TPTP Problem File: ITP208^4.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : ITP208^4 : TPTP v8.2.0. Released v8.0.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer problem Array_Time 00247_008755
% 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 : 0019_Array_Time_00247_008755 [Des22]
% Status : Theorem
% Rating : 0.67 v8.2.0, 1.00 v8.1.0
% Syntax : Number of formulae : 8580 (2574 unt; 713 typ; 0 def)
% Number of atoms : 23193 (8297 equ; 0 cnn)
% Maximal formula atoms : 20 ( 2 avg)
% Number of connectives : 127328 (1596 ~; 278 |;1294 &;114130 @)
% ( 0 <=>;10030 =>; 0 <=; 0 <~>)
% Maximal formula depth : 27 ( 7 avg)
% Number of types : 12 ( 11 usr)
% Number of type conns : 4623 (4623 >; 0 *; 0 +; 0 <<)
% Number of symbols : 705 ( 702 usr; 21 con; 0-8 aty)
% Number of variables : 23756 (2422 ^;20238 !; 419 ?;23756 :)
% ( 677 !>; 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 14:11:04.690
%------------------------------------------------------------------------------
% Could-be-implicit typings (24)
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_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_Nitpick_Ofun__box,type,
fun_box: $tType > $tType > $tType ).
thf(ty_t_Typerep_Otyperep,type,
typerep: $tType ).
thf(ty_t_String_Oliteral,type,
literal: $tType ).
thf(ty_t_Sum__Type_Osum,type,
sum_sum: $tType > $tType > $tType ).
thf(ty_t_Option_Ooption,type,
option: $tType > $tType ).
thf(ty_t_Nitpick_Oword,type,
word: $tType > $tType ).
thf(ty_t_String_Ochar,type,
char: $tType ).
thf(ty_t_Heap_Oarray,type,
array: $tType > $tType ).
thf(ty_t_List_Olist,type,
list: $tType > $tType ).
thf(ty_t_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 (689)
thf(sy_cl_Finite__Set_Ofinite,type,
finite_finite:
!>[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_Heap_Oheap,type,
heap:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Odvd,type,
dvd:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oone,type,
one:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oidom,type,
idom:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oring,type,
ring:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oplus,type,
plus:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ozero,type,
zero:
!>[A: $tType] : $o ).
thf(sy_cl_Num_Onumeral,type,
numeral:
!>[A: $tType] : $o ).
thf(sy_cl_Power_Opower,type,
power:
!>[A: $tType] : $o ).
thf(sy_cl_Fields_Ofield,type,
field:
!>[A: $tType] : $o ).
thf(sy_cl_GCD_Oring__gcd,type,
ring_gcd:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ominus,type,
minus:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oabs__if,type,
abs_if:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oring__1,type,
ring_1:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ouminus,type,
uminus:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Obot,type,
bot:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oord,type,
ord:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Otop,type,
top:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemidom,type,
semidom:
!>[A: $tType] : $o ).
thf(sy_cl_Fields_Oinverse,type,
inverse:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemiring,type,
semiring:
!>[A: $tType] : $o ).
thf(sy_cl_Nat_Oring__char__0,type,
ring_char_0:
!>[A: $tType] : $o ).
thf(sy_cl_Num_Oneg__numeral,type,
neg_numeral:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oorder,type,
order:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Ocomm__ring,type,
comm_ring:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Omult__zero,type,
mult_zero:
!>[A: $tType] : $o ).
thf(sy_cl_Typerep_Otyperep,type,
typerep2:
!>[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_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_Fields_Ofield__abs__sgn,type,
field_abs_sgn:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Osemigroup__add,type,
semigroup_add:
!>[A: $tType] : $o ).
thf(sy_cl_Num_Osemiring__numeral,type,
semiring_numeral:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemidom__divide,type,
semidom_divide:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemidom__modulo,type,
semidom_modulo:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Osemigroup__mult,type,
semigroup_mult:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Odense__order,type,
dense_order:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Ocomm__semiring__0,type,
comm_semiring_0:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Ocomm__semiring__1,type,
comm_semiring_1:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__idom,type,
linordered_idom:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__ring,type,
linordered_ring:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemiring__modulo,type,
semiring_modulo:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocomm__monoid__add,type,
comm_monoid_add:
!>[A: $tType] : $o ).
thf(sy_cl_Parity_Osemiring__parity,type,
semiring_parity:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oordered__ring__abs,type,
ordered_ring_abs:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oordered__semiring,type,
ordered_semiring:
!>[A: $tType] : $o ).
thf(sy_cl_Fields_Olinordered__field,type,
linordered_field:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oab__semigroup__add,type,
ab_semigroup_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocomm__monoid__diff,type,
comm_monoid_diff:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocomm__monoid__mult,type,
comm_monoid_mult:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oalgebraic__semidom,type,
algebraic_semidom:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemiring__1__cancel,type,
semiring_1_cancel:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oab__semigroup__mult,type,
ab_semigroup_mult:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Odistrib__lattice,type,
distrib_lattice:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Osemilattice__inf,type,
semilattice_inf:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Osemilattice__sup,type,
semilattice_sup:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Odense__linorder,type,
dense_linorder:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__semidom,type,
linordered_semidom:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oordered__semiring__0,type,
ordered_semiring_0:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__semiring,type,
linordered_semiring:
!>[A: $tType] : $o ).
thf(sy_cl_Complete__Partial__Order_Occpo,type,
comple9053668089753744459l_ccpo:
!>[A: $tType] : $o ).
thf(sy_cl_Enum_Ofinite__distrib__lattice,type,
finite8700451911770168679attice:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocancel__semigroup__add,type,
cancel_semigroup_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__group__add,type,
ordered_ab_group_add:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__semiring__1,type,
linord6961819062388156250ring_1:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Onormalization__semidom,type,
normal8620421768224518004emidom:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oordered__comm__semiring,type,
ordere2520102378445227354miring:
!>[A: $tType] : $o ).
thf(sy_cl_Bit__Operations_Osemiring__bits,type,
bit_semiring_bits:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Obounded__lattice__bot,type,
bounded_lattice_bot:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Obounded__lattice__top,type,
bounded_lattice_top:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Ocomm__semiring__1__cancel,type,
comm_s4317794764714335236cancel:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__ring__strict,type,
linord4710134922213307826strict:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocancel__comm__monoid__add,type,
cancel1802427076303600483id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oring__1__no__zero__divisors,type,
ring_15535105094025558882visors:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocancel__ab__semigroup__add,type,
cancel2418104881723323429up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Olinordered__ab__group__add,type,
linord5086331880401160121up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__comm__monoid__add,type,
ordere6911136660526730532id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Archimedean__Field_Ofloor__ceiling,type,
archim2362893244070406136eiling:
!>[A: $tType] : $o ).
thf(sy_cl_GCD_Osemiring__gcd__mult__normalize,type,
semiri6843258321239162965malize:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__group__add__abs,type,
ordere166539214618696060dd_abs:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__semigroup__add,type,
ordere6658533253407199908up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemiring__no__zero__divisors,type,
semiri3467727345109120633visors:
!>[A: $tType] : $o ).
thf(sy_cl_Boolean__Algebras_Oboolean__algebra,type,
boolea8198339166811842893lgebra:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__semiring__strict,type,
linord8928482502909563296strict:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemidom__divide__unit__factor,type,
semido2269285787275462019factor:
!>[A: $tType] : $o ).
thf(sy_cl_Euclidean__Division_Oeuclidean__ring,type,
euclid5891614535332579305n_ring:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__nonzero__semiring,type,
linord181362715937106298miring:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemiring__1__no__zero__divisors,type,
semiri2026040879449505780visors:
!>[A: $tType] : $o ).
thf(sy_cl_Bit__Operations_Oring__bit__operations,type,
bit_ri3973907225187159222ations:
!>[A: $tType] : $o ).
thf(sy_cl_Complete__Lattices_Ocomplete__lattice,type,
comple6319245703460814977attice:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Olinordered__ab__semigroup__add,type,
linord4140545234300271783up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__semiring__1__strict,type,
linord715952674999750819strict:
!>[A: $tType] : $o ).
thf(sy_cl_Archimedean__Field_Oarchimedean__field,type,
archim462609752435547400_field:
!>[A: $tType] : $o ).
thf(sy_cl_Complete__Lattices_Ocomplete__linorder,type,
comple5582772986160207858norder:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Obounded__semilattice__inf__top,type,
bounde4346867609351753570nf_top:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Obounded__semilattice__sup__bot,type,
bounde4967611905675639751up_bot:
!>[A: $tType] : $o ).
thf(sy_cl_Euclidean__Division_Oeuclidean__semiring,type,
euclid3725896446679973847miring:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
canoni5634975068530333245id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__cancel__comm__monoid__add,type,
ordere8940638589300402666id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ostrict__ordered__comm__monoid__add,type,
strict7427464778891057005id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__comm__semiring__strict,type,
linord2810124833399127020strict:
!>[A: $tType] : $o ).
thf(sy_cl_Bit__Operations_Osemiring__bit__operations,type,
bit_se359711467146920520ations:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__semigroup__add__imp__le,type,
ordere2412721322843649153imp_le:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__cancel__ab__semigroup__add,type,
ordere580206878836729694up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__cancel__comm__monoid__diff,type,
ordere1170586879665033532d_diff:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ostrict__ordered__ab__semigroup__add,type,
strict9044650504122735259up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemiring__no__zero__divisors__cancel,type,
semiri6575147826004484403cancel:
!>[A: $tType] : $o ).
thf(sy_cl_Euclidean__Division_Oeuclidean__ring__cancel,type,
euclid8851590272496341667cancel:
!>[A: $tType] : $o ).
thf(sy_cl_Divides_Ounique__euclidean__semiring__numeral,type,
unique1627219031080169319umeral:
!>[A: $tType] : $o ).
thf(sy_cl_Complete__Lattices_Ocomplete__boolean__algebra,type,
comple489889107523837845lgebra:
!>[A: $tType] : $o ).
thf(sy_cl_Complete__Lattices_Ocomplete__distrib__lattice,type,
comple592849572758109894attice:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Onormalization__semidom__multiplicative,type,
normal6328177297339901930cative:
!>[A: $tType] : $o ).
thf(sy_cl_Euclidean__Division_Oeuclidean__semiring__cancel,type,
euclid4440199948858584721cancel:
!>[A: $tType] : $o ).
thf(sy_cl_Euclidean__Division_Ounique__euclidean__semiring,type,
euclid3128863361964157862miring:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__semigroup__monoid__add__imp__le,type,
ordere1937475149494474687imp_le:
!>[A: $tType] : $o ).
thf(sy_cl_Conditionally__Complete__Lattices_Olinear__continuum,type,
condit5016429287641298734tinuum:
!>[A: $tType] : $o ).
thf(sy_cl_Euclidean__Division_Ounique__euclidean__ring__with__nat,type,
euclid8789492081693882211th_nat:
!>[A: $tType] : $o ).
thf(sy_cl_Euclidean__Division_Ounique__euclidean__semiring__with__nat,type,
euclid5411537665997757685th_nat:
!>[A: $tType] : $o ).
thf(sy_cl_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,type,
semiri1453513574482234551roduct:
!>[A: $tType] : $o ).
thf(sy_cl_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations,type,
bit_un5681908812861735899ations:
!>[A: $tType] : $o ).
thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__lattice,type,
condit1219197933456340205attice:
!>[A: $tType] : $o ).
thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__linorder,type,
condit6923001295902523014norder:
!>[A: $tType] : $o ).
thf(sy_c_Archimedean__Field_Oceiling,type,
archimedean_ceiling:
!>[A: $tType] : ( A > int ) ).
thf(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor,type,
archim6421214686448440834_floor:
!>[A: $tType] : ( A > int ) ).
thf(sy_c_Archimedean__Field_Ofrac,type,
archimedean_frac:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Archimedean__Field_Oround,type,
archimedean_round:
!>[A: $tType] : ( A > int ) ).
thf(sy_c_Array__Time_Oalloc,type,
array_alloc:
!>[A: $tType] : ( ( list @ A ) > ( heap_ext @ product_unit ) > ( product_prod @ ( array @ A ) @ ( heap_ext @ product_unit ) ) ) ).
thf(sy_c_Array__Time_Ofreeze,type,
array_freeze:
!>[A: $tType] : ( ( array @ A ) > ( heap_Time_Heap @ ( list @ A ) ) ) ).
thf(sy_c_Array__Time_Oget,type,
array_get:
!>[A: $tType] : ( ( heap_ext @ product_unit ) > ( array @ A ) > ( list @ A ) ) ).
thf(sy_c_Array__Time_Olen,type,
array_len:
!>[A: $tType] : ( ( array @ A ) > ( heap_Time_Heap @ nat ) ) ).
thf(sy_c_Array__Time_Olength,type,
array_length:
!>[A: $tType] : ( ( heap_ext @ product_unit ) > ( array @ A ) > nat ) ).
thf(sy_c_Array__Time_Omake,type,
array_make:
!>[A: $tType] : ( nat > ( nat > A ) > ( heap_Time_Heap @ ( array @ A ) ) ) ).
thf(sy_c_Array__Time_Omap__entry,type,
array_map_entry:
!>[A: $tType] : ( nat > ( A > A ) > ( array @ A ) > ( heap_Time_Heap @ ( array @ A ) ) ) ).
thf(sy_c_Array__Time_Onew,type,
array_new:
!>[A: $tType] : ( nat > A > ( heap_Time_Heap @ ( array @ A ) ) ) ).
thf(sy_c_Array__Time_Onoteq,type,
array_noteq:
!>[A: $tType,B: $tType] : ( ( array @ A ) > ( array @ B ) > $o ) ).
thf(sy_c_Array__Time_Onth,type,
array_nth:
!>[A: $tType] : ( ( array @ A ) > nat > ( heap_Time_Heap @ A ) ) ).
thf(sy_c_Array__Time_Oof__list,type,
array_of_list:
!>[A: $tType] : ( ( list @ A ) > ( heap_Time_Heap @ ( array @ A ) ) ) ).
thf(sy_c_Array__Time_Opresent,type,
array_present:
!>[A: $tType] : ( ( heap_ext @ product_unit ) > ( array @ A ) > $o ) ).
thf(sy_c_Array__Time_Oset,type,
array_set:
!>[A: $tType] : ( ( array @ A ) > ( list @ A ) > ( heap_ext @ product_unit ) > ( heap_ext @ product_unit ) ) ).
thf(sy_c_Array__Time_Oswap,type,
array_swap:
!>[A: $tType] : ( nat > A > ( array @ A ) > ( heap_Time_Heap @ A ) ) ).
thf(sy_c_Array__Time_Oupd,type,
array_upd:
!>[A: $tType] : ( nat > A > ( array @ A ) > ( heap_Time_Heap @ ( array @ A ) ) ) ).
thf(sy_c_Array__Time_Oupdate,type,
array_update:
!>[A: $tType] : ( ( array @ A ) > nat > A > ( heap_ext @ product_unit ) > ( heap_ext @ product_unit ) ) ).
thf(sy_c_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_OrelChain,type,
bNF_Ca3754400796208372196lChain:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( A > B ) > $o ) ).
thf(sy_c_BNF__Def_Orel__fun,type,
bNF_rel_fun:
!>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( A > C > $o ) > ( B > D > $o ) > ( A > B ) > ( C > D ) > $o ) ).
thf(sy_c_BNF__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_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_OordLess,type,
bNF_We4044943003108391690rdLess:
!>[A: $tType,A2: $tType] : ( set @ ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A2 @ A2 ) ) ) ) ).
thf(sy_c_BNF__Wellorder__Embedding_Oembed,type,
bNF_Wellorder_embed:
!>[A: $tType,A2: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ A2 @ A2 ) ) > ( A > A2 ) > $o ) ).
thf(sy_c_BNF__Wellorder__Embedding_OembedS,type,
bNF_Wellorder_embedS:
!>[A: $tType,A2: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ A2 @ A2 ) ) > ( A > A2 ) > $o ) ).
thf(sy_c_BNF__Wellorder__Embedding_Oiso,type,
bNF_Wellorder_iso:
!>[A: $tType,A2: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ A2 @ A2 ) ) > ( A > A2 ) > $o ) ).
thf(sy_c_BNF__Wellorder__Relation_Owo__rel,type,
bNF_Wellorder_wo_rel:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_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__BNFs_Opred__prod,type,
basic_pred_prod:
!>[A: $tType,B: $tType] : ( ( A > $o ) > ( B > $o ) > ( product_prod @ A @ B ) > $o ) ).
thf(sy_c_Binomial_Obinomial,type,
binomial: nat > nat > nat ).
thf(sy_c_Binomial_Ogbinomial,type,
gbinomial:
!>[A: $tType] : ( A > nat > A ) ).
thf(sy_c_Bit__Operations_Oand__int__rel,type,
bit_and_int_rel: ( product_prod @ int @ int ) > ( product_prod @ int @ int ) > $o ).
thf(sy_c_Bit__Operations_Oand__not__num,type,
bit_and_not_num: num > num > ( option @ num ) ).
thf(sy_c_Bit__Operations_Oconcat__bit,type,
bit_concat_bit: nat > int > int > int ).
thf(sy_c_Bit__Operations_Oor__not__num__neg,type,
bit_or_not_num_neg: num > num > num ).
thf(sy_c_Bit__Operations_Oring__bit__operations__class_Onot,type,
bit_ri4277139882892585799ns_not:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Bit__Operations_Oring__bit__operations__class_Osigned__take__bit,type,
bit_ri4674362597316999326ke_bit:
!>[A: $tType] : ( nat > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand,type,
bit_se5824344872417868541ns_and:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Odrop__bit,type,
bit_se4197421643247451524op_bit:
!>[A: $tType] : ( nat > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit,type,
bit_se8732182000553998342ip_bit:
!>[A: $tType] : ( nat > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Omask,type,
bit_se2239418461657761734s_mask:
!>[A: $tType] : ( nat > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oor,type,
bit_se1065995026697491101ons_or:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Opush__bit,type,
bit_se4730199178511100633sh_bit:
!>[A: $tType] : ( nat > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit,type,
bit_se5668285175392031749et_bit:
!>[A: $tType] : ( nat > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Otake__bit,type,
bit_se2584673776208193580ke_bit:
!>[A: $tType] : ( nat > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit,type,
bit_se2638667681897837118et_bit:
!>[A: $tType] : ( nat > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oxor,type,
bit_se5824344971392196577ns_xor:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bits__class_Obit,type,
bit_se5641148757651400278ts_bit:
!>[A: $tType] : ( A > nat > $o ) ).
thf(sy_c_Bit__Operations_Osemiring__bits__class_Opossible__bit,type,
bit_se6407376104438227557le_bit:
!>[A: $tType] : ( ( itself @ A ) > nat > $o ) ).
thf(sy_c_Bit__Operations_Otake__bit__num,type,
bit_take_bit_num: nat > num > ( option @ num ) ).
thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oand__num,type,
bit_un7362597486090784418nd_num: num > num > ( option @ num ) ).
thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_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__axioms,type,
boolea6902313364301356556axioms:
!>[A: $tType] : ( ( A > A > A ) > ( A > A > A ) > ( A > A ) > A > A > $o ) ).
thf(sy_c_Boolean__Algebras_Oabstract__boolean__algebra__sym__diff,type,
boolea3799213064322606851m_diff:
!>[A: $tType] : ( ( A > A > A ) > ( A > A > A ) > ( A > A ) > A > A > ( A > A > A ) > $o ) ).
thf(sy_c_Boolean__Algebras_Oabstract__boolean__algebra__sym__diff__axioms,type,
boolea5476839437570043046axioms:
!>[A: $tType] : ( ( A > A > A ) > ( A > A > A ) > ( A > A ) > ( A > A > A ) > $o ) ).
thf(sy_c_Code__Numeral_ONat,type,
code_Nat: code_integer > code_natural ).
thf(sy_c_Code__Numeral_ONeg,type,
code_Neg: num > code_integer ).
thf(sy_c_Code__Numeral_OPos,type,
code_Pos: num > code_integer ).
thf(sy_c_Code__Numeral_OSuc,type,
code_Suc: code_natural > code_natural ).
thf(sy_c_Code__Numeral_Obit__cut__integer,type,
code_bit_cut_integer: code_integer > ( product_prod @ code_integer @ $o ) ).
thf(sy_c_Code__Numeral_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__natural,type,
code_i5400310926305786745atural: code_natural > code_integer ).
thf(sy_c_Code__Numeral_Ointeger__of__num,type,
code_integer_of_num: num > code_integer ).
thf(sy_c_Code__Numeral_Onat__of__integer,type,
code_nat_of_integer: code_integer > nat ).
thf(sy_c_Code__Numeral_Onatural_Ocase__natural,type,
code_case_natural:
!>[T: $tType] : ( T > ( code_natural > T ) > code_natural > T ) ).
thf(sy_c_Code__Numeral_Onatural_Onat__of__natural,type,
code_nat_of_natural: code_natural > nat ).
thf(sy_c_Code__Numeral_Onatural_Onatural__of__nat,type,
code_natural_of_nat: nat > code_natural ).
thf(sy_c_Code__Numeral_Onatural_Orec__natural,type,
code_rec_natural:
!>[T: $tType] : ( T > ( code_natural > T > T ) > code_natural > T ) ).
thf(sy_c_Code__Numeral_Onatural_Orec__set__natural,type,
code_rec_set_natural:
!>[T: $tType] : ( T > ( code_natural > T > T ) > code_natural > T > $o ) ).
thf(sy_c_Code__Numeral_Onatural__of__integer,type,
code_n4118661773612635043nteger: code_integer > code_natural ).
thf(sy_c_Code__Numeral_Onegative,type,
code_negative: num > code_integer ).
thf(sy_c_Code__Numeral_Onum__of__integer,type,
code_num_of_integer: code_integer > num ).
thf(sy_c_Code__Numeral_Opcr__integer,type,
code_pcr_integer: int > code_integer > $o ).
thf(sy_c_Code__Numeral_Opcr__natural,type,
code_pcr_natural: nat > code_natural > $o ).
thf(sy_c_Code__Numeral_Opositive,type,
code_positive: num > code_integer ).
thf(sy_c_Code__Numeral_Osize__nat,type,
code_size_nat: code_natural > nat ).
thf(sy_c_Code__Numeral_Osub,type,
code_sub: num > num > code_integer ).
thf(sy_c_Complete__Lattices_OInf__class_OInf,type,
complete_Inf_Inf:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_Complete__Lattices_OSup__class_OSup,type,
complete_Sup_Sup:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_Complete__Partial__Order_Occpo_Oadmissible,type,
comple1908693960933563346ssible:
!>[A: $tType] : ( ( ( set @ A ) > A ) > ( A > A > $o ) > ( A > $o ) > $o ) ).
thf(sy_c_Complete__Partial__Order_Occpo_Ofixp,type,
comple187402453842119260l_fixp:
!>[A: $tType] : ( ( ( set @ A ) > A ) > ( A > A > $o ) > ( A > A ) > A ) ).
thf(sy_c_Complete__Partial__Order_Occpo__class_Ofixp,type,
comple115746919287870866o_fixp:
!>[A: $tType] : ( ( A > A ) > A ) ).
thf(sy_c_Complete__Partial__Order_Occpo__class_Oiterates,type,
comple6359979572994053840erates:
!>[A: $tType] : ( ( A > A ) > ( set @ A ) ) ).
thf(sy_c_Complete__Partial__Order_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_Complete__Partial__Order_Omonotone,type,
comple7038119648293358887notone:
!>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( B > B > $o ) > ( A > B ) > $o ) ).
thf(sy_c_Conditionally__Complete__Lattices_Opreorder__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_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_Divides_Oadjust__div,type,
adjust_div: ( product_prod @ int @ int ) > int ).
thf(sy_c_Divides_Oadjust__mod,type,
adjust_mod: int > int > int ).
thf(sy_c_Divides_Odivmod__nat,type,
divmod_nat: nat > nat > ( product_prod @ nat @ nat ) ).
thf(sy_c_Divides_Oeucl__rel__int,type,
eucl_rel_int: int > int > ( product_prod @ int @ int ) > $o ).
thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivides__aux,type,
unique5940410009612947441es_aux:
!>[A: $tType] : ( ( product_prod @ A @ A ) > $o ) ).
thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod,type,
unique8689654367752047608divmod:
!>[A: $tType] : ( num > num > ( product_prod @ A @ A ) ) ).
thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod__step,type,
unique1321980374590559556d_step:
!>[A: $tType] : ( num > ( product_prod @ A @ A ) > ( product_prod @ A @ A ) ) ).
thf(sy_c_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_Finite__Set_Ocard,type,
finite_card:
!>[B: $tType] : ( ( set @ B ) > nat ) ).
thf(sy_c_Finite__Set_Ofinite,type,
finite_finite2:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Finite__Set_Ofold,type,
finite_fold:
!>[A: $tType,B: $tType] : ( ( A > B > B ) > B > ( set @ A ) > B ) ).
thf(sy_c_Finite__Set_Ofolding__on,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__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__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_ONO__MATCH,type,
nO_MATCH:
!>[A: $tType,B: $tType] : ( A > B > $o ) ).
thf(sy_c_HOL_OThe,type,
the:
!>[A: $tType] : ( ( A > $o ) > A ) ).
thf(sy_c_HOL_OUniq,type,
uniq:
!>[A: $tType] : ( ( A > $o ) > $o ) ).
thf(sy_c_HOL_Oundefined,type,
undefined:
!>[A: $tType] : A ).
thf(sy_c_Heap_Oaddr__of__array,type,
addr_of_array:
!>[A: $tType] : ( ( array @ A ) > nat ) ).
thf(sy_c_Heap_Oarray_OArray,type,
array2:
!>[A: $tType] : ( nat > ( array @ A ) ) ).
thf(sy_c_Heap_Oarray_Osize__array,type,
size_array:
!>[A: $tType] : ( ( A > nat ) > ( array @ A ) > nat ) ).
thf(sy_c_Heap_Oempty,type,
empty: heap_ext @ product_unit ).
thf(sy_c_Heap_Oheap_Oarrays,type,
arrays:
!>[Z: $tType] : ( ( heap_ext @ Z ) > typerep > nat > ( list @ nat ) ) ).
thf(sy_c_Heap_Oheap_Oheap__ext,type,
heap_ext2:
!>[Z: $tType] : ( ( typerep > nat > ( list @ nat ) ) > ( typerep > nat > nat ) > nat > Z > ( heap_ext @ Z ) ) ).
thf(sy_c_Heap_Oheap_Olim,type,
lim:
!>[Z: $tType] : ( ( heap_ext @ Z ) > nat ) ).
thf(sy_c_Heap_Oheap_Olim__update,type,
lim_update:
!>[Z: $tType] : ( ( nat > nat ) > ( heap_ext @ Z ) > ( heap_ext @ Z ) ) ).
thf(sy_c_Heap__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_OHeap__lub,type,
heap_Time_Heap_lub:
!>[A: $tType] : ( ( set @ ( heap_Time_Heap @ A ) ) > ( heap_Time_Heap @ A ) ) ).
thf(sy_c_Heap__Time__Monad_OHeap__ord,type,
heap_Time_Heap_ord:
!>[A: $tType] : ( ( heap_Time_Heap @ A ) > ( heap_Time_Heap @ A ) > $o ) ).
thf(sy_c_Heap__Time__Monad_Oassert,type,
heap_Time_assert:
!>[A: $tType] : ( ( A > $o ) > A > ( heap_Time_Heap @ A ) ) ).
thf(sy_c_Heap__Time__Monad_Obind,type,
heap_Time_bind:
!>[A: $tType,B: $tType] : ( ( heap_Time_Heap @ A ) > ( A > ( heap_Time_Heap @ B ) ) > ( heap_Time_Heap @ B ) ) ).
thf(sy_c_Heap__Time__Monad_Oeffect,type,
heap_Time_effect:
!>[A: $tType] : ( ( heap_Time_Heap @ A ) > ( heap_ext @ product_unit ) > ( heap_ext @ product_unit ) > A > nat > $o ) ).
thf(sy_c_Heap__Time__Monad_Oexecute,type,
heap_Time_execute:
!>[A: $tType] : ( ( heap_Time_Heap @ A ) > ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ).
thf(sy_c_Heap__Time__Monad_Oguard,type,
heap_Time_guard:
!>[A: $tType] : ( ( ( heap_ext @ product_unit ) > $o ) > ( ( heap_ext @ product_unit ) > ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) > ( heap_Time_Heap @ A ) ) ).
thf(sy_c_Heap__Time__Monad_Oheap,type,
heap_Time_heap:
!>[A: $tType] : ( ( ( heap_ext @ product_unit ) > ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) > ( heap_Time_Heap @ A ) ) ).
thf(sy_c_Heap__Time__Monad_Olift,type,
heap_Time_lift:
!>[A: $tType,B: $tType] : ( ( A > B ) > A > ( heap_Time_Heap @ B ) ) ).
thf(sy_c_Heap__Time__Monad_Oraise,type,
heap_Time_raise:
!>[A: $tType] : ( ( list @ char ) > ( heap_Time_Heap @ A ) ) ).
thf(sy_c_Heap__Time__Monad_Oreturn,type,
heap_Time_return:
!>[A: $tType] : ( A > ( heap_Time_Heap @ A ) ) ).
thf(sy_c_Heap__Time__Monad_Osuccess,type,
heap_Time_success:
!>[A: $tType] : ( ( heap_Time_Heap @ A ) > ( heap_ext @ product_unit ) > $o ) ).
thf(sy_c_Heap__Time__Monad_Otap,type,
heap_Time_tap:
!>[A: $tType] : ( ( ( heap_ext @ product_unit ) > A ) > ( heap_Time_Heap @ A ) ) ).
thf(sy_c_Heap__Time__Monad_OtimeFrame,type,
heap_Time_timeFrame:
!>[A: $tType] : ( nat > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ).
thf(sy_c_Heap__Time__Monad_OtimeFrame__rel,type,
heap_T5500966940807335491me_rel:
!>[A: $tType] : ( ( product_prod @ nat @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) > ( product_prod @ nat @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) > $o ) ).
thf(sy_c_Heap__Time__Monad_Oureturn,type,
heap_Time_ureturn:
!>[A: $tType] : ( A > ( heap_Time_Heap @ A ) ) ).
thf(sy_c_Heap__Time__Monad_Owait,type,
heap_Time_wait: nat > ( heap_Time_Heap @ product_unit ) ).
thf(sy_c_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,type,
semilattice:
!>[A: $tType] : ( ( A > A > A ) > $o ) ).
thf(sy_c_Lattices_Osemilattice__axioms,type,
semilattice_axioms:
!>[A: $tType] : ( ( A > A > A ) > $o ) ).
thf(sy_c_Lattices_Osemilattice__neutr,type,
semilattice_neutr:
!>[A: $tType] : ( ( A > A > A ) > A > $o ) ).
thf(sy_c_Lattices_Osemilattice__neutr__order,type,
semila1105856199041335345_order:
!>[A: $tType] : ( ( A > A > A ) > A > ( A > A > $o ) > ( A > A > $o ) > $o ) ).
thf(sy_c_Lattices_Osemilattice__order,type,
semilattice_order:
!>[A: $tType] : ( ( A > A > A ) > ( A > A > $o ) > ( A > A > $o ) > $o ) ).
thf(sy_c_Lattices_Osemilattice__order__axioms,type,
semila6385135966242565138axioms:
!>[A: $tType] : ( ( A > A > A ) > ( A > A > $o ) > ( A > A > $o ) > $o ) ).
thf(sy_c_Lattices_Osup__class_Osup,type,
sup_sup:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Lattices__Big_Olinorder_OMax,type,
lattices_Max:
!>[A: $tType] : ( ( A > A > $o ) > ( set @ A ) > A ) ).
thf(sy_c_Lattices__Big_Olinorder_OMin,type,
lattices_Min:
!>[A: $tType] : ( ( A > A > $o ) > ( set @ A ) > A ) ).
thf(sy_c_Lattices__Big_Olinorder__class_OMax,type,
lattic643756798349783984er_Max:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_Lattices__Big_Olinorder__class_OMin,type,
lattic643756798350308766er_Min:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__max,type,
lattices_ord_arg_max:
!>[B: $tType,A: $tType] : ( ( B > A ) > ( B > $o ) > B ) ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__max__on,type,
lattic1883929316492267755max_on:
!>[B: $tType,A: $tType] : ( ( B > A ) > ( set @ B ) > B ) ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min,type,
lattices_ord_arg_min:
!>[B: $tType,A: $tType] : ( ( B > A ) > ( B > $o ) > B ) ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on,type,
lattic7623131987881927897min_on:
!>[B: $tType,A: $tType] : ( ( B > A ) > ( set @ B ) > B ) ).
thf(sy_c_Lattices__Big_Oord__class_Ois__arg__max,type,
lattic501386751176901750rg_max:
!>[B: $tType,A: $tType] : ( ( B > A ) > ( B > $o ) > B > $o ) ).
thf(sy_c_Lattices__Big_Oord__class_Ois__arg__min,type,
lattic501386751177426532rg_min:
!>[B: $tType,A: $tType] : ( ( B > A ) > ( B > $o ) > B > $o ) ).
thf(sy_c_Lattices__Big_Osemilattice__inf__class_OInf__fin,type,
lattic7752659483105999362nf_fin:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_Lattices__Big_Osemilattice__neutr__set,type,
lattic5652469242046573047tr_set:
!>[A: $tType] : ( ( A > A > A ) > A > $o ) ).
thf(sy_c_Lattices__Big_Osemilattice__neutr__set_OF,type,
lattic5214292709420241887eutr_F:
!>[A: $tType] : ( ( A > A > A ) > A > ( set @ A ) > A ) ).
thf(sy_c_Lattices__Big_Osemilattice__order__neutr__set,type,
lattic3600114342068043075tr_set:
!>[A: $tType] : ( ( A > A > A ) > A > ( A > A > $o ) > ( A > A > $o ) > $o ) ).
thf(sy_c_Lattices__Big_Osemilattice__order__set,type,
lattic4895041142388067077er_set:
!>[A: $tType] : ( ( A > A > A ) > ( A > A > $o ) > ( A > A > $o ) > $o ) ).
thf(sy_c_Lattices__Big_Osemilattice__set,type,
lattic149705377957585745ce_set:
!>[A: $tType] : ( ( A > A > A ) > $o ) ).
thf(sy_c_Lattices__Big_Osemilattice__set_OF,type,
lattic1715443433743089157tice_F:
!>[A: $tType] : ( ( A > A > A ) > ( set @ A ) > A ) ).
thf(sy_c_Lattices__Big_Osemilattice__sup__class_OSup__fin,type,
lattic5882676163264333800up_fin:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_Lifting_OQuotient,type,
quotient:
!>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( A > B ) > ( B > A ) > ( A > B > $o ) > $o ) ).
thf(sy_c_List_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_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_Ofilter,type,
filter:
!>[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_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_Olexord,type,
lexord:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ).
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__list__of__set,type,
linord4507533701916653071of_set:
!>[A: $tType] : ( ( set @ A ) > ( list @ A ) ) ).
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_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_Oset,type,
set2:
!>[A: $tType] : ( ( list @ A ) > ( set @ A ) ) ).
thf(sy_c_List_Olist_Osize__list,type,
size_list:
!>[A: $tType] : ( ( A > nat ) > ( list @ A ) > nat ) ).
thf(sy_c_List_Olist_Otl,type,
tl:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Olist__ex,type,
list_ex:
!>[A: $tType] : ( ( A > $o ) > ( list @ A ) > $o ) ).
thf(sy_c_List_Olist__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_Omeasures,type,
measures:
!>[A: $tType] : ( ( list @ ( A > nat ) ) > ( set @ ( product_prod @ A @ A ) ) ) ).
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__class_Olexordp,type,
ord_lexordp:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > $o ) ).
thf(sy_c_List_Oord__class_Olexordp__eq,type,
ord_lexordp_eq:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > $o ) ).
thf(sy_c_List_Oproduct,type,
product:
!>[A: $tType,B: $tType] : ( ( list @ A ) > ( list @ B ) > ( list @ ( product_prod @ A @ B ) ) ) ).
thf(sy_c_List_Oproduct__lists,type,
product_lists:
!>[A: $tType] : ( ( list @ ( list @ A ) ) > ( list @ ( list @ A ) ) ) ).
thf(sy_c_List_Oremdups,type,
remdups:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Oremdups__adj,type,
remdups_adj:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Oremove1,type,
remove1:
!>[A: $tType] : ( A > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_OremoveAll,type,
removeAll:
!>[A: $tType] : ( A > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Oreplicate,type,
replicate:
!>[A: $tType] : ( nat > A > ( list @ A ) ) ).
thf(sy_c_List_Orev,type,
rev:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Orotate,type,
rotate:
!>[A: $tType] : ( nat > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Orotate1,type,
rotate1:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Osorted__wrt,type,
sorted_wrt:
!>[A: $tType] : ( ( A > A > $o ) > ( list @ A ) > $o ) ).
thf(sy_c_List_Otake,type,
take:
!>[A: $tType] : ( nat > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_OtakeWhile,type,
takeWhile:
!>[A: $tType] : ( ( A > $o ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Otranspose,type,
transpose:
!>[A: $tType] : ( ( list @ ( list @ A ) ) > ( list @ ( list @ A ) ) ) ).
thf(sy_c_List_Oupt,type,
upt: nat > nat > ( list @ nat ) ).
thf(sy_c_List_Oupto,type,
upto: int > int > ( list @ int ) ).
thf(sy_c_List_Oupto__aux,type,
upto_aux: int > int > ( list @ int ) > ( list @ int ) ).
thf(sy_c_List_Oupto__rel,type,
upto_rel: ( product_prod @ int @ int ) > ( product_prod @ int @ int ) > $o ).
thf(sy_c_List_Ozip,type,
zip:
!>[A: $tType,B: $tType] : ( ( list @ A ) > ( list @ B ) > ( list @ ( product_prod @ A @ B ) ) ) ).
thf(sy_c_Map_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_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,
pred: nat > nat ).
thf(sy_c_Nat_Oold_Onat_Orec__nat,type,
rec_nat:
!>[T: $tType] : ( T > ( nat > T > T ) > nat > T ) ).
thf(sy_c_Nat_Oold_Onat_Orec__set__nat,type,
rec_set_nat:
!>[T: $tType] : ( T > ( nat > T > T ) > nat > T > $o ) ).
thf(sy_c_Nat_Osemiring__1__class_ONats,type,
semiring_1_Nats:
!>[A: $tType] : ( set @ A ) ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat,type,
semiring_1_of_nat:
!>[A: $tType] : ( nat > A ) ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux,type,
semiri8178284476397505188at_aux:
!>[A: $tType] : ( ( A > A ) > nat > A > A ) ).
thf(sy_c_Nat_Osize__class_Osize,type,
size_size:
!>[A: $tType] : ( A > nat ) ).
thf(sy_c_Nat__Bijection_Oint__decode,type,
nat_int_decode: nat > int ).
thf(sy_c_Nat__Bijection_Oint__encode,type,
nat_int_encode: int > nat ).
thf(sy_c_Nat__Bijection_Olist__decode,type,
nat_list_decode: nat > ( list @ nat ) ).
thf(sy_c_Nat__Bijection_Olist__decode__rel,type,
nat_list_decode_rel: nat > nat > $o ).
thf(sy_c_Nat__Bijection_Olist__encode,type,
nat_list_encode: ( list @ nat ) > nat ).
thf(sy_c_Nat__Bijection_Olist__encode__rel,type,
nat_list_encode_rel: ( list @ nat ) > ( list @ nat ) > $o ).
thf(sy_c_Nat__Bijection_Oprod__decode,type,
nat_prod_decode: nat > ( product_prod @ nat @ nat ) ).
thf(sy_c_Nat__Bijection_Oprod__decode__aux,type,
nat_prod_decode_aux: nat > nat > ( product_prod @ nat @ nat ) ).
thf(sy_c_Nat__Bijection_Oprod__decode__aux__rel,type,
nat_pr5047031295181774490ux_rel: ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) > $o ).
thf(sy_c_Nat__Bijection_Oprod__encode,type,
nat_prod_encode: ( product_prod @ nat @ nat ) > nat ).
thf(sy_c_Nat__Bijection_Oset__decode,type,
nat_set_decode: nat > ( set @ nat ) ).
thf(sy_c_Nat__Bijection_Oset__encode,type,
nat_set_encode: ( set @ nat ) > nat ).
thf(sy_c_Nat__Bijection_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_Orec__num,type,
rec_num:
!>[A: $tType] : ( A > ( num > A > A ) > ( num > A > A ) > num > A ) ).
thf(sy_c_Num_Onum_Osize__num,type,
size_num: num > nat ).
thf(sy_c_Num_Onum__of__nat,type,
num_of_nat: nat > num ).
thf(sy_c_Num_Onumeral__class_Onumeral,type,
numeral_numeral:
!>[A: $tType] : ( num > A ) ).
thf(sy_c_Num_Opow,type,
pow: num > num > num ).
thf(sy_c_Num_Opred__numeral,type,
pred_numeral: num > nat ).
thf(sy_c_Num_Oring__1__class_Oiszero,type,
ring_1_iszero:
!>[A: $tType] : ( A > $o ) ).
thf(sy_c_Num_Osqr,type,
sqr: num > num ).
thf(sy_c_Old__Datatype_OAtom,type,
old_Atom:
!>[A: $tType,B: $tType] : ( ( sum_sum @ A @ nat ) > ( set @ ( old_node @ A @ B ) ) ) ).
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_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_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_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_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_Order__Relation_Oofilter,type,
order_ofilter:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) > $o ) ).
thf(sy_c_Order__Relation_Ounder,type,
order_under:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > A > ( set @ A ) ) ).
thf(sy_c_Order__Relation_OunderS,type,
order_underS:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > A > ( set @ A ) ) ).
thf(sy_c_Order__Relation_Owell__order__on,type,
order_well_order_on:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Orderings_Obot__class_Obot,type,
bot_bot:
!>[A: $tType] : A ).
thf(sy_c_Orderings_Oord_OLeast,type,
least:
!>[A: $tType] : ( ( A > A > $o ) > ( A > $o ) > A ) ).
thf(sy_c_Orderings_Oord_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_Oordering,type,
ordering:
!>[A: $tType] : ( ( A > A > $o ) > ( A > A > $o ) > $o ) ).
thf(sy_c_Orderings_Oordering__axioms,type,
ordering_axioms:
!>[A: $tType] : ( ( A > A > $o ) > ( A > A > $o ) > $o ) ).
thf(sy_c_Orderings_Oordering__top,type,
ordering_top:
!>[A: $tType] : ( ( A > A > $o ) > ( A > A > $o ) > A > $o ) ).
thf(sy_c_Orderings_Oordering__top__axioms,type,
ordering_top_axioms:
!>[A: $tType] : ( ( A > A > $o ) > A > $o ) ).
thf(sy_c_Orderings_Opartial__preordering,type,
partial_preordering:
!>[A: $tType] : ( ( A > A > $o ) > $o ) ).
thf(sy_c_Orderings_Opreordering,type,
preordering:
!>[A: $tType] : ( ( A > A > $o ) > ( A > A > $o ) > $o ) ).
thf(sy_c_Orderings_Opreordering__axioms,type,
preordering_axioms:
!>[A: $tType] : ( ( A > A > $o ) > ( A > A > $o ) > $o ) ).
thf(sy_c_Orderings_Otop__class_Otop,type,
top_top:
!>[A: $tType] : A ).
thf(sy_c_Partial__Function_Ofun__lub,type,
partial_fun_lub:
!>[C: $tType,B: $tType,A: $tType] : ( ( ( set @ C ) > B ) > ( set @ ( A > C ) ) > A > B ) ).
thf(sy_c_Partial__Function_Ofun__ord,type,
partial_fun_ord:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > B > $o ) > ( C > A ) > ( C > B ) > $o ) ).
thf(sy_c_Partial__Function_Opartial__function__definitions,type,
partia7178651479351089652itions:
!>[A: $tType] : ( ( A > A > $o ) > ( ( set @ A ) > A ) > $o ) ).
thf(sy_c_Power_Opower_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_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_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_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_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_Olog,type,
log: code_natural > code_natural > code_natural ).
thf(sy_c_Random_Olog__rel,type,
log_rel: ( product_prod @ code_natural @ code_natural ) > ( product_prod @ code_natural @ code_natural ) > $o ).
thf(sy_c_Random_Ominus__shift,type,
minus_shift: code_natural > code_natural > code_natural > code_natural ).
thf(sy_c_Random_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__weight,type,
select_weight:
!>[A: $tType] : ( ( list @ ( product_prod @ code_natural @ A ) ) > ( product_prod @ code_natural @ code_natural ) > ( product_prod @ A @ ( product_prod @ code_natural @ code_natural ) ) ) ).
thf(sy_c_Rat_OAbs__Rat,type,
abs_Rat: ( product_prod @ int @ int ) > rat ).
thf(sy_c_Rat_OFract,type,
fract: int > int > rat ).
thf(sy_c_Rat_OFrct,type,
frct: ( product_prod @ int @ int ) > rat ).
thf(sy_c_Rat_ORep__Rat,type,
rep_Rat: rat > ( product_prod @ int @ int ) ).
thf(sy_c_Rat_Ofield__char__0__class_ORats,type,
field_char_0_Rats:
!>[A: $tType] : ( set @ A ) ).
thf(sy_c_Rat_Ofield__char__0__class_Oof__rat,type,
field_char_0_of_rat:
!>[A: $tType] : ( rat > A ) ).
thf(sy_c_Rat_Onormalize,type,
normalize: ( product_prod @ int @ int ) > ( product_prod @ int @ int ) ).
thf(sy_c_Rat_Oof__int,type,
of_int: int > rat ).
thf(sy_c_Rat_Opcr__rat,type,
pcr_rat: ( product_prod @ int @ int ) > rat > $o ).
thf(sy_c_Rat_Opositive,type,
positive: rat > $o ).
thf(sy_c_Rat_Oquotient__of,type,
quotient_of: rat > ( product_prod @ int @ int ) ).
thf(sy_c_Rat_Oratrel,type,
ratrel: ( product_prod @ int @ int ) > ( product_prod @ int @ int ) > $o ).
thf(sy_c_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_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_OPowp,type,
powp:
!>[A: $tType] : ( ( A > $o ) > ( set @ 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_Oirreflp,type,
irreflp:
!>[A: $tType] : ( ( A > A > $o ) > $o ) ).
thf(sy_c_Relation_Orelcomp,type,
relcomp:
!>[A: $tType,B: $tType,C: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ ( product_prod @ B @ C ) ) > ( set @ ( product_prod @ A @ C ) ) ) ).
thf(sy_c_Relation_Otransp,type,
transp:
!>[A: $tType] : ( ( A > A > $o ) > $o ) ).
thf(sy_c_Rings_Oalgebraic__semidom__class_Ocoprime,type,
algebr8660921524188924756oprime:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Rings_Odivide__class_Odivide,type,
divide_divide:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Rings_Odvd__class_Odvd,type,
dvd_dvd:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Rings_Omodulo__class_Omodulo,type,
modulo_modulo:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Rings_Onormalization__semidom__class_Onormalize,type,
normal6383669964737779283malize:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Rings_Ounit__factor__class_Ounit__factor,type,
unit_f5069060285200089521factor:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool,type,
zero_neq_one_of_bool:
!>[A: $tType] : ( $o > A ) ).
thf(sy_c_Set_OBall,type,
ball:
!>[A: $tType] : ( ( set @ A ) > ( A > $o ) > $o ) ).
thf(sy_c_Set_OBex,type,
bex:
!>[A: $tType] : ( ( set @ A ) > ( A > $o ) > $o ) ).
thf(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_Set_OPow,type,
pow2:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( set @ A ) ) ) ).
thf(sy_c_Set_Odisjnt,type,
disjnt:
!>[A: $tType] : ( ( set @ A ) > ( set @ A ) > $o ) ).
thf(sy_c_Set_Oimage,type,
image:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) ) ).
thf(sy_c_Set_Oinsert,type,
insert:
!>[A: $tType] : ( A > ( set @ A ) > ( set @ A ) ) ).
thf(sy_c_Set_Ois__singleton,type,
is_singleton:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
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_Ochar_OChar,type,
char2: $o > $o > $o > $o > $o > $o > $o > $o > char ).
thf(sy_c_String_Ochar_Osize__char,type,
size_char: char > nat ).
thf(sy_c_String_Ocomm__semiring__1__class_Oof__char,type,
comm_s6883823935334413003f_char:
!>[A: $tType] : ( char > A ) ).
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_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_Transfer_Obi__total,type,
bi_total:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > $o ) ).
thf(sy_c_Transitive__Closure_Ontrancl,type,
transitive_ntrancl:
!>[A: $tType] : ( nat > ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_Transitive__Closure_Ortrancl,type,
transitive_rtrancl:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_Transitive__Closure_Otrancl,type,
transitive_trancl:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_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,
typerep3: literal > ( list @ typerep ) > typerep ).
thf(sy_c_Typerep_Otyperep_Osize__typerep,type,
size_typerep: typerep > nat ).
thf(sy_c_Typerep_Otyperep__class_Otyperep,type,
typerep_typerep:
!>[A: $tType] : ( ( itself @ A ) > typerep ) ).
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_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_Zorn_Ochain__subset,type,
chain_subset:
!>[A: $tType] : ( ( set @ ( set @ A ) ) > $o ) ).
thf(sy_c_Zorn_Ochains,type,
chains:
!>[A: $tType] : ( ( set @ ( set @ A ) ) > ( set @ ( set @ ( set @ A ) ) ) ) ).
thf(sy_c_Zorn_Opred__on_Ochain,type,
pred_chain:
!>[A: $tType] : ( ( set @ A ) > ( A > A > $o ) > ( set @ A ) > $o ) ).
thf(sy_c_Zorn_Opred__on_Omaxchain,type,
pred_maxchain:
!>[A: $tType] : ( ( set @ A ) > ( A > A > $o ) > ( set @ A ) > $o ) ).
thf(sy_c_Zorn_Opred__on_Osuc,type,
pred_suc:
!>[A: $tType] : ( ( set @ A ) > ( A > A > $o ) > ( set @ A ) > ( set @ A ) ) ).
thf(sy_c_Zorn_Opred__on_Osuc__Union__closed,type,
pred_s596693808085603175closed:
!>[A: $tType] : ( ( set @ A ) > ( A > A > $o ) > ( set @ ( set @ A ) ) ) ).
thf(sy_c_Zorn_Opred__on_Osuc__Union__closedp,type,
pred_s7749564232668923593losedp:
!>[A: $tType] : ( ( set @ A ) > ( A > A > $o ) > ( set @ A ) > $o ) ).
thf(sy_c_fChoice,type,
fChoice:
!>[A: $tType] : ( ( A > $o ) > A ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_a,type,
a2: array @ a ).
thf(sy_v_h,type,
h: heap_ext @ product_unit ).
thf(sy_v_h_H,type,
h2: heap_ext @ product_unit ).
thf(sy_v_n,type,
n: nat ).
thf(sy_v_r,type,
r: nat ).
thf(sy_v_thesis,type,
thesis: $o ).
% Relevant facts (7200)
thf(fact_0_assms,axiom,
heap_Time_effect @ nat @ ( array_len @ a @ a2 ) @ h @ h2 @ r @ n ).
% assms
thf(fact_1_that,axiom,
( ( r
= ( array_length @ a @ h2 @ a2 ) )
=> ( ( h2 = h )
=> ( ( n
= ( one_one @ nat ) )
=> thesis ) ) ) ).
% that
thf(fact_2_effect__lengthI,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [H: heap_ext @ product_unit,H2: heap_ext @ product_unit,R: nat,A3: array @ A,N: nat] :
( ( H = H2 )
=> ( ( R
= ( array_length @ A @ H2 @ A3 ) )
=> ( ( N
= ( one_one @ nat ) )
=> ( heap_Time_effect @ nat @ ( array_len @ A @ A3 ) @ H2 @ H @ R @ N ) ) ) ) ) ).
% effect_lengthI
thf(fact_3_one__natural_Orsp,axiom,
( ( one_one @ nat )
= ( one_one @ nat ) ) ).
% one_natural.rsp
thf(fact_4_one__reorient,axiom,
! [A: $tType] :
( ( one @ A )
=> ! [X: A] :
( ( ( one_one @ A )
= X )
= ( X
= ( one_one @ A ) ) ) ) ).
% one_reorient
thf(fact_5_length__update,axiom,
! [B: $tType,A: $tType] :
( ( ( heap @ A )
& ( heap @ B ) )
=> ! [B2: array @ B,I: nat,V: B,H2: heap_ext @ product_unit] :
( ( array_length @ A @ ( array_update @ B @ B2 @ I @ V @ H2 ) )
= ( array_length @ A @ H2 ) ) ) ).
% length_update
thf(fact_6_euclidean__size__1,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ( ( euclid6346220572633701492n_size @ A @ ( one_one @ A ) )
= ( one_one @ nat ) ) ) ).
% euclidean_size_1
thf(fact_7_fact__1,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiring_char_0_fact @ A @ ( one_one @ nat ) )
= ( one_one @ A ) ) ) ).
% fact_1
thf(fact_8_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_9_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_10_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_11_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_12_round__1,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archimedean_round @ A @ ( one_one @ A ) )
= ( one_one @ int ) ) ) ).
% round_1
thf(fact_13_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_14_of__nat__fact,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [N: nat] :
( ( semiring_1_of_nat @ A @ ( semiring_char_0_fact @ nat @ N ) )
= ( semiring_char_0_fact @ A @ N ) ) ) ).
% of_nat_fact
thf(fact_15_euclidean__size__of__nat,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [N: nat] :
( ( euclid6346220572633701492n_size @ A @ ( semiring_1_of_nat @ A @ N ) )
= N ) ) ).
% euclidean_size_of_nat
thf(fact_16_round__of__nat,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [N: nat] :
( ( archimedean_round @ A @ ( semiring_1_of_nat @ A @ N ) )
= ( semiring_1_of_nat @ int @ N ) ) ) ).
% round_of_nat
thf(fact_17_one__integer_Orsp,axiom,
( ( one_one @ int )
= ( one_one @ int ) ) ).
% one_integer.rsp
thf(fact_18_update__swap__neqIndex,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [I: nat,I2: nat,A3: array @ A,V: A,V2: A,H2: heap_ext @ product_unit] :
( ( I != I2 )
=> ( ( array_update @ A @ A3 @ I @ V @ ( array_update @ A @ A3 @ I2 @ V2 @ H2 ) )
= ( array_update @ A @ A3 @ I2 @ V2 @ ( array_update @ A @ A3 @ I @ V @ H2 ) ) ) ) ) ).
% update_swap_neqIndex
thf(fact_19_int__ops_I2_J,axiom,
( ( semiring_1_of_nat @ int @ ( one_one @ nat ) )
= ( one_one @ int ) ) ).
% int_ops(2)
thf(fact_20_effect__deterministic_I3_J,axiom,
! [A: $tType,F: heap_Time_Heap @ A,H2: heap_ext @ product_unit,H: heap_ext @ product_unit,A3: A,N: nat,H3: heap_ext @ product_unit,B2: A,N2: nat] :
( ( heap_Time_effect @ A @ F @ H2 @ H @ A3 @ N )
=> ( ( heap_Time_effect @ A @ F @ H2 @ H3 @ B2 @ N2 )
=> ( N = N2 ) ) ) ).
% effect_deterministic(3)
thf(fact_21_effect__deterministic_I2_J,axiom,
! [A: $tType,F: heap_Time_Heap @ A,H2: heap_ext @ product_unit,H: heap_ext @ product_unit,A3: A,N: nat,H3: heap_ext @ product_unit,B2: A,N2: nat] :
( ( heap_Time_effect @ A @ F @ H2 @ H @ A3 @ N )
=> ( ( heap_Time_effect @ A @ F @ H2 @ H3 @ B2 @ N2 )
=> ( H = H3 ) ) ) ).
% effect_deterministic(2)
thf(fact_22_effect__deterministic_I1_J,axiom,
! [A: $tType,F: heap_Time_Heap @ A,H2: heap_ext @ product_unit,H: heap_ext @ product_unit,A3: A,N: nat,H3: heap_ext @ product_unit,B2: A,N2: nat] :
( ( heap_Time_effect @ A @ F @ H2 @ H @ A3 @ N )
=> ( ( heap_Time_effect @ A @ F @ H2 @ H3 @ B2 @ N2 )
=> ( A3 = B2 ) ) ) ).
% effect_deterministic(1)
thf(fact_23_effect__ifE,axiom,
! [A: $tType,C2: $o,T2: heap_Time_Heap @ A,E: heap_Time_Heap @ A,H2: heap_ext @ product_unit,H: heap_ext @ product_unit,R: A,N: nat] :
( ( heap_Time_effect @ A @ ( if @ ( heap_Time_Heap @ A ) @ C2 @ T2 @ E ) @ H2 @ H @ R @ N )
=> ( ( C2
=> ~ ( heap_Time_effect @ A @ T2 @ H2 @ H @ R @ N ) )
=> ~ ( ~ C2
=> ~ ( heap_Time_effect @ A @ E @ H2 @ H @ R @ N ) ) ) ) ).
% effect_ifE
thf(fact_24_effect__ifI,axiom,
! [A: $tType,C2: $o,T2: heap_Time_Heap @ A,H2: heap_ext @ product_unit,H: heap_ext @ product_unit,R: A,N: nat,E: heap_Time_Heap @ A] :
( ( C2
=> ( heap_Time_effect @ A @ T2 @ H2 @ H @ R @ N ) )
=> ( ( ~ C2
=> ( heap_Time_effect @ A @ E @ H2 @ H @ R @ N ) )
=> ( heap_Time_effect @ A @ ( if @ ( heap_Time_Heap @ A ) @ C2 @ T2 @ E ) @ H2 @ H @ R @ N ) ) ) ).
% effect_ifI
thf(fact_25_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_26_present__update,axiom,
! [B: $tType,A: $tType] :
( ( ( heap @ A )
& ( heap @ B ) )
=> ! [B2: array @ B,I: nat,V: B,H2: heap_ext @ product_unit] :
( ( array_present @ A @ ( array_update @ B @ B2 @ I @ V @ H2 ) )
= ( array_present @ A @ H2 ) ) ) ).
% present_update
thf(fact_27_fact__0,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiring_char_0_fact @ A @ ( zero_zero @ nat ) )
= ( one_one @ A ) ) ) ).
% fact_0
thf(fact_28_effect__tapI,axiom,
! [A: $tType,H: heap_ext @ product_unit,H2: heap_ext @ product_unit,R: A,F: ( heap_ext @ product_unit ) > A] :
( ( H = H2 )
=> ( ( R
= ( F @ H2 ) )
=> ( heap_Time_effect @ A @ ( heap_Time_tap @ A @ F ) @ H2 @ H @ R @ ( one_one @ nat ) ) ) ) ).
% effect_tapI
thf(fact_29_effect__tapE,axiom,
! [A: $tType,F: ( heap_ext @ product_unit ) > A,H2: heap_ext @ product_unit,H: heap_ext @ product_unit,R: A,N: nat] :
( ( heap_Time_effect @ A @ ( heap_Time_tap @ A @ F ) @ H2 @ H @ R @ N )
=> ~ ( ( H = H2 )
=> ( ( R
= ( F @ H2 ) )
=> ( N
!= ( one_one @ nat ) ) ) ) ) ).
% effect_tapE
thf(fact_30_update__swap__neq,axiom,
! [B: $tType,A: $tType] :
( ( ( heap @ A )
& ( heap @ B ) )
=> ! [A3: array @ A,A4: array @ B,I: nat,V: A,I2: nat,V2: B,H2: heap_ext @ product_unit] :
( ( array_noteq @ A @ B @ A3 @ A4 )
=> ( ( array_update @ A @ A3 @ I @ V @ ( array_update @ B @ A4 @ I2 @ V2 @ H2 ) )
= ( array_update @ B @ A4 @ I2 @ V2 @ ( array_update @ A @ A3 @ I @ V @ H2 ) ) ) ) ) ).
% update_swap_neq
thf(fact_31_unequal,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [A3: array @ A,A4: array @ A] :
( ( A3 != A4 )
= ( array_noteq @ A @ A @ A3 @ A4 ) ) ) ).
% unequal
thf(fact_32_division__segment__1,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ( ( euclid7384307370059645450egment @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% division_segment_1
thf(fact_33_round__0,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archimedean_round @ A @ ( zero_zero @ A ) )
= ( zero_zero @ int ) ) ) ).
% round_0
thf(fact_34_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_35_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_36_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_37_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_38_size__0,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ( ( euclid6346220572633701492n_size @ A @ ( zero_zero @ A ) )
= ( zero_zero @ nat ) ) ) ).
% size_0
thf(fact_39_nat__int__comparison_I1_J,axiom,
( ( ^ [Y: nat,Z2: nat] : Y = Z2 )
= ( ^ [A5: nat,B3: nat] :
( ( semiring_1_of_nat @ int @ A5 )
= ( semiring_1_of_nat @ int @ B3 ) ) ) ) ).
% nat_int_comparison(1)
thf(fact_40_zero__natural_Orsp,axiom,
( ( zero_zero @ nat )
= ( zero_zero @ nat ) ) ).
% zero_natural.rsp
thf(fact_41_division__segment__not__0,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [A3: A] :
( ( euclid7384307370059645450egment @ A @ A3 )
!= ( zero_zero @ A ) ) ) ).
% division_segment_not_0
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,A6: set @ A] :
( ( collect @ A
@ ^ [X2: A] : ( member @ A @ X2 @ A6 ) )
= A6 ) ).
% Collect_mem_eq
thf(fact_44_Collect__cong,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X3: A] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_45_ext,axiom,
! [B: $tType,A: $tType,F: A > B,G: A > B] :
( ! [X3: A] :
( ( F @ X3 )
= ( G @ X3 ) )
=> ( F = G ) ) ).
% ext
thf(fact_46_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_47_zero__reorient,axiom,
! [A: $tType] :
( ( zero @ A )
=> ! [X: A] :
( ( ( zero_zero @ A )
= X )
= ( X
= ( zero_zero @ A ) ) ) ) ).
% zero_reorient
thf(fact_48_noteq__sym,axiom,
! [B: $tType,A: $tType] :
( ( ( heap @ A )
& ( heap @ B ) )
=> ! [A3: array @ A,B2: array @ B] :
( ( array_noteq @ A @ B @ A3 @ B2 )
=> ( array_noteq @ B @ A @ B2 @ A3 ) ) ) ).
% noteq_sym
thf(fact_49_noteq__irrefl,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [R: array @ A] :
~ ( array_noteq @ A @ A @ R @ R ) ) ).
% noteq_irrefl
thf(fact_50_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_51_division__segment__nat__def,axiom,
( ( euclid7384307370059645450egment @ nat )
= ( ^ [N3: nat] : ( one_one @ nat ) ) ) ).
% division_segment_nat_def
thf(fact_52_division__segment__eq__iff,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A3: A,B2: A] :
( ( ( euclid7384307370059645450egment @ A @ A3 )
= ( euclid7384307370059645450egment @ A @ B2 ) )
=> ( ( ( euclid6346220572633701492n_size @ A @ A3 )
= ( euclid6346220572633701492n_size @ A @ B2 ) )
=> ( A3 = B2 ) ) ) ) ).
% division_segment_eq_iff
thf(fact_53_zero__neq__one,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ( ( zero_zero @ A )
!= ( one_one @ A ) ) ) ).
% zero_neq_one
thf(fact_54_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_55_effect__ureturnI,axiom,
! [A: $tType,H2: heap_ext @ product_unit,H: heap_ext @ product_unit,X: A] :
( ( H2 = H )
=> ( heap_Time_effect @ A @ ( heap_Time_ureturn @ A @ X ) @ H2 @ H @ X @ ( zero_zero @ nat ) ) ) ).
% effect_ureturnI
thf(fact_56_effect__ureturnE,axiom,
! [A: $tType,X: A,H2: heap_ext @ product_unit,H: heap_ext @ product_unit,R: A,N: nat] :
( ( heap_Time_effect @ A @ ( heap_Time_ureturn @ A @ X ) @ H2 @ H @ R @ N )
=> ~ ( ( R = X )
=> ( ( H = H2 )
=> ( N
!= ( zero_zero @ nat ) ) ) ) ) ).
% effect_ureturnE
thf(fact_57_effect__assertE,axiom,
! [A: $tType,P: A > $o,X: A,H2: heap_ext @ product_unit,H: heap_ext @ product_unit,R: A,N: nat] :
( ( heap_Time_effect @ A @ ( heap_Time_assert @ A @ P @ X ) @ H2 @ H @ R @ N )
=> ~ ( ( P @ X )
=> ( ( R = X )
=> ( ( H = H2 )
=> ( N
!= ( one_one @ nat ) ) ) ) ) ) ).
% effect_assertE
thf(fact_58_effect__assertI,axiom,
! [A: $tType,P: A > $o,X: A,H: heap_ext @ product_unit,H2: heap_ext @ product_unit,R: A,N: nat] :
( ( P @ X )
=> ( ( H = H2 )
=> ( ( R = X )
=> ( ( N
= ( one_one @ nat ) )
=> ( heap_Time_effect @ A @ ( heap_Time_assert @ A @ P @ X ) @ H2 @ H @ R @ N ) ) ) ) ) ).
% effect_assertI
thf(fact_59_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_60_division__segment__euclidean__size,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A3: A] :
( ( times_times @ A @ ( euclid7384307370059645450egment @ A @ A3 ) @ ( semiring_1_of_nat @ A @ ( euclid6346220572633701492n_size @ A @ A3 ) ) )
= A3 ) ) ).
% division_segment_euclidean_size
thf(fact_61_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_62_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_63_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_64_effect__returnE,axiom,
! [A: $tType,X: A,H2: heap_ext @ product_unit,H: heap_ext @ product_unit,R: A,N: nat] :
( ( heap_Time_effect @ A @ ( heap_Time_return @ A @ X ) @ H2 @ H @ R @ N )
=> ~ ( ( R = X )
=> ( ( H = H2 )
=> ( N
!= ( one_one @ nat ) ) ) ) ) ).
% effect_returnE
thf(fact_65_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_66_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_67_add_Oinverse__inverse,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A] :
( ( uminus_uminus @ A @ ( uminus_uminus @ A @ A3 ) )
= A3 ) ) ).
% add.inverse_inverse
thf(fact_68_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_69_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_70_nat_Oinject,axiom,
! [X22: nat,Y2: nat] :
( ( ( suc @ X22 )
= ( suc @ Y2 ) )
= ( X22 = Y2 ) ) ).
% nat.inject
thf(fact_71_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_72_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_73_mult__0__right,axiom,
! [M: nat] :
( ( times_times @ nat @ M @ ( zero_zero @ nat ) )
= ( zero_zero @ nat ) ) ).
% mult_0_right
thf(fact_74_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_75_le0,axiom,
! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% le0
thf(fact_76_bot__nat__0_Oextremum,axiom,
! [A3: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ A3 ) ).
% bot_nat_0.extremum
thf(fact_77_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_78_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_79_le__zero__eq,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [N: A] :
( ( ord_less_eq @ A @ N @ ( zero_zero @ A ) )
= ( N
= ( zero_zero @ A ) ) ) ) ).
% le_zero_eq
thf(fact_80_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_81_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_82_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_83_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_84_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_85_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_86_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_87_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_88_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_89_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_90_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_91_add_Oinverse__neutral,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ( ( uminus_uminus @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% add.inverse_neutral
thf(fact_92_mult__1,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A3: A] :
( ( times_times @ A @ ( one_one @ A ) @ A3 )
= A3 ) ) ).
% mult_1
thf(fact_93_mult_Oright__neutral,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A3: A] :
( ( times_times @ A @ A3 @ ( one_one @ A ) )
= A3 ) ) ).
% mult.right_neutral
thf(fact_94_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_95_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_96_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_97_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_98_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_99_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_100_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_101_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_102_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_103_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_104_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ( ord_less @ nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_105_lessI,axiom,
! [N: nat] : ( ord_less @ nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_106_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_107_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_108_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_109_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% less_nat_zero_code
thf(fact_110_neq0__conv,axiom,
! [N: nat] :
( ( N
!= ( zero_zero @ nat ) )
= ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% neq0_conv
thf(fact_111_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_112_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_113_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_114_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_115_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_116_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_117_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_118_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_119_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_120_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_121_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_122_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_123_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_124_mult__minus1,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [Z3: A] :
( ( times_times @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ Z3 )
= ( uminus_uminus @ A @ Z3 ) ) ) ).
% mult_minus1
thf(fact_125_mult__minus1__right,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [Z3: A] :
( ( times_times @ A @ Z3 @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( uminus_uminus @ A @ Z3 ) ) ) ).
% mult_minus1_right
thf(fact_126_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_127_zero__less__Suc,axiom,
! [N: nat] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_128_less__Suc0,axiom,
! [N: nat] :
( ( ord_less @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% less_Suc0
thf(fact_129_less__one,axiom,
! [N: nat] :
( ( ord_less @ nat @ N @ ( one_one @ nat ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% less_one
thf(fact_130_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_131_fact__Suc,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [N: nat] :
( ( semiring_char_0_fact @ A @ ( suc @ N ) )
= ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ N ) ) @ ( semiring_char_0_fact @ A @ N ) ) ) ) ).
% fact_Suc
thf(fact_132_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_133_zero__integer_Orsp,axiom,
( ( zero_zero @ int )
= ( zero_zero @ int ) ) ).
% zero_integer.rsp
thf(fact_134_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_135_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_136_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_137_ex__least__nat__less,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ ( zero_zero @ nat ) )
=> ? [K2: nat] :
( ( ord_less @ nat @ K2 @ N )
& ! [I3: nat] :
( ( ord_less_eq @ nat @ I3 @ K2 )
=> ~ ( P @ I3 ) )
& ( P @ ( suc @ K2 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_138_mult__le__cancel__iff1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z3: A,X: A,Y3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Z3 )
=> ( ( ord_less_eq @ A @ ( times_times @ A @ X @ Z3 ) @ ( times_times @ A @ Y3 @ Z3 ) )
= ( ord_less_eq @ A @ X @ Y3 ) ) ) ) ).
% mult_le_cancel_iff1
thf(fact_139_mult__le__cancel__iff2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z3: A,X: A,Y3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Z3 )
=> ( ( ord_less_eq @ A @ ( times_times @ A @ Z3 @ X ) @ ( times_times @ A @ Z3 @ Y3 ) )
= ( ord_less_eq @ A @ X @ Y3 ) ) ) ) ).
% mult_le_cancel_iff2
thf(fact_140_division__segment__int__def,axiom,
( ( euclid7384307370059645450egment @ int )
= ( ^ [K3: int] : ( if @ int @ ( ord_less_eq @ int @ ( zero_zero @ int ) @ K3 ) @ ( one_one @ int ) @ ( uminus_uminus @ int @ ( one_one @ int ) ) ) ) ) ).
% division_segment_int_def
thf(fact_141_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_142_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_143_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_144_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_145_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_146_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_147_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_148_mult__less__le__imp__less,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A3: A,B2: A,C2: A,D2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ D2 )
=> ( ( 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 @ D2 ) ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_149_mult__le__less__imp__less,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A3: A,B2: A,C2: A,D2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ C2 @ D2 )
=> ( ( 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 @ D2 ) ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_150_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_151_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_152_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_153_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_154_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_155_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_156_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_157_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_158_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_159_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_160_mult__strict__mono_H,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A3: A,B2: A,C2: A,D2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ C2 @ D2 )
=> ( ( 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 @ D2 ) ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_161_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_162_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_163_mult__strict__mono,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A3: A,B2: A,C2: A,D2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ C2 @ D2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ D2 ) ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_164_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_165_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_166_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_167_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_168_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_169_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_170_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_171_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_172_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_173_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_174_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_175_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_176_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_177_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_178_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_179_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_180_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_181_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_182_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_183_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_184_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_185_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_186_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_187_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_188_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_189_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_190_mult__mono_H,axiom,
! [A: $tType] :
( ( ordered_semiring @ A )
=> ! [A3: A,B2: A,C2: A,D2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ D2 )
=> ( ( 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 @ D2 ) ) ) ) ) ) ) ).
% mult_mono'
thf(fact_191_mult__mono,axiom,
! [A: $tType] :
( ( ordered_semiring @ A )
=> ! [A3: A,B2: A,C2: A,D2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ D2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ D2 ) ) ) ) ) ) ) ).
% mult_mono
thf(fact_192_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_193_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_194_verit__comp__simplify1_I1_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A] :
~ ( ord_less @ A @ A3 @ A3 ) ) ).
% verit_comp_simplify1(1)
thf(fact_195_verit__comp__simplify1_I2_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A] : ( ord_less_eq @ A @ A3 @ A3 ) ) ).
% verit_comp_simplify1(2)
thf(fact_196_verit__comp__simplify1_I3_J,axiom,
! [B: $tType] :
( ( linorder @ B )
=> ! [B4: B,A4: B] :
( ( ~ ( ord_less_eq @ B @ B4 @ A4 ) )
= ( ord_less @ B @ A4 @ B4 ) ) ) ).
% verit_comp_simplify1(3)
thf(fact_197_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_198_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_199_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_200_infinite__descent__measure,axiom,
! [A: $tType,P: A > $o,V3: A > nat,X: A] :
( ! [X3: A] :
( ~ ( P @ X3 )
=> ? [Y4: A] :
( ( ord_less @ nat @ ( V3 @ Y4 ) @ ( V3 @ X3 ) )
& ~ ( P @ Y4 ) ) )
=> ( P @ X ) ) ).
% infinite_descent_measure
thf(fact_201_transitive__stepwise__le,axiom,
! [M: nat,N: nat,R2: nat > nat > $o] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ! [X3: nat] : ( R2 @ X3 @ X3 )
=> ( ! [X3: nat,Y5: nat,Z4: nat] :
( ( R2 @ X3 @ Y5 )
=> ( ( R2 @ Y5 @ Z4 )
=> ( R2 @ X3 @ Z4 ) ) )
=> ( ! [N4: nat] : ( R2 @ N4 @ ( suc @ N4 ) )
=> ( R2 @ M @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_202_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I4: nat,J2: nat] :
( ( ord_less @ nat @ I4 @ J2 )
=> ( ord_less @ nat @ ( F @ I4 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq @ nat @ I @ J )
=> ( ord_less_eq @ nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_203_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_204_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_205_nat__induct__at__least,axiom,
! [M: nat,N: nat,P: nat > $o] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( P @ M )
=> ( ! [N4: nat] :
( ( ord_less_eq @ nat @ M @ N4 )
=> ( ( P @ N4 )
=> ( P @ ( suc @ N4 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_206_measure__induct__rule,axiom,
! [B: $tType,A: $tType] :
( ( wellorder @ B )
=> ! [F: A > B,P: A > $o,A3: A] :
( ! [X3: A] :
( ! [Y4: A] :
( ( ord_less @ B @ ( F @ Y4 ) @ ( F @ X3 ) )
=> ( P @ Y4 ) )
=> ( P @ X3 ) )
=> ( P @ A3 ) ) ) ).
% measure_induct_rule
thf(fact_207_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_208_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_209_strict__inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less @ nat @ I @ J )
=> ( ! [I4: nat] :
( ( J
= ( suc @ I4 ) )
=> ( P @ I4 ) )
=> ( ! [I4: nat] :
( ( ord_less @ nat @ I4 @ J )
=> ( ( P @ ( suc @ I4 ) )
=> ( P @ I4 ) ) )
=> ( P @ I ) ) ) ) ).
% strict_inc_induct
thf(fact_210_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_211_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_212_le__eq__less__or__eq,axiom,
( ( ord_less_eq @ nat )
= ( ^ [M2: nat,N3: nat] :
( ( ord_less @ nat @ M2 @ N3 )
| ( M2 = N3 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_213_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N4: nat] :
( ~ ( P @ N4 )
=> ? [M3: nat] :
( ( ord_less @ nat @ M3 @ N4 )
& ~ ( P @ M3 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_214_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_215_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N4: nat] :
( ! [M3: nat] :
( ( ord_less @ nat @ M3 @ N4 )
=> ( P @ M3 ) )
=> ( P @ N4 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_216_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_217_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_218_less__Suc__induct,axiom,
! [I: nat,J: nat,P: nat > nat > $o] :
( ( ord_less @ nat @ I @ J )
=> ( ! [I4: nat] : ( P @ I4 @ ( suc @ I4 ) )
=> ( ! [I4: nat,J2: nat,K2: nat] :
( ( ord_less @ nat @ I4 @ J2 )
=> ( ( ord_less @ nat @ J2 @ K2 )
=> ( ( P @ I4 @ J2 )
=> ( ( P @ J2 @ K2 )
=> ( P @ I4 @ K2 ) ) ) ) )
=> ( P @ I @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_219_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_220_full__nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N4: nat] :
( ! [M3: nat] :
( ( ord_less_eq @ nat @ ( suc @ M3 ) @ N4 )
=> ( P @ M3 ) )
=> ( P @ N4 ) )
=> ( P @ N ) ) ).
% full_nat_induct
thf(fact_221_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_222_measure__induct,axiom,
! [B: $tType,A: $tType] :
( ( wellorder @ B )
=> ! [F: A > B,P: A > $o,A3: A] :
( ! [X3: A] :
( ! [Y4: A] :
( ( ord_less @ B @ ( F @ Y4 ) @ ( F @ X3 ) )
=> ( P @ Y4 ) )
=> ( P @ X3 ) )
=> ( P @ A3 ) ) ) ).
% measure_induct
thf(fact_223_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_224_less__not__refl3,axiom,
! [S: nat,T2: nat] :
( ( ord_less @ nat @ S @ T2 )
=> ( S != T2 ) ) ).
% less_not_refl3
thf(fact_225_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_226_less__eq__Suc__le,axiom,
( ( ord_less @ nat )
= ( ^ [N3: nat] : ( ord_less_eq @ nat @ ( suc @ N3 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_227_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_228_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_229_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq @ nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_230_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ N ) ).
% less_not_refl
thf(fact_231_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_232_less__antisym,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less @ nat @ N @ M )
=> ( ( ord_less @ nat @ N @ ( suc @ M ) )
=> ( M = N ) ) ) ).
% less_antisym
thf(fact_233_Suc__less__eq2,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ ( suc @ N ) @ M )
= ( ? [M4: nat] :
( ( M
= ( suc @ M4 ) )
& ( ord_less @ nat @ N @ M4 ) ) ) ) ).
% Suc_less_eq2
thf(fact_234_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_235_All__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I5: nat] :
( ( ord_less @ nat @ I5 @ ( suc @ N ) )
=> ( P @ I5 ) ) )
= ( ( P @ N )
& ! [I5: nat] :
( ( ord_less @ nat @ I5 @ N )
=> ( P @ I5 ) ) ) ) ).
% All_less_Suc
thf(fact_236_not__less__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less @ nat @ M @ N ) )
= ( ord_less @ nat @ N @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_237_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_238_nat__less__le,axiom,
( ( ord_less @ nat )
= ( ^ [M2: nat,N3: nat] :
( ( ord_less_eq @ nat @ M2 @ N3 )
& ( M2 != N3 ) ) ) ) ).
% nat_less_le
thf(fact_239_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_240_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_241_Ex__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I5: nat] :
( ( ord_less @ nat @ I5 @ ( suc @ N ) )
& ( P @ I5 ) ) )
= ( ( P @ N )
| ? [I5: nat] :
( ( ord_less @ nat @ I5 @ N )
& ( P @ I5 ) ) ) ) ).
% Ex_less_Suc
thf(fact_242_inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( P @ J )
=> ( ! [N4: nat] :
( ( ord_less_eq @ nat @ I @ N4 )
=> ( ( ord_less @ nat @ N4 @ J )
=> ( ( P @ ( suc @ N4 ) )
=> ( P @ N4 ) ) ) )
=> ( P @ I ) ) ) ) ).
% inc_induct
thf(fact_243_dec__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( P @ I )
=> ( ! [N4: nat] :
( ( ord_less_eq @ nat @ I @ N4 )
=> ( ( ord_less @ nat @ N4 @ J )
=> ( ( P @ N4 )
=> ( P @ ( suc @ N4 ) ) ) ) )
=> ( P @ J ) ) ) ) ).
% dec_induct
thf(fact_244_Suc__inject,axiom,
! [X: nat,Y3: nat] :
( ( ( suc @ X )
= ( suc @ Y3 ) )
=> ( X = Y3 ) ) ).
% Suc_inject
thf(fact_245_less__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ( ord_less @ nat @ M @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_246_less__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less @ nat @ M @ N )
=> ( M = N ) ) ) ).
% less_SucE
thf(fact_247_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_248_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_249_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_250_Suc__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ ( suc @ M ) @ N )
=> ( ord_less @ nat @ M @ N ) ) ).
% Suc_lessD
thf(fact_251_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_252_Suc__le__D,axiom,
! [N: nat,M5: nat] :
( ( ord_less_eq @ nat @ ( suc @ N ) @ M5 )
=> ? [M6: nat] :
( M5
= ( suc @ M6 ) ) ) ).
% Suc_le_D
thf(fact_253_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ord_less_eq @ nat @ M @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_254_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_255_Suc__leI,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ( ord_less_eq @ nat @ ( suc @ M ) @ N ) ) ).
% Suc_leI
thf(fact_256_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ ( suc @ M ) @ N )
=> ( ord_less_eq @ nat @ M @ N ) ) ).
% Suc_leD
thf(fact_257_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_258_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_259_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_260_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_261_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_262_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_263_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_264_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_265_mult_Ocommute,axiom,
! [A: $tType] :
( ( ab_semigroup_mult @ A )
=> ( ( times_times @ A )
= ( ^ [A5: A,B3: A] : ( times_times @ A @ B3 @ A5 ) ) ) ) ).
% mult.commute
thf(fact_266_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_267_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_268_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_269_lift__Suc__mono__less__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F: nat > A,N: nat,M: nat] :
( ! [N4: nat] : ( ord_less @ A @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
=> ( ( ord_less @ A @ ( F @ N ) @ ( F @ M ) )
= ( ord_less @ nat @ N @ M ) ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_270_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_271_lift__Suc__antimono__le,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F: nat > A,N: nat,N2: nat] :
( ! [N4: nat] : ( ord_less_eq @ A @ ( F @ ( suc @ N4 ) ) @ ( F @ N4 ) )
=> ( ( ord_less_eq @ nat @ N @ N2 )
=> ( ord_less_eq @ A @ ( F @ N2 ) @ ( F @ N ) ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_272_lift__Suc__mono__less,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F: nat > A,N: nat,N2: nat] :
( ! [N4: nat] : ( ord_less @ A @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
=> ( ( ord_less @ nat @ N @ N2 )
=> ( ord_less @ A @ ( F @ N ) @ ( F @ N2 ) ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_273_lift__Suc__mono__le,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F: nat > A,N: nat,N2: nat] :
( ! [N4: nat] : ( ord_less_eq @ A @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
=> ( ( ord_less_eq @ nat @ N @ N2 )
=> ( ord_less_eq @ A @ ( F @ N ) @ ( F @ N2 ) ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_274_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_275_fold__if__return,axiom,
! [A: $tType,B2: $o,C2: A,D2: A] :
( ( B2
=> ( ( heap_Time_return @ A @ C2 )
= ( heap_Time_return @ A @ ( if @ A @ B2 @ C2 @ D2 ) ) ) )
& ( ~ B2
=> ( ( heap_Time_return @ A @ D2 )
= ( heap_Time_return @ A @ ( if @ A @ B2 @ C2 @ D2 ) ) ) ) ) ).
% fold_if_return
thf(fact_276_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_277_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_278_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_279_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_280_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_281_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_282_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_283_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_284_mult__less__iff1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z3: A,X: A,Y3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Z3 )
=> ( ( ord_less @ A @ ( times_times @ A @ X @ Z3 ) @ ( times_times @ A @ Y3 @ Z3 ) )
= ( ord_less @ A @ X @ Y3 ) ) ) ) ).
% mult_less_iff1
thf(fact_285_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_286_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_287_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_288_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_289_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_290_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ? [M6: nat] :
( N
= ( suc @ M6 ) ) ) ).
% gr0_implies_Suc
thf(fact_291_All__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I5: nat] :
( ( ord_less @ nat @ I5 @ ( suc @ N ) )
=> ( P @ I5 ) ) )
= ( ( P @ ( zero_zero @ nat ) )
& ! [I5: nat] :
( ( ord_less @ nat @ I5 @ N )
=> ( P @ ( suc @ I5 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_292_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
= ( ? [M2: nat] :
( N
= ( suc @ M2 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_293_Ex__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I5: nat] :
( ( ord_less @ nat @ I5 @ ( suc @ N ) )
& ( P @ I5 ) ) )
= ( ( P @ ( zero_zero @ nat ) )
| ? [I5: nat] :
( ( ord_less @ nat @ I5 @ N )
& ( P @ ( suc @ I5 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_294_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_295_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_296_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ ( zero_zero @ nat ) )
=> ? [K2: nat] :
( ( ord_less_eq @ nat @ K2 @ N )
& ! [I3: nat] :
( ( ord_less @ nat @ I3 @ K2 )
=> ~ ( P @ I3 ) )
& ( P @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_297_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_298_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_299_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_300_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_301_nat__int__comparison_I3_J,axiom,
( ( ord_less_eq @ nat )
= ( ^ [A5: nat,B3: nat] : ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ A5 ) @ ( semiring_1_of_nat @ int @ B3 ) ) ) ) ).
% nat_int_comparison(3)
thf(fact_302_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_303_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_304_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_305_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_306_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_307_fact__mono,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ord_less_eq @ A @ ( semiring_char_0_fact @ A @ M ) @ ( semiring_char_0_fact @ A @ N ) ) ) ) ).
% fact_mono
thf(fact_308_euclidean__size__mult,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [A3: A,B2: A] :
( ( euclid6346220572633701492n_size @ A @ ( times_times @ A @ A3 @ B2 ) )
= ( times_times @ nat @ ( euclid6346220572633701492n_size @ A @ A3 ) @ ( euclid6346220572633701492n_size @ A @ B2 ) ) ) ) ).
% euclidean_size_mult
thf(fact_309_round__mono,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Y3: A] :
( ( ord_less_eq @ A @ X @ Y3 )
=> ( ord_less_eq @ int @ ( archimedean_round @ A @ X ) @ ( archimedean_round @ A @ Y3 ) ) ) ) ).
% round_mono
thf(fact_310_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_311_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_312_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_313_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_314_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_315_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_316_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_317_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_318_ex__less__of__nat__mult,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X: A,Y3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X )
=> ? [N4: nat] : ( ord_less @ A @ Y3 @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N4 ) @ X ) ) ) ) ).
% ex_less_of_nat_mult
thf(fact_319_nat__induct__non__zero,axiom,
! [N: nat,P: nat > $o] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( P @ ( one_one @ nat ) )
=> ( ! [N4: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N4 )
=> ( ( P @ N4 )
=> ( P @ ( suc @ N4 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_non_zero
thf(fact_320_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_321_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_322_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_323_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_324_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_325_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_326_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_327_not__less__zero,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [N: A] :
~ ( ord_less @ A @ N @ ( zero_zero @ A ) ) ) ).
% not_less_zero
thf(fact_328_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_329_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_330_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_331_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_332_zero__le,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [X: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X ) ) ).
% zero_le
thf(fact_333_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_334_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_335_exists__least__lemma,axiom,
! [P: nat > $o] :
( ~ ( P @ ( zero_zero @ nat ) )
=> ( ? [X_1: nat] : ( P @ X_1 )
=> ? [N4: nat] :
( ~ ( P @ N4 )
& ( P @ ( suc @ N4 ) ) ) ) ) ).
% exists_least_lemma
thf(fact_336_not0__implies__Suc,axiom,
! [N: nat] :
( ( N
!= ( zero_zero @ nat ) )
=> ? [M6: nat] :
( N
= ( suc @ M6 ) ) ) ).
% not0_implies_Suc
thf(fact_337_Zero__not__Suc,axiom,
! [M: nat] :
( ( zero_zero @ nat )
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_338_Zero__neq__Suc,axiom,
! [M: nat] :
( ( zero_zero @ nat )
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_339_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= ( zero_zero @ nat ) ) ).
% Suc_neq_Zero
thf(fact_340_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N4: nat] :
( ( P @ ( suc @ N4 ) )
=> ( P @ N4 ) )
=> ( P @ ( zero_zero @ nat ) ) ) ) ).
% zero_induct
thf(fact_341_diff__induct,axiom,
! [P: nat > nat > $o,M: nat,N: nat] :
( ! [X3: nat] : ( P @ X3 @ ( zero_zero @ nat ) )
=> ( ! [Y5: nat] : ( P @ ( zero_zero @ nat ) @ ( suc @ Y5 ) )
=> ( ! [X3: nat,Y5: nat] :
( ( P @ X3 @ Y5 )
=> ( P @ ( suc @ X3 ) @ ( suc @ Y5 ) ) )
=> ( P @ M @ N ) ) ) ) ).
% diff_induct
thf(fact_342_nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ ( zero_zero @ nat ) )
=> ( ! [N4: nat] :
( ( P @ N4 )
=> ( P @ ( suc @ N4 ) ) )
=> ( P @ N ) ) ) ).
% nat_induct
thf(fact_343_old_Onat_Oexhaust,axiom,
! [Y3: nat] :
( ( Y3
!= ( zero_zero @ nat ) )
=> ~ ! [Nat3: nat] :
( Y3
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_344_nat_OdiscI,axiom,
! [Nat: nat,X22: nat] :
( ( Nat
= ( suc @ X22 ) )
=> ( Nat
!= ( zero_zero @ nat ) ) ) ).
% nat.discI
thf(fact_345_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( ( zero_zero @ nat )
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_346_old_Onat_Odistinct_I2_J,axiom,
! [Nat4: nat] :
( ( suc @ Nat4 )
!= ( zero_zero @ nat ) ) ).
% old.nat.distinct(2)
thf(fact_347_nat_Odistinct_I1_J,axiom,
! [X22: nat] :
( ( zero_zero @ nat )
!= ( suc @ X22 ) ) ).
% nat.distinct(1)
thf(fact_348_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_349_mult__0,axiom,
! [N: nat] :
( ( times_times @ nat @ ( zero_zero @ nat ) @ N )
= ( zero_zero @ nat ) ) ).
% mult_0
thf(fact_350_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_351_infinite__descent0__measure,axiom,
! [A: $tType,V3: A > nat,P: A > $o,X: A] :
( ! [X3: A] :
( ( ( V3 @ X3 )
= ( zero_zero @ nat ) )
=> ( P @ X3 ) )
=> ( ! [X3: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( V3 @ X3 ) )
=> ( ~ ( P @ X3 )
=> ? [Y4: A] :
( ( ord_less @ nat @ ( V3 @ Y4 ) @ ( V3 @ X3 ) )
& ~ ( P @ Y4 ) ) ) )
=> ( P @ X ) ) ) ).
% infinite_descent0_measure
thf(fact_352_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ ( zero_zero @ nat ) )
=> ( ! [N4: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N4 )
=> ( ~ ( P @ N4 )
=> ? [M3: nat] :
( ( ord_less @ nat @ M3 @ N4 )
& ~ ( P @ M3 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_353_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ( N
!= ( zero_zero @ nat ) ) ) ).
% gr_implies_not0
thf(fact_354_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% less_zeroE
thf(fact_355_not__less0,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% not_less0
thf(fact_356_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% not_gr0
thf(fact_357_gr0I,axiom,
! [N: nat] :
( ( N
!= ( zero_zero @ nat ) )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% gr0I
thf(fact_358_bot__nat__0_Oextremum__strict,axiom,
! [A3: nat] :
~ ( ord_less @ nat @ A3 @ ( zero_zero @ nat ) ) ).
% bot_nat_0.extremum_strict
thf(fact_359_reals__Archimedean2,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X: A] :
? [N4: nat] : ( ord_less @ A @ X @ ( semiring_1_of_nat @ A @ N4 ) ) ) ).
% reals_Archimedean2
thf(fact_360_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq @ nat @ N @ ( zero_zero @ nat ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% le_0_eq
thf(fact_361_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_362_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_363_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% less_eq_nat.simps(1)
thf(fact_364_real__arch__simple,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X: A] :
? [N4: nat] : ( ord_less_eq @ A @ X @ ( semiring_1_of_nat @ A @ N4 ) ) ) ).
% real_arch_simple
thf(fact_365_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times @ nat @ N @ ( one_one @ nat ) )
= N ) ).
% nat_mult_1_right
thf(fact_366_nat__mult__1,axiom,
! [N: nat] :
( ( times_times @ nat @ ( one_one @ nat ) @ N )
= N ) ).
% nat_mult_1
thf(fact_367_not__one__less__zero,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ~ ( ord_less @ A @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ).
% not_one_less_zero
thf(fact_368_zero__less__one,axiom,
! [A: $tType] :
( ( zero_less_one @ A )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).
% zero_less_one
thf(fact_369_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_370_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_371_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_372_fact__mono__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ord_less_eq @ nat @ ( semiring_char_0_fact @ nat @ M ) @ ( semiring_char_0_fact @ nat @ N ) ) ) ).
% fact_mono_nat
thf(fact_373_fact__ge__self,axiom,
! [N: nat] : ( ord_less_eq @ nat @ N @ ( semiring_char_0_fact @ nat @ N ) ) ).
% fact_ge_self
thf(fact_374_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_375_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_376_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_377_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_378_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_379_One__nat__def,axiom,
( ( one_one @ nat )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% One_nat_def
thf(fact_380_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_381_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_382_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_383_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_384_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_385_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_386_int__ops_I1_J,axiom,
( ( semiring_1_of_nat @ int @ ( zero_zero @ nat ) )
= ( zero_zero @ int ) ) ).
% int_ops(1)
thf(fact_387_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_388_effect__returnI,axiom,
! [A: $tType,H2: heap_ext @ product_unit,H: heap_ext @ product_unit,X: A] :
( ( H2 = H )
=> ( heap_Time_effect @ A @ ( heap_Time_return @ A @ X ) @ H2 @ H @ X @ ( one_one @ nat ) ) ) ).
% effect_returnI
thf(fact_389_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_390_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_391_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_392_compl__less__compl__iff,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y3: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ X ) @ ( uminus_uminus @ A @ Y3 ) )
= ( ord_less @ A @ Y3 @ X ) ) ) ).
% compl_less_compl_iff
thf(fact_393_compl__le__compl__iff,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y3: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ X ) @ ( uminus_uminus @ A @ Y3 ) )
= ( ord_less_eq @ A @ Y3 @ X ) ) ) ).
% compl_le_compl_iff
thf(fact_394_field__le__mult__one__interval,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y3: A] :
( ! [Z4: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Z4 )
=> ( ( ord_less @ A @ Z4 @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ Z4 @ X ) @ Y3 ) ) )
=> ( ord_less_eq @ A @ X @ Y3 ) ) ) ).
% field_le_mult_one_interval
thf(fact_395_int__cases3,axiom,
! [K: int] :
( ( K
!= ( zero_zero @ int ) )
=> ( ! [N4: nat] :
( ( K
= ( semiring_1_of_nat @ int @ N4 ) )
=> ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N4 ) )
=> ~ ! [N4: nat] :
( ( K
= ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N4 ) ) )
=> ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N4 ) ) ) ) ).
% int_cases3
thf(fact_396_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_397_int__cases4,axiom,
! [M: int] :
( ! [N4: nat] :
( M
!= ( semiring_1_of_nat @ int @ N4 ) )
=> ~ ! [N4: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N4 )
=> ( M
!= ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N4 ) ) ) ) ) ).
% int_cases4
thf(fact_398_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_399_lift__collapse,axiom,
! [A: $tType,B: $tType] :
( ( heap_Time_lift @ B @ A )
= ( ^ [F2: B > A,X2: B] : ( heap_Time_return @ A @ ( F2 @ X2 ) ) ) ) ).
% lift_collapse
thf(fact_400_boolean__algebra__class_Oboolean__algebra_Odouble__compl,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A] :
( ( uminus_uminus @ A @ ( uminus_uminus @ A @ X ) )
= X ) ) ).
% boolean_algebra_class.boolean_algebra.double_compl
thf(fact_401_boolean__algebra__class_Oboolean__algebra_Ocompl__eq__compl__iff,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y3: A] :
( ( ( uminus_uminus @ A @ X )
= ( uminus_uminus @ A @ Y3 ) )
= ( X = Y3 ) ) ) ).
% boolean_algebra_class.boolean_algebra.compl_eq_compl_iff
thf(fact_402_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_403_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_404_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B2: nat] :
( ( P @ K )
=> ( ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq @ nat @ Y5 @ B2 ) )
=> ? [X3: nat] :
( ( P @ X3 )
& ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq @ nat @ Y4 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_405_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_406_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_407_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_408_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_409_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_410_le__square,axiom,
! [M: nat] : ( ord_less_eq @ nat @ M @ ( times_times @ nat @ M @ M ) ) ).
% le_square
thf(fact_411_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq @ nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_412_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_413_le__refl,axiom,
! [N: nat] : ( ord_less_eq @ nat @ N @ N ) ).
% le_refl
thf(fact_414_le__cube,axiom,
! [M: nat] : ( ord_less_eq @ nat @ M @ ( times_times @ nat @ M @ ( times_times @ nat @ M @ M ) ) ) ).
% le_cube
thf(fact_415_less__int__code_I1_J,axiom,
~ ( ord_less @ int @ ( zero_zero @ int ) @ ( zero_zero @ int ) ) ).
% less_int_code(1)
thf(fact_416_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_417_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_418_int__one__le__iff__zero__less,axiom,
! [Z3: int] :
( ( ord_less_eq @ int @ ( one_one @ int ) @ Z3 )
= ( ord_less @ int @ ( zero_zero @ int ) @ Z3 ) ) ).
% int_one_le_iff_zero_less
thf(fact_419_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_420_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_421_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_422_negD,axiom,
! [X: int] :
( ( ord_less @ int @ X @ ( zero_zero @ int ) )
=> ? [N4: nat] :
( X
= ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ ( suc @ N4 ) ) ) ) ) ).
% negD
thf(fact_423_nat__int__comparison_I2_J,axiom,
( ( ord_less @ nat )
= ( ^ [A5: nat,B3: nat] : ( ord_less @ int @ ( semiring_1_of_nat @ int @ A5 ) @ ( semiring_1_of_nat @ int @ B3 ) ) ) ) ).
% nat_int_comparison(2)
thf(fact_424_pos__int__cases,axiom,
! [K: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ K )
=> ~ ! [N4: nat] :
( ( K
= ( semiring_1_of_nat @ int @ N4 ) )
=> ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N4 ) ) ) ).
% pos_int_cases
thf(fact_425_zero__less__imp__eq__int,axiom,
! [K: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ K )
=> ? [N4: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N4 )
& ( K
= ( semiring_1_of_nat @ int @ N4 ) ) ) ) ).
% zero_less_imp_eq_int
thf(fact_426_neg__int__cases,axiom,
! [K: int] :
( ( ord_less @ int @ K @ ( zero_zero @ int ) )
=> ~ ! [N4: nat] :
( ( K
= ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N4 ) ) )
=> ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N4 ) ) ) ).
% neg_int_cases
thf(fact_427_linordered__field__no__ub,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X4: A] :
? [X_12: A] : ( ord_less @ A @ X4 @ X_12 ) ) ).
% linordered_field_no_ub
thf(fact_428_linordered__field__no__lb,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X4: A] :
? [Y5: A] : ( ord_less @ A @ Y5 @ X4 ) ) ).
% linordered_field_no_lb
thf(fact_429_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_430_times__int__code_I2_J,axiom,
! [L: int] :
( ( times_times @ int @ ( zero_zero @ int ) @ L )
= ( zero_zero @ int ) ) ).
% times_int_code(2)
thf(fact_431_times__int__code_I1_J,axiom,
! [K: int] :
( ( times_times @ int @ K @ ( zero_zero @ int ) )
= ( zero_zero @ int ) ) ).
% times_int_code(1)
thf(fact_432_uminus__int__code_I1_J,axiom,
( ( uminus_uminus @ int @ ( zero_zero @ int ) )
= ( zero_zero @ int ) ) ).
% uminus_int_code(1)
thf(fact_433_less__eq__int__code_I1_J,axiom,
ord_less_eq @ int @ ( zero_zero @ int ) @ ( zero_zero @ int ) ).
% less_eq_int_code(1)
thf(fact_434_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_435_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_436_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_437_int__cases2,axiom,
! [Z3: int] :
( ! [N4: nat] :
( Z3
!= ( semiring_1_of_nat @ int @ N4 ) )
=> ~ ! [N4: nat] :
( Z3
!= ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N4 ) ) ) ) ).
% int_cases2
thf(fact_438_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_439_compl__le__swap2,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [Y3: A,X: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ Y3 ) @ X )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ X ) @ Y3 ) ) ) ).
% compl_le_swap2
thf(fact_440_compl__le__swap1,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [Y3: A,X: A] :
( ( ord_less_eq @ A @ Y3 @ ( uminus_uminus @ A @ X ) )
=> ( ord_less_eq @ A @ X @ ( uminus_uminus @ A @ Y3 ) ) ) ) ).
% compl_le_swap1
thf(fact_441_compl__mono,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y3: A] :
( ( ord_less_eq @ A @ X @ Y3 )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ Y3 ) @ ( uminus_uminus @ A @ X ) ) ) ) ).
% compl_mono
thf(fact_442_compl__less__swap2,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [Y3: A,X: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ Y3 ) @ X )
=> ( ord_less @ A @ ( uminus_uminus @ A @ X ) @ Y3 ) ) ) ).
% compl_less_swap2
thf(fact_443_compl__less__swap1,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [Y3: A,X: A] :
( ( ord_less @ A @ Y3 @ ( uminus_uminus @ A @ X ) )
=> ( ord_less @ A @ X @ ( uminus_uminus @ A @ Y3 ) ) ) ) ).
% compl_less_swap1
thf(fact_444_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_445_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_446_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_447_int__of__nat__induct,axiom,
! [P: int > $o,Z3: int] :
( ! [N4: nat] : ( P @ ( semiring_1_of_nat @ int @ N4 ) )
=> ( ! [N4: nat] : ( P @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ ( suc @ N4 ) ) ) )
=> ( P @ Z3 ) ) ) ).
% int_of_nat_induct
thf(fact_448_int__cases,axiom,
! [Z3: int] :
( ! [N4: nat] :
( Z3
!= ( semiring_1_of_nat @ int @ N4 ) )
=> ~ ! [N4: nat] :
( Z3
!= ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ ( suc @ N4 ) ) ) ) ) ).
% int_cases
thf(fact_449_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_450_zero__le__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
=> ? [N4: nat] :
( K
= ( semiring_1_of_nat @ int @ N4 ) ) ) ).
% zero_le_imp_eq_int
thf(fact_451_nonpos__int__cases,axiom,
! [K: int] :
( ( ord_less_eq @ int @ K @ ( zero_zero @ int ) )
=> ~ ! [N4: nat] :
( K
!= ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N4 ) ) ) ) ).
% nonpos_int_cases
thf(fact_452_nonneg__int__cases,axiom,
! [K: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
=> ~ ! [N4: nat] :
( K
!= ( semiring_1_of_nat @ int @ N4 ) ) ) ).
% nonneg_int_cases
thf(fact_453_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_454_uminus__apply,axiom,
! [B: $tType,A: $tType] :
( ( uminus @ B )
=> ( ( uminus_uminus @ ( A > B ) )
= ( ^ [A7: A > B,X2: A] : ( uminus_uminus @ B @ ( A7 @ X2 ) ) ) ) ) ).
% uminus_apply
thf(fact_455_dual__order_Orefl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A] : ( ord_less_eq @ A @ A3 @ A3 ) ) ).
% dual_order.refl
thf(fact_456_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% order_refl
thf(fact_457_imp__le__cong,axiom,
! [X: int,X5: int,P: $o,P2: $o] :
( ( X = X5 )
=> ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X5 )
=> ( P = P2 ) )
=> ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
=> P )
= ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X5 )
=> P2 ) ) ) ) ).
% imp_le_cong
thf(fact_458_conj__le__cong,axiom,
! [X: int,X5: int,P: $o,P2: $o] :
( ( X = X5 )
=> ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X5 )
=> ( P = P2 ) )
=> ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
& P )
= ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X5 )
& P2 ) ) ) ) ).
% conj_le_cong
thf(fact_459_Lattices__Big_Oex__has__greatest__nat,axiom,
! [A: $tType,P: A > $o,K: A,F: A > nat,B2: nat] :
( ( P @ K )
=> ( ! [Y5: A] :
( ( P @ Y5 )
=> ( ord_less @ nat @ ( F @ Y5 ) @ B2 ) )
=> ? [X3: A] :
( ( P @ X3 )
& ! [Y4: A] :
( ( P @ Y4 )
=> ( ord_less_eq @ nat @ ( F @ Y4 ) @ ( F @ X3 ) ) ) ) ) ) ).
% Lattices_Big.ex_has_greatest_nat
thf(fact_460_nat__descend__induct,axiom,
! [N: nat,P: nat > $o,M: nat] :
( ! [K2: nat] :
( ( ord_less @ nat @ N @ K2 )
=> ( P @ K2 ) )
=> ( ! [K2: nat] :
( ( ord_less_eq @ nat @ K2 @ N )
=> ( ! [I3: nat] :
( ( ord_less @ nat @ K2 @ I3 )
=> ( P @ I3 ) )
=> ( P @ K2 ) ) )
=> ( P @ M ) ) ) ).
% nat_descend_induct
thf(fact_461_dependent__nat__choice,axiom,
! [A: $tType,P: nat > A > $o,Q: nat > A > A > $o] :
( ? [X_1: A] : ( P @ ( zero_zero @ nat ) @ X_1 )
=> ( ! [X3: A,N4: nat] :
( ( P @ N4 @ X3 )
=> ? [Y4: A] :
( ( P @ ( suc @ N4 ) @ Y4 )
& ( Q @ N4 @ X3 @ Y4 ) ) )
=> ? [F3: nat > A] :
! [N5: nat] :
( ( P @ N5 @ ( F3 @ N5 ) )
& ( Q @ N5 @ ( F3 @ N5 ) @ ( F3 @ ( suc @ N5 ) ) ) ) ) ) ).
% dependent_nat_choice
thf(fact_462_list__decode_Ocases,axiom,
! [X: nat] :
( ( X
!= ( zero_zero @ nat ) )
=> ~ ! [N4: nat] :
( X
!= ( suc @ N4 ) ) ) ).
% list_decode.cases
thf(fact_463_order__le__imp__less__or__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y3: A] :
( ( ord_less_eq @ A @ X @ Y3 )
=> ( ( ord_less @ A @ X @ Y3 )
| ( X = Y3 ) ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_464_nle__le,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A] :
( ( ~ ( ord_less_eq @ A @ A3 @ B2 ) )
= ( ( ord_less_eq @ A @ B2 @ A3 )
& ( B2 != A3 ) ) ) ) ).
% nle_le
thf(fact_465_le__cases3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y3: A,Z3: A] :
( ( ( ord_less_eq @ A @ X @ Y3 )
=> ~ ( ord_less_eq @ A @ Y3 @ Z3 ) )
=> ( ( ( ord_less_eq @ A @ Y3 @ X )
=> ~ ( ord_less_eq @ A @ X @ Z3 ) )
=> ( ( ( ord_less_eq @ A @ X @ Z3 )
=> ~ ( ord_less_eq @ A @ Z3 @ Y3 ) )
=> ( ( ( ord_less_eq @ A @ Z3 @ Y3 )
=> ~ ( ord_less_eq @ A @ Y3 @ X ) )
=> ( ( ( ord_less_eq @ A @ Y3 @ Z3 )
=> ~ ( ord_less_eq @ A @ Z3 @ X ) )
=> ~ ( ( ord_less_eq @ A @ Z3 @ X )
=> ~ ( ord_less_eq @ A @ X @ Y3 ) ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_466_order__class_Oorder__eq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y: A,Z2: A] : Y = Z2 )
= ( ^ [X2: A,Y6: A] :
( ( ord_less_eq @ A @ X2 @ Y6 )
& ( ord_less_eq @ A @ Y6 @ X2 ) ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_467_ord__eq__le__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( A3 = B2 )
=> ( ( ord_less_eq @ A @ B2 @ C2 )
=> ( ord_less_eq @ A @ A3 @ C2 ) ) ) ) ).
% ord_eq_le_trans
thf(fact_468_ord__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( B2 = C2 )
=> ( ord_less_eq @ A @ A3 @ C2 ) ) ) ) ).
% ord_le_eq_trans
thf(fact_469_order__antisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y3: A] :
( ( ord_less_eq @ A @ X @ Y3 )
=> ( ( ord_less_eq @ A @ Y3 @ X )
=> ( X = Y3 ) ) ) ) ).
% order_antisym
thf(fact_470_order_Otrans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ C2 )
=> ( ord_less_eq @ A @ A3 @ C2 ) ) ) ) ).
% order.trans
thf(fact_471_order__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y3: A,Z3: A] :
( ( ord_less_eq @ A @ X @ Y3 )
=> ( ( ord_less_eq @ A @ Y3 @ Z3 )
=> ( ord_less_eq @ A @ X @ Z3 ) ) ) ) ).
% order_trans
thf(fact_472_linorder__wlog,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > A > $o,A3: A,B2: A] :
( ! [A8: A,B5: A] :
( ( ord_less_eq @ A @ A8 @ B5 )
=> ( P @ A8 @ B5 ) )
=> ( ! [A8: A,B5: A] :
( ( P @ B5 @ A8 )
=> ( P @ A8 @ B5 ) )
=> ( P @ A3 @ B2 ) ) ) ) ).
% linorder_wlog
thf(fact_473_dual__order_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y: A,Z2: A] : Y = Z2 )
= ( ^ [A5: A,B3: A] :
( ( ord_less_eq @ A @ B3 @ A5 )
& ( ord_less_eq @ A @ A5 @ B3 ) ) ) ) ) ).
% dual_order.eq_iff
thf(fact_474_dual__order_Oantisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B2: A,A3: A] :
( ( ord_less_eq @ A @ B2 @ A3 )
=> ( ( ord_less_eq @ A @ A3 @ B2 )
=> ( A3 = B2 ) ) ) ) ).
% dual_order.antisym
thf(fact_475_dual__order_Otrans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( ord_less_eq @ A @ B2 @ A3 )
=> ( ( ord_less_eq @ A @ C2 @ B2 )
=> ( ord_less_eq @ A @ C2 @ A3 ) ) ) ) ).
% dual_order.trans
thf(fact_476_antisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ A3 )
=> ( A3 = B2 ) ) ) ) ).
% antisym
thf(fact_477_le__funD,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F: A > B,G: A > B,X: A] :
( ( ord_less_eq @ ( A > B ) @ F @ G )
=> ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% le_funD
thf(fact_478_le__funE,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F: A > B,G: A > B,X: A] :
( ( ord_less_eq @ ( A > B ) @ F @ G )
=> ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% le_funE
thf(fact_479_le__funI,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F: A > B,G: A > B] :
( ! [X3: A] : ( ord_less_eq @ B @ ( F @ X3 ) @ ( G @ X3 ) )
=> ( ord_less_eq @ ( A > B ) @ F @ G ) ) ) ).
% le_funI
thf(fact_480_le__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ( ( ord_less_eq @ ( A > B ) )
= ( ^ [F2: A > B,G2: A > B] :
! [X2: A] : ( ord_less_eq @ B @ ( F2 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ).
% le_fun_def
thf(fact_481_Orderings_Oorder__eq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y: A,Z2: A] : Y = Z2 )
= ( ^ [A5: A,B3: A] :
( ( ord_less_eq @ A @ A5 @ B3 )
& ( ord_less_eq @ A @ B3 @ A5 ) ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_482_order__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A3: A,F: B > A,B2: B,C2: B] :
( ( ord_less_eq @ A @ A3 @ ( F @ B2 ) )
=> ( ( ord_less_eq @ B @ B2 @ C2 )
=> ( ! [X3: B,Y5: B] :
( ( ord_less_eq @ B @ X3 @ Y5 )
=> ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq @ A @ A3 @ ( F @ C2 ) ) ) ) ) ) ).
% order_subst1
thf(fact_483_order__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A3: A,B2: A,F: A > C,C2: C] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ C @ ( F @ B2 ) @ C2 )
=> ( ! [X3: A,Y5: A] :
( ( ord_less_eq @ A @ X3 @ Y5 )
=> ( ord_less_eq @ C @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq @ C @ ( F @ A3 ) @ C2 ) ) ) ) ) ).
% order_subst2
thf(fact_484_order__eq__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y3: A] :
( ( X = Y3 )
=> ( ord_less_eq @ A @ X @ Y3 ) ) ) ).
% order_eq_refl
thf(fact_485_linorder__linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y3: A] :
( ( ord_less_eq @ A @ X @ Y3 )
| ( ord_less_eq @ A @ Y3 @ X ) ) ) ).
% linorder_linear
thf(fact_486_ord__eq__le__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A3: A,F: B > A,B2: B,C2: B] :
( ( A3
= ( F @ B2 ) )
=> ( ( ord_less_eq @ B @ B2 @ C2 )
=> ( ! [X3: B,Y5: B] :
( ( ord_less_eq @ B @ X3 @ Y5 )
=> ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq @ A @ A3 @ ( F @ C2 ) ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_487_ord__le__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A3: A,B2: A,F: A > B,C2: B] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X3: A,Y5: A] :
( ( ord_less_eq @ A @ X3 @ Y5 )
=> ( ord_less_eq @ B @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq @ B @ ( F @ A3 ) @ C2 ) ) ) ) ) ).
% ord_le_eq_subst
thf(fact_488_linorder__le__cases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y3: A] :
( ~ ( ord_less_eq @ A @ X @ Y3 )
=> ( ord_less_eq @ A @ Y3 @ X ) ) ) ).
% linorder_le_cases
thf(fact_489_order__antisym__conv,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y3: A,X: A] :
( ( ord_less_eq @ A @ Y3 @ X )
=> ( ( ord_less_eq @ A @ X @ Y3 )
= ( X = Y3 ) ) ) ) ).
% order_antisym_conv
thf(fact_490_minf_I11_J,axiom,
! [C: $tType,D: $tType] :
( ( ord @ C )
=> ! [F4: D] :
? [Z4: C] :
! [X4: C] :
( ( ord_less @ C @ X4 @ Z4 )
=> ( F4 = F4 ) ) ) ).
% minf(11)
thf(fact_491_minf_I7_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ X4 @ Z4 )
=> ~ ( ord_less @ A @ T2 @ X4 ) ) ) ).
% minf(7)
thf(fact_492_minf_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ X4 @ Z4 )
=> ( ord_less @ A @ X4 @ T2 ) ) ) ).
% minf(5)
thf(fact_493_minf_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ X4 @ Z4 )
=> ( X4 != T2 ) ) ) ).
% minf(4)
thf(fact_494_minf_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ X4 @ Z4 )
=> ( X4 != T2 ) ) ) ).
% minf(3)
thf(fact_495_minf_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z5: A] :
! [X3: A] :
( ( ord_less @ A @ X3 @ Z5 )
=> ( ( P @ X3 )
= ( P2 @ X3 ) ) )
=> ( ? [Z5: A] :
! [X3: A] :
( ( ord_less @ A @ X3 @ Z5 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ X4 @ Z4 )
=> ( ( ( P @ X4 )
| ( Q @ X4 ) )
= ( ( P2 @ X4 )
| ( Q2 @ X4 ) ) ) ) ) ) ) ).
% minf(2)
thf(fact_496_minf_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z5: A] :
! [X3: A] :
( ( ord_less @ A @ X3 @ Z5 )
=> ( ( P @ X3 )
= ( P2 @ X3 ) ) )
=> ( ? [Z5: A] :
! [X3: A] :
( ( ord_less @ A @ X3 @ Z5 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ X4 @ Z4 )
=> ( ( ( P @ X4 )
& ( Q @ X4 ) )
= ( ( P2 @ X4 )
& ( Q2 @ X4 ) ) ) ) ) ) ) ).
% minf(1)
thf(fact_497_pinf_I11_J,axiom,
! [C: $tType,D: $tType] :
( ( ord @ C )
=> ! [F4: D] :
? [Z4: C] :
! [X4: C] :
( ( ord_less @ C @ Z4 @ X4 )
=> ( F4 = F4 ) ) ) ).
% pinf(11)
thf(fact_498_pinf_I7_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ Z4 @ X4 )
=> ( ord_less @ A @ T2 @ X4 ) ) ) ).
% pinf(7)
thf(fact_499_pinf_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ Z4 @ X4 )
=> ~ ( ord_less @ A @ X4 @ T2 ) ) ) ).
% pinf(5)
thf(fact_500_pinf_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ Z4 @ X4 )
=> ( X4 != T2 ) ) ) ).
% pinf(4)
thf(fact_501_pinf_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ Z4 @ X4 )
=> ( X4 != T2 ) ) ) ).
% pinf(3)
thf(fact_502_pinf_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z5: A] :
! [X3: A] :
( ( ord_less @ A @ Z5 @ X3 )
=> ( ( P @ X3 )
= ( P2 @ X3 ) ) )
=> ( ? [Z5: A] :
! [X3: A] :
( ( ord_less @ A @ Z5 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ Z4 @ X4 )
=> ( ( ( P @ X4 )
| ( Q @ X4 ) )
= ( ( P2 @ X4 )
| ( Q2 @ X4 ) ) ) ) ) ) ) ).
% pinf(2)
thf(fact_503_pinf_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z5: A] :
! [X3: A] :
( ( ord_less @ A @ Z5 @ X3 )
=> ( ( P @ X3 )
= ( P2 @ X3 ) ) )
=> ( ? [Z5: A] :
! [X3: A] :
( ( ord_less @ A @ Z5 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ Z4 @ X4 )
=> ( ( ( P @ X4 )
& ( Q @ X4 ) )
= ( ( P2 @ X4 )
& ( Q2 @ X4 ) ) ) ) ) ) ) ).
% pinf(1)
thf(fact_504_lt__ex,axiom,
! [A: $tType] :
( ( no_bot @ A )
=> ! [X: A] :
? [Y5: A] : ( ord_less @ A @ Y5 @ X ) ) ).
% lt_ex
thf(fact_505_gt__ex,axiom,
! [A: $tType] :
( ( no_top @ A )
=> ! [X: A] :
? [X_12: A] : ( ord_less @ A @ X @ X_12 ) ) ).
% gt_ex
thf(fact_506_dense,axiom,
! [A: $tType] :
( ( dense_order @ A )
=> ! [X: A,Y3: A] :
( ( ord_less @ A @ X @ Y3 )
=> ? [Z4: A] :
( ( ord_less @ A @ X @ Z4 )
& ( ord_less @ A @ Z4 @ Y3 ) ) ) ) ).
% dense
thf(fact_507_less__imp__neq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y3: A] :
( ( ord_less @ A @ X @ Y3 )
=> ( X != Y3 ) ) ) ).
% less_imp_neq
thf(fact_508_order_Oasym,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ~ ( ord_less @ A @ B2 @ A3 ) ) ) ).
% order.asym
thf(fact_509_ord__eq__less__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( A3 = B2 )
=> ( ( ord_less @ A @ B2 @ C2 )
=> ( ord_less @ A @ A3 @ C2 ) ) ) ) ).
% ord_eq_less_trans
thf(fact_510_ord__less__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( B2 = C2 )
=> ( ord_less @ A @ A3 @ C2 ) ) ) ) ).
% ord_less_eq_trans
thf(fact_511_less__induct,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P: A > $o,A3: A] :
( ! [X3: A] :
( ! [Y4: A] :
( ( ord_less @ A @ Y4 @ X3 )
=> ( P @ Y4 ) )
=> ( P @ X3 ) )
=> ( P @ A3 ) ) ) ).
% less_induct
thf(fact_512_antisym__conv3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Y3: A,X: A] :
( ~ ( ord_less @ A @ Y3 @ X )
=> ( ( ~ ( ord_less @ A @ X @ Y3 ) )
= ( X = Y3 ) ) ) ) ).
% antisym_conv3
thf(fact_513_linorder__cases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y3: A] :
( ~ ( ord_less @ A @ X @ Y3 )
=> ( ( X != Y3 )
=> ( ord_less @ A @ Y3 @ X ) ) ) ) ).
% linorder_cases
thf(fact_514_dual__order_Oasym,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B2: A,A3: A] :
( ( ord_less @ A @ B2 @ A3 )
=> ~ ( ord_less @ A @ A3 @ B2 ) ) ) ).
% dual_order.asym
thf(fact_515_dual__order_Oirrefl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A] :
~ ( ord_less @ A @ A3 @ A3 ) ) ).
% dual_order.irrefl
thf(fact_516_exists__least__iff,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ( ( ^ [P3: A > $o] :
? [X6: A] : ( P3 @ X6 ) )
= ( ^ [P4: A > $o] :
? [N3: A] :
( ( P4 @ N3 )
& ! [M2: A] :
( ( ord_less @ A @ M2 @ N3 )
=> ~ ( P4 @ M2 ) ) ) ) ) ) ).
% exists_least_iff
thf(fact_517_linorder__less__wlog,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > A > $o,A3: A,B2: A] :
( ! [A8: A,B5: A] :
( ( ord_less @ A @ A8 @ B5 )
=> ( P @ A8 @ B5 ) )
=> ( ! [A8: A] : ( P @ A8 @ A8 )
=> ( ! [A8: A,B5: A] :
( ( P @ B5 @ A8 )
=> ( P @ A8 @ B5 ) )
=> ( P @ A3 @ B2 ) ) ) ) ) ).
% linorder_less_wlog
thf(fact_518_order_Ostrict__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ B2 @ C2 )
=> ( ord_less @ A @ A3 @ C2 ) ) ) ) ).
% order.strict_trans
thf(fact_519_not__less__iff__gr__or__eq,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y3: A] :
( ( ~ ( ord_less @ A @ X @ Y3 ) )
= ( ( ord_less @ A @ Y3 @ X )
| ( X = Y3 ) ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_520_dual__order_Ostrict__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( ord_less @ A @ B2 @ A3 )
=> ( ( ord_less @ A @ C2 @ B2 )
=> ( ord_less @ A @ C2 @ A3 ) ) ) ) ).
% dual_order.strict_trans
thf(fact_521_order_Ostrict__implies__not__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( A3 != B2 ) ) ) ).
% order.strict_implies_not_eq
thf(fact_522_dual__order_Ostrict__implies__not__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B2: A,A3: A] :
( ( ord_less @ A @ B2 @ A3 )
=> ( A3 != B2 ) ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_523_linorder__neqE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y3: A] :
( ( X != Y3 )
=> ( ~ ( ord_less @ A @ X @ Y3 )
=> ( ord_less @ A @ Y3 @ X ) ) ) ) ).
% linorder_neqE
thf(fact_524_order__less__asym,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y3: A] :
( ( ord_less @ A @ X @ Y3 )
=> ~ ( ord_less @ A @ Y3 @ X ) ) ) ).
% order_less_asym
thf(fact_525_linorder__neq__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y3: A] :
( ( X != Y3 )
= ( ( ord_less @ A @ X @ Y3 )
| ( ord_less @ A @ Y3 @ X ) ) ) ) ).
% linorder_neq_iff
thf(fact_526_order__less__asym_H,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ~ ( ord_less @ A @ B2 @ A3 ) ) ) ).
% order_less_asym'
thf(fact_527_order__less__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y3: A,Z3: A] :
( ( ord_less @ A @ X @ Y3 )
=> ( ( ord_less @ A @ Y3 @ Z3 )
=> ( ord_less @ A @ X @ Z3 ) ) ) ) ).
% order_less_trans
thf(fact_528_ord__eq__less__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A3: A,F: B > A,B2: B,C2: B] :
( ( A3
= ( F @ B2 ) )
=> ( ( ord_less @ B @ B2 @ C2 )
=> ( ! [X3: B,Y5: B] :
( ( ord_less @ B @ X3 @ Y5 )
=> ( ord_less @ A @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less @ A @ A3 @ ( F @ C2 ) ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_529_ord__less__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A3: A,B2: A,F: A > B,C2: B] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X3: A,Y5: A] :
( ( ord_less @ A @ X3 @ Y5 )
=> ( ord_less @ B @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less @ B @ ( F @ A3 ) @ C2 ) ) ) ) ) ).
% ord_less_eq_subst
thf(fact_530_order__less__irrefl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A] :
~ ( ord_less @ A @ X @ X ) ) ).
% order_less_irrefl
thf(fact_531_order__less__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A3: A,F: B > A,B2: B,C2: B] :
( ( ord_less @ A @ A3 @ ( F @ B2 ) )
=> ( ( ord_less @ B @ B2 @ C2 )
=> ( ! [X3: B,Y5: B] :
( ( ord_less @ B @ X3 @ Y5 )
=> ( ord_less @ A @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less @ A @ A3 @ ( F @ C2 ) ) ) ) ) ) ).
% order_less_subst1
thf(fact_532_order__less__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A3: A,B2: A,F: A > C,C2: C] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less @ C @ ( F @ B2 ) @ C2 )
=> ( ! [X3: A,Y5: A] :
( ( ord_less @ A @ X3 @ Y5 )
=> ( ord_less @ C @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less @ C @ ( F @ A3 ) @ C2 ) ) ) ) ) ).
% order_less_subst2
thf(fact_533_order__less__not__sym,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y3: A] :
( ( ord_less @ A @ X @ Y3 )
=> ~ ( ord_less @ A @ Y3 @ X ) ) ) ).
% order_less_not_sym
thf(fact_534_order__less__imp__triv,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y3: A,P: $o] :
( ( ord_less @ A @ X @ Y3 )
=> ( ( ord_less @ A @ Y3 @ X )
=> P ) ) ) ).
% order_less_imp_triv
thf(fact_535_linorder__less__linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y3: A] :
( ( ord_less @ A @ X @ Y3 )
| ( X = Y3 )
| ( ord_less @ A @ Y3 @ X ) ) ) ).
% linorder_less_linear
thf(fact_536_order__less__imp__not__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y3: A] :
( ( ord_less @ A @ X @ Y3 )
=> ( X != Y3 ) ) ) ).
% order_less_imp_not_eq
thf(fact_537_order__less__imp__not__eq2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y3: A] :
( ( ord_less @ A @ X @ Y3 )
=> ( Y3 != X ) ) ) ).
% order_less_imp_not_eq2
thf(fact_538_order__less__imp__not__less,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y3: A] :
( ( ord_less @ A @ X @ Y3 )
=> ~ ( ord_less @ A @ Y3 @ X ) ) ) ).
% order_less_imp_not_less
thf(fact_539_fun__Compl__def,axiom,
! [B: $tType,A: $tType] :
( ( uminus @ B )
=> ( ( uminus_uminus @ ( A > B ) )
= ( ^ [A7: A > B,X2: A] : ( uminus_uminus @ B @ ( A7 @ X2 ) ) ) ) ) ).
% fun_Compl_def
thf(fact_540_ex__has__least__nat,axiom,
! [A: $tType,P: A > $o,K: A,M: A > nat] :
( ( P @ K )
=> ? [X3: A] :
( ( P @ X3 )
& ! [Y4: A] :
( ( P @ Y4 )
=> ( ord_less_eq @ nat @ ( M @ X3 ) @ ( M @ Y4 ) ) ) ) ) ).
% ex_has_least_nat
thf(fact_541_minf_I8_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ X4 @ Z4 )
=> ~ ( ord_less_eq @ A @ T2 @ X4 ) ) ) ).
% minf(8)
thf(fact_542_minf_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ X4 @ Z4 )
=> ( ord_less_eq @ A @ X4 @ T2 ) ) ) ).
% minf(6)
thf(fact_543_pinf_I8_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ Z4 @ X4 )
=> ( ord_less_eq @ A @ T2 @ X4 ) ) ) ).
% pinf(8)
thf(fact_544_pinf_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ Z4 @ X4 )
=> ~ ( ord_less_eq @ A @ X4 @ T2 ) ) ) ).
% pinf(6)
thf(fact_545_leD,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y3: A,X: A] :
( ( ord_less_eq @ A @ Y3 @ X )
=> ~ ( ord_less @ A @ X @ Y3 ) ) ) ).
% leD
thf(fact_546_leI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y3: A] :
( ~ ( ord_less @ A @ X @ Y3 )
=> ( ord_less_eq @ A @ Y3 @ X ) ) ) ).
% leI
thf(fact_547_nless__le,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B2: A] :
( ( ~ ( ord_less @ A @ A3 @ B2 ) )
= ( ~ ( ord_less_eq @ A @ A3 @ B2 )
| ( A3 = B2 ) ) ) ) ).
% nless_le
thf(fact_548_antisym__conv1,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y3: A] :
( ~ ( ord_less @ A @ X @ Y3 )
=> ( ( ord_less_eq @ A @ X @ Y3 )
= ( X = Y3 ) ) ) ) ).
% antisym_conv1
thf(fact_549_antisym__conv2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y3: A] :
( ( ord_less_eq @ A @ X @ Y3 )
=> ( ( ~ ( ord_less @ A @ X @ Y3 ) )
= ( X = Y3 ) ) ) ) ).
% antisym_conv2
thf(fact_550_dense__ge,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [Z3: A,Y3: A] :
( ! [X3: A] :
( ( ord_less @ A @ Z3 @ X3 )
=> ( ord_less_eq @ A @ Y3 @ X3 ) )
=> ( ord_less_eq @ A @ Y3 @ Z3 ) ) ) ).
% dense_ge
thf(fact_551_dense__le,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [Y3: A,Z3: A] :
( ! [X3: A] :
( ( ord_less @ A @ X3 @ Y3 )
=> ( ord_less_eq @ A @ X3 @ Z3 ) )
=> ( ord_less_eq @ A @ Y3 @ Z3 ) ) ) ).
% dense_le
thf(fact_552_less__le__not__le,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( ord_less @ A )
= ( ^ [X2: A,Y6: A] :
( ( ord_less_eq @ A @ X2 @ Y6 )
& ~ ( ord_less_eq @ A @ Y6 @ X2 ) ) ) ) ) ).
% less_le_not_le
thf(fact_553_not__le__imp__less,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Y3: A,X: A] :
( ~ ( ord_less_eq @ A @ Y3 @ X )
=> ( ord_less @ A @ X @ Y3 ) ) ) ).
% not_le_imp_less
thf(fact_554_order_Oorder__iff__strict,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A5: A,B3: A] :
( ( ord_less @ A @ A5 @ B3 )
| ( A5 = B3 ) ) ) ) ) ).
% order.order_iff_strict
thf(fact_555_order_Ostrict__iff__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less @ A )
= ( ^ [A5: A,B3: A] :
( ( ord_less_eq @ A @ A5 @ B3 )
& ( A5 != B3 ) ) ) ) ) ).
% order.strict_iff_order
thf(fact_556_order_Ostrict__trans1,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ B2 @ C2 )
=> ( ord_less @ A @ A3 @ C2 ) ) ) ) ).
% order.strict_trans1
thf(fact_557_order_Ostrict__trans2,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ C2 )
=> ( ord_less @ A @ A3 @ C2 ) ) ) ) ).
% order.strict_trans2
thf(fact_558_order_Ostrict__iff__not,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( ord_less @ A )
= ( ^ [A5: A,B3: A] :
( ( ord_less_eq @ A @ A5 @ B3 )
& ~ ( ord_less_eq @ A @ B3 @ A5 ) ) ) ) ) ).
% order.strict_iff_not
thf(fact_559_dense__ge__bounded,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [Z3: A,X: A,Y3: A] :
( ( ord_less @ A @ Z3 @ X )
=> ( ! [W: A] :
( ( ord_less @ A @ Z3 @ W )
=> ( ( ord_less @ A @ W @ X )
=> ( ord_less_eq @ A @ Y3 @ W ) ) )
=> ( ord_less_eq @ A @ Y3 @ Z3 ) ) ) ) ).
% dense_ge_bounded
thf(fact_560_dense__le__bounded,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [X: A,Y3: A,Z3: A] :
( ( ord_less @ A @ X @ Y3 )
=> ( ! [W: A] :
( ( ord_less @ A @ X @ W )
=> ( ( ord_less @ A @ W @ Y3 )
=> ( ord_less_eq @ A @ W @ Z3 ) ) )
=> ( ord_less_eq @ A @ Y3 @ Z3 ) ) ) ) ).
% dense_le_bounded
thf(fact_561_dual__order_Oorder__iff__strict,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B3: A,A5: A] :
( ( ord_less @ A @ B3 @ A5 )
| ( A5 = B3 ) ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_562_dual__order_Ostrict__iff__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less @ A )
= ( ^ [B3: A,A5: A] :
( ( ord_less_eq @ A @ B3 @ A5 )
& ( A5 != B3 ) ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_563_dual__order_Ostrict__trans1,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( ord_less_eq @ A @ B2 @ A3 )
=> ( ( ord_less @ A @ C2 @ B2 )
=> ( ord_less @ A @ C2 @ A3 ) ) ) ) ).
% dual_order.strict_trans1
thf(fact_564_dual__order_Ostrict__trans2,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( ord_less @ A @ B2 @ A3 )
=> ( ( ord_less_eq @ A @ C2 @ B2 )
=> ( ord_less @ A @ C2 @ A3 ) ) ) ) ).
% dual_order.strict_trans2
thf(fact_565_dual__order_Ostrict__iff__not,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( ord_less @ A )
= ( ^ [B3: A,A5: A] :
( ( ord_less_eq @ A @ B3 @ A5 )
& ~ ( ord_less_eq @ A @ A5 @ B3 ) ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_566_order_Ostrict__implies__order,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ord_less_eq @ A @ A3 @ B2 ) ) ) ).
% order.strict_implies_order
thf(fact_567_dual__order_Ostrict__implies__order,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B2: A,A3: A] :
( ( ord_less @ A @ B2 @ A3 )
=> ( ord_less_eq @ A @ B2 @ A3 ) ) ) ).
% dual_order.strict_implies_order
thf(fact_568_order__le__less,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [X2: A,Y6: A] :
( ( ord_less @ A @ X2 @ Y6 )
| ( X2 = Y6 ) ) ) ) ) ).
% order_le_less
thf(fact_569_order__less__le,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less @ A )
= ( ^ [X2: A,Y6: A] :
( ( ord_less_eq @ A @ X2 @ Y6 )
& ( X2 != Y6 ) ) ) ) ) ).
% order_less_le
thf(fact_570_linorder__not__le,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y3: A] :
( ( ~ ( ord_less_eq @ A @ X @ Y3 ) )
= ( ord_less @ A @ Y3 @ X ) ) ) ).
% linorder_not_le
thf(fact_571_linorder__not__less,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y3: A] :
( ( ~ ( ord_less @ A @ X @ Y3 ) )
= ( ord_less_eq @ A @ Y3 @ X ) ) ) ).
% linorder_not_less
thf(fact_572_order__less__imp__le,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y3: A] :
( ( ord_less @ A @ X @ Y3 )
=> ( ord_less_eq @ A @ X @ Y3 ) ) ) ).
% order_less_imp_le
thf(fact_573_order__le__neq__trans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( A3 != B2 )
=> ( ord_less @ A @ A3 @ B2 ) ) ) ) ).
% order_le_neq_trans
thf(fact_574_order__neq__le__trans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B2: A] :
( ( A3 != B2 )
=> ( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ord_less @ A @ A3 @ B2 ) ) ) ) ).
% order_neq_le_trans
thf(fact_575_order__le__less__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y3: A,Z3: A] :
( ( ord_less_eq @ A @ X @ Y3 )
=> ( ( ord_less @ A @ Y3 @ Z3 )
=> ( ord_less @ A @ X @ Z3 ) ) ) ) ).
% order_le_less_trans
thf(fact_576_order__less__le__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y3: A,Z3: A] :
( ( ord_less @ A @ X @ Y3 )
=> ( ( ord_less_eq @ A @ Y3 @ Z3 )
=> ( ord_less @ A @ X @ Z3 ) ) ) ) ).
% order_less_le_trans
thf(fact_577_order__le__less__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A3: A,F: B > A,B2: B,C2: B] :
( ( ord_less_eq @ A @ A3 @ ( F @ B2 ) )
=> ( ( ord_less @ B @ B2 @ C2 )
=> ( ! [X3: B,Y5: B] :
( ( ord_less @ B @ X3 @ Y5 )
=> ( ord_less @ A @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less @ A @ A3 @ ( F @ C2 ) ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_578_order__le__less__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A3: A,B2: A,F: A > C,C2: C] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less @ C @ ( F @ B2 ) @ C2 )
=> ( ! [X3: A,Y5: A] :
( ( ord_less_eq @ A @ X3 @ Y5 )
=> ( ord_less_eq @ C @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less @ C @ ( F @ A3 ) @ C2 ) ) ) ) ) ).
% order_le_less_subst2
thf(fact_579_order__less__le__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A3: A,F: B > A,B2: B,C2: B] :
( ( ord_less @ A @ A3 @ ( F @ B2 ) )
=> ( ( ord_less_eq @ B @ B2 @ C2 )
=> ( ! [X3: B,Y5: B] :
( ( ord_less_eq @ B @ X3 @ Y5 )
=> ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less @ A @ A3 @ ( F @ C2 ) ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_580_order__less__le__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A3: A,B2: A,F: A > C,C2: C] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ C @ ( F @ B2 ) @ C2 )
=> ( ! [X3: A,Y5: A] :
( ( ord_less @ A @ X3 @ Y5 )
=> ( ord_less @ C @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less @ C @ ( F @ A3 ) @ C2 ) ) ) ) ) ).
% order_less_le_subst2
thf(fact_581_linorder__le__less__linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y3: A] :
( ( ord_less_eq @ A @ X @ Y3 )
| ( ord_less @ A @ Y3 @ X ) ) ) ).
% linorder_le_less_linear
thf(fact_582_complete__interval,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [A3: A,B2: A,P: A > $o] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( P @ A3 )
=> ( ~ ( P @ B2 )
=> ? [C3: A] :
( ( ord_less_eq @ A @ A3 @ C3 )
& ( ord_less_eq @ A @ C3 @ B2 )
& ! [X4: A] :
( ( ( ord_less_eq @ A @ A3 @ X4 )
& ( ord_less @ A @ X4 @ C3 ) )
=> ( P @ X4 ) )
& ! [D3: A] :
( ! [X3: A] :
( ( ( ord_less_eq @ A @ A3 @ X3 )
& ( ord_less @ A @ X3 @ D3 ) )
=> ( P @ X3 ) )
=> ( ord_less_eq @ A @ D3 @ C3 ) ) ) ) ) ) ) ).
% complete_interval
thf(fact_583_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_584_fact__num__eq__if,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiring_char_0_fact @ A )
= ( ^ [M2: nat] :
( if @ A
@ ( M2
= ( zero_zero @ nat ) )
@ ( one_one @ A )
@ ( times_times @ A @ ( semiring_1_of_nat @ A @ M2 ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ M2 @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% fact_num_eq_if
thf(fact_585_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_586_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_587_one__less__nat__eq,axiom,
! [Z3: int] :
( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( nat2 @ Z3 ) )
= ( ord_less @ int @ ( one_one @ int ) @ Z3 ) ) ).
% one_less_nat_eq
thf(fact_588_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 ) ) )
| ? [Q3: nat] :
( ( ord_less_eq @ nat @ ( times_times @ nat @ N @ Q3 ) @ M )
& ( ord_less @ nat @ M @ ( times_times @ nat @ N @ ( suc @ Q3 ) ) )
& ( P @ Q3 ) ) ) ) ).
% split_div'
thf(fact_589_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_590_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_591_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_592_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_593_minus__apply,axiom,
! [B: $tType,A: $tType] :
( ( minus @ B )
=> ( ( minus_minus @ ( A > B ) )
= ( ^ [A7: A > B,B6: A > B,X2: A] : ( minus_minus @ B @ ( A7 @ X2 ) @ ( B6 @ X2 ) ) ) ) ) ).
% minus_apply
thf(fact_594_mod__mod__trivial,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A3: A,B2: A] :
( ( modulo_modulo @ A @ ( modulo_modulo @ A @ A3 @ B2 ) @ B2 )
= ( modulo_modulo @ A @ A3 @ B2 ) ) ) ).
% mod_mod_trivial
thf(fact_595_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_596_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_597_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_598_frac__frac,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( archimedean_frac @ A @ ( archimedean_frac @ A @ X ) )
= ( archimedean_frac @ A @ X ) ) ) ).
% frac_frac
thf(fact_599_diff__self,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A] :
( ( minus_minus @ A @ A3 @ A3 )
= ( zero_zero @ A ) ) ) ).
% diff_self
thf(fact_600_diff__0__right,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A] :
( ( minus_minus @ A @ A3 @ ( zero_zero @ A ) )
= A3 ) ) ).
% diff_0_right
thf(fact_601_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_602_diff__zero,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [A3: A] :
( ( minus_minus @ A @ A3 @ ( zero_zero @ A ) )
= A3 ) ) ).
% diff_zero
thf(fact_603_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_604_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_605_dvd__0__right,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A3: A] : ( dvd_dvd @ A @ A3 @ ( zero_zero @ A ) ) ) ).
% dvd_0_right
thf(fact_606_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_607_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_608_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_609_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_610_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_611_div__0,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A3: A] :
( ( divide_divide @ A @ ( zero_zero @ A ) @ A3 )
= ( zero_zero @ A ) ) ) ).
% div_0
thf(fact_612_times__divide__eq__right,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( times_times @ A @ A3 @ ( divide_divide @ A @ B2 @ C2 ) )
= ( divide_divide @ A @ ( times_times @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% times_divide_eq_right
thf(fact_613_divide__divide__eq__right,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( divide_divide @ A @ A3 @ ( divide_divide @ A @ B2 @ C2 ) )
= ( divide_divide @ A @ ( times_times @ A @ A3 @ C2 ) @ B2 ) ) ) ).
% divide_divide_eq_right
thf(fact_614_divide__divide__eq__left,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( divide_divide @ A @ ( divide_divide @ A @ A3 @ B2 ) @ C2 )
= ( divide_divide @ A @ A3 @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% divide_divide_eq_left
thf(fact_615_times__divide__eq__left,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [B2: A,C2: A,A3: A] :
( ( times_times @ A @ ( divide_divide @ A @ B2 @ C2 ) @ A3 )
= ( divide_divide @ A @ ( times_times @ A @ B2 @ A3 ) @ C2 ) ) ) ).
% times_divide_eq_left
thf(fact_616_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_617_div__by__1,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A3: A] :
( ( divide_divide @ A @ A3 @ ( one_one @ A ) )
= A3 ) ) ).
% div_by_1
thf(fact_618_mod__self,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [A3: A] :
( ( modulo_modulo @ A @ A3 @ A3 )
= ( zero_zero @ A ) ) ) ).
% mod_self
thf(fact_619_mod__by__0,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [A3: A] :
( ( modulo_modulo @ A @ A3 @ ( zero_zero @ A ) )
= A3 ) ) ).
% mod_by_0
thf(fact_620_mod__0,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [A3: A] :
( ( modulo_modulo @ A @ ( zero_zero @ A ) @ A3 )
= ( zero_zero @ A ) ) ) ).
% mod_0
thf(fact_621_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_622_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_623_div__minus__minus,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A3: A,B2: A] :
( ( divide_divide @ A @ ( uminus_uminus @ A @ A3 ) @ ( uminus_uminus @ A @ B2 ) )
= ( divide_divide @ A @ A3 @ B2 ) ) ) ).
% div_minus_minus
thf(fact_624_dvd__1__left,axiom,
! [K: nat] : ( dvd_dvd @ nat @ ( suc @ ( zero_zero @ nat ) ) @ K ) ).
% dvd_1_left
thf(fact_625_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_626_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_627_minus__mod__self2,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A3: A,B2: A] :
( ( modulo_modulo @ A @ ( minus_minus @ A @ A3 @ B2 ) @ B2 )
= ( modulo_modulo @ A @ A3 @ B2 ) ) ) ).
% minus_mod_self2
thf(fact_628_mod__minus__minus,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A3: A,B2: A] :
( ( modulo_modulo @ A @ ( uminus_uminus @ A @ A3 ) @ ( uminus_uminus @ A @ B2 ) )
= ( uminus_uminus @ A @ ( modulo_modulo @ A @ A3 @ B2 ) ) ) ) ).
% mod_minus_minus
thf(fact_629_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_630_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_631_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus @ nat @ ( zero_zero @ nat ) @ N )
= ( zero_zero @ nat ) ) ).
% diff_0_eq_0
thf(fact_632_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus @ nat @ M @ M )
= ( zero_zero @ nat ) ) ).
% diff_self_eq_0
thf(fact_633_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_634_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_635_sgn__0,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ( ( sgn_sgn @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% sgn_0
thf(fact_636_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_637_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_638_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_639_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_640_sgn__1,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ( ( sgn_sgn @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% sgn_1
thf(fact_641_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_642_sgn__divide,axiom,
! [A: $tType] :
( ( field_abs_sgn @ A )
=> ! [A3: A,B2: A] :
( ( sgn_sgn @ A @ ( divide_divide @ A @ A3 @ B2 ) )
= ( divide_divide @ A @ ( sgn_sgn @ A @ A3 ) @ ( sgn_sgn @ A @ B2 ) ) ) ) ).
% sgn_divide
thf(fact_643_mod__less,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ( ( modulo_modulo @ nat @ M @ N )
= M ) ) ).
% mod_less
thf(fact_644_nat__int,axiom,
! [N: nat] :
( ( nat2 @ ( semiring_1_of_nat @ int @ N ) )
= N ) ).
% nat_int
thf(fact_645_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_646_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_647_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_648_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_649_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_650_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_651_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_652_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_653_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_654_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_655_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_656_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_657_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_658_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_659_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_660_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_661_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_662_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_663_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_664_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_665_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_666_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_667_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_668_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_669_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_670_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_671_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_672_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_673_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_674_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_675_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_676_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_677_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_678_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_679_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_680_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_681_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_682_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_683_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_684_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_685_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_686_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_687_minus__mod__self1,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [B2: A,A3: A] :
( ( modulo_modulo @ A @ ( minus_minus @ A @ B2 @ A3 ) @ B2 )
= ( modulo_modulo @ A @ ( uminus_uminus @ A @ A3 ) @ B2 ) ) ) ).
% minus_mod_self1
thf(fact_688_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_689_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_690_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_691_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_692_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_693_divide__sgn,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( divide_divide @ A @ A3 @ ( sgn_sgn @ A @ B2 ) )
= ( times_times @ A @ A3 @ ( sgn_sgn @ A @ B2 ) ) ) ) ).
% divide_sgn
thf(fact_694_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus @ nat @ ( suc @ N ) @ ( one_one @ nat ) )
= N ) ).
% diff_Suc_1
thf(fact_695_mod__by__Suc__0,axiom,
! [M: nat] :
( ( modulo_modulo @ nat @ M @ ( suc @ ( zero_zero @ nat ) ) )
= ( zero_zero @ nat ) ) ).
% mod_by_Suc_0
thf(fact_696_div__by__Suc__0,axiom,
! [M: nat] :
( ( divide_divide @ nat @ M @ ( suc @ ( zero_zero @ nat ) ) )
= M ) ).
% div_by_Suc_0
thf(fact_697_div__less,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ( ( divide_divide @ nat @ M @ N )
= ( zero_zero @ nat ) ) ) ).
% div_less
thf(fact_698_ceiling__zero,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archimedean_ceiling @ A @ ( zero_zero @ A ) )
= ( zero_zero @ int ) ) ) ).
% ceiling_zero
thf(fact_699_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_700_ceiling__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archimedean_ceiling @ A @ ( one_one @ A ) )
= ( one_one @ int ) ) ) ).
% ceiling_one
thf(fact_701_ceiling__of__nat,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [N: nat] :
( ( archimedean_ceiling @ A @ ( semiring_1_of_nat @ A @ N ) )
= ( semiring_1_of_nat @ int @ N ) ) ) ).
% ceiling_of_nat
thf(fact_702_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_703_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_704_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_705_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_706_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_707_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_708_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_709_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_710_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_711_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_712_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_713_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_714_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_715_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_716_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_717_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_718_nat__1,axiom,
( ( nat2 @ ( one_one @ int ) )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% nat_1
thf(fact_719_nat__le__0,axiom,
! [Z3: int] :
( ( ord_less_eq @ int @ Z3 @ ( zero_zero @ int ) )
=> ( ( nat2 @ Z3 )
= ( zero_zero @ nat ) ) ) ).
% nat_le_0
thf(fact_720_nat__0__iff,axiom,
! [I: int] :
( ( ( nat2 @ I )
= ( zero_zero @ nat ) )
= ( ord_less_eq @ int @ I @ ( zero_zero @ int ) ) ) ).
% nat_0_iff
thf(fact_721_zless__nat__conj,axiom,
! [W2: int,Z3: int] :
( ( ord_less @ nat @ ( nat2 @ W2 ) @ ( nat2 @ Z3 ) )
= ( ( ord_less @ int @ ( zero_zero @ int ) @ Z3 )
& ( ord_less @ int @ W2 @ Z3 ) ) ) ).
% zless_nat_conj
thf(fact_722_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_723_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_724_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_725_int__nat__eq,axiom,
! [Z3: int] :
( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z3 )
=> ( ( semiring_1_of_nat @ int @ ( nat2 @ Z3 ) )
= Z3 ) )
& ( ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z3 )
=> ( ( semiring_1_of_nat @ int @ ( nat2 @ Z3 ) )
= ( zero_zero @ int ) ) ) ) ).
% int_nat_eq
thf(fact_726_nat__zminus__int,axiom,
! [N: nat] :
( ( nat2 @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N ) ) )
= ( zero_zero @ nat ) ) ).
% nat_zminus_int
thf(fact_727_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_728_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_729_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_730_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_731_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_732_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_733_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_734_zero__less__nat__eq,axiom,
! [Z3: int] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( nat2 @ Z3 ) )
= ( ord_less @ int @ ( zero_zero @ int ) @ Z3 ) ) ).
% zero_less_nat_eq
thf(fact_735_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_736_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_737_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_738_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_739_less__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ( ( ord_less @ ( A > B ) )
= ( ^ [F2: A > B,G2: A > B] :
( ( ord_less_eq @ ( A > B ) @ F2 @ G2 )
& ~ ( ord_less_eq @ ( A > B ) @ G2 @ F2 ) ) ) ) ) ).
% less_fun_def
thf(fact_740_fun__diff__def,axiom,
! [B: $tType,A: $tType] :
( ( minus @ B )
=> ( ( minus_minus @ ( A > B ) )
= ( ^ [A7: A > B,B6: A > B,X2: A] : ( minus_minus @ B @ ( A7 @ X2 ) @ ( B6 @ X2 ) ) ) ) ) ).
% fun_diff_def
thf(fact_741_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_742_dvd__eq__mod__eq__0,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ( ( dvd_dvd @ A )
= ( ^ [A5: A,B3: A] :
( ( modulo_modulo @ A @ B3 @ A5 )
= ( zero_zero @ A ) ) ) ) ) ).
% dvd_eq_mod_eq_0
thf(fact_743_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_744_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_745_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_746_dvd__diff,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [X: A,Y3: A,Z3: A] :
( ( dvd_dvd @ A @ X @ Y3 )
=> ( ( dvd_dvd @ A @ X @ Z3 )
=> ( dvd_dvd @ A @ X @ ( minus_minus @ A @ Y3 @ Z3 ) ) ) ) ) ).
% dvd_diff
thf(fact_747_dvd__refl,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: A] : ( dvd_dvd @ A @ A3 @ A3 ) ) ).
% dvd_refl
thf(fact_748_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_749_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_750_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_751_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_752_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_753_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_754_div__div__div__same,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [D2: A,B2: A,A3: A] :
( ( dvd_dvd @ A @ D2 @ B2 )
=> ( ( dvd_dvd @ A @ B2 @ A3 )
=> ( ( divide_divide @ A @ ( divide_divide @ A @ A3 @ D2 ) @ ( divide_divide @ A @ B2 @ D2 ) )
= ( divide_divide @ A @ A3 @ B2 ) ) ) ) ) ).
% div_div_div_same
thf(fact_755_diff__eq__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B2: A,C2: A,D2: A] :
( ( ( minus_minus @ A @ A3 @ B2 )
= ( minus_minus @ A @ C2 @ D2 ) )
=> ( ( A3 = B2 )
= ( C2 = D2 ) ) ) ) ).
% diff_eq_diff_eq
thf(fact_756_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_757_dvd__diff__commute,axiom,
! [A: $tType] :
( ( euclid5891614535332579305n_ring @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ ( minus_minus @ A @ C2 @ B2 ) )
= ( dvd_dvd @ A @ A3 @ ( minus_minus @ A @ B2 @ C2 ) ) ) ) ).
% dvd_diff_commute
thf(fact_758_mod__diff__eq,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( modulo_modulo @ A @ ( minus_minus @ A @ ( modulo_modulo @ A @ A3 @ C2 ) @ ( modulo_modulo @ A @ B2 @ C2 ) ) @ C2 )
= ( modulo_modulo @ A @ ( minus_minus @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% mod_diff_eq
thf(fact_759_dvd__mod,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [K: A,M: A,N: A] :
( ( dvd_dvd @ A @ K @ M )
=> ( ( dvd_dvd @ A @ K @ N )
=> ( dvd_dvd @ A @ K @ ( modulo_modulo @ A @ M @ N ) ) ) ) ) ).
% dvd_mod
thf(fact_760_mod__diff__cong,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A3: A,C2: A,A4: A,B2: A,B4: A] :
( ( ( modulo_modulo @ A @ A3 @ C2 )
= ( modulo_modulo @ A @ A4 @ C2 ) )
=> ( ( ( modulo_modulo @ A @ B2 @ C2 )
= ( modulo_modulo @ A @ B4 @ C2 ) )
=> ( ( modulo_modulo @ A @ ( minus_minus @ A @ A3 @ B2 ) @ C2 )
= ( modulo_modulo @ A @ ( minus_minus @ A @ A4 @ B4 ) @ C2 ) ) ) ) ) ).
% mod_diff_cong
thf(fact_761_mod__eq__dvd__iff,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( ( modulo_modulo @ A @ A3 @ C2 )
= ( modulo_modulo @ A @ B2 @ C2 ) )
= ( dvd_dvd @ A @ C2 @ ( minus_minus @ A @ A3 @ B2 ) ) ) ) ).
% mod_eq_dvd_iff
thf(fact_762_mod__diff__left__eq,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( modulo_modulo @ A @ ( minus_minus @ A @ ( modulo_modulo @ A @ A3 @ C2 ) @ B2 ) @ C2 )
= ( modulo_modulo @ A @ ( minus_minus @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% mod_diff_left_eq
thf(fact_763_mod__diff__right__eq,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( modulo_modulo @ A @ ( minus_minus @ A @ A3 @ ( modulo_modulo @ A @ B2 @ C2 ) ) @ C2 )
= ( modulo_modulo @ A @ ( minus_minus @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% mod_diff_right_eq
thf(fact_764_mod__mod__cancel,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [C2: A,B2: A,A3: A] :
( ( dvd_dvd @ A @ C2 @ B2 )
=> ( ( modulo_modulo @ A @ ( modulo_modulo @ A @ A3 @ B2 ) @ C2 )
= ( modulo_modulo @ A @ A3 @ C2 ) ) ) ) ).
% mod_mod_cancel
thf(fact_765_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_766_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_767_field__char__0__class_Oof__nat__div,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [M: nat,N: nat] :
( ( semiring_1_of_nat @ A @ ( divide_divide @ nat @ M @ N ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ ( modulo_modulo @ nat @ M @ N ) ) ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% field_char_0_class.of_nat_div
thf(fact_768_diff__divide__distrib,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( divide_divide @ A @ ( minus_minus @ A @ A3 @ B2 ) @ C2 )
= ( minus_minus @ A @ ( divide_divide @ A @ A3 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ).
% diff_divide_distrib
thf(fact_769_modulo__nat__def,axiom,
( ( modulo_modulo @ nat )
= ( ^ [M2: nat,N3: nat] : ( minus_minus @ nat @ M2 @ ( times_times @ nat @ ( divide_divide @ nat @ M2 @ N3 ) @ N3 ) ) ) ) ).
% modulo_nat_def
thf(fact_770_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_771_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_772_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_773_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_774_div__mult__div__if__dvd,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A3: A,D2: A,C2: A] :
( ( dvd_dvd @ A @ B2 @ A3 )
=> ( ( dvd_dvd @ A @ D2 @ C2 )
=> ( ( times_times @ A @ ( divide_divide @ A @ A3 @ B2 ) @ ( divide_divide @ A @ C2 @ D2 ) )
= ( divide_divide @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ D2 ) ) ) ) ) ) ).
% div_mult_div_if_dvd
thf(fact_775_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_776_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_777_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_778_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_779_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_780_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_781_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_782_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_783_of__nat__mod,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [M: nat,N: nat] :
( ( semiring_1_of_nat @ A @ ( modulo_modulo @ nat @ M @ N ) )
= ( modulo_modulo @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% of_nat_mod
thf(fact_784_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_785_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_786_mod__if,axiom,
( ( modulo_modulo @ nat )
= ( ^ [M2: nat,N3: nat] : ( if @ nat @ ( ord_less @ nat @ M2 @ N3 ) @ M2 @ ( modulo_modulo @ nat @ ( minus_minus @ nat @ M2 @ N3 ) @ N3 ) ) ) ) ).
% mod_if
thf(fact_787_le__mod__geq,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq @ nat @ N @ M )
=> ( ( modulo_modulo @ nat @ M @ N )
= ( modulo_modulo @ nat @ ( minus_minus @ nat @ M @ N ) @ N ) ) ) ).
% le_mod_geq
thf(fact_788_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_789_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [M: nat,N: nat] :
( ( semiring_1_of_nat @ A @ ( divide_divide @ nat @ M @ N ) )
= ( divide_divide @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% unique_euclidean_semiring_with_nat_class.of_nat_div
thf(fact_790_of__nat__dvd__iff,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [M: nat,N: nat] :
( ( dvd_dvd @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) )
= ( dvd_dvd @ nat @ M @ N ) ) ) ).
% of_nat_dvd_iff
thf(fact_791_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_792_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_793_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_794_mod__nat__eqI,axiom,
! [R: nat,N: nat,M: nat] :
( ( ord_less @ nat @ R @ N )
=> ( ( ord_less_eq @ nat @ R @ M )
=> ( ( dvd_dvd @ nat @ N @ ( minus_minus @ nat @ M @ R ) )
=> ( ( modulo_modulo @ nat @ M @ N )
= R ) ) ) ) ).
% mod_nat_eqI
thf(fact_795_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_796_of__nat__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [R: A] : ( ord_less_eq @ A @ R @ ( semiring_1_of_nat @ A @ ( nat2 @ ( archimedean_ceiling @ A @ R ) ) ) ) ) ).
% of_nat_ceiling
thf(fact_797_add__divide__eq__if__simps_I4_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z3: A,A3: A,B2: A] :
( ( ( Z3
= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ A3 @ ( divide_divide @ A @ B2 @ Z3 ) )
= A3 ) )
& ( ( Z3
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ A3 @ ( divide_divide @ A @ B2 @ Z3 ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ A3 @ Z3 ) @ B2 ) @ Z3 ) ) ) ) ) ).
% add_divide_eq_if_simps(4)
thf(fact_798_diff__frac__eq,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [Y3: A,Z3: A,X: A,W2: A] :
( ( Y3
!= ( zero_zero @ A ) )
=> ( ( Z3
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( divide_divide @ A @ X @ Y3 ) @ ( divide_divide @ A @ W2 @ Z3 ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ X @ Z3 ) @ ( times_times @ A @ W2 @ Y3 ) ) @ ( times_times @ A @ Y3 @ Z3 ) ) ) ) ) ) ).
% diff_frac_eq
thf(fact_799_diff__divide__eq__iff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z3: A,X: A,Y3: A] :
( ( Z3
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ X @ ( divide_divide @ A @ Y3 @ Z3 ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ X @ Z3 ) @ Y3 ) @ Z3 ) ) ) ) ).
% diff_divide_eq_iff
thf(fact_800_divide__diff__eq__iff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z3: A,X: A,Y3: A] :
( ( Z3
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( divide_divide @ A @ X @ Z3 ) @ Y3 )
= ( divide_divide @ A @ ( minus_minus @ A @ X @ ( times_times @ A @ Y3 @ Z3 ) ) @ Z3 ) ) ) ) ).
% divide_diff_eq_iff
thf(fact_801_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_802_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_803_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_804_dvd__div__div__eq__mult,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,C2: A,B2: A,D2: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( C2
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ A3 @ B2 )
=> ( ( dvd_dvd @ A @ C2 @ D2 )
=> ( ( ( divide_divide @ A @ B2 @ A3 )
= ( divide_divide @ A @ D2 @ C2 ) )
= ( ( times_times @ A @ B2 @ C2 )
= ( times_times @ A @ A3 @ D2 ) ) ) ) ) ) ) ) ).
% dvd_div_div_eq_mult
thf(fact_805_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_806_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_807_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_808_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_809_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_810_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_811_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_812_div__less__mono,axiom,
! [A6: nat,B7: nat,N: nat] :
( ( ord_less @ nat @ A6 @ B7 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ( modulo_modulo @ nat @ A6 @ N )
= ( zero_zero @ nat ) )
=> ( ( ( modulo_modulo @ nat @ B7 @ N )
= ( zero_zero @ nat ) )
=> ( ord_less @ nat @ ( divide_divide @ nat @ A6 @ N ) @ ( divide_divide @ nat @ B7 @ N ) ) ) ) ) ) ).
% div_less_mono
thf(fact_813_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_814_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_815_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_816_mod__mult__eq,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( modulo_modulo @ A @ ( times_times @ A @ ( modulo_modulo @ A @ A3 @ C2 ) @ ( modulo_modulo @ A @ B2 @ C2 ) ) @ C2 )
= ( modulo_modulo @ A @ ( times_times @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% mod_mult_eq
thf(fact_817_mod__mult__cong,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A3: A,C2: A,A4: A,B2: A,B4: A] :
( ( ( modulo_modulo @ A @ A3 @ C2 )
= ( modulo_modulo @ A @ A4 @ C2 ) )
=> ( ( ( modulo_modulo @ A @ B2 @ C2 )
= ( modulo_modulo @ A @ B4 @ C2 ) )
=> ( ( modulo_modulo @ A @ ( times_times @ A @ A3 @ B2 ) @ C2 )
= ( modulo_modulo @ A @ ( times_times @ A @ A4 @ B4 ) @ C2 ) ) ) ) ) ).
% mod_mult_cong
thf(fact_818_mod__mult__mult2,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( modulo_modulo @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) )
= ( times_times @ A @ ( modulo_modulo @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% mod_mult_mult2
thf(fact_819_mult__mod__right,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( times_times @ A @ C2 @ ( modulo_modulo @ A @ A3 @ B2 ) )
= ( modulo_modulo @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) ) ) ) ).
% mult_mod_right
thf(fact_820_mod__mult__left__eq,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( modulo_modulo @ A @ ( times_times @ A @ ( modulo_modulo @ A @ A3 @ C2 ) @ B2 ) @ C2 )
= ( modulo_modulo @ A @ ( times_times @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% mod_mult_left_eq
thf(fact_821_mod__mult__right__eq,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( modulo_modulo @ A @ ( times_times @ A @ A3 @ ( modulo_modulo @ A @ B2 @ C2 ) ) @ C2 )
= ( modulo_modulo @ A @ ( times_times @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% mod_mult_right_eq
thf(fact_822_diff__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B2: A,D2: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ D2 @ C2 )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ A3 @ C2 ) @ ( minus_minus @ A @ B2 @ D2 ) ) ) ) ) ).
% diff_mono
thf(fact_823_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_824_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_825_diff__eq__diff__less__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B2: A,C2: A,D2: A] :
( ( ( minus_minus @ A @ A3 @ B2 )
= ( minus_minus @ A @ C2 @ D2 ) )
=> ( ( ord_less_eq @ A @ A3 @ B2 )
= ( ord_less_eq @ A @ C2 @ D2 ) ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_826_mod__minus__eq,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A3: A,B2: A] :
( ( modulo_modulo @ A @ ( uminus_uminus @ A @ ( modulo_modulo @ A @ A3 @ B2 ) ) @ B2 )
= ( modulo_modulo @ A @ ( uminus_uminus @ A @ A3 ) @ B2 ) ) ) ).
% mod_minus_eq
thf(fact_827_mod__minus__cong,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A3: A,B2: A,A4: A] :
( ( ( modulo_modulo @ A @ A3 @ B2 )
= ( modulo_modulo @ A @ A4 @ B2 ) )
=> ( ( modulo_modulo @ A @ ( uminus_uminus @ A @ A3 ) @ B2 )
= ( modulo_modulo @ A @ ( uminus_uminus @ A @ A4 ) @ B2 ) ) ) ) ).
% mod_minus_cong
thf(fact_828_mod__minus__right,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A3: A,B2: A] :
( ( modulo_modulo @ A @ A3 @ ( uminus_uminus @ A @ B2 ) )
= ( uminus_uminus @ A @ ( modulo_modulo @ A @ ( uminus_uminus @ A @ A3 ) @ B2 ) ) ) ) ).
% mod_minus_right
thf(fact_829_eq__iff__diff__eq__0,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ( ( ^ [Y: A,Z2: A] : Y = Z2 )
= ( ^ [A5: A,B3: A] :
( ( minus_minus @ A @ A5 @ B3 )
= ( zero_zero @ A ) ) ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_830_diff__strict__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B2: A,D2: A,C2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ D2 @ C2 )
=> ( ord_less @ A @ ( minus_minus @ A @ A3 @ C2 ) @ ( minus_minus @ A @ B2 @ D2 ) ) ) ) ) ).
% diff_strict_mono
thf(fact_831_diff__eq__diff__less,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B2: A,C2: A,D2: A] :
( ( ( minus_minus @ A @ A3 @ B2 )
= ( minus_minus @ A @ C2 @ D2 ) )
=> ( ( ord_less @ A @ A3 @ B2 )
= ( ord_less @ A @ C2 @ D2 ) ) ) ) ).
% diff_eq_diff_less
thf(fact_832_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_833_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_834_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_835_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_836_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_837_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_838_inf__period_I1_J,axiom,
! [A: $tType] :
( ( ( comm_ring @ A )
& ( dvd @ A ) )
=> ! [P: A > $o,D4: A,Q: A > $o] :
( ! [X3: A,K2: A] :
( ( P @ X3 )
= ( P @ ( minus_minus @ A @ X3 @ ( times_times @ A @ K2 @ D4 ) ) ) )
=> ( ! [X3: A,K2: A] :
( ( Q @ X3 )
= ( Q @ ( minus_minus @ A @ X3 @ ( times_times @ A @ K2 @ D4 ) ) ) )
=> ! [X4: A,K4: A] :
( ( ( P @ X4 )
& ( Q @ X4 ) )
= ( ( P @ ( minus_minus @ A @ X4 @ ( times_times @ A @ K4 @ D4 ) ) )
& ( Q @ ( minus_minus @ A @ X4 @ ( times_times @ A @ K4 @ D4 ) ) ) ) ) ) ) ) ).
% inf_period(1)
thf(fact_839_inf__period_I2_J,axiom,
! [A: $tType] :
( ( ( comm_ring @ A )
& ( dvd @ A ) )
=> ! [P: A > $o,D4: A,Q: A > $o] :
( ! [X3: A,K2: A] :
( ( P @ X3 )
= ( P @ ( minus_minus @ A @ X3 @ ( times_times @ A @ K2 @ D4 ) ) ) )
=> ( ! [X3: A,K2: A] :
( ( Q @ X3 )
= ( Q @ ( minus_minus @ A @ X3 @ ( times_times @ A @ K2 @ D4 ) ) ) )
=> ! [X4: A,K4: A] :
( ( ( P @ X4 )
| ( Q @ X4 ) )
= ( ( P @ ( minus_minus @ A @ X4 @ ( times_times @ A @ K4 @ D4 ) ) )
| ( Q @ ( minus_minus @ A @ X4 @ ( times_times @ A @ K4 @ D4 ) ) ) ) ) ) ) ) ).
% inf_period(2)
thf(fact_840_dvd__field__iff,axiom,
! [A: $tType] :
( ( field @ A )
=> ( ( dvd_dvd @ A )
= ( ^ [A5: A,B3: A] :
( ( A5
= ( zero_zero @ A ) )
=> ( B3
= ( zero_zero @ A ) ) ) ) ) ) ).
% dvd_field_iff
thf(fact_841_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_842_mod__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( modulo_modulo @ nat @ ( suc @ ( modulo_modulo @ nat @ M @ N ) ) @ N )
= ( modulo_modulo @ nat @ ( suc @ M ) @ N ) ) ).
% mod_Suc_eq
thf(fact_843_mod__Suc__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( modulo_modulo @ nat @ ( suc @ ( suc @ ( modulo_modulo @ nat @ M @ N ) ) ) @ N )
= ( modulo_modulo @ nat @ ( suc @ ( suc @ M ) ) @ N ) ) ).
% mod_Suc_Suc_eq
thf(fact_844_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_845_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_846_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_847_mult__dvd__mono,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: A,B2: A,C2: A,D2: A] :
( ( dvd_dvd @ A @ A3 @ B2 )
=> ( ( dvd_dvd @ A @ C2 @ D2 )
=> ( dvd_dvd @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ D2 ) ) ) ) ) ).
% mult_dvd_mono
thf(fact_848_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_849_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_850_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_851_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_852_dvd__def,axiom,
! [A: $tType] :
( ( dvd @ A )
=> ( ( dvd_dvd @ A )
= ( ^ [B3: A,A5: A] :
? [K3: A] :
( A5
= ( times_times @ A @ B3 @ K3 ) ) ) ) ) ).
% dvd_def
thf(fact_853_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_854_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_855_one__dvd,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: A] : ( dvd_dvd @ A @ ( one_one @ A ) @ A3 ) ) ).
% one_dvd
thf(fact_856_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_857_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_858_divide__divide__eq__left_H,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( divide_divide @ A @ ( divide_divide @ A @ A3 @ B2 ) @ C2 )
= ( divide_divide @ A @ A3 @ ( times_times @ A @ C2 @ B2 ) ) ) ) ).
% divide_divide_eq_left'
thf(fact_859_divide__divide__times__eq,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [X: A,Y3: A,Z3: A,W2: A] :
( ( divide_divide @ A @ ( divide_divide @ A @ X @ Y3 ) @ ( divide_divide @ A @ Z3 @ W2 ) )
= ( divide_divide @ A @ ( times_times @ A @ X @ W2 ) @ ( times_times @ A @ Y3 @ Z3 ) ) ) ) ).
% divide_divide_times_eq
thf(fact_860_times__divide__times__eq,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [X: A,Y3: A,Z3: A,W2: A] :
( ( times_times @ A @ ( divide_divide @ A @ X @ Y3 ) @ ( divide_divide @ A @ Z3 @ W2 ) )
= ( divide_divide @ A @ ( times_times @ A @ X @ Z3 ) @ ( times_times @ A @ Y3 @ W2 ) ) ) ) ).
% times_divide_times_eq
thf(fact_861_mod__less__eq__dividend,axiom,
! [M: nat,N: nat] : ( ord_less_eq @ nat @ ( modulo_modulo @ nat @ M @ N ) @ M ) ).
% mod_less_eq_dividend
thf(fact_862_minus__divide__left,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,B2: A] :
( ( uminus_uminus @ A @ ( divide_divide @ A @ A3 @ B2 ) )
= ( divide_divide @ A @ ( uminus_uminus @ A @ A3 ) @ B2 ) ) ) ).
% minus_divide_left
thf(fact_863_minus__divide__divide,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: A,B2: A] :
( ( divide_divide @ A @ ( uminus_uminus @ A @ A3 ) @ ( uminus_uminus @ A @ B2 ) )
= ( divide_divide @ A @ A3 @ B2 ) ) ) ).
% minus_divide_divide
thf(fact_864_minus__divide__right,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: A,B2: A] :
( ( uminus_uminus @ A @ ( divide_divide @ A @ A3 @ B2 ) )
= ( divide_divide @ A @ A3 @ ( uminus_uminus @ A @ B2 ) ) ) ) ).
% minus_divide_right
thf(fact_865_div__minus__right,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A3: A,B2: A] :
( ( divide_divide @ A @ A3 @ ( uminus_uminus @ A @ B2 ) )
= ( divide_divide @ A @ ( uminus_uminus @ A @ A3 ) @ B2 ) ) ) ).
% div_minus_right
thf(fact_866_zero__induct__lemma,axiom,
! [P: nat > $o,K: nat,I: nat] :
( ( P @ K )
=> ( ! [N4: nat] :
( ( P @ ( suc @ N4 ) )
=> ( P @ N4 ) )
=> ( P @ ( minus_minus @ nat @ K @ I ) ) ) ) ).
% zero_induct_lemma
thf(fact_867_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus @ nat @ M @ ( zero_zero @ nat ) )
= M ) ).
% minus_nat.diff_0
thf(fact_868_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_869_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_870_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_871_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_872_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_873_div__le__dividend,axiom,
! [M: nat,N: nat] : ( ord_less_eq @ nat @ ( divide_divide @ nat @ M @ N ) @ M ) ).
% div_le_dividend
thf(fact_874_div__le__mono,axiom,
! [M: nat,N: nat,K: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ord_less_eq @ nat @ ( divide_divide @ nat @ M @ K ) @ ( divide_divide @ nat @ N @ K ) ) ) ).
% div_le_mono
thf(fact_875_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_876_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_877_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_878_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_879_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_880_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_881_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_882_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_883_div__mult2__eq,axiom,
! [M: nat,N: nat,Q4: nat] :
( ( divide_divide @ nat @ M @ ( times_times @ nat @ N @ Q4 ) )
= ( divide_divide @ nat @ ( divide_divide @ nat @ M @ N ) @ Q4 ) ) ).
% div_mult2_eq
thf(fact_884_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_885_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_886_uminus__dvd__conv_I2_J,axiom,
( ( dvd_dvd @ int )
= ( ^ [D5: int,T3: int] : ( dvd_dvd @ int @ D5 @ ( uminus_uminus @ int @ T3 ) ) ) ) ).
% uminus_dvd_conv(2)
thf(fact_887_uminus__dvd__conv_I1_J,axiom,
( ( dvd_dvd @ int )
= ( ^ [D5: int] : ( dvd_dvd @ int @ ( uminus_uminus @ int @ D5 ) ) ) ) ).
% uminus_dvd_conv(1)
thf(fact_888_nat__dvd__iff,axiom,
! [Z3: int,M: nat] :
( ( dvd_dvd @ nat @ ( nat2 @ Z3 ) @ M )
= ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z3 )
=> ( dvd_dvd @ int @ Z3 @ ( semiring_1_of_nat @ int @ M ) ) )
& ( ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z3 )
=> ( M
= ( zero_zero @ nat ) ) ) ) ) ).
% nat_dvd_iff
thf(fact_889_frac__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y3: A,Z3: A,X: A,W2: A] :
( ( Y3
!= ( zero_zero @ A ) )
=> ( ( Z3
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Y3 ) @ ( divide_divide @ A @ W2 @ Z3 ) )
= ( ord_less_eq @ A @ ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ X @ Z3 ) @ ( times_times @ A @ W2 @ Y3 ) ) @ ( times_times @ A @ Y3 @ Z3 ) ) @ ( zero_zero @ A ) ) ) ) ) ) ).
% frac_le_eq
thf(fact_890_frac__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y3: A,Z3: A,X: A,W2: A] :
( ( Y3
!= ( zero_zero @ A ) )
=> ( ( Z3
!= ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( divide_divide @ A @ X @ Y3 ) @ ( divide_divide @ A @ W2 @ Z3 ) )
= ( ord_less @ A @ ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ X @ Z3 ) @ ( times_times @ A @ W2 @ Y3 ) ) @ ( times_times @ A @ Y3 @ Z3 ) ) @ ( zero_zero @ A ) ) ) ) ) ) ).
% frac_less_eq
thf(fact_891_add__divide__eq__if__simps_I6_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z3: A,A3: A,B2: A] :
( ( ( Z3
= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ A3 @ Z3 ) ) @ B2 )
= ( uminus_uminus @ A @ B2 ) ) )
& ( ( Z3
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ A3 @ Z3 ) ) @ B2 )
= ( divide_divide @ A @ ( minus_minus @ A @ ( uminus_uminus @ A @ A3 ) @ ( times_times @ A @ B2 @ Z3 ) ) @ Z3 ) ) ) ) ) ).
% add_divide_eq_if_simps(6)
thf(fact_892_add__divide__eq__if__simps_I5_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z3: A,A3: A,B2: A] :
( ( ( Z3
= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( divide_divide @ A @ A3 @ Z3 ) @ B2 )
= ( uminus_uminus @ A @ B2 ) ) )
& ( ( Z3
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( divide_divide @ A @ A3 @ Z3 ) @ B2 )
= ( divide_divide @ A @ ( minus_minus @ A @ A3 @ ( times_times @ A @ B2 @ Z3 ) ) @ Z3 ) ) ) ) ) ).
% add_divide_eq_if_simps(5)
thf(fact_893_minus__divide__diff__eq__iff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z3: A,X: A,Y3: A] :
( ( Z3
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ X @ Z3 ) ) @ Y3 )
= ( divide_divide @ A @ ( minus_minus @ A @ ( uminus_uminus @ A @ X ) @ ( times_times @ A @ Y3 @ Z3 ) ) @ Z3 ) ) ) ) ).
% minus_divide_diff_eq_iff
thf(fact_894_is__unitE,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,C2: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ~ ( ( A3
!= ( zero_zero @ A ) )
=> ! [B5: A] :
( ( B5
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ B5 @ ( one_one @ A ) )
=> ( ( ( divide_divide @ A @ ( one_one @ A ) @ A3 )
= B5 )
=> ( ( ( divide_divide @ A @ ( one_one @ A ) @ B5 )
= A3 )
=> ( ( ( times_times @ A @ A3 @ B5 )
= ( one_one @ A ) )
=> ( ( divide_divide @ A @ C2 @ A3 )
!= ( times_times @ A @ C2 @ B5 ) ) ) ) ) ) ) ) ) ) ).
% is_unitE
thf(fact_895_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_896_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_897_div__if,axiom,
( ( divide_divide @ nat )
= ( ^ [M2: nat,N3: nat] :
( if @ nat
@ ( ( ord_less @ nat @ M2 @ N3 )
| ( N3
= ( zero_zero @ nat ) ) )
@ ( zero_zero @ nat )
@ ( suc @ ( divide_divide @ nat @ ( minus_minus @ nat @ M2 @ N3 ) @ N3 ) ) ) ) ) ).
% div_if
thf(fact_898_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_899_nat__zero__as__int,axiom,
( ( zero_zero @ nat )
= ( nat2 @ ( zero_zero @ int ) ) ) ).
% nat_zero_as_int
thf(fact_900_le__iff__diff__le__0,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A5: A,B3: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ A5 @ B3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% le_iff_diff_le_0
thf(fact_901_less__iff__diff__less__0,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ( ( ord_less @ A )
= ( ^ [A5: A,B3: A] : ( ord_less @ A @ ( minus_minus @ A @ A5 @ B3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% less_iff_diff_less_0
thf(fact_902_int__sgnE,axiom,
! [K: int] :
~ ! [N4: nat,L2: int] :
( K
!= ( times_times @ int @ ( sgn_sgn @ int @ L2 ) @ ( semiring_1_of_nat @ int @ N4 ) ) ) ).
% int_sgnE
thf(fact_903_ex__nat,axiom,
( ( ^ [P3: nat > $o] :
? [X6: nat] : ( P3 @ X6 ) )
= ( ^ [P4: nat > $o] :
? [X2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X2 )
& ( P4 @ ( nat2 @ X2 ) ) ) ) ) ).
% ex_nat
thf(fact_904_all__nat,axiom,
( ( ^ [P3: nat > $o] :
! [X6: nat] : ( P3 @ X6 ) )
= ( ^ [P4: nat > $o] :
! [X2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X2 )
=> ( P4 @ ( nat2 @ X2 ) ) ) ) ) ).
% all_nat
thf(fact_905_eq__nat__nat__iff,axiom,
! [Z3: int,Z6: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z3 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z6 )
=> ( ( ( nat2 @ Z3 )
= ( nat2 @ Z6 ) )
= ( Z3 = Z6 ) ) ) ) ).
% eq_nat_nat_iff
thf(fact_906_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_907_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_908_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_909_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_910_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_911_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_912_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_913_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_914_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_915_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_916_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_917_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_918_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_919_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_920_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_921_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_922_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_923_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_924_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_925_frac__eq__eq,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [Y3: A,Z3: A,X: A,W2: A] :
( ( Y3
!= ( zero_zero @ A ) )
=> ( ( Z3
!= ( zero_zero @ A ) )
=> ( ( ( divide_divide @ A @ X @ Y3 )
= ( divide_divide @ A @ W2 @ Z3 ) )
= ( ( times_times @ A @ X @ Z3 )
= ( times_times @ A @ W2 @ Y3 ) ) ) ) ) ) ).
% frac_eq_eq
thf(fact_926_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_927_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_928_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_929_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_930_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_931_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_932_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_933_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_934_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_935_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_936_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_937_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_938_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_939_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_940_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_941_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_942_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_943_mod__induct,axiom,
! [P: nat > $o,N: nat,P5: nat,M: nat] :
( ( P @ N )
=> ( ( ord_less @ nat @ N @ P5 )
=> ( ( ord_less @ nat @ M @ P5 )
=> ( ! [N4: nat] :
( ( ord_less @ nat @ N4 @ P5 )
=> ( ( P @ N4 )
=> ( P @ ( modulo_modulo @ nat @ ( suc @ N4 ) @ P5 ) ) ) )
=> ( P @ M ) ) ) ) ) ).
% mod_induct
thf(fact_944_nat__one__as__int,axiom,
( ( one_one @ nat )
= ( nat2 @ ( one_one @ int ) ) ) ).
% nat_one_as_int
thf(fact_945_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_946_mod__Suc__le__divisor,axiom,
! [M: nat,N: nat] : ( ord_less_eq @ nat @ ( modulo_modulo @ nat @ M @ ( suc @ N ) ) @ N ) ).
% mod_Suc_le_divisor
thf(fact_947_ceiling__mono,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Y3: A,X: A] :
( ( ord_less_eq @ A @ Y3 @ X )
=> ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ Y3 ) @ ( archimedean_ceiling @ A @ X ) ) ) ) ).
% ceiling_mono
thf(fact_948_ceiling__less__cancel,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Y3: A] :
( ( ord_less @ int @ ( archimedean_ceiling @ A @ X ) @ ( archimedean_ceiling @ A @ Y3 ) )
=> ( ord_less @ A @ X @ Y3 ) ) ) ).
% ceiling_less_cancel
thf(fact_949_div__mult2__eq_H,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A3: A,M: nat,N: nat] :
( ( divide_divide @ A @ A3 @ ( times_times @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) ) )
= ( divide_divide @ A @ ( divide_divide @ A @ A3 @ ( semiring_1_of_nat @ A @ M ) ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% div_mult2_eq'
thf(fact_950_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_951_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_952_diff__less__Suc,axiom,
! [M: nat,N: nat] : ( ord_less @ nat @ ( minus_minus @ nat @ M @ N ) @ ( suc @ M ) ) ).
% diff_less_Suc
thf(fact_953_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_954_Suc__div__le__mono,axiom,
! [M: nat,N: nat] : ( ord_less_eq @ nat @ ( divide_divide @ nat @ M @ N ) @ ( divide_divide @ nat @ ( suc @ M ) @ N ) ) ).
% Suc_div_le_mono
thf(fact_955_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_956_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_957_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_958_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_959_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_960_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_961_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_962_less__mult__imp__div__less,axiom,
! [M: nat,I: nat,N: nat] :
( ( ord_less @ nat @ M @ ( times_times @ nat @ I @ N ) )
=> ( ord_less @ nat @ ( divide_divide @ nat @ M @ N ) @ I ) ) ).
% less_mult_imp_div_less
thf(fact_963_times__div__less__eq__dividend,axiom,
! [N: nat,M: nat] : ( ord_less_eq @ nat @ ( times_times @ nat @ N @ ( divide_divide @ nat @ M @ N ) ) @ M ) ).
% times_div_less_eq_dividend
thf(fact_964_div__times__less__eq__dividend,axiom,
! [M: nat,N: nat] : ( ord_less_eq @ nat @ ( times_times @ nat @ ( divide_divide @ nat @ M @ N ) @ N ) @ M ) ).
% div_times_less_eq_dividend
thf(fact_965_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_966_zdvd__mono,axiom,
! [K: int,M: int,T2: int] :
( ( K
!= ( zero_zero @ int ) )
=> ( ( dvd_dvd @ int @ M @ T2 )
= ( dvd_dvd @ int @ ( times_times @ int @ K @ M ) @ ( times_times @ int @ K @ T2 ) ) ) ) ).
% zdvd_mono
thf(fact_967_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_968_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_969_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_970_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_971_fact__dvd,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N: nat,M: nat] :
( ( ord_less_eq @ nat @ N @ M )
=> ( dvd_dvd @ A @ ( semiring_char_0_fact @ A @ N ) @ ( semiring_char_0_fact @ A @ M ) ) ) ) ).
% fact_dvd
thf(fact_972_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_973_division__segment__eq__sgn,axiom,
! [K: int] :
( ( K
!= ( zero_zero @ int ) )
=> ( ( euclid7384307370059645450egment @ int @ K )
= ( sgn_sgn @ int @ K ) ) ) ).
% division_segment_eq_sgn
thf(fact_974_ceiling__ge__round,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] : ( ord_less_eq @ int @ ( archimedean_round @ A @ X ) @ ( archimedean_ceiling @ A @ X ) ) ) ).
% ceiling_ge_round
thf(fact_975_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_976_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_977_nat__mono__iff,axiom,
! [Z3: int,W2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ Z3 )
=> ( ( ord_less @ nat @ ( nat2 @ W2 ) @ ( nat2 @ Z3 ) )
= ( ord_less @ int @ W2 @ Z3 ) ) ) ).
% nat_mono_iff
thf(fact_978_frac__le,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y3: A,X: A,W2: A,Z3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y3 )
=> ( ( ord_less_eq @ A @ X @ Y3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ W2 )
=> ( ( ord_less_eq @ A @ W2 @ Z3 )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Z3 ) @ ( divide_divide @ A @ Y3 @ W2 ) ) ) ) ) ) ) ).
% frac_le
thf(fact_979_frac__less,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y3: A,W2: A,Z3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less @ A @ X @ Y3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ W2 )
=> ( ( ord_less_eq @ A @ W2 @ Z3 )
=> ( ord_less @ A @ ( divide_divide @ A @ X @ Z3 ) @ ( divide_divide @ A @ Y3 @ W2 ) ) ) ) ) ) ) ).
% frac_less
thf(fact_980_frac__less2,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y3: A,W2: A,Z3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less_eq @ A @ X @ Y3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ W2 )
=> ( ( ord_less @ A @ W2 @ Z3 )
=> ( ord_less @ A @ ( divide_divide @ A @ X @ Z3 ) @ ( divide_divide @ A @ Y3 @ W2 ) ) ) ) ) ) ) ).
% frac_less2
thf(fact_981_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_982_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_983_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_984_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_985_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_986_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_987_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_988_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_989_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_990_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_991_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_992_mult__imp__div__pos__less,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y3: A,X: A,Z3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Y3 )
=> ( ( ord_less @ A @ X @ ( times_times @ A @ Z3 @ Y3 ) )
=> ( ord_less @ A @ ( divide_divide @ A @ X @ Y3 ) @ Z3 ) ) ) ) ).
% mult_imp_div_pos_less
thf(fact_993_mult__imp__less__div__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y3: A,Z3: A,X: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Y3 )
=> ( ( ord_less @ A @ ( times_times @ A @ Z3 @ Y3 ) @ X )
=> ( ord_less @ A @ Z3 @ ( divide_divide @ A @ X @ Y3 ) ) ) ) ) ).
% mult_imp_less_div_pos
thf(fact_994_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_995_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_996_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_997_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_998_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_999_zless__nat__eq__int__zless,axiom,
! [M: nat,Z3: int] :
( ( ord_less @ nat @ M @ ( nat2 @ Z3 ) )
= ( ord_less @ int @ ( semiring_1_of_nat @ int @ M ) @ Z3 ) ) ).
% zless_nat_eq_int_zless
thf(fact_1000_int__eq__iff,axiom,
! [M: nat,Z3: int] :
( ( ( semiring_1_of_nat @ int @ M )
= Z3 )
= ( ( M
= ( nat2 @ Z3 ) )
& ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z3 ) ) ) ).
% int_eq_iff
thf(fact_1001_nat__0__le,axiom,
! [Z3: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z3 )
=> ( ( semiring_1_of_nat @ int @ ( nat2 @ Z3 ) )
= Z3 ) ) ).
% nat_0_le
thf(fact_1002_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_1003_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_1004_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_1005_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_1006_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_1007_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_1008_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_1009_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_1010_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_1011_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_1012_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_1013_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_1014_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_1015_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_1016_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_1017_div__less__iff__less__mult,axiom,
! [Q4: nat,M: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ Q4 )
=> ( ( ord_less @ nat @ ( divide_divide @ nat @ M @ Q4 ) @ N )
= ( ord_less @ nat @ M @ ( times_times @ nat @ N @ Q4 ) ) ) ) ).
% div_less_iff_less_mult
thf(fact_1018_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_1019_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_1020_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_1021_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_1022_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_1023_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_1024_zdvd__imp__le,axiom,
! [Z3: int,N: int] :
( ( dvd_dvd @ int @ Z3 @ N )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ N )
=> ( ord_less_eq @ int @ Z3 @ N ) ) ) ).
% zdvd_imp_le
thf(fact_1025_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_1026_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_1027_ex__gt__or__lt,axiom,
! [A: $tType] :
( ( condit5016429287641298734tinuum @ A )
=> ! [A3: A] :
? [B5: A] :
( ( ord_less @ A @ A3 @ B5 )
| ( ord_less @ A @ B5 @ A3 ) ) ) ).
% ex_gt_or_lt
thf(fact_1028_of__nat__euclidean__size,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A3: A] :
( ( semiring_1_of_nat @ A @ ( euclid6346220572633701492n_size @ A @ A3 ) )
= ( divide_divide @ A @ A3 @ ( euclid7384307370059645450egment @ A @ A3 ) ) ) ) ).
% of_nat_euclidean_size
thf(fact_1029_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_1030_zsgn__def,axiom,
( ( sgn_sgn @ int )
= ( ^ [I5: int] :
( if @ int
@ ( I5
= ( zero_zero @ int ) )
@ ( zero_zero @ int )
@ ( if @ int @ ( ord_less @ int @ ( zero_zero @ int ) @ I5 ) @ ( one_one @ int ) @ ( uminus_uminus @ int @ ( one_one @ int ) ) ) ) ) ) ).
% zsgn_def
thf(fact_1031_nat__less__eq__zless,axiom,
! [W2: int,Z3: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ W2 )
=> ( ( ord_less @ nat @ ( nat2 @ W2 ) @ ( nat2 @ Z3 ) )
= ( ord_less @ int @ W2 @ Z3 ) ) ) ).
% nat_less_eq_zless
thf(fact_1032_nat__le__eq__zle,axiom,
! [W2: int,Z3: int] :
( ( ( ord_less @ int @ ( zero_zero @ int ) @ W2 )
| ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z3 ) )
=> ( ( ord_less_eq @ nat @ ( nat2 @ W2 ) @ ( nat2 @ Z3 ) )
= ( ord_less_eq @ int @ W2 @ Z3 ) ) ) ).
% nat_le_eq_zle
thf(fact_1033_nat__eq__iff,axiom,
! [W2: int,M: nat] :
( ( ( nat2 @ W2 )
= M )
= ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ W2 )
=> ( W2
= ( semiring_1_of_nat @ int @ M ) ) )
& ( ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ W2 )
=> ( M
= ( zero_zero @ nat ) ) ) ) ) ).
% nat_eq_iff
thf(fact_1034_nat__eq__iff2,axiom,
! [M: nat,W2: int] :
( ( M
= ( nat2 @ W2 ) )
= ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ W2 )
=> ( W2
= ( semiring_1_of_nat @ int @ M ) ) )
& ( ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ W2 )
=> ( M
= ( zero_zero @ nat ) ) ) ) ) ).
% nat_eq_iff2
thf(fact_1035_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_1036_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_1037_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_1038_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_1039_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_1040_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_1041_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_1042_mult__imp__div__pos__le,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y3: A,X: A,Z3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Y3 )
=> ( ( ord_less_eq @ A @ X @ ( times_times @ A @ Z3 @ Y3 ) )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Y3 ) @ Z3 ) ) ) ) ).
% mult_imp_div_pos_le
thf(fact_1043_mult__imp__le__div__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y3: A,Z3: A,X: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Y3 )
=> ( ( ord_less_eq @ A @ ( times_times @ A @ Z3 @ Y3 ) @ X )
=> ( ord_less_eq @ A @ Z3 @ ( divide_divide @ A @ X @ Y3 ) ) ) ) ) ).
% mult_imp_le_div_pos
thf(fact_1044_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_1045_split__nat,axiom,
! [P: nat > $o,I: int] :
( ( P @ ( nat2 @ I ) )
= ( ! [N3: nat] :
( ( I
= ( semiring_1_of_nat @ int @ N3 ) )
=> ( P @ N3 ) )
& ( ( ord_less @ int @ I @ ( zero_zero @ int ) )
=> ( P @ ( zero_zero @ nat ) ) ) ) ) ).
% split_nat
thf(fact_1046_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_1047_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_1048_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_1049_nat__mult__distrib,axiom,
! [Z3: int,Z6: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z3 )
=> ( ( nat2 @ ( times_times @ int @ Z3 @ Z6 ) )
= ( times_times @ nat @ ( nat2 @ Z3 ) @ ( nat2 @ Z6 ) ) ) ) ).
% nat_mult_distrib
thf(fact_1050_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_1051_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_1052_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_1053_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_1054_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_1055_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_1056_div__nat__eqI,axiom,
! [N: nat,Q4: nat,M: nat] :
( ( ord_less_eq @ nat @ ( times_times @ nat @ N @ Q4 ) @ M )
=> ( ( ord_less @ nat @ M @ ( times_times @ nat @ N @ ( suc @ Q4 ) ) )
=> ( ( divide_divide @ nat @ M @ N )
= Q4 ) ) ) ).
% div_nat_eqI
thf(fact_1057_less__eq__div__iff__mult__less__eq,axiom,
! [Q4: nat,M: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ Q4 )
=> ( ( ord_less_eq @ nat @ M @ ( divide_divide @ nat @ N @ Q4 ) )
= ( ord_less_eq @ nat @ ( times_times @ nat @ M @ Q4 ) @ N ) ) ) ).
% less_eq_div_iff_mult_less_eq
thf(fact_1058_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_1059_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_1060_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_1061_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_1062_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_1063_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_1064_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_1065_nat__less__iff,axiom,
! [W2: int,M: nat] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ W2 )
=> ( ( ord_less @ nat @ ( nat2 @ W2 ) @ M )
= ( ord_less @ int @ W2 @ ( semiring_1_of_nat @ int @ M ) ) ) ) ).
% nat_less_iff
thf(fact_1066_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_1067_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_1068_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_1069_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_1070_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_1071_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_1072_nat__mult__distrib__neg,axiom,
! [Z3: int,Z6: int] :
( ( ord_less_eq @ int @ Z3 @ ( zero_zero @ int ) )
=> ( ( nat2 @ ( times_times @ int @ Z3 @ Z6 ) )
= ( times_times @ nat @ ( nat2 @ ( uminus_uminus @ int @ Z3 ) ) @ ( nat2 @ ( uminus_uminus @ int @ Z6 ) ) ) ) ) ).
% nat_mult_distrib_neg
thf(fact_1073_nat__approx__posE,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [E: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ E )
=> ~ ! [N4: nat] :
~ ( ord_less @ A @ ( divide_divide @ A @ ( one_one @ A ) @ ( semiring_1_of_nat @ A @ ( suc @ N4 ) ) ) @ E ) ) ) ).
% nat_approx_posE
thf(fact_1074_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_1075_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_1076_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_1077_choose__dvd,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [K: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ N )
=> ( dvd_dvd @ A @ ( times_times @ A @ ( semiring_char_0_fact @ A @ K ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ N @ K ) ) ) @ ( semiring_char_0_fact @ A @ N ) ) ) ) ).
% choose_dvd
thf(fact_1078_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_1079_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_1080_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_1081_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_1082_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_1083_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_1084_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_1085_mod__neg__neg__trivial,axiom,
! [K: int,L: int] :
( ( ord_less_eq @ int @ K @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ L @ K )
=> ( ( modulo_modulo @ int @ K @ L )
= K ) ) ) ).
% mod_neg_neg_trivial
thf(fact_1086_mod__pos__pos__trivial,axiom,
! [K: int,L: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
=> ( ( ord_less @ int @ K @ L )
=> ( ( modulo_modulo @ int @ K @ L )
= K ) ) ) ).
% mod_pos_pos_trivial
thf(fact_1087_div__neg__neg__trivial,axiom,
! [K: int,L: int] :
( ( ord_less_eq @ int @ K @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ L @ K )
=> ( ( divide_divide @ int @ K @ L )
= ( zero_zero @ int ) ) ) ) ).
% div_neg_neg_trivial
thf(fact_1088_div__pos__pos__trivial,axiom,
! [K: int,L: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
=> ( ( ord_less @ int @ K @ L )
=> ( ( divide_divide @ int @ K @ L )
= ( zero_zero @ int ) ) ) ) ).
% div_pos_pos_trivial
thf(fact_1089_zle__diff1__eq,axiom,
! [W2: int,Z3: int] :
( ( ord_less_eq @ int @ W2 @ ( minus_minus @ int @ Z3 @ ( one_one @ int ) ) )
= ( ord_less @ int @ W2 @ Z3 ) ) ).
% zle_diff1_eq
thf(fact_1090_dvd__antisym,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd @ nat @ M @ N )
=> ( ( dvd_dvd @ nat @ N @ M )
=> ( M = N ) ) ) ).
% dvd_antisym
thf(fact_1091_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_1092_zmod__zminus1__eq__if,axiom,
! [A3: int,B2: int] :
( ( ( ( modulo_modulo @ int @ A3 @ B2 )
= ( zero_zero @ int ) )
=> ( ( modulo_modulo @ int @ ( uminus_uminus @ int @ A3 ) @ B2 )
= ( zero_zero @ int ) ) )
& ( ( ( modulo_modulo @ int @ A3 @ B2 )
!= ( zero_zero @ int ) )
=> ( ( modulo_modulo @ int @ ( uminus_uminus @ int @ A3 ) @ B2 )
= ( minus_minus @ int @ B2 @ ( modulo_modulo @ int @ A3 @ B2 ) ) ) ) ) ).
% zmod_zminus1_eq_if
thf(fact_1093_zmod__zminus2__eq__if,axiom,
! [A3: int,B2: int] :
( ( ( ( modulo_modulo @ int @ A3 @ B2 )
= ( zero_zero @ int ) )
=> ( ( modulo_modulo @ int @ A3 @ ( uminus_uminus @ int @ B2 ) )
= ( zero_zero @ int ) ) )
& ( ( ( modulo_modulo @ int @ A3 @ B2 )
!= ( zero_zero @ int ) )
=> ( ( modulo_modulo @ int @ A3 @ ( uminus_uminus @ int @ B2 ) )
= ( minus_minus @ int @ ( modulo_modulo @ int @ A3 @ B2 ) @ B2 ) ) ) ) ).
% zmod_zminus2_eq_if
thf(fact_1094_zdiv__zminus1__eq__if,axiom,
! [B2: int,A3: int] :
( ( B2
!= ( zero_zero @ int ) )
=> ( ( ( ( modulo_modulo @ int @ A3 @ B2 )
= ( zero_zero @ int ) )
=> ( ( divide_divide @ int @ ( uminus_uminus @ int @ A3 ) @ B2 )
= ( uminus_uminus @ int @ ( divide_divide @ int @ A3 @ B2 ) ) ) )
& ( ( ( modulo_modulo @ int @ A3 @ B2 )
!= ( zero_zero @ int ) )
=> ( ( divide_divide @ int @ ( uminus_uminus @ int @ A3 ) @ B2 )
= ( minus_minus @ int @ ( uminus_uminus @ int @ ( divide_divide @ int @ A3 @ B2 ) ) @ ( one_one @ int ) ) ) ) ) ) ).
% zdiv_zminus1_eq_if
thf(fact_1095_zdiv__zminus2__eq__if,axiom,
! [B2: int,A3: int] :
( ( B2
!= ( zero_zero @ int ) )
=> ( ( ( ( modulo_modulo @ int @ A3 @ B2 )
= ( zero_zero @ int ) )
=> ( ( divide_divide @ int @ A3 @ ( uminus_uminus @ int @ B2 ) )
= ( uminus_uminus @ int @ ( divide_divide @ int @ A3 @ B2 ) ) ) )
& ( ( ( modulo_modulo @ int @ A3 @ B2 )
!= ( zero_zero @ int ) )
=> ( ( divide_divide @ int @ A3 @ ( uminus_uminus @ int @ B2 ) )
= ( minus_minus @ int @ ( uminus_uminus @ int @ ( divide_divide @ int @ A3 @ B2 ) ) @ ( one_one @ int ) ) ) ) ) ) ).
% zdiv_zminus2_eq_if
thf(fact_1096_zdiv__mono__strict,axiom,
! [A6: int,B7: int,N: int] :
( ( ord_less @ int @ A6 @ B7 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ N )
=> ( ( ( modulo_modulo @ int @ A6 @ N )
= ( zero_zero @ int ) )
=> ( ( ( modulo_modulo @ int @ B7 @ N )
= ( zero_zero @ int ) )
=> ( ord_less @ int @ ( divide_divide @ int @ A6 @ N ) @ ( divide_divide @ int @ B7 @ N ) ) ) ) ) ) ).
% zdiv_mono_strict
thf(fact_1097_mod__pos__geq,axiom,
! [L: int,K: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ L )
=> ( ( ord_less_eq @ int @ L @ K )
=> ( ( modulo_modulo @ int @ K @ L )
= ( modulo_modulo @ int @ ( minus_minus @ int @ K @ L ) @ L ) ) ) ) ).
% mod_pos_geq
thf(fact_1098_sgn__mod,axiom,
! [L: int,K: int] :
( ( L
!= ( zero_zero @ int ) )
=> ( ~ ( dvd_dvd @ int @ L @ K )
=> ( ( sgn_sgn @ int @ ( modulo_modulo @ int @ K @ L ) )
= ( sgn_sgn @ int @ L ) ) ) ) ).
% sgn_mod
thf(fact_1099_minus__int__code_I1_J,axiom,
! [K: int] :
( ( minus_minus @ int @ K @ ( zero_zero @ int ) )
= K ) ).
% minus_int_code(1)
thf(fact_1100_int__distrib_I3_J,axiom,
! [Z1: int,Z22: int,W2: int] :
( ( times_times @ int @ ( minus_minus @ int @ Z1 @ Z22 ) @ W2 )
= ( minus_minus @ int @ ( times_times @ int @ Z1 @ W2 ) @ ( times_times @ int @ Z22 @ W2 ) ) ) ).
% int_distrib(3)
thf(fact_1101_int__distrib_I4_J,axiom,
! [W2: int,Z1: int,Z22: int] :
( ( times_times @ int @ W2 @ ( minus_minus @ int @ Z1 @ Z22 ) )
= ( minus_minus @ int @ ( times_times @ int @ W2 @ Z1 ) @ ( times_times @ int @ W2 @ Z22 ) ) ) ).
% int_distrib(4)
thf(fact_1102_int__diff__cases,axiom,
! [Z3: int] :
~ ! [M6: nat,N4: nat] :
( Z3
!= ( minus_minus @ int @ ( semiring_1_of_nat @ int @ M6 ) @ ( semiring_1_of_nat @ int @ N4 ) ) ) ).
% int_diff_cases
thf(fact_1103_zmod__le__nonneg__dividend,axiom,
! [M: int,K: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ M )
=> ( ord_less_eq @ int @ ( modulo_modulo @ int @ M @ K ) @ M ) ) ).
% zmod_le_nonneg_dividend
thf(fact_1104_zmod__eq__0__iff,axiom,
! [M: int,D2: int] :
( ( ( modulo_modulo @ int @ M @ D2 )
= ( zero_zero @ int ) )
= ( ? [Q3: int] :
( M
= ( times_times @ int @ D2 @ Q3 ) ) ) ) ).
% zmod_eq_0_iff
thf(fact_1105_zmod__eq__0D,axiom,
! [M: int,D2: int] :
( ( ( modulo_modulo @ int @ M @ D2 )
= ( zero_zero @ int ) )
=> ? [Q5: int] :
( M
= ( times_times @ int @ D2 @ Q5 ) ) ) ).
% zmod_eq_0D
thf(fact_1106_zmod__zminus2__not__zero,axiom,
! [K: int,L: int] :
( ( ( modulo_modulo @ int @ K @ ( uminus_uminus @ int @ L ) )
!= ( zero_zero @ int ) )
=> ( ( modulo_modulo @ int @ K @ L )
!= ( zero_zero @ int ) ) ) ).
% zmod_zminus2_not_zero
thf(fact_1107_zmod__zminus1__not__zero,axiom,
! [K: int,L: int] :
( ( ( modulo_modulo @ int @ ( uminus_uminus @ int @ K ) @ L )
!= ( zero_zero @ int ) )
=> ( ( modulo_modulo @ int @ K @ L )
!= ( zero_zero @ int ) ) ) ).
% zmod_zminus1_not_zero
thf(fact_1108_div__neg__pos__less0,axiom,
! [A3: int,B2: int] :
( ( ord_less @ int @ A3 @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ord_less @ int @ ( divide_divide @ int @ A3 @ B2 ) @ ( zero_zero @ int ) ) ) ) ).
% div_neg_pos_less0
thf(fact_1109_neg__imp__zdiv__neg__iff,axiom,
! [B2: int,A3: int] :
( ( ord_less @ int @ B2 @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ ( divide_divide @ int @ A3 @ B2 ) @ ( zero_zero @ int ) )
= ( ord_less @ int @ ( zero_zero @ int ) @ A3 ) ) ) ).
% neg_imp_zdiv_neg_iff
thf(fact_1110_pos__imp__zdiv__neg__iff,axiom,
! [B2: int,A3: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ( ord_less @ int @ ( divide_divide @ int @ A3 @ B2 ) @ ( zero_zero @ int ) )
= ( ord_less @ int @ A3 @ ( zero_zero @ int ) ) ) ) ).
% pos_imp_zdiv_neg_iff
thf(fact_1111_zmod__int,axiom,
! [A3: nat,B2: nat] :
( ( semiring_1_of_nat @ int @ ( modulo_modulo @ nat @ A3 @ B2 ) )
= ( modulo_modulo @ int @ ( semiring_1_of_nat @ int @ A3 ) @ ( semiring_1_of_nat @ int @ B2 ) ) ) ).
% zmod_int
thf(fact_1112_zdiv__int,axiom,
! [A3: nat,B2: nat] :
( ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ A3 @ B2 ) )
= ( divide_divide @ int @ ( semiring_1_of_nat @ int @ A3 ) @ ( semiring_1_of_nat @ int @ B2 ) ) ) ).
% zdiv_int
thf(fact_1113_minus__mod__int__eq,axiom,
! [L: int,K: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ L )
=> ( ( modulo_modulo @ int @ ( uminus_uminus @ int @ K ) @ L )
= ( minus_minus @ int @ ( minus_minus @ int @ L @ ( one_one @ int ) ) @ ( modulo_modulo @ int @ ( minus_minus @ int @ K @ ( one_one @ int ) ) @ L ) ) ) ) ).
% minus_mod_int_eq
thf(fact_1114_zmod__minus1,axiom,
! [B2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ( modulo_modulo @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ B2 )
= ( minus_minus @ int @ B2 @ ( one_one @ int ) ) ) ) ).
% zmod_minus1
thf(fact_1115_neg__mod__conj,axiom,
! [B2: int,A3: int] :
( ( ord_less @ int @ B2 @ ( zero_zero @ int ) )
=> ( ( ord_less_eq @ int @ ( modulo_modulo @ int @ A3 @ B2 ) @ ( zero_zero @ int ) )
& ( ord_less @ int @ B2 @ ( modulo_modulo @ int @ A3 @ B2 ) ) ) ) ).
% neg_mod_conj
thf(fact_1116_pos__mod__conj,axiom,
! [B2: int,A3: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( modulo_modulo @ int @ A3 @ B2 ) )
& ( ord_less @ int @ ( modulo_modulo @ int @ A3 @ B2 ) @ B2 ) ) ) ).
% pos_mod_conj
thf(fact_1117_zmod__trivial__iff,axiom,
! [I: int,K: int] :
( ( ( modulo_modulo @ int @ I @ K )
= I )
= ( ( K
= ( zero_zero @ int ) )
| ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ I )
& ( ord_less @ int @ I @ K ) )
| ( ( ord_less_eq @ int @ I @ ( zero_zero @ int ) )
& ( ord_less @ int @ K @ I ) ) ) ) ).
% zmod_trivial_iff
thf(fact_1118_mod__int__pos__iff,axiom,
! [K: int,L: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( modulo_modulo @ int @ K @ L ) )
= ( ( dvd_dvd @ int @ L @ K )
| ( ( L
= ( zero_zero @ int ) )
& ( ord_less_eq @ int @ ( zero_zero @ int ) @ K ) )
| ( ord_less @ int @ ( zero_zero @ int ) @ L ) ) ) ).
% mod_int_pos_iff
thf(fact_1119_neg__mod__bound,axiom,
! [L: int,K: int] :
( ( ord_less @ int @ L @ ( zero_zero @ int ) )
=> ( ord_less @ int @ L @ ( modulo_modulo @ int @ K @ L ) ) ) ).
% neg_mod_bound
thf(fact_1120_Euclidean__Division_Opos__mod__bound,axiom,
! [L: int,K: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ L )
=> ( ord_less @ int @ ( modulo_modulo @ int @ K @ L ) @ L ) ) ).
% Euclidean_Division.pos_mod_bound
thf(fact_1121_zdiv__mono1,axiom,
! [A3: int,A4: int,B2: int] :
( ( ord_less_eq @ int @ A3 @ A4 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ord_less_eq @ int @ ( divide_divide @ int @ A3 @ B2 ) @ ( divide_divide @ int @ A4 @ B2 ) ) ) ) ).
% zdiv_mono1
thf(fact_1122_zdiv__mono2,axiom,
! [A3: int,B4: int,B2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A3 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B4 )
=> ( ( ord_less_eq @ int @ B4 @ B2 )
=> ( ord_less_eq @ int @ ( divide_divide @ int @ A3 @ B2 ) @ ( divide_divide @ int @ A3 @ B4 ) ) ) ) ) ).
% zdiv_mono2
thf(fact_1123_zdiv__eq__0__iff,axiom,
! [I: int,K: int] :
( ( ( divide_divide @ int @ I @ K )
= ( zero_zero @ int ) )
= ( ( K
= ( zero_zero @ int ) )
| ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ I )
& ( ord_less @ int @ I @ K ) )
| ( ( ord_less_eq @ int @ I @ ( zero_zero @ int ) )
& ( ord_less @ int @ K @ I ) ) ) ) ).
% zdiv_eq_0_iff
thf(fact_1124_zdiv__mono1__neg,axiom,
! [A3: int,A4: int,B2: int] :
( ( ord_less_eq @ int @ A3 @ A4 )
=> ( ( ord_less @ int @ B2 @ ( zero_zero @ int ) )
=> ( ord_less_eq @ int @ ( divide_divide @ int @ A4 @ B2 ) @ ( divide_divide @ int @ A3 @ B2 ) ) ) ) ).
% zdiv_mono1_neg
thf(fact_1125_zdiv__mono2__neg,axiom,
! [A3: int,B4: int,B2: int] :
( ( ord_less @ int @ A3 @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B4 )
=> ( ( ord_less_eq @ int @ B4 @ B2 )
=> ( ord_less_eq @ int @ ( divide_divide @ int @ A3 @ B4 ) @ ( divide_divide @ int @ A3 @ B2 ) ) ) ) ) ).
% zdiv_mono2_neg
thf(fact_1126_div__int__pos__iff,axiom,
! [K: int,L: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ K @ L ) )
= ( ( K
= ( zero_zero @ int ) )
| ( L
= ( zero_zero @ int ) )
| ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
& ( ord_less_eq @ int @ ( zero_zero @ int ) @ L ) )
| ( ( ord_less @ int @ K @ ( zero_zero @ int ) )
& ( ord_less @ int @ L @ ( zero_zero @ int ) ) ) ) ) ).
% div_int_pos_iff
thf(fact_1127_div__positive__int,axiom,
! [L: int,K: int] :
( ( ord_less_eq @ int @ L @ K )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ L )
=> ( ord_less @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ K @ L ) ) ) ) ).
% div_positive_int
thf(fact_1128_div__nonneg__neg__le0,axiom,
! [A3: int,B2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A3 )
=> ( ( ord_less @ int @ B2 @ ( zero_zero @ int ) )
=> ( ord_less_eq @ int @ ( divide_divide @ int @ A3 @ B2 ) @ ( zero_zero @ int ) ) ) ) ).
% div_nonneg_neg_le0
thf(fact_1129_div__nonpos__pos__le0,axiom,
! [A3: int,B2: int] :
( ( ord_less_eq @ int @ A3 @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ord_less_eq @ int @ ( divide_divide @ int @ A3 @ B2 ) @ ( zero_zero @ int ) ) ) ) ).
% div_nonpos_pos_le0
thf(fact_1130_pos__imp__zdiv__pos__iff,axiom,
! [K: int,I: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ K )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ I @ K ) )
= ( ord_less_eq @ int @ K @ I ) ) ) ).
% pos_imp_zdiv_pos_iff
thf(fact_1131_neg__imp__zdiv__nonneg__iff,axiom,
! [B2: int,A3: int] :
( ( ord_less @ int @ B2 @ ( zero_zero @ int ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ A3 @ B2 ) )
= ( ord_less_eq @ int @ A3 @ ( zero_zero @ int ) ) ) ) ).
% neg_imp_zdiv_nonneg_iff
thf(fact_1132_pos__imp__zdiv__nonneg__iff,axiom,
! [B2: int,A3: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ A3 @ B2 ) )
= ( ord_less_eq @ int @ ( zero_zero @ int ) @ A3 ) ) ) ).
% pos_imp_zdiv_nonneg_iff
thf(fact_1133_nonneg1__imp__zdiv__pos__iff,axiom,
! [A3: int,B2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A3 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ A3 @ B2 ) )
= ( ( ord_less_eq @ int @ B2 @ A3 )
& ( ord_less @ int @ ( zero_zero @ int ) @ B2 ) ) ) ) ).
% nonneg1_imp_zdiv_pos_iff
thf(fact_1134_int__div__less__self,axiom,
! [X: int,K: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ X )
=> ( ( ord_less @ int @ ( one_one @ int ) @ K )
=> ( ord_less @ int @ ( divide_divide @ int @ X @ K ) @ X ) ) ) ).
% int_div_less_self
thf(fact_1135_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_1136_int__le__induct,axiom,
! [I: int,K: int,P: int > $o] :
( ( ord_less_eq @ int @ I @ K )
=> ( ( P @ K )
=> ( ! [I4: int] :
( ( ord_less_eq @ int @ I4 @ K )
=> ( ( P @ I4 )
=> ( P @ ( minus_minus @ int @ I4 @ ( one_one @ int ) ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_le_induct
thf(fact_1137_int__less__induct,axiom,
! [I: int,K: int,P: int > $o] :
( ( ord_less @ int @ I @ K )
=> ( ( P @ ( minus_minus @ int @ K @ ( one_one @ int ) ) )
=> ( ! [I4: int] :
( ( ord_less @ int @ I4 @ K )
=> ( ( P @ I4 )
=> ( P @ ( minus_minus @ int @ I4 @ ( one_one @ int ) ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_less_induct
thf(fact_1138_div__eq__minus1,axiom,
! [B2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ( divide_divide @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ B2 )
= ( uminus_uminus @ int @ ( one_one @ int ) ) ) ) ).
% div_eq_minus1
thf(fact_1139_neg__mod__sign,axiom,
! [L: int,K: int] :
( ( ord_less @ int @ L @ ( zero_zero @ int ) )
=> ( ord_less_eq @ int @ ( modulo_modulo @ int @ K @ L ) @ ( zero_zero @ int ) ) ) ).
% neg_mod_sign
thf(fact_1140_Euclidean__Division_Opos__mod__sign,axiom,
! [L: int,K: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ L )
=> ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( modulo_modulo @ int @ K @ L ) ) ) ).
% Euclidean_Division.pos_mod_sign
thf(fact_1141_plusinfinity,axiom,
! [D2: int,P2: int > $o,P: int > $o] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D2 )
=> ( ! [X3: int,K2: int] :
( ( P2 @ X3 )
= ( P2 @ ( minus_minus @ int @ X3 @ ( times_times @ int @ K2 @ D2 ) ) ) )
=> ( ? [Z5: int] :
! [X3: int] :
( ( ord_less @ int @ Z5 @ X3 )
=> ( ( P @ X3 )
= ( P2 @ X3 ) ) )
=> ( ? [X_1: int] : ( P2 @ X_1 )
=> ? [X_12: int] : ( P @ X_12 ) ) ) ) ) ).
% plusinfinity
thf(fact_1142_minusinfinity,axiom,
! [D2: int,P1: int > $o,P: int > $o] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D2 )
=> ( ! [X3: int,K2: int] :
( ( P1 @ X3 )
= ( P1 @ ( minus_minus @ int @ X3 @ ( times_times @ int @ K2 @ D2 ) ) ) )
=> ( ? [Z5: int] :
! [X3: int] :
( ( ord_less @ int @ X3 @ Z5 )
=> ( ( P @ X3 )
= ( P1 @ X3 ) ) )
=> ( ? [X_1: int] : ( P1 @ X_1 )
=> ? [X_12: int] : ( P @ X_12 ) ) ) ) ) ).
% minusinfinity
thf(fact_1143_zdiv__zmult2__eq,axiom,
! [C2: int,A3: int,B2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ C2 )
=> ( ( divide_divide @ int @ A3 @ ( times_times @ int @ B2 @ C2 ) )
= ( divide_divide @ int @ ( divide_divide @ int @ A3 @ B2 ) @ C2 ) ) ) ).
% zdiv_zmult2_eq
thf(fact_1144_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_1145_nat__mod__distrib,axiom,
! [X: int,Y3: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y3 )
=> ( ( nat2 @ ( modulo_modulo @ int @ X @ Y3 ) )
= ( modulo_modulo @ nat @ ( nat2 @ X ) @ ( nat2 @ Y3 ) ) ) ) ) ).
% nat_mod_distrib
thf(fact_1146_decr__mult__lemma,axiom,
! [D2: int,P: int > $o,K: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D2 )
=> ( ! [X3: int] :
( ( P @ X3 )
=> ( P @ ( minus_minus @ int @ X3 @ D2 ) ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
=> ! [X4: int] :
( ( P @ X4 )
=> ( P @ ( minus_minus @ int @ X4 @ ( times_times @ int @ K @ D2 ) ) ) ) ) ) ) ).
% decr_mult_lemma
thf(fact_1147_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_1148_nat__diff__distrib,axiom,
! [Z6: int,Z3: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z6 )
=> ( ( ord_less_eq @ int @ Z6 @ Z3 )
=> ( ( nat2 @ ( minus_minus @ int @ Z3 @ Z6 ) )
= ( minus_minus @ nat @ ( nat2 @ Z3 ) @ ( nat2 @ Z6 ) ) ) ) ) ).
% nat_diff_distrib
thf(fact_1149_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_1150_verit__less__mono__div__int2,axiom,
! [A6: int,B7: int,N: int] :
( ( ord_less_eq @ int @ A6 @ B7 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ ( uminus_uminus @ int @ N ) )
=> ( ord_less_eq @ int @ ( divide_divide @ int @ B7 @ N ) @ ( divide_divide @ int @ A6 @ N ) ) ) ) ).
% verit_less_mono_div_int2
thf(fact_1151_nat__div__distrib_H,axiom,
! [Y3: int,X: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y3 )
=> ( ( nat2 @ ( divide_divide @ int @ X @ Y3 ) )
= ( divide_divide @ nat @ ( nat2 @ X ) @ ( nat2 @ Y3 ) ) ) ) ).
% nat_div_distrib'
thf(fact_1152_nat__div__distrib,axiom,
! [X: int,Y3: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
=> ( ( nat2 @ ( divide_divide @ int @ X @ Y3 ) )
= ( divide_divide @ nat @ ( nat2 @ X ) @ ( nat2 @ Y3 ) ) ) ) ).
% nat_div_distrib
thf(fact_1153_zdiff__int__split,axiom,
! [P: int > $o,X: nat,Y3: nat] :
( ( P @ ( semiring_1_of_nat @ int @ ( minus_minus @ nat @ X @ Y3 ) ) )
= ( ( ( ord_less_eq @ nat @ Y3 @ X )
=> ( P @ ( minus_minus @ int @ ( semiring_1_of_nat @ int @ X ) @ ( semiring_1_of_nat @ int @ Y3 ) ) ) )
& ( ( ord_less @ nat @ X @ Y3 )
=> ( P @ ( zero_zero @ int ) ) ) ) ) ).
% zdiff_int_split
thf(fact_1154_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_1155_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_1156_mod__eq__0D,axiom,
! [M: nat,D2: nat] :
( ( ( modulo_modulo @ nat @ M @ D2 )
= ( zero_zero @ nat ) )
=> ? [Q5: nat] :
( M
= ( times_times @ nat @ D2 @ Q5 ) ) ) ).
% mod_eq_0D
thf(fact_1157_mod__geq,axiom,
! [M: nat,N: nat] :
( ~ ( ord_less @ nat @ M @ N )
=> ( ( modulo_modulo @ nat @ M @ N )
= ( modulo_modulo @ nat @ ( minus_minus @ nat @ M @ N ) @ N ) ) ) ).
% mod_geq
thf(fact_1158_mod__eq__dvd__iff__nat,axiom,
! [N: nat,M: nat,Q4: nat] :
( ( ord_less_eq @ nat @ N @ M )
=> ( ( ( modulo_modulo @ nat @ M @ Q4 )
= ( modulo_modulo @ nat @ N @ Q4 ) )
= ( dvd_dvd @ nat @ Q4 @ ( minus_minus @ nat @ M @ N ) ) ) ) ).
% mod_eq_dvd_iff_nat
thf(fact_1159_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_1160_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_1161_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_1162_bezout1__nat,axiom,
! [A3: nat,B2: nat] :
? [D6: nat,X3: nat,Y5: nat] :
( ( dvd_dvd @ nat @ D6 @ A3 )
& ( dvd_dvd @ nat @ D6 @ B2 )
& ( ( ( minus_minus @ nat @ ( times_times @ nat @ A3 @ X3 ) @ ( times_times @ nat @ B2 @ Y5 ) )
= D6 )
| ( ( minus_minus @ nat @ ( times_times @ nat @ B2 @ X3 ) @ ( times_times @ nat @ A3 @ Y5 ) )
= D6 ) ) ) ).
