
% quotient_of_div
thf(fact_2936_eq__Abs__Integ,axiom,
! [Z: int] :
~ ! [X4: nat,Y4: nat] :
( Z
!= ( abs_Integ @ ( product_Pair @ nat @ nat @ X4 @ Y4 ) ) ) ).
% eq_Abs_Integ
thf(fact_2937_nat_Oabs__eq,axiom,
! [X: product_prod @ nat @ nat] :
( ( nat2 @ ( abs_Integ @ X ) )
= ( product_case_prod @ nat @ nat @ nat @ ( minus_minus @ nat ) @ X ) ) ).
% nat.abs_eq
thf(fact_2938_quotient__of__denom__pos,axiom,
! [R: rat,P4: int,Q2: int] :
( ( ( quotient_of @ R )
= ( product_Pair @ int @ int @ P4 @ Q2 ) )
=> ( ord_less @ int @ ( zero_zero @ int ) @ Q2 ) ) ).
% quotient_of_denom_pos
thf(fact_2939_zero__int__def,axiom,
( ( zero_zero @ int )
= ( abs_Integ @ ( product_Pair @ nat @ nat @ ( zero_zero @ nat ) @ ( zero_zero @ nat ) ) ) ) ).
% zero_int_def
thf(fact_2940_int__def,axiom,
( ( semiring_1_of_nat @ int )
= ( ^ [N5: nat] : ( abs_Integ @ ( product_Pair @ nat @ nat @ N5 @ ( zero_zero @ nat ) ) ) ) ) ).
% int_def
thf(fact_2941_rat__floor__code,axiom,
( ( archim6421214686448440834_floor @ rat )
= ( ^ [P5: rat] : ( product_case_prod @ int @ int @ int @ ( divide_divide @ int ) @ ( quotient_of @ P5 ) ) ) ) ).
% rat_floor_code
thf(fact_2942_rat__uminus__code,axiom,
! [P4: rat] :
( ( quotient_of @ ( uminus_uminus @ rat @ P4 ) )
= ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [A6: int] : ( product_Pair @ int @ int @ ( uminus_uminus @ int @ A6 ) )
@ ( quotient_of @ P4 ) ) ) ).
% rat_uminus_code
thf(fact_2943_rat__abs__code,axiom,
! [P4: rat] :
( ( quotient_of @ ( abs_abs @ rat @ P4 ) )
= ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [A6: int] : ( product_Pair @ int @ int @ ( abs_abs @ int @ A6 ) )
@ ( quotient_of @ P4 ) ) ) ).
% rat_abs_code
thf(fact_2944_rat__less__eq__code,axiom,
( ( ord_less_eq @ rat )
= ( ^ [P5: rat,Q4: rat] :
( product_case_prod @ int @ int @ $o
@ ^ [A6: int,C4: int] :
( product_case_prod @ int @ int @ $o
@ ^ [B5: int,D4: int] : ( ord_less_eq @ int @ ( times_times @ int @ A6 @ D4 ) @ ( times_times @ int @ C4 @ B5 ) )
@ ( quotient_of @ Q4 ) )
@ ( quotient_of @ P5 ) ) ) ) ).
% rat_less_eq_code
thf(fact_2945_rat__less__code,axiom,
( ( ord_less @ rat )
= ( ^ [P5: rat,Q4: rat] :
( product_case_prod @ int @ int @ $o
@ ^ [A6: int,C4: int] :
( product_case_prod @ int @ int @ $o
@ ^ [B5: int,D4: int] : ( ord_less @ int @ ( times_times @ int @ A6 @ D4 ) @ ( times_times @ int @ C4 @ B5 ) )
@ ( quotient_of @ Q4 ) )
@ ( quotient_of @ P5 ) ) ) ) ).
% rat_less_code
thf(fact_2946_uminus__int_Oabs__eq,axiom,
! [X: product_prod @ nat @ nat] :
( ( uminus_uminus @ int @ ( abs_Integ @ X ) )
= ( abs_Integ
@ ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [X2: nat,Y: nat] : ( product_Pair @ nat @ nat @ Y @ X2 )
@ X ) ) ) ).
% uminus_int.abs_eq
thf(fact_2947_rat__sgn__code,axiom,
! [P4: rat] :
( ( quotient_of @ ( sgn_sgn @ rat @ P4 ) )
= ( product_Pair @ int @ int @ ( sgn_sgn @ int @ ( product_fst @ int @ int @ ( quotient_of @ P4 ) ) ) @ ( one_one @ int ) ) ) ).
% rat_sgn_code
thf(fact_2948_one__int__def,axiom,
( ( one_one @ int )
= ( abs_Integ @ ( product_Pair @ nat @ nat @ ( one_one @ nat ) @ ( zero_zero @ nat ) ) ) ) ).
% one_int_def
thf(fact_2949_of__int_Oabs__eq,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X: product_prod @ nat @ nat] :
( ( ring_1_of_int @ A @ ( abs_Integ @ X ) )
= ( product_case_prod @ nat @ nat @ A
@ ^ [I2: nat,J3: nat] : ( minus_minus @ A @ ( semiring_1_of_nat @ A @ I2 ) @ ( semiring_1_of_nat @ A @ J3 ) )
@ X ) ) ) ).
% of_int.abs_eq
thf(fact_2950_less__int_Oabs__eq,axiom,
! [Xa: product_prod @ nat @ nat,X: product_prod @ nat @ nat] :
( ( ord_less @ int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X ) )
= ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X2: nat,Y: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U: nat,V6: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ X2 @ V6 ) @ ( plus_plus @ nat @ U @ Y ) ) )
@ Xa
@ X ) ) ).
% less_int.abs_eq
thf(fact_2951_less__eq__int_Oabs__eq,axiom,
! [Xa: product_prod @ nat @ nat,X: product_prod @ nat @ nat] :
( ( ord_less_eq @ int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X ) )
= ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X2: nat,Y: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U: nat,V6: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ X2 @ V6 ) @ ( plus_plus @ nat @ U @ Y ) ) )
@ Xa
@ X ) ) ).
% less_eq_int.abs_eq
thf(fact_2952_plus__int_Oabs__eq,axiom,
! [Xa: product_prod @ nat @ nat,X: product_prod @ nat @ nat] :
( ( plus_plus @ int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X ) )
= ( abs_Integ
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X2: nat,Y: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U: nat,V6: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X2 @ U ) @ ( plus_plus @ nat @ Y @ V6 ) ) )
@ Xa
@ X ) ) ) ).
% plus_int.abs_eq
thf(fact_2953_rat__times__code,axiom,
! [P4: rat,Q2: rat] :
( ( quotient_of @ ( times_times @ rat @ P4 @ Q2 ) )
= ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [A6: int,C4: int] :
( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [B5: int,D4: int] : ( normalize @ ( product_Pair @ int @ int @ ( times_times @ int @ A6 @ B5 ) @ ( times_times @ int @ C4 @ D4 ) ) )
@ ( quotient_of @ Q2 ) )
@ ( quotient_of @ P4 ) ) ) ).
% rat_times_code
thf(fact_2954_rat__divide__code,axiom,
! [P4: rat,Q2: rat] :
( ( quotient_of @ ( divide_divide @ rat @ P4 @ Q2 ) )
= ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [A6: int,C4: int] :
( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [B5: int,D4: int] : ( normalize @ ( product_Pair @ int @ int @ ( times_times @ int @ A6 @ D4 ) @ ( times_times @ int @ C4 @ B5 ) ) )
@ ( quotient_of @ Q2 ) )
@ ( quotient_of @ P4 ) ) ) ).
% rat_divide_code
thf(fact_2955_minus__int_Oabs__eq,axiom,
! [Xa: product_prod @ nat @ nat,X: product_prod @ nat @ nat] :
( ( minus_minus @ int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X ) )
= ( abs_Integ
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X2: nat,Y: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U: nat,V6: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X2 @ V6 ) @ ( plus_plus @ nat @ Y @ U ) ) )
@ Xa
@ X ) ) ) ).
% minus_int.abs_eq
thf(fact_2956_rat__plus__code,axiom,
! [P4: rat,Q2: rat] :
( ( quotient_of @ ( plus_plus @ rat @ P4 @ Q2 ) )
= ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [A6: int,C4: int] :
( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [B5: int,D4: int] : ( normalize @ ( product_Pair @ int @ int @ ( plus_plus @ int @ ( times_times @ int @ A6 @ D4 ) @ ( times_times @ int @ B5 @ C4 ) ) @ ( times_times @ int @ C4 @ D4 ) ) )
@ ( quotient_of @ Q2 ) )
@ ( quotient_of @ P4 ) ) ) ).
% rat_plus_code
thf(fact_2957_quotient__of__int,axiom,
! [A3: int] :
( ( quotient_of @ ( of_int @ A3 ) )
= ( product_Pair @ int @ int @ A3 @ ( one_one @ int ) ) ) ).
% quotient_of_int
thf(fact_2958_rat__inverse__code,axiom,
! [P4: rat] :
( ( quotient_of @ ( inverse_inverse @ rat @ P4 ) )
= ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [A6: int,B5: int] :
( if @ ( product_prod @ int @ int )
@ ( A6
= ( zero_zero @ int ) )
@ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
@ ( product_Pair @ int @ int @ ( times_times @ int @ ( sgn_sgn @ int @ A6 ) @ B5 ) @ ( abs_abs @ int @ A6 ) ) )
@ ( quotient_of @ P4 ) ) ) ).
% rat_inverse_code
thf(fact_2959_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_2960_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_2961_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_2962_inverse__zero,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ( ( inverse_inverse @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% inverse_zero
thf(fact_2963_inverse__nonzero__iff__nonzero,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A] :
( ( ( inverse_inverse @ A @ A3 )
= ( zero_zero @ A ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% inverse_nonzero_iff_nonzero
thf(fact_2964_inverse__eq__1__iff,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [X: A] :
( ( ( inverse_inverse @ A @ X )
= ( one_one @ A ) )
= ( X
= ( one_one @ A ) ) ) ) ).
% inverse_eq_1_iff
thf(fact_2965_inverse__1,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ( ( inverse_inverse @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% inverse_1
thf(fact_2966_inverse__nonnegative__iff__nonnegative,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( inverse_inverse @ A @ A3 ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% inverse_nonnegative_iff_nonnegative
thf(fact_2967_inverse__nonpositive__iff__nonpositive,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A3 ) @ ( zero_zero @ A ) )
= ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% inverse_nonpositive_iff_nonpositive
thf(fact_2968_inverse__less__iff__less,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B2 ) )
= ( ord_less @ A @ B2 @ A3 ) ) ) ) ) ).
% inverse_less_iff_less
thf(fact_2969_inverse__less__iff__less__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B2 ) )
= ( ord_less @ A @ B2 @ A3 ) ) ) ) ) ).
% inverse_less_iff_less_neg
thf(fact_2970_inverse__negative__iff__negative,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( inverse_inverse @ A @ A3 ) @ ( zero_zero @ A ) )
= ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% inverse_negative_iff_negative
thf(fact_2971_inverse__positive__iff__positive,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( inverse_inverse @ A @ A3 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% inverse_positive_iff_positive
thf(fact_2972_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_2973_inverse__le__iff__le,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B2 ) )
= ( ord_less_eq @ A @ B2 @ A3 ) ) ) ) ) ).
% inverse_le_iff_le
thf(fact_2974_inverse__le__iff__le__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B2 ) )
= ( ord_less_eq @ A @ B2 @ A3 ) ) ) ) ) ).
% inverse_le_iff_le_neg
thf(fact_2975_left__inverse,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ ( inverse_inverse @ A @ A3 ) @ A3 )
= ( one_one @ A ) ) ) ) ).
% left_inverse
thf(fact_2976_right__inverse,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ A3 @ ( inverse_inverse @ A @ A3 ) )
= ( one_one @ A ) ) ) ) ).
% right_inverse
thf(fact_2977_inverse__eq__divide__numeral,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [W: num] :
( ( inverse_inverse @ A @ ( numeral_numeral @ A @ W ) )
= ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ W ) ) ) ) ).
% inverse_eq_divide_numeral
thf(fact_2978_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_2979_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_2980_semiring__norm_I167_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [V: num,W: num,Y3: A] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) @ ( plus_plus @ A @ ( numeral_numeral @ A @ W ) @ Y3 ) )
= ( plus_plus @ A @ ( neg_numeral_sub @ A @ W @ V ) @ Y3 ) ) ) ).
% semiring_norm(167)
thf(fact_2981_semiring__norm_I166_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [V: num,W: num,Y3: A] :
( ( plus_plus @ A @ ( numeral_numeral @ A @ V ) @ ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ Y3 ) )
= ( plus_plus @ A @ ( neg_numeral_sub @ A @ V @ W ) @ Y3 ) ) ) ).
% semiring_norm(166)
thf(fact_2982_inverse__eq__divide__neg__numeral,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [W: num] :
( ( inverse_inverse @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) )
= ( divide_divide @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) ) ) ).
% inverse_eq_divide_neg_numeral
thf(fact_2983_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_2984_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_2985_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_2986_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_2987_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_2988_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_2989_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_2990_nonzero__imp__inverse__nonzero,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( inverse_inverse @ A @ A3 )
!= ( zero_zero @ A ) ) ) ) ).
% nonzero_imp_inverse_nonzero
thf(fact_2991_nonzero__inverse__inverse__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( inverse_inverse @ A @ ( inverse_inverse @ A @ A3 ) )
= A3 ) ) ) ).
% nonzero_inverse_inverse_eq
thf(fact_2992_nonzero__inverse__eq__imp__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,B2: A] :
( ( ( inverse_inverse @ A @ A3 )
= ( inverse_inverse @ A @ B2 ) )
=> ( ( A3
!= ( zero_zero @ A ) )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( A3 = B2 ) ) ) ) ) ).
% nonzero_inverse_eq_imp_eq
thf(fact_2993_inverse__zero__imp__zero,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A] :
( ( ( inverse_inverse @ A @ A3 )
= ( zero_zero @ A ) )
=> ( A3
= ( zero_zero @ A ) ) ) ) ).
% inverse_zero_imp_zero
thf(fact_2994_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_2995_inverse__less__imp__less,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B2 ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ord_less @ A @ B2 @ A3 ) ) ) ) ).
% inverse_less_imp_less
thf(fact_2996_less__imp__inverse__less,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ord_less @ A @ ( inverse_inverse @ A @ B2 ) @ ( inverse_inverse @ A @ A3 ) ) ) ) ) ).
% less_imp_inverse_less
thf(fact_2997_inverse__less__imp__less__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B2 ) )
=> ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B2 @ A3 ) ) ) ) ).
% inverse_less_imp_less_neg
thf(fact_2998_less__imp__inverse__less__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( inverse_inverse @ A @ B2 ) @ ( inverse_inverse @ A @ A3 ) ) ) ) ) ).
% less_imp_inverse_less_neg
thf(fact_2999_inverse__negative__imp__negative,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( inverse_inverse @ A @ A3 ) @ ( zero_zero @ A ) )
=> ( ( A3
!= ( zero_zero @ A ) )
=> ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ) ).
% inverse_negative_imp_negative
thf(fact_3000_inverse__positive__imp__positive,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( inverse_inverse @ A @ A3 ) )
=> ( ( A3
!= ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ) ).
% inverse_positive_imp_positive
thf(fact_3001_negative__imp__inverse__negative,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( inverse_inverse @ A @ A3 ) @ ( zero_zero @ A ) ) ) ) ).
% negative_imp_inverse_negative
thf(fact_3002_positive__imp__inverse__positive,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( inverse_inverse @ A @ A3 ) ) ) ) ).
% positive_imp_inverse_positive
thf(fact_3003_nonzero__inverse__mult__distrib,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,B2: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( ( inverse_inverse @ A @ ( times_times @ A @ A3 @ B2 ) )
= ( times_times @ A @ ( inverse_inverse @ A @ B2 ) @ ( inverse_inverse @ A @ A3 ) ) ) ) ) ) ).
% nonzero_inverse_mult_distrib
thf(fact_3004_nonzero__inverse__minus__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( inverse_inverse @ A @ ( uminus_uminus @ A @ A3 ) )
= ( uminus_uminus @ A @ ( inverse_inverse @ A @ A3 ) ) ) ) ) ).
% nonzero_inverse_minus_eq
thf(fact_3005_inverse__unique,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,B2: A] :
( ( ( times_times @ A @ A3 @ B2 )
= ( one_one @ A ) )
=> ( ( inverse_inverse @ A @ A3 )
= B2 ) ) ) ).
% inverse_unique
thf(fact_3006_inverse__eq__divide,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ( ( inverse_inverse @ A )
= ( divide_divide @ A @ ( one_one @ A ) ) ) ) ).
% inverse_eq_divide
thf(fact_3007_power__mult__power__inverse__commute,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,M: nat,N: nat] :
( ( times_times @ A @ ( power_power @ A @ X @ M ) @ ( power_power @ A @ ( inverse_inverse @ A @ X ) @ N ) )
= ( times_times @ A @ ( power_power @ A @ ( inverse_inverse @ A @ X ) @ N ) @ ( power_power @ A @ X @ M ) ) ) ) ).
% power_mult_power_inverse_commute
thf(fact_3008_power__mult__inverse__distrib,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,M: nat] :
( ( times_times @ A @ ( power_power @ A @ X @ M ) @ ( inverse_inverse @ A @ X ) )
= ( times_times @ A @ ( inverse_inverse @ A @ X ) @ ( power_power @ A @ X @ M ) ) ) ) ).
% power_mult_inverse_distrib
thf(fact_3009_nonzero__abs__inverse,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( abs_abs @ A @ ( inverse_inverse @ A @ A3 ) )
= ( inverse_inverse @ A @ ( abs_abs @ A @ A3 ) ) ) ) ) ).
% nonzero_abs_inverse
thf(fact_3010_mult__inverse__of__int__commute,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Xa: int,X: A] :
( ( times_times @ A @ ( inverse_inverse @ A @ ( ring_1_of_int @ A @ Xa ) ) @ X )
= ( times_times @ A @ X @ ( inverse_inverse @ A @ ( ring_1_of_int @ A @ Xa ) ) ) ) ) ).
% mult_inverse_of_int_commute
thf(fact_3011_prod__inversef,axiom,
! [A: $tType,B: $tType] :
( ( field @ A )
=> ! [F: B > A,A5: set @ B] :
( ( groups7121269368397514597t_prod @ B @ A @ ( comp @ A @ A @ B @ ( inverse_inverse @ A ) @ F ) @ A5 )
= ( inverse_inverse @ A @ ( groups7121269368397514597t_prod @ B @ A @ F @ A5 ) ) ) ) ).
% prod_inversef
thf(fact_3012_inverse__le__imp__le,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B2 ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ord_less_eq @ A @ B2 @ A3 ) ) ) ) ).
% inverse_le_imp_le
thf(fact_3013_le__imp__inverse__le,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ord_less_eq @ A @ ( inverse_inverse @ A @ B2 ) @ ( inverse_inverse @ A @ A3 ) ) ) ) ) ).
% le_imp_inverse_le
thf(fact_3014_inverse__le__imp__le__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B2 ) )
=> ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B2 @ A3 ) ) ) ) ).
% inverse_le_imp_le_neg
thf(fact_3015_le__imp__inverse__le__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( inverse_inverse @ A @ B2 ) @ ( inverse_inverse @ A @ A3 ) ) ) ) ) ).
% le_imp_inverse_le_neg
thf(fact_3016_inverse__le__1__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A] :
( ( ord_less_eq @ A @ ( inverse_inverse @ A @ X ) @ ( one_one @ A ) )
= ( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
| ( ord_less_eq @ A @ ( one_one @ A ) @ X ) ) ) ) ).
% inverse_le_1_iff
thf(fact_3017_one__less__inverse,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ A3 @ ( one_one @ A ) )
=> ( ord_less @ A @ ( one_one @ A ) @ ( inverse_inverse @ A @ A3 ) ) ) ) ) ).
% one_less_inverse
thf(fact_3018_one__less__inverse__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A] :
( ( ord_less @ A @ ( one_one @ A ) @ ( inverse_inverse @ A @ X ) )
= ( ( ord_less @ A @ ( zero_zero @ A ) @ X )
& ( ord_less @ A @ X @ ( one_one @ A ) ) ) ) ) ).
% one_less_inverse_iff
thf(fact_3019_inverse__add,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: A,B2: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B2 ) )
= ( times_times @ A @ ( times_times @ A @ ( plus_plus @ A @ A3 @ B2 ) @ ( inverse_inverse @ A @ A3 ) ) @ ( inverse_inverse @ A @ B2 ) ) ) ) ) ) ).
% inverse_add
thf(fact_3020_division__ring__inverse__add,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,B2: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B2 ) )
= ( times_times @ A @ ( times_times @ A @ ( inverse_inverse @ A @ A3 ) @ ( plus_plus @ A @ A3 @ B2 ) ) @ ( inverse_inverse @ A @ B2 ) ) ) ) ) ) ).
% division_ring_inverse_add
thf(fact_3021_field__class_Ofield__inverse,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ ( inverse_inverse @ A @ A3 ) @ A3 )
= ( one_one @ A ) ) ) ) ).
% field_class.field_inverse
thf(fact_3022_division__ring__inverse__diff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,B2: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B2 ) )
= ( times_times @ A @ ( times_times @ A @ ( inverse_inverse @ A @ A3 ) @ ( minus_minus @ A @ B2 @ A3 ) ) @ ( inverse_inverse @ A @ B2 ) ) ) ) ) ) ).
% division_ring_inverse_diff
thf(fact_3023_nonzero__inverse__eq__divide,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( inverse_inverse @ A @ A3 )
= ( divide_divide @ A @ ( one_one @ A ) @ A3 ) ) ) ) ).
% nonzero_inverse_eq_divide
thf(fact_3024_inverse__le__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B2 ) )
=> ( ord_less_eq @ A @ B2 @ A3 ) )
& ( ( ord_less_eq @ A @ ( times_times @ A @ A3 @ B2 ) @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ A3 @ B2 ) ) ) ) ) ).
% inverse_le_iff
thf(fact_3025_inverse__less__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B2 ) )
=> ( ord_less @ A @ B2 @ A3 ) )
& ( ( ord_less_eq @ A @ ( times_times @ A @ A3 @ B2 ) @ ( zero_zero @ A ) )
=> ( ord_less @ A @ A3 @ B2 ) ) ) ) ) ).
% inverse_less_iff
thf(fact_3026_one__le__inverse__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A] :
( ( ord_less_eq @ A @ ( one_one @ A ) @ ( inverse_inverse @ A @ X ) )
= ( ( ord_less @ A @ ( zero_zero @ A ) @ X )
& ( ord_less_eq @ A @ X @ ( one_one @ A ) ) ) ) ) ).
% one_le_inverse_iff
thf(fact_3027_inverse__less__1__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A] :
( ( ord_less @ A @ ( inverse_inverse @ A @ X ) @ ( one_one @ A ) )
= ( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
| ( ord_less @ A @ ( one_one @ A ) @ X ) ) ) ) ).
% inverse_less_1_iff
thf(fact_3028_one__le__inverse,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ A3 @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ ( inverse_inverse @ A @ A3 ) ) ) ) ) ).
% one_le_inverse
thf(fact_3029_reals__Archimedean,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X )
=> ? [N2: nat] : ( ord_less @ A @ ( inverse_inverse @ A @ ( semiring_1_of_nat @ A @ ( suc @ N2 ) ) ) @ X ) ) ) ).
% reals_Archimedean
thf(fact_3030_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_3031_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_3032_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_3033_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_3034_ex__inverse__of__nat__less,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X )
=> ? [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
& ( ord_less @ A @ ( inverse_inverse @ A @ ( semiring_1_of_nat @ A @ N2 ) ) @ X ) ) ) ) ).
% ex_inverse_of_nat_less
thf(fact_3035_power__diff__conv__inverse,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,M: nat,N: nat] :
( ( X
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ nat @ M @ N )
=> ( ( power_power @ A @ X @ ( minus_minus @ nat @ N @ M ) )
= ( times_times @ A @ ( power_power @ A @ X @ N ) @ ( power_power @ A @ ( inverse_inverse @ A @ X ) @ M ) ) ) ) ) ) ).
% power_diff_conv_inverse
thf(fact_3036_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_3037_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_3038_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_3039_floor__rat__def,axiom,
( ( archim6421214686448440834_floor @ rat )
= ( ^ [X2: rat] :
( the @ int
@ ^ [Z3: int] :
( ( ord_less_eq @ rat @ ( ring_1_of_int @ rat @ Z3 ) @ X2 )
& ( ord_less @ rat @ X2 @ ( ring_1_of_int @ rat @ ( plus_plus @ int @ Z3 @ ( one_one @ int ) ) ) ) ) ) ) ) ).
% floor_rat_def
thf(fact_3040_less__eq__int_Orep__eq,axiom,
( ( ord_less_eq @ int )
= ( ^ [X2: int,Xa3: int] :
( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [Y: nat,Z3: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U: nat,V6: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ Y @ V6 ) @ ( plus_plus @ nat @ U @ Z3 ) ) )
@ ( rep_Integ @ X2 )
@ ( rep_Integ @ Xa3 ) ) ) ) ).
% less_eq_int.rep_eq
thf(fact_3041_less__int_Orep__eq,axiom,
( ( ord_less @ int )
= ( ^ [X2: int,Xa3: int] :
( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [Y: nat,Z3: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U: nat,V6: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ Y @ V6 ) @ ( plus_plus @ nat @ U @ Z3 ) ) )
@ ( rep_Integ @ X2 )
@ ( rep_Integ @ Xa3 ) ) ) ) ).
% less_int.rep_eq
thf(fact_3042_Frct__code__post_I6_J,axiom,
! [K: num,L: num] :
( ( frct @ ( product_Pair @ int @ int @ ( numeral_numeral @ int @ K ) @ ( numeral_numeral @ int @ L ) ) )
= ( divide_divide @ rat @ ( numeral_numeral @ rat @ K ) @ ( numeral_numeral @ rat @ L ) ) ) ).
% Frct_code_post(6)
thf(fact_3043_nat_Orep__eq,axiom,
( nat2
= ( ^ [X2: int] : ( product_case_prod @ nat @ nat @ nat @ ( minus_minus @ nat ) @ ( rep_Integ @ X2 ) ) ) ) ).
% nat.rep_eq
thf(fact_3044_Frct__code__post_I1_J,axiom,
! [A3: int] :
( ( frct @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ A3 ) )
= ( zero_zero @ rat ) ) ).
% Frct_code_post(1)
thf(fact_3045_Frct__code__post_I2_J,axiom,
! [A3: int] :
( ( frct @ ( product_Pair @ int @ int @ A3 @ ( zero_zero @ int ) ) )
= ( zero_zero @ rat ) ) ).
% Frct_code_post(2)
thf(fact_3046_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_3047_Frct__code__post_I7_J,axiom,
! [A3: int,B2: int] :
( ( frct @ ( product_Pair @ int @ int @ ( uminus_uminus @ int @ A3 ) @ B2 ) )
= ( uminus_uminus @ rat @ ( frct @ ( product_Pair @ int @ int @ A3 @ B2 ) ) ) ) ).
% Frct_code_post(7)
thf(fact_3048_Frct__code__post_I8_J,axiom,
! [A3: int,B2: int] :
( ( frct @ ( product_Pair @ int @ int @ A3 @ ( uminus_uminus @ int @ B2 ) ) )
= ( uminus_uminus @ rat @ ( frct @ ( product_Pair @ int @ int @ A3 @ B2 ) ) ) ) ).
% Frct_code_post(8)
thf(fact_3049_of__int_Orep__eq,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( ( ring_1_of_int @ A )
= ( ^ [X2: 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 @ X2 ) ) ) ) ) ).
% of_int.rep_eq
thf(fact_3050_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_3051_old_Orec__unit__def,axiom,
! [T: $tType] :
( ( product_rec_unit @ T )
= ( ^ [F12: T,X2: product_unit] : ( the @ T @ ( product_rec_set_unit @ T @ F12 @ X2 ) ) ) ) ).
% old.rec_unit_def
thf(fact_3052_old_Orec__prod__def,axiom,
! [T: $tType,B: $tType,A: $tType] :
( ( product_rec_prod @ A @ B @ T )
= ( ^ [F12: A > B > T,X2: product_prod @ A @ B] : ( the @ T @ ( product_rec_set_prod @ A @ B @ T @ F12 @ X2 ) ) ) ) ).
% old.rec_prod_def
thf(fact_3053_old_Orec__nat__def,axiom,
! [T: $tType] :
( ( rec_nat @ T )
= ( ^ [F12: T,F22: nat > T > T,X2: nat] : ( the @ T @ ( rec_set_nat @ T @ F12 @ F22 @ X2 ) ) ) ) ).
% old.rec_nat_def
thf(fact_3054_the__equality,axiom,
! [A: $tType,P: A > $o,A3: A] :
( ( P @ A3 )
=> ( ! [X4: A] :
( ( P @ X4 )
=> ( X4 = A3 ) )
=> ( ( the @ A @ P )
= A3 ) ) ) ).
% the_equality
thf(fact_3055_the__eq__trivial,axiom,
! [A: $tType,A3: A] :
( ( the @ A
@ ^ [X2: A] : X2 = A3 )
= A3 ) ).
% the_eq_trivial
thf(fact_3056_the__sym__eq__trivial,axiom,
! [A: $tType,X: A] :
( ( the @ A
@ ( ^ [Y6: A,Z2: A] : Y6 = Z2
@ X ) )
= X ) ).
% the_sym_eq_trivial
thf(fact_3057_The__split__eq,axiom,
! [A: $tType,B: $tType,X: A,Y3: B] :
( ( the @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [X8: A,Y7: B] :
( ( X = X8 )
& ( Y3 = Y7 ) ) ) )
= ( product_Pair @ A @ B @ X @ Y3 ) ) ).
% The_split_eq
thf(fact_3058_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_3059_the1__equality,axiom,
! [A: $tType,P: A > $o,A3: A] :
( ? [X5: A] :
( ( P @ X5 )
& ! [Y4: A] :
( ( P @ Y4 )
=> ( Y4 = X5 ) ) )
=> ( ( P @ A3 )
=> ( ( the @ A @ P )
= A3 ) ) ) ).
% the1_equality
thf(fact_3060_the1I2,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ? [X5: A] :
( ( P @ X5 )
& ! [Y4: A] :
( ( P @ Y4 )
=> ( Y4 = X5 ) ) )
=> ( ! [X4: A] :
( ( P @ X4 )
=> ( Q @ X4 ) )
=> ( Q @ ( the @ A @ P ) ) ) ) ).
% the1I2
thf(fact_3061_If__def,axiom,
! [A: $tType] :
( ( if @ A )
= ( ^ [P3: $o,X2: A,Y: A] :
( the @ A
@ ^ [Z3: A] :
( ( P3
=> ( Z3 = X2 ) )
& ( ~ P3
=> ( Z3 = Y ) ) ) ) ) ) ).
% If_def
thf(fact_3062_theI2,axiom,
! [A: $tType,P: A > $o,A3: A,Q: A > $o] :
( ( P @ A3 )
=> ( ! [X4: A] :
( ( P @ X4 )
=> ( X4 = A3 ) )
=> ( ! [X4: A] :
( ( P @ X4 )
=> ( Q @ X4 ) )
=> ( Q @ ( the @ A @ P ) ) ) ) ) ).
% theI2
thf(fact_3063_theI_H,axiom,
! [A: $tType,P: A > $o] :
( ? [X5: A] :
( ( P @ X5 )
& ! [Y4: A] :
( ( P @ Y4 )
=> ( Y4 = X5 ) ) )
=> ( P @ ( the @ A @ P ) ) ) ).
% theI'
thf(fact_3064_theI,axiom,
! [A: $tType,P: A > $o,A3: A] :
( ( P @ A3 )
=> ( ! [X4: A] :
( ( P @ X4 )
=> ( X4 = A3 ) )
=> ( P @ ( the @ A @ P ) ) ) ) ).
% theI
thf(fact_3065_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 )
@ ^ [A6: A,B5: B] :
( product_case_prod @ A @ B @ $o
@ ^ [A9: A,B9: B] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A6 @ A9 ) @ Ra )
| ( ( A6 = A9 )
& ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ B5 @ B9 ) @ Rb ) ) ) ) ) ) ) ) ) ).
% lex_prod_def
thf(fact_3066_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 )
@ ^ [X2: nat,Y: nat] : ( product_Pair @ nat @ nat @ Y @ X2 ) ) ) ) ).
% uminus_int_def
thf(fact_3067_prod__encode__def,axiom,
( nat_prod_encode
= ( product_case_prod @ nat @ nat @ nat
@ ^ [M5: nat,N5: nat] : ( plus_plus @ nat @ ( nat_triangle @ ( plus_plus @ nat @ M5 @ N5 ) ) @ M5 ) ) ) ).
% prod_encode_def
thf(fact_3068_eq__snd__iff,axiom,
! [A: $tType,B: $tType,B2: A,P4: product_prod @ B @ A] :
( ( B2
= ( product_snd @ B @ A @ P4 ) )
= ( ? [A6: B] :
( P4
= ( product_Pair @ B @ A @ A6 @ B2 ) ) ) ) ).
% eq_snd_iff
thf(fact_3069_eq__fst__iff,axiom,
! [A: $tType,B: $tType,A3: A,P4: product_prod @ A @ B] :
( ( A3
= ( product_fst @ A @ B @ P4 ) )
= ( ? [B5: B] :
( P4
= ( product_Pair @ A @ B @ A3 @ B5 ) ) ) ) ).
% eq_fst_iff
thf(fact_3070_map__fun__apply,axiom,
! [D: $tType,A: $tType,C: $tType,B: $tType] :
( ( map_fun @ B @ C @ D @ A )
= ( ^ [F2: B > C,G2: D > A,H4: C > D,X2: B] : ( G2 @ ( H4 @ ( F2 @ X2 ) ) ) ) ) ).
% map_fun_apply
thf(fact_3071_in__lex__prod,axiom,
! [A: $tType,B: $tType,A3: A,B2: B,A7: A,B6: B,R: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ B @ B )] :
( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ ( product_Pair @ A @ B @ A7 @ B6 ) ) @ ( lex_prod @ A @ B @ R @ S ) )
= ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ A7 ) @ R )
| ( ( A3 = A7 )
& ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ B2 @ B6 ) @ S ) ) ) ) ).
% in_lex_prod
thf(fact_3072_map__fun__def,axiom,
! [B: $tType,D: $tType,A: $tType,C: $tType] :
( ( map_fun @ C @ A @ B @ D )
= ( ^ [F2: C > A,G2: B > D,H4: A > B] : ( comp @ A @ D @ C @ ( comp @ B @ D @ A @ G2 @ H4 ) @ F2 ) ) ) ).
% map_fun_def
thf(fact_3073_map__fun_Ocompositionality,axiom,
! [D: $tType,F3: $tType,C: $tType,E: $tType,B: $tType,A: $tType,F: E > C,G: D > F3,H: C > A,I: B > D,Fun: A > B] :
( ( map_fun @ E @ C @ D @ F3 @ F @ G @ ( map_fun @ C @ A @ B @ D @ H @ I @ Fun ) )
= ( map_fun @ E @ A @ B @ F3 @ ( comp @ C @ A @ E @ H @ F ) @ ( comp @ D @ F3 @ B @ G @ I ) @ Fun ) ) ).
% map_fun.compositionality
thf(fact_3074_map__fun_Ocomp,axiom,
! [E: $tType,C: $tType,A: $tType,F3: $tType,D: $tType,B: $tType,F: E > C,G: D > F3,H: C > A,I: B > D] :
( ( comp @ ( C > D ) @ ( E > F3 ) @ ( A > B ) @ ( map_fun @ E @ C @ D @ F3 @ F @ G ) @ ( map_fun @ C @ A @ B @ D @ H @ I ) )
= ( map_fun @ E @ A @ B @ F3 @ ( comp @ C @ A @ E @ H @ F ) @ ( comp @ D @ F3 @ B @ G @ I ) ) ) ).
% map_fun.comp
thf(fact_3075_sum_Osize__neq,axiom,
! [A: $tType,B: $tType,X: sum_sum @ A @ B] :
( ( size_size @ ( sum_sum @ A @ B ) @ X )
!= ( zero_zero @ nat ) ) ).
% sum.size_neq
thf(fact_3076_prod_Osize__neq,axiom,
! [A: $tType,B: $tType,X: product_prod @ A @ B] :
( ( size_size @ ( product_prod @ A @ B ) @ X )
!= ( zero_zero @ nat ) ) ).
% prod.size_neq
thf(fact_3077_le__prod__encode__1,axiom,
! [A3: nat,B2: nat] : ( ord_less_eq @ nat @ A3 @ ( nat_prod_encode @ ( product_Pair @ nat @ nat @ A3 @ B2 ) ) ) ).
% le_prod_encode_1
thf(fact_3078_le__prod__encode__2,axiom,
! [B2: nat,A3: nat] : ( ord_less_eq @ nat @ B2 @ ( nat_prod_encode @ ( product_Pair @ nat @ nat @ A3 @ B2 ) ) ) ).
% le_prod_encode_2
thf(fact_3079_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_3080_times__int__def,axiom,
( ( times_times @ int )
= ( map_fun @ int @ ( product_prod @ nat @ nat ) @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) ) @ ( int > int ) @ rep_Integ @ ( map_fun @ int @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ int @ rep_Integ @ abs_Integ )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X2: nat,Y: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U: nat,V6: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ X2 @ U ) @ ( times_times @ nat @ Y @ V6 ) ) @ ( plus_plus @ nat @ ( times_times @ nat @ X2 @ V6 ) @ ( times_times @ nat @ Y @ U ) ) ) ) ) ) ) ).
% times_int_def
thf(fact_3081_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 ) )
@ ^ [X2: nat,Y: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U: nat,V6: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X2 @ V6 ) @ ( plus_plus @ nat @ Y @ U ) ) ) ) ) ) ).
% minus_int_def
thf(fact_3082_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 ) )
@ ^ [X2: nat,Y: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U: nat,V6: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X2 @ U ) @ ( plus_plus @ nat @ Y @ V6 ) ) ) ) ) ) ).
% plus_int_def
thf(fact_3083_same__fst__def,axiom,
! [B: $tType,A: $tType] :
( ( same_fst @ A @ B )
= ( ^ [P3: A > $o,R6: A > ( set @ ( product_prod @ B @ B ) )] :
( collect @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) )
@ ( product_case_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ $o
@ ( product_case_prod @ A @ B @ ( ( product_prod @ A @ B ) > $o )
@ ^ [X8: A,Y7: B] :
( product_case_prod @ A @ B @ $o
@ ^ [X2: A,Y: B] :
( ( X8 = X2 )
& ( P3 @ X2 )
& ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y7 @ Y ) @ ( R6 @ X2 ) ) ) ) ) ) ) ) ) ).
% same_fst_def
thf(fact_3084_pred__nat__def,axiom,
( pred_nat
= ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [M5: nat,N5: nat] :
( N5
= ( suc @ M5 ) ) ) ) ) ).
% pred_nat_def
thf(fact_3085_same__fstI,axiom,
! [B: $tType,A: $tType,P: A > $o,X: A,Y8: B,Y3: B,R5: A > ( set @ ( product_prod @ B @ B ) )] :
( ( P @ X )
=> ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y8 @ Y3 ) @ ( R5 @ X ) )
=> ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y8 ) @ ( product_Pair @ A @ B @ X @ Y3 ) ) @ ( same_fst @ A @ B @ P @ R5 ) ) ) ) ).
% same_fstI
thf(fact_3086_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_3087_eq__numeral__iff__iszero_I8_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [Y3: num] :
( ( ( one_one @ A )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ Y3 ) ) )
= ( ring_1_iszero @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ one2 @ Y3 ) ) ) ) ) ).
% eq_numeral_iff_iszero(8)
thf(fact_3088_eq__numeral__iff__iszero_I7_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X: num] :
( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ X ) )
= ( one_one @ A ) )
= ( ring_1_iszero @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ X @ one2 ) ) ) ) ) ).
% eq_numeral_iff_iszero(7)
thf(fact_3089_prod_Oinsert_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [I5: set @ B,P4: B > A,I: B] :
( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ I5 )
& ( ( P4 @ X2 )
!= ( one_one @ A ) ) ) ) )
=> ( ( ( member @ B @ I @ I5 )
=> ( ( groups1962203154675924110t_prod @ B @ A @ P4 @ ( insert2 @ B @ I @ I5 ) )
= ( groups1962203154675924110t_prod @ B @ A @ P4 @ I5 ) ) )
& ( ~ ( member @ B @ I @ I5 )
=> ( ( groups1962203154675924110t_prod @ B @ A @ P4 @ ( insert2 @ B @ I @ I5 ) )
= ( times_times @ A @ ( P4 @ I ) @ ( groups1962203154675924110t_prod @ B @ A @ P4 @ I5 ) ) ) ) ) ) ) ).
% prod.insert'
thf(fact_3090_set__n__lists,axiom,
! [A: $tType,N: nat,Xs2: list @ A] :
( ( set2 @ ( list @ A ) @ ( n_lists @ A @ N @ Xs2 ) )
= ( collect @ ( list @ A )
@ ^ [Ys: list @ A] :
( ( ( size_size @ ( list @ A ) @ Ys )
= N )
& ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Ys ) @ ( set2 @ A @ Xs2 ) ) ) ) ) ).
% set_n_lists
thf(fact_3091_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 )
@ ^ [X2: nat,Y: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U: nat,V6: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ X2 @ V6 ) @ ( plus_plus @ nat @ U @ Y ) ) ) ) ) ) ).
% less_eq_int_def
thf(fact_3092_id__apply,axiom,
! [A: $tType] :
( ( id @ A )
= ( ^ [X2: A] : X2 ) ) ).
% id_apply
thf(fact_3093_fun_Omap__id,axiom,
! [A: $tType,D: $tType,T2: D > A] :
( ( comp @ A @ A @ D @ ( id @ A ) @ T2 )
= T2 ) ).
% fun.map_id
thf(fact_3094_id__comp,axiom,
! [B: $tType,A: $tType,G: A > B] :
( ( comp @ B @ B @ A @ ( id @ B ) @ G )
= G ) ).
% id_comp
thf(fact_3095_comp__id,axiom,
! [B: $tType,A: $tType,F: A > B] :
( ( comp @ A @ B @ A @ F @ ( id @ A ) )
= F ) ).
% comp_id
thf(fact_3096_apfst__id,axiom,
! [B: $tType,A: $tType] :
( ( product_apfst @ A @ A @ B @ ( id @ A ) )
= ( id @ ( product_prod @ A @ B ) ) ) ).
% apfst_id
thf(fact_3097_apsnd__id,axiom,
! [B: $tType,A: $tType] :
( ( product_apsnd @ B @ B @ A @ ( id @ B ) )
= ( id @ ( product_prod @ A @ B ) ) ) ).
% apsnd_id
thf(fact_3098_group__add__class_Ominus__comp__minus,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ( ( comp @ A @ A @ A @ ( uminus_uminus @ A ) @ ( uminus_uminus @ A ) )
= ( id @ A ) ) ) ).
% group_add_class.minus_comp_minus
thf(fact_3099_boolean__algebra__class_Ominus__comp__minus,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ( ( comp @ A @ A @ A @ ( uminus_uminus @ A ) @ ( uminus_uminus @ A ) )
= ( id @ A ) ) ) ).
% boolean_algebra_class.minus_comp_minus
thf(fact_3100_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_3101_drop__bit__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se4197421643247451524op_bit @ A @ ( zero_zero @ nat ) )
= ( id @ A ) ) ) ).
% drop_bit_0
thf(fact_3102_comp__the__Some,axiom,
! [A: $tType] :
( ( comp @ ( option @ A ) @ A @ A @ ( the2 @ A ) @ ( some @ A ) )
= ( id @ A ) ) ).
% comp_the_Some
thf(fact_3103_fun_Omap__id0,axiom,
! [A: $tType,D: $tType] :
( ( comp @ A @ A @ D @ ( id @ A ) )
= ( id @ ( D > A ) ) ) ).
% fun.map_id0
thf(fact_3104_pointfree__idE,axiom,
! [B: $tType,A: $tType,F: B > A,G: A > B,X: A] :
( ( ( comp @ B @ A @ A @ F @ G )
= ( id @ A ) )
=> ( ( F @ ( G @ X ) )
= X ) ) ).
% pointfree_idE
thf(fact_3105_comp__eq__id__dest,axiom,
! [C: $tType,B: $tType,A: $tType,A3: C > B,B2: A > C,C2: A > B,V: A] :
( ( ( comp @ C @ B @ A @ A3 @ B2 )
= ( comp @ B @ B @ A @ ( id @ B ) @ C2 ) )
=> ( ( A3 @ ( B2 @ V ) )
= ( C2 @ V ) ) ) ).
% comp_eq_id_dest
thf(fact_3106_id__def,axiom,
! [A: $tType] :
( ( id @ A )
= ( ^ [X2: A] : X2 ) ) ).
% id_def
thf(fact_3107_eq__id__iff,axiom,
! [A: $tType,F: A > A] :
( ( ! [X2: A] :
( ( F @ X2 )
= X2 ) )
= ( F
= ( id @ A ) ) ) ).
% eq_id_iff
thf(fact_3108_iszero__def,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( ( ring_1_iszero @ A )
= ( ^ [Z3: A] :
( Z3
= ( zero_zero @ A ) ) ) ) ) ).
% iszero_def
thf(fact_3109_iszero__0,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( ring_1_iszero @ A @ ( zero_zero @ A ) ) ) ).
% iszero_0
thf(fact_3110_not__iszero__1,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ~ ( ring_1_iszero @ A @ ( one_one @ A ) ) ) ).
% not_iszero_1
thf(fact_3111_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
@ ^ [X2: B] :
( ( member @ B @ X2 @ I5 )
& ( ( G @ X2 )
!= ( one_one @ A ) ) ) ) )
= ( groups1962203154675924110t_prod @ B @ A @ G @ I5 ) ) ) ).
% prod.non_neutral'
thf(fact_3112_eq__numeral__iff__iszero_I10_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [Y3: num] :
( ( ( zero_zero @ A )
= ( numeral_numeral @ A @ Y3 ) )
= ( ring_1_iszero @ A @ ( numeral_numeral @ A @ Y3 ) ) ) ) ).
% eq_numeral_iff_iszero(10)
thf(fact_3113_eq__numeral__iff__iszero_I9_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X: num] :
( ( ( numeral_numeral @ A @ X )
= ( zero_zero @ A ) )
= ( ring_1_iszero @ A @ ( numeral_numeral @ A @ X ) ) ) ) ).
% eq_numeral_iff_iszero(9)
thf(fact_3114_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_3115_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_3116_fst__diag__id,axiom,
! [A: $tType,Z: A] :
( ( comp @ ( product_prod @ A @ A ) @ A @ A @ ( product_fst @ A @ A )
@ ^ [X2: A] : ( product_Pair @ A @ A @ X2 @ X2 )
@ Z )
= ( id @ A @ Z ) ) ).
% fst_diag_id
thf(fact_3117_snd__diag__id,axiom,
! [A: $tType,Z: A] :
( ( comp @ ( product_prod @ A @ A ) @ A @ A @ ( product_snd @ A @ A )
@ ^ [X2: A] : ( product_Pair @ A @ A @ X2 @ X2 )
@ Z )
= ( id @ A @ Z ) ) ).
% snd_diag_id
thf(fact_3118_prod_Omono__neutral__cong__right_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S3: set @ B,T5: set @ B,G: B > A,H: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S3 @ T5 )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ ( minus_minus @ ( set @ B ) @ T5 @ S3 ) )
=> ( ( G @ X4 )
= ( one_one @ A ) ) )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ S3 )
=> ( ( G @ X4 )
= ( H @ X4 ) ) )
=> ( ( groups1962203154675924110t_prod @ B @ A @ G @ T5 )
= ( groups1962203154675924110t_prod @ B @ A @ H @ S3 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_right'
thf(fact_3119_prod_Omono__neutral__cong__left_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S3: set @ B,T5: set @ B,H: B > A,G: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S3 @ T5 )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ ( minus_minus @ ( set @ B ) @ T5 @ S3 ) )
=> ( ( H @ I3 )
= ( one_one @ A ) ) )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ S3 )
=> ( ( G @ X4 )
= ( H @ X4 ) ) )
=> ( ( groups1962203154675924110t_prod @ B @ A @ G @ S3 )
= ( groups1962203154675924110t_prod @ B @ A @ H @ T5 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_left'
thf(fact_3120_prod_Omono__neutral__right_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S3: set @ B,T5: set @ B,G: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S3 @ T5 )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ ( minus_minus @ ( set @ B ) @ T5 @ S3 ) )
=> ( ( G @ X4 )
= ( one_one @ A ) ) )
=> ( ( groups1962203154675924110t_prod @ B @ A @ G @ T5 )
= ( groups1962203154675924110t_prod @ B @ A @ G @ S3 ) ) ) ) ) ).
% prod.mono_neutral_right'
thf(fact_3121_prod_Omono__neutral__left_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S3: set @ B,T5: set @ B,G: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S3 @ T5 )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ ( minus_minus @ ( set @ B ) @ T5 @ S3 ) )
=> ( ( G @ X4 )
= ( one_one @ A ) ) )
=> ( ( groups1962203154675924110t_prod @ B @ A @ G @ S3 )
= ( groups1962203154675924110t_prod @ B @ A @ G @ T5 ) ) ) ) ) ).
% prod.mono_neutral_left'
thf(fact_3122_eq__numeral__iff__iszero_I12_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [Y3: num] :
( ( ( zero_zero @ A )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ Y3 ) ) )
= ( ring_1_iszero @ A @ ( numeral_numeral @ A @ Y3 ) ) ) ) ).
% eq_numeral_iff_iszero(12)
thf(fact_3123_eq__numeral__iff__iszero_I11_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X: num] :
( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ X ) )
= ( zero_zero @ A ) )
= ( ring_1_iszero @ A @ ( numeral_numeral @ A @ X ) ) ) ) ).
% eq_numeral_iff_iszero(11)
thf(fact_3124_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
@ ^ [X2: B] :
( ( member @ B @ X2 @ I5 )
& ( ( G @ X2 )
!= ( one_one @ A ) ) ) ) )
=> ( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ I5 )
& ( ( H @ X2 )
!= ( 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_3125_prod_OG__def,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ( ( groups1962203154675924110t_prod @ B @ A )
= ( ^ [P5: B > A,I6: set @ B] :
( if @ A
@ ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ I6 )
& ( ( P5 @ X2 )
!= ( one_one @ A ) ) ) ) )
@ ( groups7121269368397514597t_prod @ B @ A @ P5
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ I6 )
& ( ( P5 @ X2 )
!= ( one_one @ A ) ) ) ) )
@ ( one_one @ A ) ) ) ) ) ).
% prod.G_def
thf(fact_3126_eq__numeral__iff__iszero_I6_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [Y3: num] :
( ( ( one_one @ A )
= ( numeral_numeral @ A @ Y3 ) )
= ( ring_1_iszero @ A @ ( neg_numeral_sub @ A @ one2 @ Y3 ) ) ) ) ).
% eq_numeral_iff_iszero(6)
thf(fact_3127_eq__numeral__iff__iszero_I5_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X: num] :
( ( ( numeral_numeral @ A @ X )
= ( one_one @ A ) )
= ( ring_1_iszero @ A @ ( neg_numeral_sub @ A @ X @ one2 ) ) ) ) ).
% eq_numeral_iff_iszero(5)
thf(fact_3128_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_3129_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 )
@ ^ [X2: nat,Y: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U: nat,V6: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ X2 @ V6 ) @ ( plus_plus @ nat @ U @ Y ) ) ) ) ) ) ).
% less_int_def
thf(fact_3130_Code__Numeral_Odup__def,axiom,
( code_dup
= ( map_fun @ code_integer @ int @ int @ code_integer @ code_int_of_integer @ code_integer_of_int
@ ^ [K3: int] : ( plus_plus @ int @ K3 @ K3 ) ) ) ).
% Code_Numeral.dup_def
thf(fact_3131_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
@ ^ [X2: B] : ( zero_neq_one_of_bool @ A @ ( P @ X2 ) )
@ A5 )
= ( semiring_1_of_nat @ A @ ( finite_card @ B @ ( inf_inf @ ( set @ B ) @ A5 @ ( collect @ B @ P ) ) ) ) ) ) ) ) ).
% sum_of_bool_eq
thf(fact_3132_dual__min,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( min @ A
@ ^ [X2: A,Y: A] : ( ord_less_eq @ A @ Y @ X2 ) )
= ( ord_max @ A ) ) ) ).
% dual_min
thf(fact_3133_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_3134_prod__diff1,axiom,
! [A: $tType,B: $tType] :
( ( semidom_divide @ A )
=> ! [A5: set @ B,F: B > A,A3: B] :
( ( finite_finite2 @ B @ A5 )
=> ( ( ( F @ A3 )
!= ( zero_zero @ A ) )
=> ( ( ( member @ B @ A3 @ A5 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ F @ ( minus_minus @ ( set @ B ) @ A5 @ ( insert2 @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) )
= ( divide_divide @ A @ ( groups7121269368397514597t_prod @ B @ A @ F @ A5 ) @ ( F @ A3 ) ) ) )
& ( ~ ( member @ B @ A3 @ A5 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ F @ ( minus_minus @ ( set @ B ) @ A5 @ ( insert2 @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) )
= ( groups7121269368397514597t_prod @ B @ A @ F @ A5 ) ) ) ) ) ) ) ).
% prod_diff1
thf(fact_3135_of__nat__eq__id,axiom,
( ( semiring_1_of_nat @ nat )
= ( id @ nat ) ) ).
% of_nat_eq_id
thf(fact_3136_lessThan__0,axiom,
( ( set_ord_lessThan @ nat @ ( zero_zero @ nat ) )
= ( bot_bot @ ( set @ nat ) ) ) ).
% lessThan_0
thf(fact_3137_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_3138_min__0L,axiom,
! [N: nat] :
( ( ord_min @ nat @ ( zero_zero @ nat ) @ N )
= ( zero_zero @ nat ) ) ).
% min_0L
thf(fact_3139_min__0R,axiom,
! [N: nat] :
( ( ord_min @ nat @ N @ ( zero_zero @ nat ) )
= ( zero_zero @ nat ) ) ).
% min_0R
thf(fact_3140_set__decode__zero,axiom,
( ( nat_set_decode @ ( zero_zero @ nat ) )
= ( bot_bot @ ( set @ nat ) ) ) ).
% set_decode_zero
thf(fact_3141_set__encode__empty,axiom,
( ( nat_set_encode @ ( bot_bot @ ( set @ nat ) ) )
= ( zero_zero @ nat ) ) ).
% set_encode_empty
thf(fact_3142_singleton__conv,axiom,
! [A: $tType,A3: A] :
( ( collect @ A
@ ^ [X2: A] : X2 = A3 )
= ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% singleton_conv
thf(fact_3143_singleton__conv2,axiom,
! [A: $tType,A3: A] :
( ( collect @ A
@ ( ^ [Y6: A,Z2: A] : Y6 = Z2
@ A3 ) )
= ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% singleton_conv2
thf(fact_3144_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_3145_min__0__1_I3_J,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [X: num] :
( ( ord_min @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ X ) )
= ( zero_zero @ A ) ) ) ).
% min_0_1(3)
thf(fact_3146_min__0__1_I4_J,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [X: num] :
( ( ord_min @ A @ ( numeral_numeral @ A @ X ) @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% min_0_1(4)
thf(fact_3147_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_3148_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_3149_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_3150_min__0__1_I5_J,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [X: num] :
( ( ord_min @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ X ) )
= ( one_one @ A ) ) ) ).
% min_0_1(5)
thf(fact_3151_min__0__1_I6_J,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [X: num] :
( ( ord_min @ A @ ( numeral_numeral @ A @ X ) @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% min_0_1(6)
thf(fact_3152_card_Oempty,axiom,
! [A: $tType] :
( ( finite_card @ A @ ( bot_bot @ ( set @ A ) ) )
= ( zero_zero @ nat ) ) ).
% card.empty
thf(fact_3153_prod_Oempty_H,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [P4: B > A] :
( ( groups1962203154675924110t_prod @ B @ A @ P4 @ ( bot_bot @ ( set @ B ) ) )
= ( one_one @ A ) ) ) ).
% prod.empty'
thf(fact_3154_atMost__0,axiom,
( ( set_ord_atMost @ nat @ ( zero_zero @ nat ) )
= ( insert2 @ nat @ ( zero_zero @ nat ) @ ( bot_bot @ ( set @ nat ) ) ) ) ).
% atMost_0
thf(fact_3155_case__prod__Pair,axiom,
! [B: $tType,A: $tType] :
( ( product_case_prod @ A @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B ) )
= ( id @ ( product_prod @ A @ B ) ) ) ).
% case_prod_Pair
thf(fact_3156_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_3157_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
@ ^ [X2: B] : ( times_times @ A @ ( F @ X2 ) @ ( zero_neq_one_of_bool @ A @ ( P @ X2 ) ) )
@ A5 )
= ( groups7311177749621191930dd_sum @ B @ A @ F @ ( inf_inf @ ( set @ B ) @ A5 @ ( collect @ B @ P ) ) ) ) ) ) ).
% sum_mult_of_bool_eq
thf(fact_3158_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
@ ^ [X2: B] : ( times_times @ A @ ( zero_neq_one_of_bool @ A @ ( P @ X2 ) ) @ ( F @ X2 ) )
@ A5 )
= ( groups7311177749621191930dd_sum @ B @ A @ F @ ( inf_inf @ ( set @ B ) @ A5 @ ( collect @ B @ P ) ) ) ) ) ) ).
% sum_of_bool_mult_eq
thf(fact_3159_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_3160_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_3161_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_3162_min__diff__distrib__left,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [X: A,Y3: A,Z: A] :
( ( minus_minus @ A @ ( ord_min @ A @ X @ Y3 ) @ Z )
= ( ord_min @ A @ ( minus_minus @ A @ X @ Z ) @ ( minus_minus @ A @ Y3 @ Z ) ) ) ) ).
% min_diff_distrib_left
thf(fact_3163_of__nat__min,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [X: nat,Y3: nat] :
( ( semiring_1_of_nat @ A @ ( ord_min @ nat @ X @ Y3 ) )
= ( ord_min @ A @ ( semiring_1_of_nat @ A @ X ) @ ( semiring_1_of_nat @ A @ Y3 ) ) ) ) ).
% of_nat_min
thf(fact_3164_min__add__distrib__left,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [X: A,Y3: A,Z: A] :
( ( plus_plus @ A @ ( ord_min @ A @ X @ Y3 ) @ Z )
= ( ord_min @ A @ ( plus_plus @ A @ X @ Z ) @ ( plus_plus @ A @ Y3 @ Z ) ) ) ) ).
% min_add_distrib_left
thf(fact_3165_min__add__distrib__right,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [X: A,Y3: A,Z: A] :
( ( plus_plus @ A @ X @ ( ord_min @ A @ Y3 @ Z ) )
= ( ord_min @ A @ ( plus_plus @ A @ X @ Y3 ) @ ( plus_plus @ A @ X @ Z ) ) ) ) ).
% min_add_distrib_right
thf(fact_3166_of__int__min,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: int,Y3: int] :
( ( ring_1_of_int @ A @ ( ord_min @ int @ X @ Y3 ) )
= ( ord_min @ A @ ( ring_1_of_int @ A @ X ) @ ( ring_1_of_int @ A @ Y3 ) ) ) ) ).
% of_int_min
thf(fact_3167_Set_Oempty__def,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
= ( collect @ A
@ ^ [X2: A] : $false ) ) ).
% Set.empty_def
thf(fact_3168_Collect__conj__eq,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( collect @ A
@ ^ [X2: A] :
( ( P @ X2 )
& ( Q @ X2 ) ) )
= ( inf_inf @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_3169_Int__Collect,axiom,
! [A: $tType,X: A,A5: set @ A,P: A > $o] :
( ( member @ A @ X @ ( inf_inf @ ( set @ A ) @ A5 @ ( collect @ A @ P ) ) )
= ( ( member @ A @ X @ A5 )
& ( P @ X ) ) ) ).
% Int_Collect
thf(fact_3170_Int__def,axiom,
! [A: $tType] :
( ( inf_inf @ ( set @ A ) )
= ( ^ [A8: set @ A,B7: set @ A] :
( collect @ A
@ ^ [X2: A] :
( ( member @ A @ X2 @ A8 )
& ( member @ A @ X2 @ B7 ) ) ) ) ) ).
% Int_def
thf(fact_3171_min__def__raw,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_min @ A )
= ( ^ [A6: A,B5: A] : ( if @ A @ ( ord_less_eq @ A @ A6 @ B5 ) @ A6 @ B5 ) ) ) ) ).
% min_def_raw
thf(fact_3172_map__fun_Oidentity,axiom,
! [B: $tType,A: $tType] :
( ( map_fun @ A @ A @ B @ B
@ ^ [X2: A] : X2
@ ^ [X2: B] : X2 )
= ( id @ ( A > B ) ) ) ).
% map_fun.identity
thf(fact_3173_Collect__conv__if,axiom,
! [A: $tType,P: A > $o,A3: A] :
( ( ( P @ A3 )
=> ( ( collect @ A
@ ^ [X2: A] :
( ( X2 = A3 )
& ( P @ X2 ) ) )
= ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) )
& ( ~ ( P @ A3 )
=> ( ( collect @ A
@ ^ [X2: A] :
( ( X2 = A3 )
& ( P @ X2 ) ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Collect_conv_if
thf(fact_3174_Collect__conv__if2,axiom,
! [A: $tType,P: A > $o,A3: A] :
( ( ( P @ A3 )
=> ( ( collect @ A
@ ^ [X2: A] :
( ( A3 = X2 )
& ( P @ X2 ) ) )
= ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) )
& ( ~ ( P @ A3 )
=> ( ( collect @ A
@ ^ [X2: A] :
( ( A3 = X2 )
& ( P @ X2 ) ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Collect_conv_if2
thf(fact_3175_minus__max__eq__min,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [X: A,Y3: A] :
( ( uminus_uminus @ A @ ( ord_max @ A @ X @ Y3 ) )
= ( ord_min @ A @ ( uminus_uminus @ A @ X ) @ ( uminus_uminus @ A @ Y3 ) ) ) ) ).
% minus_max_eq_min
thf(fact_3176_minus__min__eq__max,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [X: A,Y3: A] :
( ( uminus_uminus @ A @ ( ord_min @ A @ X @ Y3 ) )
= ( ord_max @ A @ ( uminus_uminus @ A @ X ) @ ( uminus_uminus @ A @ Y3 ) ) ) ) ).
% minus_min_eq_max
thf(fact_3177_atLeastLessThan0,axiom,
! [M: nat] :
( ( set_or7035219750837199246ssThan @ nat @ M @ ( zero_zero @ nat ) )
= ( bot_bot @ ( set @ nat ) ) ) ).
% atLeastLessThan0
thf(fact_3178_lessThan__empty__iff,axiom,
! [N: nat] :
( ( ( set_ord_lessThan @ nat @ N )
= ( bot_bot @ ( set @ nat ) ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% lessThan_empty_iff
thf(fact_3179_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_3180_card__1__singletonE,axiom,
! [A: $tType,A5: set @ A] :
( ( ( finite_card @ A @ A5 )
= ( one_one @ nat ) )
=> ~ ! [X4: A] :
( A5
!= ( insert2 @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% card_1_singletonE
thf(fact_3181_sum_Ointer__restrict,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A5: set @ B,G: B > A,B4: set @ B] :
( ( finite_finite2 @ B @ A5 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( inf_inf @ ( set @ B ) @ A5 @ B4 ) )
= ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X2: B] : ( if @ A @ ( member @ B @ X2 @ B4 ) @ ( G @ X2 ) @ ( zero_zero @ A ) )
@ A5 ) ) ) ) ).
% sum.inter_restrict
thf(fact_3182_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
@ ^ [X2: B] : ( if @ A @ ( P @ X2 ) @ ( H @ X2 ) @ ( G @ X2 ) )
@ 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_3183_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
@ ^ [X2: B] : ( if @ A @ ( P @ X2 ) @ ( H @ X2 ) @ ( G @ X2 ) )
@ 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_3184_prod_Ointer__restrict,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A5: set @ B,G: B > A,B4: set @ B] :
( ( finite_finite2 @ B @ A5 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( inf_inf @ ( set @ B ) @ A5 @ B4 ) )
= ( groups7121269368397514597t_prod @ B @ A
@ ^ [X2: B] : ( if @ A @ ( member @ B @ X2 @ B4 ) @ ( G @ X2 ) @ ( one_one @ A ) )
@ A5 ) ) ) ) ).
% prod.inter_restrict
thf(fact_3185_min__mult__distrib__right,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [P4: A,X: A,Y3: A] :
( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P4 )
=> ( ( times_times @ A @ ( ord_min @ A @ X @ Y3 ) @ P4 )
= ( ord_min @ A @ ( times_times @ A @ X @ P4 ) @ ( times_times @ A @ Y3 @ P4 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P4 )
=> ( ( times_times @ A @ ( ord_min @ A @ X @ Y3 ) @ P4 )
= ( ord_max @ A @ ( times_times @ A @ X @ P4 ) @ ( times_times @ A @ Y3 @ P4 ) ) ) ) ) ) ).
% min_mult_distrib_right
thf(fact_3186_max__mult__distrib__right,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [P4: A,X: A,Y3: A] :
( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P4 )
=> ( ( times_times @ A @ ( ord_max @ A @ X @ Y3 ) @ P4 )
= ( ord_max @ A @ ( times_times @ A @ X @ P4 ) @ ( times_times @ A @ Y3 @ P4 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P4 )
=> ( ( times_times @ A @ ( ord_max @ A @ X @ Y3 ) @ P4 )
= ( ord_min @ A @ ( times_times @ A @ X @ P4 ) @ ( times_times @ A @ Y3 @ P4 ) ) ) ) ) ) ).
% max_mult_distrib_right
thf(fact_3187_min__mult__distrib__left,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [P4: A,X: A,Y3: A] :
( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P4 )
=> ( ( times_times @ A @ P4 @ ( ord_min @ A @ X @ Y3 ) )
= ( ord_min @ A @ ( times_times @ A @ P4 @ X ) @ ( times_times @ A @ P4 @ Y3 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P4 )
=> ( ( times_times @ A @ P4 @ ( ord_min @ A @ X @ Y3 ) )
= ( ord_max @ A @ ( times_times @ A @ P4 @ X ) @ ( times_times @ A @ P4 @ Y3 ) ) ) ) ) ) ).
% min_mult_distrib_left
thf(fact_3188_max__mult__distrib__left,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [P4: A,X: A,Y3: A] :
( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P4 )
=> ( ( times_times @ A @ P4 @ ( ord_max @ A @ X @ Y3 ) )
= ( ord_max @ A @ ( times_times @ A @ P4 @ X ) @ ( times_times @ A @ P4 @ Y3 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P4 )
=> ( ( times_times @ A @ P4 @ ( ord_max @ A @ X @ Y3 ) )
= ( ord_min @ A @ ( times_times @ A @ P4 @ X ) @ ( times_times @ A @ P4 @ Y3 ) ) ) ) ) ) ).
% max_mult_distrib_left
thf(fact_3189_min__divide__distrib__right,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [P4: A,X: A,Y3: A] :
( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P4 )
=> ( ( divide_divide @ A @ ( ord_min @ A @ X @ Y3 ) @ P4 )
= ( ord_min @ A @ ( divide_divide @ A @ X @ P4 ) @ ( divide_divide @ A @ Y3 @ P4 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P4 )
=> ( ( divide_divide @ A @ ( ord_min @ A @ X @ Y3 ) @ P4 )
= ( ord_max @ A @ ( divide_divide @ A @ X @ P4 ) @ ( divide_divide @ A @ Y3 @ P4 ) ) ) ) ) ) ).
% min_divide_distrib_right
thf(fact_3190_max__divide__distrib__right,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [P4: A,X: A,Y3: A] :
( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P4 )
=> ( ( divide_divide @ A @ ( ord_max @ A @ X @ Y3 ) @ P4 )
= ( ord_max @ A @ ( divide_divide @ A @ X @ P4 ) @ ( divide_divide @ A @ Y3 @ P4 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P4 )
=> ( ( divide_divide @ A @ ( ord_max @ A @ X @ Y3 ) @ P4 )
= ( ord_min @ A @ ( divide_divide @ A @ X @ P4 ) @ ( divide_divide @ A @ Y3 @ P4 ) ) ) ) ) ) ).
% max_divide_distrib_right
thf(fact_3191_min__Suc1,axiom,
! [N: nat,M: nat] :
( ( ord_min @ nat @ ( suc @ N ) @ M )
= ( case_nat @ nat @ ( zero_zero @ nat )
@ ^ [M6: nat] : ( suc @ ( ord_min @ nat @ N @ M6 ) )
@ M ) ) ).
% min_Suc1
thf(fact_3192_min__Suc2,axiom,
! [M: nat,N: nat] :
( ( ord_min @ nat @ M @ ( suc @ N ) )
= ( case_nat @ nat @ ( zero_zero @ nat )
@ ^ [M6: nat] : ( suc @ ( ord_min @ nat @ M6 @ N ) )
@ M ) ) ).
% min_Suc2
thf(fact_3193_sum_Omono__neutral__cong,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [T5: set @ B,S3: set @ B,H: B > A,G: B > A] :
( ( finite_finite2 @ B @ T5 )
=> ( ( finite_finite2 @ B @ S3 )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ ( minus_minus @ ( set @ B ) @ T5 @ S3 ) )
=> ( ( H @ I3 )
= ( zero_zero @ A ) ) )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ ( minus_minus @ ( set @ B ) @ S3 @ T5 ) )
=> ( ( G @ I3 )
= ( zero_zero @ A ) ) )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ ( inf_inf @ ( set @ B ) @ S3 @ T5 ) )
=> ( ( G @ X4 )
= ( H @ X4 ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ S3 )
= ( groups7311177749621191930dd_sum @ B @ A @ H @ T5 ) ) ) ) ) ) ) ) ).
% sum.mono_neutral_cong
thf(fact_3194_sum_OInt__Diff,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A5: set @ B,G: B > A,B4: set @ B] :
( ( finite_finite2 @ B @ A5 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ A5 )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( inf_inf @ ( set @ B ) @ A5 @ B4 ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A5 @ B4 ) ) ) ) ) ) ).
% sum.Int_Diff
thf(fact_3195_prod_Omono__neutral__cong,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [T5: set @ B,S3: set @ B,H: B > A,G: B > A] :
( ( finite_finite2 @ B @ T5 )
=> ( ( finite_finite2 @ B @ S3 )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ ( minus_minus @ ( set @ B ) @ T5 @ S3 ) )
=> ( ( H @ I3 )
= ( one_one @ A ) ) )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ ( minus_minus @ ( set @ B ) @ S3 @ T5 ) )
=> ( ( G @ I3 )
= ( one_one @ A ) ) )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ ( inf_inf @ ( set @ B ) @ S3 @ T5 ) )
=> ( ( G @ X4 )
= ( H @ X4 ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ S3 )
= ( groups7121269368397514597t_prod @ B @ A @ H @ T5 ) ) ) ) ) ) ) ) ).
% prod.mono_neutral_cong
thf(fact_3196_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_3197_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_3198_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_3199_card__Suc__eq,axiom,
! [A: $tType,A5: set @ A,K: nat] :
( ( ( finite_card @ A @ A5 )
= ( suc @ K ) )
= ( ? [B5: A,B7: set @ A] :
( ( A5
= ( insert2 @ A @ B5 @ B7 ) )
& ~ ( member @ A @ B5 @ B7 )
& ( ( finite_card @ A @ B7 )
= K )
& ( ( K
= ( zero_zero @ nat ) )
=> ( B7
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% card_Suc_eq
thf(fact_3200_card__eq__SucD,axiom,
! [A: $tType,A5: set @ A,K: nat] :
( ( ( finite_card @ A @ A5 )
= ( suc @ K ) )
=> ? [B3: A,B10: set @ A] :
( ( A5
= ( insert2 @ A @ B3 @ B10 ) )
& ~ ( member @ A @ B3 @ B10 )
& ( ( finite_card @ A @ B10 )
= K )
& ( ( K
= ( zero_zero @ nat ) )
=> ( B10
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% card_eq_SucD
thf(fact_3201_card__1__singleton__iff,axiom,
! [A: $tType,A5: set @ A] :
( ( ( finite_card @ A @ A5 )
= ( suc @ ( zero_zero @ nat ) ) )
= ( ? [X2: A] :
( A5
= ( insert2 @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% card_1_singleton_iff
thf(fact_3202_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_3203_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_3204_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_3205_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
@ ^ [A6: B] : ( divide_divide @ A @ ( F @ A6 ) @ B2 )
@ ( inf_inf @ ( set @ B ) @ A5
@ ( collect @ B
@ ^ [A6: B] : ( dvd_dvd @ A @ B2 @ ( F @ A6 ) ) ) ) )
@ ( divide_divide @ A
@ ( groups7311177749621191930dd_sum @ B @ A @ F
@ ( inf_inf @ ( set @ B ) @ A5
@ ( collect @ B
@ ^ [A6: B] :
~ ( dvd_dvd @ A @ B2 @ ( F @ A6 ) ) ) ) )
@ B2 ) ) ) ) ) ).
% sum_div_partition
thf(fact_3206_sum_Oinsert__remove,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A5: set @ B,G: B > A,X: B] :
( ( finite_finite2 @ B @ A5 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( insert2 @ B @ X @ A5 ) )
= ( plus_plus @ A @ ( G @ X ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A5 @ ( insert2 @ B @ X @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ).
% sum.insert_remove
thf(fact_3207_sum_Oremove,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A5: set @ B,X: B,G: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( member @ B @ X @ A5 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ A5 )
= ( plus_plus @ A @ ( G @ X ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A5 @ ( insert2 @ B @ X @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ).
% sum.remove
thf(fact_3208_card__Diff__singleton,axiom,
! [A: $tType,X: A,A5: set @ A] :
( ( member @ A @ X @ A5 )
=> ( ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( minus_minus @ nat @ ( finite_card @ A @ A5 ) @ ( one_one @ nat ) ) ) ) ).
% card_Diff_singleton
thf(fact_3209_card__Diff__singleton__if,axiom,
! [A: $tType,X: A,A5: set @ A] :
( ( ( member @ A @ X @ A5 )
=> ( ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( minus_minus @ nat @ ( finite_card @ A @ A5 ) @ ( one_one @ nat ) ) ) )
& ( ~ ( member @ A @ X @ A5 )
=> ( ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( finite_card @ A @ A5 ) ) ) ) ).
% card_Diff_singleton_if
thf(fact_3210_atLeast1__atMost__eq__remove0,axiom,
! [N: nat] :
( ( set_or1337092689740270186AtMost @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N )
= ( minus_minus @ ( set @ nat ) @ ( set_ord_atMost @ nat @ N ) @ ( insert2 @ nat @ ( zero_zero @ nat ) @ ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% atLeast1_atMost_eq_remove0
thf(fact_3211_atLeast1__lessThan__eq__remove0,axiom,
! [N: nat] :
( ( set_or7035219750837199246ssThan @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N )
= ( minus_minus @ ( set @ nat ) @ ( set_ord_lessThan @ nat @ N ) @ ( insert2 @ nat @ ( zero_zero @ nat ) @ ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% atLeast1_lessThan_eq_remove0
thf(fact_3212_sum_Odelta__remove,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S3: set @ B,A3: B,B2: B > A,C2: B > A] :
( ( finite_finite2 @ B @ S3 )
=> ( ( ( member @ B @ A3 @ S3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [K3: B] : ( if @ A @ ( K3 = A3 ) @ ( B2 @ K3 ) @ ( C2 @ K3 ) )
@ S3 )
= ( plus_plus @ A @ ( B2 @ A3 ) @ ( groups7311177749621191930dd_sum @ B @ A @ C2 @ ( minus_minus @ ( set @ B ) @ S3 @ ( insert2 @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) )
& ( ~ ( member @ B @ A3 @ S3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [K3: B] : ( if @ A @ ( K3 = A3 ) @ ( B2 @ K3 ) @ ( C2 @ K3 ) )
@ S3 )
= ( groups7311177749621191930dd_sum @ B @ A @ C2 @ ( minus_minus @ ( set @ B ) @ S3 @ ( insert2 @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ).
% sum.delta_remove
thf(fact_3213_prod_Odelta__remove,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S3: set @ B,A3: B,B2: B > A,C2: B > A] :
( ( finite_finite2 @ B @ S3 )
=> ( ( ( member @ B @ A3 @ S3 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K3: B] : ( if @ A @ ( K3 = A3 ) @ ( B2 @ K3 ) @ ( C2 @ K3 ) )
@ S3 )
= ( times_times @ A @ ( B2 @ A3 ) @ ( groups7121269368397514597t_prod @ B @ A @ C2 @ ( minus_minus @ ( set @ B ) @ S3 @ ( insert2 @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) )
& ( ~ ( member @ B @ A3 @ S3 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K3: B] : ( if @ A @ ( K3 = A3 ) @ ( B2 @ K3 ) @ ( C2 @ K3 ) )
@ S3 )
= ( groups7121269368397514597t_prod @ B @ A @ C2 @ ( minus_minus @ ( set @ B ) @ S3 @ ( insert2 @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ).
% prod.delta_remove
thf(fact_3214_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 )
=> ( ! [X4: C] :
( ( member @ C @ X4 @ ( minus_minus @ ( set @ C ) @ A5 @ ( insert2 @ C @ I @ ( bot_bot @ ( set @ C ) ) ) ) )
=> ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( F @ X4 ) ) )
=> ( ( finite_finite2 @ C @ A5 )
=> ( ord_less_eq @ B @ ( F @ I ) @ ( groups7311177749621191930dd_sum @ C @ B @ F @ A5 ) ) ) ) ) ) ).
% member_le_sum
thf(fact_3215_and__int_Opsimps,axiom,
! [K: int,L: int] :
( ( accp @ ( product_prod @ int @ int ) @ bit_and_int_rel @ ( product_Pair @ int @ int @ K @ L ) )
=> ( ( ( ( member @ int @ K @ ( insert2 @ int @ ( zero_zero @ int ) @ ( insert2 @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
& ( member @ int @ L @ ( insert2 @ int @ ( zero_zero @ int ) @ ( insert2 @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
=> ( ( bit_se5824344872417868541ns_and @ int @ K @ L )
= ( uminus_uminus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K )
& ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L ) ) ) ) ) )
& ( ~ ( ( member @ int @ K @ ( insert2 @ int @ ( zero_zero @ int ) @ ( insert2 @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
& ( member @ int @ L @ ( insert2 @ int @ ( zero_zero @ int ) @ ( insert2 @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
=> ( ( bit_se5824344872417868541ns_and @ int @ K @ L )
= ( plus_plus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K )
& ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L ) ) )
@ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( divide_divide @ int @ K @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% and_int.psimps
thf(fact_3216_and__int_Opelims,axiom,
! [X: int,Xa: int,Y3: int] :
( ( ( bit_se5824344872417868541ns_and @ int @ X @ Xa )
= Y3 )
=> ( ( accp @ ( product_prod @ int @ int ) @ bit_and_int_rel @ ( product_Pair @ int @ int @ X @ Xa ) )
=> ~ ( ( ( ( ( member @ int @ X @ ( insert2 @ int @ ( zero_zero @ int ) @ ( insert2 @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
& ( member @ int @ Xa @ ( insert2 @ int @ ( zero_zero @ int ) @ ( insert2 @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
=> ( Y3
= ( uminus_uminus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ X )
& ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Xa ) ) ) ) ) )
& ( ~ ( ( member @ int @ X @ ( insert2 @ int @ ( zero_zero @ int ) @ ( insert2 @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
& ( member @ int @ Xa @ ( insert2 @ int @ ( zero_zero @ int ) @ ( insert2 @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
=> ( Y3
= ( plus_plus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ X )
& ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Xa ) ) )
@ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( divide_divide @ int @ X @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ Xa @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ int @ int ) @ bit_and_int_rel @ ( product_Pair @ int @ int @ X @ Xa ) ) ) ) ) ).
% and_int.pelims
thf(fact_3217_the__elem__def,axiom,
! [A: $tType] :
( ( the_elem @ A )
= ( ^ [X9: set @ A] :
( the @ A
@ ^ [X2: A] :
( X9
= ( insert2 @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% the_elem_def
thf(fact_3218_sum__diff1_H__aux,axiom,
! [B: $tType,A: $tType] :
( ( ab_group_add @ B )
=> ! [F6: set @ A,I5: set @ A,F: A > B,I: A] :
( ( finite_finite2 @ A @ F6 )
=> ( ( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [I2: A] :
( ( member @ A @ I2 @ I5 )
& ( ( F @ I2 )
!= ( zero_zero @ B ) ) ) )
@ F6 )
=> ( ( ( member @ A @ I @ I5 )
=> ( ( groups1027152243600224163dd_sum @ A @ B @ F @ ( minus_minus @ ( set @ A ) @ I5 @ ( insert2 @ A @ I @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( minus_minus @ B @ ( groups1027152243600224163dd_sum @ A @ B @ F @ I5 ) @ ( F @ I ) ) ) )
& ( ~ ( member @ A @ I @ I5 )
=> ( ( groups1027152243600224163dd_sum @ A @ B @ F @ ( minus_minus @ ( set @ A ) @ I5 @ ( insert2 @ A @ I @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( groups1027152243600224163dd_sum @ A @ B @ F @ I5 ) ) ) ) ) ) ) ).
% sum_diff1'_aux
thf(fact_3219_card__disjoint__shuffles,axiom,
! [A: $tType,Xs2: list @ A,Ys2: list @ A] :
( ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Xs2 ) @ ( set2 @ A @ Ys2 ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_card @ ( list @ A ) @ ( shuffles @ A @ Xs2 @ Ys2 ) )
= ( binomial @ ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ ( size_size @ ( list @ A ) @ Ys2 ) ) @ ( size_size @ ( list @ A ) @ Xs2 ) ) ) ) ).
% card_disjoint_shuffles
thf(fact_3220_sum_Oempty_H,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ! [P4: B > A] :
( ( groups1027152243600224163dd_sum @ B @ A @ P4 @ ( bot_bot @ ( set @ B ) ) )
= ( zero_zero @ A ) ) ) ).
% sum.empty'
thf(fact_3221_sum_Oinsert_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [I5: set @ B,P4: B > A,I: B] :
( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ I5 )
& ( ( P4 @ X2 )
!= ( zero_zero @ A ) ) ) ) )
=> ( ( ( member @ B @ I @ I5 )
=> ( ( groups1027152243600224163dd_sum @ B @ A @ P4 @ ( insert2 @ B @ I @ I5 ) )
= ( groups1027152243600224163dd_sum @ B @ A @ P4 @ I5 ) ) )
& ( ~ ( member @ B @ I @ I5 )
=> ( ( groups1027152243600224163dd_sum @ B @ A @ P4 @ ( insert2 @ B @ I @ I5 ) )
= ( plus_plus @ A @ ( P4 @ I ) @ ( groups1027152243600224163dd_sum @ B @ A @ P4 @ I5 ) ) ) ) ) ) ) ).
% sum.insert'
thf(fact_3222_uminus__set__def,axiom,
! [A: $tType] :
( ( uminus_uminus @ ( set @ A ) )
= ( ^ [A8: set @ A] :
( collect @ A
@ ( uminus_uminus @ ( A > $o )
@ ^ [X2: A] : ( member @ A @ X2 @ A8 ) ) ) ) ) ).
% uminus_set_def
thf(fact_3223_Collect__neg__eq,axiom,
! [A: $tType,P: A > $o] :
( ( collect @ A
@ ^ [X2: A] :
~ ( P @ X2 ) )
= ( uminus_uminus @ ( set @ A ) @ ( collect @ A @ P ) ) ) ).
% Collect_neg_eq
thf(fact_3224_Compl__eq,axiom,
! [A: $tType] :
( ( uminus_uminus @ ( set @ A ) )
= ( ^ [A8: set @ A] :
( collect @ A
@ ^ [X2: A] :
~ ( member @ A @ X2 @ A8 ) ) ) ) ).
% Compl_eq
thf(fact_3225_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
@ ^ [X2: B] :
( ( member @ B @ X2 @ I5 )
& ( ( G @ X2 )
!= ( zero_zero @ A ) ) ) ) )
= ( groups1027152243600224163dd_sum @ B @ A @ G @ I5 ) ) ) ).
% sum.non_neutral'
thf(fact_3226_length__shuffles,axiom,
! [A: $tType,Zs: list @ A,Xs2: list @ A,Ys2: list @ A] :
( ( member @ ( list @ A ) @ Zs @ ( shuffles @ A @ Xs2 @ Ys2 ) )
=> ( ( size_size @ ( list @ A ) @ Zs )
= ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ ( size_size @ ( list @ A ) @ Ys2 ) ) ) ) ).
% length_shuffles
thf(fact_3227_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_3228_sum_Omono__neutral__cong__right_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S3: set @ B,T5: set @ B,G: B > A,H: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S3 @ T5 )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ ( minus_minus @ ( set @ B ) @ T5 @ S3 ) )
=> ( ( G @ X4 )
= ( zero_zero @ A ) ) )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ S3 )
=> ( ( G @ X4 )
= ( H @ X4 ) ) )
=> ( ( groups1027152243600224163dd_sum @ B @ A @ G @ T5 )
= ( groups1027152243600224163dd_sum @ B @ A @ H @ S3 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_right'
thf(fact_3229_sum_Omono__neutral__cong__left_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S3: set @ B,T5: set @ B,H: B > A,G: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S3 @ T5 )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ ( minus_minus @ ( set @ B ) @ T5 @ S3 ) )
=> ( ( H @ I3 )
= ( zero_zero @ A ) ) )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ S3 )
=> ( ( G @ X4 )
= ( H @ X4 ) ) )
=> ( ( groups1027152243600224163dd_sum @ B @ A @ G @ S3 )
= ( groups1027152243600224163dd_sum @ B @ A @ H @ T5 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_left'
thf(fact_3230_sum_Omono__neutral__right_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S3: set @ B,T5: set @ B,G: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S3 @ T5 )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ ( minus_minus @ ( set @ B ) @ T5 @ S3 ) )
=> ( ( G @ X4 )
= ( zero_zero @ A ) ) )
=> ( ( groups1027152243600224163dd_sum @ B @ A @ G @ T5 )
= ( groups1027152243600224163dd_sum @ B @ A @ G @ S3 ) ) ) ) ) ).
% sum.mono_neutral_right'
thf(fact_3231_sum_Omono__neutral__left_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S3: set @ B,T5: set @ B,G: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S3 @ T5 )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ ( minus_minus @ ( set @ B ) @ T5 @ S3 ) )
=> ( ( G @ X4 )
= ( zero_zero @ A ) ) )
=> ( ( groups1027152243600224163dd_sum @ B @ A @ G @ S3 )
= ( groups1027152243600224163dd_sum @ B @ A @ G @ T5 ) ) ) ) ) ).
% sum.mono_neutral_left'
thf(fact_3232_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
@ ^ [X2: B] :
( ( member @ B @ X2 @ I5 )
& ( ( G @ X2 )
!= ( zero_zero @ A ) ) ) ) )
=> ( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ I5 )
& ( ( H @ X2 )
!= ( 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_3233_sum_OG__def,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ( ( groups1027152243600224163dd_sum @ B @ A )
= ( ^ [P5: B > A,I6: set @ B] :
( if @ A
@ ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ I6 )
& ( ( P5 @ X2 )
!= ( zero_zero @ A ) ) ) ) )
@ ( groups7311177749621191930dd_sum @ B @ A @ P5
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ I6 )
& ( ( P5 @ X2 )
!= ( zero_zero @ A ) ) ) ) )
@ ( zero_zero @ A ) ) ) ) ) ).
% sum.G_def
thf(fact_3234_and__int_Opinduct,axiom,
! [A0: int,A1: int,P: int > int > $o] :
( ( accp @ ( product_prod @ int @ int ) @ bit_and_int_rel @ ( product_Pair @ int @ int @ A0 @ A1 ) )
=> ( ! [K2: int,L3: int] :
( ( accp @ ( product_prod @ int @ int ) @ bit_and_int_rel @ ( product_Pair @ int @ int @ K2 @ L3 ) )
=> ( ( ~ ( ( member @ int @ K2 @ ( insert2 @ int @ ( zero_zero @ int ) @ ( insert2 @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
& ( member @ int @ L3 @ ( insert2 @ int @ ( zero_zero @ int ) @ ( insert2 @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
=> ( P @ ( divide_divide @ int @ K2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) )
=> ( P @ K2 @ L3 ) ) )
=> ( P @ A0 @ A1 ) ) ) ).
% and_int.pinduct
thf(fact_3235_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 @ ( insert2 @ 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 @ ( insert2 @ A @ I @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( groups1027152243600224163dd_sum @ A @ B @ F @ I5 ) ) ) ) ) ) ).
% sum_diff1'
thf(fact_3236_upto_Opinduct,axiom,
! [A0: int,A1: int,P: int > int > $o] :
( ( accp @ ( product_prod @ int @ int ) @ upto_rel @ ( product_Pair @ int @ int @ A0 @ A1 ) )
=> ( ! [I3: int,J2: int] :
( ( accp @ ( product_prod @ int @ int ) @ upto_rel @ ( product_Pair @ int @ int @ I3 @ J2 ) )
=> ( ( ( ord_less_eq @ int @ I3 @ J2 )
=> ( P @ ( plus_plus @ int @ I3 @ ( one_one @ int ) ) @ J2 ) )
=> ( P @ I3 @ J2 ) ) )
=> ( P @ A0 @ A1 ) ) ) ).
% upto.pinduct
thf(fact_3237_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 )
@ ^ [M5: num,N5: num] : ( minus_minus @ int @ ( numeral_numeral @ int @ M5 ) @ ( numeral_numeral @ int @ N5 ) ) ) ) ).
% Code_Numeral.sub_def
thf(fact_3238_Gcd__remove0__nat,axiom,
! [M2: set @ nat] :
( ( finite_finite2 @ nat @ M2 )
=> ( ( gcd_Gcd @ nat @ M2 )
= ( gcd_Gcd @ nat @ ( minus_minus @ ( set @ nat ) @ M2 @ ( insert2 @ nat @ ( zero_zero @ nat ) @ ( bot_bot @ ( set @ nat ) ) ) ) ) ) ) ).
% Gcd_remove0_nat
thf(fact_3239_image__minus__const__atLeastLessThan__nat,axiom,
! [C2: nat,Y3: nat,X: nat] :
( ( ( ord_less @ nat @ C2 @ Y3 )
=> ( ( image2 @ nat @ nat
@ ^ [I2: nat] : ( minus_minus @ nat @ I2 @ C2 )
@ ( set_or7035219750837199246ssThan @ nat @ X @ Y3 ) )
= ( set_or7035219750837199246ssThan @ nat @ ( minus_minus @ nat @ X @ C2 ) @ ( minus_minus @ nat @ Y3 @ C2 ) ) ) )
& ( ~ ( ord_less @ nat @ C2 @ Y3 )
=> ( ( ( ord_less @ nat @ X @ Y3 )
=> ( ( image2 @ nat @ nat
@ ^ [I2: nat] : ( minus_minus @ nat @ I2 @ C2 )
@ ( set_or7035219750837199246ssThan @ nat @ X @ Y3 ) )
= ( insert2 @ nat @ ( zero_zero @ nat ) @ ( bot_bot @ ( set @ nat ) ) ) ) )
& ( ~ ( ord_less @ nat @ X @ Y3 )
=> ( ( image2 @ nat @ nat
@ ^ [I2: nat] : ( minus_minus @ nat @ I2 @ C2 )
@ ( set_or7035219750837199246ssThan @ nat @ X @ Y3 ) )
= ( bot_bot @ ( set @ nat ) ) ) ) ) ) ) ).
% image_minus_const_atLeastLessThan_nat
thf(fact_3240_dual__max,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( max @ A
@ ^ [X2: A,Y: A] : ( ord_less_eq @ A @ Y @ X2 ) )
= ( ord_min @ A ) ) ) ).
% dual_max
thf(fact_3241_image__ident,axiom,
! [A: $tType,Y9: set @ A] :
( ( image2 @ A @ A
@ ^ [X2: A] : X2
@ Y9 )
= Y9 ) ).
% image_ident
thf(fact_3242_image__id,axiom,
! [A: $tType] :
( ( image2 @ A @ A @ ( id @ A ) )
= ( id @ ( set @ A ) ) ) ).
% image_id
thf(fact_3243_image__add__0,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [S3: set @ A] :
( ( image2 @ A @ A @ ( plus_plus @ A @ ( zero_zero @ A ) ) @ S3 )
= S3 ) ) ).
% image_add_0
thf(fact_3244_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_3245_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_3246_image__add__atMost,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [C2: A,A3: A] :
( ( image2 @ A @ A @ ( plus_plus @ A @ C2 ) @ ( set_ord_atMost @ A @ A3 ) )
= ( set_ord_atMost @ A @ ( plus_plus @ A @ C2 @ A3 ) ) ) ) ).
% image_add_atMost
thf(fact_3247_Gcd__empty,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ( ( gcd_Gcd @ A @ ( bot_bot @ ( set @ A ) ) )
= ( zero_zero @ A ) ) ) ).
% Gcd_empty
thf(fact_3248_Gcd__insert,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A3: A,A5: set @ A] :
( ( gcd_Gcd @ A @ ( insert2 @ A @ A3 @ A5 ) )
= ( gcd_gcd @ A @ A3 @ ( gcd_Gcd @ A @ A5 ) ) ) ) ).
% Gcd_insert
thf(fact_3249_image__add__atLeastAtMost_H,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [K: A,I: A,J: A] :
( ( image2 @ A @ A
@ ^ [N5: A] : ( plus_plus @ A @ N5 @ K )
@ ( set_or1337092689740270186AtMost @ A @ I @ J ) )
= ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ K ) ) ) ) ).
% image_add_atLeastAtMost'
thf(fact_3250_image__minus__const__atLeastAtMost_H,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [D3: A,A3: A,B2: A] :
( ( image2 @ A @ A
@ ^ [T3: A] : ( minus_minus @ A @ T3 @ D3 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ A3 @ D3 ) @ ( minus_minus @ A @ B2 @ D3 ) ) ) ) ).
% image_minus_const_atLeastAtMost'
thf(fact_3251_image__add__atLeastLessThan_H,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [K: A,I: A,J: A] :
( ( image2 @ A @ A
@ ^ [N5: A] : ( plus_plus @ A @ N5 @ K )
@ ( set_or7035219750837199246ssThan @ A @ I @ J ) )
= ( set_or7035219750837199246ssThan @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ K ) ) ) ) ).
% image_add_atLeastLessThan'
thf(fact_3252_Gcd__2,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A3: A,B2: A] :
( ( gcd_Gcd @ A @ ( insert2 @ A @ A3 @ ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( gcd_gcd @ A @ A3 @ B2 ) ) ) ).
% Gcd_2
thf(fact_3253_image__mult__atLeastAtMost,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [D3: A,A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ D3 )
=> ( ( image2 @ A @ A @ ( times_times @ A @ D3 ) @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ D3 @ A3 ) @ ( times_times @ A @ D3 @ B2 ) ) ) ) ) ).
% image_mult_atLeastAtMost
thf(fact_3254_Gcd__0__iff,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A5: set @ A] :
( ( ( gcd_Gcd @ A @ A5 )
= ( zero_zero @ A ) )
= ( ord_less_eq @ ( set @ A ) @ A5 @ ( insert2 @ A @ ( zero_zero @ A ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% Gcd_0_iff
thf(fact_3255_image__divide__atLeastAtMost,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [D3: A,A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ D3 )
=> ( ( image2 @ A @ A
@ ^ [C4: A] : ( divide_divide @ A @ C4 @ D3 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( divide_divide @ A @ A3 @ D3 ) @ ( divide_divide @ A @ B2 @ D3 ) ) ) ) ) ).
% image_divide_atLeastAtMost
thf(fact_3256_image__comp,axiom,
! [B: $tType,A: $tType,C: $tType,F: B > A,G: C > B,R: set @ C] :
( ( image2 @ B @ A @ F @ ( image2 @ C @ B @ G @ R ) )
= ( image2 @ C @ A @ ( comp @ B @ A @ C @ F @ G ) @ R ) ) ).
% image_comp
thf(fact_3257_image__eq__imp__comp,axiom,
! [B: $tType,A: $tType,D: $tType,C: $tType,F: B > A,A5: set @ B,G: C > A,B4: set @ C,H: A > D] :
( ( ( image2 @ B @ A @ F @ A5 )
= ( image2 @ C @ A @ G @ B4 ) )
=> ( ( image2 @ B @ D @ ( comp @ A @ D @ B @ H @ F ) @ A5 )
= ( image2 @ C @ D @ ( comp @ A @ D @ C @ H @ G ) @ B4 ) ) ) ).
% image_eq_imp_comp
thf(fact_3258_Compr__image__eq,axiom,
! [A: $tType,B: $tType,F: B > A,A5: set @ B,P: A > $o] :
( ( collect @ A
@ ^ [X2: A] :
( ( member @ A @ X2 @ ( image2 @ B @ A @ F @ A5 ) )
& ( P @ X2 ) ) )
= ( image2 @ B @ A @ F
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ A5 )
& ( P @ ( F @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_3259_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
@ ^ [X2: C] : ( F @ ( G @ X2 ) )
@ A5 ) ) ).
% image_image
thf(fact_3260_imageE,axiom,
! [A: $tType,B: $tType,B2: A,F: B > A,A5: set @ B] :
( ( member @ A @ B2 @ ( image2 @ B @ A @ F @ A5 ) )
=> ~ ! [X4: B] :
( ( B2
= ( F @ X4 ) )
=> ~ ( member @ B @ X4 @ A5 ) ) ) ).
% imageE
thf(fact_3261_Gcd__mono,axiom,
! [A: $tType,B: $tType] :
( ( semiring_Gcd @ A )
=> ! [A5: set @ B,F: B > A,G: B > A] :
( ! [X4: B] :
( ( member @ B @ X4 @ A5 )
=> ( dvd_dvd @ A @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( dvd_dvd @ A @ ( gcd_Gcd @ A @ ( image2 @ B @ A @ F @ A5 ) ) @ ( gcd_Gcd @ A @ ( image2 @ B @ A @ G @ A5 ) ) ) ) ) ).
% Gcd_mono
thf(fact_3262_Gcd__greatest__nat,axiom,
! [A5: set @ nat,A3: nat] :
( ! [B3: nat] :
( ( member @ nat @ B3 @ A5 )
=> ( dvd_dvd @ nat @ A3 @ B3 ) )
=> ( dvd_dvd @ nat @ A3 @ ( gcd_Gcd @ nat @ A5 ) ) ) ).
% Gcd_greatest_nat
thf(fact_3263_Gcd__dvd__nat,axiom,
! [A3: nat,A5: set @ nat] :
( ( member @ nat @ A3 @ A5 )
=> ( dvd_dvd @ nat @ ( gcd_Gcd @ nat @ A5 ) @ A3 ) ) ).
% Gcd_dvd_nat
thf(fact_3264_Gcd__nat__eq__one,axiom,
! [N6: set @ nat] :
( ( member @ nat @ ( one_one @ nat ) @ N6 )
=> ( ( gcd_Gcd @ nat @ N6 )
= ( one_one @ nat ) ) ) ).
% Gcd_nat_eq_one
thf(fact_3265_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_3266_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_3267_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 ) )
=> ? [X4: A] :
( ( member @ A @ X4 @ A5 )
& ~ ( finite_finite2 @ A
@ ( collect @ A
@ ^ [A6: A] :
( ( member @ A @ A6 @ A5 )
& ( ( F @ A6 )
= ( F @ X4 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_3268_image__Collect__subsetI,axiom,
! [A: $tType,B: $tType,P: A > $o,F: A > B,B4: set @ B] :
( ! [X4: A] :
( ( P @ X4 )
=> ( member @ B @ ( F @ X4 ) @ B4 ) )
=> ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F @ ( collect @ A @ P ) ) @ B4 ) ) ).
% image_Collect_subsetI
thf(fact_3269_Gcd__greatest,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A5: set @ A,A3: A] :
( ! [B3: A] :
( ( member @ A @ B3 @ A5 )
=> ( dvd_dvd @ A @ A3 @ B3 ) )
=> ( dvd_dvd @ A @ A3 @ ( gcd_Gcd @ A @ A5 ) ) ) ) ).
% Gcd_greatest
thf(fact_3270_dvd__Gcd__iff,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [X: A,A5: set @ A] :
( ( dvd_dvd @ A @ X @ ( gcd_Gcd @ A @ A5 ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ A5 )
=> ( dvd_dvd @ A @ X @ X2 ) ) ) ) ) ).
% dvd_Gcd_iff
thf(fact_3271_dvd__GcdD,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [X: A,A5: set @ A,Y3: A] :
( ( dvd_dvd @ A @ X @ ( gcd_Gcd @ A @ A5 ) )
=> ( ( member @ A @ Y3 @ A5 )
=> ( dvd_dvd @ A @ X @ Y3 ) ) ) ) ).
% dvd_GcdD
thf(fact_3272_Gcd__dvd,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A3: A,A5: set @ A] :
( ( member @ A @ A3 @ A5 )
=> ( dvd_dvd @ A @ ( gcd_Gcd @ A @ A5 ) @ A3 ) ) ) ).
% Gcd_dvd
thf(fact_3273_zero__notin__Suc__image,axiom,
! [A5: set @ nat] :
~ ( member @ nat @ ( zero_zero @ nat ) @ ( image2 @ nat @ nat @ suc @ A5 ) ) ).
% zero_notin_Suc_image
thf(fact_3274_translation__Int,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A3: A,S: set @ A,T2: set @ A] :
( ( image2 @ A @ A @ ( plus_plus @ A @ A3 ) @ ( inf_inf @ ( set @ A ) @ S @ T2 ) )
= ( inf_inf @ ( set @ A ) @ ( image2 @ A @ A @ ( plus_plus @ A @ A3 ) @ S ) @ ( image2 @ A @ A @ ( plus_plus @ A @ A3 ) @ T2 ) ) ) ) ).
% translation_Int
thf(fact_3275_translation__diff,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A3: A,S: set @ A,T2: set @ A] :
( ( image2 @ A @ A @ ( plus_plus @ A @ A3 ) @ ( minus_minus @ ( set @ A ) @ S @ T2 ) )
= ( minus_minus @ ( set @ A ) @ ( image2 @ A @ A @ ( plus_plus @ A @ A3 ) @ S ) @ ( image2 @ A @ A @ ( plus_plus @ A @ A3 ) @ T2 ) ) ) ) ).
% translation_diff
thf(fact_3276_translation__Compl,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A3: A,T2: set @ A] :
( ( image2 @ A @ A @ ( plus_plus @ A @ A3 ) @ ( uminus_uminus @ ( set @ A ) @ T2 ) )
= ( uminus_uminus @ ( set @ A ) @ ( image2 @ A @ A @ ( plus_plus @ A @ A3 ) @ T2 ) ) ) ) ).
% translation_Compl
thf(fact_3277_image__constant__conv,axiom,
! [B: $tType,A: $tType,A5: set @ B,C2: A] :
( ( ( A5
= ( bot_bot @ ( set @ B ) ) )
=> ( ( image2 @ B @ A
@ ^ [X2: B] : C2
@ A5 )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( image2 @ B @ A
@ ^ [X2: B] : C2
@ A5 )
= ( insert2 @ A @ C2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% image_constant_conv
thf(fact_3278_image__constant,axiom,
! [A: $tType,B: $tType,X: A,A5: set @ A,C2: B] :
( ( member @ A @ X @ A5 )
=> ( ( image2 @ A @ B
@ ^ [X2: A] : C2
@ A5 )
= ( insert2 @ B @ C2 @ ( bot_bot @ ( set @ B ) ) ) ) ) ).
% image_constant
thf(fact_3279_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_3280_finite__conv__nat__seg__image,axiom,
! [A: $tType] :
( ( finite_finite2 @ A )
= ( ^ [A8: set @ A] :
? [N5: nat,F2: nat > A] :
( A8
= ( image2 @ nat @ A @ F2
@ ( collect @ nat
@ ^ [I2: nat] : ( ord_less @ nat @ I2 @ N5 ) ) ) ) ) ) ).
% finite_conv_nat_seg_image
thf(fact_3281_translation__subtract__Int,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A3: A,S: set @ A,T2: set @ A] :
( ( image2 @ A @ A
@ ^ [X2: A] : ( minus_minus @ A @ X2 @ A3 )
@ ( inf_inf @ ( set @ A ) @ S @ T2 ) )
= ( inf_inf @ ( set @ A )
@ ( image2 @ A @ A
@ ^ [X2: A] : ( minus_minus @ A @ X2 @ A3 )
@ S )
@ ( image2 @ A @ A
@ ^ [X2: A] : ( minus_minus @ A @ X2 @ A3 )
@ T2 ) ) ) ) ).
% translation_subtract_Int
thf(fact_3282_sum_Oimage__gen,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S3: set @ B,H: B > A,G: B > C] :
( ( finite_finite2 @ B @ S3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ H @ S3 )
= ( groups7311177749621191930dd_sum @ C @ A
@ ^ [Y: C] :
( groups7311177749621191930dd_sum @ B @ A @ H
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ S3 )
& ( ( G @ X2 )
= Y ) ) ) )
@ ( image2 @ B @ C @ G @ S3 ) ) ) ) ) ).
% sum.image_gen
thf(fact_3283_translation__subtract__diff,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A3: A,S: set @ A,T2: set @ A] :
( ( image2 @ A @ A
@ ^ [X2: A] : ( minus_minus @ A @ X2 @ A3 )
@ ( minus_minus @ ( set @ A ) @ S @ T2 ) )
= ( minus_minus @ ( set @ A )
@ ( image2 @ A @ A
@ ^ [X2: A] : ( minus_minus @ A @ X2 @ A3 )
@ S )
@ ( image2 @ A @ A
@ ^ [X2: A] : ( minus_minus @ A @ X2 @ A3 )
@ T2 ) ) ) ) ).
% translation_subtract_diff
thf(fact_3284_prod_Oimage__gen,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S3: set @ B,H: B > A,G: B > C] :
( ( finite_finite2 @ B @ S3 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ H @ S3 )
= ( groups7121269368397514597t_prod @ C @ A
@ ^ [Y: C] :
( groups7121269368397514597t_prod @ B @ A @ H
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ S3 )
& ( ( G @ X2 )
= Y ) ) ) )
@ ( image2 @ B @ C @ G @ S3 ) ) ) ) ) ).
% prod.image_gen
thf(fact_3285_translation__subtract__Compl,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A3: A,T2: set @ A] :
( ( image2 @ A @ A
@ ^ [X2: A] : ( minus_minus @ A @ X2 @ A3 )
@ ( uminus_uminus @ ( set @ A ) @ T2 ) )
= ( uminus_uminus @ ( set @ A )
@ ( image2 @ A @ A
@ ^ [X2: A] : ( minus_minus @ A @ X2 @ A3 )
@ T2 ) ) ) ) ).
% translation_subtract_Compl
thf(fact_3286_Gcd__eq__1__I,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A3: A,A5: set @ A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( member @ A @ A3 @ A5 )
=> ( ( gcd_Gcd @ A @ A5 )
= ( one_one @ A ) ) ) ) ) ).
% Gcd_eq_1_I
thf(fact_3287_Gcd__in,axiom,
! [A5: set @ nat] :
( ! [A4: nat,B3: nat] :
( ( member @ nat @ A4 @ A5 )
=> ( ( member @ nat @ B3 @ A5 )
=> ( member @ nat @ ( gcd_gcd @ nat @ A4 @ B3 ) @ A5 ) ) )
=> ( ( A5
!= ( bot_bot @ ( set @ nat ) ) )
=> ( member @ nat @ ( gcd_Gcd @ nat @ A5 ) @ A5 ) ) ) ).
% Gcd_in
thf(fact_3288_sum_Ogroup,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S3: set @ B,T5: set @ C,G: B > C,H: B > A] :
( ( finite_finite2 @ B @ S3 )
=> ( ( finite_finite2 @ C @ T5 )
=> ( ( ord_less_eq @ ( set @ C ) @ ( image2 @ B @ C @ G @ S3 ) @ T5 )
=> ( ( groups7311177749621191930dd_sum @ C @ A
@ ^ [Y: C] :
( groups7311177749621191930dd_sum @ B @ A @ H
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ S3 )
& ( ( G @ X2 )
= Y ) ) ) )
@ T5 )
= ( groups7311177749621191930dd_sum @ B @ A @ H @ S3 ) ) ) ) ) ) ).
% sum.group
thf(fact_3289_prod_Ogroup,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S3: set @ B,T5: set @ C,G: B > C,H: B > A] :
( ( finite_finite2 @ B @ S3 )
=> ( ( finite_finite2 @ C @ T5 )
=> ( ( ord_less_eq @ ( set @ C ) @ ( image2 @ B @ C @ G @ S3 ) @ T5 )
=> ( ( groups7121269368397514597t_prod @ C @ A
@ ^ [Y: C] :
( groups7121269368397514597t_prod @ B @ A @ H
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ S3 )
& ( ( G @ X2 )
= Y ) ) ) )
@ T5 )
= ( groups7121269368397514597t_prod @ B @ A @ H @ S3 ) ) ) ) ) ) ).
% prod.group
thf(fact_3290_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 )
=> ( ! [X4: B,Y4: B] :
( ( member @ B @ X4 @ A5 )
=> ( ( member @ B @ Y4 @ A5 )
=> ( ( X4 != Y4 )
=> ( ( ( H @ X4 )
= ( H @ Y4 ) )
=> ( ( G @ ( H @ X4 ) )
= ( 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_3291_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 )
=> ( ! [X4: B,Y4: B] :
( ( member @ B @ X4 @ A5 )
=> ( ( member @ B @ Y4 @ A5 )
=> ( ( X4 != Y4 )
=> ( ( ( H @ X4 )
= ( H @ Y4 ) )
=> ( ( G @ ( H @ X4 ) )
= ( 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_3292_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_3293_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_3294_atLeast0__atMost__Suc__eq__insert__0,axiom,
! [N: nat] :
( ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) )
= ( insert2 @ nat @ ( zero_zero @ nat ) @ ( image2 @ nat @ nat @ suc @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% atLeast0_atMost_Suc_eq_insert_0
thf(fact_3295_atLeast0__lessThan__Suc__eq__insert__0,axiom,
! [N: nat] :
( ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) )
= ( insert2 @ nat @ ( zero_zero @ nat ) @ ( image2 @ nat @ nat @ suc @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% atLeast0_lessThan_Suc_eq_insert_0
thf(fact_3296_atMost__Suc__eq__insert__0,axiom,
! [N: nat] :
( ( set_ord_atMost @ nat @ ( suc @ N ) )
= ( insert2 @ nat @ ( zero_zero @ nat ) @ ( image2 @ nat @ nat @ suc @ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% atMost_Suc_eq_insert_0
thf(fact_3297_lessThan__Suc__eq__insert__0,axiom,
! [N: nat] :
( ( set_ord_lessThan @ nat @ ( suc @ N ) )
= ( insert2 @ nat @ ( zero_zero @ nat ) @ ( image2 @ nat @ nat @ suc @ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% lessThan_Suc_eq_insert_0
thf(fact_3298_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_3299_image__mult__atLeastAtMost__if,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,X: A,Y3: A] :
( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( image2 @ A @ A @ ( times_times @ A @ C2 ) @ ( set_or1337092689740270186AtMost @ A @ X @ Y3 ) )
= ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ C2 @ X ) @ ( times_times @ A @ C2 @ Y3 ) ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less_eq @ A @ X @ Y3 )
=> ( ( image2 @ A @ A @ ( times_times @ A @ C2 ) @ ( set_or1337092689740270186AtMost @ A @ X @ Y3 ) )
= ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ C2 @ Y3 ) @ ( times_times @ A @ C2 @ X ) ) ) )
& ( ~ ( ord_less_eq @ A @ X @ Y3 )
=> ( ( image2 @ A @ A @ ( times_times @ A @ C2 ) @ ( set_or1337092689740270186AtMost @ A @ X @ Y3 ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% image_mult_atLeastAtMost_if
thf(fact_3300_image__mult__atLeastAtMost__if_H,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y3: A,C2: A] :
( ( ( ord_less_eq @ A @ X @ Y3 )
=> ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( image2 @ A @ A
@ ^ [X2: A] : ( times_times @ A @ X2 @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ X @ Y3 ) )
= ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ X @ C2 ) @ ( times_times @ A @ Y3 @ C2 ) ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( image2 @ A @ A
@ ^ [X2: A] : ( times_times @ A @ X2 @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ X @ Y3 ) )
= ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ Y3 @ C2 ) @ ( times_times @ A @ X @ C2 ) ) ) ) ) )
& ( ~ ( ord_less_eq @ A @ X @ Y3 )
=> ( ( image2 @ A @ A
@ ^ [X2: A] : ( times_times @ A @ X2 @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ X @ Y3 ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% image_mult_atLeastAtMost_if'
thf(fact_3301_image__affinity__atLeastAtMost,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A,M: A,C2: A] :
( ( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( image2 @ A @ A
@ ^ [X2: A] : ( plus_plus @ A @ ( times_times @ A @ M @ X2 ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B2 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
=> ( ( image2 @ A @ A
@ ^ [X2: A] : ( plus_plus @ A @ ( times_times @ A @ M @ X2 ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ ( times_times @ A @ M @ A3 ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ M @ B2 ) @ C2 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
=> ( ( image2 @ A @ A
@ ^ [X2: A] : ( plus_plus @ A @ ( times_times @ A @ M @ X2 ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ ( times_times @ A @ M @ B2 ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ M @ A3 ) @ C2 ) ) ) ) ) ) ) ) ).
% image_affinity_atLeastAtMost
thf(fact_3302_image__affinity__atLeastAtMost__diff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A,M: A,C2: A] :
( ( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( image2 @ A @ A
@ ^ [X2: A] : ( minus_minus @ A @ ( times_times @ A @ M @ X2 ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B2 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
=> ( ( image2 @ A @ A
@ ^ [X2: A] : ( minus_minus @ A @ ( times_times @ A @ M @ X2 ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ ( times_times @ A @ M @ A3 ) @ C2 ) @ ( minus_minus @ A @ ( times_times @ A @ M @ B2 ) @ C2 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
=> ( ( image2 @ A @ A
@ ^ [X2: A] : ( minus_minus @ A @ ( times_times @ A @ M @ X2 ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ ( times_times @ A @ M @ B2 ) @ C2 ) @ ( minus_minus @ A @ ( times_times @ A @ M @ A3 ) @ C2 ) ) ) ) ) ) ) ) ).
% image_affinity_atLeastAtMost_diff
thf(fact_3303_image__affinity__atLeastAtMost__div,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A,M: A,C2: A] :
( ( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( image2 @ A @ A
@ ^ [X2: A] : ( plus_plus @ A @ ( divide_divide @ A @ X2 @ M ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B2 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
=> ( ( image2 @ A @ A
@ ^ [X2: A] : ( plus_plus @ A @ ( divide_divide @ A @ X2 @ M ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ ( divide_divide @ A @ A3 @ M ) @ C2 ) @ ( plus_plus @ A @ ( divide_divide @ A @ B2 @ M ) @ C2 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
=> ( ( image2 @ A @ A
@ ^ [X2: A] : ( plus_plus @ A @ ( divide_divide @ A @ X2 @ M ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ ( divide_divide @ A @ B2 @ M ) @ C2 ) @ ( plus_plus @ A @ ( divide_divide @ A @ A3 @ M ) @ C2 ) ) ) ) ) ) ) ) ).
% image_affinity_atLeastAtMost_div
thf(fact_3304_image__affinity__atLeastAtMost__div__diff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A,M: A,C2: A] :
( ( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( image2 @ A @ A
@ ^ [X2: A] : ( minus_minus @ A @ ( divide_divide @ A @ X2 @ M ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B2 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
=> ( ( image2 @ A @ A
@ ^ [X2: A] : ( minus_minus @ A @ ( divide_divide @ A @ X2 @ M ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ ( divide_divide @ A @ A3 @ M ) @ C2 ) @ ( minus_minus @ A @ ( divide_divide @ A @ B2 @ M ) @ C2 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
=> ( ( image2 @ A @ A
@ ^ [X2: A] : ( minus_minus @ A @ ( divide_divide @ A @ X2 @ M ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ ( divide_divide @ A @ B2 @ M ) @ C2 ) @ ( minus_minus @ A @ ( divide_divide @ A @ A3 @ M ) @ C2 ) ) ) ) ) ) ) ) ).
% image_affinity_atLeastAtMost_div_diff
thf(fact_3305_sum__fun__comp,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( semiring_1 @ C )
=> ! [S3: set @ A,R5: set @ B,G: A > B,F: B > C] :
( ( finite_finite2 @ A @ S3 )
=> ( ( finite_finite2 @ B @ R5 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ G @ S3 ) @ R5 )
=> ( ( groups7311177749621191930dd_sum @ A @ C
@ ^ [X2: A] : ( F @ ( G @ X2 ) )
@ S3 )
= ( groups7311177749621191930dd_sum @ B @ C
@ ^ [Y: B] :
( times_times @ C
@ ( semiring_1_of_nat @ C
@ ( finite_card @ A
@ ( collect @ A
@ ^ [X2: A] :
( ( member @ A @ X2 @ S3 )
& ( ( G @ X2 )
= Y ) ) ) ) )
@ ( F @ Y ) )
@ R5 ) ) ) ) ) ) ).
% sum_fun_comp
thf(fact_3306_semiring__char__def,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( semiri4206861660011772517g_char @ A )
= ( ^ [Uu2: itself @ A] :
( gcd_Gcd @ nat
@ ( collect @ nat
@ ^ [N5: nat] :
( ( semiring_1_of_nat @ A @ N5 )
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% semiring_char_def
thf(fact_3307_Inf_OINF__image,axiom,
! [B: $tType,A: $tType,C: $tType,Inf: ( set @ A ) > A,G: B > A,F: C > B,A5: set @ C] :
( ( Inf @ ( image2 @ B @ A @ G @ ( image2 @ C @ B @ F @ A5 ) ) )
= ( Inf @ ( image2 @ C @ A @ ( comp @ B @ A @ C @ G @ F ) @ A5 ) ) ) ).
% Inf.INF_image
thf(fact_3308_Sup_OSUP__image,axiom,
! [B: $tType,A: $tType,C: $tType,Sup: ( set @ A ) > A,G: B > A,F: C > B,A5: set @ C] :
( ( Sup @ ( image2 @ B @ A @ G @ ( image2 @ C @ B @ F @ A5 ) ) )
= ( Sup @ ( image2 @ C @ A @ ( comp @ B @ A @ C @ G @ F ) @ A5 ) ) ) ).
% Sup.SUP_image
thf(fact_3309_card__UNION,axiom,
! [A: $tType,A5: set @ ( set @ A )] :
( ( finite_finite2 @ ( set @ A ) @ A5 )
=> ( ! [X4: set @ A] :
( ( member @ ( set @ A ) @ X4 @ A5 )
=> ( finite_finite2 @ A @ X4 ) )
=> ( ( finite_card @ A @ ( complete_Sup_Sup @ ( set @ A ) @ A5 ) )
= ( nat2
@ ( groups7311177749621191930dd_sum @ ( set @ ( set @ A ) ) @ int
@ ^ [I6: set @ ( set @ A )] : ( times_times @ int @ ( power_power @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( plus_plus @ nat @ ( finite_card @ ( set @ A ) @ I6 ) @ ( one_one @ nat ) ) ) @ ( semiring_1_of_nat @ int @ ( finite_card @ A @ ( complete_Inf_Inf @ ( set @ A ) @ I6 ) ) ) )
@ ( collect @ ( set @ ( set @ A ) )
@ ^ [I6: set @ ( set @ A )] :
( ( ord_less_eq @ ( set @ ( set @ A ) ) @ I6 @ A5 )
& ( I6
!= ( bot_bot @ ( set @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% card_UNION
thf(fact_3310_slice__eq__mask,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: nat,M: nat,A3: A] :
( ( bit_se4730199178511100633sh_bit @ A @ N @ ( bit_se2584673776208193580ke_bit @ A @ M @ ( bit_se4197421643247451524op_bit @ A @ N @ A3 ) ) )
= ( bit_se5824344872417868541ns_and @ A @ A3 @ ( 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_3311_abs__Gcd__eq,axiom,
! [K5: set @ int] :
( ( abs_abs @ int @ ( gcd_Gcd @ int @ K5 ) )
= ( gcd_Gcd @ int @ K5 ) ) ).
% abs_Gcd_eq
thf(fact_3312_Sup__apply,axiom,
! [B: $tType,A: $tType] :
( ( complete_Sup @ B )
=> ( ( complete_Sup_Sup @ ( A > B ) )
= ( ^ [A8: set @ ( A > B ),X2: A] :
( complete_Sup_Sup @ B
@ ( image2 @ ( A > B ) @ B
@ ^ [F2: A > B] : ( F2 @ X2 )
@ A8 ) ) ) ) ) ).
% Sup_apply
thf(fact_3313_Inf__apply,axiom,
! [B: $tType,A: $tType] :
( ( complete_Inf @ B )
=> ( ( complete_Inf_Inf @ ( A > B ) )
= ( ^ [A8: set @ ( A > B ),X2: A] :
( complete_Inf_Inf @ B
@ ( image2 @ ( A > B ) @ B
@ ^ [F2: A > B] : ( F2 @ X2 )
@ A8 ) ) ) ) ) ).
% Inf_apply
thf(fact_3314_UN__I,axiom,
! [B: $tType,A: $tType,A3: A,A5: set @ A,B2: B,B4: A > ( set @ B )] :
( ( member @ A @ A3 @ A5 )
=> ( ( member @ B @ B2 @ ( B4 @ A3 ) )
=> ( member @ B @ B2 @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ B4 @ A5 ) ) ) ) ) ).
% UN_I
thf(fact_3315_UN__iff,axiom,
! [A: $tType,B: $tType,B2: A,B4: B > ( set @ A ),A5: set @ B] :
( ( member @ A @ B2 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B4 @ A5 ) ) )
= ( ? [X2: B] :
( ( member @ B @ X2 @ A5 )
& ( member @ A @ B2 @ ( B4 @ X2 ) ) ) ) ) ).
% UN_iff
thf(fact_3316_Gcd__abs__eq,axiom,
! [K5: set @ int] :
( ( gcd_Gcd @ int @ ( image2 @ int @ int @ ( abs_abs @ int ) @ K5 ) )
= ( gcd_Gcd @ int @ K5 ) ) ).
% Gcd_abs_eq
thf(fact_3317_INT__I,axiom,
! [B: $tType,A: $tType,A5: set @ A,B2: B,B4: A > ( set @ B )] :
( ! [X4: A] :
( ( member @ A @ X4 @ A5 )
=> ( member @ B @ B2 @ ( B4 @ X4 ) ) )
=> ( member @ B @ B2 @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ B4 @ A5 ) ) ) ) ).
% INT_I
thf(fact_3318_INT__iff,axiom,
! [A: $tType,B: $tType,B2: A,B4: B > ( set @ A ),A5: set @ B] :
( ( member @ A @ B2 @ ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B4 @ A5 ) ) )
= ( ! [X2: B] :
( ( member @ B @ X2 @ A5 )
=> ( member @ A @ B2 @ ( B4 @ X2 ) ) ) ) ) ).
% INT_iff
thf(fact_3319_SUP__identity__eq,axiom,
! [A: $tType] :
( ( complete_Sup @ A )
=> ! [A5: set @ A] :
( ( complete_Sup_Sup @ A
@ ( image2 @ A @ A
@ ^ [X2: A] : X2
@ A5 ) )
= ( complete_Sup_Sup @ A @ A5 ) ) ) ).
% SUP_identity_eq
thf(fact_3320_SUP__apply,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( complete_Sup @ A )
=> ! [F: C > B > A,A5: set @ C,X: B] :
( ( complete_Sup_Sup @ ( B > A ) @ ( image2 @ C @ ( B > A ) @ F @ A5 ) @ X )
= ( complete_Sup_Sup @ A
@ ( image2 @ C @ A
@ ^ [Y: C] : ( F @ Y @ X )
@ A5 ) ) ) ) ).
% SUP_apply
thf(fact_3321_pair__imageI,axiom,
! [C: $tType,B: $tType,A: $tType,A3: A,B2: B,A5: set @ ( product_prod @ A @ B ),F: A > B > C] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ A5 )
=> ( member @ C @ ( F @ A3 @ B2 ) @ ( image2 @ ( product_prod @ A @ B ) @ C @ ( product_case_prod @ A @ B @ C @ F ) @ A5 ) ) ) ).
% pair_imageI
thf(fact_3322_INF__identity__eq,axiom,
! [A: $tType] :
( ( complete_Inf @ A )
=> ! [A5: set @ A] :
( ( complete_Inf_Inf @ A
@ ( image2 @ A @ A
@ ^ [X2: A] : X2
@ A5 ) )
= ( complete_Inf_Inf @ A @ A5 ) ) ) ).
% INF_identity_eq
thf(fact_3323_INF__apply,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( complete_Inf @ A )
=> ! [F: C > B > A,A5: set @ C,X: B] :
( ( complete_Inf_Inf @ ( B > A ) @ ( image2 @ C @ ( B > A ) @ F @ A5 ) @ X )
= ( complete_Inf_Inf @ A
@ ( image2 @ C @ A
@ ^ [Y: C] : ( F @ Y @ X )
@ A5 ) ) ) ) ).
% INF_apply
thf(fact_3324_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 )
@ ^ [Y: B] : C2
@ A5 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [Y: B] : C2
@ A5 ) )
= C2 ) ) ) ).
% UN_constant
thf(fact_3325_finite__UN__I,axiom,
! [B: $tType,A: $tType,A5: set @ A,B4: A > ( set @ B )] :
( ( finite_finite2 @ A @ A5 )
=> ( ! [A4: A] :
( ( member @ A @ A4 @ A5 )
=> ( finite_finite2 @ B @ ( B4 @ A4 ) ) )
=> ( finite_finite2 @ B @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ B4 @ A5 ) ) ) ) ) ).
% finite_UN_I
thf(fact_3326_finite__INT,axiom,
! [B: $tType,A: $tType,I5: set @ A,A5: A > ( set @ B )] :
( ? [X5: A] :
( ( member @ A @ X5 @ I5 )
& ( finite_finite2 @ B @ ( A5 @ X5 ) ) )
=> ( finite_finite2 @ B @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ A5 @ I5 ) ) ) ) ).
% finite_INT
thf(fact_3327_Gcd__int__eq,axiom,
! [N6: set @ nat] :
( ( gcd_Gcd @ int @ ( image2 @ nat @ int @ ( semiring_1_of_nat @ int ) @ N6 ) )
= ( semiring_1_of_nat @ int @ ( gcd_Gcd @ nat @ N6 ) ) ) ).
% Gcd_int_eq
thf(fact_3328_bit_Oconj__cancel__left,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X: A] :
( ( bit_se5824344872417868541ns_and @ A @ ( bit_ri4277139882892585799ns_not @ A @ X ) @ X )
= ( zero_zero @ A ) ) ) ).
% bit.conj_cancel_left
thf(fact_3329_bit_Oconj__cancel__right,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X: A] :
( ( bit_se5824344872417868541ns_and @ A @ X @ ( bit_ri4277139882892585799ns_not @ A @ X ) )
= ( zero_zero @ A ) ) ) ).
% bit.conj_cancel_right
thf(fact_3330_SUP__bot,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ B] :
( ( complete_Sup_Sup @ A
@ ( image2 @ B @ A
@ ^ [X2: B] : ( bot_bot @ A )
@ A5 ) )
= ( bot_bot @ A ) ) ) ).
% SUP_bot
thf(fact_3331_SUP__bot__conv_I1_J,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [B4: B > A,A5: set @ B] :
( ( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ B4 @ A5 ) )
= ( bot_bot @ A ) )
= ( ! [X2: B] :
( ( member @ B @ X2 @ A5 )
=> ( ( B4 @ X2 )
= ( bot_bot @ A ) ) ) ) ) ) ).
% SUP_bot_conv(1)
thf(fact_3332_SUP__bot__conv_I2_J,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [B4: B > A,A5: set @ B] :
( ( ( bot_bot @ A )
= ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ B4 @ A5 ) ) )
= ( ! [X2: B] :
( ( member @ B @ X2 @ A5 )
=> ( ( B4 @ X2 )
= ( bot_bot @ A ) ) ) ) ) ) ).
% SUP_bot_conv(2)
thf(fact_3333_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_3334_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_3335_UN__simps_I1_J,axiom,
! [A: $tType,B: $tType,C6: set @ B,A3: A,B4: B > ( set @ A )] :
( ( ( C6
= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [X2: B] : ( insert2 @ A @ A3 @ ( B4 @ X2 ) )
@ C6 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( C6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [X2: B] : ( insert2 @ A @ A3 @ ( B4 @ X2 ) )
@ C6 ) )
= ( insert2 @ A @ A3 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B4 @ C6 ) ) ) ) ) ) ).
% UN_simps(1)
thf(fact_3336_UN__singleton,axiom,
! [A: $tType,A5: set @ A] :
( ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ A @ ( set @ A )
@ ^ [X2: A] : ( insert2 @ A @ X2 @ ( bot_bot @ ( set @ A ) ) )
@ A5 ) )
= A5 ) ).
% UN_singleton
thf(fact_3337_INT__insert,axiom,
! [A: $tType,B: $tType,B4: B > ( set @ A ),A3: B,A5: set @ B] :
( ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B4 @ ( insert2 @ B @ A3 @ A5 ) ) )
= ( inf_inf @ ( set @ A ) @ ( B4 @ A3 ) @ ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B4 @ A5 ) ) ) ) ).
% INT_insert
thf(fact_3338_Compl__UN,axiom,
! [A: $tType,B: $tType,B4: B > ( set @ A ),A5: set @ B] :
( ( uminus_uminus @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B4 @ A5 ) ) )
= ( complete_Inf_Inf @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [X2: B] : ( uminus_uminus @ ( set @ A ) @ ( B4 @ X2 ) )
@ A5 ) ) ) ).
% Compl_UN
thf(fact_3339_Compl__INT,axiom,
! [A: $tType,B: $tType,B4: B > ( set @ A ),A5: set @ B] :
( ( uminus_uminus @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B4 @ A5 ) ) )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [X2: B] : ( uminus_uminus @ ( set @ A ) @ ( B4 @ X2 ) )
@ A5 ) ) ) ).
% Compl_INT
thf(fact_3340_Gcd__nat__abs__eq,axiom,
! [K5: set @ int] :
( ( gcd_Gcd @ nat
@ ( image2 @ int @ nat
@ ^ [K3: int] : ( nat2 @ ( abs_abs @ int @ K3 ) )
@ K5 ) )
= ( nat2 @ ( gcd_Gcd @ int @ K5 ) ) ) ).
% Gcd_nat_abs_eq
thf(fact_3341_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_3342_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_3343_bit_Odisj__cancel__right,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X: A] :
( ( bit_se1065995026697491101ons_or @ A @ X @ ( bit_ri4277139882892585799ns_not @ A @ X ) )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% bit.disj_cancel_right
thf(fact_3344_bit_Odisj__cancel__left,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X: A] :
( ( bit_se1065995026697491101ons_or @ A @ ( bit_ri4277139882892585799ns_not @ A @ X ) @ X )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% bit.disj_cancel_left
thf(fact_3345_bit_Oxor__cancel__right,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X: A] :
( ( bit_se5824344971392196577ns_xor @ A @ X @ ( bit_ri4277139882892585799ns_not @ A @ X ) )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% bit.xor_cancel_right
thf(fact_3346_bit_Oxor__cancel__left,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X: A] :
( ( bit_se5824344971392196577ns_xor @ A @ ( bit_ri4277139882892585799ns_not @ A @ X ) @ X )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% bit.xor_cancel_left
thf(fact_3347_bit_Oxor__one__right,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X: A] :
( ( bit_se5824344971392196577ns_xor @ A @ X @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( bit_ri4277139882892585799ns_not @ A @ X ) ) ) ).
% bit.xor_one_right
thf(fact_3348_bit_Oxor__one__left,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X: A] :
( ( bit_se5824344971392196577ns_xor @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ X )
= ( bit_ri4277139882892585799ns_not @ A @ X ) ) ) ).
% bit.xor_one_left
thf(fact_3349_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 ) )
= ( ! [X2: A] :
( ( ord_less @ A @ ( bot_bot @ A ) @ X2 )
=> ? [Y: B] :
( ( member @ B @ Y @ A5 )
& ( ord_less @ A @ ( F @ Y ) @ X2 ) ) ) ) ) ) ).
% INF_eq_bot_iff
thf(fact_3350_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_3351_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_3352_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 )
@ ^ [B7: set @ A] : ( inf_inf @ ( set @ A ) @ B7 @ A5 )
@ B11 ) ) ) ) ) ).
% Int_Inter_eq(2)
thf(fact_3353_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_3354_INT__extend__simps_I2_J,axiom,
! [C: $tType,D: $tType,C6: set @ D,A5: set @ C,B4: D > ( set @ C )] :
( ( ( C6
= ( bot_bot @ ( set @ D ) ) )
=> ( ( inf_inf @ ( set @ C ) @ A5 @ ( complete_Inf_Inf @ ( set @ C ) @ ( image2 @ D @ ( set @ C ) @ B4 @ C6 ) ) )
= A5 ) )
& ( ( C6
!= ( bot_bot @ ( set @ D ) ) )
=> ( ( inf_inf @ ( set @ C ) @ A5 @ ( complete_Inf_Inf @ ( set @ C ) @ ( image2 @ D @ ( set @ C ) @ B4 @ C6 ) ) )
= ( complete_Inf_Inf @ ( set @ C )
@ ( image2 @ D @ ( set @ C )
@ ^ [X2: D] : ( inf_inf @ ( set @ C ) @ A5 @ ( B4 @ X2 ) )
@ C6 ) ) ) ) ) ).
% INT_extend_simps(2)
thf(fact_3355_INT__extend__simps_I1_J,axiom,
! [B: $tType,A: $tType,C6: set @ A,A5: A > ( set @ B ),B4: set @ B] :
( ( ( C6
= ( bot_bot @ ( set @ A ) ) )
=> ( ( inf_inf @ ( set @ B ) @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ A5 @ C6 ) ) @ B4 )
= B4 ) )
& ( ( C6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( inf_inf @ ( set @ B ) @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ A5 @ C6 ) ) @ B4 )
= ( complete_Inf_Inf @ ( set @ B )
@ ( image2 @ A @ ( set @ B )
@ ^ [X2: A] : ( inf_inf @ ( set @ B ) @ ( A5 @ X2 ) @ B4 )
@ C6 ) ) ) ) ) ).
% INT_extend_simps(1)
thf(fact_3356_UNION__empty__conv_I2_J,axiom,
! [A: $tType,B: $tType,B4: B > ( set @ A ),A5: set @ B] :
( ( ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B4 @ A5 ) )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X2: B] :
( ( member @ B @ X2 @ A5 )
=> ( ( B4 @ X2 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% UNION_empty_conv(2)
thf(fact_3357_UNION__empty__conv_I1_J,axiom,
! [A: $tType,B: $tType,B4: B > ( set @ A ),A5: set @ B] :
( ( ( bot_bot @ ( set @ A ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B4 @ A5 ) ) )
= ( ! [X2: B] :
( ( member @ B @ X2 @ A5 )
=> ( ( B4 @ X2 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% UNION_empty_conv(1)
thf(fact_3358_UN__empty,axiom,
! [B: $tType,A: $tType,B4: B > ( set @ A )] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B4 @ ( bot_bot @ ( set @ B ) ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% UN_empty
thf(fact_3359_UN__empty2,axiom,
! [B: $tType,A: $tType,A5: set @ B] :
( ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [X2: B] : ( bot_bot @ ( set @ A ) )
@ A5 ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% UN_empty2
thf(fact_3360_INT__extend__simps_I4_J,axiom,
! [G3: $tType,H8: $tType,C6: set @ H8,A5: set @ G3,B4: H8 > ( set @ G3 )] :
( ( ( C6
= ( bot_bot @ ( set @ H8 ) ) )
=> ( ( minus_minus @ ( set @ G3 ) @ A5 @ ( complete_Sup_Sup @ ( set @ G3 ) @ ( image2 @ H8 @ ( set @ G3 ) @ B4 @ C6 ) ) )
= A5 ) )
& ( ( C6
!= ( bot_bot @ ( set @ H8 ) ) )
=> ( ( minus_minus @ ( set @ G3 ) @ A5 @ ( complete_Sup_Sup @ ( set @ G3 ) @ ( image2 @ H8 @ ( set @ G3 ) @ B4 @ C6 ) ) )
= ( complete_Inf_Inf @ ( set @ G3 )
@ ( image2 @ H8 @ ( set @ G3 )
@ ^ [X2: H8] : ( minus_minus @ ( set @ G3 ) @ A5 @ ( B4 @ X2 ) )
@ C6 ) ) ) ) ) ).
% INT_extend_simps(4)
thf(fact_3361_UN__extend__simps_I1_J,axiom,
! [A: $tType,B: $tType,C6: set @ B,A3: A,B4: B > ( set @ A )] :
( ( ( C6
= ( bot_bot @ ( set @ B ) ) )
=> ( ( insert2 @ A @ A3 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B4 @ C6 ) ) )
= ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) )
& ( ( C6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( insert2 @ A @ A3 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B4 @ C6 ) ) )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [X2: B] : ( insert2 @ A @ A3 @ ( B4 @ X2 ) )
@ C6 ) ) ) ) ) ).
% UN_extend_simps(1)
thf(fact_3362_INT__insert__distrib,axiom,
! [B: $tType,A: $tType,U2: A,A5: set @ A,A3: B,B4: A > ( set @ B )] :
( ( member @ A @ U2 @ A5 )
=> ( ( complete_Inf_Inf @ ( set @ B )
@ ( image2 @ A @ ( set @ B )
@ ^ [X2: A] : ( insert2 @ B @ A3 @ ( B4 @ X2 ) )
@ A5 ) )
= ( insert2 @ B @ A3 @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ B4 @ A5 ) ) ) ) ) ).
% INT_insert_distrib
thf(fact_3363_UN__insert__distrib,axiom,
! [B: $tType,A: $tType,U2: A,A5: set @ A,A3: B,B4: A > ( set @ B )] :
( ( member @ A @ U2 @ A5 )
=> ( ( complete_Sup_Sup @ ( set @ B )
@ ( image2 @ A @ ( set @ B )
@ ^ [X2: A] : ( insert2 @ B @ A3 @ ( B4 @ X2 ) )
@ A5 ) )
= ( insert2 @ B @ A3 @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ B4 @ A5 ) ) ) ) ) ).
% UN_insert_distrib
thf(fact_3364_INT__extend__simps_I5_J,axiom,
! [I7: $tType,J4: $tType,A3: I7,B4: J4 > ( set @ I7 ),C6: set @ J4] :
( ( insert2 @ I7 @ A3 @ ( complete_Inf_Inf @ ( set @ I7 ) @ ( image2 @ J4 @ ( set @ I7 ) @ B4 @ C6 ) ) )
= ( complete_Inf_Inf @ ( set @ I7 )
@ ( image2 @ J4 @ ( set @ I7 )
@ ^ [X2: J4] : ( insert2 @ I7 @ A3 @ ( B4 @ X2 ) )
@ C6 ) ) ) ).
% INT_extend_simps(5)
thf(fact_3365_UN__extend__simps_I6_J,axiom,
! [L5: $tType,K6: $tType,A5: K6 > ( set @ L5 ),C6: set @ K6,B4: set @ L5] :
( ( minus_minus @ ( set @ L5 ) @ ( complete_Sup_Sup @ ( set @ L5 ) @ ( image2 @ K6 @ ( set @ L5 ) @ A5 @ C6 ) ) @ B4 )
= ( complete_Sup_Sup @ ( set @ L5 )
@ ( image2 @ K6 @ ( set @ L5 )
@ ^ [X2: K6] : ( minus_minus @ ( set @ L5 ) @ ( A5 @ X2 ) @ B4 )
@ C6 ) ) ) ).
% UN_extend_simps(6)
thf(fact_3366_UN__extend__simps_I7_J,axiom,
! [M8: $tType,N9: $tType,A5: set @ M8,B4: N9 > ( set @ M8 ),C6: set @ N9] :
( ( minus_minus @ ( set @ M8 ) @ A5 @ ( complete_Inf_Inf @ ( set @ M8 ) @ ( image2 @ N9 @ ( set @ M8 ) @ B4 @ C6 ) ) )
= ( complete_Sup_Sup @ ( set @ M8 )
@ ( image2 @ N9 @ ( set @ M8 )
@ ^ [X2: N9] : ( minus_minus @ ( set @ M8 ) @ A5 @ ( B4 @ X2 ) )
@ C6 ) ) ) ).
% UN_extend_simps(7)
thf(fact_3367_uminus__INF,axiom,
! [A: $tType,B: $tType] :
( ( comple489889107523837845lgebra @ A )
=> ! [B4: B > A,A5: set @ B] :
( ( uminus_uminus @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ B4 @ A5 ) ) )
= ( complete_Sup_Sup @ A
@ ( image2 @ B @ A
@ ^ [X2: B] : ( uminus_uminus @ A @ ( B4 @ X2 ) )
@ A5 ) ) ) ) ).
% uminus_INF
thf(fact_3368_uminus__SUP,axiom,
! [A: $tType,B: $tType] :
( ( comple489889107523837845lgebra @ A )
=> ! [B4: B > A,A5: set @ B] :
( ( uminus_uminus @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ B4 @ A5 ) ) )
= ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [X2: B] : ( uminus_uminus @ A @ ( B4 @ X2 ) )
@ A5 ) ) ) ) ).
% uminus_SUP
thf(fact_3369_INT__subset__iff,axiom,
! [A: $tType,B: $tType,B4: set @ A,A5: B > ( set @ A ),I5: set @ B] :
( ( ord_less_eq @ ( set @ A ) @ B4 @ ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ A5 @ I5 ) ) )
= ( ! [X2: B] :
( ( member @ B @ X2 @ I5 )
=> ( ord_less_eq @ ( set @ A ) @ B4 @ ( A5 @ X2 ) ) ) ) ) ).
% INT_subset_iff
thf(fact_3370_UN__subset__iff,axiom,
! [A: $tType,B: $tType,A5: B > ( set @ A ),I5: set @ B,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ A5 @ I5 ) ) @ B4 )
= ( ! [X2: B] :
( ( member @ B @ X2 @ I5 )
=> ( ord_less_eq @ ( set @ A ) @ ( A5 @ X2 ) @ B4 ) ) ) ) ).
% UN_subset_iff
thf(fact_3371_INT__anti__mono,axiom,
! [B: $tType,A: $tType,A5: set @ A,B4: set @ A,F: A > ( set @ B ),G: A > ( set @ B )] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ A5 )
=> ( ord_less_eq @ ( set @ B ) @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_eq @ ( set @ B ) @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ F @ B4 ) ) @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ G @ A5 ) ) ) ) ) ).
% INT_anti_mono
thf(fact_3372_INT__greatest,axiom,
! [B: $tType,A: $tType,A5: set @ A,C6: set @ B,B4: A > ( set @ B )] :
( ! [X4: A] :
( ( member @ A @ X4 @ A5 )
=> ( ord_less_eq @ ( set @ B ) @ C6 @ ( B4 @ X4 ) ) )
=> ( ord_less_eq @ ( set @ B ) @ C6 @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ B4 @ A5 ) ) ) ) ).
% INT_greatest
thf(fact_3373_INT__lower,axiom,
! [B: $tType,A: $tType,A3: A,A5: set @ A,B4: A > ( set @ B )] :
( ( member @ A @ A3 @ A5 )
=> ( ord_less_eq @ ( set @ B ) @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ B4 @ A5 ) ) @ ( B4 @ A3 ) ) ) ).
% INT_lower
thf(fact_3374_UN__upper,axiom,
! [B: $tType,A: $tType,A3: A,A5: set @ A,B4: A > ( set @ B )] :
( ( member @ A @ A3 @ A5 )
=> ( ord_less_eq @ ( set @ B ) @ ( B4 @ A3 ) @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ B4 @ A5 ) ) ) ) ).
% UN_upper
thf(fact_3375_UN__least,axiom,
! [A: $tType,B: $tType,A5: set @ A,B4: A > ( set @ B ),C6: set @ B] :
( ! [X4: A] :
( ( member @ A @ X4 @ A5 )
=> ( ord_less_eq @ ( set @ B ) @ ( B4 @ X4 ) @ C6 ) )
=> ( ord_less_eq @ ( set @ B ) @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ B4 @ A5 ) ) @ C6 ) ) ).
% UN_least
thf(fact_3376_UN__mono,axiom,
! [B: $tType,A: $tType,A5: set @ A,B4: set @ A,F: A > ( set @ B ),G: A > ( set @ B )] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ A5 )
=> ( ord_less_eq @ ( set @ B ) @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( 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 @ B4 ) ) ) ) ) ).
% UN_mono
thf(fact_3377_UN__image__subset,axiom,
! [C: $tType,A: $tType,B: $tType,F: B > ( set @ A ),G: C > ( set @ B ),X: C,X7: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ F @ ( G @ X ) ) ) @ X7 )
= ( ord_less_eq @ ( set @ B ) @ ( G @ X )
@ ( collect @ B
@ ^ [X2: B] : ( ord_less_eq @ ( set @ A ) @ ( F @ X2 ) @ X7 ) ) ) ) ).
% UN_image_subset
thf(fact_3378_UN__extend__simps_I10_J,axiom,
! [V7: $tType,U4: $tType,T: $tType,B4: U4 > ( set @ V7 ),F: T > U4,A5: set @ T] :
( ( complete_Sup_Sup @ ( set @ V7 )
@ ( image2 @ T @ ( set @ V7 )
@ ^ [A6: T] : ( B4 @ ( F @ A6 ) )
@ A5 ) )
= ( complete_Sup_Sup @ ( set @ V7 ) @ ( image2 @ U4 @ ( set @ V7 ) @ B4 @ ( image2 @ T @ U4 @ F @ A5 ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_3379_UN__extend__simps_I9_J,axiom,
! [S7: $tType,R7: $tType,Q6: $tType,C6: R7 > ( set @ S7 ),B4: Q6 > ( set @ R7 ),A5: set @ Q6] :
( ( complete_Sup_Sup @ ( set @ S7 )
@ ( image2 @ Q6 @ ( set @ S7 )
@ ^ [X2: Q6] : ( complete_Sup_Sup @ ( set @ S7 ) @ ( image2 @ R7 @ ( set @ S7 ) @ C6 @ ( B4 @ X2 ) ) )
@ A5 ) )
= ( complete_Sup_Sup @ ( set @ S7 ) @ ( image2 @ R7 @ ( set @ S7 ) @ C6 @ ( complete_Sup_Sup @ ( set @ R7 ) @ ( image2 @ Q6 @ ( set @ R7 ) @ B4 @ A5 ) ) ) ) ) ).
% UN_extend_simps(9)
thf(fact_3380_UN__extend__simps_I8_J,axiom,
! [P6: $tType,O2: $tType,B4: O2 > ( set @ P6 ),A5: set @ ( set @ O2 )] :
( ( complete_Sup_Sup @ ( set @ P6 )
@ ( image2 @ ( set @ O2 ) @ ( set @ P6 )
@ ^ [Y: set @ O2] : ( complete_Sup_Sup @ ( set @ P6 ) @ ( image2 @ O2 @ ( set @ P6 ) @ B4 @ Y ) )
@ A5 ) )
= ( complete_Sup_Sup @ ( set @ P6 ) @ ( image2 @ O2 @ ( set @ P6 ) @ B4 @ ( complete_Sup_Sup @ ( set @ O2 ) @ A5 ) ) ) ) ).
% UN_extend_simps(8)
thf(fact_3381_INT__extend__simps_I10_J,axiom,
! [V7: $tType,U4: $tType,T: $tType,B4: U4 > ( set @ V7 ),F: T > U4,A5: set @ T] :
( ( complete_Inf_Inf @ ( set @ V7 )
@ ( image2 @ T @ ( set @ V7 )
@ ^ [A6: T] : ( B4 @ ( F @ A6 ) )
@ A5 ) )
= ( complete_Inf_Inf @ ( set @ V7 ) @ ( image2 @ U4 @ ( set @ V7 ) @ B4 @ ( image2 @ T @ U4 @ F @ A5 ) ) ) ) ).
% INT_extend_simps(10)
thf(fact_3382_INT__extend__simps_I9_J,axiom,
! [S7: $tType,R7: $tType,Q6: $tType,C6: R7 > ( set @ S7 ),B4: Q6 > ( set @ R7 ),A5: set @ Q6] :
( ( complete_Inf_Inf @ ( set @ S7 )
@ ( image2 @ Q6 @ ( set @ S7 )
@ ^ [X2: Q6] : ( complete_Inf_Inf @ ( set @ S7 ) @ ( image2 @ R7 @ ( set @ S7 ) @ C6 @ ( B4 @ X2 ) ) )
@ A5 ) )
= ( complete_Inf_Inf @ ( set @ S7 ) @ ( image2 @ R7 @ ( set @ S7 ) @ C6 @ ( complete_Sup_Sup @ ( set @ R7 ) @ ( image2 @ Q6 @ ( set @ R7 ) @ B4 @ A5 ) ) ) ) ) ).
% INT_extend_simps(9)
thf(fact_3383_INT__extend__simps_I8_J,axiom,
! [P6: $tType,O2: $tType,B4: O2 > ( set @ P6 ),A5: set @ ( set @ O2 )] :
( ( complete_Inf_Inf @ ( set @ P6 )
@ ( image2 @ ( set @ O2 ) @ ( set @ P6 )
@ ^ [Y: set @ O2] : ( complete_Inf_Inf @ ( set @ P6 ) @ ( image2 @ O2 @ ( set @ P6 ) @ B4 @ Y ) )
@ A5 ) )
= ( complete_Inf_Inf @ ( set @ P6 ) @ ( image2 @ O2 @ ( set @ P6 ) @ B4 @ ( complete_Sup_Sup @ ( set @ O2 ) @ A5 ) ) ) ) ).
% INT_extend_simps(8)
thf(fact_3384_INF__commute,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F: B > C > A,B4: set @ C,A5: set @ B] :
( ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [I2: B] : ( complete_Inf_Inf @ A @ ( image2 @ C @ A @ ( F @ I2 ) @ B4 ) )
@ A5 ) )
= ( complete_Inf_Inf @ A
@ ( image2 @ C @ A
@ ^ [J3: C] :
( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [I2: B] : ( F @ I2 @ J3 )
@ A5 ) )
@ B4 ) ) ) ) ).
% INF_commute
thf(fact_3385_SUP__commute,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F: B > C > A,B4: set @ C,A5: set @ B] :
( ( complete_Sup_Sup @ A
@ ( image2 @ B @ A
@ ^ [I2: B] : ( complete_Sup_Sup @ A @ ( image2 @ C @ A @ ( F @ I2 ) @ B4 ) )
@ A5 ) )
= ( complete_Sup_Sup @ A
@ ( image2 @ C @ A
@ ^ [J3: C] :
( complete_Sup_Sup @ A
@ ( image2 @ B @ A
@ ^ [I2: B] : ( F @ I2 @ J3 )
@ A5 ) )
@ B4 ) ) ) ) ).
% SUP_commute
thf(fact_3386_UN__E,axiom,
! [A: $tType,B: $tType,B2: A,B4: B > ( set @ A ),A5: set @ B] :
( ( member @ A @ B2 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B4 @ A5 ) ) )
=> ~ ! [X4: B] :
( ( member @ B @ X4 @ A5 )
=> ~ ( member @ A @ B2 @ ( B4 @ X4 ) ) ) ) ).
% UN_E
thf(fact_3387_INT__D,axiom,
! [A: $tType,B: $tType,B2: A,B4: B > ( set @ A ),A5: set @ B,A3: B] :
( ( member @ A @ B2 @ ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B4 @ A5 ) ) )
=> ( ( member @ B @ A3 @ A5 )
=> ( member @ A @ B2 @ ( B4 @ A3 ) ) ) ) ).
% INT_D
thf(fact_3388_INT__E,axiom,
! [A: $tType,B: $tType,B2: A,B4: B > ( set @ A ),A5: set @ B,A3: B] :
( ( member @ A @ B2 @ ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B4 @ A5 ) ) )
=> ( ~ ( member @ A @ B2 @ ( B4 @ A3 ) )
=> ~ ( member @ B @ A3 @ A5 ) ) ) ).
% INT_E
thf(fact_3389_image__UN,axiom,
! [A: $tType,B: $tType,C: $tType,F: B > A,B4: C > ( set @ B ),A5: set @ C] :
( ( image2 @ B @ A @ F @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ C @ ( set @ B ) @ B4 @ A5 ) ) )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ C @ ( set @ A )
@ ^ [X2: C] : ( image2 @ B @ A @ F @ ( B4 @ X2 ) )
@ A5 ) ) ) ).
% image_UN
thf(fact_3390_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
@ ^ [Y: C] : ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F @ ( G @ Y ) ) )
@ A5 ) ) ) ) ).
% SUP_UNION
thf(fact_3391_Inf__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( complete_Inf @ B )
=> ( ( complete_Inf_Inf @ ( A > B ) )
= ( ^ [A8: set @ ( A > B ),X2: A] :
( complete_Inf_Inf @ B
@ ( image2 @ ( A > B ) @ B
@ ^ [F2: A > B] : ( F2 @ X2 )
@ A8 ) ) ) ) ) ).
% Inf_fun_def
thf(fact_3392_Sup__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( complete_Sup @ B )
=> ( ( complete_Sup_Sup @ ( A > B ) )
= ( ^ [A8: set @ ( A > B ),X2: A] :
( complete_Sup_Sup @ B
@ ( image2 @ ( A > B ) @ B
@ ^ [F2: A > B] : ( F2 @ X2 )
@ A8 ) ) ) ) ) ).
% Sup_fun_def
thf(fact_3393_image__Union,axiom,
! [A: $tType,B: $tType,F: B > A,S3: set @ ( set @ B )] :
( ( image2 @ B @ A @ F @ ( complete_Sup_Sup @ ( set @ B ) @ S3 ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ ( set @ B ) @ ( set @ A ) @ ( image2 @ B @ A @ F ) @ S3 ) ) ) ).
% image_Union
thf(fact_3394_UN__UN__flatten,axiom,
! [A: $tType,B: $tType,C: $tType,C6: B > ( set @ A ),B4: 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 ) @ B4 @ A5 ) ) ) )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ C @ ( set @ A )
@ ^ [Y: C] : ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ C6 @ ( B4 @ Y ) ) )
@ A5 ) ) ) ).
% UN_UN_flatten
thf(fact_3395_Int__UN__distrib2,axiom,
! [A: $tType,C: $tType,B: $tType,A5: B > ( set @ A ),I5: set @ B,B4: C > ( set @ A ),J5: set @ C] :
( ( inf_inf @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ A5 @ I5 ) ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ C @ ( set @ A ) @ B4 @ J5 ) ) )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [I2: B] :
( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ C @ ( set @ A )
@ ^ [J3: C] : ( inf_inf @ ( set @ A ) @ ( A5 @ I2 ) @ ( B4 @ J3 ) )
@ J5 ) )
@ I5 ) ) ) ).
% Int_UN_distrib2
thf(fact_3396_Int__Inter__image,axiom,
! [A: $tType,B: $tType,A5: B > ( set @ A ),B4: B > ( set @ A ),C6: set @ B] :
( ( complete_Inf_Inf @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [X2: B] : ( inf_inf @ ( set @ A ) @ ( A5 @ X2 ) @ ( B4 @ X2 ) )
@ 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 ) @ B4 @ C6 ) ) ) ) ).
% Int_Inter_image
thf(fact_3397_INT__Int__distrib,axiom,
! [A: $tType,B: $tType,A5: B > ( set @ A ),B4: B > ( set @ A ),I5: set @ B] :
( ( complete_Inf_Inf @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [I2: B] : ( inf_inf @ ( set @ A ) @ ( A5 @ I2 ) @ ( B4 @ I2 ) )
@ I5 ) )
= ( inf_inf @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ A5 @ I5 ) ) @ ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B4 @ I5 ) ) ) ) ).
% INT_Int_distrib
thf(fact_3398_Int__UN__distrib,axiom,
! [A: $tType,B: $tType,B4: set @ A,A5: B > ( set @ A ),I5: set @ B] :
( ( inf_inf @ ( set @ A ) @ B4 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ A5 @ I5 ) ) )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [I2: B] : ( inf_inf @ ( set @ A ) @ B4 @ ( A5 @ I2 ) )
@ I5 ) ) ) ).
% Int_UN_distrib
thf(fact_3399_Int__Union2,axiom,
! [A: $tType,B4: set @ ( set @ A ),A5: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ B4 ) @ A5 )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ ( set @ A ) @ ( set @ A )
@ ^ [C7: set @ A] : ( inf_inf @ ( set @ A ) @ C7 @ A5 )
@ B4 ) ) ) ).
% Int_Union2
thf(fact_3400_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_3401_Int__Union,axiom,
! [A: $tType,A5: set @ A,B4: set @ ( set @ A )] :
( ( inf_inf @ ( set @ A ) @ A5 @ ( complete_Sup_Sup @ ( set @ A ) @ B4 ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ ( set @ A ) @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A5 ) @ B4 ) ) ) ).
% Int_Union
thf(fact_3402_UN__extend__simps_I4_J,axiom,
! [H8: $tType,G3: $tType,A5: G3 > ( set @ H8 ),C6: set @ G3,B4: set @ H8] :
( ( inf_inf @ ( set @ H8 ) @ ( complete_Sup_Sup @ ( set @ H8 ) @ ( image2 @ G3 @ ( set @ H8 ) @ A5 @ C6 ) ) @ B4 )
= ( complete_Sup_Sup @ ( set @ H8 )
@ ( image2 @ G3 @ ( set @ H8 )
@ ^ [X2: G3] : ( inf_inf @ ( set @ H8 ) @ ( A5 @ X2 ) @ B4 )
@ C6 ) ) ) ).
% UN_extend_simps(4)
thf(fact_3403_UN__extend__simps_I5_J,axiom,
! [I7: $tType,J4: $tType,A5: set @ I7,B4: J4 > ( set @ I7 ),C6: set @ J4] :
( ( inf_inf @ ( set @ I7 ) @ A5 @ ( complete_Sup_Sup @ ( set @ I7 ) @ ( image2 @ J4 @ ( set @ I7 ) @ B4 @ C6 ) ) )
= ( complete_Sup_Sup @ ( set @ I7 )
@ ( image2 @ J4 @ ( set @ I7 )
@ ^ [X2: J4] : ( inf_inf @ ( set @ I7 ) @ A5 @ ( B4 @ X2 ) )
@ C6 ) ) ) ).
% UN_extend_simps(5)
thf(fact_3404_in__Union__o__assoc,axiom,
! [B: $tType,A: $tType,C: $tType,X: A,Gset: B > ( set @ ( set @ A ) ),Gmap: C > B,A5: C] :
( ( member @ A @ X @ ( comp @ B @ ( set @ A ) @ C @ ( comp @ ( set @ ( set @ A ) ) @ ( set @ A ) @ B @ ( complete_Sup_Sup @ ( set @ A ) ) @ Gset ) @ Gmap @ A5 ) )
=> ( member @ A @ X @ ( comp @ ( set @ ( set @ A ) ) @ ( set @ A ) @ C @ ( complete_Sup_Sup @ ( set @ A ) ) @ ( comp @ B @ ( set @ ( set @ A ) ) @ C @ Gset @ Gmap ) @ A5 ) ) ) ).
% in_Union_o_assoc
thf(fact_3405_INF__image,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( complete_Inf @ A )
=> ! [G: B > A,F: C > B,A5: set @ C] :
( ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ G @ ( image2 @ C @ B @ F @ A5 ) ) )
= ( complete_Inf_Inf @ A @ ( image2 @ C @ A @ ( comp @ B @ A @ C @ G @ F ) @ A5 ) ) ) ) ).
% INF_image
thf(fact_3406_SUP__image,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( complete_Sup @ A )
=> ! [G: B > A,F: C > B,A5: set @ C] :
( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ G @ ( image2 @ C @ B @ F @ A5 ) ) )
= ( complete_Sup_Sup @ A @ ( image2 @ C @ A @ ( comp @ B @ A @ C @ G @ F ) @ A5 ) ) ) ) ).
% SUP_image
thf(fact_3407_Gcd__dvd__int,axiom,
! [A3: int,A5: set @ int] :
( ( member @ int @ A3 @ A5 )
=> ( dvd_dvd @ int @ ( gcd_Gcd @ int @ A5 ) @ A3 ) ) ).
% Gcd_dvd_int
thf(fact_3408_Gcd__greatest__int,axiom,
! [A5: set @ int,A3: int] :
( ! [B3: int] :
( ( member @ int @ B3 @ A5 )
=> ( dvd_dvd @ int @ A3 @ B3 ) )
=> ( dvd_dvd @ int @ A3 @ ( gcd_Gcd @ int @ A5 ) ) ) ).
% Gcd_greatest_int
thf(fact_3409_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_3410_INF__eqI,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ B,X: A,F: B > A] :
( ! [I3: B] :
( ( member @ B @ I3 @ A5 )
=> ( ord_less_eq @ A @ X @ ( F @ I3 ) ) )
=> ( ! [Y4: A] :
( ! [I4: B] :
( ( member @ B @ I4 @ A5 )
=> ( ord_less_eq @ A @ Y4 @ ( F @ I4 ) ) )
=> ( ord_less_eq @ A @ Y4 @ X ) )
=> ( ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F @ A5 ) )
= X ) ) ) ) ).
% INF_eqI
thf(fact_3411_INF__mono,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [B4: set @ B,A5: set @ C,F: C > A,G: B > A] :
( ! [M4: B] :
( ( member @ B @ M4 @ B4 )
=> ? [X5: C] :
( ( member @ C @ X5 @ A5 )
& ( ord_less_eq @ A @ ( F @ X5 ) @ ( G @ M4 ) ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ C @ A @ F @ A5 ) ) @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ G @ B4 ) ) ) ) ) ).
% INF_mono
thf(fact_3412_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_3413_INF__mono_H,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F: B > A,G: B > A,A5: set @ B] :
( ! [X4: B] : ( ord_less_eq @ A @ ( F @ X4 ) @ ( G @ X4 ) )
=> ( 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_3414_INF__lower2,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I: B,A5: set @ B,F: B > A,U2: A] :
( ( member @ B @ I @ A5 )
=> ( ( ord_less_eq @ A @ ( F @ I ) @ U2 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F @ A5 ) ) @ U2 ) ) ) ) ).
% INF_lower2
thf(fact_3415_le__INF__iff,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [U2: A,F: B > A,A5: set @ B] :
( ( ord_less_eq @ A @ U2 @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F @ A5 ) ) )
= ( ! [X2: B] :
( ( member @ B @ X2 @ A5 )
=> ( ord_less_eq @ A @ U2 @ ( F @ X2 ) ) ) ) ) ) ).
% le_INF_iff
thf(fact_3416_INF__greatest,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ B,U2: A,F: B > A] :
( ! [I3: B] :
( ( member @ B @ I3 @ A5 )
=> ( ord_less_eq @ A @ U2 @ ( F @ I3 ) ) )
=> ( ord_less_eq @ A @ U2 @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F @ A5 ) ) ) ) ) ).
% INF_greatest
thf(fact_3417_SUP__eqI,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ B,F: B > A,X: A] :
( ! [I3: B] :
( ( member @ B @ I3 @ A5 )
=> ( ord_less_eq @ A @ ( F @ I3 ) @ X ) )
=> ( ! [Y4: A] :
( ! [I4: B] :
( ( member @ B @ I4 @ A5 )
=> ( ord_less_eq @ A @ ( F @ I4 ) @ Y4 ) )
=> ( ord_less_eq @ A @ X @ Y4 ) )
=> ( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F @ A5 ) )
= X ) ) ) ) ).
% SUP_eqI
thf(fact_3418_SUP__mono,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ B,B4: set @ C,F: B > A,G: C > A] :
( ! [N2: B] :
( ( member @ B @ N2 @ A5 )
=> ? [X5: C] :
( ( member @ C @ X5 @ B4 )
& ( ord_less_eq @ A @ ( F @ N2 ) @ ( G @ X5 ) ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F @ A5 ) ) @ ( complete_Sup_Sup @ A @ ( image2 @ C @ A @ G @ B4 ) ) ) ) ) ).
% SUP_mono
thf(fact_3419_SUP__least,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ B,F: B > A,U2: A] :
( ! [I3: B] :
( ( member @ B @ I3 @ A5 )
=> ( ord_less_eq @ A @ ( F @ I3 ) @ U2 ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F @ A5 ) ) @ U2 ) ) ) ).
% SUP_least
thf(fact_3420_SUP__mono_H,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F: B > A,G: B > A,A5: set @ B] :
( ! [X4: B] : ( ord_less_eq @ A @ ( F @ X4 ) @ ( G @ X4 ) )
=> ( 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_3421_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_3422_SUP__le__iff,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F: B > A,A5: set @ B,U2: A] :
( ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F @ A5 ) ) @ U2 )
= ( ! [X2: B] :
( ( member @ B @ X2 @ A5 )
=> ( ord_less_eq @ A @ ( F @ X2 ) @ U2 ) ) ) ) ) ).
% SUP_le_iff
thf(fact_3423_SUP__upper2,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I: B,A5: set @ B,U2: A,F: B > A] :
( ( member @ B @ I @ A5 )
=> ( ( ord_less_eq @ A @ U2 @ ( F @ I ) )
=> ( ord_less_eq @ A @ U2 @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F @ A5 ) ) ) ) ) ) ).
% SUP_upper2
thf(fact_3424_INF__less__iff,axiom,
! [A: $tType,B: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [F: B > A,A5: set @ B,A3: A] :
( ( ord_less @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F @ A5 ) ) @ A3 )
= ( ? [X2: B] :
( ( member @ B @ X2 @ A5 )
& ( ord_less @ A @ ( F @ X2 ) @ A3 ) ) ) ) ) ).
% INF_less_iff
thf(fact_3425_less__INF__D,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [Y3: A,F: B > A,A5: set @ B,I: B] :
( ( ord_less @ A @ Y3 @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F @ A5 ) ) )
=> ( ( member @ B @ I @ A5 )
=> ( ord_less @ A @ Y3 @ ( F @ I ) ) ) ) ) ).
% less_INF_D
thf(fact_3426_less__SUP__iff,axiom,
! [A: $tType,B: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [A3: A,F: B > A,A5: set @ B] :
( ( ord_less @ A @ A3 @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F @ A5 ) ) )
= ( ? [X2: B] :
( ( member @ B @ X2 @ A5 )
& ( ord_less @ A @ A3 @ ( F @ X2 ) ) ) ) ) ) ).
% less_SUP_iff
thf(fact_3427_SUP__lessD,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F: B > A,A5: set @ B,Y3: A,I: B] :
( ( ord_less @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F @ A5 ) ) @ Y3 )
=> ( ( member @ B @ I @ A5 )
=> ( ord_less @ A @ ( F @ I ) @ Y3 ) ) ) ) ).
% SUP_lessD
thf(fact_3428_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_3429_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
@ ^ [A6: B] : ( inf_inf @ A @ ( F @ A6 ) @ ( G @ A6 ) )
@ A5 ) ) ) ) ).
% INF_inf_distrib
thf(fact_3430_image__add__int__atLeastLessThan,axiom,
! [L: int,U2: int] :
( ( image2 @ int @ int
@ ^ [X2: int] : ( plus_plus @ int @ X2 @ L )
@ ( set_or7035219750837199246ssThan @ int @ ( zero_zero @ int ) @ ( minus_minus @ int @ U2 @ L ) ) )
= ( set_or7035219750837199246ssThan @ int @ L @ U2 ) ) ).
% image_add_int_atLeastLessThan
thf(fact_3431_UN__le__add__shift,axiom,
! [A: $tType,M2: nat > ( set @ A ),K: nat,N: nat] :
( ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ nat @ ( set @ A )
@ ^ [I2: nat] : ( M2 @ ( plus_plus @ nat @ I2 @ K ) )
@ ( set_ord_atMost @ nat @ N ) ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ M2 @ ( set_or1337092689740270186AtMost @ nat @ K @ ( plus_plus @ nat @ N @ K ) ) ) ) ) ).
% UN_le_add_shift
thf(fact_3432_UN__le__add__shift__strict,axiom,
! [A: $tType,M2: nat > ( set @ A ),K: nat,N: nat] :
( ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ nat @ ( set @ A )
@ ^ [I2: nat] : ( M2 @ ( plus_plus @ nat @ I2 @ K ) )
@ ( set_ord_lessThan @ nat @ N ) ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ M2 @ ( set_or7035219750837199246ssThan @ nat @ K @ ( plus_plus @ nat @ N @ K ) ) ) ) ) ).
% UN_le_add_shift_strict
thf(fact_3433_not__diff__distrib,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A3: A,B2: A] :
( ( bit_ri4277139882892585799ns_not @ A @ ( minus_minus @ A @ A3 @ B2 ) )
= ( plus_plus @ A @ ( bit_ri4277139882892585799ns_not @ A @ A3 ) @ B2 ) ) ) ).
% not_diff_distrib
thf(fact_3434_not__add__distrib,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A3: A,B2: A] :
( ( bit_ri4277139882892585799ns_not @ A @ ( plus_plus @ A @ A3 @ B2 ) )
= ( minus_minus @ A @ ( bit_ri4277139882892585799ns_not @ A @ A3 ) @ B2 ) ) ) ).
% not_add_distrib
thf(fact_3435_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 )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ I5 )
=> ( finite_finite2 @ C @ ( A5 @ X4 ) ) )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ I5 )
=> ! [Xa4: B] :
( ( member @ B @ Xa4 @ I5 )
=> ( ( X4 != Xa4 )
=> ( ( inf_inf @ ( set @ C ) @ ( A5 @ X4 ) @ ( A5 @ Xa4 ) )
= ( bot_bot @ ( set @ C ) ) ) ) ) )
=> ( ( groups7311177749621191930dd_sum @ C @ A @ G @ ( complete_Sup_Sup @ ( set @ C ) @ ( image2 @ B @ ( set @ C ) @ A5 @ I5 ) ) )
= ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X2: B] : ( groups7311177749621191930dd_sum @ C @ A @ G @ ( A5 @ X2 ) )
@ I5 ) ) ) ) ) ) ).
% sum.UNION_disjoint
thf(fact_3436_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 )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ I5 )
=> ( finite_finite2 @ C @ ( A5 @ X4 ) ) )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ I5 )
=> ! [Xa4: B] :
( ( member @ B @ Xa4 @ I5 )
=> ( ( X4 != Xa4 )
=> ( ( inf_inf @ ( set @ C ) @ ( A5 @ X4 ) @ ( A5 @ Xa4 ) )
= ( bot_bot @ ( set @ C ) ) ) ) ) )
=> ( ( groups7121269368397514597t_prod @ C @ A @ G @ ( complete_Sup_Sup @ ( set @ C ) @ ( image2 @ B @ ( set @ C ) @ A5 @ I5 ) ) )
= ( groups7121269368397514597t_prod @ B @ A
@ ^ [X2: B] : ( groups7121269368397514597t_prod @ C @ A @ G @ ( A5 @ X2 ) )
@ I5 ) ) ) ) ) ) ).
% prod.UNION_disjoint
thf(fact_3437_None__notin__image__Some,axiom,
! [A: $tType,A5: set @ A] :
~ ( member @ ( option @ A ) @ ( none @ A ) @ ( image2 @ A @ ( option @ A ) @ ( some @ A ) @ A5 ) ) ).
% None_notin_image_Some
thf(fact_3438_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_3439_INF__superset__mono,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [B4: set @ B,A5: set @ B,F: B > A,G: B > A] :
( ( ord_less_eq @ ( set @ B ) @ B4 @ A5 )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ B4 )
=> ( ord_less_eq @ A @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F @ A5 ) ) @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ G @ B4 ) ) ) ) ) ) ).
% INF_superset_mono
thf(fact_3440_SUP__subset__mono,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ B,B4: set @ B,F: B > A,G: B > A] :
( ( ord_less_eq @ ( set @ B ) @ A5 @ B4 )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ A5 )
=> ( ord_less_eq @ A @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F @ A5 ) ) @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ G @ B4 ) ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_3441_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
@ ^ [Y: B] : C2
@ A5 ) )
= ( bot_bot @ A ) ) )
& ( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ A
@ ( image2 @ B @ A
@ ^ [Y: B] : C2
@ A5 ) )
= C2 ) ) ) ) ).
% SUP_constant
thf(fact_3442_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_3443_INF__inf__const2,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I5: set @ B,F: B > A,X: A] :
( ( I5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [I2: B] : ( inf_inf @ A @ ( F @ I2 ) @ X )
@ I5 ) )
= ( inf_inf @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F @ I5 ) ) @ X ) ) ) ) ).
% INF_inf_const2
thf(fact_3444_INF__inf__const1,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I5: set @ B,X: A,F: B > A] :
( ( I5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [I2: B] : ( inf_inf @ A @ X @ ( F @ I2 ) )
@ I5 ) )
= ( inf_inf @ A @ X @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F @ I5 ) ) ) ) ) ) ).
% INF_inf_const1
thf(fact_3445_INF__insert,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F: B > A,A3: B,A5: set @ B] :
( ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F @ ( insert2 @ B @ A3 @ A5 ) ) )
= ( inf_inf @ A @ ( F @ A3 ) @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F @ A5 ) ) ) ) ) ).
% INF_insert
thf(fact_3446_UNION__singleton__eq__range,axiom,
! [A: $tType,B: $tType,F: B > A,A5: set @ B] :
( ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [X2: B] : ( insert2 @ A @ ( F @ X2 ) @ ( bot_bot @ ( set @ A ) ) )
@ A5 ) )
= ( image2 @ B @ A @ F @ A5 ) ) ).
% UNION_singleton_eq_range
thf(fact_3447_SUP__inf__distrib2,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [F: B > A,A5: set @ B,G: C > A,B4: set @ C] :
( ( inf_inf @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F @ A5 ) ) @ ( complete_Sup_Sup @ A @ ( image2 @ C @ A @ G @ B4 ) ) )
= ( complete_Sup_Sup @ A
@ ( image2 @ B @ A
@ ^ [A6: B] :
( complete_Sup_Sup @ A
@ ( image2 @ C @ A
@ ^ [B5: C] : ( inf_inf @ A @ ( F @ A6 ) @ ( G @ B5 ) )
@ B4 ) )
@ A5 ) ) ) ) ).
% SUP_inf_distrib2
thf(fact_3448_inf__SUP,axiom,
! [A: $tType,B: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [A3: A,F: B > A,B4: set @ B] :
( ( inf_inf @ A @ A3 @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F @ B4 ) ) )
= ( complete_Sup_Sup @ A
@ ( image2 @ B @ A
@ ^ [B5: B] : ( inf_inf @ A @ A3 @ ( F @ B5 ) )
@ B4 ) ) ) ) ).
% inf_SUP
thf(fact_3449_Sup__inf,axiom,
! [A: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [B4: set @ A,A3: A] :
( ( inf_inf @ A @ ( complete_Sup_Sup @ A @ B4 ) @ A3 )
= ( complete_Sup_Sup @ A
@ ( image2 @ A @ A
@ ^ [B5: A] : ( inf_inf @ A @ B5 @ A3 )
@ B4 ) ) ) ) ).
% Sup_inf
thf(fact_3450_SUP__inf,axiom,
! [A: $tType,B: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [F: B > A,B4: set @ B,A3: A] :
( ( inf_inf @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F @ B4 ) ) @ A3 )
= ( complete_Sup_Sup @ A
@ ( image2 @ B @ A
@ ^ [B5: B] : ( inf_inf @ A @ ( F @ B5 ) @ A3 )
@ B4 ) ) ) ) ).
% SUP_inf
thf(fact_3451_Gcd__int__greater__eq__0,axiom,
! [K5: set @ int] : ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( gcd_Gcd @ int @ K5 ) ) ).
% Gcd_int_greater_eq_0
thf(fact_3452_Gcd__int__def,axiom,
( ( gcd_Gcd @ int )
= ( ^ [K7: set @ int] : ( semiring_1_of_nat @ int @ ( gcd_Gcd @ nat @ ( image2 @ int @ nat @ ( comp @ int @ nat @ int @ nat2 @ ( abs_abs @ int ) ) @ K7 ) ) ) ) ) ).
% Gcd_int_def
thf(fact_3453_card__UN__disjoint,axiom,
! [B: $tType,A: $tType,I5: set @ A,A5: A > ( set @ B )] :
( ( finite_finite2 @ A @ I5 )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ I5 )
=> ( finite_finite2 @ B @ ( A5 @ X4 ) ) )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ I5 )
=> ! [Xa4: A] :
( ( member @ A @ Xa4 @ I5 )
=> ( ( X4 != Xa4 )
=> ( ( inf_inf @ ( set @ B ) @ ( A5 @ X4 ) @ ( A5 @ Xa4 ) )
= ( 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_3454_minus__eq__not__plus__1,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( uminus_uminus @ A )
= ( ^ [A6: A] : ( plus_plus @ A @ ( bit_ri4277139882892585799ns_not @ A @ A6 ) @ ( one_one @ A ) ) ) ) ) ).
% minus_eq_not_plus_1
thf(fact_3455_minus__eq__not__minus__1,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( uminus_uminus @ A )
= ( ^ [A6: A] : ( bit_ri4277139882892585799ns_not @ A @ ( minus_minus @ A @ A6 @ ( one_one @ A ) ) ) ) ) ) ).
% minus_eq_not_minus_1
thf(fact_3456_not__eq__complement,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( bit_ri4277139882892585799ns_not @ A )
= ( ^ [A6: A] : ( minus_minus @ A @ ( uminus_uminus @ A @ A6 ) @ ( one_one @ A ) ) ) ) ) ).
% not_eq_complement
thf(fact_3457_sum_OUnion__comp,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [B4: set @ ( set @ B ),G: B > A] :
( ! [X4: set @ B] :
( ( member @ ( set @ B ) @ X4 @ B4 )
=> ( finite_finite2 @ B @ X4 ) )
=> ( ! [A13: set @ B] :
( ( member @ ( set @ B ) @ A13 @ B4 )
=> ! [A24: set @ B] :
( ( member @ ( set @ B ) @ A24 @ B4 )
=> ( ( A13 != A24 )
=> ! [X4: B] :
( ( member @ B @ X4 @ A13 )
=> ( ( member @ B @ X4 @ A24 )
=> ( ( G @ X4 )
= ( zero_zero @ A ) ) ) ) ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( complete_Sup_Sup @ ( set @ B ) @ B4 ) )
= ( comp @ ( ( set @ B ) > A ) @ ( ( set @ ( set @ B ) ) > A ) @ ( B > A ) @ ( groups7311177749621191930dd_sum @ ( set @ B ) @ A ) @ ( groups7311177749621191930dd_sum @ B @ A ) @ G @ B4 ) ) ) ) ) ).
% sum.Union_comp
thf(fact_3458_prod_OUnion__comp,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [B4: set @ ( set @ B ),G: B > A] :
( ! [X4: set @ B] :
( ( member @ ( set @ B ) @ X4 @ B4 )
=> ( finite_finite2 @ B @ X4 ) )
=> ( ! [A13: set @ B] :
( ( member @ ( set @ B ) @ A13 @ B4 )
=> ! [A24: set @ B] :
( ( member @ ( set @ B ) @ A24 @ B4 )
=> ( ( A13 != A24 )
=> ! [X4: B] :
( ( member @ B @ X4 @ A13 )
=> ( ( member @ B @ X4 @ A24 )
=> ( ( G @ X4 )
= ( one_one @ A ) ) ) ) ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( complete_Sup_Sup @ ( set @ B ) @ B4 ) )
= ( comp @ ( ( set @ B ) > A ) @ ( ( set @ ( set @ B ) ) > A ) @ ( B > A ) @ ( groups7121269368397514597t_prod @ ( set @ B ) @ A ) @ ( groups7121269368397514597t_prod @ B @ A ) @ G @ B4 ) ) ) ) ) ).
% prod.Union_comp
thf(fact_3459_sum_OUnion__disjoint,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [C6: set @ ( set @ B ),G: B > A] :
( ! [X4: set @ B] :
( ( member @ ( set @ B ) @ X4 @ C6 )
=> ( finite_finite2 @ B @ X4 ) )
=> ( ! [X4: set @ B] :
( ( member @ ( set @ B ) @ X4 @ C6 )
=> ! [Xa4: set @ B] :
( ( member @ ( set @ B ) @ Xa4 @ C6 )
=> ( ( X4 != Xa4 )
=> ( ( inf_inf @ ( set @ B ) @ X4 @ Xa4 )
= ( bot_bot @ ( set @ B ) ) ) ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( complete_Sup_Sup @ ( set @ B ) @ C6 ) )
= ( comp @ ( ( set @ B ) > A ) @ ( ( set @ ( set @ B ) ) > A ) @ ( B > A ) @ ( groups7311177749621191930dd_sum @ ( set @ B ) @ A ) @ ( groups7311177749621191930dd_sum @ B @ A ) @ G @ C6 ) ) ) ) ) ).
% sum.Union_disjoint
thf(fact_3460_prod_OUnion__disjoint,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [C6: set @ ( set @ B ),G: B > A] :
( ! [X4: set @ B] :
( ( member @ ( set @ B ) @ X4 @ C6 )
=> ( finite_finite2 @ B @ X4 ) )
=> ( ! [X4: set @ B] :
( ( member @ ( set @ B ) @ X4 @ C6 )
=> ! [Xa4: set @ B] :
( ( member @ ( set @ B ) @ Xa4 @ C6 )
=> ( ( X4 != Xa4 )
=> ( ( inf_inf @ ( set @ B ) @ X4 @ Xa4 )
= ( bot_bot @ ( set @ B ) ) ) ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( complete_Sup_Sup @ ( set @ B ) @ C6 ) )
= ( comp @ ( ( set @ B ) > A ) @ ( ( set @ ( set @ B ) ) > A ) @ ( B > A ) @ ( groups7121269368397514597t_prod @ ( set @ B ) @ A ) @ ( groups7121269368397514597t_prod @ B @ A ) @ G @ C6 ) ) ) ) ) ).
% prod.Union_disjoint
thf(fact_3461_Collect__split__mono__strong,axiom,
! [B: $tType,A: $tType,X7: set @ A,A5: set @ ( product_prod @ A @ B ),Y9: set @ B,P: A > B > $o,Q: A > B > $o] :
( ( X7
= ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ A5 ) )
=> ( ( Y9
= ( image2 @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ A5 ) )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ X7 )
=> ! [Xa4: B] :
( ( member @ B @ Xa4 @ Y9 )
=> ( ( P @ X4 @ Xa4 )
=> ( Q @ X4 @ Xa4 ) ) ) )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ A5 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ P ) ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ A5 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ Q ) ) ) ) ) ) ) ).
% Collect_split_mono_strong
thf(fact_3462_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_3463_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_3464_unset__bit__eq__and__not,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( bit_se2638667681897837118et_bit @ A )
= ( ^ [N5: nat,A6: A] : ( bit_se5824344872417868541ns_and @ A @ A6 @ ( bit_ri4277139882892585799ns_not @ A @ ( bit_se4730199178511100633sh_bit @ A @ N5 @ ( one_one @ A ) ) ) ) ) ) ) ).
% unset_bit_eq_and_not
thf(fact_3465_Sup_OSUP__identity__eq,axiom,
! [A: $tType,Sup: ( set @ A ) > A,A5: set @ A] :
( ( Sup
@ ( image2 @ A @ A
@ ^ [X2: A] : X2
@ A5 ) )
= ( Sup @ A5 ) ) ).
% Sup.SUP_identity_eq
thf(fact_3466_Inf_OINF__identity__eq,axiom,
! [A: $tType,Inf: ( set @ A ) > A,A5: set @ A] :
( ( Inf
@ ( image2 @ A @ A
@ ^ [X2: A] : X2
@ A5 ) )
= ( Inf @ A5 ) ) ).
% Inf.INF_identity_eq
thf(fact_3467_bit_Ocompl__unique,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X: A,Y3: A] :
( ( ( bit_se5824344872417868541ns_and @ A @ X @ Y3 )
= ( zero_zero @ A ) )
=> ( ( ( bit_se1065995026697491101ons_or @ A @ X @ Y3 )
= ( uminus_uminus @ A @ ( one_one @ A ) ) )
=> ( ( bit_ri4277139882892585799ns_not @ A @ X )
= Y3 ) ) ) ) ).
% bit.compl_unique
thf(fact_3468_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_3469_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_3470_bit__not__iff__eq,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A3: A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( bit_ri4277139882892585799ns_not @ A @ A3 ) @ N )
= ( ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N )
!= ( zero_zero @ A ) )
& ~ ( bit_se5641148757651400278ts_bit @ A @ A3 @ N ) ) ) ) ).
% bit_not_iff_eq
thf(fact_3471_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_3472_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_3473_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
@ ^ [X2: B] : C2
@ A5 ) )
= C2 ) ) ) ).
% cINF_const
thf(fact_3474_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
@ ^ [X2: B] : C2
@ A5 ) )
= C2 ) ) ) ).
% cSUP_const
thf(fact_3475_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_3476_UN__UN__split__split__eq,axiom,
! [A: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A5: B > C > D > E > ( set @ A ),Y9: set @ ( product_prod @ D @ E ),X7: 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 ) ) @ Y9 ) ) )
@ X7 ) )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ ( product_prod @ B @ C ) @ ( set @ A )
@ ^ [X2: product_prod @ B @ C] :
( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ ( product_prod @ D @ E ) @ ( set @ A )
@ ^ [Y: 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 ) @ Y )
@ X2 )
@ Y9 ) )
@ X7 ) ) ) ).
% UN_UN_split_split_eq
thf(fact_3477_finite__mono__strict__prefix__implies__finite__fixpoint,axiom,
! [A: $tType,F: nat > ( set @ A ),S3: set @ A] :
( ! [I3: nat] : ( ord_less_eq @ ( set @ A ) @ ( F @ I3 ) @ S3 )
=> ( ( finite_finite2 @ A @ S3 )
=> ( ? [N10: nat] :
( ! [N2: nat] :
( ( ord_less_eq @ nat @ N2 @ N10 )
=> ! [M4: nat] :
( ( ord_less_eq @ nat @ M4 @ N10 )
=> ( ( ord_less @ nat @ M4 @ N2 )
=> ( ord_less @ ( set @ A ) @ ( F @ M4 ) @ ( F @ N2 ) ) ) ) )
& ! [N2: nat] :
( ( ord_less_eq @ nat @ N10 @ N2 )
=> ( ( F @ N10 )
= ( F @ N2 ) ) ) )
=> ( ( F @ ( finite_card @ A @ S3 ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ F @ ( top_top @ ( set @ nat ) ) ) ) ) ) ) ) ).
% finite_mono_strict_prefix_implies_finite_fixpoint
thf(fact_3478_finite__option__UNIV,axiom,
! [A: $tType] :
( ( finite_finite2 @ ( option @ A ) @ ( top_top @ ( set @ ( option @ A ) ) ) )
= ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ).
% finite_option_UNIV
thf(fact_3479_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_3480_Sup__nat__empty,axiom,
( ( complete_Sup_Sup @ nat @ ( bot_bot @ ( set @ nat ) ) )
= ( zero_zero @ nat ) ) ).
% Sup_nat_empty
thf(fact_3481_Collect__const,axiom,
! [A: $tType,P: $o] :
( ( P
=> ( ( collect @ A
@ ^ [S2: A] : P )
= ( top_top @ ( set @ A ) ) ) )
& ( ~ P
=> ( ( collect @ A
@ ^ [S2: A] : P )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Collect_const
thf(fact_3482_finite__Collect__not,axiom,
! [A: $tType,P: A > $o] :
( ( finite_finite2 @ A @ ( collect @ A @ P ) )
=> ( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X2: A] :
~ ( P @ X2 ) ) )
= ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ) ).
% finite_Collect_not
thf(fact_3483_surj__plus,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A3: A] :
( ( image2 @ A @ A @ ( plus_plus @ A @ A3 ) @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ) ).
% surj_plus
thf(fact_3484_Gcd__UNIV,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ( ( gcd_Gcd @ A @ ( top_top @ ( set @ A ) ) )
= ( one_one @ A ) ) ) ).
% Gcd_UNIV
thf(fact_3485_surj__diff__right,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A3: A] :
( ( image2 @ A @ A
@ ^ [X2: A] : ( minus_minus @ A @ X2 @ A3 )
@ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ) ).
% surj_diff_right
thf(fact_3486_INF__top,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ B] :
( ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [X2: B] : ( top_top @ A )
@ A5 ) )
= ( top_top @ A ) ) ) ).
% INF_top
thf(fact_3487_INF__top__conv_I1_J,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [B4: B > A,A5: set @ B] :
( ( ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ B4 @ A5 ) )
= ( top_top @ A ) )
= ( ! [X2: B] :
( ( member @ B @ X2 @ A5 )
=> ( ( B4 @ X2 )
= ( top_top @ A ) ) ) ) ) ) ).
% INF_top_conv(1)
thf(fact_3488_INF__top__conv_I2_J,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [B4: B > A,A5: set @ B] :
( ( ( top_top @ A )
= ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ B4 @ A5 ) ) )
= ( ! [X2: B] :
( ( member @ B @ X2 @ A5 )
=> ( ( B4 @ X2 )
= ( top_top @ A ) ) ) ) ) ) ).
% INF_top_conv(2)
thf(fact_3489_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 ) )
= ( ! [X2: A] :
( ( ord_less @ A @ X2 @ ( top_top @ A ) )
=> ? [Y: B] :
( ( member @ B @ Y @ A5 )
& ( ord_less @ A @ X2 @ ( F @ Y ) ) ) ) ) ) ) ).
% SUP_eq_top_iff
thf(fact_3490_range__constant,axiom,
! [B: $tType,A: $tType,X: A] :
( ( image2 @ B @ A
@ ^ [Uu: B] : X
@ ( top_top @ ( set @ B ) ) )
= ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ).
% range_constant
thf(fact_3491_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 )
@ ^ [Y: B] : C2
@ A5 ) )
= ( top_top @ ( set @ A ) ) ) )
& ( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [Y: B] : C2
@ A5 ) )
= C2 ) ) ) ).
% INT_constant
thf(fact_3492_INT__simps_I1_J,axiom,
! [A: $tType,B: $tType,C6: set @ A,A5: A > ( set @ B ),B4: set @ B] :
( ( ( C6
= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Inf_Inf @ ( set @ B )
@ ( image2 @ A @ ( set @ B )
@ ^ [X2: A] : ( inf_inf @ ( set @ B ) @ ( A5 @ X2 ) @ B4 )
@ C6 ) )
= ( top_top @ ( set @ B ) ) ) )
& ( ( C6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Inf_Inf @ ( set @ B )
@ ( image2 @ A @ ( set @ B )
@ ^ [X2: A] : ( inf_inf @ ( set @ B ) @ ( A5 @ X2 ) @ B4 )
@ C6 ) )
= ( inf_inf @ ( set @ B ) @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ A5 @ C6 ) ) @ B4 ) ) ) ) ).
% INT_simps(1)
thf(fact_3493_INT__simps_I2_J,axiom,
! [C: $tType,D: $tType,C6: set @ D,A5: set @ C,B4: D > ( set @ C )] :
( ( ( C6
= ( bot_bot @ ( set @ D ) ) )
=> ( ( complete_Inf_Inf @ ( set @ C )
@ ( image2 @ D @ ( set @ C )
@ ^ [X2: D] : ( inf_inf @ ( set @ C ) @ A5 @ ( B4 @ X2 ) )
@ C6 ) )
= ( top_top @ ( set @ C ) ) ) )
& ( ( C6
!= ( bot_bot @ ( set @ D ) ) )
=> ( ( complete_Inf_Inf @ ( set @ C )
@ ( image2 @ D @ ( set @ C )
@ ^ [X2: D] : ( inf_inf @ ( set @ C ) @ A5 @ ( B4 @ X2 ) )
@ C6 ) )
= ( inf_inf @ ( set @ C ) @ A5 @ ( complete_Inf_Inf @ ( set @ C ) @ ( image2 @ D @ ( set @ C ) @ B4 @ C6 ) ) ) ) ) ) ).
% INT_simps(2)
thf(fact_3494_INT__simps_I3_J,axiom,
! [E: $tType,F3: $tType,C6: set @ E,A5: E > ( set @ F3 ),B4: set @ F3] :
( ( ( C6
= ( bot_bot @ ( set @ E ) ) )
=> ( ( complete_Inf_Inf @ ( set @ F3 )
@ ( image2 @ E @ ( set @ F3 )
@ ^ [X2: E] : ( minus_minus @ ( set @ F3 ) @ ( A5 @ X2 ) @ B4 )
@ C6 ) )
= ( top_top @ ( set @ F3 ) ) ) )
& ( ( C6
!= ( bot_bot @ ( set @ E ) ) )
=> ( ( complete_Inf_Inf @ ( set @ F3 )
@ ( image2 @ E @ ( set @ F3 )
@ ^ [X2: E] : ( minus_minus @ ( set @ F3 ) @ ( A5 @ X2 ) @ B4 )
@ C6 ) )
= ( minus_minus @ ( set @ F3 ) @ ( complete_Inf_Inf @ ( set @ F3 ) @ ( image2 @ E @ ( set @ F3 ) @ A5 @ C6 ) ) @ B4 ) ) ) ) ).
% INT_simps(3)
thf(fact_3495_INT__simps_I4_J,axiom,
! [G3: $tType,H8: $tType,C6: set @ H8,A5: set @ G3,B4: H8 > ( set @ G3 )] :
( ( ( C6
= ( bot_bot @ ( set @ H8 ) ) )
=> ( ( complete_Inf_Inf @ ( set @ G3 )
@ ( image2 @ H8 @ ( set @ G3 )
@ ^ [X2: H8] : ( minus_minus @ ( set @ G3 ) @ A5 @ ( B4 @ X2 ) )
@ C6 ) )
= ( top_top @ ( set @ G3 ) ) ) )
& ( ( C6
!= ( bot_bot @ ( set @ H8 ) ) )
=> ( ( complete_Inf_Inf @ ( set @ G3 )
@ ( image2 @ H8 @ ( set @ G3 )
@ ^ [X2: H8] : ( minus_minus @ ( set @ G3 ) @ A5 @ ( B4 @ X2 ) )
@ C6 ) )
= ( minus_minus @ ( set @ G3 ) @ A5 @ ( complete_Sup_Sup @ ( set @ G3 ) @ ( image2 @ H8 @ ( set @ G3 ) @ B4 @ C6 ) ) ) ) ) ) ).
% INT_simps(4)
thf(fact_3496_UNIV__option__conv,axiom,
! [A: $tType] :
( ( top_top @ ( set @ ( option @ A ) ) )
= ( insert2 @ ( option @ A ) @ ( none @ A ) @ ( image2 @ A @ ( option @ A ) @ ( some @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ).
% UNIV_option_conv
thf(fact_3497_bot__nat__def,axiom,
( ( bot_bot @ nat )
= ( zero_zero @ nat ) ) ).
% bot_nat_def
thf(fact_3498_surjD,axiom,
! [A: $tType,B: $tType,F: B > A,Y3: A] :
( ( ( image2 @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ? [X4: B] :
( Y3
= ( F @ X4 ) ) ) ).
% surjD
thf(fact_3499_surjE,axiom,
! [A: $tType,B: $tType,F: B > A,Y3: A] :
( ( ( image2 @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ~ ! [X4: B] :
( Y3
!= ( F @ X4 ) ) ) ).
% surjE
thf(fact_3500_surjI,axiom,
! [B: $tType,A: $tType,G: B > A,F: A > B] :
( ! [X4: A] :
( ( G @ ( F @ X4 ) )
= X4 )
=> ( ( image2 @ B @ A @ G @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) ) ) ).
% surjI
thf(fact_3501_surj__def,axiom,
! [B: $tType,A: $tType,F: B > A] :
( ( ( image2 @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
= ( ! [Y: A] :
? [X2: B] :
( Y
= ( F @ X2 ) ) ) ) ).
% surj_def
thf(fact_3502_UNIV__def,axiom,
! [A: $tType] :
( ( top_top @ ( set @ A ) )
= ( collect @ A
@ ^ [X2: A] : $true ) ) ).
% UNIV_def
thf(fact_3503_Inf__unit__def,axiom,
( ( complete_Inf_Inf @ product_unit )
= ( ^ [Uu2: set @ product_unit] : product_Unity ) ) ).
% Inf_unit_def
thf(fact_3504_bot__unit__def,axiom,
( ( bot_bot @ product_unit )
= product_Unity ) ).
% bot_unit_def
thf(fact_3505_Sup__unit__def,axiom,
( ( complete_Sup_Sup @ product_unit )
= ( ^ [Uu2: set @ product_unit] : product_Unity ) ) ).
% Sup_unit_def
thf(fact_3506_inf__unit__def,axiom,
( ( inf_inf @ product_unit )
= ( ^ [Uu2: product_unit,Uv2: product_unit] : product_Unity ) ) ).
% inf_unit_def
thf(fact_3507_inf__nat__def,axiom,
( ( inf_inf @ nat )
= ( ord_min @ nat ) ) ).
% inf_nat_def
thf(fact_3508_INF__INT__eq2,axiom,
! [B: $tType,C: $tType,A: $tType,R: C > ( set @ ( product_prod @ A @ B ) ),S3: set @ C] :
( ( complete_Inf_Inf @ ( A > B > $o )
@ ( image2 @ C @ ( A > B > $o )
@ ^ [I2: C,X2: A,Y: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y ) @ ( R @ I2 ) )
@ S3 ) )
= ( ^ [X2: A,Y: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y ) @ ( complete_Inf_Inf @ ( set @ ( product_prod @ A @ B ) ) @ ( image2 @ C @ ( set @ ( product_prod @ A @ B ) ) @ R @ S3 ) ) ) ) ) ).
% INF_INT_eq2
thf(fact_3509_INF__Int__eq2,axiom,
! [B: $tType,A: $tType,S3: 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 ),X2: A,Y: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y ) @ I2 )
@ S3 ) )
= ( ^ [X2: A,Y: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y ) @ ( complete_Inf_Inf @ ( set @ ( product_prod @ A @ B ) ) @ S3 ) ) ) ) ).
% INF_Int_eq2
thf(fact_3510_SUP__UN__eq2,axiom,
! [B: $tType,C: $tType,A: $tType,R: C > ( set @ ( product_prod @ A @ B ) ),S3: set @ C] :
( ( complete_Sup_Sup @ ( A > B > $o )
@ ( image2 @ C @ ( A > B > $o )
@ ^ [I2: C,X2: A,Y: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y ) @ ( R @ I2 ) )
@ S3 ) )
= ( ^ [X2: A,Y: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y ) @ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) ) @ ( image2 @ C @ ( set @ ( product_prod @ A @ B ) ) @ R @ S3 ) ) ) ) ) ).
% SUP_UN_eq2
thf(fact_3511_SUP__Sup__eq2,axiom,
! [B: $tType,A: $tType,S3: 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 ),X2: A,Y: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y ) @ I2 )
@ S3 ) )
= ( ^ [X2: A,Y: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y ) @ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) ) @ S3 ) ) ) ) ).
% SUP_Sup_eq2
thf(fact_3512_inf__Int__eq2,axiom,
! [B: $tType,A: $tType,R5: set @ ( product_prod @ A @ B ),S3: set @ ( product_prod @ A @ B )] :
( ( inf_inf @ ( A > B > $o )
@ ^ [X2: A,Y: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y ) @ R5 )
@ ^ [X2: A,Y: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y ) @ S3 ) )
= ( ^ [X2: A,Y: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y ) @ ( inf_inf @ ( set @ ( product_prod @ A @ B ) ) @ R5 @ S3 ) ) ) ) ).
% inf_Int_eq2
thf(fact_3513_bot__empty__eq2,axiom,
! [B: $tType,A: $tType] :
( ( bot_bot @ ( A > B > $o ) )
= ( ^ [X2: A,Y: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y ) @ ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% bot_empty_eq2
thf(fact_3514_inf__set__def,axiom,
! [A: $tType] :
( ( inf_inf @ ( set @ A ) )
= ( ^ [A8: set @ A,B7: set @ A] :
( collect @ A
@ ( inf_inf @ ( A > $o )
@ ^ [X2: A] : ( member @ A @ X2 @ A8 )
@ ^ [X2: A] : ( member @ A @ X2 @ B7 ) ) ) ) ) ).
% inf_set_def
thf(fact_3515_inf__Int__eq,axiom,
! [A: $tType,R5: set @ A,S3: set @ A] :
( ( inf_inf @ ( A > $o )
@ ^ [X2: A] : ( member @ A @ X2 @ R5 )
@ ^ [X2: A] : ( member @ A @ X2 @ S3 ) )
= ( ^ [X2: A] : ( member @ A @ X2 @ ( inf_inf @ ( set @ A ) @ R5 @ S3 ) ) ) ) ).
% inf_Int_eq
thf(fact_3516_SUP__UN__eq,axiom,
! [B: $tType,A: $tType,R: B > ( set @ A ),S3: set @ B] :
( ( complete_Sup_Sup @ ( A > $o )
@ ( image2 @ B @ ( A > $o )
@ ^ [I2: B,X2: A] : ( member @ A @ X2 @ ( R @ I2 ) )
@ S3 ) )
= ( ^ [X2: A] : ( member @ A @ X2 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ R @ S3 ) ) ) ) ) ).
% SUP_UN_eq
thf(fact_3517_SUP__Sup__eq,axiom,
! [A: $tType,S3: set @ ( set @ A )] :
( ( complete_Sup_Sup @ ( A > $o )
@ ( image2 @ ( set @ A ) @ ( A > $o )
@ ^ [I2: set @ A,X2: A] : ( member @ A @ X2 @ I2 )
@ S3 ) )
= ( ^ [X2: A] : ( member @ A @ X2 @ ( complete_Sup_Sup @ ( set @ A ) @ S3 ) ) ) ) ).
% SUP_Sup_eq
thf(fact_3518_INF__INT__eq,axiom,
! [B: $tType,A: $tType,R: B > ( set @ A ),S3: set @ B] :
( ( complete_Inf_Inf @ ( A > $o )
@ ( image2 @ B @ ( A > $o )
@ ^ [I2: B,X2: A] : ( member @ A @ X2 @ ( R @ I2 ) )
@ S3 ) )
= ( ^ [X2: A] : ( member @ A @ X2 @ ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ R @ S3 ) ) ) ) ) ).
% INF_INT_eq
thf(fact_3519_INF__Int__eq,axiom,
! [A: $tType,S3: set @ ( set @ A )] :
( ( complete_Inf_Inf @ ( A > $o )
@ ( image2 @ ( set @ A ) @ ( A > $o )
@ ^ [I2: set @ A,X2: A] : ( member @ A @ X2 @ I2 )
@ S3 ) )
= ( ^ [X2: A] : ( member @ A @ X2 @ ( complete_Inf_Inf @ ( set @ A ) @ S3 ) ) ) ) ).
% INF_Int_eq
thf(fact_3520_Sup__set__def,axiom,
! [A: $tType] :
( ( complete_Sup_Sup @ ( set @ A ) )
= ( ^ [A8: set @ ( set @ A )] :
( collect @ A
@ ^ [X2: A] : ( complete_Sup_Sup @ $o @ ( image2 @ ( set @ A ) @ $o @ ( member @ A @ X2 ) @ A8 ) ) ) ) ) ).
% Sup_set_def
thf(fact_3521_Inf__set__def,axiom,
! [A: $tType] :
( ( complete_Inf_Inf @ ( set @ A ) )
= ( ^ [A8: set @ ( set @ A )] :
( collect @ A
@ ^ [X2: A] : ( complete_Inf_Inf @ $o @ ( image2 @ ( set @ A ) @ $o @ ( member @ A @ X2 ) @ A8 ) ) ) ) ) ).
% Inf_set_def
thf(fact_3522_range__composition,axiom,
! [A: $tType,C: $tType,B: $tType,F: C > A,G: B > C] :
( ( image2 @ B @ A
@ ^ [X2: B] : ( F @ ( G @ X2 ) )
@ ( top_top @ ( set @ B ) ) )
= ( image2 @ C @ A @ F @ ( image2 @ B @ C @ G @ ( top_top @ ( set @ B ) ) ) ) ) ).
% range_composition
thf(fact_3523_rangeE,axiom,
! [A: $tType,B: $tType,B2: A,F: B > A] :
( ( member @ A @ B2 @ ( image2 @ B @ A @ F @ ( top_top @ ( set @ B ) ) ) )
=> ~ ! [X4: B] :
( B2
!= ( F @ X4 ) ) ) ).
% rangeE
thf(fact_3524_INTER__UNIV__conv_I1_J,axiom,
! [A: $tType,B: $tType,B4: B > ( set @ A ),A5: set @ B] :
( ( ( top_top @ ( set @ A ) )
= ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B4 @ A5 ) ) )
= ( ! [X2: B] :
( ( member @ B @ X2 @ A5 )
=> ( ( B4 @ X2 )
= ( top_top @ ( set @ A ) ) ) ) ) ) ).
% INTER_UNIV_conv(1)
thf(fact_3525_INTER__UNIV__conv_I2_J,axiom,
! [A: $tType,B: $tType,B4: B > ( set @ A ),A5: set @ B] :
( ( ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B4 @ A5 ) )
= ( top_top @ ( set @ A ) ) )
= ( ! [X2: B] :
( ( member @ B @ X2 @ A5 )
=> ( ( B4 @ X2 )
= ( top_top @ ( set @ A ) ) ) ) ) ) ).
% INTER_UNIV_conv(2)
thf(fact_3526_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_3527_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_3528_INF__SUP,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [P: C > B > A] :
( ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [Y: B] :
( complete_Sup_Sup @ A
@ ( image2 @ C @ A
@ ^ [X2: C] : ( P @ X2 @ Y )
@ ( 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
@ ^ [X2: B] : ( P @ ( F2 @ X2 ) @ X2 )
@ ( top_top @ ( set @ B ) ) ) )
@ ( top_top @ ( set @ ( B > C ) ) ) ) ) ) ) ).
% INF_SUP
thf(fact_3529_SUP__INF,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [P: C > B > A] :
( ( complete_Sup_Sup @ A
@ ( image2 @ B @ A
@ ^ [Y: B] :
( complete_Inf_Inf @ A
@ ( image2 @ C @ A
@ ^ [X2: C] : ( P @ X2 @ Y )
@ ( top_top @ ( set @ C ) ) ) )
@ ( top_top @ ( set @ B ) ) ) )
= ( complete_Inf_Inf @ A
@ ( image2 @ ( B > C ) @ A
@ ^ [X2: B > C] :
( complete_Sup_Sup @ A
@ ( image2 @ B @ A
@ ^ [Y: B] : ( P @ ( X2 @ Y ) @ Y )
@ ( top_top @ ( set @ B ) ) ) )
@ ( top_top @ ( set @ ( B > C ) ) ) ) ) ) ) ).
% SUP_INF
thf(fact_3530_inf__int__def,axiom,
( ( inf_inf @ int )
= ( ord_min @ int ) ) ).
% inf_int_def
thf(fact_3531_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_3532_surj__fun__eq,axiom,
! [B: $tType,C: $tType,A: $tType,F: B > A,X7: set @ B,G1: A > C,G22: A > C] :
( ( ( image2 @ B @ A @ F @ X7 )
= ( top_top @ ( set @ A ) ) )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ X7 )
=> ( ( comp @ A @ C @ B @ G1 @ F @ X4 )
= ( comp @ A @ C @ B @ G22 @ F @ X4 ) ) )
=> ( G1 = G22 ) ) ) ).
% surj_fun_eq
thf(fact_3533_comp__surj,axiom,
! [B: $tType,A: $tType,C: $tType,F: B > A,G: A > C] :
( ( ( image2 @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ( ( ( image2 @ A @ C @ G @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ C ) ) )
=> ( ( image2 @ B @ C @ ( comp @ A @ C @ B @ G @ F ) @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ C ) ) ) ) ) ).
% comp_surj
thf(fact_3534_fun_Oset__map,axiom,
! [B: $tType,A: $tType,D: $tType,F: A > B,V: D > A] :
( ( image2 @ D @ B @ ( comp @ A @ B @ D @ F @ V ) @ ( top_top @ ( set @ D ) ) )
= ( image2 @ A @ B @ F @ ( image2 @ D @ A @ V @ ( top_top @ ( set @ D ) ) ) ) ) ).
% fun.set_map
thf(fact_3535_fun_Omap__cong,axiom,
! [B: $tType,A: $tType,D: $tType,X: D > A,Ya: D > A,F: A > B,G: A > B] :
( ( X = Ya )
=> ( ! [Z6: A] :
( ( member @ A @ Z6 @ ( image2 @ D @ A @ Ya @ ( top_top @ ( set @ D ) ) ) )
=> ( ( F @ Z6 )
= ( G @ Z6 ) ) )
=> ( ( comp @ A @ B @ D @ F @ X )
= ( comp @ A @ B @ D @ G @ Ya ) ) ) ) ).
% fun.map_cong
thf(fact_3536_fun_Omap__cong0,axiom,
! [B: $tType,A: $tType,D: $tType,X: D > A,F: A > B,G: A > B] :
( ! [Z6: A] :
( ( member @ A @ Z6 @ ( image2 @ D @ A @ X @ ( top_top @ ( set @ D ) ) ) )
=> ( ( F @ Z6 )
= ( G @ Z6 ) ) )
=> ( ( comp @ A @ B @ D @ F @ X )
= ( comp @ A @ B @ D @ G @ X ) ) ) ).
% fun.map_cong0
thf(fact_3537_fun_Oinj__map__strong,axiom,
! [B: $tType,A: $tType,D: $tType,X: D > A,Xa: D > A,F: A > B,Fa: A > B] :
( ! [Z6: A,Za: A] :
( ( member @ A @ Z6 @ ( image2 @ D @ A @ X @ ( top_top @ ( set @ D ) ) ) )
=> ( ( member @ A @ Za @ ( image2 @ D @ A @ Xa @ ( top_top @ ( set @ D ) ) ) )
=> ( ( ( F @ Z6 )
= ( Fa @ Za ) )
=> ( Z6 = Za ) ) ) )
=> ( ( ( comp @ A @ B @ D @ F @ X )
= ( comp @ A @ B @ D @ Fa @ Xa ) )
=> ( X = Xa ) ) ) ).
% fun.inj_map_strong
thf(fact_3538_surj__id,axiom,
! [A: $tType] :
( ( image2 @ A @ A @ ( id @ A ) @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% surj_id
thf(fact_3539_finite__range__Some,axiom,
! [A: $tType] :
( ( finite_finite2 @ ( option @ A ) @ ( image2 @ A @ ( option @ A ) @ ( some @ A ) @ ( top_top @ ( set @ A ) ) ) )
= ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ).
% finite_range_Some
thf(fact_3540_Inf__INT__eq2,axiom,
! [B: $tType,A: $tType] :
( ( complete_Inf_Inf @ ( A > B > $o ) )
= ( ^ [S8: set @ ( A > B > $o ),X2: A,Y: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y ) @ ( 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 ) @ S8 ) ) ) ) ) ) ).
% Inf_INT_eq2
thf(fact_3541_Sup__SUP__eq2,axiom,
! [B: $tType,A: $tType] :
( ( complete_Sup_Sup @ ( A > B > $o ) )
= ( ^ [S8: set @ ( A > B > $o ),X2: A,Y: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y ) @ ( 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 ) @ S8 ) ) ) ) ) ) ).
% Sup_SUP_eq2
thf(fact_3542_notin__range__Some,axiom,
! [A: $tType,X: option @ A] :
( ( ~ ( member @ ( option @ A ) @ X @ ( image2 @ A @ ( option @ A ) @ ( some @ A ) @ ( top_top @ ( set @ A ) ) ) ) )
= ( X
= ( none @ A ) ) ) ).
% notin_range_Some
thf(fact_3543_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
@ ^ [X2: B] : ( F @ ( G @ X2 ) )
@ ( top_top @ ( set @ B ) ) ) ) ) ).
% finite_range_imageI
thf(fact_3544_INT__empty,axiom,
! [B: $tType,A: $tType,B4: B > ( set @ A )] :
( ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B4 @ ( bot_bot @ ( set @ B ) ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% INT_empty
thf(fact_3545_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
@ ^ [Y: B] : C2
@ A5 ) )
= ( top_top @ A ) ) )
& ( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [Y: B] : C2
@ A5 ) )
= C2 ) ) ) ) ).
% INF_constant
thf(fact_3546_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_3547_surj__Compl__image__subset,axiom,
! [A: $tType,B: $tType,F: B > A,A5: set @ B] :
( ( ( image2 @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ ( image2 @ B @ A @ F @ A5 ) ) @ ( image2 @ B @ A @ F @ ( uminus_uminus @ ( set @ B ) @ A5 ) ) ) ) ).
% surj_Compl_image_subset
thf(fact_3548_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_3549_INT__extend__simps_I3_J,axiom,
! [F3: $tType,E: $tType,C6: set @ E,A5: E > ( set @ F3 ),B4: set @ F3] :
( ( ( C6
= ( bot_bot @ ( set @ E ) ) )
=> ( ( minus_minus @ ( set @ F3 ) @ ( complete_Inf_Inf @ ( set @ F3 ) @ ( image2 @ E @ ( set @ F3 ) @ A5 @ C6 ) ) @ B4 )
= ( minus_minus @ ( set @ F3 ) @ ( top_top @ ( set @ F3 ) ) @ B4 ) ) )
& ( ( C6
!= ( bot_bot @ ( set @ E ) ) )
=> ( ( minus_minus @ ( set @ F3 ) @ ( complete_Inf_Inf @ ( set @ F3 ) @ ( image2 @ E @ ( set @ F3 ) @ A5 @ C6 ) ) @ B4 )
= ( complete_Inf_Inf @ ( set @ F3 )
@ ( image2 @ E @ ( set @ F3 )
@ ^ [X2: E] : ( minus_minus @ ( set @ F3 ) @ ( A5 @ X2 ) @ B4 )
@ C6 ) ) ) ) ) ).
% INT_extend_simps(3)
thf(fact_3550_UN__UN__finite__eq,axiom,
! [A: $tType,A5: nat > ( set @ A )] :
( ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ nat @ ( set @ A )
@ ^ [N5: nat] : ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ A5 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N5 ) ) )
@ ( top_top @ ( set @ nat ) ) ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ A5 @ ( top_top @ ( set @ nat ) ) ) ) ) ).
% UN_UN_finite_eq
thf(fact_3551_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_3552_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_3553_UN__finite2__eq,axiom,
! [A: $tType,A5: nat > ( set @ A ),B4: 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 ) @ B4 @ ( 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 ) @ B4 @ ( top_top @ ( set @ nat ) ) ) ) ) ) ).
% UN_finite2_eq
thf(fact_3554_UN__constant__eq,axiom,
! [A: $tType,B: $tType,A3: A,A5: set @ A,F: A > ( set @ B ),C2: set @ B] :
( ( member @ A @ A3 @ A5 )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ A5 )
=> ( ( F @ X4 )
= C2 ) )
=> ( ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ F @ A5 ) )
= C2 ) ) ) ).
% UN_constant_eq
thf(fact_3555_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_3556_UN__finite2__subset,axiom,
! [A: $tType,A5: nat > ( set @ A ),B4: 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 ) @ B4 @ ( 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 ) @ B4 @ ( top_top @ ( set @ nat ) ) ) ) ) ) ).
% UN_finite2_subset
thf(fact_3557_range__mod,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( image2 @ nat @ nat
@ ^ [M5: nat] : ( modulo_modulo @ nat @ M5 @ N )
@ ( top_top @ ( set @ nat ) ) )
= ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% range_mod
thf(fact_3558_INF__filter__not__bot,axiom,
! [I7: $tType,A: $tType,B4: set @ I7,F6: I7 > ( filter @ A )] :
( ! [X10: set @ I7] :
( ( ord_less_eq @ ( set @ I7 ) @ X10 @ B4 )
=> ( ( finite_finite2 @ I7 @ X10 )
=> ( ( complete_Inf_Inf @ ( filter @ A ) @ ( image2 @ I7 @ ( filter @ A ) @ F6 @ X10 ) )
!= ( bot_bot @ ( filter @ A ) ) ) ) )
=> ( ( complete_Inf_Inf @ ( filter @ A ) @ ( image2 @ I7 @ ( filter @ A ) @ F6 @ B4 ) )
!= ( bot_bot @ ( filter @ A ) ) ) ) ).
% INF_filter_not_bot
thf(fact_3559_measure__function__int,axiom,
fun_is_measure @ int @ ( comp @ int @ nat @ int @ nat2 @ ( abs_abs @ int ) ) ).
% measure_function_int
thf(fact_3560_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_3561_set__list__bind,axiom,
! [A: $tType,B: $tType,Xs2: list @ B,F: B > ( list @ A )] :
( ( set2 @ A @ ( bind @ B @ A @ Xs2 @ F ) )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [X2: B] : ( set2 @ A @ ( F @ X2 ) )
@ ( set2 @ B @ Xs2 ) ) ) ) ).
% set_list_bind
thf(fact_3562_UN__le__eq__Un0,axiom,
! [A: $tType,M2: nat > ( set @ A ),N: nat] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ M2 @ ( set_ord_atMost @ nat @ N ) ) )
= ( sup_sup @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ M2 @ ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ N ) ) ) @ ( M2 @ ( zero_zero @ nat ) ) ) ) ).
% UN_le_eq_Un0
thf(fact_3563_card__UNIV__unit,axiom,
( ( finite_card @ product_unit @ ( top_top @ ( set @ product_unit ) ) )
= ( one_one @ nat ) ) ).
% card_UNIV_unit
thf(fact_3564_Collect__const__case__prod,axiom,
! [B: $tType,A: $tType,P: $o] :
( ( P
=> ( ( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [A6: A,B5: B] : P ) )
= ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) )
& ( ~ P
=> ( ( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [A6: A,B5: B] : P ) )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% Collect_const_case_prod
thf(fact_3565_range__abs__Nats,axiom,
( ( image2 @ int @ int @ ( abs_abs @ int ) @ ( top_top @ ( set @ int ) ) )
= ( semiring_1_Nats @ int ) ) ).
% range_abs_Nats
thf(fact_3566_SUP2__I,axiom,
! [B: $tType,A: $tType,C: $tType,A3: A,A5: set @ A,B4: A > B > C > $o,B2: B,C2: C] :
( ( member @ A @ A3 @ A5 )
=> ( ( B4 @ A3 @ B2 @ C2 )
=> ( complete_Sup_Sup @ ( B > C > $o ) @ ( image2 @ A @ ( B > C > $o ) @ B4 @ A5 ) @ B2 @ C2 ) ) ) ).
% SUP2_I
thf(fact_3567_INF1__I,axiom,
! [A: $tType,B: $tType,A5: set @ A,B4: A > B > $o,B2: B] :
( ! [X4: A] :
( ( member @ A @ X4 @ A5 )
=> ( B4 @ X4 @ B2 ) )
=> ( complete_Inf_Inf @ ( B > $o ) @ ( image2 @ A @ ( B > $o ) @ B4 @ A5 ) @ B2 ) ) ).
% INF1_I
thf(fact_3568_INF2__I,axiom,
! [B: $tType,A: $tType,C: $tType,A5: set @ A,B4: A > B > C > $o,B2: B,C2: C] :
( ! [X4: A] :
( ( member @ A @ X4 @ A5 )
=> ( B4 @ X4 @ B2 @ C2 ) )
=> ( complete_Inf_Inf @ ( B > C > $o ) @ ( image2 @ A @ ( B > C > $o ) @ B4 @ A5 ) @ B2 @ C2 ) ) ).
% INF2_I
thf(fact_3569_SUP1__I,axiom,
! [A: $tType,B: $tType,A3: A,A5: set @ A,B4: A > B > $o,B2: B] :
( ( member @ A @ A3 @ A5 )
=> ( ( B4 @ A3 @ B2 )
=> ( complete_Sup_Sup @ ( B > $o ) @ ( image2 @ A @ ( B > $o ) @ B4 @ A5 ) @ B2 ) ) ) ).
% SUP1_I
thf(fact_3570_if__image__distrib,axiom,
! [A: $tType,B: $tType,P: B > $o,F: B > A,G: B > A,S3: set @ B] :
( ( image2 @ B @ A
@ ^ [X2: B] : ( if @ A @ ( P @ X2 ) @ ( F @ X2 ) @ ( G @ X2 ) )
@ S3 )
= ( sup_sup @ ( set @ A ) @ ( image2 @ B @ A @ F @ ( inf_inf @ ( set @ B ) @ S3 @ ( collect @ B @ P ) ) )
@ ( image2 @ B @ A @ G
@ ( inf_inf @ ( set @ B ) @ S3
@ ( collect @ B
@ ^ [X2: B] :
~ ( P @ X2 ) ) ) ) ) ) ).
% if_image_distrib
thf(fact_3571_UN__Un,axiom,
! [A: $tType,B: $tType,M2: B > ( set @ A ),A5: set @ B,B4: set @ B] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ M2 @ ( sup_sup @ ( set @ B ) @ A5 @ B4 ) ) )
= ( sup_sup @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ M2 @ A5 ) ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ M2 @ B4 ) ) ) ) ).
% UN_Un
thf(fact_3572_UN__simps_I3_J,axiom,
! [E: $tType,F3: $tType,C6: set @ F3,A5: set @ E,B4: F3 > ( set @ E )] :
( ( ( C6
= ( bot_bot @ ( set @ F3 ) ) )
=> ( ( complete_Sup_Sup @ ( set @ E )
@ ( image2 @ F3 @ ( set @ E )
@ ^ [X2: F3] : ( sup_sup @ ( set @ E ) @ A5 @ ( B4 @ X2 ) )
@ C6 ) )
= ( bot_bot @ ( set @ E ) ) ) )
& ( ( C6
!= ( bot_bot @ ( set @ F3 ) ) )
=> ( ( complete_Sup_Sup @ ( set @ E )
@ ( image2 @ F3 @ ( set @ E )
@ ^ [X2: F3] : ( sup_sup @ ( set @ E ) @ A5 @ ( B4 @ X2 ) )
@ C6 ) )
= ( sup_sup @ ( set @ E ) @ A5 @ ( complete_Sup_Sup @ ( set @ E ) @ ( image2 @ F3 @ ( set @ E ) @ B4 @ C6 ) ) ) ) ) ) ).
% UN_simps(3)
thf(fact_3573_UN__simps_I2_J,axiom,
! [C: $tType,D: $tType,C6: set @ C,A5: C > ( set @ D ),B4: set @ D] :
( ( ( C6
= ( bot_bot @ ( set @ C ) ) )
=> ( ( complete_Sup_Sup @ ( set @ D )
@ ( image2 @ C @ ( set @ D )
@ ^ [X2: C] : ( sup_sup @ ( set @ D ) @ ( A5 @ X2 ) @ B4 )
@ C6 ) )
= ( bot_bot @ ( set @ D ) ) ) )
& ( ( C6
!= ( bot_bot @ ( set @ C ) ) )
=> ( ( complete_Sup_Sup @ ( set @ D )
@ ( image2 @ C @ ( set @ D )
@ ^ [X2: C] : ( sup_sup @ ( set @ D ) @ ( A5 @ X2 ) @ B4 )
@ C6 ) )
= ( sup_sup @ ( set @ D ) @ ( complete_Sup_Sup @ ( set @ D ) @ ( image2 @ C @ ( set @ D ) @ A5 @ C6 ) ) @ B4 ) ) ) ) ).
% UN_simps(2)
thf(fact_3574_UN__insert,axiom,
! [A: $tType,B: $tType,B4: B > ( set @ A ),A3: B,A5: set @ B] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B4 @ ( insert2 @ B @ A3 @ A5 ) ) )
= ( sup_sup @ ( set @ A ) @ ( B4 @ A3 ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B4 @ A5 ) ) ) ) ).
% UN_insert
thf(fact_3575_INF__filter__bot__base,axiom,
! [B: $tType,A: $tType,I5: set @ A,F6: A > ( filter @ B )] :
( ! [I3: A] :
( ( member @ A @ I3 @ I5 )
=> ! [J2: A] :
( ( member @ A @ J2 @ I5 )
=> ? [X5: A] :
( ( member @ A @ X5 @ I5 )
& ( ord_less_eq @ ( filter @ B ) @ ( F6 @ X5 ) @ ( inf_inf @ ( filter @ B ) @ ( F6 @ I3 ) @ ( F6 @ J2 ) ) ) ) ) )
=> ( ( ( complete_Inf_Inf @ ( filter @ B ) @ ( image2 @ A @ ( filter @ B ) @ F6 @ I5 ) )
= ( bot_bot @ ( filter @ B ) ) )
= ( ? [X2: A] :
( ( member @ A @ X2 @ I5 )
& ( ( F6 @ X2 )
= ( bot_bot @ ( filter @ B ) ) ) ) ) ) ) ).
% INF_filter_bot_base
thf(fact_3576_SUP__union,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [M2: B > A,A5: set @ B,B4: set @ B] :
( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ M2 @ ( sup_sup @ ( set @ B ) @ A5 @ B4 ) ) )
= ( sup_sup @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ M2 @ A5 ) ) @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ M2 @ B4 ) ) ) ) ) ).
% SUP_union
thf(fact_3577_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_3578_INF1__D,axiom,
! [B: $tType,A: $tType,B4: B > A > $o,A5: set @ B,B2: A,A3: B] :
( ( complete_Inf_Inf @ ( A > $o ) @ ( image2 @ B @ ( A > $o ) @ B4 @ A5 ) @ B2 )
=> ( ( member @ B @ A3 @ A5 )
=> ( B4 @ A3 @ B2 ) ) ) ).
% INF1_D
thf(fact_3579_INF1__E,axiom,
! [A: $tType,B: $tType,B4: B > A > $o,A5: set @ B,B2: A,A3: B] :
( ( complete_Inf_Inf @ ( A > $o ) @ ( image2 @ B @ ( A > $o ) @ B4 @ A5 ) @ B2 )
=> ( ~ ( B4 @ A3 @ B2 )
=> ~ ( member @ B @ A3 @ A5 ) ) ) ).
% INF1_E
thf(fact_3580_INF2__D,axiom,
! [A: $tType,C: $tType,B: $tType,B4: C > A > B > $o,A5: set @ C,B2: A,C2: B,A3: C] :
( ( complete_Inf_Inf @ ( A > B > $o ) @ ( image2 @ C @ ( A > B > $o ) @ B4 @ A5 ) @ B2 @ C2 )
=> ( ( member @ C @ A3 @ A5 )
=> ( B4 @ A3 @ B2 @ C2 ) ) ) ).
% INF2_D
thf(fact_3581_INF2__E,axiom,
! [B: $tType,A: $tType,C: $tType,B4: C > A > B > $o,A5: set @ C,B2: A,C2: B,A3: C] :
( ( complete_Inf_Inf @ ( A > B > $o ) @ ( image2 @ C @ ( A > B > $o ) @ B4 @ A5 ) @ B2 @ C2 )
=> ( ~ ( B4 @ A3 @ B2 @ C2 )
=> ~ ( member @ C @ A3 @ A5 ) ) ) ).
% INF2_E
thf(fact_3582_SUP1__E,axiom,
! [B: $tType,A: $tType,B4: B > A > $o,A5: set @ B,B2: A] :
( ( complete_Sup_Sup @ ( A > $o ) @ ( image2 @ B @ ( A > $o ) @ B4 @ A5 ) @ B2 )
=> ~ ! [X4: B] :
( ( member @ B @ X4 @ A5 )
=> ~ ( B4 @ X4 @ B2 ) ) ) ).
% SUP1_E
thf(fact_3583_SUP2__E,axiom,
! [A: $tType,C: $tType,B: $tType,B4: C > A > B > $o,A5: set @ C,B2: A,C2: B] :
( ( complete_Sup_Sup @ ( A > B > $o ) @ ( image2 @ C @ ( A > B > $o ) @ B4 @ A5 ) @ B2 @ C2 )
=> ~ ! [X4: C] :
( ( member @ C @ X4 @ A5 )
=> ~ ( B4 @ X4 @ B2 @ C2 ) ) ) ).
% SUP2_E
thf(fact_3584_Un__eq__UN,axiom,
! [A: $tType] :
( ( sup_sup @ ( set @ A ) )
= ( ^ [A8: set @ A,B7: set @ A] :
( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ $o @ ( set @ A )
@ ^ [B5: $o] : ( if @ ( set @ A ) @ B5 @ A8 @ B7 )
@ ( top_top @ ( set @ $o ) ) ) ) ) ) ).
% Un_eq_UN
thf(fact_3585_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_3586_top__empty__eq2,axiom,
! [B: $tType,A: $tType] :
( ( top_top @ ( A > B > $o ) )
= ( ^ [X2: A,Y: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% top_empty_eq2
thf(fact_3587_top__unit__def,axiom,
( ( top_top @ product_unit )
= product_Unity ) ).
% top_unit_def
thf(fact_3588_Un__def,axiom,
! [A: $tType] :
( ( sup_sup @ ( set @ A ) )
= ( ^ [A8: set @ A,B7: set @ A] :
( collect @ A
@ ^ [X2: A] :
( ( member @ A @ X2 @ A8 )
| ( member @ A @ X2 @ B7 ) ) ) ) ) ).
% Un_def
thf(fact_3589_Collect__disj__eq,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( collect @ A
@ ^ [X2: A] :
( ( P @ X2 )
| ( Q @ X2 ) ) )
= ( sup_sup @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% Collect_disj_eq
thf(fact_3590_UNIV__unit,axiom,
( ( top_top @ ( set @ product_unit ) )
= ( insert2 @ product_unit @ product_Unity @ ( bot_bot @ ( set @ product_unit ) ) ) ) ).
% UNIV_unit
thf(fact_3591_measure__fst,axiom,
! [B: $tType,A: $tType,F: A > nat] :
( ( fun_is_measure @ A @ F )
=> ( fun_is_measure @ ( product_prod @ A @ B )
@ ^ [P5: product_prod @ A @ B] : ( F @ ( product_fst @ A @ B @ P5 ) ) ) ) ).
% measure_fst
thf(fact_3592_measure__snd,axiom,
! [A: $tType,B: $tType,F: A > nat] :
( ( fun_is_measure @ A @ F )
=> ( fun_is_measure @ ( product_prod @ B @ A )
@ ^ [P5: product_prod @ B @ A] : ( F @ ( product_snd @ B @ A @ P5 ) ) ) ) ).
% measure_snd
thf(fact_3593_insert__def,axiom,
! [A: $tType] :
( ( insert2 @ A )
= ( ^ [A6: A] :
( sup_sup @ ( set @ A )
@ ( collect @ A
@ ^ [X2: A] : X2 = A6 ) ) ) ) ).
% insert_def
thf(fact_3594_infinite__UNIV__int,axiom,
~ ( finite_finite2 @ int @ ( top_top @ ( set @ int ) ) ) ).
% infinite_UNIV_int
thf(fact_3595_Collect__imp__eq,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( collect @ A
@ ^ [X2: A] :
( ( P @ X2 )
=> ( Q @ X2 ) ) )
= ( sup_sup @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ ( collect @ A @ P ) ) @ ( collect @ A @ Q ) ) ) ).
% Collect_imp_eq
thf(fact_3596_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_3597_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
@ ^ [A6: B] : ( sup_sup @ A @ ( F @ A6 ) @ ( G @ A6 ) )
@ A5 ) ) ) ) ).
% complete_lattice_class.SUP_sup_distrib
thf(fact_3598_INF__sup__distrib2,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [F: B > A,A5: set @ B,G: C > A,B4: set @ C] :
( ( sup_sup @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F @ A5 ) ) @ ( complete_Inf_Inf @ A @ ( image2 @ C @ A @ G @ B4 ) ) )
= ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [A6: B] :
( complete_Inf_Inf @ A
@ ( image2 @ C @ A
@ ^ [B5: C] : ( sup_sup @ A @ ( F @ A6 ) @ ( G @ B5 ) )
@ B4 ) )
@ A5 ) ) ) ) ).
% INF_sup_distrib2
thf(fact_3599_sup__INF,axiom,
! [A: $tType,B: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [A3: A,F: B > A,B4: set @ B] :
( ( sup_sup @ A @ A3 @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F @ B4 ) ) )
= ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [B5: B] : ( sup_sup @ A @ A3 @ ( F @ B5 ) )
@ B4 ) ) ) ) ).
% sup_INF
thf(fact_3600_Inf__sup,axiom,
! [A: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [B4: set @ A,A3: A] :
( ( sup_sup @ A @ ( complete_Inf_Inf @ A @ B4 ) @ A3 )
= ( complete_Inf_Inf @ A
@ ( image2 @ A @ A
@ ^ [B5: A] : ( sup_sup @ A @ B5 @ A3 )
@ B4 ) ) ) ) ).
% Inf_sup
thf(fact_3601_INF__sup,axiom,
! [A: $tType,B: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [F: B > A,B4: set @ B,A3: A] :
( ( sup_sup @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F @ B4 ) ) @ A3 )
= ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [B5: B] : ( sup_sup @ A @ ( F @ B5 ) @ A3 )
@ B4 ) ) ) ) ).
% INF_sup
thf(fact_3602_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_3603_UN__Un__distrib,axiom,
! [A: $tType,B: $tType,A5: B > ( set @ A ),B4: B > ( set @ A ),I5: set @ B] :
( ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [I2: B] : ( sup_sup @ ( set @ A ) @ ( A5 @ I2 ) @ ( B4 @ 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 ) @ B4 @ I5 ) ) ) ) ).
% UN_Un_distrib
thf(fact_3604_Un__Union__image,axiom,
! [A: $tType,B: $tType,A5: B > ( set @ A ),B4: B > ( set @ A ),C6: set @ B] :
( ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [X2: B] : ( sup_sup @ ( set @ A ) @ ( A5 @ X2 ) @ ( B4 @ X2 ) )
@ 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 ) @ B4 @ C6 ) ) ) ) ).
% Un_Union_image
thf(fact_3605_INT__extend__simps_I7_J,axiom,
! [M8: $tType,N9: $tType,A5: set @ M8,B4: N9 > ( set @ M8 ),C6: set @ N9] :
( ( sup_sup @ ( set @ M8 ) @ A5 @ ( complete_Inf_Inf @ ( set @ M8 ) @ ( image2 @ N9 @ ( set @ M8 ) @ B4 @ C6 ) ) )
= ( complete_Inf_Inf @ ( set @ M8 )
@ ( image2 @ N9 @ ( set @ M8 )
@ ^ [X2: N9] : ( sup_sup @ ( set @ M8 ) @ A5 @ ( B4 @ X2 ) )
@ C6 ) ) ) ).
% INT_extend_simps(7)
thf(fact_3606_INT__extend__simps_I6_J,axiom,
! [L5: $tType,K6: $tType,A5: K6 > ( set @ L5 ),C6: set @ K6,B4: set @ L5] :
( ( sup_sup @ ( set @ L5 ) @ ( complete_Inf_Inf @ ( set @ L5 ) @ ( image2 @ K6 @ ( set @ L5 ) @ A5 @ C6 ) ) @ B4 )
= ( complete_Inf_Inf @ ( set @ L5 )
@ ( image2 @ K6 @ ( set @ L5 )
@ ^ [X2: K6] : ( sup_sup @ ( set @ L5 ) @ ( A5 @ X2 ) @ B4 )
@ C6 ) ) ) ).
% INT_extend_simps(6)
thf(fact_3607_Un__INT__distrib,axiom,
! [A: $tType,B: $tType,B4: set @ A,A5: B > ( set @ A ),I5: set @ B] :
( ( sup_sup @ ( set @ A ) @ B4 @ ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ A5 @ I5 ) ) )
= ( complete_Inf_Inf @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [I2: B] : ( sup_sup @ ( set @ A ) @ B4 @ ( A5 @ I2 ) )
@ I5 ) ) ) ).
% Un_INT_distrib
thf(fact_3608_Un__INT__distrib2,axiom,
! [A: $tType,C: $tType,B: $tType,A5: B > ( set @ A ),I5: set @ B,B4: C > ( set @ A ),J5: set @ C] :
( ( sup_sup @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ A5 @ I5 ) ) @ ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ C @ ( set @ A ) @ B4 @ J5 ) ) )
= ( complete_Inf_Inf @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [I2: B] :
( complete_Inf_Inf @ ( set @ A )
@ ( image2 @ C @ ( set @ A )
@ ^ [J3: C] : ( sup_sup @ ( set @ A ) @ ( A5 @ I2 ) @ ( B4 @ J3 ) )
@ J5 ) )
@ I5 ) ) ) ).
% Un_INT_distrib2
thf(fact_3609_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_3610_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_3611_Un__Inter,axiom,
! [A: $tType,A5: set @ A,B4: set @ ( set @ A )] :
( ( sup_sup @ ( set @ A ) @ A5 @ ( complete_Inf_Inf @ ( set @ A ) @ B4 ) )
= ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ ( set @ A ) @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A5 ) @ B4 ) ) ) ).
% Un_Inter
thf(fact_3612_card__Un__le,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] : ( ord_less_eq @ nat @ ( finite_card @ A @ ( sup_sup @ ( set @ A ) @ A5 @ B4 ) ) @ ( plus_plus @ nat @ ( finite_card @ A @ A5 ) @ ( finite_card @ A @ B4 ) ) ) ).
% card_Un_le
thf(fact_3613_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_3614_SUP__insert,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F: B > A,A3: B,A5: set @ B] :
( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F @ ( insert2 @ B @ A3 @ A5 ) ) )
= ( sup_sup @ A @ ( F @ A3 ) @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F @ A5 ) ) ) ) ) ).
% SUP_insert
thf(fact_3615_INF__union,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [M2: B > A,A5: set @ B,B4: set @ B] :
( ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ M2 @ ( sup_sup @ ( set @ B ) @ A5 @ B4 ) ) )
= ( inf_inf @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ M2 @ A5 ) ) @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ M2 @ B4 ) ) ) ) ) ).
% INF_union
thf(fact_3616_UN__extend__simps_I3_J,axiom,
! [E: $tType,F3: $tType,C6: set @ F3,A5: set @ E,B4: F3 > ( set @ E )] :
( ( ( C6
= ( bot_bot @ ( set @ F3 ) ) )
=> ( ( sup_sup @ ( set @ E ) @ A5 @ ( complete_Sup_Sup @ ( set @ E ) @ ( image2 @ F3 @ ( set @ E ) @ B4 @ C6 ) ) )
= A5 ) )
& ( ( C6
!= ( bot_bot @ ( set @ F3 ) ) )
=> ( ( sup_sup @ ( set @ E ) @ A5 @ ( complete_Sup_Sup @ ( set @ E ) @ ( image2 @ F3 @ ( set @ E ) @ B4 @ C6 ) ) )
= ( complete_Sup_Sup @ ( set @ E )
@ ( image2 @ F3 @ ( set @ E )
@ ^ [X2: F3] : ( sup_sup @ ( set @ E ) @ A5 @ ( B4 @ X2 ) )
@ C6 ) ) ) ) ) ).
% UN_extend_simps(3)
thf(fact_3617_UN__extend__simps_I2_J,axiom,
! [D: $tType,C: $tType,C6: set @ C,A5: C > ( set @ D ),B4: set @ D] :
( ( ( C6
= ( bot_bot @ ( set @ C ) ) )
=> ( ( sup_sup @ ( set @ D ) @ ( complete_Sup_Sup @ ( set @ D ) @ ( image2 @ C @ ( set @ D ) @ A5 @ C6 ) ) @ B4 )
= B4 ) )
& ( ( C6
!= ( bot_bot @ ( set @ C ) ) )
=> ( ( sup_sup @ ( set @ D ) @ ( complete_Sup_Sup @ ( set @ D ) @ ( image2 @ C @ ( set @ D ) @ A5 @ C6 ) ) @ B4 )
= ( complete_Sup_Sup @ ( set @ D )
@ ( image2 @ C @ ( set @ D )
@ ^ [X2: C] : ( sup_sup @ ( set @ D ) @ ( A5 @ X2 ) @ B4 )
@ C6 ) ) ) ) ) ).
% UN_extend_simps(2)
thf(fact_3618_INT__Un,axiom,
! [A: $tType,B: $tType,M2: B > ( set @ A ),A5: set @ B,B4: set @ B] :
( ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ M2 @ ( sup_sup @ ( set @ B ) @ A5 @ B4 ) ) )
= ( inf_inf @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ M2 @ A5 ) ) @ ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ M2 @ B4 ) ) ) ) ).