% bezout1_nat
thf(fact_1163_gcd__nat__induct,axiom,
! [P: nat > nat > $o,M: nat,N: nat] :
( ! [M6: nat] : ( P @ M6 @ ( zero_zero @ nat ) )
=> ( ! [M6: nat,N4: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N4 )
=> ( ( P @ N4 @ ( modulo_modulo @ nat @ M6 @ N4 ) )
=> ( P @ M6 @ N4 ) ) )
=> ( P @ M @ N ) ) ) ).
% gcd_nat_induct
thf(fact_1164_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_1165_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_1166_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_1167_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 ) )
=> ! [Q5: A,R3: A] :
( ( ( euclid7384307370059645450egment @ A @ R3 )
= ( euclid7384307370059645450egment @ A @ B2 ) )
=> ( ( ord_less @ nat @ ( euclid6346220572633701492n_size @ A @ R3 ) @ ( euclid6346220572633701492n_size @ A @ B2 ) )
=> ( ( R3
!= ( zero_zero @ A ) )
=> ( ( ( divide_divide @ A @ A3 @ B2 )
= Q5 )
=> ( ( ( modulo_modulo @ A @ A3 @ B2 )
= R3 )
=> ( A3
!= ( plus_plus @ A @ ( times_times @ A @ Q5 @ B2 ) @ R3 ) ) ) ) ) ) ) )
=> ( B2
= ( zero_zero @ A ) ) ) ) ) ).
% divmod_cases
thf(fact_1168_gcd__nat_Oextremum,axiom,
! [A3: nat] : ( dvd_dvd @ nat @ A3 @ ( zero_zero @ nat ) ) ).
% gcd_nat.extremum
thf(fact_1169_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_1170_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_1171_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_1172_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_1173_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus @ nat @ M @ ( zero_zero @ nat ) )
= M ) ).
% Nat.add_0_right
thf(fact_1174_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_1175_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_1176_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_1177_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_1178_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_1179_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_1180_add_Oright__neutral,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ! [A3: A] :
( ( plus_plus @ A @ A3 @ ( zero_zero @ A ) )
= A3 ) ) ).
% add.right_neutral
thf(fact_1181_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_1182_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_1183_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_1184_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_1185_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_1186_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_1187_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_1188_add__0,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ! [A3: A] :
( ( plus_plus @ A @ ( zero_zero @ A ) @ A3 )
= A3 ) ) ).
% add_0
thf(fact_1189_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_1190_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_1191_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_1192_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_1193_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_1194_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_1195_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_1196_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_1197_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_1198_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_1199_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_1200_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_1201_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_1202_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_1203_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_1204_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_1205_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_1206_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_1207_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_1208_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_1209_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_1210_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_1211_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_1212_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_1213_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_1214_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_1215_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_1216_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_1217_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_1218_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_1219_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_1220_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_1221_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_1222_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_1223_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_1224_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_1225_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_1226_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_1227_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_1228_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_1229_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_1230_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_1231_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_1232_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_1233_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_1234_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_1235_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_1236_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_1237_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_1238_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_1239_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_1240_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_1241_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_1242_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_1243_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_1244_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_1245_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_1246_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_1247_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_1248_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_1249_zle__add1__eq__le,axiom,
! [W2: int,Z3: int] :
( ( ord_less @ int @ W2 @ ( plus_plus @ int @ Z3 @ ( one_one @ int ) ) )
= ( ord_less_eq @ int @ W2 @ Z3 ) ) ).
% zle_add1_eq_le
thf(fact_1250_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_1251_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_1252_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_1253_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_1254_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_1255_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_1256_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_1257_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_1258_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_1259_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_1260_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_1261_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_1262_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_1263_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_1264_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_1265_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_1266_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_1267_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_1268_of__nat__gbinomial,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [N: nat,K: nat] :
( ( semiring_1_of_nat @ A @ ( gbinomial @ nat @ N @ K ) )
= ( gbinomial @ A @ ( semiring_1_of_nat @ A @ N ) @ K ) ) ) ).
% of_nat_gbinomial
thf(fact_1269_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_1270_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_1271_zadd__int__left,axiom,
! [M: nat,N: nat,Z3: int] :
( ( plus_plus @ int @ ( semiring_1_of_nat @ int @ M ) @ ( plus_plus @ int @ ( semiring_1_of_nat @ int @ N ) @ Z3 ) )
= ( plus_plus @ int @ ( semiring_1_of_nat @ int @ ( plus_plus @ nat @ M @ N ) ) @ Z3 ) ) ).
% zadd_int_left
thf(fact_1272_Euclid__induct,axiom,
! [P: nat > nat > $o,A3: nat,B2: nat] :
( ! [A8: nat,B5: nat] :
( ( P @ A8 @ B5 )
= ( P @ B5 @ A8 ) )
=> ( ! [A8: nat] : ( P @ A8 @ ( zero_zero @ nat ) )
=> ( ! [A8: nat,B5: nat] :
( ( P @ A8 @ B5 )
=> ( P @ A8 @ ( plus_plus @ nat @ A8 @ B5 ) ) )
=> ( P @ A3 @ B2 ) ) ) ) ).
% Euclid_induct
thf(fact_1273_of__bool__eq__iff,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ! [P5: $o,Q4: $o] :
( ( ( zero_neq_one_of_bool @ A @ P5 )
= ( zero_neq_one_of_bool @ A @ Q4 ) )
= ( P5 = Q4 ) ) ) ).
% of_bool_eq_iff
thf(fact_1274_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_1275_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_1276_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_1277_group__cancel_Oadd1,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A6: A,K: A,A3: A,B2: A] :
( ( A6
= ( plus_plus @ A @ K @ A3 ) )
=> ( ( plus_plus @ A @ A6 @ B2 )
= ( plus_plus @ A @ K @ ( plus_plus @ A @ A3 @ B2 ) ) ) ) ) ).
% group_cancel.add1
thf(fact_1278_group__cancel_Oadd2,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [B7: A,K: A,B2: A,A3: A] :
( ( B7
= ( plus_plus @ A @ K @ B2 ) )
=> ( ( plus_plus @ A @ A3 @ B7 )
= ( plus_plus @ A @ K @ ( plus_plus @ A @ A3 @ B2 ) ) ) ) ) ).
% group_cancel.add2
thf(fact_1279_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_1280_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_1281_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_1282_add_Ocommute,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A )
=> ( ( plus_plus @ A )
= ( ^ [A5: A,B3: A] : ( plus_plus @ A @ B3 @ A5 ) ) ) ) ).
% add.commute
thf(fact_1283_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_1284_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_1285_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_1286_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_1287_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_1288_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_1289_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_1290_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_1291_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_1292_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_1293_add__mono,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [A3: A,B2: A,C2: A,D2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ D2 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ C2 ) @ ( plus_plus @ A @ B2 @ D2 ) ) ) ) ) ).
% add_mono
thf(fact_1294_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_1295_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_1296_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_1297_le__iff__add,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A5: A,B3: A] :
? [C4: A] :
( B3
= ( plus_plus @ A @ A5 @ C4 ) ) ) ) ) ).
% le_iff_add
thf(fact_1298_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_1299_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_1300_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_1301_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_1302_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_1303_verit__sum__simplify,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [A3: A] :
( ( plus_plus @ A @ A3 @ ( zero_zero @ A ) )
= A3 ) ) ).
% verit_sum_simplify
thf(fact_1304_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_1305_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_1306_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_1307_add__strict__mono,axiom,
! [A: $tType] :
( ( strict9044650504122735259up_add @ A )
=> ! [A3: A,B2: A,C2: A,D2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ C2 @ D2 )
=> ( ord_less @ A @ ( plus_plus @ A @ A3 @ C2 ) @ ( plus_plus @ A @ B2 @ D2 ) ) ) ) ) ).
% add_strict_mono
thf(fact_1308_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_1309_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_1310_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_1311_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_1312_combine__common__factor,axiom,
! [A: $tType] :
( ( semiring @ A )
=> ! [A3: A,E: A,B2: A,C2: A] :
( ( plus_plus @ A @ ( times_times @ A @ A3 @ E ) @ ( plus_plus @ A @ ( times_times @ A @ B2 @ E ) @ C2 ) )
= ( plus_plus @ A @ ( times_times @ A @ ( plus_plus @ A @ A3 @ B2 ) @ E ) @ C2 ) ) ) ).
% combine_common_factor
thf(fact_1313_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_1314_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_1315_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_1316_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_1317_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_1318_group__cancel_Osub1,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A6: A,K: A,A3: A,B2: A] :
( ( A6
= ( plus_plus @ A @ K @ A3 ) )
=> ( ( minus_minus @ A @ A6 @ B2 )
= ( plus_plus @ A @ K @ ( minus_minus @ A @ A3 @ B2 ) ) ) ) ) ).
% group_cancel.sub1
thf(fact_1319_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_1320_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_1321_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_1322_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_1323_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_1324_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_1325_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_1326_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_1327_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_1328_group__cancel_Oneg1,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A6: A,K: A,A3: A] :
( ( A6
= ( plus_plus @ A @ K @ A3 ) )
=> ( ( uminus_uminus @ A @ A6 )
= ( plus_plus @ A @ ( uminus_uminus @ A @ K ) @ ( uminus_uminus @ A @ A3 ) ) ) ) ) ).
% group_cancel.neg1
thf(fact_1329_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_1330_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_1331_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_1332_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_1333_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_1334_nat__arith_Osuc1,axiom,
! [A6: nat,K: nat,A3: nat] :
( ( A6
= ( plus_plus @ nat @ K @ A3 ) )
=> ( ( suc @ A6 )
= ( plus_plus @ nat @ K @ ( suc @ A3 ) ) ) ) ).
% nat_arith.suc1
thf(fact_1335_add__Suc,axiom,
! [M: nat,N: nat] :
( ( plus_plus @ nat @ ( suc @ M ) @ N )
= ( suc @ ( plus_plus @ nat @ M @ N ) ) ) ).
% add_Suc
thf(fact_1336_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_1337_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus @ nat @ ( zero_zero @ nat ) @ N )
= N ) ).
% plus_nat.add_0
thf(fact_1338_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_1339_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_1340_mod__add__cong,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A3: A,C2: A,A4: A,B2: A,B4: A] :
( ( ( modulo_modulo @ A @ A3 @ C2 )
= ( modulo_modulo @ A @ A4 @ C2 ) )
=> ( ( ( modulo_modulo @ A @ B2 @ C2 )
= ( modulo_modulo @ A @ B4 @ C2 ) )
=> ( ( modulo_modulo @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 )
= ( modulo_modulo @ A @ ( plus_plus @ A @ A4 @ B4 ) @ C2 ) ) ) ) ) ).
% mod_add_cong
thf(fact_1341_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_1342_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_1343_plus__int__code_I2_J,axiom,
! [L: int] :
( ( plus_plus @ int @ ( zero_zero @ int ) @ L )
= L ) ).
% plus_int_code(2)
thf(fact_1344_plus__int__code_I1_J,axiom,
! [K: int] :
( ( plus_plus @ int @ K @ ( zero_zero @ int ) )
= K ) ).
% plus_int_code(1)
thf(fact_1345_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_1346_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_1347_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less @ nat @ ( plus_plus @ nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_1348_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less @ nat @ ( plus_plus @ nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_1349_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_1350_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_1351_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_1352_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_1353_nat__le__iff__add,axiom,
( ( ord_less_eq @ nat )
= ( ^ [M2: nat,N3: nat] :
? [K3: nat] :
( N3
= ( plus_plus @ nat @ M2 @ K3 ) ) ) ) ).
% nat_le_iff_add
thf(fact_1354_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_1355_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_1356_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_1357_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_1358_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq @ nat @ K @ L )
=> ? [N4: nat] :
( L
= ( plus_plus @ nat @ K @ N4 ) ) ) ).
% le_Suc_ex
thf(fact_1359_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_1360_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_1361_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq @ nat @ N @ ( plus_plus @ nat @ M @ N ) ) ).
% le_add2
thf(fact_1362_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq @ nat @ N @ ( plus_plus @ nat @ N @ M ) ) ).
% le_add1
thf(fact_1363_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_1364_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_1365_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_1366_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus @ nat @ ( plus_plus @ nat @ N @ M ) @ N )
= M ) ).
% diff_add_inverse
thf(fact_1367_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N ) @ N )
= M ) ).
% diff_add_inverse2
thf(fact_1368_left__add__mult__distrib,axiom,
! [I: nat,U: nat,J: nat,K: nat] :
( ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ K ) )
= ( plus_plus @ nat @ ( times_times @ nat @ ( plus_plus @ nat @ I @ J ) @ U ) @ K ) ) ).
% left_add_mult_distrib
thf(fact_1369_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_1370_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_1371_int__distrib_I2_J,axiom,
! [W2: int,Z1: int,Z22: int] :
( ( times_times @ int @ W2 @ ( plus_plus @ int @ Z1 @ Z22 ) )
= ( plus_plus @ int @ ( times_times @ int @ W2 @ Z1 ) @ ( times_times @ int @ W2 @ Z22 ) ) ) ).
% int_distrib(2)
thf(fact_1372_int__distrib_I1_J,axiom,
! [Z1: int,Z22: int,W2: int] :
( ( times_times @ int @ ( plus_plus @ int @ Z1 @ Z22 ) @ W2 )
= ( plus_plus @ int @ ( times_times @ int @ Z1 @ W2 ) @ ( times_times @ int @ Z22 @ W2 ) ) ) ).
% int_distrib(1)
thf(fact_1373_div__mod__decomp__int,axiom,
! [A6: int,N: int] :
( A6
= ( plus_plus @ int @ ( times_times @ int @ ( divide_divide @ int @ A6 @ N ) @ N ) @ ( modulo_modulo @ int @ A6 @ N ) ) ) ).
% div_mod_decomp_int
thf(fact_1374_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_1375_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_1376_bezout__lemma__nat,axiom,
! [D2: nat,A3: nat,B2: nat,X: nat,Y3: nat] :
( ( dvd_dvd @ nat @ D2 @ A3 )
=> ( ( dvd_dvd @ nat @ D2 @ B2 )
=> ( ( ( ( times_times @ nat @ A3 @ X )
= ( plus_plus @ nat @ ( times_times @ nat @ B2 @ Y3 ) @ D2 ) )
| ( ( times_times @ nat @ B2 @ X )
= ( plus_plus @ nat @ ( times_times @ nat @ A3 @ Y3 ) @ D2 ) ) )
=> ? [X3: nat,Y5: nat] :
( ( dvd_dvd @ nat @ D2 @ A3 )
& ( dvd_dvd @ nat @ D2 @ ( plus_plus @ nat @ A3 @ B2 ) )
& ( ( ( times_times @ nat @ A3 @ X3 )
= ( plus_plus @ nat @ ( times_times @ nat @ ( plus_plus @ nat @ A3 @ B2 ) @ Y5 ) @ D2 ) )
| ( ( times_times @ nat @ ( plus_plus @ nat @ A3 @ B2 ) @ X3 )
= ( plus_plus @ nat @ ( times_times @ nat @ A3 @ Y5 ) @ D2 ) ) ) ) ) ) ) ).
% bezout_lemma_nat
thf(fact_1377_bezout__add__nat,axiom,
! [A3: nat,B2: nat] :
? [D6: nat,X3: nat,Y5: nat] :
( ( dvd_dvd @ nat @ D6 @ A3 )
& ( dvd_dvd @ nat @ D6 @ B2 )
& ( ( ( times_times @ nat @ A3 @ X3 )
= ( plus_plus @ nat @ ( times_times @ nat @ B2 @ Y5 ) @ D6 ) )
| ( ( times_times @ nat @ B2 @ X3 )
= ( plus_plus @ nat @ ( times_times @ nat @ A3 @ Y5 ) @ D6 ) ) ) ) ).
% bezout_add_nat
thf(fact_1378_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_1379_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_1380_nat__add__distrib,axiom,
! [Z3: int,Z6: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z3 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z6 )
=> ( ( nat2 @ ( plus_plus @ int @ Z3 @ Z6 ) )
= ( plus_plus @ nat @ ( nat2 @ Z3 ) @ ( nat2 @ Z6 ) ) ) ) ) ).
% nat_add_distrib
thf(fact_1381_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_1382_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_1383_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_1384_of__bool__def,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ( ( zero_neq_one_of_bool @ A )
= ( ^ [P6: $o] : ( if @ A @ P6 @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ) ) ).
% of_bool_def
thf(fact_1385_split__of__bool,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ! [P: A > $o,P5: $o] :
( ( P @ ( zero_neq_one_of_bool @ A @ P5 ) )
= ( ( P5
=> ( P @ ( one_one @ A ) ) )
& ( ~ P5
=> ( P @ ( zero_zero @ A ) ) ) ) ) ) ).
% split_of_bool
thf(fact_1386_split__of__bool__asm,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ! [P: A > $o,P5: $o] :
( ( P @ ( zero_neq_one_of_bool @ A @ P5 ) )
= ( ~ ( ( P5
& ~ ( P @ ( one_one @ A ) ) )
| ( ~ P5
& ~ ( P @ ( zero_zero @ A ) ) ) ) ) ) ) ).
% split_of_bool_asm
thf(fact_1387_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_1388_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_1389_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_1390_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_1391_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_1392_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_1393_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_1394_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_1395_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_1396_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_1397_add__le__less__mono,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [A3: A,B2: A,C2: A,D2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ C2 @ D2 )
=> ( ord_less @ A @ ( plus_plus @ A @ A3 @ C2 ) @ ( plus_plus @ A @ B2 @ D2 ) ) ) ) ) ).
% add_le_less_mono
thf(fact_1398_add__less__le__mono,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [A3: A,B2: A,C2: A,D2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ D2 )
=> ( ord_less @ A @ ( plus_plus @ A @ A3 @ C2 ) @ ( plus_plus @ A @ B2 @ D2 ) ) ) ) ) ).
% add_less_le_mono
thf(fact_1399_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_1400_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_1401_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_1402_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_1403_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_1404_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_1405_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_1406_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_1407_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_1408_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_1409_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_1410_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_1411_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_1412_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_1413_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_1414_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_1415_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_1416_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_1417_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_1418_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_1419_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_1420_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_1421_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_1422_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_1423_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_1424_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_1425_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_1426_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_1427_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_1428_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_1429_eq__add__iff2,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A3: A,E: A,C2: A,B2: A,D2: A] :
( ( ( plus_plus @ A @ ( times_times @ A @ A3 @ E ) @ C2 )
= ( plus_plus @ A @ ( times_times @ A @ B2 @ E ) @ D2 ) )
= ( C2
= ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ B2 @ A3 ) @ E ) @ D2 ) ) ) ) ).
% eq_add_iff2
thf(fact_1430_eq__add__iff1,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A3: A,E: A,C2: A,B2: A,D2: A] :
( ( ( plus_plus @ A @ ( times_times @ A @ A3 @ E ) @ C2 )
= ( plus_plus @ A @ ( times_times @ A @ B2 @ E ) @ D2 ) )
= ( ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ A3 @ B2 ) @ E ) @ C2 )
= D2 ) ) ) ).
% eq_add_iff1
thf(fact_1431_pinf_I9_J,axiom,
! [B: $tType] :
( ( ( plus @ B )
& ( linorder @ B )
& ( dvd @ B ) )
=> ! [D2: B,S: B] :
? [Z4: B] :
! [X4: B] :
( ( ord_less @ B @ Z4 @ X4 )
=> ( ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X4 @ S ) )
= ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X4 @ S ) ) ) ) ) ).
% pinf(9)
thf(fact_1432_pinf_I10_J,axiom,
! [B: $tType] :
( ( ( plus @ B )
& ( linorder @ B )
& ( dvd @ B ) )
=> ! [D2: B,S: B] :
? [Z4: B] :
! [X4: B] :
( ( ord_less @ B @ Z4 @ X4 )
=> ( ( ~ ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X4 @ S ) ) )
= ( ~ ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X4 @ S ) ) ) ) ) ) ).
% pinf(10)
thf(fact_1433_minf_I9_J,axiom,
! [B: $tType] :
( ( ( plus @ B )
& ( linorder @ B )
& ( dvd @ B ) )
=> ! [D2: B,S: B] :
? [Z4: B] :
! [X4: B] :
( ( ord_less @ B @ X4 @ Z4 )
=> ( ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X4 @ S ) )
= ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X4 @ S ) ) ) ) ) ).
% minf(9)
thf(fact_1434_minf_I10_J,axiom,
! [B: $tType] :
( ( ( plus @ B )
& ( linorder @ B )
& ( dvd @ B ) )
=> ! [D2: B,S: B] :
? [Z4: B] :
! [X4: B] :
( ( ord_less @ B @ X4 @ Z4 )
=> ( ( ~ ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X4 @ S ) ) )
= ( ~ ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X4 @ S ) ) ) ) ) ) ).
% minf(10)
thf(fact_1435_group__cancel_Osub2,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [B7: A,K: A,B2: A,A3: A] :
( ( B7
= ( plus_plus @ A @ K @ B2 ) )
=> ( ( minus_minus @ A @ A3 @ B7 )
= ( plus_plus @ A @ ( uminus_uminus @ A @ K ) @ ( minus_minus @ A @ A3 @ B2 ) ) ) ) ) ).
% group_cancel.sub2
thf(fact_1436_diff__conv__add__uminus,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ( ( minus_minus @ A )
= ( ^ [A5: A,B3: A] : ( plus_plus @ A @ A5 @ ( uminus_uminus @ A @ B3 ) ) ) ) ) ).
% diff_conv_add_uminus
thf(fact_1437_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ( ( minus_minus @ A )
= ( ^ [A5: A,B3: A] : ( plus_plus @ A @ A5 @ ( uminus_uminus @ A @ B3 ) ) ) ) ) ).
% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_1438_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_1439_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_1440_mod__eqE,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( ( modulo_modulo @ A @ A3 @ C2 )
= ( modulo_modulo @ A @ B2 @ C2 ) )
=> ~ ! [D6: A] :
( B2
!= ( plus_plus @ A @ A3 @ ( times_times @ A @ C2 @ D6 ) ) ) ) ) ).
% mod_eqE
thf(fact_1441_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_1442_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_1443_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_1444_less__natE,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ~ ! [Q5: nat] :
( N
!= ( suc @ ( plus_plus @ nat @ M @ Q5 ) ) ) ) ).
% less_natE
thf(fact_1445_less__add__Suc1,axiom,
! [I: nat,M: nat] : ( ord_less @ nat @ I @ ( suc @ ( plus_plus @ nat @ I @ M ) ) ) ).
% less_add_Suc1
thf(fact_1446_less__add__Suc2,axiom,
! [I: nat,M: nat] : ( ord_less @ nat @ I @ ( suc @ ( plus_plus @ nat @ M @ I ) ) ) ).
% less_add_Suc2
thf(fact_1447_less__iff__Suc__add,axiom,
( ( ord_less @ nat )
= ( ^ [M2: nat,N3: nat] :
? [K3: nat] :
( N3
= ( suc @ ( plus_plus @ nat @ M2 @ K3 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_1448_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_1449_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_1450_mono__nat__linear__lb,axiom,
! [F: nat > nat,M: nat,K: nat] :
( ! [M6: nat,N4: nat] :
( ( ord_less @ nat @ M6 @ N4 )
=> ( ord_less @ nat @ ( F @ M6 ) @ ( F @ N4 ) ) )
=> ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus @ nat @ M @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1451_ex__has__greatest__nat__lemma,axiom,
! [A: $tType,P: A > $o,K: A,F: A > nat,N: nat] :
( ( P @ K )
=> ( ! [X3: A] :
( ( P @ X3 )
=> ? [Y4: A] :
( ( P @ Y4 )
& ~ ( ord_less_eq @ nat @ ( F @ Y4 ) @ ( F @ X3 ) ) ) )
=> ? [Y5: A] :
( ( P @ Y5 )
& ~ ( ord_less @ nat @ ( F @ Y5 ) @ ( plus_plus @ nat @ ( F @ K ) @ N ) ) ) ) ) ).
% ex_has_greatest_nat_lemma
thf(fact_1452_bezout__add__strong__nat,axiom,
! [A3: nat,B2: nat] :
( ( A3
!= ( zero_zero @ nat ) )
=> ? [D6: nat,X3: nat,Y5: nat] :
( ( dvd_dvd @ nat @ D6 @ A3 )
& ( dvd_dvd @ nat @ D6 @ B2 )
& ( ( times_times @ nat @ A3 @ X3 )
= ( plus_plus @ nat @ ( times_times @ nat @ B2 @ Y5 ) @ D6 ) ) ) ) ).
% bezout_add_strong_nat
thf(fact_1453_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_1454_odd__nonzero,axiom,
! [Z3: int] :
( ( plus_plus @ int @ ( plus_plus @ int @ ( one_one @ int ) @ Z3 ) @ Z3 )
!= ( zero_zero @ int ) ) ).
% odd_nonzero
thf(fact_1455_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_1456_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_1457_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_1458_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus @ nat @ ( one_one @ nat ) ) ) ).
% Suc_eq_plus1_left
thf(fact_1459_plus__1__eq__Suc,axiom,
( ( plus_plus @ nat @ ( one_one @ nat ) )
= suc ) ).
% plus_1_eq_Suc
thf(fact_1460_Suc__eq__plus1,axiom,
( suc
= ( ^ [N3: nat] : ( plus_plus @ nat @ N3 @ ( one_one @ nat ) ) ) ) ).
% Suc_eq_plus1
thf(fact_1461_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_1462_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_1463_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_1464_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_1465_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_1466_int__ge__induct,axiom,
! [K: int,I: int,P: int > $o] :
( ( ord_less_eq @ int @ K @ I )
=> ( ( P @ K )
=> ( ! [I4: int] :
( ( ord_less_eq @ int @ K @ I4 )
=> ( ( P @ I4 )
=> ( P @ ( plus_plus @ int @ I4 @ ( one_one @ int ) ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_ge_induct
thf(fact_1467_zless__add1__eq,axiom,
! [W2: int,Z3: int] :
( ( ord_less @ int @ W2 @ ( plus_plus @ int @ Z3 @ ( one_one @ int ) ) )
= ( ( ord_less @ int @ W2 @ Z3 )
| ( W2 = Z3 ) ) ) ).
% zless_add1_eq
thf(fact_1468_int__gr__induct,axiom,
! [K: int,I: int,P: int > $o] :
( ( ord_less @ int @ K @ I )
=> ( ( P @ ( plus_plus @ int @ K @ ( one_one @ int ) ) )
=> ( ! [I4: int] :
( ( ord_less @ int @ K @ I4 )
=> ( ( P @ I4 )
=> ( P @ ( plus_plus @ int @ I4 @ ( one_one @ int ) ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_gr_induct
thf(fact_1469_zle__iff__zadd,axiom,
( ( ord_less_eq @ int )
= ( ^ [W3: int,Z7: int] :
? [N3: nat] :
( Z7
= ( plus_plus @ int @ W3 @ ( semiring_1_of_nat @ int @ N3 ) ) ) ) ) ).
% zle_iff_zadd
thf(fact_1470_zdvd__period,axiom,
! [A3: int,D2: int,X: int,T2: int,C2: int] :
( ( dvd_dvd @ int @ A3 @ D2 )
=> ( ( dvd_dvd @ int @ A3 @ ( plus_plus @ int @ X @ T2 ) )
= ( dvd_dvd @ int @ A3 @ ( plus_plus @ int @ ( plus_plus @ int @ X @ ( times_times @ int @ C2 @ D2 ) ) @ T2 ) ) ) ) ).
% zdvd_period
thf(fact_1471_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_1472_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_1473_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_1474_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_1475_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_1476_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_1477_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_1478_field__le__epsilon,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y3: A] :
( ! [E2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ E2 )
=> ( ord_less_eq @ A @ X @ ( plus_plus @ A @ Y3 @ E2 ) ) )
=> ( ord_less_eq @ A @ X @ Y3 ) ) ) ).
% field_le_epsilon
thf(fact_1479_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_1480_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_1481_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_1482_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_1483_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_1484_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_1485_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_1486_discrete,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ( ( ord_less @ A )
= ( ^ [A5: A] : ( ord_less_eq @ A @ ( plus_plus @ A @ A5 @ ( one_one @ A ) ) ) ) ) ) ).
% discrete
thf(fact_1487_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_1488_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_1489_le__add__iff2,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [A3: A,E: A,C2: A,B2: A,D2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ A3 @ E ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ B2 @ E ) @ D2 ) )
= ( ord_less_eq @ A @ C2 @ ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ B2 @ A3 ) @ E ) @ D2 ) ) ) ) ).
% le_add_iff2
thf(fact_1490_le__add__iff1,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [A3: A,E: A,C2: A,B2: A,D2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ A3 @ E ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ B2 @ E ) @ D2 ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ A3 @ B2 ) @ E ) @ C2 ) @ D2 ) ) ) ).
% le_add_iff1
thf(fact_1491_less__add__iff2,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [A3: A,E: A,C2: A,B2: A,D2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ A3 @ E ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ B2 @ E ) @ D2 ) )
= ( ord_less @ A @ C2 @ ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ B2 @ A3 ) @ E ) @ D2 ) ) ) ) ).
% less_add_iff2
thf(fact_1492_less__add__iff1,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [A3: A,E: A,C2: A,B2: A,D2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ A3 @ E ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ B2 @ E ) @ D2 ) )
= ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ A3 @ B2 ) @ E ) @ C2 ) @ D2 ) ) ) ).
% less_add_iff1
thf(fact_1493_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_1494_add__divide__eq__if__simps_I2_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z3: A,A3: A,B2: A] :
( ( ( Z3
= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( divide_divide @ A @ A3 @ Z3 ) @ B2 )
= B2 ) )
& ( ( Z3
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( divide_divide @ A @ A3 @ Z3 ) @ B2 )
= ( divide_divide @ A @ ( plus_plus @ A @ A3 @ ( times_times @ A @ B2 @ Z3 ) ) @ Z3 ) ) ) ) ) ).
% add_divide_eq_if_simps(2)
thf(fact_1495_add__divide__eq__if__simps_I1_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z3: A,A3: A,B2: A] :
( ( ( Z3
= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ A3 @ ( divide_divide @ A @ B2 @ Z3 ) )
= A3 ) )
& ( ( Z3
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ A3 @ ( divide_divide @ A @ B2 @ Z3 ) )
= ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ A3 @ Z3 ) @ B2 ) @ Z3 ) ) ) ) ) ).
% add_divide_eq_if_simps(1)
thf(fact_1496_add__frac__eq,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [Y3: A,Z3: A,X: A,W2: A] :
( ( Y3
!= ( zero_zero @ A ) )
=> ( ( Z3
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( divide_divide @ A @ X @ Y3 ) @ ( divide_divide @ A @ W2 @ Z3 ) )
= ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ X @ Z3 ) @ ( times_times @ A @ W2 @ Y3 ) ) @ ( times_times @ A @ Y3 @ Z3 ) ) ) ) ) ) ).
% add_frac_eq
thf(fact_1497_add__frac__num,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [Y3: A,X: A,Z3: A] :
( ( Y3
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( divide_divide @ A @ X @ Y3 ) @ Z3 )
= ( divide_divide @ A @ ( plus_plus @ A @ X @ ( times_times @ A @ Z3 @ Y3 ) ) @ Y3 ) ) ) ) ).
% add_frac_num
thf(fact_1498_add__num__frac,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [Y3: A,Z3: A,X: A] :
( ( Y3
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ Z3 @ ( divide_divide @ A @ X @ Y3 ) )
= ( divide_divide @ A @ ( plus_plus @ A @ X @ ( times_times @ A @ Z3 @ Y3 ) ) @ Y3 ) ) ) ) ).
% add_num_frac
thf(fact_1499_add__divide__eq__iff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z3: A,X: A,Y3: A] :
( ( Z3
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ X @ ( divide_divide @ A @ Y3 @ Z3 ) )
= ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ X @ Z3 ) @ Y3 ) @ Z3 ) ) ) ) ).
% add_divide_eq_iff
thf(fact_1500_divide__add__eq__iff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z3: A,X: A,Y3: A] :
( ( Z3
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( divide_divide @ A @ X @ Z3 ) @ Y3 )
= ( divide_divide @ A @ ( plus_plus @ A @ X @ ( times_times @ A @ Y3 @ Z3 ) ) @ Z3 ) ) ) ) ).
% divide_add_eq_iff
thf(fact_1501_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_1502_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_1503_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_1504_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_1505_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_1506_inf__period_I4_J,axiom,
! [A: $tType] :
( ( ( comm_ring @ A )
& ( dvd @ A ) )
=> ! [D2: A,D4: A,T2: A] :
( ( dvd_dvd @ A @ D2 @ D4 )
=> ! [X4: A,K4: A] :
( ( ~ ( dvd_dvd @ A @ D2 @ ( plus_plus @ A @ X4 @ T2 ) ) )
= ( ~ ( dvd_dvd @ A @ D2 @ ( plus_plus @ A @ ( minus_minus @ A @ X4 @ ( times_times @ A @ K4 @ D4 ) ) @ T2 ) ) ) ) ) ) ).
% inf_period(4)
thf(fact_1507_inf__period_I3_J,axiom,
! [A: $tType] :
( ( ( comm_ring @ A )
& ( dvd @ A ) )
=> ! [D2: A,D4: A,T2: A] :
( ( dvd_dvd @ A @ D2 @ D4 )
=> ! [X4: A,K4: A] :
( ( dvd_dvd @ A @ D2 @ ( plus_plus @ A @ X4 @ T2 ) )
= ( dvd_dvd @ A @ D2 @ ( plus_plus @ A @ ( minus_minus @ A @ X4 @ ( times_times @ A @ K4 @ D4 ) ) @ T2 ) ) ) ) ) ).
% inf_period(3)
thf(fact_1508_gbinomial__of__nat__symmetric,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ N )
=> ( ( gbinomial @ A @ ( semiring_1_of_nat @ A @ N ) @ K )
= ( gbinomial @ A @ ( semiring_1_of_nat @ A @ N ) @ ( minus_minus @ nat @ N @ K ) ) ) ) ) ).
% gbinomial_of_nat_symmetric
thf(fact_1509_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_1510_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_1511_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_1512_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_1513_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_1514_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_1515_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_1516_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_1517_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 ) ) )
| ? [D5: nat] :
( ( A3
= ( plus_plus @ nat @ B2 @ D5 ) )
& ~ ( P @ D5 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_1518_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 ) ) )
& ! [D5: nat] :
( ( A3
= ( plus_plus @ nat @ B2 @ D5 ) )
=> ( P @ D5 ) ) ) ) ).
% nat_diff_split
thf(fact_1519_mod__pos__neg__trivial,axiom,
! [K: int,L: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ K )
=> ( ( ord_less_eq @ int @ ( plus_plus @ int @ K @ L ) @ ( zero_zero @ int ) )
=> ( ( modulo_modulo @ int @ K @ L )
= ( plus_plus @ int @ K @ L ) ) ) ) ).
% mod_pos_neg_trivial
thf(fact_1520_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_1521_odd__less__0__iff,axiom,
! [Z3: int] :
( ( ord_less @ int @ ( plus_plus @ int @ ( plus_plus @ int @ ( one_one @ int ) @ Z3 ) @ Z3 ) @ ( zero_zero @ int ) )
= ( ord_less @ int @ Z3 @ ( zero_zero @ int ) ) ) ).
% odd_less_0_iff
thf(fact_1522_zless__iff__Suc__zadd,axiom,
( ( ord_less @ int )
= ( ^ [W3: int,Z7: int] :
? [N3: nat] :
( Z7
= ( plus_plus @ int @ W3 @ ( semiring_1_of_nat @ int @ ( suc @ N3 ) ) ) ) ) ) ).
% zless_iff_Suc_zadd
thf(fact_1523_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_1524_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_1525_zmod__zmult2__eq,axiom,
! [C2: int,A3: int,B2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ C2 )
=> ( ( modulo_modulo @ int @ A3 @ ( times_times @ int @ B2 @ C2 ) )
= ( plus_plus @ int @ ( times_times @ int @ B2 @ ( modulo_modulo @ int @ ( divide_divide @ int @ A3 @ B2 ) @ C2 ) ) @ ( modulo_modulo @ int @ A3 @ B2 ) ) ) ) ).
% zmod_zmult2_eq
thf(fact_1526_zless__imp__add1__zle,axiom,
! [W2: int,Z3: int] :
( ( ord_less @ int @ W2 @ Z3 )
=> ( ord_less_eq @ int @ ( plus_plus @ int @ W2 @ ( one_one @ int ) ) @ Z3 ) ) ).
% zless_imp_add1_zle
thf(fact_1527_add1__zle__eq,axiom,
! [W2: int,Z3: int] :
( ( ord_less_eq @ int @ ( plus_plus @ int @ W2 @ ( one_one @ int ) ) @ Z3 )
= ( ord_less @ int @ W2 @ Z3 ) ) ).
% add1_zle_eq
thf(fact_1528_nat__eq__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ J @ I )
=> ( ( ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M )
= ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N ) )
= ( ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ I @ J ) @ U ) @ M )
= N ) ) ) ).
% nat_eq_add_iff1
thf(fact_1529_nat__eq__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M )
= ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N ) )
= ( M
= ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% nat_eq_add_iff2
thf(fact_1530_nat__le__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ J @ I )
=> ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N ) )
= ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% nat_le_add_iff1
thf(fact_1531_nat__le__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N ) )
= ( ord_less_eq @ nat @ M @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% nat_le_add_iff2
thf(fact_1532_nat__diff__add__eq1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ J @ I )
=> ( ( minus_minus @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N ) )
= ( minus_minus @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% nat_diff_add_eq1
thf(fact_1533_nat__diff__add__eq2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( minus_minus @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N ) )
= ( minus_minus @ nat @ M @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% nat_diff_add_eq2
thf(fact_1534_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_1535_int__induct,axiom,
! [P: int > $o,K: int,I: int] :
( ( P @ K )
=> ( ! [I4: int] :
( ( ord_less_eq @ int @ K @ I4 )
=> ( ( P @ I4 )
=> ( P @ ( plus_plus @ int @ I4 @ ( one_one @ int ) ) ) ) )
=> ( ! [I4: int] :
( ( ord_less_eq @ int @ I4 @ K )
=> ( ( P @ I4 )
=> ( P @ ( minus_minus @ int @ I4 @ ( one_one @ int ) ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_induct
thf(fact_1536_mod__eq__nat1E,axiom,
! [M: nat,Q4: nat,N: nat] :
( ( ( modulo_modulo @ nat @ M @ Q4 )
= ( modulo_modulo @ nat @ N @ Q4 ) )
=> ( ( ord_less_eq @ nat @ N @ M )
=> ~ ! [S2: nat] :
( M
!= ( plus_plus @ nat @ N @ ( times_times @ nat @ Q4 @ S2 ) ) ) ) ) ).
% mod_eq_nat1E
thf(fact_1537_mod__eq__nat2E,axiom,
! [M: nat,Q4: nat,N: nat] :
( ( ( modulo_modulo @ nat @ M @ Q4 )
= ( modulo_modulo @ nat @ N @ Q4 ) )
=> ( ( ord_less_eq @ nat @ M @ N )
=> ~ ! [S2: nat] :
( N
!= ( plus_plus @ nat @ M @ ( times_times @ nat @ Q4 @ S2 ) ) ) ) ) ).
% mod_eq_nat2E
thf(fact_1538_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 )
=> ? [Q5: nat] :
( X
= ( plus_plus @ nat @ Y3 @ ( times_times @ nat @ N @ Q5 ) ) ) ) ) ).
% nat_mod_eq_lemma
thf(fact_1539_mod__mult2__eq,axiom,
! [M: nat,N: nat,Q4: nat] :
( ( modulo_modulo @ nat @ M @ ( times_times @ nat @ N @ Q4 ) )
= ( plus_plus @ nat @ ( times_times @ nat @ N @ ( modulo_modulo @ nat @ ( divide_divide @ nat @ M @ N ) @ Q4 ) ) @ ( modulo_modulo @ nat @ M @ N ) ) ) ).
% mod_mult2_eq
thf(fact_1540_div__mod__decomp,axiom,
! [A6: nat,N: nat] :
( A6
= ( plus_plus @ nat @ ( times_times @ nat @ ( divide_divide @ nat @ A6 @ N ) @ N ) @ ( modulo_modulo @ nat @ A6 @ N ) ) ) ).
% div_mod_decomp
thf(fact_1541_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_1542_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_1543_convex__bound__le,axiom,
! [A: $tType] :
( ( linord6961819062388156250ring_1 @ A )
=> ! [X: A,A3: A,Y3: A,U: A,V: A] :
( ( ord_less_eq @ A @ X @ A3 )
=> ( ( ord_less_eq @ A @ Y3 @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ U )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ V )
=> ( ( ( plus_plus @ A @ U @ V )
= ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ U @ X ) @ ( times_times @ A @ V @ Y3 ) ) @ A3 ) ) ) ) ) ) ) ).
% convex_bound_le
thf(fact_1544_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_1545_add__divide__eq__if__simps_I3_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z3: A,A3: A,B2: A] :
( ( ( Z3
= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ A3 @ Z3 ) ) @ B2 )
= B2 ) )
& ( ( Z3
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ A3 @ Z3 ) ) @ B2 )
= ( divide_divide @ A @ ( plus_plus @ A @ ( uminus_uminus @ A @ A3 ) @ ( times_times @ A @ B2 @ Z3 ) ) @ Z3 ) ) ) ) ) ).
% add_divide_eq_if_simps(3)
thf(fact_1546_minus__divide__add__eq__iff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z3: A,X: A,Y3: A] :
( ( Z3
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ X @ Z3 ) ) @ Y3 )
= ( divide_divide @ A @ ( plus_plus @ A @ ( uminus_uminus @ A @ X ) @ ( times_times @ A @ Y3 @ Z3 ) ) @ Z3 ) ) ) ) ).
% minus_divide_add_eq_iff
thf(fact_1547_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_1548_add__eq__if,axiom,
( ( plus_plus @ nat )
= ( ^ [M2: nat,N3: nat] :
( if @ nat
@ ( M2
= ( zero_zero @ nat ) )
@ N3
@ ( suc @ ( plus_plus @ nat @ ( minus_minus @ nat @ M2 @ ( one_one @ nat ) ) @ N3 ) ) ) ) ) ).
% add_eq_if
thf(fact_1549_le__imp__0__less,axiom,
! [Z3: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z3 )
=> ( ord_less @ int @ ( zero_zero @ int ) @ ( plus_plus @ int @ ( one_one @ int ) @ Z3 ) ) ) ).
% le_imp_0_less
thf(fact_1550_verit__le__mono__div__int,axiom,
! [A6: int,B7: int,N: int] :
( ( ord_less @ int @ A6 @ B7 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ N )
=> ( ord_less_eq @ int
@ ( plus_plus @ int @ ( divide_divide @ int @ A6 @ N )
@ ( if @ int
@ ( ( modulo_modulo @ int @ B7 @ N )
= ( zero_zero @ int ) )
@ ( one_one @ int )
@ ( zero_zero @ int ) ) )
@ ( divide_divide @ int @ B7 @ N ) ) ) ) ).
% verit_le_mono_div_int
thf(fact_1551_nat__less__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ J @ I )
=> ( ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N ) )
= ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% nat_less_add_iff1
thf(fact_1552_nat__less__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N ) )
= ( ord_less @ nat @ M @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% nat_less_add_iff2
thf(fact_1553_incr__mult__lemma,axiom,
! [D2: int,P: int > $o,K: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D2 )
=> ( ! [X3: int] :
( ( P @ X3 )
=> ( P @ ( plus_plus @ int @ X3 @ D2 ) ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
=> ! [X4: int] :
( ( P @ X4 )
=> ( P @ ( plus_plus @ int @ X4 @ ( times_times @ int @ K @ D2 ) ) ) ) ) ) ) ).
% incr_mult_lemma
thf(fact_1554_unique__quotient__lemma__neg,axiom,
! [B2: int,Q6: int,R4: int,Q4: int,R: int] :
( ( ord_less_eq @ int @ ( plus_plus @ int @ ( times_times @ int @ B2 @ Q6 ) @ R4 ) @ ( plus_plus @ int @ ( times_times @ int @ B2 @ Q4 ) @ R ) )
=> ( ( ord_less_eq @ int @ R @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ B2 @ R )
=> ( ( ord_less @ int @ B2 @ R4 )
=> ( ord_less_eq @ int @ Q4 @ Q6 ) ) ) ) ) ).
% unique_quotient_lemma_neg
thf(fact_1555_unique__quotient__lemma,axiom,
! [B2: int,Q6: int,R4: int,Q4: int,R: int] :
( ( ord_less_eq @ int @ ( plus_plus @ int @ ( times_times @ int @ B2 @ Q6 ) @ R4 ) @ ( plus_plus @ int @ ( times_times @ int @ B2 @ Q4 ) @ R ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R4 )
=> ( ( ord_less @ int @ R4 @ B2 )
=> ( ( ord_less @ int @ R @ B2 )
=> ( ord_less_eq @ int @ Q6 @ Q4 ) ) ) ) ) ).
% unique_quotient_lemma
thf(fact_1556_zdiv__mono2__neg__lemma,axiom,
! [B2: int,Q4: int,R: int,B4: int,Q6: int,R4: int] :
( ( ( plus_plus @ int @ ( times_times @ int @ B2 @ Q4 ) @ R )
= ( plus_plus @ int @ ( times_times @ int @ B4 @ Q6 ) @ R4 ) )
=> ( ( ord_less @ int @ ( plus_plus @ int @ ( times_times @ int @ B4 @ Q6 ) @ R4 ) @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ R @ B2 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R4 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B4 )
=> ( ( ord_less_eq @ int @ B4 @ B2 )
=> ( ord_less_eq @ int @ Q6 @ Q4 ) ) ) ) ) ) ) ).
% zdiv_mono2_neg_lemma
thf(fact_1557_zdiv__mono2__lemma,axiom,
! [B2: int,Q4: int,R: int,B4: int,Q6: int,R4: int] :
( ( ( plus_plus @ int @ ( times_times @ int @ B2 @ Q4 ) @ R )
= ( plus_plus @ int @ ( times_times @ int @ B4 @ Q6 ) @ R4 ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( plus_plus @ int @ ( times_times @ int @ B4 @ Q6 ) @ R4 ) )
=> ( ( ord_less @ int @ R4 @ B4 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B4 )
=> ( ( ord_less_eq @ int @ B4 @ B2 )
=> ( ord_less_eq @ int @ Q4 @ Q6 ) ) ) ) ) ) ) ).
% zdiv_mono2_lemma
thf(fact_1558_q__pos__lemma,axiom,
! [B4: int,Q6: int,R4: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( plus_plus @ int @ ( times_times @ int @ B4 @ Q6 ) @ R4 ) )
=> ( ( ord_less @ int @ R4 @ B4 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B4 )
=> ( ord_less_eq @ int @ ( zero_zero @ int ) @ Q6 ) ) ) ) ).
% q_pos_lemma
thf(fact_1559_int__div__pos__eq,axiom,
! [A3: int,B2: int,Q4: int,R: int] :
( ( A3
= ( plus_plus @ int @ ( times_times @ int @ B2 @ Q4 ) @ R ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R )
=> ( ( ord_less @ int @ R @ B2 )
=> ( ( divide_divide @ int @ A3 @ B2 )
= Q4 ) ) ) ) ).
% int_div_pos_eq
thf(fact_1560_int__div__neg__eq,axiom,
! [A3: int,B2: int,Q4: int,R: int] :
( ( A3
= ( plus_plus @ int @ ( times_times @ int @ B2 @ Q4 ) @ R ) )
=> ( ( ord_less_eq @ int @ R @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ B2 @ R )
=> ( ( divide_divide @ int @ A3 @ B2 )
= Q4 ) ) ) ) ).
% int_div_neg_eq
thf(fact_1561_split__zdiv,axiom,
! [P: int > $o,N: int,K: int] :
( ( P @ ( divide_divide @ int @ N @ K ) )
= ( ( ( K
= ( zero_zero @ int ) )
=> ( P @ ( zero_zero @ int ) ) )
& ( ( ord_less @ int @ ( zero_zero @ int ) @ K )
=> ! [I5: int,J3: int] :
( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ J3 )
& ( ord_less @ int @ J3 @ K )
& ( N
= ( plus_plus @ int @ ( times_times @ int @ K @ I5 ) @ J3 ) ) )
=> ( P @ I5 ) ) )
& ( ( ord_less @ int @ K @ ( zero_zero @ int ) )
=> ! [I5: int,J3: int] :
( ( ( ord_less @ int @ K @ J3 )
& ( ord_less_eq @ int @ J3 @ ( zero_zero @ int ) )
& ( N
= ( plus_plus @ int @ ( times_times @ int @ K @ I5 ) @ J3 ) ) )
=> ( P @ I5 ) ) ) ) ) ).
% split_zdiv
thf(fact_1562_int__mod__pos__eq,axiom,
! [A3: int,B2: int,Q4: int,R: int] :
( ( A3
= ( plus_plus @ int @ ( times_times @ int @ B2 @ Q4 ) @ R ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R )
=> ( ( ord_less @ int @ R @ B2 )
=> ( ( modulo_modulo @ int @ A3 @ B2 )
= R ) ) ) ) ).
% int_mod_pos_eq
thf(fact_1563_int__mod__neg__eq,axiom,
! [A3: int,B2: int,Q4: int,R: int] :
( ( A3
= ( plus_plus @ int @ ( times_times @ int @ B2 @ Q4 ) @ R ) )
=> ( ( ord_less_eq @ int @ R @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ B2 @ R )
=> ( ( modulo_modulo @ int @ A3 @ B2 )
= R ) ) ) ) ).
% int_mod_neg_eq
thf(fact_1564_split__zmod,axiom,
! [P: int > $o,N: int,K: int] :
( ( P @ ( modulo_modulo @ int @ N @ K ) )
= ( ( ( K
= ( zero_zero @ int ) )
=> ( P @ N ) )
& ( ( ord_less @ int @ ( zero_zero @ int ) @ K )
=> ! [I5: int,J3: int] :
( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ J3 )
& ( ord_less @ int @ J3 @ K )
& ( N
= ( plus_plus @ int @ ( times_times @ int @ K @ I5 ) @ J3 ) ) )
=> ( P @ J3 ) ) )
& ( ( ord_less @ int @ K @ ( zero_zero @ int ) )
=> ! [I5: int,J3: int] :
( ( ( ord_less @ int @ K @ J3 )
& ( ord_less_eq @ int @ J3 @ ( zero_zero @ int ) )
& ( N
= ( plus_plus @ int @ ( times_times @ int @ K @ I5 ) @ J3 ) ) )
=> ( P @ J3 ) ) ) ) ) ).
% split_zmod
thf(fact_1565_split__pos__lemma,axiom,
! [K: int,P: int > int > $o,N: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ K )
=> ( ( P @ ( divide_divide @ int @ N @ K ) @ ( modulo_modulo @ int @ N @ K ) )
= ( ! [I5: int,J3: int] :
( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ J3 )
& ( ord_less @ int @ J3 @ K )
& ( N
= ( plus_plus @ int @ ( times_times @ int @ K @ I5 ) @ J3 ) ) )
=> ( P @ I5 @ J3 ) ) ) ) ) ).
% split_pos_lemma
thf(fact_1566_split__neg__lemma,axiom,
! [K: int,P: int > int > $o,N: int] :
( ( ord_less @ int @ K @ ( zero_zero @ int ) )
=> ( ( P @ ( divide_divide @ int @ N @ K ) @ ( modulo_modulo @ int @ N @ K ) )
= ( ! [I5: int,J3: int] :
( ( ( ord_less @ int @ K @ J3 )
& ( ord_less_eq @ int @ J3 @ ( zero_zero @ int ) )
& ( N
= ( plus_plus @ int @ ( times_times @ int @ K @ I5 ) @ J3 ) ) )
=> ( P @ I5 @ J3 ) ) ) ) ) ).
% split_neg_lemma
thf(fact_1567_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_1568_mult__eq__if,axiom,
( ( times_times @ nat )
= ( ^ [M2: nat,N3: nat] :
( if @ nat
@ ( M2
= ( zero_zero @ nat ) )
@ ( zero_zero @ nat )
@ ( plus_plus @ nat @ N3 @ ( times_times @ nat @ ( minus_minus @ nat @ M2 @ ( one_one @ nat ) ) @ N3 ) ) ) ) ) ).
% mult_eq_if
thf(fact_1569_Suc__as__int,axiom,
( suc
= ( ^ [A5: nat] : ( nat2 @ ( plus_plus @ int @ ( semiring_1_of_nat @ int @ A5 ) @ ( one_one @ int ) ) ) ) ) ).
% Suc_as_int
thf(fact_1570_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 ) )
=> ! [I5: nat,J3: nat] :
( ( ord_less @ nat @ J3 @ N )
=> ( ( M
= ( plus_plus @ nat @ ( times_times @ nat @ N @ I5 ) @ J3 ) )
=> ( P @ J3 ) ) ) ) ) ) ).
% split_mod
thf(fact_1571_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_1572_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_1573_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 ) )
=> ! [I5: nat,J3: nat] :
( ( ord_less @ nat @ J3 @ N )
=> ( ( M
= ( plus_plus @ nat @ ( times_times @ nat @ N @ I5 ) @ J3 ) )
=> ( P @ I5 ) ) ) ) ) ) ).
% split_div
thf(fact_1574_dvd__minus__add,axiom,
! [Q4: nat,N: nat,R: nat,M: nat] :
( ( ord_less_eq @ nat @ Q4 @ N )
=> ( ( ord_less_eq @ nat @ Q4 @ ( times_times @ nat @ R @ M ) )
=> ( ( dvd_dvd @ nat @ M @ ( minus_minus @ nat @ N @ Q4 ) )
= ( dvd_dvd @ nat @ M @ ( plus_plus @ nat @ N @ ( minus_minus @ nat @ ( times_times @ nat @ R @ M ) @ Q4 ) ) ) ) ) ) ).
% dvd_minus_add
thf(fact_1575_convex__bound__lt,axiom,
! [A: $tType] :
( ( linord715952674999750819strict @ A )
=> ! [X: A,A3: A,Y3: A,U: A,V: A] :
( ( ord_less @ A @ X @ A3 )
=> ( ( ord_less @ A @ Y3 @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ U )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ V )
=> ( ( ( plus_plus @ A @ U @ V )
= ( one_one @ A ) )
=> ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ U @ X ) @ ( times_times @ A @ V @ Y3 ) ) @ A3 ) ) ) ) ) ) ) ).
% convex_bound_lt
thf(fact_1576_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_1577_scaling__mono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [U: A,V: A,R: A,S: A] :
( ( ord_less_eq @ A @ U @ V )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ R )
=> ( ( ord_less_eq @ A @ R @ S )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ U @ ( divide_divide @ A @ ( times_times @ A @ R @ ( minus_minus @ A @ V @ U ) ) @ S ) ) @ V ) ) ) ) ) ).
% scaling_mono
thf(fact_1578_gbinomial__trinomial__revision,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,M: nat,A3: A] :
( ( ord_less_eq @ nat @ K @ M )
=> ( ( times_times @ A @ ( gbinomial @ A @ A3 @ M ) @ ( gbinomial @ A @ ( semiring_1_of_nat @ A @ M ) @ K ) )
= ( times_times @ A @ ( gbinomial @ A @ A3 @ K ) @ ( gbinomial @ A @ ( minus_minus @ A @ A3 @ ( semiring_1_of_nat @ A @ K ) ) @ ( minus_minus @ nat @ M @ K ) ) ) ) ) ) ).
% gbinomial_trinomial_revision
thf(fact_1579_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_1580_Suc__nat__eq__nat__zadd1,axiom,
! [Z3: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z3 )
=> ( ( suc @ ( nat2 @ Z3 ) )
= ( nat2 @ ( plus_plus @ int @ ( one_one @ int ) @ Z3 ) ) ) ) ).
% Suc_nat_eq_nat_zadd1
thf(fact_1581_div__pos__neg__trivial,axiom,
! [K: int,L: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ K )
=> ( ( ord_less_eq @ int @ ( plus_plus @ int @ K @ L ) @ ( zero_zero @ int ) )
=> ( ( divide_divide @ int @ K @ L )
= ( uminus_uminus @ int @ ( one_one @ int ) ) ) ) ) ).
% div_pos_neg_trivial
thf(fact_1582_div__pos__geq,axiom,
! [L: int,K: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ L )
=> ( ( ord_less_eq @ int @ L @ K )
=> ( ( divide_divide @ int @ K @ L )
= ( plus_plus @ int @ ( divide_divide @ int @ ( minus_minus @ int @ K @ L ) @ L ) @ ( one_one @ int ) ) ) ) ) ).
% div_pos_geq
thf(fact_1583_div__eqI,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [B2: A,R: A,Q4: 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 @ Q4 @ B2 ) @ R )
= A3 )
=> ( ( divide_divide @ A @ A3 @ B2 )
= Q4 ) ) ) ) ) ) ).
% div_eqI
thf(fact_1584_div__bounded,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [B2: A,R: A,Q4: 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 @ Q4 @ B2 ) @ R ) @ B2 )
= Q4 ) ) ) ) ) ).
% div_bounded
thf(fact_1585_mod__eqI,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [B2: A,R: A,Q4: 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 @ Q4 @ B2 ) @ R )
= A3 )
=> ( ( modulo_modulo @ A @ A3 @ B2 )
= R ) ) ) ) ) ) ).
% mod_eqI
thf(fact_1586_verit__le__mono__div,axiom,
! [A6: nat,B7: nat,N: nat] :
( ( ord_less @ nat @ A6 @ B7 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less_eq @ nat
@ ( plus_plus @ nat @ ( divide_divide @ nat @ A6 @ N )
@ ( if @ nat
@ ( ( modulo_modulo @ nat @ B7 @ N )
= ( zero_zero @ nat ) )
@ ( one_one @ nat )
@ ( zero_zero @ nat ) ) )
@ ( divide_divide @ nat @ B7 @ N ) ) ) ) ).
% verit_le_mono_div
thf(fact_1587_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_1588_dvd__productE,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [P5: A,A3: A,B2: A] :
( ( dvd_dvd @ A @ P5 @ ( times_times @ A @ A3 @ B2 ) )
=> ~ ! [X3: A,Y5: A] :
( ( P5
= ( times_times @ A @ X3 @ Y5 ) )
=> ( ( dvd_dvd @ A @ X3 @ A3 )
=> ~ ( dvd_dvd @ A @ Y5 @ B2 ) ) ) ) ) ).
% dvd_productE
thf(fact_1589_gcd__nat_Oextremum__uniqueI,axiom,
! [A3: nat] :
( ( dvd_dvd @ nat @ ( zero_zero @ nat ) @ A3 )
=> ( A3
= ( zero_zero @ nat ) ) ) ).
% gcd_nat.extremum_uniqueI
thf(fact_1590_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_1591_gcd__nat_Oextremum__unique,axiom,
! [A3: nat] :
( ( dvd_dvd @ nat @ ( zero_zero @ nat ) @ A3 )
= ( A3
= ( zero_zero @ nat ) ) ) ).
% gcd_nat.extremum_unique
thf(fact_1592_gcd__nat_Oextremum__strict,axiom,
! [A3: nat] :
~ ( ( dvd_dvd @ nat @ ( zero_zero @ nat ) @ A3 )
& ( ( zero_zero @ nat )
!= A3 ) ) ).
% gcd_nat.extremum_strict
thf(fact_1593_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_1594_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_1595_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_1596_add__scale__eq__noteq,axiom,
! [A: $tType] :
( ( semiri1453513574482234551roduct @ A )
=> ! [R: A,A3: A,B2: A,C2: A,D2: A] :
( ( R
!= ( zero_zero @ A ) )
=> ( ( ( A3 = B2 )
& ( C2 != D2 ) )
=> ( ( plus_plus @ A @ A3 @ ( times_times @ A @ R @ C2 ) )
!= ( plus_plus @ A @ B2 @ ( times_times @ A @ R @ D2 ) ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_1597_divide__int__def,axiom,
( ( divide_divide @ int )
= ( ^ [K3: int,L3: int] :
( if @ int
@ ( L3
= ( zero_zero @ int ) )
@ ( zero_zero @ int )
@ ( if @ int
@ ( ( sgn_sgn @ int @ K3 )
= ( sgn_sgn @ int @ L3 ) )
@ ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ ( nat2 @ ( abs_abs @ int @ K3 ) ) @ ( nat2 @ ( abs_abs @ int @ L3 ) ) ) )
@ ( uminus_uminus @ int
@ ( semiring_1_of_nat @ int
@ ( plus_plus @ nat @ ( divide_divide @ nat @ ( nat2 @ ( abs_abs @ int @ K3 ) ) @ ( nat2 @ ( abs_abs @ int @ L3 ) ) )
@ ( zero_neq_one_of_bool @ nat
@ ~ ( dvd_dvd @ int @ L3 @ K3 ) ) ) ) ) ) ) ) ) ).
% divide_int_def
thf(fact_1598_gbinomial__pochhammer_H,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( ( gbinomial @ A )
= ( ^ [A5: A,K3: nat] : ( divide_divide @ A @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ ( minus_minus @ A @ A5 @ ( semiring_1_of_nat @ A @ K3 ) ) @ ( one_one @ A ) ) @ K3 ) @ ( semiring_char_0_fact @ A @ K3 ) ) ) ) ) ).
% gbinomial_pochhammer'
thf(fact_1599_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_1600_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_1601_abs__0,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ( ( abs_abs @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% abs_0
thf(fact_1602_abs__zero,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ( ( abs_abs @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% abs_zero
thf(fact_1603_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_1604_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_1605_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_1606_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_1607_abs__1,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ( ( abs_abs @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% abs_1
thf(fact_1608_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_1609_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_1610_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_1611_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_1612_abs__divide,axiom,
! [A: $tType] :
( ( field_abs_sgn @ A )
=> ! [A3: A,B2: A] :
( ( abs_abs @ A @ ( divide_divide @ A @ A3 @ B2 ) )
= ( divide_divide @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B2 ) ) ) ) ).
% abs_divide
thf(fact_1613_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_1614_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_1615_pochhammer__1,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A3: A] :
( ( comm_s3205402744901411588hammer @ A @ A3 @ ( one_one @ nat ) )
= A3 ) ) ).
% pochhammer_1
thf(fact_1616_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_1617_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_1618_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_1619_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_1620_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_1621_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_1622_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_1623_pochhammer__Suc0,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A3: A] :
( ( comm_s3205402744901411588hammer @ A @ A3 @ ( suc @ ( zero_zero @ nat ) ) )
= A3 ) ) ).
% pochhammer_Suc0
thf(fact_1624_zdvd1__eq,axiom,
! [X: int] :
( ( dvd_dvd @ int @ X @ ( one_one @ int ) )
= ( ( abs_abs @ int @ X )
= ( one_one @ int ) ) ) ).
% zdvd1_eq
thf(fact_1625_abs__division__segment,axiom,
! [K: int] :
( ( abs_abs @ int @ ( euclid7384307370059645450egment @ int @ K ) )
= ( one_one @ int ) ) ).
% abs_division_segment
thf(fact_1626_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_1627_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_1628_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_1629_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_1630_zabs__less__one__iff,axiom,
! [Z3: int] :
( ( ord_less @ int @ ( abs_abs @ int @ Z3 ) @ ( one_one @ int ) )
= ( Z3
= ( zero_zero @ int ) ) ) ).
% zabs_less_one_iff
thf(fact_1631_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_1632_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_1633_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_1634_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_1635_pochhammer__of__nat,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [X: nat,N: nat] :
( ( comm_s3205402744901411588hammer @ A @ ( semiring_1_of_nat @ A @ X ) @ N )
= ( semiring_1_of_nat @ A @ ( comm_s3205402744901411588hammer @ nat @ X @ N ) ) ) ) ).
% pochhammer_of_nat
thf(fact_1636_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_1637_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_1638_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_1639_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_1640_abs__one,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( ( abs_abs @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% abs_one
thf(fact_1641_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_1642_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_1643_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_1644_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_1645_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_1646_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_1647_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_1648_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_1649_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_1650_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_1651_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_1652_abs__mult__less,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,C2: A,B2: A,D2: A] :
( ( ord_less @ A @ ( abs_abs @ A @ A3 ) @ C2 )
=> ( ( ord_less @ A @ ( abs_abs @ A @ B2 ) @ D2 )
=> ( ord_less @ A @ ( times_times @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B2 ) ) @ ( times_times @ A @ C2 @ D2 ) ) ) ) ) ).
% abs_mult_less
thf(fact_1653_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_1654_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_1655_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_1656_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_1657_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_1658_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_1659_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_1660_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_1661_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_1662_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_1663_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_1664_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_1665_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_1666_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_1667_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_1668_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_1669_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_1670_dense__eq0__I,axiom,
! [A: $tType] :
( ( ( ordere166539214618696060dd_abs @ A )
& ( dense_linorder @ A ) )
=> ! [X: A] :
( ! [E2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ E2 )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ X ) @ E2 ) )
=> ( X
= ( zero_zero @ A ) ) ) ) ).
% dense_eq0_I
thf(fact_1671_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_1672_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_1673_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_1674_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_1675_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_1676_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_1677_abs__if__raw,axiom,
! [A: $tType] :
( ( abs_if @ A )
=> ( ( abs_abs @ A )
= ( ^ [A5: A] : ( if @ A @ ( ord_less @ A @ A5 @ ( zero_zero @ A ) ) @ ( uminus_uminus @ A @ A5 ) @ A5 ) ) ) ) ).
% abs_if_raw
thf(fact_1678_abs__if,axiom,
! [A: $tType] :
( ( abs_if @ A )
=> ( ( abs_abs @ A )
= ( ^ [A5: A] : ( if @ A @ ( ord_less @ A @ A5 @ ( zero_zero @ A ) ) @ ( uminus_uminus @ A @ A5 ) @ A5 ) ) ) ) ).
% abs_if
thf(fact_1679_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_1680_abs__diff__triangle__ineq,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A,B2: A,C2: A,D2: A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ ( plus_plus @ A @ A3 @ B2 ) @ ( plus_plus @ A @ C2 @ D2 ) ) ) @ ( plus_plus @ A @ ( abs_abs @ A @ ( minus_minus @ A @ A3 @ C2 ) ) @ ( abs_abs @ A @ ( minus_minus @ A @ B2 @ D2 ) ) ) ) ) ).
% abs_diff_triangle_ineq
thf(fact_1681_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_1682_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_1683_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_1684_div__noneq__sgn__abs,axiom,
! [L: int,K: int] :
( ( L
!= ( zero_zero @ int ) )
=> ( ( ( sgn_sgn @ int @ K )
!= ( sgn_sgn @ int @ L ) )
=> ( ( divide_divide @ int @ K @ L )
= ( minus_minus @ int @ ( uminus_uminus @ int @ ( divide_divide @ int @ ( abs_abs @ int @ K ) @ ( abs_abs @ int @ L ) ) )
@ ( zero_neq_one_of_bool @ int
@ ~ ( dvd_dvd @ int @ L @ K ) ) ) ) ) ) ).
% div_noneq_sgn_abs
thf(fact_1685_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_1686_zabs__def,axiom,
( ( abs_abs @ int )
= ( ^ [I5: int] : ( if @ int @ ( ord_less @ int @ I5 @ ( zero_zero @ int ) ) @ ( uminus_uminus @ int @ I5 ) @ I5 ) ) ) ).
% zabs_def
thf(fact_1687_dvd__imp__le__int,axiom,
! [I: int,D2: int] :
( ( I
!= ( zero_zero @ int ) )
=> ( ( dvd_dvd @ int @ D2 @ I )
=> ( ord_less_eq @ int @ ( abs_abs @ int @ D2 ) @ ( abs_abs @ int @ I ) ) ) ) ).
% dvd_imp_le_int
thf(fact_1688_nat__abs__mult__distrib,axiom,
! [W2: int,Z3: int] :
( ( nat2 @ ( abs_abs @ int @ ( times_times @ int @ W2 @ Z3 ) ) )
= ( times_times @ nat @ ( nat2 @ ( abs_abs @ int @ W2 ) ) @ ( nat2 @ ( abs_abs @ int @ Z3 ) ) ) ) ).
% nat_abs_mult_distrib
thf(fact_1689_abs__mod__less,axiom,
! [L: int,K: int] :
( ( L
!= ( zero_zero @ int ) )
=> ( ord_less @ int @ ( abs_abs @ int @ ( modulo_modulo @ int @ K @ L ) ) @ ( abs_abs @ int @ L ) ) ) ).
% abs_mod_less
thf(fact_1690_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_1691_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_1692_pochhammer__product_H,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [Z3: A,N: nat,M: nat] :
( ( comm_s3205402744901411588hammer @ A @ Z3 @ ( plus_plus @ nat @ N @ M ) )
= ( times_times @ A @ ( comm_s3205402744901411588hammer @ A @ Z3 @ N ) @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ Z3 @ ( semiring_1_of_nat @ A @ N ) ) @ M ) ) ) ) ).
% pochhammer_product'
thf(fact_1693_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_1694_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_1695_div__sgn__abs__cancel,axiom,
! [V: int,K: int,L: int] :
( ( V
!= ( zero_zero @ int ) )
=> ( ( divide_divide @ int @ ( times_times @ int @ ( sgn_sgn @ int @ V ) @ ( abs_abs @ int @ K ) ) @ ( times_times @ int @ ( sgn_sgn @ int @ V ) @ ( abs_abs @ int @ L ) ) )
= ( divide_divide @ int @ ( abs_abs @ int @ K ) @ ( abs_abs @ int @ L ) ) ) ) ).
% div_sgn_abs_cancel
thf(fact_1696_div__abs__eq__div__nat,axiom,
! [K: int,L: int] :
( ( divide_divide @ int @ ( abs_abs @ int @ K ) @ ( abs_abs @ int @ L ) )
= ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ ( nat2 @ ( abs_abs @ int @ K ) ) @ ( nat2 @ ( abs_abs @ int @ L ) ) ) ) ) ).
% div_abs_eq_div_nat
thf(fact_1697_mod__abs__eq__div__nat,axiom,
! [K: int,L: int] :
( ( modulo_modulo @ int @ ( abs_abs @ int @ K ) @ ( abs_abs @ int @ L ) )
= ( semiring_1_of_nat @ int @ ( modulo_modulo @ nat @ ( nat2 @ ( abs_abs @ int @ K ) ) @ ( nat2 @ ( abs_abs @ int @ L ) ) ) ) ) ).
% mod_abs_eq_div_nat
thf(fact_1698_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_1699_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_1700_pochhammer__rec_H,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [Z3: A,N: nat] :
( ( comm_s3205402744901411588hammer @ A @ Z3 @ ( suc @ N ) )
= ( times_times @ A @ ( plus_plus @ A @ Z3 @ ( semiring_1_of_nat @ A @ N ) ) @ ( comm_s3205402744901411588hammer @ A @ Z3 @ N ) ) ) ) ).
% pochhammer_rec'
thf(fact_1701_div__dvd__sgn__abs,axiom,
! [L: int,K: int] :
( ( dvd_dvd @ int @ L @ K )
=> ( ( divide_divide @ int @ K @ L )
= ( times_times @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K ) @ ( sgn_sgn @ int @ L ) ) @ ( divide_divide @ int @ ( abs_abs @ int @ K ) @ ( abs_abs @ int @ L ) ) ) ) ) ).
% div_dvd_sgn_abs
thf(fact_1702_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_1703_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_1704_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_1705_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_1706_modulo__int__def,axiom,
( ( modulo_modulo @ int )
= ( ^ [K3: int,L3: int] :
( if @ int
@ ( L3
= ( zero_zero @ int ) )
@ K3
@ ( if @ int
@ ( ( sgn_sgn @ int @ K3 )
= ( sgn_sgn @ int @ L3 ) )
@ ( times_times @ int @ ( sgn_sgn @ int @ L3 ) @ ( semiring_1_of_nat @ int @ ( modulo_modulo @ nat @ ( nat2 @ ( abs_abs @ int @ K3 ) ) @ ( nat2 @ ( abs_abs @ int @ L3 ) ) ) ) )
@ ( times_times @ int @ ( sgn_sgn @ int @ L3 )
@ ( minus_minus @ int
@ ( times_times @ int @ ( abs_abs @ int @ L3 )
@ ( zero_neq_one_of_bool @ int
@ ~ ( dvd_dvd @ int @ L3 @ K3 ) ) )
@ ( semiring_1_of_nat @ int @ ( modulo_modulo @ nat @ ( nat2 @ ( abs_abs @ int @ K3 ) ) @ ( nat2 @ ( abs_abs @ int @ L3 ) ) ) ) ) ) ) ) ) ) ).
% modulo_int_def
thf(fact_1707_incr__lemma,axiom,
! [D2: int,Z3: int,X: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D2 )
=> ( ord_less @ int @ Z3 @ ( plus_plus @ int @ X @ ( times_times @ int @ ( plus_plus @ int @ ( abs_abs @ int @ ( minus_minus @ int @ X @ Z3 ) ) @ ( one_one @ int ) ) @ D2 ) ) ) ) ).
% incr_lemma
thf(fact_1708_decr__lemma,axiom,
! [D2: int,X: int,Z3: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D2 )
=> ( ord_less @ int @ ( minus_minus @ int @ X @ ( times_times @ int @ ( plus_plus @ int @ ( abs_abs @ int @ ( minus_minus @ int @ X @ Z3 ) ) @ ( one_one @ int ) ) @ D2 ) ) @ Z3 ) ) ).
% decr_lemma
thf(fact_1709_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_1710_pochhammer__product,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [M: nat,N: nat,Z3: A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( comm_s3205402744901411588hammer @ A @ Z3 @ N )
= ( times_times @ A @ ( comm_s3205402744901411588hammer @ A @ Z3 @ M ) @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ Z3 @ ( semiring_1_of_nat @ A @ M ) ) @ ( minus_minus @ nat @ N @ M ) ) ) ) ) ) ).
% pochhammer_product
thf(fact_1711_nat__intermed__int__val,axiom,
! [M: nat,N: nat,F: nat > int,K: int] :
( ! [I4: nat] :
( ( ( ord_less_eq @ nat @ M @ I4 )
& ( ord_less @ nat @ I4 @ N ) )
=> ( ord_less_eq @ int @ ( abs_abs @ int @ ( minus_minus @ int @ ( F @ ( suc @ I4 ) ) @ ( F @ I4 ) ) ) @ ( one_one @ int ) ) )
=> ( ( ord_less_eq @ nat @ M @ N )
=> ( ( ord_less_eq @ int @ ( F @ M ) @ K )
=> ( ( ord_less_eq @ int @ K @ ( F @ N ) )
=> ? [I4: nat] :
( ( ord_less_eq @ nat @ M @ I4 )
& ( ord_less_eq @ nat @ I4 @ N )
& ( ( F @ I4 )
= K ) ) ) ) ) ) ).
% nat_intermed_int_val
thf(fact_1712_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_1713_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_1714_crossproduct__noteq,axiom,
! [A: $tType] :
( ( semiri1453513574482234551roduct @ A )
=> ! [A3: A,B2: A,C2: A,D2: A] :
( ( ( A3 != B2 )
& ( C2 != D2 ) )
= ( ( plus_plus @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ D2 ) )
!= ( plus_plus @ A @ ( times_times @ A @ A3 @ D2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ) ).
% crossproduct_noteq
thf(fact_1715_crossproduct__eq,axiom,
! [A: $tType] :
( ( semiri1453513574482234551roduct @ A )
=> ! [W2: A,Y3: A,X: A,Z3: A] :
( ( ( plus_plus @ A @ ( times_times @ A @ W2 @ Y3 ) @ ( times_times @ A @ X @ Z3 ) )
= ( plus_plus @ A @ ( times_times @ A @ W2 @ Z3 ) @ ( times_times @ A @ X @ Y3 ) ) )
= ( ( W2 = X )
| ( Y3 = Z3 ) ) ) ) ).
% crossproduct_eq
thf(fact_1716_nat__ivt__aux,axiom,
! [N: nat,F: nat > int,K: int] :
( ! [I4: nat] :
( ( ord_less @ nat @ I4 @ N )
=> ( ord_less_eq @ int @ ( abs_abs @ int @ ( minus_minus @ int @ ( F @ ( suc @ I4 ) ) @ ( F @ I4 ) ) ) @ ( one_one @ int ) ) )
=> ( ( ord_less_eq @ int @ ( F @ ( zero_zero @ nat ) ) @ K )
=> ( ( ord_less_eq @ int @ K @ ( F @ N ) )
=> ? [I4: nat] :
( ( ord_less_eq @ nat @ I4 @ N )
& ( ( F @ I4 )
= K ) ) ) ) ) ).
% nat_ivt_aux
thf(fact_1717_nat0__intermed__int__val,axiom,
! [N: nat,F: nat > int,K: int] :
( ! [I4: nat] :
( ( ord_less @ nat @ I4 @ N )
=> ( ord_less_eq @ int @ ( abs_abs @ int @ ( minus_minus @ int @ ( F @ ( plus_plus @ nat @ I4 @ ( one_one @ nat ) ) ) @ ( F @ I4 ) ) ) @ ( one_one @ int ) ) )
=> ( ( ord_less_eq @ int @ ( F @ ( zero_zero @ nat ) ) @ K )
=> ( ( ord_less_eq @ int @ K @ ( F @ N ) )
=> ? [I4: nat] :
( ( ord_less_eq @ nat @ I4 @ N )
& ( ( F @ I4 )
= K ) ) ) ) ) ).
% nat0_intermed_int_val
thf(fact_1718_gbinomial__pochhammer,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( ( gbinomial @ A )
= ( ^ [A5: 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 @ A5 ) @ K3 ) ) @ ( semiring_char_0_fact @ A @ K3 ) ) ) ) ) ).
% gbinomial_pochhammer
thf(fact_1719_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_1720_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_1721_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_1722_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_1723_triangle__Suc,axiom,
! [N: nat] :
( ( nat_triangle @ ( suc @ N ) )
= ( plus_plus @ nat @ ( nat_triangle @ N ) @ ( suc @ N ) ) ) ).
% triangle_Suc
thf(fact_1724_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_1725_power__Suc__0,axiom,
! [N: nat] :
( ( power_power @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% power_Suc_0
thf(fact_1726_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_1727_power__one,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [N: nat] :
( ( power_power @ A @ ( one_one @ A ) @ N )
= ( one_one @ A ) ) ) ).
% power_one
thf(fact_1728_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_1729_of__nat__power,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [M: nat,N: nat] :
( ( semiring_1_of_nat @ A @ ( power_power @ nat @ M @ N ) )
= ( power_power @ A @ ( semiring_1_of_nat @ A @ M ) @ N ) ) ) ).
% of_nat_power
thf(fact_1730_of__nat__eq__of__nat__power__cancel__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [B2: nat,W2: nat,X: nat] :
( ( ( power_power @ A @ ( semiring_1_of_nat @ A @ B2 ) @ W2 )
= ( semiring_1_of_nat @ A @ X ) )
= ( ( power_power @ nat @ B2 @ W2 )
= X ) ) ) ).
% of_nat_eq_of_nat_power_cancel_iff
thf(fact_1731_of__nat__power__eq__of__nat__cancel__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [X: nat,B2: nat,W2: nat] :
( ( ( semiring_1_of_nat @ A @ X )
= ( power_power @ A @ ( semiring_1_of_nat @ A @ B2 ) @ W2 ) )
= ( X
= ( power_power @ nat @ B2 @ W2 ) ) ) ) ).
% of_nat_power_eq_of_nat_cancel_iff
thf(fact_1732_power__one__right,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A3: A] :
( ( power_power @ A @ A3 @ ( one_one @ nat ) )
= A3 ) ) ).
% power_one_right
thf(fact_1733_frac__in__Ints__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( member @ A @ ( archimedean_frac @ A @ X ) @ ( ring_1_Ints @ A ) )
= ( member @ A @ X @ ( ring_1_Ints @ A ) ) ) ) ).
% frac_in_Ints_iff
thf(fact_1734_triangle__0,axiom,
( ( nat_triangle @ ( zero_zero @ nat ) )
= ( zero_zero @ nat ) ) ).
% triangle_0
thf(fact_1735_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_1736_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_1737_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_1738_abs__power__minus,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,N: nat] :
( ( abs_abs @ A @ ( power_power @ A @ ( uminus_uminus @ A @ A3 ) @ N ) )
= ( abs_abs @ A @ ( power_power @ A @ A3 @ N ) ) ) ) ).
% abs_power_minus
thf(fact_1739_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_1740_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_1741_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_1742_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_1743_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_1744_of__nat__power__less__of__nat__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X: nat,B2: nat,W2: nat] :
( ( ord_less @ A @ ( semiring_1_of_nat @ A @ X ) @ ( power_power @ A @ ( semiring_1_of_nat @ A @ B2 ) @ W2 ) )
= ( ord_less @ nat @ X @ ( power_power @ nat @ B2 @ W2 ) ) ) ) ).
% of_nat_power_less_of_nat_cancel_iff
thf(fact_1745_of__nat__less__of__nat__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B2: nat,W2: nat,X: nat] :
( ( ord_less @ A @ ( power_power @ A @ ( semiring_1_of_nat @ A @ B2 ) @ W2 ) @ ( semiring_1_of_nat @ A @ X ) )
= ( ord_less @ nat @ ( power_power @ nat @ B2 @ W2 ) @ X ) ) ) ).
% of_nat_less_of_nat_power_cancel_iff
thf(fact_1746_of__nat__power__le__of__nat__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X: nat,B2: nat,W2: nat] :
( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ X ) @ ( power_power @ A @ ( semiring_1_of_nat @ A @ B2 ) @ W2 ) )
= ( ord_less_eq @ nat @ X @ ( power_power @ nat @ B2 @ W2 ) ) ) ) ).
% of_nat_power_le_of_nat_cancel_iff
thf(fact_1747_of__nat__le__of__nat__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B2: nat,W2: nat,X: nat] :
( ( ord_less_eq @ A @ ( power_power @ A @ ( semiring_1_of_nat @ A @ B2 ) @ W2 ) @ ( semiring_1_of_nat @ A @ X ) )
= ( ord_less_eq @ nat @ ( power_power @ nat @ B2 @ W2 ) @ X ) ) ) ).
% of_nat_le_of_nat_power_cancel_iff
thf(fact_1748_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_1749_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_1750_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_1751_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_1752_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_1753_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_1754_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_1755_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_1756_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_1757_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_1758_power__commutes,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A3: A,N: nat] :
( ( times_times @ A @ ( power_power @ A @ A3 @ N ) @ A3 )
= ( times_times @ A @ A3 @ ( power_power @ A @ A3 @ N ) ) ) ) ).
% power_commutes
thf(fact_1759_power__mult__distrib,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: A,B2: A,N: nat] :
( ( power_power @ A @ ( times_times @ A @ A3 @ B2 ) @ N )
= ( times_times @ A @ ( power_power @ A @ A3 @ N ) @ ( power_power @ A @ B2 @ N ) ) ) ) ).
% power_mult_distrib
thf(fact_1760_power__commuting__commutes,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [X: A,Y3: A,N: nat] :
( ( ( times_times @ A @ X @ Y3 )
= ( times_times @ A @ Y3 @ X ) )
=> ( ( times_times @ A @ ( power_power @ A @ X @ N ) @ Y3 )
= ( times_times @ A @ Y3 @ ( power_power @ A @ X @ N ) ) ) ) ) ).
% power_commuting_commutes
thf(fact_1761_power__mod,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A3: A,B2: A,N: nat] :
( ( modulo_modulo @ A @ ( power_power @ A @ ( modulo_modulo @ A @ A3 @ B2 ) @ N ) @ B2 )
= ( modulo_modulo @ A @ ( power_power @ A @ A3 @ N ) @ B2 ) ) ) ).
% power_mod
thf(fact_1762_power__mult,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A3: A,M: nat,N: nat] :
( ( power_power @ A @ A3 @ ( times_times @ nat @ M @ N ) )
= ( power_power @ A @ ( power_power @ A @ A3 @ M ) @ N ) ) ) ).
% power_mult
thf(fact_1763_Ints__0,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( member @ A @ ( zero_zero @ A ) @ ( ring_1_Ints @ A ) ) ) ).
% Ints_0
thf(fact_1764_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_1765_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_1766_Ints__1,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( member @ A @ ( one_one @ A ) @ ( ring_1_Ints @ A ) ) ) ).
% Ints_1
thf(fact_1767_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_1768_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_1769_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_1770_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_1771_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_1772_fact__in__Ints,axiom,
! [A: $tType] :
( ( ( semiring_char_0 @ A )
& ( ring_1 @ A ) )
=> ! [N: nat] : ( member @ A @ ( semiring_char_0_fact @ A @ N ) @ ( ring_1_Ints @ A ) ) ) ).
% fact_in_Ints
thf(fact_1773_nat__power__eq,axiom,
! [Z3: int,N: nat] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z3 )
=> ( ( nat2 @ ( power_power @ int @ Z3 @ N ) )
= ( power_power @ nat @ ( nat2 @ Z3 ) @ N ) ) ) ).
% nat_power_eq
thf(fact_1774_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_1775_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_1776_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_1777_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_1778_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_1779_power__Suc2,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A3: A,N: nat] :
( ( power_power @ A @ A3 @ ( suc @ N ) )
= ( times_times @ A @ ( power_power @ A @ A3 @ N ) @ A3 ) ) ) ).
% power_Suc2
thf(fact_1780_power__Suc,axiom,
! [A: $tType] :
( ( power @ A )
=> ! [A3: A,N: nat] :
( ( power_power @ A @ A3 @ ( suc @ N ) )
= ( times_times @ A @ A3 @ ( power_power @ A @ A3 @ N ) ) ) ) ).
% power_Suc
thf(fact_1781_power__0,axiom,
! [A: $tType] :
( ( power @ A )
=> ! [A3: A] :
( ( power_power @ A @ A3 @ ( zero_zero @ nat ) )
= ( one_one @ A ) ) ) ).
% power_0
thf(fact_1782_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_1783_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_1784_le__imp__power__dvd,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [M: nat,N: nat,A3: A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( dvd_dvd @ A @ ( power_power @ A @ A3 @ M ) @ ( power_power @ A @ A3 @ N ) ) ) ) ).
% le_imp_power_dvd
thf(fact_1785_power__le__dvd,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A3: A,N: nat,B2: A,M: nat] :
( ( dvd_dvd @ A @ ( power_power @ A @ A3 @ N ) @ B2 )
=> ( ( ord_less_eq @ nat @ M @ N )
=> ( dvd_dvd @ A @ ( power_power @ A @ A3 @ M ) @ B2 ) ) ) ) ).
% power_le_dvd
thf(fact_1786_dvd__power__le,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [X: A,Y3: A,N: nat,M: nat] :
( ( dvd_dvd @ A @ X @ Y3 )
=> ( ( ord_less_eq @ nat @ N @ M )
=> ( dvd_dvd @ A @ ( power_power @ A @ X @ N ) @ ( power_power @ A @ Y3 @ M ) ) ) ) ) ).
% dvd_power_le
thf(fact_1787_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_1788_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_1789_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_1790_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_1791_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_1792_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_1793_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_1794_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_1795_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_1796_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_1797_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_1798_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_1799_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_1800_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_1801_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_1802_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_1803_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_1804_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_1805_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_1806_fact__le__power,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N: nat] : ( ord_less_eq @ A @ ( semiring_char_0_fact @ A @ N ) @ ( semiring_1_of_nat @ A @ ( power_power @ nat @ N @ N ) ) ) ) ).
% fact_le_power
thf(fact_1807_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_1808_bounded__Max__nat,axiom,
! [P: nat > $o,X: nat,M7: nat] :
( ( P @ X )
=> ( ! [X3: nat] :
( ( P @ X3 )
=> ( ord_less_eq @ nat @ X3 @ M7 ) )
=> ~ ! [M6: nat] :
( ( P @ M6 )
=> ~ ! [X4: nat] :
( ( P @ X4 )
=> ( ord_less_eq @ nat @ X4 @ M6 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_1809_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_1810_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_1811_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_1812_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_1813_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_1814_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_1815_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_1816_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_1817_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_1818_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_1819_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_1820_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_1821_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_1822_gbinomial__ge__n__over__k__pow__k,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [K: nat,A3: A] :
( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ K ) @ A3 )
=> ( ord_less_eq @ A @ ( power_power @ A @ ( divide_divide @ A @ A3 @ ( semiring_1_of_nat @ A @ K ) ) @ K ) @ ( gbinomial @ A @ A3 @ K ) ) ) ) ).
% gbinomial_ge_n_over_k_pow_k
thf(fact_1823_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_1824_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_1825_fact__div__fact__le__pow,axiom,
! [R: nat,N: nat] :
( ( ord_less_eq @ nat @ R @ N )
=> ( ord_less_eq @ nat @ ( divide_divide @ nat @ ( semiring_char_0_fact @ nat @ N ) @ ( semiring_char_0_fact @ nat @ ( minus_minus @ nat @ N @ R ) ) ) @ ( power_power @ nat @ N @ R ) ) ) ).
% fact_div_fact_le_pow
thf(fact_1826_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_1827_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_1828_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_1829_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_1830_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_1831_power__eq__if,axiom,
! [A: $tType] :
( ( power @ A )
=> ( ( power_power @ A )
= ( ^ [P6: A,M2: nat] :
( if @ A
@ ( M2
= ( zero_zero @ nat ) )
@ ( one_one @ A )
@ ( times_times @ A @ P6 @ ( power_power @ A @ P6 @ ( minus_minus @ nat @ M2 @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% power_eq_if
thf(fact_1832_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_1833_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_1834_gbinomial__negated__upper,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( ( gbinomial @ A )
= ( ^ [A5: 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 ) @ A5 ) @ ( one_one @ A ) ) @ K3 ) ) ) ) ) ).
% gbinomial_negated_upper
thf(fact_1835_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_1836_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_1837_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_1838_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_1839_ceiling__divide__lower,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Q4: A,P5: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Q4 )
=> ( ord_less @ A @ ( times_times @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ ( divide_divide @ A @ P5 @ Q4 ) ) ) @ ( one_one @ A ) ) @ Q4 ) @ P5 ) ) ) ).
% ceiling_divide_lower
thf(fact_1840_power_Opower__eq__if,axiom,
! [A: $tType] :
( ( power2 @ A )
= ( ^ [One: A,Times: A > A > A,P6: A,M2: nat] :
( if @ A
@ ( M2
= ( zero_zero @ nat ) )
@ One
@ ( Times @ P6 @ ( power2 @ A @ One @ Times @ P6 @ ( minus_minus @ nat @ M2 @ ( one_one @ nat ) ) ) ) ) ) ) ).
% power.power_eq_if
thf(fact_1841_le__divide__eq__numeral_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [W2: num,B2: A,C2: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ ( divide_divide @ A @ B2 @ C2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C2 ) @ B2 ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B2 @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C2 ) ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% le_divide_eq_numeral(2)
thf(fact_1842_divide__le__eq__numeral_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,C2: A,W2: num] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ C2 ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ B2 @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C2 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C2 ) @ B2 ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ) ) ) ) ).
% divide_le_eq_numeral(2)
thf(fact_1843_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_1844_numeral__eq__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [M: num,N: num] :
( ( ( numeral_numeral @ A @ M )
= ( numeral_numeral @ A @ N ) )
= ( M = N ) ) ) ).
% numeral_eq_iff
thf(fact_1845_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_1846_of__int__eq__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [W2: int,Z3: int] :
( ( ( ring_1_of_int @ A @ W2 )
= ( ring_1_of_int @ A @ Z3 ) )
= ( W2 = Z3 ) ) ) ).
% of_int_eq_iff
thf(fact_1847_nat__numeral,axiom,
! [K: num] :
( ( nat2 @ ( numeral_numeral @ int @ K ) )
= ( numeral_numeral @ nat @ K ) ) ).
% nat_numeral
thf(fact_1848_numeral__le__iff,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [M: num,N: num] :
( ( ord_less_eq @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N ) )
= ( ord_less_eq @ num @ M @ N ) ) ) ).
% numeral_le_iff
thf(fact_1849_numeral__less__iff,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [M: num,N: num] :
( ( ord_less @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N ) )
= ( ord_less @ num @ M @ N ) ) ) ).
% numeral_less_iff
thf(fact_1850_mult__numeral__left__semiring__numeral,axiom,
! [A: $tType] :
( ( semiring_numeral @ A )
=> ! [V: num,W2: num,Z3: A] :
( ( times_times @ A @ ( numeral_numeral @ A @ V ) @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ Z3 ) )
= ( times_times @ A @ ( numeral_numeral @ A @ ( times_times @ num @ V @ W2 ) ) @ Z3 ) ) ) ).
% mult_numeral_left_semiring_numeral
thf(fact_1851_numeral__times__numeral,axiom,
! [A: $tType] :
( ( semiring_numeral @ A )
=> ! [M: num,N: num] :
( ( times_times @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N ) )
= ( numeral_numeral @ A @ ( times_times @ num @ M @ N ) ) ) ) ).
% numeral_times_numeral
thf(fact_1852_add__numeral__left,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ! [V: num,W2: num,Z3: A] :
( ( plus_plus @ A @ ( numeral_numeral @ A @ V ) @ ( plus_plus @ A @ ( numeral_numeral @ A @ W2 ) @ Z3 ) )
= ( plus_plus @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ V @ W2 ) ) @ Z3 ) ) ) ).
% add_numeral_left
thf(fact_1853_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_1854_neg__numeral__eq__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [M: num,N: num] :
( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( M = N ) ) ) ).
% neg_numeral_eq_iff
thf(fact_1855_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_1856_power__add__numeral,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A3: A,M: num,N: num] :
( ( times_times @ A @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ M ) ) @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ N ) ) )
= ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ ( plus_plus @ num @ M @ N ) ) ) ) ) ).
% power_add_numeral
thf(fact_1857_power__add__numeral2,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A3: A,M: num,N: num,B2: A] :
( ( times_times @ A @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ M ) ) @ ( times_times @ A @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ N ) ) @ B2 ) )
= ( times_times @ A @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ ( plus_plus @ num @ M @ N ) ) ) @ B2 ) ) ) ).
% power_add_numeral2
thf(fact_1858_of__nat__numeral,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [N: num] :
( ( semiring_1_of_nat @ A @ ( numeral_numeral @ nat @ N ) )
= ( numeral_numeral @ A @ N ) ) ) ).
% of_nat_numeral
thf(fact_1859_abs__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: num] :
( ( abs_abs @ A @ ( numeral_numeral @ A @ N ) )
= ( numeral_numeral @ A @ N ) ) ) ).
% abs_numeral
thf(fact_1860_of__int__eq__numeral__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [Z3: int,N: num] :
( ( ( ring_1_of_int @ A @ Z3 )
= ( numeral_numeral @ A @ N ) )
= ( Z3
= ( numeral_numeral @ int @ N ) ) ) ) ).
% of_int_eq_numeral_iff
thf(fact_1861_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_1862_ceiling__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V: num] :
( ( archimedean_ceiling @ A @ ( numeral_numeral @ A @ V ) )
= ( numeral_numeral @ int @ V ) ) ) ).
% ceiling_numeral
thf(fact_1863_euclidean__size__numeral,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [K: num] :
( ( euclid6346220572633701492n_size @ A @ ( numeral_numeral @ A @ K ) )
= ( numeral_numeral @ nat @ K ) ) ) ).
% euclidean_size_numeral
thf(fact_1864_of__int__fact,axiom,
! [A: $tType] :
( ( ( semiring_char_0 @ A )
& ( ring_1 @ A ) )
=> ! [N: nat] :
( ( ring_1_of_int @ A @ ( semiring_char_0_fact @ int @ N ) )
= ( semiring_char_0_fact @ A @ N ) ) ) ).
% of_int_fact
thf(fact_1865_ceiling__of__int,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z3: int] :
( ( archimedean_ceiling @ A @ ( ring_1_of_int @ A @ Z3 ) )
= Z3 ) ) ).
% ceiling_of_int
thf(fact_1866_round__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [N: num] :
( ( archimedean_round @ A @ ( numeral_numeral @ A @ N ) )
= ( numeral_numeral @ int @ N ) ) ) ).
% round_numeral
thf(fact_1867_round__of__int,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [N: int] :
( ( archimedean_round @ A @ ( ring_1_of_int @ A @ N ) )
= N ) ) ).
% round_of_int
thf(fact_1868_neg__numeral__le__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M: num,N: num] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( ord_less_eq @ num @ N @ M ) ) ) ).
% neg_numeral_le_iff
thf(fact_1869_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_1870_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_1871_neg__numeral__less__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M: num,N: num] :
( ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( ord_less @ num @ N @ M ) ) ) ).
% neg_numeral_less_iff
thf(fact_1872_right__diff__distrib__numeral,axiom,
! [A: $tType] :
( ( ( numeral @ A )
& ( ring @ A ) )
=> ! [V: num,B2: A,C2: A] :
( ( times_times @ A @ ( numeral_numeral @ A @ V ) @ ( minus_minus @ A @ B2 @ C2 ) )
= ( minus_minus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ V ) @ B2 ) @ ( times_times @ A @ ( numeral_numeral @ A @ V ) @ C2 ) ) ) ) ).
% right_diff_distrib_numeral
thf(fact_1873_left__diff__distrib__numeral,axiom,
! [A: $tType] :
( ( ( numeral @ A )
& ( ring @ A ) )
=> ! [A3: A,B2: A,V: num] :
( ( times_times @ A @ ( minus_minus @ A @ A3 @ B2 ) @ ( numeral_numeral @ A @ V ) )
= ( minus_minus @ A @ ( times_times @ A @ A3 @ ( numeral_numeral @ A @ V ) ) @ ( times_times @ A @ B2 @ ( numeral_numeral @ A @ V ) ) ) ) ) ).
% left_diff_distrib_numeral
thf(fact_1874_mult__neg__numeral__simps_I1_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [M: num,N: num] :
( ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( numeral_numeral @ A @ ( times_times @ num @ M @ N ) ) ) ) ).
% mult_neg_numeral_simps(1)
thf(fact_1875_mult__neg__numeral__simps_I2_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [M: num,N: num] :
( ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( numeral_numeral @ A @ N ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( times_times @ num @ M @ N ) ) ) ) ) ).
% mult_neg_numeral_simps(2)
thf(fact_1876_mult__neg__numeral__simps_I3_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [M: num,N: num] :
( ( times_times @ A @ ( numeral_numeral @ A @ M ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( times_times @ num @ M @ N ) ) ) ) ) ).
% mult_neg_numeral_simps(3)
thf(fact_1877_semiring__norm_I172_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [V: num,W2: num,Y3: A] :
( ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ Y3 ) )
= ( times_times @ A @ ( numeral_numeral @ A @ ( times_times @ num @ V @ W2 ) ) @ Y3 ) ) ) ).
% semiring_norm(172)
thf(fact_1878_semiring__norm_I171_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [V: num,W2: num,Y3: A] :
( ( times_times @ A @ ( numeral_numeral @ A @ V ) @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ Y3 ) )
= ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( times_times @ num @ V @ W2 ) ) ) @ Y3 ) ) ) ).
% semiring_norm(171)
thf(fact_1879_semiring__norm_I170_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [V: num,W2: num,Y3: A] :
( ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ Y3 ) )
= ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( times_times @ num @ V @ W2 ) ) ) @ Y3 ) ) ) ).
% semiring_norm(170)
thf(fact_1880_semiring__norm_I168_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [V: num,W2: num,Y3: A] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) @ ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ Y3 ) )
= ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ V @ W2 ) ) ) @ Y3 ) ) ) ).
% semiring_norm(168)
thf(fact_1881_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_1882_diff__numeral__simps_I3_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M: num,N: num] :
( ( minus_minus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( numeral_numeral @ A @ N ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ M @ N ) ) ) ) ) ).
% diff_numeral_simps(3)
thf(fact_1883_diff__numeral__simps_I2_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M: num,N: num] :
( ( minus_minus @ A @ ( numeral_numeral @ A @ M ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( numeral_numeral @ A @ ( plus_plus @ num @ M @ N ) ) ) ) ).
% diff_numeral_simps(2)
thf(fact_1884_abs__neg__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: num] :
( ( abs_abs @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( numeral_numeral @ A @ N ) ) ) ).
% abs_neg_numeral
thf(fact_1885_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_1886_of__int__0__eq__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [Z3: int] :
( ( ( zero_zero @ A )
= ( ring_1_of_int @ A @ Z3 ) )
= ( Z3
= ( zero_zero @ int ) ) ) ) ).
% of_int_0_eq_iff
thf(fact_1887_of__int__eq__0__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [Z3: int] :
( ( ( ring_1_of_int @ A @ Z3 )
= ( zero_zero @ A ) )
= ( Z3
= ( zero_zero @ int ) ) ) ) ).
% of_int_eq_0_iff
thf(fact_1888_of__int__le__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [W2: int,Z3: int] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ W2 ) @ ( ring_1_of_int @ A @ Z3 ) )
= ( ord_less_eq @ int @ W2 @ Z3 ) ) ) ).
% of_int_le_iff
thf(fact_1889_of__int__less__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [W2: int,Z3: int] :
( ( ord_less @ A @ ( ring_1_of_int @ A @ W2 ) @ ( ring_1_of_int @ A @ Z3 ) )
= ( ord_less @ int @ W2 @ Z3 ) ) ) ).
% of_int_less_iff
thf(fact_1890_of__int__eq__1__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [Z3: int] :
( ( ( ring_1_of_int @ A @ Z3 )
= ( one_one @ A ) )
= ( Z3
= ( one_one @ int ) ) ) ) ).
% of_int_eq_1_iff
thf(fact_1891_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_1892_of__int__mult,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [W2: int,Z3: int] :
( ( ring_1_of_int @ A @ ( times_times @ int @ W2 @ Z3 ) )
= ( times_times @ A @ ( ring_1_of_int @ A @ W2 ) @ ( ring_1_of_int @ A @ Z3 ) ) ) ) ).
% of_int_mult
thf(fact_1893_of__int__add,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [W2: int,Z3: int] :
( ( ring_1_of_int @ A @ ( plus_plus @ int @ W2 @ Z3 ) )
= ( plus_plus @ A @ ( ring_1_of_int @ A @ W2 ) @ ( ring_1_of_int @ A @ Z3 ) ) ) ) ).
% of_int_add
thf(fact_1894_of__int__minus,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [Z3: int] :
( ( ring_1_of_int @ A @ ( uminus_uminus @ int @ Z3 ) )
= ( uminus_uminus @ A @ ( ring_1_of_int @ A @ Z3 ) ) ) ) ).
% of_int_minus
thf(fact_1895_nat__neg__numeral,axiom,
! [K: num] :
( ( nat2 @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) )
= ( zero_zero @ nat ) ) ).
% nat_neg_numeral
thf(fact_1896_of__int__diff,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [W2: int,Z3: int] :
( ( ring_1_of_int @ A @ ( minus_minus @ int @ W2 @ Z3 ) )
= ( minus_minus @ A @ ( ring_1_of_int @ A @ W2 ) @ ( ring_1_of_int @ A @ Z3 ) ) ) ) ).
% of_int_diff
thf(fact_1897_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_1898_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_1899_diff__nat__numeral,axiom,
! [V: num,V2: num] :
( ( minus_minus @ nat @ ( numeral_numeral @ nat @ V ) @ ( numeral_numeral @ nat @ V2 ) )
= ( nat2 @ ( minus_minus @ int @ ( numeral_numeral @ int @ V ) @ ( numeral_numeral @ int @ V2 ) ) ) ) ).
% diff_nat_numeral
thf(fact_1900_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_1901_of__int__power__eq__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [X: int,B2: int,W2: nat] :
( ( ( ring_1_of_int @ A @ X )
= ( power_power @ A @ ( ring_1_of_int @ A @ B2 ) @ W2 ) )
= ( X
= ( power_power @ int @ B2 @ W2 ) ) ) ) ).
% of_int_power_eq_of_int_cancel_iff
thf(fact_1902_of__int__eq__of__int__power__cancel__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [B2: int,W2: nat,X: int] :
( ( ( power_power @ A @ ( ring_1_of_int @ A @ B2 ) @ W2 )
= ( ring_1_of_int @ A @ X ) )
= ( ( power_power @ int @ B2 @ W2 )
= X ) ) ) ).
% of_int_eq_of_int_power_cancel_iff
thf(fact_1903_of__int__power,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [Z3: int,N: nat] :
( ( ring_1_of_int @ A @ ( power_power @ int @ Z3 @ N ) )
= ( power_power @ A @ ( ring_1_of_int @ A @ Z3 ) @ N ) ) ) ).
% of_int_power
thf(fact_1904_frac__of__int,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z3: int] :
( ( archimedean_frac @ A @ ( ring_1_of_int @ A @ Z3 ) )
= ( zero_zero @ A ) ) ) ).
% frac_of_int
thf(fact_1905_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_1906_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_1907_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_1908_le__divide__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A,W2: num] :
( ( ord_less_eq @ A @ A3 @ ( divide_divide @ A @ B2 @ ( numeral_numeral @ A @ W2 ) ) )
= ( ord_less_eq @ A @ ( times_times @ A @ A3 @ ( numeral_numeral @ A @ W2 ) ) @ B2 ) ) ) ).
% le_divide_eq_numeral1(1)
thf(fact_1909_divide__le__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,W2: num,A3: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ ( numeral_numeral @ A @ W2 ) ) @ A3 )
= ( ord_less_eq @ A @ B2 @ ( times_times @ A @ A3 @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ).
% divide_le_eq_numeral1(1)
thf(fact_1910_divide__eq__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,W2: num,A3: A] :
( ( ( divide_divide @ A @ B2 @ ( numeral_numeral @ A @ W2 ) )
= A3 )
= ( ( ( ( numeral_numeral @ A @ W2 )
!= ( zero_zero @ A ) )
=> ( B2
= ( times_times @ A @ A3 @ ( numeral_numeral @ A @ W2 ) ) ) )
& ( ( ( numeral_numeral @ A @ W2 )
= ( zero_zero @ A ) )
=> ( A3
= ( zero_zero @ A ) ) ) ) ) ) ).
% divide_eq_eq_numeral1(1)
thf(fact_1911_eq__divide__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,B2: A,W2: num] :
( ( A3
= ( divide_divide @ A @ B2 @ ( numeral_numeral @ A @ W2 ) ) )
= ( ( ( ( numeral_numeral @ A @ W2 )
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ A3 @ ( numeral_numeral @ A @ W2 ) )
= B2 ) )
& ( ( ( numeral_numeral @ A @ W2 )
= ( zero_zero @ A ) )
=> ( A3
= ( zero_zero @ A ) ) ) ) ) ) ).
% eq_divide_eq_numeral1(1)
thf(fact_1912_less__divide__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A,W2: num] :
( ( ord_less @ A @ A3 @ ( divide_divide @ A @ B2 @ ( numeral_numeral @ A @ W2 ) ) )
= ( ord_less @ A @ ( times_times @ A @ A3 @ ( numeral_numeral @ A @ W2 ) ) @ B2 ) ) ) ).
% less_divide_eq_numeral1(1)
thf(fact_1913_divide__less__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,W2: num,A3: A] :
( ( ord_less @ A @ ( divide_divide @ A @ B2 @ ( numeral_numeral @ A @ W2 ) ) @ A3 )
= ( ord_less @ A @ B2 @ ( times_times @ A @ A3 @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ).
% divide_less_eq_numeral1(1)
thf(fact_1914_of__int__le__numeral__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z3: int,N: num] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z3 ) @ ( numeral_numeral @ A @ N ) )
= ( ord_less_eq @ int @ Z3 @ ( numeral_numeral @ int @ N ) ) ) ) ).
% of_int_le_numeral_iff
thf(fact_1915_of__int__numeral__le__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: num,Z3: int] :
( ( ord_less_eq @ A @ ( numeral_numeral @ A @ N ) @ ( ring_1_of_int @ A @ Z3 ) )
= ( ord_less_eq @ int @ ( numeral_numeral @ int @ N ) @ Z3 ) ) ) ).
% of_int_numeral_le_iff
thf(fact_1916_of__int__numeral__less__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: num,Z3: int] :
( ( ord_less @ A @ ( numeral_numeral @ A @ N ) @ ( ring_1_of_int @ A @ Z3 ) )
= ( ord_less @ int @ ( numeral_numeral @ int @ N ) @ Z3 ) ) ) ).
% of_int_numeral_less_iff
thf(fact_1917_of__int__less__numeral__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z3: int,N: num] :
( ( ord_less @ A @ ( ring_1_of_int @ A @ Z3 ) @ ( numeral_numeral @ A @ N ) )
= ( ord_less @ int @ Z3 @ ( numeral_numeral @ int @ N ) ) ) ) ).
% of_int_less_numeral_iff
thf(fact_1918_ceiling__le__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,V: num] :
( ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X ) @ ( numeral_numeral @ int @ V ) )
= ( ord_less_eq @ A @ X @ ( numeral_numeral @ A @ V ) ) ) ) ).
% ceiling_le_numeral
thf(fact_1919_numeral__less__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V: num,X: A] :
( ( ord_less @ int @ ( numeral_numeral @ int @ V ) @ ( archimedean_ceiling @ A @ X ) )
= ( ord_less @ A @ ( numeral_numeral @ A @ V ) @ X ) ) ) ).
% numeral_less_ceiling
thf(fact_1920_real__of__nat__eq__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [Y3: nat,X: num,N: nat] :
( ( ( semiring_1_of_nat @ A @ Y3 )
= ( power_power @ A @ ( numeral_numeral @ A @ X ) @ N ) )
= ( Y3
= ( power_power @ nat @ ( numeral_numeral @ nat @ X ) @ N ) ) ) ) ).
% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_1921_numeral__power__eq__of__nat__cancel__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [X: num,N: nat,Y3: nat] :
( ( ( power_power @ A @ ( numeral_numeral @ A @ X ) @ N )
= ( semiring_1_of_nat @ A @ Y3 ) )
= ( ( power_power @ nat @ ( numeral_numeral @ nat @ X ) @ N )
= Y3 ) ) ) ).
% numeral_power_eq_of_nat_cancel_iff
thf(fact_1922_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_1923_ceiling__neg__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V: num] :
( ( archimedean_ceiling @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) )
= ( uminus_uminus @ int @ ( numeral_numeral @ int @ V ) ) ) ) ).
% ceiling_neg_numeral
thf(fact_1924_ceiling__diff__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,V: num] :
( ( archimedean_ceiling @ A @ ( minus_minus @ A @ X @ ( numeral_numeral @ A @ V ) ) )
= ( minus_minus @ int @ ( archimedean_ceiling @ A @ X ) @ ( numeral_numeral @ int @ V ) ) ) ) ).
% ceiling_diff_numeral
thf(fact_1925_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_1926_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_1927_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_1928_ceiling__add__of__int,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Z3: int] :
( ( archimedean_ceiling @ A @ ( plus_plus @ A @ X @ ( ring_1_of_int @ A @ Z3 ) ) )
= ( plus_plus @ int @ ( archimedean_ceiling @ A @ X ) @ Z3 ) ) ) ).
% ceiling_add_of_int
thf(fact_1929_ceiling__diff__of__int,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Z3: int] :
( ( archimedean_ceiling @ A @ ( minus_minus @ A @ X @ ( ring_1_of_int @ A @ Z3 ) ) )
= ( minus_minus @ int @ ( archimedean_ceiling @ A @ X ) @ Z3 ) ) ) ).
% ceiling_diff_of_int
thf(fact_1930_round__neg__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [N: num] :
( ( archimedean_round @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) ) ) ).
% round_neg_numeral
thf(fact_1931_le__divide__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A,W2: num] :
( ( ord_less_eq @ A @ A3 @ ( divide_divide @ A @ B2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) )
= ( ord_less_eq @ A @ B2 @ ( times_times @ A @ A3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ) ).
% le_divide_eq_numeral1(2)
thf(fact_1932_divide__le__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,W2: num,A3: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) @ A3 )
= ( ord_less_eq @ A @ ( times_times @ A @ A3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) @ B2 ) ) ) ).
% divide_le_eq_numeral1(2)
thf(fact_1933_divide__eq__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,W2: num,A3: A] :
( ( ( divide_divide @ A @ B2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) )
= A3 )
= ( ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) )
!= ( zero_zero @ A ) )
=> ( B2
= ( times_times @ A @ A3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) ) )
& ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) )
= ( zero_zero @ A ) )
=> ( A3
= ( zero_zero @ A ) ) ) ) ) ) ).
% divide_eq_eq_numeral1(2)
thf(fact_1934_eq__divide__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,B2: A,W2: num] :
( ( A3
= ( divide_divide @ A @ B2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) )
= ( ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) )
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ A3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) )
= B2 ) )
& ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) )
= ( zero_zero @ A ) )
=> ( A3
= ( zero_zero @ A ) ) ) ) ) ) ).
% eq_divide_eq_numeral1(2)
thf(fact_1935_less__divide__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A,W2: num] :
( ( ord_less @ A @ A3 @ ( divide_divide @ A @ B2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) )
= ( ord_less @ A @ B2 @ ( times_times @ A @ A3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ) ).
% less_divide_eq_numeral1(2)
thf(fact_1936_divide__less__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,W2: num,A3: A] :
( ( ord_less @ A @ ( divide_divide @ A @ B2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) @ A3 )
= ( ord_less @ A @ ( times_times @ A @ A3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) @ B2 ) ) ) ).
% divide_less_eq_numeral1(2)
thf(fact_1937_dbl__dec__simps_I1_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [K: num] :
( ( neg_numeral_dbl_dec @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ K ) ) )
= ( uminus_uminus @ A @ ( neg_numeral_dbl_inc @ A @ ( numeral_numeral @ A @ K ) ) ) ) ) ).
% dbl_dec_simps(1)
thf(fact_1938_dbl__inc__simps_I1_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [K: num] :
( ( neg_numeral_dbl_inc @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ K ) ) )
= ( uminus_uminus @ A @ ( neg_numeral_dbl_dec @ A @ ( numeral_numeral @ A @ K ) ) ) ) ) ).
% dbl_inc_simps(1)
thf(fact_1939_of__int__le__0__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z3: int] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z3 ) @ ( zero_zero @ A ) )
= ( ord_less_eq @ int @ Z3 @ ( zero_zero @ int ) ) ) ) ).
% of_int_le_0_iff
thf(fact_1940_of__int__0__le__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z3: int] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( ring_1_of_int @ A @ Z3 ) )
= ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z3 ) ) ) ).
% of_int_0_le_iff
thf(fact_1941_of__int__less__0__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z3: int] :
( ( ord_less @ A @ ( ring_1_of_int @ A @ Z3 ) @ ( zero_zero @ A ) )
= ( ord_less @ int @ Z3 @ ( zero_zero @ int ) ) ) ) ).
% of_int_less_0_iff
thf(fact_1942_of__int__0__less__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z3: int] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( ring_1_of_int @ A @ Z3 ) )
= ( ord_less @ int @ ( zero_zero @ int ) @ Z3 ) ) ) ).
% of_int_0_less_iff
thf(fact_1943_of__int__1__le__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z3: int] :
( ( ord_less_eq @ A @ ( one_one @ A ) @ ( ring_1_of_int @ A @ Z3 ) )
= ( ord_less_eq @ int @ ( one_one @ int ) @ Z3 ) ) ) ).
% of_int_1_le_iff
thf(fact_1944_of__int__le__1__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z3: int] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z3 ) @ ( one_one @ A ) )
= ( ord_less_eq @ int @ Z3 @ ( one_one @ int ) ) ) ) ).
% of_int_le_1_iff
thf(fact_1945_of__int__1__less__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z3: int] :
( ( ord_less @ A @ ( one_one @ A ) @ ( ring_1_of_int @ A @ Z3 ) )
= ( ord_less @ int @ ( one_one @ int ) @ Z3 ) ) ) ).
% of_int_1_less_iff
thf(fact_1946_of__int__less__1__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z3: int] :
( ( ord_less @ A @ ( ring_1_of_int @ A @ Z3 ) @ ( one_one @ A ) )
= ( ord_less @ int @ Z3 @ ( one_one @ int ) ) ) ) ).
% of_int_less_1_iff
thf(fact_1947_of__nat__nat,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [Z3: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z3 )
=> ( ( semiring_1_of_nat @ A @ ( nat2 @ Z3 ) )
= ( ring_1_of_int @ A @ Z3 ) ) ) ) ).
% of_nat_nat
thf(fact_1948_of__int__le__of__int__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [B2: int,W2: nat,X: int] :
( ( ord_less_eq @ A @ ( power_power @ A @ ( ring_1_of_int @ A @ B2 ) @ W2 ) @ ( ring_1_of_int @ A @ X ) )
= ( ord_less_eq @ int @ ( power_power @ int @ B2 @ W2 ) @ X ) ) ) ).
% of_int_le_of_int_power_cancel_iff
thf(fact_1949_of__int__power__le__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: int,B2: int,W2: nat] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ X ) @ ( power_power @ A @ ( ring_1_of_int @ A @ B2 ) @ W2 ) )
= ( ord_less_eq @ int @ X @ ( power_power @ int @ B2 @ W2 ) ) ) ) ).
% of_int_power_le_of_int_cancel_iff
thf(fact_1950_of__int__power__less__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: int,B2: int,W2: nat] :
( ( ord_less @ A @ ( ring_1_of_int @ A @ X ) @ ( power_power @ A @ ( ring_1_of_int @ A @ B2 ) @ W2 ) )
= ( ord_less @ int @ X @ ( power_power @ int @ B2 @ W2 ) ) ) ) ).
% of_int_power_less_of_int_cancel_iff
thf(fact_1951_of__int__less__of__int__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [B2: int,W2: nat,X: int] :
( ( ord_less @ A @ ( power_power @ A @ ( ring_1_of_int @ A @ B2 ) @ W2 ) @ ( ring_1_of_int @ A @ X ) )
= ( ord_less @ int @ ( power_power @ int @ B2 @ W2 ) @ X ) ) ) ).
% of_int_less_of_int_power_cancel_iff
thf(fact_1952_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_1953_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_1954_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_1955_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_1956_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_1957_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_1958_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_1959_numeral__power__less__of__nat__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [I: num,N: nat,X: nat] :
( ( ord_less @ A @ ( power_power @ A @ ( numeral_numeral @ A @ I ) @ N ) @ ( semiring_1_of_nat @ A @ X ) )
= ( ord_less @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ I ) @ N ) @ X ) ) ) ).
% numeral_power_less_of_nat_cancel_iff
thf(fact_1960_of__nat__less__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X: nat,I: num,N: nat] :
( ( ord_less @ A @ ( semiring_1_of_nat @ A @ X ) @ ( power_power @ A @ ( numeral_numeral @ A @ I ) @ N ) )
= ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ I ) @ N ) ) ) ) ).
% of_nat_less_numeral_power_cancel_iff
thf(fact_1961_of__nat__le__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X: nat,I: num,N: nat] :
( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ X ) @ ( power_power @ A @ ( numeral_numeral @ A @ I ) @ N ) )
= ( ord_less_eq @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ I ) @ N ) ) ) ) ).
% of_nat_le_numeral_power_cancel_iff
thf(fact_1962_numeral__power__le__of__nat__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [I: num,N: nat,X: nat] :
( ( ord_less_eq @ A @ ( power_power @ A @ ( numeral_numeral @ A @ I ) @ N ) @ ( semiring_1_of_nat @ A @ X ) )
= ( ord_less_eq @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ I ) @ N ) @ X ) ) ) ).
% numeral_power_le_of_nat_cancel_iff
thf(fact_1963_ceiling__le__neg__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,V: num] :
( ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V ) ) )
= ( ord_less_eq @ A @ X @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) ) ) ) ).
% ceiling_le_neg_numeral
thf(fact_1964_neg__numeral__less__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V: num,X: A] :
( ( ord_less @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V ) ) @ ( archimedean_ceiling @ A @ X ) )
= ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) @ X ) ) ) ).
% neg_numeral_less_ceiling
thf(fact_1965_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_1966_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_1967_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_1968_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_1969_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_1970_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_1971_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_1972_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_1973_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_1974_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_1975_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_1976_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_1977_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_1978_nat__numeral__as__int,axiom,
( ( numeral_numeral @ nat )
= ( ^ [I5: num] : ( nat2 @ ( numeral_numeral @ int @ I5 ) ) ) ) ).
% nat_numeral_as_int
thf(fact_1979_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_1980_ex__le__of__int,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X: A] :
? [Z4: int] : ( ord_less_eq @ A @ X @ ( ring_1_of_int @ A @ Z4 ) ) ) ).
% ex_le_of_int
thf(fact_1981_ex__of__int__less,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X: A] :
? [Z4: int] : ( ord_less @ A @ ( ring_1_of_int @ A @ Z4 ) @ X ) ) ).
% ex_of_int_less
thf(fact_1982_ex__less__of__int,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X: A] :
? [Z4: int] : ( ord_less @ A @ X @ ( ring_1_of_int @ A @ Z4 ) ) ) ).
% ex_less_of_int
thf(fact_1983_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_1984_zero__neq__numeral,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [N: num] :
( ( zero_zero @ A )
!= ( numeral_numeral @ A @ N ) ) ) ).
% zero_neq_numeral
thf(fact_1985_numeral__neq__neg__numeral,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [M: num,N: num] :
( ( numeral_numeral @ A @ M )
!= ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) ) ) ).
% numeral_neq_neg_numeral
thf(fact_1986_neg__numeral__neq__numeral,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [M: num,N: num] :
( ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) )
!= ( numeral_numeral @ A @ N ) ) ) ).
% neg_numeral_neq_numeral
thf(fact_1987_div__mult2__numeral__eq,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A3: A,K: num,L: num] :
( ( divide_divide @ A @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ K ) ) @ ( numeral_numeral @ A @ L ) )
= ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( times_times @ num @ K @ L ) ) ) ) ) ).
% div_mult2_numeral_eq
thf(fact_1988_pochhammer__of__int,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [X: int,N: nat] :
( ( comm_s3205402744901411588hammer @ A @ ( ring_1_of_int @ A @ X ) @ N )
= ( ring_1_of_int @ A @ ( comm_s3205402744901411588hammer @ int @ X @ N ) ) ) ) ).
% pochhammer_of_int
thf(fact_1989_Ints__of__int,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [Z3: int] : ( member @ A @ ( ring_1_of_int @ A @ Z3 ) @ ( ring_1_Ints @ A ) ) ) ).
% Ints_of_int
thf(fact_1990_Ints__induct,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [Q4: A,P: A > $o] :
( ( member @ A @ Q4 @ ( ring_1_Ints @ A ) )
=> ( ! [Z4: int] : ( P @ ( ring_1_of_int @ A @ Z4 ) )
=> ( P @ Q4 ) ) ) ) ).
% Ints_induct
thf(fact_1991_Ints__cases,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [Q4: A] :
( ( member @ A @ Q4 @ ( ring_1_Ints @ A ) )
=> ~ ! [Z4: int] :
( Q4
!= ( ring_1_of_int @ A @ Z4 ) ) ) ) ).
% Ints_cases
thf(fact_1992_Ints__numeral,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [N: num] : ( member @ A @ ( numeral_numeral @ A @ N ) @ ( ring_1_Ints @ A ) ) ) ).
% Ints_numeral
thf(fact_1993_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_1994_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_1995_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_1996_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_1997_le__of__int__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] : ( ord_less_eq @ A @ X @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ X ) ) ) ) ).
% le_of_int_ceiling
thf(fact_1998_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_1999_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_2000_not__numeral__le__neg__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M: num,N: num] :
~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ M ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) ) ) ).
% not_numeral_le_neg_numeral
thf(fact_2001_neg__numeral__le__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M: num,N: num] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( numeral_numeral @ A @ N ) ) ) ).
% neg_numeral_le_numeral
thf(fact_2002_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_2003_not__numeral__less__neg__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M: num,N: num] :
~ ( ord_less @ A @ ( numeral_numeral @ A @ M ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) ) ) ).
% not_numeral_less_neg_numeral
thf(fact_2004_neg__numeral__less__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M: num,N: num] : ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( numeral_numeral @ A @ N ) ) ) ).
% neg_numeral_less_numeral
thf(fact_2005_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_2006_numeral__times__minus__swap,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [W2: num,X: A] :
( ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ ( uminus_uminus @ A @ X ) )
= ( times_times @ A @ X @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ).
% numeral_times_minus_swap
thf(fact_2007_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_2008_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_2009_power_Opower_Opower__Suc,axiom,
! [A: $tType,One2: A,Times2: A > A > A,A3: A,N: nat] :
( ( power2 @ A @ One2 @ Times2 @ A3 @ ( suc @ N ) )
= ( Times2 @ A3 @ ( power2 @ A @ One2 @ Times2 @ A3 @ N ) ) ) ).
% power.power.power_Suc
thf(fact_2010_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_2011_ceiling__le__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Z3: int] :
( ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X ) @ Z3 )
= ( ord_less_eq @ A @ X @ ( ring_1_of_int @ A @ Z3 ) ) ) ) ).
% ceiling_le_iff
thf(fact_2012_less__ceiling__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z3: int,X: A] :
( ( ord_less @ int @ Z3 @ ( archimedean_ceiling @ A @ X ) )
= ( ord_less @ A @ ( ring_1_of_int @ A @ Z3 ) @ X ) ) ) ).
% less_ceiling_iff
thf(fact_2013_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_2014_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_2015_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_2016_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_2017_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_2018_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_2019_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_2020_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_2021_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_2022_divide__eq__eq__numeral_I1_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,C2: A,W2: num] :
( ( ( divide_divide @ A @ B2 @ C2 )
= ( numeral_numeral @ A @ W2 ) )
= ( ( ( C2
!= ( zero_zero @ A ) )
=> ( B2
= ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C2 ) ) )
& ( ( C2
= ( zero_zero @ A ) )
=> ( ( numeral_numeral @ A @ W2 )
= ( zero_zero @ A ) ) ) ) ) ) ).
% divide_eq_eq_numeral(1)
thf(fact_2023_eq__divide__eq__numeral_I1_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [W2: num,B2: A,C2: A] :
( ( ( numeral_numeral @ A @ W2 )
= ( divide_divide @ A @ B2 @ C2 ) )
= ( ( ( C2
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C2 )
= B2 ) )
& ( ( C2
= ( zero_zero @ A ) )
=> ( ( numeral_numeral @ A @ W2 )
= ( zero_zero @ A ) ) ) ) ) ) ).
% eq_divide_eq_numeral(1)
thf(fact_2024_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_2025_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_2026_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_2027_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_2028_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_2029_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_2030_of__int__nonneg,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z3: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z3 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( ring_1_of_int @ A @ Z3 ) ) ) ) ).
% of_int_nonneg
thf(fact_2031_of__int__pos,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z3: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ Z3 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( ring_1_of_int @ A @ Z3 ) ) ) ) ).
% of_int_pos
thf(fact_2032_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_2033_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_2034_floor__exists1,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X: A] :
? [X3: int] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ X3 ) @ X )
& ( ord_less @ A @ X @ ( ring_1_of_int @ A @ ( plus_plus @ int @ X3 @ ( one_one @ int ) ) ) )
& ! [Y4: int] :
( ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Y4 ) @ X )
& ( ord_less @ A @ X @ ( ring_1_of_int @ A @ ( plus_plus @ int @ Y4 @ ( one_one @ int ) ) ) ) )
=> ( Y4 = X3 ) ) ) ) ).
% floor_exists1
thf(fact_2035_floor__exists,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X: A] :
? [Z4: int] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z4 ) @ X )
& ( ord_less @ A @ X @ ( ring_1_of_int @ A @ ( plus_plus @ int @ Z4 @ ( one_one @ int ) ) ) ) ) ) ).
% floor_exists
thf(fact_2036_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_2037_divide__less__eq__numeral_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,C2: A,W2: num] :
( ( ord_less @ A @ ( divide_divide @ A @ B2 @ C2 ) @ ( numeral_numeral @ A @ W2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ B2 @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C2 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C2 ) @ B2 ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ) ) ) ).
% divide_less_eq_numeral(1)
thf(fact_2038_less__divide__eq__numeral_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [W2: num,B2: A,C2: A] :
( ( ord_less @ A @ ( numeral_numeral @ A @ W2 ) @ ( divide_divide @ A @ B2 @ C2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C2 ) @ B2 ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B2 @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C2 ) ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( numeral_numeral @ A @ W2 ) @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% less_divide_eq_numeral(1)
thf(fact_2039_divide__eq__eq__numeral_I2_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,C2: A,W2: num] :
( ( ( divide_divide @ A @ B2 @ C2 )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) )
= ( ( ( C2
!= ( zero_zero @ A ) )
=> ( B2
= ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C2 ) ) )
& ( ( C2
= ( zero_zero @ A ) )
=> ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) )
= ( zero_zero @ A ) ) ) ) ) ) ).
% divide_eq_eq_numeral(2)
thf(fact_2040_eq__divide__eq__numeral_I2_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [W2: num,B2: A,C2: A] :
( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) )
= ( divide_divide @ A @ B2 @ C2 ) )
= ( ( ( C2
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C2 )
= B2 ) )
& ( ( C2
= ( zero_zero @ A ) )
=> ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) )
= ( zero_zero @ A ) ) ) ) ) ) ).
% eq_divide_eq_numeral(2)
thf(fact_2041_round__diff__minimal,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z3: A,M: int] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ Z3 @ ( ring_1_of_int @ A @ ( archimedean_round @ A @ Z3 ) ) ) ) @ ( abs_abs @ A @ ( minus_minus @ A @ Z3 @ ( ring_1_of_int @ A @ M ) ) ) ) ) ).
% round_diff_minimal
thf(fact_2042_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_2043_ceiling__unique,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z3: int,X: A] :
( ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ Z3 ) @ ( one_one @ A ) ) @ X )
=> ( ( ord_less_eq @ A @ X @ ( ring_1_of_int @ A @ Z3 ) )
=> ( ( archimedean_ceiling @ A @ X )
= Z3 ) ) ) ) ).
% ceiling_unique
thf(fact_2044_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_2045_ceiling__split,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [P: int > $o,T2: A] :
( ( P @ ( archimedean_ceiling @ A @ T2 ) )
= ( ! [I5: int] :
( ( ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ I5 ) @ ( one_one @ A ) ) @ T2 )
& ( ord_less_eq @ A @ T2 @ ( ring_1_of_int @ A @ I5 ) ) )
=> ( P @ I5 ) ) ) ) ) ).
% ceiling_split
thf(fact_2046_ceiling__less__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Z3: int] :
( ( ord_less @ int @ ( archimedean_ceiling @ A @ X ) @ Z3 )
= ( ord_less_eq @ A @ X @ ( minus_minus @ A @ ( ring_1_of_int @ A @ Z3 ) @ ( one_one @ A ) ) ) ) ) ).
% ceiling_less_iff
thf(fact_2047_le__ceiling__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z3: int,X: A] :
( ( ord_less_eq @ int @ Z3 @ ( archimedean_ceiling @ A @ X ) )
= ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ Z3 ) @ ( one_one @ A ) ) @ X ) ) ) ).
% le_ceiling_iff
thf(fact_2048_divide__le__eq__numeral_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,C2: A,W2: num] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ C2 ) @ ( numeral_numeral @ A @ W2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ B2 @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C2 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C2 ) @ B2 ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ) ) ) ).
% divide_le_eq_numeral(1)
thf(fact_2049_le__divide__eq__numeral_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [W2: num,B2: A,C2: A] :
( ( ord_less_eq @ A @ ( numeral_numeral @ A @ W2 ) @ ( divide_divide @ A @ B2 @ C2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C2 ) @ B2 ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B2 @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C2 ) ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( numeral_numeral @ A @ W2 ) @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% le_divide_eq_numeral(1)
thf(fact_2050_divide__less__eq__numeral_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,C2: A,W2: num] :
( ( ord_less @ A @ ( divide_divide @ A @ B2 @ C2 ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ B2 @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C2 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C2 ) @ B2 ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ) ) ) ) ).
% divide_less_eq_numeral(2)
thf(fact_2051_less__divide__eq__numeral_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [W2: num,B2: A,C2: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ ( divide_divide @ A @ B2 @ C2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C2 ) @ B2 ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B2 @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C2 ) ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% less_divide_eq_numeral(2)
thf(fact_2052_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_2053_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_2054_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_2055_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_2056_ceiling__divide__upper,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Q4: A,P5: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Q4 )
=> ( ord_less_eq @ A @ P5 @ ( times_times @ A @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ ( divide_divide @ A @ P5 @ Q4 ) ) ) @ Q4 ) ) ) ) ).
% ceiling_divide_upper
thf(fact_2057_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_2058_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_2059_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_2060_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_2061_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_2062_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_2063_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_2064_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_2065_floor__of__int,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z3: int] :
( ( archim6421214686448440834_floor @ A @ ( ring_1_of_int @ A @ Z3 ) )
= Z3 ) ) ).
% floor_of_int
thf(fact_2066_floor__zero,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archim6421214686448440834_floor @ A @ ( zero_zero @ A ) )
= ( zero_zero @ int ) ) ) ).
% floor_zero
thf(fact_2067_floor__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V: num] :
( ( archim6421214686448440834_floor @ A @ ( numeral_numeral @ A @ V ) )
= ( numeral_numeral @ int @ V ) ) ) ).
% floor_numeral
thf(fact_2068_floor__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archim6421214686448440834_floor @ A @ ( one_one @ A ) )
= ( one_one @ int ) ) ) ).
% floor_one
thf(fact_2069_floor__of__nat,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [N: nat] :
( ( archim6421214686448440834_floor @ A @ ( semiring_1_of_nat @ A @ N ) )
= ( semiring_1_of_nat @ int @ N ) ) ) ).
% floor_of_nat
thf(fact_2070_floor__uminus__of__int,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z3: int] :
( ( archim6421214686448440834_floor @ A @ ( uminus_uminus @ A @ ( ring_1_of_int @ A @ Z3 ) ) )
= ( uminus_uminus @ int @ Z3 ) ) ) ).
% floor_uminus_of_int
thf(fact_2071_floor__diff__of__int,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Z3: int] :
( ( archim6421214686448440834_floor @ A @ ( minus_minus @ A @ X @ ( ring_1_of_int @ A @ Z3 ) ) )
= ( minus_minus @ int @ ( archim6421214686448440834_floor @ A @ X ) @ Z3 ) ) ) ).
% floor_diff_of_int
thf(fact_2072_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_2073_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_2074_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_2075_numeral__le__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V: num,X: A] :
( ( ord_less_eq @ int @ ( numeral_numeral @ int @ V ) @ ( archim6421214686448440834_floor @ A @ X ) )
= ( ord_less_eq @ A @ ( numeral_numeral @ A @ V ) @ X ) ) ) ).
% numeral_le_floor
thf(fact_2076_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_2077_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_2078_floor__less__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,V: num] :
( ( ord_less @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( numeral_numeral @ int @ V ) )
= ( ord_less @ A @ X @ ( numeral_numeral @ A @ V ) ) ) ) ).
% floor_less_numeral
thf(fact_2079_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_2080_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_2081_floor__neg__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V: num] :
( ( archim6421214686448440834_floor @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) )
= ( uminus_uminus @ int @ ( numeral_numeral @ int @ V ) ) ) ) ).
% floor_neg_numeral
thf(fact_2082_floor__diff__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,V: num] :
( ( archim6421214686448440834_floor @ A @ ( minus_minus @ A @ X @ ( numeral_numeral @ A @ V ) ) )
= ( minus_minus @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( numeral_numeral @ int @ V ) ) ) ) ).
% floor_diff_numeral
thf(fact_2083_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_2084_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_2085_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_2086_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_2087_neg__numeral__le__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V: num,X: A] :
( ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V ) ) @ ( archim6421214686448440834_floor @ A @ X ) )
= ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) @ X ) ) ) ).
% neg_numeral_le_floor
thf(fact_2088_floor__less__neg__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,V: num] :
( ( ord_less @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V ) ) )
= ( ord_less @ A @ X @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) ) ) ) ).
% floor_less_neg_numeral
thf(fact_2089_add__inc,axiom,
! [X: num,Y3: num] :
( ( plus_plus @ num @ X @ ( inc @ Y3 ) )
= ( inc @ ( plus_plus @ num @ X @ Y3 ) ) ) ).
% add_inc
thf(fact_2090_mult__inc,axiom,
! [X: num,Y3: num] :
( ( times_times @ num @ X @ ( inc @ Y3 ) )
= ( plus_plus @ num @ ( times_times @ num @ X @ Y3 ) @ X ) ) ).
% mult_inc
thf(fact_2091_floor__mono,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Y3: A] :
( ( ord_less_eq @ A @ X @ Y3 )
=> ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( archim6421214686448440834_floor @ A @ Y3 ) ) ) ) ).
% floor_mono
thf(fact_2092_floor__less__cancel,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Y3: A] :
( ( ord_less @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( archim6421214686448440834_floor @ A @ Y3 ) )
=> ( ord_less @ A @ X @ Y3 ) ) ) ).
% floor_less_cancel
thf(fact_2093_of__int__floor__le,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] : ( ord_less_eq @ A @ ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ X ) ) @ X ) ) ).
% of_int_floor_le
thf(fact_2094_floor__le__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] : ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( archimedean_ceiling @ A @ X ) ) ) ).
% floor_le_ceiling
thf(fact_2095_floor__le__round,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] : ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( archimedean_round @ A @ X ) ) ) ).
% floor_le_round
thf(fact_2096_le__floor__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z3: int,X: A] :
( ( ord_less_eq @ int @ Z3 @ ( archim6421214686448440834_floor @ A @ X ) )
= ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z3 ) @ X ) ) ) ).
% le_floor_iff
thf(fact_2097_floor__less__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Z3: int] :
( ( ord_less @ int @ ( archim6421214686448440834_floor @ A @ X ) @ Z3 )
= ( ord_less @ A @ X @ ( ring_1_of_int @ A @ Z3 ) ) ) ) ).
% floor_less_iff
thf(fact_2098_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_2099_int__add__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z3: int,X: A] :
( ( plus_plus @ int @ Z3 @ ( archim6421214686448440834_floor @ A @ X ) )
= ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ ( ring_1_of_int @ A @ Z3 ) @ X ) ) ) ) ).
% int_add_floor
thf(fact_2100_floor__add__int,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Z3: int] :
( ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X ) @ Z3 )
= ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X @ ( ring_1_of_int @ A @ Z3 ) ) ) ) ) ).
% floor_add_int
thf(fact_2101_floor__divide__of__int__eq,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [K: int,L: int] :
( ( archim6421214686448440834_floor @ A @ ( divide_divide @ A @ ( ring_1_of_int @ A @ K ) @ ( ring_1_of_int @ A @ L ) ) )
= ( divide_divide @ int @ K @ L ) ) ) ).
% floor_divide_of_int_eq
thf(fact_2102_ceiling__def,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archimedean_ceiling @ A )
= ( ^ [X2: A] : ( uminus_uminus @ int @ ( archim6421214686448440834_floor @ A @ ( uminus_uminus @ A @ X2 ) ) ) ) ) ) ).
% ceiling_def
thf(fact_2103_floor__minus,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( archim6421214686448440834_floor @ A @ ( uminus_uminus @ A @ X ) )
= ( uminus_uminus @ int @ ( archimedean_ceiling @ A @ X ) ) ) ) ).
% floor_minus
thf(fact_2104_ceiling__minus,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( archimedean_ceiling @ A @ ( uminus_uminus @ A @ X ) )
= ( uminus_uminus @ int @ ( archim6421214686448440834_floor @ A @ X ) ) ) ) ).
% ceiling_minus
thf(fact_2105_frac__def,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archimedean_frac @ A )
= ( ^ [X2: A] : ( minus_minus @ A @ X2 @ ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ X2 ) ) ) ) ) ) ).
% frac_def
thf(fact_2106_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_2107_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_2108_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_2109_floor__divide__of__nat__eq,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [M: nat,N: nat] :
( ( archim6421214686448440834_floor @ A @ ( divide_divide @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) ) )
= ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ M @ N ) ) ) ) ).
% floor_divide_of_nat_eq
thf(fact_2110_ceiling__diff__floor__le__1,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] : ( ord_less_eq @ int @ ( minus_minus @ int @ ( archimedean_ceiling @ A @ X ) @ ( archim6421214686448440834_floor @ A @ X ) ) @ ( one_one @ int ) ) ) ).
% ceiling_diff_floor_le_1
thf(fact_2111_ceiling__altdef,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archimedean_ceiling @ A )
= ( ^ [X2: A] :
( if @ int
@ ( X2
= ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ X2 ) ) )
@ ( archim6421214686448440834_floor @ A @ X2 )
@ ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X2 ) @ ( one_one @ int ) ) ) ) ) ) ).
% ceiling_altdef
thf(fact_2112_floor__split,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [P: int > $o,T2: A] :
( ( P @ ( archim6421214686448440834_floor @ A @ T2 ) )
= ( ! [I5: int] :
( ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ I5 ) @ T2 )
& ( ord_less @ A @ T2 @ ( plus_plus @ A @ ( ring_1_of_int @ A @ I5 ) @ ( one_one @ A ) ) ) )
=> ( P @ I5 ) ) ) ) ) ).
% floor_split
thf(fact_2113_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_2114_floor__unique,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z3: int,X: A] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z3 ) @ X )
=> ( ( ord_less @ A @ X @ ( plus_plus @ A @ ( ring_1_of_int @ A @ Z3 ) @ ( one_one @ A ) ) )
=> ( ( archim6421214686448440834_floor @ A @ X )
= Z3 ) ) ) ) ).
% floor_unique
thf(fact_2115_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_2116_less__floor__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z3: int,X: A] :
( ( ord_less @ int @ Z3 @ ( archim6421214686448440834_floor @ A @ X ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ ( ring_1_of_int @ A @ Z3 ) @ ( one_one @ A ) ) @ X ) ) ) ).
% less_floor_iff
thf(fact_2117_floor__le__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Z3: int] :
( ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X ) @ Z3 )
= ( ord_less @ A @ X @ ( plus_plus @ A @ ( ring_1_of_int @ A @ Z3 ) @ ( one_one @ A ) ) ) ) ) ).
% floor_le_iff
thf(fact_2118_floor__correct,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ X ) ) @ X )
& ( ord_less @ A @ X @ ( ring_1_of_int @ A @ ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( one_one @ int ) ) ) ) ) ) ).
% floor_correct
thf(fact_2119_floor__divide__lower,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Q4: A,P5: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Q4 )
=> ( ord_less_eq @ A @ ( times_times @ A @ ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ ( divide_divide @ A @ P5 @ Q4 ) ) ) @ Q4 ) @ P5 ) ) ) ).
% floor_divide_lower
thf(fact_2120_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_2121_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_2122_floor__divide__upper,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Q4: A,P5: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Q4 )
=> ( ord_less @ A @ P5 @ ( times_times @ A @ ( plus_plus @ A @ ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ ( divide_divide @ A @ P5 @ Q4 ) ) ) @ ( one_one @ A ) ) @ Q4 ) ) ) ) ).
% floor_divide_upper
thf(fact_2123_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_2124_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_2125_cpmi,axiom,
! [D4: int,P: int > $o,P2: int > $o,B7: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
=> ( ? [Z5: int] :
! [X3: int] :
( ( ord_less @ int @ X3 @ Z5 )
=> ( ( P @ X3 )
= ( P2 @ X3 ) ) )
=> ( ! [X3: int] :
( ! [Xa: int] :
( ( member @ int @ Xa @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb: int] :
( ( member @ int @ Xb @ B7 )
=> ( X3
!= ( plus_plus @ int @ Xb @ Xa ) ) ) )
=> ( ( P @ X3 )
=> ( P @ ( minus_minus @ int @ X3 @ D4 ) ) ) )
=> ( ! [X3: int,K2: int] :
( ( P2 @ X3 )
= ( P2 @ ( minus_minus @ int @ X3 @ ( times_times @ int @ K2 @ D4 ) ) ) )
=> ( ( ? [X7: int] : ( P @ X7 ) )
= ( ? [X2: int] :
( ( member @ int @ X2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
& ( P2 @ X2 ) )
| ? [X2: int] :
( ( member @ int @ X2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
& ? [Y6: int] :
( ( member @ int @ Y6 @ B7 )
& ( P @ ( plus_plus @ int @ Y6 @ X2 ) ) ) ) ) ) ) ) ) ) ).
% cpmi
thf(fact_2126_cppi,axiom,
! [D4: int,P: int > $o,P2: int > $o,A6: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
=> ( ? [Z5: int] :
! [X3: int] :
( ( ord_less @ int @ Z5 @ X3 )
=> ( ( P @ X3 )
= ( P2 @ X3 ) ) )
=> ( ! [X3: int] :
( ! [Xa: int] :
( ( member @ int @ Xa @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb: int] :
( ( member @ int @ Xb @ A6 )
=> ( X3
!= ( minus_minus @ int @ Xb @ Xa ) ) ) )
=> ( ( P @ X3 )
=> ( P @ ( plus_plus @ int @ X3 @ D4 ) ) ) )
=> ( ! [X3: int,K2: int] :
( ( P2 @ X3 )
= ( P2 @ ( minus_minus @ int @ X3 @ ( times_times @ int @ K2 @ D4 ) ) ) )
=> ( ( ? [X7: int] : ( P @ X7 ) )
= ( ? [X2: int] :
( ( member @ int @ X2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
& ( P2 @ X2 ) )
| ? [X2: int] :
( ( member @ int @ X2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
& ? [Y6: int] :
( ( member @ int @ Y6 @ A6 )
& ( P @ ( minus_minus @ int @ Y6 @ X2 ) ) ) ) ) ) ) ) ) ) ).
% cppi
thf(fact_2127_aset_I8_J,axiom,
! [D4: int,A6: set @ int,T2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
=> ! [X4: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ A6 )
=> ( X4
!= ( minus_minus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ( ord_less_eq @ int @ T2 @ X4 )
=> ( ord_less_eq @ int @ T2 @ ( plus_plus @ int @ X4 @ D4 ) ) ) ) ) ).
% aset(8)
thf(fact_2128_aset_I6_J,axiom,
! [D4: int,T2: int,A6: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
=> ( ( member @ int @ ( plus_plus @ int @ T2 @ ( one_one @ int ) ) @ A6 )
=> ! [X4: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ A6 )
=> ( X4
!= ( minus_minus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ( ord_less_eq @ int @ X4 @ T2 )
=> ( ord_less_eq @ int @ ( plus_plus @ int @ X4 @ D4 ) @ T2 ) ) ) ) ) ).
% aset(6)
thf(fact_2129_bset_I8_J,axiom,
! [D4: int,T2: int,B7: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
=> ( ( member @ int @ ( minus_minus @ int @ T2 @ ( one_one @ int ) ) @ B7 )
=> ! [X4: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ B7 )
=> ( X4
!= ( plus_plus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ( ord_less_eq @ int @ T2 @ X4 )
=> ( ord_less_eq @ int @ T2 @ ( minus_minus @ int @ X4 @ D4 ) ) ) ) ) ) ).
% bset(8)
thf(fact_2130_atLeastatMost__subset__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A,B2: A,C2: A,D2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) @ ( set_or1337092689740270186AtMost @ A @ C2 @ D2 ) )
= ( ~ ( ord_less_eq @ A @ A3 @ B2 )
| ( ( ord_less_eq @ A @ C2 @ A3 )
& ( ord_less_eq @ A @ B2 @ D2 ) ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_2131_atLeastAtMost__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [I: A,L: A,U: A] :
( ( member @ A @ I @ ( set_or1337092689740270186AtMost @ A @ L @ U ) )
= ( ( ord_less_eq @ A @ L @ I )
& ( ord_less_eq @ A @ I @ U ) ) ) ) ).
% atLeastAtMost_iff
thf(fact_2132_Icc__eq__Icc,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,H2: A,L4: A,H: A] :
( ( ( set_or1337092689740270186AtMost @ A @ L @ H2 )
= ( set_or1337092689740270186AtMost @ A @ L4 @ H ) )
= ( ( ( L = L4 )
& ( H2 = H ) )
| ( ~ ( ord_less_eq @ A @ L @ H2 )
& ~ ( ord_less_eq @ A @ L4 @ H ) ) ) ) ) ).
% Icc_eq_Icc
thf(fact_2133_semiring__norm_I12_J,axiom,
! [N: num] :
( ( times_times @ num @ one2 @ N )
= N ) ).
% semiring_norm(12)
thf(fact_2134_semiring__norm_I11_J,axiom,
! [M: num] :
( ( times_times @ num @ M @ one2 )
= M ) ).
% semiring_norm(11)
thf(fact_2135_semiring__norm_I68_J,axiom,
! [N: num] : ( ord_less_eq @ num @ one2 @ N ) ).
% semiring_norm(68)
thf(fact_2136_semiring__norm_I75_J,axiom,
! [M: num] :
~ ( ord_less @ num @ M @ one2 ) ).
% semiring_norm(75)
thf(fact_2137_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_2138_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_2139_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_2140_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_2141_Suc__numeral,axiom,
! [N: num] :
( ( suc @ ( numeral_numeral @ nat @ N ) )
= ( numeral_numeral @ nat @ ( plus_plus @ num @ N @ one2 ) ) ) ).
% Suc_numeral
thf(fact_2142_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_2143_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_2144_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_2145_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_2146_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_2147_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_2148_add__One__commute,axiom,
! [N: num] :
( ( plus_plus @ num @ one2 @ N )
= ( plus_plus @ num @ N @ one2 ) ) ).
% add_One_commute
thf(fact_2149_num__induct,axiom,
! [P: num > $o,X: num] :
( ( P @ one2 )
=> ( ! [X3: num] :
( ( P @ X3 )
=> ( P @ ( inc @ X3 ) ) )
=> ( P @ X ) ) ) ).
% num_induct
thf(fact_2150_le__num__One__iff,axiom,
! [X: num] :
( ( ord_less_eq @ num @ X @ one2 )
= ( X = one2 ) ) ).
% le_num_One_iff
thf(fact_2151_atLeastatMost__psubset__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A,B2: A,C2: A,D2: A] :
( ( ord_less @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) @ ( set_or1337092689740270186AtMost @ A @ C2 @ D2 ) )
= ( ( ~ ( ord_less_eq @ A @ A3 @ B2 )
| ( ( ord_less_eq @ A @ C2 @ A3 )
& ( ord_less_eq @ A @ B2 @ D2 )
& ( ( ord_less @ A @ C2 @ A3 )
| ( ord_less @ A @ B2 @ D2 ) ) ) )
& ( ord_less_eq @ A @ C2 @ D2 ) ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_2152_mult__numeral__1,axiom,
! [A: $tType] :
( ( semiring_numeral @ A )
=> ! [A3: A] :
( ( times_times @ A @ ( numeral_numeral @ A @ one2 ) @ A3 )
= A3 ) ) ).
% mult_numeral_1
thf(fact_2153_mult__numeral__1__right,axiom,
! [A: $tType] :
( ( semiring_numeral @ A )
=> ! [A3: A] :
( ( times_times @ A @ A3 @ ( numeral_numeral @ A @ one2 ) )
= A3 ) ) ).
% mult_numeral_1_right
thf(fact_2154_numeral__One,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ( ( numeral_numeral @ A @ one2 )
= ( one_one @ A ) ) ) ).
% numeral_One
thf(fact_2155_divide__numeral__1,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: A] :
( ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ one2 ) )
= A3 ) ) ).
% divide_numeral_1
thf(fact_2156_numerals_I1_J,axiom,
( ( numeral_numeral @ nat @ one2 )
= ( one_one @ nat ) ) ).
% numerals(1)
thf(fact_2157_add__One,axiom,
! [X: num] :
( ( plus_plus @ num @ X @ one2 )
= ( inc @ X ) ) ).
% add_One
thf(fact_2158_bset_I1_J,axiom,
! [D4: int,B7: set @ int,P: int > $o,Q: int > $o] :
( ! [X3: int] :
( ! [Xa: int] :
( ( member @ int @ Xa @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb: int] :
( ( member @ int @ Xb @ B7 )
=> ( X3
!= ( plus_plus @ int @ Xb @ Xa ) ) ) )
=> ( ( P @ X3 )
=> ( P @ ( minus_minus @ int @ X3 @ D4 ) ) ) )
=> ( ! [X3: int] :
( ! [Xa: int] :
( ( member @ int @ Xa @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb: int] :
( ( member @ int @ Xb @ B7 )
=> ( X3
!= ( plus_plus @ int @ Xb @ Xa ) ) ) )
=> ( ( Q @ X3 )
=> ( Q @ ( minus_minus @ int @ X3 @ D4 ) ) ) )
=> ! [X4: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ B7 )
=> ( X4
!= ( plus_plus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ( ( P @ X4 )
& ( Q @ X4 ) )
=> ( ( P @ ( minus_minus @ int @ X4 @ D4 ) )
& ( Q @ ( minus_minus @ int @ X4 @ D4 ) ) ) ) ) ) ) ).
% bset(1)
thf(fact_2159_bset_I2_J,axiom,
! [D4: int,B7: set @ int,P: int > $o,Q: int > $o] :
( ! [X3: int] :
( ! [Xa: int] :
( ( member @ int @ Xa @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb: int] :
( ( member @ int @ Xb @ B7 )
=> ( X3
!= ( plus_plus @ int @ Xb @ Xa ) ) ) )
=> ( ( P @ X3 )
=> ( P @ ( minus_minus @ int @ X3 @ D4 ) ) ) )
=> ( ! [X3: int] :
( ! [Xa: int] :
( ( member @ int @ Xa @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb: int] :
( ( member @ int @ Xb @ B7 )
=> ( X3
!= ( plus_plus @ int @ Xb @ Xa ) ) ) )
=> ( ( Q @ X3 )
=> ( Q @ ( minus_minus @ int @ X3 @ D4 ) ) ) )
=> ! [X4: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ B7 )
=> ( X4
!= ( plus_plus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ( ( P @ X4 )
| ( Q @ X4 ) )
=> ( ( P @ ( minus_minus @ int @ X4 @ D4 ) )
| ( Q @ ( minus_minus @ int @ X4 @ D4 ) ) ) ) ) ) ) ).
% bset(2)
thf(fact_2160_aset_I1_J,axiom,
! [D4: int,A6: set @ int,P: int > $o,Q: int > $o] :
( ! [X3: int] :
( ! [Xa: int] :
( ( member @ int @ Xa @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb: int] :
( ( member @ int @ Xb @ A6 )
=> ( X3
!= ( minus_minus @ int @ Xb @ Xa ) ) ) )
=> ( ( P @ X3 )
=> ( P @ ( plus_plus @ int @ X3 @ D4 ) ) ) )
=> ( ! [X3: int] :
( ! [Xa: int] :
( ( member @ int @ Xa @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb: int] :
( ( member @ int @ Xb @ A6 )
=> ( X3
!= ( minus_minus @ int @ Xb @ Xa ) ) ) )
=> ( ( Q @ X3 )
=> ( Q @ ( plus_plus @ int @ X3 @ D4 ) ) ) )
=> ! [X4: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ A6 )
=> ( X4
!= ( minus_minus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ( ( P @ X4 )
& ( Q @ X4 ) )
=> ( ( P @ ( plus_plus @ int @ X4 @ D4 ) )
& ( Q @ ( plus_plus @ int @ X4 @ D4 ) ) ) ) ) ) ) ).
% aset(1)
thf(fact_2161_aset_I2_J,axiom,
! [D4: int,A6: set @ int,P: int > $o,Q: int > $o] :
( ! [X3: int] :
( ! [Xa: int] :
( ( member @ int @ Xa @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb: int] :
( ( member @ int @ Xb @ A6 )
=> ( X3
!= ( minus_minus @ int @ Xb @ Xa ) ) ) )
=> ( ( P @ X3 )
=> ( P @ ( plus_plus @ int @ X3 @ D4 ) ) ) )
=> ( ! [X3: int] :
( ! [Xa: int] :
( ( member @ int @ Xa @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb: int] :
( ( member @ int @ Xb @ A6 )
=> ( X3
!= ( minus_minus @ int @ Xb @ Xa ) ) ) )
=> ( ( Q @ X3 )
=> ( Q @ ( plus_plus @ int @ X3 @ D4 ) ) ) )
=> ! [X4: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ A6 )
=> ( X4
!= ( minus_minus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ( ( P @ X4 )
| ( Q @ X4 ) )
=> ( ( P @ ( plus_plus @ int @ X4 @ D4 ) )
| ( Q @ ( plus_plus @ int @ X4 @ D4 ) ) ) ) ) ) ) ).
% aset(2)
thf(fact_2162_mult__1s__ring__1_I1_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [B2: A] :
( ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ one2 ) ) @ B2 )
= ( uminus_uminus @ A @ B2 ) ) ) ).
% mult_1s_ring_1(1)
thf(fact_2163_mult__1s__ring__1_I2_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [B2: A] :
( ( times_times @ A @ B2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ one2 ) ) )
= ( uminus_uminus @ A @ B2 ) ) ) ).
% mult_1s_ring_1(2)
thf(fact_2164_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_2165_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_2166_numeral__1__eq__Suc__0,axiom,
( ( numeral_numeral @ nat @ one2 )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% numeral_1_eq_Suc_0
thf(fact_2167_bset_I9_J,axiom,
! [D2: int,D4: int,B7: set @ int,T2: int] :
( ( dvd_dvd @ int @ D2 @ D4 )
=> ! [X4: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ B7 )
=> ( X4
!= ( plus_plus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ( dvd_dvd @ int @ D2 @ ( plus_plus @ int @ X4 @ T2 ) )
=> ( dvd_dvd @ int @ D2 @ ( plus_plus @ int @ ( minus_minus @ int @ X4 @ D4 ) @ T2 ) ) ) ) ) ).
% bset(9)
thf(fact_2168_bset_I10_J,axiom,
! [D2: int,D4: int,B7: set @ int,T2: int] :
( ( dvd_dvd @ int @ D2 @ D4 )
=> ! [X4: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ B7 )
=> ( X4
!= ( plus_plus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ~ ( dvd_dvd @ int @ D2 @ ( plus_plus @ int @ X4 @ T2 ) )
=> ~ ( dvd_dvd @ int @ D2 @ ( plus_plus @ int @ ( minus_minus @ int @ X4 @ D4 ) @ T2 ) ) ) ) ) ).
% bset(10)
thf(fact_2169_aset_I9_J,axiom,
! [D2: int,D4: int,A6: set @ int,T2: int] :
( ( dvd_dvd @ int @ D2 @ D4 )
=> ! [X4: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ A6 )
=> ( X4
!= ( minus_minus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ( dvd_dvd @ int @ D2 @ ( plus_plus @ int @ X4 @ T2 ) )
=> ( dvd_dvd @ int @ D2 @ ( plus_plus @ int @ ( plus_plus @ int @ X4 @ D4 ) @ T2 ) ) ) ) ) ).
% aset(9)
thf(fact_2170_aset_I10_J,axiom,
! [D2: int,D4: int,A6: set @ int,T2: int] :
( ( dvd_dvd @ int @ D2 @ D4 )
=> ! [X4: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ A6 )
=> ( X4
!= ( minus_minus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ~ ( dvd_dvd @ int @ D2 @ ( plus_plus @ int @ X4 @ T2 ) )
=> ~ ( dvd_dvd @ int @ D2 @ ( plus_plus @ int @ ( plus_plus @ int @ X4 @ D4 ) @ T2 ) ) ) ) ) ).
% aset(10)
thf(fact_2171_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_2172_periodic__finite__ex,axiom,
! [D2: int,P: int > $o] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D2 )
=> ( ! [X3: int,K2: int] :
( ( P @ X3 )
= ( P @ ( minus_minus @ int @ X3 @ ( times_times @ int @ K2 @ D2 ) ) ) )
=> ( ( ? [X7: int] : ( P @ X7 ) )
= ( ? [X2: int] :
( ( member @ int @ X2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D2 ) )
& ( P @ X2 ) ) ) ) ) ) ).
% periodic_finite_ex
thf(fact_2173_aset_I7_J,axiom,
! [D4: int,A6: set @ int,T2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
=> ! [X4: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ A6 )
=> ( X4
!= ( minus_minus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ( ord_less @ int @ T2 @ X4 )
=> ( ord_less @ int @ T2 @ ( plus_plus @ int @ X4 @ D4 ) ) ) ) ) ).
% aset(7)
thf(fact_2174_aset_I5_J,axiom,
! [D4: int,T2: int,A6: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
=> ( ( member @ int @ T2 @ A6 )
=> ! [X4: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ A6 )
=> ( X4
!= ( minus_minus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ( ord_less @ int @ X4 @ T2 )
=> ( ord_less @ int @ ( plus_plus @ int @ X4 @ D4 ) @ T2 ) ) ) ) ) ).
% aset(5)
thf(fact_2175_aset_I4_J,axiom,
! [D4: int,T2: int,A6: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
=> ( ( member @ int @ T2 @ A6 )
=> ! [X4: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ A6 )
=> ( X4
!= ( minus_minus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ( X4 != T2 )
=> ( ( plus_plus @ int @ X4 @ D4 )
!= T2 ) ) ) ) ) ).
% aset(4)
thf(fact_2176_aset_I3_J,axiom,
! [D4: int,T2: int,A6: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
=> ( ( member @ int @ ( plus_plus @ int @ T2 @ ( one_one @ int ) ) @ A6 )
=> ! [X4: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ A6 )
=> ( X4
!= ( minus_minus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ( X4 = T2 )
=> ( ( plus_plus @ int @ X4 @ D4 )
= T2 ) ) ) ) ) ).
% aset(3)
thf(fact_2177_bset_I7_J,axiom,
! [D4: int,T2: int,B7: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
=> ( ( member @ int @ T2 @ B7 )
=> ! [X4: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ B7 )
=> ( X4
!= ( plus_plus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ( ord_less @ int @ T2 @ X4 )
=> ( ord_less @ int @ T2 @ ( minus_minus @ int @ X4 @ D4 ) ) ) ) ) ) ).
% bset(7)
thf(fact_2178_bset_I5_J,axiom,
! [D4: int,B7: set @ int,T2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
=> ! [X4: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ B7 )
=> ( X4
!= ( plus_plus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ( ord_less @ int @ X4 @ T2 )
=> ( ord_less @ int @ ( minus_minus @ int @ X4 @ D4 ) @ T2 ) ) ) ) ).
% bset(5)
thf(fact_2179_bset_I4_J,axiom,
! [D4: int,T2: int,B7: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
=> ( ( member @ int @ T2 @ B7 )
=> ! [X4: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ B7 )
=> ( X4
!= ( plus_plus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ( X4 != T2 )
=> ( ( minus_minus @ int @ X4 @ D4 )
!= T2 ) ) ) ) ) ).
% bset(4)
thf(fact_2180_bset_I3_J,axiom,
! [D4: int,T2: int,B7: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
=> ( ( member @ int @ ( minus_minus @ int @ T2 @ ( one_one @ int ) ) @ B7 )
=> ! [X4: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ B7 )
=> ( X4
!= ( plus_plus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ( X4 = T2 )
=> ( ( minus_minus @ int @ X4 @ D4 )
= T2 ) ) ) ) ) ).
% bset(3)
thf(fact_2181_bset_I6_J,axiom,
! [D4: int,B7: set @ int,T2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
=> ! [X4: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ B7 )
=> ( X4
!= ( plus_plus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ( ord_less_eq @ int @ X4 @ T2 )
=> ( ord_less_eq @ int @ ( minus_minus @ int @ X4 @ D4 ) @ T2 ) ) ) ) ).
% bset(6)
thf(fact_2182_pochhammer__double,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [Z3: A,N: nat] :
( ( comm_s3205402744901411588hammer @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Z3 ) @ ( 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 @ Z3 @ N ) ) @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ Z3 @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ N ) ) ) ) ).
% pochhammer_double
thf(fact_2183_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_2184_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_2185_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_2186_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_2187_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_2188_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_2189_verit__eq__simplify_I8_J,axiom,
! [X22: num,Y2: num] :
( ( ( bit0 @ X22 )
= ( bit0 @ Y2 ) )
= ( X22 = Y2 ) ) ).
% verit_eq_simplify(8)
thf(fact_2190_semiring__norm_I87_J,axiom,
! [M: num,N: num] :
( ( ( bit0 @ M )
= ( bit0 @ N ) )
= ( M = N ) ) ).
% semiring_norm(87)
thf(fact_2191_semiring__norm_I85_J,axiom,
! [M: num] :
( ( bit0 @ M )
!= one2 ) ).
% semiring_norm(85)
thf(fact_2192_semiring__norm_I83_J,axiom,
! [N: num] :
( one2
!= ( bit0 @ N ) ) ).
% semiring_norm(83)
thf(fact_2193_semiring__norm_I6_J,axiom,
! [M: num,N: num] :
( ( plus_plus @ num @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( bit0 @ ( plus_plus @ num @ M @ N ) ) ) ).
% semiring_norm(6)
thf(fact_2194_semiring__norm_I13_J,axiom,
! [M: num,N: num] :
( ( times_times @ num @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( bit0 @ ( bit0 @ ( times_times @ num @ M @ N ) ) ) ) ).
% semiring_norm(13)
thf(fact_2195_semiring__norm_I78_J,axiom,
! [M: num,N: num] :
( ( ord_less @ num @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( ord_less @ num @ M @ N ) ) ).
% semiring_norm(78)
thf(fact_2196_semiring__norm_I71_J,axiom,
! [M: num,N: num] :
( ( ord_less_eq @ num @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( ord_less_eq @ num @ M @ N ) ) ).
% semiring_norm(71)
thf(fact_2197_semiring__norm_I2_J,axiom,
( ( plus_plus @ num @ one2 @ one2 )
= ( bit0 @ one2 ) ) ).
% semiring_norm(2)
thf(fact_2198_num__double,axiom,
! [N: num] :
( ( times_times @ num @ ( bit0 @ one2 ) @ N )
= ( bit0 @ N ) ) ).
% num_double
thf(fact_2199_semiring__norm_I69_J,axiom,
! [M: num] :
~ ( ord_less_eq @ num @ ( bit0 @ M ) @ one2 ) ).
% semiring_norm(69)
thf(fact_2200_semiring__norm_I76_J,axiom,
! [N: num] : ( ord_less @ num @ one2 @ ( bit0 @ N ) ) ).
% semiring_norm(76)
thf(fact_2201_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_2202_even__mult__iff,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( times_times @ A @ A3 @ B2 ) )
= ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
| ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) ) ) ).
% even_mult_iff
thf(fact_2203_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_2204_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_2205_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_2206_power2__minus,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [A3: A] :
( ( power_power @ A @ ( uminus_uminus @ A @ A3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% power2_minus
thf(fact_2207_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_2208_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_2209_Suc__1,axiom,
( ( suc @ ( one_one @ nat ) )
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ).
% Suc_1
thf(fact_2210_even__Suc,axiom,
! [N: nat] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ N ) )
= ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ).
% even_Suc
thf(fact_2211_even__Suc__Suc__iff,axiom,
! [N: nat] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ N ) ) )
= ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ).
% even_Suc_Suc_iff
thf(fact_2212_mod2__Suc__Suc,axiom,
! [M: nat] :
( ( modulo_modulo @ nat @ ( suc @ ( suc @ M ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( modulo_modulo @ nat @ M @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% mod2_Suc_Suc
thf(fact_2213_div2__Suc__Suc,axiom,
! [M: nat] :
( ( divide_divide @ nat @ ( suc @ ( suc @ M ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( suc @ ( divide_divide @ nat @ M @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% div2_Suc_Suc
thf(fact_2214_fact__2,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiring_char_0_fact @ A @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ).
% fact_2
thf(fact_2215_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_2216_zmod__numeral__Bit0,axiom,
! [V: num,W2: num] :
( ( modulo_modulo @ int @ ( numeral_numeral @ int @ ( bit0 @ V ) ) @ ( numeral_numeral @ int @ ( bit0 @ W2 ) ) )
= ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( modulo_modulo @ int @ ( numeral_numeral @ int @ V ) @ ( numeral_numeral @ int @ W2 ) ) ) ) ).
% zmod_numeral_Bit0
thf(fact_2217_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_2218_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_2219_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_2220_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_2221_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_2222_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_2223_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_2224_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_2225_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_2226_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_2227_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_2228_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_2229_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_2230_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_2231_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_2232_Power_Oring__1__class_Opower__minus__even,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [A3: A,N: nat] :
( ( power_power @ A @ ( uminus_uminus @ A @ A3 ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
= ( power_power @ A @ A3 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% Power.ring_1_class.power_minus_even
thf(fact_2233_power__minus__odd,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [N: nat,A3: A] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( power_power @ A @ ( uminus_uminus @ A @ A3 ) @ N )
= ( uminus_uminus @ A @ ( power_power @ A @ A3 @ N ) ) ) ) ) ).
% power_minus_odd
thf(fact_2234_Parity_Oring__1__class_Opower__minus__even,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [N: nat,A3: A] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( power_power @ A @ ( uminus_uminus @ A @ A3 ) @ N )
= ( power_power @ A @ A3 @ N ) ) ) ) ).
% Parity.ring_1_class.power_minus_even
thf(fact_2235_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_2236_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_2237_half__nonnegative__int__iff,axiom,
! [K: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ K @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) )
= ( ord_less_eq @ int @ ( zero_zero @ int ) @ K ) ) ).
% half_nonnegative_int_iff
thf(fact_2238_half__negative__int__iff,axiom,
! [K: int] :
( ( ord_less @ int @ ( divide_divide @ int @ K @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( zero_zero @ int ) )
= ( ord_less @ int @ K @ ( zero_zero @ int ) ) ) ).
% half_negative_int_iff
thf(fact_2239_even__Suc__div__two,axiom,
! [N: nat] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( divide_divide @ nat @ ( suc @ N ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% even_Suc_div_two
thf(fact_2240_odd__Suc__div__two,axiom,
! [N: nat] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( divide_divide @ nat @ ( suc @ N ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( suc @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% odd_Suc_div_two
thf(fact_2241_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_2242_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_2243_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_2244_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_2245_zero__le__power__eq__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,W2: num] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ W2 ) ) )
= ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W2 ) )
| ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W2 ) )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ) ) ).
% zero_le_power_eq_numeral
thf(fact_2246_power__less__zero__eq__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,W2: num] :
( ( ord_less @ A @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ W2 ) ) @ ( zero_zero @ A ) )
= ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W2 ) )
& ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ) ).
% power_less_zero_eq_numeral
thf(fact_2247_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_2248_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_2249_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_2250_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_2251_even__of__nat,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [N: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ A @ N ) )
= ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ).
% even_of_nat
thf(fact_2252_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_2253_one__less__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less @ int @ ( one_one @ int ) @ ( archim6421214686448440834_floor @ A @ X ) )
= ( ord_less_eq @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X ) ) ) ).
% one_less_floor
thf(fact_2254_floor__le__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( one_one @ int ) )
= ( ord_less @ A @ X @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% floor_le_one
thf(fact_2255_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_2256_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_2257_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_2258_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_2259_zero__less__power__eq__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,W2: num] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ W2 ) ) )
= ( ( ( numeral_numeral @ nat @ W2 )
= ( zero_zero @ nat ) )
| ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W2 ) )
& ( A3
!= ( zero_zero @ A ) ) )
| ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W2 ) )
& ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ) ) ).
% zero_less_power_eq_numeral
thf(fact_2260_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_2261_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_2262_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_2263_power__le__zero__eq__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,W2: num] :
( ( ord_less_eq @ A @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ W2 ) ) @ ( zero_zero @ A ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( numeral_numeral @ nat @ W2 ) )
& ( ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W2 ) )
& ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) )
| ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W2 ) )
& ( A3
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% power_le_zero_eq_numeral
thf(fact_2264_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_2265_verit__eq__simplify_I10_J,axiom,
! [X22: num] :
( one2
!= ( bit0 @ X22 ) ) ).
% verit_eq_simplify(10)
thf(fact_2266_ex__nat__less,axiom,
! [N: nat,P: nat > $o] :
( ( ? [M2: nat] :
( ( ord_less_eq @ nat @ M2 @ N )
& ( P @ M2 ) ) )
= ( ? [X2: nat] :
( ( member @ nat @ X2 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
& ( P @ X2 ) ) ) ) ).
% ex_nat_less
thf(fact_2267_all__nat__less,axiom,
! [N: nat,P: nat > $o] :
( ( ! [M2: nat] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( P @ M2 ) ) )
= ( ! [X2: nat] :
( ( member @ nat @ X2 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
=> ( P @ X2 ) ) ) ) ).
% all_nat_less
thf(fact_2268_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_2269_inc_Osimps_I1_J,axiom,
( ( inc @ one2 )
= ( bit0 @ one2 ) ) ).
% inc.simps(1)
thf(fact_2270_cong__exp__iff__simps_I2_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [N: num,Q4: num] :
( ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit0 @ N ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q4 ) ) )
= ( zero_zero @ A ) )
= ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ N ) @ ( numeral_numeral @ A @ Q4 ) )
= ( zero_zero @ A ) ) ) ) ).
% cong_exp_iff_simps(2)
thf(fact_2271_numeral__Bit0__div__2,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [N: num] :
( ( divide_divide @ A @ ( numeral_numeral @ A @ ( bit0 @ N ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( numeral_numeral @ A @ N ) ) ) ).
% numeral_Bit0_div_2
thf(fact_2272_power__minus__Bit0,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X: A,K: num] :
( ( power_power @ A @ ( uminus_uminus @ A @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ K ) ) )
= ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ K ) ) ) ) ) ).
% power_minus_Bit0
thf(fact_2273_mult__2,axiom,
! [A: $tType] :
( ( semiring_numeral @ A )
=> ! [Z3: A] :
( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Z3 )
= ( plus_plus @ A @ Z3 @ Z3 ) ) ) ).
% mult_2
thf(fact_2274_mult__2__right,axiom,
! [A: $tType] :
( ( semiring_numeral @ A )
=> ! [Z3: A] :
( ( times_times @ A @ Z3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( plus_plus @ A @ Z3 @ Z3 ) ) ) ).
% mult_2_right
thf(fact_2275_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_2276_even__zero,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( zero_zero @ A ) ) ) ).
% even_zero
thf(fact_2277_evenE,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A3: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ~ ! [B5: A] :
( A3
!= ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B5 ) ) ) ) ).
% evenE
thf(fact_2278_odd__one,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( one_one @ A ) ) ) ).
% odd_one
thf(fact_2279_even__minus,axiom,
! [A: $tType] :
( ( ring_parity @ A )
=> ! [A3: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( uminus_uminus @ A @ A3 ) )
= ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) ) ).
% even_minus
thf(fact_2280_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_2281_power2__eq__square,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A3: A] :
( ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( times_times @ A @ A3 @ A3 ) ) ) ).
% power2_eq_square
thf(fact_2282_power4__eq__xxxx,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [X: A] :
( ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) )
= ( times_times @ A @ ( times_times @ A @ ( times_times @ A @ X @ X ) @ X ) @ X ) ) ) ).
% power4_eq_xxxx
thf(fact_2283_numeral__2__eq__2,axiom,
( ( numeral_numeral @ nat @ ( bit0 @ one2 ) )
= ( suc @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% numeral_2_eq_2
thf(fact_2284_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_2285_power2__eq__iff,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [X: A,Y3: A] :
( ( ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( power_power @ A @ Y3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( ( X = Y3 )
| ( X
= ( uminus_uminus @ A @ Y3 ) ) ) ) ) ).
% power2_eq_iff
thf(fact_2286_double__not__eq__Suc__double,axiom,
! [M: nat,N: nat] :
( ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M )
!= ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ).
% double_not_eq_Suc_double
thf(fact_2287_Suc__double__not__eq__double,axiom,
! [M: nat,N: nat] :
( ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
!= ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ).
% Suc_double_not_eq_double
thf(fact_2288_power__even__eq,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A3: A,N: nat] :
( ( power_power @ A @ A3 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
= ( power_power @ A @ ( power_power @ A @ A3 @ N ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% power_even_eq
thf(fact_2289_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_2290_less__exp,axiom,
! [N: nat] : ( ord_less @ nat @ N @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ).
% less_exp
thf(fact_2291_self__le__ge2__pow,axiom,
! [K: nat,M: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K )
=> ( ord_less_eq @ nat @ M @ ( power_power @ nat @ K @ M ) ) ) ).
% self_le_ge2_pow
thf(fact_2292_power2__nat__le__eq__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ ( power_power @ nat @ M @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( ord_less_eq @ nat @ M @ N ) ) ).
% power2_nat_le_eq_le
thf(fact_2293_power2__nat__le__imp__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ ( power_power @ nat @ M @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ N )
=> ( ord_less_eq @ nat @ M @ N ) ) ).
% power2_nat_le_imp_le
thf(fact_2294_even__even__mod__4__iff,axiom,
! [N: nat] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
= ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( modulo_modulo @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ).
% even_even_mod_4_iff
thf(fact_2295_div2__even__ext__nat,axiom,
! [X: nat,Y3: nat] :
( ( ( divide_divide @ nat @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( divide_divide @ nat @ Y3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
=> ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ X )
= ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Y3 ) )
=> ( X = Y3 ) ) ) ).
% div2_even_ext_nat
thf(fact_2296_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_2297_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_2298_even__two__times__div__two,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A3: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
= A3 ) ) ) ).
% even_two_times_div_two
thf(fact_2299_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_2300_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_2301_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_2302_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_2303_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_2304_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_2305_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_2306_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_2307_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_2308_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_2309_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_2310_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_2311_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_2312_abs__le__square__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y3: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ X ) @ ( abs_abs @ A @ Y3 ) )
= ( ord_less_eq @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% abs_le_square_iff
thf(fact_2313_power__mono__odd,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: nat,A3: A,B2: A] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ord_less_eq @ A @ ( power_power @ A @ A3 @ N ) @ ( power_power @ A @ B2 @ N ) ) ) ) ) ).
% power_mono_odd
thf(fact_2314_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_2315_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_2316_uminus__power__if,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [N: nat,A3: A] :
( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( power_power @ A @ ( uminus_uminus @ A @ A3 ) @ N )
= ( power_power @ A @ A3 @ N ) ) )
& ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( power_power @ A @ ( uminus_uminus @ A @ A3 ) @ N )
= ( uminus_uminus @ A @ ( power_power @ A @ A3 @ N ) ) ) ) ) ) ).
% uminus_power_if
thf(fact_2317_nat__2,axiom,
( ( nat2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) )
= ( suc @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% nat_2
thf(fact_2318_nat__induct2,axiom,
! [P: nat > $o,N: nat] :
( ( P @ ( zero_zero @ nat ) )
=> ( ( P @ ( one_one @ nat ) )
=> ( ! [N4: nat] :
( ( P @ N4 )
=> ( P @ ( plus_plus @ nat @ N4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct2
thf(fact_2319_not__exp__less__eq__0__int,axiom,
! [N: nat] :
~ ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) @ ( zero_zero @ int ) ) ).
% not_exp_less_eq_0_int
thf(fact_2320_diff__le__diff__pow,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K )
=> ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ N ) @ ( minus_minus @ nat @ ( power_power @ nat @ K @ M ) @ ( power_power @ nat @ K @ N ) ) ) ) ).
% diff_le_diff_pow
thf(fact_2321_dvd__power__iff__le,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K )
=> ( ( dvd_dvd @ nat @ ( power_power @ nat @ K @ M ) @ ( power_power @ nat @ K @ N ) )
= ( ord_less_eq @ nat @ M @ N ) ) ) ).
% dvd_power_iff_le
thf(fact_2322_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_2323_oddE,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A3: A] :
( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ~ ! [B5: A] :
( A3
!= ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B5 ) @ ( one_one @ A ) ) ) ) ) ).
% oddE
thf(fact_2324_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_2325_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_2326_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_2327_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_2328_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_2329_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_2330_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_2331_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_2332_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_2333_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_2334_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_2335_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_2336_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_2337_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_2338_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_2339_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_2340_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_2341_bits__induct,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [P: A > $o,A3: A] :
( ! [A8: A] :
( ( ( divide_divide @ A @ A8 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= A8 )
=> ( P @ A8 ) )
=> ( ! [A8: A,B5: $o] :
( ( P @ A8 )
=> ( ( ( divide_divide @ A @ ( plus_plus @ A @ ( zero_neq_one_of_bool @ A @ B5 ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A8 ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= A8 )
=> ( P @ ( plus_plus @ A @ ( zero_neq_one_of_bool @ A @ B5 ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A8 ) ) ) ) )
=> ( P @ A3 ) ) ) ) ).
% bits_induct
thf(fact_2342_minus__power__mult__self,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [A3: A,N: nat] :
( ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ A3 ) @ N ) @ ( power_power @ A @ ( uminus_uminus @ A @ A3 ) @ N ) )
= ( power_power @ A @ A3 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% minus_power_mult_self
thf(fact_2343_power__odd__eq,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A3: A,N: nat] :
( ( power_power @ A @ A3 @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) )
= ( times_times @ A @ A3 @ ( power_power @ A @ ( power_power @ A @ A3 @ N ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% power_odd_eq
thf(fact_2344_nat__bit__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ ( zero_zero @ nat ) )
=> ( ! [N4: nat] :
( ( P @ N4 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N4 )
=> ( P @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N4 ) ) ) )
=> ( ! [N4: nat] :
( ( P @ N4 )
=> ( P @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N4 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_bit_induct
thf(fact_2345_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_2346_power__mono__even,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: nat,A3: A,B2: A] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( ord_less_eq @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B2 ) )
=> ( ord_less_eq @ A @ ( power_power @ A @ A3 @ N ) @ ( power_power @ A @ B2 @ N ) ) ) ) ) ).
% power_mono_even
thf(fact_2347_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_2348_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_2349_minus__1__div__exp__eq__int,axiom,
! [N: nat] :
( ( divide_divide @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) )
= ( uminus_uminus @ int @ ( one_one @ int ) ) ) ).
% minus_1_div_exp_eq_int
thf(fact_2350_triangle__def,axiom,
( nat_triangle
= ( ^ [N3: nat] : ( divide_divide @ nat @ ( times_times @ nat @ N3 @ ( suc @ N3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% triangle_def
thf(fact_2351_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_2352_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_2353_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_2354_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_2355_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_2356_mult__exp__mod__exp__eq,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [M: nat,N: nat,A3: A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( modulo_modulo @ A @ ( times_times @ A @ A3 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
= ( times_times @ A @ ( modulo_modulo @ A @ A3 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N @ M ) ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) ) ) ) ) ).
% mult_exp_mod_exp_eq
thf(fact_2357_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_2358_exp__mod__exp,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [M: nat,N: nat] :
( ( modulo_modulo @ 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 @ ( ord_less @ nat @ M @ N ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) ) ) ) ).
% exp_mod_exp
thf(fact_2359_int__bit__induct,axiom,
! [P: int > $o,K: int] :
( ( P @ ( zero_zero @ int ) )
=> ( ( P @ ( uminus_uminus @ int @ ( one_one @ int ) ) )
=> ( ! [K2: int] :
( ( P @ K2 )
=> ( ( K2
!= ( zero_zero @ int ) )
=> ( P @ ( times_times @ int @ K2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) )
=> ( ! [K2: int] :
( ( P @ K2 )
=> ( ( K2
!= ( uminus_uminus @ int @ ( one_one @ int ) ) )
=> ( P @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ K2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) )
=> ( P @ K ) ) ) ) ) ).
% int_bit_induct
thf(fact_2360_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 )
=> ? [N4: nat] :
( ( ord_less_eq @ nat @ ( power_power @ nat @ B2 @ N4 ) @ K )
& ( ord_less @ nat @ K @ ( power_power @ nat @ B2 @ ( plus_plus @ nat @ N4 @ ( one_one @ nat ) ) ) ) ) ) ) ).
% ex_power_ivl1
thf(fact_2361_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 )
=> ? [N4: nat] :
( ( ord_less @ nat @ ( power_power @ nat @ B2 @ N4 ) @ K )
& ( ord_less_eq @ nat @ K @ ( power_power @ nat @ B2 @ ( plus_plus @ nat @ N4 @ ( one_one @ nat ) ) ) ) ) ) ) ).
% ex_power_ivl2
thf(fact_2362_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_2363_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_2364_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_2365_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_2366_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_2367_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_2368_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_2369_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_2370_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_2371_neg__zdiv__mult__2,axiom,
! [A3: int,B2: int] :
( ( ord_less_eq @ int @ A3 @ ( zero_zero @ int ) )
=> ( ( divide_divide @ int @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ B2 ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ A3 ) )
= ( divide_divide @ int @ ( plus_plus @ int @ B2 @ ( one_one @ int ) ) @ A3 ) ) ) ).
% neg_zdiv_mult_2
thf(fact_2372_pos__zdiv__mult__2,axiom,
! [A3: int,B2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A3 )
=> ( ( divide_divide @ int @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ B2 ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ A3 ) )
= ( divide_divide @ int @ B2 @ A3 ) ) ) ).
% pos_zdiv_mult_2
thf(fact_2373_even__nat__iff,axiom,
! [K: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
=> ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( nat2 @ K ) )
= ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K ) ) ) ).
% even_nat_iff
thf(fact_2374_pos__zmod__mult__2,axiom,
! [A3: int,B2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A3 )
=> ( ( modulo_modulo @ int @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ B2 ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ A3 ) )
= ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( modulo_modulo @ int @ B2 @ A3 ) ) ) ) ) ).
% pos_zmod_mult_2
thf(fact_2375_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_2376_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_2377_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_2378_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_2379_neg__zmod__mult__2,axiom,
! [A3: int,B2: int] :
( ( ord_less_eq @ int @ A3 @ ( zero_zero @ int ) )
=> ( ( modulo_modulo @ int @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ B2 ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ A3 ) )
= ( minus_minus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( modulo_modulo @ int @ ( plus_plus @ int @ B2 @ ( one_one @ int ) ) @ A3 ) ) @ ( one_one @ int ) ) ) ) ).
% neg_zmod_mult_2
thf(fact_2380_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_2381_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_2382_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_2383_psubsetI,axiom,
! [A: $tType,A6: set @ A,B7: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B7 )
=> ( ( A6 != B7 )
=> ( ord_less @ ( set @ A ) @ A6 @ B7 ) ) ) ).
% psubsetI
thf(fact_2384_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_2385_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_2386_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_2387_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_2388_Compl__subset__Compl__iff,axiom,
! [A: $tType,A6: set @ A,B7: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A6 ) @ ( uminus_uminus @ ( set @ A ) @ B7 ) )
= ( ord_less_eq @ ( set @ A ) @ B7 @ A6 ) ) ).
% Compl_subset_Compl_iff
thf(fact_2389_Compl__anti__mono,axiom,
! [A: $tType,A6: set @ A,B7: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B7 )
=> ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ B7 ) @ ( uminus_uminus @ ( set @ A ) @ A6 ) ) ) ).
% Compl_anti_mono
thf(fact_2390_unset__bit__nonnegative__int__iff,axiom,
! [N: nat,K: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se2638667681897837118et_bit @ int @ N @ K ) )
= ( ord_less_eq @ int @ ( zero_zero @ int ) @ K ) ) ).
% unset_bit_nonnegative_int_iff
thf(fact_2391_set__bit__nonnegative__int__iff,axiom,
! [N: nat,K: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se5668285175392031749et_bit @ int @ N @ K ) )
= ( ord_less_eq @ int @ ( zero_zero @ int ) @ K ) ) ).
% set_bit_nonnegative_int_iff
thf(fact_2392_flip__bit__nonnegative__int__iff,axiom,
! [N: nat,K: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se8732182000553998342ip_bit @ int @ N @ K ) )
= ( ord_less_eq @ int @ ( zero_zero @ int ) @ K ) ) ).
% flip_bit_nonnegative_int_iff
thf(fact_2393_unset__bit__negative__int__iff,axiom,
! [N: nat,K: int] :
( ( ord_less @ int @ ( bit_se2638667681897837118et_bit @ int @ N @ K ) @ ( zero_zero @ int ) )
= ( ord_less @ int @ K @ ( zero_zero @ int ) ) ) ).
% unset_bit_negative_int_iff
thf(fact_2394_set__bit__negative__int__iff,axiom,
! [N: nat,K: int] :
( ( ord_less @ int @ ( bit_se5668285175392031749et_bit @ int @ N @ K ) @ ( zero_zero @ int ) )
= ( ord_less @ int @ K @ ( zero_zero @ int ) ) ) ).
% set_bit_negative_int_iff
thf(fact_2395_flip__bit__negative__int__iff,axiom,
! [N: nat,K: int] :
( ( ord_less @ int @ ( bit_se8732182000553998342ip_bit @ int @ N @ K ) @ ( zero_zero @ int ) )
= ( ord_less @ int @ K @ ( zero_zero @ int ) ) ) ).
% flip_bit_negative_int_iff
thf(fact_2396_unset__bit__nat__def,axiom,
( ( bit_se2638667681897837118et_bit @ nat )
= ( ^ [M2: nat,N3: nat] : ( nat2 @ ( bit_se2638667681897837118et_bit @ int @ M2 @ ( semiring_1_of_nat @ int @ N3 ) ) ) ) ) ).
% unset_bit_nat_def
thf(fact_2397_unset__bit__less__eq,axiom,
! [N: nat,K: int] : ( ord_less_eq @ int @ ( bit_se2638667681897837118et_bit @ int @ N @ K ) @ K ) ).
% unset_bit_less_eq
thf(fact_2398_set__bit__greater__eq,axiom,
! [K: int,N: nat] : ( ord_less_eq @ int @ K @ ( bit_se5668285175392031749et_bit @ int @ N @ K ) ) ).
% set_bit_greater_eq
thf(fact_2399_psubset__imp__ex__mem,axiom,
! [A: $tType,A6: set @ A,B7: set @ A] :
( ( ord_less @ ( set @ A ) @ A6 @ B7 )
=> ? [B5: A] : ( member @ A @ B5 @ ( minus_minus @ ( set @ A ) @ B7 @ A6 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_2400_psubset__trans,axiom,
! [A: $tType,A6: set @ A,B7: set @ A,C6: set @ A] :
( ( ord_less @ ( set @ A ) @ A6 @ B7 )
=> ( ( ord_less @ ( set @ A ) @ B7 @ C6 )
=> ( ord_less @ ( set @ A ) @ A6 @ C6 ) ) ) ).
% psubset_trans
thf(fact_2401_psubsetD,axiom,
! [A: $tType,A6: set @ A,B7: set @ A,C2: A] :
( ( ord_less @ ( set @ A ) @ A6 @ B7 )
=> ( ( member @ A @ C2 @ A6 )
=> ( member @ A @ C2 @ B7 ) ) ) ).
% psubsetD
thf(fact_2402_psubsetE,axiom,
! [A: $tType,A6: set @ A,B7: set @ A] :
( ( ord_less @ ( set @ A ) @ A6 @ B7 )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A6 @ B7 )
=> ( ord_less_eq @ ( set @ A ) @ B7 @ A6 ) ) ) ).
% psubsetE
thf(fact_2403_psubset__eq,axiom,
! [A: $tType] :
( ( ord_less @ ( set @ A ) )
= ( ^ [A7: set @ A,B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A7 @ B6 )
& ( A7 != B6 ) ) ) ) ).
% psubset_eq
thf(fact_2404_psubset__imp__subset,axiom,
! [A: $tType,A6: set @ A,B7: set @ A] :
( ( ord_less @ ( set @ A ) @ A6 @ B7 )
=> ( ord_less_eq @ ( set @ A ) @ A6 @ B7 ) ) ).
% psubset_imp_subset
thf(fact_2405_psubset__subset__trans,axiom,
! [A: $tType,A6: set @ A,B7: set @ A,C6: set @ A] :
( ( ord_less @ ( set @ A ) @ A6 @ B7 )
=> ( ( ord_less_eq @ ( set @ A ) @ B7 @ C6 )
=> ( ord_less @ ( set @ A ) @ A6 @ C6 ) ) ) ).
% psubset_subset_trans
thf(fact_2406_subset__not__subset__eq,axiom,
! [A: $tType] :
( ( ord_less @ ( set @ A ) )
= ( ^ [A7: set @ A,B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A7 @ B6 )
& ~ ( ord_less_eq @ ( set @ A ) @ B6 @ A7 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_2407_subset__psubset__trans,axiom,
! [A: $tType,A6: set @ A,B7: set @ A,C6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B7 )
=> ( ( ord_less @ ( set @ A ) @ B7 @ C6 )
=> ( ord_less @ ( set @ A ) @ A6 @ C6 ) ) ) ).
% subset_psubset_trans
thf(fact_2408_subset__iff__psubset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A7: set @ A,B6: set @ A] :
( ( ord_less @ ( set @ A ) @ A7 @ B6 )
| ( A7 = B6 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_2409_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_2410_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_2411_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_2412_signed__take__bit__rec,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( bit_ri4674362597316999326ke_bit @ A )
= ( ^ [N3: nat,A5: A] :
( if @ A
@ ( N3
= ( zero_zero @ nat ) )
@ ( uminus_uminus @ A @ ( modulo_modulo @ A @ A5 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
@ ( plus_plus @ A @ ( modulo_modulo @ A @ A5 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_ri4674362597316999326ke_bit @ A @ ( minus_minus @ nat @ N3 @ ( one_one @ nat ) ) @ ( divide_divide @ A @ A5 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% signed_take_bit_rec
thf(fact_2413_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_2414_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_2415_concat__bit__Suc,axiom,
! [N: nat,K: int,L: int] :
( ( bit_concat_bit @ ( suc @ N ) @ K @ L )
= ( plus_plus @ int @ ( modulo_modulo @ int @ K @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_concat_bit @ N @ ( divide_divide @ int @ K @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ L ) ) ) ) ).
% concat_bit_Suc
thf(fact_2416_set__decode__Suc,axiom,
! [N: nat,X: nat] :
( ( member @ nat @ ( suc @ N ) @ ( nat_set_decode @ X ) )
= ( member @ nat @ N @ ( nat_set_decode @ ( divide_divide @ nat @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% set_decode_Suc
thf(fact_2417_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_2418_Compl__eq__Compl__iff,axiom,
! [A: $tType,A6: set @ A,B7: set @ A] :
( ( ( uminus_uminus @ ( set @ A ) @ A6 )
= ( uminus_uminus @ ( set @ A ) @ B7 ) )
= ( A6 = B7 ) ) ).
% Compl_eq_Compl_iff
thf(fact_2419_Compl__iff,axiom,
! [A: $tType,C2: A,A6: set @ A] :
( ( member @ A @ C2 @ ( uminus_uminus @ ( set @ A ) @ A6 ) )
= ( ~ ( member @ A @ C2 @ A6 ) ) ) ).
% Compl_iff
thf(fact_2420_ComplI,axiom,
! [A: $tType,C2: A,A6: set @ A] :
( ~ ( member @ A @ C2 @ A6 )
=> ( member @ A @ C2 @ ( uminus_uminus @ ( set @ A ) @ A6 ) ) ) ).
% ComplI
thf(fact_2421_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_2422_concat__bit__0,axiom,
! [K: int,L: int] :
( ( bit_concat_bit @ ( zero_zero @ nat ) @ K @ L )
= L ) ).
% concat_bit_0
thf(fact_2423_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_2424_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_2425_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_2426_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_2427_concat__bit__nonnegative__iff,axiom,
! [N: nat,K: int,L: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_concat_bit @ N @ K @ L ) )
= ( ord_less_eq @ int @ ( zero_zero @ int ) @ L ) ) ).
% concat_bit_nonnegative_iff
thf(fact_2428_concat__bit__negative__iff,axiom,
! [N: nat,K: int,L: int] :
( ( ord_less @ int @ ( bit_concat_bit @ N @ K @ L ) @ ( zero_zero @ int ) )
= ( ord_less @ int @ L @ ( zero_zero @ int ) ) ) ).
% concat_bit_negative_iff
thf(fact_2429_dbl__simps_I5_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [K: num] :
( ( neg_numeral_dbl @ A @ ( numeral_numeral @ A @ K ) )
= ( numeral_numeral @ A @ ( bit0 @ K ) ) ) ) ).
% dbl_simps(5)
thf(fact_2430_dbl__simps_I1_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [K: num] :
( ( neg_numeral_dbl @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ K ) ) )
= ( uminus_uminus @ A @ ( neg_numeral_dbl @ A @ ( numeral_numeral @ A @ K ) ) ) ) ) ).
% dbl_simps(1)
thf(fact_2431_signed__take__bit__Suc__bit0,axiom,
! [N: nat,K: num] :
( ( bit_ri4674362597316999326ke_bit @ int @ ( suc @ N ) @ ( numeral_numeral @ int @ ( bit0 @ K ) ) )
= ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N @ ( numeral_numeral @ int @ K ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ).
% signed_take_bit_Suc_bit0
thf(fact_2432_signed__take__bit__Suc__minus__bit0,axiom,
! [N: nat,K: num] :
( ( bit_ri4674362597316999326ke_bit @ int @ ( suc @ N ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ K ) ) ) )
= ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ).
% signed_take_bit_Suc_minus_bit0
thf(fact_2433_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_2434_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_2435_double__complement,axiom,
! [A: $tType,A6: set @ A] :
( ( uminus_uminus @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A6 ) )
= A6 ) ).
% double_complement
thf(fact_2436_ComplD,axiom,
! [A: $tType,C2: A,A6: set @ A] :
( ( member @ A @ C2 @ ( uminus_uminus @ ( set @ A ) @ A6 ) )
=> ~ ( member @ A @ C2 @ A6 ) ) ).
% ComplD
thf(fact_2437_signed__take__bit__mult,axiom,
! [N: nat,K: int,L: int] :
( ( bit_ri4674362597316999326ke_bit @ int @ N @ ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K ) @ ( bit_ri4674362597316999326ke_bit @ int @ N @ L ) ) )
= ( bit_ri4674362597316999326ke_bit @ int @ N @ ( times_times @ int @ K @ L ) ) ) ).
% signed_take_bit_mult
thf(fact_2438_signed__take__bit__minus,axiom,
! [N: nat,K: int] :
( ( bit_ri4674362597316999326ke_bit @ int @ N @ ( uminus_uminus @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K ) ) )
= ( bit_ri4674362597316999326ke_bit @ int @ N @ ( uminus_uminus @ int @ K ) ) ) ).
% signed_take_bit_minus
thf(fact_2439_subset__decode__imp__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ ( set @ nat ) @ ( nat_set_decode @ M ) @ ( nat_set_decode @ N ) )
=> ( ord_less_eq @ nat @ M @ N ) ) ).
% subset_decode_imp_le
thf(fact_2440_dbl__def,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( neg_numeral_dbl @ A )
= ( ^ [X2: A] : ( plus_plus @ A @ X2 @ X2 ) ) ) ) ).
% dbl_def
thf(fact_2441_signed__take__bit__int__less__exp,axiom,
! [N: nat,K: int] : ( ord_less @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ).
% signed_take_bit_int_less_exp
thf(fact_2442_signed__take__bit__int__greater__eq__self__iff,axiom,
! [K: int,N: nat] :
( ( ord_less_eq @ int @ K @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K ) )
= ( ord_less @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ).
% signed_take_bit_int_greater_eq_self_iff
thf(fact_2443_signed__take__bit__int__less__self__iff,axiom,
! [N: nat,K: int] :
( ( ord_less @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K ) @ K )
= ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) @ K ) ) ).
% signed_take_bit_int_less_self_iff
thf(fact_2444_signed__take__bit__int__greater__eq__minus__exp,axiom,
! [N: nat,K: int] : ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K ) ) ).
% signed_take_bit_int_greater_eq_minus_exp
thf(fact_2445_signed__take__bit__int__less__eq__self__iff,axiom,
! [N: nat,K: int] :
( ( ord_less_eq @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K ) @ K )
= ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) @ K ) ) ).
% signed_take_bit_int_less_eq_self_iff
thf(fact_2446_signed__take__bit__int__greater__self__iff,axiom,
! [K: int,N: nat] :
( ( ord_less @ int @ K @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K ) )
= ( ord_less @ int @ K @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% signed_take_bit_int_greater_self_iff
thf(fact_2447_signed__take__bit__int__less__eq,axiom,
! [N: nat,K: int] :
( ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) @ K )
=> ( ord_less_eq @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K ) @ ( minus_minus @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( suc @ N ) ) ) ) ) ).
% signed_take_bit_int_less_eq
thf(fact_2448_signed__take__bit__int__eq__self,axiom,
! [N: nat,K: int] :
( ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) @ K )
=> ( ( ord_less @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( bit_ri4674362597316999326ke_bit @ int @ N @ K )
= K ) ) ) ).
% signed_take_bit_int_eq_self
thf(fact_2449_signed__take__bit__int__eq__self__iff,axiom,
! [N: nat,K: int] :
( ( ( bit_ri4674362597316999326ke_bit @ int @ N @ K )
= K )
= ( ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) @ K )
& ( ord_less @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% signed_take_bit_int_eq_self_iff
thf(fact_2450_signed__take__bit__int__greater__eq,axiom,
! [K: int,N: nat] :
( ( ord_less @ int @ K @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) )
=> ( ord_less_eq @ int @ ( plus_plus @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( suc @ N ) ) ) @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K ) ) ) ).
% signed_take_bit_int_greater_eq
thf(fact_2451_signed__take__bit__Suc__minus__bit1,axiom,
! [N: nat,K: num] :
( ( bit_ri4674362597316999326ke_bit @ int @ ( suc @ N ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ K ) ) ) )
= ( plus_plus @ int @ ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N @ ( minus_minus @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) @ ( one_one @ int ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( one_one @ int ) ) ) ).
% signed_take_bit_Suc_minus_bit1
thf(fact_2452_fact__code,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiring_char_0_fact @ A )
= ( ^ [N3: nat] : ( semiring_1_of_nat @ A @ ( set_fo6178422350223883121st_nat @ nat @ ( times_times @ nat ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 @ ( one_one @ nat ) ) ) ) ) ) ).
% fact_code
thf(fact_2453_take__bit__rec,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se2584673776208193580ke_bit @ A )
= ( ^ [N3: nat,A5: A] :
( if @ A
@ ( N3
= ( zero_zero @ nat ) )
@ ( zero_zero @ A )
@ ( plus_plus @ A @ ( times_times @ A @ ( bit_se2584673776208193580ke_bit @ A @ ( minus_minus @ nat @ N3 @ ( one_one @ nat ) ) @ ( divide_divide @ A @ A5 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ A @ A5 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% take_bit_rec
thf(fact_2454_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_2455_and__int__unfold,axiom,
( ( bit_se5824344872417868541ns_and @ int )
= ( ^ [K3: int,L3: int] :
( if @ int
@ ( ( K3
= ( zero_zero @ int ) )
| ( L3
= ( zero_zero @ int ) ) )
@ ( zero_zero @ int )
@ ( if @ int
@ ( K3
= ( uminus_uminus @ int @ ( one_one @ int ) ) )
@ L3
@ ( if @ int
@ ( L3
= ( uminus_uminus @ int @ ( one_one @ int ) ) )
@ K3
@ ( plus_plus @ int @ ( times_times @ int @ ( modulo_modulo @ int @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ int @ L3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( divide_divide @ int @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ).
% and_int_unfold
thf(fact_2456_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_2457_semiring__norm_I90_J,axiom,
! [M: num,N: num] :
( ( ( bit1 @ M )
= ( bit1 @ N ) )
= ( M = N ) ) ).
% semiring_norm(90)
thf(fact_2458_verit__eq__simplify_I9_J,axiom,
! [X32: num,Y32: num] :
( ( ( bit1 @ X32 )
= ( bit1 @ Y32 ) )
= ( X32 = Y32 ) ) ).
% verit_eq_simplify(9)
thf(fact_2459_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_2460_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_2461_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_2462_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_2463_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_2464_semiring__norm_I88_J,axiom,
! [M: num,N: num] :
( ( bit0 @ M )
!= ( bit1 @ N ) ) ).
% semiring_norm(88)
thf(fact_2465_semiring__norm_I89_J,axiom,
! [M: num,N: num] :
( ( bit1 @ M )
!= ( bit0 @ N ) ) ).
% semiring_norm(89)
thf(fact_2466_semiring__norm_I84_J,axiom,
! [N: num] :
( one2
!= ( bit1 @ N ) ) ).
% semiring_norm(84)
thf(fact_2467_semiring__norm_I86_J,axiom,
! [M: num] :
( ( bit1 @ M )
!= one2 ) ).
% semiring_norm(86)
thf(fact_2468_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_2469_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_2470_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_2471_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_2472_concat__bit__of__zero__2,axiom,
! [N: nat,K: int] :
( ( bit_concat_bit @ N @ K @ ( zero_zero @ int ) )
= ( bit_se2584673776208193580ke_bit @ int @ N @ K ) ) ).
% concat_bit_of_zero_2
thf(fact_2473_semiring__norm_I73_J,axiom,
! [M: num,N: num] :
( ( ord_less_eq @ num @ ( bit1 @ M ) @ ( bit1 @ N ) )
= ( ord_less_eq @ num @ M @ N ) ) ).
% semiring_norm(73)
thf(fact_2474_semiring__norm_I80_J,axiom,
! [M: num,N: num] :
( ( ord_less @ num @ ( bit1 @ M ) @ ( bit1 @ N ) )
= ( ord_less @ num @ M @ N ) ) ).
% semiring_norm(80)
thf(fact_2475_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_2476_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_2477_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_2478_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_2479_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_2480_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_2481_of__nat__nat__take__bit__eq,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [N: nat,K: int] :
( ( semiring_1_of_nat @ A @ ( nat2 @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) ) )
= ( ring_1_of_int @ A @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) ) ) ) ).
% of_nat_nat_take_bit_eq
thf(fact_2482_and__nonnegative__int__iff,axiom,
! [K: int,L: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se5824344872417868541ns_and @ int @ K @ L ) )
= ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
| ( ord_less_eq @ int @ ( zero_zero @ int ) @ L ) ) ) ).
% and_nonnegative_int_iff
thf(fact_2483_and__negative__int__iff,axiom,
! [K: int,L: int] :
( ( ord_less @ int @ ( bit_se5824344872417868541ns_and @ int @ K @ L ) @ ( zero_zero @ int ) )
= ( ( ord_less @ int @ K @ ( zero_zero @ int ) )
& ( ord_less @ int @ L @ ( zero_zero @ int ) ) ) ) ).
% and_negative_int_iff
thf(fact_2484_semiring__norm_I9_J,axiom,
! [M: num,N: num] :
( ( plus_plus @ num @ ( bit1 @ M ) @ ( bit0 @ N ) )
= ( bit1 @ ( plus_plus @ num @ M @ N ) ) ) ).
% semiring_norm(9)
thf(fact_2485_semiring__norm_I7_J,axiom,
! [M: num,N: num] :
( ( plus_plus @ num @ ( bit0 @ M ) @ ( bit1 @ N ) )
= ( bit1 @ ( plus_plus @ num @ M @ N ) ) ) ).
% semiring_norm(7)
thf(fact_2486_semiring__norm_I14_J,axiom,
! [M: num,N: num] :
( ( times_times @ num @ ( bit0 @ M ) @ ( bit1 @ N ) )
= ( bit0 @ ( times_times @ num @ M @ ( bit1 @ N ) ) ) ) ).
% semiring_norm(14)
thf(fact_2487_semiring__norm_I15_J,axiom,
! [M: num,N: num] :
( ( times_times @ num @ ( bit1 @ M ) @ ( bit0 @ N ) )
= ( bit0 @ ( times_times @ num @ ( bit1 @ M ) @ N ) ) ) ).
% semiring_norm(15)
thf(fact_2488_semiring__norm_I81_J,axiom,
! [M: num,N: num] :
( ( ord_less @ num @ ( bit1 @ M ) @ ( bit0 @ N ) )
= ( ord_less @ num @ M @ N ) ) ).
% semiring_norm(81)
thf(fact_2489_semiring__norm_I72_J,axiom,
! [M: num,N: num] :
( ( ord_less_eq @ num @ ( bit0 @ M ) @ ( bit1 @ N ) )
= ( ord_less_eq @ num @ M @ N ) ) ).
% semiring_norm(72)
thf(fact_2490_semiring__norm_I77_J,axiom,
! [N: num] : ( ord_less @ num @ one2 @ ( bit1 @ N ) ) ).
% semiring_norm(77)
thf(fact_2491_semiring__norm_I70_J,axiom,
! [M: num] :
~ ( ord_less_eq @ num @ ( bit1 @ M ) @ one2 ) ).
% semiring_norm(70)
thf(fact_2492_dbl__inc__simps_I5_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [K: num] :
( ( neg_numeral_dbl_inc @ A @ ( numeral_numeral @ A @ K ) )
= ( numeral_numeral @ A @ ( bit1 @ K ) ) ) ) ).
% dbl_inc_simps(5)
thf(fact_2493_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_2494_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_2495_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_2496_semiring__norm_I10_J,axiom,
! [M: num,N: num] :
( ( plus_plus @ num @ ( bit1 @ M ) @ ( bit1 @ N ) )
= ( bit0 @ ( plus_plus @ num @ ( plus_plus @ num @ M @ N ) @ one2 ) ) ) ).
% semiring_norm(10)
thf(fact_2497_semiring__norm_I8_J,axiom,
! [M: num] :
( ( plus_plus @ num @ ( bit1 @ M ) @ one2 )
= ( bit0 @ ( plus_plus @ num @ M @ one2 ) ) ) ).
% semiring_norm(8)
thf(fact_2498_semiring__norm_I5_J,axiom,
! [M: num] :
( ( plus_plus @ num @ ( bit0 @ M ) @ one2 )
= ( bit1 @ M ) ) ).
% semiring_norm(5)
thf(fact_2499_semiring__norm_I4_J,axiom,
! [N: num] :
( ( plus_plus @ num @ one2 @ ( bit1 @ N ) )
= ( bit0 @ ( plus_plus @ num @ N @ one2 ) ) ) ).
% semiring_norm(4)
thf(fact_2500_semiring__norm_I3_J,axiom,
! [N: num] :
( ( plus_plus @ num @ one2 @ ( bit0 @ N ) )
= ( bit1 @ N ) ) ).
% semiring_norm(3)
thf(fact_2501_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_2502_semiring__norm_I16_J,axiom,
! [M: num,N: num] :
( ( times_times @ num @ ( bit1 @ M ) @ ( bit1 @ N ) )
= ( bit1 @ ( plus_plus @ num @ ( plus_plus @ num @ M @ N ) @ ( bit0 @ ( times_times @ num @ M @ N ) ) ) ) ) ).
% semiring_norm(16)
thf(fact_2503_semiring__norm_I74_J,axiom,
! [M: num,N: num] :
( ( ord_less_eq @ num @ ( bit1 @ M ) @ ( bit0 @ N ) )
= ( ord_less @ num @ M @ N ) ) ).
% semiring_norm(74)
thf(fact_2504_semiring__norm_I79_J,axiom,
! [M: num,N: num] :
( ( ord_less @ num @ ( bit0 @ M ) @ ( bit1 @ N ) )
= ( ord_less_eq @ num @ M @ N ) ) ).
% semiring_norm(79)
thf(fact_2505_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_2506_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_2507_and__numerals_I3_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num,Y3: num] :
( ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ ( bit0 @ X ) ) @ ( numeral_numeral @ A @ ( bit0 @ Y3 ) ) )
= ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ X ) @ ( numeral_numeral @ A @ Y3 ) ) ) ) ) ).
% and_numerals(3)
thf(fact_2508_xor__numerals_I3_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num,Y3: num] :
( ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ ( bit0 @ X ) ) @ ( numeral_numeral @ A @ ( bit0 @ Y3 ) ) )
= ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ X ) @ ( numeral_numeral @ A @ Y3 ) ) ) ) ) ).
% xor_numerals(3)
thf(fact_2509_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_2510_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_2511_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_2512_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_2513_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_2514_and__minus__numerals_I2_J,axiom,
! [N: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( one_one @ int ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) )
= ( one_one @ int ) ) ).
% and_minus_numerals(2)
thf(fact_2515_and__minus__numerals_I6_J,axiom,
! [N: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) @ ( one_one @ int ) )
= ( one_one @ int ) ) ).
% and_minus_numerals(6)
thf(fact_2516_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_2517_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_2518_and__numerals_I4_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num,Y3: num] :
( ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ ( bit0 @ X ) ) @ ( numeral_numeral @ A @ ( bit1 @ Y3 ) ) )
= ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ X ) @ ( numeral_numeral @ A @ Y3 ) ) ) ) ) ).
% and_numerals(4)
thf(fact_2519_and__numerals_I6_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num,Y3: num] :
( ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ ( bit1 @ X ) ) @ ( numeral_numeral @ A @ ( bit0 @ Y3 ) ) )
= ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ X ) @ ( numeral_numeral @ A @ Y3 ) ) ) ) ) ).
% and_numerals(6)
thf(fact_2520_xor__numerals_I7_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num,Y3: num] :
( ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ ( bit1 @ X ) ) @ ( numeral_numeral @ A @ ( bit1 @ Y3 ) ) )
= ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ X ) @ ( numeral_numeral @ A @ Y3 ) ) ) ) ) ).
% xor_numerals(7)
thf(fact_2521_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_2522_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_2523_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_2524_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_2525_and__minus__numerals_I5_J,axiom,
! [N: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) @ ( one_one @ int ) )
= ( zero_zero @ int ) ) ).
% and_minus_numerals(5)
thf(fact_2526_and__minus__numerals_I1_J,axiom,
! [N: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( one_one @ int ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) )
= ( zero_zero @ int ) ) ).
% and_minus_numerals(1)
thf(fact_2527_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_2528_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_2529_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_2530_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_2531_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_2532_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_2533_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_2534_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_2535_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_2536_take__bit__of__exp,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [M: nat,N: nat] :
( ( bit_se2584673776208193580ke_bit @ A @ M @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
= ( times_times @ A @ ( zero_neq_one_of_bool @ A @ ( ord_less @ nat @ N @ M ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% take_bit_of_exp
thf(fact_2537_take__bit__of__2,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N: nat] :
( ( bit_se2584673776208193580ke_bit @ A @ N @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( times_times @ A @ ( zero_neq_one_of_bool @ A @ ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% take_bit_of_2
thf(fact_2538_zmod__numeral__Bit1,axiom,
! [V: num,W2: num] :
( ( modulo_modulo @ int @ ( numeral_numeral @ int @ ( bit1 @ V ) ) @ ( numeral_numeral @ int @ ( bit0 @ W2 ) ) )
= ( plus_plus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( modulo_modulo @ int @ ( numeral_numeral @ int @ V ) @ ( numeral_numeral @ int @ W2 ) ) ) @ ( one_one @ int ) ) ) ).
% zmod_numeral_Bit1
thf(fact_2539_signed__take__bit__Suc__bit1,axiom,
! [N: nat,K: num] :
( ( bit_ri4674362597316999326ke_bit @ int @ ( suc @ N ) @ ( numeral_numeral @ int @ ( bit1 @ K ) ) )
= ( plus_plus @ int @ ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N @ ( numeral_numeral @ int @ K ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( one_one @ int ) ) ) ).
% signed_take_bit_Suc_bit1
thf(fact_2540_take__bit__of__nat,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,M: nat] :
( ( bit_se2584673776208193580ke_bit @ A @ N @ ( semiring_1_of_nat @ A @ M ) )
= ( semiring_1_of_nat @ A @ ( bit_se2584673776208193580ke_bit @ nat @ N @ M ) ) ) ) ).
% take_bit_of_nat
thf(fact_2541_of__nat__xor__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,N: nat] :
( ( semiring_1_of_nat @ A @ ( bit_se5824344971392196577ns_xor @ nat @ M @ N ) )
= ( bit_se5824344971392196577ns_xor @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% of_nat_xor_eq
thf(fact_2542_of__nat__and__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,N: nat] :
( ( semiring_1_of_nat @ A @ ( bit_se5824344872417868541ns_and @ nat @ M @ N ) )
= ( bit_se5824344872417868541ns_and @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% of_nat_and_eq
thf(fact_2543_take__bit__tightened,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A3: A,B2: A,M: nat] :
( ( ( bit_se2584673776208193580ke_bit @ A @ N @ A3 )
= ( bit_se2584673776208193580ke_bit @ A @ N @ B2 ) )
=> ( ( ord_less_eq @ nat @ M @ N )
=> ( ( bit_se2584673776208193580ke_bit @ A @ M @ A3 )
= ( bit_se2584673776208193580ke_bit @ A @ M @ B2 ) ) ) ) ) ).
% take_bit_tightened
thf(fact_2544_take__bit__tightened__less__eq__nat,axiom,
! [M: nat,N: nat,Q4: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ord_less_eq @ nat @ ( bit_se2584673776208193580ke_bit @ nat @ M @ Q4 ) @ ( bit_se2584673776208193580ke_bit @ nat @ N @ Q4 ) ) ) ).
% take_bit_tightened_less_eq_nat
thf(fact_2545_take__bit__nat__less__eq__self,axiom,
! [N: nat,M: nat] : ( ord_less_eq @ nat @ ( bit_se2584673776208193580ke_bit @ nat @ N @ M ) @ M ) ).
% take_bit_nat_less_eq_self
thf(fact_2546_verit__eq__simplify_I14_J,axiom,
! [X22: num,X32: num] :
( ( bit0 @ X22 )
!= ( bit1 @ X32 ) ) ).
% verit_eq_simplify(14)
thf(fact_2547_verit__eq__simplify_I12_J,axiom,
! [X32: num] :
( one2
!= ( bit1 @ X32 ) ) ).
% verit_eq_simplify(12)
thf(fact_2548_take__bit__mult,axiom,
! [N: nat,K: int,L: int] :
( ( bit_se2584673776208193580ke_bit @ int @ N @ ( times_times @ int @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) @ ( bit_se2584673776208193580ke_bit @ int @ N @ L ) ) )
= ( bit_se2584673776208193580ke_bit @ int @ N @ ( times_times @ int @ K @ L ) ) ) ).
% take_bit_mult
thf(fact_2549_take__bit__minus,axiom,
! [N: nat,K: int] :
( ( bit_se2584673776208193580ke_bit @ int @ N @ ( uminus_uminus @ int @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) ) )
= ( bit_se2584673776208193580ke_bit @ int @ N @ ( uminus_uminus @ int @ K ) ) ) ).
% take_bit_minus
thf(fact_2550_take__bit__nat__eq,axiom,
! [K: int,N: nat] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
=> ( ( bit_se2584673776208193580ke_bit @ nat @ N @ ( nat2 @ K ) )
= ( nat2 @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) ) ) ) ).
% take_bit_nat_eq
thf(fact_2551_nat__take__bit__eq,axiom,
! [K: int,N: nat] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
=> ( ( nat2 @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) )
= ( bit_se2584673776208193580ke_bit @ nat @ N @ ( nat2 @ K ) ) ) ) ).
% nat_take_bit_eq
thf(fact_2552_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_2553_num_Oexhaust,axiom,
! [Y3: num] :
( ( Y3 != one2 )
=> ( ! [X23: num] :
( Y3
!= ( bit0 @ X23 ) )
=> ~ ! [X33: num] :
( Y3
!= ( bit1 @ X33 ) ) ) ) ).
% num.exhaust
thf(fact_2554_AND__upper2_H,axiom,
! [Y3: int,Z3: int,X: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y3 )
=> ( ( ord_less_eq @ int @ Y3 @ Z3 )
=> ( ord_less_eq @ int @ ( bit_se5824344872417868541ns_and @ int @ X @ Y3 ) @ Z3 ) ) ) ).
% AND_upper2'
thf(fact_2555_AND__upper1_H,axiom,
! [Y3: int,Z3: int,Ya: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y3 )
=> ( ( ord_less_eq @ int @ Y3 @ Z3 )
=> ( ord_less_eq @ int @ ( bit_se5824344872417868541ns_and @ int @ Y3 @ Ya ) @ Z3 ) ) ) ).
% AND_upper1'
thf(fact_2556_AND__upper2,axiom,
! [Y3: int,X: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y3 )
=> ( ord_less_eq @ int @ ( bit_se5824344872417868541ns_and @ int @ X @ Y3 ) @ Y3 ) ) ).
% AND_upper2
thf(fact_2557_AND__upper1,axiom,
! [X: int,Y3: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
=> ( ord_less_eq @ int @ ( bit_se5824344872417868541ns_and @ int @ X @ Y3 ) @ X ) ) ).
% AND_upper1
thf(fact_2558_AND__lower,axiom,
! [X: int,Y3: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
=> ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se5824344872417868541ns_and @ int @ X @ Y3 ) ) ) ).
% AND_lower
thf(fact_2559_take__bit__int__less__eq__self__iff,axiom,
! [N: nat,K: int] :
( ( ord_less_eq @ int @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) @ K )
= ( ord_less_eq @ int @ ( zero_zero @ int ) @ K ) ) ).
% take_bit_int_less_eq_self_iff
thf(fact_2560_take__bit__nonnegative,axiom,
! [N: nat,K: int] : ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) ) ).
% take_bit_nonnegative
thf(fact_2561_not__take__bit__negative,axiom,
! [N: nat,K: int] :
~ ( ord_less @ int @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) @ ( zero_zero @ int ) ) ).
% not_take_bit_negative
thf(fact_2562_take__bit__int__greater__self__iff,axiom,
! [K: int,N: nat] :
( ( ord_less @ int @ K @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) )
= ( ord_less @ int @ K @ ( zero_zero @ int ) ) ) ).
% take_bit_int_greater_self_iff
thf(fact_2563_take__bit__tightened__less__eq__int,axiom,
! [M: nat,N: nat,K: int] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ord_less_eq @ int @ ( bit_se2584673776208193580ke_bit @ int @ M @ K ) @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) ) ) ).
% take_bit_tightened_less_eq_int
thf(fact_2564_signed__take__bit__eq__iff__take__bit__eq,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: nat,A3: A,B2: A] :
( ( ( bit_ri4674362597316999326ke_bit @ A @ N @ A3 )
= ( bit_ri4674362597316999326ke_bit @ A @ N @ B2 ) )
= ( ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N ) @ A3 )
= ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N ) @ B2 ) ) ) ) ).
% signed_take_bit_eq_iff_take_bit_eq
thf(fact_2565_signed__take__bit__take__bit,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [M: nat,N: nat,A3: A] :
( ( bit_ri4674362597316999326ke_bit @ A @ M @ ( bit_se2584673776208193580ke_bit @ A @ N @ A3 ) )
= ( if @ ( A > A ) @ ( ord_less_eq @ nat @ N @ M ) @ ( bit_se2584673776208193580ke_bit @ A @ N ) @ ( bit_ri4674362597316999326ke_bit @ A @ M ) @ A3 ) ) ) ).
% signed_take_bit_take_bit
thf(fact_2566_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_2567_take__bit__unset__bit__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,M: nat,A3: A] :
( ( ( ord_less_eq @ nat @ N @ M )
=> ( ( bit_se2584673776208193580ke_bit @ A @ N @ ( bit_se2638667681897837118et_bit @ A @ M @ A3 ) )
= ( bit_se2584673776208193580ke_bit @ A @ N @ A3 ) ) )
& ( ~ ( ord_less_eq @ nat @ N @ M )
=> ( ( bit_se2584673776208193580ke_bit @ A @ N @ ( bit_se2638667681897837118et_bit @ A @ M @ A3 ) )
= ( bit_se2638667681897837118et_bit @ A @ M @ ( bit_se2584673776208193580ke_bit @ A @ N @ A3 ) ) ) ) ) ) ).