% INT_Un
thf(fact_3619_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_3620_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_3621_sum_Ounion__inter,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A5: set @ B,B4: set @ B,G: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( finite_finite2 @ B @ B4 )
=> ( ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( sup_sup @ ( set @ B ) @ A5 @ B4 ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( inf_inf @ ( set @ B ) @ A5 @ B4 ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G @ A5 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ B4 ) ) ) ) ) ) ).
% sum.union_inter
thf(fact_3622_card__Un__Int,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( finite_finite2 @ A @ B4 )
=> ( ( plus_plus @ nat @ ( finite_card @ A @ A5 ) @ ( finite_card @ A @ B4 ) )
= ( plus_plus @ nat @ ( finite_card @ A @ ( sup_sup @ ( set @ A ) @ A5 @ B4 ) ) @ ( finite_card @ A @ ( inf_inf @ ( set @ A ) @ A5 @ B4 ) ) ) ) ) ) ).
% card_Un_Int
thf(fact_3623_sum_Ounion__inter__neutral,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A5: set @ B,B4: set @ B,G: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( finite_finite2 @ B @ B4 )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ ( inf_inf @ ( set @ B ) @ A5 @ B4 ) )
=> ( ( G @ X4 )
= ( zero_zero @ A ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( sup_sup @ ( set @ B ) @ A5 @ B4 ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G @ A5 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ B4 ) ) ) ) ) ) ) ).
% sum.union_inter_neutral
thf(fact_3624_sum__Un,axiom,
! [A: $tType,B: $tType] :
( ( ab_group_add @ A )
=> ! [A5: set @ B,B4: set @ B,F: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( finite_finite2 @ B @ B4 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ F @ ( sup_sup @ ( set @ B ) @ A5 @ B4 ) )
= ( minus_minus @ A @ ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F @ A5 ) @ ( groups7311177749621191930dd_sum @ B @ A @ F @ B4 ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ F @ ( inf_inf @ ( set @ B ) @ A5 @ B4 ) ) ) ) ) ) ) ).
% sum_Un
thf(fact_3625_sum_Ounion__disjoint,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A5: set @ B,B4: set @ B,G: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( finite_finite2 @ B @ B4 )
=> ( ( ( inf_inf @ ( set @ B ) @ A5 @ B4 )
= ( bot_bot @ ( set @ B ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( sup_sup @ ( set @ B ) @ A5 @ B4 ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G @ A5 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ B4 ) ) ) ) ) ) ) ).
% sum.union_disjoint
thf(fact_3626_prod_Ounion__inter__neutral,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A5: set @ B,B4: set @ B,G: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( finite_finite2 @ B @ B4 )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ ( inf_inf @ ( set @ B ) @ A5 @ B4 ) )
=> ( ( G @ X4 )
= ( one_one @ A ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( sup_sup @ ( set @ B ) @ A5 @ B4 ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G @ A5 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ B4 ) ) ) ) ) ) ) ).
% prod.union_inter_neutral
thf(fact_3627_sum_Ounion__diff2,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A5: set @ B,B4: set @ B,G: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( finite_finite2 @ B @ B4 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( sup_sup @ ( set @ B ) @ A5 @ B4 ) )
= ( plus_plus @ A @ ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A5 @ B4 ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ B4 @ A5 ) ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( inf_inf @ ( set @ B ) @ A5 @ B4 ) ) ) ) ) ) ) ).
% sum.union_diff2
thf(fact_3628_sum__Un2,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ B )
=> ! [A5: set @ A,B4: set @ A,F: A > B] :
( ( finite_finite2 @ A @ ( sup_sup @ ( set @ A ) @ A5 @ B4 ) )
=> ( ( groups7311177749621191930dd_sum @ A @ B @ F @ ( sup_sup @ ( set @ A ) @ A5 @ B4 ) )
= ( plus_plus @ B @ ( plus_plus @ B @ ( groups7311177749621191930dd_sum @ A @ B @ F @ ( minus_minus @ ( set @ A ) @ A5 @ B4 ) ) @ ( groups7311177749621191930dd_sum @ A @ B @ F @ ( minus_minus @ ( set @ A ) @ B4 @ A5 ) ) ) @ ( groups7311177749621191930dd_sum @ A @ B @ F @ ( inf_inf @ ( set @ A ) @ A5 @ B4 ) ) ) ) ) ) ).
% sum_Un2
thf(fact_3629_card__Un__disjoint,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( finite_finite2 @ A @ B4 )
=> ( ( ( inf_inf @ ( set @ A ) @ A5 @ B4 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_card @ A @ ( sup_sup @ ( set @ A ) @ A5 @ B4 ) )
= ( plus_plus @ nat @ ( finite_card @ A @ A5 ) @ ( finite_card @ A @ B4 ) ) ) ) ) ) ).
% card_Un_disjoint
thf(fact_3630_sum__Un__nat,axiom,
! [A: $tType,A5: set @ A,B4: set @ A,F: A > nat] :
( ( finite_finite2 @ A @ A5 )
=> ( ( finite_finite2 @ A @ B4 )
=> ( ( groups7311177749621191930dd_sum @ A @ nat @ F @ ( sup_sup @ ( set @ A ) @ A5 @ B4 ) )
= ( minus_minus @ nat @ ( plus_plus @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ F @ A5 ) @ ( groups7311177749621191930dd_sum @ A @ nat @ F @ B4 ) ) @ ( groups7311177749621191930dd_sum @ A @ nat @ F @ ( inf_inf @ ( set @ A ) @ A5 @ B4 ) ) ) ) ) ) ).
% sum_Un_nat
thf(fact_3631_prod__Un,axiom,
! [A: $tType,B: $tType] :
( ( field @ A )
=> ! [A5: set @ B,B4: set @ B,F: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( finite_finite2 @ B @ B4 )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ ( inf_inf @ ( set @ B ) @ A5 @ B4 ) )
=> ( ( F @ X4 )
!= ( zero_zero @ A ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ F @ ( sup_sup @ ( set @ B ) @ A5 @ B4 ) )
= ( divide_divide @ A @ ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ F @ A5 ) @ ( groups7121269368397514597t_prod @ B @ A @ F @ B4 ) ) @ ( groups7121269368397514597t_prod @ B @ A @ F @ ( inf_inf @ ( set @ B ) @ A5 @ B4 ) ) ) ) ) ) ) ) ).
% prod_Un
thf(fact_3632_sup__bot_Osemilattice__neutr__order__axioms,axiom,
! [A: $tType] :
( ( bounde4967611905675639751up_bot @ A )
=> ( semila1105856199041335345_order @ A @ ( sup_sup @ A ) @ ( bot_bot @ A )
@ ^ [X2: A,Y: A] : ( ord_less_eq @ A @ Y @ X2 )
@ ^ [X2: A,Y: A] : ( ord_less @ A @ Y @ X2 ) ) ) ).
% sup_bot.semilattice_neutr_order_axioms
thf(fact_3633_these__insert__Some,axiom,
! [A: $tType,X: A,A5: set @ ( option @ A )] :
( ( these @ A @ ( insert2 @ ( option @ A ) @ ( some @ A @ X ) @ A5 ) )
= ( insert2 @ A @ X @ ( these @ A @ A5 ) ) ) ).
% these_insert_Some
thf(fact_3634_Gcd__eq__Max,axiom,
! [M2: set @ nat] :
( ( finite_finite2 @ nat @ M2 )
=> ( ( M2
!= ( bot_bot @ ( set @ nat ) ) )
=> ( ~ ( member @ nat @ ( zero_zero @ nat ) @ M2 )
=> ( ( gcd_Gcd @ nat @ M2 )
= ( lattic643756798349783984er_Max @ nat
@ ( complete_Inf_Inf @ ( set @ nat )
@ ( image2 @ nat @ ( set @ nat )
@ ^ [M5: nat] :
( collect @ nat
@ ^ [D4: nat] : ( dvd_dvd @ nat @ D4 @ M5 ) )
@ M2 ) ) ) ) ) ) ) ).
% Gcd_eq_Max
thf(fact_3635_these__insert__None,axiom,
! [A: $tType,A5: set @ ( option @ A )] :
( ( these @ A @ ( insert2 @ ( option @ A ) @ ( none @ A ) @ A5 ) )
= ( these @ A @ A5 ) ) ).
% these_insert_None
thf(fact_3636_Some__image__these__eq,axiom,
! [A: $tType,A5: set @ ( option @ A )] :
( ( image2 @ A @ ( option @ A ) @ ( some @ A ) @ ( these @ A @ A5 ) )
= ( collect @ ( option @ A )
@ ^ [X2: option @ A] :
( ( member @ ( option @ A ) @ X2 @ A5 )
& ( X2
!= ( none @ A ) ) ) ) ) ).
% Some_image_these_eq
thf(fact_3637_these__empty,axiom,
! [A: $tType] :
( ( these @ A @ ( bot_bot @ ( set @ ( option @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% these_empty
thf(fact_3638_these__image__Some__eq,axiom,
! [A: $tType,A5: set @ A] :
( ( these @ A @ ( image2 @ A @ ( option @ A ) @ ( some @ A ) @ A5 ) )
= A5 ) ).
% these_image_Some_eq
thf(fact_3639_Max__divisors__self__nat,axiom,
! [N: nat] :
( ( N
!= ( zero_zero @ nat ) )
=> ( ( lattic643756798349783984er_Max @ nat
@ ( collect @ nat
@ ^ [D4: nat] : ( dvd_dvd @ nat @ D4 @ N ) ) )
= N ) ) ).
% Max_divisors_self_nat
thf(fact_3640_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_3641_sup__unit__def,axiom,
( ( sup_sup @ product_unit )
= ( ^ [Uu2: product_unit,Uv2: product_unit] : product_Unity ) ) ).
% sup_unit_def
thf(fact_3642_sup__nat__def,axiom,
( ( sup_sup @ nat )
= ( ord_max @ nat ) ) ).
% sup_nat_def
thf(fact_3643_sup__Un__eq2,axiom,
! [B: $tType,A: $tType,R5: set @ ( product_prod @ A @ B ),S3: set @ ( product_prod @ A @ B )] :
( ( sup_sup @ ( A > B > $o )
@ ^ [X2: A,Y: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y ) @ R5 )
@ ^ [X2: A,Y: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y ) @ S3 ) )
= ( ^ [X2: A,Y: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y ) @ ( sup_sup @ ( set @ ( product_prod @ A @ B ) ) @ R5 @ S3 ) ) ) ) ).
% sup_Un_eq2
thf(fact_3644_sup__set__def,axiom,
! [A: $tType] :
( ( sup_sup @ ( set @ A ) )
= ( ^ [A8: set @ A,B7: set @ A] :
( collect @ A
@ ( sup_sup @ ( A > $o )
@ ^ [X2: A] : ( member @ A @ X2 @ A8 )
@ ^ [X2: A] : ( member @ A @ X2 @ B7 ) ) ) ) ) ).
% sup_set_def
thf(fact_3645_sup__Un__eq,axiom,
! [A: $tType,R5: set @ A,S3: set @ A] :
( ( sup_sup @ ( A > $o )
@ ^ [X2: A] : ( member @ A @ X2 @ R5 )
@ ^ [X2: A] : ( member @ A @ X2 @ S3 ) )
= ( ^ [X2: A] : ( member @ A @ X2 @ ( sup_sup @ ( set @ A ) @ R5 @ S3 ) ) ) ) ).
% sup_Un_eq
thf(fact_3646_sup__int__def,axiom,
( ( sup_sup @ int )
= ( ord_max @ int ) ) ).
% sup_int_def
thf(fact_3647_in__these__eq,axiom,
! [A: $tType,X: A,A5: set @ ( option @ A )] :
( ( member @ A @ X @ ( these @ A @ A5 ) )
= ( member @ ( option @ A ) @ ( some @ A @ X ) @ A5 ) ) ).
% in_these_eq
thf(fact_3648_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_3649_Sup__nat__def,axiom,
( ( complete_Sup_Sup @ nat )
= ( ^ [X9: set @ nat] :
( if @ nat
@ ( X9
= ( bot_bot @ ( set @ nat ) ) )
@ ( zero_zero @ nat )
@ ( lattic643756798349783984er_Max @ nat @ X9 ) ) ) ) ).
% Sup_nat_def
thf(fact_3650_Max__add__commute,axiom,
! [B: $tType,A: $tType] :
( ( linord4140545234300271783up_add @ A )
=> ! [S3: set @ B,F: B > A,K: A] :
( ( finite_finite2 @ B @ S3 )
=> ( ( S3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( lattic643756798349783984er_Max @ A
@ ( image2 @ B @ A
@ ^ [X2: B] : ( plus_plus @ A @ ( F @ X2 ) @ K )
@ S3 ) )
= ( plus_plus @ A @ ( lattic643756798349783984er_Max @ A @ ( image2 @ B @ A @ F @ S3 ) ) @ K ) ) ) ) ) ).
% Max_add_commute
thf(fact_3651_divide__nat__def,axiom,
( ( divide_divide @ nat )
= ( ^ [M5: nat,N5: nat] :
( if @ nat
@ ( N5
= ( zero_zero @ nat ) )
@ ( zero_zero @ nat )
@ ( lattic643756798349783984er_Max @ nat
@ ( collect @ nat
@ ^ [K3: nat] : ( ord_less_eq @ nat @ ( times_times @ nat @ K3 @ N5 ) @ M5 ) ) ) ) ) ) ).
% divide_nat_def
thf(fact_3652_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
@ ^ [D4: nat] :
( ( dvd_dvd @ nat @ D4 @ M )
& ( dvd_dvd @ nat @ D4 @ N ) ) ) ) ) ) ).
% gcd_is_Max_divisors_nat
thf(fact_3653_Option_Othese__def,axiom,
! [A: $tType] :
( ( these @ A )
= ( ^ [A8: set @ ( option @ A )] :
( image2 @ ( option @ A ) @ A @ ( the2 @ A )
@ ( collect @ ( option @ A )
@ ^ [X2: option @ A] :
( ( member @ ( option @ A ) @ X2 @ A8 )
& ( X2
!= ( none @ A ) ) ) ) ) ) ) ).
% Option.these_def
thf(fact_3654_these__not__empty__eq,axiom,
! [A: $tType,B4: set @ ( option @ A )] :
( ( ( these @ A @ B4 )
!= ( bot_bot @ ( set @ A ) ) )
= ( ( B4
!= ( bot_bot @ ( set @ ( option @ A ) ) ) )
& ( B4
!= ( insert2 @ ( option @ A ) @ ( none @ A ) @ ( bot_bot @ ( set @ ( option @ A ) ) ) ) ) ) ) ).
% these_not_empty_eq
thf(fact_3655_these__empty__eq,axiom,
! [A: $tType,B4: set @ ( option @ A )] :
( ( ( these @ A @ B4 )
= ( bot_bot @ ( set @ A ) ) )
= ( ( B4
= ( bot_bot @ ( set @ ( option @ A ) ) ) )
| ( B4
= ( insert2 @ ( option @ A ) @ ( none @ A ) @ ( bot_bot @ ( set @ ( option @ A ) ) ) ) ) ) ) ).
% these_empty_eq
thf(fact_3656_dual__Min,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( lattices_Min @ A
@ ^ [X2: A,Y: A] : ( ord_less_eq @ A @ Y @ X2 ) )
= ( lattic643756798349783984er_Max @ A ) ) ) ).
% dual_Min
thf(fact_3657_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_3658_Max_Oeq__fold_H,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( lattic643756798349783984er_Max @ A )
= ( ^ [A8: set @ A] :
( the2 @ A
@ ( finite_fold @ A @ ( option @ A )
@ ^ [X2: A,Y: option @ A] : ( some @ A @ ( case_option @ A @ A @ X2 @ ( ord_max @ A @ X2 ) @ Y ) )
@ ( none @ A )
@ A8 ) ) ) ) ) ).
% Max.eq_fold'
thf(fact_3659_comp__fun__idem__on_Ocomp__comp__fun__idem__on,axiom,
! [B: $tType,A: $tType,C: $tType,S3: set @ A,F: A > B > B,G: C > A,R5: set @ C] :
( ( finite673082921795544331dem_on @ A @ B @ S3 @ F )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image2 @ C @ A @ G @ ( top_top @ ( set @ C ) ) ) @ S3 )
=> ( finite673082921795544331dem_on @ C @ B @ R5 @ ( comp @ A @ ( B > B ) @ C @ F @ G ) ) ) ) ).
% comp_fun_idem_on.comp_comp_fun_idem_on
thf(fact_3660_Max__divisors__self__int,axiom,
! [N: int] :
( ( N
!= ( zero_zero @ int ) )
=> ( ( lattic643756798349783984er_Max @ int
@ ( collect @ int
@ ^ [D4: int] : ( dvd_dvd @ int @ D4 @ N ) ) )
= ( abs_abs @ int @ N ) ) ) ).
% Max_divisors_self_int
thf(fact_3661_greaterThan__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( set_ord_greaterThan @ A )
= ( ^ [L4: A] : ( collect @ A @ ( ord_less @ A @ L4 ) ) ) ) ) ).
% greaterThan_def
thf(fact_3662_comp__fun__idem__on_Ocomp__fun__idem__on,axiom,
! [B: $tType,A: $tType,S3: set @ A,F: A > B > B,X: A] :
( ( finite673082921795544331dem_on @ A @ B @ S3 @ F )
=> ( ( member @ A @ X @ S3 )
=> ( ( comp @ B @ B @ B @ ( F @ X ) @ ( F @ X ) )
= ( F @ X ) ) ) ) ).
% comp_fun_idem_on.comp_fun_idem_on
thf(fact_3663_gcd__is__Max__divisors__int,axiom,
! [N: int,M: int] :
( ( N
!= ( zero_zero @ int ) )
=> ( ( gcd_gcd @ int @ M @ N )
= ( lattic643756798349783984er_Max @ int
@ ( collect @ int
@ ^ [D4: int] :
( ( dvd_dvd @ int @ D4 @ M )
& ( dvd_dvd @ int @ D4 @ N ) ) ) ) ) ) ).
% gcd_is_Max_divisors_int
thf(fact_3664_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_3665_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_3666_sup__SUP__fold__sup,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ B,B4: A,F: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( sup_sup @ A @ B4 @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F @ A5 ) ) )
= ( finite_fold @ B @ A @ ( comp @ A @ ( A > A ) @ B @ ( sup_sup @ A ) @ F ) @ B4 @ A5 ) ) ) ) ).
% sup_SUP_fold_sup
thf(fact_3667_inf__INF__fold__inf,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ B,B4: A,F: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( inf_inf @ A @ B4 @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F @ A5 ) ) )
= ( finite_fold @ B @ A @ ( comp @ A @ ( A > A ) @ B @ ( inf_inf @ A ) @ F ) @ B4 @ A5 ) ) ) ) ).
% inf_INF_fold_inf
thf(fact_3668_greaterThan__0,axiom,
( ( set_ord_greaterThan @ nat @ ( zero_zero @ nat ) )
= ( image2 @ nat @ nat @ suc @ ( top_top @ ( set @ nat ) ) ) ) ).
% greaterThan_0
thf(fact_3669_SUP__fold__sup,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ B,F: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F @ A5 ) )
= ( finite_fold @ B @ A @ ( comp @ A @ ( A > A ) @ B @ ( sup_sup @ A ) @ F ) @ ( bot_bot @ A ) @ A5 ) ) ) ) ).
% SUP_fold_sup
thf(fact_3670_INF__fold__inf,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ B,F: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F @ A5 ) )
= ( finite_fold @ B @ A @ ( comp @ A @ ( A > A ) @ B @ ( inf_inf @ A ) @ F ) @ ( top_top @ A ) @ A5 ) ) ) ) ).
% INF_fold_inf
thf(fact_3671_Sup__fin_Oeq__fold_H,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( lattic5882676163264333800up_fin @ A )
= ( ^ [A8: set @ A] :
( the2 @ A
@ ( finite_fold @ A @ ( option @ A )
@ ^ [X2: A,Y: option @ A] : ( some @ A @ ( case_option @ A @ A @ X2 @ ( sup_sup @ A @ X2 ) @ Y ) )
@ ( none @ A )
@ A8 ) ) ) ) ) ).
% Sup_fin.eq_fold'
thf(fact_3672_Inf__fin_Oeq__fold_H,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( lattic7752659483105999362nf_fin @ A )
= ( ^ [A8: set @ A] :
( the2 @ A
@ ( finite_fold @ A @ ( option @ A )
@ ^ [X2: A,Y: option @ A] : ( some @ A @ ( case_option @ A @ A @ X2 @ ( inf_inf @ A @ X2 ) @ Y ) )
@ ( none @ A )
@ A8 ) ) ) ) ) ).
% Inf_fin.eq_fold'
thf(fact_3673_Min_Oeq__fold_H,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( lattic643756798350308766er_Min @ A )
= ( ^ [A8: set @ A] :
( the2 @ A
@ ( finite_fold @ A @ ( option @ A )
@ ^ [X2: A,Y: option @ A] : ( some @ A @ ( case_option @ A @ A @ X2 @ ( ord_min @ A @ X2 ) @ Y ) )
@ ( none @ A )
@ A8 ) ) ) ) ) ).
% Min.eq_fold'
thf(fact_3674_fold__union__pair,axiom,
! [B: $tType,A: $tType,B4: set @ A,X: B,A5: set @ ( product_prod @ B @ A )] :
( ( finite_finite2 @ A @ B4 )
=> ( ( sup_sup @ ( set @ ( product_prod @ B @ A ) )
@ ( complete_Sup_Sup @ ( set @ ( product_prod @ B @ A ) )
@ ( image2 @ A @ ( set @ ( product_prod @ B @ A ) )
@ ^ [Y: A] : ( insert2 @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X @ Y ) @ ( bot_bot @ ( set @ ( product_prod @ B @ A ) ) ) )
@ B4 ) )
@ A5 )
= ( finite_fold @ A @ ( set @ ( product_prod @ B @ A ) )
@ ^ [Y: A] : ( insert2 @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X @ Y ) )
@ A5
@ B4 ) ) ) ).
% fold_union_pair
thf(fact_3675_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_3676_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_3677_card_Oeq__fold,axiom,
! [A: $tType] :
( ( finite_card @ A )
= ( finite_fold @ A @ nat
@ ^ [Uu: A] : suc
@ ( zero_zero @ nat ) ) ) ).
% card.eq_fold
thf(fact_3678_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_3679_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_3680_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 )
@ ^ [K3: A] : ( insert2 @ B @ ( F @ K3 ) )
@ ( bot_bot @ ( set @ B ) )
@ A5 ) ) ) ).
% image_fold_insert
thf(fact_3681_Min__add__commute,axiom,
! [B: $tType,A: $tType] :
( ( linord4140545234300271783up_add @ A )
=> ! [S3: set @ B,F: B > A,K: A] :
( ( finite_finite2 @ B @ S3 )
=> ( ( S3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( lattic643756798350308766er_Min @ A
@ ( image2 @ B @ A
@ ^ [X2: B] : ( plus_plus @ A @ ( F @ X2 ) @ K )
@ S3 ) )
= ( plus_plus @ A @ ( lattic643756798350308766er_Min @ A @ ( image2 @ B @ A @ F @ S3 ) ) @ K ) ) ) ) ) ).
% Min_add_commute
thf(fact_3682_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 )
@ ^ [X2: A,A10: set @ A] : ( if @ ( set @ A ) @ ( P @ X2 ) @ ( insert2 @ A @ X2 @ A10 ) @ A10 )
@ ( bot_bot @ ( set @ A ) )
@ A5 ) ) ) ).
% Set_filter_fold
thf(fact_3683_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 ) )
@ ^ [X2: A] : ( insert2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ X2 ) )
@ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) )
@ A5 ) ) ) ).
% Id_on_fold
thf(fact_3684_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 ) )
@ ^ [X2: A] : ( insert2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ X2 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
@ A8 ) ) ) ) ).
% Id_on_def
thf(fact_3685_dual__Max,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( lattices_Max @ A
@ ^ [X2: A,Y: A] : ( ord_less_eq @ A @ Y @ X2 ) )
= ( lattic643756798350308766er_Min @ A ) ) ) ).
% dual_Max
thf(fact_3686_Id__onI,axiom,
! [A: $tType,A3: A,A5: set @ A] :
( ( member @ A @ A3 @ A5 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ A3 ) @ ( id_on @ A @ A5 ) ) ) ).
% Id_onI
thf(fact_3687_Set_Ofilter__def,axiom,
! [A: $tType] :
( ( filter3 @ A )
= ( ^ [P3: A > $o,A8: set @ A] :
( collect @ A
@ ^ [A6: A] :
( ( member @ A @ A6 @ A8 )
& ( P3 @ A6 ) ) ) ) ) ).
% Set.filter_def
thf(fact_3688_Id__on__iff,axiom,
! [A: $tType,X: A,Y3: A,A5: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ ( id_on @ A @ A5 ) )
= ( ( X = Y3 )
& ( member @ A @ X @ A5 ) ) ) ).
% Id_on_iff
thf(fact_3689_Id__on__eqI,axiom,
! [A: $tType,A3: A,B2: A,A5: set @ A] :
( ( A3 = B2 )
=> ( ( member @ A @ A3 @ A5 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( id_on @ A @ A5 ) ) ) ) ).
% Id_on_eqI
thf(fact_3690_Id__onE,axiom,
! [A: $tType,C2: product_prod @ A @ A,A5: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ C2 @ ( id_on @ A @ A5 ) )
=> ~ ! [X4: A] :
( ( member @ A @ X4 @ A5 )
=> ( C2
!= ( product_Pair @ A @ A @ X4 @ X4 ) ) ) ) ).
% Id_onE
thf(fact_3691_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
@ ^ [X2: A,Y: A] :
( ( X2 = Y )
& ( A5 @ X2 ) ) ) ) ) ).
% Id_on_def'
thf(fact_3692_inter__Set__filter,axiom,
! [A: $tType,B4: set @ A,A5: set @ A] :
( ( finite_finite2 @ A @ B4 )
=> ( ( inf_inf @ ( set @ A ) @ A5 @ B4 )
= ( filter3 @ A
@ ^ [X2: A] : ( member @ A @ X2 @ A5 )
@ B4 ) ) ) ).
% inter_Set_filter
thf(fact_3693_comp__fun__commute__product__fold,axiom,
! [A: $tType,B: $tType,B4: set @ A] :
( ( finite_finite2 @ A @ B4 )
=> ( finite6289374366891150609ommute @ B @ ( set @ ( product_prod @ B @ A ) )
@ ^ [X2: B,Z3: set @ ( product_prod @ B @ A )] :
( finite_fold @ A @ ( set @ ( product_prod @ B @ A ) )
@ ^ [Y: A] : ( insert2 @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X2 @ Y ) )
@ Z3
@ B4 ) ) ) ).
% comp_fun_commute_product_fold
thf(fact_3694_INF__principal__finite,axiom,
! [B: $tType,A: $tType,X7: set @ A,F: A > ( set @ B )] :
( ( finite_finite2 @ A @ X7 )
=> ( ( complete_Inf_Inf @ ( filter @ B )
@ ( image2 @ A @ ( filter @ B )
@ ^ [X2: A] : ( principal @ B @ ( F @ X2 ) )
@ X7 ) )
= ( principal @ B @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ F @ X7 ) ) ) ) ) ).
% INF_principal_finite
thf(fact_3695_image__o__collect,axiom,
! [B: $tType,C: $tType,A: $tType,G: C > B,F6: 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 ) ) @ F6 ) )
= ( comp @ ( set @ C ) @ ( set @ B ) @ A @ ( image2 @ C @ B @ G ) @ ( bNF_collect @ A @ C @ F6 ) ) ) ).
% image_o_collect
thf(fact_3696_total__on__singleton,axiom,
! [A: $tType,X: A] : ( total_on @ A @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) @ ( insert2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ X ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ).
% total_on_singleton
thf(fact_3697_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_3698_comp__fun__commute__const,axiom,
! [B: $tType,A: $tType,F: B > B] :
( finite6289374366891150609ommute @ A @ B
@ ^ [Uu: A] : F ) ).
% comp_fun_commute_const
thf(fact_3699_total__onI,axiom,
! [A: $tType,A5: set @ A,R: set @ ( product_prod @ A @ A )] :
( ! [X4: A,Y4: A] :
( ( member @ A @ X4 @ A5 )
=> ( ( member @ A @ Y4 @ A5 )
=> ( ( X4 != Y4 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y4 ) @ R )
| ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ X4 ) @ R ) ) ) ) )
=> ( total_on @ A @ A5 @ R ) ) ).
% total_onI
thf(fact_3700_total__on__def,axiom,
! [A: $tType] :
( ( total_on @ A )
= ( ^ [A8: set @ A,R4: set @ ( product_prod @ A @ A )] :
! [X2: A] :
( ( member @ A @ X2 @ A8 )
=> ! [Y: A] :
( ( member @ A @ Y @ A8 )
=> ( ( X2 != Y )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y ) @ R4 )
| ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X2 ) @ R4 ) ) ) ) ) ) ) ).
% total_on_def
thf(fact_3701_comp__fun__commute__def,axiom,
! [B: $tType,A: $tType] :
( ( finite6289374366891150609ommute @ A @ B )
= ( ^ [F2: A > B > B] :
! [Y: A,X2: A] :
( ( comp @ B @ B @ B @ ( F2 @ Y ) @ ( F2 @ X2 ) )
= ( comp @ B @ B @ B @ ( F2 @ X2 ) @ ( F2 @ Y ) ) ) ) ) ).
% comp_fun_commute_def
thf(fact_3702_comp__fun__commute_Ocomp__fun__commute,axiom,
! [B: $tType,A: $tType,F: A > B > B,Y3: A,X: A] :
( ( finite6289374366891150609ommute @ A @ B @ F )
=> ( ( comp @ B @ B @ B @ ( F @ Y3 ) @ ( F @ X ) )
= ( comp @ B @ B @ B @ ( F @ X ) @ ( F @ Y3 ) ) ) ) ).
% comp_fun_commute.comp_fun_commute
thf(fact_3703_comp__fun__commute_Ointro,axiom,
! [B: $tType,A: $tType,F: A > B > B] :
( ! [Y4: A,X4: A] :
( ( comp @ B @ B @ B @ ( F @ Y4 ) @ ( F @ X4 ) )
= ( comp @ B @ B @ B @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( finite6289374366891150609ommute @ A @ B @ F ) ) ).
% comp_fun_commute.intro
thf(fact_3704_comp__fun__commute_Ocomp__comp__fun__commute,axiom,
! [B: $tType,A: $tType,C: $tType,F: A > B > B,G: C > A] :
( ( finite6289374366891150609ommute @ A @ B @ F )
=> ( finite6289374366891150609ommute @ C @ B @ ( comp @ A @ ( B > B ) @ C @ F @ G ) ) ) ).
% comp_fun_commute.comp_comp_fun_commute
thf(fact_3705_comp__fun__commute__filter__fold,axiom,
! [A: $tType,P: A > $o] :
( finite6289374366891150609ommute @ A @ ( set @ A )
@ ^ [X2: A,A10: set @ A] : ( if @ ( set @ A ) @ ( P @ X2 ) @ ( insert2 @ A @ X2 @ A10 ) @ A10 ) ) ).
% comp_fun_commute_filter_fold
thf(fact_3706_collect__def,axiom,
! [A: $tType,B: $tType] :
( ( bNF_collect @ B @ A )
= ( ^ [F7: set @ ( B > ( set @ A ) ),X2: B] :
( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ ( B > ( set @ A ) ) @ ( set @ A )
@ ^ [F2: B > ( set @ A )] : ( F2 @ X2 )
@ F7 ) ) ) ) ).
% collect_def
thf(fact_3707_comp__fun__commute__relcomp__fold,axiom,
! [A: $tType,B: $tType,C: $tType,S3: set @ ( product_prod @ A @ B )] :
( ( finite_finite2 @ ( product_prod @ A @ B ) @ S3 )
=> ( finite6289374366891150609ommute @ ( product_prod @ C @ A ) @ ( set @ ( product_prod @ C @ B ) )
@ ( product_case_prod @ C @ A @ ( ( set @ ( product_prod @ C @ B ) ) > ( set @ ( product_prod @ C @ B ) ) )
@ ^ [X2: C,Y: 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 ) ) )
@ ^ [W2: A,Z3: B,A10: set @ ( product_prod @ C @ B )] : ( if @ ( set @ ( product_prod @ C @ B ) ) @ ( Y = W2 ) @ ( insert2 @ ( product_prod @ C @ B ) @ ( product_Pair @ C @ B @ X2 @ Z3 ) @ A10 ) @ A10 ) )
@ A8
@ S3 ) ) ) ) ).
% comp_fun_commute_relcomp_fold
thf(fact_3708_collect__comp,axiom,
! [A: $tType,B: $tType,C: $tType,F6: set @ ( C > ( set @ B ) ),G: A > C] :
( ( comp @ C @ ( set @ B ) @ A @ ( bNF_collect @ C @ B @ F6 ) @ G )
= ( bNF_collect @ A @ B
@ ( image2 @ ( C > ( set @ B ) ) @ ( A > ( set @ B ) )
@ ^ [F2: C > ( set @ B )] : ( comp @ C @ ( set @ B ) @ A @ F2 @ G )
@ F6 ) ) ) ).
% collect_comp
thf(fact_3709_at__bot__def,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( at_bot @ A )
= ( complete_Inf_Inf @ ( filter @ A )
@ ( image2 @ A @ ( filter @ A )
@ ^ [K3: A] : ( principal @ A @ ( set_ord_atMost @ A @ K3 ) )
@ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% at_bot_def
thf(fact_3710_at__bot__sub,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C2: A] :
( ( at_bot @ A )
= ( complete_Inf_Inf @ ( filter @ A )
@ ( image2 @ A @ ( filter @ A )
@ ^ [K3: A] : ( principal @ A @ ( set_ord_atMost @ A @ K3 ) )
@ ( set_ord_atMost @ A @ C2 ) ) ) ) ) ).
% at_bot_sub
thf(fact_3711_insert__relcomp__union__fold,axiom,
! [C: $tType,B: $tType,A: $tType,S3: set @ ( product_prod @ A @ B ),X: product_prod @ C @ A,X7: set @ ( product_prod @ C @ B )] :
( ( finite_finite2 @ ( product_prod @ A @ B ) @ S3 )
=> ( ( sup_sup @ ( set @ ( product_prod @ C @ B ) ) @ ( relcomp @ C @ A @ B @ ( insert2 @ ( product_prod @ C @ A ) @ X @ ( bot_bot @ ( set @ ( product_prod @ C @ A ) ) ) ) @ S3 ) @ X7 )
= ( 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 ) ) )
@ ^ [W2: A,Z3: B,A10: set @ ( product_prod @ C @ B )] :
( if @ ( set @ ( product_prod @ C @ B ) )
@ ( ( product_snd @ C @ A @ X )
= W2 )
@ ( insert2 @ ( product_prod @ C @ B ) @ ( product_Pair @ C @ B @ ( product_fst @ C @ A @ X ) @ Z3 ) @ A10 )
@ A10 ) )
@ X7
@ S3 ) ) ) ).
% insert_relcomp_union_fold
thf(fact_3712_INF__set__fold,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F: B > A,Xs2: list @ B] :
( ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F @ ( set2 @ B @ Xs2 ) ) )
= ( fold @ B @ A @ ( comp @ A @ ( A > A ) @ B @ ( inf_inf @ A ) @ F ) @ Xs2 @ ( top_top @ A ) ) ) ) ).
% INF_set_fold
thf(fact_3713_relcomp_Ocases,axiom,
! [A: $tType,C: $tType,B: $tType,A1: A,A22: C,R: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ B @ C )] :
( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ A1 @ A22 ) @ ( relcomp @ A @ B @ C @ R @ S ) )
=> ~ ! [B3: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A1 @ B3 ) @ R )
=> ~ ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ B3 @ A22 ) @ S ) ) ) ).
% relcomp.cases
thf(fact_3714_relcomp_Osimps,axiom,
! [B: $tType,C: $tType,A: $tType,A1: A,A22: C,R: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ B @ C )] :
( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ A1 @ A22 ) @ ( relcomp @ A @ B @ C @ R @ S ) )
= ( ? [A6: A,B5: B,C4: C] :
( ( A1 = A6 )
& ( A22 = C4 )
& ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A6 @ B5 ) @ R )
& ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ B5 @ C4 ) @ S ) ) ) ) ).
% relcomp.simps
thf(fact_3715_relcomp_OrelcompI,axiom,
! [A: $tType,C: $tType,B: $tType,A3: A,B2: B,R: set @ ( product_prod @ A @ B ),C2: C,S: set @ ( product_prod @ B @ C )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ R )
=> ( ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ B2 @ C2 ) @ S )
=> ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ A3 @ C2 ) @ ( relcomp @ A @ B @ C @ R @ S ) ) ) ) ).
% relcomp.relcompI
thf(fact_3716_relcompE,axiom,
! [A: $tType,B: $tType,C: $tType,Xz: product_prod @ A @ B,R: set @ ( product_prod @ A @ C ),S: set @ ( product_prod @ C @ B )] :
( ( member @ ( product_prod @ A @ B ) @ Xz @ ( relcomp @ A @ C @ B @ R @ S ) )
=> ~ ! [X4: A,Y4: C,Z6: B] :
( ( Xz
= ( product_Pair @ A @ B @ X4 @ Z6 ) )
=> ( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ X4 @ Y4 ) @ R )
=> ~ ( member @ ( product_prod @ C @ B ) @ ( product_Pair @ C @ B @ Y4 @ Z6 ) @ S ) ) ) ) ).
% relcompE
thf(fact_3717_relcompEpair,axiom,
! [A: $tType,B: $tType,C: $tType,A3: A,C2: B,R: set @ ( product_prod @ A @ C ),S: set @ ( product_prod @ C @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ C2 ) @ ( relcomp @ A @ C @ B @ R @ S ) )
=> ~ ! [B3: C] :
( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ A3 @ B3 ) @ R )
=> ~ ( member @ ( product_prod @ C @ B ) @ ( product_Pair @ C @ B @ B3 @ C2 ) @ S ) ) ) ).
% relcompEpair
thf(fact_3718_fold__commute__apply,axiom,
! [A: $tType,C: $tType,B: $tType,Xs2: list @ A,H: B > C,G: A > B > B,F: A > C > C,S: B] :
( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs2 ) )
=> ( ( comp @ B @ C @ B @ H @ ( G @ X4 ) )
= ( comp @ C @ C @ B @ ( F @ X4 ) @ H ) ) )
=> ( ( H @ ( fold @ A @ B @ G @ Xs2 @ S ) )
= ( fold @ A @ C @ F @ Xs2 @ ( H @ S ) ) ) ) ).
% fold_commute_apply
thf(fact_3719_fold__commute,axiom,
! [A: $tType,C: $tType,B: $tType,Xs2: list @ A,H: B > C,G: A > B > B,F: A > C > C] :
( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs2 ) )
=> ( ( comp @ B @ C @ B @ H @ ( G @ X4 ) )
= ( comp @ C @ C @ B @ ( F @ X4 ) @ H ) ) )
=> ( ( comp @ B @ C @ B @ H @ ( fold @ A @ B @ G @ Xs2 ) )
= ( comp @ C @ C @ B @ ( fold @ A @ C @ F @ Xs2 ) @ H ) ) ) ).
% fold_commute
thf(fact_3720_comp__fun__commute__Image__fold,axiom,
! [B: $tType,A: $tType,S3: set @ A] :
( finite6289374366891150609ommute @ ( product_prod @ A @ B ) @ ( set @ B )
@ ( product_case_prod @ A @ B @ ( ( set @ B ) > ( set @ B ) )
@ ^ [X2: A,Y: B,A8: set @ B] : ( if @ ( set @ B ) @ ( member @ A @ X2 @ S3 ) @ ( insert2 @ B @ Y @ A8 ) @ A8 ) ) ) ).
% comp_fun_commute_Image_fold
thf(fact_3721_relcomp__UNION__distrib2,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,R: D > ( set @ ( product_prod @ A @ C ) ),I5: set @ D,S: 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 ) ) @ S )
= ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) )
@ ( image2 @ D @ ( set @ ( product_prod @ A @ B ) )
@ ^ [I2: D] : ( relcomp @ A @ C @ B @ ( R @ I2 ) @ S )
@ I5 ) ) ) ).
% relcomp_UNION_distrib2
thf(fact_3722_relcomp__UNION__distrib,axiom,
! [A: $tType,C: $tType,B: $tType,D: $tType,S: set @ ( product_prod @ A @ C ),R: D > ( set @ ( product_prod @ C @ B ) ),I5: set @ D] :
( ( relcomp @ A @ C @ B @ S @ ( 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 @ S @ ( R @ I2 ) )
@ I5 ) ) ) ).
% relcomp_UNION_distrib
thf(fact_3723_relcomp__fold,axiom,
! [C: $tType,B: $tType,A: $tType,R5: set @ ( product_prod @ A @ B ),S3: set @ ( product_prod @ B @ C )] :
( ( finite_finite2 @ ( product_prod @ A @ B ) @ R5 )
=> ( ( finite_finite2 @ ( product_prod @ B @ C ) @ S3 )
=> ( ( relcomp @ A @ B @ C @ R5 @ S3 )
= ( finite_fold @ ( product_prod @ A @ B ) @ ( set @ ( product_prod @ A @ C ) )
@ ( product_case_prod @ A @ B @ ( ( set @ ( product_prod @ A @ C ) ) > ( set @ ( product_prod @ A @ C ) ) )
@ ^ [X2: A,Y: 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 ) ) )
@ ^ [W2: B,Z3: C,A10: set @ ( product_prod @ A @ C )] : ( if @ ( set @ ( product_prod @ A @ C ) ) @ ( Y = W2 ) @ ( insert2 @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ X2 @ Z3 ) @ A10 ) @ A10 ) )
@ A8
@ S3 ) )
@ ( bot_bot @ ( set @ ( product_prod @ A @ C ) ) )
@ R5 ) ) ) ) ).
% relcomp_fold
thf(fact_3724_Gcd__nat__set__eq__fold,axiom,
! [Xs2: list @ nat] :
( ( gcd_Gcd @ nat @ ( set2 @ nat @ Xs2 ) )
= ( fold @ nat @ nat @ ( gcd_gcd @ nat ) @ Xs2 @ ( zero_zero @ nat ) ) ) ).
% Gcd_nat_set_eq_fold
thf(fact_3725_Gcd__int__set__eq__fold,axiom,
! [Xs2: list @ int] :
( ( gcd_Gcd @ int @ ( set2 @ int @ Xs2 ) )
= ( fold @ int @ int @ ( gcd_gcd @ int ) @ Xs2 @ ( zero_zero @ int ) ) ) ).
% Gcd_int_set_eq_fold
thf(fact_3726_comp__fun__commute__Pow__fold,axiom,
! [A: $tType] :
( finite6289374366891150609ommute @ A @ ( set @ ( set @ A ) )
@ ^ [X2: A,A8: set @ ( set @ A )] : ( sup_sup @ ( set @ ( set @ A ) ) @ A8 @ ( image2 @ ( set @ A ) @ ( set @ A ) @ ( insert2 @ A @ X2 ) @ A8 ) ) ) ).
% comp_fun_commute_Pow_fold
thf(fact_3727_Gcd__set__eq__fold,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [Xs2: list @ A] :
( ( gcd_Gcd @ A @ ( set2 @ A @ Xs2 ) )
= ( fold @ A @ A @ ( gcd_gcd @ A ) @ Xs2 @ ( zero_zero @ A ) ) ) ) ).
% Gcd_set_eq_fold
thf(fact_3728_insert__relcomp__fold,axiom,
! [C: $tType,B: $tType,A: $tType,S3: set @ ( product_prod @ A @ B ),X: product_prod @ C @ A,R5: set @ ( product_prod @ C @ A )] :
( ( finite_finite2 @ ( product_prod @ A @ B ) @ S3 )
=> ( ( relcomp @ C @ A @ B @ ( insert2 @ ( product_prod @ C @ A ) @ X @ R5 ) @ S3 )
= ( 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 ) ) )
@ ^ [W2: A,Z3: B,A10: set @ ( product_prod @ C @ B )] :
( if @ ( set @ ( product_prod @ C @ B ) )
@ ( ( product_snd @ C @ A @ X )
= W2 )
@ ( insert2 @ ( product_prod @ C @ B ) @ ( product_Pair @ C @ B @ ( product_fst @ C @ A @ X ) @ Z3 ) @ A10 )
@ A10 ) )
@ ( relcomp @ C @ A @ B @ R5 @ S3 )
@ S3 ) ) ) ).
% insert_relcomp_fold
thf(fact_3729_SUP__set__fold,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F: B > A,Xs2: list @ B] :
( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F @ ( set2 @ B @ Xs2 ) ) )
= ( fold @ B @ A @ ( comp @ A @ ( A > A ) @ B @ ( sup_sup @ A ) @ F ) @ Xs2 @ ( bot_bot @ A ) ) ) ) ).
% SUP_set_fold
thf(fact_3730_min__ext__compat,axiom,
! [A: $tType,R5: set @ ( product_prod @ A @ A ),S3: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( relcomp @ A @ A @ A @ R5 @ S3 ) @ R5 )
=> ( ord_less_eq @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( relcomp @ ( set @ A ) @ ( set @ A ) @ ( set @ A ) @ ( min_ext @ A @ R5 ) @ ( sup_sup @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( min_ext @ A @ S3 ) @ ( insert2 @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ A ) ) ) @ ( bot_bot @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) ) ) ) @ ( min_ext @ A @ R5 ) ) ) ).
% min_ext_compat
thf(fact_3731_max__ext__compat,axiom,
! [A: $tType,R5: set @ ( product_prod @ A @ A ),S3: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( relcomp @ A @ A @ A @ R5 @ S3 ) @ R5 )
=> ( ord_less_eq @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( relcomp @ ( set @ A ) @ ( set @ A ) @ ( set @ A ) @ ( max_ext @ A @ R5 ) @ ( sup_sup @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( max_ext @ A @ S3 ) @ ( insert2 @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ A ) ) ) @ ( bot_bot @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) ) ) ) @ ( max_ext @ A @ R5 ) ) ) ).
% max_ext_compat
thf(fact_3732_snd__sndOp,axiom,
! [B: $tType,A: $tType,C: $tType,P: B > C > $o,Q: C > A > $o] :
( ( product_snd @ B @ A )
= ( comp @ ( product_prod @ C @ A ) @ A @ ( product_prod @ B @ A ) @ ( product_snd @ C @ A ) @ ( bNF_sndOp @ B @ C @ A @ P @ Q ) ) ) ).
% snd_sndOp
thf(fact_3733_fst__fstOp,axiom,
! [A: $tType,B: $tType,C: $tType,P: A > C > $o,Q: C > B > $o] :
( ( product_fst @ A @ B )
= ( comp @ ( product_prod @ A @ C ) @ A @ ( product_prod @ A @ B ) @ ( product_fst @ A @ C ) @ ( bNF_fstOp @ A @ C @ B @ P @ Q ) ) ) ).
% fst_fstOp
thf(fact_3734_max__ext__additive,axiom,
! [A: $tType,A5: set @ A,B4: set @ A,R5: set @ ( product_prod @ A @ A ),C6: set @ A,D5: set @ A] :
( ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ A5 @ B4 ) @ ( max_ext @ A @ R5 ) )
=> ( ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ C6 @ D5 ) @ ( max_ext @ A @ R5 ) )
=> ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A5 @ C6 ) @ ( sup_sup @ ( set @ A ) @ B4 @ D5 ) ) @ ( max_ext @ A @ R5 ) ) ) ) ).
% max_ext_additive
thf(fact_3735_max__ext_Ocases,axiom,
! [A: $tType,A1: set @ A,A22: set @ A,R5: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ A1 @ A22 ) @ ( max_ext @ A @ R5 ) )
=> ~ ( ( finite_finite2 @ A @ A1 )
=> ( ( finite_finite2 @ A @ A22 )
=> ( ( A22
!= ( bot_bot @ ( set @ A ) ) )
=> ~ ! [X5: A] :
( ( member @ A @ X5 @ A1 )
=> ? [Xa4: A] :
( ( member @ A @ Xa4 @ A22 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ Xa4 ) @ R5 ) ) ) ) ) ) ) ).
% max_ext.cases
thf(fact_3736_max__ext_Osimps,axiom,
! [A: $tType,A1: set @ A,A22: set @ A,R5: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ A1 @ A22 ) @ ( max_ext @ A @ R5 ) )
= ( ( finite_finite2 @ A @ A1 )
& ( finite_finite2 @ A @ A22 )
& ( A22
!= ( bot_bot @ ( set @ A ) ) )
& ! [X2: A] :
( ( member @ A @ X2 @ A1 )
=> ? [Y: A] :
( ( member @ A @ Y @ A22 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y ) @ R5 ) ) ) ) ) ).
% max_ext.simps
thf(fact_3737_max__ext_Omax__extI,axiom,
! [A: $tType,X7: set @ A,Y9: set @ A,R5: set @ ( product_prod @ A @ A )] :
( ( finite_finite2 @ A @ X7 )
=> ( ( finite_finite2 @ A @ Y9 )
=> ( ( Y9
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ X7 )
=> ? [Xa2: A] :
( ( member @ A @ Xa2 @ Y9 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Xa2 ) @ R5 ) ) )
=> ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ X7 @ Y9 ) @ ( max_ext @ A @ R5 ) ) ) ) ) ) ).
% max_ext.max_extI
thf(fact_3738_sndOp__def,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( bNF_sndOp @ C @ A @ B )
= ( ^ [P3: C > A > $o,Q7: A > B > $o,Ac: product_prod @ C @ B] : ( product_Pair @ A @ B @ ( bNF_pick_middlep @ C @ A @ B @ P3 @ Q7 @ ( product_fst @ C @ B @ Ac ) @ ( product_snd @ C @ B @ Ac ) ) @ ( product_snd @ C @ B @ Ac ) ) ) ) ).
% sndOp_def
thf(fact_3739_fstOp__def,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( bNF_fstOp @ A @ B @ C )
= ( ^ [P3: A > B > $o,Q7: B > C > $o,Ac: product_prod @ A @ C] : ( product_Pair @ A @ B @ ( product_fst @ A @ C @ Ac ) @ ( bNF_pick_middlep @ A @ B @ C @ P3 @ Q7 @ ( product_fst @ A @ C @ Ac ) @ ( product_snd @ A @ C @ Ac ) ) ) ) ) ).
% fstOp_def
thf(fact_3740_max__ext__def,axiom,
! [A: $tType] :
( ( max_ext @ A )
= ( ^ [R6: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ ( set @ A ) @ ( set @ A ) )
@ ( product_case_prod @ ( set @ A ) @ ( set @ A ) @ $o
@ ( max_extp @ A
@ ^ [X2: A,Y: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y ) @ R6 ) ) ) ) ) ) ).
% max_ext_def
thf(fact_3741_Pow__fold,axiom,
! [A: $tType,A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( pow2 @ A @ A5 )
= ( finite_fold @ A @ ( set @ ( set @ A ) )
@ ^ [X2: A,A8: set @ ( set @ A )] : ( sup_sup @ ( set @ ( set @ A ) ) @ A8 @ ( image2 @ ( set @ A ) @ ( set @ A ) @ ( insert2 @ A @ X2 ) @ A8 ) )
@ ( insert2 @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ ( set @ A ) ) ) )
@ A5 ) ) ) ).
% Pow_fold
thf(fact_3742_Cantors__paradox,axiom,
! [A: $tType,A5: set @ A] :
~ ? [F8: A > ( set @ A )] :
( ( image2 @ A @ ( set @ A ) @ F8 @ A5 )
= ( pow2 @ A @ A5 ) ) ).
% Cantors_paradox
thf(fact_3743_Pow__def,axiom,
! [A: $tType] :
( ( pow2 @ A )
= ( ^ [A8: set @ A] :
( collect @ ( set @ A )
@ ^ [B7: set @ A] : ( ord_less_eq @ ( set @ A ) @ B7 @ A8 ) ) ) ) ).
% Pow_def
thf(fact_3744_Pow__INT__eq,axiom,
! [A: $tType,B: $tType,B4: B > ( set @ A ),A5: set @ B] :
( ( pow2 @ A @ ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B4 @ A5 ) ) )
= ( complete_Inf_Inf @ ( set @ ( set @ A ) )
@ ( image2 @ B @ ( set @ ( set @ A ) )
@ ^ [X2: B] : ( pow2 @ A @ ( B4 @ X2 ) )
@ A5 ) ) ) ).
% Pow_INT_eq
thf(fact_3745_UN__Pow__subset,axiom,
! [A: $tType,B: $tType,B4: B > ( set @ A ),A5: set @ B] :
( ord_less_eq @ ( set @ ( set @ A ) )
@ ( complete_Sup_Sup @ ( set @ ( set @ A ) )
@ ( image2 @ B @ ( set @ ( set @ A ) )
@ ^ [X2: B] : ( pow2 @ A @ ( B4 @ X2 ) )
@ A5 ) )
@ ( pow2 @ A @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B4 @ A5 ) ) ) ) ).
% UN_Pow_subset
thf(fact_3746_binomial__def,axiom,
( binomial
= ( ^ [N5: nat,K3: nat] :
( finite_card @ ( set @ nat )
@ ( collect @ ( set @ nat )
@ ^ [K7: set @ nat] :
( ( member @ ( set @ nat ) @ K7 @ ( pow2 @ nat @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N5 ) ) )
& ( ( finite_card @ nat @ K7 )
= K3 ) ) ) ) ) ) ).
% binomial_def
thf(fact_3747_max__extp__max__ext__eq,axiom,
! [A: $tType,R5: set @ ( product_prod @ A @ A )] :
( ( max_extp @ A
@ ^ [X2: A,Y: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y ) @ R5 ) )
= ( ^ [X2: set @ A,Y: set @ A] : ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ X2 @ Y ) @ ( max_ext @ A @ R5 ) ) ) ) ).
% max_extp_max_ext_eq
thf(fact_3748_max__extp__eq,axiom,
! [A: $tType] :
( ( max_extp @ A )
= ( ^ [R4: A > A > $o,X2: set @ A,Y: set @ A] : ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ X2 @ Y ) @ ( max_ext @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ R4 ) ) ) ) ) ) ).
% max_extp_eq
thf(fact_3749_Field__insert,axiom,
! [A: $tType,A3: A,B2: A,R: set @ ( product_prod @ A @ A )] :
( ( field2 @ A @ ( insert2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R ) )
= ( sup_sup @ ( set @ A ) @ ( insert2 @ A @ A3 @ ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( field2 @ A @ R ) ) ) ).
% Field_insert
thf(fact_3750_rotate1__length01,axiom,
! [A: $tType,Xs2: list @ A] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ ( one_one @ nat ) )
=> ( ( rotate1 @ A @ Xs2 )
= Xs2 ) ) ).
% rotate1_length01
thf(fact_3751_Pow__set_I2_J,axiom,
! [B: $tType,X: B,Xs2: list @ B] :
( ( pow2 @ B @ ( set2 @ B @ ( cons @ B @ X @ Xs2 ) ) )
= ( sup_sup @ ( set @ ( set @ B ) ) @ ( pow2 @ B @ ( set2 @ B @ Xs2 ) ) @ ( image2 @ ( set @ B ) @ ( set @ B ) @ ( insert2 @ B @ X ) @ ( pow2 @ B @ ( set2 @ B @ Xs2 ) ) ) ) ) ).
% Pow_set(2)
thf(fact_3752_FieldI1,axiom,
! [A: $tType,I: A,J: A,R5: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ I @ J ) @ R5 )
=> ( member @ A @ I @ ( field2 @ A @ R5 ) ) ) ).
% FieldI1
thf(fact_3753_FieldI2,axiom,
! [A: $tType,I: A,J: A,R5: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ I @ J ) @ R5 )
=> ( member @ A @ J @ ( field2 @ A @ R5 ) ) ) ).
% FieldI2
thf(fact_3754_fold__Cons,axiom,
! [B: $tType,A: $tType,F: A > B > B,X: A,Xs2: list @ A] :
( ( fold @ A @ B @ F @ ( cons @ A @ X @ Xs2 ) )
= ( comp @ B @ B @ B @ ( fold @ A @ B @ F @ Xs2 ) @ ( F @ X ) ) ) ).
% fold_Cons
thf(fact_3755_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_3756_Field__natLeq__on,axiom,
! [N: nat] :
( ( field2 @ nat
@ ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [X2: nat,Y: nat] :
( ( ord_less @ nat @ X2 @ N )
& ( ord_less @ nat @ Y @ N )
& ( ord_less_eq @ nat @ X2 @ Y ) ) ) ) )
= ( collect @ nat
@ ^ [X2: nat] : ( ord_less @ nat @ X2 @ N ) ) ) ).
% Field_natLeq_on
thf(fact_3757_mlex__eq,axiom,
! [A: $tType] :
( ( mlex_prod @ A )
= ( ^ [F2: A > nat,R6: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ^ [X2: A,Y: A] :
( ( ord_less @ nat @ ( F2 @ X2 ) @ ( F2 @ Y ) )
| ( ( ord_less_eq @ nat @ ( F2 @ X2 ) @ ( F2 @ Y ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y ) @ R6 ) ) ) ) ) ) ) ).
% mlex_eq
thf(fact_3758_list_Osize__gen_I2_J,axiom,
! [A: $tType,X: A > nat,X21: A,X222: list @ A] :
( ( size_list @ A @ X @ ( cons @ A @ X21 @ X222 ) )
= ( plus_plus @ nat @ ( plus_plus @ nat @ ( X @ X21 ) @ ( size_list @ A @ X @ X222 ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% list.size_gen(2)
thf(fact_3759_full__exhaustive__int_H_Ocases,axiom,
! [X: product_prod @ ( ( product_prod @ int @ ( product_unit > code_term ) ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ int @ int )] :
~ ! [F4: ( product_prod @ int @ ( product_unit > code_term ) ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ),D2: int,I3: int] :
( X
!= ( product_Pair @ ( ( product_prod @ int @ ( product_unit > code_term ) ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ int @ int ) @ F4 @ ( product_Pair @ int @ int @ D2 @ I3 ) ) ) ).
% full_exhaustive_int'.cases
thf(fact_3760_exhaustive__int_H_Ocases,axiom,
! [X: product_prod @ ( int > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ int @ int )] :
~ ! [F4: int > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ),D2: int,I3: int] :
( X
!= ( product_Pair @ ( int > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ int @ int ) @ F4 @ ( product_Pair @ int @ int @ D2 @ I3 ) ) ) ).
% exhaustive_int'.cases
thf(fact_3761_full__exhaustive__integer_H_Ocases,axiom,
! [X: product_prod @ ( ( product_prod @ code_integer @ ( product_unit > code_term ) ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ code_integer @ code_integer )] :
~ ! [F4: ( product_prod @ code_integer @ ( product_unit > code_term ) ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ),D2: code_integer,I3: code_integer] :
( X
!= ( product_Pair @ ( ( product_prod @ code_integer @ ( product_unit > code_term ) ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ code_integer @ code_integer ) @ F4 @ ( product_Pair @ code_integer @ code_integer @ D2 @ I3 ) ) ) ).
% full_exhaustive_integer'.cases
thf(fact_3762_exhaustive__integer_H_Ocases,axiom,
! [X: product_prod @ ( code_integer > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ code_integer @ code_integer )] :
~ ! [F4: code_integer > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ),D2: code_integer,I3: code_integer] :
( X
!= ( product_Pair @ ( code_integer > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ code_integer @ code_integer ) @ F4 @ ( product_Pair @ code_integer @ code_integer @ D2 @ I3 ) ) ) ).
% exhaustive_integer'.cases
thf(fact_3763_mlex__less,axiom,
! [A: $tType,F: A > nat,X: A,Y3: A,R5: set @ ( product_prod @ A @ A )] :
( ( ord_less @ nat @ ( F @ X ) @ ( F @ Y3 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ ( mlex_prod @ A @ F @ R5 ) ) ) ).
% mlex_less
thf(fact_3764_mlex__iff,axiom,
! [A: $tType,X: A,Y3: A,F: A > nat,R5: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ ( mlex_prod @ A @ F @ R5 ) )
= ( ( ord_less @ nat @ ( F @ X ) @ ( F @ Y3 ) )
| ( ( ( F @ X )
= ( F @ Y3 ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ R5 ) ) ) ) ).
% mlex_iff
thf(fact_3765_mlex__leq,axiom,
! [A: $tType,F: A > nat,X: A,Y3: A,R5: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ nat @ ( F @ X ) @ ( F @ Y3 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ R5 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ ( mlex_prod @ A @ F @ R5 ) ) ) ) ).
% mlex_leq
thf(fact_3766_relChain__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ( ( bNF_Ca3754400796208372196lChain @ A @ B )
= ( ^ [R4: set @ ( product_prod @ A @ A ),As: A > B] :
! [I2: A,J3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ I2 @ J3 ) @ R4 )
=> ( ord_less_eq @ B @ ( As @ I2 ) @ ( As @ J3 ) ) ) ) ) ) ).
% relChain_def
thf(fact_3767_natLess__def,axiom,
( bNF_Ca8459412986667044542atLess
= ( collect @ ( product_prod @ nat @ nat ) @ ( product_case_prod @ nat @ nat @ $o @ ( ord_less @ nat ) ) ) ) ).
% natLess_def
thf(fact_3768_cofinal__def,axiom,
! [A: $tType] :
( ( bNF_Ca7293521722713021262ofinal @ A )
= ( ^ [A8: set @ A,R4: set @ ( product_prod @ A @ A )] :
! [X2: A] :
( ( member @ A @ X2 @ ( field2 @ A @ R4 ) )
=> ? [Y: A] :
( ( member @ A @ Y @ A8 )
& ( X2 != Y )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y ) @ R4 ) ) ) ) ) ).
% cofinal_def
thf(fact_3769_CHAR__pos__iff,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( semiri4206861660011772517g_char @ A @ ( type2 @ A ) ) )
= ( ? [N5: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N5 )
& ( ( semiring_1_of_nat @ A @ N5 )
= ( zero_zero @ A ) ) ) ) ) ) ).
% CHAR_pos_iff
thf(fact_3770_CHAR__eq__0,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiri4206861660011772517g_char @ A @ ( type2 @ A ) )
= ( zero_zero @ nat ) ) ) ).
% CHAR_eq_0
thf(fact_3771_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_3772_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_3773_CHAR__eqI,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [C2: nat] :
( ( ( semiring_1_of_nat @ A @ C2 )
= ( zero_zero @ A ) )
=> ( ! [X4: nat] :
( ( ( semiring_1_of_nat @ A @ X4 )
= ( zero_zero @ A ) )
=> ( dvd_dvd @ nat @ C2 @ X4 ) )
=> ( ( semiri4206861660011772517g_char @ A @ ( type2 @ A ) )
= C2 ) ) ) ) ).
% CHAR_eqI
thf(fact_3774_CHAR__eq0__iff,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( ( semiri4206861660011772517g_char @ A @ ( type2 @ A ) )
= ( zero_zero @ nat ) )
= ( ! [N5: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N5 )
=> ( ( semiring_1_of_nat @ A @ N5 )
!= ( zero_zero @ A ) ) ) ) ) ) ).
% CHAR_eq0_iff
thf(fact_3775_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 ) )
=> ( ! [X4: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ X4 )
=> ( ( ord_less @ nat @ X4 @ C2 )
=> ( ( semiring_1_of_nat @ A @ X4 )
!= ( zero_zero @ A ) ) ) )
=> ( ( semiri4206861660011772517g_char @ A @ ( type2 @ A ) )
= C2 ) ) ) ) ) ).
% CHAR_eq_posI
thf(fact_3776_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_3777_bit__double__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) @ N )
= ( ( bit_se5641148757651400278ts_bit @ A @ A3 @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) )
& ( N
!= ( zero_zero @ nat ) )
& ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N ) ) ) ) ).
% bit_double_iff
thf(fact_3778_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_3779_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_3780_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_3781_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_3782_possible__bit__def,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ( ( bit_se6407376104438227557le_bit @ A )
= ( ^ [Tyrep: itself @ A,N5: nat] :
( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N5 )
!= ( zero_zero @ A ) ) ) ) ) ).
% possible_bit_def
thf(fact_3783_bit__minus__iff,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A3: A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( uminus_uminus @ A @ A3 ) @ N )
= ( ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N )
& ~ ( bit_se5641148757651400278ts_bit @ A @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) @ N ) ) ) ) ).
% bit_minus_iff
thf(fact_3784_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_3785_enumerate__simps_I2_J,axiom,
! [B: $tType,N: nat,X: B,Xs2: list @ B] :
( ( enumerate @ B @ N @ ( cons @ B @ X @ Xs2 ) )
= ( cons @ ( product_prod @ nat @ B ) @ ( product_Pair @ nat @ B @ N @ X ) @ ( enumerate @ B @ ( suc @ N ) @ Xs2 ) ) ) ).
% enumerate_simps(2)
thf(fact_3786_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_3787_fun_Oin__rel,axiom,
! [B: $tType,A: $tType,D: $tType,R5: A > B > $o,A3: D > A,B2: D > B] :
( ( bNF_rel_fun @ D @ D @ A @ B
@ ^ [Y6: D,Z2: D] : Y6 = Z2
@ R5
@ A3
@ B2 )
= ( ? [Z3: D > ( product_prod @ A @ B )] :
( ( member @ ( D > ( product_prod @ A @ B ) ) @ Z3
@ ( collect @ ( D > ( product_prod @ A @ B ) )
@ ^ [X2: D > ( product_prod @ A @ B )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( image2 @ D @ ( product_prod @ A @ B ) @ X2 @ ( top_top @ ( set @ D ) ) ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R5 ) ) ) ) )
& ( ( comp @ ( product_prod @ A @ B ) @ A @ D @ ( product_fst @ A @ B ) @ Z3 )
= A3 )
& ( ( comp @ ( product_prod @ A @ B ) @ B @ D @ ( product_snd @ A @ B ) @ Z3 )
= B2 ) ) ) ) ).
% fun.in_rel
thf(fact_3788_mono__ge2__power__minus__self,axiom,
! [K: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K )
=> ( order_mono @ nat @ nat
@ ^ [M5: nat] : ( minus_minus @ nat @ ( power_power @ nat @ K @ M5 ) @ M5 ) ) ) ).
% mono_ge2_power_minus_self
thf(fact_3789_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_3790_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_3791_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_3792_Gcd__fin_Oinsert,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,A5: set @ A] :
( ( semiring_gcd_Gcd_fin @ A @ ( insert2 @ A @ A3 @ A5 ) )
= ( gcd_gcd @ A @ A3 @ ( semiring_gcd_Gcd_fin @ A @ A5 ) ) ) ) ).
% Gcd_fin.insert
thf(fact_3793_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_3794_power__transfer,axiom,
! [A: $tType,B: $tType] :
( ( ( power @ B )
& ( power @ A ) )
=> ! [R5: A > B > $o] :
( ( R5 @ ( one_one @ A ) @ ( one_one @ B ) )
=> ( ( bNF_rel_fun @ A @ B @ ( A > A ) @ ( B > B ) @ R5 @ ( bNF_rel_fun @ A @ B @ A @ B @ R5 @ R5 ) @ ( times_times @ A ) @ ( times_times @ B ) )
=> ( bNF_rel_fun @ A @ B @ ( nat > A ) @ ( nat > B ) @ R5
@ ( bNF_rel_fun @ nat @ nat @ A @ B
@ ^ [Y6: nat,Z2: nat] : Y6 = Z2
@ R5 )
@ ( power_power @ A )
@ ( power_power @ B ) ) ) ) ) ).
% power_transfer
thf(fact_3795_transfer__rule__numeral,axiom,
! [A: $tType,B: $tType] :
( ( ( monoid_add @ B )
& ( semiring_numeral @ B )
& ( monoid_add @ A )
& ( semiring_numeral @ A ) )
=> ! [R5: A > B > $o] :
( ( R5 @ ( zero_zero @ A ) @ ( zero_zero @ B ) )
=> ( ( R5 @ ( one_one @ A ) @ ( one_one @ B ) )
=> ( ( bNF_rel_fun @ A @ B @ ( A > A ) @ ( B > B ) @ R5 @ ( bNF_rel_fun @ A @ B @ A @ B @ R5 @ R5 ) @ ( plus_plus @ A ) @ ( plus_plus @ B ) )
=> ( bNF_rel_fun @ num @ num @ A @ B
@ ^ [Y6: num,Z2: num] : Y6 = Z2
@ R5
@ ( numeral_numeral @ A )
@ ( numeral_numeral @ B ) ) ) ) ) ) ).
% transfer_rule_numeral
thf(fact_3796_sub_Orsp,axiom,
( bNF_rel_fun @ num @ num @ ( num > int ) @ ( num > int )
@ ^ [Y6: num,Z2: num] : Y6 = Z2
@ ( bNF_rel_fun @ num @ num @ int @ int
@ ^ [Y6: num,Z2: num] : Y6 = Z2
@ ^ [Y6: int,Z2: int] : Y6 = Z2 )
@ ^ [M5: num,N5: num] : ( minus_minus @ int @ ( numeral_numeral @ int @ M5 ) @ ( numeral_numeral @ int @ N5 ) )
@ ^ [M5: num,N5: num] : ( minus_minus @ int @ ( numeral_numeral @ int @ M5 ) @ ( numeral_numeral @ int @ N5 ) ) ) ).
% sub.rsp
thf(fact_3797_transfer__rule__of__int,axiom,
! [A: $tType,B: $tType] :
( ( ( ring_1 @ B )
& ( ring_1 @ A ) )
=> ! [R5: A > B > $o] :
( ( R5 @ ( zero_zero @ A ) @ ( zero_zero @ B ) )
=> ( ( R5 @ ( one_one @ A ) @ ( one_one @ B ) )
=> ( ( bNF_rel_fun @ A @ B @ ( A > A ) @ ( B > B ) @ R5 @ ( bNF_rel_fun @ A @ B @ A @ B @ R5 @ R5 ) @ ( plus_plus @ A ) @ ( plus_plus @ B ) )
=> ( ( bNF_rel_fun @ A @ B @ A @ B @ R5 @ R5 @ ( uminus_uminus @ A ) @ ( uminus_uminus @ B ) )
=> ( bNF_rel_fun @ int @ int @ A @ B
@ ^ [Y6: int,Z2: int] : Y6 = Z2
@ R5
@ ( ring_1_of_int @ A )
@ ( ring_1_of_int @ B ) ) ) ) ) ) ) ).
% transfer_rule_of_int
thf(fact_3798_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_3799_rel__fun__def__butlast,axiom,
! [B: $tType,D: $tType,C: $tType,E: $tType,F3: $tType,A: $tType,R5: A > B > $o,S3: C > E > $o,T5: D > F3 > $o,F: A > C > D,G: B > E > F3] :
( ( bNF_rel_fun @ A @ B @ ( C > D ) @ ( E > F3 ) @ R5 @ ( bNF_rel_fun @ C @ E @ D @ F3 @ S3 @ T5 ) @ F @ G )
= ( ! [X2: A,Y: B] :
( ( R5 @ X2 @ Y )
=> ( bNF_rel_fun @ C @ E @ D @ F3 @ S3 @ T5 @ ( F @ X2 ) @ ( G @ Y ) ) ) ) ) ).
% rel_fun_def_butlast
thf(fact_3800_If__transfer,axiom,
! [A: $tType,B: $tType,A5: A > B > $o] :
( bNF_rel_fun @ $o @ $o @ ( A > A > A ) @ ( B > B > B )
@ ^ [Y6: $o,Z2: $o] : Y6 = Z2
@ ( 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_3801_mono__Suc,axiom,
order_mono @ nat @ nat @ suc ).
% mono_Suc
thf(fact_3802_gcd__integer_Orsp,axiom,
( bNF_rel_fun @ int @ int @ ( int > int ) @ ( int > int )
@ ^ [Y6: int,Z2: int] : Y6 = Z2
@ ( bNF_rel_fun @ int @ int @ int @ int
@ ^ [Y6: int,Z2: int] : Y6 = Z2
@ ^ [Y6: int,Z2: int] : Y6 = Z2 )
@ ( gcd_gcd @ int )
@ ( gcd_gcd @ int ) ) ).
% gcd_integer.rsp
thf(fact_3803_mono__add,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [A3: A] : ( order_mono @ A @ A @ ( plus_plus @ A @ A3 ) ) ) ).
% mono_add
thf(fact_3804_plus__natural_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( nat > nat )
@ ^ [Y6: nat,Z2: nat] : Y6 = Z2
@ ( bNF_rel_fun @ nat @ nat @ nat @ nat
@ ^ [Y6: nat,Z2: nat] : Y6 = Z2
@ ^ [Y6: nat,Z2: nat] : Y6 = Z2 )
@ ( plus_plus @ nat )
@ ( plus_plus @ nat ) ) ).
% plus_natural.rsp
thf(fact_3805_comp__transfer,axiom,
! [A: $tType,B: $tType,E: $tType,F3: $tType,D: $tType,C: $tType,B4: A > C > $o,C6: B > D > $o,A5: E > F3 > $o] : ( bNF_rel_fun @ ( A > B ) @ ( C > D ) @ ( ( E > A ) > E > B ) @ ( ( F3 > C ) > F3 > D ) @ ( bNF_rel_fun @ A @ C @ B @ D @ B4 @ C6 ) @ ( bNF_rel_fun @ ( E > A ) @ ( F3 > C ) @ ( E > B ) @ ( F3 > D ) @ ( bNF_rel_fun @ E @ F3 @ A @ C @ A5 @ B4 ) @ ( bNF_rel_fun @ E @ F3 @ B @ D @ A5 @ C6 ) ) @ ( comp @ A @ B @ E ) @ ( comp @ C @ D @ F3 ) ) ).
% comp_transfer
thf(fact_3806_fun_Omap__transfer,axiom,
! [A: $tType,B: $tType,D: $tType,G3: $tType,F3: $tType,Rb2: A > F3 > $o,Sd: B > G3 > $o] :
( bNF_rel_fun @ ( A > B ) @ ( F3 > G3 ) @ ( ( D > A ) > D > B ) @ ( ( D > F3 ) > D > G3 ) @ ( bNF_rel_fun @ A @ F3 @ B @ G3 @ Rb2 @ Sd )
@ ( bNF_rel_fun @ ( D > A ) @ ( D > F3 ) @ ( D > B ) @ ( D > G3 )
@ ( bNF_rel_fun @ D @ D @ A @ F3
@ ^ [Y6: D,Z2: D] : Y6 = Z2
@ Rb2 )
@ ( bNF_rel_fun @ D @ D @ B @ G3
@ ^ [Y6: D,Z2: D] : Y6 = Z2
@ Sd ) )
@ ( comp @ A @ B @ D )
@ ( comp @ F3 @ G3 @ D ) ) ).
% fun.map_transfer
thf(fact_3807_fun_Orel__map_I2_J,axiom,
! [A: $tType,C: $tType,B: $tType,D: $tType,Sa: A > C > $o,X: D > A,G: B > C,Y3: D > B] :
( ( bNF_rel_fun @ D @ D @ A @ C
@ ^ [Y6: D,Z2: D] : Y6 = Z2
@ Sa
@ X
@ ( comp @ B @ C @ D @ G @ Y3 ) )
= ( bNF_rel_fun @ D @ D @ A @ B
@ ^ [Y6: D,Z2: D] : Y6 = Z2
@ ^ [X2: A,Y: B] : ( Sa @ X2 @ ( G @ Y ) )
@ X
@ Y3 ) ) ).
% fun.rel_map(2)
thf(fact_3808_fun_Orel__map_I1_J,axiom,
! [A: $tType,C: $tType,B: $tType,D: $tType,Sb: C > B > $o,I: A > C,X: D > A,Y3: D > B] :
( ( bNF_rel_fun @ D @ D @ C @ B
@ ^ [Y6: D,Z2: D] : Y6 = Z2
@ Sb
@ ( comp @ A @ C @ D @ I @ X )
@ Y3 )
= ( bNF_rel_fun @ D @ D @ A @ B
@ ^ [Y6: D,Z2: D] : Y6 = Z2
@ ^ [X2: A] : ( Sb @ ( I @ X2 ) )
@ X
@ Y3 ) ) ).
% fun.rel_map(1)
thf(fact_3809_Gcd__fin__dvd,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,A5: set @ A] :
( ( member @ A @ A3 @ A5 )
=> ( dvd_dvd @ A @ ( semiring_gcd_Gcd_fin @ A @ A5 ) @ A3 ) ) ) ).
% Gcd_fin_dvd
thf(fact_3810_dup_Orsp,axiom,
( bNF_rel_fun @ int @ int @ int @ int
@ ^ [Y6: int,Z2: int] : Y6 = Z2
@ ^ [Y6: int,Z2: int] : Y6 = Z2
@ ^ [K3: int] : ( plus_plus @ int @ K3 @ K3 )
@ ^ [K3: int] : ( plus_plus @ int @ K3 @ K3 ) ) ).
% dup.rsp
thf(fact_3811_Gcd__fin_Oin__idem,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,A5: set @ A] :
( ( member @ A @ A3 @ A5 )
=> ( ( gcd_gcd @ A @ A3 @ ( semiring_gcd_Gcd_fin @ A @ A5 ) )
= ( semiring_gcd_Gcd_fin @ A @ A5 ) ) ) ) ).
% Gcd_fin.in_idem
thf(fact_3812_mono__iff__le__Suc,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( order_mono @ nat @ A )
= ( ^ [F2: nat > A] :
! [N5: nat] : ( ord_less_eq @ A @ ( F2 @ N5 ) @ ( F2 @ ( suc @ N5 ) ) ) ) ) ) ).
% mono_iff_le_Suc
thf(fact_3813_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,A3: product_prod @ A @ B,A5: set @ ( product_prod @ A @ B ),B2: product_prod @ C @ D,B4: set @ ( product_prod @ C @ D )] :
( ( bNF_rel_fun @ A @ B @ ( C > $o ) @ ( D > $o ) @ R12
@ ( bNF_rel_fun @ C @ D @ $o @ $o @ R23
@ ^ [Y6: $o,Z2: $o] : Y6 = Z2 )
@ P
@ Q )
=> ( ( member @ ( product_prod @ A @ B ) @ A3 @ A5 )
=> ( ( member @ ( product_prod @ C @ D ) @ B2 @ B4 )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ A5 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R12 ) ) )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ C @ D ) ) @ B4 @ ( collect @ ( product_prod @ C @ D ) @ ( product_case_prod @ C @ D @ $o @ R23 ) ) )
=> ( ( P @ ( product_fst @ A @ B @ A3 ) @ ( product_fst @ C @ D @ B2 ) )
= ( Q @ ( product_snd @ A @ B @ A3 ) @ ( product_snd @ C @ D @ B2 ) ) ) ) ) ) ) ) ).
% predicate2_transferD
thf(fact_3814_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 ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ A5 )
=> ( dvd_dvd @ A @ B2 @ X2 ) ) ) ) ) ) ).
% dvd_Gcd_fin_iff
thf(fact_3815_Gcd__fin__greatest,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A5: set @ A,A3: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ! [B3: A] :
( ( member @ A @ B3 @ A5 )
=> ( dvd_dvd @ A @ A3 @ B3 ) )
=> ( dvd_dvd @ A @ A3 @ ( semiring_gcd_Gcd_fin @ A @ A5 ) ) ) ) ) ).
% Gcd_fin_greatest
thf(fact_3816_dvd__gcd__list__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [B2: A,Xs2: list @ A] :
( ( dvd_dvd @ A @ B2 @ ( semiring_gcd_Gcd_fin @ A @ ( set2 @ A @ Xs2 ) ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ Xs2 ) )
=> ( dvd_dvd @ A @ B2 @ X2 ) ) ) ) ) ).
% dvd_gcd_list_iff
thf(fact_3817_gcd__list__greatest,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [Bs: list @ A,A3: A] :
( ! [B3: A] :
( ( member @ A @ B3 @ ( set2 @ A @ Bs ) )
=> ( dvd_dvd @ A @ A3 @ B3 ) )
=> ( dvd_dvd @ A @ A3 @ ( semiring_gcd_Gcd_fin @ A @ ( set2 @ A @ Bs ) ) ) ) ) ).
% gcd_list_greatest
thf(fact_3818_Gcd__fin_Osubset,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [B4: set @ A,A5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B4 @ A5 )
=> ( ( gcd_gcd @ A @ ( semiring_gcd_Gcd_fin @ A @ B4 ) @ ( semiring_gcd_Gcd_fin @ A @ A5 ) )
= ( semiring_gcd_Gcd_fin @ A @ A5 ) ) ) ) ).
% Gcd_fin.subset
thf(fact_3819_Gcd__fin_Ounion,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A5: set @ A,B4: set @ A] :
( ( semiring_gcd_Gcd_fin @ A @ ( sup_sup @ ( set @ A ) @ A5 @ B4 ) )
= ( gcd_gcd @ A @ ( semiring_gcd_Gcd_fin @ A @ A5 ) @ ( semiring_gcd_Gcd_fin @ A @ B4 ) ) ) ) ).
% Gcd_fin.union
thf(fact_3820_mono__mult,axiom,
! [A: $tType] :
( ( ordered_semiring @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( order_mono @ A @ A @ ( times_times @ A @ A3 ) ) ) ) ).
% mono_mult
thf(fact_3821_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_3822_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
@ ^ [X2: C] : ( F @ ( A5 @ X2 ) )
@ I5 ) )
@ ( F @ ( complete_Sup_Sup @ A @ ( image2 @ C @ A @ A5 @ I5 ) ) ) ) ) ) ).
% mono_SUP
thf(fact_3823_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_3824_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
@ ^ [X2: C] : ( F @ ( A5 @ X2 ) )
@ I5 ) ) ) ) ) ).
% mono_INF
thf(fact_3825_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_3826_num__of__integer_Orsp,axiom,
( bNF_rel_fun @ int @ int @ num @ num
@ ^ [Y6: int,Z2: int] : Y6 = Z2
@ ^ [Y6: num,Z2: num] : Y6 = Z2
@ ( comp @ nat @ num @ int @ num_of_nat @ nat2 )
@ ( comp @ nat @ num @ int @ num_of_nat @ nat2 ) ) ).
% num_of_integer.rsp
thf(fact_3827_rel__fun__Collect__case__prodD,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,A5: A > B > $o,B4: C > D > $o,F: A > C,G: B > D,X7: set @ ( product_prod @ A @ B ),X: product_prod @ A @ B] :
( ( bNF_rel_fun @ A @ B @ C @ D @ A5 @ B4 @ F @ G )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ X7 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ A5 ) ) )
=> ( ( member @ ( product_prod @ A @ B ) @ X @ X7 )
=> ( B4 @ ( comp @ A @ C @ ( product_prod @ A @ B ) @ F @ ( product_fst @ A @ B ) @ X ) @ ( comp @ B @ D @ ( product_prod @ A @ B ) @ G @ ( product_snd @ A @ B ) @ X ) ) ) ) ) ).
% rel_fun_Collect_case_prodD
thf(fact_3828_Gcd__fin_Oremove,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,A5: set @ A] :
( ( member @ A @ A3 @ A5 )
=> ( ( semiring_gcd_Gcd_fin @ A @ A5 )
= ( gcd_gcd @ A @ A3 @ ( semiring_gcd_Gcd_fin @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% Gcd_fin.remove
thf(fact_3829_Gcd__fin_Oinsert__remove,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,A5: set @ A] :
( ( semiring_gcd_Gcd_fin @ A @ ( insert2 @ A @ A3 @ A5 ) )
= ( gcd_gcd @ A @ A3 @ ( semiring_gcd_Gcd_fin @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% Gcd_fin.insert_remove
thf(fact_3830_Gcd__fin_Oset__eq__fold,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [Xs2: list @ A] :
( ( semiring_gcd_Gcd_fin @ A @ ( set2 @ A @ Xs2 ) )
= ( fold @ A @ A @ ( gcd_gcd @ A ) @ Xs2 @ ( zero_zero @ A ) ) ) ) ).
% Gcd_fin.set_eq_fold
thf(fact_3831_Gcd__fin__0__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A5: set @ A] :
( ( ( semiring_gcd_Gcd_fin @ A @ A5 )
= ( zero_zero @ A ) )
= ( ( ord_less_eq @ ( set @ A ) @ A5 @ ( insert2 @ A @ ( zero_zero @ A ) @ ( bot_bot @ ( set @ A ) ) ) )
& ( finite_finite2 @ A @ A5 ) ) ) ) ).
% Gcd_fin_0_iff
thf(fact_3832_transfer__rule__of__nat,axiom,
! [A: $tType,B: $tType] :
( ( ( semiring_1 @ B )
& ( semiring_1 @ A ) )
=> ! [R5: A > B > $o] :
( ( R5 @ ( zero_zero @ A ) @ ( zero_zero @ B ) )
=> ( ( R5 @ ( one_one @ A ) @ ( one_one @ B ) )
=> ( ( bNF_rel_fun @ A @ B @ ( A > A ) @ ( B > B ) @ R5 @ ( bNF_rel_fun @ A @ B @ A @ B @ R5 @ R5 ) @ ( plus_plus @ A ) @ ( plus_plus @ B ) )
=> ( bNF_rel_fun @ nat @ nat @ A @ B
@ ^ [Y6: nat,Z2: nat] : Y6 = Z2
@ R5
@ ( semiring_1_of_nat @ A )
@ ( semiring_1_of_nat @ B ) ) ) ) ) ) ).
% transfer_rule_of_nat
thf(fact_3833_transfer__rule__of__bool,axiom,
! [A: $tType,B: $tType] :
( ( ( zero_neq_one @ B )
& ( zero_neq_one @ A ) )
=> ! [R5: A > B > $o] :
( ( R5 @ ( zero_zero @ A ) @ ( zero_zero @ B ) )
=> ( ( R5 @ ( one_one @ A ) @ ( one_one @ B ) )
=> ( bNF_rel_fun @ $o @ $o @ A @ B
@ ^ [Y6: $o,Z2: $o] : Y6 = Z2
@ R5
@ ( zero_neq_one_of_bool @ A )
@ ( zero_neq_one_of_bool @ B ) ) ) ) ) ).
% transfer_rule_of_bool
thf(fact_3834_plus__rat_Otransfer,axiom,
( bNF_rel_fun @ ( product_prod @ int @ int ) @ rat @ ( ( product_prod @ int @ int ) > ( product_prod @ int @ int ) ) @ ( rat > rat ) @ pcr_rat @ ( bNF_rel_fun @ ( product_prod @ int @ int ) @ rat @ ( product_prod @ int @ int ) @ rat @ pcr_rat @ pcr_rat )
@ ^ [X2: product_prod @ int @ int,Y: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( plus_plus @ int @ ( times_times @ int @ ( product_fst @ int @ int @ X2 ) @ ( product_snd @ int @ int @ Y ) ) @ ( times_times @ int @ ( product_fst @ int @ int @ Y ) @ ( product_snd @ int @ int @ X2 ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ X2 ) @ ( product_snd @ int @ int @ Y ) ) )
@ ( plus_plus @ rat ) ) ).
% plus_rat.transfer
thf(fact_3835_inverse__rat_Otransfer,axiom,
( bNF_rel_fun @ ( product_prod @ int @ int ) @ rat @ ( product_prod @ int @ int ) @ rat @ pcr_rat @ pcr_rat
@ ^ [X2: product_prod @ int @ int] :
( 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_inverse @ rat ) ) ).
% inverse_rat.transfer
thf(fact_3836_one__rat_Otransfer,axiom,
pcr_rat @ ( product_Pair @ int @ int @ ( one_one @ int ) @ ( one_one @ int ) ) @ ( one_one @ rat ) ).
% one_rat.transfer
thf(fact_3837_zero__rat_Otransfer,axiom,
pcr_rat @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) ) @ ( zero_zero @ rat ) ).
% zero_rat.transfer
thf(fact_3838_uminus__rat_Otransfer,axiom,
( bNF_rel_fun @ ( product_prod @ int @ int ) @ rat @ ( product_prod @ int @ int ) @ rat @ pcr_rat @ pcr_rat
@ ^ [X2: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( uminus_uminus @ int @ ( product_fst @ int @ int @ X2 ) ) @ ( product_snd @ int @ int @ X2 ) )
@ ( uminus_uminus @ rat ) ) ).
% uminus_rat.transfer
thf(fact_3839_times__rat_Otransfer,axiom,
( bNF_rel_fun @ ( product_prod @ int @ int ) @ rat @ ( ( product_prod @ int @ int ) > ( product_prod @ int @ int ) ) @ ( rat > rat ) @ pcr_rat @ ( bNF_rel_fun @ ( product_prod @ int @ int ) @ rat @ ( product_prod @ int @ int ) @ rat @ pcr_rat @ pcr_rat )
@ ^ [X2: product_prod @ int @ int,Y: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( times_times @ int @ ( product_fst @ int @ int @ X2 ) @ ( product_fst @ int @ int @ Y ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ X2 ) @ ( product_snd @ int @ int @ Y ) ) )
@ ( times_times @ rat ) ) ).
% times_rat.transfer
thf(fact_3840_o__rsp_I2_J,axiom,
! [E: $tType,F3: $tType,H8: $tType,G3: $tType,R12: E > F3 > $o] :
( bNF_rel_fun @ ( G3 > H8 ) @ ( G3 > H8 ) @ ( ( E > G3 ) > E > H8 ) @ ( ( F3 > G3 ) > F3 > H8 )
@ ^ [Y6: G3 > H8,Z2: G3 > H8] : Y6 = Z2
@ ( bNF_rel_fun @ ( E > G3 ) @ ( F3 > G3 ) @ ( E > H8 ) @ ( F3 > H8 )
@ ( bNF_rel_fun @ E @ F3 @ G3 @ G3 @ R12
@ ^ [Y6: G3,Z2: G3] : Y6 = Z2 )
@ ( bNF_rel_fun @ E @ F3 @ H8 @ H8 @ R12
@ ^ [Y6: H8,Z2: H8] : Y6 = Z2 ) )
@ ( comp @ G3 @ H8 @ E )
@ ( comp @ G3 @ H8 @ F3 ) ) ).
% o_rsp(2)
thf(fact_3841_o__rsp_I1_J,axiom,
! [A: $tType,B: $tType,E: $tType,F3: $tType,D: $tType,C: $tType,R23: A > C > $o,R32: B > D > $o,R12: E > F3 > $o] : ( bNF_rel_fun @ ( A > B ) @ ( C > D ) @ ( ( E > A ) > E > B ) @ ( ( F3 > C ) > F3 > D ) @ ( bNF_rel_fun @ A @ C @ B @ D @ R23 @ R32 ) @ ( bNF_rel_fun @ ( E > A ) @ ( F3 > C ) @ ( E > B ) @ ( F3 > D ) @ ( bNF_rel_fun @ E @ F3 @ A @ C @ R12 @ R23 ) @ ( bNF_rel_fun @ E @ F3 @ B @ D @ R12 @ R32 ) ) @ ( comp @ A @ B @ E ) @ ( comp @ C @ D @ F3 ) ) ).
% o_rsp(1)
thf(fact_3842_positive_Otransfer,axiom,
( bNF_rel_fun @ ( product_prod @ int @ int ) @ rat @ $o @ $o @ pcr_rat
@ ^ [Y6: $o,Z2: $o] : Y6 = Z2
@ ^ [X2: product_prod @ int @ int] : ( ord_less @ int @ ( zero_zero @ int ) @ ( times_times @ int @ ( product_fst @ int @ int @ X2 ) @ ( product_snd @ int @ int @ X2 ) ) )
@ positive ) ).
% positive.transfer
thf(fact_3843_in__set__enumerate__eq,axiom,
! [A: $tType,P4: product_prod @ nat @ A,N: nat,Xs2: list @ A] :
( ( member @ ( product_prod @ nat @ A ) @ P4 @ ( set2 @ ( product_prod @ nat @ A ) @ ( enumerate @ A @ N @ Xs2 ) ) )
= ( ( ord_less_eq @ nat @ N @ ( product_fst @ nat @ A @ P4 ) )
& ( ord_less @ nat @ ( product_fst @ nat @ A @ P4 ) @ ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ N ) )
& ( ( nth @ A @ Xs2 @ ( minus_minus @ nat @ ( product_fst @ nat @ A @ P4 ) @ N ) )
= ( product_snd @ nat @ A @ P4 ) ) ) ) ).
% in_set_enumerate_eq
thf(fact_3844_nth__Cons__0,axiom,
! [A: $tType,X: A,Xs2: list @ A] :
( ( nth @ A @ ( cons @ A @ X @ Xs2 ) @ ( zero_zero @ nat ) )
= X ) ).
% nth_Cons_0
thf(fact_3845_nth__Cons__numeral,axiom,
! [A: $tType,X: A,Xs2: list @ A,V: num] :
( ( nth @ A @ ( cons @ A @ X @ Xs2 ) @ ( numeral_numeral @ nat @ V ) )
= ( nth @ A @ Xs2 @ ( minus_minus @ nat @ ( numeral_numeral @ nat @ V ) @ ( one_one @ nat ) ) ) ) ).
% nth_Cons_numeral
thf(fact_3846_nth__Cons__pos,axiom,
! [A: $tType,N: nat,X: A,Xs2: list @ A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( nth @ A @ ( cons @ A @ X @ Xs2 ) @ N )
= ( nth @ A @ Xs2 @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ).
% nth_Cons_pos
thf(fact_3847_nth__enumerate__eq,axiom,
! [A: $tType,M: nat,Xs2: list @ A,N: nat] :
( ( ord_less @ nat @ M @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( nth @ ( product_prod @ nat @ A ) @ ( enumerate @ A @ N @ Xs2 ) @ M )
= ( product_Pair @ nat @ A @ ( plus_plus @ nat @ N @ M ) @ ( nth @ A @ Xs2 @ M ) ) ) ) ).
% nth_enumerate_eq
thf(fact_3848_nth__Cons,axiom,
! [A: $tType,X: A,Xs2: list @ A,N: nat] :
( ( nth @ A @ ( cons @ A @ X @ Xs2 ) @ N )
= ( case_nat @ A @ X @ ( nth @ A @ Xs2 ) @ N ) ) ).
% nth_Cons
thf(fact_3849_nth__Cons_H,axiom,
! [A: $tType,N: nat,X: A,Xs2: list @ A] :
( ( ( N
= ( zero_zero @ nat ) )
=> ( ( nth @ A @ ( cons @ A @ X @ Xs2 ) @ N )
= X ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ( ( nth @ A @ ( cons @ A @ X @ Xs2 ) @ N )
= ( nth @ A @ Xs2 @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ) ).
% nth_Cons'
thf(fact_3850_nth__equal__first__eq,axiom,
! [A: $tType,X: A,Xs2: list @ A,N: nat] :
( ~ ( member @ A @ X @ ( set2 @ A @ Xs2 ) )
=> ( ( ord_less_eq @ nat @ N @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( ( nth @ A @ ( cons @ A @ X @ Xs2 ) @ N )
= X )
= ( N
= ( zero_zero @ nat ) ) ) ) ) ).
% nth_equal_first_eq
thf(fact_3851_nth__non__equal__first__eq,axiom,
! [A: $tType,X: A,Y3: A,Xs2: list @ A,N: nat] :
( ( X != Y3 )
=> ( ( ( nth @ A @ ( cons @ A @ X @ Xs2 ) @ N )
= Y3 )
= ( ( ( nth @ A @ Xs2 @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) )
= Y3 )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% nth_non_equal_first_eq
thf(fact_3852_product__nth,axiom,
! [A: $tType,B: $tType,N: nat,Xs2: list @ A,Ys2: list @ B] :
( ( ord_less @ nat @ N @ ( times_times @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ ( size_size @ ( list @ B ) @ Ys2 ) ) )
=> ( ( nth @ ( product_prod @ A @ B ) @ ( product @ A @ B @ Xs2 @ Ys2 ) @ N )
= ( product_Pair @ A @ B @ ( nth @ A @ Xs2 @ ( divide_divide @ nat @ N @ ( size_size @ ( list @ B ) @ Ys2 ) ) ) @ ( nth @ B @ Ys2 @ ( modulo_modulo @ nat @ N @ ( size_size @ ( list @ B ) @ Ys2 ) ) ) ) ) ) ).
% product_nth
thf(fact_3853_positive__def,axiom,
( positive
= ( map_fun @ rat @ ( product_prod @ int @ int ) @ $o @ $o @ rep_Rat @ ( id @ $o )
@ ^ [X2: product_prod @ int @ int] : ( ord_less @ int @ ( zero_zero @ int ) @ ( times_times @ int @ ( product_fst @ int @ int @ X2 ) @ ( product_snd @ int @ int @ X2 ) ) ) ) ) ).
% positive_def
thf(fact_3854_remdups__adj__altdef,axiom,
! [A: $tType,Xs2: list @ A,Ys2: list @ A] :
( ( ( remdups_adj @ A @ Xs2 )
= Ys2 )
= ( ? [F2: nat > nat] :
( ( order_mono @ nat @ nat @ F2 )
& ( ( image2 @ nat @ nat @ F2 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs2 ) ) )
= ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Ys2 ) ) )
& ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( nth @ A @ Xs2 @ I2 )
= ( nth @ A @ Ys2 @ ( F2 @ I2 ) ) ) )
& ! [I2: nat] :
( ( ord_less @ nat @ ( plus_plus @ nat @ I2 @ ( one_one @ nat ) ) @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( ( nth @ A @ Xs2 @ I2 )
= ( nth @ A @ Xs2 @ ( plus_plus @ nat @ I2 @ ( one_one @ nat ) ) ) )
= ( ( F2 @ I2 )
= ( F2 @ ( plus_plus @ nat @ I2 @ ( one_one @ nat ) ) ) ) ) ) ) ) ) ).
% remdups_adj_altdef
thf(fact_3855_times__int_Otransfer,axiom,
( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) ) @ ( int > int ) @ pcr_int @ ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ ( product_prod @ nat @ nat ) @ int @ pcr_int @ pcr_int )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X2: nat,Y: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U: nat,V6: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ X2 @ U ) @ ( times_times @ nat @ Y @ V6 ) ) @ ( plus_plus @ nat @ ( times_times @ nat @ X2 @ V6 ) @ ( times_times @ nat @ Y @ U ) ) ) ) )
@ ( times_times @ int ) ) ).
% times_int.transfer
thf(fact_3856_zero__int_Otransfer,axiom,
pcr_int @ ( product_Pair @ nat @ nat @ ( zero_zero @ nat ) @ ( zero_zero @ nat ) ) @ ( zero_zero @ int ) ).
% zero_int.transfer
thf(fact_3857_int__transfer,axiom,
( bNF_rel_fun @ nat @ nat @ ( product_prod @ nat @ nat ) @ int
@ ^ [Y6: nat,Z2: nat] : Y6 = Z2
@ pcr_int
@ ^ [N5: nat] : ( product_Pair @ nat @ nat @ N5 @ ( zero_zero @ nat ) )
@ ( semiring_1_of_nat @ int ) ) ).
% int_transfer
thf(fact_3858_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 )
@ ^ [X2: nat,Y: nat] : ( product_Pair @ nat @ nat @ Y @ X2 ) )
@ ( uminus_uminus @ int ) ) ).
% uminus_int.transfer
thf(fact_3859_nat_Otransfer,axiom,
( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ nat @ nat @ pcr_int
@ ^ [Y6: nat,Z2: nat] : Y6 = Z2
@ ( product_case_prod @ nat @ nat @ nat @ ( minus_minus @ nat ) )
@ nat2 ) ).
% nat.transfer
thf(fact_3860_one__int_Otransfer,axiom,
pcr_int @ ( product_Pair @ nat @ nat @ ( one_one @ nat ) @ ( zero_zero @ nat ) ) @ ( one_one @ int ) ).
% one_int.transfer
thf(fact_3861_of__int_Otransfer,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ A @ A @ pcr_int
@ ^ [Y6: A,Z2: A] : Y6 = Z2
@ ( 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_3862_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
@ ^ [Y6: $o,Z2: $o] : Y6 = Z2 )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X2: nat,Y: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U: nat,V6: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ X2 @ V6 ) @ ( plus_plus @ nat @ U @ Y ) ) ) )
@ ( ord_less @ int ) ) ).
% less_int.transfer
thf(fact_3863_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
@ ^ [Y6: $o,Z2: $o] : Y6 = Z2 )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X2: nat,Y: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U: nat,V6: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ X2 @ V6 ) @ ( plus_plus @ nat @ U @ Y ) ) ) )
@ ( ord_less_eq @ int ) ) ).
% less_eq_int.transfer
thf(fact_3864_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 ) )
@ ^ [X2: nat,Y: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U: nat,V6: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X2 @ U ) @ ( plus_plus @ nat @ Y @ V6 ) ) ) )
@ ( plus_plus @ int ) ) ).
% plus_int.transfer
thf(fact_3865_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 ) )
@ ^ [X2: nat,Y: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U: nat,V6: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X2 @ V6 ) @ ( plus_plus @ nat @ Y @ U ) ) ) )
@ ( minus_minus @ int ) ) ).
% minus_int.transfer
thf(fact_3866_plus__rat__def,axiom,
( ( plus_plus @ rat )
= ( map_fun @ rat @ ( product_prod @ int @ int ) @ ( ( product_prod @ int @ int ) > ( product_prod @ int @ int ) ) @ ( rat > rat ) @ rep_Rat @ ( map_fun @ rat @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ rat @ rep_Rat @ abs_Rat )
@ ^ [X2: product_prod @ int @ int,Y: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( plus_plus @ int @ ( times_times @ int @ ( product_fst @ int @ int @ X2 ) @ ( product_snd @ int @ int @ Y ) ) @ ( times_times @ int @ ( product_fst @ int @ int @ Y ) @ ( product_snd @ int @ int @ X2 ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ X2 ) @ ( product_snd @ int @ int @ Y ) ) ) ) ) ).
% plus_rat_def
thf(fact_3867_inverse__rat__def,axiom,
( ( inverse_inverse @ rat )
= ( map_fun @ rat @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ rat @ rep_Rat @ abs_Rat
@ ^ [X2: product_prod @ int @ int] :
( 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_def
thf(fact_3868_times__rat__def,axiom,
( ( times_times @ rat )
= ( map_fun @ rat @ ( product_prod @ int @ int ) @ ( ( product_prod @ int @ int ) > ( product_prod @ int @ int ) ) @ ( rat > rat ) @ rep_Rat @ ( map_fun @ rat @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ rat @ rep_Rat @ abs_Rat )
@ ^ [X2: product_prod @ int @ int,Y: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( times_times @ int @ ( product_fst @ int @ int @ X2 ) @ ( product_fst @ int @ int @ Y ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ X2 ) @ ( product_snd @ int @ int @ Y ) ) ) ) ) ).
% times_rat_def
thf(fact_3869_uminus__rat__def,axiom,
( ( uminus_uminus @ rat )
= ( map_fun @ rat @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ rat @ rep_Rat @ abs_Rat
@ ^ [X2: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( uminus_uminus @ int @ ( product_fst @ int @ int @ X2 ) ) @ ( product_snd @ int @ int @ X2 ) ) ) ) ).
% uminus_rat_def
thf(fact_3870_one__rat__def,axiom,
( ( one_one @ rat )
= ( abs_Rat @ ( product_Pair @ int @ int @ ( one_one @ int ) @ ( one_one @ int ) ) ) ) ).
% one_rat_def
thf(fact_3871_zero__rat__def,axiom,
( ( zero_zero @ rat )
= ( abs_Rat @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) ) ) ) ).
% zero_rat_def
thf(fact_3872_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 )
@ ^ [X2: product_prod @ int @ int] : ( divide_divide @ A @ ( ring_1_of_int @ A @ ( product_fst @ int @ int @ X2 ) ) @ ( ring_1_of_int @ A @ ( product_snd @ int @ int @ X2 ) ) ) ) ) ) ).
% of_rat_def
thf(fact_3873_plus__rat_Oabs__eq,axiom,
! [Xa: product_prod @ int @ int,X: product_prod @ int @ int] :
( ( ratrel @ Xa @ Xa )
=> ( ( ratrel @ X @ X )
=> ( ( plus_plus @ rat @ ( abs_Rat @ Xa ) @ ( abs_Rat @ X ) )
= ( abs_Rat @ ( product_Pair @ int @ int @ ( plus_plus @ int @ ( times_times @ int @ ( product_fst @ int @ int @ Xa ) @ ( product_snd @ int @ int @ X ) ) @ ( times_times @ int @ ( product_fst @ int @ int @ X ) @ ( product_snd @ int @ int @ Xa ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ Xa ) @ ( product_snd @ int @ int @ X ) ) ) ) ) ) ) ).
% plus_rat.abs_eq
thf(fact_3874_inverse__rat_Oabs__eq,axiom,
! [X: product_prod @ int @ int] :
( ( ratrel @ X @ X )
=> ( ( inverse_inverse @ rat @ ( abs_Rat @ X ) )
= ( abs_Rat
@ ( if @ ( product_prod @ int @ int )
@ ( ( product_fst @ int @ int @ X )
= ( zero_zero @ int ) )
@ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
@ ( product_Pair @ int @ int @ ( product_snd @ int @ int @ X ) @ ( product_fst @ int @ int @ X ) ) ) ) ) ) ).
% inverse_rat.abs_eq
thf(fact_3875_horner__sum__eq__sum,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_1 @ A )
=> ( ( groups4207007520872428315er_sum @ B @ A )
= ( ^ [F2: B > A,A6: A,Xs: list @ B] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N5: nat] : ( times_times @ A @ ( F2 @ ( nth @ B @ Xs @ N5 ) ) @ ( power_power @ A @ A6 @ N5 ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ B ) @ Xs ) ) ) ) ) ) ).
% horner_sum_eq_sum
thf(fact_3876_zero__eq__of__rat__iff,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: rat] :
( ( ( zero_zero @ A )
= ( field_char_0_of_rat @ A @ A3 ) )
= ( ( zero_zero @ rat )
= A3 ) ) ) ).
% zero_eq_of_rat_iff
thf(fact_3877_of__rat__eq__0__iff,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: rat] :
( ( ( field_char_0_of_rat @ A @ A3 )
= ( zero_zero @ A ) )
= ( A3
= ( zero_zero @ rat ) ) ) ) ).
% of_rat_eq_0_iff
thf(fact_3878_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_3879_one__eq__of__rat__iff,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: rat] :
( ( ( one_one @ A )
= ( field_char_0_of_rat @ A @ A3 ) )
= ( ( one_one @ rat )
= A3 ) ) ) ).
% one_eq_of_rat_iff
thf(fact_3880_of__rat__eq__1__iff,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: rat] :
( ( ( field_char_0_of_rat @ A @ A3 )
= ( one_one @ A ) )
= ( A3
= ( one_one @ rat ) ) ) ) ).
% of_rat_eq_1_iff
thf(fact_3881_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_3882_horner__sum__simps_I2_J,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_0 @ A )
=> ! [F: B > A,A3: A,X: B,Xs2: list @ B] :
( ( groups4207007520872428315er_sum @ B @ A @ F @ A3 @ ( cons @ B @ X @ Xs2 ) )
= ( plus_plus @ A @ ( F @ X ) @ ( times_times @ A @ A3 @ ( groups4207007520872428315er_sum @ B @ A @ F @ A3 @ Xs2 ) ) ) ) ) ).
% horner_sum_simps(2)
thf(fact_3883_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_3884_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_3885_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_3886_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_3887_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_3888_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_3889_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_3890_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_3891_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_3892_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
@ ^ [A6: B] : ( field_char_0_of_rat @ A @ ( F @ A6 ) )
@ A5 ) ) ) ).
% of_rat_sum
thf(fact_3893_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
@ ^ [A6: B] : ( field_char_0_of_rat @ A @ ( F @ A6 ) )
@ A5 ) ) ) ).
% of_rat_prod
thf(fact_3894_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_3895_of__rat__add,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: rat,B2: rat] :
( ( field_char_0_of_rat @ A @ ( plus_plus @ rat @ A3 @ B2 ) )
= ( plus_plus @ A @ ( field_char_0_of_rat @ A @ A3 ) @ ( field_char_0_of_rat @ A @ B2 ) ) ) ) ).
% of_rat_add
thf(fact_3896_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_3897_Fract_Orsp,axiom,
( bNF_rel_fun @ int @ int @ ( int > ( product_prod @ int @ int ) ) @ ( int > ( product_prod @ int @ int ) )
@ ^ [Y6: int,Z2: int] : Y6 = Z2
@ ( bNF_rel_fun @ int @ int @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int )
@ ^ [Y6: int,Z2: int] : Y6 = Z2
@ ratrel )
@ ^ [A6: int,B5: int] :
( if @ ( product_prod @ int @ int )
@ ( B5
= ( zero_zero @ int ) )
@ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
@ ( product_Pair @ int @ int @ A6 @ B5 ) )
@ ^ [A6: int,B5: int] :
( if @ ( product_prod @ int @ int )
@ ( B5
= ( zero_zero @ int ) )
@ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
@ ( product_Pair @ int @ int @ A6 @ B5 ) ) ) ).
% Fract.rsp
thf(fact_3898_of__rat_Orsp,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( bNF_rel_fun @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ A @ A @ ratrel
@ ^ [Y6: A,Z2: A] : Y6 = Z2
@ ^ [X2: product_prod @ int @ int] : ( divide_divide @ A @ ( ring_1_of_int @ A @ ( product_fst @ int @ int @ X2 ) ) @ ( ring_1_of_int @ A @ ( product_snd @ int @ int @ X2 ) ) )
@ ^ [X2: product_prod @ int @ int] : ( divide_divide @ A @ ( ring_1_of_int @ A @ ( product_fst @ int @ int @ X2 ) ) @ ( ring_1_of_int @ A @ ( product_snd @ int @ int @ X2 ) ) ) ) ) ).
% of_rat.rsp
thf(fact_3899_ratrel__def,axiom,
( ratrel
= ( ^ [X2: product_prod @ int @ int,Y: product_prod @ int @ int] :
( ( ( product_snd @ int @ int @ X2 )
!= ( zero_zero @ int ) )
& ( ( product_snd @ int @ int @ Y )
!= ( zero_zero @ int ) )
& ( ( times_times @ int @ ( product_fst @ int @ int @ X2 ) @ ( product_snd @ int @ int @ Y ) )
= ( times_times @ int @ ( product_fst @ int @ int @ Y ) @ ( product_snd @ int @ int @ X2 ) ) ) ) ) ) ).
% ratrel_def
thf(fact_3900_times__rat_Orsp,axiom,
( bNF_rel_fun @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ ( ( product_prod @ int @ int ) > ( product_prod @ int @ int ) ) @ ( ( product_prod @ int @ int ) > ( product_prod @ int @ int ) ) @ ratrel @ ( bNF_rel_fun @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ ratrel @ ratrel )
@ ^ [X2: product_prod @ int @ int,Y: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( times_times @ int @ ( product_fst @ int @ int @ X2 ) @ ( product_fst @ int @ int @ Y ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ X2 ) @ ( product_snd @ int @ int @ Y ) ) )
@ ^ [X2: product_prod @ int @ int,Y: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( times_times @ int @ ( product_fst @ int @ int @ X2 ) @ ( product_fst @ int @ int @ Y ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ X2 ) @ ( product_snd @ int @ int @ Y ) ) ) ) ).
% times_rat.rsp
thf(fact_3901_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
@ ^ [X2: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( uminus_uminus @ int @ ( product_fst @ int @ int @ X2 ) ) @ ( product_snd @ int @ int @ X2 ) )
@ ^ [X2: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( uminus_uminus @ int @ ( product_fst @ int @ int @ X2 ) ) @ ( product_snd @ int @ int @ X2 ) ) ) ).
% uminus_rat.rsp
thf(fact_3902_positive_Orsp,axiom,
( bNF_rel_fun @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ $o @ $o @ ratrel
@ ^ [Y6: $o,Z2: $o] : Y6 = Z2
@ ^ [X2: product_prod @ int @ int] : ( ord_less @ int @ ( zero_zero @ int ) @ ( times_times @ int @ ( product_fst @ int @ int @ X2 ) @ ( product_snd @ int @ int @ X2 ) ) )
@ ^ [X2: product_prod @ int @ int] : ( ord_less @ int @ ( zero_zero @ int ) @ ( times_times @ int @ ( product_fst @ int @ int @ X2 ) @ ( product_snd @ int @ int @ X2 ) ) ) ) ).
% positive.rsp
thf(fact_3903_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
@ ^ [X2: product_prod @ int @ int] :
( 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 ) ) )
@ ^ [X2: product_prod @ int @ int] :
( 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.rsp
thf(fact_3904_plus__rat_Orsp,axiom,
( bNF_rel_fun @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ ( ( product_prod @ int @ int ) > ( product_prod @ int @ int ) ) @ ( ( product_prod @ int @ int ) > ( product_prod @ int @ int ) ) @ ratrel @ ( bNF_rel_fun @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ ratrel @ ratrel )
@ ^ [X2: product_prod @ int @ int,Y: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( plus_plus @ int @ ( times_times @ int @ ( product_fst @ int @ int @ X2 ) @ ( product_snd @ int @ int @ Y ) ) @ ( times_times @ int @ ( product_fst @ int @ int @ Y ) @ ( product_snd @ int @ int @ X2 ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ X2 ) @ ( product_snd @ int @ int @ Y ) ) )
@ ^ [X2: product_prod @ int @ int,Y: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( plus_plus @ int @ ( times_times @ int @ ( product_fst @ int @ int @ X2 ) @ ( product_snd @ int @ int @ Y ) ) @ ( times_times @ int @ ( product_fst @ int @ int @ Y ) @ ( product_snd @ int @ int @ X2 ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ X2 ) @ ( product_snd @ int @ int @ Y ) ) ) ) ).
% plus_rat.rsp
thf(fact_3905_uminus__rat_Oabs__eq,axiom,
! [X: product_prod @ int @ int] :
( ( ratrel @ X @ X )
=> ( ( uminus_uminus @ rat @ ( abs_Rat @ X ) )
= ( abs_Rat @ ( product_Pair @ int @ int @ ( uminus_uminus @ int @ ( product_fst @ int @ int @ X ) ) @ ( product_snd @ int @ int @ X ) ) ) ) ) ).
% uminus_rat.abs_eq
thf(fact_3906_of__rat_Otransfer,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( bNF_rel_fun @ ( product_prod @ int @ int ) @ rat @ A @ A @ pcr_rat
@ ^ [Y6: A,Z2: A] : Y6 = Z2
@ ^ [X2: product_prod @ int @ int] : ( divide_divide @ A @ ( ring_1_of_int @ A @ ( product_fst @ int @ int @ X2 ) ) @ ( ring_1_of_int @ A @ ( product_snd @ int @ int @ X2 ) ) )
@ ( field_char_0_of_rat @ A ) ) ) ).
% of_rat.transfer
thf(fact_3907_times__rat_Oabs__eq,axiom,
! [Xa: product_prod @ int @ int,X: product_prod @ int @ int] :
( ( ratrel @ Xa @ Xa )
=> ( ( ratrel @ X @ X )
=> ( ( times_times @ rat @ ( abs_Rat @ Xa ) @ ( abs_Rat @ X ) )
= ( abs_Rat @ ( product_Pair @ int @ int @ ( times_times @ int @ ( product_fst @ int @ int @ Xa ) @ ( product_fst @ int @ int @ X ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ Xa ) @ ( product_snd @ int @ int @ X ) ) ) ) ) ) ) ).
% times_rat.abs_eq
thf(fact_3908_cr__rat__def,axiom,
( cr_rat
= ( ^ [X2: product_prod @ int @ int,Y: rat] :
( ( ratrel @ X2 @ X2 )
& ( ( abs_Rat @ X2 )
= Y ) ) ) ) ).
% cr_rat_def
thf(fact_3909_horner__sum__eq__sum__funpow,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_0 @ A )
=> ( ( groups4207007520872428315er_sum @ B @ A )
= ( ^ [F2: B > A,A6: A,Xs: list @ B] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N5: nat] : ( compow @ ( A > A ) @ N5 @ ( times_times @ A @ A6 ) @ ( F2 @ ( nth @ B @ Xs @ N5 ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ B ) @ Xs ) ) ) ) ) ) ).
% horner_sum_eq_sum_funpow
thf(fact_3910_times__int_Orsp,axiom,
( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) ) @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) ) @ intrel @ ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ intrel @ intrel )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X2: nat,Y: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U: nat,V6: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ X2 @ U ) @ ( times_times @ nat @ Y @ V6 ) ) @ ( plus_plus @ nat @ ( times_times @ nat @ X2 @ V6 ) @ ( times_times @ nat @ Y @ U ) ) ) ) )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X2: nat,Y: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U: nat,V6: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ X2 @ U ) @ ( times_times @ nat @ Y @ V6 ) ) @ ( plus_plus @ nat @ ( times_times @ nat @ X2 @ V6 ) @ ( times_times @ nat @ Y @ U ) ) ) ) ) ) ).
% times_int.rsp
thf(fact_3911_nth__image,axiom,
! [A: $tType,L: nat,Xs2: list @ A] :
( ( ord_less_eq @ nat @ L @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( image2 @ nat @ A @ ( nth @ A @ Xs2 ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ L ) )
= ( set2 @ A @ ( take @ A @ L @ Xs2 ) ) ) ) ).
% nth_image
thf(fact_3912_Suc__funpow,axiom,
! [N: nat] :
( ( compow @ ( nat > nat ) @ N @ suc )
= ( plus_plus @ nat @ N ) ) ).
% Suc_funpow
thf(fact_3913_funpow__0,axiom,
! [A: $tType,F: A > A,X: A] :
( ( compow @ ( A > A ) @ ( zero_zero @ nat ) @ F @ X )
= X ) ).
% funpow_0
thf(fact_3914_id__funpow,axiom,
! [A: $tType,N: nat] :
( ( compow @ ( A > A ) @ N @ ( id @ A ) )
= ( id @ A ) ) ).
% id_funpow
thf(fact_3915_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_3916_intrel__iff,axiom,
! [X: nat,Y3: nat,U2: nat,V: nat] :
( ( intrel @ ( product_Pair @ nat @ nat @ X @ Y3 ) @ ( product_Pair @ nat @ nat @ U2 @ V ) )
= ( ( plus_plus @ nat @ X @ V )
= ( plus_plus @ nat @ U2 @ Y3 ) ) ) ).
% intrel_iff
thf(fact_3917_take__Cons__numeral,axiom,
! [A: $tType,V: num,X: A,Xs2: list @ A] :
( ( take @ A @ ( numeral_numeral @ nat @ V ) @ ( cons @ A @ X @ Xs2 ) )
= ( cons @ A @ X @ ( take @ A @ ( minus_minus @ nat @ ( numeral_numeral @ nat @ V ) @ ( one_one @ nat ) ) @ Xs2 ) ) ) ).
% take_Cons_numeral
thf(fact_3918_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_3919_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_3920_funpow__swap1,axiom,
! [A: $tType,F: A > A,N: nat,X: A] :
( ( F @ ( compow @ ( A > A ) @ N @ F @ X ) )
= ( compow @ ( A > A ) @ N @ F @ ( F @ X ) ) ) ).
% funpow_swap1
thf(fact_3921_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_3922_int_Oabs__eq__iff,axiom,
! [X: product_prod @ nat @ nat,Y3: product_prod @ nat @ nat] :
( ( ( abs_Integ @ X )
= ( abs_Integ @ Y3 ) )
= ( intrel @ X @ Y3 ) ) ).
% int.abs_eq_iff
thf(fact_3923_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
@ ^ [X2: A] : ( compow @ ( B > B ) @ ( G @ X2 ) @ ( F @ X2 ) ) ) ) ).
% comp_fun_commute.comp_fun_commute_funpow
thf(fact_3924_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_3925_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_3926_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_3927_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_3928_funpow__mono,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F: A > A,A5: A,B4: A,N: nat] :
( ( order_mono @ A @ A @ F )
=> ( ( ord_less_eq @ A @ A5 @ B4 )
=> ( ord_less_eq @ A @ ( compow @ ( A > A ) @ N @ F @ A5 ) @ ( compow @ ( A > A ) @ N @ F @ B4 ) ) ) ) ) ).
% funpow_mono
thf(fact_3929_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_3930_Kleene__iter__lpfp,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [F: A > A,P4: A,K: nat] :
( ( order_mono @ A @ A @ F )
=> ( ( ord_less_eq @ A @ ( F @ P4 ) @ P4 )
=> ( ord_less_eq @ A @ ( compow @ ( A > A ) @ K @ F @ ( bot_bot @ A ) ) @ P4 ) ) ) ) ).
% Kleene_iter_lpfp
thf(fact_3931_Kleene__iter__gpfp,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [F: A > A,P4: A,K: nat] :
( ( order_mono @ A @ A @ F )
=> ( ( ord_less_eq @ A @ P4 @ ( F @ P4 ) )
=> ( ord_less_eq @ A @ P4 @ ( compow @ ( A > A ) @ K @ F @ ( top_top @ A ) ) ) ) ) ) ).
% Kleene_iter_gpfp
thf(fact_3932_funpow__mono2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F: A > A,I: nat,J: nat,X: A,Y3: A] :
( ( order_mono @ A @ A @ F )
=> ( ( ord_less_eq @ nat @ I @ J )
=> ( ( ord_less_eq @ A @ X @ Y3 )
=> ( ( ord_less_eq @ A @ X @ ( F @ X ) )
=> ( ord_less_eq @ A @ ( compow @ ( A > A ) @ I @ F @ X ) @ ( compow @ ( A > A ) @ J @ F @ Y3 ) ) ) ) ) ) ) ).
% funpow_mono2
thf(fact_3933_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 )
@ ^ [X2: nat,Y: nat] : ( product_Pair @ nat @ nat @ Y @ X2 ) )
@ ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [X2: nat,Y: nat] : ( product_Pair @ nat @ nat @ Y @ X2 ) ) ) ).
% uminus_int.rsp
thf(fact_3934_nat_Orsp,axiom,
( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ nat @ nat @ intrel
@ ^ [Y6: nat,Z2: nat] : Y6 = Z2
@ ( product_case_prod @ nat @ nat @ nat @ ( minus_minus @ nat ) )
@ ( product_case_prod @ nat @ nat @ nat @ ( minus_minus @ nat ) ) ) ).
% nat.rsp
thf(fact_3935_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_3936_of__nat__def,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( semiring_1_of_nat @ A )
= ( ^ [N5: nat] : ( compow @ ( A > A ) @ N5 @ ( plus_plus @ A @ ( one_one @ A ) ) @ ( zero_zero @ A ) ) ) ) ) ).
% of_nat_def
thf(fact_3937_numeral__add__unfold__funpow,axiom,
! [A: $tType] :
( ( semiring_numeral @ A )
=> ! [K: num,A3: A] :
( ( plus_plus @ A @ ( numeral_numeral @ A @ K ) @ A3 )
= ( compow @ ( A > A ) @ ( numeral_numeral @ nat @ K ) @ ( plus_plus @ A @ ( one_one @ A ) ) @ A3 ) ) ) ).
% numeral_add_unfold_funpow
thf(fact_3938_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_3939_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_3940_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_3941_of__int_Orsp,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ A @ A @ intrel
@ ^ [Y6: A,Z2: A] : Y6 = Z2
@ ( 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_3942_intrel__def,axiom,
( intrel
= ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X2: nat,Y: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U: nat,V6: nat] :
( ( plus_plus @ nat @ X2 @ V6 )
= ( plus_plus @ nat @ U @ Y ) ) ) ) ) ).
% intrel_def
thf(fact_3943_numeral__unfold__funpow,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( numeral_numeral @ A )
= ( ^ [K3: num] : ( compow @ ( A > A ) @ ( numeral_numeral @ nat @ K3 ) @ ( plus_plus @ A @ ( one_one @ A ) ) @ ( zero_zero @ A ) ) ) ) ) ).
% numeral_unfold_funpow
thf(fact_3944_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
@ ^ [Y6: $o,Z2: $o] : Y6 = Z2 )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X2: nat,Y: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U: nat,V6: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ X2 @ V6 ) @ ( plus_plus @ nat @ U @ Y ) ) ) )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X2: nat,Y: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U: nat,V6: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ X2 @ V6 ) @ ( plus_plus @ nat @ U @ Y ) ) ) ) ) ).
% less_int.rsp
thf(fact_3945_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
@ ^ [Y6: $o,Z2: $o] : Y6 = Z2 )
@ intrel
@ ^ [Y6: int,Z2: int] : Y6 = Z2 ) ).
% int.rel_eq_transfer
thf(fact_3946_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
@ ^ [Y6: $o,Z2: $o] : Y6 = Z2 )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X2: nat,Y: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U: nat,V6: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ X2 @ V6 ) @ ( plus_plus @ nat @ U @ Y ) ) ) )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X2: nat,Y: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U: nat,V6: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ X2 @ V6 ) @ ( plus_plus @ nat @ U @ Y ) ) ) ) ) ).
% less_eq_int.rsp
thf(fact_3947_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 ) )
@ ^ [X2: nat,Y: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U: nat,V6: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X2 @ V6 ) @ ( plus_plus @ nat @ Y @ U ) ) ) )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X2: nat,Y: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U: nat,V6: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X2 @ V6 ) @ ( plus_plus @ nat @ Y @ U ) ) ) ) ) ).
% minus_int.rsp
thf(fact_3948_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 ) )
@ ^ [X2: nat,Y: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U: nat,V6: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X2 @ U ) @ ( plus_plus @ nat @ Y @ V6 ) ) ) )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X2: nat,Y: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U: nat,V6: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X2 @ U ) @ ( plus_plus @ nat @ Y @ V6 ) ) ) ) ) ).
% plus_int.rsp
thf(fact_3949_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_3950_relpowp__fun__conv,axiom,
! [A: $tType] :
( ( compow @ ( A > A > $o ) )
= ( ^ [N5: nat,P3: A > A > $o,X2: A,Y: A] :
? [F2: nat > A] :
( ( ( F2 @ ( zero_zero @ nat ) )
= X2 )
& ( ( F2 @ N5 )
= Y )
& ! [I2: nat] :
( ( ord_less @ nat @ I2 @ N5 )
=> ( P3 @ ( F2 @ I2 ) @ ( F2 @ ( suc @ I2 ) ) ) ) ) ) ) ).
% relpowp_fun_conv
thf(fact_3951_relpowp__1,axiom,
! [A: $tType,P: A > A > $o] :
( ( compow @ ( A > A > $o ) @ ( one_one @ nat ) @ P )
= P ) ).
% relpowp_1
thf(fact_3952_Nat_Ofunpow__code__def,axiom,
! [A: $tType] :
( ( funpow @ A )
= ( compow @ ( A > A ) ) ) ).
% Nat.funpow_code_def
thf(fact_3953_relpowp__0__I,axiom,
! [A: $tType,P: A > A > $o,X: A] : ( compow @ ( A > A > $o ) @ ( zero_zero @ nat ) @ P @ X @ X ) ).
% relpowp_0_I
thf(fact_3954_relpowp__0__E,axiom,
! [A: $tType,P: A > A > $o,X: A,Y3: A] :
( ( compow @ ( A > A > $o ) @ ( zero_zero @ nat ) @ P @ X @ Y3 )
=> ( X = Y3 ) ) ).
% relpowp_0_E
thf(fact_3955_relpowp_Osimps_I1_J,axiom,
! [A: $tType,R5: A > A > $o] :
( ( compow @ ( A > A > $o ) @ ( zero_zero @ nat ) @ R5 )
= ( ^ [Y6: A,Z2: A] : Y6 = Z2 ) ) ).
% relpowp.simps(1)
thf(fact_3956_relpowp__E,axiom,
! [A: $tType,N: nat,P: A > A > $o,X: A,Z: A] :
( ( compow @ ( A > A > $o ) @ N @ P @ X @ Z )
=> ( ( ( N
= ( zero_zero @ nat ) )
=> ( X != Z ) )
=> ~ ! [Y4: A,M4: nat] :
( ( N
= ( suc @ M4 ) )
=> ( ( compow @ ( A > A > $o ) @ M4 @ P @ X @ Y4 )
=> ~ ( P @ Y4 @ Z ) ) ) ) ) ).
% relpowp_E
thf(fact_3957_relpowp__E2,axiom,
! [A: $tType,N: nat,P: A > A > $o,X: A,Z: A] :
( ( compow @ ( A > A > $o ) @ N @ P @ X @ Z )
=> ( ( ( N
= ( zero_zero @ nat ) )
=> ( X != Z ) )
=> ~ ! [Y4: A,M4: nat] :
( ( N
= ( suc @ M4 ) )
=> ( ( P @ X @ Y4 )
=> ~ ( compow @ ( A > A > $o ) @ M4 @ P @ Y4 @ Z ) ) ) ) ) ).
% relpowp_E2
thf(fact_3958_relpow__finite__bounded1,axiom,
! [A: $tType,R5: set @ ( product_prod @ A @ A ),K: nat] :
( ( finite_finite2 @ ( product_prod @ A @ A ) @ R5 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ K @ R5 )
@ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ A ) )
@ ( image2 @ nat @ ( set @ ( product_prod @ A @ A ) )
@ ^ [N5: nat] : ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N5 @ R5 )
@ ( collect @ nat
@ ^ [N5: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N5 )
& ( ord_less_eq @ nat @ N5 @ ( finite_card @ ( product_prod @ A @ A ) @ R5 ) ) ) ) ) ) ) ) ) ).
% relpow_finite_bounded1
thf(fact_3959_lexord__take__index__conv,axiom,
! [A: $tType,X: list @ A,Y3: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Y3 ) @ ( lexord @ A @ R ) )
= ( ( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ X ) @ ( size_size @ ( list @ A ) @ Y3 ) )
& ( ( take @ A @ ( size_size @ ( list @ A ) @ X ) @ Y3 )
= X ) )
| ? [I2: nat] :
( ( ord_less @ nat @ I2 @ ( ord_min @ nat @ ( size_size @ ( list @ A ) @ X ) @ ( size_size @ ( list @ A ) @ Y3 ) ) )
& ( ( take @ A @ I2 @ X )
= ( take @ A @ I2 @ Y3 ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( nth @ A @ X @ I2 ) @ ( nth @ A @ Y3 @ I2 ) ) @ R ) ) ) ) ).
% lexord_take_index_conv
thf(fact_3960_lex__take__index,axiom,
! [A: $tType,Xs2: list @ A,Ys2: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Ys2 ) @ ( lex @ A @ R ) )
=> ~ ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Ys2 ) )
=> ( ( ( take @ A @ I3 @ Xs2 )
= ( take @ A @ I3 @ Ys2 ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( nth @ A @ Xs2 @ I3 ) @ ( nth @ A @ Ys2 @ I3 ) ) @ R ) ) ) ) ) ).
% lex_take_index
thf(fact_3961_find__Some__iff,axiom,
! [A: $tType,P: A > $o,Xs2: list @ A,X: A] :
( ( ( find @ A @ P @ Xs2 )
= ( some @ A @ X ) )
= ( ? [I2: nat] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs2 ) )
& ( P @ ( nth @ A @ Xs2 @ I2 ) )
& ( X
= ( nth @ A @ Xs2 @ I2 ) )
& ! [J3: nat] :
( ( ord_less @ nat @ J3 @ I2 )
=> ~ ( P @ ( nth @ A @ Xs2 @ J3 ) ) ) ) ) ) ).
% find_Some_iff
thf(fact_3962_relpow__1,axiom,
! [A: $tType,R5: set @ ( product_prod @ A @ A )] :
( ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( one_one @ nat ) @ R5 )
= R5 ) ).
% relpow_1
thf(fact_3963_lexord__cons__cons,axiom,
! [A: $tType,A3: A,X: list @ A,B2: A,Y3: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ A3 @ X ) @ ( cons @ A @ B2 @ Y3 ) ) @ ( lexord @ A @ R ) )
= ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R )
| ( ( A3 = B2 )
& ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Y3 ) @ ( lexord @ A @ R ) ) ) ) ) ).
% lexord_cons_cons
thf(fact_3964_finite__relpow,axiom,
! [A: $tType,R5: set @ ( product_prod @ A @ A ),N: nat] :
( ( finite_finite2 @ ( product_prod @ A @ A ) @ R5 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( finite_finite2 @ ( product_prod @ A @ A ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R5 ) ) ) ) ).
% finite_relpow
thf(fact_3965_Cons__in__lex,axiom,
! [A: $tType,X: A,Xs2: list @ A,Y3: A,Ys2: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs2 ) @ ( cons @ A @ Y3 @ Ys2 ) ) @ ( lex @ A @ R ) )
= ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ R )
& ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ A ) @ Ys2 ) ) )
| ( ( X = Y3 )
& ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Ys2 ) @ ( lex @ A @ R ) ) ) ) ) ).
% Cons_in_lex
thf(fact_3966_relpow__Suc__D2_H,axiom,
! [A: $tType,N: nat,R5: set @ ( product_prod @ A @ A ),X5: A,Y5: A,Z8: A] :
( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ Y5 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R5 ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y5 @ Z8 ) @ R5 ) )
=> ? [W3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ W3 ) @ R5 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ W3 @ Z8 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R5 ) ) ) ) ).
% relpow_Suc_D2'
thf(fact_3967_lexord__linear,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),X: list @ A,Y3: list @ A] :
( ! [A4: A,B3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ R )
| ( A4 = B3 )
| ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B3 @ A4 ) @ R ) )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Y3 ) @ ( lexord @ A @ R ) )
| ( X = Y3 )
| ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Y3 @ X ) @ ( lexord @ A @ R ) ) ) ) ).
% lexord_linear
thf(fact_3968_lexord__irreflexive,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),Xs2: list @ A] :
( ! [X4: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ X4 ) @ R )
=> ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Xs2 ) @ ( lexord @ A @ R ) ) ) ).
% lexord_irreflexive
thf(fact_3969_lexord__lex,axiom,
! [A: $tType,X: list @ A,Y3: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Y3 ) @ ( lex @ A @ R ) )
= ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Y3 ) @ ( lexord @ A @ R ) )
& ( ( size_size @ ( list @ A ) @ X )
= ( size_size @ ( list @ A ) @ Y3 ) ) ) ) ).
% lexord_lex
thf(fact_3970_relpow__Suc__E,axiom,
! [A: $tType,X: A,Z: A,N: nat,R5: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( suc @ N ) @ R5 ) )
=> ~ ! [Y4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y4 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R5 ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z ) @ R5 ) ) ) ).
% relpow_Suc_E
thf(fact_3971_relpow__Suc__I,axiom,
! [A: $tType,X: A,Y3: A,N: nat,R5: set @ ( product_prod @ A @ A ),Z: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R5 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z ) @ R5 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( suc @ N ) @ R5 ) ) ) ) ).
% relpow_Suc_I
thf(fact_3972_relpow__Suc__D2,axiom,
! [A: $tType,X: A,Z: A,N: nat,R5: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( suc @ N ) @ R5 ) )
=> ? [Y4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y4 ) @ R5 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R5 ) ) ) ) ).
% relpow_Suc_D2
thf(fact_3973_relpow__Suc__E2,axiom,
! [A: $tType,X: A,Z: A,N: nat,R5: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( suc @ N ) @ R5 ) )
=> ~ ! [Y4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y4 ) @ R5 )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R5 ) ) ) ) ).
% relpow_Suc_E2
thf(fact_3974_relpow__Suc__I2,axiom,
! [A: $tType,X: A,Y3: A,R5: set @ ( product_prod @ A @ A ),Z: A,N: nat] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ R5 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R5 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( suc @ N ) @ R5 ) ) ) ) ).
% relpow_Suc_I2
thf(fact_3975_relpow__0__I,axiom,
! [A: $tType,X: A,R5: set @ ( product_prod @ A @ A )] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ X ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( zero_zero @ nat ) @ R5 ) ) ).
% relpow_0_I
thf(fact_3976_relpow__0__E,axiom,
! [A: $tType,X: A,Y3: A,R5: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( zero_zero @ nat ) @ R5 ) )
=> ( X = Y3 ) ) ).
% relpow_0_E
thf(fact_3977_lexord__partial__trans,axiom,
! [A: $tType,Xs2: list @ A,R: set @ ( product_prod @ A @ A ),Ys2: list @ A,Zs: list @ A] :
( ! [X4: A,Y4: A,Z6: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs2 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y4 ) @ R )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z6 ) @ R )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Z6 ) @ R ) ) ) )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Ys2 ) @ ( lexord @ A @ R ) )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys2 @ Zs ) @ ( lexord @ A @ R ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Zs ) @ ( lexord @ A @ R ) ) ) ) ) ).
% lexord_partial_trans
thf(fact_3978_relpow__add,axiom,
! [A: $tType,M: nat,N: nat,R5: set @ ( product_prod @ A @ A )] :
( ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( plus_plus @ nat @ M @ N ) @ R5 )
= ( relcomp @ A @ A @ A @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ M @ R5 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R5 ) ) ) ).
% relpow_add
thf(fact_3979_relpowp__relpow__eq,axiom,
! [A: $tType,N: nat,R5: set @ ( product_prod @ A @ A )] :
( ( compow @ ( A > A > $o ) @ N
@ ^ [X2: A,Y: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y ) @ R5 ) )
= ( ^ [X2: A,Y: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R5 ) ) ) ) ).
% relpowp_relpow_eq
thf(fact_3980_relpow__E,axiom,
! [A: $tType,X: A,Z: A,N: nat,R5: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R5 ) )
=> ( ( ( N
= ( zero_zero @ nat ) )
=> ( X != Z ) )
=> ~ ! [Y4: A,M4: nat] :
( ( N
= ( suc @ M4 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y4 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ M4 @ R5 ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z ) @ R5 ) ) ) ) ) ).
% relpow_E
thf(fact_3981_relpow__E2,axiom,
! [A: $tType,X: A,Z: A,N: nat,R5: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R5 ) )
=> ( ( ( N
= ( zero_zero @ nat ) )
=> ( X != Z ) )
=> ~ ! [Y4: A,M4: nat] :
( ( N
= ( suc @ M4 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y4 ) @ R5 )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ M4 @ R5 ) ) ) ) ) ) ).
% relpow_E2
thf(fact_3982_find_Osimps_I2_J,axiom,
! [A: $tType,P: A > $o,X: A,Xs2: list @ A] :
( ( ( P @ X )
=> ( ( find @ A @ P @ ( cons @ A @ X @ Xs2 ) )
= ( some @ A @ X ) ) )
& ( ~ ( P @ X )
=> ( ( find @ A @ P @ ( cons @ A @ X @ Xs2 ) )
= ( find @ A @ P @ Xs2 ) ) ) ) ).
% find.simps(2)
thf(fact_3983_find__None__iff,axiom,
! [A: $tType,P: A > $o,Xs2: list @ A] :
( ( ( find @ A @ P @ Xs2 )
= ( none @ A ) )
= ( ~ ? [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ Xs2 ) )
& ( P @ X2 ) ) ) ) ).
% find_None_iff
thf(fact_3984_find__None__iff2,axiom,
! [A: $tType,P: A > $o,Xs2: list @ A] :
( ( ( none @ A )
= ( find @ A @ P @ Xs2 ) )
= ( ~ ? [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ Xs2 ) )
& ( P @ X2 ) ) ) ) ).
% find_None_iff2
thf(fact_3985_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_3986_relpow__fun__conv,axiom,
! [A: $tType,A3: A,B2: A,N: nat,R5: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R5 ) )
= ( ? [F2: nat > A] :
( ( ( F2 @ ( zero_zero @ nat ) )
= A3 )
& ( ( F2 @ N )
= B2 )
& ! [I2: nat] :
( ( ord_less @ nat @ I2 @ N )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( F2 @ I2 ) @ ( F2 @ ( suc @ I2 ) ) ) @ R5 ) ) ) ) ) ).
% relpow_fun_conv
thf(fact_3987_relpow__finite__bounded,axiom,
! [A: $tType,R5: set @ ( product_prod @ A @ A ),K: nat] :
( ( finite_finite2 @ ( product_prod @ A @ A ) @ R5 )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ K @ R5 )
@ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ A ) )
@ ( image2 @ nat @ ( set @ ( product_prod @ A @ A ) )
@ ^ [N5: nat] : ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N5 @ R5 )
@ ( collect @ nat
@ ^ [N5: nat] : ( ord_less_eq @ nat @ N5 @ ( finite_card @ ( product_prod @ A @ A ) @ R5 ) ) ) ) ) ) ) ).
% relpow_finite_bounded
thf(fact_3988_find__Some__iff2,axiom,
! [A: $tType,X: A,P: A > $o,Xs2: list @ A] :
( ( ( some @ A @ X )
= ( find @ A @ P @ Xs2 ) )
= ( ? [I2: nat] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs2 ) )
& ( P @ ( nth @ A @ Xs2 @ I2 ) )
& ( X
= ( nth @ A @ Xs2 @ I2 ) )
& ! [J3: nat] :
( ( ord_less @ nat @ J3 @ I2 )
=> ~ ( P @ ( nth @ A @ Xs2 @ J3 ) ) ) ) ) ) ).
% find_Some_iff2
thf(fact_3989_List_Olexordp__def,axiom,
! [A: $tType] :
( ( lexordp @ A )
= ( ^ [R4: A > A > $o,Xs: list @ A,Ys: list @ A] : ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( lexord @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ R4 ) ) ) ) ) ) ).
% List.lexordp_def
thf(fact_3990_ntrancl__def,axiom,
! [A: $tType] :
( ( transitive_ntrancl @ A )
= ( ^ [N5: nat,R6: 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 @ R6 )
@ ( collect @ nat
@ ^ [I2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ I2 )
& ( ord_less_eq @ nat @ I2 @ ( suc @ N5 ) ) ) ) ) ) ) ) ).
% ntrancl_def
thf(fact_3991_trancl__finite__eq__relpow,axiom,
! [A: $tType,R5: set @ ( product_prod @ A @ A )] :
( ( finite_finite2 @ ( product_prod @ A @ A ) @ R5 )
=> ( ( transitive_trancl @ A @ R5 )
= ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ A ) )
@ ( image2 @ nat @ ( set @ ( product_prod @ A @ A ) )
@ ^ [N5: nat] : ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N5 @ R5 )
@ ( collect @ nat
@ ^ [N5: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N5 )
& ( ord_less_eq @ nat @ N5 @ ( finite_card @ ( product_prod @ A @ A ) @ R5 ) ) ) ) ) ) ) ) ).
% trancl_finite_eq_relpow
thf(fact_3992_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_3993_ntrancl__Zero,axiom,
! [A: $tType,R5: set @ ( product_prod @ A @ A )] :
( ( transitive_ntrancl @ A @ ( zero_zero @ nat ) @ R5 )
= R5 ) ).
% ntrancl_Zero
thf(fact_3994_trancl_Ocases,axiom,
! [A: $tType,A1: A,A22: A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A1 @ A22 ) @ ( transitive_trancl @ A @ R ) )
=> ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A1 @ A22 ) @ R )
=> ~ ! [B3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A1 @ B3 ) @ ( transitive_trancl @ A @ R ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B3 @ A22 ) @ R ) ) ) ) ).
% trancl.cases
thf(fact_3995_trancl_Osimps,axiom,
! [A: $tType,A1: A,A22: A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A1 @ A22 ) @ ( transitive_trancl @ A @ R ) )
= ( ? [A6: A,B5: A] :
( ( A1 = A6 )
& ( A22 = B5 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A6 @ B5 ) @ R ) )
| ? [A6: A,B5: A,C4: A] :
( ( A1 = A6 )
& ( A22 = C4 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A6 @ B5 ) @ ( transitive_trancl @ A @ R ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B5 @ C4 ) @ R ) ) ) ) ).
% trancl.simps
thf(fact_3996_trancl_Or__into__trancl,axiom,
! [A: $tType,A3: A,B2: A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_trancl @ A @ R ) ) ) ).
% trancl.r_into_trancl
thf(fact_3997_tranclE,axiom,
! [A: $tType,A3: A,B2: A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_trancl @ A @ R ) )
=> ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R )
=> ~ ! [C3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ C3 ) @ ( transitive_trancl @ A @ R ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ C3 @ B2 ) @ R ) ) ) ) ).
% tranclE
thf(fact_3998_trancl__trans,axiom,
! [A: $tType,X: A,Y3: A,R: set @ ( product_prod @ A @ A ),Z: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ ( transitive_trancl @ A @ R ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z ) @ ( transitive_trancl @ A @ R ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z ) @ ( transitive_trancl @ A @ R ) ) ) ) ).
% trancl_trans
thf(fact_3999_trancl__induct,axiom,
! [A: $tType,A3: A,B2: A,R: set @ ( product_prod @ A @ A ),P: A > $o] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_trancl @ A @ R ) )
=> ( ! [Y4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ Y4 ) @ R )
=> ( P @ Y4 ) )
=> ( ! [Y4: A,Z6: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ Y4 ) @ ( transitive_trancl @ A @ R ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z6 ) @ R )
=> ( ( P @ Y4 )
=> ( P @ Z6 ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% trancl_induct
thf(fact_4000_r__r__into__trancl,axiom,
! [A: $tType,A3: A,B2: A,R5: set @ ( product_prod @ A @ A ),C2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R5 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ C2 ) @ R5 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ C2 ) @ ( transitive_trancl @ A @ R5 ) ) ) ) ).
% r_r_into_trancl
thf(fact_4001_converse__tranclE,axiom,
! [A: $tType,X: A,Z: A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z ) @ ( transitive_trancl @ A @ R ) )
=> ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z ) @ R )
=> ~ ! [Y4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y4 ) @ R )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z ) @ ( transitive_trancl @ A @ R ) ) ) ) ) ).
% converse_tranclE
thf(fact_4002_irrefl__trancl__rD,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),X: A,Y3: A] :
( ! [X4: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ X4 ) @ ( transitive_trancl @ A @ R ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ R )
=> ( X != Y3 ) ) ) ).
% irrefl_trancl_rD
thf(fact_4003_Transitive__Closure_Otrancl__into__trancl,axiom,
! [A: $tType,A3: A,B2: A,R: set @ ( product_prod @ A @ A ),C2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_trancl @ A @ R ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ C2 ) @ R )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ C2 ) @ ( transitive_trancl @ A @ R ) ) ) ) ).
% Transitive_Closure.trancl_into_trancl
thf(fact_4004_trancl__into__trancl2,axiom,
! [A: $tType,A3: A,B2: A,R: set @ ( product_prod @ A @ A ),C2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ C2 ) @ ( transitive_trancl @ A @ R ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ C2 ) @ ( transitive_trancl @ A @ R ) ) ) ) ).
% trancl_into_trancl2
thf(fact_4005_trancl__trans__induct,axiom,
! [A: $tType,X: A,Y3: A,R: set @ ( product_prod @ A @ A ),P: A > A > $o] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ ( transitive_trancl @ A @ R ) )
=> ( ! [X4: A,Y4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y4 ) @ R )
=> ( P @ X4 @ Y4 ) )
=> ( ! [X4: A,Y4: A,Z6: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y4 ) @ ( transitive_trancl @ A @ R ) )
=> ( ( P @ X4 @ Y4 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z6 ) @ ( transitive_trancl @ A @ R ) )
=> ( ( P @ Y4 @ Z6 )
=> ( P @ X4 @ Z6 ) ) ) ) )
=> ( P @ X @ Y3 ) ) ) ) ).
% trancl_trans_induct
thf(fact_4006_converse__trancl__induct,axiom,
! [A: $tType,A3: A,B2: A,R: set @ ( product_prod @ A @ A ),P: A > $o] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_trancl @ A @ R ) )
=> ( ! [Y4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ B2 ) @ R )
=> ( P @ Y4 ) )
=> ( ! [Y4: A,Z6: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z6 ) @ R )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z6 @ B2 ) @ ( transitive_trancl @ A @ R ) )
=> ( ( P @ Z6 )
=> ( P @ Y4 ) ) ) )
=> ( P @ A3 ) ) ) ) ).
% converse_trancl_induct
thf(fact_4007_trancl__induct2,axiom,
! [A: $tType,B: $tType,Ax: A,Ay: B,Bx: A,By: B,R: set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ),P: A > B > $o] :
( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Ax @ Ay ) @ ( product_Pair @ A @ B @ Bx @ By ) ) @ ( transitive_trancl @ ( product_prod @ A @ B ) @ R ) )
=> ( ! [A4: A,B3: B] :
( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Ax @ Ay ) @ ( product_Pair @ A @ B @ A4 @ B3 ) ) @ R )
=> ( P @ A4 @ B3 ) )
=> ( ! [A4: A,B3: B,Aa2: A,Ba: B] :
( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Ax @ Ay ) @ ( product_Pair @ A @ B @ A4 @ B3 ) ) @ ( transitive_trancl @ ( product_prod @ A @ B ) @ R ) )
=> ( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B3 ) @ ( product_Pair @ A @ B @ Aa2 @ Ba ) ) @ R )
=> ( ( P @ A4 @ B3 )
=> ( P @ Aa2 @ Ba ) ) ) )
=> ( P @ Bx @ By ) ) ) ) ).
% trancl_induct2
thf(fact_4008_finite__trancl__ntranl,axiom,
! [A: $tType,R5: set @ ( product_prod @ A @ A )] :
( ( finite_finite2 @ ( product_prod @ A @ A ) @ R5 )
=> ( ( transitive_trancl @ A @ R5 )
= ( transitive_ntrancl @ A @ ( minus_minus @ nat @ ( finite_card @ ( product_prod @ A @ A ) @ R5 ) @ ( one_one @ nat ) ) @ R5 ) ) ) ).
% finite_trancl_ntranl
thf(fact_4009_trancl__set__ntrancl,axiom,
! [A: $tType,Xs2: list @ ( product_prod @ A @ A )] :
( ( transitive_trancl @ A @ ( set2 @ ( product_prod @ A @ A ) @ Xs2 ) )
= ( transitive_ntrancl @ A @ ( minus_minus @ nat @ ( finite_card @ ( product_prod @ A @ A ) @ ( set2 @ ( product_prod @ A @ A ) @ Xs2 ) ) @ ( one_one @ nat ) ) @ ( set2 @ ( product_prod @ A @ A ) @ Xs2 ) ) ) ).
% trancl_set_ntrancl
thf(fact_4010_trancl__power,axiom,
! [A: $tType,P4: product_prod @ A @ A,R5: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ P4 @ ( transitive_trancl @ A @ R5 ) )
= ( ? [N5: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N5 )
& ( member @ ( product_prod @ A @ A ) @ P4 @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N5 @ R5 ) ) ) ) ) ).
% trancl_power
thf(fact_4011_lenlex__irreflexive,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),Xs2: list @ A] :
( ! [X4: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ X4 ) @ R )
=> ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Xs2 ) @ ( lenlex @ A @ R ) ) ) ).
% lenlex_irreflexive
thf(fact_4012_less__eq,axiom,
! [M: nat,N: nat] :
( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ M @ N ) @ ( transitive_trancl @ nat @ pred_nat ) )
= ( ord_less @ nat @ M @ N ) ) ).
% less_eq
thf(fact_4013_trancl__insert2,axiom,
! [A: $tType,A3: A,B2: A,R: set @ ( product_prod @ A @ A )] :
( ( transitive_trancl @ A @ ( insert2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R ) )
= ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ ( transitive_trancl @ A @ R )
@ ( collect @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ^ [X2: A,Y: A] :
( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ A3 ) @ ( transitive_trancl @ A @ R ) )
| ( X2 = A3 ) )
& ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ Y ) @ ( transitive_trancl @ A @ R ) )
| ( Y = B2 ) ) ) ) ) ) ) ).
% trancl_insert2
thf(fact_4014_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_4015_lenlex__conv,axiom,
! [A: $tType] :
( ( lenlex @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ ( list @ A ) @ ( list @ A ) )
@ ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
@ ^ [Xs: list @ A,Ys: list @ A] :
( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ A ) @ Ys ) )
| ( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ A ) @ Ys ) )
& ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( lex @ A @ R4 ) ) ) ) ) ) ) ) ).
% lenlex_conv
thf(fact_4016_rtrancl__finite__eq__relpow,axiom,
! [A: $tType,R5: set @ ( product_prod @ A @ A )] :
( ( finite_finite2 @ ( product_prod @ A @ A ) @ R5 )
=> ( ( transitive_rtrancl @ A @ R5 )
= ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ A ) )
@ ( image2 @ nat @ ( set @ ( product_prod @ A @ A ) )
@ ^ [N5: nat] : ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N5 @ R5 )
@ ( collect @ nat
@ ^ [N5: nat] : ( ord_less_eq @ nat @ N5 @ ( finite_card @ ( product_prod @ A @ A ) @ R5 ) ) ) ) ) ) ) ).
% rtrancl_finite_eq_relpow
thf(fact_4017_nth__rotate,axiom,
! [A: $tType,N: nat,Xs2: list @ A,M: nat] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( nth @ A @ ( rotate @ A @ M @ Xs2 ) @ N )
= ( nth @ A @ Xs2 @ ( modulo_modulo @ nat @ ( plus_plus @ nat @ M @ N ) @ ( size_size @ ( list @ A ) @ Xs2 ) ) ) ) ) ).
% nth_rotate
thf(fact_4018_nth__zip,axiom,
! [A: $tType,B: $tType,I: nat,Xs2: list @ A,Ys2: list @ B] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ B ) @ Ys2 ) )
=> ( ( nth @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs2 @ Ys2 ) @ I )
= ( product_Pair @ A @ B @ ( nth @ A @ Xs2 @ I ) @ ( nth @ B @ Ys2 @ I ) ) ) ) ) ).
% nth_zip
thf(fact_4019_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_4020_rotate0,axiom,
! [A: $tType] :
( ( rotate @ A @ ( zero_zero @ nat ) )
= ( id @ ( list @ A ) ) ) ).
% rotate0
thf(fact_4021_rotate__length01,axiom,
! [A: $tType,Xs2: list @ A,N: nat] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ ( one_one @ nat ) )
=> ( ( rotate @ A @ N @ Xs2 )
= Xs2 ) ) ).
% rotate_length01
thf(fact_4022_rotate__id,axiom,
! [A: $tType,N: nat,Xs2: list @ A] :
( ( ( modulo_modulo @ nat @ N @ ( size_size @ ( list @ A ) @ Xs2 ) )
= ( zero_zero @ nat ) )
=> ( ( rotate @ A @ N @ Xs2 )
= Xs2 ) ) ).
% rotate_id
thf(fact_4023_zip__Cons__Cons,axiom,
! [A: $tType,B: $tType,X: A,Xs2: list @ A,Y3: B,Ys2: list @ B] :
( ( zip @ A @ B @ ( cons @ A @ X @ Xs2 ) @ ( cons @ B @ Y3 @ Ys2 ) )
= ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y3 ) @ ( zip @ A @ B @ Xs2 @ Ys2 ) ) ) ).
% zip_Cons_Cons
thf(fact_4024_converse__rtrancl__into__rtrancl,axiom,
! [A: $tType,A3: A,B2: A,R: set @ ( product_prod @ A @ A ),C2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ C2 ) @ ( transitive_rtrancl @ A @ R ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ C2 ) @ ( transitive_rtrancl @ A @ R ) ) ) ) ).
% converse_rtrancl_into_rtrancl
thf(fact_4025_converse__rtrancl__induct,axiom,
! [A: $tType,A3: A,B2: A,R: set @ ( product_prod @ A @ A ),P: A > $o] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_rtrancl @ A @ R ) )
=> ( ( P @ B2 )
=> ( ! [Y4: A,Z6: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z6 ) @ R )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z6 @ B2 ) @ ( transitive_rtrancl @ A @ R ) )
=> ( ( P @ Z6 )
=> ( P @ Y4 ) ) ) )
=> ( P @ A3 ) ) ) ) ).
% converse_rtrancl_induct
thf(fact_4026_converse__rtranclE,axiom,
! [A: $tType,X: A,Z: A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z ) @ ( transitive_rtrancl @ A @ R ) )
=> ( ( X != Z )
=> ~ ! [Y4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y4 ) @ R )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z ) @ ( transitive_rtrancl @ A @ R ) ) ) ) ) ).
% converse_rtranclE
thf(fact_4027_rtrancl__induct,axiom,
! [A: $tType,A3: A,B2: A,R: set @ ( product_prod @ A @ A ),P: A > $o] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_rtrancl @ A @ R ) )
=> ( ( P @ A3 )
=> ( ! [Y4: A,Z6: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ Y4 ) @ ( transitive_rtrancl @ A @ R ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z6 ) @ R )
=> ( ( P @ Y4 )
=> ( P @ Z6 ) ) ) )
=> ( P @ B2 ) ) ) ) ).
% rtrancl_induct
thf(fact_4028_rtrancl__trans,axiom,
! [A: $tType,X: A,Y3: A,R: set @ ( product_prod @ A @ A ),Z: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ ( transitive_rtrancl @ A @ R ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z ) @ ( transitive_rtrancl @ A @ R ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z ) @ ( transitive_rtrancl @ A @ R ) ) ) ) ).
% rtrancl_trans
thf(fact_4029_rtranclE,axiom,
! [A: $tType,A3: A,B2: A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_rtrancl @ A @ R ) )
=> ( ( A3 != B2 )
=> ~ ! [Y4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ Y4 ) @ ( transitive_rtrancl @ A @ R ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ B2 ) @ R ) ) ) ) ).
% rtranclE
thf(fact_4030_rtrancl_Ortrancl__into__rtrancl,axiom,
! [A: $tType,A3: A,B2: A,R: set @ ( product_prod @ A @ A ),C2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_rtrancl @ A @ R ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ C2 ) @ R )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ C2 ) @ ( transitive_rtrancl @ A @ R ) ) ) ) ).
% rtrancl.rtrancl_into_rtrancl
thf(fact_4031_rtrancl_Ortrancl__refl,axiom,
! [A: $tType,A3: A,R: set @ ( product_prod @ A @ A )] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ A3 ) @ ( transitive_rtrancl @ A @ R ) ) ).
% rtrancl.rtrancl_refl
thf(fact_4032_rtrancl_Osimps,axiom,
! [A: $tType,A1: A,A22: A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A1 @ A22 ) @ ( transitive_rtrancl @ A @ R ) )
= ( ? [A6: A] :
( ( A1 = A6 )
& ( A22 = A6 ) )
| ? [A6: A,B5: A,C4: A] :
( ( A1 = A6 )
& ( A22 = C4 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A6 @ B5 ) @ ( transitive_rtrancl @ A @ R ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B5 @ C4 ) @ R ) ) ) ) ).
% rtrancl.simps
thf(fact_4033_rtrancl_Ocases,axiom,
! [A: $tType,A1: A,A22: A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A1 @ A22 ) @ ( transitive_rtrancl @ A @ R ) )
=> ( ( A22 != A1 )
=> ~ ! [B3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A1 @ B3 ) @ ( transitive_rtrancl @ A @ R ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B3 @ A22 ) @ R ) ) ) ) ).
% rtrancl.cases
thf(fact_4034_rotate__rotate,axiom,
! [A: $tType,M: nat,N: nat,Xs2: list @ A] :
( ( rotate @ A @ M @ ( rotate @ A @ N @ Xs2 ) )
= ( rotate @ A @ ( plus_plus @ nat @ M @ N ) @ Xs2 ) ) ).
% rotate_rotate
thf(fact_4035_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_4036_trancl__rtrancl__trancl,axiom,
! [A: $tType,A3: A,B2: A,R: set @ ( product_prod @ A @ A ),C2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_trancl @ A @ R ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ C2 ) @ ( transitive_rtrancl @ A @ R ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ C2 ) @ ( transitive_trancl @ A @ R ) ) ) ) ).
% trancl_rtrancl_trancl
thf(fact_4037_rtrancl__trancl__trancl,axiom,
! [A: $tType,X: A,Y3: A,R: set @ ( product_prod @ A @ A ),Z: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ ( transitive_rtrancl @ A @ R ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z ) @ ( transitive_trancl @ A @ R ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z ) @ ( transitive_trancl @ A @ R ) ) ) ) ).
% rtrancl_trancl_trancl
thf(fact_4038_rtrancl__into__trancl2,axiom,
! [A: $tType,A3: A,B2: A,R: set @ ( product_prod @ A @ A ),C2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ C2 ) @ ( transitive_rtrancl @ A @ R ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ C2 ) @ ( transitive_trancl @ A @ R ) ) ) ) ).
% rtrancl_into_trancl2
thf(fact_4039_rtrancl__into__trancl1,axiom,
! [A: $tType,A3: A,B2: A,R: set @ ( product_prod @ A @ A ),C2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_rtrancl @ A @ R ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ C2 ) @ R )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ C2 ) @ ( transitive_trancl @ A @ R ) ) ) ) ).
% rtrancl_into_trancl1
thf(fact_4040_rtrancl__eq__or__trancl,axiom,
! [A: $tType,X: A,Y3: A,R5: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ ( transitive_rtrancl @ A @ R5 ) )
= ( ( X = Y3 )
| ( ( X != Y3 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ ( transitive_trancl @ A @ R5 ) ) ) ) ) ).
% rtrancl_eq_or_trancl
thf(fact_4041_trancl__into__rtrancl,axiom,
! [A: $tType,A3: A,B2: A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_trancl @ A @ R ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_rtrancl @ A @ R ) ) ) ).
% trancl_into_rtrancl
thf(fact_4042_tranclD2,axiom,
! [A: $tType,X: A,Y3: A,R5: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ ( transitive_trancl @ A @ R5 ) )
=> ? [Z6: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z6 ) @ ( transitive_rtrancl @ A @ R5 ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z6 @ Y3 ) @ R5 ) ) ) ).
% tranclD2
thf(fact_4043_rtranclD,axiom,
! [A: $tType,A3: A,B2: A,R5: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_rtrancl @ A @ R5 ) )
=> ( ( A3 = B2 )
| ( ( A3 != B2 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_trancl @ A @ R5 ) ) ) ) ) ).
% rtranclD
thf(fact_4044_tranclD,axiom,
! [A: $tType,X: A,Y3: A,R5: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ ( transitive_trancl @ A @ R5 ) )
=> ? [Z6: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z6 ) @ R5 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z6 @ Y3 ) @ ( transitive_rtrancl @ A @ R5 ) ) ) ) ).
% tranclD
thf(fact_4045_rtrancl__Un__separator__converseE,axiom,
! [A: $tType,A3: A,B2: A,P: set @ ( product_prod @ A @ A ),Q: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_rtrancl @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ P @ Q ) ) )
=> ( ! [X4: A,Y4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ B2 ) @ ( transitive_rtrancl @ A @ P ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ X4 ) @ Q )
=> ( Y4 = X4 ) ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_rtrancl @ A @ P ) ) ) ) ).
% rtrancl_Un_separator_converseE
thf(fact_4046_rtrancl__Un__separatorE,axiom,
! [A: $tType,A3: A,B2: A,P: set @ ( product_prod @ A @ A ),Q: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_rtrancl @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ P @ Q ) ) )
=> ( ! [X4: A,Y4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ X4 ) @ ( transitive_rtrancl @ A @ P ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y4 ) @ Q )
=> ( X4 = Y4 ) ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_rtrancl @ A @ P ) ) ) ) ).
% rtrancl_Un_separatorE
thf(fact_4047_converse__rtrancl__induct2,axiom,
! [A: $tType,B: $tType,Ax: A,Ay: B,Bx: A,By: B,R: set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ),P: A > B > $o] :
( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Ax @ Ay ) @ ( product_Pair @ A @ B @ Bx @ By ) ) @ ( transitive_rtrancl @ ( product_prod @ A @ B ) @ R ) )
=> ( ( P @ Bx @ By )
=> ( ! [A4: A,B3: B,Aa2: A,Ba: B] :
( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B3 ) @ ( product_Pair @ A @ B @ Aa2 @ Ba ) ) @ R )
=> ( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Aa2 @ Ba ) @ ( product_Pair @ A @ B @ Bx @ By ) ) @ ( transitive_rtrancl @ ( product_prod @ A @ B ) @ R ) )
=> ( ( P @ Aa2 @ Ba )
=> ( P @ A4 @ B3 ) ) ) )
=> ( P @ Ax @ Ay ) ) ) ) ).
% converse_rtrancl_induct2
thf(fact_4048_converse__rtranclE2,axiom,
! [B: $tType,A: $tType,Xa: A,Xb: B,Za2: A,Zb: B,R: set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) )] :
( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Xa @ Xb ) @ ( product_Pair @ A @ B @ Za2 @ Zb ) ) @ ( transitive_rtrancl @ ( product_prod @ A @ B ) @ R ) )
=> ( ( ( product_Pair @ A @ B @ Xa @ Xb )
!= ( product_Pair @ A @ B @ Za2 @ Zb ) )
=> ~ ! [A4: A,B3: B] :
( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Xa @ Xb ) @ ( product_Pair @ A @ B @ A4 @ B3 ) ) @ R )
=> ~ ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B3 ) @ ( product_Pair @ A @ B @ Za2 @ Zb ) ) @ ( transitive_rtrancl @ ( product_prod @ A @ B ) @ R ) ) ) ) ) ).
% converse_rtranclE2
thf(fact_4049_rtrancl__induct2,axiom,
! [A: $tType,B: $tType,Ax: A,Ay: B,Bx: A,By: B,R: set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ),P: A > B > $o] :
( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Ax @ Ay ) @ ( product_Pair @ A @ B @ Bx @ By ) ) @ ( transitive_rtrancl @ ( product_prod @ A @ B ) @ R ) )
=> ( ( P @ Ax @ Ay )
=> ( ! [A4: A,B3: B,Aa2: A,Ba: B] :
( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Ax @ Ay ) @ ( product_Pair @ A @ B @ A4 @ B3 ) ) @ ( transitive_rtrancl @ ( product_prod @ A @ B ) @ R ) )
=> ( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B3 ) @ ( product_Pair @ A @ B @ Aa2 @ Ba ) ) @ R )
=> ( ( P @ A4 @ B3 )
=> ( P @ Aa2 @ Ba ) ) ) )
=> ( P @ Bx @ By ) ) ) ) ).
% rtrancl_induct2
thf(fact_4050_set__zip__rightD,axiom,
! [A: $tType,B: $tType,X: A,Y3: B,Xs2: list @ A,Ys2: list @ B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y3 ) @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs2 @ Ys2 ) ) )
=> ( member @ B @ Y3 @ ( set2 @ B @ Ys2 ) ) ) ).
% set_zip_rightD
thf(fact_4051_set__zip__leftD,axiom,
! [B: $tType,A: $tType,X: A,Y3: B,Xs2: list @ A,Ys2: list @ B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y3 ) @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs2 @ Ys2 ) ) )
=> ( member @ A @ X @ ( set2 @ A @ Xs2 ) ) ) ).
% set_zip_leftD
thf(fact_4052_in__set__zipE,axiom,
! [A: $tType,B: $tType,X: A,Y3: B,Xs2: list @ A,Ys2: list @ B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y3 ) @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs2 @ Ys2 ) ) )
=> ~ ( ( member @ A @ X @ ( set2 @ A @ Xs2 ) )
=> ~ ( member @ B @ Y3 @ ( set2 @ B @ Ys2 ) ) ) ) ).
% in_set_zipE
thf(fact_4053_zip__same,axiom,
! [A: $tType,A3: A,B2: A,Xs2: list @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( set2 @ ( product_prod @ A @ A ) @ ( zip @ A @ A @ Xs2 @ Xs2 ) ) )
= ( ( member @ A @ A3 @ ( set2 @ A @ Xs2 ) )
& ( A3 = B2 ) ) ) ).
% zip_same
thf(fact_4054_zip__eq__ConsE,axiom,
! [A: $tType,B: $tType,Xs2: list @ A,Ys2: list @ B,Xy2: product_prod @ A @ B,Xys: list @ ( product_prod @ A @ B )] :
( ( ( zip @ A @ B @ Xs2 @ Ys2 )
= ( cons @ ( product_prod @ A @ B ) @ Xy2 @ Xys ) )
=> ~ ! [X4: A,Xs3: list @ A] :
( ( Xs2
= ( cons @ A @ X4 @ Xs3 ) )
=> ! [Y4: B,Ys3: list @ B] :
( ( Ys2
= ( cons @ B @ Y4 @ Ys3 ) )
=> ( ( Xy2
= ( product_Pair @ A @ B @ X4 @ Y4 ) )
=> ( Xys
!= ( zip @ A @ B @ Xs3 @ Ys3 ) ) ) ) ) ) ).
% zip_eq_ConsE
thf(fact_4055_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_4056_antimono__iff__le__Suc,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( order_antimono @ nat @ A )
= ( ^ [F2: nat > A] :
! [N5: nat] : ( ord_less_eq @ A @ ( F2 @ ( suc @ N5 ) ) @ ( F2 @ N5 ) ) ) ) ) ).
% antimono_iff_le_Suc
thf(fact_4057_in__set__impl__in__set__zip2,axiom,
! [A: $tType,B: $tType,Xs2: list @ A,Ys2: list @ B,Y3: B] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ B ) @ Ys2 ) )
=> ( ( member @ B @ Y3 @ ( set2 @ B @ Ys2 ) )
=> ~ ! [X4: A] :
~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y3 ) @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs2 @ Ys2 ) ) ) ) ) ).
% in_set_impl_in_set_zip2
thf(fact_4058_in__set__impl__in__set__zip1,axiom,
! [A: $tType,B: $tType,Xs2: list @ A,Ys2: list @ B,X: A] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ B ) @ Ys2 ) )
=> ( ( member @ A @ X @ ( set2 @ A @ Xs2 ) )
=> ~ ! [Y4: B] :
~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y4 ) @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs2 @ Ys2 ) ) ) ) ) ).
% in_set_impl_in_set_zip1
thf(fact_4059_rtrancl__insert,axiom,
! [A: $tType,A3: A,B2: A,R: set @ ( product_prod @ A @ A )] :
( ( transitive_rtrancl @ A @ ( insert2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R ) )
= ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ ( transitive_rtrancl @ A @ R )
@ ( collect @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ^ [X2: A,Y: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ A3 ) @ ( transitive_rtrancl @ A @ R ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ Y ) @ ( transitive_rtrancl @ A @ R ) ) ) ) ) ) ) ).
% rtrancl_insert
thf(fact_4060_rtrancl__is__UN__relpow,axiom,
! [A: $tType] :
( ( transitive_rtrancl @ A )
= ( ^ [R6: set @ ( product_prod @ A @ A )] :
( complete_Sup_Sup @ ( set @ ( product_prod @ A @ A ) )
@ ( image2 @ nat @ ( set @ ( product_prod @ A @ A ) )
@ ^ [N5: nat] : ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N5 @ R6 )
@ ( top_top @ ( set @ nat ) ) ) ) ) ) ).
% rtrancl_is_UN_relpow
thf(fact_4061_rtrancl__imp__UN__relpow,axiom,
! [A: $tType,P4: product_prod @ A @ A,R5: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ P4 @ ( transitive_rtrancl @ A @ R5 ) )
=> ( member @ ( product_prod @ A @ A ) @ P4
@ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ A ) )
@ ( image2 @ nat @ ( set @ ( product_prod @ A @ A ) )
@ ^ [N5: nat] : ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N5 @ R5 )
@ ( top_top @ ( set @ nat ) ) ) ) ) ) ).
% rtrancl_imp_UN_relpow
thf(fact_4062_trancl__insert,axiom,
! [A: $tType,Y3: A,X: A,R: set @ ( product_prod @ A @ A )] :
( ( transitive_trancl @ A @ ( insert2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X ) @ R ) )
= ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ ( transitive_trancl @ A @ R )
@ ( collect @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ^ [A6: A,B5: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A6 @ Y3 ) @ ( transitive_rtrancl @ A @ R ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ B5 ) @ ( transitive_rtrancl @ A @ R ) ) ) ) ) ) ) ).
% trancl_insert
thf(fact_4063_set__zip,axiom,
! [B: $tType,A: $tType,Xs2: list @ A,Ys2: list @ B] :
( ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs2 @ Ys2 ) )
= ( collect @ ( product_prod @ A @ B )
@ ^ [Uu: product_prod @ A @ B] :
? [I2: nat] :
( ( Uu
= ( product_Pair @ A @ B @ ( nth @ A @ Xs2 @ I2 ) @ ( nth @ B @ Ys2 @ I2 ) ) )
& ( ord_less @ nat @ I2 @ ( ord_min @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ ( size_size @ ( list @ B ) @ Ys2 ) ) ) ) ) ) ).
% set_zip
thf(fact_4064_butlast__take,axiom,
! [A: $tType,N: nat,Xs2: list @ A] :
( ( ord_less_eq @ nat @ N @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( butlast @ A @ ( take @ A @ N @ Xs2 ) )
= ( take @ A @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ Xs2 ) ) ) ).
% butlast_take
thf(fact_4065_Image__fold,axiom,
! [B: $tType,A: $tType,R5: set @ ( product_prod @ A @ B ),S3: set @ A] :
( ( finite_finite2 @ ( product_prod @ A @ B ) @ R5 )
=> ( ( image @ A @ B @ R5 @ S3 )
= ( finite_fold @ ( product_prod @ A @ B ) @ ( set @ B )
@ ( product_case_prod @ A @ B @ ( ( set @ B ) > ( set @ B ) )
@ ^ [X2: A,Y: B,A8: set @ B] : ( if @ ( set @ B ) @ ( member @ A @ X2 @ S3 ) @ ( insert2 @ B @ Y @ A8 ) @ A8 ) )
@ ( bot_bot @ ( set @ B ) )
@ R5 ) ) ) ).
% Image_fold
thf(fact_4066_filterlim__base__iff,axiom,
! [A: $tType,C: $tType,B: $tType,D: $tType,I5: set @ A,F6: A > ( set @ B ),F: B > C,G6: D > ( set @ C ),J5: set @ D] :
( ( I5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [I3: A] :
( ( member @ A @ I3 @ I5 )
=> ! [J2: A] :
( ( member @ A @ J2 @ I5 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( F6 @ I3 ) @ ( F6 @ J2 ) )
| ( ord_less_eq @ ( set @ B ) @ ( F6 @ J2 ) @ ( F6 @ I3 ) ) ) ) )
=> ( ( filterlim @ B @ C @ F
@ ( complete_Inf_Inf @ ( filter @ C )
@ ( image2 @ D @ ( filter @ C )
@ ^ [J3: D] : ( principal @ C @ ( G6 @ J3 ) )
@ J5 ) )
@ ( complete_Inf_Inf @ ( filter @ B )
@ ( image2 @ A @ ( filter @ B )
@ ^ [I2: A] : ( principal @ B @ ( F6 @ I2 ) )
@ I5 ) ) )
= ( ! [X2: D] :
( ( member @ D @ X2 @ J5 )
=> ? [Y: A] :
( ( member @ A @ Y @ I5 )
& ! [Z3: B] :
( ( member @ B @ Z3 @ ( F6 @ Y ) )
=> ( member @ C @ ( F @ Z3 ) @ ( G6 @ X2 ) ) ) ) ) ) ) ) ) ).
% filterlim_base_iff
thf(fact_4067_ImageI,axiom,
! [B: $tType,A: $tType,A3: A,B2: B,R: set @ ( product_prod @ A @ B ),A5: set @ A] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ R )
=> ( ( member @ A @ A3 @ A5 )
=> ( member @ B @ B2 @ ( image @ A @ B @ R @ A5 ) ) ) ) ).
% ImageI
thf(fact_4068_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
@ ^ [X2: B] :
? [Y: A] :
( ( P @ Y )
& ( Q @ X2 @ Y ) ) ) )
= ( ! [Y: A] :
( ( P @ Y )
=> ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X2: B] : ( Q @ X2 @ Y ) ) ) ) ) ) ) ).
% finite_Collect_bounded_ex
thf(fact_4069_Image__singleton__iff,axiom,
! [A: $tType,B: $tType,B2: A,R: set @ ( product_prod @ B @ A ),A3: B] :
( ( member @ A @ B2 @ ( image @ B @ A @ R @ ( insert2 @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) )
= ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ A3 @ B2 ) @ R ) ) ).
% Image_singleton_iff
thf(fact_4070_length__butlast,axiom,
! [A: $tType,Xs2: list @ A] :
( ( size_size @ ( list @ A ) @ ( butlast @ A @ Xs2 ) )
= ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ ( one_one @ nat ) ) ) ).
% length_butlast
thf(fact_4071_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] :
? [X2: B] :
( ( Uu
= ( F @ X2 ) )
& ( P @ X2 ) ) ) )
= ( collect @ A
@ ^ [A6: A] :
? [X2: B] :
( ( P @ X2 )
& ( member @ A @ A6 @ ( F @ X2 ) ) ) ) ) ).
% Union_SetCompr_eq
thf(fact_4072_rev__ImageI,axiom,
! [B: $tType,A: $tType,A3: A,A5: set @ A,B2: B,R: set @ ( product_prod @ A @ B )] :
( ( member @ A @ A3 @ A5 )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ R )
=> ( member @ B @ B2 @ ( image @ A @ B @ R @ A5 ) ) ) ) ).
% rev_ImageI
thf(fact_4073_Image__iff,axiom,
! [A: $tType,B: $tType,B2: A,R: set @ ( product_prod @ B @ A ),A5: set @ B] :
( ( member @ A @ B2 @ ( image @ B @ A @ R @ A5 ) )
= ( ? [X2: B] :
( ( member @ B @ X2 @ A5 )
& ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X2 @ B2 ) @ R ) ) ) ) ).
% Image_iff
thf(fact_4074_ImageE,axiom,
! [A: $tType,B: $tType,B2: A,R: set @ ( product_prod @ B @ A ),A5: set @ B] :
( ( member @ A @ B2 @ ( image @ B @ A @ R @ A5 ) )
=> ~ ! [X4: B] :
( ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X4 @ B2 ) @ R )
=> ~ ( member @ B @ X4 @ A5 ) ) ) ).
% ImageE
thf(fact_4075_Setcompr__eq__image,axiom,
! [A: $tType,B: $tType,F: B > A,A5: set @ B] :
( ( collect @ A
@ ^ [Uu: A] :
? [X2: B] :
( ( Uu
= ( F @ X2 ) )
& ( member @ B @ X2 @ A5 ) ) )
= ( image2 @ B @ A @ F @ A5 ) ) ).
% Setcompr_eq_image
thf(fact_4076_setcompr__eq__image,axiom,
! [A: $tType,B: $tType,F: B > A,P: B > $o] :
( ( collect @ A
@ ^ [Uu: A] :
? [X2: B] :
( ( Uu
= ( F @ X2 ) )
& ( P @ X2 ) ) )
= ( image2 @ B @ A @ F @ ( collect @ B @ P ) ) ) ).
% setcompr_eq_image
thf(fact_4077_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] :
? [X2: A] :
( ( Uu
= ( F @ X2 ) )
& ( P @ X2 ) ) ) ) ) ).
% finite_image_set
thf(fact_4078_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] :
? [X2: A,Y: B] :
( ( Uu
= ( F @ X2 @ Y ) )
& ( P @ X2 )
& ( Q @ Y ) ) ) ) ) ) ).
% finite_image_set2
thf(fact_4079_filterlim__ident,axiom,
! [A: $tType,F6: filter @ A] :
( filterlim @ A @ A
@ ^ [X2: A] : X2
@ F6
@ F6 ) ).
% filterlim_ident
thf(fact_4080_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
@ ^ [X2: C] : ( G @ ( F @ X2 ) )
@ F33
@ F13 ) ) ) ).
% filterlim_compose
thf(fact_4081_finite_Omono,axiom,
! [A: $tType] :
( order_mono @ ( ( set @ A ) > $o ) @ ( ( set @ A ) > $o )
@ ^ [P5: ( set @ A ) > $o,X2: set @ A] :
( ( X2
= ( bot_bot @ ( set @ A ) ) )
| ? [A8: set @ A,A6: A] :
( ( X2
= ( insert2 @ A @ A6 @ A8 ) )
& ( P5 @ A8 ) ) ) ) ).
% finite.mono
thf(fact_4082_tranclp_Omono,axiom,
! [A: $tType,R: A > A > $o] :
( order_mono @ ( A > A > $o ) @ ( A > A > $o )
@ ^ [P5: A > A > $o,X12: A,X24: A] :
( ? [A6: A,B5: A] :
( ( X12 = A6 )
& ( X24 = B5 )
& ( R @ A6 @ B5 ) )
| ? [A6: A,B5: A,C4: A] :
( ( X12 = A6 )
& ( X24 = C4 )
& ( P5 @ A6 @ B5 )
& ( R @ B5 @ C4 ) ) ) ) ).
% tranclp.mono
thf(fact_4083_rtranclp_Omono,axiom,
! [A: $tType,R: A > A > $o] :
( order_mono @ ( A > A > $o ) @ ( A > A > $o )
@ ^ [P5: A > A > $o,X12: A,X24: A] :
( ? [A6: A] :
( ( X12 = A6 )
& ( X24 = A6 ) )
| ? [A6: A,B5: A,C4: A] :
( ( X12 = A6 )
& ( X24 = C4 )
& ( P5 @ A6 @ B5 )
& ( R @ B5 @ C4 ) ) ) ) ).
% rtranclp.mono
thf(fact_4084_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_4085_full__SetCompr__eq,axiom,
! [A: $tType,B: $tType,F: B > A] :
( ( collect @ A
@ ^ [U: A] :
? [X2: B] :
( U
= ( F @ X2 ) ) )
= ( image2 @ B @ A @ F @ ( top_top @ ( set @ B ) ) ) ) ).
% full_SetCompr_eq
thf(fact_4086_finite__inf__Sup,axiom,
! [A: $tType] :
( ( finite8700451911770168679attice @ A )
=> ! [A3: A,A5: set @ A] :
( ( inf_inf @ A @ A3 @ ( complete_Sup_Sup @ A @ A5 ) )
= ( complete_Sup_Sup @ A
@ ( collect @ A
@ ^ [Uu: A] :
? [B5: A] :
( ( Uu
= ( inf_inf @ A @ A3 @ B5 ) )
& ( member @ A @ B5 @ A5 ) ) ) ) ) ) ).
% finite_inf_Sup
thf(fact_4087_Image__UN,axiom,
! [A: $tType,B: $tType,C: $tType,R: set @ ( product_prod @ B @ A ),B4: C > ( set @ B ),A5: set @ C] :
( ( image @ B @ A @ R @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ C @ ( set @ B ) @ B4 @ A5 ) ) )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ C @ ( set @ A )
@ ^ [X2: C] : ( image @ B @ A @ R @ ( B4 @ X2 ) )
@ A5 ) ) ) ).
% Image_UN
thf(fact_4088_filterlim__INF,axiom,
! [A: $tType,B: $tType,C: $tType,F: A > B,G6: C > ( filter @ B ),B4: set @ C,F6: filter @ A] :
( ( filterlim @ A @ B @ F @ ( complete_Inf_Inf @ ( filter @ B ) @ ( image2 @ C @ ( filter @ B ) @ G6 @ B4 ) ) @ F6 )
= ( ! [X2: C] :
( ( member @ C @ X2 @ B4 )
=> ( filterlim @ A @ B @ F @ ( G6 @ X2 ) @ F6 ) ) ) ) ).
% filterlim_INF
thf(fact_4089_filterlim__INF_H,axiom,
! [C: $tType,B: $tType,A: $tType,X: A,A5: set @ A,F: B > C,F6: filter @ C,G6: A > ( filter @ B )] :
( ( member @ A @ X @ A5 )
=> ( ( filterlim @ B @ C @ F @ F6 @ ( G6 @ X ) )
=> ( filterlim @ B @ C @ F @ F6 @ ( complete_Inf_Inf @ ( filter @ B ) @ ( image2 @ A @ ( filter @ B ) @ G6 @ A5 ) ) ) ) ) ).
% filterlim_INF'
thf(fact_4090_filterlim__If,axiom,
! [B: $tType,A: $tType,F: A > B,G6: filter @ B,F6: filter @ A,P: A > $o,G: A > B] :
( ( filterlim @ A @ B @ F @ G6 @ ( inf_inf @ ( filter @ A ) @ F6 @ ( principal @ A @ ( collect @ A @ P ) ) ) )
=> ( ( filterlim @ A @ B @ G @ G6
@ ( inf_inf @ ( filter @ A ) @ F6
@ ( principal @ A
@ ( collect @ A
@ ^ [X2: A] :
~ ( P @ X2 ) ) ) ) )
=> ( filterlim @ A @ B
@ ^ [X2: A] : ( if @ B @ ( P @ X2 ) @ ( F @ X2 ) @ ( G @ X2 ) )
@ G6
@ F6 ) ) ) ).
% filterlim_If
thf(fact_4091_Image__singleton,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ B @ A ),A3: B] :
( ( image @ B @ A @ R @ ( insert2 @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) )
= ( collect @ A
@ ^ [B5: A] : ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ A3 @ B5 ) @ R ) ) ) ).
% Image_singleton
thf(fact_4092_Image__INT__subset,axiom,
! [A: $tType,B: $tType,C: $tType,R: set @ ( product_prod @ B @ A ),B4: C > ( set @ B ),A5: set @ C] :
( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ R @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ C @ ( set @ B ) @ B4 @ A5 ) ) )
@ ( complete_Inf_Inf @ ( set @ A )
@ ( image2 @ C @ ( set @ A )
@ ^ [X2: C] : ( image @ B @ A @ R @ ( B4 @ X2 ) )
@ A5 ) ) ) ).
% Image_INT_subset
thf(fact_4093_Collect__ex__eq,axiom,
! [A: $tType,B: $tType,P: A > B > $o] :
( ( collect @ A
@ ^ [X2: A] :
? [X9: B] : ( P @ X2 @ X9 ) )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [Y: B] :
( collect @ A
@ ^ [X2: A] : ( P @ X2 @ Y ) )
@ ( top_top @ ( set @ B ) ) ) ) ) ).
% Collect_ex_eq
thf(fact_4094_relcomp__unfold,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( relcomp @ A @ C @ B )
= ( ^ [R4: set @ ( product_prod @ A @ C ),S2: set @ ( product_prod @ C @ B )] :
( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [X2: A,Z3: B] :
? [Y: C] :
( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ X2 @ Y ) @ R4 )
& ( member @ ( product_prod @ C @ B ) @ ( product_Pair @ C @ B @ Y @ Z3 ) @ S2 ) ) ) ) ) ) ).
% relcomp_unfold
thf(fact_4095_filterlim__base,axiom,
! [B: $tType,A: $tType,E: $tType,D: $tType,C: $tType,J5: set @ A,I: A > C,I5: set @ C,F6: C > ( set @ D ),F: D > E,G6: A > ( set @ E )] :
( ! [M4: A,X4: B] :
( ( member @ A @ M4 @ J5 )
=> ( member @ C @ ( I @ M4 ) @ I5 ) )
=> ( ! [M4: A,X4: D] :
( ( member @ A @ M4 @ J5 )
=> ( ( member @ D @ X4 @ ( F6 @ ( I @ M4 ) ) )
=> ( member @ E @ ( F @ X4 ) @ ( G6 @ M4 ) ) ) )
=> ( filterlim @ D @ E @ F
@ ( complete_Inf_Inf @ ( filter @ E )
@ ( image2 @ A @ ( filter @ E )
@ ^ [J3: A] : ( principal @ E @ ( G6 @ J3 ) )
@ J5 ) )
@ ( complete_Inf_Inf @ ( filter @ D )
@ ( image2 @ C @ ( filter @ D )
@ ^ [I2: C] : ( principal @ D @ ( F6 @ I2 ) )
@ I5 ) ) ) ) ) ).
% filterlim_base
thf(fact_4096_Image__eq__UN,axiom,
! [A: $tType,B: $tType] :
( ( image @ B @ A )
= ( ^ [R4: set @ ( product_prod @ B @ A ),B7: set @ B] :
( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [Y: B] : ( image @ B @ A @ R4 @ ( insert2 @ B @ Y @ ( bot_bot @ ( set @ B ) ) ) )
@ B7 ) ) ) ) ).
% Image_eq_UN
thf(fact_4097_set__conv__nth,axiom,
! [A: $tType] :
( ( set2 @ A )
= ( ^ [Xs: list @ A] :
( collect @ A
@ ^ [Uu: A] :
? [I2: nat] :
( ( Uu
= ( nth @ A @ Xs @ I2 ) )
& ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) ) ).
% set_conv_nth
thf(fact_4098_inf__Sup1__distrib,axiom,
! [A: $tType] :
( ( distrib_lattice @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( inf_inf @ A @ X @ ( lattic5882676163264333800up_fin @ A @ A5 ) )
= ( lattic5882676163264333800up_fin @ A
@ ( collect @ A
@ ^ [Uu: A] :
? [A6: A] :
( ( Uu
= ( inf_inf @ A @ X @ A6 ) )
& ( member @ A @ A6 @ A5 ) ) ) ) ) ) ) ) ).
% inf_Sup1_distrib
thf(fact_4099_inf__Sup2__distrib,axiom,
! [A: $tType] :
( ( distrib_lattice @ A )
=> ! [A5: set @ A,B4: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B4 )
=> ( ( B4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( inf_inf @ A @ ( lattic5882676163264333800up_fin @ A @ A5 ) @ ( lattic5882676163264333800up_fin @ A @ B4 ) )
= ( lattic5882676163264333800up_fin @ A
@ ( collect @ A
@ ^ [Uu: A] :
? [A6: A,B5: A] :
( ( Uu
= ( inf_inf @ A @ A6 @ B5 ) )
& ( member @ A @ A6 @ A5 )
& ( member @ A @ B5 @ B4 ) ) ) ) ) ) ) ) ) ) ).
% inf_Sup2_distrib
thf(fact_4100_sup__Inf1__distrib,axiom,
! [A: $tType] :
( ( distrib_lattice @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( sup_sup @ A @ X @ ( lattic7752659483105999362nf_fin @ A @ A5 ) )
= ( lattic7752659483105999362nf_fin @ A
@ ( collect @ A
@ ^ [Uu: A] :
? [A6: A] :
( ( Uu
= ( sup_sup @ A @ X @ A6 ) )
& ( member @ A @ A6 @ A5 ) ) ) ) ) ) ) ) ).
% sup_Inf1_distrib
thf(fact_4101_sup__Inf2__distrib,axiom,
! [A: $tType] :
( ( distrib_lattice @ A )
=> ! [A5: set @ A,B4: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B4 )
=> ( ( B4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( sup_sup @ A @ ( lattic7752659483105999362nf_fin @ A @ A5 ) @ ( lattic7752659483105999362nf_fin @ A @ B4 ) )
= ( lattic7752659483105999362nf_fin @ A
@ ( collect @ A
@ ^ [Uu: A] :
? [A6: A,B5: A] :
( ( Uu
= ( sup_sup @ A @ A6 @ B5 ) )
& ( member @ A @ A6 @ A5 )
& ( member @ A @ B5 @ B4 ) ) ) ) ) ) ) ) ) ) ).
% sup_Inf2_distrib
thf(fact_4102_UN__Image,axiom,
! [A: $tType,B: $tType,C: $tType,X7: C > ( set @ ( product_prod @ B @ A ) ),I5: set @ C,S3: set @ B] :
( ( image @ B @ A @ ( complete_Sup_Sup @ ( set @ ( product_prod @ B @ A ) ) @ ( image2 @ C @ ( set @ ( product_prod @ B @ A ) ) @ X7 @ I5 ) ) @ S3 )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ C @ ( set @ A )
@ ^ [I2: C] : ( image @ B @ A @ ( X7 @ I2 ) @ S3 )
@ I5 ) ) ) ).
% UN_Image
thf(fact_4103_butlast__conv__take,axiom,
! [A: $tType] :
( ( butlast @ A )
= ( ^ [Xs: list @ A] : ( take @ A @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) ) @ Xs ) ) ) ).
% butlast_conv_take
thf(fact_4104_quotient__def,axiom,
! [A: $tType] :
( ( equiv_quotient @ A )
= ( ^ [A8: set @ A,R4: set @ ( product_prod @ A @ A )] :
( complete_Sup_Sup @ ( set @ ( set @ A ) )
@ ( image2 @ A @ ( set @ ( set @ A ) )
@ ^ [X2: A] : ( insert2 @ ( set @ A ) @ ( image @ A @ A @ R4 @ ( insert2 @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( bot_bot @ ( set @ ( set @ A ) ) ) )
@ A8 ) ) ) ) ).
% quotient_def
thf(fact_4105_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] :
? [A6: C] :
( ( Uu
= ( product_Pair @ A @ B @ ( F2 @ A6 ) @ ( G2 @ A6 ) ) )
& ( member @ C @ A6 @ A8 ) ) ) ) ) ).
% image2_def
thf(fact_4106_set__nths,axiom,
! [A: $tType,Xs2: list @ A,I5: set @ nat] :
( ( set2 @ A @ ( nths @ A @ Xs2 @ I5 ) )
= ( collect @ A
@ ^ [Uu: A] :
? [I2: nat] :
( ( Uu
= ( nth @ A @ Xs2 @ I2 ) )
& ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs2 ) )
& ( member @ nat @ I2 @ I5 ) ) ) ) ).
% set_nths
thf(fact_4107_lex__conv,axiom,
! [A: $tType] :
( ( lex @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ ( list @ A ) @ ( list @ A ) )
@ ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
@ ^ [Xs: list @ A,Ys: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ A ) @ Ys ) )
& ? [Xys2: list @ A,X2: A,Y: A,Xs4: list @ A,Ys4: list @ A] :
( ( Xs
= ( append @ A @ Xys2 @ ( cons @ A @ X2 @ Xs4 ) ) )
& ( Ys
= ( append @ A @ Xys2 @ ( cons @ A @ Y @ Ys4 ) ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y ) @ R4 ) ) ) ) ) ) ) ).
% lex_conv
thf(fact_4108_length__append,axiom,
! [A: $tType,Xs2: list @ A,Ys2: list @ A] :
( ( size_size @ ( list @ A ) @ ( append @ A @ Xs2 @ Ys2 ) )
= ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ ( size_size @ ( list @ A ) @ Ys2 ) ) ) ).
% length_append
thf(fact_4109_fold__append,axiom,
! [A: $tType,B: $tType,F: B > A > A,Xs2: list @ B,Ys2: list @ B] :
( ( fold @ B @ A @ F @ ( append @ B @ Xs2 @ Ys2 ) )
= ( comp @ A @ A @ A @ ( fold @ B @ A @ F @ Ys2 ) @ ( fold @ B @ A @ F @ Xs2 ) ) ) ).
% fold_append
thf(fact_4110_size__list__append,axiom,
! [A: $tType,F: A > nat,Xs2: list @ A,Ys2: list @ A] :
( ( size_list @ A @ F @ ( append @ A @ Xs2 @ Ys2 ) )
= ( plus_plus @ nat @ ( size_list @ A @ F @ Xs2 ) @ ( size_list @ A @ F @ Ys2 ) ) ) ).
% size_list_append
thf(fact_4111_nth__append__length__plus,axiom,
! [A: $tType,Xs2: list @ A,Ys2: list @ A,N: nat] :
( ( nth @ A @ ( append @ A @ Xs2 @ Ys2 ) @ ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ N ) )
= ( nth @ A @ Ys2 @ N ) ) ).
% nth_append_length_plus
thf(fact_4112_nths__append,axiom,
! [A: $tType,L: list @ A,L6: list @ A,A5: set @ nat] :
( ( nths @ A @ ( append @ A @ L @ L6 ) @ A5 )
= ( append @ A @ ( nths @ A @ L @ A5 )
@ ( nths @ A @ L6
@ ( collect @ nat
@ ^ [J3: nat] : ( member @ nat @ ( plus_plus @ nat @ J3 @ ( size_size @ ( list @ A ) @ L ) ) @ A5 ) ) ) ) ) ).
% nths_append
thf(fact_4113_enumerate__append__eq,axiom,
! [A: $tType,N: nat,Xs2: list @ A,Ys2: list @ A] :
( ( enumerate @ A @ N @ ( append @ A @ Xs2 @ Ys2 ) )
= ( append @ ( product_prod @ nat @ A ) @ ( enumerate @ A @ N @ Xs2 ) @ ( enumerate @ A @ ( plus_plus @ nat @ N @ ( size_size @ ( list @ A ) @ Xs2 ) ) @ Ys2 ) ) ) ).
% enumerate_append_eq
thf(fact_4114_lexord__append__leftI,axiom,
! [A: $tType,U2: list @ A,V: list @ A,R: set @ ( product_prod @ A @ A ),X: list @ A] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ U2 @ V ) @ ( lexord @ A @ R ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ X @ U2 ) @ ( append @ A @ X @ V ) ) @ ( lexord @ A @ R ) ) ) ).
% lexord_append_leftI
thf(fact_4115_lex__append__leftI,axiom,
! [A: $tType,Ys2: list @ A,Zs: list @ A,R: set @ ( product_prod @ A @ A ),Xs2: list @ A] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys2 @ Zs ) @ ( lex @ A @ R ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs2 @ Ys2 ) @ ( append @ A @ Xs2 @ Zs ) ) @ ( lex @ A @ R ) ) ) ).
% lex_append_leftI
thf(fact_4116_Pow__Compl,axiom,
! [A: $tType,A5: set @ A] :
( ( pow2 @ A @ ( uminus_uminus @ ( set @ A ) @ A5 ) )
= ( collect @ ( set @ A )
@ ^ [Uu: set @ A] :
? [B7: set @ A] :
( ( Uu
= ( uminus_uminus @ ( set @ A ) @ B7 ) )
& ( member @ ( set @ A ) @ A5 @ ( pow2 @ A @ B7 ) ) ) ) ) ).
% Pow_Compl
thf(fact_4117_lexord__append__leftD,axiom,
! [A: $tType,X: list @ A,U2: list @ A,V: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ X @ U2 ) @ ( append @ A @ X @ V ) ) @ ( lexord @ A @ R ) )
=> ( ! [A4: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ A4 ) @ R )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ U2 @ V ) @ ( lexord @ A @ R ) ) ) ) ).
% lexord_append_leftD
thf(fact_4118_lexord__append__rightI,axiom,
! [A: $tType,Y3: list @ A,X: list @ A,R: set @ ( product_prod @ A @ A )] :
( ? [B12: A,Z8: list @ A] :
( Y3
= ( cons @ A @ B12 @ Z8 ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ ( append @ A @ X @ Y3 ) ) @ ( lexord @ A @ R ) ) ) ).
% lexord_append_rightI
thf(fact_4119_lexord__sufE,axiom,
! [A: $tType,Xs2: list @ A,Zs: list @ A,Ys2: 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 @ Xs2 @ Zs ) @ ( append @ A @ Ys2 @ Qs ) ) @ ( lexord @ A @ R ) )
=> ( ( Xs2 != Ys2 )
=> ( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ A ) @ Ys2 ) )
=> ( ( ( size_size @ ( list @ A ) @ Zs )
= ( size_size @ ( list @ A ) @ Qs ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Ys2 ) @ ( lexord @ A @ R ) ) ) ) ) ) ).
% lexord_sufE
thf(fact_4120_lex__append__leftD,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),Xs2: list @ A,Ys2: list @ A,Zs: list @ A] :
( ! [X4: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ X4 ) @ R )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs2 @ Ys2 ) @ ( append @ A @ Xs2 @ Zs ) ) @ ( lex @ A @ R ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys2 @ Zs ) @ ( lex @ A @ R ) ) ) ) ).
% lex_append_leftD
thf(fact_4121_lex__append__left__iff,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),Xs2: list @ A,Ys2: list @ A,Zs: list @ A] :
( ! [X4: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ X4 ) @ R )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs2 @ Ys2 ) @ ( append @ A @ Xs2 @ Zs ) ) @ ( lex @ A @ R ) )
= ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys2 @ Zs ) @ ( lex @ A @ R ) ) ) ) ).
% lex_append_left_iff
thf(fact_4122_lex__append__rightI,axiom,
! [A: $tType,Xs2: list @ A,Ys2: 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 ) @ Xs2 @ Ys2 ) @ ( 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 @ Xs2 @ Us ) @ ( append @ A @ Ys2 @ Vs ) ) @ ( lex @ A @ R ) ) ) ) ).
% lex_append_rightI
thf(fact_4123_lenlex__append1,axiom,
! [A: $tType,Us: list @ A,Xs2: list @ A,R5: set @ ( product_prod @ A @ A ),Vs: list @ A,Ys2: list @ A] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Us @ Xs2 ) @ ( lenlex @ A @ R5 ) )
=> ( ( ( size_size @ ( list @ A ) @ Vs )
= ( size_size @ ( list @ A ) @ Ys2 ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Us @ Vs ) @ ( append @ A @ Xs2 @ Ys2 ) ) @ ( lenlex @ A @ R5 ) ) ) ) ).
% lenlex_append1
thf(fact_4124_image2__eqI,axiom,
! [A: $tType,C: $tType,B: $tType,B2: A,F: B > A,X: B,C2: C,G: B > C,A5: set @ B] :
( ( B2
= ( F @ X ) )
=> ( ( C2
= ( G @ X ) )
=> ( ( member @ B @ X @ 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_4125_lexord__append__left__rightI,axiom,
! [A: $tType,A3: A,B2: A,R: set @ ( product_prod @ A @ A ),U2: list @ A,X: list @ A,Y3: list @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ U2 @ ( cons @ A @ A3 @ X ) ) @ ( append @ A @ U2 @ ( cons @ A @ B2 @ Y3 ) ) ) @ ( lexord @ A @ R ) ) ) ).
% lexord_append_left_rightI
thf(fact_4126_lexord__same__pref__iff,axiom,
! [A: $tType,Xs2: list @ A,Ys2: list @ A,Zs: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs2 @ Ys2 ) @ ( append @ A @ Xs2 @ Zs ) ) @ ( lexord @ A @ R ) )
= ( ? [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ Xs2 ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ X2 ) @ R ) )
| ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys2 @ Zs ) @ ( lexord @ A @ R ) ) ) ) ).
% lexord_same_pref_iff
thf(fact_4127_lexord__sufI,axiom,
! [A: $tType,U2: list @ A,W: list @ A,R: set @ ( product_prod @ A @ A ),V: list @ A,Z: list @ A] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ U2 @ W ) @ ( lexord @ A @ R ) )
=> ( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ W ) @ ( size_size @ ( list @ A ) @ U2 ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ U2 @ V ) @ ( append @ A @ W @ Z ) ) @ ( lexord @ A @ R ) ) ) ) ).
% lexord_sufI
thf(fact_4128_Nats__altdef1,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( ( semiring_1_Nats @ A )
= ( collect @ A
@ ^ [Uu: A] :
? [N5: int] :
( ( Uu
= ( ring_1_of_int @ A @ N5 ) )
& ( ord_less_eq @ int @ ( zero_zero @ int ) @ N5 ) ) ) ) ) ).
% Nats_altdef1
thf(fact_4129_length__nths,axiom,
! [A: $tType,Xs2: list @ A,I5: set @ nat] :
( ( size_size @ ( list @ A ) @ ( nths @ A @ Xs2 @ I5 ) )
= ( finite_card @ nat
@ ( collect @ nat
@ ^ [I2: nat] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs2 ) )
& ( member @ nat @ I2 @ I5 ) ) ) ) ) ).
% length_nths
thf(fact_4130_horner__sum__append,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_1 @ A )
=> ! [F: B > A,A3: A,Xs2: list @ B,Ys2: list @ B] :
( ( groups4207007520872428315er_sum @ B @ A @ F @ A3 @ ( append @ B @ Xs2 @ Ys2 ) )
= ( plus_plus @ A @ ( groups4207007520872428315er_sum @ B @ A @ F @ A3 @ Xs2 ) @ ( times_times @ A @ ( power_power @ A @ A3 @ ( size_size @ ( list @ B ) @ Xs2 ) ) @ ( groups4207007520872428315er_sum @ B @ A @ F @ A3 @ Ys2 ) ) ) ) ) ).
% horner_sum_append
thf(fact_4131_lexord__def,axiom,
! [A: $tType] :
( ( lexord @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ ( list @ A ) @ ( list @ A ) )
@ ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
@ ^ [X2: list @ A,Y: list @ A] :
? [A6: A,V6: list @ A] :
( ( Y
= ( append @ A @ X2 @ ( cons @ A @ A6 @ V6 ) ) )
| ? [U: list @ A,B5: A,C4: A,W2: list @ A,Z3: list @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B5 @ C4 ) @ R4 )
& ( X2
= ( append @ A @ U @ ( cons @ A @ B5 @ W2 ) ) )
& ( Y
= ( append @ A @ U @ ( cons @ A @ C4 @ Z3 ) ) ) ) ) ) ) ) ) ).
% lexord_def
thf(fact_4132_lexn__conv,axiom,
! [A: $tType] :
( ( lexn @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A ),N5: nat] :
( collect @ ( product_prod @ ( list @ A ) @ ( list @ A ) )
@ ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
@ ^ [Xs: list @ A,Ys: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs )
= N5 )
& ( ( size_size @ ( list @ A ) @ Ys )
= N5 )
& ? [Xys2: list @ A,X2: A,Y: A,Xs4: list @ A,Ys4: list @ A] :
( ( Xs
= ( append @ A @ Xys2 @ ( cons @ A @ X2 @ Xs4 ) ) )
& ( Ys
= ( append @ A @ Xys2 @ ( cons @ A @ Y @ Ys4 ) ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y ) @ R4 ) ) ) ) ) ) ) ).
% lexn_conv
thf(fact_4133_extract__SomeE,axiom,
! [A: $tType,P: A > $o,Xs2: list @ A,Ys2: list @ A,Y3: A,Zs: list @ A] :
( ( ( extract @ A @ P @ Xs2 )
= ( some @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) @ ( product_Pair @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) @ Ys2 @ ( product_Pair @ A @ ( list @ A ) @ Y3 @ Zs ) ) ) )
=> ( ( Xs2
= ( append @ A @ Ys2 @ ( cons @ A @ Y3 @ Zs ) ) )
& ( P @ Y3 )
& ~ ? [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Ys2 ) )
& ( P @ X5 ) ) ) ) ).
% extract_SomeE
thf(fact_4134_extract__Some__iff,axiom,
! [A: $tType,P: A > $o,Xs2: list @ A,Ys2: list @ A,Y3: A,Zs: list @ A] :
( ( ( extract @ A @ P @ Xs2 )
= ( some @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) @ ( product_Pair @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) @ Ys2 @ ( product_Pair @ A @ ( list @ A ) @ Y3 @ Zs ) ) ) )
= ( ( Xs2
= ( append @ A @ Ys2 @ ( cons @ A @ Y3 @ Zs ) ) )
& ( P @ Y3 )
& ~ ? [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ Ys2 ) )
& ( P @ X2 ) ) ) ) ).
% extract_Some_iff
thf(fact_4135_set__Cons__def,axiom,
! [A: $tType] :
( ( set_Cons @ A )
= ( ^ [A8: set @ A,XS: set @ ( list @ A )] :
( collect @ ( list @ A )
@ ^ [Z3: list @ A] :
? [X2: A,Xs: list @ A] :
( ( Z3
= ( cons @ A @ X2 @ Xs ) )
& ( member @ A @ X2 @ A8 )
& ( member @ ( list @ A ) @ Xs @ XS ) ) ) ) ) ).
% set_Cons_def
thf(fact_4136_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_4137_lexn__length,axiom,
! [A: $tType,Xs2: list @ A,Ys2: list @ A,R: set @ ( product_prod @ A @ A ),N: nat] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Ys2 ) @ ( lexn @ A @ R @ N ) )
=> ( ( ( size_size @ ( list @ A ) @ Xs2 )
= N )
& ( ( size_size @ ( list @ A ) @ Ys2 )
= N ) ) ) ).
% lexn_length
thf(fact_4138_lex__def,axiom,
! [A: $tType] :
( ( lex @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A )] : ( complete_Sup_Sup @ ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( image2 @ nat @ ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( lexn @ A @ R4 ) @ ( top_top @ ( set @ nat ) ) ) ) ) ) ).
% lex_def
thf(fact_4139_listrel1__def,axiom,
! [A: $tType] :
( ( listrel1 @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ ( list @ A ) @ ( list @ A ) )
@ ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
@ ^ [Xs: list @ A,Ys: list @ A] :
? [Us2: list @ A,Z3: A,Z7: A,Vs2: list @ A] :
( ( Xs
= ( append @ A @ Us2 @ ( cons @ A @ Z3 @ Vs2 ) ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z3 @ Z7 ) @ R4 )
& ( Ys
= ( append @ A @ Us2 @ ( cons @ A @ Z7 @ Vs2 ) ) ) ) ) ) ) ) ).
% listrel1_def
thf(fact_4140_extract__Cons__code,axiom,
! [A: $tType,P: A > $o,X: A,Xs2: list @ A] :
( ( ( P @ X )
=> ( ( extract @ A @ P @ ( cons @ A @ X @ Xs2 ) )
= ( some @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) @ ( product_Pair @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) @ ( nil @ A ) @ ( product_Pair @ A @ ( list @ A ) @ X @ Xs2 ) ) ) ) )
& ( ~ ( P @ X )
=> ( ( extract @ A @ P @ ( cons @ A @ X @ Xs2 ) )
= ( 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 ) ) ) )
@ ^ [Ys: list @ A] :
( product_case_prod @ A @ ( list @ A ) @ ( option @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) )
@ ^ [Y: A,Zs2: list @ A] : ( some @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) @ ( product_Pair @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) @ ( cons @ A @ X @ Ys ) @ ( product_Pair @ A @ ( list @ A ) @ Y @ Zs2 ) ) ) ) )
@ ( extract @ A @ P @ Xs2 ) ) ) ) ) ).
% extract_Cons_code
thf(fact_4141_lenlex__append2,axiom,
! [A: $tType,R5: set @ ( product_prod @ A @ A ),Us: list @ A,Xs2: list @ A,Ys2: list @ A] :
( ( irrefl @ A @ R5 )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Us @ Xs2 ) @ ( append @ A @ Us @ Ys2 ) ) @ ( lenlex @ A @ R5 ) )
= ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Ys2 ) @ ( lenlex @ A @ R5 ) ) ) ) ).
% lenlex_append2
thf(fact_4142_lexord__same__pref__if__irrefl,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),Xs2: list @ A,Ys2: list @ A,Zs: list @ A] :
( ( irrefl @ A @ R )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs2 @ Ys2 ) @ ( append @ A @ Xs2 @ Zs ) ) @ ( lexord @ A @ R ) )
= ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys2 @ Zs ) @ ( lexord @ A @ R ) ) ) ) ).
% lexord_same_pref_if_irrefl
thf(fact_4143_length__0__conv,axiom,
! [A: $tType,Xs2: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( zero_zero @ nat ) )
= ( Xs2
= ( nil @ A ) ) ) ).
% length_0_conv
thf(fact_4144_take__eq__Nil2,axiom,
! [A: $tType,N: nat,Xs2: list @ A] :
( ( ( nil @ A )
= ( take @ A @ N @ Xs2 ) )
= ( ( N
= ( zero_zero @ nat ) )
| ( Xs2
= ( nil @ A ) ) ) ) ).
% take_eq_Nil2
thf(fact_4145_take__eq__Nil,axiom,
! [A: $tType,N: nat,Xs2: list @ A] :
( ( ( take @ A @ N @ Xs2 )
= ( nil @ A ) )
= ( ( N
= ( zero_zero @ nat ) )
| ( Xs2
= ( nil @ A ) ) ) ) ).
% take_eq_Nil
thf(fact_4146_take0,axiom,
! [A: $tType] :
( ( take @ A @ ( zero_zero @ nat ) )
= ( ^ [Xs: list @ A] : ( nil @ A ) ) ) ).
% take0
thf(fact_4147_horner__sum__simps_I1_J,axiom,
! [B: $tType,A: $tType] :
( ( comm_semiring_0 @ A )
=> ! [F: B > A,A3: A] :
( ( groups4207007520872428315er_sum @ B @ A @ F @ A3 @ ( nil @ B ) )
= ( zero_zero @ A ) ) ) ).
% horner_sum_simps(1)
thf(fact_4148_Nil__lenlex__iff1,axiom,
! [A: $tType,Ns: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ns ) @ ( lenlex @ A @ R ) )
= ( Ns
!= ( nil @ A ) ) ) ).
% Nil_lenlex_iff1
thf(fact_4149_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_4150_set__Cons__sing__Nil,axiom,
! [A: $tType,A5: set @ A] :
( ( set_Cons @ A @ A5 @ ( insert2 @ ( list @ A ) @ ( nil @ A ) @ ( bot_bot @ ( set @ ( list @ A ) ) ) ) )
= ( image2 @ A @ ( list @ A )
@ ^ [X2: A] : ( cons @ A @ X2 @ ( nil @ A ) )
@ A5 ) ) ).
% set_Cons_sing_Nil
thf(fact_4151_length__greater__0__conv,axiom,
! [A: $tType,Xs2: list @ A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs2 ) )
= ( Xs2
!= ( nil @ A ) ) ) ).
% length_greater_0_conv
thf(fact_4152_nths__singleton,axiom,
! [A: $tType,A5: set @ nat,X: A] :
( ( ( member @ nat @ ( zero_zero @ nat ) @ A5 )
=> ( ( nths @ A @ ( cons @ A @ X @ ( nil @ A ) ) @ A5 )
= ( cons @ A @ X @ ( nil @ A ) ) ) )
& ( ~ ( member @ nat @ ( zero_zero @ nat ) @ A5 )
=> ( ( nths @ A @ ( cons @ A @ X @ ( nil @ A ) ) @ A5 )
= ( nil @ A ) ) ) ) ).
% nths_singleton
thf(fact_4153_Cons__listrel1__Cons,axiom,
! [A: $tType,X: A,Xs2: list @ A,Y3: A,Ys2: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs2 ) @ ( cons @ A @ Y3 @ Ys2 ) ) @ ( listrel1 @ A @ R ) )
= ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ R )
& ( Xs2 = Ys2 ) )
| ( ( X = Y3 )
& ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Ys2 ) @ ( listrel1 @ A @ R ) ) ) ) ) ).
% Cons_listrel1_Cons
thf(fact_4154_lexord__Nil__left,axiom,
! [A: $tType,Y3: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Y3 ) @ ( lexord @ A @ R ) )
= ( ? [A6: A,X2: list @ A] :
( Y3
= ( cons @ A @ A6 @ X2 ) ) ) ) ).
% lexord_Nil_left
thf(fact_4155_not__Nil__listrel1,axiom,
! [A: $tType,Xs2: list @ A,R: set @ ( product_prod @ A @ A )] :
~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Xs2 ) @ ( listrel1 @ A @ R ) ) ).
% not_Nil_listrel1
thf(fact_4156_not__listrel1__Nil,axiom,
! [A: $tType,Xs2: list @ A,R: set @ ( product_prod @ A @ A )] :
~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ ( nil @ A ) ) @ ( listrel1 @ A @ R ) ) ).
% not_listrel1_Nil
thf(fact_4157_successively_Ocases,axiom,
! [A: $tType,X: product_prod @ ( A > A > $o ) @ ( list @ A )] :
( ! [P7: A > A > $o] :
( X
!= ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ P7 @ ( nil @ A ) ) )
=> ( ! [P7: A > A > $o,X4: A] :
( X
!= ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ P7 @ ( cons @ A @ X4 @ ( nil @ A ) ) ) )
=> ~ ! [P7: A > A > $o,X4: A,Y4: A,Xs5: list @ A] :
( X
!= ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ P7 @ ( cons @ A @ X4 @ ( cons @ A @ Y4 @ Xs5 ) ) ) ) ) ) ).
% successively.cases
thf(fact_4158_arg__min__list_Ocases,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [X: product_prod @ ( A > B ) @ ( list @ A )] :
( ! [F4: A > B,X4: A] :
( X
!= ( product_Pair @ ( A > B ) @ ( list @ A ) @ F4 @ ( cons @ A @ X4 @ ( nil @ A ) ) ) )
=> ( ! [F4: A > B,X4: A,Y4: A,Zs3: list @ A] :
( X
!= ( product_Pair @ ( A > B ) @ ( list @ A ) @ F4 @ ( cons @ A @ X4 @ ( cons @ A @ Y4 @ Zs3 ) ) ) )
=> ~ ! [A4: A > B] :
( X
!= ( product_Pair @ ( A > B ) @ ( list @ A ) @ A4 @ ( nil @ A ) ) ) ) ) ) ).
% arg_min_list.cases
thf(fact_4159_sorted__wrt_Ocases,axiom,
! [A: $tType,X: product_prod @ ( A > A > $o ) @ ( list @ A )] :
( ! [P7: A > A > $o] :
( X
!= ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ P7 @ ( nil @ A ) ) )
=> ~ ! [P7: A > A > $o,X4: A,Ys5: list @ A] :
( X
!= ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ P7 @ ( cons @ A @ X4 @ Ys5 ) ) ) ) ).
% sorted_wrt.cases
thf(fact_4160_n__lists_Osimps_I1_J,axiom,
! [A: $tType,Xs2: list @ A] :
( ( n_lists @ A @ ( zero_zero @ nat ) @ Xs2 )
= ( cons @ ( list @ A ) @ ( nil @ A ) @ ( nil @ ( list @ A ) ) ) ) ).
% n_lists.simps(1)
thf(fact_4161_irreflI,axiom,
! [A: $tType,R5: set @ ( product_prod @ A @ A )] :
( ! [A4: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ A4 ) @ R5 )
=> ( irrefl @ A @ R5 ) ) ).
% irreflI
thf(fact_4162_irrefl__def,axiom,
! [A: $tType] :
( ( irrefl @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A )] :
! [A6: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A6 @ A6 ) @ R4 ) ) ) ).
% irrefl_def
thf(fact_4163_snoc__listrel1__snoc__iff,axiom,
! [A: $tType,Xs2: list @ A,X: A,Ys2: list @ A,Y3: A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs2 @ ( cons @ A @ X @ ( nil @ A ) ) ) @ ( append @ A @ Ys2 @ ( cons @ A @ Y3 @ ( nil @ A ) ) ) ) @ ( listrel1 @ A @ R ) )
= ( ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Ys2 ) @ ( listrel1 @ A @ R ) )
& ( X = Y3 ) )
| ( ( Xs2 = Ys2 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ R ) ) ) ) ).
% snoc_listrel1_snoc_iff
thf(fact_4164_list_Osize_I3_J,axiom,
! [A: $tType] :
( ( size_size @ ( list @ A ) @ ( nil @ A ) )
= ( zero_zero @ nat ) ) ).
% list.size(3)
thf(fact_4165_splice_Ocases,axiom,
! [A: $tType,X: product_prod @ ( list @ A ) @ ( list @ A )] :
( ! [Ys5: list @ A] :
( X
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ys5 ) )
=> ~ ! [X4: A,Xs5: list @ A,Ys5: list @ A] :
( X
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X4 @ Xs5 ) @ Ys5 ) ) ) ).
% splice.cases
thf(fact_4166_shuffles_Ocases,axiom,
! [A: $tType,X: product_prod @ ( list @ A ) @ ( list @ A )] :
( ! [Ys5: list @ A] :
( X
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ys5 ) )
=> ( ! [Xs5: list @ A] :
( X
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs5 @ ( nil @ A ) ) )
=> ~ ! [X4: A,Xs5: list @ A,Y4: A,Ys5: list @ A] :
( X
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X4 @ Xs5 ) @ ( cons @ A @ Y4 @ Ys5 ) ) ) ) ) ).
% shuffles.cases
thf(fact_4167_map__tailrec__rev_Ocases,axiom,
! [A: $tType,B: $tType,X: product_prod @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) )] :
( ! [F4: A > B,Bs2: list @ B] :
( X
!= ( product_Pair @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ F4 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ Bs2 ) ) )
=> ~ ! [F4: A > B,A4: A,As2: list @ A,Bs2: list @ B] :
( X
!= ( product_Pair @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ F4 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ A4 @ As2 ) @ Bs2 ) ) ) ) ).
% map_tailrec_rev.cases
thf(fact_4168_take__0,axiom,
! [A: $tType,Xs2: list @ A] :
( ( take @ A @ ( zero_zero @ nat ) @ Xs2 )
= ( nil @ A ) ) ).
% take_0
thf(fact_4169_rtrancl__listrel1__ConsI1,axiom,
! [A: $tType,Xs2: list @ A,Ys2: list @ A,R: set @ ( product_prod @ A @ A ),X: A] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Ys2 ) @ ( transitive_rtrancl @ ( list @ A ) @ ( listrel1 @ A @ R ) ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs2 ) @ ( cons @ A @ X @ Ys2 ) ) @ ( transitive_rtrancl @ ( list @ A ) @ ( listrel1 @ A @ R ) ) ) ) ).
% rtrancl_listrel1_ConsI1
thf(fact_4170_listrel1I2,axiom,
! [A: $tType,Xs2: list @ A,Ys2: list @ A,R: set @ ( product_prod @ A @ A ),X: A] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Ys2 ) @ ( listrel1 @ A @ R ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs2 ) @ ( cons @ A @ X @ Ys2 ) ) @ ( listrel1 @ A @ R ) ) ) ).
% listrel1I2
thf(fact_4171_rtrancl__listrel1__eq__len,axiom,
! [A: $tType,X: list @ A,Y3: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Y3 ) @ ( transitive_rtrancl @ ( list @ A ) @ ( listrel1 @ A @ R ) ) )
=> ( ( size_size @ ( list @ A ) @ X )
= ( size_size @ ( list @ A ) @ Y3 ) ) ) ).
% rtrancl_listrel1_eq_len
thf(fact_4172_listrel1__eq__len,axiom,
! [A: $tType,Xs2: list @ A,Ys2: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Ys2 ) @ ( listrel1 @ A @ R ) )
=> ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ A ) @ Ys2 ) ) ) ).
% listrel1_eq_len
thf(fact_4173_append__listrel1I,axiom,
! [A: $tType,Xs2: list @ A,Ys2: list @ A,R: set @ ( product_prod @ A @ A ),Us: list @ A,Vs: list @ A] :
( ( ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Ys2 ) @ ( listrel1 @ A @ R ) )
& ( Us = Vs ) )
| ( ( Xs2 = Ys2 )
& ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Us @ Vs ) @ ( listrel1 @ A @ R ) ) ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs2 @ Us ) @ ( append @ A @ Ys2 @ Vs ) ) @ ( listrel1 @ A @ R ) ) ) ).
% append_listrel1I
thf(fact_4174_lexord__Nil__right,axiom,
! [A: $tType,X: list @ A,R: set @ ( product_prod @ A @ A )] :
~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ ( nil @ A ) ) @ ( lexord @ A @ R ) ) ).
% lexord_Nil_right
thf(fact_4175_list_Osize__gen_I1_J,axiom,
! [A: $tType,X: A > nat] :
( ( size_list @ A @ X @ ( nil @ A ) )
= ( zero_zero @ nat ) ) ).
% list.size_gen(1)
thf(fact_4176_Nil__notin__lex,axiom,
! [A: $tType,Ys2: list @ A,R: set @ ( product_prod @ A @ A )] :
~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ys2 ) @ ( lex @ A @ R ) ) ).
% Nil_notin_lex
thf(fact_4177_Nil2__notin__lex,axiom,
! [A: $tType,Xs2: list @ A,R: set @ ( product_prod @ A @ A )] :
~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ ( nil @ A ) ) @ ( lex @ A @ R ) ) ).
% Nil2_notin_lex
thf(fact_4178_Nil__lenlex__iff2,axiom,
! [A: $tType,Ns: list @ A,R: set @ ( product_prod @ A @ A )] :
~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ns @ ( nil @ A ) ) @ ( lenlex @ A @ R ) ) ).
% Nil_lenlex_iff2
thf(fact_4179_find_Osimps_I1_J,axiom,
! [A: $tType,Uu3: A > $o] :
( ( find @ A @ Uu3 @ ( nil @ A ) )
= ( none @ A ) ) ).
% find.simps(1)
thf(fact_4180_take__Cons,axiom,
! [A: $tType,N: nat,X: A,Xs2: list @ A] :
( ( take @ A @ N @ ( cons @ A @ X @ Xs2 ) )
= ( case_nat @ ( list @ A ) @ ( nil @ A )
@ ^ [M5: nat] : ( cons @ A @ X @ ( take @ A @ M5 @ Xs2 ) )
@ N ) ) ).
% take_Cons
thf(fact_4181_listrel1I1,axiom,
! [A: $tType,X: A,Y3: A,R: set @ ( product_prod @ A @ A ),Xs2: list @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ R )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs2 ) @ ( cons @ A @ Y3 @ Xs2 ) ) @ ( listrel1 @ A @ R ) ) ) ).
% listrel1I1
thf(fact_4182_Cons__listrel1E1,axiom,
! [A: $tType,X: A,Xs2: list @ A,Ys2: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs2 ) @ Ys2 ) @ ( listrel1 @ A @ R ) )
=> ( ! [Y4: A] :
( ( Ys2
= ( cons @ A @ Y4 @ Xs2 ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y4 ) @ R ) )
=> ~ ! [Zs3: list @ A] :
( ( Ys2
= ( cons @ A @ X @ Zs3 ) )
=> ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Zs3 ) @ ( listrel1 @ A @ R ) ) ) ) ) ).
% Cons_listrel1E1
thf(fact_4183_Cons__listrel1E2,axiom,
! [A: $tType,Xs2: list @ A,Y3: A,Ys2: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ ( cons @ A @ Y3 @ Ys2 ) ) @ ( listrel1 @ A @ R ) )
=> ( ! [X4: A] :
( ( Xs2
= ( cons @ A @ X4 @ Ys2 ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y3 ) @ R ) )
=> ~ ! [Zs3: list @ A] :
( ( Xs2
= ( cons @ A @ Y3 @ Zs3 ) )
=> ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Zs3 @ Ys2 ) @ ( listrel1 @ A @ R ) ) ) ) ) ).
% Cons_listrel1E2
thf(fact_4184_lexl__not__refl,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),X: list @ A] :
( ( irrefl @ A @ R )
=> ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ X ) @ ( lex @ A @ R ) ) ) ).
% lexl_not_refl
thf(fact_4185_ord_Olexordp_Omono,axiom,
! [A: $tType,Less: A > A > $o] :
( order_mono @ ( ( list @ A ) > ( list @ A ) > $o ) @ ( ( list @ A ) > ( list @ A ) > $o )
@ ^ [P5: ( list @ A ) > ( list @ A ) > $o,X12: list @ A,X24: list @ A] :
( ? [Y: A,Ys: list @ A] :
( ( X12
= ( nil @ A ) )
& ( X24
= ( cons @ A @ Y @ Ys ) ) )
| ? [X2: A,Y: A,Xs: list @ A,Ys: list @ A] :
( ( X12
= ( cons @ A @ X2 @ Xs ) )
& ( X24
= ( cons @ A @ Y @ Ys ) )
& ( Less @ X2 @ Y ) )
| ? [X2: A,Y: A,Xs: list @ A,Ys: list @ A] :
( ( X12
= ( cons @ A @ X2 @ Xs ) )
& ( X24
= ( cons @ A @ Y @ Ys ) )
& ~ ( Less @ X2 @ Y )
& ~ ( Less @ Y @ X2 )
& ( P5 @ Xs @ Ys ) ) ) ) ).
% ord.lexordp.mono
thf(fact_4186_listrel1E,axiom,
! [A: $tType,Xs2: list @ A,Ys2: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Ys2 ) @ ( listrel1 @ A @ R ) )
=> ~ ! [X4: A,Y4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y4 ) @ R )
=> ! [Us3: list @ A,Vs3: list @ A] :
( ( Xs2
= ( append @ A @ Us3 @ ( cons @ A @ X4 @ Vs3 ) ) )
=> ( Ys2
!= ( append @ A @ Us3 @ ( cons @ A @ Y4 @ Vs3 ) ) ) ) ) ) ).
% listrel1E
thf(fact_4187_listrel1I,axiom,
! [A: $tType,X: A,Y3: A,R: set @ ( product_prod @ A @ A ),Xs2: list @ A,Us: list @ A,Vs: list @ A,Ys2: list @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ R )
=> ( ( Xs2
= ( append @ A @ Us @ ( cons @ A @ X @ Vs ) ) )
=> ( ( Ys2
= ( append @ A @ Us @ ( cons @ A @ Y3 @ Vs ) ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Ys2 ) @ ( listrel1 @ A @ R ) ) ) ) ) ).
% listrel1I
thf(fact_4188_rtrancl__listrel1__ConsI2,axiom,
! [A: $tType,X: A,Y3: A,R: set @ ( product_prod @ A @ A ),Xs2: list @ A,Ys2: list @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ ( transitive_rtrancl @ A @ R ) )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Ys2 ) @ ( transitive_rtrancl @ ( list @ A ) @ ( listrel1 @ A @ R ) ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs2 ) @ ( cons @ A @ Y3 @ Ys2 ) ) @ ( transitive_rtrancl @ ( list @ A ) @ ( listrel1 @ A @ R ) ) ) ) ) ).
% rtrancl_listrel1_ConsI2
thf(fact_4189_lexordp_Omono,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( order_mono @ ( ( list @ A ) > ( list @ A ) > $o ) @ ( ( list @ A ) > ( list @ A ) > $o )
@ ^ [P5: ( list @ A ) > ( list @ A ) > $o,X12: list @ A,X24: list @ A] :
( ? [Y: A,Ys: list @ A] :
( ( X12
= ( nil @ A ) )
& ( X24
= ( cons @ A @ Y @ Ys ) ) )
| ? [X2: A,Y: A,Xs: list @ A,Ys: list @ A] :
( ( X12
= ( cons @ A @ X2 @ Xs ) )
& ( X24
= ( cons @ A @ Y @ Ys ) )
& ( ord_less @ A @ X2 @ Y ) )
| ? [X2: A,Y: A,Xs: list @ A,Ys: list @ A] :
( ( X12
= ( cons @ A @ X2 @ Xs ) )
& ( X24
= ( cons @ A @ Y @ Ys ) )
& ~ ( ord_less @ A @ X2 @ Y )
& ~ ( ord_less @ A @ Y @ X2 )
& ( P5 @ Xs @ Ys ) ) ) ) ) ).
% lexordp.mono
thf(fact_4190_remdups__adj__length__ge1,axiom,
! [A: $tType,Xs2: list @ A] :
( ( Xs2
!= ( nil @ A ) )
=> ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( size_size @ ( list @ A ) @ ( remdups_adj @ A @ Xs2 ) ) ) ) ).
% remdups_adj_length_ge1
thf(fact_4191_take__Cons_H,axiom,
! [A: $tType,N: nat,X: A,Xs2: list @ A] :
( ( ( N
= ( zero_zero @ nat ) )
=> ( ( take @ A @ N @ ( cons @ A @ X @ Xs2 ) )
= ( nil @ A ) ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ( ( take @ A @ N @ ( cons @ A @ X @ Xs2 ) )
= ( cons @ A @ X @ ( take @ A @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ Xs2 ) ) ) ) ) ).
% take_Cons'
thf(fact_4192_nths__Cons,axiom,
! [A: $tType,X: A,L: list @ A,A5: set @ nat] :
( ( nths @ A @ ( cons @ A @ X @ L ) @ A5 )
= ( append @ A @ ( if @ ( list @ A ) @ ( member @ nat @ ( zero_zero @ nat ) @ A5 ) @ ( cons @ A @ X @ ( nil @ A ) ) @ ( nil @ A ) )
@ ( nths @ A @ L
@ ( collect @ nat
@ ^ [J3: nat] : ( member @ nat @ ( suc @ J3 ) @ A5 ) ) ) ) ) ).
% nths_Cons
thf(fact_4193_listrel1p__def,axiom,
! [A: $tType] :
( ( listrel1p @ A )
= ( ^ [R4: A > A > $o,Xs: list @ A,Ys: list @ A] : ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( listrel1 @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ R4 ) ) ) ) ) ) ).
% listrel1p_def
thf(fact_4194_Succ__def,axiom,
! [A: $tType] :
( ( bNF_Greatest_Succ @ A )
= ( ^ [Kl: set @ ( list @ A ),Kl2: list @ A] :
( collect @ A
@ ^ [K3: A] : ( member @ ( list @ A ) @ ( append @ A @ Kl2 @ ( cons @ A @ K3 @ ( nil @ A ) ) ) @ Kl ) ) ) ) ).
% Succ_def
thf(fact_4195_upto_Opelims,axiom,
! [X: int,Xa: int,Y3: list @ int] :
( ( ( upto @ X @ Xa )
= Y3 )
=> ( ( accp @ ( product_prod @ int @ int ) @ upto_rel @ ( product_Pair @ int @ int @ X @ Xa ) )
=> ~ ( ( ( ( ord_less_eq @ int @ X @ Xa )
=> ( Y3
= ( cons @ int @ X @ ( upto @ ( plus_plus @ int @ X @ ( one_one @ int ) ) @ Xa ) ) ) )
& ( ~ ( ord_less_eq @ int @ X @ Xa )
=> ( Y3
= ( nil @ int ) ) ) )
=> ~ ( accp @ ( product_prod @ int @ int ) @ upto_rel @ ( product_Pair @ int @ int @ X @ Xa ) ) ) ) ) ).
% upto.pelims
thf(fact_4196_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_4197_list__encode_Oelims,axiom,
! [X: list @ nat,Y3: nat] :
( ( ( nat_list_encode @ X )
= Y3 )
=> ( ( ( X
= ( nil @ nat ) )
=> ( Y3
!= ( zero_zero @ nat ) ) )
=> ~ ! [X4: nat,Xs5: list @ nat] :
( ( X
= ( cons @ nat @ X4 @ Xs5 ) )
=> ( Y3
!= ( suc @ ( nat_prod_encode @ ( product_Pair @ nat @ nat @ X4 @ ( nat_list_encode @ Xs5 ) ) ) ) ) ) ) ) ).
% list_encode.elims
thf(fact_4198_finite__def,axiom,
! [A: $tType] :
( ( finite_finite2 @ A )
= ( complete_lattice_lfp @ ( ( set @ A ) > $o )
@ ^ [P5: ( set @ A ) > $o,X2: set @ A] :
( ( X2
= ( bot_bot @ ( set @ A ) ) )
| ? [A8: set @ A,A6: A] :
( ( X2
= ( insert2 @ A @ A6 @ A8 ) )
& ( P5 @ A8 ) ) ) ) ) ).
% finite_def
thf(fact_4199_subset__Image1__Image1__iff,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
( ( order_preorder_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( member @ A @ A3 @ ( field2 @ A @ R ) )
=> ( ( member @ A @ B2 @ ( field2 @ A @ R ) )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ R @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( image @ A @ A @ R @ ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ A3 ) @ R ) ) ) ) ) ).
% subset_Image1_Image1_iff
thf(fact_4200_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 ) )
@ ^ [X2: A] : ( equiv_quotient @ A @ ( insert2 @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) @ R )
@ A5 )
=> ( ( finite_card @ ( set @ A ) @ ( equiv_quotient @ A @ A5 @ R ) )
= ( finite_card @ A @ A5 ) ) ) ) ).
% card_quotient_disjoint
thf(fact_4201_inj__on__empty,axiom,
! [B: $tType,A: $tType,F: A > B] : ( inj_on @ A @ B @ F @ ( bot_bot @ ( set @ A ) ) ) ).
% inj_on_empty
thf(fact_4202_inj__uminus,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A5: set @ A] : ( inj_on @ A @ A @ ( uminus_uminus @ A ) @ A5 ) ) ).
% inj_uminus
thf(fact_4203_inj__mult__left,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [A3: A] :
( ( inj_on @ A @ A @ ( times_times @ A @ A3 ) @ ( top_top @ ( set @ A ) ) )
= ( A3
!= ( zero_zero @ A ) ) ) ) ).
% inj_mult_left
thf(fact_4204_inj__divide__right,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: A] :
( ( inj_on @ A @ A
@ ^ [B5: A] : ( divide_divide @ A @ B5 @ A3 )
@ ( top_top @ ( set @ A ) ) )
= ( A3
!= ( zero_zero @ A ) ) ) ) ).
% inj_divide_right
thf(fact_4205_inj__apfst,axiom,
! [B: $tType,C: $tType,A: $tType,F: A > C] :
( ( inj_on @ ( product_prod @ A @ B ) @ ( product_prod @ C @ B ) @ ( product_apfst @ A @ C @ B @ F ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) )
= ( inj_on @ A @ C @ F @ ( top_top @ ( set @ A ) ) ) ) ).
% inj_apfst
thf(fact_4206_inj__apsnd,axiom,
! [A: $tType,C: $tType,B: $tType,F: B > C] :
( ( inj_on @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) @ ( product_apsnd @ B @ C @ A @ F ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) )
= ( inj_on @ B @ C @ F @ ( top_top @ ( set @ B ) ) ) ) ).
% inj_apsnd
thf(fact_4207_inj__on__insert,axiom,
! [B: $tType,A: $tType,F: A > B,A3: A,A5: set @ A] :
( ( inj_on @ A @ B @ F @ ( insert2 @ A @ A3 @ A5 ) )
= ( ( inj_on @ A @ B @ F @ A5 )
& ~ ( member @ B @ ( F @ A3 ) @ ( image2 @ A @ B @ F @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% inj_on_insert
thf(fact_4208_inj__on__image__iff,axiom,
! [B: $tType,A: $tType,A5: set @ A,G: A > B,F: A > A] :
( ! [X4: A] :
( ( member @ A @ X4 @ A5 )
=> ! [Xa4: A] :
( ( member @ A @ Xa4 @ A5 )
=> ( ( ( G @ ( F @ X4 ) )
= ( G @ ( F @ Xa4 ) ) )
= ( ( G @ X4 )
= ( G @ Xa4 ) ) ) ) )
=> ( ( inj_on @ A @ A @ F @ A5 )
=> ( ( inj_on @ A @ B @ G @ ( image2 @ A @ A @ F @ A5 ) )
= ( inj_on @ A @ B @ G @ A5 ) ) ) ) ).
% inj_on_image_iff
thf(fact_4209_inj__on__image__Pow,axiom,
! [B: $tType,A: $tType,F: A > B,A5: set @ A] :
( ( inj_on @ A @ B @ F @ A5 )
=> ( inj_on @ ( set @ A ) @ ( set @ B ) @ ( image2 @ A @ B @ F ) @ ( pow2 @ A @ A5 ) ) ) ).
% inj_on_image_Pow
thf(fact_4210_inj__on__imageI2,axiom,
! [B: $tType,C: $tType,A: $tType,F5: C > B,F: A > C,A5: set @ A] :
( ( inj_on @ A @ B @ ( comp @ C @ B @ A @ F5 @ F ) @ A5 )
=> ( inj_on @ A @ C @ F @ A5 ) ) ).
% inj_on_imageI2
thf(fact_4211_inj__on__id,axiom,
! [A: $tType,A5: set @ A] : ( inj_on @ A @ A @ ( id @ A ) @ A5 ) ).
% inj_on_id
thf(fact_4212_finite__inverse__image__gen,axiom,
! [A: $tType,B: $tType,A5: set @ A,F: B > A,D5: set @ B] :
( ( finite_finite2 @ A @ A5 )
=> ( ( inj_on @ B @ A @ F @ D5 )
=> ( finite_finite2 @ B
@ ( collect @ B
@ ^ [J3: B] :
( ( member @ B @ J3 @ D5 )
& ( member @ A @ ( F @ J3 ) @ A5 ) ) ) ) ) ) ).
% finite_inverse_image_gen
thf(fact_4213_sorted__list__of__set_Oinj__on,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( inj_on @ A @ A
@ ^ [X2: A] : X2
@ ( top_top @ ( set @ A ) ) ) ) ).
% sorted_list_of_set.inj_on
thf(fact_4214_inj__on__diff,axiom,
! [B: $tType,A: $tType,F: A > B,A5: set @ A,B4: set @ A] :
( ( inj_on @ A @ B @ F @ A5 )
=> ( inj_on @ A @ B @ F @ ( minus_minus @ ( set @ A ) @ A5 @ B4 ) ) ) ).
% inj_on_diff
thf(fact_4215_inj__on__inverseI,axiom,
! [B: $tType,A: $tType,A5: set @ A,G: B > A,F: A > B] :
( ! [X4: A] :
( ( member @ A @ X4 @ A5 )
=> ( ( G @ ( F @ X4 ) )
= X4 ) )
=> ( inj_on @ A @ B @ F @ A5 ) ) ).
% inj_on_inverseI
thf(fact_4216_inj__on__contraD,axiom,
! [B: $tType,A: $tType,F: A > B,A5: set @ A,X: A,Y3: A] :
( ( inj_on @ A @ B @ F @ A5 )
=> ( ( X != Y3 )
=> ( ( member @ A @ X @ A5 )
=> ( ( member @ A @ Y3 @ A5 )
=> ( ( F @ X )
!= ( F @ Y3 ) ) ) ) ) ) ).
% inj_on_contraD
thf(fact_4217_inj__on__eq__iff,axiom,
! [B: $tType,A: $tType,F: A > B,A5: set @ A,X: A,Y3: A] :
( ( inj_on @ A @ B @ F @ A5 )
=> ( ( member @ A @ X @ A5 )
=> ( ( member @ A @ Y3 @ A5 )
=> ( ( ( F @ X )
= ( F @ Y3 ) )
= ( X = Y3 ) ) ) ) ) ).