% take_bit_unset_bit_eq
thf(fact_2568_take__bit__set__bit__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,M: nat,A3: A] :
( ( ( ord_less_eq @ nat @ N @ M )
=> ( ( bit_se2584673776208193580ke_bit @ A @ N @ ( bit_se5668285175392031749et_bit @ A @ M @ A3 ) )
= ( bit_se2584673776208193580ke_bit @ A @ N @ A3 ) ) )
& ( ~ ( ord_less_eq @ nat @ N @ M )
=> ( ( bit_se2584673776208193580ke_bit @ A @ N @ ( bit_se5668285175392031749et_bit @ A @ M @ A3 ) )
= ( bit_se5668285175392031749et_bit @ A @ M @ ( bit_se2584673776208193580ke_bit @ A @ N @ A3 ) ) ) ) ) ) ).
% take_bit_set_bit_eq
thf(fact_2569_take__bit__flip__bit__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,M: nat,A3: A] :
( ( ( ord_less_eq @ nat @ N @ M )
=> ( ( bit_se2584673776208193580ke_bit @ A @ N @ ( bit_se8732182000553998342ip_bit @ A @ M @ A3 ) )
= ( bit_se2584673776208193580ke_bit @ A @ N @ A3 ) ) )
& ( ~ ( ord_less_eq @ nat @ N @ M )
=> ( ( bit_se2584673776208193580ke_bit @ A @ N @ ( bit_se8732182000553998342ip_bit @ A @ M @ A3 ) )
= ( bit_se8732182000553998342ip_bit @ A @ M @ ( bit_se2584673776208193580ke_bit @ A @ N @ A3 ) ) ) ) ) ) ).
% take_bit_flip_bit_eq
thf(fact_2570_inc_Osimps_I3_J,axiom,
! [X: num] :
( ( inc @ ( bit1 @ X ) )
= ( bit0 @ ( inc @ X ) ) ) ).
% inc.simps(3)
thf(fact_2571_inc_Osimps_I2_J,axiom,
! [X: num] :
( ( inc @ ( bit0 @ X ) )
= ( bit1 @ X ) ) ).
% inc.simps(2)
thf(fact_2572_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_2573_take__bit__signed__take__bit,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [M: nat,N: nat,A3: A] :
( ( ord_less_eq @ nat @ M @ ( suc @ N ) )
=> ( ( bit_se2584673776208193580ke_bit @ A @ M @ ( bit_ri4674362597316999326ke_bit @ A @ N @ A3 ) )
= ( bit_se2584673776208193580ke_bit @ A @ M @ A3 ) ) ) ) ).
% take_bit_signed_take_bit
thf(fact_2574_and__less__eq,axiom,
! [L: int,K: int] :
( ( ord_less @ int @ L @ ( zero_zero @ int ) )
=> ( ord_less_eq @ int @ ( bit_se5824344872417868541ns_and @ int @ K @ L ) @ K ) ) ).
% and_less_eq
thf(fact_2575_AND__upper1_H_H,axiom,
! [Y3: int,Z3: int,Ya: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y3 )
=> ( ( ord_less @ int @ Y3 @ Z3 )
=> ( ord_less @ int @ ( bit_se5824344872417868541ns_and @ int @ Y3 @ Ya ) @ Z3 ) ) ) ).
% AND_upper1''
thf(fact_2576_AND__upper2_H_H,axiom,
! [Y3: int,Z3: int,X: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y3 )
=> ( ( ord_less @ int @ Y3 @ Z3 )
=> ( ord_less @ int @ ( bit_se5824344872417868541ns_and @ int @ X @ Y3 ) @ Z3 ) ) ) ).
% AND_upper2''
thf(fact_2577_eval__nat__numeral_I3_J,axiom,
! [N: num] :
( ( numeral_numeral @ nat @ ( bit1 @ N ) )
= ( suc @ ( numeral_numeral @ nat @ ( bit0 @ N ) ) ) ) ).
% eval_nat_numeral(3)
thf(fact_2578_power__minus__Bit1,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X: A,K: num] :
( ( power_power @ A @ ( uminus_uminus @ A @ X ) @ ( numeral_numeral @ nat @ ( bit1 @ K ) ) )
= ( uminus_uminus @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit1 @ K ) ) ) ) ) ) ).
% power_minus_Bit1
thf(fact_2579_take__bit__decr__eq,axiom,
! [N: nat,K: int] :
( ( ( bit_se2584673776208193580ke_bit @ int @ N @ K )
!= ( zero_zero @ int ) )
=> ( ( bit_se2584673776208193580ke_bit @ int @ N @ ( minus_minus @ int @ K @ ( one_one @ int ) ) )
= ( minus_minus @ int @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) @ ( one_one @ int ) ) ) ) ).
% take_bit_decr_eq
thf(fact_2580_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_2581_cong__exp__iff__simps_I3_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [N: num,Q4: num] :
( ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit1 @ N ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q4 ) ) )
!= ( zero_zero @ A ) ) ) ).
% cong_exp_iff_simps(3)
thf(fact_2582_numeral__Bit1__div__2,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [N: num] :
( ( divide_divide @ A @ ( numeral_numeral @ A @ ( bit1 @ N ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( numeral_numeral @ A @ N ) ) ) ).
% numeral_Bit1_div_2
thf(fact_2583_power3__eq__cube,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A3: A] :
( ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) )
= ( times_times @ A @ ( times_times @ A @ A3 @ A3 ) @ A3 ) ) ) ).
% power3_eq_cube
thf(fact_2584_numeral__3__eq__3,axiom,
( ( numeral_numeral @ nat @ ( bit1 @ one2 ) )
= ( suc @ ( suc @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% numeral_3_eq_3
thf(fact_2585_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_2586_take__bit__Suc__minus__bit1,axiom,
! [N: nat,K: num] :
( ( bit_se2584673776208193580ke_bit @ int @ ( suc @ N ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ K ) ) ) )
= ( plus_plus @ int @ ( times_times @ int @ ( bit_se2584673776208193580ke_bit @ int @ N @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( inc @ K ) ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( one_one @ int ) ) ) ).
% take_bit_Suc_minus_bit1
thf(fact_2587_take__bit__Suc__bit0,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N: nat,K: num] :
( ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N ) @ ( numeral_numeral @ A @ ( bit0 @ K ) ) )
= ( times_times @ A @ ( bit_se2584673776208193580ke_bit @ A @ N @ ( numeral_numeral @ A @ K ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% take_bit_Suc_bit0
thf(fact_2588_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_2589_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_2590_take__bit__nat__eq__self,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( bit_se2584673776208193580ke_bit @ nat @ N @ M )
= M ) ) ).
% take_bit_nat_eq_self
thf(fact_2591_take__bit__nat__less__exp,axiom,
! [N: nat,M: nat] : ( ord_less @ nat @ ( bit_se2584673776208193580ke_bit @ nat @ N @ M ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ).
% take_bit_nat_less_exp
thf(fact_2592_take__bit__nat__eq__self__iff,axiom,
! [N: nat,M: nat] :
( ( ( bit_se2584673776208193580ke_bit @ nat @ N @ M )
= M )
= ( ord_less @ nat @ M @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ).
% take_bit_nat_eq_self_iff
thf(fact_2593_cong__exp__iff__simps_I7_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [Q4: num,N: num] :
( ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ one2 ) @ ( numeral_numeral @ A @ ( bit0 @ Q4 ) ) )
= ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit1 @ N ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q4 ) ) ) )
= ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ N ) @ ( numeral_numeral @ A @ Q4 ) )
= ( zero_zero @ A ) ) ) ) ).
% cong_exp_iff_simps(7)
thf(fact_2594_cong__exp__iff__simps_I11_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M: num,Q4: num] :
( ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit1 @ M ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q4 ) ) )
= ( modulo_modulo @ A @ ( numeral_numeral @ A @ one2 ) @ ( numeral_numeral @ A @ ( bit0 @ Q4 ) ) ) )
= ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ Q4 ) )
= ( zero_zero @ A ) ) ) ) ).
% cong_exp_iff_simps(11)
thf(fact_2595_take__bit__int__less__exp,axiom,
! [N: nat,K: int] : ( ord_less @ int @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ).
% take_bit_int_less_exp
thf(fact_2596_fold__atLeastAtMost__nat_Osimps,axiom,
! [A: $tType] :
( ( set_fo6178422350223883121st_nat @ A )
= ( ^ [F2: nat > A > A,A5: nat,B3: nat,Acc: A] : ( if @ A @ ( ord_less @ nat @ B3 @ A5 ) @ Acc @ ( set_fo6178422350223883121st_nat @ A @ F2 @ ( plus_plus @ nat @ A5 @ ( one_one @ nat ) ) @ B3 @ ( F2 @ A5 @ Acc ) ) ) ) ) ).
% fold_atLeastAtMost_nat.simps
thf(fact_2597_fold__atLeastAtMost__nat_Oelims,axiom,
! [A: $tType,X: nat > A > A,Xa3: nat,Xb3: nat,Xc: A,Y3: A] :
( ( ( set_fo6178422350223883121st_nat @ A @ X @ Xa3 @ Xb3 @ Xc )
= Y3 )
=> ( ( ( ord_less @ nat @ Xb3 @ Xa3 )
=> ( Y3 = Xc ) )
& ( ~ ( ord_less @ nat @ Xb3 @ Xa3 )
=> ( Y3
= ( set_fo6178422350223883121st_nat @ A @ X @ ( plus_plus @ nat @ Xa3 @ ( one_one @ nat ) ) @ Xb3 @ ( X @ Xa3 @ Xc ) ) ) ) ) ) ).
% fold_atLeastAtMost_nat.elims
thf(fact_2598_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_2599_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_2600_num_Osize__gen_I1_J,axiom,
( ( size_num @ one2 )
= ( zero_zero @ nat ) ) ).
% num.size_gen(1)
thf(fact_2601_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_2602_take__bit__nat__less__self__iff,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ ( bit_se2584673776208193580ke_bit @ nat @ N @ M ) @ M )
= ( ord_less_eq @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ M ) ) ).
% take_bit_nat_less_self_iff
thf(fact_2603_take__bit__Suc__minus__bit0,axiom,
! [N: nat,K: num] :
( ( bit_se2584673776208193580ke_bit @ int @ ( suc @ N ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ K ) ) ) )
= ( times_times @ int @ ( bit_se2584673776208193580ke_bit @ int @ N @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ).
% take_bit_Suc_minus_bit0
thf(fact_2604_take__bit__int__less__self__iff,axiom,
! [N: nat,K: int] :
( ( ord_less @ int @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) @ K )
= ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) @ K ) ) ).
% take_bit_int_less_self_iff
thf(fact_2605_take__bit__int__greater__eq__self__iff,axiom,
! [K: int,N: nat] :
( ( ord_less_eq @ int @ K @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) )
= ( ord_less @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ).
% take_bit_int_greater_eq_self_iff
thf(fact_2606_take__bit__int__eq__self,axiom,
! [K: int,N: nat] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
=> ( ( ord_less @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( bit_se2584673776208193580ke_bit @ int @ N @ K )
= K ) ) ) ).
% take_bit_int_eq_self
thf(fact_2607_take__bit__int__eq__self__iff,axiom,
! [N: nat,K: int] :
( ( ( bit_se2584673776208193580ke_bit @ int @ N @ K )
= K )
= ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
& ( ord_less @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% take_bit_int_eq_self_iff
thf(fact_2608_signed__take__bit__eq__take__bit__shift,axiom,
( ( bit_ri4674362597316999326ke_bit @ int )
= ( ^ [N3: nat,K3: int] : ( minus_minus @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( suc @ N3 ) @ ( plus_plus @ int @ K3 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N3 ) ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ).
% signed_take_bit_eq_take_bit_shift
thf(fact_2609_take__bit__incr__eq,axiom,
! [N: nat,K: int] :
( ( ( bit_se2584673776208193580ke_bit @ int @ N @ K )
!= ( minus_minus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ int ) ) )
=> ( ( bit_se2584673776208193580ke_bit @ int @ N @ ( plus_plus @ int @ K @ ( one_one @ int ) ) )
= ( plus_plus @ int @ ( one_one @ int ) @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) ) ) ) ).
% take_bit_incr_eq
thf(fact_2610_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_2611_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_2612_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_2613_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_2614_take__bit__int__greater__eq,axiom,
! [K: int,N: nat] :
( ( ord_less @ int @ K @ ( zero_zero @ int ) )
=> ( ord_less_eq @ int @ ( plus_plus @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) ) ) ).
% take_bit_int_greater_eq
thf(fact_2615_xor__nat__unfold,axiom,
( ( bit_se5824344971392196577ns_xor @ nat )
= ( ^ [M2: nat,N3: nat] :
( if @ nat
@ ( M2
= ( zero_zero @ nat ) )
@ N3
@ ( if @ nat
@ ( N3
= ( zero_zero @ nat ) )
@ M2
@ ( plus_plus @ nat @ ( modulo_modulo @ nat @ ( plus_plus @ nat @ ( modulo_modulo @ nat @ M2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ nat @ ( divide_divide @ nat @ M2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% xor_nat_unfold
thf(fact_2616_and__int__rec,axiom,
( ( bit_se5824344872417868541ns_and @ int )
= ( ^ [K3: int,L3: int] :
( plus_plus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K3 )
& ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L3 ) ) )
@ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( divide_divide @ int @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% and_int_rec
thf(fact_2617_xor__nat__rec,axiom,
( ( bit_se5824344971392196577ns_xor @ nat )
= ( ^ [M2: nat,N3: nat] :
( plus_plus @ nat
@ ( zero_neq_one_of_bool @ nat
@ ( ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M2 ) )
!= ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) ) )
@ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ nat @ ( divide_divide @ nat @ M2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% xor_nat_rec
thf(fact_2618_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_2619_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_2620_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_2621_take__bit__minus__small__eq,axiom,
! [K: int,N: nat] :
( ( ord_less @ int @ ( zero_zero @ int ) @ K )
=> ( ( ord_less_eq @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( bit_se2584673776208193580ke_bit @ int @ N @ ( uminus_uminus @ int @ K ) )
= ( minus_minus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) @ K ) ) ) ) ).
% take_bit_minus_small_eq
thf(fact_2622_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_2623_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_2624_signed__take__bit__numeral__minus__bit1,axiom,
! [L: num,K: num] :
( ( bit_ri4674362597316999326ke_bit @ int @ ( numeral_numeral @ nat @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ K ) ) ) )
= ( plus_plus @ int @ ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( pred_numeral @ L ) @ ( minus_minus @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) @ ( one_one @ int ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( one_one @ int ) ) ) ).
% signed_take_bit_numeral_minus_bit1
thf(fact_2625_binomial__code,axiom,
( binomial
= ( ^ [N3: nat,K3: nat] : ( if @ nat @ ( ord_less @ nat @ N3 @ K3 ) @ ( zero_zero @ nat ) @ ( if @ nat @ ( ord_less @ nat @ N3 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K3 ) ) @ ( binomial @ N3 @ ( minus_minus @ nat @ N3 @ K3 ) ) @ ( divide_divide @ nat @ ( set_fo6178422350223883121st_nat @ nat @ ( times_times @ nat ) @ ( plus_plus @ nat @ ( minus_minus @ nat @ N3 @ K3 ) @ ( one_one @ nat ) ) @ N3 @ ( one_one @ nat ) ) @ ( semiring_char_0_fact @ nat @ K3 ) ) ) ) ) ) ).
% binomial_code
thf(fact_2626_signed__take__bit__numeral__bit1,axiom,
! [L: num,K: num] :
( ( bit_ri4674362597316999326ke_bit @ int @ ( numeral_numeral @ nat @ L ) @ ( numeral_numeral @ int @ ( bit1 @ K ) ) )
= ( plus_plus @ int @ ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( pred_numeral @ L ) @ ( numeral_numeral @ int @ K ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( one_one @ int ) ) ) ).
% signed_take_bit_numeral_bit1
thf(fact_2627_take__bit__numeral__minus__bit1,axiom,
! [L: num,K: num] :
( ( bit_se2584673776208193580ke_bit @ int @ ( numeral_numeral @ nat @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ K ) ) ) )
= ( plus_plus @ int @ ( times_times @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( pred_numeral @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( inc @ K ) ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( one_one @ int ) ) ) ).
% take_bit_numeral_minus_bit1
thf(fact_2628_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_2629_signed__take__bit__numeral__minus__bit0,axiom,
! [L: num,K: num] :
( ( bit_ri4674362597316999326ke_bit @ int @ ( numeral_numeral @ nat @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ K ) ) ) )
= ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( pred_numeral @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ).
% signed_take_bit_numeral_minus_bit0
thf(fact_2630_xor__nonnegative__int__iff,axiom,
! [K: int,L: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se5824344971392196577ns_xor @ int @ K @ L ) )
= ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
= ( ord_less_eq @ int @ ( zero_zero @ int ) @ L ) ) ) ).
% xor_nonnegative_int_iff
thf(fact_2631_xor__negative__int__iff,axiom,
! [K: int,L: int] :
( ( ord_less @ int @ ( bit_se5824344971392196577ns_xor @ int @ K @ L ) @ ( zero_zero @ int ) )
= ( ( ord_less @ int @ K @ ( zero_zero @ int ) )
!= ( ord_less @ int @ L @ ( zero_zero @ int ) ) ) ) ).
% xor_negative_int_iff
thf(fact_2632_binomial__Suc__n,axiom,
! [N: nat] :
( ( binomial @ ( suc @ N ) @ N )
= ( suc @ N ) ) ).
% binomial_Suc_n
thf(fact_2633_binomial__n__n,axiom,
! [N: nat] :
( ( binomial @ N @ N )
= ( one_one @ nat ) ) ).
% binomial_n_n
thf(fact_2634_binomial__1,axiom,
! [N: nat] :
( ( binomial @ N @ ( suc @ ( zero_zero @ nat ) ) )
= N ) ).
% binomial_1
thf(fact_2635_binomial__0__Suc,axiom,
! [K: nat] :
( ( binomial @ ( zero_zero @ nat ) @ ( suc @ K ) )
= ( zero_zero @ nat ) ) ).
% binomial_0_Suc
thf(fact_2636_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_2637_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_2638_pred__numeral__simps_I1_J,axiom,
( ( pred_numeral @ one2 )
= ( zero_zero @ nat ) ) ).
% pred_numeral_simps(1)
thf(fact_2639_binomial__n__0,axiom,
! [N: nat] :
( ( binomial @ N @ ( zero_zero @ nat ) )
= ( one_one @ nat ) ) ).
% binomial_n_0
thf(fact_2640_eq__numeral__Suc,axiom,
! [K: num,N: nat] :
( ( ( numeral_numeral @ nat @ K )
= ( suc @ N ) )
= ( ( pred_numeral @ K )
= N ) ) ).
% eq_numeral_Suc
thf(fact_2641_Suc__eq__numeral,axiom,
! [N: nat,K: num] :
( ( ( suc @ N )
= ( numeral_numeral @ nat @ K ) )
= ( N
= ( pred_numeral @ K ) ) ) ).
% Suc_eq_numeral
thf(fact_2642_pred__numeral__inc,axiom,
! [K: num] :
( ( pred_numeral @ ( inc @ K ) )
= ( numeral_numeral @ nat @ K ) ) ).
% pred_numeral_inc
thf(fact_2643_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_2644_less__Suc__numeral,axiom,
! [N: nat,K: num] :
( ( ord_less @ nat @ ( suc @ N ) @ ( numeral_numeral @ nat @ K ) )
= ( ord_less @ nat @ N @ ( pred_numeral @ K ) ) ) ).
% less_Suc_numeral
thf(fact_2645_less__numeral__Suc,axiom,
! [K: num,N: nat] :
( ( ord_less @ nat @ ( numeral_numeral @ nat @ K ) @ ( suc @ N ) )
= ( ord_less @ nat @ ( pred_numeral @ K ) @ N ) ) ).
% less_numeral_Suc
thf(fact_2646_le__Suc__numeral,axiom,
! [N: nat,K: num] :
( ( ord_less_eq @ nat @ ( suc @ N ) @ ( numeral_numeral @ nat @ K ) )
= ( ord_less_eq @ nat @ N @ ( pred_numeral @ K ) ) ) ).
% le_Suc_numeral
thf(fact_2647_le__numeral__Suc,axiom,
! [K: num,N: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ K ) @ ( suc @ N ) )
= ( ord_less_eq @ nat @ ( pred_numeral @ K ) @ N ) ) ).
% le_numeral_Suc
thf(fact_2648_diff__numeral__Suc,axiom,
! [K: num,N: nat] :
( ( minus_minus @ nat @ ( numeral_numeral @ nat @ K ) @ ( suc @ N ) )
= ( minus_minus @ nat @ ( pred_numeral @ K ) @ N ) ) ).
% diff_numeral_Suc
thf(fact_2649_diff__Suc__numeral,axiom,
! [N: nat,K: num] :
( ( minus_minus @ nat @ ( suc @ N ) @ ( numeral_numeral @ nat @ K ) )
= ( minus_minus @ nat @ N @ ( pred_numeral @ K ) ) ) ).
% diff_Suc_numeral
thf(fact_2650_pred__numeral__simps_I3_J,axiom,
! [K: num] :
( ( pred_numeral @ ( bit1 @ K ) )
= ( numeral_numeral @ nat @ ( bit0 @ K ) ) ) ).
% pred_numeral_simps(3)
thf(fact_2651_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_2652_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_2653_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_2654_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_2655_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_2656_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_2657_signed__take__bit__numeral__bit0,axiom,
! [L: num,K: num] :
( ( bit_ri4674362597316999326ke_bit @ int @ ( numeral_numeral @ nat @ L ) @ ( numeral_numeral @ int @ ( bit0 @ K ) ) )
= ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( pred_numeral @ L ) @ ( numeral_numeral @ int @ K ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ).
% signed_take_bit_numeral_bit0
thf(fact_2658_choose__one,axiom,
! [N: nat] :
( ( binomial @ N @ ( one_one @ nat ) )
= N ) ).
% choose_one
thf(fact_2659_binomial__eq__0,axiom,
! [N: nat,K: nat] :
( ( ord_less @ nat @ N @ K )
=> ( ( binomial @ N @ K )
= ( zero_zero @ nat ) ) ) ).
% binomial_eq_0
thf(fact_2660_Suc__times__binomial__eq,axiom,
! [N: nat,K: nat] :
( ( times_times @ nat @ ( suc @ N ) @ ( binomial @ N @ K ) )
= ( times_times @ nat @ ( binomial @ ( suc @ N ) @ ( suc @ K ) ) @ ( suc @ K ) ) ) ).
% Suc_times_binomial_eq
thf(fact_2661_Suc__times__binomial,axiom,
! [K: nat,N: nat] :
( ( times_times @ nat @ ( suc @ K ) @ ( binomial @ ( suc @ N ) @ ( suc @ K ) ) )
= ( times_times @ nat @ ( suc @ N ) @ ( binomial @ N @ K ) ) ) ).
% Suc_times_binomial
thf(fact_2662_XOR__lower,axiom,
! [X: int,Y3: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y3 )
=> ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se5824344971392196577ns_xor @ int @ X @ Y3 ) ) ) ) ).
% XOR_lower
thf(fact_2663_binomial__symmetric,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ N )
=> ( ( binomial @ N @ K )
= ( binomial @ N @ ( minus_minus @ nat @ N @ K ) ) ) ) ).
% binomial_symmetric
thf(fact_2664_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_2665_binomial__le__pow,axiom,
! [R: nat,N: nat] :
( ( ord_less_eq @ nat @ R @ N )
=> ( ord_less_eq @ nat @ ( binomial @ N @ R ) @ ( power_power @ nat @ N @ R ) ) ) ).
% binomial_le_pow
thf(fact_2666_binomial__gbinomial,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [N: nat,K: nat] :
( ( semiring_1_of_nat @ A @ ( binomial @ N @ K ) )
= ( gbinomial @ A @ ( semiring_1_of_nat @ A @ N ) @ K ) ) ) ).
% binomial_gbinomial
thf(fact_2667_numeral__eq__Suc,axiom,
( ( numeral_numeral @ nat )
= ( ^ [K3: num] : ( suc @ ( pred_numeral @ K3 ) ) ) ) ).
% numeral_eq_Suc
thf(fact_2668_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_2669_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_2670_choose__mult,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ M )
=> ( ( ord_less_eq @ nat @ M @ N )
=> ( ( times_times @ nat @ ( binomial @ N @ M ) @ ( binomial @ M @ K ) )
= ( times_times @ nat @ ( binomial @ N @ K ) @ ( binomial @ ( minus_minus @ nat @ N @ K ) @ ( minus_minus @ nat @ M @ K ) ) ) ) ) ) ).
% choose_mult
thf(fact_2671_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_2672_binomial__Suc__Suc__eq__times,axiom,
! [N: nat,K: nat] :
( ( binomial @ ( suc @ N ) @ ( suc @ K ) )
= ( divide_divide @ nat @ ( times_times @ nat @ ( suc @ N ) @ ( binomial @ N @ K ) ) @ ( suc @ K ) ) ) ).
% binomial_Suc_Suc_eq_times
thf(fact_2673_and__nat__def,axiom,
( ( bit_se5824344872417868541ns_and @ nat )
= ( ^ [M2: nat,N3: nat] : ( nat2 @ ( bit_se5824344872417868541ns_and @ int @ ( semiring_1_of_nat @ int @ M2 ) @ ( semiring_1_of_nat @ int @ N3 ) ) ) ) ) ).
% and_nat_def
thf(fact_2674_xor__nat__def,axiom,
( ( bit_se5824344971392196577ns_xor @ nat )
= ( ^ [M2: nat,N3: nat] : ( nat2 @ ( bit_se5824344971392196577ns_xor @ int @ ( semiring_1_of_nat @ int @ M2 ) @ ( semiring_1_of_nat @ int @ N3 ) ) ) ) ) ).
% xor_nat_def
thf(fact_2675_pred__numeral__def,axiom,
( pred_numeral
= ( ^ [K3: num] : ( minus_minus @ nat @ ( numeral_numeral @ nat @ K3 ) @ ( one_one @ nat ) ) ) ) ).
% pred_numeral_def
thf(fact_2676_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_2677_binomial__fact__lemma,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ N )
=> ( ( times_times @ nat @ ( times_times @ nat @ ( semiring_char_0_fact @ nat @ K ) @ ( semiring_char_0_fact @ nat @ ( minus_minus @ nat @ N @ K ) ) ) @ ( binomial @ N @ K ) )
= ( semiring_char_0_fact @ nat @ N ) ) ) ).
% binomial_fact_lemma
thf(fact_2678_fact__numeral,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [K: num] :
( ( semiring_char_0_fact @ A @ ( numeral_numeral @ nat @ K ) )
= ( times_times @ A @ ( numeral_numeral @ A @ K ) @ ( semiring_char_0_fact @ A @ ( pred_numeral @ K ) ) ) ) ) ).
% fact_numeral
thf(fact_2679_binomial__ge__n__over__k__pow__k,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [K: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ N )
=> ( ord_less_eq @ A @ ( power_power @ A @ ( divide_divide @ A @ ( semiring_1_of_nat @ A @ N ) @ ( semiring_1_of_nat @ A @ K ) ) @ K ) @ ( semiring_1_of_nat @ A @ ( binomial @ N @ K ) ) ) ) ) ).
% binomial_ge_n_over_k_pow_k
thf(fact_2680_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_2681_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_2682_binomial__le__pow2,axiom,
! [N: nat,K: nat] : ( ord_less_eq @ nat @ ( binomial @ N @ K ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ).
% binomial_le_pow2
thf(fact_2683_binomial__altdef__nat,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ N )
=> ( ( binomial @ N @ K )
= ( divide_divide @ nat @ ( semiring_char_0_fact @ nat @ N ) @ ( times_times @ nat @ ( semiring_char_0_fact @ nat @ K ) @ ( semiring_char_0_fact @ nat @ ( minus_minus @ nat @ N @ K ) ) ) ) ) ) ).
% binomial_altdef_nat
thf(fact_2684_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_2685_fact__binomial,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ N )
=> ( ( times_times @ A @ ( semiring_char_0_fact @ A @ K ) @ ( semiring_1_of_nat @ A @ ( binomial @ N @ K ) ) )
= ( divide_divide @ A @ ( semiring_char_0_fact @ A @ N ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ N @ K ) ) ) ) ) ) ).
% fact_binomial
thf(fact_2686_binomial__fact,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ N )
=> ( ( semiring_1_of_nat @ A @ ( binomial @ N @ K ) )
= ( divide_divide @ A @ ( semiring_char_0_fact @ A @ N ) @ ( times_times @ A @ ( semiring_char_0_fact @ A @ K ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ N @ K ) ) ) ) ) ) ) ).
% binomial_fact
thf(fact_2687_take__bit__numeral__bit0,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [L: num,K: num] :
( ( bit_se2584673776208193580ke_bit @ A @ ( numeral_numeral @ nat @ L ) @ ( numeral_numeral @ A @ ( bit0 @ K ) ) )
= ( times_times @ A @ ( bit_se2584673776208193580ke_bit @ A @ ( pred_numeral @ L ) @ ( numeral_numeral @ A @ K ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% take_bit_numeral_bit0
thf(fact_2688_take__bit__numeral__minus__bit0,axiom,
! [L: num,K: num] :
( ( bit_se2584673776208193580ke_bit @ int @ ( numeral_numeral @ nat @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ K ) ) ) )
= ( times_times @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( pred_numeral @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ).
% take_bit_numeral_minus_bit0
thf(fact_2689_XOR__upper,axiom,
! [X: int,N: nat,Y3: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
=> ( ( ord_less @ int @ X @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( ord_less @ int @ Y3 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) )
=> ( ord_less @ int @ ( bit_se5824344971392196577ns_xor @ int @ X @ Y3 ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ).
% XOR_upper
thf(fact_2690_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_2691_and__nat__unfold,axiom,
( ( bit_se5824344872417868541ns_and @ nat )
= ( ^ [M2: nat,N3: nat] :
( if @ nat
@ ( ( M2
= ( zero_zero @ nat ) )
| ( N3
= ( zero_zero @ nat ) ) )
@ ( zero_zero @ nat )
@ ( plus_plus @ nat @ ( times_times @ nat @ ( modulo_modulo @ nat @ M2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ nat @ ( divide_divide @ nat @ M2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ).
% and_nat_unfold
thf(fact_2692_and__nat__rec,axiom,
( ( bit_se5824344872417868541ns_and @ nat )
= ( ^ [M2: nat,N3: nat] :
( plus_plus @ nat
@ ( zero_neq_one_of_bool @ nat
@ ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M2 )
& ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) )
@ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ nat @ ( divide_divide @ nat @ M2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% and_nat_rec
thf(fact_2693_xor__int__rec,axiom,
( ( bit_se5824344971392196577ns_xor @ int )
= ( ^ [K3: int,L3: int] :
( plus_plus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K3 ) )
!= ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L3 ) ) ) )
@ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ int @ ( divide_divide @ int @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% xor_int_rec
thf(fact_2694_xor__int__unfold,axiom,
( ( bit_se5824344971392196577ns_xor @ int )
= ( ^ [K3: int,L3: int] :
( if @ int
@ ( K3
= ( uminus_uminus @ int @ ( one_one @ int ) ) )
@ ( bit_ri4277139882892585799ns_not @ int @ L3 )
@ ( if @ int
@ ( L3
= ( uminus_uminus @ int @ ( one_one @ int ) ) )
@ ( bit_ri4277139882892585799ns_not @ int @ K3 )
@ ( if @ int
@ ( K3
= ( zero_zero @ int ) )
@ L3
@ ( if @ int
@ ( L3
= ( zero_zero @ int ) )
@ K3
@ ( plus_plus @ int @ ( abs_abs @ int @ ( minus_minus @ int @ ( modulo_modulo @ int @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ int @ L3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ int @ ( divide_divide @ int @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ).
% xor_int_unfold
thf(fact_2695_signed__take__bit__eq__take__bit__minus,axiom,
( ( bit_ri4674362597316999326ke_bit @ int )
= ( ^ [N3: nat,K3: int] : ( minus_minus @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( suc @ N3 ) @ K3 ) @ ( times_times @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( suc @ N3 ) ) @ ( zero_neq_one_of_bool @ int @ ( bit_se5641148757651400278ts_bit @ int @ K3 @ N3 ) ) ) ) ) ) ).
% signed_take_bit_eq_take_bit_minus
thf(fact_2696_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_2697_take__bit__eq__mask__iff__exp__dvd,axiom,
! [N: nat,K: int] :
( ( ( bit_se2584673776208193580ke_bit @ int @ N @ K )
= ( bit_se2239418461657761734s_mask @ int @ N ) )
= ( dvd_dvd @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) @ ( plus_plus @ int @ K @ ( one_one @ int ) ) ) ) ).
% take_bit_eq_mask_iff_exp_dvd
thf(fact_2698_take__bit__Suc__from__most,axiom,
! [N: nat,K: int] :
( ( bit_se2584673776208193580ke_bit @ int @ ( suc @ N ) @ K )
= ( plus_plus @ int @ ( times_times @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) @ ( zero_neq_one_of_bool @ int @ ( bit_se5641148757651400278ts_bit @ int @ K @ N ) ) ) @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) ) ) ).
% take_bit_Suc_from_most
thf(fact_2699_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_2700_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_2701_push__bit__nonnegative__int__iff,axiom,
! [N: nat,K: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se4730199178511100633sh_bit @ int @ N @ K ) )
= ( ord_less_eq @ int @ ( zero_zero @ int ) @ K ) ) ).
% push_bit_nonnegative_int_iff
thf(fact_2702_push__bit__negative__int__iff,axiom,
! [N: nat,K: int] :
( ( ord_less @ int @ ( bit_se4730199178511100633sh_bit @ int @ N @ K ) @ ( zero_zero @ int ) )
= ( ord_less @ int @ K @ ( zero_zero @ int ) ) ) ).
% push_bit_negative_int_iff
thf(fact_2703_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_2704_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_2705_concat__bit__of__zero__1,axiom,
! [N: nat,L: int] :
( ( bit_concat_bit @ N @ ( zero_zero @ int ) @ L )
= ( bit_se4730199178511100633sh_bit @ int @ N @ L ) ) ).
% concat_bit_of_zero_1
thf(fact_2706_mask__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se2239418461657761734s_mask @ A @ ( zero_zero @ nat ) )
= ( zero_zero @ A ) ) ) ).
% mask_0
thf(fact_2707_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_2708_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_2709_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_2710_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_2711_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_2712_bit__numeral__Bit0__Suc__iff,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [M: num,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ ( bit0 @ M ) ) @ ( suc @ N ) )
= ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ M ) @ N ) ) ) ).
% bit_numeral_Bit0_Suc_iff
thf(fact_2713_push__bit__Suc__numeral,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,K: num] :
( ( bit_se4730199178511100633sh_bit @ A @ ( suc @ N ) @ ( numeral_numeral @ A @ K ) )
= ( bit_se4730199178511100633sh_bit @ A @ N @ ( numeral_numeral @ A @ ( bit0 @ K ) ) ) ) ) ).
% push_bit_Suc_numeral
thf(fact_2714_bit__numeral__Bit1__Suc__iff,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [M: num,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ ( bit1 @ M ) ) @ ( suc @ N ) )
= ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ M ) @ N ) ) ) ).
% bit_numeral_Bit1_Suc_iff
thf(fact_2715_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_2716_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_2717_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_2718_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_2719_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_2720_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_2721_not__negative__int__iff,axiom,
! [K: int] :
( ( ord_less @ int @ ( bit_ri4277139882892585799ns_not @ int @ K ) @ ( zero_zero @ int ) )
= ( ord_less_eq @ int @ ( zero_zero @ int ) @ K ) ) ).
% not_negative_int_iff
thf(fact_2722_not__nonnegative__int__iff,axiom,
! [K: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_ri4277139882892585799ns_not @ int @ K ) )
= ( ord_less @ int @ K @ ( zero_zero @ int ) ) ) ).
% not_nonnegative_int_iff
thf(fact_2723_minus__not__numeral__eq,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: num] :
( ( uminus_uminus @ A @ ( bit_ri4277139882892585799ns_not @ A @ ( numeral_numeral @ A @ N ) ) )
= ( numeral_numeral @ A @ ( inc @ N ) ) ) ) ).
% minus_not_numeral_eq
thf(fact_2724_signed__take__bit__nonnegative__iff,axiom,
! [N: nat,K: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K ) )
= ( ~ ( bit_se5641148757651400278ts_bit @ int @ K @ N ) ) ) ).
% signed_take_bit_nonnegative_iff
thf(fact_2725_signed__take__bit__negative__iff,axiom,
! [N: nat,K: int] :
( ( ord_less @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K ) @ ( zero_zero @ int ) )
= ( bit_se5641148757651400278ts_bit @ int @ K @ N ) ) ).
% signed_take_bit_negative_iff
thf(fact_2726_push__bit__Suc__minus__numeral,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: nat,K: num] :
( ( bit_se4730199178511100633sh_bit @ A @ ( suc @ N ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ K ) ) )
= ( bit_se4730199178511100633sh_bit @ A @ N @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( bit0 @ K ) ) ) ) ) ) ).
% push_bit_Suc_minus_numeral
thf(fact_2727_bit__minus__numeral__Bit0__Suc__iff,axiom,
! [W2: num,N: nat] :
( ( bit_se5641148757651400278ts_bit @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ W2 ) ) ) @ ( suc @ N ) )
= ( bit_se5641148757651400278ts_bit @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ W2 ) ) @ N ) ) ).
% bit_minus_numeral_Bit0_Suc_iff
thf(fact_2728_bit__minus__numeral__Bit1__Suc__iff,axiom,
! [W2: num,N: nat] :
( ( bit_se5641148757651400278ts_bit @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ W2 ) ) ) @ ( suc @ N ) )
= ( ~ ( bit_se5641148757651400278ts_bit @ int @ ( numeral_numeral @ int @ W2 ) @ N ) ) ) ).
% bit_minus_numeral_Bit1_Suc_iff
thf(fact_2729_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_2730_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_2731_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_2732_push__bit__Suc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A3: A] :
( ( bit_se4730199178511100633sh_bit @ A @ ( suc @ N ) @ A3 )
= ( bit_se4730199178511100633sh_bit @ A @ N @ ( times_times @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% push_bit_Suc
thf(fact_2733_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_2734_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_2735_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_2736_push__bit__minus__numeral,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [L: num,K: num] :
( ( bit_se4730199178511100633sh_bit @ A @ ( numeral_numeral @ nat @ L ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ K ) ) )
= ( bit_se4730199178511100633sh_bit @ A @ ( pred_numeral @ L ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( bit0 @ K ) ) ) ) ) ) ).
% push_bit_minus_numeral
thf(fact_2737_bit__minus__numeral__int_I1_J,axiom,
! [W2: num,N: num] :
( ( bit_se5641148757651400278ts_bit @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ W2 ) ) ) @ ( numeral_numeral @ nat @ N ) )
= ( bit_se5641148757651400278ts_bit @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ W2 ) ) @ ( pred_numeral @ N ) ) ) ).
% bit_minus_numeral_int(1)
thf(fact_2738_bit__minus__numeral__int_I2_J,axiom,
! [W2: num,N: num] :
( ( bit_se5641148757651400278ts_bit @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ W2 ) ) ) @ ( numeral_numeral @ nat @ N ) )
= ( ~ ( bit_se5641148757651400278ts_bit @ int @ ( numeral_numeral @ int @ W2 ) @ ( pred_numeral @ N ) ) ) ) ).
% bit_minus_numeral_int(2)
thf(fact_2739_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_2740_of__nat__mask__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat] :
( ( semiring_1_of_nat @ A @ ( bit_se2239418461657761734s_mask @ nat @ N ) )
= ( bit_se2239418461657761734s_mask @ A @ N ) ) ) ).
% of_nat_mask_eq
thf(fact_2741_bit__of__nat__iff__bit,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [M: nat,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( semiring_1_of_nat @ A @ M ) @ N )
= ( bit_se5641148757651400278ts_bit @ nat @ M @ N ) ) ) ).
% bit_of_nat_iff_bit
thf(fact_2742_push__bit__of__nat,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,M: nat] :
( ( bit_se4730199178511100633sh_bit @ A @ N @ ( semiring_1_of_nat @ A @ M ) )
= ( semiring_1_of_nat @ A @ ( bit_se4730199178511100633sh_bit @ nat @ N @ M ) ) ) ) ).
% push_bit_of_nat
thf(fact_2743_of__nat__push__bit,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,N: nat] :
( ( semiring_1_of_nat @ A @ ( bit_se4730199178511100633sh_bit @ nat @ M @ N ) )
= ( bit_se4730199178511100633sh_bit @ A @ M @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% of_nat_push_bit
thf(fact_2744_bit__push__bit__iff__int,axiom,
! [M: nat,K: int,N: nat] :
( ( bit_se5641148757651400278ts_bit @ int @ ( bit_se4730199178511100633sh_bit @ int @ M @ K ) @ N )
= ( ( ord_less_eq @ nat @ M @ N )
& ( bit_se5641148757651400278ts_bit @ int @ K @ ( minus_minus @ nat @ N @ M ) ) ) ) ).
% bit_push_bit_iff_int
thf(fact_2745_push__bit__minus,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: nat,A3: A] :
( ( bit_se4730199178511100633sh_bit @ A @ N @ ( uminus_uminus @ A @ A3 ) )
= ( uminus_uminus @ A @ ( bit_se4730199178511100633sh_bit @ A @ N @ A3 ) ) ) ) ).
% push_bit_minus
thf(fact_2746_less__eq__mask,axiom,
! [N: nat] : ( ord_less_eq @ nat @ N @ ( bit_se2239418461657761734s_mask @ nat @ N ) ) ).
% less_eq_mask
thf(fact_2747_bit__minus__int__iff,axiom,
! [K: int,N: nat] :
( ( bit_se5641148757651400278ts_bit @ int @ ( uminus_uminus @ int @ K ) @ N )
= ( bit_se5641148757651400278ts_bit @ int @ ( bit_ri4277139882892585799ns_not @ int @ ( minus_minus @ int @ K @ ( one_one @ int ) ) ) @ N ) ) ).
% bit_minus_int_iff
thf(fact_2748_bit__iff__and__push__bit__not__eq__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se5641148757651400278ts_bit @ A )
= ( ^ [A5: A,N3: nat] :
( ( bit_se5824344872417868541ns_and @ A @ A5 @ ( bit_se4730199178511100633sh_bit @ A @ N3 @ ( one_one @ A ) ) )
!= ( zero_zero @ A ) ) ) ) ) ).
% bit_iff_and_push_bit_not_eq_0
thf(fact_2749_unset__bit__eq__and__not,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( bit_se2638667681897837118et_bit @ A )
= ( ^ [N3: nat,A5: A] : ( bit_se5824344872417868541ns_and @ A @ A5 @ ( bit_ri4277139882892585799ns_not @ A @ ( bit_se4730199178511100633sh_bit @ A @ N3 @ ( one_one @ A ) ) ) ) ) ) ) ).
% unset_bit_eq_and_not
thf(fact_2750_unset__bit__int__def,axiom,
( ( bit_se2638667681897837118et_bit @ int )
= ( ^ [N3: nat,K3: int] : ( bit_se5824344872417868541ns_and @ int @ K3 @ ( bit_ri4277139882892585799ns_not @ int @ ( bit_se4730199178511100633sh_bit @ int @ N3 @ ( one_one @ int ) ) ) ) ) ) ).
% unset_bit_int_def
thf(fact_2751_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_2752_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_2753_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_2754_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_2755_bit__take__bit__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,A3: A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( bit_se2584673776208193580ke_bit @ A @ M @ A3 ) @ N )
= ( ( ord_less @ nat @ N @ M )
& ( bit_se5641148757651400278ts_bit @ A @ A3 @ N ) ) ) ) ).
% bit_take_bit_iff
thf(fact_2756_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_2757_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_2758_mask__nonnegative__int,axiom,
! [N: nat] : ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se2239418461657761734s_mask @ int @ N ) ) ).
% mask_nonnegative_int
thf(fact_2759_not__mask__negative__int,axiom,
! [N: nat] :
~ ( ord_less @ int @ ( bit_se2239418461657761734s_mask @ int @ N ) @ ( zero_zero @ int ) ) ).
% not_mask_negative_int
thf(fact_2760_flip__bit__nat__def,axiom,
( ( bit_se8732182000553998342ip_bit @ nat )
= ( ^ [M2: nat,N3: nat] : ( bit_se5824344971392196577ns_xor @ nat @ N3 @ ( bit_se4730199178511100633sh_bit @ nat @ M2 @ ( one_one @ nat ) ) ) ) ) ).
% flip_bit_nat_def
thf(fact_2761_flip__bit__int__def,axiom,
( ( bit_se8732182000553998342ip_bit @ int )
= ( ^ [N3: nat,K3: int] : ( bit_se5824344971392196577ns_xor @ int @ K3 @ ( bit_se4730199178511100633sh_bit @ int @ N3 @ ( one_one @ int ) ) ) ) ) ).
% flip_bit_int_def
thf(fact_2762_minus__exp__eq__not__mask,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: nat] :
( ( uminus_uminus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
= ( bit_ri4277139882892585799ns_not @ A @ ( bit_se2239418461657761734s_mask @ A @ N ) ) ) ) ).
% minus_exp_eq_not_mask
thf(fact_2763_minus__eq__not__plus__1,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( uminus_uminus @ A )
= ( ^ [A5: A] : ( plus_plus @ A @ ( bit_ri4277139882892585799ns_not @ A @ A5 ) @ ( one_one @ A ) ) ) ) ) ).
% minus_eq_not_plus_1
thf(fact_2764_minus__eq__not__minus__1,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( uminus_uminus @ A )
= ( ^ [A5: A] : ( bit_ri4277139882892585799ns_not @ A @ ( minus_minus @ A @ A5 @ ( one_one @ A ) ) ) ) ) ) ).
% minus_eq_not_minus_1
thf(fact_2765_not__eq__complement,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( bit_ri4277139882892585799ns_not @ A )
= ( ^ [A5: A] : ( minus_minus @ A @ ( uminus_uminus @ A @ A5 ) @ ( one_one @ A ) ) ) ) ) ).
% not_eq_complement
thf(fact_2766_not__int__def,axiom,
( ( bit_ri4277139882892585799ns_not @ int )
= ( ^ [K3: int] : ( minus_minus @ int @ ( uminus_uminus @ int @ K3 ) @ ( one_one @ int ) ) ) ) ).
% not_int_def
thf(fact_2767_bit__not__int__iff_H,axiom,
! [K: int,N: nat] :
( ( bit_se5641148757651400278ts_bit @ int @ ( minus_minus @ int @ ( uminus_uminus @ int @ K ) @ ( one_one @ int ) ) @ N )
= ( ~ ( bit_se5641148757651400278ts_bit @ int @ K @ N ) ) ) ).
% bit_not_int_iff'
thf(fact_2768_and__not__numerals_I1_J,axiom,
( ( bit_se5824344872417868541ns_and @ int @ ( one_one @ int ) @ ( bit_ri4277139882892585799ns_not @ int @ ( one_one @ int ) ) )
= ( zero_zero @ int ) ) ).
% and_not_numerals(1)
thf(fact_2769_minus__numeral__inc__eq,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: num] :
( ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( inc @ N ) ) )
= ( bit_ri4277139882892585799ns_not @ A @ ( numeral_numeral @ A @ N ) ) ) ) ).
% minus_numeral_inc_eq
thf(fact_2770_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_2771_flip__bit__eq__xor,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se8732182000553998342ip_bit @ A )
= ( ^ [N3: nat,A5: A] : ( bit_se5824344971392196577ns_xor @ A @ A5 @ ( bit_se4730199178511100633sh_bit @ A @ N3 @ ( one_one @ A ) ) ) ) ) ) ).
% flip_bit_eq_xor
thf(fact_2772_and__not__numerals_I4_J,axiom,
! [M: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ ( bit0 @ M ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( one_one @ int ) ) )
= ( numeral_numeral @ int @ ( bit0 @ M ) ) ) ).
% and_not_numerals(4)
thf(fact_2773_and__not__numerals_I2_J,axiom,
! [N: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( one_one @ int ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) )
= ( one_one @ int ) ) ).
% and_not_numerals(2)
thf(fact_2774_bit__imp__take__bit__positive,axiom,
! [N: nat,M: nat,K: int] :
( ( ord_less @ nat @ N @ M )
=> ( ( bit_se5641148757651400278ts_bit @ int @ K @ N )
=> ( ord_less @ int @ ( zero_zero @ int ) @ ( bit_se2584673776208193580ke_bit @ int @ M @ K ) ) ) ) ).
% bit_imp_take_bit_positive
thf(fact_2775_not__numeral__Bit0__eq,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: num] :
( ( bit_ri4277139882892585799ns_not @ A @ ( numeral_numeral @ A @ ( bit0 @ N ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( bit1 @ N ) ) ) ) ) ).
% not_numeral_Bit0_eq
thf(fact_2776_bit__concat__bit__iff,axiom,
! [M: nat,K: int,L: int,N: nat] :
( ( bit_se5641148757651400278ts_bit @ int @ ( bit_concat_bit @ M @ K @ L ) @ N )
= ( ( ( ord_less @ nat @ N @ M )
& ( bit_se5641148757651400278ts_bit @ int @ K @ N ) )
| ( ( ord_less_eq @ nat @ M @ N )
& ( bit_se5641148757651400278ts_bit @ int @ L @ ( minus_minus @ nat @ N @ M ) ) ) ) ) ).
% bit_concat_bit_iff
thf(fact_2777_push__bit__double,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A3: A] :
( ( bit_se4730199178511100633sh_bit @ A @ N @ ( times_times @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
= ( times_times @ A @ ( bit_se4730199178511100633sh_bit @ A @ N @ A3 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% push_bit_double
thf(fact_2778_signed__take__bit__eq__concat__bit,axiom,
( ( bit_ri4674362597316999326ke_bit @ int )
= ( ^ [N3: nat,K3: int] : ( bit_concat_bit @ N3 @ K3 @ ( uminus_uminus @ int @ ( zero_neq_one_of_bool @ int @ ( bit_se5641148757651400278ts_bit @ int @ K3 @ N3 ) ) ) ) ) ) ).
% signed_take_bit_eq_concat_bit
thf(fact_2779_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_2780_bit__Suc,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ A3 @ ( suc @ N ) )
= ( bit_se5641148757651400278ts_bit @ A @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ N ) ) ) ).
% bit_Suc
thf(fact_2781_take__bit__eq__mask__iff,axiom,
! [N: nat,K: int] :
( ( ( bit_se2584673776208193580ke_bit @ int @ N @ K )
= ( bit_se2239418461657761734s_mask @ int @ N ) )
= ( ( bit_se2584673776208193580ke_bit @ int @ N @ ( plus_plus @ int @ K @ ( one_one @ int ) ) )
= ( zero_zero @ int ) ) ) ).
% take_bit_eq_mask_iff
thf(fact_2782_int__bit__bound,axiom,
! [K: int] :
~ ! [N4: nat] :
( ! [M3: nat] :
( ( ord_less_eq @ nat @ N4 @ M3 )
=> ( ( bit_se5641148757651400278ts_bit @ int @ K @ M3 )
= ( bit_se5641148757651400278ts_bit @ int @ K @ N4 ) ) )
=> ~ ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N4 )
=> ( ( bit_se5641148757651400278ts_bit @ int @ K @ ( minus_minus @ nat @ N4 @ ( one_one @ nat ) ) )
= ( ~ ( bit_se5641148757651400278ts_bit @ int @ K @ N4 ) ) ) ) ) ).
% int_bit_bound
thf(fact_2783_and__not__numerals_I5_J,axiom,
! [M: num,N: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ ( bit0 @ M ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) )
= ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ M ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N ) ) ) ) ) ).
% and_not_numerals(5)
thf(fact_2784_and__not__numerals_I7_J,axiom,
! [M: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ ( bit1 @ M ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( one_one @ int ) ) )
= ( numeral_numeral @ int @ ( bit0 @ M ) ) ) ).
% and_not_numerals(7)
thf(fact_2785_and__not__numerals_I3_J,axiom,
! [N: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( one_one @ int ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) )
= ( zero_zero @ int ) ) ).
% and_not_numerals(3)
thf(fact_2786_push__bit__int__def,axiom,
( ( bit_se4730199178511100633sh_bit @ int )
= ( ^ [N3: nat,K3: int] : ( times_times @ int @ K3 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ).
% push_bit_int_def
thf(fact_2787_push__bit__nat__def,axiom,
( ( bit_se4730199178511100633sh_bit @ nat )
= ( ^ [N3: nat,M2: nat] : ( times_times @ nat @ M2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ).
% push_bit_nat_def
thf(fact_2788_push__bit__eq__mult,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se4730199178511100633sh_bit @ A )
= ( ^ [N3: nat,A5: A] : ( times_times @ A @ A5 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ) ).
% push_bit_eq_mult
thf(fact_2789_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_2790_Suc__mask__eq__exp,axiom,
! [N: nat] :
( ( suc @ ( bit_se2239418461657761734s_mask @ nat @ N ) )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ).
% Suc_mask_eq_exp
thf(fact_2791_mask__nat__less__exp,axiom,
! [N: nat] : ( ord_less @ nat @ ( bit_se2239418461657761734s_mask @ nat @ N ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ).
% mask_nat_less_exp
thf(fact_2792_and__not__numerals_I9_J,axiom,
! [M: num,N: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ ( bit1 @ M ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) )
= ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ M ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N ) ) ) ) ) ).
% and_not_numerals(9)
thf(fact_2793_and__not__numerals_I6_J,axiom,
! [M: num,N: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ ( bit0 @ M ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) )
= ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ M ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N ) ) ) ) ) ).
% and_not_numerals(6)
thf(fact_2794_push__bit__minus__one,axiom,
! [N: nat] :
( ( bit_se4730199178511100633sh_bit @ int @ N @ ( uminus_uminus @ int @ ( one_one @ int ) ) )
= ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ).
% push_bit_minus_one
thf(fact_2795_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_2796_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_2797_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_2798_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_2799_mask__int__def,axiom,
( ( bit_se2239418461657761734s_mask @ int )
= ( ^ [N3: nat] : ( minus_minus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N3 ) @ ( one_one @ int ) ) ) ) ).
% mask_int_def
thf(fact_2800_mask__nat__def,axiom,
( ( bit_se2239418461657761734s_mask @ nat )
= ( ^ [N3: nat] : ( minus_minus @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) @ ( one_one @ nat ) ) ) ) ).
% mask_nat_def
thf(fact_2801_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_2802_bit__rec,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ( ( bit_se5641148757651400278ts_bit @ A )
= ( ^ [A5: A,N3: nat] :
( ( ( N3
= ( zero_zero @ nat ) )
=> ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A5 ) )
& ( ( N3
!= ( zero_zero @ nat ) )
=> ( bit_se5641148757651400278ts_bit @ A @ ( divide_divide @ A @ A5 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( minus_minus @ nat @ N3 @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% bit_rec
thf(fact_2803_mask__eq__exp__minus__1,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se2239418461657761734s_mask @ A )
= ( ^ [N3: nat] : ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N3 ) @ ( one_one @ A ) ) ) ) ) ).
% mask_eq_exp_minus_1
thf(fact_2804_and__not__numerals_I8_J,axiom,
! [M: num,N: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ ( bit1 @ M ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) )
= ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ M ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N ) ) ) ) ) ) ).
% and_not_numerals(8)
thf(fact_2805_not__int__rec,axiom,
( ( bit_ri4277139882892585799ns_not @ int )
= ( ^ [K3: int] : ( plus_plus @ int @ ( zero_neq_one_of_bool @ int @ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K3 ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( divide_divide @ int @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% not_int_rec
thf(fact_2806_set__bit__eq,axiom,
( ( bit_se5668285175392031749et_bit @ int )
= ( ^ [N3: nat,K3: int] :
( plus_plus @ int @ K3
@ ( times_times @ int
@ ( zero_neq_one_of_bool @ int
@ ~ ( bit_se5641148757651400278ts_bit @ int @ K3 @ N3 ) )
@ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ) ).
% set_bit_eq
thf(fact_2807_unset__bit__eq,axiom,
( ( bit_se2638667681897837118et_bit @ int )
= ( ^ [N3: nat,K3: int] : ( minus_minus @ int @ K3 @ ( times_times @ int @ ( zero_neq_one_of_bool @ int @ ( bit_se5641148757651400278ts_bit @ int @ K3 @ N3 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ) ).
% unset_bit_eq
thf(fact_2808_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_2809_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_2810_or__int__rec,axiom,
( ( bit_se1065995026697491101ons_or @ int )
= ( ^ [K3: int,L3: int] :
( plus_plus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K3 )
| ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L3 ) ) )
@ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ int @ ( divide_divide @ int @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% or_int_rec
thf(fact_2811_drop__bit__numeral__minus__bit1,axiom,
! [L: num,K: num] :
( ( bit_se4197421643247451524op_bit @ int @ ( numeral_numeral @ nat @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ K ) ) ) )
= ( bit_se4197421643247451524op_bit @ int @ ( pred_numeral @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( inc @ K ) ) ) ) ) ).
% drop_bit_numeral_minus_bit1
thf(fact_2812_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_2813_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_2814_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_2815_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_2816_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_2817_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_2818_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_2819_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_2820_or__nonnegative__int__iff,axiom,
! [K: int,L: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se1065995026697491101ons_or @ int @ K @ L ) )
= ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
& ( ord_less_eq @ int @ ( zero_zero @ int ) @ L ) ) ) ).
% or_nonnegative_int_iff
thf(fact_2821_or__negative__int__iff,axiom,
! [K: int,L: int] :
( ( ord_less @ int @ ( bit_se1065995026697491101ons_or @ int @ K @ L ) @ ( zero_zero @ int ) )
= ( ( ord_less @ int @ K @ ( zero_zero @ int ) )
| ( ord_less @ int @ L @ ( zero_zero @ int ) ) ) ) ).
% or_negative_int_iff
thf(fact_2822_drop__bit__nonnegative__int__iff,axiom,
! [N: nat,K: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se4197421643247451524op_bit @ int @ N @ K ) )
= ( ord_less_eq @ int @ ( zero_zero @ int ) @ K ) ) ).
% drop_bit_nonnegative_int_iff
thf(fact_2823_drop__bit__negative__int__iff,axiom,
! [N: nat,K: int] :
( ( ord_less @ int @ ( bit_se4197421643247451524op_bit @ int @ N @ K ) @ ( zero_zero @ int ) )
= ( ord_less @ int @ K @ ( zero_zero @ int ) ) ) ).
% drop_bit_negative_int_iff
thf(fact_2824_drop__bit__minus__one,axiom,
! [N: nat] :
( ( bit_se4197421643247451524op_bit @ int @ N @ ( uminus_uminus @ int @ ( one_one @ int ) ) )
= ( uminus_uminus @ int @ ( one_one @ int ) ) ) ).
% drop_bit_minus_one
thf(fact_2825_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_2826_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_2827_drop__bit__Suc__bit0,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N: nat,K: num] :
( ( bit_se4197421643247451524op_bit @ A @ ( suc @ N ) @ ( numeral_numeral @ A @ ( bit0 @ K ) ) )
= ( bit_se4197421643247451524op_bit @ A @ N @ ( numeral_numeral @ A @ K ) ) ) ) ).
% drop_bit_Suc_bit0
thf(fact_2828_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_2829_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_2830_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_2831_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_2832_drop__bit__Suc__bit1,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N: nat,K: num] :
( ( bit_se4197421643247451524op_bit @ A @ ( suc @ N ) @ ( numeral_numeral @ A @ ( bit1 @ K ) ) )
= ( bit_se4197421643247451524op_bit @ A @ N @ ( numeral_numeral @ A @ K ) ) ) ) ).
% drop_bit_Suc_bit1
thf(fact_2833_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_2834_or__numerals_I3_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num,Y3: num] :
( ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ ( bit0 @ X ) ) @ ( numeral_numeral @ A @ ( bit0 @ Y3 ) ) )
= ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ X ) @ ( numeral_numeral @ A @ Y3 ) ) ) ) ) ).
% or_numerals(3)
thf(fact_2835_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_2836_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_2837_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_2838_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_2839_or__minus__numerals_I6_J,axiom,
! [N: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) @ ( one_one @ int ) )
= ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) ) ).
% or_minus_numerals(6)
thf(fact_2840_or__minus__numerals_I2_J,axiom,
! [N: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( one_one @ int ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) )
= ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) ) ).
% or_minus_numerals(2)
thf(fact_2841_drop__bit__Suc__minus__bit0,axiom,
! [N: nat,K: num] :
( ( bit_se4197421643247451524op_bit @ int @ ( suc @ N ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ K ) ) ) )
= ( bit_se4197421643247451524op_bit @ int @ N @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) ) ) ).
% drop_bit_Suc_minus_bit0
thf(fact_2842_drop__bit__numeral__minus__bit0,axiom,
! [L: num,K: num] :
( ( bit_se4197421643247451524op_bit @ int @ ( numeral_numeral @ nat @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ K ) ) ) )
= ( bit_se4197421643247451524op_bit @ int @ ( pred_numeral @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) ) ) ).
% drop_bit_numeral_minus_bit0
thf(fact_2843_drop__bit__Suc__minus__bit1,axiom,
! [N: nat,K: num] :
( ( bit_se4197421643247451524op_bit @ int @ ( suc @ N ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ K ) ) ) )
= ( bit_se4197421643247451524op_bit @ int @ N @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( inc @ K ) ) ) ) ) ).
% drop_bit_Suc_minus_bit1
thf(fact_2844_and__minus__minus__numerals,axiom,
! [M: num,N: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
= ( bit_ri4277139882892585799ns_not @ int @ ( bit_se1065995026697491101ons_or @ int @ ( minus_minus @ int @ ( numeral_numeral @ int @ M ) @ ( one_one @ int ) ) @ ( minus_minus @ int @ ( numeral_numeral @ int @ N ) @ ( one_one @ int ) ) ) ) ) ).
% and_minus_minus_numerals
thf(fact_2845_or__minus__minus__numerals,axiom,
! [M: num,N: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
= ( bit_ri4277139882892585799ns_not @ int @ ( bit_se5824344872417868541ns_and @ int @ ( minus_minus @ int @ ( numeral_numeral @ int @ M ) @ ( one_one @ int ) ) @ ( minus_minus @ int @ ( numeral_numeral @ int @ N ) @ ( one_one @ int ) ) ) ) ) ).
% or_minus_minus_numerals
thf(fact_2846_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_2847_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_2848_or__nat__def,axiom,
( ( bit_se1065995026697491101ons_or @ nat )
= ( ^ [M2: nat,N3: nat] : ( nat2 @ ( bit_se1065995026697491101ons_or @ int @ ( semiring_1_of_nat @ int @ M2 ) @ ( semiring_1_of_nat @ int @ N3 ) ) ) ) ) ).
% or_nat_def
thf(fact_2849_of__nat__or__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,N: nat] :
( ( semiring_1_of_nat @ A @ ( bit_se1065995026697491101ons_or @ nat @ M @ N ) )
= ( bit_se1065995026697491101ons_or @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% of_nat_or_eq
thf(fact_2850_drop__bit__of__nat,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N: nat,M: nat] :
( ( bit_se4197421643247451524op_bit @ A @ N @ ( semiring_1_of_nat @ A @ M ) )
= ( semiring_1_of_nat @ A @ ( bit_se4197421643247451524op_bit @ nat @ N @ M ) ) ) ) ).
% drop_bit_of_nat
thf(fact_2851_of__nat__drop__bit,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [M: nat,N: nat] :
( ( semiring_1_of_nat @ A @ ( bit_se4197421643247451524op_bit @ nat @ M @ N ) )
= ( bit_se4197421643247451524op_bit @ A @ M @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% of_nat_drop_bit
thf(fact_2852_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_2853_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_2854_set__bit__nat__def,axiom,
( ( bit_se5668285175392031749et_bit @ nat )
= ( ^ [M2: nat,N3: nat] : ( bit_se1065995026697491101ons_or @ nat @ N3 @ ( bit_se4730199178511100633sh_bit @ nat @ M2 @ ( one_one @ nat ) ) ) ) ) ).
% set_bit_nat_def
thf(fact_2855_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_2856_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_2857_bit__push__bit__iff__nat,axiom,
! [M: nat,Q4: nat,N: nat] :
( ( bit_se5641148757651400278ts_bit @ nat @ ( bit_se4730199178511100633sh_bit @ nat @ M @ Q4 ) @ N )
= ( ( ord_less_eq @ nat @ M @ N )
& ( bit_se5641148757651400278ts_bit @ nat @ Q4 @ ( minus_minus @ nat @ N @ M ) ) ) ) ).
% bit_push_bit_iff_nat
thf(fact_2858_OR__lower,axiom,
! [X: int,Y3: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y3 )
=> ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se1065995026697491101ons_or @ int @ X @ Y3 ) ) ) ) ).
% OR_lower
thf(fact_2859_or__greater__eq,axiom,
! [L: int,K: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ L )
=> ( ord_less_eq @ int @ K @ ( bit_se1065995026697491101ons_or @ int @ K @ L ) ) ) ).
% or_greater_eq
thf(fact_2860_set__bit__int__def,axiom,
( ( bit_se5668285175392031749et_bit @ int )
= ( ^ [N3: nat,K3: int] : ( bit_se1065995026697491101ons_or @ int @ K3 @ ( bit_se4730199178511100633sh_bit @ int @ N3 @ ( one_one @ int ) ) ) ) ) ).
% set_bit_int_def
thf(fact_2861_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_2862_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_2863_or__not__numerals_I1_J,axiom,
( ( bit_se1065995026697491101ons_or @ int @ ( one_one @ int ) @ ( bit_ri4277139882892585799ns_not @ int @ ( one_one @ int ) ) )
= ( bit_ri4277139882892585799ns_not @ int @ ( zero_zero @ int ) ) ) ).
% or_not_numerals(1)
thf(fact_2864_set__bit__eq__or,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se5668285175392031749et_bit @ A )
= ( ^ [N3: nat,A5: A] : ( bit_se1065995026697491101ons_or @ A @ A5 @ ( bit_se4730199178511100633sh_bit @ A @ N3 @ ( one_one @ A ) ) ) ) ) ) ).
% set_bit_eq_or
thf(fact_2865_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_2866_bit__iff__and__drop__bit__eq__1,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se5641148757651400278ts_bit @ A )
= ( ^ [A5: A,N3: nat] :
( ( bit_se5824344872417868541ns_and @ A @ ( bit_se4197421643247451524op_bit @ A @ N3 @ A5 ) @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ) ) ).
% bit_iff_and_drop_bit_eq_1
thf(fact_2867_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_2868_or__not__numerals_I2_J,axiom,
! [N: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( one_one @ int ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) )
= ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) ) ).
% or_not_numerals(2)
thf(fact_2869_or__not__numerals_I4_J,axiom,
! [M: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ ( bit0 @ M ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( one_one @ int ) ) )
= ( bit_ri4277139882892585799ns_not @ int @ ( one_one @ int ) ) ) ).
% or_not_numerals(4)
thf(fact_2870_bit__nat__iff,axiom,
! [K: int,N: nat] :
( ( bit_se5641148757651400278ts_bit @ nat @ ( nat2 @ K ) @ N )
= ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
& ( bit_se5641148757651400278ts_bit @ int @ K @ N ) ) ) ).
% bit_nat_iff
thf(fact_2871_num_Osize_I4_J,axiom,
( ( size_size @ num @ one2 )
= ( zero_zero @ nat ) ) ).
% num.size(4)
thf(fact_2872_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_2873_or__not__numerals_I3_J,axiom,
! [N: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( one_one @ int ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) )
= ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) ) ).
% or_not_numerals(3)
thf(fact_2874_or__not__numerals_I7_J,axiom,
! [M: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ ( bit1 @ M ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( one_one @ int ) ) )
= ( bit_ri4277139882892585799ns_not @ int @ ( zero_zero @ int ) ) ) ).
% or_not_numerals(7)
thf(fact_2875_drop__bit__Suc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A3: A] :
( ( bit_se4197421643247451524op_bit @ A @ ( suc @ N ) @ A3 )
= ( bit_se4197421643247451524op_bit @ A @ N @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% drop_bit_Suc
thf(fact_2876_mask__Suc__exp,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat] :
( ( bit_se2239418461657761734s_mask @ A @ ( suc @ N ) )
= ( bit_se1065995026697491101ons_or @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) @ ( bit_se2239418461657761734s_mask @ A @ N ) ) ) ) ).
% mask_Suc_exp
thf(fact_2877_or__not__numerals_I6_J,axiom,
! [M: num,N: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ ( bit0 @ M ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) )
= ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ M ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N ) ) ) ) ) ).
% or_not_numerals(6)
thf(fact_2878_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_2879_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_2880_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_2881_OR__upper,axiom,
! [X: int,N: nat,Y3: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
=> ( ( ord_less @ int @ X @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( ord_less @ int @ Y3 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) )
=> ( ord_less @ int @ ( bit_se1065995026697491101ons_or @ int @ X @ Y3 ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ).
% OR_upper
thf(fact_2882_or__not__numerals_I5_J,axiom,
! [M: num,N: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ ( bit0 @ M ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) )
= ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ M ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N ) ) ) ) ) ) ).
% or_not_numerals(5)
thf(fact_2883_signed__take__bit__def,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( bit_ri4674362597316999326ke_bit @ A )
= ( ^ [N3: nat,A5: A] : ( bit_se1065995026697491101ons_or @ A @ ( bit_se2584673776208193580ke_bit @ A @ N3 @ A5 ) @ ( times_times @ A @ ( zero_neq_one_of_bool @ A @ ( bit_se5641148757651400278ts_bit @ A @ A5 @ N3 ) ) @ ( bit_ri4277139882892585799ns_not @ A @ ( bit_se2239418461657761734s_mask @ A @ N3 ) ) ) ) ) ) ) ).
% signed_take_bit_def
thf(fact_2884_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_2885_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_2886_drop__bit__rec,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se4197421643247451524op_bit @ A )
= ( ^ [N3: nat,A5: A] :
( if @ A
@ ( N3
= ( zero_zero @ nat ) )
@ A5
@ ( bit_se4197421643247451524op_bit @ A @ ( minus_minus @ nat @ N3 @ ( one_one @ nat ) ) @ ( divide_divide @ A @ A5 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% drop_bit_rec
thf(fact_2887_or__not__numerals_I8_J,axiom,
! [M: num,N: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ ( bit1 @ M ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) )
= ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ M ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N ) ) ) ) ) ) ).
% or_not_numerals(8)
thf(fact_2888_or__not__numerals_I9_J,axiom,
! [M: num,N: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ ( bit1 @ M ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) )
= ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ M ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N ) ) ) ) ) ) ).
% or_not_numerals(9)
thf(fact_2889_or__nat__rec,axiom,
( ( bit_se1065995026697491101ons_or @ nat )
= ( ^ [M2: nat,N3: nat] :
( plus_plus @ nat
@ ( zero_neq_one_of_bool @ nat
@ ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M2 )
| ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) )
@ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ nat @ ( divide_divide @ nat @ M2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% or_nat_rec
thf(fact_2890_or__int__unfold,axiom,
( ( bit_se1065995026697491101ons_or @ int )
= ( ^ [K3: int,L3: int] :
( if @ int
@ ( ( K3
= ( uminus_uminus @ int @ ( one_one @ int ) ) )
| ( L3
= ( uminus_uminus @ int @ ( one_one @ int ) ) ) )
@ ( uminus_uminus @ int @ ( one_one @ int ) )
@ ( if @ int
@ ( K3
= ( zero_zero @ int ) )
@ L3
@ ( if @ int
@ ( L3
= ( zero_zero @ int ) )
@ K3
@ ( plus_plus @ int @ ( ord_max @ int @ ( modulo_modulo @ int @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ int @ L3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ int @ ( divide_divide @ int @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ).
% or_int_unfold
thf(fact_2891_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_2892_or__minus__numerals_I4_J,axiom,
! [M: num,N: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ M ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) )
= ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit_or_not_num_neg @ M @ ( bit0 @ N ) ) ) ) ) ).
% or_minus_numerals(4)
thf(fact_2893_or__minus__numerals_I8_J,axiom,
! [N: num,M: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) @ ( numeral_numeral @ int @ M ) )
= ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit_or_not_num_neg @ M @ ( bit0 @ N ) ) ) ) ) ).
% or_minus_numerals(8)
thf(fact_2894_or__nat__unfold,axiom,
( ( bit_se1065995026697491101ons_or @ nat )
= ( ^ [M2: nat,N3: nat] :
( if @ nat
@ ( M2
= ( zero_zero @ nat ) )
@ N3
@ ( if @ nat
@ ( N3
= ( zero_zero @ nat ) )
@ M2
@ ( plus_plus @ nat @ ( ord_max @ nat @ ( modulo_modulo @ nat @ M2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ nat @ ( divide_divide @ nat @ M2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% or_nat_unfold
thf(fact_2895_xor__minus__numerals_I1_J,axiom,
! [N: num,K: int] :
( ( bit_se5824344971392196577ns_xor @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) @ K )
= ( bit_ri4277139882892585799ns_not @ int @ ( bit_se5824344971392196577ns_xor @ int @ ( neg_numeral_sub @ int @ N @ one2 ) @ K ) ) ) ).
% xor_minus_numerals(1)
thf(fact_2896_max_Oright__idem,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A] :
( ( ord_max @ A @ ( ord_max @ A @ A3 @ B2 ) @ B2 )
= ( ord_max @ A @ A3 @ B2 ) ) ) ).
% max.right_idem
thf(fact_2897_max_Oleft__idem,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A] :
( ( ord_max @ A @ A3 @ ( ord_max @ A @ A3 @ B2 ) )
= ( ord_max @ A @ A3 @ B2 ) ) ) ).
% max.left_idem
thf(fact_2898_max_Oidem,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A] :
( ( ord_max @ A @ A3 @ A3 )
= A3 ) ) ).
% max.idem
thf(fact_2899_max_Oabsorb1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,A3: A] :
( ( ord_less_eq @ A @ B2 @ A3 )
=> ( ( ord_max @ A @ A3 @ B2 )
= A3 ) ) ) ).
% max.absorb1
thf(fact_2900_max_Oabsorb2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_max @ A @ A3 @ B2 )
= B2 ) ) ) ).
% max.absorb2
thf(fact_2901_max_Obounded__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,C2: A,A3: A] :
( ( ord_less_eq @ A @ ( ord_max @ A @ B2 @ C2 ) @ A3 )
= ( ( ord_less_eq @ A @ B2 @ A3 )
& ( ord_less_eq @ A @ C2 @ A3 ) ) ) ) ).
% max.bounded_iff
thf(fact_2902_max_Oabsorb3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,A3: A] :
( ( ord_less @ A @ B2 @ A3 )
=> ( ( ord_max @ A @ A3 @ B2 )
= A3 ) ) ) ).
% max.absorb3
thf(fact_2903_max_Oabsorb4,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_max @ A @ A3 @ B2 )
= B2 ) ) ) ).
% max.absorb4
thf(fact_2904_max__less__iff__conj,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y3: A,Z3: A] :
( ( ord_less @ A @ ( ord_max @ A @ X @ Y3 ) @ Z3 )
= ( ( ord_less @ A @ X @ Z3 )
& ( ord_less @ A @ Y3 @ Z3 ) ) ) ) ).
% max_less_iff_conj
thf(fact_2905_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_2906_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_2907_max__0R,axiom,
! [N: nat] :
( ( ord_max @ nat @ N @ ( zero_zero @ nat ) )
= N ) ).
% max_0R
thf(fact_2908_max__0L,axiom,
! [N: nat] :
( ( ord_max @ nat @ ( zero_zero @ nat ) @ N )
= N ) ).
% max_0L
thf(fact_2909_max__nat_Oright__neutral,axiom,
! [A3: nat] :
( ( ord_max @ nat @ A3 @ ( zero_zero @ nat ) )
= A3 ) ).
% max_nat.right_neutral
thf(fact_2910_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_2911_max__nat_Oleft__neutral,axiom,
! [A3: nat] :
( ( ord_max @ nat @ ( zero_zero @ nat ) @ A3 )
= A3 ) ).
% max_nat.left_neutral
thf(fact_2912_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_2913_max__number__of_I1_J,axiom,
! [A: $tType] :
( ( ( numeral @ A )
& ( ord @ A ) )
=> ! [U: num,V: num] :
( ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ U ) @ ( numeral_numeral @ A @ V ) )
=> ( ( ord_max @ A @ ( numeral_numeral @ A @ U ) @ ( numeral_numeral @ A @ V ) )
= ( numeral_numeral @ A @ V ) ) )
& ( ~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ U ) @ ( numeral_numeral @ A @ V ) )
=> ( ( ord_max @ A @ ( numeral_numeral @ A @ U ) @ ( numeral_numeral @ A @ V ) )
= ( numeral_numeral @ A @ U ) ) ) ) ) ).
% max_number_of(1)
thf(fact_2914_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_2915_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_2916_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_2917_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_2918_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_2919_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_2920_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_2921_diff__numeral__simps_I1_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M: num,N: num] :
( ( minus_minus @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N ) )
= ( neg_numeral_sub @ A @ M @ N ) ) ) ).
% diff_numeral_simps(1)
thf(fact_2922_sub__num__simps_I6_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [K: num,L: num] :
( ( neg_numeral_sub @ A @ ( bit0 @ K ) @ ( bit0 @ L ) )
= ( neg_numeral_dbl @ A @ ( neg_numeral_sub @ A @ K @ L ) ) ) ) ).
% sub_num_simps(6)
thf(fact_2923_sub__num__simps_I9_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [K: num,L: num] :
( ( neg_numeral_sub @ A @ ( bit1 @ K ) @ ( bit1 @ L ) )
= ( neg_numeral_dbl @ A @ ( neg_numeral_sub @ A @ K @ L ) ) ) ) ).
% sub_num_simps(9)
thf(fact_2924_max__number__of_I2_J,axiom,
! [A: $tType] :
( ( ( uminus @ A )
& ( numeral @ A )
& ( ord @ A ) )
=> ! [U: num,V: num] :
( ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ U ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) )
=> ( ( ord_max @ A @ ( numeral_numeral @ A @ U ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ U ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) )
=> ( ( ord_max @ A @ ( numeral_numeral @ A @ U ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) )
= ( numeral_numeral @ A @ U ) ) ) ) ) ).
% max_number_of(2)
thf(fact_2925_max__number__of_I3_J,axiom,
! [A: $tType] :
( ( ( uminus @ A )
& ( numeral @ A )
& ( ord @ A ) )
=> ! [U: num,V: num] :
( ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( numeral_numeral @ A @ V ) )
=> ( ( ord_max @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( numeral_numeral @ A @ V ) )
= ( numeral_numeral @ A @ V ) ) )
& ( ~ ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( numeral_numeral @ A @ V ) )
=> ( ( ord_max @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( numeral_numeral @ A @ V ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) ) ) ) ) ).
% max_number_of(3)
thf(fact_2926_max__number__of_I4_J,axiom,
! [A: $tType] :
( ( ( uminus @ A )
& ( numeral @ A )
& ( ord @ A ) )
=> ! [U: num,V: num] :
( ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) )
=> ( ( ord_max @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) )
=> ( ( ord_max @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) ) ) ) ) ).
% max_number_of(4)
thf(fact_2927_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_2928_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_2929_semiring__norm_I166_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [V: num,W2: num,Y3: A] :
( ( plus_plus @ A @ ( numeral_numeral @ A @ V ) @ ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ Y3 ) )
= ( plus_plus @ A @ ( neg_numeral_sub @ A @ V @ W2 ) @ Y3 ) ) ) ).
% semiring_norm(166)
thf(fact_2930_semiring__norm_I167_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [V: num,W2: num,Y3: A] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) @ ( plus_plus @ A @ ( numeral_numeral @ A @ W2 ) @ Y3 ) )
= ( plus_plus @ A @ ( neg_numeral_sub @ A @ W2 @ V ) @ Y3 ) ) ) ).
% semiring_norm(167)
thf(fact_2931_diff__numeral__simps_I4_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M: num,N: num] :
( ( minus_minus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( neg_numeral_sub @ A @ N @ M ) ) ) ).
% diff_numeral_simps(4)
thf(fact_2932_max__numeral__Suc,axiom,
! [K: num,N: nat] :
( ( ord_max @ nat @ ( numeral_numeral @ nat @ K ) @ ( suc @ N ) )
= ( suc @ ( ord_max @ nat @ ( pred_numeral @ K ) @ N ) ) ) ).
% max_numeral_Suc
thf(fact_2933_max__Suc__numeral,axiom,
! [N: nat,K: num] :
( ( ord_max @ nat @ ( suc @ N ) @ ( numeral_numeral @ nat @ K ) )
= ( suc @ ( ord_max @ nat @ N @ ( pred_numeral @ K ) ) ) ) ).
% max_Suc_numeral
thf(fact_2934_sub__num__simps_I7_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [K: num,L: num] :
( ( neg_numeral_sub @ A @ ( bit0 @ K ) @ ( bit1 @ L ) )
= ( neg_numeral_dbl_dec @ A @ ( neg_numeral_sub @ A @ K @ L ) ) ) ) ).
% sub_num_simps(7)
thf(fact_2935_sub__num__simps_I8_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [K: num,L: num] :
( ( neg_numeral_sub @ A @ ( bit1 @ K ) @ ( bit0 @ L ) )
= ( neg_numeral_dbl_inc @ A @ ( neg_numeral_sub @ A @ K @ L ) ) ) ) ).
% sub_num_simps(8)
thf(fact_2936_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_2937_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_2938_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_2939_sub__num__simps_I5_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [K: num] :
( ( neg_numeral_sub @ A @ ( bit1 @ K ) @ one2 )
= ( numeral_numeral @ A @ ( bit0 @ K ) ) ) ) ).
% sub_num_simps(5)
thf(fact_2940_not__minus__numeral__eq,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: num] :
( ( bit_ri4277139882892585799ns_not @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( neg_numeral_sub @ A @ N @ one2 ) ) ) ).
% not_minus_numeral_eq
thf(fact_2941_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_2942_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_2943_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_2944_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_2945_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_2946_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_2947_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_2948_sub__num__simps_I3_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [L: num] :
( ( neg_numeral_sub @ A @ one2 @ ( bit1 @ L ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( bit0 @ L ) ) ) ) ) ).
% sub_num_simps(3)
thf(fact_2949_xor__minus__numerals_I2_J,axiom,
! [K: int,N: num] :
( ( bit_se5824344971392196577ns_xor @ int @ K @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
= ( bit_ri4277139882892585799ns_not @ int @ ( bit_se5824344971392196577ns_xor @ int @ K @ ( neg_numeral_sub @ int @ N @ one2 ) ) ) ) ).
% xor_minus_numerals(2)
thf(fact_2950_max_Ostrict__coboundedI2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C2: A,B2: A,A3: A] :
( ( ord_less @ A @ C2 @ B2 )
=> ( ord_less @ A @ C2 @ ( ord_max @ A @ A3 @ B2 ) ) ) ) ).
% max.strict_coboundedI2
thf(fact_2951_max_Ostrict__coboundedI1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( ord_less @ A @ C2 @ A3 )
=> ( ord_less @ A @ C2 @ ( ord_max @ A @ A3 @ B2 ) ) ) ) ).
% max.strict_coboundedI1
thf(fact_2952_max_Ostrict__order__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less @ A )
= ( ^ [B3: A,A5: A] :
( ( A5
= ( ord_max @ A @ A5 @ B3 ) )
& ( A5 != B3 ) ) ) ) ) ).
% max.strict_order_iff
thf(fact_2953_max_Ostrict__boundedE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,C2: A,A3: A] :
( ( ord_less @ A @ ( ord_max @ A @ B2 @ C2 ) @ A3 )
=> ~ ( ( ord_less @ A @ B2 @ A3 )
=> ~ ( ord_less @ A @ C2 @ A3 ) ) ) ) ).
% max.strict_boundedE
thf(fact_2954_less__max__iff__disj,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Z3: A,X: A,Y3: A] :
( ( ord_less @ A @ Z3 @ ( ord_max @ A @ X @ Y3 ) )
= ( ( ord_less @ A @ Z3 @ X )
| ( ord_less @ A @ Z3 @ Y3 ) ) ) ) ).
% less_max_iff_disj
thf(fact_2955_max__add__distrib__left,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [X: A,Y3: A,Z3: A] :
( ( plus_plus @ A @ ( ord_max @ A @ X @ Y3 ) @ Z3 )
= ( ord_max @ A @ ( plus_plus @ A @ X @ Z3 ) @ ( plus_plus @ A @ Y3 @ Z3 ) ) ) ) ).
% max_add_distrib_left
thf(fact_2956_max__add__distrib__right,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [X: A,Y3: A,Z3: A] :
( ( plus_plus @ A @ X @ ( ord_max @ A @ Y3 @ Z3 ) )
= ( ord_max @ A @ ( plus_plus @ A @ X @ Y3 ) @ ( plus_plus @ A @ X @ Z3 ) ) ) ) ).
% max_add_distrib_right
thf(fact_2957_max__diff__distrib__left,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [X: A,Y3: A,Z3: A] :
( ( minus_minus @ A @ ( ord_max @ A @ X @ Y3 ) @ Z3 )
= ( ord_max @ A @ ( minus_minus @ A @ X @ Z3 ) @ ( minus_minus @ A @ Y3 @ Z3 ) ) ) ) ).
% max_diff_distrib_left
thf(fact_2958_abstract__boolean__algebra__sym__diff_Oconj__xor__distrib2,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,Xor: A > A > A,Y3: A,Z3: A,X: A] :
( ( boolea3799213064322606851m_diff @ A @ Conj @ Disj @ Compl @ Zero @ One2 @ Xor )
=> ( ( Conj @ ( Xor @ Y3 @ Z3 ) @ X )
= ( Xor @ ( Conj @ Y3 @ X ) @ ( Conj @ Z3 @ X ) ) ) ) ).
% abstract_boolean_algebra_sym_diff.conj_xor_distrib2
thf(fact_2959_abstract__boolean__algebra__sym__diff_Oxor__cancel__right,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,Xor: A > A > A,X: A] :
( ( boolea3799213064322606851m_diff @ A @ Conj @ Disj @ Compl @ Zero @ One2 @ Xor )
=> ( ( Xor @ X @ ( Compl @ X ) )
= One2 ) ) ).
% abstract_boolean_algebra_sym_diff.xor_cancel_right
thf(fact_2960_abstract__boolean__algebra__sym__diff_Oconj__xor__distrib,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,Xor: A > A > A,X: A,Y3: A,Z3: A] :
( ( boolea3799213064322606851m_diff @ A @ Conj @ Disj @ Compl @ Zero @ One2 @ Xor )
=> ( ( Conj @ X @ ( Xor @ Y3 @ Z3 ) )
= ( Xor @ ( Conj @ X @ Y3 ) @ ( Conj @ X @ Z3 ) ) ) ) ).
% abstract_boolean_algebra_sym_diff.conj_xor_distrib
thf(fact_2961_abstract__boolean__algebra__sym__diff_Oxor__compl__right,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,Xor: A > A > A,X: A,Y3: A] :
( ( boolea3799213064322606851m_diff @ A @ Conj @ Disj @ Compl @ Zero @ One2 @ Xor )
=> ( ( Xor @ X @ ( Compl @ Y3 ) )
= ( Compl @ ( Xor @ X @ Y3 ) ) ) ) ).
% abstract_boolean_algebra_sym_diff.xor_compl_right
thf(fact_2962_abstract__boolean__algebra__sym__diff_Oxor__cancel__left,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,Xor: A > A > A,X: A] :
( ( boolea3799213064322606851m_diff @ A @ Conj @ Disj @ Compl @ Zero @ One2 @ Xor )
=> ( ( Xor @ ( Compl @ X ) @ X )
= One2 ) ) ).
% abstract_boolean_algebra_sym_diff.xor_cancel_left
thf(fact_2963_abstract__boolean__algebra__sym__diff_Oxor__compl__left,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,Xor: A > A > A,X: A,Y3: A] :
( ( boolea3799213064322606851m_diff @ A @ Conj @ Disj @ Compl @ Zero @ One2 @ Xor )
=> ( ( Xor @ ( Compl @ X ) @ Y3 )
= ( Compl @ ( Xor @ X @ Y3 ) ) ) ) ).
% abstract_boolean_algebra_sym_diff.xor_compl_left
thf(fact_2964_abstract__boolean__algebra__sym__diff_Oxor__one__right,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,Xor: A > A > A,X: A] :
( ( boolea3799213064322606851m_diff @ A @ Conj @ Disj @ Compl @ Zero @ One2 @ Xor )
=> ( ( Xor @ X @ One2 )
= ( Compl @ X ) ) ) ).
% abstract_boolean_algebra_sym_diff.xor_one_right
thf(fact_2965_abstract__boolean__algebra__sym__diff_Oxor__left__self,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,Xor: A > A > A,X: A,Y3: A] :
( ( boolea3799213064322606851m_diff @ A @ Conj @ Disj @ Compl @ Zero @ One2 @ Xor )
=> ( ( Xor @ X @ ( Xor @ X @ Y3 ) )
= Y3 ) ) ).
% abstract_boolean_algebra_sym_diff.xor_left_self
thf(fact_2966_abstract__boolean__algebra__sym__diff_Oxor__one__left,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,Xor: A > A > A,X: A] :
( ( boolea3799213064322606851m_diff @ A @ Conj @ Disj @ Compl @ Zero @ One2 @ Xor )
=> ( ( Xor @ One2 @ X )
= ( Compl @ X ) ) ) ).
% abstract_boolean_algebra_sym_diff.xor_one_left
thf(fact_2967_abstract__boolean__algebra__sym__diff_Oxor__self,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,Xor: A > A > A,X: A] :
( ( boolea3799213064322606851m_diff @ A @ Conj @ Disj @ Compl @ Zero @ One2 @ Xor )
=> ( ( Xor @ X @ X )
= Zero ) ) ).
% abstract_boolean_algebra_sym_diff.xor_self
thf(fact_2968_abstract__boolean__algebra__sym__diff_Oxor__def2,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,Xor: A > A > A,X: A,Y3: A] :
( ( boolea3799213064322606851m_diff @ A @ Conj @ Disj @ Compl @ Zero @ One2 @ Xor )
=> ( ( Xor @ X @ Y3 )
= ( Conj @ ( Disj @ X @ Y3 ) @ ( Disj @ ( Compl @ X ) @ ( Compl @ Y3 ) ) ) ) ) ).
% abstract_boolean_algebra_sym_diff.xor_def2
thf(fact_2969_abstract__boolean__algebra__sym__diff_Oxor__def,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,Xor: A > A > A,X: A,Y3: A] :
( ( boolea3799213064322606851m_diff @ A @ Conj @ Disj @ Compl @ Zero @ One2 @ Xor )
=> ( ( Xor @ X @ Y3 )
= ( Disj @ ( Conj @ X @ ( Compl @ Y3 ) ) @ ( Conj @ ( Compl @ X ) @ Y3 ) ) ) ) ).
% abstract_boolean_algebra_sym_diff.xor_def
thf(fact_2970_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_2971_max_Oleft__commute,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( ord_max @ A @ B2 @ ( ord_max @ A @ A3 @ C2 ) )
= ( ord_max @ A @ A3 @ ( ord_max @ A @ B2 @ C2 ) ) ) ) ).
% max.left_commute
thf(fact_2972_max_Ocommute,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_max @ A )
= ( ^ [A5: A,B3: A] : ( ord_max @ A @ B3 @ A5 ) ) ) ) ).
% max.commute
thf(fact_2973_max_Oassoc,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_max @ A @ ( ord_max @ A @ A3 @ B2 ) @ C2 )
= ( ord_max @ A @ A3 @ ( ord_max @ A @ B2 @ C2 ) ) ) ) ).
% max.assoc
thf(fact_2974_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_2975_max__absorb2,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X: A,Y3: A] :
( ( ord_less_eq @ A @ X @ Y3 )
=> ( ( ord_max @ A @ X @ Y3 )
= Y3 ) ) ) ).
% max_absorb2
thf(fact_2976_max__absorb1,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y3: A,X: A] :
( ( ord_less_eq @ A @ Y3 @ X )
=> ( ( ord_max @ A @ X @ Y3 )
= X ) ) ) ).
% max_absorb1
thf(fact_2977_max_OcoboundedI2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C2: A,B2: A,A3: A] :
( ( ord_less_eq @ A @ C2 @ B2 )
=> ( ord_less_eq @ A @ C2 @ ( ord_max @ A @ A3 @ B2 ) ) ) ) ).
% max.coboundedI2
thf(fact_2978_max_OcoboundedI1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( ord_less_eq @ A @ C2 @ A3 )
=> ( ord_less_eq @ A @ C2 @ ( ord_max @ A @ A3 @ B2 ) ) ) ) ).
% max.coboundedI1
thf(fact_2979_max_Oabsorb__iff2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A5: A,B3: A] :
( ( ord_max @ A @ A5 @ B3 )
= B3 ) ) ) ) ).
% max.absorb_iff2
thf(fact_2980_max_Oabsorb__iff1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B3: A,A5: A] :
( ( ord_max @ A @ A5 @ B3 )
= A5 ) ) ) ) ).
% max.absorb_iff1
thf(fact_2981_le__max__iff__disj,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Z3: A,X: A,Y3: A] :
( ( ord_less_eq @ A @ Z3 @ ( ord_max @ A @ X @ Y3 ) )
= ( ( ord_less_eq @ A @ Z3 @ X )
| ( ord_less_eq @ A @ Z3 @ Y3 ) ) ) ) ).
% le_max_iff_disj
thf(fact_2982_max_Ocobounded2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,A3: A] : ( ord_less_eq @ A @ B2 @ ( ord_max @ A @ A3 @ B2 ) ) ) ).
% max.cobounded2
thf(fact_2983_max_Ocobounded1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A] : ( ord_less_eq @ A @ A3 @ ( ord_max @ A @ A3 @ B2 ) ) ) ).
% max.cobounded1
thf(fact_2984_max_Oorder__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B3: A,A5: A] :
( A5
= ( ord_max @ A @ A5 @ B3 ) ) ) ) ) ).
% max.order_iff
thf(fact_2985_max_OboundedI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( ord_less_eq @ A @ B2 @ A3 )
=> ( ( ord_less_eq @ A @ C2 @ A3 )
=> ( ord_less_eq @ A @ ( ord_max @ A @ B2 @ C2 ) @ A3 ) ) ) ) ).
% max.boundedI
thf(fact_2986_max_OboundedE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,C2: A,A3: A] :
( ( ord_less_eq @ A @ ( ord_max @ A @ B2 @ C2 ) @ A3 )
=> ~ ( ( ord_less_eq @ A @ B2 @ A3 )
=> ~ ( ord_less_eq @ A @ C2 @ A3 ) ) ) ) ).
% max.boundedE
thf(fact_2987_max_OorderI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A] :
( ( A3
= ( ord_max @ A @ A3 @ B2 ) )
=> ( ord_less_eq @ A @ B2 @ A3 ) ) ) ).
% max.orderI
thf(fact_2988_max_OorderE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,A3: A] :
( ( ord_less_eq @ A @ B2 @ A3 )
=> ( A3
= ( ord_max @ A @ A3 @ B2 ) ) ) ) ).
% max.orderE
thf(fact_2989_max_Omono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C2: A,A3: A,D2: A,B2: A] :
( ( ord_less_eq @ A @ C2 @ A3 )
=> ( ( ord_less_eq @ A @ D2 @ B2 )
=> ( ord_less_eq @ A @ ( ord_max @ A @ C2 @ D2 ) @ ( ord_max @ A @ A3 @ B2 ) ) ) ) ) ).
% max.mono
thf(fact_2990_max__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_max @ A )
= ( ^ [A5: A,B3: A] : ( if @ A @ ( ord_less_eq @ A @ A5 @ B3 ) @ B3 @ A5 ) ) ) ) ).
% max_def
thf(fact_2991_nat__add__max__left,axiom,
! [M: nat,N: nat,Q4: nat] :
( ( plus_plus @ nat @ ( ord_max @ nat @ M @ N ) @ Q4 )
= ( ord_max @ nat @ ( plus_plus @ nat @ M @ Q4 ) @ ( plus_plus @ nat @ N @ Q4 ) ) ) ).
% nat_add_max_left
thf(fact_2992_nat__add__max__right,axiom,
! [M: nat,N: nat,Q4: nat] :
( ( plus_plus @ nat @ M @ ( ord_max @ nat @ N @ Q4 ) )
= ( ord_max @ nat @ ( plus_plus @ nat @ M @ N ) @ ( plus_plus @ nat @ M @ Q4 ) ) ) ).
% nat_add_max_right
thf(fact_2993_nat__mult__max__left,axiom,
! [M: nat,N: nat,Q4: nat] :
( ( times_times @ nat @ ( ord_max @ nat @ M @ N ) @ Q4 )
= ( ord_max @ nat @ ( times_times @ nat @ M @ Q4 ) @ ( times_times @ nat @ N @ Q4 ) ) ) ).
% nat_mult_max_left
thf(fact_2994_nat__mult__max__right,axiom,
! [M: nat,N: nat,Q4: nat] :
( ( times_times @ nat @ M @ ( ord_max @ nat @ N @ Q4 ) )
= ( ord_max @ nat @ ( times_times @ nat @ M @ N ) @ ( times_times @ nat @ M @ Q4 ) ) ) ).
% nat_mult_max_right
thf(fact_2995_length__induct,axiom,
! [A: $tType,P: ( list @ A ) > $o,Xs: list @ A] :
( ! [Xs2: list @ A] :
( ! [Ys: list @ A] :
( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Ys ) @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( P @ Ys ) )
=> ( P @ Xs2 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_2996_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_2997_neg__numeral__class_Osub__def,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( neg_numeral_sub @ A )
= ( ^ [K3: num,L3: num] : ( minus_minus @ A @ ( numeral_numeral @ A @ K3 ) @ ( numeral_numeral @ A @ L3 ) ) ) ) ) ).
% neg_numeral_class.sub_def
thf(fact_2998_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_2999_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_3000_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_3001_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_3002_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_3003_sub__inc__One__eq,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [N: num] :
( ( neg_numeral_sub @ A @ ( inc @ N ) @ one2 )
= ( numeral_numeral @ A @ N ) ) ) ).
% sub_inc_One_eq
thf(fact_3004_minus__numeral__eq__not__sub__one,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: num] :
( ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) )
= ( bit_ri4277139882892585799ns_not @ A @ ( neg_numeral_sub @ A @ N @ one2 ) ) ) ) ).
% minus_numeral_eq_not_sub_one
thf(fact_3005_int__numeral__or__not__num__neg,axiom,
! [M: num,N: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ M ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N ) ) )
= ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit_or_not_num_neg @ M @ N ) ) ) ) ).
% int_numeral_or_not_num_neg
thf(fact_3006_int__numeral__not__or__num__neg,axiom,
! [M: num,N: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ M ) ) @ ( numeral_numeral @ int @ N ) )
= ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit_or_not_num_neg @ N @ M ) ) ) ) ).
% int_numeral_not_or_num_neg
thf(fact_3007_numeral__or__not__num__eq,axiom,
! [M: num,N: num] :
( ( numeral_numeral @ int @ ( bit_or_not_num_neg @ M @ N ) )
= ( uminus_uminus @ int @ ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ M ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N ) ) ) ) ) ).
% numeral_or_not_num_eq
thf(fact_3008_or__minus__numerals_I1_J,axiom,
! [N: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( one_one @ int ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) )
= ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit_or_not_num_neg @ one2 @ ( bitM @ N ) ) ) ) ) ).
% or_minus_numerals(1)
thf(fact_3009_or__minus__numerals_I5_J,axiom,
! [N: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) @ ( one_one @ int ) )
= ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit_or_not_num_neg @ one2 @ ( bitM @ N ) ) ) ) ) ).
% or_minus_numerals(5)
thf(fact_3010_horner__sum__of__bool__2__less,axiom,
! [Bs: list @ $o] : ( ord_less @ int @ ( groups4207007520872428315er_sum @ $o @ int @ ( zero_neq_one_of_bool @ int ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Bs ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( size_size @ ( list @ $o ) @ Bs ) ) ) ).
% horner_sum_of_bool_2_less
thf(fact_3011_or__minus__numerals_I7_J,axiom,
! [N: num,M: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) @ ( numeral_numeral @ int @ M ) )
= ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit_or_not_num_neg @ M @ ( bitM @ N ) ) ) ) ) ).
% or_minus_numerals(7)
thf(fact_3012_or__minus__numerals_I3_J,axiom,
! [M: num,N: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ M ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) )
= ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit_or_not_num_neg @ M @ ( bitM @ N ) ) ) ) ) ).
% or_minus_numerals(3)
thf(fact_3013_dbl__dec__simps_I5_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [K: num] :
( ( neg_numeral_dbl_dec @ A @ ( numeral_numeral @ A @ K ) )
= ( numeral_numeral @ A @ ( bitM @ K ) ) ) ) ).
% dbl_dec_simps(5)
thf(fact_3014_pred__numeral__simps_I2_J,axiom,
! [K: num] :
( ( pred_numeral @ ( bit0 @ K ) )
= ( numeral_numeral @ nat @ ( bitM @ K ) ) ) ).
% pred_numeral_simps(2)
thf(fact_3015_sub__num__simps_I4_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [K: num] :
( ( neg_numeral_sub @ A @ ( bit0 @ K ) @ one2 )
= ( numeral_numeral @ A @ ( bitM @ K ) ) ) ) ).
% sub_num_simps(4)
thf(fact_3016_sub__num__simps_I2_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [L: num] :
( ( neg_numeral_sub @ A @ one2 @ ( bit0 @ L ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( bitM @ L ) ) ) ) ) ).
% sub_num_simps(2)
thf(fact_3017_semiring__norm_I26_J,axiom,
( ( bitM @ one2 )
= one2 ) ).
% semiring_norm(26)
thf(fact_3018_semiring__norm_I27_J,axiom,
! [N: num] :
( ( bitM @ ( bit0 @ N ) )
= ( bit1 @ ( bitM @ N ) ) ) ).
% semiring_norm(27)
thf(fact_3019_semiring__norm_I28_J,axiom,
! [N: num] :
( ( bitM @ ( bit1 @ N ) )
= ( bit1 @ ( bit0 @ N ) ) ) ).
% semiring_norm(28)
thf(fact_3020_inc__BitM__eq,axiom,
! [N: num] :
( ( inc @ ( bitM @ N ) )
= ( bit0 @ N ) ) ).
% inc_BitM_eq
thf(fact_3021_BitM__inc__eq,axiom,
! [N: num] :
( ( bitM @ ( inc @ N ) )
= ( bit1 @ N ) ) ).
% BitM_inc_eq
thf(fact_3022_eval__nat__numeral_I2_J,axiom,
! [N: num] :
( ( numeral_numeral @ nat @ ( bit0 @ N ) )
= ( suc @ ( numeral_numeral @ nat @ ( bitM @ N ) ) ) ) ).
% eval_nat_numeral(2)
thf(fact_3023_one__plus__BitM,axiom,
! [N: num] :
( ( plus_plus @ num @ one2 @ ( bitM @ N ) )
= ( bit0 @ N ) ) ).
% one_plus_BitM
thf(fact_3024_BitM__plus__one,axiom,
! [N: num] :
( ( plus_plus @ num @ ( bitM @ N ) @ one2 )
= ( bit0 @ N ) ) ).
% BitM_plus_one
thf(fact_3025_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_3026_not__numeral__BitM__eq,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: num] :
( ( bit_ri4277139882892585799ns_not @ A @ ( numeral_numeral @ A @ ( bitM @ N ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( bit0 @ N ) ) ) ) ) ).
% not_numeral_BitM_eq
thf(fact_3027_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_3028_bit__horner__sum__bit__iff,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [Bs: list @ $o,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( groups4207007520872428315er_sum @ $o @ A @ ( zero_neq_one_of_bool @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Bs ) @ N )
= ( ( ord_less @ nat @ N @ ( size_size @ ( list @ $o ) @ Bs ) )
& ( nth @ $o @ Bs @ N ) ) ) ) ).
% bit_horner_sum_bit_iff
thf(fact_3029_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_3030_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_3031_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_3032_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_3033_num__of__nat__numeral__eq,axiom,
! [Q4: num] :
( ( num_of_nat @ ( numeral_numeral @ nat @ Q4 ) )
= Q4 ) ).
% num_of_nat_numeral_eq
thf(fact_3034_iszero__neg__numeral,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [W2: num] :
( ( ring_1_iszero @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) )
= ( ring_1_iszero @ A @ ( numeral_numeral @ A @ W2 ) ) ) ) ).
% iszero_neg_numeral
thf(fact_3035_abstract__boolean__algebra_Odisj__conj__distrib2,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,Y3: A,Z3: A,X: A] :
( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One2 )
=> ( ( Disj @ ( Conj @ Y3 @ Z3 ) @ X )
= ( Conj @ ( Disj @ Y3 @ X ) @ ( Disj @ Z3 @ X ) ) ) ) ).
% abstract_boolean_algebra.disj_conj_distrib2
thf(fact_3036_abstract__boolean__algebra_Oconj__disj__distrib2,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,Y3: A,Z3: A,X: A] :
( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One2 )
=> ( ( Conj @ ( Disj @ Y3 @ Z3 ) @ X )
= ( Disj @ ( Conj @ Y3 @ X ) @ ( Conj @ Z3 @ X ) ) ) ) ).
% abstract_boolean_algebra.conj_disj_distrib2
thf(fact_3037_abstract__boolean__algebra_Ocompl__eq__compl__iff,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,X: A,Y3: A] :
( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One2 )
=> ( ( ( Compl @ X )
= ( Compl @ Y3 ) )
= ( X = Y3 ) ) ) ).
% abstract_boolean_algebra.compl_eq_compl_iff
thf(fact_3038_abstract__boolean__algebra_Odisj__conj__distrib,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,X: A,Y3: A,Z3: A] :
( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One2 )
=> ( ( Disj @ X @ ( Conj @ Y3 @ Z3 ) )
= ( Conj @ ( Disj @ X @ Y3 ) @ ( Disj @ X @ Z3 ) ) ) ) ).
% abstract_boolean_algebra.disj_conj_distrib
thf(fact_3039_abstract__boolean__algebra_Odisj__cancel__right,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,X: A] :
( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One2 )
=> ( ( Disj @ X @ ( Compl @ X ) )
= One2 ) ) ).
% abstract_boolean_algebra.disj_cancel_right
thf(fact_3040_abstract__boolean__algebra_Oconj__disj__distrib,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,X: A,Y3: A,Z3: A] :
( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One2 )
=> ( ( Conj @ X @ ( Disj @ Y3 @ Z3 ) )
= ( Disj @ ( Conj @ X @ Y3 ) @ ( Conj @ X @ Z3 ) ) ) ) ).
% abstract_boolean_algebra.conj_disj_distrib
thf(fact_3041_abstract__boolean__algebra_Oconj__cancel__right,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,X: A] :
( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One2 )
=> ( ( Conj @ X @ ( Compl @ X ) )
= Zero ) ) ).
% abstract_boolean_algebra.conj_cancel_right
thf(fact_3042_abstract__boolean__algebra_Ocomplement__unique,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,A3: A,X: A,Y3: A] :
( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One2 )
=> ( ( ( Conj @ A3 @ X )
= Zero )
=> ( ( ( Disj @ A3 @ X )
= One2 )
=> ( ( ( Conj @ A3 @ Y3 )
= Zero )
=> ( ( ( Disj @ A3 @ Y3 )
= One2 )
=> ( X = Y3 ) ) ) ) ) ) ).
% abstract_boolean_algebra.complement_unique
thf(fact_3043_abstract__boolean__algebra_Odisj__cancel__left,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,X: A] :
( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One2 )
=> ( ( Disj @ ( Compl @ X ) @ X )
= One2 ) ) ).
% abstract_boolean_algebra.disj_cancel_left
thf(fact_3044_abstract__boolean__algebra_Oconj__cancel__left,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,X: A] :
( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One2 )
=> ( ( Conj @ ( Compl @ X ) @ X )
= Zero ) ) ).
% abstract_boolean_algebra.conj_cancel_left
thf(fact_3045_abstract__boolean__algebra_Odisj__zero__right,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,X: A] :
( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One2 )
=> ( ( Disj @ X @ Zero )
= X ) ) ).
% abstract_boolean_algebra.disj_zero_right
thf(fact_3046_abstract__boolean__algebra_Oconj__zero__right,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,X: A] :
( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One2 )
=> ( ( Conj @ X @ Zero )
= Zero ) ) ).
% abstract_boolean_algebra.conj_zero_right
thf(fact_3047_abstract__boolean__algebra_Odisj__one__right,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,X: A] :
( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One2 )
=> ( ( Disj @ X @ One2 )
= One2 ) ) ).
% abstract_boolean_algebra.disj_one_right
thf(fact_3048_abstract__boolean__algebra_Ode__Morgan__disj,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,X: A,Y3: A] :
( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One2 )
=> ( ( Compl @ ( Disj @ X @ Y3 ) )
= ( Conj @ ( Compl @ X ) @ ( Compl @ Y3 ) ) ) ) ).
% abstract_boolean_algebra.de_Morgan_disj
thf(fact_3049_abstract__boolean__algebra_Ode__Morgan__conj,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,X: A,Y3: A] :
( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One2 )
=> ( ( Compl @ ( Conj @ X @ Y3 ) )
= ( Disj @ ( Compl @ X ) @ ( Compl @ Y3 ) ) ) ) ).
% abstract_boolean_algebra.de_Morgan_conj
thf(fact_3050_abstract__boolean__algebra_Oconj__zero__left,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,X: A] :
( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One2 )
=> ( ( Conj @ Zero @ X )
= Zero ) ) ).
% abstract_boolean_algebra.conj_zero_left
thf(fact_3051_abstract__boolean__algebra_Oconj__one__right,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,X: A] :
( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One2 )
=> ( ( Conj @ X @ One2 )
= X ) ) ).
% abstract_boolean_algebra.conj_one_right
thf(fact_3052_abstract__boolean__algebra_Odisj__one__left,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,X: A] :
( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One2 )
=> ( ( Disj @ One2 @ X )
= One2 ) ) ).
% abstract_boolean_algebra.disj_one_left
thf(fact_3053_abstract__boolean__algebra_Odouble__compl,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,X: A] :
( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One2 )
=> ( ( Compl @ ( Compl @ X ) )
= X ) ) ).
% abstract_boolean_algebra.double_compl
thf(fact_3054_abstract__boolean__algebra_Ocompl__unique,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,X: A,Y3: A] :
( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One2 )
=> ( ( ( Conj @ X @ Y3 )
= Zero )
=> ( ( ( Disj @ X @ Y3 )
= One2 )
=> ( ( Compl @ X )
= Y3 ) ) ) ) ).
% abstract_boolean_algebra.compl_unique
thf(fact_3055_abstract__boolean__algebra_Ocompl__zero,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A] :
( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One2 )
=> ( ( Compl @ Zero )
= One2 ) ) ).
% abstract_boolean_algebra.compl_zero
thf(fact_3056_abstract__boolean__algebra_Ocompl__one,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A] :
( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One2 )
=> ( ( Compl @ One2 )
= Zero ) ) ).
% abstract_boolean_algebra.compl_one
thf(fact_3057_iszero__def,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( ( ring_1_iszero @ A )
= ( ^ [Z7: A] :
( Z7
= ( zero_zero @ A ) ) ) ) ) ).
% iszero_def
thf(fact_3058_iszero__0,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( ring_1_iszero @ A @ ( zero_zero @ A ) ) ) ).
% iszero_0
thf(fact_3059_not__iszero__numeral,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [W2: num] :
~ ( ring_1_iszero @ A @ ( numeral_numeral @ A @ W2 ) ) ) ).
% not_iszero_numeral
thf(fact_3060_not__iszero__1,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ~ ( ring_1_iszero @ A @ ( one_one @ A ) ) ) ).
% not_iszero_1
thf(fact_3061_eq__iff__iszero__diff,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( ( ^ [Y: A,Z2: A] : Y = Z2 )
= ( ^ [X2: A,Y6: A] : ( ring_1_iszero @ A @ ( minus_minus @ A @ X2 @ Y6 ) ) ) ) ) ).
% eq_iff_iszero_diff
thf(fact_3062_abstract__boolean__algebra__sym__diff_Oaxioms_I1_J,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,Xor: A > A > A] :
( ( boolea3799213064322606851m_diff @ A @ Conj @ Disj @ Compl @ Zero @ One2 @ Xor )
=> ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One2 ) ) ).
% abstract_boolean_algebra_sym_diff.axioms(1)
thf(fact_3063_nth__equalityI,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ A ) @ Ys2 ) )
=> ( ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ Xs @ I4 )
= ( nth @ A @ Ys2 @ I4 ) ) )
=> ( Xs = Ys2 ) ) ) ).
% nth_equalityI
thf(fact_3064_Skolem__list__nth,axiom,
! [A: $tType,K: nat,P: nat > A > $o] :
( ( ! [I5: nat] :
( ( ord_less @ nat @ I5 @ K )
=> ? [X7: A] : ( P @ I5 @ X7 ) ) )
= ( ? [Xs3: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs3 )
= K )
& ! [I5: nat] :
( ( ord_less @ nat @ I5 @ K )
=> ( P @ I5 @ ( nth @ A @ Xs3 @ I5 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_3065_list__eq__iff__nth__eq,axiom,
! [A: $tType] :
( ( ^ [Y: list @ A,Z2: list @ A] : Y = Z2 )
= ( ^ [Xs3: list @ A,Ys3: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs3 )
= ( size_size @ ( list @ A ) @ Ys3 ) )
& ! [I5: nat] :
( ( ord_less @ nat @ I5 @ ( size_size @ ( list @ A ) @ Xs3 ) )
=> ( ( nth @ A @ Xs3 @ I5 )
= ( nth @ A @ Ys3 @ I5 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_3066_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_3067_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_3068_num__of__nat_Osimps_I1_J,axiom,
( ( num_of_nat @ ( zero_zero @ nat ) )
= one2 ) ).
% num_of_nat.simps(1)
thf(fact_3069_not__iszero__Numeral1,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ~ ( ring_1_iszero @ A @ ( numeral_numeral @ A @ one2 ) ) ) ).
% not_iszero_Numeral1
thf(fact_3070_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_3071_eq__numeral__iff__iszero_I1_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X: num,Y3: num] :
( ( ( numeral_numeral @ A @ X )
= ( numeral_numeral @ A @ Y3 ) )
= ( ring_1_iszero @ A @ ( neg_numeral_sub @ A @ X @ Y3 ) ) ) ) ).
% eq_numeral_iff_iszero(1)
thf(fact_3072_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_3073_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_3074_not__iszero__neg__Numeral1,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ~ ( ring_1_iszero @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ one2 ) ) ) ) ).
% not_iszero_neg_Numeral1
thf(fact_3075_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_3076_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_3077_eq__numeral__iff__iszero_I3_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X: num,Y3: num] :
( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ X ) )
= ( numeral_numeral @ A @ Y3 ) )
= ( ring_1_iszero @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ X @ Y3 ) ) ) ) ) ).
% eq_numeral_iff_iszero(3)
thf(fact_3078_eq__numeral__iff__iszero_I2_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X: num,Y3: num] :
( ( ( numeral_numeral @ A @ X )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ Y3 ) ) )
= ( ring_1_iszero @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ X @ Y3 ) ) ) ) ) ).
% eq_numeral_iff_iszero(2)
thf(fact_3079_eq__numeral__iff__iszero_I4_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X: num,Y3: num] :
( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ X ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ Y3 ) ) )
= ( ring_1_iszero @ A @ ( neg_numeral_sub @ A @ Y3 @ X ) ) ) ) ).
% eq_numeral_iff_iszero(4)
thf(fact_3080_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_3081_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_3082_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_3083_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_3084_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_3085_nth__rotate1,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ ( rotate1 @ A @ Xs ) @ N )
= ( nth @ A @ Xs @ ( modulo_modulo @ nat @ ( suc @ N ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) ).
% nth_rotate1
thf(fact_3086_nth__update__neq,axiom,
! [B: $tType,A: $tType] :
( ( ( heap @ A )
& ( heap @ B ) )
=> ! [A3: array @ A,B2: array @ B,J: nat,V: B,H2: heap_ext @ product_unit,I: nat] :
( ( array_noteq @ A @ B @ A3 @ B2 )
=> ( ( nth @ A @ ( array_get @ A @ ( array_update @ B @ B2 @ J @ V @ H2 ) @ A3 ) @ I )
= ( nth @ A @ ( array_get @ A @ H2 @ A3 ) @ I ) ) ) ) ).
% nth_update_neq
thf(fact_3087_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_3088_set__decode__plus__power__2,axiom,
! [N: nat,Z3: nat] :
( ~ ( member @ nat @ N @ ( nat_set_decode @ Z3 ) )
=> ( ( nat_set_decode @ ( plus_plus @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ Z3 ) )
= ( insert @ nat @ N @ ( nat_set_decode @ Z3 ) ) ) ) ).
% set_decode_plus_power_2
thf(fact_3089_nth__rotate,axiom,
! [A: $tType,N: nat,Xs: list @ A,M: nat] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ ( rotate @ A @ M @ Xs ) @ N )
= ( nth @ A @ Xs @ ( modulo_modulo @ nat @ ( plus_plus @ nat @ M @ N ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) ).
% nth_rotate
thf(fact_3090_min_Oidem,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A] :
( ( ord_min @ A @ A3 @ A3 )
= A3 ) ) ).
% min.idem
thf(fact_3091_min_Oleft__idem,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A] :
( ( ord_min @ A @ A3 @ ( ord_min @ A @ A3 @ B2 ) )
= ( ord_min @ A @ A3 @ B2 ) ) ) ).
% min.left_idem
thf(fact_3092_min_Oright__idem,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A] :
( ( ord_min @ A @ ( ord_min @ A @ A3 @ B2 ) @ B2 )
= ( ord_min @ A @ A3 @ B2 ) ) ) ).
% min.right_idem
thf(fact_3093_min_Obounded__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ ( ord_min @ A @ B2 @ C2 ) )
= ( ( ord_less_eq @ A @ A3 @ B2 )
& ( ord_less_eq @ A @ A3 @ C2 ) ) ) ) ).
% min.bounded_iff
thf(fact_3094_min_Oabsorb2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,A3: A] :
( ( ord_less_eq @ A @ B2 @ A3 )
=> ( ( ord_min @ A @ A3 @ B2 )
= B2 ) ) ) ).
% min.absorb2
thf(fact_3095_min_Oabsorb1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_min @ A @ A3 @ B2 )
= A3 ) ) ) ).
% min.absorb1
thf(fact_3096_min__less__iff__conj,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Z3: A,X: A,Y3: A] :
( ( ord_less @ A @ Z3 @ ( ord_min @ A @ X @ Y3 ) )
= ( ( ord_less @ A @ Z3 @ X )
& ( ord_less @ A @ Z3 @ Y3 ) ) ) ) ).
% min_less_iff_conj
thf(fact_3097_min_Oabsorb4,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,A3: A] :
( ( ord_less @ A @ B2 @ A3 )
=> ( ( ord_min @ A @ A3 @ B2 )
= B2 ) ) ) ).
% min.absorb4
thf(fact_3098_min_Oabsorb3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_min @ A @ A3 @ B2 )
= A3 ) ) ) ).
% min.absorb3
thf(fact_3099_max__min__same_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Y3: A,X: A] :
( ( ord_max @ A @ Y3 @ ( ord_min @ A @ X @ Y3 ) )
= Y3 ) ) ).
% max_min_same(4)
thf(fact_3100_max__min__same_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y3: A] :
( ( ord_max @ A @ ( ord_min @ A @ X @ Y3 ) @ Y3 )
= Y3 ) ) ).
% max_min_same(3)
thf(fact_3101_max__min__same_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y3: A] :
( ( ord_max @ A @ ( ord_min @ A @ X @ Y3 ) @ X )
= X ) ) ).
% max_min_same(2)
thf(fact_3102_max__min__same_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y3: A] :
( ( ord_max @ A @ X @ ( ord_min @ A @ X @ Y3 ) )
= X ) ) ).
% max_min_same(1)
thf(fact_3103_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_3104_min__0L,axiom,
! [N: nat] :
( ( ord_min @ nat @ ( zero_zero @ nat ) @ N )
= ( zero_zero @ nat ) ) ).
% min_0L
thf(fact_3105_min__0R,axiom,
! [N: nat] :
( ( ord_min @ nat @ N @ ( zero_zero @ nat ) )
= ( zero_zero @ nat ) ) ).
% min_0R
thf(fact_3106_min__number__of_I1_J,axiom,
! [A: $tType] :
( ( ( numeral @ A )
& ( ord @ A ) )
=> ! [U: num,V: num] :
( ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ U ) @ ( numeral_numeral @ A @ V ) )
=> ( ( ord_min @ A @ ( numeral_numeral @ A @ U ) @ ( numeral_numeral @ A @ V ) )
= ( numeral_numeral @ A @ U ) ) )
& ( ~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ U ) @ ( numeral_numeral @ A @ V ) )
=> ( ( ord_min @ A @ ( numeral_numeral @ A @ U ) @ ( numeral_numeral @ A @ V ) )
= ( numeral_numeral @ A @ V ) ) ) ) ) ).
% min_number_of(1)
thf(fact_3107_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_3108_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_3109_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_3110_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_3111_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_3112_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_3113_rotate__Suc,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( rotate @ A @ ( suc @ N ) @ Xs )
= ( rotate1 @ A @ ( rotate @ A @ N @ Xs ) ) ) ).
% rotate_Suc
thf(fact_3114_get__update__elem__neqIndex,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [I: nat,J: nat,A3: array @ A,V: A,H2: heap_ext @ product_unit] :
( ( I != J )
=> ( ( nth @ A @ ( array_get @ A @ ( array_update @ A @ A3 @ J @ V @ H2 ) @ A3 ) @ I )
= ( nth @ A @ ( array_get @ A @ H2 @ A3 ) @ I ) ) ) ) ).
% get_update_elem_neqIndex
thf(fact_3115_min__number__of_I4_J,axiom,
! [A: $tType] :
( ( ( uminus @ A )
& ( numeral @ A )
& ( ord @ A ) )
=> ! [U: num,V: num] :
( ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) )
=> ( ( ord_min @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) )
=> ( ( ord_min @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) ) ) ) ) ).
% min_number_of(4)
thf(fact_3116_min__number__of_I3_J,axiom,
! [A: $tType] :
( ( ( uminus @ A )
& ( numeral @ A )
& ( ord @ A ) )
=> ! [U: num,V: num] :
( ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( numeral_numeral @ A @ V ) )
=> ( ( ord_min @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( numeral_numeral @ A @ V ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( numeral_numeral @ A @ V ) )
=> ( ( ord_min @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( numeral_numeral @ A @ V ) )
= ( numeral_numeral @ A @ V ) ) ) ) ) ).
% min_number_of(3)
thf(fact_3117_min__number__of_I2_J,axiom,
! [A: $tType] :
( ( ( uminus @ A )
& ( numeral @ A )
& ( ord @ A ) )
=> ! [U: num,V: num] :
( ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ U ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) )
=> ( ( ord_min @ A @ ( numeral_numeral @ A @ U ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) )
= ( numeral_numeral @ A @ U ) ) )
& ( ~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ U ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) )
=> ( ( ord_min @ A @ ( numeral_numeral @ A @ U ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) ) ) ) ) ).
% min_number_of(2)
thf(fact_3118_rotate__length01,axiom,
! [A: $tType,Xs: list @ A,N: nat] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) )
=> ( ( rotate @ A @ N @ Xs )
= Xs ) ) ).
% rotate_length01
thf(fact_3119_rotate__id,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( ( modulo_modulo @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
= ( zero_zero @ nat ) )
=> ( ( rotate @ A @ N @ Xs )
= Xs ) ) ).
% rotate_id
thf(fact_3120_min__Suc__numeral,axiom,
! [N: nat,K: num] :
( ( ord_min @ nat @ ( suc @ N ) @ ( numeral_numeral @ nat @ K ) )
= ( suc @ ( ord_min @ nat @ N @ ( pred_numeral @ K ) ) ) ) ).
% min_Suc_numeral
thf(fact_3121_min__numeral__Suc,axiom,
! [K: num,N: nat] :
( ( ord_min @ nat @ ( numeral_numeral @ nat @ K ) @ ( suc @ N ) )
= ( suc @ ( ord_min @ nat @ ( pred_numeral @ K ) @ N ) ) ) ).
% min_numeral_Suc
thf(fact_3122_rotate1__length01,axiom,
! [A: $tType,Xs: list @ A] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) )
=> ( ( rotate1 @ A @ Xs )
= Xs ) ) ).
% rotate1_length01
thf(fact_3123_min__max__distrib2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_min @ A @ A3 @ ( ord_max @ A @ B2 @ C2 ) )
= ( ord_max @ A @ ( ord_min @ A @ A3 @ B2 ) @ ( ord_min @ A @ A3 @ C2 ) ) ) ) ).
% min_max_distrib2
thf(fact_3124_min__max__distrib1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,C2: A,A3: A] :
( ( ord_min @ A @ ( ord_max @ A @ B2 @ C2 ) @ A3 )
= ( ord_max @ A @ ( ord_min @ A @ B2 @ A3 ) @ ( ord_min @ A @ C2 @ A3 ) ) ) ) ).
% min_max_distrib1
thf(fact_3125_max__min__distrib2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_max @ A @ A3 @ ( ord_min @ A @ B2 @ C2 ) )
= ( ord_min @ A @ ( ord_max @ A @ A3 @ B2 ) @ ( ord_max @ A @ A3 @ C2 ) ) ) ) ).
% max_min_distrib2
thf(fact_3126_max__min__distrib1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,C2: A,A3: A] :
( ( ord_max @ A @ ( ord_min @ A @ B2 @ C2 ) @ A3 )
= ( ord_min @ A @ ( ord_max @ A @ B2 @ A3 ) @ ( ord_max @ A @ C2 @ A3 ) ) ) ) ).
% max_min_distrib1
thf(fact_3127_min__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_min @ A )
= ( ^ [A5: A,B3: A] : ( if @ A @ ( ord_less_eq @ A @ A5 @ B3 ) @ A5 @ B3 ) ) ) ) ).
% min_def
thf(fact_3128_min_Omono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,C2: A,B2: A,D2: A] :
( ( ord_less_eq @ A @ A3 @ C2 )
=> ( ( ord_less_eq @ A @ B2 @ D2 )
=> ( ord_less_eq @ A @ ( ord_min @ A @ A3 @ B2 ) @ ( ord_min @ A @ C2 @ D2 ) ) ) ) ) ).
% min.mono
thf(fact_3129_min_OorderE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( A3
= ( ord_min @ A @ A3 @ B2 ) ) ) ) ).
% min.orderE
thf(fact_3130_min_OorderI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A] :
( ( A3
= ( ord_min @ A @ A3 @ B2 ) )
=> ( ord_less_eq @ A @ A3 @ B2 ) ) ) ).
% min.orderI
thf(fact_3131_min_OboundedE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ ( ord_min @ A @ B2 @ C2 ) )
=> ~ ( ( ord_less_eq @ A @ A3 @ B2 )
=> ~ ( ord_less_eq @ A @ A3 @ C2 ) ) ) ) ).
% min.boundedE
thf(fact_3132_min_OboundedI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ A3 @ C2 )
=> ( ord_less_eq @ A @ A3 @ ( ord_min @ A @ B2 @ C2 ) ) ) ) ) ).
% min.boundedI
thf(fact_3133_min_Oorder__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A5: A,B3: A] :
( A5
= ( ord_min @ A @ A5 @ B3 ) ) ) ) ) ).
% min.order_iff
thf(fact_3134_min_Ocobounded1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A] : ( ord_less_eq @ A @ ( ord_min @ A @ A3 @ B2 ) @ A3 ) ) ).
% min.cobounded1
thf(fact_3135_min_Ocobounded2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A] : ( ord_less_eq @ A @ ( ord_min @ A @ A3 @ B2 ) @ B2 ) ) ).
% min.cobounded2
thf(fact_3136_min_Oabsorb__iff1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A5: A,B3: A] :
( ( ord_min @ A @ A5 @ B3 )
= A5 ) ) ) ) ).
% min.absorb_iff1
thf(fact_3137_min_Oabsorb__iff2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B3: A,A5: A] :
( ( ord_min @ A @ A5 @ B3 )
= B3 ) ) ) ) ).
% min.absorb_iff2
thf(fact_3138_min_OcoboundedI1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ C2 )
=> ( ord_less_eq @ A @ ( ord_min @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% min.coboundedI1
thf(fact_3139_min_OcoboundedI2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,C2: A,A3: A] :
( ( ord_less_eq @ A @ B2 @ C2 )
=> ( ord_less_eq @ A @ ( ord_min @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% min.coboundedI2
thf(fact_3140_min__le__iff__disj,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y3: A,Z3: A] :
( ( ord_less_eq @ A @ ( ord_min @ A @ X @ Y3 ) @ Z3 )
= ( ( ord_less_eq @ A @ X @ Z3 )
| ( ord_less_eq @ A @ Y3 @ Z3 ) ) ) ) ).
% min_le_iff_disj
thf(fact_3141_min__absorb1,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X: A,Y3: A] :
( ( ord_less_eq @ A @ X @ Y3 )
=> ( ( ord_min @ A @ X @ Y3 )
= X ) ) ) ).
% min_absorb1
thf(fact_3142_min__absorb2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y3: A,X: A] :
( ( ord_less_eq @ A @ Y3 @ X )
=> ( ( ord_min @ A @ X @ Y3 )
= Y3 ) ) ) ).
% min_absorb2
thf(fact_3143_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_3144_min_Oassoc,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_min @ A @ ( ord_min @ A @ A3 @ B2 ) @ C2 )
= ( ord_min @ A @ A3 @ ( ord_min @ A @ B2 @ C2 ) ) ) ) ).
% min.assoc
thf(fact_3145_min_Ocommute,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_min @ A )
= ( ^ [A5: A,B3: A] : ( ord_min @ A @ B3 @ A5 ) ) ) ) ).
% min.commute
thf(fact_3146_min_Oleft__commute,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( ord_min @ A @ B2 @ ( ord_min @ A @ A3 @ C2 ) )
= ( ord_min @ A @ A3 @ ( ord_min @ A @ B2 @ C2 ) ) ) ) ).
% min.left_commute
thf(fact_3147_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_3148_min__diff__distrib__left,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [X: A,Y3: A,Z3: A] :
( ( minus_minus @ A @ ( ord_min @ A @ X @ Y3 ) @ Z3 )
= ( ord_min @ A @ ( minus_minus @ A @ X @ Z3 ) @ ( minus_minus @ A @ Y3 @ Z3 ) ) ) ) ).
% min_diff_distrib_left
thf(fact_3149_min__add__distrib__right,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [X: A,Y3: A,Z3: A] :
( ( plus_plus @ A @ X @ ( ord_min @ A @ Y3 @ Z3 ) )
= ( ord_min @ A @ ( plus_plus @ A @ X @ Y3 ) @ ( plus_plus @ A @ X @ Z3 ) ) ) ) ).
% min_add_distrib_right
thf(fact_3150_min__add__distrib__left,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [X: A,Y3: A,Z3: A] :
( ( plus_plus @ A @ ( ord_min @ A @ X @ Y3 ) @ Z3 )
= ( ord_min @ A @ ( plus_plus @ A @ X @ Z3 ) @ ( plus_plus @ A @ Y3 @ Z3 ) ) ) ) ).
% min_add_distrib_left
thf(fact_3151_min__less__iff__disj,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y3: A,Z3: A] :
( ( ord_less @ A @ ( ord_min @ A @ X @ Y3 ) @ Z3 )
= ( ( ord_less @ A @ X @ Z3 )
| ( ord_less @ A @ Y3 @ Z3 ) ) ) ) ).
% min_less_iff_disj
thf(fact_3152_min_Ostrict__boundedE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less @ A @ A3 @ ( ord_min @ A @ B2 @ C2 ) )
=> ~ ( ( ord_less @ A @ A3 @ B2 )
=> ~ ( ord_less @ A @ A3 @ C2 ) ) ) ) ).
% min.strict_boundedE
thf(fact_3153_min_Ostrict__order__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less @ A )
= ( ^ [A5: A,B3: A] :
( ( A5
= ( ord_min @ A @ A5 @ B3 ) )
& ( A5 != B3 ) ) ) ) ) ).
% min.strict_order_iff
thf(fact_3154_min_Ostrict__coboundedI1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( ord_less @ A @ A3 @ C2 )
=> ( ord_less @ A @ ( ord_min @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% min.strict_coboundedI1
thf(fact_3155_min_Ostrict__coboundedI2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,C2: A,A3: A] :
( ( ord_less @ A @ B2 @ C2 )
=> ( ord_less @ A @ ( ord_min @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% min.strict_coboundedI2
thf(fact_3156_nat__mult__min__left,axiom,
! [M: nat,N: nat,Q4: nat] :
( ( times_times @ nat @ ( ord_min @ nat @ M @ N ) @ Q4 )
= ( ord_min @ nat @ ( times_times @ nat @ M @ Q4 ) @ ( times_times @ nat @ N @ Q4 ) ) ) ).
% nat_mult_min_left
thf(fact_3157_nat__mult__min__right,axiom,
! [M: nat,N: nat,Q4: nat] :
( ( times_times @ nat @ M @ ( ord_min @ nat @ N @ Q4 ) )
= ( ord_min @ nat @ ( times_times @ nat @ M @ N ) @ ( times_times @ nat @ M @ Q4 ) ) ) ).
% nat_mult_min_right
thf(fact_3158_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_3159_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_3160_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_3161_atLeast0__atMost__Suc,axiom,
! [N: nat] :
( ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) )
= ( insert @ nat @ ( suc @ N ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% atLeast0_atMost_Suc
thf(fact_3162_atLeastAtMost__insertL,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( insert @ nat @ M @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ N ) )
= ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ).
% atLeastAtMost_insertL
thf(fact_3163_atLeastAtMostSuc__conv,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ ( suc @ N ) )
=> ( ( set_or1337092689740270186AtMost @ nat @ M @ ( suc @ N ) )
= ( insert @ nat @ ( suc @ N ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% atLeastAtMostSuc_conv
thf(fact_3164_Icc__eq__insert__lb__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( set_or1337092689740270186AtMost @ nat @ M @ N )
= ( insert @ nat @ M @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ N ) ) ) ) ).
% Icc_eq_insert_lb_nat
thf(fact_3165_length__def,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( ( array_length @ A )
= ( ^ [H4: heap_ext @ product_unit,A5: array @ A] : ( size_size @ ( list @ A ) @ ( array_get @ A @ H4 @ A5 ) ) ) ) ) ).
% length_def
thf(fact_3166_max__mult__distrib__left,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [P5: A,X: A,Y3: A] :
( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P5 )
=> ( ( times_times @ A @ P5 @ ( ord_max @ A @ X @ Y3 ) )
= ( ord_max @ A @ ( times_times @ A @ P5 @ X ) @ ( times_times @ A @ P5 @ Y3 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P5 )
=> ( ( times_times @ A @ P5 @ ( ord_max @ A @ X @ Y3 ) )
= ( ord_min @ A @ ( times_times @ A @ P5 @ X ) @ ( times_times @ A @ P5 @ Y3 ) ) ) ) ) ) ).
% max_mult_distrib_left
thf(fact_3167_min__mult__distrib__left,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [P5: A,X: A,Y3: A] :
( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P5 )
=> ( ( times_times @ A @ P5 @ ( ord_min @ A @ X @ Y3 ) )
= ( ord_min @ A @ ( times_times @ A @ P5 @ X ) @ ( times_times @ A @ P5 @ Y3 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P5 )
=> ( ( times_times @ A @ P5 @ ( ord_min @ A @ X @ Y3 ) )
= ( ord_max @ A @ ( times_times @ A @ P5 @ X ) @ ( times_times @ A @ P5 @ Y3 ) ) ) ) ) ) ).
% min_mult_distrib_left
thf(fact_3168_max__mult__distrib__right,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [P5: A,X: A,Y3: A] :
( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P5 )
=> ( ( times_times @ A @ ( ord_max @ A @ X @ Y3 ) @ P5 )
= ( ord_max @ A @ ( times_times @ A @ X @ P5 ) @ ( times_times @ A @ Y3 @ P5 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P5 )
=> ( ( times_times @ A @ ( ord_max @ A @ X @ Y3 ) @ P5 )
= ( ord_min @ A @ ( times_times @ A @ X @ P5 ) @ ( times_times @ A @ Y3 @ P5 ) ) ) ) ) ) ).
% max_mult_distrib_right
thf(fact_3169_min__mult__distrib__right,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [P5: A,X: A,Y3: A] :
( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P5 )
=> ( ( times_times @ A @ ( ord_min @ A @ X @ Y3 ) @ P5 )
= ( ord_min @ A @ ( times_times @ A @ X @ P5 ) @ ( times_times @ A @ Y3 @ P5 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P5 )
=> ( ( times_times @ A @ ( ord_min @ A @ X @ Y3 ) @ P5 )
= ( ord_max @ A @ ( times_times @ A @ X @ P5 ) @ ( times_times @ A @ Y3 @ P5 ) ) ) ) ) ) ).
% min_mult_distrib_right
thf(fact_3170_min__divide__distrib__right,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [P5: A,X: A,Y3: A] :
( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P5 )
=> ( ( divide_divide @ A @ ( ord_min @ A @ X @ Y3 ) @ P5 )
= ( ord_min @ A @ ( divide_divide @ A @ X @ P5 ) @ ( divide_divide @ A @ Y3 @ P5 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P5 )
=> ( ( divide_divide @ A @ ( ord_min @ A @ X @ Y3 ) @ P5 )
= ( ord_max @ A @ ( divide_divide @ A @ X @ P5 ) @ ( divide_divide @ A @ Y3 @ P5 ) ) ) ) ) ) ).
% min_divide_distrib_right
thf(fact_3171_max__divide__distrib__right,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [P5: A,X: A,Y3: A] :
( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P5 )
=> ( ( divide_divide @ A @ ( ord_max @ A @ X @ Y3 ) @ P5 )
= ( ord_max @ A @ ( divide_divide @ A @ X @ P5 ) @ ( divide_divide @ A @ Y3 @ P5 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P5 )
=> ( ( divide_divide @ A @ ( ord_max @ A @ X @ Y3 ) @ P5 )
= ( ord_min @ A @ ( divide_divide @ A @ X @ P5 ) @ ( divide_divide @ A @ Y3 @ P5 ) ) ) ) ) ) ).
% max_divide_distrib_right
thf(fact_3172_and__int_Oelims,axiom,
! [X: int,Xa3: int,Y3: int] :
( ( ( bit_se5824344872417868541ns_and @ int @ X @ Xa3 )
= Y3 )
=> ( ( ( ( member @ int @ X @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
& ( member @ int @ Xa3 @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ 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 ) ) @ Xa3 ) ) ) ) ) )
& ( ~ ( ( member @ int @ X @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
& ( member @ int @ Xa3 @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ 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 ) ) @ Xa3 ) ) )
@ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( divide_divide @ int @ X @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ Xa3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% and_int.elims
thf(fact_3173_and__int_Osimps,axiom,
( ( bit_se5824344872417868541ns_and @ int )
= ( ^ [K3: int,L3: int] :
( if @ int
@ ( ( member @ int @ K3 @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
& ( member @ int @ L3 @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
@ ( uminus_uminus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K3 )
& ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L3 ) ) ) )
@ ( plus_plus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K3 )
& ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L3 ) ) )
@ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( divide_divide @ int @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ).
% and_int.simps
thf(fact_3174_nth__Cons__pos,axiom,
! [A: $tType,N: nat,X: A,Xs: list @ A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( nth @ A @ ( cons @ A @ X @ Xs ) @ N )
= ( nth @ A @ Xs @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ).
% nth_Cons_pos
thf(fact_3175_get__set__neq,axiom,
! [B: $tType,A: $tType] :
( ( ( heap @ A )
& ( heap @ B ) )
=> ! [R: array @ A,S: array @ B,X: list @ B,H2: heap_ext @ product_unit] :
( ( array_noteq @ A @ B @ R @ S )
=> ( ( array_get @ A @ ( array_set @ B @ S @ X @ H2 ) @ R )
= ( array_get @ A @ H2 @ R ) ) ) ) ).
% get_set_neq
thf(fact_3176_divmod__step__eq,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [L: num,R: A,Q4: A] :
( ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ L ) @ R )
=> ( ( unique1321980374590559556d_step @ A @ L @ ( product_Pair @ A @ A @ Q4 @ R ) )
= ( product_Pair @ A @ A @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Q4 ) @ ( 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 @ Q4 @ R ) )
= ( product_Pair @ A @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Q4 ) @ R ) ) ) ) ) ).
% divmod_step_eq
thf(fact_3177_bot__apply,axiom,
! [C: $tType,D: $tType] :
( ( bot @ C )
=> ( ( bot_bot @ ( D > C ) )
= ( ^ [X2: D] : ( bot_bot @ C ) ) ) ) ).
% bot_apply
thf(fact_3178_set__same,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [R: array @ A,X: list @ A,Y3: list @ A,H2: heap_ext @ product_unit] :
( ( array_set @ A @ R @ X @ ( array_set @ A @ R @ Y3 @ H2 ) )
= ( array_set @ A @ R @ X @ H2 ) ) ) ).
% set_same
thf(fact_3179_max__bot2,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [X: A] :
( ( ord_max @ A @ X @ ( bot_bot @ A ) )
= X ) ) ).
% max_bot2
thf(fact_3180_max__bot,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [X: A] :
( ( ord_max @ A @ ( bot_bot @ A ) @ X )
= X ) ) ).
% max_bot
thf(fact_3181_min__bot2,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [X: A] :
( ( ord_min @ A @ X @ ( bot_bot @ A ) )
= ( bot_bot @ A ) ) ) ).
% min_bot2
thf(fact_3182_min__bot,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [X: A] :
( ( ord_min @ A @ ( bot_bot @ A ) @ X )
= ( bot_bot @ A ) ) ) ).
% min_bot
thf(fact_3183_get__set__eq,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [R: array @ A,X: list @ A,H2: heap_ext @ product_unit] :
( ( array_get @ A @ ( array_set @ A @ R @ X @ H2 ) @ R )
= X ) ) ).
% get_set_eq
thf(fact_3184_atLeastatMost__empty__iff2,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A,B2: A] :
( ( ( bot_bot @ ( set @ A ) )
= ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
= ( ~ ( ord_less_eq @ A @ A3 @ B2 ) ) ) ) ).
% atLeastatMost_empty_iff2
thf(fact_3185_atLeastatMost__empty__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A,B2: A] :
( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
= ( ~ ( ord_less_eq @ A @ A3 @ B2 ) ) ) ) ).
% atLeastatMost_empty_iff
thf(fact_3186_atLeastatMost__empty,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B2: A,A3: A] :
( ( ord_less @ A @ B2 @ A3 )
=> ( ( set_or1337092689740270186AtMost @ A @ A3 @ B2 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% atLeastatMost_empty
thf(fact_3187_nth__Cons__Suc,axiom,
! [A: $tType,X: A,Xs: list @ A,N: nat] :
( ( nth @ A @ ( cons @ A @ X @ Xs ) @ ( suc @ N ) )
= ( nth @ A @ Xs @ N ) ) ).
% nth_Cons_Suc
thf(fact_3188_nth__Cons__0,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( nth @ A @ ( cons @ A @ X @ Xs ) @ ( zero_zero @ nat ) )
= X ) ).
% nth_Cons_0
thf(fact_3189_subset__Compl__singleton,axiom,
! [A: $tType,A6: set @ A,B2: A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ ( uminus_uminus @ ( set @ A ) @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( ~ ( member @ A @ B2 @ A6 ) ) ) ).
% subset_Compl_singleton
thf(fact_3190_nth__Cons__numeral,axiom,
! [A: $tType,X: A,Xs: list @ A,V: num] :
( ( nth @ A @ ( cons @ A @ X @ Xs ) @ ( numeral_numeral @ nat @ V ) )
= ( nth @ A @ Xs @ ( minus_minus @ nat @ ( numeral_numeral @ nat @ V ) @ ( one_one @ nat ) ) ) ) ).
% nth_Cons_numeral
thf(fact_3191_bot__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( bot @ B )
=> ( ( bot_bot @ ( A > B ) )
= ( ^ [X2: A] : ( bot_bot @ B ) ) ) ) ).
% bot_fun_def
thf(fact_3192_bot_Onot__eq__extremum,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A3: A] :
( ( A3
!= ( bot_bot @ A ) )
= ( ord_less @ A @ ( bot_bot @ A ) @ A3 ) ) ) ).
% bot.not_eq_extremum
thf(fact_3193_bot_Oextremum__strict,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A3: A] :
~ ( ord_less @ A @ A3 @ ( bot_bot @ A ) ) ) ).
% bot.extremum_strict
thf(fact_3194_bot_Oextremum__uniqueI,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ A3 @ ( bot_bot @ A ) )
=> ( A3
= ( bot_bot @ A ) ) ) ) ).
% bot.extremum_uniqueI
thf(fact_3195_bot_Oextremum__unique,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ A3 @ ( bot_bot @ A ) )
= ( A3
= ( bot_bot @ A ) ) ) ) ).
% bot.extremum_unique
thf(fact_3196_bot_Oextremum,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A3: A] : ( ord_less_eq @ A @ ( bot_bot @ A ) @ A3 ) ) ).
% bot.extremum
thf(fact_3197_not__psubset__empty,axiom,
! [A: $tType,A6: set @ A] :
~ ( ord_less @ ( set @ A ) @ A6 @ ( bot_bot @ ( set @ A ) ) ) ).
% not_psubset_empty
thf(fact_3198_diff__shunt__var,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y3: A] :
( ( ( minus_minus @ A @ X @ Y3 )
= ( bot_bot @ A ) )
= ( ord_less_eq @ A @ X @ Y3 ) ) ) ).
% diff_shunt_var
thf(fact_3199_simp__from__to,axiom,
( ( set_or1337092689740270186AtMost @ int )
= ( ^ [I5: int,J3: int] : ( if @ ( set @ int ) @ ( ord_less @ int @ J3 @ I5 ) @ ( bot_bot @ ( set @ int ) ) @ ( insert @ int @ I5 @ ( set_or1337092689740270186AtMost @ int @ ( plus_plus @ int @ I5 @ ( one_one @ int ) ) @ J3 ) ) ) ) ) ).
% simp_from_to
thf(fact_3200_length__Suc__conv,axiom,
! [A: $tType,Xs: list @ A,N: nat] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( suc @ N ) )
= ( ? [Y6: A,Ys3: list @ A] :
( ( Xs
= ( cons @ A @ Y6 @ Ys3 ) )
& ( ( size_size @ ( list @ A ) @ Ys3 )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_3201_Suc__length__conv,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( ( suc @ N )
= ( size_size @ ( list @ A ) @ Xs ) )
= ( ? [Y6: A,Ys3: list @ A] :
( ( Xs
= ( cons @ A @ Y6 @ Ys3 ) )
& ( ( size_size @ ( list @ A ) @ Ys3 )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_3202_impossible__Cons,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A,X: A] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ A ) @ Ys2 ) )
=> ( Xs
!= ( cons @ A @ X @ Ys2 ) ) ) ).
% impossible_Cons
thf(fact_3203_subset__Compl__self__eq,axiom,
! [A: $tType,A6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ ( uminus_uminus @ ( set @ A ) @ A6 ) )
= ( A6
= ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_Compl_self_eq
thf(fact_3204_set__set__swap,axiom,
! [B: $tType,A: $tType] :
( ( ( heap @ A )
& ( heap @ B ) )
=> ! [R: array @ A,R4: array @ B,X: list @ A,X5: list @ B,H2: heap_ext @ product_unit] :
( ( array_noteq @ A @ B @ R @ R4 )
=> ( ( array_set @ A @ R @ X @ ( array_set @ B @ R4 @ X5 @ H2 ) )
= ( array_set @ B @ R4 @ X5 @ ( array_set @ A @ R @ X @ H2 ) ) ) ) ) ).
% set_set_swap
thf(fact_3205_Suc__le__length__iff,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( ord_less_eq @ nat @ ( suc @ N ) @ ( size_size @ ( list @ A ) @ Xs ) )
= ( ? [X2: A,Ys3: list @ A] :
( ( Xs
= ( cons @ A @ X2 @ Ys3 ) )
& ( ord_less_eq @ nat @ N @ ( size_size @ ( list @ A ) @ Ys3 ) ) ) ) ) ).
% Suc_le_length_iff
thf(fact_3206_Compl__insert,axiom,
! [A: $tType,X: A,A6: set @ A] :
( ( uminus_uminus @ ( set @ A ) @ ( insert @ A @ X @ A6 ) )
= ( minus_minus @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A6 ) @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Compl_insert
thf(fact_3207_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_3208_nth__Cons_H,axiom,
! [A: $tType,N: nat,X: A,Xs: list @ A] :
( ( ( N
= ( zero_zero @ nat ) )
=> ( ( nth @ A @ ( cons @ A @ X @ Xs ) @ N )
= X ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ( ( nth @ A @ ( cons @ A @ X @ Xs ) @ N )
= ( nth @ A @ Xs @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ) ).
% nth_Cons'
thf(fact_3209_atLeastAtMostPlus1__int__conv,axiom,
! [M: int,N: int] :
( ( ord_less_eq @ int @ M @ ( plus_plus @ int @ ( one_one @ int ) @ N ) )
=> ( ( set_or1337092689740270186AtMost @ int @ M @ ( plus_plus @ int @ ( one_one @ int ) @ N ) )
= ( insert @ int @ ( plus_plus @ int @ ( one_one @ int ) @ N ) @ ( set_or1337092689740270186AtMost @ int @ M @ N ) ) ) ) ).
% atLeastAtMostPlus1_int_conv
thf(fact_3210_psubset__insert__iff,axiom,
! [A: $tType,A6: set @ A,X: A,B7: set @ A] :
( ( ord_less @ ( set @ A ) @ A6 @ ( insert @ A @ X @ B7 ) )
= ( ( ( member @ A @ X @ B7 )
=> ( ord_less @ ( set @ A ) @ A6 @ B7 ) )
& ( ~ ( member @ A @ X @ B7 )
=> ( ( ( member @ A @ X @ A6 )
=> ( ord_less @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ B7 ) )
& ( ~ ( member @ A @ X @ A6 )
=> ( ord_less_eq @ ( set @ A ) @ A6 @ B7 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_3211_nth__non__equal__first__eq,axiom,
! [A: $tType,X: A,Y3: A,Xs: list @ A,N: nat] :
( ( X != Y3 )
=> ( ( ( nth @ A @ ( cons @ A @ X @ Xs ) @ N )
= Y3 )
= ( ( ( nth @ A @ Xs @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) )
= Y3 )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% nth_non_equal_first_eq
thf(fact_3212_horner__sum__simps_I2_J,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_0 @ A )
=> ! [F: B > A,A3: A,X: B,Xs: list @ B] :
( ( groups4207007520872428315er_sum @ B @ A @ F @ A3 @ ( cons @ B @ X @ Xs ) )
= ( plus_plus @ A @ ( F @ X ) @ ( times_times @ A @ A3 @ ( groups4207007520872428315er_sum @ B @ A @ F @ A3 @ Xs ) ) ) ) ) ).
% horner_sum_simps(2)
thf(fact_3213_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_3214_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_3215_divides__aux__eq,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [Q4: A,R: A] :
( ( unique5940410009612947441es_aux @ A @ ( product_Pair @ A @ A @ Q4 @ R ) )
= ( R
= ( zero_zero @ A ) ) ) ) ).
% divides_aux_eq
thf(fact_3216_product__nth,axiom,
! [A: $tType,B: $tType,N: nat,Xs: list @ A,Ys2: list @ B] :
( ( ord_less @ nat @ N @ ( times_times @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ B ) @ Ys2 ) ) )
=> ( ( nth @ ( product_prod @ A @ B ) @ ( product @ A @ B @ Xs @ Ys2 ) @ N )
= ( product_Pair @ A @ B @ ( nth @ A @ Xs @ ( 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_3217_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_3218_set__decode__zero,axiom,
( ( nat_set_decode @ ( zero_zero @ nat ) )
= ( bot_bot @ ( set @ nat ) ) ) ).
% set_decode_zero
thf(fact_3219_length__product,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys2: list @ B] :
( ( size_size @ ( list @ ( product_prod @ A @ B ) ) @ ( product @ A @ B @ Xs @ Ys2 ) )
= ( times_times @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ B ) @ Ys2 ) ) ) ).
% length_product
thf(fact_3220_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_3221_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_3222_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_3223_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_3224_bot__nat__def,axiom,
( ( bot_bot @ nat )
= ( zero_zero @ nat ) ) ).
% bot_nat_def
thf(fact_3225_divmod__divmod__step,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ( ( unique8689654367752047608divmod @ A )
= ( ^ [M2: num,N3: num] : ( if @ ( product_prod @ A @ A ) @ ( ord_less @ num @ M2 @ N3 ) @ ( product_Pair @ A @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ M2 ) ) @ ( unique1321980374590559556d_step @ A @ N3 @ ( unique8689654367752047608divmod @ A @ M2 @ ( bit0 @ N3 ) ) ) ) ) ) ) ).
% divmod_divmod_step
thf(fact_3226_minus__one__div__numeral,axiom,
! [N: num] :
( ( divide_divide @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( numeral_numeral @ int @ N ) )
= ( uminus_uminus @ int @ ( adjust_div @ ( unique8689654367752047608divmod @ int @ one2 @ N ) ) ) ) ).
% minus_one_div_numeral
thf(fact_3227_one__div__minus__numeral,axiom,
! [N: num] :
( ( divide_divide @ int @ ( one_one @ int ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
= ( uminus_uminus @ int @ ( adjust_div @ ( unique8689654367752047608divmod @ int @ one2 @ N ) ) ) ) ).
% one_div_minus_numeral
thf(fact_3228_neg__eucl__rel__int__mult__2,axiom,
! [B2: int,A3: int,Q4: 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 @ Q4 @ 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 @ Q4 @ ( minus_minus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ R ) @ ( one_one @ int ) ) ) ) ) ) ).
% neg_eucl_rel_int_mult_2
thf(fact_3229_Divides_Oadjust__div__eq,axiom,
! [Q4: int,R: int] :
( ( adjust_div @ ( product_Pair @ int @ int @ Q4 @ R ) )
= ( plus_plus @ int @ Q4
@ ( zero_neq_one_of_bool @ int
@ ( R
!= ( zero_zero @ int ) ) ) ) ) ).
% Divides.adjust_div_eq
thf(fact_3230_minus__numeral__div__numeral,axiom,
! [M: num,N: num] :
( ( divide_divide @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( numeral_numeral @ int @ N ) )
= ( uminus_uminus @ int @ ( adjust_div @ ( unique8689654367752047608divmod @ int @ M @ N ) ) ) ) ).
% minus_numeral_div_numeral
thf(fact_3231_numeral__div__minus__numeral,axiom,
! [M: num,N: num] :
( ( divide_divide @ int @ ( numeral_numeral @ int @ M ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
= ( uminus_uminus @ int @ ( adjust_div @ ( unique8689654367752047608divmod @ int @ M @ N ) ) ) ) ).
% numeral_div_minus_numeral
thf(fact_3232_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_3233_eucl__rel__int__dividesI,axiom,
! [L: int,K: int,Q4: int] :
( ( L
!= ( zero_zero @ int ) )
=> ( ( K
= ( times_times @ int @ Q4 @ L ) )
=> ( eucl_rel_int @ K @ L @ ( product_Pair @ int @ int @ Q4 @ ( zero_zero @ int ) ) ) ) ) ).
% eucl_rel_int_dividesI
thf(fact_3234_zminus1__lemma,axiom,
! [A3: int,B2: int,Q4: int,R: int] :
( ( eucl_rel_int @ A3 @ B2 @ ( product_Pair @ int @ int @ Q4 @ 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 @ Q4 )
@ ( minus_minus @ int @ ( uminus_uminus @ int @ Q4 ) @ ( one_one @ int ) ) )
@ ( if @ int
@ ( R
= ( zero_zero @ int ) )
@ ( zero_zero @ int )
@ ( minus_minus @ int @ B2 @ R ) ) ) ) ) ) ).
% zminus1_lemma
thf(fact_3235_eucl__rel__int__iff,axiom,
! [K: int,L: int,Q4: int,R: int] :
( ( eucl_rel_int @ K @ L @ ( product_Pair @ int @ int @ Q4 @ R ) )
= ( ( K
= ( plus_plus @ int @ ( times_times @ int @ L @ Q4 ) @ 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 ) )
=> ( Q4
= ( zero_zero @ int ) ) ) ) ) ) ) ).
% eucl_rel_int_iff
thf(fact_3236_eucl__rel__int__remainderI,axiom,
! [R: int,L: int,K: int,Q4: 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 @ Q4 @ L ) @ R ) )
=> ( eucl_rel_int @ K @ L @ ( product_Pair @ int @ int @ Q4 @ R ) ) ) ) ) ).
% eucl_rel_int_remainderI
thf(fact_3237_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 ) ) )
=> ( ! [Q5: int] :
( ( A32
= ( product_Pair @ int @ int @ Q5 @ ( zero_zero @ int ) ) )
=> ( ( A22
!= ( zero_zero @ int ) )
=> ( A1
!= ( times_times @ int @ Q5 @ A22 ) ) ) )
=> ~ ! [R3: int,Q5: int] :
( ( A32
= ( product_Pair @ int @ int @ Q5 @ R3 ) )
=> ( ( ( sgn_sgn @ int @ R3 )
= ( sgn_sgn @ int @ A22 ) )
=> ( ( ord_less @ int @ ( abs_abs @ int @ R3 ) @ ( abs_abs @ int @ A22 ) )
=> ( A1
!= ( plus_plus @ int @ ( times_times @ int @ Q5 @ A22 ) @ R3 ) ) ) ) ) ) ) ) ).
% eucl_rel_int.cases
thf(fact_3238_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 ) ) )
| ? [L3: int,K3: int,Q3: int] :
( ( A12 = K3 )
& ( A23 = L3 )
& ( A33
= ( product_Pair @ int @ int @ Q3 @ ( zero_zero @ int ) ) )
& ( L3
!= ( zero_zero @ int ) )
& ( K3
= ( times_times @ int @ Q3 @ L3 ) ) )
| ? [R5: int,L3: int,K3: int,Q3: int] :
( ( A12 = K3 )
& ( A23 = L3 )
& ( A33
= ( product_Pair @ int @ int @ Q3 @ R5 ) )
& ( ( sgn_sgn @ int @ R5 )
= ( sgn_sgn @ int @ L3 ) )
& ( ord_less @ int @ ( abs_abs @ int @ R5 ) @ ( abs_abs @ int @ L3 ) )
& ( K3
= ( plus_plus @ int @ ( times_times @ int @ Q3 @ L3 ) @ R5 ) ) ) ) ) ) ).
% eucl_rel_int.simps
thf(fact_3239_pos__eucl__rel__int__mult__2,axiom,
! [B2: int,A3: int,Q4: int,R: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ B2 )
=> ( ( eucl_rel_int @ A3 @ B2 @ ( product_Pair @ int @ int @ Q4 @ 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 @ Q4 @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ R ) ) ) ) ) ) ).
% pos_eucl_rel_int_mult_2
thf(fact_3240_and__int_Opelims,axiom,
! [X: int,Xa3: int,Y3: int] :
( ( ( bit_se5824344872417868541ns_and @ int @ X @ Xa3 )
= Y3 )
=> ( ( accp @ ( product_prod @ int @ int ) @ bit_and_int_rel @ ( product_Pair @ int @ int @ X @ Xa3 ) )
=> ~ ( ( ( ( ( member @ int @ X @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
& ( member @ int @ Xa3 @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ 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 ) ) @ Xa3 ) ) ) ) ) )
& ( ~ ( ( member @ int @ X @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
& ( member @ int @ Xa3 @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ 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 ) ) @ Xa3 ) ) )
@ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( divide_divide @ int @ X @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ Xa3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ int @ int ) @ bit_and_int_rel @ ( product_Pair @ int @ int @ X @ Xa3 ) ) ) ) ) ).
% and_int.pelims
thf(fact_3241_and__int_Opsimps,axiom,
! [K: int,L: int] :
( ( accp @ ( product_prod @ int @ int ) @ bit_and_int_rel @ ( product_Pair @ int @ int @ K @ L ) )
=> ( ( ( ( member @ int @ K @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
& ( member @ int @ L @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
=> ( ( bit_se5824344872417868541ns_and @ int @ K @ L )
= ( uminus_uminus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K )
& ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L ) ) ) ) ) )
& ( ~ ( ( member @ int @ K @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
& ( member @ int @ L @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
=> ( ( bit_se5824344872417868541ns_and @ int @ K @ L )
= ( plus_plus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K )
& ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L ) ) )
@ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( divide_divide @ int @ K @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% and_int.psimps
thf(fact_3242_length__Cons,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( size_size @ ( list @ A ) @ ( cons @ A @ X @ Xs ) )
= ( suc @ ( size_size @ ( list @ A ) @ Xs ) ) ) ).
% length_Cons
thf(fact_3243_normalize__negative,axiom,
! [Q4: int,P5: int] :
( ( ord_less @ int @ Q4 @ ( zero_zero @ int ) )
=> ( ( normalize @ ( product_Pair @ int @ int @ P5 @ Q4 ) )
= ( normalize @ ( product_Pair @ int @ int @ ( uminus_uminus @ int @ P5 ) @ ( uminus_uminus @ int @ Q4 ) ) ) ) ) ).
% normalize_negative
thf(fact_3244_upto__aux__rec,axiom,
( upto_aux
= ( ^ [I5: int,J3: int,Js: list @ int] : ( if @ ( list @ int ) @ ( ord_less @ int @ J3 @ I5 ) @ Js @ ( upto_aux @ I5 @ ( minus_minus @ int @ J3 @ ( one_one @ int ) ) @ ( cons @ int @ J3 @ Js ) ) ) ) ) ).
% upto_aux_rec
thf(fact_3245_normalize__denom__zero,axiom,
! [P5: int] :
( ( normalize @ ( product_Pair @ int @ int @ P5 @ ( zero_zero @ int ) ) )
= ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) ) ) ).
% normalize_denom_zero
thf(fact_3246_normalize__denom__pos,axiom,
! [R: product_prod @ int @ int,P5: int,Q4: int] :
( ( ( normalize @ R )
= ( product_Pair @ int @ int @ P5 @ Q4 ) )
=> ( ord_less @ int @ ( zero_zero @ int ) @ Q4 ) ) ).
% normalize_denom_pos
thf(fact_3247_normalize__crossproduct,axiom,
! [Q4: int,S: int,P5: int,R: int] :
( ( Q4
!= ( zero_zero @ int ) )
=> ( ( S
!= ( zero_zero @ int ) )
=> ( ( ( normalize @ ( product_Pair @ int @ int @ P5 @ Q4 ) )
= ( normalize @ ( product_Pair @ int @ int @ R @ S ) ) )
=> ( ( times_times @ int @ P5 @ S )
= ( times_times @ int @ R @ Q4 ) ) ) ) ) ).
% normalize_crossproduct
thf(fact_3248_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,L2: int] :
( ( accp @ ( product_prod @ int @ int ) @ bit_and_int_rel @ ( product_Pair @ int @ int @ K2 @ L2 ) )
=> ( ( ~ ( ( member @ int @ K2 @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
& ( member @ int @ L2 @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
=> ( P @ ( divide_divide @ int @ K2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) )
=> ( P @ K2 @ L2 ) ) )
=> ( P @ A0 @ A1 ) ) ) ).
% and_int.pinduct
thf(fact_3249_upto_Opinduct,axiom,
! [A0: int,A1: int,P: int > int > $o] :
( ( accp @ ( product_prod @ int @ int ) @ upto_rel @ ( product_Pair @ int @ int @ A0 @ A1 ) )
=> ( ! [I4: int,J2: int] :
( ( accp @ ( product_prod @ int @ int ) @ upto_rel @ ( product_Pair @ int @ int @ I4 @ J2 ) )
=> ( ( ( ord_less_eq @ int @ I4 @ J2 )
=> ( P @ ( plus_plus @ int @ I4 @ ( one_one @ int ) ) @ J2 ) )
=> ( P @ I4 @ J2 ) ) )
=> ( P @ A0 @ A1 ) ) ) ).
% upto.pinduct
thf(fact_3250_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_3251_bezw__0,axiom,
! [X: nat] :
( ( bezw @ X @ ( zero_zero @ nat ) )
= ( product_Pair @ int @ int @ ( one_one @ int ) @ ( zero_zero @ int ) ) ) ).
% bezw_0
thf(fact_3252_nth__zip,axiom,
! [A: $tType,B: $tType,I: nat,Xs: list @ A,Ys2: list @ B] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ B ) @ Ys2 ) )
=> ( ( nth @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys2 ) @ I )
= ( product_Pair @ A @ B @ ( nth @ A @ Xs @ I ) @ ( nth @ B @ Ys2 @ I ) ) ) ) ) ).
% nth_zip
thf(fact_3253_Gcd__0__iff,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A6: set @ A] :
( ( ( gcd_Gcd @ A @ A6 )
= ( zero_zero @ A ) )
= ( ord_less_eq @ ( set @ A ) @ A6 @ ( insert @ A @ ( zero_zero @ A ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% Gcd_0_iff
thf(fact_3254_Gcd__empty,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ( ( gcd_Gcd @ A @ ( bot_bot @ ( set @ A ) ) )
= ( zero_zero @ A ) ) ) ).
% Gcd_empty
thf(fact_3255_divmod__integer_H__def,axiom,
( ( unique8689654367752047608divmod @ code_integer )
= ( ^ [M2: num,N3: num] : ( product_Pair @ code_integer @ code_integer @ ( divide_divide @ code_integer @ ( numeral_numeral @ code_integer @ M2 ) @ ( numeral_numeral @ code_integer @ N3 ) ) @ ( modulo_modulo @ code_integer @ ( numeral_numeral @ code_integer @ M2 ) @ ( numeral_numeral @ code_integer @ N3 ) ) ) ) ) ).
% divmod_integer'_def
thf(fact_3256_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_3257_Gcd__1,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A6: set @ A] :
( ( member @ A @ ( one_one @ A ) @ A6 )
=> ( ( gcd_Gcd @ A @ A6 )
= ( one_one @ A ) ) ) ) ).
% Gcd_1
thf(fact_3258_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_3259_Gcd__eq__1__I,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A3: A,A6: set @ A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( member @ A @ A3 @ A6 )
=> ( ( gcd_Gcd @ A @ A6 )
= ( one_one @ A ) ) ) ) ) ).
% Gcd_eq_1_I
thf(fact_3260_Code__Numeral_Opositive__def,axiom,
( code_positive
= ( numeral_numeral @ code_integer ) ) ).
% Code_Numeral.positive_def
thf(fact_3261_integer__of__num__triv_I2_J,axiom,
( ( code_integer_of_num @ ( bit0 @ one2 ) )
= ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) ).
% integer_of_num_triv(2)
thf(fact_3262_card__atLeastAtMost__int,axiom,
! [L: int,U: int] :
( ( finite_card @ int @ ( set_or1337092689740270186AtMost @ int @ L @ U ) )
= ( nat2 @ ( plus_plus @ int @ ( minus_minus @ int @ U @ L ) @ ( one_one @ int ) ) ) ) ).
% card_atLeastAtMost_int
thf(fact_3263_possible__bit__def,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ( ( bit_se6407376104438227557le_bit @ A )
= ( ^ [Tyrep: itself @ A,N3: nat] :
( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N3 )
!= ( zero_zero @ A ) ) ) ) ) ).
% possible_bit_def
thf(fact_3264_abstract__boolean__algebra__sym__diff_Ointro,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,Xor: A > A > A] :
( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One2 )
=> ( ( boolea5476839437570043046axioms @ A @ Conj @ Disj @ Compl @ Xor )
=> ( boolea3799213064322606851m_diff @ A @ Conj @ Disj @ Compl @ Zero @ One2 @ Xor ) ) ) ).
% abstract_boolean_algebra_sym_diff.intro
thf(fact_3265_abstract__boolean__algebra__sym__diff__axioms_Ointro,axiom,
! [A: $tType,Xor: A > A > A,Disj: A > A > A,Conj: A > A > A,Compl: A > A] :
( ! [X3: A,Y5: A] :
( ( Xor @ X3 @ Y5 )
= ( Disj @ ( Conj @ X3 @ ( Compl @ Y5 ) ) @ ( Conj @ ( Compl @ X3 ) @ Y5 ) ) )
=> ( boolea5476839437570043046axioms @ A @ Conj @ Disj @ Compl @ Xor ) ) ).
% abstract_boolean_algebra_sym_diff_axioms.intro
thf(fact_3266_abstract__boolean__algebra__sym__diff__axioms__def,axiom,
! [A: $tType] :
( ( boolea5476839437570043046axioms @ A )
= ( ^ [Conj2: A > A > A,Disj2: A > A > A,Compl2: A > A,Xor2: A > A > A] :
! [X2: A,Y6: A] :
( ( Xor2 @ X2 @ Y6 )
= ( Disj2 @ ( Conj2 @ X2 @ ( Compl2 @ Y6 ) ) @ ( Conj2 @ ( Compl2 @ X2 ) @ Y6 ) ) ) ) ) ).
% abstract_boolean_algebra_sym_diff_axioms_def
thf(fact_3267_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_3268_possible__bit__less__imp,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [Tyrep2: itself @ A,I: nat,J: nat] :
( ( bit_se6407376104438227557le_bit @ A @ Tyrep2 @ I )
=> ( ( ord_less_eq @ nat @ J @ I )
=> ( bit_se6407376104438227557le_bit @ A @ Tyrep2 @ J ) ) ) ) ).
% possible_bit_less_imp
thf(fact_3269_integer__of__num__triv_I1_J,axiom,
( ( code_integer_of_num @ one2 )
= ( one_one @ code_integer ) ) ).
% integer_of_num_triv(1)
thf(fact_3270_integer__of__num__def,axiom,
( code_integer_of_num
= ( numeral_numeral @ code_integer ) ) ).
% integer_of_num_def
thf(fact_3271_abstract__boolean__algebra__sym__diff_Oaxioms_I2_J,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A,Xor: A > A > A] :
( ( boolea3799213064322606851m_diff @ A @ Conj @ Disj @ Compl @ Zero @ One2 @ Xor )
=> ( boolea5476839437570043046axioms @ A @ Conj @ Disj @ Compl @ Xor ) ) ).
% abstract_boolean_algebra_sym_diff.axioms(2)
thf(fact_3272_abstract__boolean__algebra__sym__diff__def,axiom,
! [A: $tType] :
( ( boolea3799213064322606851m_diff @ A )
= ( ^ [Conj2: A > A > A,Disj2: A > A > A,Compl2: A > A,Zero2: A,One: A,Xor2: A > A > A] :
( ( boolea2506097494486148201lgebra @ A @ Conj2 @ Disj2 @ Compl2 @ Zero2 @ One )
& ( boolea5476839437570043046axioms @ A @ Conj2 @ Disj2 @ Compl2 @ Xor2 ) ) ) ) ).
% abstract_boolean_algebra_sym_diff_def
thf(fact_3273_card__Diff__insert,axiom,
! [A: $tType,A3: A,A6: set @ A,B7: set @ A] :
( ( member @ A @ A3 @ A6 )
=> ( ~ ( member @ A @ A3 @ B7 )
=> ( ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ A3 @ B7 ) ) )
= ( minus_minus @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A6 @ B7 ) ) @ ( one_one @ nat ) ) ) ) ) ).
% card_Diff_insert
thf(fact_3274_card_Oempty,axiom,
! [A: $tType] :
( ( finite_card @ A @ ( bot_bot @ ( set @ A ) ) )
= ( zero_zero @ nat ) ) ).
% card.empty
thf(fact_3275_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 @ ( insert @ A @ X @ Y3 ) ) @ N ) ) ) ).
% card_insert_le_m1
thf(fact_3276_card__Diff__singleton__if,axiom,
! [A: $tType,X: A,A6: set @ A] :
( ( ( member @ A @ X @ A6 )
=> ( ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( minus_minus @ nat @ ( finite_card @ A @ A6 ) @ ( one_one @ nat ) ) ) )
& ( ~ ( member @ A @ X @ A6 )
=> ( ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( finite_card @ A @ A6 ) ) ) ) ).
% card_Diff_singleton_if
thf(fact_3277_card__Diff__singleton,axiom,
! [A: $tType,X: A,A6: set @ A] :
( ( member @ A @ X @ A6 )
=> ( ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( minus_minus @ nat @ ( finite_card @ A @ A6 ) @ ( one_one @ nat ) ) ) ) ).
% card_Diff_singleton
thf(fact_3278_card__atLeastAtMost,axiom,
! [L: nat,U: nat] :
( ( finite_card @ nat @ ( set_or1337092689740270186AtMost @ nat @ L @ U ) )
= ( minus_minus @ nat @ ( suc @ U ) @ L ) ) ).
% card_atLeastAtMost
thf(fact_3279_sgn__integer__code,axiom,
( ( sgn_sgn @ code_integer )
= ( ^ [K3: code_integer] :
( if @ code_integer
@ ( K3
= ( zero_zero @ code_integer ) )
@ ( zero_zero @ code_integer )
@ ( if @ code_integer @ ( ord_less @ code_integer @ K3 @ ( zero_zero @ code_integer ) ) @ ( uminus_uminus @ code_integer @ ( one_one @ code_integer ) ) @ ( one_one @ code_integer ) ) ) ) ) ).
% sgn_integer_code
thf(fact_3280_card__insert__le,axiom,
! [A: $tType,A6: set @ A,X: A] : ( ord_less_eq @ nat @ ( finite_card @ A @ A6 ) @ ( finite_card @ A @ ( insert @ A @ X @ A6 ) ) ) ).
% card_insert_le
thf(fact_3281_card__1__singletonE,axiom,
! [A: $tType,A6: set @ A] :
( ( ( finite_card @ A @ A6 )
= ( one_one @ nat ) )
=> ~ ! [X3: A] :
( A6
!= ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% card_1_singletonE
thf(fact_3282_card__1__singleton__iff,axiom,
! [A: $tType,A6: set @ A] :
( ( ( finite_card @ A @ A6 )
= ( suc @ ( zero_zero @ nat ) ) )
= ( ? [X2: A] :
( A6
= ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% card_1_singleton_iff
thf(fact_3283_card__eq__SucD,axiom,
! [A: $tType,A6: set @ A,K: nat] :
( ( ( finite_card @ A @ A6 )
= ( suc @ K ) )
=> ? [B5: A,B9: set @ A] :
( ( A6
= ( insert @ A @ B5 @ B9 ) )
& ~ ( member @ A @ B5 @ B9 )
& ( ( finite_card @ A @ B9 )
= K )
& ( ( K
= ( zero_zero @ nat ) )
=> ( B9
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% card_eq_SucD
thf(fact_3284_card__Suc__eq,axiom,
! [A: $tType,A6: set @ A,K: nat] :
( ( ( finite_card @ A @ A6 )
= ( suc @ K ) )
= ( ? [B3: A,B6: set @ A] :
( ( A6
= ( insert @ A @ B3 @ B6 ) )
& ~ ( member @ A @ B3 @ B6 )
& ( ( finite_card @ A @ B6 )
= K )
& ( ( K
= ( zero_zero @ nat ) )
=> ( B6
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% card_Suc_eq
thf(fact_3285_card__Diff1__le,axiom,
! [A: $tType,A6: set @ A,X: A] : ( ord_less_eq @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) @ ( finite_card @ A @ A6 ) ) ).
% card_Diff1_le
thf(fact_3286_drop__bit__exp__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,N: nat] :
( ( bit_se4197421643247451524op_bit @ A @ M @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
= ( times_times @ A
@ ( zero_neq_one_of_bool @ A
@ ( ( ord_less_eq @ nat @ M @ N )
& ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N ) ) )
@ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N @ M ) ) ) ) ) ).
% drop_bit_exp_eq
thf(fact_3287_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_3288_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_3289_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_3290_bit__minus__exp__iff,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [M: nat,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( uminus_uminus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) ) @ N )
= ( ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N )
& ( ord_less_eq @ nat @ M @ N ) ) ) ) ).
% bit_minus_exp_iff
thf(fact_3291_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_3292_plus__integer__code_I1_J,axiom,
! [K: code_integer] :
( ( plus_plus @ code_integer @ K @ ( zero_zero @ code_integer ) )
= K ) ).
% plus_integer_code(1)
thf(fact_3293_plus__integer__code_I2_J,axiom,
! [L: code_integer] :
( ( plus_plus @ code_integer @ ( zero_zero @ code_integer ) @ L )
= L ) ).
% plus_integer_code(2)
thf(fact_3294_less__integer__code_I1_J,axiom,
~ ( ord_less @ code_integer @ ( zero_zero @ code_integer ) @ ( zero_zero @ code_integer ) ) ).
% less_integer_code(1)
thf(fact_3295_abs__integer__code,axiom,
( ( abs_abs @ code_integer )
= ( ^ [K3: code_integer] : ( if @ code_integer @ ( ord_less @ code_integer @ K3 @ ( zero_zero @ code_integer ) ) @ ( uminus_uminus @ code_integer @ K3 ) @ K3 ) ) ) ).
% abs_integer_code
thf(fact_3296_uminus__integer__code_I1_J,axiom,
( ( uminus_uminus @ code_integer @ ( zero_zero @ code_integer ) )
= ( zero_zero @ code_integer ) ) ).
% uminus_integer_code(1)
thf(fact_3297_minus__integer__code_I1_J,axiom,
! [K: code_integer] :
( ( minus_minus @ code_integer @ K @ ( zero_zero @ code_integer ) )
= K ) ).
% minus_integer_code(1)
thf(fact_3298_minus__integer__code_I2_J,axiom,
! [L: code_integer] :
( ( minus_minus @ code_integer @ ( zero_zero @ code_integer ) @ L )
= ( uminus_uminus @ code_integer @ L ) ) ).
% minus_integer_code(2)
thf(fact_3299_bit__mask__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( bit_se2239418461657761734s_mask @ A @ M ) @ N )
= ( ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N )
& ( ord_less @ nat @ N @ M ) ) ) ) ).
% bit_mask_iff
thf(fact_3300_bit__of__nat__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [M: nat,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( semiring_1_of_nat @ A @ M ) @ N )
= ( ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N )
& ( bit_se5641148757651400278ts_bit @ nat @ M @ N ) ) ) ) ).
% bit_of_nat_iff
thf(fact_3301_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_3302_bit__push__bit__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,A3: A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( bit_se4730199178511100633sh_bit @ A @ M @ A3 ) @ N )
= ( ( ord_less_eq @ nat @ M @ N )
& ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N )
& ( bit_se5641148757651400278ts_bit @ A @ A3 @ ( minus_minus @ nat @ N @ M ) ) ) ) ) ).
% bit_push_bit_iff
thf(fact_3303_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_3304_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_3305_CHAR__eq0__iff,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( ( semiri4206861660011772517g_char @ A @ ( type2 @ A ) )
= ( zero_zero @ nat ) )
= ( ! [N3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ( semiring_1_of_nat @ A @ N3 )
!= ( zero_zero @ A ) ) ) ) ) ) ).
% CHAR_eq0_iff
thf(fact_3306_CHAR__eq__posI,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [C2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ C2 )
=> ( ( ( semiring_1_of_nat @ A @ C2 )
= ( zero_zero @ A ) )
=> ( ! [X3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ X3 )
=> ( ( ord_less @ nat @ X3 @ C2 )
=> ( ( semiring_1_of_nat @ A @ X3 )
!= ( zero_zero @ A ) ) ) )
=> ( ( semiri4206861660011772517g_char @ A @ ( type2 @ A ) )
= C2 ) ) ) ) ) ).
% CHAR_eq_posI
thf(fact_3307_CHAR__pos__iff,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( semiri4206861660011772517g_char @ A @ ( type2 @ A ) ) )
= ( ? [N3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
& ( ( semiring_1_of_nat @ A @ N3 )
= ( zero_zero @ A ) ) ) ) ) ) ).
% CHAR_pos_iff
thf(fact_3308_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_3309_CHAR__eqI,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [C2: nat] :
( ( ( semiring_1_of_nat @ A @ C2 )
= ( zero_zero @ A ) )
=> ( ! [X3: nat] :
( ( ( semiring_1_of_nat @ A @ X3 )
= ( zero_zero @ A ) )
=> ( dvd_dvd @ nat @ C2 @ X3 ) )
=> ( ( semiri4206861660011772517g_char @ A @ ( type2 @ A ) )
= C2 ) ) ) ) ).
% CHAR_eqI
thf(fact_3310_CHAR__eq__0,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiri4206861660011772517g_char @ A @ ( type2 @ A ) )
= ( zero_zero @ nat ) ) ) ).
% CHAR_eq_0
thf(fact_3311_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_3312_divmod__integer__def,axiom,
( code_divmod_integer
= ( ^ [K3: code_integer,L3: code_integer] : ( product_Pair @ code_integer @ code_integer @ ( divide_divide @ code_integer @ K3 @ L3 ) @ ( modulo_modulo @ code_integer @ K3 @ L3 ) ) ) ) ).
% divmod_integer_def
thf(fact_3313_divmod__abs__def,axiom,
( code_divmod_abs
= ( ^ [K3: code_integer,L3: code_integer] : ( product_Pair @ code_integer @ code_integer @ ( divide_divide @ code_integer @ ( abs_abs @ code_integer @ K3 ) @ ( abs_abs @ code_integer @ L3 ) ) @ ( modulo_modulo @ code_integer @ ( abs_abs @ code_integer @ K3 ) @ ( abs_abs @ code_integer @ L3 ) ) ) ) ) ).
% divmod_abs_def
thf(fact_3314_card__greaterThanLessThan__int,axiom,
! [L: int,U: int] :
( ( finite_card @ int @ ( set_or5935395276787703475ssThan @ int @ L @ U ) )
= ( nat2 @ ( minus_minus @ int @ U @ ( plus_plus @ int @ L @ ( one_one @ int ) ) ) ) ) ).
% card_greaterThanLessThan_int
thf(fact_3315_nth__enumerate__eq,axiom,
! [A: $tType,M: nat,Xs: list @ A,N: nat] :
( ( ord_less @ nat @ M @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ ( product_prod @ nat @ A ) @ ( enumerate @ A @ N @ Xs ) @ M )
= ( product_Pair @ nat @ A @ ( plus_plus @ nat @ N @ M ) @ ( nth @ A @ Xs @ M ) ) ) ) ).
% nth_enumerate_eq
thf(fact_3316_count__list_Osimps_I2_J,axiom,
! [A: $tType,X: A,Y3: A,Xs: list @ A] :
( ( ( X = Y3 )
=> ( ( count_list @ A @ ( cons @ A @ X @ Xs ) @ Y3 )
= ( plus_plus @ nat @ ( count_list @ A @ Xs @ Y3 ) @ ( one_one @ nat ) ) ) )
& ( ( X != Y3 )
=> ( ( count_list @ A @ ( cons @ A @ X @ Xs ) @ Y3 )
= ( count_list @ A @ Xs @ Y3 ) ) ) ) ).
% count_list.simps(2)
thf(fact_3317_greaterThanLessThan__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [I: A,L: A,U: A] :
( ( member @ A @ I @ ( set_or5935395276787703475ssThan @ A @ L @ U ) )
= ( ( ord_less @ A @ L @ I )
& ( ord_less @ A @ I @ U ) ) ) ) ).
% greaterThanLessThan_iff
thf(fact_3318_greaterThanLessThan__empty,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,K: A] :
( ( ord_less_eq @ A @ L @ K )
=> ( ( set_or5935395276787703475ssThan @ A @ K @ L )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% greaterThanLessThan_empty
thf(fact_3319_greaterThanLessThan__empty__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B2: A] :
( ( ( set_or5935395276787703475ssThan @ A @ A3 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
= ( ord_less_eq @ A @ B2 @ A3 ) ) ) ).
% greaterThanLessThan_empty_iff
thf(fact_3320_greaterThanLessThan__empty__iff2,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B2: A] :
( ( ( bot_bot @ ( set @ A ) )
= ( set_or5935395276787703475ssThan @ A @ A3 @ B2 ) )
= ( ord_less_eq @ A @ B2 @ A3 ) ) ) ).
% greaterThanLessThan_empty_iff2
thf(fact_3321_enumerate__simps_I2_J,axiom,
! [B: $tType,N: nat,X: B,Xs: list @ B] :
( ( enumerate @ B @ N @ ( cons @ B @ X @ Xs ) )
= ( cons @ ( product_prod @ nat @ B ) @ ( product_Pair @ nat @ B @ N @ X ) @ ( enumerate @ B @ ( suc @ N ) @ Xs ) ) ) ).
% enumerate_simps(2)
thf(fact_3322_greaterThanLessThan__subseteq__greaterThanLessThan,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B2: A,C2: A,D2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ A3 @ B2 ) @ ( set_or5935395276787703475ssThan @ A @ C2 @ D2 ) )
= ( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ A3 )
& ( ord_less_eq @ A @ B2 @ D2 ) ) ) ) ) ).
% greaterThanLessThan_subseteq_greaterThanLessThan
thf(fact_3323_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_3324_count__le__length,axiom,
! [A: $tType,Xs: list @ A,X: A] : ( ord_less_eq @ nat @ ( count_list @ A @ Xs @ X ) @ ( size_size @ ( list @ A ) @ Xs ) ) ).
% count_le_length
thf(fact_3325_greaterThanLessThan__subseteq__atLeastAtMost__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B2: A,C2: A,D2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ A3 @ B2 ) @ ( set_or1337092689740270186AtMost @ A @ C2 @ D2 ) )
= ( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ A3 )
& ( ord_less_eq @ A @ B2 @ D2 ) ) ) ) ) ).
% greaterThanLessThan_subseteq_atLeastAtMost_iff
thf(fact_3326_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_3327_sqr_Osimps_I3_J,axiom,
! [N: num] :
( ( sqr @ ( bit1 @ N ) )
= ( bit1 @ ( bit0 @ ( plus_plus @ num @ ( sqr @ N ) @ N ) ) ) ) ).
% sqr.simps(3)
thf(fact_3328_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_3329_prod__decode__aux_Oelims,axiom,
! [X: nat,Xa3: nat,Y3: product_prod @ nat @ nat] :
( ( ( nat_prod_decode_aux @ X @ Xa3 )
= Y3 )
=> ( ( ( ord_less_eq @ nat @ Xa3 @ X )
=> ( Y3
= ( product_Pair @ nat @ nat @ Xa3 @ ( minus_minus @ nat @ X @ Xa3 ) ) ) )
& ( ~ ( ord_less_eq @ nat @ Xa3 @ X )
=> ( Y3
= ( nat_prod_decode_aux @ ( suc @ X ) @ ( minus_minus @ nat @ Xa3 @ ( suc @ X ) ) ) ) ) ) ) ).
% prod_decode_aux.elims
thf(fact_3330_prod__decode__aux_Osimps,axiom,
( nat_prod_decode_aux
= ( ^ [K3: nat,M2: nat] : ( if @ ( product_prod @ nat @ nat ) @ ( ord_less_eq @ nat @ M2 @ K3 ) @ ( product_Pair @ nat @ nat @ M2 @ ( minus_minus @ nat @ K3 @ M2 ) ) @ ( nat_prod_decode_aux @ ( suc @ K3 ) @ ( minus_minus @ nat @ M2 @ ( suc @ K3 ) ) ) ) ) ) ).
% prod_decode_aux.simps
thf(fact_3331_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_3332_nat__of__integer__of__nat,axiom,
! [N: nat] :
( ( code_nat_of_integer @ ( semiring_1_of_nat @ code_integer @ N ) )
= N ) ).
% nat_of_integer_of_nat
thf(fact_3333_of__nat__of__integer,axiom,
! [K: code_integer] :
( ( semiring_1_of_nat @ code_integer @ ( code_nat_of_integer @ K ) )
= ( ord_max @ code_integer @ ( zero_zero @ code_integer ) @ K ) ) ).
% of_nat_of_integer
thf(fact_3334_card__greaterThanLessThan,axiom,
! [L: nat,U: nat] :
( ( finite_card @ nat @ ( set_or5935395276787703475ssThan @ nat @ L @ U ) )
= ( minus_minus @ nat @ U @ ( suc @ L ) ) ) ).
% card_greaterThanLessThan
thf(fact_3335_rat__one__code,axiom,
( ( quotient_of @ ( one_one @ rat ) )
= ( product_Pair @ int @ int @ ( one_one @ int ) @ ( one_one @ int ) ) ) ).
% rat_one_code
thf(fact_3336_rat__zero__code,axiom,
( ( quotient_of @ ( zero_zero @ rat ) )
= ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) ) ) ).
% rat_zero_code
thf(fact_3337_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_3338_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_3339_prod__decode__aux_Ocases,axiom,
! [X: product_prod @ nat @ nat] :
~ ! [K2: nat,M6: nat] :
( X
!= ( product_Pair @ nat @ nat @ K2 @ M6 ) ) ).
% prod_decode_aux.cases
thf(fact_3340_sgn__rat__def,axiom,
( ( sgn_sgn @ rat )
= ( ^ [A5: rat] :
( if @ rat
@ ( A5
= ( zero_zero @ rat ) )
@ ( zero_zero @ rat )
@ ( if @ rat @ ( ord_less @ rat @ ( zero_zero @ rat ) @ A5 ) @ ( one_one @ rat ) @ ( uminus_uminus @ rat @ ( one_one @ rat ) ) ) ) ) ) ).
% sgn_rat_def
thf(fact_3341_diff__rat__def,axiom,
( ( minus_minus @ rat )
= ( ^ [Q3: rat,R5: rat] : ( plus_plus @ rat @ Q3 @ ( uminus_uminus @ rat @ R5 ) ) ) ) ).
% diff_rat_def
thf(fact_3342_abs__rat__def,axiom,
( ( abs_abs @ rat )
= ( ^ [A5: rat] : ( if @ rat @ ( ord_less @ rat @ A5 @ ( zero_zero @ rat ) ) @ ( uminus_uminus @ rat @ A5 ) @ A5 ) ) ) ).
% abs_rat_def
thf(fact_3343_sqr_Osimps_I2_J,axiom,
! [N: num] :
( ( sqr @ ( bit0 @ N ) )
= ( bit0 @ ( bit0 @ ( sqr @ N ) ) ) ) ).
% sqr.simps(2)
thf(fact_3344_sqr_Osimps_I1_J,axiom,
( ( sqr @ one2 )
= one2 ) ).
% sqr.simps(1)
thf(fact_3345_sqr__conv__mult,axiom,
( sqr
= ( ^ [X2: num] : ( times_times @ num @ X2 @ X2 ) ) ) ).
% sqr_conv_mult
thf(fact_3346_quotient__of__denom__pos,axiom,
! [R: rat,P5: int,Q4: int] :
( ( ( quotient_of @ R )
= ( product_Pair @ int @ int @ P5 @ Q4 ) )
=> ( ord_less @ int @ ( zero_zero @ int ) @ Q4 ) ) ).
% quotient_of_denom_pos
thf(fact_3347_numeral__sqr,axiom,
! [A: $tType] :
( ( semiring_numeral @ A )
=> ! [K: num] :
( ( numeral_numeral @ A @ ( sqr @ K ) )
= ( times_times @ A @ ( numeral_numeral @ A @ K ) @ ( numeral_numeral @ A @ K ) ) ) ) ).
% numeral_sqr
thf(fact_3348_quotient__of__int,axiom,
! [A3: int] :
( ( quotient_of @ ( of_int @ A3 ) )
= ( product_Pair @ int @ int @ A3 @ ( one_one @ int ) ) ) ).
% quotient_of_int
thf(fact_3349_pow_Osimps_I3_J,axiom,
! [X: num,Y3: num] :
( ( pow @ X @ ( bit1 @ Y3 ) )
= ( times_times @ num @ ( sqr @ ( pow @ X @ Y3 ) ) @ X ) ) ).
% pow.simps(3)
thf(fact_3350_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_3351_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_3352_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_3353_prod__encode__eq,axiom,
! [X: product_prod @ nat @ nat,Y3: product_prod @ nat @ nat] :
( ( ( nat_prod_encode @ X )
= ( nat_prod_encode @ Y3 ) )
= ( X = Y3 ) ) ).
% prod_encode_eq
thf(fact_3354_less__eq__rat__def,axiom,
( ( ord_less_eq @ rat )
= ( ^ [X2: rat,Y6: rat] :
( ( ord_less @ rat @ X2 @ Y6 )
| ( X2 = Y6 ) ) ) ) ).
% less_eq_rat_def
thf(fact_3355_Frct__code__post_I9_J,axiom,
! [Q4: product_prod @ int @ int] :
( ( uminus_uminus @ rat @ ( uminus_uminus @ rat @ ( frct @ Q4 ) ) )
= ( frct @ Q4 ) ) ).
% Frct_code_post(9)
thf(fact_3356_pow_Osimps_I1_J,axiom,
! [X: num] :
( ( pow @ X @ one2 )
= X ) ).
% pow.simps(1)
thf(fact_3357_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_3358_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_3359_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_3360_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_3361_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_3362_pow_Osimps_I2_J,axiom,
! [X: num,Y3: num] :
( ( pow @ X @ ( bit0 @ Y3 ) )
= ( sqr @ ( pow @ X @ Y3 ) ) ) ).
% pow.simps(2)
thf(fact_3363_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_3364_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_3365_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_3366_rat__sgn__code,axiom,
! [P5: rat] :
( ( quotient_of @ ( sgn_sgn @ rat @ P5 ) )
= ( product_Pair @ int @ int @ ( sgn_sgn @ int @ ( product_fst @ int @ int @ ( quotient_of @ P5 ) ) ) @ ( one_one @ int ) ) ) ).
% rat_sgn_code
thf(fact_3367_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_3368_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_3369_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_3370_fst__divmod__integer,axiom,
! [K: code_integer,L: code_integer] :
( ( product_fst @ code_integer @ code_integer @ ( code_divmod_integer @ K @ L ) )
= ( divide_divide @ code_integer @ K @ L ) ) ).
% fst_divmod_integer
thf(fact_3371_fst__divmod__abs,axiom,
! [K: code_integer,L: code_integer] :
( ( product_fst @ code_integer @ code_integer @ ( code_divmod_abs @ K @ L ) )
= ( divide_divide @ code_integer @ ( abs_abs @ code_integer @ K ) @ ( abs_abs @ code_integer @ L ) ) ) ).
% fst_divmod_abs
thf(fact_3372_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_3373_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_3374_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_3375_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_3376_sorted__list__of__set__greaterThanLessThan,axiom,
! [I: nat,J: nat] :
( ( ord_less @ nat @ ( suc @ I ) @ J )
=> ( ( linord4507533701916653071of_set @ nat @ ( set_or5935395276787703475ssThan @ nat @ I @ J ) )
= ( cons @ nat @ ( suc @ I ) @ ( linord4507533701916653071of_set @ nat @ ( set_or5935395276787703475ssThan @ nat @ ( suc @ I ) @ J ) ) ) ) ) ).
% sorted_list_of_set_greaterThanLessThan
thf(fact_3377_bezw_Osimps,axiom,
( bezw
= ( ^ [X2: nat,Y6: nat] :
( if @ ( product_prod @ int @ int )
@ ( Y6
= ( zero_zero @ nat ) )
@ ( product_Pair @ int @ int @ ( one_one @ int ) @ ( zero_zero @ int ) )
@ ( product_Pair @ int @ int @ ( product_snd @ int @ int @ ( bezw @ Y6 @ ( modulo_modulo @ nat @ X2 @ Y6 ) ) ) @ ( minus_minus @ int @ ( product_fst @ int @ int @ ( bezw @ Y6 @ ( modulo_modulo @ nat @ X2 @ Y6 ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ ( bezw @ Y6 @ ( modulo_modulo @ nat @ X2 @ Y6 ) ) ) @ ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ X2 @ Y6 ) ) ) ) ) ) ) ) ).
% bezw.simps
thf(fact_3378_bezw_Oelims,axiom,
! [X: nat,Xa3: nat,Y3: product_prod @ int @ int] :
( ( ( bezw @ X @ Xa3 )
= Y3 )
=> ( ( ( Xa3
= ( zero_zero @ nat ) )
=> ( Y3
= ( product_Pair @ int @ int @ ( one_one @ int ) @ ( zero_zero @ int ) ) ) )
& ( ( Xa3
!= ( zero_zero @ nat ) )
=> ( Y3
= ( product_Pair @ int @ int @ ( product_snd @ int @ int @ ( bezw @ Xa3 @ ( modulo_modulo @ nat @ X @ Xa3 ) ) ) @ ( minus_minus @ int @ ( product_fst @ int @ int @ ( bezw @ Xa3 @ ( modulo_modulo @ nat @ X @ Xa3 ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ ( bezw @ Xa3 @ ( modulo_modulo @ nat @ X @ Xa3 ) ) ) @ ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ X @ Xa3 ) ) ) ) ) ) ) ) ) ).
% bezw.elims
thf(fact_3379_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_3380_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_3381_listrel__iff__nth,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys2: list @ B,R: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs @ Ys2 ) @ ( listrel @ A @ B @ R ) )
= ( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys2 ) )
& ! [N3: nat] :
( ( ord_less @ nat @ N3 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ ( nth @ A @ Xs @ N3 ) @ ( nth @ B @ Ys2 @ N3 ) ) @ R ) ) ) ) ).
% listrel_iff_nth
thf(fact_3382_greaterThanAtMost__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [I: A,L: A,U: A] :
( ( member @ A @ I @ ( set_or3652927894154168847AtMost @ A @ L @ U ) )
= ( ( ord_less @ A @ L @ I )
& ( ord_less_eq @ A @ I @ U ) ) ) ) ).
% greaterThanAtMost_iff
thf(fact_3383_greaterThanAtMost__empty,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,K: A] :
( ( ord_less_eq @ A @ L @ K )
=> ( ( set_or3652927894154168847AtMost @ A @ K @ L )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% greaterThanAtMost_empty
thf(fact_3384_greaterThanAtMost__empty__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [K: A,L: A] :
( ( ( set_or3652927894154168847AtMost @ A @ K @ L )
= ( bot_bot @ ( set @ A ) ) )
= ( ~ ( ord_less @ A @ K @ L ) ) ) ) ).
% greaterThanAtMost_empty_iff
thf(fact_3385_greaterThanAtMost__empty__iff2,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [K: A,L: A] :
( ( ( bot_bot @ ( set @ A ) )
= ( set_or3652927894154168847AtMost @ A @ K @ L ) )
= ( ~ ( ord_less @ A @ K @ L ) ) ) ) ).
% greaterThanAtMost_empty_iff2
thf(fact_3386_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_3387_Ioc__inj,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A,C2: A,D2: A] :
( ( ( set_or3652927894154168847AtMost @ A @ A3 @ B2 )
= ( set_or3652927894154168847AtMost @ A @ C2 @ D2 ) )
= ( ( ( ord_less_eq @ A @ B2 @ A3 )
& ( ord_less_eq @ A @ D2 @ C2 ) )
| ( ( A3 = C2 )
& ( B2 = D2 ) ) ) ) ) ).
% Ioc_inj
thf(fact_3388_atLeastSucAtMost__greaterThanAtMost,axiom,
! [L: nat,U: nat] :
( ( set_or1337092689740270186AtMost @ nat @ ( suc @ L ) @ U )
= ( set_or3652927894154168847AtMost @ nat @ L @ U ) ) ).
% atLeastSucAtMost_greaterThanAtMost
thf(fact_3389_Ioc__subset__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A,C2: A,D2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ A3 @ B2 ) @ ( set_or3652927894154168847AtMost @ A @ C2 @ D2 ) )
= ( ( ord_less_eq @ A @ B2 @ A3 )
| ( ( ord_less_eq @ A @ C2 @ A3 )
& ( ord_less_eq @ A @ B2 @ D2 ) ) ) ) ) ).
% Ioc_subset_iff
thf(fact_3390_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_3391_quotient__of__denom__pos_H,axiom,
! [R: rat] : ( ord_less @ int @ ( zero_zero @ int ) @ ( product_snd @ int @ int @ ( quotient_of @ R ) ) ) ).
% quotient_of_denom_pos'
thf(fact_3392_sorted__list__of__set__greaterThanAtMost,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq @ nat @ ( suc @ I ) @ J )
=> ( ( linord4507533701916653071of_set @ nat @ ( set_or3652927894154168847AtMost @ nat @ I @ J ) )
= ( cons @ nat @ ( suc @ I ) @ ( linord4507533701916653071of_set @ nat @ ( set_or3652927894154168847AtMost @ nat @ ( suc @ I ) @ J ) ) ) ) ) ).
% sorted_list_of_set_greaterThanAtMost
thf(fact_3393_greaterThanAtMost__subseteq__atLeastAtMost__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B2: A,C2: A,D2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ A3 @ B2 ) @ ( set_or1337092689740270186AtMost @ A @ C2 @ D2 ) )
= ( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ A3 )
& ( ord_less_eq @ A @ B2 @ D2 ) ) ) ) ) ).
% greaterThanAtMost_subseteq_atLeastAtMost_iff
thf(fact_3394_greaterThanLessThan__subseteq__greaterThanAtMost__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B2: A,C2: A,D2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ A3 @ B2 ) @ ( set_or3652927894154168847AtMost @ A @ C2 @ D2 ) )
= ( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ A3 )
& ( ord_less_eq @ A @ B2 @ D2 ) ) ) ) ) ).
% greaterThanLessThan_subseteq_greaterThanAtMost_iff
thf(fact_3395_minus__one__mod__numeral,axiom,
! [N: num] :
( ( modulo_modulo @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( numeral_numeral @ int @ N ) )
= ( adjust_mod @ ( numeral_numeral @ int @ N ) @ ( product_snd @ int @ int @ ( unique8689654367752047608divmod @ int @ one2 @ N ) ) ) ) ).
% minus_one_mod_numeral
thf(fact_3396_one__mod__minus__numeral,axiom,
! [N: num] :
( ( modulo_modulo @ int @ ( one_one @ int ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
= ( uminus_uminus @ int @ ( adjust_mod @ ( numeral_numeral @ int @ N ) @ ( product_snd @ int @ int @ ( unique8689654367752047608divmod @ int @ one2 @ N ) ) ) ) ) ).
% one_mod_minus_numeral
thf(fact_3397_numeral__mod__minus__numeral,axiom,
! [M: num,N: num] :
( ( modulo_modulo @ int @ ( numeral_numeral @ int @ M ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
= ( uminus_uminus @ int @ ( adjust_mod @ ( numeral_numeral @ int @ N ) @ ( product_snd @ int @ int @ ( unique8689654367752047608divmod @ int @ M @ N ) ) ) ) ) ).
% numeral_mod_minus_numeral
thf(fact_3398_minus__numeral__mod__numeral,axiom,
! [M: num,N: num] :
( ( modulo_modulo @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( numeral_numeral @ int @ N ) )
= ( adjust_mod @ ( numeral_numeral @ int @ N ) @ ( product_snd @ int @ int @ ( unique8689654367752047608divmod @ int @ M @ N ) ) ) ) ).
% minus_numeral_mod_numeral
thf(fact_3399_size__prod__simp,axiom,
! [B: $tType,A: $tType] :
( ( basic_BNF_size_prod @ A @ B )
= ( ^ [F2: A > nat,G2: B > nat,P6: product_prod @ A @ B] : ( plus_plus @ nat @ ( plus_plus @ nat @ ( F2 @ ( product_fst @ A @ B @ P6 ) ) @ ( G2 @ ( product_snd @ A @ B @ P6 ) ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% size_prod_simp
thf(fact_3400_snd__divmod__integer,axiom,
! [K: code_integer,L: code_integer] :
( ( product_snd @ code_integer @ code_integer @ ( code_divmod_integer @ K @ L ) )
= ( modulo_modulo @ code_integer @ K @ L ) ) ).
% snd_divmod_integer
thf(fact_3401_snd__divmod__abs,axiom,
! [K: code_integer,L: code_integer] :
( ( product_snd @ code_integer @ code_integer @ ( code_divmod_abs @ K @ L ) )
= ( modulo_modulo @ code_integer @ ( abs_abs @ code_integer @ K ) @ ( abs_abs @ code_integer @ L ) ) ) ).
% snd_divmod_abs
thf(fact_3402_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_3403_atLeastPlusOneAtMost__greaterThanAtMost__int,axiom,
! [L: int,U: int] :
( ( set_or1337092689740270186AtMost @ int @ ( plus_plus @ int @ L @ ( one_one @ int ) ) @ U )
= ( set_or3652927894154168847AtMost @ int @ L @ U ) ) ).
% atLeastPlusOneAtMost_greaterThanAtMost_int
thf(fact_3404_Divides_Oadjust__mod__def,axiom,
( adjust_mod
= ( ^ [L3: int,R5: int] :
( if @ int
@ ( R5
= ( zero_zero @ int ) )
@ ( zero_zero @ int )
@ ( minus_minus @ int @ L3 @ R5 ) ) ) ) ).
% Divides.adjust_mod_def
thf(fact_3405_bezw_Opelims,axiom,
! [X: nat,Xa3: nat,Y3: product_prod @ int @ int] :
( ( ( bezw @ X @ Xa3 )
= Y3 )
=> ( ( accp @ ( product_prod @ nat @ nat ) @ bezw_rel @ ( product_Pair @ nat @ nat @ X @ Xa3 ) )
=> ~ ( ( ( ( Xa3
= ( zero_zero @ nat ) )
=> ( Y3
= ( product_Pair @ int @ int @ ( one_one @ int ) @ ( zero_zero @ int ) ) ) )
& ( ( Xa3
!= ( zero_zero @ nat ) )
=> ( Y3
= ( product_Pair @ int @ int @ ( product_snd @ int @ int @ ( bezw @ Xa3 @ ( modulo_modulo @ nat @ X @ Xa3 ) ) ) @ ( minus_minus @ int @ ( product_fst @ int @ int @ ( bezw @ Xa3 @ ( modulo_modulo @ nat @ X @ Xa3 ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ ( bezw @ Xa3 @ ( modulo_modulo @ nat @ X @ Xa3 ) ) ) @ ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ X @ Xa3 ) ) ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ nat @ nat ) @ bezw_rel @ ( product_Pair @ nat @ nat @ X @ Xa3 ) ) ) ) ) ).
% bezw.pelims
thf(fact_3406_in__set__enumerate__eq,axiom,
! [A: $tType,P5: product_prod @ nat @ A,N: nat,Xs: list @ A] :
( ( member @ ( product_prod @ nat @ A ) @ P5 @ ( set2 @ ( product_prod @ nat @ A ) @ ( enumerate @ A @ N @ Xs ) ) )
= ( ( ord_less_eq @ nat @ N @ ( product_fst @ nat @ A @ P5 ) )
& ( ord_less @ nat @ ( product_fst @ nat @ A @ P5 ) @ ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N ) )
& ( ( nth @ A @ Xs @ ( minus_minus @ nat @ ( product_fst @ nat @ A @ P5 ) @ N ) )
= ( product_snd @ nat @ A @ P5 ) ) ) ) ).
% in_set_enumerate_eq
thf(fact_3407_list__encode_Osimps_I2_J,axiom,
! [X: nat,Xs: list @ nat] :
( ( nat_list_encode @ ( cons @ nat @ X @ Xs ) )
= ( suc @ ( nat_prod_encode @ ( product_Pair @ nat @ nat @ X @ ( nat_list_encode @ Xs ) ) ) ) ) ).
% list_encode.simps(2)
thf(fact_3408_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_3409_in__set__zip,axiom,
! [B: $tType,A: $tType,P5: product_prod @ A @ B,Xs: list @ A,Ys2: list @ B] :
( ( member @ ( product_prod @ A @ B ) @ P5 @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys2 ) ) )
= ( ? [N3: nat] :
( ( ( nth @ A @ Xs @ N3 )
= ( product_fst @ A @ B @ P5 ) )
& ( ( nth @ B @ Ys2 @ N3 )
= ( product_snd @ A @ B @ P5 ) )
& ( ord_less @ nat @ N3 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( ord_less @ nat @ N3 @ ( size_size @ ( list @ B ) @ Ys2 ) ) ) ) ) ).
% in_set_zip
thf(fact_3410_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_3411_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_3412_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_3413_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_3414_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_3415_gcd__0__left__nat,axiom,
! [X: nat] :
( ( gcd_gcd @ nat @ ( zero_zero @ nat ) @ X )
= X ) ).
% gcd_0_left_nat
thf(fact_3416_gcd__0__nat,axiom,
! [X: nat] :
( ( gcd_gcd @ nat @ X @ ( zero_zero @ nat ) )
= X ) ).
% gcd_0_nat
thf(fact_3417_gcd__nat_Oright__neutral,axiom,
! [A3: nat] :
( ( gcd_gcd @ nat @ A3 @ ( zero_zero @ nat ) )
= A3 ) ).
% gcd_nat.right_neutral
thf(fact_3418_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_3419_gcd__nat_Oleft__neutral,axiom,
! [A3: nat] :
( ( gcd_gcd @ nat @ ( zero_zero @ nat ) @ A3 )
= A3 ) ).
% gcd_nat.left_neutral
thf(fact_3420_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_3421_gcd__1__nat,axiom,
! [M: nat] :
( ( gcd_gcd @ nat @ M @ ( one_one @ nat ) )
= ( one_one @ nat ) ) ).
% gcd_1_nat
thf(fact_3422_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_3423_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_3424_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_3425_gcd__Suc__0,axiom,
! [M: nat] :
( ( gcd_gcd @ nat @ M @ ( suc @ ( zero_zero @ nat ) ) )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% gcd_Suc_0
thf(fact_3426_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_3427_count__notin,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ~ ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( count_list @ A @ Xs @ X )
= ( zero_zero @ nat ) ) ) ).
% count_notin
thf(fact_3428_list__encode__eq,axiom,
! [X: list @ nat,Y3: list @ nat] :
( ( ( nat_list_encode @ X )
= ( nat_list_encode @ Y3 ) )
= ( X = Y3 ) ) ).
% list_encode_eq
thf(fact_3429_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_3430_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_3431_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_3432_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_3433_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_3434_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_3435_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_3436_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_3437_gcd__nat_Osimps,axiom,
( ( gcd_gcd @ nat )
= ( ^ [X2: nat,Y6: nat] :
( if @ nat
@ ( Y6
= ( zero_zero @ nat ) )
@ X2
@ ( gcd_gcd @ nat @ Y6 @ ( modulo_modulo @ nat @ X2 @ Y6 ) ) ) ) ) ).
% gcd_nat.simps
thf(fact_3438_gcd__nat_Oelims,axiom,
! [X: nat,Xa3: nat,Y3: nat] :
( ( ( gcd_gcd @ nat @ X @ Xa3 )
= Y3 )
=> ( ( ( Xa3
= ( zero_zero @ nat ) )
=> ( Y3 = X ) )
& ( ( Xa3
!= ( zero_zero @ nat ) )
=> ( Y3
= ( gcd_gcd @ nat @ Xa3 @ ( modulo_modulo @ nat @ X @ Xa3 ) ) ) ) ) ) ).
% gcd_nat.elims
thf(fact_3439_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_3440_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_3441_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_3442_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_3443_bezout__nat,axiom,
! [A3: nat,B2: nat] :
( ( A3
!= ( zero_zero @ nat ) )
=> ? [X3: nat,Y5: nat] :
( ( times_times @ nat @ A3 @ X3 )
= ( plus_plus @ nat @ ( times_times @ nat @ B2 @ Y5 ) @ ( gcd_gcd @ nat @ A3 @ B2 ) ) ) ) ).
% bezout_nat
thf(fact_3444_bezout__gcd__nat_H,axiom,
! [B2: nat,A3: nat] :
? [X3: nat,Y5: nat] :
( ( ( ord_less_eq @ nat @ ( times_times @ nat @ B2 @ Y5 ) @ ( times_times @ nat @ A3 @ X3 ) )
& ( ( minus_minus @ nat @ ( times_times @ nat @ A3 @ X3 ) @ ( times_times @ nat @ B2 @ Y5 ) )
= ( gcd_gcd @ nat @ A3 @ B2 ) ) )
| ( ( ord_less_eq @ nat @ ( times_times @ nat @ A3 @ Y5 ) @ ( times_times @ nat @ B2 @ X3 ) )
& ( ( minus_minus @ nat @ ( times_times @ nat @ B2 @ X3 ) @ ( times_times @ nat @ A3 @ Y5 ) )
= ( gcd_gcd @ nat @ A3 @ B2 ) ) ) ) ).
% bezout_gcd_nat'
thf(fact_3445_length__pos__if__in__set,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ).
% length_pos_if_in_set
thf(fact_3446_card__length,axiom,
! [A: $tType,Xs: list @ A] : ( ord_less_eq @ nat @ ( finite_card @ A @ ( set2 @ A @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ).
% card_length
thf(fact_3447_all__set__conv__all__nth,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( ( ! [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
=> ( P @ X2 ) ) )
= ( ! [I5: nat] :
( ( ord_less @ nat @ I5 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( P @ ( nth @ A @ Xs @ I5 ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_3448_all__nth__imp__all__set,axiom,
! [A: $tType,Xs: list @ A,P: A > $o,X: A] :
( ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( P @ ( nth @ A @ Xs @ I4 ) ) )
=> ( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( P @ X ) ) ) ).
% all_nth_imp_all_set
thf(fact_3449_in__set__conv__nth,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
= ( ? [I5: nat] :
( ( ord_less @ nat @ I5 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( ( nth @ A @ Xs @ I5 )
= X ) ) ) ) ).
% in_set_conv_nth
thf(fact_3450_list__ball__nth,axiom,
! [A: $tType,N: nat,Xs: list @ A,P: A > $o] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( P @ X3 ) )
=> ( P @ ( nth @ A @ Xs @ N ) ) ) ) ).
% list_ball_nth
thf(fact_3451_nth__mem,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( member @ A @ ( nth @ A @ Xs @ N ) @ ( set2 @ A @ Xs ) ) ) ).
% nth_mem
thf(fact_3452_nth__equal__first__eq,axiom,
! [A: $tType,X: A,Xs: list @ A,N: nat] :
( ~ ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( ord_less_eq @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ( nth @ A @ ( cons @ A @ X @ Xs ) @ N )
= X )
= ( N
= ( zero_zero @ nat ) ) ) ) ) ).
% nth_equal_first_eq
thf(fact_3453_prod__decode__aux_Opelims,axiom,
! [X: nat,Xa3: nat,Y3: product_prod @ nat @ nat] :
( ( ( nat_prod_decode_aux @ X @ Xa3 )
= Y3 )
=> ( ( accp @ ( product_prod @ nat @ nat ) @ nat_pr5047031295181774490ux_rel @ ( product_Pair @ nat @ nat @ X @ Xa3 ) )
=> ~ ( ( ( ( ord_less_eq @ nat @ Xa3 @ X )
=> ( Y3
= ( product_Pair @ nat @ nat @ Xa3 @ ( minus_minus @ nat @ X @ Xa3 ) ) ) )
& ( ~ ( ord_less_eq @ nat @ Xa3 @ X )
=> ( Y3
= ( nat_prod_decode_aux @ ( suc @ X ) @ ( minus_minus @ nat @ Xa3 @ ( suc @ X ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ nat @ nat ) @ nat_pr5047031295181774490ux_rel @ ( product_Pair @ nat @ nat @ X @ Xa3 ) ) ) ) ) ).
% prod_decode_aux.pelims
thf(fact_3454_gcd__nat_Opelims,axiom,
! [X: nat,Xa3: nat,Y3: nat] :
( ( ( gcd_gcd @ nat @ X @ Xa3 )
= Y3 )
=> ( ( accp @ ( product_prod @ nat @ nat ) @ gcd_nat_rel @ ( product_Pair @ nat @ nat @ X @ Xa3 ) )
=> ~ ( ( ( ( Xa3
= ( zero_zero @ nat ) )
=> ( Y3 = X ) )
& ( ( Xa3
!= ( zero_zero @ nat ) )
=> ( Y3
= ( gcd_gcd @ nat @ Xa3 @ ( modulo_modulo @ nat @ X @ Xa3 ) ) ) ) )
=> ~ ( accp @ ( product_prod @ nat @ nat ) @ gcd_nat_rel @ ( product_Pair @ nat @ nat @ X @ Xa3 ) ) ) ) ) ).
% gcd_nat.pelims
thf(fact_3455_list__encode_Oelims,axiom,
! [X: list @ nat,Y3: nat] :
( ( ( nat_list_encode @ X )
= Y3 )
=> ( ( ( X
= ( nil @ nat ) )
=> ( Y3
!= ( zero_zero @ nat ) ) )
=> ~ ! [X3: nat,Xs2: list @ nat] :
( ( X
= ( cons @ nat @ X3 @ Xs2 ) )
=> ( Y3
!= ( suc @ ( nat_prod_encode @ ( product_Pair @ nat @ nat @ X3 @ ( nat_list_encode @ Xs2 ) ) ) ) ) ) ) ) ).
% list_encode.elims
thf(fact_3456_zero__rat__def,axiom,
( ( zero_zero @ rat )
= ( abs_Rat @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) ) ) ) ).
% zero_rat_def
thf(fact_3457_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_3458_gcd__1__int,axiom,
! [M: int] :
( ( gcd_gcd @ int @ M @ ( one_one @ int ) )
= ( one_one @ int ) ) ).
% gcd_1_int
thf(fact_3459_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_3460_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_3461_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_3462_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_3463_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_3464_gcd__0__left__int,axiom,
! [X: int] :
( ( gcd_gcd @ int @ ( zero_zero @ int ) @ X )
= ( abs_abs @ int @ X ) ) ).
% gcd_0_left_int
thf(fact_3465_gcd__0__int,axiom,
! [X: int] :
( ( gcd_gcd @ int @ X @ ( zero_zero @ int ) )
= ( abs_abs @ int @ X ) ) ).
% gcd_0_int
thf(fact_3466_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_3467_length__0__conv,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( zero_zero @ nat ) )
= ( Xs
= ( nil @ A ) ) ) ).
% length_0_conv
thf(fact_3468_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_3469_length__greater__0__conv,axiom,
! [A: $tType,Xs: list @ A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) )
= ( Xs
!= ( nil @ A ) ) ) ).
% length_greater_0_conv
thf(fact_3470_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_3471_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_3472_pair__less__iff1,axiom,
! [X: nat,Y3: nat,Z3: 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 @ Z3 ) ) @ fun_pair_less )
= ( ord_less @ nat @ Y3 @ Z3 ) ) ).
% pair_less_iff1
thf(fact_3473_list__encode_Ocases,axiom,
! [X: list @ nat] :
( ( X
!= ( nil @ nat ) )
=> ~ ! [X3: nat,Xs2: list @ nat] :
( X
!= ( cons @ nat @ X3 @ Xs2 ) ) ) ).
% list_encode.cases
thf(fact_3474_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_3475_bezout__int,axiom,
! [X: int,Y3: int] :
? [U2: int,V4: int] :
( ( plus_plus @ int @ ( times_times @ int @ U2 @ X ) @ ( times_times @ int @ V4 @ Y3 ) )
= ( gcd_gcd @ int @ X @ Y3 ) ) ).
% bezout_int
thf(fact_3476_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_3477_list_Osize_I3_J,axiom,
! [A: $tType] :
( ( size_size @ ( list @ A ) @ ( nil @ A ) )
= ( zero_zero @ nat ) ) ).
% list.size(3)
thf(fact_3478_list__encode_Osimps_I1_J,axiom,
( ( nat_list_encode @ ( nil @ nat ) )
= ( zero_zero @ nat ) ) ).
% list_encode.simps(1)
thf(fact_3479_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_3480_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_3481_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_3482_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_3483_gcd__unique__int,axiom,
! [D2: int,A3: int,B2: int] :
( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ D2 )
& ( dvd_dvd @ int @ D2 @ A3 )
& ( dvd_dvd @ int @ D2 @ B2 )
& ! [E3: int] :
( ( ( dvd_dvd @ int @ E3 @ A3 )
& ( dvd_dvd @ int @ E3 @ B2 ) )
=> ( dvd_dvd @ int @ E3 @ D2 ) ) )
= ( D2
= ( gcd_gcd @ int @ A3 @ B2 ) ) ) ).
% gcd_unique_int
thf(fact_3484_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_3485_gcd__code__int,axiom,
( ( gcd_gcd @ int )
= ( ^ [K3: int,L3: int] :
( abs_abs @ int
@ ( if @ int
@ ( L3
= ( zero_zero @ int ) )
@ K3
@ ( gcd_gcd @ int @ L3 @ ( modulo_modulo @ int @ ( abs_abs @ int @ K3 ) @ ( abs_abs @ int @ L3 ) ) ) ) ) ) ) ).
% gcd_code_int
thf(fact_3486_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_3487_gcd__int__def,axiom,
( ( gcd_gcd @ int )
= ( ^ [X2: int,Y6: int] : ( semiring_1_of_nat @ int @ ( gcd_gcd @ nat @ ( nat2 @ ( abs_abs @ int @ X2 ) ) @ ( nat2 @ ( abs_abs @ int @ Y6 ) ) ) ) ) ) ).
% gcd_int_def
thf(fact_3488_one__rat__def,axiom,
( ( one_one @ rat )
= ( abs_Rat @ ( product_Pair @ int @ int @ ( one_one @ int ) @ ( one_one @ int ) ) ) ) ).
% one_rat_def
thf(fact_3489_plus__rat_Oabs__eq,axiom,
! [Xa3: product_prod @ int @ int,X: product_prod @ int @ int] :
( ( ratrel @ Xa3 @ Xa3 )
=> ( ( ratrel @ X @ X )
=> ( ( plus_plus @ rat @ ( abs_Rat @ Xa3 ) @ ( abs_Rat @ X ) )
= ( abs_Rat @ ( product_Pair @ int @ int @ ( plus_plus @ int @ ( times_times @ int @ ( product_fst @ int @ int @ Xa3 ) @ ( product_snd @ int @ int @ X ) ) @ ( times_times @ int @ ( product_fst @ int @ int @ X ) @ ( product_snd @ int @ int @ Xa3 ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ Xa3 ) @ ( product_snd @ int @ int @ X ) ) ) ) ) ) ) ).
% plus_rat.abs_eq
thf(fact_3490_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_3491_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_3492_smax__insertI,axiom,
! [Y3: product_prod @ nat @ nat,Y7: set @ ( product_prod @ nat @ nat ),X: product_prod @ nat @ nat,X8: set @ ( product_prod @ nat @ nat )] :
( ( member @ ( product_prod @ nat @ nat ) @ Y3 @ Y7 )
=> ( ( 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 ) ) @ X8 @ Y7 ) @ fun_max_strict )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ ( insert @ ( product_prod @ nat @ nat ) @ X @ X8 ) @ Y7 ) @ fun_max_strict ) ) ) ) ).
% smax_insertI
thf(fact_3493_smin__insertI,axiom,
! [X: product_prod @ nat @ nat,XS: set @ ( product_prod @ nat @ nat ),Y3: product_prod @ nat @ nat,YS: set @ ( product_prod @ nat @ nat )] :
( ( member @ ( product_prod @ nat @ nat ) @ X @ XS )
=> ( ( 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 ) ) @ XS @ 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 ) ) @ XS @ ( insert @ ( product_prod @ nat @ nat ) @ Y3 @ YS ) ) @ fun_min_strict ) ) ) ) ).
% smin_insertI
thf(fact_3494_ratrel__iff,axiom,
( ratrel
= ( ^ [X2: product_prod @ int @ int,Y6: product_prod @ int @ int] :
( ( ( product_snd @ int @ int @ X2 )
!= ( zero_zero @ int ) )
& ( ( product_snd @ int @ int @ Y6 )
!= ( zero_zero @ int ) )
& ( ( times_times @ int @ ( product_fst @ int @ int @ X2 ) @ ( product_snd @ int @ int @ Y6 ) )
= ( times_times @ int @ ( product_fst @ int @ int @ Y6 ) @ ( product_snd @ int @ int @ X2 ) ) ) ) ) ) ).
% ratrel_iff
thf(fact_3495_smin__emptyI,axiom,
! [X8: set @ ( product_prod @ nat @ nat )] :
( ( X8
!= ( 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 ) ) @ X8 @ ( bot_bot @ ( set @ ( product_prod @ nat @ nat ) ) ) ) @ fun_min_strict ) ) ).
% smin_emptyI
thf(fact_3496_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_3497_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_3498_times__rat_Oabs__eq,axiom,
! [Xa3: product_prod @ int @ int,X: product_prod @ int @ int] :
( ( ratrel @ Xa3 @ Xa3 )
=> ( ( ratrel @ X @ X )
=> ( ( times_times @ rat @ ( abs_Rat @ Xa3 ) @ ( abs_Rat @ X ) )
= ( abs_Rat @ ( product_Pair @ int @ int @ ( times_times @ int @ ( product_fst @ int @ int @ Xa3 ) @ ( product_fst @ int @ int @ X ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ Xa3 ) @ ( product_snd @ int @ int @ X ) ) ) ) ) ) ) ).
% times_rat.abs_eq
thf(fact_3499_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_3500_wmin__insertI,axiom,
! [X: product_prod @ nat @ nat,XS: set @ ( product_prod @ nat @ nat ),Y3: product_prod @ nat @ nat,YS: set @ ( product_prod @ nat @ nat )] :
( ( member @ ( product_prod @ nat @ nat ) @ X @ XS )
=> ( ( 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 ) ) @ XS @ 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 ) ) @ XS @ ( insert @ ( product_prod @ nat @ nat ) @ Y3 @ YS ) ) @ fun_min_weak ) ) ) ) ).
% wmin_insertI
thf(fact_3501_wmax__insertI,axiom,
! [Y3: product_prod @ nat @ nat,YS: set @ ( product_prod @ nat @ nat ),X: product_prod @ nat @ nat,XS: 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 ) ) @ XS @ YS ) @ fun_max_weak )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ ( insert @ ( product_prod @ nat @ nat ) @ X @ XS ) @ YS ) @ fun_max_weak ) ) ) ) ).
% wmax_insertI
thf(fact_3502_positive_Oabs__eq,axiom,
! [X: product_prod @ int @ int] :
( ( ratrel @ X @ X )
=> ( ( positive @ ( abs_Rat @ X ) )
= ( ord_less @ int @ ( zero_zero @ int ) @ ( times_times @ int @ ( product_fst @ int @ int @ X ) @ ( product_snd @ int @ int @ X ) ) ) ) ) ).
% positive.abs_eq
thf(fact_3503_upto_Opelims,axiom,
! [X: int,Xa3: int,Y3: list @ int] :
( ( ( upto @ X @ Xa3 )
= Y3 )
=> ( ( accp @ ( product_prod @ int @ int ) @ upto_rel @ ( product_Pair @ int @ int @ X @ Xa3 ) )
=> ~ ( ( ( ( ord_less_eq @ int @ X @ Xa3 )
=> ( Y3
= ( cons @ int @ X @ ( upto @ ( plus_plus @ int @ X @ ( one_one @ int ) ) @ Xa3 ) ) ) )
& ( ~ ( ord_less_eq @ int @ X @ Xa3 )
=> ( Y3
= ( nil @ int ) ) ) )
=> ~ ( accp @ ( product_prod @ int @ int ) @ upto_rel @ ( product_Pair @ int @ int @ X @ Xa3 ) ) ) ) ) ).
% upto.pelims
thf(fact_3504_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_3505_upto__Nil,axiom,
! [I: int,J: int] :
( ( ( upto @ I @ J )
= ( nil @ int ) )
= ( ord_less @ int @ J @ I ) ) ).
% upto_Nil
thf(fact_3506_upto__Nil2,axiom,
! [I: int,J: int] :
( ( ( nil @ int )
= ( upto @ I @ J ) )
= ( ord_less @ int @ J @ I ) ) ).
% upto_Nil2
thf(fact_3507_upto__empty,axiom,
! [J: int,I: int] :
( ( ord_less @ int @ J @ I )
=> ( ( upto @ I @ J )
= ( nil @ int ) ) ) ).
% upto_empty
thf(fact_3508_nth__upto,axiom,
! [I: int,K: nat,J: int] :
( ( ord_less_eq @ int @ ( plus_plus @ int @ I @ ( semiring_1_of_nat @ int @ K ) ) @ J )
=> ( ( nth @ int @ ( upto @ I @ J ) @ K )
= ( plus_plus @ int @ I @ ( semiring_1_of_nat @ int @ K ) ) ) ) ).
% nth_upto
thf(fact_3509_length__upto,axiom,
! [I: int,J: int] :
( ( size_size @ ( list @ int ) @ ( upto @ I @ J ) )
= ( nat2 @ ( plus_plus @ int @ ( minus_minus @ int @ J @ I ) @ ( one_one @ int ) ) ) ) ).
% length_upto
thf(fact_3510_upto__rec__numeral_I1_J,axiom,
! [M: num,N: num] :
( ( ( ord_less_eq @ int @ ( numeral_numeral @ int @ M ) @ ( numeral_numeral @ int @ N ) )
=> ( ( upto @ ( numeral_numeral @ int @ M ) @ ( numeral_numeral @ int @ N ) )
= ( cons @ int @ ( numeral_numeral @ int @ M ) @ ( upto @ ( plus_plus @ int @ ( numeral_numeral @ int @ M ) @ ( one_one @ int ) ) @ ( numeral_numeral @ int @ N ) ) ) ) )
& ( ~ ( ord_less_eq @ int @ ( numeral_numeral @ int @ M ) @ ( numeral_numeral @ int @ N ) )
=> ( ( upto @ ( numeral_numeral @ int @ M ) @ ( numeral_numeral @ int @ N ) )
= ( nil @ int ) ) ) ) ).
% upto_rec_numeral(1)
thf(fact_3511_upto__rec__numeral_I4_J,axiom,
! [M: num,N: num] :
( ( ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
=> ( ( upto @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
= ( cons @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( upto @ ( plus_plus @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( one_one @ int ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) ) ) ) )
& ( ~ ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
=> ( ( upto @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
= ( nil @ int ) ) ) ) ).
% upto_rec_numeral(4)
thf(fact_3512_upto__rec__numeral_I3_J,axiom,
! [M: num,N: num] :
( ( ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( numeral_numeral @ int @ N ) )
=> ( ( upto @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( numeral_numeral @ int @ N ) )
= ( cons @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( upto @ ( plus_plus @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( one_one @ int ) ) @ ( numeral_numeral @ int @ N ) ) ) ) )
& ( ~ ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( numeral_numeral @ int @ N ) )
=> ( ( upto @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M ) ) @ ( numeral_numeral @ int @ N ) )
= ( nil @ int ) ) ) ) ).
% upto_rec_numeral(3)
thf(fact_3513_upto__rec__numeral_I2_J,axiom,
! [M: num,N: num] :
( ( ( ord_less_eq @ int @ ( numeral_numeral @ int @ M ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
=> ( ( upto @ ( numeral_numeral @ int @ M ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
= ( cons @ int @ ( numeral_numeral @ int @ M ) @ ( upto @ ( plus_plus @ int @ ( numeral_numeral @ int @ M ) @ ( one_one @ int ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) ) ) ) )
& ( ~ ( ord_less_eq @ int @ ( numeral_numeral @ int @ M ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
=> ( ( upto @ ( numeral_numeral @ int @ M ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
= ( nil @ int ) ) ) ) ).
% upto_rec_numeral(2)
thf(fact_3514_less__by__empty,axiom,
! [A: $tType,A6: set @ ( product_prod @ A @ A ),B7: set @ ( product_prod @ A @ A )] :
( ( A6
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ A6 @ B7 ) ) ).
% less_by_empty
thf(fact_3515_wmin__emptyI,axiom,
! [X8: 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 ) ) @ X8 @ ( bot_bot @ ( set @ ( product_prod @ nat @ nat ) ) ) ) @ fun_min_weak ) ).
% wmin_emptyI
thf(fact_3516_positive__minus,axiom,
! [X: rat] :
( ~ ( positive @ X )
=> ( ( X
!= ( zero_zero @ rat ) )
=> ( positive @ ( uminus_uminus @ rat @ X ) ) ) ) ).
% positive_minus
thf(fact_3517_greaterThanAtMost__upto,axiom,
( ( set_or3652927894154168847AtMost @ int )
= ( ^ [I5: int,J3: int] : ( set2 @ int @ ( upto @ ( plus_plus @ int @ I5 @ ( one_one @ int ) ) @ J3 ) ) ) ) ).
% greaterThanAtMost_upto
thf(fact_3518_less__rat__def,axiom,
( ( ord_less @ rat )
= ( ^ [X2: rat,Y6: rat] : ( positive @ ( minus_minus @ rat @ Y6 @ X2 ) ) ) ) ).
% less_rat_def
thf(fact_3519_upto__rec1,axiom,
! [I: int,J: int] :
( ( ord_less_eq @ int @ I @ J )
=> ( ( upto @ I @ J )
= ( cons @ int @ I @ ( upto @ ( plus_plus @ int @ I @ ( one_one @ int ) ) @ J ) ) ) ) ).
% upto_rec1
thf(fact_3520_upto_Oelims,axiom,
! [X: int,Xa3: int,Y3: list @ int] :
( ( ( upto @ X @ Xa3 )
= Y3 )
=> ( ( ( ord_less_eq @ int @ X @ Xa3 )
=> ( Y3
= ( cons @ int @ X @ ( upto @ ( plus_plus @ int @ X @ ( one_one @ int ) ) @ Xa3 ) ) ) )
& ( ~ ( ord_less_eq @ int @ X @ Xa3 )
=> ( Y3
= ( nil @ int ) ) ) ) ) ).
% upto.elims
thf(fact_3521_upto_Osimps,axiom,
( upto
= ( ^ [I5: int,J3: int] : ( if @ ( list @ int ) @ ( ord_less_eq @ int @ I5 @ J3 ) @ ( cons @ int @ I5 @ ( upto @ ( plus_plus @ int @ I5 @ ( one_one @ int ) ) @ J3 ) ) @ ( nil @ int ) ) ) ) ).
% upto.simps
thf(fact_3522_greaterThanLessThan__upto,axiom,
( ( set_or5935395276787703475ssThan @ int )
= ( ^ [I5: int,J3: int] : ( set2 @ int @ ( upto @ ( plus_plus @ int @ I5 @ ( one_one @ int ) ) @ ( minus_minus @ int @ J3 @ ( one_one @ int ) ) ) ) ) ) ).
% greaterThanLessThan_upto
thf(fact_3523_positive_Orep__eq,axiom,
( positive
= ( ^ [X2: rat] : ( ord_less @ int @ ( zero_zero @ int ) @ ( times_times @ int @ ( product_fst @ int @ int @ ( rep_Rat @ X2 ) ) @ ( product_snd @ int @ int @ ( rep_Rat @ X2 ) ) ) ) ) ) ).
% positive.rep_eq
thf(fact_3524_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_3525_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_3526_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_3527_upto__rec2,axiom,
! [I: int,J: int] :
( ( ord_less_eq @ int @ I @ J )
=> ( ( upto @ I @ J )
= ( append @ int @ ( upto @ I @ ( minus_minus @ int @ J @ ( one_one @ int ) ) ) @ ( cons @ int @ J @ ( nil @ int ) ) ) ) ) ).
% upto_rec2
thf(fact_3528_inverse__eq__iff__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,B2: A] :
( ( ( inverse_inverse @ A @ A3 )
= ( inverse_inverse @ A @ B2 ) )
= ( A3 = B2 ) ) ) ).
% inverse_eq_iff_eq
thf(fact_3529_inverse__inverse__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A] :
( ( inverse_inverse @ A @ ( inverse_inverse @ A @ A3 ) )
= A3 ) ) ).
% inverse_inverse_eq
thf(fact_3530_inverse__zero,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ( ( inverse_inverse @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% inverse_zero
thf(fact_3531_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_3532_inverse__mult__distrib,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: A,B2: A] :
( ( inverse_inverse @ A @ ( times_times @ A @ A3 @ B2 ) )
= ( times_times @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B2 ) ) ) ) ).
% inverse_mult_distrib
thf(fact_3533_inverse__1,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ( ( inverse_inverse @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% inverse_1
thf(fact_3534_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_3535_inverse__minus__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A] :
( ( inverse_inverse @ A @ ( uminus_uminus @ A @ A3 ) )
= ( uminus_uminus @ A @ ( inverse_inverse @ A @ A3 ) ) ) ) ).
% inverse_minus_eq
thf(fact_3536_inverse__divide,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: A,B2: A] :
( ( inverse_inverse @ A @ ( divide_divide @ A @ A3 @ B2 ) )
= ( divide_divide @ A @ B2 @ A3 ) ) ) ).
% inverse_divide
thf(fact_3537_abs__inverse,axiom,
! [A: $tType] :
( ( field_abs_sgn @ A )
=> ! [A3: A] :
( ( abs_abs @ A @ ( inverse_inverse @ A @ A3 ) )
= ( inverse_inverse @ A @ ( abs_abs @ A @ A3 ) ) ) ) ).
% abs_inverse
thf(fact_3538_sgn__inverse,axiom,
! [A: $tType] :
( ( field_abs_sgn @ A )
=> ! [A3: A] :
( ( sgn_sgn @ A @ ( inverse_inverse @ A @ A3 ) )
= ( inverse_inverse @ A @ ( sgn_sgn @ A @ A3 ) ) ) ) ).
% sgn_inverse
thf(fact_3539_inverse__sgn,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( inverse_inverse @ A @ ( sgn_sgn @ A @ A3 ) )
= ( sgn_sgn @ A @ A3 ) ) ) ).
% inverse_sgn
thf(fact_3540_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_3541_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_3542_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_3543_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_3544_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_3545_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_3546_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_3547_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_3548_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_3549_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_3550_inverse__eq__divide__numeral,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [W2: num] :
( ( inverse_inverse @ A @ ( numeral_numeral @ A @ W2 ) )
= ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ W2 ) ) ) ) ).
% inverse_eq_divide_numeral
thf(fact_3551_inverse__eq__divide__neg__numeral,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [W2: num] :
( ( inverse_inverse @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) )
= ( divide_divide @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ).
% inverse_eq_divide_neg_numeral
thf(fact_3552_mult__commute__imp__mult__inverse__commute,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Y3: A,X: A] :
( ( ( times_times @ A @ Y3 @ X )
= ( times_times @ A @ X @ Y3 ) )
=> ( ( times_times @ A @ ( inverse_inverse @ A @ Y3 ) @ X )
= ( times_times @ A @ X @ ( inverse_inverse @ A @ Y3 ) ) ) ) ) ).
% mult_commute_imp_mult_inverse_commute
thf(fact_3553_inverse__eq__imp__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,B2: A] :
( ( ( inverse_inverse @ A @ A3 )
= ( inverse_inverse @ A @ B2 ) )
=> ( A3 = B2 ) ) ) ).
% inverse_eq_imp_eq
thf(fact_3554_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_3555_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_3556_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_3557_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_3558_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_3559_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_3560_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_3561_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_3562_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_3563_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_3564_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_3565_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_3566_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_3567_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_3568_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_3569_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_3570_inverse__numeral__1,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ( ( inverse_inverse @ A @ ( numeral_numeral @ A @ one2 ) )
= ( numeral_numeral @ A @ one2 ) ) ) ).
% inverse_numeral_1
thf(fact_3571_divide__inverse__commute,axiom,
! [A: $tType] :
( ( field @ A )
=> ( ( divide_divide @ A )
= ( ^ [A5: A,B3: A] : ( times_times @ A @ ( inverse_inverse @ A @ B3 ) @ A5 ) ) ) ) ).
% divide_inverse_commute
thf(fact_3572_divide__inverse,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ( ( divide_divide @ A )
= ( ^ [A5: A,B3: A] : ( times_times @ A @ A5 @ ( inverse_inverse @ A @ B3 ) ) ) ) ) ).
% divide_inverse
thf(fact_3573_field__class_Ofield__divide__inverse,axiom,
! [A: $tType] :
( ( field @ A )
=> ( ( divide_divide @ A )
= ( ^ [A5: A,B3: A] : ( times_times @ A @ A5 @ ( inverse_inverse @ A @ B3 ) ) ) ) ) ).
% field_class.field_divide_inverse
thf(fact_3574_inverse__eq__divide,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ( ( inverse_inverse @ A )
= ( divide_divide @ A @ ( one_one @ A ) ) ) ) ).
% inverse_eq_divide
thf(fact_3575_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_3576_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_3577_mult__inverse__of__nat__commute,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Xa3: nat,X: A] :
( ( times_times @ A @ ( inverse_inverse @ A @ ( semiring_1_of_nat @ A @ Xa3 ) ) @ X )
= ( times_times @ A @ X @ ( inverse_inverse @ A @ ( semiring_1_of_nat @ A @ Xa3 ) ) ) ) ) ).
% mult_inverse_of_nat_commute
thf(fact_3578_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_3579_mult__inverse__of__int__commute,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Xa3: int,X: A] :
( ( times_times @ A @ ( inverse_inverse @ A @ ( ring_1_of_int @ A @ Xa3 ) ) @ X )
= ( times_times @ A @ X @ ( inverse_inverse @ A @ ( ring_1_of_int @ A @ Xa3 ) ) ) ) ) ).
% mult_inverse_of_int_commute
thf(fact_3580_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_3581_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_3582_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_3583_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_3584_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_3585_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_3586_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_3587_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_3588_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_3589_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_3590_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_3591_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_3592_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_3593_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_3594_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_3595_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_3596_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_3597_reals__Archimedean,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X )
=> ? [N4: nat] : ( ord_less @ A @ ( inverse_inverse @ A @ ( semiring_1_of_nat @ A @ ( suc @ N4 ) ) ) @ X ) ) ) ).
% reals_Archimedean
thf(fact_3598_length__append__singleton,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ( size_size @ ( list @ A ) @ ( append @ A @ Xs @ ( cons @ A @ X @ ( nil @ A ) ) ) )
= ( suc @ ( size_size @ ( list @ A ) @ Xs ) ) ) ).
% length_append_singleton
thf(fact_3599_length__Suc__conv__rev,axiom,
! [A: $tType,Xs: list @ A,N: nat] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( suc @ N ) )
= ( ? [Y6: A,Ys3: list @ A] :
( ( Xs
= ( append @ A @ Ys3 @ ( cons @ A @ Y6 @ ( nil @ A ) ) ) )
& ( ( size_size @ ( list @ A ) @ Ys3 )
= N ) ) ) ) ).
% length_Suc_conv_rev
thf(fact_3600_nth__append,axiom,
! [A: $tType,N: nat,Xs: list @ A,Ys2: list @ A] :
( ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ ( append @ A @ Xs @ Ys2 ) @ N )
= ( nth @ A @ Xs @ N ) ) )
& ( ~ ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ ( append @ A @ Xs @ Ys2 ) @ N )
= ( nth @ A @ Ys2 @ ( minus_minus @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) ) ).
% nth_append
thf(fact_3601_upto__split2,axiom,
! [I: int,J: int,K: int] :
( ( ord_less_eq @ int @ I @ J )
=> ( ( ord_less_eq @ int @ J @ K )
=> ( ( upto @ I @ K )
= ( append @ int @ ( upto @ I @ J ) @ ( upto @ ( plus_plus @ int @ J @ ( one_one @ int ) ) @ K ) ) ) ) ) ).
% upto_split2
thf(fact_3602_upto__split1,axiom,
! [I: int,J: int,K: int] :
( ( ord_less_eq @ int @ I @ J )
=> ( ( ord_less_eq @ int @ J @ K )
=> ( ( upto @ I @ K )
= ( append @ int @ ( upto @ I @ ( minus_minus @ int @ J @ ( one_one @ int ) ) ) @ ( upto @ J @ K ) ) ) ) ) ).
% upto_split1
thf(fact_3603_ex__inverse__of__nat__less,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X )
=> ? [N4: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N4 )
& ( ord_less @ A @ ( inverse_inverse @ A @ ( semiring_1_of_nat @ A @ N4 ) ) @ X ) ) ) ) ).
% ex_inverse_of_nat_less
thf(fact_3604_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_3605_horner__sum__append,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_1 @ A )
=> ! [F: B > A,A3: A,Xs: list @ B,Ys2: list @ B] :
( ( groups4207007520872428315er_sum @ B @ A @ F @ A3 @ ( append @ B @ Xs @ Ys2 ) )
= ( plus_plus @ A @ ( groups4207007520872428315er_sum @ B @ A @ F @ A3 @ Xs ) @ ( times_times @ A @ ( power_power @ A @ A3 @ ( size_size @ ( list @ B ) @ Xs ) ) @ ( groups4207007520872428315er_sum @ B @ A @ F @ A3 @ Ys2 ) ) ) ) ) ).
% horner_sum_append
thf(fact_3606_upto__split3,axiom,
! [I: int,J: int,K: int] :
( ( ord_less_eq @ int @ I @ J )
=> ( ( ord_less_eq @ int @ J @ K )
=> ( ( upto @ I @ K )
= ( append @ int @ ( upto @ I @ ( minus_minus @ int @ J @ ( one_one @ int ) ) ) @ ( cons @ int @ J @ ( upto @ ( plus_plus @ int @ J @ ( one_one @ int ) ) @ K ) ) ) ) ) ) ).
% upto_split3
thf(fact_3607_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 ) ) ) )
=> ~ ! [X3: nat,Xs2: list @ nat] :
( ( X
= ( cons @ nat @ X3 @ Xs2 ) )
=> ( ( Y3
= ( suc @ ( nat_prod_encode @ ( product_Pair @ nat @ nat @ X3 @ ( nat_list_encode @ Xs2 ) ) ) ) )
=> ~ ( accp @ ( list @ nat ) @ nat_list_encode_rel @ ( cons @ nat @ X3 @ Xs2 ) ) ) ) ) ) ) ).
% list_encode.pelims
thf(fact_3608_rp__inv__image__rp,axiom,
! [A: $tType,B: $tType,P: product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ),F: B > A] :
( ( fun_reduction_pair @ A @ P )
=> ( fun_reduction_pair @ B @ ( fun_rp_inv_image @ A @ B @ P @ F ) ) ) ).
% rp_inv_image_rp
thf(fact_3609_take__Suc__conv__app__nth,axiom,
! [A: $tType,I: nat,Xs: list @ A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( take @ A @ ( suc @ I ) @ Xs )
= ( append @ A @ ( take @ A @ I @ Xs ) @ ( cons @ A @ ( nth @ A @ Xs @ I ) @ ( nil @ A ) ) ) ) ) ).
% take_Suc_conv_app_nth
thf(fact_3610_min__strict__def,axiom,
( fun_min_strict
= ( min_ext @ ( product_prod @ nat @ nat ) @ fun_pair_less ) ) ).
% min_strict_def
thf(fact_3611_upt__rec__numeral,axiom,
! [M: num,N: num] :
( ( ( ord_less @ nat @ ( numeral_numeral @ nat @ M ) @ ( numeral_numeral @ nat @ N ) )
=> ( ( upt @ ( numeral_numeral @ nat @ M ) @ ( numeral_numeral @ nat @ N ) )
= ( cons @ nat @ ( numeral_numeral @ nat @ M ) @ ( upt @ ( suc @ ( numeral_numeral @ nat @ M ) ) @ ( numeral_numeral @ nat @ N ) ) ) ) )
& ( ~ ( ord_less @ nat @ ( numeral_numeral @ nat @ M ) @ ( numeral_numeral @ nat @ N ) )
=> ( ( upt @ ( numeral_numeral @ nat @ M ) @ ( numeral_numeral @ nat @ N ) )
= ( nil @ nat ) ) ) ) ).
% upt_rec_numeral
thf(fact_3612_take__Suc__Cons,axiom,
! [A: $tType,N: nat,X: A,Xs: list @ A] :
( ( take @ A @ ( suc @ N ) @ ( cons @ A @ X @ Xs ) )
= ( cons @ A @ X @ ( take @ A @ N @ Xs ) ) ) ).
% take_Suc_Cons
thf(fact_3613_take__eq__Nil2,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( ( nil @ A )
= ( take @ A @ N @ Xs ) )
= ( ( N
= ( zero_zero @ nat ) )
| ( Xs
= ( nil @ A ) ) ) ) ).
% take_eq_Nil2
thf(fact_3614_take__eq__Nil,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( ( take @ A @ N @ Xs )
= ( nil @ A ) )
= ( ( N
= ( zero_zero @ nat ) )
| ( Xs
= ( nil @ A ) ) ) ) ).
% take_eq_Nil
thf(fact_3615_take0,axiom,
! [A: $tType] :
( ( take @ A @ ( zero_zero @ nat ) )
= ( ^ [Xs3: list @ A] : ( nil @ A ) ) ) ).
% take0
thf(fact_3616_take__all,axiom,
! [A: $tType,Xs: list @ A,N: nat] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N )
=> ( ( take @ A @ N @ Xs )
= Xs ) ) ).
% take_all
thf(fact_3617_take__all__iff,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( ( take @ A @ N @ Xs )
= Xs )
= ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N ) ) ).
% take_all_iff
thf(fact_3618_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_3619_nth__take,axiom,
! [A: $tType,I: nat,N: nat,Xs: list @ A] :
( ( ord_less @ nat @ I @ N )
=> ( ( nth @ A @ ( take @ A @ N @ Xs ) @ I )
= ( nth @ A @ Xs @ I ) ) ) ).
% nth_take
thf(fact_3620_upt__conv__Nil,axiom,
! [J: nat,I: nat] :
( ( ord_less_eq @ nat @ J @ I )
=> ( ( upt @ I @ J )
= ( nil @ nat ) ) ) ).
% upt_conv_Nil
thf(fact_3621_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_3622_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_3623_take__Cons__numeral,axiom,
! [A: $tType,V: num,X: A,Xs: list @ A] :
( ( take @ A @ ( numeral_numeral @ nat @ V ) @ ( cons @ A @ X @ Xs ) )
= ( cons @ A @ X @ ( take @ A @ ( minus_minus @ nat @ ( numeral_numeral @ nat @ V ) @ ( one_one @ nat ) ) @ Xs ) ) ) ).
% take_Cons_numeral
thf(fact_3624_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_3625_take__0,axiom,
! [A: $tType,Xs: list @ A] :
( ( take @ A @ ( zero_zero @ nat ) @ Xs )
= ( nil @ A ) ) ).
% take_0
thf(fact_3626_atLeastAtMost__upt,axiom,
( ( set_or1337092689740270186AtMost @ nat )
= ( ^ [N3: nat,M2: nat] : ( set2 @ nat @ ( upt @ N3 @ ( suc @ M2 ) ) ) ) ) ).
% atLeastAtMost_upt
thf(fact_3627_upt__Suc,axiom,
! [I: nat,J: nat] :
( ( ( ord_less_eq @ nat @ I @ J )
=> ( ( upt @ I @ ( suc @ J ) )
= ( append @ nat @ ( upt @ I @ J ) @ ( cons @ nat @ J @ ( nil @ nat ) ) ) ) )
& ( ~ ( ord_less_eq @ nat @ I @ J )
=> ( ( upt @ I @ ( suc @ J ) )
= ( nil @ nat ) ) ) ) ).
% upt_Suc
thf(fact_3628_upt__Suc__append,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( upt @ I @ ( suc @ J ) )
= ( append @ nat @ ( upt @ I @ J ) @ ( cons @ nat @ J @ ( nil @ nat ) ) ) ) ) ).
% upt_Suc_append
thf(fact_3629_upt__conv__Cons__Cons,axiom,
! [M: nat,N: nat,Ns: list @ nat,Q4: nat] :
( ( ( cons @ nat @ M @ ( cons @ nat @ N @ Ns ) )
= ( upt @ M @ Q4 ) )
= ( ( cons @ nat @ N @ Ns )
= ( upt @ ( suc @ M ) @ Q4 ) ) ) ).
% upt_conv_Cons_Cons
thf(fact_3630_upt__0,axiom,
! [I: nat] :
( ( upt @ I @ ( zero_zero @ nat ) )
= ( nil @ nat ) ) ).
% upt_0
thf(fact_3631_greaterThanAtMost__upt,axiom,
( ( set_or3652927894154168847AtMost @ nat )
= ( ^ [N3: nat,M2: nat] : ( set2 @ nat @ ( upt @ ( suc @ N3 ) @ ( suc @ M2 ) ) ) ) ) ).
% greaterThanAtMost_upt
thf(fact_3632_greaterThanLessThan__upt,axiom,
( ( set_or5935395276787703475ssThan @ nat )
= ( ^ [N3: nat,M2: nat] : ( set2 @ nat @ ( upt @ ( suc @ N3 ) @ M2 ) ) ) ) ).
% greaterThanLessThan_upt
thf(fact_3633_set__take__subset__set__take,axiom,
! [A: $tType,M: nat,N: nat,Xs: list @ A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( take @ A @ M @ Xs ) ) @ ( set2 @ A @ ( take @ A @ N @ Xs ) ) ) ) ).
% set_take_subset_set_take
thf(fact_3634_upt__conv__Cons,axiom,
! [I: nat,J: nat] :
( ( ord_less @ nat @ I @ J )
=> ( ( upt @ I @ J )
= ( cons @ nat @ I @ ( upt @ ( suc @ I ) @ J ) ) ) ) ).
% upt_conv_Cons
thf(fact_3635_nth__take__lemma,axiom,
! [A: $tType,K: nat,Xs: list @ A,Ys2: list @ A] :
( ( ord_less_eq @ nat @ K @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ord_less_eq @ nat @ K @ ( size_size @ ( list @ A ) @ Ys2 ) )
=> ( ! [I4: nat] :
( ( ord_less @ nat @ I4 @ K )
=> ( ( nth @ A @ Xs @ I4 )
= ( nth @ A @ Ys2 @ I4 ) ) )
=> ( ( take @ A @ K @ Xs )
= ( take @ A @ K @ Ys2 ) ) ) ) ) ).
% nth_take_lemma
thf(fact_3636_upt__eq__Cons__conv,axiom,
! [I: nat,J: nat,X: nat,Xs: list @ nat] :
( ( ( upt @ I @ J )
= ( cons @ nat @ X @ Xs ) )
= ( ( ord_less @ nat @ I @ J )
& ( I = X )
& ( ( upt @ ( plus_plus @ nat @ I @ ( one_one @ nat ) ) @ J )
= Xs ) ) ) ).
% upt_eq_Cons_conv
thf(fact_3637_upt__rec,axiom,
( upt
= ( ^ [I5: nat,J3: nat] : ( if @ ( list @ nat ) @ ( ord_less @ nat @ I5 @ J3 ) @ ( cons @ nat @ I5 @ ( upt @ ( suc @ I5 ) @ J3 ) ) @ ( nil @ nat ) ) ) ) ).
% upt_rec
thf(fact_3638_take__Cons_H,axiom,
! [A: $tType,N: nat,X: A,Xs: list @ A] :
( ( ( N
= ( zero_zero @ nat ) )
=> ( ( take @ A @ N @ ( cons @ A @ X @ Xs ) )
= ( nil @ A ) ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ( ( take @ A @ N @ ( cons @ A @ X @ Xs ) )
= ( cons @ A @ X @ ( take @ A @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ Xs ) ) ) ) ) ).
% take_Cons'
thf(fact_3639_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_3640_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 ) )
| ? [I5: nat] :
( ( ord_less @ nat @ I5 @ ( ord_min @ nat @ ( size_size @ ( list @ A ) @ X ) @ ( size_size @ ( list @ A ) @ Y3 ) ) )
& ( ( take @ A @ I5 @ X )
= ( take @ A @ I5 @ Y3 ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( nth @ A @ X @ I5 ) @ ( nth @ A @ Y3 @ I5 ) ) @ R ) ) ) ) ).
% lexord_take_index_conv
thf(fact_3641_lex__take__index,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys2 ) @ ( lex @ A @ R ) )
=> ~ ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Ys2 ) )
=> ( ( ( take @ A @ I4 @ Xs )
= ( take @ A @ I4 @ Ys2 ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( nth @ A @ Xs @ I4 ) @ ( nth @ A @ Ys2 @ I4 ) ) @ R ) ) ) ) ) ).
% lex_take_index
thf(fact_3642_min__weak__def,axiom,
( fun_min_weak
= ( sup_sup @ ( set @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) ) @ ( min_ext @ ( product_prod @ nat @ nat ) @ fun_pair_leq ) @ ( insert @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ ( bot_bot @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( bot_bot @ ( set @ ( product_prod @ nat @ nat ) ) ) ) @ ( bot_bot @ ( set @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) ) ) ) ) ) ).
% min_weak_def
thf(fact_3643_sup_Oidem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: A] :
( ( sup_sup @ A @ A3 @ A3 )
= A3 ) ) ).
% sup.idem
thf(fact_3644_sup__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A] :
( ( sup_sup @ A @ X @ X )
= X ) ) ).
% sup_idem
thf(fact_3645_sup_Oleft__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: A,B2: A] :
( ( sup_sup @ A @ A3 @ ( sup_sup @ A @ A3 @ B2 ) )
= ( sup_sup @ A @ A3 @ B2 ) ) ) ).
% sup.left_idem
thf(fact_3646_sup__left__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,Y3: A] :
( ( sup_sup @ A @ X @ ( sup_sup @ A @ X @ Y3 ) )
= ( sup_sup @ A @ X @ Y3 ) ) ) ).
% sup_left_idem
thf(fact_3647_sup_Oright__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: A,B2: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ A3 @ B2 ) @ B2 )
= ( sup_sup @ A @ A3 @ B2 ) ) ) ).
% sup.right_idem
thf(fact_3648_sup__apply,axiom,
! [B: $tType,A: $tType] :
( ( semilattice_sup @ B )
=> ( ( sup_sup @ ( A > B ) )
= ( ^ [F2: A > B,G2: A > B,X2: A] : ( sup_sup @ B @ ( F2 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ).
% sup_apply
thf(fact_3649_le__sup__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,Y3: A,Z3: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ X @ Y3 ) @ Z3 )
= ( ( ord_less_eq @ A @ X @ Z3 )
& ( ord_less_eq @ A @ Y3 @ Z3 ) ) ) ) ).
% le_sup_iff
thf(fact_3650_sup_Obounded__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B2: A,C2: A,A3: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ B2 @ C2 ) @ A3 )
= ( ( ord_less_eq @ A @ B2 @ A3 )
& ( ord_less_eq @ A @ C2 @ A3 ) ) ) ) ).
% sup.bounded_iff
thf(fact_3651_sup__bot__left,axiom,
! [A: $tType] :
( ( bounde4967611905675639751up_bot @ A )
=> ! [X: A] :
( ( sup_sup @ A @ ( bot_bot @ A ) @ X )
= X ) ) ).
% sup_bot_left
thf(fact_3652_sup__bot__right,axiom,
! [A: $tType] :
( ( bounde4967611905675639751up_bot @ A )
=> ! [X: A] :
( ( sup_sup @ A @ X @ ( bot_bot @ A ) )
= X ) ) ).
% sup_bot_right
thf(fact_3653_bot__eq__sup__iff,axiom,
! [A: $tType] :
( ( bounde4967611905675639751up_bot @ A )
=> ! [X: A,Y3: A] :
( ( ( bot_bot @ A )
= ( sup_sup @ A @ X @ Y3 ) )
= ( ( X
= ( bot_bot @ A ) )
& ( Y3
= ( bot_bot @ A ) ) ) ) ) ).
% bot_eq_sup_iff
thf(fact_3654_sup__eq__bot__iff,axiom,
! [A: $tType] :
( ( bounde4967611905675639751up_bot @ A )
=> ! [X: A,Y3: A] :
( ( ( sup_sup @ A @ X @ Y3 )
= ( bot_bot @ A ) )
= ( ( X
= ( bot_bot @ A ) )
& ( Y3
= ( bot_bot @ A ) ) ) ) ) ).
% sup_eq_bot_iff
thf(fact_3655_sup__bot_Oeq__neutr__iff,axiom,
! [A: $tType] :
( ( bounde4967611905675639751up_bot @ A )
=> ! [A3: A,B2: A] :
( ( ( sup_sup @ A @ A3 @ B2 )
= ( bot_bot @ A ) )
= ( ( A3
= ( bot_bot @ A ) )
& ( B2
= ( bot_bot @ A ) ) ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_3656_sup__bot_Oleft__neutral,axiom,
! [A: $tType] :
( ( bounde4967611905675639751up_bot @ A )
=> ! [A3: A] :
( ( sup_sup @ A @ ( bot_bot @ A ) @ A3 )
= A3 ) ) ).
% sup_bot.left_neutral
thf(fact_3657_sup__bot_Oneutr__eq__iff,axiom,
! [A: $tType] :
( ( bounde4967611905675639751up_bot @ A )
=> ! [A3: A,B2: A] :
( ( ( bot_bot @ A )
= ( sup_sup @ A @ A3 @ B2 ) )
= ( ( A3
= ( bot_bot @ A ) )
& ( B2
= ( bot_bot @ A ) ) ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_3658_sup__bot_Oright__neutral,axiom,
! [A: $tType] :
( ( bounde4967611905675639751up_bot @ A )
=> ! [A3: A] :
( ( sup_sup @ A @ A3 @ ( bot_bot @ A ) )
= A3 ) ) ).
% sup_bot.right_neutral
thf(fact_3659_Compl__Diff__eq,axiom,
! [A: $tType,A6: set @ A,B7: set @ A] :
( ( uminus_uminus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A6 @ B7 ) )
= ( sup_sup @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A6 ) @ B7 ) ) ).
% Compl_Diff_eq
thf(fact_3660_nth__map,axiom,
! [B: $tType,A: $tType,N: nat,Xs: list @ A,F: A > B] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ B @ ( map @ A @ B @ F @ Xs ) @ N )
= ( F @ ( nth @ A @ Xs @ N ) ) ) ) ).
% nth_map
thf(fact_3661_inf__sup__aci_I8_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y3: A] :
( ( sup_sup @ A @ X @ ( sup_sup @ A @ X @ Y3 ) )
= ( sup_sup @ A @ X @ Y3 ) ) ) ).
% inf_sup_aci(8)
thf(fact_3662_inf__sup__aci_I7_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y3: A,Z3: A] :
( ( sup_sup @ A @ X @ ( sup_sup @ A @ Y3 @ Z3 ) )
= ( sup_sup @ A @ Y3 @ ( sup_sup @ A @ X @ Z3 ) ) ) ) ).
% inf_sup_aci(7)
thf(fact_3663_inf__sup__aci_I6_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y3: A,Z3: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ X @ Y3 ) @ Z3 )
= ( sup_sup @ A @ X @ ( sup_sup @ A @ Y3 @ Z3 ) ) ) ) ).
% inf_sup_aci(6)
thf(fact_3664_inf__sup__aci_I5_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ( ( sup_sup @ A )
= ( ^ [X2: A,Y6: A] : ( sup_sup @ A @ Y6 @ X2 ) ) ) ) ).
% inf_sup_aci(5)
thf(fact_3665_sup_Oassoc,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ A3 @ B2 ) @ C2 )
= ( sup_sup @ A @ A3 @ ( sup_sup @ A @ B2 @ C2 ) ) ) ) ).
% sup.assoc
thf(fact_3666_sup__assoc,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,Y3: A,Z3: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ X @ Y3 ) @ Z3 )
= ( sup_sup @ A @ X @ ( sup_sup @ A @ Y3 @ Z3 ) ) ) ) ).
% sup_assoc
thf(fact_3667_sup_Ocommute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( sup_sup @ A )
= ( ^ [A5: A,B3: A] : ( sup_sup @ A @ B3 @ A5 ) ) ) ) ).
% sup.commute
thf(fact_3668_sup__commute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( sup_sup @ A )
= ( ^ [X2: A,Y6: A] : ( sup_sup @ A @ Y6 @ X2 ) ) ) ) ).
% sup_commute
thf(fact_3669_sup_Oleft__commute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( sup_sup @ A @ B2 @ ( sup_sup @ A @ A3 @ C2 ) )
= ( sup_sup @ A @ A3 @ ( sup_sup @ A @ B2 @ C2 ) ) ) ) ).
% sup.left_commute
thf(fact_3670_sup__left__commute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,Y3: A,Z3: A] :
( ( sup_sup @ A @ X @ ( sup_sup @ A @ Y3 @ Z3 ) )
= ( sup_sup @ A @ Y3 @ ( sup_sup @ A @ X @ Z3 ) ) ) ) ).
% sup_left_commute
thf(fact_3671_sup__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( semilattice_sup @ B )
=> ( ( sup_sup @ ( A > B ) )
= ( ^ [F2: A > B,G2: A > B,X2: A] : ( sup_sup @ B @ ( F2 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ).
% sup_fun_def
thf(fact_3672_boolean__algebra__cancel_Osup1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A6: A,K: A,A3: A,B2: A] :
( ( A6
= ( sup_sup @ A @ K @ A3 ) )
=> ( ( sup_sup @ A @ A6 @ B2 )
= ( sup_sup @ A @ K @ ( sup_sup @ A @ A3 @ B2 ) ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_3673_boolean__algebra__cancel_Osup2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B7: A,K: A,B2: A,A3: A] :
( ( B7
= ( sup_sup @ A @ K @ B2 ) )
=> ( ( sup_sup @ A @ A3 @ B7 )
= ( sup_sup @ A @ K @ ( sup_sup @ A @ A3 @ B2 ) ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_3674_less__supI1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,A3: A,B2: A] :
( ( ord_less @ A @ X @ A3 )
=> ( ord_less @ A @ X @ ( sup_sup @ A @ A3 @ B2 ) ) ) ) ).
% less_supI1
thf(fact_3675_less__supI2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,B2: A,A3: A] :
( ( ord_less @ A @ X @ B2 )
=> ( ord_less @ A @ X @ ( sup_sup @ A @ A3 @ B2 ) ) ) ) ).
% less_supI2
thf(fact_3676_sup_Oabsorb3,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B2: A,A3: A] :
( ( ord_less @ A @ B2 @ A3 )
=> ( ( sup_sup @ A @ A3 @ B2 )
= A3 ) ) ) ).
% sup.absorb3
thf(fact_3677_sup_Oabsorb4,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( sup_sup @ A @ A3 @ B2 )
= B2 ) ) ) ).
% sup.absorb4
thf(fact_3678_sup_Ostrict__boundedE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B2: A,C2: A,A3: A] :
( ( ord_less @ A @ ( sup_sup @ A @ B2 @ C2 ) @ A3 )
=> ~ ( ( ord_less @ A @ B2 @ A3 )
=> ~ ( ord_less @ A @ C2 @ A3 ) ) ) ) ).
% sup.strict_boundedE
thf(fact_3679_sup_Ostrict__order__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( ord_less @ A )
= ( ^ [B3: A,A5: A] :
( ( A5
= ( sup_sup @ A @ A5 @ B3 ) )
& ( A5 != B3 ) ) ) ) ) ).
% sup.strict_order_iff
thf(fact_3680_sup_Ostrict__coboundedI1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( ord_less @ A @ C2 @ A3 )
=> ( ord_less @ A @ C2 @ ( sup_sup @ A @ A3 @ B2 ) ) ) ) ).
% sup.strict_coboundedI1
thf(fact_3681_sup_Ostrict__coboundedI2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [C2: A,B2: A,A3: A] :
( ( ord_less @ A @ C2 @ B2 )
=> ( ord_less @ A @ C2 @ ( sup_sup @ A @ A3 @ B2 ) ) ) ) ).
% sup.strict_coboundedI2
thf(fact_3682_sup__max,axiom,
! [A: $tType] :
( ( ( semilattice_sup @ A )
& ( linorder @ A ) )
=> ( ( sup_sup @ A )
= ( ord_max @ A ) ) ) ).
% sup_max
thf(fact_3683_sup_OcoboundedI2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [C2: A,B2: A,A3: A] :
( ( ord_less_eq @ A @ C2 @ B2 )
=> ( ord_less_eq @ A @ C2 @ ( sup_sup @ A @ A3 @ B2 ) ) ) ) ).
% sup.coboundedI2
thf(fact_3684_sup_OcoboundedI1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( ord_less_eq @ A @ C2 @ A3 )
=> ( ord_less_eq @ A @ C2 @ ( sup_sup @ A @ A3 @ B2 ) ) ) ) ).
% sup.coboundedI1
thf(fact_3685_sup_Oabsorb__iff2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A5: A,B3: A] :
( ( sup_sup @ A @ A5 @ B3 )
= B3 ) ) ) ) ).
% sup.absorb_iff2
thf(fact_3686_sup_Oabsorb__iff1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B3: A,A5: A] :
( ( sup_sup @ A @ A5 @ B3 )
= A5 ) ) ) ) ).
% sup.absorb_iff1
thf(fact_3687_sup_Ocobounded2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B2: A,A3: A] : ( ord_less_eq @ A @ B2 @ ( sup_sup @ A @ A3 @ B2 ) ) ) ).
% sup.cobounded2
thf(fact_3688_sup_Ocobounded1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: A,B2: A] : ( ord_less_eq @ A @ A3 @ ( sup_sup @ A @ A3 @ B2 ) ) ) ).
% sup.cobounded1
thf(fact_3689_sup_Oorder__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B3: A,A5: A] :
( A5
= ( sup_sup @ A @ A5 @ B3 ) ) ) ) ) ).
% sup.order_iff
thf(fact_3690_sup_OboundedI,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( ord_less_eq @ A @ B2 @ A3 )
=> ( ( ord_less_eq @ A @ C2 @ A3 )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ B2 @ C2 ) @ A3 ) ) ) ) ).
% sup.boundedI
thf(fact_3691_sup_OboundedE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B2: A,C2: A,A3: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ B2 @ C2 ) @ A3 )
=> ~ ( ( ord_less_eq @ A @ B2 @ A3 )
=> ~ ( ord_less_eq @ A @ C2 @ A3 ) ) ) ) ).
% sup.boundedE
thf(fact_3692_sup__absorb2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,Y3: A] :
( ( ord_less_eq @ A @ X @ Y3 )
=> ( ( sup_sup @ A @ X @ Y3 )
= Y3 ) ) ) ).
% sup_absorb2
thf(fact_3693_sup__absorb1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [Y3: A,X: A] :
( ( ord_less_eq @ A @ Y3 @ X )
=> ( ( sup_sup @ A @ X @ Y3 )
= X ) ) ) ).
% sup_absorb1
thf(fact_3694_sup_Oabsorb2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( sup_sup @ A @ A3 @ B2 )
= B2 ) ) ) ).
% sup.absorb2
thf(fact_3695_sup_Oabsorb1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B2: A,A3: A] :
( ( ord_less_eq @ A @ B2 @ A3 )
=> ( ( sup_sup @ A @ A3 @ B2 )
= A3 ) ) ) ).
% sup.absorb1
thf(fact_3696_sup__unique,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [F: A > A > A,X: A,Y3: A] :
( ! [X3: A,Y5: A] : ( ord_less_eq @ A @ X3 @ ( F @ X3 @ Y5 ) )
=> ( ! [X3: A,Y5: A] : ( ord_less_eq @ A @ Y5 @ ( F @ X3 @ Y5 ) )
=> ( ! [X3: A,Y5: A,Z4: A] :
( ( ord_less_eq @ A @ Y5 @ X3 )
=> ( ( ord_less_eq @ A @ Z4 @ X3 )
=> ( ord_less_eq @ A @ ( F @ Y5 @ Z4 ) @ X3 ) ) )
=> ( ( sup_sup @ A @ X @ Y3 )
= ( F @ X @ Y3 ) ) ) ) ) ) ).
% sup_unique
thf(fact_3697_sup_OorderI,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: A,B2: A] :
( ( A3
= ( sup_sup @ A @ A3 @ B2 ) )
=> ( ord_less_eq @ A @ B2 @ A3 ) ) ) ).
% sup.orderI
thf(fact_3698_sup_OorderE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B2: A,A3: A] :
( ( ord_less_eq @ A @ B2 @ A3 )
=> ( A3
= ( sup_sup @ A @ A3 @ B2 ) ) ) ) ).
% sup.orderE
thf(fact_3699_le__iff__sup,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [X2: A,Y6: A] :
( ( sup_sup @ A @ X2 @ Y6 )
= Y6 ) ) ) ) ).
% le_iff_sup
thf(fact_3700_sup__least,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [Y3: A,X: A,Z3: A] :
( ( ord_less_eq @ A @ Y3 @ X )
=> ( ( ord_less_eq @ A @ Z3 @ X )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ Y3 @ Z3 ) @ X ) ) ) ) ).
% sup_least
thf(fact_3701_sup__mono,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: A,C2: A,B2: A,D2: A] :
( ( ord_less_eq @ A @ A3 @ C2 )
=> ( ( ord_less_eq @ A @ B2 @ D2 )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ A3 @ B2 ) @ ( sup_sup @ A @ C2 @ D2 ) ) ) ) ) ).
% sup_mono
thf(fact_3702_sup_Omono,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [C2: A,A3: A,D2: A,B2: A] :
( ( ord_less_eq @ A @ C2 @ A3 )
=> ( ( ord_less_eq @ A @ D2 @ B2 )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ C2 @ D2 ) @ ( sup_sup @ A @ A3 @ B2 ) ) ) ) ) ).
% sup.mono
thf(fact_3703_le__supI2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,B2: A,A3: A] :
( ( ord_less_eq @ A @ X @ B2 )
=> ( ord_less_eq @ A @ X @ ( sup_sup @ A @ A3 @ B2 ) ) ) ) ).
% le_supI2
thf(fact_3704_le__supI1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,A3: A,B2: A] :
( ( ord_less_eq @ A @ X @ A3 )
=> ( ord_less_eq @ A @ X @ ( sup_sup @ A @ A3 @ B2 ) ) ) ) ).
% le_supI1
thf(fact_3705_sup__ge2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [Y3: A,X: A] : ( ord_less_eq @ A @ Y3 @ ( sup_sup @ A @ X @ Y3 ) ) ) ).
% sup_ge2
thf(fact_3706_sup__ge1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,Y3: A] : ( ord_less_eq @ A @ X @ ( sup_sup @ A @ X @ Y3 ) ) ) ).
% sup_ge1
thf(fact_3707_le__supI,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: A,X: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ X )
=> ( ( ord_less_eq @ A @ B2 @ X )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ A3 @ B2 ) @ X ) ) ) ) ).
% le_supI
thf(fact_3708_le__supE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: A,B2: A,X: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ A3 @ B2 ) @ X )
=> ~ ( ( ord_less_eq @ A @ A3 @ X )
=> ~ ( ord_less_eq @ A @ B2 @ X ) ) ) ) ).
% le_supE
thf(fact_3709_inf__sup__ord_I3_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y3: A] : ( ord_less_eq @ A @ X @ ( sup_sup @ A @ X @ Y3 ) ) ) ).
% inf_sup_ord(3)
thf(fact_3710_inf__sup__ord_I4_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [Y3: A,X: A] : ( ord_less_eq @ A @ Y3 @ ( sup_sup @ A @ X @ Y3 ) ) ) ).
% inf_sup_ord(4)
thf(fact_3711_boolean__algebra_Odisj__zero__right,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A] :
( ( sup_sup @ A @ X @ ( bot_bot @ A ) )
= X ) ) ).
% boolean_algebra.disj_zero_right
thf(fact_3712_ivl__disj__un__two__touch_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M: A,U: A] :
( ( ord_less_eq @ A @ L @ M )
=> ( ( ord_less_eq @ A @ M @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ M ) @ ( set_or1337092689740270186AtMost @ A @ M @ U ) )
= ( set_or1337092689740270186AtMost @ A @ L @ U ) ) ) ) ) ).
% ivl_disj_un_two_touch(4)
thf(fact_3713_ivl__disj__un__two_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M: A,U: A] :
( ( ord_less_eq @ A @ L @ M )
=> ( ( ord_less_eq @ A @ M @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ L @ M ) @ ( set_or3652927894154168847AtMost @ A @ M @ U ) )
= ( set_or3652927894154168847AtMost @ A @ L @ U ) ) ) ) ) ).
% ivl_disj_un_two(6)
thf(fact_3714_card__Un__le,axiom,
! [A: $tType,A6: set @ A,B7: set @ A] : ( ord_less_eq @ nat @ ( finite_card @ A @ ( sup_sup @ ( set @ A ) @ A6 @ B7 ) ) @ ( plus_plus @ nat @ ( finite_card @ A @ A6 ) @ ( finite_card @ A @ B7 ) ) ) ).
% card_Un_le
thf(fact_3715_ivl__disj__un__two_I8_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M: A,U: A] :
( ( ord_less_eq @ A @ L @ M )
=> ( ( ord_less_eq @ A @ M @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ M ) @ ( set_or3652927894154168847AtMost @ A @ M @ U ) )
= ( set_or1337092689740270186AtMost @ A @ L @ U ) ) ) ) ) ).
% ivl_disj_un_two(8)
thf(fact_3716_ivl__disj__un__two__touch_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M: A,U: A] :
( ( ord_less @ A @ L @ M )
=> ( ( ord_less_eq @ A @ M @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ L @ M ) @ ( set_or1337092689740270186AtMost @ A @ M @ U ) )
= ( set_or3652927894154168847AtMost @ A @ L @ U ) ) ) ) ) ).
% ivl_disj_un_two_touch(3)
thf(fact_3717_ivl__disj__un__two_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M: A,U: A] :
( ( ord_less_eq @ A @ L @ M )
=> ( ( ord_less @ A @ M @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ L @ M ) @ ( set_or5935395276787703475ssThan @ A @ M @ U ) )
= ( set_or5935395276787703475ssThan @ A @ L @ U ) ) ) ) ) ).
% ivl_disj_un_two(2)
thf(fact_3718_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_3719_lexord__sufI,axiom,
! [A: $tType,U: list @ A,W2: list @ A,R: set @ ( product_prod @ A @ A ),V: list @ A,Z3: list @ A] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ U @ W2 ) @ ( lexord @ A @ R ) )
=> ( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ W2 ) @ ( size_size @ ( list @ A ) @ U ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ U @ V ) @ ( append @ A @ W2 @ Z3 ) ) @ ( lexord @ A @ R ) ) ) ) ).
% lexord_sufI
thf(fact_3720_ivl__disj__un__singleton_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,U: A] :
( ( ord_less_eq @ A @ L @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( insert @ A @ L @ ( bot_bot @ ( set @ A ) ) ) @ ( set_or3652927894154168847AtMost @ A @ L @ U ) )
= ( set_or1337092689740270186AtMost @ A @ L @ U ) ) ) ) ).
% ivl_disj_un_singleton(5)
thf(fact_3721_ivl__disj__un__two_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M: A,U: A] :
( ( ord_less @ A @ L @ M )
=> ( ( ord_less_eq @ A @ M @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ L @ M ) @ ( set_or1337092689740270186AtMost @ A @ M @ U ) )
= ( set_or3652927894154168847AtMost @ A @ L @ U ) ) ) ) ) ).
% ivl_disj_un_two(5)
thf(fact_3722_ivl__disj__un__singleton_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,U: A] :
( ( ord_less @ A @ L @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ L @ U ) @ ( insert @ A @ U @ ( bot_bot @ ( set @ A ) ) ) )
= ( set_or3652927894154168847AtMost @ A @ L @ U ) ) ) ) ).
% ivl_disj_un_singleton(4)
thf(fact_3723_map__upt__eqI,axiom,
! [A: $tType,Xs: list @ A,N: nat,M: nat,F: nat > A] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( minus_minus @ nat @ N @ M ) )
=> ( ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ Xs @ I4 )
= ( F @ ( plus_plus @ nat @ M @ I4 ) ) ) )
=> ( ( map @ nat @ A @ F @ ( upt @ M @ N ) )
= Xs ) ) ) ).
% map_upt_eqI
thf(fact_3724_max__weak__def,axiom,
( fun_max_weak
= ( sup_sup @ ( set @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) ) @ ( max_ext @ ( product_prod @ nat @ nat ) @ fun_pair_leq ) @ ( insert @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ ( bot_bot @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( bot_bot @ ( set @ ( product_prod @ nat @ nat ) ) ) ) @ ( bot_bot @ ( set @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) ) ) ) ) ) ).
% max_weak_def
thf(fact_3725_sorted__list__of__set__atMost__Suc,axiom,
! [K: nat] :
( ( linord4507533701916653071of_set @ nat @ ( set_ord_atMost @ nat @ ( suc @ K ) ) )
= ( append @ nat @ ( linord4507533701916653071of_set @ nat @ ( set_ord_atMost @ nat @ K ) ) @ ( cons @ nat @ ( suc @ K ) @ ( nil @ nat ) ) ) ) ).
% sorted_list_of_set_atMost_Suc
thf(fact_3726_sorted__list__of__set__lessThan__Suc,axiom,
! [K: nat] :
( ( linord4507533701916653071of_set @ nat @ ( set_ord_lessThan @ nat @ ( suc @ K ) ) )
= ( append @ nat @ ( linord4507533701916653071of_set @ nat @ ( set_ord_lessThan @ nat @ K ) ) @ ( cons @ nat @ K @ ( nil @ nat ) ) ) ) ).
% sorted_list_of_set_lessThan_Suc
thf(fact_3727_id__take__nth__drop,axiom,
! [A: $tType,I: nat,Xs: list @ A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( Xs
= ( append @ A @ ( take @ A @ I @ Xs ) @ ( cons @ A @ ( nth @ A @ Xs @ I ) @ ( drop @ A @ ( suc @ I ) @ Xs ) ) ) ) ) ).
% id_take_nth_drop
thf(fact_3728_atMost__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [I: A,K: A] :
( ( member @ A @ I @ ( set_ord_atMost @ A @ K ) )
= ( ord_less_eq @ A @ I @ K ) ) ) ).
% atMost_iff
thf(fact_3729_lessThan__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [I: A,K: A] :
( ( member @ A @ I @ ( set_ord_lessThan @ A @ K ) )
= ( ord_less @ A @ I @ K ) ) ) ).
% lessThan_iff
thf(fact_3730_drop0,axiom,
! [A: $tType] :
( ( drop @ A @ ( zero_zero @ nat ) )
= ( ^ [X2: list @ A] : X2 ) ) ).
% drop0
thf(fact_3731_atMost__subset__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y3: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_ord_atMost @ A @ X ) @ ( set_ord_atMost @ A @ Y3 ) )
= ( ord_less_eq @ A @ X @ Y3 ) ) ) ).
% atMost_subset_iff
thf(fact_3732_lessThan__subset__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y3: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_ord_lessThan @ A @ X ) @ ( set_ord_lessThan @ A @ Y3 ) )
= ( ord_less_eq @ A @ X @ Y3 ) ) ) ).
% lessThan_subset_iff
thf(fact_3733_drop__Suc__Cons,axiom,
! [A: $tType,N: nat,X: A,Xs: list @ A] :
( ( drop @ A @ ( suc @ N ) @ ( cons @ A @ X @ Xs ) )
= ( drop @ A @ N @ Xs ) ) ).
% drop_Suc_Cons
thf(fact_3734_lessThan__0,axiom,
( ( set_ord_lessThan @ nat @ ( zero_zero @ nat ) )
= ( bot_bot @ ( set @ nat ) ) ) ).
% lessThan_0
thf(fact_3735_card__atMost,axiom,
! [U: nat] :
( ( finite_card @ nat @ ( set_ord_atMost @ nat @ U ) )
= ( suc @ U ) ) ).
% card_atMost
thf(fact_3736_Icc__subset__Iic__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [L: A,H2: A,H: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ H2 ) @ ( set_ord_atMost @ A @ H ) )
= ( ~ ( ord_less_eq @ A @ L @ H2 )
| ( ord_less_eq @ A @ H2 @ H ) ) ) ) ).
% Icc_subset_Iic_iff
thf(fact_3737_drop__eq__Nil2,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( ( nil @ A )
= ( drop @ A @ N @ Xs ) )
= ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N ) ) ).
% drop_eq_Nil2
thf(fact_3738_drop__eq__Nil,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( ( drop @ A @ N @ Xs )
= ( nil @ A ) )
= ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N ) ) ).
% drop_eq_Nil
thf(fact_3739_drop__all,axiom,
! [A: $tType,Xs: list @ A,N: nat] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N )
=> ( ( drop @ A @ N @ Xs )
= ( nil @ A ) ) ) ).
% drop_all
thf(fact_3740_atMost__0,axiom,
( ( set_ord_atMost @ nat @ ( zero_zero @ nat ) )
= ( insert @ nat @ ( zero_zero @ nat ) @ ( bot_bot @ ( set @ nat ) ) ) ) ).
% atMost_0
thf(fact_3741_drop__Cons__numeral,axiom,
! [A: $tType,V: num,X: A,Xs: list @ A] :
( ( drop @ A @ ( numeral_numeral @ nat @ V ) @ ( cons @ A @ X @ Xs ) )
= ( drop @ A @ ( minus_minus @ nat @ ( numeral_numeral @ nat @ V ) @ ( one_one @ nat ) ) @ Xs ) ) ).
% drop_Cons_numeral
thf(fact_3742_nth__drop,axiom,
! [A: $tType,N: nat,Xs: list @ A,I: nat] :
( ( ord_less_eq @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ ( drop @ A @ N @ Xs ) @ I )
= ( nth @ A @ Xs @ ( plus_plus @ nat @ N @ I ) ) ) ) ).
% nth_drop
thf(fact_3743_sup__nat__def,axiom,
( ( sup_sup @ nat )
= ( ord_max @ nat ) ) ).
% sup_nat_def
thf(fact_3744_Iic__subset__Iio__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_ord_atMost @ A @ A3 ) @ ( set_ord_lessThan @ A @ B2 ) )
= ( ord_less @ A @ A3 @ B2 ) ) ) ).
% Iic_subset_Iio_iff
thf(fact_3745_drop__0,axiom,
! [A: $tType,Xs: list @ A] :
( ( drop @ A @ ( zero_zero @ nat ) @ Xs )
= Xs ) ).
% drop_0
thf(fact_3746_lessThan__Suc__atMost,axiom,
! [K: nat] :
( ( set_ord_lessThan @ nat @ ( suc @ K ) )
= ( set_ord_atMost @ nat @ K ) ) ).
% lessThan_Suc_atMost
thf(fact_3747_ivl__disj__un__one_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,U: A] :
( ( ord_less_eq @ A @ L @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_ord_lessThan @ A @ L ) @ ( set_or1337092689740270186AtMost @ A @ L @ U ) )
= ( set_ord_atMost @ A @ U ) ) ) ) ).
% ivl_disj_un_one(4)
thf(fact_3748_ivl__disj__un__one_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,U: A] :
( ( ord_less @ A @ L @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_ord_atMost @ A @ L ) @ ( set_or5935395276787703475ssThan @ A @ L @ U ) )
= ( set_ord_lessThan @ A @ U ) ) ) ) ).
% ivl_disj_un_one(1)
thf(fact_3749_sup__int__def,axiom,
( ( sup_sup @ int )
= ( ord_max @ int ) ) ).
% sup_int_def
thf(fact_3750_atMost__atLeast0,axiom,
( ( set_ord_atMost @ nat )
= ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) ) ) ).
% atMost_atLeast0
thf(fact_3751_map__Suc__upt,axiom,
! [M: nat,N: nat] :
( ( map @ nat @ nat @ suc @ ( upt @ M @ N ) )
= ( upt @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% map_Suc_upt
thf(fact_3752_lessThan__Suc,axiom,
! [K: nat] :
( ( set_ord_lessThan @ nat @ ( suc @ K ) )
= ( insert @ nat @ K @ ( set_ord_lessThan @ nat @ K ) ) ) ).
% lessThan_Suc
thf(fact_3753_lessThan__empty__iff,axiom,
! [N: nat] :
( ( ( set_ord_lessThan @ nat @ N )
= ( bot_bot @ ( set @ nat ) ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% lessThan_empty_iff
thf(fact_3754_atMost__Suc,axiom,
! [K: nat] :
( ( set_ord_atMost @ nat @ ( suc @ K ) )
= ( insert @ nat @ ( suc @ K ) @ ( set_ord_atMost @ nat @ K ) ) ) ).
% atMost_Suc
thf(fact_3755_lessThan__strict__subset__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [M: A,N: A] :
( ( ord_less @ ( set @ A ) @ ( set_ord_lessThan @ A @ M ) @ ( set_ord_lessThan @ A @ N ) )
= ( ord_less @ A @ M @ N ) ) ) ).
% lessThan_strict_subset_iff
thf(fact_3756_set__drop__subset__set__drop,axiom,
! [A: $tType,N: nat,M: nat,Xs: list @ A] :
( ( ord_less_eq @ nat @ N @ M )
=> ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( drop @ A @ M @ Xs ) ) @ ( set2 @ A @ ( drop @ A @ N @ Xs ) ) ) ) ).
% set_drop_subset_set_drop
thf(fact_3757_atLeast__upt,axiom,
( ( set_ord_lessThan @ nat )
= ( ^ [N3: nat] : ( set2 @ nat @ ( upt @ ( zero_zero @ nat ) @ N3 ) ) ) ) ).
% atLeast_upt
thf(fact_3758_ivl__disj__un__one_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,U: A] :
( ( ord_less_eq @ A @ L @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_ord_atMost @ A @ L ) @ ( set_or3652927894154168847AtMost @ A @ L @ U ) )
= ( set_ord_atMost @ A @ U ) ) ) ) ).
% ivl_disj_un_one(3)
thf(fact_3759_drop__Cons_H,axiom,
! [A: $tType,N: nat,X: A,Xs: list @ A] :
( ( ( N
= ( zero_zero @ nat ) )
=> ( ( drop @ A @ N @ ( cons @ A @ X @ Xs ) )
= ( cons @ A @ X @ Xs ) ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ( ( drop @ A @ N @ ( cons @ A @ X @ Xs ) )
= ( drop @ A @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ Xs ) ) ) ) ).
% drop_Cons'
thf(fact_3760_append__eq__append__conv__if,axiom,
! [A: $tType,Xs_1: list @ A,Xs_2: list @ A,Ys_1: list @ A,Ys_2: list @ A] :
( ( ( append @ A @ Xs_1 @ Xs_2 )
= ( append @ A @ Ys_1 @ Ys_2 ) )
= ( ( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs_1 ) @ ( size_size @ ( list @ A ) @ Ys_1 ) )
=> ( ( Xs_1
= ( take @ A @ ( size_size @ ( list @ A ) @ Xs_1 ) @ Ys_1 ) )
& ( Xs_2
= ( append @ A @ ( drop @ A @ ( size_size @ ( list @ A ) @ Xs_1 ) @ Ys_1 ) @ Ys_2 ) ) ) )
& ( ~ ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs_1 ) @ ( size_size @ ( list @ A ) @ Ys_1 ) )
=> ( ( ( take @ A @ ( size_size @ ( list @ A ) @ Ys_1 ) @ Xs_1 )
= Ys_1 )
& ( ( append @ A @ ( drop @ A @ ( size_size @ ( list @ A ) @ Ys_1 ) @ Xs_1 ) @ Xs_2 )
= Ys_2 ) ) ) ) ) ).
% append_eq_append_conv_if
thf(fact_3761_atMost__upto,axiom,
( ( set_ord_atMost @ nat )
= ( ^ [N3: nat] : ( set2 @ nat @ ( upt @ ( zero_zero @ nat ) @ ( suc @ N3 ) ) ) ) ) ).
% atMost_upto
thf(fact_3762_max__strict__def,axiom,
( fun_max_strict
= ( max_ext @ ( product_prod @ nat @ nat ) @ fun_pair_less ) ) ).
% max_strict_def
thf(fact_3763_Cons__nth__drop__Suc,axiom,
! [A: $tType,I: nat,Xs: list @ A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( cons @ A @ ( nth @ A @ Xs @ I ) @ ( drop @ A @ ( suc @ I ) @ Xs ) )
= ( drop @ A @ I @ Xs ) ) ) ).
% Cons_nth_drop_Suc
thf(fact_3764_atLeast1__atMost__eq__remove0,axiom,
! [N: nat] :
( ( set_or1337092689740270186AtMost @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N )
= ( minus_minus @ ( set @ nat ) @ ( set_ord_atMost @ nat @ N ) @ ( insert @ nat @ ( zero_zero @ nat ) @ ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% atLeast1_atMost_eq_remove0
thf(fact_3765_take__hd__drop,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( append @ A @ ( take @ A @ N @ Xs ) @ ( cons @ A @ ( hd @ A @ ( drop @ A @ N @ Xs ) ) @ ( nil @ A ) ) )
= ( take @ A @ ( suc @ N ) @ Xs ) ) ) ).
% take_hd_drop
thf(fact_3766_upd__conv__take__nth__drop,axiom,
! [A: $tType,I: nat,Xs: list @ A,A3: A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( list_update @ A @ Xs @ I @ A3 )
= ( append @ A @ ( take @ A @ I @ Xs ) @ ( cons @ A @ A3 @ ( drop @ A @ ( suc @ I ) @ Xs ) ) ) ) ) ).
% upd_conv_take_nth_drop
thf(fact_3767_effect__makeE,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [N: nat,F: nat > A,H2: heap_ext @ product_unit,H: heap_ext @ product_unit,R: array @ A,N2: nat] :
( ( heap_Time_effect @ ( array @ A ) @ ( array_make @ A @ N @ F ) @ H2 @ H @ R @ N2 )
=> ~ ( ( R
= ( product_fst @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( array_alloc @ A @ ( map @ nat @ A @ F @ ( upt @ ( zero_zero @ nat ) @ N ) ) @ H2 ) ) )
=> ( ( H
= ( product_snd @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( array_alloc @ A @ ( map @ nat @ A @ F @ ( upt @ ( zero_zero @ nat ) @ N ) ) @ H2 ) ) )
=> ( ( ( array_get @ A @ H @ R )
= ( map @ nat @ A @ F @ ( upt @ ( zero_zero @ nat ) @ N ) ) )
=> ( ( array_present @ A @ H @ R )
=> ( ~ ( array_present @ A @ H2 @ R )
=> ( ( plus_plus @ nat @ N @ ( one_one @ nat ) )
!= N2 ) ) ) ) ) ) ) ) ).
% effect_makeE
thf(fact_3768_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_3769_hd__upt,axiom,
! [I: nat,J: nat] :
( ( ord_less @ nat @ I @ J )
=> ( ( hd @ nat @ ( upt @ I @ J ) )
= I ) ) ).
% hd_upt
thf(fact_3770_list__update__beyond,axiom,
! [A: $tType,Xs: list @ A,I: nat,X: A] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ I )
=> ( ( list_update @ A @ Xs @ I @ X )
= Xs ) ) ).
% list_update_beyond
thf(fact_3771_take__update__cancel,axiom,
! [A: $tType,N: nat,M: nat,Xs: list @ A,Y3: A] :
( ( ord_less_eq @ nat @ N @ M )
=> ( ( take @ A @ N @ ( list_update @ A @ Xs @ M @ Y3 ) )
= ( take @ A @ N @ Xs ) ) ) ).
% take_update_cancel
thf(fact_3772_drop__update__cancel,axiom,
! [A: $tType,N: nat,M: nat,Xs: list @ A,X: A] :
( ( ord_less @ nat @ N @ M )
=> ( ( drop @ A @ M @ ( list_update @ A @ Xs @ N @ X ) )
= ( drop @ A @ M @ Xs ) ) ) ).
% drop_update_cancel
thf(fact_3773_get__update__eq,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [A3: array @ A,I: nat,V: A,H2: heap_ext @ product_unit] :
( ( array_get @ A @ ( array_update @ A @ A3 @ I @ V @ H2 ) @ A3 )
= ( list_update @ A @ ( array_get @ A @ H2 @ A3 ) @ I @ V ) ) ) ).
% get_update_eq
thf(fact_3774_nth__list__update__eq,axiom,
! [A: $tType,I: nat,Xs: list @ A,X: A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ ( list_update @ A @ Xs @ I @ X ) @ I )
= X ) ) ).
% nth_list_update_eq
thf(fact_3775_hd__take,axiom,
! [A: $tType,J: nat,Xs: list @ A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ J )
=> ( ( hd @ A @ ( take @ A @ J @ Xs ) )
= ( hd @ A @ Xs ) ) ) ).
% hd_take
thf(fact_3776_set__swap,axiom,
! [A: $tType,I: nat,Xs: list @ A,J: nat] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( set2 @ A @ ( list_update @ A @ ( list_update @ A @ Xs @ I @ ( nth @ A @ Xs @ J ) ) @ J @ ( nth @ A @ Xs @ I ) ) )
= ( set2 @ A @ Xs ) ) ) ) ).
% set_swap
thf(fact_3777_get__alloc,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [Xs: list @ A,H2: heap_ext @ product_unit,Ys2: list @ A] :
( ( array_get @ A @ ( product_snd @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( array_alloc @ A @ Xs @ H2 ) ) @ ( product_fst @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( array_alloc @ A @ Ys2 @ H2 ) ) )
= Xs ) ) ).
% get_alloc
thf(fact_3778_set,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [Ls: list @ A,H2: heap_ext @ product_unit,New_ls: list @ A] :
( ( array_set @ A @ ( product_fst @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( array_alloc @ A @ Ls @ H2 ) ) @ New_ls @ ( product_snd @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( array_alloc @ A @ Ls @ H2 ) ) )
= ( product_snd @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( array_alloc @ A @ New_ls @ H2 ) ) ) ) ).
% set
thf(fact_3779_not__present__alloc,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [H2: heap_ext @ product_unit,Xs: list @ A] :
~ ( array_present @ A @ H2 @ ( product_fst @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( array_alloc @ A @ Xs @ H2 ) ) ) ) ).
% not_present_alloc
thf(fact_3780_present__alloc,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [Xs: list @ A,H2: heap_ext @ product_unit] : ( array_present @ A @ ( product_snd @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( array_alloc @ A @ Xs @ H2 ) ) @ ( product_fst @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( array_alloc @ A @ Xs @ H2 ) ) ) ) ).
% present_alloc
thf(fact_3781_list__update__code_I3_J,axiom,
! [A: $tType,X: A,Xs: list @ A,I: nat,Y3: A] :
( ( list_update @ A @ ( cons @ A @ X @ Xs ) @ ( suc @ I ) @ Y3 )
= ( cons @ A @ X @ ( list_update @ A @ Xs @ I @ Y3 ) ) ) ).
% list_update_code(3)
thf(fact_3782_list__update__code_I2_J,axiom,
! [A: $tType,X: A,Xs: list @ A,Y3: A] :
( ( list_update @ A @ ( cons @ A @ X @ Xs ) @ ( zero_zero @ nat ) @ Y3 )
= ( cons @ A @ Y3 @ Xs ) ) ).
% list_update_code(2)
thf(fact_3783_set__update__memI,axiom,
! [A: $tType,N: nat,Xs: list @ A,X: A] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( member @ A @ X @ ( set2 @ A @ ( list_update @ A @ Xs @ N @ X ) ) ) ) ).
% set_update_memI
thf(fact_3784_list__update__append1,axiom,
! [A: $tType,I: nat,Xs: list @ A,Ys2: list @ A,X: A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( list_update @ A @ ( append @ A @ Xs @ Ys2 ) @ I @ X )
= ( append @ A @ ( list_update @ A @ Xs @ I @ X ) @ Ys2 ) ) ) ).
% list_update_append1
thf(fact_3785_length__alloc,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [Xs: list @ A,H2: heap_ext @ product_unit,Ys2: list @ A] :
( ( array_length @ A @ ( product_snd @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( array_alloc @ A @ Xs @ H2 ) ) @ ( product_fst @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( array_alloc @ A @ Ys2 @ H2 ) ) )
= ( size_size @ ( list @ A ) @ Xs ) ) ) ).
% length_alloc
thf(fact_3786_list__update__same__conv,axiom,
! [A: $tType,I: nat,Xs: list @ A,X: A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ( list_update @ A @ Xs @ I @ X )
= Xs )
= ( ( nth @ A @ Xs @ I )
= X ) ) ) ).
% list_update_same_conv
thf(fact_3787_nth__list__update,axiom,
! [A: $tType,I: nat,Xs: list @ A,J: nat,X: A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ( I = J )
=> ( ( nth @ A @ ( list_update @ A @ Xs @ I @ X ) @ J )
= X ) )
& ( ( I != J )
=> ( ( nth @ A @ ( list_update @ A @ Xs @ I @ X ) @ J )
= ( nth @ A @ Xs @ J ) ) ) ) ) ).
% nth_list_update
thf(fact_3788_hd__conv__nth,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( hd @ A @ Xs )
= ( nth @ A @ Xs @ ( zero_zero @ nat ) ) ) ) ).
% hd_conv_nth
thf(fact_3789_drop__update__swap,axiom,
! [A: $tType,M: nat,N: nat,Xs: list @ A,X: A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( drop @ A @ M @ ( list_update @ A @ Xs @ N @ X ) )
= ( list_update @ A @ ( drop @ A @ M @ Xs ) @ ( minus_minus @ nat @ N @ M ) @ X ) ) ) ).
% drop_update_swap
thf(fact_3790_present__alloc__noteq,axiom,
! [A: $tType,B: $tType] :
( ( ( heap @ B )
& ( heap @ A ) )
=> ! [H2: heap_ext @ product_unit,A3: array @ A,Xs: list @ B] :
( ( array_present @ A @ H2 @ A3 )
=> ( array_noteq @ A @ B @ A3 @ ( product_fst @ ( array @ B ) @ ( heap_ext @ product_unit ) @ ( array_alloc @ B @ Xs @ H2 ) ) ) ) ) ).
% present_alloc_noteq
thf(fact_3791_update__def,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( ( array_update @ A )
= ( ^ [A5: array @ A,I5: nat,X2: A,H4: heap_ext @ product_unit] : ( array_set @ A @ A5 @ ( list_update @ A @ ( array_get @ A @ H4 @ A5 ) @ I5 @ X2 ) @ H4 ) ) ) ) ).
% update_def
thf(fact_3792_list__update__append,axiom,
! [A: $tType,N: nat,Xs: list @ A,Ys2: list @ A,X: A] :
( ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( list_update @ A @ ( append @ A @ Xs @ Ys2 ) @ N @ X )
= ( append @ A @ ( list_update @ A @ Xs @ N @ X ) @ Ys2 ) ) )
& ( ~ ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( list_update @ A @ ( append @ A @ Xs @ Ys2 ) @ N @ X )
= ( append @ A @ Xs @ ( list_update @ A @ Ys2 @ ( minus_minus @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) ) @ X ) ) ) ) ) ).
% list_update_append
thf(fact_3793_n__lists_Osimps_I1_J,axiom,
! [A: $tType,Xs: list @ A] :
( ( n_lists @ A @ ( zero_zero @ nat ) @ Xs )
= ( cons @ ( list @ A ) @ ( nil @ A ) @ ( nil @ ( list @ A ) ) ) ) ).
% n_lists.simps(1)
thf(fact_3794_effect__makeI,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [A3: array @ A,H: heap_ext @ product_unit,F: nat > A,N: nat,H2: heap_ext @ product_unit] :
( ( ( product_Pair @ ( array @ A ) @ ( heap_ext @ product_unit ) @ A3 @ H )
= ( array_alloc @ A @ ( map @ nat @ A @ F @ ( upt @ ( zero_zero @ nat ) @ N ) ) @ H2 ) )
=> ( heap_Time_effect @ ( array @ A ) @ ( array_make @ A @ N @ F ) @ H2 @ H @ A3 @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ).
% effect_makeI
thf(fact_3795_hd__drop__conv__nth,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( hd @ A @ ( drop @ A @ N @ Xs ) )
= ( nth @ A @ Xs @ N ) ) ) ).
% hd_drop_conv_nth
thf(fact_3796_effect__of__listE,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [Xs: list @ A,H2: heap_ext @ product_unit,H: heap_ext @ product_unit,R: array @ A,N2: nat] :
( ( heap_Time_effect @ ( array @ A ) @ ( array_of_list @ A @ Xs ) @ H2 @ H @ R @ N2 )
=> ~ ( ( R
= ( product_fst @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( array_alloc @ A @ Xs @ H2 ) ) )
=> ( ( H
= ( product_snd @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( array_alloc @ A @ Xs @ H2 ) ) )
=> ( ( ( array_get @ A @ H @ R )
= Xs )
=> ( ( array_present @ A @ H @ R )
=> ( ~ ( array_present @ A @ H2 @ R )
=> ( N2
!= ( plus_plus @ nat @ ( one_one @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) ) ) ) ) ) ).
% effect_of_listE
thf(fact_3797_effect__of__listI,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [A3: array @ A,H: heap_ext @ product_unit,Xs: list @ A,H2: heap_ext @ product_unit] :
( ( ( product_Pair @ ( array @ A ) @ ( heap_ext @ product_unit ) @ A3 @ H )
= ( array_alloc @ A @ Xs @ H2 ) )
=> ( heap_Time_effect @ ( array @ A ) @ ( array_of_list @ A @ Xs ) @ H2 @ H @ A3 @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) ).
% effect_of_listI
thf(fact_3798_set__update__distinct,axiom,
! [A: $tType,Xs: list @ A,N: nat,X: A] :
( ( distinct @ A @ Xs )
=> ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( set2 @ A @ ( list_update @ A @ Xs @ N @ X ) )
= ( insert @ A @ X @ ( minus_minus @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( insert @ A @ ( nth @ A @ Xs @ N ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% set_update_distinct
thf(fact_3799_effect__newE,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [N: nat,X: A,H2: heap_ext @ product_unit,H: heap_ext @ product_unit,R: array @ A,N2: nat] :
( ( heap_Time_effect @ ( array @ A ) @ ( array_new @ A @ N @ X ) @ H2 @ H @ R @ N2 )
=> ~ ( ( R
= ( product_fst @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( array_alloc @ A @ ( replicate @ A @ N @ X ) @ H2 ) ) )
=> ( ( H
= ( product_snd @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( array_alloc @ A @ ( replicate @ A @ N @ X ) @ H2 ) ) )
=> ( ( ( array_get @ A @ H @ R )
= ( replicate @ A @ N @ X ) )
=> ( ( array_present @ A @ H @ R )
=> ( ~ ( array_present @ A @ H2 @ R )
=> ( ( plus_plus @ nat @ N @ ( one_one @ nat ) )
!= N2 ) ) ) ) ) ) ) ) ).
% effect_newE
thf(fact_3800_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_3801_replicate__empty,axiom,
! [A: $tType,N: nat,X: A] :
( ( ( replicate @ A @ N @ X )
= ( nil @ A ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% replicate_empty
thf(fact_3802_empty__replicate,axiom,
! [A: $tType,N: nat,X: A] :
( ( ( nil @ A )
= ( replicate @ A @ N @ X ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% empty_replicate
thf(fact_3803_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_3804_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_3805_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_3806_nth__replicate,axiom,
! [A: $tType,I: nat,N: nat,X: A] :
( ( ord_less @ nat @ I @ N )
=> ( ( nth @ A @ ( replicate @ A @ N @ X ) @ I )
= X ) ) ).
% nth_replicate
thf(fact_3807_hd__replicate,axiom,
! [A: $tType,N: nat,X: A] :
( ( N
!= ( zero_zero @ nat ) )
=> ( ( hd @ A @ ( replicate @ A @ N @ X ) )
= X ) ) ).
% hd_replicate
thf(fact_3808_set__replicate,axiom,
! [A: $tType,N: nat,X: A] :
( ( N
!= ( zero_zero @ nat ) )
=> ( ( set2 @ A @ ( replicate @ A @ N @ X ) )
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% set_replicate
thf(fact_3809_distinct__swap,axiom,
! [A: $tType,I: nat,Xs: list @ A,J: nat] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( distinct @ A @ ( list_update @ A @ ( list_update @ A @ Xs @ I @ ( nth @ A @ Xs @ J ) ) @ J @ ( nth @ A @ Xs @ I ) ) )
= ( distinct @ A @ Xs ) ) ) ) ).
% distinct_swap
thf(fact_3810_replicate__Suc,axiom,
! [A: $tType,N: nat,X: A] :
( ( replicate @ A @ ( suc @ N ) @ X )
= ( cons @ A @ X @ ( replicate @ A @ N @ X ) ) ) ).
% replicate_Suc
thf(fact_3811_replicate__0,axiom,
! [A: $tType,X: A] :
( ( replicate @ A @ ( zero_zero @ nat ) @ X )
= ( nil @ A ) ) ).
% replicate_0
thf(fact_3812_distinct__conv__nth,axiom,
! [A: $tType] :
( ( distinct @ A )
= ( ^ [Xs3: list @ A] :
! [I5: nat] :
( ( ord_less @ nat @ I5 @ ( size_size @ ( list @ A ) @ Xs3 ) )
=> ! [J3: nat] :
( ( ord_less @ nat @ J3 @ ( size_size @ ( list @ A ) @ Xs3 ) )
=> ( ( I5 != J3 )
=> ( ( nth @ A @ Xs3 @ I5 )
!= ( nth @ A @ Xs3 @ J3 ) ) ) ) ) ) ) ).
% distinct_conv_nth
thf(fact_3813_nth__eq__iff__index__eq,axiom,
! [A: $tType,Xs: list @ A,I: nat,J: nat] :
( ( distinct @ A @ Xs )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ( nth @ A @ Xs @ I )
= ( nth @ A @ Xs @ J ) )
= ( I = J ) ) ) ) ) ).
% nth_eq_iff_index_eq
thf(fact_3814_effect__newI,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [A3: array @ A,H: heap_ext @ product_unit,N: nat,X: A,H2: heap_ext @ product_unit] :
( ( ( product_Pair @ ( array @ A ) @ ( heap_ext @ product_unit ) @ A3 @ H )
= ( array_alloc @ A @ ( replicate @ A @ N @ X ) @ H2 ) )
=> ( heap_Time_effect @ ( array @ A ) @ ( array_new @ A @ N @ X ) @ H2 @ H @ A3 @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ).
% effect_newI
thf(fact_3815_distinct__Ex1,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ( distinct @ A @ Xs )
=> ( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ? [X3: nat] :
( ( ord_less @ nat @ X3 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( ( nth @ A @ Xs @ X3 )
= X )
& ! [Y4: nat] :
( ( ( ord_less @ nat @ Y4 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( ( nth @ A @ Xs @ Y4 )
= X ) )
=> ( Y4 = X3 ) ) ) ) ) ).
% distinct_Ex1
thf(fact_3816_set__replicate__Suc,axiom,
! [A: $tType,N: nat,X: A] :
( ( set2 @ A @ ( replicate @ A @ ( suc @ N ) @ X ) )
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ).
% set_replicate_Suc
thf(fact_3817_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 ) )
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% set_replicate_conv_if
thf(fact_3818_Cons__replicate__eq,axiom,
! [A: $tType,X: A,Xs: list @ A,N: nat,Y3: A] :
( ( ( cons @ A @ X @ Xs )
= ( replicate @ A @ N @ Y3 ) )
= ( ( X = Y3 )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
& ( Xs
= ( replicate @ A @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ X ) ) ) ) ).
% Cons_replicate_eq
thf(fact_3819_atLeast1__lessThan__eq__remove0,axiom,
! [N: nat] :
( ( set_or7035219750837199246ssThan @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N )
= ( minus_minus @ ( set @ nat ) @ ( set_ord_lessThan @ nat @ N ) @ ( insert @ nat @ ( zero_zero @ nat ) @ ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% atLeast1_lessThan_eq_remove0
thf(fact_3820_listrel1__iff__update,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys2 ) @ ( listrel1 @ A @ R ) )
= ( ? [Y6: A,N3: nat] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( nth @ A @ Xs @ N3 ) @ Y6 ) @ R )
& ( ord_less @ nat @ N3 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( Ys2
= ( list_update @ A @ Xs @ N3 @ Y6 ) ) ) ) ) ).
% listrel1_iff_update
thf(fact_3821_Gcd__remove0__nat,axiom,
! [M7: set @ nat] :
( ( finite_finite2 @ nat @ M7 )
=> ( ( gcd_Gcd @ nat @ M7 )
= ( gcd_Gcd @ nat @ ( minus_minus @ ( set @ nat ) @ M7 @ ( insert @ nat @ ( zero_zero @ nat ) @ ( bot_bot @ ( set @ nat ) ) ) ) ) ) ) ).
% Gcd_remove0_nat
thf(fact_3822_remdups__adj__singleton__iff,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( size_size @ ( list @ A ) @ ( remdups_adj @ A @ Xs ) )
= ( suc @ ( zero_zero @ nat ) ) )
= ( ( Xs
!= ( nil @ A ) )
& ( Xs
= ( replicate @ A @ ( size_size @ ( list @ A ) @ Xs ) @ ( hd @ A @ Xs ) ) ) ) ) ).
% remdups_adj_singleton_iff
thf(fact_3823_atLeastLessThan__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [I: A,L: A,U: A] :
( ( member @ A @ I @ ( set_or7035219750837199246ssThan @ A @ L @ U ) )
= ( ( ord_less_eq @ A @ L @ I )
& ( ord_less @ A @ I @ U ) ) ) ) ).
% atLeastLessThan_iff
thf(fact_3824_infinite__Icc__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B2: A] :
( ( ~ ( finite_finite2 @ A @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) ) )
= ( ord_less @ A @ A3 @ B2 ) ) ) ).
% infinite_Icc_iff
thf(fact_3825_atLeastLessThan__empty,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B2: A,A3: A] :
( ( ord_less_eq @ A @ B2 @ A3 )
=> ( ( set_or7035219750837199246ssThan @ A @ A3 @ B2 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% atLeastLessThan_empty
thf(fact_3826_atLeastLessThan__empty__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A,B2: A] :
( ( ( set_or7035219750837199246ssThan @ A @ A3 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
= ( ~ ( ord_less @ A @ A3 @ B2 ) ) ) ) ).
% atLeastLessThan_empty_iff
thf(fact_3827_atLeastLessThan__empty__iff2,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A,B2: A] :
( ( ( bot_bot @ ( set @ A ) )
= ( set_or7035219750837199246ssThan @ A @ A3 @ B2 ) )
= ( ~ ( ord_less @ A @ A3 @ B2 ) ) ) ) ).
% atLeastLessThan_empty_iff2
thf(fact_3828_ivl__subset,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [I: A,J: A,M: A,N: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ I @ J ) @ ( set_or7035219750837199246ssThan @ A @ M @ N ) )
= ( ( ord_less_eq @ A @ J @ I )
| ( ( ord_less_eq @ A @ M @ I )
& ( ord_less_eq @ A @ J @ N ) ) ) ) ) ).
% ivl_subset
thf(fact_3829_infinite__Ico__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B2: A] :
( ( ~ ( finite_finite2 @ A @ ( set_or7035219750837199246ssThan @ A @ A3 @ B2 ) ) )
= ( ord_less @ A @ A3 @ B2 ) ) ) ).
% infinite_Ico_iff
thf(fact_3830_card_Oinfinite,axiom,
! [A: $tType,A6: set @ A] :
( ~ ( finite_finite2 @ A @ A6 )
=> ( ( finite_card @ A @ A6 )
= ( zero_zero @ nat ) ) ) ).
% card.infinite
thf(fact_3831_ivl__diff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [I: A,N: A,M: A] :
( ( ord_less_eq @ A @ I @ N )
=> ( ( minus_minus @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ I @ M ) @ ( set_or7035219750837199246ssThan @ A @ I @ N ) )
= ( set_or7035219750837199246ssThan @ A @ N @ M ) ) ) ) ).
% ivl_diff
thf(fact_3832_infinite__Ioc__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B2: A] :
( ( ~ ( finite_finite2 @ A @ ( set_or3652927894154168847AtMost @ A @ A3 @ B2 ) ) )
= ( ord_less @ A @ A3 @ B2 ) ) ) ).
% infinite_Ioc_iff
thf(fact_3833_infinite__Ioo__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B2: A] :
( ( ~ ( finite_finite2 @ A @ ( set_or5935395276787703475ssThan @ A @ A3 @ B2 ) ) )
= ( ord_less @ A @ A3 @ B2 ) ) ) ).
% infinite_Ioo_iff
thf(fact_3834_card__0__eq,axiom,
! [A: $tType,A6: set @ A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( ( finite_card @ A @ A6 )
= ( zero_zero @ nat ) )
= ( A6
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% card_0_eq
thf(fact_3835_card__insert__disjoint,axiom,
! [A: $tType,A6: set @ A,X: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ~ ( member @ A @ X @ A6 )
=> ( ( finite_card @ A @ ( insert @ A @ X @ A6 ) )
= ( suc @ ( finite_card @ A @ A6 ) ) ) ) ) ).
% card_insert_disjoint
thf(fact_3836_atLeastLessThan__singleton,axiom,
! [M: nat] :
( ( set_or7035219750837199246ssThan @ nat @ M @ ( suc @ M ) )
= ( insert @ nat @ M @ ( bot_bot @ ( set @ nat ) ) ) ) ).
% atLeastLessThan_singleton
thf(fact_3837_finite__psubset__induct,axiom,
! [A: $tType,A6: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ A6 )
=> ( ! [A9: set @ A] :
( ( finite_finite2 @ A @ A9 )
=> ( ! [B10: set @ A] :
( ( ord_less @ ( set @ A ) @ B10 @ A9 )
=> ( P @ B10 ) )
=> ( P @ A9 ) ) )
=> ( P @ A6 ) ) ) ).
% finite_psubset_induct
thf(fact_3838_atLeastLessThan__eq__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A,C2: A,D2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ C2 @ D2 )
=> ( ( ( set_or7035219750837199246ssThan @ A @ A3 @ B2 )
= ( set_or7035219750837199246ssThan @ A @ C2 @ D2 ) )
= ( ( A3 = C2 )
& ( B2 = D2 ) ) ) ) ) ) ).
% atLeastLessThan_eq_iff
thf(fact_3839_atLeastLessThan__inj_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A,C2: A,D2: A] :
( ( ( set_or7035219750837199246ssThan @ A @ A3 @ B2 )
= ( set_or7035219750837199246ssThan @ A @ C2 @ D2 ) )
=> ( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ C2 @ D2 )
=> ( A3 = C2 ) ) ) ) ) ).
% atLeastLessThan_inj(1)
thf(fact_3840_atLeastLessThan__inj_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A,C2: A,D2: A] :
( ( ( set_or7035219750837199246ssThan @ A @ A3 @ B2 )
= ( set_or7035219750837199246ssThan @ A @ C2 @ D2 ) )
=> ( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ C2 @ D2 )
=> ( B2 = D2 ) ) ) ) ) ).
% atLeastLessThan_inj(2)
thf(fact_3841_infinite__Ico,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ~ ( finite_finite2 @ A @ ( set_or7035219750837199246ssThan @ A @ A3 @ B2 ) ) ) ) ).
% infinite_Ico
thf(fact_3842_finite__nat__set__iff__bounded,axiom,
( ( finite_finite2 @ nat )
= ( ^ [N7: set @ nat] :
? [M2: nat] :
! [X2: nat] :
( ( member @ nat @ X2 @ N7 )
=> ( ord_less @ nat @ X2 @ M2 ) ) ) ) ).
% finite_nat_set_iff_bounded
thf(fact_3843_bounded__nat__set__is__finite,axiom,
! [N6: set @ nat,N: nat] :
( ! [X3: nat] :
( ( member @ nat @ X3 @ N6 )
=> ( ord_less @ nat @ X3 @ N ) )
=> ( finite_finite2 @ nat @ N6 ) ) ).
% bounded_nat_set_is_finite
thf(fact_3844_finite__nat__set__iff__bounded__le,axiom,
( ( finite_finite2 @ nat )
= ( ^ [N7: set @ nat] :
? [M2: nat] :
! [X2: nat] :
( ( member @ nat @ X2 @ N7 )
=> ( ord_less_eq @ nat @ X2 @ M2 ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_3845_finite__has__minimal2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A6: set @ A,A3: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( member @ A @ A3 @ A6 )
=> ? [X3: A] :
( ( member @ A @ X3 @ A6 )
& ( ord_less_eq @ A @ X3 @ A3 )
& ! [Xa: A] :
( ( member @ A @ Xa @ A6 )
=> ( ( ord_less_eq @ A @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_3846_finite__has__maximal2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A6: set @ A,A3: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( member @ A @ A3 @ A6 )
=> ? [X3: A] :
( ( member @ A @ X3 @ A6 )
& ( ord_less_eq @ A @ A3 @ X3 )
& ! [Xa: A] :
( ( member @ A @ Xa @ A6 )
=> ( ( ord_less_eq @ A @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_3847_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_3848_finite__set__decode,axiom,
! [N: nat] : ( finite_finite2 @ nat @ ( nat_set_decode @ N ) ) ).
% finite_set_decode
thf(fact_3849_atLeastLessThan__subset__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A,C2: A,D2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ A3 @ B2 ) @ ( set_or7035219750837199246ssThan @ A @ C2 @ D2 ) )
=> ( ( ord_less_eq @ A @ B2 @ A3 )
| ( ( ord_less_eq @ A @ C2 @ A3 )
& ( ord_less_eq @ A @ B2 @ D2 ) ) ) ) ) ).
% atLeastLessThan_subset_iff
thf(fact_3850_ivl__disj__un__two_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M: A,U: A] :
( ( ord_less_eq @ A @ L @ M )
=> ( ( ord_less_eq @ A @ M @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ L @ M ) @ ( set_or7035219750837199246ssThan @ A @ M @ U ) )
= ( set_or7035219750837199246ssThan @ A @ L @ U ) ) ) ) ) ).
% ivl_disj_un_two(3)
thf(fact_3851_ex__nat__less__eq,axiom,
! [N: nat,P: nat > $o] :
( ( ? [M2: nat] :
( ( ord_less @ nat @ M2 @ N )
& ( P @ M2 ) ) )
= ( ? [X2: nat] :
( ( member @ nat @ X2 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) )
& ( P @ X2 ) ) ) ) ).
% ex_nat_less_eq
thf(fact_3852_all__nat__less__eq,axiom,
! [N: nat,P: nat > $o] :
( ( ! [M2: nat] :
( ( ord_less @ nat @ M2 @ N )
=> ( P @ M2 ) ) )
= ( ! [X2: nat] :
( ( member @ nat @ X2 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) )
=> ( P @ X2 ) ) ) ) ).
% all_nat_less_eq
thf(fact_3853_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_3854_atLeastLessThanSuc__atLeastAtMost,axiom,
! [L: nat,U: nat] :
( ( set_or7035219750837199246ssThan @ nat @ L @ ( suc @ U ) )
= ( set_or1337092689740270186AtMost @ nat @ L @ U ) ) ).
% atLeastLessThanSuc_atLeastAtMost
thf(fact_3855_finite__has__minimal,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A6: set @ A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X3: A] :
( ( member @ A @ X3 @ A6 )
& ! [Xa: A] :
( ( member @ A @ Xa @ A6 )
=> ( ( ord_less_eq @ A @ Xa @ X3 )
=> ( X3 = Xa ) ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_3856_finite__has__maximal,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A6: set @ A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X3: A] :
( ( member @ A @ X3 @ A6 )
& ! [Xa: A] :
( ( member @ A @ Xa @ A6 )
=> ( ( ord_less_eq @ A @ X3 @ Xa )
=> ( X3 = Xa ) ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_3857_infinite__growing,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X8: set @ A] :
( ( X8
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ X8 )
=> ? [Xa: A] :
( ( member @ A @ Xa @ X8 )
& ( ord_less @ A @ X3 @ Xa ) ) )
=> ~ ( finite_finite2 @ A @ X8 ) ) ) ) ).
% infinite_growing
thf(fact_3858_ex__min__if__finite,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [S3: set @ A] :
( ( finite_finite2 @ A @ S3 )
=> ( ( S3
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X3: A] :
( ( member @ A @ X3 @ S3 )
& ~ ? [Xa: A] :
( ( member @ A @ Xa @ S3 )
& ( ord_less @ A @ Xa @ X3 ) ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_3859_infinite__Icc,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ~ ( finite_finite2 @ A @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) ) ) ) ).
% infinite_Icc
thf(fact_3860_lessThan__atLeast0,axiom,
( ( set_ord_lessThan @ nat )
= ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) ) ) ).
% lessThan_atLeast0
thf(fact_3861_card__le__if__inj__on__rel,axiom,
! [B: $tType,A: $tType,B7: set @ A,A6: set @ B,R: B > A > $o] :
( ( finite_finite2 @ A @ B7 )
=> ( ! [A8: B] :
( ( member @ B @ A8 @ A6 )
=> ? [B11: A] :
( ( member @ A @ B11 @ B7 )
& ( R @ A8 @ B11 ) ) )
=> ( ! [A13: B,A24: B,B5: A] :
( ( member @ B @ A13 @ A6 )
=> ( ( member @ B @ A24 @ A6 )
=> ( ( member @ A @ B5 @ B7 )
=> ( ( R @ A13 @ B5 )
=> ( ( R @ A24 @ B5 )
=> ( A13 = A24 ) ) ) ) ) )
=> ( ord_less_eq @ nat @ ( finite_card @ B @ A6 ) @ ( finite_card @ A @ B7 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_3862_remdups__adj__length,axiom,
! [A: $tType,Xs: list @ A] : ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ ( remdups_adj @ A @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ).
% remdups_adj_length
thf(fact_3863_infinite__Ioc,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ~ ( finite_finite2 @ A @ ( set_or3652927894154168847AtMost @ A @ A3 @ B2 ) ) ) ) ).
% infinite_Ioc
thf(fact_3864_atLeastLessThan0,axiom,
! [M: nat] :
( ( set_or7035219750837199246ssThan @ nat @ M @ ( zero_zero @ nat ) )
= ( bot_bot @ ( set @ nat ) ) ) ).
% atLeastLessThan0
thf(fact_3865_infinite__Ioo,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ~ ( finite_finite2 @ A @ ( set_or5935395276787703475ssThan @ A @ A3 @ B2 ) ) ) ) ).
% infinite_Ioo
thf(fact_3866_atLeastSucLessThan__greaterThanLessThan,axiom,
! [L: nat,U: nat] :
( ( set_or7035219750837199246ssThan @ nat @ ( suc @ L ) @ U )
= ( set_or5935395276787703475ssThan @ nat @ L @ U ) ) ).
% atLeastSucLessThan_greaterThanLessThan
thf(fact_3867_ivl__disj__un__two_I7_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M: A,U: A] :
( ( ord_less_eq @ A @ L @ M )
=> ( ( ord_less_eq @ A @ M @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ L @ M ) @ ( set_or1337092689740270186AtMost @ A @ M @ U ) )
= ( set_or1337092689740270186AtMost @ A @ L @ U ) ) ) ) ) ).
% ivl_disj_un_two(7)
thf(fact_3868_ivl__disj__un__one_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,U: A] :
( ( ord_less_eq @ A @ L @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_ord_lessThan @ A @ L ) @ ( set_or7035219750837199246ssThan @ A @ L @ U ) )
= ( set_ord_lessThan @ A @ U ) ) ) ) ).
% ivl_disj_un_one(2)
thf(fact_3869_finite__ranking__induct,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [S3: set @ B,P: ( set @ B ) > $o,F: B > A] :
( ( finite_finite2 @ B @ S3 )
=> ( ( P @ ( bot_bot @ ( set @ B ) ) )
=> ( ! [X3: B,S4: set @ B] :
( ( finite_finite2 @ B @ S4 )
=> ( ! [Y4: B] :
( ( member @ B @ Y4 @ S4 )
=> ( ord_less_eq @ A @ ( F @ Y4 ) @ ( F @ X3 ) ) )
=> ( ( P @ S4 )
=> ( P @ ( insert @ B @ X3 @ S4 ) ) ) ) )
=> ( P @ S3 ) ) ) ) ) ).
% finite_ranking_induct
thf(fact_3870_finite__linorder__max__induct,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ A6 )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [B5: A,A9: set @ A] :
( ( finite_finite2 @ A @ A9 )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ A9 )
=> ( ord_less @ A @ X4 @ B5 ) )
=> ( ( P @ A9 )
=> ( P @ ( insert @ A @ B5 @ A9 ) ) ) ) )
=> ( P @ A6 ) ) ) ) ) ).
% finite_linorder_max_induct
thf(fact_3871_finite__linorder__min__induct,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ A6 )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [B5: A,A9: set @ A] :
( ( finite_finite2 @ A @ A9 )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ A9 )
=> ( ord_less @ A @ B5 @ X4 ) )
=> ( ( P @ A9 )
=> ( P @ ( insert @ A @ B5 @ A9 ) ) ) ) )
=> ( P @ A6 ) ) ) ) ) ).
% finite_linorder_min_induct
thf(fact_3872_card__eq__0__iff,axiom,
! [A: $tType,A6: set @ A] :
( ( ( finite_card @ A @ A6 )
= ( zero_zero @ nat ) )
= ( ( A6
= ( bot_bot @ ( set @ A ) ) )
| ~ ( finite_finite2 @ A @ A6 ) ) ) ).
% card_eq_0_iff
thf(fact_3873_atLeast0__lessThan__Suc,axiom,
! [N: nat] :
( ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) )
= ( insert @ nat @ N @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% atLeast0_lessThan_Suc
thf(fact_3874_card__Suc__eq__finite,axiom,
! [A: $tType,A6: set @ A,K: nat] :
( ( ( finite_card @ A @ A6 )
= ( suc @ K ) )
= ( ? [B3: A,B6: set @ A] :
( ( A6
= ( insert @ A @ B3 @ B6 ) )
& ~ ( member @ A @ B3 @ B6 )
& ( ( finite_card @ A @ B6 )
= K )
& ( finite_finite2 @ A @ B6 ) ) ) ) ).
% card_Suc_eq_finite
thf(fact_3875_card__insert__if,axiom,
! [A: $tType,A6: set @ A,X: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( ( member @ A @ X @ A6 )
=> ( ( finite_card @ A @ ( insert @ A @ X @ A6 ) )
= ( finite_card @ A @ A6 ) ) )
& ( ~ ( member @ A @ X @ A6 )
=> ( ( finite_card @ A @ ( insert @ A @ X @ A6 ) )
= ( suc @ ( finite_card @ A @ A6 ) ) ) ) ) ) ).
% card_insert_if
thf(fact_3876_card__ge__0__finite,axiom,
! [A: $tType,A6: set @ A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( finite_card @ A @ A6 ) )
=> ( finite_finite2 @ A @ A6 ) ) ).
% card_ge_0_finite
thf(fact_3877_obtain__subset__with__card__n,axiom,
! [A: $tType,N: nat,S3: set @ A] :
( ( ord_less_eq @ nat @ N @ ( finite_card @ A @ S3 ) )
=> ~ ! [T4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ T4 @ S3 )
=> ( ( ( finite_card @ A @ T4 )
= N )
=> ~ ( finite_finite2 @ A @ T4 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_3878_finite__if__finite__subsets__card__bdd,axiom,
! [A: $tType,F4: set @ A,C6: nat] :
( ! [G3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ G3 @ F4 )
=> ( ( finite_finite2 @ A @ G3 )
=> ( ord_less_eq @ nat @ ( finite_card @ A @ G3 ) @ C6 ) ) )
=> ( ( finite_finite2 @ A @ F4 )
& ( ord_less_eq @ nat @ ( finite_card @ A @ F4 ) @ C6 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_3879_card__seteq,axiom,
! [A: $tType,B7: set @ A,A6: set @ A] :
( ( finite_finite2 @ A @ B7 )
=> ( ( ord_less_eq @ ( set @ A ) @ A6 @ B7 )
=> ( ( ord_less_eq @ nat @ ( finite_card @ A @ B7 ) @ ( finite_card @ A @ A6 ) )
=> ( A6 = B7 ) ) ) ) ).
% card_seteq
thf(fact_3880_card__mono,axiom,
! [A: $tType,B7: set @ A,A6: set @ A] :
( ( finite_finite2 @ A @ B7 )
=> ( ( ord_less_eq @ ( set @ A ) @ A6 @ B7 )
=> ( ord_less_eq @ nat @ ( finite_card @ A @ A6 ) @ ( finite_card @ A @ B7 ) ) ) ) ).
% card_mono
thf(fact_3881_card__less__sym__Diff,axiom,
! [A: $tType,A6: set @ A,B7: set @ A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( finite_finite2 @ A @ B7 )
=> ( ( ord_less @ nat @ ( finite_card @ A @ A6 ) @ ( finite_card @ A @ B7 ) )
=> ( ord_less @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A6 @ B7 ) ) @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ B7 @ A6 ) ) ) ) ) ) ).
% card_less_sym_Diff
thf(fact_3882_card__le__sym__Diff,axiom,
! [A: $tType,A6: set @ A,B7: set @ A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( finite_finite2 @ A @ B7 )
=> ( ( ord_less_eq @ nat @ ( finite_card @ A @ A6 ) @ ( finite_card @ A @ B7 ) )
=> ( ord_less_eq @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A6 @ B7 ) ) @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ B7 @ A6 ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_3883_psubset__card__mono,axiom,
! [A: $tType,B7: set @ A,A6: set @ A] :
( ( finite_finite2 @ A @ B7 )
=> ( ( ord_less @ ( set @ A ) @ A6 @ B7 )
=> ( ord_less @ nat @ ( finite_card @ A @ A6 ) @ ( finite_card @ A @ B7 ) ) ) ) ).
% psubset_card_mono
thf(fact_3884_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_3885_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A,B2: A,C2: A,D2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) @ ( set_or7035219750837199246ssThan @ A @ C2 @ D2 ) )
= ( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ A3 )
& ( ord_less @ A @ B2 @ D2 ) ) ) ) ) ).
% atLeastAtMost_subseteq_atLeastLessThan_iff
thf(fact_3886_atLeastLessThan__subseteq__atLeastAtMost__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B2: A,C2: A,D2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ A3 @ B2 ) @ ( set_or1337092689740270186AtMost @ A @ C2 @ D2 ) )
= ( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ A3 )
& ( ord_less_eq @ A @ B2 @ D2 ) ) ) ) ) ).
% atLeastLessThan_subseteq_atLeastAtMost_iff
thf(fact_3887_ivl__disj__un__two__touch_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M: A,U: A] :
( ( ord_less_eq @ A @ L @ M )
=> ( ( ord_less @ A @ M @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ M ) @ ( set_or7035219750837199246ssThan @ A @ M @ U ) )
= ( set_or7035219750837199246ssThan @ A @ L @ U ) ) ) ) ) ).
% ivl_disj_un_two_touch(2)
thf(fact_3888_greaterThanAtMost__subseteq__atLeastLessThan__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B2: A,C2: A,D2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ A3 @ B2 ) @ ( set_or7035219750837199246ssThan @ A @ C2 @ D2 ) )
= ( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ A3 )
& ( ord_less @ A @ B2 @ D2 ) ) ) ) ) ).
% greaterThanAtMost_subseteq_atLeastLessThan_iff
thf(fact_3889_greaterThanLessThan__subseteq__atLeastLessThan__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B2: A,C2: A,D2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ A3 @ B2 ) @ ( set_or7035219750837199246ssThan @ A @ C2 @ D2 ) )
= ( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ A3 )
& ( ord_less_eq @ A @ B2 @ D2 ) ) ) ) ) ).
% greaterThanLessThan_subseteq_atLeastLessThan_iff
thf(fact_3890_ivl__disj__un__two_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M: A,U: A] :
( ( ord_less @ A @ L @ M )
=> ( ( ord_less_eq @ A @ M @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ L @ M ) @ ( set_or7035219750837199246ssThan @ A @ M @ U ) )
= ( set_or5935395276787703475ssThan @ A @ L @ U ) ) ) ) ) ).
% ivl_disj_un_two(1)
thf(fact_3891_card__gt__0__iff,axiom,
! [A: $tType,A6: set @ A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( finite_card @ A @ A6 ) )
= ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
& ( finite_finite2 @ A @ A6 ) ) ) ).
% card_gt_0_iff
thf(fact_3892_card__le__Suc0__iff__eq,axiom,
! [A: $tType,A6: set @ A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( ord_less_eq @ nat @ ( finite_card @ A @ A6 ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ A6 )
=> ! [Y6: A] :
( ( member @ A @ Y6 @ A6 )
=> ( X2 = Y6 ) ) ) ) ) ) ).
% card_le_Suc0_iff_eq
thf(fact_3893_card__le__Suc__iff,axiom,
! [A: $tType,N: nat,A6: set @ A] :
( ( ord_less_eq @ nat @ ( suc @ N ) @ ( finite_card @ A @ A6 ) )
= ( ? [A5: A,B6: set @ A] :
( ( A6
= ( insert @ A @ A5 @ B6 ) )
& ~ ( member @ A @ A5 @ B6 )
& ( ord_less_eq @ nat @ N @ ( finite_card @ A @ B6 ) )
& ( finite_finite2 @ A @ B6 ) ) ) ) ).
% card_le_Suc_iff
thf(fact_3894_ivl__disj__un__two__touch_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M: A,U: A] :
( ( ord_less @ A @ L @ M )
=> ( ( ord_less @ A @ M @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ L @ M ) @ ( set_or7035219750837199246ssThan @ A @ M @ U ) )
= ( set_or5935395276787703475ssThan @ A @ L @ U ) ) ) ) ) ).
% ivl_disj_un_two_touch(1)
thf(fact_3895_finite__induct__select,axiom,
! [A: $tType,S3: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ S3 )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [T4: set @ A] :
( ( ord_less @ ( set @ A ) @ T4 @ S3 )
=> ( ( P @ T4 )
=> ? [X4: A] :
( ( member @ A @ X4 @ ( minus_minus @ ( set @ A ) @ S3 @ T4 ) )
& ( P @ ( insert @ A @ X4 @ T4 ) ) ) ) )
=> ( P @ S3 ) ) ) ) ).
% finite_induct_select
thf(fact_3896_card__psubset,axiom,
! [A: $tType,B7: set @ A,A6: set @ A] :
( ( finite_finite2 @ A @ B7 )
=> ( ( ord_less_eq @ ( set @ A ) @ A6 @ B7 )
=> ( ( ord_less @ nat @ ( finite_card @ A @ A6 ) @ ( finite_card @ A @ B7 ) )
=> ( ord_less @ ( set @ A ) @ A6 @ B7 ) ) ) ) ).
% card_psubset
thf(fact_3897_diff__card__le__card__Diff,axiom,
! [A: $tType,B7: set @ A,A6: set @ A] :
( ( finite_finite2 @ A @ B7 )
=> ( ord_less_eq @ nat @ ( minus_minus @ nat @ ( finite_card @ A @ A6 ) @ ( finite_card @ A @ B7 ) ) @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A6 @ B7 ) ) ) ) ).
% diff_card_le_card_Diff
thf(fact_3898_remdups__adj__adjacent,axiom,
! [A: $tType,I: nat,Xs: list @ A] :
( ( ord_less @ nat @ ( suc @ I ) @ ( size_size @ ( list @ A ) @ ( remdups_adj @ A @ Xs ) ) )
=> ( ( nth @ A @ ( remdups_adj @ A @ Xs ) @ I )
!= ( nth @ A @ ( remdups_adj @ A @ Xs ) @ ( suc @ I ) ) ) ) ).
% remdups_adj_adjacent
thf(fact_3899_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_3900_atLeastLessThanSuc,axiom,
! [M: nat,N: nat] :
( ( ( ord_less_eq @ nat @ M @ N )
=> ( ( set_or7035219750837199246ssThan @ nat @ M @ ( suc @ N ) )
= ( insert @ nat @ N @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) )
& ( ~ ( ord_less_eq @ nat @ M @ N )
=> ( ( set_or7035219750837199246ssThan @ nat @ M @ ( suc @ N ) )
= ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% atLeastLessThanSuc
thf(fact_3901_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_3902_ivl__disj__un__singleton_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,U: A] :
( ( ord_less_eq @ A @ L @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ L @ U ) @ ( insert @ A @ U @ ( bot_bot @ ( set @ A ) ) ) )
= ( set_or1337092689740270186AtMost @ A @ L @ U ) ) ) ) ).
% ivl_disj_un_singleton(6)
thf(fact_3903_ivl__disj__un__two_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M: A,U: A] :
( ( ord_less_eq @ A @ L @ M )
=> ( ( ord_less @ A @ M @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ M ) @ ( set_or5935395276787703475ssThan @ A @ M @ U ) )
= ( set_or7035219750837199246ssThan @ A @ L @ U ) ) ) ) ) ).
% ivl_disj_un_two(4)
thf(fact_3904_ivl__disj__un__singleton_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,U: A] :
( ( ord_less @ A @ L @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( insert @ A @ L @ ( bot_bot @ ( set @ A ) ) ) @ ( set_or5935395276787703475ssThan @ A @ L @ U ) )
= ( set_or7035219750837199246ssThan @ A @ L @ U ) ) ) ) ).
% ivl_disj_un_singleton(3)
thf(fact_3905_card_Oremove,axiom,
! [A: $tType,A6: set @ A,X: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( member @ A @ X @ A6 )
=> ( ( finite_card @ A @ A6 )
= ( suc @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% card.remove
thf(fact_3906_card_Oinsert__remove,axiom,
! [A: $tType,A6: set @ A,X: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( finite_card @ A @ ( insert @ A @ X @ A6 ) )
= ( suc @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% card.insert_remove
thf(fact_3907_card__Suc__Diff1,axiom,
! [A: $tType,A6: set @ A,X: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( member @ A @ X @ A6 )
=> ( ( suc @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) )
= ( finite_card @ A @ A6 ) ) ) ) ).
% card_Suc_Diff1
thf(fact_3908_card__Diff1__less,axiom,
! [A: $tType,A6: set @ A,X: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( member @ A @ X @ A6 )
=> ( ord_less @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) @ ( finite_card @ A @ A6 ) ) ) ) ).
% card_Diff1_less
thf(fact_3909_card__Diff2__less,axiom,
! [A: $tType,A6: set @ A,X: A,Y3: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( member @ A @ X @ A6 )
=> ( ( member @ A @ Y3 @ A6 )
=> ( ord_less @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ ( insert @ A @ Y3 @ ( bot_bot @ ( set @ A ) ) ) ) ) @ ( finite_card @ A @ A6 ) ) ) ) ) ).
% card_Diff2_less
thf(fact_3910_card__Diff1__less__iff,axiom,
! [A: $tType,A6: set @ A,X: A] :
( ( ord_less @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) @ ( finite_card @ A @ A6 ) )
= ( ( finite_finite2 @ A @ A6 )
& ( member @ A @ X @ A6 ) ) ) ).
% card_Diff1_less_iff
thf(fact_3911_remdups__adj__length__ge1,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( size_size @ ( list @ A ) @ ( remdups_adj @ A @ Xs ) ) ) ) ).
% remdups_adj_length_ge1
thf(fact_3912_atLeastLessThan__nat__numeral,axiom,
! [M: nat,K: num] :
( ( ( ord_less_eq @ nat @ M @ ( pred_numeral @ K ) )
=> ( ( set_or7035219750837199246ssThan @ nat @ M @ ( numeral_numeral @ nat @ K ) )
= ( insert @ nat @ ( pred_numeral @ K ) @ ( set_or7035219750837199246ssThan @ nat @ M @ ( pred_numeral @ K ) ) ) ) )
& ( ~ ( ord_less_eq @ nat @ M @ ( pred_numeral @ K ) )
=> ( ( set_or7035219750837199246ssThan @ nat @ M @ ( numeral_numeral @ nat @ K ) )
= ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% atLeastLessThan_nat_numeral
thf(fact_3913_set__encode__insert,axiom,
! [A6: set @ nat,N: nat] :
( ( finite_finite2 @ nat @ A6 )
=> ( ~ ( member @ nat @ N @ A6 )
=> ( ( nat_set_encode @ ( insert @ nat @ N @ A6 ) )
= ( plus_plus @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ ( nat_set_encode @ A6 ) ) ) ) ) ).
% set_encode_insert
thf(fact_3914_even__set__encode__iff,axiom,
! [A6: set @ nat] :
( ( finite_finite2 @ nat @ A6 )
=> ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( nat_set_encode @ A6 ) )
= ( ~ ( member @ nat @ ( zero_zero @ nat ) @ A6 ) ) ) ) ).
% even_set_encode_iff
thf(fact_3915_arg__min__if__finite_I2_J,axiom,
! [B: $tType,A: $tType] :
( ( order @ B )
=> ! [S3: set @ A,F: A > B] :
( ( finite_finite2 @ A @ S3 )
=> ( ( S3
!= ( bot_bot @ ( set @ A ) ) )
=> ~ ? [X4: A] :
( ( member @ A @ X4 @ S3 )
& ( ord_less @ B @ ( F @ X4 ) @ ( F @ ( lattic7623131987881927897min_on @ A @ B @ F @ S3 ) ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_3916_arg__min__least,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [S3: set @ A,Y3: A,F: A > B] :
( ( finite_finite2 @ A @ S3 )
=> ( ( S3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( member @ A @ Y3 @ S3 )
=> ( ord_less_eq @ B @ ( F @ ( lattic7623131987881927897min_on @ A @ B @ F @ S3 ) ) @ ( F @ Y3 ) ) ) ) ) ) ).
% arg_min_least
thf(fact_3917_set__decode__inverse,axiom,
! [N: nat] :
( ( nat_set_encode @ ( nat_set_decode @ N ) )
= N ) ).
% set_decode_inverse
thf(fact_3918_set__encode__empty,axiom,
( ( nat_set_encode @ ( bot_bot @ ( set @ nat ) ) )
= ( zero_zero @ nat ) ) ).
% set_encode_empty
thf(fact_3919_set__encode__inverse,axiom,
! [A6: set @ nat] :
( ( finite_finite2 @ nat @ A6 )
=> ( ( nat_set_decode @ ( nat_set_encode @ A6 ) )
= A6 ) ) ).
% set_encode_inverse
thf(fact_3920_finite__atLeastZeroLessThan__int,axiom,
! [U: int] : ( finite_finite2 @ int @ ( set_or7035219750837199246ssThan @ int @ ( zero_zero @ int ) @ U ) ) ).
% finite_atLeastZeroLessThan_int
thf(fact_3921_finite__maxlen,axiom,
! [A: $tType,M7: set @ ( list @ A )] :
( ( finite_finite2 @ ( list @ A ) @ M7 )
=> ? [N4: nat] :
! [X4: list @ A] :
( ( member @ ( list @ A ) @ X4 @ M7 )
=> ( ord_less @ nat @ ( size_size @ ( list @ A ) @ X4 ) @ N4 ) ) ) ).
% finite_maxlen
thf(fact_3922_set__encode__eq,axiom,
! [A6: set @ nat,B7: set @ nat] :
( ( finite_finite2 @ nat @ A6 )
=> ( ( finite_finite2 @ nat @ B7 )
=> ( ( ( nat_set_encode @ A6 )
= ( nat_set_encode @ B7 ) )
= ( A6 = B7 ) ) ) ) ).
% set_encode_eq
thf(fact_3923_atLeastLessThanPlusOne__atLeastAtMost__int,axiom,
! [L: int,U: int] :
( ( set_or7035219750837199246ssThan @ int @ L @ ( plus_plus @ int @ U @ ( one_one @ int ) ) )
= ( set_or1337092689740270186AtMost @ int @ L @ U ) ) ).
% atLeastLessThanPlusOne_atLeastAtMost_int
thf(fact_3924_card__atLeastZeroLessThan__int,axiom,
! [U: int] :
( ( finite_card @ int @ ( set_or7035219750837199246ssThan @ int @ ( zero_zero @ int ) @ U ) )
= ( nat2 @ U ) ) ).
% card_atLeastZeroLessThan_int
thf(fact_3925_atLeastPlusOneLessThan__greaterThanLessThan__int,axiom,
! [L: int,U: int] :
( ( set_or7035219750837199246ssThan @ int @ ( plus_plus @ int @ L @ ( one_one @ int ) ) @ U )
= ( set_or5935395276787703475ssThan @ int @ L @ U ) ) ).
% atLeastPlusOneLessThan_greaterThanLessThan_int
thf(fact_3926_set__encode__inf,axiom,
! [A6: set @ nat] :
( ~ ( finite_finite2 @ nat @ A6 )
=> ( ( nat_set_encode @ A6 )
= ( zero_zero @ nat ) ) ) ).
% set_encode_inf
thf(fact_3927_atLeastLessThan__upto,axiom,
( ( set_or7035219750837199246ssThan @ int )
= ( ^ [I5: int,J3: int] : ( set2 @ int @ ( upto @ I5 @ ( minus_minus @ int @ J3 @ ( one_one @ int ) ) ) ) ) ) ).
% atLeastLessThan_upto
thf(fact_3928_wmax__emptyI,axiom,
! [X8: set @ ( product_prod @ nat @ nat )] :
( ( finite_finite2 @ ( product_prod @ nat @ nat ) @ X8 )
=> ( 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 ) ) ) @ X8 ) @ fun_max_weak ) ) ).
% wmax_emptyI
thf(fact_3929_smax__emptyI,axiom,
! [Y7: set @ ( product_prod @ nat @ nat )] :
( ( finite_finite2 @ ( product_prod @ nat @ nat ) @ Y7 )
=> ( ( Y7
!= ( bot_bot @ ( set @ ( product_prod @ nat @ nat ) ) ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ ( bot_bot @ ( set @ ( product_prod @ nat @ nat ) ) ) @ Y7 ) @ fun_max_strict ) ) ) ).
% smax_emptyI
thf(fact_3930_arg__min__if__finite_I1_J,axiom,
! [B: $tType,A: $tType] :
( ( order @ B )
=> ! [S3: set @ A,F: A > B] :
( ( finite_finite2 @ A @ S3 )
=> ( ( S3
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ A @ ( lattic7623131987881927897min_on @ A @ B @ F @ S3 ) @ S3 ) ) ) ) ).
% arg_min_if_finite(1)
thf(fact_3931_nth__image,axiom,
! [A: $tType,L: nat,Xs: list @ A] :
( ( ord_less_eq @ nat @ L @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( image @ nat @ A @ ( nth @ A @ Xs ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ L ) )
= ( set2 @ A @ ( take @ A @ L @ Xs ) ) ) ) ).
% nth_image
thf(fact_3932_fact__split,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [K: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ N )
=> ( ( semiring_char_0_fact @ A @ N )
= ( times_times @ A @ ( semiring_1_of_nat @ A @ ( groups7121269368397514597t_prod @ nat @ nat @ suc @ ( set_or7035219750837199246ssThan @ nat @ ( minus_minus @ nat @ N @ K ) @ N ) ) ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ N @ K ) ) ) ) ) ) ).
% fact_split
thf(fact_3933_sum__list__update,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [K: nat,Xs: list @ A,X: A] :
( ( ord_less @ nat @ K @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( groups8242544230860333062m_list @ A @ ( list_update @ A @ Xs @ K @ X ) )
= ( minus_minus @ A @ ( plus_plus @ A @ ( groups8242544230860333062m_list @ A @ Xs ) @ X ) @ ( nth @ A @ Xs @ K ) ) ) ) ) ).
% sum_list_update
thf(fact_3934_set__take__disj__set__drop__if__distinct,axiom,
! [A: $tType,Vs: list @ A,I: nat,J: nat] :
( ( distinct @ A @ Vs )
=> ( ( ord_less_eq @ nat @ I @ J )
=> ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ ( take @ A @ I @ Vs ) ) @ ( set2 @ A @ ( drop @ A @ J @ Vs ) ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% set_take_disj_set_drop_if_distinct
thf(fact_3935_inf_Oidem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A] :
( ( inf_inf @ A @ A3 @ A3 )
= A3 ) ) ).
% inf.idem
thf(fact_3936_inf__idem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A] :
( ( inf_inf @ A @ X @ X )
= X ) ) ).
% inf_idem
thf(fact_3937_inf_Oleft__idem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,B2: A] :
( ( inf_inf @ A @ A3 @ ( inf_inf @ A @ A3 @ B2 ) )
= ( inf_inf @ A @ A3 @ B2 ) ) ) ).
% inf.left_idem
thf(fact_3938_inf__left__idem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A,Y3: A] :
( ( inf_inf @ A @ X @ ( inf_inf @ A @ X @ Y3 ) )
= ( inf_inf @ A @ X @ Y3 ) ) ) ).
% inf_left_idem
thf(fact_3939_inf_Oright__idem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,B2: A] :
( ( inf_inf @ A @ ( inf_inf @ A @ A3 @ B2 ) @ B2 )
= ( inf_inf @ A @ A3 @ B2 ) ) ) ).
% inf.right_idem
thf(fact_3940_inf__right__idem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A,Y3: A] :
( ( inf_inf @ A @ ( inf_inf @ A @ X @ Y3 ) @ Y3 )
= ( inf_inf @ A @ X @ Y3 ) ) ) ).
% inf_right_idem
thf(fact_3941_inf__apply,axiom,
! [B: $tType,A: $tType] :
( ( semilattice_inf @ B )
=> ( ( inf_inf @ ( A > B ) )
= ( ^ [F2: A > B,G2: A > B,X2: A] : ( inf_inf @ B @ ( F2 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ).
% inf_apply
thf(fact_3942_le__inf__iff,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A,Y3: A,Z3: A] :
( ( ord_less_eq @ A @ X @ ( inf_inf @ A @ Y3 @ Z3 ) )
= ( ( ord_less_eq @ A @ X @ Y3 )
& ( ord_less_eq @ A @ X @ Z3 ) ) ) ) ).
% le_inf_iff
thf(fact_3943_inf_Obounded__iff,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ ( inf_inf @ A @ B2 @ C2 ) )
= ( ( ord_less_eq @ A @ A3 @ B2 )
& ( ord_less_eq @ A @ A3 @ C2 ) ) ) ) ).
% inf.bounded_iff
thf(fact_3944_boolean__algebra_Oconj__zero__right,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A] :
( ( inf_inf @ A @ X @ ( bot_bot @ A ) )
= ( bot_bot @ A ) ) ) ).
% boolean_algebra.conj_zero_right
thf(fact_3945_boolean__algebra_Oconj__zero__left,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A] :
( ( inf_inf @ A @ ( bot_bot @ A ) @ X )
= ( bot_bot @ A ) ) ) ).
% boolean_algebra.conj_zero_left
thf(fact_3946_inf__bot__right,axiom,
! [A: $tType] :
( ( bounded_lattice_bot @ A )
=> ! [X: A] :
( ( inf_inf @ A @ X @ ( bot_bot @ A ) )
= ( bot_bot @ A ) ) ) ).
% inf_bot_right
thf(fact_3947_inf__bot__left,axiom,
! [A: $tType] :
( ( bounded_lattice_bot @ A )
=> ! [X: A] :
( ( inf_inf @ A @ ( bot_bot @ A ) @ X )
= ( bot_bot @ A ) ) ) ).
% inf_bot_left
thf(fact_3948_inf__sup__absorb,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y3: A] :
( ( inf_inf @ A @ X @ ( sup_sup @ A @ X @ Y3 ) )
= X ) ) ).
% inf_sup_absorb
thf(fact_3949_sup__inf__absorb,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y3: A] :
( ( sup_sup @ A @ X @ ( inf_inf @ A @ X @ Y3 ) )
= X ) ) ).
% sup_inf_absorb
thf(fact_3950_image__Suc__atLeastAtMost,axiom,
! [I: nat,J: nat] :
( ( image @ nat @ nat @ suc @ ( set_or1337092689740270186AtMost @ nat @ I @ J ) )
= ( set_or1337092689740270186AtMost @ nat @ ( suc @ I ) @ ( suc @ J ) ) ) ).
% image_Suc_atLeastAtMost
thf(fact_3951_image__Suc__atLeastLessThan,axiom,
! [I: nat,J: nat] :
( ( image @ nat @ nat @ suc @ ( set_or7035219750837199246ssThan @ nat @ I @ J ) )
= ( set_or7035219750837199246ssThan @ nat @ ( suc @ I ) @ ( suc @ J ) ) ) ).
% image_Suc_atLeastLessThan
thf(fact_3952_Ints__prod,axiom,
! [A: $tType,B: $tType] :
( ( ( comm_monoid_mult @ B )
& ( ring_1 @ B ) )
=> ! [A6: set @ A,F: A > B] :
( ! [X3: A] :
( ( member @ A @ X3 @ A6 )
=> ( member @ B @ ( F @ X3 ) @ ( ring_1_Ints @ B ) ) )
=> ( member @ B @ ( groups7121269368397514597t_prod @ A @ B @ F @ A6 ) @ ( ring_1_Ints @ B ) ) ) ) ).
% Ints_prod
thf(fact_3953_image__add__0,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [S3: set @ A] :
( ( image @ A @ A @ ( plus_plus @ A @ ( zero_zero @ A ) ) @ S3 )
= S3 ) ) ).
% image_add_0
thf(fact_3954_inf__compl__bot__left1,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y3: A] :
( ( inf_inf @ A @ ( uminus_uminus @ A @ X ) @ ( inf_inf @ A @ X @ Y3 ) )
= ( bot_bot @ A ) ) ) ).
% inf_compl_bot_left1
thf(fact_3955_inf__compl__bot__left2,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y3: A] :
( ( inf_inf @ A @ X @ ( inf_inf @ A @ ( uminus_uminus @ A @ X ) @ Y3 ) )
= ( bot_bot @ A ) ) ) ).
% inf_compl_bot_left2
thf(fact_3956_inf__compl__bot__right,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y3: A] :
( ( inf_inf @ A @ X @ ( inf_inf @ A @ Y3 @ ( uminus_uminus @ A @ X ) ) )
= ( bot_bot @ A ) ) ) ).
% inf_compl_bot_right
thf(fact_3957_boolean__algebra_Oconj__cancel__left,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A] :
( ( inf_inf @ A @ ( uminus_uminus @ A @ X ) @ X )
= ( bot_bot @ A ) ) ) ).
% boolean_algebra.conj_cancel_left
thf(fact_3958_boolean__algebra_Oconj__cancel__right,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A] :
( ( inf_inf @ A @ X @ ( uminus_uminus @ A @ X ) )
= ( bot_bot @ A ) ) ) ).
% boolean_algebra.conj_cancel_right
thf(fact_3959_boolean__algebra_Ode__Morgan__conj,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y3: A] :
( ( uminus_uminus @ A @ ( inf_inf @ A @ X @ Y3 ) )
= ( sup_sup @ A @ ( uminus_uminus @ A @ X ) @ ( uminus_uminus @ A @ Y3 ) ) ) ) ).
% boolean_algebra.de_Morgan_conj
thf(fact_3960_boolean__algebra_Ode__Morgan__disj,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y3: A] :
( ( uminus_uminus @ A @ ( sup_sup @ A @ X @ Y3 ) )
= ( inf_inf @ A @ ( uminus_uminus @ A @ X ) @ ( uminus_uminus @ A @ Y3 ) ) ) ) ).
% boolean_algebra.de_Morgan_disj
thf(fact_3961_image__uminus__atLeastAtMost,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [X: A,Y3: A] :
( ( image @ A @ A @ ( uminus_uminus @ A ) @ ( set_or1337092689740270186AtMost @ A @ X @ Y3 ) )
= ( set_or1337092689740270186AtMost @ A @ ( uminus_uminus @ A @ Y3 ) @ ( uminus_uminus @ A @ X ) ) ) ) ).
% image_uminus_atLeastAtMost
thf(fact_3962_sum__list_ONil,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ( ( groups8242544230860333062m_list @ A @ ( nil @ A ) )
= ( zero_zero @ A ) ) ) ).
% sum_list.Nil
thf(fact_3963_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_3964_image__uminus__greaterThanLessThan,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [X: A,Y3: A] :
( ( image @ A @ A @ ( uminus_uminus @ A ) @ ( set_or5935395276787703475ssThan @ A @ X @ Y3 ) )
= ( set_or5935395276787703475ssThan @ A @ ( uminus_uminus @ A @ Y3 ) @ ( uminus_uminus @ A @ X ) ) ) ) ).
% image_uminus_greaterThanLessThan
thf(fact_3965_Compl__disjoint,axiom,
! [A: $tType,A6: set @ A] :
( ( inf_inf @ ( set @ A ) @ A6 @ ( uminus_uminus @ ( set @ A ) @ A6 ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Compl_disjoint
thf(fact_3966_Compl__disjoint2,axiom,
! [A: $tType,A6: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A6 ) @ A6 )
= ( bot_bot @ ( set @ A ) ) ) ).
% Compl_disjoint2
thf(fact_3967_Diff__Compl,axiom,
! [A: $tType,A6: set @ A,B7: set @ A] :
( ( minus_minus @ ( set @ A ) @ A6 @ ( uminus_uminus @ ( set @ A ) @ B7 ) )
= ( inf_inf @ ( set @ A ) @ A6 @ B7 ) ) ).
% Diff_Compl
thf(fact_3968_image__uminus__greaterThanAtMost,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [X: A,Y3: A] :
( ( image @ A @ A @ ( uminus_uminus @ A ) @ ( set_or3652927894154168847AtMost @ A @ X @ Y3 ) )
= ( set_or7035219750837199246ssThan @ A @ ( uminus_uminus @ A @ Y3 ) @ ( uminus_uminus @ A @ X ) ) ) ) ).
% image_uminus_greaterThanAtMost
thf(fact_3969_image__uminus__atLeastLessThan,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [X: A,Y3: A] :
( ( image @ A @ A @ ( uminus_uminus @ A ) @ ( set_or7035219750837199246ssThan @ A @ X @ Y3 ) )
= ( set_or3652927894154168847AtMost @ A @ ( uminus_uminus @ A @ Y3 ) @ ( uminus_uminus @ A @ X ) ) ) ) ).
% image_uminus_atLeastLessThan
thf(fact_3970_prod_OatMost__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_atMost @ nat @ ( suc @ N ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_atMost @ nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ).
% prod.atMost_Suc
thf(fact_3971_prod_OlessThan__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_lessThan @ nat @ ( suc @ N ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_lessThan @ nat @ N ) ) @ ( G @ N ) ) ) ) ).
% prod.lessThan_Suc
thf(fact_3972_image__mult__atLeastAtMost,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [D2: A,A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ D2 )
=> ( ( image @ A @ A @ ( times_times @ A @ D2 ) @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ D2 @ A3 ) @ ( times_times @ A @ D2 @ B2 ) ) ) ) ) ).
% image_mult_atLeastAtMost
thf(fact_3973_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_3974_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_3975_distrib__imp1,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y3: A,Z3: A] :
( ! [X3: A,Y5: A,Z4: A] :
( ( inf_inf @ A @ X3 @ ( sup_sup @ A @ Y5 @ Z4 ) )
= ( sup_sup @ A @ ( inf_inf @ A @ X3 @ Y5 ) @ ( inf_inf @ A @ X3 @ Z4 ) ) )
=> ( ( sup_sup @ A @ X @ ( inf_inf @ A @ Y3 @ Z3 ) )
= ( inf_inf @ A @ ( sup_sup @ A @ X @ Y3 ) @ ( sup_sup @ A @ X @ Z3 ) ) ) ) ) ).
% distrib_imp1
thf(fact_3976_distrib__imp2,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y3: A,Z3: A] :
( ! [X3: A,Y5: A,Z4: A] :
( ( sup_sup @ A @ X3 @ ( inf_inf @ A @ Y5 @ Z4 ) )
= ( inf_inf @ A @ ( sup_sup @ A @ X3 @ Y5 ) @ ( sup_sup @ A @ X3 @ Z4 ) ) )
=> ( ( inf_inf @ A @ X @ ( sup_sup @ A @ Y3 @ Z3 ) )
= ( sup_sup @ A @ ( inf_inf @ A @ X @ Y3 ) @ ( inf_inf @ A @ X @ Z3 ) ) ) ) ) ).
% distrib_imp2
thf(fact_3977_inf__sup__distrib1,axiom,
! [A: $tType] :
( ( distrib_lattice @ A )
=> ! [X: A,Y3: A,Z3: A] :
( ( inf_inf @ A @ X @ ( sup_sup @ A @ Y3 @ Z3 ) )
= ( sup_sup @ A @ ( inf_inf @ A @ X @ Y3 ) @ ( inf_inf @ A @ X @ Z3 ) ) ) ) ).
% inf_sup_distrib1
thf(fact_3978_inf__sup__distrib2,axiom,
! [A: $tType] :
( ( distrib_lattice @ A )
=> ! [Y3: A,Z3: A,X: A] :
( ( inf_inf @ A @ ( sup_sup @ A @ Y3 @ Z3 ) @ X )
= ( sup_sup @ A @ ( inf_inf @ A @ Y3 @ X ) @ ( inf_inf @ A @ Z3 @ X ) ) ) ) ).
% inf_sup_distrib2
thf(fact_3979_sup__inf__distrib1,axiom,
! [A: $tType] :
( ( distrib_lattice @ A )
=> ! [X: A,Y3: A,Z3: A] :
( ( sup_sup @ A @ X @ ( inf_inf @ A @ Y3 @ Z3 ) )
= ( inf_inf @ A @ ( sup_sup @ A @ X @ Y3 ) @ ( sup_sup @ A @ X @ Z3 ) ) ) ) ).
% sup_inf_distrib1
thf(fact_3980_sup__inf__distrib2,axiom,
! [A: $tType] :
( ( distrib_lattice @ A )
=> ! [Y3: A,Z3: A,X: A] :
( ( sup_sup @ A @ ( inf_inf @ A @ Y3 @ Z3 ) @ X )
= ( inf_inf @ A @ ( sup_sup @ A @ Y3 @ X ) @ ( sup_sup @ A @ Z3 @ X ) ) ) ) ).
% sup_inf_distrib2
thf(fact_3981_boolean__algebra_Oconj__disj__distrib,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y3: A,Z3: A] :
( ( inf_inf @ A @ X @ ( sup_sup @ A @ Y3 @ Z3 ) )
= ( sup_sup @ A @ ( inf_inf @ A @ X @ Y3 ) @ ( inf_inf @ A @ X @ Z3 ) ) ) ) ).
% boolean_algebra.conj_disj_distrib
thf(fact_3982_boolean__algebra_Odisj__conj__distrib,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y3: A,Z3: A] :
( ( sup_sup @ A @ X @ ( inf_inf @ A @ Y3 @ Z3 ) )
= ( inf_inf @ A @ ( sup_sup @ A @ X @ Y3 ) @ ( sup_sup @ A @ X @ Z3 ) ) ) ) ).
% boolean_algebra.disj_conj_distrib
thf(fact_3983_boolean__algebra_Oconj__disj__distrib2,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [Y3: A,Z3: A,X: A] :
( ( inf_inf @ A @ ( sup_sup @ A @ Y3 @ Z3 ) @ X )
= ( sup_sup @ A @ ( inf_inf @ A @ Y3 @ X ) @ ( inf_inf @ A @ Z3 @ X ) ) ) ) ).
% boolean_algebra.conj_disj_distrib2
thf(fact_3984_boolean__algebra_Odisj__conj__distrib2,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [Y3: A,Z3: A,X: A] :
( ( sup_sup @ A @ ( inf_inf @ A @ Y3 @ Z3 ) @ X )
= ( inf_inf @ A @ ( sup_sup @ A @ Y3 @ X ) @ ( sup_sup @ A @ Z3 @ X ) ) ) ) ).
% boolean_algebra.disj_conj_distrib2
thf(fact_3985_inf__min,axiom,
! [A: $tType] :
( ( ( semilattice_inf @ A )
& ( linorder @ A ) )
=> ( ( inf_inf @ A )
= ( ord_min @ A ) ) ) ).
% inf_min
thf(fact_3986_inf__sup__aci_I4_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y3: A] :
( ( inf_inf @ A @ X @ ( inf_inf @ A @ X @ Y3 ) )
= ( inf_inf @ A @ X @ Y3 ) ) ) ).
% inf_sup_aci(4)
thf(fact_3987_inf__sup__aci_I3_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y3: A,Z3: A] :
( ( inf_inf @ A @ X @ ( inf_inf @ A @ Y3 @ Z3 ) )
= ( inf_inf @ A @ Y3 @ ( inf_inf @ A @ X @ Z3 ) ) ) ) ).
% inf_sup_aci(3)
thf(fact_3988_inf__sup__aci_I2_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y3: A,Z3: A] :
( ( inf_inf @ A @ ( inf_inf @ A @ X @ Y3 ) @ Z3 )
= ( inf_inf @ A @ X @ ( inf_inf @ A @ Y3 @ Z3 ) ) ) ) ).
% inf_sup_aci(2)
thf(fact_3989_inf__sup__aci_I1_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ( ( inf_inf @ A )
= ( ^ [X2: A,Y6: A] : ( inf_inf @ A @ Y6 @ X2 ) ) ) ) ).
% inf_sup_aci(1)
thf(fact_3990_inf_Oassoc,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( inf_inf @ A @ ( inf_inf @ A @ A3 @ B2 ) @ C2 )
= ( inf_inf @ A @ A3 @ ( inf_inf @ A @ B2 @ C2 ) ) ) ) ).
% inf.assoc
thf(fact_3991_inf__assoc,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A,Y3: A,Z3: A] :
( ( inf_inf @ A @ ( inf_inf @ A @ X @ Y3 ) @ Z3 )
= ( inf_inf @ A @ X @ ( inf_inf @ A @ Y3 @ Z3 ) ) ) ) ).
% inf_assoc
thf(fact_3992_inf_Ocommute,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( inf_inf @ A )
= ( ^ [A5: A,B3: A] : ( inf_inf @ A @ B3 @ A5 ) ) ) ) ).
% inf.commute
thf(fact_3993_inf__commute,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( inf_inf @ A )
= ( ^ [X2: A,Y6: A] : ( inf_inf @ A @ Y6 @ X2 ) ) ) ) ).
% inf_commute
thf(fact_3994_inf_Oleft__commute,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( inf_inf @ A @ B2 @ ( inf_inf @ A @ A3 @ C2 ) )
= ( inf_inf @ A @ A3 @ ( inf_inf @ A @ B2 @ C2 ) ) ) ) ).
% inf.left_commute
thf(fact_3995_inf__left__commute,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A,Y3: A,Z3: A] :
( ( inf_inf @ A @ X @ ( inf_inf @ A @ Y3 @ Z3 ) )
= ( inf_inf @ A @ Y3 @ ( inf_inf @ A @ X @ Z3 ) ) ) ) ).
% inf_left_commute
thf(fact_3996_inf__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( semilattice_inf @ B )
=> ( ( inf_inf @ ( A > B ) )
= ( ^ [F2: A > B,G2: A > B,X2: A] : ( inf_inf @ B @ ( F2 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ).
% inf_fun_def
thf(fact_3997_boolean__algebra__cancel_Oinf1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A6: A,K: A,A3: A,B2: A] :
( ( A6
= ( inf_inf @ A @ K @ A3 ) )
=> ( ( inf_inf @ A @ A6 @ B2 )
= ( inf_inf @ A @ K @ ( inf_inf @ A @ A3 @ B2 ) ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_3998_boolean__algebra__cancel_Oinf2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [B7: A,K: A,B2: A,A3: A] :
( ( B7
= ( inf_inf @ A @ K @ B2 ) )
=> ( ( inf_inf @ A @ A3 @ B7 )
= ( inf_inf @ A @ K @ ( inf_inf @ A @ A3 @ B2 ) ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_3999_less__infI1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,X: A,B2: A] :
( ( ord_less @ A @ A3 @ X )
=> ( ord_less @ A @ ( inf_inf @ A @ A3 @ B2 ) @ X ) ) ) ).
% less_infI1
thf(fact_4000_less__infI2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [B2: A,X: A,A3: A] :
( ( ord_less @ A @ B2 @ X )
=> ( ord_less @ A @ ( inf_inf @ A @ A3 @ B2 ) @ X ) ) ) ).
% less_infI2
thf(fact_4001_inf_Oabsorb3,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( inf_inf @ A @ A3 @ B2 )
= A3 ) ) ) ).
% inf.absorb3
thf(fact_4002_inf_Oabsorb4,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [B2: A,A3: A] :
( ( ord_less @ A @ B2 @ A3 )
=> ( ( inf_inf @ A @ A3 @ B2 )
= B2 ) ) ) ).
% inf.absorb4
thf(fact_4003_inf_Ostrict__boundedE,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less @ A @ A3 @ ( inf_inf @ A @ B2 @ C2 ) )
=> ~ ( ( ord_less @ A @ A3 @ B2 )
=> ~ ( ord_less @ A @ A3 @ C2 ) ) ) ) ).
% inf.strict_boundedE
thf(fact_4004_inf_Ostrict__order__iff,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( ord_less @ A )
= ( ^ [A5: A,B3: A] :
( ( A5
= ( inf_inf @ A @ A5 @ B3 ) )
& ( A5 != B3 ) ) ) ) ) ).
% inf.strict_order_iff
thf(fact_4005_inf_Ostrict__coboundedI1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( ord_less @ A @ A3 @ C2 )
=> ( ord_less @ A @ ( inf_inf @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% inf.strict_coboundedI1
thf(fact_4006_inf_Ostrict__coboundedI2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [B2: A,C2: A,A3: A] :
( ( ord_less @ A @ B2 @ C2 )
=> ( ord_less @ A @ ( inf_inf @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% inf.strict_coboundedI2
thf(fact_4007_inf_OcoboundedI2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [B2: A,C2: A,A3: A] :
( ( ord_less_eq @ A @ B2 @ C2 )
=> ( ord_less_eq @ A @ ( inf_inf @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% inf.coboundedI2
thf(fact_4008_inf_OcoboundedI1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ C2 )
=> ( ord_less_eq @ A @ ( inf_inf @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% inf.coboundedI1
thf(fact_4009_inf_Oabsorb__iff2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B3: A,A5: A] :
( ( inf_inf @ A @ A5 @ B3 )
= B3 ) ) ) ) ).
% inf.absorb_iff2
thf(fact_4010_inf_Oabsorb__iff1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A5: A,B3: A] :
( ( inf_inf @ A @ A5 @ B3 )
= A5 ) ) ) ) ).
% inf.absorb_iff1
thf(fact_4011_inf_Ocobounded2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,B2: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ A3 @ B2 ) @ B2 ) ) ).
% inf.cobounded2
thf(fact_4012_inf_Ocobounded1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,B2: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ A3 @ B2 ) @ A3 ) ) ).
% inf.cobounded1
thf(fact_4013_inf_Oorder__iff,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A5: A,B3: A] :
( A5
= ( inf_inf @ A @ A5 @ B3 ) ) ) ) ) ).
% inf.order_iff
thf(fact_4014_inf__greatest,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A,Y3: A,Z3: A] :
( ( ord_less_eq @ A @ X @ Y3 )
=> ( ( ord_less_eq @ A @ X @ Z3 )
=> ( ord_less_eq @ A @ X @ ( inf_inf @ A @ Y3 @ Z3 ) ) ) ) ) ).
% inf_greatest
thf(fact_4015_inf_OboundedI,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ A3 @ C2 )
=> ( ord_less_eq @ A @ A3 @ ( inf_inf @ A @ B2 @ C2 ) ) ) ) ) ).
% inf.boundedI
thf(fact_4016_inf_OboundedE,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ ( inf_inf @ A @ B2 @ C2 ) )
=> ~ ( ( ord_less_eq @ A @ A3 @ B2 )
=> ~ ( ord_less_eq @ A @ A3 @ C2 ) ) ) ) ).
% inf.boundedE
thf(fact_4017_inf__absorb2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [Y3: A,X: A] :
( ( ord_less_eq @ A @ Y3 @ X )
=> ( ( inf_inf @ A @ X @ Y3 )
= Y3 ) ) ) ).
% inf_absorb2
thf(fact_4018_inf__absorb1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A,Y3: A] :
( ( ord_less_eq @ A @ X @ Y3 )
=> ( ( inf_inf @ A @ X @ Y3 )
= X ) ) ) ).
% inf_absorb1
thf(fact_4019_inf_Oabsorb2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [B2: A,A3: A] :
( ( ord_less_eq @ A @ B2 @ A3 )
=> ( ( inf_inf @ A @ A3 @ B2 )
= B2 ) ) ) ).
% inf.absorb2
thf(fact_4020_inf_Oabsorb1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( inf_inf @ A @ A3 @ B2 )
= A3 ) ) ) ).
% inf.absorb1
thf(fact_4021_le__iff__inf,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [X2: A,Y6: A] :
( ( inf_inf @ A @ X2 @ Y6 )
= X2 ) ) ) ) ).
% le_iff_inf
thf(fact_4022_inf__unique,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [F: A > A > A,X: A,Y3: A] :
( ! [X3: A,Y5: A] : ( ord_less_eq @ A @ ( F @ X3 @ Y5 ) @ X3 )
=> ( ! [X3: A,Y5: A] : ( ord_less_eq @ A @ ( F @ X3 @ Y5 ) @ Y5 )
=> ( ! [X3: A,Y5: A,Z4: A] :
( ( ord_less_eq @ A @ X3 @ Y5 )
=> ( ( ord_less_eq @ A @ X3 @ Z4 )
=> ( ord_less_eq @ A @ X3 @ ( F @ Y5 @ Z4 ) ) ) )
=> ( ( inf_inf @ A @ X @ Y3 )
= ( F @ X @ Y3 ) ) ) ) ) ) ).
% inf_unique
thf(fact_4023_inf_OorderI,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,B2: A] :
( ( A3
= ( inf_inf @ A @ A3 @ B2 ) )
=> ( ord_less_eq @ A @ A3 @ B2 ) ) ) ).
% inf.orderI
thf(fact_4024_inf_OorderE,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( A3
= ( inf_inf @ A @ A3 @ B2 ) ) ) ) ).
% inf.orderE
thf(fact_4025_le__infI2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [B2: A,X: A,A3: A] :
( ( ord_less_eq @ A @ B2 @ X )
=> ( ord_less_eq @ A @ ( inf_inf @ A @ A3 @ B2 ) @ X ) ) ) ).
% le_infI2
thf(fact_4026_le__infI1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,X: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ X )
=> ( ord_less_eq @ A @ ( inf_inf @ A @ A3 @ B2 ) @ X ) ) ) ).
% le_infI1
thf(fact_4027_inf__mono,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,C2: A,B2: A,D2: A] :
( ( ord_less_eq @ A @ A3 @ C2 )
=> ( ( ord_less_eq @ A @ B2 @ D2 )
=> ( ord_less_eq @ A @ ( inf_inf @ A @ A3 @ B2 ) @ ( inf_inf @ A @ C2 @ D2 ) ) ) ) ) ).
% inf_mono
thf(fact_4028_le__infI,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A,A3: A,B2: A] :
( ( ord_less_eq @ A @ X @ A3 )
=> ( ( ord_less_eq @ A @ X @ B2 )
=> ( ord_less_eq @ A @ X @ ( inf_inf @ A @ A3 @ B2 ) ) ) ) ) ).
% le_infI
thf(fact_4029_le__infE,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A,A3: A,B2: A] :
( ( ord_less_eq @ A @ X @ ( inf_inf @ A @ A3 @ B2 ) )
=> ~ ( ( ord_less_eq @ A @ X @ A3 )
=> ~ ( ord_less_eq @ A @ X @ B2 ) ) ) ) ).
% le_infE
thf(fact_4030_inf__le2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A,Y3: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ X @ Y3 ) @ Y3 ) ) ).
% inf_le2
thf(fact_4031_inf__le1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A,Y3: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ X @ Y3 ) @ X ) ) ).
% inf_le1
thf(fact_4032_inf__sup__ord_I1_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y3: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ X @ Y3 ) @ X ) ) ).
% inf_sup_ord(1)
thf(fact_4033_inf__sup__ord_I2_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y3: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ X @ Y3 ) @ Y3 ) ) ).
% inf_sup_ord(2)
thf(fact_4034_distrib__sup__le,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y3: A,Z3: A] : ( ord_less_eq @ A @ ( sup_sup @ A @ X @ ( inf_inf @ A @ Y3 @ Z3 ) ) @ ( inf_inf @ A @ ( sup_sup @ A @ X @ Y3 ) @ ( sup_sup @ A @ X @ Z3 ) ) ) ) ).
% distrib_sup_le
thf(fact_4035_distrib__inf__le,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y3: A,Z3: A] : ( ord_less_eq @ A @ ( sup_sup @ A @ ( inf_inf @ A @ X @ Y3 ) @ ( inf_inf @ A @ X @ Z3 ) ) @ ( inf_inf @ A @ X @ ( sup_sup @ A @ Y3 @ Z3 ) ) ) ) ).
% distrib_inf_le
thf(fact_4036_diff__eq,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ( ( minus_minus @ A )
= ( ^ [X2: A,Y6: A] : ( inf_inf @ A @ X2 @ ( uminus_uminus @ A @ Y6 ) ) ) ) ) ).
% diff_eq
thf(fact_4037_inf__cancel__left1,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,A3: A,B2: A] :
( ( inf_inf @ A @ ( inf_inf @ A @ X @ A3 ) @ ( inf_inf @ A @ ( uminus_uminus @ A @ X ) @ B2 ) )
= ( bot_bot @ A ) ) ) ).
% inf_cancel_left1
thf(fact_4038_inf__cancel__left2,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,A3: A,B2: A] :
( ( inf_inf @ A @ ( inf_inf @ A @ ( uminus_uminus @ A @ X ) @ A3 ) @ ( inf_inf @ A @ X @ B2 ) )
= ( bot_bot @ A ) ) ) ).
% inf_cancel_left2
thf(fact_4039_zero__notin__Suc__image,axiom,
! [A6: set @ nat] :
~ ( member @ nat @ ( zero_zero @ nat ) @ ( image @ nat @ nat @ suc @ A6 ) ) ).
% zero_notin_Suc_image
thf(fact_4040_image__int__atLeastLessThan,axiom,
! [A3: nat,B2: nat] :
( ( image @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( set_or7035219750837199246ssThan @ nat @ A3 @ B2 ) )
= ( set_or7035219750837199246ssThan @ int @ ( semiring_1_of_nat @ int @ A3 ) @ ( semiring_1_of_nat @ int @ B2 ) ) ) ).
% image_int_atLeastLessThan
thf(fact_4041_member__le__sum__list,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [X: A,Xs: list @ A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ord_less_eq @ A @ X @ ( groups8242544230860333062m_list @ A @ Xs ) ) ) ) ).
% member_le_sum_list
thf(fact_4042_Compl__Un,axiom,
! [A: $tType,A6: set @ A,B7: set @ A] :
( ( uminus_uminus @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A6 @ B7 ) )
= ( inf_inf @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A6 ) @ ( uminus_uminus @ ( set @ A ) @ B7 ) ) ) ).
% Compl_Un
thf(fact_4043_Compl__Int,axiom,
! [A: $tType,A6: set @ A,B7: set @ A] :
( ( uminus_uminus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A6 @ B7 ) )
= ( sup_sup @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A6 ) @ ( uminus_uminus @ ( set @ A ) @ B7 ) ) ) ).
% Compl_Int
thf(fact_4044_Diff__eq,axiom,
! [A: $tType] :
( ( minus_minus @ ( set @ A ) )
= ( ^ [A7: set @ A,B6: set @ A] : ( inf_inf @ ( set @ A ) @ A7 @ ( uminus_uminus @ ( set @ A ) @ B6 ) ) ) ) ).
% Diff_eq
thf(fact_4045_prod_Oivl__cong,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( comm_monoid_mult @ A ) )
=> ! [A3: B,C2: B,B2: B,D2: B,G: B > A,H2: B > A] :
( ( A3 = C2 )
=> ( ( B2 = D2 )
=> ( ! [X3: B] :
( ( ord_less_eq @ B @ C2 @ X3 )
=> ( ( ord_less @ B @ X3 @ D2 )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( set_or7035219750837199246ssThan @ B @ A3 @ B2 ) )
= ( groups7121269368397514597t_prod @ B @ A @ H2 @ ( set_or7035219750837199246ssThan @ B @ C2 @ D2 ) ) ) ) ) ) ) ).
% prod.ivl_cong
thf(fact_4046_prod_OatLeastLessThan__concat,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M: nat,N: nat,P5: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( ord_less_eq @ nat @ N @ P5 )
=> ( ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ N @ P5 ) ) )
= ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ P5 ) ) ) ) ) ) ).
% prod.atLeastLessThan_concat
thf(fact_4047_inf__shunt,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y3: A] :
( ( ( inf_inf @ A @ X @ Y3 )
= ( bot_bot @ A ) )
= ( ord_less_eq @ A @ X @ ( uminus_uminus @ A @ Y3 ) ) ) ) ).
% inf_shunt
thf(fact_4048_shunt1,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y3: A,Z3: A] :
( ( ord_less_eq @ A @ ( inf_inf @ A @ X @ Y3 ) @ Z3 )
= ( ord_less_eq @ A @ X @ ( sup_sup @ A @ ( uminus_uminus @ A @ Y3 ) @ Z3 ) ) ) ) ).
% shunt1
thf(fact_4049_shunt2,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y3: A,Z3: A] :
( ( ord_less_eq @ A @ ( inf_inf @ A @ X @ ( uminus_uminus @ A @ Y3 ) ) @ Z3 )
= ( ord_less_eq @ A @ X @ ( sup_sup @ A @ Y3 @ Z3 ) ) ) ) ).
% shunt2
thf(fact_4050_sup__neg__inf,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [P5: A,Q4: A,R: A] :
( ( ord_less_eq @ A @ P5 @ ( sup_sup @ A @ Q4 @ R ) )
= ( ord_less_eq @ A @ ( inf_inf @ A @ P5 @ ( uminus_uminus @ A @ Q4 ) ) @ R ) ) ) ).
% sup_neg_inf
thf(fact_4051_card__image__le,axiom,
! [B: $tType,A: $tType,A6: set @ A,F: A > B] :
( ( finite_finite2 @ A @ A6 )
=> ( ord_less_eq @ nat @ ( finite_card @ B @ ( image @ A @ B @ F @ A6 ) ) @ ( finite_card @ A @ A6 ) ) ) ).
% card_image_le
thf(fact_4052_Groups__List_Osum__list__nonneg,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [Xs: list @ A] :
( ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( groups8242544230860333062m_list @ A @ Xs ) ) ) ) ).
% Groups_List.sum_list_nonneg
thf(fact_4053_sum__list__nonneg__eq__0__iff,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [Xs: list @ A] :
( ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 ) )
=> ( ( ( groups8242544230860333062m_list @ A @ Xs )
= ( zero_zero @ A ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
=> ( X2
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% sum_list_nonneg_eq_0_iff
thf(fact_4054_sum__list__nonpos,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [Xs: list @ A] :
( ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( ord_less_eq @ A @ X3 @ ( zero_zero @ A ) ) )
=> ( ord_less_eq @ A @ ( groups8242544230860333062m_list @ A @ Xs ) @ ( zero_zero @ A ) ) ) ) ).
% sum_list_nonpos
thf(fact_4055_Ioc__disjoint,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A,C2: A,D2: A] :
( ( ( inf_inf @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ A3 @ B2 ) @ ( set_or3652927894154168847AtMost @ A @ C2 @ D2 ) )
= ( bot_bot @ ( set @ A ) ) )
= ( ( ord_less_eq @ A @ B2 @ A3 )
| ( ord_less_eq @ A @ D2 @ C2 )
| ( ord_less_eq @ A @ B2 @ C2 )
| ( ord_less_eq @ A @ D2 @ A3 ) ) ) ) ).
% Ioc_disjoint
thf(fact_4056_disjoint__eq__subset__Compl,axiom,
! [A: $tType,A6: set @ A,B7: set @ A] :
( ( ( inf_inf @ ( set @ A ) @ A6 @ B7 )
= ( bot_bot @ ( set @ A ) ) )
= ( ord_less_eq @ ( set @ A ) @ A6 @ ( uminus_uminus @ ( set @ A ) @ B7 ) ) ) ).
% disjoint_eq_subset_Compl
thf(fact_4057_image__int__atLeastAtMost,axiom,
! [A3: nat,B2: nat] :
( ( image @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( set_or1337092689740270186AtMost @ nat @ A3 @ B2 ) )
= ( set_or1337092689740270186AtMost @ int @ ( semiring_1_of_nat @ int @ A3 ) @ ( semiring_1_of_nat @ int @ B2 ) ) ) ).
% image_int_atLeastAtMost
thf(fact_4058_sum__list__abs,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [Xs: list @ A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( groups8242544230860333062m_list @ A @ Xs ) ) @ ( groups8242544230860333062m_list @ A @ ( map @ A @ A @ ( abs_abs @ A ) @ Xs ) ) ) ) ).
% sum_list_abs
thf(fact_4059_sum__list__replicate,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [N: nat,C2: A] :
( ( groups8242544230860333062m_list @ A @ ( replicate @ A @ N @ C2 ) )
= ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ C2 ) ) ) ).
% sum_list_replicate
thf(fact_4060_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_4061_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_4062_prod_OatLeast__Suc__lessThan,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M: nat,N: nat,G: nat > A] :
( ( ord_less @ nat @ M @ N )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) )
= ( times_times @ A @ ( G @ M ) @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M ) @ N ) ) ) ) ) ) ).
% prod.atLeast_Suc_lessThan
thf(fact_4063_prod_OatLeast__Suc__atMost,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M: nat,N: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( times_times @ A @ ( G @ M ) @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ N ) ) ) ) ) ) ).
% prod.atLeast_Suc_atMost
thf(fact_4064_prod_Onat__ivl__Suc_H,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M: nat,N: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M @ ( suc @ N ) )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ ( suc @ N ) ) )
= ( times_times @ A @ ( G @ ( suc @ N ) ) @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ) ) ).
% prod.nat_ivl_Suc'
thf(fact_4065_prod_OatLeastLessThan__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: nat,B2: nat,G: nat > A] :
( ( ord_less_eq @ nat @ A3 @ B2 )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ A3 @ ( suc @ B2 ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ A3 @ B2 ) ) @ ( G @ B2 ) ) ) ) ) ).
% prod.atLeastLessThan_Suc
thf(fact_4066_prod_Olast__plus,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M: nat,N: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( times_times @ A @ ( G @ N ) @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ) ) ).
% prod.last_plus
thf(fact_4067_surj__card__le,axiom,
! [B: $tType,A: $tType,A6: set @ A,B7: set @ B,F: A > B] :
( ( finite_finite2 @ A @ A6 )
=> ( ( ord_less_eq @ ( set @ B ) @ B7 @ ( image @ A @ B @ F @ A6 ) )
=> ( ord_less_eq @ nat @ ( finite_card @ B @ B7 ) @ ( finite_card @ A @ A6 ) ) ) ) ).
% surj_card_le
thf(fact_4068_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_4069_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_4070_image__Suc__atMost,axiom,
! [N: nat] :
( ( image @ nat @ nat @ suc @ ( set_ord_atMost @ nat @ N ) )
= ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ ( suc @ N ) ) ) ).
% image_Suc_atMost
thf(fact_4071_Iio__Int__singleton,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,K: A] :
( ( ( ord_less @ A @ X @ K )
=> ( ( inf_inf @ ( set @ A ) @ ( set_ord_lessThan @ A @ K ) @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
& ( ~ ( ord_less @ A @ X @ K )
=> ( ( inf_inf @ ( set @ A ) @ ( set_ord_lessThan @ A @ K ) @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% Iio_Int_singleton
thf(fact_4072_image__Suc__lessThan,axiom,
! [N: nat] :
( ( image @ nat @ nat @ suc @ ( set_ord_lessThan @ nat @ N ) )
= ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ N ) ) ).
% image_Suc_lessThan
thf(fact_4073_atLeast0__atMost__Suc__eq__insert__0,axiom,
! [N: nat] :
( ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) )
= ( insert @ nat @ ( zero_zero @ nat ) @ ( image @ nat @ nat @ suc @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% atLeast0_atMost_Suc_eq_insert_0
thf(fact_4074_atLeast0__lessThan__Suc__eq__insert__0,axiom,
! [N: nat] :
( ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) )
= ( insert @ nat @ ( zero_zero @ nat ) @ ( image @ nat @ nat @ suc @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% atLeast0_lessThan_Suc_eq_insert_0
thf(fact_4075_prod_Ohead,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M: nat,N: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( times_times @ A @ ( G @ M ) @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or3652927894154168847AtMost @ nat @ M @ N ) ) ) ) ) ) ).
% prod.head
thf(fact_4076_atMost__Suc__eq__insert__0,axiom,
! [N: nat] :
( ( set_ord_atMost @ nat @ ( suc @ N ) )
= ( insert @ nat @ ( zero_zero @ nat ) @ ( image @ nat @ nat @ suc @ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% atMost_Suc_eq_insert_0
thf(fact_4077_lessThan__Suc__eq__insert__0,axiom,
! [N: nat] :
( ( set_ord_lessThan @ nat @ ( suc @ N ) )
= ( insert @ nat @ ( zero_zero @ nat ) @ ( image @ nat @ nat @ suc @ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% lessThan_Suc_eq_insert_0
thf(fact_4078_sum__list__strict__mono,axiom,
! [B: $tType,A: $tType] :
( ( ( monoid_add @ B )
& ( strict9044650504122735259up_add @ B ) )
=> ! [Xs: list @ A,F: A > B,G: A > B] :
( ( Xs
!= ( nil @ A ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( ord_less @ B @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less @ B @ ( groups8242544230860333062m_list @ B @ ( map @ A @ B @ F @ Xs ) ) @ ( groups8242544230860333062m_list @ B @ ( map @ A @ B @ G @ Xs ) ) ) ) ) ) ).
% sum_list_strict_mono
thf(fact_4079_elem__le__sum__list,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [K: nat,Ns: list @ A] :
( ( ord_less @ nat @ K @ ( size_size @ ( list @ A ) @ Ns ) )
=> ( ord_less_eq @ A @ ( nth @ A @ Ns @ K ) @ ( groups8242544230860333062m_list @ A @ Ns ) ) ) ) ).
% elem_le_sum_list
thf(fact_4080_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_4081_prod_Oub__add__nat,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M: nat,N: nat,G: nat > A,P5: 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 @ P5 ) ) )
= ( 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 @ P5 ) ) ) ) ) ) ) ).
% prod.ub_add_nat
thf(fact_4082_fact__prod__Suc,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiring_char_0_fact @ A )
= ( ^ [N3: nat] : ( semiring_1_of_nat @ A @ ( groups7121269368397514597t_prod @ nat @ nat @ suc @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N3 ) ) ) ) ) ) ).
% fact_prod_Suc
thf(fact_4083_image__atLeastZeroLessThan__int,axiom,
! [U: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ U )
=> ( ( set_or7035219750837199246ssThan @ int @ ( zero_zero @ int ) @ U )
= ( image @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( set_ord_lessThan @ nat @ ( nat2 @ U ) ) ) ) ) ).
% image_atLeastZeroLessThan_int
thf(fact_4084_image__mult__atLeastAtMost__if,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,X: A,Y3: A] :
( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( image @ 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 )
=> ( ( image @ 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 )
=> ( ( image @ A @ A @ ( times_times @ A @ C2 ) @ ( set_or1337092689740270186AtMost @ A @ X @ Y3 ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% image_mult_atLeastAtMost_if
thf(fact_4085_prod__pos__nat__iff,axiom,
! [A: $tType,A6: set @ A,F: A > nat] :
( ( finite_finite2 @ A @ A6 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( groups7121269368397514597t_prod @ A @ nat @ F @ A6 ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ A6 )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F @ X2 ) ) ) ) ) ) ).
% prod_pos_nat_iff
thf(fact_4086_prod_Oinsert,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A6: set @ B,X: B,G: B > A] :
( ( finite_finite2 @ B @ A6 )
=> ( ~ ( member @ B @ X @ A6 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( insert @ B @ X @ A6 ) )
= ( times_times @ A @ ( G @ X ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ A6 ) ) ) ) ) ) ).
% prod.insert
thf(fact_4087_prod__eq__1__iff,axiom,
! [A: $tType,A6: set @ A,F: A > nat] :
( ( finite_finite2 @ A @ A6 )
=> ( ( ( groups7121269368397514597t_prod @ A @ nat @ F @ A6 )
= ( one_one @ nat ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ A6 )
=> ( ( F @ X2 )
= ( one_one @ nat ) ) ) ) ) ) ).
% prod_eq_1_iff
thf(fact_4088_prod_Oinfinite,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A6: set @ B,G: B > A] :
( ~ ( finite_finite2 @ B @ A6 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ A6 )
= ( one_one @ A ) ) ) ) ).
% prod.infinite
thf(fact_4089_prod__zero__iff,axiom,
! [A: $tType,B: $tType] :
( ( semidom @ A )
=> ! [A6: set @ B,F: B > A] :
( ( finite_finite2 @ B @ A6 )
=> ( ( ( groups7121269368397514597t_prod @ B @ A @ F @ A6 )
= ( zero_zero @ A ) )
= ( ? [X2: B] :
( ( member @ B @ X2 @ A6 )
& ( ( F @ X2 )
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% prod_zero_iff
thf(fact_4090_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_4091_inf__nat__def,axiom,
( ( inf_inf @ nat )
= ( ord_min @ nat ) ) ).
% inf_nat_def
thf(fact_4092_inf__int__def,axiom,
( ( inf_inf @ int )
= ( ord_min @ int ) ) ).
% inf_int_def
thf(fact_4093_prod_Oneutral,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A6: set @ B,G: B > A] :
( ! [X3: B] :
( ( member @ B @ X3 @ A6 )
=> ( ( G @ X3 )
= ( one_one @ A ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ A6 )
= ( one_one @ A ) ) ) ) ).
% prod.neutral
thf(fact_4094_prod_Onot__neutral__contains__not__neutral,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: B > A,A6: set @ B] :
( ( ( groups7121269368397514597t_prod @ B @ A @ G @ A6 )
!= ( one_one @ A ) )
=> ~ ! [A8: B] :
( ( member @ B @ A8 @ A6 )
=> ( ( G @ A8 )
= ( one_one @ A ) ) ) ) ) ).
% prod.not_neutral_contains_not_neutral
thf(fact_4095_prod__nonneg,axiom,
! [A: $tType,B: $tType] :
( ( linordered_semidom @ A )
=> ! [A6: set @ B,F: B > A] :
( ! [X3: B] :
( ( member @ B @ X3 @ A6 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F @ X3 ) ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( groups7121269368397514597t_prod @ B @ A @ F @ A6 ) ) ) ) ).
% prod_nonneg
thf(fact_4096_prod__mono,axiom,
! [A: $tType,B: $tType] :
( ( linordered_semidom @ A )
=> ! [A6: set @ B,F: B > A,G: B > A] :
( ! [I4: B] :
( ( member @ B @ I4 @ A6 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F @ I4 ) )
& ( ord_less_eq @ A @ ( F @ I4 ) @ ( G @ I4 ) ) ) )
=> ( ord_less_eq @ A @ ( groups7121269368397514597t_prod @ B @ A @ F @ A6 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ A6 ) ) ) ) ).
% prod_mono
thf(fact_4097_prod__pos,axiom,
! [A: $tType,B: $tType] :
( ( linordered_semidom @ A )
=> ! [A6: set @ B,F: B > A] :
( ! [X3: B] :
( ( member @ B @ X3 @ A6 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( F @ X3 ) ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( groups7121269368397514597t_prod @ B @ A @ F @ A6 ) ) ) ) ).
% prod_pos
thf(fact_4098_prod__ge__1,axiom,
! [A: $tType,B: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [A6: set @ B,F: B > A] :
( ! [X3: B] :
( ( member @ B @ X3 @ A6 )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ ( F @ X3 ) ) )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ ( groups7121269368397514597t_prod @ B @ A @ F @ A6 ) ) ) ) ).
% prod_ge_1
thf(fact_4099_prod__zero,axiom,
! [B: $tType,A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A6: set @ B,F: B > A] :
( ( finite_finite2 @ B @ A6 )
=> ( ? [X4: B] :
( ( member @ B @ X4 @ A6 )
& ( ( F @ X4 )
= ( zero_zero @ A ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ F @ A6 )
= ( zero_zero @ A ) ) ) ) ) ).
% prod_zero
thf(fact_4100_prod__le__1,axiom,
! [B: $tType,A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [A6: set @ B,F: B > A] :
( ! [X3: B] :
( ( member @ B @ X3 @ A6 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F @ X3 ) )
& ( ord_less_eq @ A @ ( F @ X3 ) @ ( one_one @ A ) ) ) )
=> ( ord_less_eq @ A @ ( groups7121269368397514597t_prod @ B @ A @ F @ A6 ) @ ( one_one @ A ) ) ) ) ).
% prod_le_1
thf(fact_4101_prod_Orelated,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [R2: A > A > $o,S3: set @ B,H2: B > A,G: B > A] :
( ( R2 @ ( one_one @ A ) @ ( one_one @ A ) )
=> ( ! [X1: A,Y1: A,X23: A,Y22: A] :
( ( ( R2 @ X1 @ X23 )
& ( R2 @ Y1 @ Y22 ) )
=> ( R2 @ ( times_times @ A @ X1 @ Y1 ) @ ( times_times @ A @ X23 @ Y22 ) ) )
=> ( ( finite_finite2 @ B @ S3 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ S3 )
=> ( R2 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
=> ( R2 @ ( groups7121269368397514597t_prod @ B @ A @ H2 @ S3 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ S3 ) ) ) ) ) ) ) ).
% prod.related
thf(fact_4102_prod_Oinsert__if,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A6: set @ B,X: B,G: B > A] :
( ( finite_finite2 @ B @ A6 )
=> ( ( ( member @ B @ X @ A6 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( insert @ B @ X @ A6 ) )
= ( groups7121269368397514597t_prod @ B @ A @ G @ A6 ) ) )
& ( ~ ( member @ B @ X @ A6 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( insert @ B @ X @ A6 ) )
= ( times_times @ A @ ( G @ X ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ A6 ) ) ) ) ) ) ) ).
% prod.insert_if
thf(fact_4103_prod_Oreindex__bij__witness__not__neutral,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S5: set @ B,T5: set @ C,S3: set @ B,I: C > B,J: B > C,T6: set @ C,G: B > A,H2: C > A] :
( ( finite_finite2 @ B @ S5 )
=> ( ( finite_finite2 @ C @ T5 )
=> ( ! [A8: B] :
( ( member @ B @ A8 @ ( minus_minus @ ( set @ B ) @ S3 @ S5 ) )
=> ( ( I @ ( J @ A8 ) )
= A8 ) )
=> ( ! [A8: B] :
( ( member @ B @ A8 @ ( minus_minus @ ( set @ B ) @ S3 @ S5 ) )
=> ( member @ C @ ( J @ A8 ) @ ( minus_minus @ ( set @ C ) @ T6 @ T5 ) ) )
=> ( ! [B5: C] :
( ( member @ C @ B5 @ ( minus_minus @ ( set @ C ) @ T6 @ T5 ) )
=> ( ( J @ ( I @ B5 ) )
= B5 ) )
=> ( ! [B5: C] :
( ( member @ C @ B5 @ ( minus_minus @ ( set @ C ) @ T6 @ T5 ) )
=> ( member @ B @ ( I @ B5 ) @ ( minus_minus @ ( set @ B ) @ S3 @ S5 ) ) )
=> ( ! [A8: B] :
( ( member @ B @ A8 @ S5 )
=> ( ( G @ A8 )
= ( one_one @ A ) ) )
=> ( ! [B5: C] :
( ( member @ C @ B5 @ T5 )
=> ( ( H2 @ B5 )
= ( one_one @ A ) ) )
=> ( ! [A8: B] :
( ( member @ B @ A8 @ S3 )
=> ( ( H2 @ ( J @ A8 ) )
= ( G @ A8 ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ S3 )
= ( groups7121269368397514597t_prod @ C @ A @ H2 @ T6 ) ) ) ) ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_witness_not_neutral
thf(fact_4104_less__1__prod2,axiom,
! [B: $tType,A: $tType] :
( ( linordered_idom @ B )
=> ! [I6: set @ A,I: A,F: A > B] :
( ( finite_finite2 @ A @ I6 )
=> ( ( member @ A @ I @ I6 )
=> ( ( ord_less @ B @ ( one_one @ B ) @ ( F @ I ) )
=> ( ! [I4: A] :
( ( member @ A @ I4 @ I6 )
=> ( ord_less_eq @ B @ ( one_one @ B ) @ ( F @ I4 ) ) )
=> ( ord_less @ B @ ( one_one @ B ) @ ( groups7121269368397514597t_prod @ A @ B @ F @ I6 ) ) ) ) ) ) ) ).
% less_1_prod2
thf(fact_4105_less__1__prod,axiom,
! [B: $tType,A: $tType] :
( ( linordered_idom @ B )
=> ! [I6: set @ A,F: A > B] :
( ( finite_finite2 @ A @ I6 )
=> ( ( I6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [I4: A] :
( ( member @ A @ I4 @ I6 )
=> ( ord_less @ B @ ( one_one @ B ) @ ( F @ I4 ) ) )
=> ( ord_less @ B @ ( one_one @ B ) @ ( groups7121269368397514597t_prod @ A @ B @ F @ I6 ) ) ) ) ) ) ).
% less_1_prod
thf(fact_4106_prod_Osubset__diff,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [B7: set @ B,A6: set @ B,G: B > A] :
( ( ord_less_eq @ ( set @ B ) @ B7 @ A6 )
=> ( ( finite_finite2 @ B @ A6 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ A6 )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A6 @ B7 ) ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ B7 ) ) ) ) ) ) ).
% prod.subset_diff
thf(fact_4107_prod_Omono__neutral__cong__right,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [T6: set @ B,S3: set @ B,G: B > A,H2: B > A] :
( ( finite_finite2 @ B @ T6 )
=> ( ( ord_less_eq @ ( set @ B ) @ S3 @ T6 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T6 @ S3 ) )
=> ( ( G @ X3 )
= ( one_one @ A ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ S3 )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ T6 )
= ( groups7121269368397514597t_prod @ B @ A @ H2 @ S3 ) ) ) ) ) ) ) ).
% prod.mono_neutral_cong_right
thf(fact_4108_prod_Omono__neutral__cong__left,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [T6: set @ B,S3: set @ B,H2: B > A,G: B > A] :
( ( finite_finite2 @ B @ T6 )
=> ( ( ord_less_eq @ ( set @ B ) @ S3 @ T6 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T6 @ S3 ) )
=> ( ( H2 @ X3 )
= ( one_one @ A ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ S3 )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ S3 )
= ( groups7121269368397514597t_prod @ B @ A @ H2 @ T6 ) ) ) ) ) ) ) ).
% prod.mono_neutral_cong_left
thf(fact_4109_prod_Omono__neutral__right,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [T6: set @ B,S3: set @ B,G: B > A] :
( ( finite_finite2 @ B @ T6 )
=> ( ( ord_less_eq @ ( set @ B ) @ S3 @ T6 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T6 @ S3 ) )
=> ( ( G @ X3 )
= ( one_one @ A ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ T6 )
= ( groups7121269368397514597t_prod @ B @ A @ G @ S3 ) ) ) ) ) ) ).
% prod.mono_neutral_right
thf(fact_4110_prod_Omono__neutral__left,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [T6: set @ B,S3: set @ B,G: B > A] :
( ( finite_finite2 @ B @ T6 )
=> ( ( ord_less_eq @ ( set @ B ) @ S3 @ T6 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T6 @ S3 ) )
=> ( ( G @ X3 )
= ( one_one @ A ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ S3 )
= ( groups7121269368397514597t_prod @ B @ A @ G @ T6 ) ) ) ) ) ) ).
% prod.mono_neutral_left
thf(fact_4111_prod_Osame__carrierI,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [C6: set @ B,A6: set @ B,B7: set @ B,G: B > A,H2: B > A] :
( ( finite_finite2 @ B @ C6 )
=> ( ( ord_less_eq @ ( set @ B ) @ A6 @ C6 )
=> ( ( ord_less_eq @ ( set @ B ) @ B7 @ C6 )
=> ( ! [A8: B] :
( ( member @ B @ A8 @ ( minus_minus @ ( set @ B ) @ C6 @ A6 ) )
=> ( ( G @ A8 )
= ( one_one @ A ) ) )
=> ( ! [B5: B] :
( ( member @ B @ B5 @ ( minus_minus @ ( set @ B ) @ C6 @ B7 ) )
=> ( ( H2 @ B5 )
= ( one_one @ A ) ) )
=> ( ( ( groups7121269368397514597t_prod @ B @ A @ G @ C6 )
= ( groups7121269368397514597t_prod @ B @ A @ H2 @ C6 ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ A6 )
= ( groups7121269368397514597t_prod @ B @ A @ H2 @ B7 ) ) ) ) ) ) ) ) ) ).
% prod.same_carrierI
thf(fact_4112_prod_Osame__carrier,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [C6: set @ B,A6: set @ B,B7: set @ B,G: B > A,H2: B > A] :
( ( finite_finite2 @ B @ C6 )
=> ( ( ord_less_eq @ ( set @ B ) @ A6 @ C6 )
=> ( ( ord_less_eq @ ( set @ B ) @ B7 @ C6 )
=> ( ! [A8: B] :
( ( member @ B @ A8 @ ( minus_minus @ ( set @ B ) @ C6 @ A6 ) )
=> ( ( G @ A8 )
= ( one_one @ A ) ) )
=> ( ! [B5: B] :
( ( member @ B @ B5 @ ( minus_minus @ ( set @ B ) @ C6 @ B7 ) )
=> ( ( H2 @ B5 )
= ( one_one @ A ) ) )
=> ( ( ( groups7121269368397514597t_prod @ B @ A @ G @ A6 )
= ( groups7121269368397514597t_prod @ B @ A @ H2 @ B7 ) )
= ( ( groups7121269368397514597t_prod @ B @ A @ G @ C6 )
= ( groups7121269368397514597t_prod @ B @ A @ H2 @ C6 ) ) ) ) ) ) ) ) ) ).
% prod.same_carrier
thf(fact_4113_prod_Ounion__inter,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A6: set @ B,B7: set @ B,G: B > A] :
( ( finite_finite2 @ B @ A6 )
=> ( ( finite_finite2 @ B @ B7 )
=> ( ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( sup_sup @ ( set @ B ) @ A6 @ B7 ) ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( inf_inf @ ( set @ B ) @ A6 @ B7 ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G @ A6 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ B7 ) ) ) ) ) ) ).
% prod.union_inter
thf(fact_4114_prod_OInt__Diff,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A6: set @ B,G: B > A,B7: set @ B] :
( ( finite_finite2 @ B @ A6 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ A6 )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( inf_inf @ ( set @ B ) @ A6 @ B7 ) ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A6 @ B7 ) ) ) ) ) ) ).
% prod.Int_Diff
thf(fact_4115_prod_Omono__neutral__cong,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [T6: set @ B,S3: set @ B,H2: B > A,G: B > A] :
( ( finite_finite2 @ B @ T6 )
=> ( ( finite_finite2 @ B @ S3 )
=> ( ! [I4: B] :
( ( member @ B @ I4 @ ( minus_minus @ ( set @ B ) @ T6 @ S3 ) )
=> ( ( H2 @ I4 )
= ( one_one @ A ) ) )
=> ( ! [I4: B] :
( ( member @ B @ I4 @ ( minus_minus @ ( set @ B ) @ S3 @ T6 ) )
=> ( ( G @ I4 )
= ( one_one @ A ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( inf_inf @ ( set @ B ) @ S3 @ T6 ) )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ S3 )
= ( groups7121269368397514597t_prod @ B @ A @ H2 @ T6 ) ) ) ) ) ) ) ) ).
% prod.mono_neutral_cong
thf(fact_4116_prod__mono__strict,axiom,
! [A: $tType,B: $tType] :
( ( linordered_semidom @ A )
=> ! [A6: set @ B,F: B > A,G: B > A] :
( ( finite_finite2 @ B @ A6 )
=> ( ! [I4: B] :
( ( member @ B @ I4 @ A6 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F @ I4 ) )
& ( ord_less @ A @ ( F @ I4 ) @ ( G @ I4 ) ) ) )
=> ( ( A6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ord_less @ A @ ( groups7121269368397514597t_prod @ B @ A @ F @ A6 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ A6 ) ) ) ) ) ) ).
% prod_mono_strict
thf(fact_4117_prod_Oinsert__remove,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A6: set @ B,G: B > A,X: B] :
( ( finite_finite2 @ B @ A6 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( insert @ B @ X @ A6 ) )
= ( times_times @ A @ ( G @ X ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A6 @ ( insert @ B @ X @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ).
% prod.insert_remove
thf(fact_4118_prod_Oremove,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A6: set @ B,X: B,G: B > A] :
( ( finite_finite2 @ B @ A6 )
=> ( ( member @ B @ X @ A6 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ A6 )
= ( times_times @ A @ ( G @ X ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A6 @ ( insert @ B @ X @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ).
% prod.remove
thf(fact_4119_prod_Ounion__inter__neutral,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A6: set @ B,B7: set @ B,G: B > A] :
( ( finite_finite2 @ B @ A6 )
=> ( ( finite_finite2 @ B @ B7 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( inf_inf @ ( set @ B ) @ A6 @ B7 ) )
=> ( ( G @ X3 )
= ( one_one @ A ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( sup_sup @ ( set @ B ) @ A6 @ B7 ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G @ A6 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ B7 ) ) ) ) ) ) ) ).
% prod.union_inter_neutral
thf(fact_4120_prod_Ounion__disjoint,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A6: set @ B,B7: set @ B,G: B > A] :
( ( finite_finite2 @ B @ A6 )
=> ( ( finite_finite2 @ B @ B7 )
=> ( ( ( inf_inf @ ( set @ B ) @ A6 @ B7 )
= ( bot_bot @ ( set @ B ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( sup_sup @ ( set @ B ) @ A6 @ B7 ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G @ A6 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ B7 ) ) ) ) ) ) ) ).
% prod.union_disjoint
thf(fact_4121_prod_Ounion__diff2,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A6: set @ B,B7: set @ B,G: B > A] :
( ( finite_finite2 @ B @ A6 )
=> ( ( finite_finite2 @ B @ B7 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( sup_sup @ ( set @ B ) @ A6 @ B7 ) )
= ( times_times @ A @ ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A6 @ B7 ) ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ B7 @ A6 ) ) ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( inf_inf @ ( set @ B ) @ A6 @ B7 ) ) ) ) ) ) ) ).
% prod.union_diff2
thf(fact_4122_prod__mono2,axiom,
! [B: $tType,A: $tType] :
( ( linordered_idom @ B )
=> ! [B7: set @ A,A6: set @ A,F: A > B] :
( ( finite_finite2 @ A @ B7 )
=> ( ( ord_less_eq @ ( set @ A ) @ A6 @ B7 )
=> ( ! [B5: A] :
( ( member @ A @ B5 @ ( minus_minus @ ( set @ A ) @ B7 @ A6 ) )
=> ( ord_less_eq @ B @ ( one_one @ B ) @ ( F @ B5 ) ) )
=> ( ! [A8: A] :
( ( member @ A @ A8 @ A6 )
=> ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( F @ A8 ) ) )
=> ( ord_less_eq @ B @ ( groups7121269368397514597t_prod @ A @ B @ F @ A6 ) @ ( groups7121269368397514597t_prod @ A @ B @ F @ B7 ) ) ) ) ) ) ) ).
% prod_mono2
thf(fact_4123_prod__le__power,axiom,
! [B: $tType,A: $tType] :
( ( linordered_semidom @ A )
=> ! [A6: set @ B,F: B > A,N: A,K: nat] :
( ! [I4: B] :
( ( member @ B @ I4 @ A6 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F @ I4 ) )
& ( ord_less_eq @ A @ ( F @ I4 ) @ N ) ) )
=> ( ( ord_less_eq @ nat @ ( finite_card @ B @ A6 ) @ K )
=> ( ( ord_less_eq @ A @ ( one_one @ A ) @ N )
=> ( ord_less_eq @ A @ ( groups7121269368397514597t_prod @ B @ A @ F @ A6 ) @ ( power_power @ A @ N @ K ) ) ) ) ) ) ).
% prod_le_power
thf(fact_4124_prod__diff1,axiom,
! [A: $tType,B: $tType] :
( ( semidom_divide @ A )
=> ! [A6: set @ B,F: B > A,A3: B] :
( ( finite_finite2 @ B @ A6 )
=> ( ( ( F @ A3 )
!= ( zero_zero @ A ) )
=> ( ( ( member @ B @ A3 @ A6 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ F @ ( minus_minus @ ( set @ B ) @ A6 @ ( insert @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) )
= ( divide_divide @ A @ ( groups7121269368397514597t_prod @ B @ A @ F @ A6 ) @ ( F @ A3 ) ) ) )
& ( ~ ( member @ B @ A3 @ A6 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ F @ ( minus_minus @ ( set @ B ) @ A6 @ ( insert @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) )
= ( groups7121269368397514597t_prod @ B @ A @ F @ A6 ) ) ) ) ) ) ) ).
% prod_diff1
thf(fact_4125_prod__Un,axiom,
! [A: $tType,B: $tType] :
( ( field @ A )
=> ! [A6: set @ B,B7: set @ B,F: B > A] :
( ( finite_finite2 @ B @ A6 )
=> ( ( finite_finite2 @ B @ B7 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( inf_inf @ ( set @ B ) @ A6 @ B7 ) )
=> ( ( F @ X3 )
!= ( zero_zero @ A ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ F @ ( sup_sup @ ( set @ B ) @ A6 @ B7 ) )
= ( divide_divide @ A @ ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ F @ A6 ) @ ( groups7121269368397514597t_prod @ B @ A @ F @ B7 ) ) @ ( groups7121269368397514597t_prod @ B @ A @ F @ ( inf_inf @ ( set @ B ) @ A6 @ B7 ) ) ) ) ) ) ) ) ).
% prod_Un
thf(fact_4126_translation__Compl,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A3: A,T2: set @ A] :
( ( image @ A @ A @ ( plus_plus @ A @ A3 ) @ ( uminus_uminus @ ( set @ A ) @ T2 ) )
= ( uminus_uminus @ ( set @ A ) @ ( image @ A @ A @ ( plus_plus @ A @ A3 ) @ T2 ) ) ) ) ).
% translation_Compl
thf(fact_4127_remdups__adj__altdef,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A] :
( ( ( remdups_adj @ A @ Xs )
= Ys2 )
= ( ? [F2: nat > nat] :
( ( order_mono @ nat @ nat @ F2 )
& ( ( image @ nat @ nat @ F2 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) ) )
= ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Ys2 ) ) )
& ! [I5: nat] :
( ( ord_less @ nat @ I5 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ Xs @ I5 )
= ( nth @ A @ Ys2 @ ( F2 @ I5 ) ) ) )
& ! [I5: nat] :
( ( ord_less @ nat @ ( plus_plus @ nat @ I5 @ ( one_one @ nat ) ) @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ( nth @ A @ Xs @ I5 )
= ( nth @ A @ Xs @ ( plus_plus @ nat @ I5 @ ( one_one @ nat ) ) ) )
= ( ( F2 @ I5 )
= ( F2 @ ( plus_plus @ nat @ I5 @ ( one_one @ nat ) ) ) ) ) ) ) ) ) ).
% remdups_adj_altdef
thf(fact_4128_pochhammer__times__pochhammer__half,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [Z3: A,N: nat] :
( ( times_times @ A @ ( comm_s3205402744901411588hammer @ A @ Z3 @ ( suc @ N ) ) @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ Z3 @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( suc @ N ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [K3: nat] : ( plus_plus @ A @ Z3 @ ( 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_4129_butlast__take,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( ord_less_eq @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( butlast @ A @ ( take @ A @ N @ Xs ) )
= ( take @ A @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ Xs ) ) ) ).
% butlast_take
thf(fact_4130_prod_Oneutral__const,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A6: set @ B] :
( ( groups7121269368397514597t_prod @ B @ A
@ ^ [Uu: B] : ( one_one @ A )
@ A6 )
= ( one_one @ A ) ) ) ).
% prod.neutral_const
thf(fact_4131_of__int__prod,axiom,
! [A: $tType,B: $tType] :
( ( comm_ring_1 @ A )
=> ! [F: B > int,A6: set @ B] :
( ( ring_1_of_int @ A @ ( groups7121269368397514597t_prod @ B @ int @ F @ A6 ) )
= ( groups7121269368397514597t_prod @ B @ A
@ ^ [X2: B] : ( ring_1_of_int @ A @ ( F @ X2 ) )
@ A6 ) ) ) ).
% of_int_prod
thf(fact_4132_finite__Collect__less__nat,axiom,
! [K: nat] :
( finite_finite2 @ nat
@ ( collect @ nat
@ ^ [N3: nat] : ( ord_less @ nat @ N3 @ K ) ) ) ).
% finite_Collect_less_nat
thf(fact_4133_finite__Collect__le__nat,axiom,
! [K: nat] :
( finite_finite2 @ nat
@ ( collect @ nat
@ ^ [N3: nat] : ( ord_less_eq @ nat @ N3 @ K ) ) ) ).
% finite_Collect_le_nat
thf(fact_4134_card__Collect__less__nat,axiom,
! [N: nat] :
( ( finite_card @ nat
@ ( collect @ nat
@ ^ [I5: nat] : ( ord_less @ nat @ I5 @ N ) ) )
= N ) ).
% card_Collect_less_nat
thf(fact_4135_finite__interval__int1,axiom,
! [A3: int,B2: int] :
( finite_finite2 @ int
@ ( collect @ int
@ ^ [I5: int] :
( ( ord_less_eq @ int @ A3 @ I5 )
& ( ord_less_eq @ int @ I5 @ B2 ) ) ) ) ).
% finite_interval_int1
thf(fact_4136_finite__interval__int4,axiom,
! [A3: int,B2: int] :
( finite_finite2 @ int
@ ( collect @ int
@ ^ [I5: int] :
( ( ord_less @ int @ A3 @ I5 )
& ( ord_less @ int @ I5 @ B2 ) ) ) ) ).
% finite_interval_int4
thf(fact_4137_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_4138_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_4139_sum__list__0,axiom,
! [B: $tType,A: $tType] :
( ( monoid_add @ A )
=> ! [Xs: list @ B] :
( ( groups8242544230860333062m_list @ A
@ ( map @ B @ A
@ ^ [X2: B] : ( zero_zero @ A )
@ Xs ) )
= ( zero_zero @ A ) ) ) ).
% sum_list_0
thf(fact_4140_card__Collect__le__nat,axiom,
! [N: nat] :
( ( finite_card @ nat
@ ( collect @ nat
@ ^ [I5: nat] : ( ord_less_eq @ nat @ I5 @ N ) ) )
= ( suc @ N ) ) ).
% card_Collect_le_nat
thf(fact_4141_of__nat__prod,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_1 @ A )
=> ! [F: B > nat,A6: set @ B] :
( ( semiring_1_of_nat @ A @ ( groups7121269368397514597t_prod @ B @ nat @ F @ A6 ) )
= ( groups7121269368397514597t_prod @ B @ A
@ ^ [X2: B] : ( semiring_1_of_nat @ A @ ( F @ X2 ) )
@ A6 ) ) ) ).
% of_nat_prod
thf(fact_4142_finite__interval__int2,axiom,
! [A3: int,B2: int] :
( finite_finite2 @ int
@ ( collect @ int
@ ^ [I5: int] :
( ( ord_less_eq @ int @ A3 @ I5 )
& ( ord_less @ int @ I5 @ B2 ) ) ) ) ).
% finite_interval_int2
thf(fact_4143_finite__interval__int3,axiom,
! [A3: int,B2: int] :
( finite_finite2 @ int
@ ( collect @ int
@ ^ [I5: int] :
( ( ord_less @ int @ A3 @ I5 )
& ( ord_less_eq @ int @ I5 @ B2 ) ) ) ) ).
% finite_interval_int3
thf(fact_4144_length__butlast,axiom,
! [A: $tType,Xs: list @ A] :
( ( size_size @ ( list @ A ) @ ( butlast @ A @ Xs ) )
= ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) ) ) ).
% length_butlast
thf(fact_4145_image__divide__atLeastAtMost,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [D2: A,A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ D2 )
=> ( ( image @ A @ A
@ ^ [C4: A] : ( divide_divide @ A @ C4 @ D2 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( divide_divide @ A @ A3 @ D2 ) @ ( divide_divide @ A @ B2 @ D2 ) ) ) ) ) ).
% image_divide_atLeastAtMost
thf(fact_4146_Gcd__int__eq,axiom,
! [N6: set @ nat] :
( ( gcd_Gcd @ int @ ( image @ nat @ int @ ( semiring_1_of_nat @ int ) @ N6 ) )
= ( semiring_1_of_nat @ int @ ( gcd_Gcd @ nat @ N6 ) ) ) ).
% Gcd_int_eq
thf(fact_4147_prod_Ofinite__Collect__op,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [I6: set @ B,X: B > A,Y3: B > A] :
( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [I5: B] :
( ( member @ B @ I5 @ I6 )
& ( ( X @ I5 )
!= ( one_one @ A ) ) ) ) )
=> ( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [I5: B] :
( ( member @ B @ I5 @ I6 )
& ( ( Y3 @ I5 )
!= ( one_one @ A ) ) ) ) )
=> ( finite_finite2 @ B
@ ( collect @ B
@ ^ [I5: B] :
( ( member @ B @ I5 @ I6 )
& ( ( times_times @ A @ ( X @ I5 ) @ ( Y3 @ I5 ) )
!= ( one_one @ A ) ) ) ) ) ) ) ) ).
% prod.finite_Collect_op
thf(fact_4148_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
@ ^ [I5: nat] : ( G @ ( suc @ I5 ) )
@ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ).
% prod.shift_bounds_Suc_ivl
thf(fact_4149_nat__seg__image__imp__finite,axiom,
! [A: $tType,A6: set @ A,F: nat > A,N: nat] :
( ( A6
= ( image @ nat @ A @ F
@ ( collect @ nat
@ ^ [I5: nat] : ( ord_less @ nat @ I5 @ N ) ) ) )
=> ( finite_finite2 @ A @ A6 ) ) ).
% nat_seg_image_imp_finite
thf(fact_4150_finite__conv__nat__seg__image,axiom,
! [A: $tType] :
( ( finite_finite2 @ A )
= ( ^ [A7: set @ A] :
? [N3: nat,F2: nat > A] :
( A7
= ( image @ nat @ A @ F2
@ ( collect @ nat
@ ^ [I5: nat] : ( ord_less @ nat @ I5 @ N3 ) ) ) ) ) ) ).
% finite_conv_nat_seg_image
thf(fact_4151_sum_Ofinite__Collect__op,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [I6: set @ B,X: B > A,Y3: B > A] :
( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [I5: B] :
( ( member @ B @ I5 @ I6 )
& ( ( X @ I5 )
!= ( zero_zero @ A ) ) ) ) )
=> ( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [I5: B] :
( ( member @ B @ I5 @ I6 )
& ( ( Y3 @ I5 )
!= ( zero_zero @ A ) ) ) ) )
=> ( finite_finite2 @ B
@ ( collect @ B
@ ^ [I5: B] :
( ( member @ B @ I5 @ I6 )
& ( ( plus_plus @ A @ ( X @ I5 ) @ ( Y3 @ I5 ) )
!= ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_4152_prod_Ointer__filter,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A6: set @ B,G: B > A,P: B > $o] :
( ( finite_finite2 @ B @ A6 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ A6 )
& ( P @ X2 ) ) ) )
= ( groups7121269368397514597t_prod @ B @ A
@ ^ [X2: B] : ( if @ A @ ( P @ X2 ) @ ( G @ X2 ) @ ( one_one @ A ) )
@ A6 ) ) ) ) ).
% prod.inter_filter
thf(fact_4153_prod_Odistrib,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: B > A,H2: B > A,A6: set @ B] :
( ( groups7121269368397514597t_prod @ B @ A
@ ^ [X2: B] : ( times_times @ A @ ( G @ X2 ) @ ( H2 @ X2 ) )
@ A6 )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G @ A6 ) @ ( groups7121269368397514597t_prod @ B @ A @ H2 @ A6 ) ) ) ) ).
% prod.distrib
thf(fact_4154_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
@ ^ [I5: nat] : ( G @ ( suc @ I5 ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% prod.shift_bounds_cl_Suc_ivl
thf(fact_4155_mod__prod__eq,axiom,
! [B: $tType,A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [F: B > A,A3: A,A6: set @ B] :
( ( modulo_modulo @ A
@ ( groups7121269368397514597t_prod @ B @ A
@ ^ [I5: B] : ( modulo_modulo @ A @ ( F @ I5 ) @ A3 )
@ A6 )
@ A3 )
= ( modulo_modulo @ A @ ( groups7121269368397514597t_prod @ B @ A @ F @ A6 ) @ A3 ) ) ) ).
% mod_prod_eq
thf(fact_4156_int__prod,axiom,
! [B: $tType,F: B > nat,A6: set @ B] :
( ( semiring_1_of_nat @ int @ ( groups7121269368397514597t_prod @ B @ nat @ F @ A6 ) )
= ( groups7121269368397514597t_prod @ B @ int
@ ^ [X2: B] : ( semiring_1_of_nat @ int @ ( F @ X2 ) )
@ A6 ) ) ).
% int_prod
thf(fact_4157_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_4158_translation__subtract__Compl,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A3: A,T2: set @ A] :
( ( image @ A @ A
@ ^ [X2: A] : ( minus_minus @ A @ X2 @ A3 )
@ ( uminus_uminus @ ( set @ A ) @ T2 ) )
= ( uminus_uminus @ ( set @ A )
@ ( image @ A @ A
@ ^ [X2: A] : ( minus_minus @ A @ X2 @ A3 )
@ T2 ) ) ) ) ).
% translation_subtract_Compl
thf(fact_4159_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_4160_nat__less__as__int,axiom,
( ( ord_less @ nat )
= ( ^ [A5: nat,B3: nat] : ( ord_less @ int @ ( semiring_1_of_nat @ int @ A5 ) @ ( semiring_1_of_nat @ int @ B3 ) ) ) ) ).
% nat_less_as_int
thf(fact_4161_len__def,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( ( array_len @ A )
= ( ^ [A5: array @ A] :
( heap_Time_tap @ nat
@ ^ [H4: heap_ext @ product_unit] : ( array_length @ A @ H4 @ A5 ) ) ) ) ) ).
% len_def
thf(fact_4162_lambda__one,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ( ( ^ [X2: A] : X2 )
= ( times_times @ A @ ( one_one @ A ) ) ) ) ).
% lambda_one
thf(fact_4163_lambda__zero,axiom,
! [A: $tType] :
( ( mult_zero @ A )
=> ( ( ^ [H4: A] : ( zero_zero @ A ) )
= ( times_times @ A @ ( zero_zero @ A ) ) ) ) ).
% lambda_zero
thf(fact_4164_Compl__eq,axiom,
! [A: $tType] :
( ( uminus_uminus @ ( set @ A ) )
= ( ^ [A7: set @ A] :
( collect @ A
@ ^ [X2: A] :
~ ( member @ A @ X2 @ A7 ) ) ) ) ).
% Compl_eq
thf(fact_4165_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_4166_uminus__set__def,axiom,
! [A: $tType] :
( ( uminus_uminus @ ( set @ A ) )
= ( ^ [A7: set @ A] :
( collect @ A
@ ( uminus_uminus @ ( A > $o )
@ ^ [X2: A] : ( member @ A @ X2 @ A7 ) ) ) ) ) ).
% uminus_set_def
thf(fact_4167_min__of__mono,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( linorder @ B ) )
=> ! [F: A > B,M: A,N: A] :
( ( order_mono @ A @ B @ F )
=> ( ( ord_min @ B @ ( F @ M ) @ ( F @ N ) )
= ( F @ ( ord_min @ A @ M @ N ) ) ) ) ) ).
% min_of_mono
thf(fact_4168_mono__Suc,axiom,
order_mono @ nat @ nat @ suc ).
% mono_Suc
thf(fact_4169_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_4170_mono__add,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [A3: A] : ( order_mono @ A @ A @ ( plus_plus @ A @ A3 ) ) ) ).
% mono_add
thf(fact_4171_less__set__def,axiom,
! [A: $tType] :
( ( ord_less @ ( set @ A ) )
= ( ^ [A7: set @ A,B6: set @ A] :
( ord_less @ ( A > $o )
@ ^ [X2: A] : ( member @ A @ X2 @ A7 )
@ ^ [X2: A] : ( member @ A @ X2 @ B6 ) ) ) ) ).
% less_set_def
thf(fact_4172_mono__strict__invE,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( order @ B ) )
=> ! [F: A > B,X: A,Y3: A] :
( ( order_mono @ A @ B @ F )
=> ( ( ord_less @ B @ ( F @ X ) @ ( F @ Y3 ) )
=> ( ord_less @ A @ X @ Y3 ) ) ) ) ).
% mono_strict_invE
thf(fact_4173_max__of__mono,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( linorder @ B ) )
=> ! [F: A > B,M: A,N: A] :
( ( order_mono @ A @ B @ F )
=> ( ( ord_max @ B @ ( F @ M ) @ ( F @ N ) )
= ( F @ ( ord_max @ A @ M @ N ) ) ) ) ) ).
% max_of_mono
thf(fact_4174_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_4175_mono__def,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ( ( order_mono @ A @ B )
= ( ^ [F2: A > B] :
! [X2: A,Y6: A] :
( ( ord_less_eq @ A @ X2 @ Y6 )
=> ( ord_less_eq @ B @ ( F2 @ X2 ) @ ( F2 @ Y6 ) ) ) ) ) ) ).
% mono_def
thf(fact_4176_monoI,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F: A > B] :
( ! [X3: A,Y5: A] :
( ( ord_less_eq @ A @ X3 @ Y5 )
=> ( ord_less_eq @ B @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( order_mono @ A @ B @ F ) ) ) ).
% monoI
thf(fact_4177_monoE,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F: A > B,X: A,Y3: A] :
( ( order_mono @ A @ B @ F )
=> ( ( ord_less_eq @ A @ X @ Y3 )
=> ( ord_less_eq @ B @ ( F @ X ) @ ( F @ Y3 ) ) ) ) ) ).
% monoE
thf(fact_4178_monoD,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F: A > B,X: A,Y3: A] :
( ( order_mono @ A @ B @ F )
=> ( ( ord_less_eq @ A @ X @ Y3 )
=> ( ord_less_eq @ B @ ( F @ X ) @ ( F @ Y3 ) ) ) ) ) ).
% monoD
thf(fact_4179_min__def__raw,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_min @ A )
= ( ^ [A5: A,B3: A] : ( if @ A @ ( ord_less_eq @ A @ A5 @ B3 ) @ A5 @ B3 ) ) ) ) ).
% min_def_raw
thf(fact_4180_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_4181_max__def__raw,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_max @ A )
= ( ^ [A5: A,B3: A] : ( if @ A @ ( ord_less_eq @ A @ A5 @ B3 ) @ B3 @ A5 ) ) ) ) ).
% max_def_raw
thf(fact_4182_nat__leq__as__int,axiom,
( ( ord_less_eq @ nat )
= ( ^ [A5: nat,B3: nat] : ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ A5 ) @ ( semiring_1_of_nat @ int @ B3 ) ) ) ) ).
% nat_leq_as_int
thf(fact_4183_effect__LetI,axiom,
! [B: $tType,A: $tType,X: A,T2: A,F: A > ( heap_Time_Heap @ B ),H2: heap_ext @ product_unit,H: heap_ext @ product_unit,R: B,N: nat] :
( ( X = T2 )
=> ( ( heap_Time_effect @ B @ ( F @ X ) @ H2 @ H @ R @ N )
=> ( heap_Time_effect @ B @ ( F @ T2 ) @ H2 @ H @ R @ N ) ) ) ).
% effect_LetI
thf(fact_4184_lessThan__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( set_ord_lessThan @ A )
= ( ^ [U3: A] :
( collect @ A
@ ^ [X2: A] : ( ord_less @ A @ X2 @ U3 ) ) ) ) ) ).
% lessThan_def
thf(fact_4185_atMost__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( set_ord_atMost @ A )
= ( ^ [U3: A] :
( collect @ A
@ ^ [X2: A] : ( ord_less_eq @ A @ X2 @ U3 ) ) ) ) ) ).
% atMost_def
thf(fact_4186_sum__list__const__mult,axiom,
! [A: $tType,B: $tType] :
( ( semiring_0 @ A )
=> ! [C2: A,F: B > A,Xs: list @ B] :
( ( groups8242544230860333062m_list @ A
@ ( map @ B @ A
@ ^ [X2: B] : ( times_times @ A @ C2 @ ( F @ X2 ) )
@ Xs ) )
= ( times_times @ A @ C2 @ ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F @ Xs ) ) ) ) ) ).
% sum_list_const_mult
thf(fact_4187_sum__list__mult__const,axiom,
! [B: $tType,A: $tType] :
( ( semiring_0 @ A )
=> ! [F: B > A,C2: A,Xs: list @ B] :
( ( groups8242544230860333062m_list @ A
@ ( map @ B @ A
@ ^ [X2: B] : ( times_times @ A @ ( F @ X2 ) @ C2 )
@ Xs ) )
= ( times_times @ A @ ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F @ Xs ) ) @ C2 ) ) ) ).
% sum_list_mult_const
thf(fact_4188_finite__less__ub,axiom,
! [F: nat > nat,U: nat] :
( ! [N4: nat] : ( ord_less_eq @ nat @ N4 @ ( F @ N4 ) )
=> ( finite_finite2 @ nat
@ ( collect @ nat
@ ^ [N3: nat] : ( ord_less_eq @ nat @ ( F @ N3 ) @ U ) ) ) ) ).
% finite_less_ub
thf(fact_4189_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_4190_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_4191_finite__divisors__int,axiom,
! [I: int] :
( ( I
!= ( zero_zero @ int ) )
=> ( finite_finite2 @ int
@ ( collect @ int
@ ^ [D5: int] : ( dvd_dvd @ int @ D5 @ I ) ) ) ) ).
% finite_divisors_int
thf(fact_4192_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_4193_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_4194_power__numeral__even,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [Z3: A,W2: num] :
( ( power_power @ A @ Z3 @ ( numeral_numeral @ nat @ ( bit0 @ W2 ) ) )
= ( times_times @ A @ ( power_power @ A @ Z3 @ ( numeral_numeral @ nat @ W2 ) ) @ ( power_power @ A @ Z3 @ ( numeral_numeral @ nat @ W2 ) ) ) ) ) ).
% power_numeral_even
thf(fact_4195_mono__ge2__power__minus__self,axiom,
! [K: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K )
=> ( order_mono @ nat @ nat
@ ^ [M2: nat] : ( minus_minus @ nat @ ( power_power @ nat @ K @ M2 ) @ M2 ) ) ) ).
% mono_ge2_power_minus_self
thf(fact_4196_prod_Ointer__restrict,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A6: set @ B,G: B > A,B7: set @ B] :
( ( finite_finite2 @ B @ A6 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( inf_inf @ ( set @ B ) @ A6 @ B7 ) )
= ( groups7121269368397514597t_prod @ B @ A
@ ^ [X2: B] : ( if @ A @ ( member @ B @ X2 @ B7 ) @ ( G @ X2 ) @ ( one_one @ A ) )
@ A6 ) ) ) ) ).
% prod.inter_restrict
thf(fact_4197_prod_Osetdiff__irrelevant,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A6: set @ B,G: B > A] :
( ( finite_finite2 @ B @ A6 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G
@ ( minus_minus @ ( set @ B ) @ A6
@ ( collect @ B
@ ^ [X2: B] :
( ( G @ X2 )
= ( one_one @ A ) ) ) ) )
= ( groups7121269368397514597t_prod @ B @ A @ G @ A6 ) ) ) ) ).
% prod.setdiff_irrelevant
thf(fact_4198_power__numeral__odd,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [Z3: A,W2: num] :
( ( power_power @ A @ Z3 @ ( numeral_numeral @ nat @ ( bit1 @ W2 ) ) )
= ( times_times @ A @ ( times_times @ A @ Z3 @ ( power_power @ A @ Z3 @ ( numeral_numeral @ nat @ W2 ) ) ) @ ( power_power @ A @ Z3 @ ( numeral_numeral @ nat @ W2 ) ) ) ) ) ).
% power_numeral_odd
thf(fact_4199_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 )
@ ^ [L3: list @ nat] :
( ( ( size_size @ ( list @ nat ) @ L3 )
= M )
& ( ( groups8242544230860333062m_list @ nat @ L3 )
= N6 ) ) ) )
= ( plus_plus @ nat
@ ( finite_card @ ( list @ nat )
@ ( collect @ ( list @ nat )
@ ^ [L3: list @ nat] :
( ( ( size_size @ ( list @ nat ) @ L3 )
= ( minus_minus @ nat @ M @ ( one_one @ nat ) ) )
& ( ( groups8242544230860333062m_list @ nat @ L3 )
= N6 ) ) ) )
@ ( finite_card @ ( list @ nat )
@ ( collect @ ( list @ nat )
@ ^ [L3: list @ nat] :
( ( ( size_size @ ( list @ nat ) @ L3 )
= M )
& ( ( plus_plus @ nat @ ( groups8242544230860333062m_list @ nat @ L3 ) @ ( one_one @ nat ) )
= N6 ) ) ) ) ) ) ) ).
% card_length_sum_list_rec
thf(fact_4200_sum__list__mono,axiom,
! [B: $tType,A: $tType] :
( ( ( monoid_add @ B )
& ( ordere6658533253407199908up_add @ B ) )
=> ! [Xs: list @ A,F: A > B,G: A > B] :
( ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( ord_less_eq @ B @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_eq @ B @ ( groups8242544230860333062m_list @ B @ ( map @ A @ B @ F @ Xs ) ) @ ( groups8242544230860333062m_list @ B @ ( map @ A @ B @ G @ Xs ) ) ) ) ) ).
% sum_list_mono
thf(fact_4201_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_4202_nat__plus__as__int,axiom,
( ( plus_plus @ nat )
= ( ^ [A5: nat,B3: nat] : ( nat2 @ ( plus_plus @ int @ ( semiring_1_of_nat @ int @ A5 ) @ ( semiring_1_of_nat @ int @ B3 ) ) ) ) ) ).
% nat_plus_as_int
thf(fact_4203_nat__times__as__int,axiom,
( ( times_times @ nat )
= ( ^ [A5: nat,B3: nat] : ( nat2 @ ( times_times @ int @ ( semiring_1_of_nat @ int @ A5 ) @ ( semiring_1_of_nat @ int @ B3 ) ) ) ) ) ).
% nat_times_as_int
thf(fact_4204_card__length__sum__list,axiom,
! [M: nat,N6: nat] :
( ( finite_card @ ( list @ nat )
@ ( collect @ ( list @ nat )
@ ^ [L3: list @ nat] :
( ( ( size_size @ ( list @ nat ) @ L3 )
= M )
& ( ( groups8242544230860333062m_list @ nat @ L3 )
= N6 ) ) ) )
= ( binomial @ ( minus_minus @ nat @ ( plus_plus @ nat @ N6 @ M ) @ ( one_one @ nat ) ) @ N6 ) ) ).
% card_length_sum_list
thf(fact_4205_nat__minus__as__int,axiom,
( ( minus_minus @ nat )
= ( ^ [A5: nat,B3: nat] : ( nat2 @ ( minus_minus @ int @ ( semiring_1_of_nat @ int @ A5 ) @ ( semiring_1_of_nat @ int @ B3 ) ) ) ) ) ).
% nat_minus_as_int
thf(fact_4206_prod_Onat__diff__reindex,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I5: nat] : ( G @ ( minus_minus @ nat @ N @ ( suc @ I5 ) ) )
@ ( set_ord_lessThan @ nat @ N ) )
= ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% prod.nat_diff_reindex
thf(fact_4207_finite__divisors__nat,axiom,
! [M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( finite_finite2 @ nat
@ ( collect @ nat
@ ^ [D5: nat] : ( dvd_dvd @ nat @ D5 @ M ) ) ) ) ).
% finite_divisors_nat
thf(fact_4208_card__less,axiom,
! [M7: set @ nat,I: nat] :
( ( member @ nat @ ( zero_zero @ nat ) @ M7 )
=> ( ( finite_card @ nat
@ ( collect @ nat
@ ^ [K3: nat] :
( ( member @ nat @ K3 @ M7 )
& ( ord_less @ nat @ K3 @ ( suc @ I ) ) ) ) )
!= ( zero_zero @ nat ) ) ) ).
% card_less
thf(fact_4209_card__less__Suc,axiom,
! [M7: set @ nat,I: nat] :
( ( member @ nat @ ( zero_zero @ nat ) @ M7 )
=> ( ( suc
@ ( finite_card @ nat
@ ( collect @ nat
@ ^ [K3: nat] :
( ( member @ nat @ ( suc @ K3 ) @ M7 )
& ( ord_less @ nat @ K3 @ I ) ) ) ) )
= ( finite_card @ nat
@ ( collect @ nat
@ ^ [K3: nat] :
( ( member @ nat @ K3 @ M7 )
& ( ord_less @ nat @ K3 @ ( suc @ I ) ) ) ) ) ) ) ).
% card_less_Suc
thf(fact_4210_card__less__Suc2,axiom,
! [M7: set @ nat,I: nat] :
( ~ ( member @ nat @ ( zero_zero @ nat ) @ M7 )
=> ( ( finite_card @ nat
@ ( collect @ nat
@ ^ [K3: nat] :
( ( member @ nat @ ( suc @ K3 ) @ M7 )
& ( ord_less @ nat @ K3 @ I ) ) ) )
= ( finite_card @ nat
@ ( collect @ nat
@ ^ [K3: nat] :
( ( member @ nat @ K3 @ M7 )
& ( ord_less @ nat @ K3 @ ( suc @ I ) ) ) ) ) ) ) ).
% card_less_Suc2
thf(fact_4211_sum__list__Suc,axiom,
! [A: $tType,F: A > nat,Xs: list @ A] :
( ( groups8242544230860333062m_list @ nat
@ ( map @ A @ nat
@ ^ [X2: A] : ( suc @ ( F @ X2 ) )
@ Xs ) )
= ( plus_plus @ nat @ ( groups8242544230860333062m_list @ nat @ ( map @ A @ nat @ F @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ).
% sum_list_Suc
thf(fact_4212_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_4213_nat__div__as__int,axiom,
( ( divide_divide @ nat )
= ( ^ [A5: nat,B3: nat] : ( nat2 @ ( divide_divide @ int @ ( semiring_1_of_nat @ int @ A5 ) @ ( semiring_1_of_nat @ int @ B3 ) ) ) ) ) ).
% nat_div_as_int
thf(fact_4214_nat__mod__as__int,axiom,
( ( modulo_modulo @ nat )
= ( ^ [A5: nat,B3: nat] : ( nat2 @ ( modulo_modulo @ int @ ( semiring_1_of_nat @ int @ A5 ) @ ( semiring_1_of_nat @ int @ B3 ) ) ) ) ) ).
% nat_mod_as_int
thf(fact_4215_mono__invE,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( order @ B ) )
=> ! [F: A > B,X: A,Y3: A] :
( ( order_mono @ A @ B @ F )
=> ( ( ord_less @ B @ ( F @ X ) @ ( F @ Y3 ) )
=> ( ord_less_eq @ A @ X @ Y3 ) ) ) ) ).
% mono_invE
thf(fact_4216_mono__iff__le__Suc,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( order_mono @ nat @ A )
= ( ^ [F2: nat > A] :
! [N3: nat] : ( ord_less_eq @ A @ ( F2 @ N3 ) @ ( F2 @ ( suc @ N3 ) ) ) ) ) ) ).