% inj_on_eq_iff
thf(fact_4218_inj__on__cong,axiom,
! [B: $tType,A: $tType,A5: set @ A,F: A > B,G: A > B] :
( ! [A4: A] :
( ( member @ A @ A4 @ A5 )
=> ( ( F @ A4 )
= ( G @ A4 ) ) )
=> ( ( inj_on @ A @ B @ F @ A5 )
= ( inj_on @ A @ B @ G @ A5 ) ) ) ).
% inj_on_cong
thf(fact_4219_inj__on__def,axiom,
! [B: $tType,A: $tType] :
( ( inj_on @ A @ B )
= ( ^ [F2: A > B,A8: set @ A] :
! [X2: A] :
( ( member @ A @ X2 @ A8 )
=> ! [Y: A] :
( ( member @ A @ Y @ A8 )
=> ( ( ( F2 @ X2 )
= ( F2 @ Y ) )
=> ( X2 = Y ) ) ) ) ) ) ).
% inj_on_def
thf(fact_4220_inj__onI,axiom,
! [B: $tType,A: $tType,A5: set @ A,F: A > B] :
( ! [X4: A,Y4: A] :
( ( member @ A @ X4 @ A5 )
=> ( ( member @ A @ Y4 @ A5 )
=> ( ( ( F @ X4 )
= ( F @ Y4 ) )
=> ( X4 = Y4 ) ) ) )
=> ( inj_on @ A @ B @ F @ A5 ) ) ).
% inj_onI
thf(fact_4221_inj__onD,axiom,
! [B: $tType,A: $tType,F: A > B,A5: set @ A,X: A,Y3: A] :
( ( inj_on @ A @ B @ F @ A5 )
=> ( ( ( F @ X )
= ( F @ Y3 ) )
=> ( ( member @ A @ X @ A5 )
=> ( ( member @ A @ Y3 @ A5 )
=> ( X = Y3 ) ) ) ) ) ).
% inj_onD
thf(fact_4222_inj__on__add,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [A3: A,A5: set @ A] : ( inj_on @ A @ A @ ( plus_plus @ A @ A3 ) @ A5 ) ) ).
% inj_on_add
thf(fact_4223_inj__on__id2,axiom,
! [A: $tType,A5: set @ A] :
( inj_on @ A @ A
@ ^ [X2: A] : X2
@ A5 ) ).
% inj_on_id2
thf(fact_4224_inj__on__Int,axiom,
! [B: $tType,A: $tType,F: A > B,A5: set @ A,B4: set @ A] :
( ( ( inj_on @ A @ B @ F @ A5 )
| ( inj_on @ A @ B @ F @ B4 ) )
=> ( inj_on @ A @ B @ F @ ( inf_inf @ ( set @ A ) @ A5 @ B4 ) ) ) ).
% inj_on_Int
thf(fact_4225_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 )
@ ^ [X2: A,Y: C] : ( F @ X2 )
@ ( top_top @ ( set @ A ) ) ) ) ).
% inj_fun
thf(fact_4226_injD,axiom,
! [B: $tType,A: $tType,F: A > B,X: A,Y3: A] :
( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
=> ( ( ( F @ X )
= ( F @ Y3 ) )
=> ( X = Y3 ) ) ) ).
% injD
thf(fact_4227_injI,axiom,
! [B: $tType,A: $tType,F: A > B] :
( ! [X4: A,Y4: A] :
( ( ( F @ X4 )
= ( F @ Y4 ) )
=> ( X4 = Y4 ) )
=> ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) ) ) ).
% injI
thf(fact_4228_inj__eq,axiom,
! [B: $tType,A: $tType,F: A > B,X: A,Y3: A] :
( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
=> ( ( ( F @ X )
= ( F @ Y3 ) )
= ( X = Y3 ) ) ) ).
% inj_eq
thf(fact_4229_inj__def,axiom,
! [B: $tType,A: $tType,F: A > B] :
( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
= ( ! [X2: A,Y: A] :
( ( ( F @ X2 )
= ( F @ Y ) )
=> ( X2 = Y ) ) ) ) ).
% inj_def
thf(fact_4230_subset__inj__on,axiom,
! [B: $tType,A: $tType,F: A > B,B4: set @ A,A5: set @ A] :
( ( inj_on @ A @ B @ F @ B4 )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
=> ( inj_on @ A @ B @ F @ A5 ) ) ) ).
% subset_inj_on
thf(fact_4231_inj__on__subset,axiom,
! [B: $tType,A: $tType,F: A > B,A5: set @ A,B4: set @ A] :
( ( inj_on @ A @ B @ F @ A5 )
=> ( ( ord_less_eq @ ( set @ A ) @ B4 @ A5 )
=> ( inj_on @ A @ B @ F @ B4 ) ) ) ).
% inj_on_subset
thf(fact_4232_inj__on__add_H,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [A3: A,A5: set @ A] :
( inj_on @ A @ A
@ ^ [B5: A] : ( plus_plus @ A @ B5 @ A3 )
@ A5 ) ) ).
% inj_on_add'
thf(fact_4233_fun_Oinj__map,axiom,
! [B: $tType,A: $tType,D: $tType,F: A > B] :
( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
=> ( inj_on @ ( D > A ) @ ( D > B ) @ ( comp @ A @ B @ D @ F ) @ ( top_top @ ( set @ ( D > A ) ) ) ) ) ).
% fun.inj_map
thf(fact_4234_linorder__inj__onI,axiom,
! [B: $tType,A: $tType] :
( ( order @ A )
=> ! [A5: set @ A,F: A > B] :
( ! [X4: A,Y4: A] :
( ( ord_less @ A @ X4 @ Y4 )
=> ( ( member @ A @ X4 @ A5 )
=> ( ( member @ A @ Y4 @ A5 )
=> ( ( F @ X4 )
!= ( F @ Y4 ) ) ) ) )
=> ( ! [X4: A,Y4: A] :
( ( member @ A @ X4 @ A5 )
=> ( ( member @ A @ Y4 @ A5 )
=> ( ( ord_less_eq @ A @ X4 @ Y4 )
| ( ord_less_eq @ A @ Y4 @ X4 ) ) ) )
=> ( inj_on @ A @ B @ F @ A5 ) ) ) ) ).
% linorder_inj_onI
thf(fact_4235_inj__on__mult,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A3: A,A5: set @ A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( inj_on @ A @ A @ ( times_times @ A @ A3 ) @ A5 ) ) ) ).
% inj_on_mult
thf(fact_4236_list__encode_Osimps_I1_J,axiom,
( ( nat_list_encode @ ( nil @ nat ) )
= ( zero_zero @ nat ) ) ).
% list_encode.simps(1)
thf(fact_4237_linorder__injI,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ A )
=> ! [F: A > B] :
( ! [X4: A,Y4: A] :
( ( ord_less @ A @ X4 @ Y4 )
=> ( ( F @ X4 )
!= ( F @ Y4 ) ) )
=> ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) ) ) ) ).
% linorder_injI
thf(fact_4238_inj__on__image__eq__iff,axiom,
! [B: $tType,A: $tType,F: A > B,C6: set @ A,A5: set @ A,B4: set @ A] :
( ( inj_on @ A @ B @ F @ C6 )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ C6 )
=> ( ( ord_less_eq @ ( set @ A ) @ B4 @ C6 )
=> ( ( ( image2 @ A @ B @ F @ A5 )
= ( image2 @ A @ B @ F @ B4 ) )
= ( A5 = B4 ) ) ) ) ) ).
% inj_on_image_eq_iff
thf(fact_4239_inj__on__image__mem__iff,axiom,
! [B: $tType,A: $tType,F: A > B,B4: set @ A,A3: A,A5: set @ A] :
( ( inj_on @ A @ B @ F @ B4 )
=> ( ( member @ A @ A3 @ B4 )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
=> ( ( member @ B @ ( F @ A3 ) @ ( image2 @ A @ B @ F @ A5 ) )
= ( member @ A @ A3 @ A5 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_4240_inj__add__left,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [A3: A] : ( inj_on @ A @ A @ ( plus_plus @ A @ A3 ) @ ( top_top @ ( set @ A ) ) ) ) ).
% inj_add_left
thf(fact_4241_inj__img__insertE,axiom,
! [B: $tType,A: $tType,F: A > B,A5: set @ A,X: B,B4: set @ B] :
( ( inj_on @ A @ B @ F @ A5 )
=> ( ~ ( member @ B @ X @ B4 )
=> ( ( ( insert2 @ B @ X @ B4 )
= ( image2 @ A @ B @ F @ A5 ) )
=> ~ ! [X11: A,A11: set @ A] :
( ~ ( member @ A @ X11 @ A11 )
=> ( ( A5
= ( insert2 @ A @ X11 @ A11 ) )
=> ( ( X
= ( F @ X11 ) )
=> ( B4
!= ( image2 @ A @ B @ F @ A11 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_4242_inj__image__mem__iff,axiom,
! [B: $tType,A: $tType,F: A > B,A3: A,A5: set @ A] :
( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
=> ( ( member @ B @ ( F @ A3 ) @ ( image2 @ A @ B @ F @ A5 ) )
= ( member @ A @ A3 @ A5 ) ) ) ).
% inj_image_mem_iff
thf(fact_4243_inj__image__eq__iff,axiom,
! [B: $tType,A: $tType,F: A > B,A5: set @ A,B4: set @ A] :
( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
=> ( ( ( image2 @ A @ B @ F @ A5 )
= ( image2 @ A @ B @ F @ B4 ) )
= ( A5 = B4 ) ) ) ).
% inj_image_eq_iff
thf(fact_4244_range__ex1__eq,axiom,
! [B: $tType,A: $tType,F: A > B,B2: B] :
( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
=> ( ( member @ B @ B2 @ ( image2 @ A @ B @ F @ ( top_top @ ( set @ A ) ) ) )
= ( ? [X2: A] :
( ( B2
= ( F @ X2 ) )
& ! [Y: A] :
( ( B2
= ( F @ Y ) )
=> ( Y = X2 ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_4245_comp__inj__on__iff,axiom,
! [B: $tType,C: $tType,A: $tType,F: A > B,A5: set @ A,F5: B > C] :
( ( inj_on @ A @ B @ F @ A5 )
=> ( ( inj_on @ B @ C @ F5 @ ( image2 @ A @ B @ F @ A5 ) )
= ( inj_on @ A @ C @ ( comp @ B @ C @ A @ F5 @ F ) @ A5 ) ) ) ).
% comp_inj_on_iff
thf(fact_4246_inj__on__imageI,axiom,
! [B: $tType,C: $tType,A: $tType,G: C > B,F: A > C,A5: set @ A] :
( ( inj_on @ A @ B @ ( comp @ C @ B @ A @ G @ F ) @ A5 )
=> ( inj_on @ C @ B @ G @ ( image2 @ A @ C @ F @ A5 ) ) ) ).
% inj_on_imageI
thf(fact_4247_comp__inj__on,axiom,
! [B: $tType,C: $tType,A: $tType,F: A > B,A5: set @ A,G: B > C] :
( ( inj_on @ A @ B @ F @ A5 )
=> ( ( inj_on @ B @ C @ G @ ( image2 @ A @ B @ F @ A5 ) )
=> ( inj_on @ A @ C @ ( comp @ B @ C @ A @ G @ F ) @ A5 ) ) ) ).
% comp_inj_on
thf(fact_4248_inj__compose,axiom,
! [A: $tType,B: $tType,C: $tType,F: A > B,G: C > A] :
( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
=> ( ( inj_on @ C @ A @ G @ ( top_top @ ( set @ C ) ) )
=> ( inj_on @ C @ B @ ( comp @ A @ B @ C @ F @ G ) @ ( top_top @ ( set @ C ) ) ) ) ) ).
% inj_compose
thf(fact_4249_inj__on__Un__image__eq__iff,axiom,
! [B: $tType,A: $tType,F: A > B,A5: set @ A,B4: set @ A] :
( ( inj_on @ A @ B @ F @ ( sup_sup @ ( set @ A ) @ A5 @ B4 ) )
=> ( ( ( image2 @ A @ B @ F @ A5 )
= ( image2 @ A @ B @ F @ B4 ) )
= ( A5 = B4 ) ) ) ).
% inj_on_Un_image_eq_iff
thf(fact_4250_inj__on__strict__subset,axiom,
! [B: $tType,A: $tType,F: A > B,B4: set @ A,A5: set @ A] :
( ( inj_on @ A @ B @ F @ B4 )
=> ( ( ord_less @ ( set @ A ) @ A5 @ B4 )
=> ( ord_less @ ( set @ B ) @ ( image2 @ A @ B @ F @ A5 ) @ ( image2 @ A @ B @ F @ B4 ) ) ) ) ).
% inj_on_strict_subset
thf(fact_4251_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_4252_inj__diff__right,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A3: A] :
( inj_on @ A @ A
@ ^ [B5: A] : ( minus_minus @ A @ B5 @ A3 )
@ ( top_top @ ( set @ A ) ) ) ) ).
% inj_diff_right
thf(fact_4253_inj__singleton,axiom,
! [A: $tType,A5: set @ A] :
( inj_on @ A @ ( set @ A )
@ ^ [X2: A] : ( insert2 @ A @ X2 @ ( bot_bot @ ( set @ A ) ) )
@ A5 ) ).
% inj_singleton
thf(fact_4254_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_4255_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
@ ^ [X2: A] : X2
@ ( image2 @ B @ A @ G @ A5 ) )
= ( groups7311177749621191930dd_sum @ B @ A @ G @ A5 ) ) ) ) ).
% sum.image_eq
thf(fact_4256_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
@ ^ [X2: A] : X2
@ ( image2 @ B @ A @ G @ A5 ) )
= ( groups7121269368397514597t_prod @ B @ A @ G @ A5 ) ) ) ) ).
% prod.image_eq
thf(fact_4257_inj__image__subset__iff,axiom,
! [B: $tType,A: $tType,F: A > B,A5: set @ A,B4: set @ A] :
( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F @ A5 ) @ ( image2 @ A @ B @ F @ B4 ) )
= ( ord_less_eq @ ( set @ A ) @ A5 @ B4 ) ) ) ).
% inj_image_subset_iff
thf(fact_4258_inj__on__image__Int,axiom,
! [B: $tType,A: $tType,F: A > B,C6: set @ A,A5: set @ A,B4: set @ A] :
( ( inj_on @ A @ B @ F @ C6 )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ C6 )
=> ( ( ord_less_eq @ ( set @ A ) @ B4 @ C6 )
=> ( ( image2 @ A @ B @ F @ ( inf_inf @ ( set @ A ) @ A5 @ B4 ) )
= ( inf_inf @ ( set @ B ) @ ( image2 @ A @ B @ F @ A5 ) @ ( image2 @ A @ B @ F @ B4 ) ) ) ) ) ) ).
% inj_on_image_Int
thf(fact_4259_sum_Oreindex,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [H: B > C,A5: set @ B,G: C > A] :
( ( inj_on @ B @ C @ H @ A5 )
=> ( ( groups7311177749621191930dd_sum @ C @ A @ G @ ( image2 @ B @ C @ H @ A5 ) )
= ( groups7311177749621191930dd_sum @ B @ A @ ( comp @ C @ A @ B @ G @ H ) @ A5 ) ) ) ) ).
% sum.reindex
thf(fact_4260_image__Int,axiom,
! [B: $tType,A: $tType,F: A > B,A5: set @ A,B4: set @ A] :
( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
=> ( ( image2 @ A @ B @ F @ ( inf_inf @ ( set @ A ) @ A5 @ B4 ) )
= ( inf_inf @ ( set @ B ) @ ( image2 @ A @ B @ F @ A5 ) @ ( image2 @ A @ B @ F @ B4 ) ) ) ) ).
% image_Int
thf(fact_4261_inj__on__image__set__diff,axiom,
! [B: $tType,A: $tType,F: A > B,C6: set @ A,A5: set @ A,B4: set @ A] :
( ( inj_on @ A @ B @ F @ C6 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A5 @ B4 ) @ C6 )
=> ( ( ord_less_eq @ ( set @ A ) @ B4 @ C6 )
=> ( ( image2 @ A @ B @ F @ ( minus_minus @ ( set @ A ) @ A5 @ B4 ) )
= ( minus_minus @ ( set @ B ) @ ( image2 @ A @ B @ F @ A5 ) @ ( image2 @ A @ B @ F @ B4 ) ) ) ) ) ) ).
% inj_on_image_set_diff
thf(fact_4262_image__set__diff,axiom,
! [B: $tType,A: $tType,F: A > B,A5: set @ A,B4: set @ A] :
( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
=> ( ( image2 @ A @ B @ F @ ( minus_minus @ ( set @ A ) @ A5 @ B4 ) )
= ( minus_minus @ ( set @ B ) @ ( image2 @ A @ B @ F @ A5 ) @ ( image2 @ A @ B @ F @ B4 ) ) ) ) ).
% image_set_diff
thf(fact_4263_prod_Oreindex,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [H: B > C,A5: set @ B,G: C > A] :
( ( inj_on @ B @ C @ H @ A5 )
=> ( ( groups7121269368397514597t_prod @ C @ A @ G @ ( image2 @ B @ C @ H @ A5 ) )
= ( groups7121269368397514597t_prod @ B @ A @ ( comp @ C @ A @ B @ G @ H ) @ A5 ) ) ) ) ).
% prod.reindex
thf(fact_4264_fold__image,axiom,
! [C: $tType,B: $tType,A: $tType,G: A > B,A5: set @ A,F: B > C > C,Z: C] :
( ( inj_on @ A @ B @ G @ A5 )
=> ( ( finite_fold @ B @ C @ F @ Z @ ( image2 @ A @ B @ G @ A5 ) )
= ( finite_fold @ A @ C @ ( comp @ B @ ( C > C ) @ A @ F @ G ) @ Z @ A5 ) ) ) ).
% fold_image
thf(fact_4265_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_4266_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_4267_inj__image__Compl__subset,axiom,
! [B: $tType,A: $tType,F: A > B,A5: set @ A] :
( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
=> ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F @ ( uminus_uminus @ ( set @ A ) @ A5 ) ) @ ( uminus_uminus @ ( set @ B ) @ ( image2 @ A @ B @ F @ A5 ) ) ) ) ).
% inj_image_Compl_subset
thf(fact_4268_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_4269_inj__on__disjoint__Un,axiom,
! [B: $tType,A: $tType,F: A > B,A5: set @ A,G: A > B,B4: set @ A] :
( ( inj_on @ A @ B @ F @ A5 )
=> ( ( inj_on @ A @ B @ G @ B4 )
=> ( ( ( inf_inf @ ( set @ B ) @ ( image2 @ A @ B @ F @ A5 ) @ ( image2 @ A @ B @ G @ B4 ) )
= ( bot_bot @ ( set @ B ) ) )
=> ( inj_on @ A @ B
@ ^ [X2: A] : ( if @ B @ ( member @ A @ X2 @ A5 ) @ ( F @ X2 ) @ ( G @ X2 ) )
@ ( sup_sup @ ( set @ A ) @ A5 @ B4 ) ) ) ) ) ).
% inj_on_disjoint_Un
thf(fact_4270_image__INT,axiom,
! [B: $tType,A: $tType,C: $tType,F: A > B,C6: set @ A,A5: set @ C,B4: C > ( set @ A ),J: C] :
( ( inj_on @ A @ B @ F @ C6 )
=> ( ! [X4: C] :
( ( member @ C @ X4 @ A5 )
=> ( ord_less_eq @ ( set @ A ) @ ( B4 @ X4 ) @ C6 ) )
=> ( ( member @ C @ J @ A5 )
=> ( ( image2 @ A @ B @ F @ ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ C @ ( set @ A ) @ B4 @ A5 ) ) )
= ( complete_Inf_Inf @ ( set @ B )
@ ( image2 @ C @ ( set @ B )
@ ^ [X2: C] : ( image2 @ A @ B @ F @ ( B4 @ X2 ) )
@ A5 ) ) ) ) ) ) ).
% image_INT
thf(fact_4271_quotient__diff1,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A5: set @ A,A3: A] :
( ( inj_on @ A @ ( set @ ( set @ A ) )
@ ^ [A6: A] : ( equiv_quotient @ A @ ( insert2 @ A @ A6 @ ( bot_bot @ ( set @ A ) ) ) @ R )
@ A5 )
=> ( ( member @ A @ A3 @ A5 )
=> ( ( equiv_quotient @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) @ R )
= ( minus_minus @ ( set @ ( set @ A ) ) @ ( equiv_quotient @ A @ A5 @ R ) @ ( equiv_quotient @ A @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) @ R ) ) ) ) ) ).
% quotient_diff1
thf(fact_4272_subset__Image__Image__iff,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A5: set @ A,B4: set @ A] :
( ( order_preorder_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ ( field2 @ A @ R ) )
=> ( ( ord_less_eq @ ( set @ A ) @ B4 @ ( field2 @ A @ R ) )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ R @ A5 ) @ ( image @ A @ A @ R @ B4 ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ A5 )
=> ? [Y: A] :
( ( member @ A @ Y @ B4 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X2 ) @ R ) ) ) ) ) ) ) ) ).
% subset_Image_Image_iff
thf(fact_4273_inj__on__Un,axiom,
! [A: $tType,B: $tType,F: A > B,A5: set @ A,B4: set @ A] :
( ( inj_on @ A @ B @ F @ ( sup_sup @ ( set @ A ) @ A5 @ B4 ) )
= ( ( inj_on @ A @ B @ F @ A5 )
& ( inj_on @ A @ B @ F @ B4 )
& ( ( inf_inf @ ( set @ B ) @ ( image2 @ A @ B @ F @ ( minus_minus @ ( set @ A ) @ A5 @ B4 ) ) @ ( image2 @ A @ B @ F @ ( minus_minus @ ( set @ A ) @ B4 @ A5 ) ) )
= ( bot_bot @ ( set @ B ) ) ) ) ) ).
% inj_on_Un
thf(fact_4274_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_4275_list__encode_Osimps_I2_J,axiom,
! [X: nat,Xs2: list @ nat] :
( ( nat_list_encode @ ( cons @ nat @ X @ Xs2 ) )
= ( suc @ ( nat_prod_encode @ ( product_Pair @ nat @ nat @ X @ ( nat_list_encode @ Xs2 ) ) ) ) ) ).
% list_encode.simps(2)
thf(fact_4276_funpow__inj__finite,axiom,
! [A: $tType,P4: A > A,X: A] :
( ( inj_on @ A @ A @ P4 @ ( top_top @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [Y: A] :
? [N5: nat] :
( Y
= ( compow @ ( A > A ) @ N5 @ P4 @ X ) ) ) )
=> ~ ! [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( compow @ ( A > A ) @ N2 @ P4 @ X )
!= X ) ) ) ) ).
% funpow_inj_finite
thf(fact_4277_lfp__def,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ( ( complete_lattice_lfp @ A )
= ( ^ [F2: A > A] :
( complete_Inf_Inf @ A
@ ( collect @ A
@ ^ [U: A] : ( ord_less_eq @ A @ ( F2 @ U ) @ U ) ) ) ) ) ) ).
% lfp_def
thf(fact_4278_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
@ ^ [X2: A] : ( F @ ( G @ X2 ) ) ) )
= ( complete_lattice_lfp @ B
@ ^ [X2: B] : ( G @ ( F @ X2 ) ) ) ) ) ) ) ).
% lfp_rolling
thf(fact_4279_lfp__lfp,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F: A > A > A] :
( ! [X4: A,Y4: A,W3: A,Z6: A] :
( ( ord_less_eq @ A @ X4 @ Y4 )
=> ( ( ord_less_eq @ A @ W3 @ Z6 )
=> ( ord_less_eq @ A @ ( F @ X4 @ W3 ) @ ( F @ Y4 @ Z6 ) ) ) )
=> ( ( complete_lattice_lfp @ A
@ ^ [X2: A] : ( complete_lattice_lfp @ A @ ( F @ X2 ) ) )
= ( complete_lattice_lfp @ A
@ ^ [X2: A] : ( F @ X2 @ X2 ) ) ) ) ) ).
% lfp_lfp
thf(fact_4280_linear__order__on__singleton,axiom,
! [A: $tType,X: A] : ( order_679001287576687338der_on @ A @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) @ ( insert2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ X ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ).
% linear_order_on_singleton
thf(fact_4281_inj__split__Cons,axiom,
! [A: $tType,X7: set @ ( product_prod @ ( list @ A ) @ A )] :
( inj_on @ ( product_prod @ ( list @ A ) @ A ) @ ( list @ A )
@ ( product_case_prod @ ( list @ A ) @ A @ ( list @ A )
@ ^ [Xs: list @ A,N5: A] : ( cons @ A @ N5 @ Xs ) )
@ X7 ) ).
% inj_split_Cons
thf(fact_4282_lfp__induct__set,axiom,
! [A: $tType,A3: A,F: ( set @ A ) > ( set @ A ),P: A > $o] :
( ( member @ A @ A3 @ ( complete_lattice_lfp @ ( set @ A ) @ F ) )
=> ( ( order_mono @ ( set @ A ) @ ( set @ A ) @ F )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ ( F @ ( inf_inf @ ( set @ A ) @ ( complete_lattice_lfp @ ( set @ A ) @ F ) @ ( collect @ A @ P ) ) ) )
=> ( P @ X4 ) )
=> ( P @ A3 ) ) ) ) ).
% lfp_induct_set
thf(fact_4283_def__lfp__induct__set,axiom,
! [A: $tType,A5: set @ A,F: ( set @ A ) > ( set @ A ),A3: A,P: A > $o] :
( ( A5
= ( complete_lattice_lfp @ ( set @ A ) @ F ) )
=> ( ( order_mono @ ( set @ A ) @ ( set @ A ) @ F )
=> ( ( member @ A @ A3 @ A5 )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ ( F @ ( inf_inf @ ( set @ A ) @ A5 @ ( collect @ A @ P ) ) ) )
=> ( P @ X4 ) )
=> ( P @ A3 ) ) ) ) ) ).
% def_lfp_induct_set
thf(fact_4284_inj__on__convol__ident,axiom,
! [B: $tType,A: $tType,F: A > B,X7: set @ A] :
( inj_on @ A @ ( product_prod @ A @ B )
@ ^ [X2: A] : ( product_Pair @ A @ B @ X2 @ ( F @ X2 ) )
@ X7 ) ).
% inj_on_convol_ident
thf(fact_4285_inj__Some,axiom,
! [A: $tType,A5: set @ A] : ( inj_on @ A @ ( option @ A ) @ ( some @ A ) @ A5 ) ).
% inj_Some
thf(fact_4286_inj__Suc,axiom,
! [N6: set @ nat] : ( inj_on @ nat @ nat @ suc @ N6 ) ).
% inj_Suc
thf(fact_4287_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_4288_inj__on__diff__nat,axiom,
! [N6: set @ nat,K: nat] :
( ! [N2: nat] :
( ( member @ nat @ N2 @ N6 )
=> ( ord_less_eq @ nat @ K @ N2 ) )
=> ( inj_on @ nat @ nat
@ ^ [N5: nat] : ( minus_minus @ nat @ N5 @ K )
@ N6 ) ) ).
% inj_on_diff_nat
thf(fact_4289_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_4290_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
@ ^ [X2: A,Y: B] :
( Y
= ( F2 @ X2 ) ) ) )
@ ( top_top @ ( set @ ( A > B ) ) ) ) ).
% inj_graph
thf(fact_4291_finite__imp__nat__seg__image__inj__on,axiom,
! [A: $tType,A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ? [N2: nat,F4: nat > A] :
( ( A5
= ( image2 @ nat @ A @ F4
@ ( collect @ nat
@ ^ [I2: nat] : ( ord_less @ nat @ I2 @ N2 ) ) ) )
& ( inj_on @ nat @ A @ F4
@ ( collect @ nat
@ ^ [I2: nat] : ( ord_less @ nat @ I2 @ N2 ) ) ) ) ) ).
% finite_imp_nat_seg_image_inj_on
thf(fact_4292_finite__imp__inj__to__nat__seg,axiom,
! [A: $tType,A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ? [F4: A > nat,N2: nat] :
( ( ( image2 @ A @ nat @ F4 @ A5 )
= ( collect @ nat
@ ^ [I2: nat] : ( ord_less @ nat @ I2 @ N2 ) ) )
& ( inj_on @ A @ nat @ F4 @ A5 ) ) ) ).
% finite_imp_inj_to_nat_seg
thf(fact_4293_lfp__induct2,axiom,
! [A: $tType,B: $tType,A3: A,B2: B,F: ( set @ ( product_prod @ A @ B ) ) > ( set @ ( product_prod @ A @ B ) ),P: A > B > $o] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ ( complete_lattice_lfp @ ( set @ ( product_prod @ A @ B ) ) @ F ) )
=> ( ( order_mono @ ( set @ ( product_prod @ A @ B ) ) @ ( set @ ( product_prod @ A @ B ) ) @ F )
=> ( ! [A4: A,B3: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B3 ) @ ( 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 @ B3 ) )
=> ( P @ A3 @ B2 ) ) ) ) ).
% lfp_induct2
thf(fact_4294_inj__on__funpow__least,axiom,
! [A: $tType,N: nat,F: A > A,S: A] :
( ( ( compow @ ( A > A ) @ N @ F @ S )
= S )
=> ( ! [M4: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M4 )
=> ( ( ord_less @ nat @ M4 @ N )
=> ( ( compow @ ( A > A ) @ M4 @ F @ S )
!= S ) ) )
=> ( inj_on @ nat @ A
@ ^ [K3: nat] : ( compow @ ( A > A ) @ K3 @ F @ S )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% inj_on_funpow_least
thf(fact_4295_coinduct3__mono__lemma,axiom,
! [B: $tType,A: $tType] :
( ( order @ A )
=> ! [F: A > ( set @ B ),X7: set @ B,B4: set @ B] :
( ( order_mono @ A @ ( set @ B ) @ F )
=> ( order_mono @ A @ ( set @ B )
@ ^ [X2: A] : ( sup_sup @ ( set @ B ) @ ( sup_sup @ ( set @ B ) @ ( F @ X2 ) @ X7 ) @ B4 ) ) ) ) ).
% coinduct3_mono_lemma
thf(fact_4296_lfp__const,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [T2: A] :
( ( complete_lattice_lfp @ A
@ ^ [X2: A] : T2 )
= T2 ) ) ).
% lfp_const
thf(fact_4297_Total__subset__Id,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( total_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R @ ( id2 @ A ) )
=> ( ( R
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
| ? [A4: A] :
( R
= ( insert2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ A4 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ) ) ) ).
% Total_subset_Id
thf(fact_4298_ord__class_Olexordp__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_lexordp @ A )
= ( complete_lattice_lfp @ ( ( list @ A ) > ( list @ A ) > $o )
@ ^ [P5: ( list @ A ) > ( list @ A ) > $o,X12: list @ A,X24: list @ A] :
( ? [Y: A,Ys: list @ A] :
( ( X12
= ( nil @ A ) )
& ( X24
= ( cons @ A @ Y @ Ys ) ) )
| ? [X2: A,Y: A,Xs: list @ A,Ys: list @ A] :
( ( X12
= ( cons @ A @ X2 @ Xs ) )
& ( X24
= ( cons @ A @ Y @ Ys ) )
& ( ord_less @ A @ X2 @ Y ) )
| ? [X2: A,Y: A,Xs: list @ A,Ys: list @ A] :
( ( X12
= ( cons @ A @ X2 @ Xs ) )
& ( X24
= ( cons @ A @ Y @ Ys ) )
& ~ ( ord_less @ A @ X2 @ Y )
& ~ ( ord_less @ A @ Y @ X2 )
& ( P5 @ Xs @ Ys ) ) ) ) ) ) ).
% ord_class.lexordp_def
thf(fact_4299_list__encode_Opelims,axiom,
! [X: list @ nat,Y3: nat] :
( ( ( nat_list_encode @ X )
= Y3 )
=> ( ( accp @ ( list @ nat ) @ nat_list_encode_rel @ X )
=> ( ( ( X
= ( nil @ nat ) )
=> ( ( Y3
= ( zero_zero @ nat ) )
=> ~ ( accp @ ( list @ nat ) @ nat_list_encode_rel @ ( nil @ nat ) ) ) )
=> ~ ! [X4: nat,Xs5: list @ nat] :
( ( X
= ( cons @ nat @ X4 @ Xs5 ) )
=> ( ( Y3
= ( suc @ ( nat_prod_encode @ ( product_Pair @ nat @ nat @ X4 @ ( nat_list_encode @ Xs5 ) ) ) ) )
=> ~ ( accp @ ( list @ nat ) @ nat_list_encode_rel @ ( cons @ nat @ X4 @ Xs5 ) ) ) ) ) ) ) ).
% list_encode.pelims
thf(fact_4300_If__the__inv__into__f__f,axiom,
! [B: $tType,A: $tType,I: A,C6: set @ A,G: A > B,X: 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 ) @ X )
@ G
@ I )
= ( id @ A @ I ) ) ) ) ).
% If_the_inv_into_f_f
thf(fact_4301_IdI,axiom,
! [A: $tType,A3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ A3 ) @ ( id2 @ A ) ) ).
% IdI
thf(fact_4302_pair__in__Id__conv,axiom,
! [A: $tType,A3: A,B2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( id2 @ A ) )
= ( A3 = B2 ) ) ).
% pair_in_Id_conv
thf(fact_4303_the__inv__into__onto,axiom,
! [B: $tType,A: $tType,F: A > B,A5: set @ A] :
( ( inj_on @ A @ B @ F @ A5 )
=> ( ( image2 @ B @ A @ ( the_inv_into @ A @ B @ A5 @ F ) @ ( image2 @ A @ B @ F @ A5 ) )
= A5 ) ) ).
% the_inv_into_onto
thf(fact_4304_IdD,axiom,
! [A: $tType,A3: A,B2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( id2 @ A ) )
=> ( A3 = B2 ) ) ).
% IdD
thf(fact_4305_IdE,axiom,
! [A: $tType,P4: product_prod @ A @ A] :
( ( member @ ( product_prod @ A @ A ) @ P4 @ ( id2 @ A ) )
=> ~ ! [X4: A] :
( P4
!= ( product_Pair @ A @ A @ X4 @ X4 ) ) ) ).
% IdE
thf(fact_4306_the__inv__into__f__eq,axiom,
! [B: $tType,A: $tType,F: A > B,A5: set @ A,X: A,Y3: B] :
( ( inj_on @ A @ B @ F @ A5 )
=> ( ( ( F @ X )
= Y3 )
=> ( ( member @ A @ X @ A5 )
=> ( ( the_inv_into @ A @ B @ A5 @ F @ Y3 )
= X ) ) ) ) ).
% the_inv_into_f_eq
thf(fact_4307_the__inv__into__f__f,axiom,
! [B: $tType,A: $tType,F: A > B,A5: set @ A,X: A] :
( ( inj_on @ A @ B @ F @ A5 )
=> ( ( member @ A @ X @ A5 )
=> ( ( the_inv_into @ A @ B @ A5 @ F @ ( F @ X ) )
= X ) ) ) ).
% the_inv_into_f_f
thf(fact_4308_the__inv__into__def,axiom,
! [B: $tType,A: $tType] :
( ( the_inv_into @ A @ B )
= ( ^ [A8: set @ A,F2: A > B,X2: B] :
( the @ A
@ ^ [Y: A] :
( ( member @ A @ Y @ A8 )
& ( ( F2 @ Y )
= X2 ) ) ) ) ) ).
% the_inv_into_def
thf(fact_4309_relpow_Osimps_I1_J,axiom,
! [A: $tType,R5: set @ ( product_prod @ A @ A )] :
( ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( zero_zero @ nat ) @ R5 )
= ( id2 @ A ) ) ).
% relpow.simps(1)
thf(fact_4310_Id__def,axiom,
! [A: $tType] :
( ( id2 @ A )
= ( collect @ ( product_prod @ A @ A )
@ ^ [P5: product_prod @ A @ A] :
? [X2: A] :
( P5
= ( product_Pair @ A @ A @ X2 @ X2 ) ) ) ) ).
% Id_def
thf(fact_4311_Id__fstsnd__eq,axiom,
! [A: $tType] :
( ( id2 @ A )
= ( collect @ ( product_prod @ A @ A )
@ ^ [X2: product_prod @ A @ A] :
( ( product_fst @ A @ A @ X2 )
= ( product_snd @ A @ A @ X2 ) ) ) ) ).
% Id_fstsnd_eq
thf(fact_4312_f__the__inv__into__f,axiom,
! [A: $tType,B: $tType,F: A > B,A5: set @ A,Y3: B] :
( ( inj_on @ A @ B @ F @ A5 )
=> ( ( member @ B @ Y3 @ ( image2 @ A @ B @ F @ A5 ) )
=> ( ( F @ ( the_inv_into @ A @ B @ A5 @ F @ Y3 ) )
= Y3 ) ) ) ).
% f_the_inv_into_f
thf(fact_4313_inj__on__the__inv__into,axiom,
! [B: $tType,A: $tType,F: A > B,A5: set @ A] :
( ( inj_on @ A @ B @ F @ A5 )
=> ( inj_on @ B @ A @ ( the_inv_into @ A @ B @ A5 @ F ) @ ( image2 @ A @ B @ F @ A5 ) ) ) ).
% inj_on_the_inv_into
thf(fact_4314_the__inv__f__f,axiom,
! [B: $tType,A: $tType,F: A > B,X: A] :
( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
=> ( ( the_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F @ ( F @ X ) )
= X ) ) ).
% the_inv_f_f
thf(fact_4315_Linear__order__in__diff__Id,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
( ( order_679001287576687338der_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( member @ A @ A3 @ ( field2 @ A @ R ) )
=> ( ( member @ A @ B2 @ ( field2 @ A @ R ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R )
= ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ A3 ) @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R @ ( id2 @ A ) ) ) ) ) ) ) ) ).
% Linear_order_in_diff_Id
thf(fact_4316_reflcl__set__eq,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( sup_sup @ ( A > A > $o )
@ ^ [X2: A,Y: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y ) @ R )
@ ^ [Y6: A,Z2: A] : Y6 = Z2 )
= ( ^ [X2: A,Y: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y ) @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ R @ ( id2 @ A ) ) ) ) ) ).
% reflcl_set_eq
thf(fact_4317_the__inv__into__into,axiom,
! [B: $tType,A: $tType,F: A > B,A5: set @ A,X: B,B4: set @ A] :
( ( inj_on @ A @ B @ F @ A5 )
=> ( ( member @ B @ X @ ( image2 @ A @ B @ F @ A5 ) )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
=> ( member @ A @ ( the_inv_into @ A @ B @ A5 @ F @ X ) @ B4 ) ) ) ) ).
% the_inv_into_into
thf(fact_4318_the__inv__into__comp,axiom,
! [A: $tType,C: $tType,B: $tType,F: A > B,G: C > A,A5: set @ C,X: B] :
( ( inj_on @ A @ B @ F @ ( image2 @ C @ A @ G @ A5 ) )
=> ( ( inj_on @ C @ A @ G @ A5 )
=> ( ( member @ B @ X @ ( image2 @ A @ B @ F @ ( image2 @ C @ A @ G @ A5 ) ) )
=> ( ( the_inv_into @ C @ B @ A5 @ ( comp @ A @ B @ C @ F @ G ) @ X )
= ( comp @ A @ C @ B @ ( the_inv_into @ C @ A @ A5 @ G ) @ ( the_inv_into @ A @ B @ ( image2 @ C @ A @ G @ A5 ) @ F ) @ X ) ) ) ) ) ).
% the_inv_into_comp
thf(fact_4319_lexordp__conv__lexord,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_lexordp @ A )
= ( ^ [Xs: list @ A,Ys: list @ A] : ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( lexord @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ ( ord_less @ A ) ) ) ) ) ) ) ) ).
% lexordp_conv_lexord
thf(fact_4320_the__inv__f__o__f__id,axiom,
! [B: $tType,A: $tType,F: A > B,Z: A] :
( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
=> ( ( comp @ B @ A @ A @ ( the_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F ) @ F @ Z )
= ( id @ A @ Z ) ) ) ).
% the_inv_f_o_f_id
thf(fact_4321_If__the__inv__into__in__Func,axiom,
! [B: $tType,A: $tType,G: A > B,C6: set @ A,B4: set @ A,X: A] :
( ( inj_on @ A @ B @ G @ C6 )
=> ( ( ord_less_eq @ ( set @ A ) @ C6 @ ( sup_sup @ ( set @ A ) @ B4 @ ( insert2 @ A @ X @ ( 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 ) @ X )
@ ( bNF_Wellorder_Func @ B @ A @ ( top_top @ ( set @ B ) ) @ ( sup_sup @ ( set @ A ) @ B4 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% If_the_inv_into_in_Func
thf(fact_4322_ord_Olexordp__def,axiom,
! [A: $tType] :
( ( lexordp2 @ A )
= ( ^ [Less2: A > A > $o] :
( complete_lattice_lfp @ ( ( list @ A ) > ( list @ A ) > $o )
@ ^ [P5: ( list @ A ) > ( list @ A ) > $o,X12: list @ A,X24: list @ A] :
( ? [Y: A,Ys: list @ A] :
( ( X12
= ( nil @ A ) )
& ( X24
= ( cons @ A @ Y @ Ys ) ) )
| ? [X2: A,Y: A,Xs: list @ A,Ys: list @ A] :
( ( X12
= ( cons @ A @ X2 @ Xs ) )
& ( X24
= ( cons @ A @ Y @ Ys ) )
& ( Less2 @ X2 @ Y ) )
| ? [X2: A,Y: A,Xs: list @ A,Ys: list @ A] :
( ( X12
= ( cons @ A @ X2 @ Xs ) )
& ( X24
= ( cons @ A @ Y @ Ys ) )
& ~ ( Less2 @ X2 @ Y )
& ~ ( Less2 @ Y @ X2 )
& ( P5 @ Xs @ Ys ) ) ) ) ) ) ).
% ord.lexordp_def
thf(fact_4323_lists__length__Suc__eq,axiom,
! [A: $tType,A5: set @ A,N: nat] :
( ( collect @ ( list @ A )
@ ^ [Xs: list @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ A5 )
& ( ( size_size @ ( list @ A ) @ Xs )
= ( suc @ N ) ) ) )
= ( image2 @ ( product_prod @ ( list @ A ) @ A ) @ ( list @ A )
@ ( product_case_prod @ ( list @ A ) @ A @ ( list @ A )
@ ^ [Xs: list @ A,N5: A] : ( cons @ A @ N5 @ Xs ) )
@ ( product_Sigma @ ( list @ A ) @ A
@ ( collect @ ( list @ A )
@ ^ [Xs: list @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ A5 )
& ( ( size_size @ ( list @ A ) @ Xs )
= N ) ) )
@ ^ [Uu: list @ A] : A5 ) ) ) ).
% lists_length_Suc_eq
thf(fact_4324_listrel1__iff__update,axiom,
! [A: $tType,Xs2: list @ A,Ys2: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Ys2 ) @ ( listrel1 @ A @ R ) )
= ( ? [Y: A,N5: nat] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( nth @ A @ Xs2 @ N5 ) @ Y ) @ R )
& ( ord_less @ nat @ N5 @ ( size_size @ ( list @ A ) @ Xs2 ) )
& ( Ys2
= ( list_update @ A @ Xs2 @ N5 @ Y ) ) ) ) ) ).
% listrel1_iff_update
thf(fact_4325_mem__Sigma__iff,axiom,
! [B: $tType,A: $tType,A3: A,B2: B,A5: set @ A,B4: A > ( set @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ ( product_Sigma @ A @ B @ A5 @ B4 ) )
= ( ( member @ A @ A3 @ A5 )
& ( member @ B @ B2 @ ( B4 @ A3 ) ) ) ) ).
% mem_Sigma_iff
thf(fact_4326_SigmaI,axiom,
! [B: $tType,A: $tType,A3: A,A5: set @ A,B2: B,B4: A > ( set @ B )] :
( ( member @ A @ A3 @ A5 )
=> ( ( member @ B @ B2 @ ( B4 @ A3 ) )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ ( product_Sigma @ A @ B @ A5 @ B4 ) ) ) ) ).
% SigmaI
thf(fact_4327_Field__square,axiom,
! [A: $tType,X: set @ A] :
( ( field2 @ A
@ ( product_Sigma @ A @ A @ X
@ ^ [Uu: A] : X ) )
= X ) ).
% Field_square
thf(fact_4328_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
@ ^ [A6: A,B5: B] :
( ( P @ A6 )
& ( Q @ B5 ) ) ) )
= ( product_Sigma @ A @ B @ ( collect @ A @ P )
@ ^ [Uu: A] : ( collect @ B @ Q ) ) ) ).
% Collect_case_prod
thf(fact_4329_Sigma__empty1,axiom,
! [B: $tType,A: $tType,B4: A > ( set @ B )] :
( ( product_Sigma @ A @ B @ ( bot_bot @ ( set @ A ) ) @ B4 )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% Sigma_empty1
thf(fact_4330_Times__empty,axiom,
! [A: $tType,B: $tType,A5: set @ A,B4: set @ B] :
( ( ( product_Sigma @ A @ B @ A5
@ ^ [Uu: A] : B4 )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) )
= ( ( A5
= ( bot_bot @ ( set @ A ) ) )
| ( B4
= ( bot_bot @ ( set @ B ) ) ) ) ) ).
% Times_empty
thf(fact_4331_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_4332_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_4333_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_4334_finite__SigmaI,axiom,
! [B: $tType,A: $tType,A5: set @ A,B4: A > ( set @ B )] :
( ( finite_finite2 @ A @ A5 )
=> ( ! [A4: A] :
( ( member @ A @ A4 @ A5 )
=> ( finite_finite2 @ B @ ( B4 @ A4 ) ) )
=> ( finite_finite2 @ ( product_prod @ A @ B ) @ ( product_Sigma @ A @ B @ A5 @ B4 ) ) ) ) ).
% finite_SigmaI
thf(fact_4335_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_4336_fst__image__times,axiom,
! [B: $tType,A: $tType,B4: set @ B,A5: set @ A] :
( ( ( B4
= ( bot_bot @ ( set @ B ) ) )
=> ( ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B )
@ ( product_Sigma @ A @ B @ A5
@ ^ [Uu: A] : B4 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( B4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B )
@ ( product_Sigma @ A @ B @ A5
@ ^ [Uu: A] : B4 ) )
= A5 ) ) ) ).
% fst_image_times
thf(fact_4337_snd__image__times,axiom,
! [B: $tType,A: $tType,A5: set @ B,B4: set @ A] :
( ( ( A5
= ( bot_bot @ ( set @ B ) ) )
=> ( ( image2 @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A )
@ ( product_Sigma @ B @ A @ A5
@ ^ [Uu: B] : B4 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( image2 @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A )
@ ( product_Sigma @ B @ A @ A5
@ ^ [Uu: B] : B4 ) )
= B4 ) ) ) ).
% snd_image_times
thf(fact_4338_set__product,axiom,
! [B: $tType,A: $tType,Xs2: list @ A,Ys2: list @ B] :
( ( set2 @ ( product_prod @ A @ B ) @ ( product @ A @ B @ Xs2 @ Ys2 ) )
= ( product_Sigma @ A @ B @ ( set2 @ A @ Xs2 )
@ ^ [Uu: A] : ( set2 @ B @ Ys2 ) ) ) ).
% set_product
thf(fact_4339_insert__Times__insert,axiom,
! [B: $tType,A: $tType,A3: A,A5: set @ A,B2: B,B4: set @ B] :
( ( product_Sigma @ A @ B @ ( insert2 @ A @ A3 @ A5 )
@ ^ [Uu: A] : ( insert2 @ B @ B2 @ B4 ) )
= ( insert2 @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 )
@ ( sup_sup @ ( set @ ( product_prod @ A @ B ) )
@ ( product_Sigma @ A @ B @ A5
@ ^ [Uu: A] : ( insert2 @ B @ B2 @ B4 ) )
@ ( product_Sigma @ A @ B @ ( insert2 @ A @ A3 @ A5 )
@ ^ [Uu: A] : B4 ) ) ) ) ).
% insert_Times_insert
thf(fact_4340_card__SigmaI,axiom,
! [B: $tType,A: $tType,A5: set @ A,B4: A > ( set @ B )] :
( ( finite_finite2 @ A @ A5 )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ A5 )
=> ( finite_finite2 @ B @ ( B4 @ X4 ) ) )
=> ( ( finite_card @ ( product_prod @ A @ B ) @ ( product_Sigma @ A @ B @ A5 @ B4 ) )
= ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [A6: A] : ( finite_card @ B @ ( B4 @ A6 ) )
@ A5 ) ) ) ) ).
% card_SigmaI
thf(fact_4341_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_4342_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_4343_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
@ ^ [X2: A,Y: B] :
( ( P @ X2 )
& ( Q @ X2 @ Y ) ) ) )
= ( product_Sigma @ A @ B @ ( collect @ A @ P )
@ ^ [X2: A] : ( collect @ B @ ( Q @ X2 ) ) ) ) ).
% Collect_case_prod_Sigma
thf(fact_4344_Sigma__cong,axiom,
! [B: $tType,A: $tType,A5: set @ A,B4: set @ A,C6: A > ( set @ B ),D5: A > ( set @ B )] :
( ( A5 = B4 )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ B4 )
=> ( ( C6 @ X4 )
= ( D5 @ X4 ) ) )
=> ( ( product_Sigma @ A @ B @ A5 @ C6 )
= ( product_Sigma @ A @ B @ B4 @ D5 ) ) ) ) ).
% Sigma_cong
thf(fact_4345_Times__eq__cancel2,axiom,
! [A: $tType,B: $tType,X: A,C6: set @ A,A5: set @ B,B4: set @ B] :
( ( member @ A @ X @ C6 )
=> ( ( ( product_Sigma @ B @ A @ A5
@ ^ [Uu: B] : C6 )
= ( product_Sigma @ B @ A @ B4
@ ^ [Uu: B] : C6 ) )
= ( A5 = B4 ) ) ) ).
% Times_eq_cancel2
thf(fact_4346_zip__update,axiom,
! [A: $tType,B: $tType,Xs2: list @ A,I: nat,X: A,Ys2: list @ B,Y3: B] :
( ( zip @ A @ B @ ( list_update @ A @ Xs2 @ I @ X ) @ ( list_update @ B @ Ys2 @ I @ Y3 ) )
= ( list_update @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs2 @ Ys2 ) @ I @ ( product_Pair @ A @ B @ X @ Y3 ) ) ) ).
% zip_update
thf(fact_4347_SigmaE2,axiom,
! [B: $tType,A: $tType,A3: A,B2: B,A5: set @ A,B4: A > ( set @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ ( product_Sigma @ A @ B @ A5 @ B4 ) )
=> ~ ( ( member @ A @ A3 @ A5 )
=> ~ ( member @ B @ B2 @ ( B4 @ A3 ) ) ) ) ).
% SigmaE2
thf(fact_4348_SigmaD2,axiom,
! [B: $tType,A: $tType,A3: A,B2: B,A5: set @ A,B4: A > ( set @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ ( product_Sigma @ A @ B @ A5 @ B4 ) )
=> ( member @ B @ B2 @ ( B4 @ A3 ) ) ) ).
% SigmaD2
thf(fact_4349_SigmaD1,axiom,
! [B: $tType,A: $tType,A3: A,B2: B,A5: set @ A,B4: A > ( set @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ ( product_Sigma @ A @ B @ A5 @ B4 ) )
=> ( member @ A @ A3 @ A5 ) ) ).
% SigmaD1
thf(fact_4350_SigmaE,axiom,
! [A: $tType,B: $tType,C2: product_prod @ A @ B,A5: set @ A,B4: A > ( set @ B )] :
( ( member @ ( product_prod @ A @ B ) @ C2 @ ( product_Sigma @ A @ B @ A5 @ B4 ) )
=> ~ ! [X4: A] :
( ( member @ A @ X4 @ A5 )
=> ! [Y4: B] :
( ( member @ B @ Y4 @ ( B4 @ X4 ) )
=> ( C2
!= ( product_Pair @ A @ B @ X4 @ Y4 ) ) ) ) ) ).
% SigmaE
thf(fact_4351_times__eq__iff,axiom,
! [A: $tType,B: $tType,A5: set @ A,B4: set @ B,C6: set @ A,D5: set @ B] :
( ( ( product_Sigma @ A @ B @ A5
@ ^ [Uu: A] : B4 )
= ( product_Sigma @ A @ B @ C6
@ ^ [Uu: A] : D5 ) )
= ( ( ( A5 = C6 )
& ( B4 = D5 ) )
| ( ( ( A5
= ( bot_bot @ ( set @ A ) ) )
| ( B4
= ( bot_bot @ ( set @ B ) ) ) )
& ( ( C6
= ( bot_bot @ ( set @ A ) ) )
| ( D5
= ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ).
% times_eq_iff
thf(fact_4352_Sigma__Diff__distrib1,axiom,
! [B: $tType,A: $tType,I5: set @ A,J5: set @ A,C6: A > ( set @ B )] :
( ( product_Sigma @ A @ B @ ( minus_minus @ ( set @ A ) @ I5 @ J5 ) @ C6 )
= ( minus_minus @ ( set @ ( product_prod @ A @ B ) ) @ ( product_Sigma @ A @ B @ I5 @ C6 ) @ ( product_Sigma @ A @ B @ J5 @ C6 ) ) ) ).
% Sigma_Diff_distrib1
thf(fact_4353_Times__Diff__distrib1,axiom,
! [B: $tType,A: $tType,A5: set @ A,B4: set @ A,C6: set @ B] :
( ( product_Sigma @ A @ B @ ( minus_minus @ ( set @ A ) @ A5 @ B4 )
@ ^ [Uu: A] : C6 )
= ( minus_minus @ ( set @ ( product_prod @ A @ B ) )
@ ( product_Sigma @ A @ B @ A5
@ ^ [Uu: A] : C6 )
@ ( product_Sigma @ A @ B @ B4
@ ^ [Uu: A] : C6 ) ) ) ).
% Times_Diff_distrib1
thf(fact_4354_Sigma__Diff__distrib2,axiom,
! [B: $tType,A: $tType,I5: set @ A,A5: A > ( set @ B ),B4: A > ( set @ B )] :
( ( product_Sigma @ A @ B @ I5
@ ^ [I2: A] : ( minus_minus @ ( set @ B ) @ ( A5 @ I2 ) @ ( B4 @ I2 ) ) )
= ( minus_minus @ ( set @ ( product_prod @ A @ B ) ) @ ( product_Sigma @ A @ B @ I5 @ A5 ) @ ( product_Sigma @ A @ B @ I5 @ B4 ) ) ) ).
% Sigma_Diff_distrib2
thf(fact_4355_Sigma__mono,axiom,
! [B: $tType,A: $tType,A5: set @ A,C6: set @ A,B4: A > ( set @ B ),D5: A > ( set @ B )] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ C6 )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ A5 )
=> ( ord_less_eq @ ( set @ B ) @ ( B4 @ X4 ) @ ( D5 @ X4 ) ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( product_Sigma @ A @ B @ A5 @ B4 ) @ ( product_Sigma @ A @ B @ C6 @ D5 ) ) ) ) ).
% Sigma_mono
thf(fact_4356_Sigma__Int__distrib1,axiom,
! [B: $tType,A: $tType,I5: set @ A,J5: set @ A,C6: A > ( set @ B )] :
( ( product_Sigma @ A @ B @ ( inf_inf @ ( set @ A ) @ I5 @ J5 ) @ C6 )
= ( inf_inf @ ( set @ ( product_prod @ A @ B ) ) @ ( product_Sigma @ A @ B @ I5 @ C6 ) @ ( product_Sigma @ A @ B @ J5 @ C6 ) ) ) ).
% Sigma_Int_distrib1
thf(fact_4357_Sigma__Un__distrib1,axiom,
! [B: $tType,A: $tType,I5: set @ A,J5: set @ A,C6: A > ( set @ B )] :
( ( product_Sigma @ A @ B @ ( sup_sup @ ( set @ A ) @ I5 @ J5 ) @ C6 )
= ( sup_sup @ ( set @ ( product_prod @ A @ B ) ) @ ( product_Sigma @ A @ B @ I5 @ C6 ) @ ( product_Sigma @ A @ B @ J5 @ C6 ) ) ) ).
% Sigma_Un_distrib1
thf(fact_4358_list__update__code_I2_J,axiom,
! [A: $tType,X: A,Xs2: list @ A,Y3: A] :
( ( list_update @ A @ ( cons @ A @ X @ Xs2 ) @ ( zero_zero @ nat ) @ Y3 )
= ( cons @ A @ Y3 @ Xs2 ) ) ).
% list_update_code(2)
thf(fact_4359_Times__subset__cancel2,axiom,
! [A: $tType,B: $tType,X: A,C6: set @ A,A5: set @ B,B4: set @ B] :
( ( member @ A @ X @ C6 )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ B @ A ) )
@ ( product_Sigma @ B @ A @ A5
@ ^ [Uu: B] : C6 )
@ ( product_Sigma @ B @ A @ B4
@ ^ [Uu: B] : C6 ) )
= ( ord_less_eq @ ( set @ B ) @ A5 @ B4 ) ) ) ).
% Times_subset_cancel2
thf(fact_4360_mem__Times__iff,axiom,
! [A: $tType,B: $tType,X: product_prod @ A @ B,A5: set @ A,B4: set @ B] :
( ( member @ ( product_prod @ A @ B ) @ X
@ ( product_Sigma @ A @ B @ A5
@ ^ [Uu: A] : B4 ) )
= ( ( member @ A @ ( product_fst @ A @ B @ X ) @ A5 )
& ( member @ B @ ( product_snd @ A @ B @ X ) @ B4 ) ) ) ).
% mem_Times_iff
thf(fact_4361_swap__product,axiom,
! [B: $tType,A: $tType,A5: set @ B,B4: set @ A] :
( ( image2 @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B )
@ ( product_case_prod @ B @ A @ ( product_prod @ A @ B )
@ ^ [I2: B,J3: A] : ( product_Pair @ A @ B @ J3 @ I2 ) )
@ ( product_Sigma @ B @ A @ A5
@ ^ [Uu: B] : B4 ) )
= ( product_Sigma @ A @ B @ B4
@ ^ [Uu: A] : A5 ) ) ).
% swap_product
thf(fact_4362_Sigma__empty__iff,axiom,
! [B: $tType,A: $tType,I5: set @ A,X7: A > ( set @ B )] :
( ( ( product_Sigma @ A @ B @ I5 @ X7 )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ I5 )
=> ( ( X7 @ X2 )
= ( bot_bot @ ( set @ B ) ) ) ) ) ) ).
% Sigma_empty_iff
thf(fact_4363_Times__Int__Times,axiom,
! [B: $tType,A: $tType,A5: set @ A,B4: set @ B,C6: set @ A,D5: set @ B] :
( ( inf_inf @ ( set @ ( product_prod @ A @ B ) )
@ ( product_Sigma @ A @ B @ A5
@ ^ [Uu: A] : B4 )
@ ( product_Sigma @ A @ B @ C6
@ ^ [Uu: A] : D5 ) )
= ( product_Sigma @ A @ B @ ( inf_inf @ ( set @ A ) @ A5 @ C6 )
@ ^ [Uu: A] : ( inf_inf @ ( set @ B ) @ B4 @ D5 ) ) ) ).
% Times_Int_Times
thf(fact_4364_Sigma__Int__distrib2,axiom,
! [B: $tType,A: $tType,I5: set @ A,A5: A > ( set @ B ),B4: A > ( set @ B )] :
( ( product_Sigma @ A @ B @ I5
@ ^ [I2: A] : ( inf_inf @ ( set @ B ) @ ( A5 @ I2 ) @ ( B4 @ I2 ) ) )
= ( inf_inf @ ( set @ ( product_prod @ A @ B ) ) @ ( product_Sigma @ A @ B @ I5 @ A5 ) @ ( product_Sigma @ A @ B @ I5 @ B4 ) ) ) ).
% Sigma_Int_distrib2
thf(fact_4365_Times__Int__distrib1,axiom,
! [B: $tType,A: $tType,A5: set @ A,B4: set @ A,C6: set @ B] :
( ( product_Sigma @ A @ B @ ( inf_inf @ ( set @ A ) @ A5 @ B4 )
@ ^ [Uu: A] : C6 )
= ( inf_inf @ ( set @ ( product_prod @ A @ B ) )
@ ( product_Sigma @ A @ B @ A5
@ ^ [Uu: A] : C6 )
@ ( product_Sigma @ A @ B @ B4
@ ^ [Uu: A] : C6 ) ) ) ).
% Times_Int_distrib1
thf(fact_4366_infinite__cartesian__product,axiom,
! [B: $tType,A: $tType,A5: set @ A,B4: set @ B] :
( ~ ( finite_finite2 @ A @ A5 )
=> ( ~ ( finite_finite2 @ B @ B4 )
=> ~ ( finite_finite2 @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A5
@ ^ [Uu: A] : B4 ) ) ) ) ).
% infinite_cartesian_product
thf(fact_4367_finite__cartesian__product,axiom,
! [B: $tType,A: $tType,A5: set @ A,B4: set @ B] :
( ( finite_finite2 @ A @ A5 )
=> ( ( finite_finite2 @ B @ B4 )
=> ( finite_finite2 @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A5
@ ^ [Uu: A] : B4 ) ) ) ) ).
% finite_cartesian_product
thf(fact_4368_Sigma__Un__distrib2,axiom,
! [B: $tType,A: $tType,I5: set @ A,A5: A > ( set @ B ),B4: A > ( set @ B )] :
( ( product_Sigma @ A @ B @ I5
@ ^ [I2: A] : ( sup_sup @ ( set @ B ) @ ( A5 @ I2 ) @ ( B4 @ I2 ) ) )
= ( sup_sup @ ( set @ ( product_prod @ A @ B ) ) @ ( product_Sigma @ A @ B @ I5 @ A5 ) @ ( product_Sigma @ A @ B @ I5 @ B4 ) ) ) ).
% Sigma_Un_distrib2
thf(fact_4369_Times__Un__distrib1,axiom,
! [B: $tType,A: $tType,A5: set @ A,B4: set @ A,C6: set @ B] :
( ( product_Sigma @ A @ B @ ( sup_sup @ ( set @ A ) @ A5 @ B4 )
@ ^ [Uu: A] : C6 )
= ( sup_sup @ ( set @ ( product_prod @ A @ B ) )
@ ( product_Sigma @ A @ B @ A5
@ ^ [Uu: A] : C6 )
@ ( product_Sigma @ A @ B @ B4
@ ^ [Uu: A] : C6 ) ) ) ).
% Times_Un_distrib1
thf(fact_4370_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_4371_relcomp__subset__Sigma,axiom,
! [B: $tType,C: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),A5: set @ A,B4: set @ B,S: set @ ( product_prod @ B @ C ),C6: set @ C] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R
@ ( product_Sigma @ A @ B @ A5
@ ^ [Uu: A] : B4 ) )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ B @ C ) ) @ S
@ ( product_Sigma @ B @ C @ B4
@ ^ [Uu: B] : C6 ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ C ) ) @ ( relcomp @ A @ B @ C @ R @ S )
@ ( product_Sigma @ A @ C @ A5
@ ^ [Uu: A] : C6 ) ) ) ) ).
% relcomp_subset_Sigma
thf(fact_4372_Sigma__Union,axiom,
! [B: $tType,A: $tType,X7: set @ ( set @ A ),B4: A > ( set @ B )] :
( ( product_Sigma @ A @ B @ ( complete_Sup_Sup @ ( set @ A ) @ X7 ) @ B4 )
= ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) )
@ ( image2 @ ( set @ A ) @ ( set @ ( product_prod @ A @ B ) )
@ ^ [A8: set @ A] : ( product_Sigma @ A @ B @ A8 @ B4 )
@ X7 ) ) ) ).
% Sigma_Union
thf(fact_4373_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_4374_total__on__lex__prod,axiom,
! [A: $tType,B: $tType,A5: set @ A,R: set @ ( product_prod @ A @ A ),B4: set @ B,S: set @ ( product_prod @ B @ B )] :
( ( total_on @ A @ A5 @ R )
=> ( ( total_on @ B @ B4 @ S )
=> ( total_on @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A5
@ ^ [Uu: A] : B4 )
@ ( lex_prod @ A @ B @ R @ S ) ) ) ) ).
% total_on_lex_prod
thf(fact_4375_times__subset__iff,axiom,
! [A: $tType,B: $tType,A5: set @ A,C6: set @ B,B4: set @ A,D5: set @ B] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) )
@ ( product_Sigma @ A @ B @ A5
@ ^ [Uu: A] : C6 )
@ ( product_Sigma @ A @ B @ B4
@ ^ [Uu: A] : D5 ) )
= ( ( A5
= ( bot_bot @ ( set @ A ) ) )
| ( C6
= ( bot_bot @ ( set @ B ) ) )
| ( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
& ( ord_less_eq @ ( set @ B ) @ C6 @ D5 ) ) ) ) ).
% times_subset_iff
thf(fact_4376_image__paired__Times,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,F: C > A,G: D > B,A5: set @ C,B4: set @ D] :
( ( image2 @ ( product_prod @ C @ D ) @ ( product_prod @ A @ B )
@ ( product_case_prod @ C @ D @ ( product_prod @ A @ B )
@ ^ [X2: C,Y: D] : ( product_Pair @ A @ B @ ( F @ X2 ) @ ( G @ Y ) ) )
@ ( product_Sigma @ C @ D @ A5
@ ^ [Uu: C] : B4 ) )
= ( product_Sigma @ A @ B @ ( image2 @ C @ A @ F @ A5 )
@ ^ [Uu: A] : ( image2 @ D @ B @ G @ B4 ) ) ) ).
% image_paired_Times
thf(fact_4377_trancl__subset__Sigma__aux,axiom,
! [A: $tType,A3: A,B2: A,R: set @ ( product_prod @ A @ A ),A5: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( transitive_rtrancl @ A @ R ) )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R
@ ( product_Sigma @ A @ A @ A5
@ ^ [Uu: A] : A5 ) )
=> ( ( A3 = B2 )
| ( member @ A @ A3 @ A5 ) ) ) ) ).
% trancl_subset_Sigma_aux
thf(fact_4378_finite__cartesian__product__iff,axiom,
! [A: $tType,B: $tType,A5: set @ A,B4: set @ B] :
( ( finite_finite2 @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A5
@ ^ [Uu: A] : B4 ) )
= ( ( A5
= ( bot_bot @ ( set @ A ) ) )
| ( B4
= ( bot_bot @ ( set @ B ) ) )
| ( ( finite_finite2 @ A @ A5 )
& ( finite_finite2 @ B @ B4 ) ) ) ) ).
% finite_cartesian_product_iff
thf(fact_4379_finite__cartesian__productD2,axiom,
! [A: $tType,B: $tType,A5: set @ A,B4: set @ B] :
( ( finite_finite2 @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A5
@ ^ [Uu: A] : B4 ) )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( finite_finite2 @ B @ B4 ) ) ) ).
% finite_cartesian_productD2
thf(fact_4380_finite__cartesian__productD1,axiom,
! [B: $tType,A: $tType,A5: set @ A,B4: set @ B] :
( ( finite_finite2 @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A5
@ ^ [Uu: A] : B4 ) )
=> ( ( B4
!= ( bot_bot @ ( set @ B ) ) )
=> ( finite_finite2 @ A @ A5 ) ) ) ).
% finite_cartesian_productD1
thf(fact_4381_finite__SigmaI2,axiom,
! [B: $tType,A: $tType,A5: set @ A,B4: A > ( set @ B )] :
( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X2: A] :
( ( member @ A @ X2 @ A5 )
& ( ( B4 @ X2 )
!= ( bot_bot @ ( set @ B ) ) ) ) ) )
=> ( ! [A4: A] :
( ( member @ A @ A4 @ A5 )
=> ( finite_finite2 @ B @ ( B4 @ A4 ) ) )
=> ( finite_finite2 @ ( product_prod @ A @ B ) @ ( product_Sigma @ A @ B @ A5 @ B4 ) ) ) ) ).
% finite_SigmaI2
thf(fact_4382_Image__subset,axiom,
! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ B ),A5: set @ A,B4: set @ B,C6: set @ A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R
@ ( product_Sigma @ A @ B @ A5
@ ^ [Uu: A] : B4 ) )
=> ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ R @ C6 ) @ B4 ) ) ).
% Image_subset
thf(fact_4383_trancl__subset__Field2,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( transitive_trancl @ A @ R )
@ ( product_Sigma @ A @ A @ ( field2 @ A @ R )
@ ^ [Uu: A] : ( field2 @ A @ R ) ) ) ).
% trancl_subset_Field2
thf(fact_4384_fst__image__Sigma,axiom,
! [B: $tType,A: $tType,A5: set @ A,B4: A > ( set @ B )] :
( ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ ( product_Sigma @ A @ B @ A5 @ B4 ) )
= ( collect @ A
@ ^ [X2: A] :
( ( member @ A @ X2 @ A5 )
& ( ( B4 @ X2 )
!= ( bot_bot @ ( set @ B ) ) ) ) ) ) ).
% fst_image_Sigma
thf(fact_4385_finite__equiv__class,axiom,
! [A: $tType,A5: set @ A,R: set @ ( product_prod @ A @ A ),X7: 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 ) @ X7 @ ( equiv_quotient @ A @ A5 @ R ) )
=> ( finite_finite2 @ A @ X7 ) ) ) ) ).
% finite_equiv_class
thf(fact_4386_UN__Times__distrib,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,E5: C > ( set @ A ),F6: D > ( set @ B ),A5: set @ C,B4: set @ D] :
( ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) )
@ ( image2 @ ( product_prod @ C @ D ) @ ( set @ ( product_prod @ A @ B ) )
@ ( product_case_prod @ C @ D @ ( set @ ( product_prod @ A @ B ) )
@ ^ [A6: C,B5: D] :
( product_Sigma @ A @ B @ ( E5 @ A6 )
@ ^ [Uu: A] : ( F6 @ B5 ) ) )
@ ( product_Sigma @ C @ D @ A5
@ ^ [Uu: C] : B4 ) ) )
= ( product_Sigma @ A @ B @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ C @ ( set @ A ) @ E5 @ A5 ) )
@ ^ [Uu: A] : ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ D @ ( set @ B ) @ F6 @ B4 ) ) ) ) ).
% UN_Times_distrib
thf(fact_4387_list__update_Osimps_I2_J,axiom,
! [A: $tType,X: A,Xs2: list @ A,I: nat,V: A] :
( ( list_update @ A @ ( cons @ A @ X @ Xs2 ) @ I @ V )
= ( case_nat @ ( list @ A ) @ ( cons @ A @ V @ Xs2 )
@ ^ [J3: nat] : ( cons @ A @ X @ ( list_update @ A @ Xs2 @ J3 @ V ) )
@ I ) ) ).
% list_update.simps(2)
thf(fact_4388_card__cartesian__product,axiom,
! [A: $tType,B: $tType,A5: set @ A,B4: set @ B] :
( ( finite_card @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A5
@ ^ [Uu: A] : B4 ) )
= ( times_times @ nat @ ( finite_card @ A @ A5 ) @ ( finite_card @ B @ B4 ) ) ) ).
% card_cartesian_product
thf(fact_4389_sum_Ocartesian__product,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: B > C > A,B4: set @ C,A5: set @ B] :
( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X2: B] : ( groups7311177749621191930dd_sum @ C @ A @ ( G @ X2 ) @ B4 )
@ A5 )
= ( groups7311177749621191930dd_sum @ ( product_prod @ B @ C ) @ A @ ( product_case_prod @ B @ C @ A @ G )
@ ( product_Sigma @ B @ C @ A5
@ ^ [Uu: B] : B4 ) ) ) ) ).
% sum.cartesian_product
thf(fact_4390_prod_Ocartesian__product,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: B > C > A,B4: set @ C,A5: set @ B] :
( ( groups7121269368397514597t_prod @ B @ A
@ ^ [X2: B] : ( groups7121269368397514597t_prod @ C @ A @ ( G @ X2 ) @ B4 )
@ A5 )
= ( groups7121269368397514597t_prod @ ( product_prod @ B @ C ) @ A @ ( product_case_prod @ B @ C @ A @ G )
@ ( product_Sigma @ B @ C @ A5
@ ^ [Uu: B] : B4 ) ) ) ) ).
% prod.cartesian_product
thf(fact_4391_Sigma__Image,axiom,
! [A: $tType,B: $tType,A5: set @ B,B4: B > ( set @ A ),X7: set @ B] :
( ( image @ B @ A @ ( product_Sigma @ B @ A @ A5 @ B4 ) @ X7 )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B4 @ ( inf_inf @ ( set @ B ) @ X7 @ A5 ) ) ) ) ).
% Sigma_Image
thf(fact_4392_snd__image__Sigma,axiom,
! [A: $tType,B: $tType,A5: set @ B,B4: B > ( set @ A )] :
( ( image2 @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) @ ( product_Sigma @ B @ A @ A5 @ B4 ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B4 @ A5 ) ) ) ).
% snd_image_Sigma
thf(fact_4393_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_4394_card__cartesian__product__singleton,axiom,
! [A: $tType,B: $tType,X: A,A5: set @ B] :
( ( finite_card @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) )
@ ^ [Uu: A] : A5 ) )
= ( finite_card @ B @ A5 ) ) ).
% card_cartesian_product_singleton
thf(fact_4395_butlast__list__update,axiom,
! [A: $tType,K: nat,Xs2: list @ A,X: A] :
( ( ( K
= ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ ( one_one @ nat ) ) )
=> ( ( butlast @ A @ ( list_update @ A @ Xs2 @ K @ X ) )
= ( butlast @ A @ Xs2 ) ) )
& ( ( K
!= ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ ( one_one @ nat ) ) )
=> ( ( butlast @ A @ ( list_update @ A @ Xs2 @ K @ X ) )
= ( list_update @ A @ ( butlast @ A @ Xs2 ) @ K @ X ) ) ) ) ).
% butlast_list_update
thf(fact_4396_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_4397_sum_OSigma,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A5: set @ B,B4: B > ( set @ C ),G: B > C > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ A5 )
=> ( finite_finite2 @ C @ ( B4 @ X4 ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X2: B] : ( groups7311177749621191930dd_sum @ C @ A @ ( G @ X2 ) @ ( B4 @ X2 ) )
@ A5 )
= ( groups7311177749621191930dd_sum @ ( product_prod @ B @ C ) @ A @ ( product_case_prod @ B @ C @ A @ G ) @ ( product_Sigma @ B @ C @ A5 @ B4 ) ) ) ) ) ) ).
% sum.Sigma
thf(fact_4398_prod_OSigma,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A5: set @ B,B4: B > ( set @ C ),G: B > C > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ A5 )
=> ( finite_finite2 @ C @ ( B4 @ X4 ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [X2: B] : ( groups7121269368397514597t_prod @ C @ A @ ( G @ X2 ) @ ( B4 @ X2 ) )
@ A5 )
= ( groups7121269368397514597t_prod @ ( product_prod @ B @ C ) @ A @ ( product_case_prod @ B @ C @ A @ G ) @ ( product_Sigma @ B @ C @ A5 @ B4 ) ) ) ) ) ) ).
% prod.Sigma
thf(fact_4399_Sigma__def,axiom,
! [B: $tType,A: $tType] :
( ( product_Sigma @ A @ B )
= ( ^ [A8: set @ A,B7: A > ( set @ B )] :
( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) )
@ ( image2 @ A @ ( set @ ( product_prod @ A @ B ) )
@ ^ [X2: A] :
( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) )
@ ( image2 @ B @ ( set @ ( product_prod @ A @ B ) )
@ ^ [Y: B] : ( insert2 @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y ) @ ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) )
@ ( B7 @ X2 ) ) )
@ A8 ) ) ) ) ).
% Sigma_def
thf(fact_4400_product__fold,axiom,
! [B: $tType,A: $tType,A5: set @ A,B4: set @ B] :
( ( finite_finite2 @ A @ A5 )
=> ( ( finite_finite2 @ B @ B4 )
=> ( ( product_Sigma @ A @ B @ A5
@ ^ [Uu: A] : B4 )
= ( finite_fold @ A @ ( set @ ( product_prod @ A @ B ) )
@ ^ [X2: A,Z3: set @ ( product_prod @ A @ B )] :
( finite_fold @ B @ ( set @ ( product_prod @ A @ B ) )
@ ^ [Y: B] : ( insert2 @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y ) )
@ Z3
@ B4 )
@ ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) )
@ A5 ) ) ) ) ).
% product_fold
thf(fact_4401_Ex__inj__on__UNION__Sigma,axiom,
! [A: $tType,B: $tType,A5: B > ( set @ A ),I5: set @ B] :
? [F4: A > ( product_prod @ B @ A )] :
( ( inj_on @ A @ ( product_prod @ B @ A ) @ F4 @ ( 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 ) @ F4 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ A5 @ I5 ) ) ) @ ( product_Sigma @ B @ A @ I5 @ A5 ) ) ) ).
% Ex_inj_on_UNION_Sigma
thf(fact_4402_sum__mult__sum__if__inj,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( semiring_0 @ B )
=> ! [F: A > B,G: C > B,A5: set @ A,B4: set @ C] :
( ( inj_on @ ( product_prod @ A @ C ) @ B
@ ( product_case_prod @ A @ C @ B
@ ^ [A6: A,B5: C] : ( times_times @ B @ ( F @ A6 ) @ ( G @ B5 ) ) )
@ ( product_Sigma @ A @ C @ A5
@ ^ [Uu: A] : B4 ) )
=> ( ( times_times @ B @ ( groups7311177749621191930dd_sum @ A @ B @ F @ A5 ) @ ( groups7311177749621191930dd_sum @ C @ B @ G @ B4 ) )
= ( groups7311177749621191930dd_sum @ B @ B @ ( id @ B )
@ ( collect @ B
@ ^ [Uu: B] :
? [A6: A,B5: C] :
( ( Uu
= ( times_times @ B @ ( F @ A6 ) @ ( G @ B5 ) ) )
& ( member @ A @ A6 @ A5 )
& ( member @ C @ B5 @ B4 ) ) ) ) ) ) ) ).
% sum_mult_sum_if_inj
thf(fact_4403_Preorder__Restr,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A5: set @ A] :
( ( order_preorder_on @ A @ ( field2 @ A @ R ) @ R )
=> ( order_preorder_on @ A
@ ( field2 @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R
@ ( product_Sigma @ A @ A @ A5
@ ^ [Uu: A] : A5 ) ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R
@ ( product_Sigma @ A @ A @ A5
@ ^ [Uu: A] : A5 ) ) ) ) ).
% Preorder_Restr
thf(fact_4404_Linear__order__Restr,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A5: set @ A] :
( ( order_679001287576687338der_on @ A @ ( field2 @ A @ R ) @ R )
=> ( order_679001287576687338der_on @ A
@ ( field2 @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R
@ ( product_Sigma @ A @ A @ A5
@ ^ [Uu: A] : A5 ) ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R
@ ( product_Sigma @ A @ A @ A5
@ ^ [Uu: A] : A5 ) ) ) ) ).
% Linear_order_Restr
thf(fact_4405_Total__Restr,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A5: set @ A] :
( ( total_on @ A @ ( field2 @ A @ R ) @ R )
=> ( total_on @ A
@ ( field2 @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R
@ ( product_Sigma @ A @ A @ A5
@ ^ [Uu: A] : A5 ) ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R
@ ( product_Sigma @ A @ A @ A5
@ ^ [Uu: A] : A5 ) ) ) ) ).
% Total_Restr
thf(fact_4406_total__on__imp__Total__Restr,axiom,
! [A: $tType,A5: set @ A,R: set @ ( product_prod @ A @ A )] :
( ( total_on @ A @ A5 @ R )
=> ( total_on @ A
@ ( field2 @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R
@ ( product_Sigma @ A @ A @ A5
@ ^ [Uu: A] : A5 ) ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R
@ ( product_Sigma @ A @ A @ A5
@ ^ [Uu: A] : A5 ) ) ) ) ).
% total_on_imp_Total_Restr
thf(fact_4407_Restr__subset,axiom,
! [A: $tType,A5: set @ A,B4: set @ A,R: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
=> ( ( inf_inf @ ( set @ ( product_prod @ A @ A ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R
@ ( product_Sigma @ A @ A @ B4
@ ^ [Uu: A] : B4 ) )
@ ( 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_4408_Restr__Field,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R
@ ( product_Sigma @ A @ A @ ( field2 @ A @ R )
@ ^ [Uu: A] : ( field2 @ A @ R ) ) )
= R ) ).
% Restr_Field
thf(fact_4409_Field__Restr__subset,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A5: set @ A] :
( ord_less_eq @ ( set @ A )
@ ( field2 @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R
@ ( product_Sigma @ A @ A @ A5
@ ^ [Uu: A] : A5 ) ) )
@ A5 ) ).
% Field_Restr_subset
thf(fact_4410_linear__order__on__Restr,axiom,
! [A: $tType,A5: set @ A,R: set @ ( product_prod @ A @ A ),X: A] :
( ( order_679001287576687338der_on @ A @ A5 @ R )
=> ( order_679001287576687338der_on @ A @ ( inf_inf @ ( set @ A ) @ A5 @ ( order_above @ A @ R @ X ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R
@ ( product_Sigma @ A @ A @ ( order_above @ A @ R @ X )
@ ^ [Uu: A] : ( order_above @ A @ R @ X ) ) ) ) ) ).
% linear_order_on_Restr
thf(fact_4411_bsqr__def,axiom,
! [A: $tType] :
( ( bNF_Wellorder_bsqr @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ ( product_prod @ A @ A ) @ ( product_prod @ A @ A ) )
@ ( product_case_prod @ ( product_prod @ A @ A ) @ ( product_prod @ A @ A ) @ $o
@ ( product_case_prod @ A @ A @ ( ( product_prod @ A @ A ) > $o )
@ ^ [A12: A,A23: A] :
( product_case_prod @ A @ A @ $o
@ ^ [B1: A,B22: A] :
( ( ord_less_eq @ ( set @ A ) @ ( insert2 @ A @ A12 @ ( insert2 @ A @ A23 @ ( insert2 @ A @ B1 @ ( insert2 @ A @ B22 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) @ ( field2 @ A @ R4 ) )
& ( ( ( A12 = B1 )
& ( A23 = B22 ) )
| ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( bNF_We1388413361240627857o_max2 @ A @ R4 @ A12 @ A23 ) @ ( bNF_We1388413361240627857o_max2 @ A @ R4 @ B1 @ B22 ) ) @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R4 @ ( id2 @ A ) ) )
| ( ( ( bNF_We1388413361240627857o_max2 @ A @ R4 @ A12 @ A23 )
= ( bNF_We1388413361240627857o_max2 @ A @ R4 @ B1 @ B22 ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A12 @ B1 ) @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R4 @ ( id2 @ A ) ) ) )
| ( ( ( bNF_We1388413361240627857o_max2 @ A @ R4 @ A12 @ A23 )
= ( bNF_We1388413361240627857o_max2 @ A @ R4 @ B1 @ B22 ) )
& ( A12 = B1 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A23 @ B22 ) @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R4 @ ( id2 @ A ) ) ) ) ) ) ) ) ) ) ) ) ).
% bsqr_def
thf(fact_4412_Gr__incl,axiom,
! [A: $tType,B: $tType,A5: set @ A,F: A > B,B4: set @ B] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( bNF_Gr @ A @ B @ A5 @ F )
@ ( product_Sigma @ A @ B @ A5
@ ^ [Uu: A] : B4 ) )
= ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F @ A5 ) @ B4 ) ) ).
% Gr_incl
thf(fact_4413_Linear__order__wf__diff__Id,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( order_679001287576687338der_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( wf @ A @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R @ ( id2 @ A ) ) )
= ( ! [A8: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A8 @ ( field2 @ A @ R ) )
=> ( ( A8
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X2: A] :
( ( member @ A @ X2 @ A8 )
& ! [Y: A] :
( ( member @ A @ Y @ A8 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y ) @ R ) ) ) ) ) ) ) ) ).
% Linear_order_wf_diff_Id
thf(fact_4414_wf__insert,axiom,
! [A: $tType,Y3: A,X: A,R: set @ ( product_prod @ A @ A )] :
( ( wf @ A @ ( insert2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X ) @ R ) )
= ( ( wf @ A @ R )
& ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ ( transitive_rtrancl @ A @ R ) ) ) ) ).
% wf_insert
thf(fact_4415_wf__induct__rule,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),P: A > $o,A3: A] :
( ( wf @ A @ R )
=> ( ! [X4: A] :
( ! [Y5: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y5 @ X4 ) @ R )
=> ( P @ Y5 ) )
=> ( P @ X4 ) )
=> ( P @ A3 ) ) ) ).
% wf_induct_rule
thf(fact_4416_wf__eq__minimal,axiom,
! [A: $tType] :
( ( wf @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A )] :
! [Q7: set @ A] :
( ? [X2: A] : ( member @ A @ X2 @ Q7 )
=> ? [X2: A] :
( ( member @ A @ X2 @ Q7 )
& ! [Y: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X2 ) @ R4 )
=> ~ ( member @ A @ Y @ Q7 ) ) ) ) ) ) ).
% wf_eq_minimal
thf(fact_4417_wf__not__refl,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A3: A] :
( ( wf @ A @ R )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ A3 ) @ R ) ) ).
% wf_not_refl
thf(fact_4418_wf__not__sym,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A3: A,X: A] :
( ( wf @ A @ R )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ X ) @ R )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ A3 ) @ R ) ) ) ).
% wf_not_sym
thf(fact_4419_wf__irrefl,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A3: A] :
( ( wf @ A @ R )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ A3 ) @ R ) ) ).
% wf_irrefl
thf(fact_4420_wf__induct,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),P: A > $o,A3: A] :
( ( wf @ A @ R )
=> ( ! [X4: A] :
( ! [Y5: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y5 @ X4 ) @ R )
=> ( P @ Y5 ) )
=> ( P @ X4 ) )
=> ( P @ A3 ) ) ) ).
% wf_induct
thf(fact_4421_wf__asym,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A3: A,X: A] :
( ( wf @ A @ R )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ X ) @ R )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ A3 ) @ R ) ) ) ).
% wf_asym
thf(fact_4422_wfUNIVI,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ! [P7: A > $o,X4: A] :
( ! [Xa2: A] :
( ! [Y4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Xa2 ) @ R )
=> ( P7 @ Y4 ) )
=> ( P7 @ Xa2 ) )
=> ( P7 @ X4 ) )
=> ( wf @ A @ R ) ) ).
% wfUNIVI
thf(fact_4423_wfI__min,axiom,
! [A: $tType,R5: set @ ( product_prod @ A @ A )] :
( ! [X4: A,Q8: set @ A] :
( ( member @ A @ X4 @ Q8 )
=> ? [Xa2: A] :
( ( member @ A @ Xa2 @ Q8 )
& ! [Y4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Xa2 ) @ R5 )
=> ~ ( member @ A @ Y4 @ Q8 ) ) ) )
=> ( wf @ A @ R5 ) ) ).
% wfI_min
thf(fact_4424_wfE__min,axiom,
! [A: $tType,R5: set @ ( product_prod @ A @ A ),X: A,Q: set @ A] :
( ( wf @ A @ R5 )
=> ( ( member @ A @ X @ Q )
=> ~ ! [Z6: A] :
( ( member @ A @ Z6 @ Q )
=> ~ ! [Y5: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y5 @ Z6 ) @ R5 )
=> ~ ( member @ A @ Y5 @ Q ) ) ) ) ) ).
% wfE_min
thf(fact_4425_wf__def,axiom,
! [A: $tType] :
( ( wf @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A )] :
! [P3: A > $o] :
( ! [X2: A] :
( ! [Y: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X2 ) @ R4 )
=> ( P3 @ Y ) )
=> ( P3 @ X2 ) )
=> ! [X9: A] : ( P3 @ X9 ) ) ) ) ).
% wf_def
thf(fact_4426_GrD2,axiom,
! [A: $tType,B: $tType,X: A,Fx: B,A5: set @ A,F: A > B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Fx ) @ ( bNF_Gr @ A @ B @ A5 @ F ) )
=> ( ( F @ X )
= Fx ) ) ).
% GrD2
thf(fact_4427_GrD1,axiom,
! [B: $tType,A: $tType,X: A,Fx: B,A5: set @ A,F: A > B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Fx ) @ ( bNF_Gr @ A @ B @ A5 @ F ) )
=> ( member @ A @ X @ A5 ) ) ).
% GrD1
thf(fact_4428_wfE__min_H,axiom,
! [A: $tType,R5: set @ ( product_prod @ A @ A ),Q: set @ A] :
( ( wf @ A @ R5 )
=> ( ( Q
!= ( bot_bot @ ( set @ A ) ) )
=> ~ ! [Z6: A] :
( ( member @ A @ Z6 @ Q )
=> ~ ! [Y5: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y5 @ Z6 ) @ R5 )
=> ~ ( member @ A @ Y5 @ Q ) ) ) ) ) ).
% wfE_min'
thf(fact_4429_wf__no__infinite__down__chainE,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),F: nat > A] :
( ( wf @ A @ R )
=> ~ ! [K2: nat] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( F @ ( suc @ K2 ) ) @ ( F @ K2 ) ) @ R ) ) ).
% wf_no_infinite_down_chainE
thf(fact_4430_wf__iff__no__infinite__down__chain,axiom,
! [A: $tType] :
( ( wf @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A )] :
~ ? [F2: nat > A] :
! [I2: nat] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( F2 @ ( suc @ I2 ) ) @ ( F2 @ I2 ) ) @ R4 ) ) ) ).
% wf_iff_no_infinite_down_chain
thf(fact_4431_wf__int__ge__less__than,axiom,
! [D3: int] : ( wf @ int @ ( int_ge_less_than @ D3 ) ) ).
% wf_int_ge_less_than
thf(fact_4432_wf__int__ge__less__than2,axiom,
! [D3: int] : ( wf @ int @ ( int_ge_less_than2 @ D3 ) ) ).
% wf_int_ge_less_than2
thf(fact_4433_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_4434_wf__if__measure,axiom,
! [A: $tType,P: A > $o,F: A > nat,G: A > A] :
( ! [X4: A] :
( ( P @ X4 )
=> ( ord_less @ nat @ ( F @ ( G @ X4 ) ) @ ( F @ X4 ) ) )
=> ( wf @ A
@ ( collect @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ^ [Y: A,X2: A] :
( ( P @ X2 )
& ( Y
= ( G @ X2 ) ) ) ) ) ) ) ).
% wf_if_measure
thf(fact_4435_wf__less,axiom,
wf @ nat @ ( collect @ ( product_prod @ nat @ nat ) @ ( product_case_prod @ nat @ nat @ $o @ ( ord_less @ nat ) ) ) ).
% wf_less
thf(fact_4436_wf__bounded__measure,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),Ub: A > nat,F: A > nat] :
( ! [A4: A,B3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B3 @ A4 ) @ R )
=> ( ( ord_less_eq @ nat @ ( Ub @ B3 ) @ ( Ub @ A4 ) )
& ( ord_less_eq @ nat @ ( F @ B3 ) @ ( Ub @ A4 ) )
& ( ord_less @ nat @ ( F @ A4 ) @ ( F @ B3 ) ) ) )
=> ( wf @ A @ R ) ) ).
% wf_bounded_measure
thf(fact_4437_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 )
=> ( ! [X4: A,Y4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y4 ) @ ( transitive_trancl @ A @ R ) )
= ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ X4 ) @ ( transitive_rtrancl @ A @ R ) ) ) )
=> ( ( P @ K )
=> ? [X4: B] :
( ( P @ X4 )
& ! [Y5: B] :
( ( P @ Y5 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( M @ X4 ) @ ( M @ Y5 ) ) @ ( transitive_rtrancl @ A @ R ) ) ) ) ) ) ) ).
% wf_linord_ex_has_least
thf(fact_4438_wfI,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A5: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R
@ ( product_Sigma @ A @ A @ A5
@ ^ [Uu: A] : B4 ) )
=> ( ! [X4: A,P7: A > $o] :
( ! [Xa2: A] :
( ! [Y4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Xa2 ) @ R )
=> ( P7 @ Y4 ) )
=> ( P7 @ Xa2 ) )
=> ( ( member @ A @ X4 @ A5 )
=> ( ( member @ A @ X4 @ B4 )
=> ( P7 @ X4 ) ) ) )
=> ( wf @ A @ R ) ) ) ).
% wfI
thf(fact_4439_wf__eq__minimal2,axiom,
! [A: $tType] :
( ( wf @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A )] :
! [A8: set @ A] :
( ( ( ord_less_eq @ ( set @ A ) @ A8 @ ( field2 @ A @ R4 ) )
& ( A8
!= ( bot_bot @ ( set @ A ) ) ) )
=> ? [X2: A] :
( ( member @ A @ X2 @ A8 )
& ! [Y: A] :
( ( member @ A @ Y @ A8 )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X2 ) @ R4 ) ) ) ) ) ) ).
% wf_eq_minimal2
thf(fact_4440_wf__bounded__set,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),Ub: A > ( set @ B ),F: A > ( set @ B )] :
( ! [A4: A,B3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B3 @ A4 ) @ R )
=> ( ( finite_finite2 @ B @ ( Ub @ A4 ) )
& ( ord_less_eq @ ( set @ B ) @ ( Ub @ B3 ) @ ( Ub @ A4 ) )
& ( ord_less_eq @ ( set @ B ) @ ( F @ B3 ) @ ( Ub @ A4 ) )
& ( ord_less @ ( set @ B ) @ ( F @ A4 ) @ ( F @ B3 ) ) ) )
=> ( wf @ A @ R ) ) ).
% wf_bounded_set
thf(fact_4441_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] :
? [A6: A] :
( ( Uu
= ( product_Pair @ A @ B @ A6 @ ( F2 @ A6 ) ) )
& ( member @ A @ A6 @ A8 ) ) ) ) ) ).
% Gr_def
thf(fact_4442_Field__bsqr,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( field2 @ ( product_prod @ A @ A ) @ ( bNF_Wellorder_bsqr @ A @ R ) )
= ( product_Sigma @ A @ A @ ( field2 @ A @ R )
@ ^ [Uu: A] : ( field2 @ A @ R ) ) ) ).
% Field_bsqr
thf(fact_4443_above__def,axiom,
! [A: $tType] :
( ( order_above @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A ),A6: A] :
( collect @ A
@ ^ [B5: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A6 @ B5 ) @ R4 ) ) ) ) ).
% above_def
thf(fact_4444_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
@ ^ [X9: set @ A,Y10: set @ A] :
( ( ord_less @ ( set @ A ) @ X9 @ Y10 )
& ( ord_less_eq @ ( set @ A ) @ Y10 @ A5 ) ) ) ) ) ) ).
% finite_subset_wf
thf(fact_4445_reduction__pairI,axiom,
! [A: $tType,R5: set @ ( product_prod @ A @ A ),S3: set @ ( product_prod @ A @ A )] :
( ( wf @ A @ R5 )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( relcomp @ A @ A @ A @ R5 @ S3 ) @ R5 )
=> ( fun_reduction_pair @ A @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R5 @ S3 ) ) ) ) ).
% reduction_pairI
thf(fact_4446_dependent__wf__choice,axiom,
! [B: $tType,A: $tType,R5: set @ ( product_prod @ A @ A ),P: ( A > B ) > A > B > $o] :
( ( wf @ A @ R5 )
=> ( ! [F4: A > B,G4: A > B,X4: A,R2: B] :
( ! [Z8: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z8 @ X4 ) @ R5 )
=> ( ( F4 @ Z8 )
= ( G4 @ Z8 ) ) )
=> ( ( P @ F4 @ X4 @ R2 )
= ( P @ G4 @ X4 @ R2 ) ) )
=> ( ! [X4: A,F4: A > B] :
( ! [Y5: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y5 @ X4 ) @ R5 )
=> ( P @ F4 @ Y5 @ ( F4 @ Y5 ) ) )
=> ? [X_1: B] : ( P @ F4 @ X4 @ X_1 ) )
=> ? [F4: A > B] :
! [X5: A] : ( P @ F4 @ X5 @ ( F4 @ X5 ) ) ) ) ) ).
% dependent_wf_choice
thf(fact_4447_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 )
@ ^ [X2: C] : ( product_Pair @ A @ B @ ( F @ X2 ) @ ( G @ X2 ) )
@ A5 )
= ( product_Sigma @ A @ B @ ( image2 @ C @ A @ F @ A5 )
@ ^ [X2: A] : ( image2 @ C @ B @ G @ ( inf_inf @ ( set @ C ) @ ( vimage @ C @ A @ F @ ( insert2 @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) @ A5 ) ) ) ) ).
% image_split_eq_Sigma
thf(fact_4448_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,X2: D] : ( Q @ I2 @ ( F @ X2 ) ) ) ) ) ).
% mono_compose
thf(fact_4449_vimage__Collect__eq,axiom,
! [B: $tType,A: $tType,F: A > B,P: B > $o] :
( ( vimage @ A @ B @ F @ ( collect @ B @ P ) )
= ( collect @ A
@ ^ [Y: A] : ( P @ ( F @ Y ) ) ) ) ).
% vimage_Collect_eq
thf(fact_4450_vimage__ident,axiom,
! [A: $tType,Y9: set @ A] :
( ( vimage @ A @ A
@ ^ [X2: A] : X2
@ Y9 )
= Y9 ) ).
% vimage_ident
thf(fact_4451_vimage__const,axiom,
! [B: $tType,A: $tType,C2: B,A5: set @ B] :
( ( ( member @ B @ C2 @ A5 )
=> ( ( vimage @ A @ B
@ ^ [X2: A] : C2
@ A5 )
= ( top_top @ ( set @ A ) ) ) )
& ( ~ ( member @ B @ C2 @ A5 )
=> ( ( vimage @ A @ B
@ ^ [X2: A] : C2
@ A5 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% vimage_const
thf(fact_4452_vimage__id,axiom,
! [A: $tType] :
( ( vimage @ A @ A @ ( id @ A ) )
= ( id @ ( set @ A ) ) ) ).
% vimage_id
thf(fact_4453_vimage__if,axiom,
! [B: $tType,A: $tType,C2: B,A5: set @ B,D3: B,B4: set @ A] :
( ( ( member @ B @ C2 @ A5 )
=> ( ( ( member @ B @ D3 @ A5 )
=> ( ( vimage @ A @ B
@ ^ [X2: A] : ( if @ B @ ( member @ A @ X2 @ B4 ) @ C2 @ D3 )
@ A5 )
= ( top_top @ ( set @ A ) ) ) )
& ( ~ ( member @ B @ D3 @ A5 )
=> ( ( vimage @ A @ B
@ ^ [X2: A] : ( if @ B @ ( member @ A @ X2 @ B4 ) @ C2 @ D3 )
@ A5 )
= B4 ) ) ) )
& ( ~ ( member @ B @ C2 @ A5 )
=> ( ( ( member @ B @ D3 @ A5 )
=> ( ( vimage @ A @ B
@ ^ [X2: A] : ( if @ B @ ( member @ A @ X2 @ B4 ) @ C2 @ D3 )
@ A5 )
= ( uminus_uminus @ ( set @ A ) @ B4 ) ) )
& ( ~ ( member @ B @ D3 @ A5 )
=> ( ( vimage @ A @ B
@ ^ [X2: A] : ( if @ B @ ( member @ A @ X2 @ B4 ) @ C2 @ D3 )
@ A5 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% vimage_if
thf(fact_4454_set_Ocompositionality,axiom,
! [C: $tType,B: $tType,A: $tType,F: C > B,G: B > A,Set: set @ A] :
( ( vimage @ C @ B @ F @ ( vimage @ B @ A @ G @ Set ) )
= ( vimage @ C @ A @ ( comp @ B @ A @ C @ G @ F ) @ Set ) ) ).
% set.compositionality
thf(fact_4455_vimage__comp,axiom,
! [A: $tType,B: $tType,C: $tType,F: A > B,G: B > C,X: set @ C] :
( ( vimage @ A @ B @ F @ ( vimage @ B @ C @ G @ X ) )
= ( vimage @ A @ C @ ( comp @ B @ C @ A @ G @ F ) @ X ) ) ).
% vimage_comp
thf(fact_4456_vimage__def,axiom,
! [B: $tType,A: $tType] :
( ( vimage @ A @ B )
= ( ^ [F2: A > B,B7: set @ B] :
( collect @ A
@ ^ [X2: A] : ( member @ B @ ( F2 @ X2 ) @ B7 ) ) ) ) ).
% vimage_def
thf(fact_4457_surj__image__vimage__eq,axiom,
! [B: $tType,A: $tType,F: B > A,A5: set @ A] :
( ( ( image2 @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ( ( image2 @ B @ A @ F @ ( vimage @ B @ A @ F @ A5 ) )
= A5 ) ) ).
% surj_image_vimage_eq
thf(fact_4458_set_Oidentity,axiom,
! [A: $tType] :
( ( vimage @ A @ A
@ ^ [X2: A] : X2 )
= ( id @ ( set @ A ) ) ) ).
% set.identity
thf(fact_4459_vimage__UN,axiom,
! [A: $tType,B: $tType,C: $tType,F: A > B,B4: C > ( set @ B ),A5: set @ C] :
( ( vimage @ A @ B @ F @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ C @ ( set @ B ) @ B4 @ A5 ) ) )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ C @ ( set @ A )
@ ^ [X2: C] : ( vimage @ A @ B @ F @ ( B4 @ X2 ) )
@ A5 ) ) ) ).
% vimage_UN
thf(fact_4460_vimage__INT,axiom,
! [A: $tType,B: $tType,C: $tType,F: A > B,B4: C > ( set @ B ),A5: set @ C] :
( ( vimage @ A @ B @ F @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ C @ ( set @ B ) @ B4 @ A5 ) ) )
= ( complete_Inf_Inf @ ( set @ A )
@ ( image2 @ C @ ( set @ A )
@ ^ [X2: C] : ( vimage @ A @ B @ F @ ( B4 @ X2 ) )
@ A5 ) ) ) ).
% vimage_INT
thf(fact_4461_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_4462_vimage__Times,axiom,
! [A: $tType,B: $tType,C: $tType,F: A > ( product_prod @ B @ C ),A5: set @ B,B4: set @ C] :
( ( vimage @ A @ ( product_prod @ B @ C ) @ F
@ ( product_Sigma @ B @ C @ A5
@ ^ [Uu: B] : B4 ) )
= ( inf_inf @ ( set @ A ) @ ( vimage @ A @ B @ ( comp @ ( product_prod @ B @ C ) @ B @ A @ ( product_fst @ B @ C ) @ F ) @ A5 ) @ ( vimage @ A @ C @ ( comp @ ( product_prod @ B @ C ) @ C @ A @ ( product_snd @ B @ C ) @ F ) @ B4 ) ) ) ).
% vimage_Times
thf(fact_4463_set_Ocomp,axiom,
! [A: $tType,B: $tType,C: $tType,F: C > B,G: B > A] :
( ( comp @ ( set @ B ) @ ( set @ C ) @ ( set @ A ) @ ( vimage @ C @ B @ F ) @ ( vimage @ B @ A @ G ) )
= ( vimage @ C @ A @ ( comp @ B @ A @ C @ G @ F ) ) ) ).
% set.comp
thf(fact_4464_surj__vimage__empty,axiom,
! [B: $tType,A: $tType,F: B > A,A5: set @ A] :
( ( ( image2 @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ( ( ( vimage @ B @ A @ F @ A5 )
= ( bot_bot @ ( set @ B ) ) )
= ( A5
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% surj_vimage_empty
thf(fact_4465_vimage__subsetD,axiom,
! [A: $tType,B: $tType,F: B > A,B4: set @ A,A5: set @ B] :
( ( ( image2 @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ B ) @ ( vimage @ B @ A @ F @ B4 ) @ A5 )
=> ( ord_less_eq @ ( set @ A ) @ B4 @ ( image2 @ B @ A @ F @ A5 ) ) ) ) ).
% vimage_subsetD
thf(fact_4466_Pair__vimage__Sigma,axiom,
! [B: $tType,A: $tType,X: B,A5: set @ B,F: B > ( set @ A )] :
( ( ( member @ B @ X @ A5 )
=> ( ( vimage @ A @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X ) @ ( product_Sigma @ B @ A @ A5 @ F ) )
= ( F @ X ) ) )
& ( ~ ( member @ B @ X @ A5 )
=> ( ( vimage @ A @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X ) @ ( product_Sigma @ B @ A @ A5 @ F ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Pair_vimage_Sigma
thf(fact_4467_inj__vimage__image__eq,axiom,
! [B: $tType,A: $tType,F: A > B,A5: set @ A] :
( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
=> ( ( vimage @ A @ B @ F @ ( image2 @ A @ B @ F @ A5 ) )
= A5 ) ) ).
% inj_vimage_image_eq
thf(fact_4468_vimage__Suc__insert__0,axiom,
! [A5: set @ nat] :
( ( vimage @ nat @ nat @ suc @ ( insert2 @ nat @ ( zero_zero @ nat ) @ A5 ) )
= ( vimage @ nat @ nat @ suc @ A5 ) ) ).
% vimage_Suc_insert_0
thf(fact_4469_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_4470_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_4471_vimage__subsetI,axiom,
! [B: $tType,A: $tType,F: A > B,B4: set @ B,A5: set @ A] :
( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ B ) @ B4 @ ( image2 @ A @ B @ F @ A5 ) )
=> ( ord_less_eq @ ( set @ A ) @ ( vimage @ A @ B @ F @ B4 ) @ A5 ) ) ) ).
% vimage_subsetI
thf(fact_4472_vimage__eq__UN,axiom,
! [B: $tType,A: $tType] :
( ( vimage @ A @ B )
= ( ^ [F2: A > B,B7: set @ B] :
( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [Y: B] : ( vimage @ A @ B @ F2 @ ( insert2 @ B @ Y @ ( bot_bot @ ( set @ B ) ) ) )
@ B7 ) ) ) ) ).
% vimage_eq_UN
thf(fact_4473_inj__vimage__singleton,axiom,
! [B: $tType,A: $tType,F: A > B,A3: B] :
( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
=> ( ord_less_eq @ ( set @ A ) @ ( vimage @ A @ B @ F @ ( insert2 @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) )
@ ( insert2 @ A
@ ( the @ A
@ ^ [X2: A] :
( ( F @ X2 )
= A3 ) )
@ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% inj_vimage_singleton
thf(fact_4474_inj__on__vimage__singleton,axiom,
! [B: $tType,A: $tType,F: A > B,A5: set @ A,A3: B] :
( ( inj_on @ A @ B @ F @ A5 )
=> ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ ( vimage @ A @ B @ F @ ( insert2 @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) @ A5 )
@ ( insert2 @ A
@ ( the @ A
@ ^ [X2: A] :
( ( member @ A @ X2 @ A5 )
& ( ( F @ X2 )
= A3 ) ) )
@ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% inj_on_vimage_singleton
thf(fact_4475_inv__image__partition,axiom,
! [A: $tType,Xs2: list @ A,P: A > $o,Ys2: list @ A] :
( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs2 ) )
=> ( P @ X4 ) )
=> ( ! [Y4: A] :
( ( member @ A @ Y4 @ ( set2 @ A @ Ys2 ) )
=> ~ ( P @ Y4 ) )
=> ( ( vimage @ ( list @ A ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( partition @ A @ P ) @ ( insert2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Ys2 ) @ ( bot_bot @ ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ) )
= ( shuffles @ A @ Xs2 @ Ys2 ) ) ) ) ).
% inv_image_partition
thf(fact_4476_sum__list__update,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [K: nat,Xs2: list @ A,X: A] :
( ( ord_less @ nat @ K @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( groups8242544230860333062m_list @ A @ ( list_update @ A @ Xs2 @ K @ X ) )
= ( minus_minus @ A @ ( plus_plus @ A @ ( groups8242544230860333062m_list @ A @ Xs2 ) @ X ) @ ( nth @ A @ Xs2 @ K ) ) ) ) ) ).
% sum_list_update
thf(fact_4477_zip__Cons1,axiom,
! [A: $tType,B: $tType,X: A,Xs2: list @ A,Ys2: list @ B] :
( ( zip @ A @ B @ ( cons @ A @ X @ Xs2 ) @ Ys2 )
= ( case_list @ ( list @ ( product_prod @ A @ B ) ) @ B @ ( nil @ ( product_prod @ A @ B ) )
@ ^ [Y: B,Ys: list @ B] : ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ ( zip @ A @ B @ Xs2 @ Ys ) )
@ Ys2 ) ) ).
% zip_Cons1
thf(fact_4478_zip__Cons,axiom,
! [B: $tType,A: $tType,Xs2: list @ A,Y3: B,Ys2: list @ B] :
( ( zip @ A @ B @ Xs2 @ ( cons @ B @ Y3 @ Ys2 ) )
= ( case_list @ ( list @ ( product_prod @ A @ B ) ) @ A @ ( nil @ ( product_prod @ A @ B ) )
@ ^ [Z3: A,Zs2: list @ A] : ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Z3 @ Y3 ) @ ( zip @ A @ B @ Zs2 @ Ys2 ) )
@ Xs2 ) ) ).
% zip_Cons
thf(fact_4479_sum__list_ONil,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ( ( groups8242544230860333062m_list @ A @ ( nil @ A ) )
= ( zero_zero @ A ) ) ) ).
% sum_list.Nil
thf(fact_4480_sum__list__eq__0__iff,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [Ns: list @ A] :
( ( ( groups8242544230860333062m_list @ A @ Ns )
= ( zero_zero @ A ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ Ns ) )
=> ( X2
= ( zero_zero @ A ) ) ) ) ) ) ).
% sum_list_eq_0_iff
thf(fact_4481_sum__list_OCons,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ! [X: A,Xs2: list @ A] :
( ( groups8242544230860333062m_list @ A @ ( cons @ A @ X @ Xs2 ) )
= ( plus_plus @ A @ X @ ( groups8242544230860333062m_list @ A @ Xs2 ) ) ) ) ).
% sum_list.Cons
thf(fact_4482_sum__list__append,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ! [Xs2: list @ A,Ys2: list @ A] :
( ( groups8242544230860333062m_list @ A @ ( append @ A @ Xs2 @ Ys2 ) )
= ( plus_plus @ A @ ( groups8242544230860333062m_list @ A @ Xs2 ) @ ( groups8242544230860333062m_list @ A @ Ys2 ) ) ) ) ).
% sum_list_append
thf(fact_4483_list_Ocase__distrib,axiom,
! [B: $tType,C: $tType,A: $tType,H: B > C,F1: B,F23: A > ( list @ A ) > B,List: list @ A] :
( ( H @ ( case_list @ B @ A @ F1 @ F23 @ List ) )
= ( case_list @ C @ A @ ( H @ F1 )
@ ^ [X12: A,X24: list @ A] : ( H @ ( F23 @ X12 @ X24 ) )
@ List ) ) ).
% list.case_distrib
thf(fact_4484_sum__list__nonpos,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [Xs2: list @ A] :
( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs2 ) )
=> ( ord_less_eq @ A @ X4 @ ( zero_zero @ A ) ) )
=> ( ord_less_eq @ A @ ( groups8242544230860333062m_list @ A @ Xs2 ) @ ( zero_zero @ A ) ) ) ) ).
% sum_list_nonpos
thf(fact_4485_sum__list__nonneg__eq__0__iff,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [Xs2: list @ A] :
( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs2 ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ X4 ) )
=> ( ( ( groups8242544230860333062m_list @ A @ Xs2 )
= ( zero_zero @ A ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ Xs2 ) )
=> ( X2
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% sum_list_nonneg_eq_0_iff
thf(fact_4486_Groups__List_Osum__list__nonneg,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [Xs2: list @ A] :
( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs2 ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ X4 ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( groups8242544230860333062m_list @ A @ Xs2 ) ) ) ) ).
% Groups_List.sum_list_nonneg
thf(fact_4487_partition_Osimps_I1_J,axiom,
! [A: $tType,P: A > $o] :
( ( partition @ A @ P @ ( nil @ A ) )
= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) ) ) ).
% partition.simps(1)
thf(fact_4488_partition__P,axiom,
! [A: $tType,P: A > $o,Xs2: list @ A,Yes: list @ A,No: list @ A] :
( ( ( partition @ A @ P @ Xs2 )
= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes @ No ) )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Yes ) )
=> ( P @ X5 ) )
& ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ No ) )
=> ~ ( P @ X5 ) ) ) ) ).
% partition_P
thf(fact_4489_distinct__sum__list__conv__Sum,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [Xs2: list @ A] :
( ( distinct @ A @ Xs2 )
=> ( ( groups8242544230860333062m_list @ A @ Xs2 )
= ( groups7311177749621191930dd_sum @ A @ A
@ ^ [X2: A] : X2
@ ( set2 @ A @ Xs2 ) ) ) ) ) ).
% distinct_sum_list_conv_Sum
thf(fact_4490_partition_Osimps_I2_J,axiom,
! [A: $tType,P: A > $o,X: A,Xs2: list @ A] :
( ( partition @ A @ P @ ( cons @ A @ X @ Xs2 ) )
= ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) )
@ ^ [Yes2: list @ A,No2: list @ A] : ( if @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( P @ X ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Yes2 ) @ No2 ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes2 @ ( cons @ A @ X @ No2 ) ) )
@ ( partition @ A @ P @ Xs2 ) ) ) ).
% partition.simps(2)
thf(fact_4491_card__length__sum__list__rec,axiom,
! [M: nat,N6: nat] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ M )
=> ( ( finite_card @ ( list @ nat )
@ ( collect @ ( list @ nat )
@ ^ [L4: list @ nat] :
( ( ( size_size @ ( list @ nat ) @ L4 )
= M )
& ( ( groups8242544230860333062m_list @ nat @ L4 )
= N6 ) ) ) )
= ( plus_plus @ nat
@ ( finite_card @ ( list @ nat )
@ ( collect @ ( list @ nat )
@ ^ [L4: list @ nat] :
( ( ( size_size @ ( list @ nat ) @ L4 )
= ( minus_minus @ nat @ M @ ( one_one @ nat ) ) )
& ( ( groups8242544230860333062m_list @ nat @ L4 )
= N6 ) ) ) )
@ ( finite_card @ ( list @ nat )
@ ( collect @ ( list @ nat )
@ ^ [L4: list @ nat] :
( ( ( size_size @ ( list @ nat ) @ L4 )
= M )
& ( ( plus_plus @ nat @ ( groups8242544230860333062m_list @ nat @ L4 ) @ ( one_one @ nat ) )
= N6 ) ) ) ) ) ) ) ).
% card_length_sum_list_rec
thf(fact_4492_card__length__sum__list,axiom,
! [M: nat,N6: nat] :
( ( finite_card @ ( list @ nat )
@ ( collect @ ( list @ nat )
@ ^ [L4: list @ nat] :
( ( ( size_size @ ( list @ nat ) @ L4 )
= M )
& ( ( groups8242544230860333062m_list @ nat @ L4 )
= N6 ) ) ) )
= ( binomial @ ( minus_minus @ nat @ ( plus_plus @ nat @ N6 @ M ) @ ( one_one @ nat ) ) @ N6 ) ) ).
% card_length_sum_list
thf(fact_4493_remdups__adj__Cons,axiom,
! [A: $tType,X: A,Xs2: list @ A] :
( ( remdups_adj @ A @ ( cons @ A @ X @ Xs2 ) )
= ( case_list @ ( list @ A ) @ A @ ( cons @ A @ X @ ( nil @ A ) )
@ ^ [Y: A,Xs: list @ A] : ( if @ ( list @ A ) @ ( X = Y ) @ ( cons @ A @ Y @ Xs ) @ ( cons @ A @ X @ ( cons @ A @ Y @ Xs ) ) )
@ ( remdups_adj @ A @ Xs2 ) ) ) ).
% remdups_adj_Cons
thf(fact_4494_partition__set,axiom,
! [A: $tType,P: A > $o,Xs2: list @ A,Yes: list @ A,No: list @ A] :
( ( ( partition @ A @ P @ Xs2 )
= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes @ No ) )
=> ( ( sup_sup @ ( set @ A ) @ ( set2 @ A @ Yes ) @ ( set2 @ A @ No ) )
= ( set2 @ A @ Xs2 ) ) ) ).
% partition_set
thf(fact_4495_sum__list__sum__nth,axiom,
! [B: $tType] :
( ( comm_monoid_add @ B )
=> ( ( groups8242544230860333062m_list @ B )
= ( ^ [Xs: list @ B] : ( groups7311177749621191930dd_sum @ nat @ B @ ( nth @ B @ Xs ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ B ) @ Xs ) ) ) ) ) ) ).
% sum_list_sum_nth
thf(fact_4496_min__list_Osimps,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X: A,Xs2: list @ A] :
( ( min_list @ A @ ( cons @ A @ X @ Xs2 ) )
= ( case_list @ A @ A @ X
@ ^ [A6: A,List2: list @ A] : ( ord_min @ A @ X @ ( min_list @ A @ Xs2 ) )
@ Xs2 ) ) ) ).
% min_list.simps
thf(fact_4497_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
@ ^ [Xs: list @ A,Ys: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( suc @ N ) )
& ( ( size_size @ ( list @ A ) @ Ys )
= ( suc @ N ) ) ) ) ) ) ) ).
% lexn.simps(2)
thf(fact_4498_relation__of__def,axiom,
! [A: $tType] :
( ( order_relation_of @ A )
= ( ^ [P3: A > A > $o,A8: set @ A] :
( collect @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ^ [A6: A,B5: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A6 @ B5 )
@ ( product_Sigma @ A @ A @ A8
@ ^ [Uu: A] : A8 ) )
& ( P3 @ A6 @ B5 ) ) ) ) ) ) ).
% relation_of_def
thf(fact_4499_acyclic__insert,axiom,
! [A: $tType,Y3: A,X: A,R: set @ ( product_prod @ A @ A )] :
( ( transitive_acyclic @ A @ ( insert2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X ) @ R ) )
= ( ( transitive_acyclic @ A @ R )
& ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ ( transitive_rtrancl @ A @ R ) ) ) ) ).
% acyclic_insert
thf(fact_4500_map__prod__ident,axiom,
! [B: $tType,A: $tType] :
( ( product_map_prod @ A @ A @ B @ B
@ ^ [X2: A] : X2
@ ^ [Y: B] : Y )
= ( ^ [Z3: product_prod @ A @ B] : Z3 ) ) ).
% map_prod_ident
thf(fact_4501_map__prod__simp,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,F: C > A,G: D > B,A3: C,B2: D] :
( ( product_map_prod @ C @ A @ D @ B @ F @ G @ ( product_Pair @ C @ D @ A3 @ B2 ) )
= ( product_Pair @ A @ B @ ( F @ A3 ) @ ( G @ B2 ) ) ) ).
% map_prod_simp
thf(fact_4502_fst__map__prod,axiom,
! [B: $tType,A: $tType,D: $tType,C: $tType,F: C > A,G: D > B,X: product_prod @ C @ D] :
( ( product_fst @ A @ B @ ( product_map_prod @ C @ A @ D @ B @ F @ G @ X ) )
= ( F @ ( product_fst @ C @ D @ X ) ) ) ).
% fst_map_prod
thf(fact_4503_snd__map__prod,axiom,
! [B: $tType,A: $tType,D: $tType,C: $tType,F: C > B,G: D > A,X: product_prod @ C @ D] :
( ( product_snd @ B @ A @ ( product_map_prod @ C @ B @ D @ A @ F @ G @ X ) )
= ( G @ ( product_snd @ C @ D @ X ) ) ) ).
% snd_map_prod
thf(fact_4504_map__prod__imageI,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,A3: A,B2: B,R5: set @ ( product_prod @ A @ B ),F: A > C,G: B > D] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ R5 )
=> ( member @ ( product_prod @ C @ D ) @ ( product_Pair @ C @ D @ ( F @ A3 ) @ ( G @ B2 ) ) @ ( image2 @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) @ ( product_map_prod @ A @ C @ B @ D @ F @ G ) @ R5 ) ) ) ).
% map_prod_imageI
thf(fact_4505_fst__comp__map__prod,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,F: A > C,G: B > D] :
( ( comp @ ( product_prod @ C @ D ) @ C @ ( product_prod @ A @ B ) @ ( product_fst @ C @ D ) @ ( product_map_prod @ A @ C @ B @ D @ F @ G ) )
= ( comp @ A @ C @ ( product_prod @ A @ B ) @ F @ ( product_fst @ A @ B ) ) ) ).
% fst_comp_map_prod
thf(fact_4506_snd__comp__map__prod,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,F: A > D,G: B > C] :
( ( comp @ ( product_prod @ D @ C ) @ C @ ( product_prod @ A @ B ) @ ( product_snd @ D @ C ) @ ( product_map_prod @ A @ D @ B @ C @ F @ G ) )
= ( comp @ B @ C @ ( product_prod @ A @ B ) @ G @ ( product_snd @ A @ B ) ) ) ).
% snd_comp_map_prod
thf(fact_4507_map__prod_Ocomp,axiom,
! [A: $tType,C: $tType,E: $tType,F3: $tType,D: $tType,B: $tType,F: C > E,G: D > F3,H: A > C,I: B > D] :
( ( comp @ ( product_prod @ C @ D ) @ ( product_prod @ E @ F3 ) @ ( product_prod @ A @ B ) @ ( product_map_prod @ C @ E @ D @ F3 @ F @ G ) @ ( product_map_prod @ A @ C @ B @ D @ H @ I ) )
= ( product_map_prod @ A @ E @ B @ F3 @ ( comp @ C @ E @ A @ F @ H ) @ ( comp @ D @ F3 @ B @ G @ I ) ) ) ).
% map_prod.comp
thf(fact_4508_map__prod_Ocompositionality,axiom,
! [D: $tType,F3: $tType,E: $tType,C: $tType,B: $tType,A: $tType,F: C > E,G: D > F3,H: A > C,I: B > D,Prod: product_prod @ A @ B] :
( ( product_map_prod @ C @ E @ D @ F3 @ F @ G @ ( product_map_prod @ A @ C @ B @ D @ H @ I @ Prod ) )
= ( product_map_prod @ A @ E @ B @ F3 @ ( comp @ C @ E @ A @ F @ H ) @ ( comp @ D @ F3 @ B @ G @ I ) @ Prod ) ) ).
% map_prod.compositionality
thf(fact_4509_map__prod__compose,axiom,
! [D: $tType,C: $tType,A: $tType,E: $tType,F3: $tType,B: $tType,F1: E > C,F23: A > E,G1: F3 > D,G22: B > F3] :
( ( product_map_prod @ A @ C @ B @ D @ ( comp @ E @ C @ A @ F1 @ F23 ) @ ( comp @ F3 @ D @ B @ G1 @ G22 ) )
= ( comp @ ( product_prod @ E @ F3 ) @ ( product_prod @ C @ D ) @ ( product_prod @ A @ B ) @ ( product_map_prod @ E @ C @ F3 @ D @ F1 @ G1 ) @ ( product_map_prod @ A @ E @ B @ F3 @ F23 @ G22 ) ) ) ).
% map_prod_compose
thf(fact_4510_prod_Omap__comp,axiom,
! [D: $tType,F3: $tType,E: $tType,C: $tType,B: $tType,A: $tType,G1: C > E,G22: D > F3,F1: A > C,F23: B > D,V: product_prod @ A @ B] :
( ( product_map_prod @ C @ E @ D @ F3 @ G1 @ G22 @ ( product_map_prod @ A @ C @ B @ D @ F1 @ F23 @ V ) )
= ( product_map_prod @ A @ E @ B @ F3 @ ( comp @ C @ E @ A @ G1 @ F1 ) @ ( comp @ D @ F3 @ B @ G22 @ F23 ) @ V ) ) ).
% prod.map_comp
thf(fact_4511_prod_Omap__ident,axiom,
! [B: $tType,A: $tType,T2: product_prod @ A @ B] :
( ( product_map_prod @ A @ A @ B @ B
@ ^ [X2: A] : X2
@ ^ [X2: B] : X2
@ T2 )
= T2 ) ).
% prod.map_ident
thf(fact_4512_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,X: product_prod @ D @ E] :
( ( product_case_prod @ B @ C @ A @ H @ ( product_map_prod @ D @ B @ E @ C @ F @ G @ X ) )
= ( product_case_prod @ D @ E @ A
@ ^ [L4: D,R4: E] : ( H @ ( F @ L4 ) @ ( G @ R4 ) )
@ X ) ) ).
% case_prod_map_prod
thf(fact_4513_map__prod_Oidentity,axiom,
! [B: $tType,A: $tType] :
( ( product_map_prod @ A @ A @ B @ B
@ ^ [X2: A] : X2
@ ^ [X2: B] : X2 )
= ( id @ ( product_prod @ A @ B ) ) ) ).
% map_prod.identity
thf(fact_4514_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_4515_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_4516_apfst__def,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( product_apfst @ A @ C @ B )
= ( ^ [F2: A > C] : ( product_map_prod @ A @ C @ B @ B @ F2 @ ( id @ B ) ) ) ) ).
% apfst_def
thf(fact_4517_prod__fun__imageE,axiom,
! [B: $tType,A: $tType,D: $tType,C: $tType,C2: product_prod @ A @ B,F: C > A,G: D > B,R5: set @ ( product_prod @ C @ D )] :
( ( member @ ( product_prod @ A @ B ) @ C2 @ ( image2 @ ( product_prod @ C @ D ) @ ( product_prod @ A @ B ) @ ( product_map_prod @ C @ A @ D @ B @ F @ G ) @ R5 ) )
=> ~ ! [X4: C,Y4: D] :
( ( C2
= ( product_Pair @ A @ B @ ( F @ X4 ) @ ( G @ Y4 ) ) )
=> ~ ( member @ ( product_prod @ C @ D ) @ ( product_Pair @ C @ D @ X4 @ Y4 ) @ R5 ) ) ) ).
% prod_fun_imageE
thf(fact_4518_apsnd__def,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( product_apsnd @ B @ C @ A )
= ( product_map_prod @ A @ A @ B @ C @ ( id @ A ) ) ) ).
% apsnd_def
thf(fact_4519_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 )
@ ^ [X2: A,Y: B] : ( product_Pair @ C @ D @ ( F2 @ X2 ) @ ( G2 @ Y ) ) ) ) ) ).
% map_prod_def
thf(fact_4520_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
@ ^ [L4: A,R4: B] : ( F @ ( G1 @ L4 ) @ ( G22 @ R4 ) ) ) ) ).
% case_prod_o_map_prod
thf(fact_4521_acyclicI__order,axiom,
! [A: $tType,B: $tType] :
( ( preorder @ A )
=> ! [R: set @ ( product_prod @ B @ B ),F: B > A] :
( ! [A4: B,B3: B] :
( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ A4 @ B3 ) @ R )
=> ( ord_less @ A @ ( F @ B3 ) @ ( F @ A4 ) ) )
=> ( transitive_acyclic @ B @ R ) ) ) ).
% acyclicI_order
thf(fact_4522_acyclic__def,axiom,
! [A: $tType] :
( ( transitive_acyclic @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A )] :
! [X2: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ X2 ) @ ( transitive_trancl @ A @ R4 ) ) ) ) ).
% acyclic_def
thf(fact_4523_acyclicI,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ! [X4: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ X4 ) @ ( transitive_trancl @ A @ R ) )
=> ( transitive_acyclic @ A @ R ) ) ).
% acyclicI
thf(fact_4524_prod_Osize__gen__o__map,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,F: C > nat,Fa: D > nat,G: A > C,Ga: B > D] :
( ( comp @ ( product_prod @ C @ D ) @ nat @ ( product_prod @ A @ B ) @ ( basic_BNF_size_prod @ C @ D @ F @ Fa ) @ ( product_map_prod @ A @ C @ B @ D @ G @ Ga ) )
= ( basic_BNF_size_prod @ A @ B @ ( comp @ C @ nat @ A @ F @ G ) @ ( comp @ D @ nat @ B @ Fa @ Ga ) ) ) ).
% prod.size_gen_o_map
thf(fact_4525_map__prod__surj__on,axiom,
! [B: $tType,D: $tType,C: $tType,A: $tType,F: B > A,A5: set @ B,A14: set @ A,G: D > C,B4: set @ D,B13: set @ C] :
( ( ( image2 @ B @ A @ F @ A5 )
= A14 )
=> ( ( ( image2 @ D @ C @ G @ B4 )
= 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] : B4 ) )
= ( product_Sigma @ A @ C @ A14
@ ^ [Uu: A] : B13 ) ) ) ) ).
% map_prod_surj_on
thf(fact_4526_map__prod__inj__on,axiom,
! [D: $tType,B: $tType,C: $tType,A: $tType,F: A > B,A5: set @ A,G: C > D,B4: set @ C] :
( ( inj_on @ A @ B @ F @ A5 )
=> ( ( inj_on @ C @ D @ G @ B4 )
=> ( 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] : B4 ) ) ) ) ).
% map_prod_inj_on
thf(fact_4527_map__prod__surj,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,F: A > B,G: C > D] :
( ( ( image2 @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ B ) ) )
=> ( ( ( image2 @ C @ D @ G @ ( top_top @ ( set @ C ) ) )
= ( top_top @ ( set @ D ) ) )
=> ( ( image2 @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) @ ( product_map_prod @ A @ B @ C @ D @ F @ G ) @ ( top_top @ ( set @ ( product_prod @ A @ C ) ) ) )
= ( top_top @ ( set @ ( product_prod @ B @ D ) ) ) ) ) ) ).
% map_prod_surj
thf(fact_4528_min__list_Oelims,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X: list @ A,Y3: A] :
( ( ( min_list @ A @ X )
= Y3 )
=> ( ! [X4: A,Xs5: list @ A] :
( ( X
= ( cons @ A @ X4 @ Xs5 ) )
=> ( Y3
!= ( case_list @ A @ A @ X4
@ ^ [A6: A,List2: list @ A] : ( ord_min @ A @ X4 @ ( min_list @ A @ Xs5 ) )
@ Xs5 ) ) )
=> ~ ( ( X
= ( nil @ A ) )
=> ( Y3
!= ( undefined @ A ) ) ) ) ) ) ).
% min_list.elims
thf(fact_4529_min__list_Opelims,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X: list @ A,Y3: A] :
( ( ( min_list @ A @ X )
= Y3 )
=> ( ( accp @ ( list @ A ) @ ( min_list_rel @ A ) @ X )
=> ( ! [X4: A,Xs5: list @ A] :
( ( X
= ( cons @ A @ X4 @ Xs5 ) )
=> ( ( Y3
= ( case_list @ A @ A @ X4
@ ^ [A6: A,List2: list @ A] : ( ord_min @ A @ X4 @ ( min_list @ A @ Xs5 ) )
@ Xs5 ) )
=> ~ ( accp @ ( list @ A ) @ ( min_list_rel @ A ) @ ( cons @ A @ X4 @ Xs5 ) ) ) )
=> ~ ( ( X
= ( nil @ A ) )
=> ( ( Y3
= ( undefined @ A ) )
=> ~ ( accp @ ( list @ A ) @ ( min_list_rel @ A ) @ ( nil @ A ) ) ) ) ) ) ) ) ).
% min_list.pelims
thf(fact_4530_sum__list__upt,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( groups8242544230860333062m_list @ nat @ ( upt @ M @ N ) )
= ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [X2: nat] : X2
@ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ).
% sum_list_upt
thf(fact_4531_count__list_Osimps_I2_J,axiom,
! [A: $tType,X: A,Y3: A,Xs2: list @ A] :
( ( ( X = Y3 )
=> ( ( count_list @ A @ ( cons @ A @ X @ Xs2 ) @ Y3 )
= ( plus_plus @ nat @ ( count_list @ A @ Xs2 @ Y3 ) @ ( one_one @ nat ) ) ) )
& ( ( X != Y3 )
=> ( ( count_list @ A @ ( cons @ A @ X @ Xs2 ) @ Y3 )
= ( count_list @ A @ Xs2 @ Y3 ) ) ) ) ).
% count_list.simps(2)
thf(fact_4532_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_4533_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_4534_count__notin,axiom,
! [A: $tType,X: A,Xs2: list @ A] :
( ~ ( member @ A @ X @ ( set2 @ A @ Xs2 ) )
=> ( ( count_list @ A @ Xs2 @ X )
= ( zero_zero @ nat ) ) ) ).
% count_notin
thf(fact_4535_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_4536_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_4537_upt__0,axiom,
! [I: nat] :
( ( upt @ I @ ( zero_zero @ nat ) )
= ( nil @ nat ) ) ).
% upt_0
thf(fact_4538_Func__empty,axiom,
! [B: $tType,A: $tType,B4: set @ B] :
( ( bNF_Wellorder_Func @ A @ B @ ( bot_bot @ ( set @ A ) ) @ B4 )
= ( insert2 @ ( A > B )
@ ^ [X2: A] : ( undefined @ B )
@ ( bot_bot @ ( set @ ( A > B ) ) ) ) ) ).
% Func_empty
thf(fact_4539_option_Othe__def,axiom,
! [A: $tType] :
( ( the2 @ A )
= ( case_option @ A @ A @ ( undefined @ A )
@ ^ [X24: A] : X24 ) ) ).
% option.the_def
thf(fact_4540_enumerate__eq__zip,axiom,
! [A: $tType] :
( ( enumerate @ A )
= ( ^ [N5: nat,Xs: list @ A] : ( zip @ nat @ A @ ( upt @ N5 @ ( plus_plus @ nat @ N5 @ ( size_size @ ( list @ A ) @ Xs ) ) ) @ Xs ) ) ) ).
% enumerate_eq_zip
thf(fact_4541_atLeast__upt,axiom,
( ( set_ord_lessThan @ nat )
= ( ^ [N5: nat] : ( set2 @ nat @ ( upt @ ( zero_zero @ nat ) @ N5 ) ) ) ) ).
% atLeast_upt
thf(fact_4542_count__list_Osimps_I1_J,axiom,
! [A: $tType,Y3: A] :
( ( count_list @ A @ ( nil @ A ) @ Y3 )
= ( zero_zero @ nat ) ) ).
% count_list.simps(1)
thf(fact_4543_upt__eq__Cons__conv,axiom,
! [I: nat,J: nat,X: nat,Xs2: list @ nat] :
( ( ( upt @ I @ J )
= ( cons @ nat @ X @ Xs2 ) )
= ( ( ord_less @ nat @ I @ J )
& ( I = X )
& ( ( upt @ ( plus_plus @ nat @ I @ ( one_one @ nat ) ) @ J )
= Xs2 ) ) ) ).
% upt_eq_Cons_conv
thf(fact_4544_atMost__upto,axiom,
( ( set_ord_atMost @ nat )
= ( ^ [N5: nat] : ( set2 @ nat @ ( upt @ ( zero_zero @ nat ) @ ( suc @ N5 ) ) ) ) ) ).
% atMost_upto
thf(fact_4545_arg__min__list_Oelims,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [X: A > B,Xa: list @ A,Y3: A] :
( ( ( arg_min_list @ A @ B @ X @ Xa )
= Y3 )
=> ( ! [X4: A] :
( ( Xa
= ( cons @ A @ X4 @ ( nil @ A ) ) )
=> ( Y3 != X4 ) )
=> ( ! [X4: A,Y4: A,Zs3: list @ A] :
( ( Xa
= ( cons @ A @ X4 @ ( cons @ A @ Y4 @ Zs3 ) ) )
=> ( Y3
!= ( if @ A @ ( ord_less_eq @ B @ ( X @ X4 ) @ ( X @ ( arg_min_list @ A @ B @ X @ ( cons @ A @ Y4 @ Zs3 ) ) ) ) @ X4 @ ( arg_min_list @ A @ B @ X @ ( cons @ A @ Y4 @ Zs3 ) ) ) ) )
=> ~ ( ( Xa
= ( nil @ A ) )
=> ( Y3
!= ( undefined @ A ) ) ) ) ) ) ) ).
% arg_min_list.elims
thf(fact_4546_sum__list__map__eq__sum__count2,axiom,
! [A: $tType,Xs2: list @ A,X7: set @ A,F: A > nat] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs2 ) @ X7 )
=> ( ( finite_finite2 @ A @ X7 )
=> ( ( groups8242544230860333062m_list @ nat @ ( map @ A @ nat @ F @ Xs2 ) )
= ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [X2: A] : ( times_times @ nat @ ( count_list @ A @ Xs2 @ X2 ) @ ( F @ X2 ) )
@ X7 ) ) ) ) ).
% sum_list_map_eq_sum_count2
thf(fact_4547_horner__sum__bit__eq__take__bit,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: 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 @ A3 ) @ ( upt @ ( zero_zero @ nat ) @ N ) ) )
= ( bit_se2584673776208193580ke_bit @ A @ N @ A3 ) ) ) ).
% horner_sum_bit_eq_take_bit
thf(fact_4548_swap__comp__swap,axiom,
! [B: $tType,A: $tType] :
( ( comp @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A ) @ ( product_swap @ A @ B ) )
= ( id @ ( product_prod @ A @ B ) ) ) ).
% swap_comp_swap
thf(fact_4549_swap__swap,axiom,
! [B: $tType,A: $tType,P4: product_prod @ A @ B] :
( ( product_swap @ B @ A @ ( product_swap @ A @ B @ P4 ) )
= P4 ) ).
% swap_swap
thf(fact_4550_map__ident,axiom,
! [A: $tType] :
( ( map @ A @ A
@ ^ [X2: A] : X2 )
= ( ^ [Xs: list @ A] : Xs ) ) ).
% map_ident
thf(fact_4551_map__map,axiom,
! [B: $tType,A: $tType,C: $tType,F: B > A,G: C > B,Xs2: list @ C] :
( ( map @ B @ A @ F @ ( map @ C @ B @ G @ Xs2 ) )
= ( map @ C @ A @ ( comp @ B @ A @ C @ F @ G ) @ Xs2 ) ) ).
% map_map
thf(fact_4552_List_Omap_Ocompositionality,axiom,
! [B: $tType,C: $tType,A: $tType,F: B > C,G: A > B,List: list @ A] :
( ( map @ B @ C @ F @ ( map @ A @ B @ G @ List ) )
= ( map @ A @ C @ ( comp @ B @ C @ A @ F @ G ) @ List ) ) ).
% List.map.compositionality
thf(fact_4553_list_Omap__comp,axiom,
! [B: $tType,C: $tType,A: $tType,G: B > C,F: A > B,V: list @ A] :
( ( map @ B @ C @ G @ ( map @ A @ B @ F @ V ) )
= ( map @ A @ C @ ( comp @ B @ C @ A @ G @ F ) @ V ) ) ).
% list.map_comp
thf(fact_4554_swap__simp,axiom,
! [A: $tType,B: $tType,X: B,Y3: A] :
( ( product_swap @ B @ A @ ( product_Pair @ B @ A @ X @ Y3 ) )
= ( product_Pair @ A @ B @ Y3 @ X ) ) ).
% swap_simp
thf(fact_4555_case__swap,axiom,
! [A: $tType,B: $tType,C: $tType,F: C > B > A,P4: product_prod @ C @ B] :
( ( product_case_prod @ B @ C @ A
@ ^ [Y: B,X2: C] : ( F @ X2 @ Y )
@ ( product_swap @ C @ B @ P4 ) )
= ( product_case_prod @ C @ B @ A @ F @ P4 ) ) ).
% case_swap
thf(fact_4556_snd__swap,axiom,
! [B: $tType,A: $tType,X: product_prod @ A @ B] :
( ( product_snd @ B @ A @ ( product_swap @ A @ B @ X ) )
= ( product_fst @ A @ B @ X ) ) ).
% snd_swap
thf(fact_4557_fst__swap,axiom,
! [A: $tType,B: $tType,X: product_prod @ B @ A] :
( ( product_fst @ A @ B @ ( product_swap @ B @ A @ X ) )
= ( product_snd @ B @ A @ X ) ) ).
% fst_swap
thf(fact_4558_sum__list__0,axiom,
! [B: $tType,A: $tType] :
( ( monoid_add @ A )
=> ! [Xs2: list @ B] :
( ( groups8242544230860333062m_list @ A
@ ( map @ B @ A
@ ^ [X2: B] : ( zero_zero @ A )
@ Xs2 ) )
= ( zero_zero @ A ) ) ) ).
% sum_list_0
thf(fact_4559_List_Omap_Ocomp,axiom,
! [C: $tType,B: $tType,A: $tType,F: B > C,G: A > B] :
( ( comp @ ( list @ B ) @ ( list @ C ) @ ( list @ A ) @ ( map @ B @ C @ F ) @ ( map @ A @ B @ G ) )
= ( map @ A @ C @ ( comp @ B @ C @ A @ F @ G ) ) ) ).
% List.map.comp
thf(fact_4560_map__comp__map,axiom,
! [B: $tType,C: $tType,A: $tType,F: C > B,G: A > C] :
( ( comp @ ( list @ C ) @ ( list @ B ) @ ( list @ A ) @ ( map @ C @ B @ F ) @ ( map @ A @ C @ G ) )
= ( map @ A @ B @ ( comp @ C @ B @ A @ F @ G ) ) ) ).
% map_comp_map
thf(fact_4561_size__list__map,axiom,
! [A: $tType,B: $tType,F: A > nat,G: B > A,Xs2: list @ B] :
( ( size_list @ A @ F @ ( map @ B @ A @ G @ Xs2 ) )
= ( size_list @ B @ ( comp @ A @ nat @ B @ F @ G ) @ Xs2 ) ) ).
% size_list_map
thf(fact_4562_pair__in__swap__image,axiom,
! [A: $tType,B: $tType,Y3: A,X: B,A5: set @ ( product_prod @ B @ A )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Y3 @ X ) @ ( image2 @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A ) @ A5 ) )
= ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X @ Y3 ) @ A5 ) ) ).
% pair_in_swap_image
thf(fact_4563_surj__swap,axiom,
! [B: $tType,A: $tType] :
( ( image2 @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A ) @ ( top_top @ ( set @ ( product_prod @ B @ A ) ) ) )
= ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% surj_swap
thf(fact_4564_map__fst__enumerate,axiom,
! [A: $tType,N: nat,Xs2: list @ A] :
( ( map @ ( product_prod @ nat @ A ) @ nat @ ( product_fst @ nat @ A ) @ ( enumerate @ A @ N @ Xs2 ) )
= ( upt @ N @ ( plus_plus @ nat @ N @ ( size_size @ ( list @ A ) @ Xs2 ) ) ) ) ).
% map_fst_enumerate
thf(fact_4565_list_Omap__ident,axiom,
! [A: $tType,T2: list @ A] :
( ( map @ A @ A
@ ^ [X2: A] : X2
@ T2 )
= T2 ) ).
% list.map_ident
thf(fact_4566_zip__map__map,axiom,
! [B: $tType,A: $tType,C: $tType,D: $tType,F: C > A,Xs2: list @ C,G: D > B,Ys2: list @ D] :
( ( zip @ A @ B @ ( map @ C @ A @ F @ Xs2 ) @ ( map @ D @ B @ G @ Ys2 ) )
= ( map @ ( product_prod @ C @ D ) @ ( product_prod @ A @ B )
@ ( product_case_prod @ C @ D @ ( product_prod @ A @ B )
@ ^ [X2: C,Y: D] : ( product_Pair @ A @ B @ ( F @ X2 ) @ ( G @ Y ) ) )
@ ( zip @ C @ D @ Xs2 @ Ys2 ) ) ) ).
% zip_map_map
thf(fact_4567_zip__map2,axiom,
! [B: $tType,A: $tType,C: $tType,Xs2: list @ A,F: C > B,Ys2: list @ C] :
( ( zip @ A @ B @ Xs2 @ ( map @ C @ B @ F @ Ys2 ) )
= ( map @ ( product_prod @ A @ C ) @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ C @ ( product_prod @ A @ B )
@ ^ [X2: A,Y: C] : ( product_Pair @ A @ B @ X2 @ ( F @ Y ) ) )
@ ( zip @ A @ C @ Xs2 @ Ys2 ) ) ) ).
% zip_map2
thf(fact_4568_zip__map1,axiom,
! [A: $tType,C: $tType,B: $tType,F: C > A,Xs2: list @ C,Ys2: list @ B] :
( ( zip @ A @ B @ ( map @ C @ A @ F @ Xs2 ) @ Ys2 )
= ( map @ ( product_prod @ C @ B ) @ ( product_prod @ A @ B )
@ ( product_case_prod @ C @ B @ ( product_prod @ A @ B )
@ ^ [X2: C] : ( product_Pair @ A @ B @ ( F @ X2 ) ) )
@ ( zip @ C @ B @ Xs2 @ Ys2 ) ) ) ).
% zip_map1
thf(fact_4569_map__zip__map2,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,F: ( product_prod @ B @ C ) > A,Xs2: list @ B,G: D > C,Ys2: list @ D] :
( ( map @ ( product_prod @ B @ C ) @ A @ F @ ( zip @ B @ C @ Xs2 @ ( map @ D @ C @ G @ Ys2 ) ) )
= ( map @ ( product_prod @ B @ D ) @ A
@ ( product_case_prod @ B @ D @ A
@ ^ [X2: B,Y: D] : ( F @ ( product_Pair @ B @ C @ X2 @ ( G @ Y ) ) ) )
@ ( zip @ B @ D @ Xs2 @ Ys2 ) ) ) ).
% map_zip_map2
thf(fact_4570_map__zip__map,axiom,
! [B: $tType,A: $tType,D: $tType,C: $tType,F: ( product_prod @ B @ C ) > A,G: D > B,Xs2: list @ D,Ys2: list @ C] :
( ( map @ ( product_prod @ B @ C ) @ A @ F @ ( zip @ B @ C @ ( map @ D @ B @ G @ Xs2 ) @ Ys2 ) )
= ( map @ ( product_prod @ D @ C ) @ A
@ ( product_case_prod @ D @ C @ A
@ ^ [X2: D,Y: C] : ( F @ ( product_Pair @ B @ C @ ( G @ X2 ) @ Y ) ) )
@ ( zip @ D @ C @ Xs2 @ Ys2 ) ) ) ).
% map_zip_map
thf(fact_4571_map2__map__map,axiom,
! [B: $tType,A: $tType,C: $tType,D: $tType,H: B > C > A,F: D > B,Xs2: 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 @ Xs2 ) @ ( map @ D @ C @ G @ Xs2 ) ) )
= ( map @ D @ A
@ ^ [X2: D] : ( H @ ( F @ X2 ) @ ( G @ X2 ) )
@ Xs2 ) ) ).
% map2_map_map
thf(fact_4572_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 )
@ ^ [K3: nat] : ( product_Pair @ nat @ A @ K3 @ ( F @ K3 ) )
@ ( upt @ N @ M ) ) ) ).
% enumerate_map_upt
thf(fact_4573_List_Omap_Oidentity,axiom,
! [A: $tType] :
( ( map @ A @ A
@ ^ [X2: A] : X2 )
= ( id @ ( list @ A ) ) ) ).
% List.map.identity
thf(fact_4574_fold__map,axiom,
! [B: $tType,A: $tType,C: $tType,G: B > A > A,F: C > B,Xs2: list @ C] :
( ( fold @ B @ A @ G @ ( map @ C @ B @ F @ Xs2 ) )
= ( fold @ C @ A @ ( comp @ B @ ( A > A ) @ C @ G @ F ) @ Xs2 ) ) ).
% fold_map
thf(fact_4575_sum__list__mult__const,axiom,
! [B: $tType,A: $tType] :
( ( semiring_0 @ A )
=> ! [F: B > A,C2: A,Xs2: list @ B] :
( ( groups8242544230860333062m_list @ A
@ ( map @ B @ A
@ ^ [X2: B] : ( times_times @ A @ ( F @ X2 ) @ C2 )
@ Xs2 ) )
= ( times_times @ A @ ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F @ Xs2 ) ) @ C2 ) ) ) ).
% sum_list_mult_const
thf(fact_4576_sum__list__const__mult,axiom,
! [A: $tType,B: $tType] :
( ( semiring_0 @ A )
=> ! [C2: A,F: B > A,Xs2: list @ B] :
( ( groups8242544230860333062m_list @ A
@ ( map @ B @ A
@ ^ [X2: B] : ( times_times @ A @ C2 @ ( F @ X2 ) )
@ Xs2 ) )
= ( times_times @ A @ C2 @ ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F @ Xs2 ) ) ) ) ) ).
% sum_list_const_mult
thf(fact_4577_sum__list__addf,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [F: B > A,G: B > A,Xs2: list @ B] :
( ( groups8242544230860333062m_list @ A
@ ( map @ B @ A
@ ^ [X2: B] : ( plus_plus @ A @ ( F @ X2 ) @ ( G @ X2 ) )
@ Xs2 ) )
= ( plus_plus @ A @ ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F @ Xs2 ) ) @ ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ G @ Xs2 ) ) ) ) ) ).
% sum_list_addf
thf(fact_4578_sum__list__subtractf,axiom,
! [A: $tType,B: $tType] :
( ( ab_group_add @ A )
=> ! [F: B > A,G: B > A,Xs2: list @ B] :
( ( groups8242544230860333062m_list @ A
@ ( map @ B @ A
@ ^ [X2: B] : ( minus_minus @ A @ ( F @ X2 ) @ ( G @ X2 ) )
@ Xs2 ) )
= ( minus_minus @ A @ ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F @ Xs2 ) ) @ ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ G @ Xs2 ) ) ) ) ) ).
% sum_list_subtractf
thf(fact_4579_list_Osize__gen__o__map,axiom,
! [B: $tType,A: $tType,F: B > nat,G: A > B] :
( ( comp @ ( list @ B ) @ nat @ ( list @ A ) @ ( size_list @ B @ F ) @ ( map @ A @ B @ G ) )
= ( size_list @ A @ ( comp @ B @ nat @ A @ F @ G ) ) ) ).
% list.size_gen_o_map
thf(fact_4580_inj__swap,axiom,
! [B: $tType,A: $tType,A5: set @ ( product_prod @ A @ B )] : ( inj_on @ ( product_prod @ A @ B ) @ ( product_prod @ B @ A ) @ ( product_swap @ A @ B ) @ A5 ) ).
% inj_swap
thf(fact_4581_uminus__sum__list__map,axiom,
! [A: $tType,B: $tType] :
( ( ab_group_add @ A )
=> ! [F: B > A,Xs2: list @ B] :
( ( uminus_uminus @ A @ ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F @ Xs2 ) ) )
= ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ ( comp @ A @ A @ B @ ( uminus_uminus @ A ) @ F ) @ Xs2 ) ) ) ) ).
% uminus_sum_list_map
thf(fact_4582_sum__list__mono,axiom,
! [B: $tType,A: $tType] :
( ( ( monoid_add @ B )
& ( ordere6658533253407199908up_add @ B ) )
=> ! [Xs2: list @ A,F: A > B,G: A > B] :
( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs2 ) )
=> ( ord_less_eq @ B @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less_eq @ B @ ( groups8242544230860333062m_list @ B @ ( map @ A @ B @ F @ Xs2 ) ) @ ( groups8242544230860333062m_list @ B @ ( map @ A @ B @ G @ Xs2 ) ) ) ) ) ).
% sum_list_mono
thf(fact_4583_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_4584_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_4585_sum__list__triv,axiom,
! [C: $tType,B: $tType] :
( ( semiring_1 @ B )
=> ! [R: B,Xs2: list @ C] :
( ( groups8242544230860333062m_list @ B
@ ( map @ C @ B
@ ^ [X2: C] : R
@ Xs2 ) )
= ( times_times @ B @ ( semiring_1_of_nat @ B @ ( size_size @ ( list @ C ) @ Xs2 ) ) @ R ) ) ) ).
% sum_list_triv
thf(fact_4586_map__decr__upt,axiom,
! [M: nat,N: nat] :
( ( map @ nat @ nat
@ ^ [N5: nat] : ( minus_minus @ nat @ N5 @ ( suc @ ( zero_zero @ nat ) ) )
@ ( upt @ ( suc @ M ) @ ( suc @ N ) ) )
= ( upt @ M @ N ) ) ).
% map_decr_upt
thf(fact_4587_map__nth,axiom,
! [A: $tType,Xs2: list @ A] :
( ( map @ nat @ A @ ( nth @ A @ Xs2 ) @ ( upt @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs2 ) ) )
= Xs2 ) ).
% map_nth
thf(fact_4588_size__list__conv__sum__list,axiom,
! [B: $tType] :
( ( size_list @ B )
= ( ^ [F2: B > nat,Xs: list @ B] : ( plus_plus @ nat @ ( groups8242544230860333062m_list @ nat @ ( map @ B @ nat @ F2 @ Xs ) ) @ ( size_size @ ( list @ B ) @ Xs ) ) ) ) ).
% size_list_conv_sum_list
thf(fact_4589_sum__list__Suc,axiom,
! [A: $tType,F: A > nat,Xs2: list @ A] :
( ( groups8242544230860333062m_list @ nat
@ ( map @ A @ nat
@ ^ [X2: A] : ( suc @ ( F @ X2 ) )
@ Xs2 ) )
= ( plus_plus @ nat @ ( groups8242544230860333062m_list @ nat @ ( map @ A @ nat @ F @ Xs2 ) ) @ ( size_size @ ( list @ A ) @ Xs2 ) ) ) ).
% sum_list_Suc
thf(fact_4590_arg__min__list_Osimps_I2_J,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [F: A > B,X: A,Y3: A,Zs: list @ A] :
( ( arg_min_list @ A @ B @ F @ ( cons @ A @ X @ ( cons @ A @ Y3 @ Zs ) ) )
= ( if @ A @ ( ord_less_eq @ B @ ( F @ X ) @ ( F @ ( arg_min_list @ A @ B @ F @ ( cons @ A @ Y3 @ Zs ) ) ) ) @ X @ ( arg_min_list @ A @ B @ F @ ( cons @ A @ Y3 @ Zs ) ) ) ) ) ).
% arg_min_list.simps(2)
thf(fact_4591_product__swap,axiom,
! [B: $tType,A: $tType,A5: set @ B,B4: set @ A] :
( ( image2 @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A )
@ ( product_Sigma @ B @ A @ A5
@ ^ [Uu: B] : B4 ) )
= ( product_Sigma @ A @ B @ B4
@ ^ [Uu: A] : A5 ) ) ).
% product_swap
thf(fact_4592_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_4593_prod_Oswap__def,axiom,
! [B: $tType,A: $tType] :
( ( product_swap @ A @ B )
= ( ^ [P5: product_prod @ A @ B] : ( product_Pair @ B @ A @ ( product_snd @ A @ B @ P5 ) @ ( product_fst @ A @ B @ P5 ) ) ) ) ).
% prod.swap_def
thf(fact_4594_map__upt__eqI,axiom,
! [A: $tType,Xs2: list @ A,N: nat,M: nat,F: nat > A] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( minus_minus @ nat @ N @ M ) )
=> ( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( nth @ A @ Xs2 @ I3 )
= ( F @ ( plus_plus @ nat @ M @ I3 ) ) ) )
=> ( ( map @ nat @ A @ F @ ( upt @ M @ N ) )
= Xs2 ) ) ) ).
% map_upt_eqI
thf(fact_4595_sum__list__map__eq__sum__count,axiom,
! [A: $tType,F: A > nat,Xs2: list @ A] :
( ( groups8242544230860333062m_list @ nat @ ( map @ A @ nat @ F @ Xs2 ) )
= ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [X2: A] : ( times_times @ nat @ ( count_list @ A @ Xs2 @ X2 ) @ ( F @ X2 ) )
@ ( set2 @ A @ Xs2 ) ) ) ).
% sum_list_map_eq_sum_count
thf(fact_4596_transpose__rectangle,axiom,
! [A: $tType,Xs2: list @ ( list @ A ),N: nat] :
( ( ( Xs2
= ( nil @ ( list @ A ) ) )
=> ( N
= ( zero_zero @ nat ) ) )
=> ( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ ( list @ A ) ) @ Xs2 ) )
=> ( ( size_size @ ( list @ A ) @ ( nth @ ( list @ A ) @ Xs2 @ I3 ) )
= N ) )
=> ( ( transpose @ A @ Xs2 )
= ( map @ nat @ ( list @ A )
@ ^ [I2: nat] :
( map @ nat @ A
@ ^ [J3: nat] : ( nth @ A @ ( nth @ ( list @ A ) @ Xs2 @ J3 ) @ I2 )
@ ( upt @ ( zero_zero @ nat ) @ ( size_size @ ( list @ ( list @ A ) ) @ Xs2 ) ) )
@ ( upt @ ( zero_zero @ nat ) @ N ) ) ) ) ) ).
% transpose_rectangle
thf(fact_4597_sorted__wrt__less__sum__mono__lowerbound,axiom,
! [B: $tType] :
( ( ordere6911136660526730532id_add @ B )
=> ! [F: nat > B,Ns: list @ nat] :
( ! [X4: nat,Y4: nat] :
( ( ord_less_eq @ nat @ X4 @ Y4 )
=> ( ord_less_eq @ B @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ( 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_4598_map__prod__o__convol__id,axiom,
! [B: $tType,A: $tType,C: $tType,F: C > A,G: C > B,X: 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 ) @ X )
= ( bNF_convol @ C @ A @ B @ ( comp @ A @ A @ C @ ( id @ A ) @ F ) @ G @ X ) ) ).
% map_prod_o_convol_id
thf(fact_4599_map__rec,axiom,
! [A: $tType,B: $tType] :
( ( map @ B @ A )
= ( ^ [F2: B > A] :
( rec_list @ ( list @ A ) @ B @ ( nil @ A )
@ ^ [X2: B,Uu: list @ B] : ( cons @ A @ ( F2 @ X2 ) ) ) ) ) ).
% map_rec
thf(fact_4600_sorted__wrt__map,axiom,
! [A: $tType,B: $tType,R5: A > A > $o,F: B > A,Xs2: list @ B] :
( ( sorted_wrt @ A @ R5 @ ( map @ B @ A @ F @ Xs2 ) )
= ( sorted_wrt @ B
@ ^ [X2: B,Y: B] : ( R5 @ ( F @ X2 ) @ ( F @ Y ) )
@ Xs2 ) ) ).
% sorted_wrt_map
thf(fact_4601_sorted__wrt__true,axiom,
! [A: $tType,Xs2: list @ A] :
( sorted_wrt @ A
@ ^ [Uu: A,Uv: A] : $true
@ Xs2 ) ).
% sorted_wrt_true
thf(fact_4602_convol__def,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( bNF_convol @ A @ B @ C )
= ( ^ [F2: A > B,G2: A > C,A6: A] : ( product_Pair @ B @ C @ ( F2 @ A6 ) @ ( G2 @ A6 ) ) ) ) ).
% convol_def
thf(fact_4603_fst__convol_H,axiom,
! [B: $tType,A: $tType,C: $tType,F: C > A,G: C > B,X: C] :
( ( product_fst @ A @ B @ ( bNF_convol @ C @ A @ B @ F @ G @ X ) )
= ( F @ X ) ) ).
% fst_convol'
thf(fact_4604_snd__convol_H,axiom,
! [B: $tType,A: $tType,C: $tType,F: C > B,G: C > A,X: C] :
( ( product_snd @ B @ A @ ( bNF_convol @ C @ B @ A @ F @ G @ X ) )
= ( G @ X ) ) ).
% snd_convol'
thf(fact_4605_sorted__map,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F: B > A,Xs2: list @ B] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F @ Xs2 ) )
= ( sorted_wrt @ B
@ ^ [X2: B,Y: B] : ( ord_less_eq @ A @ ( F @ X2 ) @ ( F @ Y ) )
@ Xs2 ) ) ) ).
% sorted_map
thf(fact_4606_zip__same__conv__map,axiom,
! [A: $tType,Xs2: list @ A] :
( ( zip @ A @ A @ Xs2 @ Xs2 )
= ( map @ A @ ( product_prod @ A @ A )
@ ^ [X2: A] : ( product_Pair @ A @ A @ X2 @ X2 )
@ Xs2 ) ) ).
% zip_same_conv_map
thf(fact_4607_sorted__wrt01,axiom,
! [A: $tType,Xs2: list @ A,P: A > A > $o] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ ( one_one @ nat ) )
=> ( sorted_wrt @ A @ P @ Xs2 ) ) ).
% sorted_wrt01
thf(fact_4608_rec__list__Cons__imp,axiom,
! [B: $tType,A: $tType,F: ( list @ A ) > B,F1: B,F23: A > ( list @ A ) > B > B,X: A,Xs2: list @ A] :
( ( F
= ( rec_list @ B @ A @ F1 @ F23 ) )
=> ( ( F @ ( cons @ A @ X @ Xs2 ) )
= ( F23 @ X @ Xs2 @ ( F @ Xs2 ) ) ) ) ).
% rec_list_Cons_imp
thf(fact_4609_rec__list__Nil__imp,axiom,
! [A: $tType,B: $tType,F: ( list @ A ) > B,F1: B,F23: A > ( list @ A ) > B > B] :
( ( F
= ( rec_list @ B @ A @ F1 @ F23 ) )
=> ( ( F @ ( nil @ A ) )
= F1 ) ) ).
% rec_list_Nil_imp
thf(fact_4610_zip__left__commute,axiom,
! [B: $tType,A: $tType,C: $tType,Xs2: list @ A,Ys2: list @ B,Zs: list @ C] :
( ( zip @ A @ ( product_prod @ B @ C ) @ Xs2 @ ( zip @ B @ C @ Ys2 @ 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 ) )
@ ^ [Y: B] :
( product_case_prod @ A @ C @ ( product_prod @ A @ ( product_prod @ B @ C ) )
@ ^ [X2: A,Z3: C] : ( product_Pair @ A @ ( product_prod @ B @ C ) @ X2 @ ( product_Pair @ B @ C @ Y @ Z3 ) ) ) )
@ ( zip @ B @ ( product_prod @ A @ C ) @ Ys2 @ ( zip @ A @ C @ Xs2 @ Zs ) ) ) ) ).
% zip_left_commute
thf(fact_4611_zip__assoc,axiom,
! [B: $tType,A: $tType,C: $tType,Xs2: list @ A,Ys2: list @ B,Zs: list @ C] :
( ( zip @ A @ ( product_prod @ B @ C ) @ Xs2 @ ( zip @ B @ C @ Ys2 @ 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 ) ) )
@ ^ [X2: A,Y: B,Z3: C] : ( product_Pair @ A @ ( product_prod @ B @ C ) @ X2 @ ( product_Pair @ B @ C @ Y @ Z3 ) ) ) )
@ ( zip @ ( product_prod @ A @ B ) @ C @ ( zip @ A @ B @ Xs2 @ Ys2 ) @ Zs ) ) ) ).
% zip_assoc
thf(fact_4612_zip__commute,axiom,
! [B: $tType,A: $tType] :
( ( zip @ A @ B )
= ( ^ [Xs: list @ A,Ys: list @ B] :
( map @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B )
@ ( product_case_prod @ B @ A @ ( product_prod @ A @ B )
@ ^ [X2: B,Y: A] : ( product_Pair @ A @ B @ Y @ X2 ) )
@ ( zip @ B @ A @ Ys @ Xs ) ) ) ) ).
% zip_commute
thf(fact_4613_fst__convol,axiom,
! [C: $tType,B: $tType,A: $tType,F: A > B,G: A > C] :
( ( comp @ ( product_prod @ B @ C ) @ B @ A @ ( product_fst @ B @ C ) @ ( bNF_convol @ A @ B @ C @ F @ G ) )
= F ) ).
% fst_convol
thf(fact_4614_snd__convol,axiom,
! [C: $tType,B: $tType,A: $tType,F: A > C,G: A > B] :
( ( comp @ ( product_prod @ C @ B ) @ B @ A @ ( product_snd @ C @ B ) @ ( bNF_convol @ A @ C @ B @ F @ G ) )
= G ) ).
% snd_convol
thf(fact_4615_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_4616_sorted01,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs2: list @ A] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ ( one_one @ nat ) )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs2 ) ) ) ).
% sorted01
thf(fact_4617_Id__on__set,axiom,
! [A: $tType,Xs2: list @ A] :
( ( id_on @ A @ ( set2 @ A @ Xs2 ) )
= ( set2 @ ( product_prod @ A @ A )
@ ( map @ A @ ( product_prod @ A @ A )
@ ^ [X2: A] : ( product_Pair @ A @ A @ X2 @ X2 )
@ Xs2 ) ) ) ).
% Id_on_set
thf(fact_4618_product_Osimps_I2_J,axiom,
! [A: $tType,B: $tType,X: A,Xs2: list @ A,Ys2: list @ B] :
( ( product @ A @ B @ ( cons @ A @ X @ Xs2 ) @ Ys2 )
= ( append @ ( product_prod @ A @ B ) @ ( map @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X ) @ Ys2 ) @ ( product @ A @ B @ Xs2 @ Ys2 ) ) ) ).
% product.simps(2)
thf(fact_4619_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_4620_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_4621_sorted__find__Min,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs2: list @ A,P: A > $o] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs2 )
=> ( ? [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs2 ) )
& ( P @ X5 ) )
=> ( ( find @ A @ P @ Xs2 )
= ( some @ A
@ ( lattic643756798350308766er_Min @ A
@ ( collect @ A
@ ^ [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ Xs2 ) )
& ( P @ X2 ) ) ) ) ) ) ) ) ) ).
% sorted_find_Min
thf(fact_4622_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
@ ^ [X2: A,Xa3: list @ A] : ( Ga @ ( F @ X2 ) @ ( map @ A @ B @ F @ Xa3 ) ) ) ) ).
% list.rec_o_map
thf(fact_4623_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_4624_eq__key__imp__eq__value,axiom,
! [A: $tType,B: $tType,Xs2: list @ ( product_prod @ A @ B ),K: A,V1: B,V22: B] :
( ( distinct @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xs2 ) )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ V1 ) @ ( set2 @ ( product_prod @ A @ B ) @ Xs2 ) )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ V22 ) @ ( set2 @ ( product_prod @ A @ B ) @ Xs2 ) )
=> ( V1 = V22 ) ) ) ) ).
% eq_key_imp_eq_value
thf(fact_4625_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_4626_length__transpose__sorted,axiom,
! [A: $tType,Xs2: list @ ( list @ A )] :
( ( sorted_wrt @ nat @ ( ord_less_eq @ nat ) @ ( rev @ nat @ ( map @ ( list @ A ) @ nat @ ( size_size @ ( list @ A ) ) @ Xs2 ) ) )
=> ( ( ( Xs2
= ( nil @ ( list @ A ) ) )
=> ( ( size_size @ ( list @ ( list @ A ) ) @ ( transpose @ A @ Xs2 ) )
= ( zero_zero @ nat ) ) )
& ( ( Xs2
!= ( nil @ ( list @ A ) ) )
=> ( ( size_size @ ( list @ ( list @ A ) ) @ ( transpose @ A @ Xs2 ) )
= ( size_size @ ( list @ A ) @ ( nth @ ( list @ A ) @ Xs2 @ ( zero_zero @ nat ) ) ) ) ) ) ) ).
% length_transpose_sorted
thf(fact_4627_set__concat,axiom,
! [A: $tType,Xs2: list @ ( list @ A )] :
( ( set2 @ A @ ( concat @ A @ Xs2 ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ ( list @ A ) @ ( set @ A ) @ ( set2 @ A ) @ ( set2 @ ( list @ A ) @ Xs2 ) ) ) ) ).
% set_concat
thf(fact_4628_concat__map__singleton,axiom,
! [A: $tType,B: $tType,F: B > A,Xs2: list @ B] :
( ( concat @ A
@ ( map @ B @ ( list @ A )
@ ^ [X2: B] : ( cons @ A @ ( F @ X2 ) @ ( nil @ A ) )
@ Xs2 ) )
= ( map @ B @ A @ F @ Xs2 ) ) ).
% concat_map_singleton
thf(fact_4629_sorted__wrt__rev,axiom,
! [A: $tType,P: A > A > $o,Xs2: list @ A] :
( ( sorted_wrt @ A @ P @ ( rev @ A @ Xs2 ) )
= ( sorted_wrt @ A
@ ^ [X2: A,Y: A] : ( P @ Y @ X2 )
@ Xs2 ) ) ).
% sorted_wrt_rev
thf(fact_4630_fold__rev,axiom,
! [B: $tType,A: $tType,Xs2: list @ A,F: A > B > B] :
( ! [X4: A,Y4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs2 ) )
=> ( ( member @ A @ Y4 @ ( set2 @ A @ Xs2 ) )
=> ( ( comp @ B @ B @ B @ ( F @ Y4 ) @ ( F @ X4 ) )
= ( comp @ B @ B @ B @ ( F @ X4 ) @ ( F @ Y4 ) ) ) ) )
=> ( ( fold @ A @ B @ F @ ( rev @ A @ Xs2 ) )
= ( fold @ A @ B @ F @ Xs2 ) ) ) ).
% fold_rev
thf(fact_4631_fold__plus__sum__list__rev,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ! [Xs2: list @ A] :
( ( fold @ A @ A @ ( plus_plus @ A ) @ Xs2 )
= ( plus_plus @ A @ ( groups8242544230860333062m_list @ A @ ( rev @ A @ Xs2 ) ) ) ) ) ).
% fold_plus_sum_list_rev
thf(fact_4632_transpose_Osimps_I3_J,axiom,
! [A: $tType,X: A,Xs2: list @ A,Xss: list @ ( list @ A )] :
( ( transpose @ A @ ( cons @ ( list @ A ) @ ( cons @ A @ X @ Xs2 ) @ Xss ) )
= ( cons @ ( list @ A )
@ ( cons @ A @ X
@ ( concat @ A
@ ( map @ ( list @ A ) @ ( list @ A )
@ ( case_list @ ( list @ A ) @ A @ ( nil @ A )
@ ^ [H4: A,T3: list @ A] : ( cons @ A @ H4 @ ( nil @ A ) ) )
@ Xss ) ) )
@ ( transpose @ A
@ ( cons @ ( list @ A ) @ Xs2
@ ( concat @ ( list @ A )
@ ( map @ ( list @ A ) @ ( list @ ( list @ A ) )
@ ( case_list @ ( list @ ( list @ A ) ) @ A @ ( nil @ ( list @ A ) )
@ ^ [H4: A,T3: list @ A] : ( cons @ ( list @ A ) @ T3 @ ( nil @ ( list @ A ) ) ) )
@ Xss ) ) ) ) ) ) ).
% transpose.simps(3)
thf(fact_4633_transpose_Oelims,axiom,
! [A: $tType,X: list @ ( list @ A ),Y3: list @ ( list @ A )] :
( ( ( transpose @ A @ X )
= Y3 )
=> ( ( ( X
= ( nil @ ( list @ A ) ) )
=> ( Y3
!= ( nil @ ( list @ A ) ) ) )
=> ( ! [Xss2: list @ ( list @ A )] :
( ( X
= ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss2 ) )
=> ( Y3
!= ( transpose @ A @ Xss2 ) ) )
=> ~ ! [X4: A,Xs5: list @ A,Xss2: list @ ( list @ A )] :
( ( X
= ( cons @ ( list @ A ) @ ( cons @ A @ X4 @ Xs5 ) @ Xss2 ) )
=> ( Y3
!= ( cons @ ( list @ A )
@ ( cons @ A @ X4
@ ( concat @ A
@ ( map @ ( list @ A ) @ ( list @ A )
@ ( case_list @ ( list @ A ) @ A @ ( nil @ A )
@ ^ [H4: A,T3: list @ A] : ( cons @ A @ H4 @ ( nil @ A ) ) )
@ Xss2 ) ) )
@ ( transpose @ A
@ ( cons @ ( list @ A ) @ Xs5
@ ( concat @ ( list @ A )
@ ( map @ ( list @ A ) @ ( list @ ( list @ A ) )
@ ( case_list @ ( list @ ( list @ A ) ) @ A @ ( nil @ ( list @ A ) )
@ ^ [H4: A,T3: list @ A] : ( cons @ ( list @ A ) @ T3 @ ( nil @ ( list @ A ) ) ) )
@ Xss2 ) ) ) ) ) ) ) ) ) ) ).
% transpose.elims
thf(fact_4634_product__concat__map,axiom,
! [B: $tType,A: $tType] :
( ( product @ A @ B )
= ( ^ [Xs: list @ A,Ys: list @ B] :
( concat @ ( product_prod @ A @ B )
@ ( map @ A @ ( list @ ( product_prod @ A @ B ) )
@ ^ [X2: A] : ( map @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 ) @ Ys )
@ Xs ) ) ) ) ).
% product_concat_map
thf(fact_4635_rev__update,axiom,
! [A: $tType,K: nat,Xs2: list @ A,Y3: A] :
( ( ord_less @ nat @ K @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( rev @ A @ ( list_update @ A @ Xs2 @ K @ Y3 ) )
= ( list_update @ A @ ( rev @ A @ Xs2 ) @ ( minus_minus @ nat @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ K ) @ ( one_one @ nat ) ) @ Y3 ) ) ) ).
% rev_update
thf(fact_4636_set__rec,axiom,
! [A: $tType] :
( ( set2 @ A )
= ( rec_list @ ( set @ A ) @ A @ ( bot_bot @ ( set @ A ) )
@ ^ [X2: A,Uu: list @ A] : ( insert2 @ A @ X2 ) ) ) ).
% set_rec
thf(fact_4637_product__code,axiom,
! [B: $tType,A: $tType,Xs2: list @ A,Ys2: list @ B] :
( ( product_product @ A @ B @ ( set2 @ A @ Xs2 ) @ ( set2 @ B @ Ys2 ) )
= ( set2 @ ( product_prod @ A @ B )
@ ( concat @ ( product_prod @ A @ B )
@ ( map @ A @ ( list @ ( product_prod @ A @ B ) )
@ ^ [X2: A] : ( map @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 ) @ Ys2 )
@ Xs2 ) ) ) ) ).
% product_code
thf(fact_4638_transpose_Opsimps_I3_J,axiom,
! [A: $tType,X: A,Xs2: list @ A,Xss: list @ ( list @ A )] :
( ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ ( cons @ ( list @ A ) @ ( cons @ A @ X @ Xs2 ) @ Xss ) )
=> ( ( transpose @ A @ ( cons @ ( list @ A ) @ ( cons @ A @ X @ Xs2 ) @ Xss ) )
= ( cons @ ( list @ A )
@ ( cons @ A @ X
@ ( concat @ A
@ ( map @ ( list @ A ) @ ( list @ A )
@ ( case_list @ ( list @ A ) @ A @ ( nil @ A )
@ ^ [H4: A,T3: list @ A] : ( cons @ A @ H4 @ ( nil @ A ) ) )
@ Xss ) ) )
@ ( transpose @ A
@ ( cons @ ( list @ A ) @ Xs2
@ ( concat @ ( list @ A )
@ ( map @ ( list @ A ) @ ( list @ ( list @ A ) )
@ ( case_list @ ( list @ ( list @ A ) ) @ A @ ( nil @ ( list @ A ) )
@ ^ [H4: A,T3: list @ A] : ( cons @ ( list @ A ) @ T3 @ ( nil @ ( list @ A ) ) ) )
@ Xss ) ) ) ) ) ) ) ).
% transpose.psimps(3)
thf(fact_4639_nth__nth__transpose__sorted,axiom,
! [A: $tType,Xs2: list @ ( list @ A ),I: nat,J: nat] :
( ( sorted_wrt @ nat @ ( ord_less_eq @ nat ) @ ( rev @ nat @ ( map @ ( list @ A ) @ nat @ ( size_size @ ( list @ A ) ) @ Xs2 ) ) )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ ( list @ A ) ) @ ( transpose @ A @ Xs2 ) ) )
=> ( ( ord_less @ nat @ J
@ ( size_size @ ( list @ ( list @ A ) )
@ ( filter2 @ ( list @ A )
@ ^ [Ys: list @ A] : ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Ys ) )
@ Xs2 ) ) )
=> ( ( nth @ A @ ( nth @ ( list @ A ) @ ( transpose @ A @ Xs2 ) @ I ) @ J )
= ( nth @ A @ ( nth @ ( list @ A ) @ Xs2 @ J ) @ I ) ) ) ) ) ).
% nth_nth_transpose_sorted
thf(fact_4640_filter__filter,axiom,
! [A: $tType,P: A > $o,Q: A > $o,Xs2: list @ A] :
( ( filter2 @ A @ P @ ( filter2 @ A @ Q @ Xs2 ) )
= ( filter2 @ A
@ ^ [X2: A] :
( ( Q @ X2 )
& ( P @ X2 ) )
@ Xs2 ) ) ).
% filter_filter
thf(fact_4641_set__filter,axiom,
! [A: $tType,P: A > $o,Xs2: list @ A] :
( ( set2 @ A @ ( filter2 @ A @ P @ Xs2 ) )
= ( collect @ A
@ ^ [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ Xs2 ) )
& ( P @ X2 ) ) ) ) ).
% set_filter
thf(fact_4642_partition__filter__conv,axiom,
! [A: $tType] :
( ( partition @ A )
= ( ^ [F2: A > $o,Xs: list @ A] : ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( filter2 @ A @ F2 @ Xs ) @ ( filter2 @ A @ ( comp @ $o @ $o @ A @ (~) @ F2 ) @ Xs ) ) ) ) ).
% partition_filter_conv
thf(fact_4643_length__filter__map,axiom,
! [A: $tType,B: $tType,P: A > $o,F: B > A,Xs2: list @ B] :
( ( size_size @ ( list @ A ) @ ( filter2 @ A @ P @ ( map @ B @ A @ F @ Xs2 ) ) )
= ( size_size @ ( list @ B ) @ ( filter2 @ B @ ( comp @ A @ $o @ B @ P @ F ) @ Xs2 ) ) ) ).
% length_filter_map
thf(fact_4644_partition__filter2,axiom,
! [A: $tType,P: A > $o,Xs2: list @ A] :
( ( product_snd @ ( list @ A ) @ ( list @ A ) @ ( partition @ A @ P @ Xs2 ) )
= ( filter2 @ A @ ( comp @ $o @ $o @ A @ (~) @ P ) @ Xs2 ) ) ).
% partition_filter2
thf(fact_4645_partition__in__shuffles,axiom,
! [A: $tType,Xs2: list @ A,P: A > $o] :
( member @ ( list @ A ) @ Xs2
@ ( shuffles @ A @ ( filter2 @ A @ P @ Xs2 )
@ ( filter2 @ A
@ ^ [X2: A] :
~ ( P @ X2 )
@ Xs2 ) ) ) ).
% partition_in_shuffles
thf(fact_4646_filter__map,axiom,
! [A: $tType,B: $tType,P: A > $o,F: B > A,Xs2: list @ B] :
( ( filter2 @ A @ P @ ( map @ B @ A @ F @ Xs2 ) )
= ( map @ B @ A @ F @ ( filter2 @ B @ ( comp @ A @ $o @ B @ P @ F ) @ Xs2 ) ) ) ).
% filter_map
thf(fact_4647_sum__length__filter__compl,axiom,
! [A: $tType,P: A > $o,Xs2: list @ A] :
( ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ ( filter2 @ A @ P @ Xs2 ) )
@ ( size_size @ ( list @ A )
@ ( filter2 @ A
@ ^ [X2: A] :
~ ( P @ X2 )
@ Xs2 ) ) )
= ( size_size @ ( list @ A ) @ Xs2 ) ) ).
% sum_length_filter_compl
thf(fact_4648_inter__set__filter,axiom,
! [A: $tType,A5: set @ A,Xs2: list @ A] :
( ( inf_inf @ ( set @ A ) @ A5 @ ( set2 @ A @ Xs2 ) )
= ( set2 @ A
@ ( filter2 @ A
@ ^ [X2: A] : ( member @ A @ X2 @ A5 )
@ Xs2 ) ) ) ).
% inter_set_filter
thf(fact_4649_sorted__same,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [G: ( list @ A ) > A,Xs2: list @ A] :
( sorted_wrt @ A @ ( ord_less_eq @ A )
@ ( filter2 @ A
@ ^ [X2: A] :
( X2
= ( G @ Xs2 ) )
@ Xs2 ) ) ) ).
% sorted_same
thf(fact_4650_fold__filter,axiom,
! [A: $tType,B: $tType,F: B > A > A,P: B > $o,Xs2: list @ B] :
( ( fold @ B @ A @ F @ ( filter2 @ B @ P @ Xs2 ) )
= ( fold @ B @ A
@ ^ [X2: B] : ( if @ ( A > A ) @ ( P @ X2 ) @ ( F @ X2 ) @ ( id @ A ) )
@ Xs2 ) ) ).
% fold_filter
thf(fact_4651_inj__on__filter__key__eq,axiom,
! [B: $tType,A: $tType,F: A > B,Y3: A,Xs2: list @ A] :
( ( inj_on @ A @ B @ F @ ( insert2 @ A @ Y3 @ ( set2 @ A @ Xs2 ) ) )
=> ( ( filter2 @ A
@ ^ [X2: A] :
( ( F @ Y3 )
= ( F @ X2 ) )
@ Xs2 )
= ( filter2 @ A
@ ( ^ [Y6: A,Z2: A] : Y6 = Z2
@ Y3 )
@ Xs2 ) ) ) ).
% inj_on_filter_key_eq
thf(fact_4652_sorted__map__same,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F: B > A,G: ( list @ B ) > A,Xs2: list @ B] :
( sorted_wrt @ A @ ( ord_less_eq @ A )
@ ( map @ B @ A @ F
@ ( filter2 @ B
@ ^ [X2: B] :
( ( F @ X2 )
= ( G @ Xs2 ) )
@ Xs2 ) ) ) ) ).
% sorted_map_same
thf(fact_4653_sum__list__map__filter_H,axiom,
! [A: $tType,B: $tType] :
( ( monoid_add @ A )
=> ! [F: B > A,P: B > $o,Xs2: list @ B] :
( ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F @ ( filter2 @ B @ P @ Xs2 ) ) )
= ( groups8242544230860333062m_list @ A
@ ( map @ B @ A
@ ^ [X2: B] : ( if @ A @ ( P @ X2 ) @ ( F @ X2 ) @ ( zero_zero @ A ) )
@ Xs2 ) ) ) ) ).
% sum_list_map_filter'
thf(fact_4654_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 ) ) ) )
=> ( ! [X4: A,Xs5: list @ A,Xss2: list @ ( list @ A )] :
( ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ ( cons @ ( list @ A ) @ ( cons @ A @ X4 @ Xs5 ) @ Xss2 ) )
=> ( ( P
@ ( cons @ ( list @ A ) @ Xs5
@ ( concat @ ( list @ A )
@ ( map @ ( list @ A ) @ ( list @ ( list @ A ) )
@ ( case_list @ ( list @ ( list @ A ) ) @ A @ ( nil @ ( list @ A ) )
@ ^ [H4: A,T3: list @ A] : ( cons @ ( list @ A ) @ T3 @ ( nil @ ( list @ A ) ) ) )
@ Xss2 ) ) ) )
=> ( P @ ( cons @ ( list @ A ) @ ( cons @ A @ X4 @ Xs5 ) @ Xss2 ) ) ) )
=> ( P @ A0 ) ) ) ) ) ).
% transpose.pinduct
thf(fact_4655_filter__in__nths,axiom,
! [A: $tType,Xs2: list @ A,S: set @ nat] :
( ( distinct @ A @ Xs2 )
=> ( ( filter2 @ A
@ ^ [X2: A] : ( member @ A @ X2 @ ( set2 @ A @ ( nths @ A @ Xs2 @ S ) ) )
@ Xs2 )
= ( nths @ A @ Xs2 @ S ) ) ) ).
% filter_in_nths
thf(fact_4656_sum__list__map__filter,axiom,
! [A: $tType,B: $tType] :
( ( monoid_add @ A )
=> ! [Xs2: list @ B,P: B > $o,F: B > A] :
( ! [X4: B] :
( ( member @ B @ X4 @ ( set2 @ B @ Xs2 ) )
=> ( ~ ( P @ X4 )
=> ( ( F @ X4 )
= ( zero_zero @ A ) ) ) )
=> ( ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F @ ( filter2 @ B @ P @ Xs2 ) ) )
= ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F @ Xs2 ) ) ) ) ) ).
% sum_list_map_filter
thf(fact_4657_set__minus__filter__out,axiom,
! [A: $tType,Xs2: list @ A,Y3: A] :
( ( minus_minus @ ( set @ A ) @ ( set2 @ A @ Xs2 ) @ ( insert2 @ A @ Y3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( set2 @ A
@ ( filter2 @ A
@ ^ [X2: A] : X2 != Y3
@ Xs2 ) ) ) ).
% set_minus_filter_out
thf(fact_4658_filter__shuffles__disjoint1_I2_J,axiom,
! [A: $tType,Xs2: list @ A,Ys2: list @ A,Zs: list @ A] :
( ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Xs2 ) @ ( set2 @ A @ Ys2 ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( member @ ( list @ A ) @ Zs @ ( shuffles @ A @ Xs2 @ Ys2 ) )
=> ( ( filter2 @ A
@ ^ [X2: A] :
~ ( member @ A @ X2 @ ( set2 @ A @ Xs2 ) )
@ Zs )
= Ys2 ) ) ) ).
% filter_shuffles_disjoint1(2)
thf(fact_4659_filter__shuffles__disjoint1_I1_J,axiom,
! [A: $tType,Xs2: list @ A,Ys2: list @ A,Zs: list @ A] :
( ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Xs2 ) @ ( set2 @ A @ Ys2 ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( member @ ( list @ A ) @ Zs @ ( shuffles @ A @ Xs2 @ Ys2 ) )
=> ( ( filter2 @ A
@ ^ [X2: A] : ( member @ A @ X2 @ ( set2 @ A @ Xs2 ) )
@ Zs )
= Xs2 ) ) ) ).
% filter_shuffles_disjoint1(1)
thf(fact_4660_filter__shuffles__disjoint2_I2_J,axiom,
! [A: $tType,Xs2: list @ A,Ys2: list @ A,Zs: list @ A] :
( ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Xs2 ) @ ( set2 @ A @ Ys2 ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( member @ ( list @ A ) @ Zs @ ( shuffles @ A @ Xs2 @ Ys2 ) )
=> ( ( filter2 @ A
@ ^ [X2: A] :
~ ( member @ A @ X2 @ ( set2 @ A @ Ys2 ) )
@ Zs )
= Xs2 ) ) ) ).
% filter_shuffles_disjoint2(2)
thf(fact_4661_filter__shuffles__disjoint2_I1_J,axiom,
! [A: $tType,Xs2: list @ A,Ys2: list @ A,Zs: list @ A] :
( ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Xs2 ) @ ( set2 @ A @ Ys2 ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( member @ ( list @ A ) @ Zs @ ( shuffles @ A @ Xs2 @ Ys2 ) )
=> ( ( filter2 @ A
@ ^ [X2: A] : ( member @ A @ X2 @ ( set2 @ A @ Ys2 ) )
@ Zs )
= Ys2 ) ) ) ).
% filter_shuffles_disjoint2(1)
thf(fact_4662_filter__eq__nths,axiom,
! [A: $tType] :
( ( filter2 @ A )
= ( ^ [P3: A > $o,Xs: list @ A] :
( nths @ A @ Xs
@ ( collect @ nat
@ ^ [I2: nat] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( P3 @ ( nth @ A @ Xs @ I2 ) ) ) ) ) ) ) ).
% filter_eq_nths
thf(fact_4663_length__filter__conv__card,axiom,
! [A: $tType,P4: A > $o,Xs2: list @ A] :
( ( size_size @ ( list @ A ) @ ( filter2 @ A @ P4 @ Xs2 ) )
= ( finite_card @ nat
@ ( collect @ nat
@ ^ [I2: nat] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs2 ) )
& ( P4 @ ( nth @ A @ Xs2 @ I2 ) ) ) ) ) ) ).
% length_filter_conv_card
thf(fact_4664_distinct__length__filter,axiom,
! [A: $tType,Xs2: list @ A,P: A > $o] :
( ( distinct @ A @ Xs2 )
=> ( ( size_size @ ( list @ A ) @ ( filter2 @ A @ P @ Xs2 ) )
= ( finite_card @ A @ ( inf_inf @ ( set @ A ) @ ( collect @ A @ P ) @ ( set2 @ A @ Xs2 ) ) ) ) ) ).
% distinct_length_filter
thf(fact_4665_Product__Type_Oproduct__def,axiom,
! [B: $tType,A: $tType] :
( ( product_product @ A @ B )
= ( ^ [A8: set @ A,B7: set @ B] :
( product_Sigma @ A @ B @ A8
@ ^ [Uu: A] : B7 ) ) ) ).
% Product_Type.product_def
thf(fact_4666_member__product,axiom,
! [B: $tType,A: $tType,X: product_prod @ A @ B,A5: set @ A,B4: set @ B] :
( ( member @ ( product_prod @ A @ B ) @ X @ ( product_product @ A @ B @ A5 @ B4 ) )
= ( member @ ( product_prod @ A @ B ) @ X
@ ( product_Sigma @ A @ B @ A5
@ ^ [Uu: A] : B4 ) ) ) ).
% member_product
thf(fact_4667_n__lists_Osimps_I2_J,axiom,
! [A: $tType,N: nat,Xs2: list @ A] :
( ( n_lists @ A @ ( suc @ N ) @ Xs2 )
= ( concat @ ( list @ A )
@ ( map @ ( list @ A ) @ ( list @ ( list @ A ) )
@ ^ [Ys: list @ A] :
( map @ A @ ( list @ A )
@ ^ [Y: A] : ( cons @ A @ Y @ Ys )
@ Xs2 )
@ ( n_lists @ A @ N @ Xs2 ) ) ) ) ).
% n_lists.simps(2)
thf(fact_4668_nth__transpose,axiom,
! [A: $tType,I: nat,Xs2: list @ ( list @ A )] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ ( list @ A ) ) @ ( transpose @ A @ Xs2 ) ) )
=> ( ( nth @ ( list @ A ) @ ( transpose @ A @ Xs2 ) @ I )
= ( map @ ( list @ A ) @ A
@ ^ [Xs: list @ A] : ( nth @ A @ Xs @ I )
@ ( filter2 @ ( list @ A )
@ ^ [Ys: list @ A] : ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Ys ) )
@ Xs2 ) ) ) ) ).
% nth_transpose
thf(fact_4669_transpose__column__length,axiom,
! [A: $tType,Xs2: list @ ( list @ A ),I: nat] :
( ( sorted_wrt @ nat @ ( ord_less_eq @ nat ) @ ( rev @ nat @ ( map @ ( list @ A ) @ nat @ ( size_size @ ( list @ A ) ) @ Xs2 ) ) )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ ( list @ A ) ) @ Xs2 ) )
=> ( ( size_size @ ( list @ ( list @ A ) )
@ ( filter2 @ ( list @ A )
@ ^ [Ys: list @ A] : ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Ys ) )
@ ( transpose @ A @ Xs2 ) ) )
= ( size_size @ ( list @ A ) @ ( nth @ ( list @ A ) @ Xs2 @ I ) ) ) ) ) ).
% transpose_column_length
thf(fact_4670_transpose__column,axiom,
! [A: $tType,Xs2: list @ ( list @ A ),I: nat] :
( ( sorted_wrt @ nat @ ( ord_less_eq @ nat ) @ ( rev @ nat @ ( map @ ( list @ A ) @ nat @ ( size_size @ ( list @ A ) ) @ Xs2 ) ) )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ ( list @ A ) ) @ Xs2 ) )
=> ( ( map @ ( list @ A ) @ A
@ ^ [Ys: list @ A] : ( nth @ A @ Ys @ I )
@ ( filter2 @ ( list @ A )
@ ^ [Ys: list @ A] : ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Ys ) )
@ ( transpose @ A @ Xs2 ) ) )
= ( nth @ ( list @ A ) @ Xs2 @ I ) ) ) ) ).
% transpose_column
thf(fact_4671_map__filter__def,axiom,
! [B: $tType,A: $tType] :
( ( map_filter @ A @ B )
= ( ^ [F2: A > ( option @ B ),Xs: list @ A] :
( map @ A @ B @ ( comp @ ( option @ B ) @ B @ A @ ( the2 @ B ) @ F2 )
@ ( filter2 @ A
@ ^ [X2: A] :
( ( F2 @ X2 )
!= ( none @ B ) )
@ Xs ) ) ) ) ).
% map_filter_def
thf(fact_4672_product__lists_Osimps_I2_J,axiom,
! [A: $tType,Xs2: list @ A,Xss: list @ ( list @ A )] :
( ( product_lists @ A @ ( cons @ ( list @ A ) @ Xs2 @ Xss ) )
= ( concat @ ( list @ A )
@ ( map @ A @ ( list @ ( list @ A ) )
@ ^ [X2: A] : ( map @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X2 ) @ ( product_lists @ A @ Xss ) )
@ Xs2 ) ) ) ).
% product_lists.simps(2)
thf(fact_4673_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 ) )
@ ^ [H4: A,T3: list @ A] : ( cons @ ( list @ A ) @ T3 @ ( nil @ ( list @ A ) ) ) )
@ Xss ) )
= ( map @ ( list @ A ) @ ( list @ A ) @ ( tl @ A )
@ ( filter2 @ ( list @ A )
@ ^ [Ys: list @ A] :
( Ys
!= ( nil @ A ) )
@ Xss ) ) ) ).
% transpose_aux_filter_tail
thf(fact_4674_length__tl,axiom,
! [A: $tType,Xs2: list @ A] :
( ( size_size @ ( list @ A ) @ ( tl @ A @ Xs2 ) )
= ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ ( one_one @ nat ) ) ) ).
% length_tl
thf(fact_4675_tl__def,axiom,
! [A: $tType] :
( ( tl @ A )
= ( case_list @ ( list @ A ) @ A @ ( nil @ A )
@ ^ [X212: A,X223: list @ A] : X223 ) ) ).
% tl_def
thf(fact_4676_tl__append,axiom,
! [A: $tType,Xs2: list @ A,Ys2: list @ A] :
( ( tl @ A @ ( append @ A @ Xs2 @ Ys2 ) )
= ( case_list @ ( list @ A ) @ A @ ( tl @ A @ Ys2 )
@ ^ [Z3: A,Zs2: list @ A] : ( append @ A @ Zs2 @ Ys2 )
@ Xs2 ) ) ).
% tl_append
thf(fact_4677_map__filter__simps_I1_J,axiom,
! [A: $tType,B: $tType,F: B > ( option @ A ),X: B,Xs2: list @ B] :
( ( map_filter @ B @ A @ F @ ( cons @ B @ X @ Xs2 ) )
= ( case_option @ ( list @ A ) @ A @ ( map_filter @ B @ A @ F @ Xs2 )
@ ^ [Y: A] : ( cons @ A @ Y @ ( map_filter @ B @ A @ F @ Xs2 ) )
@ ( F @ X ) ) ) ).
% map_filter_simps(1)
thf(fact_4678_tl__take,axiom,
! [A: $tType,N: nat,Xs2: list @ A] :
( ( tl @ A @ ( take @ A @ N @ Xs2 ) )
= ( take @ A @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ ( tl @ A @ Xs2 ) ) ) ).
% tl_take
thf(fact_4679_Nitpick_Osize__list__simp_I2_J,axiom,
! [A: $tType] :
( ( size_size @ ( list @ A ) )
= ( ^ [Xs: list @ A] :
( if @ nat
@ ( Xs
= ( nil @ A ) )
@ ( zero_zero @ nat )
@ ( suc @ ( size_size @ ( list @ A ) @ ( tl @ A @ Xs ) ) ) ) ) ) ).
% Nitpick.size_list_simp(2)
thf(fact_4680_nths__shift__lemma__Suc,axiom,
! [A: $tType,P: nat > $o,Xs2: list @ A,Is: list @ nat] :
( ( map @ ( product_prod @ A @ nat ) @ A @ ( product_fst @ A @ nat )
@ ( filter2 @ ( product_prod @ A @ nat )
@ ^ [P5: product_prod @ A @ nat] : ( P @ ( suc @ ( product_snd @ A @ nat @ P5 ) ) )
@ ( zip @ A @ nat @ Xs2 @ Is ) ) )
= ( map @ ( product_prod @ A @ nat ) @ A @ ( product_fst @ A @ nat )
@ ( filter2 @ ( product_prod @ A @ nat )
@ ^ [P5: product_prod @ A @ nat] : ( P @ ( product_snd @ A @ nat @ P5 ) )
@ ( zip @ A @ nat @ Xs2 @ ( map @ nat @ nat @ suc @ Is ) ) ) ) ) ).
% nths_shift_lemma_Suc
thf(fact_4681_nths__shift__lemma,axiom,
! [A: $tType,A5: set @ nat,Xs2: list @ A,I: nat] :
( ( map @ ( product_prod @ A @ nat ) @ A @ ( product_fst @ A @ nat )
@ ( filter2 @ ( product_prod @ A @ nat )
@ ^ [P5: product_prod @ A @ nat] : ( member @ nat @ ( product_snd @ A @ nat @ P5 ) @ A5 )
@ ( zip @ A @ nat @ Xs2 @ ( upt @ I @ ( plus_plus @ nat @ I @ ( size_size @ ( list @ A ) @ Xs2 ) ) ) ) ) )
= ( map @ ( product_prod @ A @ nat ) @ A @ ( product_fst @ A @ nat )
@ ( filter2 @ ( product_prod @ A @ nat )
@ ^ [P5: product_prod @ A @ nat] : ( member @ nat @ ( plus_plus @ nat @ ( product_snd @ A @ nat @ P5 ) @ I ) @ A5 )
@ ( zip @ A @ nat @ Xs2 @ ( upt @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs2 ) ) ) ) ) ) ).
% nths_shift_lemma
thf(fact_4682_nths__def,axiom,
! [A: $tType] :
( ( nths @ A )
= ( ^ [Xs: list @ A,A8: set @ nat] :
( map @ ( product_prod @ A @ nat ) @ A @ ( product_fst @ A @ nat )
@ ( filter2 @ ( product_prod @ A @ nat )
@ ^ [P5: product_prod @ A @ nat] : ( member @ nat @ ( product_snd @ A @ nat @ P5 ) @ A8 )
@ ( zip @ A @ nat @ Xs @ ( upt @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) ) ) ).
% nths_def
thf(fact_4683_map__filter__map__filter,axiom,
! [A: $tType,B: $tType,F: B > A,P: B > $o,Xs2: list @ B] :
( ( map @ B @ A @ F @ ( filter2 @ B @ P @ Xs2 ) )
= ( map_filter @ B @ A
@ ^ [X2: B] : ( if @ ( option @ A ) @ ( P @ X2 ) @ ( some @ A @ ( F @ X2 ) ) @ ( none @ A ) )
@ Xs2 ) ) ).
% map_filter_map_filter
thf(fact_4684_transpose__max__length,axiom,
! [A: $tType,Xs2: list @ ( list @ A )] :
( ( foldr @ ( list @ A ) @ nat
@ ^ [Xs: list @ A] : ( ord_max @ nat @ ( size_size @ ( list @ A ) @ Xs ) )
@ ( transpose @ A @ Xs2 )
@ ( zero_zero @ nat ) )
= ( size_size @ ( list @ ( list @ A ) )
@ ( filter2 @ ( list @ A )
@ ^ [X2: list @ A] :
( X2
!= ( nil @ A ) )
@ Xs2 ) ) ) ).
% transpose_max_length
thf(fact_4685_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_4686_transpose__aux__max,axiom,
! [A: $tType,B: $tType,Xs2: list @ A,Xss: list @ ( list @ B )] :
( ( ord_max @ nat @ ( suc @ ( size_size @ ( list @ A ) @ Xs2 ) )
@ ( foldr @ ( list @ B ) @ nat
@ ^ [Xs: list @ B] : ( ord_max @ nat @ ( size_size @ ( list @ B ) @ Xs ) )
@ Xss
@ ( zero_zero @ nat ) ) )
= ( suc
@ ( ord_max @ nat @ ( size_size @ ( list @ A ) @ Xs2 )
@ ( foldr @ ( list @ B ) @ nat
@ ^ [X2: list @ B] : ( ord_max @ nat @ ( minus_minus @ nat @ ( size_size @ ( list @ B ) @ X2 ) @ ( suc @ ( zero_zero @ nat ) ) ) )
@ ( filter2 @ ( list @ B )
@ ^ [Ys: list @ B] :
( Ys
!= ( nil @ B ) )
@ Xss )
@ ( zero_zero @ nat ) ) ) ) ) ).
% transpose_aux_max
thf(fact_4687_foldr__Cons,axiom,
! [B: $tType,A: $tType,F: A > B > B,X: A,Xs2: list @ A] :
( ( foldr @ A @ B @ F @ ( cons @ A @ X @ Xs2 ) )
= ( comp @ B @ B @ B @ ( F @ X ) @ ( foldr @ A @ B @ F @ Xs2 ) ) ) ).
% foldr_Cons
thf(fact_4688_foldr__map,axiom,
! [C: $tType,B: $tType,A: $tType,G: B > A > A,F: C > B,Xs2: list @ C,A3: A] :
( ( foldr @ B @ A @ G @ ( map @ C @ B @ F @ Xs2 ) @ A3 )
= ( foldr @ C @ A @ ( comp @ B @ ( A > A ) @ C @ G @ F ) @ Xs2 @ A3 ) ) ).
% foldr_map
thf(fact_4689_foldr__filter,axiom,
! [A: $tType,B: $tType,F: B > A > A,P: B > $o,Xs2: list @ B] :
( ( foldr @ B @ A @ F @ ( filter2 @ B @ P @ Xs2 ) )
= ( foldr @ B @ A
@ ^ [X2: B] : ( if @ ( A > A ) @ ( P @ X2 ) @ ( F @ X2 ) @ ( id @ A ) )
@ Xs2 ) ) ).
% foldr_filter
thf(fact_4690_sum__list_Oeq__foldr,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ( ( groups8242544230860333062m_list @ A )
= ( ^ [Xs: list @ A] : ( foldr @ A @ A @ ( plus_plus @ A ) @ Xs @ ( zero_zero @ A ) ) ) ) ) ).
% sum_list.eq_foldr
thf(fact_4691_foldr__fold,axiom,
! [B: $tType,A: $tType,Xs2: list @ A,F: A > B > B] :
( ! [X4: A,Y4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs2 ) )
=> ( ( member @ A @ Y4 @ ( set2 @ A @ Xs2 ) )
=> ( ( comp @ B @ B @ B @ ( F @ Y4 ) @ ( F @ X4 ) )
= ( comp @ B @ B @ B @ ( F @ X4 ) @ ( F @ Y4 ) ) ) ) )
=> ( ( foldr @ A @ B @ F @ Xs2 )
= ( fold @ A @ B @ F @ Xs2 ) ) ) ).
% foldr_fold
thf(fact_4692_horner__sum__foldr,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_0 @ A )
=> ( ( groups4207007520872428315er_sum @ B @ A )
= ( ^ [F2: B > A,A6: A,Xs: list @ B] :
( foldr @ B @ A
@ ^ [X2: B,B5: A] : ( plus_plus @ A @ ( F2 @ X2 ) @ ( times_times @ A @ A6 @ B5 ) )
@ Xs
@ ( zero_zero @ A ) ) ) ) ) ).
% horner_sum_foldr
thf(fact_4693_length__transpose,axiom,
! [A: $tType,Xs2: list @ ( list @ A )] :
( ( size_size @ ( list @ ( list @ A ) ) @ ( transpose @ A @ Xs2 ) )
= ( foldr @ ( list @ A ) @ nat
@ ^ [Xs: list @ A] : ( ord_max @ nat @ ( size_size @ ( list @ A ) @ Xs ) )
@ Xs2
@ ( zero_zero @ nat ) ) ) ).
% length_transpose
thf(fact_4694_foldr__max__sorted,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs2: list @ A,Y3: A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( rev @ A @ Xs2 ) )
=> ( ( ( Xs2
= ( nil @ A ) )
=> ( ( foldr @ A @ A @ ( ord_max @ A ) @ Xs2 @ Y3 )
= Y3 ) )
& ( ( Xs2
!= ( nil @ A ) )
=> ( ( foldr @ A @ A @ ( ord_max @ A ) @ Xs2 @ Y3 )
= ( ord_max @ A @ ( nth @ A @ Xs2 @ ( zero_zero @ nat ) ) @ Y3 ) ) ) ) ) ) ).
% foldr_max_sorted
thf(fact_4695_transpose__transpose,axiom,
! [A: $tType,Xs2: list @ ( list @ A )] :
( ( sorted_wrt @ nat @ ( ord_less_eq @ nat ) @ ( rev @ nat @ ( map @ ( list @ A ) @ nat @ ( size_size @ ( list @ A ) ) @ Xs2 ) ) )
=> ( ( transpose @ A @ ( transpose @ A @ Xs2 ) )
= ( takeWhile @ ( list @ A )
@ ^ [X2: list @ A] :
( X2
!= ( nil @ A ) )
@ Xs2 ) ) ) ).
% transpose_transpose
thf(fact_4696_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_4697_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 )
@ ^ [H4: A,T3: list @ A] : ( cons @ A @ H4 @ ( nil @ A ) ) )
@ Xss ) )
= ( map @ ( list @ A ) @ A @ ( hd @ A )
@ ( filter2 @ ( list @ A )
@ ^ [Ys: list @ A] :
( Ys
!= ( nil @ A ) )
@ Xss ) ) ) ).
% transpose_aux_filter_head
thf(fact_4698_hd__take,axiom,
! [A: $tType,J: nat,Xs2: list @ A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ J )
=> ( ( hd @ A @ ( take @ A @ J @ Xs2 ) )
= ( hd @ A @ Xs2 ) ) ) ).
% hd_take
thf(fact_4699_zip__takeWhile__fst,axiom,
! [A: $tType,B: $tType,P: A > $o,Xs2: list @ A,Ys2: list @ B] :
( ( zip @ A @ B @ ( takeWhile @ A @ P @ Xs2 ) @ Ys2 )
= ( takeWhile @ ( product_prod @ A @ B ) @ ( comp @ A @ $o @ ( product_prod @ A @ B ) @ P @ ( product_fst @ A @ B ) ) @ ( zip @ A @ B @ Xs2 @ Ys2 ) ) ) ).
% zip_takeWhile_fst
thf(fact_4700_zip__takeWhile__snd,axiom,
! [A: $tType,B: $tType,Xs2: list @ A,P: B > $o,Ys2: list @ B] :
( ( zip @ A @ B @ Xs2 @ ( takeWhile @ B @ P @ Ys2 ) )
= ( takeWhile @ ( product_prod @ A @ B ) @ ( comp @ B @ $o @ ( product_prod @ A @ B ) @ P @ ( product_snd @ A @ B ) ) @ ( zip @ A @ B @ Xs2 @ Ys2 ) ) ) ).
% zip_takeWhile_snd
thf(fact_4701_hd__zip,axiom,
! [A: $tType,B: $tType,Xs2: list @ A,Ys2: list @ B] :
( ( Xs2
!= ( nil @ A ) )
=> ( ( Ys2
!= ( nil @ B ) )
=> ( ( hd @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs2 @ Ys2 ) )
= ( product_Pair @ A @ B @ ( hd @ A @ Xs2 ) @ ( hd @ B @ Ys2 ) ) ) ) ) ).
% hd_zip
thf(fact_4702_takeWhile__map,axiom,
! [A: $tType,B: $tType,P: A > $o,F: B > A,Xs2: list @ B] :
( ( takeWhile @ A @ P @ ( map @ B @ A @ F @ Xs2 ) )
= ( map @ B @ A @ F @ ( takeWhile @ B @ ( comp @ A @ $o @ B @ P @ F ) @ Xs2 ) ) ) ).
% takeWhile_map
thf(fact_4703_hd__def,axiom,
! [A: $tType] :
( ( hd @ A )
= ( case_list @ A @ A @ ( undefined @ A )
@ ^ [X212: A,X223: list @ A] : X212 ) ) ).
% hd_def
thf(fact_4704_hd__conv__nth,axiom,
! [A: $tType,Xs2: list @ A] :
( ( Xs2
!= ( nil @ A ) )
=> ( ( hd @ A @ Xs2 )
= ( nth @ A @ Xs2 @ ( zero_zero @ nat ) ) ) ) ).
% hd_conv_nth
thf(fact_4705_Nitpick_Osize__list__simp_I1_J,axiom,
! [A: $tType] :
( ( size_list @ A )
= ( ^ [F2: A > nat,Xs: list @ A] :
( if @ nat
@ ( Xs
= ( nil @ A ) )
@ ( zero_zero @ nat )
@ ( suc @ ( plus_plus @ nat @ ( F2 @ ( hd @ A @ Xs ) ) @ ( size_list @ A @ F2 @ ( tl @ A @ Xs ) ) ) ) ) ) ) ).
% Nitpick.size_list_simp(1)
thf(fact_4706_filter__equals__takeWhile__sorted__rev,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F: B > A,Xs2: list @ B,T2: A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( rev @ A @ ( map @ B @ A @ F @ Xs2 ) ) )
=> ( ( filter2 @ B
@ ^ [X2: B] : ( ord_less @ A @ T2 @ ( F @ X2 ) )
@ Xs2 )
= ( takeWhile @ B
@ ^ [X2: B] : ( ord_less @ A @ T2 @ ( F @ X2 ) )
@ Xs2 ) ) ) ) ).
% filter_equals_takeWhile_sorted_rev
thf(fact_4707_extract__def,axiom,
! [A: $tType] :
( ( extract @ A )
= ( ^ [P3: A > $o,Xs: 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 ) ) ) )
@ ^ [Y: A,Ys: list @ A] : ( some @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) @ ( product_Pair @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) @ ( takeWhile @ A @ ( comp @ $o @ $o @ A @ (~) @ P3 ) @ Xs ) @ ( product_Pair @ A @ ( list @ A ) @ Y @ Ys ) ) )
@ ( dropWhile @ A @ ( comp @ $o @ $o @ A @ (~) @ P3 ) @ Xs ) ) ) ) ).
% extract_def
thf(fact_4708_map__of__eq__Some__iff,axiom,
! [B: $tType,A: $tType,Xys: list @ ( product_prod @ A @ B ),X: A,Y3: B] :
( ( distinct @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xys ) )
=> ( ( ( map_of @ A @ B @ Xys @ X )
= ( some @ B @ Y3 ) )
= ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y3 ) @ ( set2 @ ( product_prod @ A @ B ) @ Xys ) ) ) ) ).
% map_of_eq_Some_iff
thf(fact_4709_Some__eq__map__of__iff,axiom,
! [B: $tType,A: $tType,Xys: list @ ( product_prod @ A @ B ),Y3: B,X: A] :
( ( distinct @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xys ) )
=> ( ( ( some @ B @ Y3 )
= ( map_of @ A @ B @ Xys @ X ) )
= ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y3 ) @ ( set2 @ ( product_prod @ A @ B ) @ Xys ) ) ) ) ).
% Some_eq_map_of_iff
thf(fact_4710_map__of__eq__empty__iff,axiom,
! [B: $tType,A: $tType,Xys: list @ ( product_prod @ A @ B )] :
( ( ( map_of @ A @ B @ Xys )
= ( ^ [X2: A] : ( none @ B ) ) )
= ( Xys
= ( nil @ ( product_prod @ A @ B ) ) ) ) ).
% map_of_eq_empty_iff
thf(fact_4711_empty__eq__map__of__iff,axiom,
! [B: $tType,A: $tType,Xys: list @ ( product_prod @ A @ B )] :
( ( ( ^ [X2: A] : ( none @ B ) )
= ( map_of @ A @ B @ Xys ) )
= ( Xys
= ( nil @ ( product_prod @ A @ B ) ) ) ) ).
% empty_eq_map_of_iff
thf(fact_4712_map__of__zip__is__None,axiom,
! [B: $tType,A: $tType,Xs2: list @ A,Ys2: list @ B,X: A] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ B ) @ Ys2 ) )
=> ( ( ( map_of @ A @ B @ ( zip @ A @ B @ Xs2 @ Ys2 ) @ X )
= ( none @ B ) )
= ( ~ ( member @ A @ X @ ( set2 @ A @ Xs2 ) ) ) ) ) ).
% map_of_zip_is_None
thf(fact_4713_map__of__is__SomeI,axiom,
! [A: $tType,B: $tType,Xys: list @ ( product_prod @ A @ B ),X: A,Y3: B] :
( ( distinct @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xys ) )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y3 ) @ ( set2 @ ( product_prod @ A @ B ) @ Xys ) )
=> ( ( map_of @ A @ B @ Xys @ X )
= ( some @ B @ Y3 ) ) ) ) ).
% map_of_is_SomeI
thf(fact_4714_dropWhile__map,axiom,
! [A: $tType,B: $tType,P: A > $o,F: B > A,Xs2: list @ B] :
( ( dropWhile @ A @ P @ ( map @ B @ A @ F @ Xs2 ) )
= ( map @ B @ A @ F @ ( dropWhile @ B @ ( comp @ A @ $o @ B @ P @ F ) @ Xs2 ) ) ) ).
% dropWhile_map
thf(fact_4715_map__of__Cons__code_I1_J,axiom,
! [B: $tType,A: $tType,K: B] :
( ( map_of @ B @ A @ ( nil @ ( product_prod @ B @ A ) ) @ K )
= ( none @ A ) ) ).
% map_of_Cons_code(1)
thf(fact_4716_map__of_Osimps_I1_J,axiom,
! [B: $tType,A: $tType] :
( ( map_of @ A @ B @ ( nil @ ( product_prod @ A @ B ) ) )
= ( ^ [X2: A] : ( none @ B ) ) ) ).
% map_of.simps(1)
thf(fact_4717_remdups__adj__Cons_H,axiom,
! [A: $tType,X: A,Xs2: list @ A] :
( ( remdups_adj @ A @ ( cons @ A @ X @ Xs2 ) )
= ( cons @ A @ X
@ ( remdups_adj @ A
@ ( dropWhile @ A
@ ^ [Y: A] : Y = X
@ Xs2 ) ) ) ) ).
% remdups_adj_Cons'
thf(fact_4718_weak__map__of__SomeI,axiom,
! [A: $tType,B: $tType,K: A,X: B,L: list @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ X ) @ ( set2 @ ( product_prod @ A @ B ) @ L ) )
=> ? [X4: B] :
( ( map_of @ A @ B @ L @ K )
= ( some @ B @ X4 ) ) ) ).
% weak_map_of_SomeI
thf(fact_4719_map__of__SomeD,axiom,
! [A: $tType,B: $tType,Xs2: list @ ( product_prod @ B @ A ),K: B,Y3: A] :
( ( ( map_of @ B @ A @ Xs2 @ K )
= ( some @ A @ Y3 ) )
=> ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ K @ Y3 ) @ ( set2 @ ( product_prod @ B @ A ) @ Xs2 ) ) ) ).
% map_of_SomeD
thf(fact_4720_map__of__Cons__code_I2_J,axiom,
! [C: $tType,B: $tType,L: B,K: B,V: C,Ps: list @ ( product_prod @ B @ C )] :
( ( ( L = K )
=> ( ( map_of @ B @ C @ ( cons @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ L @ V ) @ Ps ) @ K )
= ( some @ C @ V ) ) )
& ( ( L != K )
=> ( ( map_of @ B @ C @ ( cons @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ L @ V ) @ Ps ) @ K )
= ( map_of @ B @ C @ Ps @ K ) ) ) ) ).
% map_of_Cons_code(2)
thf(fact_4721_map__of__filter__in,axiom,
! [B: $tType,A: $tType,Xs2: list @ ( product_prod @ B @ A ),K: B,Z: A,P: B > A > $o] :
( ( ( map_of @ B @ A @ Xs2 @ K )
= ( some @ A @ Z ) )
=> ( ( P @ K @ Z )
=> ( ( map_of @ B @ A @ ( filter2 @ ( product_prod @ B @ A ) @ ( product_case_prod @ B @ A @ $o @ P ) @ Xs2 ) @ K )
= ( some @ A @ Z ) ) ) ) ).
% map_of_filter_in
thf(fact_4722_map__of__zip__is__Some,axiom,
! [A: $tType,B: $tType,Xs2: list @ A,Ys2: list @ B,X: A] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ B ) @ Ys2 ) )
=> ( ( member @ A @ X @ ( set2 @ A @ Xs2 ) )
= ( ? [Y: B] :
( ( map_of @ A @ B @ ( zip @ A @ B @ Xs2 @ Ys2 ) @ X )
= ( some @ B @ Y ) ) ) ) ) ).
% map_of_zip_is_Some
thf(fact_4723_map__of__eq__None__iff,axiom,
! [A: $tType,B: $tType,Xys: list @ ( product_prod @ B @ A ),X: B] :
( ( ( map_of @ B @ A @ Xys @ X )
= ( none @ A ) )
= ( ~ ( member @ B @ X @ ( image2 @ ( product_prod @ B @ A ) @ B @ ( product_fst @ B @ A ) @ ( set2 @ ( product_prod @ B @ A ) @ Xys ) ) ) ) ) ).
% map_of_eq_None_iff
thf(fact_4724_remdups__adj__append__dropWhile,axiom,
! [A: $tType,Xs2: list @ A,Y3: A,Ys2: list @ A] :
( ( remdups_adj @ A @ ( append @ A @ Xs2 @ ( cons @ A @ Y3 @ Ys2 ) ) )
= ( append @ A @ ( remdups_adj @ A @ ( append @ A @ Xs2 @ ( cons @ A @ Y3 @ ( nil @ A ) ) ) )
@ ( remdups_adj @ A
@ ( dropWhile @ A
@ ^ [X2: A] : X2 = Y3
@ Ys2 ) ) ) ) ).
% remdups_adj_append_dropWhile
thf(fact_4725_tl__remdups__adj,axiom,
! [A: $tType,Ys2: list @ A] :
( ( Ys2
!= ( nil @ A ) )
=> ( ( tl @ A @ ( remdups_adj @ A @ Ys2 ) )
= ( remdups_adj @ A
@ ( dropWhile @ A
@ ^ [X2: A] :
( X2
= ( hd @ A @ Ys2 ) )
@ ( tl @ A @ Ys2 ) ) ) ) ) ).
% tl_remdups_adj
thf(fact_4726_map__of__zip__map,axiom,
! [B: $tType,A: $tType,Xs2: list @ A,F: A > B] :
( ( map_of @ A @ B @ ( zip @ A @ B @ Xs2 @ ( map @ A @ B @ F @ Xs2 ) ) )
= ( ^ [X2: A] : ( if @ ( option @ B ) @ ( member @ A @ X2 @ ( set2 @ A @ Xs2 ) ) @ ( some @ B @ ( F @ X2 ) ) @ ( none @ B ) ) ) ) ).
% map_of_zip_map
thf(fact_4727_dropWhile__nth,axiom,
! [A: $tType,J: nat,P: A > $o,Xs2: list @ A] :
( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ ( dropWhile @ A @ P @ Xs2 ) ) )
=> ( ( nth @ A @ ( dropWhile @ A @ P @ Xs2 ) @ J )
= ( nth @ A @ Xs2 @ ( plus_plus @ nat @ J @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P @ Xs2 ) ) ) ) ) ) ).
% dropWhile_nth
thf(fact_4728_find__dropWhile,axiom,
! [A: $tType] :
( ( find @ A )
= ( ^ [P3: A > $o,Xs: list @ A] :
( case_list @ ( option @ A ) @ A @ ( none @ A )
@ ^ [X2: A,Xa3: list @ A] : ( some @ A @ X2 )
@ ( dropWhile @ A @ ( comp @ $o @ $o @ A @ (~) @ P3 ) @ Xs ) ) ) ) ).
% find_dropWhile
thf(fact_4729_dropWhile__neq__rev,axiom,
! [A: $tType,Xs2: list @ A,X: A] :
( ( distinct @ A @ Xs2 )
=> ( ( member @ A @ X @ ( set2 @ A @ Xs2 ) )
=> ( ( dropWhile @ A
@ ^ [Y: A] : Y != X
@ ( rev @ A @ Xs2 ) )
= ( cons @ A @ X
@ ( rev @ A
@ ( takeWhile @ A
@ ^ [Y: A] : Y != X
@ Xs2 ) ) ) ) ) ) ).
% dropWhile_neq_rev
thf(fact_4730_takeWhile__neq__rev,axiom,
! [A: $tType,Xs2: list @ A,X: A] :
( ( distinct @ A @ Xs2 )
=> ( ( member @ A @ X @ ( set2 @ A @ Xs2 ) )
=> ( ( takeWhile @ A
@ ^ [Y: A] : Y != X
@ ( rev @ A @ Xs2 ) )
= ( rev @ A
@ ( tl @ A
@ ( dropWhile @ A
@ ^ [Y: A] : Y != X
@ Xs2 ) ) ) ) ) ) ).
% takeWhile_neq_rev
thf(fact_4731_map__of__zip__nth,axiom,
! [A: $tType,B: $tType,Xs2: list @ A,Ys2: list @ B,I: nat] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ B ) @ Ys2 ) )
=> ( ( distinct @ A @ Xs2 )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ B ) @ Ys2 ) )
=> ( ( map_of @ A @ B @ ( zip @ A @ B @ Xs2 @ Ys2 ) @ ( nth @ A @ Xs2 @ I ) )
= ( some @ B @ ( nth @ B @ Ys2 @ I ) ) ) ) ) ) ).
% map_of_zip_nth
thf(fact_4732_set__map__of__compr,axiom,
! [B: $tType,A: $tType,Xs2: list @ ( product_prod @ A @ B )] :
( ( distinct @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xs2 ) )
=> ( ( set2 @ ( product_prod @ A @ B ) @ Xs2 )
= ( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [K3: A,V6: B] :
( ( map_of @ A @ B @ Xs2 @ K3 )
= ( some @ B @ V6 ) ) ) ) ) ) ).
% set_map_of_compr
thf(fact_4733_map__of__mapk__SomeI,axiom,
! [A: $tType,B: $tType,C: $tType,F: A > B,T2: list @ ( product_prod @ A @ C ),K: A,X: C] :
( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
=> ( ( ( map_of @ A @ C @ T2 @ K )
= ( some @ C @ X ) )
=> ( ( map_of @ B @ C
@ ( map @ ( product_prod @ A @ C ) @ ( product_prod @ B @ C )
@ ( product_case_prod @ A @ C @ ( product_prod @ B @ C )
@ ^ [K3: A] : ( product_Pair @ B @ C @ ( F @ K3 ) ) )
@ T2 )
@ ( F @ K ) )
= ( some @ C @ X ) ) ) ) ).
% map_of_mapk_SomeI
thf(fact_4734_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 )
@ ^ [K3: A] : ( product_Pair @ A @ B @ K3 @ ( F @ K3 ) )
@ Ks ) )
= ( restrict_map @ A @ B @ ( comp @ B @ ( option @ B ) @ A @ ( some @ B ) @ F ) @ ( set2 @ A @ Ks ) ) ) ).
% map_of_map_restrict
thf(fact_4735_convol__image__vimage2p,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,F: C > A,G: D > B,R5: A > B > $o] : ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( image2 @ ( product_prod @ C @ D ) @ ( product_prod @ A @ B ) @ ( bNF_convol @ ( product_prod @ C @ D ) @ A @ B @ ( comp @ C @ A @ ( product_prod @ C @ D ) @ F @ ( product_fst @ C @ D ) ) @ ( comp @ D @ B @ ( product_prod @ C @ D ) @ G @ ( product_snd @ C @ D ) ) ) @ ( collect @ ( product_prod @ C @ D ) @ ( product_case_prod @ C @ D @ $o @ ( bNF_vimage2p @ C @ A @ D @ B @ $o @ F @ G @ R5 ) ) ) ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R5 ) ) ) ).
% convol_image_vimage2p
thf(fact_4736_enumerate__replicate__eq,axiom,
! [A: $tType,N: nat,M: nat,A3: A] :
( ( enumerate @ A @ N @ ( replicate @ A @ M @ A3 ) )
= ( map @ nat @ ( product_prod @ nat @ A )
@ ^ [Q4: nat] : ( product_Pair @ nat @ A @ Q4 @ A3 )
@ ( upt @ N @ ( plus_plus @ nat @ N @ M ) ) ) ) ).
% enumerate_replicate_eq
thf(fact_4737_replicate__eq__replicate,axiom,
! [A: $tType,M: nat,X: A,N: nat,Y3: A] :
( ( ( replicate @ A @ M @ X )
= ( replicate @ A @ N @ Y3 ) )
= ( ( M = N )
& ( ( M
!= ( zero_zero @ nat ) )
=> ( X = Y3 ) ) ) ) ).
% replicate_eq_replicate
thf(fact_4738_restrict__out,axiom,
! [A: $tType,B: $tType,X: A,A5: set @ A,M: A > ( option @ B )] :
( ~ ( member @ A @ X @ A5 )
=> ( ( restrict_map @ A @ B @ M @ A5 @ X )
= ( none @ B ) ) ) ).
% restrict_out
thf(fact_4739_restrict__map__empty,axiom,
! [B: $tType,A: $tType,D5: set @ A] :
( ( restrict_map @ A @ B
@ ^ [X2: A] : ( none @ B )
@ D5 )
= ( ^ [X2: A] : ( none @ B ) ) ) ).
% restrict_map_empty
thf(fact_4740_replicate__empty,axiom,
! [A: $tType,N: nat,X: A] :
( ( ( replicate @ A @ N @ X )
= ( nil @ A ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% replicate_empty
thf(fact_4741_empty__replicate,axiom,
! [A: $tType,N: nat,X: A] :
( ( ( nil @ A )
= ( replicate @ A @ N @ X ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% empty_replicate
thf(fact_4742_Ball__set__replicate,axiom,
! [A: $tType,N: nat,A3: A,P: A > $o] :
( ( ! [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ ( replicate @ A @ N @ A3 ) ) )
=> ( P @ X2 ) ) )
= ( ( P @ A3 )
| ( N
= ( zero_zero @ nat ) ) ) ) ).
% Ball_set_replicate
thf(fact_4743_Bex__set__replicate,axiom,
! [A: $tType,N: nat,A3: A,P: A > $o] :
( ( ? [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ ( replicate @ A @ N @ A3 ) ) )
& ( P @ X2 ) ) )
= ( ( P @ A3 )
& ( N
!= ( zero_zero @ nat ) ) ) ) ).
% Bex_set_replicate
thf(fact_4744_in__set__replicate,axiom,
! [A: $tType,X: A,N: nat,Y3: A] :
( ( member @ A @ X @ ( set2 @ A @ ( replicate @ A @ N @ Y3 ) ) )
= ( ( X = Y3 )
& ( N
!= ( zero_zero @ nat ) ) ) ) ).
% in_set_replicate
thf(fact_4745_restrict__map__to__empty,axiom,
! [B: $tType,A: $tType,M: A > ( option @ B )] :
( ( restrict_map @ A @ B @ M @ ( bot_bot @ ( set @ A ) ) )
= ( ^ [X2: A] : ( none @ B ) ) ) ).
% restrict_map_to_empty
thf(fact_4746_hd__replicate,axiom,
! [A: $tType,N: nat,X: A] :
( ( N
!= ( zero_zero @ nat ) )
=> ( ( hd @ A @ ( replicate @ A @ N @ X ) )
= X ) ) ).
% hd_replicate
thf(fact_4747_tl__replicate,axiom,
! [A: $tType,N: nat,X: A] :
( ( tl @ A @ ( replicate @ A @ N @ X ) )
= ( replicate @ A @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ X ) ) ).
% tl_replicate
thf(fact_4748_zip__replicate,axiom,
! [A: $tType,B: $tType,I: nat,X: A,J: nat,Y3: B] :
( ( zip @ A @ B @ ( replicate @ A @ I @ X ) @ ( replicate @ B @ J @ Y3 ) )
= ( replicate @ ( product_prod @ A @ B ) @ ( ord_min @ nat @ I @ J ) @ ( product_Pair @ A @ B @ X @ Y3 ) ) ) ).
% zip_replicate
thf(fact_4749_set__replicate,axiom,
! [A: $tType,N: nat,X: A] :
( ( N
!= ( zero_zero @ nat ) )
=> ( ( set2 @ A @ ( replicate @ A @ N @ X ) )
= ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% set_replicate
thf(fact_4750_vimage2p__mono,axiom,
! [A: $tType,D: $tType,B: $tType,C: $tType,F: A > B,G: C > D,R5: B > D > $o,X: A,Y3: C,S3: B > D > $o] :
( ( bNF_vimage2p @ A @ B @ C @ D @ $o @ F @ G @ R5 @ X @ Y3 )
=> ( ( ord_less_eq @ ( B > D > $o ) @ R5 @ S3 )
=> ( bNF_vimage2p @ A @ B @ C @ D @ $o @ F @ G @ S3 @ X @ Y3 ) ) ) ).
% vimage2p_mono
thf(fact_4751_vimage2p__rel__fun,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,F: A > C,G: B > D,R5: C > D > $o] : ( bNF_rel_fun @ A @ B @ C @ D @ ( bNF_vimage2p @ A @ C @ B @ D @ $o @ F @ G @ R5 ) @ R5 @ F @ G ) ).
% vimage2p_rel_fun
thf(fact_4752_vimage2p__refl,axiom,
! [A: $tType,B: $tType,R5: A > A > $o,F: B > A,X: B] :
( ! [X4: A] : ( R5 @ X4 @ X4 )
=> ( bNF_vimage2p @ B @ A @ B @ A @ $o @ F @ F @ R5 @ X @ X ) ) ).
% vimage2p_refl
thf(fact_4753_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,R6: D > E > C,X2: A,Y: B] : ( R6 @ ( F2 @ X2 ) @ ( G2 @ Y ) ) ) ) ).
% vimage2p_def
thf(fact_4754_restrict__map__def,axiom,
! [B: $tType,A: $tType] :
( ( restrict_map @ A @ B )
= ( ^ [M5: A > ( option @ B ),A8: set @ A,X2: A] : ( if @ ( option @ B ) @ ( member @ A @ X2 @ A8 ) @ ( M5 @ X2 ) @ ( none @ B ) ) ) ) ).
% restrict_map_def
thf(fact_4755_replicate__0,axiom,
! [A: $tType,X: A] :
( ( replicate @ A @ ( zero_zero @ nat ) @ X )
= ( nil @ A ) ) ).
% replicate_0
thf(fact_4756_replicate__add,axiom,
! [A: $tType,N: nat,M: nat,X: A] :
( ( replicate @ A @ ( plus_plus @ nat @ N @ M ) @ X )
= ( append @ A @ ( replicate @ A @ N @ X ) @ ( replicate @ A @ M @ X ) ) ) ).
% replicate_add
thf(fact_4757_vimage2p__comp,axiom,
! [E: $tType,D: $tType,F3: $tType,A: $tType,C: $tType,B: $tType,G3: $tType,F1: F3 > A,F23: D > F3,G1: G3 > B,G22: E > G3] :
( ( bNF_vimage2p @ D @ A @ E @ B @ C @ ( comp @ F3 @ A @ D @ F1 @ F23 ) @ ( comp @ G3 @ B @ E @ G1 @ G22 ) )
= ( comp @ ( F3 > G3 > C ) @ ( D > E > C ) @ ( A > B > C ) @ ( bNF_vimage2p @ D @ F3 @ E @ G3 @ C @ F23 @ G22 ) @ ( bNF_vimage2p @ F3 @ A @ G3 @ B @ C @ F1 @ G1 ) ) ) ).
% vimage2p_comp
thf(fact_4758_vimage2p__id,axiom,
! [C: $tType,B: $tType,A: $tType,R5: A > B > C] :
( ( bNF_vimage2p @ A @ A @ B @ B @ C @ ( id @ A ) @ ( id @ B ) @ R5 )
= R5 ) ).
% vimage2p_id
thf(fact_4759_map__replicate__const,axiom,
! [B: $tType,A: $tType,K: A,Lst: list @ B] :
( ( map @ B @ A
@ ^ [X2: B] : K
@ Lst )
= ( replicate @ A @ ( size_size @ ( list @ B ) @ Lst ) @ K ) ) ).
% map_replicate_const
thf(fact_4760_replicate__length__filter,axiom,
! [A: $tType,X: A,Xs2: list @ A] :
( ( replicate @ A
@ ( size_size @ ( list @ A )
@ ( filter2 @ A
@ ( ^ [Y6: A,Z2: A] : Y6 = Z2
@ X )
@ Xs2 ) )
@ X )
= ( filter2 @ A
@ ( ^ [Y6: A,Z2: A] : Y6 = Z2
@ X )
@ Xs2 ) ) ).
% replicate_length_filter
thf(fact_4761_map__replicate__trivial,axiom,
! [A: $tType,X: A,I: nat] :
( ( map @ nat @ A
@ ^ [I2: nat] : X
@ ( upt @ ( zero_zero @ nat ) @ I ) )
= ( replicate @ A @ I @ X ) ) ).
% map_replicate_trivial
thf(fact_4762_set__replicate__conv__if,axiom,
! [A: $tType,N: nat,X: A] :
( ( ( N
= ( zero_zero @ nat ) )
=> ( ( set2 @ A @ ( replicate @ A @ N @ X ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ( ( set2 @ A @ ( replicate @ A @ N @ X ) )
= ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% set_replicate_conv_if
thf(fact_4763_remdups__adj__replicate,axiom,
! [A: $tType,N: nat,X: A] :
( ( ( N
= ( zero_zero @ nat ) )
=> ( ( remdups_adj @ A @ ( replicate @ A @ N @ X ) )
= ( nil @ A ) ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ( ( remdups_adj @ A @ ( replicate @ A @ N @ X ) )
= ( cons @ A @ X @ ( nil @ A ) ) ) ) ) ).
% remdups_adj_replicate
thf(fact_4764_zip__replicate1,axiom,
! [A: $tType,B: $tType,N: nat,X: A,Ys2: list @ B] :
( ( zip @ A @ B @ ( replicate @ A @ N @ X ) @ Ys2 )
= ( map @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X ) @ ( take @ B @ N @ Ys2 ) ) ) ).
% zip_replicate1
thf(fact_4765_zip__replicate2,axiom,
! [B: $tType,A: $tType,Xs2: list @ A,N: nat,Y3: B] :
( ( zip @ A @ B @ Xs2 @ ( replicate @ B @ N @ Y3 ) )
= ( map @ A @ ( product_prod @ A @ B )
@ ^ [X2: A] : ( product_Pair @ A @ B @ X2 @ Y3 )
@ ( take @ A @ N @ Xs2 ) ) ) ).
% zip_replicate2
thf(fact_4766_Cons__replicate__eq,axiom,
! [A: $tType,X: A,Xs2: list @ A,N: nat,Y3: A] :
( ( ( cons @ A @ X @ Xs2 )
= ( replicate @ A @ N @ Y3 ) )
= ( ( X = Y3 )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
& ( Xs2
= ( replicate @ A @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ X ) ) ) ) ).
% Cons_replicate_eq
thf(fact_4767_comm__append__is__replicate,axiom,
! [A: $tType,Xs2: list @ A,Ys2: list @ A] :
( ( Xs2
!= ( nil @ A ) )
=> ( ( Ys2
!= ( nil @ A ) )
=> ( ( ( append @ A @ Xs2 @ Ys2 )
= ( append @ A @ Ys2 @ Xs2 ) )
=> ? [N2: nat,Zs3: list @ A] :
( ( ord_less @ nat @ ( one_one @ nat ) @ N2 )
& ( ( concat @ A @ ( replicate @ ( list @ A ) @ N2 @ Zs3 ) )
= ( append @ A @ Xs2 @ Ys2 ) ) ) ) ) ) ).
% comm_append_is_replicate
thf(fact_4768_remdups__adj__singleton__iff,axiom,
! [A: $tType,Xs2: list @ A] :
( ( ( size_size @ ( list @ A ) @ ( remdups_adj @ A @ Xs2 ) )
= ( suc @ ( zero_zero @ nat ) ) )
= ( ( Xs2
!= ( nil @ A ) )
& ( Xs2
= ( replicate @ A @ ( size_size @ ( list @ A ) @ Xs2 ) @ ( hd @ A @ Xs2 ) ) ) ) ) ).
% remdups_adj_singleton_iff
thf(fact_4769_sorted__insort__is__snoc,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs2: list @ A,A3: A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs2 )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs2 ) )
=> ( ord_less_eq @ A @ X4 @ A3 ) )
=> ( ( linorder_insort_key @ A @ A
@ ^ [X2: A] : X2
@ A3
@ Xs2 )
= ( append @ A @ Xs2 @ ( cons @ A @ A3 @ ( nil @ A ) ) ) ) ) ) ) ).
% sorted_insort_is_snoc
thf(fact_4770_list_Oin__rel,axiom,
! [B: $tType,A: $tType] :
( ( list_all2 @ A @ B )
= ( ^ [R6: A > B > $o,A6: list @ A,B5: list @ B] :
? [Z3: list @ ( product_prod @ A @ B )] :
( ( member @ ( list @ ( product_prod @ A @ B ) ) @ Z3
@ ( collect @ ( list @ ( product_prod @ A @ B ) )
@ ^ [X2: list @ ( product_prod @ A @ B )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( set2 @ ( product_prod @ A @ B ) @ X2 ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R6 ) ) ) ) )
& ( ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Z3 )
= A6 )
& ( ( map @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ Z3 )
= B5 ) ) ) ) ).
% list.in_rel
thf(fact_4771_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,Xs: list @ B,K3: A] :
( ( filter2 @ B
@ ^ [Y: B] :
( ( F2 @ Y )
= K3 )
@ ( Sk @ F2 @ Xs ) )
= ( filter2 @ B
@ ^ [Y: B] :
( ( F2 @ Y )
= K3 )
@ Xs ) ) ) ) ) ).
% stable_sort_key_def
thf(fact_4772_comp__fun__commute__insort,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( finite6289374366891150609ommute @ A @ ( list @ A )
@ ( linorder_insort_key @ A @ A
@ ^ [X2: A] : X2 ) ) ) ).
% comp_fun_commute_insort
thf(fact_4773_insort__left__comm,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y3: A,Xs2: list @ A] :
( ( linorder_insort_key @ A @ A
@ ^ [X2: A] : X2
@ X
@ ( linorder_insort_key @ A @ A
@ ^ [X2: A] : X2
@ Y3
@ Xs2 ) )
= ( linorder_insort_key @ A @ A
@ ^ [X2: A] : X2
@ Y3
@ ( linorder_insort_key @ A @ A
@ ^ [X2: A] : X2
@ X
@ Xs2 ) ) ) ) ).
% insort_left_comm
thf(fact_4774_list_Odisc__transfer_I1_J,axiom,
! [A: $tType,B: $tType,R5: A > B > $o] :
( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ $o @ $o @ ( list_all2 @ A @ B @ R5 )
@ ^ [Y6: $o,Z2: $o] : Y6 = Z2
@ ^ [List2: list @ A] :
( List2
= ( nil @ A ) )
@ ^ [List2: list @ B] :
( List2
= ( nil @ B ) ) ) ).
% list.disc_transfer(1)
thf(fact_4775_list_Odisc__transfer_I2_J,axiom,
! [A: $tType,B: $tType,R5: A > B > $o] :
( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ $o @ $o @ ( list_all2 @ A @ B @ R5 )
@ ^ [Y6: $o,Z2: $o] : Y6 = Z2
@ ^ [List2: list @ A] :
( List2
!= ( nil @ A ) )
@ ^ [List2: list @ B] :
( List2
!= ( nil @ B ) ) ) ).
% list.disc_transfer(2)
thf(fact_4776_list_Orel__map_I2_J,axiom,
! [A: $tType,C: $tType,B: $tType,Sa: A > C > $o,X: list @ A,G: B > C,Y3: list @ B] :
( ( list_all2 @ A @ C @ Sa @ X @ ( map @ B @ C @ G @ Y3 ) )
= ( list_all2 @ A @ B
@ ^ [X2: A,Y: B] : ( Sa @ X2 @ ( G @ Y ) )
@ X
@ Y3 ) ) ).
% list.rel_map(2)
thf(fact_4777_list_Orel__map_I1_J,axiom,
! [A: $tType,C: $tType,B: $tType,Sb: C > B > $o,I: A > C,X: list @ A,Y3: list @ B] :
( ( list_all2 @ C @ B @ Sb @ ( map @ A @ C @ I @ X ) @ Y3 )
= ( list_all2 @ A @ B
@ ^ [X2: A] : ( Sb @ ( I @ X2 ) )
@ X
@ Y3 ) ) ).
% list.rel_map(1)
thf(fact_4778_list__all2__map1,axiom,
! [C: $tType,A: $tType,B: $tType,P: A > B > $o,F: C > A,As3: list @ C,Bs: list @ B] :
( ( list_all2 @ A @ B @ P @ ( map @ C @ A @ F @ As3 ) @ Bs )
= ( list_all2 @ C @ B
@ ^ [X2: C] : ( P @ ( F @ X2 ) )
@ As3
@ Bs ) ) ).
% list_all2_map1
thf(fact_4779_list__all2__map2,axiom,
! [A: $tType,B: $tType,C: $tType,P: A > B > $o,As3: list @ A,F: C > B,Bs: list @ C] :
( ( list_all2 @ A @ B @ P @ As3 @ ( map @ C @ B @ F @ Bs ) )
= ( list_all2 @ A @ C
@ ^ [X2: A,Y: C] : ( P @ X2 @ ( F @ Y ) )
@ As3
@ Bs ) ) ).
% list_all2_map2
thf(fact_4780_sorted__list__of__set_Ofold__insort__key_Ocomp__fun__commute__on,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Y3: A,X: A] :
( ( comp @ ( list @ A ) @ ( list @ A ) @ ( list @ A )
@ ( linorder_insort_key @ A @ A
@ ^ [X2: A] : X2
@ Y3 )
@ ( linorder_insort_key @ A @ A
@ ^ [X2: A] : X2
@ X ) )
= ( comp @ ( list @ A ) @ ( list @ A ) @ ( list @ A )
@ ( linorder_insort_key @ A @ A
@ ^ [X2: A] : X2
@ X )
@ ( linorder_insort_key @ A @ A
@ ^ [X2: A] : X2
@ Y3 ) ) ) ) ).
% sorted_list_of_set.fold_insort_key.comp_fun_commute_on
thf(fact_4781_sorted__insort,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Xs2: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A )
@ ( linorder_insort_key @ A @ A
@ ^ [X2: A] : X2
@ X
@ Xs2 ) )
= ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs2 ) ) ) ).
% sorted_insort
thf(fact_4782_product__lists__set,axiom,
! [A: $tType,Xss: list @ ( list @ A )] :
( ( set2 @ ( list @ A ) @ ( product_lists @ A @ Xss ) )
= ( collect @ ( list @ A )
@ ^ [Xs: list @ A] :
( list_all2 @ A @ ( list @ A )
@ ^ [X2: A,Ys: list @ A] : ( member @ A @ X2 @ ( set2 @ A @ Ys ) )
@ Xs
@ Xss ) ) ) ).
% product_lists_set
thf(fact_4783_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_4784_horner__sum__transfer,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType] :
( ( ( comm_semiring_0 @ B )
& ( comm_semiring_0 @ A ) )
=> ! [A5: A > B > $o,B4: C > D > $o] :
( ( A5 @ ( zero_zero @ A ) @ ( zero_zero @ B ) )
=> ( ( bNF_rel_fun @ A @ B @ ( A > A ) @ ( B > B ) @ A5 @ ( bNF_rel_fun @ A @ B @ A @ B @ A5 @ A5 ) @ ( plus_plus @ A ) @ ( plus_plus @ B ) )
=> ( ( bNF_rel_fun @ A @ B @ ( A > A ) @ ( B > B ) @ A5 @ ( bNF_rel_fun @ A @ B @ A @ B @ A5 @ A5 ) @ ( times_times @ A ) @ ( times_times @ B ) )
=> ( bNF_rel_fun @ ( C > A ) @ ( D > B ) @ ( A > ( list @ C ) > A ) @ ( B > ( list @ D ) > B ) @ ( bNF_rel_fun @ C @ D @ A @ B @ B4 @ A5 ) @ ( bNF_rel_fun @ A @ B @ ( ( list @ C ) > A ) @ ( ( list @ D ) > B ) @ A5 @ ( bNF_rel_fun @ ( list @ C ) @ ( list @ D ) @ A @ B @ ( list_all2 @ C @ D @ B4 ) @ A5 ) ) @ ( groups4207007520872428315er_sum @ C @ A ) @ ( groups4207007520872428315er_sum @ D @ B ) ) ) ) ) ) ).
% horner_sum_transfer
thf(fact_4785_sorted__list__of__set_Osorted__key__list__of__set__insert__remove,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( linord4507533701916653071of_set @ A @ ( insert2 @ A @ X @ A5 ) )
= ( linorder_insort_key @ A @ A
@ ^ [X2: A] : X2
@ X
@ ( linord4507533701916653071of_set @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_insert_remove
thf(fact_4786_insort__key__remove1,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [A3: B,Xs2: list @ B,F: B > A] :
( ( member @ B @ A3 @ ( set2 @ B @ Xs2 ) )
=> ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F @ Xs2 ) )
=> ( ( ( hd @ B
@ ( filter2 @ B
@ ^ [X2: B] :
( ( F @ A3 )
= ( F @ X2 ) )
@ Xs2 ) )
= A3 )
=> ( ( linorder_insort_key @ B @ A @ F @ A3 @ ( remove1 @ B @ A3 @ Xs2 ) )
= Xs2 ) ) ) ) ) ).
% insort_key_remove1
thf(fact_4787_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_4788_prod__list_ONil,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ( ( groups5270119922927024881d_list @ A @ ( nil @ A ) )
= ( one_one @ A ) ) ) ).
% prod_list.Nil
thf(fact_4789_sorted__list__of__set_Osorted__key__list__of__set__insert,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ~ ( member @ A @ X @ A5 )
=> ( ( linord4507533701916653071of_set @ A @ ( insert2 @ A @ X @ A5 ) )
= ( linorder_insort_key @ A @ A
@ ^ [X2: A] : X2
@ X
@ ( linord4507533701916653071of_set @ A @ A5 ) ) ) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_insert
thf(fact_4790_prod__list__zero__iff,axiom,
! [A: $tType] :
( ( ( semiring_1 @ A )
& ( semiri3467727345109120633visors @ A ) )
=> ! [Xs2: list @ A] :
( ( ( groups5270119922927024881d_list @ A @ Xs2 )
= ( zero_zero @ A ) )
= ( member @ A @ ( zero_zero @ A ) @ ( set2 @ A @ Xs2 ) ) ) ) ).
% prod_list_zero_iff
thf(fact_4791_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
@ ^ [X2: A] : X2 )
@ ( nil @ A ) ) ) ) ).
% sorted_list_of_set.fold_insort_key.eq_fold
thf(fact_4792_fold__remove1__split,axiom,
! [B: $tType,A: $tType,Xs2: list @ A,F: A > B > B,X: A] :
( ! [X4: A,Y4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs2 ) )
=> ( ( member @ A @ Y4 @ ( set2 @ A @ Xs2 ) )
=> ( ( comp @ B @ B @ B @ ( F @ X4 ) @ ( F @ Y4 ) )
= ( comp @ B @ B @ B @ ( F @ Y4 ) @ ( F @ X4 ) ) ) ) )
=> ( ( member @ A @ X @ ( set2 @ A @ Xs2 ) )
=> ( ( fold @ A @ B @ F @ Xs2 )
= ( comp @ B @ B @ B @ ( fold @ A @ B @ F @ ( remove1 @ A @ X @ Xs2 ) ) @ ( F @ X ) ) ) ) ) ).
% fold_remove1_split
thf(fact_4793_prod__list_Oeq__foldr,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ( ( groups5270119922927024881d_list @ A )
= ( ^ [Xs: list @ A] : ( foldr @ A @ A @ ( times_times @ A ) @ Xs @ ( one_one @ A ) ) ) ) ) ).
% prod_list.eq_foldr
thf(fact_4794_length__remove1,axiom,
! [A: $tType,X: A,Xs2: list @ A] :
( ( ( member @ A @ X @ ( set2 @ A @ Xs2 ) )
=> ( ( size_size @ ( list @ A ) @ ( remove1 @ A @ X @ Xs2 ) )
= ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ ( one_one @ nat ) ) ) )
& ( ~ ( member @ A @ X @ ( set2 @ A @ Xs2 ) )
=> ( ( size_size @ ( list @ A ) @ ( remove1 @ A @ X @ Xs2 ) )
= ( size_size @ ( list @ A ) @ Xs2 ) ) ) ) ).
% length_remove1
thf(fact_4795_sum__list__map__remove1,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [X: B,Xs2: list @ B,F: B > A] :
( ( member @ B @ X @ ( set2 @ B @ Xs2 ) )
=> ( ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F @ Xs2 ) )
= ( plus_plus @ A @ ( F @ X ) @ ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F @ ( remove1 @ B @ X @ Xs2 ) ) ) ) ) ) ) ).
% sum_list_map_remove1
thf(fact_4796_insort__remove1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,Xs2: list @ A] :
( ( member @ A @ A3 @ ( set2 @ A @ Xs2 ) )
=> ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs2 )
=> ( ( linorder_insort_key @ A @ A
@ ^ [X2: A] : X2
@ A3
@ ( remove1 @ A @ A3 @ Xs2 ) )
= Xs2 ) ) ) ) ).
% insort_remove1
thf(fact_4797_sorted__list__of__set_Ofold__insort__key_Oremove,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( member @ A @ X @ A5 )
=> ( ( linord4507533701916653071of_set @ A @ A5 )
= ( linorder_insort_key @ A @ A
@ ^ [X2: A] : X2
@ X
@ ( linord4507533701916653071of_set @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ).
% sorted_list_of_set.fold_insort_key.remove
thf(fact_4798_sorted__list__of__set__def,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( linord4507533701916653071of_set @ A )
= ( linord144544945434240204of_set @ A @ A
@ ^ [X2: A] : X2 ) ) ) ).
% sorted_list_of_set_def
thf(fact_4799_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_4800_restrict__upd__same,axiom,
! [B: $tType,A: $tType,M: A > ( option @ B ),X: A,Y3: B] :
( ( restrict_map @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M @ X @ ( some @ B @ Y3 ) ) @ ( uminus_uminus @ ( set @ A ) @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( restrict_map @ A @ B @ M @ ( uminus_uminus @ ( set @ A ) @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% restrict_upd_same
thf(fact_4801_fun__upd__upd,axiom,
! [A: $tType,B: $tType,F: A > B,X: A,Y3: B,Z: B] :
( ( fun_upd @ A @ B @ ( fun_upd @ A @ B @ F @ X @ Y3 ) @ X @ Z )
= ( fun_upd @ A @ B @ F @ X @ Z ) ) ).
% fun_upd_upd
thf(fact_4802_fun__upd__triv,axiom,
! [B: $tType,A: $tType,F: A > B,X: A] :
( ( fun_upd @ A @ B @ F @ X @ ( F @ X ) )
= F ) ).
% fun_upd_triv
thf(fact_4803_fun__upd__apply,axiom,
! [A: $tType,B: $tType] :
( ( fun_upd @ B @ A )
= ( ^ [F2: B > A,X2: B,Y: A,Z3: B] : ( if @ A @ ( Z3 = X2 ) @ Y @ ( F2 @ Z3 ) ) ) ) ).
% fun_upd_apply
thf(fact_4804_empty__upd__none,axiom,
! [B: $tType,A: $tType,X: A] :
( ( fun_upd @ A @ ( option @ B )
@ ^ [X2: A] : ( none @ B )
@ X
@ ( none @ B ) )
= ( ^ [X2: A] : ( none @ B ) ) ) ).
% empty_upd_none
thf(fact_4805_image__map__upd,axiom,
! [B: $tType,A: $tType,X: A,A5: set @ A,M: A > ( option @ B ),Y3: B] :
( ~ ( member @ A @ X @ A5 )
=> ( ( image2 @ A @ ( option @ B ) @ ( fun_upd @ A @ ( option @ B ) @ M @ X @ ( some @ B @ Y3 ) ) @ A5 )
= ( image2 @ A @ ( option @ B ) @ M @ A5 ) ) ) ).
% image_map_upd
thf(fact_4806_fun__upd__None__restrict,axiom,
! [B: $tType,A: $tType,X: A,D5: set @ A,M: A > ( option @ B )] :
( ( ( member @ A @ X @ D5 )
=> ( ( fun_upd @ A @ ( option @ B ) @ ( restrict_map @ A @ B @ M @ D5 ) @ X @ ( none @ B ) )
= ( restrict_map @ A @ B @ M @ ( minus_minus @ ( set @ A ) @ D5 @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) )
& ( ~ ( member @ A @ X @ D5 )
=> ( ( fun_upd @ A @ ( option @ B ) @ ( restrict_map @ A @ B @ M @ D5 ) @ X @ ( none @ B ) )
= ( restrict_map @ A @ B @ M @ D5 ) ) ) ) ).
% fun_upd_None_restrict
thf(fact_4807_map__upd__Some__unfold,axiom,
! [B: $tType,A: $tType,M: B > ( option @ A ),A3: B,B2: A,X: B,Y3: A] :
( ( ( fun_upd @ B @ ( option @ A ) @ M @ A3 @ ( some @ A @ B2 ) @ X )
= ( some @ A @ Y3 ) )
= ( ( ( X = A3 )
& ( B2 = Y3 ) )
| ( ( X != A3 )
& ( ( M @ X )
= ( some @ A @ Y3 ) ) ) ) ) ).
% map_upd_Some_unfold
thf(fact_4808_map__upd__triv,axiom,
! [A: $tType,B: $tType,T2: B > ( option @ A ),K: B,X: A] :
( ( ( T2 @ K )
= ( some @ A @ X ) )
=> ( ( fun_upd @ B @ ( option @ A ) @ T2 @ K @ ( some @ A @ X ) )
= T2 ) ) ).
% map_upd_triv
thf(fact_4809_map__upd__eqD1,axiom,
! [A: $tType,B: $tType,M: A > ( option @ B ),A3: A,X: B,N: A > ( option @ B ),Y3: B] :
( ( ( fun_upd @ A @ ( option @ B ) @ M @ A3 @ ( some @ B @ X ) )
= ( fun_upd @ A @ ( option @ B ) @ N @ A3 @ ( some @ B @ Y3 ) ) )
=> ( X = Y3 ) ) ).
% map_upd_eqD1
thf(fact_4810_fun__upd__def,axiom,
! [B: $tType,A: $tType] :
( ( fun_upd @ A @ B )
= ( ^ [F2: A > B,A6: A,B5: B,X2: A] : ( if @ B @ ( X2 = A6 ) @ B5 @ ( F2 @ X2 ) ) ) ) ).
% fun_upd_def
thf(fact_4811_fun__upd__eqD,axiom,
! [A: $tType,B: $tType,F: A > B,X: A,Y3: B,G: A > B,Z: B] :
( ( ( fun_upd @ A @ B @ F @ X @ Y3 )
= ( fun_upd @ A @ B @ G @ X @ Z ) )
=> ( Y3 = Z ) ) ).
% fun_upd_eqD
thf(fact_4812_fun__upd__idem,axiom,
! [A: $tType,B: $tType,F: B > A,X: B,Y3: A] :
( ( ( F @ X )
= Y3 )
=> ( ( fun_upd @ B @ A @ F @ X @ Y3 )
= F ) ) ).
% fun_upd_idem
thf(fact_4813_fun__upd__same,axiom,
! [B: $tType,A: $tType,F: B > A,X: B,Y3: A] :
( ( fun_upd @ B @ A @ F @ X @ Y3 @ X )
= Y3 ) ).
% fun_upd_same
thf(fact_4814_fun__upd__other,axiom,
! [B: $tType,A: $tType,Z: A,X: A,F: A > B,Y3: B] :
( ( Z != X )
=> ( ( fun_upd @ A @ B @ F @ X @ Y3 @ Z )
= ( F @ Z ) ) ) ).
% fun_upd_other
thf(fact_4815_fun__upd__twist,axiom,
! [A: $tType,B: $tType,A3: A,C2: A,M: A > B,B2: B,D3: B] :
( ( A3 != C2 )
=> ( ( fun_upd @ A @ B @ ( fun_upd @ A @ B @ M @ A3 @ B2 ) @ C2 @ D3 )
= ( fun_upd @ A @ B @ ( fun_upd @ A @ B @ M @ C2 @ D3 ) @ A3 @ B2 ) ) ) ).
% fun_upd_twist
thf(fact_4816_fun__upd__idem__iff,axiom,
! [A: $tType,B: $tType,F: A > B,X: A,Y3: B] :
( ( ( fun_upd @ A @ B @ F @ X @ Y3 )
= F )
= ( ( F @ X )
= Y3 ) ) ).
% fun_upd_idem_iff
thf(fact_4817_fun__upd__comp,axiom,
! [A: $tType,B: $tType,C: $tType,F: C > B,G: A > C,X: A,Y3: C] :
( ( comp @ C @ B @ A @ F @ ( fun_upd @ A @ C @ G @ X @ Y3 ) )
= ( fun_upd @ A @ B @ ( comp @ C @ B @ A @ F @ G ) @ X @ ( F @ Y3 ) ) ) ).
% fun_upd_comp
thf(fact_4818_finite__update__induct,axiom,
! [B: $tType,A: $tType,F: A > B,C2: B,P: ( A > B ) > $o] :
( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [A6: A] :
( ( F @ A6 )
!= C2 ) ) )
=> ( ( P
@ ^ [A6: A] : C2 )
=> ( ! [A4: A,B3: B,F4: A > B] :
( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [C4: A] :
( ( F4 @ C4 )
!= C2 ) ) )
=> ( ( ( F4 @ A4 )
= C2 )
=> ( ( B3 != C2 )
=> ( ( P @ F4 )
=> ( P @ ( fun_upd @ A @ B @ F4 @ A4 @ B3 ) ) ) ) ) )
=> ( P @ F ) ) ) ) ).
% finite_update_induct
thf(fact_4819_map__upd__nonempty,axiom,
! [B: $tType,A: $tType,T2: A > ( option @ B ),K: A,X: B] :
( ( fun_upd @ A @ ( option @ B ) @ T2 @ K @ ( some @ B @ X ) )
!= ( ^ [X2: A] : ( none @ B ) ) ) ).
% map_upd_nonempty
thf(fact_4820_inj__on__fun__updI,axiom,
! [B: $tType,A: $tType,F: A > B,A5: set @ A,Y3: B,X: A] :
( ( inj_on @ A @ B @ F @ A5 )
=> ( ~ ( member @ B @ Y3 @ ( image2 @ A @ B @ F @ A5 ) )
=> ( inj_on @ A @ B @ ( fun_upd @ A @ B @ F @ X @ Y3 ) @ A5 ) ) ) ).
% inj_on_fun_updI
thf(fact_4821_finite__range__updI,axiom,
! [A: $tType,B: $tType,F: B > ( option @ A ),A3: B,B2: A] :
( ( finite_finite2 @ ( option @ A ) @ ( image2 @ B @ ( option @ A ) @ F @ ( top_top @ ( set @ B ) ) ) )
=> ( finite_finite2 @ ( option @ A ) @ ( image2 @ B @ ( option @ A ) @ ( fun_upd @ B @ ( option @ A ) @ F @ A3 @ ( some @ A @ B2 ) ) @ ( top_top @ ( set @ B ) ) ) ) ) ).
% finite_range_updI
thf(fact_4822_map__of__zip__upd,axiom,
! [A: $tType,B: $tType,Ys2: list @ B,Xs2: list @ A,Zs: list @ B,X: A,Y3: B,Z: B] :
( ( ( size_size @ ( list @ B ) @ Ys2 )
= ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( ( size_size @ ( list @ B ) @ Zs )
= ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ~ ( member @ A @ X @ ( set2 @ A @ Xs2 ) )
=> ( ( ( fun_upd @ A @ ( option @ B ) @ ( map_of @ A @ B @ ( zip @ A @ B @ Xs2 @ Ys2 ) ) @ X @ ( some @ B @ Y3 ) )
= ( fun_upd @ A @ ( option @ B ) @ ( map_of @ A @ B @ ( zip @ A @ B @ Xs2 @ Zs ) ) @ X @ ( some @ B @ Z ) ) )
=> ( ( map_of @ A @ B @ ( zip @ A @ B @ Xs2 @ Ys2 ) )
= ( map_of @ A @ B @ ( zip @ A @ B @ Xs2 @ Zs ) ) ) ) ) ) ) ).
% map_of_zip_upd
thf(fact_4823_restrict__complement__singleton__eq,axiom,
! [A: $tType,B: $tType,F: A > ( option @ B ),X: A] :
( ( restrict_map @ A @ B @ F @ ( uminus_uminus @ ( set @ A ) @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( fun_upd @ A @ ( option @ B ) @ F @ X @ ( none @ B ) ) ) ).
% restrict_complement_singleton_eq
thf(fact_4824_fun__upd__image,axiom,
! [A: $tType,B: $tType,X: B,A5: set @ B,F: B > A,Y3: A] :
( ( ( member @ B @ X @ A5 )
=> ( ( image2 @ B @ A @ ( fun_upd @ B @ A @ F @ X @ Y3 ) @ A5 )
= ( insert2 @ A @ Y3 @ ( image2 @ B @ A @ F @ ( minus_minus @ ( set @ B ) @ A5 @ ( insert2 @ B @ X @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) )
& ( ~ ( member @ B @ X @ A5 )
=> ( ( image2 @ B @ A @ ( fun_upd @ B @ A @ F @ X @ Y3 ) @ A5 )
= ( image2 @ B @ A @ F @ A5 ) ) ) ) ).
% fun_upd_image
thf(fact_4825_map__of_Osimps_I2_J,axiom,
! [B: $tType,A: $tType,P4: product_prod @ A @ B,Ps: list @ ( product_prod @ A @ B )] :
( ( map_of @ A @ B @ ( cons @ ( product_prod @ A @ B ) @ P4 @ Ps ) )
= ( fun_upd @ A @ ( option @ B ) @ ( map_of @ A @ B @ Ps ) @ ( product_fst @ A @ B @ P4 ) @ ( some @ B @ ( product_snd @ A @ B @ P4 ) ) ) ) ).
% map_of.simps(2)
thf(fact_4826_map__upds__append1,axiom,
! [B: $tType,A: $tType,Xs2: list @ A,Ys2: list @ B,M: A > ( option @ B ),X: A] :
( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ ( size_size @ ( list @ B ) @ Ys2 ) )
=> ( ( map_upds @ A @ B @ M @ ( append @ A @ Xs2 @ ( cons @ A @ X @ ( nil @ A ) ) ) @ Ys2 )
= ( fun_upd @ A @ ( option @ B ) @ ( map_upds @ A @ B @ M @ Xs2 @ Ys2 ) @ X @ ( some @ B @ ( nth @ B @ Ys2 @ ( size_size @ ( list @ A ) @ Xs2 ) ) ) ) ) ) ).
% map_upds_append1
thf(fact_4827_graph__map__upd,axiom,
! [A: $tType,B: $tType,M: A > ( option @ B ),K: A,V: B] :
( ( graph @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M @ K @ ( some @ B @ V ) ) )
= ( insert2 @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ V ) @ ( graph @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M @ K @ ( none @ B ) ) ) ) ) ).
% graph_map_upd
thf(fact_4828_ran__map__upd,axiom,
! [A: $tType,B: $tType,M: B > ( option @ A ),A3: B,B2: A] :
( ( ( M @ A3 )
= ( none @ A ) )
=> ( ( ran @ B @ A @ ( fun_upd @ B @ ( option @ A ) @ M @ A3 @ ( some @ A @ B2 ) ) )
= ( insert2 @ A @ B2 @ ( ran @ B @ A @ M ) ) ) ) ).
% ran_map_upd
thf(fact_4829_ran__empty,axiom,
! [B: $tType,A: $tType] :
( ( ran @ B @ A
@ ^ [X2: B] : ( none @ A ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% ran_empty
thf(fact_4830_graph__empty,axiom,
! [B: $tType,A: $tType] :
( ( graph @ A @ B
@ ^ [X2: A] : ( none @ B ) )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% graph_empty
thf(fact_4831_map__upds__Cons,axiom,
! [A: $tType,B: $tType,M: A > ( option @ B ),A3: A,As3: list @ A,B2: B,Bs: list @ B] :
( ( map_upds @ A @ B @ M @ ( cons @ A @ A3 @ As3 ) @ ( cons @ B @ B2 @ Bs ) )
= ( map_upds @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M @ A3 @ ( some @ B @ B2 ) ) @ As3 @ Bs ) ) ).
% map_upds_Cons
thf(fact_4832_map__upds__twist,axiom,
! [A: $tType,B: $tType,A3: A,As3: list @ A,M: A > ( option @ B ),B2: B,Bs: list @ B] :
( ~ ( member @ A @ A3 @ ( set2 @ A @ As3 ) )
=> ( ( map_upds @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M @ A3 @ ( some @ B @ B2 ) ) @ As3 @ Bs )
= ( fun_upd @ A @ ( option @ B ) @ ( map_upds @ A @ B @ M @ As3 @ Bs ) @ A3 @ ( some @ B @ B2 ) ) ) ) ).
% map_upds_twist
thf(fact_4833_ranI,axiom,
! [A: $tType,B: $tType,M: B > ( option @ A ),A3: B,B2: A] :
( ( ( M @ A3 )
= ( some @ A @ B2 ) )
=> ( member @ A @ B2 @ ( ran @ B @ A @ M ) ) ) ).
% ranI
thf(fact_4834_in__graphI,axiom,
! [A: $tType,B: $tType,M: B > ( option @ A ),K: B,V: A] :
( ( ( M @ K )
= ( some @ A @ V ) )
=> ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ K @ V ) @ ( graph @ B @ A @ M ) ) ) ).
% in_graphI
thf(fact_4835_in__graphD,axiom,
! [A: $tType,B: $tType,K: A,V: B,M: A > ( option @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ V ) @ ( graph @ A @ B @ M ) )
=> ( ( M @ K )
= ( some @ B @ V ) ) ) ).
% in_graphD
thf(fact_4836_graph__restrictD_I1_J,axiom,
! [B: $tType,A: $tType,K: A,V: B,M: A > ( option @ B ),A5: set @ A] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ V ) @ ( graph @ A @ B @ ( restrict_map @ A @ B @ M @ A5 ) ) )
=> ( member @ A @ K @ A5 ) ) ).
% graph_restrictD(1)
thf(fact_4837_ran__restrictD,axiom,
! [B: $tType,A: $tType,Y3: A,M: B > ( option @ A ),A5: set @ B] :
( ( member @ A @ Y3 @ ( ran @ B @ A @ ( restrict_map @ B @ A @ M @ A5 ) ) )
=> ? [X4: B] :
( ( member @ B @ X4 @ A5 )
& ( ( M @ X4 )
= ( some @ A @ Y3 ) ) ) ) ).
% ran_restrictD
thf(fact_4838_ran__def,axiom,
! [B: $tType,A: $tType] :
( ( ran @ A @ B )
= ( ^ [M5: A > ( option @ B )] :
( collect @ B
@ ^ [B5: B] :
? [A6: A] :
( ( M5 @ A6 )
= ( some @ B @ B5 ) ) ) ) ) ).
% ran_def
thf(fact_4839_graph__restrictD_I2_J,axiom,
! [A: $tType,B: $tType,K: A,V: B,M: A > ( option @ B ),A5: set @ A] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ V ) @ ( graph @ A @ B @ ( restrict_map @ A @ B @ M @ A5 ) ) )
=> ( ( M @ K )
= ( some @ B @ V ) ) ) ).
% graph_restrictD(2)
thf(fact_4840_graph__def,axiom,
! [B: $tType,A: $tType] :
( ( graph @ A @ B )
= ( ^ [M5: A > ( option @ B )] :
( collect @ ( product_prod @ A @ B )
@ ^ [Uu: product_prod @ A @ B] :
? [A6: A,B5: B] :
( ( Uu
= ( product_Pair @ A @ B @ A6 @ B5 ) )
& ( ( M5 @ A6 )
= ( some @ B @ B5 ) ) ) ) ) ) ).
% graph_def
thf(fact_4841_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 )
@ ^ [E4: product_prod @ A @ B] :
( ( member @ ( product_prod @ A @ B ) @ E4 @ ( graph @ A @ B @ M ) )
& ( ( product_fst @ A @ B @ E4 )
!= K ) ) ) ) ).
% graph_fun_upd_None
thf(fact_4842_map__upd__upds__conv__if,axiom,
! [A: $tType,B: $tType,X: A,Ys2: list @ B,Xs2: list @ A,F: A > ( option @ B ),Y3: B] :
( ( ( member @ A @ X @ ( set2 @ A @ ( take @ A @ ( size_size @ ( list @ B ) @ Ys2 ) @ Xs2 ) ) )
=> ( ( map_upds @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ F @ X @ ( some @ B @ Y3 ) ) @ Xs2 @ Ys2 )
= ( map_upds @ A @ B @ F @ Xs2 @ Ys2 ) ) )
& ( ~ ( member @ A @ X @ ( set2 @ A @ ( take @ A @ ( size_size @ ( list @ B ) @ Ys2 ) @ Xs2 ) ) )
=> ( ( map_upds @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ F @ X @ ( some @ B @ Y3 ) ) @ Xs2 @ Ys2 )
= ( fun_upd @ A @ ( option @ B ) @ ( map_upds @ A @ B @ F @ Xs2 @ Ys2 ) @ X @ ( some @ B @ Y3 ) ) ) ) ) ).
% map_upd_upds_conv_if
thf(fact_4843_ran__map__upd__Some,axiom,
! [B: $tType,A: $tType,M: B > ( option @ A ),X: B,Y3: A,Z: A] :
( ( ( M @ X )
= ( some @ A @ Y3 ) )
=> ( ( inj_on @ B @ ( option @ A ) @ M @ ( dom @ B @ A @ M ) )
=> ( ~ ( member @ A @ Z @ ( ran @ B @ A @ M ) )
=> ( ( ran @ B @ A @ ( fun_upd @ B @ ( option @ A ) @ M @ X @ ( some @ A @ Z ) ) )
= ( sup_sup @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ ( ran @ B @ A @ M ) @ ( insert2 @ A @ Y3 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( insert2 @ A @ Z @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% ran_map_upd_Some
thf(fact_4844_map__upds__fold__map__upd,axiom,
! [B: $tType,A: $tType] :
( ( map_upds @ A @ B )
= ( ^ [M5: A > ( option @ B ),Ks2: list @ A,Vs2: list @ B] :
( foldl @ ( A > ( option @ B ) ) @ ( product_prod @ A @ B )
@ ^ [N5: A > ( option @ B )] :
( product_case_prod @ A @ B @ ( A > ( option @ B ) )
@ ^ [K3: A,V6: B] : ( fun_upd @ A @ ( option @ B ) @ N5 @ K3 @ ( some @ B @ V6 ) ) )
@ M5
@ ( zip @ A @ B @ Ks2 @ Vs2 ) ) ) ) ).
% map_upds_fold_map_upd
thf(fact_4845_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_4846_image__add__greaterThanAtMost,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( image2 @ A @ A @ ( plus_plus @ A @ C2 ) @ ( set_or3652927894154168847AtMost @ A @ A3 @ B2 ) )
= ( set_or3652927894154168847AtMost @ A @ ( plus_plus @ A @ C2 @ A3 ) @ ( plus_plus @ A @ C2 @ B2 ) ) ) ) ).
% image_add_greaterThanAtMost
thf(fact_4847_dom__eq__empty__conv,axiom,
! [B: $tType,A: $tType,F: A > ( option @ B )] :
( ( ( dom @ A @ B @ F )
= ( bot_bot @ ( set @ A ) ) )
= ( F
= ( ^ [X2: A] : ( none @ B ) ) ) ) ).
% dom_eq_empty_conv
thf(fact_4848_fun__upd__None__if__notin__dom,axiom,
! [B: $tType,A: $tType,K: A,M: A > ( option @ B )] :
( ~ ( member @ A @ K @ ( dom @ A @ B @ M ) )
=> ( ( fun_upd @ A @ ( option @ B ) @ M @ K @ ( none @ B ) )
= M ) ) ).
% fun_upd_None_if_notin_dom
thf(fact_4849_dom__empty,axiom,
! [B: $tType,A: $tType] :
( ( dom @ A @ B
@ ^ [X2: A] : ( none @ B ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% dom_empty
thf(fact_4850_dom__const,axiom,
! [B: $tType,A: $tType,F: A > B] :
( ( dom @ A @ B
@ ^ [X2: A] : ( some @ B @ ( F @ X2 ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% dom_const
thf(fact_4851_dom__fun__upd,axiom,
! [B: $tType,A: $tType,Y3: option @ B,F: A > ( option @ B ),X: A] :
( ( ( Y3
= ( none @ B ) )
=> ( ( dom @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ F @ X @ Y3 ) )
= ( minus_minus @ ( set @ A ) @ ( dom @ A @ B @ F ) @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) )
& ( ( Y3
!= ( none @ B ) )
=> ( ( dom @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ F @ X @ Y3 ) )
= ( insert2 @ A @ X @ ( dom @ A @ B @ F ) ) ) ) ) ).
% dom_fun_upd
thf(fact_4852_domI,axiom,
! [A: $tType,B: $tType,M: B > ( option @ A ),A3: B,B2: A] :
( ( ( M @ A3 )
= ( some @ A @ B2 ) )
=> ( member @ B @ A3 @ ( dom @ B @ A @ M ) ) ) ).
% domI
thf(fact_4853_domD,axiom,
! [A: $tType,B: $tType,A3: A,M: A > ( option @ B )] :
( ( member @ A @ A3 @ ( dom @ A @ B @ M ) )
=> ? [B3: B] :
( ( M @ A3 )
= ( some @ B @ B3 ) ) ) ).
% domD
thf(fact_4854_dom__def,axiom,
! [B: $tType,A: $tType] :
( ( dom @ A @ B )
= ( ^ [M5: A > ( option @ B )] :
( collect @ A
@ ^ [A6: A] :
( ( M5 @ A6 )
!= ( none @ B ) ) ) ) ) ).
% dom_def
thf(fact_4855_domIff,axiom,
! [A: $tType,B: $tType,A3: A,M: A > ( option @ B )] :
( ( member @ A @ A3 @ ( dom @ A @ B @ M ) )
= ( ( M @ A3 )
!= ( none @ B ) ) ) ).
% domIff
thf(fact_4856_insert__dom,axiom,
! [A: $tType,B: $tType,F: B > ( option @ A ),X: B,Y3: A] :
( ( ( F @ X )
= ( some @ A @ Y3 ) )
=> ( ( insert2 @ B @ X @ ( dom @ B @ A @ F ) )
= ( dom @ B @ A @ F ) ) ) ).
% insert_dom
thf(fact_4857_foldl__map,axiom,
! [A: $tType,B: $tType,C: $tType,G: A > B > A,A3: A,F: C > B,Xs2: list @ C] :
( ( foldl @ A @ B @ G @ A3 @ ( map @ C @ B @ F @ Xs2 ) )
= ( foldl @ A @ C
@ ^ [A6: A,X2: C] : ( G @ A6 @ ( F @ X2 ) )
@ A3
@ Xs2 ) ) ).
% foldl_map
thf(fact_4858_dom__if,axiom,
! [B: $tType,A: $tType,P: A > $o,F: A > ( option @ B ),G: A > ( option @ B )] :
( ( dom @ A @ B
@ ^ [X2: A] : ( if @ ( option @ B ) @ ( P @ X2 ) @ ( F @ X2 ) @ ( G @ X2 ) ) )
= ( 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
@ ^ [X2: A] :
~ ( P @ X2 ) ) ) ) ) ).
% dom_if
thf(fact_4859_foldl__conv__fold,axiom,
! [B: $tType,A: $tType] :
( ( foldl @ A @ B )
= ( ^ [F2: A > B > A,S2: A,Xs: list @ B] :
( fold @ B @ A
@ ^ [X2: B,T3: A] : ( F2 @ T3 @ X2 )
@ Xs
@ S2 ) ) ) ).
% foldl_conv_fold
thf(fact_4860_finite__map__freshness,axiom,
! [A: $tType,B: $tType,F: A > ( option @ B )] :
( ( finite_finite2 @ A @ ( dom @ A @ B @ F ) )
=> ( ~ ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) )
=> ? [X4: A] :
( ( F @ X4 )
= ( none @ B ) ) ) ) ).
% finite_map_freshness
thf(fact_4861_dom__minus,axiom,
! [A: $tType,B: $tType,F: B > ( option @ A ),X: B,A5: set @ B] :
( ( ( F @ X )
= ( none @ A ) )
=> ( ( minus_minus @ ( set @ B ) @ ( dom @ B @ A @ F ) @ ( insert2 @ B @ X @ A5 ) )
= ( minus_minus @ ( set @ B ) @ ( dom @ B @ A @ F ) @ A5 ) ) ) ).
% dom_minus
thf(fact_4862_finite__set__of__finite__maps,axiom,
! [B: $tType,A: $tType,A5: set @ A,B4: set @ B] :
( ( finite_finite2 @ A @ A5 )
=> ( ( finite_finite2 @ B @ B4 )
=> ( finite_finite2 @ ( A > ( option @ B ) )
@ ( collect @ ( A > ( option @ B ) )
@ ^ [M5: A > ( option @ B )] :
( ( ( dom @ A @ B @ M5 )
= A5 )
& ( ord_less_eq @ ( set @ B ) @ ( ran @ A @ B @ M5 ) @ B4 ) ) ) ) ) ) ).
% finite_set_of_finite_maps
thf(fact_4863_foldl__conv__foldr,axiom,
! [B: $tType,A: $tType] :
( ( foldl @ A @ B )
= ( ^ [F2: A > B > A,A6: A,Xs: list @ B] :
( foldr @ B @ A
@ ^ [X2: B,Y: A] : ( F2 @ Y @ X2 )
@ ( rev @ B @ Xs )
@ A6 ) ) ) ).
% foldl_conv_foldr
thf(fact_4864_foldr__conv__foldl,axiom,
! [A: $tType,B: $tType] :
( ( foldr @ B @ A )
= ( ^ [F2: B > A > A,Xs: list @ B,A6: A] :
( foldl @ A @ B
@ ^ [X2: A,Y: B] : ( F2 @ Y @ X2 )
@ A6
@ ( rev @ B @ Xs ) ) ) ) ).
% foldr_conv_foldl
thf(fact_4865_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_4866_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
@ ^ [X2: A] : ( none @ B ) )
=> ( ! [K2: A,V5: B,M4: A > ( option @ B )] :
( ( finite_finite2 @ A @ ( dom @ A @ B @ M4 ) )
=> ( ~ ( member @ A @ K2 @ ( dom @ A @ B @ M4 ) )
=> ( ( P @ M4 )
=> ( P @ ( fun_upd @ A @ ( option @ B ) @ M4 @ K2 @ ( some @ B @ V5 ) ) ) ) ) )
=> ( P @ M ) ) ) ) ).
% finite_Map_induct
thf(fact_4867_graph__eq__to__snd__dom,axiom,
! [B: $tType,A: $tType] :
( ( graph @ A @ B )
= ( ^ [M5: A > ( option @ B )] :
( image2 @ A @ ( product_prod @ A @ B )
@ ^ [X2: A] : ( product_Pair @ A @ B @ X2 @ ( the2 @ B @ ( M5 @ X2 ) ) )
@ ( dom @ A @ B @ M5 ) ) ) ) ).
% graph_eq_to_snd_dom
thf(fact_4868_dom__eq__singleton__conv,axiom,
! [A: $tType,B: $tType,F: A > ( option @ B ),X: A] :
( ( ( dom @ A @ B @ F )
= ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( ? [V6: B] :
( F
= ( fun_upd @ A @ ( option @ B )
@ ^ [X2: A] : ( none @ B )
@ X
@ ( some @ B @ V6 ) ) ) ) ) ).
% dom_eq_singleton_conv
thf(fact_4869_map__of__map__keys,axiom,
! [B: $tType,A: $tType,Xs2: list @ A,M: A > ( option @ B )] :
( ( ( set2 @ A @ Xs2 )
= ( dom @ A @ B @ M ) )
=> ( ( map_of @ A @ B
@ ( map @ A @ ( product_prod @ A @ B )
@ ^ [K3: A] : ( product_Pair @ A @ B @ K3 @ ( the2 @ B @ ( M @ K3 ) ) )
@ Xs2 ) )
= M ) ) ).
% map_of_map_keys
thf(fact_4870_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
@ ^ [A6: A] : ( member @ A @ A6 @ ( minus_minus @ ( set @ A ) @ A5 @ ( dom @ A @ B @ G ) ) ) ) )
@ ( collect @ A
@ ^ [A6: A] : ( member @ A @ A6 @ ( inf_inf @ ( set @ A ) @ A5 @ ( dom @ A @ B @ G ) ) ) ) ) ) ).
% dom_override_on
thf(fact_4871_listrel__iff__nth,axiom,
! [A: $tType,B: $tType,Xs2: list @ A,Ys2: list @ B,R: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs2 @ Ys2 ) @ ( listrel @ A @ B @ R ) )
= ( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ B ) @ Ys2 ) )
& ! [N5: nat] :
( ( ord_less @ nat @ N5 @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ ( nth @ A @ Xs2 @ N5 ) @ ( nth @ B @ Ys2 @ N5 ) ) @ R ) ) ) ) ).
% listrel_iff_nth
thf(fact_4872_last__list__update,axiom,
! [A: $tType,Xs2: list @ A,K: nat,X: A] :
( ( Xs2
!= ( nil @ A ) )
=> ( ( ( K
= ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ ( one_one @ nat ) ) )
=> ( ( last @ A @ ( list_update @ A @ Xs2 @ K @ X ) )
= X ) )
& ( ( K
!= ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ ( one_one @ nat ) ) )
=> ( ( last @ A @ ( list_update @ A @ Xs2 @ K @ X ) )
= ( last @ A @ Xs2 ) ) ) ) ) ).
% last_list_update
thf(fact_4873_override__on__apply__notin,axiom,
! [B: $tType,A: $tType,A3: A,A5: set @ A,F: A > B,G: A > B] :
( ~ ( member @ A @ A3 @ A5 )
=> ( ( override_on @ A @ B @ F @ G @ A5 @ A3 )
= ( F @ A3 ) ) ) ).
% override_on_apply_notin
thf(fact_4874_override__on__apply__in,axiom,
! [B: $tType,A: $tType,A3: A,A5: set @ A,F: A > B,G: A > B] :
( ( member @ A @ A3 @ A5 )
=> ( ( override_on @ A @ B @ F @ G @ A5 @ A3 )
= ( G @ A3 ) ) ) ).
% override_on_apply_in
thf(fact_4875_override__on__emptyset,axiom,
! [B: $tType,A: $tType,F: A > B,G: A > B] :
( ( override_on @ A @ B @ F @ G @ ( bot_bot @ ( set @ A ) ) )
= F ) ).
% override_on_emptyset
thf(fact_4876_last__replicate,axiom,
! [A: $tType,N: nat,X: A] :
( ( N
!= ( zero_zero @ nat ) )
=> ( ( last @ A @ ( replicate @ A @ N @ X ) )
= X ) ) ).
% last_replicate
thf(fact_4877_listrel__rtrancl__refl,axiom,
! [A: $tType,Xs2: list @ A,R: set @ ( product_prod @ A @ A )] : ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Xs2 ) @ ( listrel @ A @ A @ ( transitive_rtrancl @ A @ R ) ) ) ).
% listrel_rtrancl_refl
thf(fact_4878_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_4879_override__on__def,axiom,
! [B: $tType,A: $tType] :
( ( override_on @ A @ B )
= ( ^ [F2: A > B,G2: A > B,A8: set @ A,A6: A] : ( if @ B @ ( member @ A @ A6 @ A8 ) @ ( G2 @ A6 ) @ ( F2 @ A6 ) ) ) ) ).
% override_on_def
thf(fact_4880_last__zip,axiom,
! [A: $tType,B: $tType,Xs2: list @ A,Ys2: list @ B] :
( ( Xs2
!= ( nil @ A ) )
=> ( ( Ys2
!= ( nil @ B ) )
=> ( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ B ) @ Ys2 ) )
=> ( ( last @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs2 @ Ys2 ) )
= ( product_Pair @ A @ B @ ( last @ A @ Xs2 ) @ ( last @ B @ Ys2 ) ) ) ) ) ) ).
% last_zip
thf(fact_4881_listrel_ONil,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B )] : ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ ( nil @ B ) ) @ ( listrel @ A @ B @ R ) ) ).
% listrel.Nil
thf(fact_4882_listrel__Nil1,axiom,
! [A: $tType,B: $tType,Xs2: list @ B,R: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ Xs2 ) @ ( listrel @ A @ B @ R ) )
=> ( Xs2
= ( nil @ B ) ) ) ).
% listrel_Nil1
thf(fact_4883_listrel__Nil2,axiom,
! [B: $tType,A: $tType,Xs2: list @ A,R: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs2 @ ( nil @ B ) ) @ ( listrel @ A @ B @ R ) )
=> ( Xs2
= ( nil @ A ) ) ) ).
% listrel_Nil2
thf(fact_4884_listrel__eq__len,axiom,
! [A: $tType,B: $tType,Xs2: list @ A,Ys2: list @ B,R: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs2 @ Ys2 ) @ ( listrel @ A @ B @ R ) )
=> ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ B ) @ Ys2 ) ) ) ).
% listrel_eq_len
thf(fact_4885_listrel__rtrancl__trans,axiom,
! [A: $tType,Xs2: list @ A,Ys2: list @ A,R: set @ ( product_prod @ A @ A ),Zs: list @ A] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Ys2 ) @ ( listrel @ A @ A @ ( transitive_rtrancl @ A @ R ) ) )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys2 @ Zs ) @ ( listrel @ A @ A @ ( transitive_rtrancl @ A @ R ) ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Zs ) @ ( listrel @ A @ A @ ( transitive_rtrancl @ A @ R ) ) ) ) ) ).
% listrel_rtrancl_trans
thf(fact_4886_listrel_OCons,axiom,
! [B: $tType,A: $tType,X: A,Y3: B,R: set @ ( product_prod @ A @ B ),Xs2: list @ A,Ys2: list @ B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y3 ) @ R )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs2 @ Ys2 ) @ ( listrel @ A @ B @ R ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ X @ Xs2 ) @ ( cons @ B @ Y3 @ Ys2 ) ) @ ( listrel @ A @ B @ R ) ) ) ) ).
% listrel.Cons
thf(fact_4887_listrel__Cons1,axiom,
! [B: $tType,A: $tType,Y3: A,Ys2: list @ A,Xs2: list @ B,R: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ Y3 @ Ys2 ) @ Xs2 ) @ ( listrel @ A @ B @ R ) )
=> ~ ! [Y4: B,Ys5: list @ B] :
( ( Xs2
= ( cons @ B @ Y4 @ Ys5 ) )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Y3 @ Y4 ) @ R )
=> ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Ys2 @ Ys5 ) @ ( listrel @ A @ B @ R ) ) ) ) ) ).
% listrel_Cons1
thf(fact_4888_listrel__Cons2,axiom,
! [B: $tType,A: $tType,Xs2: list @ A,Y3: B,Ys2: list @ B,R: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs2 @ ( cons @ B @ Y3 @ Ys2 ) ) @ ( listrel @ A @ B @ R ) )
=> ~ ! [X4: A,Xs5: list @ A] :
( ( Xs2
= ( cons @ A @ X4 @ Xs5 ) )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y3 ) @ R )
=> ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs5 @ Ys2 ) @ ( listrel @ A @ B @ R ) ) ) ) ) ).
% listrel_Cons2
thf(fact_4889_listrel__reflcl__if__listrel1,axiom,
! [A: $tType,Xs2: list @ A,Ys2: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Ys2 ) @ ( listrel1 @ A @ R ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Ys2 ) @ ( listrel @ A @ A @ ( transitive_rtrancl @ A @ R ) ) ) ) ).
% listrel_reflcl_if_listrel1
thf(fact_4890_override__on__insert,axiom,
! [B: $tType,A: $tType,F: A > B,G: A > B,X: A,X7: set @ A] :
( ( override_on @ A @ B @ F @ G @ ( insert2 @ A @ X @ X7 ) )
= ( fun_upd @ A @ B @ ( override_on @ A @ B @ F @ G @ X7 ) @ X @ ( G @ X ) ) ) ).
% override_on_insert
thf(fact_4891_override__on__insert_H,axiom,
! [B: $tType,A: $tType,F: A > B,G: A > B,X: A,X7: set @ A] :
( ( override_on @ A @ B @ F @ G @ ( insert2 @ A @ X @ X7 ) )
= ( override_on @ A @ B @ ( fun_upd @ A @ B @ F @ X @ ( G @ X ) ) @ G @ X7 ) ) ).
% override_on_insert'
thf(fact_4892_rtrancl__listrel1__if__listrel,axiom,
! [A: $tType,Xs2: list @ A,Ys2: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Ys2 ) @ ( listrel @ A @ A @ R ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Ys2 ) @ ( transitive_rtrancl @ ( list @ A ) @ ( listrel1 @ A @ R ) ) ) ) ).
% rtrancl_listrel1_if_listrel
thf(fact_4893_takeWhile__not__last,axiom,
! [A: $tType,Xs2: list @ A] :
( ( distinct @ A @ Xs2 )
=> ( ( takeWhile @ A
@ ^ [Y: A] :
( Y
!= ( last @ A @ Xs2 ) )
@ Xs2 )
= ( butlast @ A @ Xs2 ) ) ) ).
% takeWhile_not_last
thf(fact_4894_listrel_Ocases,axiom,
! [B: $tType,A: $tType,A1: list @ A,A22: list @ B,R: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ A1 @ A22 ) @ ( listrel @ A @ B @ R ) )
=> ( ( ( A1
= ( nil @ A ) )
=> ( A22
!= ( nil @ B ) ) )
=> ~ ! [X4: A,Y4: B,Xs5: list @ A] :
( ( A1
= ( cons @ A @ X4 @ Xs5 ) )
=> ! [Ys5: list @ B] :
( ( A22
= ( cons @ B @ Y4 @ Ys5 ) )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y4 ) @ R )
=> ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs5 @ Ys5 ) @ ( listrel @ A @ B @ R ) ) ) ) ) ) ) ).
% listrel.cases
thf(fact_4895_listrel_Osimps,axiom,
! [B: $tType,A: $tType,A1: list @ A,A22: list @ B,R: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ A1 @ A22 ) @ ( listrel @ A @ B @ R ) )
= ( ( ( A1
= ( nil @ A ) )
& ( A22
= ( nil @ B ) ) )
| ? [X2: A,Y: B,Xs: list @ A,Ys: list @ B] :
( ( A1
= ( cons @ A @ X2 @ Xs ) )
& ( A22
= ( cons @ B @ Y @ Ys ) )
& ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y ) @ R )
& ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs @ Ys ) @ ( listrel @ A @ B @ R ) ) ) ) ) ).
% listrel.simps
thf(fact_4896_remdups__adj__append_H_H,axiom,
! [A: $tType,Xs2: list @ A,Ys2: list @ A] :
( ( Xs2
!= ( nil @ A ) )
=> ( ( remdups_adj @ A @ ( append @ A @ Xs2 @ Ys2 ) )
= ( append @ A @ ( remdups_adj @ A @ Xs2 )
@ ( remdups_adj @ A
@ ( dropWhile @ A
@ ^ [Y: A] :
( Y
= ( last @ A @ Xs2 ) )
@ Ys2 ) ) ) ) ) ).
% remdups_adj_append''
thf(fact_4897_last__conv__nth,axiom,
! [A: $tType,Xs2: list @ A] :
( ( Xs2
!= ( nil @ A ) )
=> ( ( last @ A @ Xs2 )
= ( nth @ A @ Xs2 @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ ( one_one @ nat ) ) ) ) ) ).
% last_conv_nth
thf(fact_4898_listrel__iff__zip,axiom,
! [B: $tType,A: $tType,Xs2: list @ A,Ys2: list @ B,R: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs2 @ Ys2 ) @ ( listrel @ A @ B @ R ) )
= ( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ B ) @ Ys2 ) )
& ! [X2: product_prod @ A @ B] :
( ( member @ ( product_prod @ A @ B ) @ X2 @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs2 @ Ys2 ) ) )
=> ( product_case_prod @ A @ B @ $o
@ ^ [Y: A,Z3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Y @ Z3 ) @ R )
@ X2 ) ) ) ) ).
% listrel_iff_zip
thf(fact_4899_listrel__def,axiom,
! [B: $tType,A: $tType] :
( ( listrel @ A @ B )
= ( ^ [R4: set @ ( product_prod @ A @ B )] :
( collect @ ( product_prod @ ( list @ A ) @ ( list @ B ) )
@ ( product_case_prod @ ( list @ A ) @ ( list @ B ) @ $o
@ ( listrelp @ A @ B
@ ^ [X2: A,Y: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y ) @ R4 ) ) ) ) ) ) ).
% listrel_def
thf(fact_4900_listrelp__listrel__eq,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B )] :
( ( listrelp @ A @ B
@ ^ [X2: A,Y: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y ) @ R ) )
= ( ^ [X2: list @ A,Y: list @ B] : ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ X2 @ Y ) @ ( listrel @ A @ B @ R ) ) ) ) ).
% listrelp_listrel_eq
thf(fact_4901_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_4902_Ball__fold,axiom,
! [A: $tType,A5: set @ A,P: A > $o] :
( ( finite_finite2 @ A @ A5 )
=> ( ( ! [X2: A] :
( ( member @ A @ X2 @ A5 )
=> ( P @ X2 ) ) )
= ( finite_fold @ A @ $o
@ ^ [K3: A,S2: $o] :
( S2
& ( P @ K3 ) )
@ $true
@ A5 ) ) ) ).
% Ball_fold
thf(fact_4903_Ball__image__comp,axiom,
! [A: $tType,B: $tType,F: B > A,A5: set @ B,G: A > $o] :
( ( ! [X2: A] :
( ( member @ A @ X2 @ ( image2 @ B @ A @ F @ A5 ) )
=> ( G @ X2 ) ) )
= ( ! [X2: B] :
( ( member @ B @ X2 @ A5 )
=> ( comp @ A @ $o @ B @ G @ F @ X2 ) ) ) ) ).
% Ball_image_comp
thf(fact_4904_INTER__eq,axiom,
! [B: $tType,A: $tType,B4: B > ( set @ A ),A5: set @ B] :
( ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B4 @ A5 ) )
= ( collect @ A
@ ^ [Y: A] :
! [X2: B] :
( ( member @ B @ X2 @ A5 )
=> ( member @ A @ Y @ ( B4 @ X2 ) ) ) ) ) ).
% INTER_eq
thf(fact_4905_Collect__ball__eq,axiom,
! [A: $tType,B: $tType,A5: set @ B,P: A > B > $o] :
( ( collect @ A
@ ^ [X2: A] :
! [Y: B] :
( ( member @ B @ Y @ A5 )
=> ( P @ X2 @ Y ) ) )
= ( complete_Inf_Inf @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [Y: B] :
( collect @ A
@ ^ [X2: A] : ( P @ X2 @ Y ) )
@ A5 ) ) ) ).
% Collect_ball_eq
thf(fact_4906_irrefl__distinct,axiom,
! [A: $tType] :
( ( irrefl @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A )] :
! [X2: product_prod @ A @ A] :
( ( member @ ( product_prod @ A @ A ) @ X2 @ R4 )
=> ( product_case_prod @ A @ A @ $o
@ ^ [A6: A,B5: A] : A6 != B5
@ X2 ) ) ) ) ).
% irrefl_distinct
thf(fact_4907_Ball__comp__iff,axiom,
! [C: $tType,B: $tType,A: $tType,A5: B > ( set @ C ),F: C > $o,G: A > B] :
( ( comp @ B @ $o @ A
@ ^ [X2: B] :
! [Y: C] :
( ( member @ C @ Y @ ( A5 @ X2 ) )
=> ( F @ Y ) )
@ G )
= ( ^ [X2: A] :
! [Y: C] :
( ( member @ C @ Y @ ( comp @ B @ ( set @ C ) @ A @ A5 @ G @ X2 ) )
=> ( F @ Y ) ) ) ) ).
% Ball_comp_iff
thf(fact_4908_Inf__eq__Sup,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ( ( complete_Inf_Inf @ A )
= ( ^ [A8: set @ A] :
( complete_Sup_Sup @ A
@ ( collect @ A
@ ^ [B5: A] :
! [X2: A] :
( ( member @ A @ X2 @ A8 )
=> ( ord_less_eq @ A @ B5 @ X2 ) ) ) ) ) ) ) ).
% Inf_eq_Sup
thf(fact_4909_Sup__eq__Inf,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ( ( complete_Sup_Sup @ A )
= ( ^ [A8: set @ A] :
( complete_Inf_Inf @ A
@ ( collect @ A
@ ^ [B5: A] :
! [X2: A] :
( ( member @ A @ X2 @ A8 )
=> ( ord_less_eq @ A @ X2 @ B5 ) ) ) ) ) ) ) ).
% Sup_eq_Inf
thf(fact_4910_takeWhile__append,axiom,
! [A: $tType,Xs2: list @ A,P: A > $o,Ys2: list @ A] :
( ( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs2 ) )
=> ( P @ X4 ) )
=> ( ( takeWhile @ A @ P @ ( append @ A @ Xs2 @ Ys2 ) )
= ( append @ A @ Xs2 @ ( takeWhile @ A @ P @ Ys2 ) ) ) )
& ( ~ ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs2 ) )
=> ( P @ X5 ) )
=> ( ( takeWhile @ A @ P @ ( append @ A @ Xs2 @ Ys2 ) )
= ( takeWhile @ A @ P @ Xs2 ) ) ) ) ).
% takeWhile_append
thf(fact_4911_dropWhile__append,axiom,
! [A: $tType,Xs2: list @ A,P: A > $o,Ys2: list @ A] :
( ( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs2 ) )
=> ( P @ X4 ) )
=> ( ( dropWhile @ A @ P @ ( append @ A @ Xs2 @ Ys2 ) )
= ( dropWhile @ A @ P @ Ys2 ) ) )
& ( ~ ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs2 ) )
=> ( P @ X5 ) )
=> ( ( dropWhile @ A @ P @ ( append @ A @ Xs2 @ Ys2 ) )
= ( append @ A @ ( dropWhile @ A @ P @ Xs2 ) @ Ys2 ) ) ) ) ).
% dropWhile_append
thf(fact_4912_list__eq__iff__zip__eq,axiom,
! [A: $tType] :
( ( ^ [Y6: list @ A,Z2: list @ A] : Y6 = Z2 )
= ( ^ [Xs: list @ A,Ys: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ A ) @ Ys ) )
& ! [X2: product_prod @ A @ A] :
( ( member @ ( product_prod @ A @ A ) @ X2 @ ( set2 @ ( product_prod @ A @ A ) @ ( zip @ A @ A @ Xs @ Ys ) ) )
=> ( product_case_prod @ A @ A @ $o
@ ^ [Y6: A,Z2: A] : Y6 = Z2
@ X2 ) ) ) ) ) ).
% list_eq_iff_zip_eq
thf(fact_4913_concat__injective,axiom,
! [A: $tType,Xs2: list @ ( list @ A ),Ys2: list @ ( list @ A )] :
( ( ( concat @ A @ Xs2 )
= ( concat @ A @ Ys2 ) )
=> ( ( ( size_size @ ( list @ ( list @ A ) ) @ Xs2 )
= ( size_size @ ( list @ ( list @ A ) ) @ Ys2 ) )
=> ( ! [X4: product_prod @ ( list @ A ) @ ( list @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ X4 @ ( set2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( zip @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Ys2 ) ) )
=> ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
@ ^ [Y: list @ A,Z3: list @ A] :
( ( size_size @ ( list @ A ) @ Y )
= ( size_size @ ( list @ A ) @ Z3 ) )
@ X4 ) )
=> ( Xs2 = Ys2 ) ) ) ) ).
% concat_injective
thf(fact_4914_concat__eq__concat__iff,axiom,
! [A: $tType,Xs2: list @ ( list @ A ),Ys2: list @ ( list @ A )] :
( ! [X4: product_prod @ ( list @ A ) @ ( list @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ X4 @ ( set2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( zip @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Ys2 ) ) )
=> ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
@ ^ [Y: list @ A,Z3: list @ A] :
( ( size_size @ ( list @ A ) @ Y )
= ( size_size @ ( list @ A ) @ Z3 ) )
@ X4 ) )
=> ( ( ( size_size @ ( list @ ( list @ A ) ) @ Xs2 )
= ( size_size @ ( list @ ( list @ A ) ) @ Ys2 ) )
=> ( ( ( concat @ A @ Xs2 )
= ( concat @ A @ Ys2 ) )
= ( Xs2 = Ys2 ) ) ) ) ).
% concat_eq_concat_iff
thf(fact_4915_list__all2__iff,axiom,
! [B: $tType,A: $tType] :
( ( list_all2 @ A @ B )
= ( ^ [P3: A > B > $o,Xs: list @ A,Ys: list @ B] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
& ! [X2: product_prod @ A @ B] :
( ( member @ ( product_prod @ A @ B ) @ X2 @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) ) )
=> ( product_case_prod @ A @ B @ $o @ P3 @ X2 ) ) ) ) ) ).
% list_all2_iff
thf(fact_4916_Under__def,axiom,
! [A: $tType] :
( ( order_Under @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A ),A8: set @ A] :
( collect @ A
@ ^ [B5: A] :
( ( member @ A @ B5 @ ( field2 @ A @ R4 ) )
& ! [X2: A] :
( ( member @ A @ X2 @ A8 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B5 @ X2 ) @ R4 ) ) ) ) ) ) ).
% Under_def
thf(fact_4917_UnderS__def,axiom,
! [A: $tType] :
( ( order_UnderS @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A ),A8: set @ A] :
( collect @ A
@ ^ [B5: A] :
( ( member @ A @ B5 @ ( field2 @ A @ R4 ) )
& ! [X2: A] :
( ( member @ A @ X2 @ A8 )
=> ( ( B5 != X2 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B5 @ X2 ) @ R4 ) ) ) ) ) ) ) ).
% UnderS_def
thf(fact_4918_Above__def,axiom,
! [A: $tType] :
( ( order_Above @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A ),A8: set @ A] :
( collect @ A
@ ^ [B5: A] :
( ( member @ A @ B5 @ ( field2 @ A @ R4 ) )
& ! [X2: A] :
( ( member @ A @ X2 @ A8 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ B5 ) @ R4 ) ) ) ) ) ) ).
% Above_def
thf(fact_4919_Inter__eq,axiom,
! [A: $tType] :
( ( complete_Inf_Inf @ ( set @ A ) )
= ( ^ [A8: set @ ( set @ A )] :
( collect @ A
@ ^ [X2: A] :
! [Y: set @ A] :
( ( member @ ( set @ A ) @ Y @ A8 )
=> ( member @ A @ X2 @ Y ) ) ) ) ) ).
% Inter_eq
thf(fact_4920_Sup__int__def,axiom,
( ( complete_Sup_Sup @ int )
= ( ^ [X9: set @ int] :
( the @ int
@ ^ [X2: int] :
( ( member @ int @ X2 @ X9 )
& ! [Y: int] :
( ( member @ int @ Y @ X9 )
=> ( ord_less_eq @ int @ Y @ X2 ) ) ) ) ) ) ).
% Sup_int_def
thf(fact_4921_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 ) )
& ! [X2: set @ A] :
( ( member @ ( set @ A ) @ X2 @ A5 )
=> ( member @ A @ ( F2 @ X2 ) @ X2 ) ) ) ) ) ) ) ) ).
% Inf_Sup
thf(fact_4922_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 ) )
& ! [X2: set @ A] :
( ( member @ ( set @ A ) @ X2 @ A5 )
=> ( member @ A @ ( F2 @ X2 ) @ X2 ) ) ) ) ) ) ) ) ).
% Sup_Inf
thf(fact_4923_Inf__filter__def,axiom,
! [A: $tType] :
( ( complete_Inf_Inf @ ( filter @ A ) )
= ( ^ [S8: set @ ( filter @ A )] :
( complete_Sup_Sup @ ( filter @ A )
@ ( collect @ ( filter @ A )
@ ^ [F7: filter @ A] :
! [X2: filter @ A] :
( ( member @ ( filter @ A ) @ X2 @ S8 )
=> ( ord_less_eq @ ( filter @ A ) @ F7 @ X2 ) ) ) ) ) ) ).
% Inf_filter_def
thf(fact_4924_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 ) )
& ! [X2: set @ A] :
( ( member @ ( set @ A ) @ X2 @ A5 )
=> ( member @ A @ ( F2 @ X2 ) @ X2 ) ) ) ) ) ) ) ) ).
% finite_Inf_Sup
thf(fact_4925_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 ) )
& ! [X2: set @ A] :
( ( member @ ( set @ A ) @ X2 @ A5 )
=> ( member @ A @ ( F2 @ X2 ) @ X2 ) ) ) ) ) )
@ ( complete_Inf_Inf @ A @ ( image2 @ ( set @ A ) @ A @ ( complete_Sup_Sup @ A ) @ A5 ) ) ) ) ).
% Sup_Inf_le
thf(fact_4926_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 ) )
& ! [X2: set @ A] :
( ( member @ ( set @ A ) @ X2 @ A5 )
=> ( member @ A @ ( F2 @ X2 ) @ X2 ) ) ) ) ) ) ) ) ).
% Inf_Sup_le
thf(fact_4927_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
@ ^ [B7: set @ B] : ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ G @ B7 ) )
@ A5 ) )
= ( complete_Sup_Sup @ A
@ ( image2 @ ( set @ B ) @ A
@ ^ [B7: set @ B] : ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ G @ B7 ) )
@ ( collect @ ( set @ B )
@ ^ [Uu: set @ B] :
? [F2: ( set @ B ) > B] :
( ( Uu
= ( image2 @ ( set @ B ) @ B @ F2 @ A5 ) )
& ! [X2: set @ B] :
( ( member @ ( set @ B ) @ X2 @ A5 )
=> ( member @ B @ ( F2 @ X2 ) @ X2 ) ) ) ) ) ) ) ) ).
% INF_SUP_set
thf(fact_4928_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
@ ^ [X2: set @ B] : ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ G @ X2 ) )
@ A5 ) )
= ( complete_Inf_Inf @ A
@ ( image2 @ ( set @ B ) @ A
@ ^ [X2: set @ B] : ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ G @ X2 ) )
@ ( collect @ ( set @ B )
@ ^ [Uu: set @ B] :
? [F2: ( set @ B ) > B] :
( ( Uu
= ( image2 @ ( set @ B ) @ B @ F2 @ A5 ) )
& ! [X2: set @ B] :
( ( member @ ( set @ B ) @ X2 @ A5 )
=> ( member @ B @ ( F2 @ X2 ) @ X2 ) ) ) ) ) ) ) ) ).
% SUP_INF_set
thf(fact_4929_min__ext__def,axiom,
! [A: $tType] :
( ( min_ext @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ ( set @ A ) @ ( set @ A ) )
@ ^ [Uu: product_prod @ ( set @ A ) @ ( set @ A )] :
? [X9: set @ A,Y10: set @ A] :
( ( Uu
= ( product_Pair @ ( set @ A ) @ ( set @ A ) @ X9 @ Y10 ) )
& ( X9
!= ( bot_bot @ ( set @ A ) ) )
& ! [X2: A] :
( ( member @ A @ X2 @ Y10 )
=> ? [Y: A] :
( ( member @ A @ Y @ X9 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X2 ) @ R4 ) ) ) ) ) ) ) ).
% min_ext_def
thf(fact_4930_iteratesp_Omono,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [F: A > A] :
( order_mono @ ( A > $o ) @ ( A > $o )
@ ^ [P5: A > $o,X2: A] :
( ? [Y: A] :
( ( X2
= ( F @ Y ) )
& ( P5 @ Y ) )
| ? [M9: set @ A] :
( ( X2
= ( complete_Sup_Sup @ A @ M9 ) )
& ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ M9 )
& ! [Y: A] :
( ( member @ A @ Y @ M9 )
=> ( P5 @ Y ) ) ) ) ) ) ).
% iteratesp.mono
thf(fact_4931_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] :
! [X2: product_prod @ A @ A] :
( ( member @ ( product_prod @ A @ A ) @ X2 @ R13 )
=> ( product_case_prod @ A @ A @ $o
@ ^ [Y13: A,Z12: A] :
! [Y: product_prod @ B @ B] :
( ( member @ ( product_prod @ B @ B ) @ Y @ R24 )
=> ( product_case_prod @ B @ B @ $o
@ ^ [Y23: B,Z23: B] :
( ( F2 @ Y13 @ Y23 )
= ( F2 @ Z12 @ Z23 ) )
@ Y ) )
@ X2 ) ) ) ) ).
% congruent2_def
thf(fact_4932_finite__Collect__bex,axiom,
! [B: $tType,A: $tType,A5: set @ A,Q: B > A > $o] :
( ( finite_finite2 @ A @ A5 )
=> ( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X2: B] :
? [Y: A] :
( ( member @ A @ Y @ A5 )
& ( Q @ X2 @ Y ) ) ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ A5 )
=> ( finite_finite2 @ B
@ ( collect @ B
@ ^ [Y: B] : ( Q @ Y @ X2 ) ) ) ) ) ) ) ).
% finite_Collect_bex
thf(fact_4933_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
@ ^ [Y: A] :
? [X2: B] :
( ( member @ B @ X2 @ A5 )
& ( P @ X2 @ Y ) ) ) ) ).
% Image_Collect_case_prod
thf(fact_4934_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_4935_image__def,axiom,
! [B: $tType,A: $tType] :
( ( image2 @ A @ B )
= ( ^ [F2: A > B,A8: set @ A] :
( collect @ B
@ ^ [Y: B] :
? [X2: A] :
( ( member @ A @ X2 @ A8 )
& ( Y
= ( F2 @ X2 ) ) ) ) ) ) ).
% image_def
thf(fact_4936_Bex__def__raw,axiom,
! [A: $tType] :
( ( bex @ A )
= ( ^ [A8: set @ A,P3: A > $o] :
? [X2: A] :
( ( member @ A @ X2 @ A8 )
& ( P3 @ X2 ) ) ) ) ).
% Bex_def_raw
thf(fact_4937_Union__eq,axiom,
! [A: $tType] :
( ( complete_Sup_Sup @ ( set @ A ) )
= ( ^ [A8: set @ ( set @ A )] :
( collect @ A
@ ^ [X2: A] :
? [Y: set @ A] :
( ( member @ ( set @ A ) @ Y @ A8 )
& ( member @ A @ X2 @ Y ) ) ) ) ) ).
% Union_eq
thf(fact_4938_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
@ ^ [X2: A] :
( ( member @ A @ X2 @ A5 )
& ( P @ X2 ) ) ) ) ) ).
% chain_compr
thf(fact_4939_UNION__eq,axiom,
! [B: $tType,A: $tType,B4: B > ( set @ A ),A5: set @ B] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B4 @ A5 ) )
= ( collect @ A
@ ^ [Y: A] :
? [X2: B] :
( ( member @ B @ X2 @ A5 )
& ( member @ A @ Y @ ( B4 @ X2 ) ) ) ) ) ).
% UNION_eq
thf(fact_4940_Collect__bex__eq,axiom,
! [A: $tType,B: $tType,A5: set @ B,P: A > B > $o] :
( ( collect @ A
@ ^ [X2: A] :
? [Y: B] :
( ( member @ B @ Y @ A5 )
& ( P @ X2 @ Y ) ) )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [Y: B] :
( collect @ A
@ ^ [X2: A] : ( P @ X2 @ Y ) )
@ A5 ) ) ) ).
% Collect_bex_eq
thf(fact_4941_Image__def,axiom,
! [B: $tType,A: $tType] :
( ( image @ A @ B )
= ( ^ [R4: set @ ( product_prod @ A @ B ),S2: set @ A] :
( collect @ B
@ ^ [Y: B] :
? [X2: A] :
( ( member @ A @ X2 @ S2 )
& ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y ) @ R4 ) ) ) ) ) ).
% Image_def
thf(fact_4942_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
@ ^ [Y: A] :
? [X2: A] :
( ( member @ A @ X2 @ A5 )
& ( ( F @ X2 )
= ( F @ Y ) ) ) ) ) ).
% vimage_image_eq
thf(fact_4943_congruent2I_H,axiom,
! [C: $tType,B: $tType,A: $tType,R1: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B ),F: A > B > C] :
( ! [Y12: A,Z13: A,Y22: B,Z24: B] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y12 @ Z13 ) @ R1 )
=> ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y22 @ Z24 ) @ R22 )
=> ( ( F @ Y12 @ Y22 )
= ( F @ Z13 @ Z24 ) ) ) )
=> ( equiv_congruent2 @ A @ B @ C @ R1 @ R22 @ F ) ) ).
% congruent2I'
thf(fact_4944_congruent2D,axiom,
! [A: $tType,C: $tType,B: $tType,R1: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B ),F: A > B > C,Y1: A,Z1: A,Y2: B,Z22: B] :
( ( equiv_congruent2 @ A @ B @ C @ R1 @ R22 @ F )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y1 @ Z1 ) @ R1 )
=> ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y2 @ Z22 ) @ R22 )
=> ( ( F @ Y1 @ Y2 )
= ( F @ Z1 @ Z22 ) ) ) ) ) ).
% congruent2D
thf(fact_4945_Bex__fold,axiom,
! [A: $tType,A5: set @ A,P: A > $o] :
( ( finite_finite2 @ A @ A5 )
=> ( ( ? [X2: A] :
( ( member @ A @ X2 @ A5 )
& ( P @ X2 ) ) )
= ( finite_fold @ A @ $o
@ ^ [K3: A,S2: $o] :
( S2
| ( P @ K3 ) )
@ $false
@ A5 ) ) ) ).
% Bex_fold
thf(fact_4946_nths__nths,axiom,
! [A: $tType,Xs2: list @ A,A5: set @ nat,B4: set @ nat] :
( ( nths @ A @ ( nths @ A @ Xs2 @ A5 ) @ B4 )
= ( nths @ A @ Xs2
@ ( collect @ nat
@ ^ [I2: nat] :
( ( member @ nat @ I2 @ A5 )
& ( member @ nat
@ ( finite_card @ nat
@ ( collect @ nat
@ ^ [I8: nat] :
( ( member @ nat @ I8 @ A5 )
& ( ord_less @ nat @ I8 @ I2 ) ) ) )
@ B4 ) ) ) ) ) ).
% nths_nths
thf(fact_4947_max__extp_Ocases,axiom,
! [A: $tType,R5: A > A > $o,A1: set @ A,A22: set @ A] :
( ( max_extp @ A @ R5 @ A1 @ A22 )
=> ~ ( ( finite_finite2 @ A @ A1 )
=> ( ( finite_finite2 @ A @ A22 )
=> ( ( A22
!= ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) )
=> ~ ! [X5: A] :
( ( member @ A @ X5 @ A1 )
=> ? [Xa4: A] :
( ( member @ A @ Xa4 @ A22 )
& ( R5 @ X5 @ Xa4 ) ) ) ) ) ) ) ).
% max_extp.cases
thf(fact_4948_max__extp_Osimps,axiom,
! [A: $tType] :
( ( max_extp @ A )
= ( ^ [R6: A > A > $o,A12: set @ A,A23: set @ A] :
( ( finite_finite2 @ A @ A12 )
& ( finite_finite2 @ A @ A23 )
& ( A23
!= ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) )
& ! [X2: A] :
( ( member @ A @ X2 @ A12 )
=> ? [Y: A] :
( ( member @ A @ Y @ A23 )
& ( R6 @ X2 @ Y ) ) ) ) ) ) ).
% max_extp.simps
thf(fact_4949_max__extp_Omax__extI,axiom,
! [A: $tType,X7: set @ A,Y9: set @ A,R5: A > A > $o] :
( ( finite_finite2 @ A @ X7 )
=> ( ( finite_finite2 @ A @ Y9 )
=> ( ( Y9
!= ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ X7 )
=> ? [Xa2: A] :
( ( member @ A @ Xa2 @ Y9 )
& ( R5 @ X4 @ Xa2 ) ) )
=> ( max_extp @ A @ R5 @ X7 @ Y9 ) ) ) ) ) ).
% max_extp.max_extI
thf(fact_4950_map__project__def,axiom,
! [B: $tType,A: $tType] :
( ( map_project @ A @ B )
= ( ^ [F2: A > ( option @ B ),A8: set @ A] :
( collect @ B
@ ^ [B5: B] :
? [X2: A] :
( ( member @ A @ X2 @ A8 )
& ( ( F2 @ X2 )
= ( some @ B @ B5 ) ) ) ) ) ) ).
% map_project_def
thf(fact_4951_iteratesp__def,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ( ( comple7512665784863727008ratesp @ A )
= ( ^ [F2: A > A] :
( complete_lattice_lfp @ ( A > $o )
@ ^ [P5: A > $o,X2: A] :
( ? [Y: A] :
( ( X2
= ( F2 @ Y ) )
& ( P5 @ Y ) )
| ? [M9: set @ A] :
( ( X2
= ( complete_Sup_Sup @ A @ M9 ) )
& ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ M9 )
& ! [Y: A] :
( ( member @ A @ Y @ M9 )
=> ( P5 @ Y ) ) ) ) ) ) ) ) ).
% iteratesp_def
thf(fact_4952_max__ext__eq,axiom,
! [A: $tType] :
( ( max_ext @ A )
= ( ^ [R6: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ ( set @ A ) @ ( set @ A ) )
@ ( product_case_prod @ ( set @ A ) @ ( set @ A ) @ $o
@ ^ [X9: set @ A,Y10: set @ A] :
( ( finite_finite2 @ A @ X9 )
& ( finite_finite2 @ A @ Y10 )
& ( Y10
!= ( bot_bot @ ( set @ A ) ) )
& ! [X2: A] :
( ( member @ A @ X2 @ X9 )
=> ? [Y: A] :
( ( member @ A @ Y @ Y10 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y ) @ R6 ) ) ) ) ) ) ) ) ).
% max_ext_eq
thf(fact_4953_Ball__def__raw,axiom,
! [A: $tType] :
( ( ball @ A )
= ( ^ [A8: set @ A,P3: A > $o] :
! [X2: A] :
( ( member @ A @ X2 @ A8 )
=> ( P3 @ X2 ) ) ) ) ).
% Ball_def_raw
thf(fact_4954_rel__fun__eq__rel,axiom,
! [C: $tType,B: $tType,A: $tType,R5: B > C > $o] :
( ( bNF_rel_fun @ A @ A @ B @ C
@ ^ [Y6: A,Z2: A] : Y6 = Z2
@ R5 )
= ( ^ [F2: A > B,G2: A > C] :
! [X2: A] : ( R5 @ ( F2 @ X2 ) @ ( G2 @ X2 ) ) ) ) ).
% rel_fun_eq_rel
thf(fact_4955_rel__fun__def,axiom,
! [B: $tType,D: $tType,C: $tType,A: $tType] :
( ( bNF_rel_fun @ A @ C @ B @ D )
= ( ^ [A8: A > C > $o,B7: B > D > $o,F2: A > B,G2: C > D] :
! [X2: A,Y: C] :
( ( A8 @ X2 @ Y )
=> ( B7 @ ( F2 @ X2 ) @ ( G2 @ Y ) ) ) ) ) ).
% rel_fun_def
thf(fact_4956_finite__set__of__finite__funs,axiom,
! [B: $tType,A: $tType,A5: set @ A,B4: set @ B,D3: B] :
( ( finite_finite2 @ A @ A5 )
=> ( ( finite_finite2 @ B @ B4 )
=> ( finite_finite2 @ ( A > B )
@ ( collect @ ( A > B )
@ ^ [F2: A > B] :
! [X2: A] :
( ( ( member @ A @ X2 @ A5 )
=> ( member @ B @ ( F2 @ X2 ) @ B4 ) )
& ( ~ ( member @ A @ X2 @ A5 )
=> ( ( F2 @ X2 )
= D3 ) ) ) ) ) ) ) ).
% finite_set_of_finite_funs
thf(fact_4957_Collect__all__eq,axiom,
! [A: $tType,B: $tType,P: A > B > $o] :
( ( collect @ A
@ ^ [X2: A] :
! [X9: B] : ( P @ X2 @ X9 ) )
= ( complete_Inf_Inf @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [Y: B] :
( collect @ A
@ ^ [X2: A] : ( P @ X2 @ Y ) )
@ ( top_top @ ( set @ B ) ) ) ) ) ).
% Collect_all_eq
thf(fact_4958_Func__def,axiom,
! [B: $tType,A: $tType] :
( ( bNF_Wellorder_Func @ A @ B )
= ( ^ [A8: set @ A,B7: set @ B] :
( collect @ ( A > B )
@ ^ [F2: A > B] :
( ! [X2: A] :
( ( member @ A @ X2 @ A8 )
=> ( member @ B @ ( F2 @ X2 ) @ B7 ) )
& ! [A6: A] :
( ~ ( member @ A @ A6 @ A8 )
=> ( ( F2 @ A6 )
= ( undefined @ B ) ) ) ) ) ) ) ).
% Func_def
thf(fact_4959_Greatest__def,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( order_Greatest @ A )
= ( ^ [P3: A > $o] :
( the @ A
@ ^ [X2: A] :
( ( P3 @ X2 )
& ! [Y: A] :
( ( P3 @ Y )
=> ( ord_less_eq @ A @ Y @ X2 ) ) ) ) ) ) ) ).
% Greatest_def
thf(fact_4960_ord_OLeast__def,axiom,
! [A: $tType] :
( ( least @ A )
= ( ^ [Less_eq: A > A > $o,P3: A > $o] :
( the @ A
@ ^ [X2: A] :
( ( P3 @ X2 )
& ! [Y: A] :
( ( P3 @ Y )
=> ( Less_eq @ X2 @ Y ) ) ) ) ) ) ).
% ord.Least_def
thf(fact_4961_transfer__bforall__def,axiom,
! [A: $tType] :
( ( transfer_bforall @ A )
= ( ^ [P3: A > $o,Q7: A > $o] :
! [X2: A] :
( ( P3 @ X2 )
=> ( Q7 @ X2 ) ) ) ) ).
% transfer_bforall_def
thf(fact_4962_GreatestI__nat,axiom,
! [P: nat > $o,K: nat,B2: nat] :
( ( P @ K )
=> ( ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq @ nat @ Y4 @ B2 ) )
=> ( P @ ( order_Greatest @ nat @ P ) ) ) ) ).
% GreatestI_nat
thf(fact_4963_Greatest__le__nat,axiom,
! [P: nat > $o,K: nat,B2: nat] :
( ( P @ K )
=> ( ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq @ nat @ Y4 @ B2 ) )
=> ( ord_less_eq @ nat @ K @ ( order_Greatest @ nat @ P ) ) ) ) ).
% Greatest_le_nat
thf(fact_4964_GreatestI__ex__nat,axiom,
! [P: nat > $o,B2: nat] :
( ? [X_1: nat] : ( P @ X_1 )
=> ( ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq @ nat @ Y4 @ B2 ) )
=> ( P @ ( order_Greatest @ nat @ P ) ) ) ) ).
% GreatestI_ex_nat
thf(fact_4965_congruent__def,axiom,
! [B: $tType,A: $tType] :
( ( equiv_congruent @ A @ B )
= ( ^ [R4: set @ ( product_prod @ A @ A ),F2: A > B] :
! [X2: product_prod @ A @ A] :
( ( member @ ( product_prod @ A @ A ) @ X2 @ R4 )
=> ( product_case_prod @ A @ A @ $o
@ ^ [Y: A,Z3: A] :
( ( F2 @ Y )
= ( F2 @ Z3 ) )
@ X2 ) ) ) ) ).
% congruent_def
thf(fact_4966_type__copy__set__map0,axiom,
! [A: $tType,B: $tType,D: $tType,E: $tType,C: $tType,F3: $tType,Rep: A > B,Abs: B > A,S3: B > ( set @ D ),M2: C > B,F: E > D,S6: C > ( set @ E ),G: F3 > C] :
( ( type_definition @ A @ B @ Rep @ Abs @ ( top_top @ ( set @ B ) ) )
=> ( ( ( comp @ B @ ( set @ D ) @ C @ S3 @ M2 )
= ( comp @ ( set @ E ) @ ( set @ D ) @ C @ ( image2 @ E @ D @ F ) @ S6 ) )
=> ( ( comp @ A @ ( set @ D ) @ F3 @ ( comp @ B @ ( set @ D ) @ A @ S3 @ Rep ) @ ( comp @ C @ A @ F3 @ ( comp @ B @ A @ C @ Abs @ M2 ) @ G ) )
= ( comp @ ( set @ E ) @ ( set @ D ) @ F3 @ ( image2 @ E @ D @ F ) @ ( comp @ C @ ( set @ E ) @ F3 @ S6 @ G ) ) ) ) ) ).
% type_copy_set_map0
thf(fact_4967_irreflp__irrefl__eq,axiom,
! [A: $tType,R5: set @ ( product_prod @ A @ A )] :
( ( irreflp @ A
@ ^ [A6: A,B5: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A6 @ B5 ) @ R5 ) )
= ( irrefl @ A @ R5 ) ) ).
% irreflp_irrefl_eq
thf(fact_4968_type__copy__obj__one__point__absE,axiom,
! [A: $tType,B: $tType,Rep: A > B,Abs: B > A,S: A] :
( ( type_definition @ A @ B @ Rep @ Abs @ ( top_top @ ( set @ B ) ) )
=> ~ ! [X4: B] :
( S
!= ( Abs @ X4 ) ) ) ).
% type_copy_obj_one_point_absE
thf(fact_4969_type__copy__map__comp0__undo,axiom,
! [E: $tType,A: $tType,C: $tType,B: $tType,D: $tType,F3: $tType,Rep: A > B,Abs: B > A,Rep2: C > D,Abs2: D > C,Rep3: E > F3,Abs3: F3 > E,M2: F3 > D,M1: B > D,M22: F3 > B] :
( ( type_definition @ A @ B @ Rep @ Abs @ ( top_top @ ( set @ B ) ) )
=> ( ( type_definition @ C @ D @ Rep2 @ Abs2 @ ( top_top @ ( set @ D ) ) )
=> ( ( type_definition @ E @ F3 @ Rep3 @ Abs3 @ ( top_top @ ( set @ F3 ) ) )
=> ( ( ( comp @ F3 @ C @ E @ ( comp @ D @ C @ F3 @ Abs2 @ M2 ) @ Rep3 )
= ( comp @ A @ C @ E @ ( comp @ B @ C @ A @ ( comp @ D @ C @ B @ Abs2 @ M1 ) @ Rep ) @ ( comp @ F3 @ A @ E @ ( comp @ B @ A @ F3 @ Abs @ M22 ) @ Rep3 ) ) )
=> ( ( comp @ B @ D @ F3 @ M1 @ M22 )
= M2 ) ) ) ) ) ).
% type_copy_map_comp0_undo
thf(fact_4970_type__copy__map__comp0,axiom,
! [F3: $tType,D: $tType,B: $tType,A: $tType,C: $tType,E: $tType,Rep: A > B,Abs: B > A,M2: C > D,M1: B > D,M22: C > B,F: D > F3,G: E > C] :
( ( type_definition @ A @ B @ Rep @ Abs @ ( top_top @ ( set @ B ) ) )
=> ( ( M2
= ( comp @ B @ D @ C @ M1 @ M22 ) )
=> ( ( comp @ C @ F3 @ E @ ( comp @ D @ F3 @ C @ F @ M2 ) @ G )
= ( comp @ A @ F3 @ E @ ( comp @ B @ F3 @ A @ ( comp @ D @ F3 @ B @ F @ M1 ) @ Rep ) @ ( comp @ C @ A @ E @ ( comp @ B @ A @ C @ Abs @ M22 ) @ G ) ) ) ) ) ).
% type_copy_map_comp0
thf(fact_4971_type__copy__wit,axiom,
! [A: $tType,C: $tType,B: $tType,Rep: A > B,Abs: B > A,X: C,S3: B > ( set @ C ),Y3: B] :
( ( type_definition @ A @ B @ Rep @ Abs @ ( top_top @ ( set @ B ) ) )
=> ( ( member @ C @ X @ ( comp @ B @ ( set @ C ) @ A @ S3 @ Rep @ ( Abs @ Y3 ) ) )
=> ( member @ C @ X @ ( S3 @ Y3 ) ) ) ) ).
% type_copy_wit
thf(fact_4972_congruentD,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),F: A > B,Y3: A,Z: A] :
( ( equiv_congruent @ A @ B @ R @ F )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z ) @ R )
=> ( ( F @ Y3 )
= ( F @ Z ) ) ) ) ).
% congruentD
thf(fact_4973_congruentI,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),F: A > B] :
( ! [Y4: A,Z6: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z6 ) @ R )
=> ( ( F @ Y4 )
= ( F @ Z6 ) ) )
=> ( equiv_congruent @ A @ B @ R @ F ) ) ).
% congruentI
thf(fact_4974_irreflp__greater,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( irreflp @ A
@ ^ [X2: A,Y: A] : ( ord_less @ A @ Y @ X2 ) ) ) ).
% irreflp_greater
thf(fact_4975_type__copy__map__id0,axiom,
! [B: $tType,A: $tType,Rep: A > B,Abs: B > A,M2: B > B] :
( ( type_definition @ A @ B @ Rep @ Abs @ ( top_top @ ( set @ B ) ) )
=> ( ( M2
= ( id @ B ) )
=> ( ( comp @ B @ A @ A @ ( comp @ B @ A @ B @ Abs @ M2 ) @ Rep )
= ( id @ A ) ) ) ) ).
% type_copy_map_id0
thf(fact_4976_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_4977_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_4978_Abs__transfer,axiom,
! [B: $tType,A: $tType,C: $tType,D: $tType,Rep1: A > B,Abs1: B > A,Rep22: C > D,Abs22: D > C,R5: 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 @ R5 @ ( bNF_vimage2p @ A @ B @ C @ D @ $o @ Rep1 @ Rep22 @ R5 ) @ Abs1 @ Abs22 ) ) ) ).
% Abs_transfer
thf(fact_4979_typedef__rep__transfer,axiom,
! [A: $tType,B: $tType,Rep: B > A,Abs: A > B,A5: set @ A,T5: A > B > $o] :
( ( type_definition @ B @ A @ Rep @ Abs @ A5 )
=> ( ( T5
= ( ^ [X2: A,Y: B] :
( X2
= ( Rep @ Y ) ) ) )
=> ( bNF_rel_fun @ A @ B @ A @ A @ T5
@ ^ [Y6: A,Z2: A] : Y6 = Z2
@ ^ [X2: A] : X2
@ Rep ) ) ) ).
% typedef_rep_transfer
thf(fact_4980_curr__in,axiom,
! [A: $tType,B: $tType,C: $tType,F: ( product_prod @ A @ B ) > C,A5: set @ A,B4: 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] : B4 )
@ 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 @ B4 @ C6 ) ) ) ) ).
% curr_in
thf(fact_4981_curr__surj,axiom,
! [C: $tType,B: $tType,A: $tType,G: A > B > C,A5: set @ A,B4: set @ B,C6: set @ C] :
( ( member @ ( A > B > C ) @ G @ ( bNF_Wellorder_Func @ A @ ( B > C ) @ A5 @ ( bNF_Wellorder_Func @ B @ C @ B4 @ C6 ) ) )
=> ? [X4: ( product_prod @ A @ B ) > C] :
( ( member @ ( ( product_prod @ A @ B ) > C ) @ X4
@ ( bNF_Wellorder_Func @ ( product_prod @ A @ B ) @ C
@ ( product_Sigma @ A @ B @ A5
@ ^ [Uu: A] : B4 )
@ C6 ) )
& ( ( bNF_Wellorder_curr @ A @ B @ C @ A5 @ X4 )
= G ) ) ) ).
% curr_surj
thf(fact_4982_curr__def,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( bNF_Wellorder_curr @ A @ B @ C )
= ( ^ [A8: set @ A,F2: ( product_prod @ A @ B ) > C,A6: A] :
( if @ ( B > C ) @ ( member @ A @ A6 @ A8 )
@ ^ [B5: B] : ( F2 @ ( product_Pair @ A @ B @ A6 @ B5 ) )
@ ( undefined @ ( B > C ) ) ) ) ) ).
% curr_def
thf(fact_4983_curr__inj,axiom,
! [C: $tType,B: $tType,A: $tType,F1: ( product_prod @ A @ B ) > C,A5: set @ A,B4: set @ B,C6: set @ C,F23: ( 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] : B4 )
@ C6 ) )
=> ( ( member @ ( ( product_prod @ A @ B ) > C ) @ F23
@ ( bNF_Wellorder_Func @ ( product_prod @ A @ B ) @ C
@ ( product_Sigma @ A @ B @ A5
@ ^ [Uu: A] : B4 )
@ C6 ) )
=> ( ( ( bNF_Wellorder_curr @ A @ B @ C @ A5 @ F1 )
= ( bNF_Wellorder_curr @ A @ B @ C @ A5 @ F23 ) )
= ( F1 = F23 ) ) ) ) ).
% curr_inj
thf(fact_4984_rel__pred__comp__def,axiom,
! [B: $tType,A: $tType] :
( ( rel_pred_comp @ A @ B )
= ( ^ [R6: A > B > $o,P3: B > $o,X2: A] :
? [Y: B] :
( ( R6 @ X2 @ Y )
& ( P3 @ Y ) ) ) ) ).
% rel_pred_comp_def
thf(fact_4985_bij__betw__curr,axiom,
! [A: $tType,B: $tType,C: $tType,A5: set @ A,B4: 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] : B4 )
@ C6 )
@ ( bNF_Wellorder_Func @ A @ ( B > C ) @ A5 @ ( bNF_Wellorder_Func @ B @ C @ B4 @ C6 ) ) ) ).
% bij_betw_curr
thf(fact_4986_init__seg__of__def,axiom,
! [A: $tType] :
( ( init_seg_of @ A )
= ( collect @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) )
@ ( product_case_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ $o
@ ^ [R4: set @ ( product_prod @ A @ A ),S2: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R4 @ S2 )
& ! [A6: A,B5: A,C4: A] :
( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A6 @ B5 ) @ S2 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B5 @ C4 ) @ R4 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A6 @ B5 ) @ R4 ) ) ) ) ) ) ).
% init_seg_of_def
thf(fact_4987_bij__betw__id,axiom,
! [A: $tType,A5: set @ A] : ( bij_betw @ A @ A @ ( id @ A ) @ A5 @ A5 ) ).
% bij_betw_id
thf(fact_4988_bij__betw__add,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [A3: A,A5: set @ A,B4: set @ A] :
( ( bij_betw @ A @ A @ ( plus_plus @ A @ A3 ) @ A5 @ B4 )
= ( ( image2 @ A @ A @ ( plus_plus @ A @ A3 ) @ A5 )
= B4 ) ) ) ).
% bij_betw_add
thf(fact_4989_bij__betw__imp__surj__on,axiom,
! [A: $tType,B: $tType,F: A > B,A5: set @ A,B4: set @ B] :
( ( bij_betw @ A @ B @ F @ A5 @ B4 )
=> ( ( image2 @ A @ B @ F @ A5 )
= B4 ) ) ).
% bij_betw_imp_surj_on
thf(fact_4990_bij__betw__trans,axiom,
! [A: $tType,B: $tType,C: $tType,F: A > B,A5: set @ A,B4: set @ B,G: B > C,C6: set @ C] :
( ( bij_betw @ A @ B @ F @ A5 @ B4 )
=> ( ( bij_betw @ B @ C @ G @ B4 @ C6 )
=> ( bij_betw @ A @ C @ ( comp @ B @ C @ A @ G @ F ) @ A5 @ C6 ) ) ) ).
% bij_betw_trans
thf(fact_4991_bij__betw__comp__iff,axiom,
! [A: $tType,B: $tType,C: $tType,F: A > B,A5: set @ A,A14: set @ B,F5: B > C,A15: set @ C] :
( ( bij_betw @ A @ B @ F @ A5 @ A14 )
=> ( ( bij_betw @ B @ C @ F5 @ A14 @ A15 )
= ( bij_betw @ A @ C @ ( comp @ B @ C @ A @ F5 @ F ) @ A5 @ A15 ) ) ) ).
% bij_betw_comp_iff
thf(fact_4992_bij__betw__id__iff,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( bij_betw @ A @ A @ ( id @ A ) @ A5 @ B4 )
= ( A5 = B4 ) ) ).
% bij_betw_id_iff
thf(fact_4993_bij__betw__iff__bijections,axiom,
! [B: $tType,A: $tType] :
( ( bij_betw @ A @ B )
= ( ^ [F2: A > B,A8: set @ A,B7: set @ B] :
? [G2: B > A] :
( ! [X2: A] :
( ( member @ A @ X2 @ A8 )
=> ( ( member @ B @ ( F2 @ X2 ) @ B7 )
& ( ( G2 @ ( F2 @ X2 ) )
= X2 ) ) )
& ! [X2: B] :
( ( member @ B @ X2 @ B7 )
=> ( ( member @ A @ ( G2 @ X2 ) @ A8 )
& ( ( F2 @ ( G2 @ X2 ) )
= X2 ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_4994_bij__betw__apply,axiom,
! [A: $tType,B: $tType,F: A > B,A5: set @ A,B4: set @ B,A3: A] :
( ( bij_betw @ A @ B @ F @ A5 @ B4 )
=> ( ( member @ A @ A3 @ A5 )
=> ( member @ B @ ( F @ A3 ) @ B4 ) ) ) ).
% bij_betw_apply
thf(fact_4995_bij__betw__cong,axiom,
! [A: $tType,B: $tType,A5: set @ A,F: A > B,G: A > B,A14: set @ B] :
( ! [A4: A] :
( ( member @ A @ A4 @ A5 )
=> ( ( F @ A4 )
= ( G @ A4 ) ) )
=> ( ( bij_betw @ A @ B @ F @ A5 @ A14 )
= ( bij_betw @ A @ B @ G @ A5 @ A14 ) ) ) ).
% bij_betw_cong
thf(fact_4996_bij__betw__inv,axiom,
! [B: $tType,A: $tType,F: A > B,A5: set @ A,B4: set @ B] :
( ( bij_betw @ A @ B @ F @ A5 @ B4 )
=> ? [G4: B > A] : ( bij_betw @ B @ A @ G4 @ B4 @ A5 ) ) ).
% bij_betw_inv
thf(fact_4997_bij__betwE,axiom,
! [A: $tType,B: $tType,F: A > B,A5: set @ A,B4: set @ B] :
( ( bij_betw @ A @ B @ F @ A5 @ B4 )
=> ! [X5: A] :
( ( member @ A @ X5 @ A5 )
=> ( member @ B @ ( F @ X5 ) @ B4 ) ) ) ).
% bij_betwE
thf(fact_4998_bij__iff,axiom,
! [A: $tType,B: $tType,F: A > B] :
( ( bij_betw @ A @ B @ F @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
= ( ! [X2: B] :
? [Y: A] :
( ( ( F @ Y )
= X2 )
& ! [Z3: A] :
( ( ( F @ Z3 )
= X2 )
=> ( Z3 = Y ) ) ) ) ) ).
% bij_iff
thf(fact_4999_bij__pointE,axiom,
! [B: $tType,A: $tType,F: A > B,Y3: B] :
( ( bij_betw @ A @ B @ F @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
=> ~ ! [X4: A] :
( ( Y3
= ( F @ X4 ) )
=> ~ ! [X14: A] :
( ( Y3
= ( F @ X14 ) )
=> ( X14 = X4 ) ) ) ) ).
% bij_pointE
thf(fact_5000_involuntory__imp__bij,axiom,
! [A: $tType,F: A > A] :
( ! [X4: A] :
( ( F @ ( F @ X4 ) )
= X4 )
=> ( bij_betw @ A @ A @ F @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ A ) ) ) ) ).
% involuntory_imp_bij
thf(fact_5001_bij__betw__funpow,axiom,
! [A: $tType,F: A > A,S3: set @ A,N: nat] :
( ( bij_betw @ A @ A @ F @ S3 @ S3 )
=> ( bij_betw @ A @ A @ ( compow @ ( A > A ) @ N @ F ) @ S3 @ S3 ) ) ).
% bij_betw_funpow
thf(fact_5002_bij__betw__the__inv__into,axiom,
! [B: $tType,A: $tType,F: A > B,A5: set @ A,B4: set @ B] :
( ( bij_betw @ A @ B @ F @ A5 @ B4 )
=> ( bij_betw @ B @ A @ ( the_inv_into @ A @ B @ A5 @ F ) @ B4 @ A5 ) ) ).
% bij_betw_the_inv_into
thf(fact_5003_f__the__inv__into__f__bij__betw,axiom,
! [A: $tType,B: $tType,F: A > B,A5: set @ A,B4: set @ B,X: B] :
( ( bij_betw @ A @ B @ F @ A5 @ B4 )
=> ( ( ( bij_betw @ A @ B @ F @ A5 @ B4 )
=> ( member @ B @ X @ B4 ) )
=> ( ( F @ ( the_inv_into @ A @ B @ A5 @ F @ X ) )
= X ) ) ) ).
% f_the_inv_into_f_bij_betw
thf(fact_5004_bij__betw__imp__inj__on,axiom,
! [B: $tType,A: $tType,F: A > B,A5: set @ A,B4: set @ B] :
( ( bij_betw @ A @ B @ F @ A5 @ B4 )
=> ( inj_on @ A @ B @ F @ A5 ) ) ).
% bij_betw_imp_inj_on
thf(fact_5005_sum_Oreindex__bij__betw,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comm_monoid_add @ A )
=> ! [H: B > C,S3: set @ B,T5: set @ C,G: C > A] :
( ( bij_betw @ B @ C @ H @ S3 @ T5 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X2: B] : ( G @ ( H @ X2 ) )
@ S3 )
= ( groups7311177749621191930dd_sum @ C @ A @ G @ T5 ) ) ) ) ).
% sum.reindex_bij_betw
thf(fact_5006_prod_Oreindex__bij__betw,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comm_monoid_mult @ A )
=> ! [H: B > C,S3: set @ B,T5: set @ C,G: C > A] :
( ( bij_betw @ B @ C @ H @ S3 @ T5 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [X2: B] : ( G @ ( H @ X2 ) )
@ S3 )
= ( groups7121269368397514597t_prod @ C @ A @ G @ T5 ) ) ) ) ).
% prod.reindex_bij_betw
thf(fact_5007_bij__betw__empty2,axiom,
! [B: $tType,A: $tType,F: A > B,A5: set @ A] :
( ( bij_betw @ A @ B @ F @ A5 @ ( bot_bot @ ( set @ B ) ) )
=> ( A5
= ( bot_bot @ ( set @ A ) ) ) ) ).
% bij_betw_empty2
thf(fact_5008_bij__betw__empty1,axiom,
! [A: $tType,B: $tType,F: A > B,A5: set @ B] :
( ( bij_betw @ A @ B @ F @ ( bot_bot @ ( set @ A ) ) @ A5 )
=> ( A5
= ( bot_bot @ ( set @ B ) ) ) ) ).
% bij_betw_empty1
thf(fact_5009_bij__betw__subset,axiom,
! [A: $tType,B: $tType,F: A > B,A5: set @ A,A14: set @ B,B4: set @ A,B13: set @ B] :
( ( bij_betw @ A @ B @ F @ A5 @ A14 )
=> ( ( ord_less_eq @ ( set @ A ) @ B4 @ A5 )
=> ( ( ( image2 @ A @ B @ F @ B4 )
= B13 )
=> ( bij_betw @ A @ B @ F @ B4 @ B13 ) ) ) ) ).
% bij_betw_subset
thf(fact_5010_bij__betw__byWitness,axiom,
! [A: $tType,B: $tType,A5: set @ A,F5: B > A,F: A > B,A14: set @ B] :
( ! [X4: A] :
( ( member @ A @ X4 @ A5 )
=> ( ( F5 @ ( F @ X4 ) )
= X4 ) )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ A14 )
=> ( ( F @ ( F5 @ X4 ) )
= X4 ) )
=> ( ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F @ A5 ) @ A14 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image2 @ B @ A @ F5 @ A14 ) @ A5 )
=> ( bij_betw @ A @ B @ F @ A5 @ A14 ) ) ) ) ) ).
% bij_betw_byWitness
thf(fact_5011_bij__betw__imp__surj,axiom,
! [A: $tType,B: $tType,F: A > B,A5: set @ A] :
( ( bij_betw @ A @ B @ F @ A5 @ ( top_top @ ( set @ B ) ) )
=> ( ( image2 @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ B ) ) ) ) ).
% bij_betw_imp_surj
thf(fact_5012_bij__is__surj,axiom,
! [A: $tType,B: $tType,F: A > B] :
( ( bij_betw @ A @ B @ F @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
=> ( ( image2 @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ B ) ) ) ) ).
% bij_is_surj
thf(fact_5013_bij__uminus,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ( bij_betw @ A @ A @ ( uminus_uminus @ A ) @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ A ) ) ) ) ).
% bij_uminus
thf(fact_5014_bij__comp,axiom,
! [A: $tType,B: $tType,C: $tType,F: A > B,G: B > C] :
( ( bij_betw @ A @ B @ F @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
=> ( ( bij_betw @ B @ C @ G @ ( top_top @ ( set @ B ) ) @ ( top_top @ ( set @ C ) ) )
=> ( bij_betw @ A @ C @ ( comp @ B @ C @ A @ G @ F ) @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ C ) ) ) ) ) ).
% bij_comp
thf(fact_5015_bij__betw__def,axiom,
! [B: $tType,A: $tType] :
( ( bij_betw @ A @ B )
= ( ^ [F2: A > B,A8: set @ A,B7: set @ B] :
( ( inj_on @ A @ B @ F2 @ A8 )
& ( ( image2 @ A @ B @ F2 @ A8 )
= B7 ) ) ) ) ).
% bij_betw_def
thf(fact_5016_bij__betw__imageI,axiom,
! [A: $tType,B: $tType,F: A > B,A5: set @ A,B4: set @ B] :
( ( inj_on @ A @ B @ F @ A5 )
=> ( ( ( image2 @ A @ B @ F @ A5 )
= B4 )
=> ( bij_betw @ A @ B @ F @ A5 @ B4 ) ) ) ).
% bij_betw_imageI
thf(fact_5017_inj__on__imp__bij__betw,axiom,
! [B: $tType,A: $tType,F: A > B,A5: set @ A] :
( ( inj_on @ A @ B @ F @ A5 )
=> ( bij_betw @ A @ B @ F @ A5 @ ( image2 @ A @ B @ F @ A5 ) ) ) ).
% inj_on_imp_bij_betw
thf(fact_5018_bij__is__inj,axiom,
! [B: $tType,A: $tType,F: A > B] :
( ( bij_betw @ A @ B @ F @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
=> ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) ) ) ).
% bij_is_inj
thf(fact_5019_bij__id,axiom,
! [A: $tType] : ( bij_betw @ A @ A @ ( id @ A ) @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ A ) ) ) ).
% bij_id
thf(fact_5020_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_5021_trans__init__seg__of,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ A @ A ),T2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R @ S ) @ ( init_seg_of @ A ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ S @ T2 ) @ ( init_seg_of @ A ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R @ T2 ) @ ( init_seg_of @ A ) ) ) ) ).
% trans_init_seg_of
thf(fact_5022_antisym__init__seg__of,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R @ S ) @ ( init_seg_of @ A ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ S @ R ) @ ( init_seg_of @ A ) )
=> ( R = S ) ) ) ).
% antisym_init_seg_of
thf(fact_5023_refl__on__init__seg__of,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] : ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R @ R ) @ ( init_seg_of @ A ) ) ).
% refl_on_init_seg_of
thf(fact_5024_bij__betw__comp__iff2,axiom,
! [C: $tType,A: $tType,B: $tType,F5: A > B,A14: set @ A,A15: set @ B,F: C > A,A5: set @ C] :
( ( bij_betw @ A @ B @ F5 @ A14 @ A15 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image2 @ C @ A @ F @ A5 ) @ A14 )
=> ( ( bij_betw @ C @ A @ F @ A5 @ A14 )
= ( bij_betw @ C @ B @ ( comp @ A @ B @ C @ F5 @ F ) @ A5 @ A15 ) ) ) ) ).
% bij_betw_comp_iff2
thf(fact_5025_notIn__Un__bij__betw3,axiom,
! [A: $tType,B: $tType,B2: A,A5: set @ A,F: A > B,A14: set @ B] :
( ~ ( member @ A @ B2 @ A5 )
=> ( ~ ( member @ B @ ( F @ B2 ) @ A14 )
=> ( ( bij_betw @ A @ B @ F @ A5 @ A14 )
= ( bij_betw @ A @ B @ F @ ( sup_sup @ ( set @ A ) @ A5 @ ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( sup_sup @ ( set @ B ) @ A14 @ ( insert2 @ B @ ( F @ B2 ) @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ).
% notIn_Un_bij_betw3
thf(fact_5026_notIn__Un__bij__betw,axiom,
! [A: $tType,B: $tType,B2: A,A5: set @ A,F: A > B,A14: set @ B] :
( ~ ( member @ A @ B2 @ A5 )
=> ( ~ ( member @ B @ ( F @ B2 ) @ A14 )
=> ( ( bij_betw @ A @ B @ F @ A5 @ A14 )
=> ( bij_betw @ A @ B @ F @ ( sup_sup @ ( set @ A ) @ A5 @ ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( sup_sup @ ( set @ B ) @ A14 @ ( insert2 @ B @ ( F @ B2 ) @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ).
% notIn_Un_bij_betw
thf(fact_5027_bijI,axiom,
! [A: $tType,B: $tType,F: A > B] :
( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
=> ( ( ( image2 @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ B ) ) )
=> ( bij_betw @ A @ B @ F @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) ) ) ) ).
% bijI
thf(fact_5028_bij__def,axiom,
! [A: $tType,B: $tType,F: A > B] :
( ( bij_betw @ A @ B @ F @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
= ( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
& ( ( image2 @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ B ) ) ) ) ) ).
% bij_def
thf(fact_5029_bij__betw__partition,axiom,
! [A: $tType,B: $tType,F: A > B,A5: set @ A,C6: set @ A,B4: set @ B,D5: set @ B] :
( ( bij_betw @ A @ B @ F @ ( sup_sup @ ( set @ A ) @ A5 @ C6 ) @ ( sup_sup @ ( set @ B ) @ B4 @ D5 ) )
=> ( ( bij_betw @ A @ B @ F @ C6 @ D5 )
=> ( ( ( inf_inf @ ( set @ A ) @ A5 @ C6 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( inf_inf @ ( set @ B ) @ B4 @ D5 )
= ( bot_bot @ ( set @ B ) ) )
=> ( bij_betw @ A @ B @ F @ A5 @ B4 ) ) ) ) ) ).
% bij_betw_partition
thf(fact_5030_bij__betw__combine,axiom,
! [A: $tType,B: $tType,F: A > B,A5: set @ A,B4: set @ B,C6: set @ A,D5: set @ B] :
( ( bij_betw @ A @ B @ F @ A5 @ B4 )
=> ( ( bij_betw @ A @ B @ F @ C6 @ D5 )
=> ( ( ( inf_inf @ ( set @ B ) @ B4 @ D5 )
= ( bot_bot @ ( set @ B ) ) )
=> ( bij_betw @ A @ B @ F @ ( sup_sup @ ( set @ A ) @ A5 @ C6 ) @ ( sup_sup @ ( set @ B ) @ B4 @ D5 ) ) ) ) ) ).
% bij_betw_combine
thf(fact_5031_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_5032_bij__image__Compl__eq,axiom,
! [B: $tType,A: $tType,F: A > B,A5: set @ A] :
( ( bij_betw @ A @ B @ F @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
=> ( ( image2 @ A @ B @ F @ ( uminus_uminus @ ( set @ A ) @ A5 ) )
= ( uminus_uminus @ ( set @ B ) @ ( image2 @ A @ B @ F @ A5 ) ) ) ) ).
% bij_image_Compl_eq
thf(fact_5033_bij__betw__disjoint__Un,axiom,
! [A: $tType,B: $tType,F: A > B,A5: set @ A,C6: set @ B,G: A > B,B4: set @ A,D5: set @ B] :
( ( bij_betw @ A @ B @ F @ A5 @ C6 )
=> ( ( bij_betw @ A @ B @ G @ B4 @ D5 )
=> ( ( ( inf_inf @ ( set @ A ) @ A5 @ B4 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( inf_inf @ ( set @ B ) @ C6 @ D5 )
= ( bot_bot @ ( set @ B ) ) )
=> ( bij_betw @ A @ B
@ ^ [X2: A] : ( if @ B @ ( member @ A @ X2 @ A5 ) @ ( F @ X2 ) @ ( G @ X2 ) )
@ ( sup_sup @ ( set @ A ) @ A5 @ B4 )
@ ( sup_sup @ ( set @ B ) @ C6 @ D5 ) ) ) ) ) ) ).
% bij_betw_disjoint_Un
thf(fact_5034_bij__betw__UNION__chain,axiom,
! [B: $tType,C: $tType,A: $tType,I5: set @ A,A5: A > ( set @ B ),F: B > C,A14: 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 ) @ ( A14 @ 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 ) @ A14 @ I5 ) ) ) ) ) ).
% bij_betw_UNION_chain
thf(fact_5035_vimage__subset__eq,axiom,
! [B: $tType,A: $tType,F: A > B,B4: set @ B,A5: set @ A] :
( ( bij_betw @ A @ B @ F @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ ( vimage @ A @ B @ F @ B4 ) @ A5 )
= ( ord_less_eq @ ( set @ B ) @ B4 @ ( image2 @ A @ B @ F @ A5 ) ) ) ) ).
% vimage_subset_eq
thf(fact_5036_sum_Oreindex__bij__betw__not__neutral,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comm_monoid_add @ A )
=> ! [S6: set @ B,T4: set @ C,H: B > C,S3: set @ B,T5: set @ C,G: C > A] :
( ( finite_finite2 @ B @ S6 )
=> ( ( finite_finite2 @ C @ T4 )
=> ( ( bij_betw @ B @ C @ H @ ( minus_minus @ ( set @ B ) @ S3 @ S6 ) @ ( minus_minus @ ( set @ C ) @ T5 @ T4 ) )
=> ( ! [A4: B] :
( ( member @ B @ A4 @ S6 )
=> ( ( G @ ( H @ A4 ) )
= ( zero_zero @ A ) ) )
=> ( ! [B3: C] :
( ( member @ C @ B3 @ T4 )
=> ( ( G @ B3 )
= ( zero_zero @ A ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X2: B] : ( G @ ( H @ X2 ) )
@ S3 )
= ( groups7311177749621191930dd_sum @ C @ A @ G @ T5 ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_betw_not_neutral
thf(fact_5037_prod_Oreindex__bij__betw__not__neutral,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S6: set @ B,T4: set @ C,H: B > C,S3: set @ B,T5: set @ C,G: C > A] :
( ( finite_finite2 @ B @ S6 )
=> ( ( finite_finite2 @ C @ T4 )
=> ( ( bij_betw @ B @ C @ H @ ( minus_minus @ ( set @ B ) @ S3 @ S6 ) @ ( minus_minus @ ( set @ C ) @ T5 @ T4 ) )
=> ( ! [A4: B] :
( ( member @ B @ A4 @ S6 )
=> ( ( G @ ( H @ A4 ) )
= ( one_one @ A ) ) )
=> ( ! [B3: C] :
( ( member @ C @ B3 @ T4 )
=> ( ( G @ B3 )
= ( one_one @ A ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [X2: B] : ( G @ ( H @ X2 ) )
@ S3 )
= ( groups7121269368397514597t_prod @ C @ A @ G @ T5 ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_betw_not_neutral
thf(fact_5038_initial__segment__of__Diff,axiom,
! [A: $tType,P4: set @ ( product_prod @ A @ A ),Q2: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ P4 @ Q2 ) @ ( init_seg_of @ A ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ P4 @ S ) @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ Q2 @ S ) ) @ ( init_seg_of @ A ) ) ) ).
% initial_segment_of_Diff
thf(fact_5039_bij__image__INT,axiom,
! [B: $tType,A: $tType,C: $tType,F: A > B,B4: 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 ) @ B4 @ A5 ) ) )
= ( complete_Inf_Inf @ ( set @ B )
@ ( image2 @ C @ ( set @ B )
@ ^ [X2: C] : ( image2 @ A @ B @ F @ ( B4 @ X2 ) )
@ A5 ) ) ) ) ).
% bij_image_INT
thf(fact_5040_Chains__init__seg__of__Union,axiom,
! [A: $tType,R5: set @ ( set @ ( product_prod @ A @ A ) ),R: set @ ( product_prod @ A @ A )] :
( ( member @ ( set @ ( set @ ( product_prod @ A @ A ) ) ) @ R5 @ ( chains @ ( set @ ( product_prod @ A @ A ) ) @ ( init_seg_of @ A ) ) )
=> ( ( member @ ( set @ ( product_prod @ A @ A ) ) @ R @ R5 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R @ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ A ) ) @ R5 ) ) @ ( init_seg_of @ A ) ) ) ) ).
% Chains_init_seg_of_Union
thf(fact_5041_minus__coset__filter,axiom,
! [A: $tType,A5: set @ A,Xs2: list @ A] :
( ( minus_minus @ ( set @ A ) @ A5 @ ( coset @ A @ Xs2 ) )
= ( set2 @ A
@ ( filter2 @ A
@ ^ [X2: A] : ( member @ A @ X2 @ A5 )
@ Xs2 ) ) ) ).
% minus_coset_filter
thf(fact_5042_flat__lub__def,axiom,
! [A: $tType] :
( ( partial_flat_lub @ A )
= ( ^ [B5: A,A8: set @ A] :
( if @ A @ ( ord_less_eq @ ( set @ A ) @ A8 @ ( insert2 @ A @ B5 @ ( bot_bot @ ( set @ A ) ) ) ) @ B5
@ ( the @ A
@ ^ [X2: A] : ( member @ A @ X2 @ ( minus_minus @ ( set @ A ) @ A8 @ ( insert2 @ A @ B5 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% flat_lub_def
thf(fact_5043_bij__betw__Suc,axiom,
! [M2: set @ nat,N6: set @ nat] :
( ( bij_betw @ nat @ nat @ suc @ M2 @ N6 )
= ( ( image2 @ nat @ nat @ suc @ M2 )
= N6 ) ) ).
% bij_betw_Suc
thf(fact_5044_bij__swap,axiom,
! [A: $tType,B: $tType] : ( bij_betw @ ( product_prod @ A @ B ) @ ( product_prod @ B @ A ) @ ( product_swap @ A @ B ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) @ ( top_top @ ( set @ ( product_prod @ B @ A ) ) ) ) ).
% bij_swap
thf(fact_5045_ex__bij__betw__finite__nat,axiom,
! [A: $tType,M2: set @ A] :
( ( finite_finite2 @ A @ M2 )
=> ? [H6: A > nat] : ( bij_betw @ A @ nat @ H6 @ M2 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( finite_card @ A @ M2 ) ) ) ) ).
% ex_bij_betw_finite_nat
thf(fact_5046_ex__bij__betw__nat__finite,axiom,
! [A: $tType,M2: set @ A] :
( ( finite_finite2 @ A @ M2 )
=> ? [H6: nat > A] : ( bij_betw @ nat @ A @ H6 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( finite_card @ A @ M2 ) ) @ M2 ) ) ).
% ex_bij_betw_nat_finite
thf(fact_5047_ex__bij__betw__nat__finite__1,axiom,
! [A: $tType,M2: set @ A] :
( ( finite_finite2 @ A @ M2 )
=> ? [H6: nat > A] : ( bij_betw @ nat @ A @ H6 @ ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ ( finite_card @ A @ M2 ) ) @ M2 ) ) ).
% ex_bij_betw_nat_finite_1
thf(fact_5048_union__coset__filter,axiom,
! [A: $tType,Xs2: list @ A,A5: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( coset @ A @ Xs2 ) @ A5 )
= ( coset @ A
@ ( filter2 @ A
@ ^ [X2: A] :
~ ( member @ A @ X2 @ A5 )
@ Xs2 ) ) ) ).
% union_coset_filter
thf(fact_5049_Chains__def,axiom,
! [A: $tType] :
( ( chains @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A )] :
( collect @ ( set @ A )
@ ^ [C7: set @ A] :
! [X2: A] :
( ( member @ A @ X2 @ C7 )
=> ! [Y: A] :
( ( member @ A @ Y @ C7 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y ) @ R4 )
| ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X2 ) @ R4 ) ) ) ) ) ) ) ).
% Chains_def
thf(fact_5050_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_5051_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_5052_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_5053_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_5054_Chains__inits__DiffI,axiom,
! [A: $tType,R5: set @ ( set @ ( product_prod @ A @ A ) ),S: set @ ( product_prod @ A @ A )] :
( ( member @ ( set @ ( set @ ( product_prod @ A @ A ) ) ) @ R5 @ ( chains @ ( set @ ( product_prod @ A @ A ) ) @ ( init_seg_of @ A ) ) )
=> ( member @ ( set @ ( set @ ( product_prod @ A @ A ) ) )
@ ( collect @ ( set @ ( product_prod @ A @ A ) )
@ ^ [Uu: set @ ( product_prod @ A @ A )] :
? [R4: set @ ( product_prod @ A @ A )] :
( ( Uu
= ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R4 @ S ) )
& ( member @ ( set @ ( product_prod @ A @ A ) ) @ R4 @ R5 ) ) )
@ ( chains @ ( set @ ( product_prod @ A @ A ) ) @ ( init_seg_of @ A ) ) ) ) ).
% Chains_inits_DiffI
thf(fact_5055_Chains__subset,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ord_less_eq @ ( set @ ( set @ A ) ) @ ( chains @ A @ R )
@ ( collect @ ( set @ A )
@ ( pred_chain @ A @ ( top_top @ ( set @ A ) )
@ ^ [X2: A,Y: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y ) @ R ) ) ) ) ).
% Chains_subset
thf(fact_5056_arg__min__inj__eq,axiom,
! [B: $tType,A: $tType] :
( ( order @ B )
=> ! [F: A > B,P: A > $o,A3: A] :
( ( inj_on @ A @ B @ F @ ( collect @ A @ P ) )
=> ( ( P @ A3 )
=> ( ! [Y4: A] :
( ( P @ Y4 )
=> ( ord_less_eq @ B @ ( F @ A3 ) @ ( F @ Y4 ) ) )
=> ( ( lattices_ord_arg_min @ A @ B @ F @ P )
= A3 ) ) ) ) ) ).
% arg_min_inj_eq
thf(fact_5057_in__measures_I2_J,axiom,
! [A: $tType,X: A,Y3: A,F: A > nat,Fs: list @ ( A > nat )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ ( measures @ A @ ( cons @ ( A > nat ) @ F @ Fs ) ) )
= ( ( ord_less @ nat @ ( F @ X ) @ ( F @ Y3 ) )
| ( ( ( F @ X )
= ( F @ Y3 ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ ( measures @ A @ Fs ) ) ) ) ) ).
% in_measures(2)
thf(fact_5058_in__measures_I1_J,axiom,
! [A: $tType,X: A,Y3: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ ( measures @ A @ ( nil @ ( A > nat ) ) ) ) ).
% in_measures(1)
thf(fact_5059_measures__less,axiom,
! [A: $tType,F: A > nat,X: A,Y3: A,Fs: list @ ( A > nat )] :
( ( ord_less @ nat @ ( F @ X ) @ ( F @ Y3 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ ( measures @ A @ ( cons @ ( A > nat ) @ F @ Fs ) ) ) ) ).
% measures_less
thf(fact_5060_measures__lesseq,axiom,
! [A: $tType,F: A > nat,X: A,Y3: A,Fs: list @ ( A > nat )] :
( ( ord_less_eq @ nat @ ( F @ X ) @ ( F @ Y3 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ ( measures @ A @ Fs ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ ( measures @ A @ ( cons @ ( A > nat ) @ F @ Fs ) ) ) ) ) ).
% measures_lesseq
thf(fact_5061_Chains__subset_H,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( refl_on @ A @ ( top_top @ ( set @ A ) ) @ R )
=> ( ord_less_eq @ ( set @ ( set @ A ) )
@ ( collect @ ( set @ A )
@ ( pred_chain @ A @ ( top_top @ ( set @ A ) )
@ ^ [X2: A,Y: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y ) @ R ) ) )
@ ( chains @ A @ R ) ) ) ).
% Chains_subset'
thf(fact_5062_refl__on__singleton,axiom,
! [A: $tType,X: A] : ( refl_on @ A @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) @ ( insert2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ X ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ).
% refl_on_singleton
thf(fact_5063_UN__equiv__class,axiom,
! [B: $tType,A: $tType,A5: set @ A,R: set @ ( product_prod @ A @ A ),F: A > ( set @ B ),A3: A] :
( ( equiv_equiv @ A @ A5 @ R )
=> ( ( equiv_congruent @ A @ ( set @ B ) @ R @ F )
=> ( ( member @ A @ A3 @ A5 )
=> ( ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ F @ ( image @ A @ A @ R @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) )
= ( F @ A3 ) ) ) ) ) ).
% UN_equiv_class
thf(fact_5064_refl__onD,axiom,
! [A: $tType,A5: set @ A,R: set @ ( product_prod @ A @ A ),A3: A] :
( ( refl_on @ A @ A5 @ R )
=> ( ( member @ A @ A3 @ A5 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ A3 ) @ R ) ) ) ).
% refl_onD
thf(fact_5065_refl__onD1,axiom,
! [A: $tType,A5: set @ A,R: set @ ( product_prod @ A @ A ),X: A,Y3: A] :
( ( refl_on @ A @ A5 @ R )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ R )
=> ( member @ A @ X @ A5 ) ) ) ).
% refl_onD1
thf(fact_5066_refl__onD2,axiom,
! [A: $tType,A5: set @ A,R: set @ ( product_prod @ A @ A ),X: A,Y3: A] :
( ( refl_on @ A @ A5 @ R )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ R )
=> ( member @ A @ Y3 @ A5 ) ) ) ).
% refl_onD2
thf(fact_5067_refl__on__domain,axiom,
! [A: $tType,A5: set @ A,R: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
( ( refl_on @ A @ A5 @ R )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R )
=> ( ( member @ A @ A3 @ A5 )
& ( member @ A @ B2 @ A5 ) ) ) ) ).
% refl_on_domain
thf(fact_5068_quotient__eqI,axiom,
! [A: $tType,A5: set @ A,R: set @ ( product_prod @ A @ A ),X7: set @ A,Y9: set @ A,X: A,Y3: A] :
( ( equiv_equiv @ A @ A5 @ R )
=> ( ( member @ ( set @ A ) @ X7 @ ( equiv_quotient @ A @ A5 @ R ) )
=> ( ( member @ ( set @ A ) @ Y9 @ ( equiv_quotient @ A @ A5 @ R ) )
=> ( ( member @ A @ X @ X7 )
=> ( ( member @ A @ Y3 @ Y9 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ R )
=> ( X7 = Y9 ) ) ) ) ) ) ) ).
% quotient_eqI
thf(fact_5069_quotient__eq__iff,axiom,
! [A: $tType,A5: set @ A,R: set @ ( product_prod @ A @ A ),X7: set @ A,Y9: set @ A,X: A,Y3: A] :
( ( equiv_equiv @ A @ A5 @ R )
=> ( ( member @ ( set @ A ) @ X7 @ ( equiv_quotient @ A @ A5 @ R ) )
=> ( ( member @ ( set @ A ) @ Y9 @ ( equiv_quotient @ A @ A5 @ R ) )
=> ( ( member @ A @ X @ X7 )
=> ( ( member @ A @ Y3 @ Y9 )
=> ( ( X7 = Y9 )
= ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ R ) ) ) ) ) ) ) ).
% quotient_eq_iff
thf(fact_5070_in__quotient__imp__closed,axiom,
! [A: $tType,A5: set @ A,R: set @ ( product_prod @ A @ A ),X7: set @ A,X: A,Y3: A] :
( ( equiv_equiv @ A @ A5 @ R )
=> ( ( member @ ( set @ A ) @ X7 @ ( equiv_quotient @ A @ A5 @ R ) )
=> ( ( member @ A @ X @ X7 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ R )
=> ( member @ A @ Y3 @ X7 ) ) ) ) ) ).
% in_quotient_imp_closed
thf(fact_5071_congruent2__commuteI,axiom,
! [B: $tType,A: $tType,A5: set @ A,R: set @ ( product_prod @ A @ A ),F: A > A > B] :
( ( equiv_equiv @ A @ A5 @ R )
=> ( ! [Y4: A,Z6: A] :
( ( member @ A @ Y4 @ A5 )
=> ( ( member @ A @ Z6 @ A5 )
=> ( ( F @ Y4 @ Z6 )
= ( F @ Z6 @ Y4 ) ) ) )
=> ( ! [Y4: A,Z6: A,W3: A] :
( ( member @ A @ W3 @ A5 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z6 ) @ R )
=> ( ( F @ W3 @ Y4 )
= ( F @ W3 @ Z6 ) ) ) )
=> ( equiv_congruent2 @ A @ A @ B @ R @ R @ F ) ) ) ) ).
% congruent2_commuteI
thf(fact_5072_congruent2I,axiom,
! [C: $tType,B: $tType,A: $tType,A16: set @ A,R1: set @ ( product_prod @ A @ A ),A25: set @ B,R22: set @ ( product_prod @ B @ B ),F: A > B > C] :
( ( equiv_equiv @ A @ A16 @ R1 )
=> ( ( equiv_equiv @ B @ A25 @ R22 )
=> ( ! [Y4: A,Z6: A,W3: B] :
( ( member @ B @ W3 @ A25 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z6 ) @ R1 )
=> ( ( F @ Y4 @ W3 )
= ( F @ Z6 @ W3 ) ) ) )
=> ( ! [Y4: B,Z6: B,W3: A] :
( ( member @ A @ W3 @ A16 )
=> ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y4 @ Z6 ) @ R22 )
=> ( ( F @ W3 @ Y4 )
= ( F @ W3 @ Z6 ) ) ) )
=> ( equiv_congruent2 @ A @ B @ C @ R1 @ R22 @ F ) ) ) ) ) ).
% congruent2I
thf(fact_5073_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_5074_UN__equiv__class__type2,axiom,
! [A: $tType,B: $tType,C: $tType,A16: set @ A,R1: set @ ( product_prod @ A @ A ),A25: set @ B,R22: set @ ( product_prod @ B @ B ),F: A > B > ( set @ C ),X15: set @ A,X25: set @ B,B4: set @ ( set @ C )] :
( ( equiv_equiv @ A @ A16 @ R1 )
=> ( ( equiv_equiv @ B @ A25 @ R22 )
=> ( ( equiv_congruent2 @ A @ B @ ( set @ C ) @ R1 @ R22 @ F )
=> ( ( member @ ( set @ A ) @ X15 @ ( equiv_quotient @ A @ A16 @ R1 ) )
=> ( ( member @ ( set @ B ) @ X25 @ ( equiv_quotient @ B @ A25 @ R22 ) )
=> ( ! [X13: A,X23: B] :
( ( member @ A @ X13 @ A16 )
=> ( ( member @ B @ X23 @ A25 )
=> ( member @ ( set @ C ) @ ( F @ X13 @ X23 ) @ B4 ) ) )
=> ( 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 ) )
@ X15 ) )
@ B4 ) ) ) ) ) ) ) ).
% UN_equiv_class_type2
thf(fact_5075_UN__equiv__class__type,axiom,
! [A: $tType,B: $tType,A5: set @ A,R: set @ ( product_prod @ A @ A ),F: A > ( set @ B ),X7: set @ A,B4: set @ ( set @ B )] :
( ( equiv_equiv @ A @ A5 @ R )
=> ( ( equiv_congruent @ A @ ( set @ B ) @ R @ F )
=> ( ( member @ ( set @ A ) @ X7 @ ( equiv_quotient @ A @ A5 @ R ) )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ A5 )
=> ( member @ ( set @ B ) @ ( F @ X4 ) @ B4 ) )
=> ( member @ ( set @ B ) @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ F @ X7 ) ) @ B4 ) ) ) ) ) ).
% UN_equiv_class_type
thf(fact_5076_refl__on__def,axiom,
! [A: $tType] :
( ( refl_on @ A )
= ( ^ [A8: set @ A,R4: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R4
@ ( product_Sigma @ A @ A @ A8
@ ^ [Uu: A] : A8 ) )
& ! [X2: A] :
( ( member @ A @ X2 @ A8 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ X2 ) @ R4 ) ) ) ) ) ).
% refl_on_def
thf(fact_5077_refl__onI,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 ) )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ A5 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ X4 ) @ R ) )
=> ( refl_on @ A @ A5 @ R ) ) ) ).
% refl_onI
thf(fact_5078_Refl__Restr,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A5: set @ A] :
( ( refl_on @ A @ ( field2 @ A @ R ) @ R )
=> ( refl_on @ A
@ ( field2 @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R
@ ( product_Sigma @ A @ A @ A5
@ ^ [Uu: A] : A5 ) ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R
@ ( product_Sigma @ A @ A @ A5
@ ^ [Uu: A] : A5 ) ) ) ) ).
% Refl_Restr
thf(fact_5079_refl__on__def_H,axiom,
! [A: $tType] :
( ( refl_on @ A )
= ( ^ [A8: set @ A,R4: set @ ( product_prod @ A @ A )] :
( ! [X2: product_prod @ A @ A] :
( ( member @ ( product_prod @ A @ A ) @ X2 @ R4 )
=> ( product_case_prod @ A @ A @ $o
@ ^ [Y: A,Z3: A] :
( ( member @ A @ Y @ A8 )
& ( member @ A @ Z3 @ A8 ) )
@ X2 ) )
& ! [X2: A] :
( ( member @ A @ X2 @ A8 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ X2 ) @ R4 ) ) ) ) ) ).
% refl_on_def'
thf(fact_5080_equiv__class__eq__iff,axiom,
! [A: $tType,A5: set @ A,R: set @ ( product_prod @ A @ A ),X: A,Y3: A] :
( ( equiv_equiv @ A @ A5 @ R )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ R )
= ( ( ( image @ A @ A @ R @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( image @ A @ A @ R @ ( insert2 @ A @ Y3 @ ( bot_bot @ ( set @ A ) ) ) ) )
& ( member @ A @ X @ A5 )
& ( member @ A @ Y3 @ A5 ) ) ) ) ).
% equiv_class_eq_iff
thf(fact_5081_eq__equiv__class__iff,axiom,
! [A: $tType,A5: set @ A,R: set @ ( product_prod @ A @ A ),X: A,Y3: A] :
( ( equiv_equiv @ A @ A5 @ R )
=> ( ( member @ A @ X @ A5 )
=> ( ( member @ A @ Y3 @ A5 )
=> ( ( ( image @ A @ A @ R @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( image @ A @ A @ R @ ( insert2 @ A @ Y3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ R ) ) ) ) ) ).
% eq_equiv_class_iff
thf(fact_5082_equiv__class__eq,axiom,
! [A: $tType,A5: set @ A,R: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
( ( equiv_equiv @ A @ A5 @ R )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R )
=> ( ( image @ A @ A @ R @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( image @ A @ A @ R @ ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% equiv_class_eq
thf(fact_5083_eq__equiv__class,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A3: A,B2: A,A5: set @ A] :
( ( ( image @ A @ A @ R @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( image @ A @ A @ R @ ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( ( equiv_equiv @ A @ A5 @ R )
=> ( ( member @ A @ B2 @ A5 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R ) ) ) ) ).
% eq_equiv_class
thf(fact_5084_eq__equiv__class__iff2,axiom,
! [A: $tType,A5: set @ A,R: set @ ( product_prod @ A @ A ),X: A,Y3: A] :
( ( equiv_equiv @ A @ A5 @ R )
=> ( ( member @ A @ X @ A5 )
=> ( ( member @ A @ Y3 @ A5 )
=> ( ( ( equiv_quotient @ A @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) @ R )
= ( equiv_quotient @ A @ ( insert2 @ A @ Y3 @ ( bot_bot @ ( set @ A ) ) ) @ R ) )
= ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ R ) ) ) ) ) ).
% eq_equiv_class_iff2
thf(fact_5085_UN__equiv__class__inject,axiom,
! [B: $tType,A: $tType,A5: set @ A,R: set @ ( product_prod @ A @ A ),F: A > ( set @ B ),X7: set @ A,Y9: set @ A] :
( ( equiv_equiv @ A @ A5 @ R )
=> ( ( equiv_congruent @ A @ ( set @ B ) @ R @ F )
=> ( ( ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ F @ X7 ) )
= ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ F @ Y9 ) ) )
=> ( ( member @ ( set @ A ) @ X7 @ ( equiv_quotient @ A @ A5 @ R ) )
=> ( ( member @ ( set @ A ) @ Y9 @ ( equiv_quotient @ A @ A5 @ R ) )
=> ( ! [X4: A,Y4: A] :
( ( member @ A @ X4 @ A5 )
=> ( ( member @ A @ Y4 @ A5 )
=> ( ( ( F @ X4 )
= ( F @ Y4 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y4 ) @ R ) ) ) )
=> ( X7 = Y9 ) ) ) ) ) ) ) ).
% UN_equiv_class_inject
thf(fact_5086_Refl__Field__Restr2,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A5: set @ A] :
( ( refl_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ ( field2 @ A @ R ) )
=> ( ( field2 @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R
@ ( product_Sigma @ A @ A @ A5
@ ^ [Uu: A] : A5 ) ) )
= A5 ) ) ) ).
% Refl_Field_Restr2
thf(fact_5087_Refl__Field__Restr,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A5: set @ A] :
( ( refl_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( field2 @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R
@ ( product_Sigma @ A @ A @ A5
@ ^ [Uu: A] : A5 ) ) )
= ( inf_inf @ ( set @ A ) @ ( field2 @ A @ R ) @ A5 ) ) ) ).
% Refl_Field_Restr
thf(fact_5088_refines__equiv__image__eq,axiom,
! [A: $tType,R5: set @ ( product_prod @ A @ A ),S3: set @ ( product_prod @ A @ A ),A5: set @ A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R5 @ S3 )
=> ( ( equiv_equiv @ A @ A5 @ R5 )
=> ( ( equiv_equiv @ A @ A5 @ S3 )
=> ( ( image2 @ ( set @ A ) @ ( set @ A ) @ ( image @ A @ A @ S3 ) @ ( equiv_quotient @ A @ A5 @ R5 ) )
= ( equiv_quotient @ A @ A5 @ S3 ) ) ) ) ) ).
% refines_equiv_image_eq
thf(fact_5089_subset__equiv__class,axiom,
! [A: $tType,A5: set @ A,R: set @ ( product_prod @ A @ A ),B2: A,A3: A] :
( ( equiv_equiv @ A @ A5 @ R )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ R @ ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( image @ A @ A @ R @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( ( member @ A @ B2 @ A5 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R ) ) ) ) ).
% subset_equiv_class
thf(fact_5090_equiv__class__subset,axiom,
! [A: $tType,A5: set @ A,R: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
( ( equiv_equiv @ A @ A5 @ R )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R )
=> ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ R @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( image @ A @ A @ R @ ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% equiv_class_subset
thf(fact_5091_equiv__class__nondisjoint,axiom,
! [A: $tType,A5: set @ A,R: set @ ( product_prod @ A @ A ),X: A,A3: A,B2: A] :
( ( equiv_equiv @ A @ A5 @ R )
=> ( ( member @ A @ X @ ( inf_inf @ ( set @ A ) @ ( image @ A @ A @ R @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( image @ A @ A @ R @ ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R ) ) ) ).
% equiv_class_nondisjoint
thf(fact_5092_in__quotient__imp__in__rel,axiom,
! [A: $tType,A5: set @ A,R: set @ ( product_prod @ A @ A ),X7: set @ A,X: A,Y3: A] :
( ( equiv_equiv @ A @ A5 @ R )
=> ( ( member @ ( set @ A ) @ X7 @ ( equiv_quotient @ A @ A5 @ R ) )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert2 @ A @ X @ ( insert2 @ A @ Y3 @ ( bot_bot @ ( set @ A ) ) ) ) @ X7 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ R ) ) ) ) ).
% in_quotient_imp_in_rel
thf(fact_5093_Chains__alt__def,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( refl_on @ A @ ( top_top @ ( set @ A ) ) @ R )
=> ( ( chains @ A @ R )
= ( collect @ ( set @ A )
@ ( pred_chain @ A @ ( top_top @ ( set @ A ) )
@ ^ [X2: A,Y: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y ) @ R ) ) ) ) ) ).
% Chains_alt_def
thf(fact_5094_UN__equiv__class2,axiom,
! [A: $tType,C: $tType,B: $tType,A16: set @ A,R1: set @ ( product_prod @ A @ A ),A25: set @ B,R22: set @ ( product_prod @ B @ B ),F: A > B > ( set @ C ),A1: A,A22: B] :
( ( equiv_equiv @ A @ A16 @ R1 )
=> ( ( equiv_equiv @ B @ A25 @ R22 )
=> ( ( equiv_congruent2 @ A @ B @ ( set @ C ) @ R1 @ R22 @ F )
=> ( ( member @ A @ A1 @ A16 )
=> ( ( member @ B @ A22 @ A25 )
=> ( ( 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 @ ( insert2 @ B @ A22 @ ( bot_bot @ ( set @ B ) ) ) ) ) )
@ ( image @ A @ A @ R1 @ ( insert2 @ A @ A1 @ ( bot_bot @ ( set @ A ) ) ) ) ) )
= ( F @ A1 @ A22 ) ) ) ) ) ) ) ).
% UN_equiv_class2
thf(fact_5095_congruent2__implies__congruent__UN,axiom,
! [B: $tType,C: $tType,A: $tType,A16: set @ A,R1: set @ ( product_prod @ A @ A ),A25: set @ B,R22: set @ ( product_prod @ B @ B ),F: A > B > ( set @ C ),A3: B] :
( ( equiv_equiv @ A @ A16 @ R1 )
=> ( ( equiv_equiv @ B @ A25 @ R22 )
=> ( ( equiv_congruent2 @ A @ B @ ( set @ C ) @ R1 @ R22 @ F )
=> ( ( member @ B @ A3 @ A25 )
=> ( equiv_congruent @ A @ ( set @ C ) @ R1
@ ^ [X12: A] : ( complete_Sup_Sup @ ( set @ C ) @ ( image2 @ B @ ( set @ C ) @ ( F @ X12 ) @ ( image @ B @ B @ R22 @ ( insert2 @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ) ).
% congruent2_implies_congruent_UN
thf(fact_5096_proj__iff,axiom,
! [A: $tType,A5: set @ A,R: set @ ( product_prod @ A @ A ),X: A,Y3: A] :
( ( equiv_equiv @ A @ A5 @ R )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert2 @ A @ X @ ( insert2 @ A @ Y3 @ ( bot_bot @ ( set @ A ) ) ) ) @ A5 )
=> ( ( ( equiv_proj @ A @ A @ R @ X )
= ( equiv_proj @ A @ A @ R @ Y3 ) )
= ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ R ) ) ) ) ).
% proj_iff
thf(fact_5097_disjnt__equiv__class,axiom,
! [A: $tType,A5: set @ A,R: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
( ( equiv_equiv @ A @ A5 @ R )
=> ( ( disjnt @ A @ ( image @ A @ A @ R @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( image @ A @ A @ R @ ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R ) ) ) ) ).
% disjnt_equiv_class
thf(fact_5098_equiv__proj,axiom,
! [A: $tType,A5: set @ A,R5: set @ ( product_prod @ A @ A ),Z: product_prod @ A @ A] :
( ( equiv_equiv @ A @ A5 @ R5 )
=> ( ( member @ ( product_prod @ A @ A ) @ Z @ R5 )
=> ( ( comp @ A @ ( set @ A ) @ ( product_prod @ A @ A ) @ ( equiv_proj @ A @ A @ R5 ) @ ( product_fst @ A @ A ) @ Z )
= ( comp @ A @ ( set @ A ) @ ( product_prod @ A @ A ) @ ( equiv_proj @ A @ A @ R5 ) @ ( product_snd @ A @ A ) @ Z ) ) ) ) ).
% equiv_proj
thf(fact_5099_disjnt__Times2__iff,axiom,
! [B: $tType,A: $tType,A5: set @ A,C6: set @ B,B4: set @ A] :
( ( disjnt @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A5
@ ^ [Uu: A] : C6 )
@ ( product_Sigma @ A @ B @ B4
@ ^ [Uu: A] : C6 ) )
= ( ( C6
= ( bot_bot @ ( set @ B ) ) )
| ( disjnt @ A @ A5 @ B4 ) ) ) ).
% disjnt_Times2_iff
thf(fact_5100_disjnt__Times1__iff,axiom,
! [A: $tType,B: $tType,C6: set @ A,A5: set @ B,B4: set @ B] :
( ( disjnt @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ C6
@ ^ [Uu: A] : A5 )
@ ( product_Sigma @ A @ B @ C6
@ ^ [Uu: A] : B4 ) )
= ( ( C6
= ( bot_bot @ ( set @ A ) ) )
| ( disjnt @ B @ A5 @ B4 ) ) ) ).
% disjnt_Times1_iff
thf(fact_5101_disjnt__Sigma__iff,axiom,
! [B: $tType,A: $tType,A5: set @ A,C6: A > ( set @ B ),B4: set @ A] :
( ( disjnt @ ( product_prod @ A @ B ) @ ( product_Sigma @ A @ B @ A5 @ C6 ) @ ( product_Sigma @ A @ B @ B4 @ C6 ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ ( inf_inf @ ( set @ A ) @ A5 @ B4 ) )
=> ( ( C6 @ X2 )
= ( bot_bot @ ( set @ B ) ) ) )
| ( disjnt @ A @ A5 @ B4 ) ) ) ).
% disjnt_Sigma_iff
thf(fact_5102_disjoint__UN__iff,axiom,
! [A: $tType,B: $tType,A5: set @ A,B4: B > ( set @ A ),I5: set @ B] :
( ( disjnt @ A @ A5 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B4 @ I5 ) ) )
= ( ! [X2: B] :
( ( member @ B @ X2 @ I5 )
=> ( disjnt @ A @ A5 @ ( B4 @ X2 ) ) ) ) ) ).
% disjoint_UN_iff
thf(fact_5103_card__Un__disjnt,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( finite_finite2 @ A @ B4 )
=> ( ( disjnt @ A @ A5 @ B4 )
=> ( ( finite_card @ A @ ( sup_sup @ ( set @ A ) @ A5 @ B4 ) )
= ( plus_plus @ nat @ ( finite_card @ A @ A5 ) @ ( finite_card @ A @ B4 ) ) ) ) ) ) ).
% card_Un_disjnt
thf(fact_5104_sum__card__image,axiom,
! [B: $tType,A: $tType,A5: set @ A,F: A > ( set @ B )] :
( ( finite_finite2 @ A @ A5 )
=> ( ( pairwise @ A
@ ^ [S2: A,T3: A] : ( disjnt @ B @ ( F @ S2 ) @ ( F @ T3 ) )
@ A5 )
=> ( ( groups7311177749621191930dd_sum @ ( set @ B ) @ nat @ ( finite_card @ B ) @ ( image2 @ A @ ( set @ B ) @ F @ A5 ) )
= ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [A6: A] : ( finite_card @ B @ ( F @ A6 ) )
@ A5 ) ) ) ) ).
% sum_card_image
thf(fact_5105_pair__lessI2,axiom,
! [A3: nat,B2: nat,S: nat,T2: nat] :
( ( ord_less_eq @ nat @ A3 @ B2 )
=> ( ( ord_less @ nat @ S @ 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 @ A3 @ S ) @ ( product_Pair @ nat @ nat @ B2 @ T2 ) ) @ fun_pair_less ) ) ) ).
% pair_lessI2
thf(fact_5106_power__int__def,axiom,
! [A: $tType] :
( ( ( inverse @ A )
& ( power @ A ) )
=> ( ( power_int @ A )
= ( ^ [X2: A,N5: int] : ( if @ A @ ( ord_less_eq @ int @ ( zero_zero @ int ) @ N5 ) @ ( power_power @ A @ X2 @ ( nat2 @ N5 ) ) @ ( power_power @ A @ ( inverse_inverse @ A @ X2 ) @ ( nat2 @ ( uminus_uminus @ int @ N5 ) ) ) ) ) ) ) ).
% power_int_def
thf(fact_5107_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_5108_power__int__mult__numeral,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,M: num,N: num] :
( ( power_int @ A @ ( power_int @ A @ X @ ( numeral_numeral @ int @ M ) ) @ ( numeral_numeral @ int @ N ) )
= ( power_int @ A @ X @ ( numeral_numeral @ int @ ( times_times @ num @ M @ N ) ) ) ) ) ).
% power_int_mult_numeral
thf(fact_5109_power__int__1__right,axiom,
! [A: $tType] :
( ( ( inverse @ A )
& ( monoid_mult @ A ) )
=> ! [Y3: A] :
( ( power_int @ A @ Y3 @ ( one_one @ int ) )
= Y3 ) ) ).
% power_int_1_right
thf(fact_5110_power__int__sgn,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,N: int] :
( ( sgn_sgn @ A @ ( power_int @ A @ A3 @ N ) )
= ( power_int @ A @ ( sgn_sgn @ A @ A3 ) @ N ) ) ) ).
% power_int_sgn
thf(fact_5111_power__int__mult__distrib__numeral1,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [W: num,Y3: A,M: int] :
( ( power_int @ A @ ( times_times @ A @ ( numeral_numeral @ A @ W ) @ Y3 ) @ M )
= ( times_times @ A @ ( power_int @ A @ ( numeral_numeral @ A @ W ) @ M ) @ ( power_int @ A @ Y3 @ M ) ) ) ) ).
% power_int_mult_distrib_numeral1
thf(fact_5112_power__int__mult__distrib__numeral2,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [X: A,W: num,M: int] :
( ( power_int @ A @ ( times_times @ A @ X @ ( numeral_numeral @ A @ W ) ) @ M )
= ( times_times @ A @ ( power_int @ A @ X @ M ) @ ( power_int @ A @ ( numeral_numeral @ A @ W ) @ M ) ) ) ) ).
% power_int_mult_distrib_numeral2
thf(fact_5113_power__int__eq__0__iff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,N: int] :
( ( ( power_int @ A @ X @ N )
= ( zero_zero @ A ) )
= ( ( X
= ( zero_zero @ A ) )
& ( N
!= ( zero_zero @ int ) ) ) ) ) ).
% power_int_eq_0_iff
thf(fact_5114_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_5115_power__int__0__right,axiom,
! [B: $tType] :
( ( ( inverse @ B )
& ( power @ B ) )
=> ! [X: B] :
( ( power_int @ B @ X @ ( zero_zero @ int ) )
= ( one_one @ B ) ) ) ).
% power_int_0_right
thf(fact_5116_abs__power__int__minus,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,N: int] :
( ( abs_abs @ A @ ( power_int @ A @ ( uminus_uminus @ A @ A3 ) @ N ) )
= ( abs_abs @ A @ ( power_int @ A @ A3 @ N ) ) ) ) ).
% abs_power_int_minus
thf(fact_5117_power__int__of__nat,axiom,
! [A: $tType] :
( ( ( inverse @ A )
& ( power @ A ) )
=> ! [X: A,N: nat] :
( ( power_int @ A @ X @ ( semiring_1_of_nat @ int @ N ) )
= ( power_power @ A @ X @ N ) ) ) ).
% power_int_of_nat
thf(fact_5118_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_5119_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_5120_power__int__numeral,axiom,
! [A: $tType] :
( ( ( inverse @ A )
& ( power @ A ) )
=> ! [X: A,N: num] :
( ( power_int @ A @ X @ ( numeral_numeral @ int @ N ) )
= ( power_power @ A @ X @ ( numeral_numeral @ nat @ N ) ) ) ) ).
% power_int_numeral
thf(fact_5121_power__int__add__numeral,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,M: num,N: num] :
( ( times_times @ A @ ( power_int @ A @ X @ ( numeral_numeral @ int @ M ) ) @ ( power_int @ A @ X @ ( numeral_numeral @ int @ N ) ) )
= ( power_int @ A @ X @ ( numeral_numeral @ int @ ( plus_plus @ num @ M @ N ) ) ) ) ) ).
% power_int_add_numeral
thf(fact_5122_power__int__add__numeral2,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,M: num,N: num,B2: A] :
( ( times_times @ A @ ( power_int @ A @ X @ ( numeral_numeral @ int @ M ) ) @ ( times_times @ A @ ( power_int @ A @ X @ ( numeral_numeral @ int @ N ) ) @ B2 ) )
= ( times_times @ A @ ( power_int @ A @ X @ ( numeral_numeral @ int @ ( plus_plus @ num @ M @ N ) ) ) @ B2 ) ) ) ).
% power_int_add_numeral2
thf(fact_5123_power__int__minus1__right,axiom,
! [A: $tType] :
( ( ( inverse @ A )
& ( monoid_mult @ A ) )
=> ! [Y3: A] :
( ( power_int @ A @ Y3 @ ( uminus_uminus @ int @ ( one_one @ int ) ) )
= ( inverse_inverse @ A @ Y3 ) ) ) ).
% power_int_minus1_right
thf(fact_5124_pair__less__iff1,axiom,
! [X: nat,Y3: nat,Z: nat] :
( ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ X @ Y3 ) @ ( product_Pair @ nat @ nat @ X @ Z ) ) @ fun_pair_less )
= ( ord_less @ nat @ Y3 @ Z ) ) ).
% pair_less_iff1
thf(fact_5125_power__int__mono__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A,N: int] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ N )
=> ( ( ord_less_eq @ A @ ( power_int @ A @ A3 @ N ) @ ( power_int @ A @ B2 @ N ) )
= ( ord_less_eq @ A @ A3 @ B2 ) ) ) ) ) ) ).
% power_int_mono_iff
thf(fact_5126_zero__less__power__int,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,N: int] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( power_int @ A @ X @ N ) ) ) ) ).
% zero_less_power_int
thf(fact_5127_zero__le__power__int,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,N: int] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_int @ A @ X @ N ) ) ) ) ).
% zero_le_power_int
thf(fact_5128_pairwise__trivial,axiom,
! [A: $tType,I5: set @ A] :
( pairwise @ A
@ ^ [I2: A,J3: A] : J3 != I2
@ I5 ) ).
% pairwise_trivial
thf(fact_5129_power__int__abs,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,N: int] :
( ( abs_abs @ A @ ( power_int @ A @ A3 @ N ) )
= ( power_int @ A @ ( abs_abs @ A @ A3 ) @ N ) ) ) ).
% power_int_abs
thf(fact_5130_power__int__mult,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,M: int,N: int] :
( ( power_int @ A @ X @ ( times_times @ int @ M @ N ) )
= ( power_int @ A @ ( power_int @ A @ X @ M ) @ N ) ) ) ).
% power_int_mult
thf(fact_5131_power__int__divide__distrib,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [X: A,Y3: A,M: int] :
( ( power_int @ A @ ( divide_divide @ A @ X @ Y3 ) @ M )
= ( divide_divide @ A @ ( power_int @ A @ X @ M ) @ ( power_int @ A @ Y3 @ M ) ) ) ) ).
% power_int_divide_distrib
thf(fact_5132_power__int__commutes,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,N: int] :
( ( times_times @ A @ ( power_int @ A @ X @ N ) @ X )
= ( times_times @ A @ X @ ( power_int @ A @ X @ N ) ) ) ) ).
% power_int_commutes
thf(fact_5133_power__int__mult__distrib,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [X: A,Y3: A,M: int] :
( ( power_int @ A @ ( times_times @ A @ X @ Y3 ) @ M )
= ( times_times @ A @ ( power_int @ A @ X @ M ) @ ( power_int @ A @ Y3 @ M ) ) ) ) ).
% power_int_mult_distrib
thf(fact_5134_power__int__inverse,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,N: int] :
( ( power_int @ A @ ( inverse_inverse @ A @ X ) @ N )
= ( inverse_inverse @ A @ ( power_int @ A @ X @ N ) ) ) ) ).
% power_int_inverse
thf(fact_5135_power__int__not__zero,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,N: int] :
( ( ( X
!= ( zero_zero @ A ) )
| ( N
= ( zero_zero @ int ) ) )
=> ( ( power_int @ A @ X @ N )
!= ( zero_zero @ A ) ) ) ) ).
% power_int_not_zero
thf(fact_5136_power__int__one__over,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,N: int] :
( ( power_int @ A @ ( divide_divide @ A @ ( one_one @ A ) @ X ) @ N )
= ( divide_divide @ A @ ( one_one @ A ) @ ( power_int @ A @ X @ N ) ) ) ) ).
% power_int_one_over
thf(fact_5137_power__int__minus,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,N: int] :
( ( power_int @ A @ X @ ( uminus_uminus @ int @ N ) )
= ( inverse_inverse @ A @ ( power_int @ A @ X @ N ) ) ) ) ).
% power_int_minus
thf(fact_5138_pairwise__image,axiom,
! [A: $tType,B: $tType,R: A > A > $o,F: B > A,S: set @ B] :
( ( pairwise @ A @ R @ ( image2 @ B @ A @ F @ S ) )
= ( pairwise @ B
@ ^ [X2: B,Y: B] :
( ( ( F @ X2 )
!= ( F @ Y ) )
=> ( R @ ( F @ X2 ) @ ( F @ Y ) ) )
@ S ) ) ).
% pairwise_image
thf(fact_5139_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_5140_power__int__increasing,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [N: int,N6: int,A3: A] :
( ( ord_less_eq @ int @ N @ N6 )
=> ( ( ord_less_eq @ A @ ( one_one @ A ) @ A3 )
=> ( ord_less_eq @ A @ ( power_int @ A @ A3 @ N ) @ ( power_int @ A @ A3 @ N6 ) ) ) ) ) ).
% power_int_increasing
thf(fact_5141_power__int__strict__increasing,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [N: int,N6: int,A3: A] :
( ( ord_less @ int @ N @ N6 )
=> ( ( ord_less @ A @ ( one_one @ A ) @ A3 )
=> ( ord_less @ A @ ( power_int @ A @ A3 @ N ) @ ( power_int @ A @ A3 @ N6 ) ) ) ) ) ).
% power_int_strict_increasing
thf(fact_5142_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_5143_power__int__diff,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [X: A,M: int,N: int] :
( ( ( X
!= ( zero_zero @ A ) )
| ( M != N ) )
=> ( ( power_int @ A @ X @ ( minus_minus @ int @ M @ N ) )
= ( divide_divide @ A @ ( power_int @ A @ X @ M ) @ ( power_int @ A @ X @ N ) ) ) ) ) ).
% power_int_diff
thf(fact_5144_power__int__minus__one__diff__commute,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: int,B2: int] :
( ( power_int @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( minus_minus @ int @ A3 @ B2 ) )
= ( power_int @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( minus_minus @ int @ B2 @ A3 ) ) ) ) ).
% power_int_minus_one_diff_commute
thf(fact_5145_power__int__strict__decreasing,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [N: int,N6: int,A3: A] :
( ( ord_less @ int @ N @ N6 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ A3 @ ( one_one @ A ) )
=> ( ord_less @ A @ ( power_int @ A @ A3 @ N6 ) @ ( power_int @ A @ A3 @ N ) ) ) ) ) ) ).
% power_int_strict_decreasing
thf(fact_5146_power__int__mono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y3: A,N: int] :
( ( ord_less_eq @ A @ X @ Y3 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ord_less_eq @ A @ ( power_int @ A @ X @ N ) @ ( power_int @ A @ Y3 @ N ) ) ) ) ) ) ).
% power_int_mono
thf(fact_5147_power__int__strict__antimono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A,N: int] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ int @ N @ ( zero_zero @ int ) )
=> ( ord_less @ A @ ( power_int @ A @ B2 @ N ) @ ( power_int @ A @ A3 @ N ) ) ) ) ) ) ).
% power_int_strict_antimono
thf(fact_5148_one__le__power__int,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,N: int] :
( ( ord_less_eq @ A @ ( one_one @ A ) @ X )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ ( power_int @ A @ X @ N ) ) ) ) ) ).
% one_le_power_int
thf(fact_5149_one__less__power__int,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,N: int] :
( ( ord_less @ A @ ( one_one @ A ) @ A3 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ N )
=> ( ord_less @ A @ ( one_one @ A ) @ ( power_int @ A @ A3 @ N ) ) ) ) ) ).
% one_less_power_int
thf(fact_5150_power__int__add,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,M: int,N: int] :
( ( ( X
!= ( zero_zero @ A ) )
| ( ( plus_plus @ int @ M @ N )
!= ( zero_zero @ int ) ) )
=> ( ( power_int @ A @ X @ ( plus_plus @ int @ M @ N ) )
= ( times_times @ A @ ( power_int @ A @ X @ M ) @ ( power_int @ A @ X @ N ) ) ) ) ) ).
% power_int_add
thf(fact_5151_power__int__minus__left__distrib,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( division_ring @ A )
& ( one @ B )
& ( uminus @ B ) )
=> ! [X: C,A3: A,N: int] :
( ( nO_MATCH @ B @ C @ ( uminus_uminus @ B @ ( one_one @ B ) ) @ X )
=> ( ( power_int @ A @ ( uminus_uminus @ A @ A3 ) @ N )
= ( times_times @ A @ ( power_int @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N ) @ ( power_int @ A @ A3 @ N ) ) ) ) ) ).
% power_int_minus_left_distrib
thf(fact_5152_pair__lessI1,axiom,
! [A3: nat,B2: nat,S: nat,T2: nat] :
( ( ord_less @ nat @ A3 @ B2 )
=> ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ A3 @ S ) @ ( product_Pair @ nat @ nat @ B2 @ T2 ) ) @ fun_pair_less ) ) ).
% pair_lessI1
thf(fact_5153_power__int__strict__mono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A,N: int] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ N )
=> ( ord_less @ A @ ( power_int @ A @ A3 @ N ) @ ( power_int @ A @ B2 @ N ) ) ) ) ) ) ).
% power_int_strict_mono
thf(fact_5154_power__int__antimono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A,N: int] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ int @ N @ ( zero_zero @ int ) )
=> ( ord_less_eq @ A @ ( power_int @ A @ B2 @ N ) @ ( power_int @ A @ A3 @ N ) ) ) ) ) ) ).
% power_int_antimono
thf(fact_5155_power__int__decreasing,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [N: int,N6: int,A3: A] :
( ( ord_less_eq @ int @ N @ N6 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ A3 @ ( one_one @ A ) )
=> ( ( ( A3
!= ( zero_zero @ A ) )
| ( N6
!= ( zero_zero @ int ) )
| ( N
= ( zero_zero @ int ) ) )
=> ( ord_less_eq @ A @ ( power_int @ A @ A3 @ N6 ) @ ( power_int @ A @ A3 @ N ) ) ) ) ) ) ) ).
% power_int_decreasing
thf(fact_5156_power__int__le__one,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,N: int] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N )
=> ( ( ord_less_eq @ A @ X @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( power_int @ A @ X @ N ) @ ( one_one @ A ) ) ) ) ) ) ).
% power_int_le_one
thf(fact_5157_power__int__le__imp__le__exp,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,M: int,N: int] :
( ( ord_less @ A @ ( one_one @ A ) @ X )
=> ( ( ord_less_eq @ A @ ( power_int @ A @ X @ M ) @ ( power_int @ A @ X @ N ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N )
=> ( ord_less_eq @ int @ M @ N ) ) ) ) ) ).
% power_int_le_imp_le_exp
thf(fact_5158_power__int__le__imp__less__exp,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,M: int,N: int] :
( ( ord_less @ A @ ( one_one @ A ) @ X )
=> ( ( ord_less @ A @ ( power_int @ A @ X @ M ) @ ( power_int @ A @ X @ N ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N )
=> ( ord_less @ int @ M @ N ) ) ) ) ) ).
% power_int_le_imp_less_exp
thf(fact_5159_power__int__minus__mult,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [X: A,N: int] :
( ( ( X
!= ( zero_zero @ A ) )
| ( N
!= ( zero_zero @ int ) ) )
=> ( ( times_times @ A @ ( power_int @ A @ X @ ( minus_minus @ int @ N @ ( one_one @ int ) ) ) @ X )
= ( power_int @ A @ X @ N ) ) ) ) ).
% power_int_minus_mult
thf(fact_5160_power__int__add__1,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,M: int] :
( ( ( X
!= ( zero_zero @ A ) )
| ( M
!= ( uminus_uminus @ int @ ( one_one @ int ) ) ) )
=> ( ( power_int @ A @ X @ ( plus_plus @ int @ M @ ( one_one @ int ) ) )
= ( times_times @ A @ ( power_int @ A @ X @ M ) @ X ) ) ) ) ).
% power_int_add_1
thf(fact_5161_power__int__add__1_H,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,M: int] :
( ( ( X
!= ( zero_zero @ A ) )
| ( M
!= ( uminus_uminus @ int @ ( one_one @ int ) ) ) )
=> ( ( power_int @ A @ X @ ( plus_plus @ int @ M @ ( one_one @ int ) ) )
= ( times_times @ A @ X @ ( power_int @ A @ X @ M ) ) ) ) ) ).
% power_int_add_1'
thf(fact_5162_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_5163_pair__leqI2,axiom,
! [A3: nat,B2: nat,S: nat,T2: nat] :
( ( ord_less_eq @ nat @ A3 @ B2 )
=> ( ( ord_less_eq @ nat @ S @ 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 @ A3 @ S ) @ ( product_Pair @ nat @ nat @ B2 @ T2 ) ) @ fun_pair_leq ) ) ) ).
% pair_leqI2
thf(fact_5164_pair__leqI1,axiom,
! [A3: nat,B2: nat,S: nat,T2: nat] :
( ( ord_less @ nat @ A3 @ B2 )
=> ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ A3 @ S ) @ ( product_Pair @ nat @ nat @ B2 @ T2 ) ) @ fun_pair_leq ) ) ).
% pair_leqI1
thf(fact_5165_wmax__insertI,axiom,
! [Y3: product_prod @ nat @ nat,YS: set @ ( product_prod @ nat @ nat ),X: product_prod @ nat @ nat,XS2: set @ ( product_prod @ nat @ nat )] :
( ( member @ ( product_prod @ nat @ nat ) @ Y3 @ YS )
=> ( ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ X @ Y3 ) @ fun_pair_leq )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ XS2 @ YS ) @ fun_max_weak )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ ( insert2 @ ( product_prod @ nat @ nat ) @ X @ XS2 ) @ YS ) @ fun_max_weak ) ) ) ) ).
% wmax_insertI
thf(fact_5166_wmin__insertI,axiom,
! [X: product_prod @ nat @ nat,XS2: set @ ( product_prod @ nat @ nat ),Y3: product_prod @ nat @ nat,YS: set @ ( product_prod @ nat @ nat )] :
( ( member @ ( product_prod @ nat @ nat ) @ X @ XS2 )
=> ( ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ X @ Y3 ) @ fun_pair_leq )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ XS2 @ YS ) @ fun_min_weak )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ XS2 @ ( insert2 @ ( product_prod @ nat @ nat ) @ Y3 @ YS ) ) @ fun_min_weak ) ) ) ) ).
% wmin_insertI
thf(fact_5167_smin__insertI,axiom,
! [X: product_prod @ nat @ nat,XS2: set @ ( product_prod @ nat @ nat ),Y3: product_prod @ nat @ nat,YS: set @ ( product_prod @ nat @ nat )] :
( ( member @ ( product_prod @ nat @ nat ) @ X @ XS2 )
=> ( ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ X @ Y3 ) @ fun_pair_less )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ XS2 @ YS ) @ fun_min_strict )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ XS2 @ ( insert2 @ ( product_prod @ nat @ nat ) @ Y3 @ YS ) ) @ fun_min_strict ) ) ) ) ).
% smin_insertI
thf(fact_5168_min__rpair__set,axiom,
fun_reduction_pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) ) @ ( set @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) ) @ fun_min_strict @ fun_min_weak ) ).
% min_rpair_set
thf(fact_5169_wmax__emptyI,axiom,
! [X7: set @ ( product_prod @ nat @ nat )] :
( ( finite_finite2 @ ( product_prod @ nat @ nat ) @ X7 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ ( bot_bot @ ( set @ ( product_prod @ nat @ nat ) ) ) @ X7 ) @ fun_max_weak ) ) ).
% wmax_emptyI
thf(fact_5170_smin__emptyI,axiom,
! [X7: set @ ( product_prod @ nat @ nat )] :
( ( X7
!= ( bot_bot @ ( set @ ( product_prod @ nat @ nat ) ) ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ X7 @ ( bot_bot @ ( set @ ( product_prod @ nat @ nat ) ) ) ) @ fun_min_strict ) ) ).
% smin_emptyI
thf(fact_5171_wmin__emptyI,axiom,
! [X7: set @ ( product_prod @ nat @ nat )] : ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ X7 @ ( bot_bot @ ( set @ ( product_prod @ nat @ nat ) ) ) ) @ fun_min_weak ) ).
% wmin_emptyI
thf(fact_5172_min__weak__def,axiom,
( fun_min_weak
= ( sup_sup @ ( set @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) ) @ ( min_ext @ ( product_prod @ nat @ nat ) @ fun_pair_leq ) @ ( insert2 @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ ( bot_bot @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( bot_bot @ ( set @ ( product_prod @ nat @ nat ) ) ) ) @ ( bot_bot @ ( set @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) ) ) ) ) ) ).
% min_weak_def
thf(fact_5173_max__weak__def,axiom,
( fun_max_weak
= ( sup_sup @ ( set @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) ) @ ( max_ext @ ( product_prod @ nat @ nat ) @ fun_pair_leq ) @ ( insert2 @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ ( bot_bot @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( bot_bot @ ( set @ ( product_prod @ nat @ nat ) ) ) ) @ ( bot_bot @ ( set @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) ) ) ) ) ) ).
% max_weak_def
thf(fact_5174_smax__insertI,axiom,
! [Y3: product_prod @ nat @ nat,Y9: set @ ( product_prod @ nat @ nat ),X: product_prod @ nat @ nat,X7: set @ ( product_prod @ nat @ nat )] :
( ( member @ ( product_prod @ nat @ nat ) @ Y3 @ Y9 )
=> ( ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ X @ Y3 ) @ fun_pair_less )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ X7 @ Y9 ) @ fun_max_strict )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ ( insert2 @ ( product_prod @ nat @ nat ) @ X @ X7 ) @ Y9 ) @ fun_max_strict ) ) ) ) ).
% smax_insertI
thf(fact_5175_wo__rel_Ocases__Total3,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A3: A,B2: A,Phi: A > A > $o] :
( ( bNF_Wellorder_wo_rel @ A @ R )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert2 @ A @ A3 @ ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( field2 @ A @ R ) )
=> ( ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R @ ( id2 @ A ) ) )
| ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ A3 ) @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R @ ( id2 @ A ) ) ) )
=> ( Phi @ A3 @ B2 ) )
=> ( ( ( A3 = B2 )
=> ( Phi @ A3 @ B2 ) )
=> ( Phi @ A3 @ B2 ) ) ) ) ) ).
% wo_rel.cases_Total3
thf(fact_5176_assert__def,axiom,
! [A: $tType] :
( ( heap_Time_assert @ A )
= ( ^ [P3: A > $o,X2: A] : ( if @ ( heap_Time_Heap @ A ) @ ( P3 @ X2 ) @ ( heap_Time_return @ A @ X2 ) @ ( 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_5177_wo__rel_Omax2__def,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
( ( bNF_Wellorder_wo_rel @ A @ R )
=> ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R )
=> ( ( bNF_We1388413361240627857o_max2 @ A @ R @ A3 @ B2 )
= B2 ) )
& ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R )
=> ( ( bNF_We1388413361240627857o_max2 @ A @ R @ A3 @ B2 )
= A3 ) ) ) ) ).
% wo_rel.max2_def
thf(fact_5178_wo__rel_OTOTALS,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( bNF_Wellorder_wo_rel @ A @ R )
=> ! [X5: A] :
( ( member @ A @ X5 @ ( field2 @ A @ R ) )
=> ! [Xa2: A] :
( ( member @ A @ Xa2 @ ( field2 @ A @ R ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ Xa2 ) @ R )
| ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Xa2 @ X5 ) @ R ) ) ) ) ) ).
% wo_rel.TOTALS
thf(fact_5179_wo__rel_Owell__order__induct,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),P: A > $o,A3: A] :
( ( bNF_Wellorder_wo_rel @ A @ R )
=> ( ! [X4: A] :
( ! [Y5: A] :
( ( ( Y5 != X4 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y5 @ X4 ) @ R ) )
=> ( P @ Y5 ) )
=> ( P @ X4 ) )
=> ( P @ A3 ) ) ) ).
% wo_rel.well_order_induct
thf(fact_5180_well__order__induct__imp,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),P: A > $o,A3: A] :
( ( bNF_Wellorder_wo_rel @ A @ R )
=> ( ! [X4: A] :
( ! [Y5: A] :
( ( ( Y5 != X4 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y5 @ X4 ) @ R ) )
=> ( ( member @ A @ Y5 @ ( field2 @ A @ R ) )
=> ( P @ Y5 ) ) )
=> ( ( member @ A @ X4 @ ( field2 @ A @ R ) )
=> ( P @ X4 ) ) )
=> ( ( member @ A @ A3 @ ( field2 @ A @ R ) )
=> ( P @ A3 ) ) ) ) ).
% well_order_induct_imp
thf(fact_5181_wo__rel_Omax2__greater,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
( ( bNF_Wellorder_wo_rel @ A @ R )
=> ( ( member @ A @ A3 @ ( field2 @ A @ R ) )
=> ( ( member @ A @ B2 @ ( field2 @ A @ R ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ ( bNF_We1388413361240627857o_max2 @ A @ R @ A3 @ B2 ) ) @ R )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ ( bNF_We1388413361240627857o_max2 @ A @ R @ A3 @ B2 ) ) @ R ) ) ) ) ) ).
% wo_rel.max2_greater
thf(fact_5182_wo__rel_Omax2__equals2,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
( ( bNF_Wellorder_wo_rel @ A @ R )
=> ( ( member @ A @ A3 @ ( field2 @ A @ R ) )
=> ( ( member @ A @ B2 @ ( field2 @ A @ R ) )
=> ( ( ( bNF_We1388413361240627857o_max2 @ A @ R @ A3 @ B2 )
= B2 )
= ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R ) ) ) ) ) ).
% wo_rel.max2_equals2
thf(fact_5183_wo__rel_Omax2__equals1,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
( ( bNF_Wellorder_wo_rel @ A @ R )
=> ( ( member @ A @ A3 @ ( field2 @ A @ R ) )
=> ( ( member @ A @ B2 @ ( field2 @ A @ R ) )
=> ( ( ( bNF_We1388413361240627857o_max2 @ A @ R @ A3 @ B2 )
= A3 )
= ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ A3 ) @ R ) ) ) ) ) ).
% wo_rel.max2_equals1
thf(fact_5184_wo__rel_Ocases__Total,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A3: A,B2: A,Phi: A > A > $o] :
( ( bNF_Wellorder_wo_rel @ A @ R )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert2 @ A @ A3 @ ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( field2 @ A @ R ) )
=> ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R )
=> ( Phi @ A3 @ B2 ) )
=> ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ A3 ) @ R )
=> ( Phi @ A3 @ B2 ) )
=> ( Phi @ A3 @ B2 ) ) ) ) ) ).
% wo_rel.cases_Total
thf(fact_5185_smax__emptyI,axiom,
! [Y9: set @ ( product_prod @ nat @ nat )] :
( ( finite_finite2 @ ( product_prod @ nat @ nat ) @ Y9 )
=> ( ( Y9
!= ( bot_bot @ ( set @ ( product_prod @ nat @ nat ) ) ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ ( bot_bot @ ( set @ ( product_prod @ nat @ nat ) ) ) @ Y9 ) @ fun_max_strict ) ) ) ).
% smax_emptyI
thf(fact_5186_natLeq__on__wo__rel,axiom,
! [N: nat] :
( bNF_Wellorder_wo_rel @ nat
@ ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [X2: nat,Y: nat] :
( ( ord_less @ nat @ X2 @ N )
& ( ord_less @ nat @ Y @ N )
& ( ord_less_eq @ nat @ X2 @ Y ) ) ) ) ) ).
% natLeq_on_wo_rel
thf(fact_5187_wo__rel_Omax2__greater__among,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
( ( bNF_Wellorder_wo_rel @ A @ R )
=> ( ( member @ A @ A3 @ ( field2 @ A @ R ) )
=> ( ( member @ A @ B2 @ ( field2 @ A @ R ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ ( bNF_We1388413361240627857o_max2 @ A @ R @ A3 @ B2 ) ) @ R )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ ( bNF_We1388413361240627857o_max2 @ A @ R @ A3 @ B2 ) ) @ R )
& ( member @ A @ ( bNF_We1388413361240627857o_max2 @ A @ R @ A3 @ B2 ) @ ( insert2 @ A @ A3 @ ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% wo_rel.max2_greater_among
thf(fact_5188_max__rpair__set,axiom,
fun_reduction_pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) ) @ ( set @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) ) @ fun_max_strict @ fun_max_weak ) ).
% max_rpair_set
thf(fact_5189_char__of__def,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ( ( unique5772411509450598832har_of @ A )
= ( ^ [N5: A] :
( char2
@ ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N5 )
@ ( bit_se5641148757651400278ts_bit @ A @ N5 @ ( one_one @ nat ) )
@ ( bit_se5641148757651400278ts_bit @ A @ N5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
@ ( bit_se5641148757651400278ts_bit @ A @ N5 @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) )
@ ( bit_se5641148757651400278ts_bit @ A @ N5 @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) )
@ ( bit_se5641148757651400278ts_bit @ A @ N5 @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ one2 ) ) ) )
@ ( bit_se5641148757651400278ts_bit @ A @ N5 @ ( numeral_numeral @ nat @ ( bit0 @ ( bit1 @ one2 ) ) ) )
@ ( bit_se5641148757651400278ts_bit @ A @ N5 @ ( numeral_numeral @ nat @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ).
% char_of_def
thf(fact_5190_char_Osize_I2_J,axiom,
! [X1: $o,X22: $o,X32: $o,X42: $o,X52: $o,X62: $o,X72: $o,X82: $o] :
( ( size_size @ char @ ( char2 @ X1 @ X22 @ X32 @ X42 @ X52 @ X62 @ X72 @ X82 ) )
= ( zero_zero @ nat ) ) ).
% char.size(2)
thf(fact_5191_wo__rel_Oequals__minim,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),B4: set @ A,A3: A] :
( ( bNF_Wellorder_wo_rel @ A @ R )
=> ( ( ord_less_eq @ ( set @ A ) @ B4 @ ( field2 @ A @ R ) )
=> ( ( member @ A @ A3 @ B4 )
=> ( ! [B3: A] :
( ( member @ A @ B3 @ B4 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B3 ) @ R ) )
=> ( A3
= ( bNF_We6954850376910717587_minim @ A @ R @ B4 ) ) ) ) ) ) ).
% wo_rel.equals_minim
thf(fact_5192_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_5193_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_5194_wo__rel_Ominim__least,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),B4: set @ A,B2: A] :
( ( bNF_Wellorder_wo_rel @ A @ R )
=> ( ( ord_less_eq @ ( set @ A ) @ B4 @ ( field2 @ A @ R ) )
=> ( ( member @ A @ B2 @ B4 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( bNF_We6954850376910717587_minim @ A @ R @ B4 ) @ B2 ) @ R ) ) ) ) ).
% wo_rel.minim_least
thf(fact_5195_char_Osize__gen,axiom,
! [X1: $o,X22: $o,X32: $o,X42: $o,X52: $o,X62: $o,X72: $o,X82: $o] :
( ( size_char @ ( char2 @ X1 @ X22 @ X32 @ X42 @ X52 @ X62 @ X72 @ X82 ) )
= ( zero_zero @ nat ) ) ).
% char.size_gen
thf(fact_5196_wo__rel_Ominim__def,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A5: set @ A] :
( ( bNF_Wellorder_wo_rel @ A @ R )
=> ( ( bNF_We6954850376910717587_minim @ A @ R @ A5 )
= ( the @ A @ ( bNF_We4791949203932849705sMinim @ A @ R @ A5 ) ) ) ) ).
% wo_rel.minim_def
thf(fact_5197_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_5198_char__of__comp__of__char,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ( ( comp @ A @ char @ char @ ( unique5772411509450598832har_of @ A ) @ ( comm_s6883823935334413003f_char @ A ) )
= ( id @ char ) ) ) ).
% char_of_comp_of_char
thf(fact_5199_wo__rel_OisMinim__def,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A5: set @ A,B2: A] :
( ( bNF_Wellorder_wo_rel @ A @ R )
=> ( ( bNF_We4791949203932849705sMinim @ A @ R @ A5 @ B2 )
= ( ( member @ A @ B2 @ A5 )
& ! [X2: A] :
( ( member @ A @ X2 @ A5 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ X2 ) @ R ) ) ) ) ) ).
% wo_rel.isMinim_def
thf(fact_5200_bot_Oordering__top__axioms,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ( ordering_top @ A
@ ^ [X2: A,Y: A] : ( ord_less_eq @ A @ Y @ X2 )
@ ^ [X2: A,Y: A] : ( ord_less @ A @ Y @ X2 )
@ ( bot_bot @ A ) ) ) ).
% bot.ordering_top_axioms
thf(fact_5201_eventually__INF__base,axiom,
! [B: $tType,A: $tType,B4: set @ A,F6: A > ( filter @ B ),P: B > $o] :
( ( B4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [A4: A] :
( ( member @ A @ A4 @ B4 )
=> ! [B3: A] :
( ( member @ A @ B3 @ B4 )
=> ? [X5: A] :
( ( member @ A @ X5 @ B4 )
& ( ord_less_eq @ ( filter @ B ) @ ( F6 @ X5 ) @ ( inf_inf @ ( filter @ B ) @ ( F6 @ A4 ) @ ( F6 @ B3 ) ) ) ) ) )
=> ( ( eventually @ B @ P @ ( complete_Inf_Inf @ ( filter @ B ) @ ( image2 @ A @ ( filter @ B ) @ F6 @ B4 ) ) )
= ( ? [X2: A] :
( ( member @ A @ X2 @ B4 )
& ( eventually @ B @ P @ ( F6 @ X2 ) ) ) ) ) ) ) ).
% eventually_INF_base
thf(fact_5202_euclidean__size__times__nonunit,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [A3: A,B2: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( ~ ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ord_less @ nat @ ( euclid6346220572633701492n_size @ A @ B2 ) @ ( euclid6346220572633701492n_size @ A @ ( times_times @ A @ A3 @ B2 ) ) ) ) ) ) ) ).
% euclidean_size_times_nonunit
thf(fact_5203_eventually__const,axiom,
! [A: $tType,F6: filter @ A,P: $o] :
( ( F6
!= ( bot_bot @ ( filter @ A ) ) )
=> ( ( eventually @ A
@ ^ [X2: A] : P
@ F6 )
= P ) ) ).
% eventually_const
thf(fact_5204_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_5205_size__0,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ( ( euclid6346220572633701492n_size @ A @ ( zero_zero @ A ) )
= ( zero_zero @ nat ) ) ) ).
% size_0
thf(fact_5206_euclidean__size__1,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ( ( euclid6346220572633701492n_size @ A @ ( one_one @ A ) )
= ( one_one @ nat ) ) ) ).
% euclidean_size_1
thf(fact_5207_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_5208_False__imp__not__eventually,axiom,
! [A: $tType,P: A > $o,Net: filter @ A] :
( ! [X4: A] :
~ ( P @ X4 )
=> ( ( Net
!= ( bot_bot @ ( filter @ A ) ) )
=> ~ ( eventually @ A @ P @ Net ) ) ) ).
% False_imp_not_eventually
thf(fact_5209_eventually__const__iff,axiom,
! [A: $tType,P: $o,F6: filter @ A] :
( ( eventually @ A
@ ^ [X2: A] : P
@ F6 )
= ( P
| ( F6
= ( bot_bot @ ( filter @ A ) ) ) ) ) ).
% eventually_const_iff
thf(fact_5210_trivial__limit__def,axiom,
! [A: $tType,F6: filter @ A] :
( ( F6
= ( bot_bot @ ( filter @ A ) ) )
= ( eventually @ A
@ ^ [X2: A] : $false
@ F6 ) ) ).
% trivial_limit_def
thf(fact_5211_eventually__ex,axiom,
! [B: $tType,A: $tType,P: A > B > $o,F6: filter @ A] :
( ( eventually @ A
@ ^ [X2: A] :
? [X9: B] : ( P @ X2 @ X9 )
@ F6 )
= ( ? [Y10: A > B] :
( eventually @ A
@ ^ [X2: A] : ( P @ X2 @ ( Y10 @ X2 ) )
@ F6 ) ) ) ).
% eventually_ex
thf(fact_5212_eventually__all__finite,axiom,
! [B: $tType,A: $tType] :
( ( finite_finite @ B )
=> ! [P: A > B > $o,Net: filter @ A] :
( ! [Y4: B] :
( eventually @ A
@ ^ [X2: A] : ( P @ X2 @ Y4 )
@ Net )
=> ( eventually @ A
@ ^ [X2: A] :
! [X9: B] : ( P @ X2 @ X9 )
@ Net ) ) ) ).
% eventually_all_finite
thf(fact_5213_eventually__compose__filterlim,axiom,
! [A: $tType,B: $tType,P: A > $o,F6: filter @ A,F: B > A,G6: filter @ B] :
( ( eventually @ A @ P @ F6 )
=> ( ( filterlim @ B @ A @ F @ F6 @ G6 )
=> ( eventually @ B
@ ^ [X2: B] : ( P @ ( F @ X2 ) )
@ G6 ) ) ) ).
% eventually_compose_filterlim
thf(fact_5214_filterlim__cong,axiom,
! [A: $tType,B: $tType,F13: filter @ A,F14: filter @ A,F24: filter @ B,F25: filter @ B,F: B > A,G: B > A] :
( ( F13 = F14 )
=> ( ( F24 = F25 )
=> ( ( eventually @ B
@ ^ [X2: B] :
( ( F @ X2 )
= ( G @ X2 ) )
@ F24 )
=> ( ( filterlim @ B @ A @ F @ F13 @ F24 )
= ( filterlim @ B @ A @ G @ F14 @ F25 ) ) ) ) ) ).
% filterlim_cong
thf(fact_5215_filterlim__iff,axiom,
! [B: $tType,A: $tType] :
( ( filterlim @ A @ B )
= ( ^ [F2: A > B,F26: filter @ B,F15: filter @ A] :
! [P3: B > $o] :
( ( eventually @ B @ P3 @ F26 )
=> ( eventually @ A
@ ^ [X2: A] : ( P3 @ ( F2 @ X2 ) )
@ F15 ) ) ) ) ).
% filterlim_iff
thf(fact_5216_eventually__frequently__const__simps_I6_J,axiom,
! [A: $tType,C6: $o,P: A > $o,F6: filter @ A] :
( ( eventually @ A
@ ^ [X2: A] :
( C6
=> ( P @ X2 ) )
@ F6 )
= ( C6
=> ( eventually @ A @ P @ F6 ) ) ) ).
% eventually_frequently_const_simps(6)
thf(fact_5217_eventually__frequently__const__simps_I4_J,axiom,
! [A: $tType,C6: $o,P: A > $o,F6: filter @ A] :
( ( eventually @ A
@ ^ [X2: A] :
( C6
| ( P @ X2 ) )
@ F6 )
= ( C6
| ( eventually @ A @ P @ F6 ) ) ) ).
% eventually_frequently_const_simps(4)
thf(fact_5218_eventually__frequently__const__simps_I3_J,axiom,
! [A: $tType,P: A > $o,C6: $o,F6: filter @ A] :
( ( eventually @ A
@ ^ [X2: A] :
( ( P @ X2 )
| C6 )
@ F6 )
= ( ( eventually @ A @ P @ F6 )
| C6 ) ) ).
% eventually_frequently_const_simps(3)
thf(fact_5219_eventually__mp,axiom,
! [A: $tType,P: A > $o,Q: A > $o,F6: filter @ A] :
( ( eventually @ A
@ ^ [X2: A] :
( ( P @ X2 )
=> ( Q @ X2 ) )
@ F6 )
=> ( ( eventually @ A @ P @ F6 )
=> ( eventually @ A @ Q @ F6 ) ) ) ).
% eventually_mp
thf(fact_5220_eventually__True,axiom,
! [A: $tType,F6: filter @ A] :
( eventually @ A
@ ^ [X2: A] : $true
@ F6 ) ).
% eventually_True
thf(fact_5221_eventually__conj,axiom,
! [A: $tType,P: A > $o,F6: filter @ A,Q: A > $o] :
( ( eventually @ A @ P @ F6 )
=> ( ( eventually @ A @ Q @ F6 )
=> ( eventually @ A
@ ^ [X2: A] :
( ( P @ X2 )
& ( Q @ X2 ) )
@ F6 ) ) ) ).
% eventually_conj
thf(fact_5222_eventually__elim2,axiom,
! [A: $tType,P: A > $o,F6: filter @ A,Q: A > $o,R5: A > $o] :
( ( eventually @ A @ P @ F6 )
=> ( ( eventually @ A @ Q @ F6 )
=> ( ! [I3: A] :
( ( P @ I3 )
=> ( ( Q @ I3 )
=> ( R5 @ I3 ) ) )
=> ( eventually @ A @ R5 @ F6 ) ) ) ) ).
% eventually_elim2
thf(fact_5223_eventually__subst,axiom,
! [A: $tType,P: A > $o,Q: A > $o,F6: filter @ A] :
( ( eventually @ A
@ ^ [N5: A] :
( ( P @ N5 )
= ( Q @ N5 ) )
@ F6 )
=> ( ( eventually @ A @ P @ F6 )
= ( eventually @ A @ Q @ F6 ) ) ) ).
% eventually_subst
thf(fact_5224_eventually__rev__mp,axiom,
! [A: $tType,P: A > $o,F6: filter @ A,Q: A > $o] :
( ( eventually @ A @ P @ F6 )
=> ( ( eventually @ A
@ ^ [X2: A] :
( ( P @ X2 )
=> ( Q @ X2 ) )
@ F6 )
=> ( eventually @ A @ Q @ F6 ) ) ) ).
% eventually_rev_mp
thf(fact_5225_eventually__conj__iff,axiom,
! [A: $tType,P: A > $o,Q: A > $o,F6: filter @ A] :
( ( eventually @ A
@ ^ [X2: A] :
( ( P @ X2 )
& ( Q @ X2 ) )
@ F6 )
= ( ( eventually @ A @ P @ F6 )
& ( eventually @ A @ Q @ F6 ) ) ) ).
% eventually_conj_iff
thf(fact_5226_not__eventually__impI,axiom,
! [A: $tType,P: A > $o,F6: filter @ A,Q: A > $o] :
( ( eventually @ A @ P @ F6 )
=> ( ~ ( eventually @ A @ Q @ F6 )
=> ~ ( eventually @ A
@ ^ [X2: A] :
( ( P @ X2 )
=> ( Q @ X2 ) )
@ F6 ) ) ) ).
% not_eventually_impI
thf(fact_5227_eventually__at__bot__not__equal,axiom,
! [A: $tType] :
( ( ( linorder @ A )
& ( no_bot @ A ) )
=> ! [C2: A] :
( eventually @ A
@ ^ [X2: A] : X2 != C2
@ ( at_bot @ A ) ) ) ).
% eventually_at_bot_not_equal
thf(fact_5228_eventually__ball__finite,axiom,
! [A: $tType,B: $tType,A5: set @ A,P: B > A > $o,Net: filter @ B] :
( ( finite_finite2 @ A @ A5 )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ A5 )
=> ( eventually @ B
@ ^ [Y: B] : ( P @ Y @ X4 )
@ Net ) )
=> ( eventually @ B
@ ^ [X2: B] :
! [Y: A] :
( ( member @ A @ Y @ A5 )
=> ( P @ X2 @ Y ) )
@ Net ) ) ) ).
% eventually_ball_finite
thf(fact_5229_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
@ ^ [X2: B] :
! [Y: A] :
( ( member @ A @ Y @ A5 )
=> ( P @ X2 @ Y ) )
@ Net )
= ( ! [X2: A] :
( ( member @ A @ X2 @ A5 )
=> ( eventually @ B
@ ^ [Y: B] : ( P @ Y @ X2 )
@ Net ) ) ) ) ) ).
% eventually_ball_finite_distrib
thf(fact_5230_dvd__euclidean__size__eq__imp__dvd,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [A3: A,B2: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( ( euclid6346220572633701492n_size @ A @ A3 )
= ( euclid6346220572633701492n_size @ A @ B2 ) )
=> ( ( dvd_dvd @ A @ B2 @ A3 )
=> ( dvd_dvd @ A @ A3 @ B2 ) ) ) ) ) ).
% dvd_euclidean_size_eq_imp_dvd
thf(fact_5231_euclidean__size__unit,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [A3: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( euclid6346220572633701492n_size @ A @ A3 )
= ( euclid6346220572633701492n_size @ A @ ( one_one @ A ) ) ) ) ) ).
% euclidean_size_unit
thf(fact_5232_eventually__le__at__bot,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C2: A] :
( eventually @ A
@ ^ [X2: A] : ( ord_less_eq @ A @ X2 @ C2 )
@ ( at_bot @ A ) ) ) ).
% eventually_le_at_bot
thf(fact_5233_eventually__gt__at__bot,axiom,
! [A: $tType] :
( ( unboun7993243217541854897norder @ A )
=> ! [C2: A] :
( eventually @ A
@ ^ [X2: A] : ( ord_less @ A @ X2 @ C2 )
@ ( at_bot @ A ) ) ) ).
% eventually_gt_at_bot
thf(fact_5234_filterlim__mono__eventually,axiom,
! [B: $tType,A: $tType,F: A > B,F6: filter @ B,G6: filter @ A,F9: filter @ B,G7: filter @ A,F5: A > B] :
( ( filterlim @ A @ B @ F @ F6 @ G6 )
=> ( ( ord_less_eq @ ( filter @ B ) @ F6 @ F9 )
=> ( ( ord_less_eq @ ( filter @ A ) @ G7 @ G6 )
=> ( ( eventually @ A
@ ^ [X2: A] :
( ( F @ X2 )
= ( F5 @ X2 ) )
@ G7 )
=> ( filterlim @ A @ B @ F5 @ F9 @ G7 ) ) ) ) ) ).
% filterlim_mono_eventually
thf(fact_5235_filterlim__principal,axiom,
! [B: $tType,A: $tType,F: A > B,S3: set @ B,F6: filter @ A] :
( ( filterlim @ A @ B @ F @ ( principal @ B @ S3 ) @ F6 )
= ( eventually @ A
@ ^ [X2: A] : ( member @ B @ ( F @ X2 ) @ S3 )
@ F6 ) ) ).
% filterlim_principal
thf(fact_5236_le__principal,axiom,
! [A: $tType,F6: filter @ A,A5: set @ A] :
( ( ord_less_eq @ ( filter @ A ) @ F6 @ ( principal @ A @ A5 ) )
= ( eventually @ A
@ ^ [X2: A] : ( member @ A @ X2 @ A5 )
@ F6 ) ) ).
% le_principal
thf(fact_5237_eventually__INF1,axiom,
! [B: $tType,A: $tType,I: A,I5: set @ A,P: B > $o,F6: A > ( filter @ B )] :
( ( member @ A @ I @ I5 )
=> ( ( eventually @ B @ P @ ( F6 @ I ) )
=> ( eventually @ B @ P @ ( complete_Inf_Inf @ ( filter @ B ) @ ( image2 @ A @ ( filter @ B ) @ F6 @ I5 ) ) ) ) ) ).
% eventually_INF1
thf(fact_5238_eventually__inf__principal,axiom,
! [A: $tType,P: A > $o,F6: filter @ A,S: set @ A] :
( ( eventually @ A @ P @ ( inf_inf @ ( filter @ A ) @ F6 @ ( principal @ A @ S ) ) )
= ( eventually @ A
@ ^ [X2: A] :
( ( member @ A @ X2 @ S )
=> ( P @ X2 ) )
@ F6 ) ) ).
% eventually_inf_principal
thf(fact_5239_unit__iff__euclidean__size,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [A3: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
= ( ( ( euclid6346220572633701492n_size @ A @ A3 )
= ( euclid6346220572633701492n_size @ A @ ( one_one @ A ) ) )
& ( A3
!= ( zero_zero @ A ) ) ) ) ) ).
% unit_iff_euclidean_size
thf(fact_5240_eventually__INF__finite,axiom,
! [B: $tType,A: $tType,A5: set @ A,P: B > $o,F6: A > ( filter @ B )] :
( ( finite_finite2 @ A @ A5 )
=> ( ( eventually @ B @ P @ ( complete_Inf_Inf @ ( filter @ B ) @ ( image2 @ A @ ( filter @ B ) @ F6 @ A5 ) ) )
= ( ? [Q7: A > B > $o] :
( ! [X2: A] :
( ( member @ A @ X2 @ A5 )
=> ( eventually @ B @ ( Q7 @ X2 ) @ ( F6 @ X2 ) ) )
& ! [Y: B] :
( ! [X2: A] :
( ( member @ A @ X2 @ A5 )
=> ( Q7 @ X2 @ Y ) )
=> ( P @ Y ) ) ) ) ) ) ).
% eventually_INF_finite
thf(fact_5241_size__mult__mono,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [B2: A,A3: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ord_less_eq @ nat @ ( euclid6346220572633701492n_size @ A @ A3 ) @ ( euclid6346220572633701492n_size @ A @ ( times_times @ A @ A3 @ B2 ) ) ) ) ) ).
% size_mult_mono
thf(fact_5242_size__mult__mono_H,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [B2: A,A3: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ord_less_eq @ nat @ ( euclid6346220572633701492n_size @ A @ A3 ) @ ( euclid6346220572633701492n_size @ A @ ( times_times @ A @ B2 @ A3 ) ) ) ) ) ).
% size_mult_mono'
thf(fact_5243_euclidean__size__times__unit,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( euclid6346220572633701492n_size @ A @ ( times_times @ A @ A3 @ B2 ) )
= ( euclid6346220572633701492n_size @ A @ B2 ) ) ) ) ).
% euclidean_size_times_unit
thf(fact_5244_dvd__proper__imp__size__less,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ B2 )
=> ( ~ ( dvd_dvd @ A @ B2 @ A3 )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( ord_less @ nat @ ( euclid6346220572633701492n_size @ A @ A3 ) @ ( euclid6346220572633701492n_size @ A @ B2 ) ) ) ) ) ) ).
% dvd_proper_imp_size_less
thf(fact_5245_dvd__imp__size__le,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ B2 )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( ord_less_eq @ nat @ ( euclid6346220572633701492n_size @ A @ A3 ) @ ( euclid6346220572633701492n_size @ A @ B2 ) ) ) ) ) ).
% dvd_imp_size_le
thf(fact_5246_mod__size__less,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [B2: A,A3: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ord_less @ nat @ ( euclid6346220572633701492n_size @ A @ ( modulo_modulo @ A @ A3 @ B2 ) ) @ ( euclid6346220572633701492n_size @ A @ B2 ) ) ) ) ).
% mod_size_less
thf(fact_5247_filterlim__at__bot__le,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ B )
=> ! [F: A > B,F6: filter @ A,C2: B] :
( ( filterlim @ A @ B @ F @ ( at_bot @ B ) @ F6 )
= ( ! [Z9: B] :
( ( ord_less_eq @ B @ Z9 @ C2 )
=> ( eventually @ A
@ ^ [X2: A] : ( ord_less_eq @ B @ ( F @ X2 ) @ Z9 )
@ F6 ) ) ) ) ) ).
% filterlim_at_bot_le
thf(fact_5248_filterlim__at__bot,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ B )
=> ! [F: A > B,F6: filter @ A] :
( ( filterlim @ A @ B @ F @ ( at_bot @ B ) @ F6 )
= ( ! [Z9: B] :
( eventually @ A
@ ^ [X2: A] : ( ord_less_eq @ B @ ( F @ X2 ) @ Z9 )
@ F6 ) ) ) ) ).
% filterlim_at_bot
thf(fact_5249_filterlim__at__bot__dense,axiom,
! [A: $tType,B: $tType] :
( ( ( dense_linorder @ B )
& ( no_bot @ B ) )
=> ! [F: A > B,F6: filter @ A] :
( ( filterlim @ A @ B @ F @ ( at_bot @ B ) @ F6 )
= ( ! [Z9: B] :
( eventually @ A
@ ^ [X2: A] : ( ord_less @ B @ ( F @ X2 ) @ Z9 )
@ F6 ) ) ) ) ).
% filterlim_at_bot_dense
thf(fact_5250_gcd__nat_Oordering__top__axioms,axiom,
( ordering_top @ nat @ ( dvd_dvd @ nat )
@ ^ [M5: nat,N5: nat] :
( ( dvd_dvd @ nat @ M5 @ N5 )
& ( M5 != N5 ) )
@ ( zero_zero @ nat ) ) ).
% gcd_nat.ordering_top_axioms
thf(fact_5251_euclidean__size__int__def,axiom,
( ( euclid6346220572633701492n_size @ int )
= ( comp @ int @ nat @ int @ nat2 @ ( abs_abs @ int ) ) ) ).
% euclidean_size_int_def
thf(fact_5252_eventually__INF,axiom,
! [A: $tType,B: $tType,P: A > $o,F6: B > ( filter @ A ),B4: set @ B] :
( ( eventually @ A @ P @ ( complete_Inf_Inf @ ( filter @ A ) @ ( image2 @ B @ ( filter @ A ) @ F6 @ B4 ) ) )
= ( ? [X9: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ X9 @ B4 )
& ( finite_finite2 @ B @ X9 )
& ( eventually @ A @ P @ ( complete_Inf_Inf @ ( filter @ A ) @ ( image2 @ B @ ( filter @ A ) @ F6 @ X9 ) ) ) ) ) ) ).
% eventually_INF
thf(fact_5253_filterlim__at__bot__lt,axiom,
! [A: $tType,B: $tType] :
( ( unboun7993243217541854897norder @ B )
=> ! [F: A > B,F6: filter @ A,C2: B] :
( ( filterlim @ A @ B @ F @ ( at_bot @ B ) @ F6 )
= ( ! [Z9: B] :
( ( ord_less @ B @ Z9 @ C2 )
=> ( eventually @ A
@ ^ [X2: A] : ( ord_less_eq @ B @ ( F @ X2 ) @ Z9 )
@ F6 ) ) ) ) ) ).
% filterlim_at_bot_lt
thf(fact_5254_bot__nat__0_Oordering__top__axioms,axiom,
( ordering_top @ nat
@ ^ [X2: nat,Y: nat] : ( ord_less_eq @ nat @ Y @ X2 )
@ ^ [X2: nat,Y: nat] : ( ord_less @ nat @ Y @ X2 )
@ ( zero_zero @ nat ) ) ).
% bot_nat_0.ordering_top_axioms
thf(fact_5255_map__filter__on__comp,axiom,
! [A: $tType,C: $tType,B: $tType,G: B > A,Y9: set @ B,X7: set @ A,F6: filter @ B,F: A > C] :
( ( ord_less_eq @ ( set @ A ) @ ( image2 @ B @ A @ G @ Y9 ) @ X7 )
=> ( ( eventually @ B
@ ^ [X2: B] : ( member @ B @ X2 @ Y9 )
@ F6 )
=> ( ( map_filter_on @ A @ C @ X7 @ F @ ( map_filter_on @ B @ A @ Y9 @ G @ F6 ) )
= ( map_filter_on @ B @ C @ Y9 @ ( comp @ A @ C @ B @ F @ G ) @ F6 ) ) ) ) ).
% map_filter_on_comp
thf(fact_5256_divmod__cases,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [B2: A,A3: A] :
( ( ( B2
!= ( zero_zero @ A ) )
=> ( ( ( modulo_modulo @ A @ A3 @ B2 )
= ( zero_zero @ A ) )
=> ( A3
!= ( times_times @ A @ ( divide_divide @ A @ A3 @ B2 ) @ B2 ) ) ) )
=> ( ( ( B2
!= ( zero_zero @ A ) )
=> ! [Q3: A,R2: A] :
( ( ( euclid7384307370059645450egment @ A @ R2 )
= ( euclid7384307370059645450egment @ A @ B2 ) )
=> ( ( ord_less @ nat @ ( euclid6346220572633701492n_size @ A @ R2 ) @ ( euclid6346220572633701492n_size @ A @ B2 ) )
=> ( ( R2
!= ( zero_zero @ A ) )
=> ( ( ( divide_divide @ A @ A3 @ B2 )
= Q3 )
=> ( ( ( modulo_modulo @ A @ A3 @ B2 )
= R2 )
=> ( A3
!= ( plus_plus @ A @ ( times_times @ A @ Q3 @ B2 ) @ R2 ) ) ) ) ) ) ) )
=> ( B2
= ( zero_zero @ A ) ) ) ) ) ).
% divmod_cases
thf(fact_5257_mod__eqI,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [B2: A,R: A,Q2: A,A3: 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 )
= A3 )
=> ( ( modulo_modulo @ A @ A3 @ B2 )
= R ) ) ) ) ) ) ).
% mod_eqI
thf(fact_5258_division__segment__1,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ( ( euclid7384307370059645450egment @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% division_segment_1
thf(fact_5259_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_5260_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_5261_division__segment__nat__def,axiom,
( ( euclid7384307370059645450egment @ nat )
= ( ^ [N5: nat] : ( one_one @ nat ) ) ) ).
% division_segment_nat_def
thf(fact_5262_division__segment__not__0,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [A3: A] :
( ( euclid7384307370059645450egment @ A @ A3 )
!= ( zero_zero @ A ) ) ) ).
% division_segment_not_0
thf(fact_5263_division__segment__mult,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [A3: A,B2: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( ( euclid7384307370059645450egment @ A @ ( times_times @ A @ A3 @ B2 ) )
= ( times_times @ A @ ( euclid7384307370059645450egment @ A @ A3 ) @ ( euclid7384307370059645450egment @ A @ B2 ) ) ) ) ) ) ).
% division_segment_mult
thf(fact_5264_is__unit__division__segment,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [A3: A] : ( dvd_dvd @ A @ ( euclid7384307370059645450egment @ A @ A3 ) @ ( one_one @ A ) ) ) ).
% is_unit_division_segment
thf(fact_5265_eventually__map__filter__on,axiom,
! [B: $tType,A: $tType,X7: set @ A,F6: filter @ A,P: B > $o,F: A > B] :
( ( eventually @ A
@ ^ [X2: A] : ( member @ A @ X2 @ X7 )
@ F6 )
=> ( ( eventually @ B @ P @ ( map_filter_on @ A @ B @ X7 @ F @ F6 ) )
= ( eventually @ A
@ ^ [X2: A] :
( ( P @ ( F @ X2 ) )
& ( member @ A @ X2 @ X7 ) )
@ F6 ) ) ) ).
% eventually_map_filter_on
thf(fact_5266_division__segment__mod,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [B2: A,A3: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ~ ( dvd_dvd @ A @ B2 @ A3 )
=> ( ( euclid7384307370059645450egment @ A @ ( modulo_modulo @ A @ A3 @ B2 ) )
= ( euclid7384307370059645450egment @ A @ B2 ) ) ) ) ) ).
% division_segment_mod
thf(fact_5267_unique__euclidean__semiring__class_Odiv__eq__0__iff,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [A3: A,B2: A] :
( ( ( euclid7384307370059645450egment @ A @ A3 )
= ( euclid7384307370059645450egment @ A @ B2 ) )
=> ( ( ( divide_divide @ A @ A3 @ B2 )
= ( zero_zero @ A ) )
= ( ( ord_less @ nat @ ( euclid6346220572633701492n_size @ A @ A3 ) @ ( euclid6346220572633701492n_size @ A @ B2 ) )
| ( B2
= ( zero_zero @ A ) ) ) ) ) ) ).
% unique_euclidean_semiring_class.div_eq_0_iff
thf(fact_5268_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_5269_div__eqI,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [B2: A,R: A,Q2: A,A3: 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 )
= A3 )
=> ( ( divide_divide @ A @ A3 @ B2 )
= Q2 ) ) ) ) ) ) ).
% div_eqI
thf(fact_5270_eventually__cofinite,axiom,
! [A: $tType,P: A > $o] :
( ( eventually @ A @ P @ ( cofinite @ A ) )
= ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X2: A] :
~ ( P @ X2 ) ) ) ) ).
% eventually_cofinite
thf(fact_5271_Sup__filter__def,axiom,
! [A: $tType] :
( ( complete_Sup_Sup @ ( filter @ A ) )
= ( ^ [S8: set @ ( filter @ A )] :
( abs_filter @ A
@ ^ [P3: A > $o] :
! [X2: filter @ A] :
( ( member @ ( filter @ A ) @ X2 @ S8 )
=> ( eventually @ A @ P3 @ X2 ) ) ) ) ) ).
% Sup_filter_def
thf(fact_5272_filterlim__finite__subsets__at__top,axiom,
! [A: $tType,B: $tType,F: A > ( set @ B ),A5: set @ B,F6: filter @ A] :
( ( filterlim @ A @ ( set @ B ) @ F @ ( finite5375528669736107172at_top @ B @ A5 ) @ F6 )
= ( ! [X9: set @ B] :
( ( ( finite_finite2 @ B @ X9 )
& ( ord_less_eq @ ( set @ B ) @ X9 @ A5 ) )
=> ( eventually @ A
@ ^ [Y: A] :
( ( finite_finite2 @ B @ ( F @ Y ) )
& ( ord_less_eq @ ( set @ B ) @ X9 @ ( F @ Y ) )
& ( ord_less_eq @ ( set @ B ) @ ( F @ Y ) @ A5 ) )
@ F6 ) ) ) ) ).
% filterlim_finite_subsets_at_top
thf(fact_5273_cofinite__def,axiom,
! [A: $tType] :
( ( cofinite @ A )
= ( abs_filter @ A
@ ^ [P3: A > $o] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X2: A] :
~ ( P3 @ X2 ) ) ) ) ) ).
% cofinite_def
thf(fact_5274_sup__filter__def,axiom,
! [A: $tType] :
( ( sup_sup @ ( filter @ A ) )
= ( ^ [F7: filter @ A,F10: filter @ A] :
( abs_filter @ A
@ ^ [P3: A > $o] :
( ( eventually @ A @ P3 @ F7 )
& ( eventually @ A @ P3 @ F10 ) ) ) ) ) ).
% sup_filter_def
thf(fact_5275_bot__filter__def,axiom,
! [A: $tType] :
( ( bot_bot @ ( filter @ A ) )
= ( abs_filter @ A
@ ^ [P3: A > $o] : $true ) ) ).
% bot_filter_def
thf(fact_5276_top__filter__def,axiom,
! [A: $tType] :
( ( top_top @ ( filter @ A ) )
= ( abs_filter @ A
@ ^ [P2: A > $o] :
! [X3: A] : ( P2 @ X3 ) ) ) ).
% top_filter_def
thf(fact_5277_principal__def,axiom,
! [A: $tType] :
( ( principal @ A )
= ( ^ [S8: set @ A] : ( abs_filter @ A @ ( ball @ A @ S8 ) ) ) ) ).
% principal_def
thf(fact_5278_map__filter__on__def,axiom,
! [B: $tType,A: $tType] :
( ( map_filter_on @ A @ B )
= ( ^ [X9: set @ A,F2: A > B,F7: filter @ A] :
( abs_filter @ B
@ ^ [P3: B > $o] :
( eventually @ A
@ ^ [X2: A] :
( ( P3 @ ( F2 @ X2 ) )
& ( member @ A @ X2 @ X9 ) )
@ F7 ) ) ) ) ).
% map_filter_on_def
thf(fact_5279_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 ) )
@ ^ [X9: set @ A] :
( principal @ ( set @ A )
@ ( collect @ ( set @ A )
@ ^ [Y10: set @ A] :
( ( finite_finite2 @ A @ Y10 )
& ( ord_less_eq @ ( set @ A ) @ X9 @ Y10 )
& ( ord_less_eq @ ( set @ A ) @ Y10 @ A8 ) ) ) )
@ ( collect @ ( set @ A )
@ ^ [X9: set @ A] :
( ( finite_finite2 @ A @ X9 )
& ( ord_less_eq @ ( set @ A ) @ X9 @ A8 ) ) ) ) ) ) ) ).
% finite_subsets_at_top_def
thf(fact_5280_inf__filter__def,axiom,
! [A: $tType] :
( ( inf_inf @ ( filter @ A ) )
= ( ^ [F7: filter @ A,F10: filter @ A] :
( abs_filter @ A
@ ^ [P3: A > $o] :
? [Q7: A > $o,R6: A > $o] :
( ( eventually @ A @ Q7 @ F7 )
& ( eventually @ A @ R6 @ F10 )
& ! [X2: A] :
( ( ( Q7 @ X2 )
& ( R6 @ X2 ) )
=> ( P3 @ X2 ) ) ) ) ) ) ).
% inf_filter_def
thf(fact_5281_filterlim__at__top__gt,axiom,
! [A: $tType,B: $tType] :
( ( unboun7993243217541854897norder @ B )
=> ! [F: A > B,F6: filter @ A,C2: B] :
( ( filterlim @ A @ B @ F @ ( at_top @ B ) @ F6 )
= ( ! [Z9: B] :
( ( ord_less @ B @ C2 @ Z9 )
=> ( eventually @ A
@ ^ [X2: A] : ( ord_less_eq @ B @ Z9 @ ( F @ X2 ) )
@ F6 ) ) ) ) ) ).
% filterlim_at_top_gt
thf(fact_5282_coinduct3,axiom,
! [A: $tType,F: ( set @ A ) > ( set @ A ),A3: A,X7: set @ A] :
( ( order_mono @ ( set @ A ) @ ( set @ A ) @ F )
=> ( ( member @ A @ A3 @ X7 )
=> ( ( ord_less_eq @ ( set @ A ) @ X7
@ ( F
@ ( complete_lattice_lfp @ ( set @ A )
@ ^ [X2: set @ A] : ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ ( F @ X2 ) @ X7 ) @ ( complete_lattice_gfp @ ( set @ A ) @ F ) ) ) ) )
=> ( member @ A @ A3 @ ( complete_lattice_gfp @ ( set @ A ) @ F ) ) ) ) ) ).
% coinduct3
thf(fact_5283_def__coinduct3,axiom,
! [A: $tType,A5: set @ A,F: ( set @ A ) > ( set @ A ),A3: A,X7: set @ A] :
( ( A5
= ( complete_lattice_gfp @ ( set @ A ) @ F ) )
=> ( ( order_mono @ ( set @ A ) @ ( set @ A ) @ F )
=> ( ( member @ A @ A3 @ X7 )
=> ( ( ord_less_eq @ ( set @ A ) @ X7
@ ( F
@ ( complete_lattice_lfp @ ( set @ A )
@ ^ [X2: set @ A] : ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ ( F @ X2 ) @ X7 ) @ A5 ) ) ) )
=> ( member @ A @ A3 @ A5 ) ) ) ) ) ).
% def_coinduct3
thf(fact_5284_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_5285_eventually__sequentially__seg,axiom,
! [P: nat > $o,K: nat] :
( ( eventually @ nat
@ ^ [N5: nat] : ( P @ ( plus_plus @ nat @ N5 @ K ) )
@ ( at_top @ nat ) )
= ( eventually @ nat @ P @ ( at_top @ nat ) ) ) ).
% eventually_sequentially_seg
thf(fact_5286_filterlim__sequentially__Suc,axiom,
! [A: $tType,F: nat > A,F6: filter @ A] :
( ( filterlim @ nat @ A
@ ^ [X2: nat] : ( F @ ( suc @ X2 ) )
@ F6
@ ( at_top @ nat ) )
= ( filterlim @ nat @ A @ F @ F6 @ ( at_top @ nat ) ) ) ).
% filterlim_sequentially_Suc
thf(fact_5287_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
@ ^ [X2: A] : ( F @ ( G @ X2 ) ) ) )
= ( complete_lattice_gfp @ B
@ ^ [X2: B] : ( G @ ( F @ X2 ) ) ) ) ) ) ) ).
% gfp_rolling
thf(fact_5288_le__sequentially,axiom,
! [F6: filter @ nat] :
( ( ord_less_eq @ ( filter @ nat ) @ F6 @ ( at_top @ nat ) )
= ( ! [N11: nat] : ( eventually @ nat @ ( ord_less_eq @ nat @ N11 ) @ F6 ) ) ) ).
% le_sequentially
thf(fact_5289_gfp__gfp,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F: A > A > A] :
( ! [X4: A,Y4: A,W3: A,Z6: A] :
( ( ord_less_eq @ A @ X4 @ Y4 )
=> ( ( ord_less_eq @ A @ W3 @ Z6 )
=> ( ord_less_eq @ A @ ( F @ X4 @ W3 ) @ ( F @ Y4 @ Z6 ) ) ) )
=> ( ( complete_lattice_gfp @ A
@ ^ [X2: A] : ( complete_lattice_gfp @ A @ ( F @ X2 ) ) )
= ( complete_lattice_gfp @ A
@ ^ [X2: A] : ( F @ X2 @ X2 ) ) ) ) ) ).
% gfp_gfp
thf(fact_5290_gfp__const,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [T2: A] :
( ( complete_lattice_gfp @ A
@ ^ [X2: A] : T2 )
= T2 ) ) ).
% gfp_const
thf(fact_5291_eventually__at__top__not__equal,axiom,
! [A: $tType] :
( ( ( linorder @ A )
& ( no_top @ A ) )
=> ! [C2: A] :
( eventually @ A
@ ^ [X2: A] : X2 != C2
@ ( at_top @ A ) ) ) ).
% eventually_at_top_not_equal
thf(fact_5292_gfp__def,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ( ( complete_lattice_gfp @ A )
= ( ^ [F2: A > A] :
( complete_Sup_Sup @ A
@ ( collect @ A
@ ^ [U: A] : ( ord_less_eq @ A @ U @ ( F2 @ U ) ) ) ) ) ) ) ).
% gfp_def
thf(fact_5293_def__Collect__coinduct,axiom,
! [A: $tType,A5: set @ A,P: ( set @ A ) > A > $o,A3: A,X7: set @ A] :
( ( A5
= ( complete_lattice_gfp @ ( set @ A )
@ ^ [W2: set @ A] : ( collect @ A @ ( P @ W2 ) ) ) )
=> ( ( order_mono @ ( set @ A ) @ ( set @ A )
@ ^ [W2: set @ A] : ( collect @ A @ ( P @ W2 ) ) )
=> ( ( member @ A @ A3 @ X7 )
=> ( ! [Z6: A] :
( ( member @ A @ Z6 @ X7 )
=> ( P @ ( sup_sup @ ( set @ A ) @ X7 @ A5 ) @ Z6 ) )
=> ( member @ A @ A3 @ A5 ) ) ) ) ) ).
% def_Collect_coinduct
thf(fact_5294_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_5295_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_5296_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_5297_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_5298_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_5299_eventually__all__ge__at__top,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o] :
( ( eventually @ A @ P @ ( at_top @ A ) )
=> ( eventually @ A
@ ^ [X2: A] :
! [Y: A] :
( ( ord_less_eq @ A @ X2 @ Y )
=> ( P @ Y ) )
@ ( at_top @ A ) ) ) ) ).
% eventually_all_ge_at_top
thf(fact_5300_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_5301_filterlim__at__top,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ B )
=> ! [F: A > B,F6: filter @ A] :
( ( filterlim @ A @ B @ F @ ( at_top @ B ) @ F6 )
= ( ! [Z9: B] :
( eventually @ A
@ ^ [X2: A] : ( ord_less_eq @ B @ Z9 @ ( F @ X2 ) )
@ F6 ) ) ) ) ).
% filterlim_at_top
thf(fact_5302_filterlim__at__top__ge,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ B )
=> ! [F: A > B,F6: filter @ A,C2: B] :
( ( filterlim @ A @ B @ F @ ( at_top @ B ) @ F6 )
= ( ! [Z9: B] :
( ( ord_less_eq @ B @ C2 @ Z9 )
=> ( eventually @ A
@ ^ [X2: A] : ( ord_less_eq @ B @ Z9 @ ( F @ X2 ) )
@ F6 ) ) ) ) ) ).
% filterlim_at_top_ge
thf(fact_5303_filterlim__at__top__mono,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F: B > A,F6: filter @ B,G: B > A] :
( ( filterlim @ B @ A @ F @ ( at_top @ A ) @ F6 )
=> ( ( eventually @ B
@ ^ [X2: B] : ( ord_less_eq @ A @ ( F @ X2 ) @ ( G @ X2 ) )
@ F6 )
=> ( filterlim @ B @ A @ G @ ( at_top @ A ) @ F6 ) ) ) ) ).
% filterlim_at_top_mono
thf(fact_5304_filterlim__at__top__dense,axiom,
! [A: $tType,B: $tType] :
( ( unboun7993243217541854897norder @ B )
=> ! [F: A > B,F6: filter @ A] :
( ( filterlim @ A @ B @ F @ ( at_top @ B ) @ F6 )
= ( ! [Z9: B] :
( eventually @ A
@ ^ [X2: A] : ( ord_less @ B @ Z9 @ ( F @ X2 ) )
@ F6 ) ) ) ) ).
% filterlim_at_top_dense
thf(fact_5305_coinduct3__lemma,axiom,
! [A: $tType,X7: set @ A,F: ( set @ A ) > ( set @ A )] :
( ( ord_less_eq @ ( set @ A ) @ X7
@ ( F
@ ( complete_lattice_lfp @ ( set @ A )
@ ^ [X2: set @ A] : ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ ( F @ X2 ) @ X7 ) @ ( complete_lattice_gfp @ ( set @ A ) @ F ) ) ) ) )
=> ( ( order_mono @ ( set @ A ) @ ( set @ A ) @ F )
=> ( ord_less_eq @ ( set @ A )
@ ( complete_lattice_lfp @ ( set @ A )
@ ^ [X2: set @ A] : ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ ( F @ X2 ) @ X7 ) @ ( complete_lattice_gfp @ ( set @ A ) @ F ) ) )
@ ( F
@ ( complete_lattice_lfp @ ( set @ A )
@ ^ [X2: set @ A] : ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ ( F @ X2 ) @ X7 ) @ ( complete_lattice_gfp @ ( set @ A ) @ F ) ) ) ) ) ) ) ).
% coinduct3_lemma
thf(fact_5306_at__top__def,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( at_top @ A )
= ( complete_Inf_Inf @ ( filter @ A )
@ ( image2 @ A @ ( filter @ A )
@ ^ [K3: A] : ( principal @ A @ ( set_ord_atLeast @ A @ K3 ) )
@ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% at_top_def
thf(fact_5307_filtercomap__def,axiom,
! [B: $tType,A: $tType] :
( ( filtercomap @ A @ B )
= ( ^ [F2: A > B,F7: filter @ B] :
( abs_filter @ A
@ ^ [P3: A > $o] :
? [Q7: B > $o] :
( ( eventually @ B @ Q7 @ F7 )
& ! [X2: A] :
( ( Q7 @ ( F2 @ X2 ) )
=> ( P3 @ X2 ) ) ) ) ) ) ).
% filtercomap_def
thf(fact_5308_at__top__sub,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C2: A] :
( ( at_top @ A )
= ( complete_Inf_Inf @ ( filter @ A )
@ ( image2 @ A @ ( filter @ A )
@ ^ [K3: A] : ( principal @ A @ ( set_ord_atLeast @ A @ K3 ) )
@ ( set_ord_atLeast @ A @ C2 ) ) ) ) ) ).
% at_top_sub
thf(fact_5309_atLeast__0,axiom,
( ( set_ord_atLeast @ nat @ ( zero_zero @ nat ) )
= ( top_top @ ( set @ nat ) ) ) ).
% atLeast_0
thf(fact_5310_eventually__filtercomapI,axiom,
! [B: $tType,A: $tType,P: A > $o,F6: filter @ A,F: B > A] :
( ( eventually @ A @ P @ F6 )
=> ( eventually @ B
@ ^ [X2: B] : ( P @ ( F @ X2 ) )
@ ( filtercomap @ B @ A @ F @ F6 ) ) ) ).
% eventually_filtercomapI
thf(fact_5311_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_5312_eventually__False__sequentially,axiom,
~ ( eventually @ nat
@ ^ [N5: nat] : $false
@ ( at_top @ nat ) ) ).
% eventually_False_sequentially
thf(fact_5313_atLeast__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( set_ord_atLeast @ A )
= ( ^ [L4: A] : ( collect @ A @ ( ord_less_eq @ A @ L4 ) ) ) ) ) ).
% atLeast_def
thf(fact_5314_filtercomap__ident,axiom,
! [A: $tType,F6: filter @ A] :
( ( filtercomap @ A @ A
@ ^ [X2: A] : X2
@ F6 )
= F6 ) ).
% filtercomap_ident
thf(fact_5315_filtercomap__filtercomap,axiom,
! [A: $tType,B: $tType,C: $tType,F: A > B,G: B > C,F6: filter @ C] :
( ( filtercomap @ A @ B @ F @ ( filtercomap @ B @ C @ G @ F6 ) )
= ( filtercomap @ A @ C
@ ^ [X2: A] : ( G @ ( F @ X2 ) )
@ F6 ) ) ).
% filtercomap_filtercomap
thf(fact_5316_filterlim__filtercomap__iff,axiom,
! [C: $tType,B: $tType,A: $tType,F: A > B,G: B > C,G6: filter @ C,F6: filter @ A] :
( ( filterlim @ A @ B @ F @ ( filtercomap @ B @ C @ G @ G6 ) @ F6 )
= ( filterlim @ A @ C @ ( comp @ B @ C @ A @ G @ F ) @ G6 @ F6 ) ) ).
% filterlim_filtercomap_iff
thf(fact_5317_filtercomap__INF,axiom,
! [A: $tType,B: $tType,C: $tType,F: A > B,F6: C > ( filter @ B ),B4: set @ C] :
( ( filtercomap @ A @ B @ F @ ( complete_Inf_Inf @ ( filter @ B ) @ ( image2 @ C @ ( filter @ B ) @ F6 @ B4 ) ) )
= ( complete_Inf_Inf @ ( filter @ A )
@ ( image2 @ C @ ( filter @ A )
@ ^ [B5: C] : ( filtercomap @ A @ B @ F @ ( F6 @ B5 ) )
@ B4 ) ) ) ).
% filtercomap_INF
thf(fact_5318_filtercomap__SUP,axiom,
! [A: $tType,C: $tType,B: $tType,F: A > C,F6: B > ( filter @ C ),B4: set @ B] :
( ord_less_eq @ ( filter @ A )
@ ( complete_Sup_Sup @ ( filter @ A )
@ ( image2 @ B @ ( filter @ A )
@ ^ [B5: B] : ( filtercomap @ A @ C @ F @ ( F6 @ B5 ) )
@ B4 ) )
@ ( filtercomap @ A @ C @ F @ ( complete_Sup_Sup @ ( filter @ C ) @ ( image2 @ B @ ( filter @ C ) @ F6 @ B4 ) ) ) ) ).
% filtercomap_SUP
thf(fact_5319_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_5320_AboveS__def,axiom,
! [A: $tType] :
( ( order_AboveS @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A ),A8: set @ A] :
( collect @ A
@ ^ [B5: A] :
( ( member @ A @ B5 @ ( field2 @ A @ R4 ) )
& ! [X2: A] :
( ( member @ A @ X2 @ A8 )
=> ( ( B5 != X2 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ B5 ) @ R4 ) ) ) ) ) ) ) ).
% AboveS_def
thf(fact_5321_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_5322_cSUP__UNION,axiom,
! [B: $tType,D: $tType,C: $tType] :
( ( condit1219197933456340205attice @ B )
=> ! [A5: set @ C,B4: C > ( set @ D ),F: D > B] :
( ( A5
!= ( bot_bot @ ( set @ C ) ) )
=> ( ! [X4: C] :
( ( member @ C @ X4 @ A5 )
=> ( ( B4 @ X4 )
!= ( bot_bot @ ( set @ D ) ) ) )
=> ( ( condit941137186595557371_above @ B
@ ( complete_Sup_Sup @ ( set @ B )
@ ( image2 @ C @ ( set @ B )
@ ^ [X2: C] : ( image2 @ D @ B @ F @ ( B4 @ X2 ) )
@ A5 ) ) )
=> ( ( complete_Sup_Sup @ B @ ( image2 @ D @ B @ F @ ( complete_Sup_Sup @ ( set @ D ) @ ( image2 @ C @ ( set @ D ) @ B4 @ A5 ) ) ) )
= ( complete_Sup_Sup @ B
@ ( image2 @ C @ B
@ ^ [X2: C] : ( complete_Sup_Sup @ B @ ( image2 @ D @ B @ F @ ( B4 @ X2 ) ) )
@ A5 ) ) ) ) ) ) ) ).
% cSUP_UNION
thf(fact_5323_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
@ ^ [X2: B] : ( sup_sup @ A @ ( F @ X2 ) @ ( G @ X2 ) )
@ A5 ) )
= ( ( condit941137186595557371_above @ A @ ( image2 @ B @ A @ F @ A5 ) )
& ( condit941137186595557371_above @ A @ ( image2 @ B @ A @ G @ A5 ) ) ) ) ) ).
% bdd_above_image_sup
thf(fact_5324_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 ) ) )
= ( ! [X2: B] :
( ( member @ B @ X2 @ I5 )
=> ( condit941137186595557371_above @ A @ ( A5 @ X2 ) ) ) ) ) ) ) ).
% bdd_above_UN
thf(fact_5325_cSUP__lessD,axiom,
! [B: $tType,A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [F: B > A,A5: set @ B,Y3: A,I: B] :
( ( condit941137186595557371_above @ A @ ( image2 @ B @ A @ F @ A5 ) )
=> ( ( ord_less @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F @ A5 ) ) @ Y3 )
=> ( ( member @ B @ I @ A5 )
=> ( ord_less @ A @ ( F @ I ) @ Y3 ) ) ) ) ) ).
% cSUP_lessD
thf(fact_5326_prod__decode__def,axiom,
( nat_prod_decode
= ( nat_prod_decode_aux @ ( zero_zero @ nat ) ) ) ).
% prod_decode_def
thf(fact_5327_less__cSUP__iff,axiom,
! [A: $tType,B: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [A5: set @ B,F: B > A,A3: A] :
( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit941137186595557371_above @ A @ ( image2 @ B @ A @ F @ A5 ) )
=> ( ( ord_less @ A @ A3 @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F @ A5 ) ) )
= ( ? [X2: B] :
( ( member @ B @ X2 @ A5 )
& ( ord_less @ A @ A3 @ ( F @ X2 ) ) ) ) ) ) ) ) ).
% less_cSUP_iff
thf(fact_5328_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
@ ^ [A6: B] : ( sup_sup @ A @ ( F @ A6 ) @ ( G @ A6 ) )
@ A5 ) ) ) ) ) ) ) ).
% conditionally_complete_lattice_class.SUP_sup_distrib
thf(fact_5329_cSup__cInf,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [S3: set @ A] :
( ( S3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit941137186595557371_above @ A @ S3 )
=> ( ( complete_Sup_Sup @ A @ S3 )
= ( complete_Inf_Inf @ A
@ ( collect @ A
@ ^ [X2: A] :
! [Y: A] :
( ( member @ A @ Y @ S3 )
=> ( ord_less_eq @ A @ Y @ X2 ) ) ) ) ) ) ) ) ).
% cSup_cInf
thf(fact_5330_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
@ ^ [X2: C] : ( F @ ( A5 @ X2 ) )
@ I5 ) )
@ ( F @ ( complete_Sup_Sup @ A @ ( image2 @ C @ A @ A5 @ I5 ) ) ) ) ) ) ) ) ).
% mono_cSUP
thf(fact_5331_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_5332_wo__rel_Osuc__greater,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),B4: set @ A,B2: A] :
( ( bNF_Wellorder_wo_rel @ A @ R )
=> ( ( ord_less_eq @ ( set @ A ) @ B4 @ ( field2 @ A @ R ) )
=> ( ( ( order_AboveS @ A @ R @ B4 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( member @ A @ B2 @ B4 )
=> ( ( ( bNF_Wellorder_wo_suc @ A @ R @ B4 )
!= B2 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ ( bNF_Wellorder_wo_suc @ A @ R @ B4 ) ) @ R ) ) ) ) ) ) ).
% wo_rel.suc_greater
thf(fact_5333_list__decode_Oelims,axiom,
! [X: nat,Y3: list @ nat] :
( ( ( nat_list_decode @ X )
= Y3 )
=> ( ( ( X
= ( zero_zero @ nat ) )
=> ( Y3
!= ( nil @ nat ) ) )
=> ~ ! [N2: nat] :
( ( X
= ( suc @ N2 ) )
=> ( Y3
!= ( product_case_prod @ nat @ nat @ ( list @ nat )
@ ^ [X2: nat,Y: nat] : ( cons @ nat @ X2 @ ( nat_list_decode @ Y ) )
@ ( nat_prod_decode @ N2 ) ) ) ) ) ) ).
% list_decode.elims
thf(fact_5334_wo__rel_Oequals__suc__AboveS,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),B4: set @ A,A3: A] :
( ( bNF_Wellorder_wo_rel @ A @ R )
=> ( ( ord_less_eq @ ( set @ A ) @ B4 @ ( field2 @ A @ R ) )
=> ( ( member @ A @ A3 @ ( order_AboveS @ A @ R @ B4 ) )
=> ( ! [A17: A] :
( ( member @ A @ A17 @ ( order_AboveS @ A @ R @ B4 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ A17 ) @ R ) )
=> ( A3
= ( bNF_Wellorder_wo_suc @ A @ R @ B4 ) ) ) ) ) ) ).
% wo_rel.equals_suc_AboveS
thf(fact_5335_list__decode_Osimps_I1_J,axiom,
( ( nat_list_decode @ ( zero_zero @ nat ) )
= ( nil @ nat ) ) ).
% list_decode.simps(1)
thf(fact_5336_wo__rel_Osuc__least__AboveS,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A3: A,B4: set @ A] :
( ( bNF_Wellorder_wo_rel @ A @ R )
=> ( ( member @ A @ A3 @ ( order_AboveS @ A @ R @ B4 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( bNF_Wellorder_wo_suc @ A @ R @ B4 ) @ A3 ) @ R ) ) ) ).
% wo_rel.suc_least_AboveS
thf(fact_5337_list__decode_Osimps_I2_J,axiom,
! [N: nat] :
( ( nat_list_decode @ ( suc @ N ) )
= ( product_case_prod @ nat @ nat @ ( list @ nat )
@ ^ [X2: nat,Y: nat] : ( cons @ nat @ X2 @ ( nat_list_decode @ Y ) )
@ ( nat_prod_decode @ N ) ) ) ).
% list_decode.simps(2)
thf(fact_5338_wo__rel_Osuc__ofilter__in,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A5: set @ A,B2: A] :
( ( bNF_Wellorder_wo_rel @ A @ R )
=> ( ( order_ofilter @ A @ R @ A5 )
=> ( ( ( order_AboveS @ A @ R @ A5 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ ( bNF_Wellorder_wo_suc @ A @ R @ A5 ) ) @ R )
=> ( ( B2
!= ( bNF_Wellorder_wo_suc @ A @ R @ A5 ) )
=> ( member @ A @ B2 @ A5 ) ) ) ) ) ) ).
% wo_rel.suc_ofilter_in
thf(fact_5339_list__decode_Opelims,axiom,
! [X: nat,Y3: list @ nat] :
( ( ( nat_list_decode @ X )
= Y3 )
=> ( ( accp @ nat @ nat_list_decode_rel @ X )
=> ( ( ( X
= ( zero_zero @ nat ) )
=> ( ( Y3
= ( nil @ nat ) )
=> ~ ( accp @ nat @ nat_list_decode_rel @ ( zero_zero @ nat ) ) ) )
=> ~ ! [N2: nat] :
( ( X
= ( suc @ N2 ) )
=> ( ( Y3
= ( product_case_prod @ nat @ nat @ ( list @ nat )
@ ^ [X2: nat,Y: nat] : ( cons @ nat @ X2 @ ( nat_list_decode @ Y ) )
@ ( nat_prod_decode @ N2 ) ) )
=> ~ ( accp @ nat @ nat_list_decode_rel @ ( suc @ N2 ) ) ) ) ) ) ) ).
% list_decode.pelims
thf(fact_5340_cINF__UNION,axiom,
! [B: $tType,D: $tType,C: $tType] :
( ( condit1219197933456340205attice @ B )
=> ! [A5: set @ C,B4: C > ( set @ D ),F: D > B] :
( ( A5
!= ( bot_bot @ ( set @ C ) ) )
=> ( ! [X4: C] :
( ( member @ C @ X4 @ A5 )
=> ( ( B4 @ X4 )
!= ( bot_bot @ ( set @ D ) ) ) )
=> ( ( condit1013018076250108175_below @ B
@ ( complete_Sup_Sup @ ( set @ B )
@ ( image2 @ C @ ( set @ B )
@ ^ [X2: C] : ( image2 @ D @ B @ F @ ( B4 @ X2 ) )
@ A5 ) ) )
=> ( ( complete_Inf_Inf @ B @ ( image2 @ D @ B @ F @ ( complete_Sup_Sup @ ( set @ D ) @ ( image2 @ C @ ( set @ D ) @ B4 @ A5 ) ) ) )
= ( complete_Inf_Inf @ B
@ ( image2 @ C @ B
@ ^ [X2: C] : ( complete_Inf_Inf @ B @ ( image2 @ D @ B @ F @ ( B4 @ X2 ) ) )
@ A5 ) ) ) ) ) ) ) ).
% cINF_UNION
thf(fact_5341_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
@ ^ [X2: B] : ( inf_inf @ A @ ( F @ X2 ) @ ( G @ X2 ) )
@ A5 ) )
= ( ( condit1013018076250108175_below @ A @ ( image2 @ B @ A @ F @ A5 ) )
& ( condit1013018076250108175_below @ A @ ( image2 @ B @ A @ G @ A5 ) ) ) ) ) ).
% bdd_below_image_inf
thf(fact_5342_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 ) ) )
= ( ! [X2: B] :
( ( member @ B @ X2 @ I5 )
=> ( condit1013018076250108175_below @ A @ ( A5 @ X2 ) ) ) ) ) ) ) ).
% bdd_below_UN
thf(fact_5343_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_5344_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 ) )
=> ( ! [X5: nat,Y5: nat] :
( ( ( product_Pair @ nat @ nat @ X5 @ Y5 )
= ( nat_prod_decode @ N2 ) )
=> ( P @ Y5 ) )
=> ( P @ ( suc @ N2 ) ) ) )
=> ( P @ A0 ) ) ) ) ).
% list_decode.pinduct
thf(fact_5345_less__cINF__D,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [F: B > A,A5: set @ B,Y3: A,I: B] :
( ( condit1013018076250108175_below @ A @ ( image2 @ B @ A @ F @ A5 ) )
=> ( ( ord_less @ A @ Y3 @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F @ A5 ) ) )
=> ( ( member @ B @ I @ A5 )
=> ( ord_less @ A @ Y3 @ ( F @ I ) ) ) ) ) ) ).
% less_cINF_D
thf(fact_5346_wo__rel_Oofilter__UNION,axiom,
! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ A ),I5: set @ B,A5: B > ( set @ A )] :
( ( bNF_Wellorder_wo_rel @ A @ R )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ I5 )
=> ( order_ofilter @ A @ R @ ( A5 @ I3 ) ) )
=> ( order_ofilter @ A @ R @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ A5 @ I5 ) ) ) ) ) ).
% wo_rel.ofilter_UNION
thf(fact_5347_cINF__less__iff,axiom,
! [A: $tType,B: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [A5: set @ B,F: B > A,A3: A] :
( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit1013018076250108175_below @ A @ ( image2 @ B @ A @ F @ A5 ) )
=> ( ( ord_less @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F @ A5 ) ) @ A3 )
= ( ? [X2: B] :
( ( member @ B @ X2 @ A5 )
& ( ord_less @ A @ ( F @ X2 ) @ A3 ) ) ) ) ) ) ) ).
% cINF_less_iff
thf(fact_5348_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
@ ^ [A6: B] : ( inf_inf @ A @ ( F @ A6 ) @ ( G @ A6 ) )
@ A5 ) ) ) ) ) ) ) ).
% cINF_inf_distrib
thf(fact_5349_cSUP__eq__cINF__D,axiom,
! [B: $tType,C: $tType] :
( ( condit1219197933456340205attice @ B )
=> ! [F: C > B,A5: set @ C,A3: 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 @ A3 @ A5 )
=> ( ( F @ A3 )
= ( complete_Inf_Inf @ B @ ( image2 @ C @ B @ F @ A5 ) ) ) ) ) ) ) ) ).
% cSUP_eq_cINF_D
thf(fact_5350_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 )
@ ^ [X2: nat,Y: nat] : ( cons @ nat @ X2 @ ( nat_list_decode @ Y ) )
@ ( nat_prod_decode @ N ) ) ) ) ).
% list_decode.psimps(2)
thf(fact_5351_cInf__cSup,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [S3: set @ A] :
( ( S3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit1013018076250108175_below @ A @ S3 )
=> ( ( complete_Inf_Inf @ A @ S3 )
= ( complete_Sup_Sup @ A
@ ( collect @ A
@ ^ [X2: A] :
! [Y: A] :
( ( member @ A @ Y @ S3 )
=> ( ord_less_eq @ A @ X2 @ Y ) ) ) ) ) ) ) ) ).
% cInf_cSup
thf(fact_5352_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
@ ^ [X2: C] : ( F @ ( A5 @ X2 ) )
@ I5 ) ) ) ) ) ) ) ).