% mono_iff_le_Suc
thf(fact_4217_image__add__int__atLeastLessThan,axiom,
! [L: int,U: int] :
( ( image @ int @ int
@ ^ [X2: int] : ( plus_plus @ int @ X2 @ L )
@ ( set_or7035219750837199246ssThan @ int @ ( zero_zero @ int ) @ ( minus_minus @ int @ U @ L ) ) )
= ( set_or7035219750837199246ssThan @ int @ L @ U ) ) ).
% image_add_int_atLeastLessThan
thf(fact_4218_mono__inf,axiom,
! [B: $tType,A: $tType] :
( ( ( semilattice_inf @ A )
& ( semilattice_inf @ B ) )
=> ! [F: A > B,A6: A,B7: A] :
( ( order_mono @ A @ B @ F )
=> ( ord_less_eq @ B @ ( F @ ( inf_inf @ A @ A6 @ B7 ) ) @ ( inf_inf @ B @ ( F @ A6 ) @ ( F @ B7 ) ) ) ) ) ).
% mono_inf
thf(fact_4219_map__replicate__trivial,axiom,
! [A: $tType,X: A,I: nat] :
( ( map @ nat @ A
@ ^ [I5: nat] : X
@ ( upt @ ( zero_zero @ nat ) @ I ) )
= ( replicate @ A @ I @ X ) ) ).
% map_replicate_trivial
thf(fact_4220_mono__sup,axiom,
! [B: $tType,A: $tType] :
( ( ( semilattice_sup @ A )
& ( semilattice_sup @ B ) )
=> ! [F: A > B,A6: A,B7: A] :
( ( order_mono @ A @ B @ F )
=> ( ord_less_eq @ B @ ( sup_sup @ B @ ( F @ A6 ) @ ( F @ B7 ) ) @ ( F @ ( sup_sup @ A @ A6 @ B7 ) ) ) ) ) ).
% mono_sup
thf(fact_4221_semiring__char__def,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( semiri4206861660011772517g_char @ A )
= ( ^ [Uu2: itself @ A] :
( gcd_Gcd @ nat
@ ( collect @ nat
@ ^ [N3: nat] :
( ( semiring_1_of_nat @ A @ N3 )
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% semiring_char_def
thf(fact_4222_map__add__upt,axiom,
! [N: nat,M: nat] :
( ( map @ nat @ nat
@ ^ [I5: nat] : ( plus_plus @ nat @ I5 @ N )
@ ( upt @ ( zero_zero @ nat ) @ M ) )
= ( upt @ N @ ( plus_plus @ nat @ M @ N ) ) ) ).
% map_add_upt
thf(fact_4223_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_4224_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
@ ^ [I5: nat] : ( G @ ( suc @ I5 ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ) ) ).
% prod.Suc_reindex_ivl
thf(fact_4225_prod_OIf__cases,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A6: set @ B,P: B > $o,H2: B > A,G: B > A] :
( ( finite_finite2 @ B @ A6 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [X2: B] : ( if @ A @ ( P @ X2 ) @ ( H2 @ X2 ) @ ( G @ X2 ) )
@ A6 )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ H2 @ ( inf_inf @ ( set @ B ) @ A6 @ ( collect @ B @ P ) ) ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( inf_inf @ ( set @ B ) @ A6 @ ( uminus_uminus @ ( set @ B ) @ ( collect @ B @ P ) ) ) ) ) ) ) ) ).
% prod.If_cases
thf(fact_4226_sum__list__triv,axiom,
! [C: $tType,B: $tType] :
( ( semiring_1 @ B )
=> ! [R: B,Xs: list @ C] :
( ( groups8242544230860333062m_list @ B
@ ( map @ C @ B
@ ^ [X2: C] : R
@ Xs ) )
= ( times_times @ B @ ( semiring_1_of_nat @ B @ ( size_size @ ( list @ C ) @ Xs ) ) @ R ) ) ) ).
% sum_list_triv
thf(fact_4227_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
@ ^ [I5: nat] : ( G @ ( suc @ I5 ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ) ).
% prod.atMost_Suc_shift
thf(fact_4228_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
@ ^ [I5: nat] : ( G @ ( suc @ I5 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% prod.lessThan_Suc_shift
thf(fact_4229_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
@ ^ [I5: nat] : ( G @ ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N ) @ ( suc @ I5 ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ N @ M ) ) ) ) ).
% prod.atLeastLessThan_rev
thf(fact_4230_prod_Onested__swap,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: nat > nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I5: nat] : ( groups7121269368397514597t_prod @ nat @ A @ ( A3 @ I5 ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ I5 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [J3: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I5: nat] : ( A3 @ I5 @ J3 )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J3 ) @ N ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% prod.nested_swap
thf(fact_4231_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_4232_prod_Onat__group,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,K: nat,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A
@ ^ [M2: nat] : ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( times_times @ nat @ M2 @ K ) @ ( plus_plus @ nat @ ( times_times @ nat @ M2 @ K ) @ K ) ) )
@ ( set_ord_lessThan @ nat @ N ) )
= ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_lessThan @ nat @ ( times_times @ nat @ N @ K ) ) ) ) ) ).
% prod.nat_group
thf(fact_4233_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
@ ^ [I5: nat] : ( F @ ( suc @ I5 ) )
@ ( upt @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% map_upt_Suc
thf(fact_4234_prod_Onested__swap_H,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: nat > nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I5: nat] : ( groups7121269368397514597t_prod @ nat @ A @ ( A3 @ I5 ) @ ( set_ord_lessThan @ nat @ I5 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [J3: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I5: nat] : ( A3 @ I5 @ J3 )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J3 ) @ N ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% prod.nested_swap'
thf(fact_4235_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
@ ^ [A5: nat] : ( times_times @ A @ ( F @ A5 ) )
@ A3
@ B2
@ ( one_one @ A ) ) ) ) ).
% prod_atLeastAtMost_code
thf(fact_4236_map__nth,axiom,
! [A: $tType,Xs: list @ A] :
( ( map @ nat @ A @ ( nth @ A @ Xs ) @ ( upt @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) ) )
= Xs ) ).
% map_nth
thf(fact_4237_ratrel__def,axiom,
( ratrel
= ( ^ [X2: product_prod @ int @ int,Y6: product_prod @ int @ int] :
( ( ( product_snd @ int @ int @ X2 )
!= ( zero_zero @ int ) )
& ( ( product_snd @ int @ int @ Y6 )
!= ( zero_zero @ int ) )
& ( ( times_times @ int @ ( product_fst @ int @ int @ X2 ) @ ( product_snd @ int @ int @ Y6 ) )
= ( times_times @ int @ ( product_fst @ int @ int @ Y6 ) @ ( product_snd @ int @ int @ X2 ) ) ) ) ) ) ).
% ratrel_def
thf(fact_4238_fact__prod,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiring_char_0_fact @ A )
= ( ^ [N3: nat] :
( semiring_1_of_nat @ A
@ ( groups7121269368397514597t_prod @ nat @ nat
@ ^ [X2: nat] : X2
@ ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ N3 ) ) ) ) ) ) ).
% fact_prod
thf(fact_4239_map__decr__upt,axiom,
! [M: nat,N: nat] :
( ( map @ nat @ nat
@ ^ [N3: nat] : ( minus_minus @ nat @ N3 @ ( suc @ ( zero_zero @ nat ) ) )
@ ( upt @ ( suc @ M ) @ ( suc @ N ) ) )
= ( upt @ M @ N ) ) ).
% map_decr_upt
thf(fact_4240_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 @ ( insert @ 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 @ ( insert @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ).
% prod.delta_remove
thf(fact_4241_diff__nat__eq__if,axiom,
! [Z6: int,Z3: int] :
( ( ( ord_less @ int @ Z6 @ ( zero_zero @ int ) )
=> ( ( minus_minus @ nat @ ( nat2 @ Z3 ) @ ( nat2 @ Z6 ) )
= ( nat2 @ Z3 ) ) )
& ( ~ ( ord_less @ int @ Z6 @ ( zero_zero @ int ) )
=> ( ( minus_minus @ nat @ ( nat2 @ Z3 ) @ ( nat2 @ Z6 ) )
= ( if @ nat @ ( ord_less @ int @ ( minus_minus @ int @ Z3 @ Z6 ) @ ( zero_zero @ int ) ) @ ( zero_zero @ nat ) @ ( nat2 @ ( minus_minus @ int @ Z3 @ Z6 ) ) ) ) ) ) ).
% diff_nat_eq_if
thf(fact_4242_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
@ ^ [I5: nat] : ( G @ ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N ) @ I5 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ N ) @ M ) ) ) ) ).
% prod.atLeastLessThan_rev_at_least_Suc_atMost
thf(fact_4243_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
@ ^ [I5: nat] : ( G @ ( suc @ I5 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% prod.atMost_shift
thf(fact_4244_pochhammer__prod,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ( ( comm_s3205402744901411588hammer @ A )
= ( ^ [A5: A,N3: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I5: nat] : ( plus_plus @ A @ A5 @ ( semiring_1_of_nat @ A @ I5 ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N3 ) ) ) ) ) ).
% pochhammer_prod
thf(fact_4245_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_4246_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_4247_finite__lists__length__le,axiom,
! [A: $tType,A6: set @ A,N: nat] :
( ( finite_finite2 @ A @ A6 )
=> ( finite_finite2 @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Xs3: list @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs3 ) @ A6 )
& ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs3 ) @ N ) ) ) ) ) ).
% finite_lists_length_le
thf(fact_4248_mono__image__least,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [F: A > B,M: A,N: A,M5: B,N2: B] :
( ( order_mono @ A @ B @ F )
=> ( ( ( image @ A @ B @ F @ ( set_or7035219750837199246ssThan @ A @ M @ N ) )
= ( set_or7035219750837199246ssThan @ B @ M5 @ N2 ) )
=> ( ( ord_less @ A @ M @ N )
=> ( ( F @ M )
= M5 ) ) ) ) ) ).
% mono_image_least
thf(fact_4249_fact__prod__rev,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiring_char_0_fact @ A )
= ( ^ [N3: nat] : ( semiring_1_of_nat @ A @ ( groups7121269368397514597t_prod @ nat @ nat @ ( minus_minus @ nat @ N3 ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N3 ) ) ) ) ) ) ).
% fact_prod_rev
thf(fact_4250_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_4251_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 )
=> ( ( image @ 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 )
=> ( ( image @ 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 )
=> ( ( image @ A @ A
@ ^ [X2: A] : ( times_times @ A @ X2 @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ X @ Y3 ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% image_mult_atLeastAtMost_if'
thf(fact_4252_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 ) ) )
=> ( ( image @ 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 )
=> ( ( image @ 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 )
=> ( ( image @ 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_4253_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 ) ) )
=> ( ( image @ 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 )
=> ( ( image @ 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 )
=> ( ( image @ 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_4254_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 ) ) )
=> ( ( image @ 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 )
=> ( ( image @ 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 )
=> ( ( image @ 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_4255_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 ) ) )
=> ( ( image @ 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 )
=> ( ( image @ 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 )
=> ( ( image @ 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_4256_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_4257_pochhammer__Suc__prod,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A3: A,N: nat] :
( ( comm_s3205402744901411588hammer @ A @ A3 @ ( suc @ N ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I5: nat] : ( plus_plus @ A @ A3 @ ( semiring_1_of_nat @ A @ I5 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% pochhammer_Suc_prod
thf(fact_4258_pochhammer__prod__rev,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ( ( comm_s3205402744901411588hammer @ A )
= ( ^ [A5: A,N3: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I5: nat] : ( plus_plus @ A @ A5 @ ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ N3 @ I5 ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ N3 ) ) ) ) ) ).
% pochhammer_prod_rev
thf(fact_4259_set__decode__def,axiom,
( nat_set_decode
= ( ^ [X2: nat] :
( collect @ nat
@ ^ [N3: nat] :
~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ X2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) ) ) ) ) ).
% set_decode_def
thf(fact_4260_image__minus__const__atLeastLessThan__nat,axiom,
! [C2: nat,Y3: nat,X: nat] :
( ( ( ord_less @ nat @ C2 @ Y3 )
=> ( ( image @ nat @ nat
@ ^ [I5: nat] : ( minus_minus @ nat @ I5 @ 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 )
=> ( ( image @ nat @ nat
@ ^ [I5: nat] : ( minus_minus @ nat @ I5 @ C2 )
@ ( set_or7035219750837199246ssThan @ nat @ X @ Y3 ) )
= ( insert @ nat @ ( zero_zero @ nat ) @ ( bot_bot @ ( set @ nat ) ) ) ) )
& ( ~ ( ord_less @ nat @ X @ Y3 )
=> ( ( image @ nat @ nat
@ ^ [I5: nat] : ( minus_minus @ nat @ I5 @ C2 )
@ ( set_or7035219750837199246ssThan @ nat @ X @ Y3 ) )
= ( bot_bot @ ( set @ nat ) ) ) ) ) ) ) ).
% image_minus_const_atLeastLessThan_nat
thf(fact_4261_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_4262_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
@ ^ [I5: nat] : ( times_times @ A @ ( G @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I5 ) ) @ ( G @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I5 ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% prod.in_pairs
thf(fact_4263_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
@ ^ [I5: nat] : ( times_times @ A @ ( G @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I5 ) ) @ ( G @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I5 ) ) ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% prod.in_pairs_0
thf(fact_4264_gbinomial__altdef__of__nat,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( ( gbinomial @ A )
= ( ^ [A5: A,K3: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I5: nat] : ( divide_divide @ A @ ( minus_minus @ A @ A5 @ ( semiring_1_of_nat @ A @ I5 ) ) @ ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ K3 @ I5 ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K3 ) ) ) ) ) ).
% gbinomial_altdef_of_nat
thf(fact_4265_signed__take__bit__code,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( bit_ri4674362597316999326ke_bit @ A )
= ( ^ [N3: nat,A5: A] : ( if @ A @ ( bit_se5641148757651400278ts_bit @ A @ ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N3 ) @ A5 ) @ N3 ) @ ( plus_plus @ A @ ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N3 ) @ A5 ) @ ( bit_se4730199178511100633sh_bit @ A @ ( suc @ N3 ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) @ ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N3 ) @ A5 ) ) ) ) ) ).
% signed_take_bit_code
thf(fact_4266_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
@ ^ [I5: nat] : ( plus_plus @ A @ A3 @ ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ N @ I5 ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% pochhammer_Suc_prod_rev
thf(fact_4267_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
@ ^ [I5: nat] : ( divide_divide @ A @ ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ N @ I5 ) ) @ ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ K @ I5 ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K ) ) ) ) ) ).
% binomial_altdef_of_nat
thf(fact_4268_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
@ ^ [I5: nat] : ( minus_minus @ A @ A3 @ ( semiring_1_of_nat @ A @ I5 ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K ) ) ) ) ).
% gbinomial_mult_fact
thf(fact_4269_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
@ ^ [I5: nat] : ( minus_minus @ A @ A3 @ ( semiring_1_of_nat @ A @ I5 ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K ) ) ) ) ).
% gbinomial_mult_fact'
thf(fact_4270_gbinomial__prod__rev,axiom,
! [A: $tType] :
( ( ( semiring_char_0 @ A )
& ( semidom_divide @ A ) )
=> ( ( gbinomial @ A )
= ( ^ [A5: A,K3: nat] :
( divide_divide @ A
@ ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I5: nat] : ( minus_minus @ A @ A5 @ ( semiring_1_of_nat @ A @ I5 ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K3 ) )
@ ( semiring_char_0_fact @ A @ K3 ) ) ) ) ) ).
% gbinomial_prod_rev
thf(fact_4271_nth__butlast,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ ( butlast @ A @ Xs ) ) )
=> ( ( nth @ A @ ( butlast @ A @ Xs ) @ N )
= ( nth @ A @ Xs @ N ) ) ) ).
% nth_butlast
thf(fact_4272_take__butlast,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( take @ A @ N @ ( butlast @ A @ Xs ) )
= ( take @ A @ N @ Xs ) ) ) ).
% take_butlast
thf(fact_4273_prod_Ozero__middle,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [P5: nat,K: nat,G: nat > A,H2: nat > A] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ P5 )
=> ( ( ord_less_eq @ nat @ K @ P5 )
=> ( ( groups7121269368397514597t_prod @ nat @ A
@ ^ [J3: nat] : ( if @ A @ ( ord_less @ nat @ J3 @ K ) @ ( G @ J3 ) @ ( if @ A @ ( J3 = K ) @ ( one_one @ A ) @ ( H2 @ ( minus_minus @ nat @ J3 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) )
@ ( set_ord_atMost @ nat @ P5 ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [J3: nat] : ( if @ A @ ( ord_less @ nat @ J3 @ K ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
@ ( set_ord_atMost @ nat @ ( minus_minus @ nat @ P5 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) ) ).
% prod.zero_middle
thf(fact_4274_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
@ ^ [I5: nat] : ( minus_minus @ A @ A3 @ ( semiring_1_of_nat @ A @ I5 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ K ) )
@ ( semiring_char_0_fact @ A @ ( suc @ K ) ) ) ) ) ).
% gbinomial_Suc
thf(fact_4275_pochhammer__code,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ( ( comm_s3205402744901411588hammer @ A )
= ( ^ [A5: A,N3: nat] :
( if @ A
@ ( N3
= ( zero_zero @ nat ) )
@ ( one_one @ A )
@ ( set_fo6178422350223883121st_nat @ A
@ ^ [O: nat] : ( times_times @ A @ ( plus_plus @ A @ A5 @ ( semiring_1_of_nat @ A @ O ) ) )
@ ( zero_zero @ nat )
@ ( minus_minus @ nat @ N3 @ ( one_one @ nat ) )
@ ( one_one @ A ) ) ) ) ) ) ).
% pochhammer_code
thf(fact_4276_normalize__def,axiom,
( normalize
= ( ^ [P6: product_prod @ int @ int] :
( if @ ( product_prod @ int @ int ) @ ( ord_less @ int @ ( zero_zero @ int ) @ ( product_snd @ int @ int @ P6 ) ) @ ( product_Pair @ int @ int @ ( divide_divide @ int @ ( product_fst @ int @ int @ P6 ) @ ( gcd_gcd @ int @ ( product_fst @ int @ int @ P6 ) @ ( product_snd @ int @ int @ P6 ) ) ) @ ( divide_divide @ int @ ( product_snd @ int @ int @ P6 ) @ ( gcd_gcd @ int @ ( product_fst @ int @ int @ P6 ) @ ( product_snd @ int @ int @ P6 ) ) ) )
@ ( if @ ( product_prod @ int @ int )
@ ( ( product_snd @ int @ int @ P6 )
= ( zero_zero @ int ) )
@ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
@ ( product_Pair @ int @ int @ ( divide_divide @ int @ ( product_fst @ int @ int @ P6 ) @ ( uminus_uminus @ int @ ( gcd_gcd @ int @ ( product_fst @ int @ int @ P6 ) @ ( product_snd @ int @ int @ P6 ) ) ) ) @ ( divide_divide @ int @ ( product_snd @ int @ int @ P6 ) @ ( uminus_uminus @ int @ ( gcd_gcd @ int @ ( product_fst @ int @ int @ P6 ) @ ( product_snd @ int @ int @ P6 ) ) ) ) ) ) ) ) ) ).
% normalize_def
thf(fact_4277_butlast__conv__take,axiom,
! [A: $tType] :
( ( butlast @ A )
= ( ^ [Xs3: list @ A] : ( take @ A @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs3 ) @ ( one_one @ nat ) ) @ Xs3 ) ) ) ).
% butlast_conv_take
thf(fact_4278_butlast__list__update,axiom,
! [A: $tType,K: nat,Xs: list @ A,X: A] :
( ( ( K
= ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) ) )
=> ( ( butlast @ A @ ( list_update @ A @ Xs @ K @ X ) )
= ( butlast @ A @ Xs ) ) )
& ( ( K
!= ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) ) )
=> ( ( butlast @ A @ ( list_update @ A @ Xs @ K @ X ) )
= ( list_update @ A @ ( butlast @ A @ Xs ) @ K @ X ) ) ) ) ).
% butlast_list_update
thf(fact_4279_gbinomial__code,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( ( gbinomial @ A )
= ( ^ [A5: A,K3: nat] :
( if @ A
@ ( K3
= ( zero_zero @ nat ) )
@ ( one_one @ A )
@ ( divide_divide @ A
@ ( set_fo6178422350223883121st_nat @ A
@ ^ [L3: nat] : ( times_times @ A @ ( minus_minus @ A @ A5 @ ( semiring_1_of_nat @ A @ L3 ) ) )
@ ( zero_zero @ nat )
@ ( minus_minus @ nat @ K3 @ ( one_one @ nat ) )
@ ( one_one @ A ) )
@ ( semiring_char_0_fact @ A @ K3 ) ) ) ) ) ) ).
% gbinomial_code
thf(fact_4280_card__lists__distinct__length__eq_H,axiom,
! [A: $tType,K: nat,A6: set @ A] :
( ( ord_less @ nat @ K @ ( finite_card @ A @ A6 ) )
=> ( ( finite_card @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Xs3: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs3 )
= K )
& ( distinct @ A @ Xs3 )
& ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs3 ) @ A6 ) ) ) )
= ( groups7121269368397514597t_prod @ nat @ nat
@ ^ [X2: nat] : X2
@ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ ( minus_minus @ nat @ ( finite_card @ A @ A6 ) @ K ) @ ( one_one @ nat ) ) @ ( finite_card @ A @ A6 ) ) ) ) ) ).
% card_lists_distinct_length_eq'
thf(fact_4281_card__lists__distinct__length__eq,axiom,
! [A: $tType,A6: set @ A,K: nat] :
( ( finite_finite2 @ A @ A6 )
=> ( ( ord_less_eq @ nat @ K @ ( finite_card @ A @ A6 ) )
=> ( ( finite_card @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Xs3: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs3 )
= K )
& ( distinct @ A @ Xs3 )
& ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs3 ) @ A6 ) ) ) )
= ( groups7121269368397514597t_prod @ nat @ nat
@ ^ [X2: nat] : X2
@ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ ( minus_minus @ nat @ ( finite_card @ A @ A6 ) @ K ) @ ( one_one @ nat ) ) @ ( finite_card @ A @ A6 ) ) ) ) ) ) ).
% card_lists_distinct_length_eq
thf(fact_4282_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 )
@ ^ [R5: code_integer,S6: code_integer] :
( product_Pair @ code_integer @ $o @ ( if @ code_integer @ ( ord_less @ code_integer @ ( zero_zero @ code_integer ) @ K3 ) @ R5 @ ( minus_minus @ code_integer @ ( uminus_uminus @ code_integer @ R5 ) @ S6 ) )
@ ( S6
= ( one_one @ code_integer ) ) )
@ ( code_divmod_abs @ K3 @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% bit_cut_integer_code
thf(fact_4283_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_4284_swap__def,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( ( array_swap @ A )
= ( ^ [I5: nat,X2: A,A5: array @ A] :
( heap_Time_guard @ A
@ ^ [H4: heap_ext @ product_unit] : ( ord_less @ nat @ I5 @ ( array_length @ A @ H4 @ A5 ) )
@ ^ [H4: heap_ext @ product_unit] : ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( nth @ A @ ( array_get @ A @ H4 @ A5 ) @ I5 ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ ( array_update @ A @ A5 @ I5 @ X2 @ H4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% swap_def
thf(fact_4285_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
@ ^ [I5: nat] : ( if @ A @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I5 ) @ ( semiring_1_of_nat @ A @ ( binomial @ N @ I5 ) ) @ ( zero_zero @ A ) )
@ ( set_ord_atMost @ nat @ N ) ) )
= ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% choose_even_sum
thf(fact_4286_integer__of__int__eq__of__int,axiom,
( code_integer_of_int
= ( ring_1_of_int @ code_integer ) ) ).
% integer_of_int_eq_of_int
thf(fact_4287_Ints__sum,axiom,
! [A: $tType,B: $tType] :
( ( ring_1 @ B )
=> ! [A6: set @ A,F: A > B] :
( ! [X3: A] :
( ( member @ A @ X3 @ A6 )
=> ( member @ B @ ( F @ X3 ) @ ( ring_1_Ints @ B ) ) )
=> ( member @ B @ ( groups7311177749621191930dd_sum @ A @ B @ F @ A6 ) @ ( ring_1_Ints @ B ) ) ) ) ).
% Ints_sum
thf(fact_4288_sum_Oneutral__const,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A6: set @ B] :
( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [Uu: B] : ( zero_zero @ A )
@ A6 )
= ( zero_zero @ A ) ) ) ).
% sum.neutral_const
thf(fact_4289_of__nat__sum,axiom,
! [A: $tType,B: $tType] :
( ( semiring_1 @ A )
=> ! [F: B > nat,A6: set @ B] :
( ( semiring_1_of_nat @ A @ ( groups7311177749621191930dd_sum @ B @ nat @ F @ A6 ) )
= ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X2: B] : ( semiring_1_of_nat @ A @ ( F @ X2 ) )
@ A6 ) ) ) ).
% of_nat_sum
thf(fact_4290_of__int__sum,axiom,
! [A: $tType,B: $tType] :
( ( ring_1 @ A )
=> ! [F: B > int,A6: set @ B] :
( ( ring_1_of_int @ A @ ( groups7311177749621191930dd_sum @ B @ int @ F @ A6 ) )
= ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X2: B] : ( ring_1_of_int @ A @ ( F @ X2 ) )
@ A6 ) ) ) ).
% of_int_sum
thf(fact_4291_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_4292_sum__eq__0__iff,axiom,
! [A: $tType,B: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [F4: set @ B,F: B > A] :
( ( finite_finite2 @ B @ F4 )
=> ( ( ( groups7311177749621191930dd_sum @ B @ A @ F @ F4 )
= ( zero_zero @ A ) )
= ( ! [X2: B] :
( ( member @ B @ X2 @ F4 )
=> ( ( F @ X2 )
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% sum_eq_0_iff
thf(fact_4293_sum_Oinfinite,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A6: set @ B,G: B > A] :
( ~ ( finite_finite2 @ B @ A6 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ A6 )
= ( zero_zero @ A ) ) ) ) ).
% sum.infinite
thf(fact_4294_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_4295_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_4296_sum__abs,axiom,
! [B: $tType,A: $tType] :
( ( ordere166539214618696060dd_abs @ B )
=> ! [F: A > B,A6: set @ A] :
( ord_less_eq @ B @ ( abs_abs @ B @ ( groups7311177749621191930dd_sum @ A @ B @ F @ A6 ) )
@ ( groups7311177749621191930dd_sum @ A @ B
@ ^ [I5: A] : ( abs_abs @ B @ ( F @ I5 ) )
@ A6 ) ) ) ).
% sum_abs
thf(fact_4297_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_4298_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_4299_sum__abs__ge__zero,axiom,
! [B: $tType,A: $tType] :
( ( ordere166539214618696060dd_abs @ B )
=> ! [F: A > B,A6: set @ A] :
( ord_less_eq @ B @ ( zero_zero @ B )
@ ( groups7311177749621191930dd_sum @ A @ B
@ ^ [I5: A] : ( abs_abs @ B @ ( F @ I5 ) )
@ A6 ) ) ) ).
% sum_abs_ge_zero
thf(fact_4300_sum__constant,axiom,
! [B: $tType,A: $tType] :
( ( semiring_1 @ A )
=> ! [Y3: A,A6: set @ B] :
( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X2: B] : Y3
@ A6 )
= ( times_times @ A @ ( semiring_1_of_nat @ A @ ( finite_card @ B @ A6 ) ) @ Y3 ) ) ) ).
% sum_constant
thf(fact_4301_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_4302_sum__of__bool__mult__eq,axiom,
! [A: $tType,B: $tType] :
( ( semiring_1 @ A )
=> ! [A6: set @ B,P: B > $o,F: B > A] :
( ( finite_finite2 @ B @ A6 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X2: B] : ( times_times @ A @ ( zero_neq_one_of_bool @ A @ ( P @ X2 ) ) @ ( F @ X2 ) )
@ A6 )
= ( groups7311177749621191930dd_sum @ B @ A @ F @ ( inf_inf @ ( set @ B ) @ A6 @ ( collect @ B @ P ) ) ) ) ) ) ).
% sum_of_bool_mult_eq
thf(fact_4303_sum__mult__of__bool__eq,axiom,
! [A: $tType,B: $tType] :
( ( semiring_1 @ A )
=> ! [A6: set @ B,F: B > A,P: B > $o] :
( ( finite_finite2 @ B @ A6 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X2: B] : ( times_times @ A @ ( F @ X2 ) @ ( zero_neq_one_of_bool @ A @ ( P @ X2 ) ) )
@ A6 )
= ( groups7311177749621191930dd_sum @ B @ A @ F @ ( inf_inf @ ( set @ B ) @ A6 @ ( collect @ B @ P ) ) ) ) ) ) ).
% sum_mult_of_bool_eq
thf(fact_4304_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_4305_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_4306_sum__of__bool__eq,axiom,
! [A: $tType,B: $tType] :
( ( semiring_1 @ A )
=> ! [A6: set @ B,P: B > $o] :
( ( finite_finite2 @ B @ A6 )
=> ( ( finite_finite2 @ B @ A6 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X2: B] : ( zero_neq_one_of_bool @ A @ ( P @ X2 ) )
@ A6 )
= ( semiring_1_of_nat @ A @ ( finite_card @ B @ ( inf_inf @ ( set @ B ) @ A6 @ ( collect @ B @ P ) ) ) ) ) ) ) ) ).
% sum_of_bool_eq
thf(fact_4307_sum__zero__power,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A6: set @ nat,C2: nat > A] :
( ( ( ( finite_finite2 @ nat @ A6 )
& ( member @ nat @ ( zero_zero @ nat ) @ A6 ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I5: nat] : ( times_times @ A @ ( C2 @ I5 ) @ ( power_power @ A @ ( zero_zero @ A ) @ I5 ) )
@ A6 )
= ( C2 @ ( zero_zero @ nat ) ) ) )
& ( ~ ( ( finite_finite2 @ nat @ A6 )
& ( member @ nat @ ( zero_zero @ nat ) @ A6 ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I5: nat] : ( times_times @ A @ ( C2 @ I5 ) @ ( power_power @ A @ ( zero_zero @ A ) @ I5 ) )
@ A6 )
= ( zero_zero @ A ) ) ) ) ) ).
% sum_zero_power
thf(fact_4308_sum__zero__power_H,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A6: set @ nat,C2: nat > A,D2: nat > A] :
( ( ( ( finite_finite2 @ nat @ A6 )
& ( member @ nat @ ( zero_zero @ nat ) @ A6 ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I5: nat] : ( divide_divide @ A @ ( times_times @ A @ ( C2 @ I5 ) @ ( power_power @ A @ ( zero_zero @ A ) @ I5 ) ) @ ( D2 @ I5 ) )
@ A6 )
= ( divide_divide @ A @ ( C2 @ ( zero_zero @ nat ) ) @ ( D2 @ ( zero_zero @ nat ) ) ) ) )
& ( ~ ( ( finite_finite2 @ nat @ A6 )
& ( member @ nat @ ( zero_zero @ nat ) @ A6 ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I5: nat] : ( divide_divide @ A @ ( times_times @ A @ ( C2 @ I5 ) @ ( power_power @ A @ ( zero_zero @ A ) @ I5 ) ) @ ( D2 @ I5 ) )
@ A6 )
= ( zero_zero @ A ) ) ) ) ) ).
% sum_zero_power'
thf(fact_4309_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 )
@ ^ [Q3: A,R5: A] : ( product_Pair @ A @ A @ Q3 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ R5 ) )
@ ( unique8689654367752047608divmod @ A @ M @ N ) ) ) ) ).
% divmod_algorithm_code(5)
thf(fact_4310_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 )
@ ^ [Q3: A,R5: A] : ( product_Pair @ A @ A @ Q3 @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ R5 ) @ ( one_one @ A ) ) )
@ ( unique8689654367752047608divmod @ A @ M @ N ) ) ) ) ).
% divmod_algorithm_code(6)
thf(fact_4311_sum__mono,axiom,
! [A: $tType,B: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [K5: set @ B,F: B > A,G: B > A] :
( ! [I4: B] :
( ( member @ B @ I4 @ K5 )
=> ( ord_less_eq @ A @ ( F @ I4 ) @ ( G @ I4 ) ) )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F @ K5 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ K5 ) ) ) ) ).
% sum_mono
thf(fact_4312_sum__negf,axiom,
! [A: $tType,B: $tType] :
( ( ab_group_add @ A )
=> ! [F: B > A,A6: set @ B] :
( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X2: B] : ( uminus_uminus @ A @ ( F @ X2 ) )
@ A6 )
= ( uminus_uminus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F @ A6 ) ) ) ) ).
% sum_negf
thf(fact_4313_sum__distrib__left,axiom,
! [A: $tType,B: $tType] :
( ( semiring_0 @ A )
=> ! [R: A,F: B > A,A6: set @ B] :
( ( times_times @ A @ R @ ( groups7311177749621191930dd_sum @ B @ A @ F @ A6 ) )
= ( groups7311177749621191930dd_sum @ B @ A
@ ^ [N3: B] : ( times_times @ A @ R @ ( F @ N3 ) )
@ A6 ) ) ) ).
% sum_distrib_left
thf(fact_4314_sum__distrib__right,axiom,
! [A: $tType,B: $tType] :
( ( semiring_0 @ A )
=> ! [F: B > A,A6: set @ B,R: A] :
( ( times_times @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F @ A6 ) @ R )
= ( groups7311177749621191930dd_sum @ B @ A
@ ^ [N3: B] : ( times_times @ A @ ( F @ N3 ) @ R )
@ A6 ) ) ) ).
% sum_distrib_right
thf(fact_4315_sum__product,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( semiring_0 @ B )
=> ! [F: A > B,A6: set @ A,G: C > B,B7: set @ C] :
( ( times_times @ B @ ( groups7311177749621191930dd_sum @ A @ B @ F @ A6 ) @ ( groups7311177749621191930dd_sum @ C @ B @ G @ B7 ) )
= ( groups7311177749621191930dd_sum @ A @ B
@ ^ [I5: A] :
( groups7311177749621191930dd_sum @ C @ B
@ ^ [J3: C] : ( times_times @ B @ ( F @ I5 ) @ ( G @ J3 ) )
@ B7 )
@ A6 ) ) ) ).
% sum_product
thf(fact_4316_mod__sum__eq,axiom,
! [B: $tType,A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [F: B > A,A3: A,A6: set @ B] :
( ( modulo_modulo @ A
@ ( groups7311177749621191930dd_sum @ B @ A
@ ^ [I5: B] : ( modulo_modulo @ A @ ( F @ I5 ) @ A3 )
@ A6 )
@ A3 )
= ( modulo_modulo @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F @ A6 ) @ A3 ) ) ) ).
% mod_sum_eq
thf(fact_4317_unset__bit__integer_Oabs__eq,axiom,
! [Xa3: nat,X: int] :
( ( bit_se2638667681897837118et_bit @ code_integer @ Xa3 @ ( code_integer_of_int @ X ) )
= ( code_integer_of_int @ ( bit_se2638667681897837118et_bit @ int @ Xa3 @ X ) ) ) ).
% unset_bit_integer.abs_eq
thf(fact_4318_bit__integer_Oabs__eq,axiom,
! [X: int] :
( ( bit_se5641148757651400278ts_bit @ code_integer @ ( code_integer_of_int @ X ) )
= ( bit_se5641148757651400278ts_bit @ int @ X ) ) ).
% bit_integer.abs_eq
thf(fact_4319_or__integer_Oabs__eq,axiom,
! [Xa3: int,X: int] :
( ( bit_se1065995026697491101ons_or @ code_integer @ ( code_integer_of_int @ Xa3 ) @ ( code_integer_of_int @ X ) )
= ( code_integer_of_int @ ( bit_se1065995026697491101ons_or @ int @ Xa3 @ X ) ) ) ).
% or_integer.abs_eq
thf(fact_4320_and__integer_Oabs__eq,axiom,
! [Xa3: int,X: int] :
( ( bit_se5824344872417868541ns_and @ code_integer @ ( code_integer_of_int @ Xa3 ) @ ( code_integer_of_int @ X ) )
= ( code_integer_of_int @ ( bit_se5824344872417868541ns_and @ int @ Xa3 @ X ) ) ) ).
% and_integer.abs_eq
thf(fact_4321_euclidean__size__integer_Oabs__eq,axiom,
! [X: int] :
( ( euclid6346220572633701492n_size @ code_integer @ ( code_integer_of_int @ X ) )
= ( euclid6346220572633701492n_size @ int @ X ) ) ).
% euclidean_size_integer.abs_eq
thf(fact_4322_not__integer_Oabs__eq,axiom,
! [X: int] :
( ( bit_ri4277139882892585799ns_not @ code_integer @ ( code_integer_of_int @ X ) )
= ( code_integer_of_int @ ( bit_ri4277139882892585799ns_not @ int @ X ) ) ) ).
% not_integer.abs_eq
thf(fact_4323_take__bit__integer_Oabs__eq,axiom,
! [Xa3: nat,X: int] :
( ( bit_se2584673776208193580ke_bit @ code_integer @ Xa3 @ ( code_integer_of_int @ X ) )
= ( code_integer_of_int @ ( bit_se2584673776208193580ke_bit @ int @ Xa3 @ X ) ) ) ).
% take_bit_integer.abs_eq
thf(fact_4324_xor__integer_Oabs__eq,axiom,
! [Xa3: int,X: int] :
( ( bit_se5824344971392196577ns_xor @ code_integer @ ( code_integer_of_int @ Xa3 ) @ ( code_integer_of_int @ X ) )
= ( code_integer_of_int @ ( bit_se5824344971392196577ns_xor @ int @ Xa3 @ X ) ) ) ).
% xor_integer.abs_eq
thf(fact_4325_times__integer_Oabs__eq,axiom,
! [Xa3: int,X: int] :
( ( times_times @ code_integer @ ( code_integer_of_int @ Xa3 ) @ ( code_integer_of_int @ X ) )
= ( code_integer_of_int @ ( times_times @ int @ Xa3 @ X ) ) ) ).
% times_integer.abs_eq
thf(fact_4326_push__bit__integer_Oabs__eq,axiom,
! [Xa3: nat,X: int] :
( ( bit_se4730199178511100633sh_bit @ code_integer @ Xa3 @ ( code_integer_of_int @ X ) )
= ( code_integer_of_int @ ( bit_se4730199178511100633sh_bit @ int @ Xa3 @ X ) ) ) ).
% push_bit_integer.abs_eq
thf(fact_4327_drop__bit__integer_Oabs__eq,axiom,
! [Xa3: nat,X: int] :
( ( bit_se4197421643247451524op_bit @ code_integer @ Xa3 @ ( code_integer_of_int @ X ) )
= ( code_integer_of_int @ ( bit_se4197421643247451524op_bit @ int @ Xa3 @ X ) ) ) ).
% drop_bit_integer.abs_eq
thf(fact_4328_mask__integer_Oabs__eq,axiom,
( ( bit_se2239418461657761734s_mask @ code_integer )
= ( ^ [X2: nat] : ( code_integer_of_int @ ( bit_se2239418461657761734s_mask @ int @ X2 ) ) ) ) ).
% mask_integer.abs_eq
thf(fact_4329_split__cong,axiom,
! [C: $tType,B: $tType,A: $tType,Q4: product_prod @ A @ B,F: A > B > C,G: A > B > C,P5: product_prod @ A @ B] :
( ! [X3: A,Y5: B] :
( ( ( product_Pair @ A @ B @ X3 @ Y5 )
= Q4 )
=> ( ( F @ X3 @ Y5 )
= ( G @ X3 @ Y5 ) ) )
=> ( ( P5 = Q4 )
=> ( ( product_case_prod @ A @ B @ C @ F @ P5 )
= ( product_case_prod @ A @ B @ C @ G @ Q4 ) ) ) ) ).
% split_cong
thf(fact_4330_set__bit__integer_Oabs__eq,axiom,
! [Xa3: nat,X: int] :
( ( bit_se5668285175392031749et_bit @ code_integer @ Xa3 @ ( code_integer_of_int @ X ) )
= ( code_integer_of_int @ ( bit_se5668285175392031749et_bit @ int @ Xa3 @ X ) ) ) ).
% set_bit_integer.abs_eq
thf(fact_4331_flip__bit__integer_Oabs__eq,axiom,
! [Xa3: nat,X: int] :
( ( bit_se8732182000553998342ip_bit @ code_integer @ Xa3 @ ( code_integer_of_int @ X ) )
= ( code_integer_of_int @ ( bit_se8732182000553998342ip_bit @ int @ Xa3 @ X ) ) ) ).
% flip_bit_integer.abs_eq
thf(fact_4332_division__segment__integer_Oabs__eq,axiom,
! [X: int] :
( ( euclid7384307370059645450egment @ code_integer @ ( code_integer_of_int @ X ) )
= ( code_integer_of_int @ ( euclid7384307370059645450egment @ int @ X ) ) ) ).
% division_segment_integer.abs_eq
thf(fact_4333_sum_Oneutral,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A6: set @ B,G: B > A] :
( ! [X3: B] :
( ( member @ B @ X3 @ A6 )
=> ( ( G @ X3 )
= ( zero_zero @ A ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ A6 )
= ( zero_zero @ A ) ) ) ) ).
% sum.neutral
thf(fact_4334_sum_Onot__neutral__contains__not__neutral,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: B > A,A6: set @ B] :
( ( ( groups7311177749621191930dd_sum @ B @ A @ G @ A6 )
!= ( zero_zero @ A ) )
=> ~ ! [A8: B] :
( ( member @ B @ A8 @ A6 )
=> ( ( G @ A8 )
= ( zero_zero @ A ) ) ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_4335_sum__nonpos,axiom,
! [B: $tType,A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A6: set @ B,F: B > A] :
( ! [X3: B] :
( ( member @ B @ X3 @ A6 )
=> ( ord_less_eq @ A @ ( F @ X3 ) @ ( zero_zero @ A ) ) )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F @ A6 ) @ ( zero_zero @ A ) ) ) ) ).
% sum_nonpos
thf(fact_4336_sum__nonneg,axiom,
! [A: $tType,B: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A6: set @ B,F: B > A] :
( ! [X3: B] :
( ( member @ B @ X3 @ A6 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F @ X3 ) ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( groups7311177749621191930dd_sum @ B @ A @ F @ A6 ) ) ) ) ).
% sum_nonneg
thf(fact_4337_sum__mono__inv,axiom,
! [A: $tType,I7: $tType] :
( ( ordere8940638589300402666id_add @ A )
=> ! [F: I7 > A,I6: set @ I7,G: I7 > A,I: I7] :
( ( ( groups7311177749621191930dd_sum @ I7 @ A @ F @ I6 )
= ( groups7311177749621191930dd_sum @ I7 @ A @ G @ I6 ) )
=> ( ! [I4: I7] :
( ( member @ I7 @ I4 @ I6 )
=> ( ord_less_eq @ A @ ( F @ I4 ) @ ( G @ I4 ) ) )
=> ( ( member @ I7 @ I @ I6 )
=> ( ( finite_finite2 @ I7 @ I6 )
=> ( ( F @ I )
= ( G @ I ) ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_4338_sum__cong__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A6: set @ nat,F: nat > A,G: nat > A] :
( ~ ( member @ nat @ ( zero_zero @ nat ) @ A6 )
=> ( ! [X3: nat] :
( ( member @ nat @ ( suc @ X3 ) @ A6 )
=> ( ( F @ ( suc @ X3 ) )
= ( G @ ( suc @ X3 ) ) ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ F @ A6 )
= ( groups7311177749621191930dd_sum @ nat @ A @ G @ A6 ) ) ) ) ) ).
% sum_cong_Suc
thf(fact_4339_zero__integer__def,axiom,
( ( zero_zero @ code_integer )
= ( code_integer_of_int @ ( zero_zero @ int ) ) ) ).
% zero_integer_def
thf(fact_4340_less__integer_Oabs__eq,axiom,
! [Xa3: int,X: int] :
( ( ord_less @ code_integer @ ( code_integer_of_int @ Xa3 ) @ ( code_integer_of_int @ X ) )
= ( ord_less @ int @ Xa3 @ X ) ) ).
% less_integer.abs_eq
thf(fact_4341_plus__integer_Oabs__eq,axiom,
! [Xa3: int,X: int] :
( ( plus_plus @ code_integer @ ( code_integer_of_int @ Xa3 ) @ ( code_integer_of_int @ X ) )
= ( code_integer_of_int @ ( plus_plus @ int @ Xa3 @ X ) ) ) ).
% plus_integer.abs_eq
thf(fact_4342_uminus__integer_Oabs__eq,axiom,
! [X: int] :
( ( uminus_uminus @ code_integer @ ( code_integer_of_int @ X ) )
= ( code_integer_of_int @ ( uminus_uminus @ int @ X ) ) ) ).
% uminus_integer.abs_eq
thf(fact_4343_sum_Ointer__filter,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A6: set @ B,G: B > A,P: B > $o] :
( ( finite_finite2 @ B @ A6 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ A6 )
& ( P @ X2 ) ) ) )
= ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X2: B] : ( if @ A @ ( P @ X2 ) @ ( G @ X2 ) @ ( zero_zero @ A ) )
@ A6 ) ) ) ) ).
% sum.inter_filter
thf(fact_4344_one__integer__def,axiom,
( ( one_one @ code_integer )
= ( code_integer_of_int @ ( one_one @ int ) ) ) ).
% one_integer_def
thf(fact_4345_less__eq__integer_Oabs__eq,axiom,
! [Xa3: int,X: int] :
( ( ord_less_eq @ code_integer @ ( code_integer_of_int @ Xa3 ) @ ( code_integer_of_int @ X ) )
= ( ord_less_eq @ int @ Xa3 @ X ) ) ).
% less_eq_integer.abs_eq
thf(fact_4346_minus__integer_Oabs__eq,axiom,
! [Xa3: int,X: int] :
( ( minus_minus @ code_integer @ ( code_integer_of_int @ Xa3 ) @ ( code_integer_of_int @ X ) )
= ( code_integer_of_int @ ( minus_minus @ int @ Xa3 @ X ) ) ) ).
% minus_integer.abs_eq
thf(fact_4347_abs__integer_Oabs__eq,axiom,
! [X: int] :
( ( abs_abs @ code_integer @ ( code_integer_of_int @ X ) )
= ( code_integer_of_int @ ( abs_abs @ int @ X ) ) ) ).
% abs_integer.abs_eq
thf(fact_4348_divide__integer_Oabs__eq,axiom,
! [Xa3: int,X: int] :
( ( divide_divide @ code_integer @ ( code_integer_of_int @ Xa3 ) @ ( code_integer_of_int @ X ) )
= ( code_integer_of_int @ ( divide_divide @ int @ Xa3 @ X ) ) ) ).
% divide_integer.abs_eq
thf(fact_4349_nat__of__integer_Oabs__eq,axiom,
! [X: int] :
( ( code_nat_of_integer @ ( code_integer_of_int @ X ) )
= ( nat2 @ X ) ) ).
% nat_of_integer.abs_eq
thf(fact_4350_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
@ ^ [I5: nat] : ( G @ ( suc @ I5 ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% sum.shift_bounds_cl_Suc_ivl
thf(fact_4351_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
@ ^ [I5: nat] : ( G @ ( suc @ I5 ) )
@ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ).
% sum.shift_bounds_Suc_ivl
thf(fact_4352_modulo__integer_Oabs__eq,axiom,
! [Xa3: int,X: int] :
( ( modulo_modulo @ code_integer @ ( code_integer_of_int @ Xa3 ) @ ( code_integer_of_int @ X ) )
= ( code_integer_of_int @ ( modulo_modulo @ int @ Xa3 @ X ) ) ) ).
% modulo_integer.abs_eq
thf(fact_4353_sgn__integer_Oabs__eq,axiom,
! [X: int] :
( ( sgn_sgn @ code_integer @ ( code_integer_of_int @ X ) )
= ( code_integer_of_int @ ( sgn_sgn @ int @ X ) ) ) ).
% sgn_integer.abs_eq
thf(fact_4354_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 ) ),H2: heap_ext @ product_unit,H: heap_ext @ product_unit,R: A,N: nat] :
( ( heap_Time_effect @ A @ ( heap_Time_guard @ A @ P @ F ) @ H2 @ H @ R @ N )
=> ~ ( ( H
= ( product_fst @ ( heap_ext @ product_unit ) @ nat @ ( product_snd @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( F @ H2 ) ) ) )
=> ( ( N
= ( product_snd @ ( heap_ext @ product_unit ) @ nat @ ( product_snd @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( F @ H2 ) ) ) )
=> ( ( R
= ( product_fst @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( F @ H2 ) ) )
=> ~ ( P @ H2 ) ) ) ) ) ).
% effect_guardE
thf(fact_4355_effect__guardI,axiom,
! [A: $tType,P: ( heap_ext @ product_unit ) > $o,H2: heap_ext @ product_unit,H: heap_ext @ product_unit,F: ( heap_ext @ product_unit ) > ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ),N: nat,R: A] :
( ( P @ H2 )
=> ( ( H
= ( product_fst @ ( heap_ext @ product_unit ) @ nat @ ( product_snd @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( F @ H2 ) ) ) )
=> ( ( N
= ( product_snd @ ( heap_ext @ product_unit ) @ nat @ ( product_snd @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( F @ H2 ) ) ) )
=> ( ( R
= ( product_fst @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( F @ H2 ) ) )
=> ( heap_Time_effect @ A @ ( heap_Time_guard @ A @ P @ F ) @ H2 @ H @ R @ N ) ) ) ) ) ).
% effect_guardI
thf(fact_4356_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 )
=> ( ! [X3: C] :
( ( member @ C @ X3 @ T2 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( G @ X3 ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ S )
=> ? [Xa: C] :
( ( member @ C @ Xa @ T2 )
& ( ( I @ Xa )
= X3 )
& ( ord_less_eq @ A @ ( F @ X3 ) @ ( G @ Xa ) ) ) )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F @ S ) @ ( groups7311177749621191930dd_sum @ C @ A @ G @ T2 ) ) ) ) ) ) ) ).
% sum_le_included
thf(fact_4357_sum__nonneg__eq__0__iff,axiom,
! [A: $tType,B: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A6: set @ B,F: B > A] :
( ( finite_finite2 @ B @ A6 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ A6 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F @ X3 ) ) )
=> ( ( ( groups7311177749621191930dd_sum @ B @ A @ F @ A6 )
= ( zero_zero @ A ) )
= ( ! [X2: B] :
( ( member @ B @ X2 @ A6 )
=> ( ( F @ X2 )
= ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_4358_sum__strict__mono__ex1,axiom,
! [A: $tType,I7: $tType] :
( ( ordere8940638589300402666id_add @ A )
=> ! [A6: set @ I7,F: I7 > A,G: I7 > A] :
( ( finite_finite2 @ I7 @ A6 )
=> ( ! [X3: I7] :
( ( member @ I7 @ X3 @ A6 )
=> ( ord_less_eq @ A @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ? [X4: I7] :
( ( member @ I7 @ X4 @ A6 )
& ( ord_less @ A @ ( F @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less @ A @ ( groups7311177749621191930dd_sum @ I7 @ A @ F @ A6 ) @ ( groups7311177749621191930dd_sum @ I7 @ A @ G @ A6 ) ) ) ) ) ) ).
% sum_strict_mono_ex1
thf(fact_4359_sum_Orelated,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [R2: A > A > $o,S3: set @ B,H2: B > A,G: B > A] :
( ( R2 @ ( zero_zero @ A ) @ ( zero_zero @ A ) )
=> ( ! [X1: A,Y1: A,X23: A,Y22: A] :
( ( ( R2 @ X1 @ X23 )
& ( R2 @ Y1 @ Y22 ) )
=> ( R2 @ ( plus_plus @ A @ X1 @ Y1 ) @ ( plus_plus @ A @ X23 @ Y22 ) ) )
=> ( ( finite_finite2 @ B @ S3 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ S3 )
=> ( R2 @ ( H2 @ X3 ) @ ( G @ X3 ) ) )
=> ( R2 @ ( groups7311177749621191930dd_sum @ B @ A @ H2 @ S3 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ S3 ) ) ) ) ) ) ) ).
% sum.related
thf(fact_4360_sum__strict__mono,axiom,
! [A: $tType,B: $tType] :
( ( strict7427464778891057005id_add @ A )
=> ! [A6: set @ B,F: B > A,G: B > A] :
( ( finite_finite2 @ B @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ A6 )
=> ( ord_less @ A @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F @ A6 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ A6 ) ) ) ) ) ) ).
% sum_strict_mono
thf(fact_4361_sum_Oivl__cong,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( comm_monoid_add @ A ) )
=> ! [A3: B,C2: B,B2: B,D2: B,G: B > A,H2: B > A] :
( ( A3 = C2 )
=> ( ( B2 = D2 )
=> ( ! [X3: B] :
( ( ord_less_eq @ B @ C2 @ X3 )
=> ( ( ord_less @ B @ X3 @ D2 )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( set_or7035219750837199246ssThan @ B @ A3 @ B2 ) )
= ( groups7311177749621191930dd_sum @ B @ A @ H2 @ ( set_or7035219750837199246ssThan @ B @ C2 @ D2 ) ) ) ) ) ) ) ).
% sum.ivl_cong
thf(fact_4362_sum_Oreindex__bij__witness__not__neutral,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comm_monoid_add @ A )
=> ! [S5: set @ B,T5: set @ C,S3: set @ B,I: C > B,J: B > C,T6: set @ C,G: B > A,H2: C > A] :
( ( finite_finite2 @ B @ S5 )
=> ( ( finite_finite2 @ C @ T5 )
=> ( ! [A8: B] :
( ( member @ B @ A8 @ ( minus_minus @ ( set @ B ) @ S3 @ S5 ) )
=> ( ( I @ ( J @ A8 ) )
= A8 ) )
=> ( ! [A8: B] :
( ( member @ B @ A8 @ ( minus_minus @ ( set @ B ) @ S3 @ S5 ) )
=> ( member @ C @ ( J @ A8 ) @ ( minus_minus @ ( set @ C ) @ T6 @ T5 ) ) )
=> ( ! [B5: C] :
( ( member @ C @ B5 @ ( minus_minus @ ( set @ C ) @ T6 @ T5 ) )
=> ( ( J @ ( I @ B5 ) )
= B5 ) )
=> ( ! [B5: C] :
( ( member @ C @ B5 @ ( minus_minus @ ( set @ C ) @ T6 @ T5 ) )
=> ( member @ B @ ( I @ B5 ) @ ( minus_minus @ ( set @ B ) @ S3 @ S5 ) ) )
=> ( ! [A8: B] :
( ( member @ B @ A8 @ S5 )
=> ( ( G @ A8 )
= ( zero_zero @ A ) ) )
=> ( ! [B5: C] :
( ( member @ C @ B5 @ T5 )
=> ( ( H2 @ B5 )
= ( zero_zero @ A ) ) )
=> ( ! [A8: B] :
( ( member @ B @ A8 @ S3 )
=> ( ( H2 @ ( J @ A8 ) )
= ( G @ A8 ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ S3 )
= ( groups7311177749621191930dd_sum @ C @ A @ H2 @ T6 ) ) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
thf(fact_4363_sum_OatLeastLessThan__concat,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M: nat,N: nat,P5: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( ord_less_eq @ nat @ N @ P5 )
=> ( ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ N @ P5 ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ P5 ) ) ) ) ) ) ).
% sum.atLeastLessThan_concat
thf(fact_4364_sum__diff__nat__ivl,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [M: nat,N: nat,P5: nat,F: nat > A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( ord_less_eq @ nat @ N @ P5 )
=> ( ( minus_minus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F @ ( set_or7035219750837199246ssThan @ nat @ M @ P5 ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ F @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ F @ ( set_or7035219750837199246ssThan @ nat @ N @ P5 ) ) ) ) ) ) ).
% sum_diff_nat_ivl
thf(fact_4365_sum__nonneg__0,axiom,
! [B: $tType,A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [S: set @ B,F: B > A,I: B] :
( ( finite_finite2 @ B @ S )
=> ( ! [I4: B] :
( ( member @ B @ I4 @ S )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F @ I4 ) ) )
=> ( ( ( groups7311177749621191930dd_sum @ B @ A @ F @ S )
= ( zero_zero @ A ) )
=> ( ( member @ B @ I @ S )
=> ( ( F @ I )
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_4366_sum__nonneg__leq__bound,axiom,
! [B: $tType,A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [S: set @ B,F: B > A,B7: A,I: B] :
( ( finite_finite2 @ B @ S )
=> ( ! [I4: B] :
( ( member @ B @ I4 @ S )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F @ I4 ) ) )
=> ( ( ( groups7311177749621191930dd_sum @ B @ A @ F @ S )
= B7 )
=> ( ( member @ B @ I @ S )
=> ( ord_less_eq @ A @ ( F @ I ) @ B7 ) ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_4367_sum_Ointer__restrict,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A6: set @ B,G: B > A,B7: set @ B] :
( ( finite_finite2 @ B @ A6 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( inf_inf @ ( set @ B ) @ A6 @ B7 ) )
= ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X2: B] : ( if @ A @ ( member @ B @ X2 @ B7 ) @ ( G @ X2 ) @ ( zero_zero @ A ) )
@ A6 ) ) ) ) ).
% sum.inter_restrict
thf(fact_4368_sum_Osetdiff__irrelevant,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A6: set @ B,G: B > A] :
( ( finite_finite2 @ B @ A6 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G
@ ( minus_minus @ ( set @ B ) @ A6
@ ( collect @ B
@ ^ [X2: B] :
( ( G @ X2 )
= ( zero_zero @ A ) ) ) ) )
= ( groups7311177749621191930dd_sum @ B @ A @ G @ A6 ) ) ) ) ).
% sum.setdiff_irrelevant
thf(fact_4369_sum__power__add,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X: A,M: nat,I6: set @ nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I5: nat] : ( power_power @ A @ X @ ( plus_plus @ nat @ M @ I5 ) )
@ I6 )
= ( times_times @ A @ ( power_power @ A @ X @ M ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ I6 ) ) ) ) ).
% sum_power_add
thf(fact_4370_sum_Onat__diff__reindex,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I5: nat] : ( G @ ( minus_minus @ nat @ N @ ( suc @ I5 ) ) )
@ ( set_ord_lessThan @ nat @ N ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% sum.nat_diff_reindex
thf(fact_4371_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_4372_sum__pos2,axiom,
! [A: $tType,B: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [I6: set @ B,I: B,F: B > A] :
( ( finite_finite2 @ B @ I6 )
=> ( ( member @ B @ I @ I6 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( F @ I ) )
=> ( ! [I4: B] :
( ( member @ B @ I4 @ I6 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F @ I4 ) ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( groups7311177749621191930dd_sum @ B @ A @ F @ I6 ) ) ) ) ) ) ) ).
% sum_pos2
thf(fact_4373_sum__pos,axiom,
! [A: $tType,B: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [I6: set @ B,F: B > A] :
( ( finite_finite2 @ B @ I6 )
=> ( ( I6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [I4: B] :
( ( member @ B @ I4 @ I6 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( F @ I4 ) ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( groups7311177749621191930dd_sum @ B @ A @ F @ I6 ) ) ) ) ) ) ).
% sum_pos
thf(fact_4374_sum_Omono__neutral__cong__right,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [T6: set @ B,S3: set @ B,G: B > A,H2: B > A] :
( ( finite_finite2 @ B @ T6 )
=> ( ( ord_less_eq @ ( set @ B ) @ S3 @ T6 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T6 @ S3 ) )
=> ( ( G @ X3 )
= ( zero_zero @ A ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ S3 )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ T6 )
= ( groups7311177749621191930dd_sum @ B @ A @ H2 @ S3 ) ) ) ) ) ) ) ).
% sum.mono_neutral_cong_right
thf(fact_4375_sum_Omono__neutral__cong__left,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [T6: set @ B,S3: set @ B,H2: B > A,G: B > A] :
( ( finite_finite2 @ B @ T6 )
=> ( ( ord_less_eq @ ( set @ B ) @ S3 @ T6 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T6 @ S3 ) )
=> ( ( H2 @ X3 )
= ( zero_zero @ A ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ S3 )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ S3 )
= ( groups7311177749621191930dd_sum @ B @ A @ H2 @ T6 ) ) ) ) ) ) ) ).
% sum.mono_neutral_cong_left
thf(fact_4376_sum_Omono__neutral__right,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [T6: set @ B,S3: set @ B,G: B > A] :
( ( finite_finite2 @ B @ T6 )
=> ( ( ord_less_eq @ ( set @ B ) @ S3 @ T6 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T6 @ S3 ) )
=> ( ( G @ X3 )
= ( zero_zero @ A ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ T6 )
= ( groups7311177749621191930dd_sum @ B @ A @ G @ S3 ) ) ) ) ) ) ).
% sum.mono_neutral_right
thf(fact_4377_sum_Omono__neutral__left,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [T6: set @ B,S3: set @ B,G: B > A] :
( ( finite_finite2 @ B @ T6 )
=> ( ( ord_less_eq @ ( set @ B ) @ S3 @ T6 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T6 @ S3 ) )
=> ( ( G @ X3 )
= ( zero_zero @ A ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ S3 )
= ( groups7311177749621191930dd_sum @ B @ A @ G @ T6 ) ) ) ) ) ) ).
% sum.mono_neutral_left
thf(fact_4378_sum_Osame__carrierI,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [C6: set @ B,A6: set @ B,B7: set @ B,G: B > A,H2: B > A] :
( ( finite_finite2 @ B @ C6 )
=> ( ( ord_less_eq @ ( set @ B ) @ A6 @ C6 )
=> ( ( ord_less_eq @ ( set @ B ) @ B7 @ C6 )
=> ( ! [A8: B] :
( ( member @ B @ A8 @ ( minus_minus @ ( set @ B ) @ C6 @ A6 ) )
=> ( ( G @ A8 )
= ( zero_zero @ A ) ) )
=> ( ! [B5: B] :
( ( member @ B @ B5 @ ( minus_minus @ ( set @ B ) @ C6 @ B7 ) )
=> ( ( H2 @ B5 )
= ( zero_zero @ A ) ) )
=> ( ( ( groups7311177749621191930dd_sum @ B @ A @ G @ C6 )
= ( groups7311177749621191930dd_sum @ B @ A @ H2 @ C6 ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ A6 )
= ( groups7311177749621191930dd_sum @ B @ A @ H2 @ B7 ) ) ) ) ) ) ) ) ) ).
% sum.same_carrierI
thf(fact_4379_sum_Osame__carrier,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [C6: set @ B,A6: set @ B,B7: set @ B,G: B > A,H2: B > A] :
( ( finite_finite2 @ B @ C6 )
=> ( ( ord_less_eq @ ( set @ B ) @ A6 @ C6 )
=> ( ( ord_less_eq @ ( set @ B ) @ B7 @ C6 )
=> ( ! [A8: B] :
( ( member @ B @ A8 @ ( minus_minus @ ( set @ B ) @ C6 @ A6 ) )
=> ( ( G @ A8 )
= ( zero_zero @ A ) ) )
=> ( ! [B5: B] :
( ( member @ B @ B5 @ ( minus_minus @ ( set @ B ) @ C6 @ B7 ) )
=> ( ( H2 @ B5 )
= ( zero_zero @ A ) ) )
=> ( ( ( groups7311177749621191930dd_sum @ B @ A @ G @ A6 )
= ( groups7311177749621191930dd_sum @ B @ A @ H2 @ B7 ) )
= ( ( groups7311177749621191930dd_sum @ B @ A @ G @ C6 )
= ( groups7311177749621191930dd_sum @ B @ A @ H2 @ C6 ) ) ) ) ) ) ) ) ) ).
% sum.same_carrier
thf(fact_4380_sum__bounded__below,axiom,
! [A: $tType,B: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( semiring_1 @ A ) )
=> ! [A6: set @ B,K5: A,F: B > A] :
( ! [I4: B] :
( ( member @ B @ I4 @ A6 )
=> ( ord_less_eq @ A @ K5 @ ( F @ I4 ) ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( finite_card @ B @ A6 ) ) @ K5 ) @ ( groups7311177749621191930dd_sum @ B @ A @ F @ A6 ) ) ) ) ).
% sum_bounded_below
thf(fact_4381_sum__bounded__above,axiom,
! [B: $tType,A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( semiring_1 @ A ) )
=> ! [A6: set @ B,F: B > A,K5: A] :
( ! [I4: B] :
( ( member @ B @ I4 @ A6 )
=> ( ord_less_eq @ A @ ( F @ I4 ) @ K5 ) )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F @ A6 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( finite_card @ B @ A6 ) ) @ K5 ) ) ) ) ).
% sum_bounded_above
thf(fact_4382_sum_Omono__neutral__cong,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [T6: set @ B,S3: set @ B,H2: B > A,G: B > A] :
( ( finite_finite2 @ B @ T6 )
=> ( ( finite_finite2 @ B @ S3 )
=> ( ! [I4: B] :
( ( member @ B @ I4 @ ( minus_minus @ ( set @ B ) @ T6 @ S3 ) )
=> ( ( H2 @ I4 )
= ( zero_zero @ A ) ) )
=> ( ! [I4: B] :
( ( member @ B @ I4 @ ( minus_minus @ ( set @ B ) @ S3 @ T6 ) )
=> ( ( G @ I4 )
= ( zero_zero @ A ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( inf_inf @ ( set @ B ) @ S3 @ T6 ) )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ S3 )
= ( groups7311177749621191930dd_sum @ B @ A @ H2 @ T6 ) ) ) ) ) ) ) ) ).
% sum.mono_neutral_cong
thf(fact_4383_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_4384_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_4385_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_4386_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_4387_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_4388_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_4389_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_4390_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_4391_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_4392_sum_OIf__cases,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A6: set @ B,P: B > $o,H2: B > A,G: B > A] :
( ( finite_finite2 @ B @ A6 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X2: B] : ( if @ A @ ( P @ X2 ) @ ( H2 @ X2 ) @ ( G @ X2 ) )
@ A6 )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ H2 @ ( inf_inf @ ( set @ B ) @ A6 @ ( collect @ B @ P ) ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( inf_inf @ ( set @ B ) @ A6 @ ( uminus_uminus @ ( set @ B ) @ ( collect @ B @ P ) ) ) ) ) ) ) ) ).
% sum.If_cases
thf(fact_4393_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
@ ^ [I5: nat] : ( G @ ( suc @ I5 ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ) ) ).
% sum.Suc_reindex_ivl
thf(fact_4394_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
@ ^ [I5: nat] : ( minus_minus @ A @ ( F @ ( suc @ I5 ) ) @ ( F @ I5 ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( minus_minus @ A @ ( F @ ( suc @ N ) ) @ ( F @ M ) ) ) ) ) ).
% sum_Suc_diff
thf(fact_4395_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
@ ^ [I5: nat] : ( minus_minus @ A @ ( F @ ( suc @ I5 ) ) @ ( F @ I5 ) )
@ ( set_or7035219750837199246ssThan @ nat @ M @ N ) )
= ( minus_minus @ A @ ( F @ N ) @ ( F @ M ) ) ) ) ) ).
% sum_Suc_diff'
thf(fact_4396_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
@ ^ [I5: nat] : ( G @ ( suc @ I5 ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ) ).
% sum.atMost_Suc_shift
thf(fact_4397_sum__telescope,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [F: nat > A,I: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I5: nat] : ( minus_minus @ A @ ( F @ I5 ) @ ( F @ ( suc @ I5 ) ) )
@ ( set_ord_atMost @ nat @ I ) )
= ( minus_minus @ A @ ( F @ ( zero_zero @ nat ) ) @ ( F @ ( suc @ I ) ) ) ) ) ).
% sum_telescope
thf(fact_4398_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_4399_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
@ ^ [I5: nat] : ( G @ ( suc @ I5 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% sum.lessThan_Suc_shift
thf(fact_4400_sum__lessThan__telescope_H,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [F: nat > A,M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N3: nat] : ( minus_minus @ A @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
@ ( set_ord_lessThan @ nat @ M ) )
= ( minus_minus @ A @ ( F @ ( zero_zero @ nat ) ) @ ( F @ M ) ) ) ) ).
% sum_lessThan_telescope'
thf(fact_4401_sum__lessThan__telescope,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [F: nat > A,M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N3: nat] : ( minus_minus @ A @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
@ ( set_ord_lessThan @ nat @ M ) )
= ( minus_minus @ A @ ( F @ M ) @ ( F @ ( zero_zero @ nat ) ) ) ) ) ).
% sum_lessThan_telescope
thf(fact_4402_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
@ ^ [I5: nat] : ( G @ ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N ) @ ( suc @ I5 ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ N @ M ) ) ) ) ).
% sum.atLeastLessThan_rev
thf(fact_4403_sum_Onested__swap,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: nat > nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I5: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ ( A3 @ I5 ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ I5 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J3: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I5: nat] : ( A3 @ I5 @ J3 )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J3 ) @ N ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% sum.nested_swap
thf(fact_4404_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_4405_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_4406_sum_Onat__group,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,K: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [M2: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( times_times @ nat @ M2 @ K ) @ ( plus_plus @ nat @ ( times_times @ nat @ M2 @ K ) @ K ) ) )
@ ( set_ord_lessThan @ nat @ N ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_lessThan @ nat @ ( times_times @ nat @ N @ K ) ) ) ) ) ).
% sum.nat_group
thf(fact_4407_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_4408_sum_Onested__swap_H,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: nat > nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I5: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ ( A3 @ I5 ) @ ( set_ord_lessThan @ nat @ I5 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J3: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I5: nat] : ( A3 @ I5 @ J3 )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J3 ) @ N ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% sum.nested_swap'
thf(fact_4409_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_4410_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_4411_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
@ ^ [A5: nat] : ( plus_plus @ A @ ( F @ A5 ) )
@ A3
@ B2
@ ( zero_zero @ A ) ) ) ) ).
% sum_atLeastAtMost_code
thf(fact_4412_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_4413_sum__mono2,axiom,
! [A: $tType,B: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [B7: set @ B,A6: set @ B,F: B > A] :
( ( finite_finite2 @ B @ B7 )
=> ( ( ord_less_eq @ ( set @ B ) @ A6 @ B7 )
=> ( ! [B5: B] :
( ( member @ B @ B5 @ ( minus_minus @ ( set @ B ) @ B7 @ A6 ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F @ B5 ) ) )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F @ A6 ) @ ( groups7311177749621191930dd_sum @ B @ A @ F @ B7 ) ) ) ) ) ) ).
% sum_mono2
thf(fact_4414_sum_Ounion__inter__neutral,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A6: set @ B,B7: set @ B,G: B > A] :
( ( finite_finite2 @ B @ A6 )
=> ( ( finite_finite2 @ B @ B7 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( inf_inf @ ( set @ B ) @ A6 @ B7 ) )
=> ( ( G @ X3 )
= ( zero_zero @ A ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( sup_sup @ ( set @ B ) @ A6 @ B7 ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G @ A6 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ B7 ) ) ) ) ) ) ) ).
% sum.union_inter_neutral
thf(fact_4415_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_4416_sum_Oub__add__nat,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M: nat,N: nat,G: nat > A,P5: 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 @ P5 ) ) )
= ( 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 @ P5 ) ) ) ) ) ) ) ).
% sum.ub_add_nat
thf(fact_4417_sum__div__partition,axiom,
! [B: $tType,A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A6: set @ B,F: B > A,B2: A] :
( ( finite_finite2 @ B @ A6 )
=> ( ( divide_divide @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F @ A6 ) @ B2 )
= ( plus_plus @ A
@ ( groups7311177749621191930dd_sum @ B @ A
@ ^ [A5: B] : ( divide_divide @ A @ ( F @ A5 ) @ B2 )
@ ( inf_inf @ ( set @ B ) @ A6
@ ( collect @ B
@ ^ [A5: B] : ( dvd_dvd @ A @ B2 @ ( F @ A5 ) ) ) ) )
@ ( divide_divide @ A
@ ( groups7311177749621191930dd_sum @ B @ A @ F
@ ( inf_inf @ ( set @ B ) @ A6
@ ( collect @ B
@ ^ [A5: B] :
~ ( dvd_dvd @ A @ B2 @ ( F @ A5 ) ) ) ) )
@ B2 ) ) ) ) ) ).
% sum_div_partition
thf(fact_4418_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_4419_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_4420_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_4421_set__encode__def,axiom,
( nat_set_encode
= ( groups7311177749621191930dd_sum @ nat @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% set_encode_def
thf(fact_4422_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
@ ^ [I5: nat] : ( G @ ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N ) @ I5 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ N ) @ M ) ) ) ) ).
% sum.atLeastLessThan_rev_at_least_Suc_atMost
thf(fact_4423_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
@ ^ [I5: nat] : ( G @ ( suc @ I5 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% sum.atMost_shift
thf(fact_4424_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_4425_sum__list__sum__nth,axiom,
! [B: $tType] :
( ( comm_monoid_add @ B )
=> ( ( groups8242544230860333062m_list @ B )
= ( ^ [Xs3: list @ B] : ( groups7311177749621191930dd_sum @ nat @ B @ ( nth @ B @ Xs3 ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ B ) @ Xs3 ) ) ) ) ) ) ).
% sum_list_sum_nth
thf(fact_4426_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_4427_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_4428_sum__strict__mono2,axiom,
! [B: $tType,A: $tType] :
( ( ordere8940638589300402666id_add @ B )
=> ! [B7: set @ A,A6: set @ A,B2: A,F: A > B] :
( ( finite_finite2 @ A @ B7 )
=> ( ( ord_less_eq @ ( set @ A ) @ A6 @ B7 )
=> ( ( member @ A @ B2 @ ( minus_minus @ ( set @ A ) @ B7 @ A6 ) )
=> ( ( ord_less @ B @ ( zero_zero @ B ) @ ( F @ B2 ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ B7 )
=> ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( F @ X3 ) ) )
=> ( ord_less @ B @ ( groups7311177749621191930dd_sum @ A @ B @ F @ A6 ) @ ( groups7311177749621191930dd_sum @ A @ B @ F @ B7 ) ) ) ) ) ) ) ) ).
% sum_strict_mono2
thf(fact_4429_member__le__sum,axiom,
! [B: $tType,C: $tType] :
( ( ( ordere6911136660526730532id_add @ B )
& ( semiring_1 @ B ) )
=> ! [I: C,A6: set @ C,F: C > B] :
( ( member @ C @ I @ A6 )
=> ( ! [X3: C] :
( ( member @ C @ X3 @ ( minus_minus @ ( set @ C ) @ A6 @ ( insert @ C @ I @ ( bot_bot @ ( set @ C ) ) ) ) )
=> ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( F @ X3 ) ) )
=> ( ( finite_finite2 @ C @ A6 )
=> ( ord_less_eq @ B @ ( F @ I ) @ ( groups7311177749621191930dd_sum @ C @ B @ F @ A6 ) ) ) ) ) ) ).
% member_le_sum
thf(fact_4430_sum__bounded__above__divide,axiom,
! [B: $tType,A: $tType] :
( ( linordered_field @ A )
=> ! [A6: set @ B,F: B > A,K5: A] :
( ! [I4: B] :
( ( member @ B @ I4 @ A6 )
=> ( ord_less_eq @ A @ ( F @ I4 ) @ ( divide_divide @ A @ K5 @ ( semiring_1_of_nat @ A @ ( finite_card @ B @ A6 ) ) ) ) )
=> ( ( finite_finite2 @ B @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F @ A6 ) @ K5 ) ) ) ) ) ).
% sum_bounded_above_divide
thf(fact_4431_sum__bounded__above__strict,axiom,
! [B: $tType,A: $tType] :
( ( ( ordere8940638589300402666id_add @ A )
& ( semiring_1 @ A ) )
=> ! [A6: set @ B,F: B > A,K5: A] :
( ! [I4: B] :
( ( member @ B @ I4 @ A6 )
=> ( ord_less @ A @ ( F @ I4 ) @ K5 ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( finite_card @ B @ A6 ) )
=> ( ord_less @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F @ A6 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( finite_card @ B @ A6 ) ) @ K5 ) ) ) ) ) ).
% sum_bounded_above_strict
thf(fact_4432_convex__sum__bound__le,axiom,
! [A: $tType,B: $tType] :
( ( linordered_idom @ B )
=> ! [I6: set @ A,X: A > B,A3: A > B,B2: B,Delta: B] :
( ! [I4: A] :
( ( member @ A @ I4 @ I6 )
=> ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( X @ I4 ) ) )
=> ( ( ( groups7311177749621191930dd_sum @ A @ B @ X @ I6 )
= ( one_one @ B ) )
=> ( ! [I4: A] :
( ( member @ A @ I4 @ I6 )
=> ( ord_less_eq @ B @ ( abs_abs @ B @ ( minus_minus @ B @ ( A3 @ I4 ) @ B2 ) ) @ Delta ) )
=> ( ord_less_eq @ B
@ ( abs_abs @ B
@ ( minus_minus @ B
@ ( groups7311177749621191930dd_sum @ A @ B
@ ^ [I5: A] : ( times_times @ B @ ( A3 @ I5 ) @ ( X @ I5 ) )
@ I6 )
@ B2 ) )
@ Delta ) ) ) ) ) ).
% convex_sum_bound_le
thf(fact_4433_sum__fun__comp,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( semiring_1 @ C )
=> ! [S3: set @ A,R2: set @ B,G: A > B,F: B > C] :
( ( finite_finite2 @ A @ S3 )
=> ( ( finite_finite2 @ B @ R2 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ G @ S3 ) @ R2 )
=> ( ( groups7311177749621191930dd_sum @ A @ C
@ ^ [X2: A] : ( F @ ( G @ X2 ) )
@ S3 )
= ( groups7311177749621191930dd_sum @ B @ C
@ ^ [Y6: B] :
( times_times @ C
@ ( semiring_1_of_nat @ C
@ ( finite_card @ A
@ ( collect @ A
@ ^ [X2: A] :
( ( member @ A @ X2 @ S3 )
& ( ( G @ X2 )
= Y6 ) ) ) ) )
@ ( F @ Y6 ) )
@ R2 ) ) ) ) ) ) ).
% sum_fun_comp
thf(fact_4434_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_4435_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_4436_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_4437_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
@ ^ [P6: nat] : ( times_times @ A @ ( power_power @ A @ X @ P6 ) @ ( power_power @ A @ Y3 @ ( minus_minus @ nat @ N @ P6 ) ) )
@ ( set_ord_lessThan @ nat @ ( suc @ N ) ) ) ) ) ) ).
% diff_power_eq_sum
thf(fact_4438_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
@ ^ [I5: nat] : ( times_times @ A @ ( power_power @ A @ Y3 @ ( minus_minus @ nat @ N @ ( suc @ I5 ) ) ) @ ( power_power @ A @ X @ I5 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% power_diff_sumr2
thf(fact_4439_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_4440_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_4441_take__bit__sum,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ( ( bit_se2584673776208193580ke_bit @ A )
= ( ^ [N3: nat,A5: A] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( bit_se4730199178511100633sh_bit @ A @ K3 @ ( zero_neq_one_of_bool @ A @ ( bit_se5641148757651400278ts_bit @ A @ A5 @ K3 ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N3 ) ) ) ) ) ).
% take_bit_sum
thf(fact_4442_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_4443_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
@ ^ [Q3: nat] : ( ord_less @ nat @ Q3 @ N ) ) ) ) ) ).
% mask_eq_sum_exp
thf(fact_4444_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_4445_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
@ ^ [I5: nat] : ( plus_plus @ A @ ( G @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I5 ) ) @ ( G @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I5 ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% sum.in_pairs
thf(fact_4446_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
@ ^ [I5: nat] : ( power_power @ A @ X @ ( minus_minus @ nat @ N @ ( suc @ I5 ) ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% one_diff_power_eq'
thf(fact_4447_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
@ ^ [I5: nat] : ( plus_plus @ A @ ( G @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I5 ) ) @ ( G @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I5 ) ) ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% sum.in_pairs_0
thf(fact_4448_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_4449_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_4450_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_4451_horner__sum__eq__sum,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_1 @ A )
=> ( ( groups4207007520872428315er_sum @ B @ A )
= ( ^ [F2: B > A,A5: A,Xs3: list @ B] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N3: nat] : ( times_times @ A @ ( F2 @ ( nth @ B @ Xs3 @ N3 ) ) @ ( power_power @ A @ A5 @ N3 ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ B ) @ Xs3 ) ) ) ) ) ) ).
% horner_sum_eq_sum
thf(fact_4452_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_4453_sum_Ozero__middle,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [P5: nat,K: nat,G: nat > A,H2: nat > A] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ P5 )
=> ( ( ord_less_eq @ nat @ K @ P5 )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J3: nat] : ( if @ A @ ( ord_less @ nat @ J3 @ K ) @ ( G @ J3 ) @ ( if @ A @ ( J3 = K ) @ ( zero_zero @ A ) @ ( H2 @ ( minus_minus @ nat @ J3 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) )
@ ( set_ord_atMost @ nat @ P5 ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J3: nat] : ( if @ A @ ( ord_less @ nat @ J3 @ K ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
@ ( set_ord_atMost @ nat @ ( minus_minus @ nat @ P5 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) ) ).
% sum.zero_middle
thf(fact_4454_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
@ ^ [Q3: nat] : ( ord_less @ nat @ Q3 @ N ) ) ) ) ).
% mask_eq_sum_exp_nat
thf(fact_4455_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_4456_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_4457_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_4458_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_4459_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_4460_card__lists__length__le,axiom,
! [A: $tType,A6: set @ A,N: nat] :
( ( finite_finite2 @ A @ A6 )
=> ( ( finite_card @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Xs3: list @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs3 ) @ A6 )
& ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs3 ) @ N ) ) ) )
= ( groups7311177749621191930dd_sum @ nat @ nat @ ( power_power @ nat @ ( finite_card @ A @ A6 ) ) @ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% card_lists_length_le
thf(fact_4461_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_4462_double__arith__series,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A3: A,D2: A,N: nat] :
( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I5: nat] : ( plus_plus @ A @ A3 @ ( times_times @ A @ ( semiring_1_of_nat @ A @ I5 ) @ D2 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) )
= ( times_times @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N ) @ ( one_one @ A ) ) @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ D2 ) ) ) ) ) ).
% double_arith_series
thf(fact_4463_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_4464_choose__alternating__linear__sum,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [N: nat] :
( ( N
!= ( one_one @ nat ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I5: nat] : ( times_times @ A @ ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ I5 ) @ ( semiring_1_of_nat @ A @ I5 ) ) @ ( semiring_1_of_nat @ A @ ( binomial @ N @ I5 ) ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) ) ) ) ).
% choose_alternating_linear_sum
thf(fact_4465_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_4466_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_4467_arith__series__nat,axiom,
! [A3: nat,D2: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [I5: nat] : ( plus_plus @ nat @ A3 @ ( times_times @ nat @ I5 @ D2 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
= ( divide_divide @ nat @ ( times_times @ nat @ ( suc @ N ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ A3 ) @ ( times_times @ nat @ N @ D2 ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% arith_series_nat
thf(fact_4468_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_4469_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_4470_choose__linear__sum,axiom,
! [N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [I5: nat] : ( times_times @ nat @ I5 @ ( binomial @ N @ I5 ) )
@ ( 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_4471_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_4472_arith__series,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A3: A,D2: A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I5: nat] : ( plus_plus @ A @ A3 @ ( times_times @ A @ ( semiring_1_of_nat @ A @ I5 ) @ D2 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
= ( divide_divide @ A @ ( times_times @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N ) @ ( one_one @ A ) ) @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ D2 ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% arith_series
thf(fact_4473_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_4474_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_4475_choose__alternating__sum,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I5: nat] : ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ I5 ) @ ( semiring_1_of_nat @ A @ ( binomial @ N @ I5 ) ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) ) ) ) ).
% choose_alternating_sum
thf(fact_4476_divmod__step__def,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ( ( unique1321980374590559556d_step @ A )
= ( ^ [L3: num] :
( product_case_prod @ A @ A @ ( product_prod @ A @ A )
@ ^ [Q3: A,R5: A] : ( if @ ( product_prod @ A @ A ) @ ( ord_less_eq @ A @ ( numeral_numeral @ A @ L3 ) @ R5 ) @ ( product_Pair @ A @ A @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Q3 ) @ ( one_one @ A ) ) @ ( minus_minus @ A @ R5 @ ( numeral_numeral @ A @ L3 ) ) ) @ ( product_Pair @ A @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Q3 ) @ R5 ) ) ) ) ) ) ).
% divmod_step_def
thf(fact_4477_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_4478_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_4479_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_4480_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_4481_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_4482_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
@ ^ [I5: nat] :
( if @ A
@ ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I5 )
@ ( semiring_1_of_nat @ A @ ( binomial @ N @ I5 ) )
@ ( zero_zero @ A ) )
@ ( set_ord_atMost @ nat @ N ) ) )
= ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% choose_odd_sum
thf(fact_4483_map__entry__def,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( ( array_map_entry @ A )
= ( ^ [I5: nat,F2: A > A,A5: array @ A] :
( heap_Time_guard @ ( array @ A )
@ ^ [H4: heap_ext @ product_unit] : ( ord_less @ nat @ I5 @ ( array_length @ A @ H4 @ A5 ) )
@ ^ [H4: heap_ext @ product_unit] : ( product_Pair @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ A5 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ ( array_update @ A @ A5 @ I5 @ ( F2 @ ( nth @ A @ ( array_get @ A @ H4 @ A5 ) @ I5 ) ) @ H4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% map_entry_def
thf(fact_4484_Array__Time_Onth__def,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( ( array_nth @ A )
= ( ^ [A5: array @ A,I5: nat] :
( heap_Time_guard @ A
@ ^ [H4: heap_ext @ product_unit] : ( ord_less @ nat @ I5 @ ( array_length @ A @ H4 @ A5 ) )
@ ^ [H4: heap_ext @ product_unit] : ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( nth @ A @ ( array_get @ A @ H4 @ A5 ) @ I5 ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H4 @ ( one_one @ nat ) ) ) ) ) ) ) ).
% Array_Time.nth_def
thf(fact_4485_upd__def,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( ( array_upd @ A )
= ( ^ [I5: nat,X2: A,A5: array @ A] :
( heap_Time_guard @ ( array @ A )
@ ^ [H4: heap_ext @ product_unit] : ( ord_less @ nat @ I5 @ ( array_length @ A @ H4 @ A5 ) )
@ ^ [H4: heap_ext @ product_unit] : ( product_Pair @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ A5 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ ( array_update @ A @ A5 @ I5 @ X2 @ H4 ) @ ( one_one @ nat ) ) ) ) ) ) ) ).
% upd_def
thf(fact_4486_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
@ ^ [L3: code_integer,J3: code_integer] :
( if @ nat
@ ( J3
= ( zero_zero @ code_integer ) )
@ ( plus_plus @ nat @ ( code_nat_of_integer @ L3 ) @ ( code_nat_of_integer @ L3 ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( code_nat_of_integer @ L3 ) @ ( code_nat_of_integer @ L3 ) ) @ ( one_one @ nat ) ) )
@ ( code_divmod_integer @ K3 @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% nat_of_integer_code
thf(fact_4487_int__sum,axiom,
! [B: $tType,F: B > nat,A6: set @ B] :
( ( semiring_1_of_nat @ int @ ( groups7311177749621191930dd_sum @ B @ nat @ F @ A6 ) )
= ( groups7311177749621191930dd_sum @ B @ int
@ ^ [X2: B] : ( semiring_1_of_nat @ int @ ( F @ X2 ) )
@ A6 ) ) ).
% int_sum
thf(fact_4488_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
@ ^ [I5: nat,J3: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ I5 @ J3 ) @ N ) ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I5: nat] : ( G @ I5 @ ( minus_minus @ nat @ K3 @ I5 ) )
@ ( set_ord_atMost @ nat @ K3 ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% sum.triangle_reindex_eq
thf(fact_4489_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
@ ^ [I5: nat,J3: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ I5 @ J3 ) @ N ) ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I5: nat] : ( G @ I5 @ ( minus_minus @ nat @ K3 @ I5 ) )
@ ( set_ord_atMost @ nat @ K3 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% sum.triangle_reindex
thf(fact_4490_sum__subtractf__nat,axiom,
! [A: $tType,A6: set @ A,G: A > nat,F: A > nat] :
( ! [X3: A] :
( ( member @ A @ X3 @ A6 )
=> ( ord_less_eq @ nat @ ( G @ X3 ) @ ( F @ X3 ) ) )
=> ( ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [X2: A] : ( minus_minus @ nat @ ( F @ X2 ) @ ( G @ X2 ) )
@ A6 )
= ( minus_minus @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ F @ A6 ) @ ( groups7311177749621191930dd_sum @ A @ nat @ G @ A6 ) ) ) ) ).
% sum_subtractf_nat
thf(fact_4491_card__eq__sum,axiom,
! [A: $tType] :
( ( finite_card @ A )
= ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [X2: A] : ( one_one @ nat ) ) ) ).
% card_eq_sum
thf(fact_4492_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
@ ^ [I5: nat,J3: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ I5 @ J3 ) @ N ) ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [K3: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I5: nat] : ( G @ I5 @ ( minus_minus @ nat @ K3 @ I5 ) )
@ ( set_ord_atMost @ nat @ K3 ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% prod.triangle_reindex_eq
thf(fact_4493_sum__eq__Suc0__iff,axiom,
! [A: $tType,A6: set @ A,F: A > nat] :
( ( finite_finite2 @ A @ A6 )
=> ( ( ( groups7311177749621191930dd_sum @ A @ nat @ F @ A6 )
= ( suc @ ( zero_zero @ nat ) ) )
= ( ? [X2: A] :
( ( member @ A @ X2 @ A6 )
& ( ( F @ X2 )
= ( suc @ ( zero_zero @ nat ) ) )
& ! [Y6: A] :
( ( member @ A @ Y6 @ A6 )
=> ( ( X2 != Y6 )
=> ( ( F @ Y6 )
= ( zero_zero @ nat ) ) ) ) ) ) ) ) ).
% sum_eq_Suc0_iff
thf(fact_4494_sum__SucD,axiom,
! [A: $tType,F: A > nat,A6: set @ A,N: nat] :
( ( ( groups7311177749621191930dd_sum @ A @ nat @ F @ A6 )
= ( suc @ N ) )
=> ? [X3: A] :
( ( member @ A @ X3 @ A6 )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F @ X3 ) ) ) ) ).
% sum_SucD
thf(fact_4495_sum__eq__1__iff,axiom,
! [A: $tType,A6: set @ A,F: A > nat] :
( ( finite_finite2 @ A @ A6 )
=> ( ( ( groups7311177749621191930dd_sum @ A @ nat @ F @ A6 )
= ( one_one @ nat ) )
= ( ? [X2: A] :
( ( member @ A @ X2 @ A6 )
& ( ( F @ X2 )
= ( one_one @ nat ) )
& ! [Y6: A] :
( ( member @ A @ Y6 @ A6 )
=> ( ( X2 != Y6 )
=> ( ( F @ Y6 )
= ( zero_zero @ nat ) ) ) ) ) ) ) ) ).
% sum_eq_1_iff
thf(fact_4496_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
@ ^ [I5: nat,J3: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ I5 @ J3 ) @ N ) ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [K3: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I5: nat] : ( G @ I5 @ ( minus_minus @ nat @ K3 @ I5 ) )
@ ( set_ord_atMost @ nat @ K3 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% prod.triangle_reindex
thf(fact_4497_sum__Suc,axiom,
! [A: $tType,F: A > nat,A6: set @ A] :
( ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [X2: A] : ( suc @ ( F @ X2 ) )
@ A6 )
= ( plus_plus @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ F @ A6 ) @ ( finite_card @ A @ A6 ) ) ) ).
% sum_Suc
thf(fact_4498_sum__multicount,axiom,
! [A: $tType,B: $tType,S3: set @ A,T6: set @ B,R2: A > B > $o,K: nat] :
( ( finite_finite2 @ A @ S3 )
=> ( ( finite_finite2 @ B @ T6 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ T6 )
=> ( ( finite_card @ A
@ ( collect @ A
@ ^ [I5: A] :
( ( member @ A @ I5 @ S3 )
& ( R2 @ I5 @ X3 ) ) ) )
= K ) )
=> ( ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [I5: A] :
( finite_card @ B
@ ( collect @ B
@ ^ [J3: B] :
( ( member @ B @ J3 @ T6 )
& ( R2 @ I5 @ J3 ) ) ) )
@ S3 )
= ( times_times @ nat @ K @ ( finite_card @ B @ T6 ) ) ) ) ) ) ).
% sum_multicount
thf(fact_4499_sum__diff__distrib,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [Q: A > nat,P: A > nat,N: A] :
( ! [X3: A] : ( ord_less_eq @ nat @ ( Q @ X3 ) @ ( P @ X3 ) )
=> ( ( minus_minus @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ P @ ( set_ord_lessThan @ A @ N ) ) @ ( groups7311177749621191930dd_sum @ A @ nat @ Q @ ( set_ord_lessThan @ A @ N ) ) )
= ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [X2: A] : ( minus_minus @ nat @ ( P @ X2 ) @ ( Q @ X2 ) )
@ ( set_ord_lessThan @ A @ N ) ) ) ) ) ).
% sum_diff_distrib
thf(fact_4500_prod__encode__def,axiom,
( nat_prod_encode
= ( product_case_prod @ nat @ nat @ nat
@ ^ [M2: nat,N3: nat] : ( plus_plus @ nat @ ( nat_triangle @ ( plus_plus @ nat @ M2 @ N3 ) ) @ M2 ) ) ) ).
% prod_encode_def
thf(fact_4501_rat__less__code,axiom,
( ( ord_less @ rat )
= ( ^ [P6: rat,Q3: rat] :
( product_case_prod @ int @ int @ $o
@ ^ [A5: int,C4: int] :
( product_case_prod @ int @ int @ $o
@ ^ [B3: int,D5: int] : ( ord_less @ int @ ( times_times @ int @ A5 @ D5 ) @ ( times_times @ int @ C4 @ B3 ) )
@ ( quotient_of @ Q3 ) )
@ ( quotient_of @ P6 ) ) ) ) ).
% rat_less_code
thf(fact_4502_rat__less__eq__code,axiom,
( ( ord_less_eq @ rat )
= ( ^ [P6: rat,Q3: rat] :
( product_case_prod @ int @ int @ $o
@ ^ [A5: int,C4: int] :
( product_case_prod @ int @ int @ $o
@ ^ [B3: int,D5: int] : ( ord_less_eq @ int @ ( times_times @ int @ A5 @ D5 ) @ ( times_times @ int @ C4 @ B3 ) )
@ ( quotient_of @ Q3 ) )
@ ( quotient_of @ P6 ) ) ) ) ).
% rat_less_eq_code
thf(fact_4503_rat__uminus__code,axiom,
! [P5: rat] :
( ( quotient_of @ ( uminus_uminus @ rat @ P5 ) )
= ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [A5: int] : ( product_Pair @ int @ int @ ( uminus_uminus @ int @ A5 ) )
@ ( quotient_of @ P5 ) ) ) ).
% rat_uminus_code
thf(fact_4504_lenlex__conv,axiom,
! [A: $tType] :
( ( lenlex @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ ( list @ A ) @ ( list @ A ) )
@ ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
@ ^ [Xs3: list @ A,Ys3: list @ A] :
( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Xs3 ) @ ( size_size @ ( list @ A ) @ Ys3 ) )
| ( ( ( size_size @ ( list @ A ) @ Xs3 )
= ( size_size @ ( list @ A ) @ Ys3 ) )
& ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs3 @ Ys3 ) @ ( lex @ A @ R5 ) ) ) ) ) ) ) ) ).
% lenlex_conv
thf(fact_4505_Divides_Oadjust__div__def,axiom,
( adjust_div
= ( product_case_prod @ int @ int @ int
@ ^ [Q3: int,R5: int] :
( plus_plus @ int @ Q3
@ ( zero_neq_one_of_bool @ int
@ ( R5
!= ( zero_zero @ int ) ) ) ) ) ) ).
% Divides.adjust_div_def
thf(fact_4506_sum__list__map__eq__sum__count,axiom,
! [A: $tType,F: A > nat,Xs: list @ A] :
( ( groups8242544230860333062m_list @ nat @ ( map @ A @ nat @ F @ Xs ) )
= ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [X2: A] : ( times_times @ nat @ ( count_list @ A @ Xs @ X2 ) @ ( F @ X2 ) )
@ ( set2 @ A @ Xs ) ) ) ).
% sum_list_map_eq_sum_count
thf(fact_4507_rat__divide__code,axiom,
! [P5: rat,Q4: rat] :
( ( quotient_of @ ( divide_divide @ rat @ P5 @ Q4 ) )
= ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [A5: int,C4: int] :
( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [B3: int,D5: int] : ( normalize @ ( product_Pair @ int @ int @ ( times_times @ int @ A5 @ D5 ) @ ( times_times @ int @ C4 @ B3 ) ) )
@ ( quotient_of @ Q4 ) )
@ ( quotient_of @ P5 ) ) ) ).
% rat_divide_code
thf(fact_4508_rat__times__code,axiom,
! [P5: rat,Q4: rat] :
( ( quotient_of @ ( times_times @ rat @ P5 @ Q4 ) )
= ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [A5: int,C4: int] :
( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [B3: int,D5: int] : ( normalize @ ( product_Pair @ int @ int @ ( times_times @ int @ A5 @ B3 ) @ ( times_times @ int @ C4 @ D5 ) ) )
@ ( quotient_of @ Q4 ) )
@ ( quotient_of @ P5 ) ) ) ).
% rat_times_code
thf(fact_4509_rat__plus__code,axiom,
! [P5: rat,Q4: rat] :
( ( quotient_of @ ( plus_plus @ rat @ P5 @ Q4 ) )
= ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [A5: int,C4: int] :
( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [B3: int,D5: int] : ( normalize @ ( product_Pair @ int @ int @ ( plus_plus @ int @ ( times_times @ int @ A5 @ D5 ) @ ( times_times @ int @ B3 @ C4 ) ) @ ( times_times @ int @ C4 @ D5 ) ) )
@ ( quotient_of @ Q4 ) )
@ ( quotient_of @ P5 ) ) ) ).
% rat_plus_code
thf(fact_4510_rat__minus__code,axiom,
! [P5: rat,Q4: rat] :
( ( quotient_of @ ( minus_minus @ rat @ P5 @ Q4 ) )
= ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [A5: int,C4: int] :
( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [B3: int,D5: int] : ( normalize @ ( product_Pair @ int @ int @ ( minus_minus @ int @ ( times_times @ int @ A5 @ D5 ) @ ( times_times @ int @ B3 @ C4 ) ) @ ( times_times @ int @ C4 @ D5 ) ) )
@ ( quotient_of @ Q4 ) )
@ ( quotient_of @ P5 ) ) ) ).
% rat_minus_code
thf(fact_4511_sum__list__map__eq__sum__count2,axiom,
! [A: $tType,Xs: list @ A,X8: set @ A,F: A > nat] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ X8 )
=> ( ( finite_finite2 @ A @ X8 )
=> ( ( groups8242544230860333062m_list @ nat @ ( map @ A @ nat @ F @ Xs ) )
= ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [X2: A] : ( times_times @ nat @ ( count_list @ A @ Xs @ X2 ) @ ( F @ X2 ) )
@ X8 ) ) ) ) ).
% sum_list_map_eq_sum_count2
thf(fact_4512_divmod__step__nat__def,axiom,
( ( unique1321980374590559556d_step @ nat )
= ( ^ [L3: num] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [Q3: nat,R5: nat] : ( if @ ( product_prod @ nat @ nat ) @ ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ L3 ) @ R5 ) @ ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Q3 ) @ ( one_one @ nat ) ) @ ( minus_minus @ nat @ R5 @ ( numeral_numeral @ nat @ L3 ) ) ) @ ( product_Pair @ nat @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Q3 ) @ R5 ) ) ) ) ) ).
% divmod_step_nat_def
thf(fact_4513_rat__inverse__code,axiom,
! [P5: rat] :
( ( quotient_of @ ( inverse_inverse @ rat @ P5 ) )
= ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [A5: int,B3: int] :
( if @ ( product_prod @ int @ int )
@ ( A5
= ( zero_zero @ int ) )
@ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
@ ( product_Pair @ int @ int @ ( times_times @ int @ ( sgn_sgn @ int @ A5 ) @ B3 ) @ ( abs_abs @ int @ A5 ) ) )
@ ( quotient_of @ P5 ) ) ) ).
% rat_inverse_code
thf(fact_4514_divmod__step__int__def,axiom,
( ( unique1321980374590559556d_step @ int )
= ( ^ [L3: num] :
( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [Q3: int,R5: int] : ( if @ ( product_prod @ int @ int ) @ ( ord_less_eq @ int @ ( numeral_numeral @ int @ L3 ) @ R5 ) @ ( product_Pair @ int @ int @ ( plus_plus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Q3 ) @ ( one_one @ int ) ) @ ( minus_minus @ int @ R5 @ ( numeral_numeral @ int @ L3 ) ) ) @ ( product_Pair @ int @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Q3 ) @ R5 ) ) ) ) ) ).
% divmod_step_int_def
thf(fact_4515_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_4516_divmod__step__integer__def,axiom,
( ( unique1321980374590559556d_step @ code_integer )
= ( ^ [L3: num] :
( product_case_prod @ code_integer @ code_integer @ ( product_prod @ code_integer @ code_integer )
@ ^ [Q3: code_integer,R5: code_integer] : ( if @ ( product_prod @ code_integer @ code_integer ) @ ( ord_less_eq @ code_integer @ ( numeral_numeral @ code_integer @ L3 ) @ R5 ) @ ( product_Pair @ code_integer @ code_integer @ ( plus_plus @ code_integer @ ( times_times @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) @ Q3 ) @ ( one_one @ code_integer ) ) @ ( minus_minus @ code_integer @ R5 @ ( numeral_numeral @ code_integer @ L3 ) ) ) @ ( product_Pair @ code_integer @ code_integer @ ( times_times @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) @ Q3 ) @ R5 ) ) ) ) ) ).
% divmod_step_integer_def
thf(fact_4517_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
@ ^ [L3: code_integer,J3: code_integer] :
( if @ num
@ ( J3
= ( zero_zero @ code_integer ) )
@ ( plus_plus @ num @ ( code_num_of_integer @ L3 ) @ ( code_num_of_integer @ L3 ) )
@ ( plus_plus @ num @ ( plus_plus @ num @ ( code_num_of_integer @ L3 ) @ ( code_num_of_integer @ L3 ) ) @ one2 ) )
@ ( code_divmod_integer @ K3 @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% num_of_integer_code
thf(fact_4518_Array__Time_Omake__def,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( ( array_make @ A )
= ( ^ [N3: nat,F2: nat > A] :
( heap_Time_heap @ ( array @ A )
@ ^ [H4: heap_ext @ product_unit] :
( product_case_prod @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( product_prod @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )
@ ^ [R5: array @ A,H5: heap_ext @ product_unit] : ( product_Pair @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R5 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H5 @ ( plus_plus @ nat @ N3 @ ( one_one @ nat ) ) ) )
@ ( array_alloc @ A @ ( map @ nat @ A @ F2 @ ( upt @ ( zero_zero @ nat ) @ N3 ) ) @ H4 ) ) ) ) ) ) ).
% Array_Time.make_def
thf(fact_4519_new__def,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( ( array_new @ A )
= ( ^ [N3: nat,X2: A] :
( heap_Time_heap @ ( array @ A )
@ ^ [H4: heap_ext @ product_unit] :
( product_case_prod @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( product_prod @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )
@ ^ [R5: array @ A,H5: heap_ext @ product_unit] : ( product_Pair @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R5 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H5 @ ( plus_plus @ nat @ N3 @ ( one_one @ nat ) ) ) )
@ ( array_alloc @ A @ ( replicate @ A @ N3 @ X2 ) @ H4 ) ) ) ) ) ) ).
% new_def
thf(fact_4520_of__list__def,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( ( array_of_list @ A )
= ( ^ [Xs3: list @ A] :
( heap_Time_heap @ ( array @ A )
@ ^ [H4: heap_ext @ product_unit] :
( product_case_prod @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( product_prod @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )
@ ^ [R5: array @ A,H5: heap_ext @ product_unit] : ( product_Pair @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R5 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H5 @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( size_size @ ( list @ A ) @ Xs3 ) ) ) )
@ ( array_alloc @ A @ Xs3 @ H4 ) ) ) ) ) ) ).
% of_list_def
thf(fact_4521_num__of__integer_Oabs__eq,axiom,
! [X: int] :
( ( code_num_of_integer @ ( code_integer_of_int @ X ) )
= ( num_of_nat @ ( nat2 @ X ) ) ) ).
% num_of_integer.abs_eq
thf(fact_4522_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_4523_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_4524_effect__heapE,axiom,
! [A: $tType,F: ( heap_ext @ product_unit ) > ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ),H2: heap_ext @ product_unit,H: heap_ext @ product_unit,R: A,N: nat] :
( ( heap_Time_effect @ A @ ( heap_Time_heap @ A @ F ) @ H2 @ H @ R @ N )
=> ~ ( ( H
= ( product_fst @ ( heap_ext @ product_unit ) @ nat @ ( product_snd @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( F @ H2 ) ) ) )
=> ( ( N
= ( product_snd @ ( heap_ext @ product_unit ) @ nat @ ( product_snd @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( F @ H2 ) ) ) )
=> ( R
!= ( product_fst @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( F @ H2 ) ) ) ) ) ) ).
% effect_heapE
thf(fact_4525_effect__heapI,axiom,
! [A: $tType,N: nat,F: ( heap_ext @ product_unit ) > ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ),H2: heap_ext @ product_unit,H: 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 @ H2 ) ) ) )
=> ( ( H
= ( product_fst @ ( heap_ext @ product_unit ) @ nat @ ( product_snd @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( F @ H2 ) ) ) )
=> ( ( R
= ( product_fst @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( F @ H2 ) ) )
=> ( heap_Time_effect @ A @ ( heap_Time_heap @ A @ F ) @ H2 @ H @ R @ N ) ) ) ) ).
% effect_heapI
thf(fact_4526_divmod__nat__if,axiom,
( divmod_nat
= ( ^ [M2: nat,N3: nat] :
( if @ ( product_prod @ nat @ nat )
@ ( ( N3
= ( zero_zero @ nat ) )
| ( ord_less @ nat @ M2 @ N3 ) )
@ ( product_Pair @ nat @ nat @ ( zero_zero @ nat ) @ M2 )
@ ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [Q3: nat] : ( product_Pair @ nat @ nat @ ( suc @ Q3 ) )
@ ( divmod_nat @ ( minus_minus @ nat @ M2 @ N3 ) @ N3 ) ) ) ) ) ).
% divmod_nat_if
thf(fact_4527_int__ge__less__than__def,axiom,
( int_ge_less_than
= ( ^ [D5: int] :
( collect @ ( product_prod @ int @ int )
@ ( product_case_prod @ int @ int @ $o
@ ^ [Z8: int,Z7: int] :
( ( ord_less_eq @ int @ D5 @ Z8 )
& ( ord_less @ int @ Z8 @ Z7 ) ) ) ) ) ) ).
% int_ge_less_than_def
thf(fact_4528_int__ge__less__than2__def,axiom,
( int_ge_less_than2
= ( ^ [D5: int] :
( collect @ ( product_prod @ int @ int )
@ ( product_case_prod @ int @ int @ $o
@ ^ [Z8: int,Z7: int] :
( ( ord_less_eq @ int @ D5 @ Z7 )
& ( ord_less @ int @ Z8 @ Z7 ) ) ) ) ) ) ).
% int_ge_less_than2_def
thf(fact_4529_divmod__integer__code,axiom,
( code_divmod_integer
= ( ^ [K3: code_integer,L3: 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 ) @ L3 )
@ ( if @ ( product_prod @ code_integer @ code_integer ) @ ( ord_less @ code_integer @ ( zero_zero @ code_integer ) @ K3 ) @ ( code_divmod_abs @ K3 @ L3 )
@ ( product_case_prod @ code_integer @ code_integer @ ( product_prod @ code_integer @ code_integer )
@ ^ [R5: code_integer,S6: code_integer] :
( if @ ( product_prod @ code_integer @ code_integer )
@ ( S6
= ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( uminus_uminus @ code_integer @ R5 ) @ ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( minus_minus @ code_integer @ ( uminus_uminus @ code_integer @ R5 ) @ ( one_one @ code_integer ) ) @ ( minus_minus @ code_integer @ L3 @ S6 ) ) )
@ ( code_divmod_abs @ K3 @ L3 ) ) )
@ ( if @ ( product_prod @ code_integer @ code_integer )
@ ( L3
= ( 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 @ L3 )
@ ( product_case_prod @ code_integer @ code_integer @ ( product_prod @ code_integer @ code_integer )
@ ^ [R5: code_integer,S6: code_integer] :
( if @ ( product_prod @ code_integer @ code_integer )
@ ( S6
= ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( uminus_uminus @ code_integer @ R5 ) @ ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( minus_minus @ code_integer @ ( uminus_uminus @ code_integer @ R5 ) @ ( one_one @ code_integer ) ) @ ( minus_minus @ code_integer @ ( uminus_uminus @ code_integer @ L3 ) @ S6 ) ) )
@ ( code_divmod_abs @ K3 @ L3 ) ) ) ) ) ) ) ) ) ).
% divmod_integer_code
thf(fact_4530_freeze__def,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( ( array_freeze @ A )
= ( ^ [A5: array @ A] :
( heap_Time_heap @ ( list @ A )
@ ^ [H4: heap_ext @ product_unit] : ( product_Pair @ ( list @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( array_get @ A @ H4 @ A5 ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H4 @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( array_length @ A @ H4 @ A5 ) ) ) ) ) ) ) ) ).
% freeze_def
thf(fact_4531_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
@ ^ [L3: code_integer,J3: code_integer] :
( if @ int
@ ( J3
= ( zero_zero @ code_integer ) )
@ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( code_int_of_integer @ L3 ) )
@ ( plus_plus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( code_int_of_integer @ L3 ) ) @ ( one_one @ int ) ) )
@ ( code_divmod_integer @ K3 @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% int_of_integer_code
thf(fact_4532_of__nat__code,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( semiring_1_of_nat @ A )
= ( ^ [N3: nat] :
( semiri8178284476397505188at_aux @ A
@ ^ [I5: A] : ( plus_plus @ A @ I5 @ ( one_one @ A ) )
@ N3
@ ( zero_zero @ A ) ) ) ) ) ).
% of_nat_code
thf(fact_4533_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_4534_int__of__integer__inverse,axiom,
! [X: code_integer] :
( ( code_integer_of_int @ ( code_int_of_integer @ X ) )
= X ) ).
% int_of_integer_inverse
thf(fact_4535_int__of__integer__integer__of__int,axiom,
! [K: int] :
( ( code_int_of_integer @ ( code_integer_of_int @ K ) )
= K ) ).
% int_of_integer_integer_of_int
thf(fact_4536_integer__of__int__int__of__integer,axiom,
! [K: code_integer] :
( ( code_integer_of_int @ ( code_int_of_integer @ K ) )
= K ) ).
% integer_of_int_int_of_integer
thf(fact_4537_int__of__integer__sub,axiom,
! [K: num,L: num] :
( ( code_int_of_integer @ ( neg_numeral_sub @ code_integer @ K @ L ) )
= ( neg_numeral_sub @ int @ K @ L ) ) ).
% int_of_integer_sub
thf(fact_4538_int__of__integer__min,axiom,
! [K: code_integer,L: code_integer] :
( ( code_int_of_integer @ ( ord_min @ code_integer @ K @ L ) )
= ( ord_min @ int @ ( code_int_of_integer @ K ) @ ( code_int_of_integer @ L ) ) ) ).
% int_of_integer_min
thf(fact_4539_of__int__integer__of,axiom,
! [K: code_integer] :
( ( ring_1_of_int @ code_integer @ ( code_int_of_integer @ K ) )
= K ) ).
% of_int_integer_of
thf(fact_4540_int__of__integer__of__int,axiom,
! [K: int] :
( ( code_int_of_integer @ ( ring_1_of_int @ code_integer @ K ) )
= K ) ).
% int_of_integer_of_int
thf(fact_4541_zero__integer_Orep__eq,axiom,
( ( code_int_of_integer @ ( zero_zero @ code_integer ) )
= ( zero_zero @ int ) ) ).
% zero_integer.rep_eq
thf(fact_4542_int__of__integer__numeral,axiom,
! [K: num] :
( ( code_int_of_integer @ ( numeral_numeral @ code_integer @ K ) )
= ( numeral_numeral @ int @ K ) ) ).
% int_of_integer_numeral
thf(fact_4543_plus__integer_Orep__eq,axiom,
! [X: code_integer,Xa3: code_integer] :
( ( code_int_of_integer @ ( plus_plus @ code_integer @ X @ Xa3 ) )
= ( plus_plus @ int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa3 ) ) ) ).
% plus_integer.rep_eq
thf(fact_4544_uminus__integer_Orep__eq,axiom,
! [X: code_integer] :
( ( code_int_of_integer @ ( uminus_uminus @ code_integer @ X ) )
= ( uminus_uminus @ int @ ( code_int_of_integer @ X ) ) ) ).
% uminus_integer.rep_eq
thf(fact_4545_times__integer_Orep__eq,axiom,
! [X: code_integer,Xa3: code_integer] :
( ( code_int_of_integer @ ( times_times @ code_integer @ X @ Xa3 ) )
= ( times_times @ int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa3 ) ) ) ).
% times_integer.rep_eq
thf(fact_4546_one__integer_Orep__eq,axiom,
( ( code_int_of_integer @ ( one_one @ code_integer ) )
= ( one_one @ int ) ) ).
% one_integer.rep_eq
thf(fact_4547_minus__integer_Orep__eq,axiom,
! [X: code_integer,Xa3: code_integer] :
( ( code_int_of_integer @ ( minus_minus @ code_integer @ X @ Xa3 ) )
= ( minus_minus @ int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa3 ) ) ) ).
% minus_integer.rep_eq
thf(fact_4548_abs__integer_Orep__eq,axiom,
! [X: code_integer] :
( ( code_int_of_integer @ ( abs_abs @ code_integer @ X ) )
= ( abs_abs @ int @ ( code_int_of_integer @ X ) ) ) ).
% abs_integer.rep_eq
thf(fact_4549_divide__integer_Orep__eq,axiom,
! [X: code_integer,Xa3: code_integer] :
( ( code_int_of_integer @ ( divide_divide @ code_integer @ X @ Xa3 ) )
= ( divide_divide @ int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa3 ) ) ) ).
% divide_integer.rep_eq
thf(fact_4550_modulo__integer_Orep__eq,axiom,
! [X: code_integer,Xa3: code_integer] :
( ( code_int_of_integer @ ( modulo_modulo @ code_integer @ X @ Xa3 ) )
= ( modulo_modulo @ int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa3 ) ) ) ).
% modulo_integer.rep_eq
thf(fact_4551_sgn__integer_Orep__eq,axiom,
! [X: code_integer] :
( ( code_int_of_integer @ ( sgn_sgn @ code_integer @ X ) )
= ( sgn_sgn @ int @ ( code_int_of_integer @ X ) ) ) ).
% sgn_integer.rep_eq
thf(fact_4552_int__of__integer__of__nat,axiom,
! [N: nat] :
( ( code_int_of_integer @ ( semiring_1_of_nat @ code_integer @ N ) )
= ( semiring_1_of_nat @ int @ N ) ) ).
% int_of_integer_of_nat
thf(fact_4553_division__segment__integer_Orep__eq,axiom,
! [X: code_integer] :
( ( code_int_of_integer @ ( euclid7384307370059645450egment @ code_integer @ X ) )
= ( euclid7384307370059645450egment @ int @ ( code_int_of_integer @ X ) ) ) ).
% division_segment_integer.rep_eq
thf(fact_4554_euclidean__size__integer_Orep__eq,axiom,
( ( euclid6346220572633701492n_size @ code_integer )
= ( ^ [X2: code_integer] : ( euclid6346220572633701492n_size @ int @ ( code_int_of_integer @ X2 ) ) ) ) ).
% euclidean_size_integer.rep_eq
thf(fact_4555_int__of__integer__max,axiom,
! [K: code_integer,L: code_integer] :
( ( code_int_of_integer @ ( ord_max @ code_integer @ K @ L ) )
= ( ord_max @ int @ ( code_int_of_integer @ K ) @ ( code_int_of_integer @ L ) ) ) ).
% int_of_integer_max
thf(fact_4556_integer__eq__iff,axiom,
( ( ^ [Y: code_integer,Z2: code_integer] : Y = Z2 )
= ( ^ [K3: code_integer,L3: code_integer] :
( ( code_int_of_integer @ K3 )
= ( code_int_of_integer @ L3 ) ) ) ) ).
% integer_eq_iff
thf(fact_4557_integer__eqI,axiom,
! [K: code_integer,L: code_integer] :
( ( ( code_int_of_integer @ K )
= ( code_int_of_integer @ L ) )
=> ( K = L ) ) ).
% integer_eqI
thf(fact_4558_int__of__integer__inject,axiom,
! [X: code_integer,Y3: code_integer] :
( ( ( code_int_of_integer @ X )
= ( code_int_of_integer @ Y3 ) )
= ( X = Y3 ) ) ).
% int_of_integer_inject
thf(fact_4559_less__integer_Orep__eq,axiom,
( ( ord_less @ code_integer )
= ( ^ [X2: code_integer,Xa4: code_integer] : ( ord_less @ int @ ( code_int_of_integer @ X2 ) @ ( code_int_of_integer @ Xa4 ) ) ) ) ).
% less_integer.rep_eq
thf(fact_4560_integer__less__iff,axiom,
( ( ord_less @ code_integer )
= ( ^ [K3: code_integer,L3: code_integer] : ( ord_less @ int @ ( code_int_of_integer @ K3 ) @ ( code_int_of_integer @ L3 ) ) ) ) ).
% integer_less_iff
thf(fact_4561_integer__less__eq__iff,axiom,
( ( ord_less_eq @ code_integer )
= ( ^ [K3: code_integer,L3: code_integer] : ( ord_less_eq @ int @ ( code_int_of_integer @ K3 ) @ ( code_int_of_integer @ L3 ) ) ) ) ).
% integer_less_eq_iff
thf(fact_4562_less__eq__integer_Orep__eq,axiom,
( ( ord_less_eq @ code_integer )
= ( ^ [X2: code_integer,Xa4: code_integer] : ( ord_less_eq @ int @ ( code_int_of_integer @ X2 ) @ ( code_int_of_integer @ Xa4 ) ) ) ) ).
% less_eq_integer.rep_eq
thf(fact_4563_nat__of__integer_Orep__eq,axiom,
( code_nat_of_integer
= ( ^ [X2: code_integer] : ( nat2 @ ( code_int_of_integer @ X2 ) ) ) ) ).
% nat_of_integer.rep_eq
thf(fact_4564_take__bit__integer_Orep__eq,axiom,
! [X: nat,Xa3: code_integer] :
( ( code_int_of_integer @ ( bit_se2584673776208193580ke_bit @ code_integer @ X @ Xa3 ) )
= ( bit_se2584673776208193580ke_bit @ int @ X @ ( code_int_of_integer @ Xa3 ) ) ) ).
% take_bit_integer.rep_eq
thf(fact_4565_not__integer_Orep__eq,axiom,
! [X: code_integer] :
( ( code_int_of_integer @ ( bit_ri4277139882892585799ns_not @ code_integer @ X ) )
= ( bit_ri4277139882892585799ns_not @ int @ ( code_int_of_integer @ X ) ) ) ).
% not_integer.rep_eq
thf(fact_4566_and__integer_Orep__eq,axiom,
! [X: code_integer,Xa3: code_integer] :
( ( code_int_of_integer @ ( bit_se5824344872417868541ns_and @ code_integer @ X @ Xa3 ) )
= ( bit_se5824344872417868541ns_and @ int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa3 ) ) ) ).
% and_integer.rep_eq
thf(fact_4567_bit__integer_Orep__eq,axiom,
( ( bit_se5641148757651400278ts_bit @ code_integer )
= ( ^ [X2: code_integer] : ( bit_se5641148757651400278ts_bit @ int @ ( code_int_of_integer @ X2 ) ) ) ) ).
% bit_integer.rep_eq
thf(fact_4568_or__integer_Orep__eq,axiom,
! [X: code_integer,Xa3: code_integer] :
( ( code_int_of_integer @ ( bit_se1065995026697491101ons_or @ code_integer @ X @ Xa3 ) )
= ( bit_se1065995026697491101ons_or @ int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa3 ) ) ) ).
% or_integer.rep_eq
thf(fact_4569_xor__integer_Orep__eq,axiom,
! [X: code_integer,Xa3: code_integer] :
( ( code_int_of_integer @ ( bit_se5824344971392196577ns_xor @ code_integer @ X @ Xa3 ) )
= ( bit_se5824344971392196577ns_xor @ int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa3 ) ) ) ).
% xor_integer.rep_eq
thf(fact_4570_push__bit__integer_Orep__eq,axiom,
! [X: nat,Xa3: code_integer] :
( ( code_int_of_integer @ ( bit_se4730199178511100633sh_bit @ code_integer @ X @ Xa3 ) )
= ( bit_se4730199178511100633sh_bit @ int @ X @ ( code_int_of_integer @ Xa3 ) ) ) ).
% push_bit_integer.rep_eq
thf(fact_4571_drop__bit__integer_Orep__eq,axiom,
! [X: nat,Xa3: code_integer] :
( ( code_int_of_integer @ ( bit_se4197421643247451524op_bit @ code_integer @ X @ Xa3 ) )
= ( bit_se4197421643247451524op_bit @ int @ X @ ( code_int_of_integer @ Xa3 ) ) ) ).
% drop_bit_integer.rep_eq
thf(fact_4572_mask__integer_Orep__eq,axiom,
! [X: nat] :
( ( code_int_of_integer @ ( bit_se2239418461657761734s_mask @ code_integer @ X ) )
= ( bit_se2239418461657761734s_mask @ int @ X ) ) ).
% mask_integer.rep_eq
thf(fact_4573_unset__bit__integer_Orep__eq,axiom,
! [X: nat,Xa3: code_integer] :
( ( code_int_of_integer @ ( bit_se2638667681897837118et_bit @ code_integer @ X @ Xa3 ) )
= ( bit_se2638667681897837118et_bit @ int @ X @ ( code_int_of_integer @ Xa3 ) ) ) ).
% unset_bit_integer.rep_eq
thf(fact_4574_set__bit__integer_Orep__eq,axiom,
! [X: nat,Xa3: code_integer] :
( ( code_int_of_integer @ ( bit_se5668285175392031749et_bit @ code_integer @ X @ Xa3 ) )
= ( bit_se5668285175392031749et_bit @ int @ X @ ( code_int_of_integer @ Xa3 ) ) ) ).
% set_bit_integer.rep_eq
thf(fact_4575_flip__bit__integer_Orep__eq,axiom,
! [X: nat,Xa3: code_integer] :
( ( code_int_of_integer @ ( bit_se8732182000553998342ip_bit @ code_integer @ X @ Xa3 ) )
= ( bit_se8732182000553998342ip_bit @ int @ X @ ( code_int_of_integer @ Xa3 ) ) ) ).
% flip_bit_integer.rep_eq
thf(fact_4576_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_4577_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_4578_num__of__integer_Orep__eq,axiom,
( code_num_of_integer
= ( ^ [X2: code_integer] : ( num_of_nat @ ( nat2 @ ( code_int_of_integer @ X2 ) ) ) ) ) ).
% num_of_integer.rep_eq
thf(fact_4579_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_4580_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,Y6: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( plus_plus @ int @ ( times_times @ int @ ( product_fst @ int @ int @ X2 ) @ ( product_snd @ int @ int @ Y6 ) ) @ ( times_times @ int @ ( product_fst @ int @ int @ Y6 ) @ ( product_snd @ int @ int @ X2 ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ X2 ) @ ( product_snd @ int @ int @ Y6 ) ) ) ) ) ).
% plus_rat_def
thf(fact_4581_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,Y6: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( times_times @ int @ ( product_fst @ int @ int @ X2 ) @ ( product_fst @ int @ int @ Y6 ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ X2 ) @ ( product_snd @ int @ int @ Y6 ) ) ) ) ) ).
% times_rat_def
thf(fact_4582_transpose__rectangle,axiom,
! [A: $tType,Xs: list @ ( list @ A ),N: nat] :
( ( ( Xs
= ( nil @ ( list @ A ) ) )
=> ( N
= ( zero_zero @ nat ) ) )
=> ( ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ ( list @ A ) ) @ Xs ) )
=> ( ( size_size @ ( list @ A ) @ ( nth @ ( list @ A ) @ Xs @ I4 ) )
= N ) )
=> ( ( transpose @ A @ Xs )
= ( map @ nat @ ( list @ A )
@ ^ [I5: nat] :
( map @ nat @ A
@ ^ [J3: nat] : ( nth @ A @ ( nth @ ( list @ A ) @ Xs @ J3 ) @ I5 )
@ ( upt @ ( zero_zero @ nat ) @ ( size_size @ ( list @ ( list @ A ) ) @ Xs ) ) )
@ ( upt @ ( zero_zero @ nat ) @ N ) ) ) ) ) ).
% transpose_rectangle
thf(fact_4583_sorted__wrt__less__sum__mono__lowerbound,axiom,
! [B: $tType] :
( ( ordere6911136660526730532id_add @ B )
=> ! [F: nat > B,Ns: list @ nat] :
( ! [X3: nat,Y5: nat] :
( ( ord_less_eq @ nat @ X3 @ Y5 )
=> ( ord_less_eq @ B @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ( 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_4584_strict__sorted__imp__sorted,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( sorted_wrt @ A @ ( ord_less @ A ) @ Xs )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs ) ) ) ).
% strict_sorted_imp_sorted
thf(fact_4585_sorted2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y3: A,Zs: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( cons @ A @ X @ ( cons @ A @ Y3 @ Zs ) ) )
= ( ( ord_less_eq @ A @ X @ Y3 )
& ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( cons @ A @ Y3 @ Zs ) ) ) ) ) ).
% sorted2
thf(fact_4586_sorted0,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( nil @ A ) ) ) ).
% sorted0
thf(fact_4587_strict__sorted__simps_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( sorted_wrt @ A @ ( ord_less @ A ) @ ( nil @ A ) ) ) ).
% strict_sorted_simps(1)
thf(fact_4588_strict__sorted__equal,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,Ys2: list @ A] :
( ( sorted_wrt @ A @ ( ord_less @ A ) @ Xs )
=> ( ( sorted_wrt @ A @ ( ord_less @ A ) @ Ys2 )
=> ( ( ( set2 @ A @ Ys2 )
= ( set2 @ A @ Xs ) )
=> ( Ys2 = Xs ) ) ) ) ) ).
% strict_sorted_equal
thf(fact_4589_sorted__take,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,N: nat] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( take @ A @ N @ Xs ) ) ) ) ).
% sorted_take
thf(fact_4590_sorted__replicate,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [N: nat,X: A] : ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( replicate @ A @ N @ X ) ) ) ).
% sorted_replicate
thf(fact_4591_sorted__drop,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,N: nat] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( drop @ A @ N @ Xs ) ) ) ) ).
% sorted_drop
thf(fact_4592_sorted__remdups__adj,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( remdups_adj @ A @ Xs ) ) ) ) ).
% sorted_remdups_adj
thf(fact_4593_sorted__wrt__upt,axiom,
! [M: nat,N: nat] : ( sorted_wrt @ nat @ ( ord_less @ nat ) @ ( upt @ M @ N ) ) ).
% sorted_wrt_upt
thf(fact_4594_sorted__upt,axiom,
! [M: nat,N: nat] : ( sorted_wrt @ nat @ ( ord_less_eq @ nat ) @ ( upt @ M @ N ) ) ).
% sorted_upt
thf(fact_4595_sorted__list__of__set_Osorted__sorted__key__list__of__set,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A] : ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( linord4507533701916653071of_set @ A @ A6 ) ) ) ).
% sorted_list_of_set.sorted_sorted_key_list_of_set
thf(fact_4596_sorted__list__of__set_Ostrict__sorted__key__list__of__set,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A] : ( sorted_wrt @ A @ ( ord_less @ A ) @ ( linord4507533701916653071of_set @ A @ A6 ) ) ) ).
% sorted_list_of_set.strict_sorted_key_list_of_set
thf(fact_4597_sorted__map,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F: B > A,Xs: list @ B] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F @ Xs ) )
= ( sorted_wrt @ B
@ ^ [X2: B,Y6: B] : ( ord_less_eq @ A @ ( F @ X2 ) @ ( F @ Y6 ) )
@ Xs ) ) ) ).
% sorted_map
thf(fact_4598_sorted1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A] : ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( cons @ A @ X @ ( nil @ A ) ) ) ) ).
% sorted1
thf(fact_4599_sorted__simps_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Ys2: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( cons @ A @ X @ Ys2 ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ Ys2 ) )
=> ( ord_less_eq @ A @ X @ X2 ) )
& ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Ys2 ) ) ) ) ).
% sorted_simps(2)
thf(fact_4600_strict__sorted__simps_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Ys2: list @ A] :
( ( sorted_wrt @ A @ ( ord_less @ A ) @ ( cons @ A @ X @ Ys2 ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ Ys2 ) )
=> ( ord_less @ A @ X @ X2 ) )
& ( sorted_wrt @ A @ ( ord_less @ A ) @ Ys2 ) ) ) ) ).
% strict_sorted_simps(2)
thf(fact_4601_sorted__append,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,Ys2: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( append @ A @ Xs @ Ys2 ) )
= ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
& ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Ys2 )
& ! [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
=> ! [Y6: A] :
( ( member @ A @ Y6 @ ( set2 @ A @ Ys2 ) )
=> ( ord_less_eq @ A @ X2 @ Y6 ) ) ) ) ) ) ).
% sorted_append
thf(fact_4602_strict__sorted__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: list @ A] :
( ( sorted_wrt @ A @ ( ord_less @ A ) @ L )
= ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ L )
& ( distinct @ A @ L ) ) ) ) ).
% strict_sorted_iff
thf(fact_4603_sorted__distinct__set__unique,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,Ys2: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( ( distinct @ A @ Xs )
=> ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Ys2 )
=> ( ( distinct @ A @ Ys2 )
=> ( ( ( set2 @ A @ Xs )
= ( set2 @ A @ Ys2 ) )
=> ( Xs = Ys2 ) ) ) ) ) ) ) ).
% sorted_distinct_set_unique
thf(fact_4604_sorted__wrt01,axiom,
! [A: $tType,Xs: list @ A,P: A > A > $o] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) )
=> ( sorted_wrt @ A @ P @ Xs ) ) ).
% sorted_wrt01
thf(fact_4605_sorted__wrt__iff__nth__less,axiom,
! [A: $tType] :
( ( sorted_wrt @ A )
= ( ^ [P4: A > A > $o,Xs3: list @ A] :
! [I5: nat,J3: nat] :
( ( ord_less @ nat @ I5 @ J3 )
=> ( ( ord_less @ nat @ J3 @ ( size_size @ ( list @ A ) @ Xs3 ) )
=> ( P4 @ ( nth @ A @ Xs3 @ I5 ) @ ( nth @ A @ Xs3 @ J3 ) ) ) ) ) ) ).
% sorted_wrt_iff_nth_less
thf(fact_4606_sorted__wrt__nth__less,axiom,
! [A: $tType,P: A > A > $o,Xs: list @ A,I: nat,J: nat] :
( ( sorted_wrt @ A @ P @ Xs )
=> ( ( ord_less @ nat @ I @ J )
=> ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( P @ ( nth @ A @ Xs @ I ) @ ( nth @ A @ Xs @ J ) ) ) ) ) ).
% sorted_wrt_nth_less
thf(fact_4607_sorted__butlast,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( butlast @ A @ Xs ) ) ) ) ) ).
% sorted_butlast
thf(fact_4608_sorted__enumerate,axiom,
! [A: $tType,N: nat,Xs: list @ A] : ( sorted_wrt @ nat @ ( ord_less_eq @ nat ) @ ( map @ ( product_prod @ nat @ A ) @ nat @ ( product_fst @ nat @ A ) @ ( enumerate @ A @ N @ Xs ) ) ) ).
% sorted_enumerate
thf(fact_4609_sorted01,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs ) ) ) ).
% sorted01
thf(fact_4610_finite__sorted__distinct__unique,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A] :
( ( finite_finite2 @ A @ A6 )
=> ? [X3: list @ A] :
( ( ( set2 @ A @ X3 )
= A6 )
& ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ X3 )
& ( distinct @ A @ X3 )
& ! [Y4: list @ A] :
( ( ( ( set2 @ A @ Y4 )
= A6 )
& ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Y4 )
& ( distinct @ A @ Y4 ) )
=> ( Y4 = X3 ) ) ) ) ) ).
% finite_sorted_distinct_unique
thf(fact_4611_sorted__iff__nth__mono__less,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
= ( ! [I5: nat,J3: nat] :
( ( ord_less @ nat @ I5 @ J3 )
=> ( ( ord_less @ nat @ J3 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ord_less_eq @ A @ ( nth @ A @ Xs @ I5 ) @ ( nth @ A @ Xs @ J3 ) ) ) ) ) ) ) ).
% sorted_iff_nth_mono_less
thf(fact_4612_sorted__list__of__set_Oidem__if__sorted__distinct,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( ( distinct @ A @ Xs )
=> ( ( linord4507533701916653071of_set @ A @ ( set2 @ A @ Xs ) )
= Xs ) ) ) ) ).
% sorted_list_of_set.idem_if_sorted_distinct
thf(fact_4613_sorted__list__of__set_Ofinite__set__strict__sorted,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A] :
( ( finite_finite2 @ A @ A6 )
=> ~ ! [L2: list @ A] :
( ( sorted_wrt @ A @ ( ord_less @ A ) @ L2 )
=> ( ( ( set2 @ A @ L2 )
= A6 )
=> ( ( size_size @ ( list @ A ) @ L2 )
!= ( finite_card @ A @ A6 ) ) ) ) ) ) ).
% sorted_list_of_set.finite_set_strict_sorted
thf(fact_4614_sorted__iff__nth__Suc,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
= ( ! [I5: nat] :
( ( ord_less @ nat @ ( suc @ I5 ) @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ord_less_eq @ A @ ( nth @ A @ Xs @ I5 ) @ ( nth @ A @ Xs @ ( suc @ I5 ) ) ) ) ) ) ) ).
% sorted_iff_nth_Suc
thf(fact_4615_sorted__nth__mono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,I: nat,J: nat] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( ( ord_less_eq @ nat @ I @ J )
=> ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ord_less_eq @ A @ ( nth @ A @ Xs @ I ) @ ( nth @ A @ Xs @ J ) ) ) ) ) ) ).
% sorted_nth_mono
thf(fact_4616_sorted__iff__nth__mono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
= ( ! [I5: nat,J3: nat] :
( ( ord_less_eq @ nat @ I5 @ J3 )
=> ( ( ord_less @ nat @ J3 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ord_less_eq @ A @ ( nth @ A @ Xs @ I5 ) @ ( nth @ A @ Xs @ J3 ) ) ) ) ) ) ) ).
% sorted_iff_nth_mono
thf(fact_4617_sorted__wrt__less__idx,axiom,
! [Ns: list @ nat,I: nat] :
( ( sorted_wrt @ nat @ ( ord_less @ nat ) @ Ns )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ nat ) @ Ns ) )
=> ( ord_less_eq @ nat @ I @ ( nth @ nat @ Ns @ I ) ) ) ) ).
% sorted_wrt_less_idx
thf(fact_4618_sorted__list__of__set_Osorted__key__list__of__set__unique,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A,L: list @ A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( ( sorted_wrt @ A @ ( ord_less @ A ) @ L )
& ( ( set2 @ A @ L )
= A6 )
& ( ( size_size @ ( list @ A ) @ L )
= ( finite_card @ A @ A6 ) ) )
= ( ( linord4507533701916653071of_set @ A @ A6 )
= L ) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_unique
thf(fact_4619_length__transpose__sorted,axiom,
! [A: $tType,Xs: list @ ( list @ A )] :
( ( sorted_wrt @ nat @ ( ord_less_eq @ nat ) @ ( rev @ nat @ ( map @ ( list @ A ) @ nat @ ( size_size @ ( list @ A ) ) @ Xs ) ) )
=> ( ( ( Xs
= ( nil @ ( list @ A ) ) )
=> ( ( size_size @ ( list @ ( list @ A ) ) @ ( transpose @ A @ Xs ) )
= ( zero_zero @ nat ) ) )
& ( ( Xs
!= ( nil @ ( list @ A ) ) )
=> ( ( size_size @ ( list @ ( list @ A ) ) @ ( transpose @ A @ Xs ) )
= ( size_size @ ( list @ A ) @ ( nth @ ( list @ A ) @ Xs @ ( zero_zero @ nat ) ) ) ) ) ) ) ).
% length_transpose_sorted
thf(fact_4620_sum__diff1_H__aux,axiom,
! [B: $tType,A: $tType] :
( ( ab_group_add @ B )
=> ! [F4: set @ A,I6: set @ A,F: A > B,I: A] :
( ( finite_finite2 @ A @ F4 )
=> ( ( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [I5: A] :
( ( member @ A @ I5 @ I6 )
& ( ( F @ I5 )
!= ( zero_zero @ B ) ) ) )
@ F4 )
=> ( ( ( member @ A @ I @ I6 )
=> ( ( groups1027152243600224163dd_sum @ A @ B @ F @ ( minus_minus @ ( set @ A ) @ I6 @ ( insert @ A @ I @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( minus_minus @ B @ ( groups1027152243600224163dd_sum @ A @ B @ F @ I6 ) @ ( F @ I ) ) ) )
& ( ~ ( member @ A @ I @ I6 )
=> ( ( groups1027152243600224163dd_sum @ A @ B @ F @ ( minus_minus @ ( set @ A ) @ I6 @ ( insert @ A @ I @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( groups1027152243600224163dd_sum @ A @ B @ F @ I6 ) ) ) ) ) ) ) ).
% sum_diff1'_aux
thf(fact_4621_finite__mono__remains__stable__implies__strict__prefix,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F: nat > A] :
( ( finite_finite2 @ A @ ( image @ nat @ A @ F @ ( top_top @ ( set @ nat ) ) ) )
=> ( ( order_mono @ nat @ A @ F )
=> ( ! [N4: nat] :
( ( ( F @ N4 )
= ( F @ ( suc @ N4 ) ) )
=> ( ( F @ ( suc @ N4 ) )
= ( F @ ( suc @ ( suc @ N4 ) ) ) ) )
=> ? [N8: nat] :
( ! [N5: nat] :
( ( ord_less_eq @ nat @ N5 @ N8 )
=> ! [M3: nat] :
( ( ord_less_eq @ nat @ M3 @ N8 )
=> ( ( ord_less @ nat @ M3 @ N5 )
=> ( ord_less @ A @ ( F @ M3 ) @ ( F @ N5 ) ) ) ) )
& ! [N5: nat] :
( ( ord_less_eq @ nat @ N8 @ N5 )
=> ( ( F @ N8 )
= ( F @ N5 ) ) ) ) ) ) ) ) ).
% finite_mono_remains_stable_implies_strict_prefix
thf(fact_4622_top__apply,axiom,
! [C: $tType,D: $tType] :
( ( top @ C )
=> ( ( top_top @ ( D > C ) )
= ( ^ [X2: D] : ( top_top @ C ) ) ) ) ).
% top_apply
thf(fact_4623_inf__top__left,axiom,
! [A: $tType] :
( ( bounde4346867609351753570nf_top @ A )
=> ! [X: A] :
( ( inf_inf @ A @ ( top_top @ A ) @ X )
= X ) ) ).
% inf_top_left
thf(fact_4624_inf__top__right,axiom,
! [A: $tType] :
( ( bounde4346867609351753570nf_top @ A )
=> ! [X: A] :
( ( inf_inf @ A @ X @ ( top_top @ A ) )
= X ) ) ).
% inf_top_right
thf(fact_4625_inf__eq__top__iff,axiom,
! [A: $tType] :
( ( bounde4346867609351753570nf_top @ A )
=> ! [X: A,Y3: A] :
( ( ( inf_inf @ A @ X @ Y3 )
= ( top_top @ A ) )
= ( ( X
= ( top_top @ A ) )
& ( Y3
= ( top_top @ A ) ) ) ) ) ).
% inf_eq_top_iff
thf(fact_4626_top__eq__inf__iff,axiom,
! [A: $tType] :
( ( bounde4346867609351753570nf_top @ A )
=> ! [X: A,Y3: A] :
( ( ( top_top @ A )
= ( inf_inf @ A @ X @ Y3 ) )
= ( ( X
= ( top_top @ A ) )
& ( Y3
= ( top_top @ A ) ) ) ) ) ).
% top_eq_inf_iff
thf(fact_4627_inf__top_Oeq__neutr__iff,axiom,
! [A: $tType] :
( ( bounde4346867609351753570nf_top @ A )
=> ! [A3: A,B2: A] :
( ( ( inf_inf @ A @ A3 @ B2 )
= ( top_top @ A ) )
= ( ( A3
= ( top_top @ A ) )
& ( B2
= ( top_top @ A ) ) ) ) ) ).
% inf_top.eq_neutr_iff
thf(fact_4628_inf__top_Oleft__neutral,axiom,
! [A: $tType] :
( ( bounde4346867609351753570nf_top @ A )
=> ! [A3: A] :
( ( inf_inf @ A @ ( top_top @ A ) @ A3 )
= A3 ) ) ).
% inf_top.left_neutral
thf(fact_4629_inf__top_Oneutr__eq__iff,axiom,
! [A: $tType] :
( ( bounde4346867609351753570nf_top @ A )
=> ! [A3: A,B2: A] :
( ( ( top_top @ A )
= ( inf_inf @ A @ A3 @ B2 ) )
= ( ( A3
= ( top_top @ A ) )
& ( B2
= ( top_top @ A ) ) ) ) ) ).
% inf_top.neutr_eq_iff
thf(fact_4630_inf__top_Oright__neutral,axiom,
! [A: $tType] :
( ( bounde4346867609351753570nf_top @ A )
=> ! [A3: A] :
( ( inf_inf @ A @ A3 @ ( top_top @ A ) )
= A3 ) ) ).
% inf_top.right_neutral
thf(fact_4631_boolean__algebra_Odisj__one__right,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A] :
( ( sup_sup @ A @ X @ ( top_top @ A ) )
= ( top_top @ A ) ) ) ).
% boolean_algebra.disj_one_right
thf(fact_4632_boolean__algebra_Odisj__one__left,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A] :
( ( sup_sup @ A @ ( top_top @ A ) @ X )
= ( top_top @ A ) ) ) ).
% boolean_algebra.disj_one_left
thf(fact_4633_sup__top__right,axiom,
! [A: $tType] :
( ( bounded_lattice_top @ A )
=> ! [X: A] :
( ( sup_sup @ A @ X @ ( top_top @ A ) )
= ( top_top @ A ) ) ) ).
% sup_top_right
thf(fact_4634_sup__top__left,axiom,
! [A: $tType] :
( ( bounded_lattice_top @ A )
=> ! [X: A] :
( ( sup_sup @ A @ ( top_top @ A ) @ X )
= ( top_top @ A ) ) ) ).
% sup_top_left
thf(fact_4635_max__top,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [X: A] :
( ( ord_max @ A @ ( top_top @ A ) @ X )
= ( top_top @ A ) ) ) ).
% max_top
thf(fact_4636_max__top2,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [X: A] :
( ( ord_max @ A @ X @ ( top_top @ A ) )
= ( top_top @ A ) ) ) ).
% max_top2
thf(fact_4637_min__top2,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [X: A] :
( ( ord_min @ A @ X @ ( top_top @ A ) )
= X ) ) ).
% min_top2
thf(fact_4638_min__top,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [X: A] :
( ( ord_min @ A @ ( top_top @ A ) @ X )
= X ) ) ).
% min_top
thf(fact_4639_boolean__algebra_Ocompl__zero,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ( ( uminus_uminus @ A @ ( bot_bot @ A ) )
= ( top_top @ A ) ) ) ).
% boolean_algebra.compl_zero
thf(fact_4640_boolean__algebra_Ocompl__one,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ( ( uminus_uminus @ A @ ( top_top @ A ) )
= ( bot_bot @ A ) ) ) ).
% boolean_algebra.compl_one
thf(fact_4641_boolean__algebra_Odisj__cancel__right,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A] :
( ( sup_sup @ A @ X @ ( uminus_uminus @ A @ X ) )
= ( top_top @ A ) ) ) ).
% boolean_algebra.disj_cancel_right
thf(fact_4642_boolean__algebra_Odisj__cancel__left,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A] :
( ( sup_sup @ A @ ( uminus_uminus @ A @ X ) @ X )
= ( top_top @ A ) ) ) ).
% boolean_algebra.disj_cancel_left
thf(fact_4643_sup__compl__top__left2,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y3: A] :
( ( sup_sup @ A @ X @ ( sup_sup @ A @ ( uminus_uminus @ A @ X ) @ Y3 ) )
= ( top_top @ A ) ) ) ).
% sup_compl_top_left2
thf(fact_4644_sup__compl__top__left1,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y3: A] :
( ( sup_sup @ A @ ( uminus_uminus @ A @ X ) @ ( sup_sup @ A @ X @ Y3 ) )
= ( top_top @ A ) ) ) ).
% sup_compl_top_left1
thf(fact_4645_finite__compl,axiom,
! [A: $tType,A6: set @ A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( finite_finite2 @ A @ ( uminus_uminus @ ( set @ A ) @ A6 ) )
= ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ) ).
% finite_compl
thf(fact_4646_Gcd__UNIV,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ( ( gcd_Gcd @ A @ ( top_top @ ( set @ A ) ) )
= ( one_one @ A ) ) ) ).
% Gcd_UNIV
thf(fact_4647_sum_Oempty_H,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ! [P5: B > A] :
( ( groups1027152243600224163dd_sum @ B @ A @ P5 @ ( bot_bot @ ( set @ B ) ) )
= ( zero_zero @ A ) ) ) ).
% sum.empty'
thf(fact_4648_sum_Oinsert_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [I6: set @ B,P5: B > A,I: B] :
( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ I6 )
& ( ( P5 @ X2 )
!= ( zero_zero @ A ) ) ) ) )
=> ( ( ( member @ B @ I @ I6 )
=> ( ( groups1027152243600224163dd_sum @ B @ A @ P5 @ ( insert @ B @ I @ I6 ) )
= ( groups1027152243600224163dd_sum @ B @ A @ P5 @ I6 ) ) )
& ( ~ ( member @ B @ I @ I6 )
=> ( ( groups1027152243600224163dd_sum @ B @ A @ P5 @ ( insert @ B @ I @ I6 ) )
= ( plus_plus @ A @ ( P5 @ I ) @ ( groups1027152243600224163dd_sum @ B @ A @ P5 @ I6 ) ) ) ) ) ) ) ).
% sum.insert'
thf(fact_4649_boolean__algebra_Oconj__one__right,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A] :
( ( inf_inf @ A @ X @ ( top_top @ A ) )
= X ) ) ).
% boolean_algebra.conj_one_right
thf(fact_4650_top__greatest,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A3: A] : ( ord_less_eq @ A @ A3 @ ( top_top @ A ) ) ) ).
% top_greatest
thf(fact_4651_top_Oextremum__unique,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( top_top @ A ) @ A3 )
= ( A3
= ( top_top @ A ) ) ) ) ).
% top.extremum_unique
thf(fact_4652_top_Oextremum__uniqueI,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( top_top @ A ) @ A3 )
=> ( A3
= ( top_top @ A ) ) ) ) ).
% top.extremum_uniqueI
thf(fact_4653_top_Oextremum__strict,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A3: A] :
~ ( ord_less @ A @ ( top_top @ A ) @ A3 ) ) ).
% top.extremum_strict
thf(fact_4654_top_Onot__eq__extremum,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A3: A] :
( ( A3
!= ( top_top @ A ) )
= ( ord_less @ A @ A3 @ ( top_top @ A ) ) ) ) ).
% top.not_eq_extremum
thf(fact_4655_sum_Onon__neutral_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: B > A,I6: set @ B] :
( ( groups1027152243600224163dd_sum @ B @ A @ G
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ I6 )
& ( ( G @ X2 )
!= ( zero_zero @ A ) ) ) ) )
= ( groups1027152243600224163dd_sum @ B @ A @ G @ I6 ) ) ) ).
% sum.non_neutral'
thf(fact_4656_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_4657_sorted__upto,axiom,
! [M: int,N: int] : ( sorted_wrt @ int @ ( ord_less_eq @ int ) @ ( upto @ M @ N ) ) ).
% sorted_upto
thf(fact_4658_sorted__wrt__upto,axiom,
! [I: int,J: int] : ( sorted_wrt @ int @ ( ord_less @ int ) @ ( upto @ I @ J ) ) ).
% sorted_wrt_upto
thf(fact_4659_sup__cancel__left2,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,A3: A,B2: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ ( uminus_uminus @ A @ X ) @ A3 ) @ ( sup_sup @ A @ X @ B2 ) )
= ( top_top @ A ) ) ) ).
% sup_cancel_left2
thf(fact_4660_sup__cancel__left1,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,A3: A,B2: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ X @ A3 ) @ ( sup_sup @ A @ ( uminus_uminus @ A @ X ) @ B2 ) )
= ( top_top @ A ) ) ) ).
% sup_cancel_left1
thf(fact_4661_Compl__empty__eq,axiom,
! [A: $tType] :
( ( uminus_uminus @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% Compl_empty_eq
thf(fact_4662_Compl__UNIV__eq,axiom,
! [A: $tType] :
( ( uminus_uminus @ ( set @ A ) @ ( top_top @ ( set @ A ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Compl_UNIV_eq
thf(fact_4663_Compl__partition2,axiom,
! [A: $tType,A6: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A6 ) @ A6 )
= ( top_top @ ( set @ A ) ) ) ).
% Compl_partition2
thf(fact_4664_Compl__partition,axiom,
! [A: $tType,A6: set @ A] :
( ( sup_sup @ ( set @ A ) @ A6 @ ( uminus_uminus @ ( set @ A ) @ A6 ) )
= ( top_top @ ( set @ A ) ) ) ).
% Compl_partition
thf(fact_4665_Compl__eq__Diff__UNIV,axiom,
! [A: $tType] :
( ( uminus_uminus @ ( set @ A ) )
= ( minus_minus @ ( set @ A ) @ ( top_top @ ( set @ A ) ) ) ) ).
% Compl_eq_Diff_UNIV
thf(fact_4666_sup__shunt,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y3: A] :
( ( ( sup_sup @ A @ X @ Y3 )
= ( top_top @ A ) )
= ( ord_less_eq @ A @ ( uminus_uminus @ A @ X ) @ Y3 ) ) ) ).
% sup_shunt
thf(fact_4667_boolean__algebra_Ocomplement__unique,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [A3: A,X: A,Y3: A] :
( ( ( inf_inf @ A @ A3 @ X )
= ( bot_bot @ A ) )
=> ( ( ( sup_sup @ A @ A3 @ X )
= ( top_top @ A ) )
=> ( ( ( inf_inf @ A @ A3 @ Y3 )
= ( bot_bot @ A ) )
=> ( ( ( sup_sup @ A @ A3 @ Y3 )
= ( top_top @ A ) )
=> ( X = Y3 ) ) ) ) ) ) ).
% boolean_algebra.complement_unique
thf(fact_4668_surj__Compl__image__subset,axiom,
! [A: $tType,B: $tType,F: B > A,A6: set @ B] :
( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ ( image @ B @ A @ F @ A6 ) ) @ ( image @ B @ A @ F @ ( uminus_uminus @ ( set @ B ) @ A6 ) ) ) ) ).
% surj_Compl_image_subset
thf(fact_4669_sum_Omono__neutral__left_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S3: set @ B,T6: set @ B,G: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S3 @ T6 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T6 @ S3 ) )
=> ( ( G @ X3 )
= ( zero_zero @ A ) ) )
=> ( ( groups1027152243600224163dd_sum @ B @ A @ G @ S3 )
= ( groups1027152243600224163dd_sum @ B @ A @ G @ T6 ) ) ) ) ) ).
% sum.mono_neutral_left'
thf(fact_4670_sum_Omono__neutral__right_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S3: set @ B,T6: set @ B,G: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S3 @ T6 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T6 @ S3 ) )
=> ( ( G @ X3 )
= ( zero_zero @ A ) ) )
=> ( ( groups1027152243600224163dd_sum @ B @ A @ G @ T6 )
= ( groups1027152243600224163dd_sum @ B @ A @ G @ S3 ) ) ) ) ) ).
% sum.mono_neutral_right'
thf(fact_4671_sum_Omono__neutral__cong__left_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S3: set @ B,T6: set @ B,H2: B > A,G: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S3 @ T6 )
=> ( ! [I4: B] :
( ( member @ B @ I4 @ ( minus_minus @ ( set @ B ) @ T6 @ S3 ) )
=> ( ( H2 @ I4 )
= ( zero_zero @ A ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ S3 )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups1027152243600224163dd_sum @ B @ A @ G @ S3 )
= ( groups1027152243600224163dd_sum @ B @ A @ H2 @ T6 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_left'
thf(fact_4672_sum_Omono__neutral__cong__right_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S3: set @ B,T6: set @ B,G: B > A,H2: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S3 @ T6 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T6 @ S3 ) )
=> ( ( G @ X3 )
= ( zero_zero @ A ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ S3 )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups1027152243600224163dd_sum @ B @ A @ G @ T6 )
= ( groups1027152243600224163dd_sum @ B @ A @ H2 @ S3 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_right'
thf(fact_4673_boolean__algebra__class_Oboolean__algebra_Ocompl__unique,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y3: A] :
( ( ( inf_inf @ A @ X @ Y3 )
= ( bot_bot @ A ) )
=> ( ( ( sup_sup @ A @ X @ Y3 )
= ( top_top @ A ) )
=> ( ( uminus_uminus @ A @ X )
= Y3 ) ) ) ) ).
% boolean_algebra_class.boolean_algebra.compl_unique
thf(fact_4674_sum_Odistrib_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [I6: set @ B,G: B > A,H2: B > A] :
( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ I6 )
& ( ( G @ X2 )
!= ( zero_zero @ A ) ) ) ) )
=> ( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ I6 )
& ( ( H2 @ X2 )
!= ( zero_zero @ A ) ) ) ) )
=> ( ( groups1027152243600224163dd_sum @ B @ A
@ ^ [I5: B] : ( plus_plus @ A @ ( G @ I5 ) @ ( H2 @ I5 ) )
@ I6 )
= ( plus_plus @ A @ ( groups1027152243600224163dd_sum @ B @ A @ G @ I6 ) @ ( groups1027152243600224163dd_sum @ B @ A @ H2 @ I6 ) ) ) ) ) ) ).
% sum.distrib'
thf(fact_4675_sum_OG__def,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ( ( groups1027152243600224163dd_sum @ B @ A )
= ( ^ [P6: B > A,I8: set @ B] :
( if @ A
@ ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ I8 )
& ( ( P6 @ X2 )
!= ( zero_zero @ A ) ) ) ) )
@ ( groups7311177749621191930dd_sum @ B @ A @ P6
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ I8 )
& ( ( P6 @ X2 )
!= ( zero_zero @ A ) ) ) ) )
@ ( zero_zero @ A ) ) ) ) ) ).
% sum.G_def
thf(fact_4676_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_4677_sorted__transpose,axiom,
! [A: $tType,Xs: list @ ( list @ A )] : ( sorted_wrt @ nat @ ( ord_less_eq @ nat ) @ ( rev @ nat @ ( map @ ( list @ A ) @ nat @ ( size_size @ ( list @ A ) ) @ ( transpose @ A @ Xs ) ) ) ) ).
% sorted_transpose
thf(fact_4678_rev__nth,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ ( rev @ A @ Xs ) @ N )
= ( nth @ A @ Xs @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( suc @ N ) ) ) ) ) ).
% rev_nth
thf(fact_4679_rev__update,axiom,
! [A: $tType,K: nat,Xs: list @ A,Y3: A] :
( ( ord_less @ nat @ K @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( rev @ A @ ( list_update @ A @ Xs @ K @ Y3 ) )
= ( list_update @ A @ ( rev @ A @ Xs ) @ ( minus_minus @ nat @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ K ) @ ( one_one @ nat ) ) @ Y3 ) ) ) ).
% rev_update
thf(fact_4680_boolean__algebra_Oabstract__boolean__algebra__axioms,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ( boolea2506097494486148201lgebra @ A @ ( inf_inf @ A ) @ ( sup_sup @ A ) @ ( uminus_uminus @ A ) @ ( bot_bot @ A ) @ ( top_top @ A ) ) ) ).
% boolean_algebra.abstract_boolean_algebra_axioms
thf(fact_4681_card__range__greater__zero,axiom,
! [A: $tType,B: $tType,F: B > A] :
( ( finite_finite2 @ A @ ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) ) )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ ( finite_card @ A @ ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) ) ) ) ) ).
% card_range_greater_zero
thf(fact_4682_sorted__rev__iff__nth__Suc,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( rev @ A @ Xs ) )
= ( ! [I5: nat] :
( ( ord_less @ nat @ ( suc @ I5 ) @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ord_less_eq @ A @ ( nth @ A @ Xs @ ( suc @ I5 ) ) @ ( nth @ A @ Xs @ I5 ) ) ) ) ) ) ).
% sorted_rev_iff_nth_Suc
thf(fact_4683_sorted__rev__iff__nth__mono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( rev @ A @ Xs ) )
= ( ! [I5: nat,J3: nat] :
( ( ord_less_eq @ nat @ I5 @ J3 )
=> ( ( ord_less @ nat @ J3 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ord_less_eq @ A @ ( nth @ A @ Xs @ J3 ) @ ( nth @ A @ Xs @ I5 ) ) ) ) ) ) ) ).
% sorted_rev_iff_nth_mono
thf(fact_4684_sorted__rev__nth__mono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,I: nat,J: nat] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( rev @ A @ Xs ) )
=> ( ( ord_less_eq @ nat @ I @ J )
=> ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ord_less_eq @ A @ ( nth @ A @ Xs @ J ) @ ( nth @ A @ Xs @ I ) ) ) ) ) ) ).
% sorted_rev_nth_mono
thf(fact_4685_range__mod,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( image @ nat @ nat
@ ^ [M2: nat] : ( modulo_modulo @ nat @ M2 @ N )
@ ( top_top @ ( set @ nat ) ) )
= ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% range_mod
thf(fact_4686_sum__diff1_H,axiom,
! [B: $tType,A: $tType] :
( ( ab_group_add @ B )
=> ! [I6: set @ A,F: A > B,I: A] :
( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [I5: A] :
( ( member @ A @ I5 @ I6 )
& ( ( F @ I5 )
!= ( zero_zero @ B ) ) ) ) )
=> ( ( ( member @ A @ I @ I6 )
=> ( ( groups1027152243600224163dd_sum @ A @ B @ F @ ( minus_minus @ ( set @ A ) @ I6 @ ( insert @ A @ I @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( minus_minus @ B @ ( groups1027152243600224163dd_sum @ A @ B @ F @ I6 ) @ ( F @ I ) ) ) )
& ( ~ ( member @ A @ I @ I6 )
=> ( ( groups1027152243600224163dd_sum @ A @ B @ F @ ( minus_minus @ ( set @ A ) @ I6 @ ( insert @ A @ I @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( groups1027152243600224163dd_sum @ A @ B @ F @ I6 ) ) ) ) ) ) ).
% sum_diff1'
thf(fact_4687_nth__nth__transpose__sorted,axiom,
! [A: $tType,Xs: list @ ( list @ A ),I: nat,J: nat] :
( ( sorted_wrt @ nat @ ( ord_less_eq @ nat ) @ ( rev @ nat @ ( map @ ( list @ A ) @ nat @ ( size_size @ ( list @ A ) ) @ Xs ) ) )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ ( list @ A ) ) @ ( transpose @ A @ Xs ) ) )
=> ( ( ord_less @ nat @ J
@ ( size_size @ ( list @ ( list @ A ) )
@ ( filter @ ( list @ A )
@ ^ [Ys3: list @ A] : ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Ys3 ) )
@ Xs ) ) )
=> ( ( nth @ A @ ( nth @ ( list @ A ) @ ( transpose @ A @ Xs ) @ I ) @ J )
= ( nth @ A @ ( nth @ ( list @ A ) @ Xs @ J ) @ I ) ) ) ) ) ).
% nth_nth_transpose_sorted
thf(fact_4688_transpose__column,axiom,
! [A: $tType,Xs: list @ ( list @ A ),I: nat] :
( ( sorted_wrt @ nat @ ( ord_less_eq @ nat ) @ ( rev @ nat @ ( map @ ( list @ A ) @ nat @ ( size_size @ ( list @ A ) ) @ Xs ) ) )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ ( list @ A ) ) @ Xs ) )
=> ( ( map @ ( list @ A ) @ A
@ ^ [Ys3: list @ A] : ( nth @ A @ Ys3 @ I )
@ ( filter @ ( list @ A )
@ ^ [Ys3: list @ A] : ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Ys3 ) )
@ ( transpose @ A @ Xs ) ) )
= ( nth @ ( list @ A ) @ Xs @ I ) ) ) ) ).
% transpose_column
thf(fact_4689_transpose__column__length,axiom,
! [A: $tType,Xs: list @ ( list @ A ),I: nat] :
( ( sorted_wrt @ nat @ ( ord_less_eq @ nat ) @ ( rev @ nat @ ( map @ ( list @ A ) @ nat @ ( size_size @ ( list @ A ) ) @ Xs ) ) )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ ( list @ A ) ) @ Xs ) )
=> ( ( size_size @ ( list @ ( list @ A ) )
@ ( filter @ ( list @ A )
@ ^ [Ys3: list @ A] : ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Ys3 ) )
@ ( transpose @ A @ Xs ) ) )
= ( size_size @ ( list @ A ) @ ( nth @ ( list @ A ) @ Xs @ I ) ) ) ) ) ).
% transpose_column_length
thf(fact_4690_card__UNIV__unit,axiom,
( ( finite_card @ product_unit @ ( top_top @ ( set @ product_unit ) ) )
= ( one_one @ nat ) ) ).
% card_UNIV_unit
thf(fact_4691_infinite__UNIV__int,axiom,
~ ( finite_finite2 @ int @ ( top_top @ ( set @ int ) ) ) ).
% infinite_UNIV_int
thf(fact_4692_integer__of__int__cases,axiom,
! [X: code_integer] :
~ ! [Y5: int] :
( ( X
= ( code_integer_of_int @ Y5 ) )
=> ~ ( member @ int @ Y5 @ ( top_top @ ( set @ int ) ) ) ) ).
% integer_of_int_cases
thf(fact_4693_integer__of__int__induct,axiom,
! [P: code_integer > $o,X: code_integer] :
( ! [Y5: int] :
( ( member @ int @ Y5 @ ( top_top @ ( set @ int ) ) )
=> ( P @ ( code_integer_of_int @ Y5 ) ) )
=> ( P @ X ) ) ).
% integer_of_int_induct
thf(fact_4694_integer__of__int__inject,axiom,
! [X: int,Y3: int] :
( ( member @ int @ X @ ( top_top @ ( set @ int ) ) )
=> ( ( member @ int @ Y3 @ ( top_top @ ( set @ int ) ) )
=> ( ( ( code_integer_of_int @ X )
= ( code_integer_of_int @ Y3 ) )
= ( X = Y3 ) ) ) ) ).
% integer_of_int_inject
thf(fact_4695_int__of__integer,axiom,
! [X: code_integer] : ( member @ int @ ( code_int_of_integer @ X ) @ ( top_top @ ( set @ int ) ) ) ).
% int_of_integer
thf(fact_4696_int__of__integer__cases,axiom,
! [Y3: int] :
( ( member @ int @ Y3 @ ( top_top @ ( set @ int ) ) )
=> ~ ! [X3: code_integer] :
( Y3
!= ( code_int_of_integer @ X3 ) ) ) ).
% int_of_integer_cases
thf(fact_4697_int__of__integer__induct,axiom,
! [Y3: int,P: int > $o] :
( ( member @ int @ Y3 @ ( top_top @ ( set @ int ) ) )
=> ( ! [X3: code_integer] : ( P @ ( code_int_of_integer @ X3 ) )
=> ( P @ Y3 ) ) ) ).
% int_of_integer_induct
thf(fact_4698_length__filter__le,axiom,
! [A: $tType,P: A > $o,Xs: list @ A] : ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ ( filter @ A @ P @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ).
% length_filter_le
thf(fact_4699_sorted__same,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [G: ( list @ A ) > A,Xs: list @ A] :
( sorted_wrt @ A @ ( ord_less_eq @ A )
@ ( filter @ A
@ ^ [X2: A] :
( X2
= ( G @ Xs ) )
@ Xs ) ) ) ).
% sorted_same
thf(fact_4700_integer__of__int__inverse,axiom,
! [Y3: int] :
( ( member @ int @ Y3 @ ( top_top @ ( set @ int ) ) )
=> ( ( code_int_of_integer @ ( code_integer_of_int @ Y3 ) )
= Y3 ) ) ).
% integer_of_int_inverse
thf(fact_4701_length__filter__less,axiom,
! [A: $tType,X: A,Xs: list @ A,P: A > $o] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ~ ( P @ X )
=> ( ord_less @ nat @ ( size_size @ ( list @ A ) @ ( filter @ A @ P @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ).
% length_filter_less
thf(fact_4702_surj__list__encode,axiom,
( ( image @ ( list @ nat ) @ nat @ nat_list_encode @ ( top_top @ ( set @ ( list @ nat ) ) ) )
= ( top_top @ ( set @ nat ) ) ) ).
% surj_list_encode
thf(fact_4703_surj__prod__encode,axiom,
( ( image @ ( product_prod @ nat @ nat ) @ nat @ nat_prod_encode @ ( top_top @ ( set @ ( product_prod @ nat @ nat ) ) ) )
= ( top_top @ ( set @ nat ) ) ) ).
% surj_prod_encode
thf(fact_4704_sorted__filter,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F: B > A,Xs: list @ B,P: B > $o] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F @ Xs ) )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F @ ( filter @ B @ P @ Xs ) ) ) ) ) ).
% sorted_filter
thf(fact_4705_sorted__map__same,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F: B > A,G: ( list @ B ) > A,Xs: list @ B] :
( sorted_wrt @ A @ ( ord_less_eq @ A )
@ ( map @ B @ A @ F
@ ( filter @ B
@ ^ [X2: B] :
( ( F @ X2 )
= ( G @ Xs ) )
@ Xs ) ) ) ) ).
% sorted_map_same
thf(fact_4706_sum__list__map__filter_H,axiom,
! [A: $tType,B: $tType] :
( ( monoid_add @ A )
=> ! [F: B > A,P: B > $o,Xs: list @ B] :
( ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F @ ( filter @ B @ P @ Xs ) ) )
= ( groups8242544230860333062m_list @ A
@ ( map @ B @ A
@ ^ [X2: B] : ( if @ A @ ( P @ X2 ) @ ( F @ X2 ) @ ( zero_zero @ A ) )
@ Xs ) ) ) ) ).
% sum_list_map_filter'
thf(fact_4707_sum__list__filter__le__nat,axiom,
! [A: $tType,F: A > nat,P: A > $o,Xs: list @ A] : ( ord_less_eq @ nat @ ( groups8242544230860333062m_list @ nat @ ( map @ A @ nat @ F @ ( filter @ A @ P @ Xs ) ) ) @ ( groups8242544230860333062m_list @ nat @ ( map @ A @ nat @ F @ Xs ) ) ) ).
% sum_list_filter_le_nat
thf(fact_4708_sum__list__map__filter,axiom,
! [A: $tType,B: $tType] :
( ( monoid_add @ A )
=> ! [Xs: list @ B,P: B > $o,F: B > A] :
( ! [X3: B] :
( ( member @ B @ X3 @ ( set2 @ B @ Xs ) )
=> ( ~ ( P @ X3 )
=> ( ( F @ X3 )
= ( zero_zero @ A ) ) ) )
=> ( ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F @ ( filter @ B @ P @ Xs ) ) )
= ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F @ Xs ) ) ) ) ) ).
% sum_list_map_filter
thf(fact_4709_Ints__def,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( ( ring_1_Ints @ A )
= ( image @ int @ A @ ( ring_1_of_int @ A ) @ ( top_top @ ( set @ int ) ) ) ) ) ).
% Ints_def
thf(fact_4710_int__in__range__abs,axiom,
! [N: nat] : ( member @ int @ ( semiring_1_of_nat @ int @ N ) @ ( image @ int @ int @ ( abs_abs @ int ) @ ( top_top @ ( set @ int ) ) ) ) ).
% int_in_range_abs
thf(fact_4711_length__filter__conv__card,axiom,
! [A: $tType,P5: A > $o,Xs: list @ A] :
( ( size_size @ ( list @ A ) @ ( filter @ A @ P5 @ Xs ) )
= ( finite_card @ nat
@ ( collect @ nat
@ ^ [I5: nat] :
( ( ord_less @ nat @ I5 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( P5 @ ( nth @ A @ Xs @ I5 ) ) ) ) ) ) ).
% length_filter_conv_card
thf(fact_4712_nth__transpose,axiom,
! [A: $tType,I: nat,Xs: list @ ( list @ A )] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ ( list @ A ) ) @ ( transpose @ A @ Xs ) ) )
=> ( ( nth @ ( list @ A ) @ ( transpose @ A @ Xs ) @ I )
= ( map @ ( list @ A ) @ A
@ ^ [Xs3: list @ A] : ( nth @ A @ Xs3 @ I )
@ ( filter @ ( list @ A )
@ ^ [Ys3: list @ A] : ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Ys3 ) )
@ Xs ) ) ) ) ).
% nth_transpose
thf(fact_4713_transpose__max__length,axiom,
! [A: $tType,Xs: list @ ( list @ A )] :
( ( foldr @ ( list @ A ) @ nat
@ ^ [Xs3: list @ A] : ( ord_max @ nat @ ( size_size @ ( list @ A ) @ Xs3 ) )
@ ( transpose @ A @ Xs )
@ ( zero_zero @ nat ) )
= ( size_size @ ( list @ ( list @ A ) )
@ ( filter @ ( list @ A )
@ ^ [X2: list @ A] :
( X2
!= ( nil @ A ) )
@ Xs ) ) ) ).
% transpose_max_length
thf(fact_4714_transpose__aux__max,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Xss: list @ ( list @ B )] :
( ( ord_max @ nat @ ( suc @ ( size_size @ ( list @ A ) @ Xs ) )
@ ( foldr @ ( list @ B ) @ nat
@ ^ [Xs3: list @ B] : ( ord_max @ nat @ ( size_size @ ( list @ B ) @ Xs3 ) )
@ Xss
@ ( zero_zero @ nat ) ) )
= ( suc
@ ( ord_max @ nat @ ( size_size @ ( list @ A ) @ Xs )
@ ( foldr @ ( list @ B ) @ nat
@ ^ [X2: list @ B] : ( ord_max @ nat @ ( minus_minus @ nat @ ( size_size @ ( list @ B ) @ X2 ) @ ( suc @ ( zero_zero @ nat ) ) ) )
@ ( filter @ ( list @ B )
@ ^ [Ys3: list @ B] :
( Ys3
!= ( nil @ B ) )
@ Xss )
@ ( zero_zero @ nat ) ) ) ) ) ).
% transpose_aux_max
thf(fact_4715_transpose__transpose,axiom,
! [A: $tType,Xs: list @ ( list @ A )] :
( ( sorted_wrt @ nat @ ( ord_less_eq @ nat ) @ ( rev @ nat @ ( map @ ( list @ A ) @ nat @ ( size_size @ ( list @ A ) ) @ Xs ) ) )
=> ( ( transpose @ A @ ( transpose @ A @ Xs ) )
= ( takeWhile @ ( list @ A )
@ ^ [X2: list @ A] :
( X2
!= ( nil @ A ) )
@ Xs ) ) ) ).
% transpose_transpose
thf(fact_4716_length__takeWhile__le,axiom,
! [A: $tType,P: A > $o,Xs: list @ A] : ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ).
% length_takeWhile_le
thf(fact_4717_sorted__takeWhile,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,P: A > $o] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( takeWhile @ A @ P @ Xs ) ) ) ) ).
% sorted_takeWhile
thf(fact_4718_takeWhile__nth,axiom,
! [A: $tType,J: nat,P: A > $o,Xs: list @ A] :
( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P @ Xs ) ) )
=> ( ( nth @ A @ ( takeWhile @ A @ P @ Xs ) @ J )
= ( nth @ A @ Xs @ J ) ) ) ).
% takeWhile_nth
thf(fact_4719_nth__length__takeWhile,axiom,
! [A: $tType,P: A > $o,Xs: list @ A] :
( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) )
=> ~ ( P @ ( nth @ A @ Xs @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P @ Xs ) ) ) ) ) ).
% nth_length_takeWhile
thf(fact_4720_sum__list_Oeq__foldr,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ( ( groups8242544230860333062m_list @ A )
= ( ^ [Xs3: list @ A] : ( foldr @ A @ A @ ( plus_plus @ A ) @ Xs3 @ ( zero_zero @ A ) ) ) ) ) ).
% sum_list.eq_foldr
thf(fact_4721_length__takeWhile__less__P__nth,axiom,
! [A: $tType,J: nat,P: A > $o,Xs: list @ A] :
( ! [I4: nat] :
( ( ord_less @ nat @ I4 @ J )
=> ( P @ ( nth @ A @ Xs @ I4 ) ) )
=> ( ( ord_less_eq @ nat @ J @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ord_less_eq @ nat @ J @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P @ Xs ) ) ) ) ) ).
% length_takeWhile_less_P_nth
thf(fact_4722_takeWhile__eq__take__P__nth,axiom,
! [A: $tType,N: nat,Xs: list @ A,P: A > $o] :
( ! [I4: nat] :
( ( ord_less @ nat @ I4 @ N )
=> ( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( P @ ( nth @ A @ Xs @ I4 ) ) ) )
=> ( ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ~ ( P @ ( nth @ A @ Xs @ N ) ) )
=> ( ( takeWhile @ A @ P @ Xs )
= ( take @ A @ N @ Xs ) ) ) ) ).
% takeWhile_eq_take_P_nth
thf(fact_4723_nths__shift__lemma__Suc,axiom,
! [A: $tType,P: nat > $o,Xs: list @ A,Is: list @ nat] :
( ( map @ ( product_prod @ A @ nat ) @ A @ ( product_fst @ A @ nat )
@ ( filter @ ( product_prod @ A @ nat )
@ ^ [P6: product_prod @ A @ nat] : ( P @ ( suc @ ( product_snd @ A @ nat @ P6 ) ) )
@ ( zip @ A @ nat @ Xs @ Is ) ) )
= ( map @ ( product_prod @ A @ nat ) @ A @ ( product_fst @ A @ nat )
@ ( filter @ ( product_prod @ A @ nat )
@ ^ [P6: product_prod @ A @ nat] : ( P @ ( product_snd @ A @ nat @ P6 ) )
@ ( zip @ A @ nat @ Xs @ ( map @ nat @ nat @ suc @ Is ) ) ) ) ) ).
% nths_shift_lemma_Suc
thf(fact_4724_horner__sum__foldr,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_0 @ A )
=> ( ( groups4207007520872428315er_sum @ B @ A )
= ( ^ [F2: B > A,A5: A,Xs3: list @ B] :
( foldr @ B @ A
@ ^ [X2: B,B3: A] : ( plus_plus @ A @ ( F2 @ X2 ) @ ( times_times @ A @ A5 @ B3 ) )
@ Xs3
@ ( zero_zero @ A ) ) ) ) ) ).
% horner_sum_foldr
thf(fact_4725_nths__shift__lemma,axiom,
! [A: $tType,A6: set @ nat,Xs: list @ A,I: nat] :
( ( map @ ( product_prod @ A @ nat ) @ A @ ( product_fst @ A @ nat )
@ ( filter @ ( product_prod @ A @ nat )
@ ^ [P6: product_prod @ A @ nat] : ( member @ nat @ ( product_snd @ A @ nat @ P6 ) @ A6 )
@ ( zip @ A @ nat @ Xs @ ( upt @ I @ ( plus_plus @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) )
= ( map @ ( product_prod @ A @ nat ) @ A @ ( product_fst @ A @ nat )
@ ( filter @ ( product_prod @ A @ nat )
@ ^ [P6: product_prod @ A @ nat] : ( member @ nat @ ( plus_plus @ nat @ ( product_snd @ A @ nat @ P6 ) @ I ) @ A6 )
@ ( zip @ A @ nat @ Xs @ ( upt @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) ) ).
% nths_shift_lemma
thf(fact_4726_filter__equals__takeWhile__sorted__rev,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F: B > A,Xs: list @ B,T2: A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( rev @ A @ ( map @ B @ A @ F @ Xs ) ) )
=> ( ( filter @ B
@ ^ [X2: B] : ( ord_less @ A @ T2 @ ( F @ X2 ) )
@ Xs )
= ( takeWhile @ B
@ ^ [X2: B] : ( ord_less @ A @ T2 @ ( F @ X2 ) )
@ Xs ) ) ) ) ).
% filter_equals_takeWhile_sorted_rev
thf(fact_4727_length__transpose,axiom,
! [A: $tType,Xs: list @ ( list @ A )] :
( ( size_size @ ( list @ ( list @ A ) ) @ ( transpose @ A @ Xs ) )
= ( foldr @ ( list @ A ) @ nat
@ ^ [Xs3: list @ A] : ( ord_max @ nat @ ( size_size @ ( list @ A ) @ Xs3 ) )
@ Xs
@ ( zero_zero @ nat ) ) ) ).
% length_transpose
thf(fact_4728_foldr__max__sorted,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,Y3: A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( rev @ A @ Xs ) )
=> ( ( ( Xs
= ( nil @ A ) )
=> ( ( foldr @ A @ A @ ( ord_max @ A ) @ Xs @ Y3 )
= Y3 ) )
& ( ( Xs
!= ( nil @ A ) )
=> ( ( foldr @ A @ A @ ( ord_max @ A ) @ Xs @ Y3 )
= ( ord_max @ A @ ( nth @ A @ Xs @ ( zero_zero @ nat ) ) @ Y3 ) ) ) ) ) ) ).
% foldr_max_sorted
thf(fact_4729_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_4730_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_4731_divmod__integer__eq__cases,axiom,
( code_divmod_integer
= ( ^ [K3: code_integer,L3: 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 )
@ ( L3
= ( 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 ) @ L3
@ ( if @ ( product_prod @ code_integer @ code_integer )
@ ( ( sgn_sgn @ code_integer @ K3 )
= ( sgn_sgn @ code_integer @ L3 ) )
@ ( code_divmod_abs @ K3 @ L3 )
@ ( product_case_prod @ code_integer @ code_integer @ ( product_prod @ code_integer @ code_integer )
@ ^ [R5: code_integer,S6: code_integer] :
( if @ ( product_prod @ code_integer @ code_integer )
@ ( S6
= ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( uminus_uminus @ code_integer @ R5 ) @ ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( minus_minus @ code_integer @ ( uminus_uminus @ code_integer @ R5 ) @ ( one_one @ code_integer ) ) @ ( minus_minus @ code_integer @ ( abs_abs @ code_integer @ L3 ) @ S6 ) ) )
@ ( code_divmod_abs @ K3 @ L3 ) ) ) ) ) ) ) ) ).
% divmod_integer_eq_cases
thf(fact_4732_old_Onat_Osimps_I7_J,axiom,
! [T: $tType,F1: T,F22: nat > T > T,Nat: nat] :
( ( rec_nat @ T @ F1 @ F22 @ ( suc @ Nat ) )
= ( F22 @ Nat @ ( rec_nat @ T @ F1 @ F22 @ Nat ) ) ) ).
% old.nat.simps(7)
thf(fact_4733_old_Onat_Osimps_I6_J,axiom,
! [T: $tType,F1: T,F22: nat > T > T] :
( ( rec_nat @ T @ F1 @ F22 @ ( zero_zero @ nat ) )
= F1 ) ).
% old.nat.simps(6)
thf(fact_4734_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_4735_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_4736_comp__cong,axiom,
! [C: $tType,B: $tType,D: $tType,A: $tType,E4: $tType,F: B > A,G: C > B,X: C,F5: D > A,G4: E4 > D,X5: E4] :
( ( ( F @ ( G @ X ) )
= ( F5 @ ( G4 @ X5 ) ) )
=> ( ( comp @ B @ A @ C @ F @ G @ X )
= ( comp @ D @ A @ E4 @ F5 @ G4 @ X5 ) ) ) ).
% comp_cong
thf(fact_4737_nat_Ocase__distrib,axiom,
! [B: $tType,A: $tType,H2: A > B,F1: A,F22: nat > A,Nat: nat] :
( ( H2 @ ( case_nat @ A @ F1 @ F22 @ Nat ) )
= ( case_nat @ B @ ( H2 @ F1 )
@ ^ [X2: nat] : ( H2 @ ( F22 @ X2 ) )
@ Nat ) ) ).
% nat.case_distrib
thf(fact_4738_old_Onat_Osimps_I5_J,axiom,
! [A: $tType,F1: A,F22: nat > A,X22: nat] :
( ( case_nat @ A @ F1 @ F22 @ ( suc @ X22 ) )
= ( F22 @ X22 ) ) ).
% old.nat.simps(5)
thf(fact_4739_old_Onat_Osimps_I4_J,axiom,
! [A: $tType,F1: A,F22: nat > A] :
( ( case_nat @ A @ F1 @ F22 @ ( zero_zero @ nat ) )
= F1 ) ).
% old.nat.simps(4)
thf(fact_4740_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_4741_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_4742_rec__nat__Suc__imp,axiom,
! [A: $tType,F: nat > A,F1: A,F22: nat > A > A,N: nat] :
( ( F
= ( rec_nat @ A @ F1 @ F22 ) )
=> ( ( F @ ( suc @ N ) )
= ( F22 @ N @ ( F @ N ) ) ) ) ).
% rec_nat_Suc_imp
thf(fact_4743_rec__nat__0__imp,axiom,
! [A: $tType,F: nat > A,F1: A,F22: nat > A > A] :
( ( F
= ( rec_nat @ A @ F1 @ F22 ) )
=> ( ( F @ ( zero_zero @ nat ) )
= F1 ) ) ).
% rec_nat_0_imp
thf(fact_4744_sum__comp__morphism,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( ( comm_monoid_add @ B )
& ( comm_monoid_add @ A ) )
=> ! [H2: B > A,G: C > B,A6: set @ C] :
( ( ( H2 @ ( zero_zero @ B ) )
= ( zero_zero @ A ) )
=> ( ! [X3: B,Y5: B] :
( ( H2 @ ( plus_plus @ B @ X3 @ Y5 ) )
= ( plus_plus @ A @ ( H2 @ X3 ) @ ( H2 @ Y5 ) ) )
=> ( ( groups7311177749621191930dd_sum @ C @ A @ ( comp @ B @ A @ C @ H2 @ G ) @ A6 )
= ( H2 @ ( groups7311177749621191930dd_sum @ C @ B @ G @ A6 ) ) ) ) ) ) ).
% sum_comp_morphism
thf(fact_4745_uminus__sum__list__map,axiom,
! [A: $tType,B: $tType] :
( ( ab_group_add @ A )
=> ! [F: B > A,Xs: list @ B] :
( ( uminus_uminus @ A @ ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F @ Xs ) ) )
= ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ ( comp @ A @ A @ B @ ( uminus_uminus @ A ) @ F ) @ Xs ) ) ) ) ).
% uminus_sum_list_map
thf(fact_4746_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_4747_max__Suc1,axiom,
! [N: nat,M: nat] :
( ( ord_max @ nat @ ( suc @ N ) @ M )
= ( case_nat @ nat @ ( suc @ N )
@ ^ [M4: nat] : ( suc @ ( ord_max @ nat @ N @ M4 ) )
@ M ) ) ).
% max_Suc1
thf(fact_4748_max__Suc2,axiom,
! [M: nat,N: nat] :
( ( ord_max @ nat @ M @ ( suc @ N ) )
= ( case_nat @ nat @ ( suc @ N )
@ ^ [M4: nat] : ( suc @ ( ord_max @ nat @ M4 @ N ) )
@ M ) ) ).
% max_Suc2
thf(fact_4749_sum_Oreindex__nontrivial,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A6: set @ B,H2: B > C,G: C > A] :
( ( finite_finite2 @ B @ A6 )
=> ( ! [X3: B,Y5: B] :
( ( member @ B @ X3 @ A6 )
=> ( ( member @ B @ Y5 @ A6 )
=> ( ( X3 != Y5 )
=> ( ( ( H2 @ X3 )
= ( H2 @ Y5 ) )
=> ( ( G @ ( H2 @ X3 ) )
= ( zero_zero @ A ) ) ) ) ) )
=> ( ( groups7311177749621191930dd_sum @ C @ A @ G @ ( image @ B @ C @ H2 @ A6 ) )
= ( groups7311177749621191930dd_sum @ B @ A @ ( comp @ C @ A @ B @ G @ H2 ) @ A6 ) ) ) ) ) ).
% sum.reindex_nontrivial
thf(fact_4750_prod_Oreindex__nontrivial,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A6: set @ B,H2: B > C,G: C > A] :
( ( finite_finite2 @ B @ A6 )
=> ( ! [X3: B,Y5: B] :
( ( member @ B @ X3 @ A6 )
=> ( ( member @ B @ Y5 @ A6 )
=> ( ( X3 != Y5 )
=> ( ( ( H2 @ X3 )
= ( H2 @ Y5 ) )
=> ( ( G @ ( H2 @ X3 ) )
= ( one_one @ A ) ) ) ) ) )
=> ( ( groups7121269368397514597t_prod @ C @ A @ G @ ( image @ B @ C @ H2 @ A6 ) )
= ( groups7121269368397514597t_prod @ B @ A @ ( comp @ C @ A @ B @ G @ H2 ) @ A6 ) ) ) ) ) ).
% prod.reindex_nontrivial
thf(fact_4751_sum__image__le,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( ordere6911136660526730532id_add @ B )
=> ! [I6: set @ C,G: A > B,F: C > A] :
( ( finite_finite2 @ C @ I6 )
=> ( ! [I4: C] :
( ( member @ C @ I4 @ I6 )
=> ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( G @ ( F @ I4 ) ) ) )
=> ( ord_less_eq @ B @ ( groups7311177749621191930dd_sum @ A @ B @ G @ ( image @ C @ A @ F @ I6 ) ) @ ( groups7311177749621191930dd_sum @ C @ B @ ( comp @ A @ B @ C @ G @ F ) @ I6 ) ) ) ) ) ).
% sum_image_le
thf(fact_4752_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_4753_min__Suc1,axiom,
! [N: nat,M: nat] :
( ( ord_min @ nat @ ( suc @ N ) @ M )
= ( case_nat @ nat @ ( zero_zero @ nat )
@ ^ [M4: nat] : ( suc @ ( ord_min @ nat @ N @ M4 ) )
@ M ) ) ).
% min_Suc1
thf(fact_4754_min__Suc2,axiom,
! [M: nat,N: nat] :
( ( ord_min @ nat @ M @ ( suc @ N ) )
= ( case_nat @ nat @ ( zero_zero @ nat )
@ ^ [M4: nat] : ( suc @ ( ord_min @ nat @ M4 @ N ) )
@ M ) ) ).
% min_Suc2
thf(fact_4755_Nitpick_Ocase__nat__unfold,axiom,
! [A: $tType] :
( ( case_nat @ A )
= ( ^ [X2: A,F2: nat > A,N3: nat] :
( if @ A
@ ( N3
= ( zero_zero @ nat ) )
@ X2
@ ( F2 @ ( minus_minus @ nat @ N3 @ ( one_one @ nat ) ) ) ) ) ) ).
% Nitpick.case_nat_unfold
thf(fact_4756_nths__def,axiom,
! [A: $tType] :
( ( nths @ A )
= ( ^ [Xs3: list @ A,A7: set @ nat] :
( map @ ( product_prod @ A @ nat ) @ A @ ( product_fst @ A @ nat )
@ ( filter @ ( product_prod @ A @ nat )
@ ^ [P6: product_prod @ A @ nat] : ( member @ nat @ ( product_snd @ A @ nat @ P6 ) @ A7 )
@ ( zip @ A @ nat @ Xs3 @ ( upt @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs3 ) ) ) ) ) ) ) ).
% nths_def
thf(fact_4757_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_4758_nat_Osplit__sels_I2_J,axiom,
! [A: $tType,P: A > $o,F1: A,F22: nat > A,Nat: nat] :
( ( P @ ( case_nat @ A @ F1 @ F22 @ Nat ) )
= ( ~ ( ( ( Nat
= ( zero_zero @ nat ) )
& ~ ( P @ F1 ) )
| ( ( Nat
= ( suc @ ( pred @ Nat ) ) )
& ~ ( P @ ( F22 @ ( pred @ Nat ) ) ) ) ) ) ) ).
% nat.split_sels(2)
thf(fact_4759_nths__singleton,axiom,
! [A: $tType,A6: set @ nat,X: A] :
( ( ( member @ nat @ ( zero_zero @ nat ) @ A6 )
=> ( ( nths @ A @ ( cons @ A @ X @ ( nil @ A ) ) @ A6 )
= ( cons @ A @ X @ ( nil @ A ) ) ) )
& ( ~ ( member @ nat @ ( zero_zero @ nat ) @ A6 )
=> ( ( nths @ A @ ( cons @ A @ X @ ( nil @ A ) ) @ A6 )
= ( nil @ A ) ) ) ) ).
% nths_singleton
thf(fact_4760_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_4761_nths__all,axiom,
! [A: $tType,Xs: list @ A,I6: set @ nat] :
( ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( member @ nat @ I4 @ I6 ) )
=> ( ( nths @ A @ Xs @ I6 )
= Xs ) ) ).
% nths_all
thf(fact_4762_sorted__nths,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,I6: set @ nat] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( nths @ A @ Xs @ I6 ) ) ) ) ).
% sorted_nths
thf(fact_4763_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_4764_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_4765_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_4766_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_4767_lift__def,axiom,
! [A: $tType,B: $tType] :
( ( heap_Time_lift @ A @ B )
= ( comp @ B @ ( heap_Time_Heap @ B ) @ A @ ( heap_Time_return @ B ) ) ) ).
% lift_def
thf(fact_4768_drop__eq__nths,axiom,
! [A: $tType] :
( ( drop @ A )
= ( ^ [N3: nat,Xs3: list @ A] : ( nths @ A @ Xs3 @ ( collect @ nat @ ( ord_less_eq @ nat @ N3 ) ) ) ) ) ).
% drop_eq_nths
thf(fact_4769_euclidean__size__int__def,axiom,
( ( euclid6346220572633701492n_size @ int )
= ( comp @ int @ nat @ int @ nat2 @ ( abs_abs @ int ) ) ) ).
% euclidean_size_int_def
thf(fact_4770_pred__def,axiom,
( pred
= ( case_nat @ nat @ ( zero_zero @ nat )
@ ^ [X24: nat] : X24 ) ) ).
% pred_def
thf(fact_4771_length__nths,axiom,
! [A: $tType,Xs: list @ A,I6: set @ nat] :
( ( size_size @ ( list @ A ) @ ( nths @ A @ Xs @ I6 ) )
= ( finite_card @ nat
@ ( collect @ nat
@ ^ [I5: nat] :
( ( ord_less @ nat @ I5 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( member @ nat @ I5 @ I6 ) ) ) ) ) ).
% length_nths
thf(fact_4772_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_4773_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_4774_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_4775_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_4776_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_4777_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_4778_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
@ ^ [N3: nat] : ( minus_minus @ nat @ N3 @ ( 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_4779_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
@ ^ [N3: nat] : ( minus_minus @ nat @ N3 @ ( 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_4780_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
@ ^ [N3: nat] : ( minus_minus @ nat @ N3 @ ( 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_4781_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
@ ^ [N3: nat] : ( minus_minus @ nat @ N3 @ ( 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_4782_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_4783_filter__eq__nths,axiom,
! [A: $tType] :
( ( filter @ A )
= ( ^ [P4: A > $o,Xs3: list @ A] :
( nths @ A @ Xs3
@ ( collect @ nat
@ ^ [I5: nat] :
( ( ord_less @ nat @ I5 @ ( size_size @ ( list @ A ) @ Xs3 ) )
& ( P4 @ ( nth @ A @ Xs3 @ I5 ) ) ) ) ) ) ) ).
% filter_eq_nths
thf(fact_4784_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_4785_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_4786_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_4787_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_4788_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_4789_Gcd__int__def,axiom,
( ( gcd_Gcd @ int )
= ( ^ [K6: set @ int] : ( semiring_1_of_nat @ int @ ( gcd_Gcd @ nat @ ( image @ int @ nat @ ( comp @ int @ nat @ int @ nat2 @ ( abs_abs @ int ) ) @ K6 ) ) ) ) ) ).
% Gcd_int_def
thf(fact_4790_nths__Cons,axiom,
! [A: $tType,X: A,L: list @ A,A6: set @ nat] :
( ( nths @ A @ ( cons @ A @ X @ L ) @ A6 )
= ( append @ A @ ( if @ ( list @ A ) @ ( member @ nat @ ( zero_zero @ nat ) @ A6 ) @ ( cons @ A @ X @ ( nil @ A ) ) @ ( nil @ A ) )
@ ( nths @ A @ L
@ ( collect @ nat
@ ^ [J3: nat] : ( member @ nat @ ( suc @ J3 ) @ A6 ) ) ) ) ) ).
% nths_Cons
thf(fact_4791_nat_Osplit__sels_I1_J,axiom,
! [A: $tType,P: A > $o,F1: A,F22: nat > A,Nat: nat] :
( ( P @ ( case_nat @ A @ F1 @ F22 @ Nat ) )
= ( ( ( Nat
= ( zero_zero @ nat ) )
=> ( P @ F1 ) )
& ( ( Nat
= ( suc @ ( pred @ Nat ) ) )
=> ( P @ ( F22 @ ( pred @ Nat ) ) ) ) ) ) ).
% nat.split_sels(1)
thf(fact_4792_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_4793_finite__mono__strict__prefix__implies__finite__fixpoint,axiom,
! [A: $tType,F: nat > ( set @ A ),S3: set @ A] :
( ! [I4: nat] : ( ord_less_eq @ ( set @ A ) @ ( F @ I4 ) @ S3 )
=> ( ( finite_finite2 @ A @ S3 )
=> ( ? [N9: nat] :
( ! [N4: nat] :
( ( ord_less_eq @ nat @ N4 @ N9 )
=> ! [M6: nat] :
( ( ord_less_eq @ nat @ M6 @ N9 )
=> ( ( ord_less @ nat @ M6 @ N4 )
=> ( ord_less @ ( set @ A ) @ ( F @ M6 ) @ ( F @ N4 ) ) ) ) )
& ! [N4: nat] :
( ( ord_less_eq @ nat @ N9 @ N4 )
=> ( ( F @ N9 )
= ( F @ N4 ) ) ) )
=> ( ( F @ ( finite_card @ A @ S3 ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ F @ ( top_top @ ( set @ nat ) ) ) ) ) ) ) ) ).
% finite_mono_strict_prefix_implies_finite_fixpoint
thf(fact_4794_range__abs__Nats,axiom,
( ( image @ int @ int @ ( abs_abs @ int ) @ ( top_top @ ( set @ int ) ) )
= ( semiring_1_Nats @ int ) ) ).
% range_abs_Nats
thf(fact_4795_Sup__atLeastAtMost,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X: A,Y3: A] :
( ( ord_less_eq @ A @ X @ Y3 )
=> ( ( complete_Sup_Sup @ A @ ( set_or1337092689740270186AtMost @ A @ X @ Y3 ) )
= Y3 ) ) ) ).
% Sup_atLeastAtMost
thf(fact_4796_cSup__atLeastAtMost,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [Y3: A,X: A] :
( ( ord_less_eq @ A @ Y3 @ X )
=> ( ( complete_Sup_Sup @ A @ ( set_or1337092689740270186AtMost @ A @ Y3 @ X ) )
= X ) ) ) ).
% cSup_atLeastAtMost
thf(fact_4797_cSup__atLeastLessThan,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( dense_linorder @ A ) )
=> ! [Y3: A,X: A] :
( ( ord_less @ A @ Y3 @ X )
=> ( ( complete_Sup_Sup @ A @ ( set_or7035219750837199246ssThan @ A @ Y3 @ X ) )
= X ) ) ) ).
% cSup_atLeastLessThan
thf(fact_4798_Sup__atLeastLessThan,axiom,
! [A: $tType] :
( ( ( comple6319245703460814977attice @ A )
& ( dense_linorder @ A ) )
=> ! [X: A,Y3: A] :
( ( ord_less @ A @ X @ Y3 )
=> ( ( complete_Sup_Sup @ A @ ( set_or7035219750837199246ssThan @ A @ X @ Y3 ) )
= Y3 ) ) ) ).
% Sup_atLeastLessThan
thf(fact_4799_cSup__greaterThanAtMost,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [Y3: A,X: A] :
( ( ord_less @ A @ Y3 @ X )
=> ( ( complete_Sup_Sup @ A @ ( set_or3652927894154168847AtMost @ A @ Y3 @ X ) )
= X ) ) ) ).
% cSup_greaterThanAtMost
thf(fact_4800_Sup__greaterThanAtMost,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X: A,Y3: A] :
( ( ord_less @ A @ X @ Y3 )
=> ( ( complete_Sup_Sup @ A @ ( set_or3652927894154168847AtMost @ A @ X @ Y3 ) )
= Y3 ) ) ) ).
% Sup_greaterThanAtMost
thf(fact_4801_Sup__greaterThanLessThan,axiom,
! [A: $tType] :
( ( ( comple6319245703460814977attice @ A )
& ( dense_linorder @ A ) )
=> ! [X: A,Y3: A] :
( ( ord_less @ A @ X @ Y3 )
=> ( ( complete_Sup_Sup @ A @ ( set_or5935395276787703475ssThan @ A @ X @ Y3 ) )
= Y3 ) ) ) ).
% Sup_greaterThanLessThan
thf(fact_4802_cSup__greaterThanLessThan,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( dense_linorder @ A ) )
=> ! [Y3: A,X: A] :
( ( ord_less @ A @ Y3 @ X )
=> ( ( complete_Sup_Sup @ A @ ( set_or5935395276787703475ssThan @ A @ Y3 @ X ) )
= X ) ) ) ).
% cSup_greaterThanLessThan
thf(fact_4803_Nats__0,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( member @ A @ ( zero_zero @ A ) @ ( semiring_1_Nats @ A ) ) ) ).
% Nats_0
thf(fact_4804_cSup__eq,axiom,
! [A: $tType] :
( ( ( condit1219197933456340205attice @ A )
& ( no_bot @ A ) )
=> ! [X8: set @ A,A3: A] :
( ! [X3: A] :
( ( member @ A @ X3 @ X8 )
=> ( ord_less_eq @ A @ X3 @ A3 ) )
=> ( ! [Y5: A] :
( ! [X4: A] :
( ( member @ A @ X4 @ X8 )
=> ( ord_less_eq @ A @ X4 @ Y5 ) )
=> ( ord_less_eq @ A @ A3 @ Y5 ) )
=> ( ( complete_Sup_Sup @ A @ X8 )
= A3 ) ) ) ) ).
% cSup_eq
thf(fact_4805_cSup__eq__maximum,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [Z3: A,X8: set @ A] :
( ( member @ A @ Z3 @ X8 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ X8 )
=> ( ord_less_eq @ A @ X3 @ Z3 ) )
=> ( ( complete_Sup_Sup @ A @ X8 )
= Z3 ) ) ) ) ).
% cSup_eq_maximum
thf(fact_4806_Nats__1,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( member @ A @ ( one_one @ A ) @ ( semiring_1_Nats @ A ) ) ) ).
% Nats_1
thf(fact_4807_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_4808_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_4809_Nats__numeral,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [W2: num] : ( member @ A @ ( numeral_numeral @ A @ W2 ) @ ( semiring_1_Nats @ A ) ) ) ).
% Nats_numeral
thf(fact_4810_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_4811_Nats__induct,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [X: A,P: A > $o] :
( ( member @ A @ X @ ( semiring_1_Nats @ A ) )
=> ( ! [N4: nat] : ( P @ ( semiring_1_of_nat @ A @ N4 ) )
=> ( P @ X ) ) ) ) ).
% Nats_induct
thf(fact_4812_Nats__cases,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [X: A] :
( ( member @ A @ X @ ( semiring_1_Nats @ A ) )
=> ~ ! [N4: nat] :
( X
!= ( semiring_1_of_nat @ A @ N4 ) ) ) ) ).
% Nats_cases
thf(fact_4813_fact__in__Nats,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [N: nat] : ( member @ A @ ( semiring_char_0_fact @ A @ N ) @ ( semiring_1_Nats @ A ) ) ) ).
% fact_in_Nats
thf(fact_4814_cSup__eq__non__empty,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X8: set @ A,A3: A] :
( ( X8
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ X8 )
=> ( ord_less_eq @ A @ X3 @ A3 ) )
=> ( ! [Y5: A] :
( ! [X4: A] :
( ( member @ A @ X4 @ X8 )
=> ( ord_less_eq @ A @ X4 @ Y5 ) )
=> ( ord_less_eq @ A @ A3 @ Y5 ) )
=> ( ( complete_Sup_Sup @ A @ X8 )
= A3 ) ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_4815_cSup__least,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X8: set @ A,Z3: A] :
( ( X8
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ X8 )
=> ( ord_less_eq @ A @ X3 @ Z3 ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ X8 ) @ Z3 ) ) ) ) ).
% cSup_least
thf(fact_4816_le__cSup__finite,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X8: set @ A,X: A] :
( ( finite_finite2 @ A @ X8 )
=> ( ( member @ A @ X @ X8 )
=> ( ord_less_eq @ A @ X @ ( complete_Sup_Sup @ A @ X8 ) ) ) ) ) ).
% le_cSup_finite
thf(fact_4817_less__cSupD,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X8: set @ A,Z3: A] :
( ( X8
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ Z3 @ ( complete_Sup_Sup @ A @ X8 ) )
=> ? [X3: A] :
( ( member @ A @ X3 @ X8 )
& ( ord_less @ A @ Z3 @ X3 ) ) ) ) ) ).
% less_cSupD
thf(fact_4818_less__cSupE,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [Y3: A,X8: set @ A] :
( ( ord_less @ A @ Y3 @ ( complete_Sup_Sup @ A @ X8 ) )
=> ( ( X8
!= ( bot_bot @ ( set @ A ) ) )
=> ~ ! [X3: A] :
( ( member @ A @ X3 @ X8 )
=> ~ ( ord_less @ A @ Y3 @ X3 ) ) ) ) ) ).
% less_cSupE
thf(fact_4819_finite__imp__Sup__less,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X8: set @ A,X: A,A3: A] :
( ( finite_finite2 @ A @ X8 )
=> ( ( member @ A @ X @ X8 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ X8 )
=> ( ord_less @ A @ X3 @ A3 ) )
=> ( ord_less @ A @ ( complete_Sup_Sup @ A @ X8 ) @ A3 ) ) ) ) ) ).
% finite_imp_Sup_less
thf(fact_4820_inf__Sup,axiom,
! [A: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [A3: A,B7: set @ A] :
( ( inf_inf @ A @ A3 @ ( complete_Sup_Sup @ A @ B7 ) )
= ( complete_Sup_Sup @ A @ ( image @ A @ A @ ( inf_inf @ A @ A3 ) @ B7 ) ) ) ) ).
% inf_Sup
thf(fact_4821_Sup__inf__eq__bot__iff,axiom,
! [A: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [B7: set @ A,A3: A] :
( ( ( inf_inf @ A @ ( complete_Sup_Sup @ A @ B7 ) @ A3 )
= ( bot_bot @ A ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ B7 )
=> ( ( inf_inf @ A @ X2 @ A3 )
= ( bot_bot @ A ) ) ) ) ) ) ).
% Sup_inf_eq_bot_iff
thf(fact_4822_finite__subset__Union,axiom,
! [A: $tType,A6: set @ A,B12: set @ ( set @ A )] :
( ( finite_finite2 @ A @ A6 )
=> ( ( ord_less_eq @ ( set @ A ) @ A6 @ ( complete_Sup_Sup @ ( set @ A ) @ B12 ) )
=> ~ ! [F6: set @ ( set @ A )] :
( ( finite_finite2 @ ( set @ A ) @ F6 )
=> ( ( ord_less_eq @ ( set @ ( set @ A ) ) @ F6 @ B12 )
=> ~ ( ord_less_eq @ ( set @ A ) @ A6 @ ( complete_Sup_Sup @ ( set @ A ) @ F6 ) ) ) ) ) ) ).
% finite_subset_Union
thf(fact_4823_card__Union__le__sum__card,axiom,
! [A: $tType,U4: set @ ( set @ A )] : ( ord_less_eq @ nat @ ( finite_card @ A @ ( complete_Sup_Sup @ ( set @ A ) @ U4 ) ) @ ( groups7311177749621191930dd_sum @ ( set @ A ) @ nat @ ( finite_card @ A ) @ U4 ) ) ).
% card_Union_le_sum_card
thf(fact_4824_SUP__inf,axiom,
! [A: $tType,B: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [F: B > A,B7: set @ B,A3: A] :
( ( inf_inf @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ B7 ) ) @ A3 )
= ( complete_Sup_Sup @ A
@ ( image @ B @ A
@ ^ [B3: B] : ( inf_inf @ A @ ( F @ B3 ) @ A3 )
@ B7 ) ) ) ) ).
% SUP_inf
thf(fact_4825_Sup__inf,axiom,
! [A: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [B7: set @ A,A3: A] :
( ( inf_inf @ A @ ( complete_Sup_Sup @ A @ B7 ) @ A3 )
= ( complete_Sup_Sup @ A
@ ( image @ A @ A
@ ^ [B3: A] : ( inf_inf @ A @ B3 @ A3 )
@ B7 ) ) ) ) ).
% Sup_inf
thf(fact_4826_inf__SUP,axiom,
! [A: $tType,B: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [A3: A,F: B > A,B7: set @ B] :
( ( inf_inf @ A @ A3 @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ B7 ) ) )
= ( complete_Sup_Sup @ A
@ ( image @ B @ A
@ ^ [B3: B] : ( inf_inf @ A @ A3 @ ( F @ B3 ) )
@ B7 ) ) ) ) ).
% inf_SUP
thf(fact_4827_SUP__inf__distrib2,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [F: B > A,A6: set @ B,G: C > A,B7: set @ C] :
( ( inf_inf @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A6 ) ) @ ( complete_Sup_Sup @ A @ ( image @ C @ A @ G @ B7 ) ) )
= ( complete_Sup_Sup @ A
@ ( image @ B @ A
@ ^ [A5: B] :
( complete_Sup_Sup @ A
@ ( image @ C @ A
@ ^ [B3: C] : ( inf_inf @ A @ ( F @ A5 ) @ ( G @ B3 ) )
@ B7 ) )
@ A6 ) ) ) ) ).
% SUP_inf_distrib2
thf(fact_4828_cSUP__least,axiom,
! [B: $tType,A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A6: set @ B,F: B > A,M7: A] :
( ( A6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ A6 )
=> ( ord_less_eq @ A @ ( F @ X3 ) @ M7 ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A6 ) ) @ M7 ) ) ) ) ).
% cSUP_least
thf(fact_4829_finite__Sup__less__iff,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X8: set @ A,A3: A] :
( ( finite_finite2 @ A @ X8 )
=> ( ( X8
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ ( complete_Sup_Sup @ A @ X8 ) @ A3 )
= ( ! [X2: A] :
( ( member @ A @ X2 @ X8 )
=> ( ord_less @ A @ X2 @ A3 ) ) ) ) ) ) ) ).
% finite_Sup_less_iff
thf(fact_4830_cSup__abs__le,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( linordered_idom @ A ) )
=> ! [S3: set @ A,A3: A] :
( ( S3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ S3 )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ X3 ) @ A3 ) )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ ( complete_Sup_Sup @ A @ S3 ) ) @ A3 ) ) ) ) ).
% cSup_abs_le
thf(fact_4831_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_4832_sum_OUnion__comp,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [B7: set @ ( set @ B ),G: B > A] :
( ! [X3: set @ B] :
( ( member @ ( set @ B ) @ X3 @ B7 )
=> ( finite_finite2 @ B @ X3 ) )
=> ( ! [A14: set @ B] :
( ( member @ ( set @ B ) @ A14 @ B7 )
=> ! [A25: set @ B] :
( ( member @ ( set @ B ) @ A25 @ B7 )
=> ( ( A14 != A25 )
=> ! [X3: B] :
( ( member @ B @ X3 @ A14 )
=> ( ( member @ B @ X3 @ A25 )
=> ( ( G @ X3 )
= ( zero_zero @ A ) ) ) ) ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( complete_Sup_Sup @ ( set @ B ) @ B7 ) )
= ( comp @ ( ( set @ B ) > A ) @ ( ( set @ ( set @ B ) ) > A ) @ ( B > A ) @ ( groups7311177749621191930dd_sum @ ( set @ B ) @ A ) @ ( groups7311177749621191930dd_sum @ B @ A ) @ G @ B7 ) ) ) ) ) ).
% sum.Union_comp
thf(fact_4833_prod_OUnion__comp,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [B7: set @ ( set @ B ),G: B > A] :
( ! [X3: set @ B] :
( ( member @ ( set @ B ) @ X3 @ B7 )
=> ( finite_finite2 @ B @ X3 ) )
=> ( ! [A14: set @ B] :
( ( member @ ( set @ B ) @ A14 @ B7 )
=> ! [A25: set @ B] :
( ( member @ ( set @ B ) @ A25 @ B7 )
=> ( ( A14 != A25 )
=> ! [X3: B] :
( ( member @ B @ X3 @ A14 )
=> ( ( member @ B @ X3 @ A25 )
=> ( ( G @ X3 )
= ( one_one @ A ) ) ) ) ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( complete_Sup_Sup @ ( set @ B ) @ B7 ) )
= ( comp @ ( ( set @ B ) > A ) @ ( ( set @ ( set @ B ) ) > A ) @ ( B > A ) @ ( groups7121269368397514597t_prod @ ( set @ B ) @ A ) @ ( groups7121269368397514597t_prod @ B @ A ) @ G @ B7 ) ) ) ) ) ).
% prod.Union_comp
thf(fact_4834_card__Union__le__sum__card__weak,axiom,
! [A: $tType,U4: set @ ( set @ A )] :
( ! [X3: set @ A] :
( ( member @ ( set @ A ) @ X3 @ U4 )
=> ( finite_finite2 @ A @ X3 ) )
=> ( ord_less_eq @ nat @ ( finite_card @ A @ ( complete_Sup_Sup @ ( set @ A ) @ U4 ) ) @ ( groups7311177749621191930dd_sum @ ( set @ A ) @ nat @ ( finite_card @ A ) @ U4 ) ) ) ).
% card_Union_le_sum_card_weak
thf(fact_4835_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_4836_UN__UN__finite__eq,axiom,
! [A: $tType,A6: nat > ( set @ A )] :
( ( complete_Sup_Sup @ ( set @ A )
@ ( image @ nat @ ( set @ A )
@ ^ [N3: nat] : ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ A6 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N3 ) ) )
@ ( top_top @ ( set @ nat ) ) ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ A6 @ ( top_top @ ( set @ nat ) ) ) ) ) ).
% UN_UN_finite_eq
thf(fact_4837_cSup__asclose,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( linordered_idom @ A ) )
=> ! [S3: set @ A,L: A,E: A] :
( ( S3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ S3 )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X3 @ L ) ) @ E ) )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ ( complete_Sup_Sup @ A @ S3 ) @ L ) ) @ E ) ) ) ) ).
% cSup_asclose
thf(fact_4838_card__UN__le,axiom,
! [B: $tType,A: $tType,I6: set @ A,A6: A > ( set @ B )] :
( ( finite_finite2 @ A @ I6 )
=> ( ord_less_eq @ nat @ ( finite_card @ B @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ A6 @ I6 ) ) )
@ ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [I5: A] : ( finite_card @ B @ ( A6 @ I5 ) )
@ I6 ) ) ) ).
% card_UN_le
thf(fact_4839_UN__finite__subset,axiom,
! [A: $tType,A6: nat > ( set @ A ),C6: set @ A] :
( ! [N4: nat] : ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ A6 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N4 ) ) ) @ C6 )
=> ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ A6 @ ( top_top @ ( set @ nat ) ) ) ) @ C6 ) ) ).
% UN_finite_subset
thf(fact_4840_UN__finite2__eq,axiom,
! [A: $tType,A6: nat > ( set @ A ),B7: nat > ( set @ A ),K: nat] :
( ! [N4: nat] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ A6 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N4 ) ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ B7 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( plus_plus @ nat @ N4 @ K ) ) ) ) )
=> ( ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ A6 @ ( top_top @ ( set @ nat ) ) ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ B7 @ ( top_top @ ( set @ nat ) ) ) ) ) ) ).
% UN_finite2_eq
thf(fact_4841_Nats__def,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( semiring_1_Nats @ A )
= ( image @ nat @ A @ ( semiring_1_of_nat @ A ) @ ( top_top @ ( set @ nat ) ) ) ) ) ).
% Nats_def
thf(fact_4842_card__partition,axiom,
! [A: $tType,C6: set @ ( set @ A ),K: nat] :
( ( finite_finite2 @ ( set @ A ) @ C6 )
=> ( ( finite_finite2 @ A @ ( complete_Sup_Sup @ ( set @ A ) @ C6 ) )
=> ( ! [C3: set @ A] :
( ( member @ ( set @ A ) @ C3 @ C6 )
=> ( ( finite_card @ A @ C3 )
= K ) )
=> ( ! [C1: set @ A,C22: set @ A] :
( ( member @ ( set @ A ) @ C1 @ C6 )
=> ( ( member @ ( set @ A ) @ C22 @ C6 )
=> ( ( C1 != C22 )
=> ( ( inf_inf @ ( set @ A ) @ C1 @ C22 )
= ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( ( times_times @ nat @ K @ ( finite_card @ ( set @ A ) @ C6 ) )
= ( finite_card @ A @ ( complete_Sup_Sup @ ( set @ A ) @ C6 ) ) ) ) ) ) ) ).
% card_partition
thf(fact_4843_Nats__altdef2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( ( semiring_1_Nats @ A )
= ( collect @ A
@ ^ [N3: A] :
( ( member @ A @ N3 @ ( ring_1_Ints @ A ) )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ N3 ) ) ) ) ) ).
% Nats_altdef2
thf(fact_4844_UN__finite2__subset,axiom,
! [A: $tType,A6: nat > ( set @ A ),B7: nat > ( set @ A ),K: nat] :
( ! [N4: nat] : ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ A6 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N4 ) ) ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ B7 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( plus_plus @ nat @ N4 @ K ) ) ) ) )
=> ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ A6 @ ( top_top @ ( set @ nat ) ) ) ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ B7 @ ( top_top @ ( set @ nat ) ) ) ) ) ) ).
% UN_finite2_subset
thf(fact_4845_UN__le__eq__Un0,axiom,
! [A: $tType,M7: nat > ( set @ A ),N: nat] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ M7 @ ( set_ord_atMost @ nat @ N ) ) )
= ( sup_sup @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ M7 @ ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ N ) ) ) @ ( M7 @ ( zero_zero @ nat ) ) ) ) ).
% UN_le_eq_Un0
thf(fact_4846_SUP__eq__top__iff,axiom,
! [B: $tType,A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [F: B > A,A6: set @ B] :
( ( ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A6 ) )
= ( top_top @ A ) )
= ( ! [X2: A] :
( ( ord_less @ A @ X2 @ ( top_top @ A ) )
=> ? [Y6: B] :
( ( member @ B @ Y6 @ A6 )
& ( ord_less @ A @ X2 @ ( F @ Y6 ) ) ) ) ) ) ) ).
% SUP_eq_top_iff
thf(fact_4847_Sup__eq__top__iff,axiom,
! [A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [A6: set @ A] :
( ( ( complete_Sup_Sup @ A @ A6 )
= ( top_top @ A ) )
= ( ! [X2: A] :
( ( ord_less @ A @ X2 @ ( top_top @ A ) )
=> ? [Y6: A] :
( ( member @ A @ Y6 @ A6 )
& ( ord_less @ A @ X2 @ Y6 ) ) ) ) ) ) ).
% Sup_eq_top_iff
thf(fact_4848_mono__SUP,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ( comple6319245703460814977attice @ A )
& ( comple6319245703460814977attice @ B ) )
=> ! [F: A > B,A6: C > A,I6: set @ C] :
( ( order_mono @ A @ B @ F )
=> ( ord_less_eq @ B
@ ( complete_Sup_Sup @ B
@ ( image @ C @ B
@ ^ [X2: C] : ( F @ ( A6 @ X2 ) )
@ I6 ) )
@ ( F @ ( complete_Sup_Sup @ A @ ( image @ C @ A @ A6 @ I6 ) ) ) ) ) ) ).
% mono_SUP
thf(fact_4849_Sup__nat__empty,axiom,
( ( complete_Sup_Sup @ nat @ ( bot_bot @ ( set @ nat ) ) )
= ( zero_zero @ nat ) ) ).
% Sup_nat_empty
thf(fact_4850_Sup__upper2,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [U: A,A6: set @ A,V: A] :
( ( member @ A @ U @ A6 )
=> ( ( ord_less_eq @ A @ V @ U )
=> ( ord_less_eq @ A @ V @ ( complete_Sup_Sup @ A @ A6 ) ) ) ) ) ).
% Sup_upper2
thf(fact_4851_Sup__le__iff,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A6: set @ A,B2: A] :
( ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A6 ) @ B2 )
= ( ! [X2: A] :
( ( member @ A @ X2 @ A6 )
=> ( ord_less_eq @ A @ X2 @ B2 ) ) ) ) ) ).
% Sup_le_iff
thf(fact_4852_Sup__upper,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X: A,A6: set @ A] :
( ( member @ A @ X @ A6 )
=> ( ord_less_eq @ A @ X @ ( complete_Sup_Sup @ A @ A6 ) ) ) ) ).
% Sup_upper
thf(fact_4853_Sup__least,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A6: set @ A,Z3: A] :
( ! [X3: A] :
( ( member @ A @ X3 @ A6 )
=> ( ord_less_eq @ A @ X3 @ Z3 ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A6 ) @ Z3 ) ) ) ).
% Sup_least
thf(fact_4854_Sup__mono,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A6: set @ A,B7: set @ A] :
( ! [A8: A] :
( ( member @ A @ A8 @ A6 )
=> ? [X4: A] :
( ( member @ A @ X4 @ B7 )
& ( ord_less_eq @ A @ A8 @ X4 ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A6 ) @ ( complete_Sup_Sup @ A @ B7 ) ) ) ) ).
% Sup_mono
thf(fact_4855_Sup__eqI,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A6: set @ A,X: A] :
( ! [Y5: A] :
( ( member @ A @ Y5 @ A6 )
=> ( ord_less_eq @ A @ Y5 @ X ) )
=> ( ! [Y5: A] :
( ! [Z5: A] :
( ( member @ A @ Z5 @ A6 )
=> ( ord_less_eq @ A @ Z5 @ Y5 ) )
=> ( ord_less_eq @ A @ X @ Y5 ) )
=> ( ( complete_Sup_Sup @ A @ A6 )
= X ) ) ) ) ).
% Sup_eqI
thf(fact_4856_less__Sup__iff,axiom,
! [A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [A3: A,S3: set @ A] :
( ( ord_less @ A @ A3 @ ( complete_Sup_Sup @ A @ S3 ) )
= ( ? [X2: A] :
( ( member @ A @ X2 @ S3 )
& ( ord_less @ A @ A3 @ X2 ) ) ) ) ) ).
% less_Sup_iff
thf(fact_4857_le__Sup__iff,axiom,
! [A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [X: A,A6: set @ A] :
( ( ord_less_eq @ A @ X @ ( complete_Sup_Sup @ A @ A6 ) )
= ( ! [Y6: A] :
( ( ord_less @ A @ Y6 @ X )
=> ? [X2: A] :
( ( member @ A @ X2 @ A6 )
& ( ord_less @ A @ Y6 @ X2 ) ) ) ) ) ) ).
% le_Sup_iff
thf(fact_4858_SUP__eq,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A6: set @ B,B7: set @ C,F: B > A,G: C > A] :
( ! [I4: B] :
( ( member @ B @ I4 @ A6 )
=> ? [X4: C] :
( ( member @ C @ X4 @ B7 )
& ( ord_less_eq @ A @ ( F @ I4 ) @ ( G @ X4 ) ) ) )
=> ( ! [J2: C] :
( ( member @ C @ J2 @ B7 )
=> ? [X4: B] :
( ( member @ B @ X4 @ A6 )
& ( ord_less_eq @ A @ ( G @ J2 ) @ ( F @ X4 ) ) ) )
=> ( ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A6 ) )
= ( complete_Sup_Sup @ A @ ( image @ C @ A @ G @ B7 ) ) ) ) ) ) ).
% SUP_eq
thf(fact_4859_less__eq__Sup,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A6: set @ A,U: A] :
( ! [V4: A] :
( ( member @ A @ V4 @ A6 )
=> ( ord_less_eq @ A @ U @ V4 ) )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ A @ U @ ( complete_Sup_Sup @ A @ A6 ) ) ) ) ) ).
% less_eq_Sup
thf(fact_4860_Sup__subset__mono,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A6: set @ A,B7: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B7 )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A6 ) @ ( complete_Sup_Sup @ A @ B7 ) ) ) ) ).
% Sup_subset_mono
thf(fact_4861_SUP__upper2,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I: B,A6: set @ B,U: A,F: B > A] :
( ( member @ B @ I @ A6 )
=> ( ( ord_less_eq @ A @ U @ ( F @ I ) )
=> ( ord_less_eq @ A @ U @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A6 ) ) ) ) ) ) ).
% SUP_upper2
thf(fact_4862_SUP__le__iff,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F: B > A,A6: set @ B,U: A] :
( ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A6 ) ) @ U )
= ( ! [X2: B] :
( ( member @ B @ X2 @ A6 )
=> ( ord_less_eq @ A @ ( F @ X2 ) @ U ) ) ) ) ) ).
% SUP_le_iff
thf(fact_4863_SUP__upper,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I: B,A6: set @ B,F: B > A] :
( ( member @ B @ I @ A6 )
=> ( ord_less_eq @ A @ ( F @ I ) @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A6 ) ) ) ) ) ).
% SUP_upper
thf(fact_4864_SUP__mono_H,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F: B > A,G: B > A,A6: set @ B] :
( ! [X3: B] : ( ord_less_eq @ A @ ( F @ X3 ) @ ( G @ X3 ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A6 ) ) @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ G @ A6 ) ) ) ) ) ).
% SUP_mono'
thf(fact_4865_SUP__least,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A6: set @ B,F: B > A,U: A] :
( ! [I4: B] :
( ( member @ B @ I4 @ A6 )
=> ( ord_less_eq @ A @ ( F @ I4 ) @ U ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A6 ) ) @ U ) ) ) ).