% mono_cINF
thf(fact_5353_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_5354_ofilterIncl__def,axiom,
! [A: $tType] :
( ( bNF_We413866401316099525erIncl @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ ( set @ A ) @ ( set @ A ) )
@ ( product_case_prod @ ( set @ A ) @ ( set @ A ) @ $o
@ ^ [A8: set @ A,B7: set @ A] :
( ( order_ofilter @ A @ R4 @ A8 )
& ( A8
!= ( field2 @ A @ R4 ) )
& ( order_ofilter @ A @ R4 @ B7 )
& ( B7
!= ( field2 @ A @ R4 ) )
& ( ord_less @ ( set @ A ) @ A8 @ B7 ) ) ) ) ) ) ).
% ofilterIncl_def
thf(fact_5355_wo__rel_Oofilter__under__UNION,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A5: set @ A] :
( ( bNF_Wellorder_wo_rel @ A @ R )
=> ( ( order_ofilter @ A @ R @ A5 )
=> ( A5
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ A @ ( set @ A ) @ ( order_under @ A @ R ) @ A5 ) ) ) ) ) ).
% wo_rel.ofilter_under_UNION
thf(fact_5356_bsqr__max2,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A1: A,A22: A,B14: A,B23: A] :
( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( member @ ( product_prod @ ( product_prod @ A @ A ) @ ( product_prod @ A @ A ) ) @ ( product_Pair @ ( product_prod @ A @ A ) @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A1 @ A22 ) @ ( product_Pair @ A @ A @ B14 @ B23 ) ) @ ( bNF_Wellorder_bsqr @ A @ R ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( bNF_We1388413361240627857o_max2 @ A @ R @ A1 @ A22 ) @ ( bNF_We1388413361240627857o_max2 @ A @ R @ B14 @ B23 ) ) @ R ) ) ) ).
% bsqr_max2
thf(fact_5357_under__def,axiom,
! [A: $tType] :
( ( order_under @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A ),A6: A] :
( collect @ A
@ ^ [B5: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B5 @ A6 ) @ R4 ) ) ) ) ).
% under_def
thf(fact_5358_well__order__on__domain,axiom,
! [A: $tType,A5: set @ A,R: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
( ( order_well_order_on @ A @ A5 @ R )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R )
=> ( ( member @ A @ A3 @ A5 )
& ( member @ A @ B2 @ A5 ) ) ) ) ).
% well_order_on_domain
thf(fact_5359_ofilter__Restr__under,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A5: set @ A,A3: A] :
( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( order_ofilter @ A @ R @ A5 )
=> ( ( member @ A @ A3 @ A5 )
=> ( ( order_under @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R
@ ( product_Sigma @ A @ A @ A5
@ ^ [Uu: A] : A5 ) )
@ A3 )
= ( order_under @ A @ R @ A3 ) ) ) ) ) ).
% ofilter_Restr_under
thf(fact_5360_Well__order__Restr,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A5: set @ A] :
( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
=> ( order_well_order_on @ A
@ ( field2 @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R
@ ( product_Sigma @ A @ A @ A5
@ ^ [Uu: A] : A5 ) ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R
@ ( product_Sigma @ A @ A @ A5
@ ^ [Uu: A] : A5 ) ) ) ) ).
% Well_order_Restr
thf(fact_5361_natLeq__on__well__order__on,axiom,
! [N: nat] :
( order_well_order_on @ nat
@ ( collect @ nat
@ ^ [X2: nat] : ( ord_less @ nat @ X2 @ N ) )
@ ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [X2: nat,Y: nat] :
( ( ord_less @ nat @ X2 @ N )
& ( ord_less @ nat @ Y @ N )
& ( ord_less_eq @ nat @ X2 @ Y ) ) ) ) ) ).
% natLeq_on_well_order_on
thf(fact_5362_bsqr__ofilter,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),D5: set @ ( product_prod @ A @ A )] :
( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( order_ofilter @ ( product_prod @ A @ A ) @ ( bNF_Wellorder_bsqr @ A @ R ) @ D5 )
=> ( ( ord_less @ ( set @ ( product_prod @ A @ A ) ) @ D5
@ ( product_Sigma @ A @ A @ ( field2 @ A @ R )
@ ^ [Uu: A] : ( field2 @ A @ R ) ) )
=> ( ~ ? [A4: A] :
( ( field2 @ A @ R )
= ( order_under @ A @ R @ A4 ) )
=> ? [A18: set @ A] :
( ( order_ofilter @ A @ R @ A18 )
& ( ord_less @ ( set @ A ) @ A18 @ ( field2 @ A @ R ) )
& ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ D5
@ ( product_Sigma @ A @ A @ A18
@ ^ [Uu: A] : A18 ) ) ) ) ) ) ) ).
% bsqr_ofilter
thf(fact_5363_well__order__on__Restr,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A5: set @ A] :
( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ ( field2 @ A @ R ) )
=> ( order_well_order_on @ A @ A5
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R
@ ( product_Sigma @ A @ A @ A5
@ ^ [Uu: A] : A5 ) ) ) ) ) ).
% well_order_on_Restr
thf(fact_5364_Field__Restr__ofilter,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A5: set @ A] :
( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( order_ofilter @ A @ R @ A5 )
=> ( ( field2 @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R
@ ( product_Sigma @ A @ A @ A5
@ ^ [Uu: A] : A5 ) ) )
= A5 ) ) ) ).
% Field_Restr_ofilter
thf(fact_5365_natLeq__on__Well__order,axiom,
! [N: nat] :
( order_well_order_on @ nat
@ ( field2 @ nat
@ ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [X2: nat,Y: nat] :
( ( ord_less @ nat @ X2 @ N )
& ( ord_less @ nat @ Y @ N )
& ( ord_less_eq @ nat @ X2 @ Y ) ) ) ) )
@ ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [X2: nat,Y: nat] :
( ( ord_less @ nat @ X2 @ N )
& ( ord_less @ nat @ Y @ N )
& ( ord_less_eq @ nat @ X2 @ Y ) ) ) ) ) ).
% natLeq_on_Well_order
thf(fact_5366_Linear__order__Well__order__iff,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( order_679001287576687338der_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
= ( ! [A8: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A8 @ ( field2 @ A @ R ) )
=> ( ( A8
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X2: A] :
( ( member @ A @ X2 @ A8 )
& ! [Y: A] :
( ( member @ A @ Y @ A8 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y ) @ R ) ) ) ) ) ) ) ) ).
% Linear_order_Well_order_iff
thf(fact_5367_ofilter__Restr__subset,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A5: set @ A,B4: set @ A] :
( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( order_ofilter @ A @ R @ A5 )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
=> ( order_ofilter @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R
@ ( product_Sigma @ A @ A @ B4
@ ^ [Uu: A] : B4 ) )
@ A5 ) ) ) ) ).
% ofilter_Restr_subset
thf(fact_5368_ofilter__Restr__Int,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A5: set @ A,B4: set @ A] :
( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( order_ofilter @ A @ R @ A5 )
=> ( order_ofilter @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R
@ ( product_Sigma @ A @ A @ B4
@ ^ [Uu: A] : B4 ) )
@ ( inf_inf @ ( set @ A ) @ A5 @ B4 ) ) ) ) ).
% ofilter_Restr_Int
thf(fact_5369_UNION__inj__on__ofilter,axiom,
! [C: $tType,A: $tType,B: $tType,R: set @ ( product_prod @ A @ A ),I5: set @ B,A5: B > ( set @ A ),F: A > C] :
( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ I5 )
=> ( order_ofilter @ A @ R @ ( A5 @ I3 ) ) )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ I5 )
=> ( inj_on @ A @ C @ F @ ( A5 @ I3 ) ) )
=> ( inj_on @ A @ C @ F @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ A5 @ I5 ) ) ) ) ) ) ).
% UNION_inj_on_ofilter
thf(fact_5370_ofilter__subset__embedS,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A5: set @ A,B4: set @ A] :
( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( order_ofilter @ A @ R @ A5 )
=> ( ( order_ofilter @ A @ R @ B4 )
=> ( ( ord_less @ ( set @ A ) @ A5 @ B4 )
= ( bNF_Wellorder_embedS @ A @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R
@ ( product_Sigma @ A @ A @ A5
@ ^ [Uu: A] : A5 ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R
@ ( product_Sigma @ A @ A @ B4
@ ^ [Uu: A] : B4 ) )
@ ( id @ A ) ) ) ) ) ) ).
% ofilter_subset_embedS
thf(fact_5371_ofilter__subset__ordLess,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A5: set @ A,B4: set @ A] :
( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( order_ofilter @ A @ R @ A5 )
=> ( ( order_ofilter @ A @ R @ B4 )
=> ( ( ord_less @ ( set @ A ) @ A5 @ B4 )
= ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R
@ ( product_Sigma @ A @ A @ A5
@ ^ [Uu: A] : A5 ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R
@ ( product_Sigma @ A @ A @ B4
@ ^ [Uu: A] : B4 ) ) )
@ ( bNF_We4044943003108391690rdLess @ A @ A ) ) ) ) ) ) ).
% ofilter_subset_ordLess
thf(fact_5372_ordLess__irreflexive,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
~ ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R @ R ) @ ( bNF_We4044943003108391690rdLess @ A @ A ) ) ).
% ordLess_irreflexive
thf(fact_5373_ordLess__transitive,axiom,
! [B: $tType,C: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R3: set @ ( product_prod @ B @ B ),R8: set @ ( product_prod @ C @ C )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R3 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R3 @ R8 ) @ ( bNF_We4044943003108391690rdLess @ B @ C ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R @ R8 ) @ ( bNF_We4044943003108391690rdLess @ A @ C ) ) ) ) ).
% ordLess_transitive
thf(fact_5374_ordLess__def,axiom,
! [A2: $tType,A: $tType] :
( ( bNF_We4044943003108391690rdLess @ A @ A2 )
= ( collect @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A2 @ A2 ) ) )
@ ( product_case_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A2 @ A2 ) ) @ $o
@ ^ [R4: set @ ( product_prod @ A @ A ),R9: set @ ( product_prod @ A2 @ A2 )] :
( ( order_well_order_on @ A @ ( field2 @ A @ R4 ) @ R4 )
& ( order_well_order_on @ A2 @ ( field2 @ A2 @ R9 ) @ R9 )
& ? [X9: A > A2] : ( bNF_Wellorder_embedS @ A @ A2 @ R4 @ R9 @ X9 ) ) ) ) ) ).
% ordLess_def
thf(fact_5375_finite__ordLess__infinite,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R3: set @ ( product_prod @ B @ B )] :
( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( order_well_order_on @ B @ ( field2 @ B @ R3 ) @ R3 )
=> ( ( finite_finite2 @ A @ ( field2 @ A @ R ) )
=> ( ~ ( finite_finite2 @ B @ ( field2 @ B @ R3 ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R3 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) ) ) ) ) ) ).
% finite_ordLess_infinite
thf(fact_5376_ofilter__ordLess,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A5: set @ A] :
( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( order_ofilter @ A @ R @ A5 )
=> ( ( ord_less @ ( set @ A ) @ A5 @ ( field2 @ A @ R ) )
= ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R
@ ( product_Sigma @ A @ A @ A5
@ ^ [Uu: A] : A5 ) )
@ R )
@ ( bNF_We4044943003108391690rdLess @ A @ A ) ) ) ) ) ).
% ofilter_ordLess
thf(fact_5377_ofilter__subset__embedS__iso,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A5: set @ A,B4: set @ A] :
( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( order_ofilter @ A @ R @ A5 )
=> ( ( order_ofilter @ A @ R @ B4 )
=> ( ( ( ord_less @ ( set @ A ) @ A5 @ B4 )
= ( bNF_Wellorder_embedS @ A @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R
@ ( product_Sigma @ A @ A @ A5
@ ^ [Uu: A] : A5 ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R
@ ( product_Sigma @ A @ A @ B4
@ ^ [Uu: A] : B4 ) )
@ ( id @ A ) ) )
& ( ( A5 = B4 )
= ( bNF_Wellorder_iso @ A @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R
@ ( product_Sigma @ A @ A @ A5
@ ^ [Uu: A] : A5 ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R
@ ( product_Sigma @ A @ A @ B4
@ ^ [Uu: A] : B4 ) )
@ ( id @ A ) ) ) ) ) ) ) ).
% ofilter_subset_embedS_iso
thf(fact_5378_underS__Restr__ordLess,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A3: A] :
( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( ( field2 @ A @ R )
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R
@ ( product_Sigma @ A @ A @ ( order_underS @ A @ R @ A3 )
@ ^ [Uu: A] : ( order_underS @ A @ R @ A3 ) ) )
@ R )
@ ( bNF_We4044943003108391690rdLess @ A @ A ) ) ) ) ).
% underS_Restr_ordLess
thf(fact_5379_ofilter__embed,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A5: set @ A] :
( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( order_ofilter @ A @ R @ A5 )
= ( ( ord_less_eq @ ( set @ A ) @ A5 @ ( field2 @ A @ R ) )
& ( bNF_Wellorder_embed @ A @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R
@ ( product_Sigma @ A @ A @ A5
@ ^ [Uu: A] : A5 ) )
@ R
@ ( id @ A ) ) ) ) ) ).
% ofilter_embed
thf(fact_5380_iso__comp__embed,axiom,
! [C: $tType,B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R3: set @ ( product_prod @ B @ B ),F: A > B,R8: set @ ( product_prod @ C @ C ),F5: B > C] :
( ( bNF_Wellorder_iso @ A @ B @ R @ R3 @ F )
=> ( ( bNF_Wellorder_embed @ B @ C @ R3 @ R8 @ F5 )
=> ( bNF_Wellorder_embed @ A @ C @ R @ R8 @ ( comp @ B @ C @ A @ F5 @ F ) ) ) ) ).
% iso_comp_embed
thf(fact_5381_embed__comp__iso,axiom,
! [C: $tType,B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R3: set @ ( product_prod @ B @ B ),F: A > B,R8: set @ ( product_prod @ C @ C ),F5: B > C] :
( ( bNF_Wellorder_embed @ A @ B @ R @ R3 @ F )
=> ( ( bNF_Wellorder_iso @ B @ C @ R3 @ R8 @ F5 )
=> ( bNF_Wellorder_embed @ A @ C @ R @ R8 @ ( comp @ B @ C @ A @ F5 @ F ) ) ) ) ).
% embed_comp_iso
thf(fact_5382_comp__embed,axiom,
! [C: $tType,B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R3: set @ ( product_prod @ B @ B ),F: A > B,R8: set @ ( product_prod @ C @ C ),F5: B > C] :
( ( bNF_Wellorder_embed @ A @ B @ R @ R3 @ F )
=> ( ( bNF_Wellorder_embed @ B @ C @ R3 @ R8 @ F5 )
=> ( bNF_Wellorder_embed @ A @ C @ R @ R8 @ ( comp @ B @ C @ A @ F5 @ F ) ) ) ) ).
% comp_embed
thf(fact_5383_comp__iso,axiom,
! [C: $tType,B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R3: set @ ( product_prod @ B @ B ),F: A > B,R8: set @ ( product_prod @ C @ C ),F5: B > C] :
( ( bNF_Wellorder_iso @ A @ B @ R @ R3 @ F )
=> ( ( bNF_Wellorder_iso @ B @ C @ R3 @ R8 @ F5 )
=> ( bNF_Wellorder_iso @ A @ C @ R @ R8 @ ( comp @ B @ C @ A @ F5 @ F ) ) ) ) ).
% comp_iso
thf(fact_5384_iso__defs_I2_J,axiom,
! [A2: $tType,A: $tType] :
( ( bNF_Wellorder_iso @ A @ A2 )
= ( ^ [R4: set @ ( product_prod @ A @ A ),R9: set @ ( product_prod @ A2 @ A2 ),F2: A > A2] :
( ( bNF_Wellorder_embed @ A @ A2 @ R4 @ R9 @ F2 )
& ( bij_betw @ A @ A2 @ F2 @ ( field2 @ A @ R4 ) @ ( field2 @ A2 @ R9 ) ) ) ) ) ).
% iso_defs(2)
thf(fact_5385_iso__forward,axiom,
! [A: $tType,B: $tType,X: A,Y3: A,R: set @ ( product_prod @ A @ A ),R3: set @ ( product_prod @ B @ B ),F: A > B] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ R )
=> ( ( bNF_Wellorder_iso @ A @ B @ R @ R3 @ F )
=> ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F @ X ) @ ( F @ Y3 ) ) @ R3 ) ) ) ).
% iso_forward
thf(fact_5386_underS__def,axiom,
! [A: $tType] :
( ( order_underS @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A ),A6: A] :
( collect @ A
@ ^ [B5: A] :
( ( B5 != A6 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B5 @ A6 ) @ R4 ) ) ) ) ) ).
% underS_def
thf(fact_5387_underS__I,axiom,
! [A: $tType,I: A,J: A,R5: set @ ( product_prod @ A @ A )] :
( ( I != J )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ I @ J ) @ R5 )
=> ( member @ A @ I @ ( order_underS @ A @ R5 @ J ) ) ) ) ).
% underS_I
thf(fact_5388_underS__E,axiom,
! [A: $tType,I: A,R5: set @ ( product_prod @ A @ A ),J: A] :
( ( member @ A @ I @ ( order_underS @ A @ R5 @ J ) )
=> ( ( I != J )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ I @ J ) @ R5 ) ) ) ).
% underS_E
thf(fact_5389_ordLess__not__embed,axiom,
! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ A ),R3: set @ ( product_prod @ B @ B )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R3 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) )
=> ~ ? [X_1: B > A] : ( bNF_Wellorder_embed @ B @ A @ R3 @ R @ X_1 ) ) ).
% ordLess_not_embed
thf(fact_5390_embed__implies__iso__Restr,axiom,
! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ A ),R3: set @ ( product_prod @ B @ B ),F: B > A] :
( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( order_well_order_on @ B @ ( field2 @ B @ R3 ) @ R3 )
=> ( ( bNF_Wellorder_embed @ B @ A @ R3 @ R @ F )
=> ( bNF_Wellorder_iso @ B @ A @ R3
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R
@ ( product_Sigma @ A @ A @ ( image2 @ B @ A @ F @ ( field2 @ B @ R3 ) )
@ ^ [Uu: A] : ( image2 @ B @ A @ F @ ( field2 @ B @ R3 ) ) ) )
@ F ) ) ) ) ).
% embed_implies_iso_Restr
thf(fact_5391_embedS__comp__iso,axiom,
! [C: $tType,B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R3: set @ ( product_prod @ B @ B ),F: A > B,R8: set @ ( product_prod @ C @ C ),F5: B > C] :
( ( bNF_Wellorder_embedS @ A @ B @ R @ R3 @ F )
=> ( ( bNF_Wellorder_iso @ B @ C @ R3 @ R8 @ F5 )
=> ( bNF_Wellorder_embedS @ A @ C @ R @ R8 @ ( comp @ B @ C @ A @ F5 @ F ) ) ) ) ).
% embedS_comp_iso
thf(fact_5392_iso__iff2,axiom,
! [B: $tType,A: $tType] :
( ( bNF_Wellorder_iso @ A @ B )
= ( ^ [R4: set @ ( product_prod @ A @ A ),R9: set @ ( product_prod @ B @ B ),F2: A > B] :
( ( bij_betw @ A @ B @ F2 @ ( field2 @ A @ R4 ) @ ( field2 @ B @ R9 ) )
& ! [X2: A] :
( ( member @ A @ X2 @ ( field2 @ A @ R4 ) )
=> ! [Y: A] :
( ( member @ A @ Y @ ( field2 @ A @ R4 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y ) @ R4 )
= ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F2 @ X2 ) @ ( F2 @ Y ) ) @ R9 ) ) ) ) ) ) ) ).
% iso_iff2
thf(fact_5393_BNF__Wellorder__Constructions_OordLess__Field,axiom,
! [A: $tType,B: $tType,R1: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B ),F: A > B] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R1 @ R22 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) )
=> ( ( bNF_Wellorder_embed @ A @ B @ R1 @ R22 @ F )
=> ( ( image2 @ A @ B @ F @ ( field2 @ A @ R1 ) )
!= ( field2 @ B @ R22 ) ) ) ) ).
% BNF_Wellorder_Constructions.ordLess_Field
thf(fact_5394_embedS__defs_I2_J,axiom,
! [A2: $tType,A: $tType] :
( ( bNF_Wellorder_embedS @ A @ A2 )
= ( ^ [R4: set @ ( product_prod @ A @ A ),R9: set @ ( product_prod @ A2 @ A2 ),F2: A > A2] :
( ( bNF_Wellorder_embed @ A @ A2 @ R4 @ R9 @ F2 )
& ~ ( bij_betw @ A @ A2 @ F2 @ ( field2 @ A @ R4 ) @ ( field2 @ A2 @ R9 ) ) ) ) ) ).
% embedS_defs(2)
thf(fact_5395_ordLess__iff,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R3: set @ ( product_prod @ B @ B )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R3 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) )
= ( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
& ( order_well_order_on @ B @ ( field2 @ B @ R3 ) @ R3 )
& ~ ? [X9: B > A] : ( bNF_Wellorder_embed @ B @ A @ R3 @ R @ X9 ) ) ) ).
% ordLess_iff
thf(fact_5396_embedS__comp__embed,axiom,
! [C: $tType,B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R3: set @ ( product_prod @ B @ B ),F: A > B,R8: set @ ( product_prod @ C @ C ),F5: B > C] :
( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( order_well_order_on @ B @ ( field2 @ B @ R3 ) @ R3 )
=> ( ( bNF_Wellorder_embedS @ A @ B @ R @ R3 @ F )
=> ( ( bNF_Wellorder_embed @ B @ C @ R3 @ R8 @ F5 )
=> ( bNF_Wellorder_embedS @ A @ C @ R @ R8 @ ( comp @ B @ C @ A @ F5 @ F ) ) ) ) ) ) ).
% embedS_comp_embed
thf(fact_5397_embed__comp__embedS,axiom,
! [C: $tType,B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R3: set @ ( product_prod @ B @ B ),F: A > B,R8: set @ ( product_prod @ C @ C ),F5: B > C] :
( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( order_well_order_on @ B @ ( field2 @ B @ R3 ) @ R3 )
=> ( ( bNF_Wellorder_embed @ A @ B @ R @ R3 @ F )
=> ( ( bNF_Wellorder_embedS @ B @ C @ R3 @ R8 @ F5 )
=> ( bNF_Wellorder_embedS @ A @ C @ R @ R8 @ ( comp @ B @ C @ A @ F5 @ F ) ) ) ) ) ) ).
% embed_comp_embedS
thf(fact_5398_underS__incl__iff,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
( ( order_679001287576687338der_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( member @ A @ A3 @ ( field2 @ A @ R ) )
=> ( ( member @ A @ B2 @ ( field2 @ A @ R ) )
=> ( ( ord_less_eq @ ( set @ A ) @ ( order_underS @ A @ R @ A3 ) @ ( order_underS @ A @ R @ B2 ) )
= ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R ) ) ) ) ) ).
% underS_incl_iff
thf(fact_5399_embed__defs_I2_J,axiom,
! [A2: $tType,A: $tType] :
( ( bNF_Wellorder_embed @ A @ A2 )
= ( ^ [R4: set @ ( product_prod @ A @ A ),R9: set @ ( product_prod @ A2 @ A2 ),F2: A > A2] :
! [X2: A] :
( ( member @ A @ X2 @ ( field2 @ A @ R4 ) )
=> ( bij_betw @ A @ A2 @ F2 @ ( order_under @ A @ R4 @ X2 ) @ ( order_under @ A2 @ R9 @ ( F2 @ X2 ) ) ) ) ) ) ).
% embed_defs(2)
thf(fact_5400_iso__comp__embedS,axiom,
! [C: $tType,B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R3: set @ ( product_prod @ B @ B ),F: A > B,R8: set @ ( product_prod @ C @ C ),F5: B > C] :
( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( order_well_order_on @ B @ ( field2 @ B @ R3 ) @ R3 )
=> ( ( bNF_Wellorder_iso @ A @ B @ R @ R3 @ F )
=> ( ( bNF_Wellorder_embedS @ B @ C @ R3 @ R8 @ F5 )
=> ( bNF_Wellorder_embedS @ A @ C @ R @ R8 @ ( comp @ B @ C @ A @ F5 @ F ) ) ) ) ) ) ).
% iso_comp_embedS
thf(fact_5401_embed__ordLess__ofilterIncl,axiom,
! [B: $tType,A: $tType,C: $tType,R1: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B ),R33: set @ ( product_prod @ C @ C ),F132: A > C,F232: B > C] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R1 @ R22 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R22 @ R33 ) @ ( bNF_We4044943003108391690rdLess @ B @ C ) )
=> ( ( bNF_Wellorder_embed @ A @ C @ R1 @ R33 @ F132 )
=> ( ( bNF_Wellorder_embed @ B @ C @ R22 @ R33 @ F232 )
=> ( member @ ( product_prod @ ( set @ C ) @ ( set @ C ) ) @ ( product_Pair @ ( set @ C ) @ ( set @ C ) @ ( image2 @ A @ C @ F132 @ ( field2 @ A @ R1 ) ) @ ( image2 @ B @ C @ F232 @ ( field2 @ B @ R22 ) ) ) @ ( bNF_We413866401316099525erIncl @ C @ R33 ) ) ) ) ) ) ).
% embed_ordLess_ofilterIncl
thf(fact_5402_ofilter__subset__embed,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A5: set @ A,B4: set @ A] :
( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( order_ofilter @ A @ R @ A5 )
=> ( ( order_ofilter @ A @ R @ B4 )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
= ( bNF_Wellorder_embed @ A @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R
@ ( product_Sigma @ A @ A @ A5
@ ^ [Uu: A] : A5 ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R
@ ( product_Sigma @ A @ A @ B4
@ ^ [Uu: A] : B4 ) )
@ ( id @ A ) ) ) ) ) ) ).
% ofilter_subset_embed
thf(fact_5403_ordLess__iff__ordIso__Restr,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R3: set @ ( product_prod @ B @ B )] :
( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( order_well_order_on @ B @ ( field2 @ B @ R3 ) @ R3 )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ R ) @ ( bNF_We4044943003108391690rdLess @ B @ A ) )
= ( ? [X2: A] :
( ( member @ A @ X2 @ ( field2 @ A @ R ) )
& ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R3
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R
@ ( product_Sigma @ A @ A @ ( order_underS @ A @ R @ X2 )
@ ^ [Uu: A] : ( order_underS @ A @ R @ X2 ) ) ) )
@ ( bNF_Wellorder_ordIso @ B @ A ) ) ) ) ) ) ) ).
% ordLess_iff_ordIso_Restr
thf(fact_5404_ordLeq__iff__ordLess__Restr,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R3: set @ ( product_prod @ B @ B )] :
( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( order_well_order_on @ B @ ( field2 @ B @ R3 ) @ R3 )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R3 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ ( field2 @ A @ R ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R
@ ( product_Sigma @ A @ A @ ( order_underS @ A @ R @ X2 )
@ ^ [Uu: A] : ( order_underS @ A @ R @ X2 ) ) )
@ R3 )
@ ( bNF_We4044943003108391690rdLess @ A @ B ) ) ) ) ) ) ) ).
% ordLeq_iff_ordLess_Restr
thf(fact_5405_ofilter__subset__ordLeq,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A5: set @ A,B4: set @ A] :
( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( order_ofilter @ A @ R @ A5 )
=> ( ( order_ofilter @ A @ R @ B4 )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
= ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R
@ ( product_Sigma @ A @ A @ A5
@ ^ [Uu: A] : A5 ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R
@ ( product_Sigma @ A @ A @ B4
@ ^ [Uu: A] : B4 ) ) )
@ ( bNF_Wellorder_ordLeq @ A @ A ) ) ) ) ) ) ).
% ofilter_subset_ordLeq
thf(fact_5406_internalize__ordLeq,axiom,
! [A: $tType,B: $tType,R3: set @ ( product_prod @ A @ A ),R: set @ ( product_prod @ B @ B )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R3 @ R ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
= ( ? [P5: set @ ( product_prod @ B @ B )] :
( ( ord_less_eq @ ( set @ B ) @ ( field2 @ B @ P5 ) @ ( field2 @ B @ R ) )
& ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R3 @ P5 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
& ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P5 @ R ) @ ( bNF_Wellorder_ordLeq @ B @ B ) ) ) ) ) ).
% internalize_ordLeq
thf(fact_5407_ordLeq__ordIso__trans,axiom,
! [B: $tType,C: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R3: set @ ( product_prod @ B @ B ),R8: set @ ( product_prod @ C @ C )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R3 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R3 @ R8 ) @ ( bNF_Wellorder_ordIso @ B @ C ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R @ R8 ) @ ( bNF_Wellorder_ordLeq @ A @ C ) ) ) ) ).
% ordLeq_ordIso_trans
thf(fact_5408_ordIso__ordLeq__trans,axiom,
! [B: $tType,C: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R3: set @ ( product_prod @ B @ B ),R8: set @ ( product_prod @ C @ C )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R3 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R3 @ R8 ) @ ( bNF_Wellorder_ordLeq @ B @ C ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R @ R8 ) @ ( bNF_Wellorder_ordLeq @ A @ C ) ) ) ) ).
% ordIso_ordLeq_trans
thf(fact_5409_ordLeq__transitive,axiom,
! [B: $tType,C: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R3: set @ ( product_prod @ B @ B ),R8: set @ ( product_prod @ C @ C )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R3 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R3 @ R8 ) @ ( bNF_Wellorder_ordLeq @ B @ C ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R @ R8 ) @ ( bNF_Wellorder_ordLeq @ A @ C ) ) ) ) ).
% ordLeq_transitive
thf(fact_5410_ordIso__transitive,axiom,
! [B: $tType,C: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R3: set @ ( product_prod @ B @ B ),R8: set @ ( product_prod @ C @ C )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R3 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R3 @ R8 ) @ ( bNF_Wellorder_ordIso @ B @ C ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R @ R8 ) @ ( bNF_Wellorder_ordIso @ A @ C ) ) ) ) ).
% ordIso_transitive
thf(fact_5411_ordIso__imp__ordLeq,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R3: set @ ( product_prod @ B @ B )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R3 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R3 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ) ).
% ordIso_imp_ordLeq
thf(fact_5412_ordIso__iff__ordLeq,axiom,
! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ A ),R3: set @ ( product_prod @ B @ B )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R3 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
= ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R3 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
& ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ R ) @ ( bNF_Wellorder_ordLeq @ B @ A ) ) ) ) ).
% ordIso_iff_ordLeq
thf(fact_5413_ordIso__symmetric,axiom,
! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ A ),R3: set @ ( product_prod @ B @ B )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R3 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ R ) @ ( bNF_Wellorder_ordIso @ B @ A ) ) ) ).
% ordIso_symmetric
thf(fact_5414_ordLeq__iff__ordLess__or__ordIso,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R3: set @ ( product_prod @ B @ B )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R3 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
= ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R3 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) )
| ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R3 ) @ ( bNF_Wellorder_ordIso @ A @ B ) ) ) ) ).
% ordLeq_iff_ordLess_or_ordIso
thf(fact_5415_ordLess__ordLeq__trans,axiom,
! [B: $tType,C: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R3: set @ ( product_prod @ B @ B ),R8: set @ ( product_prod @ C @ C )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R3 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R3 @ R8 ) @ ( bNF_Wellorder_ordLeq @ B @ C ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R @ R8 ) @ ( bNF_We4044943003108391690rdLess @ A @ C ) ) ) ) ).
% ordLess_ordLeq_trans
thf(fact_5416_ordLeq__ordLess__trans,axiom,
! [B: $tType,C: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R3: set @ ( product_prod @ B @ B ),R8: set @ ( product_prod @ C @ C )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R3 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R3 @ R8 ) @ ( bNF_We4044943003108391690rdLess @ B @ C ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R @ R8 ) @ ( bNF_We4044943003108391690rdLess @ A @ C ) ) ) ) ).
% ordLeq_ordLess_trans
thf(fact_5417_ordLess__imp__ordLeq,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R3: set @ ( product_prod @ B @ B )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R3 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R3 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ) ).
% ordLess_imp_ordLeq
thf(fact_5418_not__ordLess__ordLeq,axiom,
! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ A ),R3: set @ ( product_prod @ B @ B )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R3 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) )
=> ~ ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ R ) @ ( bNF_Wellorder_ordLeq @ B @ A ) ) ) ).
% not_ordLess_ordLeq
thf(fact_5419_ordLess__ordIso__trans,axiom,
! [B: $tType,C: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R3: set @ ( product_prod @ B @ B ),R8: set @ ( product_prod @ C @ C )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R3 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R3 @ R8 ) @ ( bNF_Wellorder_ordIso @ B @ C ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R @ R8 ) @ ( bNF_We4044943003108391690rdLess @ A @ C ) ) ) ) ).
% ordLess_ordIso_trans
thf(fact_5420_ordIso__ordLess__trans,axiom,
! [B: $tType,C: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R3: set @ ( product_prod @ B @ B ),R8: set @ ( product_prod @ C @ C )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R3 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R3 @ R8 ) @ ( bNF_We4044943003108391690rdLess @ B @ C ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R @ R8 ) @ ( bNF_We4044943003108391690rdLess @ A @ C ) ) ) ) ).
% ordIso_ordLess_trans
thf(fact_5421_not__ordLess__ordIso,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R3: set @ ( product_prod @ B @ B )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R3 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) )
=> ~ ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R3 ) @ ( bNF_Wellorder_ordIso @ A @ B ) ) ) ).
% not_ordLess_ordIso
thf(fact_5422_finite__well__order__on__ordIso,axiom,
! [A: $tType,A5: set @ A,R: set @ ( product_prod @ A @ A ),R3: set @ ( product_prod @ A @ A )] :
( ( finite_finite2 @ A @ A5 )
=> ( ( order_well_order_on @ A @ A5 @ R )
=> ( ( order_well_order_on @ A @ A5 @ R3 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R @ R3 ) @ ( bNF_Wellorder_ordIso @ A @ A ) ) ) ) ) ).
% finite_well_order_on_ordIso
thf(fact_5423_exists__minim__Well__order,axiom,
! [A: $tType,R5: set @ ( set @ ( product_prod @ A @ A ) )] :
( ( R5
!= ( bot_bot @ ( set @ ( set @ ( product_prod @ A @ A ) ) ) ) )
=> ( ! [X4: set @ ( product_prod @ A @ A )] :
( ( member @ ( set @ ( product_prod @ A @ A ) ) @ X4 @ R5 )
=> ( order_well_order_on @ A @ ( field2 @ A @ X4 ) @ X4 ) )
=> ? [X4: set @ ( product_prod @ A @ A )] :
( ( member @ ( set @ ( product_prod @ A @ A ) ) @ X4 @ R5 )
& ! [Xa2: set @ ( product_prod @ A @ A )] :
( ( member @ ( set @ ( product_prod @ A @ A ) ) @ Xa2 @ R5 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ X4 @ Xa2 ) @ ( bNF_Wellorder_ordLeq @ A @ A ) ) ) ) ) ) ).
% exists_minim_Well_order
thf(fact_5424_ordLeq__total,axiom,
! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ A ),R3: set @ ( product_prod @ B @ B )] :
( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( order_well_order_on @ B @ ( field2 @ B @ R3 ) @ R3 )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R3 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
| ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ R ) @ ( bNF_Wellorder_ordLeq @ B @ A ) ) ) ) ) ).
% ordLeq_total
thf(fact_5425_ordLeq__reflexive,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R @ R ) @ ( bNF_Wellorder_ordLeq @ A @ A ) ) ) ).
% ordLeq_reflexive
thf(fact_5426_ordLeq__Well__order__simp,axiom,
! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ A ),R3: set @ ( product_prod @ B @ B )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R3 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
& ( order_well_order_on @ B @ ( field2 @ B @ R3 ) @ R3 ) ) ) ).
% ordLeq_Well_order_simp
thf(fact_5427_ordIso__reflexive,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R @ R ) @ ( bNF_Wellorder_ordIso @ A @ A ) ) ) ).
% ordIso_reflexive
thf(fact_5428_Well__order__iso__copy,axiom,
! [B: $tType,A: $tType,A5: set @ A,R: set @ ( product_prod @ A @ A ),F: A > B,A14: set @ B] :
( ( order_well_order_on @ A @ A5 @ R )
=> ( ( bij_betw @ A @ B @ F @ A5 @ A14 )
=> ? [R10: set @ ( product_prod @ B @ B )] :
( ( order_well_order_on @ B @ A14 @ R10 )
& ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R10 ) @ ( bNF_Wellorder_ordIso @ A @ B ) ) ) ) ) ).
% Well_order_iso_copy
thf(fact_5429_internalize__ordLess,axiom,
! [A: $tType,B: $tType,R3: set @ ( product_prod @ A @ A ),R: set @ ( product_prod @ B @ B )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R3 @ R ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) )
= ( ? [P5: set @ ( product_prod @ B @ B )] :
( ( ord_less @ ( set @ B ) @ ( field2 @ B @ P5 ) @ ( field2 @ B @ R ) )
& ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R3 @ P5 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
& ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P5 @ R ) @ ( bNF_We4044943003108391690rdLess @ B @ B ) ) ) ) ) ).
% internalize_ordLess
thf(fact_5430_not__ordLess__iff__ordLeq,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R3: set @ ( product_prod @ B @ B )] :
( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( order_well_order_on @ B @ ( field2 @ B @ R3 ) @ R3 )
=> ( ( ~ ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ R ) @ ( bNF_We4044943003108391690rdLess @ B @ A ) ) )
= ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R3 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ) ) ) ).
% not_ordLess_iff_ordLeq
thf(fact_5431_not__ordLeq__iff__ordLess,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R3: set @ ( product_prod @ B @ B )] :
( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( order_well_order_on @ B @ ( field2 @ B @ R3 ) @ R3 )
=> ( ( ~ ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ R ) @ ( bNF_Wellorder_ordLeq @ B @ A ) ) )
= ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R3 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) ) ) ) ) ).
% not_ordLeq_iff_ordLess
thf(fact_5432_ordLess__or__ordLeq,axiom,
! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ A ),R3: set @ ( product_prod @ B @ B )] :
( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( order_well_order_on @ B @ ( field2 @ B @ R3 ) @ R3 )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R3 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) )
| ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ R ) @ ( bNF_Wellorder_ordLeq @ B @ A ) ) ) ) ) ).
% ordLess_or_ordLeq
thf(fact_5433_ordLeq__def,axiom,
! [A2: $tType,A: $tType] :
( ( bNF_Wellorder_ordLeq @ A @ A2 )
= ( collect @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A2 @ A2 ) ) )
@ ( product_case_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A2 @ A2 ) ) @ $o
@ ^ [R4: set @ ( product_prod @ A @ A ),R9: set @ ( product_prod @ A2 @ A2 )] :
( ( order_well_order_on @ A @ ( field2 @ A @ R4 ) @ R4 )
& ( order_well_order_on @ A2 @ ( field2 @ A2 @ R9 ) @ R9 )
& ? [X9: A > A2] : ( bNF_Wellorder_embed @ A @ A2 @ R4 @ R9 @ X9 ) ) ) ) ) ).
% ordLeq_def
thf(fact_5434_ordIso__def,axiom,
! [A2: $tType,A: $tType] :
( ( bNF_Wellorder_ordIso @ A @ A2 )
= ( collect @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A2 @ A2 ) ) )
@ ( product_case_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A2 @ A2 ) ) @ $o
@ ^ [R4: set @ ( product_prod @ A @ A ),R9: set @ ( product_prod @ A2 @ A2 )] :
( ( order_well_order_on @ A @ ( field2 @ A @ R4 ) @ R4 )
& ( order_well_order_on @ A2 @ ( field2 @ A2 @ R9 ) @ R9 )
& ? [X9: A > A2] : ( bNF_Wellorder_iso @ A @ A2 @ R4 @ R9 @ X9 ) ) ) ) ) ).
% ordIso_def
thf(fact_5435_comp__set__bd__Union__o__collect,axiom,
! [C: $tType,B: $tType,A: $tType,X: C,X7: set @ ( C > ( set @ ( set @ A ) ) ),Hbd: set @ ( product_prod @ B @ B )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) )
@ ( bNF_Ca6860139660246222851ard_of @ A
@ ( complete_Sup_Sup @ ( set @ A )
@ ( complete_Sup_Sup @ ( set @ ( set @ A ) )
@ ( image2 @ ( C > ( set @ ( set @ A ) ) ) @ ( set @ ( set @ A ) )
@ ^ [F2: C > ( set @ ( set @ A ) )] : ( F2 @ X )
@ X7 ) ) ) )
@ Hbd )
@ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ ( comp @ ( set @ ( set @ A ) ) @ ( set @ A ) @ C @ ( complete_Sup_Sup @ ( set @ A ) ) @ ( bNF_collect @ C @ ( set @ A ) @ X7 ) @ X ) ) @ Hbd ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ) ).
% comp_set_bd_Union_o_collect
thf(fact_5436_dir__image__ordIso,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),F: A > B] :
( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( inj_on @ A @ B @ F @ ( field2 @ A @ R ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ ( bNF_We2720479622203943262_image @ A @ B @ R @ F ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) ) ) ) ).
% dir_image_ordIso
thf(fact_5437_map__comp__def,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( map_comp @ B @ C @ A )
= ( ^ [F2: B > ( option @ C ),G2: A > ( option @ B ),K3: A] : ( case_option @ ( option @ C ) @ B @ ( none @ C ) @ F2 @ ( G2 @ K3 ) ) ) ) ).
% map_comp_def
thf(fact_5438_map__comp__simps_I1_J,axiom,
! [B: $tType,A: $tType,C: $tType,M23: B > ( option @ A ),K: B,M12: A > ( option @ C )] :
( ( ( M23 @ K )
= ( none @ A ) )
=> ( ( map_comp @ A @ C @ B @ M12 @ M23 @ K )
= ( none @ C ) ) ) ).
% map_comp_simps(1)
thf(fact_5439_map__comp__simps_I2_J,axiom,
! [B: $tType,C: $tType,A: $tType,M23: B > ( option @ A ),K: B,K8: A,M12: A > ( option @ C )] :
( ( ( M23 @ K )
= ( some @ A @ K8 ) )
=> ( ( map_comp @ A @ C @ B @ M12 @ M23 @ K )
= ( M12 @ K8 ) ) ) ).
% map_comp_simps(2)
thf(fact_5440_map__comp__empty_I1_J,axiom,
! [C: $tType,B: $tType,A: $tType,M: C > ( option @ B )] :
( ( map_comp @ C @ B @ A @ M
@ ^ [X2: A] : ( none @ C ) )
= ( ^ [X2: A] : ( none @ B ) ) ) ).
% map_comp_empty(1)
thf(fact_5441_map__comp__empty_I2_J,axiom,
! [B: $tType,D: $tType,C: $tType,M: C > ( option @ B )] :
( ( map_comp @ B @ D @ C
@ ^ [X2: B] : ( none @ D )
@ M )
= ( ^ [X2: C] : ( none @ D ) ) ) ).
% map_comp_empty(2)
thf(fact_5442_card__of__Times2,axiom,
! [A: $tType,B: $tType,A5: set @ A,B4: set @ B] :
( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B4 )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A5
@ ^ [Uu: A] : B4 ) ) )
@ ( bNF_Wellorder_ordLeq @ B @ ( product_prod @ A @ B ) ) ) ) ).
% card_of_Times2
thf(fact_5443_card__of__Times1,axiom,
! [A: $tType,B: $tType,A5: set @ A,B4: set @ B] :
( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B4 )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ B @ A )
@ ( product_Sigma @ B @ A @ B4
@ ^ [Uu: B] : A5 ) ) )
@ ( bNF_Wellorder_ordLeq @ B @ ( product_prod @ B @ A ) ) ) ) ).
% card_of_Times1
thf(fact_5444_card__of__Times__same__infinite,axiom,
! [A: $tType,A5: set @ A] :
( ~ ( finite_finite2 @ A @ A5 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ A ) @ ( product_prod @ A @ A ) ) ) @ ( set @ ( product_prod @ A @ A ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ A ) @ ( product_prod @ A @ A ) ) ) @ ( set @ ( product_prod @ A @ A ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ A )
@ ( product_Sigma @ A @ A @ A5
@ ^ [Uu: A] : A5 ) )
@ ( bNF_Ca6860139660246222851ard_of @ A @ A5 ) )
@ ( bNF_Wellorder_ordIso @ ( product_prod @ A @ A ) @ A ) ) ) ).
% card_of_Times_same_infinite
thf(fact_5445_card__of__Times3,axiom,
! [A: $tType,A5: set @ A] :
( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ A ) @ ( product_prod @ A @ A ) ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ A ) @ ( product_prod @ A @ A ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A5 )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ A )
@ ( product_Sigma @ A @ A @ A5
@ ^ [Uu: A] : A5 ) ) )
@ ( bNF_Wellorder_ordLeq @ A @ ( product_prod @ A @ A ) ) ) ).
% card_of_Times3
thf(fact_5446_card__of__Sigma__mono1,axiom,
! [C: $tType,B: $tType,A: $tType,I5: set @ A,A5: A > ( set @ B ),B4: A > ( set @ C )] :
( ! [X4: A] :
( ( member @ A @ X4 @ I5 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ ( A5 @ X4 ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ ( B4 @ X4 ) ) ) @ ( bNF_Wellorder_ordLeq @ B @ C ) ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ A @ C ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ A @ C ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ B ) @ ( product_Sigma @ A @ B @ I5 @ A5 ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ C ) @ ( product_Sigma @ A @ C @ I5 @ B4 ) ) ) @ ( bNF_Wellorder_ordLeq @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) ) ) ) ).
% card_of_Sigma_mono1
thf(fact_5447_card__of__Times__mono1,axiom,
! [B: $tType,C: $tType,A: $tType,A5: set @ A,B4: set @ B,C6: set @ C] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A5 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B4 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ A @ C ) ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ A @ C ) ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ C )
@ ( product_Sigma @ A @ C @ A5
@ ^ [Uu: A] : C6 ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ B @ C )
@ ( product_Sigma @ B @ C @ B4
@ ^ [Uu: B] : C6 ) ) )
@ ( bNF_Wellorder_ordLeq @ ( product_prod @ A @ C ) @ ( product_prod @ B @ C ) ) ) ) ).
% card_of_Times_mono1
thf(fact_5448_card__of__Times__mono2,axiom,
! [B: $tType,A: $tType,C: $tType,A5: set @ A,B4: set @ B,C6: set @ C] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A5 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B4 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ C @ A ) @ ( product_prod @ C @ A ) ) ) @ ( set @ ( product_prod @ ( product_prod @ C @ B ) @ ( product_prod @ C @ B ) ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ C @ A ) @ ( product_prod @ C @ A ) ) ) @ ( set @ ( product_prod @ ( product_prod @ C @ B ) @ ( product_prod @ C @ B ) ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ C @ A )
@ ( product_Sigma @ C @ A @ C6
@ ^ [Uu: C] : A5 ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ C @ B )
@ ( product_Sigma @ C @ B @ C6
@ ^ [Uu: C] : B4 ) ) )
@ ( bNF_Wellorder_ordLeq @ ( product_prod @ C @ A ) @ ( product_prod @ C @ B ) ) ) ) ).
% card_of_Times_mono2
thf(fact_5449_card__of__Times__commute,axiom,
! [B: $tType,A: $tType,A5: set @ A,B4: set @ B] :
( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A5
@ ^ [Uu: A] : B4 ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ B @ A )
@ ( product_Sigma @ B @ A @ B4
@ ^ [Uu: B] : A5 ) ) )
@ ( bNF_Wellorder_ordIso @ ( product_prod @ A @ B ) @ ( product_prod @ B @ A ) ) ) ).
% card_of_Times_commute
thf(fact_5450_Func__Times__Range,axiom,
! [C: $tType,B: $tType,A: $tType,A5: set @ A,B4: set @ B,C6: set @ C] :
( member @ ( product_prod @ ( set @ ( product_prod @ ( A > ( product_prod @ B @ C ) ) @ ( A > ( product_prod @ B @ C ) ) ) ) @ ( set @ ( product_prod @ ( product_prod @ ( A > B ) @ ( A > C ) ) @ ( product_prod @ ( A > B ) @ ( A > C ) ) ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ ( A > ( product_prod @ B @ C ) ) @ ( A > ( product_prod @ B @ C ) ) ) ) @ ( set @ ( product_prod @ ( product_prod @ ( A > B ) @ ( A > C ) ) @ ( product_prod @ ( A > B ) @ ( A > C ) ) ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( A > ( product_prod @ B @ C ) )
@ ( bNF_Wellorder_Func @ A @ ( product_prod @ B @ C ) @ A5
@ ( product_Sigma @ B @ C @ B4
@ ^ [Uu: B] : C6 ) ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ ( A > B ) @ ( A > C ) )
@ ( product_Sigma @ ( A > B ) @ ( A > C ) @ ( bNF_Wellorder_Func @ A @ B @ A5 @ B4 )
@ ^ [Uu: A > B] : ( bNF_Wellorder_Func @ A @ C @ A5 @ C6 ) ) ) )
@ ( bNF_Wellorder_ordIso @ ( A > ( product_prod @ B @ C ) ) @ ( product_prod @ ( A > B ) @ ( A > C ) ) ) ) ).
% Func_Times_Range
thf(fact_5451_card__of__bool,axiom,
! [A: $tType,A1: A,A22: A] :
( ( A1 != A22 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ $o @ $o ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ $o @ $o ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ $o @ ( top_top @ ( set @ $o ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ ( insert2 @ A @ A1 @ ( insert2 @ A @ A22 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) @ ( bNF_Wellorder_ordIso @ $o @ A ) ) ) ).
% card_of_bool
thf(fact_5452_infinite__iff__card__of__nat,axiom,
! [A: $tType,A5: set @ A] :
( ( ~ ( finite_finite2 @ A @ A5 ) )
= ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ nat @ ( top_top @ ( set @ nat ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A5 ) ) @ ( bNF_Wellorder_ordLeq @ nat @ A ) ) ) ).
% infinite_iff_card_of_nat
thf(fact_5453_card__of__UNION__Sigma,axiom,
! [B: $tType,A: $tType,A5: B > ( set @ A ),I5: set @ B] : ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ A5 @ I5 ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ B @ A ) @ ( product_Sigma @ B @ A @ I5 @ A5 ) ) ) @ ( bNF_Wellorder_ordLeq @ A @ ( product_prod @ B @ A ) ) ) ).
% card_of_UNION_Sigma
thf(fact_5454_card__of__Pow__Func,axiom,
! [A: $tType,A5: set @ A] : ( member @ ( product_prod @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( set @ ( product_prod @ ( A > $o ) @ ( A > $o ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( set @ ( product_prod @ ( A > $o ) @ ( A > $o ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( set @ A ) @ ( pow2 @ A @ A5 ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( A > $o ) @ ( bNF_Wellorder_Func @ A @ $o @ A5 @ ( top_top @ ( set @ $o ) ) ) ) ) @ ( bNF_Wellorder_ordIso @ ( set @ A ) @ ( A > $o ) ) ) ).
% card_of_Pow_Func
thf(fact_5455_card__of__Pow,axiom,
! [A: $tType,A5: set @ A] : ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A5 ) @ ( bNF_Ca6860139660246222851ard_of @ ( set @ A ) @ ( pow2 @ A @ A5 ) ) ) @ ( bNF_We4044943003108391690rdLess @ A @ ( set @ A ) ) ) ).
% card_of_Pow
thf(fact_5456_card__of__Sigma__ordLeq__infinite,axiom,
! [A: $tType,C: $tType,B: $tType,B4: set @ A,I5: set @ B,A5: B > ( set @ C )] :
( ~ ( finite_finite2 @ A @ B4 )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ I5 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ B4 ) ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ I5 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ ( A5 @ X4 ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ B4 ) ) @ ( bNF_Wellorder_ordLeq @ C @ A ) ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ B @ C ) @ ( product_Sigma @ B @ C @ I5 @ A5 ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ B4 ) ) @ ( bNF_Wellorder_ordLeq @ ( product_prod @ B @ C ) @ A ) ) ) ) ) ).
% card_of_Sigma_ordLeq_infinite
thf(fact_5457_card__of__refl,axiom,
! [A: $tType,A5: set @ A] : ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A5 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A5 ) ) @ ( bNF_Wellorder_ordIso @ A @ A ) ) ).
% card_of_refl
thf(fact_5458_map__comp__Some__iff,axiom,
! [A: $tType,C: $tType,B: $tType,M12: B > ( option @ A ),M23: C > ( option @ B ),K: C,V: A] :
( ( ( map_comp @ B @ A @ C @ M12 @ M23 @ K )
= ( some @ A @ V ) )
= ( ? [K9: B] :
( ( ( M23 @ K )
= ( some @ B @ K9 ) )
& ( ( M12 @ K9 )
= ( some @ A @ V ) ) ) ) ) ).
% map_comp_Some_iff
thf(fact_5459_card__of__Times__infinite,axiom,
! [A: $tType,B: $tType,A5: set @ A,B4: set @ B] :
( ~ ( finite_finite2 @ A @ A5 )
=> ( ( B4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B4 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A5 ) ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ A @ A ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ A @ A ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A5
@ ^ [Uu: A] : B4 ) )
@ ( bNF_Ca6860139660246222851ard_of @ A @ A5 ) )
@ ( bNF_Wellorder_ordIso @ ( product_prod @ A @ B ) @ A ) )
& ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) @ ( set @ ( product_prod @ A @ A ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) @ ( set @ ( product_prod @ A @ A ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ B @ A )
@ ( product_Sigma @ B @ A @ B4
@ ^ [Uu: B] : A5 ) )
@ ( bNF_Ca6860139660246222851ard_of @ A @ A5 ) )
@ ( bNF_Wellorder_ordIso @ ( product_prod @ B @ A ) @ A ) ) ) ) ) ) ).
% card_of_Times_infinite
thf(fact_5460_card__of__Times__infinite__simps_I1_J,axiom,
! [B: $tType,A: $tType,A5: set @ A,B4: set @ B] :
( ~ ( finite_finite2 @ A @ A5 )
=> ( ( B4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B4 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A5 ) ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ A @ A ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ A @ A ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A5
@ ^ [Uu: A] : B4 ) )
@ ( bNF_Ca6860139660246222851ard_of @ A @ A5 ) )
@ ( bNF_Wellorder_ordIso @ ( product_prod @ A @ B ) @ A ) ) ) ) ) ).
% card_of_Times_infinite_simps(1)
thf(fact_5461_card__of__Times__infinite__simps_I2_J,axiom,
! [B: $tType,A: $tType,A5: set @ A,B4: set @ B] :
( ~ ( finite_finite2 @ A @ A5 )
=> ( ( B4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B4 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A5 ) ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A5 )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A5
@ ^ [Uu: A] : B4 ) ) )
@ ( bNF_Wellorder_ordIso @ A @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% card_of_Times_infinite_simps(2)
thf(fact_5462_card__of__Times__infinite__simps_I3_J,axiom,
! [A: $tType,B: $tType,A5: set @ A,B4: set @ B] :
( ~ ( finite_finite2 @ A @ A5 )
=> ( ( B4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B4 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A5 ) ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) @ ( set @ ( product_prod @ A @ A ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) @ ( set @ ( product_prod @ A @ A ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ B @ A )
@ ( product_Sigma @ B @ A @ B4
@ ^ [Uu: B] : A5 ) )
@ ( bNF_Ca6860139660246222851ard_of @ A @ A5 ) )
@ ( bNF_Wellorder_ordIso @ ( product_prod @ B @ A ) @ A ) ) ) ) ) ).
% card_of_Times_infinite_simps(3)
thf(fact_5463_card__of__Times__infinite__simps_I4_J,axiom,
! [B: $tType,A: $tType,A5: set @ A,B4: set @ B] :
( ~ ( finite_finite2 @ A @ A5 )
=> ( ( B4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B4 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A5 ) ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A5 )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ B @ A )
@ ( product_Sigma @ B @ A @ B4
@ ^ [Uu: B] : A5 ) ) )
@ ( bNF_Wellorder_ordIso @ A @ ( product_prod @ B @ A ) ) ) ) ) ) ).
% card_of_Times_infinite_simps(4)
thf(fact_5464_card__of__Func__Times,axiom,
! [C: $tType,B: $tType,A: $tType,A5: set @ A,B4: set @ B,C6: set @ C] :
( member @ ( product_prod @ ( set @ ( product_prod @ ( ( product_prod @ A @ B ) > C ) @ ( ( product_prod @ A @ B ) > C ) ) ) @ ( set @ ( product_prod @ ( A > B > C ) @ ( A > B > C ) ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ ( ( product_prod @ A @ B ) > C ) @ ( ( product_prod @ A @ B ) > C ) ) ) @ ( set @ ( product_prod @ ( A > B > C ) @ ( A > B > C ) ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( ( product_prod @ A @ B ) > C )
@ ( bNF_Wellorder_Func @ ( product_prod @ A @ B ) @ C
@ ( product_Sigma @ A @ B @ A5
@ ^ [Uu: A] : B4 )
@ C6 ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( A > B > C ) @ ( bNF_Wellorder_Func @ A @ ( B > C ) @ A5 @ ( bNF_Wellorder_Func @ B @ C @ B4 @ C6 ) ) ) )
@ ( bNF_Wellorder_ordIso @ ( ( product_prod @ A @ B ) > C ) @ ( A > B > C ) ) ) ).
% card_of_Func_Times
thf(fact_5465_card__of__image,axiom,
! [B: $tType,A: $tType,F: B > A,A5: set @ B] : ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ ( image2 @ B @ A @ F @ A5 ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ A5 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ).
% card_of_image
thf(fact_5466_card__of__empty,axiom,
! [B: $tType,A: $tType,A5: set @ B] : ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ ( bot_bot @ ( set @ A ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ A5 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ).
% card_of_empty
thf(fact_5467_card__of__empty3,axiom,
! [B: $tType,A: $tType,A5: set @ A] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A5 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ ( bot_bot @ ( set @ B ) ) ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( A5
= ( bot_bot @ ( set @ A ) ) ) ) ).
% card_of_empty3
thf(fact_5468_card__of__ordLeq__infinite,axiom,
! [A: $tType,B: $tType,A5: set @ A,B4: set @ B] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A5 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B4 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ~ ( finite_finite2 @ A @ A5 )
=> ~ ( finite_finite2 @ B @ B4 ) ) ) ).
% card_of_ordLeq_infinite
thf(fact_5469_card__of__ordLeq__finite,axiom,
! [B: $tType,A: $tType,A5: set @ A,B4: set @ B] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A5 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B4 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( finite_finite2 @ B @ B4 )
=> ( finite_finite2 @ A @ A5 ) ) ) ).
% card_of_ordLeq_finite
thf(fact_5470_card__of__mono1,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A5 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ B4 ) ) @ ( bNF_Wellorder_ordLeq @ A @ A ) ) ) ).
% card_of_mono1
thf(fact_5471_card__of__empty2,axiom,
! [B: $tType,A: $tType,A5: set @ A] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A5 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ ( bot_bot @ ( set @ B ) ) ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( A5
= ( bot_bot @ ( set @ A ) ) ) ) ).
% card_of_empty2
thf(fact_5472_card__of__empty__ordIso,axiom,
! [B: $tType,A: $tType] : ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ ( bot_bot @ ( set @ A ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ ( bot_bot @ ( set @ B ) ) ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) ) ).
% card_of_empty_ordIso
thf(fact_5473_card__of__mono2,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R3: set @ ( product_prod @ B @ B )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R3 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ ( field2 @ A @ R ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ ( field2 @ B @ R3 ) ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ) ).
% card_of_mono2
thf(fact_5474_card__of__ordIso__finite,axiom,
! [A: $tType,B: $tType,A5: set @ A,B4: set @ B] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A5 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B4 ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( ( finite_finite2 @ A @ A5 )
= ( finite_finite2 @ B @ B4 ) ) ) ).
% card_of_ordIso_finite
thf(fact_5475_card__of__cong,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R3: set @ ( product_prod @ B @ B )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R3 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ ( field2 @ A @ R ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ ( field2 @ B @ R3 ) ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) ) ) ).
% card_of_cong
thf(fact_5476_card__of__ordLeqI,axiom,
! [B: $tType,A: $tType,F: A > B,A5: set @ A,B4: set @ B] :
( ( inj_on @ A @ B @ F @ A5 )
=> ( ! [A4: A] :
( ( member @ A @ A4 @ A5 )
=> ( member @ B @ ( F @ A4 ) @ B4 ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A5 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B4 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ) ) ).
% card_of_ordLeqI
thf(fact_5477_dir__image__def,axiom,
! [A2: $tType,A: $tType] :
( ( bNF_We2720479622203943262_image @ A @ A2 )
= ( ^ [R4: set @ ( product_prod @ A @ A ),F2: A > A2] :
( collect @ ( product_prod @ A2 @ A2 )
@ ^ [Uu: product_prod @ A2 @ A2] :
? [A6: A,B5: A] :
( ( Uu
= ( product_Pair @ A2 @ A2 @ ( F2 @ A6 ) @ ( F2 @ B5 ) ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A6 @ B5 ) @ R4 ) ) ) ) ) ).
% dir_image_def
thf(fact_5478_ex__bij__betw,axiom,
! [B: $tType,A: $tType,A5: set @ A,R: set @ ( product_prod @ B @ B )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A5 ) @ R ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ? [F4: B > A,B10: set @ B] : ( bij_betw @ B @ A @ F4 @ B10 @ A5 ) ) ).
% ex_bij_betw
thf(fact_5479_card__of__least,axiom,
! [A: $tType,A5: set @ A,R: set @ ( product_prod @ A @ A )] :
( ( order_well_order_on @ A @ A5 @ R )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A5 ) @ R ) @ ( bNF_Wellorder_ordLeq @ A @ A ) ) ) ).
% card_of_least
thf(fact_5480_BNF__Cardinal__Order__Relation_OordLess__Field,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R3: set @ ( product_prod @ B @ B )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R3 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ ( field2 @ A @ R ) ) @ R3 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) ) ) ).
% BNF_Cardinal_Order_Relation.ordLess_Field
thf(fact_5481_card__of__ordIso,axiom,
! [B: $tType,A: $tType,A5: set @ A,B4: set @ B] :
( ( ? [F2: A > B] : ( bij_betw @ A @ B @ F2 @ A5 @ B4 ) )
= ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A5 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B4 ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) ) ) ).
% card_of_ordIso
thf(fact_5482_card__of__ordIsoI,axiom,
! [B: $tType,A: $tType,F: A > B,A5: set @ A,B4: set @ B] :
( ( bij_betw @ A @ B @ F @ A5 @ B4 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A5 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B4 ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) ) ) ).
% card_of_ordIsoI
thf(fact_5483_type__copy__set__bd,axiom,
! [A: $tType,D: $tType,C: $tType,B: $tType,S3: A > ( set @ B ),Bd: set @ ( product_prod @ C @ C ),Rep: D > A,X: D] :
( ! [Y4: A] : ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ ( S3 @ Y4 ) ) @ Bd ) @ ( bNF_Wellorder_ordLeq @ B @ C ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ ( comp @ A @ ( set @ B ) @ D @ S3 @ Rep @ X ) ) @ Bd ) @ ( bNF_Wellorder_ordLeq @ B @ C ) ) ) ).
% type_copy_set_bd
thf(fact_5484_card__of__ordLeq2,axiom,
! [B: $tType,A: $tType,A5: set @ A,B4: set @ B] :
( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ? [G2: B > A] :
( ( image2 @ B @ A @ G2 @ B4 )
= A5 ) )
= ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A5 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B4 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ) ) ).
% card_of_ordLeq2
thf(fact_5485_surj__imp__ordLeq,axiom,
! [B: $tType,A: $tType,B4: set @ A,F: B > A,A5: set @ B] :
( ( ord_less_eq @ ( set @ A ) @ B4 @ ( image2 @ B @ A @ F @ A5 ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ B4 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ A5 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ) ).
% surj_imp_ordLeq
thf(fact_5486_card__of__singl__ordLeq,axiom,
! [A: $tType,B: $tType,A5: set @ A,B2: B] :
( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ ( insert2 @ B @ B2 @ ( bot_bot @ ( set @ B ) ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A5 ) ) @ ( bNF_Wellorder_ordLeq @ B @ A ) ) ) ).
% card_of_singl_ordLeq
thf(fact_5487_card__of__ordLess2,axiom,
! [A: $tType,B: $tType,B4: set @ A,A5: set @ B] :
( ( B4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ~ ? [F2: B > A] :
( ( image2 @ B @ A @ F2 @ A5 )
= B4 ) )
= ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ A5 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ B4 ) ) @ ( bNF_We4044943003108391690rdLess @ B @ A ) ) ) ) ).
% card_of_ordLess2
thf(fact_5488_internalize__card__of__ordLeq2,axiom,
! [A: $tType,B: $tType,A5: set @ A,C6: set @ B] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A5 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ C6 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
= ( ? [B7: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ B7 @ C6 )
& ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A5 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B7 ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
& ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B7 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ C6 ) ) @ ( bNF_Wellorder_ordLeq @ B @ B ) ) ) ) ) ).
% internalize_card_of_ordLeq2
thf(fact_5489_card__of__Field__ordLess,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ ( field2 @ A @ R ) ) @ R ) @ ( bNF_Wellorder_ordLeq @ A @ A ) ) ) ).
% card_of_Field_ordLess
thf(fact_5490_card__of__Func__UNIV,axiom,
! [B: $tType,A: $tType,B4: set @ B] :
( member @ ( product_prod @ ( set @ ( product_prod @ ( A > B ) @ ( A > B ) ) ) @ ( set @ ( product_prod @ ( A > B ) @ ( A > B ) ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ ( A > B ) @ ( A > B ) ) ) @ ( set @ ( product_prod @ ( A > B ) @ ( A > B ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( A > B ) @ ( bNF_Wellorder_Func @ A @ B @ ( top_top @ ( set @ A ) ) @ B4 ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( A > B )
@ ( collect @ ( A > B )
@ ^ [F2: A > B] : ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) ) @ B4 ) ) ) )
@ ( bNF_Wellorder_ordIso @ ( A > B ) @ ( A > B ) ) ) ).
% card_of_Func_UNIV
thf(fact_5491_map__comp__None__iff,axiom,
! [A: $tType,C: $tType,B: $tType,M12: B > ( option @ A ),M23: C > ( option @ B ),K: C] :
( ( ( map_comp @ B @ A @ C @ M12 @ M23 @ K )
= ( none @ A ) )
= ( ( ( M23 @ K )
= ( none @ B ) )
| ? [K9: B] :
( ( ( M23 @ K )
= ( some @ B @ K9 ) )
& ( ( M12 @ K9 )
= ( none @ A ) ) ) ) ) ).
% map_comp_None_iff
thf(fact_5492_ordLeq__Times__mono1,axiom,
! [B: $tType,C: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R3: set @ ( product_prod @ B @ B ),C6: set @ C] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R3 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ A @ C ) ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ A @ C ) ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ C )
@ ( product_Sigma @ A @ C @ ( field2 @ A @ R )
@ ^ [Uu: A] : C6 ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ B @ C )
@ ( product_Sigma @ B @ C @ ( field2 @ B @ R3 )
@ ^ [Uu: B] : C6 ) ) )
@ ( bNF_Wellorder_ordLeq @ ( product_prod @ A @ C ) @ ( product_prod @ B @ C ) ) ) ) ).
% ordLeq_Times_mono1
thf(fact_5493_ordLeq__Times__mono2,axiom,
! [B: $tType,A: $tType,C: $tType,R: set @ ( product_prod @ A @ A ),R3: set @ ( product_prod @ B @ B ),A5: set @ C] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R3 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ C @ A ) @ ( product_prod @ C @ A ) ) ) @ ( set @ ( product_prod @ ( product_prod @ C @ B ) @ ( product_prod @ C @ B ) ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ C @ A ) @ ( product_prod @ C @ A ) ) ) @ ( set @ ( product_prod @ ( product_prod @ C @ B ) @ ( product_prod @ C @ B ) ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ C @ A )
@ ( product_Sigma @ C @ A @ A5
@ ^ [Uu: C] : ( field2 @ A @ R ) ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ C @ B )
@ ( product_Sigma @ C @ B @ A5
@ ^ [Uu: C] : ( field2 @ B @ R3 ) ) ) )
@ ( bNF_Wellorder_ordLeq @ ( product_prod @ C @ A ) @ ( product_prod @ C @ B ) ) ) ) ).
% ordLeq_Times_mono2
thf(fact_5494_card__of__ordLeq,axiom,
! [B: $tType,A: $tType,A5: set @ A,B4: set @ B] :
( ( ? [F2: A > B] :
( ( inj_on @ A @ B @ F2 @ A5 )
& ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F2 @ A5 ) @ B4 ) ) )
= ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A5 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B4 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ) ).
% card_of_ordLeq
thf(fact_5495_internalize__card__of__ordLeq,axiom,
! [A: $tType,B: $tType,A5: set @ A,R: set @ ( product_prod @ B @ B )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A5 ) @ R ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
= ( ? [B7: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ B7 @ ( field2 @ B @ R ) )
& ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A5 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B7 ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
& ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B7 ) @ R ) @ ( bNF_Wellorder_ordLeq @ B @ B ) ) ) ) ) ).
% internalize_card_of_ordLeq
thf(fact_5496_card__of__ordLess,axiom,
! [A: $tType,B: $tType,A5: set @ A,B4: set @ B] :
( ( ~ ? [F2: A > B] :
( ( inj_on @ A @ B @ F2 @ A5 )
& ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F2 @ A5 ) @ B4 ) ) )
= ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B4 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A5 ) ) @ ( bNF_We4044943003108391690rdLess @ B @ A ) ) ) ).
% card_of_ordLess
thf(fact_5497_card__of__UNION__ordLeq__infinite,axiom,
! [B: $tType,A: $tType,C: $tType,B4: set @ A,I5: set @ B,A5: B > ( set @ C )] :
( ~ ( finite_finite2 @ A @ B4 )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ I5 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ B4 ) ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ I5 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ ( A5 @ X4 ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ B4 ) ) @ ( bNF_Wellorder_ordLeq @ C @ A ) ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ ( complete_Sup_Sup @ ( set @ C ) @ ( image2 @ B @ ( set @ C ) @ A5 @ I5 ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ B4 ) ) @ ( bNF_Wellorder_ordLeq @ C @ A ) ) ) ) ) ).
% card_of_UNION_ordLeq_infinite
thf(fact_5498_regularCard__def,axiom,
! [A: $tType] :
( ( bNF_Ca7133664381575040944arCard @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A )] :
! [K7: set @ A] :
( ( ( ord_less_eq @ ( set @ A ) @ K7 @ ( field2 @ A @ R4 ) )
& ( bNF_Ca7293521722713021262ofinal @ A @ K7 @ R4 ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ K7 ) @ R4 ) @ ( bNF_Wellorder_ordIso @ A @ A ) ) ) ) ) ).
% regularCard_def
thf(fact_5499_card__of__ordIso__subst,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( A5 = B4 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A5 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ B4 ) ) @ ( bNF_Wellorder_ordIso @ A @ A ) ) ) ).
% card_of_ordIso_subst
thf(fact_5500_Card__order__iff__Restr__underS,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
= ( ! [X2: A] :
( ( member @ A @ X2 @ ( field2 @ A @ R ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R
@ ( product_Sigma @ A @ A @ ( order_underS @ A @ R @ X2 )
@ ^ [Uu: A] : ( order_underS @ A @ R @ X2 ) ) )
@ ( bNF_Ca6860139660246222851ard_of @ A @ ( field2 @ A @ R ) ) )
@ ( bNF_We4044943003108391690rdLess @ A @ A ) ) ) ) ) ) ).
% Card_order_iff_Restr_underS
thf(fact_5501_Card__order__trans,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),X: A,Y3: A,Z: A] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( X != Y3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ R )
=> ( ( Y3 != Z )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z ) @ R )
=> ( ( X != Z )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z ) @ R ) ) ) ) ) ) ) ).
% Card_order_trans
thf(fact_5502_ordLeq__refl,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R @ R ) @ ( bNF_Wellorder_ordLeq @ A @ A ) ) ) ).
% ordLeq_refl
thf(fact_5503_ordIso__refl,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R @ R ) @ ( bNF_Wellorder_ordIso @ A @ A ) ) ) ).
% ordIso_refl
thf(fact_5504_card__order__on__ordIso,axiom,
! [A: $tType,A5: set @ A,R: set @ ( product_prod @ A @ A ),R3: set @ ( product_prod @ A @ A )] :
( ( bNF_Ca8970107618336181345der_on @ A @ A5 @ R )
=> ( ( bNF_Ca8970107618336181345der_on @ A @ A5 @ R3 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R @ R3 ) @ ( bNF_Wellorder_ordIso @ A @ A ) ) ) ) ).
% card_order_on_ordIso
thf(fact_5505_infinite__Card__order__limit,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A3: A] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ~ ( finite_finite2 @ A @ ( field2 @ A @ R ) )
=> ( ( member @ A @ A3 @ ( field2 @ A @ R ) )
=> ? [X4: A] :
( ( member @ A @ X4 @ ( field2 @ A @ R ) )
& ( A3 != X4 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ X4 ) @ R ) ) ) ) ) ).
% infinite_Card_order_limit
thf(fact_5506_dir__image,axiom,
! [B: $tType,A: $tType,F: A > B,R: set @ ( product_prod @ A @ A )] :
( ! [X4: A,Y4: A] :
( ( ( F @ X4 )
= ( F @ Y4 ) )
= ( X4 = Y4 ) )
=> ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ ( bNF_We2720479622203943262_image @ A @ B @ R @ F ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) ) ) ) ).
% dir_image
thf(fact_5507_Card__order__ordIso2,axiom,
! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ A ),R3: set @ ( product_prod @ B @ B )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R3 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R3 ) @ R3 ) ) ) ).
% Card_order_ordIso2
thf(fact_5508_Card__order__ordIso,axiom,
! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ A ),R3: set @ ( product_prod @ B @ B )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ R ) @ ( bNF_Wellorder_ordIso @ B @ A ) )
=> ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R3 ) @ R3 ) ) ) ).
% Card_order_ordIso
thf(fact_5509_card__of__unique,axiom,
! [A: $tType,A5: set @ A,R: set @ ( product_prod @ A @ A )] :
( ( bNF_Ca8970107618336181345der_on @ A @ A5 @ R )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R @ ( bNF_Ca6860139660246222851ard_of @ A @ A5 ) ) @ ( bNF_Wellorder_ordIso @ A @ A ) ) ) ).
% card_of_unique
thf(fact_5510_card__order__on__def,axiom,
! [A: $tType] :
( ( bNF_Ca8970107618336181345der_on @ A )
= ( ^ [A8: set @ A,R4: set @ ( product_prod @ A @ A )] :
( ( order_well_order_on @ A @ A8 @ R4 )
& ! [R9: set @ ( product_prod @ A @ A )] :
( ( order_well_order_on @ A @ A8 @ R9 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R4 @ R9 ) @ ( bNF_Wellorder_ordLeq @ A @ A ) ) ) ) ) ) ).
% card_order_on_def
thf(fact_5511_Card__order__Times2,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),A5: set @ B] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) @ R
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ B @ A )
@ ( product_Sigma @ B @ A @ A5
@ ^ [Uu: B] : ( field2 @ A @ R ) ) ) )
@ ( bNF_Wellorder_ordLeq @ A @ ( product_prod @ B @ A ) ) ) ) ) ).
% Card_order_Times2
thf(fact_5512_Card__order__Times1,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),B4: set @ B] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( B4
!= ( bot_bot @ ( set @ B ) ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ R
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ ( field2 @ A @ R )
@ ^ [Uu: A] : B4 ) ) )
@ ( bNF_Wellorder_ordLeq @ A @ ( product_prod @ A @ B ) ) ) ) ) ).
% Card_order_Times1
thf(fact_5513_Card__order__Times__same__infinite,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ~ ( finite_finite2 @ A @ ( field2 @ A @ R ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ A ) @ ( product_prod @ A @ A ) ) ) @ ( set @ ( product_prod @ A @ A ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ A ) @ ( product_prod @ A @ A ) ) ) @ ( set @ ( product_prod @ A @ A ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ A )
@ ( product_Sigma @ A @ A @ ( field2 @ A @ R )
@ ^ [Uu: A] : ( field2 @ A @ R ) ) )
@ R )
@ ( bNF_Wellorder_ordLeq @ ( product_prod @ A @ A ) @ A ) ) ) ) ).
% Card_order_Times_same_infinite
thf(fact_5514_Card__order__Pow,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ R @ ( bNF_Ca6860139660246222851ard_of @ ( set @ A ) @ ( pow2 @ A @ ( field2 @ A @ R ) ) ) ) @ ( bNF_We4044943003108391690rdLess @ A @ ( set @ A ) ) ) ) ).
% Card_order_Pow
thf(fact_5515_Card__order__iff__ordLeq__card__of,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
= ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R @ ( bNF_Ca6860139660246222851ard_of @ A @ ( field2 @ A @ R ) ) ) @ ( bNF_Wellorder_ordLeq @ A @ A ) ) ) ).
% Card_order_iff_ordLeq_card_of
thf(fact_5516_card__of__Field__ordIso,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ ( field2 @ A @ R ) ) @ R ) @ ( bNF_Wellorder_ordIso @ A @ A ) ) ) ).
% card_of_Field_ordIso
thf(fact_5517_Card__order__iff__ordIso__card__of,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
= ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R @ ( bNF_Ca6860139660246222851ard_of @ A @ ( field2 @ A @ R ) ) ) @ ( bNF_Wellorder_ordIso @ A @ A ) ) ) ).
% Card_order_iff_ordIso_card_of
thf(fact_5518_ordIso__card__of__imp__Card__order,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),A5: set @ B] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ ( bNF_Ca6860139660246222851ard_of @ B @ A5 ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R ) ) ).
% ordIso_card_of_imp_Card_order
thf(fact_5519_exists__minim__Card__order,axiom,
! [A: $tType,R5: set @ ( set @ ( product_prod @ A @ A ) )] :
( ( R5
!= ( bot_bot @ ( set @ ( set @ ( product_prod @ A @ A ) ) ) ) )
=> ( ! [X4: set @ ( product_prod @ A @ A )] :
( ( member @ ( set @ ( product_prod @ A @ A ) ) @ X4 @ R5 )
=> ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ X4 ) @ X4 ) )
=> ? [X4: set @ ( product_prod @ A @ A )] :
( ( member @ ( set @ ( product_prod @ A @ A ) ) @ X4 @ R5 )
& ! [Xa2: set @ ( product_prod @ A @ A )] :
( ( member @ ( set @ ( product_prod @ A @ A ) ) @ Xa2 @ R5 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ X4 @ Xa2 ) @ ( bNF_Wellorder_ordLeq @ A @ A ) ) ) ) ) ) ).
% exists_minim_Card_order
thf(fact_5520_Card__order__empty,axiom,
! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ A )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ ( bot_bot @ ( set @ B ) ) ) @ R ) @ ( bNF_Wellorder_ordLeq @ B @ A ) ) ) ).
% Card_order_empty
thf(fact_5521_card__of__ordIso__finite__Field,axiom,
! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ A ),A5: set @ B] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ ( bNF_Ca6860139660246222851ard_of @ B @ A5 ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( ( finite_finite2 @ A @ ( field2 @ A @ R ) )
= ( finite_finite2 @ B @ A5 ) ) ) ) ).
% card_of_ordIso_finite_Field
thf(fact_5522_card__of__underS,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A3: A] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( member @ A @ A3 @ ( field2 @ A @ R ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ ( order_underS @ A @ R @ A3 ) ) @ R ) @ ( bNF_We4044943003108391690rdLess @ A @ A ) ) ) ) ).
% card_of_underS
thf(fact_5523_regularCard__UNION,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),As4: A > ( set @ B ),B4: set @ B] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( bNF_Ca7133664381575040944arCard @ A @ R )
=> ( ( bNF_Ca3754400796208372196lChain @ A @ ( set @ B ) @ R @ As4 )
=> ( ( ord_less_eq @ ( set @ B ) @ B4 @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ As4 @ ( field2 @ A @ R ) ) ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B4 ) @ R ) @ ( bNF_We4044943003108391690rdLess @ B @ A ) )
=> ? [X4: A] :
( ( member @ A @ X4 @ ( field2 @ A @ R ) )
& ( ord_less_eq @ ( set @ B ) @ B4 @ ( As4 @ X4 ) ) ) ) ) ) ) ) ).
% regularCard_UNION
thf(fact_5524_Card__order__singl__ordLeq,axiom,
! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ A ),B2: B] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( ( field2 @ A @ R )
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ ( insert2 @ B @ B2 @ ( bot_bot @ ( set @ B ) ) ) ) @ R ) @ ( bNF_Wellorder_ordLeq @ B @ A ) ) ) ) ).
% Card_order_singl_ordLeq
thf(fact_5525_card__of__Un__ordLeq__infinite__Field,axiom,
! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ A ),A5: set @ B,B4: set @ B] :
( ~ ( finite_finite2 @ A @ ( field2 @ A @ R ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ A5 ) @ R ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B4 ) @ R ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ ( sup_sup @ ( set @ B ) @ A5 @ B4 ) ) @ R ) @ ( bNF_Wellorder_ordLeq @ B @ A ) ) ) ) ) ) ).
% card_of_Un_ordLeq_infinite_Field
thf(fact_5526_card__of__empty1,axiom,
! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ A )] :
( ( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
| ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ ( bot_bot @ ( set @ B ) ) ) @ R ) @ ( bNF_Wellorder_ordLeq @ B @ A ) ) ) ).
% card_of_empty1
thf(fact_5527_Card__order__Times__infinite,axiom,
! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ A ),P4: set @ ( product_prod @ B @ B )] :
( ~ ( finite_finite2 @ A @ ( field2 @ A @ R ) )
=> ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( ( field2 @ B @ P4 )
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ P4 @ R ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ A @ A ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ A @ A ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ ( field2 @ A @ R )
@ ^ [Uu: A] : ( field2 @ B @ P4 ) ) )
@ R )
@ ( bNF_Wellorder_ordIso @ ( product_prod @ A @ B ) @ A ) )
& ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) @ ( set @ ( product_prod @ A @ A ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) @ ( set @ ( product_prod @ A @ A ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ B @ A )
@ ( product_Sigma @ B @ A @ ( field2 @ B @ P4 )
@ ^ [Uu: B] : ( field2 @ A @ R ) ) )
@ R )
@ ( bNF_Wellorder_ordIso @ ( product_prod @ B @ A ) @ A ) ) ) ) ) ) ) ).
% Card_order_Times_infinite
thf(fact_5528_card__of__Sigma__ordLeq__infinite__Field,axiom,
! [A: $tType,C: $tType,B: $tType,R: set @ ( product_prod @ A @ A ),I5: set @ B,A5: B > ( set @ C )] :
( ~ ( finite_finite2 @ A @ ( field2 @ A @ R ) )
=> ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ I5 ) @ R ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ I5 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ ( A5 @ X4 ) ) @ R ) @ ( bNF_Wellorder_ordLeq @ C @ A ) ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ B @ C ) @ ( product_Sigma @ B @ C @ I5 @ A5 ) ) @ R ) @ ( bNF_Wellorder_ordLeq @ ( product_prod @ B @ C ) @ A ) ) ) ) ) ) ).
% card_of_Sigma_ordLeq_infinite_Field
thf(fact_5529_card__of__Times__ordLeq__infinite__Field,axiom,
! [A: $tType,C: $tType,B: $tType,R: set @ ( product_prod @ A @ A ),A5: set @ B,B4: set @ C] :
( ~ ( finite_finite2 @ A @ ( field2 @ A @ R ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ A5 ) @ R ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ B4 ) @ R ) @ ( bNF_Wellorder_ordLeq @ C @ A ) )
=> ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) ) @ ( set @ ( product_prod @ A @ A ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) ) @ ( set @ ( product_prod @ A @ A ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ B @ C )
@ ( product_Sigma @ B @ C @ A5
@ ^ [Uu: B] : B4 ) )
@ R )
@ ( bNF_Wellorder_ordLeq @ ( product_prod @ B @ C ) @ A ) ) ) ) ) ) ).
% card_of_Times_ordLeq_infinite_Field
thf(fact_5530_card__of__UNION__ordLeq__infinite__Field,axiom,
! [B: $tType,A: $tType,C: $tType,R: set @ ( product_prod @ A @ A ),I5: set @ B,A5: B > ( set @ C )] :
( ~ ( finite_finite2 @ A @ ( field2 @ A @ R ) )
=> ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ I5 ) @ R ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ I5 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ ( A5 @ X4 ) ) @ R ) @ ( bNF_Wellorder_ordLeq @ C @ A ) ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ ( complete_Sup_Sup @ ( set @ C ) @ ( image2 @ B @ ( set @ C ) @ A5 @ I5 ) ) ) @ R ) @ ( bNF_Wellorder_ordLeq @ C @ A ) ) ) ) ) ) ).
% card_of_UNION_ordLeq_infinite_Field
thf(fact_5531_ex__toCard__pred,axiom,
! [B: $tType,A: $tType,A5: set @ A,R: set @ ( product_prod @ B @ B )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A5 ) @ R ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R ) @ R )
=> ? [X_12: A > B] : ( bNF_Gr1419584066657907630d_pred @ A @ B @ A5 @ R @ X_12 ) ) ) ).
% ex_toCard_pred
thf(fact_5532_cardSuc__UNION,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),As4: ( set @ A ) > ( set @ B ),B4: set @ B] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ~ ( finite_finite2 @ A @ ( field2 @ A @ R ) )
=> ( ( bNF_Ca3754400796208372196lChain @ ( set @ A ) @ ( set @ B ) @ ( bNF_Ca8387033319878233205ardSuc @ A @ R ) @ As4 )
=> ( ( ord_less_eq @ ( set @ B ) @ B4 @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ ( set @ A ) @ ( set @ B ) @ As4 @ ( field2 @ ( set @ A ) @ ( bNF_Ca8387033319878233205ardSuc @ A @ R ) ) ) ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B4 ) @ R ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ? [X4: set @ A] :
( ( member @ ( set @ A ) @ X4 @ ( field2 @ ( set @ A ) @ ( bNF_Ca8387033319878233205ardSuc @ A @ R ) ) )
& ( ord_less_eq @ ( set @ B ) @ B4 @ ( As4 @ X4 ) ) ) ) ) ) ) ) ).
% cardSuc_UNION
thf(fact_5533_isCardSuc__def,axiom,
! [A: $tType] :
( ( bNF_Ca6246979054910435723ardSuc @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A ),R9: set @ ( product_prod @ ( set @ A ) @ ( set @ A ) )] :
( ( bNF_Ca8970107618336181345der_on @ ( set @ A ) @ ( field2 @ ( set @ A ) @ R9 ) @ R9 )
& ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ R4 @ R9 ) @ ( bNF_We4044943003108391690rdLess @ A @ ( set @ A ) ) )
& ! [R11: set @ ( product_prod @ ( set @ A ) @ ( set @ A ) )] :
( ( ( bNF_Ca8970107618336181345der_on @ ( set @ A ) @ ( field2 @ ( set @ A ) @ R11 ) @ R11 )
& ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ R4 @ R11 ) @ ( bNF_We4044943003108391690rdLess @ A @ ( set @ A ) ) ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ R9 @ R11 ) @ ( bNF_Wellorder_ordLeq @ ( set @ A ) @ ( set @ A ) ) ) ) ) ) ) ).
% isCardSuc_def
thf(fact_5534_cardSuc__ordLeq,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ R @ ( bNF_Ca8387033319878233205ardSuc @ A @ R ) ) @ ( bNF_Wellorder_ordLeq @ A @ ( set @ A ) ) ) ) ).
% cardSuc_ordLeq
thf(fact_5535_cardSuc__greater,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ R @ ( bNF_Ca8387033319878233205ardSuc @ A @ R ) ) @ ( bNF_We4044943003108391690rdLess @ A @ ( set @ A ) ) ) ) ).
% cardSuc_greater
thf(fact_5536_cardSuc__ordLess__ordLeq,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R3: set @ ( product_prod @ B @ B )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R3 ) @ R3 )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R3 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) )
= ( member @ ( product_prod @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca8387033319878233205ardSuc @ A @ R ) @ R3 ) @ ( bNF_Wellorder_ordLeq @ ( set @ A ) @ B ) ) ) ) ) ).
% cardSuc_ordLess_ordLeq
thf(fact_5537_cardSuc__least,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R3: set @ ( product_prod @ B @ B )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R3 ) @ R3 )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R3 ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca8387033319878233205ardSuc @ A @ R ) @ R3 ) @ ( bNF_Wellorder_ordLeq @ ( set @ A ) @ B ) ) ) ) ) ).
% cardSuc_least
thf(fact_5538_cardSuc__mono__ordLeq,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R3: set @ ( product_prod @ B @ B )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R3 ) @ R3 )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( set @ ( product_prod @ ( set @ B ) @ ( set @ B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( set @ ( product_prod @ ( set @ B ) @ ( set @ B ) ) ) @ ( bNF_Ca8387033319878233205ardSuc @ A @ R ) @ ( bNF_Ca8387033319878233205ardSuc @ B @ R3 ) ) @ ( bNF_Wellorder_ordLeq @ ( set @ A ) @ ( set @ B ) ) )
= ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R3 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ) ) ) ).
% cardSuc_mono_ordLeq
thf(fact_5539_cardSuc__invar__ordIso,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R3: set @ ( product_prod @ B @ B )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R3 ) @ R3 )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( set @ ( product_prod @ ( set @ B ) @ ( set @ B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( set @ ( product_prod @ ( set @ B ) @ ( set @ B ) ) ) @ ( bNF_Ca8387033319878233205ardSuc @ A @ R ) @ ( bNF_Ca8387033319878233205ardSuc @ B @ R3 ) ) @ ( bNF_Wellorder_ordIso @ ( set @ A ) @ ( set @ B ) ) )
= ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R3 ) @ ( bNF_Wellorder_ordIso @ A @ B ) ) ) ) ) ).
% cardSuc_invar_ordIso
thf(fact_5540_cardSuc__least__aux,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),R3: set @ ( product_prod @ ( set @ A ) @ ( set @ A ) )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( bNF_Ca8970107618336181345der_on @ ( set @ A ) @ ( field2 @ ( set @ A ) @ R3 ) @ R3 )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ R @ R3 ) @ ( bNF_We4044943003108391690rdLess @ A @ ( set @ A ) ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( bNF_Ca8387033319878233205ardSuc @ A @ R ) @ R3 ) @ ( bNF_Wellorder_ordLeq @ ( set @ A ) @ ( set @ A ) ) ) ) ) ) ).
% cardSuc_least_aux
thf(fact_5541_cardSuc__ordLeq__ordLess,axiom,
! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ A ),R3: set @ ( product_prod @ B @ B )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R3 ) @ R3 )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ R3 @ ( bNF_Ca8387033319878233205ardSuc @ A @ R ) ) @ ( bNF_We4044943003108391690rdLess @ B @ ( set @ A ) ) )
= ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ R ) @ ( bNF_Wellorder_ordLeq @ B @ A ) ) ) ) ) ).
% cardSuc_ordLeq_ordLess
thf(fact_5542_toCard__pred__toCard,axiom,
! [A: $tType,B: $tType,A5: set @ A,R: set @ ( product_prod @ B @ B )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A5 ) @ R ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R ) @ R )
=> ( bNF_Gr1419584066657907630d_pred @ A @ B @ A5 @ R @ ( bNF_Greatest_toCard @ A @ B @ A5 @ R ) ) ) ) ).
% toCard_pred_toCard
thf(fact_5543_cardSuc__UNION__Cinfinite,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),As4: ( set @ A ) > ( set @ B ),B4: set @ B] :
( ( ( bNF_Ca4139267488887388095finite @ A @ R )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R ) )
=> ( ( bNF_Ca3754400796208372196lChain @ ( set @ A ) @ ( set @ B ) @ ( bNF_Ca8387033319878233205ardSuc @ A @ R ) @ As4 )
=> ( ( ord_less_eq @ ( set @ B ) @ B4 @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ ( set @ A ) @ ( set @ B ) @ As4 @ ( field2 @ ( set @ A ) @ ( bNF_Ca8387033319878233205ardSuc @ A @ R ) ) ) ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B4 ) @ R ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ? [X4: set @ A] :
( ( member @ ( set @ A ) @ X4 @ ( field2 @ ( set @ A ) @ ( bNF_Ca8387033319878233205ardSuc @ A @ R ) ) )
& ( ord_less_eq @ ( set @ B ) @ B4 @ ( As4 @ X4 ) ) ) ) ) ) ) ).
% cardSuc_UNION_Cinfinite
thf(fact_5544_toCard__inj,axiom,
! [B: $tType,A: $tType,A5: set @ A,R: set @ ( product_prod @ B @ B ),X: A,Y3: A] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A5 ) @ R ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R ) @ R )
=> ( ( member @ A @ X @ A5 )
=> ( ( member @ A @ Y3 @ A5 )
=> ( ( ( bNF_Greatest_toCard @ A @ B @ A5 @ R @ X )
= ( bNF_Greatest_toCard @ A @ B @ A5 @ R @ Y3 ) )
= ( X = Y3 ) ) ) ) ) ) ).
% toCard_inj
thf(fact_5545_cinfinite__mono,axiom,
! [A: $tType,B: $tType,R1: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R1 @ R22 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( bNF_Ca4139267488887388095finite @ A @ R1 )
=> ( bNF_Ca4139267488887388095finite @ B @ R22 ) ) ) ).
% cinfinite_mono
thf(fact_5546_Cinfinite__limit2,axiom,
! [A: $tType,X1: A,R: set @ ( product_prod @ A @ A ),X22: A] :
( ( member @ A @ X1 @ ( field2 @ A @ R ) )
=> ( ( member @ A @ X22 @ ( field2 @ A @ R ) )
=> ( ( ( bNF_Ca4139267488887388095finite @ A @ R )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R ) )
=> ? [X4: A] :
( ( member @ A @ X4 @ ( field2 @ A @ R ) )
& ( X1 != X4 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X1 @ X4 ) @ R )
& ( X22 != X4 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X22 @ X4 ) @ R ) ) ) ) ) ).
% Cinfinite_limit2
thf(fact_5547_Cinfinite__limit,axiom,
! [A: $tType,X: A,R: set @ ( product_prod @ A @ A )] :
( ( member @ A @ X @ ( field2 @ A @ R ) )
=> ( ( ( bNF_Ca4139267488887388095finite @ A @ R )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R ) )
=> ? [X4: A] :
( ( member @ A @ X4 @ ( field2 @ A @ R ) )
& ( X != X4 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ X4 ) @ R ) ) ) ) ).
% Cinfinite_limit
thf(fact_5548_Cinfinite__cong,axiom,
! [A: $tType,B: $tType,R1: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R1 @ R22 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( ( ( bNF_Ca4139267488887388095finite @ A @ R1 )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R1 ) @ R1 ) )
=> ( ( bNF_Ca4139267488887388095finite @ B @ R22 )
& ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R22 ) @ R22 ) ) ) ) ).
% Cinfinite_cong
thf(fact_5549_Cinfinite__limit__finite,axiom,
! [A: $tType,X7: set @ A,R: set @ ( product_prod @ A @ A )] :
( ( finite_finite2 @ A @ X7 )
=> ( ( ord_less_eq @ ( set @ A ) @ X7 @ ( field2 @ A @ R ) )
=> ( ( ( bNF_Ca4139267488887388095finite @ A @ R )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R ) )
=> ? [X4: A] :
( ( member @ A @ X4 @ ( field2 @ A @ R ) )
& ! [Xa2: A] :
( ( member @ A @ Xa2 @ X7 )
=> ( ( Xa2 != X4 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Xa2 @ X4 ) @ R ) ) ) ) ) ) ) ).
% Cinfinite_limit_finite
thf(fact_5550_Un__Cinfinite__bound,axiom,
! [B: $tType,A: $tType,A5: set @ A,R: set @ ( product_prod @ B @ B ),B4: set @ A] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A5 ) @ R ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ B4 ) @ R ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( ( bNF_Ca4139267488887388095finite @ B @ R )
& ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R ) @ R ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ ( sup_sup @ ( set @ A ) @ A5 @ B4 ) ) @ R ) @ ( bNF_Wellorder_ordLeq @ A @ B ) ) ) ) ) ).
% Un_Cinfinite_bound
thf(fact_5551_UNION__Cinfinite__bound,axiom,
! [A: $tType,B: $tType,C: $tType,I5: set @ A,R: set @ ( product_prod @ B @ B ),A5: A > ( set @ C )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ I5 ) @ R ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ I5 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ ( A5 @ X4 ) ) @ R ) @ ( bNF_Wellorder_ordLeq @ C @ B ) ) )
=> ( ( ( bNF_Ca4139267488887388095finite @ B @ R )
& ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R ) @ R ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ ( complete_Sup_Sup @ ( set @ C ) @ ( image2 @ A @ ( set @ C ) @ A5 @ I5 ) ) ) @ R ) @ ( bNF_Wellorder_ordLeq @ C @ B ) ) ) ) ) ).
% UNION_Cinfinite_bound
thf(fact_5552_fromCard__toCard,axiom,
! [B: $tType,A: $tType,A5: set @ A,R: set @ ( product_prod @ B @ B ),B2: A] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A5 ) @ R ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R ) @ R )
=> ( ( member @ A @ B2 @ A5 )
=> ( ( bNF_Gr5436034075474128252omCard @ A @ B @ A5 @ R @ ( bNF_Greatest_toCard @ A @ B @ A5 @ R @ B2 ) )
= B2 ) ) ) ) ).
% fromCard_toCard
thf(fact_5553_Cfinite__ordLess__Cinfinite,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ B @ B )] :
( ( ( bNF_Cardinal_cfinite @ A @ R )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R ) )
=> ( ( ( bNF_Ca4139267488887388095finite @ B @ S )
& ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ S ) @ S ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ S ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) ) ) ) ).
% Cfinite_ordLess_Cinfinite
thf(fact_5554_comp__single__set__bd,axiom,
! [B: $tType,D: $tType,A: $tType,E: $tType,C: $tType,Fbd: set @ ( product_prod @ A @ A ),Fset: B > ( set @ C ),Gset: D > ( set @ B ),Gbd: set @ ( product_prod @ E @ E ),X: D] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ Fbd ) @ Fbd )
=> ( ! [X4: B] : ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ ( Fset @ X4 ) ) @ Fbd ) @ ( bNF_Wellorder_ordLeq @ C @ A ) )
=> ( ! [X4: D] : ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ E @ E ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ E @ E ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ ( Gset @ X4 ) ) @ Gbd ) @ ( bNF_Wellorder_ordLeq @ B @ E ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ ( product_prod @ E @ A ) @ ( product_prod @ E @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ ( product_prod @ E @ A ) @ ( product_prod @ E @ A ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ ( complete_Sup_Sup @ ( set @ C ) @ ( image2 @ B @ ( set @ C ) @ Fset @ ( Gset @ X ) ) ) ) @ ( bNF_Cardinal_cprod @ E @ A @ Gbd @ Fbd ) ) @ ( bNF_Wellorder_ordLeq @ C @ ( product_prod @ E @ A ) ) ) ) ) ) ).
% comp_single_set_bd
thf(fact_5555_cprod__com,axiom,
! [B: $tType,A: $tType,P1: set @ ( product_prod @ A @ A ),P22: set @ ( product_prod @ B @ B )] : ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) @ ( bNF_Cardinal_cprod @ A @ B @ P1 @ P22 ) @ ( bNF_Cardinal_cprod @ B @ A @ P22 @ P1 ) ) @ ( bNF_Wellorder_ordIso @ ( product_prod @ A @ B ) @ ( product_prod @ B @ A ) ) ) ).
% cprod_com
thf(fact_5556_Cfinite__cprod__Cinfinite,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ B @ B )] :
( ( ( bNF_Cardinal_cfinite @ A @ R )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R ) )
=> ( ( ( bNF_Ca4139267488887388095finite @ B @ S )
& ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ S ) @ S ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Cardinal_cprod @ A @ B @ R @ S ) @ S ) @ ( bNF_Wellorder_ordLeq @ ( product_prod @ A @ B ) @ B ) ) ) ) ).
% Cfinite_cprod_Cinfinite
thf(fact_5557_cprod__def,axiom,
! [B: $tType,A: $tType] :
( ( bNF_Cardinal_cprod @ A @ B )
= ( ^ [R13: set @ ( product_prod @ A @ A ),R24: set @ ( product_prod @ B @ B )] :
( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ ( field2 @ A @ R13 )
@ ^ [Uu: A] : ( field2 @ B @ R24 ) ) ) ) ) ).
% cprod_def
thf(fact_5558_cprod__mono,axiom,
! [D: $tType,B: $tType,C: $tType,A: $tType,P1: set @ ( product_prod @ A @ A ),R1: set @ ( product_prod @ B @ B ),P22: set @ ( product_prod @ C @ C ),R22: set @ ( product_prod @ D @ D )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P1 @ R1 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ D @ D ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ D @ D ) ) @ P22 @ R22 ) @ ( bNF_Wellorder_ordLeq @ C @ D ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ A @ C ) ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ D ) @ ( product_prod @ B @ D ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ A @ C ) ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ D ) @ ( product_prod @ B @ D ) ) ) @ ( bNF_Cardinal_cprod @ A @ C @ P1 @ P22 ) @ ( bNF_Cardinal_cprod @ B @ D @ R1 @ R22 ) ) @ ( bNF_Wellorder_ordLeq @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) ) ) ) ) ).
% cprod_mono
thf(fact_5559_cprod__mono1,axiom,
! [B: $tType,C: $tType,A: $tType,P1: set @ ( product_prod @ A @ A ),R1: set @ ( product_prod @ B @ B ),Q2: set @ ( product_prod @ C @ C )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P1 @ R1 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ A @ C ) ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ A @ C ) ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) ) @ ( bNF_Cardinal_cprod @ A @ C @ P1 @ Q2 ) @ ( bNF_Cardinal_cprod @ B @ C @ R1 @ Q2 ) ) @ ( bNF_Wellorder_ordLeq @ ( product_prod @ A @ C ) @ ( product_prod @ B @ C ) ) ) ) ).
% cprod_mono1
thf(fact_5560_cprod__mono2,axiom,
! [B: $tType,A: $tType,C: $tType,P22: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B ),Q2: set @ ( product_prod @ C @ C )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P22 @ R22 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ C @ A ) @ ( product_prod @ C @ A ) ) ) @ ( set @ ( product_prod @ ( product_prod @ C @ B ) @ ( product_prod @ C @ B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ C @ A ) @ ( product_prod @ C @ A ) ) ) @ ( set @ ( product_prod @ ( product_prod @ C @ B ) @ ( product_prod @ C @ B ) ) ) @ ( bNF_Cardinal_cprod @ C @ A @ Q2 @ P22 ) @ ( bNF_Cardinal_cprod @ C @ B @ Q2 @ R22 ) ) @ ( bNF_Wellorder_ordLeq @ ( product_prod @ C @ A ) @ ( product_prod @ C @ B ) ) ) ) ).
% cprod_mono2
thf(fact_5561_cprod__cong,axiom,
! [D: $tType,B: $tType,C: $tType,A: $tType,P1: set @ ( product_prod @ A @ A ),R1: set @ ( product_prod @ B @ B ),P22: set @ ( product_prod @ C @ C ),R22: set @ ( product_prod @ D @ D )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P1 @ R1 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ D @ D ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ D @ D ) ) @ P22 @ R22 ) @ ( bNF_Wellorder_ordIso @ C @ D ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ A @ C ) ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ D ) @ ( product_prod @ B @ D ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ A @ C ) ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ D ) @ ( product_prod @ B @ D ) ) ) @ ( bNF_Cardinal_cprod @ A @ C @ P1 @ P22 ) @ ( bNF_Cardinal_cprod @ B @ D @ R1 @ R22 ) ) @ ( bNF_Wellorder_ordIso @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) ) ) ) ) ).
% cprod_cong
thf(fact_5562_cprod__cong1,axiom,
! [B: $tType,C: $tType,A: $tType,P1: set @ ( product_prod @ A @ A ),R1: set @ ( product_prod @ B @ B ),P22: set @ ( product_prod @ C @ C )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P1 @ R1 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ A @ C ) ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ A @ C ) ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) ) @ ( bNF_Cardinal_cprod @ A @ C @ P1 @ P22 ) @ ( bNF_Cardinal_cprod @ B @ C @ R1 @ P22 ) ) @ ( bNF_Wellorder_ordIso @ ( product_prod @ A @ C ) @ ( product_prod @ B @ C ) ) ) ) ).
% cprod_cong1
thf(fact_5563_cprod__cong2,axiom,
! [B: $tType,A: $tType,C: $tType,P22: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B ),Q2: set @ ( product_prod @ C @ C )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P22 @ R22 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ C @ A ) @ ( product_prod @ C @ A ) ) ) @ ( set @ ( product_prod @ ( product_prod @ C @ B ) @ ( product_prod @ C @ B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ C @ A ) @ ( product_prod @ C @ A ) ) ) @ ( set @ ( product_prod @ ( product_prod @ C @ B ) @ ( product_prod @ C @ B ) ) ) @ ( bNF_Cardinal_cprod @ C @ A @ Q2 @ P22 ) @ ( bNF_Cardinal_cprod @ C @ B @ Q2 @ R22 ) ) @ ( bNF_Wellorder_ordIso @ ( product_prod @ C @ A ) @ ( product_prod @ C @ B ) ) ) ) ).
% cprod_cong2
thf(fact_5564_cprod__infinite,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( ( bNF_Ca4139267488887388095finite @ A @ R )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ A ) @ ( product_prod @ A @ A ) ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ A ) @ ( product_prod @ A @ A ) ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Cardinal_cprod @ A @ A @ R @ R ) @ R ) @ ( bNF_Wellorder_ordIso @ ( product_prod @ A @ A ) @ A ) ) ) ).
% cprod_infinite
thf(fact_5565_cprod__cinfinite__bound,axiom,
! [B: $tType,C: $tType,A: $tType,P4: set @ ( product_prod @ A @ A ),R: set @ ( product_prod @ B @ B ),Q2: set @ ( product_prod @ C @ C )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P4 @ R ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ B @ B ) ) @ Q2 @ R ) @ ( bNF_Wellorder_ordLeq @ C @ B ) )
=> ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ P4 ) @ P4 )
=> ( ( bNF_Ca8970107618336181345der_on @ C @ ( field2 @ C @ Q2 ) @ Q2 )
=> ( ( ( bNF_Ca4139267488887388095finite @ B @ R )
& ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R ) @ R ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ A @ C ) ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ A @ C ) ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Cardinal_cprod @ A @ C @ P4 @ Q2 ) @ R ) @ ( bNF_Wellorder_ordLeq @ ( product_prod @ A @ C ) @ B ) ) ) ) ) ) ) ).
% cprod_cinfinite_bound
thf(fact_5566_cprod__dup,axiom,
! [A: $tType,C: $tType,B: $tType,R: set @ ( product_prod @ A @ A ),P4: set @ ( product_prod @ B @ B ),P8: set @ ( product_prod @ C @ C )] :
( ( bNF_Ca4139267488887388095finite @ A @ R )
=> ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ A ) @ ( product_prod @ A @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ A ) @ ( product_prod @ A @ A ) ) ) @ ( bNF_Cardinal_cprod @ B @ C @ P4 @ P8 ) @ ( bNF_Cardinal_cprod @ A @ A @ R @ R ) ) @ ( bNF_Wellorder_ordIso @ ( product_prod @ B @ C ) @ ( product_prod @ A @ A ) ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Cardinal_cprod @ B @ C @ P4 @ P8 ) @ R ) @ ( bNF_Wellorder_ordIso @ ( product_prod @ B @ C ) @ A ) ) ) ) ) ).
% cprod_dup
thf(fact_5567_card__of__Csum__Times_H,axiom,
! [A: $tType,C: $tType,B: $tType,R: set @ ( product_prod @ A @ A ),I5: set @ B,A5: B > ( set @ C )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ I5 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ ( A5 @ X4 ) ) @ R ) @ ( bNF_Wellorder_ordLeq @ C @ A ) ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ B @ C ) @ ( product_prod @ B @ C ) ) ) @ ( set @ ( product_prod @ ( product_prod @ B @ A ) @ ( product_prod @ B @ A ) ) )
@ ( bNF_Cardinal_Csum @ B @ C @ ( bNF_Ca6860139660246222851ard_of @ B @ I5 )
@ ^ [I2: B] : ( bNF_Ca6860139660246222851ard_of @ C @ ( A5 @ I2 ) ) )
@ ( bNF_Cardinal_cprod @ B @ A @ ( bNF_Ca6860139660246222851ard_of @ B @ I5 ) @ R ) )
@ ( bNF_Wellorder_ordLeq @ ( product_prod @ B @ C ) @ ( product_prod @ B @ A ) ) ) ) ) ).
% card_of_Csum_Times'
thf(fact_5568_card__of__Csum__Times,axiom,
! [C: $tType,B: $tType,A: $tType,I5: set @ A,A5: A > ( set @ B ),B4: set @ C] :
( ! [X4: A] :
( ( member @ A @ X4 @ I5 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ C @ C ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ ( A5 @ X4 ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ B4 ) ) @ ( bNF_Wellorder_ordLeq @ B @ C ) ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ A @ C ) ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ A @ C ) ) )
@ ( bNF_Cardinal_Csum @ A @ B @ ( bNF_Ca6860139660246222851ard_of @ A @ I5 )
@ ^ [I2: A] : ( bNF_Ca6860139660246222851ard_of @ B @ ( A5 @ I2 ) ) )
@ ( bNF_Cardinal_cprod @ A @ C @ ( bNF_Ca6860139660246222851ard_of @ A @ I5 ) @ ( bNF_Ca6860139660246222851ard_of @ C @ B4 ) ) )
@ ( bNF_Wellorder_ordLeq @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) ) ) ) ).
% card_of_Csum_Times
thf(fact_5569_cprod__infinite1_H,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),P4: set @ ( product_prod @ B @ B )] :
( ( ( bNF_Ca4139267488887388095finite @ A @ R )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R ) )
=> ( ( ~ ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P4 @ ( bNF_Cardinal_czero @ B ) ) @ ( bNF_Wellorder_ordIso @ B @ B ) )
& ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ P4 ) @ P4 ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ P4 @ R ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Cardinal_cprod @ A @ B @ R @ P4 ) @ R ) @ ( bNF_Wellorder_ordIso @ ( product_prod @ A @ B ) @ A ) ) ) ) ) ).
% cprod_infinite1'
thf(fact_5570_czero__ordIso,axiom,
! [B: $tType,A: $tType] : ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Cardinal_czero @ A ) @ ( bNF_Cardinal_czero @ B ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) ) ).
% czero_ordIso
thf(fact_5571_cinfinite__not__czero,axiom,
! [A: $tType,B: $tType,R: set @ ( product_prod @ B @ B )] :
( ( bNF_Ca4139267488887388095finite @ B @ R )
=> ~ ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R @ ( bNF_Cardinal_czero @ A ) ) @ ( bNF_Wellorder_ordIso @ B @ A ) ) ) ).
% cinfinite_not_czero
thf(fact_5572_czeroE,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ ( bNF_Cardinal_czero @ B ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( ( field2 @ A @ R )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% czeroE
thf(fact_5573_card__of__ordIso__czero__iff__empty,axiom,
! [B: $tType,A: $tType,A5: set @ A] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A5 ) @ ( bNF_Cardinal_czero @ B ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
= ( A5
= ( bot_bot @ ( set @ A ) ) ) ) ).
% card_of_ordIso_czero_iff_empty
thf(fact_5574_czeroI,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( ( field2 @ A @ R )
= ( bot_bot @ ( set @ A ) ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ ( bNF_Cardinal_czero @ B ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) ) ) ) ).
% czeroI
thf(fact_5575_Cnotzero__imp__not__empty,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( ~ ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R @ ( bNF_Cardinal_czero @ A ) ) @ ( bNF_Wellorder_ordIso @ A @ A ) )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R ) )
=> ( ( field2 @ A @ R )
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% Cnotzero_imp_not_empty
thf(fact_5576_Cnotzero__mono,axiom,
! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ A ),Q2: set @ ( product_prod @ B @ B )] :
( ( ~ ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R @ ( bNF_Cardinal_czero @ A ) ) @ ( bNF_Wellorder_ordIso @ A @ A ) )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R ) )
=> ( ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ Q2 ) @ Q2 )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ Q2 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ~ ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) @ Q2 @ ( bNF_Cardinal_czero @ B ) ) @ ( bNF_Wellorder_ordIso @ B @ B ) )
& ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ Q2 ) @ Q2 ) ) ) ) ) ).
% Cnotzero_mono
thf(fact_5577_Cinfinite__Cnotzero,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( ( bNF_Ca4139267488887388095finite @ A @ R )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R ) )
=> ( ~ ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R @ ( bNF_Cardinal_czero @ A ) ) @ ( bNF_Wellorder_ordIso @ A @ A ) )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R ) ) ) ).
% Cinfinite_Cnotzero
thf(fact_5578_Cnotzero__UNIV,axiom,
! [A: $tType] :
( ~ ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ ( top_top @ ( set @ A ) ) ) @ ( bNF_Cardinal_czero @ A ) ) @ ( bNF_Wellorder_ordIso @ A @ A ) )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ ( bNF_Ca6860139660246222851ard_of @ A @ ( top_top @ ( set @ A ) ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ ( top_top @ ( set @ A ) ) ) ) ) ).
% Cnotzero_UNIV
thf(fact_5579_cinfinite__cprod2,axiom,
! [A: $tType,B: $tType,R1: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B )] :
( ( ~ ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R1 @ ( bNF_Cardinal_czero @ A ) ) @ ( bNF_Wellorder_ordIso @ A @ A ) )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R1 ) @ R1 ) )
=> ( ( ( bNF_Ca4139267488887388095finite @ B @ R22 )
& ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R22 ) @ R22 ) )
=> ( bNF_Ca4139267488887388095finite @ ( product_prod @ A @ B ) @ ( bNF_Cardinal_cprod @ A @ B @ R1 @ R22 ) ) ) ) ).
% cinfinite_cprod2
thf(fact_5580_Csum__def,axiom,
! [B: $tType,A: $tType] :
( ( bNF_Cardinal_Csum @ A @ B )
= ( ^ [R4: set @ ( product_prod @ A @ A ),Rs: A > ( set @ ( product_prod @ B @ B ) )] :
( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ ( field2 @ A @ R4 )
@ ^ [I2: A] : ( field2 @ B @ ( Rs @ I2 ) ) ) ) ) ) ).
% Csum_def
thf(fact_5581_SIGMA__CSUM,axiom,
! [B: $tType,A: $tType,I5: set @ A,As4: A > ( set @ B )] :
( ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ B ) @ ( product_Sigma @ A @ B @ I5 @ As4 ) )
= ( bNF_Cardinal_Csum @ A @ B @ ( bNF_Ca6860139660246222851ard_of @ A @ I5 )
@ ^ [I2: A] : ( bNF_Ca6860139660246222851ard_of @ B @ ( As4 @ I2 ) ) ) ) ).
% SIGMA_CSUM
thf(fact_5582_ordLeq__cprod2,axiom,
! [A: $tType,B: $tType,P1: set @ ( product_prod @ A @ A ),P22: set @ ( product_prod @ B @ B )] :
( ( ~ ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ P1 @ ( bNF_Cardinal_czero @ A ) ) @ ( bNF_Wellorder_ordIso @ A @ A ) )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ P1 ) @ P1 ) )
=> ( ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ P22 ) @ P22 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ P22 @ ( bNF_Cardinal_cprod @ A @ B @ P1 @ P22 ) ) @ ( bNF_Wellorder_ordLeq @ B @ ( product_prod @ A @ B ) ) ) ) ) ).
% ordLeq_cprod2
thf(fact_5583_Cinfinite__cprod2,axiom,
! [A: $tType,B: $tType,R1: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B )] :
( ( ~ ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R1 @ ( bNF_Cardinal_czero @ A ) ) @ ( bNF_Wellorder_ordIso @ A @ A ) )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R1 ) @ R1 ) )
=> ( ( ( bNF_Ca4139267488887388095finite @ B @ R22 )
& ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R22 ) @ R22 ) )
=> ( ( bNF_Ca4139267488887388095finite @ ( product_prod @ A @ B ) @ ( bNF_Cardinal_cprod @ A @ B @ R1 @ R22 ) )
& ( bNF_Ca8970107618336181345der_on @ ( product_prod @ A @ B ) @ ( field2 @ ( product_prod @ A @ B ) @ ( bNF_Cardinal_cprod @ A @ B @ R1 @ R22 ) ) @ ( bNF_Cardinal_cprod @ A @ B @ R1 @ R22 ) ) ) ) ) ).
% Cinfinite_cprod2
thf(fact_5584_cone__ordLeq__Cnotzero,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( ~ ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R @ ( bNF_Cardinal_czero @ A ) ) @ ( bNF_Wellorder_ordIso @ A @ A ) )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ product_unit @ product_unit ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ product_unit @ product_unit ) ) @ ( set @ ( product_prod @ A @ A ) ) @ bNF_Cardinal_cone @ R ) @ ( bNF_Wellorder_ordLeq @ product_unit @ A ) ) ) ).
% cone_ordLeq_Cnotzero
thf(fact_5585_cexp__mono,axiom,
! [E: $tType,F3: $tType,B: $tType,D: $tType,A: $tType,C: $tType,P1: set @ ( product_prod @ A @ A ),R1: set @ ( product_prod @ B @ B ),P22: set @ ( product_prod @ C @ C ),R22: set @ ( product_prod @ D @ D )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P1 @ R1 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ D @ D ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ D @ D ) ) @ P22 @ R22 ) @ ( bNF_Wellorder_ordLeq @ C @ D ) )
=> ( ( ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ E @ E ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ E @ E ) ) @ P22 @ ( bNF_Cardinal_czero @ E ) ) @ ( bNF_Wellorder_ordIso @ C @ E ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ D @ D ) ) @ ( set @ ( product_prod @ F3 @ F3 ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ D @ D ) ) @ ( set @ ( product_prod @ F3 @ F3 ) ) @ R22 @ ( bNF_Cardinal_czero @ F3 ) ) @ ( bNF_Wellorder_ordIso @ D @ F3 ) ) )
=> ( ( bNF_Ca8970107618336181345der_on @ C @ ( field2 @ C @ P22 ) @ P22 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( C > A ) @ ( C > A ) ) ) @ ( set @ ( product_prod @ ( D > B ) @ ( D > B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( C > A ) @ ( C > A ) ) ) @ ( set @ ( product_prod @ ( D > B ) @ ( D > B ) ) ) @ ( bNF_Cardinal_cexp @ A @ C @ P1 @ P22 ) @ ( bNF_Cardinal_cexp @ B @ D @ R1 @ R22 ) ) @ ( bNF_Wellorder_ordLeq @ ( C > A ) @ ( D > B ) ) ) ) ) ) ) ).
% cexp_mono
thf(fact_5586_cexp__mono2,axiom,
! [D: $tType,E: $tType,B: $tType,C: $tType,A: $tType,P22: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B ),Q2: set @ ( product_prod @ C @ C )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P22 @ R22 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( bNF_Ca8970107618336181345der_on @ C @ ( field2 @ C @ Q2 ) @ Q2 )
=> ( ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ D @ D ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ D @ D ) ) @ P22 @ ( bNF_Cardinal_czero @ D ) ) @ ( bNF_Wellorder_ordIso @ A @ D ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ E @ E ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ E @ E ) ) @ R22 @ ( bNF_Cardinal_czero @ E ) ) @ ( bNF_Wellorder_ordIso @ B @ E ) ) )
=> ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ P22 ) @ P22 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( A > C ) @ ( A > C ) ) ) @ ( set @ ( product_prod @ ( B > C ) @ ( B > C ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( A > C ) @ ( A > C ) ) ) @ ( set @ ( product_prod @ ( B > C ) @ ( B > C ) ) ) @ ( bNF_Cardinal_cexp @ C @ A @ Q2 @ P22 ) @ ( bNF_Cardinal_cexp @ C @ B @ Q2 @ R22 ) ) @ ( bNF_Wellorder_ordLeq @ ( A > C ) @ ( B > C ) ) ) ) ) ) ) ).
% cexp_mono2
thf(fact_5587_cexp__cone,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_unit > A ) @ ( product_unit > A ) ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( product_unit > A ) @ ( product_unit > A ) ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Cardinal_cexp @ A @ product_unit @ R @ bNF_Cardinal_cone ) @ R ) @ ( bNF_Wellorder_ordIso @ ( product_unit > A ) @ A ) ) ) ).
% cexp_cone
thf(fact_5588_cexp__cprod,axiom,
! [A: $tType,C: $tType,B: $tType,R1: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ C @ C ),R33: set @ ( product_prod @ B @ B )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R1 ) @ R1 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( B > C > A ) @ ( B > C > A ) ) ) @ ( set @ ( product_prod @ ( ( product_prod @ C @ B ) > A ) @ ( ( product_prod @ C @ B ) > A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( B > C > A ) @ ( B > C > A ) ) ) @ ( set @ ( product_prod @ ( ( product_prod @ C @ B ) > A ) @ ( ( product_prod @ C @ B ) > A ) ) ) @ ( bNF_Cardinal_cexp @ ( C > A ) @ B @ ( bNF_Cardinal_cexp @ A @ C @ R1 @ R22 ) @ R33 ) @ ( bNF_Cardinal_cexp @ A @ ( product_prod @ C @ B ) @ R1 @ ( bNF_Cardinal_cprod @ C @ B @ R22 @ R33 ) ) ) @ ( bNF_Wellorder_ordIso @ ( B > C > A ) @ ( ( product_prod @ C @ B ) > A ) ) ) ) ).
% cexp_cprod
thf(fact_5589_cone__not__czero,axiom,
! [A: $tType] :
~ ( member @ ( product_prod @ ( set @ ( product_prod @ product_unit @ product_unit ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ product_unit @ product_unit ) ) @ ( set @ ( product_prod @ A @ A ) ) @ bNF_Cardinal_cone @ ( bNF_Cardinal_czero @ A ) ) @ ( bNF_Wellorder_ordIso @ product_unit @ A ) ) ).
% cone_not_czero
thf(fact_5590_cprod__cexp,axiom,
! [C: $tType,B: $tType,A: $tType,R: set @ ( product_prod @ B @ B ),S: set @ ( product_prod @ C @ C ),T2: set @ ( product_prod @ A @ A )] : ( member @ ( product_prod @ ( set @ ( product_prod @ ( A > ( product_prod @ B @ C ) ) @ ( A > ( product_prod @ B @ C ) ) ) ) @ ( set @ ( product_prod @ ( product_prod @ ( A > B ) @ ( A > C ) ) @ ( product_prod @ ( A > B ) @ ( A > C ) ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( A > ( product_prod @ B @ C ) ) @ ( A > ( product_prod @ B @ C ) ) ) ) @ ( set @ ( product_prod @ ( product_prod @ ( A > B ) @ ( A > C ) ) @ ( product_prod @ ( A > B ) @ ( A > C ) ) ) ) @ ( bNF_Cardinal_cexp @ ( product_prod @ B @ C ) @ A @ ( bNF_Cardinal_cprod @ B @ C @ R @ S ) @ T2 ) @ ( bNF_Cardinal_cprod @ ( A > B ) @ ( A > C ) @ ( bNF_Cardinal_cexp @ B @ A @ R @ T2 ) @ ( bNF_Cardinal_cexp @ C @ A @ S @ T2 ) ) ) @ ( bNF_Wellorder_ordIso @ ( A > ( product_prod @ B @ C ) ) @ ( product_prod @ ( A > B ) @ ( A > C ) ) ) ) ).
% cprod_cexp
thf(fact_5591_cexp__cprod__ordLeq,axiom,
! [A: $tType,B: $tType,C: $tType,R1: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B ),R33: set @ ( product_prod @ C @ C )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R1 ) @ R1 )
=> ( ( ( bNF_Ca4139267488887388095finite @ B @ R22 )
& ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R22 ) @ R22 ) )
=> ( ( ~ ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ C @ C ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ C @ C ) ) @ R33 @ ( bNF_Cardinal_czero @ C ) ) @ ( bNF_Wellorder_ordIso @ C @ C ) )
& ( bNF_Ca8970107618336181345der_on @ C @ ( field2 @ C @ R33 ) @ R33 ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R33 @ R22 ) @ ( bNF_Wellorder_ordLeq @ C @ B ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( C > B > A ) @ ( C > B > A ) ) ) @ ( set @ ( product_prod @ ( B > A ) @ ( B > A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( C > B > A ) @ ( C > B > A ) ) ) @ ( set @ ( product_prod @ ( B > A ) @ ( B > A ) ) ) @ ( bNF_Cardinal_cexp @ ( B > A ) @ C @ ( bNF_Cardinal_cexp @ A @ B @ R1 @ R22 ) @ R33 ) @ ( bNF_Cardinal_cexp @ A @ B @ R1 @ R22 ) ) @ ( bNF_Wellorder_ordIso @ ( C > B > A ) @ ( B > A ) ) ) ) ) ) ) ).
% cexp_cprod_ordLeq
thf(fact_5592_cexp__mono_H,axiom,
! [B: $tType,D: $tType,A: $tType,C: $tType,P1: set @ ( product_prod @ A @ A ),R1: set @ ( product_prod @ B @ B ),P22: set @ ( product_prod @ C @ C ),R22: set @ ( product_prod @ D @ D )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P1 @ R1 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ D @ D ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ D @ D ) ) @ P22 @ R22 ) @ ( bNF_Wellorder_ordLeq @ C @ D ) )
=> ( ( ( ( field2 @ C @ P22 )
= ( bot_bot @ ( set @ C ) ) )
=> ( ( field2 @ D @ R22 )
= ( bot_bot @ ( set @ D ) ) ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( C > A ) @ ( C > A ) ) ) @ ( set @ ( product_prod @ ( D > B ) @ ( D > B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( C > A ) @ ( C > A ) ) ) @ ( set @ ( product_prod @ ( D > B ) @ ( D > B ) ) ) @ ( bNF_Cardinal_cexp @ A @ C @ P1 @ P22 ) @ ( bNF_Cardinal_cexp @ B @ D @ R1 @ R22 ) ) @ ( bNF_Wellorder_ordLeq @ ( C > A ) @ ( D > B ) ) ) ) ) ) ).
% cexp_mono'
thf(fact_5593_cexp__mono1,axiom,
! [B: $tType,A: $tType,C: $tType,P1: set @ ( product_prod @ A @ A ),R1: set @ ( product_prod @ B @ B ),Q2: set @ ( product_prod @ C @ C )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P1 @ R1 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( bNF_Ca8970107618336181345der_on @ C @ ( field2 @ C @ Q2 ) @ Q2 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( C > A ) @ ( C > A ) ) ) @ ( set @ ( product_prod @ ( C > B ) @ ( C > B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( C > A ) @ ( C > A ) ) ) @ ( set @ ( product_prod @ ( C > B ) @ ( C > B ) ) ) @ ( bNF_Cardinal_cexp @ A @ C @ P1 @ Q2 ) @ ( bNF_Cardinal_cexp @ B @ C @ R1 @ Q2 ) ) @ ( bNF_Wellorder_ordLeq @ ( C > A ) @ ( C > B ) ) ) ) ) ).
% cexp_mono1
thf(fact_5594_cexp__mono2_H,axiom,
! [B: $tType,C: $tType,A: $tType,P22: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B ),Q2: set @ ( product_prod @ C @ C )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P22 @ R22 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( bNF_Ca8970107618336181345der_on @ C @ ( field2 @ C @ Q2 ) @ Q2 )
=> ( ( ( ( field2 @ A @ P22 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( field2 @ B @ R22 )
= ( bot_bot @ ( set @ B ) ) ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( A > C ) @ ( A > C ) ) ) @ ( set @ ( product_prod @ ( B > C ) @ ( B > C ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( A > C ) @ ( A > C ) ) ) @ ( set @ ( product_prod @ ( B > C ) @ ( B > C ) ) ) @ ( bNF_Cardinal_cexp @ C @ A @ Q2 @ P22 ) @ ( bNF_Cardinal_cexp @ C @ B @ Q2 @ R22 ) ) @ ( bNF_Wellorder_ordLeq @ ( A > C ) @ ( B > C ) ) ) ) ) ) ).
% cexp_mono2'
thf(fact_5595_ordLeq__cexp1,axiom,
! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ A ),Q2: set @ ( product_prod @ B @ B )] :
( ( ~ ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R @ ( bNF_Cardinal_czero @ A ) ) @ ( bNF_Wellorder_ordIso @ A @ A ) )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R ) )
=> ( ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ Q2 ) @ Q2 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ ( A > B ) @ ( A > B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ ( A > B ) @ ( A > B ) ) ) @ Q2 @ ( bNF_Cardinal_cexp @ B @ A @ Q2 @ R ) ) @ ( bNF_Wellorder_ordLeq @ B @ ( A > B ) ) ) ) ) ).
% ordLeq_cexp1
thf(fact_5596_cexp__cong2,axiom,
! [B: $tType,C: $tType,A: $tType,P22: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B ),Q2: set @ ( product_prod @ C @ C )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P22 @ R22 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( ( bNF_Ca8970107618336181345der_on @ C @ ( field2 @ C @ Q2 ) @ Q2 )
=> ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ P22 ) @ P22 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( A > C ) @ ( A > C ) ) ) @ ( set @ ( product_prod @ ( B > C ) @ ( B > C ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( A > C ) @ ( A > C ) ) ) @ ( set @ ( product_prod @ ( B > C ) @ ( B > C ) ) ) @ ( bNF_Cardinal_cexp @ C @ A @ Q2 @ P22 ) @ ( bNF_Cardinal_cexp @ C @ B @ Q2 @ R22 ) ) @ ( bNF_Wellorder_ordIso @ ( A > C ) @ ( B > C ) ) ) ) ) ) ).
% cexp_cong2
thf(fact_5597_cexp__cong1,axiom,
! [B: $tType,A: $tType,C: $tType,P1: set @ ( product_prod @ A @ A ),R1: set @ ( product_prod @ B @ B ),Q2: set @ ( product_prod @ C @ C )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P1 @ R1 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( ( bNF_Ca8970107618336181345der_on @ C @ ( field2 @ C @ Q2 ) @ Q2 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( C > A ) @ ( C > A ) ) ) @ ( set @ ( product_prod @ ( C > B ) @ ( C > B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( C > A ) @ ( C > A ) ) ) @ ( set @ ( product_prod @ ( C > B ) @ ( C > B ) ) ) @ ( bNF_Cardinal_cexp @ A @ C @ P1 @ Q2 ) @ ( bNF_Cardinal_cexp @ B @ C @ R1 @ Q2 ) ) @ ( bNF_Wellorder_ordIso @ ( C > A ) @ ( C > B ) ) ) ) ) ).
% cexp_cong1
thf(fact_5598_cexp__cong,axiom,
! [B: $tType,D: $tType,A: $tType,C: $tType,P1: set @ ( product_prod @ A @ A ),R1: set @ ( product_prod @ B @ B ),P22: set @ ( product_prod @ C @ C ),R22: set @ ( product_prod @ D @ D )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P1 @ R1 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ D @ D ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ D @ D ) ) @ P22 @ R22 ) @ ( bNF_Wellorder_ordIso @ C @ D ) )
=> ( ( bNF_Ca8970107618336181345der_on @ D @ ( field2 @ D @ R22 ) @ R22 )
=> ( ( bNF_Ca8970107618336181345der_on @ C @ ( field2 @ C @ P22 ) @ P22 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( C > A ) @ ( C > A ) ) ) @ ( set @ ( product_prod @ ( D > B ) @ ( D > B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( C > A ) @ ( C > A ) ) ) @ ( set @ ( product_prod @ ( D > B ) @ ( D > B ) ) ) @ ( bNF_Cardinal_cexp @ A @ C @ P1 @ P22 ) @ ( bNF_Cardinal_cexp @ B @ D @ R1 @ R22 ) ) @ ( bNF_Wellorder_ordIso @ ( C > A ) @ ( D > B ) ) ) ) ) ) ) ).
% cexp_cong
thf(fact_5599_cexp__mono2__Cnotzero,axiom,
! [B: $tType,C: $tType,A: $tType,P22: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B ),Q2: set @ ( product_prod @ C @ C )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P22 @ R22 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( bNF_Ca8970107618336181345der_on @ C @ ( field2 @ C @ Q2 ) @ Q2 )
=> ( ( ~ ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ P22 @ ( bNF_Cardinal_czero @ A ) ) @ ( bNF_Wellorder_ordIso @ A @ A ) )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ P22 ) @ P22 ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( A > C ) @ ( A > C ) ) ) @ ( set @ ( product_prod @ ( B > C ) @ ( B > C ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( A > C ) @ ( A > C ) ) ) @ ( set @ ( product_prod @ ( B > C ) @ ( B > C ) ) ) @ ( bNF_Cardinal_cexp @ C @ A @ Q2 @ P22 ) @ ( bNF_Cardinal_cexp @ C @ B @ Q2 @ R22 ) ) @ ( bNF_Wellorder_ordLeq @ ( A > C ) @ ( B > C ) ) ) ) ) ) ).
% cexp_mono2_Cnotzero
thf(fact_5600_ordLeq__cexp2,axiom,
! [A: $tType,B: $tType,Q2: set @ ( product_prod @ A @ A ),R: set @ ( product_prod @ B @ B )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ $o @ $o ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ $o @ $o ) ) @ ( set @ ( product_prod @ A @ A ) ) @ bNF_Cardinal_ctwo @ Q2 ) @ ( bNF_Wellorder_ordLeq @ $o @ A ) )
=> ( ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R ) @ R )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ ( B > A ) @ ( B > A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ ( B > A ) @ ( B > A ) ) ) @ R @ ( bNF_Cardinal_cexp @ A @ B @ Q2 @ R ) ) @ ( bNF_Wellorder_ordLeq @ B @ ( B > A ) ) ) ) ) ).
% ordLeq_cexp2
thf(fact_5601_Cfinite__cexp__Cinfinite,axiom,
! [A: $tType,B: $tType,S: set @ ( product_prod @ A @ A ),T2: set @ ( product_prod @ B @ B )] :
( ( ( bNF_Cardinal_cfinite @ A @ S )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ S ) @ S ) )
=> ( ( ( bNF_Ca4139267488887388095finite @ B @ T2 )
& ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ T2 ) @ T2 ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( B > A ) @ ( B > A ) ) ) @ ( set @ ( product_prod @ ( B > $o ) @ ( B > $o ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( B > A ) @ ( B > A ) ) ) @ ( set @ ( product_prod @ ( B > $o ) @ ( B > $o ) ) ) @ ( bNF_Cardinal_cexp @ A @ B @ S @ T2 ) @ ( bNF_Cardinal_cexp @ $o @ B @ bNF_Cardinal_ctwo @ T2 ) ) @ ( bNF_Wellorder_ordLeq @ ( B > A ) @ ( B > $o ) ) ) ) ) ).
% Cfinite_cexp_Cinfinite
thf(fact_5602_cinfinite__cexp,axiom,
! [A: $tType,B: $tType,Q2: set @ ( product_prod @ A @ A ),R: set @ ( product_prod @ B @ B )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ $o @ $o ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ $o @ $o ) ) @ ( set @ ( product_prod @ A @ A ) ) @ bNF_Cardinal_ctwo @ Q2 ) @ ( bNF_Wellorder_ordLeq @ $o @ A ) )
=> ( ( ( bNF_Ca4139267488887388095finite @ B @ R )
& ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R ) @ R ) )
=> ( bNF_Ca4139267488887388095finite @ ( B > A ) @ ( bNF_Cardinal_cexp @ A @ B @ Q2 @ R ) ) ) ) ).
% cinfinite_cexp
thf(fact_5603_ctwo__Cnotzero,axiom,
( ~ ( member @ ( product_prod @ ( set @ ( product_prod @ $o @ $o ) ) @ ( set @ ( product_prod @ $o @ $o ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ $o @ $o ) ) @ ( set @ ( product_prod @ $o @ $o ) ) @ bNF_Cardinal_ctwo @ ( bNF_Cardinal_czero @ $o ) ) @ ( bNF_Wellorder_ordIso @ $o @ $o ) )
& ( bNF_Ca8970107618336181345der_on @ $o @ ( field2 @ $o @ bNF_Cardinal_ctwo ) @ bNF_Cardinal_ctwo ) ) ).
% ctwo_Cnotzero
thf(fact_5604_ordLess__ctwo__cexp,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( A > $o ) @ ( A > $o ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( A > $o ) @ ( A > $o ) ) ) @ R @ ( bNF_Cardinal_cexp @ $o @ A @ bNF_Cardinal_ctwo @ R ) ) @ ( bNF_We4044943003108391690rdLess @ A @ ( A > $o ) ) ) ) ).
% ordLess_ctwo_cexp
thf(fact_5605_ctwo__not__czero,axiom,
! [A: $tType] :
~ ( member @ ( product_prod @ ( set @ ( product_prod @ $o @ $o ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ $o @ $o ) ) @ ( set @ ( product_prod @ A @ A ) ) @ bNF_Cardinal_ctwo @ ( bNF_Cardinal_czero @ A ) ) @ ( bNF_Wellorder_ordIso @ $o @ A ) ) ).
% ctwo_not_czero
thf(fact_5606_ctwo__ordLess__Cinfinite,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( ( bNF_Ca4139267488887388095finite @ A @ R )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ $o @ $o ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ $o @ $o ) ) @ ( set @ ( product_prod @ A @ A ) ) @ bNF_Cardinal_ctwo @ R ) @ ( bNF_We4044943003108391690rdLess @ $o @ A ) ) ) ).
% ctwo_ordLess_Cinfinite
thf(fact_5607_ctwo__ordLeq__Cinfinite,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( ( bNF_Ca4139267488887388095finite @ A @ R )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ $o @ $o ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ $o @ $o ) ) @ ( set @ ( product_prod @ A @ A ) ) @ bNF_Cardinal_ctwo @ R ) @ ( bNF_Wellorder_ordLeq @ $o @ A ) ) ) ).
% ctwo_ordLeq_Cinfinite
thf(fact_5608_Cinfinite__cexp,axiom,
! [A: $tType,B: $tType,Q2: set @ ( product_prod @ A @ A ),R: set @ ( product_prod @ B @ B )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ $o @ $o ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ $o @ $o ) ) @ ( set @ ( product_prod @ A @ A ) ) @ bNF_Cardinal_ctwo @ Q2 ) @ ( bNF_Wellorder_ordLeq @ $o @ A ) )
=> ( ( ( bNF_Ca4139267488887388095finite @ B @ R )
& ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R ) @ R ) )
=> ( ( bNF_Ca4139267488887388095finite @ ( B > A ) @ ( bNF_Cardinal_cexp @ A @ B @ Q2 @ R ) )
& ( bNF_Ca8970107618336181345der_on @ ( B > A ) @ ( field2 @ ( B > A ) @ ( bNF_Cardinal_cexp @ A @ B @ Q2 @ R ) ) @ ( bNF_Cardinal_cexp @ A @ B @ Q2 @ R ) ) ) ) ) ).
% Cinfinite_cexp
thf(fact_5609_card__of__Plus__Times__aux,axiom,
! [B: $tType,A: $tType,A1: A,A22: A,A5: set @ A,B4: set @ B] :
( ( ( A1 != A22 )
& ( ord_less_eq @ ( set @ A ) @ ( insert2 @ A @ A1 @ ( insert2 @ A @ A22 @ ( bot_bot @ ( set @ A ) ) ) ) @ A5 ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A5 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B4 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ A5 @ B4 ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A5
@ ^ [Uu: A] : B4 ) ) )
@ ( bNF_Wellorder_ordLeq @ ( sum_sum @ A @ B ) @ ( product_prod @ A @ B ) ) ) ) ) ).
% card_of_Plus_Times_aux
thf(fact_5610_natLeq__ordLeq__cinfinite,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( ( bNF_Ca4139267488887388095finite @ A @ R )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ A @ A ) ) @ bNF_Ca8665028551170535155natLeq @ R ) @ ( bNF_Wellorder_ordLeq @ nat @ A ) ) ) ).
% natLeq_ordLeq_cinfinite
thf(fact_5611_cprod__cexp__csum__cexp__Cinfinite,axiom,
! [C: $tType,B: $tType,A: $tType,T2: set @ ( product_prod @ A @ A ),R: set @ ( product_prod @ B @ B ),S: set @ ( product_prod @ C @ C )] :
( ( ( bNF_Ca4139267488887388095finite @ A @ T2 )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ T2 ) @ T2 ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( A > ( product_prod @ B @ C ) ) @ ( A > ( product_prod @ B @ C ) ) ) ) @ ( set @ ( product_prod @ ( A > ( sum_sum @ B @ C ) ) @ ( A > ( sum_sum @ B @ C ) ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( A > ( product_prod @ B @ C ) ) @ ( A > ( product_prod @ B @ C ) ) ) ) @ ( set @ ( product_prod @ ( A > ( sum_sum @ B @ C ) ) @ ( A > ( sum_sum @ B @ C ) ) ) ) @ ( bNF_Cardinal_cexp @ ( product_prod @ B @ C ) @ A @ ( bNF_Cardinal_cprod @ B @ C @ R @ S ) @ T2 ) @ ( bNF_Cardinal_cexp @ ( sum_sum @ B @ C ) @ A @ ( bNF_Cardinal_csum @ B @ C @ R @ S ) @ T2 ) ) @ ( bNF_Wellorder_ordLeq @ ( A > ( product_prod @ B @ C ) ) @ ( A > ( sum_sum @ B @ C ) ) ) ) ) ).
% cprod_cexp_csum_cexp_Cinfinite
thf(fact_5612_Plus__csum,axiom,
! [B: $tType,A: $tType,A5: set @ A,B4: set @ B] : ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ A5 @ B4 ) ) @ ( bNF_Cardinal_csum @ A @ B @ ( bNF_Ca6860139660246222851ard_of @ A @ A5 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B4 ) ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) ).
% Plus_csum
thf(fact_5613_card__of__Plus1,axiom,
! [B: $tType,A: $tType,A5: set @ A,B4: set @ B] : ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A5 ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ A5 @ B4 ) ) ) @ ( bNF_Wellorder_ordLeq @ A @ ( sum_sum @ A @ B ) ) ) ).
% card_of_Plus1
thf(fact_5614_card__of__Plus2,axiom,
! [B: $tType,A: $tType,B4: set @ A,A5: set @ B] : ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ A ) @ ( sum_sum @ B @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ A ) @ ( sum_sum @ B @ A ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ B4 ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ A ) @ ( sum_Plus @ B @ A @ A5 @ B4 ) ) ) @ ( bNF_Wellorder_ordLeq @ A @ ( sum_sum @ B @ A ) ) ) ).
% card_of_Plus2
thf(fact_5615_card__of__Plus__assoc,axiom,
! [C: $tType,B: $tType,A: $tType,A5: set @ A,B4: set @ B,C6: set @ C] : ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ ( sum_sum @ A @ B ) @ C ) @ ( sum_sum @ ( sum_sum @ A @ B ) @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ ( sum_sum @ B @ C ) ) @ ( sum_sum @ A @ ( sum_sum @ B @ C ) ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ ( sum_sum @ A @ B ) @ C ) @ ( sum_sum @ ( sum_sum @ A @ B ) @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ ( sum_sum @ B @ C ) ) @ ( sum_sum @ A @ ( sum_sum @ B @ C ) ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ ( sum_sum @ A @ B ) @ C ) @ ( sum_Plus @ ( sum_sum @ A @ B ) @ C @ ( sum_Plus @ A @ B @ A5 @ B4 ) @ C6 ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ ( sum_sum @ B @ C ) ) @ ( sum_Plus @ A @ ( sum_sum @ B @ C ) @ A5 @ ( sum_Plus @ B @ C @ B4 @ C6 ) ) ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ ( sum_sum @ A @ B ) @ C ) @ ( sum_sum @ A @ ( sum_sum @ B @ C ) ) ) ) ).
% card_of_Plus_assoc
thf(fact_5616_card__of__Plus__commute,axiom,
! [B: $tType,A: $tType,A5: set @ A,B4: set @ B] : ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ A ) @ ( sum_sum @ B @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ A ) @ ( sum_sum @ B @ A ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ A5 @ B4 ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ A ) @ ( sum_Plus @ B @ A @ B4 @ A5 ) ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ B ) @ ( sum_sum @ B @ A ) ) ) ).
% card_of_Plus_commute
thf(fact_5617_card__of__Times__Plus__distrib,axiom,
! [C: $tType,B: $tType,A: $tType,A5: set @ A,B4: set @ B,C6: set @ C] :
( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ ( sum_sum @ B @ C ) ) @ ( product_prod @ A @ ( sum_sum @ B @ C ) ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) ) @ ( sum_sum @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) ) ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ ( sum_sum @ B @ C ) ) @ ( product_prod @ A @ ( sum_sum @ B @ C ) ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) ) @ ( sum_sum @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) ) ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ ( sum_sum @ B @ C ) )
@ ( product_Sigma @ A @ ( sum_sum @ B @ C ) @ A5
@ ^ [Uu: A] : ( sum_Plus @ B @ C @ B4 @ C6 ) ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) )
@ ( sum_Plus @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C )
@ ( product_Sigma @ A @ B @ A5
@ ^ [Uu: A] : B4 )
@ ( product_Sigma @ A @ C @ A5
@ ^ [Uu: A] : C6 ) ) ) )
@ ( bNF_Wellorder_ordIso @ ( product_prod @ A @ ( sum_sum @ B @ C ) ) @ ( sum_sum @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) ) ) ) ).
% card_of_Times_Plus_distrib
thf(fact_5618_card__of__Plus__Times__bool,axiom,
! [A: $tType,A5: set @ A] :
( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ A ) @ ( sum_sum @ A @ A ) ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ $o ) @ ( product_prod @ A @ $o ) ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ A ) @ ( sum_sum @ A @ A ) ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ $o ) @ ( product_prod @ A @ $o ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ A ) @ ( sum_Plus @ A @ A @ A5 @ A5 ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ $o )
@ ( product_Sigma @ A @ $o @ A5
@ ^ [Uu: A] : ( top_top @ ( set @ $o ) ) ) ) )
@ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ A ) @ ( product_prod @ A @ $o ) ) ) ).
% card_of_Plus_Times_bool
thf(fact_5619_ctwo__ordLess__natLeq,axiom,
member @ ( product_prod @ ( set @ ( product_prod @ $o @ $o ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ $o @ $o ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ bNF_Cardinal_ctwo @ bNF_Ca8665028551170535155natLeq ) @ ( bNF_We4044943003108391690rdLess @ $o @ nat ) ).
% ctwo_ordLess_natLeq
thf(fact_5620_csum__assoc,axiom,
! [C: $tType,B: $tType,A: $tType,P1: set @ ( product_prod @ A @ A ),P22: set @ ( product_prod @ B @ B ),P32: set @ ( product_prod @ C @ C )] : ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ ( sum_sum @ A @ B ) @ C ) @ ( sum_sum @ ( sum_sum @ A @ B ) @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ ( sum_sum @ B @ C ) ) @ ( sum_sum @ A @ ( sum_sum @ B @ C ) ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ ( sum_sum @ A @ B ) @ C ) @ ( sum_sum @ ( sum_sum @ A @ B ) @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ ( sum_sum @ B @ C ) ) @ ( sum_sum @ A @ ( sum_sum @ B @ C ) ) ) ) @ ( bNF_Cardinal_csum @ ( sum_sum @ A @ B ) @ C @ ( bNF_Cardinal_csum @ A @ B @ P1 @ P22 ) @ P32 ) @ ( bNF_Cardinal_csum @ A @ ( sum_sum @ B @ C ) @ P1 @ ( bNF_Cardinal_csum @ B @ C @ P22 @ P32 ) ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ ( sum_sum @ A @ B ) @ C ) @ ( sum_sum @ A @ ( sum_sum @ B @ C ) ) ) ) ).
% csum_assoc
thf(fact_5621_csum__csum,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,R1: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B ),R33: set @ ( product_prod @ C @ C ),R42: set @ ( product_prod @ D @ D )] : ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ ( sum_sum @ A @ B ) @ ( sum_sum @ C @ D ) ) @ ( sum_sum @ ( sum_sum @ A @ B ) @ ( sum_sum @ C @ D ) ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ D ) ) @ ( sum_sum @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ D ) ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ ( sum_sum @ A @ B ) @ ( sum_sum @ C @ D ) ) @ ( sum_sum @ ( sum_sum @ A @ B ) @ ( sum_sum @ C @ D ) ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ D ) ) @ ( sum_sum @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ D ) ) ) ) @ ( bNF_Cardinal_csum @ ( sum_sum @ A @ B ) @ ( sum_sum @ C @ D ) @ ( bNF_Cardinal_csum @ A @ B @ R1 @ R22 ) @ ( bNF_Cardinal_csum @ C @ D @ R33 @ R42 ) ) @ ( bNF_Cardinal_csum @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ D ) @ ( bNF_Cardinal_csum @ A @ C @ R1 @ R33 ) @ ( bNF_Cardinal_csum @ B @ D @ R22 @ R42 ) ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ ( sum_sum @ A @ B ) @ ( sum_sum @ C @ D ) ) @ ( sum_sum @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ D ) ) ) ) ).
% csum_csum
thf(fact_5622_csum__com,axiom,
! [B: $tType,A: $tType,P1: set @ ( product_prod @ A @ A ),P22: set @ ( product_prod @ B @ B )] : ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ A ) @ ( sum_sum @ B @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ A ) @ ( sum_sum @ B @ A ) ) ) @ ( bNF_Cardinal_csum @ A @ B @ P1 @ P22 ) @ ( bNF_Cardinal_csum @ B @ A @ P22 @ P1 ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ B ) @ ( sum_sum @ B @ A ) ) ) ).
% csum_com
thf(fact_5623_cprod__csum__distrib1,axiom,
! [C: $tType,B: $tType,A: $tType,R1: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B ),R33: set @ ( product_prod @ C @ C )] : ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) ) @ ( sum_sum @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) ) ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ ( sum_sum @ B @ C ) ) @ ( product_prod @ A @ ( sum_sum @ B @ C ) ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) ) @ ( sum_sum @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) ) ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ ( sum_sum @ B @ C ) ) @ ( product_prod @ A @ ( sum_sum @ B @ C ) ) ) ) @ ( bNF_Cardinal_csum @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) @ ( bNF_Cardinal_cprod @ A @ B @ R1 @ R22 ) @ ( bNF_Cardinal_cprod @ A @ C @ R1 @ R33 ) ) @ ( bNF_Cardinal_cprod @ A @ ( sum_sum @ B @ C ) @ R1 @ ( bNF_Cardinal_csum @ B @ C @ R22 @ R33 ) ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) ) @ ( product_prod @ A @ ( sum_sum @ B @ C ) ) ) ) ).
% cprod_csum_distrib1
thf(fact_5624_natLeq__underS__less,axiom,
! [N: nat] :
( ( order_underS @ nat @ bNF_Ca8665028551170535155natLeq @ N )
= ( collect @ nat
@ ^ [X2: nat] : ( ord_less @ nat @ X2 @ N ) ) ) ).
% natLeq_underS_less
thf(fact_5625_card__of__Plus__empty2,axiom,
! [B: $tType,A: $tType,A5: set @ A] : ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ A ) @ ( sum_sum @ B @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ A ) @ ( sum_sum @ B @ A ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A5 ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ A ) @ ( sum_Plus @ B @ A @ ( bot_bot @ ( set @ B ) ) @ A5 ) ) ) @ ( bNF_Wellorder_ordIso @ A @ ( sum_sum @ B @ A ) ) ) ).
% card_of_Plus_empty2
thf(fact_5626_card__of__Plus__empty1,axiom,
! [B: $tType,A: $tType,A5: set @ A] : ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A5 ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ A5 @ ( bot_bot @ ( set @ B ) ) ) ) ) @ ( bNF_Wellorder_ordIso @ A @ ( sum_sum @ A @ B ) ) ) ).
% card_of_Plus_empty1
thf(fact_5627_card__of__Un__Plus__ordLeq,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] : ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ A ) @ ( sum_sum @ A @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ A ) @ ( sum_sum @ A @ A ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ ( sup_sup @ ( set @ A ) @ A5 @ B4 ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ A ) @ ( sum_Plus @ A @ A @ A5 @ B4 ) ) ) @ ( bNF_Wellorder_ordLeq @ A @ ( sum_sum @ A @ A ) ) ) ).
% card_of_Un_Plus_ordLeq
thf(fact_5628_Card__order__Plus1,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),B4: set @ B] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ R @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ ( field2 @ A @ R ) @ B4 ) ) ) @ ( bNF_Wellorder_ordLeq @ A @ ( sum_sum @ A @ B ) ) ) ) ).
% Card_order_Plus1
thf(fact_5629_Card__order__Plus2,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),A5: set @ B] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ A ) @ ( sum_sum @ B @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ A ) @ ( sum_sum @ B @ A ) ) ) @ R @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ A ) @ ( sum_Plus @ B @ A @ A5 @ ( field2 @ A @ R ) ) ) ) @ ( bNF_Wellorder_ordLeq @ A @ ( sum_sum @ B @ A ) ) ) ) ).
% Card_order_Plus2
thf(fact_5630_Un__csum,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] : ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ A ) @ ( sum_sum @ A @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ A ) @ ( sum_sum @ A @ A ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ ( sup_sup @ ( set @ A ) @ A5 @ B4 ) ) @ ( bNF_Cardinal_csum @ A @ A @ ( bNF_Ca6860139660246222851ard_of @ A @ A5 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ B4 ) ) ) @ ( bNF_Wellorder_ordLeq @ A @ ( sum_sum @ A @ A ) ) ) ).
% Un_csum
thf(fact_5631_ordLeq__csum2,axiom,
! [B: $tType,A: $tType,P22: set @ ( product_prod @ A @ A ),P1: set @ ( product_prod @ B @ B )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ P22 ) @ P22 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ A ) @ ( sum_sum @ B @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ A ) @ ( sum_sum @ B @ A ) ) ) @ P22 @ ( bNF_Cardinal_csum @ B @ A @ P1 @ P22 ) ) @ ( bNF_Wellorder_ordLeq @ A @ ( sum_sum @ B @ A ) ) ) ) ).
% ordLeq_csum2
thf(fact_5632_ordLeq__csum1,axiom,
! [B: $tType,A: $tType,P1: set @ ( product_prod @ A @ A ),P22: set @ ( product_prod @ B @ B )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ P1 ) @ P1 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ P1 @ ( bNF_Cardinal_csum @ A @ B @ P1 @ P22 ) ) @ ( bNF_Wellorder_ordLeq @ A @ ( sum_sum @ A @ B ) ) ) ) ).
% ordLeq_csum1
thf(fact_5633_csum__mono,axiom,
! [D: $tType,B: $tType,C: $tType,A: $tType,P1: set @ ( product_prod @ A @ A ),R1: set @ ( product_prod @ B @ B ),P22: set @ ( product_prod @ C @ C ),R22: set @ ( product_prod @ D @ D )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P1 @ R1 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ D @ D ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ D @ D ) ) @ P22 @ R22 ) @ ( bNF_Wellorder_ordLeq @ C @ D ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ D ) @ ( sum_sum @ B @ D ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ D ) @ ( sum_sum @ B @ D ) ) ) @ ( bNF_Cardinal_csum @ A @ C @ P1 @ P22 ) @ ( bNF_Cardinal_csum @ B @ D @ R1 @ R22 ) ) @ ( bNF_Wellorder_ordLeq @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ D ) ) ) ) ) ).
% csum_mono
thf(fact_5634_csum__mono1,axiom,
! [B: $tType,C: $tType,A: $tType,P1: set @ ( product_prod @ A @ A ),R1: set @ ( product_prod @ B @ B ),Q2: set @ ( product_prod @ C @ C )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P1 @ R1 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) @ ( bNF_Cardinal_csum @ A @ C @ P1 @ Q2 ) @ ( bNF_Cardinal_csum @ B @ C @ R1 @ Q2 ) ) @ ( bNF_Wellorder_ordLeq @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ C ) ) ) ) ).
% csum_mono1
thf(fact_5635_csum__mono2,axiom,
! [B: $tType,A: $tType,C: $tType,P22: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B ),Q2: set @ ( product_prod @ C @ C )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P22 @ R22 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ C @ A ) @ ( sum_sum @ C @ A ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ C @ B ) @ ( sum_sum @ C @ B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ C @ A ) @ ( sum_sum @ C @ A ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ C @ B ) @ ( sum_sum @ C @ B ) ) ) @ ( bNF_Cardinal_csum @ C @ A @ Q2 @ P22 ) @ ( bNF_Cardinal_csum @ C @ B @ Q2 @ R22 ) ) @ ( bNF_Wellorder_ordLeq @ ( sum_sum @ C @ A ) @ ( sum_sum @ C @ B ) ) ) ) ).
% csum_mono2
thf(fact_5636_csum__cong,axiom,
! [D: $tType,B: $tType,C: $tType,A: $tType,P1: set @ ( product_prod @ A @ A ),R1: set @ ( product_prod @ B @ B ),P22: set @ ( product_prod @ C @ C ),R22: set @ ( product_prod @ D @ D )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P1 @ R1 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ D @ D ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ D @ D ) ) @ P22 @ R22 ) @ ( bNF_Wellorder_ordIso @ C @ D ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ D ) @ ( sum_sum @ B @ D ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ D ) @ ( sum_sum @ B @ D ) ) ) @ ( bNF_Cardinal_csum @ A @ C @ P1 @ P22 ) @ ( bNF_Cardinal_csum @ B @ D @ R1 @ R22 ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ D ) ) ) ) ) ).
% csum_cong
thf(fact_5637_csum__cong1,axiom,
! [B: $tType,C: $tType,A: $tType,P1: set @ ( product_prod @ A @ A ),R1: set @ ( product_prod @ B @ B ),Q2: set @ ( product_prod @ C @ C )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P1 @ R1 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) @ ( bNF_Cardinal_csum @ A @ C @ P1 @ Q2 ) @ ( bNF_Cardinal_csum @ B @ C @ R1 @ Q2 ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ C ) ) ) ) ).
% csum_cong1
thf(fact_5638_csum__cong2,axiom,
! [B: $tType,A: $tType,C: $tType,P22: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B ),Q2: set @ ( product_prod @ C @ C )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P22 @ R22 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ C @ A ) @ ( sum_sum @ C @ A ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ C @ B ) @ ( sum_sum @ C @ B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ C @ A ) @ ( sum_sum @ C @ A ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ C @ B ) @ ( sum_sum @ C @ B ) ) ) @ ( bNF_Cardinal_csum @ C @ A @ Q2 @ P22 ) @ ( bNF_Cardinal_csum @ C @ B @ Q2 @ R22 ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ C @ A ) @ ( sum_sum @ C @ B ) ) ) ) ).
% csum_cong2
thf(fact_5639_cprod__csum__cexp,axiom,
! [B: $tType,A: $tType,R1: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B )] : ( member @ ( product_prod @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ ( $o > ( sum_sum @ A @ B ) ) @ ( $o > ( sum_sum @ A @ B ) ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( set @ ( product_prod @ ( $o > ( sum_sum @ A @ B ) ) @ ( $o > ( sum_sum @ A @ B ) ) ) ) @ ( bNF_Cardinal_cprod @ A @ B @ R1 @ R22 ) @ ( bNF_Cardinal_cexp @ ( sum_sum @ A @ B ) @ $o @ ( bNF_Cardinal_csum @ A @ B @ R1 @ R22 ) @ bNF_Cardinal_ctwo ) ) @ ( bNF_Wellorder_ordLeq @ ( product_prod @ A @ B ) @ ( $o > ( sum_sum @ A @ B ) ) ) ) ).
% cprod_csum_cexp
thf(fact_5640_csum__Cfinite__cexp__Cinfinite,axiom,
! [B: $tType,A: $tType,C: $tType,R: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ B @ B ),T2: set @ ( product_prod @ C @ C )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( ( bNF_Cardinal_cfinite @ B @ S )
& ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ S ) @ S ) )
=> ( ( ( bNF_Ca4139267488887388095finite @ C @ T2 )
& ( bNF_Ca8970107618336181345der_on @ C @ ( field2 @ C @ T2 ) @ T2 ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( C > ( sum_sum @ A @ B ) ) @ ( C > ( sum_sum @ A @ B ) ) ) ) @ ( set @ ( product_prod @ ( C > ( sum_sum @ A @ $o ) ) @ ( C > ( sum_sum @ A @ $o ) ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( C > ( sum_sum @ A @ B ) ) @ ( C > ( sum_sum @ A @ B ) ) ) ) @ ( set @ ( product_prod @ ( C > ( sum_sum @ A @ $o ) ) @ ( C > ( sum_sum @ A @ $o ) ) ) ) @ ( bNF_Cardinal_cexp @ ( sum_sum @ A @ B ) @ C @ ( bNF_Cardinal_csum @ A @ B @ R @ S ) @ T2 ) @ ( bNF_Cardinal_cexp @ ( sum_sum @ A @ $o ) @ C @ ( bNF_Cardinal_csum @ A @ $o @ R @ bNF_Cardinal_ctwo ) @ T2 ) ) @ ( bNF_Wellorder_ordLeq @ ( C > ( sum_sum @ A @ B ) ) @ ( C > ( sum_sum @ A @ $o ) ) ) ) ) ) ) ).
% csum_Cfinite_cexp_Cinfinite
thf(fact_5641_card__Plus,axiom,
! [A: $tType,B: $tType,A5: set @ A,B4: set @ B] :
( ( finite_finite2 @ A @ A5 )
=> ( ( finite_finite2 @ B @ B4 )
=> ( ( finite_card @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ A5 @ B4 ) )
= ( plus_plus @ nat @ ( finite_card @ A @ A5 ) @ ( finite_card @ B @ B4 ) ) ) ) ) ).
% card_Plus
thf(fact_5642_ordLeq__Plus__mono2,axiom,
! [B: $tType,A: $tType,C: $tType,R: set @ ( product_prod @ A @ A ),R3: set @ ( product_prod @ B @ B ),A5: set @ C] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R3 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ C @ A ) @ ( sum_sum @ C @ A ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ C @ B ) @ ( sum_sum @ C @ B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ C @ A ) @ ( sum_sum @ C @ A ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ C @ B ) @ ( sum_sum @ C @ B ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ C @ A ) @ ( sum_Plus @ C @ A @ A5 @ ( field2 @ A @ R ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ C @ B ) @ ( sum_Plus @ C @ B @ A5 @ ( field2 @ B @ R3 ) ) ) ) @ ( bNF_Wellorder_ordLeq @ ( sum_sum @ C @ A ) @ ( sum_sum @ C @ B ) ) ) ) ).
% ordLeq_Plus_mono2
thf(fact_5643_ordLeq__Plus__mono1,axiom,
! [B: $tType,C: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R3: set @ ( product_prod @ B @ B ),C6: set @ C] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R3 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ C ) @ ( sum_Plus @ A @ C @ ( field2 @ A @ R ) @ C6 ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ C ) @ ( sum_Plus @ B @ C @ ( field2 @ B @ R3 ) @ C6 ) ) ) @ ( bNF_Wellorder_ordLeq @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ C ) ) ) ) ).
% ordLeq_Plus_mono1
thf(fact_5644_ordLeq__Plus__mono,axiom,
! [D: $tType,B: $tType,C: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R3: set @ ( product_prod @ B @ B ),P4: set @ ( product_prod @ C @ C ),P8: set @ ( product_prod @ D @ D )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R3 ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ D @ D ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ D @ D ) ) @ P4 @ P8 ) @ ( bNF_Wellorder_ordLeq @ C @ D ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ D ) @ ( sum_sum @ B @ D ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ D ) @ ( sum_sum @ B @ D ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ C ) @ ( sum_Plus @ A @ C @ ( field2 @ A @ R ) @ ( field2 @ C @ P4 ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ D ) @ ( sum_Plus @ B @ D @ ( field2 @ B @ R3 ) @ ( field2 @ D @ P8 ) ) ) ) @ ( bNF_Wellorder_ordLeq @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ D ) ) ) ) ) ).
% ordLeq_Plus_mono
thf(fact_5645_ordIso__Plus__cong2,axiom,
! [B: $tType,A: $tType,C: $tType,R: set @ ( product_prod @ A @ A ),R3: set @ ( product_prod @ B @ B ),A5: set @ C] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R3 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ C @ A ) @ ( sum_sum @ C @ A ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ C @ B ) @ ( sum_sum @ C @ B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ C @ A ) @ ( sum_sum @ C @ A ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ C @ B ) @ ( sum_sum @ C @ B ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ C @ A ) @ ( sum_Plus @ C @ A @ A5 @ ( field2 @ A @ R ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ C @ B ) @ ( sum_Plus @ C @ B @ A5 @ ( field2 @ B @ R3 ) ) ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ C @ A ) @ ( sum_sum @ C @ B ) ) ) ) ).
% ordIso_Plus_cong2
thf(fact_5646_ordIso__Plus__cong1,axiom,
! [B: $tType,C: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R3: set @ ( product_prod @ B @ B ),C6: set @ C] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R3 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ C ) @ ( sum_Plus @ A @ C @ ( field2 @ A @ R ) @ C6 ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ C ) @ ( sum_Plus @ B @ C @ ( field2 @ B @ R3 ) @ C6 ) ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ C ) ) ) ) ).
% ordIso_Plus_cong1
thf(fact_5647_ordIso__Plus__cong,axiom,
! [D: $tType,B: $tType,C: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),R3: set @ ( product_prod @ B @ B ),P4: set @ ( product_prod @ C @ C ),P8: set @ ( product_prod @ D @ D )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ R @ R3 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ D @ D ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ D @ D ) ) @ P4 @ P8 ) @ ( bNF_Wellorder_ordIso @ C @ D ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ D ) @ ( sum_sum @ B @ D ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ D ) @ ( sum_sum @ B @ D ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ C ) @ ( sum_Plus @ A @ C @ ( field2 @ A @ R ) @ ( field2 @ C @ P4 ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ D ) @ ( sum_Plus @ B @ D @ ( field2 @ B @ R3 ) @ ( field2 @ D @ P8 ) ) ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ D ) ) ) ) ) ).
% ordIso_Plus_cong
thf(fact_5648_card__of__Plus__mono,axiom,
! [D: $tType,B: $tType,C: $tType,A: $tType,A5: set @ A,B4: set @ B,C6: set @ C,D5: set @ D] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A5 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B4 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ D @ D ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ D @ D ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ C6 ) @ ( bNF_Ca6860139660246222851ard_of @ D @ D5 ) ) @ ( bNF_Wellorder_ordLeq @ C @ D ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ D ) @ ( sum_sum @ B @ D ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ D ) @ ( sum_sum @ B @ D ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ C ) @ ( sum_Plus @ A @ C @ A5 @ C6 ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ D ) @ ( sum_Plus @ B @ D @ B4 @ D5 ) ) ) @ ( bNF_Wellorder_ordLeq @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ D ) ) ) ) ) ).
% card_of_Plus_mono
thf(fact_5649_card__of__Plus__mono1,axiom,
! [B: $tType,C: $tType,A: $tType,A5: set @ A,B4: set @ B,C6: set @ C] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A5 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B4 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ C ) @ ( sum_Plus @ A @ C @ A5 @ C6 ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ C ) @ ( sum_Plus @ B @ C @ B4 @ C6 ) ) ) @ ( bNF_Wellorder_ordLeq @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ C ) ) ) ) ).
% card_of_Plus_mono1
thf(fact_5650_card__of__Plus__mono2,axiom,
! [B: $tType,A: $tType,C: $tType,A5: set @ A,B4: set @ B,C6: set @ C] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A5 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B4 ) ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ C @ A ) @ ( sum_sum @ C @ A ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ C @ B ) @ ( sum_sum @ C @ B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ C @ A ) @ ( sum_sum @ C @ A ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ C @ B ) @ ( sum_sum @ C @ B ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ C @ A ) @ ( sum_Plus @ C @ A @ C6 @ A5 ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ C @ B ) @ ( sum_Plus @ C @ B @ C6 @ B4 ) ) ) @ ( bNF_Wellorder_ordLeq @ ( sum_sum @ C @ A ) @ ( sum_sum @ C @ B ) ) ) ) ).
% card_of_Plus_mono2
thf(fact_5651_card__of__Plus__cong,axiom,
! [D: $tType,B: $tType,C: $tType,A: $tType,A5: set @ A,B4: set @ B,C6: set @ C,D5: set @ D] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A5 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B4 ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ D @ D ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ D @ D ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ C6 ) @ ( bNF_Ca6860139660246222851ard_of @ D @ D5 ) ) @ ( bNF_Wellorder_ordIso @ C @ D ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ D ) @ ( sum_sum @ B @ D ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ D ) @ ( sum_sum @ B @ D ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ C ) @ ( sum_Plus @ A @ C @ A5 @ C6 ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ D ) @ ( sum_Plus @ B @ D @ B4 @ D5 ) ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ D ) ) ) ) ) ).
% card_of_Plus_cong
thf(fact_5652_card__of__Plus__cong1,axiom,
! [B: $tType,C: $tType,A: $tType,A5: set @ A,B4: set @ B,C6: set @ C] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A5 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B4 ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ C ) @ ( sum_Plus @ A @ C @ A5 @ C6 ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ C ) @ ( sum_Plus @ B @ C @ B4 @ C6 ) ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ C ) ) ) ) ).
% card_of_Plus_cong1
thf(fact_5653_card__of__Plus__cong2,axiom,
! [B: $tType,A: $tType,C: $tType,A5: set @ A,B4: set @ B,C6: set @ C] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A5 ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B4 ) ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ C @ A ) @ ( sum_sum @ C @ A ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ C @ B ) @ ( sum_sum @ C @ B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ C @ A ) @ ( sum_sum @ C @ A ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ C @ B ) @ ( sum_sum @ C @ B ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ C @ A ) @ ( sum_Plus @ C @ A @ C6 @ A5 ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ C @ B ) @ ( sum_Plus @ C @ B @ C6 @ B4 ) ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ C @ A ) @ ( sum_sum @ C @ B ) ) ) ) ).
% card_of_Plus_cong2
thf(fact_5654_csum__dup,axiom,
! [A: $tType,C: $tType,B: $tType,R: set @ ( product_prod @ A @ A ),P4: set @ ( product_prod @ B @ B ),P8: set @ ( product_prod @ C @ C )] :
( ( bNF_Ca4139267488887388095finite @ A @ R )
=> ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ A ) @ ( sum_sum @ A @ A ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ A ) @ ( sum_sum @ A @ A ) ) ) @ ( bNF_Cardinal_csum @ B @ C @ P4 @ P8 ) @ ( bNF_Cardinal_csum @ A @ A @ R @ R ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ B @ C ) @ ( sum_sum @ A @ A ) ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Cardinal_csum @ B @ C @ P4 @ P8 ) @ R ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ B @ C ) @ A ) ) ) ) ) ).
% csum_dup
thf(fact_5655_natLeq__def,axiom,
( bNF_Ca8665028551170535155natLeq
= ( collect @ ( product_prod @ nat @ nat ) @ ( product_case_prod @ nat @ nat @ $o @ ( ord_less_eq @ nat ) ) ) ) ).
% natLeq_def
thf(fact_5656_card__Plus__conv__if,axiom,
! [B: $tType,A: $tType,A5: set @ A,B4: set @ B] :
( ( ( ( finite_finite2 @ A @ A5 )
& ( finite_finite2 @ B @ B4 ) )
=> ( ( finite_card @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ A5 @ B4 ) )
= ( plus_plus @ nat @ ( finite_card @ A @ A5 ) @ ( finite_card @ B @ B4 ) ) ) )
& ( ~ ( ( finite_finite2 @ A @ A5 )
& ( finite_finite2 @ B @ B4 ) )
=> ( ( finite_card @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ A5 @ B4 ) )
= ( zero_zero @ nat ) ) ) ) ).
% card_Plus_conv_if
thf(fact_5657_card__of__Plus__infinite,axiom,
! [A: $tType,B: $tType,A5: set @ A,B4: set @ B] :
( ~ ( finite_finite2 @ A @ A5 )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B4 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A5 ) ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ A5 @ B4 ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A5 ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ B ) @ A ) )
& ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ B @ A ) @ ( sum_sum @ B @ A ) ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ B @ A ) @ ( sum_sum @ B @ A ) ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ A ) @ ( sum_Plus @ B @ A @ B4 @ A5 ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A5 ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ B @ A ) @ A ) ) ) ) ) ).
% card_of_Plus_infinite
thf(fact_5658_card__of__Plus__infinite1,axiom,
! [B: $tType,A: $tType,A5: set @ A,B4: set @ B] :
( ~ ( finite_finite2 @ A @ A5 )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B4 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A5 ) ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ A5 @ B4 ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A5 ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ B ) @ A ) ) ) ) ).
% card_of_Plus_infinite1
thf(fact_5659_card__of__Plus__infinite2,axiom,
! [A: $tType,B: $tType,A5: set @ A,B4: set @ B] :
( ~ ( finite_finite2 @ A @ A5 )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ B4 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A5 ) ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ B @ A ) @ ( sum_sum @ B @ A ) ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ B @ A ) @ ( sum_sum @ B @ A ) ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ A ) @ ( sum_Plus @ B @ A @ B4 @ A5 ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A5 ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ B @ A ) @ A ) ) ) ) ).
% card_of_Plus_infinite2
thf(fact_5660_card__of__Plus__ordLess__infinite,axiom,
! [A: $tType,C: $tType,B: $tType,C6: set @ A,A5: set @ B,B4: set @ C] :
( ~ ( finite_finite2 @ A @ C6 )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ A5 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ C6 ) ) @ ( bNF_We4044943003108391690rdLess @ B @ A ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ B4 ) @ ( bNF_Ca6860139660246222851ard_of @ A @ C6 ) ) @ ( bNF_We4044943003108391690rdLess @ C @ A ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ C ) @ ( sum_Plus @ B @ C @ A5 @ B4 ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ C6 ) ) @ ( bNF_We4044943003108391690rdLess @ ( sum_sum @ B @ C ) @ A ) ) ) ) ) ).
% card_of_Plus_ordLess_infinite
thf(fact_5661_finite__iff__ordLess__natLeq,axiom,
! [A: $tType] :
( ( finite_finite2 @ A )
= ( ^ [A8: set @ A] : ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ ( bNF_Ca6860139660246222851ard_of @ A @ A8 ) @ bNF_Ca8665028551170535155natLeq ) @ ( bNF_We4044943003108391690rdLess @ A @ nat ) ) ) ) ).
% finite_iff_ordLess_natLeq
thf(fact_5662_Restr__natLeq2,axiom,
! [N: nat] :
( ( inf_inf @ ( set @ ( product_prod @ nat @ nat ) ) @ bNF_Ca8665028551170535155natLeq
@ ( product_Sigma @ nat @ nat @ ( order_underS @ nat @ bNF_Ca8665028551170535155natLeq @ N )
@ ^ [Uu: nat] : ( order_underS @ nat @ bNF_Ca8665028551170535155natLeq @ N ) ) )
= ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [X2: nat,Y: nat] :
( ( ord_less @ nat @ X2 @ N )
& ( ord_less @ nat @ Y @ N )
& ( ord_less_eq @ nat @ X2 @ Y ) ) ) ) ) ).
% Restr_natLeq2
thf(fact_5663_Restr__natLeq,axiom,
! [N: nat] :
( ( inf_inf @ ( set @ ( product_prod @ nat @ nat ) ) @ bNF_Ca8665028551170535155natLeq
@ ( product_Sigma @ nat @ nat
@ ( collect @ nat
@ ^ [X2: nat] : ( ord_less @ nat @ X2 @ N ) )
@ ^ [Uu: nat] :
( collect @ nat
@ ^ [X2: nat] : ( ord_less @ nat @ X2 @ N ) ) ) )
= ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [X2: nat,Y: nat] :
( ( ord_less @ nat @ X2 @ N )
& ( ord_less @ nat @ Y @ N )
& ( ord_less_eq @ nat @ X2 @ Y ) ) ) ) ) ).
% Restr_natLeq
thf(fact_5664_Card__order__Plus__infinite,axiom,
! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ A ),P4: set @ ( product_prod @ B @ B )] :
( ~ ( finite_finite2 @ A @ ( field2 @ A @ R ) )
=> ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ P4 @ R ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ ( field2 @ A @ R ) @ ( field2 @ B @ P4 ) ) ) @ R ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ B ) @ A ) )
& ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ B @ A ) @ ( sum_sum @ B @ A ) ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ B @ A ) @ ( sum_sum @ B @ A ) ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ A ) @ ( sum_Plus @ B @ A @ ( field2 @ B @ P4 ) @ ( field2 @ A @ R ) ) ) @ R ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ B @ A ) @ A ) ) ) ) ) ) ).
% Card_order_Plus_infinite
thf(fact_5665_csum__cinfinite__bound,axiom,
! [B: $tType,C: $tType,A: $tType,P4: set @ ( product_prod @ A @ A ),R: set @ ( product_prod @ B @ B ),Q2: set @ ( product_prod @ C @ C )] :
( ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P4 @ R ) @ ( bNF_Wellorder_ordLeq @ A @ B ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ B @ B ) ) @ Q2 @ R ) @ ( bNF_Wellorder_ordLeq @ C @ B ) )
=> ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ P4 ) @ P4 )
=> ( ( bNF_Ca8970107618336181345der_on @ C @ ( field2 @ C @ Q2 ) @ Q2 )
=> ( ( ( bNF_Ca4139267488887388095finite @ B @ R )
& ( bNF_Ca8970107618336181345der_on @ B @ ( field2 @ B @ R ) @ R ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ C ) @ ( sum_sum @ A @ C ) ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Cardinal_csum @ A @ C @ P4 @ Q2 ) @ R ) @ ( bNF_Wellorder_ordLeq @ ( sum_sum @ A @ C ) @ B ) ) ) ) ) ) ) ).
% csum_cinfinite_bound
thf(fact_5666_csum__absorb2_H,axiom,
! [A: $tType,B: $tType,R22: set @ ( product_prod @ A @ A ),R1: set @ ( product_prod @ B @ B )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R22 ) @ R22 )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R1 @ R22 ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( ( ( bNF_Ca4139267488887388095finite @ B @ R1 )
| ( bNF_Ca4139267488887388095finite @ A @ R22 ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ B @ A ) @ ( sum_sum @ B @ A ) ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ B @ A ) @ ( sum_sum @ B @ A ) ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Cardinal_csum @ B @ A @ R1 @ R22 ) @ R22 ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ B @ A ) @ A ) ) ) ) ) ).
% csum_absorb2'
thf(fact_5667_csum__absorb1_H,axiom,
! [B: $tType,A: $tType,R22: set @ ( product_prod @ A @ A ),R1: set @ ( product_prod @ B @ B )] :
( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R22 ) @ R22 )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R1 @ R22 ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( ( ( bNF_Ca4139267488887388095finite @ B @ R1 )
| ( bNF_Ca4139267488887388095finite @ A @ R22 ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Cardinal_csum @ A @ B @ R22 @ R1 ) @ R22 ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ B ) @ A ) ) ) ) ) ).
% csum_absorb1'
thf(fact_5668_csum__absorb1,axiom,
! [B: $tType,A: $tType,R22: set @ ( product_prod @ A @ A ),R1: set @ ( product_prod @ B @ B )] :
( ( ( bNF_Ca4139267488887388095finite @ A @ R22 )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R22 ) @ R22 ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R1 @ R22 ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Cardinal_csum @ A @ B @ R22 @ R1 ) @ R22 ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ B ) @ A ) ) ) ) ).
% csum_absorb1
thf(fact_5669_card__of__nat,axiom,
member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ ( bNF_Ca6860139660246222851ard_of @ nat @ ( top_top @ ( set @ nat ) ) ) @ bNF_Ca8665028551170535155natLeq ) @ ( bNF_Wellorder_ordIso @ nat @ nat ) ).
% card_of_nat
thf(fact_5670_csum__Cnotzero1,axiom,
! [A: $tType,B: $tType,R1: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B )] :
( ( ~ ( member @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R1 @ ( bNF_Cardinal_czero @ A ) ) @ ( bNF_Wellorder_ordIso @ A @ A ) )
& ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R1 ) @ R1 ) )
=> ( ~ ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( bNF_Cardinal_csum @ A @ B @ R1 @ R22 ) @ ( bNF_Cardinal_czero @ ( sum_sum @ A @ B ) ) ) @ ( bNF_Wellorder_ordIso @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) )
& ( bNF_Ca8970107618336181345der_on @ ( sum_sum @ A @ B ) @ ( field2 @ ( sum_sum @ A @ B ) @ ( bNF_Cardinal_csum @ A @ B @ R1 @ R22 ) ) @ ( bNF_Cardinal_csum @ A @ B @ R1 @ R22 ) ) ) ) ).
% csum_Cnotzero1
thf(fact_5671_card__of__Plus__Times,axiom,
! [B: $tType,A: $tType,A1: A,A22: A,A5: set @ A,B14: B,B23: B,B4: set @ B] :
( ( ( A1 != A22 )
& ( ord_less_eq @ ( set @ A ) @ ( insert2 @ A @ A1 @ ( insert2 @ A @ A22 @ ( bot_bot @ ( set @ A ) ) ) ) @ A5 ) )
=> ( ( ( B14 != B23 )
& ( ord_less_eq @ ( set @ B ) @ ( insert2 @ B @ B14 @ ( insert2 @ B @ B23 @ ( bot_bot @ ( set @ B ) ) ) ) @ B4 ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) )
@ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ A @ B ) @ ( sum_sum @ A @ B ) ) ) @ ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ A5 @ B4 ) )
@ ( bNF_Ca6860139660246222851ard_of @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A5
@ ^ [Uu: A] : B4 ) ) )
@ ( bNF_Wellorder_ordLeq @ ( sum_sum @ A @ B ) @ ( product_prod @ A @ B ) ) ) ) ) ).
% card_of_Plus_Times
thf(fact_5672_card__of__Field__natLeq,axiom,
member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ ( bNF_Ca6860139660246222851ard_of @ nat @ ( field2 @ nat @ bNF_Ca8665028551170535155natLeq ) ) @ bNF_Ca8665028551170535155natLeq ) @ ( bNF_Wellorder_ordIso @ nat @ nat ) ).
% card_of_Field_natLeq
thf(fact_5673_card__of__Plus__ordLeq__infinite__Field,axiom,
! [A: $tType,C: $tType,B: $tType,R: set @ ( product_prod @ A @ A ),A5: set @ B,B4: set @ C] :
( ~ ( finite_finite2 @ A @ ( field2 @ A @ R ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ A5 ) @ R ) @ ( bNF_Wellorder_ordLeq @ B @ A ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ B4 ) @ R ) @ ( bNF_Wellorder_ordLeq @ C @ A ) )
=> ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ C ) @ ( sum_Plus @ B @ C @ A5 @ B4 ) ) @ R ) @ ( bNF_Wellorder_ordLeq @ ( sum_sum @ B @ C ) @ A ) ) ) ) ) ) ).
% card_of_Plus_ordLeq_infinite_Field
thf(fact_5674_card__of__Plus__ordLess__infinite__Field,axiom,
! [A: $tType,C: $tType,B: $tType,R: set @ ( product_prod @ A @ A ),A5: set @ B,B4: set @ C] :
( ~ ( finite_finite2 @ A @ ( field2 @ A @ R ) )
=> ( ( bNF_Ca8970107618336181345der_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ B @ A5 ) @ R ) @ ( bNF_We4044943003108391690rdLess @ B @ A ) )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ C @ C ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ C @ B4 ) @ R ) @ ( bNF_We4044943003108391690rdLess @ C @ A ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( sum_sum @ B @ C ) @ ( sum_sum @ B @ C ) ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca6860139660246222851ard_of @ ( sum_sum @ B @ C ) @ ( sum_Plus @ B @ C @ A5 @ B4 ) ) @ R ) @ ( bNF_We4044943003108391690rdLess @ ( sum_sum @ B @ C ) @ A ) ) ) ) ) ) ).
% card_of_Plus_ordLess_infinite_Field
thf(fact_5675_infinite__iff__natLeq__ordLeq,axiom,
! [A: $tType,A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
!= ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ A @ A ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ A @ A ) ) @ bNF_Ca8665028551170535155natLeq @ ( bNF_Ca6860139660246222851ard_of @ A @ A5 ) ) @ ( bNF_Wellorder_ordLeq @ nat @ A ) ) ) ).
% infinite_iff_natLeq_ordLeq
thf(fact_5676_filterlim__INF__INF,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,J5: set @ A,I5: set @ B,F: D > C,F6: B > ( filter @ D ),G6: A > ( filter @ C )] :
( ! [M4: A] :
( ( member @ A @ M4 @ J5 )
=> ? [X5: B] :
( ( member @ B @ X5 @ I5 )
& ( ord_less_eq @ ( filter @ C ) @ ( filtermap @ D @ C @ F @ ( F6 @ X5 ) ) @ ( G6 @ M4 ) ) ) )
=> ( filterlim @ D @ C @ F @ ( complete_Inf_Inf @ ( filter @ C ) @ ( image2 @ A @ ( filter @ C ) @ G6 @ J5 ) ) @ ( complete_Inf_Inf @ ( filter @ D ) @ ( image2 @ B @ ( filter @ D ) @ F6 @ I5 ) ) ) ) ).
% filterlim_INF_INF
thf(fact_5677_num__of__integer_Otransfer,axiom,
( bNF_rel_fun @ int @ code_integer @ num @ num @ code_pcr_integer
@ ^ [Y6: num,Z2: num] : Y6 = Z2
@ ( comp @ nat @ num @ int @ num_of_nat @ nat2 )
@ code_num_of_integer ) ).
% num_of_integer.transfer
thf(fact_5678_subset__singleton__iff__Uniq,axiom,
! [A: $tType,A5: set @ A] :
( ( ? [A6: A] : ( ord_less_eq @ ( set @ A ) @ A5 @ ( insert2 @ A @ A6 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( uniq @ A
@ ^ [X2: A] : ( member @ A @ X2 @ A5 ) ) ) ).
% subset_singleton_iff_Uniq
thf(fact_5679_filtermap__id_H,axiom,
! [A: $tType] :
( ( filtermap @ A @ A
@ ^ [X2: A] : X2 )
= ( ^ [F7: filter @ A] : F7 ) ) ).
% filtermap_id'
thf(fact_5680_eventually__filtermap,axiom,
! [A: $tType,B: $tType,P: A > $o,F: B > A,F6: filter @ B] :
( ( eventually @ A @ P @ ( filtermap @ B @ A @ F @ F6 ) )
= ( eventually @ B
@ ^ [X2: B] : ( P @ ( F @ X2 ) )
@ F6 ) ) ).
% eventually_filtermap
thf(fact_5681_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
@ ^ [X2: C] : ( F @ ( G @ X2 ) )
@ F13
@ F24 ) ) ).
% filterlim_filtermap
thf(fact_5682_filtermap__filtermap,axiom,
! [A: $tType,B: $tType,C: $tType,F: B > A,G: C > B,F6: filter @ C] :
( ( filtermap @ B @ A @ F @ ( filtermap @ C @ B @ G @ F6 ) )
= ( filtermap @ C @ A
@ ^ [X2: C] : ( F @ ( G @ X2 ) )
@ F6 ) ) ).
% filtermap_filtermap
thf(fact_5683_filtermap__ident,axiom,
! [A: $tType,F6: filter @ A] :
( ( filtermap @ A @ A
@ ^ [X2: A] : X2
@ F6 )
= F6 ) ).
% filtermap_ident
thf(fact_5684_alt__ex1E_H,axiom,
! [A: $tType,P: A > $o] :
( ? [X5: A] :
( ( P @ X5 )
& ! [Y4: A] :
( ( P @ Y4 )
=> ( Y4 = X5 ) ) )
=> ~ ( ? [X_12: A] : ( P @ X_12 )
=> ~ ( uniq @ A @ P ) ) ) ).
% alt_ex1E'
thf(fact_5685_ex1__iff__ex__Uniq,axiom,
! [A: $tType] :
( ( ex1 @ A )
= ( ^ [P3: A > $o] :
( ? [X9: A] : ( P3 @ X9 )
& ( uniq @ A @ P3 ) ) ) ) ).
% ex1_iff_ex_Uniq
thf(fact_5686_integer_Oid__abs__transfer,axiom,
( bNF_rel_fun @ int @ int @ int @ code_integer
@ ^ [Y6: int,Z2: int] : Y6 = Z2
@ code_pcr_integer
@ ^ [X2: int] : X2
@ code_integer_of_int ) ).
% integer.id_abs_transfer
thf(fact_5687_integer_Orep__transfer,axiom,
( bNF_rel_fun @ int @ code_integer @ int @ int @ code_pcr_integer
@ ^ [Y6: int,Z2: int] : Y6 = Z2
@ ^ [X2: int] : X2
@ code_int_of_integer ) ).
% integer.rep_transfer
thf(fact_5688_inj__on__iff__Uniq,axiom,
! [B: $tType,A: $tType] :
( ( inj_on @ A @ B )
= ( ^ [F2: A > B,A8: set @ A] :
! [X2: A] :
( ( member @ A @ X2 @ A8 )
=> ( uniq @ A
@ ^ [Y: A] :
( ( member @ A @ Y @ A8 )
& ( ( F2 @ X2 )
= ( F2 @ Y ) ) ) ) ) ) ) ).
% inj_on_iff_Uniq
thf(fact_5689_pairwise__disjnt__iff,axiom,
! [A: $tType,A19: set @ ( set @ A )] :
( ( pairwise @ ( set @ A ) @ ( disjnt @ A ) @ A19 )
= ( ! [X2: A] :
( uniq @ ( set @ A )
@ ^ [X9: set @ A] :
( ( member @ ( set @ A ) @ X9 @ A19 )
& ( member @ A @ X2 @ X9 ) ) ) ) ) ).
% pairwise_disjnt_iff
thf(fact_5690_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_5691_the1__equality_H,axiom,
! [A: $tType,P: A > $o,A3: A] :
( ( uniq @ A @ P )
=> ( ( P @ A3 )
=> ( ( the @ A @ P )
= A3 ) ) ) ).
% the1_equality'
thf(fact_5692_dup_Otransfer,axiom,
( bNF_rel_fun @ int @ code_integer @ int @ code_integer @ code_pcr_integer @ code_pcr_integer
@ ^ [K3: int] : ( plus_plus @ int @ K3 @ K3 )
@ code_dup ) ).
% dup.transfer
thf(fact_5693_filtermap__fun__inverse,axiom,
! [B: $tType,A: $tType,G: A > B,F6: filter @ B,G6: filter @ A,F: B > A] :
( ( filterlim @ A @ B @ G @ F6 @ G6 )
=> ( ( filterlim @ B @ A @ F @ G6 @ F6 )
=> ( ( eventually @ A
@ ^ [X2: A] :
( ( F @ ( G @ X2 ) )
= X2 )
@ G6 )
=> ( ( filtermap @ B @ A @ F @ F6 )
= G6 ) ) ) ) ).
% filtermap_fun_inverse
thf(fact_5694_filtermap__SUP,axiom,
! [A: $tType,B: $tType,C: $tType,F: B > A,F6: C > ( filter @ B ),B4: set @ C] :
( ( filtermap @ B @ A @ F @ ( complete_Sup_Sup @ ( filter @ B ) @ ( image2 @ C @ ( filter @ B ) @ F6 @ B4 ) ) )
= ( complete_Sup_Sup @ ( filter @ A )
@ ( image2 @ C @ ( filter @ A )
@ ^ [B5: C] : ( filtermap @ B @ A @ F @ ( F6 @ B5 ) )
@ B4 ) ) ) ).
% filtermap_SUP
thf(fact_5695_filtermap__def,axiom,
! [B: $tType,A: $tType] :
( ( filtermap @ A @ B )
= ( ^ [F2: A > B,F7: filter @ A] :
( abs_filter @ B
@ ^ [P3: B > $o] :
( eventually @ A
@ ^ [X2: A] : ( P3 @ ( F2 @ X2 ) )
@ F7 ) ) ) ) ).
% filtermap_def
thf(fact_5696_sub_Otransfer,axiom,
( bNF_rel_fun @ num @ num @ ( num > int ) @ ( num > code_integer )
@ ^ [Y6: num,Z2: num] : Y6 = Z2
@ ( bNF_rel_fun @ num @ num @ int @ code_integer
@ ^ [Y6: num,Z2: num] : Y6 = Z2
@ code_pcr_integer )
@ ^ [M5: num,N5: num] : ( minus_minus @ int @ ( numeral_numeral @ int @ M5 ) @ ( numeral_numeral @ int @ N5 ) )
@ code_sub ) ).
% sub.transfer
thf(fact_5697_filtermap__INF,axiom,
! [A: $tType,B: $tType,C: $tType,F: B > A,F6: C > ( filter @ B ),B4: set @ C] :
( ord_less_eq @ ( filter @ A ) @ ( filtermap @ B @ A @ F @ ( complete_Inf_Inf @ ( filter @ B ) @ ( image2 @ C @ ( filter @ B ) @ F6 @ B4 ) ) )
@ ( complete_Inf_Inf @ ( filter @ A )
@ ( image2 @ C @ ( filter @ A )
@ ^ [B5: C] : ( filtermap @ B @ A @ F @ ( F6 @ B5 ) )
@ B4 ) ) ) ).
% filtermap_INF
thf(fact_5698_strict__sorted__equal__Uniq,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A] :
( uniq @ ( list @ A )
@ ^ [Xs: list @ A] :
( ( sorted_wrt @ A @ ( ord_less @ A ) @ Xs )
& ( ( set2 @ A @ Xs )
= A5 ) ) ) ) ).
% strict_sorted_equal_Uniq
thf(fact_5699_prod__filter__principal__singleton2,axiom,
! [B: $tType,A: $tType,F6: filter @ A,X: B] :
( ( prod_filter @ A @ B @ F6 @ ( principal @ B @ ( insert2 @ B @ X @ ( bot_bot @ ( set @ B ) ) ) ) )
= ( filtermap @ A @ ( product_prod @ A @ B )
@ ^ [A6: A] : ( product_Pair @ A @ B @ A6 @ X )
@ F6 ) ) ).
% prod_filter_principal_singleton2
thf(fact_5700_Max_Osemilattice__order__set__axioms,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( lattic4895041142388067077er_set @ A @ ( ord_max @ A )
@ ^ [X2: A,Y: A] : ( ord_less_eq @ A @ Y @ X2 )
@ ^ [X2: A,Y: A] : ( ord_less @ A @ Y @ X2 ) ) ) ).
% Max.semilattice_order_set_axioms
thf(fact_5701_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_5702_prod__filter__assoc,axiom,
! [A: $tType,B: $tType,C: $tType,F6: filter @ A,G6: filter @ B,H9: filter @ C] :
( ( prod_filter @ ( product_prod @ A @ B ) @ C @ ( prod_filter @ A @ B @ F6 @ G6 ) @ H9 )
= ( filtermap @ ( product_prod @ A @ ( product_prod @ B @ C ) ) @ ( product_prod @ ( product_prod @ A @ B ) @ C )
@ ( product_case_prod @ A @ ( product_prod @ B @ C ) @ ( product_prod @ ( product_prod @ A @ B ) @ C )
@ ^ [X2: A] :
( product_case_prod @ B @ C @ ( product_prod @ ( product_prod @ A @ B ) @ C )
@ ^ [Y: B] : ( product_Pair @ ( product_prod @ A @ B ) @ C @ ( product_Pair @ A @ B @ X2 @ Y ) ) ) )
@ ( prod_filter @ A @ ( product_prod @ B @ C ) @ F6 @ ( prod_filter @ B @ C @ G6 @ H9 ) ) ) ) ).
% prod_filter_assoc
thf(fact_5703_bounded__quasi__semilattice__set_OF_Ocong,axiom,
! [A: $tType] :
( ( bounde2362111253966948842tice_F @ A )
= ( bounde2362111253966948842tice_F @ A ) ) ).
% bounded_quasi_semilattice_set.F.cong
thf(fact_5704_eventually__prod__same,axiom,
! [A: $tType,P: ( product_prod @ A @ A ) > $o,F6: filter @ A] :
( ( eventually @ ( product_prod @ A @ A ) @ P @ ( prod_filter @ A @ A @ F6 @ F6 ) )
= ( ? [Q7: A > $o] :
( ( eventually @ A @ Q7 @ F6 )
& ! [X2: A,Y: A] :
( ( Q7 @ X2 )
=> ( ( Q7 @ Y )
=> ( P @ ( product_Pair @ A @ A @ X2 @ Y ) ) ) ) ) ) ) ).
% eventually_prod_same
thf(fact_5705_eventually__prod__filter,axiom,
! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,F6: filter @ A,G6: filter @ B] :
( ( eventually @ ( product_prod @ A @ B ) @ P @ ( prod_filter @ A @ B @ F6 @ G6 ) )
= ( ? [Pf: A > $o,Pg: B > $o] :
( ( eventually @ A @ Pf @ F6 )
& ( eventually @ B @ Pg @ G6 )
& ! [X2: A,Y: B] :
( ( Pf @ X2 )
=> ( ( Pg @ Y )
=> ( P @ ( product_Pair @ A @ B @ X2 @ Y ) ) ) ) ) ) ) ).
% eventually_prod_filter
thf(fact_5706_filterlim__Pair,axiom,
! [C: $tType,B: $tType,A: $tType,F: A > B,G6: filter @ B,F6: filter @ A,G: A > C,H9: filter @ C] :
( ( filterlim @ A @ B @ F @ G6 @ F6 )
=> ( ( filterlim @ A @ C @ G @ H9 @ F6 )
=> ( filterlim @ A @ ( product_prod @ B @ C )
@ ^ [X2: A] : ( product_Pair @ B @ C @ ( F @ X2 ) @ ( G @ X2 ) )
@ ( prod_filter @ B @ C @ G6 @ H9 )
@ F6 ) ) ) ).
% filterlim_Pair
thf(fact_5707_filtermap__Pair,axiom,
! [A: $tType,B: $tType,C: $tType,F: C > A,G: C > B,F6: filter @ C] :
( ord_less_eq @ ( filter @ ( product_prod @ A @ B ) )
@ ( filtermap @ C @ ( product_prod @ A @ B )
@ ^ [X2: C] : ( product_Pair @ A @ B @ ( F @ X2 ) @ ( G @ X2 ) )
@ F6 )
@ ( prod_filter @ A @ B @ ( filtermap @ C @ A @ F @ F6 ) @ ( filtermap @ C @ B @ G @ F6 ) ) ) ).
% filtermap_Pair
thf(fact_5708_principal__prod__principal,axiom,
! [B: $tType,A: $tType,A5: set @ A,B4: set @ B] :
( ( prod_filter @ A @ B @ ( principal @ A @ A5 ) @ ( principal @ B @ B4 ) )
= ( principal @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A5
@ ^ [Uu: A] : B4 ) ) ) ).
% principal_prod_principal
thf(fact_5709_prod__filter__def,axiom,
! [B: $tType,A: $tType] :
( ( prod_filter @ A @ B )
= ( ^ [F7: filter @ A,G8: filter @ B] :
( complete_Inf_Inf @ ( filter @ ( product_prod @ A @ B ) )
@ ( image2 @ ( product_prod @ ( A > $o ) @ ( B > $o ) ) @ ( filter @ ( product_prod @ A @ B ) )
@ ( product_case_prod @ ( A > $o ) @ ( B > $o ) @ ( filter @ ( product_prod @ A @ B ) )
@ ^ [P3: A > $o,Q7: B > $o] :
( principal @ ( product_prod @ A @ B )
@ ( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [X2: A,Y: B] :
( ( P3 @ X2 )
& ( Q7 @ Y ) ) ) ) ) )
@ ( collect @ ( product_prod @ ( A > $o ) @ ( B > $o ) )
@ ( product_case_prod @ ( A > $o ) @ ( B > $o ) @ $o
@ ^ [P3: A > $o,Q7: B > $o] :
( ( eventually @ A @ P3 @ F7 )
& ( eventually @ B @ Q7 @ G8 ) ) ) ) ) ) ) ) ).
% prod_filter_def
thf(fact_5710_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 ) ) )
= ( ? [N11: nat] :
! [M5: nat] :
( ( ord_less_eq @ nat @ N11 @ M5 )
=> ! [N5: nat] :
( ( ord_less_eq @ nat @ N11 @ N5 )
=> ( P @ ( product_Pair @ nat @ nat @ N5 @ M5 ) ) ) ) ) ) ).
% eventually_prod_sequentially
thf(fact_5711_eventually__prodI,axiom,
! [A: $tType,B: $tType,P: A > $o,F6: filter @ A,Q: B > $o,G6: filter @ B] :
( ( eventually @ A @ P @ F6 )
=> ( ( eventually @ B @ Q @ G6 )
=> ( eventually @ ( product_prod @ A @ B )
@ ^ [X2: product_prod @ A @ B] :
( ( P @ ( product_fst @ A @ B @ X2 ) )
& ( Q @ ( product_snd @ A @ B @ X2 ) ) )
@ ( prod_filter @ A @ B @ F6 @ G6 ) ) ) ) ).
% eventually_prodI
thf(fact_5712_eventually__prod2,axiom,
! [A: $tType,B: $tType,A5: filter @ A,P: B > $o,B4: filter @ B] :
( ( A5
!= ( bot_bot @ ( filter @ A ) ) )
=> ( ( eventually @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [X2: A] : P )
@ ( prod_filter @ A @ B @ A5 @ B4 ) )
= ( eventually @ B @ P @ B4 ) ) ) ).
% eventually_prod2
thf(fact_5713_eventually__prod1,axiom,
! [A: $tType,B: $tType,B4: filter @ A,P: B > $o,A5: filter @ B] :
( ( B4
!= ( bot_bot @ ( filter @ A ) ) )
=> ( ( eventually @ ( product_prod @ B @ A )
@ ( product_case_prod @ B @ A @ $o
@ ^ [X2: B,Y: A] : ( P @ X2 ) )
@ ( prod_filter @ B @ A @ A5 @ B4 ) )
= ( eventually @ B @ P @ A5 ) ) ) ).
% eventually_prod1
thf(fact_5714_prod__filter__INF,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,I5: set @ A,J5: set @ B,A5: A > ( filter @ C ),B4: B > ( filter @ D )] :
( ( I5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( J5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( prod_filter @ C @ D @ ( complete_Inf_Inf @ ( filter @ C ) @ ( image2 @ A @ ( filter @ C ) @ A5 @ I5 ) ) @ ( complete_Inf_Inf @ ( filter @ D ) @ ( image2 @ B @ ( filter @ D ) @ B4 @ J5 ) ) )
= ( 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 ) @ ( B4 @ J3 ) )
@ J5 ) )
@ I5 ) ) ) ) ) ).
% prod_filter_INF
thf(fact_5715_prod__filter__INF1,axiom,
! [B: $tType,C: $tType,A: $tType,I5: set @ A,A5: A > ( filter @ B ),B4: filter @ C] :
( ( I5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( prod_filter @ B @ C @ ( complete_Inf_Inf @ ( filter @ B ) @ ( image2 @ A @ ( filter @ B ) @ A5 @ I5 ) ) @ B4 )
= ( complete_Inf_Inf @ ( filter @ ( product_prod @ B @ C ) )
@ ( image2 @ A @ ( filter @ ( product_prod @ B @ C ) )
@ ^ [I2: A] : ( prod_filter @ B @ C @ ( A5 @ I2 ) @ B4 )
@ I5 ) ) ) ) ).
% prod_filter_INF1
thf(fact_5716_prod__filter__INF2,axiom,
! [B: $tType,C: $tType,A: $tType,J5: set @ A,A5: filter @ B,B4: A > ( filter @ C )] :
( ( J5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( prod_filter @ B @ C @ A5 @ ( complete_Inf_Inf @ ( filter @ C ) @ ( image2 @ A @ ( filter @ C ) @ B4 @ J5 ) ) )
= ( complete_Inf_Inf @ ( filter @ ( product_prod @ B @ C ) )
@ ( image2 @ A @ ( filter @ ( product_prod @ B @ C ) )
@ ^ [I2: A] : ( prod_filter @ B @ C @ A5 @ ( B4 @ I2 ) )
@ J5 ) ) ) ) ).
% prod_filter_INF2
thf(fact_5717_prod__filter__principal__singleton,axiom,
! [A: $tType,B: $tType,X: A,F6: filter @ B] :
( ( prod_filter @ A @ B @ ( principal @ A @ ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ F6 )
= ( filtermap @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X ) @ F6 ) ) ).
% prod_filter_principal_singleton
thf(fact_5718_Sup__fin_Osemilattice__order__set__axioms,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( lattic4895041142388067077er_set @ A @ ( sup_sup @ A )
@ ^ [X2: A,Y: A] : ( ord_less_eq @ A @ Y @ X2 )
@ ^ [X2: A,Y: A] : ( ord_less @ A @ Y @ X2 ) ) ) ).
% Sup_fin.semilattice_order_set_axioms
thf(fact_5719_bounded__quasi__semilattice__set_Oinsert__remove,axiom,
! [A: $tType,F: A > A > A,Top: A,Bot: A,Normalize: A > A,A3: A,A5: set @ A] :
( ( bounde6485984586167503788ce_set @ A @ F @ Top @ Bot @ Normalize )
=> ( ( bounde2362111253966948842tice_F @ A @ F @ Top @ Bot @ ( insert2 @ A @ A3 @ A5 ) )
= ( F @ A3 @ ( bounde2362111253966948842tice_F @ A @ F @ Top @ Bot @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% bounded_quasi_semilattice_set.insert_remove
thf(fact_5720_bounded__quasi__semilattice__set_Oremove,axiom,
! [A: $tType,F: A > A > A,Top: A,Bot: A,Normalize: A > A,A3: A,A5: set @ A] :
( ( bounde6485984586167503788ce_set @ A @ F @ Top @ Bot @ Normalize )
=> ( ( member @ A @ A3 @ A5 )
=> ( ( bounde2362111253966948842tice_F @ A @ F @ Top @ Bot @ A5 )
= ( F @ A3 @ ( bounde2362111253966948842tice_F @ A @ F @ Top @ Bot @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% bounded_quasi_semilattice_set.remove
thf(fact_5721_arg__min__list_Opelims,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [X: A > B,Xa: list @ A,Y3: A] :
( ( ( arg_min_list @ A @ B @ X @ Xa )
= Y3 )
=> ( ( accp @ ( product_prod @ ( A > B ) @ ( list @ A ) ) @ ( arg_min_list_rel @ A @ B ) @ ( product_Pair @ ( A > B ) @ ( list @ A ) @ X @ Xa ) )
=> ( ! [X4: A] :
( ( Xa
= ( cons @ A @ X4 @ ( nil @ A ) ) )
=> ( ( Y3 = X4 )
=> ~ ( accp @ ( product_prod @ ( A > B ) @ ( list @ A ) ) @ ( arg_min_list_rel @ A @ B ) @ ( product_Pair @ ( A > B ) @ ( list @ A ) @ X @ ( cons @ A @ X4 @ ( nil @ A ) ) ) ) ) )
=> ( ! [X4: A,Y4: A,Zs3: list @ A] :
( ( Xa
= ( cons @ A @ X4 @ ( cons @ A @ Y4 @ Zs3 ) ) )
=> ( ( Y3
= ( if @ A @ ( ord_less_eq @ B @ ( X @ X4 ) @ ( X @ ( arg_min_list @ A @ B @ X @ ( cons @ A @ Y4 @ Zs3 ) ) ) ) @ X4 @ ( arg_min_list @ A @ B @ X @ ( cons @ A @ Y4 @ Zs3 ) ) ) )
=> ~ ( accp @ ( product_prod @ ( A > B ) @ ( list @ A ) ) @ ( arg_min_list_rel @ A @ B ) @ ( product_Pair @ ( A > B ) @ ( list @ A ) @ X @ ( cons @ A @ X4 @ ( cons @ A @ Y4 @ Zs3 ) ) ) ) ) )
=> ~ ( ( Xa
= ( nil @ A ) )
=> ( ( Y3
= ( undefined @ A ) )
=> ~ ( accp @ ( product_prod @ ( A > B ) @ ( list @ A ) ) @ ( arg_min_list_rel @ A @ B ) @ ( product_Pair @ ( A > B ) @ ( list @ A ) @ X @ ( nil @ A ) ) ) ) ) ) ) ) ) ) ).
% arg_min_list.pelims
thf(fact_5722_bounded__quasi__semilattice__set_Oin__idem,axiom,
! [A: $tType,F: A > A > A,Top: A,Bot: A,Normalize: A > A,A3: A,A5: set @ A] :
( ( bounde6485984586167503788ce_set @ A @ F @ Top @ Bot @ Normalize )
=> ( ( member @ A @ A3 @ A5 )
=> ( ( F @ A3 @ ( bounde2362111253966948842tice_F @ A @ F @ Top @ Bot @ A5 ) )
= ( bounde2362111253966948842tice_F @ A @ F @ Top @ Bot @ A5 ) ) ) ) ).
% bounded_quasi_semilattice_set.in_idem
thf(fact_5723_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_5724_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_5725_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_5726_bounded__quasi__semilattice__set_Osubset,axiom,
! [A: $tType,F: A > A > A,Top: A,Bot: A,Normalize: A > A,B4: set @ A,A5: set @ A] :
( ( bounde6485984586167503788ce_set @ A @ F @ Top @ Bot @ Normalize )
=> ( ( ord_less_eq @ ( set @ A ) @ B4 @ A5 )
=> ( ( F @ ( bounde2362111253966948842tice_F @ A @ F @ Top @ Bot @ B4 ) @ ( bounde2362111253966948842tice_F @ A @ F @ Top @ Bot @ A5 ) )
= ( bounde2362111253966948842tice_F @ A @ F @ Top @ Bot @ A5 ) ) ) ) ).
% bounded_quasi_semilattice_set.subset
thf(fact_5727_bounded__quasi__semilattice__set_Oinsert,axiom,
! [A: $tType,F: A > A > A,Top: A,Bot: A,Normalize: A > A,A3: A,A5: set @ A] :
( ( bounde6485984586167503788ce_set @ A @ F @ Top @ Bot @ Normalize )
=> ( ( bounde2362111253966948842tice_F @ A @ F @ Top @ Bot @ ( insert2 @ A @ A3 @ A5 ) )
= ( F @ A3 @ ( bounde2362111253966948842tice_F @ A @ F @ Top @ Bot @ A5 ) ) ) ) ).
% bounded_quasi_semilattice_set.insert
thf(fact_5728_bounded__quasi__semilattice__set_Ounion,axiom,
! [A: $tType,F: A > A > A,Top: A,Bot: A,Normalize: A > A,A5: set @ A,B4: set @ A] :
( ( bounde6485984586167503788ce_set @ A @ F @ Top @ Bot @ Normalize )
=> ( ( bounde2362111253966948842tice_F @ A @ F @ Top @ Bot @ ( sup_sup @ ( set @ A ) @ A5 @ B4 ) )
= ( F @ ( bounde2362111253966948842tice_F @ A @ F @ Top @ Bot @ A5 ) @ ( bounde2362111253966948842tice_F @ A @ F @ Top @ Bot @ B4 ) ) ) ) ).
% bounded_quasi_semilattice_set.union
thf(fact_5729_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_5730_bounded__quasi__semilattice__set_Oset__eq__fold,axiom,
! [A: $tType,F: A > A > A,Top: A,Bot: A,Normalize: A > A,Xs2: list @ A] :
( ( bounde6485984586167503788ce_set @ A @ F @ Top @ Bot @ Normalize )
=> ( ( bounde2362111253966948842tice_F @ A @ F @ Top @ Bot @ ( set2 @ A @ Xs2 ) )
= ( fold @ A @ A @ F @ Xs2 @ Top ) ) ) ).
% bounded_quasi_semilattice_set.set_eq_fold
thf(fact_5731_sorted__wrt_Opelims_I2_J,axiom,
! [A: $tType,X: A > A > $o,Xa: list @ A] :
( ( sorted_wrt @ A @ X @ Xa )
=> ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( sorted_wrt_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ Xa ) )
=> ( ( ( Xa
= ( nil @ A ) )
=> ~ ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( sorted_wrt_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ ( nil @ A ) ) ) )
=> ~ ! [X4: A,Ys5: list @ A] :
( ( Xa
= ( cons @ A @ X4 @ Ys5 ) )
=> ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( sorted_wrt_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ ( cons @ A @ X4 @ Ys5 ) ) )
=> ~ ( ! [Xa2: A] :
( ( member @ A @ Xa2 @ ( set2 @ A @ Ys5 ) )
=> ( X @ X4 @ Xa2 ) )
& ( sorted_wrt @ A @ X @ Ys5 ) ) ) ) ) ) ) ).
% sorted_wrt.pelims(2)
thf(fact_5732_sorted__wrt_Opelims_I1_J,axiom,
! [A: $tType,X: A > A > $o,Xa: list @ A,Y3: $o] :
( ( ( sorted_wrt @ A @ X @ Xa )
= Y3 )
=> ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( sorted_wrt_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ Xa ) )
=> ( ( ( Xa
= ( nil @ A ) )
=> ( Y3
=> ~ ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( sorted_wrt_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ ( nil @ A ) ) ) ) )
=> ~ ! [X4: A,Ys5: list @ A] :
( ( Xa
= ( cons @ A @ X4 @ Ys5 ) )
=> ( ( Y3
= ( ! [Y: A] :
( ( member @ A @ Y @ ( set2 @ A @ Ys5 ) )
=> ( X @ X4 @ Y ) )
& ( sorted_wrt @ A @ X @ Ys5 ) ) )
=> ~ ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( sorted_wrt_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ ( cons @ A @ X4 @ Ys5 ) ) ) ) ) ) ) ) ).
% sorted_wrt.pelims(1)
thf(fact_5733_sorted__wrt_Opelims_I3_J,axiom,
! [A: $tType,X: A > A > $o,Xa: list @ A] :
( ~ ( sorted_wrt @ A @ X @ Xa )
=> ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( sorted_wrt_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ Xa ) )
=> ~ ! [X4: A,Ys5: list @ A] :
( ( Xa
= ( cons @ A @ X4 @ Ys5 ) )
=> ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( sorted_wrt_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ ( cons @ A @ X4 @ Ys5 ) ) )
=> ( ! [Xa4: A] :
( ( member @ A @ Xa4 @ ( set2 @ A @ Ys5 ) )
=> ( X @ X4 @ Xa4 ) )
& ( sorted_wrt @ A @ X @ Ys5 ) ) ) ) ) ) ).
% sorted_wrt.pelims(3)
thf(fact_5734_gen__length__def,axiom,
! [A: $tType] :
( ( gen_length @ A )
= ( ^ [N5: nat,Xs: list @ A] : ( plus_plus @ nat @ N5 @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ).
% gen_length_def
thf(fact_5735_ex__is__arg__min__if__finite,axiom,
! [B: $tType,A: $tType] :
( ( order @ B )
=> ! [S3: set @ A,F: A > B] :
( ( finite_finite2 @ A @ S3 )
=> ( ( S3
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X_12: A] :
( lattic501386751177426532rg_min @ A @ B @ F
@ ^ [X2: A] : ( member @ A @ X2 @ S3 )
@ X_12 ) ) ) ) ).
% ex_is_arg_min_if_finite
thf(fact_5736_relImage__def,axiom,
! [A: $tType,B: $tType] :
( ( bNF_Gr4221423524335903396lImage @ B @ A )
= ( ^ [R6: 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 ) @ R6 ) ) ) ) ) ).
% relImage_def
thf(fact_5737_length__code,axiom,
! [A: $tType] :
( ( size_size @ ( list @ A ) )
= ( gen_length @ A @ ( zero_zero @ nat ) ) ) ).
% length_code
thf(fact_5738_relImage__relInvImage,axiom,
! [B: $tType,A: $tType,R5: set @ ( product_prod @ A @ A ),F: B > A,A5: set @ B] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R5
@ ( 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 @ R5 @ F ) @ F )
= R5 ) ) ).
% relImage_relInvImage
thf(fact_5739_relImage__Gr,axiom,
! [B: $tType,A: $tType,R5: set @ ( product_prod @ A @ A ),A5: set @ A,F: A > B] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R5
@ ( product_Sigma @ A @ A @ A5
@ ^ [Uu: A] : A5 ) )
=> ( ( bNF_Gr4221423524335903396lImage @ A @ B @ R5 @ F )
= ( relcomp @ B @ A @ B @ ( converse @ A @ B @ ( bNF_Gr @ A @ B @ A5 @ F ) ) @ ( relcomp @ A @ A @ B @ R5 @ ( bNF_Gr @ A @ B @ A5 @ F ) ) ) ) ) ).
% relImage_Gr
thf(fact_5740_card__def,axiom,
! [B: $tType] :
( ( finite_card @ B )
= ( finite_folding_F @ B @ nat
@ ^ [Uu: B] : suc
@ ( zero_zero @ nat ) ) ) ).
% card_def
thf(fact_5741_converse__iff,axiom,
! [A: $tType,B: $tType,A3: A,B2: B,R: set @ ( product_prod @ B @ A )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ ( converse @ B @ A @ R ) )
= ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ B2 @ A3 ) @ R ) ) ).
% converse_iff
thf(fact_5742_trancl__converseD,axiom,
! [A: $tType,X: A,Y3: A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ ( transitive_trancl @ A @ ( converse @ A @ A @ R ) ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ ( converse @ A @ A @ ( transitive_trancl @ A @ R ) ) ) ) ).
% trancl_converseD
thf(fact_5743_trancl__converseI,axiom,
! [A: $tType,X: A,Y3: A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ ( converse @ A @ A @ ( transitive_trancl @ A @ R ) ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ ( transitive_trancl @ A @ ( converse @ A @ A @ R ) ) ) ) ).
% trancl_converseI
thf(fact_5744_rtrancl__converseD,axiom,
! [A: $tType,X: A,Y3: A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ ( transitive_rtrancl @ A @ ( converse @ A @ A @ R ) ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X ) @ ( transitive_rtrancl @ A @ R ) ) ) ).
% rtrancl_converseD
thf(fact_5745_rtrancl__converseI,axiom,
! [A: $tType,Y3: A,X: A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X ) @ ( transitive_rtrancl @ A @ R ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ ( transitive_rtrancl @ A @ ( converse @ A @ A @ R ) ) ) ) ).
% rtrancl_converseI
thf(fact_5746_converse_Ocases,axiom,
! [B: $tType,A: $tType,A1: B,A22: A,R: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ A1 @ A22 ) @ ( converse @ A @ B @ R ) )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A22 @ A1 ) @ R ) ) ).
% converse.cases
thf(fact_5747_converse_Osimps,axiom,
! [B: $tType,A: $tType,A1: B,A22: A,R: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ A1 @ A22 ) @ ( converse @ A @ B @ R ) )
= ( ? [A6: A,B5: B] :
( ( A1 = B5 )
& ( A22 = A6 )
& ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A6 @ B5 ) @ R ) ) ) ) ).
% converse.simps
thf(fact_5748_converseD,axiom,
! [A: $tType,B: $tType,A3: A,B2: B,R: set @ ( product_prod @ B @ A )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ ( converse @ B @ A @ R ) )
=> ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ B2 @ A3 ) @ R ) ) ).
% converseD
thf(fact_5749_converseE,axiom,
! [A: $tType,B: $tType,Yx: product_prod @ A @ B,R: set @ ( product_prod @ B @ A )] :
( ( member @ ( product_prod @ A @ B ) @ Yx @ ( converse @ B @ A @ R ) )
=> ~ ! [X4: B,Y4: A] :
( ( Yx
= ( product_Pair @ A @ B @ Y4 @ X4 ) )
=> ~ ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X4 @ Y4 ) @ R ) ) ) ).
% converseE
thf(fact_5750_converseI,axiom,
! [B: $tType,A: $tType,A3: A,B2: B,R: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ R )
=> ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ B2 @ A3 ) @ ( converse @ A @ B @ R ) ) ) ).
% converseI
thf(fact_5751_in__listrel1__converse,axiom,
! [A: $tType,X: list @ A,Y3: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Y3 ) @ ( listrel1 @ A @ ( converse @ A @ A @ R ) ) )
= ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Y3 ) @ ( converse @ ( list @ A ) @ ( list @ A ) @ ( listrel1 @ A @ R ) ) ) ) ).
% in_listrel1_converse
thf(fact_5752_converse__Times,axiom,
! [B: $tType,A: $tType,A5: set @ B,B4: set @ A] :
( ( converse @ B @ A
@ ( product_Sigma @ B @ A @ A5
@ ^ [Uu: B] : B4 ) )
= ( product_Sigma @ A @ B @ B4
@ ^ [Uu: A] : A5 ) ) ).
% converse_Times
thf(fact_5753_converse__unfold,axiom,
! [A: $tType,B: $tType] :
( ( converse @ B @ A )
= ( ^ [R4: set @ ( product_prod @ B @ A )] :
( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [Y: A,X2: B] : ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X2 @ Y ) @ R4 ) ) ) ) ) ).
% converse_unfold
thf(fact_5754_converse__UNION,axiom,
! [B: $tType,A: $tType,C: $tType,R: C > ( set @ ( product_prod @ B @ A ) ),S3: set @ C] :
( ( converse @ B @ A @ ( complete_Sup_Sup @ ( set @ ( product_prod @ B @ A ) ) @ ( image2 @ C @ ( set @ ( product_prod @ B @ A ) ) @ R @ S3 ) ) )
= ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) )
@ ( image2 @ C @ ( set @ ( product_prod @ A @ B ) )
@ ^ [X2: C] : ( converse @ B @ A @ ( R @ X2 ) )
@ S3 ) ) ) ).
% converse_UNION
thf(fact_5755_converse__INTER,axiom,
! [B: $tType,A: $tType,C: $tType,R: C > ( set @ ( product_prod @ B @ A ) ),S3: set @ C] :
( ( converse @ B @ A @ ( complete_Inf_Inf @ ( set @ ( product_prod @ B @ A ) ) @ ( image2 @ C @ ( set @ ( product_prod @ B @ A ) ) @ R @ S3 ) ) )
= ( complete_Inf_Inf @ ( set @ ( product_prod @ A @ B ) )
@ ( image2 @ C @ ( set @ ( product_prod @ A @ B ) )
@ ^ [X2: C] : ( converse @ B @ A @ ( R @ X2 ) )
@ S3 ) ) ) ).
% converse_INTER
thf(fact_5756_relInvImage__Gr,axiom,
! [A: $tType,B: $tType,R5: set @ ( product_prod @ A @ A ),B4: set @ A,A5: set @ B,F: B > A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R5
@ ( product_Sigma @ A @ A @ B4
@ ^ [Uu: A] : B4 ) )
=> ( ( bNF_Gr7122648621184425601vImage @ B @ A @ A5 @ R5 @ F )
= ( relcomp @ B @ A @ B @ ( bNF_Gr @ B @ A @ A5 @ F ) @ ( relcomp @ A @ A @ B @ R5 @ ( converse @ B @ A @ ( bNF_Gr @ B @ A @ A5 @ F ) ) ) ) ) ) ).
% relInvImage_Gr
thf(fact_5757_relInvImage__def,axiom,
! [B: $tType,A: $tType] :
( ( bNF_Gr7122648621184425601vImage @ A @ B )
= ( ^ [A8: set @ A,R6: 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 ) ) @ R6 ) ) ) ) ) ).
% relInvImage_def
thf(fact_5758_irrefl__tranclI,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),X: A] :
( ( ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ ( converse @ A @ A @ R ) @ ( transitive_rtrancl @ A @ R ) )
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ X ) @ ( transitive_trancl @ A @ R ) ) ) ).
% irrefl_tranclI
thf(fact_5759_Image__INT__eq,axiom,
! [A: $tType,B: $tType,C: $tType,R: set @ ( product_prod @ B @ A ),A5: set @ C,B4: 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 ) @ B4 @ A5 ) ) )
= ( complete_Inf_Inf @ ( set @ A )
@ ( image2 @ C @ ( set @ A )
@ ^ [X2: C] : ( image @ B @ A @ R @ ( B4 @ X2 ) )
@ A5 ) ) ) ) ) ).
% Image_INT_eq
thf(fact_5760_trans__wf__iff,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( trans @ A @ R )
=> ( ( wf @ A @ R )
= ( ! [A6: A] :
( wf @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R
@ ( product_Sigma @ A @ A @ ( image @ A @ A @ ( converse @ A @ A @ R ) @ ( insert2 @ A @ A6 @ ( bot_bot @ ( set @ A ) ) ) )
@ ^ [Uu: A] : ( image @ A @ A @ ( converse @ A @ A @ R ) @ ( insert2 @ A @ A6 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ).
% trans_wf_iff
thf(fact_5761_ord__to__filter__compat,axiom,
! [A: $tType,R0: set @ ( product_prod @ A @ A )] :
( bNF_Wellorder_compat @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ A )
@ ( inf_inf @ ( set @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) ) @ ( bNF_We4044943003108391690rdLess @ A @ A )
@ ( product_Sigma @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( image @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( converse @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_We4044943003108391690rdLess @ A @ A ) ) @ ( insert2 @ ( set @ ( product_prod @ A @ A ) ) @ R0 @ ( bot_bot @ ( set @ ( set @ ( product_prod @ A @ A ) ) ) ) ) )
@ ^ [Uu: set @ ( product_prod @ A @ A )] : ( image @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( converse @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_We4044943003108391690rdLess @ A @ A ) ) @ ( insert2 @ ( set @ ( product_prod @ A @ A ) ) @ R0 @ ( bot_bot @ ( set @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ) ) )
@ ( bNF_We413866401316099525erIncl @ A @ R0 )
@ ( bNF_We8469521843155493636filter @ A @ R0 ) ) ).
% ord_to_filter_compat
thf(fact_5762_single__valuedD,axiom,
! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ B ),X: A,Y3: B,Z: B] :
( ( single_valued @ A @ B @ R )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y3 ) @ R )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Z ) @ R )
=> ( Y3 = Z ) ) ) ) ).
% single_valuedD
thf(fact_5763_single__valuedI,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B )] :
( ! [X4: A,Y4: B,Z6: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y4 ) @ R )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Z6 ) @ R )
=> ( Y4 = Z6 ) ) )
=> ( single_valued @ A @ B @ R ) ) ).
% single_valuedI
thf(fact_5764_single__valued__def,axiom,
! [B: $tType,A: $tType] :
( ( single_valued @ A @ B )
= ( ^ [R4: set @ ( product_prod @ A @ B )] :
! [X2: A,Y: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y ) @ R4 )
=> ! [Z3: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Z3 ) @ R4 )
=> ( Y = Z3 ) ) ) ) ) ).
% single_valued_def
thf(fact_5765_transD,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),X: A,Y3: A,Z: A] :
( ( trans @ A @ R )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ R )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z ) @ R )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z ) @ R ) ) ) ) ).
% transD
thf(fact_5766_transE,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),X: A,Y3: A,Z: A] :
( ( trans @ A @ R )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ R )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z ) @ R )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z ) @ R ) ) ) ) ).
% transE
thf(fact_5767_transI,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ! [X4: A,Y4: A,Z6: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y4 ) @ R )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z6 ) @ R )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Z6 ) @ R ) ) )
=> ( trans @ A @ R ) ) ).
% transI
thf(fact_5768_trans__def,axiom,
! [A: $tType] :
( ( trans @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A )] :
! [X2: A,Y: A,Z3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y ) @ R4 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z3 ) @ R4 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Z3 ) @ R4 ) ) ) ) ) ).
% trans_def
thf(fact_5769_lexord__trans,axiom,
! [A: $tType,X: list @ A,Y3: list @ A,R: set @ ( product_prod @ A @ A ),Z: list @ A] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Y3 ) @ ( lexord @ A @ R ) )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Y3 @ Z ) @ ( lexord @ A @ R ) )
=> ( ( trans @ A @ R )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Z ) @ ( lexord @ A @ R ) ) ) ) ) ).
% lexord_trans
thf(fact_5770_compat__def,axiom,
! [A2: $tType,A: $tType] :
( ( bNF_Wellorder_compat @ A @ A2 )
= ( ^ [R4: set @ ( product_prod @ A @ A ),R9: set @ ( product_prod @ A2 @ A2 ),F2: A > A2] :
! [A6: A,B5: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A6 @ B5 ) @ R4 )
=> ( member @ ( product_prod @ A2 @ A2 ) @ ( product_Pair @ A2 @ A2 @ ( F2 @ A6 ) @ ( F2 @ B5 ) ) @ R9 ) ) ) ) ).
% compat_def
thf(fact_5771_lenlex__trans,axiom,
! [A: $tType,X: list @ A,Y3: list @ A,R: set @ ( product_prod @ A @ A ),Z: list @ A] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Y3 ) @ ( lenlex @ A @ R ) )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Y3 @ Z ) @ ( lenlex @ A @ R ) )
=> ( ( trans @ A @ R )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Z ) @ ( lenlex @ A @ R ) ) ) ) ) ).
% lenlex_trans
thf(fact_5772_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_5773_single__valued__confluent,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),X: A,Y3: A,Z: A] :
( ( single_valued @ A @ A @ R )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ ( transitive_rtrancl @ A @ R ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z ) @ ( transitive_rtrancl @ A @ R ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z ) @ ( transitive_rtrancl @ A @ R ) )
| ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z @ Y3 ) @ ( transitive_rtrancl @ A @ R ) ) ) ) ) ) ).
% single_valued_confluent
thf(fact_5774_under__incr,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
( ( trans @ A @ R )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R )
=> ( ord_less_eq @ ( set @ A ) @ ( order_under @ A @ R @ A3 ) @ ( order_under @ A @ R @ B2 ) ) ) ) ).
% under_incr
thf(fact_5775_trans__singleton,axiom,
! [A: $tType,A3: A] : ( trans @ A @ ( insert2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ A3 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ).
% trans_singleton
thf(fact_5776_trans__join,axiom,
! [A: $tType] :
( ( trans @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A )] :
! [X2: product_prod @ A @ A] :
( ( member @ ( product_prod @ A @ A ) @ X2 @ R4 )
=> ( product_case_prod @ A @ A @ $o
@ ^ [Y: A,Y13: A] :
! [Z3: product_prod @ A @ A] :
( ( member @ ( product_prod @ A @ A ) @ Z3 @ R4 )
=> ( product_case_prod @ A @ A @ $o
@ ^ [Y23: A,Aa3: A] :
( ( Y13 = Y23 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Aa3 ) @ R4 ) )
@ Z3 ) )
@ X2 ) ) ) ) ).
% trans_join
thf(fact_5777_wf__finite__segments,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( irrefl @ A @ R )
=> ( ( trans @ A @ R )
=> ( ! [X4: A] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [Y: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X4 ) @ R ) ) )
=> ( wf @ A @ R ) ) ) ) ).
% wf_finite_segments
thf(fact_5778_ord__to__filter__def,axiom,
! [A: $tType] :
( ( bNF_We8469521843155493636filter @ A )
= ( ^ [R02: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ A @ A )] : ( image2 @ A @ A @ ( fChoice @ ( A > A ) @ ( bNF_Wellorder_embed @ A @ A @ R4 @ R02 ) ) @ ( field2 @ A @ R4 ) ) ) ) ).
% ord_to_filter_def
thf(fact_5779_underS__incr,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
( ( trans @ A @ R )
=> ( ( antisym @ A @ R )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R )
=> ( ord_less_eq @ ( set @ A ) @ ( order_underS @ A @ R @ A3 ) @ ( order_underS @ A @ R @ B2 ) ) ) ) ) ).
% underS_incr
thf(fact_5780_less__than__iff,axiom,
! [X: nat,Y3: nat] :
( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ X @ Y3 ) @ less_than )
= ( ord_less @ nat @ X @ Y3 ) ) ).
% less_than_iff
thf(fact_5781_some__equality,axiom,
! [A: $tType,P: A > $o,A3: A] :
( ( P @ A3 )
=> ( ! [X4: A] :
( ( P @ X4 )
=> ( X4 = A3 ) )
=> ( ( fChoice @ A @ P )
= A3 ) ) ) ).
% some_equality
thf(fact_5782_some__eq__trivial,axiom,
! [A: $tType,X: A] :
( ( fChoice @ A
@ ^ [Y: A] : Y = X )
= X ) ).
% some_eq_trivial
thf(fact_5783_some__sym__eq__trivial,axiom,
! [A: $tType,X: A] :
( ( fChoice @ A
@ ( ^ [Y6: A,Z2: A] : Y6 = Z2
@ X ) )
= X ) ).
% some_sym_eq_trivial
thf(fact_5784_antisymD,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
( ( antisym @ A @ R )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ A3 ) @ R )
=> ( A3 = B2 ) ) ) ) ).
% antisymD
thf(fact_5785_antisymI,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ! [X4: A,Y4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y4 ) @ R )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ X4 ) @ R )
=> ( X4 = Y4 ) ) )
=> ( antisym @ A @ R ) ) ).
% antisymI
thf(fact_5786_antisym__def,axiom,
! [A: $tType] :
( ( antisym @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A )] :
! [X2: A,Y: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y ) @ R4 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X2 ) @ R4 )
=> ( X2 = Y ) ) ) ) ) ).
% antisym_def
thf(fact_5787_verit__sko__ex_H,axiom,
! [A: $tType,P: A > $o,A5: $o] :
( ( ( P @ ( fChoice @ A @ P ) )
= A5 )
=> ( ( ? [X9: A] : ( P @ X9 ) )
= A5 ) ) ).
% verit_sko_ex'
thf(fact_5788_verit__sko__forall,axiom,
! [A: $tType] :
( ( ^ [P2: A > $o] :
! [X3: A] : ( P2 @ X3 ) )
= ( ^ [P3: A > $o] :
( P3
@ ( fChoice @ A
@ ^ [X2: A] :
~ ( P3 @ X2 ) ) ) ) ) ).
% verit_sko_forall
thf(fact_5789_verit__sko__forall_H,axiom,
! [A: $tType,P: A > $o,A5: $o] :
( ( ( P
@ ( fChoice @ A
@ ^ [X2: A] :
~ ( P @ X2 ) ) )
= A5 )
=> ( ( ! [X9: A] : ( P @ X9 ) )
= A5 ) ) ).
% verit_sko_forall'
thf(fact_5790_verit__sko__forall_H_H,axiom,
! [A: $tType,B4: A,A5: A,P: A > $o] :
( ( B4 = A5 )
=> ( ( ( fChoice @ A @ P )
= A5 )
= ( ( fChoice @ A @ P )
= B4 ) ) ) ).
% verit_sko_forall''
thf(fact_5791_verit__sko__ex__indirect,axiom,
! [A: $tType,X: A,P: A > $o] :
( ( X
= ( fChoice @ A @ P ) )
=> ( ( ? [X9: A] : ( P @ X9 ) )
= ( P @ X ) ) ) ).
% verit_sko_ex_indirect
thf(fact_5792_verit__sko__ex__indirect2,axiom,
! [A: $tType,X: A,P: A > $o,P9: A > $o] :
( ( X
= ( fChoice @ A @ P ) )
=> ( ! [X4: A] :
( ( P @ X4 )
= ( P9 @ X4 ) )
=> ( ( ? [X9: A] : ( P9 @ X9 ) )
= ( P @ X ) ) ) ) ).
% verit_sko_ex_indirect2
thf(fact_5793_verit__sko__forall__indirect,axiom,
! [A: $tType,X: A,P: A > $o] :
( ( X
= ( fChoice @ A
@ ^ [X2: A] :
~ ( P @ X2 ) ) )
=> ( ( ! [X9: A] : ( P @ X9 ) )
= ( P @ X ) ) ) ).
% verit_sko_forall_indirect
thf(fact_5794_verit__sko__forall__indirect2,axiom,
! [A: $tType,X: A,P: A > $o,P9: A > $o] :
( ( X
= ( fChoice @ A
@ ^ [X2: A] :
~ ( P @ X2 ) ) )
=> ( ! [X4: A] :
( ( P @ X4 )
= ( P9 @ X4 ) )
=> ( ( ! [X9: A] : ( P9 @ X9 ) )
= ( P @ X ) ) ) ) ).
% verit_sko_forall_indirect2
thf(fact_5795_someI2,axiom,
! [A: $tType,P: A > $o,A3: A,Q: A > $o] :
( ( P @ A3 )
=> ( ! [X4: A] :
( ( P @ X4 )
=> ( Q @ X4 ) )
=> ( Q @ ( fChoice @ A @ P ) ) ) ) ).
% someI2
thf(fact_5796_someI__ex,axiom,
! [A: $tType,P: A > $o] :
( ? [X_1: A] : ( P @ X_1 )
=> ( P @ ( fChoice @ A @ P ) ) ) ).
% someI_ex
thf(fact_5797_someI2__ex,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ? [X_1: A] : ( P @ X_1 )
=> ( ! [X4: A] :
( ( P @ X4 )
=> ( Q @ X4 ) )
=> ( Q @ ( fChoice @ A @ P ) ) ) ) ).
% someI2_ex
thf(fact_5798_someI2__bex,axiom,
! [A: $tType,A5: set @ A,P: A > $o,Q: A > $o] :
( ? [X5: A] :
( ( member @ A @ X5 @ A5 )
& ( P @ X5 ) )
=> ( ! [X4: A] :
( ( ( member @ A @ X4 @ A5 )
& ( P @ X4 ) )
=> ( Q @ X4 ) )
=> ( Q
@ ( fChoice @ A
@ ^ [X2: A] :
( ( member @ A @ X2 @ A5 )
& ( P @ X2 ) ) ) ) ) ) ).
% someI2_bex
thf(fact_5799_some__eq__ex,axiom,
! [A: $tType,P: A > $o] :
( ( P @ ( fChoice @ A @ P ) )
= ( ? [X9: A] : ( P @ X9 ) ) ) ).
% some_eq_ex
thf(fact_5800_some1__equality,axiom,
! [A: $tType,P: A > $o,A3: A] :
( ? [X5: A] :
( ( P @ X5 )
& ! [Y4: A] :
( ( P @ Y4 )
=> ( Y4 = X5 ) ) )
=> ( ( P @ A3 )
=> ( ( fChoice @ A @ P )
= A3 ) ) ) ).
% some1_equality
thf(fact_5801_some__in__eq,axiom,
! [A: $tType,A5: set @ A] :
( ( member @ A
@ ( fChoice @ A
@ ^ [X2: A] : ( member @ A @ X2 @ A5 ) )
@ A5 )
= ( A5
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% some_in_eq
thf(fact_5802_pick__middlep__def,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( bNF_pick_middlep @ B @ A @ C )
= ( ^ [P3: B > A > $o,Q7: A > C > $o,A6: B,C4: C] :
( fChoice @ A
@ ^ [B5: A] :
( ( P3 @ A6 @ B5 )
& ( Q7 @ B5 @ C4 ) ) ) ) ) ).
% pick_middlep_def
thf(fact_5803_equiv__Eps__preserves,axiom,
! [A: $tType,A5: set @ A,R: set @ ( product_prod @ A @ A ),X7: set @ A] :
( ( equiv_equiv @ A @ A5 @ R )
=> ( ( member @ ( set @ A ) @ X7 @ ( equiv_quotient @ A @ A5 @ R ) )
=> ( member @ A
@ ( fChoice @ A
@ ^ [X2: A] : ( member @ A @ X2 @ X7 ) )
@ A5 ) ) ) ).
% equiv_Eps_preserves
thf(fact_5804_equiv__Eps__in,axiom,
! [A: $tType,A5: set @ A,R: set @ ( product_prod @ A @ A ),X7: set @ A] :
( ( equiv_equiv @ A @ A5 @ R )
=> ( ( member @ ( set @ A ) @ X7 @ ( equiv_quotient @ A @ A5 @ R ) )
=> ( member @ A
@ ( fChoice @ A
@ ^ [X2: A] : ( member @ A @ X2 @ X7 ) )
@ X7 ) ) ) ).
% equiv_Eps_in
thf(fact_5805_antisym__Restr,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A5: set @ A] :
( ( antisym @ A @ R )
=> ( antisym @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R
@ ( product_Sigma @ A @ A @ A5
@ ^ [Uu: A] : A5 ) ) ) ) ).
% antisym_Restr
thf(fact_5806_arg__min__def,axiom,
! [A: $tType,B: $tType] :
( ( ord @ A )
=> ( ( lattices_ord_arg_min @ B @ A )
= ( ^ [F2: B > A,P3: B > $o] : ( fChoice @ B @ ( lattic501386751177426532rg_min @ B @ A @ F2 @ P3 ) ) ) ) ) ).
% arg_min_def
thf(fact_5807_toCard__def,axiom,
! [B: $tType,A: $tType] :
( ( bNF_Greatest_toCard @ A @ B )
= ( ^ [A8: set @ A,R4: set @ ( product_prod @ B @ B )] : ( fChoice @ ( A > B ) @ ( bNF_Gr1419584066657907630d_pred @ A @ B @ A8 @ R4 ) ) ) ) ).
% toCard_def
thf(fact_5808_proj__Eps,axiom,
! [A: $tType,A5: set @ A,R: set @ ( product_prod @ A @ A ),X7: set @ A] :
( ( equiv_equiv @ A @ A5 @ R )
=> ( ( member @ ( set @ A ) @ X7 @ ( equiv_quotient @ A @ A5 @ R ) )
=> ( ( equiv_proj @ A @ A @ R
@ ( fChoice @ A
@ ^ [X2: A] : ( member @ A @ X2 @ X7 ) ) )
= X7 ) ) ) ).