% SUP_least
thf(fact_4866_SUP__mono,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A6: set @ B,B7: set @ C,F: B > A,G: C > A] :
( ! [N4: B] :
( ( member @ B @ N4 @ A6 )
=> ? [X4: C] :
( ( member @ C @ X4 @ B7 )
& ( ord_less_eq @ A @ ( F @ N4 ) @ ( G @ X4 ) ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A6 ) ) @ ( complete_Sup_Sup @ A @ ( image @ C @ A @ G @ B7 ) ) ) ) ) ).
% SUP_mono
thf(fact_4867_SUP__eqI,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A6: set @ B,F: B > A,X: A] :
( ! [I4: B] :
( ( member @ B @ I4 @ A6 )
=> ( ord_less_eq @ A @ ( F @ I4 ) @ X ) )
=> ( ! [Y5: A] :
( ! [I3: B] :
( ( member @ B @ I3 @ A6 )
=> ( ord_less_eq @ A @ ( F @ I3 ) @ Y5 ) )
=> ( ord_less_eq @ A @ X @ Y5 ) )
=> ( ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A6 ) )
= X ) ) ) ) ).
% SUP_eqI
thf(fact_4868_SUP__lessD,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F: B > A,A6: set @ B,Y3: A,I: B] :
( ( ord_less @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A6 ) ) @ Y3 )
=> ( ( member @ B @ I @ A6 )
=> ( ord_less @ A @ ( F @ I ) @ Y3 ) ) ) ) ).
% SUP_lessD
thf(fact_4869_less__SUP__iff,axiom,
! [A: $tType,B: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [A3: A,F: B > A,A6: set @ B] :
( ( ord_less @ A @ A3 @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A6 ) ) )
= ( ? [X2: B] :
( ( member @ B @ X2 @ A6 )
& ( ord_less @ A @ A3 @ ( F @ X2 ) ) ) ) ) ) ).
% less_SUP_iff
thf(fact_4870_le__SUP__iff,axiom,
! [B: $tType,A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [X: A,F: B > A,A6: set @ B] :
( ( ord_less_eq @ A @ X @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A6 ) ) )
= ( ! [Y6: A] :
( ( ord_less @ A @ Y6 @ X )
=> ? [X2: B] :
( ( member @ B @ X2 @ A6 )
& ( ord_less @ A @ Y6 @ ( F @ X2 ) ) ) ) ) ) ) ).
% le_SUP_iff
thf(fact_4871_SUP__eq__iff,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I6: set @ B,C2: A,F: B > A] :
( ( I6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [I4: B] :
( ( member @ B @ I4 @ I6 )
=> ( ord_less_eq @ A @ C2 @ ( F @ I4 ) ) )
=> ( ( ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ I6 ) )
= C2 )
= ( ! [X2: B] :
( ( member @ B @ X2 @ I6 )
=> ( ( F @ X2 )
= C2 ) ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_4872_Sup__inter__less__eq,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A6: set @ A,B7: set @ A] : ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( inf_inf @ ( set @ A ) @ A6 @ B7 ) ) @ ( inf_inf @ A @ ( complete_Sup_Sup @ A @ A6 ) @ ( complete_Sup_Sup @ A @ B7 ) ) ) ) ).
% Sup_inter_less_eq
thf(fact_4873_SUP__subset__mono,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A6: set @ B,B7: set @ B,F: B > A,G: B > A] :
( ( ord_less_eq @ ( set @ B ) @ A6 @ B7 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ A6 )
=> ( ord_less_eq @ A @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A6 ) ) @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ G @ B7 ) ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_4874_mono__Sup,axiom,
! [B: $tType,A: $tType] :
( ( ( comple6319245703460814977attice @ A )
& ( comple6319245703460814977attice @ B ) )
=> ! [F: A > B,A6: set @ A] :
( ( order_mono @ A @ B @ F )
=> ( ord_less_eq @ B @ ( complete_Sup_Sup @ B @ ( image @ A @ B @ F @ A6 ) ) @ ( F @ ( complete_Sup_Sup @ A @ A6 ) ) ) ) ) ).
% mono_Sup
thf(fact_4875_card__UNION,axiom,
! [A: $tType,A6: set @ ( set @ A )] :
( ( finite_finite2 @ ( set @ A ) @ A6 )
=> ( ! [X3: set @ A] :
( ( member @ ( set @ A ) @ X3 @ A6 )
=> ( finite_finite2 @ A @ X3 ) )
=> ( ( finite_card @ A @ ( complete_Sup_Sup @ ( set @ A ) @ A6 ) )
= ( nat2
@ ( groups7311177749621191930dd_sum @ ( set @ ( set @ A ) ) @ int
@ ^ [I8: set @ ( set @ A )] : ( times_times @ int @ ( power_power @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( plus_plus @ nat @ ( finite_card @ ( set @ A ) @ I8 ) @ ( one_one @ nat ) ) ) @ ( semiring_1_of_nat @ int @ ( finite_card @ A @ ( complete_Inf_Inf @ ( set @ A ) @ I8 ) ) ) )
@ ( collect @ ( set @ ( set @ A ) )
@ ^ [I8: set @ ( set @ A )] :
( ( ord_less_eq @ ( set @ ( set @ A ) ) @ I8 @ A6 )
& ( I8
!= ( bot_bot @ ( set @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% card_UNION
thf(fact_4876_surj__int__encode,axiom,
( ( image @ int @ nat @ nat_int_encode @ ( top_top @ ( set @ int ) ) )
= ( top_top @ ( set @ nat ) ) ) ).
% surj_int_encode
thf(fact_4877_horner__sum__eq__sum__funpow,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_0 @ A )
=> ( ( groups4207007520872428315er_sum @ B @ A )
= ( ^ [F2: B > A,A5: A,Xs3: list @ B] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N3: nat] : ( compow @ ( A > A ) @ N3 @ ( times_times @ A @ A5 ) @ ( F2 @ ( nth @ B @ Xs3 @ N3 ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ B ) @ Xs3 ) ) ) ) ) ) ).
% horner_sum_eq_sum_funpow
thf(fact_4878_Suc__funpow,axiom,
! [N: nat] :
( ( compow @ ( nat > nat ) @ N @ suc )
= ( plus_plus @ nat @ N ) ) ).
% Suc_funpow
thf(fact_4879_funpow__0,axiom,
! [A: $tType,F: A > A,X: A] :
( ( compow @ ( A > A ) @ ( zero_zero @ nat ) @ F @ X )
= X ) ).
% funpow_0
thf(fact_4880_Inf__eq__bot__iff,axiom,
! [A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [A6: set @ A] :
( ( ( complete_Inf_Inf @ A @ A6 )
= ( bot_bot @ A ) )
= ( ! [X2: A] :
( ( ord_less @ A @ ( bot_bot @ A ) @ X2 )
=> ? [Y6: A] :
( ( member @ A @ Y6 @ A6 )
& ( ord_less @ A @ Y6 @ X2 ) ) ) ) ) ) ).
% Inf_eq_bot_iff
thf(fact_4881_Inf__atLeastAtMost,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X: A,Y3: A] :
( ( ord_less_eq @ A @ X @ Y3 )
=> ( ( complete_Inf_Inf @ A @ ( set_or1337092689740270186AtMost @ A @ X @ Y3 ) )
= X ) ) ) ).
% Inf_atLeastAtMost
thf(fact_4882_cInf__atLeastAtMost,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [Y3: A,X: A] :
( ( ord_less_eq @ A @ Y3 @ X )
=> ( ( complete_Inf_Inf @ A @ ( set_or1337092689740270186AtMost @ A @ Y3 @ X ) )
= Y3 ) ) ) ).
% cInf_atLeastAtMost
thf(fact_4883_cInf__atLeastLessThan,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [Y3: A,X: A] :
( ( ord_less @ A @ Y3 @ X )
=> ( ( complete_Inf_Inf @ A @ ( set_or7035219750837199246ssThan @ A @ Y3 @ X ) )
= Y3 ) ) ) ).
% cInf_atLeastLessThan
thf(fact_4884_Inf__atLeastLessThan,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X: A,Y3: A] :
( ( ord_less @ A @ X @ Y3 )
=> ( ( complete_Inf_Inf @ A @ ( set_or7035219750837199246ssThan @ A @ X @ Y3 ) )
= X ) ) ) ).
% Inf_atLeastLessThan
thf(fact_4885_surj__fn,axiom,
! [A: $tType,F: A > A,N: nat] :
( ( ( image @ A @ A @ F @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) )
=> ( ( image @ A @ A @ ( compow @ ( A > A ) @ N @ F ) @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ) ).
% surj_fn
thf(fact_4886_cInf__greaterThanAtMost,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( dense_linorder @ A ) )
=> ! [Y3: A,X: A] :
( ( ord_less @ A @ Y3 @ X )
=> ( ( complete_Inf_Inf @ A @ ( set_or3652927894154168847AtMost @ A @ Y3 @ X ) )
= Y3 ) ) ) ).
% cInf_greaterThanAtMost
thf(fact_4887_Inf__greaterThanAtMost,axiom,
! [A: $tType] :
( ( ( comple6319245703460814977attice @ A )
& ( dense_linorder @ A ) )
=> ! [X: A,Y3: A] :
( ( ord_less @ A @ X @ Y3 )
=> ( ( complete_Inf_Inf @ A @ ( set_or3652927894154168847AtMost @ A @ X @ Y3 ) )
= X ) ) ) ).
% Inf_greaterThanAtMost
thf(fact_4888_Inf__greaterThanLessThan,axiom,
! [A: $tType] :
( ( ( comple6319245703460814977attice @ A )
& ( dense_linorder @ A ) )
=> ! [X: A,Y3: A] :
( ( ord_less @ A @ X @ Y3 )
=> ( ( complete_Inf_Inf @ A @ ( set_or5935395276787703475ssThan @ A @ X @ Y3 ) )
= X ) ) ) ).
% Inf_greaterThanLessThan
thf(fact_4889_cInf__greaterThanLessThan,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( dense_linorder @ A ) )
=> ! [Y3: A,X: A] :
( ( ord_less @ A @ Y3 @ X )
=> ( ( complete_Inf_Inf @ A @ ( set_or5935395276787703475ssThan @ A @ Y3 @ X ) )
= Y3 ) ) ) ).
% cInf_greaterThanLessThan
thf(fact_4890_INF__eq__bot__iff,axiom,
! [B: $tType,A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [F: B > A,A6: set @ B] :
( ( ( complete_Inf_Inf @ A @ ( image @ B @ A @ F @ A6 ) )
= ( bot_bot @ A ) )
= ( ! [X2: A] :
( ( ord_less @ A @ ( bot_bot @ A ) @ X2 )
=> ? [Y6: B] :
( ( member @ B @ Y6 @ A6 )
& ( ord_less @ A @ ( F @ Y6 ) @ X2 ) ) ) ) ) ) ).
% INF_eq_bot_iff
thf(fact_4891_Compl__UN,axiom,
! [A: $tType,B: $tType,B7: B > ( set @ A ),A6: set @ B] :
( ( uminus_uminus @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B7 @ A6 ) ) )
= ( complete_Inf_Inf @ ( set @ A )
@ ( image @ B @ ( set @ A )
@ ^ [X2: B] : ( uminus_uminus @ ( set @ A ) @ ( B7 @ X2 ) )
@ A6 ) ) ) ).
% Compl_UN
thf(fact_4892_Compl__INT,axiom,
! [A: $tType,B: $tType,B7: B > ( set @ A ),A6: set @ B] :
( ( uminus_uminus @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B7 @ A6 ) ) )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image @ B @ ( set @ A )
@ ^ [X2: B] : ( uminus_uminus @ ( set @ A ) @ ( B7 @ X2 ) )
@ A6 ) ) ) ).
% Compl_INT
thf(fact_4893_Inf__atMostLessThan,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X: A] :
( ( ord_less @ A @ ( top_top @ A ) @ X )
=> ( ( complete_Inf_Inf @ A @ ( set_ord_lessThan @ A @ X ) )
= ( bot_bot @ A ) ) ) ) ).
% Inf_atMostLessThan
thf(fact_4894_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_4895_Inf__greatest,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A6: set @ A,Z3: A] :
( ! [X3: A] :
( ( member @ A @ X3 @ A6 )
=> ( ord_less_eq @ A @ Z3 @ X3 ) )
=> ( ord_less_eq @ A @ Z3 @ ( complete_Inf_Inf @ A @ A6 ) ) ) ) ).
% Inf_greatest
thf(fact_4896_le__Inf__iff,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [B2: A,A6: set @ A] :
( ( ord_less_eq @ A @ B2 @ ( complete_Inf_Inf @ A @ A6 ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ A6 )
=> ( ord_less_eq @ A @ B2 @ X2 ) ) ) ) ) ).
% le_Inf_iff
thf(fact_4897_Inf__lower2,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [U: A,A6: set @ A,V: A] :
( ( member @ A @ U @ A6 )
=> ( ( ord_less_eq @ A @ U @ V )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A6 ) @ V ) ) ) ) ).
% Inf_lower2
thf(fact_4898_Inf__lower,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X: A,A6: set @ A] :
( ( member @ A @ X @ A6 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A6 ) @ X ) ) ) ).
% Inf_lower
thf(fact_4899_Inf__mono,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [B7: set @ A,A6: set @ A] :
( ! [B5: A] :
( ( member @ A @ B5 @ B7 )
=> ? [X4: A] :
( ( member @ A @ X4 @ A6 )
& ( ord_less_eq @ A @ X4 @ B5 ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A6 ) @ ( complete_Inf_Inf @ A @ B7 ) ) ) ) ).
% Inf_mono
thf(fact_4900_Inf__eqI,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A6: set @ A,X: A] :
( ! [I4: A] :
( ( member @ A @ I4 @ A6 )
=> ( ord_less_eq @ A @ X @ I4 ) )
=> ( ! [Y5: A] :
( ! [I3: A] :
( ( member @ A @ I3 @ A6 )
=> ( ord_less_eq @ A @ Y5 @ I3 ) )
=> ( ord_less_eq @ A @ Y5 @ X ) )
=> ( ( complete_Inf_Inf @ A @ A6 )
= X ) ) ) ) ).
% Inf_eqI
thf(fact_4901_cInf__eq__minimum,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [Z3: A,X8: set @ A] :
( ( member @ A @ Z3 @ X8 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ X8 )
=> ( ord_less_eq @ A @ Z3 @ X3 ) )
=> ( ( complete_Inf_Inf @ A @ X8 )
= Z3 ) ) ) ) ).
% cInf_eq_minimum
thf(fact_4902_cInf__eq,axiom,
! [A: $tType] :
( ( ( condit1219197933456340205attice @ A )
& ( no_top @ A ) )
=> ! [X8: set @ A,A3: A] :
( ! [X3: A] :
( ( member @ A @ X3 @ X8 )
=> ( ord_less_eq @ A @ A3 @ X3 ) )
=> ( ! [Y5: A] :
( ! [X4: A] :
( ( member @ A @ X4 @ X8 )
=> ( ord_less_eq @ A @ Y5 @ X4 ) )
=> ( ord_less_eq @ A @ Y5 @ A3 ) )
=> ( ( complete_Inf_Inf @ A @ X8 )
= A3 ) ) ) ) ).
% cInf_eq
thf(fact_4903_Inf__less__iff,axiom,
! [A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [S3: set @ A,A3: A] :
( ( ord_less @ A @ ( complete_Inf_Inf @ A @ S3 ) @ A3 )
= ( ? [X2: A] :
( ( member @ A @ X2 @ S3 )
& ( ord_less @ A @ X2 @ A3 ) ) ) ) ) ).
% Inf_less_iff
thf(fact_4904_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_4905_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_4906_int__encode__eq,axiom,
! [X: int,Y3: int] :
( ( ( nat_int_encode @ X )
= ( nat_int_encode @ Y3 ) )
= ( X = Y3 ) ) ).
% int_encode_eq
thf(fact_4907_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_4908_funpow__mod__eq,axiom,
! [A: $tType,N: nat,F: A > A,X: A,M: nat] :
( ( ( compow @ ( A > A ) @ N @ F @ X )
= X )
=> ( ( compow @ ( A > A ) @ ( modulo_modulo @ nat @ M @ N ) @ F @ X )
= ( compow @ ( A > A ) @ M @ F @ X ) ) ) ).
% funpow_mod_eq
thf(fact_4909_Inf__le__iff,axiom,
! [A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [A6: set @ A,X: A] :
( ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A6 ) @ X )
= ( ! [Y6: A] :
( ( ord_less @ A @ X @ Y6 )
=> ? [X2: A] :
( ( member @ A @ X2 @ A6 )
& ( ord_less @ A @ X2 @ Y6 ) ) ) ) ) ) ).
% Inf_le_iff
thf(fact_4910_INF__eq,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A6: set @ B,B7: set @ C,G: C > A,F: B > A] :
( ! [I4: B] :
( ( member @ B @ I4 @ A6 )
=> ? [X4: C] :
( ( member @ C @ X4 @ B7 )
& ( ord_less_eq @ A @ ( G @ X4 ) @ ( F @ I4 ) ) ) )
=> ( ! [J2: C] :
( ( member @ C @ J2 @ B7 )
=> ? [X4: B] :
( ( member @ B @ X4 @ A6 )
& ( ord_less_eq @ A @ ( F @ X4 ) @ ( G @ J2 ) ) ) )
=> ( ( complete_Inf_Inf @ A @ ( image @ B @ A @ F @ A6 ) )
= ( complete_Inf_Inf @ A @ ( image @ C @ A @ G @ B7 ) ) ) ) ) ) ).
% INF_eq
thf(fact_4911_cInf__eq__non__empty,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X8: set @ A,A3: A] :
( ( X8
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ X8 )
=> ( ord_less_eq @ A @ A3 @ X3 ) )
=> ( ! [Y5: A] :
( ! [X4: A] :
( ( member @ A @ X4 @ X8 )
=> ( ord_less_eq @ A @ Y5 @ X4 ) )
=> ( ord_less_eq @ A @ Y5 @ A3 ) )
=> ( ( complete_Inf_Inf @ A @ X8 )
= A3 ) ) ) ) ) ).
% cInf_eq_non_empty
thf(fact_4912_cInf__greatest,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X8: set @ A,Z3: A] :
( ( X8
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ X8 )
=> ( ord_less_eq @ A @ Z3 @ X3 ) )
=> ( ord_less_eq @ A @ Z3 @ ( complete_Inf_Inf @ A @ X8 ) ) ) ) ) ).
% cInf_greatest
thf(fact_4913_Inf__less__eq,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A6: set @ A,U: A] :
( ! [V4: A] :
( ( member @ A @ V4 @ A6 )
=> ( ord_less_eq @ A @ V4 @ U ) )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A6 ) @ U ) ) ) ) ).
% Inf_less_eq
thf(fact_4914_cInf__le__finite,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X8: set @ A,X: A] :
( ( finite_finite2 @ A @ X8 )
=> ( ( member @ A @ X @ X8 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ X8 ) @ X ) ) ) ) ).
% cInf_le_finite
thf(fact_4915_cInf__lessD,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X8: set @ A,Z3: A] :
( ( X8
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ ( complete_Inf_Inf @ A @ X8 ) @ Z3 )
=> ? [X3: A] :
( ( member @ A @ X3 @ X8 )
& ( ord_less @ A @ X3 @ Z3 ) ) ) ) ) ).
% cInf_lessD
thf(fact_4916_Inf__superset__mono,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [B7: set @ A,A6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B7 @ A6 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A6 ) @ ( complete_Inf_Inf @ A @ B7 ) ) ) ) ).
% Inf_superset_mono
thf(fact_4917_finite__imp__less__Inf,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X8: set @ A,X: A,A3: A] :
( ( finite_finite2 @ A @ X8 )
=> ( ( member @ A @ X @ X8 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ X8 )
=> ( ord_less @ A @ A3 @ X3 ) )
=> ( ord_less @ A @ A3 @ ( complete_Inf_Inf @ A @ X8 ) ) ) ) ) ) ).
% finite_imp_less_Inf
thf(fact_4918_funpow__times__power,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [F: A > nat,X: A] :
( ( compow @ ( A > A ) @ ( F @ X ) @ ( times_times @ A @ X ) )
= ( times_times @ A @ ( power_power @ A @ X @ ( F @ X ) ) ) ) ) ).
% funpow_times_power
thf(fact_4919_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_4920_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_4921_sup__Inf,axiom,
! [A: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [A3: A,B7: set @ A] :
( ( sup_sup @ A @ A3 @ ( complete_Inf_Inf @ A @ B7 ) )
= ( complete_Inf_Inf @ A @ ( image @ A @ A @ ( sup_sup @ A @ A3 ) @ B7 ) ) ) ) ).
% sup_Inf
thf(fact_4922_Inf__sup__eq__top__iff,axiom,
! [A: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [B7: set @ A,A3: A] :
( ( ( sup_sup @ A @ ( complete_Inf_Inf @ A @ B7 ) @ A3 )
= ( top_top @ A ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ B7 )
=> ( ( sup_sup @ A @ X2 @ A3 )
= ( top_top @ A ) ) ) ) ) ) ).
% Inf_sup_eq_top_iff
thf(fact_4923_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_4924_funpow__mono,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F: A > A,A6: A,B7: A,N: nat] :
( ( order_mono @ A @ A @ F )
=> ( ( ord_less_eq @ A @ A6 @ B7 )
=> ( ord_less_eq @ A @ ( compow @ ( A > A ) @ N @ F @ A6 ) @ ( compow @ ( A > A ) @ N @ F @ B7 ) ) ) ) ) ).
% funpow_mono
thf(fact_4925_INF__greatest,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A6: set @ B,U: A,F: B > A] :
( ! [I4: B] :
( ( member @ B @ I4 @ A6 )
=> ( ord_less_eq @ A @ U @ ( F @ I4 ) ) )
=> ( ord_less_eq @ A @ U @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F @ A6 ) ) ) ) ) ).
% INF_greatest
thf(fact_4926_le__INF__iff,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [U: A,F: B > A,A6: set @ B] :
( ( ord_less_eq @ A @ U @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F @ A6 ) ) )
= ( ! [X2: B] :
( ( member @ B @ X2 @ A6 )
=> ( ord_less_eq @ A @ U @ ( F @ X2 ) ) ) ) ) ) ).
% le_INF_iff
thf(fact_4927_INF__lower2,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I: B,A6: set @ B,F: B > A,U: A] :
( ( member @ B @ I @ A6 )
=> ( ( ord_less_eq @ A @ ( F @ I ) @ U )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F @ A6 ) ) @ U ) ) ) ) ).
% INF_lower2
thf(fact_4928_INF__mono_H,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F: B > A,G: B > A,A6: set @ B] :
( ! [X3: B] : ( ord_less_eq @ A @ ( F @ X3 ) @ ( G @ X3 ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F @ A6 ) ) @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ G @ A6 ) ) ) ) ) ).
% INF_mono'
thf(fact_4929_INF__lower,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I: B,A6: set @ B,F: B > A] :
( ( member @ B @ I @ A6 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F @ A6 ) ) @ ( F @ I ) ) ) ) ).
% INF_lower
thf(fact_4930_INF__mono,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [B7: set @ B,A6: set @ C,F: C > A,G: B > A] :
( ! [M6: B] :
( ( member @ B @ M6 @ B7 )
=> ? [X4: C] :
( ( member @ C @ X4 @ A6 )
& ( ord_less_eq @ A @ ( F @ X4 ) @ ( G @ M6 ) ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image @ C @ A @ F @ A6 ) ) @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ G @ B7 ) ) ) ) ) ).
% INF_mono
thf(fact_4931_INF__eqI,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A6: set @ B,X: A,F: B > A] :
( ! [I4: B] :
( ( member @ B @ I4 @ A6 )
=> ( ord_less_eq @ A @ X @ ( F @ I4 ) ) )
=> ( ! [Y5: A] :
( ! [I3: B] :
( ( member @ B @ I3 @ A6 )
=> ( ord_less_eq @ A @ Y5 @ ( F @ I3 ) ) )
=> ( ord_less_eq @ A @ Y5 @ X ) )
=> ( ( complete_Inf_Inf @ A @ ( image @ B @ A @ F @ A6 ) )
= X ) ) ) ) ).
% INF_eqI
thf(fact_4932_less__INF__D,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [Y3: A,F: B > A,A6: set @ B,I: B] :
( ( ord_less @ A @ Y3 @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F @ A6 ) ) )
=> ( ( member @ B @ I @ A6 )
=> ( ord_less @ A @ Y3 @ ( F @ I ) ) ) ) ) ).
% less_INF_D
thf(fact_4933_INF__less__iff,axiom,
! [A: $tType,B: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [F: B > A,A6: set @ B,A3: A] :
( ( ord_less @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F @ A6 ) ) @ A3 )
= ( ? [X2: B] :
( ( member @ B @ X2 @ A6 )
& ( ord_less @ A @ ( F @ X2 ) @ A3 ) ) ) ) ) ).
% INF_less_iff
thf(fact_4934_INF__sup,axiom,
! [A: $tType,B: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [F: B > A,B7: set @ B,A3: A] :
( ( sup_sup @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F @ B7 ) ) @ A3 )
= ( complete_Inf_Inf @ A
@ ( image @ B @ A
@ ^ [B3: B] : ( sup_sup @ A @ ( F @ B3 ) @ A3 )
@ B7 ) ) ) ) ).
% INF_sup
thf(fact_4935_Inf__sup,axiom,
! [A: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [B7: set @ A,A3: A] :
( ( sup_sup @ A @ ( complete_Inf_Inf @ A @ B7 ) @ A3 )
= ( complete_Inf_Inf @ A
@ ( image @ A @ A
@ ^ [B3: A] : ( sup_sup @ A @ B3 @ A3 )
@ B7 ) ) ) ) ).
% Inf_sup
thf(fact_4936_sup__INF,axiom,
! [A: $tType,B: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [A3: A,F: B > A,B7: set @ B] :
( ( sup_sup @ A @ A3 @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F @ B7 ) ) )
= ( complete_Inf_Inf @ A
@ ( image @ B @ A
@ ^ [B3: B] : ( sup_sup @ A @ A3 @ ( F @ B3 ) )
@ B7 ) ) ) ) ).
% sup_INF
thf(fact_4937_INF__sup__distrib2,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [F: B > A,A6: set @ B,G: C > A,B7: set @ C] :
( ( sup_sup @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F @ A6 ) ) @ ( complete_Inf_Inf @ A @ ( image @ C @ A @ G @ B7 ) ) )
= ( complete_Inf_Inf @ A
@ ( image @ B @ A
@ ^ [A5: B] :
( complete_Inf_Inf @ A
@ ( image @ C @ A
@ ^ [B3: C] : ( sup_sup @ A @ ( F @ A5 ) @ ( G @ B3 ) )
@ B7 ) )
@ A6 ) ) ) ) ).
% INF_sup_distrib2
thf(fact_4938_INF__le__iff,axiom,
! [B: $tType,A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [F: B > A,A6: set @ B,X: A] :
( ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F @ A6 ) ) @ X )
= ( ! [Y6: A] :
( ( ord_less @ A @ X @ Y6 )
=> ? [X2: B] :
( ( member @ B @ X2 @ A6 )
& ( ord_less @ A @ ( F @ X2 ) @ Y6 ) ) ) ) ) ) ).
% INF_le_iff
thf(fact_4939_cINF__greatest,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A6: set @ B,M: A,F: B > A] :
( ( A6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ A6 )
=> ( ord_less_eq @ A @ M @ ( F @ X3 ) ) )
=> ( ord_less_eq @ A @ M @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F @ A6 ) ) ) ) ) ) ).
% cINF_greatest
thf(fact_4940_INF__eq__iff,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I6: set @ B,F: B > A,C2: A] :
( ( I6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [I4: B] :
( ( member @ B @ I4 @ I6 )
=> ( ord_less_eq @ A @ ( F @ I4 ) @ C2 ) )
=> ( ( ( complete_Inf_Inf @ A @ ( image @ B @ A @ F @ I6 ) )
= C2 )
= ( ! [X2: B] :
( ( member @ B @ X2 @ I6 )
=> ( ( F @ X2 )
= C2 ) ) ) ) ) ) ) ).
% INF_eq_iff
thf(fact_4941_finite__less__Inf__iff,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X8: set @ A,A3: A] :
( ( finite_finite2 @ A @ X8 )
=> ( ( X8
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ A3 @ ( complete_Inf_Inf @ A @ X8 ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ X8 )
=> ( ord_less @ A @ A3 @ X2 ) ) ) ) ) ) ) ).
% finite_less_Inf_iff
thf(fact_4942_Inf__le__Sup,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A6: set @ A] :
( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A6 ) @ ( complete_Sup_Sup @ A @ A6 ) ) ) ) ).
% Inf_le_Sup
thf(fact_4943_cInf__abs__ge,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( linordered_idom @ A ) )
=> ! [S3: set @ A,A3: A] :
( ( S3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ S3 )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ X3 ) @ A3 ) )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ ( complete_Inf_Inf @ A @ S3 ) ) @ A3 ) ) ) ) ).
% cInf_abs_ge
thf(fact_4944_uminus__Sup,axiom,
! [A: $tType] :
( ( comple489889107523837845lgebra @ A )
=> ! [A6: set @ A] :
( ( uminus_uminus @ A @ ( complete_Sup_Sup @ A @ A6 ) )
= ( complete_Inf_Inf @ A @ ( image @ A @ A @ ( uminus_uminus @ A ) @ A6 ) ) ) ) ).
% uminus_Sup
thf(fact_4945_uminus__Inf,axiom,
! [A: $tType] :
( ( comple489889107523837845lgebra @ A )
=> ! [A6: set @ A] :
( ( uminus_uminus @ A @ ( complete_Inf_Inf @ A @ A6 ) )
= ( complete_Sup_Sup @ A @ ( image @ A @ A @ ( uminus_uminus @ A ) @ A6 ) ) ) ) ).
% uminus_Inf
thf(fact_4946_less__eq__Inf__inter,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A6: set @ A,B7: set @ A] : ( ord_less_eq @ A @ ( sup_sup @ A @ ( complete_Inf_Inf @ A @ A6 ) @ ( complete_Inf_Inf @ A @ B7 ) ) @ ( complete_Inf_Inf @ A @ ( inf_inf @ ( set @ A ) @ A6 @ B7 ) ) ) ) ).
% less_eq_Inf_inter
thf(fact_4947_Kleene__iter__lpfp,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [F: A > A,P5: A,K: nat] :
( ( order_mono @ A @ A @ F )
=> ( ( ord_less_eq @ A @ ( F @ P5 ) @ P5 )
=> ( ord_less_eq @ A @ ( compow @ ( A > A ) @ K @ F @ ( bot_bot @ A ) ) @ P5 ) ) ) ) ).
% Kleene_iter_lpfp
thf(fact_4948_Kleene__iter__gpfp,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [F: A > A,P5: A,K: nat] :
( ( order_mono @ A @ A @ F )
=> ( ( ord_less_eq @ A @ P5 @ ( F @ P5 ) )
=> ( ord_less_eq @ A @ P5 @ ( compow @ ( A > A ) @ K @ F @ ( top_top @ A ) ) ) ) ) ) ).
% Kleene_iter_gpfp
thf(fact_4949_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_4950_INF__superset__mono,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [B7: set @ B,A6: set @ B,F: B > A,G: B > A] :
( ( ord_less_eq @ ( set @ B ) @ B7 @ A6 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ B7 )
=> ( ord_less_eq @ A @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F @ A6 ) ) @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ G @ B7 ) ) ) ) ) ) ).
% INF_superset_mono
thf(fact_4951_uminus__INF,axiom,
! [A: $tType,B: $tType] :
( ( comple489889107523837845lgebra @ A )
=> ! [B7: B > A,A6: set @ B] :
( ( uminus_uminus @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ B7 @ A6 ) ) )
= ( complete_Sup_Sup @ A
@ ( image @ B @ A
@ ^ [X2: B] : ( uminus_uminus @ A @ ( B7 @ X2 ) )
@ A6 ) ) ) ) ).
% uminus_INF
thf(fact_4952_uminus__SUP,axiom,
! [A: $tType,B: $tType] :
( ( comple489889107523837845lgebra @ A )
=> ! [B7: B > A,A6: set @ B] :
( ( uminus_uminus @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ B7 @ A6 ) ) )
= ( complete_Inf_Inf @ A
@ ( image @ B @ A
@ ^ [X2: B] : ( uminus_uminus @ A @ ( B7 @ X2 ) )
@ A6 ) ) ) ) ).
% uminus_SUP
thf(fact_4953_mono__INF,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ( comple6319245703460814977attice @ A )
& ( comple6319245703460814977attice @ B ) )
=> ! [F: A > B,A6: C > A,I6: set @ C] :
( ( order_mono @ A @ B @ F )
=> ( ord_less_eq @ B @ ( F @ ( complete_Inf_Inf @ A @ ( image @ C @ A @ A6 @ I6 ) ) )
@ ( complete_Inf_Inf @ B
@ ( image @ C @ B
@ ^ [X2: C] : ( F @ ( A6 @ X2 ) )
@ I6 ) ) ) ) ) ).
% mono_INF
thf(fact_4954_mono__Inf,axiom,
! [B: $tType,A: $tType] :
( ( ( comple6319245703460814977attice @ A )
& ( comple6319245703460814977attice @ B ) )
=> ! [F: A > B,A6: set @ A] :
( ( order_mono @ A @ B @ F )
=> ( ord_less_eq @ B @ ( F @ ( complete_Inf_Inf @ A @ A6 ) ) @ ( complete_Inf_Inf @ B @ ( image @ A @ B @ F @ A6 ) ) ) ) ) ).
% mono_Inf
thf(fact_4955_of__nat__def,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( semiring_1_of_nat @ A )
= ( ^ [N3: nat] : ( compow @ ( A > A ) @ N3 @ ( plus_plus @ A @ ( one_one @ A ) ) @ ( zero_zero @ A ) ) ) ) ) ).
% of_nat_def
thf(fact_4956_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_4957_INF__le__SUP,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A6: set @ B,F: B > A] :
( ( A6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F @ A6 ) ) @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A6 ) ) ) ) ) ).
% INF_le_SUP
thf(fact_4958_mono__funpow,axiom,
! [A: $tType] :
( ( ( lattice @ A )
& ( order_bot @ A ) )
=> ! [Q: A > A] :
( ( order_mono @ A @ A @ Q )
=> ( order_mono @ nat @ A
@ ^ [I5: nat] : ( compow @ ( A > A ) @ I5 @ Q @ ( bot_bot @ A ) ) ) ) ) ).
% mono_funpow
thf(fact_4959_cInf__asclose,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( linordered_idom @ A ) )
=> ! [S3: set @ A,L: A,E: A] :
( ( S3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ S3 )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X3 @ L ) ) @ E ) )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ ( complete_Inf_Inf @ A @ S3 ) @ L ) ) @ E ) ) ) ) ).
% cInf_asclose
thf(fact_4960_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_4961_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_4962_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_4963_SUP__INF,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [P: C > B > A] :
( ( complete_Sup_Sup @ A
@ ( image @ B @ A
@ ^ [Y6: B] :
( complete_Inf_Inf @ A
@ ( image @ C @ A
@ ^ [X2: C] : ( P @ X2 @ Y6 )
@ ( top_top @ ( set @ C ) ) ) )
@ ( top_top @ ( set @ B ) ) ) )
= ( complete_Inf_Inf @ A
@ ( image @ ( B > C ) @ A
@ ^ [X2: B > C] :
( complete_Sup_Sup @ A
@ ( image @ B @ A
@ ^ [Y6: B] : ( P @ ( X2 @ Y6 ) @ Y6 )
@ ( top_top @ ( set @ B ) ) ) )
@ ( top_top @ ( set @ ( B > C ) ) ) ) ) ) ) ).
% SUP_INF
thf(fact_4964_INF__SUP,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [P: C > B > A] :
( ( complete_Inf_Inf @ A
@ ( image @ B @ A
@ ^ [Y6: B] :
( complete_Sup_Sup @ A
@ ( image @ C @ A
@ ^ [X2: C] : ( P @ X2 @ Y6 )
@ ( top_top @ ( set @ C ) ) ) )
@ ( top_top @ ( set @ B ) ) ) )
= ( complete_Sup_Sup @ A
@ ( image @ ( B > C ) @ A
@ ^ [F2: B > C] :
( complete_Inf_Inf @ A
@ ( image @ B @ A
@ ^ [X2: B] : ( P @ ( F2 @ X2 ) @ X2 )
@ ( top_top @ ( set @ B ) ) ) )
@ ( top_top @ ( set @ ( B > C ) ) ) ) ) ) ) ).
% INF_SUP
thf(fact_4965_relpowp__fun__conv,axiom,
! [A: $tType] :
( ( compow @ ( A > A > $o ) )
= ( ^ [N3: nat,P4: A > A > $o,X2: A,Y6: A] :
? [F2: nat > A] :
( ( ( F2 @ ( zero_zero @ nat ) )
= X2 )
& ( ( F2 @ N3 )
= Y6 )
& ! [I5: nat] :
( ( ord_less @ nat @ I5 @ N3 )
=> ( P4 @ ( F2 @ I5 ) @ ( F2 @ ( suc @ I5 ) ) ) ) ) ) ) ).
% relpowp_fun_conv
thf(fact_4966_relpowp__1,axiom,
! [A: $tType,P: A > A > $o] :
( ( compow @ ( A > A > $o ) @ ( one_one @ nat ) @ P )
= P ) ).
% relpowp_1
thf(fact_4967_Nat_Ofunpow__code__def,axiom,
! [A: $tType] :
( ( funpow @ A )
= ( compow @ ( A > A ) ) ) ).
% Nat.funpow_code_def
thf(fact_4968_Inf__int__def,axiom,
( ( complete_Inf_Inf @ int )
= ( ^ [X7: set @ int] : ( uminus_uminus @ int @ ( complete_Sup_Sup @ int @ ( image @ int @ int @ ( uminus_uminus @ int ) @ X7 ) ) ) ) ) ).
% Inf_int_def
thf(fact_4969_relpowp__Suc__I2,axiom,
! [A: $tType,P: A > A > $o,X: A,Y3: A,N: nat,Z3: A] :
( ( P @ X @ Y3 )
=> ( ( compow @ ( A > A > $o ) @ N @ P @ Y3 @ Z3 )
=> ( compow @ ( A > A > $o ) @ ( suc @ N ) @ P @ X @ Z3 ) ) ) ).
% relpowp_Suc_I2
thf(fact_4970_relpowp__Suc__E2,axiom,
! [A: $tType,N: nat,P: A > A > $o,X: A,Z3: A] :
( ( compow @ ( A > A > $o ) @ ( suc @ N ) @ P @ X @ Z3 )
=> ~ ! [Y5: A] :
( ( P @ X @ Y5 )
=> ~ ( compow @ ( A > A > $o ) @ N @ P @ Y5 @ Z3 ) ) ) ).
% relpowp_Suc_E2
thf(fact_4971_relpowp__Suc__D2,axiom,
! [A: $tType,N: nat,P: A > A > $o,X: A,Z3: A] :
( ( compow @ ( A > A > $o ) @ ( suc @ N ) @ P @ X @ Z3 )
=> ? [Y5: A] :
( ( P @ X @ Y5 )
& ( compow @ ( A > A > $o ) @ N @ P @ Y5 @ Z3 ) ) ) ).
% relpowp_Suc_D2
thf(fact_4972_relpowp__Suc__I,axiom,
! [A: $tType,N: nat,P: A > A > $o,X: A,Y3: A,Z3: A] :
( ( compow @ ( A > A > $o ) @ N @ P @ X @ Y3 )
=> ( ( P @ Y3 @ Z3 )
=> ( compow @ ( A > A > $o ) @ ( suc @ N ) @ P @ X @ Z3 ) ) ) ).
% relpowp_Suc_I
thf(fact_4973_relpowp__Suc__E,axiom,
! [A: $tType,N: nat,P: A > A > $o,X: A,Z3: A] :
( ( compow @ ( A > A > $o ) @ ( suc @ N ) @ P @ X @ Z3 )
=> ~ ! [Y5: A] :
( ( compow @ ( A > A > $o ) @ N @ P @ X @ Y5 )
=> ~ ( P @ Y5 @ Z3 ) ) ) ).
% relpowp_Suc_E
thf(fact_4974_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_4975_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_4976_relpowp_Osimps_I1_J,axiom,
! [A: $tType,R2: A > A > $o] :
( ( compow @ ( A > A > $o ) @ ( zero_zero @ nat ) @ R2 )
= ( ^ [Y: A,Z2: A] : Y = Z2 ) ) ).
% relpowp.simps(1)
thf(fact_4977_relpowp__E,axiom,
! [A: $tType,N: nat,P: A > A > $o,X: A,Z3: A] :
( ( compow @ ( A > A > $o ) @ N @ P @ X @ Z3 )
=> ( ( ( N
= ( zero_zero @ nat ) )
=> ( X != Z3 ) )
=> ~ ! [Y5: A,M6: nat] :
( ( N
= ( suc @ M6 ) )
=> ( ( compow @ ( A > A > $o ) @ M6 @ P @ X @ Y5 )
=> ~ ( P @ Y5 @ Z3 ) ) ) ) ) ).
% relpowp_E
thf(fact_4978_relpowp__E2,axiom,
! [A: $tType,N: nat,P: A > A > $o,X: A,Z3: A] :
( ( compow @ ( A > A > $o ) @ N @ P @ X @ Z3 )
=> ( ( ( N
= ( zero_zero @ nat ) )
=> ( X != Z3 ) )
=> ~ ! [Y5: A,M6: nat] :
( ( N
= ( suc @ M6 ) )
=> ( ( P @ X @ Y5 )
=> ~ ( compow @ ( A > A > $o ) @ M6 @ P @ Y5 @ Z3 ) ) ) ) ) ).
% relpowp_E2
thf(fact_4979_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_4980_set__zip,axiom,
! [B: $tType,A: $tType,Xs: list @ A,Ys2: list @ B] :
( ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys2 ) )
= ( collect @ ( product_prod @ A @ B )
@ ^ [Uu: product_prod @ A @ B] :
? [I5: nat] :
( ( Uu
= ( product_Pair @ A @ B @ ( nth @ A @ Xs @ I5 ) @ ( nth @ B @ Ys2 @ I5 ) ) )
& ( ord_less @ nat @ I5 @ ( ord_min @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ B ) @ Ys2 ) ) ) ) ) ) ).
% set_zip
thf(fact_4981_measure__function__int,axiom,
fun_is_measure @ int @ ( comp @ int @ nat @ int @ nat2 @ ( abs_abs @ int ) ) ).
% measure_function_int
thf(fact_4982_surj__int__decode,axiom,
( ( image @ nat @ int @ nat_int_decode @ ( top_top @ ( set @ nat ) ) )
= ( top_top @ ( set @ int ) ) ) ).
% surj_int_decode
thf(fact_4983_int__decode__inverse,axiom,
! [N: nat] :
( ( nat_int_encode @ ( nat_int_decode @ N ) )
= N ) ).
% int_decode_inverse
thf(fact_4984_int__encode__inverse,axiom,
! [X: int] :
( ( nat_int_decode @ ( nat_int_encode @ X ) )
= X ) ).
% int_encode_inverse
thf(fact_4985_measure__snd,axiom,
! [A: $tType,B: $tType,F: A > nat] :
( ( fun_is_measure @ A @ F )
=> ( fun_is_measure @ ( product_prod @ B @ A )
@ ^ [P6: product_prod @ B @ A] : ( F @ ( product_snd @ B @ A @ P6 ) ) ) ) ).
% measure_snd
thf(fact_4986_measure__fst,axiom,
! [B: $tType,A: $tType,F: A > nat] :
( ( fun_is_measure @ A @ F )
=> ( fun_is_measure @ ( product_prod @ A @ B )
@ ^ [P6: product_prod @ A @ B] : ( F @ ( product_fst @ A @ B @ P6 ) ) ) ) ).
% measure_fst
thf(fact_4987_lexordp_Omono,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( order_mono @ ( ( list @ A ) > ( list @ A ) > $o ) @ ( ( list @ A ) > ( list @ A ) > $o )
@ ^ [P6: ( list @ A ) > ( list @ A ) > $o,X12: list @ A,X24: list @ A] :
( ? [Y6: A,Ys3: list @ A] :
( ( X12
= ( nil @ A ) )
& ( X24
= ( cons @ A @ Y6 @ Ys3 ) ) )
| ? [X2: A,Y6: A,Xs3: list @ A,Ys3: list @ A] :
( ( X12
= ( cons @ A @ X2 @ Xs3 ) )
& ( X24
= ( cons @ A @ Y6 @ Ys3 ) )
& ( ord_less @ A @ X2 @ Y6 ) )
| ? [X2: A,Y6: A,Xs3: list @ A,Ys3: list @ A] :
( ( X12
= ( cons @ A @ X2 @ Xs3 ) )
& ( X24
= ( cons @ A @ Y6 @ Ys3 ) )
& ~ ( ord_less @ A @ X2 @ Y6 )
& ~ ( ord_less @ A @ Y6 @ X2 )
& ( P6 @ Xs3 @ Ys3 ) ) ) ) ) ).
% lexordp.mono
thf(fact_4988_is__measure_Osimps,axiom,
! [A: $tType] :
( ( fun_is_measure @ A )
= ( ^ [A5: A > nat] :
? [X7: A > nat] :
( ^ [Y: A > nat,Z2: A > nat] : Y = Z2
@ A5
@ X7 ) ) ) ).
% is_measure.simps
thf(fact_4989_is__measure__trivial,axiom,
! [A: $tType,F: A > nat] : ( fun_is_measure @ A @ F ) ).
% is_measure_trivial
thf(fact_4990_int__decode__eq,axiom,
! [X: nat,Y3: nat] :
( ( ( nat_int_decode @ X )
= ( nat_int_decode @ Y3 ) )
= ( X = Y3 ) ) ).
% int_decode_eq
thf(fact_4991_bot2E,axiom,
! [A: $tType,B: $tType,X: A,Y3: B] :
~ ( bot_bot @ ( A > B > $o ) @ X @ Y3 ) ).
% bot2E
thf(fact_4992_measure__size,axiom,
! [A: $tType] :
( ( size @ A )
=> ( fun_is_measure @ A @ ( size_size @ A ) ) ) ).
% measure_size
thf(fact_4993_set__conv__nth,axiom,
! [A: $tType] :
( ( set2 @ A )
= ( ^ [Xs3: list @ A] :
( collect @ A
@ ^ [Uu: A] :
? [I5: nat] :
( ( Uu
= ( nth @ A @ Xs3 @ I5 ) )
& ( ord_less @ nat @ I5 @ ( size_size @ ( list @ A ) @ Xs3 ) ) ) ) ) ) ).
% set_conv_nth
thf(fact_4994_set__nths,axiom,
! [A: $tType,Xs: list @ A,I6: set @ nat] :
( ( set2 @ A @ ( nths @ A @ Xs @ I6 ) )
= ( collect @ A
@ ^ [Uu: A] :
? [I5: nat] :
( ( Uu
= ( nth @ A @ Xs @ I5 ) )
& ( ord_less @ nat @ I5 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( member @ nat @ I5 @ I6 ) ) ) ) ).
% set_nths
thf(fact_4995_sup__bot_Osemilattice__neutr__order__axioms,axiom,
! [A: $tType] :
( ( bounde4967611905675639751up_bot @ A )
=> ( semila1105856199041335345_order @ A @ ( sup_sup @ A ) @ ( bot_bot @ A )
@ ^ [X2: A,Y6: A] : ( ord_less_eq @ A @ Y6 @ X2 )
@ ^ [X2: A,Y6: A] : ( ord_less @ A @ Y6 @ X2 ) ) ) ).
% sup_bot.semilattice_neutr_order_axioms
thf(fact_4996_max__nat_Osemilattice__neutr__order__axioms,axiom,
( semila1105856199041335345_order @ nat @ ( ord_max @ nat ) @ ( zero_zero @ nat )
@ ^ [X2: nat,Y6: nat] : ( ord_less_eq @ nat @ Y6 @ X2 )
@ ^ [X2: nat,Y6: nat] : ( ord_less @ nat @ Y6 @ X2 ) ) ).
% max_nat.semilattice_neutr_order_axioms
thf(fact_4997_times__int_Oabs__eq,axiom,
! [Xa3: product_prod @ nat @ nat,X: product_prod @ nat @ nat] :
( ( times_times @ int @ ( abs_Integ @ Xa3 ) @ ( abs_Integ @ X ) )
= ( abs_Integ
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X2: nat,Y6: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U3: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ X2 @ U3 ) @ ( times_times @ nat @ Y6 @ V5 ) ) @ ( plus_plus @ nat @ ( times_times @ nat @ X2 @ V5 ) @ ( times_times @ nat @ Y6 @ U3 ) ) ) )
@ Xa3
@ X ) ) ) ).
% times_int.abs_eq
thf(fact_4998_int_Oabs__induct,axiom,
! [P: int > $o,X: int] :
( ! [Y5: product_prod @ nat @ nat] : ( P @ ( abs_Integ @ Y5 ) )
=> ( P @ X ) ) ).
% int.abs_induct
thf(fact_4999_semilattice__neutr__order_Oneutr__eq__iff,axiom,
! [A: $tType,F: A > A > A,Z3: A,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A] :
( ( semila1105856199041335345_order @ A @ F @ Z3 @ Less_eq @ Less )
=> ( ( Z3
= ( F @ A3 @ B2 ) )
= ( ( A3 = Z3 )
& ( B2 = Z3 ) ) ) ) ).
% semilattice_neutr_order.neutr_eq_iff
thf(fact_5000_semilattice__neutr__order_Oeq__neutr__iff,axiom,
! [A: $tType,F: A > A > A,Z3: A,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A] :
( ( semila1105856199041335345_order @ A @ F @ Z3 @ Less_eq @ Less )
=> ( ( ( F @ A3 @ B2 )
= Z3 )
= ( ( A3 = Z3 )
& ( B2 = Z3 ) ) ) ) ).
% semilattice_neutr_order.eq_neutr_iff
thf(fact_5001_eq__Abs__Integ,axiom,
! [Z3: int] :
~ ! [X3: nat,Y5: nat] :
( Z3
!= ( abs_Integ @ ( product_Pair @ nat @ nat @ X3 @ Y5 ) ) ) ).
% eq_Abs_Integ
thf(fact_5002_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 )
@ ^ [I5: A,X2: D] : ( Q @ I5 @ ( F @ X2 ) ) ) ) ) ).
% mono_compose
thf(fact_5003_zero__int__def,axiom,
( ( zero_zero @ int )
= ( abs_Integ @ ( product_Pair @ nat @ nat @ ( zero_zero @ nat ) @ ( zero_zero @ nat ) ) ) ) ).
% zero_int_def
thf(fact_5004_int__def,axiom,
( ( semiring_1_of_nat @ int )
= ( ^ [N3: nat] : ( abs_Integ @ ( product_Pair @ nat @ nat @ N3 @ ( zero_zero @ nat ) ) ) ) ) ).
% int_def
thf(fact_5005_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_5006_Nats__altdef1,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( ( semiring_1_Nats @ A )
= ( collect @ A
@ ^ [Uu: A] :
? [N3: int] :
( ( Uu
= ( ring_1_of_int @ A @ N3 ) )
& ( ord_less_eq @ int @ ( zero_zero @ int ) @ N3 ) ) ) ) ) ).
% Nats_altdef1
thf(fact_5007_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,Y6: nat] : ( product_Pair @ nat @ nat @ Y6 @ X2 )
@ X ) ) ) ).
% uminus_int.abs_eq
thf(fact_5008_one__int__def,axiom,
( ( one_one @ int )
= ( abs_Integ @ ( product_Pair @ nat @ nat @ ( one_one @ nat ) @ ( zero_zero @ nat ) ) ) ) ).
% one_int_def
thf(fact_5009_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
@ ^ [I5: nat,J3: nat] : ( minus_minus @ A @ ( semiring_1_of_nat @ A @ I5 ) @ ( semiring_1_of_nat @ A @ J3 ) )
@ X ) ) ) ).
% of_int.abs_eq
thf(fact_5010_gcd__nat_Osemilattice__neutr__order__axioms,axiom,
( semila1105856199041335345_order @ nat @ ( gcd_gcd @ nat ) @ ( zero_zero @ nat ) @ ( dvd_dvd @ nat )
@ ^ [M2: nat,N3: nat] :
( ( dvd_dvd @ nat @ M2 @ N3 )
& ( M2 != N3 ) ) ) ).
% gcd_nat.semilattice_neutr_order_axioms
thf(fact_5011_less__int_Oabs__eq,axiom,
! [Xa3: product_prod @ nat @ nat,X: product_prod @ nat @ nat] :
( ( ord_less @ int @ ( abs_Integ @ Xa3 ) @ ( abs_Integ @ X ) )
= ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X2: nat,Y6: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U3: nat,V5: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ X2 @ V5 ) @ ( plus_plus @ nat @ U3 @ Y6 ) ) )
@ Xa3
@ X ) ) ).
% less_int.abs_eq
thf(fact_5012_less__eq__int_Oabs__eq,axiom,
! [Xa3: product_prod @ nat @ nat,X: product_prod @ nat @ nat] :
( ( ord_less_eq @ int @ ( abs_Integ @ Xa3 ) @ ( abs_Integ @ X ) )
= ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X2: nat,Y6: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U3: nat,V5: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ X2 @ V5 ) @ ( plus_plus @ nat @ U3 @ Y6 ) ) )
@ Xa3
@ X ) ) ).
% less_eq_int.abs_eq
thf(fact_5013_plus__int_Oabs__eq,axiom,
! [Xa3: product_prod @ nat @ nat,X: product_prod @ nat @ nat] :
( ( plus_plus @ int @ ( abs_Integ @ Xa3 ) @ ( abs_Integ @ X ) )
= ( abs_Integ
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X2: nat,Y6: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U3: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X2 @ U3 ) @ ( plus_plus @ nat @ Y6 @ V5 ) ) )
@ Xa3
@ X ) ) ) ).
% plus_int.abs_eq
thf(fact_5014_minus__int_Oabs__eq,axiom,
! [Xa3: product_prod @ nat @ nat,X: product_prod @ nat @ nat] :
( ( minus_minus @ int @ ( abs_Integ @ Xa3 ) @ ( abs_Integ @ X ) )
= ( abs_Integ
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X2: nat,Y6: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U3: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X2 @ V5 ) @ ( plus_plus @ nat @ Y6 @ U3 ) ) )
@ Xa3
@ X ) ) ) ).
% minus_int.abs_eq
thf(fact_5015_inf__top_Osemilattice__neutr__order__axioms,axiom,
! [A: $tType] :
( ( bounde4346867609351753570nf_top @ A )
=> ( semila1105856199041335345_order @ A @ ( inf_inf @ A ) @ ( top_top @ A ) @ ( ord_less_eq @ A ) @ ( ord_less @ A ) ) ) ).
% inf_top.semilattice_neutr_order_axioms
thf(fact_5016_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 )
@ ^ [Q3: 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 @ Q3 ) ) )
@ ( bit_take_bit_num @ ( numeral_numeral @ nat @ M ) @ N ) ) ) ).
% take_bit_numeral_minus_numeral_int
thf(fact_5017_prod_Oinsert_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [I6: set @ B,P5: B > A,I: B] :
( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ I6 )
& ( ( P5 @ X2 )
!= ( one_one @ A ) ) ) ) )
=> ( ( ( member @ B @ I @ I6 )
=> ( ( groups1962203154675924110t_prod @ B @ A @ P5 @ ( insert @ B @ I @ I6 ) )
= ( groups1962203154675924110t_prod @ B @ A @ P5 @ I6 ) ) )
& ( ~ ( member @ B @ I @ I6 )
=> ( ( groups1962203154675924110t_prod @ B @ A @ P5 @ ( insert @ B @ I @ I6 ) )
= ( times_times @ A @ ( P5 @ I ) @ ( groups1962203154675924110t_prod @ B @ A @ P5 @ I6 ) ) ) ) ) ) ) ).
% prod.insert'
thf(fact_5018_sorted__insort__is__snoc,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,A3: A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( ord_less_eq @ A @ X3 @ A3 ) )
=> ( ( linorder_insort_key @ A @ A
@ ^ [X2: A] : X2
@ A3
@ Xs )
= ( append @ A @ Xs @ ( cons @ A @ A3 @ ( nil @ A ) ) ) ) ) ) ) ).
% sorted_insort_is_snoc
thf(fact_5019_length__insort,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F: B > A,X: B,Xs: list @ B] :
( ( size_size @ ( list @ B ) @ ( linorder_insort_key @ B @ A @ F @ X @ Xs ) )
= ( suc @ ( size_size @ ( list @ B ) @ Xs ) ) ) ) ).
% length_insort
thf(fact_5020_prod_Oempty_H,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [P5: B > A] :
( ( groups1962203154675924110t_prod @ B @ A @ P5 @ ( bot_bot @ ( set @ B ) ) )
= ( one_one @ A ) ) ) ).
% prod.empty'
thf(fact_5021_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_5022_prod_Onon__neutral_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: B > A,I6: set @ B] :
( ( groups1962203154675924110t_prod @ B @ A @ G
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ I6 )
& ( ( G @ X2 )
!= ( one_one @ A ) ) ) ) )
= ( groups1962203154675924110t_prod @ B @ A @ G @ I6 ) ) ) ).
% prod.non_neutral'
thf(fact_5023_insort__key_Osimps_I2_J,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F: B > A,X: B,Y3: B,Ys2: list @ B] :
( ( ( ord_less_eq @ A @ ( F @ X ) @ ( F @ Y3 ) )
=> ( ( linorder_insort_key @ B @ A @ F @ X @ ( cons @ B @ Y3 @ Ys2 ) )
= ( cons @ B @ X @ ( cons @ B @ Y3 @ Ys2 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( F @ X ) @ ( F @ Y3 ) )
=> ( ( linorder_insort_key @ B @ A @ F @ X @ ( cons @ B @ Y3 @ Ys2 ) )
= ( cons @ B @ Y3 @ ( linorder_insort_key @ B @ A @ F @ X @ Ys2 ) ) ) ) ) ) ).
% insort_key.simps(2)
thf(fact_5024_sorted__insort,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Xs: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A )
@ ( linorder_insort_key @ A @ A
@ ^ [X2: A] : X2
@ X
@ Xs ) )
= ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs ) ) ) ).
% sorted_insort
thf(fact_5025_prod_Odistrib__triv_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [I6: set @ B,G: B > A,H2: B > A] :
( ( finite_finite2 @ B @ I6 )
=> ( ( groups1962203154675924110t_prod @ B @ A
@ ^ [I5: B] : ( times_times @ A @ ( G @ I5 ) @ ( H2 @ I5 ) )
@ I6 )
= ( times_times @ A @ ( groups1962203154675924110t_prod @ B @ A @ G @ I6 ) @ ( groups1962203154675924110t_prod @ B @ A @ H2 @ I6 ) ) ) ) ) ).
% prod.distrib_triv'
thf(fact_5026_insort__is__Cons,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ B,F: B > A,A3: B] :
( ! [X3: B] :
( ( member @ B @ X3 @ ( set2 @ B @ Xs ) )
=> ( ord_less_eq @ A @ ( F @ A3 ) @ ( F @ X3 ) ) )
=> ( ( linorder_insort_key @ B @ A @ F @ A3 @ Xs )
= ( cons @ B @ A3 @ Xs ) ) ) ) ).
% insort_is_Cons
thf(fact_5027_sorted__insort__key,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F: B > A,X: B,Xs: list @ B] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F @ ( linorder_insort_key @ B @ A @ F @ X @ Xs ) ) )
= ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F @ Xs ) ) ) ) ).
% sorted_insort_key
thf(fact_5028_prod_Omono__neutral__cong__right_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S3: set @ B,T6: set @ B,G: B > A,H2: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S3 @ T6 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T6 @ S3 ) )
=> ( ( G @ X3 )
= ( one_one @ A ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ S3 )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups1962203154675924110t_prod @ B @ A @ G @ T6 )
= ( groups1962203154675924110t_prod @ B @ A @ H2 @ S3 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_right'
thf(fact_5029_prod_Omono__neutral__cong__left_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S3: set @ B,T6: set @ B,H2: B > A,G: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S3 @ T6 )
=> ( ! [I4: B] :
( ( member @ B @ I4 @ ( minus_minus @ ( set @ B ) @ T6 @ S3 ) )
=> ( ( H2 @ I4 )
= ( one_one @ A ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ S3 )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) )
=> ( ( groups1962203154675924110t_prod @ B @ A @ G @ S3 )
= ( groups1962203154675924110t_prod @ B @ A @ H2 @ T6 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_left'
thf(fact_5030_prod_Omono__neutral__right_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S3: set @ B,T6: set @ B,G: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S3 @ T6 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T6 @ S3 ) )
=> ( ( G @ X3 )
= ( one_one @ A ) ) )
=> ( ( groups1962203154675924110t_prod @ B @ A @ G @ T6 )
= ( groups1962203154675924110t_prod @ B @ A @ G @ S3 ) ) ) ) ) ).
% prod.mono_neutral_right'
thf(fact_5031_prod_Omono__neutral__left_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S3: set @ B,T6: set @ B,G: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S3 @ T6 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T6 @ S3 ) )
=> ( ( G @ X3 )
= ( one_one @ A ) ) )
=> ( ( groups1962203154675924110t_prod @ B @ A @ G @ S3 )
= ( groups1962203154675924110t_prod @ B @ A @ G @ T6 ) ) ) ) ) ).
% prod.mono_neutral_left'
thf(fact_5032_prod_Odistrib_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [I6: set @ B,G: B > A,H2: B > A] :
( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ I6 )
& ( ( G @ X2 )
!= ( one_one @ A ) ) ) ) )
=> ( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ I6 )
& ( ( H2 @ X2 )
!= ( one_one @ A ) ) ) ) )
=> ( ( groups1962203154675924110t_prod @ B @ A
@ ^ [I5: B] : ( times_times @ A @ ( G @ I5 ) @ ( H2 @ I5 ) )
@ I6 )
= ( times_times @ A @ ( groups1962203154675924110t_prod @ B @ A @ G @ I6 ) @ ( groups1962203154675924110t_prod @ B @ A @ H2 @ I6 ) ) ) ) ) ) ).
% prod.distrib'
thf(fact_5033_prod_OG__def,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ( ( groups1962203154675924110t_prod @ B @ A )
= ( ^ [P6: B > A,I8: set @ B] :
( if @ A
@ ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ I8 )
& ( ( P6 @ X2 )
!= ( one_one @ A ) ) ) ) )
@ ( groups7121269368397514597t_prod @ B @ A @ P6
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ I8 )
& ( ( P6 @ X2 )
!= ( one_one @ A ) ) ) ) )
@ ( one_one @ A ) ) ) ) ) ).
% prod.G_def
thf(fact_5034_filter__insort,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F: B > A,Xs: list @ B,P: B > $o,X: B] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F @ Xs ) )
=> ( ( P @ X )
=> ( ( filter @ B @ P @ ( linorder_insort_key @ B @ A @ F @ X @ Xs ) )
= ( linorder_insort_key @ B @ A @ F @ X @ ( filter @ B @ P @ Xs ) ) ) ) ) ) ).
% filter_insort
thf(fact_5035_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_5036_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_5037_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_5038_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_5039_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_5040_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_5041_insort__key__remove1,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [A3: B,Xs: list @ B,F: B > A] :
( ( member @ B @ A3 @ ( set2 @ B @ Xs ) )
=> ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F @ Xs ) )
=> ( ( ( hd @ B
@ ( filter @ B
@ ^ [X2: B] :
( ( F @ A3 )
= ( F @ X2 ) )
@ Xs ) )
= A3 )
=> ( ( linorder_insort_key @ B @ A @ F @ A3 @ ( remove1 @ B @ A3 @ Xs ) )
= Xs ) ) ) ) ) ).
% insort_key_remove1
thf(fact_5042_less__eq__int_Orep__eq,axiom,
( ( ord_less_eq @ int )
= ( ^ [X2: int,Xa4: int] :
( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [Y6: nat,Z7: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U3: nat,V5: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ Y6 @ V5 ) @ ( plus_plus @ nat @ U3 @ Z7 ) ) )
@ ( rep_Integ @ X2 )
@ ( rep_Integ @ Xa4 ) ) ) ) ).
% less_eq_int.rep_eq
thf(fact_5043_less__int_Orep__eq,axiom,
( ( ord_less @ int )
= ( ^ [X2: int,Xa4: int] :
( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [Y6: nat,Z7: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U3: nat,V5: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ Y6 @ V5 ) @ ( plus_plus @ nat @ U3 @ Z7 ) ) )
@ ( rep_Integ @ X2 )
@ ( rep_Integ @ Xa4 ) ) ) ) ).
% less_int.rep_eq
thf(fact_5044_sorted__remove1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,A3: A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( remove1 @ A @ A3 @ Xs ) ) ) ) ).
% sorted_remove1
thf(fact_5045_sorted__map__remove1,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F: B > A,Xs: list @ B,X: B] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F @ Xs ) )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F @ ( remove1 @ B @ X @ Xs ) ) ) ) ) ).
% sorted_map_remove1
thf(fact_5046_length__remove1,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( size_size @ ( list @ A ) @ ( remove1 @ A @ X @ Xs ) )
= ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) ) ) )
& ( ~ ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( size_size @ ( list @ A ) @ ( remove1 @ A @ X @ Xs ) )
= ( size_size @ ( list @ A ) @ Xs ) ) ) ) ).
% length_remove1
thf(fact_5047_nat_Orep__eq,axiom,
( nat2
= ( ^ [X2: int] : ( product_case_prod @ nat @ nat @ nat @ ( minus_minus @ nat ) @ ( rep_Integ @ X2 ) ) ) ) ).
% nat.rep_eq
thf(fact_5048_insort__remove1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,Xs: list @ A] :
( ( member @ A @ A3 @ ( set2 @ A @ Xs ) )
=> ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( ( linorder_insort_key @ A @ A
@ ^ [X2: A] : X2
@ A3
@ ( remove1 @ A @ A3 @ Xs ) )
= Xs ) ) ) ) ).
% insort_remove1
thf(fact_5049_of__int_Orep__eq,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( ( ring_1_of_int @ A )
= ( ^ [X2: int] :
( product_case_prod @ nat @ nat @ A
@ ^ [I5: nat,J3: nat] : ( minus_minus @ A @ ( semiring_1_of_nat @ A @ I5 ) @ ( semiring_1_of_nat @ A @ J3 ) )
@ ( rep_Integ @ X2 ) ) ) ) ) ).
% of_int.rep_eq
thf(fact_5050_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,Y6: nat] : ( product_Pair @ nat @ nat @ Y6 @ X2 ) ) ) ) ).
% uminus_int_def
thf(fact_5051_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_5052_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_5053_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,Y6: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U3: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ X2 @ U3 ) @ ( times_times @ nat @ Y6 @ V5 ) ) @ ( plus_plus @ nat @ ( times_times @ nat @ X2 @ V5 ) @ ( times_times @ nat @ Y6 @ U3 ) ) ) ) ) ) ) ).
% times_int_def
thf(fact_5054_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,Y6: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U3: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X2 @ V5 ) @ ( plus_plus @ nat @ Y6 @ U3 ) ) ) ) ) ) ).
% minus_int_def
thf(fact_5055_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,Y6: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U3: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X2 @ U3 ) @ ( plus_plus @ nat @ Y6 @ V5 ) ) ) ) ) ) ).
% plus_int_def
thf(fact_5056_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 ),S7: 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 @ S7 @ F2 ) ) ) ) ).
% rp_inv_image_def
thf(fact_5057_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
@ ^ [I5: nat,J3: nat] : ( minus_minus @ A @ ( semiring_1_of_nat @ A @ I5 ) @ ( semiring_1_of_nat @ A @ J3 ) ) ) ) ) ) ).
% ring_1_class.of_int_def
thf(fact_5058_card__Min__le__sum,axiom,
! [A: $tType,A6: set @ A,F: A > nat] :
( ( finite_finite2 @ A @ A6 )
=> ( ord_less_eq @ nat @ ( times_times @ nat @ ( finite_card @ A @ A6 ) @ ( lattic643756798350308766er_Min @ nat @ ( image @ A @ nat @ F @ A6 ) ) ) @ ( groups7311177749621191930dd_sum @ A @ nat @ F @ A6 ) ) ) ).
% card_Min_le_sum
thf(fact_5059_of__nat__eq__id,axiom,
( ( semiring_1_of_nat @ nat )
= ( id @ nat ) ) ).
% of_nat_eq_id
thf(fact_5060_id__funpow,axiom,
! [A: $tType,N: nat] :
( ( compow @ ( A > A ) @ N @ ( id @ A ) )
= ( id @ A ) ) ).
% id_funpow
thf(fact_5061_rotate0,axiom,
! [A: $tType] :
( ( rotate @ A @ ( zero_zero @ nat ) )
= ( id @ ( list @ A ) ) ) ).
% rotate0
thf(fact_5062_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_5063_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_5064_Min__singleton,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A] :
( ( lattic643756798350308766er_Min @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= X ) ) ).
% Min_singleton
thf(fact_5065_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_5066_drop__bit__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se4197421643247451524op_bit @ A @ ( zero_zero @ nat ) )
= ( id @ A ) ) ) ).
% drop_bit_0
thf(fact_5067_Min_Obounded__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A,X: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ X @ ( lattic643756798350308766er_Min @ A @ A6 ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ A6 )
=> ( ord_less_eq @ A @ X @ X2 ) ) ) ) ) ) ) ).
% Min.bounded_iff
thf(fact_5068_Min__gr__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A,X: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ X @ ( lattic643756798350308766er_Min @ A @ A6 ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ A6 )
=> ( ord_less @ A @ X @ X2 ) ) ) ) ) ) ) ).
% Min_gr_iff
thf(fact_5069_Min__const,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ B,C2: A] :
( ( finite_finite2 @ B @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( lattic643756798350308766er_Min @ A
@ ( image @ B @ A
@ ^ [Uu: B] : C2
@ A6 ) )
= C2 ) ) ) ) ).
% Min_const
thf(fact_5070_Min__insert,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A,X: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798350308766er_Min @ A @ ( insert @ A @ X @ A6 ) )
= ( ord_min @ A @ X @ ( lattic643756798350308766er_Min @ A @ A6 ) ) ) ) ) ) ).
% Min_insert
thf(fact_5071_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_5072_Min_OcoboundedI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A,A3: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( member @ A @ A3 @ A6 )
=> ( ord_less_eq @ A @ ( lattic643756798350308766er_Min @ A @ A6 ) @ A3 ) ) ) ) ).
% Min.coboundedI
thf(fact_5073_Min__eqI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A,X: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ! [Y5: A] :
( ( member @ A @ Y5 @ A6 )
=> ( ord_less_eq @ A @ X @ Y5 ) )
=> ( ( member @ A @ X @ A6 )
=> ( ( lattic643756798350308766er_Min @ A @ A6 )
= X ) ) ) ) ) ).
% Min_eqI
thf(fact_5074_Min__le,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A,X: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( member @ A @ X @ A6 )
=> ( ord_less_eq @ A @ ( lattic643756798350308766er_Min @ A @ A6 ) @ X ) ) ) ) ).
% Min_le
thf(fact_5075_Min__in,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ A @ ( lattic643756798350308766er_Min @ A @ A6 ) @ A6 ) ) ) ) ).
% Min_in
thf(fact_5076_Min_Oin__idem,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A,X: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( member @ A @ X @ A6 )
=> ( ( ord_min @ A @ X @ ( lattic643756798350308766er_Min @ A @ A6 ) )
= ( lattic643756798350308766er_Min @ A @ A6 ) ) ) ) ) ).
% Min.in_idem
thf(fact_5077_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,Y6: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U3: nat,V5: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ X2 @ V5 ) @ ( plus_plus @ nat @ U3 @ Y6 ) ) ) ) ) ) ).
% less_int_def
thf(fact_5078_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,Y6: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U3: nat,V5: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ X2 @ V5 ) @ ( plus_plus @ nat @ U3 @ Y6 ) ) ) ) ) ) ).
% less_eq_int_def
thf(fact_5079_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_5080_Min_OboundedI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A,X: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [A8: A] :
( ( member @ A @ A8 @ A6 )
=> ( ord_less_eq @ A @ X @ A8 ) )
=> ( ord_less_eq @ A @ X @ ( lattic643756798350308766er_Min @ A @ A6 ) ) ) ) ) ) ).
% Min.boundedI
thf(fact_5081_Min_OboundedE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A,X: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ X @ ( lattic643756798350308766er_Min @ A @ A6 ) )
=> ! [A10: A] :
( ( member @ A @ A10 @ A6 )
=> ( ord_less_eq @ A @ X @ A10 ) ) ) ) ) ) ).
% Min.boundedE
thf(fact_5082_eq__Min__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A,M: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( M
= ( lattic643756798350308766er_Min @ A @ A6 ) )
= ( ( member @ A @ M @ A6 )
& ! [X2: A] :
( ( member @ A @ X2 @ A6 )
=> ( ord_less_eq @ A @ M @ X2 ) ) ) ) ) ) ) ).
% eq_Min_iff
thf(fact_5083_Min__le__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A,X: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ ( lattic643756798350308766er_Min @ A @ A6 ) @ X )
= ( ? [X2: A] :
( ( member @ A @ X2 @ A6 )
& ( ord_less_eq @ A @ X2 @ X ) ) ) ) ) ) ) ).
% Min_le_iff
thf(fact_5084_Min__eq__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A,M: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( lattic643756798350308766er_Min @ A @ A6 )
= M )
= ( ( member @ A @ M @ A6 )
& ! [X2: A] :
( ( member @ A @ X2 @ A6 )
=> ( ord_less_eq @ A @ M @ X2 ) ) ) ) ) ) ) ).
% Min_eq_iff
thf(fact_5085_Min__less__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A,X: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ ( lattic643756798350308766er_Min @ A @ A6 ) @ X )
= ( ? [X2: A] :
( ( member @ A @ X2 @ A6 )
& ( ord_less @ A @ X2 @ X ) ) ) ) ) ) ) ).
% Min_less_iff
thf(fact_5086_Min__insert2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A,A3: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ! [B5: A] :
( ( member @ A @ B5 @ A6 )
=> ( ord_less_eq @ A @ A3 @ B5 ) )
=> ( ( lattic643756798350308766er_Min @ A @ ( insert @ A @ A3 @ A6 ) )
= A3 ) ) ) ) ).
% Min_insert2
thf(fact_5087_Min__Inf,axiom,
! [A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [A6: set @ A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798350308766er_Min @ A @ A6 )
= ( complete_Inf_Inf @ A @ A6 ) ) ) ) ) ).
% Min_Inf
thf(fact_5088_Min_Osubset__imp,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A,B7: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B7 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B7 )
=> ( ord_less_eq @ A @ ( lattic643756798350308766er_Min @ A @ B7 ) @ ( lattic643756798350308766er_Min @ A @ A6 ) ) ) ) ) ) ).
% Min.subset_imp
thf(fact_5089_Min__antimono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [M7: set @ A,N6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ M7 @ N6 )
=> ( ( M7
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ N6 )
=> ( ord_less_eq @ A @ ( lattic643756798350308766er_Min @ A @ N6 ) @ ( lattic643756798350308766er_Min @ A @ M7 ) ) ) ) ) ) ).
% Min_antimono
thf(fact_5090_hom__Min__commute,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [H2: A > A,N6: set @ A] :
( ! [X3: A,Y5: A] :
( ( H2 @ ( ord_min @ A @ X3 @ Y5 ) )
= ( ord_min @ A @ ( H2 @ X3 ) @ ( H2 @ Y5 ) ) )
=> ( ( finite_finite2 @ A @ N6 )
=> ( ( N6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( H2 @ ( lattic643756798350308766er_Min @ A @ N6 ) )
= ( lattic643756798350308766er_Min @ A @ ( image @ A @ A @ H2 @ N6 ) ) ) ) ) ) ) ).
% hom_Min_commute
thf(fact_5091_Min_Osubset,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A,B7: set @ A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( B7
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ B7 @ A6 )
=> ( ( ord_min @ A @ ( lattic643756798350308766er_Min @ A @ B7 ) @ ( lattic643756798350308766er_Min @ A @ A6 ) )
= ( lattic643756798350308766er_Min @ A @ A6 ) ) ) ) ) ) ).
% Min.subset
thf(fact_5092_Min_Oclosed,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A,Y5: A] : ( member @ A @ ( ord_min @ A @ X3 @ Y5 ) @ ( insert @ A @ X3 @ ( insert @ A @ Y5 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( member @ A @ ( lattic643756798350308766er_Min @ A @ A6 ) @ A6 ) ) ) ) ) ).
% Min.closed
thf(fact_5093_Min_Oinsert__not__elem,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A,X: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ~ ( member @ A @ X @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798350308766er_Min @ A @ ( insert @ A @ X @ A6 ) )
= ( ord_min @ A @ X @ ( lattic643756798350308766er_Min @ A @ A6 ) ) ) ) ) ) ) ).
% Min.insert_not_elem
thf(fact_5094_mono__Min__commute,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( linorder @ B ) )
=> ! [F: A > B,A6: set @ A] :
( ( order_mono @ A @ B @ F )
=> ( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( F @ ( lattic643756798350308766er_Min @ A @ A6 ) )
= ( lattic643756798350308766er_Min @ B @ ( image @ A @ B @ F @ A6 ) ) ) ) ) ) ) ).
% mono_Min_commute
thf(fact_5095_Min_Ounion,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A,B7: set @ A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B7 )
=> ( ( B7
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798350308766er_Min @ A @ ( sup_sup @ ( set @ A ) @ A6 @ B7 ) )
= ( ord_min @ A @ ( lattic643756798350308766er_Min @ A @ A6 ) @ ( lattic643756798350308766er_Min @ A @ B7 ) ) ) ) ) ) ) ) ).
% Min.union
thf(fact_5096_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
@ ( image @ B @ A
@ ^ [X2: B] : ( plus_plus @ A @ ( F @ X2 ) @ K )
@ S3 ) )
= ( plus_plus @ A @ ( lattic643756798350308766er_Min @ A @ ( image @ B @ A @ F @ S3 ) ) @ K ) ) ) ) ) ).
% Min_add_commute
thf(fact_5097_Min_Oinsert__remove,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A,X: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798350308766er_Min @ A @ ( insert @ A @ X @ A6 ) )
= X ) )
& ( ( ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798350308766er_Min @ A @ ( insert @ A @ X @ A6 ) )
= ( ord_min @ A @ X @ ( lattic643756798350308766er_Min @ A @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% Min.insert_remove
thf(fact_5098_Min_Oremove,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A,X: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( member @ A @ X @ A6 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798350308766er_Min @ A @ A6 )
= X ) )
& ( ( ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798350308766er_Min @ A @ A6 )
= ( ord_min @ A @ X @ ( lattic643756798350308766er_Min @ A @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ) ).
% Min.remove
thf(fact_5099_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_5100_dual__Max,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( lattices_Max @ A
@ ^ [X2: A,Y6: A] : ( ord_less_eq @ A @ Y6 @ X2 ) )
= ( lattic643756798350308766er_Min @ A ) ) ) ).
% dual_Max
thf(fact_5101_dual__max,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( max @ A
@ ^ [X2: A,Y6: A] : ( ord_less_eq @ A @ Y6 @ X2 ) )
= ( ord_min @ A ) ) ) ).
% dual_max
thf(fact_5102_euclidean__size__nat__def,axiom,
( ( euclid6346220572633701492n_size @ nat )
= ( id @ nat ) ) ).
% euclidean_size_nat_def
thf(fact_5103_linorder_OMax_Ocong,axiom,
! [A: $tType] :
( ( lattices_Max @ A )
= ( lattices_Max @ A ) ) ).
% linorder.Max.cong
thf(fact_5104_ord_Omax__def,axiom,
! [A: $tType] :
( ( max @ A )
= ( ^ [Less_eq2: A > A > $o,A5: A,B3: A] : ( if @ A @ ( Less_eq2 @ A5 @ B3 ) @ B3 @ A5 ) ) ) ).
% ord.max_def
thf(fact_5105_ord_Omax_Ocong,axiom,
! [A: $tType] :
( ( max @ A )
= ( max @ A ) ) ).
% ord.max.cong
thf(fact_5106_dual__min,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( min @ A
@ ^ [X2: A,Y6: A] : ( ord_less_eq @ A @ Y6 @ X2 ) )
= ( ord_max @ A ) ) ) ).
% dual_min
thf(fact_5107_sum__le__card__Max,axiom,
! [A: $tType,A6: set @ A,F: A > nat] :
( ( finite_finite2 @ A @ A6 )
=> ( ord_less_eq @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ F @ A6 ) @ ( times_times @ nat @ ( finite_card @ A @ A6 ) @ ( lattic643756798349783984er_Max @ nat @ ( image @ A @ nat @ F @ A6 ) ) ) ) ) ).
% sum_le_card_Max
thf(fact_5108_euclidean__size__integer__def,axiom,
( ( euclid6346220572633701492n_size @ code_integer )
= ( map_fun @ code_integer @ int @ nat @ nat @ code_int_of_integer @ ( id @ nat ) @ ( euclid6346220572633701492n_size @ int ) ) ) ).
% euclidean_size_integer_def
thf(fact_5109_Max__singleton,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A] :
( ( lattic643756798349783984er_Max @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= X ) ) ).
% Max_singleton
thf(fact_5110_Max__divisors__self__nat,axiom,
! [N: nat] :
( ( N
!= ( zero_zero @ nat ) )
=> ( ( lattic643756798349783984er_Max @ nat
@ ( collect @ nat
@ ^ [D5: nat] : ( dvd_dvd @ nat @ D5 @ N ) ) )
= N ) ) ).
% Max_divisors_self_nat
thf(fact_5111_Max_Obounded__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A,X: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ ( lattic643756798349783984er_Max @ A @ A6 ) @ X )
= ( ! [X2: A] :
( ( member @ A @ X2 @ A6 )
=> ( ord_less_eq @ A @ X2 @ X ) ) ) ) ) ) ) ).
% Max.bounded_iff
thf(fact_5112_Max__less__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A,X: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ ( lattic643756798349783984er_Max @ A @ A6 ) @ X )
= ( ! [X2: A] :
( ( member @ A @ X2 @ A6 )
=> ( ord_less @ A @ X2 @ X ) ) ) ) ) ) ) ).
% Max_less_iff
thf(fact_5113_Max__const,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ B,C2: A] :
( ( finite_finite2 @ B @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( lattic643756798349783984er_Max @ A
@ ( image @ B @ A
@ ^ [Uu: B] : C2
@ A6 ) )
= C2 ) ) ) ) ).
% Max_const
thf(fact_5114_Max__insert,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A,X: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ ( insert @ A @ X @ A6 ) )
= ( ord_max @ A @ X @ ( lattic643756798349783984er_Max @ A @ A6 ) ) ) ) ) ) ).
% Max_insert
thf(fact_5115_minus__Max__eq__Min,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [S3: set @ A] :
( ( finite_finite2 @ A @ S3 )
=> ( ( S3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( uminus_uminus @ A @ ( lattic643756798349783984er_Max @ A @ S3 ) )
= ( lattic643756798350308766er_Min @ A @ ( image @ A @ A @ ( uminus_uminus @ A ) @ S3 ) ) ) ) ) ) ).
% minus_Max_eq_Min
thf(fact_5116_minus__Min__eq__Max,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [S3: set @ A] :
( ( finite_finite2 @ A @ S3 )
=> ( ( S3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( uminus_uminus @ A @ ( lattic643756798350308766er_Min @ A @ S3 ) )
= ( lattic643756798349783984er_Max @ A @ ( image @ A @ A @ ( uminus_uminus @ A ) @ S3 ) ) ) ) ) ) ).
% minus_Min_eq_Max
thf(fact_5117_ord_Omin_Ocong,axiom,
! [A: $tType] :
( ( min @ A )
= ( min @ A ) ) ).
% ord.min.cong
thf(fact_5118_ord_Omin__def,axiom,
! [A: $tType] :
( ( min @ A )
= ( ^ [Less_eq2: A > A > $o,A5: A,B3: A] : ( if @ A @ ( Less_eq2 @ A5 @ B3 ) @ A5 @ B3 ) ) ) ).
% ord.min_def
thf(fact_5119_Max_OcoboundedI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A,A3: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( member @ A @ A3 @ A6 )
=> ( ord_less_eq @ A @ A3 @ ( lattic643756798349783984er_Max @ A @ A6 ) ) ) ) ) ).
% Max.coboundedI
thf(fact_5120_Max__eq__if,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A,B7: set @ A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( finite_finite2 @ A @ B7 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ A6 )
=> ? [Xa: A] :
( ( member @ A @ Xa @ B7 )
& ( ord_less_eq @ A @ X3 @ Xa ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ B7 )
=> ? [Xa: A] :
( ( member @ A @ Xa @ A6 )
& ( ord_less_eq @ A @ X3 @ Xa ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ A6 )
= ( lattic643756798349783984er_Max @ A @ B7 ) ) ) ) ) ) ) ).
% Max_eq_if
thf(fact_5121_Max__eqI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A,X: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ! [Y5: A] :
( ( member @ A @ Y5 @ A6 )
=> ( ord_less_eq @ A @ Y5 @ X ) )
=> ( ( member @ A @ X @ A6 )
=> ( ( lattic643756798349783984er_Max @ A @ A6 )
= X ) ) ) ) ) ).
% Max_eqI
thf(fact_5122_Max__ge,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A,X: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( member @ A @ X @ A6 )
=> ( ord_less_eq @ A @ X @ ( lattic643756798349783984er_Max @ A @ A6 ) ) ) ) ) ).
% Max_ge
thf(fact_5123_Max__in,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ A @ ( lattic643756798349783984er_Max @ A @ A6 ) @ A6 ) ) ) ) ).
% Max_in
thf(fact_5124_Max_Oin__idem,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A,X: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( member @ A @ X @ A6 )
=> ( ( ord_max @ A @ X @ ( lattic643756798349783984er_Max @ A @ A6 ) )
= ( lattic643756798349783984er_Max @ A @ A6 ) ) ) ) ) ).
% Max.in_idem
thf(fact_5125_Max_OboundedI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A,X: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [A8: A] :
( ( member @ A @ A8 @ A6 )
=> ( ord_less_eq @ A @ A8 @ X ) )
=> ( ord_less_eq @ A @ ( lattic643756798349783984er_Max @ A @ A6 ) @ X ) ) ) ) ) ).
% Max.boundedI
thf(fact_5126_Max_OboundedE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A,X: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ ( lattic643756798349783984er_Max @ A @ A6 ) @ X )
=> ! [A10: A] :
( ( member @ A @ A10 @ A6 )
=> ( ord_less_eq @ A @ A10 @ X ) ) ) ) ) ) ).
% Max.boundedE
thf(fact_5127_eq__Max__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A,M: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( M
= ( lattic643756798349783984er_Max @ A @ A6 ) )
= ( ( member @ A @ M @ A6 )
& ! [X2: A] :
( ( member @ A @ X2 @ A6 )
=> ( ord_less_eq @ A @ X2 @ M ) ) ) ) ) ) ) ).
% eq_Max_iff
thf(fact_5128_Max__ge__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A,X: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ X @ ( lattic643756798349783984er_Max @ A @ A6 ) )
= ( ? [X2: A] :
( ( member @ A @ X2 @ A6 )
& ( ord_less_eq @ A @ X @ X2 ) ) ) ) ) ) ) ).
% Max_ge_iff
thf(fact_5129_Max__eq__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A,M: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( lattic643756798349783984er_Max @ A @ A6 )
= M )
= ( ( member @ A @ M @ A6 )
& ! [X2: A] :
( ( member @ A @ X2 @ A6 )
=> ( ord_less_eq @ A @ X2 @ M ) ) ) ) ) ) ) ).
% Max_eq_iff
thf(fact_5130_Max__gr__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A,X: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ X @ ( lattic643756798349783984er_Max @ A @ A6 ) )
= ( ? [X2: A] :
( ( member @ A @ X2 @ A6 )
& ( ord_less @ A @ X @ X2 ) ) ) ) ) ) ) ).
% Max_gr_iff
thf(fact_5131_Max__insert2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A,A3: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ! [B5: A] :
( ( member @ A @ B5 @ A6 )
=> ( ord_less_eq @ A @ B5 @ A3 ) )
=> ( ( lattic643756798349783984er_Max @ A @ ( insert @ A @ A3 @ A6 ) )
= A3 ) ) ) ) ).
% Max_insert2
thf(fact_5132_Max__Sup,axiom,
! [A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [A6: set @ A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ A6 )
= ( complete_Sup_Sup @ A @ A6 ) ) ) ) ) ).
% Max_Sup
thf(fact_5133_Max_Osubset__imp,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A,B7: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B7 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B7 )
=> ( ord_less_eq @ A @ ( lattic643756798349783984er_Max @ A @ A6 ) @ ( lattic643756798349783984er_Max @ A @ B7 ) ) ) ) ) ) ).
% Max.subset_imp
thf(fact_5134_Max__mono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [M7: set @ A,N6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ M7 @ N6 )
=> ( ( M7
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ N6 )
=> ( ord_less_eq @ A @ ( lattic643756798349783984er_Max @ A @ M7 ) @ ( lattic643756798349783984er_Max @ A @ N6 ) ) ) ) ) ) ).
% Max_mono
thf(fact_5135_hom__Max__commute,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [H2: A > A,N6: set @ A] :
( ! [X3: A,Y5: A] :
( ( H2 @ ( ord_max @ A @ X3 @ Y5 ) )
= ( ord_max @ A @ ( H2 @ X3 ) @ ( H2 @ Y5 ) ) )
=> ( ( finite_finite2 @ A @ N6 )
=> ( ( N6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( H2 @ ( lattic643756798349783984er_Max @ A @ N6 ) )
= ( lattic643756798349783984er_Max @ A @ ( image @ A @ A @ H2 @ N6 ) ) ) ) ) ) ) ).
% hom_Max_commute
thf(fact_5136_Max_Osubset,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A,B7: set @ A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( B7
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ B7 @ A6 )
=> ( ( ord_max @ A @ ( lattic643756798349783984er_Max @ A @ B7 ) @ ( lattic643756798349783984er_Max @ A @ A6 ) )
= ( lattic643756798349783984er_Max @ A @ A6 ) ) ) ) ) ) ).
% Max.subset
thf(fact_5137_Max_Oclosed,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A,Y5: A] : ( member @ A @ ( ord_max @ A @ X3 @ Y5 ) @ ( insert @ A @ X3 @ ( insert @ A @ Y5 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( member @ A @ ( lattic643756798349783984er_Max @ A @ A6 ) @ A6 ) ) ) ) ) ).
% Max.closed
thf(fact_5138_Max_Oinsert__not__elem,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A,X: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ~ ( member @ A @ X @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ ( insert @ A @ X @ A6 ) )
= ( ord_max @ A @ X @ ( lattic643756798349783984er_Max @ A @ A6 ) ) ) ) ) ) ) ).
% Max.insert_not_elem
thf(fact_5139_mono__Max__commute,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( linorder @ B ) )
=> ! [F: A > B,A6: set @ A] :
( ( order_mono @ A @ B @ F )
=> ( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( F @ ( lattic643756798349783984er_Max @ A @ A6 ) )
= ( lattic643756798349783984er_Max @ B @ ( image @ A @ B @ F @ A6 ) ) ) ) ) ) ) ).
% mono_Max_commute
thf(fact_5140_Max_Ounion,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A,B7: set @ A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B7 )
=> ( ( B7
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ ( sup_sup @ ( set @ A ) @ A6 @ B7 ) )
= ( ord_max @ A @ ( lattic643756798349783984er_Max @ A @ A6 ) @ ( lattic643756798349783984er_Max @ A @ B7 ) ) ) ) ) ) ) ) ).
% Max.union
thf(fact_5141_Sup__nat__def,axiom,
( ( complete_Sup_Sup @ nat )
= ( ^ [X7: set @ nat] :
( if @ nat
@ ( X7
= ( bot_bot @ ( set @ nat ) ) )
@ ( zero_zero @ nat )
@ ( lattic643756798349783984er_Max @ nat @ X7 ) ) ) ) ).
% Sup_nat_def
thf(fact_5142_card__le__Suc__Max,axiom,
! [S3: set @ nat] :
( ( finite_finite2 @ nat @ S3 )
=> ( ord_less_eq @ nat @ ( finite_card @ nat @ S3 ) @ ( suc @ ( lattic643756798349783984er_Max @ nat @ S3 ) ) ) ) ).
% card_le_Suc_Max
thf(fact_5143_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
@ ( image @ B @ A
@ ^ [X2: B] : ( plus_plus @ A @ ( F @ X2 ) @ K )
@ S3 ) )
= ( plus_plus @ A @ ( lattic643756798349783984er_Max @ A @ ( image @ B @ A @ F @ S3 ) ) @ K ) ) ) ) ) ).
% Max_add_commute
thf(fact_5144_divide__nat__def,axiom,
( ( divide_divide @ nat )
= ( ^ [M2: nat,N3: nat] :
( if @ nat
@ ( N3
= ( zero_zero @ nat ) )
@ ( zero_zero @ nat )
@ ( lattic643756798349783984er_Max @ nat
@ ( collect @ nat
@ ^ [K3: nat] : ( ord_less_eq @ nat @ ( times_times @ nat @ K3 @ N3 ) @ M2 ) ) ) ) ) ) ).
% divide_nat_def
thf(fact_5145_gcd__is__Max__divisors__nat,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( gcd_gcd @ nat @ M @ N )
= ( lattic643756798349783984er_Max @ nat
@ ( collect @ nat
@ ^ [D5: nat] :
( ( dvd_dvd @ nat @ D5 @ M )
& ( dvd_dvd @ nat @ D5 @ N ) ) ) ) ) ) ).
% gcd_is_Max_divisors_nat
thf(fact_5146_Max_Oinsert__remove,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A,X: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ ( insert @ A @ X @ A6 ) )
= X ) )
& ( ( ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ ( insert @ A @ X @ A6 ) )
= ( ord_max @ A @ X @ ( lattic643756798349783984er_Max @ A @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% Max.insert_remove
thf(fact_5147_Max_Oremove,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A,X: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( member @ A @ X @ A6 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ A6 )
= X ) )
& ( ( ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ A6 )
= ( ord_max @ A @ X @ ( lattic643756798349783984er_Max @ A @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ) ).
% Max.remove
thf(fact_5148_nat__of__integer__def,axiom,
( code_nat_of_integer
= ( map_fun @ code_integer @ int @ nat @ nat @ code_int_of_integer @ ( id @ nat ) @ nat2 ) ) ).
% nat_of_integer_def
thf(fact_5149_Gcd__eq__Max,axiom,
! [M7: set @ nat] :
( ( finite_finite2 @ nat @ M7 )
=> ( ( M7
!= ( bot_bot @ ( set @ nat ) ) )
=> ( ~ ( member @ nat @ ( zero_zero @ nat ) @ M7 )
=> ( ( gcd_Gcd @ nat @ M7 )
= ( lattic643756798349783984er_Max @ nat
@ ( complete_Inf_Inf @ ( set @ nat )
@ ( image @ nat @ ( set @ nat )
@ ^ [M2: nat] :
( collect @ nat
@ ^ [D5: nat] : ( dvd_dvd @ nat @ D5 @ M2 ) )
@ M7 ) ) ) ) ) ) ) ).
% Gcd_eq_Max
thf(fact_5150_dual__Min,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( lattices_Min @ A
@ ^ [X2: A,Y6: A] : ( ord_less_eq @ A @ Y6 @ X2 ) )
= ( lattic643756798349783984er_Max @ A ) ) ) ).
% dual_Min
thf(fact_5151_mask__integer__def,axiom,
( ( bit_se2239418461657761734s_mask @ code_integer )
= ( map_fun @ nat @ nat @ int @ code_integer @ ( id @ nat ) @ code_integer_of_int @ ( bit_se2239418461657761734s_mask @ int ) ) ) ).
% mask_integer_def
thf(fact_5152_positive_Orsp,axiom,
( bNF_rel_fun @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ $o @ $o @ ratrel
@ ^ [Y: $o,Z2: $o] : Y = 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_5153_Max__divisors__self__int,axiom,
! [N: int] :
( ( N
!= ( zero_zero @ int ) )
=> ( ( lattic643756798349783984er_Max @ int
@ ( collect @ int
@ ^ [D5: int] : ( dvd_dvd @ int @ D5 @ N ) ) )
= ( abs_abs @ int @ N ) ) ) ).
% Max_divisors_self_int
thf(fact_5154_power__transfer,axiom,
! [A: $tType,B: $tType] :
( ( ( power @ B )
& ( power @ A ) )
=> ! [R2: A > B > $o] :
( ( R2 @ ( one_one @ A ) @ ( one_one @ B ) )
=> ( ( bNF_rel_fun @ A @ B @ ( A > A ) @ ( B > B ) @ R2 @ ( bNF_rel_fun @ A @ B @ A @ B @ R2 @ R2 ) @ ( times_times @ A ) @ ( times_times @ B ) )
=> ( bNF_rel_fun @ A @ B @ ( nat > A ) @ ( nat > B ) @ R2
@ ( bNF_rel_fun @ nat @ nat @ A @ B
@ ^ [Y: nat,Z2: nat] : Y = Z2
@ R2 )
@ ( power_power @ A )
@ ( power_power @ B ) ) ) ) ) ).
% power_transfer
thf(fact_5155_linorder_OMin_Ocong,axiom,
! [A: $tType] :
( ( lattices_Min @ A )
= ( lattices_Min @ A ) ) ).
% linorder.Min.cong
thf(fact_5156_transfer__rule__numeral,axiom,
! [A: $tType,B: $tType] :
( ( ( monoid_add @ B )
& ( semiring_numeral @ B )
& ( monoid_add @ A )
& ( semiring_numeral @ A ) )
=> ! [R2: A > B > $o] :
( ( R2 @ ( zero_zero @ A ) @ ( zero_zero @ B ) )
=> ( ( R2 @ ( one_one @ A ) @ ( one_one @ B ) )
=> ( ( bNF_rel_fun @ A @ B @ ( A > A ) @ ( B > B ) @ R2 @ ( bNF_rel_fun @ A @ B @ A @ B @ R2 @ R2 ) @ ( plus_plus @ A ) @ ( plus_plus @ B ) )
=> ( bNF_rel_fun @ num @ num @ A @ B
@ ^ [Y: num,Z2: num] : Y = Z2
@ R2
@ ( numeral_numeral @ A )
@ ( numeral_numeral @ B ) ) ) ) ) ) ).
% transfer_rule_numeral
thf(fact_5157_transfer__rule__of__int,axiom,
! [A: $tType,B: $tType] :
( ( ( ring_1 @ B )
& ( ring_1 @ A ) )
=> ! [R2: A > B > $o] :
( ( R2 @ ( zero_zero @ A ) @ ( zero_zero @ B ) )
=> ( ( R2 @ ( one_one @ A ) @ ( one_one @ B ) )
=> ( ( bNF_rel_fun @ A @ B @ ( A > A ) @ ( B > B ) @ R2 @ ( bNF_rel_fun @ A @ B @ A @ B @ R2 @ R2 ) @ ( plus_plus @ A ) @ ( plus_plus @ B ) )
=> ( ( bNF_rel_fun @ A @ B @ A @ B @ R2 @ R2 @ ( uminus_uminus @ A ) @ ( uminus_uminus @ B ) )
=> ( bNF_rel_fun @ int @ int @ A @ B
@ ^ [Y: int,Z2: int] : Y = Z2
@ R2
@ ( ring_1_of_int @ A )
@ ( ring_1_of_int @ B ) ) ) ) ) ) ) ).
% transfer_rule_of_int
thf(fact_5158_gcd__is__Max__divisors__int,axiom,
! [N: int,M: int] :
( ( N
!= ( zero_zero @ int ) )
=> ( ( gcd_gcd @ int @ M @ N )
= ( lattic643756798349783984er_Max @ int
@ ( collect @ int
@ ^ [D5: int] :
( ( dvd_dvd @ int @ D5 @ M )
& ( dvd_dvd @ int @ D5 @ N ) ) ) ) ) ) ).
% gcd_is_Max_divisors_int
thf(fact_5159_transfer__rule__of__nat,axiom,
! [A: $tType,B: $tType] :
( ( ( semiring_1 @ B )
& ( semiring_1 @ A ) )
=> ! [R2: A > B > $o] :
( ( R2 @ ( zero_zero @ A ) @ ( zero_zero @ B ) )
=> ( ( R2 @ ( one_one @ A ) @ ( one_one @ B ) )
=> ( ( bNF_rel_fun @ A @ B @ ( A > A ) @ ( B > B ) @ R2 @ ( bNF_rel_fun @ A @ B @ A @ B @ R2 @ R2 ) @ ( plus_plus @ A ) @ ( plus_plus @ B ) )
=> ( bNF_rel_fun @ nat @ nat @ A @ B
@ ^ [Y: nat,Z2: nat] : Y = Z2
@ R2
@ ( semiring_1_of_nat @ A )
@ ( semiring_1_of_nat @ B ) ) ) ) ) ) ).
% transfer_rule_of_nat
thf(fact_5160_integer__of__nat__def,axiom,
( code_integer_of_nat
= ( map_fun @ nat @ nat @ int @ code_integer @ ( id @ nat ) @ code_integer_of_int @ ( semiring_1_of_nat @ int ) ) ) ).
% integer_of_nat_def
thf(fact_5161_antimono__funpow,axiom,
! [A: $tType] :
( ( ( lattice @ A )
& ( order_top @ A ) )
=> ! [Q: A > A] :
( ( order_mono @ A @ A @ Q )
=> ( order_antimono @ nat @ A
@ ^ [I5: nat] : ( compow @ ( A > A ) @ I5 @ Q @ ( top_top @ A ) ) ) ) ) ).
% antimono_funpow
thf(fact_5162_nat__of__integer__integer__of__nat,axiom,
! [N: nat] :
( ( code_nat_of_integer @ ( code_integer_of_nat @ N ) )
= N ) ).
% nat_of_integer_integer_of_nat
thf(fact_5163_integer__of__nat_Orep__eq,axiom,
! [X: nat] :
( ( code_int_of_integer @ ( code_integer_of_nat @ X ) )
= ( semiring_1_of_nat @ int @ X ) ) ).
% integer_of_nat.rep_eq
thf(fact_5164_int__of__integer__integer__of__nat,axiom,
! [N: nat] :
( ( code_int_of_integer @ ( code_integer_of_nat @ N ) )
= ( semiring_1_of_nat @ int @ N ) ) ).
% int_of_integer_integer_of_nat
thf(fact_5165_euclidean__size__integer_Orsp,axiom,
( bNF_rel_fun @ int @ int @ nat @ nat
@ ^ [Y: int,Z2: int] : Y = Z2
@ ^ [Y: nat,Z2: nat] : Y = Z2
@ ( euclid6346220572633701492n_size @ int )
@ ( euclid6346220572633701492n_size @ int ) ) ).
% euclidean_size_integer.rsp
thf(fact_5166_euclidean__size__natural_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ nat @ nat
@ ^ [Y: nat,Z2: nat] : Y = Z2
@ ^ [Y: nat,Z2: nat] : Y = Z2
@ ( euclid6346220572633701492n_size @ nat )
@ ( euclid6346220572633701492n_size @ nat ) ) ).
% euclidean_size_natural.rsp
thf(fact_5167_dup_Orsp,axiom,
( bNF_rel_fun @ int @ int @ int @ int
@ ^ [Y: int,Z2: int] : Y = Z2
@ ^ [Y: int,Z2: int] : Y = Z2
@ ^ [K3: int] : ( plus_plus @ int @ K3 @ K3 )
@ ^ [K3: int] : ( plus_plus @ int @ K3 @ K3 ) ) ).
% dup.rsp
thf(fact_5168_sub_Orsp,axiom,
( bNF_rel_fun @ num @ num @ ( num > int ) @ ( num > int )
@ ^ [Y: num,Z2: num] : Y = Z2
@ ( bNF_rel_fun @ num @ num @ int @ int
@ ^ [Y: num,Z2: num] : Y = Z2
@ ^ [Y: int,Z2: int] : Y = Z2 )
@ ^ [M2: num,N3: num] : ( minus_minus @ int @ ( numeral_numeral @ int @ M2 ) @ ( numeral_numeral @ int @ N3 ) )
@ ^ [M2: num,N3: num] : ( minus_minus @ int @ ( numeral_numeral @ int @ M2 ) @ ( numeral_numeral @ int @ N3 ) ) ) ).
% sub.rsp
thf(fact_5169_less__eq__integer_Orsp,axiom,
( bNF_rel_fun @ int @ int @ ( int > $o ) @ ( int > $o )
@ ^ [Y: int,Z2: int] : Y = Z2
@ ( bNF_rel_fun @ int @ int @ $o @ $o
@ ^ [Y: int,Z2: int] : Y = Z2
@ ^ [Y: $o,Z2: $o] : Y = Z2 )
@ ( ord_less_eq @ int )
@ ( ord_less_eq @ int ) ) ).
% less_eq_integer.rsp
thf(fact_5170_less__eq__natural_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ ( nat > $o ) @ ( nat > $o )
@ ^ [Y: nat,Z2: nat] : Y = Z2
@ ( bNF_rel_fun @ nat @ nat @ $o @ $o
@ ^ [Y: nat,Z2: nat] : Y = Z2
@ ^ [Y: $o,Z2: $o] : Y = Z2 )
@ ( ord_less_eq @ nat )
@ ( ord_less_eq @ nat ) ) ).
% less_eq_natural.rsp
thf(fact_5171_less__integer_Orsp,axiom,
( bNF_rel_fun @ int @ int @ ( int > $o ) @ ( int > $o )
@ ^ [Y: int,Z2: int] : Y = Z2
@ ( bNF_rel_fun @ int @ int @ $o @ $o
@ ^ [Y: int,Z2: int] : Y = Z2
@ ^ [Y: $o,Z2: $o] : Y = Z2 )
@ ( ord_less @ int )
@ ( ord_less @ int ) ) ).
% less_integer.rsp
thf(fact_5172_less__natural_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ ( nat > $o ) @ ( nat > $o )
@ ^ [Y: nat,Z2: nat] : Y = Z2
@ ( bNF_rel_fun @ nat @ nat @ $o @ $o
@ ^ [Y: nat,Z2: nat] : Y = Z2
@ ^ [Y: $o,Z2: $o] : Y = Z2 )
@ ( ord_less @ nat )
@ ( ord_less @ nat ) ) ).
% less_natural.rsp
thf(fact_5173_uminus__integer_Orsp,axiom,
( bNF_rel_fun @ int @ int @ int @ int
@ ^ [Y: int,Z2: int] : Y = Z2
@ ^ [Y: int,Z2: int] : Y = Z2
@ ( uminus_uminus @ int )
@ ( uminus_uminus @ int ) ) ).
% uminus_integer.rsp
thf(fact_5174_times__natural_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( nat > nat )
@ ^ [Y: nat,Z2: nat] : Y = Z2
@ ( bNF_rel_fun @ nat @ nat @ nat @ nat
@ ^ [Y: nat,Z2: nat] : Y = Z2
@ ^ [Y: nat,Z2: nat] : Y = Z2 )
@ ( times_times @ nat )
@ ( times_times @ nat ) ) ).
% times_natural.rsp
thf(fact_5175_times__integer_Orsp,axiom,
( bNF_rel_fun @ int @ int @ ( int > int ) @ ( int > int )
@ ^ [Y: int,Z2: int] : Y = Z2
@ ( bNF_rel_fun @ int @ int @ int @ int
@ ^ [Y: int,Z2: int] : Y = Z2
@ ^ [Y: int,Z2: int] : Y = Z2 )
@ ( times_times @ int )
@ ( times_times @ int ) ) ).
% times_integer.rsp
thf(fact_5176_natural__of__integer_Orsp,axiom,
( bNF_rel_fun @ int @ int @ nat @ nat
@ ^ [Y: int,Z2: int] : Y = Z2
@ ^ [Y: nat,Z2: nat] : Y = Z2
@ nat2
@ nat2 ) ).
% natural_of_integer.rsp
thf(fact_5177_Suc_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ nat @ nat
@ ^ [Y: nat,Z2: nat] : Y = Z2
@ ^ [Y: nat,Z2: nat] : Y = Z2
@ suc
@ suc ) ).
% Suc.rsp
thf(fact_5178_modulo__integer_Orsp,axiom,
( bNF_rel_fun @ int @ int @ ( int > int ) @ ( int > int )
@ ^ [Y: int,Z2: int] : Y = Z2
@ ( bNF_rel_fun @ int @ int @ int @ int
@ ^ [Y: int,Z2: int] : Y = Z2
@ ^ [Y: int,Z2: int] : Y = Z2 )
@ ( modulo_modulo @ int )
@ ( modulo_modulo @ int ) ) ).
% modulo_integer.rsp
thf(fact_5179_modulo__natural_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( nat > nat )
@ ^ [Y: nat,Z2: nat] : Y = Z2
@ ( bNF_rel_fun @ nat @ nat @ nat @ nat
@ ^ [Y: nat,Z2: nat] : Y = Z2
@ ^ [Y: nat,Z2: nat] : Y = Z2 )
@ ( modulo_modulo @ nat )
@ ( modulo_modulo @ nat ) ) ).
% modulo_natural.rsp
thf(fact_5180_sgn__integer_Orsp,axiom,
( bNF_rel_fun @ int @ int @ int @ int
@ ^ [Y: int,Z2: int] : Y = Z2
@ ^ [Y: int,Z2: int] : Y = Z2
@ ( sgn_sgn @ int )
@ ( sgn_sgn @ int ) ) ).
% sgn_integer.rsp
thf(fact_5181_plus__natural_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( nat > nat )
@ ^ [Y: nat,Z2: nat] : Y = Z2
@ ( bNF_rel_fun @ nat @ nat @ nat @ nat
@ ^ [Y: nat,Z2: nat] : Y = Z2
@ ^ [Y: nat,Z2: nat] : Y = Z2 )
@ ( plus_plus @ nat )
@ ( plus_plus @ nat ) ) ).
% plus_natural.rsp
thf(fact_5182_plus__integer_Orsp,axiom,
( bNF_rel_fun @ int @ int @ ( int > int ) @ ( int > int )
@ ^ [Y: int,Z2: int] : Y = Z2
@ ( bNF_rel_fun @ int @ int @ int @ int
@ ^ [Y: int,Z2: int] : Y = Z2
@ ^ [Y: int,Z2: int] : Y = Z2 )
@ ( plus_plus @ int )
@ ( plus_plus @ int ) ) ).
% plus_integer.rsp
thf(fact_5183_divide__natural_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( nat > nat )
@ ^ [Y: nat,Z2: nat] : Y = Z2
@ ( bNF_rel_fun @ nat @ nat @ nat @ nat
@ ^ [Y: nat,Z2: nat] : Y = Z2
@ ^ [Y: nat,Z2: nat] : Y = Z2 )
@ ( divide_divide @ nat )
@ ( divide_divide @ nat ) ) ).
% divide_natural.rsp
thf(fact_5184_divide__integer_Orsp,axiom,
( bNF_rel_fun @ int @ int @ ( int > int ) @ ( int > int )
@ ^ [Y: int,Z2: int] : Y = Z2
@ ( bNF_rel_fun @ int @ int @ int @ int
@ ^ [Y: int,Z2: int] : Y = Z2
@ ^ [Y: int,Z2: int] : Y = Z2 )
@ ( divide_divide @ int )
@ ( divide_divide @ int ) ) ).
% divide_integer.rsp
thf(fact_5185_abs__integer_Orsp,axiom,
( bNF_rel_fun @ int @ int @ int @ int
@ ^ [Y: int,Z2: int] : Y = Z2
@ ^ [Y: int,Z2: int] : Y = Z2
@ ( abs_abs @ int )
@ ( abs_abs @ int ) ) ).
% abs_integer.rsp
thf(fact_5186_minus__integer_Orsp,axiom,
( bNF_rel_fun @ int @ int @ ( int > int ) @ ( int > int )
@ ^ [Y: int,Z2: int] : Y = Z2
@ ( bNF_rel_fun @ int @ int @ int @ int
@ ^ [Y: int,Z2: int] : Y = Z2
@ ^ [Y: int,Z2: int] : Y = Z2 )
@ ( minus_minus @ int )
@ ( minus_minus @ int ) ) ).
% minus_integer.rsp
thf(fact_5187_minus__natural_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( nat > nat )
@ ^ [Y: nat,Z2: nat] : Y = Z2
@ ( bNF_rel_fun @ nat @ nat @ nat @ nat
@ ^ [Y: nat,Z2: nat] : Y = Z2
@ ^ [Y: nat,Z2: nat] : Y = Z2 )
@ ( minus_minus @ nat )
@ ( minus_minus @ nat ) ) ).
% minus_natural.rsp
thf(fact_5188_integer__of__natural_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ int @ int
@ ^ [Y: nat,Z2: nat] : Y = Z2
@ ^ [Y: int,Z2: int] : Y = Z2
@ ( semiring_1_of_nat @ int )
@ ( semiring_1_of_nat @ int ) ) ).
% integer_of_natural.rsp
thf(fact_5189_push__bit__natural_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( nat > nat )
@ ^ [Y: nat,Z2: nat] : Y = Z2
@ ( bNF_rel_fun @ nat @ nat @ nat @ nat
@ ^ [Y: nat,Z2: nat] : Y = Z2
@ ^ [Y: nat,Z2: nat] : Y = Z2 )
@ ( bit_se4730199178511100633sh_bit @ nat )
@ ( bit_se4730199178511100633sh_bit @ nat ) ) ).
% push_bit_natural.rsp
thf(fact_5190_push__bit__integer_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ ( int > int ) @ ( int > int )
@ ^ [Y: nat,Z2: nat] : Y = Z2
@ ( bNF_rel_fun @ int @ int @ int @ int
@ ^ [Y: int,Z2: int] : Y = Z2
@ ^ [Y: int,Z2: int] : Y = Z2 )
@ ( bit_se4730199178511100633sh_bit @ int )
@ ( bit_se4730199178511100633sh_bit @ int ) ) ).
% push_bit_integer.rsp
thf(fact_5191_flip__bit__integer_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ ( int > int ) @ ( int > int )
@ ^ [Y: nat,Z2: nat] : Y = Z2
@ ( bNF_rel_fun @ int @ int @ int @ int
@ ^ [Y: int,Z2: int] : Y = Z2
@ ^ [Y: int,Z2: int] : Y = Z2 )
@ ( bit_se8732182000553998342ip_bit @ int )
@ ( bit_se8732182000553998342ip_bit @ int ) ) ).
% flip_bit_integer.rsp
thf(fact_5192_flip__bit__natural_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( nat > nat )
@ ^ [Y: nat,Z2: nat] : Y = Z2
@ ( bNF_rel_fun @ nat @ nat @ nat @ nat
@ ^ [Y: nat,Z2: nat] : Y = Z2
@ ^ [Y: nat,Z2: nat] : Y = Z2 )
@ ( bit_se8732182000553998342ip_bit @ nat )
@ ( bit_se8732182000553998342ip_bit @ nat ) ) ).
% flip_bit_natural.rsp
thf(fact_5193_take__bit__natural_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( nat > nat )
@ ^ [Y: nat,Z2: nat] : Y = Z2
@ ( bNF_rel_fun @ nat @ nat @ nat @ nat
@ ^ [Y: nat,Z2: nat] : Y = Z2
@ ^ [Y: nat,Z2: nat] : Y = Z2 )
@ ( bit_se2584673776208193580ke_bit @ nat )
@ ( bit_se2584673776208193580ke_bit @ nat ) ) ).
% take_bit_natural.rsp
thf(fact_5194_take__bit__integer_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ ( int > int ) @ ( int > int )
@ ^ [Y: nat,Z2: nat] : Y = Z2
@ ( bNF_rel_fun @ int @ int @ int @ int
@ ^ [Y: int,Z2: int] : Y = Z2
@ ^ [Y: int,Z2: int] : Y = Z2 )
@ ( bit_se2584673776208193580ke_bit @ int )
@ ( bit_se2584673776208193580ke_bit @ int ) ) ).
% take_bit_integer.rsp
thf(fact_5195_and__natural_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( nat > nat )
@ ^ [Y: nat,Z2: nat] : Y = Z2
@ ( bNF_rel_fun @ nat @ nat @ nat @ nat
@ ^ [Y: nat,Z2: nat] : Y = Z2
@ ^ [Y: nat,Z2: nat] : Y = Z2 )
@ ( bit_se5824344872417868541ns_and @ nat )
@ ( bit_se5824344872417868541ns_and @ nat ) ) ).
% and_natural.rsp
thf(fact_5196_and__integer_Orsp,axiom,
( bNF_rel_fun @ int @ int @ ( int > int ) @ ( int > int )
@ ^ [Y: int,Z2: int] : Y = Z2
@ ( bNF_rel_fun @ int @ int @ int @ int
@ ^ [Y: int,Z2: int] : Y = Z2
@ ^ [Y: int,Z2: int] : Y = Z2 )
@ ( bit_se5824344872417868541ns_and @ int )
@ ( bit_se5824344872417868541ns_and @ int ) ) ).
% and_integer.rsp
thf(fact_5197_unset__bit__natural_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( nat > nat )
@ ^ [Y: nat,Z2: nat] : Y = Z2
@ ( bNF_rel_fun @ nat @ nat @ nat @ nat
@ ^ [Y: nat,Z2: nat] : Y = Z2
@ ^ [Y: nat,Z2: nat] : Y = Z2 )
@ ( bit_se2638667681897837118et_bit @ nat )
@ ( bit_se2638667681897837118et_bit @ nat ) ) ).
% unset_bit_natural.rsp
thf(fact_5198_unset__bit__integer_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ ( int > int ) @ ( int > int )
@ ^ [Y: nat,Z2: nat] : Y = Z2
@ ( bNF_rel_fun @ int @ int @ int @ int
@ ^ [Y: int,Z2: int] : Y = Z2
@ ^ [Y: int,Z2: int] : Y = Z2 )
@ ( bit_se2638667681897837118et_bit @ int )
@ ( bit_se2638667681897837118et_bit @ int ) ) ).
% unset_bit_integer.rsp
thf(fact_5199_xor__integer_Orsp,axiom,
( bNF_rel_fun @ int @ int @ ( int > int ) @ ( int > int )
@ ^ [Y: int,Z2: int] : Y = Z2
@ ( bNF_rel_fun @ int @ int @ int @ int
@ ^ [Y: int,Z2: int] : Y = Z2
@ ^ [Y: int,Z2: int] : Y = Z2 )
@ ( bit_se5824344971392196577ns_xor @ int )
@ ( bit_se5824344971392196577ns_xor @ int ) ) ).
% xor_integer.rsp
thf(fact_5200_xor__natural_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( nat > nat )
@ ^ [Y: nat,Z2: nat] : Y = Z2
@ ( bNF_rel_fun @ nat @ nat @ nat @ nat
@ ^ [Y: nat,Z2: nat] : Y = Z2
@ ^ [Y: nat,Z2: nat] : Y = Z2 )
@ ( bit_se5824344971392196577ns_xor @ nat )
@ ( bit_se5824344971392196577ns_xor @ nat ) ) ).
% xor_natural.rsp
thf(fact_5201_set__bit__integer_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ ( int > int ) @ ( int > int )
@ ^ [Y: nat,Z2: nat] : Y = Z2
@ ( bNF_rel_fun @ int @ int @ int @ int
@ ^ [Y: int,Z2: int] : Y = Z2
@ ^ [Y: int,Z2: int] : Y = Z2 )
@ ( bit_se5668285175392031749et_bit @ int )
@ ( bit_se5668285175392031749et_bit @ int ) ) ).
% set_bit_integer.rsp
thf(fact_5202_set__bit__natural_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( nat > nat )
@ ^ [Y: nat,Z2: nat] : Y = Z2
@ ( bNF_rel_fun @ nat @ nat @ nat @ nat
@ ^ [Y: nat,Z2: nat] : Y = Z2
@ ^ [Y: nat,Z2: nat] : Y = Z2 )
@ ( bit_se5668285175392031749et_bit @ nat )
@ ( bit_se5668285175392031749et_bit @ nat ) ) ).
% set_bit_natural.rsp
thf(fact_5203_drop__bit__natural_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( nat > nat )
@ ^ [Y: nat,Z2: nat] : Y = Z2
@ ( bNF_rel_fun @ nat @ nat @ nat @ nat
@ ^ [Y: nat,Z2: nat] : Y = Z2
@ ^ [Y: nat,Z2: nat] : Y = Z2 )
@ ( bit_se4197421643247451524op_bit @ nat )
@ ( bit_se4197421643247451524op_bit @ nat ) ) ).
% drop_bit_natural.rsp
thf(fact_5204_drop__bit__integer_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ ( int > int ) @ ( int > int )
@ ^ [Y: nat,Z2: nat] : Y = Z2
@ ( bNF_rel_fun @ int @ int @ int @ int
@ ^ [Y: int,Z2: int] : Y = Z2
@ ^ [Y: int,Z2: int] : Y = Z2 )
@ ( bit_se4197421643247451524op_bit @ int )
@ ( bit_se4197421643247451524op_bit @ int ) ) ).
% drop_bit_integer.rsp
thf(fact_5205_or__natural_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( nat > nat )
@ ^ [Y: nat,Z2: nat] : Y = Z2
@ ( bNF_rel_fun @ nat @ nat @ nat @ nat
@ ^ [Y: nat,Z2: nat] : Y = Z2
@ ^ [Y: nat,Z2: nat] : Y = Z2 )
@ ( bit_se1065995026697491101ons_or @ nat )
@ ( bit_se1065995026697491101ons_or @ nat ) ) ).
% or_natural.rsp
thf(fact_5206_or__integer_Orsp,axiom,
( bNF_rel_fun @ int @ int @ ( int > int ) @ ( int > int )
@ ^ [Y: int,Z2: int] : Y = Z2
@ ( bNF_rel_fun @ int @ int @ int @ int
@ ^ [Y: int,Z2: int] : Y = Z2
@ ^ [Y: int,Z2: int] : Y = Z2 )
@ ( bit_se1065995026697491101ons_or @ int )
@ ( bit_se1065995026697491101ons_or @ int ) ) ).
% or_integer.rsp
thf(fact_5207_bit__integer_Orsp,axiom,
( bNF_rel_fun @ int @ int @ ( nat > $o ) @ ( nat > $o )
@ ^ [Y: int,Z2: int] : Y = Z2
@ ^ [Y: nat > $o,Z2: nat > $o] : Y = Z2
@ ( bit_se5641148757651400278ts_bit @ int )
@ ( bit_se5641148757651400278ts_bit @ int ) ) ).
% bit_integer.rsp
thf(fact_5208_not__integer_Orsp,axiom,
( bNF_rel_fun @ int @ int @ int @ int
@ ^ [Y: int,Z2: int] : Y = Z2
@ ^ [Y: int,Z2: int] : Y = Z2
@ ( bit_ri4277139882892585799ns_not @ int )
@ ( bit_ri4277139882892585799ns_not @ int ) ) ).
% not_integer.rsp
thf(fact_5209_bit__natural_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ ( nat > $o ) @ ( nat > $o )
@ ^ [Y: nat,Z2: nat] : Y = Z2
@ ^ [Y: nat > $o,Z2: nat > $o] : Y = Z2
@ ( bit_se5641148757651400278ts_bit @ nat )
@ ( bit_se5641148757651400278ts_bit @ nat ) ) ).
% bit_natural.rsp
thf(fact_5210_mask__integer_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ int @ int
@ ^ [Y: nat,Z2: nat] : Y = Z2
@ ^ [Y: int,Z2: int] : Y = Z2
@ ( bit_se2239418461657761734s_mask @ int )
@ ( bit_se2239418461657761734s_mask @ int ) ) ).
% mask_integer.rsp
thf(fact_5211_mask__natural_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ nat @ nat
@ ^ [Y: nat,Z2: nat] : Y = Z2
@ ^ [Y: nat,Z2: nat] : Y = Z2
@ ( bit_se2239418461657761734s_mask @ nat )
@ ( bit_se2239418461657761734s_mask @ nat ) ) ).
% mask_natural.rsp
thf(fact_5212_division__segment__integer_Orsp,axiom,
( bNF_rel_fun @ int @ int @ int @ int
@ ^ [Y: int,Z2: int] : Y = Z2
@ ^ [Y: int,Z2: int] : Y = Z2
@ ( euclid7384307370059645450egment @ int )
@ ( euclid7384307370059645450egment @ int ) ) ).
% division_segment_integer.rsp
thf(fact_5213_division__segment__natural_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ nat @ nat
@ ^ [Y: nat,Z2: nat] : Y = Z2
@ ^ [Y: nat,Z2: nat] : Y = Z2
@ ( euclid7384307370059645450egment @ nat )
@ ( euclid7384307370059645450egment @ nat ) ) ).
% division_segment_natural.rsp
thf(fact_5214_Fract_Orsp,axiom,
( bNF_rel_fun @ int @ int @ ( int > ( product_prod @ int @ int ) ) @ ( int > ( product_prod @ int @ int ) )
@ ^ [Y: int,Z2: int] : Y = Z2
@ ( bNF_rel_fun @ int @ int @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int )
@ ^ [Y: int,Z2: int] : Y = Z2
@ ratrel )
@ ^ [A5: int,B3: int] :
( if @ ( product_prod @ int @ int )
@ ( B3
= ( zero_zero @ int ) )
@ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
@ ( product_Pair @ int @ int @ A5 @ B3 ) )
@ ^ [A5: int,B3: int] :
( if @ ( product_prod @ int @ int )
@ ( B3
= ( zero_zero @ int ) )
@ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
@ ( product_Pair @ int @ int @ A5 @ B3 ) ) ) ).
% Fract.rsp
thf(fact_5215_num__of__integer_Orsp,axiom,
( bNF_rel_fun @ int @ int @ num @ num
@ ^ [Y: int,Z2: int] : Y = Z2
@ ^ [Y: num,Z2: num] : Y = Z2
@ ( comp @ nat @ num @ int @ num_of_nat @ nat2 )
@ ( comp @ nat @ num @ int @ num_of_nat @ nat2 ) ) ).
% num_of_integer.rsp
thf(fact_5216_antimonoD,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F: A > B,X: A,Y3: A] :
( ( order_antimono @ A @ B @ F )
=> ( ( ord_less_eq @ A @ X @ Y3 )
=> ( ord_less_eq @ B @ ( F @ Y3 ) @ ( F @ X ) ) ) ) ) ).
% antimonoD
thf(fact_5217_antimonoE,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F: A > B,X: A,Y3: A] :
( ( order_antimono @ A @ B @ F )
=> ( ( ord_less_eq @ A @ X @ Y3 )
=> ( ord_less_eq @ B @ ( F @ Y3 ) @ ( F @ X ) ) ) ) ) ).
% antimonoE
thf(fact_5218_antimonoI,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F: A > B] :
( ! [X3: A,Y5: A] :
( ( ord_less_eq @ A @ X3 @ Y5 )
=> ( ord_less_eq @ B @ ( F @ Y5 ) @ ( F @ X3 ) ) )
=> ( order_antimono @ A @ B @ F ) ) ) ).
% antimonoI
thf(fact_5219_antimono__def,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ( ( order_antimono @ A @ B )
= ( ^ [F2: A > B] :
! [X2: A,Y6: A] :
( ( ord_less_eq @ A @ X2 @ Y6 )
=> ( ord_less_eq @ B @ ( F2 @ Y6 ) @ ( F2 @ X2 ) ) ) ) ) ) ).
% antimono_def
thf(fact_5220_transfer__rule__of__bool,axiom,
! [A: $tType,B: $tType] :
( ( ( zero_neq_one @ B )
& ( zero_neq_one @ A ) )
=> ! [R2: A > B > $o] :
( ( R2 @ ( zero_zero @ A ) @ ( zero_zero @ B ) )
=> ( ( R2 @ ( one_one @ A ) @ ( one_one @ B ) )
=> ( bNF_rel_fun @ $o @ $o @ A @ B
@ ^ [Y: $o,Z2: $o] : Y = Z2
@ R2
@ ( zero_neq_one_of_bool @ A )
@ ( zero_neq_one_of_bool @ B ) ) ) ) ) ).
% transfer_rule_of_bool
thf(fact_5221_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,Y6: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( times_times @ int @ ( product_fst @ int @ int @ X2 ) @ ( product_fst @ int @ int @ Y6 ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ X2 ) @ ( product_snd @ int @ int @ Y6 ) ) )
@ ^ [X2: product_prod @ int @ int,Y6: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( times_times @ int @ ( product_fst @ int @ int @ X2 ) @ ( product_fst @ int @ int @ Y6 ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ X2 ) @ ( product_snd @ int @ int @ Y6 ) ) ) ) ).
% times_rat.rsp
thf(fact_5222_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,Y6: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( plus_plus @ int @ ( times_times @ int @ ( product_fst @ int @ int @ X2 ) @ ( product_snd @ int @ int @ Y6 ) ) @ ( times_times @ int @ ( product_fst @ int @ int @ Y6 ) @ ( product_snd @ int @ int @ X2 ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ X2 ) @ ( product_snd @ int @ int @ Y6 ) ) )
@ ^ [X2: product_prod @ int @ int,Y6: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( plus_plus @ int @ ( times_times @ int @ ( product_fst @ int @ int @ X2 ) @ ( product_snd @ int @ int @ Y6 ) ) @ ( times_times @ int @ ( product_fst @ int @ int @ Y6 ) @ ( product_snd @ int @ int @ X2 ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ X2 ) @ ( product_snd @ int @ int @ Y6 ) ) ) ) ).
% plus_rat.rsp
thf(fact_5223_integer__of__nat__eq__of__nat,axiom,
( code_integer_of_nat
= ( semiring_1_of_nat @ code_integer ) ) ).
% integer_of_nat_eq_of_nat
thf(fact_5224_antimono__iff__le__Suc,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( order_antimono @ nat @ A )
= ( ^ [F2: nat > A] :
! [N3: nat] : ( ord_less_eq @ A @ ( F2 @ ( suc @ N3 ) ) @ ( F2 @ N3 ) ) ) ) ) ).
% antimono_iff_le_Suc
thf(fact_5225_integer__of__nat__0,axiom,
( ( code_integer_of_nat @ ( zero_zero @ nat ) )
= ( zero_zero @ code_integer ) ) ).
% integer_of_nat_0
thf(fact_5226_integer__of__nat_Oabs__eq,axiom,
( code_integer_of_nat
= ( ^ [X2: nat] : ( code_integer_of_int @ ( semiring_1_of_nat @ int @ X2 ) ) ) ) ).
% integer_of_nat.abs_eq
thf(fact_5227_integer__of__nat__numeral,axiom,
! [N: num] :
( ( code_integer_of_nat @ ( numeral_numeral @ nat @ N ) )
= ( numeral_numeral @ code_integer @ N ) ) ).
% integer_of_nat_numeral
thf(fact_5228_min__of__antimono,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( linorder @ B ) )
=> ! [F: A > B,X: A,Y3: A] :
( ( order_antimono @ A @ B @ F )
=> ( ( ord_min @ B @ ( F @ X ) @ ( F @ Y3 ) )
= ( F @ ( ord_max @ A @ X @ Y3 ) ) ) ) ) ).
% min_of_antimono
thf(fact_5229_max__of__antimono,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( linorder @ B ) )
=> ! [F: A > B,X: A,Y3: A] :
( ( order_antimono @ A @ B @ F )
=> ( ( ord_max @ B @ ( F @ X ) @ ( F @ Y3 ) )
= ( F @ ( ord_min @ A @ X @ Y3 ) ) ) ) ) ).
% max_of_antimono
thf(fact_5230_integer__of__nat__1,axiom,
( ( code_integer_of_nat @ ( one_one @ nat ) )
= ( one_one @ code_integer ) ) ).
% integer_of_nat_1
thf(fact_5231_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_5232_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_5233_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,Y6: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( plus_plus @ int @ ( times_times @ int @ ( product_fst @ int @ int @ X2 ) @ ( product_snd @ int @ int @ Y6 ) ) @ ( times_times @ int @ ( product_fst @ int @ int @ Y6 ) @ ( product_snd @ int @ int @ X2 ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ X2 ) @ ( product_snd @ int @ int @ Y6 ) ) )
@ ( plus_plus @ rat ) ) ).
% plus_rat.transfer
thf(fact_5234_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_5235_positive_Otransfer,axiom,
( bNF_rel_fun @ ( product_prod @ int @ int ) @ rat @ $o @ $o @ pcr_rat
@ ^ [Y: $o,Z2: $o] : Y = 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_5236_one__rat_Otransfer,axiom,
pcr_rat @ ( product_Pair @ int @ int @ ( one_one @ int ) @ ( one_one @ int ) ) @ ( one_one @ rat ) ).
% one_rat.transfer
thf(fact_5237_zero__rat_Otransfer,axiom,
pcr_rat @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) ) @ ( zero_zero @ rat ) ).
% zero_rat.transfer
thf(fact_5238_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_5239_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,Y6: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( times_times @ int @ ( product_fst @ int @ int @ X2 ) @ ( product_fst @ int @ int @ Y6 ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ X2 ) @ ( product_snd @ int @ int @ Y6 ) ) )
@ ( times_times @ rat ) ) ).
% times_rat.transfer
thf(fact_5240_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,Y6: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U3: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ X2 @ U3 ) @ ( times_times @ nat @ Y6 @ V5 ) ) @ ( plus_plus @ nat @ ( times_times @ nat @ X2 @ V5 ) @ ( times_times @ nat @ Y6 @ U3 ) ) ) ) )
@ ( times_times @ int ) ) ).
% times_int.transfer
thf(fact_5241_num_Orec__transfer,axiom,
! [A: $tType,B: $tType,S3: A > B > $o] :
( bNF_rel_fun @ A @ B @ ( ( num > A > A ) > ( num > A > A ) > num > A ) @ ( ( num > B > B ) > ( num > B > B ) > num > B ) @ S3
@ ( bNF_rel_fun @ ( num > A > A ) @ ( num > B > B ) @ ( ( num > A > A ) > num > A ) @ ( ( num > B > B ) > num > B )
@ ( bNF_rel_fun @ num @ num @ ( A > A ) @ ( B > B )
@ ^ [Y: num,Z2: num] : Y = Z2
@ ( bNF_rel_fun @ A @ B @ A @ B @ S3 @ S3 ) )
@ ( bNF_rel_fun @ ( num > A > A ) @ ( num > B > B ) @ ( num > A ) @ ( num > B )
@ ( bNF_rel_fun @ num @ num @ ( A > A ) @ ( B > B )
@ ^ [Y: num,Z2: num] : Y = Z2
@ ( bNF_rel_fun @ A @ B @ A @ B @ S3 @ S3 ) )
@ ( bNF_rel_fun @ num @ num @ A @ B
@ ^ [Y: num,Z2: num] : Y = Z2
@ S3 ) ) )
@ ( rec_num @ A )
@ ( rec_num @ B ) ) ).
% num.rec_transfer
thf(fact_5242_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_5243_verit__eq__simplify_I20_J,axiom,
! [A: $tType,F1: A,F22: num > A > A,F32: num > A > A,X22: num] :
( ( rec_num @ A @ F1 @ F22 @ F32 @ ( bit0 @ X22 ) )
= ( F22 @ X22 @ ( rec_num @ A @ F1 @ F22 @ F32 @ X22 ) ) ) ).
% verit_eq_simplify(20)
thf(fact_5244_verit__eq__simplify_I19_J,axiom,
! [A: $tType,F1: A,F22: num > A > A,F32: num > A > A] :
( ( rec_num @ A @ F1 @ F22 @ F32 @ one2 )
= F1 ) ).
% verit_eq_simplify(19)
thf(fact_5245_verit__eq__simplify_I21_J,axiom,
! [A: $tType,F1: A,F22: num > A > A,F32: num > A > A,X32: num] :
( ( rec_num @ A @ F1 @ F22 @ F32 @ ( bit1 @ X32 ) )
= ( F32 @ X32 @ ( rec_num @ A @ F1 @ F22 @ F32 @ X32 ) ) ) ).
% verit_eq_simplify(21)
thf(fact_5246_zero__int_Otransfer,axiom,
pcr_int @ ( product_Pair @ nat @ nat @ ( zero_zero @ nat ) @ ( zero_zero @ nat ) ) @ ( zero_zero @ int ) ).
% zero_int.transfer
thf(fact_5247_int__transfer,axiom,
( bNF_rel_fun @ nat @ nat @ ( product_prod @ nat @ nat ) @ int
@ ^ [Y: nat,Z2: nat] : Y = Z2
@ pcr_int
@ ^ [N3: nat] : ( product_Pair @ nat @ nat @ N3 @ ( zero_zero @ nat ) )
@ ( semiring_1_of_nat @ int ) ) ).
% int_transfer
thf(fact_5248_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,Y6: nat] : ( product_Pair @ nat @ nat @ Y6 @ X2 ) )
@ ( uminus_uminus @ int ) ) ).
% uminus_int.transfer
thf(fact_5249_nat_Otransfer,axiom,
( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ nat @ nat @ pcr_int
@ ^ [Y: nat,Z2: nat] : Y = Z2
@ ( product_case_prod @ nat @ nat @ nat @ ( minus_minus @ nat ) )
@ nat2 ) ).
% nat.transfer
thf(fact_5250_one__int_Otransfer,axiom,
pcr_int @ ( product_Pair @ nat @ nat @ ( one_one @ nat ) @ ( zero_zero @ nat ) ) @ ( one_one @ int ) ).
% one_int.transfer
thf(fact_5251_of__int_Otransfer,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ A @ A @ pcr_int
@ ^ [Y: A,Z2: A] : Y = Z2
@ ( product_case_prod @ nat @ nat @ A
@ ^ [I5: nat,J3: nat] : ( minus_minus @ A @ ( semiring_1_of_nat @ A @ I5 ) @ ( semiring_1_of_nat @ A @ J3 ) ) )
@ ( ring_1_of_int @ A ) ) ) ).
% of_int.transfer
thf(fact_5252_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
@ ^ [Y: $o,Z2: $o] : Y = Z2 )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X2: nat,Y6: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U3: nat,V5: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ X2 @ V5 ) @ ( plus_plus @ nat @ U3 @ Y6 ) ) ) )
@ ( ord_less @ int ) ) ).
% less_int.transfer
thf(fact_5253_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
@ ^ [Y: $o,Z2: $o] : Y = Z2 )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X2: nat,Y6: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U3: nat,V5: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ X2 @ V5 ) @ ( plus_plus @ nat @ U3 @ Y6 ) ) ) )
@ ( ord_less_eq @ int ) ) ).
% less_eq_int.transfer
thf(fact_5254_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,Y6: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U3: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X2 @ U3 ) @ ( plus_plus @ nat @ Y6 @ V5 ) ) ) )
@ ( plus_plus @ int ) ) ).
% plus_int.transfer
thf(fact_5255_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,Y6: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U3: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X2 @ V5 ) @ ( plus_plus @ nat @ Y6 @ U3 ) ) ) )
@ ( minus_minus @ int ) ) ).
% minus_int.transfer
thf(fact_5256_num_Ocase__transfer,axiom,
! [A: $tType,B: $tType,S3: A > B > $o] :
( bNF_rel_fun @ A @ B @ ( ( num > A ) > ( num > A ) > num > A ) @ ( ( num > B ) > ( num > B ) > num > B ) @ S3
@ ( bNF_rel_fun @ ( num > A ) @ ( num > B ) @ ( ( num > A ) > num > A ) @ ( ( num > B ) > num > B )
@ ( bNF_rel_fun @ num @ num @ A @ B
@ ^ [Y: num,Z2: num] : Y = Z2
@ S3 )
@ ( bNF_rel_fun @ ( num > A ) @ ( num > B ) @ ( num > A ) @ ( num > B )
@ ( bNF_rel_fun @ num @ num @ A @ B
@ ^ [Y: num,Z2: num] : Y = Z2
@ S3 )
@ ( bNF_rel_fun @ num @ num @ A @ B
@ ^ [Y: num,Z2: num] : Y = Z2
@ S3 ) ) )
@ ( case_num @ A )
@ ( case_num @ B ) ) ).
% num.case_transfer
thf(fact_5257_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,Y6: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U3: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ X2 @ U3 ) @ ( times_times @ nat @ Y6 @ V5 ) ) @ ( plus_plus @ nat @ ( times_times @ nat @ X2 @ V5 ) @ ( times_times @ nat @ Y6 @ U3 ) ) ) ) )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X2: nat,Y6: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U3: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ X2 @ U3 ) @ ( times_times @ nat @ Y6 @ V5 ) ) @ ( plus_plus @ nat @ ( times_times @ nat @ X2 @ V5 ) @ ( times_times @ nat @ Y6 @ U3 ) ) ) ) ) ) ).
% times_int.rsp
thf(fact_5258_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_5259_intrel__iff,axiom,
! [X: nat,Y3: nat,U: nat,V: nat] :
( ( intrel @ ( product_Pair @ nat @ nat @ X @ Y3 ) @ ( product_Pair @ nat @ nat @ U @ V ) )
= ( ( plus_plus @ nat @ X @ V )
= ( plus_plus @ nat @ U @ Y3 ) ) ) ).
% intrel_iff
thf(fact_5260_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
@ ^ [Y: $o,Z2: $o] : Y = Z2 )
@ intrel
@ ^ [Y: int,Z2: int] : Y = Z2 ) ).
% int.rel_eq_transfer
thf(fact_5261_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_5262_num_Ocase__distrib,axiom,
! [B: $tType,A: $tType,H2: A > B,F1: A,F22: num > A,F32: num > A,Num: num] :
( ( H2 @ ( case_num @ A @ F1 @ F22 @ F32 @ Num ) )
= ( case_num @ B @ ( H2 @ F1 )
@ ^ [X2: num] : ( H2 @ ( F22 @ X2 ) )
@ ^ [X2: num] : ( H2 @ ( F32 @ X2 ) )
@ Num ) ) ).
% num.case_distrib
thf(fact_5263_verit__eq__simplify_I17_J,axiom,
! [A: $tType,F1: A,F22: num > A,F32: num > A,X22: num] :
( ( case_num @ A @ F1 @ F22 @ F32 @ ( bit0 @ X22 ) )
= ( F22 @ X22 ) ) ).
% verit_eq_simplify(17)
thf(fact_5264_verit__eq__simplify_I16_J,axiom,
! [A: $tType,F1: A,F22: num > A,F32: num > A] :
( ( case_num @ A @ F1 @ F22 @ F32 @ one2 )
= F1 ) ).
% verit_eq_simplify(16)
thf(fact_5265_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_5266_verit__eq__simplify_I18_J,axiom,
! [A: $tType,F1: A,F22: num > A,F32: num > A,X32: num] :
( ( case_num @ A @ F1 @ F22 @ F32 @ ( bit1 @ X32 ) )
= ( F32 @ X32 ) ) ).
% verit_eq_simplify(18)
thf(fact_5267_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_5268_size__list__estimation,axiom,
! [A: $tType,X: A,Xs: list @ A,Y3: nat,F: A > nat] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( ord_less @ nat @ Y3 @ ( F @ X ) )
=> ( ord_less @ nat @ Y3 @ ( size_list @ A @ F @ Xs ) ) ) ) ).
% size_list_estimation
thf(fact_5269_size__list__pointwise,axiom,
! [A: $tType,Xs: list @ A,F: A > nat,G: A > nat] :
( ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( ord_less_eq @ nat @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_eq @ nat @ ( size_list @ A @ F @ Xs ) @ ( size_list @ A @ G @ Xs ) ) ) ).
% size_list_pointwise
thf(fact_5270_size__list__estimation_H,axiom,
! [A: $tType,X: A,Xs: list @ A,Y3: nat,F: A > nat] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( ord_less_eq @ nat @ Y3 @ ( F @ X ) )
=> ( ord_less_eq @ nat @ Y3 @ ( size_list @ A @ F @ Xs ) ) ) ) ).
% size_list_estimation'
thf(fact_5271_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,Y6: nat] : ( product_Pair @ nat @ nat @ Y6 @ X2 ) )
@ ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [X2: nat,Y6: nat] : ( product_Pair @ nat @ nat @ Y6 @ X2 ) ) ) ).
% uminus_int.rsp
thf(fact_5272_nat_Orsp,axiom,
( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ nat @ nat @ intrel
@ ^ [Y: nat,Z2: nat] : Y = Z2
@ ( product_case_prod @ nat @ nat @ nat @ ( minus_minus @ nat ) )
@ ( product_case_prod @ nat @ nat @ nat @ ( minus_minus @ nat ) ) ) ).
% nat.rsp
thf(fact_5273_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_5274_of__int_Orsp,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ A @ A @ intrel
@ ^ [Y: A,Z2: A] : Y = Z2
@ ( product_case_prod @ nat @ nat @ A
@ ^ [I5: nat,J3: nat] : ( minus_minus @ A @ ( semiring_1_of_nat @ A @ I5 ) @ ( semiring_1_of_nat @ A @ J3 ) ) )
@ ( product_case_prod @ nat @ nat @ A
@ ^ [I5: nat,J3: nat] : ( minus_minus @ A @ ( semiring_1_of_nat @ A @ I5 ) @ ( semiring_1_of_nat @ A @ J3 ) ) ) ) ) ).
% of_int.rsp
thf(fact_5275_intrel__def,axiom,
( intrel
= ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X2: nat,Y6: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U3: nat,V5: nat] :
( ( plus_plus @ nat @ X2 @ V5 )
= ( plus_plus @ nat @ U3 @ Y6 ) ) ) ) ) ).
% intrel_def
thf(fact_5276_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
@ ^ [Y: $o,Z2: $o] : Y = Z2 )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X2: nat,Y6: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U3: nat,V5: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ X2 @ V5 ) @ ( plus_plus @ nat @ U3 @ Y6 ) ) ) )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X2: nat,Y6: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U3: nat,V5: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ X2 @ V5 ) @ ( plus_plus @ nat @ U3 @ Y6 ) ) ) ) ) ).
% less_int.rsp
thf(fact_5277_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
@ ^ [Y: $o,Z2: $o] : Y = Z2 )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X2: nat,Y6: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U3: nat,V5: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ X2 @ V5 ) @ ( plus_plus @ nat @ U3 @ Y6 ) ) ) )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X2: nat,Y6: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U3: nat,V5: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ X2 @ V5 ) @ ( plus_plus @ nat @ U3 @ Y6 ) ) ) ) ) ).
% less_eq_int.rsp
thf(fact_5278_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,Y6: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U3: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X2 @ U3 ) @ ( plus_plus @ nat @ Y6 @ V5 ) ) ) )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X2: nat,Y6: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U3: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X2 @ U3 ) @ ( plus_plus @ nat @ Y6 @ V5 ) ) ) ) ) ).
% plus_int.rsp
thf(fact_5279_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,Y6: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U3: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X2 @ V5 ) @ ( plus_plus @ nat @ Y6 @ U3 ) ) ) )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X2: nat,Y6: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U3: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X2 @ V5 ) @ ( plus_plus @ nat @ Y6 @ U3 ) ) ) ) ) ).
% minus_int.rsp
thf(fact_5280_Nitpick_Osize__list__simp_I1_J,axiom,
! [A: $tType] :
( ( size_list @ A )
= ( ^ [F2: A > nat,Xs3: list @ A] :
( if @ nat
@ ( Xs3
= ( nil @ A ) )
@ ( zero_zero @ nat )
@ ( suc @ ( plus_plus @ nat @ ( F2 @ ( hd @ A @ Xs3 ) ) @ ( size_list @ A @ F2 @ ( tl @ A @ Xs3 ) ) ) ) ) ) ) ).
% Nitpick.size_list_simp(1)
thf(fact_5281_execute__make,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [N: nat,F: nat > A,H2: heap_ext @ product_unit] :
( ( heap_Time_execute @ ( array @ A ) @ ( array_make @ A @ N @ F ) @ H2 )
= ( some @ ( product_prod @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )
@ ( product_case_prod @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( product_prod @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )
@ ^ [R5: array @ A,H5: heap_ext @ product_unit] : ( product_Pair @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R5 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H5 @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) )
@ ( array_alloc @ A @ ( map @ nat @ A @ F @ ( upt @ ( zero_zero @ nat ) @ N ) ) @ H2 ) ) ) ) ) ).
% execute_make
thf(fact_5282_relpow__finite__bounded1,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),K: nat] :
( ( finite_finite2 @ ( product_prod @ A @ A ) @ R2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ K @ R2 )
@ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ A ) )
@ ( image @ nat @ ( set @ ( product_prod @ A @ A ) )
@ ^ [N3: nat] : ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N3 @ R2 )
@ ( collect @ nat
@ ^ [N3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
& ( ord_less_eq @ nat @ N3 @ ( finite_card @ ( product_prod @ A @ A ) @ R2 ) ) ) ) ) ) ) ) ) ).
% relpow_finite_bounded1
thf(fact_5283_tl__upt,axiom,
! [M: nat,N: nat] :
( ( tl @ nat @ ( upt @ M @ N ) )
= ( upt @ ( suc @ M ) @ N ) ) ).
% tl_upt
thf(fact_5284_relpow__1,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( one_one @ nat ) @ R2 )
= R2 ) ).
% relpow_1
thf(fact_5285_length__tl,axiom,
! [A: $tType,Xs: list @ A] :
( ( size_size @ ( list @ A ) @ ( tl @ A @ Xs ) )
= ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) ) ) ).
% length_tl
thf(fact_5286_finite__relpow,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),N: nat] :
( ( finite_finite2 @ ( product_prod @ A @ A ) @ R2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( finite_finite2 @ ( product_prod @ A @ A ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R2 ) ) ) ) ).
% finite_relpow
thf(fact_5287_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_5288_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_5289_execute__guard_I2_J,axiom,
! [A: $tType,P: ( heap_ext @ product_unit ) > $o,H2: heap_ext @ product_unit,F: ( heap_ext @ product_unit ) > ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )] :
( ( P @ H2 )
=> ( ( heap_Time_execute @ A @ ( heap_Time_guard @ A @ P @ F ) @ H2 )
= ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( F @ H2 ) ) ) ) ).
% execute_guard(2)
thf(fact_5290_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_5291_effectI,axiom,
! [A: $tType,C2: heap_Time_Heap @ A,H2: heap_ext @ product_unit,R: A,H: heap_ext @ product_unit,N: nat] :
( ( ( heap_Time_execute @ A @ C2 @ H2 )
= ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H @ N ) ) ) )
=> ( heap_Time_effect @ A @ C2 @ H2 @ H @ R @ N ) ) ).
% effectI
thf(fact_5292_effect__def,axiom,
! [A: $tType] :
( ( heap_Time_effect @ A )
= ( ^ [C4: heap_Time_Heap @ A,H4: heap_ext @ product_unit,H5: heap_ext @ product_unit,R5: A,N3: 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 ) @ R5 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H5 @ N3 ) ) ) ) ) ) ).
% effect_def
thf(fact_5293_sorted__tl,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( tl @ A @ Xs ) ) ) ) ).
% sorted_tl
thf(fact_5294_relpow__Suc__E,axiom,
! [A: $tType,X: A,Z3: A,N: nat,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z3 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( suc @ N ) @ R2 ) )
=> ~ ! [Y5: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y5 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R2 ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y5 @ Z3 ) @ R2 ) ) ) ).
% relpow_Suc_E
thf(fact_5295_relpow__Suc__I,axiom,
! [A: $tType,X: A,Y3: A,N: nat,R2: set @ ( product_prod @ A @ A ),Z3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R2 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z3 ) @ R2 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z3 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( suc @ N ) @ R2 ) ) ) ) ).
% relpow_Suc_I
thf(fact_5296_relpow__Suc__D2,axiom,
! [A: $tType,X: A,Z3: A,N: nat,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z3 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( suc @ N ) @ R2 ) )
=> ? [Y5: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y5 ) @ R2 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y5 @ Z3 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R2 ) ) ) ) ).
% relpow_Suc_D2
thf(fact_5297_relpow__Suc__E2,axiom,
! [A: $tType,X: A,Z3: A,N: nat,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z3 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( suc @ N ) @ R2 ) )
=> ~ ! [Y5: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y5 ) @ R2 )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y5 @ Z3 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R2 ) ) ) ) ).
% relpow_Suc_E2
thf(fact_5298_relpow__Suc__I2,axiom,
! [A: $tType,X: A,Y3: A,R2: set @ ( product_prod @ A @ A ),Z3: A,N: nat] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z3 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R2 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z3 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( suc @ N ) @ R2 ) ) ) ) ).
% relpow_Suc_I2
thf(fact_5299_relpow__0__I,axiom,
! [A: $tType,X: A,R2: set @ ( product_prod @ A @ A )] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ X ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( zero_zero @ nat ) @ R2 ) ) ).
% relpow_0_I
thf(fact_5300_relpow__0__E,axiom,
! [A: $tType,X: A,Y3: A,R2: 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 ) @ R2 ) )
=> ( X = Y3 ) ) ).
% relpow_0_E
thf(fact_5301_take__tl,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( take @ A @ N @ ( tl @ A @ Xs ) )
= ( tl @ A @ ( take @ A @ ( suc @ N ) @ Xs ) ) ) ).
% take_tl
thf(fact_5302_drop__Suc,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( drop @ A @ ( suc @ N ) @ Xs )
= ( drop @ A @ N @ ( tl @ A @ Xs ) ) ) ).
% drop_Suc
thf(fact_5303_execute__assert_I1_J,axiom,
! [A: $tType,P: A > $o,X: A,H2: heap_ext @ product_unit] :
( ( P @ X )
=> ( ( heap_Time_execute @ A @ ( heap_Time_assert @ A @ P @ X ) @ H2 )
= ( 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 @ ( one_one @ nat ) ) ) ) ) ) ).
% execute_assert(1)
thf(fact_5304_execute__tap,axiom,
! [A: $tType,F: ( heap_ext @ product_unit ) > A,H2: heap_ext @ product_unit] :
( ( heap_Time_execute @ A @ ( heap_Time_tap @ A @ F ) @ H2 )
= ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( F @ H2 ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H2 @ ( one_one @ nat ) ) ) ) ) ).
% execute_tap
thf(fact_5305_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_5306_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_5307_relpow__E,axiom,
! [A: $tType,X: A,Z3: A,N: nat,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z3 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R2 ) )
=> ( ( ( N
= ( zero_zero @ nat ) )
=> ( X != Z3 ) )
=> ~ ! [Y5: A,M6: nat] :
( ( N
= ( suc @ M6 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y5 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ M6 @ R2 ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y5 @ Z3 ) @ R2 ) ) ) ) ) ).
% relpow_E
thf(fact_5308_relpow__E2,axiom,
! [A: $tType,X: A,Z3: A,N: nat,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z3 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R2 ) )
=> ( ( ( N
= ( zero_zero @ nat ) )
=> ( X != Z3 ) )
=> ~ ! [Y5: A,M6: nat] :
( ( N
= ( suc @ M6 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y5 ) @ R2 )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y5 @ Z3 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ M6 @ R2 ) ) ) ) ) ) ).
% relpow_E2
thf(fact_5309_tl__take,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( tl @ A @ ( take @ A @ N @ Xs ) )
= ( take @ A @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ ( tl @ A @ Xs ) ) ) ).
% tl_take
thf(fact_5310_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_5311_relpow__finite__bounded,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),K: nat] :
( ( finite_finite2 @ ( product_prod @ A @ A ) @ R2 )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ K @ R2 )
@ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ A ) )
@ ( image @ nat @ ( set @ ( product_prod @ A @ A ) )
@ ^ [N3: nat] : ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N3 @ R2 )
@ ( collect @ nat
@ ^ [N3: nat] : ( ord_less_eq @ nat @ N3 @ ( finite_card @ ( product_prod @ A @ A ) @ R2 ) ) ) ) ) ) ) ).
% relpow_finite_bounded
thf(fact_5312_Nitpick_Osize__list__simp_I2_J,axiom,
! [A: $tType] :
( ( size_size @ ( list @ A ) )
= ( ^ [Xs3: list @ A] :
( if @ nat
@ ( Xs3
= ( nil @ A ) )
@ ( zero_zero @ nat )
@ ( suc @ ( size_size @ ( list @ A ) @ ( tl @ A @ Xs3 ) ) ) ) ) ) ).
% Nitpick.size_list_simp(2)
thf(fact_5313_nth__tl,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ ( tl @ A @ Xs ) ) )
=> ( ( nth @ A @ ( tl @ A @ Xs ) @ N )
= ( nth @ A @ Xs @ ( suc @ N ) ) ) ) ).
% nth_tl
thf(fact_5314_relpow__fun__conv,axiom,
! [A: $tType,A3: A,B2: A,N: nat,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R2 ) )
= ( ? [F2: nat > A] :
( ( ( F2 @ ( zero_zero @ nat ) )
= A3 )
& ( ( F2 @ N )
= B2 )
& ! [I5: nat] :
( ( ord_less @ nat @ I5 @ N )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( F2 @ I5 ) @ ( F2 @ ( suc @ I5 ) ) ) @ R2 ) ) ) ) ) ).
% relpow_fun_conv
thf(fact_5315_execute__of__list,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [Xs: list @ A,H2: heap_ext @ product_unit] :
( ( heap_Time_execute @ ( array @ A ) @ ( array_of_list @ A @ Xs ) @ H2 )
= ( some @ ( product_prod @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )
@ ( product_case_prod @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( product_prod @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )
@ ^ [R5: array @ A,H5: heap_ext @ product_unit] : ( product_Pair @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R5 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H5 @ ( plus_plus @ nat @ ( one_one @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) )
@ ( array_alloc @ A @ Xs @ H2 ) ) ) ) ) ).
% execute_of_list
thf(fact_5316_execute__new,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [N: nat,X: A,H2: heap_ext @ product_unit] :
( ( heap_Time_execute @ ( array @ A ) @ ( array_new @ A @ N @ X ) @ H2 )
= ( some @ ( product_prod @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )
@ ( product_case_prod @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( product_prod @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )
@ ^ [R5: array @ A,H5: heap_ext @ product_unit] : ( product_Pair @ ( array @ A ) @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R5 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H5 @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) )
@ ( array_alloc @ A @ ( replicate @ A @ N @ X ) @ H2 ) ) ) ) ) ).
% execute_new
thf(fact_5317_take__Suc,axiom,
! [A: $tType,Xs: list @ A,N: nat] :
( ( Xs
!= ( nil @ A ) )
=> ( ( take @ A @ ( suc @ N ) @ Xs )
= ( cons @ A @ ( hd @ A @ Xs ) @ ( take @ A @ N @ ( tl @ A @ Xs ) ) ) ) ) ).
% take_Suc
thf(fact_5318_ntrancl__def,axiom,
! [A: $tType] :
( ( transitive_ntrancl @ A )
= ( ^ [N3: nat,R6: set @ ( product_prod @ A @ A )] :
( complete_Sup_Sup @ ( set @ ( product_prod @ A @ A ) )
@ ( image @ nat @ ( set @ ( product_prod @ A @ A ) )
@ ^ [I5: nat] : ( compow @ ( set @ ( product_prod @ A @ A ) ) @ I5 @ R6 )
@ ( collect @ nat
@ ^ [I5: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ I5 )
& ( ord_less_eq @ nat @ I5 @ ( suc @ N3 ) ) ) ) ) ) ) ) ).
% ntrancl_def
thf(fact_5319_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_5320_trancl__finite__eq__relpow,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( finite_finite2 @ ( product_prod @ A @ A ) @ R2 )
=> ( ( transitive_trancl @ A @ R2 )
= ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ A ) )
@ ( image @ nat @ ( set @ ( product_prod @ A @ A ) )
@ ^ [N3: nat] : ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N3 @ R2 )
@ ( collect @ nat
@ ^ [N3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
& ( ord_less_eq @ nat @ N3 @ ( finite_card @ ( product_prod @ A @ A ) @ R2 ) ) ) ) ) ) ) ) ).
% trancl_finite_eq_relpow
thf(fact_5321_ntrancl__Zero,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( transitive_ntrancl @ A @ ( zero_zero @ nat ) @ R2 )
= R2 ) ).
% ntrancl_Zero
thf(fact_5322_take__bit__num__simps_I2_J,axiom,
! [N: nat] :
( ( bit_take_bit_num @ ( suc @ N ) @ one2 )
= ( some @ num @ one2 ) ) ).
% take_bit_num_simps(2)
thf(fact_5323_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_5324_execute__len,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [A3: array @ A,H2: heap_ext @ product_unit] :
( ( heap_Time_execute @ nat @ ( array_len @ A @ A3 ) @ H2 )
= ( some @ ( product_prod @ nat @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ nat @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ ( array_length @ A @ H2 @ A3 ) @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H2 @ ( one_one @ nat ) ) ) ) ) ) ).
% execute_len
thf(fact_5325_finite__trancl__ntranl,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( finite_finite2 @ ( product_prod @ A @ A ) @ R2 )
=> ( ( transitive_trancl @ A @ R2 )
= ( transitive_ntrancl @ A @ ( minus_minus @ nat @ ( finite_card @ ( product_prod @ A @ A ) @ R2 ) @ ( one_one @ nat ) ) @ R2 ) ) ) ).
% finite_trancl_ntranl
thf(fact_5326_trancl__set__ntrancl,axiom,
! [A: $tType,Xs: list @ ( product_prod @ A @ A )] :
( ( transitive_trancl @ A @ ( set2 @ ( product_prod @ A @ A ) @ Xs ) )
= ( transitive_ntrancl @ A @ ( minus_minus @ nat @ ( finite_card @ ( product_prod @ A @ A ) @ ( set2 @ ( product_prod @ A @ A ) @ Xs ) ) @ ( one_one @ nat ) ) @ ( set2 @ ( product_prod @ A @ A ) @ Xs ) ) ) ).
% trancl_set_ntrancl
thf(fact_5327_trancl__power,axiom,
! [A: $tType,P5: product_prod @ A @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ P5 @ ( transitive_trancl @ A @ R2 ) )
= ( ? [N3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
& ( member @ ( product_prod @ A @ A ) @ P5 @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N3 @ R2 ) ) ) ) ) ).
% trancl_power
thf(fact_5328_option_Osize__neq,axiom,
! [A: $tType,X: option @ A] :
( ( size_size @ ( option @ A ) @ X )
!= ( zero_zero @ nat ) ) ).
% option.size_neq
thf(fact_5329_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_5330_sorted__find__Min,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,P: A > $o] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( ? [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
& ( P @ X4 ) )
=> ( ( find @ A @ P @ Xs )
= ( some @ A
@ ( lattic643756798350308766er_Min @ A
@ ( collect @ A
@ ^ [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
& ( P @ X2 ) ) ) ) ) ) ) ) ) ).
% sorted_find_Min
thf(fact_5331_timeFrame_Osimps_I1_J,axiom,
! [A: $tType,N: nat,R: A,H2: heap_ext @ product_unit,N2: 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 @ H2 @ N2 ) ) ) )
= ( 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 @ ( plus_plus @ nat @ N @ N2 ) ) ) ) ) ).
% timeFrame.simps(1)
thf(fact_5332_timeFrame__zero,axiom,
! [A: $tType,H2: option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )] :
( ( heap_Time_timeFrame @ A @ ( zero_zero @ nat ) @ H2 )
= H2 ) ).
% timeFrame_zero
thf(fact_5333_timeFrame__assoc,axiom,
! [A: $tType,N: nat,N2: nat,F: option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )] :
( ( heap_Time_timeFrame @ A @ N @ ( heap_Time_timeFrame @ A @ N2 @ F ) )
= ( heap_Time_timeFrame @ A @ ( plus_plus @ nat @ N @ N2 ) @ F ) ) ).
% timeFrame_assoc
thf(fact_5334_find__Some__iff2,axiom,
! [A: $tType,X: A,P: A > $o,Xs: list @ A] :
( ( ( some @ A @ X )
= ( find @ A @ P @ Xs ) )
= ( ? [I5: nat] :
( ( ord_less @ nat @ I5 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( P @ ( nth @ A @ Xs @ I5 ) )
& ( X
= ( nth @ A @ Xs @ I5 ) )
& ! [J3: nat] :
( ( ord_less @ nat @ J3 @ I5 )
=> ~ ( P @ ( nth @ A @ Xs @ J3 ) ) ) ) ) ) ).
% find_Some_iff2
thf(fact_5335_find__Some__iff,axiom,
! [A: $tType,P: A > $o,Xs: list @ A,X: A] :
( ( ( find @ A @ P @ Xs )
= ( some @ A @ X ) )
= ( ? [I5: nat] :
( ( ord_less @ nat @ I5 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( P @ ( nth @ A @ Xs @ I5 ) )
& ( X
= ( nth @ A @ Xs @ I5 ) )
& ! [J3: nat] :
( ( ord_less @ nat @ J3 @ I5 )
=> ~ ( P @ ( nth @ A @ Xs @ J3 ) ) ) ) ) ) ).
% find_Some_iff
thf(fact_5336_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_5337_execute__bind_I1_J,axiom,
! [A: $tType,B: $tType,F: heap_Time_Heap @ A,H2: heap_ext @ product_unit,X: A,H: heap_ext @ product_unit,N: nat,G: A > ( heap_Time_Heap @ B )] :
( ( ( heap_Time_execute @ A @ F @ H2 )
= ( 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 @ N ) ) ) )
=> ( ( heap_Time_execute @ B @ ( heap_Time_bind @ A @ B @ F @ G ) @ H2 )
= ( heap_Time_timeFrame @ B @ N @ ( heap_Time_execute @ B @ ( G @ X ) @ H ) ) ) ) ).
% execute_bind(1)
thf(fact_5338_rtrancl__finite__eq__relpow,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( finite_finite2 @ ( product_prod @ A @ A ) @ R2 )
=> ( ( transitive_rtrancl @ A @ R2 )
= ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ A ) )
@ ( image @ nat @ ( set @ ( product_prod @ A @ A ) )
@ ^ [N3: nat] : ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N3 @ R2 )
@ ( collect @ nat
@ ^ [N3: nat] : ( ord_less_eq @ nat @ N3 @ ( finite_card @ ( product_prod @ A @ A ) @ R2 ) ) ) ) ) ) ) ).
% rtrancl_finite_eq_relpow
thf(fact_5339_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_5340_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_5341_Heap__execute,axiom,
! [A: $tType,F: heap_Time_Heap @ A] :
( ( heap_Time_Heap2 @ A @ ( heap_Time_execute @ A @ F ) )
= F ) ).
% Heap_execute
thf(fact_5342_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_5343_distrib__if__bind,axiom,
! [A: $tType,B: $tType,B2: $o,C2: heap_Time_Heap @ B,D2: heap_Time_Heap @ B,F: B > ( heap_Time_Heap @ A )] :
( ( B2
=> ( ( heap_Time_bind @ B @ A @ ( if @ ( heap_Time_Heap @ B ) @ B2 @ C2 @ D2 ) @ F )
= ( heap_Time_bind @ B @ A @ C2 @ F ) ) )
& ( ~ B2
=> ( ( heap_Time_bind @ B @ A @ ( if @ ( heap_Time_Heap @ B ) @ B2 @ C2 @ D2 ) @ F )
= ( heap_Time_bind @ B @ A @ D2 @ F ) ) ) ) ).
% distrib_if_bind
thf(fact_5344_Heap_Oexhaust,axiom,
! [A: $tType,Y3: heap_Time_Heap @ A] :
~ ! [X3: ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )] :
( Y3
!= ( heap_Time_Heap2 @ A @ X3 ) ) ).
% Heap.exhaust
thf(fact_5345_assert__cong,axiom,
! [B: $tType,A: $tType,P: A > $o,P2: A > $o,F: A > ( heap_Time_Heap @ B ),F5: A > ( heap_Time_Heap @ B ),X: A] :
( ( P = P2 )
=> ( ! [X3: A] :
( ( P2 @ X3 )
=> ( ( F @ X3 )
= ( F5 @ X3 ) ) )
=> ( ( heap_Time_bind @ A @ B @ ( heap_Time_assert @ A @ P @ X ) @ F )
= ( heap_Time_bind @ A @ B @ ( heap_Time_assert @ A @ P2 @ X ) @ F5 ) ) ) ) ).
% assert_cong
thf(fact_5346_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_5347_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_5348_effect__bindE,axiom,
! [B: $tType,A: $tType,F: heap_Time_Heap @ B,G: B > ( heap_Time_Heap @ A ),H2: heap_ext @ product_unit,H3: heap_ext @ product_unit,R4: A,N: nat] :
( ( heap_Time_effect @ A @ ( heap_Time_bind @ B @ A @ F @ G ) @ H2 @ H3 @ R4 @ N )
=> ~ ! [H7: heap_ext @ product_unit,R3: B,N1: nat] :
( ( heap_Time_effect @ B @ F @ H2 @ H7 @ R3 @ N1 )
=> ! [N22: nat] :
( ( heap_Time_effect @ A @ ( G @ R3 ) @ H7 @ H3 @ R4 @ N22 )
=> ( N
!= ( plus_plus @ nat @ N1 @ N22 ) ) ) ) ) ).
% effect_bindE
thf(fact_5349_effect__bindI,axiom,
! [B: $tType,A: $tType,F: heap_Time_Heap @ A,H2: heap_ext @ product_unit,H: heap_ext @ product_unit,R: A,N: nat,G: A > ( heap_Time_Heap @ B ),H3: heap_ext @ product_unit,R4: B,N2: nat] :
( ( heap_Time_effect @ A @ F @ H2 @ H @ R @ N )
=> ( ( heap_Time_effect @ B @ ( G @ R ) @ H @ H3 @ R4 @ N2 )
=> ( heap_Time_effect @ B @ ( heap_Time_bind @ A @ B @ F @ G ) @ H2 @ H3 @ R4 @ ( plus_plus @ nat @ N @ N2 ) ) ) ) ).
% effect_bindI
thf(fact_5350_bind__lift,axiom,
! [A: $tType,B: $tType,F: heap_Time_Heap @ B,G: B > A] :
( ( heap_Time_bind @ B @ A @ F @ ( heap_Time_lift @ B @ A @ G ) )
= ( heap_Time_bind @ B @ A @ F
@ ^ [X2: B] : ( heap_Time_return @ A @ ( G @ X2 ) ) ) ) ).
% bind_lift
thf(fact_5351_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_5352_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_5353_execute__bind__eq__SomeI,axiom,
! [A: $tType,B: $tType,F: heap_Time_Heap @ A,H2: heap_ext @ product_unit,X: A,H: heap_ext @ product_unit,N: nat,G: A > ( heap_Time_Heap @ B ),Y3: B,H3: heap_ext @ product_unit,N2: nat] :
( ( ( heap_Time_execute @ A @ F @ H2 )
= ( 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 @ N ) ) ) )
=> ( ( ( heap_Time_execute @ B @ ( G @ X ) @ 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 @ H3 @ N2 ) ) ) )
=> ( ( heap_Time_execute @ B @ ( heap_Time_bind @ A @ B @ F @ G ) @ 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 @ H3 @ ( plus_plus @ nat @ N @ N2 ) ) ) ) ) ) ) ).
% execute_bind_eq_SomeI
thf(fact_5354_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_5355_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_5356_Heap_Osize__gen,axiom,
! [A: $tType,Xa3: A > nat,X: ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )] :
( ( heap_Time_size_Heap @ A @ Xa3 @ ( heap_Time_Heap2 @ A @ X ) )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% Heap.size_gen
thf(fact_5357_success__bind__executeI,axiom,
! [A: $tType,B: $tType,F: heap_Time_Heap @ A,H2: heap_ext @ product_unit,X: A,H: heap_ext @ product_unit,N: nat,G: A > ( heap_Time_Heap @ B )] :
( ( ( heap_Time_execute @ A @ F @ H2 )
= ( 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 @ N ) ) ) )
=> ( ( heap_Time_success @ B @ ( G @ X ) @ H )
=> ( heap_Time_success @ B @ ( heap_Time_bind @ A @ B @ F @ G ) @ H2 ) ) ) ).
% success_bind_executeI
thf(fact_5358_take__bit__num__def,axiom,
( bit_take_bit_num
= ( ^ [N3: nat,M2: num] :
( if @ ( option @ num )
@ ( ( bit_se2584673776208193580ke_bit @ nat @ N3 @ ( numeral_numeral @ nat @ M2 ) )
= ( zero_zero @ nat ) )
@ ( none @ num )
@ ( some @ num @ ( num_of_nat @ ( bit_se2584673776208193580ke_bit @ nat @ N3 @ ( numeral_numeral @ nat @ M2 ) ) ) ) ) ) ) ).
% take_bit_num_def
thf(fact_5359_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_5360_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_5361_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_5362_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_5363_power__int__mult__distrib__numeral2,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [X: A,W2: num,M: int] :
( ( power_int @ A @ ( times_times @ A @ X @ ( numeral_numeral @ A @ W2 ) ) @ M )
= ( times_times @ A @ ( power_int @ A @ X @ M ) @ ( power_int @ A @ ( numeral_numeral @ A @ W2 ) @ M ) ) ) ) ).
% power_int_mult_distrib_numeral2
thf(fact_5364_power__int__mult__distrib__numeral1,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [W2: num,Y3: A,M: int] :
( ( power_int @ A @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ Y3 ) @ M )
= ( times_times @ A @ ( power_int @ A @ ( numeral_numeral @ A @ W2 ) @ M ) @ ( power_int @ A @ Y3 @ M ) ) ) ) ).
% power_int_mult_distrib_numeral1
thf(fact_5365_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_5366_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_5367_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_5368_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_5369_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_5370_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_5371_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_5372_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_5373_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_5374_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_5375_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_5376_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_5377_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_5378_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_5379_power__int__minus__left__even,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [N: int,A3: A] :
( ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N )
=> ( ( power_int @ A @ ( uminus_uminus @ A @ A3 ) @ N )
= ( power_int @ A @ A3 @ N ) ) ) ) ).
% power_int_minus_left_even
thf(fact_5380_power__int__minus__left__odd,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [N: int,A3: A] :
( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N )
=> ( ( power_int @ A @ ( uminus_uminus @ A @ A3 ) @ N )
= ( uminus_uminus @ A @ ( power_int @ A @ A3 @ N ) ) ) ) ) ).
% power_int_minus_left_odd
thf(fact_5381_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 )
@ ^ [Q3: num] : ( some @ num @ ( bit0 @ Q3 ) )
@ ( bit_take_bit_num @ N @ M ) ) ) ).
% take_bit_num_simps(3)
thf(fact_5382_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_5383_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_5384_success__LetI,axiom,
! [A: $tType,B: $tType,X: A,T2: A,F: A > ( heap_Time_Heap @ B ),H2: heap_ext @ product_unit] :
( ( X = T2 )
=> ( ( heap_Time_success @ B @ ( F @ X ) @ H2 )
=> ( heap_Time_success @ B @ ( F @ T2 ) @ H2 ) ) ) ).
% success_LetI
thf(fact_5385_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_5386_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_5387_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_5388_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_5389_success__ifI,axiom,
! [A: $tType,C2: $o,T2: heap_Time_Heap @ A,H2: heap_ext @ product_unit,E: heap_Time_Heap @ A] :
( ( C2
=> ( heap_Time_success @ A @ T2 @ H2 ) )
=> ( ( ~ C2
=> ( heap_Time_success @ A @ E @ H2 ) )
=> ( heap_Time_success @ A @ ( if @ ( heap_Time_Heap @ A ) @ C2 @ T2 @ E ) @ H2 ) ) ) ).
% success_ifI
thf(fact_5390_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_5391_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_5392_effect__success,axiom,
! [A: $tType,C2: heap_Time_Heap @ A,H2: heap_ext @ product_unit,H: heap_ext @ product_unit,R: A,N: nat] :
( ( heap_Time_effect @ A @ C2 @ H2 @ H @ R @ N )
=> ( heap_Time_success @ A @ C2 @ H2 ) ) ).
% effect_success
thf(fact_5393_success__effectE,axiom,
! [A: $tType,C2: heap_Time_Heap @ A,H2: heap_ext @ product_unit] :
( ( heap_Time_success @ A @ C2 @ H2 )
=> ~ ! [R3: A,H7: heap_ext @ product_unit,N4: nat] :
~ ( heap_Time_effect @ A @ C2 @ H2 @ H7 @ R3 @ N4 ) ) ).
% success_effectE
thf(fact_5394_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_5395_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_5396_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_5397_success__returnI,axiom,
! [A: $tType,X: A,H2: heap_ext @ product_unit] : ( heap_Time_success @ A @ ( heap_Time_return @ A @ X ) @ H2 ) ).
% success_returnI
thf(fact_5398_success__heapI,axiom,
! [A: $tType,F: ( heap_ext @ product_unit ) > ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ),H2: heap_ext @ product_unit] : ( heap_Time_success @ A @ ( heap_Time_heap @ A @ F ) @ H2 ) ).
% success_heapI
thf(fact_5399_success__guardI,axiom,
! [A: $tType,P: ( heap_ext @ product_unit ) > $o,H2: heap_ext @ product_unit,F: ( heap_ext @ product_unit ) > ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )] :
( ( P @ H2 )
=> ( heap_Time_success @ A @ ( heap_Time_guard @ A @ P @ F ) @ H2 ) ) ).
% success_guardI
thf(fact_5400_success__assertI,axiom,
! [A: $tType,P: A > $o,X: A,H2: heap_ext @ product_unit] :
( ( P @ X )
=> ( heap_Time_success @ A @ ( heap_Time_assert @ A @ P @ X ) @ H2 ) ) ).
% success_assertI
thf(fact_5401_success__ureturnI,axiom,
! [A: $tType,X: A,H2: heap_ext @ product_unit] : ( heap_Time_success @ A @ ( heap_Time_ureturn @ A @ X ) @ H2 ) ).
% success_ureturnI
thf(fact_5402_success__makeI,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [N: nat,F: nat > A,H2: heap_ext @ product_unit] : ( heap_Time_success @ ( array @ A ) @ ( array_make @ A @ N @ F ) @ H2 ) ) ).
% success_makeI
thf(fact_5403_success__of__listI,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [Xs: list @ A,H2: heap_ext @ product_unit] : ( heap_Time_success @ ( array @ A ) @ ( array_of_list @ A @ Xs ) @ H2 ) ) ).
% success_of_listI
thf(fact_5404_success__newI,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [N: nat,X: A,H2: heap_ext @ product_unit] : ( heap_Time_success @ ( array @ A ) @ ( array_new @ A @ N @ X ) @ H2 ) ) ).
% success_newI
thf(fact_5405_success__tapI,axiom,
! [A: $tType,F: ( heap_ext @ product_unit ) > A,H2: heap_ext @ product_unit] : ( heap_Time_success @ A @ ( heap_Time_tap @ A @ F ) @ H2 ) ).
% success_tapI
thf(fact_5406_success__lenI,axiom,
! [A: $tType] :
( ( heap @ A )
=> ! [A3: array @ A,H2: heap_ext @ product_unit] : ( heap_Time_success @ nat @ ( array_len @ A @ A3 ) @ H2 ) ) ).
% success_lenI
thf(fact_5407_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_5408_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_5409_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_5410_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_5411_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_5412_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_5413_success__bind__effectI,axiom,
! [A: $tType,B: $tType,F: heap_Time_Heap @ A,H2: heap_ext @ product_unit,H: heap_ext @ product_unit,X: A,N: nat,G: A > ( heap_Time_Heap @ B )] :
( ( heap_Time_effect @ A @ F @ H2 @ H @ X @ N )
=> ( ( heap_Time_success @ B @ ( G @ X ) @ H )
=> ( heap_Time_success @ B @ ( heap_Time_bind @ A @ B @ F @ G ) @ H2 ) ) ) ).
% success_bind_effectI
thf(fact_5414_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_5415_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_5416_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_5417_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_5418_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_5419_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_5420_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_5421_option_Osize_I3_J,axiom,
! [A: $tType] :
( ( size_size @ ( option @ A ) @ ( none @ A ) )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% option.size(3)
thf(fact_5422_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_5423_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_5424_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_5425_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_5426_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_5427_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_5428_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_5429_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_5430_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_5431_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_5432_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_5433_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_5434_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_5435_power__int__minus__left,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [N: int,A3: A] :
( ( ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N )
=> ( ( power_int @ A @ ( uminus_uminus @ A @ A3 ) @ N )
= ( power_int @ A @ A3 @ N ) ) )
& ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N )
=> ( ( power_int @ A @ ( uminus_uminus @ A @ A3 ) @ N )
= ( uminus_uminus @ A @ ( power_int @ A @ A3 @ N ) ) ) ) ) ) ).
% power_int_minus_left
thf(fact_5436_successE,axiom,
! [A: $tType,F: heap_Time_Heap @ A,H2: heap_ext @ product_unit] :
( ( heap_Time_success @ A @ F @ H2 )
=> ~ ! [R3: A,H7: product_prod @ ( heap_ext @ product_unit ) @ nat] :
( ( heap_Time_execute @ A @ F @ H2 )
!= ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R3 @ H7 ) ) ) ) ).
% successE
thf(fact_5437_and__not__num__eq__None__iff,axiom,
! [M: num,N: num] :
( ( ( bit_and_not_num @ M @ N )
= ( none @ num ) )
= ( ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ M ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N ) ) )
= ( zero_zero @ int ) ) ) ).
% and_not_num_eq_None_iff
thf(fact_5438_power__int__def,axiom,
! [A: $tType] :
( ( ( inverse @ A )
& ( power @ A ) )
=> ( ( power_int @ A )
= ( ^ [X2: A,N3: int] : ( if @ A @ ( ord_less_eq @ int @ ( zero_zero @ int ) @ N3 ) @ ( power_power @ A @ X2 @ ( nat2 @ N3 ) ) @ ( power_power @ A @ ( inverse_inverse @ A @ X2 ) @ ( nat2 @ ( uminus_uminus @ int @ N3 ) ) ) ) ) ) ) ).
% power_int_def
thf(fact_5439_execute__bind__success,axiom,
! [B: $tType,A: $tType,F: heap_Time_Heap @ A,H2: heap_ext @ product_unit,G: A > ( heap_Time_Heap @ B )] :
( ( heap_Time_success @ A @ F @ H2 )
=> ( ( heap_Time_execute @ B @ ( heap_Time_bind @ A @ B @ F @ G ) @ H2 )
= ( 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 @ H2 ) ) ) ) @ ( 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 @ 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 @ F @ H2 ) ) ) ) ) ) ) ) ).
% execute_bind_success
thf(fact_5440_effectE,axiom,
! [A: $tType,C2: heap_Time_Heap @ A,H2: heap_ext @ product_unit,H: heap_ext @ product_unit,R: A,N: nat] :
( ( heap_Time_effect @ A @ C2 @ H2 @ H @ 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 @ H2 ) ) ) )
=> ( ( 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 @ C2 @ H2 ) ) ) ) )
=> ( ( 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 @ H2 ) ) ) ) )
=> ~ ( heap_Time_success @ A @ C2 @ H2 ) ) ) ) ) ).
% effectE
thf(fact_5441_in__finite__psubset,axiom,
! [A: $tType,A6: set @ A,B7: set @ A] :
( ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ A6 @ B7 ) @ ( finite_psubset @ A ) )
= ( ( ord_less @ ( set @ A ) @ A6 @ B7 )
& ( finite_finite2 @ A @ B7 ) ) ) ).
% in_finite_psubset
thf(fact_5442_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_5443_execute__bind_I2_J,axiom,
! [A: $tType,B: $tType,F: heap_Time_Heap @ A,H2: heap_ext @ product_unit,G: A > ( heap_Time_Heap @ B )] :
( ( ( heap_Time_execute @ A @ F @ H2 )
= ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) )
=> ( ( heap_Time_execute @ B @ ( heap_Time_bind @ A @ B @ F @ G ) @ H2 )
= ( none @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ).
% execute_bind(2)
thf(fact_5444_successI,axiom,
! [A: $tType,F: heap_Time_Heap @ A,H2: heap_ext @ product_unit] :
( ( ( heap_Time_execute @ A @ F @ H2 )
!= ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) )
=> ( heap_Time_success @ A @ F @ H2 ) ) ).
% successI
thf(fact_5445_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_5446_execute__guard_I1_J,axiom,
! [A: $tType,P: ( heap_ext @ product_unit ) > $o,H2: heap_ext @ product_unit,F: ( heap_ext @ product_unit ) > ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )] :
( ~ ( P @ H2 )
=> ( ( heap_Time_execute @ A @ ( heap_Time_guard @ A @ P @ F ) @ H2 )
= ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ).
% execute_guard(1)
thf(fact_5447_execute__assert_I2_J,axiom,
! [A: $tType,P: A > $o,X: A,H2: heap_ext @ product_unit] :
( ~ ( P @ X )
=> ( ( heap_Time_execute @ A @ ( heap_Time_assert @ A @ P @ X ) @ H2 )
= ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ).
% execute_assert(2)
thf(fact_5448_finite__psubset__def,axiom,
! [A: $tType] :
( ( finite_psubset @ A )
= ( collect @ ( product_prod @ ( set @ A ) @ ( set @ A ) )
@ ( product_case_prod @ ( set @ A ) @ ( set @ A ) @ $o
@ ^ [A7: set @ A,B6: set @ A] :
( ( ord_less @ ( set @ A ) @ A7 @ B6 )
& ( finite_finite2 @ A @ B6 ) ) ) ) ) ).
% finite_psubset_def
thf(fact_5449_Heap__cases,axiom,
! [A: $tType,F: heap_Time_Heap @ A,H2: heap_ext @ product_unit] :
( ! [X3: A,H7: product_prod @ ( heap_ext @ product_unit ) @ nat] :
( ( heap_Time_execute @ A @ F @ H2 )
!= ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ X3 @ H7 ) ) )
=> ( ( heap_Time_execute @ A @ F @ H2 )
= ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ).
% Heap_cases
thf(fact_5450_timeFrame_Ocases,axiom,
! [A: $tType,X: product_prod @ nat @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )] :
( ! [N4: nat,R3: A,H6: heap_ext @ product_unit,N10: nat] :
( X
!= ( product_Pair @ nat @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ N4 @ ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R3 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H6 @ N10 ) ) ) ) )
=> ~ ! [N4: nat] :
( X
!= ( product_Pair @ nat @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ N4 @ ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ) ).
% timeFrame.cases
thf(fact_5451_Max_Oinfinite,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A] :
( ~ ( finite_finite2 @ A @ A6 )
=> ( ( lattic643756798349783984er_Max @ A @ A6 )
= ( the2 @ A @ ( none @ A ) ) ) ) ) ).
% Max.infinite
thf(fact_5452_Min_Oinfinite,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A] :
( ~ ( finite_finite2 @ A @ A6 )
=> ( ( lattic643756798350308766er_Min @ A @ A6 )
= ( the2 @ A @ ( none @ A ) ) ) ) ) ).
% Min.infinite
thf(fact_5453_guard__def,axiom,
! [A: $tType] :
( ( heap_Time_guard @ A )
= ( ^ [P4: ( heap_ext @ product_unit ) > $o,F2: ( heap_ext @ product_unit ) > ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) )] :
( heap_Time_Heap2 @ A
@ ^ [H4: heap_ext @ product_unit] : ( if @ ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) @ ( P4 @ 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_5454_timeFrame_Oelims,axiom,
! [A: $tType,X: nat,Xa3: 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 @ Xa3 )
= Y3 )
=> ( ! [R3: A,H6: heap_ext @ product_unit,N10: nat] :
( ( Xa3
= ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R3 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H6 @ N10 ) ) ) )
=> ( Y3
!= ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R3 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H6 @ ( plus_plus @ nat @ X @ N10 ) ) ) ) ) )
=> ~ ( ( Xa3
= ( 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_5455_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_5456_pred__nat__def,axiom,
( pred_nat
= ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [M2: nat,N3: nat] :
( N3
= ( suc @ M2 ) ) ) ) ) ).
% pred_nat_def
thf(fact_5457_Min_Oeq__fold_H,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( lattic643756798350308766er_Min @ A )
= ( ^ [A7: set @ A] :
( the2 @ A
@ ( finite_fold @ A @ ( option @ A )
@ ^ [X2: A,Y6: option @ A] : ( some @ A @ ( case_option @ A @ A @ X2 @ ( ord_min @ A @ X2 ) @ Y6 ) )
@ ( none @ A )
@ A7 ) ) ) ) ) ).
% Min.eq_fold'
thf(fact_5458_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_5459_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_5460_Max_Oeq__fold,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A,X: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( lattic643756798349783984er_Max @ A @ ( insert @ A @ X @ A6 ) )
= ( finite_fold @ A @ A @ ( ord_max @ A ) @ X @ A6 ) ) ) ) ).
% Max.eq_fold
thf(fact_5461_Min_Oeq__fold,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A,X: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( lattic643756798350308766er_Min @ A @ ( insert @ A @ X @ A6 ) )
= ( finite_fold @ A @ A @ ( ord_min @ A ) @ X @ A6 ) ) ) ) ).
% Min.eq_fold
thf(fact_5462_execute__bind__case,axiom,
! [A: $tType,B: $tType,F: heap_Time_Heap @ B,G: B > ( heap_Time_Heap @ A ),H2: heap_ext @ product_unit] :
( ( heap_Time_execute @ A @ ( heap_Time_bind @ B @ A @ F @ G ) @ H2 )
= ( 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 ) ) )
@ ^ [H5: heap_ext @ product_unit,N3: nat] : ( heap_Time_timeFrame @ A @ N3 @ ( heap_Time_execute @ A @ ( G @ X2 ) @ H5 ) ) ) )
@ ( heap_Time_execute @ B @ F @ H2 ) ) ) ).
% execute_bind_case
thf(fact_5463_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_5464_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_5465_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 ) ) )
@ ^ [R5: A] :
( product_case_prod @ ( heap_ext @ product_unit ) @ nat @ ( option @ ( product_prod @ B @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) )
@ ^ [H5: heap_ext @ product_unit,N3: nat] : ( heap_Time_timeFrame @ B @ N3 @ ( heap_Time_execute @ B @ ( G2 @ R5 ) @ H5 ) ) ) )
@ ( heap_Time_execute @ A @ F2 @ H4 ) ) ) ) ) ).
% Heap_Time_Monad.bind_def
thf(fact_5466_Max_Oeq__fold_H,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( lattic643756798349783984er_Max @ A )
= ( ^ [A7: set @ A] :
( the2 @ A
@ ( finite_fold @ A @ ( option @ A )
@ ^ [X2: A,Y6: option @ A] : ( some @ A @ ( case_option @ A @ A @ X2 @ ( ord_max @ A @ X2 ) @ Y6 ) )
@ ( none @ A )
@ A7 ) ) ) ) ) ).
% Max.eq_fold'
thf(fact_5467_Sup__fin_Oeq__fold_H,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( lattic5882676163264333800up_fin @ A )
= ( ^ [A7: set @ A] :
( the2 @ A
@ ( finite_fold @ A @ ( option @ A )
@ ^ [X2: A,Y6: option @ A] : ( some @ A @ ( case_option @ A @ A @ X2 @ ( sup_sup @ A @ X2 ) @ Y6 ) )
@ ( none @ A )
@ A7 ) ) ) ) ) ).
% Sup_fin.eq_fold'
thf(fact_5468_Inf__fin_Oeq__fold_H,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( lattic7752659483105999362nf_fin @ A )
= ( ^ [A7: set @ A] :
( the2 @ A
@ ( finite_fold @ A @ ( option @ A )
@ ^ [X2: A,Y6: option @ A] : ( some @ A @ ( case_option @ A @ A @ X2 @ ( inf_inf @ A @ X2 ) @ Y6 ) )
@ ( none @ A )
@ A7 ) ) ) ) ) ).
% Inf_fin.eq_fold'
thf(fact_5469_timeFrame_Opelims,axiom,
! [A: $tType,X: nat,Xa3: 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 @ Xa3 )
= 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 @ Xa3 ) )
=> ( ! [R3: A,H6: heap_ext @ product_unit,N10: nat] :
( ( Xa3
= ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R3 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H6 @ N10 ) ) ) )
=> ( ( Y3
= ( some @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) @ ( product_Pair @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) @ R3 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H6 @ ( plus_plus @ nat @ X @ N10 ) ) ) ) )
=> ~ ( 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 ) @ R3 @ ( product_Pair @ ( heap_ext @ product_unit ) @ nat @ H6 @ N10 ) ) ) ) ) ) )
=> ~ ( ( Xa3
= ( 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_5470_Inf__fin_Osingleton,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A] :
( ( lattic7752659483105999362nf_fin @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= X ) ) ).
% Inf_fin.singleton
thf(fact_5471_Sup__fin_Osingleton,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A] :
( ( lattic5882676163264333800up_fin @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= X ) ) ).
% Sup_fin.singleton
thf(fact_5472_inf__Sup__absorb,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [A6: set @ A,A3: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( member @ A @ A3 @ A6 )
=> ( ( inf_inf @ A @ A3 @ ( lattic5882676163264333800up_fin @ A @ A6 ) )
= A3 ) ) ) ) ).
% inf_Sup_absorb
thf(fact_5473_sup__Inf__absorb,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [A6: set @ A,A3: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( member @ A @ A3 @ A6 )
=> ( ( sup_sup @ A @ ( lattic7752659483105999362nf_fin @ A @ A6 ) @ A3 )
= A3 ) ) ) ) ).
% sup_Inf_absorb
thf(fact_5474_Inf__fin_Oinsert,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A6: set @ A,X: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic7752659483105999362nf_fin @ A @ ( insert @ A @ X @ A6 ) )
= ( inf_inf @ A @ X @ ( lattic7752659483105999362nf_fin @ A @ A6 ) ) ) ) ) ) ).
% Inf_fin.insert
thf(fact_5475_Sup__fin_Oinsert,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A6: set @ A,X: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic5882676163264333800up_fin @ A @ ( insert @ A @ X @ A6 ) )
= ( sup_sup @ A @ X @ ( lattic5882676163264333800up_fin @ A @ A6 ) ) ) ) ) ) ).
% Sup_fin.insert
thf(fact_5476_Inf__fin__Min,axiom,
! [A: $tType] :
( ( ( semilattice_inf @ A )
& ( linorder @ A ) )
=> ( ( lattic7752659483105999362nf_fin @ A )
= ( lattic643756798350308766er_Min @ A ) ) ) ).
% Inf_fin_Min
thf(fact_5477_Sup__fin__Max,axiom,
! [A: $tType] :
( ( ( semilattice_sup @ A )
& ( linorder @ A ) )
=> ( ( lattic5882676163264333800up_fin @ A )
= ( lattic643756798349783984er_Max @ A ) ) ) ).
% Sup_fin_Max
thf(fact_5478_Inf__fin__le__Sup__fin,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [A6: set @ A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ A @ ( lattic7752659483105999362nf_fin @ A @ A6 ) @ ( lattic5882676163264333800up_fin @ A @ A6 ) ) ) ) ) ).
% Inf_fin_le_Sup_fin
thf(fact_5479_Sup__fin_OcoboundedI,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A6: set @ A,A3: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( member @ A @ A3 @ A6 )
=> ( ord_less_eq @ A @ A3 @ ( lattic5882676163264333800up_fin @ A @ A6 ) ) ) ) ) ).
% Sup_fin.coboundedI
thf(fact_5480_Inf__fin_OcoboundedI,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A6: set @ A,A3: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( member @ A @ A3 @ A6 )
=> ( ord_less_eq @ A @ ( lattic7752659483105999362nf_fin @ A @ A6 ) @ A3 ) ) ) ) ).
% Inf_fin.coboundedI
thf(fact_5481_Inf__fin_Oin__idem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A6: set @ A,X: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( member @ A @ X @ A6 )
=> ( ( inf_inf @ A @ X @ ( lattic7752659483105999362nf_fin @ A @ A6 ) )
= ( lattic7752659483105999362nf_fin @ A @ A6 ) ) ) ) ) ).
% Inf_fin.in_idem
thf(fact_5482_Sup__fin_Oin__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A6: set @ A,X: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( member @ A @ X @ A6 )
=> ( ( sup_sup @ A @ X @ ( lattic5882676163264333800up_fin @ A @ A6 ) )
= ( lattic5882676163264333800up_fin @ A @ A6 ) ) ) ) ) ).
% Sup_fin.in_idem
thf(fact_5483_Sup__fin_Obounded__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A6: set @ A,X: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ ( lattic5882676163264333800up_fin @ A @ A6 ) @ X )
= ( ! [X2: A] :
( ( member @ A @ X2 @ A6 )
=> ( ord_less_eq @ A @ X2 @ X ) ) ) ) ) ) ) ).
% Sup_fin.bounded_iff
thf(fact_5484_Inf__fin_Obounded__iff,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A6: set @ A,X: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ X @ ( lattic7752659483105999362nf_fin @ A @ A6 ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ A6 )
=> ( ord_less_eq @ A @ X @ X2 ) ) ) ) ) ) ) ).
% Inf_fin.bounded_iff
thf(fact_5485_Sup__fin_OboundedI,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A6: set @ A,X: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [A8: A] :
( ( member @ A @ A8 @ A6 )
=> ( ord_less_eq @ A @ A8 @ X ) )
=> ( ord_less_eq @ A @ ( lattic5882676163264333800up_fin @ A @ A6 ) @ X ) ) ) ) ) ).
% Sup_fin.boundedI
thf(fact_5486_Sup__fin_OboundedE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A6: set @ A,X: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ ( lattic5882676163264333800up_fin @ A @ A6 ) @ X )
=> ! [A10: A] :
( ( member @ A @ A10 @ A6 )
=> ( ord_less_eq @ A @ A10 @ X ) ) ) ) ) ) ).
% Sup_fin.boundedE
thf(fact_5487_Inf__fin_OboundedI,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A6: set @ A,X: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [A8: A] :
( ( member @ A @ A8 @ A6 )
=> ( ord_less_eq @ A @ X @ A8 ) )
=> ( ord_less_eq @ A @ X @ ( lattic7752659483105999362nf_fin @ A @ A6 ) ) ) ) ) ) ).
% Inf_fin.boundedI
thf(fact_5488_Inf__fin_OboundedE,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A6: set @ A,X: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ X @ ( lattic7752659483105999362nf_fin @ A @ A6 ) )
=> ! [A10: A] :
( ( member @ A @ A10 @ A6 )
=> ( ord_less_eq @ A @ X @ A10 ) ) ) ) ) ) ).
% Inf_fin.boundedE
thf(fact_5489_Sup__fin__Sup,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A6: set @ A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic5882676163264333800up_fin @ A @ A6 )
= ( complete_Sup_Sup @ A @ A6 ) ) ) ) ) ).
% Sup_fin_Sup
thf(fact_5490_card_Oeq__fold,axiom,
! [A: $tType] :
( ( finite_card @ A )
= ( finite_fold @ A @ nat
@ ^ [Uu: A] : suc
@ ( zero_zero @ nat ) ) ) ).
% card.eq_fold
thf(fact_5491_Inf__fin__Inf,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A6: set @ A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic7752659483105999362nf_fin @ A @ A6 )
= ( complete_Inf_Inf @ A @ A6 ) ) ) ) ) ).
% Inf_fin_Inf
thf(fact_5492_Inf__fin_Oinfinite,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A6: set @ A] :
( ~ ( finite_finite2 @ A @ A6 )
=> ( ( lattic7752659483105999362nf_fin @ A @ A6 )
= ( the2 @ A @ ( none @ A ) ) ) ) ) ).
% Inf_fin.infinite
thf(fact_5493_Sup__fin_Oinfinite,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A6: set @ A] :
( ~ ( finite_finite2 @ A @ A6 )
=> ( ( lattic5882676163264333800up_fin @ A @ A6 )
= ( the2 @ A @ ( none @ A ) ) ) ) ) ).
% Sup_fin.infinite
thf(fact_5494_Sup__fin_Osubset__imp,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A6: set @ A,B7: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B7 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B7 )
=> ( ord_less_eq @ A @ ( lattic5882676163264333800up_fin @ A @ A6 ) @ ( lattic5882676163264333800up_fin @ A @ B7 ) ) ) ) ) ) ).
% Sup_fin.subset_imp
thf(fact_5495_Inf__fin_Osubset__imp,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A6: set @ A,B7: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B7 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B7 )
=> ( ord_less_eq @ A @ ( lattic7752659483105999362nf_fin @ A @ B7 ) @ ( lattic7752659483105999362nf_fin @ A @ A6 ) ) ) ) ) ) ).
% Inf_fin.subset_imp
thf(fact_5496_Inf__fin_Ohom__commute,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [H2: A > A,N6: set @ A] :
( ! [X3: A,Y5: A] :
( ( H2 @ ( inf_inf @ A @ X3 @ Y5 ) )
= ( inf_inf @ A @ ( H2 @ X3 ) @ ( H2 @ Y5 ) ) )
=> ( ( finite_finite2 @ A @ N6 )
=> ( ( N6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( H2 @ ( lattic7752659483105999362nf_fin @ A @ N6 ) )
= ( lattic7752659483105999362nf_fin @ A @ ( image @ A @ A @ H2 @ N6 ) ) ) ) ) ) ) ).
% Inf_fin.hom_commute
thf(fact_5497_Sup__fin_Ohom__commute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [H2: A > A,N6: set @ A] :
( ! [X3: A,Y5: A] :
( ( H2 @ ( sup_sup @ A @ X3 @ Y5 ) )
= ( sup_sup @ A @ ( H2 @ X3 ) @ ( H2 @ Y5 ) ) )
=> ( ( finite_finite2 @ A @ N6 )
=> ( ( N6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( H2 @ ( lattic5882676163264333800up_fin @ A @ N6 ) )
= ( lattic5882676163264333800up_fin @ A @ ( image @ A @ A @ H2 @ N6 ) ) ) ) ) ) ) ).
% Sup_fin.hom_commute
thf(fact_5498_Inf__fin_Osubset,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A6: set @ A,B7: set @ A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( B7
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ B7 @ A6 )
=> ( ( inf_inf @ A @ ( lattic7752659483105999362nf_fin @ A @ B7 ) @ ( lattic7752659483105999362nf_fin @ A @ A6 ) )
= ( lattic7752659483105999362nf_fin @ A @ A6 ) ) ) ) ) ) ).
% Inf_fin.subset
thf(fact_5499_Sup__fin_Osubset,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A6: set @ A,B7: set @ A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( B7
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ B7 @ A6 )
=> ( ( sup_sup @ A @ ( lattic5882676163264333800up_fin @ A @ B7 ) @ ( lattic5882676163264333800up_fin @ A @ A6 ) )
= ( lattic5882676163264333800up_fin @ A @ A6 ) ) ) ) ) ) ).
% Sup_fin.subset
thf(fact_5500_Inf__fin_Oinsert__not__elem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A6: set @ A,X: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ~ ( member @ A @ X @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic7752659483105999362nf_fin @ A @ ( insert @ A @ X @ A6 ) )
= ( inf_inf @ A @ X @ ( lattic7752659483105999362nf_fin @ A @ A6 ) ) ) ) ) ) ) ).
% Inf_fin.insert_not_elem
thf(fact_5501_Inf__fin_Oclosed,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A6: set @ A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A,Y5: A] : ( member @ A @ ( inf_inf @ A @ X3 @ Y5 ) @ ( insert @ A @ X3 @ ( insert @ A @ Y5 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( member @ A @ ( lattic7752659483105999362nf_fin @ A @ A6 ) @ A6 ) ) ) ) ) ).
% Inf_fin.closed
thf(fact_5502_Sup__fin_Oclosed,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A6: set @ A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A,Y5: A] : ( member @ A @ ( sup_sup @ A @ X3 @ Y5 ) @ ( insert @ A @ X3 @ ( insert @ A @ Y5 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( member @ A @ ( lattic5882676163264333800up_fin @ A @ A6 ) @ A6 ) ) ) ) ) ).
% Sup_fin.closed
thf(fact_5503_Sup__fin_Oinsert__not__elem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A6: set @ A,X: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ~ ( member @ A @ X @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic5882676163264333800up_fin @ A @ ( insert @ A @ X @ A6 ) )
= ( sup_sup @ A @ X @ ( lattic5882676163264333800up_fin @ A @ A6 ) ) ) ) ) ) ) ).
% Sup_fin.insert_not_elem
thf(fact_5504_Inf__fin_Ounion,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A6: set @ A,B7: set @ A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B7 )
=> ( ( B7
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic7752659483105999362nf_fin @ A @ ( sup_sup @ ( set @ A ) @ A6 @ B7 ) )
= ( inf_inf @ A @ ( lattic7752659483105999362nf_fin @ A @ A6 ) @ ( lattic7752659483105999362nf_fin @ A @ B7 ) ) ) ) ) ) ) ) ).
% Inf_fin.union
thf(fact_5505_Sup__fin_Ounion,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A6: set @ A,B7: set @ A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B7 )
=> ( ( B7
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic5882676163264333800up_fin @ A @ ( sup_sup @ ( set @ A ) @ A6 @ B7 ) )
= ( sup_sup @ A @ ( lattic5882676163264333800up_fin @ A @ A6 ) @ ( lattic5882676163264333800up_fin @ A @ B7 ) ) ) ) ) ) ) ) ).
% Sup_fin.union
thf(fact_5506_Inf__fin_Oeq__fold,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A6: set @ A,X: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( lattic7752659483105999362nf_fin @ A @ ( insert @ A @ X @ A6 ) )
= ( finite_fold @ A @ A @ ( inf_inf @ A ) @ X @ A6 ) ) ) ) ).
% Inf_fin.eq_fold
thf(fact_5507_Sup__fin_Oeq__fold,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A6: set @ A,X: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( lattic5882676163264333800up_fin @ A @ ( insert @ A @ X @ A6 ) )
= ( finite_fold @ A @ A @ ( sup_sup @ A ) @ X @ A6 ) ) ) ) ).
% Sup_fin.eq_fold
thf(fact_5508_inf__Sup1__distrib,axiom,
! [A: $tType] :
( ( distrib_lattice @ A )
=> ! [A6: set @ A,X: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( inf_inf @ A @ X @ ( lattic5882676163264333800up_fin @ A @ A6 ) )
= ( lattic5882676163264333800up_fin @ A
@ ( collect @ A
@ ^ [Uu: A] :
? [A5: A] :
( ( Uu
= ( inf_inf @ A @ X @ A5 ) )
& ( member @ A @ A5 @ A6 ) ) ) ) ) ) ) ) ).
% inf_Sup1_distrib
thf(fact_5509_inf__Sup2__distrib,axiom,
! [A: $tType] :
( ( distrib_lattice @ A )
=> ! [A6: set @ A,B7: set @ A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B7 )
=> ( ( B7
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( inf_inf @ A @ ( lattic5882676163264333800up_fin @ A @ A6 ) @ ( lattic5882676163264333800up_fin @ A @ B7 ) )
= ( lattic5882676163264333800up_fin @ A
@ ( collect @ A
@ ^ [Uu: A] :
? [A5: A,B3: A] :
( ( Uu
= ( inf_inf @ A @ A5 @ B3 ) )
& ( member @ A @ A5 @ A6 )
& ( member @ A @ B3 @ B7 ) ) ) ) ) ) ) ) ) ) ).
% inf_Sup2_distrib
thf(fact_5510_sup__Inf2__distrib,axiom,
! [A: $tType] :
( ( distrib_lattice @ A )
=> ! [A6: set @ A,B7: set @ A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B7 )
=> ( ( B7
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( sup_sup @ A @ ( lattic7752659483105999362nf_fin @ A @ A6 ) @ ( lattic7752659483105999362nf_fin @ A @ B7 ) )
= ( lattic7752659483105999362nf_fin @ A
@ ( collect @ A
@ ^ [Uu: A] :
? [A5: A,B3: A] :
( ( Uu
= ( sup_sup @ A @ A5 @ B3 ) )
& ( member @ A @ A5 @ A6 )
& ( member @ A @ B3 @ B7 ) ) ) ) ) ) ) ) ) ) ).
% sup_Inf2_distrib
thf(fact_5511_sup__Inf1__distrib,axiom,
! [A: $tType] :
( ( distrib_lattice @ A )
=> ! [A6: set @ A,X: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( sup_sup @ A @ X @ ( lattic7752659483105999362nf_fin @ A @ A6 ) )
= ( lattic7752659483105999362nf_fin @ A
@ ( collect @ A
@ ^ [Uu: A] :
? [A5: A] :
( ( Uu
= ( sup_sup @ A @ X @ A5 ) )
& ( member @ A @ A5 @ A6 ) ) ) ) ) ) ) ) ).
% sup_Inf1_distrib
thf(fact_5512_Inf__fin_Oinsert__remove,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A6: set @ A,X: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic7752659483105999362nf_fin @ A @ ( insert @ A @ X @ A6 ) )
= X ) )
& ( ( ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic7752659483105999362nf_fin @ A @ ( insert @ A @ X @ A6 ) )
= ( inf_inf @ A @ X @ ( lattic7752659483105999362nf_fin @ A @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% Inf_fin.insert_remove
thf(fact_5513_Inf__fin_Oremove,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A6: set @ A,X: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( member @ A @ X @ A6 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic7752659483105999362nf_fin @ A @ A6 )
= X ) )
& ( ( ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic7752659483105999362nf_fin @ A @ A6 )
= ( inf_inf @ A @ X @ ( lattic7752659483105999362nf_fin @ A @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ) ).
% Inf_fin.remove
thf(fact_5514_Sup__fin_Oinsert__remove,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A6: set @ A,X: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic5882676163264333800up_fin @ A @ ( insert @ A @ X @ A6 ) )
= X ) )
& ( ( ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic5882676163264333800up_fin @ A @ ( insert @ A @ X @ A6 ) )
= ( sup_sup @ A @ X @ ( lattic5882676163264333800up_fin @ A @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% Sup_fin.insert_remove
thf(fact_5515_Sup__fin_Oremove,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A6: set @ A,X: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( member @ A @ X @ A6 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic5882676163264333800up_fin @ A @ A6 )
= X ) )
& ( ( ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic5882676163264333800up_fin @ A @ A6 )
= ( sup_sup @ A @ X @ ( lattic5882676163264333800up_fin @ A @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ) ).
% Sup_fin.remove
thf(fact_5516_raise__def,axiom,
! [A: $tType] :
( ( heap_Time_raise @ A )
= ( ^ [S6: list @ char] :
( heap_Time_Heap2 @ A
@ ^ [Uu: heap_ext @ product_unit] : ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ) ).
% raise_def
thf(fact_5517_Gcd__fin_Oeq__fold,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ( ( semiring_gcd_Gcd_fin @ A )
= ( ^ [A7: set @ A] : ( if @ A @ ( finite_finite2 @ A @ A7 ) @ ( finite_fold @ A @ A @ ( gcd_gcd @ A ) @ ( zero_zero @ A ) @ A7 ) @ ( one_one @ A ) ) ) ) ) ).
% Gcd_fin.eq_fold
thf(fact_5518_funpow__inj__finite,axiom,
! [A: $tType,P5: A > A,X: A] :
( ( inj_on @ A @ A @ P5 @ ( top_top @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [Y6: A] :
? [N3: nat] :
( Y6
= ( compow @ ( A > A ) @ N3 @ P5 @ X ) ) ) )
=> ~ ! [N4: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N4 )
=> ( ( compow @ ( A > A ) @ N4 @ P5 @ X )
!= X ) ) ) ) ).
% funpow_inj_finite
thf(fact_5519_inj__uminus,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A6: set @ A] : ( inj_on @ A @ A @ ( uminus_uminus @ A ) @ A6 ) ) ).
% inj_uminus
thf(fact_5520_raise__bind,axiom,
! [B: $tType,A: $tType,E: list @ char,F: B > ( heap_Time_Heap @ A )] :
( ( heap_Time_bind @ B @ A @ ( heap_Time_raise @ B @ E ) @ F )
= ( heap_Time_raise @ A @ E ) ) ).
% raise_bind
thf(fact_5521_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_5522_Gcd__fin_Oinfinite,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A6: set @ A] :
( ~ ( finite_finite2 @ A @ A6 )
=> ( ( semiring_gcd_Gcd_fin @ A @ A6 )
= ( one_one @ A ) ) ) ) ).
% Gcd_fin.infinite
thf(fact_5523_is__unit__Gcd__fin__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A6: set @ A] :
( ( dvd_dvd @ A @ ( semiring_gcd_Gcd_fin @ A @ A6 ) @ ( one_one @ A ) )
= ( ( semiring_gcd_Gcd_fin @ A @ A6 )
= ( one_one @ A ) ) ) ) ).
% is_unit_Gcd_fin_iff
thf(fact_5524_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_5525_inj__divide__right,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: A] :
( ( inj_on @ A @ A
@ ^ [B3: A] : ( divide_divide @ A @ B3 @ A3 )
@ ( top_top @ ( set @ A ) ) )
= ( A3
!= ( zero_zero @ A ) ) ) ) ).
% inj_divide_right
thf(fact_5526_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_5527_subset__image__inj,axiom,
! [A: $tType,B: $tType,S3: set @ A,F: B > A,T6: set @ B] :
( ( ord_less_eq @ ( set @ A ) @ S3 @ ( image @ B @ A @ F @ T6 ) )
= ( ? [U5: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ U5 @ T6 )
& ( inj_on @ B @ A @ F @ U5 )
& ( S3
= ( image @ B @ A @ F @ U5 ) ) ) ) ) ).
% subset_image_inj
thf(fact_5528_inj__list__encode,axiom,
! [A6: set @ ( list @ nat )] : ( inj_on @ ( list @ nat ) @ nat @ nat_list_encode @ A6 ) ).
% inj_list_encode
thf(fact_5529_inj__prod__encode,axiom,
! [A6: set @ ( product_prod @ nat @ nat )] : ( inj_on @ ( product_prod @ nat @ nat ) @ nat @ nat_prod_encode @ A6 ) ).
% inj_prod_encode
thf(fact_5530_inj__Suc,axiom,
! [N6: set @ nat] : ( inj_on @ nat @ nat @ suc @ N6 ) ).
% inj_Suc
thf(fact_5531_inj__on__of__nat,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [N6: set @ nat] : ( inj_on @ nat @ A @ ( semiring_1_of_nat @ A ) @ N6 ) ) ).
% inj_on_of_nat
thf(fact_5532_inj__int__encode,axiom,
! [A6: set @ int] : ( inj_on @ int @ nat @ nat_int_encode @ A6 ) ).
% inj_int_encode
thf(fact_5533_inj__int__decode,axiom,
! [A6: set @ nat] : ( inj_on @ nat @ int @ nat_int_decode @ A6 ) ).
% inj_int_decode
thf(fact_5534_linorder__inj__onI,axiom,
! [B: $tType,A: $tType] :
( ( order @ A )
=> ! [A6: set @ A,F: A > B] :
( ! [X3: A,Y5: A] :
( ( ord_less @ A @ X3 @ Y5 )
=> ( ( member @ A @ X3 @ A6 )
=> ( ( member @ A @ Y5 @ A6 )
=> ( ( F @ X3 )
!= ( F @ Y5 ) ) ) ) )
=> ( ! [X3: A,Y5: A] :
( ( member @ A @ X3 @ A6 )
=> ( ( member @ A @ Y5 @ A6 )
=> ( ( ord_less_eq @ A @ X3 @ Y5 )
| ( ord_less_eq @ A @ Y5 @ X3 ) ) ) )
=> ( inj_on @ A @ B @ F @ A6 ) ) ) ) ).
% linorder_inj_onI
thf(fact_5535_inj__on__mult,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A3: A,A6: set @ A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( inj_on @ A @ A @ ( times_times @ A @ A3 ) @ A6 ) ) ) ).
% inj_on_mult
thf(fact_5536_linorder__injI,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ A )
=> ! [F: A > B] :
( ! [X3: A,Y5: A] :
( ( ord_less @ A @ X3 @ Y5 )
=> ( ( F @ X3 )
!= ( F @ Y5 ) ) )
=> ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) ) ) ) ).
% linorder_injI
thf(fact_5537_inj__on__strict__subset,axiom,
! [B: $tType,A: $tType,F: A > B,B7: set @ A,A6: set @ A] :
( ( inj_on @ A @ B @ F @ B7 )
=> ( ( ord_less @ ( set @ A ) @ A6 @ B7 )
=> ( ord_less @ ( set @ B ) @ ( image @ A @ B @ F @ A6 ) @ ( image @ A @ B @ F @ B7 ) ) ) ) ).
% inj_on_strict_subset
thf(fact_5538_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_5539_inj__on__diff__nat,axiom,
! [N6: set @ nat,K: nat] :
( ! [N4: nat] :
( ( member @ nat @ N4 @ N6 )
=> ( ord_less_eq @ nat @ K @ N4 ) )
=> ( inj_on @ nat @ nat
@ ^ [N3: nat] : ( minus_minus @ nat @ N3 @ K )
@ N6 ) ) ).
% inj_on_diff_nat
thf(fact_5540_inj__on__set__encode,axiom,
inj_on @ ( set @ nat ) @ nat @ nat_set_encode @ ( collect @ ( set @ nat ) @ ( finite_finite2 @ nat ) ) ).
% inj_on_set_encode
thf(fact_5541_inj__on__iff__surj,axiom,
! [A: $tType,B: $tType,A6: set @ A,A11: set @ B] :
( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ? [F2: A > B] :
( ( inj_on @ A @ B @ F2 @ A6 )
& ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F2 @ A6 ) @ A11 ) ) )
= ( ? [G2: B > A] :
( ( image @ B @ A @ G2 @ A11 )
= A6 ) ) ) ) ).
% inj_on_iff_surj
thf(fact_5542_pigeonhole,axiom,
! [A: $tType,B: $tType,F: B > A,A6: set @ B] :
( ( ord_less @ nat @ ( finite_card @ A @ ( image @ B @ A @ F @ A6 ) ) @ ( finite_card @ B @ A6 ) )
=> ~ ( inj_on @ B @ A @ F @ A6 ) ) ).
% pigeonhole
thf(fact_5543_effect__raiseE,axiom,
! [A: $tType,X: list @ char,H2: heap_ext @ product_unit,H: heap_ext @ product_unit,R: A,N: nat] :
~ ( heap_Time_effect @ A @ ( heap_Time_raise @ A @ X ) @ H2 @ H @ R @ N ) ).
% effect_raiseE
thf(fact_5544_finite__imp__nat__seg__image__inj__on,axiom,
! [A: $tType,A6: set @ A] :
( ( finite_finite2 @ A @ A6 )
=> ? [N4: nat,F3: nat > A] :
( ( A6
= ( image @ nat @ A @ F3
@ ( collect @ nat
@ ^ [I5: nat] : ( ord_less @ nat @ I5 @ N4 ) ) ) )
& ( inj_on @ nat @ A @ F3
@ ( collect @ nat
@ ^ [I5: nat] : ( ord_less @ nat @ I5 @ N4 ) ) ) ) ) ).
% finite_imp_nat_seg_image_inj_on
thf(fact_5545_finite__imp__inj__to__nat__seg,axiom,
! [A: $tType,A6: set @ A] :
( ( finite_finite2 @ A @ A6 )
=> ? [F3: A > nat,N4: nat] :
( ( ( image @ A @ nat @ F3 @ A6 )
= ( collect @ nat
@ ^ [I5: nat] : ( ord_less @ nat @ I5 @ N4 ) ) )
& ( inj_on @ A @ nat @ F3 @ A6 ) ) ) ).
% finite_imp_inj_to_nat_seg
thf(fact_5546_inj__image__Compl__subset,axiom,
! [B: $tType,A: $tType,F: A > B,A6: set @ A] :
( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
=> ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F @ ( uminus_uminus @ ( set @ A ) @ A6 ) ) @ ( uminus_uminus @ ( set @ B ) @ ( image @ A @ B @ F @ A6 ) ) ) ) ).
% inj_image_Compl_subset
thf(fact_5547_inj__on__nth,axiom,
! [A: $tType,Xs: list @ A,I6: set @ nat] :
( ( distinct @ A @ Xs )
=> ( ! [X3: nat] :
( ( member @ nat @ X3 @ I6 )
=> ( ord_less @ nat @ X3 @ ( size_size @ ( list @ A ) @ Xs ) ) )
=> ( inj_on @ nat @ A @ ( nth @ A @ Xs ) @ I6 ) ) ) ).
% inj_on_nth
thf(fact_5548_infinite__iff__countable__subset,axiom,
! [A: $tType,S3: set @ A] :
( ( ~ ( finite_finite2 @ A @ S3 ) )
= ( ? [F2: nat > A] :
( ( inj_on @ nat @ A @ F2 @ ( top_top @ ( set @ nat ) ) )
& ( ord_less_eq @ ( set @ A ) @ ( image @ nat @ A @ F2 @ ( top_top @ ( set @ nat ) ) ) @ S3 ) ) ) ) ).
% infinite_iff_countable_subset
thf(fact_5549_infinite__countable__subset,axiom,
! [A: $tType,S3: set @ A] :
( ~ ( finite_finite2 @ A @ S3 )
=> ? [F3: nat > A] :
( ( inj_on @ nat @ A @ F3 @ ( top_top @ ( set @ nat ) ) )
& ( ord_less_eq @ ( set @ A ) @ ( image @ nat @ A @ F3 @ ( top_top @ ( set @ nat ) ) ) @ S3 ) ) ) ).
% infinite_countable_subset
thf(fact_5550_inj__on__funpow__least,axiom,
! [A: $tType,N: nat,F: A > A,S: A] :
( ( ( compow @ ( A > A ) @ N @ F @ S )
= S )
=> ( ! [M6: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M6 )
=> ( ( ord_less @ nat @ M6 @ N )
=> ( ( compow @ ( A > A ) @ M6 @ 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_5551_inj__on__iff__card__le,axiom,
! [A: $tType,B: $tType,A6: set @ A,B7: set @ B] :
( ( finite_finite2 @ A @ A6 )
=> ( ( finite_finite2 @ B @ B7 )
=> ( ( ? [F2: A > B] :
( ( inj_on @ A @ B @ F2 @ A6 )
& ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F2 @ A6 ) @ B7 ) ) )
= ( ord_less_eq @ nat @ ( finite_card @ A @ A6 ) @ ( finite_card @ B @ B7 ) ) ) ) ) ).
% inj_on_iff_card_le
thf(fact_5552_card__inj__on__le,axiom,
! [A: $tType,B: $tType,F: A > B,A6: set @ A,B7: set @ B] :
( ( inj_on @ A @ B @ F @ A6 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F @ A6 ) @ B7 )
=> ( ( finite_finite2 @ B @ B7 )
=> ( ord_less_eq @ nat @ ( finite_card @ A @ A6 ) @ ( finite_card @ B @ B7 ) ) ) ) ) ).
% card_inj_on_le
thf(fact_5553_card__le__inj,axiom,
! [B: $tType,A: $tType,A6: set @ A,B7: set @ B] :
( ( finite_finite2 @ A @ A6 )
=> ( ( finite_finite2 @ B @ B7 )
=> ( ( ord_less_eq @ nat @ ( finite_card @ A @ A6 ) @ ( finite_card @ B @ B7 ) )
=> ? [F3: A > B] :
( ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F3 @ A6 ) @ B7 )
& ( inj_on @ A @ B @ F3 @ A6 ) ) ) ) ) ).
% card_le_inj
thf(fact_5554_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_5555_map__sorted__distinct__set__unique,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F: B > A,Xs: list @ B,Ys2: list @ B] :
( ( inj_on @ B @ A @ F @ ( sup_sup @ ( set @ B ) @ ( set2 @ B @ Xs ) @ ( set2 @ B @ Ys2 ) ) )
=> ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F @ Xs ) )
=> ( ( distinct @ A @ ( map @ B @ A @ F @ Xs ) )
=> ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F @ Ys2 ) )
=> ( ( distinct @ A @ ( map @ B @ A @ F @ Ys2 ) )
=> ( ( ( set2 @ B @ Xs )
= ( set2 @ B @ Ys2 ) )
=> ( Xs = Ys2 ) ) ) ) ) ) ) ) ).
% map_sorted_distinct_set_unique
thf(fact_5556_Gcd__fin__0__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A6: set @ A] :
( ( ( semiring_gcd_Gcd_fin @ A @ A6 )
= ( zero_zero @ A ) )
= ( ( ord_less_eq @ ( set @ A ) @ A6 @ ( insert @ A @ ( zero_zero @ A ) @ ( bot_bot @ ( set @ A ) ) ) )
& ( finite_finite2 @ A @ A6 ) ) ) ) ).
% Gcd_fin_0_iff
thf(fact_5557_greaterThan__Suc,axiom,
! [K: nat] :
( ( set_ord_greaterThan @ nat @ ( suc @ K ) )
= ( minus_minus @ ( set @ nat ) @ ( set_ord_greaterThan @ nat @ K ) @ ( insert @ nat @ ( suc @ K ) @ ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% greaterThan_Suc
thf(fact_5558_last__list__update,axiom,
! [A: $tType,Xs: list @ A,K: nat,X: A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( ( K
= ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) ) )
=> ( ( last @ A @ ( list_update @ A @ Xs @ K @ X ) )
= X ) )
& ( ( K
!= ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) ) )
=> ( ( last @ A @ ( list_update @ A @ Xs @ K @ X ) )
= ( last @ A @ Xs ) ) ) ) ) ).
% last_list_update
thf(fact_5559_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_5560_greaterThan__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [I: A,K: A] :
( ( member @ A @ I @ ( set_ord_greaterThan @ A @ K ) )
= ( ord_less @ A @ K @ I ) ) ) ).
% greaterThan_iff
thf(fact_5561_greaterThan__subset__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y3: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_ord_greaterThan @ A @ X ) @ ( set_ord_greaterThan @ A @ Y3 ) )
= ( ord_less_eq @ A @ Y3 @ X ) ) ) ).
% greaterThan_subset_iff
thf(fact_5562_last__replicate,axiom,
! [A: $tType,N: nat,X: A] :
( ( N
!= ( zero_zero @ nat ) )
=> ( ( last @ A @ ( replicate @ A @ N @ X ) )
= X ) ) ).
% last_replicate
thf(fact_5563_Compl__greaterThan,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [K: A] :
( ( uminus_uminus @ ( set @ A ) @ ( set_ord_greaterThan @ A @ K ) )
= ( set_ord_atMost @ A @ K ) ) ) ).
% Compl_greaterThan
thf(fact_5564_Compl__atMost,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [K: A] :
( ( uminus_uminus @ ( set @ A ) @ ( set_ord_atMost @ A @ K ) )
= ( set_ord_greaterThan @ A @ K ) ) ) ).
% Compl_atMost
thf(fact_5565_Sup__greaterThanAtLeast,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X: A] :
( ( ord_less @ A @ X @ ( top_top @ A ) )
=> ( ( complete_Sup_Sup @ A @ ( set_ord_greaterThan @ A @ X ) )
= ( top_top @ A ) ) ) ) ).
% Sup_greaterThanAtLeast
thf(fact_5566_image__uminus__greaterThan,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [X: A] :
( ( image @ A @ A @ ( uminus_uminus @ A ) @ ( set_ord_greaterThan @ A @ X ) )
= ( set_ord_lessThan @ A @ ( uminus_uminus @ A @ X ) ) ) ) ).
% image_uminus_greaterThan
thf(fact_5567_image__uminus__lessThan,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [X: A] :
( ( image @ A @ A @ ( uminus_uminus @ A ) @ ( set_ord_lessThan @ A @ X ) )
= ( set_ord_greaterThan @ A @ ( uminus_uminus @ A @ X ) ) ) ) ).
% image_uminus_lessThan
thf(fact_5568_last__drop,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( last @ A @ ( drop @ A @ N @ Xs ) )
= ( last @ A @ Xs ) ) ) ).
% last_drop
thf(fact_5569_greaterThan__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( set_ord_greaterThan @ A )
= ( ^ [L3: A] : ( collect @ A @ ( ord_less @ A @ L3 ) ) ) ) ) ).
% greaterThan_def
thf(fact_5570_ivl__disj__un__one_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,U: A] :
( ( ord_less_eq @ A @ L @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ L @ U ) @ ( set_ord_greaterThan @ A @ U ) )
= ( set_ord_greaterThan @ A @ L ) ) ) ) ).
% ivl_disj_un_one(5)
thf(fact_5571_greaterThan__0,axiom,
( ( set_ord_greaterThan @ nat @ ( zero_zero @ nat ) )
= ( image @ nat @ nat @ suc @ ( top_top @ ( set @ nat ) ) ) ) ).
% greaterThan_0
thf(fact_5572_last__conv__nth,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( last @ A @ Xs )
= ( nth @ A @ Xs @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) ) ) ) ) ).
% last_conv_nth
thf(fact_5573_interval__cases,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [S3: set @ A] :
( ! [A8: A,B5: A,X3: A] :
( ( member @ A @ A8 @ S3 )
=> ( ( member @ A @ B5 @ S3 )
=> ( ( ord_less_eq @ A @ A8 @ X3 )
=> ( ( ord_less_eq @ A @ X3 @ B5 )
=> ( member @ A @ X3 @ S3 ) ) ) ) )
=> ? [A8: A,B5: A] :
( ( S3
= ( bot_bot @ ( set @ A ) ) )
| ( S3
= ( top_top @ ( set @ A ) ) )
| ( S3
= ( set_ord_lessThan @ A @ B5 ) )
| ( S3
= ( set_ord_atMost @ A @ B5 ) )
| ( S3
= ( set_ord_greaterThan @ A @ A8 ) )
| ( S3
= ( set_ord_atLeast @ A @ A8 ) )
| ( S3
= ( set_or5935395276787703475ssThan @ A @ A8 @ B5 ) )
| ( S3
= ( set_or3652927894154168847AtMost @ A @ A8 @ B5 ) )
| ( S3
= ( set_or7035219750837199246ssThan @ A @ A8 @ B5 ) )
| ( S3
= ( set_or1337092689740270186AtMost @ A @ A8 @ B5 ) ) ) ) ) ).
% interval_cases
thf(fact_5574_sum__mult__sum__if__inj,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( semiring_0 @ B )
=> ! [F: A > B,G: C > B,A6: set @ A,B7: set @ C] :
( ( inj_on @ ( product_prod @ A @ C ) @ B
@ ( product_case_prod @ A @ C @ B
@ ^ [A5: A,B3: C] : ( times_times @ B @ ( F @ A5 ) @ ( G @ B3 ) ) )
@ ( product_Sigma @ A @ C @ A6
@ ^ [Uu: A] : B7 ) )
=> ( ( times_times @ B @ ( groups7311177749621191930dd_sum @ A @ B @ F @ A6 ) @ ( groups7311177749621191930dd_sum @ C @ B @ G @ B7 ) )
= ( groups7311177749621191930dd_sum @ B @ B @ ( id @ B )
@ ( collect @ B
@ ^ [Uu: B] :
? [A5: A,B3: C] :
( ( Uu
= ( times_times @ B @ ( F @ A5 ) @ ( G @ B3 ) ) )
& ( member @ A @ A5 @ A6 )
& ( member @ C @ B3 @ B7 ) ) ) ) ) ) ) ).
% sum_mult_sum_if_inj
thf(fact_5575_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 )
=> ( ! [Y5: A] :
( ( P @ Y5 )
=> ( ord_less_eq @ B @ ( F @ A3 ) @ ( F @ Y5 ) ) )
=> ( ( lattices_ord_arg_min @ A @ B @ F @ P )
= A3 ) ) ) ) ) ).
% arg_min_inj_eq
thf(fact_5576_atLeast__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [I: A,K: A] :
( ( member @ A @ I @ ( set_ord_atLeast @ A @ K ) )
= ( ord_less_eq @ A @ K @ I ) ) ) ).
% atLeast_iff
thf(fact_5577_atLeast__0,axiom,
( ( set_ord_atLeast @ nat @ ( zero_zero @ nat ) )
= ( top_top @ ( set @ nat ) ) ) ).
% atLeast_0
thf(fact_5578_atLeast__subset__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y3: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_ord_atLeast @ A @ X ) @ ( set_ord_atLeast @ A @ Y3 ) )
= ( ord_less_eq @ A @ Y3 @ X ) ) ) ).
% atLeast_subset_iff
thf(fact_5579_Compl__atLeast,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [K: A] :
( ( uminus_uminus @ ( set @ A ) @ ( set_ord_atLeast @ A @ K ) )
= ( set_ord_lessThan @ A @ K ) ) ) ).
% Compl_atLeast
thf(fact_5580_Compl__lessThan,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [K: A] :
( ( uminus_uminus @ ( set @ A ) @ ( set_ord_lessThan @ A @ K ) )
= ( set_ord_atLeast @ A @ K ) ) ) ).
% Compl_lessThan
thf(fact_5581_Icc__subset__Ici__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [L: A,H2: A,L4: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ H2 ) @ ( set_ord_atLeast @ A @ L4 ) )
= ( ~ ( ord_less_eq @ A @ L @ H2 )
| ( ord_less_eq @ A @ L4 @ L ) ) ) ) ).
% Icc_subset_Ici_iff
thf(fact_5582_image__uminus__atMost,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [X: A] :
( ( image @ A @ A @ ( uminus_uminus @ A ) @ ( set_ord_atMost @ A @ X ) )
= ( set_ord_atLeast @ A @ ( uminus_uminus @ A @ X ) ) ) ) ).
% image_uminus_atMost
thf(fact_5583_image__uminus__atLeast,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [X: A] :
( ( image @ A @ A @ ( uminus_uminus @ A ) @ ( set_ord_atLeast @ A @ X ) )
= ( set_ord_atMost @ A @ ( uminus_uminus @ A @ X ) ) ) ) ).
% image_uminus_atLeast
thf(fact_5584_arg__min__nat__lemma,axiom,
! [A: $tType,P: A > $o,K: A,M: A > nat] :
( ( P @ K )
=> ( ( P @ ( lattices_ord_arg_min @ A @ nat @ M @ P ) )
& ! [Y4: A] :
( ( P @ Y4 )
=> ( ord_less_eq @ nat @ ( M @ ( lattices_ord_arg_min @ A @ nat @ M @ P ) ) @ ( M @ Y4 ) ) ) ) ) ).
% arg_min_nat_lemma
thf(fact_5585_arg__min__nat__le,axiom,
! [A: $tType,P: A > $o,X: A,M: A > nat] :
( ( P @ X )
=> ( ord_less_eq @ nat @ ( M @ ( lattices_ord_arg_min @ A @ nat @ M @ P ) ) @ ( M @ X ) ) ) ).
% arg_min_nat_le
thf(fact_5586_atLeast__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( set_ord_atLeast @ A )
= ( ^ [L3: A] : ( collect @ A @ ( ord_less_eq @ A @ L3 ) ) ) ) ) ).
% atLeast_def
thf(fact_5587_arg__min__equality,axiom,
! [A: $tType,C: $tType] :
( ( order @ A )
=> ! [P: C > $o,K: C,F: C > A] :
( ( P @ K )
=> ( ! [X3: C] :
( ( P @ X3 )
=> ( ord_less_eq @ A @ ( F @ K ) @ ( F @ X3 ) ) )
=> ( ( F @ ( lattices_ord_arg_min @ C @ A @ F @ P ) )
= ( F @ K ) ) ) ) ) ).
% arg_min_equality
thf(fact_5588_arg__minI,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [P: A > $o,X: A,F: A > B,Q: A > $o] :
( ( P @ X )
=> ( ! [Y5: A] :
( ( P @ Y5 )
=> ~ ( ord_less @ B @ ( F @ Y5 ) @ ( F @ X ) ) )
=> ( ! [X3: A] :
( ( P @ X3 )
=> ( ! [Y4: A] :
( ( P @ Y4 )
=> ~ ( ord_less @ B @ ( F @ Y4 ) @ ( F @ X3 ) ) )
=> ( Q @ X3 ) ) )
=> ( Q @ ( lattices_ord_arg_min @ A @ B @ F @ P ) ) ) ) ) ) ).
% arg_minI
thf(fact_5589_atLeast__Suc__greaterThan,axiom,
! [K: nat] :
( ( set_ord_atLeast @ nat @ ( suc @ K ) )
= ( set_ord_greaterThan @ nat @ K ) ) ).
% atLeast_Suc_greaterThan
thf(fact_5590_Ex__inj__on__UNION__Sigma,axiom,
! [A: $tType,B: $tType,A6: B > ( set @ A ),I6: set @ B] :
? [F3: A > ( product_prod @ B @ A )] :
( ( inj_on @ A @ ( product_prod @ B @ A ) @ F3 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ A6 @ I6 ) ) )
& ( ord_less_eq @ ( set @ ( product_prod @ B @ A ) ) @ ( image @ A @ ( product_prod @ B @ A ) @ F3 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ A6 @ I6 ) ) ) @ ( product_Sigma @ B @ A @ I6 @ A6 ) ) ) ).
% Ex_inj_on_UNION_Sigma
thf(fact_5591_ivl__disj__un__one_I8_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,U: A] :
( ( ord_less_eq @ A @ L @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ L @ U ) @ ( set_ord_atLeast @ A @ U ) )
= ( set_ord_atLeast @ A @ L ) ) ) ) ).
% ivl_disj_un_one(8)
thf(fact_5592_Ici__subset__Ioi__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_ord_atLeast @ A @ A3 ) @ ( set_ord_greaterThan @ A @ B2 ) )
= ( ord_less @ A @ B2 @ A3 ) ) ) ).
% Ici_subset_Ioi_iff
thf(fact_5593_card__cartesian__product,axiom,
! [A: $tType,B: $tType,A6: set @ A,B7: set @ B] :
( ( finite_card @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A6
@ ^ [Uu: A] : B7 ) )
= ( times_times @ nat @ ( finite_card @ A @ A6 ) @ ( finite_card @ B @ B7 ) ) ) ).
% card_cartesian_product
thf(fact_5594_ivl__disj__un__one_I7_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,U: A] :
( ( ord_less_eq @ A @ L @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ U ) @ ( set_ord_greaterThan @ A @ U ) )
= ( set_ord_atLeast @ A @ L ) ) ) ) ).
% ivl_disj_un_one(7)
thf(fact_5595_ivl__disj__un__one_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,U: A] :
( ( ord_less @ A @ L @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ L @ U ) @ ( set_ord_atLeast @ A @ U ) )
= ( set_ord_greaterThan @ A @ L ) ) ) ) ).
% ivl_disj_un_one(6)
thf(fact_5596_lists__length__Suc__eq,axiom,
! [A: $tType,A6: set @ A,N: nat] :
( ( collect @ ( list @ A )
@ ^ [Xs3: list @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs3 ) @ A6 )
& ( ( size_size @ ( list @ A ) @ Xs3 )
= ( suc @ N ) ) ) )
= ( image @ ( product_prod @ ( list @ A ) @ A ) @ ( list @ A )
@ ( product_case_prod @ ( list @ A ) @ A @ ( list @ A )
@ ^ [Xs3: list @ A,N3: A] : ( cons @ A @ N3 @ Xs3 ) )
@ ( product_Sigma @ ( list @ A ) @ A
@ ( collect @ ( list @ A )
@ ^ [Xs3: list @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs3 ) @ A6 )
& ( ( size_size @ ( list @ A ) @ Xs3 )
= N ) ) )
@ ^ [Uu: list @ A] : A6 ) ) ) ).
% lists_length_Suc_eq
thf(fact_5597_arg__min__on__def,axiom,
! [A: $tType,B: $tType] :
( ( ord @ A )
=> ( ( lattic7623131987881927897min_on @ B @ A )
= ( ^ [F2: B > A,S7: set @ B] :
( lattices_ord_arg_min @ B @ A @ F2
@ ^ [X2: B] : ( member @ B @ X2 @ S7 ) ) ) ) ) ).
% arg_min_on_def
thf(fact_5598_atLeast__Suc,axiom,
! [K: nat] :
( ( set_ord_atLeast @ nat @ ( suc @ K ) )
= ( minus_minus @ ( set @ nat ) @ ( set_ord_atLeast @ nat @ K ) @ ( insert @ nat @ K @ ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% atLeast_Suc
thf(fact_5599_Compl__Times__UNIV1,axiom,
! [B: $tType,A: $tType,A6: set @ B] :
( ( uminus_uminus @ ( set @ ( product_prod @ A @ B ) )
@ ( product_Sigma @ A @ B @ ( top_top @ ( set @ A ) )
@ ^ [Uu: A] : A6 ) )
= ( product_Sigma @ A @ B @ ( top_top @ ( set @ A ) )
@ ^ [Uu: A] : ( uminus_uminus @ ( set @ B ) @ A6 ) ) ) ).
% Compl_Times_UNIV1
thf(fact_5600_Compl__Times__UNIV2,axiom,
! [B: $tType,A: $tType,A6: set @ A] :
( ( uminus_uminus @ ( set @ ( product_prod @ A @ B ) )
@ ( product_Sigma @ A @ B @ A6
@ ^ [Uu: A] : ( top_top @ ( set @ B ) ) ) )
= ( product_Sigma @ A @ B @ ( uminus_uminus @ ( set @ A ) @ A6 )
@ ^ [Uu: A] : ( top_top @ ( set @ B ) ) ) ) ).
% Compl_Times_UNIV2
thf(fact_5601_mult__inj__if__coprime__nat,axiom,
! [B: $tType,A: $tType,F: A > nat,A6: set @ A,G: B > nat,B7: set @ B] :
( ( inj_on @ A @ nat @ F @ A6 )
=> ( ( inj_on @ B @ nat @ G @ B7 )
=> ( ! [A8: A,B5: B] :
( ( member @ A @ A8 @ A6 )
=> ( ( member @ B @ B5 @ B7 )
=> ( algebr8660921524188924756oprime @ nat @ ( F @ A8 ) @ ( G @ B5 ) ) ) )
=> ( inj_on @ ( product_prod @ A @ B ) @ nat
@ ( product_case_prod @ A @ B @ nat
@ ^ [A5: A,B3: B] : ( times_times @ nat @ ( F @ A5 ) @ ( G @ B3 ) ) )
@ ( product_Sigma @ A @ B @ A6
@ ^ [Uu: A] : B7 ) ) ) ) ) ).
% mult_inj_if_coprime_nat
thf(fact_5602_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_5603_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_5604_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_5605_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_5606_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_5607_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_5608_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_5609_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_5610_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_5611_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_5612_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_5613_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_5614_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_5615_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_5616_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_5617_arg__min__natI,axiom,
! [A: $tType,P: A > $o,K: A,M: A > nat] :
( ( P @ K )
=> ( P @ ( lattices_ord_arg_min @ A @ nat @ M @ P ) ) ) ).
% arg_min_natI
thf(fact_5618_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_5619_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_5620_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_5621_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_5622_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_5623_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_5624_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_5625_coprime__Suc__0__left,axiom,
! [N: nat] : ( algebr8660921524188924756oprime @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N ) ).
% coprime_Suc_0_left
thf(fact_5626_coprime__Suc__0__right,axiom,
! [N: nat] : ( algebr8660921524188924756oprime @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ).
% coprime_Suc_0_right
thf(fact_5627_coprime__1__left,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A] : ( algebr8660921524188924756oprime @ A @ ( one_one @ A ) @ A3 ) ) ).
% coprime_1_left
thf(fact_5628_coprime__1__right,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A] : ( algebr8660921524188924756oprime @ A @ A3 @ ( one_one @ A ) ) ) ).
% coprime_1_right
thf(fact_5629_coprime__Suc__left__nat,axiom,
! [N: nat] : ( algebr8660921524188924756oprime @ nat @ ( suc @ N ) @ N ) ).
% coprime_Suc_left_nat
thf(fact_5630_coprime__Suc__right__nat,axiom,
! [N: nat] : ( algebr8660921524188924756oprime @ nat @ N @ ( suc @ N ) ) ).
% coprime_Suc_right_nat
thf(fact_5631_coprime__commute,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ( ( algebr8660921524188924756oprime @ A )
= ( ^ [B3: A,A5: A] : ( algebr8660921524188924756oprime @ A @ A5 @ B3 ) ) ) ) ).
% coprime_commute
thf(fact_5632_coprime__divisors,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,C2: A,B2: A,D2: A] :
( ( dvd_dvd @ A @ A3 @ C2 )
=> ( ( dvd_dvd @ A @ B2 @ D2 )
=> ( ( algebr8660921524188924756oprime @ A @ C2 @ D2 )
=> ( algebr8660921524188924756oprime @ A @ A3 @ B2 ) ) ) ) ) ).
% coprime_divisors
thf(fact_5633_coprime__crossproduct__nat,axiom,
! [A3: nat,D2: nat,B2: nat,C2: nat] :
( ( algebr8660921524188924756oprime @ nat @ A3 @ D2 )
=> ( ( algebr8660921524188924756oprime @ nat @ B2 @ C2 )
=> ( ( ( times_times @ nat @ A3 @ C2 )
= ( times_times @ nat @ B2 @ D2 ) )
= ( ( A3 = B2 )
& ( C2 = D2 ) ) ) ) ) ).
% coprime_crossproduct_nat
thf(fact_5634_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_5635_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_5636_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_5637_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_5638_coprime__imp__coprime,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C2: A,D2: A,A3: A,B2: A] :
( ( algebr8660921524188924756oprime @ A @ C2 @ D2 )
=> ( ! [E2: A] :
( ~ ( dvd_dvd @ A @ E2 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ E2 @ A3 )
=> ( ( dvd_dvd @ A @ E2 @ B2 )
=> ( dvd_dvd @ A @ E2 @ C2 ) ) ) )
=> ( ! [E2: A] :
( ~ ( dvd_dvd @ A @ E2 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ E2 @ A3 )
=> ( ( dvd_dvd @ A @ E2 @ B2 )
=> ( dvd_dvd @ A @ E2 @ D2 ) ) ) )
=> ( algebr8660921524188924756oprime @ A @ A3 @ B2 ) ) ) ) ) ).
% coprime_imp_coprime
thf(fact_5639_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_5640_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_5641_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_5642_coprime__def,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ( ( algebr8660921524188924756oprime @ A )
= ( ^ [A5: A,B3: A] :
! [C4: A] :
( ( dvd_dvd @ A @ C4 @ A5 )
=> ( ( dvd_dvd @ A @ C4 @ B3 )
=> ( dvd_dvd @ A @ C4 @ ( one_one @ A ) ) ) ) ) ) ) ).
% coprime_def
thf(fact_5643_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_5644_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_5645_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_5646_coprime__iff__gcd__eq__1,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ( ( algebr8660921524188924756oprime @ A )
= ( ^ [A5: A,B3: A] :
( ( gcd_gcd @ A @ A5 @ B3 )
= ( one_one @ A ) ) ) ) ) ).
% coprime_iff_gcd_eq_1
thf(fact_5647_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_5648_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_5649_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_5650_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_5651_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_5652_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_5653_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_5654_gcd__coprime__exists,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A] :
( ( ( gcd_gcd @ A @ A3 @ B2 )
!= ( zero_zero @ A ) )
=> ? [A15: A,B8: A] :
( ( A3
= ( times_times @ A @ A15 @ ( gcd_gcd @ A @ A3 @ B2 ) ) )
& ( B2
= ( times_times @ A @ B8 @ ( gcd_gcd @ A @ A3 @ B2 ) ) )
& ( algebr8660921524188924756oprime @ A @ A15 @ B8 ) ) ) ) ).
% gcd_coprime_exists
thf(fact_5655_gcd__coprime,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A,A4: A,B4: A] :
( ( ( gcd_gcd @ A @ A3 @ B2 )
!= ( zero_zero @ A ) )
=> ( ( A3
= ( times_times @ A @ A4 @ ( gcd_gcd @ A @ A3 @ B2 ) ) )
=> ( ( B2
= ( times_times @ A @ B4 @ ( gcd_gcd @ A @ A3 @ B2 ) ) )
=> ( algebr8660921524188924756oprime @ A @ A4 @ B4 ) ) ) ) ) ).
% gcd_coprime
thf(fact_5656_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_5657_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_5658_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_5659_set__encode__vimage__Suc,axiom,
! [A6: set @ nat] :
( ( nat_set_encode @ ( vimage @ nat @ nat @ suc @ A6 ) )
= ( divide_divide @ nat @ ( nat_set_encode @ A6 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% set_encode_vimage_Suc
thf(fact_5660_floor__rat__def,axiom,
( ( archim6421214686448440834_floor @ rat )
= ( ^ [X2: rat] :
( the @ int
@ ^ [Z7: int] :
( ( ord_less_eq @ rat @ ( ring_1_of_int @ rat @ Z7 ) @ X2 )
& ( ord_less @ rat @ X2 @ ( ring_1_of_int @ rat @ ( plus_plus @ int @ Z7 @ ( one_one @ int ) ) ) ) ) ) ) ) ).
% floor_rat_def
thf(fact_5661_sorted__insort__insert__key,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F: B > A,Xs: list @ B,X: B] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F @ Xs ) )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F @ ( linord329482645794927042rt_key @ B @ A @ F @ X @ Xs ) ) ) ) ) ).
% sorted_insort_insert_key
thf(fact_5662_coprime__abs__left__iff,axiom,
! [K: int,L: int] :
( ( algebr8660921524188924756oprime @ int @ ( abs_abs @ int @ K ) @ L )
= ( algebr8660921524188924756oprime @ int @ K @ L ) ) ).
% coprime_abs_left_iff
thf(fact_5663_coprime__abs__right__iff,axiom,
! [K: int,L: int] :
( ( algebr8660921524188924756oprime @ int @ K @ ( abs_abs @ int @ L ) )
= ( algebr8660921524188924756oprime @ int @ K @ L ) ) ).
% coprime_abs_right_iff
thf(fact_5664_coprime__int__iff,axiom,
! [M: nat,N: nat] :
( ( algebr8660921524188924756oprime @ int @ ( semiring_1_of_nat @ int @ M ) @ ( semiring_1_of_nat @ int @ N ) )
= ( algebr8660921524188924756oprime @ nat @ M @ N ) ) ).
% coprime_int_iff
thf(fact_5665_normalize__stable,axiom,
! [Q4: int,P5: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ Q4 )
=> ( ( algebr8660921524188924756oprime @ int @ P5 @ Q4 )
=> ( ( normalize @ ( product_Pair @ int @ int @ P5 @ Q4 ) )
= ( product_Pair @ int @ int @ P5 @ Q4 ) ) ) ) ).
% normalize_stable
thf(fact_5666_coprime__nat__abs__left__iff,axiom,
! [K: int,N: nat] :
( ( algebr8660921524188924756oprime @ nat @ ( nat2 @ ( abs_abs @ int @ K ) ) @ N )
= ( algebr8660921524188924756oprime @ int @ K @ ( semiring_1_of_nat @ int @ N ) ) ) ).
% coprime_nat_abs_left_iff
thf(fact_5667_coprime__nat__abs__right__iff,axiom,
! [N: nat,K: int] :
( ( algebr8660921524188924756oprime @ nat @ N @ ( nat2 @ ( abs_abs @ int @ K ) ) )
= ( algebr8660921524188924756oprime @ int @ ( semiring_1_of_nat @ int @ N ) @ K ) ) ).
% coprime_nat_abs_right_iff
thf(fact_5668_vimage__if,axiom,
! [B: $tType,A: $tType,C2: B,A6: set @ B,D2: B,B7: set @ A] :
( ( ( member @ B @ C2 @ A6 )
=> ( ( ( member @ B @ D2 @ A6 )
=> ( ( vimage @ A @ B
@ ^ [X2: A] : ( if @ B @ ( member @ A @ X2 @ B7 ) @ C2 @ D2 )
@ A6 )
= ( top_top @ ( set @ A ) ) ) )
& ( ~ ( member @ B @ D2 @ A6 )
=> ( ( vimage @ A @ B
@ ^ [X2: A] : ( if @ B @ ( member @ A @ X2 @ B7 ) @ C2 @ D2 )
@ A6 )
= B7 ) ) ) )
& ( ~ ( member @ B @ C2 @ A6 )
=> ( ( ( member @ B @ D2 @ A6 )
=> ( ( vimage @ A @ B
@ ^ [X2: A] : ( if @ B @ ( member @ A @ X2 @ B7 ) @ C2 @ D2 )
@ A6 )
= ( uminus_uminus @ ( set @ A ) @ B7 ) ) )
& ( ~ ( member @ B @ D2 @ A6 )
=> ( ( vimage @ A @ B
@ ^ [X2: A] : ( if @ B @ ( member @ A @ X2 @ B7 ) @ C2 @ D2 )
@ A6 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% vimage_if
thf(fact_5669_vimage__Compl,axiom,
! [A: $tType,B: $tType,F: A > B,A6: set @ B] :
( ( vimage @ A @ B @ F @ ( uminus_uminus @ ( set @ B ) @ A6 ) )
= ( uminus_uminus @ ( set @ A ) @ ( vimage @ A @ B @ F @ A6 ) ) ) ).
% vimage_Compl
thf(fact_5670_coprime__crossproduct__int,axiom,
! [A3: int,D2: int,B2: int,C2: int] :
( ( algebr8660921524188924756oprime @ int @ A3 @ D2 )
=> ( ( 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 @ D2 ) ) )
= ( ( ( abs_abs @ int @ A3 )
= ( abs_abs @ int @ B2 ) )
& ( ( abs_abs @ int @ C2 )
= ( abs_abs @ int @ D2 ) ) ) ) ) ) ).
% coprime_crossproduct_int
thf(fact_5671_finite__vimage__Suc__iff,axiom,
! [F4: set @ nat] :
( ( finite_finite2 @ nat @ ( vimage @ nat @ nat @ suc @ F4 ) )
= ( finite_finite2 @ nat @ F4 ) ) ).
% finite_vimage_Suc_iff
thf(fact_5672_Nitpick_OThe__psimp,axiom,
! [A: $tType,P: A > $o,X: A] :
( ( P
= ( ^ [Y: A,Z2: A] : Y = Z2
@ X ) )
=> ( ( the @ A @ P )
= X ) ) ).
% Nitpick.The_psimp
thf(fact_5673_vimage__Suc__insert__Suc,axiom,
! [N: nat,A6: set @ nat] :
( ( vimage @ nat @ nat @ suc @ ( insert @ nat @ ( suc @ N ) @ A6 ) )
= ( insert @ nat @ N @ ( vimage @ nat @ nat @ suc @ A6 ) ) ) ).
% vimage_Suc_insert_Suc
thf(fact_5674_vimage__Suc__insert__0,axiom,
! [A6: set @ nat] :
( ( vimage @ nat @ nat @ suc @ ( insert @ nat @ ( zero_zero @ nat ) @ A6 ) )
= ( vimage @ nat @ nat @ suc @ A6 ) ) ).
% vimage_Suc_insert_0
thf(fact_5675_coprime__common__divisor__int,axiom,
! [A3: int,B2: int,X: int] :
( ( algebr8660921524188924756oprime @ int @ A3 @ B2 )
=> ( ( dvd_dvd @ int @ X @ A3 )
=> ( ( dvd_dvd @ int @ X @ B2 )
=> ( ( abs_abs @ int @ X )
= ( one_one @ int ) ) ) ) ) ).
% coprime_common_divisor_int
thf(fact_5676_sorted__insort__insert,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,X: A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A )
@ ( linord329482645794927042rt_key @ A @ A
@ ^ [X2: A] : X2
@ X
@ Xs ) ) ) ) ).
% sorted_insort_insert
thf(fact_5677_card__vimage__inj__on__le,axiom,
! [A: $tType,B: $tType,F: A > B,D4: set @ A,A6: set @ B] :
( ( inj_on @ A @ B @ F @ D4 )
=> ( ( finite_finite2 @ B @ A6 )
=> ( ord_less_eq @ nat @ ( finite_card @ A @ ( inf_inf @ ( set @ A ) @ ( vimage @ A @ B @ F @ A6 ) @ D4 ) ) @ ( finite_card @ B @ A6 ) ) ) ) ).
% card_vimage_inj_on_le
thf(fact_5678_set__decode__div__2,axiom,
! [X: nat] :
( ( nat_set_decode @ ( divide_divide @ nat @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( vimage @ nat @ nat @ suc @ ( nat_set_decode @ X ) ) ) ).
% set_decode_div_2
thf(fact_5679_old_Orec__nat__def,axiom,
! [T: $tType] :
( ( rec_nat @ T )
= ( ^ [F12: T,F23: nat > T > T,X2: nat] : ( the @ T @ ( rec_set_nat @ T @ F12 @ F23 @ X2 ) ) ) ) ).
% old.rec_nat_def
thf(fact_5680_rat__floor__lemma,axiom,
! [A3: int,B2: int] :
( ( ord_less_eq @ rat @ ( ring_1_of_int @ rat @ ( divide_divide @ int @ A3 @ B2 ) ) @ ( fract @ A3 @ B2 ) )
& ( ord_less @ rat @ ( fract @ A3 @ B2 ) @ ( ring_1_of_int @ rat @ ( plus_plus @ int @ ( divide_divide @ int @ A3 @ B2 ) @ ( one_one @ int ) ) ) ) ) ).
% rat_floor_lemma
thf(fact_5681_sgn__rat,axiom,
! [A3: int,B2: int] :
( ( sgn_sgn @ rat @ ( fract @ A3 @ B2 ) )
= ( ring_1_of_int @ rat @ ( times_times @ int @ ( sgn_sgn @ int @ A3 ) @ ( sgn_sgn @ int @ B2 ) ) ) ) ).
% sgn_rat
thf(fact_5682_minus__rat__cancel,axiom,
! [A3: int,B2: int] :
( ( fract @ ( uminus_uminus @ int @ A3 ) @ ( uminus_uminus @ int @ B2 ) )
= ( fract @ A3 @ B2 ) ) ).
% minus_rat_cancel
thf(fact_5683_minus__rat,axiom,
! [A3: int,B2: int] :
( ( uminus_uminus @ rat @ ( fract @ A3 @ B2 ) )
= ( fract @ ( uminus_uminus @ int @ A3 ) @ B2 ) ) ).
% minus_rat
thf(fact_5684_divide__rat,axiom,
! [A3: int,B2: int,C2: int,D2: int] :
( ( divide_divide @ rat @ ( fract @ A3 @ B2 ) @ ( fract @ C2 @ D2 ) )
= ( fract @ ( times_times @ int @ A3 @ D2 ) @ ( times_times @ int @ B2 @ C2 ) ) ) ).
% divide_rat
thf(fact_5685_mult__rat,axiom,
! [A3: int,B2: int,C2: int,D2: int] :
( ( times_times @ rat @ ( fract @ A3 @ B2 ) @ ( fract @ C2 @ D2 ) )
= ( fract @ ( times_times @ int @ A3 @ C2 ) @ ( times_times @ int @ B2 @ D2 ) ) ) ).
% mult_rat
thf(fact_5686_less__rat,axiom,
! [B2: int,D2: int,A3: int,C2: int] :
( ( B2
!= ( zero_zero @ int ) )
=> ( ( D2
!= ( zero_zero @ int ) )
=> ( ( ord_less @ rat @ ( fract @ A3 @ B2 ) @ ( fract @ C2 @ D2 ) )
= ( ord_less @ int @ ( times_times @ int @ ( times_times @ int @ A3 @ D2 ) @ ( times_times @ int @ B2 @ D2 ) ) @ ( times_times @ int @ ( times_times @ int @ C2 @ B2 ) @ ( times_times @ int @ B2 @ D2 ) ) ) ) ) ) ).
% less_rat
thf(fact_5687_le__rat,axiom,
! [B2: int,D2: int,A3: int,C2: int] :
( ( B2
!= ( zero_zero @ int ) )
=> ( ( D2
!= ( zero_zero @ int ) )
=> ( ( ord_less_eq @ rat @ ( fract @ A3 @ B2 ) @ ( fract @ C2 @ D2 ) )
= ( ord_less_eq @ int @ ( times_times @ int @ ( times_times @ int @ A3 @ D2 ) @ ( times_times @ int @ B2 @ D2 ) ) @ ( times_times @ int @ ( times_times @ int @ C2 @ B2 ) @ ( times_times @ int @ B2 @ D2 ) ) ) ) ) ) ).
% le_rat
thf(fact_5688_add__rat,axiom,
! [B2: int,D2: int,A3: int,C2: int] :
( ( B2
!= ( zero_zero @ int ) )
=> ( ( D2
!= ( zero_zero @ int ) )
=> ( ( plus_plus @ rat @ ( fract @ A3 @ B2 ) @ ( fract @ C2 @ D2 ) )
= ( fract @ ( plus_plus @ int @ ( times_times @ int @ A3 @ D2 ) @ ( times_times @ int @ C2 @ B2 ) ) @ ( times_times @ int @ B2 @ D2 ) ) ) ) ) ).
% add_rat
thf(fact_5689_diff__rat,axiom,
! [B2: int,D2: int,A3: int,C2: int] :
( ( B2
!= ( zero_zero @ int ) )
=> ( ( D2
!= ( zero_zero @ int ) )
=> ( ( minus_minus @ rat @ ( fract @ A3 @ B2 ) @ ( fract @ C2 @ D2 ) )
= ( fract @ ( minus_minus @ int @ ( times_times @ int @ A3 @ D2 ) @ ( times_times @ int @ C2 @ B2 ) ) @ ( times_times @ int @ B2 @ D2 ) ) ) ) ) ).
% diff_rat
thf(fact_5690_Rat__cases,axiom,
! [Q4: rat] :
~ ! [A8: int,B5: int] :
( ( Q4
= ( fract @ A8 @ B5 ) )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B5 )
=> ~ ( algebr8660921524188924756oprime @ int @ A8 @ B5 ) ) ) ).
% Rat_cases
thf(fact_5691_Rat__induct,axiom,
! [P: rat > $o,Q4: rat] :
( ! [A8: int,B5: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B5 )
=> ( ( algebr8660921524188924756oprime @ int @ A8 @ B5 )
=> ( P @ ( fract @ A8 @ B5 ) ) ) )
=> ( P @ Q4 ) ) ).
% Rat_induct
thf(fact_5692_Fract__of__nat__eq,axiom,
! [K: nat] :
( ( fract @ ( semiring_1_of_nat @ int @ K ) @ ( one_one @ int ) )
= ( semiring_1_of_nat @ rat @ K ) ) ).
% Fract_of_nat_eq
thf(fact_5693_eq__rat_I3_J,axiom,
! [A3: int,C2: int] :
( ( fract @ ( zero_zero @ int ) @ A3 )
= ( fract @ ( zero_zero @ int ) @ C2 ) ) ).
% eq_rat(3)
thf(fact_5694_eq__rat_I1_J,axiom,
! [B2: int,D2: int,A3: int,C2: int] :
( ( B2
!= ( zero_zero @ int ) )
=> ( ( D2
!= ( zero_zero @ int ) )
=> ( ( ( fract @ A3 @ B2 )
= ( fract @ C2 @ D2 ) )
= ( ( times_times @ int @ A3 @ D2 )
= ( times_times @ int @ C2 @ B2 ) ) ) ) ) ).
% eq_rat(1)
thf(fact_5695_mult__rat__cancel,axiom,
! [C2: int,A3: int,B2: int] :
( ( C2
!= ( zero_zero @ int ) )
=> ( ( fract @ ( times_times @ int @ C2 @ A3 ) @ ( times_times @ int @ C2 @ B2 ) )
= ( fract @ A3 @ B2 ) ) ) ).
% mult_rat_cancel
thf(fact_5696_eq__rat_I2_J,axiom,
! [A3: int] :
( ( fract @ A3 @ ( zero_zero @ int ) )
= ( fract @ ( zero_zero @ int ) @ ( one_one @ int ) ) ) ).
% eq_rat(2)
thf(fact_5697_Rat__induct__pos,axiom,
! [P: rat > $o,Q4: rat] :
( ! [A8: int,B5: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B5 )
=> ( P @ ( fract @ A8 @ B5 ) ) )
=> ( P @ Q4 ) ) ).
% Rat_induct_pos
thf(fact_5698_rat__number__collapse_I6_J,axiom,
! [K: int] :
( ( fract @ K @ ( zero_zero @ int ) )
= ( zero_zero @ rat ) ) ).
% rat_number_collapse(6)
thf(fact_5699_rat__number__collapse_I1_J,axiom,
! [K: int] :
( ( fract @ ( zero_zero @ int ) @ K )
= ( zero_zero @ rat ) ) ).
% rat_number_collapse(1)
thf(fact_5700_One__rat__def,axiom,
( ( one_one @ rat )
= ( fract @ ( one_one @ int ) @ ( one_one @ int ) ) ) ).
% One_rat_def
thf(fact_5701_Fract__of__int__eq,axiom,
! [K: int] :
( ( fract @ K @ ( one_one @ int ) )
= ( ring_1_of_int @ rat @ K ) ) ).
% Fract_of_int_eq
thf(fact_5702_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_5703_Rat__cases__nonzero,axiom,
! [Q4: rat] :
( ! [A8: int,B5: int] :
( ( Q4
= ( fract @ A8 @ B5 ) )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B5 )
=> ( ( A8
!= ( zero_zero @ int ) )
=> ~ ( algebr8660921524188924756oprime @ int @ A8 @ B5 ) ) ) )
=> ( Q4
= ( zero_zero @ rat ) ) ) ).
% Rat_cases_nonzero
thf(fact_5704_Zero__rat__def,axiom,
( ( zero_zero @ rat )
= ( fract @ ( zero_zero @ int ) @ ( one_one @ int ) ) ) ).
% Zero_rat_def
thf(fact_5705_rat__number__expand_I3_J,axiom,
( ( numeral_numeral @ rat )
= ( ^ [K3: num] : ( fract @ ( numeral_numeral @ int @ K3 ) @ ( one_one @ int ) ) ) ) ).
% rat_number_expand(3)
thf(fact_5706_rat__number__collapse_I3_J,axiom,
! [W2: num] :
( ( fract @ ( numeral_numeral @ int @ W2 ) @ ( one_one @ int ) )
= ( numeral_numeral @ rat @ W2 ) ) ).
% rat_number_collapse(3)
thf(fact_5707_quotient__of__unique,axiom,
! [R: rat] :
? [X3: product_prod @ int @ int] :
( ( R
= ( fract @ ( product_fst @ int @ int @ X3 ) @ ( product_snd @ int @ int @ X3 ) ) )
& ( ord_less @ int @ ( zero_zero @ int ) @ ( product_snd @ int @ int @ X3 ) )
& ( algebr8660921524188924756oprime @ int @ ( product_fst @ int @ int @ X3 ) @ ( product_snd @ int @ int @ X3 ) )
& ! [Y4: product_prod @ int @ int] :
( ( ( R
= ( fract @ ( product_fst @ int @ int @ Y4 ) @ ( product_snd @ int @ int @ Y4 ) ) )
& ( ord_less @ int @ ( zero_zero @ int ) @ ( product_snd @ int @ int @ Y4 ) )
& ( algebr8660921524188924756oprime @ int @ ( product_fst @ int @ int @ Y4 ) @ ( product_snd @ int @ int @ Y4 ) ) )
=> ( Y4 = X3 ) ) ) ).
% quotient_of_unique
thf(fact_5708_Fract_Oabs__eq,axiom,
( fract
= ( ^ [Xa4: 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 @ Xa4 @ X2 ) ) ) ) ) ).
% Fract.abs_eq
thf(fact_5709_positive__rat,axiom,
! [A3: int,B2: int] :
( ( positive @ ( fract @ A3 @ B2 ) )
= ( ord_less @ int @ ( zero_zero @ int ) @ ( times_times @ int @ A3 @ B2 ) ) ) ).
% positive_rat
thf(fact_5710_Fract__less__zero__iff,axiom,
! [B2: int,A3: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ( ord_less @ rat @ ( fract @ A3 @ B2 ) @ ( zero_zero @ rat ) )
= ( ord_less @ int @ A3 @ ( zero_zero @ int ) ) ) ) ).
% Fract_less_zero_iff
thf(fact_5711_zero__less__Fract__iff,axiom,
! [B2: int,A3: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ( ord_less @ rat @ ( zero_zero @ rat ) @ ( fract @ A3 @ B2 ) )
= ( ord_less @ int @ ( zero_zero @ int ) @ A3 ) ) ) ).
% zero_less_Fract_iff
thf(fact_5712_one__less__Fract__iff,axiom,
! [B2: int,A3: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ( ord_less @ rat @ ( one_one @ rat ) @ ( fract @ A3 @ B2 ) )
= ( ord_less @ int @ B2 @ A3 ) ) ) ).
% one_less_Fract_iff
thf(fact_5713_Fract__less__one__iff,axiom,
! [B2: int,A3: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ( ord_less @ rat @ ( fract @ A3 @ B2 ) @ ( one_one @ rat ) )
= ( ord_less @ int @ A3 @ B2 ) ) ) ).
% Fract_less_one_iff
thf(fact_5714_rat__number__collapse_I5_J,axiom,
( ( fract @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( one_one @ int ) )
= ( uminus_uminus @ rat @ ( one_one @ rat ) ) ) ).
% rat_number_collapse(5)
thf(fact_5715_Fract_Otransfer,axiom,
( bNF_rel_fun @ int @ int @ ( int > ( product_prod @ int @ int ) ) @ ( int > rat )
@ ^ [Y: int,Z2: int] : Y = Z2
@ ( bNF_rel_fun @ int @ int @ ( product_prod @ int @ int ) @ rat
@ ^ [Y: int,Z2: int] : Y = Z2
@ pcr_rat )
@ ^ [A5: int,B3: int] :
( if @ ( product_prod @ int @ int )
@ ( B3
= ( zero_zero @ int ) )
@ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
@ ( product_Pair @ int @ int @ A5 @ B3 ) )
@ fract ) ).
% Fract.transfer
thf(fact_5716_Fract__add__one,axiom,
! [N: int,M: int] :
( ( N
!= ( zero_zero @ int ) )
=> ( ( fract @ ( plus_plus @ int @ M @ N ) @ N )
= ( plus_plus @ rat @ ( fract @ M @ N ) @ ( one_one @ rat ) ) ) ) ).
% Fract_add_one
thf(fact_5717_Fract__le__zero__iff,axiom,
! [B2: int,A3: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ( ord_less_eq @ rat @ ( fract @ A3 @ B2 ) @ ( zero_zero @ rat ) )
= ( ord_less_eq @ int @ A3 @ ( zero_zero @ int ) ) ) ) ).
% Fract_le_zero_iff
thf(fact_5718_zero__le__Fract__iff,axiom,
! [B2: int,A3: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ( ord_less_eq @ rat @ ( zero_zero @ rat ) @ ( fract @ A3 @ B2 ) )
= ( ord_less_eq @ int @ ( zero_zero @ int ) @ A3 ) ) ) ).
% zero_le_Fract_iff
thf(fact_5719_one__le__Fract__iff,axiom,
! [B2: int,A3: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ( ord_less_eq @ rat @ ( one_one @ rat ) @ ( fract @ A3 @ B2 ) )
= ( ord_less_eq @ int @ B2 @ A3 ) ) ) ).
% one_le_Fract_iff
thf(fact_5720_Fract__le__one__iff,axiom,
! [B2: int,A3: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ( ord_less_eq @ rat @ ( fract @ A3 @ B2 ) @ ( one_one @ rat ) )
= ( ord_less_eq @ int @ A3 @ B2 ) ) ) ).
% Fract_le_one_iff
thf(fact_5721_rat__number__expand_I5_J,axiom,
! [K: num] :
( ( uminus_uminus @ rat @ ( numeral_numeral @ rat @ K ) )
= ( fract @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) @ ( one_one @ int ) ) ) ).
% rat_number_expand(5)
thf(fact_5722_rat__number__collapse_I4_J,axiom,
! [W2: num] :
( ( fract @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ W2 ) ) @ ( one_one @ int ) )
= ( uminus_uminus @ rat @ ( numeral_numeral @ rat @ W2 ) ) ) ).
% rat_number_collapse(4)
thf(fact_5723_Rats__cases_H,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [X: A] :
( ( member @ A @ X @ ( field_char_0_Rats @ A ) )
=> ~ ! [A8: int,B5: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B5 )
=> ( ( algebr8660921524188924756oprime @ int @ A8 @ B5 )
=> ( X
!= ( divide_divide @ A @ ( ring_1_of_int @ A @ A8 ) @ ( ring_1_of_int @ A @ B5 ) ) ) ) ) ) ) ).
% Rats_cases'
thf(fact_5724_horner__sum__transfer,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType] :
( ( ( comm_semiring_0 @ B )
& ( comm_semiring_0 @ A ) )
=> ! [A6: A > B > $o,B7: C > D > $o] :
( ( A6 @ ( zero_zero @ A ) @ ( zero_zero @ B ) )
=> ( ( bNF_rel_fun @ A @ B @ ( A > A ) @ ( B > B ) @ A6 @ ( bNF_rel_fun @ A @ B @ A @ B @ A6 @ A6 ) @ ( plus_plus @ A ) @ ( plus_plus @ B ) )
=> ( ( bNF_rel_fun @ A @ B @ ( A > A ) @ ( B > B ) @ A6 @ ( bNF_rel_fun @ A @ B @ A @ B @ A6 @ A6 ) @ ( 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 @ B7 @ A6 ) @ ( bNF_rel_fun @ A @ B @ ( ( list @ C ) > A ) @ ( ( list @ D ) > B ) @ A6 @ ( bNF_rel_fun @ ( list @ C ) @ ( list @ D ) @ A @ B @ ( list_all2 @ C @ D @ B7 ) @ A6 ) ) @ ( groups4207007520872428315er_sum @ C @ A ) @ ( groups4207007520872428315er_sum @ D @ B ) ) ) ) ) ) ).
% horner_sum_transfer
thf(fact_5725_of__rat__neg__numeral__eq,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [W2: num] :
( ( field_char_0_of_rat @ A @ ( uminus_uminus @ rat @ ( numeral_numeral @ rat @ W2 ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) ) ).
% of_rat_neg_numeral_eq
thf(fact_5726_Rats__minus__iff,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A] :
( ( member @ A @ ( uminus_uminus @ A @ A3 ) @ ( field_char_0_Rats @ A ) )
= ( member @ A @ A3 @ ( field_char_0_Rats @ A ) ) ) ) ).
% Rats_minus_iff
thf(fact_5727_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_5728_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_5729_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_5730_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_5731_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_5732_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_5733_of__rat__of__nat__eq,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [N: nat] :
( ( field_char_0_of_rat @ A @ ( semiring_1_of_nat @ rat @ N ) )
= ( semiring_1_of_nat @ A @ N ) ) ) ).
% of_rat_of_nat_eq
thf(fact_5734_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_5735_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_5736_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_5737_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_5738_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_5739_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_5740_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_5741_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_5742_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_5743_Rats__mult,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,B2: A] :
( ( member @ A @ A3 @ ( field_char_0_Rats @ A ) )
=> ( ( member @ A @ B2 @ ( field_char_0_Rats @ A ) )
=> ( member @ A @ ( times_times @ A @ A3 @ B2 ) @ ( field_char_0_Rats @ A ) ) ) ) ) ).
% Rats_mult
thf(fact_5744_Rats__0,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( member @ A @ ( zero_zero @ A ) @ ( field_char_0_Rats @ A ) ) ) ).
% Rats_0
thf(fact_5745_Rats__1,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( member @ A @ ( one_one @ A ) @ ( field_char_0_Rats @ A ) ) ) ).
% Rats_1
thf(fact_5746_Rats__of__nat,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [N: nat] : ( member @ A @ ( semiring_1_of_nat @ A @ N ) @ ( field_char_0_Rats @ A ) ) ) ).
% Rats_of_nat
thf(fact_5747_of__rat__minus,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: rat] :
( ( field_char_0_of_rat @ A @ ( uminus_uminus @ rat @ A3 ) )
= ( uminus_uminus @ A @ ( field_char_0_of_rat @ A @ A3 ) ) ) ) ).
% of_rat_minus
thf(fact_5748_of__rat__less,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [R: rat,S: rat] :
( ( ord_less @ A @ ( field_char_0_of_rat @ A @ R ) @ ( field_char_0_of_rat @ A @ S ) )
= ( ord_less @ rat @ R @ S ) ) ) ).
% of_rat_less
thf(fact_5749_of__rat__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [R: rat,S: rat] :
( ( ord_less_eq @ A @ ( field_char_0_of_rat @ A @ R ) @ ( field_char_0_of_rat @ A @ S ) )
= ( ord_less_eq @ rat @ R @ S ) ) ) ).
% of_rat_less_eq
thf(fact_5750_of__rat__mult,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: rat,B2: rat] :
( ( field_char_0_of_rat @ A @ ( times_times @ rat @ A3 @ B2 ) )
= ( times_times @ A @ ( field_char_0_of_rat @ A @ A3 ) @ ( field_char_0_of_rat @ A @ B2 ) ) ) ) ).
% of_rat_mult
thf(fact_5751_list__all2__conv__all__nth,axiom,
! [B: $tType,A: $tType] :
( ( list_all2 @ A @ B )
= ( ^ [P4: A > B > $o,Xs3: list @ A,Ys3: list @ B] :
( ( ( size_size @ ( list @ A ) @ Xs3 )
= ( size_size @ ( list @ B ) @ Ys3 ) )
& ! [I5: nat] :
( ( ord_less @ nat @ I5 @ ( size_size @ ( list @ A ) @ Xs3 ) )
=> ( P4 @ ( nth @ A @ Xs3 @ I5 ) @ ( nth @ B @ Ys3 @ I5 ) ) ) ) ) ) ).
% list_all2_conv_all_nth
thf(fact_5752_list__all2__all__nthI,axiom,
! [A: $tType,B: $tType,A3: list @ A,B2: list @ B,P: A > B > $o] :
( ( ( size_size @ ( list @ A ) @ A3 )
= ( size_size @ ( list @ B ) @ B2 ) )
=> ( ! [N4: nat] :
( ( ord_less @ nat @ N4 @ ( size_size @ ( list @ A ) @ A3 ) )
=> ( P @ ( nth @ A @ A3 @ N4 ) @ ( nth @ B @ B2 @ N4 ) ) )
=> ( list_all2 @ A @ B @ P @ A3 @ B2 ) ) ) ).
% list_all2_all_nthI
thf(fact_5753_list__all2__nthD2,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Xs: list @ A,Ys2: list @ B,P5: nat] :
( ( list_all2 @ A @ B @ P @ Xs @ Ys2 )
=> ( ( ord_less @ nat @ P5 @ ( size_size @ ( list @ B ) @ Ys2 ) )
=> ( P @ ( nth @ A @ Xs @ P5 ) @ ( nth @ B @ Ys2 @ P5 ) ) ) ) ).
% list_all2_nthD2
thf(fact_5754_list__all2__nthD,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Xs: list @ A,Ys2: list @ B,P5: nat] :
( ( list_all2 @ A @ B @ P @ Xs @ Ys2 )
=> ( ( ord_less @ nat @ P5 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( P @ ( nth @ A @ Xs @ P5 ) @ ( nth @ B @ Ys2 @ P5 ) ) ) ) ).
% list_all2_nthD
thf(fact_5755_of__rat__rat,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [B2: int,A3: int] :
( ( B2
!= ( zero_zero @ int ) )
=> ( ( field_char_0_of_rat @ A @ ( fract @ A3 @ B2 ) )
= ( divide_divide @ A @ ( ring_1_of_int @ A @ A3 ) @ ( ring_1_of_int @ A @ B2 ) ) ) ) ) ).
% of_rat_rat
thf(fact_5756_sum__list__transfer,axiom,
! [A: $tType,B: $tType] :
( ( ( monoid_add @ B )
& ( monoid_add @ A ) )
=> ! [A6: A > B > $o] :
( ( A6 @ ( zero_zero @ A ) @ ( zero_zero @ B ) )
=> ( ( bNF_rel_fun @ A @ B @ ( A > A ) @ ( B > B ) @ A6 @ ( bNF_rel_fun @ A @ B @ A @ B @ A6 @ A6 ) @ ( plus_plus @ A ) @ ( plus_plus @ B ) )
=> ( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ A @ B @ ( list_all2 @ A @ B @ A6 ) @ A6 @ ( groups8242544230860333062m_list @ A ) @ ( groups8242544230860333062m_list @ B ) ) ) ) ) ).
% sum_list_transfer
thf(fact_5757_prod__list__transfer,axiom,
! [A: $tType,B: $tType] :
( ( ( monoid_mult @ B )
& ( monoid_mult @ A ) )
=> ! [A6: A > B > $o] :
( ( A6 @ ( one_one @ A ) @ ( one_one @ B ) )
=> ( ( bNF_rel_fun @ A @ B @ ( A > A ) @ ( B > B ) @ A6 @ ( bNF_rel_fun @ A @ B @ A @ B @ A6 @ A6 ) @ ( times_times @ A ) @ ( times_times @ B ) )
=> ( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ A @ B @ ( list_all2 @ A @ B @ A6 ) @ A6 @ ( groups5270119922927024881d_list @ A ) @ ( groups5270119922927024881d_list @ B ) ) ) ) ) ).
% prod_list_transfer
thf(fact_5758_bot_Oordering__top__axioms,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ( ordering_top @ A
@ ^ [X2: A,Y6: A] : ( ord_less_eq @ A @ Y6 @ X2 )
@ ^ [X2: A,Y6: A] : ( ord_less @ A @ Y6 @ X2 )
@ ( bot_bot @ A ) ) ) ).
% bot.ordering_top_axioms
thf(fact_5759_dropWhile__nth,axiom,
! [A: $tType,J: nat,P: A > $o,Xs: list @ A] :
( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ ( dropWhile @ A @ P @ Xs ) ) )
=> ( ( nth @ A @ ( dropWhile @ A @ P @ Xs ) @ J )
= ( nth @ A @ Xs @ ( plus_plus @ nat @ J @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P @ Xs ) ) ) ) ) ) ).
% dropWhile_nth
thf(fact_5760_prod__list_OCons,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [X: A,Xs: list @ A] :
( ( groups5270119922927024881d_list @ A @ ( cons @ A @ X @ Xs ) )
= ( times_times @ A @ X @ ( groups5270119922927024881d_list @ A @ Xs ) ) ) ) ).
% prod_list.Cons
thf(fact_5761_prod__list_ONil,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ( ( groups5270119922927024881d_list @ A @ ( nil @ A ) )
= ( one_one @ A ) ) ) ).
% prod_list.Nil
thf(fact_5762_prod__list_Oappend,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [Xs: list @ A,Ys2: list @ A] :
( ( groups5270119922927024881d_list @ A @ ( append @ A @ Xs @ Ys2 ) )
= ( times_times @ A @ ( groups5270119922927024881d_list @ A @ Xs ) @ ( groups5270119922927024881d_list @ A @ Ys2 ) ) ) ) ).
% prod_list.append
thf(fact_5763_ordering__top_Oextremum__uniqueI,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,Top: A,A3: A] :
( ( ordering_top @ A @ Less_eq @ Less @ Top )
=> ( ( Less_eq @ Top @ A3 )
=> ( A3 = Top ) ) ) ).
% ordering_top.extremum_uniqueI
thf(fact_5764_ordering__top_Onot__eq__extremum,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,Top: A,A3: A] :
( ( ordering_top @ A @ Less_eq @ Less @ Top )
=> ( ( A3 != Top )
= ( Less @ A3 @ Top ) ) ) ).
% ordering_top.not_eq_extremum
thf(fact_5765_ordering__top_Oextremum__unique,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,Top: A,A3: A] :
( ( ordering_top @ A @ Less_eq @ Less @ Top )
=> ( ( Less_eq @ Top @ A3 )
= ( A3 = Top ) ) ) ).
% ordering_top.extremum_unique
thf(fact_5766_ordering__top_Oextremum__strict,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,Top: A,A3: A] :
( ( ordering_top @ A @ Less_eq @ Less @ Top )
=> ~ ( Less @ Top @ A3 ) ) ).
% ordering_top.extremum_strict
thf(fact_5767_ordering__top_Oextremum,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,Top: A,A3: A] :
( ( ordering_top @ A @ Less_eq @ Less @ Top )
=> ( Less_eq @ A3 @ Top ) ) ).
% ordering_top.extremum
thf(fact_5768_length__dropWhile__le,axiom,
! [A: $tType,P: A > $o,Xs: list @ A] : ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ ( dropWhile @ A @ P @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ).
% length_dropWhile_le
thf(fact_5769_sorted__dropWhile,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,P: A > $o] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( dropWhile @ A @ P @ Xs ) ) ) ) ).
% sorted_dropWhile
thf(fact_5770_prod__list__zero__iff,axiom,
! [A: $tType] :
( ( ( semiring_1 @ A )
& ( semiri3467727345109120633visors @ A ) )
=> ! [Xs: list @ A] :
( ( ( groups5270119922927024881d_list @ A @ Xs )
= ( zero_zero @ A ) )
= ( member @ A @ ( zero_zero @ A ) @ ( set2 @ A @ Xs ) ) ) ) ).
% prod_list_zero_iff
thf(fact_5771_gcd__nat_Oordering__top__axioms,axiom,
( ordering_top @ nat @ ( dvd_dvd @ nat )
@ ^ [M2: nat,N3: nat] :
( ( dvd_dvd @ nat @ M2 @ N3 )
& ( M2 != N3 ) )
@ ( zero_zero @ nat ) ) ).
% gcd_nat.ordering_top_axioms
thf(fact_5772_prod__list_Oeq__foldr,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ( ( groups5270119922927024881d_list @ A )
= ( ^ [Xs3: list @ A] : ( foldr @ A @ A @ ( times_times @ A ) @ Xs3 @ ( one_one @ A ) ) ) ) ) ).
% prod_list.eq_foldr
thf(fact_5773_top_Oordering__top__axioms,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ( ordering_top @ A @ ( ord_less_eq @ A ) @ ( ord_less @ A ) @ ( top_top @ A ) ) ) ).
% top.ordering_top_axioms
thf(fact_5774_bot__nat__0_Oordering__top__axioms,axiom,
( ordering_top @ nat
@ ^ [X2: nat,Y6: nat] : ( ord_less_eq @ nat @ Y6 @ X2 )
@ ^ [X2: nat,Y6: nat] : ( ord_less @ nat @ Y6 @ X2 )
@ ( zero_zero @ nat ) ) ).
% bot_nat_0.ordering_top_axioms
thf(fact_5775_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,Y6: A] :
( ( ord_less @ nat @ ( F2 @ X2 ) @ ( F2 @ Y6 ) )
| ( ( ord_less_eq @ nat @ ( F2 @ X2 ) @ ( F2 @ Y6 ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y6 ) @ R6 ) ) ) ) ) ) ) ).
% mlex_eq
thf(fact_5776_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
@ ^ [Y6: A] :
( ( member @ A @ Y6 @ S3 )
& ( ( F @ Y6 )
= ( lattic643756798350308766er_Min @ B @ ( image @ A @ B @ F @ S3 ) ) ) ) ) ) ) ) ).
% arg_min_SOME_Min
thf(fact_5777_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_5778_prod__decode__eq,axiom,
! [X: nat,Y3: nat] :
( ( ( nat_prod_decode @ X )
= ( nat_prod_decode @ Y3 ) )
= ( X = Y3 ) ) ).
% prod_decode_eq
thf(fact_5779_some__equality,axiom,
! [A: $tType,P: A > $o,A3: A] :
( ( P @ A3 )
=> ( ! [X3: A] :
( ( P @ X3 )
=> ( X3 = A3 ) )
=> ( ( fChoice @ A @ P )
= A3 ) ) ) ).
% some_equality
thf(fact_5780_some__eq__trivial,axiom,
! [A: $tType,X: A] :
( ( fChoice @ A
@ ^ [Y6: A] : Y6 = X )
= X ) ).
% some_eq_trivial
thf(fact_5781_some__sym__eq__trivial,axiom,
! [A: $tType,X: A] :
( ( fChoice @ A
@ ( ^ [Y: A,Z2: A] : Y = Z2
@ X ) )
= X ) ).
% some_sym_eq_trivial
thf(fact_5782_Eps__case__prod__eq,axiom,
! [A: $tType,B: $tType,X: A,Y3: B] :
( ( fChoice @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [X9: A,Y8: B] :
( ( X = X9 )
& ( Y3 = Y8 ) ) ) )
= ( product_Pair @ A @ B @ X @ Y3 ) ) ).
% Eps_case_prod_eq
thf(fact_5783_prod__decode__inverse,axiom,
! [N: nat] :
( ( nat_prod_encode @ ( nat_prod_decode @ N ) )
= N ) ).
% prod_decode_inverse
thf(fact_5784_prod__encode__inverse,axiom,
! [X: product_prod @ nat @ nat] :
( ( nat_prod_decode @ ( nat_prod_encode @ X ) )
= X ) ).
% prod_encode_inverse
thf(fact_5785_inj__prod__decode,axiom,
! [A6: set @ nat] : ( inj_on @ nat @ ( product_prod @ nat @ nat ) @ nat_prod_decode @ A6 ) ).
% inj_prod_decode
thf(fact_5786_someI2,axiom,
! [A: $tType,P: A > $o,A3: A,Q: A > $o] :
( ( P @ A3 )
=> ( ! [X3: A] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( Q @ ( fChoice @ A @ P ) ) ) ) ).
% someI2
thf(fact_5787_someI__ex,axiom,
! [A: $tType,P: A > $o] :
( ? [X_1: A] : ( P @ X_1 )
=> ( P @ ( fChoice @ A @ P ) ) ) ).
% someI_ex
thf(fact_5788_someI2__ex,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ? [X_1: A] : ( P @ X_1 )
=> ( ! [X3: A] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( Q @ ( fChoice @ A @ P ) ) ) ) ).
% someI2_ex
thf(fact_5789_someI2__bex,axiom,
! [A: $tType,A6: set @ A,P: A > $o,Q: A > $o] :
( ? [X4: A] :
( ( member @ A @ X4 @ A6 )
& ( P @ X4 ) )
=> ( ! [X3: A] :
( ( ( member @ A @ X3 @ A6 )
& ( P @ X3 ) )
=> ( Q @ X3 ) )
=> ( Q
@ ( fChoice @ A
@ ^ [X2: A] :
( ( member @ A @ X2 @ A6 )
& ( P @ X2 ) ) ) ) ) ) ).
% someI2_bex
thf(fact_5790_some__eq__ex,axiom,
! [A: $tType,P: A > $o] :
( ( P @ ( fChoice @ A @ P ) )
= ( ? [X7: A] : ( P @ X7 ) ) ) ).
% some_eq_ex
thf(fact_5791_some1__equality,axiom,
! [A: $tType,P: A > $o,A3: A] :
( ? [X4: A] :
( ( P @ X4 )
& ! [Y5: A] :
( ( P @ Y5 )
=> ( Y5 = X4 ) ) )
=> ( ( P @ A3 )
=> ( ( fChoice @ A @ P )
= A3 ) ) ) ).
% some1_equality
thf(fact_5792_verit__sko__ex_H,axiom,
! [A: $tType,P: A > $o,A6: $o] :
( ( ( P @ ( fChoice @ A @ P ) )
= A6 )
=> ( ( ? [X7: A] : ( P @ X7 ) )
= A6 ) ) ).
% verit_sko_ex'
thf(fact_5793_verit__sko__forall,axiom,
! [A: $tType] :
( ( ^ [P3: A > $o] :
! [X6: A] : ( P3 @ X6 ) )
= ( ^ [P4: A > $o] :
( P4
@ ( fChoice @ A
@ ^ [X2: A] :
~ ( P4 @ X2 ) ) ) ) ) ).
% verit_sko_forall
thf(fact_5794_verit__sko__forall_H,axiom,
! [A: $tType,P: A > $o,A6: $o] :
( ( ( P
@ ( fChoice @ A
@ ^ [X2: A] :
~ ( P @ X2 ) ) )
= A6 )
=> ( ( ! [X7: A] : ( P @ X7 ) )
= A6 ) ) ).
% verit_sko_forall'
thf(fact_5795_verit__sko__forall_H_H,axiom,
! [A: $tType,B7: A,A6: A,P: A > $o] :
( ( B7 = A6 )
=> ( ( ( fChoice @ A @ P )
= A6 )
= ( ( fChoice @ A @ P )
= B7 ) ) ) ).
% verit_sko_forall''
thf(fact_5796_verit__sko__ex__indirect,axiom,
! [A: $tType,X: A,P: A > $o] :
( ( X
= ( fChoice @ A @ P ) )
=> ( ( ? [X7: A] : ( P @ X7 ) )
= ( P @ X ) ) ) ).
% verit_sko_ex_indirect
thf(fact_5797_verit__sko__ex__indirect2,axiom,
! [A: $tType,X: A,P: A > $o,P2: A > $o] :
( ( X
= ( fChoice @ A @ P ) )
=> ( ! [X3: A] :
( ( P @ X3 )
= ( P2 @ X3 ) )
=> ( ( ? [X7: A] : ( P2 @ X7 ) )
= ( P @ X ) ) ) ) ).
% verit_sko_ex_indirect2
thf(fact_5798_verit__sko__forall__indirect,axiom,
! [A: $tType,X: A,P: A > $o] :
( ( X
= ( fChoice @ A
@ ^ [X2: A] :
~ ( P @ X2 ) ) )
=> ( ( ! [X7: A] : ( P @ X7 ) )
= ( P @ X ) ) ) ).
% verit_sko_forall_indirect
thf(fact_5799_verit__sko__forall__indirect2,axiom,
! [A: $tType,X: A,P: A > $o,P2: A > $o] :
( ( X
= ( fChoice @ A
@ ^ [X2: A] :
~ ( P @ X2 ) ) )
=> ( ! [X3: A] :
( ( P @ X3 )
= ( P2 @ X3 ) )
=> ( ( ! [X7: A] : ( P2 @ X7 ) )
= ( P @ X ) ) ) ) ).
% verit_sko_forall_indirect2
thf(fact_5800_Nitpick_OEps__psimp,axiom,
! [A: $tType,P: A > $o,X: A,Y3: A] :
( ( P @ X )
=> ( ~ ( P @ Y3 )
=> ( ( ( fChoice @ A @ P )
= Y3 )
=> ( ( fChoice @ A @ P )
= X ) ) ) ) ).
% Nitpick.Eps_psimp
thf(fact_5801_someI,axiom,
! [A: $tType,P: A > $o,X: A] :
( ( P @ X )
=> ( P @ ( fChoice @ A @ P ) ) ) ).
% someI
thf(fact_5802_Eps__cong,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X3: A] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( fChoice @ A @ P )
= ( fChoice @ A @ Q ) ) ) ).
% Eps_cong
thf(fact_5803_tfl__some,axiom,
! [A: $tType,P7: A > $o,X4: A] :
( ( P7 @ X4 )
=> ( P7 @ ( fChoice @ A @ P7 ) ) ) ).
% tfl_some
thf(fact_5804_some__eq__imp,axiom,
! [A: $tType,P: A > $o,A3: A,B2: A] :
( ( ( fChoice @ A @ P )
= A3 )
=> ( ( P @ B2 )
=> ( P @ A3 ) ) ) ).
% some_eq_imp
thf(fact_5805_some__in__eq,axiom,
! [A: $tType,A6: set @ A] :
( ( member @ A
@ ( fChoice @ A
@ ^ [X2: A] : ( member @ A @ X2 @ A6 ) )
@ A6 )
= ( A6
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% some_in_eq
thf(fact_5806_exE__some,axiom,
! [A: $tType,P: A > $o,C2: A] :
( ? [X_1: A] : ( P @ X_1 )
=> ( ( C2
= ( fChoice @ A @ P ) )
=> ( P @ C2 ) ) ) ).
% exE_some
thf(fact_5807_split__paired__Eps,axiom,
! [B: $tType,A: $tType] :
( ( fChoice @ ( product_prod @ A @ B ) )
= ( ^ [P4: ( product_prod @ A @ B ) > $o] :
( fChoice @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [A5: A,B3: B] : ( P4 @ ( product_Pair @ A @ B @ A5 @ B3 ) ) ) ) ) ) ).
% split_paired_Eps
thf(fact_5808_prod__decode__def,axiom,
( nat_prod_decode
= ( nat_prod_decode_aux @ ( zero_zero @ nat ) ) ) ).
% prod_decode_def
thf(fact_5809_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_5810_surj__prod__decode,axiom,
( ( image @ nat @ ( product_prod @ nat @ nat ) @ nat_prod_decode @ ( top_top @ ( set @ nat ) ) )
= ( top_top @ ( set @ ( product_prod @ nat @ nat ) ) ) ) ).
% surj_prod_decode
thf(fact_5811_mlex__less,axiom,
! [A: $tType,F: A > nat,X: A,Y3: A,R2: 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 @ R2 ) ) ) ).
% mlex_less
thf(fact_5812_mlex__iff,axiom,
! [A: $tType,X: A,Y3: A,F: A > nat,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ ( mlex_prod @ A @ F @ R2 ) )
= ( ( ord_less @ nat @ ( F @ X ) @ ( F @ Y3 ) )
| ( ( ( F @ X )
= ( F @ Y3 ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ R2 ) ) ) ) ).
% mlex_iff
thf(fact_5813_mlex__leq,axiom,
! [A: $tType,F: A > nat,X: A,Y3: A,R2: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ nat @ ( F @ X ) @ ( F @ Y3 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ R2 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ ( mlex_prod @ A @ F @ R2 ) ) ) ) ).
% mlex_leq
thf(fact_5814_list__decode_Oelims,axiom,
! [X: nat,Y3: list @ nat] :
( ( ( nat_list_decode @ X )
= Y3 )
=> ( ( ( X
= ( zero_zero @ nat ) )
=> ( Y3
!= ( nil @ nat ) ) )
=> ~ ! [N4: nat] :
( ( X
= ( suc @ N4 ) )
=> ( Y3
!= ( product_case_prod @ nat @ nat @ ( list @ nat )
@ ^ [X2: nat,Y6: nat] : ( cons @ nat @ X2 @ ( nat_list_decode @ Y6 ) )
@ ( nat_prod_decode @ N4 ) ) ) ) ) ) ).
% list_decode.elims
thf(fact_5815_arg__min__def,axiom,
! [A: $tType,B: $tType] :
( ( ord @ A )
=> ( ( lattices_ord_arg_min @ B @ A )
= ( ^ [F2: B > A,P4: B > $o] : ( fChoice @ B @ ( lattic501386751177426532rg_min @ B @ A @ F2 @ P4 ) ) ) ) ) ).
% arg_min_def
thf(fact_5816_list__decode_Osimps_I2_J,axiom,
! [N: nat] :
( ( nat_list_decode @ ( suc @ N ) )
= ( product_case_prod @ nat @ nat @ ( list @ nat )
@ ^ [X2: nat,Y6: nat] : ( cons @ nat @ X2 @ ( nat_list_decode @ Y6 ) )
@ ( nat_prod_decode @ N ) ) ) ).
% list_decode.simps(2)
thf(fact_5817_list__encode__inverse,axiom,
! [X: list @ nat] :
( ( nat_list_decode @ ( nat_list_encode @ X ) )
= X ) ).
% list_encode_inverse
thf(fact_5818_list__decode__inverse,axiom,
! [N: nat] :
( ( nat_list_encode @ ( nat_list_decode @ N ) )
= N ) ).
% list_decode_inverse
thf(fact_5819_list__decode__eq,axiom,
! [X: nat,Y3: nat] :
( ( ( nat_list_decode @ X )
= ( nat_list_decode @ Y3 ) )
= ( X = Y3 ) ) ).
% list_decode_eq
thf(fact_5820_inj__list__decode,axiom,
! [A6: set @ nat] : ( inj_on @ nat @ ( list @ nat ) @ nat_list_decode @ A6 ) ).
% inj_list_decode
thf(fact_5821_is__arg__min__antimono,axiom,
! [B: $tType,A: $tType] :
( ( order @ B )
=> ! [F: A > B,P: A > $o,X: A,Y3: A] :
( ( lattic501386751177426532rg_min @ A @ B @ F @ P @ X )
=> ( ( ord_less_eq @ B @ ( F @ Y3 ) @ ( F @ X ) )
=> ( ( P @ Y3 )
=> ( lattic501386751177426532rg_min @ A @ B @ F @ P @ Y3 ) ) ) ) ) ).
% is_arg_min_antimono
thf(fact_5822_is__arg__min__linorder,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ( ( lattic501386751177426532rg_min @ A @ B )
= ( ^ [F2: A > B,P4: A > $o,X2: A] :
( ( P4 @ X2 )
& ! [Y6: A] :
( ( P4 @ Y6 )
=> ( ord_less_eq @ B @ ( F2 @ X2 ) @ ( F2 @ Y6 ) ) ) ) ) ) ) ).
% is_arg_min_linorder
thf(fact_5823_is__arg__min__def,axiom,
! [A: $tType,B: $tType] :
( ( ord @ A )
=> ( ( lattic501386751177426532rg_min @ B @ A )
= ( ^ [F2: B > A,P4: B > $o,X2: B] :
( ( P4 @ X2 )
& ~ ? [Y6: B] :
( ( P4 @ Y6 )
& ( ord_less @ A @ ( F2 @ Y6 ) @ ( F2 @ X2 ) ) ) ) ) ) ) ).
% is_arg_min_def
thf(fact_5824_list__decode_Osimps_I1_J,axiom,
( ( nat_list_decode @ ( zero_zero @ nat ) )
= ( nil @ nat ) ) ).
% list_decode.simps(1)
thf(fact_5825_is__arg__min__arg__min__nat,axiom,
! [A: $tType,P: A > $o,X: A,M: A > nat] :
( ( P @ X )
=> ( lattic501386751177426532rg_min @ A @ nat @ M @ P @ ( lattices_ord_arg_min @ A @ nat @ M @ P ) ) ) ).
% is_arg_min_arg_min_nat
thf(fact_5826_surj__list__decode,axiom,
( ( image @ nat @ ( list @ nat ) @ nat_list_decode @ ( top_top @ ( set @ nat ) ) )
= ( top_top @ ( set @ ( list @ nat ) ) ) ) ).
% surj_list_decode
thf(fact_5827_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_5828_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 ) ) ) )
=> ~ ! [N4: nat] :
( ( X
= ( suc @ N4 ) )
=> ( ( Y3
= ( product_case_prod @ nat @ nat @ ( list @ nat )
@ ^ [X2: nat,Y6: nat] : ( cons @ nat @ X2 @ ( nat_list_decode @ Y6 ) )
@ ( nat_prod_decode @ N4 ) ) )
=> ~ ( accp @ nat @ nat_list_decode_rel @ ( suc @ N4 ) ) ) ) ) ) ) ).
% list_decode.pelims
thf(fact_5829_mono__cSUP,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ( condit1219197933456340205attice @ A )
& ( condit1219197933456340205attice @ B ) )
=> ! [F: A > B,A6: C > A,I6: set @ C] :
( ( order_mono @ A @ B @ F )
=> ( ( condit941137186595557371_above @ A @ ( image @ C @ A @ A6 @ I6 ) )
=> ( ( I6
!= ( bot_bot @ ( set @ C ) ) )
=> ( ord_less_eq @ B
@ ( complete_Sup_Sup @ B
@ ( image @ C @ B
@ ^ [X2: C] : ( F @ ( A6 @ X2 ) )
@ I6 ) )
@ ( F @ ( complete_Sup_Sup @ A @ ( image @ C @ A @ A6 @ I6 ) ) ) ) ) ) ) ) ).
% mono_cSUP
thf(fact_5830_mono__cSup,axiom,
! [B: $tType,A: $tType] :
( ( ( condit1219197933456340205attice @ A )
& ( condit1219197933456340205attice @ B ) )
=> ! [F: A > B,A6: set @ A] :
( ( order_mono @ A @ B @ F )
=> ( ( condit941137186595557371_above @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ B @ ( complete_Sup_Sup @ B @ ( image @ A @ B @ F @ A6 ) ) @ ( F @ ( complete_Sup_Sup @ A @ A6 ) ) ) ) ) ) ) ).
% mono_cSup
thf(fact_5831_bdd__above_OI,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A6: set @ A,M7: A] :
( ! [X3: A] :
( ( member @ A @ X3 @ A6 )
=> ( ord_less_eq @ A @ X3 @ M7 ) )
=> ( condit941137186595557371_above @ A @ A6 ) ) ) ).
% bdd_above.I
thf(fact_5832_bdd__above_OI2,axiom,
! [A: $tType,B: $tType] :
( ( preorder @ A )
=> ! [A6: set @ B,F: B > A,M7: A] :
( ! [X3: B] :
( ( member @ B @ X3 @ A6 )
=> ( ord_less_eq @ A @ ( F @ X3 ) @ M7 ) )
=> ( condit941137186595557371_above @ A @ ( image @ B @ A @ F @ A6 ) ) ) ) ).
% bdd_above.I2
thf(fact_5833_bdd__above_OE,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A6: set @ A] :
( ( condit941137186595557371_above @ A @ A6 )
=> ~ ! [M8: A] :
~ ! [X4: A] :
( ( member @ A @ X4 @ A6 )
=> ( ord_less_eq @ A @ X4 @ M8 ) ) ) ) ).
% bdd_above.E
thf(fact_5834_bdd__above_Ounfold,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( condit941137186595557371_above @ A )
= ( ^ [A7: set @ A] :
? [M9: A] :
! [X2: A] :
( ( member @ A @ X2 @ A7 )
=> ( ord_less_eq @ A @ X2 @ M9 ) ) ) ) ) ).
% bdd_above.unfold
thf(fact_5835_cSup__upper2,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X: A,X8: set @ A,Y3: A] :
( ( member @ A @ X @ X8 )
=> ( ( ord_less_eq @ A @ Y3 @ X )
=> ( ( condit941137186595557371_above @ A @ X8 )
=> ( ord_less_eq @ A @ Y3 @ ( complete_Sup_Sup @ A @ X8 ) ) ) ) ) ) ).
% cSup_upper2
thf(fact_5836_cSup__upper,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X: A,X8: set @ A] :
( ( member @ A @ X @ X8 )
=> ( ( condit941137186595557371_above @ A @ X8 )
=> ( ord_less_eq @ A @ X @ ( complete_Sup_Sup @ A @ X8 ) ) ) ) ) ).
% cSup_upper
thf(fact_5837_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_5838_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 ) ) )
=> ( ! [N4: nat] :
( ( accp @ nat @ nat_list_decode_rel @ ( suc @ N4 ) )
=> ( ! [X4: nat,Y4: nat] :
( ( ( product_Pair @ nat @ nat @ X4 @ Y4 )
= ( nat_prod_decode @ N4 ) )
=> ( P @ Y4 ) )
=> ( P @ ( suc @ N4 ) ) ) )
=> ( P @ A0 ) ) ) ) ).
% list_decode.pinduct
thf(fact_5839_cSUP__upper,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X: B,A6: set @ B,F: B > A] :
( ( member @ B @ X @ A6 )
=> ( ( condit941137186595557371_above @ A @ ( image @ B @ A @ F @ A6 ) )
=> ( ord_less_eq @ A @ ( F @ X ) @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A6 ) ) ) ) ) ) ).
% cSUP_upper
thf(fact_5840_cSUP__upper2,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [F: B > A,A6: set @ B,X: B,U: A] :
( ( condit941137186595557371_above @ A @ ( image @ B @ A @ F @ A6 ) )
=> ( ( member @ B @ X @ A6 )
=> ( ( ord_less_eq @ A @ U @ ( F @ X ) )
=> ( ord_less_eq @ A @ U @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A6 ) ) ) ) ) ) ) ).
% cSUP_upper2
thf(fact_5841_cSup__le__iff,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [S3: set @ A,A3: A] :
( ( S3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit941137186595557371_above @ A @ S3 )
=> ( ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ S3 ) @ A3 )
= ( ! [X2: A] :
( ( member @ A @ X2 @ S3 )
=> ( ord_less_eq @ A @ X2 @ A3 ) ) ) ) ) ) ) ).
% cSup_le_iff
thf(fact_5842_cSup__mono,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [B7: set @ A,A6: set @ A] :
( ( B7
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit941137186595557371_above @ A @ A6 )
=> ( ! [B5: A] :
( ( member @ A @ B5 @ B7 )
=> ? [X4: A] :
( ( member @ A @ X4 @ A6 )
& ( ord_less_eq @ A @ B5 @ X4 ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ B7 ) @ ( complete_Sup_Sup @ A @ A6 ) ) ) ) ) ) ).
% cSup_mono
thf(fact_5843_less__cSup__iff,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X8: set @ A,Y3: A] :
( ( X8
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit941137186595557371_above @ A @ X8 )
=> ( ( ord_less @ A @ Y3 @ ( complete_Sup_Sup @ A @ X8 ) )
= ( ? [X2: A] :
( ( member @ A @ X2 @ X8 )
& ( ord_less @ A @ Y3 @ X2 ) ) ) ) ) ) ) ).
% less_cSup_iff
thf(fact_5844_cSUP__lessD,axiom,
! [B: $tType,A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [F: B > A,A6: set @ B,Y3: A,I: B] :
( ( condit941137186595557371_above @ A @ ( image @ B @ A @ F @ A6 ) )
=> ( ( ord_less @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A6 ) ) @ Y3 )
=> ( ( member @ B @ I @ A6 )
=> ( ord_less @ A @ ( F @ I ) @ Y3 ) ) ) ) ) ).
% cSUP_lessD
thf(fact_5845_cSUP__le__iff,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A6: set @ B,F: B > A,U: A] :
( ( A6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit941137186595557371_above @ A @ ( image @ B @ A @ F @ A6 ) )
=> ( ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A6 ) ) @ U )
= ( ! [X2: B] :
( ( member @ B @ X2 @ A6 )
=> ( ord_less_eq @ A @ ( F @ X2 ) @ U ) ) ) ) ) ) ) ).
% cSUP_le_iff
thf(fact_5846_cSUP__mono,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A6: set @ B,G: C > A,B7: set @ C,F: B > A] :
( ( A6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit941137186595557371_above @ A @ ( image @ C @ A @ G @ B7 ) )
=> ( ! [N4: B] :
( ( member @ B @ N4 @ A6 )
=> ? [X4: C] :
( ( member @ C @ X4 @ B7 )
& ( ord_less_eq @ A @ ( F @ N4 ) @ ( G @ X4 ) ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A6 ) ) @ ( complete_Sup_Sup @ A @ ( image @ C @ A @ G @ B7 ) ) ) ) ) ) ) ).
% cSUP_mono
thf(fact_5847_cSup__subset__mono,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A6: set @ A,B7: set @ A] :
( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit941137186595557371_above @ A @ B7 )
=> ( ( ord_less_eq @ ( set @ A ) @ A6 @ B7 )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A6 ) @ ( complete_Sup_Sup @ A @ B7 ) ) ) ) ) ) ).
% cSup_subset_mono
thf(fact_5848_less__cSUP__iff,axiom,
! [A: $tType,B: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [A6: set @ B,F: B > A,A3: A] :
( ( A6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit941137186595557371_above @ A @ ( image @ B @ A @ F @ A6 ) )
=> ( ( ord_less @ A @ A3 @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A6 ) ) )
= ( ? [X2: B] :
( ( member @ B @ X2 @ A6 )
& ( ord_less @ A @ A3 @ ( F @ X2 ) ) ) ) ) ) ) ) ).
% less_cSUP_iff
thf(fact_5849_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,Y6: nat] : ( cons @ nat @ X2 @ ( nat_list_decode @ Y6 ) )
@ ( nat_prod_decode @ N ) ) ) ) ).
% list_decode.psimps(2)
thf(fact_5850_cSUP__subset__mono,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A6: set @ B,G: B > A,B7: set @ B,F: B > A] :
( ( A6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit941137186595557371_above @ A @ ( image @ B @ A @ G @ B7 ) )
=> ( ( ord_less_eq @ ( set @ B ) @ A6 @ B7 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ A6 )
=> ( ord_less_eq @ A @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A6 ) ) @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ G @ B7 ) ) ) ) ) ) ) ) ).
% cSUP_subset_mono
thf(fact_5851_cSup__inter__less__eq,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A6: set @ A,B7: set @ A] :
( ( condit941137186595557371_above @ A @ A6 )
=> ( ( condit941137186595557371_above @ A @ B7 )
=> ( ( ( inf_inf @ ( set @ A ) @ A6 @ B7 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( inf_inf @ ( set @ A ) @ A6 @ B7 ) ) @ ( sup_sup @ A @ ( complete_Sup_Sup @ A @ A6 ) @ ( complete_Sup_Sup @ A @ B7 ) ) ) ) ) ) ) ).
% cSup_inter_less_eq
thf(fact_5852_division__segment__integer__def,axiom,
( ( euclid7384307370059645450egment @ code_integer )
= ( map_fun @ code_integer @ int @ int @ code_integer @ code_int_of_integer @ code_integer_of_int @ ( euclid7384307370059645450egment @ int ) ) ) ).
% division_segment_integer_def
thf(fact_5853_mono__cInf,axiom,
! [B: $tType,A: $tType] :
( ( ( condit1219197933456340205attice @ A )
& ( condit1219197933456340205attice @ B ) )
=> ! [F: A > B,A6: set @ A] :
( ( order_mono @ A @ B @ F )
=> ( ( condit1013018076250108175_below @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ B @ ( F @ ( complete_Inf_Inf @ A @ A6 ) ) @ ( complete_Inf_Inf @ B @ ( image @ A @ B @ F @ A6 ) ) ) ) ) ) ) ).
% mono_cInf
thf(fact_5854_mono__cINF,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ( condit1219197933456340205attice @ A )
& ( condit1219197933456340205attice @ B ) )
=> ! [F: A > B,A6: C > A,I6: set @ C] :
( ( order_mono @ A @ B @ F )
=> ( ( condit1013018076250108175_below @ A @ ( image @ C @ A @ A6 @ I6 ) )
=> ( ( I6
!= ( bot_bot @ ( set @ C ) ) )
=> ( ord_less_eq @ B @ ( F @ ( complete_Inf_Inf @ A @ ( image @ C @ A @ A6 @ I6 ) ) )
@ ( complete_Inf_Inf @ B
@ ( image @ C @ B
@ ^ [X2: C] : ( F @ ( A6 @ X2 ) )
@ I6 ) ) ) ) ) ) ) ).
% mono_cINF
thf(fact_5855_bdd__belowI,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A6: set @ A,M: A] :
( ! [X3: A] :
( ( member @ A @ X3 @ A6 )
=> ( ord_less_eq @ A @ M @ X3 ) )
=> ( condit1013018076250108175_below @ A @ A6 ) ) ) ).
% bdd_belowI
thf(fact_5856_bdd__below_OI,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A6: set @ A,M7: A] :
( ! [X3: A] :
( ( member @ A @ X3 @ A6 )
=> ( ord_less_eq @ A @ M7 @ X3 ) )
=> ( condit1013018076250108175_below @ A @ A6 ) ) ) ).
% bdd_below.I
thf(fact_5857_bdd__below__uminus,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [X8: set @ A] :
( ( condit1013018076250108175_below @ A @ ( image @ A @ A @ ( uminus_uminus @ A ) @ X8 ) )
= ( condit941137186595557371_above @ A @ X8 ) ) ) ).
% bdd_below_uminus
thf(fact_5858_bdd__above__uminus,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [X8: set @ A] :
( ( condit941137186595557371_above @ A @ ( image @ A @ A @ ( uminus_uminus @ A ) @ X8 ) )
= ( condit1013018076250108175_below @ A @ X8 ) ) ) ).
% bdd_above_uminus
thf(fact_5859_bdd__below_OI2,axiom,
! [A: $tType,B: $tType] :
( ( preorder @ A )
=> ! [A6: set @ B,M7: A,F: B > A] :
( ! [X3: B] :
( ( member @ B @ X3 @ A6 )
=> ( ord_less_eq @ A @ M7 @ ( F @ X3 ) ) )
=> ( condit1013018076250108175_below @ A @ ( image @ B @ A @ F @ A6 ) ) ) ) ).
% bdd_below.I2
thf(fact_5860_bdd__belowI2,axiom,
! [A: $tType,B: $tType] :
( ( preorder @ A )
=> ! [A6: set @ B,M: A,F: B > A] :
( ! [X3: B] :
( ( member @ B @ X3 @ A6 )
=> ( ord_less_eq @ A @ M @ ( F @ X3 ) ) )
=> ( condit1013018076250108175_below @ A @ ( image @ B @ A @ F @ A6 ) ) ) ) ).
% bdd_belowI2
thf(fact_5861_cInf__lower2,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X: A,X8: set @ A,Y3: A] :
( ( member @ A @ X @ X8 )
=> ( ( ord_less_eq @ A @ X @ Y3 )
=> ( ( condit1013018076250108175_below @ A @ X8 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ X8 ) @ Y3 ) ) ) ) ) ).
% cInf_lower2
thf(fact_5862_cInf__lower,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X: A,X8: set @ A] :
( ( member @ A @ X @ X8 )
=> ( ( condit1013018076250108175_below @ A @ X8 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ X8 ) @ X ) ) ) ) ).
% cInf_lower
thf(fact_5863_bdd__below_Ounfold,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( condit1013018076250108175_below @ A )
= ( ^ [A7: set @ A] :
? [M9: A] :
! [X2: A] :
( ( member @ A @ X2 @ A7 )
=> ( ord_less_eq @ A @ M9 @ X2 ) ) ) ) ) ).
% bdd_below.unfold
thf(fact_5864_bdd__below_OE,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A6: set @ A] :
( ( condit1013018076250108175_below @ A @ A6 )
=> ~ ! [M8: A] :
~ ! [X4: A] :
( ( member @ A @ X4 @ A6 )
=> ( ord_less_eq @ A @ M8 @ X4 ) ) ) ) ).
% bdd_below.E
thf(fact_5865_cINF__lower2,axiom,
! [B: $tType,A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [F: B > A,A6: set @ B,X: B,U: A] :
( ( condit1013018076250108175_below @ A @ ( image @ B @ A @ F @ A6 ) )
=> ( ( member @ B @ X @ A6 )
=> ( ( ord_less_eq @ A @ ( F @ X ) @ U )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F @ A6 ) ) @ U ) ) ) ) ) ).
% cINF_lower2
thf(fact_5866_cINF__lower,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [F: B > A,A6: set @ B,X: B] :
( ( condit1013018076250108175_below @ A @ ( image @ B @ A @ F @ A6 ) )
=> ( ( member @ B @ X @ A6 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F @ A6 ) ) @ ( F @ X ) ) ) ) ) ).
% cINF_lower
thf(fact_5867_le__cInf__iff,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [S3: set @ A,A3: A] :
( ( S3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit1013018076250108175_below @ A @ S3 )
=> ( ( ord_less_eq @ A @ A3 @ ( complete_Inf_Inf @ A @ S3 ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ S3 )
=> ( ord_less_eq @ A @ A3 @ X2 ) ) ) ) ) ) ) ).
% le_cInf_iff
thf(fact_5868_cInf__mono,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [B7: set @ A,A6: set @ A] :
( ( B7
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit1013018076250108175_below @ A @ A6 )
=> ( ! [B5: A] :
( ( member @ A @ B5 @ B7 )
=> ? [X4: A] :
( ( member @ A @ X4 @ A6 )
& ( ord_less_eq @ A @ X4 @ B5 ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A6 ) @ ( complete_Inf_Inf @ A @ B7 ) ) ) ) ) ) ).
% cInf_mono
thf(fact_5869_cInf__less__iff,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X8: set @ A,Y3: A] :
( ( X8
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit1013018076250108175_below @ A @ X8 )
=> ( ( ord_less @ A @ ( complete_Inf_Inf @ A @ X8 ) @ Y3 )
= ( ? [X2: A] :
( ( member @ A @ X2 @ X8 )
& ( ord_less @ A @ X2 @ Y3 ) ) ) ) ) ) ) ).
% cInf_less_iff
thf(fact_5870_less__cINF__D,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [F: B > A,A6: set @ B,Y3: A,I: B] :
( ( condit1013018076250108175_below @ A @ ( image @ B @ A @ F @ A6 ) )
=> ( ( ord_less @ A @ Y3 @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F @ A6 ) ) )
=> ( ( member @ B @ I @ A6 )
=> ( ord_less @ A @ Y3 @ ( F @ I ) ) ) ) ) ) ).
% less_cINF_D
thf(fact_5871_cINF__mono,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [B7: set @ B,F: C > A,A6: set @ C,G: B > A] :
( ( B7
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit1013018076250108175_below @ A @ ( image @ C @ A @ F @ A6 ) )
=> ( ! [M6: B] :
( ( member @ B @ M6 @ B7 )
=> ? [X4: C] :
( ( member @ C @ X4 @ A6 )
& ( ord_less_eq @ A @ ( F @ X4 ) @ ( G @ M6 ) ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image @ C @ A @ F @ A6 ) ) @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ G @ B7 ) ) ) ) ) ) ) ).
% cINF_mono
thf(fact_5872_le__cINF__iff,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A6: set @ B,F: B > A,U: A] :
( ( A6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit1013018076250108175_below @ A @ ( image @ B @ A @ F @ A6 ) )
=> ( ( ord_less_eq @ A @ U @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F @ A6 ) ) )
= ( ! [X2: B] :
( ( member @ B @ X2 @ A6 )
=> ( ord_less_eq @ A @ U @ ( F @ X2 ) ) ) ) ) ) ) ) ).
% le_cINF_iff
thf(fact_5873_cInf__superset__mono,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A6: set @ A,B7: set @ A] :
( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit1013018076250108175_below @ A @ B7 )
=> ( ( ord_less_eq @ ( set @ A ) @ A6 @ B7 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ B7 ) @ ( complete_Inf_Inf @ A @ A6 ) ) ) ) ) ) ).
% cInf_superset_mono
thf(fact_5874_cINF__less__iff,axiom,
! [A: $tType,B: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [A6: set @ B,F: B > A,A3: A] :
( ( A6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit1013018076250108175_below @ A @ ( image @ B @ A @ F @ A6 ) )
=> ( ( ord_less @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F @ A6 ) ) @ A3 )
= ( ? [X2: B] :
( ( member @ B @ X2 @ A6 )
& ( ord_less @ A @ ( F @ X2 ) @ A3 ) ) ) ) ) ) ) ).
% cINF_less_iff
thf(fact_5875_uminus__integer__def,axiom,
( ( uminus_uminus @ code_integer )
= ( map_fun @ code_integer @ int @ int @ code_integer @ code_int_of_integer @ code_integer_of_int @ ( uminus_uminus @ int ) ) ) ).
% uminus_integer_def
thf(fact_5876_abs__integer__def,axiom,
( ( abs_abs @ code_integer )
= ( map_fun @ code_integer @ int @ int @ code_integer @ code_int_of_integer @ code_integer_of_int @ ( abs_abs @ int ) ) ) ).
% abs_integer_def
thf(fact_5877_sgn__integer__def,axiom,
( ( sgn_sgn @ code_integer )
= ( map_fun @ code_integer @ int @ int @ code_integer @ code_int_of_integer @ code_integer_of_int @ ( sgn_sgn @ int ) ) ) ).
% sgn_integer_def
thf(fact_5878_cINF__superset__mono,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A6: set @ B,G: B > A,B7: set @ B,F: B > A] :
( ( A6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit1013018076250108175_below @ A @ ( image @ B @ A @ G @ B7 ) )
=> ( ( ord_less_eq @ ( set @ B ) @ A6 @ B7 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ B7 )
=> ( ord_less_eq @ A @ ( G @ X3 ) @ ( F @ X3 ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ G @ B7 ) ) @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F @ A6 ) ) ) ) ) ) ) ) ).
% cINF_superset_mono
thf(fact_5879_less__eq__cInf__inter,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A6: set @ A,B7: set @ A] :
( ( condit1013018076250108175_below @ A @ A6 )
=> ( ( condit1013018076250108175_below @ A @ B7 )
=> ( ( ( inf_inf @ ( set @ A ) @ A6 @ B7 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ A @ ( inf_inf @ A @ ( complete_Inf_Inf @ A @ A6 ) @ ( complete_Inf_Inf @ A @ B7 ) ) @ ( complete_Inf_Inf @ A @ ( inf_inf @ ( set @ A ) @ A6 @ B7 ) ) ) ) ) ) ) ).
% less_eq_cInf_inter
thf(fact_5880_cInf__le__cSup,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A6: set @ A] :
( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit941137186595557371_above @ A @ A6 )
=> ( ( condit1013018076250108175_below @ A @ A6 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A6 ) @ ( complete_Sup_Sup @ A @ A6 ) ) ) ) ) ) ).
% cInf_le_cSup
thf(fact_5881_not__integer__def,axiom,
( ( bit_ri4277139882892585799ns_not @ code_integer )
= ( map_fun @ code_integer @ int @ int @ code_integer @ code_int_of_integer @ code_integer_of_int @ ( bit_ri4277139882892585799ns_not @ int ) ) ) ).
% not_integer_def
thf(fact_5882_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_5883_flip__bit__integer__def,axiom,
( ( bit_se8732182000553998342ip_bit @ code_integer )
= ( map_fun @ nat @ nat @ ( int > int ) @ ( code_integer > code_integer ) @ ( id @ nat ) @ ( map_fun @ code_integer @ int @ int @ code_integer @ code_int_of_integer @ code_integer_of_int ) @ ( bit_se8732182000553998342ip_bit @ int ) ) ) ).
% flip_bit_integer_def
thf(fact_5884_set__bit__integer__def,axiom,
( ( bit_se5668285175392031749et_bit @ code_integer )
= ( map_fun @ nat @ nat @ ( int > int ) @ ( code_integer > code_integer ) @ ( id @ nat ) @ ( map_fun @ code_integer @ int @ int @ code_integer @ code_int_of_integer @ code_integer_of_int ) @ ( bit_se5668285175392031749et_bit @ int ) ) ) ).
% set_bit_integer_def
thf(fact_5885_Code__Numeral_Odup__code_I1_J,axiom,
( ( code_dup @ ( zero_zero @ code_integer ) )
= ( zero_zero @ code_integer ) ) ).
% Code_Numeral.dup_code(1)
thf(fact_5886_dup_Orep__eq,axiom,
! [X: code_integer] :
( ( code_int_of_integer @ ( code_dup @ X ) )
= ( plus_plus @ int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ X ) ) ) ).
% dup.rep_eq
thf(fact_5887_dup_Oabs__eq,axiom,
! [X: int] :
( ( code_dup @ ( code_integer_of_int @ X ) )
= ( code_integer_of_int @ ( plus_plus @ int @ X @ X ) ) ) ).
% dup.abs_eq
thf(fact_5888_take__bit__integer__def,axiom,
( ( bit_se2584673776208193580ke_bit @ code_integer )
= ( map_fun @ nat @ nat @ ( int > int ) @ ( code_integer > code_integer ) @ ( id @ nat ) @ ( map_fun @ code_integer @ int @ int @ code_integer @ code_int_of_integer @ code_integer_of_int ) @ ( bit_se2584673776208193580ke_bit @ int ) ) ) ).
% take_bit_integer_def
thf(fact_5889_drop__bit__integer__def,axiom,
( ( bit_se4197421643247451524op_bit @ code_integer )
= ( map_fun @ nat @ nat @ ( int > int ) @ ( code_integer > code_integer ) @ ( id @ nat ) @ ( map_fun @ code_integer @ int @ int @ code_integer @ code_int_of_integer @ code_integer_of_int ) @ ( bit_se4197421643247451524op_bit @ int ) ) ) ).
% drop_bit_integer_def
thf(fact_5890_push__bit__integer__def,axiom,
( ( bit_se4730199178511100633sh_bit @ code_integer )
= ( map_fun @ nat @ nat @ ( int > int ) @ ( code_integer > code_integer ) @ ( id @ nat ) @ ( map_fun @ code_integer @ int @ int @ code_integer @ code_int_of_integer @ code_integer_of_int ) @ ( bit_se4730199178511100633sh_bit @ int ) ) ) ).
% push_bit_integer_def
thf(fact_5891_unset__bit__integer__def,axiom,
( ( bit_se2638667681897837118et_bit @ code_integer )
= ( map_fun @ nat @ nat @ ( int > int ) @ ( code_integer > code_integer ) @ ( id @ nat ) @ ( map_fun @ code_integer @ int @ int @ code_integer @ code_int_of_integer @ code_integer_of_int ) @ ( bit_se2638667681897837118et_bit @ int ) ) ) ).
% unset_bit_integer_def
thf(fact_5892_Code__Numeral_Osub__code_I9_J,axiom,
! [M: num,N: num] :
( ( code_sub @ ( bit0 @ M ) @ ( bit1 @ N ) )
= ( minus_minus @ code_integer @ ( code_dup @ ( code_sub @ M @ N ) ) @ ( one_one @ code_integer ) ) ) ).
% Code_Numeral.sub_code(9)
thf(fact_5893_Code__Numeral_Osub__code_I8_J,axiom,
! [M: num,N: num] :
( ( code_sub @ ( bit1 @ M ) @ ( bit0 @ N ) )
= ( plus_plus @ code_integer @ ( code_dup @ ( code_sub @ M @ N ) ) @ ( one_one @ code_integer ) ) ) ).
% Code_Numeral.sub_code(8)
thf(fact_5894_xor__integer__def,axiom,
( ( bit_se5824344971392196577ns_xor @ code_integer )
= ( map_fun @ code_integer @ int @ ( int > int ) @ ( code_integer > code_integer ) @ code_int_of_integer @ ( map_fun @ code_integer @ int @ int @ code_integer @ code_int_of_integer @ code_integer_of_int ) @ ( bit_se5824344971392196577ns_xor @ int ) ) ) ).
% xor_integer_def
thf(fact_5895_Code__Numeral_Osub__code_I1_J,axiom,
( ( code_sub @ one2 @ one2 )
= ( zero_zero @ code_integer ) ) ).
% Code_Numeral.sub_code(1)
thf(fact_5896_Code__Numeral_Osub__code_I6_J,axiom,
! [M: num,N: num] :
( ( code_sub @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( code_dup @ ( code_sub @ M @ N ) ) ) ).
% Code_Numeral.sub_code(6)
thf(fact_5897_Code__Numeral_Osub__code_I7_J,axiom,
! [M: num,N: num] :
( ( code_sub @ ( bit1 @ M ) @ ( bit1 @ N ) )
= ( code_dup @ ( code_sub @ M @ N ) ) ) ).
% Code_Numeral.sub_code(7)
thf(fact_5898_sub_Orep__eq,axiom,
! [X: num,Xa3: num] :
( ( code_int_of_integer @ ( code_sub @ X @ Xa3 ) )
= ( minus_minus @ int @ ( numeral_numeral @ int @ X ) @ ( numeral_numeral @ int @ Xa3 ) ) ) ).
% sub.rep_eq
thf(fact_5899_sub_Oabs__eq,axiom,
( code_sub
= ( ^ [Xa4: num,X2: num] : ( code_integer_of_int @ ( minus_minus @ int @ ( numeral_numeral @ int @ Xa4 ) @ ( numeral_numeral @ int @ X2 ) ) ) ) ) ).
% sub.abs_eq
thf(fact_5900_plus__integer__def,axiom,
( ( plus_plus @ code_integer )
= ( map_fun @ code_integer @ int @ ( int > int ) @ ( code_integer > code_integer ) @ code_int_of_integer @ ( map_fun @ code_integer @ int @ int @ code_integer @ code_int_of_integer @ code_integer_of_int ) @ ( plus_plus @ int ) ) ) ).
% plus_integer_def
thf(fact_5901_times__integer__def,axiom,
( ( times_times @ code_integer )
= ( map_fun @ code_integer @ int @ ( int > int ) @ ( code_integer > code_integer ) @ code_int_of_integer @ ( map_fun @ code_integer @ int @ int @ code_integer @ code_int_of_integer @ code_integer_of_int ) @ ( times_times @ int ) ) ) ).
% times_integer_def
thf(fact_5902_minus__integer__def,axiom,
( ( minus_minus @ code_integer )
= ( map_fun @ code_integer @ int @ ( int > int ) @ ( code_integer > code_integer ) @ code_int_of_integer @ ( map_fun @ code_integer @ int @ int @ code_integer @ code_int_of_integer @ code_integer_of_int ) @ ( minus_minus @ int ) ) ) ).
% minus_integer_def
thf(fact_5903_divide__integer__def,axiom,
( ( divide_divide @ code_integer )
= ( map_fun @ code_integer @ int @ ( int > int ) @ ( code_integer > code_integer ) @ code_int_of_integer @ ( map_fun @ code_integer @ int @ int @ code_integer @ code_int_of_integer @ code_integer_of_int ) @ ( divide_divide @ int ) ) ) ).
% divide_integer_def
thf(fact_5904_modulo__integer__def,axiom,
( ( modulo_modulo @ code_integer )
= ( map_fun @ code_integer @ int @ ( int > int ) @ ( code_integer > code_integer ) @ code_int_of_integer @ ( map_fun @ code_integer @ int @ int @ code_integer @ code_int_of_integer @ code_integer_of_int ) @ ( modulo_modulo @ int ) ) ) ).
% modulo_integer_def
thf(fact_5905_and__integer__def,axiom,
( ( bit_se5824344872417868541ns_and @ code_integer )
= ( map_fun @ code_integer @ int @ ( int > int ) @ ( code_integer > code_integer ) @ code_int_of_integer @ ( map_fun @ code_integer @ int @ int @ code_integer @ code_int_of_integer @ code_integer_of_int ) @ ( bit_se5824344872417868541ns_and @ int ) ) ) ).
% and_integer_def
thf(fact_5906_or__integer__def,axiom,
( ( bit_se1065995026697491101ons_or @ code_integer )
= ( map_fun @ code_integer @ int @ ( int > int ) @ ( code_integer > code_integer ) @ code_int_of_integer @ ( map_fun @ code_integer @ int @ int @ code_integer @ code_int_of_integer @ code_integer_of_int ) @ ( bit_se1065995026697491101ons_or @ int ) ) ) ).
% or_integer_def
thf(fact_5907_Sup__int__def,axiom,
( ( complete_Sup_Sup @ int )
= ( ^ [X7: set @ int] :
( the @ int
@ ^ [X2: int] :
( ( member @ int @ X2 @ X7 )
& ! [Y6: int] :
( ( member @ int @ Y6 @ X7 )
=> ( ord_less_eq @ int @ Y6 @ X2 ) ) ) ) ) ) ).
% Sup_int_def
thf(fact_5908_lexordp__conv__lexord,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_lexordp @ A )
= ( ^ [Xs3: list @ A,Ys3: list @ A] : ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs3 @ Ys3 ) @ ( lexord @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ ( ord_less @ A ) ) ) ) ) ) ) ) ).
% lexordp_conv_lexord
thf(fact_5909_sorted__wrt__iff__nth__Suc__transp,axiom,
! [A: $tType,P: A > A > $o,Xs: list @ A] :
( ( transp @ A @ P )
=> ( ( sorted_wrt @ A @ P @ Xs )
= ( ! [I5: nat] :
( ( ord_less @ nat @ ( suc @ I5 ) @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( P @ ( nth @ A @ Xs @ I5 ) @ ( nth @ A @ Xs @ ( suc @ I5 ) ) ) ) ) ) ) ).
% sorted_wrt_iff_nth_Suc_transp
thf(fact_5910_lexordp__simps_I3_J,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X: A,Xs: list @ A,Y3: A,Ys2: list @ A] :
( ( ord_lexordp @ A @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y3 @ Ys2 ) )
= ( ( ord_less @ A @ X @ Y3 )
| ( ~ ( ord_less @ A @ Y3 @ X )
& ( ord_lexordp @ A @ Xs @ Ys2 ) ) ) ) ) ).
% lexordp_simps(3)
thf(fact_5911_lexordp__irreflexive,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [Xs: list @ A] :
( ! [X3: A] :
~ ( ord_less @ A @ X3 @ X3 )
=> ~ ( ord_lexordp @ A @ Xs @ Xs ) ) ) ).
% lexordp_irreflexive
thf(fact_5912_lexordp_OCons,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X: A,Y3: A,Xs: list @ A,Ys2: list @ A] :
( ( ord_less @ A @ X @ Y3 )
=> ( ord_lexordp @ A @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y3 @ Ys2 ) ) ) ) ).
% lexordp.Cons
thf(fact_5913_lexordp_OCons__eq,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X: A,Y3: A,Xs: list @ A,Ys2: list @ A] :
( ~ ( ord_less @ A @ X @ Y3 )
=> ( ~ ( ord_less @ A @ Y3 @ X )
=> ( ( ord_lexordp @ A @ Xs @ Ys2 )
=> ( ord_lexordp @ A @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y3 @ Ys2 ) ) ) ) ) ) ).
% lexordp.Cons_eq
thf(fact_5914_lexordp__append__leftD,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [Xs: list @ A,Us: list @ A,Vs: list @ A] :
( ( ord_lexordp @ A @ ( append @ A @ Xs @ Us ) @ ( append @ A @ Xs @ Vs ) )
=> ( ! [A8: A] :
~ ( ord_less @ A @ A8 @ A8 )
=> ( ord_lexordp @ A @ Us @ Vs ) ) ) ) ).
% lexordp_append_leftD
thf(fact_5915_Sup__eq__Inf,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ( ( complete_Sup_Sup @ A )
= ( ^ [A7: set @ A] :
( complete_Inf_Inf @ A
@ ( collect @ A
@ ^ [B3: A] :
! [X2: A] :
( ( member @ A @ X2 @ A7 )
=> ( ord_less_eq @ A @ X2 @ B3 ) ) ) ) ) ) ) ).
% Sup_eq_Inf
thf(fact_5916_Inf__eq__Sup,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ( ( complete_Inf_Inf @ A )
= ( ^ [A7: set @ A] :
( complete_Sup_Sup @ A
@ ( collect @ A
@ ^ [B3: A] :
! [X2: A] :
( ( member @ A @ X2 @ A7 )
=> ( ord_less_eq @ A @ B3 @ X2 ) ) ) ) ) ) ) ).
% Inf_eq_Sup
thf(fact_5917_lexordp__induct,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,Ys2: list @ A,P: ( list @ A ) > ( list @ A ) > $o] :
( ( ord_lexordp @ A @ Xs @ Ys2 )
=> ( ! [Y5: A,Ys4: list @ A] : ( P @ ( nil @ A ) @ ( cons @ A @ Y5 @ Ys4 ) )
=> ( ! [X3: A,Xs2: list @ A,Y5: A,Ys4: list @ A] :
( ( ord_less @ A @ X3 @ Y5 )
=> ( P @ ( cons @ A @ X3 @ Xs2 ) @ ( cons @ A @ Y5 @ Ys4 ) ) )
=> ( ! [X3: A,Xs2: list @ A,Ys4: list @ A] :
( ( ord_lexordp @ A @ Xs2 @ Ys4 )
=> ( ( P @ Xs2 @ Ys4 )
=> ( P @ ( cons @ A @ X3 @ Xs2 ) @ ( cons @ A @ X3 @ Ys4 ) ) ) )
=> ( P @ Xs @ Ys2 ) ) ) ) ) ) ).
% lexordp_induct
thf(fact_5918_lexordp__cases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,Ys2: list @ A] :
( ( ord_lexordp @ A @ Xs @ Ys2 )
=> ( ( ( Xs
= ( nil @ A ) )
=> ! [Y5: A,Ys5: list @ A] :
( Ys2
!= ( cons @ A @ Y5 @ Ys5 ) ) )
=> ( ! [X3: A] :
( ? [Xs4: list @ A] :
( Xs
= ( cons @ A @ X3 @ Xs4 ) )
=> ! [Y5: A] :
( ? [Ys5: list @ A] :
( Ys2
= ( cons @ A @ Y5 @ Ys5 ) )
=> ~ ( ord_less @ A @ X3 @ Y5 ) ) )
=> ~ ! [X3: A,Xs4: list @ A] :
( ( Xs
= ( cons @ A @ X3 @ Xs4 ) )
=> ! [Ys5: list @ A] :
( ( Ys2
= ( cons @ A @ X3 @ Ys5 ) )
=> ~ ( ord_lexordp @ A @ Xs4 @ Ys5 ) ) ) ) ) ) ) ).
% lexordp_cases
thf(fact_5919_lexordp_Osimps,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_lexordp @ A )
= ( ^ [A12: list @ A,A23: list @ A] :
( ? [Y6: A,Ys3: list @ A] :
( ( A12
= ( nil @ A ) )
& ( A23
= ( cons @ A @ Y6 @ Ys3 ) ) )
| ? [X2: A,Y6: A,Xs3: list @ A,Ys3: list @ A] :
( ( A12
= ( cons @ A @ X2 @ Xs3 ) )
& ( A23
= ( cons @ A @ Y6 @ Ys3 ) )
& ( ord_less @ A @ X2 @ Y6 ) )
| ? [X2: A,Y6: A,Xs3: list @ A,Ys3: list @ A] :
( ( A12
= ( cons @ A @ X2 @ Xs3 ) )
& ( A23
= ( cons @ A @ Y6 @ Ys3 ) )
& ~ ( ord_less @ A @ X2 @ Y6 )
& ~ ( ord_less @ A @ Y6 @ X2 )
& ( ord_lexordp @ A @ Xs3 @ Ys3 ) ) ) ) ) ) ).
% lexordp.simps
thf(fact_5920_lexordp_Ocases,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A1: list @ A,A22: list @ A] :
( ( ord_lexordp @ A @ A1 @ A22 )
=> ( ( ( A1
= ( nil @ A ) )
=> ! [Y5: A,Ys4: list @ A] :
( A22
!= ( cons @ A @ Y5 @ Ys4 ) ) )
=> ( ! [X3: A] :
( ? [Xs2: list @ A] :
( A1
= ( cons @ A @ X3 @ Xs2 ) )
=> ! [Y5: A] :
( ? [Ys4: list @ A] :
( A22
= ( cons @ A @ Y5 @ Ys4 ) )
=> ~ ( ord_less @ A @ X3 @ Y5 ) ) )
=> ~ ! [X3: A,Y5: A,Xs2: list @ A] :
( ( A1
= ( cons @ A @ X3 @ Xs2 ) )
=> ! [Ys4: list @ A] :
( ( A22
= ( cons @ A @ Y5 @ Ys4 ) )
=> ( ~ ( ord_less @ A @ X3 @ Y5 )
=> ( ~ ( ord_less @ A @ Y5 @ X3 )
=> ~ ( ord_lexordp @ A @ Xs2 @ Ys4 ) ) ) ) ) ) ) ) ) ).
% lexordp.cases
thf(fact_5921_lexordp__append__left__rightI,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X: A,Y3: A,Us: list @ A,Xs: list @ A,Ys2: list @ A] :
( ( ord_less @ A @ X @ Y3 )
=> ( ord_lexordp @ A @ ( append @ A @ Us @ ( cons @ A @ X @ Xs ) ) @ ( append @ A @ Us @ ( cons @ A @ Y3 @ Ys2 ) ) ) ) ) ).
% lexordp_append_left_rightI
thf(fact_5922_lexordp__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_lexordp @ A )
= ( ^ [Xs3: list @ A,Ys3: list @ A] :
( ? [X2: A,Vs2: list @ A] :
( Ys3
= ( append @ A @ Xs3 @ ( cons @ A @ X2 @ Vs2 ) ) )
| ? [Us2: list @ A,A5: A,B3: A,Vs2: list @ A,Ws: list @ A] :
( ( ord_less @ A @ A5 @ B3 )
& ( Xs3
= ( append @ A @ Us2 @ ( cons @ A @ A5 @ Vs2 ) ) )
& ( Ys3
= ( append @ A @ Us2 @ ( cons @ A @ B3 @ Ws ) ) ) ) ) ) ) ) ).
% lexordp_iff
thf(fact_5923_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] :
! [Y6: A] :
( ( member @ A @ Y6 @ S3 )
=> ( ord_less_eq @ A @ X2 @ Y6 ) ) ) ) ) ) ) ) ).
% cInf_cSup
thf(fact_5924_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] :
! [Y6: A] :
( ( member @ A @ Y6 @ S3 )
=> ( ord_less_eq @ A @ Y6 @ X2 ) ) ) ) ) ) ) ) ).
% cSup_cInf
thf(fact_5925_transp__gr,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( transp @ A
@ ^ [X2: A,Y6: A] : ( ord_less @ A @ Y6 @ X2 ) ) ) ).
% transp_gr
thf(fact_5926_transp__ge,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( transp @ A
@ ^ [X2: A,Y6: A] : ( ord_less_eq @ A @ Y6 @ X2 ) ) ) ).
% transp_ge
thf(fact_5927_ord__class_Olexordp__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_lexordp @ A )
= ( complete_lattice_lfp @ ( ( list @ A ) > ( list @ A ) > $o )
@ ^ [P6: ( list @ A ) > ( list @ A ) > $o,X12: list @ A,X24: list @ A] :
( ? [Y6: A,Ys3: list @ A] :
( ( X12
= ( nil @ A ) )
& ( X24
= ( cons @ A @ Y6 @ Ys3 ) ) )
| ? [X2: A,Y6: A,Xs3: list @ A,Ys3: list @ A] :
( ( X12
= ( cons @ A @ X2 @ Xs3 ) )
& ( X24
= ( cons @ A @ Y6 @ Ys3 ) )
& ( ord_less @ A @ X2 @ Y6 ) )
| ? [X2: A,Y6: A,Xs3: list @ A,Ys3: list @ A] :
( ( X12
= ( cons @ A @ X2 @ Xs3 ) )
& ( X24
= ( cons @ A @ Y6 @ Ys3 ) )
& ~ ( ord_less @ A @ X2 @ Y6 )
& ~ ( ord_less @ A @ Y6 @ X2 )
& ( P6 @ Xs3 @ Ys3 ) ) ) ) ) ) ).
% ord_class.lexordp_def
thf(fact_5928_Sup__Inf,axiom,
! [A: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [A6: set @ ( set @ A )] :
( ( complete_Sup_Sup @ A @ ( image @ ( set @ A ) @ A @ ( complete_Inf_Inf @ A ) @ A6 ) )
= ( complete_Inf_Inf @ A
@ ( image @ ( set @ A ) @ A @ ( complete_Sup_Sup @ A )
@ ( collect @ ( set @ A )
@ ^ [Uu: set @ A] :
? [F2: ( set @ A ) > A] :
( ( Uu
= ( image @ ( set @ A ) @ A @ F2 @ A6 ) )
& ! [X2: set @ A] :
( ( member @ ( set @ A ) @ X2 @ A6 )
=> ( member @ A @ ( F2 @ X2 ) @ X2 ) ) ) ) ) ) ) ) ).
% Sup_Inf
thf(fact_5929_Inf__Sup__le,axiom,
! [A: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [A6: set @ ( set @ A )] :
( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image @ ( set @ A ) @ A @ ( complete_Sup_Sup @ A ) @ A6 ) )
@ ( complete_Sup_Sup @ A
@ ( image @ ( set @ A ) @ A @ ( complete_Inf_Inf @ A )
@ ( collect @ ( set @ A )
@ ^ [Uu: set @ A] :
? [F2: ( set @ A ) > A] :
( ( Uu
= ( image @ ( set @ A ) @ A @ F2 @ A6 ) )
& ! [X2: set @ A] :
( ( member @ ( set @ A ) @ X2 @ A6 )
=> ( member @ A @ ( F2 @ X2 ) @ X2 ) ) ) ) ) ) ) ) ).
% Inf_Sup_le
thf(fact_5930_Sup__Inf__le,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A6: set @ ( set @ A )] :
( ord_less_eq @ A
@ ( complete_Sup_Sup @ A
@ ( image @ ( set @ A ) @ A @ ( complete_Inf_Inf @ A )
@ ( collect @ ( set @ A )
@ ^ [Uu: set @ A] :
? [F2: ( set @ A ) > A] :
( ( Uu
= ( image @ ( set @ A ) @ A @ F2 @ A6 ) )
& ! [X2: set @ A] :
( ( member @ ( set @ A ) @ X2 @ A6 )
=> ( member @ A @ ( F2 @ X2 ) @ X2 ) ) ) ) ) )
@ ( complete_Inf_Inf @ A @ ( image @ ( set @ A ) @ A @ ( complete_Sup_Sup @ A ) @ A6 ) ) ) ) ).
% Sup_Inf_le
thf(fact_5931_finite__Inf__Sup,axiom,
! [A: $tType] :
( ( finite8700451911770168679attice @ A )
=> ! [A6: set @ ( set @ A )] :
( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image @ ( set @ A ) @ A @ ( complete_Sup_Sup @ A ) @ A6 ) )
@ ( complete_Sup_Sup @ A
@ ( image @ ( set @ A ) @ A @ ( complete_Inf_Inf @ A )
@ ( collect @ ( set @ A )
@ ^ [Uu: set @ A] :
? [F2: ( set @ A ) > A] :
( ( Uu
= ( image @ ( set @ A ) @ A @ F2 @ A6 ) )
& ! [X2: set @ A] :
( ( member @ ( set @ A ) @ X2 @ A6 )
=> ( member @ A @ ( F2 @ X2 ) @ X2 ) ) ) ) ) ) ) ) ).
% finite_Inf_Sup
thf(fact_5932_SUP__INF__set,axiom,
! [A: $tType,B: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [G: B > A,A6: set @ ( set @ B )] :
( ( complete_Sup_Sup @ A
@ ( image @ ( set @ B ) @ A
@ ^ [X2: set @ B] : ( complete_Inf_Inf @ A @ ( image @ B @ A @ G @ X2 ) )
@ A6 ) )
= ( complete_Inf_Inf @ A
@ ( image @ ( set @ B ) @ A
@ ^ [X2: set @ B] : ( complete_Sup_Sup @ A @ ( image @ B @ A @ G @ X2 ) )
@ ( collect @ ( set @ B )
@ ^ [Uu: set @ B] :
? [F2: ( set @ B ) > B] :
( ( Uu
= ( image @ ( set @ B ) @ B @ F2 @ A6 ) )
& ! [X2: set @ B] :
( ( member @ ( set @ B ) @ X2 @ A6 )
=> ( member @ B @ ( F2 @ X2 ) @ X2 ) ) ) ) ) ) ) ) ).
% SUP_INF_set
thf(fact_5933_INF__SUP__set,axiom,
! [A: $tType,B: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [G: B > A,A6: set @ ( set @ B )] :
( ( complete_Inf_Inf @ A
@ ( image @ ( set @ B ) @ A
@ ^ [B6: set @ B] : ( complete_Sup_Sup @ A @ ( image @ B @ A @ G @ B6 ) )
@ A6 ) )
= ( complete_Sup_Sup @ A
@ ( image @ ( set @ B ) @ A
@ ^ [B6: set @ B] : ( complete_Inf_Inf @ A @ ( image @ B @ A @ G @ B6 ) )
@ ( collect @ ( set @ B )
@ ^ [Uu: set @ B] :
? [F2: ( set @ B ) > B] :
( ( Uu
= ( image @ ( set @ B ) @ B @ F2 @ A6 ) )
& ! [X2: set @ B] :
( ( member @ ( set @ B ) @ X2 @ A6 )
=> ( member @ B @ ( F2 @ X2 ) @ X2 ) ) ) ) ) ) ) ) ).
% INF_SUP_set
thf(fact_5934_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_5935_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_5936_transp__le,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( transp @ A @ ( ord_less_eq @ A ) ) ) ).
% transp_le
thf(fact_5937_transp__less,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( transp @ A @ ( ord_less @ A ) ) ) ).
% transp_less
thf(fact_5938_lfp__ordinal__induct,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F: A > A,P: A > $o] :
( ( order_mono @ A @ A @ F )
=> ( ! [S4: A] :
( ( P @ S4 )
=> ( ( ord_less_eq @ A @ S4 @ ( complete_lattice_lfp @ A @ F ) )
=> ( P @ ( F @ S4 ) ) ) )
=> ( ! [M8: set @ A] :
( ! [X4: A] :
( ( member @ A @ X4 @ M8 )
=> ( P @ X4 ) )
=> ( P @ ( complete_Sup_Sup @ A @ M8 ) ) )
=> ( P @ ( complete_lattice_lfp @ A @ F ) ) ) ) ) ) ).
% lfp_ordinal_induct
thf(fact_5939_def__lfp__induct,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A6: A,F: A > A,P: A] :
( ( A6
= ( complete_lattice_lfp @ A @ F ) )
=> ( ( order_mono @ A @ A @ F )
=> ( ( ord_less_eq @ A @ ( F @ ( inf_inf @ A @ A6 @ P ) ) @ P )
=> ( ord_less_eq @ A @ A6 @ P ) ) ) ) ) ).
% def_lfp_induct
thf(fact_5940_lfp__induct,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F: A > A,P: A] :
( ( order_mono @ A @ A @ F )
=> ( ( ord_less_eq @ A @ ( F @ ( inf_inf @ A @ ( complete_lattice_lfp @ A @ F ) @ P ) ) @ P )
=> ( ord_less_eq @ A @ ( complete_lattice_lfp @ A @ F ) @ P ) ) ) ) ).
% lfp_induct
thf(fact_5941_le__rel__bool__arg__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_less_eq @ ( $o > A ) )
= ( ^ [X7: $o > A,Y9: $o > A] :
( ( ord_less_eq @ A @ ( X7 @ $false ) @ ( Y9 @ $false ) )
& ( ord_less_eq @ A @ ( X7 @ $true ) @ ( Y9 @ $true ) ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_5942_lfp__mono,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F: A > A,G: A > A] :
( ! [Z9: A] : ( ord_less_eq @ A @ ( F @ Z9 ) @ ( G @ Z9 ) )
=> ( ord_less_eq @ A @ ( complete_lattice_lfp @ A @ F ) @ ( complete_lattice_lfp @ A @ G ) ) ) ) ).
% lfp_mono
thf(fact_5943_lfp__lowerbound,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F: A > A,A6: A] :
( ( ord_less_eq @ A @ ( F @ A6 ) @ A6 )
=> ( ord_less_eq @ A @ ( complete_lattice_lfp @ A @ F ) @ A6 ) ) ) ).
% lfp_lowerbound
thf(fact_5944_lfp__greatest,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F: A > A,A6: A] :
( ! [U2: A] :
( ( ord_less_eq @ A @ ( F @ U2 ) @ U2 )
=> ( ord_less_eq @ A @ A6 @ U2 ) )
=> ( ord_less_eq @ A @ A6 @ ( complete_lattice_lfp @ A @ F ) ) ) ) ).
% lfp_greatest
thf(fact_5945_lfp__lfp,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F: A > A > A] :
( ! [X3: A,Y5: A,W: A,Z4: A] :
( ( ord_less_eq @ A @ X3 @ Y5 )
=> ( ( ord_less_eq @ A @ W @ Z4 )
=> ( ord_less_eq @ A @ ( F @ X3 @ W ) @ ( F @ Y5 @ Z4 ) ) ) )
=> ( ( 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_5946_lfp__eqI,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F4: A > A,X: A] :
( ( order_mono @ A @ A @ F4 )
=> ( ( ( F4 @ X )
= X )
=> ( ! [Z4: A] :
( ( ( F4 @ Z4 )
= Z4 )
=> ( ord_less_eq @ A @ X @ Z4 ) )
=> ( ( complete_lattice_lfp @ A @ F4 )
= X ) ) ) ) ) ).
% lfp_eqI
thf(fact_5947_lfp__def,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ( ( complete_lattice_lfp @ A )
= ( ^ [F2: A > A] :
( complete_Inf_Inf @ A
@ ( collect @ A
@ ^ [U3: A] : ( ord_less_eq @ A @ ( F2 @ U3 ) @ U3 ) ) ) ) ) ) ).
% lfp_def
thf(fact_5948_char__of__take__bit__eq,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N: nat,M: A] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) @ N )
=> ( ( unique5772411509450598832har_of @ A @ ( bit_se2584673776208193580ke_bit @ A @ N @ M ) )
= ( unique5772411509450598832har_of @ A @ M ) ) ) ) ).
% char_of_take_bit_eq
thf(fact_5949_num__of__integer_Otransfer,axiom,
( bNF_rel_fun @ int @ code_integer @ num @ num @ code_pcr_integer
@ ^ [Y: num,Z2: num] : Y = Z2
@ ( comp @ nat @ num @ int @ num_of_nat @ nat2 )
@ code_num_of_integer ) ).
% num_of_integer.transfer
thf(fact_5950_Max_Osemilattice__order__set__axioms,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( lattic4895041142388067077er_set @ A @ ( ord_max @ A )
@ ^ [X2: A,Y6: A] : ( ord_less_eq @ A @ Y6 @ X2 )
@ ^ [X2: A,Y6: A] : ( ord_less @ A @ Y6 @ X2 ) ) ) ).
% Max.semilattice_order_set_axioms
thf(fact_5951_char__of__nat,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N: nat] :
( ( unique5772411509450598832har_of @ A @ ( semiring_1_of_nat @ A @ N ) )
= ( unique5772411509450598832har_of @ nat @ N ) ) ) ).
% char_of_nat
thf(fact_5952_integer_Oid__abs__transfer,axiom,
( bNF_rel_fun @ int @ int @ int @ code_integer
@ ^ [Y: int,Z2: int] : Y = Z2
@ code_pcr_integer
@ ^ [X2: int] : X2
@ code_integer_of_int ) ).
% integer.id_abs_transfer
thf(fact_5953_integer_Orep__transfer,axiom,
( bNF_rel_fun @ int @ code_integer @ int @ int @ code_pcr_integer
@ ^ [Y: int,Z2: int] : Y = Z2
@ ^ [X2: int] : X2
@ code_int_of_integer ) ).
% integer.rep_transfer
thf(fact_5954_zero__integer_Otransfer,axiom,
code_pcr_integer @ ( zero_zero @ int ) @ ( zero_zero @ code_integer ) ).
% zero_integer.transfer
thf(fact_5955_less__integer_Otransfer,axiom,
( bNF_rel_fun @ int @ code_integer @ ( int > $o ) @ ( code_integer > $o ) @ code_pcr_integer
@ ( bNF_rel_fun @ int @ code_integer @ $o @ $o @ code_pcr_integer
@ ^ [Y: $o,Z2: $o] : Y = Z2 )
@ ( ord_less @ int )
@ ( ord_less @ code_integer ) ) ).
% less_integer.transfer
thf(fact_5956_plus__integer_Otransfer,axiom,
bNF_rel_fun @ int @ code_integer @ ( int > int ) @ ( code_integer > code_integer ) @ code_pcr_integer @ ( bNF_rel_fun @ int @ code_integer @ int @ code_integer @ code_pcr_integer @ code_pcr_integer ) @ ( plus_plus @ int ) @ ( plus_plus @ code_integer ) ).
% plus_integer.transfer
thf(fact_5957_uminus__integer_Otransfer,axiom,
bNF_rel_fun @ int @ code_integer @ int @ code_integer @ code_pcr_integer @ code_pcr_integer @ ( uminus_uminus @ int ) @ ( uminus_uminus @ code_integer ) ).
% uminus_integer.transfer
thf(fact_5958_times__integer_Otransfer,axiom,
bNF_rel_fun @ int @ code_integer @ ( int > int ) @ ( code_integer > code_integer ) @ code_pcr_integer @ ( bNF_rel_fun @ int @ code_integer @ int @ code_integer @ code_pcr_integer @ code_pcr_integer ) @ ( times_times @ int ) @ ( times_times @ code_integer ) ).
% times_integer.transfer
thf(fact_5959_one__integer_Otransfer,axiom,
code_pcr_integer @ ( one_one @ int ) @ ( one_one @ code_integer ) ).
% one_integer.transfer
thf(fact_5960_less__eq__integer_Otransfer,axiom,
( bNF_rel_fun @ int @ code_integer @ ( int > $o ) @ ( code_integer > $o ) @ code_pcr_integer
@ ( bNF_rel_fun @ int @ code_integer @ $o @ $o @ code_pcr_integer
@ ^ [Y: $o,Z2: $o] : Y = Z2 )
@ ( ord_less_eq @ int )
@ ( ord_less_eq @ code_integer ) ) ).
% less_eq_integer.transfer
thf(fact_5961_minus__integer_Otransfer,axiom,
bNF_rel_fun @ int @ code_integer @ ( int > int ) @ ( code_integer > code_integer ) @ code_pcr_integer @ ( bNF_rel_fun @ int @ code_integer @ int @ code_integer @ code_pcr_integer @ code_pcr_integer ) @ ( minus_minus @ int ) @ ( minus_minus @ code_integer ) ).
% minus_integer.transfer
thf(fact_5962_abs__integer_Otransfer,axiom,
bNF_rel_fun @ int @ code_integer @ int @ code_integer @ code_pcr_integer @ code_pcr_integer @ ( abs_abs @ int ) @ ( abs_abs @ code_integer ) ).
% abs_integer.transfer
thf(fact_5963_integer__of__nat_Otransfer,axiom,
( bNF_rel_fun @ nat @ nat @ int @ code_integer
@ ^ [Y: nat,Z2: nat] : Y = Z2
@ code_pcr_integer
@ ( semiring_1_of_nat @ int )
@ code_integer_of_nat ) ).
% integer_of_nat.transfer
thf(fact_5964_divide__integer_Otransfer,axiom,
bNF_rel_fun @ int @ code_integer @ ( int > int ) @ ( code_integer > code_integer ) @ code_pcr_integer @ ( bNF_rel_fun @ int @ code_integer @ int @ code_integer @ code_pcr_integer @ code_pcr_integer ) @ ( divide_divide @ int ) @ ( divide_divide @ code_integer ) ).
% divide_integer.transfer
thf(fact_5965_nat__of__integer_Otransfer,axiom,
( bNF_rel_fun @ int @ code_integer @ nat @ nat @ code_pcr_integer
@ ^ [Y: nat,Z2: nat] : Y = Z2
@ nat2
@ code_nat_of_integer ) ).
% nat_of_integer.transfer
thf(fact_5966_modulo__integer_Otransfer,axiom,
bNF_rel_fun @ int @ code_integer @ ( int > int ) @ ( code_integer > code_integer ) @ code_pcr_integer @ ( bNF_rel_fun @ int @ code_integer @ int @ code_integer @ code_pcr_integer @ code_pcr_integer ) @ ( modulo_modulo @ int ) @ ( modulo_modulo @ code_integer ) ).
% modulo_integer.transfer
thf(fact_5967_sgn__integer_Otransfer,axiom,
bNF_rel_fun @ int @ code_integer @ int @ code_integer @ code_pcr_integer @ code_pcr_integer @ ( sgn_sgn @ int ) @ ( sgn_sgn @ code_integer ) ).
% sgn_integer.transfer
thf(fact_5968_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_5969_take__bit__integer_Otransfer,axiom,
( bNF_rel_fun @ nat @ nat @ ( int > int ) @ ( code_integer > code_integer )
@ ^ [Y: nat,Z2: nat] : Y = Z2
@ ( bNF_rel_fun @ int @ code_integer @ int @ code_integer @ code_pcr_integer @ code_pcr_integer )
@ ( bit_se2584673776208193580ke_bit @ int )
@ ( bit_se2584673776208193580ke_bit @ code_integer ) ) ).
% take_bit_integer.transfer
thf(fact_5970_not__integer_Otransfer,axiom,
bNF_rel_fun @ int @ code_integer @ int @ code_integer @ code_pcr_integer @ code_pcr_integer @ ( bit_ri4277139882892585799ns_not @ int ) @ ( bit_ri4277139882892585799ns_not @ code_integer ) ).
% not_integer.transfer
thf(fact_5971_and__integer_Otransfer,axiom,
bNF_rel_fun @ int @ code_integer @ ( int > int ) @ ( code_integer > code_integer ) @ code_pcr_integer @ ( bNF_rel_fun @ int @ code_integer @ int @ code_integer @ code_pcr_integer @ code_pcr_integer ) @ ( bit_se5824344872417868541ns_and @ int ) @ ( bit_se5824344872417868541ns_and @ code_integer ) ).
% and_integer.transfer
thf(fact_5972_bit__integer_Otransfer,axiom,
( bNF_rel_fun @ int @ code_integer @ ( nat > $o ) @ ( nat > $o ) @ code_pcr_integer
@ ^ [Y: nat > $o,Z2: nat > $o] : Y = Z2
@ ( bit_se5641148757651400278ts_bit @ int )
@ ( bit_se5641148757651400278ts_bit @ code_integer ) ) ).
% bit_integer.transfer
thf(fact_5973_or__integer_Otransfer,axiom,
bNF_rel_fun @ int @ code_integer @ ( int > int ) @ ( code_integer > code_integer ) @ code_pcr_integer @ ( bNF_rel_fun @ int @ code_integer @ int @ code_integer @ code_pcr_integer @ code_pcr_integer ) @ ( bit_se1065995026697491101ons_or @ int ) @ ( bit_se1065995026697491101ons_or @ code_integer ) ).
% or_integer.transfer
thf(fact_5974_xor__integer_Otransfer,axiom,
bNF_rel_fun @ int @ code_integer @ ( int > int ) @ ( code_integer > code_integer ) @ code_pcr_integer @ ( bNF_rel_fun @ int @ code_integer @ int @ code_integer @ code_pcr_integer @ code_pcr_integer ) @ ( bit_se5824344971392196577ns_xor @ int ) @ ( bit_se5824344971392196577ns_xor @ code_integer ) ).
% xor_integer.transfer
thf(fact_5975_push__bit__integer_Otransfer,axiom,
( bNF_rel_fun @ nat @ nat @ ( int > int ) @ ( code_integer > code_integer )
@ ^ [Y: nat,Z2: nat] : Y = Z2
@ ( bNF_rel_fun @ int @ code_integer @ int @ code_integer @ code_pcr_integer @ code_pcr_integer )
@ ( bit_se4730199178511100633sh_bit @ int )
@ ( bit_se4730199178511100633sh_bit @ code_integer ) ) ).
% push_bit_integer.transfer
thf(fact_5976_drop__bit__integer_Otransfer,axiom,
( bNF_rel_fun @ nat @ nat @ ( int > int ) @ ( code_integer > code_integer )
@ ^ [Y: nat,Z2: nat] : Y = Z2
@ ( bNF_rel_fun @ int @ code_integer @ int @ code_integer @ code_pcr_integer @ code_pcr_integer )
@ ( bit_se4197421643247451524op_bit @ int )
@ ( bit_se4197421643247451524op_bit @ code_integer ) ) ).
% drop_bit_integer.transfer
thf(fact_5977_mask__integer_Otransfer,axiom,
( bNF_rel_fun @ nat @ nat @ int @ code_integer
@ ^ [Y: nat,Z2: nat] : Y = Z2
@ code_pcr_integer
@ ( bit_se2239418461657761734s_mask @ int )
@ ( bit_se2239418461657761734s_mask @ code_integer ) ) ).
% mask_integer.transfer
thf(fact_5978_unset__bit__integer_Otransfer,axiom,
( bNF_rel_fun @ nat @ nat @ ( int > int ) @ ( code_integer > code_integer )
@ ^ [Y: nat,Z2: nat] : Y = Z2
@ ( bNF_rel_fun @ int @ code_integer @ int @ code_integer @ code_pcr_integer @ code_pcr_integer )
@ ( bit_se2638667681897837118et_bit @ int )
@ ( bit_se2638667681897837118et_bit @ code_integer ) ) ).
% unset_bit_integer.transfer
thf(fact_5979_set__bit__integer_Otransfer,axiom,
( bNF_rel_fun @ nat @ nat @ ( int > int ) @ ( code_integer > code_integer )
@ ^ [Y: nat,Z2: nat] : Y = Z2
@ ( bNF_rel_fun @ int @ code_integer @ int @ code_integer @ code_pcr_integer @ code_pcr_integer )
@ ( bit_se5668285175392031749et_bit @ int )
@ ( bit_se5668285175392031749et_bit @ code_integer ) ) ).
% set_bit_integer.transfer
thf(fact_5980_flip__bit__integer_Otransfer,axiom,
( bNF_rel_fun @ nat @ nat @ ( int > int ) @ ( code_integer > code_integer )
@ ^ [Y: nat,Z2: nat] : Y = Z2
@ ( bNF_rel_fun @ int @ code_integer @ int @ code_integer @ code_pcr_integer @ code_pcr_integer )
@ ( bit_se8732182000553998342ip_bit @ int )
@ ( bit_se8732182000553998342ip_bit @ code_integer ) ) ).
% flip_bit_integer.transfer
thf(fact_5981_division__segment__integer_Otransfer,axiom,
bNF_rel_fun @ int @ code_integer @ int @ code_integer @ code_pcr_integer @ code_pcr_integer @ ( euclid7384307370059645450egment @ int ) @ ( euclid7384307370059645450egment @ code_integer ) ).
% division_segment_integer.transfer
thf(fact_5982_euclidean__size__integer_Otransfer,axiom,
( bNF_rel_fun @ int @ code_integer @ nat @ nat @ code_pcr_integer
@ ^ [Y: nat,Z2: nat] : Y = Z2
@ ( euclid6346220572633701492n_size @ int )
@ ( euclid6346220572633701492n_size @ code_integer ) ) ).
% euclidean_size_integer.transfer
thf(fact_5983_sub_Otransfer,axiom,
( bNF_rel_fun @ num @ num @ ( num > int ) @ ( num > code_integer )
@ ^ [Y: num,Z2: num] : Y = Z2
@ ( bNF_rel_fun @ num @ num @ int @ code_integer
@ ^ [Y: num,Z2: num] : Y = Z2
@ code_pcr_integer )
@ ^ [M2: num,N3: num] : ( minus_minus @ int @ ( numeral_numeral @ int @ M2 ) @ ( numeral_numeral @ int @ N3 ) )
@ code_sub ) ).
% sub.transfer
thf(fact_5984_Inf__fin_Osemilattice__order__set__axioms,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( lattic4895041142388067077er_set @ A @ ( inf_inf @ A ) @ ( ord_less_eq @ A ) @ ( ord_less @ A ) ) ) ).
% Inf_fin.semilattice_order_set_axioms
thf(fact_5985_Min_Osemilattice__order__set__axioms,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( lattic4895041142388067077er_set @ A @ ( ord_min @ A ) @ ( ord_less_eq @ A ) @ ( ord_less @ A ) ) ) ).
% Min.semilattice_order_set_axioms
thf(fact_5986_UNIV__char__of__nat,axiom,
( ( top_top @ ( set @ char ) )
= ( image @ 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_5987_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_5988_Sup__fin_Osemilattice__order__set__axioms,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( lattic4895041142388067077er_set @ A @ ( sup_sup @ A )
@ ^ [X2: A,Y6: A] : ( ord_less_eq @ A @ Y6 @ X2 )
@ ^ [X2: A,Y6: A] : ( ord_less @ A @ Y6 @ X2 ) ) ) ).
% Sup_fin.semilattice_order_set_axioms
thf(fact_5989_char__of__def,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ( ( unique5772411509450598832har_of @ A )
= ( ^ [N3: A] :
( char2
@ ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N3 )
@ ( bit_se5641148757651400278ts_bit @ A @ N3 @ ( one_one @ nat ) )
@ ( bit_se5641148757651400278ts_bit @ A @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
@ ( bit_se5641148757651400278ts_bit @ A @ N3 @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) )
@ ( bit_se5641148757651400278ts_bit @ A @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) )
@ ( bit_se5641148757651400278ts_bit @ A @ N3 @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ one2 ) ) ) )
@ ( bit_se5641148757651400278ts_bit @ A @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ ( bit1 @ one2 ) ) ) )
@ ( bit_se5641148757651400278ts_bit @ A @ N3 @ ( numeral_numeral @ nat @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ).
% char_of_def
thf(fact_5990_range__nat__of__char,axiom,
( ( image @ 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_5991_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_5992_char_Osize_I2_J,axiom,
! [X13: $o,X22: $o,X32: $o,X42: $o,X52: $o,X62: $o,X72: $o,X82: $o] :
( ( size_size @ char @ ( char2 @ X13 @ X22 @ X32 @ X42 @ X52 @ X62 @ X72 @ X82 ) )
= ( zero_zero @ nat ) ) ).
% char.size(2)
thf(fact_5993_nat__of__char__less__256,axiom,
! [C2: char] : ( ord_less @ nat @ ( comm_s6883823935334413003f_char @ nat @ C2 ) @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% nat_of_char_less_256
thf(fact_5994_assert__def,axiom,
! [A: $tType] :
( ( heap_Time_assert @ A )
= ( ^ [P4: A > $o,X2: A] : ( if @ ( heap_Time_Heap @ A ) @ ( P4 @ 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_5995_char_Osize__gen,axiom,
! [X13: $o,X22: $o,X32: $o,X42: $o,X52: $o,X62: $o,X72: $o,X82: $o] :
( ( size_char @ ( char2 @ X13 @ X22 @ X32 @ X42 @ X52 @ X62 @ X72 @ X82 ) )
= ( zero_zero @ nat ) ) ).
% char.size_gen
thf(fact_5996_n__lists_Osimps_I2_J,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( n_lists @ A @ ( suc @ N ) @ Xs )
= ( concat @ ( list @ A )
@ ( map @ ( list @ A ) @ ( list @ ( list @ A ) )
@ ^ [Ys3: list @ A] :
( map @ A @ ( list @ A )
@ ^ [Y6: A] : ( cons @ A @ Y6 @ Ys3 )
@ Xs )
@ ( n_lists @ A @ N @ Xs ) ) ) ) ).
% n_lists.simps(2)
thf(fact_5997_comm__append__is__replicate,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( Ys2
!= ( nil @ A ) )
=> ( ( ( append @ A @ Xs @ Ys2 )
= ( append @ A @ Ys2 @ Xs ) )
=> ? [N4: nat,Zs2: list @ A] :
( ( ord_less @ nat @ ( one_one @ nat ) @ N4 )
& ( ( concat @ A @ ( replicate @ ( list @ A ) @ N4 @ Zs2 ) )
= ( append @ A @ Xs @ Ys2 ) ) ) ) ) ) ).
% comm_append_is_replicate
thf(fact_5998_strict__sorted__equal__Uniq,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A6: set @ A] :
( uniq @ ( list @ A )
@ ^ [Xs3: list @ A] :
( ( sorted_wrt @ A @ ( ord_less @ A ) @ Xs3 )
& ( ( set2 @ A @ Xs3 )
= A6 ) ) ) ) ).
% strict_sorted_equal_Uniq
thf(fact_5999_integer__of__natural__of__nat,axiom,
! [N: nat] :
( ( code_i5400310926305786745atural @ ( semiring_1_of_nat @ code_natural @ N ) )
= ( semiring_1_of_nat @ code_integer @ N ) ) ).
% integer_of_natural_of_nat
thf(fact_6000_integer__of__natural__of__integer,axiom,
! [K: code_integer] :
( ( code_i5400310926305786745atural @ ( code_n4118661773612635043nteger @ K ) )
= ( ord_max @ code_integer @ ( zero_zero @ code_integer ) @ K ) ) ).
% integer_of_natural_of_integer
thf(fact_6001_natural__of__integer__of__natural,axiom,
! [N: code_natural] :
( ( code_n4118661773612635043nteger @ ( code_i5400310926305786745atural @ N ) )
= N ) ).
% natural_of_integer_of_natural
thf(fact_6002_card__def,axiom,
! [B: $tType] :
( ( finite_card @ B )
= ( finite_folding_F @ B @ nat
@ ^ [Uu: B] : suc
@ ( zero_zero @ nat ) ) ) ).
% card_def
thf(fact_6003_length__code,axiom,
! [A: $tType] :
( ( size_size @ ( list @ A ) )
= ( gen_length @ A @ ( zero_zero @ nat ) ) ) ).
% length_code
thf(fact_6004_gen__length__code_I2_J,axiom,
! [B: $tType,N: nat,X: B,Xs: list @ B] :
( ( gen_length @ B @ N @ ( cons @ B @ X @ Xs ) )
= ( gen_length @ B @ ( suc @ N ) @ Xs ) ) ).
% gen_length_code(2)
thf(fact_6005_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_6006_Code__Numeral_Osub__code_I4_J,axiom,
! [N: num] :
( ( code_sub @ one2 @ ( bit0 @ N ) )
= ( code_Neg @ ( bitM @ N ) ) ) ).
% Code_Numeral.sub_code(4)
thf(fact_6007_Code__Numeral_Odup__code_I3_J,axiom,
! [N: num] :
( ( code_dup @ ( code_Neg @ N ) )
= ( code_Neg @ ( bit0 @ N ) ) ) ).
% Code_Numeral.dup_code(3)
thf(fact_6008_less__integer__code_I7_J,axiom,
! [K: num] : ( ord_less @ code_integer @ ( code_Neg @ K ) @ ( zero_zero @ code_integer ) ) ).
% less_integer_code(7)
thf(fact_6009_less__integer__code_I3_J,axiom,
! [L: num] :
~ ( ord_less @ code_integer @ ( zero_zero @ code_integer ) @ ( code_Neg @ L ) ) ).
% less_integer_code(3)
thf(fact_6010_plus__integer__code_I6_J,axiom,
! [M: num,N: num] :
( ( plus_plus @ code_integer @ ( code_Neg @ M ) @ ( code_Neg @ N ) )
= ( code_Neg @ ( plus_plus @ num @ M @ N ) ) ) ).
% plus_integer_code(6)
thf(fact_6011_less__eq__integer__code_I7_J,axiom,
! [K: num] : ( ord_less_eq @ code_integer @ ( code_Neg @ K ) @ ( zero_zero @ code_integer ) ) ).
% less_eq_integer_code(7)
thf(fact_6012_less__eq__integer__code_I3_J,axiom,
! [L: num] :
~ ( ord_less_eq @ code_integer @ ( zero_zero @ code_integer ) @ ( code_Neg @ L ) ) ).
% less_eq_integer_code(3)
thf(fact_6013_less__integer__code_I9_J,axiom,
! [K: num,L: num] :
( ( ord_less @ code_integer @ ( code_Neg @ K ) @ ( code_Neg @ L ) )
= ( ord_less @ num @ L @ K ) ) ).
% less_integer_code(9)
thf(fact_6014_less__eq__integer__code_I9_J,axiom,
! [K: num,L: num] :
( ( ord_less_eq @ code_integer @ ( code_Neg @ K ) @ ( code_Neg @ L ) )
= ( ord_less_eq @ num @ L @ K ) ) ).
% less_eq_integer_code(9)
thf(fact_6015_minus__integer__code_I6_J,axiom,
! [M: num,N: num] :
( ( minus_minus @ code_integer @ ( code_Neg @ M ) @ ( code_Neg @ N ) )
= ( code_sub @ N @ M ) ) ).
% minus_integer_code(6)
thf(fact_6016_divmod__abs__code_I2_J,axiom,
! [K: num,L: num] :
( ( code_divmod_abs @ ( code_Neg @ K ) @ ( code_Neg @ L ) )
= ( unique8689654367752047608divmod @ code_integer @ K @ L ) ) ).
% divmod_abs_code(2)
thf(fact_6017_Code__Numeral_Osub__code_I5_J,axiom,
! [N: num] :
( ( code_sub @ one2 @ ( bit1 @ N ) )
= ( code_Neg @ ( bit0 @ N ) ) ) ).
% Code_Numeral.sub_code(5)
thf(fact_6018_pair__less__def,axiom,
( fun_pair_less
= ( lex_prod @ nat @ nat @ less_than @ less_than ) ) ).
% pair_less_def
thf(fact_6019_Gcd__int__set__eq__fold,axiom,
! [Xs: list @ int] :
( ( gcd_Gcd @ int @ ( set2 @ int @ Xs ) )
= ( fold @ int @ int @ ( gcd_gcd @ int ) @ Xs @ ( zero_zero @ int ) ) ) ).
% Gcd_int_set_eq_fold
thf(fact_6020_Gcd__set__eq__fold,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [Xs: list @ A] :
( ( gcd_Gcd @ A @ ( set2 @ A @ Xs ) )
= ( fold @ A @ A @ ( gcd_gcd @ A ) @ Xs @ ( zero_zero @ A ) ) ) ) ).
% Gcd_set_eq_fold
thf(fact_6021_Gcd__fin_Oset__eq__fold,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [Xs: list @ A] :
( ( semiring_gcd_Gcd_fin @ A @ ( set2 @ A @ Xs ) )
= ( fold @ A @ A @ ( gcd_gcd @ A ) @ Xs @ ( zero_zero @ A ) ) ) ) ).
% Gcd_fin.set_eq_fold
thf(fact_6022_mono__transfer,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ D )
& ( order @ C )
& ( order @ A ) )
=> ! [A6: A > B > $o,B7: C > D > $o] :
( ( bi_total @ A @ B @ A6 )
=> ( ( bNF_rel_fun @ A @ B @ ( A > $o ) @ ( B > $o ) @ A6
@ ( bNF_rel_fun @ A @ B @ $o @ $o @ A6
@ ^ [Y: $o,Z2: $o] : Y = Z2 )
@ ( ord_less_eq @ A )
@ ( ord_less_eq @ B ) )
=> ( ( bNF_rel_fun @ C @ D @ ( C > $o ) @ ( D > $o ) @ B7
@ ( bNF_rel_fun @ C @ D @ $o @ $o @ B7
@ ^ [Y: $o,Z2: $o] : Y = Z2 )
@ ( ord_less_eq @ C )
@ ( ord_less_eq @ D ) )
=> ( bNF_rel_fun @ ( A > C ) @ ( B > D ) @ $o @ $o @ ( bNF_rel_fun @ A @ B @ C @ D @ A6 @ B7 )
@ ^ [Y: $o,Z2: $o] : Y = Z2
@ ( order_mono @ A @ C )
@ ( order_mono @ B @ D ) ) ) ) ) ) ).
% mono_transfer
thf(fact_6023_binomial__def,axiom,
( binomial
= ( ^ [N3: nat,K3: nat] :
( finite_card @ ( set @ nat )
@ ( collect @ ( set @ nat )
@ ^ [K6: set @ nat] :
( ( member @ ( set @ nat ) @ K6 @ ( pow2 @ nat @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N3 ) ) )
& ( ( finite_card @ nat @ K6 )
= K3 ) ) ) ) ) ) ).
% binomial_def
thf(fact_6024_Pow__Compl,axiom,
! [A: $tType,A6: set @ A] :
( ( pow2 @ A @ ( uminus_uminus @ ( set @ A ) @ A6 ) )
= ( collect @ ( set @ A )
@ ^ [Uu: set @ A] :
? [B6: set @ A] :
( ( Uu
= ( uminus_uminus @ ( set @ A ) @ B6 ) )
& ( member @ ( set @ A ) @ A6 @ ( pow2 @ A @ B6 ) ) ) ) ) ).
% Pow_Compl
thf(fact_6025_int_Obi__total,axiom,
bi_total @ ( product_prod @ nat @ nat ) @ int @ pcr_int ).
% int.bi_total
thf(fact_6026_integer_Obi__total,axiom,
bi_total @ int @ code_integer @ code_pcr_integer ).
% integer.bi_total
thf(fact_6027_Gcd__nat__set__eq__fold,axiom,
! [Xs: list @ nat] :
( ( gcd_Gcd @ nat @ ( set2 @ nat @ Xs ) )
= ( fold @ nat @ nat @ ( gcd_gcd @ nat ) @ Xs @ ( zero_zero @ nat ) ) ) ).
% Gcd_nat_set_eq_fold
thf(fact_6028_Heap__lub__empty,axiom,
! [A: $tType] :
( ( heap_Time_Heap_lub @ A @ ( bot_bot @ ( set @ ( heap_Time_Heap @ A ) ) ) )
= ( heap_Time_Heap2 @ A
@ ^ [X2: heap_ext @ product_unit] : ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) ) ).
% Heap_lub_empty
thf(fact_6029_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_6030_gfp__gfp,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F: A > A > A] :
( ! [X3: A,Y5: A,W: A,Z4: A] :
( ( ord_less_eq @ A @ X3 @ Y5 )
=> ( ( ord_less_eq @ A @ W @ Z4 )
=> ( ord_less_eq @ A @ ( F @ X3 @ W ) @ ( F @ Y5 @ Z4 ) ) ) )
=> ( ( 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_6031_gfp__mono,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F: A > A,G: A > A] :
( ! [Z9: A] : ( ord_less_eq @ A @ ( F @ Z9 ) @ ( G @ Z9 ) )
=> ( ord_less_eq @ A @ ( complete_lattice_gfp @ A @ F ) @ ( complete_lattice_gfp @ A @ G ) ) ) ) ).
% gfp_mono
thf(fact_6032_gfp__upperbound,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X8: A,F: A > A] :
( ( ord_less_eq @ A @ X8 @ ( F @ X8 ) )
=> ( ord_less_eq @ A @ X8 @ ( complete_lattice_gfp @ A @ F ) ) ) ) ).
% gfp_upperbound
thf(fact_6033_gfp__least,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F: A > A,X8: A] :
( ! [U2: A] :
( ( ord_less_eq @ A @ U2 @ ( F @ U2 ) )
=> ( ord_less_eq @ A @ U2 @ X8 ) )
=> ( ord_less_eq @ A @ ( complete_lattice_gfp @ A @ F ) @ X8 ) ) ) ).
% gfp_least
thf(fact_6034_gfp__eqI,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F4: A > A,X: A] :
( ( order_mono @ A @ A @ F4 )
=> ( ( ( F4 @ X )
= X )
=> ( ! [Z4: A] :
( ( ( F4 @ Z4 )
= Z4 )
=> ( ord_less_eq @ A @ Z4 @ X ) )
=> ( ( complete_lattice_gfp @ A @ F4 )
= X ) ) ) ) ) ).
% gfp_eqI
thf(fact_6035_gfp__def,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ( ( complete_lattice_gfp @ A )
= ( ^ [F2: A > A] :
( complete_Sup_Sup @ A
@ ( collect @ A
@ ^ [U3: A] : ( ord_less_eq @ A @ U3 @ ( F2 @ U3 ) ) ) ) ) ) ) ).
% gfp_def
thf(fact_6036_coinduct,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F: A > A,X8: A] :
( ( order_mono @ A @ A @ F )
=> ( ( ord_less_eq @ A @ X8 @ ( F @ ( sup_sup @ A @ X8 @ ( complete_lattice_gfp @ A @ F ) ) ) )
=> ( ord_less_eq @ A @ X8 @ ( complete_lattice_gfp @ A @ F ) ) ) ) ) ).
% coinduct
thf(fact_6037_def__coinduct,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A6: A,F: A > A,X8: A] :
( ( A6
= ( complete_lattice_gfp @ A @ F ) )
=> ( ( order_mono @ A @ A @ F )
=> ( ( ord_less_eq @ A @ X8 @ ( F @ ( sup_sup @ A @ X8 @ A6 ) ) )
=> ( ord_less_eq @ A @ X8 @ A6 ) ) ) ) ) ).
% def_coinduct
thf(fact_6038_coinduct__lemma,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X8: A,F: A > A] :
( ( ord_less_eq @ A @ X8 @ ( F @ ( sup_sup @ A @ X8 @ ( complete_lattice_gfp @ A @ F ) ) ) )
=> ( ( order_mono @ A @ A @ F )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ X8 @ ( complete_lattice_gfp @ A @ F ) ) @ ( F @ ( sup_sup @ A @ X8 @ ( complete_lattice_gfp @ A @ F ) ) ) ) ) ) ) ).
% coinduct_lemma
thf(fact_6039_gfp__ordinal__induct,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F: A > A,P: A > $o] :
( ( order_mono @ A @ A @ F )
=> ( ! [S4: A] :
( ( P @ S4 )
=> ( ( ord_less_eq @ A @ ( complete_lattice_gfp @ A @ F ) @ S4 )
=> ( P @ ( F @ S4 ) ) ) )
=> ( ! [M8: set @ A] :
( ! [X4: A] :
( ( member @ A @ X4 @ M8 )
=> ( P @ X4 ) )
=> ( P @ ( complete_Inf_Inf @ A @ M8 ) ) )
=> ( P @ ( complete_lattice_gfp @ A @ F ) ) ) ) ) ) ).
% gfp_ordinal_induct
thf(fact_6040_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_6041_lfp__le__gfp,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F: A > A] :
( ( order_mono @ A @ A @ F )
=> ( ord_less_eq @ A @ ( complete_lattice_lfp @ A @ F ) @ ( complete_lattice_gfp @ A @ F ) ) ) ) ).
% lfp_le_gfp
thf(fact_6042_Code__Numeral_Osub__code_I2_J,axiom,
! [M: num] :
( ( code_sub @ ( bit0 @ M ) @ one2 )
= ( code_Pos @ ( bitM @ M ) ) ) ).
% Code_Numeral.sub_code(2)
thf(fact_6043_is__singleton__altdef,axiom,
! [A: $tType] :
( ( is_singleton @ A )
= ( ^ [A7: set @ A] :
( ( finite_card @ A @ A7 )
= ( one_one @ nat ) ) ) ) ).
% is_singleton_altdef
thf(fact_6044_Code__Numeral_OPos__def,axiom,
( code_Pos
= ( numeral_numeral @ code_integer ) ) ).
% Code_Numeral.Pos_def
thf(fact_6045_Pos__fold_I2_J,axiom,
! [K: num] :
( ( numeral_numeral @ code_integer @ ( bit0 @ K ) )
= ( code_Pos @ ( bit0 @ K ) ) ) ).
% Pos_fold(2)
thf(fact_6046_Pos__fold_I1_J,axiom,
( ( numeral_numeral @ code_integer @ one2 )
= ( code_Pos @ one2 ) ) ).
% Pos_fold(1)
thf(fact_6047_Pos__fold_I3_J,axiom,
! [K: num] :
( ( numeral_numeral @ code_integer @ ( bit1 @ K ) )
= ( code_Pos @ ( bit1 @ K ) ) ) ).
% Pos_fold(3)
thf(fact_6048_one__integer__code,axiom,
( ( one_one @ code_integer )
= ( code_Pos @ one2 ) ) ).
% one_integer_code
thf(fact_6049_Code__Numeral_Odup__code_I2_J,axiom,
! [N: num] :
( ( code_dup @ ( code_Pos @ N ) )
= ( code_Pos @ ( bit0 @ N ) ) ) ).
% Code_Numeral.dup_code(2)
thf(fact_6050_less__integer__code_I4_J,axiom,
! [K: num] :
~ ( ord_less @ code_integer @ ( code_Pos @ K ) @ ( zero_zero @ code_integer ) ) ).
% less_integer_code(4)
thf(fact_6051_less__integer__code_I2_J,axiom,
! [L: num] : ( ord_less @ code_integer @ ( zero_zero @ code_integer ) @ ( code_Pos @ L ) ) ).
% less_integer_code(2)
thf(fact_6052_plus__integer__code_I3_J,axiom,
! [M: num,N: num] :
( ( plus_plus @ code_integer @ ( code_Pos @ M ) @ ( code_Pos @ N ) )
= ( code_Pos @ ( plus_plus @ num @ M @ N ) ) ) ).
% plus_integer_code(3)
thf(fact_6053_less__eq__integer__code_I4_J,axiom,
! [K: num] :
~ ( ord_less_eq @ code_integer @ ( code_Pos @ K ) @ ( zero_zero @ code_integer ) ) ).
% less_eq_integer_code(4)
thf(fact_6054_less__eq__integer__code_I2_J,axiom,
! [L: num] : ( ord_less_eq @ code_integer @ ( zero_zero @ code_integer ) @ ( code_Pos @ L ) ) ).
% less_eq_integer_code(2)
thf(fact_6055_Code__Numeral_ONeg__def,axiom,
( code_Neg
= ( ^ [N3: num] : ( uminus_uminus @ code_integer @ ( code_Pos @ N3 ) ) ) ) ).
% Code_Numeral.Neg_def
thf(fact_6056_uminus__integer__code_I2_J,axiom,
! [M: num] :
( ( uminus_uminus @ code_integer @ ( code_Pos @ M ) )
= ( code_Neg @ M ) ) ).
% uminus_integer_code(2)
thf(fact_6057_uminus__integer__code_I3_J,axiom,
! [M: num] :
( ( uminus_uminus @ code_integer @ ( code_Neg @ M ) )
= ( code_Pos @ M ) ) ).
% uminus_integer_code(3)
thf(fact_6058_less__integer__code_I8_J,axiom,
! [K: num,L: num] : ( ord_less @ code_integer @ ( code_Neg @ K ) @ ( code_Pos @ L ) ) ).
% less_integer_code(8)
thf(fact_6059_less__integer__code_I6_J,axiom,
! [K: num,L: num] :
~ ( ord_less @ code_integer @ ( code_Pos @ K ) @ ( code_Neg @ L ) ) ).
% less_integer_code(6)
thf(fact_6060_times__integer__code_I3_J,axiom,
! [M: num,N: num] :
( ( times_times @ code_integer @ ( code_Pos @ M ) @ ( code_Pos @ N ) )
= ( code_Pos @ ( times_times @ num @ M @ N ) ) ) ).
% times_integer_code(3)
thf(fact_6061_less__eq__integer__code_I8_J,axiom,
! [K: num,L: num] : ( ord_less_eq @ code_integer @ ( code_Neg @ K ) @ ( code_Pos @ L ) ) ).
% less_eq_integer_code(8)
thf(fact_6062_less__eq__integer__code_I6_J,axiom,
! [K: num,L: num] :
~ ( ord_less_eq @ code_integer @ ( code_Pos @ K ) @ ( code_Neg @ L ) ) ).
% less_eq_integer_code(6)
thf(fact_6063_less__integer__code_I5_J,axiom,
! [K: num,L: num] :
( ( ord_less @ code_integer @ ( code_Pos @ K ) @ ( code_Pos @ L ) )
= ( ord_less @ num @ K @ L ) ) ).
% less_integer_code(5)
thf(fact_6064_less__eq__integer__code_I5_J,axiom,
! [K: num,L: num] :
( ( ord_less_eq @ code_integer @ ( code_Pos @ K ) @ ( code_Pos @ L ) )
= ( ord_less_eq @ num @ K @ L ) ) ).
% less_eq_integer_code(5)
thf(fact_6065_minus__integer__code_I3_J,axiom,
! [M: num,N: num] :
( ( minus_minus @ code_integer @ ( code_Pos @ M ) @ ( code_Pos @ N ) )
= ( code_sub @ M @ N ) ) ).
% minus_integer_code(3)
thf(fact_6066_divmod__abs__code_I1_J,axiom,
! [K: num,L: num] :
( ( code_divmod_abs @ ( code_Pos @ K ) @ ( code_Pos @ L ) )
= ( unique8689654367752047608divmod @ code_integer @ K @ L ) ) ).
% divmod_abs_code(1)
thf(fact_6067_minus__integer__code_I5_J,axiom,
! [M: num,N: num] :
( ( minus_minus @ code_integer @ ( code_Neg @ M ) @ ( code_Pos @ N ) )
= ( code_Neg @ ( plus_plus @ num @ M @ N ) ) ) ).
% minus_integer_code(5)
thf(fact_6068_minus__integer__code_I4_J,axiom,
! [M: num,N: num] :
( ( minus_minus @ code_integer @ ( code_Pos @ M ) @ ( code_Neg @ N ) )
= ( code_Pos @ ( plus_plus @ num @ M @ N ) ) ) ).
% minus_integer_code(4)
thf(fact_6069_times__integer__code_I6_J,axiom,
! [M: num,N: num] :
( ( times_times @ code_integer @ ( code_Neg @ M ) @ ( code_Neg @ N ) )
= ( code_Pos @ ( times_times @ num @ M @ N ) ) ) ).
% times_integer_code(6)
thf(fact_6070_times__integer__code_I5_J,axiom,
! [M: num,N: num] :
( ( times_times @ code_integer @ ( code_Neg @ M ) @ ( code_Pos @ N ) )
= ( code_Neg @ ( times_times @ num @ M @ N ) ) ) ).
% times_integer_code(5)
thf(fact_6071_times__integer__code_I4_J,axiom,
! [M: num,N: num] :
( ( times_times @ code_integer @ ( code_Pos @ M ) @ ( code_Neg @ N ) )
= ( code_Neg @ ( times_times @ num @ M @ N ) ) ) ).
% times_integer_code(4)
thf(fact_6072_plus__integer__code_I4_J,axiom,
! [M: num,N: num] :
( ( plus_plus @ code_integer @ ( code_Pos @ M ) @ ( code_Neg @ N ) )
= ( code_sub @ M @ N ) ) ).
% plus_integer_code(4)
thf(fact_6073_plus__integer__code_I5_J,axiom,
! [M: num,N: num] :
( ( plus_plus @ code_integer @ ( code_Neg @ M ) @ ( code_Pos @ N ) )
= ( code_sub @ N @ M ) ) ).
% plus_integer_code(5)
thf(fact_6074_divmod__abs__code_I3_J,axiom,
! [K: num,L: num] :
( ( code_divmod_abs @ ( code_Neg @ K ) @ ( code_Pos @ L ) )
= ( unique8689654367752047608divmod @ code_integer @ K @ L ) ) ).
% divmod_abs_code(3)
thf(fact_6075_divmod__abs__code_I4_J,axiom,
! [K: num,L: num] :
( ( code_divmod_abs @ ( code_Pos @ K ) @ ( code_Neg @ L ) )
= ( unique8689654367752047608divmod @ code_integer @ K @ L ) ) ).
% divmod_abs_code(4)
thf(fact_6076_Code__Numeral_Osub__code_I3_J,axiom,
! [M: num] :
( ( code_sub @ ( bit1 @ M ) @ one2 )
= ( code_Pos @ ( bit0 @ M ) ) ) ).
% Code_Numeral.sub_code(3)
thf(fact_6077_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_6078_semilattice__order__set_Osubset__imp,axiom,
! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A6: set @ A,B7: set @ A] :
( ( lattic4895041142388067077er_set @ A @ F @ Less_eq @ Less )
=> ( ( ord_less_eq @ ( set @ A ) @ A6 @ B7 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B7 )
=> ( Less_eq @ ( lattic1715443433743089157tice_F @ A @ F @ B7 ) @ ( lattic1715443433743089157tice_F @ A @ F @ A6 ) ) ) ) ) ) ).
% semilattice_order_set.subset_imp
thf(fact_6079_semilattice__set_OF_Ocong,axiom,
! [A: $tType] :
( ( lattic1715443433743089157tice_F @ A )
= ( lattic1715443433743089157tice_F @ A ) ) ).
% semilattice_set.F.cong
thf(fact_6080_Inf__fin__def,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( lattic7752659483105999362nf_fin @ A )
= ( lattic1715443433743089157tice_F @ A @ ( inf_inf @ A ) ) ) ) ).
% Inf_fin_def
thf(fact_6081_Sup__fin__def,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( lattic5882676163264333800up_fin @ A )
= ( lattic1715443433743089157tice_F @ A @ ( sup_sup @ A ) ) ) ) ).
% Sup_fin_def
thf(fact_6082_Max__def,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( lattic643756798349783984er_Max @ A )
= ( lattic1715443433743089157tice_F @ A @ ( ord_max @ A ) ) ) ) ).
% Max_def
thf(fact_6083_Min__def,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( lattic643756798350308766er_Min @ A )
= ( lattic1715443433743089157tice_F @ A @ ( ord_min @ A ) ) ) ) ).
% Min_def
thf(fact_6084_semilattice__order__set_OcoboundedI,axiom,
! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A6: set @ A,A3: A] :
( ( lattic4895041142388067077er_set @ A @ F @ Less_eq @ Less )
=> ( ( finite_finite2 @ A @ A6 )
=> ( ( member @ A @ A3 @ A6 )
=> ( Less_eq @ ( lattic1715443433743089157tice_F @ A @ F @ A6 ) @ A3 ) ) ) ) ).
% semilattice_order_set.coboundedI
thf(fact_6085_semilattice__order__set_OboundedE,axiom,
! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A6: set @ A,X: A] :
( ( lattic4895041142388067077er_set @ A @ F @ Less_eq @ Less )
=> ( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( Less_eq @ X @ ( lattic1715443433743089157tice_F @ A @ F @ A6 ) )
=> ! [A10: A] :
( ( member @ A @ A10 @ A6 )
=> ( Less_eq @ X @ A10 ) ) ) ) ) ) ).
% semilattice_order_set.boundedE
thf(fact_6086_semilattice__order__set_OboundedI,axiom,
! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A6: set @ A,X: A] :
( ( lattic4895041142388067077er_set @ A @ F @ Less_eq @ Less )
=> ( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [A8: A] :
( ( member @ A @ A8 @ A6 )
=> ( Less_eq @ X @ A8 ) )
=> ( Less_eq @ X @ ( lattic1715443433743089157tice_F @ A @ F @ A6 ) ) ) ) ) ) ).
% semilattice_order_set.boundedI
thf(fact_6087_semilattice__order__set_Obounded__iff,axiom,
! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A6: set @ A,X: A] :
( ( lattic4895041142388067077er_set @ A @ F @ Less_eq @ Less )
=> ( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( Less_eq @ X @ ( lattic1715443433743089157tice_F @ A @ F @ A6 ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ A6 )
=> ( Less_eq @ X @ X2 ) ) ) ) ) ) ) ).
% semilattice_order_set.bounded_iff
thf(fact_6088_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_6089_semilattice__set_Oeq__fold_H,axiom,
! [A: $tType,F: A > A > A,A6: set @ A] :
( ( lattic149705377957585745ce_set @ A @ F )
=> ( ( lattic1715443433743089157tice_F @ A @ F @ A6 )
= ( the2 @ A
@ ( finite_fold @ A @ ( option @ A )
@ ^ [X2: A,Y6: option @ A] : ( some @ A @ ( case_option @ A @ A @ X2 @ ( F @ X2 ) @ Y6 ) )
@ ( none @ A )
@ A6 ) ) ) ) ).
% semilattice_set.eq_fold'
thf(fact_6090_semilattice__set_Oinsert__remove,axiom,
! [A: $tType,F: A > A > A,A6: set @ A,X: A] :
( ( lattic149705377957585745ce_set @ A @ F )
=> ( ( finite_finite2 @ A @ A6 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic1715443433743089157tice_F @ A @ F @ ( insert @ A @ X @ A6 ) )
= X ) )
& ( ( ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic1715443433743089157tice_F @ A @ F @ ( insert @ A @ X @ A6 ) )
= ( F @ X @ ( lattic1715443433743089157tice_F @ A @ F @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% semilattice_set.insert_remove
thf(fact_6091_Min_Osemilattice__set__axioms,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( lattic149705377957585745ce_set @ A @ ( ord_min @ A ) ) ) ).
% Min.semilattice_set_axioms
thf(fact_6092_Max_Osemilattice__set__axioms,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( lattic149705377957585745ce_set @ A @ ( ord_max @ A ) ) ) ).
% Max.semilattice_set_axioms
thf(fact_6093_Inf__fin_Osemilattice__set__axioms,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( lattic149705377957585745ce_set @ A @ ( inf_inf @ A ) ) ) ).
% Inf_fin.semilattice_set_axioms
thf(fact_6094_Sup__fin_Osemilattice__set__axioms,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( lattic149705377957585745ce_set @ A @ ( sup_sup @ A ) ) ) ).
% Sup_fin.semilattice_set_axioms
thf(fact_6095_semilattice__order__set_Oaxioms_I2_J,axiom,
! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o] :
( ( lattic4895041142388067077er_set @ A @ F @ Less_eq @ Less )
=> ( lattic149705377957585745ce_set @ A @ F ) ) ).
% semilattice_order_set.axioms(2)
thf(fact_6096_semilattice__set_Oin__idem,axiom,
! [A: $tType,F: A > A > A,A6: set @ A,X: A] :
( ( lattic149705377957585745ce_set @ A @ F )
=> ( ( finite_finite2 @ A @ A6 )
=> ( ( member @ A @ X @ A6 )
=> ( ( F @ X @ ( lattic1715443433743089157tice_F @ A @ F @ A6 ) )
= ( lattic1715443433743089157tice_F @ A @ F @ A6 ) ) ) ) ) ).
% semilattice_set.in_idem
thf(fact_6097_semilattice__set_Osingleton,axiom,
! [A: $tType,F: A > A > A,X: A] :
( ( lattic149705377957585745ce_set @ A @ F )
=> ( ( lattic1715443433743089157tice_F @ A @ F @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= X ) ) ).
% semilattice_set.singleton
thf(fact_6098_semilattice__set_Ohom__commute,axiom,
! [A: $tType,F: A > A > A,H2: A > A,N6: set @ A] :
( ( lattic149705377957585745ce_set @ A @ F )
=> ( ! [X3: A,Y5: A] :
( ( H2 @ ( F @ X3 @ Y5 ) )
= ( F @ ( H2 @ X3 ) @ ( H2 @ Y5 ) ) )
=> ( ( finite_finite2 @ A @ N6 )
=> ( ( N6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( H2 @ ( lattic1715443433743089157tice_F @ A @ F @ N6 ) )
= ( lattic1715443433743089157tice_F @ A @ F @ ( image @ A @ A @ H2 @ N6 ) ) ) ) ) ) ) ).
% semilattice_set.hom_commute
thf(fact_6099_semilattice__set_Osubset,axiom,
! [A: $tType,F: A > A > A,A6: set @ A,B7: set @ A] :
( ( lattic149705377957585745ce_set @ A @ F )
=> ( ( finite_finite2 @ A @ A6 )
=> ( ( B7
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ B7 @ A6 )
=> ( ( F @ ( lattic1715443433743089157tice_F @ A @ F @ B7 ) @ ( lattic1715443433743089157tice_F @ A @ F @ A6 ) )
= ( lattic1715443433743089157tice_F @ A @ F @ A6 ) ) ) ) ) ) ).
% semilattice_set.subset
thf(fact_6100_semilattice__set_Oinsert__not__elem,axiom,
! [A: $tType,F: A > A > A,A6: set @ A,X: A] :
( ( lattic149705377957585745ce_set @ A @ F )
=> ( ( finite_finite2 @ A @ A6 )
=> ( ~ ( member @ A @ X @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic1715443433743089157tice_F @ A @ F @ ( insert @ A @ X @ A6 ) )
= ( F @ X @ ( lattic1715443433743089157tice_F @ A @ F @ A6 ) ) ) ) ) ) ) ).
% semilattice_set.insert_not_elem
thf(fact_6101_semilattice__set_Oinsert,axiom,
! [A: $tType,F: A > A > A,A6: set @ A,X: A] :
( ( lattic149705377957585745ce_set @ A @ F )
=> ( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic1715443433743089157tice_F @ A @ F @ ( insert @ A @ X @ A6 ) )
= ( F @ X @ ( lattic1715443433743089157tice_F @ A @ F @ A6 ) ) ) ) ) ) ).
% semilattice_set.insert
thf(fact_6102_semilattice__set_Oclosed,axiom,
! [A: $tType,F: A > A > A,A6: set @ A] :
( ( lattic149705377957585745ce_set @ A @ F )
=> ( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A,Y5: A] : ( member @ A @ ( F @ X3 @ Y5 ) @ ( insert @ A @ X3 @ ( insert @ A @ Y5 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( member @ A @ ( lattic1715443433743089157tice_F @ A @ F @ A6 ) @ A6 ) ) ) ) ) ).
% semilattice_set.closed
thf(fact_6103_semilattice__set_Ounion,axiom,
! [A: $tType,F: A > A > A,A6: set @ A,B7: set @ A] :
( ( lattic149705377957585745ce_set @ A @ F )
=> ( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B7 )
=> ( ( B7
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic1715443433743089157tice_F @ A @ F @ ( sup_sup @ ( set @ A ) @ A6 @ B7 ) )
= ( F @ ( lattic1715443433743089157tice_F @ A @ F @ A6 ) @ ( lattic1715443433743089157tice_F @ A @ F @ B7 ) ) ) ) ) ) ) ) ).
% semilattice_set.union
thf(fact_6104_semilattice__set_Oeq__fold,axiom,
! [A: $tType,F: A > A > A,A6: set @ A,X: A] :
( ( lattic149705377957585745ce_set @ A @ F )
=> ( ( finite_finite2 @ A @ A6 )
=> ( ( lattic1715443433743089157tice_F @ A @ F @ ( insert @ A @ X @ A6 ) )
= ( finite_fold @ A @ A @ F @ X @ A6 ) ) ) ) ).
% semilattice_set.eq_fold
thf(fact_6105_semilattice__set_Oinfinite,axiom,
! [A: $tType,F: A > A > A,A6: set @ A] :
( ( lattic149705377957585745ce_set @ A @ F )
=> ( ~ ( finite_finite2 @ A @ A6 )
=> ( ( lattic1715443433743089157tice_F @ A @ F @ A6 )
= ( the2 @ A @ ( none @ A ) ) ) ) ) ).
% semilattice_set.infinite
thf(fact_6106_semilattice__set_Oremove,axiom,
! [A: $tType,F: A > A > A,A6: set @ A,X: A] :
( ( lattic149705377957585745ce_set @ A @ F )
=> ( ( finite_finite2 @ A @ A6 )
=> ( ( member @ A @ X @ A6 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic1715443433743089157tice_F @ A @ F @ A6 )
= X ) )
& ( ( ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic1715443433743089157tice_F @ A @ F @ A6 )
= ( F @ X @ ( lattic1715443433743089157tice_F @ A @ F @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ) ).
% semilattice_set.remove
thf(fact_6107_Least__mono,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F: A > B,S3: set @ A] :
( ( order_mono @ A @ B @ F )
=> ( ? [X4: A] :
( ( member @ A @ X4 @ S3 )
& ! [Xa2: A] :
( ( member @ A @ Xa2 @ S3 )
=> ( ord_less_eq @ A @ X4 @ Xa2 ) ) )
=> ( ( ord_Least @ B
@ ^ [Y6: B] : ( member @ B @ Y6 @ ( image @ A @ B @ F @ S3 ) ) )
= ( F
@ ( ord_Least @ A
@ ^ [X2: A] : ( member @ A @ X2 @ S3 ) ) ) ) ) ) ) ).
% Least_mono
thf(fact_6108_cr__int__def,axiom,
( cr_int
= ( ^ [X2: product_prod @ nat @ nat] :
( ^ [Y: int,Z2: int] : Y = Z2
@ ( abs_Integ @ X2 ) ) ) ) ).
% cr_int_def
thf(fact_6109_Least__eq__0,axiom,
! [P: nat > $o] :
( ( P @ ( zero_zero @ nat ) )
=> ( ( ord_Least @ nat @ P )
= ( zero_zero @ nat ) ) ) ).
% Least_eq_0
thf(fact_6110_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_6111_Least1I,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P: A > $o] :
( ? [X4: A] :
( ( P @ X4 )
& ! [Y5: A] :
( ( P @ Y5 )
=> ( ord_less_eq @ A @ X4 @ Y5 ) )
& ! [Y5: A] :
( ( ( P @ Y5 )
& ! [Ya2: A] :
( ( P @ Ya2 )
=> ( ord_less_eq @ A @ Y5 @ Ya2 ) ) )
=> ( Y5 = X4 ) ) )
=> ( P @ ( ord_Least @ A @ P ) ) ) ) ).
% Least1I
thf(fact_6112_Least1__le,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P: A > $o,Z3: A] :
( ? [X4: A] :
( ( P @ X4 )
& ! [Y5: A] :
( ( P @ Y5 )
=> ( ord_less_eq @ A @ X4 @ Y5 ) )
& ! [Y5: A] :
( ( ( P @ Y5 )
& ! [Ya2: A] :
( ( P @ Ya2 )
=> ( ord_less_eq @ A @ Y5 @ Ya2 ) ) )
=> ( Y5 = X4 ) ) )
=> ( ( P @ Z3 )
=> ( ord_less_eq @ A @ ( ord_Least @ A @ P ) @ Z3 ) ) ) ) ).
% Least1_le
thf(fact_6113_LeastI2__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P: A > $o,X: A,Q: A > $o] :
( ( P @ X )
=> ( ! [Y5: A] :
( ( P @ Y5 )
=> ( ord_less_eq @ A @ X @ Y5 ) )
=> ( ! [X3: A] :
( ( P @ X3 )
=> ( ! [Y4: A] :
( ( P @ Y4 )
=> ( ord_less_eq @ A @ X3 @ Y4 ) )
=> ( Q @ X3 ) ) )
=> ( Q @ ( ord_Least @ A @ P ) ) ) ) ) ) ).
% LeastI2_order
thf(fact_6114_Least__equality,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P: A > $o,X: A] :
( ( P @ X )
=> ( ! [Y5: A] :
( ( P @ Y5 )
=> ( ord_less_eq @ A @ X @ Y5 ) )
=> ( ( ord_Least @ A @ P )
= X ) ) ) ) ).
% Least_equality
thf(fact_6115_LeastI2__wellorder,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P: A > $o,A3: A,Q: A > $o] :
( ( P @ A3 )
=> ( ! [A8: A] :
( ( P @ A8 )
=> ( ! [B11: A] :
( ( P @ B11 )
=> ( ord_less_eq @ A @ A8 @ B11 ) )
=> ( Q @ A8 ) ) )
=> ( Q @ ( ord_Least @ A @ P ) ) ) ) ) ).
% LeastI2_wellorder
thf(fact_6116_LeastI2__wellorder__ex,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P: A > $o,Q: A > $o] :
( ? [X_1: A] : ( P @ X_1 )
=> ( ! [A8: A] :
( ( P @ A8 )
=> ( ! [B11: A] :
( ( P @ B11 )
=> ( ord_less_eq @ A @ A8 @ B11 ) )
=> ( Q @ A8 ) ) )
=> ( Q @ ( ord_Least @ A @ P ) ) ) ) ) ).
% LeastI2_wellorder_ex
thf(fact_6117_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_6118_LeastI,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P: A > $o,K: A] :
( ( P @ K )
=> ( P @ ( ord_Least @ A @ P ) ) ) ) ).
% LeastI
thf(fact_6119_LeastI2,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P: A > $o,A3: A,Q: A > $o] :
( ( P @ A3 )
=> ( ! [X3: A] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( Q @ ( ord_Least @ A @ P ) ) ) ) ) ).
% LeastI2
thf(fact_6120_LeastI__ex,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P: A > $o] :
( ? [X_1: A] : ( P @ X_1 )
=> ( P @ ( ord_Least @ A @ P ) ) ) ) ).
% LeastI_ex
thf(fact_6121_LeastI2__ex,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P: A > $o,Q: A > $o] :
( ? [X_1: A] : ( P @ X_1 )
=> ( ! [X3: A] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( Q @ ( ord_Least @ A @ P ) ) ) ) ) ).
% LeastI2_ex
thf(fact_6122_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_6123_Least__Suc,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ ( zero_zero @ nat ) )
=> ( ( ord_Least @ nat @ P )
= ( suc
@ ( ord_Least @ nat
@ ^ [M2: nat] : ( P @ ( suc @ M2 ) ) ) ) ) ) ) ).
% Least_Suc
thf(fact_6124_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_6125_int_Opcr__cr__eq,axiom,
pcr_int = cr_int ).
% int.pcr_cr_eq
thf(fact_6126_Quotient__int,axiom,
quotient @ ( product_prod @ nat @ nat ) @ int @ intrel @ abs_Integ @ rep_Integ @ cr_int ).
% Quotient_int
thf(fact_6127_list__ex__length,axiom,
! [A: $tType] :
( ( list_ex @ A )
= ( ^ [P4: A > $o,Xs3: list @ A] :
? [N3: nat] :
( ( ord_less @ nat @ N3 @ ( size_size @ ( list @ A ) @ Xs3 ) )
& ( P4 @ ( nth @ A @ Xs3 @ N3 ) ) ) ) ) ).
% list_ex_length
thf(fact_6128_map__of__zip__nth,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys2: list @ B,I: nat] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys2 ) )
=> ( ( distinct @ A @ Xs )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ B ) @ Ys2 ) )
=> ( ( map_of @ A @ B @ ( zip @ A @ B @ Xs @ Ys2 ) @ ( nth @ A @ Xs @ I ) )
= ( some @ B @ ( nth @ B @ Ys2 @ I ) ) ) ) ) ) ).
% map_of_zip_nth
thf(fact_6129_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_6130_semilattice__neutr__order_Oaxioms_I1_J,axiom,
! [A: $tType,F: A > A > A,Z3: A,Less_eq: A > A > $o,Less: A > A > $o] :
( ( semila1105856199041335345_order @ A @ F @ Z3 @ Less_eq @ Less )
=> ( semilattice_neutr @ A @ F @ Z3 ) ) ).
% semilattice_neutr_order.axioms(1)
thf(fact_6131_sup__bot_Osemilattice__neutr__axioms,axiom,
! [A: $tType] :
( ( bounde4967611905675639751up_bot @ A )
=> ( semilattice_neutr @ A @ ( sup_sup @ A ) @ ( bot_bot @ A ) ) ) ).
% sup_bot.semilattice_neutr_axioms
thf(fact_6132_inf__top_Osemilattice__neutr__axioms,axiom,
! [A: $tType] :
( ( bounde4346867609351753570nf_top @ A )
=> ( semilattice_neutr @ A @ ( inf_inf @ A ) @ ( top_top @ A ) ) ) ).
% inf_top.semilattice_neutr_axioms
thf(fact_6133_gcd__nat_Osemilattice__neutr__axioms,axiom,
semilattice_neutr @ nat @ ( gcd_gcd @ nat ) @ ( zero_zero @ nat ) ).
% gcd_nat.semilattice_neutr_axioms
thf(fact_6134_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_6135_max__nat_Osemilattice__neutr__axioms,axiom,
semilattice_neutr @ nat @ ( ord_max @ nat ) @ ( zero_zero @ nat ) ).
% max_nat.semilattice_neutr_axioms
thf(fact_6136_map__upds__list__update2__drop,axiom,
! [A: $tType,B: $tType,Xs: list @ A,I: nat,M: A > ( option @ B ),Ys2: list @ B,Y3: B] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ I )
=> ( ( map_upds @ A @ B @ M @ Xs @ ( list_update @ B @ Ys2 @ I @ Y3 ) )
= ( map_upds @ A @ B @ M @ Xs @ Ys2 ) ) ) ).
% map_upds_list_update2_drop
thf(fact_6137_natLess__def,axiom,
( bNF_Ca8459412986667044542atLess
= ( collect @ ( product_prod @ nat @ nat ) @ ( product_case_prod @ nat @ nat @ $o @ ( ord_less @ nat ) ) ) ) ).
% natLess_def
thf(fact_6138_relChain__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ( ( bNF_Ca3754400796208372196lChain @ A @ B )
= ( ^ [R5: set @ ( product_prod @ A @ A ),As: A > B] :
! [I5: A,J3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ I5 @ J3 ) @ R5 )
=> ( ord_less_eq @ B @ ( As @ I5 ) @ ( As @ J3 ) ) ) ) ) ) ).
% relChain_def
thf(fact_6139_map__upds__append1,axiom,
! [B: $tType,A: $tType,Xs: list @ A,Ys2: list @ B,M: A > ( option @ B ),X: A] :
( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ B ) @ Ys2 ) )
=> ( ( map_upds @ A @ B @ M @ ( append @ A @ Xs @ ( cons @ A @ X @ ( nil @ A ) ) ) @ Ys2 )
= ( fun_upd @ A @ ( option @ B ) @ ( map_upds @ A @ B @ M @ Xs @ Ys2 ) @ X @ ( some @ B @ ( nth @ B @ Ys2 @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) ) ).
% map_upds_append1
thf(fact_6140_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 ) @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( restrict_map @ A @ B @ M @ ( uminus_uminus @ ( set @ A ) @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% restrict_upd_same
thf(fact_6141_restrict__complement__singleton__eq,axiom,
! [A: $tType,B: $tType,F: A > ( option @ B ),X: A] :
( ( restrict_map @ A @ B @ F @ ( uminus_uminus @ ( set @ A ) @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( fun_upd @ A @ ( option @ B ) @ F @ X @ ( none @ B ) ) ) ).
% restrict_complement_singleton_eq
thf(fact_6142_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,Y6: nat] :
( ( ord_less @ nat @ X2 @ N )
& ( ord_less @ nat @ Y6 @ N )
& ( ord_less_eq @ nat @ X2 @ Y6 ) ) ) ) ) ).
% Restr_natLeq
thf(fact_6143_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_6144_natLeq__def,axiom,
( bNF_Ca8665028551170535155natLeq
= ( collect @ ( product_prod @ nat @ nat ) @ ( product_case_prod @ nat @ nat @ $o @ ( ord_less_eq @ nat ) ) ) ) ).
% natLeq_def
thf(fact_6145_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,Y6: nat] :
( ( ord_less @ nat @ X2 @ N )
& ( ord_less @ nat @ Y6 @ N )
& ( ord_less_eq @ nat @ X2 @ Y6 ) ) ) ) ) ).
% Restr_natLeq2
thf(fact_6146_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_6147_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_6148_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_6149_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_6150_inv__into__f__f,axiom,
! [B: $tType,A: $tType,F: A > B,A6: set @ A,X: A] :
( ( inj_on @ A @ B @ F @ A6 )
=> ( ( member @ A @ X @ A6 )
=> ( ( hilbert_inv_into @ A @ B @ A6 @ F @ ( F @ X ) )
= X ) ) ) ).
% inv_into_f_f
thf(fact_6151_inv__identity,axiom,
! [A: $tType] :
( ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) )
@ ^ [A5: A] : A5 )
= ( ^ [A5: A] : A5 ) ) ).
% inv_identity
thf(fact_6152_inv__id,axiom,
! [A: $tType] :
( ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ ( id @ A ) )
= ( id @ A ) ) ).
% inv_id
thf(fact_6153_inv__into__image__cancel,axiom,
! [B: $tType,A: $tType,F: A > B,A6: set @ A,S3: set @ A] :
( ( inj_on @ A @ B @ F @ A6 )
=> ( ( ord_less_eq @ ( set @ A ) @ S3 @ A6 )
=> ( ( image @ B @ A @ ( hilbert_inv_into @ A @ B @ A6 @ F ) @ ( image @ A @ B @ F @ S3 ) )
= S3 ) ) ) ).
% inv_into_image_cancel
thf(fact_6154_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_6155_inv__into__def,axiom,
! [B: $tType,A: $tType] :
( ( hilbert_inv_into @ A @ B )
= ( ^ [A7: set @ A,F2: A > B,X2: B] :
( fChoice @ A
@ ^ [Y6: A] :
( ( member @ A @ Y6 @ A7 )
& ( ( F2 @ Y6 )
= X2 ) ) ) ) ) ).
% inv_into_def
thf(fact_6156_inv__into__def2,axiom,
! [B: $tType,A: $tType] :
( ( hilbert_inv_into @ A @ B )
= ( ^ [A7: set @ A,F2: A > B,X2: B] :
( fChoice @ A
@ ^ [Y6: A] :
( ( member @ A @ Y6 @ A7 )
& ( ( F2 @ Y6 )
= X2 ) ) ) ) ) ).
% inv_into_def2
thf(fact_6157_group_Oleft__cancel,axiom,
! [A: $tType,F: A > A > A,Z3: A,Inverse: A > A,A3: A,B2: A,C2: A] :
( ( group @ A @ F @ Z3 @ Inverse )
=> ( ( ( F @ A3 @ B2 )
= ( F @ A3 @ C2 ) )
= ( B2 = C2 ) ) ) ).
% group.left_cancel
thf(fact_6158_group_Oleft__inverse,axiom,
! [A: $tType,F: A > A > A,Z3: A,Inverse: A > A,A3: A] :
( ( group @ A @ F @ Z3 @ Inverse )
=> ( ( F @ ( Inverse @ A3 ) @ A3 )
= Z3 ) ) ).
% group.left_inverse
thf(fact_6159_group_Oright__cancel,axiom,
! [A: $tType,F: A > A > A,Z3: A,Inverse: A > A,B2: A,A3: A,C2: A] :
( ( group @ A @ F @ Z3 @ Inverse )
=> ( ( ( F @ B2 @ A3 )
= ( F @ C2 @ A3 ) )
= ( B2 = C2 ) ) ) ).
% group.right_cancel
thf(fact_6160_group_Oright__inverse,axiom,
! [A: $tType,F: A > A > A,Z3: A,Inverse: A > A,A3: A] :
( ( group @ A @ F @ Z3 @ Inverse )
=> ( ( F @ A3 @ ( Inverse @ A3 ) )
= Z3 ) ) ).
% group.right_inverse
thf(fact_6161_group_Oinverse__unique,axiom,
! [A: $tType,F: A > A > A,Z3: A,Inverse: A > A,A3: A,B2: A] :
( ( group @ A @ F @ Z3 @ Inverse )
=> ( ( ( F @ A3 @ B2 )
= Z3 )
=> ( ( Inverse @ A3 )
= B2 ) ) ) ).
% group.inverse_unique
thf(fact_6162_group_Oinverse__inverse,axiom,
! [A: $tType,F: A > A > A,Z3: A,Inverse: A > A,A3: A] :
( ( group @ A @ F @ Z3 @ Inverse )
=> ( ( Inverse @ ( Inverse @ A3 ) )
= A3 ) ) ).
% group.inverse_inverse
thf(fact_6163_group_Oinverse__neutral,axiom,
! [A: $tType,F: A > A > A,Z3: A,Inverse: A > A] :
( ( group @ A @ F @ Z3 @ Inverse )
=> ( ( Inverse @ Z3 )
= Z3 ) ) ).
% group.inverse_neutral
thf(fact_6164_group_Ogroup__left__neutral,axiom,
! [A: $tType,F: A > A > A,Z3: A,Inverse: A > A,A3: A] :
( ( group @ A @ F @ Z3 @ Inverse )
=> ( ( F @ Z3 @ A3 )
= A3 ) ) ).
% group.group_left_neutral
thf(fact_6165_group_Oinverse__distrib__swap,axiom,
! [A: $tType,F: A > A > A,Z3: A,Inverse: A > A,A3: A,B2: A] :
( ( group @ A @ F @ Z3 @ Inverse )
=> ( ( Inverse @ ( F @ A3 @ B2 ) )
= ( F @ ( Inverse @ B2 ) @ ( Inverse @ A3 ) ) ) ) ).
% group.inverse_distrib_swap
thf(fact_6166_inv__into__injective,axiom,
! [A: $tType,B: $tType,A6: set @ A,F: A > B,X: B,Y3: B] :
( ( ( hilbert_inv_into @ A @ B @ A6 @ F @ X )
= ( hilbert_inv_into @ A @ B @ A6 @ F @ Y3 ) )
=> ( ( member @ B @ X @ ( image @ A @ B @ F @ A6 ) )
=> ( ( member @ B @ Y3 @ ( image @ A @ B @ F @ A6 ) )
=> ( X = Y3 ) ) ) ) ).
% inv_into_injective
thf(fact_6167_inv__into__into,axiom,
! [A: $tType,B: $tType,X: A,F: B > A,A6: set @ B] :
( ( member @ A @ X @ ( image @ B @ A @ F @ A6 ) )
=> ( member @ B @ ( hilbert_inv_into @ B @ A @ A6 @ F @ X ) @ A6 ) ) ).
% inv_into_into
thf(fact_6168_f__inv__into__f,axiom,
! [B: $tType,A: $tType,Y3: A,F: B > A,A6: set @ B] :
( ( member @ A @ Y3 @ ( image @ B @ A @ F @ A6 ) )
=> ( ( F @ ( hilbert_inv_into @ B @ A @ A6 @ F @ Y3 ) )
= Y3 ) ) ).
% f_inv_into_f
thf(fact_6169_surj__f__inv__f,axiom,
! [B: $tType,A: $tType,F: B > A,Y3: A] :
( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ( ( F @ ( hilbert_inv_into @ B @ A @ ( top_top @ ( set @ B ) ) @ F @ Y3 ) )
= Y3 ) ) ).
% surj_f_inv_f
thf(fact_6170_surj__iff__all,axiom,
! [B: $tType,A: $tType,F: B > A] :
( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
= ( ! [X2: A] :
( ( F @ ( hilbert_inv_into @ B @ A @ ( top_top @ ( set @ B ) ) @ F @ X2 ) )
= X2 ) ) ) ).
% surj_iff_all
thf(fact_6171_image__f__inv__f,axiom,
! [B: $tType,A: $tType,F: B > A,A6: set @ A] :
( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ( ( image @ B @ A @ F @ ( image @ A @ B @ ( hilbert_inv_into @ B @ A @ ( top_top @ ( set @ B ) ) @ F ) @ A6 ) )
= A6 ) ) ).
% image_f_inv_f
thf(fact_6172_surj__imp__inv__eq,axiom,
! [B: $tType,A: $tType,F: B > A,G: A > B] :
( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ( ! [X3: B] :
( ( G @ ( F @ X3 ) )
= X3 )
=> ( ( hilbert_inv_into @ B @ A @ ( top_top @ ( set @ B ) ) @ F )
= G ) ) ) ).
% surj_imp_inv_eq
thf(fact_6173_inv__into__f__eq,axiom,
! [B: $tType,A: $tType,F: A > B,A6: set @ A,X: A,Y3: B] :
( ( inj_on @ A @ B @ F @ A6 )
=> ( ( member @ A @ X @ A6 )
=> ( ( ( F @ X )
= Y3 )
=> ( ( hilbert_inv_into @ A @ B @ A6 @ F @ Y3 )
= X ) ) ) ) ).
% inv_into_f_eq
thf(fact_6174_inv__f__f,axiom,
! [B: $tType,A: $tType,F: A > B,X: A] :
( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
=> ( ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F @ ( F @ X ) )
= X ) ) ).
% inv_f_f
thf(fact_6175_inv__f__eq,axiom,
! [B: $tType,A: $tType,F: A > B,X: A,Y3: B] :
( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
=> ( ( ( F @ X )
= Y3 )
=> ( ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F @ Y3 )
= X ) ) ) ).
% inv_f_eq
thf(fact_6176_inj__imp__inv__eq,axiom,
! [A: $tType,B: $tType,F: A > B,G: B > A] :
( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
=> ( ! [X3: B] :
( ( F @ ( G @ X3 ) )
= X3 )
=> ( ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F )
= G ) ) ) ).
% inj_imp_inv_eq
thf(fact_6177_inv__equality,axiom,
! [A: $tType,B: $tType,G: B > A,F: A > B] :
( ! [X3: A] :
( ( G @ ( F @ X3 ) )
= X3 )
=> ( ! [Y5: B] :
( ( F @ ( G @ Y5 ) )
= Y5 )
=> ( ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F )
= G ) ) ) ).
% inv_equality
thf(fact_6178_image__inv__into__cancel,axiom,
! [B: $tType,A: $tType,F: B > A,A6: set @ B,A11: set @ A,B13: set @ A] :
( ( ( image @ B @ A @ F @ A6 )
= A11 )
=> ( ( ord_less_eq @ ( set @ A ) @ B13 @ A11 )
=> ( ( image @ B @ A @ F @ ( image @ A @ B @ ( hilbert_inv_into @ B @ A @ A6 @ F ) @ B13 ) )
= B13 ) ) ) ).
% image_inv_into_cancel
thf(fact_6179_inv__def,axiom,
! [B: $tType,A: $tType,F: B > A] :
( ( hilbert_inv_into @ B @ A @ ( top_top @ ( set @ B ) ) @ F )
= ( ^ [Y6: A] :
( fChoice @ B
@ ^ [X2: B] :
( ( F @ X2 )
= Y6 ) ) ) ) ).
% inv_def
thf(fact_6180_inj__on__inv__into,axiom,
! [B: $tType,A: $tType,B7: set @ A,F: B > A,A6: set @ B] :
( ( ord_less_eq @ ( set @ A ) @ B7 @ ( image @ B @ A @ F @ A6 ) )
=> ( inj_on @ A @ B @ ( hilbert_inv_into @ B @ A @ A6 @ F ) @ B7 ) ) ).
% inj_on_inv_into
thf(fact_6181_inj__transfer,axiom,
! [B: $tType,A: $tType,F: A > B,P: A > $o,X: A] :
( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
=> ( ! [Y5: B] :
( ( member @ B @ Y5 @ ( image @ A @ B @ F @ ( top_top @ ( set @ A ) ) ) )
=> ( P @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F @ Y5 ) ) )
=> ( P @ X ) ) ) ).
% inj_transfer
thf(fact_6182_image__inv__f__f,axiom,
! [B: $tType,A: $tType,F: A > B,A6: set @ A] :
( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
=> ( ( image @ B @ A @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F ) @ ( image @ A @ B @ F @ A6 ) )
= A6 ) ) ).
% image_inv_f_f
thf(fact_6183_inj__imp__surj__inv,axiom,
! [B: $tType,A: $tType,F: A > B] :
( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
=> ( ( image @ B @ A @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F ) @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) ) ) ).
% inj_imp_surj_inv
thf(fact_6184_surj__imp__inj__inv,axiom,
! [B: $tType,A: $tType,F: B > A] :
( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ( inj_on @ A @ B @ ( hilbert_inv_into @ B @ A @ ( top_top @ ( set @ B ) ) @ F ) @ ( top_top @ ( set @ A ) ) ) ) ).
% surj_imp_inj_inv
thf(fact_6185_inv__into__comp,axiom,
! [A: $tType,C: $tType,B: $tType,F: A > B,G: C > A,A6: set @ C,X: B] :
( ( inj_on @ A @ B @ F @ ( image @ C @ A @ G @ A6 ) )
=> ( ( inj_on @ C @ A @ G @ A6 )
=> ( ( member @ B @ X @ ( image @ A @ B @ F @ ( image @ C @ A @ G @ A6 ) ) )
=> ( ( hilbert_inv_into @ C @ B @ A6 @ ( comp @ A @ B @ C @ F @ G ) @ X )
= ( comp @ A @ C @ B @ ( hilbert_inv_into @ C @ A @ A6 @ G ) @ ( hilbert_inv_into @ A @ B @ ( image @ C @ A @ G @ A6 ) @ F ) @ X ) ) ) ) ) ).
% inv_into_comp
thf(fact_6186_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_6187_surj__iff,axiom,
! [B: $tType,A: $tType,F: B > A] :
( ( ( image @ 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_6188_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_6189_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_6190_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_6191_bijection_Oeq__iff,axiom,
! [A: $tType,F: A > A,A3: A,B2: A] :
( ( hilbert_bijection @ A @ F )
=> ( ( ( F @ A3 )
= ( F @ B2 ) )
= ( A3 = B2 ) ) ) ).
% bijection.eq_iff
thf(fact_6192_bijection_OeqI,axiom,
! [A: $tType,F: A > A,A3: A,B2: A] :
( ( hilbert_bijection @ A @ F )
=> ( ( ( F @ A3 )
= ( F @ B2 ) )
=> ( A3 = B2 ) ) ) ).
% bijection.eqI
thf(fact_6193_bijection_Osurj,axiom,
! [A: $tType,F: A > A] :
( ( hilbert_bijection @ A @ F )
=> ( ( image @ A @ A @ F @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ) ).
% bijection.surj
thf(fact_6194_bijection_Oinj,axiom,
! [A: $tType,F: A > A] :
( ( hilbert_bijection @ A @ F )
=> ( inj_on @ A @ A @ F @ ( top_top @ ( set @ A ) ) ) ) ).
% bijection.inj
thf(fact_6195_bijection_Oinv__right__eq__iff,axiom,
! [A: $tType,F: A > A,B2: A,A3: A] :
( ( hilbert_bijection @ A @ F )
=> ( ( B2
= ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F @ A3 ) )
= ( ( F @ B2 )
= A3 ) ) ) ).
% bijection.inv_right_eq_iff
thf(fact_6196_bijection_Oinv__left__eq__iff,axiom,
! [A: $tType,F: A > A,A3: A,B2: A] :
( ( hilbert_bijection @ A @ F )
=> ( ( ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F @ A3 )
= B2 )
= ( ( F @ B2 )
= A3 ) ) ) ).
% bijection.inv_left_eq_iff
thf(fact_6197_bijection_Oeq__inv__iff,axiom,
! [A: $tType,F: A > A,A3: A,B2: A] :
( ( hilbert_bijection @ A @ F )
=> ( ( ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F @ A3 )
= ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F @ B2 ) )
= ( A3 = B2 ) ) ) ).
% bijection.eq_inv_iff
thf(fact_6198_bijection_Oinv__right,axiom,
! [A: $tType,F: A > A,A3: A] :
( ( hilbert_bijection @ A @ F )
=> ( ( F @ ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F @ A3 ) )
= A3 ) ) ).
% bijection.inv_right
thf(fact_6199_bijection_Oinv__left,axiom,
! [A: $tType,F: A > A,A3: A] :
( ( hilbert_bijection @ A @ F )
=> ( ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F @ ( F @ A3 ) )
= A3 ) ) ).
% bijection.inv_left
thf(fact_6200_bijection_Oeq__invI,axiom,
! [A: $tType,F: A > A,A3: A,B2: A] :
( ( hilbert_bijection @ A @ F )
=> ( ( ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F @ A3 )
= ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F @ B2 ) )
=> ( A3 = B2 ) ) ) ).
% bijection.eq_invI
thf(fact_6201_bijection_Osurj__inv,axiom,
! [A: $tType,F: A > A] :
( ( hilbert_bijection @ A @ F )
=> ( ( image @ A @ A @ ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F ) @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ) ).
% bijection.surj_inv
thf(fact_6202_bijection_Oinj__inv,axiom,
! [A: $tType,F: A > A] :
( ( hilbert_bijection @ A @ F )
=> ( inj_on @ A @ A @ ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F ) @ ( top_top @ ( set @ A ) ) ) ) ).
% bijection.inj_inv
thf(fact_6203_iteratesp_Omono,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [F: A > A] :
( order_mono @ ( A > $o ) @ ( A > $o )
@ ^ [P6: A > $o,X2: A] :
( ? [Y6: A] :
( ( X2
= ( F @ Y6 ) )
& ( P6 @ Y6 ) )
| ? [M9: set @ A] :
( ( X2
= ( complete_Sup_Sup @ A @ M9 ) )
& ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ M9 )
& ! [Y6: A] :
( ( member @ A @ Y6 @ M9 )
=> ( P6 @ Y6 ) ) ) ) ) ) ).
% iteratesp.mono
thf(fact_6204_finite__subset__wf,axiom,
! [A: $tType,A6: set @ A] :
( ( finite_finite2 @ A @ A6 )
=> ( wf @ ( set @ A )
@ ( collect @ ( product_prod @ ( set @ A ) @ ( set @ A ) )
@ ( product_case_prod @ ( set @ A ) @ ( set @ A ) @ $o
@ ^ [X7: set @ A,Y9: set @ A] :
( ( ord_less @ ( set @ A ) @ X7 @ Y9 )
& ( ord_less_eq @ ( set @ A ) @ Y9 @ A6 ) ) ) ) ) ) ).
% finite_subset_wf
thf(fact_6205_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_6206_ccpo__Sup__least,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [A6: set @ A,Z3: A] :
( ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ A6 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ A6 )
=> ( ord_less_eq @ A @ X3 @ Z3 ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A6 ) @ Z3 ) ) ) ) ).
% ccpo_Sup_least
thf(fact_6207_ccpo__Sup__upper,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [A6: set @ A,X: A] :
( ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ A6 )
=> ( ( member @ A @ X @ A6 )
=> ( ord_less_eq @ A @ X @ ( complete_Sup_Sup @ A @ A6 ) ) ) ) ) ).
% ccpo_Sup_upper
thf(fact_6208_wf__iff__no__infinite__down__chain,axiom,
! [A: $tType] :
( ( wf @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] :
~ ? [F2: nat > A] :
! [I5: nat] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( F2 @ ( suc @ I5 ) ) @ ( F2 @ I5 ) ) @ R5 ) ) ) ).
% wf_iff_no_infinite_down_chain
thf(fact_6209_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_6210_chain__singleton,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [X: A] : ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% chain_singleton
thf(fact_6211_wf__if__measure,axiom,
! [A: $tType,P: A > $o,F: A > nat,G: A > A] :
( ! [X3: A] :
( ( P @ X3 )
=> ( ord_less @ nat @ ( F @ ( G @ X3 ) ) @ ( F @ X3 ) ) )
=> ( wf @ A
@ ( collect @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ^ [Y6: A,X2: A] :
( ( P @ X2 )
& ( Y6
= ( G @ X2 ) ) ) ) ) ) ) ).
% wf_if_measure
thf(fact_6212_wf__bounded__measure,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),Ub: A > nat,F: A > nat] :
( ! [A8: A,B5: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B5 @ A8 ) @ R )
=> ( ( ord_less_eq @ nat @ ( Ub @ B5 ) @ ( Ub @ A8 ) )
& ( ord_less_eq @ nat @ ( F @ B5 ) @ ( Ub @ A8 ) )
& ( ord_less @ nat @ ( F @ A8 ) @ ( F @ B5 ) ) ) )
=> ( wf @ A @ R ) ) ).
% wf_bounded_measure
thf(fact_6213_wf__linord__ex__has__least,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),P: B > $o,K: B,M: B > A] :
( ( wf @ A @ R )
=> ( ! [X3: A,Y5: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y5 ) @ ( transitive_trancl @ A @ R ) )
= ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y5 @ X3 ) @ ( transitive_rtrancl @ A @ R ) ) ) )
=> ( ( P @ K )
=> ? [X3: B] :
( ( P @ X3 )
& ! [Y4: B] :
( ( P @ Y4 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( M @ X3 ) @ ( M @ Y4 ) ) @ ( transitive_rtrancl @ A @ R ) ) ) ) ) ) ) ).
% wf_linord_ex_has_least
thf(fact_6214_reduction__pair__lemma,axiom,
! [A: $tType,P: product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ),R2: set @ ( product_prod @ A @ A ),S3: set @ ( product_prod @ A @ A )] :
( ( fun_reduction_pair @ A @ P )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ ( product_fst @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ P ) )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ S3 @ ( product_snd @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ P ) )
=> ( ( wf @ A @ S3 )
=> ( wf @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ S3 ) ) ) ) ) ) ).
% reduction_pair_lemma
thf(fact_6215_wf__bounded__set,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),Ub: A > ( set @ B ),F: A > ( set @ B )] :
( ! [A8: A,B5: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B5 @ A8 ) @ R )
=> ( ( finite_finite2 @ B @ ( Ub @ A8 ) )
& ( ord_less_eq @ ( set @ B ) @ ( Ub @ B5 ) @ ( Ub @ A8 ) )
& ( ord_less_eq @ ( set @ B ) @ ( F @ B5 ) @ ( Ub @ A8 ) )
& ( ord_less @ ( set @ B ) @ ( F @ A8 ) @ ( F @ B5 ) ) ) )
=> ( wf @ A @ R ) ) ).
% wf_bounded_set
thf(fact_6216_in__chain__finite,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [A6: set @ A] :
( ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ A6 )
=> ( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ A @ ( complete_Sup_Sup @ A @ A6 ) @ A6 ) ) ) ) ) ).
% in_chain_finite
thf(fact_6217_iteratesp__def,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ( ( comple7512665784863727008ratesp @ A )
= ( ^ [F2: A > A] :
( complete_lattice_lfp @ ( A > $o )
@ ^ [P6: A > $o,X2: A] :
( ? [Y6: A] :
( ( X2
= ( F2 @ Y6 ) )
& ( P6 @ Y6 ) )
| ? [M9: set @ A] :
( ( X2
= ( complete_Sup_Sup @ A @ M9 ) )
& ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ M9 )
& ! [Y6: A] :
( ( member @ A @ Y6 @ M9 )
=> ( P6 @ Y6 ) ) ) ) ) ) ) ) ).
% iteratesp_def
thf(fact_6218_wf__int__ge__less__than2,axiom,
! [D2: int] : ( wf @ int @ ( int_ge_less_than2 @ D2 ) ) ).
% wf_int_ge_less_than2
thf(fact_6219_wf__int__ge__less__than,axiom,
! [D2: int] : ( wf @ int @ ( int_ge_less_than @ D2 ) ) ).
% wf_int_ge_less_than
thf(fact_6220_wf__pair__less,axiom,
wf @ ( product_prod @ nat @ nat ) @ fun_pair_less ).
% wf_pair_less
thf(fact_6221_wf__less,axiom,
wf @ nat @ ( collect @ ( product_prod @ nat @ nat ) @ ( product_case_prod @ nat @ nat @ $o @ ( ord_less @ nat ) ) ) ).
% wf_less
thf(fact_6222_iteratesp_Osimps,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ( ( comple7512665784863727008ratesp @ A )
= ( ^ [F2: A > A,A5: A] :
( ? [X2: A] :
( ( A5
= ( F2 @ X2 ) )
& ( comple7512665784863727008ratesp @ A @ F2 @ X2 ) )
| ? [M9: set @ A] :
( ( A5
= ( complete_Sup_Sup @ A @ M9 ) )
& ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ M9 )
& ! [X2: A] :
( ( member @ A @ X2 @ M9 )
=> ( comple7512665784863727008ratesp @ A @ F2 @ X2 ) ) ) ) ) ) ) ).
% iteratesp.simps
thf(fact_6223_iteratesp_Ocases,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [F: A > A,A3: A] :
( ( comple7512665784863727008ratesp @ A @ F @ A3 )
=> ( ! [X3: A] :
( ( A3
= ( F @ X3 ) )
=> ~ ( comple7512665784863727008ratesp @ A @ F @ X3 ) )
=> ~ ! [M8: set @ A] :
( ( A3
= ( complete_Sup_Sup @ A @ M8 ) )
=> ( ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ M8 )
=> ~ ! [X4: A] :
( ( member @ A @ X4 @ M8 )
=> ( comple7512665784863727008ratesp @ A @ F @ X4 ) ) ) ) ) ) ) ).
% iteratesp.cases
thf(fact_6224_iteratesp_OSup,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [M7: set @ A,F: A > A] :
( ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ M7 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ M7 )
=> ( comple7512665784863727008ratesp @ A @ F @ X3 ) )
=> ( comple7512665784863727008ratesp @ A @ F @ ( complete_Sup_Sup @ A @ M7 ) ) ) ) ) ).
% iteratesp.Sup
thf(fact_6225_reduction__pair__def,axiom,
! [A: $tType] :
( ( fun_reduction_pair @ A )
= ( ^ [P4: product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) )] :
( ( wf @ A @ ( product_fst @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ P4 ) )
& ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( relcomp @ A @ A @ A @ ( product_fst @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ P4 ) @ ( product_snd @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ P4 ) ) @ ( product_fst @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ P4 ) ) ) ) ) ).
% reduction_pair_def
thf(fact_6226_admissible__chfin,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [P: A > $o] :
( ! [S4: set @ A] :
( ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ S4 )
=> ( finite_finite2 @ A @ S4 ) )
=> ( comple1908693960933563346ssible @ A @ ( complete_Sup_Sup @ A ) @ ( ord_less_eq @ A ) @ P ) ) ) ).
% admissible_chfin
thf(fact_6227_reduction__pairI,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),S3: set @ ( product_prod @ A @ A )] :
( ( wf @ A @ R2 )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( relcomp @ A @ A @ A @ R2 @ S3 ) @ R2 )
=> ( fun_reduction_pair @ A @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ S3 ) ) ) ) ).
% reduction_pairI
thf(fact_6228_wf__no__loop,axiom,
! [B: $tType,R2: set @ ( product_prod @ B @ B )] :
( ( ( relcomp @ B @ B @ B @ R2 @ R2 )
= ( bot_bot @ ( set @ ( product_prod @ B @ B ) ) ) )
=> ( wf @ B @ R2 ) ) ).