% proj_Eps
thf(fact_5809_fromCard__def,axiom,
! [B: $tType,A: $tType] :
( ( bNF_Gr5436034075474128252omCard @ A @ B )
= ( ^ [A8: set @ A,R4: set @ ( product_prod @ B @ B ),K3: B] :
( fChoice @ A
@ ^ [B5: A] :
( ( member @ A @ B5 @ A8 )
& ( ( bNF_Greatest_toCard @ A @ B @ A8 @ R4 @ B5 )
= K3 ) ) ) ) ) ).
% fromCard_def
thf(fact_5810_arg__min__SOME__Min,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [S3: set @ A,F: A > B] :
( ( finite_finite2 @ A @ S3 )
=> ( ( lattic7623131987881927897min_on @ A @ B @ F @ S3 )
= ( fChoice @ A
@ ^ [Y: A] :
( ( member @ A @ Y @ S3 )
& ( ( F @ Y )
= ( lattic643756798350308766er_Min @ B @ ( image2 @ A @ B @ F @ S3 ) ) ) ) ) ) ) ) ).
% arg_min_SOME_Min
thf(fact_5811_lenlex__def,axiom,
! [A: $tType] :
( ( lenlex @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A )] :
( inv_image @ ( product_prod @ nat @ ( list @ A ) ) @ ( list @ A ) @ ( lex_prod @ nat @ ( list @ A ) @ less_than @ ( lex @ A @ R4 ) )
@ ^ [Xs: list @ A] : ( product_Pair @ nat @ ( list @ A ) @ ( size_size @ ( list @ A ) @ Xs ) @ Xs ) ) ) ) ).
% lenlex_def
thf(fact_5812_mlex__prod__def,axiom,
! [A: $tType] :
( ( mlex_prod @ A )
= ( ^ [F2: A > nat,R6: set @ ( product_prod @ A @ A )] :
( inv_image @ ( product_prod @ nat @ A ) @ A @ ( lex_prod @ nat @ A @ less_than @ R6 )
@ ^ [X2: A] : ( product_Pair @ nat @ A @ ( F2 @ X2 ) @ X2 ) ) ) ) ).
% mlex_prod_def
thf(fact_5813_in__inv__image,axiom,
! [A: $tType,B: $tType,X: A,Y3: A,R: set @ ( product_prod @ B @ B ),F: A > B] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ ( inv_image @ B @ A @ R @ F ) )
= ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F @ X ) @ ( F @ Y3 ) ) @ R ) ) ).
% in_inv_image
thf(fact_5814_Eps__case__prod__eq,axiom,
! [A: $tType,B: $tType,X: A,Y3: B] :
( ( fChoice @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [X8: A,Y7: B] :
( ( X = X8 )
& ( Y3 = Y7 ) ) ) )
= ( product_Pair @ A @ B @ X @ Y3 ) ) ).
% Eps_case_prod_eq
thf(fact_5815_split__paired__Eps,axiom,
! [B: $tType,A: $tType] :
( ( fChoice @ ( product_prod @ A @ B ) )
= ( ^ [P3: ( product_prod @ A @ B ) > $o] :
( fChoice @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [A6: A,B5: B] : ( P3 @ ( product_Pair @ A @ B @ A6 @ B5 ) ) ) ) ) ) ).
% split_paired_Eps
thf(fact_5816_card__of__def,axiom,
! [A: $tType] :
( ( bNF_Ca6860139660246222851ard_of @ A )
= ( ^ [A8: set @ A] : ( fChoice @ ( set @ ( product_prod @ A @ A ) ) @ ( bNF_Ca8970107618336181345der_on @ A @ A8 ) ) ) ) ).
% card_of_def
thf(fact_5817_cardSuc__def,axiom,
! [A: $tType] :
( ( bNF_Ca8387033319878233205ardSuc @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A )] : ( fChoice @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( bNF_Ca6246979054910435723ardSuc @ A @ R4 ) ) ) ) ).
% cardSuc_def
thf(fact_5818_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_5819_inv__image__def,axiom,
! [A: $tType,B: $tType] :
( ( inv_image @ B @ A )
= ( ^ [R4: set @ ( product_prod @ B @ B ),F2: A > B] :
( collect @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ^ [X2: A,Y: A] : ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F2 @ X2 ) @ ( F2 @ Y ) ) @ R4 ) ) ) ) ) ).
% inv_image_def
thf(fact_5820_arg__min__on__def,axiom,
! [A: $tType,B: $tType] :
( ( ord @ A )
=> ( ( lattic7623131987881927897min_on @ B @ A )
= ( ^ [F2: B > A,S8: set @ B] :
( lattices_ord_arg_min @ B @ A @ F2
@ ^ [X2: B] : ( member @ B @ X2 @ S8 ) ) ) ) ) ).
% arg_min_on_def
thf(fact_5821_rp__inv__image__def,axiom,
! [B: $tType,A: $tType] :
( ( fun_rp_inv_image @ A @ B )
= ( product_case_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( ( B > A ) > ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) ) )
@ ^ [R6: set @ ( product_prod @ A @ A ),S8: set @ ( product_prod @ A @ A ),F2: B > A] : ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( inv_image @ A @ B @ R6 @ F2 ) @ ( inv_image @ A @ B @ S8 @ F2 ) ) ) ) ).
% rp_inv_image_def
thf(fact_5822_measures__def,axiom,
! [A: $tType] :
( ( measures @ A )
= ( ^ [Fs2: list @ ( A > nat )] :
( inv_image @ ( list @ nat ) @ A @ ( lex @ nat @ less_than )
@ ^ [A6: A] :
( map @ ( A > nat ) @ nat
@ ^ [F2: A > nat] : ( F2 @ A6 )
@ Fs2 ) ) ) ) ).
% measures_def
thf(fact_5823_rel__filter_Ocases,axiom,
! [A: $tType,B: $tType,R5: A > B > $o,F6: filter @ A,G6: filter @ B] :
( ( rel_filter @ A @ B @ R5 @ F6 @ G6 )
=> ~ ! [Z10: filter @ ( product_prod @ A @ B )] :
( ( eventually @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R5 ) @ Z10 )
=> ( ( ( map_filter_on @ ( product_prod @ A @ B ) @ A @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R5 ) ) @ ( product_fst @ A @ B ) @ Z10 )
= F6 )
=> ( ( map_filter_on @ ( product_prod @ A @ B ) @ B @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R5 ) ) @ ( product_snd @ A @ B ) @ Z10 )
!= G6 ) ) ) ) ).
% rel_filter.cases
thf(fact_5824_rel__filter_Ointros,axiom,
! [A: $tType,B: $tType,R5: A > B > $o,Z5: filter @ ( product_prod @ A @ B ),F6: filter @ A,G6: filter @ B] :
( ( eventually @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R5 ) @ Z5 )
=> ( ( ( map_filter_on @ ( product_prod @ A @ B ) @ A @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R5 ) ) @ ( product_fst @ A @ B ) @ Z5 )
= F6 )
=> ( ( ( map_filter_on @ ( product_prod @ A @ B ) @ B @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R5 ) ) @ ( product_snd @ A @ B ) @ Z5 )
= G6 )
=> ( rel_filter @ A @ B @ R5 @ F6 @ G6 ) ) ) ) ).
% rel_filter.intros
thf(fact_5825_rel__filter_Osimps,axiom,
! [B: $tType,A: $tType] :
( ( rel_filter @ A @ B )
= ( ^ [R6: A > B > $o,F7: filter @ A,G8: filter @ B] :
? [Z9: filter @ ( product_prod @ A @ B )] :
( ( eventually @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R6 ) @ Z9 )
& ( ( map_filter_on @ ( product_prod @ A @ B ) @ A @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R6 ) ) @ ( product_fst @ A @ B ) @ Z9 )
= F7 )
& ( ( map_filter_on @ ( product_prod @ A @ B ) @ B @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R6 ) ) @ ( product_snd @ A @ B ) @ Z9 )
= G8 ) ) ) ) ).
% rel_filter.simps
thf(fact_5826_pred__on_Onot__maxchain__Some,axiom,
! [A: $tType,A5: set @ A,P: A > A > $o,C6: set @ A] :
( ( pred_chain @ A @ A5 @ P @ C6 )
=> ( ~ ( pred_maxchain @ A @ A5 @ P @ C6 )
=> ( ( pred_chain @ A @ A5 @ P
@ ( fChoice @ ( set @ A )
@ ^ [D6: set @ A] :
( ( pred_chain @ A @ A5 @ P @ D6 )
& ( ord_less @ ( set @ A ) @ C6 @ D6 ) ) ) )
& ( ord_less @ ( set @ A ) @ C6
@ ( fChoice @ ( set @ A )
@ ^ [D6: set @ A] :
( ( pred_chain @ A @ A5 @ P @ D6 )
& ( ord_less @ ( set @ A ) @ C6 @ D6 ) ) ) ) ) ) ) ).
% pred_on.not_maxchain_Some
thf(fact_5827_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
@ ^ [X2: D] : ( P @ ( G @ X2 ) ) )
@ ( comp @ C @ B @ D @ ( comp @ A @ B @ C @ Rep @ H ) @ G ) ) ) ) ).
% type_copy_vimage2p_Grp_Abs
thf(fact_5828_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
@ ^ [X2: C] : ( P @ ( F @ X2 ) ) )
@ ( comp @ D @ A @ C @ ( comp @ B @ A @ D @ Abs @ H ) @ F ) ) ) ) ).
% type_copy_vimage2p_Grp_Rep
thf(fact_5829_Grp__def,axiom,
! [B: $tType,A: $tType] :
( ( bNF_Grp @ A @ B )
= ( ^ [A8: set @ A,F2: A > B,A6: A,B5: B] :
( ( B5
= ( F2 @ A6 ) )
& ( member @ A @ A6 @ A8 ) ) ) ) ).
% Grp_def
thf(fact_5830_subset_Onot__maxchain__Some,axiom,
! [A: $tType,A5: set @ ( set @ A ),C6: set @ ( set @ A )] :
( ( pred_chain @ ( set @ A ) @ A5 @ ( ord_less @ ( set @ A ) ) @ C6 )
=> ( ~ ( pred_maxchain @ ( set @ A ) @ A5 @ ( ord_less @ ( set @ A ) ) @ C6 )
=> ( ( pred_chain @ ( set @ A ) @ A5 @ ( ord_less @ ( set @ A ) )
@ ( fChoice @ ( set @ ( set @ A ) )
@ ^ [D6: set @ ( set @ A )] :
( ( pred_chain @ ( set @ A ) @ A5 @ ( ord_less @ ( set @ A ) ) @ D6 )
& ( ord_less @ ( set @ ( set @ A ) ) @ C6 @ D6 ) ) ) )
& ( ord_less @ ( set @ ( set @ A ) ) @ C6
@ ( fChoice @ ( set @ ( set @ A ) )
@ ^ [D6: set @ ( set @ A )] :
( ( pred_chain @ ( set @ A ) @ A5 @ ( ord_less @ ( set @ A ) ) @ D6 )
& ( ord_less @ ( set @ ( set @ A ) ) @ C6 @ D6 ) ) ) ) ) ) ) ).
% subset.not_maxchain_Some
thf(fact_5831_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 )
@ ^ [X2: list @ A] : ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ X2 ) @ A5 ) )
@ ( map @ A @ B @ F ) ) ) ).
% list.rel_Grp
thf(fact_5832_fun_Orel__Grp,axiom,
! [D: $tType,B: $tType,A: $tType,A5: set @ A,F: A > B] :
( ( bNF_rel_fun @ D @ D @ A @ B
@ ^ [Y6: D,Z2: D] : Y6 = Z2
@ ( bNF_Grp @ A @ B @ A5 @ F ) )
= ( bNF_Grp @ ( D > A ) @ ( D > B )
@ ( collect @ ( D > A )
@ ^ [X2: D > A] : ( ord_less_eq @ ( set @ A ) @ ( image2 @ D @ A @ X2 @ ( top_top @ ( set @ D ) ) ) @ A5 ) )
@ ( comp @ A @ B @ D @ F ) ) ) ).
% fun.rel_Grp
thf(fact_5833_eq__le__Grp__id__iff,axiom,
! [A: $tType,R5: A > $o] :
( ( ord_less_eq @ ( A > A > $o )
@ ^ [Y6: A,Z2: A] : Y6 = Z2
@ ( bNF_Grp @ A @ A @ ( collect @ A @ R5 ) @ ( id @ A ) ) )
= ( ! [X9: A] : ( R5 @ X9 ) ) ) ).
% eq_le_Grp_id_iff
thf(fact_5834_Collect__case__prod__Grp__eqD,axiom,
! [B: $tType,A: $tType,Z: product_prod @ A @ B,A5: set @ A,F: A > B] :
( ( member @ ( product_prod @ A @ B ) @ Z @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ ( bNF_Grp @ A @ B @ A5 @ F ) ) ) )
=> ( ( comp @ A @ B @ ( product_prod @ A @ B ) @ F @ ( product_fst @ A @ B ) @ Z )
= ( product_snd @ A @ B @ Z ) ) ) ).
% Collect_case_prod_Grp_eqD
thf(fact_5835_pred__on_Osuc__def,axiom,
! [A: $tType] :
( ( pred_suc @ A )
= ( ^ [A8: set @ A,P3: A > A > $o,C7: set @ A] :
( if @ ( set @ A )
@ ( ~ ( pred_chain @ A @ A8 @ P3 @ C7 )
| ( pred_maxchain @ A @ A8 @ P3 @ C7 ) )
@ C7
@ ( fChoice @ ( set @ A )
@ ^ [D6: set @ A] :
( ( pred_chain @ A @ A8 @ P3 @ D6 )
& ( ord_less @ ( set @ A ) @ C7 @ D6 ) ) ) ) ) ) ).
% pred_on.suc_def
thf(fact_5836_flip__pred,axiom,
! [A: $tType,B: $tType,A5: set @ ( product_prod @ A @ B ),R5: 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 @ R5 ) ) ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ B @ A ) )
@ ( image2 @ ( product_prod @ A @ B ) @ ( product_prod @ B @ A )
@ ( product_case_prod @ A @ B @ ( product_prod @ B @ A )
@ ^ [X2: A,Y: B] : ( product_Pair @ B @ A @ Y @ X2 ) )
@ A5 )
@ ( collect @ ( product_prod @ B @ A ) @ ( product_case_prod @ B @ A @ $o @ R5 ) ) ) ) ).
% flip_pred
thf(fact_5837_sorted__list__of__set__sort__remdups,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs2: list @ A] :
( ( linord4507533701916653071of_set @ A @ ( set2 @ A @ Xs2 ) )
= ( linorder_sort_key @ A @ A
@ ^ [X2: A] : X2
@ ( remdups @ A @ Xs2 ) ) ) ) ).
% sorted_list_of_set_sort_remdups
thf(fact_5838_conversep__noteq,axiom,
! [A: $tType] :
( ( conversep @ A @ A
@ ^ [X2: A,Y: A] : X2 != Y )
= ( ^ [X2: A,Y: A] : X2 != Y ) ) ).
% conversep_noteq
thf(fact_5839_sort__upt,axiom,
! [M: nat,N: nat] :
( ( linorder_sort_key @ nat @ nat
@ ^ [X2: nat] : X2
@ ( upt @ M @ N ) )
= ( upt @ M @ N ) ) ).
% sort_upt
thf(fact_5840_sort__upto,axiom,
! [I: int,J: int] :
( ( linorder_sort_key @ int @ int
@ ^ [X2: int] : X2
@ ( upto @ I @ J ) )
= ( upto @ I @ J ) ) ).
% sort_upto
thf(fact_5841_acyclicP__converse,axiom,
! [A: $tType,R: A > A > $o] :
( ( transitive_acyclic @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ ( conversep @ A @ A @ R ) ) ) )
= ( transitive_acyclic @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ R ) ) ) ) ).
% acyclicP_converse
thf(fact_5842_subset_Osuc__def,axiom,
! [A: $tType,A5: set @ ( set @ A ),C6: set @ ( set @ A )] :
( ( ( ~ ( pred_chain @ ( set @ A ) @ A5 @ ( ord_less @ ( set @ A ) ) @ C6 )
| ( pred_maxchain @ ( set @ A ) @ A5 @ ( ord_less @ ( set @ A ) ) @ C6 ) )
=> ( ( pred_suc @ ( set @ A ) @ A5 @ ( ord_less @ ( set @ A ) ) @ C6 )
= C6 ) )
& ( ~ ( ~ ( pred_chain @ ( set @ A ) @ A5 @ ( ord_less @ ( set @ A ) ) @ C6 )
| ( pred_maxchain @ ( set @ A ) @ A5 @ ( ord_less @ ( set @ A ) ) @ C6 ) )
=> ( ( pred_suc @ ( set @ A ) @ A5 @ ( ord_less @ ( set @ A ) ) @ C6 )
= ( fChoice @ ( set @ ( set @ A ) )
@ ^ [D6: set @ ( set @ A )] :
( ( pred_chain @ ( set @ A ) @ A5 @ ( ord_less @ ( set @ A ) ) @ D6 )
& ( ord_less @ ( set @ ( set @ A ) ) @ C6 @ D6 ) ) ) ) ) ) ).
% subset.suc_def
thf(fact_5843_sort__key__const,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [C2: B,Xs2: list @ A] :
( ( linorder_sort_key @ A @ B
@ ^ [X2: A] : C2
@ Xs2 )
= Xs2 ) ) ).
% sort_key_const
thf(fact_5844_sort__key__stable,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [F: A > B,K: B,Xs2: list @ A] :
( ( filter2 @ A
@ ^ [Y: A] :
( ( F @ Y )
= K )
@ ( linorder_sort_key @ A @ B @ F @ Xs2 ) )
= ( filter2 @ A
@ ^ [Y: A] :
( ( F @ Y )
= K )
@ Xs2 ) ) ) ).
% sort_key_stable
thf(fact_5845_conversep__converse__eq,axiom,
! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ B )] :
( ( conversep @ A @ B
@ ^ [X2: A,Y: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y ) @ R ) )
= ( ^ [X2: B,Y: A] : ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X2 @ Y ) @ ( converse @ A @ B @ R ) ) ) ) ).
% conversep_converse_eq
thf(fact_5846_converse__def,axiom,
! [B: $tType,A: $tType] :
( ( converse @ A @ B )
= ( ^ [R4: set @ ( product_prod @ A @ B )] :
( collect @ ( product_prod @ B @ A )
@ ( product_case_prod @ B @ A @ $o
@ ( conversep @ A @ B
@ ^ [X2: A,Y: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y ) @ R4 ) ) ) ) ) ) ).
% converse_def
thf(fact_5847_sorted__sort,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs2: list @ A] :
( sorted_wrt @ A @ ( ord_less_eq @ A )
@ ( linorder_sort_key @ A @ A
@ ^ [X2: A] : X2
@ Xs2 ) ) ) ).
% sorted_sort
thf(fact_5848_sorted__sort__id,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs2: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs2 )
=> ( ( linorder_sort_key @ A @ A
@ ^ [X2: A] : X2
@ Xs2 )
= Xs2 ) ) ) ).
% sorted_sort_id
thf(fact_5849_sort__conv__fold,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs2: list @ A] :
( ( linorder_sort_key @ A @ A
@ ^ [X2: A] : X2
@ Xs2 )
= ( fold @ A @ ( list @ A )
@ ( linorder_insort_key @ A @ A
@ ^ [X2: A] : X2 )
@ Xs2
@ ( nil @ A ) ) ) ) ).
% sort_conv_fold
thf(fact_5850_Bleast__code,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs2: list @ A,P: A > $o] :
( ( bleast @ A @ ( set2 @ A @ Xs2 ) @ P )
= ( case_list @ A @ A @ ( abort_Bleast @ A @ ( set2 @ A @ Xs2 ) @ P )
@ ^ [X2: A,Xs: list @ A] : X2
@ ( filter2 @ A @ P
@ ( linorder_sort_key @ A @ A
@ ^ [X2: A] : X2
@ Xs2 ) ) ) ) ) ).
% Bleast_code
thf(fact_5851_fun_Orel__compp__Grp,axiom,
! [D: $tType,B: $tType,A: $tType,R5: A > B > $o] :
( ( bNF_rel_fun @ D @ D @ A @ B
@ ^ [Y6: D,Z2: D] : Y6 = Z2
@ R5 )
= ( relcompp @ ( D > A ) @ ( D > ( product_prod @ A @ B ) ) @ ( D > B )
@ ( conversep @ ( D > ( product_prod @ A @ B ) ) @ ( D > A )
@ ( bNF_Grp @ ( D > ( product_prod @ A @ B ) ) @ ( D > A )
@ ( collect @ ( D > ( product_prod @ A @ B ) )
@ ^ [X2: D > ( product_prod @ A @ B )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( image2 @ D @ ( product_prod @ A @ B ) @ X2 @ ( top_top @ ( set @ D ) ) ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R5 ) ) ) )
@ ( comp @ ( product_prod @ A @ B ) @ A @ D @ ( product_fst @ A @ B ) ) ) )
@ ( bNF_Grp @ ( D > ( product_prod @ A @ B ) ) @ ( D > B )
@ ( collect @ ( D > ( product_prod @ A @ B ) )
@ ^ [X2: D > ( product_prod @ A @ B )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( image2 @ D @ ( product_prod @ A @ B ) @ X2 @ ( top_top @ ( set @ D ) ) ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R5 ) ) ) )
@ ( comp @ ( product_prod @ A @ B ) @ B @ D @ ( product_snd @ A @ B ) ) ) ) ) ).
% fun.rel_compp_Grp
thf(fact_5852_list_Orel__compp__Grp,axiom,
! [B: $tType,A: $tType] :
( ( list_all2 @ A @ B )
= ( ^ [R6: A > B > $o] :
( relcompp @ ( list @ A ) @ ( list @ ( product_prod @ A @ B ) ) @ ( list @ B )
@ ( conversep @ ( list @ ( product_prod @ A @ B ) ) @ ( list @ A )
@ ( bNF_Grp @ ( list @ ( product_prod @ A @ B ) ) @ ( list @ A )
@ ( collect @ ( list @ ( product_prod @ A @ B ) )
@ ^ [X2: list @ ( product_prod @ A @ B )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( set2 @ ( product_prod @ A @ B ) @ X2 ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R6 ) ) ) )
@ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) ) ) )
@ ( bNF_Grp @ ( list @ ( product_prod @ A @ B ) ) @ ( list @ B )
@ ( collect @ ( list @ ( product_prod @ A @ B ) )
@ ^ [X2: list @ ( product_prod @ A @ B )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( set2 @ ( product_prod @ A @ B ) @ X2 ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R6 ) ) ) )
@ ( map @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) ) ) ) ) ) ).
% list.rel_compp_Grp
thf(fact_5853_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_5854_OO__def,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( relcompp @ A @ C @ B )
= ( ^ [R6: A > C > $o,S8: C > B > $o,X2: A,Z3: B] :
? [Y: C] :
( ( R6 @ X2 @ Y )
& ( S8 @ Y @ Z3 ) ) ) ) ).
% OO_def
thf(fact_5855_relcompp__relcomp__eq,axiom,
! [C: $tType,B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ B @ C )] :
( ( relcompp @ A @ B @ C
@ ^ [X2: A,Y: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y ) @ R )
@ ^ [X2: B,Y: C] : ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ X2 @ Y ) @ S ) )
= ( ^ [X2: A,Y: C] : ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ X2 @ Y ) @ ( relcomp @ A @ B @ C @ R @ S ) ) ) ) ).
% relcompp_relcomp_eq
thf(fact_5856_relcompp__SUP__distrib,axiom,
! [A: $tType,C: $tType,B: $tType,D: $tType,S: A > C > $o,R: D > C > B > $o,I5: set @ D] :
( ( relcompp @ A @ C @ B @ S @ ( 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 @ S @ ( R @ I2 ) )
@ I5 ) ) ) ).
% relcompp_SUP_distrib
thf(fact_5857_relcompp__SUP__distrib2,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,R: D > A > C > $o,I5: set @ D,S: C > B > $o] :
( ( relcompp @ A @ C @ B @ ( complete_Sup_Sup @ ( A > C > $o ) @ ( image2 @ D @ ( A > C > $o ) @ R @ I5 ) ) @ S )
= ( complete_Sup_Sup @ ( A > B > $o )
@ ( image2 @ D @ ( A > B > $o )
@ ^ [I2: D] : ( relcompp @ A @ C @ B @ ( R @ I2 ) @ S )
@ I5 ) ) ) ).
% relcompp_SUP_distrib2
thf(fact_5858_relcomp__def,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( relcomp @ A @ B @ C )
= ( ^ [R4: set @ ( product_prod @ A @ B ),S2: set @ ( product_prod @ B @ C )] :
( collect @ ( product_prod @ A @ C )
@ ( product_case_prod @ A @ C @ $o
@ ( relcompp @ A @ B @ C
@ ^ [X2: A,Y: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y ) @ R4 )
@ ^ [X2: B,Y: C] : ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ X2 @ Y ) @ S2 ) ) ) ) ) ) ).
% relcomp_def
thf(fact_5859_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 ) )
= ( ^ [X2: A,Y: B] :
? [Z3: C] :
( ( member @ C @ Z3 @ A5 )
& ( ( F @ Z3 )
= X2 )
& ( ( G @ Z3 )
= Y ) ) ) ) ).
% OO_Grp_alt
thf(fact_5860_chains__alt__def,axiom,
! [A: $tType] :
( ( chains2 @ A )
= ( ^ [A8: set @ ( set @ A )] : ( collect @ ( set @ ( set @ A ) ) @ ( pred_chain @ ( set @ A ) @ A8 @ ( ord_less @ ( set @ A ) ) ) ) ) ) ).
% chains_alt_def
thf(fact_5861_single__valuedp__single__valued__eq,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B )] :
( ( single_valuedp @ A @ B
@ ^ [X2: A,Y: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y ) @ R ) )
= ( single_valued @ A @ B @ R ) ) ).
% single_valuedp_single_valued_eq
thf(fact_5862_univ__def,axiom,
! [A: $tType,B: $tType] :
( ( bNF_Greatest_univ @ B @ A )
= ( ^ [F2: B > A,X9: set @ B] :
( F2
@ ( fChoice @ B
@ ^ [X2: B] : ( member @ B @ X2 @ X9 ) ) ) ) ) ).
% univ_def
thf(fact_5863_single__valuedp__iff__Uniq,axiom,
! [B: $tType,A: $tType] :
( ( single_valuedp @ A @ B )
= ( ^ [R4: A > B > $o] :
! [X2: A] : ( uniq @ B @ ( R4 @ X2 ) ) ) ) ).
% single_valuedp_iff_Uniq
thf(fact_5864_chains__def,axiom,
! [A: $tType] :
( ( chains2 @ A )
= ( ^ [A8: set @ ( set @ A )] :
( collect @ ( set @ ( set @ A ) )
@ ^ [C7: set @ ( set @ A )] :
( ( ord_less_eq @ ( set @ ( set @ A ) ) @ C7 @ A8 )
& ( chain_subset @ A @ C7 ) ) ) ) ) ).
% chains_def
thf(fact_5865_prod__set__simps_I1_J,axiom,
! [B: $tType,A: $tType,X: A,Y3: B] :
( ( basic_fsts @ A @ B @ ( product_Pair @ A @ B @ X @ Y3 ) )
= ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ).
% prod_set_simps(1)
thf(fact_5866_prod__set__simps_I2_J,axiom,
! [A: $tType,B: $tType,X: A,Y3: B] :
( ( basic_snds @ A @ B @ ( product_Pair @ A @ B @ X @ Y3 ) )
= ( insert2 @ B @ Y3 @ ( bot_bot @ ( set @ B ) ) ) ) ).
% prod_set_simps(2)
thf(fact_5867_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 ) )
@ ^ [X2: product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ C ) ) @ ( basic_fsts @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) @ X2 ) @ ( collect @ ( product_prod @ A @ C ) @ ( product_case_prod @ A @ C @ $o @ R14 ) ) )
& ( ord_less_eq @ ( set @ ( product_prod @ B @ D ) ) @ ( basic_snds @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) @ X2 ) @ ( collect @ ( product_prod @ B @ D ) @ ( product_case_prod @ B @ D @ $o @ 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 ) )
@ ^ [X2: product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ C ) ) @ ( basic_fsts @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) @ X2 ) @ ( collect @ ( product_prod @ A @ C ) @ ( product_case_prod @ A @ C @ $o @ R14 ) ) )
& ( ord_less_eq @ ( set @ ( product_prod @ B @ D ) ) @ ( basic_snds @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) @ X2 ) @ ( collect @ ( product_prod @ B @ D ) @ ( product_case_prod @ B @ D @ $o @ 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_5868_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,A6: product_prod @ A @ B,B5: product_prod @ C @ D] :
? [Z3: product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D )] :
( ( member @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) ) @ Z3
@ ( collect @ ( product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) )
@ ^ [X2: product_prod @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ C ) ) @ ( basic_fsts @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) @ X2 ) @ ( collect @ ( product_prod @ A @ C ) @ ( product_case_prod @ A @ C @ $o @ R14 ) ) )
& ( ord_less_eq @ ( set @ ( product_prod @ B @ D ) ) @ ( basic_snds @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) @ X2 ) @ ( collect @ ( product_prod @ B @ D ) @ ( product_case_prod @ B @ D @ $o @ R25 ) ) ) ) ) )
& ( ( product_map_prod @ ( product_prod @ A @ C ) @ A @ ( product_prod @ B @ D ) @ B @ ( product_fst @ A @ C ) @ ( product_fst @ B @ D ) @ Z3 )
= A6 )
& ( ( product_map_prod @ ( product_prod @ A @ C ) @ C @ ( product_prod @ B @ D ) @ D @ ( product_snd @ A @ C ) @ ( product_snd @ B @ D ) @ Z3 )
= B5 ) ) ) ) ).
% prod.in_rel
thf(fact_5869_prod_Orel__Grp,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,A16: set @ A,F1: A > C,A25: set @ B,F23: B > D] :
( ( basic_rel_prod @ A @ C @ B @ D @ ( bNF_Grp @ A @ C @ A16 @ F1 ) @ ( bNF_Grp @ B @ D @ A25 @ F23 ) )
= ( bNF_Grp @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D )
@ ( collect @ ( product_prod @ A @ B )
@ ^ [X2: product_prod @ A @ B] :
( ( ord_less_eq @ ( set @ A ) @ ( basic_fsts @ A @ B @ X2 ) @ A16 )
& ( ord_less_eq @ ( set @ B ) @ ( basic_snds @ A @ B @ X2 ) @ A25 ) ) )
@ ( product_map_prod @ A @ C @ B @ D @ F1 @ F23 ) ) ) ).
% prod.rel_Grp
thf(fact_5870_rel__prod__inject,axiom,
! [B: $tType,A: $tType,C: $tType,D: $tType,R12: A > B > $o,R23: C > D > $o,A3: A,B2: C,C2: B,D3: D] :
( ( basic_rel_prod @ A @ B @ C @ D @ R12 @ R23 @ ( product_Pair @ A @ C @ A3 @ B2 ) @ ( product_Pair @ B @ D @ C2 @ D3 ) )
= ( ( R12 @ A3 @ C2 )
& ( R23 @ B2 @ D3 ) ) ) ).
% rel_prod_inject
thf(fact_5871_fst__transfer,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,A5: A > C > $o,B4: B > D > $o] : ( bNF_rel_fun @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) @ A @ C @ ( basic_rel_prod @ A @ C @ B @ D @ A5 @ B4 ) @ A5 @ ( product_fst @ A @ B ) @ ( product_fst @ C @ D ) ) ).
% fst_transfer
thf(fact_5872_snd__transfer,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,A5: A > C > $o,B4: B > D > $o] : ( bNF_rel_fun @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) @ B @ D @ ( basic_rel_prod @ A @ C @ B @ D @ A5 @ B4 ) @ B4 @ ( product_snd @ A @ B ) @ ( product_snd @ C @ D ) ) ).
% snd_transfer
thf(fact_5873_rel__prod_Ointros,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,R12: A > B > $o,A3: A,B2: B,R23: C > D > $o,C2: C,D3: D] :
( ( R12 @ A3 @ B2 )
=> ( ( R23 @ C2 @ D3 )
=> ( basic_rel_prod @ A @ B @ C @ D @ R12 @ R23 @ ( product_Pair @ A @ C @ A3 @ C2 ) @ ( product_Pair @ B @ D @ B2 @ D3 ) ) ) ) ).
% rel_prod.intros
thf(fact_5874_rel__prod_Osimps,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType] :
( ( basic_rel_prod @ A @ B @ C @ D )
= ( ^ [R14: A > B > $o,R25: C > D > $o,A12: product_prod @ A @ C,A23: product_prod @ B @ D] :
? [A6: A,B5: B,C4: C,D4: D] :
( ( A12
= ( product_Pair @ A @ C @ A6 @ C4 ) )
& ( A23
= ( product_Pair @ B @ D @ B5 @ D4 ) )
& ( R14 @ A6 @ B5 )
& ( R25 @ C4 @ D4 ) ) ) ) ).
% rel_prod.simps
thf(fact_5875_rel__prod_Ocases,axiom,
! [B: $tType,A: $tType,C: $tType,D: $tType,R12: A > B > $o,R23: C > D > $o,A1: product_prod @ A @ C,A22: product_prod @ B @ D] :
( ( basic_rel_prod @ A @ B @ C @ D @ R12 @ R23 @ A1 @ A22 )
=> ~ ! [A4: A,B3: B,C3: C] :
( ( A1
= ( product_Pair @ A @ C @ A4 @ C3 ) )
=> ! [D2: D] :
( ( A22
= ( product_Pair @ B @ D @ B3 @ D2 ) )
=> ( ( R12 @ A4 @ B3 )
=> ~ ( R23 @ C3 @ D2 ) ) ) ) ) ).
% rel_prod.cases
thf(fact_5876_prod_Orel__map_I2_J,axiom,
! [A: $tType,B: $tType,E: $tType,F3: $tType,D: $tType,C: $tType,S1a: A > E > $o,S2a: B > F3 > $o,X: product_prod @ A @ B,G1: C > E,G22: D > F3,Y3: product_prod @ C @ D] :
( ( basic_rel_prod @ A @ E @ B @ F3 @ S1a @ S2a @ X @ ( product_map_prod @ C @ E @ D @ F3 @ G1 @ G22 @ Y3 ) )
= ( basic_rel_prod @ A @ C @ B @ D
@ ^ [X2: A,Y: C] : ( S1a @ X2 @ ( G1 @ Y ) )
@ ^ [X2: B,Y: D] : ( S2a @ X2 @ ( G22 @ Y ) )
@ X
@ Y3 ) ) ).
% prod.rel_map(2)
thf(fact_5877_prod_Orel__map_I1_J,axiom,
! [A: $tType,B: $tType,E: $tType,F3: $tType,D: $tType,C: $tType,S1b: E > C > $o,S2b: F3 > D > $o,I1: A > E,I22: B > F3,X: product_prod @ A @ B,Y3: product_prod @ C @ D] :
( ( basic_rel_prod @ E @ C @ F3 @ D @ S1b @ S2b @ ( product_map_prod @ A @ E @ B @ F3 @ I1 @ I22 @ X ) @ Y3 )
= ( basic_rel_prod @ A @ C @ B @ D
@ ^ [X2: A] : ( S1b @ ( I1 @ X2 ) )
@ ^ [X2: B] : ( S2b @ ( I22 @ X2 ) )
@ X
@ Y3 ) ) ).
% prod.rel_map(1)
thf(fact_5878_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 )
@ ^ [A6: A,B5: B] :
( product_case_prod @ C @ D @ $o
@ ^ [C4: C,D4: D] :
( ( R14 @ A6 @ C4 )
& ( R25 @ B5 @ D4 ) ) ) ) ) ) ).
% rel_prod_conv
thf(fact_5879_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
@ ^ [Y6: nat,Z2: nat] : Y6 = Z2
@ ^ [Y6: nat,Z2: nat] : Y6 = Z2 )
@ pcr_int
@ ^ [X2: product_prod @ nat @ nat] : X2
@ abs_Integ ) ).
% int.id_abs_transfer
thf(fact_5880_convol__transfer,axiom,
! [A: $tType,B: $tType,E: $tType,F3: $tType,D: $tType,C: $tType,R5: A > C > $o,S3: B > D > $o,T5: E > F3 > $o] : ( bNF_rel_fun @ ( A > B ) @ ( C > D ) @ ( ( A > E ) > A > ( product_prod @ B @ E ) ) @ ( ( C > F3 ) > C > ( product_prod @ D @ F3 ) ) @ ( bNF_rel_fun @ A @ C @ B @ D @ R5 @ S3 ) @ ( bNF_rel_fun @ ( A > E ) @ ( C > F3 ) @ ( A > ( product_prod @ B @ E ) ) @ ( C > ( product_prod @ D @ F3 ) ) @ ( bNF_rel_fun @ A @ C @ E @ F3 @ R5 @ T5 ) @ ( bNF_rel_fun @ A @ C @ ( product_prod @ B @ E ) @ ( product_prod @ D @ F3 ) @ R5 @ ( basic_rel_prod @ B @ D @ E @ F3 @ S3 @ T5 ) ) ) @ ( bNF_convol @ A @ B @ E ) @ ( bNF_convol @ C @ D @ F3 ) ) ).
% convol_transfer
thf(fact_5881_Pair__transfer,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,A5: A > B > $o,B4: 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 ) @ B4 @ ( basic_rel_prod @ A @ B @ C @ D @ A5 @ B4 ) ) @ ( product_Pair @ A @ C ) @ ( product_Pair @ B @ D ) ) ).
% Pair_transfer
thf(fact_5882_case__prod__transfer,axiom,
! [A: $tType,B: $tType,C: $tType,F3: $tType,E: $tType,D: $tType,A5: A > D > $o,B4: B > E > $o,C6: C > F3 > $o] : ( bNF_rel_fun @ ( A > B > C ) @ ( D > E > F3 ) @ ( ( product_prod @ A @ B ) > C ) @ ( ( product_prod @ D @ E ) > F3 ) @ ( bNF_rel_fun @ A @ D @ ( B > C ) @ ( E > F3 ) @ A5 @ ( bNF_rel_fun @ B @ E @ C @ F3 @ B4 @ C6 ) ) @ ( bNF_rel_fun @ ( product_prod @ A @ B ) @ ( product_prod @ D @ E ) @ C @ F3 @ ( basic_rel_prod @ A @ D @ B @ E @ A5 @ B4 ) @ C6 ) @ ( product_case_prod @ A @ B @ C ) @ ( product_case_prod @ D @ E @ F3 ) ) ).
% case_prod_transfer
thf(fact_5883_Partial__order__Restr,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A5: set @ A] :
( ( order_7125193373082350890der_on @ A @ ( field2 @ A @ R ) @ R )
=> ( order_7125193373082350890der_on @ A
@ ( field2 @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R
@ ( product_Sigma @ A @ A @ A5
@ ^ [Uu: A] : A5 ) ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R
@ ( product_Sigma @ A @ A @ A5
@ ^ [Uu: A] : A5 ) ) ) ) ).
% Partial_order_Restr
thf(fact_5884_comp__fun__commute__on_Ocomp__comp__fun__commute__on,axiom,
! [B: $tType,A: $tType,C: $tType,S3: set @ A,F: A > B > B,G: C > A,R5: set @ C] :
( ( finite4664212375090638736ute_on @ A @ B @ S3 @ F )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image2 @ C @ A @ G @ ( top_top @ ( set @ C ) ) ) @ S3 )
=> ( finite4664212375090638736ute_on @ C @ B @ R5 @ ( comp @ A @ ( B > B ) @ C @ F @ G ) ) ) ) ).
% comp_fun_commute_on.comp_comp_fun_commute_on
thf(fact_5885_is__singleton__altdef,axiom,
! [A: $tType] :
( ( is_singleton @ A )
= ( ^ [A8: set @ A] :
( ( finite_card @ A @ A8 )
= ( one_one @ nat ) ) ) ) ).
% is_singleton_altdef
thf(fact_5886_comp__fun__commute__on_Ocomp__fun__commute__on__funpow,axiom,
! [B: $tType,A: $tType,S3: set @ A,F: A > B > B,G: A > nat] :
( ( finite4664212375090638736ute_on @ A @ B @ S3 @ F )
=> ( finite4664212375090638736ute_on @ A @ B @ S3
@ ^ [X2: A] : ( compow @ ( B > B ) @ ( G @ X2 ) @ ( F @ X2 ) ) ) ) ).
% comp_fun_commute_on.comp_fun_commute_on_funpow
thf(fact_5887_comp__fun__commute__on__def,axiom,
! [B: $tType,A: $tType] :
( ( finite4664212375090638736ute_on @ A @ B )
= ( ^ [S8: set @ A,F2: A > B > B] :
! [X2: A,Y: A] :
( ( member @ A @ X2 @ S8 )
=> ( ( member @ A @ Y @ S8 )
=> ( ( comp @ B @ B @ B @ ( F2 @ Y ) @ ( F2 @ X2 ) )
= ( comp @ B @ B @ B @ ( F2 @ X2 ) @ ( F2 @ Y ) ) ) ) ) ) ) ).
% comp_fun_commute_on_def
thf(fact_5888_comp__fun__commute__on_Ocomp__fun__commute__on,axiom,
! [B: $tType,A: $tType,S3: set @ A,F: A > B > B,X: A,Y3: A] :
( ( finite4664212375090638736ute_on @ A @ B @ S3 @ F )
=> ( ( member @ A @ X @ S3 )
=> ( ( member @ A @ Y3 @ S3 )
=> ( ( comp @ B @ B @ B @ ( F @ Y3 ) @ ( F @ X ) )
= ( comp @ B @ B @ B @ ( F @ X ) @ ( F @ Y3 ) ) ) ) ) ) ).
% comp_fun_commute_on.comp_fun_commute_on
thf(fact_5889_comp__fun__commute__on_Ocommute__left__comp,axiom,
! [A: $tType,B: $tType,C: $tType,S3: set @ A,F: A > B > B,X: A,Y3: A,G: C > B] :
( ( finite4664212375090638736ute_on @ A @ B @ S3 @ F )
=> ( ( member @ A @ X @ S3 )
=> ( ( member @ A @ Y3 @ S3 )
=> ( ( comp @ B @ B @ C @ ( F @ Y3 ) @ ( comp @ B @ B @ C @ ( F @ X ) @ G ) )
= ( comp @ B @ B @ C @ ( F @ X ) @ ( comp @ B @ B @ C @ ( F @ Y3 ) @ G ) ) ) ) ) ) ).
% comp_fun_commute_on.commute_left_comp
thf(fact_5890_comp__fun__commute__on_Ointro,axiom,
! [B: $tType,A: $tType,S3: set @ A,F: A > B > B] :
( ! [X4: A,Y4: A] :
( ( member @ A @ X4 @ S3 )
=> ( ( member @ A @ Y4 @ S3 )
=> ( ( comp @ B @ B @ B @ ( F @ Y4 ) @ ( F @ X4 ) )
= ( comp @ B @ B @ B @ ( F @ X4 ) @ ( F @ Y4 ) ) ) ) )
=> ( finite4664212375090638736ute_on @ A @ B @ S3 @ F ) ) ).
% comp_fun_commute_on.intro
thf(fact_5891_Zorns__po__lemma,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( order_7125193373082350890der_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ! [C8: set @ A] :
( ( member @ ( set @ A ) @ C8 @ ( chains @ A @ R ) )
=> ? [X5: A] :
( ( member @ A @ X5 @ ( field2 @ A @ R ) )
& ! [Xa4: A] :
( ( member @ A @ Xa4 @ C8 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Xa4 @ X5 ) @ R ) ) ) )
=> ? [X4: A] :
( ( member @ A @ X4 @ ( field2 @ A @ R ) )
& ! [Xa2: A] :
( ( member @ A @ Xa2 @ ( field2 @ A @ R ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Xa2 ) @ R )
=> ( Xa2 = X4 ) ) ) ) ) ) ).
% Zorns_po_lemma
thf(fact_5892_pcr__int__def,axiom,
( pcr_int
= ( relcompp @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ int
@ ( basic_rel_prod @ nat @ nat @ nat @ nat
@ ^ [Y6: nat,Z2: nat] : Y6 = Z2
@ ^ [Y6: nat,Z2: nat] : Y6 = Z2 )
@ cr_int ) ) ).
% pcr_int_def
thf(fact_5893_lcm__code__integer,axiom,
( ( gcd_lcm @ code_integer )
= ( ^ [A6: code_integer,B5: code_integer] : ( divide_divide @ code_integer @ ( times_times @ code_integer @ ( abs_abs @ code_integer @ A6 ) @ ( abs_abs @ code_integer @ B5 ) ) @ ( gcd_gcd @ code_integer @ A6 @ B5 ) ) ) ) ).
% lcm_code_integer
thf(fact_5894_Least__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_Least @ A )
= ( ^ [P3: A > $o] :
( the @ A
@ ^ [X2: A] :
( ( P3 @ X2 )
& ! [Y: A] :
( ( P3 @ Y )
=> ( ord_less_eq @ A @ X2 @ Y ) ) ) ) ) ) ) ).
% Least_def
thf(fact_5895_Least__eq__0,axiom,
! [P: nat > $o] :
( ( P @ ( zero_zero @ nat ) )
=> ( ( ord_Least @ nat @ P )
= ( zero_zero @ nat ) ) ) ).
% Least_eq_0
thf(fact_5896_lcm__left__idem,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A] :
( ( gcd_lcm @ A @ A3 @ ( gcd_lcm @ A @ A3 @ B2 ) )
= ( gcd_lcm @ A @ A3 @ B2 ) ) ) ).
% lcm_left_idem
thf(fact_5897_lcm__right__idem,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A] :
( ( gcd_lcm @ A @ ( gcd_lcm @ A @ A3 @ B2 ) @ B2 )
= ( gcd_lcm @ A @ A3 @ B2 ) ) ) ).
% lcm_right_idem
thf(fact_5898_lcm_Obottom__right__bottom,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A] :
( ( gcd_lcm @ A @ A3 @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% lcm.bottom_right_bottom
thf(fact_5899_lcm_Obottom__left__bottom,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A] :
( ( gcd_lcm @ A @ ( zero_zero @ A ) @ A3 )
= ( zero_zero @ A ) ) ) ).
% lcm.bottom_left_bottom
thf(fact_5900_lcm__neg1,axiom,
! [A: $tType] :
( ( ring_gcd @ A )
=> ! [A3: A,B2: A] :
( ( gcd_lcm @ A @ ( uminus_uminus @ A @ A3 ) @ B2 )
= ( gcd_lcm @ A @ A3 @ B2 ) ) ) ).
% lcm_neg1
thf(fact_5901_lcm__neg2,axiom,
! [A: $tType] :
( ( ring_gcd @ A )
=> ! [A3: A,B2: A] :
( ( gcd_lcm @ A @ A3 @ ( uminus_uminus @ A @ B2 ) )
= ( gcd_lcm @ A @ A3 @ B2 ) ) ) ).
% lcm_neg2
thf(fact_5902_lcm__least__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ ( gcd_lcm @ A @ A3 @ B2 ) @ C2 )
= ( ( dvd_dvd @ A @ A3 @ C2 )
& ( dvd_dvd @ A @ B2 @ C2 ) ) ) ) ).
% lcm_least_iff
thf(fact_5903_dvd__lcm2,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [B2: A,A3: A] : ( dvd_dvd @ A @ B2 @ ( gcd_lcm @ A @ A3 @ B2 ) ) ) ).
% dvd_lcm2
thf(fact_5904_dvd__lcm1,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A] : ( dvd_dvd @ A @ A3 @ ( gcd_lcm @ A @ A3 @ B2 ) ) ) ).
% dvd_lcm1
thf(fact_5905_lcm__neg__numeral__2,axiom,
! [A: $tType] :
( ( ring_gcd @ A )
=> ! [A3: A,N: num] :
( ( gcd_lcm @ A @ A3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( gcd_lcm @ A @ A3 @ ( numeral_numeral @ A @ N ) ) ) ) ).
% lcm_neg_numeral_2
thf(fact_5906_lcm__neg__numeral__1,axiom,
! [A: $tType] :
( ( ring_gcd @ A )
=> ! [N: num,A3: A] :
( ( gcd_lcm @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) @ A3 )
= ( gcd_lcm @ A @ ( numeral_numeral @ A @ N ) @ A3 ) ) ) ).
% lcm_neg_numeral_1
thf(fact_5907_lcm__eq__1__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A] :
( ( ( gcd_lcm @ A @ A3 @ B2 )
= ( one_one @ A ) )
= ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
& ( dvd_dvd @ A @ B2 @ ( one_one @ A ) ) ) ) ) ).
% lcm_eq_1_iff
thf(fact_5908_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_5909_gcd__dvd__lcm,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A] : ( dvd_dvd @ A @ ( gcd_gcd @ A @ A3 @ B2 ) @ ( gcd_lcm @ A @ A3 @ B2 ) ) ) ).
% gcd_dvd_lcm
thf(fact_5910_LeastI,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P: A > $o,K: A] :
( ( P @ K )
=> ( P @ ( ord_Least @ A @ P ) ) ) ) ).
% LeastI
thf(fact_5911_lcm__least,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ C2 )
=> ( ( dvd_dvd @ A @ B2 @ C2 )
=> ( dvd_dvd @ A @ ( gcd_lcm @ A @ A3 @ B2 ) @ C2 ) ) ) ) ).
% lcm_least
thf(fact_5912_lcm__dvdD2,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ ( gcd_lcm @ A @ A3 @ B2 ) @ C2 )
=> ( dvd_dvd @ A @ B2 @ C2 ) ) ) ).
% lcm_dvdD2
thf(fact_5913_lcm__dvdD1,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ ( gcd_lcm @ A @ A3 @ B2 ) @ C2 )
=> ( dvd_dvd @ A @ A3 @ C2 ) ) ) ).
% lcm_dvdD1
thf(fact_5914_dvd__lcmI2,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ C2 )
=> ( dvd_dvd @ A @ A3 @ ( gcd_lcm @ A @ B2 @ C2 ) ) ) ) ).
% dvd_lcmI2
thf(fact_5915_dvd__lcmI1,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A3 @ B2 )
=> ( dvd_dvd @ A @ A3 @ ( gcd_lcm @ A @ B2 @ C2 ) ) ) ) ).
% dvd_lcmI1
thf(fact_5916_lcm__mono,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,C2: A,B2: A,D3: A] :
( ( dvd_dvd @ A @ A3 @ C2 )
=> ( ( dvd_dvd @ A @ B2 @ D3 )
=> ( dvd_dvd @ A @ ( gcd_lcm @ A @ A3 @ B2 ) @ ( gcd_lcm @ A @ C2 @ D3 ) ) ) ) ) ).
% lcm_mono
thf(fact_5917_lcm_Oassoc,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( gcd_lcm @ A @ ( gcd_lcm @ A @ A3 @ B2 ) @ C2 )
= ( gcd_lcm @ A @ A3 @ ( gcd_lcm @ A @ B2 @ C2 ) ) ) ) ).
% lcm.assoc
thf(fact_5918_lcm_Ocommute,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ( ( gcd_lcm @ A )
= ( ^ [A6: A,B5: A] : ( gcd_lcm @ A @ B5 @ A6 ) ) ) ) ).
% lcm.commute
thf(fact_5919_lcm_Oleft__commute,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( gcd_lcm @ A @ B2 @ ( gcd_lcm @ A @ A3 @ C2 ) )
= ( gcd_lcm @ A @ A3 @ ( gcd_lcm @ A @ B2 @ C2 ) ) ) ) ).
% lcm.left_commute
thf(fact_5920_lcm__eq__0__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A] :
( ( ( gcd_lcm @ A @ A3 @ B2 )
= ( zero_zero @ A ) )
= ( ( A3
= ( zero_zero @ A ) )
| ( B2
= ( zero_zero @ A ) ) ) ) ) ).
% lcm_eq_0_iff
thf(fact_5921_zero__eq__lcm__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A] :
( ( ( zero_zero @ A )
= ( gcd_lcm @ A @ A3 @ B2 ) )
= ( ( A3
= ( zero_zero @ A ) )
| ( B2
= ( zero_zero @ A ) ) ) ) ) ).
% zero_eq_lcm_iff
thf(fact_5922_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_5923_Least__Suc2,axiom,
! [P: nat > $o,N: nat,Q: nat > $o,M: nat] :
( ( P @ N )
=> ( ( Q @ M )
=> ( ~ ( P @ ( zero_zero @ nat ) )
=> ( ! [K2: nat] :
( ( P @ ( suc @ K2 ) )
= ( Q @ K2 ) )
=> ( ( ord_Least @ nat @ P )
= ( suc @ ( ord_Least @ nat @ Q ) ) ) ) ) ) ) ).
% Least_Suc2
thf(fact_5924_lcm__mult__unit1,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( gcd_lcm @ A @ ( times_times @ A @ B2 @ A3 ) @ C2 )
= ( gcd_lcm @ A @ B2 @ C2 ) ) ) ) ).
% lcm_mult_unit1
thf(fact_5925_lcm__mult__unit2,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( gcd_lcm @ A @ B2 @ ( times_times @ A @ C2 @ A3 ) )
= ( gcd_lcm @ A @ B2 @ C2 ) ) ) ) ).
% lcm_mult_unit2
thf(fact_5926_lcm__div__unit1,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( gcd_lcm @ A @ ( divide_divide @ A @ B2 @ A3 ) @ C2 )
= ( gcd_lcm @ A @ B2 @ C2 ) ) ) ) ).
% lcm_div_unit1
thf(fact_5927_lcm__div__unit2,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( gcd_lcm @ A @ B2 @ ( divide_divide @ A @ C2 @ A3 ) )
= ( gcd_lcm @ A @ B2 @ C2 ) ) ) ) ).
% lcm_div_unit2
thf(fact_5928_Least__Suc,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ ( zero_zero @ nat ) )
=> ( ( ord_Least @ nat @ P )
= ( suc
@ ( ord_Least @ nat
@ ^ [M5: nat] : ( P @ ( suc @ M5 ) ) ) ) ) ) ) ).
% Least_Suc
thf(fact_5929_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_5930_Bleast__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( bleast @ A )
= ( ^ [S8: set @ A,P3: A > $o] :
( ord_Least @ A
@ ^ [X2: A] :
( ( member @ A @ X2 @ S8 )
& ( P3 @ X2 ) ) ) ) ) ) ).
% Bleast_def
thf(fact_5931_abort__Bleast__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( abort_Bleast @ A )
= ( ^ [S8: set @ A,P3: A > $o] :
( ord_Least @ A
@ ^ [X2: A] :
( ( member @ A @ X2 @ S8 )
& ( P3 @ X2 ) ) ) ) ) ) ).
% abort_Bleast_def
thf(fact_5932_int_Opcr__cr__eq,axiom,
pcr_int = cr_int ).
% int.pcr_cr_eq
thf(fact_5933_cr__int__def,axiom,
( cr_int
= ( ^ [X2: product_prod @ nat @ nat] :
( ^ [Y6: int,Z2: int] : Y6 = Z2
@ ( abs_Integ @ X2 ) ) ) ) ).
% cr_int_def
thf(fact_5934_Least__mono,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F: A > B,S3: set @ A] :
( ( order_mono @ A @ B @ F )
=> ( ? [X5: A] :
( ( member @ A @ X5 @ S3 )
& ! [Xa4: A] :
( ( member @ A @ Xa4 @ S3 )
=> ( ord_less_eq @ A @ X5 @ Xa4 ) ) )
=> ( ( ord_Least @ B
@ ^ [Y: B] : ( member @ B @ Y @ ( image2 @ A @ B @ F @ S3 ) ) )
= ( F
@ ( ord_Least @ A
@ ^ [X2: A] : ( member @ A @ X2 @ S3 ) ) ) ) ) ) ) ).
% Least_mono
thf(fact_5935_Lcm__fin_Oeq__fold,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ( ( semiring_gcd_Lcm_fin @ A )
= ( ^ [A8: set @ A] : ( if @ A @ ( finite_finite2 @ A @ A8 ) @ ( finite_fold @ A @ A @ ( gcd_lcm @ A ) @ ( one_one @ A ) @ A8 ) @ ( zero_zero @ A ) ) ) ) ) ).
% Lcm_fin.eq_fold
thf(fact_5936_lcm__integer__def,axiom,
( ( gcd_lcm @ 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_lcm @ int ) ) ) ).
% lcm_integer_def
thf(fact_5937_Quotient__int,axiom,
quotient @ ( product_prod @ nat @ nat ) @ int @ intrel @ abs_Integ @ rep_Integ @ cr_int ).
% Quotient_int
thf(fact_5938_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_5939_lcm__proj2__if__dvd__nat,axiom,
! [X: nat,Y3: nat] :
( ( dvd_dvd @ nat @ X @ Y3 )
=> ( ( gcd_lcm @ nat @ X @ Y3 )
= Y3 ) ) ).
% lcm_proj2_if_dvd_nat
thf(fact_5940_lcm__proj1__if__dvd__nat,axiom,
! [X: nat,Y3: nat] :
( ( dvd_dvd @ nat @ X @ Y3 )
=> ( ( gcd_lcm @ nat @ Y3 @ X )
= Y3 ) ) ).
% lcm_proj1_if_dvd_nat
thf(fact_5941_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_5942_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_5943_lcm__1__iff__nat,axiom,
! [M: nat,N: nat] :
( ( ( gcd_lcm @ nat @ M @ N )
= ( suc @ ( zero_zero @ nat ) ) )
= ( ( M
= ( suc @ ( zero_zero @ nat ) ) )
& ( N
= ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% lcm_1_iff_nat
thf(fact_5944_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_5945_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_5946_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_5947_lcm__abs1__int,axiom,
! [X: int,Y3: int] :
( ( gcd_lcm @ int @ ( abs_abs @ int @ X ) @ Y3 )
= ( gcd_lcm @ int @ X @ Y3 ) ) ).
% lcm_abs1_int
thf(fact_5948_lcm__abs2__int,axiom,
! [X: int,Y3: int] :
( ( gcd_lcm @ int @ X @ ( abs_abs @ int @ Y3 ) )
= ( gcd_lcm @ int @ X @ Y3 ) ) ).
% lcm_abs2_int
thf(fact_5949_Lcm__fin_Oinfinite,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A5: set @ A] :
( ~ ( finite_finite2 @ A @ A5 )
=> ( ( semiring_gcd_Lcm_fin @ A @ A5 )
= ( zero_zero @ A ) ) ) ) ).
% Lcm_fin.infinite
thf(fact_5950_Lcm__fin_Oempty,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ( ( semiring_gcd_Lcm_fin @ A @ ( bot_bot @ ( set @ A ) ) )
= ( one_one @ A ) ) ) ).
% Lcm_fin.empty
thf(fact_5951_is__unit__Lcm__fin__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A5: set @ A] :
( ( dvd_dvd @ A @ ( semiring_gcd_Lcm_fin @ A @ A5 ) @ ( one_one @ A ) )
= ( ( semiring_gcd_Lcm_fin @ A @ A5 )
= ( one_one @ A ) ) ) ) ).
% is_unit_Lcm_fin_iff
thf(fact_5952_lcm__1__iff__int,axiom,
! [M: int,N: int] :
( ( ( gcd_lcm @ int @ M @ N )
= ( one_one @ int ) )
= ( ( ( M
= ( one_one @ int ) )
| ( M
= ( uminus_uminus @ int @ ( one_one @ int ) ) ) )
& ( ( N
= ( one_one @ int ) )
| ( N
= ( uminus_uminus @ int @ ( one_one @ int ) ) ) ) ) ) ).
% lcm_1_iff_int
thf(fact_5953_Lcm__fin_Oinsert,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,A5: set @ A] :
( ( semiring_gcd_Lcm_fin @ A @ ( insert2 @ A @ A3 @ A5 ) )
= ( gcd_lcm @ A @ A3 @ ( semiring_gcd_Lcm_fin @ A @ A5 ) ) ) ) ).
% Lcm_fin.insert
thf(fact_5954_lcm__proj2__if__dvd__int,axiom,
! [X: int,Y3: int] :
( ( dvd_dvd @ int @ X @ Y3 )
=> ( ( gcd_lcm @ int @ X @ Y3 )
= ( abs_abs @ int @ Y3 ) ) ) ).
% lcm_proj2_if_dvd_int
thf(fact_5955_lcm__proj1__if__dvd__int,axiom,
! [X: int,Y3: int] :
( ( dvd_dvd @ int @ X @ Y3 )
=> ( ( gcd_lcm @ int @ Y3 @ X )
= ( abs_abs @ int @ Y3 ) ) ) ).
% lcm_proj1_if_dvd_int
thf(fact_5956_lcm__proj2__iff__int,axiom,
! [M: int,N: int] :
( ( ( gcd_lcm @ int @ M @ N )
= ( abs_abs @ int @ N ) )
= ( dvd_dvd @ int @ M @ N ) ) ).
% lcm_proj2_iff_int
thf(fact_5957_lcm__proj1__iff__int,axiom,
! [M: int,N: int] :
( ( ( gcd_lcm @ int @ M @ N )
= ( abs_abs @ int @ M ) )
= ( dvd_dvd @ int @ N @ M ) ) ).
% lcm_proj1_iff_int
thf(fact_5958_lcm__nat__abs__right__eq,axiom,
! [N: nat,K: int] :
( ( gcd_lcm @ nat @ N @ ( nat2 @ ( abs_abs @ int @ K ) ) )
= ( nat2 @ ( gcd_lcm @ int @ ( semiring_1_of_nat @ int @ N ) @ K ) ) ) ).
% lcm_nat_abs_right_eq
thf(fact_5959_lcm__nat__abs__left__eq,axiom,
! [K: int,N: nat] :
( ( gcd_lcm @ nat @ ( nat2 @ ( abs_abs @ int @ K ) ) @ N )
= ( nat2 @ ( gcd_lcm @ int @ K @ ( semiring_1_of_nat @ int @ N ) ) ) ) ).
% lcm_nat_abs_left_eq
thf(fact_5960_lcm__ge__0__int,axiom,
! [X: int,Y3: int] : ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( gcd_lcm @ int @ X @ Y3 ) ) ).
% lcm_ge_0_int
thf(fact_5961_Inf__nat__def,axiom,
( ( complete_Inf_Inf @ nat )
= ( ^ [X9: set @ nat] :
( ord_Least @ nat
@ ^ [N5: nat] : ( member @ nat @ N5 @ X9 ) ) ) ) ).
% Inf_nat_def
thf(fact_5962_prod__gcd__lcm__nat,axiom,
( ( times_times @ nat )
= ( ^ [M5: nat,N5: nat] : ( times_times @ nat @ ( gcd_gcd @ nat @ M5 @ N5 ) @ ( gcd_lcm @ nat @ M5 @ N5 ) ) ) ) ).
% prod_gcd_lcm_nat
thf(fact_5963_lcm__neg1__int,axiom,
! [X: int,Y3: int] :
( ( gcd_lcm @ int @ ( uminus_uminus @ int @ X ) @ Y3 )
= ( gcd_lcm @ int @ X @ Y3 ) ) ).
% lcm_neg1_int
thf(fact_5964_lcm__neg2__int,axiom,
! [X: int,Y3: int] :
( ( gcd_lcm @ int @ X @ ( uminus_uminus @ int @ Y3 ) )
= ( gcd_lcm @ int @ X @ Y3 ) ) ).
% lcm_neg2_int
thf(fact_5965_dvd__lcm__I2__int,axiom,
! [I: int,N: int,M: int] :
( ( dvd_dvd @ int @ I @ N )
=> ( dvd_dvd @ int @ I @ ( gcd_lcm @ int @ M @ N ) ) ) ).
% dvd_lcm_I2_int
thf(fact_5966_dvd__lcm__I1__int,axiom,
! [I: int,M: int,N: int] :
( ( dvd_dvd @ int @ I @ M )
=> ( dvd_dvd @ int @ I @ ( gcd_lcm @ int @ M @ N ) ) ) ).
% dvd_lcm_I1_int
thf(fact_5967_lcm__unique__nat,axiom,
! [A3: nat,D3: nat,B2: nat] :
( ( ( dvd_dvd @ nat @ A3 @ D3 )
& ( dvd_dvd @ nat @ B2 @ D3 )
& ! [E4: nat] :
( ( ( dvd_dvd @ nat @ A3 @ E4 )
& ( dvd_dvd @ nat @ B2 @ E4 ) )
=> ( dvd_dvd @ nat @ D3 @ E4 ) ) )
= ( D3
= ( gcd_lcm @ nat @ A3 @ B2 ) ) ) ).
% lcm_unique_nat
thf(fact_5968_dvd__lcm__I2__nat,axiom,
! [K: nat,N: nat,M: nat] :
( ( dvd_dvd @ nat @ K @ N )
=> ( dvd_dvd @ nat @ K @ ( gcd_lcm @ nat @ M @ N ) ) ) ).
% dvd_lcm_I2_nat
thf(fact_5969_dvd__lcm__I1__nat,axiom,
! [K: nat,M: nat,N: nat] :
( ( dvd_dvd @ nat @ K @ M )
=> ( dvd_dvd @ nat @ K @ ( gcd_lcm @ nat @ M @ N ) ) ) ).
% dvd_lcm_I1_nat
thf(fact_5970_Lcm__fin_Oin__idem,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,A5: set @ A] :
( ( member @ A @ A3 @ A5 )
=> ( ( gcd_lcm @ A @ A3 @ ( semiring_gcd_Lcm_fin @ A @ A5 ) )
= ( semiring_gcd_Lcm_fin @ A @ A5 ) ) ) ) ).
% Lcm_fin.in_idem
thf(fact_5971_lcm__int__def,axiom,
( ( gcd_lcm @ int )
= ( ^ [X2: int,Y: int] : ( semiring_1_of_nat @ int @ ( gcd_lcm @ nat @ ( nat2 @ ( abs_abs @ int @ X2 ) ) @ ( nat2 @ ( abs_abs @ int @ Y ) ) ) ) ) ) ).
% lcm_int_def
thf(fact_5972_lcm__pos__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ ( gcd_lcm @ nat @ M @ N ) ) ) ) ).
% lcm_pos_nat
thf(fact_5973_lcm__pos__int,axiom,
! [M: int,N: int] :
( ( M
!= ( zero_zero @ int ) )
=> ( ( N
!= ( zero_zero @ int ) )
=> ( ord_less @ int @ ( zero_zero @ int ) @ ( gcd_lcm @ int @ M @ N ) ) ) ) ).
% lcm_pos_int
thf(fact_5974_Quotient__cr__rel,axiom,
! [B: $tType,A: $tType,R5: A > A > $o,Abs: A > B,Rep: B > A,T5: A > B > $o] :
( ( quotient @ A @ B @ R5 @ Abs @ Rep @ T5 )
=> ( T5
= ( ^ [X2: A,Y: B] :
( ( R5 @ X2 @ X2 )
& ( ( Abs @ X2 )
= Y ) ) ) ) ) ).
% Quotient_cr_rel
thf(fact_5975_Quotient__def,axiom,
! [B: $tType,A: $tType] :
( ( quotient @ A @ B )
= ( ^ [R6: A > A > $o,Abs4: A > B,Rep4: B > A,T6: A > B > $o] :
( ! [A6: B] :
( ( Abs4 @ ( Rep4 @ A6 ) )
= A6 )
& ! [A6: B] : ( R6 @ ( Rep4 @ A6 ) @ ( Rep4 @ A6 ) )
& ! [R4: A,S2: A] :
( ( R6 @ R4 @ S2 )
= ( ( R6 @ R4 @ R4 )
& ( R6 @ S2 @ S2 )
& ( ( Abs4 @ R4 )
= ( Abs4 @ S2 ) ) ) )
& ( T6
= ( ^ [X2: A,Y: B] :
( ( R6 @ X2 @ X2 )
& ( ( Abs4 @ X2 )
= Y ) ) ) ) ) ) ) ).
% Quotient_def
thf(fact_5976_QuotientI,axiom,
! [A: $tType,B: $tType,Abs: B > A,Rep: A > B,R5: B > B > $o,T5: B > A > $o] :
( ! [A4: A] :
( ( Abs @ ( Rep @ A4 ) )
= A4 )
=> ( ! [A4: A] : ( R5 @ ( Rep @ A4 ) @ ( Rep @ A4 ) )
=> ( ! [R2: B,S5: B] :
( ( R5 @ R2 @ S5 )
= ( ( R5 @ R2 @ R2 )
& ( R5 @ S5 @ S5 )
& ( ( Abs @ R2 )
= ( Abs @ S5 ) ) ) )
=> ( ( T5
= ( ^ [X2: B,Y: A] :
( ( R5 @ X2 @ X2 )
& ( ( Abs @ X2 )
= Y ) ) ) )
=> ( quotient @ B @ A @ R5 @ Abs @ Rep @ T5 ) ) ) ) ) ).
% QuotientI
thf(fact_5977_dvd__Lcm__fin,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,A5: set @ A] :
( ( member @ A @ A3 @ A5 )
=> ( dvd_dvd @ A @ A3 @ ( semiring_gcd_Lcm_fin @ A @ A5 ) ) ) ) ).
% dvd_Lcm_fin
thf(fact_5978_Quotient__crel__typecopy,axiom,
! [B: $tType,A: $tType,Abs: A > B,Rep: B > A,T5: A > B > $o] :
( ( quotient @ A @ B
@ ^ [Y6: A,Z2: A] : Y6 = Z2
@ Abs
@ Rep
@ T5 )
=> ( T5
= ( ^ [X2: A,Y: B] :
( X2
= ( Rep @ Y ) ) ) ) ) ).
% Quotient_crel_typecopy
thf(fact_5979_Lcm__fin__0__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( ( semiring_gcd_Lcm_fin @ A @ A5 )
= ( zero_zero @ A ) )
= ( member @ A @ ( zero_zero @ A ) @ A5 ) ) ) ) ).
% Lcm_fin_0_iff
thf(fact_5980_Lcm__fin__least,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A5: set @ A,A3: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ! [B3: A] :
( ( member @ A @ B3 @ A5 )
=> ( dvd_dvd @ A @ B3 @ A3 ) )
=> ( dvd_dvd @ A @ ( semiring_gcd_Lcm_fin @ A @ A5 ) @ A3 ) ) ) ) ).
% Lcm_fin_least
thf(fact_5981_Lcm__fin__dvd__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A5: set @ A,B2: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( dvd_dvd @ A @ ( semiring_gcd_Lcm_fin @ A @ A5 ) @ B2 )
= ( ! [X2: A] :
( ( member @ A @ X2 @ A5 )
=> ( dvd_dvd @ A @ X2 @ B2 ) ) ) ) ) ) ).
% Lcm_fin_dvd_iff
thf(fact_5982_UNIV__typedef__to__Quotient,axiom,
! [A: $tType,B: $tType,Rep: A > B,Abs: B > A,T5: B > A > $o] :
( ( type_definition @ A @ B @ Rep @ Abs @ ( top_top @ ( set @ B ) ) )
=> ( ( T5
= ( ^ [X2: B,Y: A] :
( X2
= ( Rep @ Y ) ) ) )
=> ( quotient @ B @ A
@ ^ [Y6: B,Z2: B] : Y6 = Z2
@ Abs
@ Rep
@ T5 ) ) ) ).
% UNIV_typedef_to_Quotient
thf(fact_5983_lcm__list__least,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [Bs: list @ A,A3: A] :
( ! [B3: A] :
( ( member @ A @ B3 @ ( set2 @ A @ Bs ) )
=> ( dvd_dvd @ A @ B3 @ A3 ) )
=> ( dvd_dvd @ A @ ( semiring_gcd_Lcm_fin @ A @ ( set2 @ A @ Bs ) ) @ A3 ) ) ) ).
% lcm_list_least
thf(fact_5984_lcm__list__dvd__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [Xs2: list @ A,B2: A] :
( ( dvd_dvd @ A @ ( semiring_gcd_Lcm_fin @ A @ ( set2 @ A @ Xs2 ) ) @ B2 )
= ( ! [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ Xs2 ) )
=> ( dvd_dvd @ A @ X2 @ B2 ) ) ) ) ) ).
% lcm_list_dvd_iff
thf(fact_5985_lcm__integer_Orsp,axiom,
( bNF_rel_fun @ int @ int @ ( int > int ) @ ( int > int )
@ ^ [Y6: int,Z2: int] : Y6 = Z2
@ ( bNF_rel_fun @ int @ int @ int @ int
@ ^ [Y6: int,Z2: int] : Y6 = Z2
@ ^ [Y6: int,Z2: int] : Y6 = Z2 )
@ ( gcd_lcm @ int )
@ ( gcd_lcm @ int ) ) ).
% lcm_integer.rsp
thf(fact_5986_Lcm__fin_Osubset,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [B4: set @ A,A5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B4 @ A5 )
=> ( ( gcd_lcm @ A @ ( semiring_gcd_Lcm_fin @ A @ B4 ) @ ( semiring_gcd_Lcm_fin @ A @ A5 ) )
= ( semiring_gcd_Lcm_fin @ A @ A5 ) ) ) ) ).
% Lcm_fin.subset
thf(fact_5987_lcm__integer_Orep__eq,axiom,
! [X: code_integer,Xa: code_integer] :
( ( code_int_of_integer @ ( gcd_lcm @ code_integer @ X @ Xa ) )
= ( gcd_lcm @ int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa ) ) ) ).
% lcm_integer.rep_eq
thf(fact_5988_lcm__integer_Oabs__eq,axiom,
! [Xa: int,X: int] :
( ( gcd_lcm @ code_integer @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X ) )
= ( code_integer_of_int @ ( gcd_lcm @ int @ Xa @ X ) ) ) ).
% lcm_integer.abs_eq
thf(fact_5989_Lcm__fin_Ounion,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A5: set @ A,B4: set @ A] :
( ( semiring_gcd_Lcm_fin @ A @ ( sup_sup @ ( set @ A ) @ A5 @ B4 ) )
= ( gcd_lcm @ A @ ( semiring_gcd_Lcm_fin @ A @ A5 ) @ ( semiring_gcd_Lcm_fin @ A @ B4 ) ) ) ) ).
% Lcm_fin.union
thf(fact_5990_lcm__cases__int,axiom,
! [X: int,Y3: int,P: int > $o] :
( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y3 )
=> ( P @ ( gcd_lcm @ int @ X @ Y3 ) ) ) )
=> ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
=> ( ( ord_less_eq @ int @ Y3 @ ( zero_zero @ int ) )
=> ( P @ ( gcd_lcm @ int @ X @ ( uminus_uminus @ int @ Y3 ) ) ) ) )
=> ( ( ( ord_less_eq @ int @ X @ ( zero_zero @ int ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y3 )
=> ( P @ ( gcd_lcm @ int @ ( uminus_uminus @ int @ X ) @ Y3 ) ) ) )
=> ( ( ( ord_less_eq @ int @ X @ ( zero_zero @ int ) )
=> ( ( ord_less_eq @ int @ Y3 @ ( zero_zero @ int ) )
=> ( P @ ( gcd_lcm @ int @ ( uminus_uminus @ int @ X ) @ ( uminus_uminus @ int @ Y3 ) ) ) ) )
=> ( P @ ( gcd_lcm @ int @ X @ Y3 ) ) ) ) ) ) ).
% lcm_cases_int
thf(fact_5991_lcm__unique__int,axiom,
! [D3: int,A3: int,B2: int] :
( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ D3 )
& ( dvd_dvd @ int @ A3 @ D3 )
& ( dvd_dvd @ int @ B2 @ D3 )
& ! [E4: int] :
( ( ( dvd_dvd @ int @ A3 @ E4 )
& ( dvd_dvd @ int @ B2 @ E4 ) )
=> ( dvd_dvd @ int @ D3 @ E4 ) ) )
= ( D3
= ( gcd_lcm @ int @ A3 @ B2 ) ) ) ).
% lcm_unique_int
thf(fact_5992_lcm__nat__def,axiom,
( ( gcd_lcm @ nat )
= ( ^ [X2: nat,Y: nat] : ( divide_divide @ nat @ ( times_times @ nat @ X2 @ Y ) @ ( gcd_gcd @ nat @ X2 @ Y ) ) ) ) ).
% lcm_nat_def
thf(fact_5993_Quotient__compose,axiom,
! [A: $tType,C: $tType,B: $tType,R12: A > A > $o,Abs1: A > B,Rep1: B > A,T1: A > B > $o,R23: B > B > $o,Abs22: B > C,Rep22: C > B,T22: B > C > $o] :
( ( quotient @ A @ B @ R12 @ Abs1 @ Rep1 @ T1 )
=> ( ( quotient @ B @ C @ R23 @ Abs22 @ Rep22 @ T22 )
=> ( quotient @ A @ C @ ( relcompp @ A @ B @ A @ T1 @ ( relcompp @ B @ B @ A @ R23 @ ( conversep @ A @ B @ T1 ) ) ) @ ( comp @ B @ C @ A @ Abs22 @ Abs1 ) @ ( comp @ B @ A @ C @ Rep1 @ Rep22 ) @ ( relcompp @ A @ B @ C @ T1 @ T22 ) ) ) ) ).
% Quotient_compose
thf(fact_5994_prod__gcd__lcm__int,axiom,
! [M: int,N: int] :
( ( times_times @ int @ ( abs_abs @ int @ M ) @ ( abs_abs @ int @ N ) )
= ( times_times @ int @ ( gcd_gcd @ int @ M @ N ) @ ( gcd_lcm @ int @ M @ N ) ) ) ).
% prod_gcd_lcm_int
thf(fact_5995_Lcm__fin__1__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A5: set @ A] :
( ( ( semiring_gcd_Lcm_fin @ A @ A5 )
= ( one_one @ A ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ A5 )
=> ( dvd_dvd @ A @ X2 @ ( one_one @ A ) ) )
& ( finite_finite2 @ A @ A5 ) ) ) ) ).
% Lcm_fin_1_iff
thf(fact_5996_lcm__altdef__int,axiom,
( ( gcd_lcm @ int )
= ( ^ [A6: int,B5: int] : ( divide_divide @ int @ ( times_times @ int @ ( abs_abs @ int @ A6 ) @ ( abs_abs @ int @ B5 ) ) @ ( gcd_gcd @ int @ A6 @ B5 ) ) ) ) ).
% lcm_altdef_int
thf(fact_5997_Lcm__fin_Oremove,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,A5: set @ A] :
( ( member @ A @ A3 @ A5 )
=> ( ( semiring_gcd_Lcm_fin @ A @ A5 )
= ( gcd_lcm @ A @ A3 @ ( semiring_gcd_Lcm_fin @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% Lcm_fin.remove
thf(fact_5998_Lcm__fin_Oinsert__remove,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,A5: set @ A] :
( ( semiring_gcd_Lcm_fin @ A @ ( insert2 @ A @ A3 @ A5 ) )
= ( gcd_lcm @ A @ A3 @ ( semiring_gcd_Lcm_fin @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% Lcm_fin.insert_remove
thf(fact_5999_Lcm__fin_Oset__eq__fold,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [Xs2: list @ A] :
( ( semiring_gcd_Lcm_fin @ A @ ( set2 @ A @ Xs2 ) )
= ( fold @ A @ A @ ( gcd_lcm @ A ) @ Xs2 @ ( one_one @ A ) ) ) ) ).
% Lcm_fin.set_eq_fold
thf(fact_6000_lcm__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_lcm @ int ) @ ( gcd_lcm @ code_integer ) ).
% lcm_integer.transfer
thf(fact_6001_Lcm__fin__def,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ( ( semiring_gcd_Lcm_fin @ A )
= ( bounde2362111253966948842tice_F @ A @ ( gcd_lcm @ A ) @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ) ).
% Lcm_fin_def
thf(fact_6002_Lcm__int__set__eq__fold,axiom,
! [Xs2: list @ int] :
( ( gcd_Lcm @ int @ ( set2 @ int @ Xs2 ) )
= ( fold @ int @ int @ ( gcd_lcm @ int ) @ Xs2 @ ( one_one @ int ) ) ) ).
% Lcm_int_set_eq_fold
thf(fact_6003_Lcm__eq__Max__nat,axiom,
! [M2: set @ nat] :
( ( finite_finite2 @ nat @ M2 )
=> ( ( M2
!= ( bot_bot @ ( set @ nat ) ) )
=> ( ~ ( member @ nat @ ( zero_zero @ nat ) @ M2 )
=> ( ! [M4: nat,N2: nat] :
( ( member @ nat @ M4 @ M2 )
=> ( ( member @ nat @ N2 @ M2 )
=> ( member @ nat @ ( gcd_lcm @ nat @ M4 @ N2 ) @ M2 ) ) )
=> ( ( gcd_Lcm @ nat @ M2 )
= ( lattic643756798349783984er_Max @ nat @ M2 ) ) ) ) ) ) ).
% Lcm_eq_Max_nat
thf(fact_6004_map__of__map,axiom,
! [B: $tType,C: $tType,A: $tType,F: C > B,Xs2: list @ ( product_prod @ A @ C )] :
( ( map_of @ A @ B
@ ( map @ ( product_prod @ A @ C ) @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ C @ ( product_prod @ A @ B )
@ ^ [K3: A,V6: C] : ( product_Pair @ A @ B @ K3 @ ( F @ V6 ) ) )
@ Xs2 ) )
= ( comp @ ( option @ C ) @ ( option @ B ) @ A @ ( map_option @ C @ B @ F ) @ ( map_of @ A @ C @ Xs2 ) ) ) ).
% map_of_map
thf(fact_6005_None__eq__map__option__iff,axiom,
! [A: $tType,B: $tType,F: B > A,X: option @ B] :
( ( ( none @ A )
= ( map_option @ B @ A @ F @ X ) )
= ( X
= ( none @ B ) ) ) ).
% None_eq_map_option_iff
thf(fact_6006_map__option__is__None,axiom,
! [A: $tType,B: $tType,F: B > A,Opt: option @ B] :
( ( ( map_option @ B @ A @ F @ Opt )
= ( none @ A ) )
= ( Opt
= ( none @ B ) ) ) ).
% map_option_is_None
thf(fact_6007_option_Omap__disc__iff,axiom,
! [B: $tType,A: $tType,F: A > B,A3: option @ A] :
( ( ( map_option @ A @ B @ F @ A3 )
= ( none @ B ) )
= ( A3
= ( none @ A ) ) ) ).
% option.map_disc_iff
thf(fact_6008_map__option__eq__Some,axiom,
! [A: $tType,B: $tType,F: B > A,Xo: option @ B,Y3: A] :
( ( ( map_option @ B @ A @ F @ Xo )
= ( some @ A @ Y3 ) )
= ( ? [Z3: B] :
( ( Xo
= ( some @ B @ Z3 ) )
& ( ( F @ Z3 )
= Y3 ) ) ) ) ).
% map_option_eq_Some
thf(fact_6009_Lcm__eq__0__I__nat,axiom,
! [A5: set @ nat] :
( ( member @ nat @ ( zero_zero @ nat ) @ A5 )
=> ( ( gcd_Lcm @ nat @ A5 )
= ( zero_zero @ nat ) ) ) ).
% Lcm_eq_0_I_nat
thf(fact_6010_abs__Lcm__eq,axiom,
! [K5: set @ int] :
( ( abs_abs @ int @ ( gcd_Lcm @ int @ K5 ) )
= ( gcd_Lcm @ int @ K5 ) ) ).
% abs_Lcm_eq
thf(fact_6011_Lcm__UNIV,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ( ( gcd_Lcm @ A @ ( top_top @ ( set @ A ) ) )
= ( zero_zero @ A ) ) ) ).
% Lcm_UNIV
thf(fact_6012_Lcm__empty,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ( ( gcd_Lcm @ A @ ( bot_bot @ ( set @ A ) ) )
= ( one_one @ A ) ) ) ).
% Lcm_empty
thf(fact_6013_Lcm__1__iff,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A5: set @ A] :
( ( ( gcd_Lcm @ A @ A5 )
= ( one_one @ A ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ A5 )
=> ( dvd_dvd @ A @ X2 @ ( one_one @ A ) ) ) ) ) ) ).
% Lcm_1_iff
thf(fact_6014_Lcm__insert,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A3: A,A5: set @ A] :
( ( gcd_Lcm @ A @ ( insert2 @ A @ A3 @ A5 ) )
= ( gcd_lcm @ A @ A3 @ ( gcd_Lcm @ A @ A5 ) ) ) ) ).
% Lcm_insert
thf(fact_6015_case__map__option,axiom,
! [B: $tType,A: $tType,C: $tType,G: A,H: B > A,F: C > B,X: option @ C] :
( ( case_option @ A @ B @ G @ H @ ( map_option @ C @ B @ F @ X ) )
= ( case_option @ A @ C @ G @ ( comp @ B @ A @ C @ H @ F ) @ X ) ) ).
% case_map_option
thf(fact_6016_Lcm__fin__eq__Lcm,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( semiring_gcd_Lcm_fin @ A @ A5 )
= ( gcd_Lcm @ A @ A5 ) ) ) ) ).
% Lcm_fin_eq_Lcm
thf(fact_6017_Lcm__0__iff__nat,axiom,
! [A5: set @ nat] :
( ( finite_finite2 @ nat @ A5 )
=> ( ( ( gcd_Lcm @ nat @ A5 )
= ( zero_zero @ nat ) )
= ( member @ nat @ ( zero_zero @ nat ) @ A5 ) ) ) ).
% Lcm_0_iff_nat
thf(fact_6018_dom__map__option__comp,axiom,
! [B: $tType,C: $tType,A: $tType,G: C > B,M: A > ( option @ C )] :
( ( dom @ A @ B @ ( comp @ ( option @ C ) @ ( option @ B ) @ A @ ( map_option @ C @ B @ G ) @ M ) )
= ( dom @ A @ C @ M ) ) ).
% dom_map_option_comp
thf(fact_6019_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 )
@ ^ [X2: A] : ( none @ C ) )
= ( ^ [X2: A] : ( none @ B ) ) ) ).
% map_option_o_empty
thf(fact_6020_Lcm__2,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A3: A,B2: A] :
( ( gcd_Lcm @ A @ ( insert2 @ A @ A3 @ ( insert2 @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( gcd_lcm @ A @ A3 @ B2 ) ) ) ).
% Lcm_2
thf(fact_6021_Lcm__abs__eq,axiom,
! [K5: set @ int] :
( ( gcd_Lcm @ int @ ( image2 @ int @ int @ ( abs_abs @ int ) @ K5 ) )
= ( gcd_Lcm @ int @ K5 ) ) ).
% Lcm_abs_eq
thf(fact_6022_map__option__o__map__upd,axiom,
! [A: $tType,B: $tType,C: $tType,F: C > B,M: A > ( option @ C ),A3: A,B2: C] :
( ( comp @ ( option @ C ) @ ( option @ B ) @ A @ ( map_option @ C @ B @ F ) @ ( fun_upd @ A @ ( option @ C ) @ M @ A3 @ ( some @ C @ B2 ) ) )
= ( fun_upd @ A @ ( option @ B ) @ ( comp @ ( option @ C ) @ ( option @ B ) @ A @ ( map_option @ C @ B @ F ) @ M ) @ A3 @ ( some @ B @ ( F @ B2 ) ) ) ) ).
% map_option_o_map_upd
thf(fact_6023_Lcm__int__eq,axiom,
! [N6: set @ nat] :
( ( gcd_Lcm @ int @ ( image2 @ nat @ int @ ( semiring_1_of_nat @ int ) @ N6 ) )
= ( semiring_1_of_nat @ int @ ( gcd_Lcm @ nat @ N6 ) ) ) ).
% Lcm_int_eq
thf(fact_6024_Lcm__nat__abs__eq,axiom,
! [K5: set @ int] :
( ( gcd_Lcm @ nat
@ ( image2 @ int @ nat
@ ^ [K3: int] : ( nat2 @ ( abs_abs @ int @ K3 ) )
@ K5 ) )
= ( nat2 @ ( gcd_Lcm @ int @ K5 ) ) ) ).
% Lcm_nat_abs_eq
thf(fact_6025_Lcm__nat__insert,axiom,
! [N: nat,M2: set @ nat] :
( ( gcd_Lcm @ nat @ ( insert2 @ nat @ N @ M2 ) )
= ( gcd_lcm @ nat @ N @ ( gcd_Lcm @ nat @ M2 ) ) ) ).
% Lcm_nat_insert
thf(fact_6026_option_Osimps_I8_J,axiom,
! [A: $tType,B: $tType,F: A > B] :
( ( map_option @ A @ B @ F @ ( none @ A ) )
= ( none @ B ) ) ).
% option.simps(8)
thf(fact_6027_Lcm__least__int,axiom,
! [A5: set @ int,A3: int] :
( ! [B3: int] :
( ( member @ int @ B3 @ A5 )
=> ( dvd_dvd @ int @ B3 @ A3 ) )
=> ( dvd_dvd @ int @ ( gcd_Lcm @ int @ A5 ) @ A3 ) ) ).
% Lcm_least_int
thf(fact_6028_dvd__Lcm__int,axiom,
! [M: int,M2: set @ int] :
( ( member @ int @ M @ M2 )
=> ( dvd_dvd @ int @ M @ ( gcd_Lcm @ int @ M2 ) ) ) ).
% dvd_Lcm_int
thf(fact_6029_Lcm__dvd__iff,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A5: set @ A,X: A] :
( ( dvd_dvd @ A @ ( gcd_Lcm @ A @ A5 ) @ X )
= ( ! [X2: A] :
( ( member @ A @ X2 @ A5 )
=> ( dvd_dvd @ A @ X2 @ X ) ) ) ) ) ).
% Lcm_dvd_iff
thf(fact_6030_Lcm__least,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A5: set @ A,A3: A] :
( ! [B3: A] :
( ( member @ A @ B3 @ A5 )
=> ( dvd_dvd @ A @ B3 @ A3 ) )
=> ( dvd_dvd @ A @ ( gcd_Lcm @ A @ A5 ) @ A3 ) ) ) ).
% Lcm_least
thf(fact_6031_Lcm__dvdD,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A5: set @ A,X: A,Y3: A] :
( ( dvd_dvd @ A @ ( gcd_Lcm @ A @ A5 ) @ X )
=> ( ( member @ A @ Y3 @ A5 )
=> ( dvd_dvd @ A @ Y3 @ X ) ) ) ) ).
% Lcm_dvdD
thf(fact_6032_dvd__Lcm,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A3: A,A5: set @ A] :
( ( member @ A @ A3 @ A5 )
=> ( dvd_dvd @ A @ A3 @ ( gcd_Lcm @ A @ A5 ) ) ) ) ).
% dvd_Lcm
thf(fact_6033_option_Omap__ident,axiom,
! [A: $tType,T2: option @ A] :
( ( map_option @ A @ A
@ ^ [X2: A] : X2
@ T2 )
= T2 ) ).
% option.map_ident
thf(fact_6034_dvd__Lcm__nat,axiom,
! [M: nat,M2: set @ nat] :
( ( member @ nat @ M @ M2 )
=> ( dvd_dvd @ nat @ M @ ( gcd_Lcm @ nat @ M2 ) ) ) ).
% dvd_Lcm_nat
thf(fact_6035_Lcm__dvd__nat,axiom,
! [M2: set @ nat,N: nat] :
( ! [X4: nat] :
( ( member @ nat @ X4 @ M2 )
=> ( dvd_dvd @ nat @ X4 @ N ) )
=> ( dvd_dvd @ nat @ ( gcd_Lcm @ nat @ M2 ) @ N ) ) ).
% Lcm_dvd_nat
thf(fact_6036_Lcm__eq__0__I,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A5: set @ A] :
( ( member @ A @ ( zero_zero @ A ) @ A5 )
=> ( ( gcd_Lcm @ A @ A5 )
= ( zero_zero @ A ) ) ) ) ).
% Lcm_eq_0_I
thf(fact_6037_map__option__cong,axiom,
! [B: $tType,A: $tType,X: option @ A,Y3: option @ A,F: A > B,G: A > B] :
( ( X = Y3 )
=> ( ! [A4: A] :
( ( Y3
= ( some @ A @ A4 ) )
=> ( ( F @ A4 )
= ( G @ A4 ) ) )
=> ( ( map_option @ A @ B @ F @ X )
= ( map_option @ A @ B @ G @ Y3 ) ) ) ) ).
% map_option_cong
thf(fact_6038_option_Osimps_I9_J,axiom,
! [B: $tType,A: $tType,F: A > B,X22: A] :
( ( map_option @ A @ B @ F @ ( some @ A @ X22 ) )
= ( some @ B @ ( F @ X22 ) ) ) ).
% option.simps(9)
thf(fact_6039_dom__map__option,axiom,
! [B: $tType,C: $tType,A: $tType,F: A > C > B,M: A > ( option @ C )] :
( ( dom @ A @ B
@ ^ [K3: A] : ( map_option @ C @ B @ ( F @ K3 ) @ ( M @ K3 ) ) )
= ( dom @ A @ C @ M ) ) ).
% dom_map_option
thf(fact_6040_map__option_Oidentity,axiom,
! [A: $tType] :
( ( map_option @ A @ A
@ ^ [X2: A] : X2 )
= ( id @ ( option @ A ) ) ) ).
% map_option.identity
thf(fact_6041_option_Omap__id,axiom,
! [A: $tType,T2: option @ A] :
( ( map_option @ A @ A @ ( id @ A ) @ T2 )
= T2 ) ).
% option.map_id
thf(fact_6042_option_Omap__id0,axiom,
! [A: $tType] :
( ( map_option @ A @ A @ ( id @ A ) )
= ( id @ ( option @ A ) ) ) ).
% option.map_id0
thf(fact_6043_map__option_Ocomp,axiom,
! [C: $tType,B: $tType,A: $tType,F: B > C,G: A > B] :
( ( comp @ ( option @ B ) @ ( option @ C ) @ ( option @ A ) @ ( map_option @ B @ C @ F ) @ ( map_option @ A @ B @ G ) )
= ( map_option @ A @ C @ ( comp @ B @ C @ A @ F @ G ) ) ) ).
% map_option.comp
thf(fact_6044_option_Omap__comp,axiom,
! [B: $tType,C: $tType,A: $tType,G: B > C,F: A > B,V: option @ A] :
( ( map_option @ B @ C @ G @ ( map_option @ A @ B @ F @ V ) )
= ( map_option @ A @ C @ ( comp @ B @ C @ A @ G @ F ) @ V ) ) ).
% option.map_comp
thf(fact_6045_map__option_Ocompositionality,axiom,
! [B: $tType,C: $tType,A: $tType,F: B > C,G: A > B,Option: option @ A] :
( ( map_option @ B @ C @ F @ ( map_option @ A @ B @ G @ Option ) )
= ( map_option @ A @ C @ ( comp @ B @ C @ A @ F @ G ) @ Option ) ) ).
% map_option.compositionality
thf(fact_6046_option_Omap__sel,axiom,
! [B: $tType,A: $tType,A3: option @ A,F: A > B] :
( ( A3
!= ( none @ A ) )
=> ( ( the2 @ B @ ( map_option @ A @ B @ F @ A3 ) )
= ( F @ ( the2 @ A @ A3 ) ) ) ) ).
% option.map_sel
thf(fact_6047_Lcm__int__greater__eq__0,axiom,
! [K5: set @ int] : ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( gcd_Lcm @ int @ K5 ) ) ).
% Lcm_int_greater_eq_0
thf(fact_6048_Lcm__0__iff,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( ( gcd_Lcm @ A @ A5 )
= ( zero_zero @ A ) )
= ( member @ A @ ( zero_zero @ A ) @ A5 ) ) ) ) ).
% Lcm_0_iff
thf(fact_6049_Lcm__0__iff_H,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A5: set @ A] :
( ( ( gcd_Lcm @ A @ A5 )
= ( zero_zero @ A ) )
= ( ~ ? [L4: A] :
( ( L4
!= ( zero_zero @ A ) )
& ! [X2: A] :
( ( member @ A @ X2 @ A5 )
=> ( dvd_dvd @ A @ X2 @ L4 ) ) ) ) ) ) ).
% Lcm_0_iff'
thf(fact_6050_Lcm__no__multiple,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A5: set @ A] :
( ! [M4: A] :
( ( M4
!= ( zero_zero @ A ) )
=> ? [X5: A] :
( ( member @ A @ X5 @ A5 )
& ~ ( dvd_dvd @ A @ X5 @ M4 ) ) )
=> ( ( gcd_Lcm @ A @ A5 )
= ( zero_zero @ A ) ) ) ) ).
% Lcm_no_multiple
thf(fact_6051_Lcm__nat__infinite,axiom,
! [M2: set @ nat] :
( ~ ( finite_finite2 @ nat @ M2 )
=> ( ( gcd_Lcm @ nat @ M2 )
= ( zero_zero @ nat ) ) ) ).
% Lcm_nat_infinite
thf(fact_6052_Lcm__subset,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A5: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B4 )
=> ( dvd_dvd @ A @ ( gcd_Lcm @ A @ A5 ) @ ( gcd_Lcm @ A @ B4 ) ) ) ) ).
% Lcm_subset
thf(fact_6053_Lcm__nat__empty,axiom,
( ( gcd_Lcm @ nat @ ( bot_bot @ ( set @ nat ) ) )
= ( one_one @ nat ) ) ).
% Lcm_nat_empty
thf(fact_6054_Lcm__Un,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A5: set @ A,B4: set @ A] :
( ( gcd_Lcm @ A @ ( sup_sup @ ( set @ A ) @ A5 @ B4 ) )
= ( gcd_lcm @ A @ ( gcd_Lcm @ A @ A5 ) @ ( gcd_Lcm @ A @ B4 ) ) ) ) ).
% Lcm_Un
thf(fact_6055_ran__map__option,axiom,
! [A: $tType,C: $tType,B: $tType,F: C > A,M: B > ( option @ C )] :
( ( ran @ B @ A
@ ^ [X2: B] : ( map_option @ C @ A @ F @ ( M @ X2 ) ) )
= ( image2 @ C @ A @ F @ ( ran @ B @ C @ M ) ) ) ).
% ran_map_option
thf(fact_6056_Lcm__in__lcm__closed__set__nat,axiom,
! [M2: set @ nat] :
( ( finite_finite2 @ nat @ M2 )
=> ( ( M2
!= ( bot_bot @ ( set @ nat ) ) )
=> ( ! [M4: nat,N2: nat] :
( ( member @ nat @ M4 @ M2 )
=> ( ( member @ nat @ N2 @ M2 )
=> ( member @ nat @ ( gcd_lcm @ nat @ M4 @ N2 ) @ M2 ) ) )
=> ( member @ nat @ ( gcd_Lcm @ nat @ M2 ) @ M2 ) ) ) ) ).
% Lcm_in_lcm_closed_set_nat
thf(fact_6057_Lcm__mono,axiom,
! [A: $tType,B: $tType] :
( ( semiring_Gcd @ A )
=> ! [A5: set @ B,F: B > A,G: B > A] :
( ! [X4: B] :
( ( member @ B @ X4 @ A5 )
=> ( dvd_dvd @ A @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( dvd_dvd @ A @ ( gcd_Lcm @ A @ ( image2 @ B @ A @ F @ A5 ) ) @ ( gcd_Lcm @ A @ ( image2 @ B @ A @ G @ A5 ) ) ) ) ) ).
% Lcm_mono
thf(fact_6058_option_Osize__gen__o__map,axiom,
! [B: $tType,A: $tType,F: B > nat,G: A > B] :
( ( comp @ ( option @ B ) @ nat @ ( option @ A ) @ ( size_option @ B @ F ) @ ( map_option @ A @ B @ G ) )
= ( size_option @ A @ ( comp @ B @ nat @ A @ F @ G ) ) ) ).
% option.size_gen_o_map
thf(fact_6059_Gcd__nat__def,axiom,
( ( gcd_Gcd @ nat )
= ( ^ [M9: set @ nat] :
( gcd_Lcm @ nat
@ ( collect @ nat
@ ^ [D4: nat] :
! [X2: nat] :
( ( member @ nat @ X2 @ M9 )
=> ( dvd_dvd @ nat @ D4 @ X2 ) ) ) ) ) ) ).
% Gcd_nat_def
thf(fact_6060_Lcm__int__def,axiom,
( ( gcd_Lcm @ int )
= ( ^ [K7: set @ int] : ( semiring_1_of_nat @ int @ ( gcd_Lcm @ nat @ ( image2 @ int @ nat @ ( comp @ int @ nat @ int @ nat2 @ ( abs_abs @ int ) ) @ K7 ) ) ) ) ) ).
% Lcm_int_def
thf(fact_6061_map__option__case,axiom,
! [A: $tType,B: $tType] :
( ( map_option @ B @ A )
= ( ^ [F2: B > A] :
( case_option @ ( option @ A ) @ B @ ( none @ A )
@ ^ [X2: B] : ( some @ A @ ( F2 @ X2 ) ) ) ) ) ).
% map_option_case
thf(fact_6062_option_Oinj__map,axiom,
! [B: $tType,A: $tType,F: A > B] :
( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
=> ( inj_on @ ( option @ A ) @ ( option @ B ) @ ( map_option @ A @ B @ F ) @ ( top_top @ ( set @ ( option @ A ) ) ) ) ) ).
% option.inj_map
thf(fact_6063_Lcm__no__units,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ( ( gcd_Lcm @ A )
= ( ^ [A8: set @ A] :
( gcd_Lcm @ A
@ ( minus_minus @ ( set @ A ) @ A8
@ ( collect @ A
@ ^ [A6: A] : ( dvd_dvd @ A @ A6 @ ( one_one @ A ) ) ) ) ) ) ) ) ).
% Lcm_no_units
thf(fact_6064_Gcd__Lcm,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ( ( gcd_Gcd @ A )
= ( ^ [A8: set @ A] :
( gcd_Lcm @ A
@ ( collect @ A
@ ^ [B5: A] :
! [X2: A] :
( ( member @ A @ X2 @ A8 )
=> ( dvd_dvd @ A @ B5 @ X2 ) ) ) ) ) ) ) ).
% Gcd_Lcm
thf(fact_6065_Lcm__Gcd,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ( ( gcd_Lcm @ A )
= ( ^ [A8: set @ A] :
( gcd_Gcd @ A
@ ( collect @ A
@ ^ [B5: A] :
! [X2: A] :
( ( member @ A @ X2 @ A8 )
=> ( dvd_dvd @ A @ X2 @ B5 ) ) ) ) ) ) ) ).
% Lcm_Gcd
thf(fact_6066_Lcm__set__eq__fold,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [Xs2: list @ A] :
( ( gcd_Lcm @ A @ ( set2 @ A @ Xs2 ) )
= ( fold @ A @ A @ ( gcd_lcm @ A ) @ Xs2 @ ( one_one @ A ) ) ) ) ).
% Lcm_set_eq_fold
thf(fact_6067_Lcm__nat__set__eq__fold,axiom,
! [Xs2: list @ nat] :
( ( gcd_Lcm @ nat @ ( set2 @ nat @ Xs2 ) )
= ( fold @ nat @ nat @ ( gcd_lcm @ nat ) @ Xs2 @ ( one_one @ nat ) ) ) ).
% Lcm_nat_set_eq_fold
thf(fact_6068_Lcm__nat__def,axiom,
( ( gcd_Lcm @ nat )
= ( ^ [M9: set @ nat] : ( if @ nat @ ( finite_finite2 @ nat @ M9 ) @ ( lattic5214292709420241887eutr_F @ nat @ ( gcd_lcm @ nat ) @ ( one_one @ nat ) @ M9 ) @ ( zero_zero @ nat ) ) ) ) ).
% Lcm_nat_def
thf(fact_6069_aboveS__def,axiom,
! [A: $tType] :
( ( order_aboveS @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A ),A6: A] :
( collect @ A
@ ^ [B5: A] :
( ( B5 != A6 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A6 @ B5 ) @ R4 ) ) ) ) ) ).
% aboveS_def
thf(fact_6070_option_Orec__o__map,axiom,
! [B: $tType,C: $tType,A: $tType,G: C,Ga: B > C,F: A > B] :
( ( comp @ ( option @ B ) @ C @ ( option @ A ) @ ( rec_option @ C @ B @ G @ Ga ) @ ( map_option @ A @ B @ F ) )
= ( rec_option @ C @ A @ G
@ ^ [X2: A] : ( Ga @ ( F @ X2 ) ) ) ) ).
% option.rec_o_map
thf(fact_6071_prod__list__def,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ( ( groups5270119922927024881d_list @ A )
= ( groups_monoid_F @ A @ ( times_times @ A ) @ ( one_one @ A ) ) ) ) ).
% prod_list_def
thf(fact_6072_option_Osimps_I6_J,axiom,
! [A: $tType,C: $tType,F1: C,F23: A > C] :
( ( rec_option @ C @ A @ F1 @ F23 @ ( none @ A ) )
= F1 ) ).
% option.simps(6)
thf(fact_6073_option_Osimps_I7_J,axiom,
! [C: $tType,A: $tType,F1: C,F23: A > C,X22: A] :
( ( rec_option @ C @ A @ F1 @ F23 @ ( some @ A @ X22 ) )
= ( F23 @ X22 ) ) ).
% option.simps(7)
thf(fact_6074_sum__list__def,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ( ( groups8242544230860333062m_list @ A )
= ( groups_monoid_F @ A @ ( plus_plus @ A ) @ ( zero_zero @ A ) ) ) ) ).
% sum_list_def
thf(fact_6075_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,X2: A] : ( G2 @ ( product_fst @ B @ C @ ( F2 @ X2 ) ) @ ( product_snd @ B @ C @ ( F2 @ X2 ) ) ) ) ) ).
% scomp_unfold
thf(fact_6076_normalize__stable,axiom,
! [Q2: int,P4: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ Q2 )
=> ( ( algebr8660921524188924756oprime @ int @ P4 @ Q2 )
=> ( ( normalize @ ( product_Pair @ int @ int @ P4 @ Q2 ) )
= ( product_Pair @ int @ int @ P4 @ Q2 ) ) ) ) ).
% normalize_stable
thf(fact_6077_coprime__mult__right__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( algebr8660921524188924756oprime @ A @ C2 @ ( times_times @ A @ A3 @ B2 ) )
= ( ( algebr8660921524188924756oprime @ A @ C2 @ A3 )
& ( algebr8660921524188924756oprime @ A @ C2 @ B2 ) ) ) ) ).
% coprime_mult_right_iff
thf(fact_6078_coprime__mult__left__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( algebr8660921524188924756oprime @ A @ ( times_times @ A @ A3 @ B2 ) @ C2 )
= ( ( algebr8660921524188924756oprime @ A @ A3 @ C2 )
& ( algebr8660921524188924756oprime @ A @ B2 @ C2 ) ) ) ) ).
% coprime_mult_left_iff
thf(fact_6079_coprime__minus__right__iff,axiom,
! [A: $tType] :
( ( ring_gcd @ A )
=> ! [A3: A,B2: A] :
( ( algebr8660921524188924756oprime @ A @ A3 @ ( uminus_uminus @ A @ B2 ) )
= ( algebr8660921524188924756oprime @ A @ A3 @ B2 ) ) ) ).
% coprime_minus_right_iff
thf(fact_6080_coprime__minus__left__iff,axiom,
! [A: $tType] :
( ( ring_gcd @ A )
=> ! [A3: A,B2: A] :
( ( algebr8660921524188924756oprime @ A @ ( uminus_uminus @ A @ A3 ) @ B2 )
= ( algebr8660921524188924756oprime @ A @ A3 @ B2 ) ) ) ).
% coprime_minus_left_iff
thf(fact_6081_scomp__apply,axiom,
! [A: $tType,D: $tType,C: $tType,B: $tType] :
( ( product_scomp @ B @ C @ D @ A )
= ( ^ [F2: B > ( product_prod @ C @ D ),G2: C > D > A,X2: B] : ( product_case_prod @ C @ D @ A @ G2 @ ( F2 @ X2 ) ) ) ) ).
% scomp_apply
thf(fact_6082_coprime__self,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A] :
( ( algebr8660921524188924756oprime @ A @ A3 @ A3 )
= ( dvd_dvd @ A @ A3 @ ( one_one @ A ) ) ) ) ).
% coprime_self
thf(fact_6083_coprime__mod__left__iff,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [B2: A,A3: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( algebr8660921524188924756oprime @ A @ ( modulo_modulo @ A @ A3 @ B2 ) @ B2 )
= ( algebr8660921524188924756oprime @ A @ A3 @ B2 ) ) ) ) ).
% coprime_mod_left_iff
thf(fact_6084_coprime__mod__right__iff,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [A3: A,B2: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( algebr8660921524188924756oprime @ A @ A3 @ ( modulo_modulo @ A @ B2 @ A3 ) )
= ( algebr8660921524188924756oprime @ A @ A3 @ B2 ) ) ) ) ).
% coprime_mod_right_iff
thf(fact_6085_coprime__power__left__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,N: nat,B2: A] :
( ( algebr8660921524188924756oprime @ A @ ( power_power @ A @ A3 @ N ) @ B2 )
= ( ( algebr8660921524188924756oprime @ A @ A3 @ B2 )
| ( N
= ( zero_zero @ nat ) ) ) ) ) ).
% coprime_power_left_iff
thf(fact_6086_coprime__power__right__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A,N: nat] :
( ( algebr8660921524188924756oprime @ A @ A3 @ ( power_power @ A @ B2 @ N ) )
= ( ( algebr8660921524188924756oprime @ A @ A3 @ B2 )
| ( N
= ( zero_zero @ nat ) ) ) ) ) ).
% coprime_power_right_iff
thf(fact_6087_coprime__imp__gcd__eq__1,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A] :
( ( algebr8660921524188924756oprime @ A @ A3 @ B2 )
=> ( ( gcd_gcd @ A @ A3 @ B2 )
= ( one_one @ A ) ) ) ) ).
% coprime_imp_gcd_eq_1
thf(fact_6088_coprime__0__left__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A] :
( ( algebr8660921524188924756oprime @ A @ ( zero_zero @ A ) @ A3 )
= ( dvd_dvd @ A @ A3 @ ( one_one @ A ) ) ) ) ).
% coprime_0_left_iff
thf(fact_6089_coprime__0__right__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A] :
( ( algebr8660921524188924756oprime @ A @ A3 @ ( zero_zero @ A ) )
= ( dvd_dvd @ A @ A3 @ ( one_one @ A ) ) ) ) ).
% coprime_0_right_iff
thf(fact_6090_coprime__mult__self__right__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( algebr8660921524188924756oprime @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) )
= ( ( dvd_dvd @ A @ C2 @ ( one_one @ A ) )
& ( algebr8660921524188924756oprime @ A @ A3 @ B2 ) ) ) ) ).
% coprime_mult_self_right_iff
thf(fact_6091_coprime__mult__self__left__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( algebr8660921524188924756oprime @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( ( dvd_dvd @ A @ C2 @ ( one_one @ A ) )
& ( algebr8660921524188924756oprime @ A @ A3 @ B2 ) ) ) ) ).
% coprime_mult_self_left_iff
thf(fact_6092_is__unit__gcd,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ ( gcd_gcd @ A @ A3 @ B2 ) @ ( one_one @ A ) )
= ( algebr8660921524188924756oprime @ A @ A3 @ B2 ) ) ) ).
% is_unit_gcd
thf(fact_6093_prod__list__coprime__right,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [Xs2: list @ A,A3: A] :
( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs2 ) )
=> ( algebr8660921524188924756oprime @ A @ A3 @ X4 ) )
=> ( algebr8660921524188924756oprime @ A @ A3 @ ( groups5270119922927024881d_list @ A @ Xs2 ) ) ) ) ).
% prod_list_coprime_right
thf(fact_6094_prod__list__coprime__left,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [Xs2: list @ A,A3: A] :
( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs2 ) )
=> ( algebr8660921524188924756oprime @ A @ X4 @ A3 ) )
=> ( algebr8660921524188924756oprime @ A @ ( groups5270119922927024881d_list @ A @ Xs2 ) @ A3 ) ) ) ).
% prod_list_coprime_left
thf(fact_6095_normalize__coprime,axiom,
! [R: product_prod @ int @ int,P4: int,Q2: int] :
( ( ( normalize @ R )
= ( product_Pair @ int @ int @ P4 @ Q2 ) )
=> ( algebr8660921524188924756oprime @ int @ P4 @ Q2 ) ) ).
% normalize_coprime
thf(fact_6096_scomp__def,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,X2: A] : ( product_case_prod @ B @ C @ D @ G2 @ ( F2 @ X2 ) ) ) ) ).
% scomp_def
thf(fact_6097_gcd__mult__right__right__cancel,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( algebr8660921524188924756oprime @ A @ A3 @ C2 )
=> ( ( gcd_gcd @ A @ A3 @ ( times_times @ A @ B2 @ C2 ) )
= ( gcd_gcd @ A @ A3 @ B2 ) ) ) ) ).
% gcd_mult_right_right_cancel
thf(fact_6098_gcd__mult__right__left__cancel,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( algebr8660921524188924756oprime @ A @ A3 @ C2 )
=> ( ( gcd_gcd @ A @ A3 @ ( times_times @ A @ C2 @ B2 ) )
= ( gcd_gcd @ A @ A3 @ B2 ) ) ) ) ).
% gcd_mult_right_left_cancel
thf(fact_6099_gcd__mult__left__right__cancel,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [B2: A,C2: A,A3: A] :
( ( algebr8660921524188924756oprime @ A @ B2 @ C2 )
=> ( ( gcd_gcd @ A @ ( times_times @ A @ A3 @ C2 ) @ B2 )
= ( gcd_gcd @ A @ A3 @ B2 ) ) ) ) ).
% gcd_mult_left_right_cancel
thf(fact_6100_gcd__mult__left__left__cancel,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [B2: A,C2: A,A3: A] :
( ( algebr8660921524188924756oprime @ A @ B2 @ C2 )
=> ( ( gcd_gcd @ A @ ( times_times @ A @ C2 @ A3 ) @ B2 )
= ( gcd_gcd @ A @ A3 @ B2 ) ) ) ) ).
% gcd_mult_left_left_cancel
thf(fact_6101_gcd__eq__1__imp__coprime,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A] :
( ( ( gcd_gcd @ A @ A3 @ B2 )
= ( one_one @ A ) )
=> ( algebr8660921524188924756oprime @ A @ A3 @ B2 ) ) ) ).
% gcd_eq_1_imp_coprime
thf(fact_6102_coprime__iff__gcd__eq__1,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ( ( algebr8660921524188924756oprime @ A )
= ( ^ [A6: A,B5: A] :
( ( gcd_gcd @ A @ A6 @ B5 )
= ( one_one @ A ) ) ) ) ) ).
% coprime_iff_gcd_eq_1
thf(fact_6103_is__unit__right__imp__coprime,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A3: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( algebr8660921524188924756oprime @ A @ A3 @ B2 ) ) ) ).
% is_unit_right_imp_coprime
thf(fact_6104_is__unit__left__imp__coprime,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( algebr8660921524188924756oprime @ A @ A3 @ B2 ) ) ) ).
% is_unit_left_imp_coprime
thf(fact_6105_coprime__common__divisor,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( algebr8660921524188924756oprime @ A @ A3 @ B2 )
=> ( ( dvd_dvd @ A @ C2 @ A3 )
=> ( ( dvd_dvd @ A @ C2 @ B2 )
=> ( dvd_dvd @ A @ C2 @ ( one_one @ A ) ) ) ) ) ) ).
% coprime_common_divisor
thf(fact_6106_coprime__absorb__right,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [Y3: A,X: A] :
( ( dvd_dvd @ A @ Y3 @ X )
=> ( ( algebr8660921524188924756oprime @ A @ X @ Y3 )
= ( dvd_dvd @ A @ Y3 @ ( one_one @ A ) ) ) ) ) ).
% coprime_absorb_right
thf(fact_6107_coprime__imp__coprime,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C2: A,D3: A,A3: A,B2: A] :
( ( algebr8660921524188924756oprime @ A @ C2 @ D3 )
=> ( ! [E3: A] :
( ~ ( dvd_dvd @ A @ E3 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ E3 @ A3 )
=> ( ( dvd_dvd @ A @ E3 @ B2 )
=> ( dvd_dvd @ A @ E3 @ C2 ) ) ) )
=> ( ! [E3: A] :
( ~ ( dvd_dvd @ A @ E3 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ E3 @ A3 )
=> ( ( dvd_dvd @ A @ E3 @ B2 )
=> ( dvd_dvd @ A @ E3 @ D3 ) ) ) )
=> ( algebr8660921524188924756oprime @ A @ A3 @ B2 ) ) ) ) ) ).
% coprime_imp_coprime
thf(fact_6108_coprime__absorb__left,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [X: A,Y3: A] :
( ( dvd_dvd @ A @ X @ Y3 )
=> ( ( algebr8660921524188924756oprime @ A @ X @ Y3 )
= ( dvd_dvd @ A @ X @ ( one_one @ A ) ) ) ) ) ).
% coprime_absorb_left
thf(fact_6109_not__coprimeI,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( dvd_dvd @ A @ C2 @ A3 )
=> ( ( dvd_dvd @ A @ C2 @ B2 )
=> ( ~ ( dvd_dvd @ A @ C2 @ ( one_one @ A ) )
=> ~ ( algebr8660921524188924756oprime @ A @ A3 @ B2 ) ) ) ) ) ).
% not_coprimeI
thf(fact_6110_not__coprimeE,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A] :
( ~ ( algebr8660921524188924756oprime @ A @ A3 @ B2 )
=> ~ ! [C3: A] :
( ( dvd_dvd @ A @ C3 @ A3 )
=> ( ( dvd_dvd @ A @ C3 @ B2 )
=> ( dvd_dvd @ A @ C3 @ ( one_one @ A ) ) ) ) ) ) ).
% not_coprimeE
thf(fact_6111_coprime__def,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ( ( algebr8660921524188924756oprime @ A )
= ( ^ [A6: A,B5: A] :
! [C4: A] :
( ( dvd_dvd @ A @ C4 @ A6 )
=> ( ( dvd_dvd @ A @ C4 @ B5 )
=> ( dvd_dvd @ A @ C4 @ ( one_one @ A ) ) ) ) ) ) ) ).
% coprime_def
thf(fact_6112_coprimeI,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A] :
( ! [C3: A] :
( ( dvd_dvd @ A @ C3 @ A3 )
=> ( ( dvd_dvd @ A @ C3 @ B2 )
=> ( dvd_dvd @ A @ C3 @ ( one_one @ A ) ) ) )
=> ( algebr8660921524188924756oprime @ A @ A3 @ B2 ) ) ) ).
% coprimeI
thf(fact_6113_coprime__add__one__right,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A] : ( algebr8660921524188924756oprime @ A @ A3 @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) ) ) ).
% coprime_add_one_right
thf(fact_6114_coprime__add__one__left,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A] : ( algebr8660921524188924756oprime @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ A3 ) ) ).
% coprime_add_one_left
thf(fact_6115_coprime__diff__one__left,axiom,
! [A: $tType] :
( ( ring_gcd @ A )
=> ! [A3: A] : ( algebr8660921524188924756oprime @ A @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) @ A3 ) ) ).
% coprime_diff_one_left
thf(fact_6116_coprime__doff__one__right,axiom,
! [A: $tType] :
( ( ring_gcd @ A )
=> ! [A3: A] : ( algebr8660921524188924756oprime @ A @ A3 @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) ) ) ).
% coprime_doff_one_right
thf(fact_6117_coprime__1__right,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A] : ( algebr8660921524188924756oprime @ A @ A3 @ ( one_one @ A ) ) ) ).
% coprime_1_right
thf(fact_6118_coprime__1__left,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A] : ( algebr8660921524188924756oprime @ A @ ( one_one @ A ) @ A3 ) ) ).
% coprime_1_left
thf(fact_6119_coprime__divisors,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,C2: A,B2: A,D3: A] :
( ( dvd_dvd @ A @ A3 @ C2 )
=> ( ( dvd_dvd @ A @ B2 @ D3 )
=> ( ( algebr8660921524188924756oprime @ A @ C2 @ D3 )
=> ( algebr8660921524188924756oprime @ A @ A3 @ B2 ) ) ) ) ) ).
% coprime_divisors
thf(fact_6120_coprime__commute,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ( ( algebr8660921524188924756oprime @ A )
= ( ^ [B5: A,A6: A] : ( algebr8660921524188924756oprime @ A @ A6 @ B5 ) ) ) ) ).
% coprime_commute
thf(fact_6121_scomp__scomp,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,F3: $tType,E: $tType,F: A > ( product_prod @ E @ F3 ),G: E > F3 > ( product_prod @ C @ D ),H: C > D > B] :
( ( product_scomp @ A @ C @ D @ B @ ( product_scomp @ A @ E @ F3 @ ( product_prod @ C @ D ) @ F @ G ) @ H )
= ( product_scomp @ A @ E @ F3 @ B @ F
@ ^ [X2: E] : ( product_scomp @ F3 @ C @ D @ B @ ( G @ X2 ) @ H ) ) ) ).
% scomp_scomp
thf(fact_6122_prod__coprime__left,axiom,
! [B: $tType,A: $tType] :
( ( semiring_gcd @ A )
=> ! [A5: set @ B,F: B > A,A3: A] :
( ! [I3: B] :
( ( member @ B @ I3 @ A5 )
=> ( algebr8660921524188924756oprime @ A @ ( F @ I3 ) @ A3 ) )
=> ( algebr8660921524188924756oprime @ A @ ( groups7121269368397514597t_prod @ B @ A @ F @ A5 ) @ A3 ) ) ) ).
% prod_coprime_left
thf(fact_6123_prod__coprime__right,axiom,
! [A: $tType,B: $tType] :
( ( semiring_gcd @ A )
=> ! [A5: set @ B,A3: A,F: B > A] :
( ! [I3: B] :
( ( member @ B @ I3 @ A5 )
=> ( algebr8660921524188924756oprime @ A @ A3 @ ( F @ I3 ) ) )
=> ( algebr8660921524188924756oprime @ A @ A3 @ ( groups7121269368397514597t_prod @ B @ A @ F @ A5 ) ) ) ) ).
% prod_coprime_right
thf(fact_6124_divides__mult,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ C2 )
=> ( ( dvd_dvd @ A @ B2 @ C2 )
=> ( ( algebr8660921524188924756oprime @ A @ A3 @ B2 )
=> ( dvd_dvd @ A @ ( times_times @ A @ A3 @ B2 ) @ C2 ) ) ) ) ) ).
% divides_mult
thf(fact_6125_coprime__dvd__mult__left__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( algebr8660921524188924756oprime @ A @ A3 @ C2 )
=> ( ( dvd_dvd @ A @ A3 @ ( times_times @ A @ B2 @ C2 ) )
= ( dvd_dvd @ A @ A3 @ B2 ) ) ) ) ).
% coprime_dvd_mult_left_iff
thf(fact_6126_coprime__dvd__mult__right__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( algebr8660921524188924756oprime @ A @ A3 @ C2 )
=> ( ( dvd_dvd @ A @ A3 @ ( times_times @ A @ C2 @ B2 ) )
= ( dvd_dvd @ A @ A3 @ B2 ) ) ) ) ).
% coprime_dvd_mult_right_iff
thf(fact_6127_mult__mod__cancel__left,axiom,
! [A: $tType] :
( ( ( euclid8851590272496341667cancel @ A )
& ( semiring_gcd @ A ) )
=> ! [N: A,A3: A,M: A,B2: A] :
( ( ( modulo_modulo @ A @ ( times_times @ A @ N @ A3 ) @ M )
= ( modulo_modulo @ A @ ( times_times @ A @ N @ B2 ) @ M ) )
=> ( ( algebr8660921524188924756oprime @ A @ M @ N )
=> ( ( modulo_modulo @ A @ A3 @ M )
= ( modulo_modulo @ A @ B2 @ M ) ) ) ) ) ).
% mult_mod_cancel_left
thf(fact_6128_mult__mod__cancel__right,axiom,
! [A: $tType] :
( ( ( euclid8851590272496341667cancel @ A )
& ( semiring_gcd @ A ) )
=> ! [A3: A,N: A,M: A,B2: A] :
( ( ( modulo_modulo @ A @ ( times_times @ A @ A3 @ N ) @ M )
= ( modulo_modulo @ A @ ( times_times @ A @ B2 @ N ) @ M ) )
=> ( ( algebr8660921524188924756oprime @ A @ M @ N )
=> ( ( modulo_modulo @ A @ A3 @ M )
= ( modulo_modulo @ A @ B2 @ M ) ) ) ) ) ).
% mult_mod_cancel_right
thf(fact_6129_coprime__crossproduct__int,axiom,
! [A3: int,D3: int,B2: int,C2: int] :
( ( algebr8660921524188924756oprime @ int @ A3 @ D3 )
=> ( ( algebr8660921524188924756oprime @ int @ B2 @ C2 )
=> ( ( ( times_times @ int @ ( abs_abs @ int @ A3 ) @ ( abs_abs @ int @ C2 ) )
= ( times_times @ int @ ( abs_abs @ int @ B2 ) @ ( abs_abs @ int @ D3 ) ) )
= ( ( ( abs_abs @ int @ A3 )
= ( abs_abs @ int @ B2 ) )
& ( ( abs_abs @ int @ C2 )
= ( abs_abs @ int @ D3 ) ) ) ) ) ) ).
% coprime_crossproduct_int
thf(fact_6130_Pair__scomp,axiom,
! [A: $tType,B: $tType,C: $tType,X: C,F: C > A > B] :
( ( product_scomp @ A @ C @ A @ B @ ( product_Pair @ C @ A @ X ) @ F )
= ( F @ X ) ) ).
% Pair_scomp
thf(fact_6131_scomp__Pair,axiom,
! [C: $tType,B: $tType,A: $tType,X: A > ( product_prod @ B @ C )] :
( ( product_scomp @ A @ B @ C @ ( product_prod @ B @ C ) @ X @ ( product_Pair @ B @ C ) )
= X ) ).
% scomp_Pair
thf(fact_6132_quotient__of__coprime,axiom,
! [R: rat,P4: int,Q2: int] :
( ( ( quotient_of @ R )
= ( product_Pair @ int @ int @ P4 @ Q2 ) )
=> ( algebr8660921524188924756oprime @ int @ P4 @ Q2 ) ) ).
% quotient_of_coprime
thf(fact_6133_invertible__coprime,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ( modulo_modulo @ A @ ( times_times @ A @ A3 @ B2 ) @ C2 )
= ( one_one @ A ) )
=> ( algebr8660921524188924756oprime @ A @ A3 @ C2 ) ) ) ).
% invertible_coprime
thf(fact_6134_gcd__coprime__exists,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A] :
( ( ( gcd_gcd @ A @ A3 @ B2 )
!= ( zero_zero @ A ) )
=> ? [A17: A,B8: A] :
( ( A3
= ( times_times @ A @ A17 @ ( gcd_gcd @ A @ A3 @ B2 ) ) )
& ( B2
= ( times_times @ A @ B8 @ ( gcd_gcd @ A @ A3 @ B2 ) ) )
& ( algebr8660921524188924756oprime @ A @ A17 @ B8 ) ) ) ) ).
% gcd_coprime_exists
thf(fact_6135_gcd__coprime,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A,A7: A,B6: A] :
( ( ( gcd_gcd @ A @ A3 @ B2 )
!= ( zero_zero @ A ) )
=> ( ( A3
= ( times_times @ A @ A7 @ ( gcd_gcd @ A @ A3 @ B2 ) ) )
=> ( ( B2
= ( times_times @ A @ B6 @ ( gcd_gcd @ A @ A3 @ B2 ) ) )
=> ( algebr8660921524188924756oprime @ A @ A7 @ B6 ) ) ) ) ) ).
% gcd_coprime
thf(fact_6136_div__gcd__coprime,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A] :
( ( ( A3
!= ( zero_zero @ A ) )
| ( B2
!= ( zero_zero @ A ) ) )
=> ( algebr8660921524188924756oprime @ A @ ( divide_divide @ A @ A3 @ ( gcd_gcd @ A @ A3 @ B2 ) ) @ ( divide_divide @ A @ B2 @ ( gcd_gcd @ A @ A3 @ B2 ) ) ) ) ) ).
% div_gcd_coprime
thf(fact_6137_quotient__of__def,axiom,
( quotient_of
= ( ^ [X2: rat] :
( the @ ( product_prod @ int @ int )
@ ^ [Pair: product_prod @ int @ int] :
( ( X2
= ( 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_6138_iterate_Osimps,axiom,
! [B: $tType,A: $tType] :
( ( iterate @ B @ A )
= ( ^ [K3: code_natural,F2: B > A > ( product_prod @ B @ A ),X2: B] :
( if @ ( A > ( product_prod @ B @ A ) )
@ ( K3
= ( zero_zero @ code_natural ) )
@ ( product_Pair @ B @ A @ X2 )
@ ( product_scomp @ A @ B @ A @ ( product_prod @ B @ A ) @ ( F2 @ X2 ) @ ( iterate @ B @ A @ ( minus_minus @ code_natural @ K3 @ ( one_one @ code_natural ) ) @ F2 ) ) ) ) ) ).
% iterate.simps
thf(fact_6139_coprime__common__divisor__nat,axiom,
! [A3: nat,B2: nat,X: nat] :
( ( algebr8660921524188924756oprime @ nat @ A3 @ B2 )
=> ( ( dvd_dvd @ nat @ X @ A3 )
=> ( ( dvd_dvd @ nat @ X @ B2 )
=> ( X
= ( one_one @ nat ) ) ) ) ) ).
% coprime_common_divisor_nat
thf(fact_6140_coprime__Suc__left__nat,axiom,
! [N: nat] : ( algebr8660921524188924756oprime @ nat @ ( suc @ N ) @ N ) ).
% coprime_Suc_left_nat
thf(fact_6141_coprime__Suc__right__nat,axiom,
! [N: nat] : ( algebr8660921524188924756oprime @ nat @ N @ ( suc @ N ) ) ).
% coprime_Suc_right_nat
thf(fact_6142_coprime__crossproduct__nat,axiom,
! [A3: nat,D3: nat,B2: nat,C2: nat] :
( ( algebr8660921524188924756oprime @ nat @ A3 @ D3 )
=> ( ( algebr8660921524188924756oprime @ nat @ B2 @ C2 )
=> ( ( ( times_times @ nat @ A3 @ C2 )
= ( times_times @ nat @ B2 @ D3 ) )
= ( ( A3 = B2 )
& ( C2 = D3 ) ) ) ) ) ).
% coprime_crossproduct_nat
thf(fact_6143_coprime__Suc__0__left,axiom,
! [N: nat] : ( algebr8660921524188924756oprime @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N ) ).
% coprime_Suc_0_left
thf(fact_6144_coprime__Suc__0__right,axiom,
! [N: nat] : ( algebr8660921524188924756oprime @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ).
% coprime_Suc_0_right
thf(fact_6145_quotient__of__eq,axiom,
! [A3: int,B2: int,P4: int,Q2: int] :
( ( ( quotient_of @ ( fract @ A3 @ B2 ) )
= ( product_Pair @ int @ int @ P4 @ Q2 ) )
=> ( ( fract @ P4 @ Q2 )
= ( fract @ A3 @ B2 ) ) ) ).
% quotient_of_eq
thf(fact_6146_normalize__eq,axiom,
! [A3: int,B2: int,P4: int,Q2: int] :
( ( ( normalize @ ( product_Pair @ int @ int @ A3 @ B2 ) )
= ( product_Pair @ int @ int @ P4 @ Q2 ) )
=> ( ( fract @ P4 @ Q2 )
= ( fract @ A3 @ B2 ) ) ) ).
% normalize_eq
thf(fact_6147_quotient__of__Fract,axiom,
! [A3: int,B2: int] :
( ( quotient_of @ ( fract @ A3 @ B2 ) )
= ( normalize @ ( product_Pair @ int @ int @ A3 @ B2 ) ) ) ).
% quotient_of_Fract
thf(fact_6148_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_6149_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_6150_Fract_Oabs__eq,axiom,
( fract
= ( ^ [Xa3: int,X2: int] :
( abs_Rat
@ ( if @ ( product_prod @ int @ int )
@ ( X2
= ( zero_zero @ int ) )
@ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
@ ( product_Pair @ int @ int @ Xa3 @ X2 ) ) ) ) ) ).
% Fract.abs_eq
thf(fact_6151_Fract_Otransfer,axiom,
( bNF_rel_fun @ int @ int @ ( int > ( product_prod @ int @ int ) ) @ ( int > rat )
@ ^ [Y6: int,Z2: int] : Y6 = Z2
@ ( bNF_rel_fun @ int @ int @ ( product_prod @ int @ int ) @ rat
@ ^ [Y6: int,Z2: int] : Y6 = Z2
@ pcr_rat )
@ ^ [A6: int,B5: int] :
( if @ ( product_prod @ int @ int )
@ ( B5
= ( zero_zero @ int ) )
@ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
@ ( product_Pair @ int @ int @ A6 @ B5 ) )
@ fract ) ).
% Fract.transfer
thf(fact_6152_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 )
@ ^ [A6: int,B5: int] :
( if @ ( product_prod @ int @ int )
@ ( B5
= ( zero_zero @ int ) )
@ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
@ ( product_Pair @ int @ int @ A6 @ B5 ) ) ) ) ).
% Fract_def
thf(fact_6153_mult__inj__if__coprime__nat,axiom,
! [B: $tType,A: $tType,F: A > nat,A5: set @ A,G: B > nat,B4: set @ B] :
( ( inj_on @ A @ nat @ F @ A5 )
=> ( ( inj_on @ B @ nat @ G @ B4 )
=> ( ! [A4: A,B3: B] :
( ( member @ A @ A4 @ A5 )
=> ( ( member @ B @ B3 @ B4 )
=> ( algebr8660921524188924756oprime @ nat @ ( F @ A4 ) @ ( G @ B3 ) ) ) )
=> ( inj_on @ ( product_prod @ A @ B ) @ nat
@ ( product_case_prod @ A @ B @ nat
@ ^ [A6: A,B5: B] : ( times_times @ nat @ ( F @ A6 ) @ ( G @ B5 ) ) )
@ ( product_Sigma @ A @ B @ A5
@ ^ [Uu: A] : B4 ) ) ) ) ) ).
% mult_inj_if_coprime_nat
thf(fact_6154_iterate_Oelims,axiom,
! [A: $tType,B: $tType,X: code_natural,Xa: B > A > ( product_prod @ B @ A ),Xb: B,Y3: A > ( product_prod @ B @ A )] :
( ( ( iterate @ B @ A @ X @ Xa @ Xb )
= Y3 )
=> ( ( ( X
= ( zero_zero @ code_natural ) )
=> ( Y3
= ( product_Pair @ B @ A @ Xb ) ) )
& ( ( X
!= ( zero_zero @ code_natural ) )
=> ( Y3
= ( product_scomp @ A @ B @ A @ ( product_prod @ B @ A ) @ ( Xa @ Xb ) @ ( iterate @ B @ A @ ( minus_minus @ code_natural @ X @ ( one_one @ code_natural ) ) @ Xa ) ) ) ) ) ) ).
% iterate.elims
thf(fact_6155_iterate_Opelims,axiom,
! [A: $tType,B: $tType,X: code_natural,Xa: B > A > ( product_prod @ B @ A ),Xb: B,Y3: A > ( product_prod @ B @ A )] :
( ( ( iterate @ B @ A @ X @ Xa @ Xb )
= Y3 )
=> ( ( accp @ ( product_prod @ code_natural @ ( product_prod @ ( B > A > ( product_prod @ B @ A ) ) @ B ) ) @ ( iterate_rel @ B @ A ) @ ( product_Pair @ code_natural @ ( product_prod @ ( B > A > ( product_prod @ B @ A ) ) @ B ) @ X @ ( product_Pair @ ( B > A > ( product_prod @ B @ A ) ) @ B @ Xa @ Xb ) ) )
=> ~ ( ( ( ( X
= ( zero_zero @ code_natural ) )
=> ( Y3
= ( product_Pair @ B @ A @ Xb ) ) )
& ( ( X
!= ( zero_zero @ code_natural ) )
=> ( Y3
= ( product_scomp @ A @ B @ A @ ( product_prod @ B @ A ) @ ( Xa @ Xb ) @ ( iterate @ B @ A @ ( minus_minus @ code_natural @ X @ ( one_one @ code_natural ) ) @ Xa ) ) ) ) )
=> ~ ( accp @ ( product_prod @ code_natural @ ( product_prod @ ( B > A > ( product_prod @ B @ A ) ) @ B ) ) @ ( iterate_rel @ B @ A ) @ ( product_Pair @ code_natural @ ( product_prod @ ( B > A > ( product_prod @ B @ A ) ) @ B ) @ X @ ( product_Pair @ ( B > A > ( product_prod @ B @ A ) ) @ B @ Xa @ Xb ) ) ) ) ) ) ).
% iterate.pelims
thf(fact_6156_Lcm__coprime_H,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A5: set @ A] :
( ( ( finite_card @ A @ A5 )
!= ( zero_zero @ nat ) )
=> ( ! [A4: A,B3: A] :
( ( member @ A @ A4 @ A5 )
=> ( ( member @ A @ B3 @ A5 )
=> ( ( A4 != B3 )
=> ( algebr8660921524188924756oprime @ A @ A4 @ B3 ) ) ) )
=> ( ( gcd_Lcm @ A @ A5 )
= ( normal6383669964737779283malize @ A
@ ( groups7121269368397514597t_prod @ A @ A
@ ^ [X2: A] : X2
@ A5 ) ) ) ) ) ) ).
% Lcm_coprime'
thf(fact_6157_normalize__idem,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A] :
( ( normal6383669964737779283malize @ A @ ( normal6383669964737779283malize @ A @ A3 ) )
= ( normal6383669964737779283malize @ A @ A3 ) ) ) ).
% normalize_idem
thf(fact_6158_of__int__eq__id,axiom,
( ( ring_1_of_int @ int )
= ( id @ int ) ) ).
% of_int_eq_id
thf(fact_6159_lcm_Onormalize__bottom,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ( ( normal6383669964737779283malize @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% lcm.normalize_bottom
thf(fact_6160_normalize__0,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ( ( normal6383669964737779283malize @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% normalize_0
thf(fact_6161_normalize__eq__0__iff,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A] :
( ( ( normal6383669964737779283malize @ A @ A3 )
= ( zero_zero @ A ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% normalize_eq_0_iff
thf(fact_6162_normalize__mult__normalize__right,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A,B2: A] :
( ( normal6383669964737779283malize @ A @ ( times_times @ A @ A3 @ ( normal6383669964737779283malize @ A @ B2 ) ) )
= ( normal6383669964737779283malize @ A @ ( times_times @ A @ A3 @ B2 ) ) ) ) ).
% normalize_mult_normalize_right
thf(fact_6163_normalize__mult__normalize__left,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A,B2: A] :
( ( normal6383669964737779283malize @ A @ ( times_times @ A @ ( normal6383669964737779283malize @ A @ A3 ) @ B2 ) )
= ( normal6383669964737779283malize @ A @ ( times_times @ A @ A3 @ B2 ) ) ) ) ).
% normalize_mult_normalize_left
thf(fact_6164_gcd_Onormalize__bottom,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ( ( normal6383669964737779283malize @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% gcd.normalize_bottom
thf(fact_6165_normalize__1,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ( ( normal6383669964737779283malize @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% normalize_1
thf(fact_6166_dvd__normalize__iff,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ ( normal6383669964737779283malize @ A @ B2 ) )
= ( dvd_dvd @ A @ A3 @ B2 ) ) ) ).
% dvd_normalize_iff
thf(fact_6167_normalize__dvd__iff,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ ( normal6383669964737779283malize @ A @ A3 ) @ B2 )
= ( dvd_dvd @ A @ A3 @ B2 ) ) ) ).
% normalize_dvd_iff
thf(fact_6168_gcd_Oidem__normalize,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A] :
( ( gcd_gcd @ A @ A3 @ A3 )
= ( normal6383669964737779283malize @ A @ A3 ) ) ) ).
% gcd.idem_normalize
thf(fact_6169_gcd_Onormalize__idem,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A] :
( ( normal6383669964737779283malize @ A @ ( gcd_gcd @ A @ A3 @ B2 ) )
= ( gcd_gcd @ A @ A3 @ B2 ) ) ) ).
% gcd.normalize_idem
thf(fact_6170_gcd_Onormalize__left__idem,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A] :
( ( gcd_gcd @ A @ ( normal6383669964737779283malize @ A @ A3 ) @ B2 )
= ( gcd_gcd @ A @ A3 @ B2 ) ) ) ).
% gcd.normalize_left_idem
thf(fact_6171_gcd_Onormalize__right__idem,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A] :
( ( gcd_gcd @ A @ A3 @ ( normal6383669964737779283malize @ A @ B2 ) )
= ( gcd_gcd @ A @ A3 @ B2 ) ) ) ).
% gcd.normalize_right_idem
thf(fact_6172_lcm_Oidem__normalize,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A] :
( ( gcd_lcm @ A @ A3 @ A3 )
= ( normal6383669964737779283malize @ A @ A3 ) ) ) ).
% lcm.idem_normalize
thf(fact_6173_lcm_Onormalize__idem,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A] :
( ( normal6383669964737779283malize @ A @ ( gcd_lcm @ A @ A3 @ B2 ) )
= ( gcd_lcm @ A @ A3 @ B2 ) ) ) ).
% lcm.normalize_idem
thf(fact_6174_lcm_Onormalize__left__idem,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A] :
( ( gcd_lcm @ A @ ( normal6383669964737779283malize @ A @ A3 ) @ B2 )
= ( gcd_lcm @ A @ A3 @ B2 ) ) ) ).
% lcm.normalize_left_idem
thf(fact_6175_lcm_Onormalize__right__idem,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A] :
( ( gcd_lcm @ A @ A3 @ ( normal6383669964737779283malize @ A @ B2 ) )
= ( gcd_lcm @ A @ A3 @ B2 ) ) ) ).
% lcm.normalize_right_idem
thf(fact_6176_coprime__normalize__right__iff,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A,B2: A] :
( ( algebr8660921524188924756oprime @ A @ A3 @ ( normal6383669964737779283malize @ A @ B2 ) )
= ( algebr8660921524188924756oprime @ A @ A3 @ B2 ) ) ) ).
% coprime_normalize_right_iff
thf(fact_6177_coprime__normalize__left__iff,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A,B2: A] :
( ( algebr8660921524188924756oprime @ A @ ( normal6383669964737779283malize @ A @ A3 ) @ B2 )
= ( algebr8660921524188924756oprime @ A @ A3 @ B2 ) ) ) ).
% coprime_normalize_left_iff
thf(fact_6178_normalize__Lcm,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A5: set @ A] :
( ( normal6383669964737779283malize @ A @ ( gcd_Lcm @ A @ A5 ) )
= ( gcd_Lcm @ A @ A5 ) ) ) ).
% normalize_Lcm
thf(fact_6179_normalize__Gcd,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A5: set @ A] :
( ( normal6383669964737779283malize @ A @ ( gcd_Gcd @ A @ A5 ) )
= ( gcd_Gcd @ A @ A5 ) ) ) ).
% normalize_Gcd
thf(fact_6180_Lcm__fin_Onormalize,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A5: set @ A] :
( ( normal6383669964737779283malize @ A @ ( semiring_gcd_Lcm_fin @ A @ A5 ) )
= ( semiring_gcd_Lcm_fin @ A @ A5 ) ) ) ).
% Lcm_fin.normalize
thf(fact_6181_Gcd__fin_Onormalize,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A5: set @ A] :
( ( normal6383669964737779283malize @ A @ ( semiring_gcd_Gcd_fin @ A @ A5 ) )
= ( semiring_gcd_Gcd_fin @ A @ A5 ) ) ) ).
% Gcd_fin.normalize
thf(fact_6182_gcd_Otop__right__normalize,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A] :
( ( gcd_gcd @ A @ A3 @ ( zero_zero @ A ) )
= ( normal6383669964737779283malize @ A @ A3 ) ) ) ).
% gcd.top_right_normalize
thf(fact_6183_gcd_Otop__left__normalize,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A] :
( ( gcd_gcd @ A @ ( zero_zero @ A ) @ A3 )
= ( normal6383669964737779283malize @ A @ A3 ) ) ) ).
% gcd.top_left_normalize
thf(fact_6184_lcm_Otop__right__normalize,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A] :
( ( gcd_lcm @ A @ A3 @ ( one_one @ A ) )
= ( normal6383669964737779283malize @ A @ A3 ) ) ) ).
% lcm.top_right_normalize
thf(fact_6185_lcm_Otop__left__normalize,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A] :
( ( gcd_lcm @ A @ ( one_one @ A ) @ A3 )
= ( normal6383669964737779283malize @ A @ A3 ) ) ) ).
% lcm.top_left_normalize
thf(fact_6186_Gcd__image__normalize,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A5: set @ A] :
( ( gcd_Gcd @ A @ ( image2 @ A @ A @ ( normal6383669964737779283malize @ A ) @ A5 ) )
= ( gcd_Gcd @ A @ A5 ) ) ) ).
% Gcd_image_normalize
thf(fact_6187_normalize__mult__unit__right,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [B2: A,A3: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( normal6383669964737779283malize @ A @ ( times_times @ A @ A3 @ B2 ) )
= ( normal6383669964737779283malize @ A @ A3 ) ) ) ) ).
% normalize_mult_unit_right
thf(fact_6188_normalize__mult__unit__left,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( normal6383669964737779283malize @ A @ ( times_times @ A @ A3 @ B2 ) )
= ( normal6383669964737779283malize @ A @ B2 ) ) ) ) ).
% normalize_mult_unit_left
thf(fact_6189_gcd__mult__lcm,axiom,
! [A: $tType] :
( ( semiri6843258321239162965malize @ A )
=> ! [A3: A,B2: A] :
( ( times_times @ A @ ( gcd_gcd @ A @ A3 @ B2 ) @ ( gcd_lcm @ A @ A3 @ B2 ) )
= ( times_times @ A @ ( normal6383669964737779283malize @ A @ A3 ) @ ( normal6383669964737779283malize @ A @ B2 ) ) ) ) ).
% gcd_mult_lcm
thf(fact_6190_lcm__mult__gcd,axiom,
! [A: $tType] :
( ( semiri6843258321239162965malize @ A )
=> ! [A3: A,B2: A] :
( ( times_times @ A @ ( gcd_lcm @ A @ A3 @ B2 ) @ ( gcd_gcd @ A @ A3 @ B2 ) )
= ( times_times @ A @ ( normal6383669964737779283malize @ A @ A3 ) @ ( normal6383669964737779283malize @ A @ B2 ) ) ) ) ).
% lcm_mult_gcd
thf(fact_6191_Lcm__singleton,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A3: A] :
( ( gcd_Lcm @ A @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( normal6383669964737779283malize @ A @ A3 ) ) ) ).
% Lcm_singleton
thf(fact_6192_Gcd__singleton,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A3: A] :
( ( gcd_Gcd @ A @ ( insert2 @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( normal6383669964737779283malize @ A @ A3 ) ) ) ).
% Gcd_singleton
thf(fact_6193_coprime__crossproduct,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,D3: A,B2: A,C2: A] :
( ( algebr8660921524188924756oprime @ A @ A3 @ D3 )
=> ( ( algebr8660921524188924756oprime @ A @ B2 @ C2 )
=> ( ( ( times_times @ A @ ( normal6383669964737779283malize @ A @ A3 ) @ ( normal6383669964737779283malize @ A @ C2 ) )
= ( times_times @ A @ ( normal6383669964737779283malize @ A @ B2 ) @ ( normal6383669964737779283malize @ A @ D3 ) ) )
= ( ( ( normal6383669964737779283malize @ A @ A3 )
= ( normal6383669964737779283malize @ A @ B2 ) )
& ( ( normal6383669964737779283malize @ A @ C2 )
= ( normal6383669964737779283malize @ A @ D3 ) ) ) ) ) ) ) ).
% coprime_crossproduct
thf(fact_6194_iterate_Ocases,axiom,
! [A: $tType,B: $tType,X: product_prod @ code_natural @ ( product_prod @ ( B > A > ( product_prod @ B @ A ) ) @ B )] :
~ ! [K2: code_natural,F4: B > A > ( product_prod @ B @ A ),X4: B] :
( X
!= ( product_Pair @ code_natural @ ( product_prod @ ( B > A > ( product_prod @ B @ A ) ) @ B ) @ K2 @ ( product_Pair @ ( B > A > ( product_prod @ B @ A ) ) @ B @ F4 @ X4 ) ) ) ).
% iterate.cases
thf(fact_6195_lcm__mult__distrib_H,axiom,
! [A: $tType] :
( ( semiri6843258321239162965malize @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( times_times @ A @ ( normal6383669964737779283malize @ A @ C2 ) @ ( gcd_lcm @ A @ A3 @ B2 ) )
= ( gcd_lcm @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) ) ) ) ).
% lcm_mult_distrib'
thf(fact_6196_lcm__mult__right,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( gcd_lcm @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) )
= ( normal6383669964737779283malize @ A @ ( times_times @ A @ ( gcd_lcm @ A @ B2 @ A3 ) @ C2 ) ) ) ) ).
% lcm_mult_right
thf(fact_6197_lcm__mult__left,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( gcd_lcm @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( normal6383669964737779283malize @ A @ ( times_times @ A @ C2 @ ( gcd_lcm @ A @ A3 @ B2 ) ) ) ) ) ).
% lcm_mult_left
thf(fact_6198_lcm__proj2__if__dvd,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ B2 )
=> ( ( gcd_lcm @ A @ A3 @ B2 )
= ( normal6383669964737779283malize @ A @ B2 ) ) ) ) ).
% lcm_proj2_if_dvd
thf(fact_6199_lcm__proj1__if__dvd,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [B2: A,A3: A] :
( ( dvd_dvd @ A @ B2 @ A3 )
=> ( ( gcd_lcm @ A @ A3 @ B2 )
= ( normal6383669964737779283malize @ A @ A3 ) ) ) ) ).
% lcm_proj1_if_dvd
thf(fact_6200_lcm__proj2__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [M: A,N: A] :
( ( ( gcd_lcm @ A @ M @ N )
= ( normal6383669964737779283malize @ A @ N ) )
= ( dvd_dvd @ A @ M @ N ) ) ) ).
% lcm_proj2_iff
thf(fact_6201_lcm__proj1__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [M: A,N: A] :
( ( ( gcd_lcm @ A @ M @ N )
= ( normal6383669964737779283malize @ A @ M ) )
= ( dvd_dvd @ A @ N @ M ) ) ) ).
% lcm_proj1_iff
thf(fact_6202_lcm__unique,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,D3: A,B2: A] :
( ( ( dvd_dvd @ A @ A3 @ D3 )
& ( dvd_dvd @ A @ B2 @ D3 )
& ( ( normal6383669964737779283malize @ A @ D3 )
= D3 )
& ! [E4: A] :
( ( ( dvd_dvd @ A @ A3 @ E4 )
& ( dvd_dvd @ A @ B2 @ E4 ) )
=> ( dvd_dvd @ A @ D3 @ E4 ) ) )
= ( D3
= ( gcd_lcm @ A @ A3 @ B2 ) ) ) ) ).
% lcm_unique
thf(fact_6203_lcmI,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ C2 )
=> ( ( dvd_dvd @ A @ B2 @ C2 )
=> ( ! [D2: A] :
( ( dvd_dvd @ A @ A3 @ D2 )
=> ( ( dvd_dvd @ A @ B2 @ D2 )
=> ( dvd_dvd @ A @ C2 @ D2 ) ) )
=> ( ( ( normal6383669964737779283malize @ A @ C2 )
= C2 )
=> ( C2
= ( gcd_lcm @ A @ A3 @ B2 ) ) ) ) ) ) ) ).
% lcmI
thf(fact_6204_gcd__proj2__if__dvd,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [B2: A,A3: A] :
( ( dvd_dvd @ A @ B2 @ A3 )
=> ( ( gcd_gcd @ A @ A3 @ B2 )
= ( normal6383669964737779283malize @ A @ B2 ) ) ) ) ).
% gcd_proj2_if_dvd
thf(fact_6205_gcd__proj1__if__dvd,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ B2 )
=> ( ( gcd_gcd @ A @ A3 @ B2 )
= ( normal6383669964737779283malize @ A @ A3 ) ) ) ) ).
% gcd_proj1_if_dvd
thf(fact_6206_gcd__proj2__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [M: A,N: A] :
( ( ( gcd_gcd @ A @ M @ N )
= ( normal6383669964737779283malize @ A @ N ) )
= ( dvd_dvd @ A @ N @ M ) ) ) ).
% gcd_proj2_iff
thf(fact_6207_gcd__proj1__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [M: A,N: A] :
( ( ( gcd_gcd @ A @ M @ N )
= ( normal6383669964737779283malize @ A @ M ) )
= ( dvd_dvd @ A @ M @ N ) ) ) ).
% gcd_proj1_iff
thf(fact_6208_gcd__unique,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [D3: A,A3: A,B2: A] :
( ( ( dvd_dvd @ A @ D3 @ A3 )
& ( dvd_dvd @ A @ D3 @ B2 )
& ( ( normal6383669964737779283malize @ A @ D3 )
= D3 )
& ! [E4: A] :
( ( ( dvd_dvd @ A @ E4 @ A3 )
& ( dvd_dvd @ A @ E4 @ B2 ) )
=> ( dvd_dvd @ A @ E4 @ D3 ) ) )
= ( D3
= ( gcd_gcd @ A @ A3 @ B2 ) ) ) ) ).
% gcd_unique
thf(fact_6209_gcdI,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( dvd_dvd @ A @ C2 @ A3 )
=> ( ( dvd_dvd @ A @ C2 @ B2 )
=> ( ! [D2: A] :
( ( dvd_dvd @ A @ D2 @ A3 )
=> ( ( dvd_dvd @ A @ D2 @ B2 )
=> ( dvd_dvd @ A @ D2 @ C2 ) ) )
=> ( ( ( normal6383669964737779283malize @ A @ C2 )
= C2 )
=> ( C2
= ( gcd_gcd @ A @ A3 @ B2 ) ) ) ) ) ) ) ).
% gcdI
thf(fact_6210_gcd__mult__distrib_H,axiom,
! [A: $tType] :
( ( semiri6843258321239162965malize @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( times_times @ A @ ( normal6383669964737779283malize @ A @ C2 ) @ ( gcd_gcd @ A @ A3 @ B2 ) )
= ( gcd_gcd @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) ) ) ) ).
% gcd_mult_distrib'
thf(fact_6211_gcd__mult__right,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( gcd_gcd @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) )
= ( normal6383669964737779283malize @ A @ ( times_times @ A @ ( gcd_gcd @ A @ B2 @ A3 ) @ C2 ) ) ) ) ).
% gcd_mult_right
thf(fact_6212_gcd__mult__left,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( gcd_gcd @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( normal6383669964737779283malize @ A @ ( times_times @ A @ C2 @ ( gcd_gcd @ A @ A3 @ B2 ) ) ) ) ) ).
% gcd_mult_left
thf(fact_6213_associatedI,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ B2 )
=> ( ( dvd_dvd @ A @ B2 @ A3 )
=> ( ( normal6383669964737779283malize @ A @ A3 )
= ( normal6383669964737779283malize @ A @ B2 ) ) ) ) ) ).
% associatedI
thf(fact_6214_associatedD1,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A,B2: A] :
( ( ( normal6383669964737779283malize @ A @ A3 )
= ( normal6383669964737779283malize @ A @ B2 ) )
=> ( dvd_dvd @ A @ A3 @ B2 ) ) ) ).
% associatedD1
thf(fact_6215_associatedD2,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A,B2: A] :
( ( ( normal6383669964737779283malize @ A @ A3 )
= ( normal6383669964737779283malize @ A @ B2 ) )
=> ( dvd_dvd @ A @ B2 @ A3 ) ) ) ).
% associatedD2
thf(fact_6216_associated__eqI,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ B2 )
=> ( ( dvd_dvd @ A @ B2 @ A3 )
=> ( ( ( normal6383669964737779283malize @ A @ A3 )
= A3 )
=> ( ( ( normal6383669964737779283malize @ A @ B2 )
= B2 )
=> ( A3 = B2 ) ) ) ) ) ) ).
% associated_eqI
thf(fact_6217_associated__iff__dvd,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A,B2: A] :
( ( ( normal6383669964737779283malize @ A @ A3 )
= ( normal6383669964737779283malize @ A @ B2 ) )
= ( ( dvd_dvd @ A @ A3 @ B2 )
& ( dvd_dvd @ A @ B2 @ A3 ) ) ) ) ).
% associated_iff_dvd
thf(fact_6218_normalize__mult,axiom,
! [A: $tType] :
( ( normal6328177297339901930cative @ A )
=> ! [A3: A,B2: A] :
( ( normal6383669964737779283malize @ A @ ( times_times @ A @ A3 @ B2 ) )
= ( times_times @ A @ ( normal6383669964737779283malize @ A @ A3 ) @ ( normal6383669964737779283malize @ A @ B2 ) ) ) ) ).
% normalize_mult
thf(fact_6219_dvd__normalize__div,axiom,
! [A: $tType] :
( ( normal6328177297339901930cative @ A )
=> ! [B2: A,A3: A] :
( ( dvd_dvd @ A @ B2 @ A3 )
=> ( ( normal6383669964737779283malize @ A @ ( divide_divide @ A @ A3 @ B2 ) )
= ( divide_divide @ A @ ( normal6383669964737779283malize @ A @ A3 ) @ ( normal6383669964737779283malize @ A @ B2 ) ) ) ) ) ).
% dvd_normalize_div
thf(fact_6220_normalize__idem__imp__is__unit__iff,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A] :
( ( ( normal6383669964737779283malize @ A @ A3 )
= A3 )
=> ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
= ( A3
= ( one_one @ A ) ) ) ) ) ).
% normalize_idem_imp_is_unit_iff
thf(fact_6221_is__unit__normalize,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( normal6383669964737779283malize @ A @ A3 )
= ( one_one @ A ) ) ) ) ).
% is_unit_normalize
thf(fact_6222_normalize__1__iff,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A] :
( ( ( normal6383669964737779283malize @ A @ A3 )
= ( one_one @ A ) )
= ( dvd_dvd @ A @ A3 @ ( one_one @ A ) ) ) ) ).
% normalize_1_iff
thf(fact_6223_associated__unit,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A,B2: A] :
( ( ( normal6383669964737779283malize @ A @ A3 )
= ( normal6383669964737779283malize @ A @ B2 ) )
=> ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( dvd_dvd @ A @ B2 @ ( one_one @ A ) ) ) ) ) ).
% associated_unit
thf(fact_6224_gcd__exp__weak,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,N: nat,B2: A] :
( ( gcd_gcd @ A @ ( power_power @ A @ A3 @ N ) @ ( power_power @ A @ B2 @ N ) )
= ( normal6383669964737779283malize @ A @ ( power_power @ A @ ( gcd_gcd @ A @ A3 @ B2 ) @ N ) ) ) ) ).
% gcd_exp_weak
thf(fact_6225_Gcd__eqI,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A3: A,A5: set @ A] :
( ( ( normal6383669964737779283malize @ A @ A3 )
= A3 )
=> ( ! [B3: A] :
( ( member @ A @ B3 @ A5 )
=> ( dvd_dvd @ A @ A3 @ B3 ) )
=> ( ! [C3: A] :
( ! [B12: A] :
( ( member @ A @ B12 @ A5 )
=> ( dvd_dvd @ A @ C3 @ B12 ) )
=> ( dvd_dvd @ A @ C3 @ A3 ) )
=> ( ( gcd_Gcd @ A @ A5 )
= A3 ) ) ) ) ) ).
% Gcd_eqI
thf(fact_6226_GcdI,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A5: set @ A,B2: A] :
( ! [A4: A] :
( ( member @ A @ A4 @ A5 )
=> ( dvd_dvd @ A @ B2 @ A4 ) )
=> ( ! [C3: A] :
( ! [A20: A] :
( ( member @ A @ A20 @ A5 )
=> ( dvd_dvd @ A @ C3 @ A20 ) )
=> ( dvd_dvd @ A @ C3 @ B2 ) )
=> ( ( ( normal6383669964737779283malize @ A @ B2 )
= B2 )
=> ( B2
= ( gcd_Gcd @ A @ A5 ) ) ) ) ) ) ).
% GcdI
thf(fact_6227_LcmI,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A5: set @ A,B2: A] :
( ! [A4: A] :
( ( member @ A @ A4 @ A5 )
=> ( dvd_dvd @ A @ A4 @ B2 ) )
=> ( ! [C3: A] :
( ! [A20: A] :
( ( member @ A @ A20 @ A5 )
=> ( dvd_dvd @ A @ A20 @ C3 ) )
=> ( dvd_dvd @ A @ B2 @ C3 ) )
=> ( ( ( normal6383669964737779283malize @ A @ B2 )
= B2 )
=> ( B2
= ( gcd_Lcm @ A @ A5 ) ) ) ) ) ) ).
% LcmI
thf(fact_6228_Lcm__eqI,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A3: A,A5: set @ A] :
( ( ( normal6383669964737779283malize @ A @ A3 )
= A3 )
=> ( ! [B3: A] :
( ( member @ A @ B3 @ A5 )
=> ( dvd_dvd @ A @ B3 @ A3 ) )
=> ( ! [C3: A] :
( ! [B12: A] :
( ( member @ A @ B12 @ A5 )
=> ( dvd_dvd @ A @ B12 @ C3 ) )
=> ( dvd_dvd @ A @ A3 @ C3 ) )
=> ( ( gcd_Lcm @ A @ A5 )
= A3 ) ) ) ) ) ).
% Lcm_eqI
thf(fact_6229_lcm__gcd__prod,axiom,
! [A: $tType] :
( ( semiri6843258321239162965malize @ A )
=> ! [A3: A,B2: A] :
( ( times_times @ A @ ( gcd_lcm @ A @ A3 @ B2 ) @ ( gcd_gcd @ A @ A3 @ B2 ) )
= ( normal6383669964737779283malize @ A @ ( times_times @ A @ A3 @ B2 ) ) ) ) ).
% lcm_gcd_prod
thf(fact_6230_Gcd__mult,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [C2: A,A5: set @ A] :
( ( gcd_Gcd @ A @ ( image2 @ A @ A @ ( times_times @ A @ C2 ) @ A5 ) )
= ( normal6383669964737779283malize @ A @ ( times_times @ A @ C2 @ ( gcd_Gcd @ A @ A5 ) ) ) ) ) ).
% Gcd_mult
thf(fact_6231_lcm__coprime,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A] :
( ( algebr8660921524188924756oprime @ A @ A3 @ B2 )
=> ( ( gcd_lcm @ A @ A3 @ B2 )
= ( normal6383669964737779283malize @ A @ ( times_times @ A @ A3 @ B2 ) ) ) ) ) ).
% lcm_coprime
thf(fact_6232_Lcm__mult,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A5: set @ A,C2: A] :
( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( gcd_Lcm @ A @ ( image2 @ A @ A @ ( times_times @ A @ C2 ) @ A5 ) )
= ( normal6383669964737779283malize @ A @ ( times_times @ A @ C2 @ ( gcd_Lcm @ A @ A5 ) ) ) ) ) ) ).
% Lcm_mult
thf(fact_6233_lcm__gcd,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ( ( gcd_lcm @ A )
= ( ^ [A6: A,B5: A] : ( normal6383669964737779283malize @ A @ ( divide_divide @ A @ ( times_times @ A @ A6 @ B5 ) @ ( gcd_gcd @ A @ A6 @ B5 ) ) ) ) ) ) ).
% lcm_gcd
thf(fact_6234_Lcm__fin__mult,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A5: set @ A,B2: A] :
( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( semiring_gcd_Lcm_fin @ A @ ( image2 @ A @ A @ ( times_times @ A @ B2 ) @ A5 ) )
= ( normal6383669964737779283malize @ A @ ( times_times @ A @ B2 @ ( semiring_gcd_Lcm_fin @ A @ A5 ) ) ) ) ) ) ).
% Lcm_fin_mult
thf(fact_6235_Gcd__fin__mult,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A5: set @ A,B2: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( semiring_gcd_Gcd_fin @ A @ ( image2 @ A @ A @ ( times_times @ A @ B2 ) @ A5 ) )
= ( normal6383669964737779283malize @ A @ ( times_times @ A @ B2 @ ( semiring_gcd_Gcd_fin @ A @ A5 ) ) ) ) ) ) ).
% Gcd_fin_mult
thf(fact_6236_Gcd__fin_Obounded__quasi__semilattice__set__axioms,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ( bounde6485984586167503788ce_set @ A @ ( gcd_gcd @ A ) @ ( zero_zero @ A ) @ ( one_one @ A ) @ ( normal6383669964737779283malize @ A ) ) ) ).
% Gcd_fin.bounded_quasi_semilattice_set_axioms
thf(fact_6237_Lcm__fin_Obounded__quasi__semilattice__set__axioms,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ( bounde6485984586167503788ce_set @ A @ ( gcd_lcm @ A ) @ ( one_one @ A ) @ ( zero_zero @ A ) @ ( normal6383669964737779283malize @ A ) ) ) ).
% Lcm_fin.bounded_quasi_semilattice_set_axioms
thf(fact_6238_gcd__lcm,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( ( gcd_gcd @ A @ A3 @ B2 )
= ( normal6383669964737779283malize @ A @ ( divide_divide @ A @ ( times_times @ A @ A3 @ B2 ) @ ( gcd_lcm @ A @ A3 @ B2 ) ) ) ) ) ) ) ).
% gcd_lcm
thf(fact_6239_Lcm__coprime,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [A4: A,B3: A] :
( ( member @ A @ A4 @ A5 )
=> ( ( member @ A @ B3 @ A5 )
=> ( ( A4 != B3 )
=> ( algebr8660921524188924756oprime @ A @ A4 @ B3 ) ) ) )
=> ( ( gcd_Lcm @ A @ A5 )
= ( normal6383669964737779283malize @ A
@ ( groups7121269368397514597t_prod @ A @ A
@ ^ [X2: A] : X2
@ A5 ) ) ) ) ) ) ) ).
% Lcm_coprime
thf(fact_6240_select__weight__cons__zero,axiom,
! [A: $tType,X: A,Xs2: list @ ( product_prod @ code_natural @ A )] :
( ( select_weight @ A @ ( cons @ ( product_prod @ code_natural @ A ) @ ( product_Pair @ code_natural @ A @ ( zero_zero @ code_natural ) @ X ) @ Xs2 ) )
= ( select_weight @ A @ Xs2 ) ) ).
% select_weight_cons_zero
thf(fact_6241_lcm_Obounded__quasi__semilattice__axioms,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ( bounde8507323023520639062attice @ A @ ( gcd_lcm @ A ) @ ( one_one @ A ) @ ( zero_zero @ A ) @ ( normal6383669964737779283malize @ A ) ) ) ).
% lcm.bounded_quasi_semilattice_axioms
thf(fact_6242_bounded__quasi__semilattice__set__def,axiom,
! [A: $tType] :
( ( bounde6485984586167503788ce_set @ A )
= ( bounde8507323023520639062attice @ A ) ) ).
% bounded_quasi_semilattice_set_def
thf(fact_6243_bounded__quasi__semilattice__set_Oaxioms,axiom,
! [A: $tType,F: A > A > A,Top: A,Bot: A,Normalize: A > A] :
( ( bounde6485984586167503788ce_set @ A @ F @ Top @ Bot @ Normalize )
=> ( bounde8507323023520639062attice @ A @ F @ Top @ Bot @ Normalize ) ) ).
% bounded_quasi_semilattice_set.axioms
thf(fact_6244_bounded__quasi__semilattice__set_Ointro,axiom,
! [A: $tType,F: A > A > A,Top: A,Bot: A,Normalize: A > A] :
( ( bounde8507323023520639062attice @ A @ F @ Top @ Bot @ Normalize )
=> ( bounde6485984586167503788ce_set @ A @ F @ Top @ Bot @ Normalize ) ) ).
% bounded_quasi_semilattice_set.intro
thf(fact_6245_bounded__quasi__semilattice_Oleft__idem,axiom,
! [A: $tType,F: A > A > A,Top: A,Bot: A,Normalize: A > A,A3: A,B2: A] :
( ( bounde8507323023520639062attice @ A @ F @ Top @ Bot @ Normalize )
=> ( ( F @ A3 @ ( F @ A3 @ B2 ) )
= ( F @ A3 @ B2 ) ) ) ).
% bounded_quasi_semilattice.left_idem
thf(fact_6246_bounded__quasi__semilattice_Oright__idem,axiom,
! [A: $tType,F: A > A > A,Top: A,Bot: A,Normalize: A > A,A3: A,B2: A] :
( ( bounde8507323023520639062attice @ A @ F @ Top @ Bot @ Normalize )
=> ( ( F @ ( F @ A3 @ B2 ) @ B2 )
= ( F @ A3 @ B2 ) ) ) ).
% bounded_quasi_semilattice.right_idem
thf(fact_6247_bounded__quasi__semilattice_Onormalize__top,axiom,
! [A: $tType,F: A > A > A,Top: A,Bot: A,Normalize: A > A] :
( ( bounde8507323023520639062attice @ A @ F @ Top @ Bot @ Normalize )
=> ( ( Normalize @ Top )
= Top ) ) ).
% bounded_quasi_semilattice.normalize_top
thf(fact_6248_bounded__quasi__semilattice_Oidem__normalize,axiom,
! [A: $tType,F: A > A > A,Top: A,Bot: A,Normalize: A > A,A3: A] :
( ( bounde8507323023520639062attice @ A @ F @ Top @ Bot @ Normalize )
=> ( ( F @ A3 @ A3 )
= ( Normalize @ A3 ) ) ) ).
% bounded_quasi_semilattice.idem_normalize
thf(fact_6249_bounded__quasi__semilattice_Onormalize__idem,axiom,
! [A: $tType,F: A > A > A,Top: A,Bot: A,Normalize: A > A,A3: A,B2: A] :
( ( bounde8507323023520639062attice @ A @ F @ Top @ Bot @ Normalize )
=> ( ( Normalize @ ( F @ A3 @ B2 ) )
= ( F @ A3 @ B2 ) ) ) ).
% bounded_quasi_semilattice.normalize_idem
thf(fact_6250_bounded__quasi__semilattice_Onormalize__bottom,axiom,
! [A: $tType,F: A > A > A,Top: A,Bot: A,Normalize: A > A] :
( ( bounde8507323023520639062attice @ A @ F @ Top @ Bot @ Normalize )
=> ( ( Normalize @ Bot )
= Bot ) ) ).
% bounded_quasi_semilattice.normalize_bottom
thf(fact_6251_bounded__quasi__semilattice_Obottom__left__bottom,axiom,
! [A: $tType,F: A > A > A,Top: A,Bot: A,Normalize: A > A,A3: A] :
( ( bounde8507323023520639062attice @ A @ F @ Top @ Bot @ Normalize )
=> ( ( F @ Bot @ A3 )
= Bot ) ) ).
% bounded_quasi_semilattice.bottom_left_bottom
thf(fact_6252_bounded__quasi__semilattice_Otop__left__normalize,axiom,
! [A: $tType,F: A > A > A,Top: A,Bot: A,Normalize: A > A,A3: A] :
( ( bounde8507323023520639062attice @ A @ F @ Top @ Bot @ Normalize )
=> ( ( F @ Top @ A3 )
= ( Normalize @ A3 ) ) ) ).
% bounded_quasi_semilattice.top_left_normalize
thf(fact_6253_bounded__quasi__semilattice_Obottom__right__bottom,axiom,
! [A: $tType,F: A > A > A,Top: A,Bot: A,Normalize: A > A,A3: A] :
( ( bounde8507323023520639062attice @ A @ F @ Top @ Bot @ Normalize )
=> ( ( F @ A3 @ Bot )
= Bot ) ) ).
% bounded_quasi_semilattice.bottom_right_bottom
thf(fact_6254_bounded__quasi__semilattice_Onormalize__left__idem,axiom,
! [A: $tType,F: A > A > A,Top: A,Bot: A,Normalize: A > A,A3: A,B2: A] :
( ( bounde8507323023520639062attice @ A @ F @ Top @ Bot @ Normalize )
=> ( ( F @ ( Normalize @ A3 ) @ B2 )
= ( F @ A3 @ B2 ) ) ) ).
% bounded_quasi_semilattice.normalize_left_idem
thf(fact_6255_bounded__quasi__semilattice_Otop__right__normalize,axiom,
! [A: $tType,F: A > A > A,Top: A,Bot: A,Normalize: A > A,A3: A] :
( ( bounde8507323023520639062attice @ A @ F @ Top @ Bot @ Normalize )
=> ( ( F @ A3 @ Top )
= ( Normalize @ A3 ) ) ) ).
% bounded_quasi_semilattice.top_right_normalize
thf(fact_6256_bounded__quasi__semilattice_Onormalize__right__idem,axiom,
! [A: $tType,F: A > A > A,Top: A,Bot: A,Normalize: A > A,A3: A,B2: A] :
( ( bounde8507323023520639062attice @ A @ F @ Top @ Bot @ Normalize )
=> ( ( F @ A3 @ ( Normalize @ B2 ) )
= ( F @ A3 @ B2 ) ) ) ).
% bounded_quasi_semilattice.normalize_right_idem
thf(fact_6257_select__weight__drop__zero,axiom,
! [A: $tType,Xs2: list @ ( product_prod @ code_natural @ A )] :
( ( select_weight @ A
@ ( filter2 @ ( product_prod @ code_natural @ A )
@ ( product_case_prod @ code_natural @ A @ $o
@ ^ [K3: code_natural,Uu: A] : ( ord_less @ code_natural @ ( zero_zero @ code_natural ) @ K3 ) )
@ Xs2 ) )
= ( select_weight @ A @ Xs2 ) ) ).
% select_weight_drop_zero
thf(fact_6258_gcd_Obounded__quasi__semilattice__axioms,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ( bounde8507323023520639062attice @ A @ ( gcd_gcd @ A ) @ ( zero_zero @ A ) @ ( one_one @ A ) @ ( normal6383669964737779283malize @ A ) ) ) ).
% gcd.bounded_quasi_semilattice_axioms
thf(fact_6259_select__weight__select,axiom,
! [A: $tType,Xs2: list @ A] :
( ( Xs2
!= ( nil @ A ) )
=> ( ( select_weight @ A @ ( map @ A @ ( product_prod @ code_natural @ A ) @ ( product_Pair @ code_natural @ A @ ( one_one @ code_natural ) ) @ Xs2 ) )
= ( select @ A @ Xs2 ) ) ) ).
% select_weight_select
thf(fact_6260_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_6261_plus__natural_Orep__eq,axiom,
! [X: code_natural,Xa: code_natural] :
( ( code_nat_of_natural @ ( plus_plus @ code_natural @ X @ Xa ) )
= ( plus_plus @ nat @ ( code_nat_of_natural @ X ) @ ( code_nat_of_natural @ Xa ) ) ) ).
% plus_natural.rep_eq
thf(fact_6262_zero__natural_Orep__eq,axiom,
( ( code_nat_of_natural @ ( zero_zero @ code_natural ) )
= ( zero_zero @ nat ) ) ).
% zero_natural.rep_eq
thf(fact_6263_one__natural_Orep__eq,axiom,
( ( code_nat_of_natural @ ( one_one @ code_natural ) )
= ( one_one @ nat ) ) ).
% one_natural.rep_eq
thf(fact_6264_pick__drop__zero,axiom,
! [A: $tType,Xs2: list @ ( product_prod @ code_natural @ A )] :
( ( pick @ A
@ ( filter2 @ ( product_prod @ code_natural @ A )
@ ( product_case_prod @ code_natural @ A @ $o
@ ^ [K3: code_natural,Uu: A] : ( ord_less @ code_natural @ ( zero_zero @ code_natural ) @ K3 ) )
@ Xs2 ) )
= ( pick @ A @ Xs2 ) ) ).
% pick_drop_zero
thf(fact_6265_pick__same,axiom,
! [A: $tType,L: nat,Xs2: list @ A] :
( ( ord_less @ nat @ L @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( pick @ A @ ( map @ A @ ( product_prod @ code_natural @ A ) @ ( product_Pair @ code_natural @ A @ ( one_one @ code_natural ) ) @ Xs2 ) @ ( code_natural_of_nat @ L ) )
= ( nth @ A @ Xs2 @ L ) ) ) ).
% pick_same
thf(fact_6266_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_6267_one__natural__def,axiom,
( ( one_one @ code_natural )
= ( code_natural_of_nat @ ( one_one @ nat ) ) ) ).
% one_natural_def
thf(fact_6268_zero__natural__def,axiom,
( ( zero_zero @ code_natural )
= ( code_natural_of_nat @ ( zero_zero @ nat ) ) ) ).
% zero_natural_def
thf(fact_6269_plus__natural_Oabs__eq,axiom,
! [Xa: nat,X: nat] :
( ( plus_plus @ code_natural @ ( code_natural_of_nat @ Xa ) @ ( code_natural_of_nat @ X ) )
= ( code_natural_of_nat @ ( plus_plus @ nat @ Xa @ X ) ) ) ).
% plus_natural.abs_eq
thf(fact_6270_select__def,axiom,
! [A: $tType] :
( ( select @ A )
= ( ^ [Xs: 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 ) @ Xs ) ) )
@ ^ [K3: code_natural] : ( product_Pair @ A @ ( product_prod @ code_natural @ code_natural ) @ ( nth @ A @ Xs @ ( code_nat_of_natural @ K3 ) ) ) ) ) ) ).
% select_def
thf(fact_6271_select__weight__def,axiom,
! [A: $tType] :
( ( select_weight @ A )
= ( ^ [Xs: 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 ) @ Xs ) ) )
@ ^ [K3: code_natural] : ( product_Pair @ A @ ( product_prod @ code_natural @ code_natural ) @ ( pick @ A @ Xs @ K3 ) ) ) ) ) ).
% select_weight_def
thf(fact_6272_nth__drop,axiom,
! [A: $tType,N: nat,Xs2: list @ A,I: nat] :
( ( ord_less_eq @ nat @ N @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( nth @ A @ ( drop @ A @ N @ Xs2 ) @ I )
= ( nth @ A @ Xs2 @ ( plus_plus @ nat @ N @ I ) ) ) ) ).
% nth_drop
thf(fact_6273_antisymp__antisym__eq,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( antisymp @ A
@ ^ [X2: A,Y: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y ) @ R ) )
= ( antisym @ A @ R ) ) ).
% antisymp_antisym_eq
thf(fact_6274_drop0,axiom,
! [A: $tType] :
( ( drop @ A @ ( zero_zero @ nat ) )
= ( ^ [X2: list @ A] : X2 ) ) ).
% drop0
thf(fact_6275_drop__drop,axiom,
! [A: $tType,N: nat,M: nat,Xs2: list @ A] :
( ( drop @ A @ N @ ( drop @ A @ M @ Xs2 ) )
= ( drop @ A @ ( plus_plus @ nat @ N @ M ) @ Xs2 ) ) ).
% drop_drop
thf(fact_6276_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_6277_drop__Cons__numeral,axiom,
! [A: $tType,V: num,X: A,Xs2: list @ A] :
( ( drop @ A @ ( numeral_numeral @ nat @ V ) @ ( cons @ A @ X @ Xs2 ) )
= ( drop @ A @ ( minus_minus @ nat @ ( numeral_numeral @ nat @ V ) @ ( one_one @ nat ) ) @ Xs2 ) ) ).
% drop_Cons_numeral
thf(fact_6278_drop__0,axiom,
! [A: $tType,Xs2: list @ A] :
( ( drop @ A @ ( zero_zero @ nat ) @ Xs2 )
= Xs2 ) ).
% drop_0
thf(fact_6279_take__drop,axiom,
! [A: $tType,N: nat,M: nat,Xs2: list @ A] :
( ( take @ A @ N @ ( drop @ A @ M @ Xs2 ) )
= ( drop @ A @ M @ ( take @ A @ ( plus_plus @ nat @ N @ M ) @ Xs2 ) ) ) ).
% take_drop
thf(fact_6280_drop__eq__nths,axiom,
! [A: $tType] :
( ( drop @ A )
= ( ^ [N5: nat,Xs: list @ A] : ( nths @ A @ Xs @ ( collect @ nat @ ( ord_less_eq @ nat @ N5 ) ) ) ) ) ).
% drop_eq_nths
thf(fact_6281_take__add,axiom,
! [A: $tType,I: nat,J: nat,Xs2: list @ A] :
( ( take @ A @ ( plus_plus @ nat @ I @ J ) @ Xs2 )
= ( append @ A @ ( take @ A @ I @ Xs2 ) @ ( take @ A @ J @ ( drop @ A @ I @ Xs2 ) ) ) ) ).
% take_add
thf(fact_6282_drop__Cons,axiom,
! [A: $tType,N: nat,X: A,Xs2: list @ A] :
( ( drop @ A @ N @ ( cons @ A @ X @ Xs2 ) )
= ( case_nat @ ( list @ A ) @ ( cons @ A @ X @ Xs2 )
@ ^ [M5: nat] : ( drop @ A @ M5 @ Xs2 )
@ N ) ) ).
% drop_Cons
thf(fact_6283_nths__drop,axiom,
! [A: $tType,N: nat,Xs2: list @ A,I5: set @ nat] :
( ( nths @ A @ ( drop @ A @ N @ Xs2 ) @ I5 )
= ( nths @ A @ Xs2 @ ( image2 @ nat @ nat @ ( plus_plus @ nat @ N ) @ I5 ) ) ) ).
% nths_drop
thf(fact_6284_drop__Cons_H,axiom,
! [A: $tType,N: nat,X: A,Xs2: list @ A] :
( ( ( N
= ( zero_zero @ nat ) )
=> ( ( drop @ A @ N @ ( cons @ A @ X @ Xs2 ) )
= ( cons @ A @ X @ Xs2 ) ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ( ( drop @ A @ N @ ( cons @ A @ X @ Xs2 ) )
= ( drop @ A @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ Xs2 ) ) ) ) ).
% drop_Cons'
thf(fact_6285_insort__insert__insort,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Xs2: list @ A] :
( ~ ( member @ A @ X @ ( set2 @ A @ Xs2 ) )
=> ( ( linord329482645794927042rt_key @ A @ A
@ ^ [X2: A] : X2
@ X
@ Xs2 )
= ( linorder_insort_key @ A @ A
@ ^ [X2: A] : X2
@ X
@ Xs2 ) ) ) ) ).
% insort_insert_insort
thf(fact_6286_Fpow__Pow__finite,axiom,
! [A: $tType] :
( ( finite_Fpow @ A )
= ( ^ [A8: set @ A] : ( inf_inf @ ( set @ ( set @ A ) ) @ ( pow2 @ A @ A8 ) @ ( collect @ ( set @ A ) @ ( finite_finite2 @ A ) ) ) ) ) ).
% Fpow_Pow_finite
thf(fact_6287_insort__insert__triv,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Xs2: list @ A] :
( ( member @ A @ X @ ( set2 @ A @ Xs2 ) )
=> ( ( linord329482645794927042rt_key @ A @ A
@ ^ [X2: A] : X2
@ X
@ Xs2 )
= Xs2 ) ) ) ).
% insort_insert_triv
thf(fact_6288_Fpow__def,axiom,
! [A: $tType] :
( ( finite_Fpow @ A )
= ( ^ [A8: set @ A] :
( collect @ ( set @ A )
@ ^ [X9: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ X9 @ A8 )
& ( finite_finite2 @ A @ X9 ) ) ) ) ) ).
% Fpow_def
thf(fact_6289_set__insort__insert,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Xs2: list @ A] :
( ( set2 @ A
@ ( linord329482645794927042rt_key @ A @ A
@ ^ [X2: A] : X2
@ X
@ Xs2 ) )
= ( insert2 @ A @ X @ ( set2 @ A @ Xs2 ) ) ) ) ).
% set_insort_insert
thf(fact_6290_sorted__insort__insert,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs2: list @ A,X: A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs2 )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A )
@ ( linord329482645794927042rt_key @ A @ A
@ ^ [X2: A] : X2
@ X
@ Xs2 ) ) ) ) ).
% sorted_insort_insert
thf(fact_6291_bind__singleton__conv__image,axiom,
! [A: $tType,B: $tType,A5: set @ B,F: B > A] :
( ( bind4 @ B @ A @ A5
@ ^ [X2: B] : ( insert2 @ A @ ( F @ X2 ) @ ( bot_bot @ ( set @ A ) ) ) )
= ( image2 @ B @ A @ F @ A5 ) ) ).
% bind_singleton_conv_image
thf(fact_6292_and_Osemilattice__neutr__axioms,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( semilattice_neutr @ A @ ( bit_se5824344872417868541ns_and @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% and.semilattice_neutr_axioms
thf(fact_6293_nonempty__bind__const,axiom,
! [A: $tType,B: $tType,A5: set @ A,B4: set @ B] :
( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( bind4 @ A @ B @ A5
@ ^ [Uu: A] : B4 )
= B4 ) ) ).
% nonempty_bind_const
thf(fact_6294_bind__const,axiom,
! [B: $tType,A: $tType,A5: set @ B,B4: set @ A] :
( ( ( A5
= ( bot_bot @ ( set @ B ) ) )
=> ( ( bind4 @ B @ A @ A5
@ ^ [Uu: B] : B4 )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( bind4 @ B @ A @ A5
@ ^ [Uu: B] : B4 )
= B4 ) ) ) ).
% bind_const
thf(fact_6295_Set_Obind__bind,axiom,
! [C: $tType,B: $tType,A: $tType,A5: set @ A,B4: A > ( set @ C ),C6: C > ( set @ B )] :
( ( bind4 @ C @ B @ ( bind4 @ A @ C @ A5 @ B4 ) @ C6 )
= ( bind4 @ A @ B @ A5
@ ^ [X2: A] : ( bind4 @ C @ B @ ( B4 @ X2 ) @ C6 ) ) ) ).
% Set.bind_bind
thf(fact_6296_bind__image,axiom,
! [A: $tType,B: $tType,C: $tType,F: C > B,A5: set @ C,G: B > ( set @ A )] :
( ( bind4 @ B @ A @ ( image2 @ C @ B @ F @ A5 ) @ G )
= ( bind4 @ C @ A @ A5 @ ( comp @ B @ ( set @ A ) @ C @ G @ F ) ) ) ).
% bind_image
thf(fact_6297_gcd__nat_Osemilattice__neutr__axioms,axiom,
semilattice_neutr @ nat @ ( gcd_gcd @ nat ) @ ( zero_zero @ nat ) ).
% gcd_nat.semilattice_neutr_axioms
thf(fact_6298_or_Osemilattice__neutr__axioms,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( semilattice_neutr @ A @ ( bit_se1065995026697491101ons_or @ A ) @ ( zero_zero @ A ) ) ) ).
% or.semilattice_neutr_axioms
thf(fact_6299_max__nat_Osemilattice__neutr__axioms,axiom,
semilattice_neutr @ nat @ ( ord_max @ nat ) @ ( zero_zero @ nat ) ).
% max_nat.semilattice_neutr_axioms
thf(fact_6300_image__bind,axiom,
! [A: $tType,B: $tType,C: $tType,F: B > A,A5: set @ C,G: C > ( set @ B )] :
( ( image2 @ B @ A @ F @ ( bind4 @ C @ B @ A5 @ G ) )
= ( bind4 @ C @ A @ A5 @ ( comp @ ( set @ B ) @ ( set @ A ) @ C @ ( image2 @ B @ A @ F ) @ G ) ) ) ).
% image_bind
thf(fact_6301_Set_Obind__def,axiom,
! [B: $tType,A: $tType] :
( ( bind4 @ A @ B )
= ( ^ [A8: set @ A,F2: A > ( set @ B )] :
( collect @ B
@ ^ [X2: B] :
? [Y: set @ B] :
( ( member @ ( set @ B ) @ Y @ ( image2 @ A @ ( set @ B ) @ F2 @ A8 ) )
& ( member @ B @ X2 @ Y ) ) ) ) ) ).
% Set.bind_def
thf(fact_6302_fun_Opred__set,axiom,
! [A: $tType,D: $tType,P: A > $o] :
( ( basic_pred_fun @ D @ A
@ ^ [Uu: D] : $true
@ P )
= ( ^ [X2: D > A] :
! [Y: A] :
( ( member @ A @ Y @ ( image2 @ D @ A @ X2 @ ( top_top @ ( set @ D ) ) ) )
=> ( P @ Y ) ) ) ) ).
% fun.pred_set
thf(fact_6303_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 ) )
@ ^ [X2: A] : X2 ) ) ).
% sorted_list_of_set.folding_insort_key_axioms
thf(fact_6304_fun_Omap__cong__pred,axiom,
! [B: $tType,A: $tType,D: $tType,X: D > A,Ya: D > A,F: A > B,G: A > B] :
( ( X = Ya )
=> ( ( basic_pred_fun @ D @ A
@ ^ [Uu: D] : $true
@ ^ [Z3: A] :
( ( F @ Z3 )
= ( G @ Z3 ) )
@ Ya )
=> ( ( comp @ A @ B @ D @ F @ X )
= ( comp @ A @ B @ D @ G @ Ya ) ) ) ) ).
% fun.map_cong_pred
thf(fact_6305_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_6306_pred__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( basic_pred_fun @ A @ B )
= ( ^ [A8: A > $o,B7: B > $o,F2: A > B] :
! [X2: A] :
( ( A8 @ X2 )
=> ( B7 @ ( F2 @ X2 ) ) ) ) ) ).
% pred_fun_def
thf(fact_6307_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_6308_pred__fun__True__id,axiom,
! [A: $tType,B: $tType,C: $tType,P4: B > $o,F: C > B] :
( ( nO_MATCH @ ( A > A ) @ ( B > $o ) @ ( id @ A ) @ P4 )
=> ( ( basic_pred_fun @ C @ B
@ ^ [X2: C] : $true
@ P4
@ F )
= ( basic_pred_fun @ C @ $o
@ ^ [X2: C] : $true
@ ( id @ $o )
@ ( comp @ B @ $o @ C @ P4 @ F ) ) ) ) ).
% pred_fun_True_id
thf(fact_6309_fun_Opred__mono__strong,axiom,
! [A: $tType,D: $tType,P: A > $o,X: D > A,Pa: A > $o] :
( ( basic_pred_fun @ D @ A
@ ^ [Uu: D] : $true
@ P
@ X )
=> ( ! [Z6: A] :
( ( member @ A @ Z6 @ ( image2 @ D @ A @ X @ ( top_top @ ( set @ D ) ) ) )
=> ( ( P @ Z6 )
=> ( Pa @ Z6 ) ) )
=> ( basic_pred_fun @ D @ A
@ ^ [Uu: D] : $true
@ Pa
@ X ) ) ) ).
% fun.pred_mono_strong
thf(fact_6310_fun_Opred__cong,axiom,
! [A: $tType,D: $tType,X: D > A,Ya: D > A,P: A > $o,Pa: A > $o] :
( ( X = Ya )
=> ( ! [Z6: A] :
( ( member @ A @ Z6 @ ( image2 @ D @ A @ Ya @ ( top_top @ ( set @ D ) ) ) )
=> ( ( P @ Z6 )
= ( Pa @ Z6 ) ) )
=> ( ( basic_pred_fun @ D @ A
@ ^ [Uu: D] : $true
@ P
@ X )
= ( basic_pred_fun @ D @ A
@ ^ [Uu: D] : $true
@ Pa
@ Ya ) ) ) ) ).
% fun.pred_cong
thf(fact_6311_fun_Opred__transfer,axiom,
! [A: $tType,B: $tType,D: $tType,R5: A > B > $o] :
( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ ( ( D > A ) > $o ) @ ( ( D > B ) > $o )
@ ( bNF_rel_fun @ A @ B @ $o @ $o @ R5
@ ^ [Y6: $o,Z2: $o] : Y6 = Z2 )
@ ( bNF_rel_fun @ ( D > A ) @ ( D > B ) @ $o @ $o
@ ( bNF_rel_fun @ D @ D @ A @ B
@ ^ [Y6: D,Z2: D] : Y6 = Z2
@ R5 )
@ ^ [Y6: $o,Z2: $o] : Y6 = Z2 )
@ ( basic_pred_fun @ D @ A
@ ^ [Uu: D] : $true )
@ ( basic_pred_fun @ D @ B
@ ^ [Uu: D] : $true ) ) ).
% fun.pred_transfer
thf(fact_6312_fun_Opred__map,axiom,
! [B: $tType,A: $tType,D: $tType,Q: B > $o,F: A > B,X: D > A] :
( ( basic_pred_fun @ D @ B
@ ^ [Uu: D] : $true
@ Q
@ ( comp @ A @ B @ D @ F @ X ) )
= ( basic_pred_fun @ D @ A
@ ^ [Uu: D] : $true
@ ( comp @ B @ $o @ A @ Q @ F )
@ X ) ) ).
% fun.pred_map
thf(fact_6313_folding__insort__key_Oinsort__key__commute,axiom,
! [A: $tType,B: $tType,Less_eq2: A > A > $o,Less: A > A > $o,S3: set @ B,F: B > A,X: B,Y3: B] :
( ( folding_insort_key @ A @ B @ Less_eq2 @ Less @ S3 @ F )
=> ( ( member @ B @ X @ S3 )
=> ( ( member @ B @ Y3 @ S3 )
=> ( ( comp @ ( list @ B ) @ ( list @ B ) @ ( list @ B ) @ ( insort_key @ A @ B @ Less_eq2 @ F @ Y3 ) @ ( insort_key @ A @ B @ Less_eq2 @ F @ X ) )
= ( comp @ ( list @ B ) @ ( list @ B ) @ ( list @ B ) @ ( insort_key @ A @ B @ Less_eq2 @ F @ X ) @ ( insort_key @ A @ B @ Less_eq2 @ F @ Y3 ) ) ) ) ) ) ).
% folding_insort_key.insort_key_commute
thf(fact_6314_o__prs_I2_J,axiom,
! [F3: $tType,E: $tType,C: $tType,D: $tType,A: $tType,B: $tType,H8: $tType,G3: $tType,R12: A > A > $o,Abs1: A > B,Rep1: B > A,R23: C > C > $o,Abs22: C > D,Rep22: D > C,R32: E > E > $o,Abs32: E > F3,Rep32: F3 > E] :
( ( quotient3 @ A @ B @ R12 @ Abs1 @ Rep1 )
=> ( ( quotient3 @ C @ D @ R23 @ Abs22 @ Rep22 )
=> ( ( quotient3 @ E @ F3 @ R32 @ Abs32 @ Rep32 )
=> ( ( map_fun @ ( G3 > H8 ) @ ( G3 > H8 ) @ ( ( A > G3 ) > A > H8 ) @ ( ( B > G3 ) > B > H8 ) @ ( id @ ( G3 > H8 ) ) @ ( map_fun @ ( B > G3 ) @ ( A > G3 ) @ ( A > H8 ) @ ( B > H8 ) @ ( map_fun @ A @ B @ G3 @ G3 @ Abs1 @ ( id @ G3 ) ) @ ( map_fun @ B @ A @ H8 @ H8 @ Rep1 @ ( id @ H8 ) ) ) @ ( comp @ G3 @ H8 @ A ) )
= ( comp @ G3 @ H8 @ B ) ) ) ) ) ).
% o_prs(2)
thf(fact_6315_lambda__prs,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,R12: A > A > $o,Abs1: A > B,Rep1: B > A,R23: C > C > $o,Abs22: C > D,Rep22: D > C,F: B > D] :
( ( quotient3 @ A @ B @ R12 @ Abs1 @ Rep1 )
=> ( ( quotient3 @ C @ D @ R23 @ Abs22 @ Rep22 )
=> ( ( map_fun @ B @ A @ C @ D @ Rep1 @ Abs22
@ ^ [X2: A] : ( Rep22 @ ( F @ ( Abs1 @ X2 ) ) ) )
= F ) ) ) ).
% lambda_prs
thf(fact_6316_lambda__prs1,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,R12: A > A > $o,Abs1: A > B,Rep1: B > A,R23: C > C > $o,Abs22: C > D,Rep22: D > C,F: B > D] :
( ( quotient3 @ A @ B @ R12 @ Abs1 @ Rep1 )
=> ( ( quotient3 @ C @ D @ R23 @ Abs22 @ Rep22 )
=> ( ( map_fun @ B @ A @ C @ D @ Rep1 @ Abs22 @ ( map_fun @ A @ B @ D @ C @ Abs1 @ Rep22 @ F ) )
= F ) ) ) ).
% lambda_prs1
thf(fact_6317_o__prs_I1_J,axiom,
! [C: $tType,E: $tType,A: $tType,B: $tType,F3: $tType,D: $tType,R12: A > A > $o,Abs1: A > B,Rep1: B > A,R23: C > C > $o,Abs22: C > D,Rep22: D > C,R32: E > E > $o,Abs32: E > F3,Rep32: F3 > E] :
( ( quotient3 @ A @ B @ R12 @ Abs1 @ Rep1 )
=> ( ( quotient3 @ C @ D @ R23 @ Abs22 @ Rep22 )
=> ( ( quotient3 @ E @ F3 @ R32 @ Abs32 @ Rep32 )
=> ( ( map_fun @ ( D > F3 ) @ ( C > E ) @ ( ( A > C ) > A > E ) @ ( ( B > D ) > B > F3 ) @ ( map_fun @ C @ D @ F3 @ E @ Abs22 @ Rep32 ) @ ( map_fun @ ( B > D ) @ ( A > C ) @ ( A > E ) @ ( B > F3 ) @ ( map_fun @ A @ B @ D @ C @ Abs1 @ Rep22 ) @ ( map_fun @ B @ A @ E @ F3 @ Rep1 @ Abs32 ) ) @ ( comp @ C @ E @ A ) )
= ( comp @ D @ F3 @ B ) ) ) ) ) ).
% o_prs(1)
thf(fact_6318_abs__o__rep,axiom,
! [A: $tType,B: $tType,R5: A > A > $o,Abs: A > B,Rep: B > A] :
( ( quotient3 @ A @ B @ R5 @ Abs @ Rep )
=> ( ( comp @ A @ B @ B @ Abs @ Rep )
= ( id @ B ) ) ) ).
% abs_o_rep
thf(fact_6319_Quotient3__to__Quotient,axiom,
! [B: $tType,A: $tType,R5: A > A > $o,Abs: A > B,Rep: B > A,T5: A > B > $o] :
( ( quotient3 @ A @ B @ R5 @ Abs @ Rep )
=> ( ( T5
= ( ^ [X2: A,Y: B] :
( ( R5 @ X2 @ X2 )
& ( ( Abs @ X2 )
= Y ) ) ) )
=> ( quotient @ A @ B @ R5 @ Abs @ Rep @ T5 ) ) ) ).
% Quotient3_to_Quotient
thf(fact_6320_OOO__quotient3,axiom,
! [A: $tType,B: $tType,C: $tType,R12: A > A > $o,Abs1: A > B,Rep1: B > A,R23: B > B > $o,Abs22: B > C,Rep22: C > B,R26: A > A > $o] :
( ( quotient3 @ A @ B @ R12 @ Abs1 @ Rep1 )
=> ( ( quotient3 @ B @ C @ R23 @ Abs22 @ Rep22 )
=> ( ! [X4: A,Y4: A] :
( ( R26 @ X4 @ Y4 )
=> ( ( R12 @ X4 @ X4 )
=> ( ( R12 @ Y4 @ Y4 )
=> ( R23 @ ( Abs1 @ X4 ) @ ( Abs1 @ Y4 ) ) ) ) )
=> ( ! [X4: B,Y4: B] :
( ( R23 @ X4 @ Y4 )
=> ( R26 @ ( Rep1 @ X4 ) @ ( Rep1 @ Y4 ) ) )
=> ( quotient3 @ A @ C @ ( relcompp @ A @ A @ A @ R12 @ ( relcompp @ A @ A @ A @ R26 @ R12 ) ) @ ( comp @ B @ C @ A @ Abs22 @ Abs1 ) @ ( comp @ B @ A @ C @ Rep1 @ Rep22 ) ) ) ) ) ) ).
% OOO_quotient3
thf(fact_6321_OOO__eq__quotient3,axiom,
! [A: $tType,B: $tType,C: $tType,R12: A > A > $o,Abs1: A > B,Rep1: B > A,Abs22: B > C,Rep22: C > B] :
( ( quotient3 @ A @ B @ R12 @ Abs1 @ Rep1 )
=> ( ( quotient3 @ B @ C
@ ^ [Y6: B,Z2: B] : Y6 = Z2
@ Abs22
@ Rep22 )
=> ( quotient3 @ A @ C
@ ( relcompp @ A @ A @ A @ R12
@ ( relcompp @ A @ A @ A
@ ^ [Y6: A,Z2: A] : Y6 = Z2
@ R12 ) )
@ ( comp @ B @ C @ A @ Abs22 @ Abs1 )
@ ( comp @ B @ A @ C @ Rep1 @ Rep22 ) ) ) ) ).
% OOO_eq_quotient3
thf(fact_6322_Quotient3__int,axiom,
quotient3 @ ( product_prod @ nat @ nat ) @ int @ intrel @ abs_Integ @ rep_Integ ).
% Quotient3_int
thf(fact_6323_String_Oless__literal__def,axiom,
( ( ord_less @ literal )
= ( map_fun @ literal @ ( list @ char ) @ ( ( list @ char ) > $o ) @ ( literal > $o ) @ explode @ ( map_fun @ literal @ ( list @ char ) @ $o @ $o @ explode @ ( id @ $o ) )
@ ( lexordp2 @ char
@ ^ [C4: char,D4: char] : ( ord_less @ nat @ ( comm_s6883823935334413003f_char @ nat @ C4 ) @ ( comm_s6883823935334413003f_char @ nat @ D4 ) ) ) ) ) ).
% String.less_literal_def
thf(fact_6324_Range__insert,axiom,
! [A: $tType,B: $tType,A3: B,B2: A,R: set @ ( product_prod @ B @ A )] :
( ( range2 @ B @ A @ ( insert2 @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ A3 @ B2 ) @ R ) )
= ( insert2 @ A @ B2 @ ( range2 @ B @ A @ R ) ) ) ).
% Range_insert
thf(fact_6325_Range__Collect__case__prod,axiom,
! [B: $tType,A: $tType,P: B > A > $o] :
( ( range2 @ B @ A @ ( collect @ ( product_prod @ B @ A ) @ ( product_case_prod @ B @ A @ $o @ P ) ) )
= ( collect @ A
@ ^ [Y: A] :
? [X2: B] : ( P @ X2 @ Y ) ) ) ).
% Range_Collect_case_prod
thf(fact_6326_Range_Ocases,axiom,
! [B: $tType,A: $tType,A3: B,R: set @ ( product_prod @ A @ B )] :
( ( member @ B @ A3 @ ( range2 @ A @ B @ R ) )
=> ~ ! [A4: A] :
~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ A3 ) @ R ) ) ).
% Range.cases
thf(fact_6327_Range_Osimps,axiom,
! [B: $tType,A: $tType,A3: B,R: set @ ( product_prod @ A @ B )] :
( ( member @ B @ A3 @ ( range2 @ A @ B @ R ) )
= ( ? [A6: A,B5: B] :
( ( A3 = B5 )
& ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A6 @ B5 ) @ R ) ) ) ) ).
% Range.simps
thf(fact_6328_Range_Ointros,axiom,
! [B: $tType,A: $tType,A3: A,B2: B,R: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ R )
=> ( member @ B @ B2 @ ( range2 @ A @ B @ R ) ) ) ).
% Range.intros
thf(fact_6329_RangeE,axiom,
! [A: $tType,B: $tType,B2: A,R: set @ ( product_prod @ B @ A )] :
( ( member @ A @ B2 @ ( range2 @ B @ A @ R ) )
=> ~ ! [A4: B] :
~ ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ A4 @ B2 ) @ R ) ) ).
% RangeE
thf(fact_6330_Range__iff,axiom,
! [A: $tType,B: $tType,A3: A,R: set @ ( product_prod @ B @ A )] :
( ( member @ A @ A3 @ ( range2 @ B @ A @ R ) )
= ( ? [Y: B] : ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ Y @ A3 ) @ R ) ) ) ).
% Range_iff
thf(fact_6331_Range__Union,axiom,
! [A: $tType,B: $tType,S3: set @ ( set @ ( product_prod @ B @ A ) )] :
( ( range2 @ B @ A @ ( complete_Sup_Sup @ ( set @ ( product_prod @ B @ A ) ) @ S3 ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ ( set @ ( product_prod @ B @ A ) ) @ ( set @ A ) @ ( range2 @ B @ A ) @ S3 ) ) ) ).
% Range_Union
thf(fact_6332_less__literal_Orep__eq,axiom,
( ( ord_less @ literal )
= ( ^ [X2: literal,Xa3: literal] :
( lexordp2 @ char
@ ^ [C4: char,D4: char] : ( ord_less @ nat @ ( comm_s6883823935334413003f_char @ nat @ C4 ) @ ( comm_s6883823935334413003f_char @ nat @ D4 ) )
@ ( explode @ X2 )
@ ( explode @ Xa3 ) ) ) ) ).
% less_literal.rep_eq
thf(fact_6333_String_Oless__eq__literal__def,axiom,
( ( ord_less_eq @ literal )
= ( map_fun @ literal @ ( list @ char ) @ ( ( list @ char ) > $o ) @ ( literal > $o ) @ explode @ ( map_fun @ literal @ ( list @ char ) @ $o @ $o @ explode @ ( id @ $o ) )
@ ( lexordp_eq @ char
@ ^ [C4: char,D4: char] : ( ord_less @ nat @ ( comm_s6883823935334413003f_char @ nat @ C4 ) @ ( comm_s6883823935334413003f_char @ nat @ D4 ) ) ) ) ) ).
% String.less_eq_literal_def
thf(fact_6334_wf__UN,axiom,
! [B: $tType,A: $tType,I5: set @ A,R: A > ( set @ ( product_prod @ B @ B ) )] :
( ! [I3: A] :
( ( member @ A @ I3 @ I5 )
=> ( wf @ B @ ( R @ I3 ) ) )
=> ( ! [I3: A,J2: A] :
( ( member @ A @ I3 @ I5 )
=> ( ( member @ A @ J2 @ I5 )
=> ( ( ( R @ I3 )
!= ( R @ J2 ) )
=> ( ( inf_inf @ ( set @ B ) @ ( domain @ B @ B @ ( R @ I3 ) ) @ ( range2 @ B @ B @ ( R @ J2 ) ) )
= ( bot_bot @ ( set @ B ) ) ) ) ) )
=> ( wf @ B @ ( complete_Sup_Sup @ ( set @ ( product_prod @ B @ B ) ) @ ( image2 @ A @ ( set @ ( product_prod @ B @ B ) ) @ R @ I5 ) ) ) ) ) ).
% wf_UN
thf(fact_6335_Domain__Collect__case__prod,axiom,
! [B: $tType,A: $tType,P: A > B > $o] :
( ( domain @ A @ B @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ P ) ) )
= ( collect @ A
@ ^ [X2: A] :
? [X9: B] : ( P @ X2 @ X9 ) ) ) ).
% Domain_Collect_case_prod
thf(fact_6336_Domain__insert,axiom,
! [B: $tType,A: $tType,A3: A,B2: B,R: set @ ( product_prod @ A @ B )] :
( ( domain @ A @ B @ ( insert2 @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ R ) )
= ( insert2 @ A @ A3 @ ( domain @ A @ B @ R ) ) ) ).
% Domain_insert
thf(fact_6337_Domain__iff,axiom,
! [A: $tType,B: $tType,A3: A,R: set @ ( product_prod @ A @ B )] :
( ( member @ A @ A3 @ ( domain @ A @ B @ R ) )
= ( ? [Y: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ Y ) @ R ) ) ) ).
% Domain_iff
thf(fact_6338_DomainE,axiom,
! [B: $tType,A: $tType,A3: A,R: set @ ( product_prod @ A @ B )] :
( ( member @ A @ A3 @ ( domain @ A @ B @ R ) )
=> ~ ! [B3: B] :
~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B3 ) @ R ) ) ).
% DomainE
thf(fact_6339_Domain_ODomainI,axiom,
! [B: $tType,A: $tType,A3: A,B2: B,R: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B2 ) @ R )
=> ( member @ A @ A3 @ ( domain @ A @ B @ R ) ) ) ).
% Domain.DomainI
thf(fact_6340_Domain_Osimps,axiom,
! [B: $tType,A: $tType,A3: A,R: set @ ( product_prod @ A @ B )] :
( ( member @ A @ A3 @ ( domain @ A @ B @ R ) )
= ( ? [A6: A,B5: B] :
( ( A3 = A6 )
& ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A6 @ B5 ) @ R ) ) ) ) ).
% Domain.simps
thf(fact_6341_Domain_Ocases,axiom,
! [B: $tType,A: $tType,A3: A,R: set @ ( product_prod @ A @ B )] :
( ( member @ A @ A3 @ ( domain @ A @ B @ R ) )
=> ~ ! [B3: B] :
~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B3 ) @ R ) ) ).
% Domain.cases
thf(fact_6342_Not__Domain__rtrancl,axiom,
! [A: $tType,X: A,R5: set @ ( product_prod @ A @ A ),Y3: A] :
( ~ ( member @ A @ X @ ( domain @ A @ A @ R5 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ ( transitive_rtrancl @ A @ R5 ) )
= ( X = Y3 ) ) ) ).
% Not_Domain_rtrancl
thf(fact_6343_Domain__unfold,axiom,
! [B: $tType,A: $tType] :
( ( domain @ A @ B )
= ( ^ [R4: set @ ( product_prod @ A @ B )] :
( collect @ A
@ ^ [X2: A] :
? [Y: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y ) @ R4 ) ) ) ) ).
% Domain_unfold
thf(fact_6344_Domain__Union,axiom,
! [B: $tType,A: $tType,S3: set @ ( set @ ( product_prod @ A @ B ) )] :
( ( domain @ A @ B @ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) ) @ S3 ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ ( set @ ( product_prod @ A @ B ) ) @ ( set @ A ) @ ( domain @ A @ B ) @ S3 ) ) ) ).
% Domain_Union
thf(fact_6345_less__eq__literal_Orep__eq,axiom,
( ( ord_less_eq @ literal )
= ( ^ [X2: literal,Xa3: literal] :
( lexordp_eq @ char
@ ^ [C4: char,D4: char] : ( ord_less @ nat @ ( comm_s6883823935334413003f_char @ nat @ C4 ) @ ( comm_s6883823935334413003f_char @ nat @ D4 ) )
@ ( explode @ X2 )
@ ( explode @ Xa3 ) ) ) ) ).
% less_eq_literal.rep_eq
thf(fact_6346_less__eq__literal_Otransfer,axiom,
( bNF_rel_fun @ ( list @ char ) @ literal @ ( ( list @ char ) > $o ) @ ( literal > $o ) @ pcr_literal
@ ( bNF_rel_fun @ ( list @ char ) @ literal @ $o @ $o @ pcr_literal
@ ^ [Y6: $o,Z2: $o] : Y6 = Z2 )
@ ( lexordp_eq @ char
@ ^ [C4: char,D4: char] : ( ord_less @ nat @ ( comm_s6883823935334413003f_char @ nat @ C4 ) @ ( comm_s6883823935334413003f_char @ nat @ D4 ) ) )
@ ( ord_less_eq @ literal ) ) ).
% less_eq_literal.transfer
thf(fact_6347_less__literal_Otransfer,axiom,
( bNF_rel_fun @ ( list @ char ) @ literal @ ( ( list @ char ) > $o ) @ ( literal > $o ) @ pcr_literal
@ ( bNF_rel_fun @ ( list @ char ) @ literal @ $o @ $o @ pcr_literal
@ ^ [Y6: $o,Z2: $o] : Y6 = Z2 )
@ ( lexordp2 @ char
@ ^ [C4: char,D4: char] : ( ord_less @ nat @ ( comm_s6883823935334413003f_char @ nat @ C4 ) @ ( comm_s6883823935334413003f_char @ nat @ D4 ) ) )
@ ( ord_less @ literal ) ) ).
% less_literal.transfer
thf(fact_6348_literal_Orep__transfer,axiom,
( bNF_rel_fun @ ( list @ char ) @ literal @ ( list @ char ) @ ( list @ char ) @ pcr_literal
@ ( list_all2 @ char @ char
@ ^ [Y6: char,Z2: char] : Y6 = Z2 )
@ ^ [X2: list @ char] : X2
@ explode ) ).
% literal.rep_transfer
thf(fact_6349_Range__def,axiom,
! [B: $tType,A: $tType] :
( ( range2 @ A @ B )
= ( ^ [R4: set @ ( product_prod @ A @ B )] :
( collect @ B
@ ( rangep @ A @ B
@ ^ [X2: A,Y: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y ) @ R4 ) ) ) ) ) ).
% Range_def
thf(fact_6350_Rangep__Range__eq,axiom,
! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ B )] :
( ( rangep @ A @ B
@ ^ [X2: A,Y: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y ) @ R ) )
= ( ^ [X2: B] : ( member @ B @ X2 @ ( range2 @ A @ B @ R ) ) ) ) ).
% Rangep_Range_eq
thf(fact_6351_String_Ocr__literal__def,axiom,
( cr_literal
= ( ^ [X2: list @ char,Y: literal] :
( X2
= ( explode @ Y ) ) ) ) ).
% String.cr_literal_def
thf(fact_6352_map__add__map__of__foldr,axiom,
! [B: $tType,A: $tType,M: A > ( option @ B ),Ps: list @ ( product_prod @ A @ B )] :
( ( map_add @ A @ B @ M @ ( map_of @ A @ B @ Ps ) )
= ( foldr @ ( product_prod @ A @ B ) @ ( A > ( option @ B ) )
@ ( product_case_prod @ A @ B @ ( ( A > ( option @ B ) ) > A > ( option @ B ) )
@ ^ [K3: A,V6: B,M5: A > ( option @ B )] : ( fun_upd @ A @ ( option @ B ) @ M5 @ K3 @ ( some @ B @ V6 ) ) )
@ Ps
@ M ) ) ).
% map_add_map_of_foldr
thf(fact_6353_map__add__None,axiom,
! [B: $tType,A: $tType,M: B > ( option @ A ),N: B > ( option @ A ),K: B] :
( ( ( map_add @ B @ A @ M @ N @ K )
= ( none @ A ) )
= ( ( ( N @ K )
= ( none @ A ) )
& ( ( M @ K )
= ( none @ A ) ) ) ) ).
% map_add_None
thf(fact_6354_map__add__eq__empty__iff,axiom,
! [B: $tType,A: $tType,F: A > ( option @ B ),G: A > ( option @ B )] :
( ( ( map_add @ A @ B @ F @ G )
= ( ^ [X2: A] : ( none @ B ) ) )
= ( ( F
= ( ^ [X2: A] : ( none @ B ) ) )
& ( G
= ( ^ [X2: A] : ( none @ B ) ) ) ) ) ).
% map_add_eq_empty_iff
thf(fact_6355_map__add__find__right,axiom,
! [B: $tType,A: $tType,N: B > ( option @ A ),K: B,Xx: A,M: B > ( option @ A )] :
( ( ( N @ K )
= ( some @ A @ Xx ) )
=> ( ( map_add @ B @ A @ M @ N @ K )
= ( some @ A @ Xx ) ) ) ).
% map_add_find_right
thf(fact_6356_empty__eq__map__add__iff,axiom,
! [B: $tType,A: $tType,F: A > ( option @ B ),G: A > ( option @ B )] :
( ( ( ^ [X2: A] : ( none @ B ) )
= ( map_add @ A @ B @ F @ G ) )
= ( ( F
= ( ^ [X2: A] : ( none @ B ) ) )
& ( G
= ( ^ [X2: A] : ( none @ B ) ) ) ) ) ).
% empty_eq_map_add_iff
thf(fact_6357_map__add__empty,axiom,
! [B: $tType,A: $tType,M: A > ( option @ B )] :
( ( map_add @ A @ B @ M
@ ^ [X2: A] : ( none @ B ) )
= M ) ).
% map_add_empty
thf(fact_6358_empty__map__add,axiom,
! [B: $tType,A: $tType,M: A > ( option @ B )] :
( ( map_add @ A @ B
@ ^ [X2: A] : ( none @ B )
@ M )
= M ) ).
% empty_map_add
thf(fact_6359_map__add__upd,axiom,
! [A: $tType,B: $tType,F: A > ( option @ B ),G: A > ( option @ B ),X: A,Y3: B] :
( ( map_add @ A @ B @ F @ ( fun_upd @ A @ ( option @ B ) @ G @ X @ ( some @ B @ Y3 ) ) )
= ( fun_upd @ A @ ( option @ B ) @ ( map_add @ A @ B @ F @ G ) @ X @ ( some @ B @ Y3 ) ) ) ).
% map_add_upd
thf(fact_6360_map__add__Some__iff,axiom,
! [B: $tType,A: $tType,M: B > ( option @ A ),N: B > ( option @ A ),K: B,X: A] :
( ( ( map_add @ B @ A @ M @ N @ K )
= ( some @ A @ X ) )
= ( ( ( N @ K )
= ( some @ A @ X ) )
| ( ( ( N @ K )
= ( none @ A ) )
& ( ( M @ K )
= ( some @ A @ X ) ) ) ) ) ).
% map_add_Some_iff
thf(fact_6361_map__add__SomeD,axiom,
! [B: $tType,A: $tType,M: B > ( option @ A ),N: B > ( option @ A ),K: B,X: A] :
( ( ( map_add @ B @ A @ M @ N @ K )
= ( some @ A @ X ) )
=> ( ( ( N @ K )
= ( some @ A @ X ) )
| ( ( ( N @ K )
= ( none @ A ) )
& ( ( M @ K )
= ( some @ A @ X ) ) ) ) ) ).
% map_add_SomeD
thf(fact_6362_map__add__def,axiom,
! [B: $tType,A: $tType] :
( ( map_add @ A @ B )
= ( ^ [M13: A > ( option @ B ),M24: A > ( option @ B ),X2: A] : ( case_option @ ( option @ B ) @ B @ ( M13 @ X2 ) @ ( some @ B ) @ ( M24 @ X2 ) ) ) ) ).
% map_add_def
thf(fact_6363_map__add__upd__left,axiom,
! [A: $tType,B: $tType,M: A,E22: A > ( option @ B ),E1: A > ( option @ B ),U1: B] :
( ~ ( member @ A @ M @ ( dom @ A @ B @ E22 ) )
=> ( ( map_add @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ E1 @ M @ ( some @ B @ U1 ) ) @ E22 )
= ( fun_upd @ A @ ( option @ B ) @ ( map_add @ A @ B @ E1 @ E22 ) @ M @ ( some @ B @ U1 ) ) ) ) ).
% map_add_upd_left
thf(fact_6364_inv__o__cancel,axiom,
! [B: $tType,A: $tType,F: A > B] :
( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
=> ( ( comp @ B @ A @ A @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F ) @ F )
= ( id @ A ) ) ) ).
% inv_o_cancel
thf(fact_6365_cr__natural__def,axiom,
( code_cr_natural
= ( ^ [X2: nat,Y: code_natural] :
( X2
= ( code_nat_of_natural @ Y ) ) ) ) ).
% cr_natural_def
thf(fact_6366_inv__identity,axiom,
! [A: $tType] :
( ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) )
@ ^ [A6: A] : A6 )
= ( ^ [A6: A] : A6 ) ) ).
% inv_identity
thf(fact_6367_o__inv__o__cancel,axiom,
! [B: $tType,C: $tType,A: $tType,F: A > B,G: A > C] :
( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
=> ( ( comp @ B @ C @ A @ ( comp @ A @ C @ B @ G @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F ) ) @ F )
= G ) ) ).
% o_inv_o_cancel
thf(fact_6368_inv__def,axiom,
! [B: $tType,A: $tType,F: B > A] :
( ( hilbert_inv_into @ B @ A @ ( top_top @ ( set @ B ) ) @ F )
= ( ^ [Y: A] :
( fChoice @ B
@ ^ [X2: B] :
( ( F @ X2 )
= Y ) ) ) ) ).
% inv_def
thf(fact_6369_inv__into__def2,axiom,
! [B: $tType,A: $tType] :
( ( hilbert_inv_into @ A @ B )
= ( ^ [A8: set @ A,F2: A > B,X2: B] :
( fChoice @ A
@ ^ [Y: A] :
( ( member @ A @ Y @ A8 )
& ( ( F2 @ Y )
= X2 ) ) ) ) ) ).
% inv_into_def2
thf(fact_6370_inv__into__def,axiom,
! [B: $tType,A: $tType] :
( ( hilbert_inv_into @ A @ B )
= ( ^ [A8: set @ A,F2: A > B,X2: B] :
( fChoice @ A
@ ^ [Y: A] :
( ( member @ A @ Y @ A8 )
& ( ( F2 @ Y )
= X2 ) ) ) ) ) ).
% inv_into_def
thf(fact_6371_inv__into__comp,axiom,
! [A: $tType,C: $tType,B: $tType,F: A > B,G: C > A,A5: set @ C,X: B] :
( ( inj_on @ A @ B @ F @ ( image2 @ C @ A @ G @ A5 ) )
=> ( ( inj_on @ C @ A @ G @ A5 )
=> ( ( member @ B @ X @ ( image2 @ A @ B @ F @ ( image2 @ C @ A @ G @ A5 ) ) )
=> ( ( hilbert_inv_into @ C @ B @ A5 @ ( comp @ A @ B @ C @ F @ G ) @ X )
= ( comp @ A @ C @ B @ ( hilbert_inv_into @ C @ A @ A5 @ G ) @ ( hilbert_inv_into @ A @ B @ ( image2 @ C @ A @ G @ A5 ) @ F ) @ X ) ) ) ) ) ).
% inv_into_comp
thf(fact_6372_inv__unique__comp,axiom,
! [B: $tType,A: $tType,F: B > A,G: A > B] :
( ( ( comp @ B @ A @ A @ F @ G )
= ( id @ A ) )
=> ( ( ( comp @ A @ B @ B @ G @ F )
= ( id @ B ) )
=> ( ( hilbert_inv_into @ B @ A @ ( top_top @ ( set @ B ) ) @ F )
= G ) ) ) ).
% inv_unique_comp
thf(fact_6373_o__inv__distrib,axiom,
! [C: $tType,B: $tType,A: $tType,F: A > B,G: C > A] :
( ( bij_betw @ A @ B @ F @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
=> ( ( bij_betw @ C @ A @ G @ ( top_top @ ( set @ C ) ) @ ( top_top @ ( set @ A ) ) )
=> ( ( hilbert_inv_into @ C @ B @ ( top_top @ ( set @ C ) ) @ ( comp @ A @ B @ C @ F @ G ) )
= ( comp @ A @ C @ B @ ( hilbert_inv_into @ C @ A @ ( top_top @ ( set @ C ) ) @ G ) @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F ) ) ) ) ) ).
% o_inv_distrib
thf(fact_6374_iso__backward,axiom,
! [A: $tType,B: $tType,X: A,Y3: A,R3: set @ ( product_prod @ A @ A ),R: set @ ( product_prod @ B @ B ),F: B > A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ R3 )
=> ( ( bNF_Wellorder_iso @ B @ A @ R @ R3 @ F )
=> ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( hilbert_inv_into @ B @ A @ ( field2 @ B @ R ) @ F @ X ) @ ( hilbert_inv_into @ B @ A @ ( field2 @ B @ R ) @ F @ Y3 ) ) @ R ) ) ) ).
% iso_backward
thf(fact_6375_bij__image__Collect__eq,axiom,
! [A: $tType,B: $tType,F: A > B,P: A > $o] :
( ( bij_betw @ A @ B @ F @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
=> ( ( image2 @ A @ B @ F @ ( collect @ A @ P ) )
= ( collect @ B
@ ^ [Y: B] : ( P @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F @ Y ) ) ) ) ) ).
% bij_image_Collect_eq
thf(fact_6376_surj__iff,axiom,
! [B: $tType,A: $tType,F: B > A] :
( ( ( image2 @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
= ( ( comp @ B @ A @ A @ F @ ( hilbert_inv_into @ B @ A @ ( top_top @ ( set @ B ) ) @ F ) )
= ( id @ A ) ) ) ).
% surj_iff
thf(fact_6377_inj__iff,axiom,
! [B: $tType,A: $tType,F: A > B] :
( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
= ( ( comp @ B @ A @ A @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F ) @ F )
= ( id @ A ) ) ) ).
% inj_iff
thf(fact_6378_inv__fn__o__fn__is__id,axiom,
! [A: $tType,F: A > A,N: nat] :
( ( bij_betw @ A @ A @ F @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ A ) ) )
=> ( ( comp @ A @ A @ A @ ( compow @ ( A > A ) @ N @ ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F ) ) @ ( compow @ ( A > A ) @ N @ F ) )
= ( ^ [X2: A] : X2 ) ) ) ).
% inv_fn_o_fn_is_id
thf(fact_6379_fn__o__inv__fn__is__id,axiom,
! [A: $tType,F: A > A,N: nat] :
( ( bij_betw @ A @ A @ F @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ A ) ) )
=> ( ( comp @ A @ A @ A @ ( compow @ ( A > A ) @ N @ F ) @ ( compow @ ( A > A ) @ N @ ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F ) ) )
= ( ^ [X2: A] : X2 ) ) ) ).
% fn_o_inv_fn_is_id
thf(fact_6380_bijection_Oinv__comp__right,axiom,
! [A: $tType,F: A > A] :
( ( hilbert_bijection @ A @ F )
=> ( ( comp @ A @ A @ A @ F @ ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F ) )
= ( id @ A ) ) ) ).
% bijection.inv_comp_right
thf(fact_6381_bijection_Oinv__comp__left,axiom,
! [A: $tType,F: A > A] :
( ( hilbert_bijection @ A @ F )
=> ( ( comp @ A @ A @ A @ ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F ) @ F )
= ( id @ A ) ) ) ).
% bijection.inv_comp_left
thf(fact_6382_subseqs_Osimps_I2_J,axiom,
! [A: $tType,X: A,Xs2: list @ A] :
( ( subseqs @ A @ ( cons @ A @ X @ Xs2 ) )
= ( append @ ( list @ A ) @ ( map @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X ) @ ( subseqs @ A @ Xs2 ) ) @ ( subseqs @ A @ Xs2 ) ) ) ).
% subseqs.simps(2)
thf(fact_6383_splice__replicate,axiom,
! [A: $tType,M: nat,X: A,N: nat] :
( ( splice @ A @ ( replicate @ A @ M @ X ) @ ( replicate @ A @ N @ X ) )
= ( replicate @ A @ ( plus_plus @ nat @ M @ N ) @ X ) ) ).
% splice_replicate
thf(fact_6384_length__splice,axiom,
! [A: $tType,Xs2: list @ A,Ys2: list @ A] :
( ( size_size @ ( list @ A ) @ ( splice @ A @ Xs2 @ Ys2 ) )
= ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ ( size_size @ ( list @ A ) @ Ys2 ) ) ) ).
% length_splice
thf(fact_6385_and_Omonoid__axioms,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( monoid @ A @ ( bit_se5824344872417868541ns_and @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% and.monoid_axioms
thf(fact_6386_cr__integer__def,axiom,
( code_cr_integer
= ( ^ [X2: int,Y: code_integer] :
( X2
= ( code_int_of_integer @ Y ) ) ) ) ).
% cr_integer_def
thf(fact_6387_monoid_Oright__neutral,axiom,
! [A: $tType,F: A > A > A,Z: A,A3: A] :
( ( monoid @ A @ F @ Z )
=> ( ( F @ A3 @ Z )
= A3 ) ) ).
% monoid.right_neutral
thf(fact_6388_monoid_Oleft__neutral,axiom,
! [A: $tType,F: A > A > A,Z: A,A3: A] :
( ( monoid @ A @ F @ Z )
=> ( ( F @ Z @ A3 )
= A3 ) ) ).
% monoid.left_neutral
thf(fact_6389_add_Omonoid__axioms,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ( monoid @ A @ ( plus_plus @ A ) @ ( zero_zero @ A ) ) ) ).
% add.monoid_axioms
thf(fact_6390_mult_Omonoid__axioms,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ( monoid @ A @ ( times_times @ A ) @ ( one_one @ A ) ) ) ).
% mult.monoid_axioms
thf(fact_6391_gcd__nat_Omonoid__axioms,axiom,
monoid @ nat @ ( gcd_gcd @ nat ) @ ( zero_zero @ nat ) ).
% gcd_nat.monoid_axioms
thf(fact_6392_or_Omonoid__axioms,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( monoid @ A @ ( bit_se1065995026697491101ons_or @ A ) @ ( zero_zero @ A ) ) ) ).
% or.monoid_axioms
thf(fact_6393_xor_Omonoid__axioms,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( monoid @ A @ ( bit_se5824344971392196577ns_xor @ A ) @ ( zero_zero @ A ) ) ) ).
% xor.monoid_axioms
thf(fact_6394_max__nat_Omonoid__axioms,axiom,
monoid @ nat @ ( ord_max @ nat ) @ ( zero_zero @ nat ) ).
% max_nat.monoid_axioms
thf(fact_6395_ndepth__Push__Node__aux,axiom,
! [A: $tType,I: nat,F: nat > ( sum_sum @ A @ nat ),K: nat] :
( ( ( case_nat @ ( sum_sum @ A @ nat ) @ ( sum_Inr @ nat @ A @ ( suc @ I ) ) @ F @ K )
= ( sum_Inr @ nat @ A @ ( zero_zero @ nat ) ) )
=> ( ord_less_eq @ nat
@ ( suc
@ ( ord_Least @ nat
@ ^ [X2: nat] :
( ( F @ X2 )
= ( sum_Inr @ nat @ A @ ( zero_zero @ nat ) ) ) ) )
@ K ) ) ).
% ndepth_Push_Node_aux
thf(fact_6396_Finite__Set_Ofold__def,axiom,
! [B: $tType,A: $tType] :
( ( finite_fold @ A @ B )
= ( ^ [F2: A > B > B,Z3: B,A8: set @ A] : ( if @ B @ ( finite_finite2 @ A @ A8 ) @ ( the @ B @ ( finite_fold_graph @ A @ B @ F2 @ Z3 @ A8 ) ) @ Z3 ) ) ) ).
% Finite_Set.fold_def
thf(fact_6397_sum_Osize_I4_J,axiom,
! [B: $tType,A: $tType,X22: B] :
( ( size_size @ ( sum_sum @ A @ B ) @ ( sum_Inr @ B @ A @ X22 ) )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% sum.size(4)
thf(fact_6398_fold__graph__image,axiom,
! [C: $tType,B: $tType,A: $tType,G: A > B,A5: set @ A,F: B > C > C,Z: C] :
( ( inj_on @ A @ B @ G @ A5 )
=> ( ( finite_fold_graph @ B @ C @ F @ Z @ ( image2 @ A @ B @ G @ A5 ) )
= ( finite_fold_graph @ A @ C @ ( comp @ B @ ( C > C ) @ A @ F @ G ) @ Z @ A5 ) ) ) ).
% fold_graph_image
thf(fact_6399_sum_Osize__gen_I2_J,axiom,
! [A: $tType,B: $tType,Xa: A > nat,X: B > nat,X22: B] :
( ( basic_BNF_size_sum @ A @ B @ Xa @ X @ ( sum_Inr @ B @ A @ X22 ) )
= ( plus_plus @ nat @ ( X @ X22 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% sum.size_gen(2)
thf(fact_6400_Node__def,axiom,
! [A: $tType,B: $tType] :
( ( old_Node @ B @ A )
= ( collect @ ( product_prod @ ( nat > ( sum_sum @ B @ nat ) ) @ ( sum_sum @ A @ nat ) )
@ ^ [P5: product_prod @ ( nat > ( sum_sum @ B @ nat ) ) @ ( sum_sum @ A @ nat )] :
? [F2: nat > ( sum_sum @ B @ nat ),X2: sum_sum @ A @ nat,K3: nat] :
( ( P5
= ( product_Pair @ ( nat > ( sum_sum @ B @ nat ) ) @ ( sum_sum @ A @ nat ) @ F2 @ X2 ) )
& ( ( F2 @ K3 )
= ( sum_Inr @ nat @ B @ ( zero_zero @ nat ) ) ) ) ) ) ).
% Node_def
thf(fact_6401_Node__K0__I,axiom,
! [B: $tType,A: $tType,A3: sum_sum @ B @ nat] :
( member @ ( product_prod @ ( nat > ( sum_sum @ A @ nat ) ) @ ( sum_sum @ B @ nat ) )
@ ( product_Pair @ ( nat > ( sum_sum @ A @ nat ) ) @ ( sum_sum @ B @ nat )
@ ^ [K3: nat] : ( sum_Inr @ nat @ A @ ( zero_zero @ nat ) )
@ A3 )
@ ( old_Node @ A @ B ) ) ).
% Node_K0_I
thf(fact_6402_Push__neq__K0,axiom,
! [A: $tType,K: nat,F: nat > ( sum_sum @ A @ nat )] :
( ( old_Push @ A @ ( sum_Inr @ nat @ A @ ( suc @ K ) ) @ F )
!= ( ^ [Z3: nat] : ( sum_Inr @ nat @ A @ ( zero_zero @ nat ) ) ) ) ).
% Push_neq_K0
thf(fact_6403_sum__set__simps_I4_J,axiom,
! [E: $tType,A: $tType,X: A] :
( ( basic_setr @ E @ A @ ( sum_Inr @ A @ E @ X ) )
= ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ).
% sum_set_simps(4)
thf(fact_6404_Push__def,axiom,
! [B: $tType] :
( ( old_Push @ B )
= ( case_nat @ ( sum_sum @ B @ nat ) ) ) ).
% Push_def
thf(fact_6405_prod_OPlus,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A5: set @ B,B4: set @ C,G: ( sum_sum @ B @ C ) > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( finite_finite2 @ C @ B4 )
=> ( ( groups7121269368397514597t_prod @ ( sum_sum @ B @ C ) @ A @ G @ ( sum_Plus @ B @ C @ A5 @ B4 ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ ( comp @ ( sum_sum @ B @ C ) @ A @ B @ G @ ( sum_Inl @ B @ C ) ) @ A5 ) @ ( groups7121269368397514597t_prod @ C @ A @ ( comp @ ( sum_sum @ B @ C ) @ A @ C @ G @ ( sum_Inr @ C @ B ) ) @ B4 ) ) ) ) ) ) ).
% prod.Plus
thf(fact_6406_sum_OPlus,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comm_monoid_add @ A )
=> ! [A5: set @ B,B4: set @ C,G: ( sum_sum @ B @ C ) > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( finite_finite2 @ C @ B4 )
=> ( ( groups7311177749621191930dd_sum @ ( sum_sum @ B @ C ) @ A @ G @ ( sum_Plus @ B @ C @ A5 @ B4 ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ ( comp @ ( sum_sum @ B @ C ) @ A @ B @ G @ ( sum_Inl @ B @ C ) ) @ A5 ) @ ( groups7311177749621191930dd_sum @ C @ A @ ( comp @ ( sum_sum @ B @ C ) @ A @ C @ G @ ( sum_Inr @ C @ B ) ) @ B4 ) ) ) ) ) ) ).
% sum.Plus
thf(fact_6407_Inr__Inl__False,axiom,
! [B: $tType,A: $tType,X: B,Y3: A] :
( ( sum_Inr @ B @ A @ X )
!= ( sum_Inl @ A @ B @ Y3 ) ) ).
% Inr_Inl_False
thf(fact_6408_Inl__Inr__False,axiom,
! [A: $tType,B: $tType,X: A,Y3: B] :
( ( sum_Inl @ A @ B @ X )
!= ( sum_Inr @ B @ A @ Y3 ) ) ).
% Inl_Inr_False
thf(fact_6409_sum__set__simps_I3_J,axiom,
! [A: $tType,D: $tType,X: A] :
( ( basic_setr @ A @ D @ ( sum_Inl @ A @ D @ X ) )
= ( bot_bot @ ( set @ D ) ) ) ).
% sum_set_simps(3)
thf(fact_6410_obj__sumE__f,axiom,
! [A: $tType,C: $tType,B: $tType,S: B,F: ( sum_sum @ A @ C ) > B,P: $o] :
( ! [X4: A] :
( ( S
= ( F @ ( sum_Inl @ A @ C @ X4 ) ) )
=> P )
=> ( ! [X4: C] :
( ( S
= ( F @ ( sum_Inr @ C @ A @ X4 ) ) )
=> P )
=> ! [X5: sum_sum @ A @ C] :
( ( S
= ( F @ X5 ) )
=> P ) ) ) ).
% obj_sumE_f
thf(fact_6411_sum_Osize_I3_J,axiom,
! [A: $tType,B: $tType,X1: A] :
( ( size_size @ ( sum_sum @ A @ B ) @ ( sum_Inl @ A @ B @ X1 ) )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% sum.size(3)
thf(fact_6412_sum_Osize__gen_I1_J,axiom,
! [B: $tType,A: $tType,Xa: A > nat,X: B > nat,X1: A] :
( ( basic_BNF_size_sum @ A @ B @ Xa @ X @ ( sum_Inl @ A @ B @ X1 ) )
= ( plus_plus @ nat @ ( Xa @ X1 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% sum.size_gen(1)
thf(fact_6413_sum__set__simps_I1_J,axiom,
! [B: $tType,A: $tType,X: A] :
( ( basic_setl @ A @ B @ ( sum_Inl @ A @ B @ X ) )
= ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ).
% sum_set_simps(1)
thf(fact_6414_ndepth__def,axiom,
! [B: $tType,A: $tType] :
( ( old_ndepth @ A @ B )
= ( ^ [N5: old_node @ A @ B] :
( product_case_prod @ ( nat > ( sum_sum @ B @ nat ) ) @ ( sum_sum @ A @ nat ) @ nat
@ ^ [F2: nat > ( sum_sum @ B @ nat ),X2: sum_sum @ A @ nat] :
( ord_Least @ nat
@ ^ [K3: nat] :
( ( F2 @ K3 )
= ( sum_Inr @ nat @ B @ ( zero_zero @ nat ) ) ) )
@ ( old_Rep_Node @ A @ B @ N5 ) ) ) ) ).
% ndepth_def
thf(fact_6415_sum__set__simps_I2_J,axiom,
! [A: $tType,C: $tType,X: A] :
( ( basic_setl @ C @ A @ ( sum_Inr @ A @ C @ X ) )
= ( bot_bot @ ( set @ C ) ) ) ).
% sum_set_simps(2)
thf(fact_6416_ndepth__K0,axiom,
! [A: $tType,B: $tType,X: sum_sum @ A @ nat] :
( ( old_ndepth @ A @ B
@ ( old_Abs_Node @ B @ A
@ ( product_Pair @ ( nat > ( sum_sum @ B @ nat ) ) @ ( sum_sum @ A @ nat )
@ ^ [K3: nat] : ( sum_Inr @ nat @ B @ ( zero_zero @ nat ) )
@ X ) ) )
= ( zero_zero @ nat ) ) ).
% ndepth_K0
thf(fact_6417_integer__of__nat__1,axiom,
( ( code_integer_of_nat @ ( one_one @ nat ) )
= ( one_one @ code_integer ) ) ).
% integer_of_nat_1
thf(fact_6418_integer__of__nat__0,axiom,
( ( code_integer_of_nat @ ( zero_zero @ nat ) )
= ( zero_zero @ code_integer ) ) ).
% integer_of_nat_0
thf(fact_6419_Atom__def,axiom,
! [B: $tType,A: $tType] :
( ( old_Atom @ A @ B )
= ( ^ [X2: sum_sum @ A @ nat] :
( insert2 @ ( old_node @ A @ B )
@ ( old_Abs_Node @ B @ A
@ ( product_Pair @ ( nat > ( sum_sum @ B @ nat ) ) @ ( sum_sum @ A @ nat )
@ ^ [K3: nat] : ( sum_Inr @ nat @ B @ ( zero_zero @ nat ) )
@ X2 ) )
@ ( bot_bot @ ( set @ ( old_node @ A @ B ) ) ) ) ) ) ).
% Atom_def
thf(fact_6420_Push__Node__def,axiom,
! [A: $tType,B: $tType] :
( ( old_Push_Node @ B @ A )
= ( ^ [N5: sum_sum @ B @ nat,X2: old_node @ A @ B] : ( old_Abs_Node @ B @ A @ ( product_apfst @ ( nat > ( sum_sum @ B @ nat ) ) @ ( nat > ( sum_sum @ B @ nat ) ) @ ( sum_sum @ A @ nat ) @ ( old_Push @ B @ N5 ) @ ( old_Rep_Node @ A @ B @ X2 ) ) ) ) ) ).
% Push_Node_def
thf(fact_6421_Lim__def,axiom,
! [A: $tType,B: $tType] :
( ( old_Lim @ B @ A )
= ( ^ [F2: B > ( set @ ( old_node @ A @ B ) )] :
( complete_Sup_Sup @ ( set @ ( old_node @ A @ B ) )
@ ( collect @ ( set @ ( old_node @ A @ B ) )
@ ^ [Z3: set @ ( old_node @ A @ B )] :
? [X2: B] :
( Z3
= ( image2 @ ( old_node @ A @ B ) @ ( old_node @ A @ B ) @ ( old_Push_Node @ B @ A @ ( sum_Inl @ B @ nat @ X2 ) ) @ ( F2 @ X2 ) ) ) ) ) ) ) ).
% Lim_def
thf(fact_6422_Scons__def,axiom,
! [B: $tType,A: $tType] :
( ( old_Scons @ A @ B )
= ( ^ [M9: set @ ( old_node @ A @ B ),N11: set @ ( old_node @ A @ B )] : ( sup_sup @ ( set @ ( old_node @ A @ B ) ) @ ( image2 @ ( old_node @ A @ B ) @ ( old_node @ A @ B ) @ ( old_Push_Node @ B @ A @ ( sum_Inr @ nat @ B @ ( one_one @ nat ) ) ) @ M9 ) @ ( image2 @ ( old_node @ A @ B ) @ ( old_node @ A @ B ) @ ( old_Push_Node @ B @ A @ ( sum_Inr @ nat @ B @ ( suc @ ( one_one @ nat ) ) ) ) @ N11 ) ) ) ) ).
% Scons_def
thf(fact_6423_Scons__UN1__y,axiom,
! [A: $tType,B: $tType,C: $tType,M2: set @ ( old_node @ A @ B ),F: C > ( set @ ( old_node @ A @ B ) )] :
( ( old_Scons @ A @ B @ M2 @ ( complete_Sup_Sup @ ( set @ ( old_node @ A @ B ) ) @ ( image2 @ C @ ( set @ ( old_node @ A @ B ) ) @ F @ ( top_top @ ( set @ C ) ) ) ) )
= ( complete_Sup_Sup @ ( set @ ( old_node @ A @ B ) )
@ ( image2 @ C @ ( set @ ( old_node @ A @ B ) )
@ ^ [X2: C] : ( old_Scons @ A @ B @ M2 @ ( F @ X2 ) )
@ ( top_top @ ( set @ C ) ) ) ) ) ).
% Scons_UN1_y
thf(fact_6424_Scons__UN1__x,axiom,
! [B: $tType,A: $tType,C: $tType,F: C > ( set @ ( old_node @ A @ B ) ),M2: set @ ( old_node @ A @ B )] :
( ( old_Scons @ A @ B @ ( complete_Sup_Sup @ ( set @ ( old_node @ A @ B ) ) @ ( image2 @ C @ ( set @ ( old_node @ A @ B ) ) @ F @ ( top_top @ ( set @ C ) ) ) ) @ M2 )
= ( complete_Sup_Sup @ ( set @ ( old_node @ A @ B ) )
@ ( image2 @ C @ ( set @ ( old_node @ A @ B ) )
@ ^ [X2: C] : ( old_Scons @ A @ B @ ( F @ X2 ) @ M2 )
@ ( top_top @ ( set @ C ) ) ) ) ) ).
% Scons_UN1_x
thf(fact_6425_Split__def,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( old_Split @ A @ B @ C )
= ( ^ [C4: ( set @ ( old_node @ A @ B ) ) > ( set @ ( old_node @ A @ B ) ) > C,M9: set @ ( old_node @ A @ B )] :
( the @ C
@ ^ [U: C] :
? [X2: set @ ( old_node @ A @ B ),Y: set @ ( old_node @ A @ B )] :
( ( M9
= ( old_Scons @ A @ B @ X2 @ Y ) )
& ( U
= ( C4 @ X2 @ Y ) ) ) ) ) ) ).
% Split_def
thf(fact_6426_Leaf__def,axiom,
! [B: $tType,A: $tType] :
( ( old_Leaf @ A @ B )
= ( comp @ ( sum_sum @ A @ nat ) @ ( set @ ( old_node @ A @ B ) ) @ A @ ( old_Atom @ A @ B ) @ ( sum_Inl @ A @ nat ) ) ) ).
% Leaf_def
thf(fact_6427_ntrunc__UN1,axiom,
! [A: $tType,B: $tType,C: $tType,K: nat,F: C > ( set @ ( old_node @ A @ B ) )] :
( ( old_ntrunc @ A @ B @ K @ ( complete_Sup_Sup @ ( set @ ( old_node @ A @ B ) ) @ ( image2 @ C @ ( set @ ( old_node @ A @ B ) ) @ F @ ( top_top @ ( set @ C ) ) ) ) )
= ( complete_Sup_Sup @ ( set @ ( old_node @ A @ B ) )
@ ( image2 @ C @ ( set @ ( old_node @ A @ B ) )
@ ^ [X2: C] : ( old_ntrunc @ A @ B @ K @ ( F @ X2 ) )
@ ( top_top @ ( set @ C ) ) ) ) ) ).
% ntrunc_UN1
thf(fact_6428_In1__UN1,axiom,
! [A: $tType,B: $tType,C: $tType,F: C > ( set @ ( old_node @ A @ B ) )] :
( ( old_In1 @ A @ B @ ( complete_Sup_Sup @ ( set @ ( old_node @ A @ B ) ) @ ( image2 @ C @ ( set @ ( old_node @ A @ B ) ) @ F @ ( top_top @ ( set @ C ) ) ) ) )
= ( complete_Sup_Sup @ ( set @ ( old_node @ A @ B ) )
@ ( image2 @ C @ ( set @ ( old_node @ A @ B ) )
@ ^ [X2: C] : ( old_In1 @ A @ B @ ( F @ X2 ) )
@ ( top_top @ ( set @ C ) ) ) ) ) ).
% In1_UN1
thf(fact_6429_ntrunc__0,axiom,
! [B: $tType,A: $tType,M2: set @ ( old_node @ A @ B )] :
( ( old_ntrunc @ A @ B @ ( zero_zero @ nat ) @ M2 )
= ( bot_bot @ ( set @ ( old_node @ A @ B ) ) ) ) ).
% ntrunc_0
thf(fact_6430_ntrunc__one__In1,axiom,
! [B: $tType,A: $tType,M2: set @ ( old_node @ A @ B )] :
( ( old_ntrunc @ A @ B @ ( suc @ ( zero_zero @ nat ) ) @ ( old_In1 @ A @ B @ M2 ) )
= ( bot_bot @ ( set @ ( old_node @ A @ B ) ) ) ) ).
% ntrunc_one_In1
thf(fact_6431_ntrunc__o__equality,axiom,
! [C: $tType,B: $tType,A: $tType,H1: A > ( set @ ( old_node @ B @ C ) ),H22: A > ( set @ ( old_node @ B @ C ) )] :
( ! [K2: nat] :
( ( comp @ ( set @ ( old_node @ B @ C ) ) @ ( set @ ( old_node @ B @ C ) ) @ A @ ( old_ntrunc @ B @ C @ K2 ) @ H1 )
= ( comp @ ( set @ ( old_node @ B @ C ) ) @ ( set @ ( old_node @ B @ C ) ) @ A @ ( old_ntrunc @ B @ C @ K2 ) @ H22 ) )
=> ( H1 = H22 ) ) ).
% ntrunc_o_equality
thf(fact_6432_ntrunc__def,axiom,
! [A: $tType,B: $tType] :
( ( old_ntrunc @ A @ B )
= ( ^ [K3: nat,N11: set @ ( old_node @ A @ B )] :
( collect @ ( old_node @ A @ B )
@ ^ [N5: old_node @ A @ B] :
( ( member @ ( old_node @ A @ B ) @ N5 @ N11 )
& ( ord_less @ nat @ ( old_ndepth @ A @ B @ N5 ) @ K3 ) ) ) ) ) ).
% ntrunc_def
thf(fact_6433_ntrunc__one__In0,axiom,
! [B: $tType,A: $tType,M2: set @ ( old_node @ A @ B )] :
( ( old_ntrunc @ A @ B @ ( suc @ ( zero_zero @ nat ) ) @ ( old_In0 @ A @ B @ M2 ) )
= ( bot_bot @ ( set @ ( old_node @ A @ B ) ) ) ) ).
% ntrunc_one_In0
thf(fact_6434_In1__def,axiom,
! [A: $tType,B: $tType] :
( ( old_In1 @ A @ B )
= ( old_Scons @ A @ B @ ( old_Numb @ A @ B @ ( one_one @ nat ) ) ) ) ).
% In1_def
thf(fact_6435_In0__def,axiom,
! [A: $tType,B: $tType] :
( ( old_In0 @ A @ B )
= ( old_Scons @ A @ B @ ( old_Numb @ A @ B @ ( zero_zero @ nat ) ) ) ) ).
% In0_def
thf(fact_6436_Numb__def,axiom,
! [B: $tType,A: $tType] :
( ( old_Numb @ A @ B )
= ( comp @ ( sum_sum @ A @ nat ) @ ( set @ ( old_node @ A @ B ) ) @ nat @ ( old_Atom @ A @ B ) @ ( sum_Inr @ nat @ A ) ) ) ).
% Numb_def
thf(fact_6437_In0__UN1,axiom,
! [A: $tType,B: $tType,C: $tType,F: C > ( set @ ( old_node @ A @ B ) )] :
( ( old_In0 @ A @ B @ ( complete_Sup_Sup @ ( set @ ( old_node @ A @ B ) ) @ ( image2 @ C @ ( set @ ( old_node @ A @ B ) ) @ F @ ( top_top @ ( set @ C ) ) ) ) )
= ( complete_Sup_Sup @ ( set @ ( old_node @ A @ B ) )
@ ( image2 @ C @ ( set @ ( old_node @ A @ B ) )
@ ^ [X2: C] : ( old_In0 @ A @ B @ ( F @ X2 ) )
@ ( top_top @ ( set @ C ) ) ) ) ) ).
% In0_UN1
thf(fact_6438_Case__def,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( old_Case @ A @ B @ C )
= ( ^ [C4: ( set @ ( old_node @ A @ B ) ) > C,D4: ( set @ ( old_node @ A @ B ) ) > C,M9: set @ ( old_node @ A @ B )] :
( the @ C
@ ^ [U: C] :
( ? [X2: set @ ( old_node @ A @ B )] :
( ( M9
= ( old_In0 @ A @ B @ X2 ) )
& ( U
= ( C4 @ X2 ) ) )
| ? [Y: set @ ( old_node @ A @ B )] :
( ( M9
= ( old_In1 @ A @ B @ Y ) )
& ( U
= ( D4 @ Y ) ) ) ) ) ) ) ).
% Case_def
thf(fact_6439_dsum__def,axiom,
! [B: $tType,A: $tType] :
( ( old_dsum @ A @ B )
= ( ^ [R4: set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ),S2: set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) )] :
( sup_sup @ ( set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) )
@ ( complete_Sup_Sup @ ( set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) )
@ ( image2 @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) @ ( set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) )
@ ( product_case_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) )
@ ^ [X2: set @ ( old_node @ A @ B ),X8: set @ ( old_node @ A @ B )] : ( insert2 @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) @ ( product_Pair @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ ( old_In0 @ A @ B @ X2 ) @ ( old_In0 @ A @ B @ X8 ) ) @ ( bot_bot @ ( set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) ) ) ) )
@ R4 ) )
@ ( complete_Sup_Sup @ ( set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) )
@ ( image2 @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) @ ( set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) )
@ ( product_case_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) )
@ ^ [Y: set @ ( old_node @ A @ B ),Y7: set @ ( old_node @ A @ B )] : ( insert2 @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) @ ( product_Pair @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ ( old_In1 @ A @ B @ Y ) @ ( old_In1 @ A @ B @ Y7 ) ) @ ( bot_bot @ ( set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) ) ) ) )
@ S2 ) ) ) ) ) ).
% dsum_def
thf(fact_6440_dsum__In1I,axiom,
! [B: $tType,A: $tType,N6: set @ ( old_node @ A @ B ),N12: set @ ( old_node @ A @ B ),S: set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ),R: set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) )] :
( ( member @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) @ ( product_Pair @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ N6 @ N12 ) @ S )
=> ( member @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) @ ( product_Pair @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ ( old_In1 @ A @ B @ N6 ) @ ( old_In1 @ A @ B @ N12 ) ) @ ( old_dsum @ A @ B @ R @ S ) ) ) ).
% dsum_In1I
thf(fact_6441_dsum__In0I,axiom,
! [B: $tType,A: $tType,M2: set @ ( old_node @ A @ B ),M10: set @ ( old_node @ A @ B ),R: set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ),S: set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) )] :
( ( member @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) @ ( product_Pair @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ M2 @ M10 ) @ R )
=> ( member @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) @ ( product_Pair @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ ( old_In0 @ A @ B @ M2 ) @ ( old_In0 @ A @ B @ M10 ) ) @ ( old_dsum @ A @ B @ R @ S ) ) ) ).
% dsum_In0I
thf(fact_6442_dsumE,axiom,
! [B: $tType,A: $tType,W: product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ),R: set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ),S: set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) )] :
( ( member @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) @ W @ ( old_dsum @ A @ B @ R @ S ) )
=> ( ! [X4: set @ ( old_node @ A @ B ),X11: set @ ( old_node @ A @ B )] :
( ( member @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) @ ( product_Pair @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ X4 @ X11 ) @ R )
=> ( W
!= ( product_Pair @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ ( old_In0 @ A @ B @ X4 ) @ ( old_In0 @ A @ B @ X11 ) ) ) )
=> ~ ! [Y4: set @ ( old_node @ A @ B ),Y11: set @ ( old_node @ A @ B )] :
( ( member @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) @ ( product_Pair @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ Y4 @ Y11 ) @ S )
=> ( W
!= ( product_Pair @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ ( old_In1 @ A @ B @ Y4 ) @ ( old_In1 @ A @ B @ Y11 ) ) ) ) ) ) ).
% dsumE
thf(fact_6443_dprod__def,axiom,
! [B: $tType,A: $tType] :
( ( old_dprod @ A @ B )
= ( ^ [R4: set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ),S2: set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) )] :
( complete_Sup_Sup @ ( set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) )
@ ( image2 @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) @ ( set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) )
@ ( product_case_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) )
@ ^ [X2: set @ ( old_node @ A @ B ),X8: set @ ( old_node @ A @ B )] :
( complete_Sup_Sup @ ( set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) )
@ ( image2 @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) @ ( set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) )
@ ( product_case_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) )
@ ^ [Y: set @ ( old_node @ A @ B ),Y7: set @ ( old_node @ A @ B )] : ( insert2 @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) @ ( product_Pair @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ ( old_Scons @ A @ B @ X2 @ Y ) @ ( old_Scons @ A @ B @ X8 @ Y7 ) ) @ ( bot_bot @ ( set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) ) ) ) )
@ S2 ) ) )
@ R4 ) ) ) ) ).
% dprod_def
thf(fact_6444_dprodI,axiom,
! [B: $tType,A: $tType,M2: set @ ( old_node @ A @ B ),M10: set @ ( old_node @ A @ B ),R: set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ),N6: set @ ( old_node @ A @ B ),N12: set @ ( old_node @ A @ B ),S: set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) )] :
( ( member @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) @ ( product_Pair @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ M2 @ M10 ) @ R )
=> ( ( member @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) @ ( product_Pair @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ N6 @ N12 ) @ S )
=> ( member @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) @ ( product_Pair @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ ( old_Scons @ A @ B @ M2 @ N6 ) @ ( old_Scons @ A @ B @ M10 @ N12 ) ) @ ( old_dprod @ A @ B @ R @ S ) ) ) ) ).
% dprodI
thf(fact_6445_dprodE,axiom,
! [B: $tType,A: $tType,C2: product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ),R: set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ),S: set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) )] :
( ( member @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) @ C2 @ ( old_dprod @ A @ B @ R @ S ) )
=> ~ ! [X4: set @ ( old_node @ A @ B ),Y4: set @ ( old_node @ A @ B ),X11: set @ ( old_node @ A @ B )] :
( ( member @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) @ ( product_Pair @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ X4 @ X11 ) @ R )
=> ! [Y11: set @ ( old_node @ A @ B )] :
( ( member @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) @ ( product_Pair @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ Y4 @ Y11 ) @ S )
=> ( C2
!= ( product_Pair @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ ( old_Scons @ A @ B @ X4 @ Y4 ) @ ( old_Scons @ A @ B @ X11 @ Y11 ) ) ) ) ) ) ).
% dprodE
thf(fact_6446_Powp__Pow__eq,axiom,
! [A: $tType,A5: set @ A] :
( ( powp @ A
@ ^ [X2: A] : ( member @ A @ X2 @ A5 ) )
= ( ^ [X2: set @ A] : ( member @ ( set @ A ) @ X2 @ ( pow2 @ A @ A5 ) ) ) ) ).
% Powp_Pow_eq
thf(fact_6447_combine__options__def,axiom,
! [A: $tType] :
( ( combine_options @ A )
= ( ^ [F2: A > A > A,X2: option @ A,Y: option @ A] :
( case_option @ ( option @ A ) @ A @ Y
@ ^ [Z3: A] :
( case_option @ ( option @ A ) @ A @ ( some @ A @ Z3 )
@ ^ [Aa3: A] : ( some @ A @ ( F2 @ Z3 @ Aa3 ) )
@ Y )
@ X2 ) ) ) ).
% combine_options_def
thf(fact_6448_combine__options__simps_I2_J,axiom,
! [A: $tType,F: A > A > A,X: option @ A] :
( ( combine_options @ A @ F @ X @ ( none @ A ) )
= X ) ).
% combine_options_simps(2)
thf(fact_6449_combine__options__simps_I1_J,axiom,
! [A: $tType,F: A > A > A,Y3: option @ A] :
( ( combine_options @ A @ F @ ( none @ A ) @ Y3 )
= Y3 ) ).
% combine_options_simps(1)
thf(fact_6450_combine__options__simps_I3_J,axiom,
! [A: $tType,F: A > A > A,A3: A,B2: A] :
( ( combine_options @ A @ F @ ( some @ A @ A3 ) @ ( some @ A @ B2 ) )
= ( some @ A @ ( F @ A3 @ B2 ) ) ) ).
% combine_options_simps(3)
thf(fact_6451_combine__options__left__commute,axiom,
! [A: $tType,F: A > A > A,Y3: option @ A,X: option @ A,Z: option @ A] :
( ! [X4: A,Y4: A] :
( ( F @ X4 @ Y4 )
= ( F @ Y4 @ X4 ) )
=> ( ! [X4: A,Y4: A,Z6: A] :
( ( F @ ( F @ X4 @ Y4 ) @ Z6 )
= ( F @ X4 @ ( F @ Y4 @ Z6 ) ) )
=> ( ( combine_options @ A @ F @ Y3 @ ( combine_options @ A @ F @ X @ Z ) )
= ( combine_options @ A @ F @ X @ ( combine_options @ A @ F @ Y3 @ Z ) ) ) ) ) ).
% combine_options_left_commute
thf(fact_6452_combine__options__commute,axiom,
! [A: $tType,F: A > A > A,X: option @ A,Y3: option @ A] :
( ! [X4: A,Y4: A] :
( ( F @ X4 @ Y4 )
= ( F @ Y4 @ X4 ) )
=> ( ( combine_options @ A @ F @ X @ Y3 )
= ( combine_options @ A @ F @ Y3 @ X ) ) ) ).
% combine_options_commute
thf(fact_6453_combine__options__assoc,axiom,
! [A: $tType,F: A > A > A,X: option @ A,Y3: option @ A,Z: option @ A] :
( ! [X4: A,Y4: A,Z6: A] :
( ( F @ ( F @ X4 @ Y4 ) @ Z6 )
= ( F @ X4 @ ( F @ Y4 @ Z6 ) ) )
=> ( ( combine_options @ A @ F @ ( combine_options @ A @ F @ X @ Y3 ) @ Z )
= ( combine_options @ A @ F @ X @ ( combine_options @ A @ F @ Y3 @ Z ) ) ) ) ).
% combine_options_assoc
thf(fact_6454_Powp__def,axiom,
! [A: $tType] :
( ( powp @ A )
= ( ^ [A8: A > $o,B7: set @ A] :
! [X2: A] :
( ( member @ A @ X2 @ B7 )
=> ( A8 @ X2 ) ) ) ) ).
% Powp_def
thf(fact_6455_shuffles_Opelims,axiom,
! [A: $tType,X: list @ A,Xa: list @ A,Y3: set @ ( list @ A )] :
( ( ( shuffles @ A @ X @ Xa )
= Y3 )
=> ( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( shuffles_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Xa ) )
=> ( ( ( X
= ( nil @ A ) )
=> ( ( Y3
= ( insert2 @ ( list @ A ) @ Xa @ ( bot_bot @ ( set @ ( list @ A ) ) ) ) )
=> ~ ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( shuffles_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Xa ) ) ) )
=> ( ( ( Xa
= ( nil @ A ) )
=> ( ( Y3
= ( insert2 @ ( list @ A ) @ X @ ( bot_bot @ ( set @ ( list @ A ) ) ) ) )
=> ~ ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( shuffles_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ ( nil @ A ) ) ) ) )
=> ~ ! [X4: A,Xs5: list @ A] :
( ( X
= ( cons @ A @ X4 @ Xs5 ) )
=> ! [Y4: A,Ys5: list @ A] :
( ( Xa
= ( cons @ A @ Y4 @ Ys5 ) )
=> ( ( Y3
= ( sup_sup @ ( set @ ( list @ A ) ) @ ( image2 @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X4 ) @ ( shuffles @ A @ Xs5 @ ( cons @ A @ Y4 @ Ys5 ) ) ) @ ( image2 @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ Y4 ) @ ( shuffles @ A @ ( cons @ A @ X4 @ Xs5 ) @ Ys5 ) ) ) )
=> ~ ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( shuffles_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X4 @ Xs5 ) @ ( cons @ A @ Y4 @ Ys5 ) ) ) ) ) ) ) ) ) ) ).
% shuffles.pelims
thf(fact_6456_shuffles_Opsimps_I3_J,axiom,
! [A: $tType,X: A,Xs2: list @ A,Y3: A,Ys2: list @ A] :
( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( shuffles_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs2 ) @ ( cons @ A @ Y3 @ Ys2 ) ) )
=> ( ( shuffles @ A @ ( cons @ A @ X @ Xs2 ) @ ( cons @ A @ Y3 @ Ys2 ) )
= ( sup_sup @ ( set @ ( list @ A ) ) @ ( image2 @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X ) @ ( shuffles @ A @ Xs2 @ ( cons @ A @ Y3 @ Ys2 ) ) ) @ ( image2 @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ Y3 ) @ ( shuffles @ A @ ( cons @ A @ X @ Xs2 ) @ Ys2 ) ) ) ) ) ).
% shuffles.psimps(3)
thf(fact_6457_shuffles_Opinduct,axiom,
! [A: $tType,A0: list @ A,A1: list @ A,P: ( list @ A ) > ( list @ A ) > $o] :
( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( shuffles_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ A0 @ A1 ) )
=> ( ! [Ys5: list @ A] :
( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( shuffles_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ys5 ) )
=> ( P @ ( nil @ A ) @ Ys5 ) )
=> ( ! [Xs5: list @ A] :
( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( shuffles_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs5 @ ( nil @ A ) ) )
=> ( P @ Xs5 @ ( nil @ A ) ) )
=> ( ! [X4: A,Xs5: list @ A,Y4: A,Ys5: list @ A] :
( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( shuffles_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X4 @ Xs5 ) @ ( cons @ A @ Y4 @ Ys5 ) ) )
=> ( ( P @ Xs5 @ ( cons @ A @ Y4 @ Ys5 ) )
=> ( ( P @ ( cons @ A @ X4 @ Xs5 ) @ Ys5 )
=> ( P @ ( cons @ A @ X4 @ Xs5 ) @ ( cons @ A @ Y4 @ Ys5 ) ) ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ) ).
% shuffles.pinduct
thf(fact_6458_shuffles_Opsimps_I1_J,axiom,
! [A: $tType,Ys2: list @ A] :
( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( shuffles_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ys2 ) )
=> ( ( shuffles @ A @ ( nil @ A ) @ Ys2 )
= ( insert2 @ ( list @ A ) @ Ys2 @ ( bot_bot @ ( set @ ( list @ A ) ) ) ) ) ) ).
% shuffles.psimps(1)
thf(fact_6459_shuffles_Opsimps_I2_J,axiom,
! [A: $tType,Xs2: list @ A] :
( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( shuffles_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ ( nil @ A ) ) )
=> ( ( shuffles @ A @ Xs2 @ ( nil @ A ) )
= ( insert2 @ ( list @ A ) @ Xs2 @ ( bot_bot @ ( set @ ( list @ A ) ) ) ) ) ) ).
% shuffles.psimps(2)
thf(fact_6460_splice_Opelims,axiom,
! [A: $tType,X: list @ A,Xa: list @ A,Y3: list @ A] :
( ( ( splice @ A @ X @ Xa )
= Y3 )
=> ( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( splice_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Xa ) )
=> ( ( ( X
= ( nil @ A ) )
=> ( ( Y3 = Xa )
=> ~ ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( splice_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Xa ) ) ) )
=> ~ ! [X4: A,Xs5: list @ A] :
( ( X
= ( cons @ A @ X4 @ Xs5 ) )
=> ( ( Y3
= ( cons @ A @ X4 @ ( splice @ A @ Xa @ Xs5 ) ) )
=> ~ ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( splice_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X4 @ Xs5 ) @ Xa ) ) ) ) ) ) ) ).
% splice.pelims
thf(fact_6461_splice_Opsimps_I1_J,axiom,
! [A: $tType,Ys2: list @ A] :
( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( splice_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ys2 ) )
=> ( ( splice @ A @ ( nil @ A ) @ Ys2 )
= Ys2 ) ) ).
% splice.psimps(1)
thf(fact_6462_splice_Opinduct,axiom,
! [A: $tType,A0: list @ A,A1: list @ A,P: ( list @ A ) > ( list @ A ) > $o] :
( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( splice_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ A0 @ A1 ) )
=> ( ! [Ys5: list @ A] :
( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( splice_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ys5 ) )
=> ( P @ ( nil @ A ) @ Ys5 ) )
=> ( ! [X4: A,Xs5: list @ A,Ys5: list @ A] :
( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( splice_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X4 @ Xs5 ) @ Ys5 ) )
=> ( ( P @ Ys5 @ Xs5 )
=> ( P @ ( cons @ A @ X4 @ Xs5 ) @ Ys5 ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% splice.pinduct
thf(fact_6463_splice_Opsimps_I2_J,axiom,
! [A: $tType,X: A,Xs2: list @ A,Ys2: list @ A] :
( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( splice_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs2 ) @ Ys2 ) )
=> ( ( splice @ A @ ( cons @ A @ X @ Xs2 ) @ Ys2 )
= ( cons @ A @ X @ ( splice @ A @ Ys2 @ Xs2 ) ) ) ) ).
% splice.psimps(2)
thf(fact_6464_equivp__equiv,axiom,
! [A: $tType,A5: set @ ( product_prod @ A @ A )] :
( ( equiv_equiv @ A @ ( top_top @ ( set @ A ) ) @ A5 )
= ( equiv_equivp @ A
@ ^ [X2: A,Y: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y ) @ A5 ) ) ) ).
% equivp_equiv
thf(fact_6465_Shift__def,axiom,
! [A: $tType] :
( ( bNF_Greatest_Shift @ A )
= ( ^ [Kl: set @ ( list @ A ),K3: A] :
( collect @ ( list @ A )
@ ^ [Kl2: list @ A] : ( member @ ( list @ A ) @ ( cons @ A @ K3 @ Kl2 ) @ Kl ) ) ) ) ).
% Shift_def
thf(fact_6466_int__equivp,axiom,
equiv_equivp @ ( product_prod @ nat @ nat ) @ intrel ).
% int_equivp
thf(fact_6467_Quotient__crel__quotient,axiom,
! [B: $tType,A: $tType,R5: A > A > $o,Abs: A > B,Rep: B > A,T5: A > B > $o] :
( ( quotient @ A @ B @ R5 @ Abs @ Rep @ T5 )
=> ( ( equiv_equivp @ A @ R5 )
=> ( T5
= ( ^ [X2: A] :
( ^ [Y6: B,Z2: B] : Y6 = Z2
@ ( Abs @ X2 ) ) ) ) ) ) ).
% Quotient_crel_quotient
thf(fact_6468_Quotient3__to__Quotient__equivp,axiom,
! [B: $tType,A: $tType,R5: A > A > $o,Abs: A > B,Rep: B > A,T5: A > B > $o] :
( ( quotient3 @ A @ B @ R5 @ Abs @ Rep )
=> ( ( T5
= ( ^ [X2: A] :
( ^ [Y6: B,Z2: B] : Y6 = Z2
@ ( Abs @ X2 ) ) ) )
=> ( ( equiv_equivp @ A @ R5 )
=> ( quotient @ A @ B @ R5 @ Abs @ Rep @ T5 ) ) ) ) ).
% Quotient3_to_Quotient_equivp
thf(fact_6469_cut__def,axiom,
! [B: $tType,A: $tType] :
( ( cut @ A @ B )
= ( ^ [F2: A > B,R6: set @ ( product_prod @ A @ A ),X2: A,Y: A] : ( if @ B @ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X2 ) @ R6 ) @ ( F2 @ Y ) @ ( undefined @ B ) ) ) ) ).
% cut_def
thf(fact_6470_map__conv__bind__option,axiom,
! [A: $tType,B: $tType] :
( ( map_option @ B @ A )
= ( ^ [F2: B > A,X2: option @ B] : ( bind2 @ B @ A @ X2 @ ( comp @ A @ ( option @ A ) @ B @ ( some @ A ) @ F2 ) ) ) ) ).
% map_conv_bind_option
thf(fact_6471_bind__assoc,axiom,
! [B: $tType,A: $tType,C: $tType,X: option @ C,F: C > ( option @ B ),G: B > ( option @ A )] :
( ( bind2 @ B @ A @ ( bind2 @ C @ B @ X @ F ) @ G )
= ( bind2 @ C @ A @ X
@ ^ [Y: C] : ( bind2 @ B @ A @ ( F @ Y ) @ G ) ) ) ).
% bind_assoc
thf(fact_6472_bind__runit,axiom,
! [A: $tType,X: option @ A] :
( ( bind2 @ A @ A @ X @ ( some @ A ) )
= X ) ).
% bind_runit
thf(fact_6473_bind__rzero,axiom,
! [B: $tType,A: $tType,X: option @ B] :
( ( bind2 @ B @ A @ X
@ ^ [X2: B] : ( none @ A ) )
= ( none @ A ) ) ).
% bind_rzero
thf(fact_6474_bind_Obind__lunit,axiom,
! [B: $tType,A: $tType,X: A,F: A > ( option @ B )] :
( ( bind2 @ A @ B @ ( some @ A @ X ) @ F )
= ( F @ X ) ) ).
% bind.bind_lunit
thf(fact_6475_Option_Obind__cong,axiom,
! [B: $tType,A: $tType,X: option @ A,Y3: option @ A,F: A > ( option @ B ),G: A > ( option @ B )] :
( ( X = Y3 )
=> ( ! [A4: A] :
( ( Y3
= ( some @ A @ A4 ) )
=> ( ( F @ A4 )
= ( G @ A4 ) ) )
=> ( ( bind2 @ A @ B @ X @ F )
= ( bind2 @ A @ B @ Y3 @ G ) ) ) ) ).
% Option.bind_cong
thf(fact_6476_bind__eq__Some__conv,axiom,
! [A: $tType,B: $tType,F: option @ B,G: B > ( option @ A ),X: A] :
( ( ( bind2 @ B @ A @ F @ G )
= ( some @ A @ X ) )
= ( ? [Y: B] :
( ( F
= ( some @ B @ Y ) )
& ( ( G @ Y )
= ( some @ A @ X ) ) ) ) ) ).
% bind_eq_Some_conv
thf(fact_6477_bind__option__cong__code,axiom,
! [B: $tType,A: $tType,X: option @ A,Y3: option @ A,F: A > ( option @ B )] :
( ( X = Y3 )
=> ( ( bind2 @ A @ B @ X @ F )
= ( bind2 @ A @ B @ Y3 @ F ) ) ) ).
% bind_option_cong_code
thf(fact_6478_bind_Obind__lzero,axiom,
! [A: $tType,B: $tType,F: A > ( option @ B )] :
( ( bind2 @ A @ B @ ( none @ A ) @ F )
= ( none @ B ) ) ).
% bind.bind_lzero
thf(fact_6479_cuts__eq,axiom,
! [B: $tType,A: $tType,F: A > B,R5: set @ ( product_prod @ A @ A ),X: A,G: A > B] :
( ( ( cut @ A @ B @ F @ R5 @ X )
= ( cut @ A @ B @ G @ R5 @ X ) )
= ( ! [Y: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X ) @ R5 )
=> ( ( F @ Y )
= ( G @ Y ) ) ) ) ) ).
% cuts_eq
thf(fact_6480_cut__apply,axiom,
! [B: $tType,A: $tType,X: A,A3: A,R5: set @ ( product_prod @ A @ A ),F: A > B] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ A3 ) @ R5 )
=> ( ( cut @ A @ B @ F @ R5 @ A3 @ X )
= ( F @ X ) ) ) ).
% cut_apply
thf(fact_6481_bind__split__asm,axiom,
! [A: $tType,B: $tType,P: ( option @ A ) > $o,M: option @ B,F: B > ( option @ A )] :
( ( P @ ( bind2 @ B @ A @ M @ F ) )
= ( ~ ( ( ( M
= ( none @ B ) )
& ~ ( P @ ( none @ A ) ) )
| ? [X2: B] :
( ( M
= ( some @ B @ X2 ) )
& ~ ( P @ ( F @ X2 ) ) ) ) ) ) ).
% bind_split_asm
thf(fact_6482_bind__split,axiom,
! [A: $tType,B: $tType,P: ( option @ A ) > $o,M: option @ B,F: B > ( option @ A )] :
( ( P @ ( bind2 @ B @ A @ M @ F ) )
= ( ( ( M
= ( none @ B ) )
=> ( P @ ( none @ A ) ) )
& ! [V6: B] :
( ( M
= ( some @ B @ V6 ) )
=> ( P @ ( F @ V6 ) ) ) ) ) ).
% bind_split
thf(fact_6483_bind__eq__None__conv,axiom,
! [B: $tType,A: $tType,A3: option @ B,F: B > ( option @ A )] :
( ( ( bind2 @ B @ A @ A3 @ F )
= ( none @ A ) )
= ( ( A3
= ( none @ B ) )
| ( ( F @ ( the2 @ B @ A3 ) )
= ( none @ A ) ) ) ) ).
% bind_eq_None_conv
thf(fact_6484_bind__map__option,axiom,
! [A: $tType,B: $tType,C: $tType,F: C > B,X: option @ C,G: B > ( option @ A )] :
( ( bind2 @ B @ A @ ( map_option @ C @ B @ F @ X ) @ G )
= ( bind2 @ C @ A @ X @ ( comp @ B @ ( option @ A ) @ C @ G @ F ) ) ) ).
% bind_map_option
thf(fact_6485_map__option__bind,axiom,
! [A: $tType,B: $tType,C: $tType,F: B > A,X: option @ C,G: C > ( option @ B )] :
( ( map_option @ B @ A @ F @ ( bind2 @ C @ B @ X @ G ) )
= ( bind2 @ C @ A @ X @ ( comp @ ( option @ B ) @ ( option @ A ) @ C @ ( map_option @ B @ A @ F ) @ G ) ) ) ).
% map_option_bind
thf(fact_6486_adm__lemma,axiom,
! [B: $tType,A: $tType,R5: set @ ( product_prod @ A @ A ),F6: ( A > B ) > A > B] :
( adm_wf @ A @ B @ R5
@ ^ [F2: A > B,X2: A] : ( F6 @ ( cut @ A @ B @ F2 @ R5 @ X2 ) @ X2 ) ) ).
% adm_lemma
thf(fact_6487_set__bind__option,axiom,
! [A: $tType,B: $tType,X: option @ B,F: B > ( option @ A )] :
( ( set_option @ A @ ( bind2 @ B @ A @ X @ F ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ ( comp @ ( option @ A ) @ ( set @ A ) @ B @ ( set_option @ A ) @ F ) @ ( set_option @ B @ X ) ) ) ) ).
% set_bind_option
thf(fact_6488_elem__set,axiom,
! [A: $tType,X: A,Xo: option @ A] :
( ( member @ A @ X @ ( set_option @ A @ Xo ) )
= ( Xo
= ( some @ A @ X ) ) ) ).
% elem_set
thf(fact_6489_set__empty__eq,axiom,
! [A: $tType,Xo: option @ A] :
( ( ( set_option @ A @ Xo )
= ( bot_bot @ ( set @ A ) ) )
= ( Xo
= ( none @ A ) ) ) ).
% set_empty_eq
thf(fact_6490_bind__option__cong,axiom,
! [B: $tType,A: $tType,X: option @ A,Y3: option @ A,F: A > ( option @ B ),G: A > ( option @ B )] :
( ( X = Y3 )
=> ( ! [Z6: A] :
( ( member @ A @ Z6 @ ( set_option @ A @ Y3 ) )
=> ( ( F @ Z6 )
= ( G @ Z6 ) ) )
=> ( ( bind2 @ A @ B @ X @ F )
= ( bind2 @ A @ B @ Y3 @ G ) ) ) ) ).
% bind_option_cong
thf(fact_6491_option_Oset__cases,axiom,
! [A: $tType,E2: A,A3: option @ A] :
( ( member @ A @ E2 @ ( set_option @ A @ A3 ) )
=> ( A3
= ( some @ A @ E2 ) ) ) ).
% option.set_cases
thf(fact_6492_option_Oset__intros,axiom,
! [A: $tType,X22: A] : ( member @ A @ X22 @ ( set_option @ A @ ( some @ A @ X22 ) ) ) ).
% option.set_intros
thf(fact_6493_ospec,axiom,
! [A: $tType,A5: option @ A,P: A > $o,X: A] :
( ! [X4: A] :
( ( member @ A @ X4 @ ( set_option @ A @ A5 ) )
=> ( P @ X4 ) )
=> ( ( A5
= ( some @ A @ X ) )
=> ( P @ X ) ) ) ).
% ospec
thf(fact_6494_map__option__idI,axiom,
! [A: $tType,X: option @ A,F: A > A] :
( ! [Y4: A] :
( ( member @ A @ Y4 @ ( set_option @ A @ X ) )
=> ( ( F @ Y4 )
= Y4 ) )
=> ( ( map_option @ A @ A @ F @ X )
= X ) ) ).
% map_option_idI
thf(fact_6495_option_Oinj__map__strong,axiom,
! [B: $tType,A: $tType,X: option @ A,Xa: option @ A,F: A > B,Fa: A > B] :
( ! [Z6: A,Za: A] :
( ( member @ A @ Z6 @ ( set_option @ A @ X ) )
=> ( ( member @ A @ Za @ ( set_option @ A @ Xa ) )
=> ( ( ( F @ Z6 )
= ( Fa @ Za ) )
=> ( Z6 = Za ) ) ) )
=> ( ( ( map_option @ A @ B @ F @ X )
= ( map_option @ A @ B @ Fa @ Xa ) )
=> ( X = Xa ) ) ) ).
% option.inj_map_strong
thf(fact_6496_option_Omap__cong0,axiom,
! [B: $tType,A: $tType,X: option @ A,F: A > B,G: A > B] :
( ! [Z6: A] :
( ( member @ A @ Z6 @ ( set_option @ A @ X ) )
=> ( ( F @ Z6 )
= ( G @ Z6 ) ) )
=> ( ( map_option @ A @ B @ F @ X )
= ( map_option @ A @ B @ G @ X ) ) ) ).
% option.map_cong0
thf(fact_6497_option_Omap__cong,axiom,
! [B: $tType,A: $tType,X: option @ A,Ya: option @ A,F: A > B,G: A > B] :
( ( X = Ya )
=> ( ! [Z6: A] :
( ( member @ A @ Z6 @ ( set_option @ A @ Ya ) )
=> ( ( F @ Z6 )
= ( G @ Z6 ) ) )
=> ( ( map_option @ A @ B @ F @ X )
= ( map_option @ A @ B @ G @ Ya ) ) ) ) ).
% option.map_cong
thf(fact_6498_adm__wf__def,axiom,
! [B: $tType,A: $tType] :
( ( adm_wf @ A @ B )
= ( ^ [R6: set @ ( product_prod @ A @ A ),F7: ( A > B ) > A > B] :
! [F2: A > B,G2: A > B,X2: A] :
( ! [Z3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z3 @ X2 ) @ R6 )
=> ( ( F2 @ Z3 )
= ( G2 @ Z3 ) ) )
=> ( ( F7 @ F2 @ X2 )
= ( F7 @ G2 @ X2 ) ) ) ) ) ).
% adm_wf_def
thf(fact_6499_option_Osimps_I14_J,axiom,
! [A: $tType] :
( ( set_option @ A @ ( none @ A ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% option.simps(14)
thf(fact_6500_option_Oset__map,axiom,
! [B: $tType,A: $tType,F: A > B,V: option @ A] :
( ( set_option @ B @ ( map_option @ A @ B @ F @ V ) )
= ( image2 @ A @ B @ F @ ( set_option @ A @ V ) ) ) ).
% option.set_map
thf(fact_6501_option_Oset__sel,axiom,
! [A: $tType,A3: option @ A] :
( ( A3
!= ( none @ A ) )
=> ( member @ A @ ( the2 @ A @ A3 ) @ ( set_option @ A @ A3 ) ) ) ).
% option.set_sel
thf(fact_6502_option_Osimps_I15_J,axiom,
! [A: $tType,X22: A] :
( ( set_option @ A @ ( some @ A @ X22 ) )
= ( insert2 @ A @ X22 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% option.simps(15)
thf(fact_6503_option_Orel__Grp,axiom,
! [B: $tType,A: $tType,A5: set @ A,F: A > B] :
( ( rel_option @ A @ B @ ( bNF_Grp @ A @ B @ A5 @ F ) )
= ( bNF_Grp @ ( option @ A ) @ ( option @ B )
@ ( collect @ ( option @ A )
@ ^ [X2: option @ A] : ( ord_less_eq @ ( set @ A ) @ ( set_option @ A @ X2 ) @ A5 ) )
@ ( map_option @ A @ B @ F ) ) ) ).
% option.rel_Grp
thf(fact_6504_option_Orel__compp__Grp,axiom,
! [B: $tType,A: $tType] :
( ( rel_option @ A @ B )
= ( ^ [R6: A > B > $o] :
( relcompp @ ( option @ A ) @ ( option @ ( product_prod @ A @ B ) ) @ ( option @ B )
@ ( conversep @ ( option @ ( product_prod @ A @ B ) ) @ ( option @ A )
@ ( bNF_Grp @ ( option @ ( product_prod @ A @ B ) ) @ ( option @ A )
@ ( collect @ ( option @ ( product_prod @ A @ B ) )
@ ^ [X2: option @ ( product_prod @ A @ B )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( set_option @ ( product_prod @ A @ B ) @ X2 ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R6 ) ) ) )
@ ( map_option @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) ) ) )
@ ( bNF_Grp @ ( option @ ( product_prod @ A @ B ) ) @ ( option @ B )
@ ( collect @ ( option @ ( product_prod @ A @ B ) )
@ ^ [X2: option @ ( product_prod @ A @ B )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( set_option @ ( product_prod @ A @ B ) @ X2 ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R6 ) ) ) )
@ ( map_option @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) ) ) ) ) ) ).
% option.rel_compp_Grp
thf(fact_6505_rel__option__None1,axiom,
! [A: $tType,B: $tType,P: A > B > $o,X: option @ B] :
( ( rel_option @ A @ B @ P @ ( none @ A ) @ X )
= ( X
= ( none @ B ) ) ) ).
% rel_option_None1
thf(fact_6506_rel__option__None2,axiom,
! [B: $tType,A: $tType,P: A > B > $o,X: option @ A] :
( ( rel_option @ A @ B @ P @ X @ ( none @ B ) )
= ( X
= ( none @ A ) ) ) ).
% rel_option_None2
thf(fact_6507_option_Orel__cong,axiom,
! [A: $tType,B: $tType,X: option @ A,Ya: option @ A,Y3: option @ B,Xa: option @ B,R5: A > B > $o,Ra2: A > B > $o] :
( ( X = Ya )
=> ( ( Y3 = Xa )
=> ( ! [Z6: A,Yb: B] :
( ( member @ A @ Z6 @ ( set_option @ A @ Ya ) )
=> ( ( member @ B @ Yb @ ( set_option @ B @ Xa ) )
=> ( ( R5 @ Z6 @ Yb )
= ( Ra2 @ Z6 @ Yb ) ) ) )
=> ( ( rel_option @ A @ B @ R5 @ X @ Y3 )
= ( rel_option @ A @ B @ Ra2 @ Ya @ Xa ) ) ) ) ) ).
% option.rel_cong
thf(fact_6508_option_Orel__mono__strong,axiom,
! [A: $tType,B: $tType,R5: A > B > $o,X: option @ A,Y3: option @ B,Ra2: A > B > $o] :
( ( rel_option @ A @ B @ R5 @ X @ Y3 )
=> ( ! [Z6: A,Yb: B] :
( ( member @ A @ Z6 @ ( set_option @ A @ X ) )
=> ( ( member @ B @ Yb @ ( set_option @ B @ Y3 ) )
=> ( ( R5 @ Z6 @ Yb )
=> ( Ra2 @ Z6 @ Yb ) ) ) )
=> ( rel_option @ A @ B @ Ra2 @ X @ Y3 ) ) ) ).
% option.rel_mono_strong
thf(fact_6509_option_Orel__refl__strong,axiom,
! [A: $tType,X: option @ A,Ra2: A > A > $o] :
( ! [Z6: A] :
( ( member @ A @ Z6 @ ( set_option @ A @ X ) )
=> ( Ra2 @ Z6 @ Z6 ) )
=> ( rel_option @ A @ A @ Ra2 @ X @ X ) ) ).
% option.rel_refl_strong
thf(fact_6510_rel__option__reflI,axiom,
! [A: $tType,Y3: option @ A,P: A > A > $o] :
( ! [X4: A] :
( ( member @ A @ X4 @ ( set_option @ A @ Y3 ) )
=> ( P @ X4 @ X4 ) )
=> ( rel_option @ A @ A @ P @ Y3 @ Y3 ) ) ).
% rel_option_reflI
thf(fact_6511_option_Orec__transfer,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,S3: C > D > $o,R5: A > B > $o] : ( bNF_rel_fun @ C @ D @ ( ( A > C ) > ( option @ A ) > C ) @ ( ( B > D ) > ( option @ B ) > D ) @ S3 @ ( bNF_rel_fun @ ( A > C ) @ ( B > D ) @ ( ( option @ A ) > C ) @ ( ( option @ B ) > D ) @ ( bNF_rel_fun @ A @ B @ C @ D @ R5 @ S3 ) @ ( bNF_rel_fun @ ( option @ A ) @ ( option @ B ) @ C @ D @ ( rel_option @ A @ B @ R5 ) @ S3 ) ) @ ( rec_option @ C @ A ) @ ( rec_option @ D @ B ) ) ).
% option.rec_transfer
thf(fact_6512_option_Omap__transfer,axiom,
! [A: $tType,B: $tType,F3: $tType,E: $tType,Rb2: A > E > $o,Sd: B > F3 > $o] : ( bNF_rel_fun @ ( A > B ) @ ( E > F3 ) @ ( ( option @ A ) > ( option @ B ) ) @ ( ( option @ E ) > ( option @ F3 ) ) @ ( bNF_rel_fun @ A @ E @ B @ F3 @ Rb2 @ Sd ) @ ( bNF_rel_fun @ ( option @ A ) @ ( option @ E ) @ ( option @ B ) @ ( option @ F3 ) @ ( rel_option @ A @ E @ Rb2 ) @ ( rel_option @ B @ F3 @ Sd ) ) @ ( map_option @ A @ B ) @ ( map_option @ E @ F3 ) ) ).
% option.map_transfer
thf(fact_6513_option_Orel__transfer,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,Sa: A > C > $o,Sc: B > D > $o] :
( bNF_rel_fun @ ( A > B > $o ) @ ( C > D > $o ) @ ( ( option @ A ) > ( option @ B ) > $o ) @ ( ( option @ C ) > ( option @ D ) > $o )
@ ( bNF_rel_fun @ A @ C @ ( B > $o ) @ ( D > $o ) @ Sa
@ ( bNF_rel_fun @ B @ D @ $o @ $o @ Sc
@ ^ [Y6: $o,Z2: $o] : Y6 = Z2 ) )
@ ( bNF_rel_fun @ ( option @ A ) @ ( option @ C ) @ ( ( option @ B ) > $o ) @ ( ( option @ D ) > $o ) @ ( rel_option @ A @ C @ Sa )
@ ( bNF_rel_fun @ ( option @ B ) @ ( option @ D ) @ $o @ $o @ ( rel_option @ B @ D @ Sc )
@ ^ [Y6: $o,Z2: $o] : Y6 = Z2 ) )
@ ( rel_option @ A @ B )
@ ( rel_option @ C @ D ) ) ).
% option.rel_transfer
thf(fact_6514_option_Octr__transfer_I2_J,axiom,
! [A: $tType,B: $tType,R5: A > B > $o] : ( bNF_rel_fun @ A @ B @ ( option @ A ) @ ( option @ B ) @ R5 @ ( rel_option @ A @ B @ R5 ) @ ( some @ A ) @ ( some @ B ) ) ).
% option.ctr_transfer(2)
thf(fact_6515_option_Odisc__transfer_I1_J,axiom,
! [A: $tType,B: $tType,R5: A > B > $o] :
( bNF_rel_fun @ ( option @ A ) @ ( option @ B ) @ $o @ $o @ ( rel_option @ A @ B @ R5 )
@ ^ [Y6: $o,Z2: $o] : Y6 = Z2
@ ^ [Option2: option @ A] :
( Option2
= ( none @ A ) )
@ ^ [Option2: option @ B] :
( Option2
= ( none @ B ) ) ) ).
% option.disc_transfer(1)
thf(fact_6516_option_Odisc__transfer_I2_J,axiom,
! [A: $tType,B: $tType,R5: A > B > $o] :
( bNF_rel_fun @ ( option @ A ) @ ( option @ B ) @ $o @ $o @ ( rel_option @ A @ B @ R5 )
@ ^ [Y6: $o,Z2: $o] : Y6 = Z2
@ ^ [Option2: option @ A] :
( Option2
!= ( none @ A ) )
@ ^ [Option2: option @ B] :
( Option2
!= ( none @ B ) ) ) ).
% option.disc_transfer(2)
thf(fact_6517_option_Ocase__transfer,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,S3: C > D > $o,R5: A > B > $o] : ( bNF_rel_fun @ C @ D @ ( ( A > C ) > ( option @ A ) > C ) @ ( ( B > D ) > ( option @ B ) > D ) @ S3 @ ( bNF_rel_fun @ ( A > C ) @ ( B > D ) @ ( ( option @ A ) > C ) @ ( ( option @ B ) > D ) @ ( bNF_rel_fun @ A @ B @ C @ D @ R5 @ S3 ) @ ( bNF_rel_fun @ ( option @ A ) @ ( option @ B ) @ C @ D @ ( rel_option @ A @ B @ R5 ) @ S3 ) ) @ ( case_option @ C @ A ) @ ( case_option @ D @ B ) ) ).
% option.case_transfer
thf(fact_6518_option_Orel__mono,axiom,
! [B: $tType,A: $tType,R5: A > B > $o,Ra2: A > B > $o] :
( ( ord_less_eq @ ( A > B > $o ) @ R5 @ Ra2 )
=> ( ord_less_eq @ ( ( option @ A ) > ( option @ B ) > $o ) @ ( rel_option @ A @ B @ R5 ) @ ( rel_option @ A @ B @ Ra2 ) ) ) ).
% option.rel_mono
thf(fact_6519_option_OQuotient,axiom,
! [B: $tType,A: $tType,R5: A > A > $o,Abs: A > B,Rep: B > A,T5: A > B > $o] :
( ( quotient @ A @ B @ R5 @ Abs @ Rep @ T5 )
=> ( quotient @ ( option @ A ) @ ( option @ B ) @ ( rel_option @ A @ A @ R5 ) @ ( map_option @ A @ B @ Abs ) @ ( map_option @ B @ A @ Rep ) @ ( rel_option @ A @ B @ T5 ) ) ) ).
% option.Quotient
thf(fact_6520_option_Orel__map_I2_J,axiom,
! [A: $tType,C: $tType,B: $tType,Sa: A > C > $o,X: option @ A,G: B > C,Y3: option @ B] :
( ( rel_option @ A @ C @ Sa @ X @ ( map_option @ B @ C @ G @ Y3 ) )
= ( rel_option @ A @ B
@ ^ [X2: A,Y: B] : ( Sa @ X2 @ ( G @ Y ) )
@ X
@ Y3 ) ) ).
% option.rel_map(2)
thf(fact_6521_option_Orel__map_I1_J,axiom,
! [A: $tType,C: $tType,B: $tType,Sb: C > B > $o,I: A > C,X: option @ A,Y3: option @ B] :
( ( rel_option @ C @ B @ Sb @ ( map_option @ A @ C @ I @ X ) @ Y3 )
= ( rel_option @ A @ B
@ ^ [X2: A] : ( Sb @ ( I @ X2 ) )
@ X
@ Y3 ) ) ).
% option.rel_map(1)
thf(fact_6522_option_Orel__compp,axiom,
! [A: $tType,C: $tType,B: $tType,R5: A > B > $o,S3: B > C > $o] :
( ( rel_option @ A @ C @ ( relcompp @ A @ B @ C @ R5 @ S3 ) )
= ( relcompp @ ( option @ A ) @ ( option @ B ) @ ( option @ C ) @ ( rel_option @ A @ B @ R5 ) @ ( rel_option @ B @ C @ S3 ) ) ) ).
% option.rel_compp
thf(fact_6523_option_Orel__flip,axiom,
! [A: $tType,B: $tType,R5: A > B > $o,A3: option @ B,B2: option @ A] :
( ( rel_option @ B @ A @ ( conversep @ A @ B @ R5 ) @ A3 @ B2 )
= ( rel_option @ A @ B @ R5 @ B2 @ A3 ) ) ).
% option.rel_flip
thf(fact_6524_option_Orel__conversep,axiom,
! [B: $tType,A: $tType,R5: A > B > $o] :
( ( rel_option @ B @ A @ ( conversep @ A @ B @ R5 ) )
= ( conversep @ ( option @ A ) @ ( option @ B ) @ ( rel_option @ A @ B @ R5 ) ) ) ).
% option.rel_conversep
thf(fact_6525_option_Orel__sel,axiom,
! [B: $tType,A: $tType] :
( ( rel_option @ A @ B )
= ( ^ [R6: A > B > $o,A6: option @ A,B5: option @ B] :
( ( ( A6
= ( none @ A ) )
= ( B5
= ( none @ B ) ) )
& ( ( A6
!= ( none @ A ) )
=> ( ( B5
!= ( none @ B ) )
=> ( R6 @ ( the2 @ A @ A6 ) @ ( the2 @ B @ B5 ) ) ) ) ) ) ) ).
% option.rel_sel
thf(fact_6526_option_Orel__inject_I2_J,axiom,
! [A: $tType,B: $tType,R5: A > B > $o,X22: A,Y2: B] :
( ( rel_option @ A @ B @ R5 @ ( some @ A @ X22 ) @ ( some @ B @ Y2 ) )
= ( R5 @ X22 @ Y2 ) ) ).
% option.rel_inject(2)
thf(fact_6527_option_Orel__intros_I2_J,axiom,
! [A: $tType,B: $tType,R5: A > B > $o,X22: A,Y2: B] :
( ( R5 @ X22 @ Y2 )
=> ( rel_option @ A @ B @ R5 @ ( some @ A @ X22 ) @ ( some @ B @ Y2 ) ) ) ).
% option.rel_intros(2)
thf(fact_6528_option__rel__Some1,axiom,
! [A: $tType,B: $tType,A5: A > B > $o,X: A,Y3: option @ B] :
( ( rel_option @ A @ B @ A5 @ ( some @ A @ X ) @ Y3 )
= ( ? [Y7: B] :
( ( Y3
= ( some @ B @ Y7 ) )
& ( A5 @ X @ Y7 ) ) ) ) ).
% option_rel_Some1
thf(fact_6529_option__rel__Some2,axiom,
! [B: $tType,A: $tType,A5: A > B > $o,X: option @ A,Y3: B] :
( ( rel_option @ A @ B @ A5 @ X @ ( some @ B @ Y3 ) )
= ( ? [X8: A] :
( ( X
= ( some @ A @ X8 ) )
& ( A5 @ X8 @ Y3 ) ) ) ) ).
% option_rel_Some2
thf(fact_6530_option_Orel__refl,axiom,
! [B: $tType,Ra2: B > B > $o,X: option @ B] :
( ! [X4: B] : ( Ra2 @ X4 @ X4 )
=> ( rel_option @ B @ B @ Ra2 @ X @ X ) ) ).
% option.rel_refl
thf(fact_6531_option_Orel__eq,axiom,
! [A: $tType] :
( ( rel_option @ A @ A
@ ^ [Y6: A,Z2: A] : Y6 = Z2 )
= ( ^ [Y6: option @ A,Z2: option @ A] : Y6 = Z2 ) ) ).
% option.rel_eq
thf(fact_6532_option_Octr__transfer_I1_J,axiom,
! [A: $tType,B: $tType,R5: A > B > $o] : ( rel_option @ A @ B @ R5 @ ( none @ A ) @ ( none @ B ) ) ).
% option.ctr_transfer(1)
thf(fact_6533_option_Orel__distinct_I2_J,axiom,
! [A: $tType,B: $tType,R5: A > B > $o,Y2: A] :
~ ( rel_option @ A @ B @ R5 @ ( some @ A @ Y2 ) @ ( none @ B ) ) ).
% option.rel_distinct(2)
thf(fact_6534_option_Orel__distinct_I1_J,axiom,
! [A: $tType,B: $tType,R5: A > B > $o,Y2: B] :
~ ( rel_option @ A @ B @ R5 @ ( none @ A ) @ ( some @ B @ Y2 ) ) ).
% option.rel_distinct(1)
thf(fact_6535_option_Orel__cases,axiom,
! [A: $tType,B: $tType,R5: A > B > $o,A3: option @ A,B2: option @ B] :
( ( rel_option @ A @ B @ R5 @ A3 @ B2 )
=> ( ( ( A3
= ( none @ A ) )
=> ( B2
!= ( none @ B ) ) )
=> ~ ! [X4: A] :
( ( A3
= ( some @ A @ X4 ) )
=> ! [Y4: B] :
( ( B2
= ( some @ B @ Y4 ) )
=> ~ ( R5 @ X4 @ Y4 ) ) ) ) ) ).
% option.rel_cases
thf(fact_6536_option_Orel__induct,axiom,
! [A: $tType,B: $tType,R5: A > B > $o,X: option @ A,Y3: option @ B,Q: ( option @ A ) > ( option @ B ) > $o] :
( ( rel_option @ A @ B @ R5 @ X @ Y3 )
=> ( ( Q @ ( none @ A ) @ ( none @ B ) )
=> ( ! [A26: A,B24: B] :
( ( R5 @ A26 @ B24 )
=> ( Q @ ( some @ A @ A26 ) @ ( some @ B @ B24 ) ) )
=> ( Q @ X @ Y3 ) ) ) ) ).
% option.rel_induct
thf(fact_6537_rel__option__inf,axiom,
! [B: $tType,A: $tType,A5: A > B > $o,B4: A > B > $o] :
( ( inf_inf @ ( ( option @ A ) > ( option @ B ) > $o ) @ ( rel_option @ A @ B @ A5 ) @ ( rel_option @ A @ B @ B4 ) )
= ( rel_option @ A @ B @ ( inf_inf @ ( A > B > $o ) @ A5 @ B4 ) ) ) ).
% rel_option_inf
thf(fact_6538_rel__option__iff,axiom,
! [B: $tType,A: $tType] :
( ( rel_option @ A @ B )
= ( ^ [R6: A > B > $o,X2: option @ A,Y: option @ B] :
( product_case_prod @ ( option @ A ) @ ( option @ B ) @ $o
@ ^ [A6: option @ A,B5: option @ B] :
( case_option @ $o @ A
@ ( case_option @ $o @ B @ $true
@ ^ [C4: B] : $false
@ B5 )
@ ^ [Z3: A] : ( case_option @ $o @ B @ $false @ ( R6 @ Z3 ) @ B5 )
@ A6 )
@ ( product_Pair @ ( option @ A ) @ ( option @ B ) @ X2 @ Y ) ) ) ) ).
% rel_option_iff
thf(fact_6539_option__bind__transfer,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,A5: A > B > $o,B4: C > D > $o] : ( bNF_rel_fun @ ( option @ A ) @ ( option @ B ) @ ( ( A > ( option @ C ) ) > ( option @ C ) ) @ ( ( B > ( option @ D ) ) > ( option @ D ) ) @ ( rel_option @ A @ B @ A5 ) @ ( bNF_rel_fun @ ( A > ( option @ C ) ) @ ( B > ( option @ D ) ) @ ( option @ C ) @ ( option @ D ) @ ( bNF_rel_fun @ A @ B @ ( option @ C ) @ ( option @ D ) @ A5 @ ( rel_option @ C @ D @ B4 ) ) @ ( rel_option @ C @ D @ B4 ) ) @ ( bind2 @ A @ C ) @ ( bind2 @ B @ D ) ) ).
% option_bind_transfer
thf(fact_6540_option_Oin__rel,axiom,
! [B: $tType,A: $tType] :
( ( rel_option @ A @ B )
= ( ^ [R6: A > B > $o,A6: option @ A,B5: option @ B] :
? [Z3: option @ ( product_prod @ A @ B )] :
( ( member @ ( option @ ( product_prod @ A @ B ) ) @ Z3
@ ( collect @ ( option @ ( product_prod @ A @ B ) )
@ ^ [X2: option @ ( product_prod @ A @ B )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( set_option @ ( product_prod @ A @ B ) @ X2 ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R6 ) ) ) ) )
& ( ( map_option @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Z3 )
= A6 )
& ( ( map_option @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ Z3 )
= B5 ) ) ) ) ).
% option.in_rel
thf(fact_6541_listrel__subset,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 @ ( list @ A ) @ ( list @ A ) ) ) @ ( listrel @ A @ A @ R )
@ ( product_Sigma @ ( list @ A ) @ ( list @ A ) @ ( lists @ A @ A5 )
@ ^ [Uu: list @ A] : ( lists @ A @ A5 ) ) ) ) ).
% listrel_subset
thf(fact_6542_normalize__div,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A] :
( ( divide_divide @ A @ ( normal6383669964737779283malize @ A @ A3 ) @ A3 )
= ( divide_divide @ A @ ( one_one @ A ) @ ( unit_f5069060285200089521factor @ A @ A3 ) ) ) ) ).
% normalize_div
thf(fact_6543_unit__factor__simps_I1_J,axiom,
( ( unit_f5069060285200089521factor @ nat @ ( zero_zero @ nat ) )
= ( zero_zero @ nat ) ) ).
% unit_factor_simps(1)
thf(fact_6544_unit__factor__idem,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A] :
( ( unit_f5069060285200089521factor @ A @ ( unit_f5069060285200089521factor @ A @ A3 ) )
= ( unit_f5069060285200089521factor @ A @ A3 ) ) ) ).
% unit_factor_idem
thf(fact_6545_unit__factor__simps_I2_J,axiom,
! [N: nat] :
( ( unit_f5069060285200089521factor @ nat @ ( suc @ N ) )
= ( one_one @ nat ) ) ).
% unit_factor_simps(2)
thf(fact_6546_unit__factor__0,axiom,
! [A: $tType] :
( ( semido2269285787275462019factor @ A )
=> ( ( unit_f5069060285200089521factor @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% unit_factor_0
thf(fact_6547_unit__factor__eq__0__iff,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A] :
( ( ( unit_f5069060285200089521factor @ A @ A3 )
= ( zero_zero @ A ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% unit_factor_eq_0_iff
thf(fact_6548_unit__factor__1,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ( ( unit_f5069060285200089521factor @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% unit_factor_1
thf(fact_6549_normalize__mult__unit__factor,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A] :
( ( times_times @ A @ ( normal6383669964737779283malize @ A @ A3 ) @ ( unit_f5069060285200089521factor @ A @ A3 ) )
= A3 ) ) ).
% normalize_mult_unit_factor
thf(fact_6550_unit__factor__mult__normalize,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A] :
( ( times_times @ A @ ( unit_f5069060285200089521factor @ A @ A3 ) @ ( normal6383669964737779283malize @ A @ A3 ) )
= A3 ) ) ).
% unit_factor_mult_normalize
thf(fact_6551_div__normalize,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A] :
( ( divide_divide @ A @ A3 @ ( normal6383669964737779283malize @ A @ A3 ) )
= ( unit_f5069060285200089521factor @ A @ A3 ) ) ) ).
% div_normalize
thf(fact_6552_div__unit__factor,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A] :
( ( divide_divide @ A @ A3 @ ( unit_f5069060285200089521factor @ A @ A3 ) )
= ( normal6383669964737779283malize @ A @ A3 ) ) ) ).
% div_unit_factor
thf(fact_6553_inv__unit__factor__eq__0__iff,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A] :
( ( ( divide_divide @ A @ ( one_one @ A ) @ ( unit_f5069060285200089521factor @ A @ A3 ) )
= ( zero_zero @ A ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% inv_unit_factor_eq_0_iff
thf(fact_6554_unit__factor__mult__unit__left,axiom,
! [A: $tType] :
( ( semido2269285787275462019factor @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( unit_f5069060285200089521factor @ A @ ( times_times @ A @ A3 @ B2 ) )
= ( times_times @ A @ A3 @ ( unit_f5069060285200089521factor @ A @ B2 ) ) ) ) ) ).
% unit_factor_mult_unit_left
thf(fact_6555_unit__factor__mult__unit__right,axiom,
! [A: $tType] :
( ( semido2269285787275462019factor @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( unit_f5069060285200089521factor @ A @ ( times_times @ A @ B2 @ A3 ) )
= ( times_times @ A @ ( unit_f5069060285200089521factor @ A @ B2 ) @ A3 ) ) ) ) ).
% unit_factor_mult_unit_right
thf(fact_6556_mult__one__div__unit__factor,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A,B2: A] :
( ( times_times @ A @ A3 @ ( divide_divide @ A @ ( one_one @ A ) @ ( unit_f5069060285200089521factor @ A @ B2 ) ) )
= ( divide_divide @ A @ A3 @ ( unit_f5069060285200089521factor @ A @ B2 ) ) ) ) ).
% mult_one_div_unit_factor
thf(fact_6557_unit__factor__lcm,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A] :
( ( ( ( A3
= ( zero_zero @ A ) )
| ( B2
= ( zero_zero @ A ) ) )
=> ( ( unit_f5069060285200089521factor @ A @ ( gcd_lcm @ A @ A3 @ B2 ) )
= ( zero_zero @ A ) ) )
& ( ~ ( ( A3
= ( zero_zero @ A ) )
| ( B2
= ( zero_zero @ A ) ) )
=> ( ( unit_f5069060285200089521factor @ A @ ( gcd_lcm @ A @ A3 @ B2 ) )
= ( one_one @ A ) ) ) ) ) ).
% unit_factor_lcm
thf(fact_6558_normalize__unit__factor,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( normal6383669964737779283malize @ A @ ( unit_f5069060285200089521factor @ A @ A3 ) )
= ( one_one @ A ) ) ) ) ).
% normalize_unit_factor
thf(fact_6559_unit__factor__normalize,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( unit_f5069060285200089521factor @ A @ ( normal6383669964737779283malize @ A @ A3 ) )
= ( one_one @ A ) ) ) ) ).
% unit_factor_normalize
thf(fact_6560_normalize__unit__factor__eqI,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A,B2: A] :
( ( ( normal6383669964737779283malize @ A @ A3 )
= ( normal6383669964737779283malize @ A @ B2 ) )
=> ( ( ( unit_f5069060285200089521factor @ A @ A3 )
= ( unit_f5069060285200089521factor @ A @ B2 ) )
=> ( A3 = B2 ) ) ) ) ).
% normalize_unit_factor_eqI
thf(fact_6561_unit__factor__nat__def,axiom,
( ( unit_f5069060285200089521factor @ nat )
= ( ^ [N5: nat] :
( if @ nat
@ ( N5
= ( zero_zero @ nat ) )
@ ( zero_zero @ nat )
@ ( one_one @ nat ) ) ) ) ).
% unit_factor_nat_def
thf(fact_6562_mult__gcd__left,axiom,
! [A: $tType] :
( ( semiri6843258321239162965malize @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( times_times @ A @ C2 @ ( gcd_gcd @ A @ A3 @ B2 ) )
= ( times_times @ A @ ( unit_f5069060285200089521factor @ A @ C2 ) @ ( gcd_gcd @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) ) ) ) ) ).
% mult_gcd_left
thf(fact_6563_mult__gcd__right,axiom,
! [A: $tType] :
( ( semiri6843258321239162965malize @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( times_times @ A @ ( gcd_gcd @ A @ A3 @ B2 ) @ C2 )
= ( times_times @ A @ ( gcd_gcd @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) @ ( unit_f5069060285200089521factor @ A @ C2 ) ) ) ) ).
% mult_gcd_right
thf(fact_6564_gcd__mult__distrib,axiom,
! [A: $tType] :
( ( semiri6843258321239162965malize @ A )
=> ! [K: A,A3: A,B2: A] :
( ( times_times @ A @ K @ ( gcd_gcd @ A @ A3 @ B2 ) )
= ( times_times @ A @ ( gcd_gcd @ A @ ( times_times @ A @ K @ A3 ) @ ( times_times @ A @ K @ B2 ) ) @ ( unit_f5069060285200089521factor @ A @ K ) ) ) ) ).
% gcd_mult_distrib
thf(fact_6565_unit__factor__dvd,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A,B2: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( dvd_dvd @ A @ ( unit_f5069060285200089521factor @ A @ A3 ) @ B2 ) ) ) ).
% unit_factor_dvd
thf(fact_6566_unit__factor__mult,axiom,
! [A: $tType] :
( ( normal6328177297339901930cative @ A )
=> ! [A3: A,B2: A] :
( ( unit_f5069060285200089521factor @ A @ ( times_times @ A @ A3 @ B2 ) )
= ( times_times @ A @ ( unit_f5069060285200089521factor @ A @ A3 ) @ ( unit_f5069060285200089521factor @ A @ B2 ) ) ) ) ).
% unit_factor_mult
thf(fact_6567_unit__factor__self,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A] : ( dvd_dvd @ A @ ( unit_f5069060285200089521factor @ A @ A3 ) @ A3 ) ) ).
% unit_factor_self
thf(fact_6568_dvd__unit__factor__div,axiom,
! [A: $tType] :
( ( normal6328177297339901930cative @ A )
=> ! [B2: A,A3: A] :
( ( dvd_dvd @ A @ B2 @ A3 )
=> ( ( unit_f5069060285200089521factor @ A @ ( divide_divide @ A @ A3 @ B2 ) )
= ( divide_divide @ A @ ( unit_f5069060285200089521factor @ A @ A3 ) @ ( unit_f5069060285200089521factor @ A @ B2 ) ) ) ) ) ).
% dvd_unit_factor_div
thf(fact_6569_is__unit__unit__factor,axiom,
! [A: $tType] :
( ( semido2269285787275462019factor @ A )
=> ! [A3: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( unit_f5069060285200089521factor @ A @ A3 )
= A3 ) ) ) ).
% is_unit_unit_factor
thf(fact_6570_mult__lcm__left,axiom,
! [A: $tType] :
( ( semiri6843258321239162965malize @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( times_times @ A @ C2 @ ( gcd_lcm @ A @ A3 @ B2 ) )
= ( times_times @ A @ ( unit_f5069060285200089521factor @ A @ C2 ) @ ( gcd_lcm @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) ) ) ) ) ).
% mult_lcm_left
thf(fact_6571_mult__lcm__right,axiom,
! [A: $tType] :
( ( semiri6843258321239162965malize @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( times_times @ A @ ( gcd_lcm @ A @ A3 @ B2 ) @ C2 )
= ( times_times @ A @ ( gcd_lcm @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) @ ( unit_f5069060285200089521factor @ A @ C2 ) ) ) ) ).
% mult_lcm_right
thf(fact_6572_lcm__mult__distrib,axiom,
! [A: $tType] :
( ( semiri6843258321239162965malize @ A )
=> ! [K: A,A3: A,B2: A] :
( ( times_times @ A @ K @ ( gcd_lcm @ A @ A3 @ B2 ) )
= ( times_times @ A @ ( gcd_lcm @ A @ ( times_times @ A @ K @ A3 ) @ ( times_times @ A @ K @ B2 ) ) @ ( unit_f5069060285200089521factor @ A @ K ) ) ) ) ).
% lcm_mult_distrib
thf(fact_6573_lists__eq__set,axiom,
! [A: $tType] :
( ( lists @ A )
= ( ^ [A8: set @ A] :
( collect @ ( list @ A )
@ ^ [Xs: list @ A] : ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ A8 ) ) ) ) ).
% lists_eq_set
thf(fact_6574_unit__factor__is__unit,axiom,
! [A: $tType] :
( ( semido2269285787275462019factor @ A )
=> ! [A3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( dvd_dvd @ A @ ( unit_f5069060285200089521factor @ A @ A3 ) @ ( one_one @ A ) ) ) ) ).
% unit_factor_is_unit
thf(fact_6575_unit__factor__gcd,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A] :
( ( ( ( A3
= ( zero_zero @ A ) )
& ( B2
= ( zero_zero @ A ) ) )
=> ( ( unit_f5069060285200089521factor @ A @ ( gcd_gcd @ A @ A3 @ B2 ) )
= ( zero_zero @ A ) ) )
& ( ~ ( ( A3
= ( zero_zero @ A ) )
& ( B2
= ( zero_zero @ A ) ) )
=> ( ( unit_f5069060285200089521factor @ A @ ( gcd_gcd @ A @ A3 @ B2 ) )
= ( one_one @ A ) ) ) ) ) ).
% unit_factor_gcd
thf(fact_6576_coprime__crossproduct_H,axiom,
! [A: $tType] :
( ( semiri6843258321239162965malize @ A )
=> ! [B2: A,D3: A,A3: A,C2: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( ( unit_f5069060285200089521factor @ A @ B2 )
= ( unit_f5069060285200089521factor @ A @ D3 ) )
=> ( ( algebr8660921524188924756oprime @ A @ A3 @ B2 )
=> ( ( algebr8660921524188924756oprime @ A @ C2 @ D3 )
=> ( ( ( times_times @ A @ A3 @ D3 )
= ( times_times @ A @ B2 @ C2 ) )
= ( ( A3 = C2 )
& ( B2 = D3 ) ) ) ) ) ) ) ) ).
% coprime_crossproduct'
thf(fact_6577_unit__factor__Lcm,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A5: set @ A] :
( ( ( ( gcd_Lcm @ A @ A5 )
= ( zero_zero @ A ) )
=> ( ( unit_f5069060285200089521factor @ A @ ( gcd_Lcm @ A @ A5 ) )
= ( zero_zero @ A ) ) )
& ( ( ( gcd_Lcm @ A @ A5 )
!= ( zero_zero @ A ) )
=> ( ( unit_f5069060285200089521factor @ A @ ( gcd_Lcm @ A @ A5 ) )
= ( one_one @ A ) ) ) ) ) ).
% unit_factor_Lcm
thf(fact_6578_unit__factor__Gcd,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A5: set @ A] :
( ( ( ( gcd_Gcd @ A @ A5 )
= ( zero_zero @ A ) )
=> ( ( unit_f5069060285200089521factor @ A @ ( gcd_Gcd @ A @ A5 ) )
= ( zero_zero @ A ) ) )
& ( ( ( gcd_Gcd @ A @ A5 )
!= ( zero_zero @ A ) )
=> ( ( unit_f5069060285200089521factor @ A @ ( gcd_Gcd @ A @ A5 ) )
= ( one_one @ A ) ) ) ) ) ).
% unit_factor_Gcd
thf(fact_6579_normalize__idem__imp__unit__factor__eq,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A] :
( ( ( normal6383669964737779283malize @ A @ A3 )
= A3 )
=> ( ( unit_f5069060285200089521factor @ A @ A3 )
= ( zero_neq_one_of_bool @ A
@ ( A3
!= ( zero_zero @ A ) ) ) ) ) ) ).
% normalize_idem_imp_unit_factor_eq
thf(fact_6580_unit__factor__Lcm__fin,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A5: set @ A] :
( ( unit_f5069060285200089521factor @ A @ ( semiring_gcd_Lcm_fin @ A @ A5 ) )
= ( zero_neq_one_of_bool @ A
@ ( ( semiring_gcd_Lcm_fin @ A @ A5 )
!= ( zero_zero @ A ) ) ) ) ) ).
% unit_factor_Lcm_fin
thf(fact_6581_unit__factor__Gcd__fin,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A5: set @ A] :
( ( unit_f5069060285200089521factor @ A @ ( semiring_gcd_Gcd_fin @ A @ A5 ) )
= ( zero_neq_one_of_bool @ A
@ ( ( semiring_gcd_Gcd_fin @ A @ A5 )
!= ( zero_zero @ A ) ) ) ) ) ).
% unit_factor_Gcd_fin
thf(fact_6582_pred__option__parametric,axiom,
! [A: $tType,B: $tType,A5: A > B > $o] :
( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ ( ( option @ A ) > $o ) @ ( ( option @ B ) > $o )
@ ( bNF_rel_fun @ A @ B @ $o @ $o @ A5
@ ^ [Y6: $o,Z2: $o] : Y6 = Z2 )
@ ( bNF_rel_fun @ ( option @ A ) @ ( option @ B ) @ $o @ $o @ ( rel_option @ A @ B @ A5 )
@ ^ [Y6: $o,Z2: $o] : Y6 = Z2 )
@ ( pred_option @ A )
@ ( pred_option @ B ) ) ).
% pred_option_parametric
thf(fact_6583_option_Opred__transfer,axiom,
! [A: $tType,B: $tType,R5: A > B > $o] :
( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ ( ( option @ A ) > $o ) @ ( ( option @ B ) > $o )
@ ( bNF_rel_fun @ A @ B @ $o @ $o @ R5
@ ^ [Y6: $o,Z2: $o] : Y6 = Z2 )
@ ( bNF_rel_fun @ ( option @ A ) @ ( option @ B ) @ $o @ $o @ ( rel_option @ A @ B @ R5 )
@ ^ [Y6: $o,Z2: $o] : Y6 = Z2 )
@ ( pred_option @ A )
@ ( pred_option @ B ) ) ).
% option.pred_transfer
thf(fact_6584_option_Opred__inject_I2_J,axiom,
! [A: $tType,P: A > $o,A3: A] :
( ( pred_option @ A @ P @ ( some @ A @ A3 ) )
= ( P @ A3 ) ) ).
% option.pred_inject(2)
thf(fact_6585_option_Opred__mono,axiom,
! [A: $tType,P: A > $o,Pa: A > $o] :
( ( ord_less_eq @ ( A > $o ) @ P @ Pa )
=> ( ord_less_eq @ ( ( option @ A ) > $o ) @ ( pred_option @ A @ P ) @ ( pred_option @ A @ Pa ) ) ) ).
% option.pred_mono
thf(fact_6586_option_Opred__inject_I1_J,axiom,
! [A: $tType,P: A > $o] : ( pred_option @ A @ P @ ( none @ A ) ) ).
% option.pred_inject(1)
thf(fact_6587_option_Opred__True,axiom,
! [A: $tType] :
( ( pred_option @ A
@ ^ [Uu: A] : $true )
= ( ^ [Uu: option @ A] : $true ) ) ).
% option.pred_True
thf(fact_6588_option_Omap__cong__pred,axiom,
! [B: $tType,A: $tType,X: option @ A,Ya: option @ A,F: A > B,G: A > B] :
( ( X = Ya )
=> ( ( pred_option @ A
@ ^ [Z3: A] :
( ( F @ Z3 )
= ( G @ Z3 ) )
@ Ya )
=> ( ( map_option @ A @ B @ F @ X )
= ( map_option @ A @ B @ G @ Ya ) ) ) ) ).
% option.map_cong_pred
thf(fact_6589_option_Opred__mono__strong,axiom,
! [A: $tType,P: A > $o,X: option @ A,Pa: A > $o] :
( ( pred_option @ A @ P @ X )
=> ( ! [Z6: A] :
( ( member @ A @ Z6 @ ( set_option @ A @ X ) )
=> ( ( P @ Z6 )
=> ( Pa @ Z6 ) ) )
=> ( pred_option @ A @ Pa @ X ) ) ) ).
% option.pred_mono_strong
thf(fact_6590_option_Opred__cong,axiom,
! [A: $tType,X: option @ A,Ya: option @ A,P: A > $o,Pa: A > $o] :
( ( X = Ya )
=> ( ! [Z6: A] :
( ( member @ A @ Z6 @ ( set_option @ A @ Ya ) )
=> ( ( P @ Z6 )
= ( Pa @ Z6 ) ) )
=> ( ( pred_option @ A @ P @ X )
= ( pred_option @ A @ Pa @ Ya ) ) ) ) ).
% option.pred_cong
thf(fact_6591_option_Opred__set,axiom,
! [A: $tType] :
( ( pred_option @ A )
= ( ^ [P3: A > $o,X2: option @ A] :
! [Y: A] :
( ( member @ A @ Y @ ( set_option @ A @ X2 ) )
=> ( P3 @ Y ) ) ) ) ).
% option.pred_set
thf(fact_6592_option_Opred__map,axiom,
! [B: $tType,A: $tType,Q: B > $o,F: A > B,X: option @ A] :
( ( pred_option @ B @ Q @ ( map_option @ A @ B @ F @ X ) )
= ( pred_option @ A @ ( comp @ B @ $o @ A @ Q @ F ) @ X ) ) ).
% option.pred_map
thf(fact_6593_ID_Oin__rel,axiom,
! [B: $tType,A: $tType] :
( ( bNF_id_bnf @ ( A > B > $o ) )
= ( ^ [R6: A > B > $o,A6: A,B5: B] :
? [Z3: product_prod @ A @ B] :
( ( member @ ( product_prod @ A @ B ) @ Z3
@ ( collect @ ( product_prod @ A @ B )
@ ^ [X2: product_prod @ A @ B] : ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( insert2 @ ( product_prod @ A @ B ) @ X2 @ ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R6 ) ) ) ) )
& ( ( bNF_id_bnf @ ( ( product_prod @ A @ B ) > A ) @ ( product_fst @ A @ B ) @ Z3 )
= A6 )
& ( ( bNF_id_bnf @ ( ( product_prod @ A @ B ) > B ) @ ( product_snd @ A @ B ) @ Z3 )
= B5 ) ) ) ) ).
% ID.in_rel
thf(fact_6594_map__le__imp__upd__le,axiom,
! [A: $tType,B: $tType,M12: A > ( option @ B ),M23: A > ( option @ B ),X: A,Y3: B] :
( ( map_le @ A @ B @ M12 @ M23 )
=> ( map_le @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M12 @ X @ ( none @ B ) ) @ ( fun_upd @ A @ ( option @ B ) @ M23 @ X @ ( some @ B @ Y3 ) ) ) ) ).
% map_le_imp_upd_le
thf(fact_6595_ID_Omap__cong__pred,axiom,
! [B: $tType,A: $tType,X: A,Ya: A,F: A > B,G: A > B] :
( ( X = Ya )
=> ( ( bNF_id_bnf @ ( A > $o )
@ ^ [Z3: A] :
( ( F @ Z3 )
= ( G @ Z3 ) )
@ Ya )
=> ( ( bNF_id_bnf @ ( A > B ) @ F @ X )
= ( bNF_id_bnf @ ( A > B ) @ G @ Ya ) ) ) ) ).
% ID.map_cong_pred
thf(fact_6596_map__le__empty,axiom,
! [B: $tType,A: $tType,G: A > ( option @ B )] :
( map_le @ A @ B
@ ^ [X2: A] : ( none @ B )
@ G ) ).
% map_le_empty
thf(fact_6597_ID_Opred__map,axiom,
! [B: $tType,A: $tType,Q: B > $o,F: A > B,X: A] :
( ( bNF_id_bnf @ ( B > $o ) @ Q @ ( bNF_id_bnf @ ( A > B ) @ F @ X ) )
= ( bNF_id_bnf @ ( A > $o ) @ ( comp @ B @ $o @ A @ Q @ F ) @ X ) ) ).
% ID.pred_map
thf(fact_6598_upd__None__map__le,axiom,
! [B: $tType,A: $tType,F: A > ( option @ B ),X: A] : ( map_le @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ F @ X @ ( none @ B ) ) @ F ) ).
% upd_None_map_le
thf(fact_6599_Basic__BNF__LFPs_Octor__rec__o__map,axiom,
! [B: $tType,C: $tType,A: $tType,F: C > B,G: A > C] :
( ( comp @ C @ B @ A @ ( basic_BNF_ctor_rec @ ( C > B ) @ F ) @ G )
= ( basic_BNF_ctor_rec @ ( A > B ) @ ( comp @ C @ B @ A @ F @ ( comp @ A @ C @ A @ ( comp @ C @ C @ A @ ( bNF_id_bnf @ C ) @ G ) @ ( bNF_id_bnf @ A ) ) ) ) ) ).
% Basic_BNF_LFPs.ctor_rec_o_map
thf(fact_6600_Basic__BNF__LFPs_Octor__rec__def__alt,axiom,
! [B: $tType,A: $tType,F: A > B] :
( F
= ( basic_BNF_ctor_rec @ ( A > B ) @ ( comp @ A @ B @ A @ F @ ( bNF_id_bnf @ A ) ) ) ) ).
% Basic_BNF_LFPs.ctor_rec_def_alt
thf(fact_6601_ID_Opred__True,axiom,
! [A: $tType] :
( ( bNF_id_bnf @ ( A > $o )
@ ^ [Uu: A] : $true )
= ( ^ [Uu: A] : $true ) ) ).
% ID.pred_True
thf(fact_6602_ctor__rec__unique,axiom,
! [B: $tType,A: $tType,G: A > A,F: A > B,S: A > B] :
( ( G
= ( id @ A ) )
=> ( ( ( comp @ A @ B @ A @ F @ ( basic_BNF_xtor @ A ) )
= ( comp @ A @ B @ A @ S @ ( comp @ A @ A @ A @ ( comp @ A @ A @ A @ ( bNF_id_bnf @ A ) @ G ) @ ( bNF_id_bnf @ A ) ) ) )
=> ( F
= ( basic_BNF_ctor_rec @ ( A > B ) @ S ) ) ) ) ).
% ctor_rec_unique
thf(fact_6603_Basic__BNF__LFPs_Octor__rec,axiom,
! [B: $tType,A: $tType,G: A > A,F: A > B,X: A] :
( ( G
= ( id @ A ) )
=> ( ( basic_BNF_ctor_rec @ ( A > B ) @ F @ ( basic_BNF_xtor @ A @ X ) )
= ( F @ ( comp @ A @ A @ A @ ( comp @ A @ A @ A @ ( bNF_id_bnf @ A ) @ G ) @ ( bNF_id_bnf @ A ) @ X ) ) ) ) ).
% Basic_BNF_LFPs.ctor_rec
thf(fact_6604_Basic__BNF__LFPs_OPair__def__alt,axiom,
! [B: $tType,A: $tType] :
( ( product_Pair @ A @ B )
= ( ^ [A6: A,B5: B] : ( basic_BNF_xtor @ ( product_prod @ A @ B ) @ ( bNF_id_bnf @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A6 @ B5 ) ) ) ) ) ).
% Basic_BNF_LFPs.Pair_def_alt
thf(fact_6605_xtor__map__unique,axiom,
! [B: $tType,A: $tType,U2: A > B,F: A > B] :
( ( ( comp @ A @ B @ A @ U2 @ ( basic_BNF_xtor @ A ) )
= ( comp @ B @ B @ A @ ( basic_BNF_xtor @ B ) @ F ) )
=> ( U2 = F ) ) ).
% xtor_map_unique
thf(fact_6606_mk__less__def,axiom,
! [A: $tType] :
( ( partial_mk_less @ A )
= ( ^ [R6: A > A > $o,X2: A,Y: A] :
( ( R6 @ X2 @ Y )
& ~ ( R6 @ Y @ X2 ) ) ) ) ).
% mk_less_def
thf(fact_6607_img__ord__def,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( partial_img_ord @ A @ C @ B )
= ( ^ [F2: A > C,Ord2: C > C > B,X2: A,Y: A] : ( Ord2 @ ( F2 @ X2 ) @ ( F2 @ Y ) ) ) ) ).
% img_ord_def
thf(fact_6608_and_Ocomm__monoid__axioms,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( comm_monoid @ A @ ( bit_se5824344872417868541ns_and @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% and.comm_monoid_axioms
thf(fact_6609_Literal_Otransfer,axiom,
( bNF_rel_fun @ $o @ $o @ ( $o > $o > $o > $o > $o > $o > ( list @ char ) > ( list @ char ) ) @ ( $o > $o > $o > $o > $o > $o > literal > literal )
@ ^ [Y6: $o,Z2: $o] : Y6 = Z2
@ ( bNF_rel_fun @ $o @ $o @ ( $o > $o > $o > $o > $o > ( list @ char ) > ( list @ char ) ) @ ( $o > $o > $o > $o > $o > literal > literal )
@ ^ [Y6: $o,Z2: $o] : Y6 = Z2
@ ( bNF_rel_fun @ $o @ $o @ ( $o > $o > $o > $o > ( list @ char ) > ( list @ char ) ) @ ( $o > $o > $o > $o > literal > literal )
@ ^ [Y6: $o,Z2: $o] : Y6 = Z2
@ ( bNF_rel_fun @ $o @ $o @ ( $o > $o > $o > ( list @ char ) > ( list @ char ) ) @ ( $o > $o > $o > literal > literal )
@ ^ [Y6: $o,Z2: $o] : Y6 = Z2
@ ( bNF_rel_fun @ $o @ $o @ ( $o > $o > ( list @ char ) > ( list @ char ) ) @ ( $o > $o > literal > literal )
@ ^ [Y6: $o,Z2: $o] : Y6 = Z2
@ ( bNF_rel_fun @ $o @ $o @ ( $o > ( list @ char ) > ( list @ char ) ) @ ( $o > literal > literal )
@ ^ [Y6: $o,Z2: $o] : Y6 = Z2
@ ( bNF_rel_fun @ $o @ $o @ ( ( list @ char ) > ( list @ char ) ) @ ( literal > literal )
@ ^ [Y6: $o,Z2: $o] : Y6 = Z2
@ ( bNF_rel_fun @ ( list @ char ) @ literal @ ( list @ char ) @ literal @ pcr_literal @ pcr_literal ) ) ) ) ) ) )
@ ^ [B0: $o,B1: $o,B22: $o,B32: $o,B42: $o,B52: $o,B62: $o] : ( cons @ char @ ( char2 @ B0 @ B1 @ B22 @ B32 @ B42 @ B52 @ B62 @ $false ) )
@ literal2 ) ).
% Literal.transfer
thf(fact_6610_comm__monoid_Ocomm__neutral,axiom,
! [A: $tType,F: A > A > A,Z: A,A3: A] :
( ( comm_monoid @ A @ F @ Z )
=> ( ( F @ A3 @ Z )
= A3 ) ) ).
% comm_monoid.comm_neutral
thf(fact_6611_add_Ocomm__monoid__axioms,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ( comm_monoid @ A @ ( plus_plus @ A ) @ ( zero_zero @ A ) ) ) ).
% add.comm_monoid_axioms
thf(fact_6612_mult_Ocomm__monoid__axioms,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ( comm_monoid @ A @ ( times_times @ A ) @ ( one_one @ A ) ) ) ).
% mult.comm_monoid_axioms
thf(fact_6613_or_Ocomm__monoid__axioms,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( comm_monoid @ A @ ( bit_se1065995026697491101ons_or @ A ) @ ( zero_zero @ A ) ) ) ).
% or.comm_monoid_axioms
thf(fact_6614_gcd__nat_Ocomm__monoid__axioms,axiom,
comm_monoid @ nat @ ( gcd_gcd @ nat ) @ ( zero_zero @ nat ) ).
% gcd_nat.comm_monoid_axioms
thf(fact_6615_xor_Ocomm__monoid__axioms,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( comm_monoid @ A @ ( bit_se5824344971392196577ns_xor @ A ) @ ( zero_zero @ A ) ) ) ).
% xor.comm_monoid_axioms
thf(fact_6616_max__nat_Ocomm__monoid__axioms,axiom,
comm_monoid @ nat @ ( ord_max @ nat ) @ ( zero_zero @ nat ) ).
% max_nat.comm_monoid_axioms
thf(fact_6617_String_OLiteral__def,axiom,
( literal2
= ( map_fun @ $o @ $o @ ( $o > $o > $o > $o > $o > $o > ( list @ char ) > ( list @ char ) ) @ ( $o > $o > $o > $o > $o > $o > literal > literal ) @ ( id @ $o ) @ ( map_fun @ $o @ $o @ ( $o > $o > $o > $o > $o > ( list @ char ) > ( list @ char ) ) @ ( $o > $o > $o > $o > $o > literal > literal ) @ ( id @ $o ) @ ( map_fun @ $o @ $o @ ( $o > $o > $o > $o > ( list @ char ) > ( list @ char ) ) @ ( $o > $o > $o > $o > literal > literal ) @ ( id @ $o ) @ ( map_fun @ $o @ $o @ ( $o > $o > $o > ( list @ char ) > ( list @ char ) ) @ ( $o > $o > $o > literal > literal ) @ ( id @ $o ) @ ( map_fun @ $o @ $o @ ( $o > $o > ( list @ char ) > ( list @ char ) ) @ ( $o > $o > literal > literal ) @ ( id @ $o ) @ ( map_fun @ $o @ $o @ ( $o > ( list @ char ) > ( list @ char ) ) @ ( $o > literal > literal ) @ ( id @ $o ) @ ( map_fun @ $o @ $o @ ( ( list @ char ) > ( list @ char ) ) @ ( literal > literal ) @ ( id @ $o ) @ ( map_fun @ literal @ ( list @ char ) @ ( list @ char ) @ literal @ explode @ abs_literal ) ) ) ) ) ) )
@ ^ [B0: $o,B1: $o,B22: $o,B32: $o,B42: $o,B52: $o,B62: $o] : ( cons @ char @ ( char2 @ B0 @ B1 @ B22 @ B32 @ B42 @ B52 @ B62 @ $false ) ) ) ) ).
% String.Literal_def
thf(fact_6618_sum__set__defs_I1_J,axiom,
! [A: $tType,B: $tType] :
( ( basic_setl @ A @ B )
= ( sum_case_sum @ A @ ( set @ A ) @ B
@ ^ [Z3: A] : ( insert2 @ A @ Z3 @ ( bot_bot @ ( set @ A ) ) )
@ ^ [B5: B] : ( bot_bot @ ( set @ A ) ) ) ) ).
% sum_set_defs(1)
thf(fact_6619_o__case__sum,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,H: D > C,F: A > D,G: B > D] :
( ( comp @ D @ C @ ( sum_sum @ A @ B ) @ H @ ( sum_case_sum @ A @ D @ B @ F @ G ) )
= ( sum_case_sum @ A @ C @ B @ ( comp @ D @ C @ A @ H @ F ) @ ( comp @ D @ C @ B @ H @ G ) ) ) ).
% o_case_sum
thf(fact_6620_disjE__realizer,axiom,
! [C: $tType,B: $tType,A: $tType,P: A > $o,Q: B > $o,X: sum_sum @ A @ B,R5: C > $o,F: A > C,G: B > C] :
( ( sum_case_sum @ A @ $o @ B @ P @ Q @ X )
=> ( ! [P10: A] :
( ( P @ P10 )
=> ( R5 @ ( F @ P10 ) ) )
=> ( ! [Q3: B] :
( ( Q @ Q3 )
=> ( R5 @ ( G @ Q3 ) ) )
=> ( R5 @ ( sum_case_sum @ A @ C @ B @ F @ G @ X ) ) ) ) ) ).
% disjE_realizer
thf(fact_6621_case__sum__step_I2_J,axiom,
! [E: $tType,A: $tType,C: $tType,B: $tType,F: E > A,F5: B > A,G5: C > A,P4: sum_sum @ B @ C] :
( ( sum_case_sum @ E @ A @ ( sum_sum @ B @ C ) @ F @ ( sum_case_sum @ B @ A @ C @ F5 @ G5 ) @ ( sum_Inr @ ( sum_sum @ B @ C ) @ E @ P4 ) )
= ( sum_case_sum @ B @ A @ C @ F5 @ G5 @ P4 ) ) ).
% case_sum_step(2)
thf(fact_6622_case__sum__step_I1_J,axiom,
! [D: $tType,A: $tType,C: $tType,B: $tType,F5: B > A,G5: C > A,G: D > A,P4: sum_sum @ B @ C] :
( ( sum_case_sum @ ( sum_sum @ B @ C ) @ A @ D @ ( sum_case_sum @ B @ A @ C @ F5 @ G5 ) @ G @ ( sum_Inl @ ( sum_sum @ B @ C ) @ D @ P4 ) )
= ( sum_case_sum @ B @ A @ C @ F5 @ G5 @ P4 ) ) ).
% case_sum_step(1)
thf(fact_6623_case__sum__if,axiom,
! [B: $tType,A: $tType,C: $tType,P4: $o,F: B > A,G: C > A,X: B,Y3: C] :
( ( P4
=> ( ( sum_case_sum @ B @ A @ C @ F @ G @ ( if @ ( sum_sum @ B @ C ) @ P4 @ ( sum_Inl @ B @ C @ X ) @ ( sum_Inr @ C @ B @ Y3 ) ) )
= ( F @ X ) ) )
& ( ~ P4
=> ( ( sum_case_sum @ B @ A @ C @ F @ G @ ( if @ ( sum_sum @ B @ C ) @ P4 @ ( sum_Inl @ B @ C @ X ) @ ( sum_Inr @ C @ B @ Y3 ) ) )
= ( G @ Y3 ) ) ) ) ).
% case_sum_if
thf(fact_6624_case__sum__o__inj_I2_J,axiom,
! [A: $tType,B: $tType,C: $tType,F: A > B,G: C > B] :
( ( comp @ ( sum_sum @ A @ C ) @ B @ C @ ( sum_case_sum @ A @ B @ C @ F @ G ) @ ( sum_Inr @ C @ A ) )
= G ) ).
% case_sum_o_inj(2)
thf(fact_6625_case__sum__o__inj_I1_J,axiom,
! [C: $tType,B: $tType,A: $tType,F: A > B,G: C > B] :
( ( comp @ ( sum_sum @ A @ C ) @ B @ A @ ( sum_case_sum @ A @ B @ C @ F @ G ) @ ( sum_Inl @ A @ C ) )
= F ) ).
% case_sum_o_inj(1)
thf(fact_6626_case__sum__expand__Inr,axiom,
! [B: $tType,C: $tType,A: $tType,F: ( sum_sum @ A @ C ) > B,G: A > B,X: sum_sum @ A @ C] :
( ( ( comp @ ( sum_sum @ A @ C ) @ B @ A @ F @ ( sum_Inl @ A @ C ) )
= G )
=> ( ( F @ X )
= ( sum_case_sum @ A @ B @ C @ G @ ( comp @ ( sum_sum @ A @ C ) @ B @ C @ F @ ( sum_Inr @ C @ A ) ) @ X ) ) ) ).
% case_sum_expand_Inr
thf(fact_6627_case__sum__expand__Inr_H,axiom,
! [B: $tType,C: $tType,A: $tType,F: ( sum_sum @ A @ C ) > B,G: A > B,H: C > B] :
( ( ( comp @ ( sum_sum @ A @ C ) @ B @ A @ F @ ( sum_Inl @ A @ C ) )
= G )
=> ( ( H
= ( comp @ ( sum_sum @ A @ C ) @ B @ C @ F @ ( sum_Inr @ C @ A ) ) )
= ( ( sum_case_sum @ A @ B @ C @ G @ H )
= F ) ) ) ).
% case_sum_expand_Inr'
thf(fact_6628_case__sum__expand__Inr__pointfree,axiom,
! [B: $tType,C: $tType,A: $tType,F: ( sum_sum @ A @ C ) > B,G: A > B] :
( ( ( comp @ ( sum_sum @ A @ C ) @ B @ A @ F @ ( sum_Inl @ A @ C ) )
= G )
=> ( ( sum_case_sum @ A @ B @ C @ G @ ( comp @ ( sum_sum @ A @ C ) @ B @ C @ F @ ( sum_Inr @ C @ A ) ) )
= F ) ) ).
% case_sum_expand_Inr_pointfree
thf(fact_6629_sum__set__defs_I2_J,axiom,
! [C: $tType,D: $tType] :
( ( basic_setr @ C @ D )
= ( sum_case_sum @ C @ ( set @ D ) @ D
@ ^ [A6: C] : ( bot_bot @ ( set @ D ) )
@ ^ [Z3: D] : ( insert2 @ D @ Z3 @ ( bot_bot @ ( set @ D ) ) ) ) ) ).
% sum_set_defs(2)
thf(fact_6630_case__sum__KK,axiom,
! [C: $tType,B: $tType,A: $tType,A3: C] :
( ( sum_case_sum @ A @ C @ B
@ ^ [X2: A] : A3
@ ^ [X2: B] : A3 )
= ( ^ [X2: sum_sum @ A @ B] : A3 ) ) ).
% case_sum_KK
thf(fact_6631_surjective__sum,axiom,
! [C: $tType,B: $tType,A: $tType,F: ( sum_sum @ A @ B ) > C] :
( ( sum_case_sum @ A @ C @ B
@ ^ [X2: A] : ( F @ ( sum_Inl @ A @ B @ X2 ) )
@ ^ [Y: B] : ( F @ ( sum_Inr @ B @ A @ Y ) ) )
= F ) ).
% surjective_sum
thf(fact_6632_map__option__o__case__sum,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,F: D > C,G: A > ( option @ D ),H: B > ( option @ D )] :
( ( comp @ ( option @ D ) @ ( option @ C ) @ ( sum_sum @ A @ B ) @ ( map_option @ D @ C @ F ) @ ( sum_case_sum @ A @ ( option @ D ) @ B @ G @ H ) )
= ( sum_case_sum @ A @ ( option @ C ) @ B @ ( comp @ ( option @ D ) @ ( option @ C ) @ A @ ( map_option @ D @ C @ F ) @ G ) @ ( comp @ ( option @ D ) @ ( option @ C ) @ B @ ( map_option @ D @ C @ F ) @ H ) ) ) ).
% map_option_o_case_sum
thf(fact_6633_sum_Ocase__distrib,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,H: C > D,F1: A > C,F23: B > C,Sum: sum_sum @ A @ B] :
( ( H @ ( sum_case_sum @ A @ C @ B @ F1 @ F23 @ Sum ) )
= ( sum_case_sum @ A @ D @ B
@ ^ [X2: A] : ( H @ ( F1 @ X2 ) )
@ ^ [X2: B] : ( H @ ( F23 @ X2 ) )
@ Sum ) ) ).
% sum.case_distrib
thf(fact_6634_sum__encode__def,axiom,
( nat_sum_encode
= ( sum_case_sum @ nat @ nat @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
@ ^ [B5: nat] : ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ B5 ) ) ) ) ).
% sum_encode_def
thf(fact_6635_projl__def,axiom,
! [A: $tType,B: $tType] :
( ( sum_projl @ A @ B )
= ( sum_case_sum @ A @ A @ B
@ ^ [X12: A] : X12
@ ^ [Uu2: B] : ( undefined @ A ) ) ) ).
% projl_def
thf(fact_6636_projr__def,axiom,
! [A: $tType,B: $tType] :
( ( sum_projr @ A @ B )
= ( sum_case_sum @ A @ B @ B
@ ^ [Uu2: A] : ( undefined @ B )
@ ^ [X24: B] : X24 ) ) ).
% projr_def
thf(fact_6637_old_Orec__sum__def,axiom,
! [B: $tType,T: $tType,A: $tType] :
( ( sum_rec_sum @ A @ T @ B )
= ( ^ [F12: A > T,F22: B > T,X2: sum_sum @ A @ B] : ( the @ T @ ( sum_rec_set_sum @ A @ T @ B @ F12 @ F22 @ X2 ) ) ) ) ).
% old.rec_sum_def
thf(fact_6638_nat__of__num__code_I3_J,axiom,
! [N: num] :
( ( nat_of_num @ ( bit1 @ N ) )
= ( suc @ ( plus_plus @ nat @ ( nat_of_num @ N ) @ ( nat_of_num @ N ) ) ) ) ).
% nat_of_num_code(3)
thf(fact_6639_sorted__list__of__set_Ofold__insort__key_Ofolding__on__axioms,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( finite_folding_on @ A @ ( list @ A ) @ ( top_top @ ( set @ A ) )
@ ( linorder_insort_key @ A @ A
@ ^ [X2: A] : X2 ) ) ) ).
% sorted_list_of_set.fold_insort_key.folding_on_axioms
thf(fact_6640_nat__of__num__code_I1_J,axiom,
( ( nat_of_num @ one2 )
= ( one_one @ nat ) ) ).
% nat_of_num_code(1)
thf(fact_6641_nat__of__num_Osimps_I2_J,axiom,
! [X: num] :
( ( nat_of_num @ ( bit0 @ X ) )
= ( plus_plus @ nat @ ( nat_of_num @ X ) @ ( nat_of_num @ X ) ) ) ).
% nat_of_num.simps(2)
thf(fact_6642_nat__of__num__neq__0,axiom,
! [X: num] :
( ( nat_of_num @ X )
!= ( zero_zero @ nat ) ) ).
% nat_of_num_neq_0
thf(fact_6643_nat__of__num__pos,axiom,
! [X: num] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( nat_of_num @ X ) ) ).
% nat_of_num_pos
thf(fact_6644_folding__on__def,axiom,
! [B: $tType,A: $tType] :
( ( finite_folding_on @ A @ B )
= ( ^ [S8: set @ A,F2: A > B > B] :
! [X2: A,Y: A] :
( ( member @ A @ X2 @ S8 )
=> ( ( member @ A @ Y @ S8 )
=> ( ( comp @ B @ B @ B @ ( F2 @ Y ) @ ( F2 @ X2 ) )
= ( comp @ B @ B @ B @ ( F2 @ X2 ) @ ( F2 @ Y ) ) ) ) ) ) ) ).
% folding_on_def
thf(fact_6645_folding__on_Ocomp__fun__commute__on,axiom,
! [B: $tType,A: $tType,S3: set @ A,F: A > B > B,X: A,Y3: A] :
( ( finite_folding_on @ A @ B @ S3 @ F )
=> ( ( member @ A @ X @ S3 )
=> ( ( member @ A @ Y3 @ S3 )
=> ( ( comp @ B @ B @ B @ ( F @ Y3 ) @ ( F @ X ) )
= ( comp @ B @ B @ B @ ( F @ X ) @ ( F @ Y3 ) ) ) ) ) ) ).
% folding_on.comp_fun_commute_on
thf(fact_6646_folding__on_Ointro,axiom,
! [B: $tType,A: $tType,S3: set @ A,F: A > B > B] :
( ! [X4: A,Y4: A] :
( ( member @ A @ X4 @ S3 )
=> ( ( member @ A @ Y4 @ S3 )
=> ( ( comp @ B @ B @ B @ ( F @ Y4 ) @ ( F @ X4 ) )
= ( comp @ B @ B @ B @ ( F @ X4 ) @ ( F @ Y4 ) ) ) ) )
=> ( finite_folding_on @ A @ B @ S3 @ F ) ) ).
% folding_on.intro
thf(fact_6647_nat__of__num__add,axiom,
! [X: num,Y3: num] :
( ( nat_of_num @ ( plus_plus @ num @ X @ Y3 ) )
= ( plus_plus @ nat @ ( nat_of_num @ X ) @ ( nat_of_num @ Y3 ) ) ) ).
% nat_of_num_add
thf(fact_6648_nat__of__num__code_I2_J,axiom,
! [N: num] :
( ( nat_of_num @ ( bit0 @ N ) )
= ( plus_plus @ nat @ ( nat_of_num @ N ) @ ( nat_of_num @ N ) ) ) ).
% nat_of_num_code(2)
thf(fact_6649_nat__of__num_Osimps_I1_J,axiom,
( ( nat_of_num @ one2 )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% nat_of_num.simps(1)
thf(fact_6650_nat__of__num_Osimps_I3_J,axiom,
! [X: num] :
( ( nat_of_num @ ( bit1 @ X ) )
= ( suc @ ( plus_plus @ nat @ ( nat_of_num @ X ) @ ( nat_of_num @ X ) ) ) ) ).
% nat_of_num.simps(3)
thf(fact_6651_num__of__nat__inverse,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( nat_of_num @ ( num_of_nat @ N ) )
= N ) ) ).
% num_of_nat_inverse
thf(fact_6652_plus__num__def,axiom,
( ( plus_plus @ num )
= ( ^ [M5: num,N5: num] : ( num_of_nat @ ( plus_plus @ nat @ ( nat_of_num @ M5 ) @ ( nat_of_num @ N5 ) ) ) ) ) ).
% plus_num_def
thf(fact_6653_snds__def,axiom,
! [B: $tType,A: $tType] :
( ( basic_snds @ A @ B )
= ( ^ [P5: product_prod @ A @ B] : ( collect @ B @ ( basic_sndsp @ A @ B @ P5 ) ) ) ) ).
% snds_def
thf(fact_6654_setr__def,axiom,
! [B: $tType,A: $tType] :
( ( basic_setr @ A @ B )
= ( ^ [S2: sum_sum @ A @ B] : ( collect @ B @ ( basic_setrp @ A @ B @ S2 ) ) ) ) ).
% setr_def
thf(fact_6655_card_Ofolding__on__axioms,axiom,
! [A: $tType] :
( finite_folding_on @ A @ nat @ ( top_top @ ( set @ A ) )
@ ^ [Uu: A] : suc ) ).
% card.folding_on_axioms
thf(fact_6656_setl__def,axiom,
! [B: $tType,A: $tType] :
( ( basic_setl @ A @ B )
= ( ^ [S2: sum_sum @ A @ B] : ( collect @ A @ ( basic_setlp @ A @ B @ S2 ) ) ) ) ).
% setl_def
thf(fact_6657_fsts__def,axiom,
! [B: $tType,A: $tType] :
( ( basic_fsts @ A @ B )
= ( ^ [P5: product_prod @ A @ B] : ( collect @ A @ ( basic_fstsp @ A @ B @ P5 ) ) ) ) ).
% fsts_def
thf(fact_6658_prod_Ocomm__monoid__list__set__axioms,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ( groups4802862169904069756st_set @ A @ ( times_times @ A ) @ ( one_one @ A ) ) ) ).
% prod.comm_monoid_list_set_axioms
thf(fact_6659_prod_H__def,axiom,
! [C: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ( ( groups1962203154675924110t_prod @ C @ A )
= ( groups_comm_monoid_G @ A @ C @ ( times_times @ A ) @ ( one_one @ A ) ) ) ) ).
% prod'_def
thf(fact_6660_sum_H__def,axiom,
! [C: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ( ( groups1027152243600224163dd_sum @ C @ A )
= ( groups_comm_monoid_G @ A @ C @ ( plus_plus @ A ) @ ( zero_zero @ A ) ) ) ) ).
% sum'_def
thf(fact_6661_sum_Ocomm__monoid__list__set__axioms,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ( groups4802862169904069756st_set @ A @ ( plus_plus @ A ) @ ( zero_zero @ A ) ) ) ).
% sum.comm_monoid_list_set_axioms
thf(fact_6662_Heap_Osize__gen__o__map,axiom,
! [B: $tType,A: $tType,F: B > nat,G: A > B] :
( ( comp @ ( heap_Time_Heap @ B ) @ nat @ ( heap_Time_Heap @ A ) @ ( heap_Time_size_Heap @ B @ F ) @ ( heap_Time_map_Heap @ A @ B @ G ) )
= ( heap_Time_size_Heap @ A @ ( comp @ B @ nat @ A @ F @ G ) ) ) ).
% Heap.size_gen_o_map
thf(fact_6663_max_Osemilattice__order__axioms,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( semilattice_order @ A @ ( ord_max @ A )
@ ^ [X2: A,Y: A] : ( ord_less_eq @ A @ Y @ X2 )
@ ^ [X2: A,Y: A] : ( ord_less @ A @ Y @ X2 ) ) ) ).
% max.semilattice_order_axioms
thf(fact_6664_Heap_Omap__ident,axiom,
! [A: $tType,T2: heap_Time_Heap @ A] :
( ( heap_Time_map_Heap @ A @ A
@ ^ [X2: A] : X2
@ T2 )
= T2 ) ).
% Heap.map_ident
thf(fact_6665_Heap_Omap__id,axiom,
! [A: $tType,T2: heap_Time_Heap @ A] :
( ( heap_Time_map_Heap @ A @ A @ ( id @ A ) @ T2 )
= T2 ) ).
% Heap.map_id
thf(fact_6666_Heap_Omap__id0,axiom,
! [A: $tType] :
( ( heap_Time_map_Heap @ A @ A @ ( id @ A ) )
= ( id @ ( heap_Time_Heap @ A ) ) ) ).
% Heap.map_id0
thf(fact_6667_Heap_Omap__comp,axiom,
! [B: $tType,C: $tType,A: $tType,G: B > C,F: A > B,V: heap_Time_Heap @ A] :
( ( heap_Time_map_Heap @ B @ C @ G @ ( heap_Time_map_Heap @ A @ B @ F @ V ) )
= ( heap_Time_map_Heap @ A @ C @ ( comp @ B @ C @ A @ G @ F ) @ V ) ) ).
% Heap.map_comp
thf(fact_6668_Heap_Omap,axiom,
! [B: $tType,A: $tType,F: A > B,X: ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )] :
( ( heap_Time_map_Heap @ A @ B @ F @ ( heap_Time_Heap2 @ A @ X ) )
= ( heap_Time_Heap2 @ B @ ( comp @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ ( option @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ ( heap_ext @ product_unit ) @ ( map_option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_map_prod @ A @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ F @ ( id @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) @ X ) ) ) ).
% Heap.map
thf(fact_6669_Heap_Oinj__map,axiom,
! [B: $tType,A: $tType,F: A > B] :
( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
=> ( inj_on @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ B ) @ ( heap_Time_map_Heap @ A @ B @ F ) @ ( top_top @ ( set @ ( heap_Time_Heap @ A ) ) ) ) ) ).
% Heap.inj_map
thf(fact_6670_gcd__nat_Osemilattice__order__axioms,axiom,
( semilattice_order @ nat @ ( gcd_gcd @ nat ) @ ( dvd_dvd @ nat )
@ ^ [M5: nat,N5: nat] :
( ( dvd_dvd @ nat @ M5 @ N5 )
& ( M5 != N5 ) ) ) ).
% gcd_nat.semilattice_order_axioms
thf(fact_6671_sup_Osemilattice__order__axioms,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( semilattice_order @ A @ ( sup_sup @ A )
@ ^ [X2: A,Y: A] : ( ord_less_eq @ A @ Y @ X2 )
@ ^ [X2: A,Y: A] : ( ord_less @ A @ Y @ X2 ) ) ) ).
% sup.semilattice_order_axioms
thf(fact_6672_Heap_Orec__o__map,axiom,
! [B: $tType,C: $tType,A: $tType,G: ( ( heap_ext @ product_unit ) > ( option @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) > C,F: A > B] :
( ( comp @ ( heap_Time_Heap @ B ) @ C @ ( heap_Time_Heap @ A ) @ ( heap_Time_rec_Heap @ B @ C @ G ) @ ( heap_Time_map_Heap @ A @ B @ F ) )
= ( heap_Time_rec_Heap @ A @ C
@ ^ [X2: ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )] : ( G @ ( comp @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ ( option @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ ( heap_ext @ product_unit ) @ ( map_option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_map_prod @ A @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ F @ ( id @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) @ X2 ) ) ) ) ).
% Heap.rec_o_map
thf(fact_6673_Heap_Oset,axiom,
! [A: $tType,X: ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )] :
( ( heap_Time_set_Heap @ A @ ( heap_Time_Heap2 @ A @ X ) )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ ( set @ A )
@ ^ [Uu2: option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )] : ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( set @ A ) @ ( basic_fsts @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( set_option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ Uu2 ) ) )
@ ( image2 @ ( heap_ext @ product_unit ) @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ X @ ( top_top @ ( set @ ( heap_ext @ product_unit ) ) ) ) ) ) ) ).
% Heap.set
thf(fact_6674_Heap_Omap__cong,axiom,
! [B: $tType,A: $tType,X: heap_Time_Heap @ A,Ya: heap_Time_Heap @ A,F: A > B,G: A > B] :
( ( X = Ya )
=> ( ! [Z6: A] :
( ( member @ A @ Z6 @ ( heap_Time_set_Heap @ A @ Ya ) )
=> ( ( F @ Z6 )
= ( G @ Z6 ) ) )
=> ( ( heap_Time_map_Heap @ A @ B @ F @ X )
= ( heap_Time_map_Heap @ A @ B @ G @ Ya ) ) ) ) ).
% Heap.map_cong
thf(fact_6675_Heap_Omap__cong0,axiom,
! [B: $tType,A: $tType,X: heap_Time_Heap @ A,F: A > B,G: A > B] :
( ! [Z6: A] :
( ( member @ A @ Z6 @ ( heap_Time_set_Heap @ A @ X ) )
=> ( ( F @ Z6 )
= ( G @ Z6 ) ) )
=> ( ( heap_Time_map_Heap @ A @ B @ F @ X )
= ( heap_Time_map_Heap @ A @ B @ G @ X ) ) ) ).
% Heap.map_cong0
thf(fact_6676_Heap_Oinj__map__strong,axiom,
! [B: $tType,A: $tType,X: heap_Time_Heap @ A,Xa: heap_Time_Heap @ A,F: A > B,Fa: A > B] :
( ! [Z6: A,Za: A] :
( ( member @ A @ Z6 @ ( heap_Time_set_Heap @ A @ X ) )
=> ( ( member @ A @ Za @ ( heap_Time_set_Heap @ A @ Xa ) )
=> ( ( ( F @ Z6 )
= ( Fa @ Za ) )
=> ( Z6 = Za ) ) ) )
=> ( ( ( heap_Time_map_Heap @ A @ B @ F @ X )
= ( heap_Time_map_Heap @ A @ B @ Fa @ Xa ) )
=> ( X = Xa ) ) ) ).
% Heap.inj_map_strong
thf(fact_6677_Heap_Oset__cases,axiom,
! [A: $tType,E2: A,A3: heap_Time_Heap @ A] :
( ( member @ A @ E2 @ ( heap_Time_set_Heap @ A @ A3 ) )
=> ~ ! [Z6: ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )] :
( ( A3
= ( heap_Time_Heap2 @ A @ Z6 ) )
=> ! [Xa4: option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )] :
( ( member @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ Xa4 @ ( image2 @ ( heap_ext @ product_unit ) @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ Z6 @ ( top_top @ ( set @ ( heap_ext @ product_unit ) ) ) ) )
=> ! [Xb2: product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat )] :
( ( member @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ Xb2 @ ( set_option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ Xa4 ) )
=> ~ ( member @ A @ E2 @ ( basic_fsts @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ Xb2 ) ) ) ) ) ) ).
% Heap.set_cases
thf(fact_6678_Heap_Oset__intros,axiom,
! [A: $tType,Y3: option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ),X: ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ),Ya: product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ),Yb2: A] :
( ( member @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ Y3 @ ( image2 @ ( heap_ext @ product_unit ) @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ X @ ( top_top @ ( set @ ( heap_ext @ product_unit ) ) ) ) )
=> ( ( member @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ Ya @ ( set_option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ Y3 ) )
=> ( ( member @ A @ Yb2 @ ( basic_fsts @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ Ya ) )
=> ( member @ A @ Yb2 @ ( heap_Time_set_Heap @ A @ ( heap_Time_Heap2 @ A @ X ) ) ) ) ) ) ).
% Heap.set_intros
thf(fact_6679_Heap_Orec,axiom,
! [C: $tType,A: $tType,F: ( ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) > C,X: ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )] :
( ( heap_Time_rec_Heap @ A @ C @ F @ ( heap_Time_Heap2 @ A @ X ) )
= ( F @ X ) ) ).
% Heap.rec
thf(fact_6680_Heap_Oset__map,axiom,
! [B: $tType,A: $tType,F: A > B,V: heap_Time_Heap @ A] :
( ( heap_Time_set_Heap @ B @ ( heap_Time_map_Heap @ A @ B @ F @ V ) )
= ( image2 @ A @ B @ F @ ( heap_Time_set_Heap @ A @ V ) ) ) ).
% Heap.set_map
thf(fact_6681_Heap_Orel__Grp,axiom,
! [B: $tType,A: $tType,A5: set @ A,F: A > B] :
( ( heap_Time_rel_Heap @ A @ B @ ( bNF_Grp @ A @ B @ A5 @ F ) )
= ( bNF_Grp @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ B )
@ ( collect @ ( heap_Time_Heap @ A )
@ ^ [X2: heap_Time_Heap @ A] : ( ord_less_eq @ ( set @ A ) @ ( heap_Time_set_Heap @ A @ X2 ) @ A5 ) )
@ ( heap_Time_map_Heap @ A @ B @ F ) ) ) ).
% Heap.rel_Grp
thf(fact_6682_Heap_Oin__rel,axiom,
! [B: $tType,A: $tType] :
( ( heap_Time_rel_Heap @ A @ B )
= ( ^ [R6: A > B > $o,A6: heap_Time_Heap @ A,B5: heap_Time_Heap @ B] :
? [Z3: heap_Time_Heap @ ( product_prod @ A @ B )] :
( ( member @ ( heap_Time_Heap @ ( product_prod @ A @ B ) ) @ Z3
@ ( collect @ ( heap_Time_Heap @ ( product_prod @ A @ B ) )
@ ^ [X2: heap_Time_Heap @ ( product_prod @ A @ B )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( heap_Time_set_Heap @ ( product_prod @ A @ B ) @ X2 ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R6 ) ) ) ) )
& ( ( heap_Time_map_Heap @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Z3 )
= A6 )
& ( ( heap_Time_map_Heap @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ Z3 )
= B5 ) ) ) ) ).
% Heap.in_rel
thf(fact_6683_Heap_Orel__refl__strong,axiom,
! [A: $tType,X: heap_Time_Heap @ A,Ra2: A > A > $o] :
( ! [Z6: A] :
( ( member @ A @ Z6 @ ( heap_Time_set_Heap @ A @ X ) )
=> ( Ra2 @ Z6 @ Z6 ) )
=> ( heap_Time_rel_Heap @ A @ A @ Ra2 @ X @ X ) ) ).
% Heap.rel_refl_strong
thf(fact_6684_Heap_Orel__mono__strong,axiom,
! [A: $tType,B: $tType,R5: A > B > $o,X: heap_Time_Heap @ A,Y3: heap_Time_Heap @ B,Ra2: A > B > $o] :
( ( heap_Time_rel_Heap @ A @ B @ R5 @ X @ Y3 )
=> ( ! [Z6: A,Yb: B] :
( ( member @ A @ Z6 @ ( heap_Time_set_Heap @ A @ X ) )
=> ( ( member @ B @ Yb @ ( heap_Time_set_Heap @ B @ Y3 ) )
=> ( ( R5 @ Z6 @ Yb )
=> ( Ra2 @ Z6 @ Yb ) ) ) )
=> ( heap_Time_rel_Heap @ A @ B @ Ra2 @ X @ Y3 ) ) ) ).
% Heap.rel_mono_strong
thf(fact_6685_Heap_Orel__cong,axiom,
! [A: $tType,B: $tType,X: heap_Time_Heap @ A,Ya: heap_Time_Heap @ A,Y3: heap_Time_Heap @ B,Xa: heap_Time_Heap @ B,R5: A > B > $o,Ra2: A > B > $o] :
( ( X = Ya )
=> ( ( Y3 = Xa )
=> ( ! [Z6: A,Yb: B] :
( ( member @ A @ Z6 @ ( heap_Time_set_Heap @ A @ Ya ) )
=> ( ( member @ B @ Yb @ ( heap_Time_set_Heap @ B @ Xa ) )
=> ( ( R5 @ Z6 @ Yb )
= ( Ra2 @ Z6 @ Yb ) ) ) )
=> ( ( heap_Time_rel_Heap @ A @ B @ R5 @ X @ Y3 )
= ( heap_Time_rel_Heap @ A @ B @ Ra2 @ Ya @ Xa ) ) ) ) ) ).
% Heap.rel_cong
thf(fact_6686_Heap_Orec__transfer,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,R5: A > B > $o,S3: C > D > $o] :
( bNF_rel_fun @ ( ( ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) > C ) @ ( ( ( heap_ext @ product_unit ) > ( option @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) > D ) @ ( ( heap_Time_Heap @ A ) > C ) @ ( ( heap_Time_Heap @ B ) > D )
@ ( bNF_rel_fun @ ( ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) @ ( ( heap_ext @ product_unit ) > ( option @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) @ C @ D
@ ( bNF_rel_fun @ ( heap_ext @ product_unit ) @ ( heap_ext @ product_unit ) @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ ( option @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )
@ ^ [Y6: heap_ext @ product_unit,Z2: heap_ext @ product_unit] : Y6 = Z2
@ ( rel_option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )
@ ( basic_rel_prod @ A @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R5
@ ^ [Y6: product_prod @ ( heap_ext @ product_unit ) @ nat,Z2: product_prod @ ( heap_ext @ product_unit ) @ nat] : Y6 = Z2 ) ) )
@ S3 )
@ ( bNF_rel_fun @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ B ) @ C @ D @ ( heap_Time_rel_Heap @ A @ B @ R5 ) @ S3 )
@ ( heap_Time_rec_Heap @ A @ C )
@ ( heap_Time_rec_Heap @ B @ D ) ) ).
% Heap.rec_transfer
thf(fact_6687_Heap_Orel__cases,axiom,
! [A: $tType,B: $tType,R5: A > B > $o,A3: heap_Time_Heap @ A,B2: heap_Time_Heap @ B] :
( ( heap_Time_rel_Heap @ A @ B @ R5 @ A3 @ B2 )
=> ~ ! [Xa4: ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )] :
( ( A3
= ( heap_Time_Heap2 @ A @ Xa4 ) )
=> ! [Ya2: ( heap_ext @ product_unit ) > ( option @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )] :
( ( B2
= ( heap_Time_Heap2 @ B @ Ya2 ) )
=> ~ ( bNF_rel_fun @ ( heap_ext @ product_unit ) @ ( heap_ext @ product_unit ) @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ ( option @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )
@ ^ [Y6: heap_ext @ product_unit,Z2: heap_ext @ product_unit] : Y6 = Z2
@ ( rel_option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )
@ ( basic_rel_prod @ A @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R5
@ ^ [Y6: product_prod @ ( heap_ext @ product_unit ) @ nat,Z2: product_prod @ ( heap_ext @ product_unit ) @ nat] : Y6 = Z2 ) )
@ Xa4
@ Ya2 ) ) ) ) ).
% Heap.rel_cases
thf(fact_6688_Heap_Orel__induct,axiom,
! [A: $tType,B: $tType,R5: A > B > $o,X: heap_Time_Heap @ A,Y3: heap_Time_Heap @ B,Q: ( heap_Time_Heap @ A ) > ( heap_Time_Heap @ B ) > $o] :
( ( heap_Time_rel_Heap @ A @ B @ R5 @ X @ Y3 )
=> ( ! [A4: ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ),B3: ( heap_ext @ product_unit ) > ( option @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )] :
( ( bNF_rel_fun @ ( heap_ext @ product_unit ) @ ( heap_ext @ product_unit ) @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ ( option @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )
@ ^ [Y6: heap_ext @ product_unit,Z2: heap_ext @ product_unit] : Y6 = Z2
@ ( rel_option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )
@ ( basic_rel_prod @ A @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R5
@ ^ [Y6: product_prod @ ( heap_ext @ product_unit ) @ nat,Z2: product_prod @ ( heap_ext @ product_unit ) @ nat] : Y6 = Z2 ) )
@ A4
@ B3 )
=> ( Q @ ( heap_Time_Heap2 @ A @ A4 ) @ ( heap_Time_Heap2 @ B @ B3 ) ) )
=> ( Q @ X @ Y3 ) ) ) ).
% Heap.rel_induct
thf(fact_6689_Heap_Orel__inject,axiom,
! [A: $tType,B: $tType,R5: A > B > $o,X: ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ),Y3: ( heap_ext @ product_unit ) > ( option @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )] :
( ( heap_Time_rel_Heap @ A @ B @ R5 @ ( heap_Time_Heap2 @ A @ X ) @ ( heap_Time_Heap2 @ B @ Y3 ) )
= ( bNF_rel_fun @ ( heap_ext @ product_unit ) @ ( heap_ext @ product_unit ) @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ ( option @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )
@ ^ [Y6: heap_ext @ product_unit,Z2: heap_ext @ product_unit] : Y6 = Z2
@ ( rel_option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )
@ ( basic_rel_prod @ A @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R5
@ ^ [Y6: product_prod @ ( heap_ext @ product_unit ) @ nat,Z2: product_prod @ ( heap_ext @ product_unit ) @ nat] : Y6 = Z2 ) )
@ X
@ Y3 ) ) ).
% Heap.rel_inject
thf(fact_6690_Heap_Orel__intros,axiom,
! [A: $tType,B: $tType,R5: A > B > $o,X: ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ),Y3: ( heap_ext @ product_unit ) > ( option @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )] :
( ( bNF_rel_fun @ ( heap_ext @ product_unit ) @ ( heap_ext @ product_unit ) @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ ( option @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )
@ ^ [Y6: heap_ext @ product_unit,Z2: heap_ext @ product_unit] : Y6 = Z2
@ ( rel_option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )
@ ( basic_rel_prod @ A @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R5
@ ^ [Y6: product_prod @ ( heap_ext @ product_unit ) @ nat,Z2: product_prod @ ( heap_ext @ product_unit ) @ nat] : Y6 = Z2 ) )
@ X
@ Y3 )
=> ( heap_Time_rel_Heap @ A @ B @ R5 @ ( heap_Time_Heap2 @ A @ X ) @ ( heap_Time_Heap2 @ B @ Y3 ) ) ) ).
% Heap.rel_intros
thf(fact_6691_Heap_Octr__transfer,axiom,
! [A: $tType,B: $tType,R5: A > B > $o] :
( bNF_rel_fun @ ( ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) @ ( ( heap_ext @ product_unit ) > ( option @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ B )
@ ( bNF_rel_fun @ ( heap_ext @ product_unit ) @ ( heap_ext @ product_unit ) @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ ( option @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )
@ ^ [Y6: heap_ext @ product_unit,Z2: heap_ext @ product_unit] : Y6 = Z2
@ ( rel_option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )
@ ( basic_rel_prod @ A @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R5
@ ^ [Y6: product_prod @ ( heap_ext @ product_unit ) @ nat,Z2: product_prod @ ( heap_ext @ product_unit ) @ nat] : Y6 = Z2 ) ) )
@ ( heap_Time_rel_Heap @ A @ B @ R5 )
@ ( heap_Time_Heap2 @ A )
@ ( heap_Time_Heap2 @ B ) ) ).
% Heap.ctr_transfer
thf(fact_6692_Heap_Orel__transfer,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,Sa: A > C > $o,Sc: B > D > $o] :
( bNF_rel_fun @ ( A > B > $o ) @ ( C > D > $o ) @ ( ( heap_Time_Heap @ A ) > ( heap_Time_Heap @ B ) > $o ) @ ( ( heap_Time_Heap @ C ) > ( heap_Time_Heap @ D ) > $o )
@ ( bNF_rel_fun @ A @ C @ ( B > $o ) @ ( D > $o ) @ Sa
@ ( bNF_rel_fun @ B @ D @ $o @ $o @ Sc
@ ^ [Y6: $o,Z2: $o] : Y6 = Z2 ) )
@ ( bNF_rel_fun @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ C ) @ ( ( heap_Time_Heap @ B ) > $o ) @ ( ( heap_Time_Heap @ D ) > $o ) @ ( heap_Time_rel_Heap @ A @ C @ Sa )
@ ( bNF_rel_fun @ ( heap_Time_Heap @ B ) @ ( heap_Time_Heap @ D ) @ $o @ $o @ ( heap_Time_rel_Heap @ B @ D @ Sc )
@ ^ [Y6: $o,Z2: $o] : Y6 = Z2 ) )
@ ( heap_Time_rel_Heap @ A @ B )
@ ( heap_Time_rel_Heap @ C @ D ) ) ).
% Heap.rel_transfer
thf(fact_6693_Heap_Orel__mono,axiom,
! [B: $tType,A: $tType,R5: A > B > $o,Ra2: A > B > $o] :
( ( ord_less_eq @ ( A > B > $o ) @ R5 @ Ra2 )
=> ( ord_less_eq @ ( ( heap_Time_Heap @ A ) > ( heap_Time_Heap @ B ) > $o ) @ ( heap_Time_rel_Heap @ A @ B @ R5 ) @ ( heap_Time_rel_Heap @ A @ B @ Ra2 ) ) ) ).
% Heap.rel_mono
thf(fact_6694_Heap_Orel__compp,axiom,
! [A: $tType,C: $tType,B: $tType,R5: A > B > $o,S3: B > C > $o] :
( ( heap_Time_rel_Heap @ A @ C @ ( relcompp @ A @ B @ C @ R5 @ S3 ) )
= ( relcompp @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ B ) @ ( heap_Time_Heap @ C ) @ ( heap_Time_rel_Heap @ A @ B @ R5 ) @ ( heap_Time_rel_Heap @ B @ C @ S3 ) ) ) ).
% Heap.rel_compp
thf(fact_6695_Heap_Orel__conversep,axiom,
! [B: $tType,A: $tType,R5: A > B > $o] :
( ( heap_Time_rel_Heap @ B @ A @ ( conversep @ A @ B @ R5 ) )
= ( conversep @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ B ) @ ( heap_Time_rel_Heap @ A @ B @ R5 ) ) ) ).
% Heap.rel_conversep
thf(fact_6696_Heap_Orel__flip,axiom,
! [A: $tType,B: $tType,R5: A > B > $o,A3: heap_Time_Heap @ B,B2: heap_Time_Heap @ A] :
( ( heap_Time_rel_Heap @ B @ A @ ( conversep @ A @ B @ R5 ) @ A3 @ B2 )
= ( heap_Time_rel_Heap @ A @ B @ R5 @ B2 @ A3 ) ) ).
% Heap.rel_flip
thf(fact_6697_Heap_Orel__eq,axiom,
! [A: $tType] :
( ( heap_Time_rel_Heap @ A @ A
@ ^ [Y6: A,Z2: A] : Y6 = Z2 )
= ( ^ [Y6: heap_Time_Heap @ A,Z2: heap_Time_Heap @ A] : Y6 = Z2 ) ) ).
% Heap.rel_eq
thf(fact_6698_Heap_Orel__refl,axiom,
! [B: $tType,Ra2: B > B > $o,X: heap_Time_Heap @ B] :
( ! [X4: B] : ( Ra2 @ X4 @ X4 )
=> ( heap_Time_rel_Heap @ B @ B @ Ra2 @ X @ X ) ) ).
% Heap.rel_refl
thf(fact_6699_Heap_Orel__map_I1_J,axiom,
! [A: $tType,C: $tType,B: $tType,Sb: C > B > $o,I: A > C,X: heap_Time_Heap @ A,Y3: heap_Time_Heap @ B] :
( ( heap_Time_rel_Heap @ C @ B @ Sb @ ( heap_Time_map_Heap @ A @ C @ I @ X ) @ Y3 )
= ( heap_Time_rel_Heap @ A @ B
@ ^ [X2: A] : ( Sb @ ( I @ X2 ) )
@ X
@ Y3 ) ) ).
% Heap.rel_map(1)
thf(fact_6700_Heap_Orel__map_I2_J,axiom,
! [A: $tType,C: $tType,B: $tType,Sa: A > C > $o,X: heap_Time_Heap @ A,G: B > C,Y3: heap_Time_Heap @ B] :
( ( heap_Time_rel_Heap @ A @ C @ Sa @ X @ ( heap_Time_map_Heap @ B @ C @ G @ Y3 ) )
= ( heap_Time_rel_Heap @ A @ B
@ ^ [X2: A,Y: B] : ( Sa @ X2 @ ( G @ Y ) )
@ X
@ Y3 ) ) ).
% Heap.rel_map(2)
thf(fact_6701_Heap_OQuotient,axiom,
! [B: $tType,A: $tType,R5: A > A > $o,Abs: A > B,Rep: B > A,T5: A > B > $o] :
( ( quotient @ A @ B @ R5 @ Abs @ Rep @ T5 )
=> ( quotient @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ B ) @ ( heap_Time_rel_Heap @ A @ A @ R5 ) @ ( heap_Time_map_Heap @ A @ B @ Abs ) @ ( heap_Time_map_Heap @ B @ A @ Rep ) @ ( heap_Time_rel_Heap @ A @ B @ T5 ) ) ) ).
% Heap.Quotient
thf(fact_6702_Heap_Omap__transfer,axiom,
! [A: $tType,B: $tType,F3: $tType,E: $tType,Rb2: A > E > $o,Sd: B > F3 > $o] : ( bNF_rel_fun @ ( A > B ) @ ( E > F3 ) @ ( ( heap_Time_Heap @ A ) > ( heap_Time_Heap @ B ) ) @ ( ( heap_Time_Heap @ E ) > ( heap_Time_Heap @ F3 ) ) @ ( bNF_rel_fun @ A @ E @ B @ F3 @ Rb2 @ Sd ) @ ( bNF_rel_fun @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ E ) @ ( heap_Time_Heap @ B ) @ ( heap_Time_Heap @ F3 ) @ ( heap_Time_rel_Heap @ A @ E @ Rb2 ) @ ( heap_Time_rel_Heap @ B @ F3 @ Sd ) ) @ ( heap_Time_map_Heap @ A @ B ) @ ( heap_Time_map_Heap @ E @ F3 ) ) ).
% Heap.map_transfer
thf(fact_6703_Heap_Orel__compp__Grp,axiom,
! [B: $tType,A: $tType] :
( ( heap_Time_rel_Heap @ A @ B )
= ( ^ [R6: A > B > $o] :
( relcompp @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ ( product_prod @ A @ B ) ) @ ( heap_Time_Heap @ B )
@ ( conversep @ ( heap_Time_Heap @ ( product_prod @ A @ B ) ) @ ( heap_Time_Heap @ A )
@ ( bNF_Grp @ ( heap_Time_Heap @ ( product_prod @ A @ B ) ) @ ( heap_Time_Heap @ A )
@ ( collect @ ( heap_Time_Heap @ ( product_prod @ A @ B ) )
@ ^ [X2: heap_Time_Heap @ ( product_prod @ A @ B )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( heap_Time_set_Heap @ ( product_prod @ A @ B ) @ X2 ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R6 ) ) ) )
@ ( heap_Time_map_Heap @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) ) ) )
@ ( bNF_Grp @ ( heap_Time_Heap @ ( product_prod @ A @ B ) ) @ ( heap_Time_Heap @ B )
@ ( collect @ ( heap_Time_Heap @ ( product_prod @ A @ B ) )
@ ^ [X2: heap_Time_Heap @ ( product_prod @ A @ B )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( heap_Time_set_Heap @ ( product_prod @ A @ B ) @ X2 ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R6 ) ) ) )
@ ( heap_Time_map_Heap @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) ) ) ) ) ) ).
% Heap.rel_compp_Grp
thf(fact_6704_those_Osimps_I2_J,axiom,
! [A: $tType,X: option @ A,Xs2: list @ ( option @ A )] :
( ( those @ A @ ( cons @ ( option @ A ) @ X @ Xs2 ) )
= ( case_option @ ( option @ ( list @ A ) ) @ A @ ( none @ ( list @ A ) )
@ ^ [Y: A] : ( map_option @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ Y ) @ ( those @ A @ Xs2 ) )
@ X ) ) ).
% those.simps(2)
thf(fact_6705_bdd__below__primitive__def,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( condit1013018076250108175_below @ A )
= ( condit16957441358409770ng_bdd @ A
@ ^ [X2: A,Y: A] : ( ord_less_eq @ A @ Y @ X2 ) ) ) ) ).
% bdd_below_primitive_def
thf(fact_6706_Heap_Ocase__transfer,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,R5: A > B > $o,S3: C > D > $o] :
( bNF_rel_fun @ ( ( ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) > C ) @ ( ( ( heap_ext @ product_unit ) > ( option @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) > D ) @ ( ( heap_Time_Heap @ A ) > C ) @ ( ( heap_Time_Heap @ B ) > D )
@ ( bNF_rel_fun @ ( ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) @ ( ( heap_ext @ product_unit ) > ( option @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) @ C @ D
@ ( bNF_rel_fun @ ( heap_ext @ product_unit ) @ ( heap_ext @ product_unit ) @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ ( option @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )
@ ^ [Y6: heap_ext @ product_unit,Z2: heap_ext @ product_unit] : Y6 = Z2
@ ( rel_option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )
@ ( basic_rel_prod @ A @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R5
@ ^ [Y6: product_prod @ ( heap_ext @ product_unit ) @ nat,Z2: product_prod @ ( heap_ext @ product_unit ) @ nat] : Y6 = Z2 ) ) )
@ S3 )
@ ( bNF_rel_fun @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ B ) @ C @ D @ ( heap_Time_rel_Heap @ A @ B @ R5 ) @ S3 )
@ ( heap_Time_case_Heap @ A @ C )
@ ( heap_Time_case_Heap @ B @ D ) ) ).
% Heap.case_transfer
thf(fact_6707_Heap_Opred__transfer,axiom,
! [A: $tType,B: $tType,R5: A > B > $o] :
( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ ( ( heap_Time_Heap @ A ) > $o ) @ ( ( heap_Time_Heap @ B ) > $o )
@ ( bNF_rel_fun @ A @ B @ $o @ $o @ R5
@ ^ [Y6: $o,Z2: $o] : Y6 = Z2 )
@ ( bNF_rel_fun @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ B ) @ $o @ $o @ ( heap_Time_rel_Heap @ A @ B @ R5 )
@ ^ [Y6: $o,Z2: $o] : Y6 = Z2 )
@ ( heap_Time_pred_Heap @ A )
@ ( heap_Time_pred_Heap @ B ) ) ).
% Heap.pred_transfer
thf(fact_6708_Heap_Opred__mono,axiom,
! [A: $tType,P: A > $o,Pa: A > $o] :
( ( ord_less_eq @ ( A > $o ) @ P @ Pa )
=> ( ord_less_eq @ ( ( heap_Time_Heap @ A ) > $o ) @ ( heap_Time_pred_Heap @ A @ P ) @ ( heap_Time_pred_Heap @ A @ Pa ) ) ) ).
% Heap.pred_mono
thf(fact_6709_Heap_Opred__True,axiom,
! [A: $tType] :
( ( heap_Time_pred_Heap @ A
@ ^ [Uu: A] : $true )
= ( ^ [Uu: heap_Time_Heap @ A] : $true ) ) ).
% Heap.pred_True
thf(fact_6710_Heap_Ocase__distrib,axiom,
! [C: $tType,B: $tType,A: $tType,H: B > C,F: ( ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) > B,Heap: heap_Time_Heap @ A] :
( ( H @ ( heap_Time_case_Heap @ A @ B @ F @ Heap ) )
= ( heap_Time_case_Heap @ A @ C
@ ^ [X2: ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )] : ( H @ ( F @ X2 ) )
@ Heap ) ) ).
% Heap.case_distrib
thf(fact_6711_Heap_Omap__cong__pred,axiom,
! [B: $tType,A: $tType,X: heap_Time_Heap @ A,Ya: heap_Time_Heap @ A,F: A > B,G: A > B] :
( ( X = Ya )
=> ( ( heap_Time_pred_Heap @ A
@ ^ [Z3: A] :
( ( F @ Z3 )
= ( G @ Z3 ) )
@ Ya )
=> ( ( heap_Time_map_Heap @ A @ B @ F @ X )
= ( heap_Time_map_Heap @ A @ B @ G @ Ya ) ) ) ) ).
% Heap.map_cong_pred
thf(fact_6712_Heap_Opred__cong,axiom,
! [A: $tType,X: heap_Time_Heap @ A,Ya: heap_Time_Heap @ A,P: A > $o,Pa: A > $o] :
( ( X = Ya )
=> ( ! [Z6: A] :
( ( member @ A @ Z6 @ ( heap_Time_set_Heap @ A @ Ya ) )
=> ( ( P @ Z6 )
= ( Pa @ Z6 ) ) )
=> ( ( heap_Time_pred_Heap @ A @ P @ X )
= ( heap_Time_pred_Heap @ A @ Pa @ Ya ) ) ) ) ).
% Heap.pred_cong
thf(fact_6713_Heap_Opred__mono__strong,axiom,
! [A: $tType,P: A > $o,X: heap_Time_Heap @ A,Pa: A > $o] :
( ( heap_Time_pred_Heap @ A @ P @ X )
=> ( ! [Z6: A] :
( ( member @ A @ Z6 @ ( heap_Time_set_Heap @ A @ X ) )
=> ( ( P @ Z6 )
=> ( Pa @ Z6 ) ) )
=> ( heap_Time_pred_Heap @ A @ Pa @ X ) ) ) ).
% Heap.pred_mono_strong
thf(fact_6714_Heap_Ocase,axiom,
! [B: $tType,A: $tType,F: ( ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) > B,X: ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )] :
( ( heap_Time_case_Heap @ A @ B @ F @ ( heap_Time_Heap2 @ A @ X ) )
= ( F @ X ) ) ).
% Heap.case
thf(fact_6715_Heap_Opred__set,axiom,
! [A: $tType] :
( ( heap_Time_pred_Heap @ A )
= ( ^ [P3: A > $o,X2: heap_Time_Heap @ A] :
! [Y: A] :
( ( member @ A @ Y @ ( heap_Time_set_Heap @ A @ X2 ) )
=> ( P3 @ Y ) ) ) ) ).
% Heap.pred_set
thf(fact_6716_Heap_Opred__map,axiom,
! [B: $tType,A: $tType,Q: B > $o,F: A > B,X: heap_Time_Heap @ A] :
( ( heap_Time_pred_Heap @ B @ Q @ ( heap_Time_map_Heap @ A @ B @ F @ X ) )
= ( heap_Time_pred_Heap @ A @ ( comp @ B @ $o @ A @ Q @ F ) @ X ) ) ).
% Heap.pred_map
thf(fact_6717_Heap_Opred__inject,axiom,
! [A: $tType,P: A > $o,A3: ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )] :
( ( heap_Time_pred_Heap @ A @ P @ ( heap_Time_Heap2 @ A @ A3 ) )
= ( basic_pred_fun @ ( heap_ext @ product_unit ) @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )
@ ^ [Uu: heap_ext @ product_unit] : $true
@ ( pred_option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( basic_pred_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ P @ ( top_top @ ( ( product_prod @ ( heap_ext @ product_unit ) @ nat ) > $o ) ) ) )
@ A3 ) ) ).
% Heap.pred_inject
thf(fact_6718_bdd__below_Opreordering__bdd__axioms,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( condit622319405099724424ng_bdd @ A
@ ^ [X2: A,Y: A] : ( ord_less_eq @ A @ Y @ X2 )
@ ^ [X2: A,Y: A] : ( ord_less @ A @ Y @ X2 ) ) ) ).
% bdd_below.preordering_bdd_axioms
thf(fact_6719_pred__prod__inject,axiom,
! [A: $tType,B: $tType,P12: A > $o,P23: B > $o,A3: A,B2: B] :
( ( basic_pred_prod @ A @ B @ P12 @ P23 @ ( product_Pair @ A @ B @ A3 @ B2 ) )
= ( ( P12 @ A3 )
& ( P23 @ B2 ) ) ) ).
% pred_prod_inject
thf(fact_6720_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_6721_pred__prod_Ointros,axiom,
! [A: $tType,B: $tType,P12: A > $o,A3: A,P23: B > $o,B2: B] :
( ( P12 @ A3 )
=> ( ( P23 @ B2 )
=> ( basic_pred_prod @ A @ B @ P12 @ P23 @ ( product_Pair @ A @ B @ A3 @ B2 ) ) ) ) ).
% pred_prod.intros
thf(fact_6722_pred__prod_Osimps,axiom,
! [B: $tType,A: $tType] :
( ( basic_pred_prod @ A @ B )
= ( ^ [P13: A > $o,P24: B > $o,A6: product_prod @ A @ B] :
? [B5: A,C4: B] :
( ( A6
= ( product_Pair @ A @ B @ B5 @ C4 ) )
& ( P13 @ B5 )
& ( P24 @ C4 ) ) ) ) ).
% pred_prod.simps
thf(fact_6723_pred__prod_Ocases,axiom,
! [A: $tType,B: $tType,P12: A > $o,P23: B > $o,A3: product_prod @ A @ B] :
( ( basic_pred_prod @ A @ B @ P12 @ P23 @ A3 )
=> ~ ! [A4: A,B3: B] :
( ( A3
= ( product_Pair @ A @ B @ A4 @ B3 ) )
=> ( ( P12 @ A4 )
=> ~ ( P23 @ B3 ) ) ) ) ).
% pred_prod.cases
thf(fact_6724_pred__prod__split,axiom,
! [B: $tType,A: $tType,P: $o > $o,Q: A > $o,R5: B > $o,Xy2: product_prod @ A @ B] :
( ( P @ ( basic_pred_prod @ A @ B @ Q @ R5 @ Xy2 ) )
= ( ! [X2: A,Y: B] :
( ( Xy2
= ( product_Pair @ A @ B @ X2 @ Y ) )
=> ( P
@ ( ( Q @ X2 )
& ( R5 @ Y ) ) ) ) ) ) ).
% pred_prod_split
thf(fact_6725_prod_Omap__cong__pred,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,X: product_prod @ A @ B,Ya: product_prod @ A @ B,F1: A > C,G1: A > C,F23: B > D,G22: B > D] :
( ( X = Ya )
=> ( ( basic_pred_prod @ A @ B
@ ^ [Z12: A] :
( ( F1 @ Z12 )
= ( G1 @ Z12 ) )
@ ^ [Z23: B] :
( ( F23 @ Z23 )
= ( G22 @ Z23 ) )
@ Ya )
=> ( ( product_map_prod @ A @ C @ B @ D @ F1 @ F23 @ X )
= ( product_map_prod @ A @ C @ B @ D @ G1 @ G22 @ Ya ) ) ) ) ).
% prod.map_cong_pred
thf(fact_6726_top__conj_I2_J,axiom,
! [A: $tType,P: $o,X: A] :
( ( P
& ( top_top @ ( A > $o ) @ X ) )
= P ) ).
% top_conj(2)
thf(fact_6727_top__conj_I1_J,axiom,
! [A: $tType,X: A,P: $o] :
( ( ( top_top @ ( A > $o ) @ X )
& P )
= P ) ).
% top_conj(1)
thf(fact_6728_prod_Opred__map,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,Q12: C > $o,Q23: D > $o,F1: A > C,F23: B > D,X: product_prod @ A @ B] :
( ( basic_pred_prod @ C @ D @ Q12 @ Q23 @ ( product_map_prod @ A @ C @ B @ D @ F1 @ F23 @ X ) )
= ( basic_pred_prod @ A @ B @ ( comp @ C @ $o @ A @ Q12 @ F1 ) @ ( comp @ D @ $o @ B @ Q23 @ F23 ) @ X ) ) ).
% prod.pred_map
thf(fact_6729_wfP__acyclicP,axiom,
! [A: $tType,R: A > A > $o] :
( ( wfP @ A @ R )
=> ( transitive_acyclic @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ R ) ) ) ) ).
% wfP_acyclicP
thf(fact_6730_wfP__empty,axiom,
! [A: $tType] :
( wfP @ A
@ ^ [X2: A,Y: A] : $false ) ).
% wfP_empty
thf(fact_6731_wfP__wf__eq,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( wfP @ A
@ ^ [X2: A,Y: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y ) @ R ) )
= ( wf @ A @ R ) ) ).
% wfP_wf_eq
thf(fact_6732_wfP__def,axiom,
! [A: $tType] :
( ( wfP @ A )
= ( ^ [R4: A > A > $o] : ( wf @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ R4 ) ) ) ) ) ).
% wfP_def
thf(fact_6733_int__decode__def,axiom,
( nat_int_decode
= ( ^ [N5: nat] :
( sum_case_sum @ nat @ int @ nat @ ( semiring_1_of_nat @ int )
@ ^ [B5: nat] : ( minus_minus @ int @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ B5 ) ) @ ( one_one @ int ) )
@ ( nat_sum_decode @ N5 ) ) ) ) ).
% int_decode_def
thf(fact_6734_to__nat__def,axiom,
! [A: $tType] :
( ( countable @ A )
=> ( ( to_nat @ A )
= ( fChoice @ ( A > nat )
@ ^ [F2: A > nat] : ( inj_on @ A @ nat @ F2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% to_nat_def
thf(fact_6735_nat__to__rat__surj__def,axiom,
( nat_to_rat_surj
= ( ^ [N5: nat] :
( product_case_prod @ nat @ nat @ rat
@ ^ [A6: nat,B5: nat] : ( fract @ ( nat_int_decode @ A6 ) @ ( nat_int_decode @ B5 ) )
@ ( nat_prod_decode @ N5 ) ) ) ) ).
% nat_to_rat_surj_def
thf(fact_6736_nth__item_Oelims,axiom,
! [A: $tType] :
( ( countable @ A )
=> ! [X: nat,Y3: set @ ( old_node @ A @ product_unit )] :
( ( ( nth_item @ A @ X )
= Y3 )
=> ( ( ( X
= ( zero_zero @ nat ) )
=> ( Y3
!= ( undefined @ ( set @ ( old_node @ A @ product_unit ) ) ) ) )
=> ~ ! [N2: nat] :
( ( X
= ( suc @ N2 ) )
=> ( Y3
!= ( sum_case_sum @ nat @ ( set @ ( old_node @ A @ product_unit ) ) @ nat
@ ^ [I2: nat] :
( sum_case_sum @ nat @ ( set @ ( old_node @ A @ product_unit ) ) @ nat
@ ^ [J3: nat] : ( old_In0 @ A @ product_unit @ ( nth_item @ A @ J3 ) )
@ ^ [J3: nat] : ( old_In1 @ A @ product_unit @ ( nth_item @ A @ J3 ) )
@ ( nat_sum_decode @ I2 ) )
@ ^ [I2: nat] :
( sum_case_sum @ nat @ ( set @ ( old_node @ A @ product_unit ) ) @ nat
@ ^ [J3: nat] : ( old_Leaf @ A @ product_unit @ ( from_nat @ A @ J3 ) )
@ ^ [J3: nat] :
( product_case_prod @ nat @ nat @ ( set @ ( old_node @ A @ product_unit ) )
@ ^ [A6: nat,B5: nat] : ( old_Scons @ A @ product_unit @ ( nth_item @ A @ A6 ) @ ( nth_item @ A @ B5 ) )
@ ( nat_prod_decode @ J3 ) )
@ ( nat_sum_decode @ I2 ) )
@ ( nat_sum_decode @ N2 ) ) ) ) ) ) ) ).
% nth_item.elims
thf(fact_6737_nth__item_Osimps_I1_J,axiom,
! [A: $tType] :
( ( countable @ A )
=> ( ( nth_item @ A @ ( zero_zero @ nat ) )
= ( undefined @ ( set @ ( old_node @ A @ product_unit ) ) ) ) ) ).
% nth_item.simps(1)
thf(fact_6738_nth__item_Osimps_I2_J,axiom,
! [A: $tType] :
( ( countable @ A )
=> ! [N: nat] :
( ( nth_item @ A @ ( suc @ N ) )
= ( sum_case_sum @ nat @ ( set @ ( old_node @ A @ product_unit ) ) @ nat
@ ^ [I2: nat] :
( sum_case_sum @ nat @ ( set @ ( old_node @ A @ product_unit ) ) @ nat
@ ^ [J3: nat] : ( old_In0 @ A @ product_unit @ ( nth_item @ A @ J3 ) )
@ ^ [J3: nat] : ( old_In1 @ A @ product_unit @ ( nth_item @ A @ J3 ) )
@ ( nat_sum_decode @ I2 ) )
@ ^ [I2: nat] :
( sum_case_sum @ nat @ ( set @ ( old_node @ A @ product_unit ) ) @ nat
@ ^ [J3: nat] : ( old_Leaf @ A @ product_unit @ ( from_nat @ A @ J3 ) )
@ ^ [J3: nat] :
( product_case_prod @ nat @ nat @ ( set @ ( old_node @ A @ product_unit ) )
@ ^ [A6: nat,B5: nat] : ( old_Scons @ A @ product_unit @ ( nth_item @ A @ A6 ) @ ( nth_item @ A @ B5 ) )
@ ( nat_prod_decode @ J3 ) )
@ ( nat_sum_decode @ I2 ) )
@ ( nat_sum_decode @ N ) ) ) ) ).
% nth_item.simps(2)
thf(fact_6739_nth__item_Opelims,axiom,
! [A: $tType] :
( ( countable @ A )
=> ! [X: nat,Y3: set @ ( old_node @ A @ product_unit )] :
( ( ( nth_item @ A @ X )
= Y3 )
=> ( ( accp @ nat @ nth_item_rel @ X )
=> ( ( ( X
= ( zero_zero @ nat ) )
=> ( ( Y3
= ( undefined @ ( set @ ( old_node @ A @ product_unit ) ) ) )
=> ~ ( accp @ nat @ nth_item_rel @ ( zero_zero @ nat ) ) ) )
=> ~ ! [N2: nat] :
( ( X
= ( suc @ N2 ) )
=> ( ( Y3
= ( sum_case_sum @ nat @ ( set @ ( old_node @ A @ product_unit ) ) @ nat
@ ^ [I2: nat] :
( sum_case_sum @ nat @ ( set @ ( old_node @ A @ product_unit ) ) @ nat
@ ^ [J3: nat] : ( old_In0 @ A @ product_unit @ ( nth_item @ A @ J3 ) )
@ ^ [J3: nat] : ( old_In1 @ A @ product_unit @ ( nth_item @ A @ J3 ) )
@ ( nat_sum_decode @ I2 ) )
@ ^ [I2: nat] :
( sum_case_sum @ nat @ ( set @ ( old_node @ A @ product_unit ) ) @ nat
@ ^ [J3: nat] : ( old_Leaf @ A @ product_unit @ ( from_nat @ A @ J3 ) )
@ ^ [J3: nat] :
( product_case_prod @ nat @ nat @ ( set @ ( old_node @ A @ product_unit ) )
@ ^ [A6: nat,B5: nat] : ( old_Scons @ A @ product_unit @ ( nth_item @ A @ A6 ) @ ( nth_item @ A @ B5 ) )
@ ( nat_prod_decode @ J3 ) )
@ ( nat_sum_decode @ I2 ) )
@ ( nat_sum_decode @ N2 ) ) )
=> ~ ( accp @ nat @ nth_item_rel @ ( suc @ N2 ) ) ) ) ) ) ) ) ).
% nth_item.pelims
thf(fact_6740_nth__item_Opsimps_I2_J,axiom,
! [A: $tType] :
( ( countable @ A )
=> ! [N: nat] :
( ( accp @ nat @ nth_item_rel @ ( suc @ N ) )
=> ( ( nth_item @ A @ ( suc @ N ) )
= ( sum_case_sum @ nat @ ( set @ ( old_node @ A @ product_unit ) ) @ nat
@ ^ [I2: nat] :
( sum_case_sum @ nat @ ( set @ ( old_node @ A @ product_unit ) ) @ nat
@ ^ [J3: nat] : ( old_In0 @ A @ product_unit @ ( nth_item @ A @ J3 ) )
@ ^ [J3: nat] : ( old_In1 @ A @ product_unit @ ( nth_item @ A @ J3 ) )
@ ( nat_sum_decode @ I2 ) )
@ ^ [I2: nat] :
( sum_case_sum @ nat @ ( set @ ( old_node @ A @ product_unit ) ) @ nat
@ ^ [J3: nat] : ( old_Leaf @ A @ product_unit @ ( from_nat @ A @ J3 ) )
@ ^ [J3: nat] :
( product_case_prod @ nat @ nat @ ( set @ ( old_node @ A @ product_unit ) )
@ ^ [A6: nat,B5: nat] : ( old_Scons @ A @ product_unit @ ( nth_item @ A @ A6 ) @ ( nth_item @ A @ B5 ) )
@ ( nat_prod_decode @ J3 ) )
@ ( nat_sum_decode @ I2 ) )
@ ( nat_sum_decode @ N ) ) ) ) ) ).
% nth_item.psimps(2)
thf(fact_6741_nth__item_Opsimps_I1_J,axiom,
! [A: $tType] :
( ( countable @ A )
=> ( ( accp @ nat @ nth_item_rel @ ( zero_zero @ nat ) )
=> ( ( nth_item @ A @ ( zero_zero @ nat ) )
= ( undefined @ ( set @ ( old_node @ A @ product_unit ) ) ) ) ) ) ).
% nth_item.psimps(1)
thf(fact_6742_Rats__eq__range__of__rat__o__nat__to__rat__surj,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( ( field_char_0_Rats @ A )
= ( image2 @ nat @ A @ ( comp @ rat @ A @ nat @ ( field_char_0_of_rat @ A ) @ nat_to_rat_surj ) @ ( top_top @ ( set @ nat ) ) ) ) ) ).
% Rats_eq_range_of_rat_o_nat_to_rat_surj
thf(fact_6743_trancl__def,axiom,
! [A: $tType] :
( ( transitive_trancl @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ( transitive_tranclp @ A
@ ^ [X2: A,Y: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y ) @ R4 ) ) ) ) ) ) ).
% trancl_def
thf(fact_6744_tranclp__trancl__eq,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( transitive_tranclp @ A
@ ^ [X2: A,Y: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y ) @ R ) )
= ( ^ [X2: A,Y: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y ) @ ( transitive_trancl @ A @ R ) ) ) ) ).
% tranclp_trancl_eq
thf(fact_6745_tranclp__induct2,axiom,
! [A: $tType,B: $tType,R: ( product_prod @ A @ B ) > ( product_prod @ A @ B ) > $o,Ax: A,Ay: B,Bx: A,By: B,P: A > B > $o] :
( ( transitive_tranclp @ ( product_prod @ A @ B ) @ R @ ( product_Pair @ A @ B @ Ax @ Ay ) @ ( product_Pair @ A @ B @ Bx @ By ) )
=> ( ! [A4: A,B3: B] :
( ( R @ ( product_Pair @ A @ B @ Ax @ Ay ) @ ( product_Pair @ A @ B @ A4 @ B3 ) )
=> ( P @ A4 @ B3 ) )
=> ( ! [A4: A,B3: B,Aa2: A,Ba: B] :
( ( transitive_tranclp @ ( product_prod @ A @ B ) @ R @ ( product_Pair @ A @ B @ Ax @ Ay ) @ ( product_Pair @ A @ B @ A4 @ B3 ) )
=> ( ( R @ ( product_Pair @ A @ B @ A4 @ B3 ) @ ( product_Pair @ A @ B @ Aa2 @ Ba ) )
=> ( ( P @ A4 @ B3 )
=> ( P @ Aa2 @ Ba ) ) ) )
=> ( P @ Bx @ By ) ) ) ) ).
% tranclp_induct2
thf(fact_6746_less__nat__rel,axiom,
( ( ord_less @ nat )
= ( transitive_tranclp @ nat
@ ^ [M5: nat,N5: nat] :
( N5
= ( suc @ M5 ) ) ) ) ).
% less_nat_rel
thf(fact_6747_Rats__add,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,B2: A] :
( ( member @ A @ A3 @ ( field_char_0_Rats @ A ) )
=> ( ( member @ A @ B2 @ ( field_char_0_Rats @ A ) )
=> ( member @ A @ ( plus_plus @ A @ A3 @ B2 ) @ ( field_char_0_Rats @ A ) ) ) ) ) ).
% Rats_add
thf(fact_6748_Rats__1,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( member @ A @ ( one_one @ A ) @ ( field_char_0_Rats @ A ) ) ) ).
% Rats_1
thf(fact_6749_Rats__0,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( member @ A @ ( zero_zero @ A ) @ ( field_char_0_Rats @ A ) ) ) ).
% Rats_0
thf(fact_6750_tranclp__power,axiom,
! [A: $tType] :
( ( transitive_tranclp @ A )
= ( ^ [P3: A > A > $o,X2: A,Y: A] :
? [N5: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N5 )
& ( compow @ ( A > A > $o ) @ N5 @ P3 @ X2 @ Y ) ) ) ) ).
% tranclp_power
thf(fact_6751_tranclp__def,axiom,
! [A: $tType] :
( ( transitive_tranclp @ A )
= ( ^ [R4: A > A > $o] :
( complete_lattice_lfp @ ( A > A > $o )
@ ^ [P5: A > A > $o,X12: A,X24: A] :
( ? [A6: A,B5: A] :
( ( X12 = A6 )
& ( X24 = B5 )
& ( R4 @ A6 @ B5 ) )
| ? [A6: A,B5: A,C4: A] :
( ( X12 = A6 )
& ( X24 = C4 )
& ( P5 @ A6 @ B5 )
& ( R4 @ B5 @ C4 ) ) ) ) ) ) ).
% tranclp_def
thf(fact_6752_Nitpick_Otranclp__unfold,axiom,
! [A: $tType] :
( ( transitive_tranclp @ A )
= ( ^ [R4: A > A > $o,A6: A,B5: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A6 @ B5 ) @ ( transitive_trancl @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ R4 ) ) ) ) ) ) ).
% Nitpick.tranclp_unfold
thf(fact_6753_semilattice__set_Oeq__fold_H,axiom,
! [A: $tType,F: A > A > A,A5: set @ A] :
( ( lattic149705377957585745ce_set @ A @ F )
=> ( ( lattic1715443433743089157tice_F @ A @ F @ A5 )
= ( the2 @ A
@ ( finite_fold @ A @ ( option @ A )
@ ^ [X2: A,Y: option @ A] : ( some @ A @ ( case_option @ A @ A @ X2 @ ( F @ X2 ) @ Y ) )
@ ( none @ A )
@ A5 ) ) ) ) ).
% semilattice_set.eq_fold'
thf(fact_6754_reflp__refl__eq,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( reflp @ A
@ ^ [X2: A,Y: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y ) @ R ) )
= ( refl_on @ A @ ( top_top @ ( set @ A ) ) @ R ) ) ).
% reflp_refl_eq
thf(fact_6755_Quotient__id__abs__transfer,axiom,
! [B: $tType,A: $tType,R5: A > A > $o,Abs: A > B,Rep: B > A,T5: A > B > $o] :
( ( quotient @ A @ B @ R5 @ Abs @ Rep @ T5 )
=> ( ( reflp @ A @ R5 )
=> ( bNF_rel_fun @ A @ A @ A @ B
@ ^ [Y6: A,Z2: A] : Y6 = Z2
@ T5
@ ^ [X2: A] : X2
@ Abs ) ) ) ).
% Quotient_id_abs_transfer
thf(fact_6756_Heap_Orel__reflp,axiom,
! [A: $tType,R5: A > A > $o] :
( ( reflp @ A @ R5 )
=> ( reflp @ ( heap_Time_Heap @ A ) @ ( heap_Time_rel_Heap @ A @ A @ R5 ) ) ) ).
% Heap.rel_reflp
thf(fact_6757_semilattice__set_Oinfinite,axiom,
! [A: $tType,F: A > A > A,A5: set @ A] :
( ( lattic149705377957585745ce_set @ A @ F )
=> ( ~ ( finite_finite2 @ A @ A5 )
=> ( ( lattic1715443433743089157tice_F @ A @ F @ A5 )
= ( the2 @ A @ ( none @ A ) ) ) ) ) ).
% semilattice_set.infinite
thf(fact_6758_option_Orel__reflp,axiom,
! [A: $tType,R5: A > A > $o] :
( ( reflp @ A @ R5 )
=> ( reflp @ ( option @ A ) @ ( rel_option @ A @ A @ R5 ) ) ) ).
% option.rel_reflp
thf(fact_6759_image2p__def,axiom,
! [D: $tType,B: $tType,A: $tType,C: $tType] :
( ( bNF_Greatest_image2p @ C @ A @ D @ B )
= ( ^ [F2: C > A,G2: D > B,R6: C > D > $o,X2: A,Y: B] :
? [X8: C,Y7: D] :
( ( R6 @ X8 @ Y7 )
& ( ( F2 @ X8 )
= X2 )
& ( ( G2 @ Y7 )
= Y ) ) ) ) ).
% image2p_def
thf(fact_6760_small__lazy_H_Ocases,axiom,
! [X: product_prod @ int @ int] :
~ ! [D2: int,I3: int] :
( X
!= ( product_Pair @ int @ int @ D2 @ I3 ) ) ).
% small_lazy'.cases
thf(fact_6761_old_Orec__bool__def,axiom,
! [T: $tType] :
( ( product_rec_bool @ T )
= ( ^ [F12: T,F22: T,X2: $o] : ( the @ T @ ( product_rec_set_bool @ T @ F12 @ F22 @ X2 ) ) ) ) ).
% old.rec_bool_def
thf(fact_6762_lexord__asymmetric,axiom,
! [A: $tType,R5: set @ ( product_prod @ A @ A ),A3: list @ A,B2: list @ A] :
( ( asym @ A @ R5 )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ A3 @ B2 ) @ ( lexord @ A @ R5 ) )
=> ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ B2 @ A3 ) @ ( lexord @ A @ R5 ) ) ) ) ).
% lexord_asymmetric
thf(fact_6763_old_Obool_Osimps_I6_J,axiom,
! [T: $tType,F1: T,F23: T] :
( ( product_rec_bool @ T @ F1 @ F23 @ $false )
= F23 ) ).
% old.bool.simps(6)
thf(fact_6764_old_Obool_Osimps_I5_J,axiom,
! [T: $tType,F1: T,F23: T] :
( ( product_rec_bool @ T @ F1 @ F23 @ $true )
= F1 ) ).
% old.bool.simps(5)
thf(fact_6765_asym_Ocases,axiom,
! [A: $tType,A3: set @ ( product_prod @ A @ A )] :
( ( asym @ A @ A3 )
=> ! [A20: A,B12: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A20 @ B12 ) @ A3 )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B12 @ A20 ) @ A3 ) ) ) ).
% asym.cases
thf(fact_6766_asym_Osimps,axiom,
! [A: $tType] :
( ( asym @ A )
= ( ^ [A6: set @ ( product_prod @ A @ A )] :
? [R6: set @ ( product_prod @ A @ A )] :
( ( A6 = R6 )
& ! [X2: A,Y: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y ) @ R6 )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X2 ) @ R6 ) ) ) ) ) ).
% asym.simps
thf(fact_6767_asym_Ointros,axiom,
! [A: $tType,R5: set @ ( product_prod @ A @ A )] :
( ! [A4: A,B3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ R5 )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B3 @ A4 ) @ R5 ) )
=> ( asym @ A @ R5 ) ) ).
% asym.intros
thf(fact_6768_asymD,axiom,
! [A: $tType,R5: set @ ( product_prod @ A @ A ),X: A,Y3: A] :
( ( asym @ A @ R5 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ R5 )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X ) @ R5 ) ) ) ).
% asymD
thf(fact_6769_asym__iff,axiom,
! [A: $tType] :
( ( asym @ A )
= ( ^ [R6: set @ ( product_prod @ A @ A )] :
! [X2: A,Y: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y ) @ R6 )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X2 ) @ R6 ) ) ) ) ).
% asym_iff
thf(fact_6770_asymp__asym__eq,axiom,
! [A: $tType,R5: set @ ( product_prod @ A @ A )] :
( ( asymp @ A
@ ^ [A6: A,B5: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A6 @ B5 ) @ R5 ) )
= ( asym @ A @ R5 ) ) ).
% asymp_asym_eq
thf(fact_6771_of__int__code_I1_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [K: num] :
( ( ring_1_of_int @ A @ ( neg @ K ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ K ) ) ) ) ).
% of_int_code(1)
thf(fact_6772_less__eq__int__code_I3_J,axiom,
! [L: num] :
~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( neg @ L ) ) ).
% less_eq_int_code(3)
thf(fact_6773_less__eq__int__code_I7_J,axiom,
! [K: num] : ( ord_less_eq @ int @ ( neg @ K ) @ ( zero_zero @ int ) ) ).
% less_eq_int_code(7)
thf(fact_6774_less__int__code_I3_J,axiom,
! [L: num] :
~ ( ord_less @ int @ ( zero_zero @ int ) @ ( neg @ L ) ) ).
% less_int_code(3)
thf(fact_6775_less__int__code_I7_J,axiom,
! [K: num] : ( ord_less @ int @ ( neg @ K ) @ ( zero_zero @ int ) ) ).
% less_int_code(7)
thf(fact_6776_nat__code_I1_J,axiom,
! [K: num] :
( ( nat2 @ ( neg @ K ) )
= ( zero_zero @ nat ) ) ).
% nat_code(1)
thf(fact_6777_plus__int__code_I6_J,axiom,
! [M: num,N: num] :
( ( plus_plus @ int @ ( neg @ M ) @ ( neg @ N ) )
= ( neg @ ( plus_plus @ num @ M @ N ) ) ) ).
% plus_int_code(6)
thf(fact_6778_asymp__greater,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( asymp @ A
@ ^ [X2: A,Y: A] : ( ord_less @ A @ Y @ X2 ) ) ) ).
% asymp_greater
thf(fact_6779_less__eq__int__code_I9_J,axiom,
! [K: num,L: num] :
( ( ord_less_eq @ int @ ( neg @ K ) @ ( neg @ L ) )
= ( ord_less_eq @ num @ L @ K ) ) ).
% less_eq_int_code(9)
thf(fact_6780_less__int__code_I9_J,axiom,
! [K: num,L: num] :
( ( ord_less @ int @ ( neg @ K ) @ ( neg @ L ) )
= ( ord_less @ num @ L @ K ) ) ).
% less_int_code(9)
thf(fact_6781_Int_Osub__code_I4_J,axiom,
! [N: num] :
( ( sub @ one2 @ ( bit0 @ N ) )
= ( neg @ ( bitM @ N ) ) ) ).
% Int.sub_code(4)
thf(fact_6782_Int_Osub__code_I5_J,axiom,
! [N: num] :
( ( sub @ one2 @ ( bit1 @ N ) )
= ( neg @ ( bit0 @ N ) ) ) ).
% Int.sub_code(5)
thf(fact_6783_Int_Osub__code_I1_J,axiom,
( ( sub @ one2 @ one2 )
= ( zero_zero @ int ) ) ).
% Int.sub_code(1)
thf(fact_6784_Int_Osub__def,axiom,
( sub
= ( ^ [M5: num,N5: num] : ( minus_minus @ int @ ( numeral_numeral @ int @ M5 ) @ ( numeral_numeral @ int @ N5 ) ) ) ) ).
% Int.sub_def
thf(fact_6785_minus__int__code_I6_J,axiom,
! [M: num,N: num] :
( ( minus_minus @ int @ ( neg @ M ) @ ( neg @ N ) )
= ( sub @ N @ M ) ) ).
% minus_int_code(6)
thf(fact_6786_Int_Osub__code_I9_J,axiom,
! [M: num,N: num] :
( ( sub @ ( bit0 @ M ) @ ( bit1 @ N ) )
= ( minus_minus @ int @ ( dup @ ( sub @ M @ N ) ) @ ( one_one @ int ) ) ) ).
% Int.sub_code(9)
thf(fact_6787_Int_Osub__code_I8_J,axiom,
! [M: num,N: num] :
( ( sub @ ( bit1 @ M ) @ ( bit0 @ N ) )
= ( plus_plus @ int @ ( dup @ ( sub @ M @ N ) ) @ ( one_one @ int ) ) ) ).
% Int.sub_code(8)
thf(fact_6788_Int_Odup__code_I1_J,axiom,
( ( dup @ ( zero_zero @ int ) )
= ( zero_zero @ int ) ) ).
% Int.dup_code(1)
thf(fact_6789_Int_Odup__def,axiom,
( dup
= ( ^ [K3: int] : ( plus_plus @ int @ K3 @ K3 ) ) ) ).
% Int.dup_def
thf(fact_6790_Int_Odup__code_I3_J,axiom,
! [N: num] :
( ( dup @ ( neg @ N ) )
= ( neg @ ( bit0 @ N ) ) ) ).
% Int.dup_code(3)
thf(fact_6791_Int_Osub__code_I6_J,axiom,
! [M: num,N: num] :
( ( sub @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( dup @ ( sub @ M @ N ) ) ) ).
% Int.sub_code(6)
thf(fact_6792_Int_Osub__code_I7_J,axiom,
! [M: num,N: num] :
( ( sub @ ( bit1 @ M ) @ ( bit1 @ N ) )
= ( dup @ ( sub @ M @ N ) ) ) ).
% Int.sub_code(7)
thf(fact_6793_Int_Osub__code_I2_J,axiom,
! [M: num] :
( ( sub @ ( bit0 @ M ) @ one2 )
= ( pos @ ( bitM @ M ) ) ) ).
% Int.sub_code(2)
thf(fact_6794_Int_Osub__code_I3_J,axiom,
! [M: num] :
( ( sub @ ( bit1 @ M ) @ one2 )
= ( pos @ ( bit0 @ M ) ) ) ).
% Int.sub_code(3)
thf(fact_6795_Int_ONeg__def,axiom,
( neg
= ( ^ [N5: num] : ( uminus_uminus @ int @ ( pos @ N5 ) ) ) ) ).
% Int.Neg_def
thf(fact_6796_uminus__int__code_I2_J,axiom,
! [M: num] :
( ( uminus_uminus @ int @ ( pos @ M ) )
= ( neg @ M ) ) ).
% uminus_int_code(2)
thf(fact_6797_uminus__int__code_I3_J,axiom,
! [M: num] :
( ( uminus_uminus @ int @ ( neg @ M ) )
= ( pos @ M ) ) ).
% uminus_int_code(3)
thf(fact_6798_less__eq__int__code_I8_J,axiom,
! [K: num,L: num] : ( ord_less_eq @ int @ ( neg @ K ) @ ( pos @ L ) ) ).
% less_eq_int_code(8)
thf(fact_6799_less__eq__int__code_I6_J,axiom,
! [K: num,L: num] :
~ ( ord_less_eq @ int @ ( pos @ K ) @ ( neg @ L ) ) ).
% less_eq_int_code(6)
thf(fact_6800_less__int__code_I6_J,axiom,
! [K: num,L: num] :
~ ( ord_less @ int @ ( pos @ K ) @ ( neg @ L ) ) ).
% less_int_code(6)
thf(fact_6801_less__int__code_I8_J,axiom,
! [K: num,L: num] : ( ord_less @ int @ ( neg @ K ) @ ( pos @ L ) ) ).
% less_int_code(8)
thf(fact_6802_Int_OPos__def,axiom,
( pos
= ( numeral_numeral @ int ) ) ).
% Int.Pos_def
thf(fact_6803_one__int__code,axiom,
( ( one_one @ int )
= ( pos @ one2 ) ) ).
% one_int_code
thf(fact_6804_of__int__code_I3_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [K: num] :
( ( ring_1_of_int @ A @ ( pos @ K ) )
= ( numeral_numeral @ A @ K ) ) ) ).
% of_int_code(3)
thf(fact_6805_less__int__code_I4_J,axiom,
! [K: num] :
~ ( ord_less @ int @ ( pos @ K ) @ ( zero_zero @ int ) ) ).
% less_int_code(4)
thf(fact_6806_less__int__code_I2_J,axiom,
! [L: num] : ( ord_less @ int @ ( zero_zero @ int ) @ ( pos @ L ) ) ).
% less_int_code(2)
thf(fact_6807_plus__int__code_I3_J,axiom,
! [M: num,N: num] :
( ( plus_plus @ int @ ( pos @ M ) @ ( pos @ N ) )
= ( pos @ ( plus_plus @ num @ M @ N ) ) ) ).
% plus_int_code(3)
thf(fact_6808_less__eq__int__code_I4_J,axiom,
! [K: num] :
~ ( ord_less_eq @ int @ ( pos @ K ) @ ( zero_zero @ int ) ) ).
% less_eq_int_code(4)
thf(fact_6809_less__eq__int__code_I2_J,axiom,
! [L: num] : ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( pos @ L ) ) ).
% less_eq_int_code(2)
thf(fact_6810_times__int__code_I3_J,axiom,
! [M: num,N: num] :
( ( times_times @ int @ ( pos @ M ) @ ( pos @ N ) )
= ( pos @ ( times_times @ num @ M @ N ) ) ) ).
% times_int_code(3)
thf(fact_6811_nat__code_I3_J,axiom,
! [K: num] :
( ( nat2 @ ( pos @ K ) )
= ( nat_of_num @ K ) ) ).
% nat_code(3)
thf(fact_6812_less__eq__int__code_I5_J,axiom,
! [K: num,L: num] :
( ( ord_less_eq @ int @ ( pos @ K ) @ ( pos @ L ) )
= ( ord_less_eq @ num @ K @ L ) ) ).
% less_eq_int_code(5)
thf(fact_6813_less__int__code_I5_J,axiom,
! [K: num,L: num] :
( ( ord_less @ int @ ( pos @ K ) @ ( pos @ L ) )
= ( ord_less @ num @ K @ L ) ) ).
% less_int_code(5)
thf(fact_6814_minus__int__code_I3_J,axiom,
! [M: num,N: num] :
( ( minus_minus @ int @ ( pos @ M ) @ ( pos @ N ) )
= ( sub @ M @ N ) ) ).
% minus_int_code(3)
thf(fact_6815_Int_Odup__code_I2_J,axiom,
! [N: num] :
( ( dup @ ( pos @ N ) )
= ( pos @ ( bit0 @ N ) ) ) ).
% Int.dup_code(2)
thf(fact_6816_minus__int__code_I5_J,axiom,
! [M: num,N: num] :
( ( minus_minus @ int @ ( neg @ M ) @ ( pos @ N ) )
= ( neg @ ( plus_plus @ num @ M @ N ) ) ) ).
% minus_int_code(5)
thf(fact_6817_minus__int__code_I4_J,axiom,
! [M: num,N: num] :
( ( minus_minus @ int @ ( pos @ M ) @ ( neg @ N ) )
= ( pos @ ( plus_plus @ num @ M @ N ) ) ) ).
% minus_int_code(4)
thf(fact_6818_times__int__code_I4_J,axiom,
! [M: num,N: num] :
( ( times_times @ int @ ( pos @ M ) @ ( neg @ N ) )
= ( neg @ ( times_times @ num @ M @ N ) ) ) ).
% times_int_code(4)
thf(fact_6819_times__int__code_I5_J,axiom,
! [M: num,N: num] :
( ( times_times @ int @ ( neg @ M ) @ ( pos @ N ) )
= ( neg @ ( times_times @ num @ M @ N ) ) ) ).
% times_int_code(5)
thf(fact_6820_times__int__code_I6_J,axiom,
! [M: num,N: num] :
( ( times_times @ int @ ( neg @ M ) @ ( neg @ N ) )
= ( pos @ ( times_times @ num @ M @ N ) ) ) ).
% times_int_code(6)
thf(fact_6821_plus__int__code_I5_J,axiom,
! [M: num,N: num] :
( ( plus_plus @ int @ ( neg @ M ) @ ( pos @ N ) )
= ( sub @ N @ M ) ) ).
% plus_int_code(5)
thf(fact_6822_plus__int__code_I4_J,axiom,
! [M: num,N: num] :
( ( plus_plus @ int @ ( pos @ M ) @ ( neg @ N ) )
= ( sub @ M @ N ) ) ).
% plus_int_code(4)
thf(fact_6823_rtranclp__imp__Sup__relpowp,axiom,
! [A: $tType,P: A > A > $o,X: A,Y3: A] :
( ( transitive_rtranclp @ A @ P @ X @ Y3 )
=> ( complete_Sup_Sup @ ( A > A > $o )
@ ( image2 @ nat @ ( A > A > $o )
@ ^ [N5: nat] : ( compow @ ( A > A > $o ) @ N5 @ P )
@ ( top_top @ ( set @ nat ) ) )
@ X
@ Y3 ) ) ).
% rtranclp_imp_Sup_relpowp
thf(fact_6824_rtranclp__is__Sup__relpowp,axiom,
! [A: $tType] :
( ( transitive_rtranclp @ A )
= ( ^ [P3: A > A > $o] :
( complete_Sup_Sup @ ( A > A > $o )
@ ( image2 @ nat @ ( A > A > $o )
@ ^ [N5: nat] : ( compow @ ( A > A > $o ) @ N5 @ P3 )
@ ( top_top @ ( set @ nat ) ) ) ) ) ) ).
% rtranclp_is_Sup_relpowp
thf(fact_6825_rtranclp__induct2,axiom,
! [A: $tType,B: $tType,R: ( product_prod @ A @ B ) > ( product_prod @ A @ B ) > $o,Ax: A,Ay: B,Bx: A,By: B,P: A > B > $o] :
( ( transitive_rtranclp @ ( product_prod @ A @ B ) @ R @ ( product_Pair @ A @ B @ Ax @ Ay ) @ ( product_Pair @ A @ B @ Bx @ By ) )
=> ( ( P @ Ax @ Ay )
=> ( ! [A4: A,B3: B,Aa2: A,Ba: B] :
( ( transitive_rtranclp @ ( product_prod @ A @ B ) @ R @ ( product_Pair @ A @ B @ Ax @ Ay ) @ ( product_Pair @ A @ B @ A4 @ B3 ) )
=> ( ( R @ ( product_Pair @ A @ B @ A4 @ B3 ) @ ( product_Pair @ A @ B @ Aa2 @ Ba ) )
=> ( ( P @ A4 @ B3 )
=> ( P @ Aa2 @ Ba ) ) ) )
=> ( P @ Bx @ By ) ) ) ) ).
% rtranclp_induct2
thf(fact_6826_converse__rtranclpE2,axiom,
! [A: $tType,B: $tType,R: ( product_prod @ A @ B ) > ( product_prod @ A @ B ) > $o,Xa: A,Xb: B,Za2: A,Zb: B] :
( ( transitive_rtranclp @ ( product_prod @ A @ B ) @ R @ ( product_Pair @ A @ B @ Xa @ Xb ) @ ( product_Pair @ A @ B @ Za2 @ Zb ) )
=> ( ( ( product_Pair @ A @ B @ Xa @ Xb )
!= ( product_Pair @ A @ B @ Za2 @ Zb ) )
=> ~ ! [A4: A,B3: B] :
( ( R @ ( product_Pair @ A @ B @ Xa @ Xb ) @ ( product_Pair @ A @ B @ A4 @ B3 ) )
=> ~ ( transitive_rtranclp @ ( product_prod @ A @ B ) @ R @ ( product_Pair @ A @ B @ A4 @ B3 ) @ ( product_Pair @ A @ B @ Za2 @ Zb ) ) ) ) ) ).
% converse_rtranclpE2
thf(fact_6827_converse__rtranclp__induct2,axiom,
! [A: $tType,B: $tType,R: ( product_prod @ A @ B ) > ( product_prod @ A @ B ) > $o,Ax: A,Ay: B,Bx: A,By: B,P: A > B > $o] :
( ( transitive_rtranclp @ ( product_prod @ A @ B ) @ R @ ( product_Pair @ A @ B @ Ax @ Ay ) @ ( product_Pair @ A @ B @ Bx @ By ) )
=> ( ( P @ Bx @ By )
=> ( ! [A4: A,B3: B,Aa2: A,Ba: B] :
( ( R @ ( product_Pair @ A @ B @ A4 @ B3 ) @ ( product_Pair @ A @ B @ Aa2 @ Ba ) )
=> ( ( transitive_rtranclp @ ( product_prod @ A @ B ) @ R @ ( product_Pair @ A @ B @ Aa2 @ Ba ) @ ( product_Pair @ A @ B @ Bx @ By ) )
=> ( ( P @ Aa2 @ Ba )
=> ( P @ A4 @ B3 ) ) ) )
=> ( P @ Ax @ Ay ) ) ) ) ).
% converse_rtranclp_induct2
thf(fact_6828_Transitive__Closure_Ortranclp__rtrancl__eq,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( transitive_rtranclp @ A
@ ^ [X2: A,Y: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y ) @ R ) )
= ( ^ [X2: A,Y: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y ) @ ( transitive_rtrancl @ A @ R ) ) ) ) ).
% Transitive_Closure.rtranclp_rtrancl_eq
thf(fact_6829_rtranclp__r__diff__Id,axiom,
! [A: $tType,R: A > A > $o] :
( ( transitive_rtranclp @ A
@ ( inf_inf @ ( A > A > $o ) @ R
@ ^ [X2: A,Y: A] : X2 != Y ) )
= ( transitive_rtranclp @ A @ R ) ) ).
% rtranclp_r_diff_Id
thf(fact_6830_Enum_Ortranclp__rtrancl__eq,axiom,
! [A: $tType] :
( ( transitive_rtranclp @ A )
= ( ^ [R4: A > A > $o,X2: A,Y: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y ) @ ( transitive_rtrancl @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ R4 ) ) ) ) ) ) ).
% Enum.rtranclp_rtrancl_eq
thf(fact_6831_rtrancl__def,axiom,
! [A: $tType] :
( ( transitive_rtrancl @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ( transitive_rtranclp @ A
@ ^ [X2: A,Y: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y ) @ R4 ) ) ) ) ) ) ).
% rtrancl_def
thf(fact_6832_rtranclp__def,axiom,
! [A: $tType] :
( ( transitive_rtranclp @ A )
= ( ^ [R4: A > A > $o] :
( complete_lattice_lfp @ ( A > A > $o )
@ ^ [P5: A > A > $o,X12: A,X24: A] :
( ? [A6: A] :
( ( X12 = A6 )
& ( X24 = A6 ) )
| ? [A6: A,B5: A,C4: A] :
( ( X12 = A6 )
& ( X24 = C4 )
& ( P5 @ A6 @ B5 )
& ( R4 @ B5 @ C4 ) ) ) ) ) ) ).
% rtranclp_def
thf(fact_6833_is__none__bind,axiom,
! [A: $tType,B: $tType,F: option @ B,G: B > ( option @ A )] :
( ( is_none @ A @ ( bind2 @ B @ A @ F @ G ) )
= ( ( is_none @ B @ F )
| ( is_none @ A @ ( G @ ( the2 @ B @ F ) ) ) ) ) ).
% is_none_bind
thf(fact_6834_prod__list_Ocomm__monoid__list__axioms,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ( groups1828464146339083142d_list @ A @ ( times_times @ A ) @ ( one_one @ A ) ) ) ).
% prod_list.comm_monoid_list_axioms
thf(fact_6835_is__none__code_I1_J,axiom,
! [A: $tType] : ( is_none @ A @ ( none @ A ) ) ).
% is_none_code(1)
thf(fact_6836_is__none__code_I2_J,axiom,
! [B: $tType,X: B] :
~ ( is_none @ B @ ( some @ B @ X ) ) ).
% is_none_code(2)
thf(fact_6837_is__none__map__option,axiom,
! [A: $tType,B: $tType,F: B > A,X: option @ B] :
( ( is_none @ A @ ( map_option @ B @ A @ F @ X ) )
= ( is_none @ B @ X ) ) ).
% is_none_map_option
thf(fact_6838_is__none__simps_I2_J,axiom,
! [B: $tType,X: B] :
~ ( is_none @ B @ ( some @ B @ X ) ) ).
% is_none_simps(2)
thf(fact_6839_Option_Ois__none__def,axiom,
! [A: $tType] :
( ( is_none @ A )
= ( ^ [X2: option @ A] :
( X2
= ( none @ A ) ) ) ) ).
% Option.is_none_def
thf(fact_6840_is__none__simps_I1_J,axiom,
! [A: $tType] : ( is_none @ A @ ( none @ A ) ) ).
% is_none_simps(1)
thf(fact_6841_rel__optionI,axiom,
! [A: $tType,B: $tType,X: option @ A,Y3: option @ B,P: A > B > $o] :
( ( ( is_none @ A @ X )
= ( is_none @ B @ Y3 ) )
=> ( ( ~ ( is_none @ A @ X )
=> ( ~ ( is_none @ B @ Y3 )
=> ( P @ ( the2 @ A @ X ) @ ( the2 @ B @ Y3 ) ) ) )
=> ( rel_option @ A @ B @ P @ X @ Y3 ) ) ) ).
% rel_optionI
thf(fact_6842_rel__option__unfold,axiom,
! [B: $tType,A: $tType] :
( ( rel_option @ A @ B )
= ( ^ [R6: A > B > $o,X2: option @ A,Y: option @ B] :
( ( ( is_none @ A @ X2 )
= ( is_none @ B @ Y ) )
& ( ~ ( is_none @ A @ X2 )
=> ( ~ ( is_none @ B @ Y )
=> ( R6 @ ( the2 @ A @ X2 ) @ ( the2 @ B @ Y ) ) ) ) ) ) ) ).
% rel_option_unfold
thf(fact_6843_the__map__option,axiom,
! [B: $tType,A: $tType,X: option @ A,F: A > B] :
( ~ ( is_none @ A @ X )
=> ( ( the2 @ B @ ( map_option @ A @ B @ F @ X ) )
= ( F @ ( the2 @ A @ X ) ) ) ) ).
% the_map_option
thf(fact_6844_sum__list_Ocomm__monoid__list__axioms,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ( groups1828464146339083142d_list @ A @ ( plus_plus @ A ) @ ( zero_zero @ A ) ) ) ).
% sum_list.comm_monoid_list_axioms
thf(fact_6845_group_Oaxioms_I2_J,axiom,
! [A: $tType,F: A > A > A,Z: A,Inverse: A > A] :
( ( group @ A @ F @ Z @ Inverse )
=> ( group_axioms @ A @ F @ Z @ Inverse ) ) ).
% group.axioms(2)
thf(fact_6846_strict__mono__inv,axiom,
! [A: $tType,B: $tType] :
( ( ( linorder @ B )
& ( linorder @ A ) )
=> ! [F: A > B,G: B > A] :
( ( order_strict_mono @ A @ B @ F )
=> ( ( ( image2 @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ B ) ) )
=> ( ! [X4: A] :
( ( G @ ( F @ X4 ) )
= X4 )
=> ( order_strict_mono @ B @ A @ G ) ) ) ) ) ).
% strict_mono_inv
thf(fact_6847_strict__mono__imp__increasing,axiom,
! [F: nat > nat,N: nat] :
( ( order_strict_mono @ nat @ nat @ F )
=> ( ord_less_eq @ nat @ N @ ( F @ N ) ) ) ).
% strict_mono_imp_increasing
thf(fact_6848_strict__mono__imp__inj__on,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( order @ B ) )
=> ! [F: A > B,A5: set @ A] :
( ( order_strict_mono @ A @ B @ F )
=> ( inj_on @ A @ B @ F @ A5 ) ) ) ).
% strict_mono_imp_inj_on
thf(fact_6849_group__axioms__def,axiom,
! [A: $tType] :
( ( group_axioms @ A )
= ( ^ [F2: A > A > A,Z3: A,Inverse2: A > A] :
( ! [A6: A] :
( ( F2 @ Z3 @ A6 )
= A6 )
& ! [A6: A] :
( ( F2 @ ( Inverse2 @ A6 ) @ A6 )
= Z3 ) ) ) ) ).
% group_axioms_def
thf(fact_6850_group__axioms_Ointro,axiom,
! [A: $tType,F: A > A > A,Z: A,Inverse: A > A] :
( ! [A4: A] :
( ( F @ Z @ A4 )
= A4 )
=> ( ! [A4: A] :
( ( F @ ( Inverse @ A4 ) @ A4 )
= Z )
=> ( group_axioms @ A @ F @ Z @ Inverse ) ) ) ).
% group_axioms.intro
thf(fact_6851_group_Ointro,axiom,
! [A: $tType,F: A > A > A,Z: A,Inverse: A > A] :
( ( semigroup @ A @ F )
=> ( ( group_axioms @ A @ F @ Z @ Inverse )
=> ( group @ A @ F @ Z @ Inverse ) ) ) ).
% group.intro
thf(fact_6852_group__def,axiom,
! [A: $tType] :
( ( group @ A )
= ( ^ [F2: A > A > A,Z3: A,Inverse2: A > A] :
( ( semigroup @ A @ F2 )
& ( group_axioms @ A @ F2 @ Z3 @ Inverse2 ) ) ) ) ).
% group_def
thf(fact_6853_mult_Osemigroup__axioms,axiom,
! [A: $tType] :
( ( semigroup_mult @ A )
=> ( semigroup @ A @ ( times_times @ A ) ) ) ).
% mult.semigroup_axioms
thf(fact_6854_gcd_Osemigroup__axioms,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ( semigroup @ A @ ( gcd_gcd @ A ) ) ) ).
% gcd.semigroup_axioms
thf(fact_6855_semigroup_Oassoc,axiom,
! [A: $tType,F: A > A > A,A3: A,B2: A,C2: A] :
( ( semigroup @ A @ F )
=> ( ( F @ ( F @ A3 @ B2 ) @ C2 )
= ( F @ A3 @ ( F @ B2 @ C2 ) ) ) ) ).
% semigroup.assoc
thf(fact_6856_semigroup_Ointro,axiom,
! [A: $tType,F: A > A > A] :
( ! [A4: A,B3: A,C3: A] :
( ( F @ ( F @ A4 @ B3 ) @ C3 )
= ( F @ A4 @ ( F @ B3 @ C3 ) ) )
=> ( semigroup @ A @ F ) ) ).
% semigroup.intro
thf(fact_6857_semigroup__def,axiom,
! [A: $tType] :
( ( semigroup @ A )
= ( ^ [F2: A > A > A] :
! [A6: A,B5: A,C4: A] :
( ( F2 @ ( F2 @ A6 @ B5 ) @ C4 )
= ( F2 @ A6 @ ( F2 @ B5 @ C4 ) ) ) ) ) ).
% semigroup_def
thf(fact_6858_group_Oaxioms_I1_J,axiom,
! [A: $tType,F: A > A > A,Z: A,Inverse: A > A] :
( ( group @ A @ F @ Z @ Inverse )
=> ( semigroup @ A @ F ) ) ).
% group.axioms(1)
thf(fact_6859_lcm_Osemigroup__axioms,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ( semigroup @ A @ ( gcd_lcm @ A ) ) ) ).
% lcm.semigroup_axioms
thf(fact_6860_add_Osemigroup__axioms,axiom,
! [A: $tType] :
( ( semigroup_add @ A )
=> ( semigroup @ A @ ( plus_plus @ A ) ) ) ).
% add.semigroup_axioms
thf(fact_6861_monoid_Oaxioms_I1_J,axiom,
! [A: $tType,F: A > A > A,Z: A] :
( ( monoid @ A @ F @ Z )
=> ( semigroup @ A @ F ) ) ).
% monoid.axioms(1)
thf(fact_6862_monoid_Ointro,axiom,
! [A: $tType,F: A > A > A,Z: A] :
( ( semigroup @ A @ F )
=> ( ( monoid_axioms @ A @ F @ Z )
=> ( monoid @ A @ F @ Z ) ) ) ).
% monoid.intro
thf(fact_6863_monoid__def,axiom,
! [A: $tType] :
( ( monoid @ A )
= ( ^ [F2: A > A > A,Z3: A] :
( ( semigroup @ A @ F2 )
& ( monoid_axioms @ A @ F2 @ Z3 ) ) ) ) ).
% monoid_def
thf(fact_6864_monoid__axioms__def,axiom,
! [A: $tType] :
( ( monoid_axioms @ A )
= ( ^ [F2: A > A > A,Z3: A] :
( ! [A6: A] :
( ( F2 @ Z3 @ A6 )
= A6 )
& ! [A6: A] :
( ( F2 @ A6 @ Z3 )
= A6 ) ) ) ) ).
% monoid_axioms_def
thf(fact_6865_monoid__axioms_Ointro,axiom,
! [A: $tType,F: A > A > A,Z: A] :
( ! [A4: A] :
( ( F @ Z @ A4 )
= A4 )
=> ( ! [A4: A] :
( ( F @ A4 @ Z )
= A4 )
=> ( monoid_axioms @ A @ F @ Z ) ) ) ).
% monoid_axioms.intro
thf(fact_6866_monoid_Oaxioms_I2_J,axiom,
! [A: $tType,F: A > A > A,Z: A] :
( ( monoid @ A @ F @ Z )
=> ( monoid_axioms @ A @ F @ Z ) ) ).
% monoid.axioms(2)
thf(fact_6867_rec__natural__def,axiom,
! [T: $tType] :
( ( code_rec_natural @ T )
= ( ^ [F12: T,F22: code_natural > T > T,X2: code_natural] : ( the @ T @ ( code_rec_set_natural @ T @ F12 @ F22 @ X2 ) ) ) ) ).
% rec_natural_def
thf(fact_6868_dsum__Sigma,axiom,
! [B: $tType,A: $tType,A5: set @ ( set @ ( old_node @ A @ B ) ),B4: set @ ( set @ ( old_node @ A @ B ) ),C6: set @ ( set @ ( old_node @ A @ B ) ),D5: set @ ( set @ ( old_node @ A @ B ) )] :
( ord_less_eq @ ( set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) )
@ ( old_dsum @ A @ B
@ ( product_Sigma @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ A5
@ ^ [Uu: set @ ( old_node @ A @ B )] : B4 )
@ ( product_Sigma @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ C6
@ ^ [Uu: set @ ( old_node @ A @ B )] : D5 ) )
@ ( product_Sigma @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ ( old_usum @ A @ B @ A5 @ C6 )
@ ^ [Uu: set @ ( old_node @ A @ B )] : ( old_usum @ A @ B @ B4 @ D5 ) ) ) ).
% dsum_Sigma
thf(fact_6869_dsum__subset__Sigma,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ),A5: set @ ( set @ ( old_node @ A @ B ) ),B4: set @ ( set @ ( old_node @ A @ B ) ),S: set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ),C6: set @ ( set @ ( old_node @ A @ B ) ),D5: set @ ( set @ ( old_node @ A @ B ) )] :
( ( ord_less_eq @ ( set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) ) @ R
@ ( product_Sigma @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ A5
@ ^ [Uu: set @ ( old_node @ A @ B )] : B4 ) )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) ) @ S
@ ( product_Sigma @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ C6
@ ^ [Uu: set @ ( old_node @ A @ B )] : D5 ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) ) @ ( old_dsum @ A @ B @ R @ S )
@ ( product_Sigma @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ ( old_usum @ A @ B @ A5 @ C6 )
@ ^ [Uu: set @ ( old_node @ A @ B )] : ( old_usum @ A @ B @ B4 @ D5 ) ) ) ) ) ).
% dsum_subset_Sigma
thf(fact_6870_case__natural__def,axiom,
! [T: $tType] :
( ( code_case_natural @ T )
= ( ^ [F12: T,F22: code_natural > T] :
( code_rec_natural @ T @ F12
@ ^ [X12: code_natural,X24: T] : ( F22 @ X12 ) ) ) ) ).
% case_natural_def
thf(fact_6871_dprod__subset__Sigma,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ),A5: set @ ( set @ ( old_node @ A @ B ) ),B4: set @ ( set @ ( old_node @ A @ B ) ),S: set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ),C6: set @ ( set @ ( old_node @ A @ B ) ),D5: set @ ( set @ ( old_node @ A @ B ) )] :
( ( ord_less_eq @ ( set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) ) @ R
@ ( product_Sigma @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ A5
@ ^ [Uu: set @ ( old_node @ A @ B )] : B4 ) )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) ) @ S
@ ( product_Sigma @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ C6
@ ^ [Uu: set @ ( old_node @ A @ B )] : D5 ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) ) @ ( old_dprod @ A @ B @ R @ S )
@ ( product_Sigma @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ ( old_uprod @ A @ B @ A5 @ C6 )
@ ^ [Uu: set @ ( old_node @ A @ B )] : ( old_uprod @ A @ B @ B4 @ D5 ) ) ) ) ) ).
% dprod_subset_Sigma
thf(fact_6872_uprod__def,axiom,
! [B: $tType,A: $tType] :
( ( old_uprod @ A @ B )
= ( ^ [A8: set @ ( set @ ( old_node @ A @ B ) ),B7: set @ ( set @ ( old_node @ A @ B ) )] :
( complete_Sup_Sup @ ( set @ ( set @ ( old_node @ A @ B ) ) )
@ ( image2 @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( set @ ( old_node @ A @ B ) ) )
@ ^ [X2: set @ ( old_node @ A @ B )] :
( complete_Sup_Sup @ ( set @ ( set @ ( old_node @ A @ B ) ) )
@ ( image2 @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( set @ ( old_node @ A @ B ) ) )
@ ^ [Y: set @ ( old_node @ A @ B )] : ( insert2 @ ( set @ ( old_node @ A @ B ) ) @ ( old_Scons @ A @ B @ X2 @ Y ) @ ( bot_bot @ ( set @ ( set @ ( old_node @ A @ B ) ) ) ) )
@ B7 ) )
@ A8 ) ) ) ) ).
% uprod_def
thf(fact_6873_dprod__subset__Sigma2,axiom,
! [B: $tType,A: $tType,A5: set @ ( set @ ( old_node @ A @ B ) ),B4: ( set @ ( old_node @ A @ B ) ) > ( set @ ( set @ ( old_node @ A @ B ) ) ),C6: set @ ( set @ ( old_node @ A @ B ) ),D5: ( set @ ( old_node @ A @ B ) ) > ( set @ ( set @ ( old_node @ A @ B ) ) )] :
( ord_less_eq @ ( set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) ) @ ( old_dprod @ A @ B @ ( product_Sigma @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ A5 @ B4 ) @ ( product_Sigma @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ C6 @ D5 ) )
@ ( product_Sigma @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ ( old_uprod @ A @ B @ A5 @ C6 )
@ ( old_Split @ A @ B @ ( set @ ( set @ ( old_node @ A @ B ) ) )
@ ^ [X2: set @ ( old_node @ A @ B ),Y: set @ ( old_node @ A @ B )] : ( old_uprod @ A @ B @ ( B4 @ X2 ) @ ( D5 @ Y ) ) ) ) ) ).
% dprod_subset_Sigma2
thf(fact_6874_dprod__Sigma,axiom,
! [B: $tType,A: $tType,A5: set @ ( set @ ( old_node @ A @ B ) ),B4: set @ ( set @ ( old_node @ A @ B ) ),C6: set @ ( set @ ( old_node @ A @ B ) ),D5: set @ ( set @ ( old_node @ A @ B ) )] :
( ord_less_eq @ ( set @ ( product_prod @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) ) )
@ ( old_dprod @ A @ B
@ ( product_Sigma @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ A5
@ ^ [Uu: set @ ( old_node @ A @ B )] : B4 )
@ ( product_Sigma @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ C6
@ ^ [Uu: set @ ( old_node @ A @ B )] : D5 ) )
@ ( product_Sigma @ ( set @ ( old_node @ A @ B ) ) @ ( set @ ( old_node @ A @ B ) ) @ ( old_uprod @ A @ B @ A5 @ C6 )
@ ^ [Uu: set @ ( old_node @ A @ B )] : ( old_uprod @ A @ B @ B4 @ D5 ) ) ) ).
% dprod_Sigma
thf(fact_6875_Abs__int__inject,axiom,
! [X: set @ ( product_prod @ nat @ nat ),Y3: set @ ( product_prod @ nat @ nat )] :
( ( member @ ( set @ ( product_prod @ nat @ nat ) ) @ X
@ ( collect @ ( set @ ( product_prod @ nat @ nat ) )
@ ^ [C4: set @ ( product_prod @ nat @ nat )] :
? [X2: product_prod @ nat @ nat] :
( ( intrel @ X2 @ X2 )
& ( C4
= ( collect @ ( product_prod @ nat @ nat ) @ ( intrel @ X2 ) ) ) ) ) )
=> ( ( member @ ( set @ ( product_prod @ nat @ nat ) ) @ Y3
@ ( collect @ ( set @ ( product_prod @ nat @ nat ) )
@ ^ [C4: set @ ( product_prod @ nat @ nat )] :
? [X2: product_prod @ nat @ nat] :
( ( intrel @ X2 @ X2 )
& ( C4
= ( collect @ ( product_prod @ nat @ nat ) @ ( intrel @ X2 ) ) ) ) ) )
=> ( ( ( abs_int @ X )
= ( abs_int @ Y3 ) )
= ( X = Y3 ) ) ) ) ).
% Abs_int_inject
thf(fact_6876_Abs__int__induct,axiom,
! [P: int > $o,X: int] :
( ! [Y4: set @ ( product_prod @ nat @ nat )] :
( ( member @ ( set @ ( product_prod @ nat @ nat ) ) @ Y4
@ ( collect @ ( set @ ( product_prod @ nat @ nat ) )
@ ^ [C4: set @ ( product_prod @ nat @ nat )] :
? [X2: product_prod @ nat @ nat] :
( ( intrel @ X2 @ X2 )
& ( C4
= ( collect @ ( product_prod @ nat @ nat ) @ ( intrel @ X2 ) ) ) ) ) )
=> ( P @ ( abs_int @ Y4 ) ) )
=> ( P @ X ) ) ).
% Abs_int_induct
thf(fact_6877_Abs__int__cases,axiom,
! [X: int] :
~ ! [Y4: set @ ( product_prod @ nat @ nat )] :
( ( X
= ( abs_int @ Y4 ) )
=> ~ ( member @ ( set @ ( product_prod @ nat @ nat ) ) @ Y4
@ ( collect @ ( set @ ( product_prod @ nat @ nat ) )
@ ^ [C4: set @ ( product_prod @ nat @ nat )] :
? [X2: product_prod @ nat @ nat] :
( ( intrel @ X2 @ X2 )
& ( C4
= ( collect @ ( product_prod @ nat @ nat ) @ ( intrel @ X2 ) ) ) ) ) ) ) ).
% Abs_int_cases
thf(fact_6878_Abs__Integ__def,axiom,
( abs_Integ
= ( quot_abs @ ( product_prod @ nat @ nat ) @ int @ intrel @ abs_int ) ) ).
% Abs_Integ_def
thf(fact_6879_Abs__int__inverse,axiom,
! [Y3: set @ ( product_prod @ nat @ nat )] :
( ( member @ ( set @ ( product_prod @ nat @ nat ) ) @ Y3
@ ( collect @ ( set @ ( product_prod @ nat @ nat ) )
@ ^ [C4: set @ ( product_prod @ nat @ nat )] :
? [X2: product_prod @ nat @ nat] :
( ( intrel @ X2 @ X2 )
& ( C4
= ( collect @ ( product_prod @ nat @ nat ) @ ( intrel @ X2 ) ) ) ) ) )
=> ( ( rep_int @ ( abs_int @ Y3 ) )
= Y3 ) ) ).
% Abs_int_inverse
thf(fact_6880_Rep__int__inject,axiom,
! [X: int,Y3: int] :
( ( ( rep_int @ X )
= ( rep_int @ Y3 ) )
= ( X = Y3 ) ) ).
% Rep_int_inject
thf(fact_6881_Rep__int__inverse,axiom,
! [X: int] :
( ( abs_int @ ( rep_int @ X ) )
= X ) ).
% Rep_int_inverse
thf(fact_6882_Rep__int__induct,axiom,
! [Y3: set @ ( product_prod @ nat @ nat ),P: ( set @ ( product_prod @ nat @ nat ) ) > $o] :
( ( member @ ( set @ ( product_prod @ nat @ nat ) ) @ Y3
@ ( collect @ ( set @ ( product_prod @ nat @ nat ) )
@ ^ [C4: set @ ( product_prod @ nat @ nat )] :
? [X2: product_prod @ nat @ nat] :
( ( intrel @ X2 @ X2 )
& ( C4
= ( collect @ ( product_prod @ nat @ nat ) @ ( intrel @ X2 ) ) ) ) ) )
=> ( ! [X4: int] : ( P @ ( rep_int @ X4 ) )
=> ( P @ Y3 ) ) ) ).
% Rep_int_induct
thf(fact_6883_Rep__int__cases,axiom,
! [Y3: set @ ( product_prod @ nat @ nat )] :
( ( member @ ( set @ ( product_prod @ nat @ nat ) ) @ Y3
@ ( collect @ ( set @ ( product_prod @ nat @ nat ) )
@ ^ [C4: set @ ( product_prod @ nat @ nat )] :
? [X2: product_prod @ nat @ nat] :
( ( intrel @ X2 @ X2 )
& ( C4
= ( collect @ ( product_prod @ nat @ nat ) @ ( intrel @ X2 ) ) ) ) ) )
=> ~ ! [X4: int] :
( Y3
!= ( rep_int @ X4 ) ) ) ).
% Rep_int_cases
thf(fact_6884_Rep__int,axiom,
! [X: int] :
( member @ ( set @ ( product_prod @ nat @ nat ) ) @ ( rep_int @ X )
@ ( collect @ ( set @ ( product_prod @ nat @ nat ) )
@ ^ [C4: set @ ( product_prod @ nat @ nat )] :
? [X2: product_prod @ nat @ nat] :
( ( intrel @ X2 @ X2 )
& ( C4
= ( collect @ ( product_prod @ nat @ nat ) @ ( intrel @ X2 ) ) ) ) ) ) ).
% Rep_int
thf(fact_6885_type__definition__int,axiom,
( type_definition @ int @ ( set @ ( product_prod @ nat @ nat ) ) @ rep_int @ abs_int
@ ( collect @ ( set @ ( product_prod @ nat @ nat ) )
@ ^ [C4: set @ ( product_prod @ nat @ nat )] :
? [X2: product_prod @ nat @ nat] :
( ( intrel @ X2 @ X2 )
& ( C4
= ( collect @ ( product_prod @ nat @ nat ) @ ( intrel @ X2 ) ) ) ) ) ) ).
% type_definition_int
thf(fact_6886_Rep__Integ__def,axiom,
( rep_Integ
= ( quot_rep @ int @ ( product_prod @ nat @ nat ) @ rep_int ) ) ).
% Rep_Integ_def
thf(fact_6887_less__eq__literal_Oabs__eq,axiom,
! [Xa: list @ char,X: list @ char] :
( ( bNF_eq_onp @ ( list @ char )
@ ^ [Cs: list @ char] :
! [X2: char] :
( ( member @ char @ X2 @ ( set2 @ char @ Cs ) )
=> ~ ( digit7 @ X2 ) )
@ Xa
@ Xa )
=> ( ( bNF_eq_onp @ ( list @ char )
@ ^ [Cs: list @ char] :
! [X2: char] :
( ( member @ char @ X2 @ ( set2 @ char @ Cs ) )
=> ~ ( digit7 @ X2 ) )
@ X
@ X )
=> ( ( ord_less_eq @ literal @ ( abs_literal @ Xa ) @ ( abs_literal @ X ) )
= ( lexordp_eq @ char
@ ^ [C4: char,D4: char] : ( ord_less @ nat @ ( comm_s6883823935334413003f_char @ nat @ C4 ) @ ( comm_s6883823935334413003f_char @ nat @ D4 ) )
@ Xa
@ X ) ) ) ) ).
% less_eq_literal.abs_eq
thf(fact_6888_fun_Opred__rel,axiom,
! [A: $tType,D: $tType,P: A > $o,X: D > A] :
( ( basic_pred_fun @ D @ A
@ ^ [Uu: D] : $true
@ P
@ X )
= ( bNF_rel_fun @ D @ D @ A @ A
@ ^ [Y6: D,Z2: D] : Y6 = Z2
@ ( bNF_eq_onp @ A @ P )
@ X
@ X ) ) ).
% fun.pred_rel
thf(fact_6889_plus__literal_Orsp,axiom,
( bNF_rel_fun @ ( list @ char ) @ ( list @ char ) @ ( ( list @ char ) > ( list @ char ) ) @ ( ( list @ char ) > ( list @ char ) )
@ ( bNF_eq_onp @ ( list @ char )
@ ^ [Cs: list @ char] :
! [X2: char] :
( ( member @ char @ X2 @ ( set2 @ char @ Cs ) )
=> ~ ( digit7 @ X2 ) ) )
@ ( bNF_rel_fun @ ( list @ char ) @ ( list @ char ) @ ( list @ char ) @ ( list @ char )
@ ( bNF_eq_onp @ ( list @ char )
@ ^ [Cs: list @ char] :
! [X2: char] :
( ( member @ char @ X2 @ ( set2 @ char @ Cs ) )
=> ~ ( digit7 @ X2 ) ) )
@ ( bNF_eq_onp @ ( list @ char )
@ ^ [Cs: list @ char] :
! [X2: char] :
( ( member @ char @ X2 @ ( set2 @ char @ Cs ) )
=> ~ ( digit7 @ X2 ) ) ) )
@ ( append @ char )
@ ( append @ char ) ) ).
% plus_literal.rsp
thf(fact_6890_less__eq__literal_Orsp,axiom,
( bNF_rel_fun @ ( list @ char ) @ ( list @ char ) @ ( ( list @ char ) > $o ) @ ( ( list @ char ) > $o )
@ ( bNF_eq_onp @ ( list @ char )
@ ^ [Cs: list @ char] :
! [X2: char] :
( ( member @ char @ X2 @ ( set2 @ char @ Cs ) )
=> ~ ( digit7 @ X2 ) ) )
@ ( bNF_rel_fun @ ( list @ char ) @ ( list @ char ) @ $o @ $o
@ ( bNF_eq_onp @ ( list @ char )
@ ^ [Cs: list @ char] :
! [X2: char] :
( ( member @ char @ X2 @ ( set2 @ char @ Cs ) )
=> ~ ( digit7 @ X2 ) ) )
@ ^ [Y6: $o,Z2: $o] : Y6 = Z2 )
@ ( lexordp_eq @ char
@ ^ [C4: char,D4: char] : ( ord_less @ nat @ ( comm_s6883823935334413003f_char @ nat @ C4 ) @ ( comm_s6883823935334413003f_char @ nat @ D4 ) ) )
@ ( lexordp_eq @ char
@ ^ [C4: char,D4: char] : ( ord_less @ nat @ ( comm_s6883823935334413003f_char @ nat @ C4 ) @ ( comm_s6883823935334413003f_char @ nat @ D4 ) ) ) ) ).
% less_eq_literal.rsp
thf(fact_6891_asciis__of__literal_Orsp,axiom,
( bNF_rel_fun @ ( list @ char ) @ ( list @ char ) @ ( list @ code_integer ) @ ( list @ code_integer )
@ ( bNF_eq_onp @ ( list @ char )
@ ^ [Cs: list @ char] :
! [X2: char] :
( ( member @ char @ X2 @ ( set2 @ char @ Cs ) )
=> ~ ( digit7 @ X2 ) ) )
@ ^ [Y6: list @ code_integer,Z2: list @ code_integer] : Y6 = Z2
@ ( map @ char @ code_integer @ ( comm_s6883823935334413003f_char @ code_integer ) )
@ ( map @ char @ code_integer @ ( comm_s6883823935334413003f_char @ code_integer ) ) ) ).
% asciis_of_literal.rsp
thf(fact_6892_rel__fun__eq__eq__onp,axiom,
! [B: $tType,A: $tType,P: B > $o] :
( ( bNF_rel_fun @ A @ A @ B @ B
@ ^ [Y6: A,Z2: A] : Y6 = Z2
@ ( bNF_eq_onp @ B @ P ) )
= ( bNF_eq_onp @ ( A > B )
@ ^ [F2: A > B] :
! [X2: A] : ( P @ ( F2 @ X2 ) ) ) ) ).
% rel_fun_eq_eq_onp
thf(fact_6893_size__literal_Orsp,axiom,
( bNF_rel_fun @ ( list @ char ) @ ( list @ char ) @ nat @ nat
@ ( bNF_eq_onp @ ( list @ char )
@ ^ [Cs: list @ char] :
! [X2: char] :
( ( member @ char @ X2 @ ( set2 @ char @ Cs ) )
=> ~ ( digit7 @ X2 ) ) )
@ ^ [Y6: nat,Z2: nat] : Y6 = Z2
@ ( size_size @ ( list @ char ) )
@ ( size_size @ ( list @ char ) ) ) ).
% size_literal.rsp
thf(fact_6894_Literal_Orsp,axiom,
( bNF_rel_fun @ $o @ $o @ ( $o > $o > $o > $o > $o > $o > ( list @ char ) > ( list @ char ) ) @ ( $o > $o > $o > $o > $o > $o > ( list @ char ) > ( list @ char ) )
@ ^ [Y6: $o,Z2: $o] : Y6 = Z2
@ ( bNF_rel_fun @ $o @ $o @ ( $o > $o > $o > $o > $o > ( list @ char ) > ( list @ char ) ) @ ( $o > $o > $o > $o > $o > ( list @ char ) > ( list @ char ) )
@ ^ [Y6: $o,Z2: $o] : Y6 = Z2
@ ( bNF_rel_fun @ $o @ $o @ ( $o > $o > $o > $o > ( list @ char ) > ( list @ char ) ) @ ( $o > $o > $o > $o > ( list @ char ) > ( list @ char ) )
@ ^ [Y6: $o,Z2: $o] : Y6 = Z2
@ ( bNF_rel_fun @ $o @ $o @ ( $o > $o > $o > ( list @ char ) > ( list @ char ) ) @ ( $o > $o > $o > ( list @ char ) > ( list @ char ) )
@ ^ [Y6: $o,Z2: $o] : Y6 = Z2
@ ( bNF_rel_fun @ $o @ $o @ ( $o > $o > ( list @ char ) > ( list @ char ) ) @ ( $o > $o > ( list @ char ) > ( list @ char ) )
@ ^ [Y6: $o,Z2: $o] : Y6 = Z2
@ ( bNF_rel_fun @ $o @ $o @ ( $o > ( list @ char ) > ( list @ char ) ) @ ( $o > ( list @ char ) > ( list @ char ) )
@ ^ [Y6: $o,Z2: $o] : Y6 = Z2
@ ( bNF_rel_fun @ $o @ $o @ ( ( list @ char ) > ( list @ char ) ) @ ( ( list @ char ) > ( list @ char ) )
@ ^ [Y6: $o,Z2: $o] : Y6 = Z2
@ ( bNF_rel_fun @ ( list @ char ) @ ( list @ char ) @ ( list @ char ) @ ( list @ char )
@ ( bNF_eq_onp @ ( list @ char )
@ ^ [Cs: list @ char] :
! [X2: char] :
( ( member @ char @ X2 @ ( set2 @ char @ Cs ) )
=> ~ ( digit7 @ X2 ) ) )
@ ( bNF_eq_onp @ ( list @ char )
@ ^ [Cs: list @ char] :
! [X2: char] :
( ( member @ char @ X2 @ ( set2 @ char @ Cs ) )
=> ~ ( digit7 @ X2 ) ) ) ) ) ) ) ) ) )
@ ^ [B0: $o,B1: $o,B22: $o,B32: $o,B42: $o,B52: $o,B62: $o] : ( cons @ char @ ( char2 @ B0 @ B1 @ B22 @ B32 @ B42 @ B52 @ B62 @ $false ) )
@ ^ [B0: $o,B1: $o,B22: $o,B32: $o,B42: $o,B52: $o,B62: $o] : ( cons @ char @ ( char2 @ B0 @ B1 @ B22 @ B32 @ B42 @ B52 @ B62 @ $false ) ) ) ).
% Literal.rsp
thf(fact_6895_less__literal_Orsp,axiom,
( bNF_rel_fun @ ( list @ char ) @ ( list @ char ) @ ( ( list @ char ) > $o ) @ ( ( list @ char ) > $o )
@ ( bNF_eq_onp @ ( list @ char )
@ ^ [Cs: list @ char] :
! [X2: char] :
( ( member @ char @ X2 @ ( set2 @ char @ Cs ) )
=> ~ ( digit7 @ X2 ) ) )
@ ( bNF_rel_fun @ ( list @ char ) @ ( list @ char ) @ $o @ $o
@ ( bNF_eq_onp @ ( list @ char )
@ ^ [Cs: list @ char] :
! [X2: char] :
( ( member @ char @ X2 @ ( set2 @ char @ Cs ) )
=> ~ ( digit7 @ X2 ) ) )
@ ^ [Y6: $o,Z2: $o] : Y6 = Z2 )
@ ( lexordp2 @ char
@ ^ [C4: char,D4: char] : ( ord_less @ nat @ ( comm_s6883823935334413003f_char @ nat @ C4 ) @ ( comm_s6883823935334413003f_char @ nat @ D4 ) ) )
@ ( lexordp2 @ char
@ ^ [C4: char,D4: char] : ( ord_less @ nat @ ( comm_s6883823935334413003f_char @ nat @ C4 ) @ ( comm_s6883823935334413003f_char @ nat @ D4 ) ) ) ) ).
% less_literal.rsp
thf(fact_6896_Heap_Orel__eq__onp,axiom,
! [A: $tType,P: A > $o] :
( ( heap_Time_rel_Heap @ A @ A @ ( bNF_eq_onp @ A @ P ) )
= ( bNF_eq_onp @ ( heap_Time_Heap @ A ) @ ( heap_Time_pred_Heap @ A @ P ) ) ) ).
% Heap.rel_eq_onp
thf(fact_6897_Heap_Opred__rel,axiom,
! [A: $tType] :
( ( heap_Time_pred_Heap @ A )
= ( ^ [P3: A > $o,X2: heap_Time_Heap @ A] : ( heap_Time_rel_Heap @ A @ A @ ( bNF_eq_onp @ A @ P3 ) @ X2 @ X2 ) ) ) ).
% Heap.pred_rel
thf(fact_6898_plus__literal_Oabs__eq,axiom,
! [Xa: list @ char,X: list @ char] :
( ( bNF_eq_onp @ ( list @ char )
@ ^ [Cs: list @ char] :
! [X2: char] :
( ( member @ char @ X2 @ ( set2 @ char @ Cs ) )
=> ~ ( digit7 @ X2 ) )
@ Xa
@ Xa )
=> ( ( bNF_eq_onp @ ( list @ char )
@ ^ [Cs: list @ char] :
! [X2: char] :
( ( member @ char @ X2 @ ( set2 @ char @ Cs ) )
=> ~ ( digit7 @ X2 ) )
@ X
@ X )
=> ( ( plus_plus @ literal @ ( abs_literal @ Xa ) @ ( abs_literal @ X ) )
= ( abs_literal @ ( append @ char @ Xa @ X ) ) ) ) ) ).
% plus_literal.abs_eq
thf(fact_6899_literal_OAbs__literal__cases,axiom,
! [X: literal] :
~ ! [Y4: list @ char] :
( ( X
= ( abs_literal @ Y4 ) )
=> ~ ( member @ ( list @ char ) @ Y4
@ ( collect @ ( list @ char )
@ ^ [Cs: list @ char] :
! [X2: char] :
( ( member @ char @ X2 @ ( set2 @ char @ Cs ) )
=> ~ ( digit7 @ X2 ) ) ) ) ) ).
% literal.Abs_literal_cases
thf(fact_6900_literal_OAbs__literal__induct,axiom,
! [P: literal > $o,X: literal] :
( ! [Y4: list @ char] :
( ( member @ ( list @ char ) @ Y4
@ ( collect @ ( list @ char )
@ ^ [Cs: list @ char] :
! [X2: char] :
( ( member @ char @ X2 @ ( set2 @ char @ Cs ) )
=> ~ ( digit7 @ X2 ) ) ) )
=> ( P @ ( abs_literal @ Y4 ) ) )
=> ( P @ X ) ) ).
% literal.Abs_literal_induct
thf(fact_6901_literal_OAbs__literal__inject,axiom,
! [X: list @ char,Y3: list @ char] :
( ( member @ ( list @ char ) @ X
@ ( collect @ ( list @ char )
@ ^ [Cs: list @ char] :
! [X2: char] :
( ( member @ char @ X2 @ ( set2 @ char @ Cs ) )
=> ~ ( digit7 @ X2 ) ) ) )
=> ( ( member @ ( list @ char ) @ Y3
@ ( collect @ ( list @ char )
@ ^ [Cs: list @ char] :
! [X2: char] :
( ( member @ char @ X2 @ ( set2 @ char @ Cs ) )
=> ~ ( digit7 @ X2 ) ) ) )
=> ( ( ( abs_literal @ X )
= ( abs_literal @ Y3 ) )
= ( X = Y3 ) ) ) ) ).
% literal.Abs_literal_inject
thf(fact_6902_size__literal_Oabs__eq,axiom,
! [X: list @ char] :
( ( bNF_eq_onp @ ( list @ char )
@ ^ [Cs: list @ char] :
! [X2: char] :
( ( member @ char @ X2 @ ( set2 @ char @ Cs ) )
=> ~ ( digit7 @ X2 ) )
@ X
@ X )
=> ( ( size_size @ literal @ ( abs_literal @ X ) )
= ( size_size @ ( list @ char ) @ X ) ) ) ).
% size_literal.abs_eq
thf(fact_6903_literal_Oexplode,axiom,
! [X: literal] :
( member @ ( list @ char ) @ ( explode @ X )
@ ( collect @ ( list @ char )
@ ^ [Cs: list @ char] :
! [X2: char] :
( ( member @ char @ X2 @ ( set2 @ char @ Cs ) )
=> ~ ( digit7 @ X2 ) ) ) ) ).
% literal.explode
thf(fact_6904_literal_Oexplode__cases,axiom,
! [Y3: list @ char] :
( ( member @ ( list @ char ) @ Y3
@ ( collect @ ( list @ char )
@ ^ [Cs: list @ char] :
! [X2: char] :
( ( member @ char @ X2 @ ( set2 @ char @ Cs ) )
=> ~ ( digit7 @ X2 ) ) ) )
=> ~ ! [X4: literal] :
( Y3
!= ( explode @ X4 ) ) ) ).
% literal.explode_cases
thf(fact_6905_literal_Oexplode__induct,axiom,
! [Y3: list @ char,P: ( list @ char ) > $o] :
( ( member @ ( list @ char ) @ Y3
@ ( collect @ ( list @ char )
@ ^ [Cs: list @ char] :
! [X2: char] :
( ( member @ char @ X2 @ ( set2 @ char @ Cs ) )
=> ~ ( digit7 @ X2 ) ) ) )
=> ( ! [X4: literal] : ( P @ ( explode @ X4 ) )
=> ( P @ Y3 ) ) ) ).
% literal.explode_induct
thf(fact_6906_zero__literal_Orsp,axiom,
( bNF_eq_onp @ ( list @ char )
@ ^ [Cs: list @ char] :
! [X2: char] :
( ( member @ char @ X2 @ ( set2 @ char @ Cs ) )
=> ~ ( digit7 @ X2 ) )
@ ( nil @ char )
@ ( nil @ char ) ) ).
% zero_literal.rsp
thf(fact_6907_eq__onp__def,axiom,
! [A: $tType] :
( ( bNF_eq_onp @ A )
= ( ^ [R6: A > $o,X2: A,Y: A] :
( ( R6 @ X2 )
& ( X2 = Y ) ) ) ) ).
% eq_onp_def
thf(fact_6908_eq__onp__True,axiom,
! [A: $tType] :
( ( bNF_eq_onp @ A
@ ^ [Uu: A] : $true )
= ( ^ [Y6: A,Z2: A] : Y6 = Z2 ) ) ).
% eq_onp_True
thf(fact_6909_eq__onp__live__step,axiom,
! [A: $tType,X: $o,Y3: $o,P: A > $o,A3: A] :
( ( X = Y3 )
=> ( ( ( bNF_eq_onp @ A @ P @ A3 @ A3 )
& X )
= ( ( P @ A3 )
& Y3 ) ) ) ).
% eq_onp_live_step
thf(fact_6910_Quotient__crel__typedef,axiom,
! [B: $tType,A: $tType,P: A > $o,Abs: A > B,Rep: B > A,T5: A > B > $o] :
( ( quotient @ A @ B @ ( bNF_eq_onp @ A @ P ) @ Abs @ Rep @ T5 )
=> ( T5
= ( ^ [X2: A,Y: B] :
( X2
= ( Rep @ Y ) ) ) ) ) ).
% Quotient_crel_typedef
thf(fact_6911_String_OQuotient__literal,axiom,
( quotient @ ( list @ char ) @ literal
@ ( bNF_eq_onp @ ( list @ char )
@ ^ [Cs: list @ char] :
! [X2: char] :
( ( member @ char @ X2 @ ( set2 @ char @ Cs ) )
=> ~ ( digit7 @ X2 ) ) )
@ abs_literal
@ explode
@ cr_literal ) ).
% String.Quotient_literal
thf(fact_6912_Literal_Oabs__eq,axiom,
! [X: list @ char,Xg: $o,Xf: $o,Xe: $o,Xd: $o,Xc: $o,Xb: $o,Xa: $o] :
( ( bNF_eq_onp @ ( list @ char )
@ ^ [Cs: list @ char] :
! [X2: char] :
( ( member @ char @ X2 @ ( set2 @ char @ Cs ) )
=> ~ ( digit7 @ X2 ) )
@ X
@ X )
=> ( ( literal2 @ Xg @ Xf @ Xe @ Xd @ Xc @ Xb @ Xa @ ( abs_literal @ X ) )
= ( abs_literal @ ( cons @ char @ ( char2 @ Xg @ Xf @ Xe @ Xd @ Xc @ Xb @ Xa @ $false ) @ X ) ) ) ) ).
% Literal.abs_eq
thf(fact_6913_option_Opred__rel,axiom,
! [A: $tType] :
( ( pred_option @ A )
= ( ^ [P3: A > $o,X2: option @ A] : ( rel_option @ A @ A @ ( bNF_eq_onp @ A @ P3 ) @ X2 @ X2 ) ) ) ).
% option.pred_rel
thf(fact_6914_option_Orel__eq__onp,axiom,
! [A: $tType,P: A > $o] :
( ( rel_option @ A @ A @ ( bNF_eq_onp @ A @ P ) )
= ( bNF_eq_onp @ ( option @ A ) @ ( pred_option @ A @ P ) ) ) ).
% option.rel_eq_onp
thf(fact_6915_fun_Orel__eq__onp,axiom,
! [D: $tType,A: $tType,P: A > $o] :
( ( bNF_rel_fun @ D @ D @ A @ A
@ ^ [Y6: D,Z2: D] : Y6 = Z2
@ ( 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_6916_typedef__to__Quotient,axiom,
! [A: $tType,B: $tType,Rep: A > B,Abs: B > A,S3: set @ B,T5: B > A > $o] :
( ( type_definition @ A @ B @ Rep @ Abs @ S3 )
=> ( ( T5
= ( ^ [X2: B,Y: A] :
( X2
= ( Rep @ Y ) ) ) )
=> ( quotient @ B @ A
@ ( bNF_eq_onp @ B
@ ^ [X2: B] : ( member @ B @ X2 @ S3 ) )
@ Abs
@ Rep
@ T5 ) ) ) ).
% typedef_to_Quotient
thf(fact_6917_open__typedef__to__Quotient,axiom,
! [A: $tType,B: $tType,Rep: A > B,Abs: B > A,P: B > $o,T5: B > A > $o] :
( ( type_definition @ A @ B @ Rep @ Abs @ ( collect @ B @ P ) )
=> ( ( T5
= ( ^ [X2: B,Y: A] :
( X2
= ( Rep @ Y ) ) ) )
=> ( quotient @ B @ A @ ( bNF_eq_onp @ B @ P ) @ Abs @ Rep @ T5 ) ) ) ).
% open_typedef_to_Quotient
thf(fact_6918_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_6919_rel__fun__eq__onp__rel,axiom,
! [C: $tType,B: $tType,A: $tType,R5: A > $o,S3: B > C > $o] :
( ( bNF_rel_fun @ A @ A @ B @ C @ ( bNF_eq_onp @ A @ R5 ) @ S3 )
= ( ^ [F2: A > B,G2: A > C] :
! [X2: A] :
( ( R5 @ X2 )
=> ( S3 @ ( F2 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ).
% rel_fun_eq_onp_rel
thf(fact_6920_literal_OAbs__literal__inverse,axiom,
! [Y3: list @ char] :
( ( member @ ( list @ char ) @ Y3
@ ( collect @ ( list @ char )
@ ^ [Cs: list @ char] :
! [X2: char] :
( ( member @ char @ X2 @ ( set2 @ char @ Cs ) )
=> ~ ( digit7 @ X2 ) ) ) )
=> ( ( explode @ ( abs_literal @ Y3 ) )
= Y3 ) ) ).
% literal.Abs_literal_inverse
thf(fact_6921_literal_Otype__definition__literal,axiom,
( type_definition @ literal @ ( list @ char ) @ explode @ abs_literal
@ ( collect @ ( list @ char )
@ ^ [Cs: list @ char] :
! [X2: char] :
( ( member @ char @ X2 @ ( set2 @ char @ Cs ) )
=> ~ ( digit7 @ X2 ) ) ) ) ).
% literal.type_definition_literal
thf(fact_6922_less__literal_Oabs__eq,axiom,
! [Xa: list @ char,X: list @ char] :
( ( bNF_eq_onp @ ( list @ char )
@ ^ [Cs: list @ char] :
! [X2: char] :
( ( member @ char @ X2 @ ( set2 @ char @ Cs ) )
=> ~ ( digit7 @ X2 ) )
@ Xa
@ Xa )
=> ( ( bNF_eq_onp @ ( list @ char )
@ ^ [Cs: list @ char] :
! [X2: char] :
( ( member @ char @ X2 @ ( set2 @ char @ Cs ) )
=> ~ ( digit7 @ X2 ) )
@ X
@ X )
=> ( ( ord_less @ literal @ ( abs_literal @ Xa ) @ ( abs_literal @ X ) )
= ( lexordp2 @ char
@ ^ [C4: char,D4: char] : ( ord_less @ nat @ ( comm_s6883823935334413003f_char @ nat @ C4 ) @ ( comm_s6883823935334413003f_char @ nat @ D4 ) )
@ Xa
@ X ) ) ) ) ).
% less_literal.abs_eq
thf(fact_6923_asciis__of__literal_Oabs__eq,axiom,
! [X: list @ char] :
( ( bNF_eq_onp @ ( list @ char )
@ ^ [Cs: list @ char] :
! [X2: char] :
( ( member @ char @ X2 @ ( set2 @ char @ Cs ) )
=> ~ ( digit7 @ X2 ) )
@ X
@ X )
=> ( ( asciis_of_literal @ ( abs_literal @ X ) )
= ( map @ char @ code_integer @ ( comm_s6883823935334413003f_char @ code_integer ) @ X ) ) ) ).
% asciis_of_literal.abs_eq
thf(fact_6924_literal__of__asciis_Orsp,axiom,
( bNF_rel_fun @ ( list @ code_integer ) @ ( list @ code_integer ) @ ( list @ char ) @ ( list @ char )
@ ^ [Y6: list @ code_integer,Z2: list @ code_integer] : Y6 = Z2
@ ( bNF_eq_onp @ ( list @ char )
@ ^ [Cs: list @ char] :
! [X2: char] :
( ( member @ char @ X2 @ ( set2 @ char @ Cs ) )
=> ~ ( digit7 @ X2 ) ) )
@ ( map @ code_integer @ char @ ( comp @ char @ char @ code_integer @ ascii_of @ ( unique5772411509450598832har_of @ code_integer ) ) )
@ ( map @ code_integer @ char @ ( comp @ char @ char @ code_integer @ ascii_of @ ( unique5772411509450598832har_of @ code_integer ) ) ) ) ).
% literal_of_asciis.rsp
thf(fact_6925_implode_Orsp,axiom,
( bNF_rel_fun @ ( list @ char ) @ ( list @ char ) @ ( list @ char ) @ ( list @ char )
@ ^ [Y6: list @ char,Z2: list @ char] : Y6 = Z2
@ ( bNF_eq_onp @ ( list @ char )
@ ^ [Cs: list @ char] :
! [X2: char] :
( ( member @ char @ X2 @ ( set2 @ char @ Cs ) )
=> ~ ( digit7 @ X2 ) ) )
@ ( map @ char @ char @ ascii_of )
@ ( map @ char @ char @ ascii_of ) ) ).
% implode.rsp
thf(fact_6926_literal__of__asciis_Otransfer,axiom,
( bNF_rel_fun @ ( list @ code_integer ) @ ( list @ code_integer ) @ ( list @ char ) @ literal
@ ^ [Y6: list @ code_integer,Z2: list @ code_integer] : Y6 = Z2
@ pcr_literal
@ ( map @ code_integer @ char @ ( comp @ char @ char @ code_integer @ ascii_of @ ( unique5772411509450598832har_of @ code_integer ) ) )
@ literal_of_asciis ) ).
% literal_of_asciis.transfer
thf(fact_6927_String_Oliteral__of__asciis__def,axiom,
( literal_of_asciis
= ( map_fun @ ( list @ code_integer ) @ ( list @ code_integer ) @ ( list @ char ) @ literal @ ( id @ ( list @ code_integer ) ) @ abs_literal @ ( map @ code_integer @ char @ ( comp @ char @ char @ code_integer @ ascii_of @ ( unique5772411509450598832har_of @ code_integer ) ) ) ) ) ).
% String.literal_of_asciis_def
thf(fact_6928_literal__of__asciis_Orep__eq,axiom,
! [X: list @ code_integer] :
( ( explode @ ( literal_of_asciis @ X ) )
= ( map @ code_integer @ char @ ( comp @ char @ char @ code_integer @ ascii_of @ ( unique5772411509450598832har_of @ code_integer ) ) @ X ) ) ).
% literal_of_asciis.rep_eq
thf(fact_6929_literal__of__asciis_Oabs__eq,axiom,
( literal_of_asciis
= ( ^ [X2: list @ code_integer] : ( abs_literal @ ( map @ code_integer @ char @ ( comp @ char @ char @ code_integer @ ascii_of @ ( unique5772411509450598832har_of @ code_integer ) ) @ X2 ) ) ) ) ).
% literal_of_asciis.abs_eq
thf(fact_6930_String_Oliteral__of__asciis__Cons,axiom,
! [K: code_integer,Ks: list @ code_integer] :
( ( literal_of_asciis @ ( cons @ code_integer @ K @ Ks ) )
= ( case_char @ literal
@ ^ [B0: $o,B1: $o,B22: $o,B32: $o,B42: $o,B52: $o,B62: $o,B72: $o] : ( literal2 @ B0 @ B1 @ B22 @ B32 @ B42 @ B52 @ B62 @ ( literal_of_asciis @ Ks ) )
@ ( unique5772411509450598832har_of @ code_integer @ K ) ) ) ).
% String.literal_of_asciis_Cons
thf(fact_6931_literal_Odomain,axiom,
( ( domainp @ ( list @ char ) @ literal @ pcr_literal )
= ( ^ [X2: list @ char] :
? [Y: list @ char] :
( ( list_all2 @ char @ char
@ ^ [Y6: char,Z2: char] : Y6 = Z2
@ X2
@ Y )
& ! [Z3: char] :
( ( member @ char @ Z3 @ ( set2 @ char @ Y ) )
=> ~ ( digit7 @ Z3 ) ) ) ) ) ).
% literal.domain
thf(fact_6932_fun_ODomainp__rel,axiom,
! [C: $tType,B: $tType,A: $tType,R5: A > B > $o] :
( ( domainp @ ( C > A ) @ ( C > B )
@ ( bNF_rel_fun @ C @ C @ A @ B
@ ^ [Y6: C,Z2: C] : Y6 = Z2
@ R5 ) )
= ( basic_pred_fun @ C @ A
@ ^ [Uu: C] : $true
@ ( domainp @ A @ B @ R5 ) ) ) ).
% fun.Domainp_rel
thf(fact_6933_pcr__Domainp,axiom,
! [B: $tType,A: $tType,C: $tType,B4: A > B > $o,P: A > $o,A5: C > A > $o] :
( ( ( domainp @ A @ B @ B4 )
= P )
=> ( ( domainp @ C @ B @ ( relcompp @ C @ A @ B @ A5 @ B4 ) )
= ( ^ [X2: C] :
? [Y: A] :
( ( A5 @ X2 @ Y )
& ( P @ Y ) ) ) ) ) ).
% pcr_Domainp
thf(fact_6934_option_ODomainp__rel,axiom,
! [B: $tType,A: $tType,R5: A > B > $o] :
( ( domainp @ ( option @ A ) @ ( option @ B ) @ ( rel_option @ A @ B @ R5 ) )
= ( pred_option @ A @ ( domainp @ A @ B @ R5 ) ) ) ).
% option.Domainp_rel
thf(fact_6935_Domainp__Domain__eq,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B )] :
( ( domainp @ A @ B
@ ^ [X2: A,Y: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y ) @ R ) )
= ( ^ [X2: A] : ( member @ A @ X2 @ ( domain @ A @ B @ R ) ) ) ) ).
% Domainp_Domain_eq
thf(fact_6936_Domain__def,axiom,
! [B: $tType,A: $tType] :
( ( domain @ A @ B )
= ( ^ [R4: set @ ( product_prod @ A @ B )] :
( collect @ A
@ ( domainp @ A @ B
@ ^ [X2: A,Y: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y ) @ R4 ) ) ) ) ) ).
% Domain_def
thf(fact_6937_char_Ocase__distrib,axiom,
! [A: $tType,B: $tType,H: A > B,F: $o > $o > $o > $o > $o > $o > $o > $o > A,Char: char] :
( ( H @ ( case_char @ A @ F @ Char ) )
= ( case_char @ B
@ ^ [X12: $o,X24: $o,X33: $o,X43: $o,X53: $o,X63: $o,X73: $o,X83: $o] : ( H @ ( F @ X12 @ X24 @ X33 @ X43 @ X53 @ X63 @ X73 @ X83 ) )
@ Char ) ) ).
% char.case_distrib
thf(fact_6938_Heap_ODomainp__rel,axiom,
! [B: $tType,A: $tType,R5: A > B > $o] :
( ( domainp @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ B ) @ ( heap_Time_rel_Heap @ A @ B @ R5 ) )
= ( heap_Time_pred_Heap @ A @ ( domainp @ A @ B @ R5 ) ) ) ).
% Heap.Domainp_rel
thf(fact_6939_literal_Odomain__eq,axiom,
( ( domainp @ ( list @ char ) @ literal @ pcr_literal )
= ( ^ [Cs: list @ char] :
! [X2: char] :
( ( member @ char @ X2 @ ( set2 @ char @ Cs ) )
=> ~ ( digit7 @ X2 ) ) ) ) ).
% literal.domain_eq
thf(fact_6940_literal_Odomain__par__left__total,axiom,
! [P9: ( list @ char ) > $o] :
( ( left_total @ ( list @ char ) @ ( list @ char )
@ ( list_all2 @ char @ char
@ ^ [Y6: char,Z2: char] : Y6 = Z2 ) )
=> ( ( bNF_rel_fun @ ( list @ char ) @ ( list @ char ) @ $o @ $o
@ ( list_all2 @ char @ char
@ ^ [Y6: char,Z2: char] : Y6 = Z2 )
@ ^ [Y6: $o,Z2: $o] : Y6 = Z2
@ P9
@ ^ [Cs: list @ char] :
! [X2: char] :
( ( member @ char @ X2 @ ( set2 @ char @ Cs ) )
=> ~ ( digit7 @ X2 ) ) )
=> ( ( domainp @ ( list @ char ) @ literal @ pcr_literal )
= P9 ) ) ) ).
% literal.domain_par_left_total
thf(fact_6941_right__total__relcompp__transfer,axiom,
! [C: $tType,A: $tType,E: $tType,F3: $tType,B: $tType,D: $tType,B4: A > B > $o,A5: C > D > $o,C6: E > F3 > $o] :
( ( right_total @ A @ B @ B4 )
=> ( bNF_rel_fun @ ( C > A > $o ) @ ( D > B > $o ) @ ( ( A > E > $o ) > C > E > $o ) @ ( ( B > F3 > $o ) > D > F3 > $o )
@ ( bNF_rel_fun @ C @ D @ ( A > $o ) @ ( B > $o ) @ A5
@ ( bNF_rel_fun @ A @ B @ $o @ $o @ B4
@ ^ [Y6: $o,Z2: $o] : Y6 = Z2 ) )
@ ( bNF_rel_fun @ ( A > E > $o ) @ ( B > F3 > $o ) @ ( C > E > $o ) @ ( D > F3 > $o )
@ ( bNF_rel_fun @ A @ B @ ( E > $o ) @ ( F3 > $o ) @ B4
@ ( bNF_rel_fun @ E @ F3 @ $o @ $o @ C6
@ ^ [Y6: $o,Z2: $o] : Y6 = Z2 ) )
@ ( bNF_rel_fun @ C @ D @ ( E > $o ) @ ( F3 > $o ) @ A5
@ ( bNF_rel_fun @ E @ F3 @ $o @ $o @ C6
@ ^ [Y6: $o,Z2: $o] : Y6 = Z2 ) ) )
@ ^ [R6: C > A > $o,S8: A > E > $o,X2: C,Z3: E] :
? [Y: A] :
( ( member @ A @ Y @ ( collect @ A @ ( domainp @ A @ B @ B4 ) ) )
& ( R6 @ X2 @ Y )
& ( S8 @ Y @ Z3 ) )
@ ( relcompp @ D @ B @ F3 ) ) ) ).
% right_total_relcompp_transfer
thf(fact_6942_rat_Odomain__par,axiom,
! [DR1: int > $o,DR2: int > $o,P25: ( product_prod @ int @ int ) > $o] :
( ( ( domainp @ int @ int
@ ^ [Y6: int,Z2: int] : Y6 = Z2 )
= DR1 )
=> ( ( ( domainp @ int @ int
@ ^ [Y6: int,Z2: int] : Y6 = Z2 )
= DR2 )
=> ( ( bNF_rel_fun @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ $o @ $o
@ ( basic_rel_prod @ int @ int @ int @ int
@ ^ [Y6: int,Z2: int] : Y6 = Z2
@ ^ [Y6: int,Z2: int] : Y6 = Z2 )
@ ^ [Y6: $o,Z2: $o] : Y6 = Z2
@ P25
@ ^ [X2: product_prod @ int @ int] : ( ratrel @ X2 @ X2 ) )
=> ( ( domainp @ ( product_prod @ int @ int ) @ rat @ pcr_rat )
= ( inf_inf @ ( ( product_prod @ int @ int ) > $o ) @ ( basic_pred_prod @ int @ int @ DR1 @ DR2 ) @ P25 ) ) ) ) ) ).
% rat.domain_par
thf(fact_6943_digit7__def,axiom,
( digit7
= ( case_char @ $o
@ ^ [X12: $o,X24: $o,X33: $o,X43: $o,X53: $o,X63: $o,X73: $o,X83: $o] : X83 ) ) ).
% digit7_def
thf(fact_6944_int_Odomain,axiom,
! [DR1: nat > $o,DR2: nat > $o] :
( ( ( domainp @ nat @ nat
@ ^ [Y6: nat,Z2: nat] : Y6 = Z2 )
= DR1 )
=> ( ( ( domainp @ nat @ nat
@ ^ [Y6: nat,Z2: nat] : Y6 = Z2 )
= DR2 )
=> ( ( domainp @ ( product_prod @ nat @ nat ) @ int @ pcr_int )
= ( basic_pred_prod @ nat @ nat @ DR1 @ DR2 ) ) ) ) ).
% int.domain
thf(fact_6945_typedef__right__total,axiom,
! [B: $tType,A: $tType,Rep: B > A,Abs: A > B,A5: set @ A,T5: A > B > $o] :
( ( type_definition @ B @ A @ Rep @ Abs @ A5 )
=> ( ( T5
= ( ^ [X2: A,Y: B] :
( X2
= ( Rep @ Y ) ) ) )
=> ( right_total @ A @ B @ T5 ) ) ) ).
% typedef_right_total
thf(fact_6946_Heap_Oleft__total__rel,axiom,
! [B: $tType,A: $tType,R5: A > B > $o] :
( ( left_total @ A @ B @ R5 )
=> ( left_total @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ B ) @ ( heap_Time_rel_Heap @ A @ B @ R5 ) ) ) ).
% Heap.left_total_rel
thf(fact_6947_Heap_Oright__total__rel,axiom,
! [B: $tType,A: $tType,R5: A > B > $o] :
( ( right_total @ A @ B @ R5 )
=> ( right_total @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ B ) @ ( heap_Time_rel_Heap @ A @ B @ R5 ) ) ) ).
% Heap.right_total_rel
thf(fact_6948_int_Oright__total,axiom,
right_total @ ( product_prod @ nat @ nat ) @ int @ pcr_int ).
% int.right_total
thf(fact_6949_option_Oleft__total__rel,axiom,
! [B: $tType,A: $tType,R5: A > B > $o] :
( ( left_total @ A @ B @ R5 )
=> ( left_total @ ( option @ A ) @ ( option @ B ) @ ( rel_option @ A @ B @ R5 ) ) ) ).
% option.left_total_rel
thf(fact_6950_option_Oright__total__rel,axiom,
! [B: $tType,A: $tType,R5: A > B > $o] :
( ( right_total @ A @ B @ R5 )
=> ( right_total @ ( option @ A ) @ ( option @ B ) @ ( rel_option @ A @ B @ R5 ) ) ) ).
% option.right_total_rel
thf(fact_6951_rat_Odomain,axiom,
( ( domainp @ ( product_prod @ int @ int ) @ rat @ pcr_rat )
= ( ^ [X2: product_prod @ int @ int] :
? [Y: product_prod @ int @ int] :
( ( basic_rel_prod @ int @ int @ int @ int
@ ^ [Y6: int,Z2: int] : Y6 = Z2
@ ^ [Y6: int,Z2: int] : Y6 = Z2
@ X2
@ Y )
& ( ratrel @ Y @ Y ) ) ) ) ).
% rat.domain
thf(fact_6952_right__total__Domainp__transfer,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,B4: A > B > $o,A5: C > D > $o] :
( ( right_total @ A @ B @ B4 )
=> ( bNF_rel_fun @ ( C > A > $o ) @ ( D > B > $o ) @ ( C > $o ) @ ( D > $o )
@ ( bNF_rel_fun @ C @ D @ ( A > $o ) @ ( B > $o ) @ A5
@ ( bNF_rel_fun @ A @ B @ $o @ $o @ B4
@ ^ [Y6: $o,Z2: $o] : Y6 = Z2 ) )
@ ( bNF_rel_fun @ C @ D @ $o @ $o @ A5
@ ^ [Y6: $o,Z2: $o] : Y6 = Z2 )
@ ^ [T6: C > A > $o,X2: C] :
? [Y: A] :
( ( member @ A @ Y @ ( collect @ A @ ( domainp @ A @ B @ B4 ) ) )
& ( T6 @ X2 @ Y ) )
@ ( domainp @ D @ B ) ) ) ).
% right_total_Domainp_transfer
thf(fact_6953_literal_Odomain__par,axiom,
! [DR: char > $o,P25: ( list @ char ) > $o] :
( ( ( domainp @ char @ char
@ ^ [Y6: char,Z2: char] : Y6 = Z2 )
= DR )
=> ( ( bNF_rel_fun @ ( list @ char ) @ ( list @ char ) @ $o @ $o
@ ( list_all2 @ char @ char
@ ^ [Y6: char,Z2: char] : Y6 = Z2 )
@ ^ [Y6: $o,Z2: $o] : Y6 = Z2
@ P25
@ ^ [Cs: list @ char] :
! [X2: char] :
( ( member @ char @ X2 @ ( set2 @ char @ Cs ) )
=> ~ ( digit7 @ X2 ) ) )
=> ( ( domainp @ ( list @ char ) @ literal @ pcr_literal )
= ( inf_inf @ ( ( list @ char ) > $o ) @ ( list_all @ char @ DR ) @ P25 ) ) ) ) ).
% literal.domain_par
thf(fact_6954_right__total__fun__eq__transfer,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,A5: A > B > $o,B4: C > D > $o] :
( ( right_total @ A @ B @ A5 )
=> ( ( bi_unique @ C @ D @ B4 )
=> ( bNF_rel_fun @ ( A > C ) @ ( B > D ) @ ( ( A > C ) > $o ) @ ( ( B > D ) > $o ) @ ( bNF_rel_fun @ A @ B @ C @ D @ A5 @ B4 )
@ ( bNF_rel_fun @ ( A > C ) @ ( B > D ) @ $o @ $o @ ( bNF_rel_fun @ A @ B @ C @ D @ A5 @ B4 )
@ ^ [Y6: $o,Z2: $o] : Y6 = Z2 )
@ ^ [F2: A > C,G2: A > C] :
! [X2: A] :
( ( member @ A @ X2 @ ( collect @ A @ ( domainp @ A @ B @ A5 ) ) )
=> ( ( F2 @ X2 )
= ( G2 @ X2 ) ) )
@ ^ [Y6: B > D,Z2: B > D] : Y6 = Z2 ) ) ) ).
% right_total_fun_eq_transfer
thf(fact_6955_char_Odisc__eq__case,axiom,
! [Char: char] :
( case_char @ $o
@ ^ [Uu: $o,Uv: $o,Uw: $o,Ux: $o,Uy: $o,Uz: $o,Va: $o,Vb: $o] : $true
@ Char ) ).
% char.disc_eq_case
thf(fact_6956_int_Oleft__total,axiom,
left_total @ ( product_prod @ nat @ nat ) @ int @ pcr_int ).
% int.left_total
thf(fact_6957_list_Opred__map,axiom,
! [B: $tType,A: $tType,Q: B > $o,F: A > B,X: list @ A] :
( ( list_all @ B @ Q @ ( map @ A @ B @ F @ X ) )
= ( list_all @ A @ ( comp @ B @ $o @ A @ Q @ F ) @ X ) ) ).
% list.pred_map
thf(fact_6958_list_Omap__cong__pred,axiom,
! [B: $tType,A: $tType,X: list @ A,Ya: list @ A,F: A > B,G: A > B] :
( ( X = Ya )
=> ( ( list_all @ A
@ ^ [Z3: A] :
( ( F @ Z3 )
= ( G @ Z3 ) )
@ Ya )
=> ( ( map @ A @ B @ F @ X )
= ( map @ A @ B @ G @ Ya ) ) ) ) ).
% list.map_cong_pred
thf(fact_6959_option_Obi__unique__rel,axiom,
! [B: $tType,A: $tType,R5: A > B > $o] :
( ( bi_unique @ A @ B @ R5 )
=> ( bi_unique @ ( option @ A ) @ ( option @ B ) @ ( rel_option @ A @ B @ R5 ) ) ) ).
% option.bi_unique_rel
thf(fact_6960_bi__unique__iff,axiom,
! [B: $tType,A: $tType] :
( ( bi_unique @ A @ B )
= ( ^ [R6: A > B > $o] :
( ! [Z3: B] :
( uniq @ A
@ ^ [X2: A] : ( R6 @ X2 @ Z3 ) )
& ! [Z3: A] : ( uniq @ B @ ( R6 @ Z3 ) ) ) ) ) ).
% bi_unique_iff
thf(fact_6961_list_Opred__True,axiom,
! [A: $tType] :
( ( list_all @ A
@ ^ [Uu: A] : $true )
= ( ^ [Uu: list @ A] : $true ) ) ).
% list.pred_True
thf(fact_6962_Heap_Obi__unique__rel,axiom,
! [B: $tType,A: $tType,R5: A > B > $o] :
( ( bi_unique @ A @ B @ R5 )
=> ( bi_unique @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ B ) @ ( heap_Time_rel_Heap @ A @ B @ R5 ) ) ) ).
% Heap.bi_unique_rel
thf(fact_6963_typedef__bi__unique,axiom,
! [B: $tType,A: $tType,Rep: B > A,Abs: A > B,A5: set @ A,T5: A > B > $o] :
( ( type_definition @ B @ A @ Rep @ Abs @ A5 )
=> ( ( T5
= ( ^ [X2: A,Y: B] :
( X2
= ( Rep @ Y ) ) ) )
=> ( bi_unique @ A @ B @ T5 ) ) ) ).
% typedef_bi_unique
thf(fact_6964_rat_Odomain__par__left__total,axiom,
! [P9: ( product_prod @ int @ int ) > $o] :
( ( left_total @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int )
@ ( basic_rel_prod @ int @ int @ int @ int
@ ^ [Y6: int,Z2: int] : Y6 = Z2
@ ^ [Y6: int,Z2: int] : Y6 = Z2 ) )
=> ( ( bNF_rel_fun @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ $o @ $o
@ ( basic_rel_prod @ int @ int @ int @ int
@ ^ [Y6: int,Z2: int] : Y6 = Z2
@ ^ [Y6: int,Z2: int] : Y6 = Z2 )
@ ^ [Y6: $o,Z2: $o] : Y6 = Z2
@ P9
@ ^ [X2: product_prod @ int @ int] : ( ratrel @ X2 @ X2 ) )
=> ( ( domainp @ ( product_prod @ int @ int ) @ rat @ pcr_rat )
= P9 ) ) ) ).
% rat.domain_par_left_total
thf(fact_6965_list__ex1__simps_I2_J,axiom,
! [A: $tType,P: A > $o,X: A,Xs2: list @ A] :
( ( list_ex1 @ A @ P @ ( cons @ A @ X @ Xs2 ) )
= ( ( ( P @ X )
=> ( list_all @ A
@ ^ [Y: A] :
( ~ ( P @ Y )
| ( X = Y ) )
@ Xs2 ) )
& ( ~ ( P @ X )
=> ( list_ex1 @ A @ P @ Xs2 ) ) ) ) ).
% list_ex1_simps(2)
thf(fact_6966_Abs__rat__inject,axiom,
! [X: set @ ( product_prod @ int @ int ),Y3: set @ ( product_prod @ int @ int )] :
( ( member @ ( set @ ( product_prod @ int @ int ) ) @ X
@ ( collect @ ( set @ ( product_prod @ int @ int ) )
@ ^ [C4: set @ ( product_prod @ int @ int )] :
? [X2: product_prod @ int @ int] :
( ( ratrel @ X2 @ X2 )
& ( C4
= ( collect @ ( product_prod @ int @ int ) @ ( ratrel @ X2 ) ) ) ) ) )
=> ( ( member @ ( set @ ( product_prod @ int @ int ) ) @ Y3
@ ( collect @ ( set @ ( product_prod @ int @ int ) )
@ ^ [C4: set @ ( product_prod @ int @ int )] :
? [X2: product_prod @ int @ int] :
( ( ratrel @ X2 @ X2 )
& ( C4
= ( collect @ ( product_prod @ int @ int ) @ ( ratrel @ X2 ) ) ) ) ) )
=> ( ( ( abs_rat @ X )
= ( abs_rat @ Y3 ) )
= ( X = Y3 ) ) ) ) ).
% Abs_rat_inject
thf(fact_6967_Abs__rat__cases,axiom,
! [X: rat] :
~ ! [Y4: set @ ( product_prod @ int @ int )] :
( ( X
= ( abs_rat @ Y4 ) )
=> ~ ( member @ ( set @ ( product_prod @ int @ int ) ) @ Y4
@ ( collect @ ( set @ ( product_prod @ int @ int ) )
@ ^ [C4: set @ ( product_prod @ int @ int )] :
? [X2: product_prod @ int @ int] :
( ( ratrel @ X2 @ X2 )
& ( C4
= ( collect @ ( product_prod @ int @ int ) @ ( ratrel @ X2 ) ) ) ) ) ) ) ).
% Abs_rat_cases
thf(fact_6968_Abs__rat__induct,axiom,
! [P: rat > $o,X: rat] :
( ! [Y4: set @ ( product_prod @ int @ int )] :
( ( member @ ( set @ ( product_prod @ int @ int ) ) @ Y4
@ ( collect @ ( set @ ( product_prod @ int @ int ) )
@ ^ [C4: set @ ( product_prod @ int @ int )] :
? [X2: product_prod @ int @ int] :
( ( ratrel @ X2 @ X2 )
& ( C4
= ( collect @ ( product_prod @ int @ int ) @ ( ratrel @ X2 ) ) ) ) ) )
=> ( P @ ( abs_rat @ Y4 ) ) )
=> ( P @ X ) ) ).
% Abs_rat_induct
thf(fact_6969_Abs__rat__inverse,axiom,
! [Y3: set @ ( product_prod @ int @ int )] :
( ( member @ ( set @ ( product_prod @ int @ int ) ) @ Y3
@ ( collect @ ( set @ ( product_prod @ int @ int ) )
@ ^ [C4: set @ ( product_prod @ int @ int )] :
? [X2: product_prod @ int @ int] :
( ( ratrel @ X2 @ X2 )
& ( C4
= ( collect @ ( product_prod @ int @ int ) @ ( ratrel @ X2 ) ) ) ) ) )
=> ( ( rep_rat @ ( abs_rat @ Y3 ) )
= Y3 ) ) ).
% Abs_rat_inverse
thf(fact_6970_type__definition__rat,axiom,
( type_definition @ rat @ ( set @ ( product_prod @ int @ int ) ) @ rep_rat @ abs_rat
@ ( collect @ ( set @ ( product_prod @ int @ int ) )
@ ^ [C4: set @ ( product_prod @ int @ int )] :
? [X2: product_prod @ int @ int] :
( ( ratrel @ X2 @ X2 )
& ( C4
= ( collect @ ( product_prod @ int @ int ) @ ( ratrel @ X2 ) ) ) ) ) ) ).
% type_definition_rat
thf(fact_6971_Rep__rat__induct,axiom,
! [Y3: set @ ( product_prod @ int @ int ),P: ( set @ ( product_prod @ int @ int ) ) > $o] :
( ( member @ ( set @ ( product_prod @ int @ int ) ) @ Y3
@ ( collect @ ( set @ ( product_prod @ int @ int ) )
@ ^ [C4: set @ ( product_prod @ int @ int )] :
? [X2: product_prod @ int @ int] :
( ( ratrel @ X2 @ X2 )
& ( C4
= ( collect @ ( product_prod @ int @ int ) @ ( ratrel @ X2 ) ) ) ) ) )
=> ( ! [X4: rat] : ( P @ ( rep_rat @ X4 ) )
=> ( P @ Y3 ) ) ) ).
% Rep_rat_induct
thf(fact_6972_Rep__rat__cases,axiom,
! [Y3: set @ ( product_prod @ int @ int )] :
( ( member @ ( set @ ( product_prod @ int @ int ) ) @ Y3
@ ( collect @ ( set @ ( product_prod @ int @ int ) )
@ ^ [C4: set @ ( product_prod @ int @ int )] :
? [X2: product_prod @ int @ int] :
( ( ratrel @ X2 @ X2 )
& ( C4
= ( collect @ ( product_prod @ int @ int ) @ ( ratrel @ X2 ) ) ) ) ) )
=> ~ ! [X4: rat] :
( Y3
!= ( rep_rat @ X4 ) ) ) ).
% Rep_rat_cases
thf(fact_6973_Rep__rat,axiom,
! [X: rat] :
( member @ ( set @ ( product_prod @ int @ int ) ) @ ( rep_rat @ X )
@ ( collect @ ( set @ ( product_prod @ int @ int ) )
@ ^ [C4: set @ ( product_prod @ int @ int )] :
? [X2: product_prod @ int @ int] :
( ( ratrel @ X2 @ X2 )
& ( C4
= ( collect @ ( product_prod @ int @ int ) @ ( ratrel @ X2 ) ) ) ) ) ) ).
% Rep_rat
thf(fact_6974_quot__type_Orep__def,axiom,
! [B: $tType,A: $tType,R5: A > A > $o,Abs: ( set @ A ) > B,Rep: B > ( set @ A ),A3: B] :
( ( quot_type @ A @ B @ R5 @ Abs @ Rep )
=> ( ( quot_rep @ B @ A @ Rep @ A3 )
= ( fChoice @ A
@ ^ [X2: A] : ( member @ A @ X2 @ ( Rep @ A3 ) ) ) ) ) ).
% quot_type.rep_def
thf(fact_6975_vimage__right__total__transfer,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,B4: A > B > $o,A5: C > D > $o] :
( ( bi_unique @ A @ B @ B4 )
=> ( ( right_total @ C @ D @ A5 )
=> ( bNF_rel_fun @ ( C > A ) @ ( D > B ) @ ( ( set @ A ) > ( set @ C ) ) @ ( ( set @ B ) > ( set @ D ) ) @ ( bNF_rel_fun @ C @ D @ A @ B @ A5 @ B4 ) @ ( bNF_rel_fun @ ( set @ A ) @ ( set @ B ) @ ( set @ C ) @ ( set @ D ) @ ( bNF_rel_set @ A @ B @ B4 ) @ ( bNF_rel_set @ C @ D @ A5 ) )
@ ^ [F2: C > A,X9: set @ A] : ( inf_inf @ ( set @ C ) @ ( vimage @ C @ A @ F2 @ X9 ) @ ( collect @ C @ ( domainp @ C @ D @ A5 ) ) )
@ ( vimage @ D @ B ) ) ) ) ).
% vimage_right_total_transfer
thf(fact_6976_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
@ ^ [Y6: $o,Z2: $o] : Y6 = Z2 )
@ ( bNF_rel_set @ A @ B @ A5 )
@ ^ [P3: A > $o] :
( collect @ A
@ ^ [X2: A] :
( ( P3 @ X2 )
& ( domainp @ A @ B @ A5 @ X2 ) ) )
@ ( collect @ B ) ) ) ).
% right_total_Collect_transfer
thf(fact_6977_Heap_Oset__transfer,axiom,
! [A: $tType,B: $tType,R5: A > B > $o] : ( bNF_rel_fun @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ B ) @ ( set @ A ) @ ( set @ B ) @ ( heap_Time_rel_Heap @ A @ B @ R5 ) @ ( bNF_rel_set @ A @ B @ R5 ) @ ( heap_Time_set_Heap @ A ) @ ( heap_Time_set_Heap @ B ) ) ).
% Heap.set_transfer
thf(fact_6978_option_Oset__transfer,axiom,
! [A: $tType,B: $tType,R5: A > B > $o] : ( bNF_rel_fun @ ( option @ A ) @ ( option @ B ) @ ( set @ A ) @ ( set @ B ) @ ( rel_option @ A @ B @ R5 ) @ ( bNF_rel_set @ A @ B @ R5 ) @ ( set_option @ A ) @ ( set_option @ B ) ) ).
% option.set_transfer
thf(fact_6979_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_6980_UNION__transfer,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,A5: A > B > $o,B4: C > D > $o] :
( bNF_rel_fun @ ( set @ A ) @ ( set @ B ) @ ( ( A > ( set @ C ) ) > ( set @ C ) ) @ ( ( B > ( set @ D ) ) > ( set @ D ) ) @ ( bNF_rel_set @ A @ B @ A5 ) @ ( bNF_rel_fun @ ( A > ( set @ C ) ) @ ( B > ( set @ D ) ) @ ( set @ C ) @ ( set @ D ) @ ( bNF_rel_fun @ A @ B @ ( set @ C ) @ ( set @ D ) @ A5 @ ( bNF_rel_set @ C @ D @ B4 ) ) @ ( bNF_rel_set @ C @ D @ B4 ) )
@ ^ [A8: set @ A,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_6981_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] :
! [X2: A] :
( ( member @ A @ X2 @ A8 )
=> ( P @ X2 ) ) ) ) ).
% set_relator_eq_onp
thf(fact_6982_rel__set__def,axiom,
! [B: $tType,A: $tType] :
( ( bNF_rel_set @ A @ B )
= ( ^ [R6: A > B > $o,A8: set @ A,B7: set @ B] :
( ! [X2: A] :
( ( member @ A @ X2 @ A8 )
=> ? [Y: B] :
( ( member @ B @ Y @ B7 )
& ( R6 @ X2 @ Y ) ) )
& ! [X2: B] :
( ( member @ B @ X2 @ B7 )
=> ? [Y: A] :
( ( member @ A @ Y @ A8 )
& ( R6 @ Y @ X2 ) ) ) ) ) ) ).
% rel_set_def
thf(fact_6983_quot__type_Osome__collect,axiom,
! [B: $tType,A: $tType,R5: A > A > $o,Abs: ( set @ A ) > B,Rep: B > ( set @ A ),R: A] :
( ( quot_type @ A @ B @ R5 @ Abs @ Rep )
=> ( ( R5 @ R @ R )
=> ( ( R5
@ ( fChoice @ A
@ ^ [X2: A] : ( member @ A @ X2 @ ( collect @ A @ ( R5 @ R ) ) ) ) )
= ( R5 @ R ) ) ) ) ).
% quot_type.some_collect
thf(fact_6984_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 )
@ ^ [S8: set @ ( set @ A )] : ( inf_inf @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ S8 ) @ ( collect @ A @ ( domainp @ A @ B @ A5 ) ) )
@ ( complete_Inf_Inf @ ( set @ B ) ) ) ) ) ).
% right_total_Inter_transfer
thf(fact_6985_fun_Oset__transfer,axiom,
! [A: $tType,B: $tType,D: $tType,R5: A > B > $o] :
( bNF_rel_fun @ ( D > A ) @ ( D > B ) @ ( set @ A ) @ ( set @ B )
@ ( bNF_rel_fun @ D @ D @ A @ B
@ ^ [Y6: D,Z2: D] : Y6 = Z2
@ R5 )
@ ( bNF_rel_set @ A @ B @ R5 )
@ ^ [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_6986_SUP__parametric,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( complete_Sup @ C )
=> ! [R5: A > B > $o] :
( bNF_rel_fun @ ( set @ A ) @ ( set @ B ) @ ( ( A > C ) > C ) @ ( ( B > C ) > C ) @ ( bNF_rel_set @ A @ B @ R5 )
@ ( bNF_rel_fun @ ( A > C ) @ ( B > C ) @ C @ C
@ ( bNF_rel_fun @ A @ B @ C @ C @ R5
@ ^ [Y6: C,Z2: C] : Y6 = Z2 )
@ ^ [Y6: C,Z2: C] : Y6 = Z2 )
@ ^ [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_6987_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
@ ^ [Y6: C,Z2: C] : Y6 = Z2 )
@ ^ [Y6: C,Z2: C] : Y6 = Z2 )
@ ^ [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_6988_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 )
@ ^ [S8: set @ A] : ( inf_inf @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ S8 ) @ ( collect @ A @ ( domainp @ A @ B @ A5 ) ) )
@ ( uminus_uminus @ ( set @ B ) ) ) ) ) ).
% right_total_Compl_transfer
thf(fact_6989_open__typedef__to__part__equivp,axiom,
! [A: $tType,B: $tType,Rep: A > B,Abs: B > A,P: B > $o] :
( ( type_definition @ A @ B @ Rep @ Abs @ ( collect @ B @ P ) )
=> ( equiv_part_equivp @ B @ ( bNF_eq_onp @ B @ P ) ) ) ).
% open_typedef_to_part_equivp
thf(fact_6990_typedef__to__part__equivp,axiom,
! [A: $tType,B: $tType,Rep: A > B,Abs: B > A,S3: set @ B] :
( ( type_definition @ A @ B @ Rep @ Abs @ S3 )
=> ( equiv_part_equivp @ B
@ ( bNF_eq_onp @ B
@ ^ [X2: B] : ( member @ B @ X2 @ S3 ) ) ) ) ).
% typedef_to_part_equivp
thf(fact_6991_part__equivp__typedef,axiom,
! [A: $tType,R5: A > A > $o] :
( ( equiv_part_equivp @ A @ R5 )
=> ? [D2: set @ A] :
( member @ ( set @ A ) @ D2
@ ( collect @ ( set @ A )
@ ^ [C4: set @ A] :
? [X2: A] :
( ( R5 @ X2 @ X2 )
& ( C4
= ( collect @ A @ ( R5 @ X2 ) ) ) ) ) ) ) ).
% part_equivp_typedef
thf(fact_6992_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_6993_bounded__quasi__semilattice_Oaxioms_I2_J,axiom,
! [A: $tType,F: A > A > A,Top: A,Bot: A,Normalize: A > A] :
( ( bounde8507323023520639062attice @ A @ F @ Top @ Bot @ Normalize )
=> ( bounde5338082073960204281axioms @ A @ F @ Top @ Bot @ Normalize ) ) ).
% bounded_quasi_semilattice.axioms(2)
thf(fact_6994_Heap_Obi__total__rel,axiom,
! [B: $tType,A: $tType,R5: A > B > $o] :
( ( bi_total @ A @ B @ R5 )
=> ( bi_total @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ B ) @ ( heap_Time_rel_Heap @ A @ B @ R5 ) ) ) ).
% Heap.bi_total_rel
thf(fact_6995_bounded__quasi__semilattice__axioms_Ointro,axiom,
! [A: $tType,F: A > A > A,Normalize: A > A,Top: A,Bot: A] :
( ! [A4: A] :
( ( F @ A4 @ A4 )
= ( Normalize @ A4 ) )
=> ( ! [A4: A,B3: A] :
( ( F @ ( Normalize @ A4 ) @ B3 )
= ( F @ A4 @ B3 ) )
=> ( ! [A4: A,B3: A] :
( ( Normalize @ ( F @ A4 @ B3 ) )
= ( F @ A4 @ B3 ) )
=> ( ( ( Normalize @ Top )
= Top )
=> ( ( ( Normalize @ Bot )
= Bot )
=> ( ! [A4: A] :
( ( F @ Top @ A4 )
= ( Normalize @ A4 ) )
=> ( ! [A4: A] :
( ( F @ Bot @ A4 )
= Bot )
=> ( bounde5338082073960204281axioms @ A @ F @ Top @ Bot @ Normalize ) ) ) ) ) ) ) ) ).
% bounded_quasi_semilattice_axioms.intro
thf(fact_6996_bounded__quasi__semilattice__axioms__def,axiom,
! [A: $tType] :
( ( bounde5338082073960204281axioms @ A )
= ( ^ [F2: A > A > A,Top2: A,Bot2: A,Normalize2: A > A] :
( ! [A6: A] :
( ( F2 @ A6 @ A6 )
= ( Normalize2 @ A6 ) )
& ! [A6: A,B5: A] :
( ( F2 @ ( Normalize2 @ A6 ) @ B5 )
= ( F2 @ A6 @ B5 ) )
& ! [A6: A,B5: A] :
( ( Normalize2 @ ( F2 @ A6 @ B5 ) )
= ( F2 @ A6 @ B5 ) )
& ( ( Normalize2 @ Top2 )
= Top2 )
& ( ( Normalize2 @ Bot2 )
= Bot2 )
& ! [A6: A] :
( ( F2 @ Top2 @ A6 )
= ( Normalize2 @ A6 ) )
& ! [A6: A] :
( ( F2 @ Bot2 @ A6 )
= Bot2 ) ) ) ) ).
% bounded_quasi_semilattice_axioms_def
thf(fact_6997_int_Obi__total,axiom,
bi_total @ ( product_prod @ nat @ nat ) @ int @ pcr_int ).
% int.bi_total
thf(fact_6998_option_Obi__total__rel,axiom,
! [B: $tType,A: $tType,R5: A > B > $o] :
( ( bi_total @ A @ B @ R5 )
=> ( bi_total @ ( option @ A ) @ ( option @ B ) @ ( rel_option @ A @ B @ R5 ) ) ) ).
% option.bi_total_rel
thf(fact_6999_bounded__quasi__semilattice__def,axiom,
! [A: $tType] :
( ( bounde8507323023520639062attice @ A )
= ( ^ [F2: A > A > A,Top2: A,Bot2: A,Normalize2: A > A] :
( ( abel_semigroup @ A @ F2 )
& ( bounde5338082073960204281axioms @ A @ F2 @ Top2 @ Bot2 @ Normalize2 ) ) ) ) ).
% bounded_quasi_semilattice_def
thf(fact_7000_bounded__quasi__semilattice_Ointro,axiom,
! [A: $tType,F: A > A > A,Top: A,Bot: A,Normalize: A > A] :
( ( abel_semigroup @ A @ F )
=> ( ( bounde5338082073960204281axioms @ A @ F @ Top @ Bot @ Normalize )
=> ( bounde8507323023520639062attice @ A @ F @ Top @ Bot @ Normalize ) ) ) ).
% bounded_quasi_semilattice.intro
thf(fact_7001_abel__semigroup_Oaxioms_I1_J,axiom,
! [A: $tType,F: A > A > A] :
( ( abel_semigroup @ A @ F )
=> ( semigroup @ A @ F ) ) ).
% abel_semigroup.axioms(1)
thf(fact_7002_bounded__quasi__semilattice_Oaxioms_I1_J,axiom,
! [A: $tType,F: A > A > A,Top: A,Bot: A,Normalize: A > A] :
( ( bounde8507323023520639062attice @ A @ F @ Top @ Bot @ Normalize )
=> ( abel_semigroup @ A @ F ) ) ).
% bounded_quasi_semilattice.axioms(1)
thf(fact_7003_lcm_Oabel__semigroup__axioms,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ( abel_semigroup @ A @ ( gcd_lcm @ A ) ) ) ).
% lcm.abel_semigroup_axioms
thf(fact_7004_abel__semigroup_Oleft__commute,axiom,
! [A: $tType,F: A > A > A,B2: A,A3: A,C2: A] :
( ( abel_semigroup @ A @ F )
=> ( ( F @ B2 @ ( F @ A3 @ C2 ) )
= ( F @ A3 @ ( F @ B2 @ C2 ) ) ) ) ).
% abel_semigroup.left_commute
thf(fact_7005_abel__semigroup_Ocommute,axiom,
! [A: $tType,F: A > A > A,A3: A,B2: A] :
( ( abel_semigroup @ A @ F )
=> ( ( F @ A3 @ B2 )
= ( F @ B2 @ A3 ) ) ) ).
% abel_semigroup.commute
thf(fact_7006_gcd_Oabel__semigroup__axioms,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ( abel_semigroup @ A @ ( gcd_gcd @ A ) ) ) ).
% gcd.abel_semigroup_axioms
thf(fact_7007_mult_Oabel__semigroup__axioms,axiom,
! [A: $tType] :
( ( ab_semigroup_mult @ A )
=> ( abel_semigroup @ A @ ( times_times @ A ) ) ) ).
% mult.abel_semigroup_axioms
thf(fact_7008_comm__monoid_Oaxioms_I1_J,axiom,
! [A: $tType,F: A > A > A,Z: A] :
( ( comm_monoid @ A @ F @ Z )
=> ( abel_semigroup @ A @ F ) ) ).
% comm_monoid.axioms(1)
thf(fact_7009_add_Oabel__semigroup__axioms,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A )
=> ( abel_semigroup @ A @ ( plus_plus @ A ) ) ) ).
% add.abel_semigroup_axioms
thf(fact_7010_comm__monoid__def,axiom,
! [A: $tType] :
( ( comm_monoid @ A )
= ( ^ [F2: A > A > A,Z3: A] :
( ( abel_semigroup @ A @ F2 )
& ( comm_monoid_axioms @ A @ F2 @ Z3 ) ) ) ) ).
% comm_monoid_def
thf(fact_7011_comm__monoid_Ointro,axiom,
! [A: $tType,F: A > A > A,Z: A] :
( ( abel_semigroup @ A @ F )
=> ( ( comm_monoid_axioms @ A @ F @ Z )
=> ( comm_monoid @ A @ F @ Z ) ) ) ).
% comm_monoid.intro
thf(fact_7012_comm__monoid__axioms_Ointro,axiom,
! [A: $tType,F: A > A > A,Z: A] :
( ! [A4: A] :
( ( F @ A4 @ Z )
= A4 )
=> ( comm_monoid_axioms @ A @ F @ Z ) ) ).
% comm_monoid_axioms.intro
thf(fact_7013_comm__monoid__axioms__def,axiom,
! [A: $tType] :
( ( comm_monoid_axioms @ A )
= ( ^ [F2: A > A > A,Z3: A] :
! [A6: A] :
( ( F2 @ A6 @ Z3 )
= A6 ) ) ) ).
% comm_monoid_axioms_def
thf(fact_7014_comm__monoid_Oaxioms_I2_J,axiom,
! [A: $tType,F: A > A > A,Z: A] :
( ( comm_monoid @ A @ F @ Z )
=> ( comm_monoid_axioms @ A @ F @ Z ) ) ).
% comm_monoid.axioms(2)
thf(fact_7015_abel__semigroup__def,axiom,
! [A: $tType] :
( ( abel_semigroup @ A )
= ( ^ [F2: A > A > A] :
( ( semigroup @ A @ F2 )
& ( abel_s757365448890700780axioms @ A @ F2 ) ) ) ) ).
% abel_semigroup_def
thf(fact_7016_abel__semigroup_Ointro,axiom,
! [A: $tType,F: A > A > A] :
( ( semigroup @ A @ F )
=> ( ( abel_s757365448890700780axioms @ A @ F )
=> ( abel_semigroup @ A @ F ) ) ) ).
% abel_semigroup.intro
thf(fact_7017_abel__semigroup__axioms__def,axiom,
! [A: $tType] :
( ( abel_s757365448890700780axioms @ A )
= ( ^ [F2: A > A > A] :
! [A6: A,B5: A] :
( ( F2 @ A6 @ B5 )
= ( F2 @ B5 @ A6 ) ) ) ) ).
% abel_semigroup_axioms_def
thf(fact_7018_abel__semigroup__axioms_Ointro,axiom,
! [A: $tType,F: A > A > A] :
( ! [A4: A,B3: A] :
( ( F @ A4 @ B3 )
= ( F @ B3 @ A4 ) )
=> ( abel_s757365448890700780axioms @ A @ F ) ) ).
% abel_semigroup_axioms.intro
thf(fact_7019_abel__semigroup_Oaxioms_I2_J,axiom,
! [A: $tType,F: A > A > A] :
( ( abel_semigroup @ A @ F )
=> ( abel_s757365448890700780axioms @ A @ F ) ) ).
% abel_semigroup.axioms(2)
thf(fact_7020_dual__order_Oordering__axioms,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ordering @ A
@ ^ [X2: A,Y: A] : ( ord_less_eq @ A @ Y @ X2 )
@ ^ [X2: A,Y: A] : ( ord_less @ A @ Y @ X2 ) ) ) ).
% dual_order.ordering_axioms
thf(fact_7021_frequently__cofinite,axiom,
! [A: $tType,P: A > $o] :
( ( frequently @ A @ P @ ( cofinite @ A ) )
= ( ~ ( finite_finite2 @ A @ ( collect @ A @ P ) ) ) ) ).
% frequently_cofinite
thf(fact_7022_frequently__const,axiom,
! [A: $tType,F6: filter @ A,P: $o] :
( ( F6
!= ( bot_bot @ ( filter @ A ) ) )
=> ( ( frequently @ A
@ ^ [X2: A] : P
@ F6 )
= P ) ) ).
% frequently_const
thf(fact_7023_frequently__bex__finite__distrib,axiom,
! [B: $tType,A: $tType,A5: set @ A,P: B > A > $o,F6: filter @ B] :
( ( finite_finite2 @ A @ A5 )
=> ( ( frequently @ B
@ ^ [X2: B] :
? [Y: A] :
( ( member @ A @ Y @ A5 )
& ( P @ X2 @ Y ) )
@ F6 )
= ( ? [X2: A] :
( ( member @ A @ X2 @ A5 )
& ( frequently @ B
@ ^ [Y: B] : ( P @ Y @ X2 )
@ F6 ) ) ) ) ) ).
% frequently_bex_finite_distrib
thf(fact_7024_frequently__bex__finite,axiom,
! [A: $tType,B: $tType,A5: set @ A,P: B > A > $o,F6: filter @ B] :
( ( finite_finite2 @ A @ A5 )
=> ( ( frequently @ B
@ ^ [X2: B] :
? [Y: A] :
( ( member @ A @ Y @ A5 )
& ( P @ X2 @ Y ) )
@ F6 )
=> ? [X4: A] :
( ( member @ A @ X4 @ A5 )
& ( frequently @ B
@ ^ [Y: B] : ( P @ Y @ X4 )
@ F6 ) ) ) ) ).
% frequently_bex_finite
thf(fact_7025_gcd__nat_Oordering__axioms,axiom,
( ordering @ nat @ ( dvd_dvd @ nat )
@ ^ [M5: nat,N5: nat] :
( ( dvd_dvd @ nat @ M5 @ N5 )
& ( M5 != N5 ) ) ) ).
% gcd_nat.ordering_axioms
thf(fact_7026_frequently__imp__iff,axiom,
! [A: $tType,P: A > $o,Q: A > $o,F6: filter @ A] :
( ( frequently @ A
@ ^ [X2: A] :
( ( P @ X2 )
=> ( Q @ X2 ) )
@ F6 )
= ( ( eventually @ A @ P @ F6 )
=> ( frequently @ A @ Q @ F6 ) ) ) ).
% frequently_imp_iff
thf(fact_7027_frequently__rev__mp,axiom,
! [A: $tType,P: A > $o,F6: filter @ A,Q: A > $o] :
( ( frequently @ A @ P @ F6 )
=> ( ( eventually @ A
@ ^ [X2: A] :
( ( P @ X2 )
=> ( Q @ X2 ) )
@ F6 )
=> ( frequently @ A @ Q @ F6 ) ) ) ).
% frequently_rev_mp
thf(fact_7028_not__frequently,axiom,
! [A: $tType,P: A > $o,F6: filter @ A] :
( ( ~ ( frequently @ A @ P @ F6 ) )
= ( eventually @ A
@ ^ [X2: A] :
~ ( P @ X2 )
@ F6 ) ) ).
% not_frequently
thf(fact_7029_not__eventually,axiom,
! [A: $tType,P: A > $o,F6: filter @ A] :
( ( ~ ( eventually @ A @ P @ F6 ) )
= ( frequently @ A
@ ^ [X2: A] :
~ ( P @ X2 )
@ F6 ) ) ).
% not_eventually
thf(fact_7030_frequently__def,axiom,
! [A: $tType] :
( ( frequently @ A )
= ( ^ [P3: A > $o,F7: filter @ A] :
~ ( eventually @ A
@ ^ [X2: A] :
~ ( P3 @ X2 )
@ F7 ) ) ) ).
% frequently_def
thf(fact_7031_frequently__mp,axiom,
! [A: $tType,P: A > $o,Q: A > $o,F6: filter @ A] :
( ( eventually @ A
@ ^ [X2: A] :
( ( P @ X2 )
=> ( Q @ X2 ) )
@ F6 )
=> ( ( frequently @ A @ P @ F6 )
=> ( frequently @ A @ Q @ F6 ) ) ) ).
% frequently_mp
thf(fact_7032_eventually__frequently__const__simps_I5_J,axiom,
! [A: $tType,P: A > $o,C6: $o,F6: filter @ A] :
( ( eventually @ A
@ ^ [X2: A] :
( ( P @ X2 )
=> C6 )
@ F6 )
= ( ( frequently @ A @ P @ F6 )
=> C6 ) ) ).
% eventually_frequently_const_simps(5)
thf(fact_7033_frequently__all,axiom,
! [B: $tType,A: $tType,P: A > B > $o,F6: filter @ A] :
( ( frequently @ A
@ ^ [X2: A] :
! [X9: B] : ( P @ X2 @ X9 )
@ F6 )
= ( ! [Y10: A > B] :
( frequently @ A
@ ^ [X2: A] : ( P @ X2 @ ( Y10 @ X2 ) )
@ F6 ) ) ) ).
% frequently_all
thf(fact_7034_ordering__dualI,axiom,
! [A: $tType,Less_eq2: A > A > $o,Less: A > A > $o] :
( ( ordering @ A
@ ^ [A6: A,B5: A] : ( Less_eq2 @ B5 @ A6 )
@ ^ [A6: A,B5: A] : ( Less @ B5 @ A6 ) )
=> ( ordering @ A @ Less_eq2 @ Less ) ) ).
% ordering_dualI
thf(fact_7035_eventually__frequently__const__simps_I2_J,axiom,
! [A: $tType,C6: $o,P: A > $o,F6: filter @ A] :
( ( frequently @ A
@ ^ [X2: A] :
( C6
& ( P @ X2 ) )
@ F6 )
= ( C6
& ( frequently @ A @ P @ F6 ) ) ) ).
% eventually_frequently_const_simps(2)
thf(fact_7036_eventually__frequently__const__simps_I1_J,axiom,
! [A: $tType,P: A > $o,C6: $o,F6: filter @ A] :
( ( frequently @ A
@ ^ [X2: A] :
( ( P @ X2 )
& C6 )
@ F6 )
= ( ( frequently @ A @ P @ F6 )
& C6 ) ) ).
% eventually_frequently_const_simps(1)
thf(fact_7037_frequently__ex,axiom,
! [A: $tType,P: A > $o,F6: filter @ A] :
( ( frequently @ A @ P @ F6 )
=> ? [X_12: A] : ( P @ X_12 ) ) ).
% frequently_ex
thf(fact_7038_frequently__disj,axiom,
! [A: $tType,P: A > $o,F6: filter @ A,Q: A > $o] :
( ( frequently @ A @ P @ F6 )
=> ( ( frequently @ A @ Q @ F6 )
=> ( frequently @ A
@ ^ [X2: A] :
( ( P @ X2 )
| ( Q @ X2 ) )
@ F6 ) ) ) ).
% frequently_disj
thf(fact_7039_frequently__elim1,axiom,
! [A: $tType,P: A > $o,F6: filter @ A,Q: A > $o] :
( ( frequently @ A @ P @ F6 )
=> ( ! [I3: A] :
( ( P @ I3 )
=> ( Q @ I3 ) )
=> ( frequently @ A @ Q @ F6 ) ) ) ).
% frequently_elim1
thf(fact_7040_frequently__disj__iff,axiom,
! [A: $tType,P: A > $o,Q: A > $o,F6: filter @ A] :
( ( frequently @ A
@ ^ [X2: A] :
( ( P @ X2 )
| ( Q @ X2 ) )
@ F6 )
= ( ( frequently @ A @ P @ F6 )
| ( frequently @ A @ Q @ F6 ) ) ) ).
% frequently_disj_iff
thf(fact_7041_not__frequently__False,axiom,
! [A: $tType,F6: filter @ A] :
~ ( frequently @ A
@ ^ [X2: A] : $false
@ F6 ) ).
% not_frequently_False
thf(fact_7042_frequently__const__iff,axiom,
! [A: $tType,P: $o,F6: filter @ A] :
( ( frequently @ A
@ ^ [X2: A] : P
@ F6 )
= ( P
& ( F6
!= ( bot_bot @ ( filter @ A ) ) ) ) ) ).
% frequently_const_iff
thf(fact_7043_eventually__frequentlyE,axiom,
! [A: $tType,P: A > $o,F6: filter @ A,Q: A > $o] :
( ( eventually @ A @ P @ F6 )
=> ( ( eventually @ A
@ ^ [X2: A] :
( ~ ( P @ X2 )
| ( Q @ X2 ) )
@ F6 )
=> ( ( F6
!= ( bot_bot @ ( filter @ A ) ) )
=> ( frequently @ A @ Q @ F6 ) ) ) ) ).
% eventually_frequentlyE
thf(fact_7044_prod__list_Omonoid__list__axioms,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ( groups_monoid_list @ A @ ( times_times @ A ) @ ( one_one @ A ) ) ) ).
% prod_list.monoid_list_axioms
thf(fact_7045_case__sum__o__map__sum__id,axiom,
! [A: $tType,B: $tType,C: $tType,G: B > A,F: C > A,X: sum_sum @ C @ B] :
( ( comp @ ( sum_sum @ A @ B ) @ A @ ( sum_sum @ C @ B ) @ ( sum_case_sum @ A @ A @ B @ ( id @ A ) @ G ) @ ( sum_map_sum @ C @ A @ B @ B @ F @ ( id @ B ) ) @ X )
= ( sum_case_sum @ C @ A @ B @ ( comp @ C @ A @ C @ F @ ( id @ C ) ) @ G @ X ) ) ).
% case_sum_o_map_sum_id
thf(fact_7046_map__sum_Oidentity,axiom,
! [B: $tType,A: $tType] :
( ( sum_map_sum @ A @ A @ B @ B
@ ^ [X2: A] : X2
@ ^ [X2: B] : X2 )
= ( id @ ( sum_sum @ A @ B ) ) ) ).
% map_sum.identity
thf(fact_7047_sum_Omap__ident,axiom,
! [B: $tType,A: $tType,T2: sum_sum @ A @ B] :
( ( sum_map_sum @ A @ A @ B @ B
@ ^ [X2: A] : X2
@ ^ [X2: B] : X2
@ T2 )
= T2 ) ).
% sum.map_ident
thf(fact_7048_map__sum__if__distrib__then_I2_J,axiom,
! [H8: $tType,F3: $tType,G3: $tType,E: $tType,E2: $o,F: E > F3,G: G3 > H8,X: G3,Y3: sum_sum @ E @ G3] :
( ( E2
=> ( ( sum_map_sum @ E @ F3 @ G3 @ H8 @ F @ G @ ( if @ ( sum_sum @ E @ G3 ) @ E2 @ ( sum_Inr @ G3 @ E @ X ) @ Y3 ) )
= ( sum_Inr @ H8 @ F3 @ ( G @ X ) ) ) )
& ( ~ E2
=> ( ( sum_map_sum @ E @ F3 @ G3 @ H8 @ F @ G @ ( if @ ( sum_sum @ E @ G3 ) @ E2 @ ( sum_Inr @ G3 @ E @ X ) @ Y3 ) )
= ( sum_map_sum @ E @ F3 @ G3 @ H8 @ F @ G @ Y3 ) ) ) ) ).
% map_sum_if_distrib_then(2)
thf(fact_7049_map__sum__if__distrib__else_I2_J,axiom,
! [E: $tType,F3: $tType,H8: $tType,G3: $tType,E2: $o,F: E > F3,G: G3 > H8,X: sum_sum @ E @ G3,Y3: G3] :
( ( E2
=> ( ( sum_map_sum @ E @ F3 @ G3 @ H8 @ F @ G @ ( if @ ( sum_sum @ E @ G3 ) @ E2 @ X @ ( sum_Inr @ G3 @ E @ Y3 ) ) )
= ( sum_map_sum @ E @ F3 @ G3 @ H8 @ F @ G @ X ) ) )
& ( ~ E2
=> ( ( sum_map_sum @ E @ F3 @ G3 @ H8 @ F @ G @ ( if @ ( sum_sum @ E @ G3 ) @ E2 @ X @ ( sum_Inr @ G3 @ E @ Y3 ) ) )
= ( sum_Inr @ H8 @ F3 @ ( G @ Y3 ) ) ) ) ) ).
% map_sum_if_distrib_else(2)
thf(fact_7050_map__sum__if__distrib__then_I1_J,axiom,
! [D: $tType,B: $tType,C: $tType,A: $tType,E2: $o,F: A > B,G: C > D,X: A,Y3: sum_sum @ A @ C] :
( ( E2
=> ( ( sum_map_sum @ A @ B @ C @ D @ F @ G @ ( if @ ( sum_sum @ A @ C ) @ E2 @ ( sum_Inl @ A @ C @ X ) @ Y3 ) )
= ( sum_Inl @ B @ D @ ( F @ X ) ) ) )
& ( ~ E2
=> ( ( sum_map_sum @ A @ B @ C @ D @ F @ G @ ( if @ ( sum_sum @ A @ C ) @ E2 @ ( sum_Inl @ A @ C @ X ) @ Y3 ) )
= ( sum_map_sum @ A @ B @ C @ D @ F @ G @ Y3 ) ) ) ) ).
% map_sum_if_distrib_then(1)
thf(fact_7051_map__sum__if__distrib__else_I1_J,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,E2: $o,F: A > B,G: C > D,X: sum_sum @ A @ C,Y3: A] :
( ( E2
=> ( ( sum_map_sum @ A @ B @ C @ D @ F @ G @ ( if @ ( sum_sum @ A @ C ) @ E2 @ X @ ( sum_Inl @ A @ C @ Y3 ) ) )
= ( sum_map_sum @ A @ B @ C @ D @ F @ G @ X ) ) )
& ( ~ E2
=> ( ( sum_map_sum @ A @ B @ C @ D @ F @ G @ ( if @ ( sum_sum @ A @ C ) @ E2 @ X @ ( sum_Inl @ A @ C @ Y3 ) ) )
= ( sum_Inl @ B @ D @ ( F @ Y3 ) ) ) ) ) ).
% map_sum_if_distrib_else(1)
thf(fact_7052_map__sum_Ocomp,axiom,
! [A: $tType,C: $tType,E: $tType,F3: $tType,D: $tType,B: $tType,F: C > E,G: D > F3,H: A > C,I: B > D] :
( ( comp @ ( sum_sum @ C @ D ) @ ( sum_sum @ E @ F3 ) @ ( sum_sum @ A @ B ) @ ( sum_map_sum @ C @ E @ D @ F3 @ F @ G ) @ ( sum_map_sum @ A @ C @ B @ D @ H @ I ) )
= ( sum_map_sum @ A @ E @ B @ F3 @ ( comp @ C @ E @ A @ F @ H ) @ ( comp @ D @ F3 @ B @ G @ I ) ) ) ).
% map_sum.comp
thf(fact_7053_map__sum_Ocompositionality,axiom,
! [D: $tType,F3: $tType,E: $tType,C: $tType,B: $tType,A: $tType,F: C > E,G: D > F3,H: A > C,I: B > D,Sum: sum_sum @ A @ B] :
( ( sum_map_sum @ C @ E @ D @ F3 @ F @ G @ ( sum_map_sum @ A @ C @ B @ D @ H @ I @ Sum ) )
= ( sum_map_sum @ A @ E @ B @ F3 @ ( comp @ C @ E @ A @ F @ H ) @ ( comp @ D @ F3 @ B @ G @ I ) @ Sum ) ) ).
% map_sum.compositionality
thf(fact_7054_sum_Omap__comp,axiom,
! [D: $tType,F3: $tType,E: $tType,C: $tType,B: $tType,A: $tType,G1: C > E,G22: D > F3,F1: A > C,F23: B > D,V: sum_sum @ A @ B] :
( ( sum_map_sum @ C @ E @ D @ F3 @ G1 @ G22 @ ( sum_map_sum @ A @ C @ B @ D @ F1 @ F23 @ V ) )
= ( sum_map_sum @ A @ E @ B @ F3 @ ( comp @ C @ E @ A @ G1 @ F1 ) @ ( comp @ D @ F3 @ B @ G22 @ F23 ) @ V ) ) ).
% sum.map_comp
thf(fact_7055_map__sum__o__inj_I1_J,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,F: A > B,G: D > C] :
( ( comp @ ( sum_sum @ A @ D ) @ ( sum_sum @ B @ C ) @ A @ ( sum_map_sum @ A @ B @ D @ C @ F @ G ) @ ( sum_Inl @ A @ D ) )
= ( comp @ B @ ( sum_sum @ B @ C ) @ A @ ( sum_Inl @ B @ C ) @ F ) ) ).
% map_sum_o_inj(1)
thf(fact_7056_map__sum__o__inj_I2_J,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,F: A > B,G: D > C] :
( ( comp @ ( sum_sum @ A @ D ) @ ( sum_sum @ B @ C ) @ D @ ( sum_map_sum @ A @ B @ D @ C @ F @ G ) @ ( sum_Inr @ D @ A ) )
= ( comp @ C @ ( sum_sum @ B @ C ) @ D @ ( sum_Inr @ C @ B ) @ G ) ) ).
% map_sum_o_inj(2)
thf(fact_7057_case__sum__o__map__sum,axiom,
! [A: $tType,D: $tType,C: $tType,E: $tType,B: $tType,F: D > C,G: E > C,H1: A > D,H22: B > E] :
( ( comp @ ( sum_sum @ D @ E ) @ C @ ( sum_sum @ A @ B ) @ ( sum_case_sum @ D @ C @ E @ F @ G ) @ ( sum_map_sum @ A @ D @ B @ E @ H1 @ H22 ) )
= ( sum_case_sum @ A @ C @ B @ ( comp @ D @ C @ A @ F @ H1 ) @ ( comp @ E @ C @ B @ G @ H22 ) ) ) ).
% case_sum_o_map_sum
thf(fact_7058_case__sum__map__sum,axiom,
! [C: $tType,A: $tType,B: $tType,E: $tType,D: $tType,L: B > A,R: C > A,F: D > B,G: E > C,X: sum_sum @ D @ E] :
( ( sum_case_sum @ B @ A @ C @ L @ R @ ( sum_map_sum @ D @ B @ E @ C @ F @ G @ X ) )
= ( sum_case_sum @ D @ A @ E @ ( comp @ B @ A @ D @ L @ F ) @ ( comp @ C @ A @ E @ R @ G ) @ X ) ) ).
% case_sum_map_sum
thf(fact_7059_sum_Osize__gen__o__map,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,F: C > nat,Fa: D > nat,G: A > C,Ga: B > D] :
( ( comp @ ( sum_sum @ C @ D ) @ nat @ ( sum_sum @ A @ B ) @ ( basic_BNF_size_sum @ C @ D @ F @ Fa ) @ ( sum_map_sum @ A @ C @ B @ D @ G @ Ga ) )
= ( basic_BNF_size_sum @ A @ B @ ( comp @ C @ nat @ A @ F @ G ) @ ( comp @ D @ nat @ B @ Fa @ Ga ) ) ) ).
% sum.size_gen_o_map
thf(fact_7060_sum__list_Omonoid__list__axioms,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ( groups_monoid_list @ A @ ( plus_plus @ A ) @ ( zero_zero @ A ) ) ) ).
% sum_list.monoid_list_axioms
thf(fact_7061_sum_Orel__Grp,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,A16: set @ A,F1: A > C,A25: set @ B,F23: B > D] :
( ( bNF_rel_sum @ A @ C @ B @ D @ ( bNF_Grp @ A @ C @ A16 @ F1 ) @ ( bNF_Grp @ B @ D @ A25 @ F23 ) )
= ( bNF_Grp @ ( sum_sum @ A @ B ) @ ( sum_sum @ C @ D )
@ ( collect @ ( sum_sum @ A @ B )
@ ^ [X2: sum_sum @ A @ B] :
( ( ord_less_eq @ ( set @ A ) @ ( basic_setl @ A @ B @ X2 ) @ A16 )
& ( ord_less_eq @ ( set @ B ) @ ( basic_setr @ A @ B @ X2 ) @ A25 ) ) )
@ ( sum_map_sum @ A @ C @ B @ D @ F1 @ F23 ) ) ) ).
% sum.rel_Grp
thf(fact_7062_sum_Oin__rel,axiom,
! [B: $tType,D: $tType,C: $tType,A: $tType] :
( ( bNF_rel_sum @ A @ C @ B @ D )
= ( ^ [R14: A > C > $o,R25: B > D > $o,A6: sum_sum @ A @ B,B5: sum_sum @ C @ D] :
? [Z3: sum_sum @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D )] :
( ( member @ ( sum_sum @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) ) @ Z3
@ ( collect @ ( sum_sum @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) )
@ ^ [X2: sum_sum @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ C ) ) @ ( basic_setl @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) @ X2 ) @ ( collect @ ( product_prod @ A @ C ) @ ( product_case_prod @ A @ C @ $o @ R14 ) ) )
& ( ord_less_eq @ ( set @ ( product_prod @ B @ D ) ) @ ( basic_setr @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) @ X2 ) @ ( collect @ ( product_prod @ B @ D ) @ ( product_case_prod @ B @ D @ $o @ R25 ) ) ) ) ) )
& ( ( sum_map_sum @ ( product_prod @ A @ C ) @ A @ ( product_prod @ B @ D ) @ B @ ( product_fst @ A @ C ) @ ( product_fst @ B @ D ) @ Z3 )
= A6 )
& ( ( sum_map_sum @ ( product_prod @ A @ C ) @ C @ ( product_prod @ B @ D ) @ D @ ( product_snd @ A @ C ) @ ( product_snd @ B @ D ) @ Z3 )
= B5 ) ) ) ) ).
% sum.in_rel
thf(fact_7063_sum_Orel__map_I1_J,axiom,
! [A: $tType,B: $tType,E: $tType,F3: $tType,D: $tType,C: $tType,S1b: E > C > $o,S2b: F3 > D > $o,I1: A > E,I22: B > F3,X: sum_sum @ A @ B,Y3: sum_sum @ C @ D] :
( ( bNF_rel_sum @ E @ C @ F3 @ D @ S1b @ S2b @ ( sum_map_sum @ A @ E @ B @ F3 @ I1 @ I22 @ X ) @ Y3 )
= ( bNF_rel_sum @ A @ C @ B @ D
@ ^ [X2: A] : ( S1b @ ( I1 @ X2 ) )
@ ^ [X2: B] : ( S2b @ ( I22 @ X2 ) )
@ X
@ Y3 ) ) ).
% sum.rel_map(1)
thf(fact_7064_sum_Orel__map_I2_J,axiom,
! [A: $tType,B: $tType,E: $tType,F3: $tType,D: $tType,C: $tType,S1a: A > E > $o,S2a: B > F3 > $o,X: sum_sum @ A @ B,G1: C > E,G22: D > F3,Y3: sum_sum @ C @ D] :
( ( bNF_rel_sum @ A @ E @ B @ F3 @ S1a @ S2a @ X @ ( sum_map_sum @ C @ E @ D @ F3 @ G1 @ G22 @ Y3 ) )
= ( bNF_rel_sum @ A @ C @ B @ D
@ ^ [X2: A,Y: C] : ( S1a @ X2 @ ( G1 @ Y ) )
@ ^ [X2: B,Y: D] : ( S2a @ X2 @ ( G22 @ Y ) )
@ X
@ Y3 ) ) ).
% sum.rel_map(2)
thf(fact_7065_case__sum__transfer,axiom,
! [A: $tType,B: $tType,E: $tType,F3: $tType,D: $tType,C: $tType,R5: A > C > $o,T5: B > D > $o,S3: E > F3 > $o] : ( bNF_rel_fun @ ( A > B ) @ ( C > D ) @ ( ( E > B ) > ( sum_sum @ A @ E ) > B ) @ ( ( F3 > D ) > ( sum_sum @ C @ F3 ) > D ) @ ( bNF_rel_fun @ A @ C @ B @ D @ R5 @ T5 ) @ ( bNF_rel_fun @ ( E > B ) @ ( F3 > D ) @ ( ( sum_sum @ A @ E ) > B ) @ ( ( sum_sum @ C @ F3 ) > D ) @ ( bNF_rel_fun @ E @ F3 @ B @ D @ S3 @ T5 ) @ ( bNF_rel_fun @ ( sum_sum @ A @ E ) @ ( sum_sum @ C @ F3 ) @ B @ D @ ( bNF_rel_sum @ A @ C @ E @ F3 @ R5 @ S3 ) @ T5 ) ) @ ( sum_case_sum @ A @ B @ E ) @ ( sum_case_sum @ C @ D @ F3 ) ) ).
% case_sum_transfer
thf(fact_7066_Inr__transfer,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,T5: A > B > $o,S3: C > D > $o] : ( bNF_rel_fun @ A @ B @ ( sum_sum @ C @ A ) @ ( sum_sum @ D @ B ) @ T5 @ ( bNF_rel_sum @ C @ D @ A @ B @ S3 @ T5 ) @ ( sum_Inr @ A @ C ) @ ( sum_Inr @ B @ D ) ) ).
% Inr_transfer
thf(fact_7067_Inl__transfer,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,S3: A > B > $o,T5: C > D > $o] : ( bNF_rel_fun @ A @ B @ ( sum_sum @ A @ C ) @ ( sum_sum @ B @ D ) @ S3 @ ( bNF_rel_sum @ A @ B @ C @ D @ S3 @ T5 ) @ ( sum_Inl @ A @ C ) @ ( sum_Inl @ B @ D ) ) ).
% Inl_transfer
thf(fact_7068_sum_Orel__compp__Grp,axiom,
! [B: $tType,D: $tType,C: $tType,A: $tType] :
( ( bNF_rel_sum @ A @ C @ B @ D )
= ( ^ [R14: A > C > $o,R25: B > D > $o] :
( relcompp @ ( sum_sum @ A @ B ) @ ( sum_sum @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) ) @ ( sum_sum @ C @ D )
@ ( conversep @ ( sum_sum @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) ) @ ( sum_sum @ A @ B )
@ ( bNF_Grp @ ( sum_sum @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) ) @ ( sum_sum @ A @ B )
@ ( collect @ ( sum_sum @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) )
@ ^ [X2: sum_sum @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ C ) ) @ ( basic_setl @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) @ X2 ) @ ( collect @ ( product_prod @ A @ C ) @ ( product_case_prod @ A @ C @ $o @ R14 ) ) )
& ( ord_less_eq @ ( set @ ( product_prod @ B @ D ) ) @ ( basic_setr @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) @ X2 ) @ ( collect @ ( product_prod @ B @ D ) @ ( product_case_prod @ B @ D @ $o @ R25 ) ) ) ) )
@ ( sum_map_sum @ ( product_prod @ A @ C ) @ A @ ( product_prod @ B @ D ) @ B @ ( product_fst @ A @ C ) @ ( product_fst @ B @ D ) ) ) )
@ ( bNF_Grp @ ( sum_sum @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) ) @ ( sum_sum @ C @ D )
@ ( collect @ ( sum_sum @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) )
@ ^ [X2: sum_sum @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ C ) ) @ ( basic_setl @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) @ X2 ) @ ( collect @ ( product_prod @ A @ C ) @ ( product_case_prod @ A @ C @ $o @ R14 ) ) )
& ( ord_less_eq @ ( set @ ( product_prod @ B @ D ) ) @ ( basic_setr @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) @ X2 ) @ ( collect @ ( product_prod @ B @ D ) @ ( product_case_prod @ B @ D @ $o @ R25 ) ) ) ) )
@ ( sum_map_sum @ ( product_prod @ A @ C ) @ C @ ( product_prod @ B @ D ) @ D @ ( product_snd @ A @ C ) @ ( product_snd @ B @ D ) ) ) ) ) ) ).
% sum.rel_compp_Grp
thf(fact_7069_typerep_Osize__neq,axiom,
! [X: typerep] :
( ( size_size @ typerep @ X )
!= ( zero_zero @ nat ) ) ).
% typerep.size_neq
thf(fact_7070_dual__order_Opreordering__axioms,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( preordering @ A
@ ^ [X2: A,Y: A] : ( ord_less_eq @ A @ Y @ X2 )
@ ^ [X2: A,Y: A] : ( ord_less @ A @ Y @ X2 ) ) ) ).
% dual_order.preordering_axioms
thf(fact_7071_preordering__dualI,axiom,
! [A: $tType,Less_eq2: A > A > $o,Less: A > A > $o] :
( ( preordering @ A
@ ^ [A6: A,B5: A] : ( Less_eq2 @ B5 @ A6 )
@ ^ [A6: A,B5: A] : ( Less @ B5 @ A6 ) )
=> ( preordering @ A @ Less_eq2 @ Less ) ) ).
% preordering_dualI
thf(fact_7072_gcd__nat_Opreordering__axioms,axiom,
( preordering @ nat @ ( dvd_dvd @ nat )
@ ^ [M5: nat,N5: nat] :
( ( dvd_dvd @ nat @ M5 @ N5 )
& ( M5 != N5 ) ) ) ).
% gcd_nat.preordering_axioms
thf(fact_7073_typerep_Osize_I2_J,axiom,
! [X1: literal,X22: list @ typerep] :
( ( size_size @ typerep @ ( typerep2 @ X1 @ X22 ) )
= ( plus_plus @ nat @ ( size_list @ typerep @ ( size_size @ typerep ) @ X22 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% typerep.size(2)
thf(fact_7074_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 )
@ ^ [X2: A] :
( ( P @ X2 )
| ( Q @ X2 ) ) ) ) ) ) ).
% admissible_disj
thf(fact_7075_admissible__True,axiom,
! [A: $tType,Lub: ( set @ A ) > A,Ord: A > A > $o] :
( comple1908693960933563346ssible @ A @ Lub @ Ord
@ ^ [X2: A] : $true ) ).
% admissible_True
thf(fact_7076_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
@ ^ [X2: A] :
( ( P @ X2 )
& ( Q @ X2 ) ) ) ) ) ).
% admissible_conj
thf(fact_7077_admissible__const,axiom,
! [A: $tType,Lub: ( set @ A ) > A,Ord: A > A > $o,T2: $o] :
( comple1908693960933563346ssible @ A @ Lub @ Ord
@ ^ [X2: A] : T2 ) ).
% admissible_const
thf(fact_7078_admissible__all,axiom,
! [A: $tType,B: $tType,Lub: ( set @ B ) > B,Ord: B > B > $o,P: B > A > $o] :
( ! [Y4: A] :
( comple1908693960933563346ssible @ B @ Lub @ Ord
@ ^ [X2: B] : ( P @ X2 @ Y4 ) )
=> ( comple1908693960933563346ssible @ B @ Lub @ Ord
@ ^ [X2: B] :
! [X9: A] : ( P @ X2 @ X9 ) ) ) ).
% admissible_all
thf(fact_7079_admissible__ball,axiom,
! [A: $tType,B: $tType,A5: set @ A,Lub: ( set @ B ) > B,Ord: B > B > $o,P: B > A > $o] :
( ! [Y4: A] :
( ( member @ A @ Y4 @ A5 )
=> ( comple1908693960933563346ssible @ B @ Lub @ Ord
@ ^ [X2: B] : ( P @ X2 @ Y4 ) ) )
=> ( comple1908693960933563346ssible @ B @ Lub @ Ord
@ ^ [X2: B] :
! [Y: A] :
( ( member @ A @ Y @ A5 )
=> ( P @ X2 @ Y ) ) ) ) ).
% admissible_ball
thf(fact_7080_typerep_Osize__gen,axiom,
! [X1: literal,X22: list @ typerep] :
( ( size_typerep @ ( typerep2 @ X1 @ X22 ) )
= ( plus_plus @ nat @ ( size_list @ typerep @ size_typerep @ X22 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% typerep.size_gen
thf(fact_7081_typerep_Orec,axiom,
! [A: $tType,F: literal > ( list @ ( product_prod @ typerep @ A ) ) > A,X1: literal,X22: list @ typerep] :
( ( rec_typerep @ A @ F @ ( typerep2 @ X1 @ X22 ) )
= ( F @ X1
@ ( map @ typerep @ ( product_prod @ typerep @ A )
@ ^ [Typerep: typerep] : ( product_Pair @ typerep @ A @ Typerep @ ( rec_typerep @ A @ F @ Typerep ) )
@ X22 ) ) ) ).
% typerep.rec
thf(fact_7082_comm__monoid__set_Ozero__middle,axiom,
! [A: $tType,F: A > A > A,Z: A,P4: nat,K: nat,G: nat > A,H: nat > A] :
( ( groups778175481326437816id_set @ A @ F @ Z )
=> ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ P4 )
=> ( ( ord_less_eq @ nat @ K @ P4 )
=> ( ( groups_comm_monoid_F @ A @ nat @ F @ Z
@ ^ [J3: nat] : ( if @ A @ ( ord_less @ nat @ J3 @ K ) @ ( G @ J3 ) @ ( if @ A @ ( J3 = K ) @ Z @ ( H @ ( minus_minus @ nat @ J3 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) )
@ ( set_ord_atMost @ nat @ P4 ) )
= ( groups_comm_monoid_F @ A @ nat @ F @ Z
@ ^ [J3: nat] : ( if @ A @ ( ord_less @ nat @ J3 @ K ) @ ( G @ J3 ) @ ( H @ J3 ) )
@ ( set_ord_atMost @ nat @ ( minus_minus @ nat @ P4 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) ) ).
% comm_monoid_set.zero_middle
thf(fact_7083_comm__monoid__set_Oinsert_H,axiom,
! [A: $tType,B: $tType,F: A > A > A,Z: A,I5: set @ B,P4: B > A,I: B] :
( ( groups778175481326437816id_set @ A @ F @ Z )
=> ( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ I5 )
& ( ( P4 @ X2 )
!= Z ) ) ) )
=> ( ( ( member @ B @ I @ I5 )
=> ( ( groups_comm_monoid_G @ A @ B @ F @ Z @ P4 @ ( insert2 @ B @ I @ I5 ) )
= ( groups_comm_monoid_G @ A @ B @ F @ Z @ P4 @ I5 ) ) )
& ( ~ ( member @ B @ I @ I5 )
=> ( ( groups_comm_monoid_G @ A @ B @ F @ Z @ P4 @ ( insert2 @ B @ I @ I5 ) )
= ( F @ ( P4 @ I ) @ ( groups_comm_monoid_G @ A @ B @ F @ Z @ P4 @ I5 ) ) ) ) ) ) ) ).
% comm_monoid_set.insert'
thf(fact_7084_comm__monoid__set_OUNION__disjoint,axiom,
! [A: $tType,C: $tType,B: $tType,F: A > A > A,Z: A,I5: set @ B,A5: B > ( set @ C ),G: C > A] :
( ( groups778175481326437816id_set @ A @ F @ Z )
=> ( ( finite_finite2 @ B @ I5 )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ I5 )
=> ( finite_finite2 @ C @ ( A5 @ X4 ) ) )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ I5 )
=> ! [Xa4: B] :
( ( member @ B @ Xa4 @ I5 )
=> ( ( X4 != Xa4 )
=> ( ( inf_inf @ ( set @ C ) @ ( A5 @ X4 ) @ ( A5 @ Xa4 ) )
= ( bot_bot @ ( set @ C ) ) ) ) ) )
=> ( ( groups_comm_monoid_F @ A @ C @ F @ Z @ G @ ( complete_Sup_Sup @ ( set @ C ) @ ( image2 @ B @ ( set @ C ) @ A5 @ I5 ) ) )
= ( groups_comm_monoid_F @ A @ B @ F @ Z
@ ^ [X2: B] : ( groups_comm_monoid_F @ A @ C @ F @ Z @ G @ ( A5 @ X2 ) )
@ I5 ) ) ) ) ) ) ).
% comm_monoid_set.UNION_disjoint
thf(fact_7085_comm__monoid__set_OIf__cases,axiom,
! [A: $tType,B: $tType,F: A > A > A,Z: A,A5: set @ B,P: B > $o,H: B > A,G: B > A] :
( ( groups778175481326437816id_set @ A @ F @ Z )
=> ( ( finite_finite2 @ B @ A5 )
=> ( ( groups_comm_monoid_F @ A @ B @ F @ Z
@ ^ [X2: B] : ( if @ A @ ( P @ X2 ) @ ( H @ X2 ) @ ( G @ X2 ) )
@ A5 )
= ( F @ ( groups_comm_monoid_F @ A @ B @ F @ Z @ H @ ( inf_inf @ ( set @ B ) @ A5 @ ( collect @ B @ P ) ) ) @ ( groups_comm_monoid_F @ A @ B @ F @ Z @ G @ ( inf_inf @ ( set @ B ) @ A5 @ ( uminus_uminus @ ( set @ B ) @ ( collect @ B @ P ) ) ) ) ) ) ) ) ).
% comm_monoid_set.If_cases
thf(fact_7086_comm__monoid__set_Ointer__restrict,axiom,
! [A: $tType,B: $tType,F: A > A > A,Z: A,A5: set @ B,G: B > A,B4: set @ B] :
( ( groups778175481326437816id_set @ A @ F @ Z )
=> ( ( finite_finite2 @ B @ A5 )
=> ( ( groups_comm_monoid_F @ A @ B @ F @ Z @ G @ ( inf_inf @ ( set @ B ) @ A5 @ B4 ) )
= ( groups_comm_monoid_F @ A @ B @ F @ Z
@ ^ [X2: B] : ( if @ A @ ( member @ B @ X2 @ B4 ) @ ( G @ X2 ) @ Z )
@ A5 ) ) ) ) ).
% comm_monoid_set.inter_restrict
thf(fact_7087_comm__monoid__set_Oub__add__nat,axiom,
! [A: $tType,F: A > A > A,Z: A,M: nat,N: nat,G: nat > A,P4: nat] :
( ( groups778175481326437816id_set @ A @ F @ Z )
=> ( ( ord_less_eq @ nat @ M @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) )
=> ( ( groups_comm_monoid_F @ A @ nat @ F @ Z @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ ( plus_plus @ nat @ N @ P4 ) ) )
= ( F @ ( groups_comm_monoid_F @ A @ nat @ F @ Z @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) @ ( groups_comm_monoid_F @ A @ nat @ F @ Z @ G @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) @ ( plus_plus @ nat @ N @ P4 ) ) ) ) ) ) ) ).
% comm_monoid_set.ub_add_nat
thf(fact_7088_comm__monoid__add__class_Osum__def,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ( ( groups7311177749621191930dd_sum @ B @ A )
= ( groups_comm_monoid_F @ A @ B @ ( plus_plus @ A ) @ ( zero_zero @ A ) ) ) ) ).
% comm_monoid_add_class.sum_def
thf(fact_7089_comm__monoid__set_Onat__group,axiom,
! [A: $tType,F: A > A > A,Z: A,G: nat > A,K: nat,N: nat] :
( ( groups778175481326437816id_set @ A @ F @ Z )
=> ( ( groups_comm_monoid_F @ A @ nat @ F @ Z
@ ^ [M5: nat] : ( groups_comm_monoid_F @ A @ nat @ F @ Z @ G @ ( set_or7035219750837199246ssThan @ nat @ ( times_times @ nat @ M5 @ K ) @ ( plus_plus @ nat @ ( times_times @ nat @ M5 @ K ) @ K ) ) )
@ ( set_ord_lessThan @ nat @ N ) )
= ( groups_comm_monoid_F @ A @ nat @ F @ Z @ G @ ( set_ord_lessThan @ nat @ ( times_times @ nat @ N @ K ) ) ) ) ) ).
% comm_monoid_set.nat_group
thf(fact_7090_comm__monoid__set_OatLeastLessThan__rev,axiom,
! [A: $tType,F: A > A > A,Z: A,G: nat > A,N: nat,M: nat] :
( ( groups778175481326437816id_set @ A @ F @ Z )
=> ( ( groups_comm_monoid_F @ A @ nat @ F @ Z @ G @ ( set_or7035219750837199246ssThan @ nat @ N @ M ) )
= ( groups_comm_monoid_F @ A @ nat @ F @ Z
@ ^ [I2: nat] : ( G @ ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N ) @ ( suc @ I2 ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ N @ M ) ) ) ) ).
% comm_monoid_set.atLeastLessThan_rev
thf(fact_7091_sum_Ocomm__monoid__set__axioms,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ( groups778175481326437816id_set @ A @ ( plus_plus @ A ) @ ( zero_zero @ A ) ) ) ).
% sum.comm_monoid_set_axioms
thf(fact_7092_comm__monoid__set_Oshift__bounds__nat__ivl,axiom,
! [A: $tType,F: A > A > A,Z: A,G: nat > A,M: nat,K: nat,N: nat] :
( ( groups778175481326437816id_set @ A @ F @ Z )
=> ( ( groups_comm_monoid_F @ A @ nat @ F @ Z @ G @ ( set_or7035219750837199246ssThan @ nat @ ( plus_plus @ nat @ M @ K ) @ ( plus_plus @ nat @ N @ K ) ) )
= ( groups_comm_monoid_F @ A @ nat @ F @ Z
@ ^ [I2: nat] : ( G @ ( plus_plus @ nat @ I2 @ K ) )
@ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ).
% comm_monoid_set.shift_bounds_nat_ivl
thf(fact_7093_comm__monoid__set_Oshift__bounds__cl__nat__ivl,axiom,
! [A: $tType,F: A > A > A,Z: A,G: nat > A,M: nat,K: nat,N: nat] :
( ( groups778175481326437816id_set @ A @ F @ Z )
=> ( ( groups_comm_monoid_F @ A @ nat @ F @ Z @ G @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ M @ K ) @ ( plus_plus @ nat @ N @ K ) ) )
= ( groups_comm_monoid_F @ A @ nat @ F @ Z
@ ^ [I2: nat] : ( G @ ( plus_plus @ nat @ I2 @ K ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% comm_monoid_set.shift_bounds_cl_nat_ivl
thf(fact_7094_comm__monoid__set_OatLeastAtMost__rev,axiom,
! [A: $tType,F: A > A > A,Z: A,G: nat > A,N: nat,M: nat] :
( ( groups778175481326437816id_set @ A @ F @ Z )
=> ( ( groups_comm_monoid_F @ A @ nat @ F @ Z @ G @ ( set_or1337092689740270186AtMost @ nat @ N @ M ) )
= ( groups_comm_monoid_F @ A @ nat @ F @ Z
@ ^ [I2: nat] : ( G @ ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N ) @ I2 ) )
@ ( set_or1337092689740270186AtMost @ nat @ N @ M ) ) ) ) ).
% comm_monoid_set.atLeastAtMost_rev
thf(fact_7095_comm__monoid__set_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
! [A: $tType,F: A > A > A,Z: A,G: nat > A,N: nat,M: nat] :
( ( groups778175481326437816id_set @ A @ F @ Z )
=> ( ( groups_comm_monoid_F @ A @ nat @ F @ Z @ G @ ( set_or7035219750837199246ssThan @ nat @ N @ M ) )
= ( groups_comm_monoid_F @ A @ nat @ F @ Z
@ ^ [I2: nat] : ( G @ ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N ) @ I2 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ N ) @ M ) ) ) ) ).
% comm_monoid_set.atLeastLessThan_rev_at_least_Suc_atMost
thf(fact_7096_comm__monoid__mult__class_Oprod__def,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ( ( groups7121269368397514597t_prod @ B @ A )
= ( groups_comm_monoid_F @ A @ B @ ( times_times @ A ) @ ( one_one @ A ) ) ) ) ).
% comm_monoid_mult_class.prod_def
thf(fact_7097_comm__monoid__set_Onested__swap,axiom,
! [A: $tType,F: A > A > A,Z: A,A3: nat > nat > A,N: nat] :
( ( groups778175481326437816id_set @ A @ F @ Z )
=> ( ( groups_comm_monoid_F @ A @ nat @ F @ Z
@ ^ [I2: nat] : ( groups_comm_monoid_F @ A @ nat @ F @ Z @ ( A3 @ I2 ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ I2 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
= ( groups_comm_monoid_F @ A @ nat @ F @ Z
@ ^ [J3: nat] :
( groups_comm_monoid_F @ A @ nat @ F @ Z
@ ^ [I2: nat] : ( A3 @ I2 @ J3 )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J3 ) @ N ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% comm_monoid_set.nested_swap
thf(fact_7098_comm__monoid__set_Ofinite__Collect__op,axiom,
! [A: $tType,B: $tType,F: A > A > A,Z: A,I5: set @ B,X: B > A,Y3: B > A] :
( ( groups778175481326437816id_set @ A @ F @ Z )
=> ( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [I2: B] :
( ( member @ B @ I2 @ I5 )
& ( ( X @ I2 )
!= Z ) ) ) )
=> ( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [I2: B] :
( ( member @ B @ I2 @ I5 )
& ( ( Y3 @ I2 )
!= Z ) ) ) )
=> ( finite_finite2 @ B
@ ( collect @ B
@ ^ [I2: B] :
( ( member @ B @ I2 @ I5 )
& ( ( F @ ( X @ I2 ) @ ( Y3 @ I2 ) )
!= Z ) ) ) ) ) ) ) ).
% comm_monoid_set.finite_Collect_op
thf(fact_7099_comm__monoid__set_Odelta,axiom,
! [B: $tType,A: $tType,F: A > A > A,Z: A,S3: set @ B,A3: B,B2: B > A] :
( ( groups778175481326437816id_set @ A @ F @ Z )
=> ( ( finite_finite2 @ B @ S3 )
=> ( ( ( member @ B @ A3 @ S3 )
=> ( ( groups_comm_monoid_F @ A @ B @ F @ Z
@ ^ [K3: B] : ( if @ A @ ( K3 = A3 ) @ ( B2 @ K3 ) @ Z )
@ S3 )
= ( B2 @ A3 ) ) )
& ( ~ ( member @ B @ A3 @ S3 )
=> ( ( groups_comm_monoid_F @ A @ B @ F @ Z
@ ^ [K3: B] : ( if @ A @ ( K3 = A3 ) @ ( B2 @ K3 ) @ Z )
@ S3 )
= Z ) ) ) ) ) ).
% comm_monoid_set.delta
thf(fact_7100_comm__monoid__set_Odelta_H,axiom,
! [B: $tType,A: $tType,F: A > A > A,Z: A,S3: set @ B,A3: B,B2: B > A] :
( ( groups778175481326437816id_set @ A @ F @ Z )
=> ( ( finite_finite2 @ B @ S3 )
=> ( ( ( member @ B @ A3 @ S3 )
=> ( ( groups_comm_monoid_F @ A @ B @ F @ Z
@ ^ [K3: B] : ( if @ A @ ( A3 = K3 ) @ ( B2 @ K3 ) @ Z )
@ S3 )
= ( B2 @ A3 ) ) )
& ( ~ ( member @ B @ A3 @ S3 )
=> ( ( groups_comm_monoid_F @ A @ B @ F @ Z
@ ^ [K3: B] : ( if @ A @ ( A3 = K3 ) @ ( B2 @ K3 ) @ Z )
@ S3 )
= Z ) ) ) ) ) ).
% comm_monoid_set.delta'
thf(fact_7101_comm__monoid__set_Ointer__filter,axiom,
! [A: $tType,B: $tType,F: A > A > A,Z: A,A5: set @ B,G: B > A,P: B > $o] :
( ( groups778175481326437816id_set @ A @ F @ Z )
=> ( ( finite_finite2 @ B @ A5 )
=> ( ( groups_comm_monoid_F @ A @ B @ F @ Z @ G
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ A5 )
& ( P @ X2 ) ) ) )
= ( groups_comm_monoid_F @ A @ B @ F @ Z
@ ^ [X2: B] : ( if @ A @ ( P @ X2 ) @ ( G @ X2 ) @ Z )
@ A5 ) ) ) ) ).
% comm_monoid_set.inter_filter
thf(fact_7102_comm__monoid__set_Oswap__restrict,axiom,
! [A: $tType,B: $tType,C: $tType,F: A > A > A,Z: A,A5: set @ B,B4: set @ C,G: B > C > A,R5: B > C > $o] :
( ( groups778175481326437816id_set @ A @ F @ Z )
=> ( ( finite_finite2 @ B @ A5 )
=> ( ( finite_finite2 @ C @ B4 )
=> ( ( groups_comm_monoid_F @ A @ B @ F @ Z
@ ^ [X2: B] :
( groups_comm_monoid_F @ A @ C @ F @ Z @ ( G @ X2 )
@ ( collect @ C
@ ^ [Y: C] :
( ( member @ C @ Y @ B4 )
& ( R5 @ X2 @ Y ) ) ) )
@ A5 )
= ( groups_comm_monoid_F @ A @ C @ F @ Z
@ ^ [Y: C] :
( groups_comm_monoid_F @ A @ B @ F @ Z
@ ^ [X2: B] : ( G @ X2 @ Y )
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ A5 )
& ( R5 @ X2 @ Y ) ) ) )
@ B4 ) ) ) ) ) ).
% comm_monoid_set.swap_restrict
thf(fact_7103_comm__monoid__set_Osetdiff__irrelevant,axiom,
! [A: $tType,B: $tType,F: A > A > A,Z: A,A5: set @ B,G: B > A] :
( ( groups778175481326437816id_set @ A @ F @ Z )
=> ( ( finite_finite2 @ B @ A5 )
=> ( ( groups_comm_monoid_F @ A @ B @ F @ Z @ G
@ ( minus_minus @ ( set @ B ) @ A5
@ ( collect @ B
@ ^ [X2: B] :
( ( G @ X2 )
= Z ) ) ) )
= ( groups_comm_monoid_F @ A @ B @ F @ Z @ G @ A5 ) ) ) ) ).
% comm_monoid_set.setdiff_irrelevant
thf(fact_7104_comm__monoid__set_Oreindex__bij__betw__not__neutral,axiom,
! [B: $tType,A: $tType,C: $tType,F: A > A > A,Z: A,S6: set @ B,T4: set @ C,H: B > C,S3: set @ B,T5: set @ C,G: C > A] :
( ( groups778175481326437816id_set @ A @ F @ Z )
=> ( ( finite_finite2 @ B @ S6 )
=> ( ( finite_finite2 @ C @ T4 )
=> ( ( bij_betw @ B @ C @ H @ ( minus_minus @ ( set @ B ) @ S3 @ S6 ) @ ( minus_minus @ ( set @ C ) @ T5 @ T4 ) )
=> ( ! [A4: B] :
( ( member @ B @ A4 @ S6 )
=> ( ( G @ ( H @ A4 ) )
= Z ) )
=> ( ! [B3: C] :
( ( member @ C @ B3 @ T4 )
=> ( ( G @ B3 )
= Z ) )
=> ( ( groups_comm_monoid_F @ A @ B @ F @ Z
@ ^ [X2: B] : ( G @ ( H @ X2 ) )
@ S3 )
= ( groups_comm_monoid_F @ A @ C @ F @ Z @ G @ T5 ) ) ) ) ) ) ) ) ).
% comm_monoid_set.reindex_bij_betw_not_neutral
thf(fact_7105_prod_Ocomm__monoid__set__axioms,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ( groups778175481326437816id_set @ A @ ( times_times @ A ) @ ( one_one @ A ) ) ) ).
% prod.comm_monoid_set_axioms
thf(fact_7106_comm__monoid__set_Oreindex__bij__betw,axiom,
! [B: $tType,A: $tType,C: $tType,F: A > A > A,Z: A,H: B > C,S3: set @ B,T5: set @ C,G: C > A] :
( ( groups778175481326437816id_set @ A @ F @ Z )
=> ( ( bij_betw @ B @ C @ H @ S3 @ T5 )
=> ( ( groups_comm_monoid_F @ A @ B @ F @ Z
@ ^ [X2: B] : ( G @ ( H @ X2 ) )
@ S3 )
= ( groups_comm_monoid_F @ A @ C @ F @ Z @ G @ T5 ) ) ) ) ).
% comm_monoid_set.reindex_bij_betw
thf(fact_7107_comm__monoid__set_Oshift__bounds__Suc__ivl,axiom,
! [A: $tType,F: A > A > A,Z: A,G: nat > A,M: nat,N: nat] :
( ( groups778175481326437816id_set @ A @ F @ Z )
=> ( ( groups_comm_monoid_F @ A @ nat @ F @ Z @ G @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M ) @ ( suc @ N ) ) )
= ( groups_comm_monoid_F @ A @ nat @ F @ Z
@ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
@ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ).
% comm_monoid_set.shift_bounds_Suc_ivl
thf(fact_7108_comm__monoid__set_Oswap,axiom,
! [A: $tType,B: $tType,C: $tType,F: A > A > A,Z: A,G: B > C > A,B4: set @ C,A5: set @ B] :
( ( groups778175481326437816id_set @ A @ F @ Z )
=> ( ( groups_comm_monoid_F @ A @ B @ F @ Z
@ ^ [I2: B] : ( groups_comm_monoid_F @ A @ C @ F @ Z @ ( G @ I2 ) @ B4 )
@ A5 )
= ( groups_comm_monoid_F @ A @ C @ F @ Z
@ ^ [J3: C] :
( groups_comm_monoid_F @ A @ B @ F @ Z
@ ^ [I2: B] : ( G @ I2 @ J3 )
@ A5 )
@ B4 ) ) ) ).
% comm_monoid_set.swap
thf(fact_7109_comm__monoid__set_Odistrib,axiom,
! [A: $tType,B: $tType,F: A > A > A,Z: A,G: B > A,H: B > A,A5: set @ B] :
( ( groups778175481326437816id_set @ A @ F @ Z )
=> ( ( groups_comm_monoid_F @ A @ B @ F @ Z
@ ^ [X2: B] : ( F @ ( G @ X2 ) @ ( H @ X2 ) )
@ A5 )
= ( F @ ( groups_comm_monoid_F @ A @ B @ F @ Z @ G @ A5 ) @ ( groups_comm_monoid_F @ A @ B @ F @ Z @ H @ A5 ) ) ) ) ).
% comm_monoid_set.distrib
thf(fact_7110_comm__monoid__set_Oneutral__const,axiom,
! [B: $tType,A: $tType,F: A > A > A,Z: A,A5: set @ B] :
( ( groups778175481326437816id_set @ A @ F @ Z )
=> ( ( groups_comm_monoid_F @ A @ B @ F @ Z
@ ^ [Uu: B] : Z
@ A5 )
= Z ) ) ).
% comm_monoid_set.neutral_const
thf(fact_7111_comm__monoid__set_Oshift__bounds__cl__Suc__ivl,axiom,
! [A: $tType,F: A > A > A,Z: A,G: nat > A,M: nat,N: nat] :
( ( groups778175481326437816id_set @ A @ F @ Z )
=> ( ( groups_comm_monoid_F @ A @ nat @ F @ Z @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ ( suc @ N ) ) )
= ( groups_comm_monoid_F @ A @ nat @ F @ Z
@ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% comm_monoid_set.shift_bounds_cl_Suc_ivl
thf(fact_7112_comm__monoid__set_OatLeast0__lessThan__Suc,axiom,
! [A: $tType,F: A > A > A,Z: A,G: nat > A,N: nat] :
( ( groups778175481326437816id_set @ A @ F @ Z )
=> ( ( groups_comm_monoid_F @ A @ nat @ F @ Z @ G @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
= ( F @ ( groups_comm_monoid_F @ A @ nat @ F @ Z @ G @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) @ ( G @ N ) ) ) ) ).
% comm_monoid_set.atLeast0_lessThan_Suc
thf(fact_7113_comm__monoid__set_OatLeast0__atMost__Suc,axiom,
! [A: $tType,F: A > A > A,Z: A,G: nat > A,N: nat] :
( ( groups778175481326437816id_set @ A @ F @ Z )
=> ( ( groups_comm_monoid_F @ A @ nat @ F @ Z @ G @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
= ( F @ ( groups_comm_monoid_F @ A @ nat @ F @ Z @ G @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ).
% comm_monoid_set.atLeast0_atMost_Suc
thf(fact_7114_comm__monoid__set_OatMost__Suc__shift,axiom,
! [A: $tType,F: A > A > A,Z: A,G: nat > A,N: nat] :
( ( groups778175481326437816id_set @ A @ F @ Z )
=> ( ( groups_comm_monoid_F @ A @ nat @ F @ Z @ G @ ( set_ord_atMost @ nat @ ( suc @ N ) ) )
= ( F @ ( G @ ( zero_zero @ nat ) )
@ ( groups_comm_monoid_F @ A @ nat @ F @ Z
@ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ) ).
% comm_monoid_set.atMost_Suc_shift
thf(fact_7115_comm__monoid__set_OlessThan__Suc__shift,axiom,
! [A: $tType,F: A > A > A,Z: A,G: nat > A,N: nat] :
( ( groups778175481326437816id_set @ A @ F @ Z )
=> ( ( groups_comm_monoid_F @ A @ nat @ F @ Z @ G @ ( set_ord_lessThan @ nat @ ( suc @ N ) ) )
= ( F @ ( G @ ( zero_zero @ nat ) )
@ ( groups_comm_monoid_F @ A @ nat @ F @ Z
@ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% comm_monoid_set.lessThan_Suc_shift
thf(fact_7116_comm__monoid__set_Onat__diff__reindex,axiom,
! [A: $tType,F: A > A > A,Z: A,G: nat > A,N: nat] :
( ( groups778175481326437816id_set @ A @ F @ Z )
=> ( ( groups_comm_monoid_F @ A @ nat @ F @ Z
@ ^ [I2: nat] : ( G @ ( minus_minus @ nat @ N @ ( suc @ I2 ) ) )
@ ( set_ord_lessThan @ nat @ N ) )
= ( groups_comm_monoid_F @ A @ nat @ F @ Z @ G @ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% comm_monoid_set.nat_diff_reindex
thf(fact_7117_comm__monoid__set_OatLeast1__atMost__eq,axiom,
! [A: $tType,F: A > A > A,Z: A,G: nat > A,N: nat] :
( ( groups778175481326437816id_set @ A @ F @ Z )
=> ( ( groups_comm_monoid_F @ A @ nat @ F @ Z @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N ) )
= ( groups_comm_monoid_F @ A @ nat @ F @ Z
@ ^ [K3: nat] : ( G @ ( suc @ K3 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% comm_monoid_set.atLeast1_atMost_eq
thf(fact_7118_comm__monoid__set_OatMost__shift,axiom,
! [A: $tType,F: A > A > A,Z: A,G: nat > A,N: nat] :
( ( groups778175481326437816id_set @ A @ F @ Z )
=> ( ( groups_comm_monoid_F @ A @ nat @ F @ Z @ G @ ( set_ord_atMost @ nat @ N ) )
= ( F @ ( G @ ( zero_zero @ nat ) )
@ ( groups_comm_monoid_F @ A @ nat @ F @ Z
@ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% comm_monoid_set.atMost_shift
thf(fact_7119_comm__monoid__set_Onested__swap_H,axiom,
! [A: $tType,F: A > A > A,Z: A,A3: nat > nat > A,N: nat] :
( ( groups778175481326437816id_set @ A @ F @ Z )
=> ( ( groups_comm_monoid_F @ A @ nat @ F @ Z
@ ^ [I2: nat] : ( groups_comm_monoid_F @ A @ nat @ F @ Z @ ( A3 @ I2 ) @ ( set_ord_lessThan @ nat @ I2 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( groups_comm_monoid_F @ A @ nat @ F @ Z
@ ^ [J3: nat] :
( groups_comm_monoid_F @ A @ nat @ F @ Z
@ ^ [I2: nat] : ( A3 @ I2 @ J3 )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J3 ) @ N ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% comm_monoid_set.nested_swap'
thf(fact_7120_comm__monoid__set_Onon__neutral_H,axiom,
! [A: $tType,B: $tType,F: A > A > A,Z: A,G: B > A,I5: set @ B] :
( ( groups778175481326437816id_set @ A @ F @ Z )
=> ( ( groups_comm_monoid_G @ A @ B @ F @ Z @ G
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ I5 )
& ( ( G @ X2 )
!= Z ) ) ) )
= ( groups_comm_monoid_G @ A @ B @ F @ Z @ G @ I5 ) ) ) ).
% comm_monoid_set.non_neutral'
thf(fact_7121_comm__monoid__set_Odistrib__triv_H,axiom,
! [A: $tType,B: $tType,F: A > A > A,Z: A,I5: set @ B,G: B > A,H: B > A] :
( ( groups778175481326437816id_set @ A @ F @ Z )
=> ( ( finite_finite2 @ B @ I5 )
=> ( ( groups_comm_monoid_G @ A @ B @ F @ Z
@ ^ [I2: B] : ( F @ ( G @ I2 ) @ ( H @ I2 ) )
@ I5 )
= ( F @ ( groups_comm_monoid_G @ A @ B @ F @ Z @ G @ I5 ) @ ( groups_comm_monoid_G @ A @ B @ F @ Z @ H @ I5 ) ) ) ) ) ).
% comm_monoid_set.distrib_triv'
thf(fact_7122_comm__monoid__set_Odistrib_H,axiom,
! [A: $tType,B: $tType,F: A > A > A,Z: A,I5: set @ B,G: B > A,H: B > A] :
( ( groups778175481326437816id_set @ A @ F @ Z )
=> ( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ I5 )
& ( ( G @ X2 )
!= Z ) ) ) )
=> ( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ I5 )
& ( ( H @ X2 )
!= Z ) ) ) )
=> ( ( groups_comm_monoid_G @ A @ B @ F @ Z
@ ^ [I2: B] : ( F @ ( G @ I2 ) @ ( H @ I2 ) )
@ I5 )
= ( F @ ( groups_comm_monoid_G @ A @ B @ F @ Z @ G @ I5 ) @ ( groups_comm_monoid_G @ A @ B @ F @ Z @ H @ I5 ) ) ) ) ) ) ).
% comm_monoid_set.distrib'
thf(fact_7123_comm__monoid__set_OG__def,axiom,
! [B: $tType,A: $tType,F: A > A > A,Z: A,I5: set @ B,P4: B > A] :
( ( groups778175481326437816id_set @ A @ F @ Z )
=> ( ( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ I5 )
& ( ( P4 @ X2 )
!= Z ) ) ) )
=> ( ( groups_comm_monoid_G @ A @ B @ F @ Z @ P4 @ I5 )
= ( groups_comm_monoid_F @ A @ B @ F @ Z @ P4
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ I5 )
& ( ( P4 @ X2 )
!= Z ) ) ) ) ) )
& ( ~ ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ I5 )
& ( ( P4 @ X2 )
!= Z ) ) ) )
=> ( ( groups_comm_monoid_G @ A @ B @ F @ Z @ P4 @ I5 )
= Z ) ) ) ) ).
% comm_monoid_set.G_def
thf(fact_7124_comm__monoid__set_OSuc__reindex__ivl,axiom,
! [A: $tType,F: A > A > A,Z: A,M: nat,N: nat,G: nat > A] :
( ( groups778175481326437816id_set @ A @ F @ Z )
=> ( ( ord_less_eq @ nat @ M @ N )
=> ( ( F @ ( groups_comm_monoid_F @ A @ nat @ F @ Z @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) )
= ( F @ ( G @ M )
@ ( groups_comm_monoid_F @ A @ nat @ F @ Z
@ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ) ) ).
% comm_monoid_set.Suc_reindex_ivl
thf(fact_7125_comm__monoid__set_OSigma,axiom,
! [A: $tType,C: $tType,B: $tType,F: A > A > A,Z: A,A5: set @ B,B4: B > ( set @ C ),G: B > C > A] :
( ( groups778175481326437816id_set @ A @ F @ Z )
=> ( ( finite_finite2 @ B @ A5 )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ A5 )
=> ( finite_finite2 @ C @ ( B4 @ X4 ) ) )
=> ( ( groups_comm_monoid_F @ A @ B @ F @ Z
@ ^ [X2: B] : ( groups_comm_monoid_F @ A @ C @ F @ Z @ ( G @ X2 ) @ ( B4 @ X2 ) )
@ A5 )
= ( groups_comm_monoid_F @ A @ ( product_prod @ B @ C ) @ F @ Z @ ( product_case_prod @ B @ C @ A @ G ) @ ( product_Sigma @ B @ C @ A5 @ B4 ) ) ) ) ) ) ).
% comm_monoid_set.Sigma
thf(fact_7126_comm__monoid__set_Ocartesian__product,axiom,
! [A: $tType,C: $tType,B: $tType,F: A > A > A,Z: A,G: B > C > A,B4: set @ C,A5: set @ B] :
( ( groups778175481326437816id_set @ A @ F @ Z )
=> ( ( groups_comm_monoid_F @ A @ B @ F @ Z
@ ^ [X2: B] : ( groups_comm_monoid_F @ A @ C @ F @ Z @ ( G @ X2 ) @ B4 )
@ A5 )
= ( groups_comm_monoid_F @ A @ ( product_prod @ B @ C ) @ F @ Z @ ( product_case_prod @ B @ C @ A @ G )
@ ( product_Sigma @ B @ C @ A5
@ ^ [Uu: B] : B4 ) ) ) ) ).
% comm_monoid_set.cartesian_product
thf(fact_7127_comm__monoid__set_Ogroup,axiom,
! [C: $tType,A: $tType,B: $tType,F: A > A > A,Z: A,S3: set @ B,T5: set @ C,G: B > C,H: B > A] :
( ( groups778175481326437816id_set @ A @ F @ Z )
=> ( ( finite_finite2 @ B @ S3 )
=> ( ( finite_finite2 @ C @ T5 )
=> ( ( ord_less_eq @ ( set @ C ) @ ( image2 @ B @ C @ G @ S3 ) @ T5 )
=> ( ( groups_comm_monoid_F @ A @ C @ F @ Z
@ ^ [Y: C] :
( groups_comm_monoid_F @ A @ B @ F @ Z @ H
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ S3 )
& ( ( G @ X2 )
= Y ) ) ) )
@ T5 )
= ( groups_comm_monoid_F @ A @ B @ F @ Z @ H @ S3 ) ) ) ) ) ) ).
% comm_monoid_set.group
thf(fact_7128_comm__monoid__set_Odelta__remove,axiom,
! [A: $tType,B: $tType,F: A > A > A,Z: A,S3: set @ B,A3: B,B2: B > A,C2: B > A] :
( ( groups778175481326437816id_set @ A @ F @ Z )
=> ( ( finite_finite2 @ B @ S3 )
=> ( ( ( member @ B @ A3 @ S3 )
=> ( ( groups_comm_monoid_F @ A @ B @ F @ Z
@ ^ [K3: B] : ( if @ A @ ( K3 = A3 ) @ ( B2 @ K3 ) @ ( C2 @ K3 ) )
@ S3 )
= ( F @ ( B2 @ A3 ) @ ( groups_comm_monoid_F @ A @ B @ F @ Z @ C2 @ ( minus_minus @ ( set @ B ) @ S3 @ ( insert2 @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) )
& ( ~ ( member @ B @ A3 @ S3 )
=> ( ( groups_comm_monoid_F @ A @ B @ F @ Z
@ ^ [K3: B] : ( if @ A @ ( K3 = A3 ) @ ( B2 @ K3 ) @ ( C2 @ K3 ) )
@ S3 )
= ( groups_comm_monoid_F @ A @ B @ F @ Z @ C2 @ ( minus_minus @ ( set @ B ) @ S3 @ ( insert2 @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ).
% comm_monoid_set.delta_remove
thf(fact_7129_comm__monoid__set_Oimage__eq,axiom,
! [A: $tType,B: $tType,F: A > A > A,Z: A,G: B > A,A5: set @ B] :
( ( groups778175481326437816id_set @ A @ F @ Z )
=> ( ( inj_on @ B @ A @ G @ A5 )
=> ( ( groups_comm_monoid_F @ A @ A @ F @ Z
@ ^ [X2: A] : X2
@ ( image2 @ B @ A @ G @ A5 ) )
= ( groups_comm_monoid_F @ A @ B @ F @ Z @ G @ A5 ) ) ) ) ).
% comm_monoid_set.image_eq
thf(fact_7130_comm__monoid__set_Oimage__gen,axiom,
! [A: $tType,C: $tType,B: $tType,F: A > A > A,Z: A,S3: set @ B,H: B > A,G: B > C] :
( ( groups778175481326437816id_set @ A @ F @ Z )
=> ( ( finite_finite2 @ B @ S3 )
=> ( ( groups_comm_monoid_F @ A @ B @ F @ Z @ H @ S3 )
= ( groups_comm_monoid_F @ A @ C @ F @ Z
@ ^ [Y: C] :
( groups_comm_monoid_F @ A @ B @ F @ Z @ H
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ S3 )
& ( ( G @ X2 )
= Y ) ) ) )
@ ( image2 @ B @ C @ G @ S3 ) ) ) ) ) ).
% comm_monoid_set.image_gen
thf(fact_7131_comm__monoid__set_Oin__pairs,axiom,
! [A: $tType,F: A > A > A,Z: A,G: nat > A,M: nat,N: nat] :
( ( groups778175481326437816id_set @ A @ F @ Z )
=> ( ( groups_comm_monoid_F @ A @ nat @ F @ Z @ G @ ( set_or1337092689740270186AtMost @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) )
= ( groups_comm_monoid_F @ A @ nat @ F @ Z
@ ^ [I2: nat] : ( F @ ( 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 ) ) ) ) ).
% comm_monoid_set.in_pairs
thf(fact_7132_comm__monoid__set_Oin__pairs__0,axiom,
! [A: $tType,F: A > A > A,Z: A,G: nat > A,N: nat] :
( ( groups778175481326437816id_set @ A @ F @ Z )
=> ( ( groups_comm_monoid_F @ A @ nat @ F @ Z @ G @ ( set_ord_atMost @ nat @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) )
= ( groups_comm_monoid_F @ A @ nat @ F @ Z
@ ^ [I2: nat] : ( F @ ( 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 ) ) ) ) ).
% comm_monoid_set.in_pairs_0
thf(fact_7133_comm__monoid__set_OUnion__disjoint,axiom,
! [A: $tType,B: $tType,F: A > A > A,Z: A,C6: set @ ( set @ B ),G: B > A] :
( ( groups778175481326437816id_set @ A @ F @ Z )
=> ( ! [X4: set @ B] :
( ( member @ ( set @ B ) @ X4 @ C6 )
=> ( finite_finite2 @ B @ X4 ) )
=> ( ! [X4: set @ B] :
( ( member @ ( set @ B ) @ X4 @ C6 )
=> ! [Xa4: set @ B] :
( ( member @ ( set @ B ) @ Xa4 @ C6 )
=> ( ( X4 != Xa4 )
=> ( ( inf_inf @ ( set @ B ) @ X4 @ Xa4 )
= ( bot_bot @ ( set @ B ) ) ) ) ) )
=> ( ( groups_comm_monoid_F @ A @ B @ F @ Z @ G @ ( complete_Sup_Sup @ ( set @ B ) @ C6 ) )
= ( comp @ ( ( set @ B ) > A ) @ ( ( set @ ( set @ B ) ) > A ) @ ( B > A ) @ ( groups_comm_monoid_F @ A @ ( set @ B ) @ F @ Z ) @ ( groups_comm_monoid_F @ A @ B @ F @ Z ) @ G @ C6 ) ) ) ) ) ).
% comm_monoid_set.Union_disjoint
thf(fact_7134_comm__monoid__set_Oreindex,axiom,
! [C: $tType,A: $tType,B: $tType,F: A > A > A,Z: A,H: B > C,A5: set @ B,G: C > A] :
( ( groups778175481326437816id_set @ A @ F @ Z )
=> ( ( inj_on @ B @ C @ H @ A5 )
=> ( ( groups_comm_monoid_F @ A @ C @ F @ Z @ G @ ( image2 @ B @ C @ H @ A5 ) )
= ( groups_comm_monoid_F @ A @ B @ F @ Z @ ( comp @ C @ A @ B @ G @ H ) @ A5 ) ) ) ) ).
% comm_monoid_set.reindex
thf(fact_7135_comm__monoid__set_Oreindex__nontrivial,axiom,
! [C: $tType,A: $tType,B: $tType,F: A > A > A,Z: A,A5: set @ B,H: B > C,G: C > A] :
( ( groups778175481326437816id_set @ A @ F @ Z )
=> ( ( finite_finite2 @ B @ A5 )
=> ( ! [X4: B,Y4: B] :
( ( member @ B @ X4 @ A5 )
=> ( ( member @ B @ Y4 @ A5 )
=> ( ( X4 != Y4 )
=> ( ( ( H @ X4 )
= ( H @ Y4 ) )
=> ( ( G @ ( H @ X4 ) )
= Z ) ) ) ) )
=> ( ( groups_comm_monoid_F @ A @ C @ F @ Z @ G @ ( image2 @ B @ C @ H @ A5 ) )
= ( groups_comm_monoid_F @ A @ B @ F @ Z @ ( comp @ C @ A @ B @ G @ H ) @ A5 ) ) ) ) ) ).
% comm_monoid_set.reindex_nontrivial
thf(fact_7136_comm__monoid__set_OUnion__comp,axiom,
! [A: $tType,B: $tType,F: A > A > A,Z: A,B4: set @ ( set @ B ),G: B > A] :
( ( groups778175481326437816id_set @ A @ F @ Z )
=> ( ! [X4: set @ B] :
( ( member @ ( set @ B ) @ X4 @ B4 )
=> ( finite_finite2 @ B @ X4 ) )
=> ( ! [A13: set @ B] :
( ( member @ ( set @ B ) @ A13 @ B4 )
=> ! [A24: set @ B] :
( ( member @ ( set @ B ) @ A24 @ B4 )
=> ( ( A13 != A24 )
=> ! [X4: B] :
( ( member @ B @ X4 @ A13 )
=> ( ( member @ B @ X4 @ A24 )
=> ( ( G @ X4 )
= Z ) ) ) ) ) )
=> ( ( groups_comm_monoid_F @ A @ B @ F @ Z @ G @ ( complete_Sup_Sup @ ( set @ B ) @ B4 ) )
= ( comp @ ( ( set @ B ) > A ) @ ( ( set @ ( set @ B ) ) > A ) @ ( B > A ) @ ( groups_comm_monoid_F @ A @ ( set @ B ) @ F @ Z ) @ ( groups_comm_monoid_F @ A @ B @ F @ Z ) @ G @ B4 ) ) ) ) ) ).
% comm_monoid_set.Union_comp
thf(fact_7137_comm__monoid__set_OatLeastAtMost__shift__0,axiom,
! [A: $tType,F: A > A > A,Z: A,M: nat,N: nat,G: nat > A] :
( ( groups778175481326437816id_set @ A @ F @ Z )
=> ( ( ord_less_eq @ nat @ M @ N )
=> ( ( groups_comm_monoid_F @ A @ nat @ F @ Z @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( groups_comm_monoid_F @ A @ nat @ F @ Z @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ M ) ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N @ M ) ) ) ) ) ) ).
% comm_monoid_set.atLeastAtMost_shift_0
thf(fact_7138_comm__monoid__set_OPlus,axiom,
! [B: $tType,A: $tType,C: $tType,F: A > A > A,Z: A,A5: set @ B,B4: set @ C,G: ( sum_sum @ B @ C ) > A] :
( ( groups778175481326437816id_set @ A @ F @ Z )
=> ( ( finite_finite2 @ B @ A5 )
=> ( ( finite_finite2 @ C @ B4 )
=> ( ( groups_comm_monoid_F @ A @ ( sum_sum @ B @ C ) @ F @ Z @ G @ ( sum_Plus @ B @ C @ A5 @ B4 ) )
= ( F @ ( groups_comm_monoid_F @ A @ B @ F @ Z @ ( comp @ ( sum_sum @ B @ C ) @ A @ B @ G @ ( sum_Inl @ B @ C ) ) @ A5 ) @ ( groups_comm_monoid_F @ A @ C @ F @ Z @ ( comp @ ( sum_sum @ B @ C ) @ A @ C @ G @ ( sum_Inr @ C @ B ) ) @ B4 ) ) ) ) ) ) ).
% comm_monoid_set.Plus
thf(fact_7139_comm__monoid__set_OatLeastLessThan__shift__0,axiom,
! [A: $tType,F: A > A > A,Z: A,G: nat > A,M: nat,N: nat] :
( ( groups778175481326437816id_set @ A @ F @ Z )
=> ( ( groups_comm_monoid_F @ A @ nat @ F @ Z @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) )
= ( groups_comm_monoid_F @ A @ nat @ F @ Z @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ M ) ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N @ M ) ) ) ) ) ).
% comm_monoid_set.atLeastLessThan_shift_0
thf(fact_7140_comm__monoid__set_OatLeastLessThan__shift__bounds,axiom,
! [A: $tType,F: A > A > A,Z: A,G: nat > A,M: nat,K: nat,N: nat] :
( ( groups778175481326437816id_set @ A @ F @ Z )
=> ( ( groups_comm_monoid_F @ A @ nat @ F @ Z @ G @ ( set_or7035219750837199246ssThan @ nat @ ( plus_plus @ nat @ M @ K ) @ ( plus_plus @ nat @ N @ K ) ) )
= ( groups_comm_monoid_F @ A @ nat @ F @ Z @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ K ) ) @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ).
% comm_monoid_set.atLeastLessThan_shift_bounds
thf(fact_7141_comm__monoid__set_OatLeastAtMost__shift__bounds,axiom,
! [A: $tType,F: A > A > A,Z: A,G: nat > A,M: nat,K: nat,N: nat] :
( ( groups778175481326437816id_set @ A @ F @ Z )
=> ( ( groups_comm_monoid_F @ A @ nat @ F @ Z @ G @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ M @ K ) @ ( plus_plus @ nat @ N @ K ) ) )
= ( groups_comm_monoid_F @ A @ nat @ F @ Z @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ K ) ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% comm_monoid_set.atLeastAtMost_shift_bounds
thf(fact_7142_comm__monoid__set_OatLeastLessThan__reindex,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F: A > A > A,Z: A,H: nat > B,M: nat,N: nat,G: B > A] :
( ( groups778175481326437816id_set @ A @ F @ Z )
=> ( ( bij_betw @ nat @ B @ H @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) @ ( set_or7035219750837199246ssThan @ B @ ( H @ M ) @ ( H @ N ) ) )
=> ( ( groups_comm_monoid_F @ A @ B @ F @ Z @ G @ ( set_or7035219750837199246ssThan @ B @ ( H @ M ) @ ( H @ N ) ) )
= ( groups_comm_monoid_F @ A @ nat @ F @ Z @ ( comp @ B @ A @ nat @ G @ H ) @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ) ) ).
% comm_monoid_set.atLeastLessThan_reindex
thf(fact_7143_comm__monoid__set_OatLeastAtMost__reindex,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F: A > A > A,Z: A,H: nat > B,M: nat,N: nat,G: B > A] :
( ( groups778175481326437816id_set @ A @ F @ Z )
=> ( ( bij_betw @ nat @ B @ H @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) @ ( set_or1337092689740270186AtMost @ B @ ( H @ M ) @ ( H @ N ) ) )
=> ( ( groups_comm_monoid_F @ A @ B @ F @ Z @ G @ ( set_or1337092689740270186AtMost @ B @ ( H @ M ) @ ( H @ N ) ) )
= ( groups_comm_monoid_F @ A @ nat @ F @ Z @ ( comp @ B @ A @ nat @ G @ H ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ) ) ).
% comm_monoid_set.atLeastAtMost_reindex
thf(fact_7144_comm__monoid__set_OatLeast0__atMost__Suc__shift,axiom,
! [A: $tType,F: A > A > A,Z: A,G: nat > A,N: nat] :
( ( groups778175481326437816id_set @ A @ F @ Z )
=> ( ( groups_comm_monoid_F @ A @ nat @ F @ Z @ G @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
= ( F @ ( G @ ( zero_zero @ nat ) ) @ ( groups_comm_monoid_F @ A @ nat @ F @ Z @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ) ).
% comm_monoid_set.atLeast0_atMost_Suc_shift
thf(fact_7145_comm__monoid__set_OatLeast0__lessThan__Suc__shift,axiom,
! [A: $tType,F: A > A > A,Z: A,G: nat > A,N: nat] :
( ( groups778175481326437816id_set @ A @ F @ Z )
=> ( ( groups_comm_monoid_F @ A @ nat @ F @ Z @ G @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
= ( F @ ( G @ ( zero_zero @ nat ) ) @ ( groups_comm_monoid_F @ A @ nat @ F @ Z @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ) ).
% comm_monoid_set.atLeast0_lessThan_Suc_shift
thf(fact_7146_comm__monoid__set_OatLeast__Suc__lessThan__Suc__shift,axiom,
! [A: $tType,F: A > A > A,Z: A,G: nat > A,M: nat,N: nat] :
( ( groups778175481326437816id_set @ A @ F @ Z )
=> ( ( groups_comm_monoid_F @ A @ nat @ F @ Z @ G @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M ) @ ( suc @ N ) ) )
= ( groups_comm_monoid_F @ A @ nat @ F @ Z @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ).
% comm_monoid_set.atLeast_Suc_lessThan_Suc_shift
thf(fact_7147_comm__monoid__set_OatLeast__Suc__atMost__Suc__shift,axiom,
! [A: $tType,F: A > A > A,Z: A,G: nat > A,M: nat,N: nat] :
( ( groups778175481326437816id_set @ A @ F @ Z )
=> ( ( groups_comm_monoid_F @ A @ nat @ F @ Z @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ ( suc @ N ) ) )
= ( groups_comm_monoid_F @ A @ nat @ F @ Z @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% comm_monoid_set.atLeast_Suc_atMost_Suc_shift
thf(fact_7148_comm__monoid__set_Oeq__fold,axiom,
! [A: $tType,B: $tType,F: A > A > A,Z: A,G: B > A,A5: set @ B] :
( ( groups778175481326437816id_set @ A @ F @ Z )
=> ( ( groups_comm_monoid_F @ A @ B @ F @ Z @ G @ A5 )
= ( finite_fold @ B @ A @ ( comp @ A @ ( A > A ) @ B @ F @ G ) @ Z @ A5 ) ) ) ).
% comm_monoid_set.eq_fold
thf(fact_7149_comm__monoid__set_OatLeast__lessThan__pred__shift,axiom,
! [A: $tType,F: A > A > A,Z: A,G: nat > A,M: nat,N: nat] :
( ( groups778175481326437816id_set @ A @ F @ Z )
=> ( ( groups_comm_monoid_F @ A @ nat @ F @ Z
@ ( comp @ nat @ A @ nat @ G
@ ^ [N5: nat] : ( minus_minus @ nat @ N5 @ ( suc @ ( zero_zero @ nat ) ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M ) @ ( suc @ N ) ) )
= ( groups_comm_monoid_F @ A @ nat @ F @ Z @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ).
% comm_monoid_set.atLeast_lessThan_pred_shift
thf(fact_7150_comm__monoid__set_OatLeast__atMost__pred__shift,axiom,
! [A: $tType,F: A > A > A,Z: A,G: nat > A,M: nat,N: nat] :
( ( groups778175481326437816id_set @ A @ F @ Z )
=> ( ( groups_comm_monoid_F @ A @ nat @ F @ Z
@ ( comp @ nat @ A @ nat @ G
@ ^ [N5: nat] : ( minus_minus @ nat @ N5 @ ( suc @ ( zero_zero @ nat ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ ( suc @ N ) ) )
= ( groups_comm_monoid_F @ A @ nat @ F @ Z @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% comm_monoid_set.atLeast_atMost_pred_shift
thf(fact_7151_comm__monoid__set_Otriangle__reindex,axiom,
! [A: $tType,F: A > A > A,Z: A,G: nat > nat > A,N: nat] :
( ( groups778175481326437816id_set @ A @ F @ Z )
=> ( ( groups_comm_monoid_F @ A @ ( product_prod @ nat @ nat ) @ F @ Z @ ( 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 ) ) ) )
= ( groups_comm_monoid_F @ A @ nat @ F @ Z
@ ^ [K3: nat] :
( groups_comm_monoid_F @ A @ nat @ F @ Z
@ ^ [I2: nat] : ( G @ I2 @ ( minus_minus @ nat @ K3 @ I2 ) )
@ ( set_ord_atMost @ nat @ K3 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% comm_monoid_set.triangle_reindex
thf(fact_7152_comm__monoid__set_Otriangle__reindex__eq,axiom,
! [A: $tType,F: A > A > A,Z: A,G: nat > nat > A,N: nat] :
( ( groups778175481326437816id_set @ A @ F @ Z )
=> ( ( groups_comm_monoid_F @ A @ ( product_prod @ nat @ nat ) @ F @ Z @ ( 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 ) ) ) )
= ( groups_comm_monoid_F @ A @ nat @ F @ Z
@ ^ [K3: nat] :
( groups_comm_monoid_F @ A @ nat @ F @ Z
@ ^ [I2: nat] : ( G @ I2 @ ( minus_minus @ nat @ K3 @ I2 ) )
@ ( set_ord_atMost @ nat @ K3 ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% comm_monoid_set.triangle_reindex_eq
thf(fact_7153_comm__monoid__set_OatLeast__int__atMost__int__shift,axiom,
! [A: $tType,F: A > A > A,Z: A,G: int > A,M: nat,N: nat] :
( ( groups778175481326437816id_set @ A @ F @ Z )
=> ( ( groups_comm_monoid_F @ A @ int @ F @ Z @ G @ ( set_or1337092689740270186AtMost @ int @ ( semiring_1_of_nat @ int @ M ) @ ( semiring_1_of_nat @ int @ N ) ) )
= ( groups_comm_monoid_F @ A @ nat @ F @ Z @ ( comp @ int @ A @ nat @ G @ ( semiring_1_of_nat @ int ) ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% comm_monoid_set.atLeast_int_atMost_int_shift
thf(fact_7154_comm__monoid__set_OatLeast__int__lessThan__int__shift,axiom,
! [A: $tType,F: A > A > A,Z: A,G: int > A,M: nat,N: nat] :
( ( groups778175481326437816id_set @ A @ F @ Z )
=> ( ( groups_comm_monoid_F @ A @ int @ F @ Z @ G @ ( set_or7035219750837199246ssThan @ int @ ( semiring_1_of_nat @ int @ M ) @ ( semiring_1_of_nat @ int @ N ) ) )
= ( groups_comm_monoid_F @ A @ nat @ F @ Z @ ( comp @ int @ A @ nat @ G @ ( semiring_1_of_nat @ int ) ) @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ).
% comm_monoid_set.atLeast_int_lessThan_int_shift
thf(fact_7155_digit6__def,axiom,
( digit6
= ( case_char @ $o
@ ^ [X12: $o,X24: $o,X33: $o,X43: $o,X53: $o,X63: $o,X73: $o,X83: $o] : X73 ) ) ).
% digit6_def
thf(fact_7156_digit5__def,axiom,
( digit5
= ( case_char @ $o
@ ^ [X12: $o,X24: $o,X33: $o,X43: $o,X53: $o,X63: $o,X73: $o,X83: $o] : X63 ) ) ).
% digit5_def
thf(fact_7157_digit0__def,axiom,
( digit0
= ( case_char @ $o
@ ^ [X12: $o,X24: $o,X33: $o,X43: $o,X53: $o,X63: $o,X73: $o,X83: $o] : X12 ) ) ).
% digit0_def
thf(fact_7158_digit1__def,axiom,
( digit1
= ( case_char @ $o
@ ^ [X12: $o,X24: $o,X33: $o,X43: $o,X53: $o,X63: $o,X73: $o,X83: $o] : X24 ) ) ).
% digit1_def
thf(fact_7159_digit4__def,axiom,
( digit4
= ( case_char @ $o
@ ^ [X12: $o,X24: $o,X33: $o,X43: $o,X53: $o,X63: $o,X73: $o,X83: $o] : X53 ) ) ).
% digit4_def
thf(fact_7160_digit3__def,axiom,
( digit3
= ( case_char @ $o
@ ^ [X12: $o,X24: $o,X33: $o,X43: $o,X53: $o,X63: $o,X73: $o,X83: $o] : X43 ) ) ).
% digit3_def
thf(fact_7161_digit2__def,axiom,
( digit2
= ( case_char @ $o
@ ^ [X12: $o,X24: $o,X33: $o,X43: $o,X53: $o,X63: $o,X73: $o,X83: $o] : X33 ) ) ).
% digit2_def
thf(fact_7162_accp__eq__acc,axiom,
! [A: $tType] :
( ( accp @ A )
= ( ^ [R4: A > A > $o,X2: A] : ( member @ A @ X2 @ ( acc @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ R4 ) ) ) ) ) ) ).
% accp_eq_acc
thf(fact_7163_acc__downwards,axiom,
! [A: $tType,A3: A,R: set @ ( product_prod @ A @ A ),B2: A] :
( ( member @ A @ A3 @ ( acc @ A @ R ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ A3 ) @ ( transitive_rtrancl @ A @ R ) )
=> ( member @ A @ B2 @ ( acc @ A @ R ) ) ) ) ).
% acc_downwards
thf(fact_7164_acc__downwards__aux,axiom,
! [A: $tType,B2: A,A3: A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ A3 ) @ ( transitive_rtrancl @ A @ R ) )
=> ( ( member @ A @ A3 @ ( acc @ A @ R ) )
=> ( member @ A @ B2 @ ( acc @ A @ R ) ) ) ) ).
% acc_downwards_aux
thf(fact_7165_acc_Ocases,axiom,
! [A: $tType,A3: A,R: set @ ( product_prod @ A @ A )] :
( ( member @ A @ A3 @ ( acc @ A @ R ) )
=> ! [Y5: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y5 @ A3 ) @ R )
=> ( member @ A @ Y5 @ ( acc @ A @ R ) ) ) ) ).
% acc.cases
thf(fact_7166_acc_Osimps,axiom,
! [A: $tType,A3: A,R: set @ ( product_prod @ A @ A )] :
( ( member @ A @ A3 @ ( acc @ A @ R ) )
= ( ? [X2: A] :
( ( A3 = X2 )
& ! [Y: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X2 ) @ R )
=> ( member @ A @ Y @ ( acc @ A @ R ) ) ) ) ) ) ).
% acc.simps
thf(fact_7167_acc_Ointros,axiom,
! [A: $tType,X: A,R: set @ ( product_prod @ A @ A )] :
( ! [Y4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ X ) @ R )
=> ( member @ A @ Y4 @ ( acc @ A @ R ) ) )
=> ( member @ A @ X @ ( acc @ A @ R ) ) ) ).
% acc.intros
thf(fact_7168_acc__induct,axiom,
! [A: $tType,A3: A,R: set @ ( product_prod @ A @ A ),P: A > $o] :
( ( member @ A @ A3 @ ( acc @ A @ R ) )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ ( acc @ A @ R ) )
=> ( ! [Y5: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y5 @ X4 ) @ R )
=> ( P @ Y5 ) )
=> ( P @ X4 ) ) )
=> ( P @ A3 ) ) ) ).
% acc_induct
thf(fact_7169_acc__downward,axiom,
! [A: $tType,B2: A,R: set @ ( product_prod @ A @ A ),A3: A] :
( ( member @ A @ B2 @ ( acc @ A @ R ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R )
=> ( member @ A @ A3 @ ( acc @ A @ R ) ) ) ) ).
% acc_downward
thf(fact_7170_not__acc__down,axiom,
! [A: $tType,X: A,R5: set @ ( product_prod @ A @ A )] :
( ~ ( member @ A @ X @ ( acc @ A @ R5 ) )
=> ~ ! [Z6: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z6 @ X ) @ R5 )
=> ( member @ A @ Z6 @ ( acc @ A @ R5 ) ) ) ) ).
% not_acc_down
thf(fact_7171_acc__induct__rule,axiom,
! [A: $tType,A3: A,R: set @ ( product_prod @ A @ A ),P: A > $o] :
( ( member @ A @ A3 @ ( acc @ A @ R ) )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ ( acc @ A @ R ) )
=> ( ! [Y5: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y5 @ X4 ) @ R )
=> ( P @ Y5 ) )
=> ( P @ X4 ) ) )
=> ( P @ A3 ) ) ) ).
% acc_induct_rule
thf(fact_7172_acc__subset__induct,axiom,
! [A: $tType,D5: set @ A,R5: set @ ( product_prod @ A @ A ),X: A,P: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ D5 @ ( acc @ A @ R5 ) )
=> ( ! [X4: A,Z6: A] :
( ( member @ A @ X4 @ D5 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z6 @ X4 ) @ R5 )
=> ( member @ A @ Z6 @ D5 ) ) )
=> ( ( member @ A @ X @ D5 )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ D5 )
=> ( ! [Z8: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z8 @ X4 ) @ R5 )
=> ( P @ Z8 ) )
=> ( P @ X4 ) ) )
=> ( P @ X ) ) ) ) ) ).
% acc_subset_induct
thf(fact_7173_accp__acc__eq,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( accp @ A
@ ^ [X2: A,Y: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y ) @ R ) )
= ( ^ [X2: A] : ( member @ A @ X2 @ ( acc @ A @ R ) ) ) ) ).
% accp_acc_eq
thf(fact_7174_acc__def,axiom,
! [A: $tType] :
( ( acc @ A )
= ( ^ [R4: set @ ( product_prod @ A @ A )] :
( collect @ A
@ ( accp @ A
@ ^ [X2: A,Y: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y ) @ R4 ) ) ) ) ) ).
% acc_def
thf(fact_7175_log_Opelims,axiom,
! [X: code_natural,Xa: code_natural,Y3: code_natural] :
( ( ( log @ X @ Xa )
= Y3 )
=> ( ( accp @ ( product_prod @ code_natural @ code_natural ) @ log_rel @ ( product_Pair @ code_natural @ code_natural @ X @ Xa ) )
=> ~ ( ( ( ( ( ord_less_eq @ code_natural @ X @ ( one_one @ code_natural ) )
| ( ord_less @ code_natural @ Xa @ X ) )
=> ( Y3
= ( one_one @ code_natural ) ) )
& ( ~ ( ( ord_less_eq @ code_natural @ X @ ( one_one @ code_natural ) )
| ( ord_less @ code_natural @ Xa @ X ) )
=> ( Y3
= ( plus_plus @ code_natural @ ( one_one @ code_natural ) @ ( log @ X @ ( divide_divide @ code_natural @ Xa @ X ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ code_natural @ code_natural ) @ log_rel @ ( product_Pair @ code_natural @ code_natural @ X @ Xa ) ) ) ) ) ).
% log.pelims
thf(fact_7176_map__tailrec__rev_Opelims,axiom,
! [A: $tType,B: $tType,X: A > B,Xa: list @ A,Xb: list @ B,Y3: list @ B] :
( ( ( map_tailrec_rev @ A @ B @ X @ Xa @ Xb )
= Y3 )
=> ( ( accp @ ( product_prod @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) @ ( map_tailrec_rev_rel @ A @ B ) @ ( product_Pair @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xa @ Xb ) ) )
=> ( ( ( Xa
= ( nil @ A ) )
=> ( ( Y3 = Xb )
=> ~ ( accp @ ( product_prod @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) @ ( map_tailrec_rev_rel @ A @ B ) @ ( product_Pair @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ Xb ) ) ) ) )
=> ~ ! [A4: A,As2: list @ A] :
( ( Xa
= ( cons @ A @ A4 @ As2 ) )
=> ( ( Y3
= ( map_tailrec_rev @ A @ B @ X @ As2 @ ( cons @ B @ ( X @ A4 ) @ Xb ) ) )
=> ~ ( accp @ ( product_prod @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) @ ( map_tailrec_rev_rel @ A @ B ) @ ( product_Pair @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ A4 @ As2 ) @ Xb ) ) ) ) ) ) ) ) ).
% map_tailrec_rev.pelims
thf(fact_7177_log_Ocases,axiom,
! [X: product_prod @ code_natural @ code_natural] :
~ ! [B3: code_natural,I3: code_natural] :
( X
!= ( product_Pair @ code_natural @ code_natural @ B3 @ I3 ) ) ).
% log.cases
thf(fact_7178_full__exhaustive__natural_H_Ocases,axiom,
! [X: product_prod @ ( ( product_prod @ code_natural @ ( product_unit > code_term ) ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ code_natural @ code_natural )] :
~ ! [F4: ( product_prod @ code_natural @ ( product_unit > code_term ) ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ),D2: code_natural,I3: code_natural] :
( X
!= ( product_Pair @ ( ( product_prod @ code_natural @ ( product_unit > code_term ) ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ code_natural @ code_natural ) @ F4 @ ( product_Pair @ code_natural @ code_natural @ D2 @ I3 ) ) ) ).
% full_exhaustive_natural'.cases
thf(fact_7179_full__exhaustive__fun_H_Ocases,axiom,
! [A: $tType,B: $tType] :
( ( ( quickc3360725361186068524ustive @ B )
& ( cl_HOL_Oequal @ A )
& ( quickc3360725361186068524ustive @ A ) )
=> ! [X: product_prod @ ( ( product_prod @ ( A > B ) @ ( product_unit > code_term ) ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ code_natural @ code_natural )] :
~ ! [F4: ( product_prod @ ( A > B ) @ ( product_unit > code_term ) ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ),I3: code_natural,D2: code_natural] :
( X
!= ( product_Pair @ ( ( product_prod @ ( A > B ) @ ( product_unit > code_term ) ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ code_natural @ code_natural ) @ F4 @ ( product_Pair @ code_natural @ code_natural @ I3 @ D2 ) ) ) ) ).
% full_exhaustive_fun'.cases
thf(fact_7180_exhaustive__natural_H_Ocases,axiom,
! [X: product_prod @ ( code_natural > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ code_natural @ code_natural )] :
~ ! [F4: code_natural > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ),D2: code_natural,I3: code_natural] :
( X
!= ( product_Pair @ ( code_natural > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ code_natural @ code_natural ) @ F4 @ ( product_Pair @ code_natural @ code_natural @ D2 @ I3 ) ) ) ).
% exhaustive_natural'.cases
thf(fact_7181_exhaustive__fun_H_Ocases,axiom,
! [A: $tType,B: $tType] :
( ( ( quickc658316121487927005ustive @ B )
& ( cl_HOL_Oequal @ A )
& ( quickc658316121487927005ustive @ A ) )
=> ! [X: product_prod @ ( ( A > B ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ code_natural @ code_natural )] :
~ ! [F4: ( A > B ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ),I3: code_natural,D2: code_natural] :
( X
!= ( product_Pair @ ( ( A > B ) > ( option @ ( product_prod @ $o @ ( list @ code_term ) ) ) ) @ ( product_prod @ code_natural @ code_natural ) @ F4 @ ( product_Pair @ code_natural @ code_natural @ I3 @ D2 ) ) ) ) ).
% exhaustive_fun'.cases
thf(fact_7182_Lazy__Sequence_Oiterate__upto_Ocases,axiom,
! [A: $tType,X: product_prod @ ( code_natural > A ) @ ( product_prod @ code_natural @ code_natural )] :
~ ! [F4: code_natural > A,N2: code_natural,M4: code_natural] :
( X
!= ( product_Pair @ ( code_natural > A ) @ ( product_prod @ code_natural @ code_natural ) @ F4 @ ( product_Pair @ code_natural @ code_natural @ N2 @ M4 ) ) ) ).
% Lazy_Sequence.iterate_upto.cases
thf(fact_7183_Predicate_Oiterate__upto_Opinduct,axiom,
! [A: $tType,A0: code_natural > A,A1: code_natural,A22: code_natural,P: ( code_natural > A ) > code_natural > code_natural > $o] :
( ( accp @ ( product_prod @ ( code_natural > A ) @ ( product_prod @ code_natural @ code_natural ) ) @ ( iterate_upto_rel @ A ) @ ( product_Pair @ ( code_natural > A ) @ ( product_prod @ code_natural @ code_natural ) @ A0 @ ( product_Pair @ code_natural @ code_natural @ A1 @ A22 ) ) )
=> ( ! [F4: 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 ) @ F4 @ ( product_Pair @ code_natural @ code_natural @ N2 @ M4 ) ) )
=> ( ! [X5: product_unit] :
( ~ ( ord_less @ code_natural @ M4 @ N2 )
=> ( P @ F4 @ ( plus_plus @ code_natural @ N2 @ ( one_one @ code_natural ) ) @ M4 ) )
=> ( P @ F4 @ N2 @ M4 ) ) )
=> ( P @ A0 @ A1 @ A22 ) ) ) ).
% Predicate.iterate_upto.pinduct
thf(fact_7184_sum_Opred__map,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,Q12: C > $o,Q23: D > $o,F1: A > C,F23: B > D,X: sum_sum @ A @ B] :
( ( basic_pred_sum @ C @ D @ Q12 @ Q23 @ ( sum_map_sum @ A @ C @ B @ D @ F1 @ F23 @ X ) )
= ( basic_pred_sum @ A @ B @ ( comp @ C @ $o @ A @ Q12 @ F1 ) @ ( comp @ D @ $o @ B @ Q23 @ F23 ) @ X ) ) ).
% sum.pred_map
thf(fact_7185_sum_Omap__cong__pred,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,X: sum_sum @ A @ B,Ya: sum_sum @ A @ B,F1: A > C,G1: A > C,F23: B > D,G22: B > D] :
( ( X = Ya )
=> ( ( basic_pred_sum @ A @ B
@ ^ [Z12: A] :
( ( F1 @ Z12 )
= ( G1 @ Z12 ) )
@ ^ [Z23: B] :
( ( F23 @ Z23 )
= ( G22 @ Z23 ) )
@ Ya )
=> ( ( sum_map_sum @ A @ C @ B @ D @ F1 @ F23 @ X )
= ( sum_map_sum @ A @ C @ B @ D @ G1 @ G22 @ Ya ) ) ) ) ).
% sum.map_cong_pred
thf(fact_7186_sum_Opred__True,axiom,
! [B: $tType,A: $tType] :
( ( basic_pred_sum @ A @ B
@ ^ [Uu: A] : $true
@ ^ [Uu: B] : $true )
= ( ^ [Uu: sum_sum @ A @ B] : $true ) ) ).
% sum.pred_True
thf(fact_7187_Predicate_Oiterate__upto_Opsimps,axiom,
! [A: $tType,F: code_natural > A,N: code_natural,M: 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 ) @ F @ ( product_Pair @ code_natural @ code_natural @ N @ M ) ) )
=> ( ( iterate_upto @ A @ F @ N @ M )
= ( seq2 @ A
@ ^ [U: product_unit] : ( if @ ( seq @ A ) @ ( ord_less @ code_natural @ M @ N ) @ ( empty @ A ) @ ( insert @ A @ ( F @ N ) @ ( iterate_upto @ A @ F @ ( plus_plus @ code_natural @ N @ ( one_one @ code_natural ) ) @ M ) ) ) ) ) ) ).
% Predicate.iterate_upto.psimps
thf(fact_7188_Predicate_Oiterate__upto_Opelims,axiom,
! [A: $tType,X: code_natural > A,Xa: code_natural,Xb: code_natural,Y3: pred @ A] :
( ( ( iterate_upto @ A @ X @ Xa @ Xb )
= Y3 )
=> ( ( 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 ) @ X @ ( product_Pair @ code_natural @ code_natural @ Xa @ Xb ) ) )
=> ~ ( ( Y3
= ( seq2 @ A
@ ^ [U: product_unit] : ( if @ ( seq @ A ) @ ( ord_less @ code_natural @ Xb @ Xa ) @ ( empty @ A ) @ ( insert @ A @ ( X @ Xa ) @ ( iterate_upto @ A @ X @ ( plus_plus @ code_natural @ Xa @ ( one_one @ code_natural ) ) @ Xb ) ) ) ) )
=> ~ ( accp @ ( product_prod @ ( code_natural > A ) @ ( product_prod @ code_natural @ code_natural ) ) @ ( iterate_upto_rel @ A ) @ ( product_Pair @ ( code_natural > A ) @ ( product_prod @ code_natural @ code_natural ) @ X @ ( product_Pair @ code_natural @ code_natural @ Xa @ Xb ) ) ) ) ) ) ).
% Predicate.iterate_upto.pelims
thf(fact_7189_bot__set__code,axiom,
! [A: $tType] :
( ( bot_bot @ ( pred @ A ) )
= ( seq2 @ A
@ ^ [U: product_unit] : ( empty @ A ) ) ) ).
% bot_set_code
thf(fact_7190_Predicate_Oiterate__upto_Osimps,axiom,
! [A: $tType] :
( ( iterate_upto @ A )
= ( ^ [F2: code_natural > A,N5: code_natural,M5: code_natural] :
( seq2 @ A
@ ^ [U: product_unit] : ( if @ ( seq @ A ) @ ( ord_less @ code_natural @ M5 @ N5 ) @ ( empty @ A ) @ ( insert @ A @ ( F2 @ N5 ) @ ( iterate_upto @ A @ F2 @ ( plus_plus @ code_natural @ N5 @ ( one_one @ code_natural ) ) @ M5 ) ) ) ) ) ) ).
% Predicate.iterate_upto.simps
thf(fact_7191_Predicate_Oiterate__upto_Oelims,axiom,
! [A: $tType,X: code_natural > A,Xa: code_natural,Xb: code_natural,Y3: pred @ A] :
( ( ( iterate_upto @ A @ X @ Xa @ Xb )
= Y3 )
=> ( Y3
= ( seq2 @ A
@ ^ [U: product_unit] : ( if @ ( seq @ A ) @ ( ord_less @ code_natural @ Xb @ Xa ) @ ( empty @ A ) @ ( insert @ A @ ( X @ Xa ) @ ( iterate_upto @ A @ X @ ( plus_plus @ code_natural @ Xa @ ( one_one @ code_natural ) ) @ Xb ) ) ) ) ) ) ).
% Predicate.iterate_upto.elims
thf(fact_7192_Predicate_Osingle__code,axiom,
! [A: $tType] :
( ( single @ A )
= ( ^ [X2: A] :
( seq2 @ A
@ ^ [U: product_unit] : ( insert @ A @ X2 @ ( bot_bot @ ( pred @ A ) ) ) ) ) ) ).
% Predicate.single_code
thf(fact_7193_of__pred__code,axiom,
! [A: $tType,F: product_unit > ( seq @ A )] :
( ( set_of_pred @ A @ ( seq2 @ A @ F ) )
= ( case_seq @ ( set @ A ) @ A @ ( bot_bot @ ( set @ A ) )
@ ^ [X2: A,P3: pred @ A] : ( insert2 @ A @ X2 @ ( set_of_pred @ A @ P3 ) )
@ ^ [P3: pred @ A,Xq: seq @ A] : ( sup_sup @ ( set @ A ) @ ( set_of_pred @ A @ P3 ) @ ( set_of_seq @ A @ Xq ) )
@ ( F @ product_Unity ) ) ) ).
% of_pred_code
thf(fact_7194_seq_Ocase__distrib,axiom,
! [B: $tType,C: $tType,A: $tType,H: B > C,F1: B,F23: A > ( pred @ A ) > B,F32: ( pred @ A ) > ( seq @ A ) > B,Seq: seq @ A] :
( ( H @ ( case_seq @ B @ A @ F1 @ F23 @ F32 @ Seq ) )
= ( case_seq @ C @ A @ ( H @ F1 )
@ ^ [X12: A,X24: pred @ A] : ( H @ ( F23 @ X12 @ X24 ) )
@ ^ [X12: pred @ A,X24: seq @ A] : ( H @ ( F32 @ X12 @ X24 ) )
@ Seq ) ) ).
% seq.case_distrib
thf(fact_7195_set__of__seq__def,axiom,
! [A: $tType] :
( ( set_of_seq @ A )
= ( comp @ ( pred @ A ) @ ( set @ A ) @ ( seq @ A ) @ ( set_of_pred @ A ) @ ( pred_of_seq @ A ) ) ) ).
% set_of_seq_def
thf(fact_7196_eval__SUP,axiom,
! [A: $tType,B: $tType,F: B > ( pred @ A ),A5: set @ B] :
( ( eval @ A @ ( complete_Sup_Sup @ ( pred @ A ) @ ( image2 @ B @ ( pred @ A ) @ F @ A5 ) ) )
= ( complete_Sup_Sup @ ( A > $o ) @ ( image2 @ B @ ( A > $o ) @ ( comp @ ( pred @ A ) @ ( A > $o ) @ B @ ( eval @ A ) @ F ) @ A5 ) ) ) ).
% eval_SUP
thf(fact_7197_eval__INF,axiom,
! [A: $tType,B: $tType,F: B > ( pred @ A ),A5: set @ B] :
( ( eval @ A @ ( complete_Inf_Inf @ ( pred @ A ) @ ( image2 @ B @ ( pred @ A ) @ F @ A5 ) ) )
= ( complete_Inf_Inf @ ( A > $o ) @ ( image2 @ B @ ( A > $o ) @ ( comp @ ( pred @ A ) @ ( A > $o ) @ B @ ( eval @ A ) @ F ) @ A5 ) ) ) ).
% eval_INF
thf(fact_7198_set__of__pred__def,axiom,
! [A: $tType] :
( ( set_of_pred @ A )
= ( comp @ ( A > $o ) @ ( set @ A ) @ ( pred @ A ) @ ( collect @ A ) @ ( eval @ A ) ) ) ).
% set_of_pred_def
thf(fact_7199_eval__map,axiom,
! [A: $tType,B: $tType,F: B > A,P: pred @ B] :
( ( eval @ A @ ( map2 @ B @ A @ F @ P ) )
= ( complete_Sup_Sup @ ( A > $o )
@ ( image2 @ B @ ( A > $o )
@ ^ [X2: B] :
( ^ [Y6: A,Z2: A] : Y6 = Z2
@ ( F @ X2 ) )
@ ( collect @ B @ ( eval @ B @ P ) ) ) ) ) ).
% eval_map
thf(fact_7200_less__eq__pred__code,axiom,
! [A: $tType,F: product_unit > ( seq @ A ),Q: pred @ A] :
( ( ord_less_eq @ ( pred @ A ) @ ( seq2 @ A @ F ) @ Q )
= ( case_seq @ $o @ A @ $true
@ ^ [X2: A,P3: pred @ A] :
( ( eval @ A @ Q @ X2 )
& ( ord_less_eq @ ( pred @ A ) @ P3 @ Q ) )
@ ^ [P3: pred @ A,Xq: seq @ A] :
( ( ord_less_eq @ ( pred @ A ) @ P3 @ Q )
& ( contained @ A @ Xq @ Q ) )
@ ( F @ product_Unity ) ) ) ).
% less_eq_pred_code
thf(fact_7201_Predicate_Omap_Ocompositionality,axiom,
! [B: $tType,C: $tType,A: $tType,F: B > C,G: A > B,Pred: pred @ A] :
( ( map2 @ B @ C @ F @ ( map2 @ A @ B @ G @ Pred ) )
= ( map2 @ A @ C @ ( comp @ B @ C @ A @ F @ G ) @ Pred ) ) ).
% Predicate.map.compositionality
thf(fact_7202_Predicate_Omap_Ocomp,axiom,
! [C: $tType,B: $tType,A: $tType,F: B > C,G: A > B] :
( ( comp @ ( pred @ B ) @ ( pred @ C ) @ ( pred @ A ) @ ( map2 @ B @ C @ F ) @ ( map2 @ A @ B @ G ) )
= ( map2 @ A @ C @ ( comp @ B @ C @ A @ F @ G ) ) ) ).
% Predicate.map.comp
thf(fact_7203_Predicate_Omap_Oidentity,axiom,
! [A: $tType] :
( ( map2 @ A @ A
@ ^ [X2: A] : X2 )
= ( id @ ( pred @ A ) ) ) ).
% Predicate.map.identity
thf(fact_7204_eval__bind,axiom,
! [A: $tType,B: $tType,P: pred @ B,F: B > ( pred @ A )] :
( ( eval @ A @ ( bind3 @ B @ A @ P @ F ) )
= ( eval @ A @ ( complete_Sup_Sup @ ( pred @ A ) @ ( image2 @ B @ ( pred @ A ) @ F @ ( collect @ B @ ( eval @ B @ P ) ) ) ) ) ) ).
% eval_bind
thf(fact_7205_Predicate_Othe__def,axiom,
! [A: $tType] :
( ( the3 @ A )
= ( ^ [A8: pred @ A] : ( the @ A @ ( eval @ A @ A8 ) ) ) ) ).
% Predicate.the_def
thf(fact_7206_Predicate_Obind__bind,axiom,
! [B: $tType,A: $tType,C: $tType,P: pred @ C,Q: C > ( pred @ B ),R5: B > ( pred @ A )] :
( ( bind3 @ B @ A @ ( bind3 @ C @ B @ P @ Q ) @ R5 )
= ( bind3 @ C @ A @ P
@ ^ [X2: C] : ( bind3 @ B @ A @ ( Q @ X2 ) @ R5 ) ) ) ).
% Predicate.bind_bind
thf(fact_7207_Sup__bind,axiom,
! [A: $tType,B: $tType,A5: set @ ( pred @ B ),F: B > ( pred @ A )] :
( ( bind3 @ B @ A @ ( complete_Sup_Sup @ ( pred @ B ) @ A5 ) @ F )
= ( complete_Sup_Sup @ ( pred @ A )
@ ( image2 @ ( pred @ B ) @ ( pred @ A )
@ ^ [X2: pred @ B] : ( bind3 @ B @ A @ X2 @ F )
@ A5 ) ) ) ).
% Sup_bind
thf(fact_7208_Predicate_Omap__def,axiom,
! [B: $tType,A: $tType] :
( ( map2 @ A @ B )
= ( ^ [F2: A > B,P3: pred @ A] : ( bind3 @ A @ B @ P3 @ ( comp @ B @ ( pred @ B ) @ A @ ( single @ B ) @ F2 ) ) ) ) ).
% Predicate.map_def
thf(fact_7209_Predicate_Obind__def,axiom,
! [B: $tType,A: $tType] :
( ( bind3 @ A @ B )
= ( ^ [P3: pred @ A,F2: A > ( pred @ B )] : ( complete_Sup_Sup @ ( pred @ B ) @ ( image2 @ A @ ( pred @ B ) @ F2 @ ( collect @ A @ ( eval @ A @ P3 ) ) ) ) ) ) ).
% Predicate.bind_def
thf(fact_7210_the__eqI,axiom,
! [A: $tType,P: pred @ A,X: A] :
( ( ( the @ A @ ( eval @ A @ P ) )
= X )
=> ( ( the3 @ A @ P )
= X ) ) ).
% the_eqI
thf(fact_7211_bind__code,axiom,
! [B: $tType,A: $tType,G: product_unit > ( seq @ B ),F: B > ( pred @ A )] :
( ( bind3 @ B @ A @ ( seq2 @ B @ G ) @ F )
= ( seq2 @ A
@ ^ [U: product_unit] : ( apply @ B @ A @ F @ ( G @ product_Unity ) ) ) ) ).
% bind_code
thf(fact_7212_sup__code,axiom,
! [A: $tType,F: product_unit > ( seq @ A ),G: product_unit > ( seq @ A )] :
( ( sup_sup @ ( pred @ A ) @ ( seq2 @ A @ F ) @ ( seq2 @ A @ G ) )
= ( seq2 @ A
@ ^ [U: product_unit] :
( case_seq @ ( seq @ A ) @ A @ ( G @ product_Unity )
@ ^ [X2: A,P3: pred @ A] : ( insert @ A @ X2 @ ( sup_sup @ ( pred @ A ) @ P3 @ ( seq2 @ A @ G ) ) )
@ ^ [P3: pred @ A,Xq: seq @ A] : ( adjunct @ A @ ( seq2 @ A @ G ) @ ( join @ A @ P3 @ Xq ) )
@ ( F @ product_Unity ) ) ) ) ).
% sup_code
thf(fact_7213_Random__Pred_Ounion__def,axiom,
! [A: $tType] :
( ( random_union @ A )
= ( ^ [R14: ( product_prod @ code_natural @ code_natural ) > ( product_prod @ ( pred @ A ) @ ( product_prod @ code_natural @ code_natural ) ),R25: ( product_prod @ code_natural @ code_natural ) > ( product_prod @ ( pred @ A ) @ ( product_prod @ code_natural @ code_natural ) ),S2: product_prod @ code_natural @ code_natural] :
( product_case_prod @ ( pred @ A ) @ ( product_prod @ code_natural @ code_natural ) @ ( product_prod @ ( pred @ A ) @ ( product_prod @ code_natural @ code_natural ) )
@ ^ [P13: pred @ A,S9: product_prod @ code_natural @ code_natural] :
( product_case_prod @ ( pred @ A ) @ ( product_prod @ code_natural @ code_natural ) @ ( product_prod @ ( pred @ A ) @ ( product_prod @ code_natural @ code_natural ) )
@ ^ [P24: pred @ A] : ( product_Pair @ ( pred @ A ) @ ( product_prod @ code_natural @ code_natural ) @ ( sup_sup @ ( pred @ A ) @ P13 @ P24 ) )
@ ( R25 @ S9 ) )
@ ( R14 @ S2 ) ) ) ) ).
% Random_Pred.union_def
thf(fact_7214_eval__def,axiom,
! [A: $tType] :
( ( eval @ A )
= ( case_pred @ A @ ( A > $o )
@ ^ [X2: A > $o] : X2 ) ) ).
% eval_def
thf(fact_7215_pred_Ocase__distrib,axiom,
! [C: $tType,B: $tType,A: $tType,H: B > C,F: ( A > $o ) > B,Pred: pred @ A] :
( ( H @ ( case_pred @ A @ B @ F @ Pred ) )
= ( case_pred @ A @ C
@ ^ [X2: A > $o] : ( H @ ( F @ X2 ) )
@ Pred ) ) ).
% pred.case_distrib
thf(fact_7216_Random__Pred_Oiterate__upto__def,axiom,
! [A: $tType] :
( ( random_iterate_upto @ A )
= ( ^ [F2: code_natural > A,N5: code_natural,M5: code_natural] : ( product_Pair @ ( pred @ A ) @ ( product_prod @ code_natural @ code_natural ) @ ( iterate_upto @ A @ F2 @ N5 @ M5 ) ) ) ) ).
% Random_Pred.iterate_upto_def
thf(fact_7217_Random__Pred_Oempty__def,axiom,
! [A: $tType] :
( ( random_empty @ A )
= ( product_Pair @ ( pred @ A ) @ ( product_prod @ code_natural @ code_natural ) @ ( bot_bot @ ( pred @ A ) ) ) ) ).
% Random_Pred.empty_def
thf(fact_7218_Random__Pred_ORandom__def,axiom,
! [A: $tType] :
( ( random_Random @ A )
= ( ^ [G2: ( product_prod @ code_natural @ code_natural ) > ( product_prod @ ( product_prod @ A @ ( product_unit > code_term ) ) @ ( product_prod @ code_natural @ code_natural ) )] : ( product_scomp @ ( product_prod @ code_natural @ code_natural ) @ ( product_prod @ A @ ( product_unit > code_term ) ) @ ( product_prod @ code_natural @ code_natural ) @ ( product_prod @ ( pred @ A ) @ ( product_prod @ code_natural @ code_natural ) ) @ G2 @ ( comp @ ( pred @ A ) @ ( ( product_prod @ code_natural @ code_natural ) > ( product_prod @ ( pred @ A ) @ ( product_prod @ code_natural @ code_natural ) ) ) @ ( product_prod @ A @ ( product_unit > code_term ) ) @ ( product_Pair @ ( pred @ A ) @ ( product_prod @ code_natural @ code_natural ) ) @ ( comp @ A @ ( pred @ A ) @ ( product_prod @ A @ ( product_unit > code_term ) ) @ ( single @ A ) @ ( product_fst @ A @ ( product_unit > code_term ) ) ) ) ) ) ) ).
% Random_Pred.Random_def
thf(fact_7219_Random__Pred_Obind__def,axiom,
! [B: $tType,A: $tType] :
( ( random_bind @ A @ B )
= ( ^ [R6: ( product_prod @ code_natural @ code_natural ) > ( product_prod @ ( pred @ A ) @ ( product_prod @ code_natural @ code_natural ) ),F2: A > ( product_prod @ code_natural @ code_natural ) > ( product_prod @ ( pred @ B ) @ ( product_prod @ code_natural @ code_natural ) ),S2: product_prod @ code_natural @ code_natural] :
( product_case_prod @ ( pred @ A ) @ ( product_prod @ code_natural @ code_natural ) @ ( product_prod @ ( pred @ B ) @ ( product_prod @ code_natural @ code_natural ) )
@ ^ [P3: pred @ A,S9: product_prod @ code_natural @ code_natural] :
( product_case_prod @ ( product_prod @ code_natural @ code_natural ) @ ( product_prod @ code_natural @ code_natural ) @ ( product_prod @ ( pred @ B ) @ ( product_prod @ code_natural @ code_natural ) )
@ ^ [S1: product_prod @ code_natural @ code_natural] :
( product_Pair @ ( pred @ B ) @ ( product_prod @ code_natural @ code_natural )
@ ( bind3 @ A @ B @ P3
@ ^ [A6: A] : ( product_fst @ ( pred @ B ) @ ( product_prod @ code_natural @ code_natural ) @ ( F2 @ A6 @ S1 ) ) ) )
@ ( split_seed @ S9 ) )
@ ( R6 @ S2 ) ) ) ) ).
% Random_Pred.bind_def
thf(fact_7220_Random__Pred_Osingle__def,axiom,
! [A: $tType] :
( ( random_single @ A )
= ( ^ [X2: A] : ( product_Pair @ ( pred @ A ) @ ( product_prod @ code_natural @ code_natural ) @ ( single @ A @ X2 ) ) ) ) ).
% Random_Pred.single_def
thf(fact_7221_Random__Pred_Onot__randompred__def,axiom,
( random6974930770145893639ompred
= ( ^ [P3: ( product_prod @ code_natural @ code_natural ) > ( product_prod @ ( pred @ product_unit ) @ ( product_prod @ code_natural @ code_natural ) ),S2: product_prod @ code_natural @ code_natural] :
( product_case_prod @ ( pred @ product_unit ) @ ( product_prod @ code_natural @ code_natural ) @ ( product_prod @ ( pred @ product_unit ) @ ( product_prod @ code_natural @ code_natural ) )
@ ^ [P11: pred @ product_unit,S9: product_prod @ code_natural @ code_natural] : ( if @ ( product_prod @ ( pred @ product_unit ) @ ( product_prod @ code_natural @ code_natural ) ) @ ( eval @ product_unit @ P11 @ product_Unity ) @ ( product_Pair @ ( pred @ product_unit ) @ ( product_prod @ code_natural @ code_natural ) @ ( bot_bot @ ( pred @ product_unit ) ) @ S9 ) @ ( product_Pair @ ( pred @ product_unit ) @ ( product_prod @ code_natural @ code_natural ) @ ( single @ product_unit @ product_Unity ) @ S9 ) )
@ ( P3 @ S2 ) ) ) ) ).
% Random_Pred.not_randompred_def
thf(fact_7222_Random__Pred_Omap__def,axiom,
! [B: $tType,A: $tType] :
( ( random_map @ A @ B )
= ( ^ [F2: A > B,P3: ( product_prod @ code_natural @ code_natural ) > ( product_prod @ ( pred @ A ) @ ( product_prod @ code_natural @ code_natural ) )] : ( random_bind @ A @ B @ P3 @ ( comp @ B @ ( ( product_prod @ code_natural @ code_natural ) > ( product_prod @ ( pred @ B ) @ ( product_prod @ code_natural @ code_natural ) ) ) @ A @ ( random_single @ B ) @ F2 ) ) ) ) ).
% Random_Pred.map_def
% Type constructors (689)
thf(tcon_Product__Type_Ounit___Lattices_Obounded__lattice,axiom,
bounded_lattice @ product_unit ).
thf(tcon_Product__Type_Ounit___Enum_Oenum,axiom,
enum @ product_unit ).
thf(tcon_Product__Type_Oprod___Enum_Oenum_1,axiom,
! [A21: $tType,A27: $tType] :
( ( ( enum @ A21 )
& ( enum @ A27 ) )
=> ( enum @ ( product_prod @ A21 @ A27 ) ) ) ).
thf(tcon_Predicate_Opred___Lattices_Obounded__lattice_2,axiom,
! [A21: $tType] : ( bounded_lattice @ ( pred @ A21 ) ) ).
thf(tcon_Option_Ooption___Enum_Oenum_3,axiom,
! [A21: $tType] :
( ( enum @ A21 )
=> ( enum @ ( option @ A21 ) ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Obounded__lattice_4,axiom,
! [A21: $tType] : ( bounded_lattice @ ( filter @ A21 ) ) ).
thf(tcon_Sum__Type_Osum___Enum_Oenum_5,axiom,
! [A21: $tType,A27: $tType] :
( ( ( enum @ A21 )
& ( enum @ A27 ) )
=> ( enum @ ( sum_sum @ A21 @ A27 ) ) ) ).
thf(tcon_String_Ochar___Enum_Oenum_6,axiom,
enum @ char ).
thf(tcon_HOL_Obool___Lattices_Obounded__lattice_7,axiom,
bounded_lattice @ $o ).
thf(tcon_HOL_Obool___Enum_Oenum_8,axiom,
enum @ $o ).
thf(tcon_Set_Oset___Lattices_Obounded__lattice_9,axiom,
! [A21: $tType] : ( bounded_lattice @ ( set @ A21 ) ) ).
thf(tcon_Set_Oset___Enum_Oenum_10,axiom,
! [A21: $tType] :
( ( enum @ A21 )
=> ( enum @ ( set @ A21 ) ) ) ).
thf(tcon_fun___Lattices_Obounded__lattice_11,axiom,
! [A21: $tType,A27: $tType] :
( ( bounded_lattice @ A27 )
=> ( bounded_lattice @ ( A21 > A27 ) ) ) ).
thf(tcon_fun___Enum_Oenum_12,axiom,
! [A21: $tType,A27: $tType] :
( ( ( enum @ A21 )
& ( enum @ A27 ) )
=> ( enum @ ( A21 > A27 ) ) ) ).
thf(tcon_fun___Conditionally__Complete__Lattices_Oconditionally__complete__lattice,axiom,
! [A21: $tType,A27: $tType] :
( ( comple6319245703460814977attice @ A27 )
=> ( condit1219197933456340205attice @ ( A21 > A27 ) ) ) ).
thf(tcon_fun___Complete__Lattices_Ocomplete__distrib__lattice,axiom,
! [A21: $tType,A27: $tType] :
( ( comple592849572758109894attice @ A27 )
=> ( comple592849572758109894attice @ ( A21 > A27 ) ) ) ).
thf(tcon_fun___Complete__Lattices_Ocomplete__boolean__algebra,axiom,
! [A21: $tType,A27: $tType] :
( ( comple489889107523837845lgebra @ A27 )
=> ( comple489889107523837845lgebra @ ( A21 > A27 ) ) ) ).
thf(tcon_fun___Quickcheck__Exhaustive_Ofull__exhaustive,axiom,
! [A21: $tType,A27: $tType] :
( ( ( cl_HOL_Oequal @ A21 )
& ( quickc3360725361186068524ustive @ A21 )
& ( quickc3360725361186068524ustive @ A27 ) )
=> ( quickc3360725361186068524ustive @ ( A21 > A27 ) ) ) ).
thf(tcon_fun___Lattices_Obounded__semilattice__sup__bot,axiom,
! [A21: $tType,A27: $tType] :
( ( bounded_lattice @ A27 )
=> ( bounde4967611905675639751up_bot @ ( A21 > A27 ) ) ) ).
thf(tcon_fun___Complete__Lattices_Ocomplete__lattice,axiom,
! [A21: $tType,A27: $tType] :
( ( comple6319245703460814977attice @ A27 )
=> ( comple6319245703460814977attice @ ( A21 > A27 ) ) ) ).
thf(tcon_fun___Quickcheck__Exhaustive_Oexhaustive,axiom,
! [A21: $tType,A27: $tType] :
( ( ( cl_HOL_Oequal @ A21 )
& ( quickc658316121487927005ustive @ A21 )
& ( quickc658316121487927005ustive @ A27 ) )
=> ( quickc658316121487927005ustive @ ( A21 > A27 ) ) ) ).
thf(tcon_fun___Boolean__Algebras_Oboolean__algebra,axiom,
! [A21: $tType,A27: $tType] :
( ( boolea8198339166811842893lgebra @ A27 )
=> ( boolea8198339166811842893lgebra @ ( A21 > A27 ) ) ) ).
thf(tcon_fun___Complete__Partial__Order_Occpo,axiom,
! [A21: $tType,A27: $tType] :
( ( comple6319245703460814977attice @ A27 )
=> ( comple9053668089753744459l_ccpo @ ( A21 > A27 ) ) ) ).
thf(tcon_fun___Lattices_Osemilattice__sup,axiom,
! [A21: $tType,A27: $tType] :
( ( semilattice_sup @ A27 )
=> ( semilattice_sup @ ( A21 > A27 ) ) ) ).
thf(tcon_fun___Lattices_Osemilattice__inf,axiom,
! [A21: $tType,A27: $tType] :
( ( semilattice_inf @ A27 )
=> ( semilattice_inf @ ( A21 > A27 ) ) ) ).
thf(tcon_fun___Lattices_Odistrib__lattice,axiom,
! [A21: $tType,A27: $tType] :
( ( distrib_lattice @ A27 )
=> ( distrib_lattice @ ( A21 > A27 ) ) ) ).
thf(tcon_fun___Complete__Lattices_OSup,axiom,
! [A21: $tType,A27: $tType] :
( ( complete_Sup @ A27 )
=> ( complete_Sup @ ( A21 > A27 ) ) ) ).
thf(tcon_fun___Complete__Lattices_OInf,axiom,
! [A21: $tType,A27: $tType] :
( ( complete_Inf @ A27 )
=> ( complete_Inf @ ( A21 > A27 ) ) ) ).
thf(tcon_fun___Orderings_Oorder__top,axiom,
! [A21: $tType,A27: $tType] :
( ( order_top @ A27 )
=> ( order_top @ ( A21 > A27 ) ) ) ).
thf(tcon_fun___Orderings_Oorder__bot,axiom,
! [A21: $tType,A27: $tType] :
( ( order_bot @ A27 )
=> ( order_bot @ ( A21 > A27 ) ) ) ).
thf(tcon_fun___Countable_Ocountable,axiom,
! [A21: $tType,A27: $tType] :
( ( ( finite_finite @ A21 )
& ( countable @ A27 ) )
=> ( countable @ ( A21 > A27 ) ) ) ).
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A21: $tType,A27: $tType] :
( ( preorder @ A27 )
=> ( preorder @ ( A21 > A27 ) ) ) ).
thf(tcon_fun___Finite__Set_Ofinite,axiom,
! [A21: $tType,A27: $tType] :
( ( ( finite_finite @ A21 )
& ( finite_finite @ A27 ) )
=> ( finite_finite @ ( A21 > A27 ) ) ) ).
thf(tcon_fun___Lattices_Olattice,axiom,
! [A21: $tType,A27: $tType] :
( ( lattice @ A27 )
=> ( lattice @ ( A21 > A27 ) ) ) ).
thf(tcon_fun___Orderings_Oorder,axiom,
! [A21: $tType,A27: $tType] :
( ( order @ A27 )
=> ( order @ ( A21 > A27 ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A21: $tType,A27: $tType] :
( ( ord @ A27 )
=> ( ord @ ( A21 > A27 ) ) ) ).
thf(tcon_fun___Groups_Ouminus,axiom,
! [A21: $tType,A27: $tType] :
( ( uminus @ A27 )
=> ( uminus @ ( A21 > A27 ) ) ) ).
thf(tcon_fun___HOL_Oequal,axiom,
! [A21: $tType,A27: $tType] :
( ( ( enum @ A21 )
& ( cl_HOL_Oequal @ A27 ) )
=> ( cl_HOL_Oequal @ ( A21 > A27 ) ) ) ).
thf(tcon_Int_Oint___Conditionally__Complete__Lattices_Oconditionally__complete__linorder,axiom,
condit6923001295902523014norder @ int ).
thf(tcon_Int_Oint___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_13,axiom,
condit1219197933456340205attice @ int ).
thf(tcon_Int_Oint___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations,axiom,
bit_un5681908812861735899ations @ int ).
thf(tcon_Int_Oint___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,axiom,
semiri1453513574482234551roduct @ int ).
thf(tcon_Int_Oint___Euclidean__Division_Ounique__euclidean__semiring__with__nat,axiom,
euclid5411537665997757685th_nat @ int ).
thf(tcon_Int_Oint___Euclidean__Division_Ounique__euclidean__ring__with__nat,axiom,
euclid8789492081693882211th_nat @ int ).
thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__monoid__add__imp__le,axiom,
ordere1937475149494474687imp_le @ int ).
thf(tcon_Int_Oint___Euclidean__Division_Ounique__euclidean__semiring,axiom,
euclid3128863361964157862miring @ int ).
thf(tcon_Int_Oint___Euclidean__Division_Oeuclidean__semiring__cancel,axiom,
euclid4440199948858584721cancel @ int ).
thf(tcon_Int_Oint___Rings_Onormalization__semidom__multiplicative,axiom,
normal6328177297339901930cative @ int ).
thf(tcon_Int_Oint___Divides_Ounique__euclidean__semiring__numeral,axiom,
unique1627219031080169319umeral @ int ).
thf(tcon_Int_Oint___Euclidean__Division_Oeuclidean__ring__cancel,axiom,
euclid8851590272496341667cancel @ int ).
thf(tcon_Int_Oint___Rings_Osemiring__no__zero__divisors__cancel,axiom,
semiri6575147826004484403cancel @ int ).
thf(tcon_Int_Oint___Groups_Ostrict__ordered__ab__semigroup__add,axiom,
strict9044650504122735259up_add @ int ).
thf(tcon_Int_Oint___Groups_Oordered__cancel__ab__semigroup__add,axiom,
ordere580206878836729694up_add @ int ).
thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
ordere2412721322843649153imp_le @ int ).
thf(tcon_Int_Oint___Bit__Operations_Osemiring__bit__operations,axiom,
bit_se359711467146920520ations @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__comm__semiring__strict,axiom,
linord2810124833399127020strict @ int ).
thf(tcon_Int_Oint___Quickcheck__Exhaustive_Ofull__exhaustive_14,axiom,
quickc3360725361186068524ustive @ int ).
thf(tcon_Int_Oint___Groups_Ostrict__ordered__comm__monoid__add,axiom,
strict7427464778891057005id_add @ int ).
thf(tcon_Int_Oint___Groups_Oordered__cancel__comm__monoid__add,axiom,
ordere8940638589300402666id_add @ int ).
thf(tcon_Int_Oint___Euclidean__Division_Oeuclidean__semiring,axiom,
euclid3725896446679973847miring @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__semiring__1__strict,axiom,
linord715952674999750819strict @ int ).
thf(tcon_Int_Oint___Groups_Olinordered__ab__semigroup__add,axiom,
linord4140545234300271783up_add @ int ).
thf(tcon_Int_Oint___Bit__Operations_Oring__bit__operations,axiom,
bit_ri3973907225187159222ations @ int ).
thf(tcon_Int_Oint___Rings_Osemiring__1__no__zero__divisors,axiom,
semiri2026040879449505780visors @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__nonzero__semiring,axiom,
linord181362715937106298miring @ int ).
thf(tcon_Int_Oint___Rings_Osemidom__divide__unit__factor,axiom,
semido2269285787275462019factor @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__semiring__strict,axiom,
linord8928482502909563296strict @ int ).
thf(tcon_Int_Oint___Quickcheck__Exhaustive_Oexhaustive_15,axiom,
quickc658316121487927005ustive @ int ).
thf(tcon_Int_Oint___Rings_Osemiring__no__zero__divisors,axiom,
semiri3467727345109120633visors @ int ).
thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__add,axiom,
ordere6658533253407199908up_add @ int ).
thf(tcon_Int_Oint___Groups_Oordered__ab__group__add__abs,axiom,
ordere166539214618696060dd_abs @ int ).
thf(tcon_Int_Oint___GCD_Osemiring__gcd__mult__normalize,axiom,
semiri6843258321239162965malize @ int ).
thf(tcon_Int_Oint___Groups_Oordered__comm__monoid__add,axiom,
ordere6911136660526730532id_add @ int ).
thf(tcon_Int_Oint___Groups_Olinordered__ab__group__add,axiom,
linord5086331880401160121up_add @ int ).
thf(tcon_Int_Oint___Groups_Ocancel__ab__semigroup__add,axiom,
cancel2418104881723323429up_add @ int ).
thf(tcon_Int_Oint___Rings_Oring__1__no__zero__divisors,axiom,
ring_15535105094025558882visors @ int ).
thf(tcon_Int_Oint___Groups_Ocancel__comm__monoid__add,axiom,
cancel1802427076303600483id_add @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__ring__strict,axiom,
linord4710134922213307826strict @ int ).
thf(tcon_Int_Oint___Rings_Ocomm__semiring__1__cancel,axiom,
comm_s4317794764714335236cancel @ int ).
thf(tcon_Int_Oint___Bit__Operations_Osemiring__bits,axiom,
bit_semiring_bits @ int ).
thf(tcon_Int_Oint___Rings_Oordered__comm__semiring,axiom,
ordere2520102378445227354miring @ int ).
thf(tcon_Int_Oint___Rings_Onormalization__semidom,axiom,
normal8620421768224518004emidom @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__semiring__1,axiom,
linord6961819062388156250ring_1 @ int ).
thf(tcon_Int_Oint___Groups_Oordered__ab__group__add,axiom,
ordered_ab_group_add @ int ).
thf(tcon_Int_Oint___Groups_Ocancel__semigroup__add,axiom,
cancel_semigroup_add @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__semiring,axiom,
linordered_semiring @ int ).
thf(tcon_Int_Oint___Rings_Oordered__semiring__0,axiom,
ordered_semiring_0 @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__semidom,axiom,
linordered_semidom @ int ).
thf(tcon_Int_Oint___Lattices_Osemilattice__sup_16,axiom,
semilattice_sup @ int ).
thf(tcon_Int_Oint___Lattices_Osemilattice__inf_17,axiom,
semilattice_inf @ int ).
thf(tcon_Int_Oint___Lattices_Odistrib__lattice_18,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_19,axiom,
complete_Sup @ int ).
thf(tcon_Int_Oint___Complete__Lattices_OInf_20,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___Countable_Ocountable_21,axiom,
countable @ 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_22,axiom,
preorder @ int ).
thf(tcon_Int_Oint___Orderings_Olinorder,axiom,
linorder @ int ).
thf(tcon_Int_Oint___Groups_Omonoid__mult,axiom,
monoid_mult @ int ).
thf(tcon_Int_Oint___Rings_Oidom__modulo,axiom,
idom_modulo @ int ).
thf(tcon_Int_Oint___Rings_Oidom__divide,axiom,
idom_divide @ int ).
thf(tcon_Int_Oint___Rings_Ocomm__ring__1,axiom,
comm_ring_1 @ int ).
thf(tcon_Int_Oint___Groups_Omonoid__add,axiom,
monoid_add @ int ).
thf(tcon_Int_Oint___Rings_Osemiring__1,axiom,
semiring_1 @ int ).
thf(tcon_Int_Oint___Rings_Osemiring__0,axiom,
semiring_0 @ int ).
thf(tcon_Int_Oint___Orderings_Ono__top,axiom,
no_top @ int ).
thf(tcon_Int_Oint___Orderings_Ono__bot,axiom,
no_bot @ int ).
thf(tcon_Int_Oint___Lattices_Olattice_23,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_24,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_25,axiom,
ord @ int ).
thf(tcon_Int_Oint___Groups_Ouminus_26,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___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___HOL_Oequal_27,axiom,
cl_HOL_Oequal @ int ).
thf(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_28,axiom,
condit6923001295902523014norder @ nat ).
thf(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_29,axiom,
condit1219197933456340205attice @ nat ).
thf(tcon_Nat_Onat___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_30,axiom,
bit_un5681908812861735899ations @ nat ).
thf(tcon_Nat_Onat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_31,axiom,
semiri1453513574482234551roduct @ nat ).
thf(tcon_Nat_Onat___Euclidean__Division_Ounique__euclidean__semiring__with__nat_32,axiom,
euclid5411537665997757685th_nat @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_33,axiom,
ordere1937475149494474687imp_le @ nat ).
thf(tcon_Nat_Onat___Euclidean__Division_Ounique__euclidean__semiring_34,axiom,
euclid3128863361964157862miring @ nat ).
thf(tcon_Nat_Onat___Euclidean__Division_Oeuclidean__semiring__cancel_35,axiom,
euclid4440199948858584721cancel @ nat ).
thf(tcon_Nat_Onat___Rings_Onormalization__semidom__multiplicative_36,axiom,
normal6328177297339901930cative @ nat ).
thf(tcon_Nat_Onat___Divides_Ounique__euclidean__semiring__numeral_37,axiom,
unique1627219031080169319umeral @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors__cancel_38,axiom,
semiri6575147826004484403cancel @ nat ).
thf(tcon_Nat_Onat___Groups_Ostrict__ordered__ab__semigroup__add_39,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_40,axiom,
ordere580206878836729694up_add @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le_41,axiom,
ordere2412721322843649153imp_le @ nat ).
thf(tcon_Nat_Onat___Bit__Operations_Osemiring__bit__operations_42,axiom,
bit_se359711467146920520ations @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__comm__semiring__strict_43,axiom,
linord2810124833399127020strict @ nat ).
thf(tcon_Nat_Onat___Quickcheck__Exhaustive_Ofull__exhaustive_44,axiom,
quickc3360725361186068524ustive @ nat ).
thf(tcon_Nat_Onat___Groups_Ostrict__ordered__comm__monoid__add_45,axiom,
strict7427464778891057005id_add @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__cancel__comm__monoid__add_46,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_47,axiom,
euclid3725896446679973847miring @ nat ).
thf(tcon_Nat_Onat___Groups_Olinordered__ab__semigroup__add_48,axiom,
linord4140545234300271783up_add @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__1__no__zero__divisors_49,axiom,
semiri2026040879449505780visors @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring_50,axiom,
linord181362715937106298miring @ nat ).
thf(tcon_Nat_Onat___Rings_Osemidom__divide__unit__factor_51,axiom,
semido2269285787275462019factor @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__semiring__strict_52,axiom,
linord8928482502909563296strict @ nat ).
thf(tcon_Nat_Onat___Quickcheck__Exhaustive_Oexhaustive_53,axiom,
quickc658316121487927005ustive @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors_54,axiom,
semiri3467727345109120633visors @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add_55,axiom,
ordere6658533253407199908up_add @ nat ).
thf(tcon_Nat_Onat___GCD_Osemiring__gcd__mult__normalize_56,axiom,
semiri6843258321239162965malize @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__comm__monoid__add_57,axiom,
ordere6911136660526730532id_add @ nat ).
thf(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add_58,axiom,
cancel2418104881723323429up_add @ nat ).
thf(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add_59,axiom,
cancel1802427076303600483id_add @ nat ).
thf(tcon_Nat_Onat___Rings_Ocomm__semiring__1__cancel_60,axiom,
comm_s4317794764714335236cancel @ nat ).
thf(tcon_Nat_Onat___Bit__Operations_Osemiring__bits_61,axiom,
bit_semiring_bits @ nat ).
thf(tcon_Nat_Onat___Rings_Oordered__comm__semiring_62,axiom,
ordere2520102378445227354miring @ nat ).
thf(tcon_Nat_Onat___Rings_Onormalization__semidom_63,axiom,
normal8620421768224518004emidom @ nat ).
thf(tcon_Nat_Onat___Groups_Ocancel__semigroup__add_64,axiom,
cancel_semigroup_add @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__semiring_65,axiom,
linordered_semiring @ nat ).
thf(tcon_Nat_Onat___Rings_Oordered__semiring__0_66,axiom,
ordered_semiring_0 @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__semidom_67,axiom,
linordered_semidom @ nat ).
thf(tcon_Nat_Onat___Lattices_Osemilattice__sup_68,axiom,
semilattice_sup @ nat ).
thf(tcon_Nat_Onat___Lattices_Osemilattice__inf_69,axiom,
semilattice_inf @ nat ).
thf(tcon_Nat_Onat___Lattices_Odistrib__lattice_70,axiom,
distrib_lattice @ nat ).
thf(tcon_Nat_Onat___Groups_Oab__semigroup__mult_71,axiom,
ab_semigroup_mult @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__1__cancel_72,axiom,
semiring_1_cancel @ nat ).
thf(tcon_Nat_Onat___Rings_Oalgebraic__semidom_73,axiom,
algebraic_semidom @ nat ).
thf(tcon_Nat_Onat___Groups_Ocomm__monoid__mult_74,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_75,axiom,
ab_semigroup_add @ nat ).
thf(tcon_Nat_Onat___Rings_Oordered__semiring_76,axiom,
ordered_semiring @ nat ).
thf(tcon_Nat_Onat___Parity_Osemiring__parity_77,axiom,
semiring_parity @ nat ).
thf(tcon_Nat_Onat___Groups_Ocomm__monoid__add_78,axiom,
comm_monoid_add @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__modulo_79,axiom,
semiring_modulo @ nat ).
thf(tcon_Nat_Onat___Rings_Ocomm__semiring__1_80,axiom,
comm_semiring_1 @ nat ).
thf(tcon_Nat_Onat___Rings_Ocomm__semiring__0_81,axiom,
comm_semiring_0 @ nat ).
thf(tcon_Nat_Onat___Groups_Osemigroup__mult_82,axiom,
semigroup_mult @ nat ).
thf(tcon_Nat_Onat___Complete__Lattices_OSup_83,axiom,
complete_Sup @ nat ).
thf(tcon_Nat_Onat___Complete__Lattices_OInf_84,axiom,
complete_Inf @ nat ).
thf(tcon_Nat_Onat___Rings_Osemidom__modulo_85,axiom,
semidom_modulo @ nat ).
thf(tcon_Nat_Onat___Rings_Osemidom__divide_86,axiom,
semidom_divide @ nat ).
thf(tcon_Nat_Onat___Num_Osemiring__numeral_87,axiom,
semiring_numeral @ nat ).
thf(tcon_Nat_Onat___Groups_Osemigroup__add_88,axiom,
semigroup_add @ nat ).
thf(tcon_Nat_Onat___Rings_Ozero__less__one_89,axiom,
zero_less_one @ nat ).
thf(tcon_Nat_Onat___Rings_Ocomm__semiring_90,axiom,
comm_semiring @ nat ).
thf(tcon_Nat_Onat___Orderings_Owellorder,axiom,
wellorder @ nat ).
thf(tcon_Nat_Onat___Orderings_Oorder__bot_91,axiom,
order_bot @ nat ).
thf(tcon_Nat_Onat___Nat_Osemiring__char__0_92,axiom,
semiring_char_0 @ nat ).
thf(tcon_Nat_Onat___Countable_Ocountable_93,axiom,
countable @ nat ).
thf(tcon_Nat_Onat___Rings_Ozero__neq__one_94,axiom,
zero_neq_one @ nat ).
thf(tcon_Nat_Onat___Orderings_Opreorder_95,axiom,
preorder @ nat ).
thf(tcon_Nat_Onat___Orderings_Olinorder_96,axiom,
linorder @ nat ).
thf(tcon_Nat_Onat___Groups_Omonoid__mult_97,axiom,
monoid_mult @ nat ).
thf(tcon_Nat_Onat___Groups_Omonoid__add_98,axiom,
monoid_add @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__1_99,axiom,
semiring_1 @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__0_100,axiom,
semiring_0 @ nat ).
thf(tcon_Nat_Onat___Orderings_Ono__top_101,axiom,
no_top @ nat ).
thf(tcon_Nat_Onat___Lattices_Olattice_102,axiom,
lattice @ nat ).
thf(tcon_Nat_Onat___GCD_Osemiring__gcd_103,axiom,
semiring_gcd @ nat ).
thf(tcon_Nat_Onat___GCD_Osemiring__Gcd_104,axiom,
semiring_Gcd @ nat ).
thf(tcon_Nat_Onat___Rings_Omult__zero_105,axiom,
mult_zero @ nat ).
thf(tcon_Nat_Onat___Orderings_Oorder_106,axiom,
order @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring_107,axiom,
semiring @ nat ).
thf(tcon_Nat_Onat___Rings_Osemidom_108,axiom,
semidom @ nat ).
thf(tcon_Nat_Onat___Orderings_Oord_109,axiom,
ord @ nat ).
thf(tcon_Nat_Onat___Power_Opower_110,axiom,
power @ nat ).
thf(tcon_Nat_Onat___Num_Onumeral_111,axiom,
numeral @ nat ).
thf(tcon_Nat_Onat___Groups_Ozero_112,axiom,
zero @ nat ).
thf(tcon_Nat_Onat___Groups_Oplus_113,axiom,
plus @ nat ).
thf(tcon_Nat_Onat___Groups_Oone_114,axiom,
one @ nat ).
thf(tcon_Nat_Onat___Rings_Odvd_115,axiom,
dvd @ nat ).
thf(tcon_Nat_Onat___HOL_Oequal_116,axiom,
cl_HOL_Oequal @ nat ).
thf(tcon_Nat_Onat___Nat_Osize,axiom,
size @ nat ).
thf(tcon_Num_Onum___Quickcheck__Exhaustive_Ofull__exhaustive_117,axiom,
quickc3360725361186068524ustive @ num ).
thf(tcon_Num_Onum___Orderings_Opreorder_118,axiom,
preorder @ num ).
thf(tcon_Num_Onum___Orderings_Olinorder_119,axiom,
linorder @ num ).
thf(tcon_Num_Onum___Orderings_Oorder_120,axiom,
order @ num ).
thf(tcon_Num_Onum___Orderings_Oord_121,axiom,
ord @ num ).
thf(tcon_Num_Onum___Groups_Oplus_122,axiom,
plus @ num ).
thf(tcon_Num_Onum___HOL_Oequal_123,axiom,
cl_HOL_Oequal @ num ).
thf(tcon_Num_Onum___Nat_Osize_124,axiom,
size @ num ).
thf(tcon_Rat_Orat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_125,axiom,
semiri1453513574482234551roduct @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_126,axiom,
ordere1937475149494474687imp_le @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__no__zero__divisors__cancel_127,axiom,
semiri6575147826004484403cancel @ rat ).
thf(tcon_Rat_Orat___Groups_Ostrict__ordered__ab__semigroup__add_128,axiom,
strict9044650504122735259up_add @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__cancel__ab__semigroup__add_129,axiom,
ordere580206878836729694up_add @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__add__imp__le_130,axiom,
ordere2412721322843649153imp_le @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__comm__semiring__strict_131,axiom,
linord2810124833399127020strict @ rat ).
thf(tcon_Rat_Orat___Quickcheck__Exhaustive_Ofull__exhaustive_132,axiom,
quickc3360725361186068524ustive @ rat ).
thf(tcon_Rat_Orat___Groups_Ostrict__ordered__comm__monoid__add_133,axiom,
strict7427464778891057005id_add @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__cancel__comm__monoid__add_134,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_135,axiom,
linord715952674999750819strict @ rat ).
thf(tcon_Rat_Orat___Orderings_Ounbounded__dense__linorder,axiom,
unboun7993243217541854897norder @ rat ).
thf(tcon_Rat_Orat___Groups_Olinordered__ab__semigroup__add_136,axiom,
linord4140545234300271783up_add @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__1__no__zero__divisors_137,axiom,
semiri2026040879449505780visors @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__nonzero__semiring_138,axiom,
linord181362715937106298miring @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__semiring__strict_139,axiom,
linord8928482502909563296strict @ rat ).
thf(tcon_Rat_Orat___Quickcheck__Exhaustive_Oexhaustive_140,axiom,
quickc658316121487927005ustive @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__no__zero__divisors_141,axiom,
semiri3467727345109120633visors @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__add_142,axiom,
ordere6658533253407199908up_add @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__ab__group__add__abs_143,axiom,
ordere166539214618696060dd_abs @ rat ).
thf(tcon_Rat_Orat___Archimedean__Field_Ofloor__ceiling,axiom,
archim2362893244070406136eiling @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__comm__monoid__add_144,axiom,
ordere6911136660526730532id_add @ rat ).
thf(tcon_Rat_Orat___Groups_Olinordered__ab__group__add_145,axiom,
linord5086331880401160121up_add @ rat ).
thf(tcon_Rat_Orat___Groups_Ocancel__ab__semigroup__add_146,axiom,
cancel2418104881723323429up_add @ rat ).
thf(tcon_Rat_Orat___Rings_Oring__1__no__zero__divisors_147,axiom,
ring_15535105094025558882visors @ rat ).
thf(tcon_Rat_Orat___Groups_Ocancel__comm__monoid__add_148,axiom,
cancel1802427076303600483id_add @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__ring__strict_149,axiom,
linord4710134922213307826strict @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__semiring__1__cancel_150,axiom,
comm_s4317794764714335236cancel @ rat ).
thf(tcon_Rat_Orat___Rings_Oordered__comm__semiring_151,axiom,
ordere2520102378445227354miring @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__semiring__1_152,axiom,
linord6961819062388156250ring_1 @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__ab__group__add_153,axiom,
ordered_ab_group_add @ rat ).
thf(tcon_Rat_Orat___Groups_Ocancel__semigroup__add_154,axiom,
cancel_semigroup_add @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__semiring_155,axiom,
linordered_semiring @ rat ).
thf(tcon_Rat_Orat___Rings_Oordered__semiring__0_156,axiom,
ordered_semiring_0 @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__semidom_157,axiom,
linordered_semidom @ rat ).
thf(tcon_Rat_Orat___Orderings_Odense__linorder,axiom,
dense_linorder @ rat ).
thf(tcon_Rat_Orat___Lattices_Osemilattice__sup_158,axiom,
semilattice_sup @ rat ).
thf(tcon_Rat_Orat___Lattices_Osemilattice__inf_159,axiom,
semilattice_inf @ rat ).
thf(tcon_Rat_Orat___Lattices_Odistrib__lattice_160,axiom,
distrib_lattice @ rat ).
thf(tcon_Rat_Orat___Groups_Oab__semigroup__mult_161,axiom,
ab_semigroup_mult @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__1__cancel_162,axiom,
semiring_1_cancel @ rat ).
thf(tcon_Rat_Orat___Groups_Ocomm__monoid__mult_163,axiom,
comm_monoid_mult @ rat ).
thf(tcon_Rat_Orat___Groups_Oab__semigroup__add_164,axiom,
ab_semigroup_add @ rat ).
thf(tcon_Rat_Orat___Fields_Olinordered__field,axiom,
linordered_field @ rat ).
thf(tcon_Rat_Orat___Rings_Oordered__semiring_165,axiom,
ordered_semiring @ rat ).
thf(tcon_Rat_Orat___Rings_Oordered__ring__abs_166,axiom,
ordered_ring_abs @ rat ).
thf(tcon_Rat_Orat___Groups_Ocomm__monoid__add_167,axiom,
comm_monoid_add @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__ring_168,axiom,
linordered_ring @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__idom_169,axiom,
linordered_idom @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__semiring__1_170,axiom,
comm_semiring_1 @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__semiring__0_171,axiom,
comm_semiring_0 @ rat ).
thf(tcon_Rat_Orat___Groups_Osemigroup__mult_172,axiom,
semigroup_mult @ rat ).
thf(tcon_Rat_Orat___Rings_Osemidom__divide_173,axiom,
semidom_divide @ rat ).
thf(tcon_Rat_Orat___Num_Osemiring__numeral_174,axiom,
semiring_numeral @ rat ).
thf(tcon_Rat_Orat___Groups_Osemigroup__add_175,axiom,
semigroup_add @ rat ).
thf(tcon_Rat_Orat___Fields_Odivision__ring,axiom,
division_ring @ rat ).
thf(tcon_Rat_Orat___Rings_Ozero__less__one_176,axiom,
zero_less_one @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__semiring_177,axiom,
comm_semiring @ rat ).
thf(tcon_Rat_Orat___Nat_Osemiring__char__0_178,axiom,
semiring_char_0 @ rat ).
thf(tcon_Rat_Orat___Groups_Oab__group__add_179,axiom,
ab_group_add @ rat ).
thf(tcon_Rat_Orat___Fields_Ofield__char__0,axiom,
field_char_0 @ rat ).
thf(tcon_Rat_Orat___Countable_Ocountable_180,axiom,
countable @ rat ).
thf(tcon_Rat_Orat___Rings_Ozero__neq__one_181,axiom,
zero_neq_one @ rat ).
thf(tcon_Rat_Orat___Rings_Oordered__ring_182,axiom,
ordered_ring @ rat ).
thf(tcon_Rat_Orat___Rings_Oidom__abs__sgn_183,axiom,
idom_abs_sgn @ rat ).
thf(tcon_Rat_Orat___Orderings_Opreorder_184,axiom,
preorder @ rat ).
thf(tcon_Rat_Orat___Orderings_Olinorder_185,axiom,
linorder @ rat ).
thf(tcon_Rat_Orat___Groups_Omonoid__mult_186,axiom,
monoid_mult @ rat ).
thf(tcon_Rat_Orat___Rings_Oidom__divide_187,axiom,
idom_divide @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__ring__1_188,axiom,
comm_ring_1 @ rat ).
thf(tcon_Rat_Orat___Groups_Omonoid__add_189,axiom,
monoid_add @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__1_190,axiom,
semiring_1 @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__0_191,axiom,
semiring_0 @ rat ).
thf(tcon_Rat_Orat___Orderings_Ono__top_192,axiom,
no_top @ rat ).
thf(tcon_Rat_Orat___Orderings_Ono__bot_193,axiom,
no_bot @ rat ).
thf(tcon_Rat_Orat___Lattices_Olattice_194,axiom,
lattice @ rat ).
thf(tcon_Rat_Orat___Groups_Ogroup__add_195,axiom,
group_add @ rat ).
thf(tcon_Rat_Orat___Rings_Omult__zero_196,axiom,
mult_zero @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__ring_197,axiom,
comm_ring @ rat ).
thf(tcon_Rat_Orat___Orderings_Oorder_198,axiom,
order @ rat ).
thf(tcon_Rat_Orat___Num_Oneg__numeral_199,axiom,
neg_numeral @ rat ).
thf(tcon_Rat_Orat___Nat_Oring__char__0_200,axiom,
ring_char_0 @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring_201,axiom,
semiring @ rat ).
thf(tcon_Rat_Orat___Fields_Oinverse,axiom,
inverse @ rat ).
thf(tcon_Rat_Orat___Rings_Osemidom_202,axiom,
semidom @ rat ).
thf(tcon_Rat_Orat___Orderings_Oord_203,axiom,
ord @ rat ).
thf(tcon_Rat_Orat___Groups_Ouminus_204,axiom,
uminus @ rat ).
thf(tcon_Rat_Orat___Rings_Oring__1_205,axiom,
ring_1 @ rat ).
thf(tcon_Rat_Orat___Rings_Oabs__if_206,axiom,
abs_if @ rat ).
thf(tcon_Rat_Orat___Fields_Ofield,axiom,
field @ rat ).
thf(tcon_Rat_Orat___Power_Opower_207,axiom,
power @ rat ).
thf(tcon_Rat_Orat___Num_Onumeral_208,axiom,
numeral @ rat ).
thf(tcon_Rat_Orat___Groups_Ozero_209,axiom,
zero @ rat ).
thf(tcon_Rat_Orat___Groups_Oplus_210,axiom,
plus @ rat ).
thf(tcon_Rat_Orat___Rings_Oring_211,axiom,
ring @ rat ).
thf(tcon_Rat_Orat___Rings_Oidom_212,axiom,
idom @ rat ).
thf(tcon_Rat_Orat___Groups_Oone_213,axiom,
one @ rat ).
thf(tcon_Rat_Orat___Rings_Odvd_214,axiom,
dvd @ rat ).
thf(tcon_Rat_Orat___HOL_Oequal_215,axiom,
cl_HOL_Oequal @ rat ).
thf(tcon_Set_Oset___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_216,axiom,
! [A21: $tType] : ( condit1219197933456340205attice @ ( set @ A21 ) ) ).
thf(tcon_Set_Oset___Complete__Lattices_Ocomplete__distrib__lattice_217,axiom,
! [A21: $tType] : ( comple592849572758109894attice @ ( set @ A21 ) ) ).
thf(tcon_Set_Oset___Complete__Lattices_Ocomplete__boolean__algebra_218,axiom,
! [A21: $tType] : ( comple489889107523837845lgebra @ ( set @ A21 ) ) ).
thf(tcon_Set_Oset___Quickcheck__Exhaustive_Ofull__exhaustive_219,axiom,
! [A21: $tType] :
( ( quickc3360725361186068524ustive @ A21 )
=> ( quickc3360725361186068524ustive @ ( set @ A21 ) ) ) ).
thf(tcon_Set_Oset___Lattices_Obounded__semilattice__sup__bot_220,axiom,
! [A21: $tType] : ( bounde4967611905675639751up_bot @ ( set @ A21 ) ) ).
thf(tcon_Set_Oset___Complete__Lattices_Ocomplete__lattice_221,axiom,
! [A21: $tType] : ( comple6319245703460814977attice @ ( set @ A21 ) ) ).
thf(tcon_Set_Oset___Quickcheck__Exhaustive_Oexhaustive_222,axiom,
! [A21: $tType] :
( ( quickc658316121487927005ustive @ A21 )
=> ( quickc658316121487927005ustive @ ( set @ A21 ) ) ) ).
thf(tcon_Set_Oset___Boolean__Algebras_Oboolean__algebra_223,axiom,
! [A21: $tType] : ( boolea8198339166811842893lgebra @ ( set @ A21 ) ) ).
thf(tcon_Set_Oset___Complete__Partial__Order_Occpo_224,axiom,
! [A21: $tType] : ( comple9053668089753744459l_ccpo @ ( set @ A21 ) ) ).
thf(tcon_Set_Oset___Lattices_Osemilattice__sup_225,axiom,
! [A21: $tType] : ( semilattice_sup @ ( set @ A21 ) ) ).
thf(tcon_Set_Oset___Lattices_Osemilattice__inf_226,axiom,
! [A21: $tType] : ( semilattice_inf @ ( set @ A21 ) ) ).
thf(tcon_Set_Oset___Lattices_Odistrib__lattice_227,axiom,
! [A21: $tType] : ( distrib_lattice @ ( set @ A21 ) ) ).
thf(tcon_Set_Oset___Complete__Lattices_OSup_228,axiom,
! [A21: $tType] : ( complete_Sup @ ( set @ A21 ) ) ).
thf(tcon_Set_Oset___Complete__Lattices_OInf_229,axiom,
! [A21: $tType] : ( complete_Inf @ ( set @ A21 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder__top_230,axiom,
! [A21: $tType] : ( order_top @ ( set @ A21 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder__bot_231,axiom,
! [A21: $tType] : ( order_bot @ ( set @ A21 ) ) ).
thf(tcon_Set_Oset___Countable_Ocountable_232,axiom,
! [A21: $tType] :
( ( finite_finite @ A21 )
=> ( countable @ ( set @ A21 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Opreorder_233,axiom,
! [A21: $tType] : ( preorder @ ( set @ A21 ) ) ).
thf(tcon_Set_Oset___Finite__Set_Ofinite_234,axiom,
! [A21: $tType] :
( ( finite_finite @ A21 )
=> ( finite_finite @ ( set @ A21 ) ) ) ).
thf(tcon_Set_Oset___Lattices_Olattice_235,axiom,
! [A21: $tType] : ( lattice @ ( set @ A21 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_236,axiom,
! [A21: $tType] : ( order @ ( set @ A21 ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_237,axiom,
! [A21: $tType] : ( ord @ ( set @ A21 ) ) ).
thf(tcon_Set_Oset___Groups_Ouminus_238,axiom,
! [A21: $tType] : ( uminus @ ( set @ A21 ) ) ).
thf(tcon_Set_Oset___HOL_Oequal_239,axiom,
! [A21: $tType] :
( ( cl_HOL_Oequal @ A21 )
=> ( cl_HOL_Oequal @ ( set @ A21 ) ) ) ).
thf(tcon_HOL_Obool___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_240,axiom,
condit1219197933456340205attice @ $o ).
thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__distrib__lattice_241,axiom,
comple592849572758109894attice @ $o ).
thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__boolean__algebra_242,axiom,
comple489889107523837845lgebra @ $o ).
thf(tcon_HOL_Obool___Quickcheck__Exhaustive_Ofull__exhaustive_243,axiom,
quickc3360725361186068524ustive @ $o ).
thf(tcon_HOL_Obool___Lattices_Obounded__semilattice__sup__bot_244,axiom,
bounde4967611905675639751up_bot @ $o ).
thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__lattice_245,axiom,
comple6319245703460814977attice @ $o ).
thf(tcon_HOL_Obool___Boolean__Algebras_Oboolean__algebra_246,axiom,
boolea8198339166811842893lgebra @ $o ).
thf(tcon_HOL_Obool___Complete__Partial__Order_Occpo_247,axiom,
comple9053668089753744459l_ccpo @ $o ).
thf(tcon_HOL_Obool___Lattices_Osemilattice__sup_248,axiom,
semilattice_sup @ $o ).
thf(tcon_HOL_Obool___Lattices_Osemilattice__inf_249,axiom,
semilattice_inf @ $o ).
thf(tcon_HOL_Obool___Lattices_Odistrib__lattice_250,axiom,
distrib_lattice @ $o ).
thf(tcon_HOL_Obool___Complete__Lattices_OSup_251,axiom,
complete_Sup @ $o ).
thf(tcon_HOL_Obool___Complete__Lattices_OInf_252,axiom,
complete_Inf @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder__top_253,axiom,
order_top @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder__bot_254,axiom,
order_bot @ $o ).
thf(tcon_HOL_Obool___Countable_Ocountable_255,axiom,
countable @ $o ).
thf(tcon_HOL_Obool___Orderings_Opreorder_256,axiom,
preorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Olinorder_257,axiom,
linorder @ $o ).
thf(tcon_HOL_Obool___Finite__Set_Ofinite_258,axiom,
finite_finite @ $o ).
thf(tcon_HOL_Obool___Lattices_Olattice_259,axiom,
lattice @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder_260,axiom,
order @ $o ).
thf(tcon_HOL_Obool___Orderings_Oord_261,axiom,
ord @ $o ).
thf(tcon_HOL_Obool___Groups_Ouminus_262,axiom,
uminus @ $o ).
thf(tcon_HOL_Obool___HOL_Oequal_263,axiom,
cl_HOL_Oequal @ $o ).
thf(tcon_List_Olist___Quickcheck__Exhaustive_Ofull__exhaustive_264,axiom,
! [A21: $tType] :
( ( quickc3360725361186068524ustive @ A21 )
=> ( quickc3360725361186068524ustive @ ( list @ A21 ) ) ) ).
thf(tcon_List_Olist___Countable_Ocountable_265,axiom,
! [A21: $tType] :
( ( countable @ A21 )
=> ( countable @ ( list @ A21 ) ) ) ).
thf(tcon_List_Olist___HOL_Oequal_266,axiom,
! [A21: $tType] : ( cl_HOL_Oequal @ ( list @ A21 ) ) ).
thf(tcon_List_Olist___Nat_Osize_267,axiom,
! [A21: $tType] : ( size @ ( list @ A21 ) ) ).
thf(tcon_String_Ochar___Quickcheck__Exhaustive_Ofull__exhaustive_268,axiom,
quickc3360725361186068524ustive @ char ).
thf(tcon_String_Ochar___Countable_Ocountable_269,axiom,
countable @ char ).
thf(tcon_String_Ochar___Finite__Set_Ofinite_270,axiom,
finite_finite @ char ).
thf(tcon_String_Ochar___HOL_Oequal_271,axiom,
cl_HOL_Oequal @ char ).
thf(tcon_String_Ochar___Nat_Osize_272,axiom,
size @ char ).
thf(tcon_Sum__Type_Osum___Quickcheck__Exhaustive_Ofull__exhaustive_273,axiom,
! [A21: $tType,A27: $tType] :
( ( ( quickc3360725361186068524ustive @ A21 )
& ( quickc3360725361186068524ustive @ A27 ) )
=> ( quickc3360725361186068524ustive @ ( sum_sum @ A21 @ A27 ) ) ) ).
thf(tcon_Sum__Type_Osum___Countable_Ocountable_274,axiom,
! [A21: $tType,A27: $tType] :
( ( ( countable @ A21 )
& ( countable @ A27 ) )
=> ( countable @ ( sum_sum @ A21 @ A27 ) ) ) ).
thf(tcon_Sum__Type_Osum___Finite__Set_Ofinite_275,axiom,
! [A21: $tType,A27: $tType] :
( ( ( finite_finite @ A21 )
& ( finite_finite @ A27 ) )
=> ( finite_finite @ ( sum_sum @ A21 @ A27 ) ) ) ).
thf(tcon_Sum__Type_Osum___HOL_Oequal_276,axiom,
! [A21: $tType,A27: $tType] : ( cl_HOL_Oequal @ ( sum_sum @ A21 @ A27 ) ) ).
thf(tcon_Sum__Type_Osum___Nat_Osize_277,axiom,
! [A21: $tType,A27: $tType] : ( size @ ( sum_sum @ A21 @ A27 ) ) ).
thf(tcon_Filter_Ofilter___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_278,axiom,
! [A21: $tType] : ( condit1219197933456340205attice @ ( filter @ A21 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Obounded__semilattice__sup__bot_279,axiom,
! [A21: $tType] : ( bounde4967611905675639751up_bot @ ( filter @ A21 ) ) ).
thf(tcon_Filter_Ofilter___Complete__Lattices_Ocomplete__lattice_280,axiom,
! [A21: $tType] : ( comple6319245703460814977attice @ ( filter @ A21 ) ) ).
thf(tcon_Filter_Ofilter___Complete__Partial__Order_Occpo_281,axiom,
! [A21: $tType] : ( comple9053668089753744459l_ccpo @ ( filter @ A21 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Osemilattice__sup_282,axiom,
! [A21: $tType] : ( semilattice_sup @ ( filter @ A21 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Osemilattice__inf_283,axiom,
! [A21: $tType] : ( semilattice_inf @ ( filter @ A21 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Odistrib__lattice_284,axiom,
! [A21: $tType] : ( distrib_lattice @ ( filter @ A21 ) ) ).
thf(tcon_Filter_Ofilter___Complete__Lattices_OSup_285,axiom,
! [A21: $tType] : ( complete_Sup @ ( filter @ A21 ) ) ).
thf(tcon_Filter_Ofilter___Complete__Lattices_OInf_286,axiom,
! [A21: $tType] : ( complete_Inf @ ( filter @ A21 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Oorder__top_287,axiom,
! [A21: $tType] : ( order_top @ ( filter @ A21 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Oorder__bot_288,axiom,
! [A21: $tType] : ( order_bot @ ( filter @ A21 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Opreorder_289,axiom,
! [A21: $tType] : ( preorder @ ( filter @ A21 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Olattice_290,axiom,
! [A21: $tType] : ( lattice @ ( filter @ A21 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Oorder_291,axiom,
! [A21: $tType] : ( order @ ( filter @ A21 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Oord_292,axiom,
! [A21: $tType] : ( ord @ ( filter @ A21 ) ) ).
thf(tcon_Filter_Ofilter___HOL_Oequal_293,axiom,
! [A21: $tType] :
( ( cl_HOL_Oequal @ A21 )
=> ( cl_HOL_Oequal @ ( filter @ A21 ) ) ) ).
thf(tcon_Option_Ooption___Quickcheck__Exhaustive_Ofull__exhaustive_294,axiom,
! [A21: $tType] :
( ( quickc3360725361186068524ustive @ A21 )
=> ( quickc3360725361186068524ustive @ ( option @ A21 ) ) ) ).
thf(tcon_Option_Ooption___Countable_Ocountable_295,axiom,
! [A21: $tType] :
( ( countable @ A21 )
=> ( countable @ ( option @ A21 ) ) ) ).
thf(tcon_Option_Ooption___Finite__Set_Ofinite_296,axiom,
! [A21: $tType] :
( ( finite_finite @ A21 )
=> ( finite_finite @ ( option @ A21 ) ) ) ).
thf(tcon_Option_Ooption___HOL_Oequal_297,axiom,
! [A21: $tType] : ( cl_HOL_Oequal @ ( option @ A21 ) ) ).
thf(tcon_Option_Ooption___Nat_Osize_298,axiom,
! [A21: $tType] : ( size @ ( option @ A21 ) ) ).
thf(tcon_Predicate_Oseq___HOL_Oequal_299,axiom,
! [A21: $tType] : ( cl_HOL_Oequal @ ( seq @ A21 ) ) ).
thf(tcon_Predicate_Opred___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_300,axiom,
! [A21: $tType] : ( condit1219197933456340205attice @ ( pred @ A21 ) ) ).
thf(tcon_Predicate_Opred___Complete__Lattices_Ocomplete__distrib__lattice_301,axiom,
! [A21: $tType] : ( comple592849572758109894attice @ ( pred @ A21 ) ) ).
thf(tcon_Predicate_Opred___Complete__Lattices_Ocomplete__boolean__algebra_302,axiom,
! [A21: $tType] : ( comple489889107523837845lgebra @ ( pred @ A21 ) ) ).
thf(tcon_Predicate_Opred___Lattices_Obounded__semilattice__sup__bot_303,axiom,
! [A21: $tType] : ( bounde4967611905675639751up_bot @ ( pred @ A21 ) ) ).
thf(tcon_Predicate_Opred___Complete__Lattices_Ocomplete__lattice_304,axiom,
! [A21: $tType] : ( comple6319245703460814977attice @ ( pred @ A21 ) ) ).
thf(tcon_Predicate_Opred___Boolean__Algebras_Oboolean__algebra_305,axiom,
! [A21: $tType] : ( boolea8198339166811842893lgebra @ ( pred @ A21 ) ) ).
thf(tcon_Predicate_Opred___Complete__Partial__Order_Occpo_306,axiom,
! [A21: $tType] : ( comple9053668089753744459l_ccpo @ ( pred @ A21 ) ) ).
thf(tcon_Predicate_Opred___Lattices_Osemilattice__sup_307,axiom,
! [A21: $tType] : ( semilattice_sup @ ( pred @ A21 ) ) ).
thf(tcon_Predicate_Opred___Lattices_Osemilattice__inf_308,axiom,
! [A21: $tType] : ( semilattice_inf @ ( pred @ A21 ) ) ).
thf(tcon_Predicate_Opred___Lattices_Odistrib__lattice_309,axiom,
! [A21: $tType] : ( distrib_lattice @ ( pred @ A21 ) ) ).
thf(tcon_Predicate_Opred___Complete__Lattices_OSup_310,axiom,
! [A21: $tType] : ( complete_Sup @ ( pred @ A21 ) ) ).
thf(tcon_Predicate_Opred___Complete__Lattices_OInf_311,axiom,
! [A21: $tType] : ( complete_Inf @ ( pred @ A21 ) ) ).
thf(tcon_Predicate_Opred___Orderings_Oorder__top_312,axiom,
! [A21: $tType] : ( order_top @ ( pred @ A21 ) ) ).
thf(tcon_Predicate_Opred___Orderings_Oorder__bot_313,axiom,
! [A21: $tType] : ( order_bot @ ( pred @ A21 ) ) ).
thf(tcon_Predicate_Opred___Orderings_Opreorder_314,axiom,
! [A21: $tType] : ( preorder @ ( pred @ A21 ) ) ).
thf(tcon_Predicate_Opred___Lattices_Olattice_315,axiom,
! [A21: $tType] : ( lattice @ ( pred @ A21 ) ) ).
thf(tcon_Predicate_Opred___Orderings_Oorder_316,axiom,
! [A21: $tType] : ( order @ ( pred @ A21 ) ) ).
thf(tcon_Predicate_Opred___Orderings_Oord_317,axiom,
! [A21: $tType] : ( ord @ ( pred @ A21 ) ) ).
thf(tcon_Predicate_Opred___Groups_Ouminus_318,axiom,
! [A21: $tType] : ( uminus @ ( pred @ A21 ) ) ).
thf(tcon_Predicate_Opred___HOL_Oequal_319,axiom,
! [A21: $tType] : ( cl_HOL_Oequal @ ( pred @ A21 ) ) ).
thf(tcon_String_Oliteral___Groups_Osemigroup__add_320,axiom,
semigroup_add @ literal ).
thf(tcon_String_Oliteral___Countable_Ocountable_321,axiom,
countable @ literal ).
thf(tcon_String_Oliteral___Orderings_Opreorder_322,axiom,
preorder @ literal ).
thf(tcon_String_Oliteral___Orderings_Olinorder_323,axiom,
linorder @ literal ).
thf(tcon_String_Oliteral___Groups_Omonoid__add_324,axiom,
monoid_add @ literal ).
thf(tcon_String_Oliteral___Orderings_Oorder_325,axiom,
order @ literal ).
thf(tcon_String_Oliteral___Orderings_Oord_326,axiom,
ord @ literal ).
thf(tcon_String_Oliteral___Groups_Ozero_327,axiom,
zero @ literal ).
thf(tcon_String_Oliteral___Groups_Oplus_328,axiom,
plus @ literal ).
thf(tcon_String_Oliteral___HOL_Oequal_329,axiom,
cl_HOL_Oequal @ literal ).
thf(tcon_String_Oliteral___Nat_Osize_330,axiom,
size @ literal ).
thf(tcon_Typerep_Otyperep___Countable_Ocountable_331,axiom,
countable @ typerep ).
thf(tcon_Typerep_Otyperep___HOL_Oequal_332,axiom,
cl_HOL_Oequal @ typerep ).
thf(tcon_Typerep_Otyperep___Nat_Osize_333,axiom,
size @ typerep ).
thf(tcon_Product__Type_Oprod___Quickcheck__Exhaustive_Ofull__exhaustive_334,axiom,
! [A21: $tType,A27: $tType] :
( ( ( quickc3360725361186068524ustive @ A21 )
& ( quickc3360725361186068524ustive @ A27 ) )
=> ( quickc3360725361186068524ustive @ ( product_prod @ A21 @ A27 ) ) ) ).
thf(tcon_Product__Type_Oprod___Quickcheck__Exhaustive_Oexhaustive_335,axiom,
! [A21: $tType,A27: $tType] :
( ( ( quickc658316121487927005ustive @ A21 )
& ( quickc658316121487927005ustive @ A27 ) )
=> ( quickc658316121487927005ustive @ ( product_prod @ A21 @ A27 ) ) ) ).
thf(tcon_Product__Type_Oprod___Countable_Ocountable_336,axiom,
! [A21: $tType,A27: $tType] :
( ( ( countable @ A21 )
& ( countable @ A27 ) )
=> ( countable @ ( product_prod @ A21 @ A27 ) ) ) ).
thf(tcon_Product__Type_Oprod___Finite__Set_Ofinite_337,axiom,
! [A21: $tType,A27: $tType] :
( ( ( finite_finite @ A21 )
& ( finite_finite @ A27 ) )
=> ( finite_finite @ ( product_prod @ A21 @ A27 ) ) ) ).
thf(tcon_Product__Type_Oprod___HOL_Oequal_338,axiom,
! [A21: $tType,A27: $tType] : ( cl_HOL_Oequal @ ( product_prod @ A21 @ A27 ) ) ).
thf(tcon_Product__Type_Oprod___Nat_Osize_339,axiom,
! [A21: $tType,A27: $tType] : ( size @ ( product_prod @ A21 @ A27 ) ) ).
thf(tcon_Product__Type_Ounit___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_340,axiom,
condit6923001295902523014norder @ product_unit ).
thf(tcon_Product__Type_Ounit___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_341,axiom,
condit1219197933456340205attice @ product_unit ).
thf(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__distrib__lattice_342,axiom,
comple592849572758109894attice @ product_unit ).
thf(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__boolean__algebra_343,axiom,
comple489889107523837845lgebra @ product_unit ).
thf(tcon_Product__Type_Ounit___Quickcheck__Exhaustive_Ofull__exhaustive_344,axiom,
quickc3360725361186068524ustive @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Obounded__semilattice__sup__bot_345,axiom,
bounde4967611905675639751up_bot @ product_unit ).
thf(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__linorder,axiom,
comple5582772986160207858norder @ product_unit ).
thf(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__lattice_346,axiom,
comple6319245703460814977attice @ product_unit ).
thf(tcon_Product__Type_Ounit___Boolean__Algebras_Oboolean__algebra_347,axiom,
boolea8198339166811842893lgebra @ product_unit ).
thf(tcon_Product__Type_Ounit___Complete__Partial__Order_Occpo_348,axiom,
comple9053668089753744459l_ccpo @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Osemilattice__sup_349,axiom,
semilattice_sup @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Osemilattice__inf_350,axiom,
semilattice_inf @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Odistrib__lattice_351,axiom,
distrib_lattice @ product_unit ).
thf(tcon_Product__Type_Ounit___Complete__Lattices_OSup_352,axiom,
complete_Sup @ product_unit ).
thf(tcon_Product__Type_Ounit___Complete__Lattices_OInf_353,axiom,
complete_Inf @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Owellorder_354,axiom,
wellorder @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oorder__top_355,axiom,
order_top @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oorder__bot_356,axiom,
order_bot @ product_unit ).
thf(tcon_Product__Type_Ounit___Countable_Ocountable_357,axiom,
countable @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Opreorder_358,axiom,
preorder @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Olinorder_359,axiom,
linorder @ product_unit ).
thf(tcon_Product__Type_Ounit___Finite__Set_Ofinite_360,axiom,
finite_finite @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Olattice_361,axiom,
lattice @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oorder_362,axiom,
order @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oord_363,axiom,
ord @ product_unit ).
thf(tcon_Product__Type_Ounit___Groups_Ouminus_364,axiom,
uminus @ product_unit ).
thf(tcon_Product__Type_Ounit___HOL_Oequal_365,axiom,
cl_HOL_Oequal @ product_unit ).
thf(tcon_Heap_Oheap_Oheap__ext___HOL_Oequal_366,axiom,
! [A21: $tType] : ( cl_HOL_Oequal @ ( heap_ext @ A21 ) ) ).
thf(tcon_Code__Numeral_Ointeger___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_367,axiom,
bit_un5681908812861735899ations @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_368,axiom,
semiri1453513574482234551roduct @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__semiring__with__nat_369,axiom,
euclid5411537665997757685th_nat @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__ring__with__nat_370,axiom,
euclid8789492081693882211th_nat @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__monoid__add__imp__le_371,axiom,
ordere1937475149494474687imp_le @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__semiring_372,axiom,
euclid3128863361964157862miring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__semiring__cancel_373,axiom,
euclid4440199948858584721cancel @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Divides_Ounique__euclidean__semiring__numeral_374,axiom,
unique1627219031080169319umeral @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__ring__cancel_375,axiom,
euclid8851590272496341667cancel @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__no__zero__divisors__cancel_376,axiom,
semiri6575147826004484403cancel @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ostrict__ordered__ab__semigroup__add_377,axiom,
strict9044650504122735259up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__cancel__ab__semigroup__add_378,axiom,
ordere580206878836729694up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__add__imp__le_379,axiom,
ordere2412721322843649153imp_le @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Bit__Operations_Osemiring__bit__operations_380,axiom,
bit_se359711467146920520ations @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__comm__semiring__strict_381,axiom,
linord2810124833399127020strict @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Quickcheck__Exhaustive_Ofull__exhaustive_382,axiom,
quickc3360725361186068524ustive @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ostrict__ordered__comm__monoid__add_383,axiom,
strict7427464778891057005id_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__cancel__comm__monoid__add_384,axiom,
ordere8940638589300402666id_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__semiring_385,axiom,
euclid3725896446679973847miring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__1__strict_386,axiom,
linord715952674999750819strict @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Olinordered__ab__semigroup__add_387,axiom,
linord4140545234300271783up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Bit__Operations_Oring__bit__operations_388,axiom,
bit_ri3973907225187159222ations @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1__no__zero__divisors_389,axiom,
semiri2026040879449505780visors @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__nonzero__semiring_390,axiom,
linord181362715937106298miring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__strict_391,axiom,
linord8928482502909563296strict @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Quickcheck__Exhaustive_Oexhaustive_392,axiom,
quickc658316121487927005ustive @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__no__zero__divisors_393,axiom,
semiri3467727345109120633visors @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__add_394,axiom,
ordere6658533253407199908up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__group__add__abs_395,axiom,
ordere166539214618696060dd_abs @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__comm__monoid__add_396,axiom,
ordere6911136660526730532id_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Olinordered__ab__group__add_397,axiom,
linord5086331880401160121up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ocancel__ab__semigroup__add_398,axiom,
cancel2418104881723323429up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oring__1__no__zero__divisors_399,axiom,
ring_15535105094025558882visors @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ocancel__comm__monoid__add_400,axiom,
cancel1802427076303600483id_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__ring__strict_401,axiom,
linord4710134922213307826strict @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__1__cancel_402,axiom,
comm_s4317794764714335236cancel @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Bit__Operations_Osemiring__bits_403,axiom,
bit_semiring_bits @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__comm__semiring_404,axiom,
ordere2520102378445227354miring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__1_405,axiom,
linord6961819062388156250ring_1 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__group__add_406,axiom,
ordered_ab_group_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ocancel__semigroup__add_407,axiom,
cancel_semigroup_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring_408,axiom,
linordered_semiring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__semiring__0_409,axiom,
ordered_semiring_0 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semidom_410,axiom,
linordered_semidom @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oab__semigroup__mult_411,axiom,
ab_semigroup_mult @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1__cancel_412,axiom,
semiring_1_cancel @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oalgebraic__semidom_413,axiom,
algebraic_semidom @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ocomm__monoid__mult_414,axiom,
comm_monoid_mult @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oab__semigroup__add_415,axiom,
ab_semigroup_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__semiring_416,axiom,
ordered_semiring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__ring__abs_417,axiom,
ordered_ring_abs @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Parity_Osemiring__parity_418,axiom,
semiring_parity @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ocomm__monoid__add_419,axiom,
comm_monoid_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__modulo_420,axiom,
semiring_modulo @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__ring_421,axiom,
linordered_ring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__idom_422,axiom,
linordered_idom @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__1_423,axiom,
comm_semiring_1 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__0_424,axiom,
comm_semiring_0 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Osemigroup__mult_425,axiom,
semigroup_mult @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemidom__modulo_426,axiom,
semidom_modulo @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemidom__divide_427,axiom,
semidom_divide @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Num_Osemiring__numeral_428,axiom,
semiring_numeral @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Osemigroup__add_429,axiom,
semigroup_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ozero__less__one_430,axiom,
zero_less_one @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring_431,axiom,
comm_semiring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Nat_Osemiring__char__0_432,axiom,
semiring_char_0 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oab__group__add_433,axiom,
ab_group_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ozero__neq__one_434,axiom,
zero_neq_one @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__ring_435,axiom,
ordered_ring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oidom__abs__sgn_436,axiom,
idom_abs_sgn @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Parity_Oring__parity_437,axiom,
ring_parity @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Orderings_Opreorder_438,axiom,
preorder @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Orderings_Olinorder_439,axiom,
linorder @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Omonoid__mult_440,axiom,
monoid_mult @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oidom__modulo_441,axiom,
idom_modulo @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oidom__divide_442,axiom,
idom_divide @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__ring__1_443,axiom,
comm_ring_1 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Omonoid__add_444,axiom,
monoid_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1_445,axiom,
semiring_1 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__0_446,axiom,
semiring_0 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ogroup__add_447,axiom,
group_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Omult__zero_448,axiom,
mult_zero @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__ring_449,axiom,
comm_ring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Orderings_Oorder_450,axiom,
order @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Num_Oneg__numeral_451,axiom,
neg_numeral @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Nat_Oring__char__0_452,axiom,
ring_char_0 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring_453,axiom,
semiring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemidom_454,axiom,
semidom @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Orderings_Oord_455,axiom,
ord @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ouminus_456,axiom,
uminus @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oring__1_457,axiom,
ring_1 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oabs__if_458,axiom,
abs_if @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Power_Opower_459,axiom,
power @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Num_Onumeral_460,axiom,
numeral @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ozero_461,axiom,
zero @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oplus_462,axiom,
plus @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oring_463,axiom,
ring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oidom_464,axiom,
idom @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oone_465,axiom,
one @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Odvd_466,axiom,
dvd @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___HOL_Oequal_467,axiom,
cl_HOL_Oequal @ code_integer ).
thf(tcon_Code__Numeral_Onatural___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_468,axiom,
bit_un5681908812861735899ations @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Euclidean__Division_Ounique__euclidean__semiring__with__nat_469,axiom,
euclid5411537665997757685th_nat @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Oordered__ab__semigroup__monoid__add__imp__le_470,axiom,
ordere1937475149494474687imp_le @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Euclidean__Division_Ounique__euclidean__semiring_471,axiom,
euclid3128863361964157862miring @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Euclidean__Division_Oeuclidean__semiring__cancel_472,axiom,
euclid4440199948858584721cancel @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Osemiring__no__zero__divisors__cancel_473,axiom,
semiri6575147826004484403cancel @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Ostrict__ordered__ab__semigroup__add_474,axiom,
strict9044650504122735259up_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Oordered__cancel__ab__semigroup__add_475,axiom,
ordere580206878836729694up_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Oordered__ab__semigroup__add__imp__le_476,axiom,
ordere2412721322843649153imp_le @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Bit__Operations_Osemiring__bit__operations_477,axiom,
bit_se359711467146920520ations @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Olinordered__comm__semiring__strict_478,axiom,
linord2810124833399127020strict @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Quickcheck__Exhaustive_Ofull__exhaustive_479,axiom,
quickc3360725361186068524ustive @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Ostrict__ordered__comm__monoid__add_480,axiom,
strict7427464778891057005id_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Oordered__cancel__comm__monoid__add_481,axiom,
ordere8940638589300402666id_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Euclidean__Division_Oeuclidean__semiring_482,axiom,
euclid3725896446679973847miring @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Olinordered__ab__semigroup__add_483,axiom,
linord4140545234300271783up_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Osemiring__1__no__zero__divisors_484,axiom,
semiri2026040879449505780visors @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Olinordered__nonzero__semiring_485,axiom,
linord181362715937106298miring @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Olinordered__semiring__strict_486,axiom,
linord8928482502909563296strict @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Quickcheck__Exhaustive_Oexhaustive_487,axiom,
quickc658316121487927005ustive @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Osemiring__no__zero__divisors_488,axiom,
semiri3467727345109120633visors @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Oordered__ab__semigroup__add_489,axiom,
ordere6658533253407199908up_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Oordered__comm__monoid__add_490,axiom,
ordere6911136660526730532id_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Ocancel__ab__semigroup__add_491,axiom,
cancel2418104881723323429up_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Ocancel__comm__monoid__add_492,axiom,
cancel1802427076303600483id_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring__1__cancel_493,axiom,
comm_s4317794764714335236cancel @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Bit__Operations_Osemiring__bits_494,axiom,
bit_semiring_bits @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Oordered__comm__semiring_495,axiom,
ordere2520102378445227354miring @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Ocancel__semigroup__add_496,axiom,
cancel_semigroup_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Olinordered__semiring_497,axiom,
linordered_semiring @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Oordered__semiring__0_498,axiom,
ordered_semiring_0 @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Olinordered__semidom_499,axiom,
linordered_semidom @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Oab__semigroup__mult_500,axiom,
ab_semigroup_mult @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Osemiring__1__cancel_501,axiom,
semiring_1_cancel @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Oalgebraic__semidom_502,axiom,
algebraic_semidom @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Ocomm__monoid__mult_503,axiom,
comm_monoid_mult @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Ocomm__monoid__diff_504,axiom,
comm_monoid_diff @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Oab__semigroup__add_505,axiom,
ab_semigroup_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Oordered__semiring_506,axiom,
ordered_semiring @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Parity_Osemiring__parity_507,axiom,
semiring_parity @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Ocomm__monoid__add_508,axiom,
comm_monoid_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Osemiring__modulo_509,axiom,
semiring_modulo @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring__1_510,axiom,
comm_semiring_1 @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring__0_511,axiom,
comm_semiring_0 @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Osemigroup__mult_512,axiom,
semigroup_mult @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Osemidom__modulo_513,axiom,
semidom_modulo @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Osemidom__divide_514,axiom,
semidom_divide @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Num_Osemiring__numeral_515,axiom,
semiring_numeral @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Osemigroup__add_516,axiom,
semigroup_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Ozero__less__one_517,axiom,
zero_less_one @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring_518,axiom,
comm_semiring @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Nat_Osemiring__char__0_519,axiom,
semiring_char_0 @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Ozero__neq__one_520,axiom,
zero_neq_one @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Orderings_Opreorder_521,axiom,
preorder @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Orderings_Olinorder_522,axiom,
linorder @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Omonoid__mult_523,axiom,
monoid_mult @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Omonoid__add_524,axiom,
monoid_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Osemiring__1_525,axiom,
semiring_1 @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Osemiring__0_526,axiom,
semiring_0 @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Omult__zero_527,axiom,
mult_zero @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Orderings_Oorder_528,axiom,
order @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Osemiring_529,axiom,
semiring @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Osemidom_530,axiom,
semidom @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Orderings_Oord_531,axiom,
ord @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Power_Opower_532,axiom,
power @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Num_Onumeral_533,axiom,
numeral @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Ozero_534,axiom,
zero @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Oplus_535,axiom,
plus @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Oone_536,axiom,
one @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Odvd_537,axiom,
dvd @ code_natural ).
thf(tcon_Code__Numeral_Onatural___HOL_Oequal_538,axiom,
cl_HOL_Oequal @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Nat_Osize_539,axiom,
size @ code_natural ).
thf(tcon_Heap__Time__Monad_OHeap___HOL_Oequal_540,axiom,
! [A21: $tType] : ( cl_HOL_Oequal @ ( heap_Time_Heap @ A21 ) ) ).
thf(tcon_Heap__Time__Monad_OHeap___Nat_Osize_541,axiom,
! [A21: $tType] : ( size @ ( heap_Time_Heap @ A21 ) ) ).
thf(tcon_Record_Otuple__isomorphism___Quickcheck__Exhaustive_Ofull__exhaustive_542,axiom,
! [A21: $tType,A27: $tType,A28: $tType] :
( ( ( cl_HOL_Oequal @ A21 )
& ( quickc3360725361186068524ustive @ A21 )
& ( quickc3360725361186068524ustive @ A27 )
& ( quickc3360725361186068524ustive @ A28 ) )
=> ( quickc3360725361186068524ustive @ ( tuple_isomorphism @ A21 @ A27 @ A28 ) ) ) ).
thf(tcon_Record_Otuple__isomorphism___HOL_Oequal_543,axiom,
! [A21: $tType,A27: $tType,A28: $tType] : ( cl_HOL_Oequal @ ( tuple_isomorphism @ A21 @ A27 @ A28 ) ) ).
thf(tcon_Record_Otuple__isomorphism___Nat_Osize_544,axiom,
! [A21: $tType,A27: $tType,A28: $tType] : ( size @ ( tuple_isomorphism @ A21 @ A27 @ A28 ) ) ).
% Helper facts (4)
thf(help_If_3_1_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_T,axiom,
! [A: $tType,X: A,Y3: A] :
( ( if @ A @ $false @ X @ Y3 )
= Y3 ) ).
thf(help_If_1_1_T,axiom,
! [A: $tType,X: A,Y3: A] :
( ( if @ A @ $true @ X @ Y3 )
= X ) ).
thf(help_fChoice_1_1_T,axiom,
! [A: $tType,P: A > $o] :
( ( P @ ( fChoice @ A @ P ) )
= ( ? [X9: A] : ( P @ X9 ) ) ) ).
% Conjectures (2)
thf(conj_0,hypothesis,
p @ x ).
thf(conj_1,conjecture,
heap_Time_success @ a @ ( heap_Time_assert @ a @ p @ x ) @ h ).
%------------------------------------------------------------------------------