% wf_no_loop
thf(fact_6229_union__comp__emptyL,axiom,
! [A: $tType,A6: set @ ( product_prod @ A @ A ),C6: set @ ( product_prod @ A @ A ),B7: set @ ( product_prod @ A @ A )] :
( ( ( relcomp @ A @ A @ A @ A6 @ C6 )
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
=> ( ( ( relcomp @ A @ A @ A @ B7 @ C6 )
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
=> ( ( relcomp @ A @ A @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ A6 @ B7 ) @ C6 )
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ) ).
% union_comp_emptyL
thf(fact_6230_union__comp__emptyR,axiom,
! [A: $tType,A6: set @ ( product_prod @ A @ A ),B7: set @ ( product_prod @ A @ A ),C6: set @ ( product_prod @ A @ A )] :
( ( ( relcomp @ A @ A @ A @ A6 @ B7 )
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
=> ( ( ( relcomp @ A @ A @ A @ A6 @ C6 )
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
=> ( ( relcomp @ A @ A @ A @ A6 @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ B7 @ C6 ) )
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ) ).
% union_comp_emptyR
thf(fact_6231_relpow_Osimps_I2_J,axiom,
! [A: $tType,N: nat,R2: set @ ( product_prod @ A @ A )] :
( ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( suc @ N ) @ R2 )
= ( relcomp @ A @ A @ A @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R2 ) @ R2 ) ) ).
% relpow.simps(2)
thf(fact_6232_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_6233_max__ext__compat,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),S3: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( relcomp @ A @ A @ A @ R2 @ S3 ) @ R2 )
=> ( ord_less_eq @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( relcomp @ ( set @ A ) @ ( set @ A ) @ ( set @ A ) @ ( max_ext @ A @ R2 ) @ ( sup_sup @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( max_ext @ A @ S3 ) @ ( insert @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ A ) ) ) @ ( bot_bot @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) ) ) ) @ ( max_ext @ A @ R2 ) ) ) ).
% max_ext_compat
thf(fact_6234_min__ext__compat,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),S3: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( relcomp @ A @ A @ A @ R2 @ S3 ) @ R2 )
=> ( ord_less_eq @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( relcomp @ ( set @ A ) @ ( set @ A ) @ ( set @ A ) @ ( min_ext @ A @ R2 ) @ ( sup_sup @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( min_ext @ A @ S3 ) @ ( insert @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ A ) ) ) @ ( bot_bot @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) ) ) ) @ ( min_ext @ A @ R2 ) ) ) ).
% min_ext_compat
thf(fact_6235_ntrancl__Suc,axiom,
! [A: $tType,N: nat,R2: set @ ( product_prod @ A @ A )] :
( ( transitive_ntrancl @ A @ ( suc @ N ) @ R2 )
= ( relcomp @ A @ A @ A @ ( transitive_ntrancl @ A @ N @ R2 ) @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ ( id2 @ A ) @ R2 ) ) ) ).
% ntrancl_Suc
thf(fact_6236_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_6237_monoid_Oright__neutral,axiom,
! [A: $tType,F: A > A > A,Z3: A,A3: A] :
( ( monoid @ A @ F @ Z3 )
=> ( ( F @ A3 @ Z3 )
= A3 ) ) ).
% monoid.right_neutral
thf(fact_6238_monoid_Oleft__neutral,axiom,
! [A: $tType,F: A > A > A,Z3: A,A3: A] :
( ( monoid @ A @ F @ Z3 )
=> ( ( F @ Z3 @ A3 )
= A3 ) ) ).
% monoid.left_neutral
thf(fact_6239_relpow_Osimps_I1_J,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( zero_zero @ nat ) @ R2 )
= ( id2 @ A ) ) ).
% relpow.simps(1)
thf(fact_6240_add_Omonoid__axioms,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ( monoid @ A @ ( plus_plus @ A ) @ ( zero_zero @ A ) ) ) ).
% add.monoid_axioms
thf(fact_6241_mult_Omonoid__axioms,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ( monoid @ A @ ( times_times @ A ) @ ( one_one @ A ) ) ) ).
% mult.monoid_axioms
thf(fact_6242_sup__bot_Omonoid__axioms,axiom,
! [A: $tType] :
( ( bounde4967611905675639751up_bot @ A )
=> ( monoid @ A @ ( sup_sup @ A ) @ ( bot_bot @ A ) ) ) ).
% sup_bot.monoid_axioms
thf(fact_6243_inf__top_Omonoid__axioms,axiom,
! [A: $tType] :
( ( bounde4346867609351753570nf_top @ A )
=> ( monoid @ A @ ( inf_inf @ A ) @ ( top_top @ A ) ) ) ).
% inf_top.monoid_axioms
thf(fact_6244_gcd__nat_Omonoid__axioms,axiom,
monoid @ nat @ ( gcd_gcd @ nat ) @ ( zero_zero @ nat ) ).
% gcd_nat.monoid_axioms
thf(fact_6245_or_Omonoid__axioms,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( monoid @ A @ ( bit_se1065995026697491101ons_or @ A ) @ ( zero_zero @ A ) ) ) ).
% or.monoid_axioms
thf(fact_6246_xor_Omonoid__axioms,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( monoid @ A @ ( bit_se5824344971392196577ns_xor @ A ) @ ( zero_zero @ A ) ) ) ).
% xor.monoid_axioms
thf(fact_6247_max__nat_Omonoid__axioms,axiom,
monoid @ nat @ ( ord_max @ nat ) @ ( zero_zero @ nat ) ).
% max_nat.monoid_axioms
thf(fact_6248_pair__leq__def,axiom,
( fun_pair_leq
= ( sup_sup @ ( set @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) ) @ fun_pair_less @ ( id2 @ ( product_prod @ nat @ nat ) ) ) ) ).
% pair_leq_def
thf(fact_6249_cr__integer__def,axiom,
( code_cr_integer
= ( ^ [X2: int,Y6: code_integer] :
( X2
= ( code_int_of_integer @ Y6 ) ) ) ) ).
% cr_integer_def
thf(fact_6250_bit__integer__def,axiom,
( ( bit_se5641148757651400278ts_bit @ code_integer )
= ( map_fun @ code_integer @ int @ ( nat > $o ) @ ( nat > $o ) @ code_int_of_integer @ ( id @ ( nat > $o ) ) @ ( bit_se5641148757651400278ts_bit @ int ) ) ) ).
% bit_integer_def
thf(fact_6251_integer_Opcr__cr__eq,axiom,
code_pcr_integer = code_cr_integer ).
% integer.pcr_cr_eq
thf(fact_6252_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_6253_iterates_OSup,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [M7: set @ A,F: A > A] :
( ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ M7 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ M7 )
=> ( member @ A @ X3 @ ( comple6359979572994053840erates @ A @ F ) ) )
=> ( member @ A @ ( complete_Sup_Sup @ A @ M7 ) @ ( comple6359979572994053840erates @ A @ F ) ) ) ) ) ).
% iterates.Sup
thf(fact_6254_iterates_Osimps,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [A3: A,F: A > A] :
( ( member @ A @ A3 @ ( comple6359979572994053840erates @ A @ F ) )
= ( ? [X2: A] :
( ( A3
= ( F @ X2 ) )
& ( member @ A @ X2 @ ( comple6359979572994053840erates @ A @ F ) ) )
| ? [M9: set @ A] :
( ( A3
= ( complete_Sup_Sup @ A @ M9 ) )
& ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ M9 )
& ! [X2: A] :
( ( member @ A @ X2 @ M9 )
=> ( member @ A @ X2 @ ( comple6359979572994053840erates @ A @ F ) ) ) ) ) ) ) ).
% iterates.simps
thf(fact_6255_iterates_Ocases,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [A3: A,F: A > A] :
( ( member @ A @ A3 @ ( comple6359979572994053840erates @ A @ F ) )
=> ( ! [X3: A] :
( ( A3
= ( F @ X3 ) )
=> ~ ( member @ A @ X3 @ ( comple6359979572994053840erates @ A @ F ) ) )
=> ~ ! [M8: set @ A] :
( ( A3
= ( complete_Sup_Sup @ A @ M8 ) )
=> ( ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ M8 )
=> ~ ! [X4: A] :
( ( member @ A @ X4 @ M8 )
=> ( member @ A @ X4 @ ( comple6359979572994053840erates @ A @ F ) ) ) ) ) ) ) ) ).
% iterates.cases
thf(fact_6256_arg__min__list_Oelims,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [X: A > B,Xa3: list @ A,Y3: A] :
( ( ( arg_min_list @ A @ B @ X @ Xa3 )
= Y3 )
=> ( ! [X3: A] :
( ( Xa3
= ( cons @ A @ X3 @ ( nil @ A ) ) )
=> ( Y3 != X3 ) )
=> ( ! [X3: A,Y5: A,Zs2: list @ A] :
( ( Xa3
= ( cons @ A @ X3 @ ( cons @ A @ Y5 @ Zs2 ) ) )
=> ( Y3
!= ( if @ A @ ( ord_less_eq @ B @ ( X @ X3 ) @ ( X @ ( arg_min_list @ A @ B @ X @ ( cons @ A @ Y5 @ Zs2 ) ) ) ) @ X3 @ ( arg_min_list @ A @ B @ X @ ( cons @ A @ Y5 @ Zs2 ) ) ) ) )
=> ~ ( ( Xa3
= ( nil @ A ) )
=> ( Y3
!= ( undefined @ A ) ) ) ) ) ) ) ).
% arg_min_list.elims
thf(fact_6257_chain__iterates,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [F: A > A] :
( ( comple7038119648293358887notone @ A @ A @ ( ord_less_eq @ A ) @ ( ord_less_eq @ A ) @ F )
=> ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ ( comple6359979572994053840erates @ A @ F ) ) ) ) ).
% chain_iterates
thf(fact_6258_iterates__le__f,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [X: A,F: A > A] :
( ( member @ A @ X @ ( comple6359979572994053840erates @ A @ F ) )
=> ( ( comple7038119648293358887notone @ A @ A @ ( ord_less_eq @ A ) @ ( ord_less_eq @ A ) @ F )
=> ( ord_less_eq @ A @ X @ ( F @ X ) ) ) ) ) ).
% iterates_le_f
thf(fact_6259_fixp__induct,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [P: A > $o,F: A > A] :
( ( comple1908693960933563346ssible @ A @ ( complete_Sup_Sup @ A ) @ ( ord_less_eq @ A ) @ P )
=> ( ( comple7038119648293358887notone @ A @ A @ ( ord_less_eq @ A ) @ ( ord_less_eq @ A ) @ F )
=> ( ( P @ ( complete_Sup_Sup @ A @ ( bot_bot @ ( set @ A ) ) ) )
=> ( ! [X3: A] :
( ( P @ X3 )
=> ( P @ ( F @ X3 ) ) )
=> ( P @ ( comple115746919287870866o_fixp @ A @ F ) ) ) ) ) ) ) ).
% fixp_induct
thf(fact_6260_iterates__fixp,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [F: A > A] :
( ( comple7038119648293358887notone @ A @ A @ ( ord_less_eq @ A ) @ ( ord_less_eq @ A ) @ F )
=> ( member @ A @ ( comple115746919287870866o_fixp @ A @ F ) @ ( comple6359979572994053840erates @ A @ F ) ) ) ) ).
% iterates_fixp
thf(fact_6261_fixp__unfold,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [F: A > A] :
( ( comple7038119648293358887notone @ A @ A @ ( ord_less_eq @ A ) @ ( ord_less_eq @ A ) @ F )
=> ( ( comple115746919287870866o_fixp @ A @ F )
= ( F @ ( comple115746919287870866o_fixp @ A @ F ) ) ) ) ) ).
% fixp_unfold
thf(fact_6262_fixp__lowerbound,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [F: A > A,Z3: A] :
( ( comple7038119648293358887notone @ A @ A @ ( ord_less_eq @ A ) @ ( ord_less_eq @ A ) @ F )
=> ( ( ord_less_eq @ A @ ( F @ Z3 ) @ Z3 )
=> ( ord_less_eq @ A @ ( comple115746919287870866o_fixp @ A @ F ) @ Z3 ) ) ) ) ).
% fixp_lowerbound
thf(fact_6263_arg__min__list_Opelims,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [X: A > B,Xa3: list @ A,Y3: A] :
( ( ( arg_min_list @ A @ B @ X @ Xa3 )
= Y3 )
=> ( ( accp @ ( product_prod @ ( A > B ) @ ( list @ A ) ) @ ( arg_min_list_rel @ A @ B ) @ ( product_Pair @ ( A > B ) @ ( list @ A ) @ X @ Xa3 ) )
=> ( ! [X3: A] :
( ( Xa3
= ( cons @ A @ X3 @ ( nil @ A ) ) )
=> ( ( Y3 = X3 )
=> ~ ( accp @ ( product_prod @ ( A > B ) @ ( list @ A ) ) @ ( arg_min_list_rel @ A @ B ) @ ( product_Pair @ ( A > B ) @ ( list @ A ) @ X @ ( cons @ A @ X3 @ ( nil @ A ) ) ) ) ) )
=> ( ! [X3: A,Y5: A,Zs2: list @ A] :
( ( Xa3
= ( cons @ A @ X3 @ ( cons @ A @ Y5 @ Zs2 ) ) )
=> ( ( Y3
= ( if @ A @ ( ord_less_eq @ B @ ( X @ X3 ) @ ( X @ ( arg_min_list @ A @ B @ X @ ( cons @ A @ Y5 @ Zs2 ) ) ) ) @ X3 @ ( arg_min_list @ A @ B @ X @ ( cons @ A @ Y5 @ Zs2 ) ) ) )
=> ~ ( accp @ ( product_prod @ ( A > B ) @ ( list @ A ) ) @ ( arg_min_list_rel @ A @ B ) @ ( product_Pair @ ( A > B ) @ ( list @ A ) @ X @ ( cons @ A @ X3 @ ( cons @ A @ Y5 @ Zs2 ) ) ) ) ) )
=> ~ ( ( Xa3
= ( 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_6264_sorted__sort__key,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F: B > A,Xs: list @ B] : ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F @ ( linorder_sort_key @ B @ A @ F @ Xs ) ) ) ) ).
% sorted_sort_key
thf(fact_6265_sorted__sort,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( sorted_wrt @ A @ ( ord_less_eq @ A )
@ ( linorder_sort_key @ A @ A
@ ^ [X2: A] : X2
@ Xs ) ) ) ).
% sorted_sort
thf(fact_6266_sorted__sort__id,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( ( linorder_sort_key @ A @ A
@ ^ [X2: A] : X2
@ Xs )
= Xs ) ) ) ).
% sorted_sort_id
thf(fact_6267_length__removeAll__less,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ord_less @ nat @ ( size_size @ ( list @ A ) @ ( removeAll @ A @ X @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ).
% length_removeAll_less
thf(fact_6268_card__Plus__conv__if,axiom,
! [B: $tType,A: $tType,A6: set @ A,B7: set @ B] :
( ( ( ( finite_finite2 @ A @ A6 )
& ( finite_finite2 @ B @ B7 ) )
=> ( ( finite_card @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ A6 @ B7 ) )
= ( plus_plus @ nat @ ( finite_card @ A @ A6 ) @ ( finite_card @ B @ B7 ) ) ) )
& ( ~ ( ( finite_finite2 @ A @ A6 )
& ( finite_finite2 @ B @ B7 ) )
=> ( ( finite_card @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ A6 @ B7 ) )
= ( zero_zero @ nat ) ) ) ) ).
% card_Plus_conv_if
thf(fact_6269_length__removeAll__less__eq,axiom,
! [A: $tType,X: A,Xs: list @ A] : ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ ( removeAll @ A @ X @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ).
% length_removeAll_less_eq
thf(fact_6270_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_6271_lcm__altdef__int,axiom,
( ( gcd_lcm @ int )
= ( ^ [A5: int,B3: int] : ( divide_divide @ int @ ( times_times @ int @ ( abs_abs @ int @ A5 ) @ ( abs_abs @ int @ B3 ) ) @ ( gcd_gcd @ int @ A5 @ B3 ) ) ) ) ).
% lcm_altdef_int
thf(fact_6272_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_6273_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_6274_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_6275_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_6276_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_6277_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_6278_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_6279_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_6280_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_6281_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_6282_semilattice__neutr_Oaxioms_I2_J,axiom,
! [A: $tType,F: A > A > A,Z3: A] :
( ( semilattice_neutr @ A @ F @ Z3 )
=> ( comm_monoid @ A @ F @ Z3 ) ) ).
% semilattice_neutr.axioms(2)
thf(fact_6283_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_6284_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_6285_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_6286_comm__monoid_Ocomm__neutral,axiom,
! [A: $tType,F: A > A > A,Z3: A,A3: A] :
( ( comm_monoid @ A @ F @ Z3 )
=> ( ( F @ A3 @ Z3 )
= A3 ) ) ).
% comm_monoid.comm_neutral
thf(fact_6287_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_6288_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_6289_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_6290_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_6291_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_6292_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_6293_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_6294_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_6295_sup__bot_Ocomm__monoid__axioms,axiom,
! [A: $tType] :
( ( bounde4967611905675639751up_bot @ A )
=> ( comm_monoid @ A @ ( sup_sup @ A ) @ ( bot_bot @ A ) ) ) ).
% sup_bot.comm_monoid_axioms
thf(fact_6296_inf__top_Ocomm__monoid__axioms,axiom,
! [A: $tType] :
( ( bounde4346867609351753570nf_top @ A )
=> ( comm_monoid @ A @ ( inf_inf @ A ) @ ( top_top @ A ) ) ) ).
% inf_top.comm_monoid_axioms
thf(fact_6297_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_6298_gcd__nat_Ocomm__monoid__axioms,axiom,
comm_monoid @ nat @ ( gcd_gcd @ nat ) @ ( zero_zero @ nat ) ).
% gcd_nat.comm_monoid_axioms
thf(fact_6299_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_6300_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_6301_max__nat_Ocomm__monoid__axioms,axiom,
comm_monoid @ nat @ ( ord_max @ nat ) @ ( zero_zero @ nat ) ).
% max_nat.comm_monoid_axioms
thf(fact_6302_lcm__unique__int,axiom,
! [D2: int,A3: int,B2: int] :
( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ D2 )
& ( dvd_dvd @ int @ A3 @ D2 )
& ( dvd_dvd @ int @ B2 @ D2 )
& ! [E3: int] :
( ( ( dvd_dvd @ int @ A3 @ E3 )
& ( dvd_dvd @ int @ B2 @ E3 ) )
=> ( dvd_dvd @ int @ D2 @ E3 ) ) )
= ( D2
= ( gcd_lcm @ int @ A3 @ B2 ) ) ) ).
% lcm_unique_int
thf(fact_6303_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_6304_Lcm__int__set__eq__fold,axiom,
! [Xs: list @ int] :
( ( gcd_Lcm @ int @ ( set2 @ int @ Xs ) )
= ( fold @ int @ int @ ( gcd_lcm @ int ) @ Xs @ ( one_one @ int ) ) ) ).
% Lcm_int_set_eq_fold
thf(fact_6305_Lcm__fin_Oeq__fold,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ( ( semiring_gcd_Lcm_fin @ A )
= ( ^ [A7: set @ A] : ( if @ A @ ( finite_finite2 @ A @ A7 ) @ ( finite_fold @ A @ A @ ( gcd_lcm @ A ) @ ( one_one @ A ) @ A7 ) @ ( zero_zero @ A ) ) ) ) ) ).
% Lcm_fin.eq_fold
thf(fact_6306_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_6307_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_6308_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_6309_Lcm__UNIV,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ( ( gcd_Lcm @ A @ ( top_top @ ( set @ A ) ) )
= ( zero_zero @ A ) ) ) ).
% Lcm_UNIV
thf(fact_6310_Lcm__empty,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ( ( gcd_Lcm @ A @ ( bot_bot @ ( set @ A ) ) )
= ( one_one @ A ) ) ) ).
% Lcm_empty
thf(fact_6311_Lcm__1__iff,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A6: set @ A] :
( ( ( gcd_Lcm @ A @ A6 )
= ( one_one @ A ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ A6 )
=> ( dvd_dvd @ A @ X2 @ ( one_one @ A ) ) ) ) ) ) ).
% Lcm_1_iff
thf(fact_6312_Lcm__fin_Oinfinite,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A6: set @ A] :
( ~ ( finite_finite2 @ A @ A6 )
=> ( ( semiring_gcd_Lcm_fin @ A @ A6 )
= ( zero_zero @ A ) ) ) ) ).
% Lcm_fin.infinite
thf(fact_6313_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_6314_is__unit__Lcm__fin__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A6: set @ A] :
( ( dvd_dvd @ A @ ( semiring_gcd_Lcm_fin @ A @ A6 ) @ ( one_one @ A ) )
= ( ( semiring_gcd_Lcm_fin @ A @ A6 )
= ( one_one @ A ) ) ) ) ).
% is_unit_Lcm_fin_iff
thf(fact_6315_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_6316_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_6317_prod__gcd__lcm__nat,axiom,
( ( times_times @ nat )
= ( ^ [M2: nat,N3: nat] : ( times_times @ nat @ ( gcd_gcd @ nat @ M2 @ N3 ) @ ( gcd_lcm @ nat @ M2 @ N3 ) ) ) ) ).
% prod_gcd_lcm_nat
thf(fact_6318_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_6319_Lcm__no__multiple,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A6: set @ A] :
( ! [M6: A] :
( ( M6
!= ( zero_zero @ A ) )
=> ? [X4: A] :
( ( member @ A @ X4 @ A6 )
& ~ ( dvd_dvd @ A @ X4 @ M6 ) ) )
=> ( ( gcd_Lcm @ A @ A6 )
= ( zero_zero @ A ) ) ) ) ).
% Lcm_no_multiple
thf(fact_6320_Lcm__0__iff_H,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A6: set @ A] :
( ( ( gcd_Lcm @ A @ A6 )
= ( zero_zero @ A ) )
= ( ~ ? [L3: A] :
( ( L3
!= ( zero_zero @ A ) )
& ! [X2: A] :
( ( member @ A @ X2 @ A6 )
=> ( dvd_dvd @ A @ X2 @ L3 ) ) ) ) ) ) ).
% Lcm_0_iff'
thf(fact_6321_Lcm__eq__0__I,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A6: set @ A] :
( ( member @ A @ ( zero_zero @ A ) @ A6 )
=> ( ( gcd_Lcm @ A @ A6 )
= ( zero_zero @ A ) ) ) ) ).
% Lcm_eq_0_I
thf(fact_6322_Lcm__0__iff,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A6: set @ A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( ( gcd_Lcm @ A @ A6 )
= ( zero_zero @ A ) )
= ( member @ A @ ( zero_zero @ A ) @ A6 ) ) ) ) ).
% Lcm_0_iff
thf(fact_6323_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_6324_Lcm__fin__0__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A6: set @ A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( ( semiring_gcd_Lcm_fin @ A @ A6 )
= ( zero_zero @ A ) )
= ( member @ A @ ( zero_zero @ A ) @ A6 ) ) ) ) ).
% Lcm_fin_0_iff
thf(fact_6325_lcm__nat__def,axiom,
( ( gcd_lcm @ nat )
= ( ^ [X2: nat,Y6: nat] : ( divide_divide @ nat @ ( times_times @ nat @ X2 @ Y6 ) @ ( gcd_gcd @ nat @ X2 @ Y6 ) ) ) ) ).
% lcm_nat_def
thf(fact_6326_Lcm__fin__1__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A6: set @ A] :
( ( ( semiring_gcd_Lcm_fin @ A @ A6 )
= ( one_one @ A ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ A6 )
=> ( dvd_dvd @ A @ X2 @ ( one_one @ A ) ) )
& ( finite_finite2 @ A @ A6 ) ) ) ) ).
% Lcm_fin_1_iff
thf(fact_6327_Lcm__no__units,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ( ( gcd_Lcm @ A )
= ( ^ [A7: set @ A] :
( gcd_Lcm @ A
@ ( minus_minus @ ( set @ A ) @ A7
@ ( collect @ A
@ ^ [A5: A] : ( dvd_dvd @ A @ A5 @ ( one_one @ A ) ) ) ) ) ) ) ) ).
% Lcm_no_units
thf(fact_6328_lcm__int__def,axiom,
( ( gcd_lcm @ int )
= ( ^ [X2: int,Y6: int] : ( semiring_1_of_nat @ int @ ( gcd_lcm @ nat @ ( nat2 @ ( abs_abs @ int @ X2 ) ) @ ( nat2 @ ( abs_abs @ int @ Y6 ) ) ) ) ) ) ).
% lcm_int_def
thf(fact_6329_Lcm__set__eq__fold,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [Xs: list @ A] :
( ( gcd_Lcm @ A @ ( set2 @ A @ Xs ) )
= ( fold @ A @ A @ ( gcd_lcm @ A ) @ Xs @ ( one_one @ A ) ) ) ) ).
% Lcm_set_eq_fold
thf(fact_6330_Lcm__fin_Oset__eq__fold,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [Xs: list @ A] :
( ( semiring_gcd_Lcm_fin @ A @ ( set2 @ A @ Xs ) )
= ( fold @ A @ A @ ( gcd_lcm @ A ) @ Xs @ ( one_one @ A ) ) ) ) ).
% Lcm_fin.set_eq_fold
thf(fact_6331_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_6332_Lcm__int__def,axiom,
( ( gcd_Lcm @ int )
= ( ^ [K6: set @ int] : ( semiring_1_of_nat @ int @ ( gcd_Lcm @ nat @ ( image @ int @ nat @ ( comp @ int @ nat @ int @ nat2 @ ( abs_abs @ int ) ) @ K6 ) ) ) ) ) ).
% Lcm_int_def
thf(fact_6333_Lcm__int__eq,axiom,
! [N6: set @ nat] :
( ( gcd_Lcm @ int @ ( image @ nat @ int @ ( semiring_1_of_nat @ int ) @ N6 ) )
= ( semiring_1_of_nat @ int @ ( gcd_Lcm @ nat @ N6 ) ) ) ).
% Lcm_int_eq
thf(fact_6334_Lcm__eq__0__I__nat,axiom,
! [A6: set @ nat] :
( ( member @ nat @ ( zero_zero @ nat ) @ A6 )
=> ( ( gcd_Lcm @ nat @ A6 )
= ( zero_zero @ nat ) ) ) ).
% Lcm_eq_0_I_nat
thf(fact_6335_Lcm__0__iff__nat,axiom,
! [A6: set @ nat] :
( ( finite_finite2 @ nat @ A6 )
=> ( ( ( gcd_Lcm @ nat @ A6 )
= ( zero_zero @ nat ) )
= ( member @ nat @ ( zero_zero @ nat ) @ A6 ) ) ) ).
% Lcm_0_iff_nat
thf(fact_6336_Lcm__nat__infinite,axiom,
! [M7: set @ nat] :
( ~ ( finite_finite2 @ nat @ M7 )
=> ( ( gcd_Lcm @ nat @ M7 )
= ( zero_zero @ nat ) ) ) ).
% Lcm_nat_infinite
thf(fact_6337_Lcm__nat__empty,axiom,
( ( gcd_Lcm @ nat @ ( bot_bot @ ( set @ nat ) ) )
= ( one_one @ nat ) ) ).
% Lcm_nat_empty
thf(fact_6338_Lcm__nat__set__eq__fold,axiom,
! [Xs: list @ nat] :
( ( gcd_Lcm @ nat @ ( set2 @ nat @ Xs ) )
= ( fold @ nat @ nat @ ( gcd_lcm @ nat ) @ Xs @ ( one_one @ nat ) ) ) ).
% Lcm_nat_set_eq_fold
thf(fact_6339_Lcm__eq__Max__nat,axiom,
! [M7: set @ nat] :
( ( finite_finite2 @ nat @ M7 )
=> ( ( M7
!= ( bot_bot @ ( set @ nat ) ) )
=> ( ~ ( member @ nat @ ( zero_zero @ nat ) @ M7 )
=> ( ! [M6: nat,N4: nat] :
( ( member @ nat @ M6 @ M7 )
=> ( ( member @ nat @ N4 @ M7 )
=> ( member @ nat @ ( gcd_lcm @ nat @ M6 @ N4 ) @ M7 ) ) )
=> ( ( gcd_Lcm @ nat @ M7 )
= ( lattic643756798349783984er_Max @ nat @ M7 ) ) ) ) ) ) ).
% Lcm_eq_Max_nat
thf(fact_6340_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_6341_Lcm__coprime_H,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A6: set @ A] :
( ( ( finite_card @ A @ A6 )
!= ( zero_zero @ nat ) )
=> ( ! [A8: A,B5: A] :
( ( member @ A @ A8 @ A6 )
=> ( ( member @ A @ B5 @ A6 )
=> ( ( A8 != B5 )
=> ( algebr8660921524188924756oprime @ A @ A8 @ B5 ) ) ) )
=> ( ( gcd_Lcm @ A @ A6 )
= ( normal6383669964737779283malize @ A
@ ( groups7121269368397514597t_prod @ A @ A
@ ^ [X2: A] : X2
@ A6 ) ) ) ) ) ) ).
% Lcm_coprime'
thf(fact_6342_normalize__idem,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A] :
( ( normal6383669964737779283malize @ A @ ( normal6383669964737779283malize @ A @ A3 ) )
= ( normal6383669964737779283malize @ A @ A3 ) ) ) ).
% normalize_idem
thf(fact_6343_lcm_Onormalize__bottom,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ( ( normal6383669964737779283malize @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% lcm.normalize_bottom
thf(fact_6344_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_6345_normalize__0,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ( ( normal6383669964737779283malize @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% normalize_0
thf(fact_6346_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_6347_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_6348_gcd_Onormalize__bottom,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ( ( normal6383669964737779283malize @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% gcd.normalize_bottom
thf(fact_6349_normalize__1,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ( ( normal6383669964737779283malize @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% normalize_1
thf(fact_6350_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_6351_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_6352_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_6353_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_6354_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_6355_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_6356_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_6357_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_6358_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_6359_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_6360_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_6361_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_6362_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_6363_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_6364_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_6365_normalize__nat__def,axiom,
( ( normal6383669964737779283malize @ nat )
= ( id @ nat ) ) ).
% normalize_nat_def
thf(fact_6366_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_6367_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_6368_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_6369_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_6370_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_6371_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_6372_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_6373_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_6374_normalize__int__def,axiom,
( ( normal6383669964737779283malize @ int )
= ( abs_abs @ int ) ) ).
% normalize_int_def
thf(fact_6375_semilattice__neutr__set_OF_Ocong,axiom,
! [A: $tType] :
( ( lattic5214292709420241887eutr_F @ A )
= ( lattic5214292709420241887eutr_F @ A ) ) ).
% semilattice_neutr_set.F.cong
thf(fact_6376_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_6377_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_6378_associatedD2,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A,B2: A] :
( ( ( normal6383669964737779283malize @ A @ A3 )
= ( normal6383669964737779283malize @ A @ B2 ) )
=> ( dvd_dvd @ A @ B2 @ A3 ) ) ) ).
% associatedD2
thf(fact_6379_associatedD1,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A,B2: A] :
( ( ( normal6383669964737779283malize @ A @ A3 )
= ( normal6383669964737779283malize @ A @ B2 ) )
=> ( dvd_dvd @ A @ A3 @ B2 ) ) ) ).
% associatedD1
thf(fact_6380_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_6381_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_6382_coprime__crossproduct,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,D2: A,B2: A,C2: A] :
( ( algebr8660921524188924756oprime @ A @ A3 @ D2 )
=> ( ( algebr8660921524188924756oprime @ A @ B2 @ C2 )
=> ( ( ( times_times @ A @ ( normal6383669964737779283malize @ A @ A3 ) @ ( normal6383669964737779283malize @ A @ C2 ) )
= ( times_times @ A @ ( normal6383669964737779283malize @ A @ B2 ) @ ( normal6383669964737779283malize @ A @ D2 ) ) )
= ( ( ( normal6383669964737779283malize @ A @ A3 )
= ( normal6383669964737779283malize @ A @ B2 ) )
& ( ( normal6383669964737779283malize @ A @ C2 )
= ( normal6383669964737779283malize @ A @ D2 ) ) ) ) ) ) ) ).
% coprime_crossproduct
thf(fact_6383_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_6384_Gcd__mult,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [C2: A,A6: set @ A] :
( ( gcd_Gcd @ A @ ( image @ A @ A @ ( times_times @ A @ C2 ) @ A6 ) )
= ( normal6383669964737779283malize @ A @ ( times_times @ A @ C2 @ ( gcd_Gcd @ A @ A6 ) ) ) ) ) ).
% Gcd_mult
thf(fact_6385_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_6386_Lcm__mult,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A6: set @ A,C2: A] :
( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( gcd_Lcm @ A @ ( image @ A @ A @ ( times_times @ A @ C2 ) @ A6 ) )
= ( normal6383669964737779283malize @ A @ ( times_times @ A @ C2 @ ( gcd_Lcm @ A @ A6 ) ) ) ) ) ) ).
% Lcm_mult
thf(fact_6387_lcm__gcd,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ( ( gcd_lcm @ A )
= ( ^ [A5: A,B3: A] : ( normal6383669964737779283malize @ A @ ( divide_divide @ A @ ( times_times @ A @ A5 @ B3 ) @ ( gcd_gcd @ A @ A5 @ B3 ) ) ) ) ) ) ).
% lcm_gcd
thf(fact_6388_Lcm__fin__mult,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A6: set @ A,B2: A] :
( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( semiring_gcd_Lcm_fin @ A @ ( image @ A @ A @ ( times_times @ A @ B2 ) @ A6 ) )
= ( normal6383669964737779283malize @ A @ ( times_times @ A @ B2 @ ( semiring_gcd_Lcm_fin @ A @ A6 ) ) ) ) ) ) ).
% Lcm_fin_mult
thf(fact_6389_Gcd__fin__mult,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A6: set @ A,B2: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( semiring_gcd_Gcd_fin @ A @ ( image @ A @ A @ ( times_times @ A @ B2 ) @ A6 ) )
= ( normal6383669964737779283malize @ A @ ( times_times @ A @ B2 @ ( semiring_gcd_Gcd_fin @ A @ A6 ) ) ) ) ) ) ).
% Gcd_fin_mult
thf(fact_6390_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_6391_semilattice__neutr__set_Oinsert__remove,axiom,
! [A: $tType,F: A > A > A,Z3: A,A6: set @ A,X: A] :
( ( lattic5652469242046573047tr_set @ A @ F @ Z3 )
=> ( ( finite_finite2 @ A @ A6 )
=> ( ( lattic5214292709420241887eutr_F @ A @ F @ Z3 @ ( insert @ A @ X @ A6 ) )
= ( F @ X @ ( lattic5214292709420241887eutr_F @ A @ F @ Z3 @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% semilattice_neutr_set.insert_remove
thf(fact_6392_semilattice__neutr__set_Oremove,axiom,
! [A: $tType,F: A > A > A,Z3: A,A6: set @ A,X: A] :
( ( lattic5652469242046573047tr_set @ A @ F @ Z3 )
=> ( ( finite_finite2 @ A @ A6 )
=> ( ( member @ A @ X @ A6 )
=> ( ( lattic5214292709420241887eutr_F @ A @ F @ Z3 @ A6 )
= ( F @ X @ ( lattic5214292709420241887eutr_F @ A @ F @ Z3 @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ).
% semilattice_neutr_set.remove
thf(fact_6393_semilattice__neutr__set_Ointro,axiom,
! [A: $tType,F: A > A > A,Z3: A] :
( ( semilattice_neutr @ A @ F @ Z3 )
=> ( lattic5652469242046573047tr_set @ A @ F @ Z3 ) ) ).
% semilattice_neutr_set.intro
thf(fact_6394_semilattice__neutr__set_Oaxioms,axiom,
! [A: $tType,F: A > A > A,Z3: A] :
( ( lattic5652469242046573047tr_set @ A @ F @ Z3 )
=> ( semilattice_neutr @ A @ F @ Z3 ) ) ).
% semilattice_neutr_set.axioms
thf(fact_6395_semilattice__neutr__set__def,axiom,
! [A: $tType] :
( ( lattic5652469242046573047tr_set @ A )
= ( semilattice_neutr @ A ) ) ).
% semilattice_neutr_set_def
thf(fact_6396_semilattice__neutr__set_Oempty,axiom,
! [A: $tType,F: A > A > A,Z3: A] :
( ( lattic5652469242046573047tr_set @ A @ F @ Z3 )
=> ( ( lattic5214292709420241887eutr_F @ A @ F @ Z3 @ ( bot_bot @ ( set @ A ) ) )
= Z3 ) ) ).
% semilattice_neutr_set.empty
thf(fact_6397_semilattice__neutr__set_Oin__idem,axiom,
! [A: $tType,F: A > A > A,Z3: A,A6: set @ A,X: A] :
( ( lattic5652469242046573047tr_set @ A @ F @ Z3 )
=> ( ( finite_finite2 @ A @ A6 )
=> ( ( member @ A @ X @ A6 )
=> ( ( F @ X @ ( lattic5214292709420241887eutr_F @ A @ F @ Z3 @ A6 ) )
= ( lattic5214292709420241887eutr_F @ A @ F @ Z3 @ A6 ) ) ) ) ) ).
% semilattice_neutr_set.in_idem
thf(fact_6398_semilattice__neutr__set_Oinfinite,axiom,
! [A: $tType,F: A > A > A,Z3: A,A6: set @ A] :
( ( lattic5652469242046573047tr_set @ A @ F @ Z3 )
=> ( ~ ( finite_finite2 @ A @ A6 )
=> ( ( lattic5214292709420241887eutr_F @ A @ F @ Z3 @ A6 )
= Z3 ) ) ) ).
% semilattice_neutr_set.infinite
thf(fact_6399_semilattice__neutr__set_Oeq__fold,axiom,
! [A: $tType,F: A > A > A,Z3: A,A6: set @ A] :
( ( lattic5652469242046573047tr_set @ A @ F @ Z3 )
=> ( ( lattic5214292709420241887eutr_F @ A @ F @ Z3 @ A6 )
= ( finite_fold @ A @ A @ F @ Z3 @ A6 ) ) ) ).
% semilattice_neutr_set.eq_fold
thf(fact_6400_semilattice__neutr__set_Osubset,axiom,
! [A: $tType,F: A > A > A,Z3: A,A6: set @ A,B7: set @ A] :
( ( lattic5652469242046573047tr_set @ A @ F @ Z3 )
=> ( ( finite_finite2 @ A @ A6 )
=> ( ( ord_less_eq @ ( set @ A ) @ B7 @ A6 )
=> ( ( F @ ( lattic5214292709420241887eutr_F @ A @ F @ Z3 @ B7 ) @ ( lattic5214292709420241887eutr_F @ A @ F @ Z3 @ A6 ) )
= ( lattic5214292709420241887eutr_F @ A @ F @ Z3 @ A6 ) ) ) ) ) ).
% semilattice_neutr_set.subset
thf(fact_6401_semilattice__neutr__set_Oinsert,axiom,
! [A: $tType,F: A > A > A,Z3: A,A6: set @ A,X: A] :
( ( lattic5652469242046573047tr_set @ A @ F @ Z3 )
=> ( ( finite_finite2 @ A @ A6 )
=> ( ( lattic5214292709420241887eutr_F @ A @ F @ Z3 @ ( insert @ A @ X @ A6 ) )
= ( F @ X @ ( lattic5214292709420241887eutr_F @ A @ F @ Z3 @ A6 ) ) ) ) ) ).
% semilattice_neutr_set.insert
thf(fact_6402_semilattice__neutr__set_Ounion,axiom,
! [A: $tType,F: A > A > A,Z3: A,A6: set @ A,B7: set @ A] :
( ( lattic5652469242046573047tr_set @ A @ F @ Z3 )
=> ( ( finite_finite2 @ A @ A6 )
=> ( ( finite_finite2 @ A @ B7 )
=> ( ( lattic5214292709420241887eutr_F @ A @ F @ Z3 @ ( sup_sup @ ( set @ A ) @ A6 @ B7 ) )
= ( F @ ( lattic5214292709420241887eutr_F @ A @ F @ Z3 @ A6 ) @ ( lattic5214292709420241887eutr_F @ A @ F @ Z3 @ B7 ) ) ) ) ) ) ).
% semilattice_neutr_set.union
thf(fact_6403_semilattice__neutr__set_Oclosed,axiom,
! [A: $tType,F: A > A > A,Z3: A,A6: set @ A] :
( ( lattic5652469242046573047tr_set @ A @ F @ Z3 )
=> ( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A,Y5: A] : ( member @ A @ ( F @ X3 @ Y5 ) @ ( insert @ A @ X3 @ ( insert @ A @ Y5 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( member @ A @ ( lattic5214292709420241887eutr_F @ A @ F @ Z3 @ A6 ) @ A6 ) ) ) ) ) ).
% semilattice_neutr_set.closed
thf(fact_6404_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_6405_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_6406_semilattice__order__neutr__set_Osubset__imp,axiom,
! [A: $tType,F: A > A > A,Z3: A,Less_eq: A > A > $o,Less: A > A > $o,A6: set @ A,B7: set @ A] :
( ( lattic3600114342068043075tr_set @ A @ F @ Z3 @ Less_eq @ Less )
=> ( ( ord_less_eq @ ( set @ A ) @ A6 @ B7 )
=> ( ( finite_finite2 @ A @ B7 )
=> ( Less_eq @ ( lattic5214292709420241887eutr_F @ A @ F @ Z3 @ B7 ) @ ( lattic5214292709420241887eutr_F @ A @ F @ Z3 @ A6 ) ) ) ) ) ).
% semilattice_order_neutr_set.subset_imp
thf(fact_6407_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_6408_semilattice__order__neutr__set_Oaxioms_I2_J,axiom,
! [A: $tType,F: A > A > A,Z3: A,Less_eq: A > A > $o,Less: A > A > $o] :
( ( lattic3600114342068043075tr_set @ A @ F @ Z3 @ Less_eq @ Less )
=> ( lattic5652469242046573047tr_set @ A @ F @ Z3 ) ) ).
% semilattice_order_neutr_set.axioms(2)
thf(fact_6409_semilattice__order__neutr__set_Oaxioms_I1_J,axiom,
! [A: $tType,F: A > A > A,Z3: A,Less_eq: A > A > $o,Less: A > A > $o] :
( ( lattic3600114342068043075tr_set @ A @ F @ Z3 @ Less_eq @ Less )
=> ( semila1105856199041335345_order @ A @ F @ Z3 @ Less_eq @ Less ) ) ).
% semilattice_order_neutr_set.axioms(1)
thf(fact_6410_semilattice__order__neutr__set_OboundedE,axiom,
! [A: $tType,F: A > A > A,Z3: A,Less_eq: A > A > $o,Less: A > A > $o,A6: set @ A,X: A] :
( ( lattic3600114342068043075tr_set @ A @ F @ Z3 @ Less_eq @ Less )
=> ( ( finite_finite2 @ A @ A6 )
=> ( ( Less_eq @ X @ ( lattic5214292709420241887eutr_F @ A @ F @ Z3 @ A6 ) )
=> ! [A10: A] :
( ( member @ A @ A10 @ A6 )
=> ( Less_eq @ X @ A10 ) ) ) ) ) ).
% semilattice_order_neutr_set.boundedE
thf(fact_6411_semilattice__order__neutr__set_OboundedI,axiom,
! [A: $tType,F: A > A > A,Z3: A,Less_eq: A > A > $o,Less: A > A > $o,A6: set @ A,X: A] :
( ( lattic3600114342068043075tr_set @ A @ F @ Z3 @ Less_eq @ Less )
=> ( ( finite_finite2 @ A @ A6 )
=> ( ! [A8: A] :
( ( member @ A @ A8 @ A6 )
=> ( Less_eq @ X @ A8 ) )
=> ( Less_eq @ X @ ( lattic5214292709420241887eutr_F @ A @ F @ Z3 @ A6 ) ) ) ) ) ).
% semilattice_order_neutr_set.boundedI
thf(fact_6412_semilattice__order__neutr__set_OcoboundedI,axiom,
! [A: $tType,F: A > A > A,Z3: A,Less_eq: A > A > $o,Less: A > A > $o,A6: set @ A,A3: A] :
( ( lattic3600114342068043075tr_set @ A @ F @ Z3 @ Less_eq @ Less )
=> ( ( finite_finite2 @ A @ A6 )
=> ( ( member @ A @ A3 @ A6 )
=> ( Less_eq @ ( lattic5214292709420241887eutr_F @ A @ F @ Z3 @ A6 ) @ A3 ) ) ) ) ).
% semilattice_order_neutr_set.coboundedI
thf(fact_6413_semilattice__order__neutr__set_Obounded__iff,axiom,
! [A: $tType,F: A > A > A,Z3: A,Less_eq: A > A > $o,Less: A > A > $o,A6: set @ A,X: A] :
( ( lattic3600114342068043075tr_set @ A @ F @ Z3 @ Less_eq @ Less )
=> ( ( finite_finite2 @ A @ A6 )
=> ( ( Less_eq @ X @ ( lattic5214292709420241887eutr_F @ A @ F @ Z3 @ A6 ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ A6 )
=> ( Less_eq @ X @ X2 ) ) ) ) ) ) ).
% semilattice_order_neutr_set.bounded_iff
thf(fact_6414_semilattice__order__neutr__set__def,axiom,
! [A: $tType] :
( ( lattic3600114342068043075tr_set @ A )
= ( ^ [F2: A > A > A,Z7: A,Less_eq2: A > A > $o,Less2: A > A > $o] :
( ( semila1105856199041335345_order @ A @ F2 @ Z7 @ Less_eq2 @ Less2 )
& ( lattic5652469242046573047tr_set @ A @ F2 @ Z7 ) ) ) ) ).
% semilattice_order_neutr_set_def
thf(fact_6415_semilattice__order__neutr__set_Ointro,axiom,
! [A: $tType,F: A > A > A,Z3: A,Less_eq: A > A > $o,Less: A > A > $o] :
( ( semila1105856199041335345_order @ A @ F @ Z3 @ Less_eq @ Less )
=> ( ( lattic5652469242046573047tr_set @ A @ F @ Z3 )
=> ( lattic3600114342068043075tr_set @ A @ F @ Z3 @ Less_eq @ Less ) ) ) ).
% semilattice_order_neutr_set.intro
thf(fact_6416_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_6417_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_6418_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_6419_unit__factor__simps_I1_J,axiom,
( ( unit_f5069060285200089521factor @ nat @ ( zero_zero @ nat ) )
= ( zero_zero @ nat ) ) ).
% unit_factor_simps(1)
thf(fact_6420_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_6421_unit__factor__simps_I2_J,axiom,
! [N: nat] :
( ( unit_f5069060285200089521factor @ nat @ ( suc @ N ) )
= ( one_one @ nat ) ) ).
% unit_factor_simps(2)
thf(fact_6422_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_6423_unit__factor__0,axiom,
! [A: $tType] :
( ( semido2269285787275462019factor @ A )
=> ( ( unit_f5069060285200089521factor @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% unit_factor_0
thf(fact_6424_unit__factor__1,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ( ( unit_f5069060285200089521factor @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% unit_factor_1
thf(fact_6425_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_6426_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_6427_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_6428_div__normalize,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A] :
( ( divide_divide @ A @ A3 @ ( normal6383669964737779283malize @ A @ A3 ) )
= ( unit_f5069060285200089521factor @ A @ A3 ) ) ) ).
% div_normalize
thf(fact_6429_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_6430_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_6431_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_6432_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_6433_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_6434_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_6435_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_6436_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_6437_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_6438_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_6439_unit__factor__nat__def,axiom,
( ( unit_f5069060285200089521factor @ nat )
= ( ^ [N3: nat] :
( if @ nat
@ ( N3
= ( zero_zero @ nat ) )
@ ( zero_zero @ nat )
@ ( one_one @ nat ) ) ) ) ).
% unit_factor_nat_def
thf(fact_6440_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_6441_unit__factor__self,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A] : ( dvd_dvd @ A @ ( unit_f5069060285200089521factor @ A @ A3 ) @ A3 ) ) ).
% unit_factor_self
thf(fact_6442_unit__factor__int__def,axiom,
( ( unit_f5069060285200089521factor @ int )
= ( sgn_sgn @ int ) ) ).
% unit_factor_int_def
thf(fact_6443_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_6444_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_6445_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_6446_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_6447_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_6448_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_6449_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_6450_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_6451_coprime__crossproduct_H,axiom,
! [A: $tType] :
( ( semiri6843258321239162965malize @ A )
=> ! [B2: A,D2: A,A3: A,C2: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( ( unit_f5069060285200089521factor @ A @ B2 )
= ( unit_f5069060285200089521factor @ A @ D2 ) )
=> ( ( algebr8660921524188924756oprime @ A @ A3 @ B2 )
=> ( ( algebr8660921524188924756oprime @ A @ C2 @ D2 )
=> ( ( ( times_times @ A @ A3 @ D2 )
= ( times_times @ A @ B2 @ C2 ) )
= ( ( A3 = C2 )
& ( B2 = D2 ) ) ) ) ) ) ) ) ).
% coprime_crossproduct'
thf(fact_6452_unit__factor__Lcm,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A6: set @ A] :
( ( ( ( gcd_Lcm @ A @ A6 )
= ( zero_zero @ A ) )
=> ( ( unit_f5069060285200089521factor @ A @ ( gcd_Lcm @ A @ A6 ) )
= ( zero_zero @ A ) ) )
& ( ( ( gcd_Lcm @ A @ A6 )
!= ( zero_zero @ A ) )
=> ( ( unit_f5069060285200089521factor @ A @ ( gcd_Lcm @ A @ A6 ) )
= ( one_one @ A ) ) ) ) ) ).
% unit_factor_Lcm
thf(fact_6453_unit__factor__Gcd,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A6: set @ A] :
( ( ( ( gcd_Gcd @ A @ A6 )
= ( zero_zero @ A ) )
=> ( ( unit_f5069060285200089521factor @ A @ ( gcd_Gcd @ A @ A6 ) )
= ( zero_zero @ A ) ) )
& ( ( ( gcd_Gcd @ A @ A6 )
!= ( zero_zero @ A ) )
=> ( ( unit_f5069060285200089521factor @ A @ ( gcd_Gcd @ A @ A6 ) )
= ( one_one @ A ) ) ) ) ) ).
% unit_factor_Gcd
thf(fact_6454_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_6455_unit__factor__Lcm__fin,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A6: set @ A] :
( ( unit_f5069060285200089521factor @ A @ ( semiring_gcd_Lcm_fin @ A @ A6 ) )
= ( zero_neq_one_of_bool @ A
@ ( ( semiring_gcd_Lcm_fin @ A @ A6 )
!= ( zero_zero @ A ) ) ) ) ) ).
% unit_factor_Lcm_fin
thf(fact_6456_unit__factor__Gcd__fin,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A6: set @ A] :
( ( unit_f5069060285200089521factor @ A @ ( semiring_gcd_Gcd_fin @ A @ A6 ) )
= ( zero_neq_one_of_bool @ A
@ ( ( semiring_gcd_Gcd_fin @ A @ A6 )
!= ( zero_zero @ A ) ) ) ) ) ).
% unit_factor_Gcd_fin
thf(fact_6457_times__num__def,axiom,
( ( times_times @ num )
= ( ^ [M2: num,N3: num] : ( num_of_nat @ ( times_times @ nat @ ( nat_of_num @ M2 ) @ ( nat_of_num @ N3 ) ) ) ) ) ).
% times_num_def
thf(fact_6458_nat__of__num__pos,axiom,
! [X: num] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( nat_of_num @ X ) ) ).
% nat_of_num_pos
thf(fact_6459_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_6460_is__num__normalize_I5_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [X: A] :
( ( neg_numeral_is_num @ A @ X )
=> ( neg_numeral_is_num @ A @ ( uminus_uminus @ A @ X ) ) ) ) ).
% is_num_normalize(5)
thf(fact_6461_is__num__numeral,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [K: num] : ( neg_numeral_is_num @ A @ ( numeral_numeral @ A @ K ) ) ) ).
% is_num_numeral
thf(fact_6462_nat__of__num__neq__0,axiom,
! [X: num] :
( ( nat_of_num @ X )
!= ( zero_zero @ nat ) ) ).
% nat_of_num_neq_0
thf(fact_6463_num__eq__iff,axiom,
( ( ^ [Y: num,Z2: num] : Y = Z2 )
= ( ^ [X2: num,Y6: num] :
( ( nat_of_num @ X2 )
= ( nat_of_num @ Y6 ) ) ) ) ).
% num_eq_iff
thf(fact_6464_nat__of__num__numeral,axiom,
( nat_of_num
= ( numeral_numeral @ nat ) ) ).
% nat_of_num_numeral
thf(fact_6465_nat__of__num__inverse,axiom,
! [X: num] :
( ( num_of_nat @ ( nat_of_num @ X ) )
= X ) ).
% nat_of_num_inverse
thf(fact_6466_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_6467_is__num__add__left__commute,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [X: A,Y3: A,Z3: A] :
( ( neg_numeral_is_num @ A @ X )
=> ( ( neg_numeral_is_num @ A @ Y3 )
=> ( ( plus_plus @ A @ X @ ( plus_plus @ A @ Y3 @ Z3 ) )
= ( plus_plus @ A @ Y3 @ ( plus_plus @ A @ X @ Z3 ) ) ) ) ) ) ).
% is_num_add_left_commute
thf(fact_6468_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_6469_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_6470_nat__of__num__code_I1_J,axiom,
( ( nat_of_num @ one2 )
= ( one_one @ nat ) ) ).
% nat_of_num_code(1)
thf(fact_6471_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_6472_nat__of__num__mult,axiom,
! [X: num,Y3: num] :
( ( nat_of_num @ ( times_times @ num @ X @ Y3 ) )
= ( times_times @ nat @ ( nat_of_num @ X ) @ ( nat_of_num @ Y3 ) ) ) ).
% nat_of_num_mult
thf(fact_6473_nat__of__num__inc,axiom,
! [X: num] :
( ( nat_of_num @ ( inc @ X ) )
= ( suc @ ( nat_of_num @ X ) ) ) ).
% nat_of_num_inc
thf(fact_6474_less__num__def,axiom,
( ( ord_less @ num )
= ( ^ [M2: num,N3: num] : ( ord_less @ nat @ ( nat_of_num @ M2 ) @ ( nat_of_num @ N3 ) ) ) ) ).
% less_num_def
thf(fact_6475_less__eq__num__def,axiom,
( ( ord_less_eq @ num )
= ( ^ [M2: num,N3: num] : ( ord_less_eq @ nat @ ( nat_of_num @ M2 ) @ ( nat_of_num @ N3 ) ) ) ) ).
% less_eq_num_def
thf(fact_6476_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_6477_nat__of__num__sqr,axiom,
! [X: num] :
( ( nat_of_num @ ( sqr @ X ) )
= ( times_times @ nat @ ( nat_of_num @ X ) @ ( nat_of_num @ X ) ) ) ).
% nat_of_num_sqr
thf(fact_6478_nat__of__num_Osimps_I1_J,axiom,
( ( nat_of_num @ one2 )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% nat_of_num.simps(1)
thf(fact_6479_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_6480_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_6481_plus__num__def,axiom,
( ( plus_plus @ num )
= ( ^ [M2: num,N3: num] : ( num_of_nat @ ( plus_plus @ nat @ ( nat_of_num @ M2 ) @ ( nat_of_num @ N3 ) ) ) ) ) ).
% plus_num_def
thf(fact_6482_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_6483_is__num_Ocases,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [A3: A] :
( ( neg_numeral_is_num @ A @ A3 )
=> ( ( A3
!= ( one_one @ A ) )
=> ( ! [X3: A] :
( ( A3
= ( uminus_uminus @ A @ X3 ) )
=> ~ ( neg_numeral_is_num @ A @ X3 ) )
=> ~ ! [X3: A,Y5: A] :
( ( A3
= ( plus_plus @ A @ X3 @ Y5 ) )
=> ( ( neg_numeral_is_num @ A @ X3 )
=> ~ ( neg_numeral_is_num @ A @ Y5 ) ) ) ) ) ) ) ).
% is_num.cases
thf(fact_6484_is__num_Osimps,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( neg_numeral_is_num @ A )
= ( ^ [A5: A] :
( ( A5
= ( one_one @ A ) )
| ? [X2: A] :
( ( A5
= ( uminus_uminus @ A @ X2 ) )
& ( neg_numeral_is_num @ A @ X2 ) )
| ? [X2: A,Y6: A] :
( ( A5
= ( plus_plus @ A @ X2 @ Y6 ) )
& ( neg_numeral_is_num @ A @ X2 )
& ( neg_numeral_is_num @ A @ Y6 ) ) ) ) ) ) ).
% is_num.simps
thf(fact_6485_arg__max__nat__lemma,axiom,
! [A: $tType,P: A > $o,K: A,F: A > nat,B2: nat] :
( ( P @ K )
=> ( ! [Y5: A] :
( ( P @ Y5 )
=> ( ord_less @ nat @ ( F @ Y5 ) @ B2 ) )
=> ( ( P @ ( lattices_ord_arg_max @ A @ nat @ F @ P ) )
& ! [Y4: A] :
( ( P @ Y4 )
=> ( ord_less_eq @ nat @ ( F @ Y4 ) @ ( F @ ( lattices_ord_arg_max @ A @ nat @ F @ P ) ) ) ) ) ) ) ).
% arg_max_nat_lemma
thf(fact_6486_arg__max__nat__le,axiom,
! [A: $tType,P: A > $o,X: A,F: A > nat,B2: nat] :
( ( P @ X )
=> ( ! [Y5: A] :
( ( P @ Y5 )
=> ( ord_less @ nat @ ( F @ Y5 ) @ B2 ) )
=> ( ord_less_eq @ nat @ ( F @ X ) @ ( F @ ( lattices_ord_arg_max @ A @ nat @ F @ P ) ) ) ) ) ).
% arg_max_nat_le
thf(fact_6487_arg__max__natI,axiom,
! [A: $tType,P: A > $o,K: A,F: A > nat,B2: nat] :
( ( P @ K )
=> ( ! [Y5: A] :
( ( P @ Y5 )
=> ( ord_less @ nat @ ( F @ Y5 ) @ B2 ) )
=> ( P @ ( lattices_ord_arg_max @ A @ nat @ F @ P ) ) ) ) ).
% arg_max_natI
thf(fact_6488_arg__max__equality,axiom,
! [A: $tType,C: $tType] :
( ( order @ A )
=> ! [P: C > $o,K: C,F: C > A] :
( ( P @ K )
=> ( ! [X3: C] :
( ( P @ X3 )
=> ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ K ) ) )
=> ( ( F @ ( lattices_ord_arg_max @ C @ A @ F @ P ) )
= ( F @ K ) ) ) ) ) ).
% arg_max_equality
thf(fact_6489_arg__maxI,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [P: A > $o,X: A,F: A > B,Q: A > $o] :
( ( P @ X )
=> ( ! [Y5: A] :
( ( P @ Y5 )
=> ~ ( ord_less @ B @ ( F @ X ) @ ( F @ Y5 ) ) )
=> ( ! [X3: A] :
( ( P @ X3 )
=> ( ! [Y4: A] :
( ( P @ Y4 )
=> ~ ( ord_less @ B @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( Q @ X3 ) ) )
=> ( Q @ ( lattices_ord_arg_max @ A @ B @ F @ P ) ) ) ) ) ) ).
% arg_maxI
thf(fact_6490_arg__max__on__def,axiom,
! [A: $tType,B: $tType] :
( ( ord @ A )
=> ( ( lattic1883929316492267755max_on @ B @ A )
= ( ^ [F2: B > A,S7: set @ B] :
( lattices_ord_arg_max @ B @ A @ F2
@ ^ [X2: B] : ( member @ B @ X2 @ S7 ) ) ) ) ) ).
% arg_max_on_def
thf(fact_6491_arg__max__def,axiom,
! [A: $tType,B: $tType] :
( ( ord @ A )
=> ( ( lattices_ord_arg_max @ B @ A )
= ( ^ [F2: B > A,P4: B > $o] : ( fChoice @ B @ ( lattic501386751176901750rg_max @ B @ A @ F2 @ P4 ) ) ) ) ) ).
% arg_max_def
thf(fact_6492_is__arg__max__linorder,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ( ( lattic501386751176901750rg_max @ A @ B )
= ( ^ [F2: A > B,P4: A > $o,X2: A] :
( ( P4 @ X2 )
& ! [Y6: A] :
( ( P4 @ Y6 )
=> ( ord_less_eq @ B @ ( F2 @ Y6 ) @ ( F2 @ X2 ) ) ) ) ) ) ) ).
% is_arg_max_linorder
thf(fact_6493_is__arg__max__def,axiom,
! [A: $tType,B: $tType] :
( ( ord @ A )
=> ( ( lattic501386751176901750rg_max @ B @ A )
= ( ^ [F2: B > A,P4: B > $o,X2: B] :
( ( P4 @ X2 )
& ~ ? [Y6: B] :
( ( P4 @ Y6 )
& ( ord_less @ A @ ( F2 @ X2 ) @ ( F2 @ Y6 ) ) ) ) ) ) ) ).
% is_arg_max_def
thf(fact_6494_int__of__integer__of__natural,axiom,
! [N: code_natural] :
( ( code_int_of_integer @ ( code_i5400310926305786745atural @ N ) )
= ( semiring_1_of_nat @ int @ ( code_nat_of_natural @ N ) ) ) ).
% int_of_integer_of_natural
thf(fact_6495_integer__of__natural_Orep__eq,axiom,
! [X: code_natural] :
( ( code_int_of_integer @ ( code_i5400310926305786745atural @ X ) )
= ( semiring_1_of_nat @ int @ ( code_nat_of_natural @ X ) ) ) ).
% integer_of_natural.rep_eq
thf(fact_6496_zero__natural_Orep__eq,axiom,
( ( code_nat_of_natural @ ( zero_zero @ code_natural ) )
= ( zero_zero @ nat ) ) ).
% zero_natural.rep_eq
thf(fact_6497_plus__natural_Orep__eq,axiom,
! [X: code_natural,Xa3: code_natural] :
( ( code_nat_of_natural @ ( plus_plus @ code_natural @ X @ Xa3 ) )
= ( plus_plus @ nat @ ( code_nat_of_natural @ X ) @ ( code_nat_of_natural @ Xa3 ) ) ) ).
% plus_natural.rep_eq
thf(fact_6498_nat__of__natural__numeral,axiom,
! [K: num] :
( ( code_nat_of_natural @ ( numeral_numeral @ code_natural @ K ) )
= ( numeral_numeral @ nat @ K ) ) ).
% nat_of_natural_numeral
thf(fact_6499_minus__natural_Orep__eq,axiom,
! [X: code_natural,Xa3: code_natural] :
( ( code_nat_of_natural @ ( minus_minus @ code_natural @ X @ Xa3 ) )
= ( minus_minus @ nat @ ( code_nat_of_natural @ X ) @ ( code_nat_of_natural @ Xa3 ) ) ) ).
% minus_natural.rep_eq
thf(fact_6500_times__natural_Orep__eq,axiom,
! [X: code_natural,Xa3: code_natural] :
( ( code_nat_of_natural @ ( times_times @ code_natural @ X @ Xa3 ) )
= ( times_times @ nat @ ( code_nat_of_natural @ X ) @ ( code_nat_of_natural @ Xa3 ) ) ) ).
% times_natural.rep_eq
thf(fact_6501_one__natural_Orep__eq,axiom,
( ( code_nat_of_natural @ ( one_one @ code_natural ) )
= ( one_one @ nat ) ) ).
% one_natural.rep_eq
thf(fact_6502_modulo__natural_Orep__eq,axiom,
! [X: code_natural,Xa3: code_natural] :
( ( code_nat_of_natural @ ( modulo_modulo @ code_natural @ X @ Xa3 ) )
= ( modulo_modulo @ nat @ ( code_nat_of_natural @ X ) @ ( code_nat_of_natural @ Xa3 ) ) ) ).
% modulo_natural.rep_eq
thf(fact_6503_divide__natural_Orep__eq,axiom,
! [X: code_natural,Xa3: code_natural] :
( ( code_nat_of_natural @ ( divide_divide @ code_natural @ X @ Xa3 ) )
= ( divide_divide @ nat @ ( code_nat_of_natural @ X ) @ ( code_nat_of_natural @ Xa3 ) ) ) ).
% divide_natural.rep_eq
thf(fact_6504_nat__of__natural__min,axiom,
! [K: code_natural,L: code_natural] :
( ( code_nat_of_natural @ ( ord_min @ code_natural @ K @ L ) )
= ( ord_min @ nat @ ( code_nat_of_natural @ K ) @ ( code_nat_of_natural @ L ) ) ) ).
% nat_of_natural_min
thf(fact_6505_nat__of__natural__max,axiom,
! [K: code_natural,L: code_natural] :
( ( code_nat_of_natural @ ( ord_max @ code_natural @ K @ L ) )
= ( ord_max @ nat @ ( code_nat_of_natural @ K ) @ ( code_nat_of_natural @ L ) ) ) ).
% nat_of_natural_max
thf(fact_6506_division__segment__natural_Orep__eq,axiom,
! [X: code_natural] :
( ( code_nat_of_natural @ ( euclid7384307370059645450egment @ code_natural @ X ) )
= ( euclid7384307370059645450egment @ nat @ ( code_nat_of_natural @ X ) ) ) ).
% division_segment_natural.rep_eq
thf(fact_6507_euclidean__size__natural_Orep__eq,axiom,
( ( euclid6346220572633701492n_size @ code_natural )
= ( ^ [X2: code_natural] : ( euclid6346220572633701492n_size @ nat @ ( code_nat_of_natural @ X2 ) ) ) ) ).
% euclidean_size_natural.rep_eq
thf(fact_6508_of__nat__of__natural,axiom,
! [N: code_natural] :
( ( semiring_1_of_nat @ code_natural @ ( code_nat_of_natural @ N ) )
= N ) ).
% of_nat_of_natural
thf(fact_6509_nat__of__natural__of__nat,axiom,
! [N: nat] :
( ( code_nat_of_natural @ ( semiring_1_of_nat @ code_natural @ N ) )
= N ) ).
% nat_of_natural_of_nat
thf(fact_6510_nat__of__natural__induct,axiom,
! [Y3: nat,P: nat > $o] :
( ( member @ nat @ Y3 @ ( top_top @ ( set @ nat ) ) )
=> ( ! [X3: code_natural] : ( P @ ( code_nat_of_natural @ X3 ) )
=> ( P @ Y3 ) ) ) ).
% nat_of_natural_induct
thf(fact_6511_nat__of__natural__cases,axiom,
! [Y3: nat] :
( ( member @ nat @ Y3 @ ( top_top @ ( set @ nat ) ) )
=> ~ ! [X3: code_natural] :
( Y3
!= ( code_nat_of_natural @ X3 ) ) ) ).
% nat_of_natural_cases
thf(fact_6512_nat__of__natural,axiom,
! [X: code_natural] : ( member @ nat @ ( code_nat_of_natural @ X ) @ ( top_top @ ( set @ nat ) ) ) ).
% nat_of_natural
thf(fact_6513_set__bit__natural_Orep__eq,axiom,
! [X: nat,Xa3: code_natural] :
( ( code_nat_of_natural @ ( bit_se5668285175392031749et_bit @ code_natural @ X @ Xa3 ) )
= ( bit_se5668285175392031749et_bit @ nat @ X @ ( code_nat_of_natural @ Xa3 ) ) ) ).
% set_bit_natural.rep_eq
thf(fact_6514_unset__bit__natural_Orep__eq,axiom,
! [X: nat,Xa3: code_natural] :
( ( code_nat_of_natural @ ( bit_se2638667681897837118et_bit @ code_natural @ X @ Xa3 ) )
= ( bit_se2638667681897837118et_bit @ nat @ X @ ( code_nat_of_natural @ Xa3 ) ) ) ).
% unset_bit_natural.rep_eq
thf(fact_6515_flip__bit__natural_Orep__eq,axiom,
! [X: nat,Xa3: code_natural] :
( ( code_nat_of_natural @ ( bit_se8732182000553998342ip_bit @ code_natural @ X @ Xa3 ) )
= ( bit_se8732182000553998342ip_bit @ nat @ X @ ( code_nat_of_natural @ Xa3 ) ) ) ).
% flip_bit_natural.rep_eq
thf(fact_6516_drop__bit__natural_Orep__eq,axiom,
! [X: nat,Xa3: code_natural] :
( ( code_nat_of_natural @ ( bit_se4197421643247451524op_bit @ code_natural @ X @ Xa3 ) )
= ( bit_se4197421643247451524op_bit @ nat @ X @ ( code_nat_of_natural @ Xa3 ) ) ) ).
% drop_bit_natural.rep_eq
thf(fact_6517_bit__natural_Orep__eq,axiom,
( ( bit_se5641148757651400278ts_bit @ code_natural )
= ( ^ [X2: code_natural] : ( bit_se5641148757651400278ts_bit @ nat @ ( code_nat_of_natural @ X2 ) ) ) ) ).
% bit_natural.rep_eq
thf(fact_6518_take__bit__natural_Orep__eq,axiom,
! [X: nat,Xa3: code_natural] :
( ( code_nat_of_natural @ ( bit_se2584673776208193580ke_bit @ code_natural @ X @ Xa3 ) )
= ( bit_se2584673776208193580ke_bit @ nat @ X @ ( code_nat_of_natural @ Xa3 ) ) ) ).
% take_bit_natural.rep_eq
thf(fact_6519_or__natural_Orep__eq,axiom,
! [X: code_natural,Xa3: code_natural] :
( ( code_nat_of_natural @ ( bit_se1065995026697491101ons_or @ code_natural @ X @ Xa3 ) )
= ( bit_se1065995026697491101ons_or @ nat @ ( code_nat_of_natural @ X ) @ ( code_nat_of_natural @ Xa3 ) ) ) ).
% or_natural.rep_eq
thf(fact_6520_xor__natural_Orep__eq,axiom,
! [X: code_natural,Xa3: code_natural] :
( ( code_nat_of_natural @ ( bit_se5824344971392196577ns_xor @ code_natural @ X @ Xa3 ) )
= ( bit_se5824344971392196577ns_xor @ nat @ ( code_nat_of_natural @ X ) @ ( code_nat_of_natural @ Xa3 ) ) ) ).
% xor_natural.rep_eq
thf(fact_6521_and__natural_Orep__eq,axiom,
! [X: code_natural,Xa3: code_natural] :
( ( code_nat_of_natural @ ( bit_se5824344872417868541ns_and @ code_natural @ X @ Xa3 ) )
= ( bit_se5824344872417868541ns_and @ nat @ ( code_nat_of_natural @ X ) @ ( code_nat_of_natural @ Xa3 ) ) ) ).
% and_natural.rep_eq
thf(fact_6522_mask__natural_Orep__eq,axiom,
! [X: nat] :
( ( code_nat_of_natural @ ( bit_se2239418461657761734s_mask @ code_natural @ X ) )
= ( bit_se2239418461657761734s_mask @ nat @ X ) ) ).
% mask_natural.rep_eq
thf(fact_6523_push__bit__natural_Orep__eq,axiom,
! [X: nat,Xa3: code_natural] :
( ( code_nat_of_natural @ ( bit_se4730199178511100633sh_bit @ code_natural @ X @ Xa3 ) )
= ( bit_se4730199178511100633sh_bit @ nat @ X @ ( code_nat_of_natural @ Xa3 ) ) ) ).
% push_bit_natural.rep_eq
thf(fact_6524_natural__eq__iff,axiom,
( ( ^ [Y: code_natural,Z2: code_natural] : Y = Z2 )
= ( ^ [M2: code_natural,N3: code_natural] :
( ( code_nat_of_natural @ M2 )
= ( code_nat_of_natural @ N3 ) ) ) ) ).
% natural_eq_iff
thf(fact_6525_natural__eqI,axiom,
! [M: code_natural,N: code_natural] :
( ( ( code_nat_of_natural @ M )
= ( code_nat_of_natural @ N ) )
=> ( M = N ) ) ).
% natural_eqI
thf(fact_6526_nat__of__natural__inject,axiom,
! [X: code_natural,Y3: code_natural] :
( ( ( code_nat_of_natural @ X )
= ( code_nat_of_natural @ Y3 ) )
= ( X = Y3 ) ) ).
% nat_of_natural_inject
thf(fact_6527_less__natural_Orep__eq,axiom,
( ( ord_less @ code_natural )
= ( ^ [X2: code_natural,Xa4: code_natural] : ( ord_less @ nat @ ( code_nat_of_natural @ X2 ) @ ( code_nat_of_natural @ Xa4 ) ) ) ) ).
% less_natural.rep_eq
thf(fact_6528_less__eq__natural_Orep__eq,axiom,
( ( ord_less_eq @ code_natural )
= ( ^ [X2: code_natural,Xa4: code_natural] : ( ord_less_eq @ nat @ ( code_nat_of_natural @ X2 ) @ ( code_nat_of_natural @ Xa4 ) ) ) ) ).
% less_eq_natural.rep_eq
thf(fact_6529_natural__of__integer_Orep__eq,axiom,
! [X: code_integer] :
( ( code_nat_of_natural @ ( code_n4118661773612635043nteger @ X ) )
= ( nat2 @ ( code_int_of_integer @ X ) ) ) ).
% natural_of_integer.rep_eq
thf(fact_6530_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_6531_Nat_Orep__eq,axiom,
! [X: code_integer] :
( ( code_nat_of_natural @ ( code_Nat @ X ) )
= ( nat2 @ ( code_int_of_integer @ X ) ) ) ).
% Nat.rep_eq
thf(fact_6532_Code__Numeral_Osize__nat__def,axiom,
code_size_nat = code_nat_of_natural ).
% Code_Numeral.size_nat_def
thf(fact_6533_natural__zero__minus__one,axiom,
( ( minus_minus @ code_natural @ ( zero_zero @ code_natural ) @ ( one_one @ code_natural ) )
= ( zero_zero @ code_natural ) ) ).
% natural_zero_minus_one
thf(fact_6534_log_Osimps,axiom,
( log
= ( ^ [B3: code_natural,I5: code_natural] :
( if @ code_natural
@ ( ( ord_less_eq @ code_natural @ B3 @ ( one_one @ code_natural ) )
| ( ord_less @ code_natural @ I5 @ B3 ) )
@ ( one_one @ code_natural )
@ ( plus_plus @ code_natural @ ( one_one @ code_natural ) @ ( log @ B3 @ ( divide_divide @ code_natural @ I5 @ B3 ) ) ) ) ) ) ).
% log.simps
thf(fact_6535_log_Oelims,axiom,
! [X: code_natural,Xa3: code_natural,Y3: code_natural] :
( ( ( log @ X @ Xa3 )
= Y3 )
=> ( ( ( ( ord_less_eq @ code_natural @ X @ ( one_one @ code_natural ) )
| ( ord_less @ code_natural @ Xa3 @ X ) )
=> ( Y3
= ( one_one @ code_natural ) ) )
& ( ~ ( ( ord_less_eq @ code_natural @ X @ ( one_one @ code_natural ) )
| ( ord_less @ code_natural @ Xa3 @ X ) )
=> ( Y3
= ( plus_plus @ code_natural @ ( one_one @ code_natural ) @ ( log @ X @ ( divide_divide @ code_natural @ Xa3 @ X ) ) ) ) ) ) ) ).
% log.elims
thf(fact_6536_minus__shift__def,axiom,
( minus_shift
= ( ^ [R5: code_natural,K3: code_natural,L3: code_natural] : ( if @ code_natural @ ( ord_less @ code_natural @ K3 @ L3 ) @ ( minus_minus @ code_natural @ ( plus_plus @ code_natural @ R5 @ K3 ) @ L3 ) @ ( minus_minus @ code_natural @ K3 @ L3 ) ) ) ) ).
% minus_shift_def
thf(fact_6537_natural_Osimps_I4_J,axiom,
! [T: $tType,F1: T,F22: code_natural > T] :
( ( code_case_natural @ T @ F1 @ F22 @ ( zero_zero @ code_natural ) )
= F1 ) ).
% natural.simps(4)
thf(fact_6538_range,axiom,
! [K: code_natural,S: product_prod @ code_natural @ code_natural] :
( ( ord_less @ code_natural @ ( zero_zero @ code_natural ) @ K )
=> ( ord_less @ code_natural @ ( product_fst @ code_natural @ ( product_prod @ code_natural @ code_natural ) @ ( range @ K @ S ) ) @ K ) ) ).
% range
thf(fact_6539_select__weight__member,axiom,
! [A: $tType,Xs: list @ ( product_prod @ code_natural @ A ),S: product_prod @ code_natural @ code_natural] :
( ( ord_less @ code_natural @ ( zero_zero @ code_natural ) @ ( groups8242544230860333062m_list @ code_natural @ ( map @ ( product_prod @ code_natural @ A ) @ code_natural @ ( product_fst @ code_natural @ A ) @ Xs ) ) )
=> ( member @ A @ ( product_fst @ A @ ( product_prod @ code_natural @ code_natural ) @ ( select_weight @ A @ Xs @ S ) ) @ ( set2 @ A @ ( map @ ( product_prod @ code_natural @ A ) @ A @ ( product_snd @ code_natural @ A ) @ Xs ) ) ) ) ).
% select_weight_member
thf(fact_6540_select__weight__drop__zero,axiom,
! [A: $tType,Xs: list @ ( product_prod @ code_natural @ A )] :
( ( select_weight @ A
@ ( filter @ ( 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 ) )
@ Xs ) )
= ( select_weight @ A @ Xs ) ) ).
% select_weight_drop_zero
thf(fact_6541_pick__member,axiom,
! [A: $tType,I: code_natural,Xs: list @ ( product_prod @ code_natural @ A )] :
( ( ord_less @ code_natural @ I @ ( groups8242544230860333062m_list @ code_natural @ ( map @ ( product_prod @ code_natural @ A ) @ code_natural @ ( product_fst @ code_natural @ A ) @ Xs ) ) )
=> ( member @ A @ ( pick @ A @ Xs @ I ) @ ( set2 @ A @ ( map @ ( product_prod @ code_natural @ A ) @ A @ ( product_snd @ code_natural @ A ) @ Xs ) ) ) ) ).
% pick_member
thf(fact_6542_pick_Osimps,axiom,
! [A: $tType,I: code_natural,X: product_prod @ code_natural @ A,Xs: list @ ( product_prod @ code_natural @ A )] :
( ( ( ord_less @ code_natural @ I @ ( product_fst @ code_natural @ A @ X ) )
=> ( ( pick @ A @ ( cons @ ( product_prod @ code_natural @ A ) @ X @ Xs ) @ I )
= ( product_snd @ code_natural @ A @ X ) ) )
& ( ~ ( ord_less @ code_natural @ I @ ( product_fst @ code_natural @ A @ X ) )
=> ( ( pick @ A @ ( cons @ ( product_prod @ code_natural @ A ) @ X @ Xs ) @ I )
= ( pick @ A @ Xs @ ( minus_minus @ code_natural @ I @ ( product_fst @ code_natural @ A @ X ) ) ) ) ) ) ).
% pick.simps
thf(fact_6543_pick__drop__zero,axiom,
! [A: $tType,Xs: list @ ( product_prod @ code_natural @ A )] :
( ( pick @ A
@ ( filter @ ( 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 ) )
@ Xs ) )
= ( pick @ A @ Xs ) ) ).
% pick_drop_zero
thf(fact_6544_log_Opelims,axiom,
! [X: code_natural,Xa3: code_natural,Y3: code_natural] :
( ( ( log @ X @ Xa3 )
= Y3 )
=> ( ( accp @ ( product_prod @ code_natural @ code_natural ) @ log_rel @ ( product_Pair @ code_natural @ code_natural @ X @ Xa3 ) )
=> ~ ( ( ( ( ( ord_less_eq @ code_natural @ X @ ( one_one @ code_natural ) )
| ( ord_less @ code_natural @ Xa3 @ X ) )
=> ( Y3
= ( one_one @ code_natural ) ) )
& ( ~ ( ( ord_less_eq @ code_natural @ X @ ( one_one @ code_natural ) )
| ( ord_less @ code_natural @ Xa3 @ X ) )
=> ( Y3
= ( plus_plus @ code_natural @ ( one_one @ code_natural ) @ ( log @ X @ ( divide_divide @ code_natural @ Xa3 @ X ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ code_natural @ code_natural ) @ log_rel @ ( product_Pair @ code_natural @ code_natural @ X @ Xa3 ) ) ) ) ) ).
% log.pelims
thf(fact_6545_bit__natural__def,axiom,
( ( bit_se5641148757651400278ts_bit @ code_natural )
= ( map_fun @ code_natural @ nat @ ( nat > $o ) @ ( nat > $o ) @ code_nat_of_natural @ ( id @ ( nat > $o ) ) @ ( bit_se5641148757651400278ts_bit @ nat ) ) ) ).
% bit_natural_def
thf(fact_6546_pick__same,axiom,
! [A: $tType,L: nat,Xs: list @ A] :
( ( ord_less @ nat @ L @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( pick @ A @ ( map @ A @ ( product_prod @ code_natural @ A ) @ ( product_Pair @ code_natural @ A @ ( one_one @ code_natural ) ) @ Xs ) @ ( code_natural_of_nat @ L ) )
= ( nth @ A @ Xs @ L ) ) ) ).
% pick_same
thf(fact_6547_euclidean__size__natural__def,axiom,
( ( euclid6346220572633701492n_size @ code_natural )
= ( map_fun @ code_natural @ nat @ nat @ nat @ code_nat_of_natural @ ( id @ nat ) @ ( euclid6346220572633701492n_size @ nat ) ) ) ).
% euclidean_size_natural_def
thf(fact_6548_natural__of__nat__of__natural__inverse,axiom,
! [N: code_natural] :
( ( code_natural_of_nat @ ( code_nat_of_natural @ N ) )
= N ) ).
% natural_of_nat_of_natural_inverse
thf(fact_6549_nat__of__natural__of__nat__inverse,axiom,
! [N: nat] :
( ( code_nat_of_natural @ ( code_natural_of_nat @ N ) )
= N ) ).
% nat_of_natural_of_nat_inverse
thf(fact_6550_nat__of__natural__inverse,axiom,
! [X: code_natural] :
( ( code_natural_of_nat @ ( code_nat_of_natural @ X ) )
= X ) ).
% nat_of_natural_inverse
thf(fact_6551_natural__of__nat__of__nat,axiom,
( code_natural_of_nat
= ( semiring_1_of_nat @ code_natural ) ) ).
% natural_of_nat_of_nat
thf(fact_6552_natural__of__nat__inverse,axiom,
! [Y3: nat] :
( ( member @ nat @ Y3 @ ( top_top @ ( set @ nat ) ) )
=> ( ( code_nat_of_natural @ ( code_natural_of_nat @ Y3 ) )
= Y3 ) ) ).
% natural_of_nat_inverse
thf(fact_6553_natural__of__nat__cases,axiom,
! [X: code_natural] :
~ ! [Y5: nat] :
( ( X
= ( code_natural_of_nat @ Y5 ) )
=> ~ ( member @ nat @ Y5 @ ( top_top @ ( set @ nat ) ) ) ) ).
% natural_of_nat_cases
thf(fact_6554_natural__of__nat__induct,axiom,
! [P: code_natural > $o,X: code_natural] :
( ! [Y5: nat] :
( ( member @ nat @ Y5 @ ( top_top @ ( set @ nat ) ) )
=> ( P @ ( code_natural_of_nat @ Y5 ) ) )
=> ( P @ X ) ) ).
% natural_of_nat_induct
thf(fact_6555_natural__of__nat__inject,axiom,
! [X: nat,Y3: nat] :
( ( member @ nat @ X @ ( top_top @ ( set @ nat ) ) )
=> ( ( member @ nat @ Y3 @ ( top_top @ ( set @ nat ) ) )
=> ( ( ( code_natural_of_nat @ X )
= ( code_natural_of_nat @ Y3 ) )
= ( X = Y3 ) ) ) ) ).
% natural_of_nat_inject
thf(fact_6556_minus__natural_Oabs__eq,axiom,
! [Xa3: nat,X: nat] :
( ( minus_minus @ code_natural @ ( code_natural_of_nat @ Xa3 ) @ ( code_natural_of_nat @ X ) )
= ( code_natural_of_nat @ ( minus_minus @ nat @ Xa3 @ X ) ) ) ).
% minus_natural.abs_eq
thf(fact_6557_one__natural__def,axiom,
( ( one_one @ code_natural )
= ( code_natural_of_nat @ ( one_one @ nat ) ) ) ).
% one_natural_def
thf(fact_6558_divide__natural_Oabs__eq,axiom,
! [Xa3: nat,X: nat] :
( ( divide_divide @ code_natural @ ( code_natural_of_nat @ Xa3 ) @ ( code_natural_of_nat @ X ) )
= ( code_natural_of_nat @ ( divide_divide @ nat @ Xa3 @ X ) ) ) ).
% divide_natural.abs_eq
thf(fact_6559_plus__natural_Oabs__eq,axiom,
! [Xa3: nat,X: nat] :
( ( plus_plus @ code_natural @ ( code_natural_of_nat @ Xa3 ) @ ( code_natural_of_nat @ X ) )
= ( code_natural_of_nat @ ( plus_plus @ nat @ Xa3 @ X ) ) ) ).
% plus_natural.abs_eq
thf(fact_6560_zero__natural__def,axiom,
( ( zero_zero @ code_natural )
= ( code_natural_of_nat @ ( zero_zero @ nat ) ) ) ).
% zero_natural_def
thf(fact_6561_less__natural_Oabs__eq,axiom,
! [Xa3: nat,X: nat] :
( ( ord_less @ code_natural @ ( code_natural_of_nat @ Xa3 ) @ ( code_natural_of_nat @ X ) )
= ( ord_less @ nat @ Xa3 @ X ) ) ).
% less_natural.abs_eq
thf(fact_6562_less__eq__natural_Oabs__eq,axiom,
! [Xa3: nat,X: nat] :
( ( ord_less_eq @ code_natural @ ( code_natural_of_nat @ Xa3 ) @ ( code_natural_of_nat @ X ) )
= ( ord_less_eq @ nat @ Xa3 @ X ) ) ).
% less_eq_natural.abs_eq
thf(fact_6563_times__natural_Oabs__eq,axiom,
! [Xa3: nat,X: nat] :
( ( times_times @ code_natural @ ( code_natural_of_nat @ Xa3 ) @ ( code_natural_of_nat @ X ) )
= ( code_natural_of_nat @ ( times_times @ nat @ Xa3 @ X ) ) ) ).
% times_natural.abs_eq
thf(fact_6564_modulo__natural_Oabs__eq,axiom,
! [Xa3: nat,X: nat] :
( ( modulo_modulo @ code_natural @ ( code_natural_of_nat @ Xa3 ) @ ( code_natural_of_nat @ X ) )
= ( code_natural_of_nat @ ( modulo_modulo @ nat @ Xa3 @ X ) ) ) ).
% modulo_natural.abs_eq
thf(fact_6565_take__bit__natural_Oabs__eq,axiom,
! [Xa3: nat,X: nat] :
( ( bit_se2584673776208193580ke_bit @ code_natural @ Xa3 @ ( code_natural_of_nat @ X ) )
= ( code_natural_of_nat @ ( bit_se2584673776208193580ke_bit @ nat @ Xa3 @ X ) ) ) ).
% take_bit_natural.abs_eq
thf(fact_6566_xor__natural_Oabs__eq,axiom,
! [Xa3: nat,X: nat] :
( ( bit_se5824344971392196577ns_xor @ code_natural @ ( code_natural_of_nat @ Xa3 ) @ ( code_natural_of_nat @ X ) )
= ( code_natural_of_nat @ ( bit_se5824344971392196577ns_xor @ nat @ Xa3 @ X ) ) ) ).
% xor_natural.abs_eq
thf(fact_6567_or__natural_Oabs__eq,axiom,
! [Xa3: nat,X: nat] :
( ( bit_se1065995026697491101ons_or @ code_natural @ ( code_natural_of_nat @ Xa3 ) @ ( code_natural_of_nat @ X ) )
= ( code_natural_of_nat @ ( bit_se1065995026697491101ons_or @ nat @ Xa3 @ X ) ) ) ).
% or_natural.abs_eq
thf(fact_6568_and__natural_Oabs__eq,axiom,
! [Xa3: nat,X: nat] :
( ( bit_se5824344872417868541ns_and @ code_natural @ ( code_natural_of_nat @ Xa3 ) @ ( code_natural_of_nat @ X ) )
= ( code_natural_of_nat @ ( bit_se5824344872417868541ns_and @ nat @ Xa3 @ X ) ) ) ).
% and_natural.abs_eq
thf(fact_6569_bit__natural_Oabs__eq,axiom,
! [X: nat] :
( ( bit_se5641148757651400278ts_bit @ code_natural @ ( code_natural_of_nat @ X ) )
= ( bit_se5641148757651400278ts_bit @ nat @ X ) ) ).
% bit_natural.abs_eq
thf(fact_6570_push__bit__natural_Oabs__eq,axiom,
! [Xa3: nat,X: nat] :
( ( bit_se4730199178511100633sh_bit @ code_natural @ Xa3 @ ( code_natural_of_nat @ X ) )
= ( code_natural_of_nat @ ( bit_se4730199178511100633sh_bit @ nat @ Xa3 @ X ) ) ) ).
% push_bit_natural.abs_eq
thf(fact_6571_mask__natural_Oabs__eq,axiom,
( ( bit_se2239418461657761734s_mask @ code_natural )
= ( ^ [X2: nat] : ( code_natural_of_nat @ ( bit_se2239418461657761734s_mask @ nat @ X2 ) ) ) ) ).
% mask_natural.abs_eq
thf(fact_6572_drop__bit__natural_Oabs__eq,axiom,
! [Xa3: nat,X: nat] :
( ( bit_se4197421643247451524op_bit @ code_natural @ Xa3 @ ( code_natural_of_nat @ X ) )
= ( code_natural_of_nat @ ( bit_se4197421643247451524op_bit @ nat @ Xa3 @ X ) ) ) ).
% drop_bit_natural.abs_eq
thf(fact_6573_euclidean__size__natural_Oabs__eq,axiom,
! [X: nat] :
( ( euclid6346220572633701492n_size @ code_natural @ ( code_natural_of_nat @ X ) )
= ( euclid6346220572633701492n_size @ nat @ X ) ) ).
% euclidean_size_natural.abs_eq
thf(fact_6574_flip__bit__natural_Oabs__eq,axiom,
! [Xa3: nat,X: nat] :
( ( bit_se8732182000553998342ip_bit @ code_natural @ Xa3 @ ( code_natural_of_nat @ X ) )
= ( code_natural_of_nat @ ( bit_se8732182000553998342ip_bit @ nat @ Xa3 @ X ) ) ) ).
% flip_bit_natural.abs_eq
thf(fact_6575_unset__bit__natural_Oabs__eq,axiom,
! [Xa3: nat,X: nat] :
( ( bit_se2638667681897837118et_bit @ code_natural @ Xa3 @ ( code_natural_of_nat @ X ) )
= ( code_natural_of_nat @ ( bit_se2638667681897837118et_bit @ nat @ Xa3 @ X ) ) ) ).
% unset_bit_natural.abs_eq
thf(fact_6576_division__segment__natural_Oabs__eq,axiom,
! [X: nat] :
( ( euclid7384307370059645450egment @ code_natural @ ( code_natural_of_nat @ X ) )
= ( code_natural_of_nat @ ( euclid7384307370059645450egment @ nat @ X ) ) ) ).
% division_segment_natural.abs_eq
thf(fact_6577_set__bit__natural_Oabs__eq,axiom,
! [Xa3: nat,X: nat] :
( ( bit_se5668285175392031749et_bit @ code_natural @ Xa3 @ ( code_natural_of_nat @ X ) )
= ( code_natural_of_nat @ ( bit_se5668285175392031749et_bit @ nat @ Xa3 @ X ) ) ) ).
% set_bit_natural.abs_eq
thf(fact_6578_integer__of__natural_Oabs__eq,axiom,
! [X: nat] :
( ( code_i5400310926305786745atural @ ( code_natural_of_nat @ X ) )
= ( code_integer_of_int @ ( semiring_1_of_nat @ int @ X ) ) ) ).
% integer_of_natural.abs_eq
thf(fact_6579_natural__of__integer_Oabs__eq,axiom,
! [X: int] :
( ( code_n4118661773612635043nteger @ ( code_integer_of_int @ X ) )
= ( code_natural_of_nat @ ( nat2 @ X ) ) ) ).
% natural_of_integer.abs_eq
thf(fact_6580_Nat_Oabs__eq,axiom,
! [X: int] :
( ( code_Nat @ ( code_integer_of_int @ X ) )
= ( code_natural_of_nat @ ( nat2 @ X ) ) ) ).
% Nat.abs_eq
thf(fact_6581_mask__natural__def,axiom,
( ( bit_se2239418461657761734s_mask @ code_natural )
= ( map_fun @ nat @ nat @ nat @ code_natural @ ( id @ nat ) @ code_natural_of_nat @ ( bit_se2239418461657761734s_mask @ nat ) ) ) ).
% mask_natural_def
thf(fact_6582_case__natural__def,axiom,
! [T: $tType] :
( ( code_case_natural @ T )
= ( ^ [F12: T,F23: code_natural > T] :
( code_rec_natural @ T @ F12
@ ^ [X12: code_natural,X24: T] : ( F23 @ X12 ) ) ) ) ).
% case_natural_def
thf(fact_6583_natural_Osimps_I6_J,axiom,
! [T: $tType,F1: T,F22: code_natural > T > T] :
( ( code_rec_natural @ T @ F1 @ F22 @ ( zero_zero @ code_natural ) )
= F1 ) ).
% natural.simps(6)
thf(fact_6584_rec__natural__def,axiom,
! [T: $tType] :
( ( code_rec_natural @ T )
= ( ^ [F12: T,F23: code_natural > T > T,X2: code_natural] : ( the @ T @ ( code_rec_set_natural @ T @ F12 @ F23 @ X2 ) ) ) ) ).
% rec_natural_def
thf(fact_6585_Nat__def,axiom,
( code_Nat
= ( map_fun @ code_integer @ int @ nat @ code_natural @ code_int_of_integer @ code_natural_of_nat @ nat2 ) ) ).
% Nat_def
thf(fact_6586_natural__of__integer__def,axiom,
( code_n4118661773612635043nteger
= ( map_fun @ code_integer @ int @ nat @ code_natural @ code_int_of_integer @ code_natural_of_nat @ nat2 ) ) ).
% natural_of_integer_def
thf(fact_6587_division__segment__natural__def,axiom,
( ( euclid7384307370059645450egment @ code_natural )
= ( map_fun @ code_natural @ nat @ nat @ code_natural @ code_nat_of_natural @ code_natural_of_nat @ ( euclid7384307370059645450egment @ nat ) ) ) ).
% division_segment_natural_def
thf(fact_6588_integer__of__natural__def,axiom,
( code_i5400310926305786745atural
= ( map_fun @ code_natural @ nat @ int @ code_integer @ code_nat_of_natural @ code_integer_of_int @ ( semiring_1_of_nat @ int ) ) ) ).
% integer_of_natural_def
thf(fact_6589_set__bit__natural__def,axiom,
( ( bit_se5668285175392031749et_bit @ code_natural )
= ( map_fun @ nat @ nat @ ( nat > nat ) @ ( code_natural > code_natural ) @ ( id @ nat ) @ ( map_fun @ code_natural @ nat @ nat @ code_natural @ code_nat_of_natural @ code_natural_of_nat ) @ ( bit_se5668285175392031749et_bit @ nat ) ) ) ).
% set_bit_natural_def
thf(fact_6590_unset__bit__natural__def,axiom,
( ( bit_se2638667681897837118et_bit @ code_natural )
= ( map_fun @ nat @ nat @ ( nat > nat ) @ ( code_natural > code_natural ) @ ( id @ nat ) @ ( map_fun @ code_natural @ nat @ nat @ code_natural @ code_nat_of_natural @ code_natural_of_nat ) @ ( bit_se2638667681897837118et_bit @ nat ) ) ) ).
% unset_bit_natural_def
thf(fact_6591_take__bit__natural__def,axiom,
( ( bit_se2584673776208193580ke_bit @ code_natural )
= ( map_fun @ nat @ nat @ ( nat > nat ) @ ( code_natural > code_natural ) @ ( id @ nat ) @ ( map_fun @ code_natural @ nat @ nat @ code_natural @ code_nat_of_natural @ code_natural_of_nat ) @ ( bit_se2584673776208193580ke_bit @ nat ) ) ) ).
% take_bit_natural_def
thf(fact_6592_push__bit__natural__def,axiom,
( ( bit_se4730199178511100633sh_bit @ code_natural )
= ( map_fun @ nat @ nat @ ( nat > nat ) @ ( code_natural > code_natural ) @ ( id @ nat ) @ ( map_fun @ code_natural @ nat @ nat @ code_natural @ code_nat_of_natural @ code_natural_of_nat ) @ ( bit_se4730199178511100633sh_bit @ nat ) ) ) ).
% push_bit_natural_def
thf(fact_6593_drop__bit__natural__def,axiom,
( ( bit_se4197421643247451524op_bit @ code_natural )
= ( map_fun @ nat @ nat @ ( nat > nat ) @ ( code_natural > code_natural ) @ ( id @ nat ) @ ( map_fun @ code_natural @ nat @ nat @ code_natural @ code_nat_of_natural @ code_natural_of_nat ) @ ( bit_se4197421643247451524op_bit @ nat ) ) ) ).
% drop_bit_natural_def
thf(fact_6594_flip__bit__natural__def,axiom,
( ( bit_se8732182000553998342ip_bit @ code_natural )
= ( map_fun @ nat @ nat @ ( nat > nat ) @ ( code_natural > code_natural ) @ ( id @ nat ) @ ( map_fun @ code_natural @ nat @ nat @ code_natural @ code_nat_of_natural @ code_natural_of_nat ) @ ( bit_se8732182000553998342ip_bit @ nat ) ) ) ).
% flip_bit_natural_def
thf(fact_6595_and__natural__def,axiom,
( ( bit_se5824344872417868541ns_and @ code_natural )
= ( map_fun @ code_natural @ nat @ ( nat > nat ) @ ( code_natural > code_natural ) @ code_nat_of_natural @ ( map_fun @ code_natural @ nat @ nat @ code_natural @ code_nat_of_natural @ code_natural_of_nat ) @ ( bit_se5824344872417868541ns_and @ nat ) ) ) ).
% and_natural_def
thf(fact_6596_xor__natural__def,axiom,
( ( bit_se5824344971392196577ns_xor @ code_natural )
= ( map_fun @ code_natural @ nat @ ( nat > nat ) @ ( code_natural > code_natural ) @ code_nat_of_natural @ ( map_fun @ code_natural @ nat @ nat @ code_natural @ code_nat_of_natural @ code_natural_of_nat ) @ ( bit_se5824344971392196577ns_xor @ nat ) ) ) ).
% xor_natural_def
thf(fact_6597_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_6598_minus__natural__def,axiom,
( ( minus_minus @ code_natural )
= ( map_fun @ code_natural @ nat @ ( nat > nat ) @ ( code_natural > code_natural ) @ code_nat_of_natural @ ( map_fun @ code_natural @ nat @ nat @ code_natural @ code_nat_of_natural @ code_natural_of_nat ) @ ( minus_minus @ nat ) ) ) ).
% minus_natural_def
thf(fact_6599_divide__natural__def,axiom,
( ( divide_divide @ code_natural )
= ( map_fun @ code_natural @ nat @ ( nat > nat ) @ ( code_natural > code_natural ) @ code_nat_of_natural @ ( map_fun @ code_natural @ nat @ nat @ code_natural @ code_nat_of_natural @ code_natural_of_nat ) @ ( divide_divide @ nat ) ) ) ).
% divide_natural_def
thf(fact_6600_times__natural__def,axiom,
( ( times_times @ code_natural )
= ( map_fun @ code_natural @ nat @ ( nat > nat ) @ ( code_natural > code_natural ) @ code_nat_of_natural @ ( map_fun @ code_natural @ nat @ nat @ code_natural @ code_nat_of_natural @ code_natural_of_nat ) @ ( times_times @ nat ) ) ) ).
% times_natural_def
thf(fact_6601_modulo__natural__def,axiom,
( ( modulo_modulo @ code_natural )
= ( map_fun @ code_natural @ nat @ ( nat > nat ) @ ( code_natural > code_natural ) @ code_nat_of_natural @ ( map_fun @ code_natural @ nat @ nat @ code_natural @ code_nat_of_natural @ code_natural_of_nat ) @ ( modulo_modulo @ nat ) ) ) ).
% modulo_natural_def
thf(fact_6602_or__natural__def,axiom,
( ( bit_se1065995026697491101ons_or @ code_natural )
= ( map_fun @ code_natural @ nat @ ( nat > nat ) @ ( code_natural > code_natural ) @ code_nat_of_natural @ ( map_fun @ code_natural @ nat @ nat @ code_natural @ code_nat_of_natural @ code_natural_of_nat ) @ ( bit_se1065995026697491101ons_or @ nat ) ) ) ).
% or_natural_def
thf(fact_6603_Code__Numeral_OSuc__def,axiom,
( code_Suc
= ( map_fun @ code_natural @ nat @ nat @ code_natural @ code_nat_of_natural @ code_natural_of_nat @ suc ) ) ).
% Code_Numeral.Suc_def
thf(fact_6604_cr__natural__def,axiom,
( code_cr_natural
= ( ^ [X2: nat,Y6: code_natural] :
( X2
= ( code_nat_of_natural @ Y6 ) ) ) ) ).
% cr_natural_def
thf(fact_6605_natural_Oinject,axiom,
! [Natural: code_natural,Natural2: code_natural] :
( ( ( code_Suc @ Natural )
= ( code_Suc @ Natural2 ) )
= ( Natural = Natural2 ) ) ).
% natural.inject
thf(fact_6606_natural_Osimps_I7_J,axiom,
! [T: $tType,F1: T,F22: code_natural > T > T,Natural: code_natural] :
( ( code_rec_natural @ T @ F1 @ F22 @ ( code_Suc @ Natural ) )
= ( F22 @ Natural @ ( code_rec_natural @ T @ F1 @ F22 @ Natural ) ) ) ).
% natural.simps(7)
thf(fact_6607_natural_Osimps_I5_J,axiom,
! [T: $tType,F1: T,F22: code_natural > T,Natural: code_natural] :
( ( code_case_natural @ T @ F1 @ F22 @ ( code_Suc @ Natural ) )
= ( F22 @ Natural ) ) ).
% natural.simps(5)
thf(fact_6608_Suc_Orep__eq,axiom,
! [X: code_natural] :
( ( code_nat_of_natural @ ( code_Suc @ X ) )
= ( suc @ ( code_nat_of_natural @ X ) ) ) ).
% Suc.rep_eq
thf(fact_6609_natural_Oexhaust,axiom,
! [Y3: code_natural] :
( ( Y3
!= ( zero_zero @ code_natural ) )
=> ~ ! [Natural3: code_natural] :
( Y3
!= ( code_Suc @ Natural3 ) ) ) ).
% natural.exhaust
thf(fact_6610_natural_Odistinct_I1_J,axiom,
! [Natural2: code_natural] :
( ( zero_zero @ code_natural )
!= ( code_Suc @ Natural2 ) ) ).
% natural.distinct(1)
thf(fact_6611_natural_Odistinct_I2_J,axiom,
! [Natural4: code_natural] :
( ( code_Suc @ Natural4 )
!= ( zero_zero @ code_natural ) ) ).
% natural.distinct(2)
thf(fact_6612_Suc_Oabs__eq,axiom,
! [X: nat] :
( ( code_Suc @ ( code_natural_of_nat @ X ) )
= ( code_natural_of_nat @ ( suc @ X ) ) ) ).
% Suc.abs_eq
thf(fact_6613_Suc__natural__minus__one,axiom,
! [N: code_natural] :
( ( minus_minus @ code_natural @ ( code_Suc @ N ) @ ( one_one @ code_natural ) )
= N ) ).
% Suc_natural_minus_one
thf(fact_6614_Nat_Otransfer,axiom,
bNF_rel_fun @ int @ code_integer @ nat @ code_natural @ code_pcr_integer @ code_pcr_natural @ nat2 @ code_Nat ).
% Nat.transfer
thf(fact_6615_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_6616_Suc_Otransfer,axiom,
bNF_rel_fun @ nat @ code_natural @ nat @ code_natural @ code_pcr_natural @ code_pcr_natural @ suc @ code_Suc ).
% Suc.transfer
thf(fact_6617_natural_Opcr__cr__eq,axiom,
code_pcr_natural = code_cr_natural ).
% natural.pcr_cr_eq
thf(fact_6618_natural_Oid__abs__transfer,axiom,
( bNF_rel_fun @ nat @ nat @ nat @ code_natural
@ ^ [Y: nat,Z2: nat] : Y = Z2
@ code_pcr_natural
@ ^ [X2: nat] : X2
@ code_natural_of_nat ) ).
% natural.id_abs_transfer
thf(fact_6619_less__natural_Otransfer,axiom,
( bNF_rel_fun @ nat @ code_natural @ ( nat > $o ) @ ( code_natural > $o ) @ code_pcr_natural
@ ( bNF_rel_fun @ nat @ code_natural @ $o @ $o @ code_pcr_natural
@ ^ [Y: $o,Z2: $o] : Y = Z2 )
@ ( ord_less @ nat )
@ ( ord_less @ code_natural ) ) ).
% less_natural.transfer
thf(fact_6620_plus__natural_Otransfer,axiom,
bNF_rel_fun @ nat @ code_natural @ ( nat > nat ) @ ( code_natural > code_natural ) @ code_pcr_natural @ ( bNF_rel_fun @ nat @ code_natural @ nat @ code_natural @ code_pcr_natural @ code_pcr_natural ) @ ( plus_plus @ nat ) @ ( plus_plus @ code_natural ) ).
% plus_natural.transfer
thf(fact_6621_minus__natural_Otransfer,axiom,
bNF_rel_fun @ nat @ code_natural @ ( nat > nat ) @ ( code_natural > code_natural ) @ code_pcr_natural @ ( bNF_rel_fun @ nat @ code_natural @ nat @ code_natural @ code_pcr_natural @ code_pcr_natural ) @ ( minus_minus @ nat ) @ ( minus_minus @ code_natural ) ).
% minus_natural.transfer
thf(fact_6622_zero__natural_Otransfer,axiom,
code_pcr_natural @ ( zero_zero @ nat ) @ ( zero_zero @ code_natural ) ).
% zero_natural.transfer
thf(fact_6623_one__natural_Otransfer,axiom,
code_pcr_natural @ ( one_one @ nat ) @ ( one_one @ code_natural ) ).
% one_natural.transfer
thf(fact_6624_natural_Obi__total,axiom,
bi_total @ nat @ code_natural @ code_pcr_natural ).
% natural.bi_total
thf(fact_6625_natural_Orep__transfer,axiom,
( bNF_rel_fun @ nat @ code_natural @ nat @ nat @ code_pcr_natural
@ ^ [Y: nat,Z2: nat] : Y = Z2
@ ^ [X2: nat] : X2
@ code_nat_of_natural ) ).
% natural.rep_transfer
thf(fact_6626_less__eq__natural_Otransfer,axiom,
( bNF_rel_fun @ nat @ code_natural @ ( nat > $o ) @ ( code_natural > $o ) @ code_pcr_natural
@ ( bNF_rel_fun @ nat @ code_natural @ $o @ $o @ code_pcr_natural
@ ^ [Y: $o,Z2: $o] : Y = Z2 )
@ ( ord_less_eq @ nat )
@ ( ord_less_eq @ code_natural ) ) ).
% less_eq_natural.transfer
thf(fact_6627_divide__natural_Otransfer,axiom,
bNF_rel_fun @ nat @ code_natural @ ( nat > nat ) @ ( code_natural > code_natural ) @ code_pcr_natural @ ( bNF_rel_fun @ nat @ code_natural @ nat @ code_natural @ code_pcr_natural @ code_pcr_natural ) @ ( divide_divide @ nat ) @ ( divide_divide @ code_natural ) ).
% divide_natural.transfer
thf(fact_6628_times__natural_Otransfer,axiom,
bNF_rel_fun @ nat @ code_natural @ ( nat > nat ) @ ( code_natural > code_natural ) @ code_pcr_natural @ ( bNF_rel_fun @ nat @ code_natural @ nat @ code_natural @ code_pcr_natural @ code_pcr_natural ) @ ( times_times @ nat ) @ ( times_times @ code_natural ) ).
% times_natural.transfer
thf(fact_6629_modulo__natural_Otransfer,axiom,
bNF_rel_fun @ nat @ code_natural @ ( nat > nat ) @ ( code_natural > code_natural ) @ code_pcr_natural @ ( bNF_rel_fun @ nat @ code_natural @ nat @ code_natural @ code_pcr_natural @ code_pcr_natural ) @ ( modulo_modulo @ nat ) @ ( modulo_modulo @ code_natural ) ).
% modulo_natural.transfer
thf(fact_6630_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 ) ) ) )
=> ( ! [F3: nat > A > A,A8: nat,B5: nat,Acc3: A] :
( ( accp @ ( product_prod @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) ) @ ( set_fo1817059534552279752at_rel @ A ) @ ( product_Pair @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) @ F3 @ ( product_Pair @ nat @ ( product_prod @ nat @ A ) @ A8 @ ( product_Pair @ nat @ A @ B5 @ Acc3 ) ) ) )
=> ( ( ~ ( ord_less @ nat @ B5 @ A8 )
=> ( P @ F3 @ ( plus_plus @ nat @ A8 @ ( one_one @ nat ) ) @ B5 @ ( F3 @ A8 @ Acc3 ) ) )
=> ( P @ F3 @ A8 @ B5 @ Acc3 ) ) )
=> ( P @ A0 @ A1 @ A22 @ A32 ) ) ) ).
% fold_atLeastAtMost_nat.pinduct
thf(fact_6631_take__bit__natural_Otransfer,axiom,
( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( code_natural > code_natural )
@ ^ [Y: nat,Z2: nat] : Y = Z2
@ ( bNF_rel_fun @ nat @ code_natural @ nat @ code_natural @ code_pcr_natural @ code_pcr_natural )
@ ( bit_se2584673776208193580ke_bit @ nat )
@ ( bit_se2584673776208193580ke_bit @ code_natural ) ) ).
% take_bit_natural.transfer
thf(fact_6632_or__natural_Otransfer,axiom,
bNF_rel_fun @ nat @ code_natural @ ( nat > nat ) @ ( code_natural > code_natural ) @ code_pcr_natural @ ( bNF_rel_fun @ nat @ code_natural @ nat @ code_natural @ code_pcr_natural @ code_pcr_natural ) @ ( bit_se1065995026697491101ons_or @ nat ) @ ( bit_se1065995026697491101ons_or @ code_natural ) ).
% or_natural.transfer
thf(fact_6633_xor__natural_Otransfer,axiom,
bNF_rel_fun @ nat @ code_natural @ ( nat > nat ) @ ( code_natural > code_natural ) @ code_pcr_natural @ ( bNF_rel_fun @ nat @ code_natural @ nat @ code_natural @ code_pcr_natural @ code_pcr_natural ) @ ( bit_se5824344971392196577ns_xor @ nat ) @ ( bit_se5824344971392196577ns_xor @ code_natural ) ).
% xor_natural.transfer
thf(fact_6634_and__natural_Otransfer,axiom,
bNF_rel_fun @ nat @ code_natural @ ( nat > nat ) @ ( code_natural > code_natural ) @ code_pcr_natural @ ( bNF_rel_fun @ nat @ code_natural @ nat @ code_natural @ code_pcr_natural @ code_pcr_natural ) @ ( bit_se5824344872417868541ns_and @ nat ) @ ( bit_se5824344872417868541ns_and @ code_natural ) ).
% and_natural.transfer
thf(fact_6635_bit__natural_Otransfer,axiom,
( bNF_rel_fun @ nat @ code_natural @ ( nat > $o ) @ ( nat > $o ) @ code_pcr_natural
@ ^ [Y: nat > $o,Z2: nat > $o] : Y = Z2
@ ( bit_se5641148757651400278ts_bit @ nat )
@ ( bit_se5641148757651400278ts_bit @ code_natural ) ) ).
% bit_natural.transfer
thf(fact_6636_mask__natural_Otransfer,axiom,
( bNF_rel_fun @ nat @ nat @ nat @ code_natural
@ ^ [Y: nat,Z2: nat] : Y = Z2
@ code_pcr_natural
@ ( bit_se2239418461657761734s_mask @ nat )
@ ( bit_se2239418461657761734s_mask @ code_natural ) ) ).
% mask_natural.transfer
thf(fact_6637_push__bit__natural_Otransfer,axiom,
( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( code_natural > code_natural )
@ ^ [Y: nat,Z2: nat] : Y = Z2
@ ( bNF_rel_fun @ nat @ code_natural @ nat @ code_natural @ code_pcr_natural @ code_pcr_natural )
@ ( bit_se4730199178511100633sh_bit @ nat )
@ ( bit_se4730199178511100633sh_bit @ code_natural ) ) ).
% push_bit_natural.transfer
thf(fact_6638_drop__bit__natural_Otransfer,axiom,
( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( code_natural > code_natural )
@ ^ [Y: nat,Z2: nat] : Y = Z2
@ ( bNF_rel_fun @ nat @ code_natural @ nat @ code_natural @ code_pcr_natural @ code_pcr_natural )
@ ( bit_se4197421643247451524op_bit @ nat )
@ ( bit_se4197421643247451524op_bit @ code_natural ) ) ).
% drop_bit_natural.transfer
thf(fact_6639_set__bit__natural_Otransfer,axiom,
( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( code_natural > code_natural )
@ ^ [Y: nat,Z2: nat] : Y = Z2
@ ( bNF_rel_fun @ nat @ code_natural @ nat @ code_natural @ code_pcr_natural @ code_pcr_natural )
@ ( bit_se5668285175392031749et_bit @ nat )
@ ( bit_se5668285175392031749et_bit @ code_natural ) ) ).
% set_bit_natural.transfer
thf(fact_6640_division__segment__natural_Otransfer,axiom,
bNF_rel_fun @ nat @ code_natural @ nat @ code_natural @ code_pcr_natural @ code_pcr_natural @ ( euclid7384307370059645450egment @ nat ) @ ( euclid7384307370059645450egment @ code_natural ) ).
% division_segment_natural.transfer
thf(fact_6641_unset__bit__natural_Otransfer,axiom,
( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( code_natural > code_natural )
@ ^ [Y: nat,Z2: nat] : Y = Z2
@ ( bNF_rel_fun @ nat @ code_natural @ nat @ code_natural @ code_pcr_natural @ code_pcr_natural )
@ ( bit_se2638667681897837118et_bit @ nat )
@ ( bit_se2638667681897837118et_bit @ code_natural ) ) ).
% unset_bit_natural.transfer
thf(fact_6642_flip__bit__natural_Otransfer,axiom,
( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( code_natural > code_natural )
@ ^ [Y: nat,Z2: nat] : Y = Z2
@ ( bNF_rel_fun @ nat @ code_natural @ nat @ code_natural @ code_pcr_natural @ code_pcr_natural )
@ ( bit_se8732182000553998342ip_bit @ nat )
@ ( bit_se8732182000553998342ip_bit @ code_natural ) ) ).
% flip_bit_natural.transfer
thf(fact_6643_euclidean__size__natural_Otransfer,axiom,
( bNF_rel_fun @ nat @ code_natural @ nat @ nat @ code_pcr_natural
@ ^ [Y: nat,Z2: nat] : Y = Z2
@ ( euclid6346220572633701492n_size @ nat )
@ ( euclid6346220572633701492n_size @ code_natural ) ) ).
% euclidean_size_natural.transfer
thf(fact_6644_integer__of__natural_Otransfer,axiom,
bNF_rel_fun @ nat @ code_natural @ int @ code_integer @ code_pcr_natural @ code_pcr_integer @ ( semiring_1_of_nat @ int ) @ code_i5400310926305786745atural ).
% integer_of_natural.transfer
thf(fact_6645_natural__of__integer_Otransfer,axiom,
bNF_rel_fun @ int @ code_integer @ nat @ code_natural @ code_pcr_integer @ code_pcr_natural @ nat2 @ code_n4118661773612635043nteger ).
% natural_of_integer.transfer
thf(fact_6646_fold__atLeastAtMost__nat_Opelims,axiom,
! [A: $tType,X: nat > A > A,Xa3: nat,Xb3: nat,Xc: A,Y3: A] :
( ( ( set_fo6178422350223883121st_nat @ A @ X @ Xa3 @ Xb3 @ 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 ) @ Xa3 @ ( product_Pair @ nat @ A @ Xb3 @ Xc ) ) ) )
=> ~ ( ( ( ( ord_less @ nat @ Xb3 @ Xa3 )
=> ( Y3 = Xc ) )
& ( ~ ( ord_less @ nat @ Xb3 @ Xa3 )
=> ( Y3
= ( set_fo6178422350223883121st_nat @ A @ X @ ( plus_plus @ nat @ Xa3 @ ( one_one @ nat ) ) @ Xb3 @ ( X @ Xa3 @ 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 ) @ Xa3 @ ( product_Pair @ nat @ A @ Xb3 @ Xc ) ) ) ) ) ) ) ).
% fold_atLeastAtMost_nat.pelims
thf(fact_6647_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_6648_prod_Oreindex__bij__betw__not__neutral,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S5: set @ B,T5: set @ C,H2: B > C,S3: set @ B,T6: set @ C,G: C > A] :
( ( finite_finite2 @ B @ S5 )
=> ( ( finite_finite2 @ C @ T5 )
=> ( ( bij_betw @ B @ C @ H2 @ ( minus_minus @ ( set @ B ) @ S3 @ S5 ) @ ( minus_minus @ ( set @ C ) @ T6 @ T5 ) )
=> ( ! [A8: B] :
( ( member @ B @ A8 @ S5 )
=> ( ( G @ ( H2 @ A8 ) )
= ( one_one @ A ) ) )
=> ( ! [B5: C] :
( ( member @ C @ B5 @ T5 )
=> ( ( G @ B5 )
= ( one_one @ A ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [X2: B] : ( G @ ( H2 @ X2 ) )
@ S3 )
= ( groups7121269368397514597t_prod @ C @ A @ G @ T6 ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_betw_not_neutral
thf(fact_6649_bij__betw__of__nat,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [N6: set @ nat,A6: set @ A] :
( ( bij_betw @ nat @ A @ ( semiring_1_of_nat @ A ) @ N6 @ A6 )
= ( ( image @ nat @ A @ ( semiring_1_of_nat @ A ) @ N6 )
= A6 ) ) ) ).
% bij_betw_of_nat
thf(fact_6650_bij__betw__Suc,axiom,
! [M7: set @ nat,N6: set @ nat] :
( ( bij_betw @ nat @ nat @ suc @ M7 @ N6 )
= ( ( image @ nat @ nat @ suc @ M7 )
= N6 ) ) ).
% bij_betw_Suc
thf(fact_6651_bijection__def,axiom,
! [A: $tType] :
( ( hilbert_bijection @ A )
= ( ^ [F2: A > A] : ( bij_betw @ A @ A @ F2 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ A ) ) ) ) ) ).
% bijection_def
thf(fact_6652_bijection_Ointro,axiom,
! [A: $tType,F: A > A] :
( ( bij_betw @ A @ A @ F @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ A ) ) )
=> ( hilbert_bijection @ A @ F ) ) ).
% bijection.intro
thf(fact_6653_bijection_Obij,axiom,
! [A: $tType,F: A > A] :
( ( hilbert_bijection @ A @ F )
=> ( bij_betw @ A @ A @ F @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ A ) ) ) ) ).
% bijection.bij
thf(fact_6654_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_6655_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_6656_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_6657_bij__imp__bij__inv,axiom,
! [B: $tType,A: $tType,F: A > B] :
( ( bij_betw @ A @ B @ F @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
=> ( bij_betw @ B @ A @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F ) @ ( top_top @ ( set @ B ) ) @ ( top_top @ ( set @ A ) ) ) ) ).
% bij_imp_bij_inv
thf(fact_6658_bij__inv__eq__iff,axiom,
! [A: $tType,B: $tType,P5: A > B,X: A,Y3: B] :
( ( bij_betw @ A @ B @ P5 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
=> ( ( X
= ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ P5 @ Y3 ) )
= ( ( P5 @ X )
= Y3 ) ) ) ).
% bij_inv_eq_iff
thf(fact_6659_inv__inv__eq,axiom,
! [B: $tType,A: $tType,F: A > B] :
( ( bij_betw @ A @ B @ F @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
=> ( ( hilbert_inv_into @ B @ A @ ( top_top @ ( set @ B ) ) @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F ) )
= F ) ) ).
% inv_inv_eq
thf(fact_6660_bij__betw__inv__into__right,axiom,
! [A: $tType,B: $tType,F: A > B,A6: set @ A,A11: set @ B,A4: B] :
( ( bij_betw @ A @ B @ F @ A6 @ A11 )
=> ( ( member @ B @ A4 @ A11 )
=> ( ( F @ ( hilbert_inv_into @ A @ B @ A6 @ F @ A4 ) )
= A4 ) ) ) ).
% bij_betw_inv_into_right
thf(fact_6661_bij__betw__inv__into__left,axiom,
! [B: $tType,A: $tType,F: A > B,A6: set @ A,A11: set @ B,A3: A] :
( ( bij_betw @ A @ B @ F @ A6 @ A11 )
=> ( ( member @ A @ A3 @ A6 )
=> ( ( hilbert_inv_into @ A @ B @ A6 @ F @ ( F @ A3 ) )
= A3 ) ) ) ).
% bij_betw_inv_into_left
thf(fact_6662_inv__into__inv__into__eq,axiom,
! [B: $tType,A: $tType,F: A > B,A6: set @ A,A11: set @ B,A3: A] :
( ( bij_betw @ A @ B @ F @ A6 @ A11 )
=> ( ( member @ A @ A3 @ A6 )
=> ( ( hilbert_inv_into @ B @ A @ A11 @ ( hilbert_inv_into @ A @ B @ A6 @ F ) @ A3 )
= ( F @ A3 ) ) ) ) ).
% inv_into_inv_into_eq
thf(fact_6663_bij__betw__inv__into,axiom,
! [B: $tType,A: $tType,F: A > B,A6: set @ A,B7: set @ B] :
( ( bij_betw @ A @ B @ F @ A6 @ B7 )
=> ( bij_betw @ B @ A @ ( hilbert_inv_into @ A @ B @ A6 @ F ) @ B7 @ A6 ) ) ).
% bij_betw_inv_into
thf(fact_6664_bij__betw__inv__into__subset,axiom,
! [B: $tType,A: $tType,F: A > B,A6: set @ A,A11: set @ B,B7: set @ A,B13: set @ B] :
( ( bij_betw @ A @ B @ F @ A6 @ A11 )
=> ( ( ord_less_eq @ ( set @ A ) @ B7 @ A6 )
=> ( ( ( image @ A @ B @ F @ B7 )
= B13 )
=> ( bij_betw @ B @ A @ ( hilbert_inv_into @ A @ B @ A6 @ F ) @ B13 @ B7 ) ) ) ) ).
% bij_betw_inv_into_subset
thf(fact_6665_bij__image__Compl__eq,axiom,
! [B: $tType,A: $tType,F: A > B,A6: set @ A] :
( ( bij_betw @ A @ B @ F @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
=> ( ( image @ A @ B @ F @ ( uminus_uminus @ ( set @ A ) @ A6 ) )
= ( uminus_uminus @ ( set @ B ) @ ( image @ A @ B @ F @ A6 ) ) ) ) ).
% bij_image_Compl_eq
thf(fact_6666_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_6667_inv__fn,axiom,
! [A: $tType,F: A > A,N: nat] :
( ( bij_betw @ A @ A @ F @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ A ) ) )
=> ( ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ ( compow @ ( A > A ) @ N @ F ) )
= ( compow @ ( A > A ) @ N @ ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F ) ) ) ) ).
% inv_fn
thf(fact_6668_mono__inv,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( linorder @ B ) )
=> ! [F: A > B] :
( ( order_mono @ A @ B @ F )
=> ( ( bij_betw @ A @ B @ F @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
=> ( order_mono @ B @ A @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F ) ) ) ) ) ).
% mono_inv
thf(fact_6669_bijection_Obij__inv,axiom,
! [A: $tType,F: A > A] :
( ( hilbert_bijection @ A @ F )
=> ( bij_betw @ A @ A @ ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F ) @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ A ) ) ) ) ).
% bijection.bij_inv
thf(fact_6670_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 ) ) )
=> ( ( image @ A @ B @ F @ ( collect @ A @ P ) )
= ( collect @ B
@ ^ [Y6: B] : ( P @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F @ Y6 ) ) ) ) ) ).
% bij_image_Collect_eq
thf(fact_6671_bij__int__decode,axiom,
bij_betw @ nat @ int @ nat_int_decode @ ( top_top @ ( set @ nat ) ) @ ( top_top @ ( set @ int ) ) ).
% bij_int_decode
thf(fact_6672_bij__int__encode,axiom,
bij_betw @ int @ nat @ nat_int_encode @ ( top_top @ ( set @ int ) ) @ ( top_top @ ( set @ nat ) ) ).
% bij_int_encode
thf(fact_6673_infinite__imp__bij__betw2,axiom,
! [A: $tType,A6: set @ A,A3: A] :
( ~ ( finite_finite2 @ A @ A6 )
=> ? [H6: A > A] : ( bij_betw @ A @ A @ H6 @ A6 @ ( sup_sup @ ( set @ A ) @ A6 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% infinite_imp_bij_betw2
thf(fact_6674_infinite__imp__bij__betw,axiom,
! [A: $tType,A6: set @ A,A3: A] :
( ~ ( finite_finite2 @ A @ A6 )
=> ? [H6: A > A] : ( bij_betw @ A @ A @ H6 @ A6 @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% infinite_imp_bij_betw
thf(fact_6675_ex__bij__betw__finite__nat,axiom,
! [A: $tType,M7: set @ A] :
( ( finite_finite2 @ A @ M7 )
=> ? [H6: A > nat] : ( bij_betw @ A @ nat @ H6 @ M7 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( finite_card @ A @ M7 ) ) ) ) ).
% ex_bij_betw_finite_nat
thf(fact_6676_ex__bij__betw__nat__finite,axiom,
! [A: $tType,M7: set @ A] :
( ( finite_finite2 @ A @ M7 )
=> ? [H6: nat > A] : ( bij_betw @ nat @ A @ H6 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( finite_card @ A @ M7 ) ) @ M7 ) ) ).
% ex_bij_betw_nat_finite
thf(fact_6677_mono__bij__Inf,axiom,
! [B: $tType,A: $tType] :
( ( ( comple5582772986160207858norder @ A )
& ( comple5582772986160207858norder @ B ) )
=> ! [F: A > B,A6: set @ A] :
( ( order_mono @ A @ B @ F )
=> ( ( bij_betw @ A @ B @ F @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
=> ( ( F @ ( complete_Inf_Inf @ A @ A6 ) )
= ( complete_Inf_Inf @ B @ ( image @ A @ B @ F @ A6 ) ) ) ) ) ) ).
% mono_bij_Inf
thf(fact_6678_bij__prod__decode,axiom,
bij_betw @ nat @ ( product_prod @ nat @ nat ) @ nat_prod_decode @ ( top_top @ ( set @ nat ) ) @ ( top_top @ ( set @ ( product_prod @ nat @ nat ) ) ) ).
% bij_prod_decode
thf(fact_6679_inj__imp__bij__betw__inv,axiom,
! [B: $tType,A: $tType,F: A > B,M7: set @ A] :
( ( inj_on @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
=> ( bij_betw @ B @ A @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F ) @ ( image @ A @ B @ F @ M7 ) @ M7 ) ) ).
% inj_imp_bij_betw_inv
thf(fact_6680_ex__bij__betw__nat__finite__1,axiom,
! [A: $tType,M7: set @ A] :
( ( finite_finite2 @ A @ M7 )
=> ? [H6: nat > A] : ( bij_betw @ nat @ A @ H6 @ ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ ( finite_card @ A @ M7 ) ) @ M7 ) ) ).
% ex_bij_betw_nat_finite_1
thf(fact_6681_bij__list__decode,axiom,
bij_betw @ nat @ ( list @ nat ) @ nat_list_decode @ ( top_top @ ( set @ nat ) ) @ ( top_top @ ( set @ ( list @ nat ) ) ) ).
% bij_list_decode
thf(fact_6682_bij__list__encode,axiom,
bij_betw @ ( list @ nat ) @ nat @ nat_list_encode @ ( top_top @ ( set @ ( list @ nat ) ) ) @ ( top_top @ ( set @ nat ) ) ).
% bij_list_encode
thf(fact_6683_bij__prod__encode,axiom,
bij_betw @ ( product_prod @ nat @ nat ) @ nat @ nat_prod_encode @ ( top_top @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( top_top @ ( set @ nat ) ) ).
% bij_prod_encode
thf(fact_6684_bij__vimage__eq__inv__image,axiom,
! [A: $tType,B: $tType,F: A > B,A6: set @ B] :
( ( bij_betw @ A @ B @ F @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
=> ( ( vimage @ A @ B @ F @ A6 )
= ( image @ B @ A @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F ) @ A6 ) ) ) ).
% bij_vimage_eq_inv_image
thf(fact_6685_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_6686_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_6687_sum_Oreindex__bij__betw__not__neutral,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comm_monoid_add @ A )
=> ! [S5: set @ B,T5: set @ C,H2: B > C,S3: set @ B,T6: set @ C,G: C > A] :
( ( finite_finite2 @ B @ S5 )
=> ( ( finite_finite2 @ C @ T5 )
=> ( ( bij_betw @ B @ C @ H2 @ ( minus_minus @ ( set @ B ) @ S3 @ S5 ) @ ( minus_minus @ ( set @ C ) @ T6 @ T5 ) )
=> ( ! [A8: B] :
( ( member @ B @ A8 @ S5 )
=> ( ( G @ ( H2 @ A8 ) )
= ( zero_zero @ A ) ) )
=> ( ! [B5: C] :
( ( member @ C @ B5 @ T5 )
=> ( ( G @ B5 )
= ( zero_zero @ A ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X2: B] : ( G @ ( H2 @ X2 ) )
@ S3 )
= ( groups7311177749621191930dd_sum @ C @ A @ G @ T6 ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_betw_not_neutral
thf(fact_6688_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_6689_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_6690_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,Y6: nat] :
( ( ord_less @ nat @ X2 @ N )
& ( ord_less @ nat @ Y6 @ N )
& ( ord_less_eq @ nat @ X2 @ Y6 ) ) ) ) ) ).
% natLeq_on_wo_rel
thf(fact_6691_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_6692_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_6693_max_Osemilattice__order__axioms,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( semilattice_order @ A @ ( ord_max @ A )
@ ^ [X2: A,Y6: A] : ( ord_less_eq @ A @ Y6 @ X2 )
@ ^ [X2: A,Y6: A] : ( ord_less @ A @ Y6 @ X2 ) ) ) ).
% max.semilattice_order_axioms
thf(fact_6694_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_6695_semilattice__order__set_Oaxioms_I1_J,axiom,
! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o] :
( ( lattic4895041142388067077er_set @ A @ F @ Less_eq @ Less )
=> ( semilattice_order @ A @ F @ Less_eq @ Less ) ) ).
% semilattice_order_set.axioms(1)
thf(fact_6696_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_6697_semilattice__neutr__order_Oaxioms_I2_J,axiom,
! [A: $tType,F: A > A > A,Z3: A,Less_eq: A > A > $o,Less: A > A > $o] :
( ( semila1105856199041335345_order @ A @ F @ Z3 @ Less_eq @ Less )
=> ( semilattice_order @ A @ F @ Less_eq @ Less ) ) ).
% semilattice_neutr_order.axioms(2)
thf(fact_6698_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_6699_semilattice__order_Omono,axiom,
! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A3: A,C2: A,B2: A,D2: A] :
( ( semilattice_order @ A @ F @ Less_eq @ Less )
=> ( ( Less_eq @ A3 @ C2 )
=> ( ( Less_eq @ B2 @ D2 )
=> ( Less_eq @ ( F @ A3 @ B2 ) @ ( F @ C2 @ D2 ) ) ) ) ) ).
% semilattice_order.mono
thf(fact_6700_semilattice__order_OorderE,axiom,
! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A] :
( ( semilattice_order @ A @ F @ Less_eq @ Less )
=> ( ( Less_eq @ A3 @ B2 )
=> ( A3
= ( F @ A3 @ B2 ) ) ) ) ).
% semilattice_order.orderE
thf(fact_6701_semilattice__order_OorderI,axiom,
! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A] :
( ( semilattice_order @ A @ F @ Less_eq @ Less )
=> ( ( A3
= ( F @ A3 @ B2 ) )
=> ( Less_eq @ A3 @ B2 ) ) ) ).
% semilattice_order.orderI
thf(fact_6702_semilattice__order_Oabsorb1,axiom,
! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A] :
( ( semilattice_order @ A @ F @ Less_eq @ Less )
=> ( ( Less_eq @ A3 @ B2 )
=> ( ( F @ A3 @ B2 )
= A3 ) ) ) ).
% semilattice_order.absorb1
thf(fact_6703_semilattice__order_Oabsorb2,axiom,
! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,B2: A,A3: A] :
( ( semilattice_order @ A @ F @ Less_eq @ Less )
=> ( ( Less_eq @ B2 @ A3 )
=> ( ( F @ A3 @ B2 )
= B2 ) ) ) ).
% semilattice_order.absorb2
thf(fact_6704_semilattice__order_Oabsorb3,axiom,
! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A] :
( ( semilattice_order @ A @ F @ Less_eq @ Less )
=> ( ( Less @ A3 @ B2 )
=> ( ( F @ A3 @ B2 )
= A3 ) ) ) ).
% semilattice_order.absorb3
thf(fact_6705_semilattice__order_Oabsorb4,axiom,
! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,B2: A,A3: A] :
( ( semilattice_order @ A @ F @ Less_eq @ Less )
=> ( ( Less @ B2 @ A3 )
=> ( ( F @ A3 @ B2 )
= B2 ) ) ) ).
% semilattice_order.absorb4
thf(fact_6706_semilattice__order_OboundedE,axiom,
! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A,C2: A] :
( ( semilattice_order @ A @ F @ Less_eq @ Less )
=> ( ( Less_eq @ A3 @ ( F @ B2 @ C2 ) )
=> ~ ( ( Less_eq @ A3 @ B2 )
=> ~ ( Less_eq @ A3 @ C2 ) ) ) ) ).
% semilattice_order.boundedE
thf(fact_6707_semilattice__order_OboundedI,axiom,
! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A,C2: A] :
( ( semilattice_order @ A @ F @ Less_eq @ Less )
=> ( ( Less_eq @ A3 @ B2 )
=> ( ( Less_eq @ A3 @ C2 )
=> ( Less_eq @ A3 @ ( F @ B2 @ C2 ) ) ) ) ) ).
% semilattice_order.boundedI
thf(fact_6708_semilattice__order_Oorder__iff,axiom,
! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A] :
( ( semilattice_order @ A @ F @ Less_eq @ Less )
=> ( ( Less_eq @ A3 @ B2 )
= ( A3
= ( F @ A3 @ B2 ) ) ) ) ).
% semilattice_order.order_iff
thf(fact_6709_semilattice__order_Ocobounded1,axiom,
! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A] :
( ( semilattice_order @ A @ F @ Less_eq @ Less )
=> ( Less_eq @ ( F @ A3 @ B2 ) @ A3 ) ) ).
% semilattice_order.cobounded1
thf(fact_6710_semilattice__order_Ocobounded2,axiom,
! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A] :
( ( semilattice_order @ A @ F @ Less_eq @ Less )
=> ( Less_eq @ ( F @ A3 @ B2 ) @ B2 ) ) ).
% semilattice_order.cobounded2
thf(fact_6711_semilattice__order_Oabsorb__iff1,axiom,
! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A] :
( ( semilattice_order @ A @ F @ Less_eq @ Less )
=> ( ( Less_eq @ A3 @ B2 )
= ( ( F @ A3 @ B2 )
= A3 ) ) ) ).
% semilattice_order.absorb_iff1
thf(fact_6712_semilattice__order_Oabsorb__iff2,axiom,
! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,B2: A,A3: A] :
( ( semilattice_order @ A @ F @ Less_eq @ Less )
=> ( ( Less_eq @ B2 @ A3 )
= ( ( F @ A3 @ B2 )
= B2 ) ) ) ).
% semilattice_order.absorb_iff2
thf(fact_6713_semilattice__order_Obounded__iff,axiom,
! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A,C2: A] :
( ( semilattice_order @ A @ F @ Less_eq @ Less )
=> ( ( Less_eq @ A3 @ ( F @ B2 @ C2 ) )
= ( ( Less_eq @ A3 @ B2 )
& ( Less_eq @ A3 @ C2 ) ) ) ) ).
% semilattice_order.bounded_iff
thf(fact_6714_semilattice__order_OcoboundedI1,axiom,
! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A3: A,C2: A,B2: A] :
( ( semilattice_order @ A @ F @ Less_eq @ Less )
=> ( ( Less_eq @ A3 @ C2 )
=> ( Less_eq @ ( F @ A3 @ B2 ) @ C2 ) ) ) ).
% semilattice_order.coboundedI1
thf(fact_6715_semilattice__order_OcoboundedI2,axiom,
! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,B2: A,C2: A,A3: A] :
( ( semilattice_order @ A @ F @ Less_eq @ Less )
=> ( ( Less_eq @ B2 @ C2 )
=> ( Less_eq @ ( F @ A3 @ B2 ) @ C2 ) ) ) ).
% semilattice_order.coboundedI2
thf(fact_6716_semilattice__order_Ostrict__boundedE,axiom,
! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A,C2: A] :
( ( semilattice_order @ A @ F @ Less_eq @ Less )
=> ( ( Less @ A3 @ ( F @ B2 @ C2 ) )
=> ~ ( ( Less @ A3 @ B2 )
=> ~ ( Less @ A3 @ C2 ) ) ) ) ).
% semilattice_order.strict_boundedE
thf(fact_6717_semilattice__order_Ostrict__order__iff,axiom,
! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A] :
( ( semilattice_order @ A @ F @ Less_eq @ Less )
=> ( ( Less @ A3 @ B2 )
= ( ( A3
= ( F @ A3 @ B2 ) )
& ( A3 != B2 ) ) ) ) ).
% semilattice_order.strict_order_iff
thf(fact_6718_semilattice__order_Ostrict__coboundedI1,axiom,
! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A3: A,C2: A,B2: A] :
( ( semilattice_order @ A @ F @ Less_eq @ Less )
=> ( ( Less @ A3 @ C2 )
=> ( Less @ ( F @ A3 @ B2 ) @ C2 ) ) ) ).
% semilattice_order.strict_coboundedI1
thf(fact_6719_semilattice__order_Ostrict__coboundedI2,axiom,
! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,B2: A,C2: A,A3: A] :
( ( semilattice_order @ A @ F @ Less_eq @ Less )
=> ( ( Less @ B2 @ C2 )
=> ( Less @ ( F @ A3 @ B2 ) @ C2 ) ) ) ).
% semilattice_order.strict_coboundedI2
thf(fact_6720_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_6721_Heap_Omap__id0,axiom,
! [A: $tType] :
( ( heap_Time_map_Heap @ A @ A @ ( id @ A ) )
= ( id @ ( heap_Time_Heap @ A ) ) ) ).
% Heap.map_id0
thf(fact_6722_semilattice__order__set__def,axiom,
! [A: $tType] :
( ( lattic4895041142388067077er_set @ A )
= ( ^ [F2: A > A > A,Less_eq2: A > A > $o,Less2: A > A > $o] :
( ( semilattice_order @ A @ F2 @ Less_eq2 @ Less2 )
& ( lattic149705377957585745ce_set @ A @ F2 ) ) ) ) ).
% semilattice_order_set_def
thf(fact_6723_semilattice__order__set_Ointro,axiom,
! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o] :
( ( semilattice_order @ A @ F @ Less_eq @ Less )
=> ( ( lattic149705377957585745ce_set @ A @ F )
=> ( lattic4895041142388067077er_set @ A @ F @ Less_eq @ Less ) ) ) ).
% semilattice_order_set.intro
thf(fact_6724_semilattice__neutr__order__def,axiom,
! [A: $tType] :
( ( semila1105856199041335345_order @ A )
= ( ^ [F2: A > A > A,Z7: A,Less_eq2: A > A > $o,Less2: A > A > $o] :
( ( semilattice_neutr @ A @ F2 @ Z7 )
& ( semilattice_order @ A @ F2 @ Less_eq2 @ Less2 ) ) ) ) ).
% semilattice_neutr_order_def
thf(fact_6725_semilattice__neutr__order_Ointro,axiom,
! [A: $tType,F: A > A > A,Z3: A,Less_eq: A > A > $o,Less: A > A > $o] :
( ( semilattice_neutr @ A @ F @ Z3 )
=> ( ( semilattice_order @ A @ F @ Less_eq @ Less )
=> ( semila1105856199041335345_order @ A @ F @ Z3 @ Less_eq @ Less ) ) ) ).
% semilattice_neutr_order.intro
thf(fact_6726_inf_Osemilattice__order__axioms,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( semilattice_order @ A @ ( inf_inf @ A ) @ ( ord_less_eq @ A ) @ ( ord_less @ A ) ) ) ).
% inf.semilattice_order_axioms
thf(fact_6727_min_Osemilattice__order__axioms,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( semilattice_order @ A @ ( ord_min @ A ) @ ( ord_less_eq @ A ) @ ( ord_less @ A ) ) ) ).
% min.semilattice_order_axioms
thf(fact_6728_sup_Osemilattice__order__axioms,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( semilattice_order @ A @ ( sup_sup @ A )
@ ^ [X2: A,Y6: A] : ( ord_less_eq @ A @ Y6 @ X2 )
@ ^ [X2: A,Y6: A] : ( ord_less @ A @ Y6 @ X2 ) ) ) ).
% sup.semilattice_order_axioms
thf(fact_6729_lexordp__eq_Osimps,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_lexordp_eq @ A )
= ( ^ [A12: list @ A,A23: list @ A] :
( ? [Ys3: list @ A] :
( ( A12
= ( nil @ A ) )
& ( A23 = Ys3 ) )
| ? [X2: A,Y6: A,Xs3: list @ A,Ys3: list @ A] :
( ( A12
= ( cons @ A @ X2 @ Xs3 ) )
& ( A23
= ( cons @ A @ Y6 @ Ys3 ) )
& ( ord_less @ A @ X2 @ Y6 ) )
| ? [X2: A,Y6: A,Xs3: list @ A,Ys3: list @ A] :
( ( A12
= ( cons @ A @ X2 @ Xs3 ) )
& ( A23
= ( cons @ A @ Y6 @ Ys3 ) )
& ~ ( ord_less @ A @ X2 @ Y6 )
& ~ ( ord_less @ A @ Y6 @ X2 )
& ( ord_lexordp_eq @ A @ Xs3 @ Ys3 ) ) ) ) ) ) ).
% lexordp_eq.simps
thf(fact_6730_lexordp__eq__simps_I4_J,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X: A,Xs: list @ A,Y3: A,Ys2: list @ A] :
( ( ord_lexordp_eq @ A @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y3 @ Ys2 ) )
= ( ( ord_less @ A @ X @ Y3 )
| ( ~ ( ord_less @ A @ Y3 @ X )
& ( ord_lexordp_eq @ A @ Xs @ Ys2 ) ) ) ) ) ).
% lexordp_eq_simps(4)
thf(fact_6731_lexordp__eq_OCons,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X: A,Y3: A,Xs: list @ A,Ys2: list @ A] :
( ( ord_less @ A @ X @ Y3 )
=> ( ord_lexordp_eq @ A @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y3 @ Ys2 ) ) ) ) ).
% lexordp_eq.Cons
thf(fact_6732_lexordp__eq_OCons__eq,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X: A,Y3: A,Xs: list @ A,Ys2: list @ A] :
( ~ ( ord_less @ A @ X @ Y3 )
=> ( ~ ( ord_less @ A @ Y3 @ X )
=> ( ( ord_lexordp_eq @ A @ Xs @ Ys2 )
=> ( ord_lexordp_eq @ A @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y3 @ Ys2 ) ) ) ) ) ) ).
% lexordp_eq.Cons_eq
thf(fact_6733_lexordp__eq_Ocases,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A1: list @ A,A22: list @ A] :
( ( ord_lexordp_eq @ A @ A1 @ A22 )
=> ( ( A1
!= ( nil @ A ) )
=> ( ! [X3: A] :
( ? [Xs2: list @ A] :
( A1
= ( cons @ A @ X3 @ Xs2 ) )
=> ! [Y5: A] :
( ? [Ys4: list @ A] :
( A22
= ( cons @ A @ Y5 @ Ys4 ) )
=> ~ ( ord_less @ A @ X3 @ Y5 ) ) )
=> ~ ! [X3: A,Y5: A,Xs2: list @ A] :
( ( A1
= ( cons @ A @ X3 @ Xs2 ) )
=> ! [Ys4: list @ A] :
( ( A22
= ( cons @ A @ Y5 @ Ys4 ) )
=> ( ~ ( ord_less @ A @ X3 @ Y5 )
=> ( ~ ( ord_less @ A @ Y5 @ X3 )
=> ~ ( ord_lexordp_eq @ A @ Xs2 @ Ys4 ) ) ) ) ) ) ) ) ) ).
% lexordp_eq.cases
thf(fact_6734_bdd__below__primitive__def,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( condit1013018076250108175_below @ A )
= ( condit16957441358409770ng_bdd @ A
@ ^ [X2: A,Y6: A] : ( ord_less_eq @ A @ Y6 @ X2 ) ) ) ) ).
% bdd_below_primitive_def
thf(fact_6735_Quotient__natural,axiom,
( quotient @ nat @ code_natural
@ ^ [Y: nat,Z2: nat] : Y = Z2
@ code_natural_of_nat
@ code_nat_of_natural
@ code_cr_natural ) ).
% Quotient_natural
thf(fact_6736_bdd__above__primitive__def,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( condit941137186595557371_above @ A )
= ( condit16957441358409770ng_bdd @ A @ ( ord_less_eq @ A ) ) ) ) ).
% bdd_above_primitive_def
thf(fact_6737_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_6738_bdd__below_Opreordering__bdd__axioms,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( condit622319405099724424ng_bdd @ A
@ ^ [X2: A,Y6: A] : ( ord_less_eq @ A @ Y6 @ X2 )
@ ^ [X2: A,Y6: A] : ( ord_less @ A @ Y6 @ X2 ) ) ) ).
% bdd_below.preordering_bdd_axioms
thf(fact_6739_bdd__above_Opreordering__bdd__axioms,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( condit622319405099724424ng_bdd @ A @ ( ord_less_eq @ A ) @ ( ord_less @ A ) ) ) ).
% bdd_above.preordering_bdd_axioms
thf(fact_6740_Quotient__integer,axiom,
( quotient @ int @ code_integer
@ ^ [Y: int,Z2: int] : Y = Z2
@ code_integer_of_int
@ code_int_of_integer
@ code_cr_integer ) ).
% Quotient_integer
thf(fact_6741_type__definition__integer,axiom,
type_definition @ code_integer @ int @ code_int_of_integer @ code_integer_of_int @ ( top_top @ ( set @ int ) ) ).
% type_definition_integer
thf(fact_6742_finite__subset__Union__chain,axiom,
! [A: $tType,A6: set @ A,B12: set @ ( set @ A ),A16: set @ ( set @ A )] :
( ( finite_finite2 @ A @ A6 )
=> ( ( ord_less_eq @ ( set @ A ) @ A6 @ ( complete_Sup_Sup @ ( set @ A ) @ B12 ) )
=> ( ( B12
!= ( bot_bot @ ( set @ ( set @ A ) ) ) )
=> ( ( pred_chain @ ( set @ A ) @ A16 @ ( ord_less @ ( set @ A ) ) @ B12 )
=> ~ ! [B9: set @ A] :
( ( member @ ( set @ A ) @ B9 @ B12 )
=> ~ ( ord_less_eq @ ( set @ A ) @ A6 @ B9 ) ) ) ) ) ) ).
% finite_subset_Union_chain
thf(fact_6743_wait__def,axiom,
( heap_Time_wait
= ( ^ [N3: 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 @ N3 ) ) ) ) ) ) ).
% wait_def
thf(fact_6744_subset_Ochain__empty,axiom,
! [A: $tType,A6: set @ ( set @ A )] : ( pred_chain @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ ( bot_bot @ ( set @ ( set @ A ) ) ) ) ).
% subset.chain_empty
thf(fact_6745_subset__chain__def,axiom,
! [A: $tType,A16: set @ ( set @ A ),C7: set @ ( set @ A )] :
( ( pred_chain @ ( set @ A ) @ A16 @ ( ord_less @ ( set @ A ) ) @ C7 )
= ( ( ord_less_eq @ ( set @ ( set @ A ) ) @ C7 @ A16 )
& ! [X2: set @ A] :
( ( member @ ( set @ A ) @ X2 @ C7 )
=> ! [Y6: set @ A] :
( ( member @ ( set @ A ) @ Y6 @ C7 )
=> ( ( ord_less_eq @ ( set @ A ) @ X2 @ Y6 )
| ( ord_less_eq @ ( set @ A ) @ Y6 @ X2 ) ) ) ) ) ) ).
% subset_chain_def
thf(fact_6746_subset__Zorn,axiom,
! [A: $tType,A6: set @ ( set @ A )] :
( ! [C8: set @ ( set @ A )] :
( ( pred_chain @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ C8 )
=> ? [X4: set @ A] :
( ( member @ ( set @ A ) @ X4 @ A6 )
& ! [Xa2: set @ A] :
( ( member @ ( set @ A ) @ Xa2 @ C8 )
=> ( ord_less_eq @ ( set @ A ) @ Xa2 @ X4 ) ) ) )
=> ? [X3: set @ A] :
( ( member @ ( set @ A ) @ X3 @ A6 )
& ! [Xa: set @ A] :
( ( member @ ( set @ A ) @ Xa @ A6 )
=> ( ( ord_less_eq @ ( set @ A ) @ X3 @ Xa )
=> ( Xa = X3 ) ) ) ) ) ).
% subset_Zorn
thf(fact_6747_subset__chain__insert,axiom,
! [A: $tType,A16: set @ ( set @ A ),B7: set @ A,B12: set @ ( set @ A )] :
( ( pred_chain @ ( set @ A ) @ A16 @ ( ord_less @ ( set @ A ) ) @ ( insert @ ( set @ A ) @ B7 @ B12 ) )
= ( ( member @ ( set @ A ) @ B7 @ A16 )
& ! [X2: set @ A] :
( ( member @ ( set @ A ) @ X2 @ B12 )
=> ( ( ord_less_eq @ ( set @ A ) @ X2 @ B7 )
| ( ord_less_eq @ ( set @ A ) @ B7 @ X2 ) ) )
& ( pred_chain @ ( set @ A ) @ A16 @ ( ord_less @ ( set @ A ) ) @ B12 ) ) ) ).
% subset_chain_insert
thf(fact_6748_subset__Zorn_H,axiom,
! [A: $tType,A6: set @ ( set @ A )] :
( ! [C8: set @ ( set @ A )] :
( ( pred_chain @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ C8 )
=> ( member @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ C8 ) @ A6 ) )
=> ? [X3: set @ A] :
( ( member @ ( set @ A ) @ X3 @ A6 )
& ! [Xa: set @ A] :
( ( member @ ( set @ A ) @ Xa @ A6 )
=> ( ( ord_less_eq @ ( set @ A ) @ X3 @ Xa )
=> ( Xa = X3 ) ) ) ) ) ).
% subset_Zorn'
thf(fact_6749_subset_Ochain__extend,axiom,
! [A: $tType,A6: set @ ( set @ A ),C6: set @ ( set @ A ),Z3: set @ A] :
( ( pred_chain @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ C6 )
=> ( ( member @ ( set @ A ) @ Z3 @ A6 )
=> ( ! [X3: set @ A] :
( ( member @ ( set @ A ) @ X3 @ C6 )
=> ( sup_sup @ ( ( set @ A ) > ( set @ A ) > $o ) @ ( ord_less @ ( set @ A ) )
@ ^ [Y: set @ A,Z2: set @ A] : Y = Z2
@ X3
@ Z3 ) )
=> ( pred_chain @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ ( sup_sup @ ( set @ ( set @ A ) ) @ ( insert @ ( set @ A ) @ Z3 @ ( bot_bot @ ( set @ ( set @ A ) ) ) ) @ C6 ) ) ) ) ) ).
% subset.chain_extend
thf(fact_6750_subset_Ochain__total,axiom,
! [A: $tType,A6: set @ ( set @ A ),C6: set @ ( set @ A ),X: set @ A,Y3: set @ A] :
( ( pred_chain @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ C6 )
=> ( ( member @ ( set @ A ) @ X @ C6 )
=> ( ( member @ ( set @ A ) @ Y3 @ C6 )
=> ( ( sup_sup @ ( ( set @ A ) > ( set @ A ) > $o ) @ ( ord_less @ ( set @ A ) )
@ ^ [Y: set @ A,Z2: set @ A] : Y = Z2
@ X
@ Y3 )
| ( sup_sup @ ( ( set @ A ) > ( set @ A ) > $o ) @ ( ord_less @ ( set @ A ) )
@ ^ [Y: set @ A,Z2: set @ A] : Y = Z2
@ Y3
@ X ) ) ) ) ) ).
% subset.chain_total
thf(fact_6751_subset_OchainI,axiom,
! [A: $tType,C6: set @ ( set @ A ),A6: set @ ( set @ A )] :
( ( ord_less_eq @ ( set @ ( set @ A ) ) @ C6 @ A6 )
=> ( ! [X3: set @ A,Y5: set @ A] :
( ( member @ ( set @ A ) @ X3 @ C6 )
=> ( ( member @ ( set @ A ) @ Y5 @ C6 )
=> ( ( sup_sup @ ( ( set @ A ) > ( set @ A ) > $o ) @ ( ord_less @ ( set @ A ) )
@ ^ [Y: set @ A,Z2: set @ A] : Y = Z2
@ X3
@ Y5 )
| ( sup_sup @ ( ( set @ A ) > ( set @ A ) > $o ) @ ( ord_less @ ( set @ A ) )
@ ^ [Y: set @ A,Z2: set @ A] : Y = Z2
@ Y5
@ X3 ) ) ) )
=> ( pred_chain @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ C6 ) ) ) ).
% subset.chainI
thf(fact_6752_subset_Ochain__def,axiom,
! [A: $tType,A6: set @ ( set @ A ),C6: set @ ( set @ A )] :
( ( pred_chain @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ C6 )
= ( ( ord_less_eq @ ( set @ ( set @ A ) ) @ C6 @ A6 )
& ! [X2: set @ A] :
( ( member @ ( set @ A ) @ X2 @ C6 )
=> ! [Y6: set @ A] :
( ( member @ ( set @ A ) @ Y6 @ C6 )
=> ( ( sup_sup @ ( ( set @ A ) > ( set @ A ) > $o ) @ ( ord_less @ ( set @ A ) )
@ ^ [Y: set @ A,Z2: set @ A] : Y = Z2
@ X2
@ Y6 )
| ( sup_sup @ ( ( set @ A ) > ( set @ A ) > $o ) @ ( ord_less @ ( set @ A ) )
@ ^ [Y: set @ A,Z2: set @ A] : Y = Z2
@ Y6
@ X2 ) ) ) ) ) ) ).
% subset.chain_def
thf(fact_6753_uminus__unit__def,axiom,
( ( uminus_uminus @ product_unit )
= ( ^ [Uu2: product_unit] : product_Unity ) ) ).
% uminus_unit_def
thf(fact_6754_subset__Zorn__nonempty,axiom,
! [A: $tType,A16: set @ ( set @ A )] :
( ( A16
!= ( bot_bot @ ( set @ ( set @ A ) ) ) )
=> ( ! [C9: set @ ( set @ A )] :
( ( C9
!= ( bot_bot @ ( set @ ( set @ A ) ) ) )
=> ( ( pred_chain @ ( set @ A ) @ A16 @ ( ord_less @ ( set @ A ) ) @ C9 )
=> ( member @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ C9 ) @ A16 ) ) )
=> ? [X3: set @ A] :
( ( member @ ( set @ A ) @ X3 @ A16 )
& ! [Xa: set @ A] :
( ( member @ ( set @ A ) @ Xa @ A16 )
=> ( ( ord_less_eq @ ( set @ A ) @ X3 @ Xa )
=> ( Xa = X3 ) ) ) ) ) ) ).
% subset_Zorn_nonempty
thf(fact_6755_dependent__wellorder__choice,axiom,
! [B: $tType,A: $tType] :
( ( wellorder @ A )
=> ! [P: ( A > B ) > A > B > $o] :
( ! [R3: B,F3: A > B,G5: A > B,X3: A] :
( ! [Y4: A] :
( ( ord_less @ A @ Y4 @ X3 )
=> ( ( F3 @ Y4 )
= ( G5 @ Y4 ) ) )
=> ( ( P @ F3 @ X3 @ R3 )
= ( P @ G5 @ X3 @ R3 ) ) )
=> ( ! [X3: A,F3: A > B] :
( ! [Y4: A] :
( ( ord_less @ A @ Y4 @ X3 )
=> ( P @ F3 @ Y4 @ ( F3 @ Y4 ) ) )
=> ? [X_1: B] : ( P @ F3 @ X3 @ X_1 ) )
=> ? [F3: A > B] :
! [X4: A] : ( P @ F3 @ X4 @ ( F3 @ X4 ) ) ) ) ) ).
% dependent_wellorder_choice
thf(fact_6756_Union__in__chain,axiom,
! [A: $tType,B12: set @ ( set @ A ),A16: set @ ( set @ A )] :
( ( finite_finite2 @ ( set @ A ) @ B12 )
=> ( ( B12
!= ( bot_bot @ ( set @ ( set @ A ) ) ) )
=> ( ( pred_chain @ ( set @ A ) @ A16 @ ( ord_less @ ( set @ A ) ) @ B12 )
=> ( member @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ B12 ) @ B12 ) ) ) ) ).
% Union_in_chain
thf(fact_6757_Inter__in__chain,axiom,
! [A: $tType,B12: set @ ( set @ A ),A16: set @ ( set @ A )] :
( ( finite_finite2 @ ( set @ A ) @ B12 )
=> ( ( B12
!= ( bot_bot @ ( set @ ( set @ A ) ) ) )
=> ( ( pred_chain @ ( set @ A ) @ A16 @ ( ord_less @ ( set @ A ) ) @ B12 )
=> ( member @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ B12 ) @ B12 ) ) ) ) ).
% Inter_in_chain
thf(fact_6758_chain__subset__alt__def,axiom,
! [A: $tType] :
( ( chain_subset @ A )
= ( pred_chain @ ( set @ A ) @ ( top_top @ ( set @ ( set @ A ) ) ) @ ( ord_less @ ( set @ A ) ) ) ) ).
% chain_subset_alt_def
thf(fact_6759_chains__alt__def,axiom,
! [A: $tType] :
( ( chains @ A )
= ( ^ [A7: set @ ( set @ A )] : ( collect @ ( set @ ( set @ A ) ) @ ( pred_chain @ ( set @ A ) @ A7 @ ( ord_less @ ( set @ A ) ) ) ) ) ) ).
% chains_alt_def
thf(fact_6760_sup2CI,axiom,
! [A: $tType,B: $tType,B7: A > B > $o,X: A,Y3: B,A6: A > B > $o] :
( ( ~ ( B7 @ X @ Y3 )
=> ( A6 @ X @ Y3 ) )
=> ( sup_sup @ ( A > B > $o ) @ A6 @ B7 @ X @ Y3 ) ) ).
% sup2CI
thf(fact_6761_sup2I2,axiom,
! [A: $tType,B: $tType,B7: A > B > $o,X: A,Y3: B,A6: A > B > $o] :
( ( B7 @ X @ Y3 )
=> ( sup_sup @ ( A > B > $o ) @ A6 @ B7 @ X @ Y3 ) ) ).
% sup2I2
thf(fact_6762_sup2I1,axiom,
! [A: $tType,B: $tType,A6: A > B > $o,X: A,Y3: B,B7: A > B > $o] :
( ( A6 @ X @ Y3 )
=> ( sup_sup @ ( A > B > $o ) @ A6 @ B7 @ X @ Y3 ) ) ).
% sup2I1
thf(fact_6763_sup2E,axiom,
! [A: $tType,B: $tType,A6: A > B > $o,B7: A > B > $o,X: A,Y3: B] :
( ( sup_sup @ ( A > B > $o ) @ A6 @ B7 @ X @ Y3 )
=> ( ~ ( A6 @ X @ Y3 )
=> ( B7 @ X @ Y3 ) ) ) ).
% sup2E
thf(fact_6764_pred__on_Onot__maxchain__Some,axiom,
! [A: $tType,A6: set @ A,P: A > A > $o,C6: set @ A] :
( ( pred_chain @ A @ A6 @ P @ C6 )
=> ( ~ ( pred_maxchain @ A @ A6 @ P @ C6 )
=> ( ( pred_chain @ A @ A6 @ P
@ ( fChoice @ ( set @ A )
@ ^ [D7: set @ A] :
( ( pred_chain @ A @ A6 @ P @ D7 )
& ( ord_less @ ( set @ A ) @ C6 @ D7 ) ) ) )
& ( ord_less @ ( set @ A ) @ C6
@ ( fChoice @ ( set @ A )
@ ^ [D7: set @ A] :
( ( pred_chain @ A @ A6 @ P @ D7 )
& ( ord_less @ ( set @ A ) @ C6 @ D7 ) ) ) ) ) ) ) ).
% pred_on.not_maxchain_Some
thf(fact_6765_subset__maxchain__max,axiom,
! [A: $tType,A6: set @ ( set @ A ),C6: set @ ( set @ A ),X8: set @ A] :
( ( pred_maxchain @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ C6 )
=> ( ( member @ ( set @ A ) @ X8 @ A6 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ C6 ) @ X8 )
=> ( ( complete_Sup_Sup @ ( set @ A ) @ C6 )
= X8 ) ) ) ) ).
% subset_maxchain_max
thf(fact_6766_subset_OHausdorff,axiom,
! [A: $tType,A6: set @ ( set @ A )] :
? [X_12: set @ ( set @ A )] : ( pred_maxchain @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ X_12 ) ).
% subset.Hausdorff
thf(fact_6767_subset_Omaxchain__def,axiom,
! [A: $tType,A6: set @ ( set @ A ),C6: set @ ( set @ A )] :
( ( pred_maxchain @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ C6 )
= ( ( pred_chain @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ C6 )
& ~ ? [S7: set @ ( set @ A )] :
( ( pred_chain @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ S7 )
& ( ord_less @ ( set @ ( set @ A ) ) @ C6 @ S7 ) ) ) ) ).
% subset.maxchain_def
thf(fact_6768_subset_Omaxchain__imp__chain,axiom,
! [A: $tType,A6: set @ ( set @ A ),C6: set @ ( set @ A )] :
( ( pred_maxchain @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ C6 )
=> ( pred_chain @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ C6 ) ) ).
% subset.maxchain_imp_chain
thf(fact_6769_subset_Onot__maxchain__Some,axiom,
! [A: $tType,A6: set @ ( set @ A ),C6: set @ ( set @ A )] :
( ( pred_chain @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ C6 )
=> ( ~ ( pred_maxchain @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ C6 )
=> ( ( pred_chain @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) )
@ ( fChoice @ ( set @ ( set @ A ) )
@ ^ [D7: set @ ( set @ A )] :
( ( pred_chain @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ D7 )
& ( ord_less @ ( set @ ( set @ A ) ) @ C6 @ D7 ) ) ) )
& ( ord_less @ ( set @ ( set @ A ) ) @ C6
@ ( fChoice @ ( set @ ( set @ A ) )
@ ^ [D7: set @ ( set @ A )] :
( ( pred_chain @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ D7 )
& ( ord_less @ ( set @ ( set @ A ) ) @ C6 @ D7 ) ) ) ) ) ) ) ).
% subset.not_maxchain_Some
thf(fact_6770_pred__on_Omaxchain__def,axiom,
! [A: $tType] :
( ( pred_maxchain @ A )
= ( ^ [A7: set @ A,P4: A > A > $o,C10: set @ A] :
( ( pred_chain @ A @ A7 @ P4 @ C10 )
& ~ ? [S7: set @ A] :
( ( pred_chain @ A @ A7 @ P4 @ S7 )
& ( ord_less @ ( set @ A ) @ C10 @ S7 ) ) ) ) ) ).
% pred_on.maxchain_def
thf(fact_6771_pred__on_Osuc__def,axiom,
! [A: $tType] :
( ( pred_suc @ A )
= ( ^ [A7: set @ A,P4: A > A > $o,C10: set @ A] :
( if @ ( set @ A )
@ ( ~ ( pred_chain @ A @ A7 @ P4 @ C10 )
| ( pred_maxchain @ A @ A7 @ P4 @ C10 ) )
@ C10
@ ( fChoice @ ( set @ A )
@ ^ [D7: set @ A] :
( ( pred_chain @ A @ A7 @ P4 @ D7 )
& ( ord_less @ ( set @ A ) @ C10 @ D7 ) ) ) ) ) ) ).
% pred_on.suc_def
thf(fact_6772_subset_Osuc__def,axiom,
! [A: $tType,A6: set @ ( set @ A ),C6: set @ ( set @ A )] :
( ( ( ~ ( pred_chain @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ C6 )
| ( pred_maxchain @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ C6 ) )
=> ( ( pred_suc @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ C6 )
= C6 ) )
& ( ~ ( ~ ( pred_chain @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ C6 )
| ( pred_maxchain @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ C6 ) )
=> ( ( pred_suc @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ C6 )
= ( fChoice @ ( set @ ( set @ A ) )
@ ^ [D7: set @ ( set @ A )] :
( ( pred_chain @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ D7 )
& ( ord_less @ ( set @ ( set @ A ) ) @ C6 @ D7 ) ) ) ) ) ) ).
% subset.suc_def
thf(fact_6773_subset_Onot__chain__suc,axiom,
! [A: $tType,A6: set @ ( set @ A ),X8: set @ ( set @ A )] :
( ~ ( pred_chain @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ X8 )
=> ( ( pred_suc @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ X8 )
= X8 ) ) ).
% subset.not_chain_suc
thf(fact_6774_subset_Omaxchain__suc,axiom,
! [A: $tType,A6: set @ ( set @ A ),X8: set @ ( set @ A )] :
( ( pred_maxchain @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ X8 )
=> ( ( pred_suc @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ X8 )
= X8 ) ) ).
% subset.maxchain_suc
thf(fact_6775_subset_Ochain__suc,axiom,
! [A: $tType,A6: set @ ( set @ A ),X8: set @ ( set @ A )] :
( ( pred_chain @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ X8 )
=> ( pred_chain @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ ( pred_suc @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ X8 ) ) ) ).
% subset.chain_suc
thf(fact_6776_subset_Osubset__suc,axiom,
! [A: $tType,X8: set @ ( set @ A ),Y7: set @ ( set @ A ),A6: set @ ( set @ A )] :
( ( ord_less_eq @ ( set @ ( set @ A ) ) @ X8 @ Y7 )
=> ( ord_less_eq @ ( set @ ( set @ A ) ) @ X8 @ ( pred_suc @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ Y7 ) ) ) ).
% subset.subset_suc
thf(fact_6777_subset_Osuc__subset,axiom,
! [A: $tType,X8: set @ ( set @ A ),A6: set @ ( set @ A )] : ( ord_less_eq @ ( set @ ( set @ A ) ) @ X8 @ ( pred_suc @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ X8 ) ) ).
% subset.suc_subset
thf(fact_6778_subset_Osuc__in__carrier,axiom,
! [A: $tType,X8: set @ ( set @ A ),A6: set @ ( set @ A )] :
( ( ord_less_eq @ ( set @ ( set @ A ) ) @ X8 @ A6 )
=> ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( pred_suc @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ X8 ) @ A6 ) ) ).
% subset.suc_in_carrier
thf(fact_6779_subset_Ochain__sucD,axiom,
! [A: $tType,A6: set @ ( set @ A ),X8: set @ ( set @ A )] :
( ( pred_chain @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ X8 )
=> ( ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( pred_suc @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ X8 ) @ A6 )
& ( pred_chain @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ ( pred_suc @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ X8 ) ) ) ) ).
% subset.chain_sucD
thf(fact_6780_subset_Osuc__not__equals,axiom,
! [A: $tType,A6: set @ ( set @ A ),C6: set @ ( set @ A )] :
( ( pred_chain @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ C6 )
=> ( ~ ( pred_maxchain @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ C6 )
=> ( ( pred_suc @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ C6 )
!= C6 ) ) ) ).
% subset.suc_not_equals
thf(fact_6781_alloc__def,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( ( array_alloc @ A )
= ( ^ [Xs3: list @ A,H4: heap_ext @ product_unit] :
( product_Pair @ ( array @ A ) @ ( heap_ext @ product_unit ) @ ( array2 @ A @ ( lim @ product_unit @ H4 ) )
@ ( array_set @ A @ ( array2 @ A @ ( lim @ product_unit @ H4 ) ) @ Xs3
@ ( lim_update @ product_unit
@ ^ [Uu: nat] : ( plus_plus @ nat @ ( lim @ product_unit @ H4 ) @ ( one_one @ nat ) )
@ H4 ) ) ) ) ) ) ).
% alloc_def
thf(fact_6782_Code__Numeral_Osub__def,axiom,
( code_sub
= ( map_fun @ num @ num @ ( num > int ) @ ( num > code_integer ) @ ( id @ num ) @ ( map_fun @ num @ num @ int @ code_integer @ ( id @ num ) @ code_integer_of_int )
@ ^ [M2: num,N3: num] : ( minus_minus @ int @ ( numeral_numeral @ int @ M2 ) @ ( numeral_numeral @ int @ N3 ) ) ) ) ).
% Code_Numeral.sub_def
thf(fact_6783_present__def,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( ( array_present @ A )
= ( ^ [H4: heap_ext @ product_unit,A5: array @ A] : ( ord_less @ nat @ ( addr_of_array @ A @ A5 ) @ ( lim @ product_unit @ H4 ) ) ) ) ) ).
% present_def
thf(fact_6784_array_Osize_I2_J,axiom,
! [A: $tType,X: nat] :
( ( size_size @ ( array @ A ) @ ( array2 @ A @ X ) )
= ( zero_zero @ nat ) ) ).
% array.size(2)
thf(fact_6785_array_Osize__gen,axiom,
! [A: $tType,Xa3: A > nat,X: nat] :
( ( size_array @ A @ Xa3 @ ( array2 @ A @ X ) )
= ( zero_zero @ nat ) ) ).
% array.size_gen
thf(fact_6786_noteq__def,axiom,
! [B: $tType,A: $tType] :
( ( ( heap @ A )
& ( heap @ B ) )
=> ( ( array_noteq @ A @ B )
= ( ^ [R5: array @ A,S6: array @ B] :
( ( ( typerep_typerep @ A @ ( type2 @ A ) )
!= ( typerep_typerep @ B @ ( type2 @ B ) ) )
| ( ( addr_of_array @ A @ R5 )
!= ( addr_of_array @ B @ S6 ) ) ) ) ) ) ).
% noteq_def
thf(fact_6787_get__def,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( ( array_get @ A )
= ( ^ [H4: heap_ext @ product_unit,A5: array @ A] : ( map @ nat @ A @ ( from_nat @ A ) @ ( arrays @ product_unit @ H4 @ ( typerep_typerep @ A @ ( type2 @ A ) ) @ ( addr_of_array @ A @ A5 ) ) ) ) ) ) ).
% get_def
thf(fact_6788_Heap_Oempty__def,axiom,
( empty
= ( heap_ext2 @ product_unit
@ ^ [Uu: typerep,Uv: nat] : ( nil @ nat )
@ ^ [Uu: typerep,Uv: nat] : ( zero_zero @ nat )
@ ( zero_zero @ nat )
@ product_Unity ) ) ).
% Heap.empty_def
thf(fact_6789_coset__def,axiom,
! [A: $tType] :
( ( coset @ A )
= ( ^ [Xs3: list @ A] : ( uminus_uminus @ ( set @ A ) @ ( set2 @ A @ Xs3 ) ) ) ) ).
% coset_def
thf(fact_6790_compl__coset,axiom,
! [A: $tType,Xs: list @ A] :
( ( uminus_uminus @ ( set @ A ) @ ( coset @ A @ Xs ) )
= ( set2 @ A @ Xs ) ) ).
% compl_coset
thf(fact_6791_Nitpick_Otranclp__unfold,axiom,
! [A: $tType] :
( ( transitive_tranclp @ A )
= ( ^ [R5: A > A > $o,A5: A,B3: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A5 @ B3 ) @ ( transitive_trancl @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ R5 ) ) ) ) ) ) ).
% Nitpick.tranclp_unfold
thf(fact_6792_Field__natLeq__on,axiom,
! [N: nat] :
( ( field2 @ nat
@ ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [X2: nat,Y6: nat] :
( ( ord_less @ nat @ X2 @ N )
& ( ord_less @ nat @ Y6 @ N )
& ( ord_less_eq @ nat @ X2 @ Y6 ) ) ) ) )
= ( collect @ nat
@ ^ [X2: nat] : ( ord_less @ nat @ X2 @ N ) ) ) ).
% Field_natLeq_on
thf(fact_6793_less__nat__rel,axiom,
( ( ord_less @ nat )
= ( transitive_tranclp @ nat
@ ^ [M2: nat,N3: nat] :
( N3
= ( suc @ M2 ) ) ) ) ).
% less_nat_rel
thf(fact_6794_tranclp__power,axiom,
! [A: $tType] :
( ( transitive_tranclp @ A )
= ( ^ [P4: A > A > $o,X2: A,Y6: A] :
? [N3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
& ( compow @ ( A > A > $o ) @ N3 @ P4 @ X2 @ Y6 ) ) ) ) ).
% tranclp_power
thf(fact_6795_underS__Field3,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A3: A] :
( ( ( field2 @ A @ R )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less @ ( set @ A ) @ ( order_underS @ A @ R @ A3 ) @ ( field2 @ A @ R ) ) ) ).
% underS_Field3
thf(fact_6796_underS__Field2,axiom,
! [A: $tType,A3: A,R: set @ ( product_prod @ A @ A )] :
( ( member @ A @ A3 @ ( field2 @ A @ R ) )
=> ( ord_less @ ( set @ A ) @ ( order_underS @ A @ R @ A3 ) @ ( field2 @ A @ R ) ) ) ).
% underS_Field2
thf(fact_6797_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,Y6: nat] :
( ( ord_less @ nat @ X2 @ N )
& ( ord_less @ nat @ Y6 @ N )
& ( ord_less_eq @ nat @ X2 @ Y6 ) ) ) ) )
@ ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [X2: nat,Y6: nat] :
( ( ord_less @ nat @ X2 @ N )
& ( ord_less @ nat @ Y6 @ N )
& ( ord_less_eq @ nat @ X2 @ Y6 ) ) ) ) ) ).
% natLeq_on_Well_order
thf(fact_6798_Quotient3__int,axiom,
quotient3 @ ( product_prod @ nat @ nat ) @ int @ intrel @ abs_Integ @ rep_Integ ).
% Quotient3_int
thf(fact_6799_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,Y6: nat] :
( ( ord_less @ nat @ X2 @ N )
& ( ord_less @ nat @ Y6 @ N )
& ( ord_less_eq @ nat @ X2 @ Y6 ) ) ) ) ) ).
% natLeq_on_well_order_on
thf(fact_6800_embedS__Field,axiom,
! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B ),F: A > B] :
( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( bNF_Wellorder_embedS @ A @ B @ R @ R4 @ F )
=> ( ord_less @ ( set @ B ) @ ( image @ A @ B @ F @ ( field2 @ A @ R ) ) @ ( field2 @ B @ R4 ) ) ) ) ).
% embedS_Field
thf(fact_6801_list__all__length,axiom,
! [A: $tType] :
( ( list_all @ A )
= ( ^ [P4: A > $o,Xs3: list @ A] :
! [N3: nat] :
( ( ord_less @ nat @ N3 @ ( size_size @ ( list @ A ) @ Xs3 ) )
=> ( P4 @ ( nth @ A @ Xs3 @ N3 ) ) ) ) ) ).
% list_all_length
thf(fact_6802_ofilter__subset__embedS,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A6: set @ A,B7: set @ A] :
( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( order_ofilter @ A @ R @ A6 )
=> ( ( order_ofilter @ A @ R @ B7 )
=> ( ( ord_less @ ( set @ A ) @ A6 @ B7 )
= ( bNF_Wellorder_embedS @ A @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R
@ ( product_Sigma @ A @ A @ A6
@ ^ [Uu: A] : A6 ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R
@ ( product_Sigma @ A @ A @ B7
@ ^ [Uu: A] : B7 ) )
@ ( id @ A ) ) ) ) ) ) ).
% ofilter_subset_embedS
thf(fact_6803_embedS__iff,axiom,
! [A: $tType,B: $tType,R: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ B @ B ),F: A > B] :
( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( bNF_Wellorder_embed @ A @ B @ R @ R4 @ F )
=> ( ( bNF_Wellorder_embedS @ A @ B @ R @ R4 @ F )
= ( ord_less @ ( set @ B ) @ ( image @ A @ B @ F @ ( field2 @ A @ R ) ) @ ( field2 @ B @ R4 ) ) ) ) ) ).
% embedS_iff
thf(fact_6804_ofilterIncl__def,axiom,
! [A: $tType] :
( ( bNF_We413866401316099525erIncl @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ ( set @ A ) @ ( set @ A ) )
@ ( product_case_prod @ ( set @ A ) @ ( set @ A ) @ $o
@ ^ [A7: set @ A,B6: set @ A] :
( ( order_ofilter @ A @ R5 @ A7 )
& ( A7
!= ( field2 @ A @ R5 ) )
& ( order_ofilter @ A @ R5 @ B6 )
& ( B6
!= ( field2 @ A @ R5 ) )
& ( ord_less @ ( set @ A ) @ A7 @ B6 ) ) ) ) ) ) ).
% ofilterIncl_def
thf(fact_6805_ofilter__subset__embedS__iso,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A6: set @ A,B7: set @ A] :
( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( order_ofilter @ A @ R @ A6 )
=> ( ( order_ofilter @ A @ R @ B7 )
=> ( ( ( ord_less @ ( set @ A ) @ A6 @ B7 )
= ( bNF_Wellorder_embedS @ A @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R
@ ( product_Sigma @ A @ A @ A6
@ ^ [Uu: A] : A6 ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R
@ ( product_Sigma @ A @ A @ B7
@ ^ [Uu: A] : B7 ) )
@ ( id @ A ) ) )
& ( ( A6 = B7 )
= ( bNF_Wellorder_iso @ A @ A
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R
@ ( product_Sigma @ A @ A @ A6
@ ^ [Uu: A] : A6 ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R
@ ( product_Sigma @ A @ A @ B7
@ ^ [Uu: A] : B7 ) )
@ ( id @ A ) ) ) ) ) ) ) ).
% ofilter_subset_embedS_iso
thf(fact_6806_ofilter__ordLess,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A6: set @ A] :
( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( order_ofilter @ A @ R @ A6 )
=> ( ( ord_less @ ( set @ A ) @ A6 @ ( 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 @ A6
@ ^ [Uu: A] : A6 ) )
@ R )
@ ( bNF_We4044943003108391690rdLess @ A @ A ) ) ) ) ) ).
% ofilter_ordLess
thf(fact_6807_ofilter__subset__ordLess,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A6: set @ A,B7: set @ A] :
( ( order_well_order_on @ A @ ( field2 @ A @ R ) @ R )
=> ( ( order_ofilter @ A @ R @ A6 )
=> ( ( order_ofilter @ A @ R @ B7 )
=> ( ( ord_less @ ( set @ A ) @ A6 @ B7 )
= ( 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 @ A6
@ ^ [Uu: A] : A6 ) )
@ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ R
@ ( product_Sigma @ A @ A @ B7
@ ^ [Uu: A] : B7 ) ) )
@ ( bNF_We4044943003108391690rdLess @ A @ A ) ) ) ) ) ) ).
% ofilter_subset_ordLess
thf(fact_6808_bsqr__ofilter,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),D4: 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 ) @ D4 )
=> ( ( ord_less @ ( set @ ( product_prod @ A @ A ) ) @ D4
@ ( product_Sigma @ A @ A @ ( field2 @ A @ R )
@ ^ [Uu: A] : ( field2 @ A @ R ) ) )
=> ( ~ ? [A8: A] :
( ( field2 @ A @ R )
= ( order_under @ A @ R @ A8 ) )
=> ? [A9: set @ A] :
( ( order_ofilter @ A @ R @ A9 )
& ( ord_less @ ( set @ A ) @ A9 @ ( field2 @ A @ R ) )
& ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ D4
@ ( product_Sigma @ A @ A @ A9
@ ^ [Uu: A] : A9 ) ) ) ) ) ) ) ).
% bsqr_ofilter
thf(fact_6809_internalize__ordLess,axiom,
! [A: $tType,B: $tType,R4: 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 ) ) @ R4 @ R ) @ ( bNF_We4044943003108391690rdLess @ A @ B ) )
= ( ? [P6: set @ ( product_prod @ B @ B )] :
( ( ord_less @ ( set @ B ) @ ( field2 @ B @ P6 ) @ ( 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 ) ) @ R4 @ P6 ) @ ( bNF_Wellorder_ordIso @ A @ B ) )
& ( member @ ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) @ P6 @ R ) @ ( bNF_We4044943003108391690rdLess @ B @ B ) ) ) ) ) ).
% internalize_ordLess
thf(fact_6810_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_6811_type__definition__natural,axiom,
type_definition @ code_natural @ nat @ code_nat_of_natural @ code_natural_of_nat @ ( top_top @ ( set @ nat ) ) ).
% type_definition_natural
thf(fact_6812_less__int__code_I3_J,axiom,
! [L: num] :
~ ( ord_less @ int @ ( zero_zero @ int ) @ ( neg @ L ) ) ).
% less_int_code(3)
thf(fact_6813_less__int__code_I7_J,axiom,
! [K: num] : ( ord_less @ int @ ( neg @ K ) @ ( zero_zero @ int ) ) ).
% less_int_code(7)
thf(fact_6814_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_6815_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_6816_nat__code_I1_J,axiom,
! [K: num] :
( ( nat2 @ ( neg @ K ) )
= ( zero_zero @ nat ) ) ).
% nat_code(1)
thf(fact_6817_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_6818_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_6819_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_6820_Int_Osub__code_I4_J,axiom,
! [N: num] :
( ( sub @ one2 @ ( bit0 @ N ) )
= ( neg @ ( bitM @ N ) ) ) ).
% Int.sub_code(4)
thf(fact_6821_Int_Osub__code_I5_J,axiom,
! [N: num] :
( ( sub @ one2 @ ( bit1 @ N ) )
= ( neg @ ( bit0 @ N ) ) ) ).
% Int.sub_code(5)
thf(fact_6822_Int_Osub__code_I1_J,axiom,
( ( sub @ one2 @ one2 )
= ( zero_zero @ int ) ) ).
% Int.sub_code(1)
thf(fact_6823_Int_Osub__def,axiom,
( sub
= ( ^ [M2: num,N3: num] : ( minus_minus @ int @ ( numeral_numeral @ int @ M2 ) @ ( numeral_numeral @ int @ N3 ) ) ) ) ).
% Int.sub_def
thf(fact_6824_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_6825_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_6826_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_6827_Int_Odup__code_I1_J,axiom,
( ( dup @ ( zero_zero @ int ) )
= ( zero_zero @ int ) ) ).
% Int.dup_code(1)
thf(fact_6828_Int_Odup__def,axiom,
( dup
= ( ^ [K3: int] : ( plus_plus @ int @ K3 @ K3 ) ) ) ).
% Int.dup_def
thf(fact_6829_Int_Odup__code_I3_J,axiom,
! [N: num] :
( ( dup @ ( neg @ N ) )
= ( neg @ ( bit0 @ N ) ) ) ).
% Int.dup_code(3)
thf(fact_6830_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_6831_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_6832_Int_Osub__code_I2_J,axiom,
! [M: num] :
( ( sub @ ( bit0 @ M ) @ one2 )
= ( pos @ ( bitM @ M ) ) ) ).
% Int.sub_code(2)
thf(fact_6833_Int_Osub__code_I3_J,axiom,
! [M: num] :
( ( sub @ ( bit1 @ M ) @ one2 )
= ( pos @ ( bit0 @ M ) ) ) ).
% Int.sub_code(3)
thf(fact_6834_less__int__code_I8_J,axiom,
! [K: num,L: num] : ( ord_less @ int @ ( neg @ K ) @ ( pos @ L ) ) ).
% less_int_code(8)
thf(fact_6835_less__int__code_I6_J,axiom,
! [K: num,L: num] :
~ ( ord_less @ int @ ( pos @ K ) @ ( neg @ L ) ) ).
% less_int_code(6)
thf(fact_6836_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_6837_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_6838_Int_ONeg__def,axiom,
( neg
= ( ^ [N3: num] : ( uminus_uminus @ int @ ( pos @ N3 ) ) ) ) ).
% Int.Neg_def
thf(fact_6839_uminus__int__code_I2_J,axiom,
! [M: num] :
( ( uminus_uminus @ int @ ( pos @ M ) )
= ( neg @ M ) ) ).
% uminus_int_code(2)
thf(fact_6840_uminus__int__code_I3_J,axiom,
! [M: num] :
( ( uminus_uminus @ int @ ( neg @ M ) )
= ( pos @ M ) ) ).
% uminus_int_code(3)
thf(fact_6841_nat__code_I3_J,axiom,
! [K: num] :
( ( nat2 @ ( pos @ K ) )
= ( nat_of_num @ K ) ) ).
% nat_code(3)
thf(fact_6842_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_6843_Int_OPos__def,axiom,
( pos
= ( numeral_numeral @ int ) ) ).
% Int.Pos_def
thf(fact_6844_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_6845_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_6846_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_6847_one__int__code,axiom,
( ( one_one @ int )
= ( pos @ one2 ) ) ).
% one_int_code
thf(fact_6848_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_6849_less__int__code_I4_J,axiom,
! [K: num] :
~ ( ord_less @ int @ ( pos @ K ) @ ( zero_zero @ int ) ) ).
% less_int_code(4)
thf(fact_6850_less__int__code_I2_J,axiom,
! [L: num] : ( ord_less @ int @ ( zero_zero @ int ) @ ( pos @ L ) ) ).
% less_int_code(2)
thf(fact_6851_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_6852_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_6853_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_6854_Int_Odup__code_I2_J,axiom,
! [N: num] :
( ( dup @ ( pos @ N ) )
= ( pos @ ( bit0 @ N ) ) ) ).
% Int.dup_code(2)
thf(fact_6855_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_6856_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_6857_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_6858_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_6859_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_6860_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_6861_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_6862_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_6863_Least__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_Least @ A )
= ( ^ [P4: A > $o] :
( the @ A
@ ^ [X2: A] :
( ( P4 @ X2 )
& ! [Y6: A] :
( ( P4 @ Y6 )
=> ( ord_less_eq @ A @ X2 @ Y6 ) ) ) ) ) ) ) ).
% Least_def
thf(fact_6864_Ball__def__raw,axiom,
! [A: $tType] :
( ( ball @ A )
= ( ^ [A7: set @ A,P4: A > $o] :
! [X2: A] :
( ( member @ A @ X2 @ A7 )
=> ( P4 @ X2 ) ) ) ) ).
% Ball_def_raw
thf(fact_6865_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_6866_Greatest__def,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( order_Greatest @ A )
= ( ^ [P4: A > $o] :
( the @ A
@ ^ [X2: A] :
( ( P4 @ X2 )
& ! [Y6: A] :
( ( P4 @ Y6 )
=> ( ord_less_eq @ A @ Y6 @ X2 ) ) ) ) ) ) ) ).
% Greatest_def
thf(fact_6867_ord_OLeast__def,axiom,
! [A: $tType] :
( ( least @ A )
= ( ^ [Less_eq2: A > A > $o,P4: A > $o] :
( the @ A
@ ^ [X2: A] :
( ( P4 @ X2 )
& ! [Y6: A] :
( ( P4 @ Y6 )
=> ( Less_eq2 @ X2 @ Y6 ) ) ) ) ) ) ).
% ord.Least_def
thf(fact_6868_ord_OLeast_Ocong,axiom,
! [A: $tType] :
( ( least @ A )
= ( least @ A ) ) ).
% ord.Least.cong
thf(fact_6869_GreatestI__nat,axiom,
! [P: nat > $o,K: nat,B2: nat] :
( ( P @ K )
=> ( ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq @ nat @ Y5 @ B2 ) )
=> ( P @ ( order_Greatest @ nat @ P ) ) ) ) ).
% GreatestI_nat
thf(fact_6870_Greatest__le__nat,axiom,
! [P: nat > $o,K: nat,B2: nat] :
( ( P @ K )
=> ( ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq @ nat @ Y5 @ B2 ) )
=> ( ord_less_eq @ nat @ K @ ( order_Greatest @ nat @ P ) ) ) ) ).
% Greatest_le_nat
thf(fact_6871_GreatestI__ex__nat,axiom,
! [P: nat > $o,B2: nat] :
( ? [X_1: nat] : ( P @ X_1 )
=> ( ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq @ nat @ Y5 @ B2 ) )
=> ( P @ ( order_Greatest @ nat @ P ) ) ) ) ).
% GreatestI_ex_nat
thf(fact_6872_GreatestI2__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P: A > $o,X: A,Q: A > $o] :
( ( P @ X )
=> ( ! [Y5: A] :
( ( P @ Y5 )
=> ( ord_less_eq @ A @ Y5 @ X ) )
=> ( ! [X3: A] :
( ( P @ X3 )
=> ( ! [Y4: A] :
( ( P @ Y4 )
=> ( ord_less_eq @ A @ Y4 @ X3 ) )
=> ( Q @ X3 ) ) )
=> ( Q @ ( order_Greatest @ A @ P ) ) ) ) ) ) ).
% GreatestI2_order
thf(fact_6873_Greatest__equality,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P: A > $o,X: A] :
( ( P @ X )
=> ( ! [Y5: A] :
( ( P @ Y5 )
=> ( ord_less_eq @ A @ Y5 @ X ) )
=> ( ( order_Greatest @ A @ P )
= X ) ) ) ) ).
% Greatest_equality
thf(fact_6874_sum__encode__def,axiom,
( nat_sum_encode
= ( sum_case_sum @ nat @ nat @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
@ ^ [B3: nat] : ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ B3 ) ) ) ) ).
% sum_encode_def
thf(fact_6875_disjnt__ge__max,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Y7: set @ A,X8: set @ A] :
( ( finite_finite2 @ A @ Y7 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ X8 )
=> ( ord_less @ A @ ( lattic643756798349783984er_Max @ A @ Y7 ) @ X3 ) )
=> ( disjnt @ A @ X8 @ Y7 ) ) ) ) ).
% disjnt_ge_max
thf(fact_6876_inj__sum__encode,axiom,
! [A6: set @ ( sum_sum @ nat @ nat )] : ( inj_on @ ( sum_sum @ nat @ nat ) @ nat @ nat_sum_encode @ A6 ) ).
% inj_sum_encode
thf(fact_6877_sum__encode__eq,axiom,
! [X: sum_sum @ nat @ nat,Y3: sum_sum @ nat @ nat] :
( ( ( nat_sum_encode @ X )
= ( nat_sum_encode @ Y3 ) )
= ( X = Y3 ) ) ).
% sum_encode_eq
thf(fact_6878_bij__sum__encode,axiom,
bij_betw @ ( sum_sum @ nat @ nat ) @ nat @ nat_sum_encode @ ( top_top @ ( set @ ( sum_sum @ nat @ nat ) ) ) @ ( top_top @ ( set @ nat ) ) ).
% bij_sum_encode
thf(fact_6879_surj__sum__encode,axiom,
( ( image @ ( sum_sum @ nat @ nat ) @ nat @ nat_sum_encode @ ( top_top @ ( set @ ( sum_sum @ nat @ nat ) ) ) )
= ( top_top @ ( set @ nat ) ) ) ).
% surj_sum_encode
thf(fact_6880_int__encode__def,axiom,
( nat_int_encode
= ( ^ [I5: int] : ( nat_sum_encode @ ( if @ ( sum_sum @ nat @ nat ) @ ( ord_less_eq @ int @ ( zero_zero @ int ) @ I5 ) @ ( sum_Inl @ nat @ nat @ ( nat2 @ I5 ) ) @ ( sum_Inr @ nat @ nat @ ( nat2 @ ( minus_minus @ int @ ( uminus_uminus @ int @ I5 ) @ ( one_one @ int ) ) ) ) ) ) ) ) ).
% int_encode_def
thf(fact_6881_ordering__top_Oaxioms_I2_J,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,Top: A] :
( ( ordering_top @ A @ Less_eq @ Less @ Top )
=> ( ordering_top_axioms @ A @ Less_eq @ Top ) ) ).
% ordering_top.axioms(2)
thf(fact_6882_prod_OPlus,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A6: set @ B,B7: set @ C,G: ( sum_sum @ B @ C ) > A] :
( ( finite_finite2 @ B @ A6 )
=> ( ( finite_finite2 @ C @ B7 )
=> ( ( groups7121269368397514597t_prod @ ( sum_sum @ B @ C ) @ A @ G @ ( sum_Plus @ B @ C @ A6 @ B7 ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ ( comp @ ( sum_sum @ B @ C ) @ A @ B @ G @ ( sum_Inl @ B @ C ) ) @ A6 ) @ ( groups7121269368397514597t_prod @ C @ A @ ( comp @ ( sum_sum @ B @ C ) @ A @ C @ G @ ( sum_Inr @ C @ B ) ) @ B7 ) ) ) ) ) ) ).
% prod.Plus
thf(fact_6883_ordering__top__axioms__def,axiom,
! [A: $tType] :
( ( ordering_top_axioms @ A )
= ( ^ [Less_eq2: A > A > $o,Top2: A] :
! [A5: A] : ( Less_eq2 @ A5 @ Top2 ) ) ) ).
% ordering_top_axioms_def
thf(fact_6884_ordering__top__axioms_Ointro,axiom,
! [A: $tType,Less_eq: A > A > $o,Top: A] :
( ! [A8: A] : ( Less_eq @ A8 @ Top )
=> ( ordering_top_axioms @ A @ Less_eq @ Top ) ) ).
% ordering_top_axioms.intro
thf(fact_6885_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_6886_le__sum__encode__Inl,axiom,
! [X: nat,Y3: nat] :
( ( ord_less_eq @ nat @ X @ Y3 )
=> ( ord_less_eq @ nat @ X @ ( nat_sum_encode @ ( sum_Inl @ nat @ nat @ Y3 ) ) ) ) ).
% le_sum_encode_Inl
thf(fact_6887_le__sum__encode__Inr,axiom,
! [X: nat,Y3: nat] :
( ( ord_less_eq @ nat @ X @ Y3 )
=> ( ord_less_eq @ nat @ X @ ( nat_sum_encode @ ( sum_Inr @ nat @ nat @ Y3 ) ) ) ) ).
% le_sum_encode_Inr
thf(fact_6888_sum_Osize_I3_J,axiom,
! [A: $tType,B: $tType,X13: A] :
( ( size_size @ ( sum_sum @ A @ B ) @ ( sum_Inl @ A @ B @ X13 ) )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% sum.size(3)
thf(fact_6889_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_6890_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_6891_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_6892_sum_Osize__gen_I1_J,axiom,
! [B: $tType,A: $tType,Xa3: A > nat,X: B > nat,X13: A] :
( ( basic_BNF_size_sum @ A @ B @ Xa3 @ X @ ( sum_Inl @ A @ B @ X13 ) )
= ( plus_plus @ nat @ ( Xa3 @ X13 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% sum.size_gen(1)
thf(fact_6893_sum_Osize__gen_I2_J,axiom,
! [A: $tType,B: $tType,Xa3: A > nat,X: B > nat,X22: B] :
( ( basic_BNF_size_sum @ A @ B @ Xa3 @ X @ ( sum_Inr @ B @ A @ X22 ) )
= ( plus_plus @ nat @ ( X @ X22 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% sum.size_gen(2)
thf(fact_6894_sum__decode__def,axiom,
( nat_sum_decode
= ( ^ [N3: nat] : ( if @ ( sum_sum @ nat @ nat ) @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) @ ( sum_Inl @ nat @ nat @ ( divide_divide @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( sum_Inr @ nat @ nat @ ( divide_divide @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% sum_decode_def
thf(fact_6895_sum__decode__inverse,axiom,
! [N: nat] :
( ( nat_sum_encode @ ( nat_sum_decode @ N ) )
= N ) ).
% sum_decode_inverse
thf(fact_6896_sum__encode__inverse,axiom,
! [X: sum_sum @ nat @ nat] :
( ( nat_sum_decode @ ( nat_sum_encode @ X ) )
= X ) ).
% sum_encode_inverse
thf(fact_6897_sum__decode__eq,axiom,
! [X: nat,Y3: nat] :
( ( ( nat_sum_decode @ X )
= ( nat_sum_decode @ Y3 ) )
= ( X = Y3 ) ) ).
% sum_decode_eq
thf(fact_6898_inj__sum__decode,axiom,
! [A6: set @ nat] : ( inj_on @ nat @ ( sum_sum @ nat @ nat ) @ nat_sum_decode @ A6 ) ).
% inj_sum_decode
thf(fact_6899_surj__sum__decode,axiom,
( ( image @ nat @ ( sum_sum @ nat @ nat ) @ nat_sum_decode @ ( top_top @ ( set @ nat ) ) )
= ( top_top @ ( set @ ( sum_sum @ nat @ nat ) ) ) ) ).
% surj_sum_decode
thf(fact_6900_bij__sum__decode,axiom,
bij_betw @ nat @ ( sum_sum @ nat @ nat ) @ nat_sum_decode @ ( top_top @ ( set @ nat ) ) @ ( top_top @ ( set @ ( sum_sum @ nat @ nat ) ) ) ).
% bij_sum_decode
thf(fact_6901_int__decode__def,axiom,
( nat_int_decode
= ( ^ [N3: nat] :
( sum_case_sum @ nat @ int @ nat @ ( semiring_1_of_nat @ int )
@ ^ [B3: nat] : ( minus_minus @ int @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ B3 ) ) @ ( one_one @ int ) )
@ ( nat_sum_decode @ N3 ) ) ) ) ).
% int_decode_def
thf(fact_6902_Node__def,axiom,
! [A: $tType,B: $tType] :
( ( old_Node @ B @ A )
= ( collect @ ( product_prod @ ( nat > ( sum_sum @ B @ nat ) ) @ ( sum_sum @ A @ nat ) )
@ ^ [P6: product_prod @ ( nat > ( sum_sum @ B @ nat ) ) @ ( sum_sum @ A @ nat )] :
? [F2: nat > ( sum_sum @ B @ nat ),X2: sum_sum @ A @ nat,K3: nat] :
( ( P6
= ( 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_6903_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_6904_Push__neq__K0,axiom,
! [A: $tType,K: nat,F: nat > ( sum_sum @ A @ nat )] :
( ( old_Push @ A @ ( sum_Inr @ nat @ A @ ( suc @ K ) ) @ F )
!= ( ^ [Z7: nat] : ( sum_Inr @ nat @ A @ ( zero_zero @ nat ) ) ) ) ).
% Push_neq_K0
thf(fact_6905_ndepth__def,axiom,
! [B: $tType,A: $tType] :
( ( old_ndepth @ A @ B )
= ( ^ [N3: 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 @ N3 ) ) ) ) ).
% ndepth_def
thf(fact_6906_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_6907_ndepth__Push__Node,axiom,
! [B: $tType,A: $tType,I: nat,N: old_node @ A @ B] :
( ( old_ndepth @ A @ B @ ( old_Push_Node @ B @ A @ ( sum_Inr @ nat @ B @ ( suc @ I ) ) @ N ) )
= ( suc @ ( old_ndepth @ A @ B @ N ) ) ) ).
% ndepth_Push_Node
thf(fact_6908_Atom__def,axiom,
! [B: $tType,A: $tType] :
( ( old_Atom @ A @ B )
= ( ^ [X2: sum_sum @ A @ nat] :
( insert @ ( 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_6909_Scons__def,axiom,
! [B: $tType,A: $tType] :
( ( old_Scons @ A @ B )
= ( ^ [M9: set @ ( old_node @ A @ B ),N7: set @ ( old_node @ A @ B )] : ( sup_sup @ ( set @ ( old_node @ A @ B ) ) @ ( image @ ( old_node @ A @ B ) @ ( old_node @ A @ B ) @ ( old_Push_Node @ B @ A @ ( sum_Inr @ nat @ B @ ( one_one @ nat ) ) ) @ M9 ) @ ( image @ ( old_node @ A @ B ) @ ( old_node @ A @ B ) @ ( old_Push_Node @ B @ A @ ( sum_Inr @ nat @ B @ ( suc @ ( one_one @ nat ) ) ) ) @ N7 ) ) ) ) ).
% Scons_def
thf(fact_6910_ntrunc__0,axiom,
! [B: $tType,A: $tType,M7: set @ ( old_node @ A @ B )] :
( ( old_ntrunc @ A @ B @ ( zero_zero @ nat ) @ M7 )
= ( bot_bot @ ( set @ ( old_node @ A @ B ) ) ) ) ).
% ntrunc_0
thf(fact_6911_ntrunc__Atom,axiom,
! [B: $tType,A: $tType,K: nat,A3: sum_sum @ A @ nat] :
( ( old_ntrunc @ A @ B @ ( suc @ K ) @ ( old_Atom @ A @ B @ A3 ) )
= ( old_Atom @ A @ B @ A3 ) ) ).
% ntrunc_Atom
thf(fact_6912_ntrunc__Scons,axiom,
! [B: $tType,A: $tType,K: nat,M7: set @ ( old_node @ A @ B ),N6: set @ ( old_node @ A @ B )] :
( ( old_ntrunc @ A @ B @ ( suc @ K ) @ ( old_Scons @ A @ B @ M7 @ N6 ) )
= ( old_Scons @ A @ B @ ( old_ntrunc @ A @ B @ K @ M7 ) @ ( old_ntrunc @ A @ B @ K @ N6 ) ) ) ).
% ntrunc_Scons
thf(fact_6913_ntrunc__def,axiom,
! [A: $tType,B: $tType] :
( ( old_ntrunc @ A @ B )
= ( ^ [K3: nat,N7: set @ ( old_node @ A @ B )] :
( collect @ ( old_node @ A @ B )
@ ^ [N3: old_node @ A @ B] :
( ( member @ ( old_node @ A @ B ) @ N3 @ N7 )
& ( ord_less @ nat @ ( old_ndepth @ A @ B @ N3 ) @ K3 ) ) ) ) ) ).
% ntrunc_def
thf(fact_6914_ntrunc__one__In0,axiom,
! [B: $tType,A: $tType,M7: set @ ( old_node @ A @ B )] :
( ( old_ntrunc @ A @ B @ ( suc @ ( zero_zero @ nat ) ) @ ( old_In0 @ A @ B @ M7 ) )
= ( bot_bot @ ( set @ ( old_node @ A @ B ) ) ) ) ).
% ntrunc_one_In0
thf(fact_6915_ntrunc__one__In1,axiom,
! [B: $tType,A: $tType,M7: set @ ( old_node @ A @ B )] :
( ( old_ntrunc @ A @ B @ ( suc @ ( zero_zero @ nat ) ) @ ( old_In1 @ A @ B @ M7 ) )
= ( bot_bot @ ( set @ ( old_node @ A @ B ) ) ) ) ).
% ntrunc_one_In1
thf(fact_6916_ntrunc__In0,axiom,
! [B: $tType,A: $tType,K: nat,M7: set @ ( old_node @ A @ B )] :
( ( old_ntrunc @ A @ B @ ( suc @ ( suc @ K ) ) @ ( old_In0 @ A @ B @ M7 ) )
= ( old_In0 @ A @ B @ ( old_ntrunc @ A @ B @ ( suc @ K ) @ M7 ) ) ) ).
% ntrunc_In0
thf(fact_6917_ntrunc__In1,axiom,
! [B: $tType,A: $tType,K: nat,M7: set @ ( old_node @ A @ B )] :
( ( old_ntrunc @ A @ B @ ( suc @ ( suc @ K ) ) @ ( old_In1 @ A @ B @ M7 ) )
= ( old_In1 @ A @ B @ ( old_ntrunc @ A @ B @ ( suc @ K ) @ M7 ) ) ) ).
% ntrunc_In1
thf(fact_6918_ntrunc__Leaf,axiom,
! [B: $tType,A: $tType,K: nat,A3: A] :
( ( old_ntrunc @ A @ B @ ( suc @ K ) @ ( old_Leaf @ A @ B @ A3 ) )
= ( old_Leaf @ A @ B @ A3 ) ) ).
% ntrunc_Leaf
thf(fact_6919_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_6920_ntrunc__Numb,axiom,
! [A: $tType,B: $tType,K: nat,I: nat] :
( ( old_ntrunc @ A @ B @ ( suc @ K ) @ ( old_Numb @ A @ B @ I ) )
= ( old_Numb @ A @ B @ I ) ) ).
% ntrunc_Numb
thf(fact_6921_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_6922_ordering__top_Ointro,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,Top: A] :
( ( ordering @ A @ Less_eq @ Less )
=> ( ( ordering_top_axioms @ A @ Less_eq @ Top )
=> ( ordering_top @ A @ Less_eq @ Less @ Top ) ) ) ).
% ordering_top.intro
thf(fact_6923_ordering__top__def,axiom,
! [A: $tType] :
( ( ordering_top @ A )
= ( ^ [Less_eq2: A > A > $o,Less2: A > A > $o,Top2: A] :
( ( ordering @ A @ Less_eq2 @ Less2 )
& ( ordering_top_axioms @ A @ Less_eq2 @ Top2 ) ) ) ) ).
% ordering_top_def
thf(fact_6924_ordering__top_Oaxioms_I1_J,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,Top: A] :
( ( ordering_top @ A @ Less_eq @ Less @ Top )
=> ( ordering @ A @ Less_eq @ Less ) ) ).
% ordering_top.axioms(1)
thf(fact_6925_ordering__strictI,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o] :
( ! [A8: A,B5: A] :
( ( Less_eq @ A8 @ B5 )
= ( ( Less @ A8 @ B5 )
| ( A8 = B5 ) ) )
=> ( ! [A8: A,B5: A] :
( ( Less @ A8 @ B5 )
=> ~ ( Less @ B5 @ A8 ) )
=> ( ! [A8: A] :
~ ( Less @ A8 @ A8 )
=> ( ! [A8: A,B5: A,C3: A] :
( ( Less @ A8 @ B5 )
=> ( ( Less @ B5 @ C3 )
=> ( Less @ A8 @ C3 ) ) )
=> ( ordering @ A @ Less_eq @ Less ) ) ) ) ) ).
% ordering_strictI
thf(fact_6926_ordering_Onot__eq__order__implies__strict,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A] :
( ( ordering @ A @ Less_eq @ Less )
=> ( ( A3 != B2 )
=> ( ( Less_eq @ A3 @ B2 )
=> ( Less @ A3 @ B2 ) ) ) ) ).
% ordering.not_eq_order_implies_strict
thf(fact_6927_ordering_Ostrict__implies__not__eq,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A] :
( ( ordering @ A @ Less_eq @ Less )
=> ( ( Less @ A3 @ B2 )
=> ( A3 != B2 ) ) ) ).
% ordering.strict_implies_not_eq
thf(fact_6928_ordering_Ostrict__iff__order,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A] :
( ( ordering @ A @ Less_eq @ Less )
=> ( ( Less @ A3 @ B2 )
= ( ( Less_eq @ A3 @ B2 )
& ( A3 != B2 ) ) ) ) ).
% ordering.strict_iff_order
thf(fact_6929_ordering_Oorder__iff__strict,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A] :
( ( ordering @ A @ Less_eq @ Less )
=> ( ( Less_eq @ A3 @ B2 )
= ( ( Less @ A3 @ B2 )
| ( A3 = B2 ) ) ) ) ).
% ordering.order_iff_strict
thf(fact_6930_ordering_Oantisym,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A] :
( ( ordering @ A @ Less_eq @ Less )
=> ( ( Less_eq @ A3 @ B2 )
=> ( ( Less_eq @ B2 @ A3 )
=> ( A3 = B2 ) ) ) ) ).
% ordering.antisym
thf(fact_6931_ordering_Oeq__iff,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A] :
( ( ordering @ A @ Less_eq @ Less )
=> ( ( A3 = B2 )
= ( ( Less_eq @ A3 @ B2 )
& ( Less_eq @ B2 @ A3 ) ) ) ) ).
% ordering.eq_iff
thf(fact_6932_ordering__dualI,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o] :
( ( ordering @ A
@ ^ [A5: A,B3: A] : ( Less_eq @ B3 @ A5 )
@ ^ [A5: A,B3: A] : ( Less @ B3 @ A5 ) )
=> ( ordering @ A @ Less_eq @ Less ) ) ).
% ordering_dualI
thf(fact_6933_order_Oordering__axioms,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ordering @ A @ ( ord_less_eq @ A ) @ ( ord_less @ A ) ) ) ).
% order.ordering_axioms
thf(fact_6934_dual__order_Oordering__axioms,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ordering @ A
@ ^ [X2: A,Y6: A] : ( ord_less_eq @ A @ Y6 @ X2 )
@ ^ [X2: A,Y6: A] : ( ord_less @ A @ Y6 @ X2 ) ) ) ).
% dual_order.ordering_axioms
thf(fact_6935_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 ) ) ) ) )
=> ~ ! [N4: nat] :
( ( X
= ( suc @ N4 ) )
=> ( Y3
!= ( sum_case_sum @ nat @ ( set @ ( old_node @ A @ product_unit ) ) @ nat
@ ^ [I5: 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 @ I5 ) )
@ ^ [I5: 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 ) )
@ ^ [A5: nat,B3: nat] : ( old_Scons @ A @ product_unit @ ( nth_item @ A @ A5 ) @ ( nth_item @ A @ B3 ) )
@ ( nat_prod_decode @ J3 ) )
@ ( nat_sum_decode @ I5 ) )
@ ( nat_sum_decode @ N4 ) ) ) ) ) ) ) ).
% nth_item.elims
thf(fact_6936_group_Oaxioms_I2_J,axiom,
! [A: $tType,F: A > A > A,Z3: A,Inverse: A > A] :
( ( group @ A @ F @ Z3 @ Inverse )
=> ( group_axioms @ A @ F @ Z3 @ Inverse ) ) ).
% group.axioms(2)
thf(fact_6937_group__axioms_Ointro,axiom,
! [A: $tType,F: A > A > A,Z3: A,Inverse: A > A] :
( ! [A8: A] :
( ( F @ Z3 @ A8 )
= A8 )
=> ( ! [A8: A] :
( ( F @ ( Inverse @ A8 ) @ A8 )
= Z3 )
=> ( group_axioms @ A @ F @ Z3 @ Inverse ) ) ) ).
% group_axioms.intro
thf(fact_6938_group__axioms__def,axiom,
! [A: $tType] :
( ( group_axioms @ A )
= ( ^ [F2: A > A > A,Z7: A,Inverse2: A > A] :
( ! [A5: A] :
( ( F2 @ Z7 @ A5 )
= A5 )
& ! [A5: A] :
( ( F2 @ ( Inverse2 @ A5 ) @ A5 )
= Z7 ) ) ) ) ).
% group_axioms_def
thf(fact_6939_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_6940_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
@ ^ [I5: 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 @ I5 ) )
@ ^ [I5: 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 ) )
@ ^ [A5: nat,B3: nat] : ( old_Scons @ A @ product_unit @ ( nth_item @ A @ A5 ) @ ( nth_item @ A @ B3 ) )
@ ( nat_prod_decode @ J3 ) )
@ ( nat_sum_decode @ I5 ) )
@ ( nat_sum_decode @ N ) ) ) ) ).
% nth_item.simps(2)
thf(fact_6941_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 ) ) ) )
=> ~ ! [N4: nat] :
( ( X
= ( suc @ N4 ) )
=> ( ( Y3
= ( sum_case_sum @ nat @ ( set @ ( old_node @ A @ product_unit ) ) @ nat
@ ^ [I5: 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 @ I5 ) )
@ ^ [I5: 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 ) )
@ ^ [A5: nat,B3: nat] : ( old_Scons @ A @ product_unit @ ( nth_item @ A @ A5 ) @ ( nth_item @ A @ B3 ) )
@ ( nat_prod_decode @ J3 ) )
@ ( nat_sum_decode @ I5 ) )
@ ( nat_sum_decode @ N4 ) ) )
=> ~ ( accp @ nat @ nth_item_rel @ ( suc @ N4 ) ) ) ) ) ) ) ) ).
% nth_item.pelims
thf(fact_6942_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
@ ^ [I5: 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 @ I5 ) )
@ ^ [I5: 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 ) )
@ ^ [A5: nat,B3: nat] : ( old_Scons @ A @ product_unit @ ( nth_item @ A @ A5 ) @ ( nth_item @ A @ B3 ) )
@ ( nat_prod_decode @ J3 ) )
@ ( nat_sum_decode @ I5 ) )
@ ( nat_sum_decode @ N ) ) ) ) ) ).
% nth_item.psimps(2)
thf(fact_6943_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_6944_group_Ointro,axiom,
! [A: $tType,F: A > A > A,Z3: A,Inverse: A > A] :
( ( semigroup @ A @ F )
=> ( ( group_axioms @ A @ F @ Z3 @ Inverse )
=> ( group @ A @ F @ Z3 @ Inverse ) ) ) ).
% group.intro
thf(fact_6945_group__def,axiom,
! [A: $tType] :
( ( group @ A )
= ( ^ [F2: A > A > A,Z7: A,Inverse2: A > A] :
( ( semigroup @ A @ F2 )
& ( group_axioms @ A @ F2 @ Z7 @ Inverse2 ) ) ) ) ).
% group_def
thf(fact_6946_sup_Osemigroup__axioms,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( semigroup @ A @ ( sup_sup @ A ) ) ) ).
% sup.semigroup_axioms
thf(fact_6947_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_6948_semigroup_Ointro,axiom,
! [A: $tType,F: A > A > A] :
( ! [A8: A,B5: A,C3: A] :
( ( F @ ( F @ A8 @ B5 ) @ C3 )
= ( F @ A8 @ ( F @ B5 @ C3 ) ) )
=> ( semigroup @ A @ F ) ) ).
% semigroup.intro
thf(fact_6949_semigroup__def,axiom,
! [A: $tType] :
( ( semigroup @ A )
= ( ^ [F2: A > A > A] :
! [A5: A,B3: A,C4: A] :
( ( F2 @ ( F2 @ A5 @ B3 ) @ C4 )
= ( F2 @ A5 @ ( F2 @ B3 @ C4 ) ) ) ) ) ).
% semigroup_def
thf(fact_6950_group_Oaxioms_I1_J,axiom,
! [A: $tType,F: A > A > A,Z3: A,Inverse: A > A] :
( ( group @ A @ F @ Z3 @ Inverse )
=> ( semigroup @ A @ F ) ) ).
% group.axioms(1)
thf(fact_6951_monoid_Oaxioms_I1_J,axiom,
! [A: $tType,F: A > A > A,Z3: A] :
( ( monoid @ A @ F @ Z3 )
=> ( semigroup @ A @ F ) ) ).
% monoid.axioms(1)
thf(fact_6952_min_Osemigroup__axioms,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( semigroup @ A @ ( ord_min @ A ) ) ) ).
% min.semigroup_axioms
thf(fact_6953_max_Osemigroup__axioms,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( semigroup @ A @ ( ord_max @ A ) ) ) ).
% max.semigroup_axioms
thf(fact_6954_add_Osemigroup__axioms,axiom,
! [A: $tType] :
( ( semigroup_add @ A )
=> ( semigroup @ A @ ( plus_plus @ A ) ) ) ).
% add.semigroup_axioms
thf(fact_6955_mult_Osemigroup__axioms,axiom,
! [A: $tType] :
( ( semigroup_mult @ A )
=> ( semigroup @ A @ ( times_times @ A ) ) ) ).
% mult.semigroup_axioms
thf(fact_6956_inf_Osemigroup__axioms,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( semigroup @ A @ ( inf_inf @ A ) ) ) ).
% inf.semigroup_axioms
thf(fact_6957_monoid_Ointro,axiom,
! [A: $tType,F: A > A > A,Z3: A] :
( ( semigroup @ A @ F )
=> ( ( monoid_axioms @ A @ F @ Z3 )
=> ( monoid @ A @ F @ Z3 ) ) ) ).
% monoid.intro
thf(fact_6958_monoid__def,axiom,
! [A: $tType] :
( ( monoid @ A )
= ( ^ [F2: A > A > A,Z7: A] :
( ( semigroup @ A @ F2 )
& ( monoid_axioms @ A @ F2 @ Z7 ) ) ) ) ).
% monoid_def
thf(fact_6959_monoid__axioms__def,axiom,
! [A: $tType] :
( ( monoid_axioms @ A )
= ( ^ [F2: A > A > A,Z7: A] :
( ! [A5: A] :
( ( F2 @ Z7 @ A5 )
= A5 )
& ! [A5: A] :
( ( F2 @ A5 @ Z7 )
= A5 ) ) ) ) ).
% monoid_axioms_def
thf(fact_6960_monoid__axioms_Ointro,axiom,
! [A: $tType,F: A > A > A,Z3: A] :
( ! [A8: A] :
( ( F @ Z3 @ A8 )
= A8 )
=> ( ! [A8: A] :
( ( F @ A8 @ Z3 )
= A8 )
=> ( monoid_axioms @ A @ F @ Z3 ) ) ) ).
% monoid_axioms.intro
thf(fact_6961_monoid_Oaxioms_I2_J,axiom,
! [A: $tType,F: A > A > A,Z3: A] :
( ( monoid @ A @ F @ Z3 )
=> ( monoid_axioms @ A @ F @ Z3 ) ) ).
% monoid.axioms(2)
thf(fact_6962_nths__nths,axiom,
! [A: $tType,Xs: list @ A,A6: set @ nat,B7: set @ nat] :
( ( nths @ A @ ( nths @ A @ Xs @ A6 ) @ B7 )
= ( nths @ A @ Xs
@ ( collect @ nat
@ ^ [I5: nat] :
( ( member @ nat @ I5 @ A6 )
& ( member @ nat
@ ( finite_card @ nat
@ ( collect @ nat
@ ^ [I9: nat] :
( ( member @ nat @ I9 @ A6 )
& ( ord_less @ nat @ I9 @ I5 ) ) ) )
@ B7 ) ) ) ) ) ).
% nths_nths
thf(fact_6963_ordering_Oaxioms_I2_J,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o] :
( ( ordering @ A @ Less_eq @ Less )
=> ( ordering_axioms @ A @ Less_eq @ Less ) ) ).
% ordering.axioms(2)
thf(fact_6964_ordering__axioms_Ointro,axiom,
! [A: $tType,Less: A > A > $o,Less_eq: A > A > $o] :
( ! [A8: A,B5: A] :
( ( Less @ A8 @ B5 )
= ( ( Less_eq @ A8 @ B5 )
& ( A8 != B5 ) ) )
=> ( ! [A8: A,B5: A] :
( ( Less_eq @ A8 @ B5 )
=> ( ( Less_eq @ B5 @ A8 )
=> ( A8 = B5 ) ) )
=> ( ordering_axioms @ A @ Less_eq @ Less ) ) ) ).
% ordering_axioms.intro
thf(fact_6965_ordering__axioms__def,axiom,
! [A: $tType] :
( ( ordering_axioms @ A )
= ( ^ [Less_eq2: A > A > $o,Less2: A > A > $o] :
( ! [A5: A,B3: A] :
( ( Less2 @ A5 @ B3 )
= ( ( Less_eq2 @ A5 @ B3 )
& ( A5 != B3 ) ) )
& ! [A5: A,B3: A] :
( ( Less_eq2 @ A5 @ B3 )
=> ( ( Less_eq2 @ B3 @ A5 )
=> ( A5 = B3 ) ) ) ) ) ) ).
% ordering_axioms_def
thf(fact_6966_Bex__def__raw,axiom,
! [A: $tType] :
( ( bex @ A )
= ( ^ [A7: set @ A,P4: A > $o] :
? [X2: A] :
( ( member @ A @ X2 @ A7 )
& ( P4 @ X2 ) ) ) ) ).
% Bex_def_raw
thf(fact_6967_ordering__def,axiom,
! [A: $tType] :
( ( ordering @ A )
= ( ^ [Less_eq2: A > A > $o,Less2: A > A > $o] :
( ( partial_preordering @ A @ Less_eq2 )
& ( ordering_axioms @ A @ Less_eq2 @ Less2 ) ) ) ) ).
% ordering_def
thf(fact_6968_ordering_Ointro,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o] :
( ( partial_preordering @ A @ Less_eq )
=> ( ( ordering_axioms @ A @ Less_eq @ Less )
=> ( ordering @ A @ Less_eq @ Less ) ) ) ).
% ordering.intro
thf(fact_6969_dual__order_Opartial__preordering__axioms,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( partial_preordering @ A
@ ^ [X2: A,Y6: A] : ( ord_less_eq @ A @ Y6 @ X2 ) ) ) ).
% dual_order.partial_preordering_axioms
thf(fact_6970_order_Opartial__preordering__axioms,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( partial_preordering @ A @ ( ord_less_eq @ A ) ) ) ).
% order.partial_preordering_axioms
thf(fact_6971_partial__preordering__def,axiom,
! [A: $tType] :
( ( partial_preordering @ A )
= ( ^ [Less_eq2: A > A > $o] :
( ! [A5: A] : ( Less_eq2 @ A5 @ A5 )
& ! [A5: A,B3: A,C4: A] :
( ( Less_eq2 @ A5 @ B3 )
=> ( ( Less_eq2 @ B3 @ C4 )
=> ( Less_eq2 @ A5 @ C4 ) ) ) ) ) ) ).
% partial_preordering_def
thf(fact_6972_partial__preordering_Otrans,axiom,
! [A: $tType,Less_eq: A > A > $o,A3: A,B2: A,C2: A] :
( ( partial_preordering @ A @ Less_eq )
=> ( ( Less_eq @ A3 @ B2 )
=> ( ( Less_eq @ B2 @ C2 )
=> ( Less_eq @ A3 @ C2 ) ) ) ) ).
% partial_preordering.trans
thf(fact_6973_partial__preordering_Ointro,axiom,
! [A: $tType,Less_eq: A > A > $o] :
( ! [A8: A] : ( Less_eq @ A8 @ A8 )
=> ( ! [A8: A,B5: A,C3: A] :
( ( Less_eq @ A8 @ B5 )
=> ( ( Less_eq @ B5 @ C3 )
=> ( Less_eq @ A8 @ C3 ) ) )
=> ( partial_preordering @ A @ Less_eq ) ) ) ).
% partial_preordering.intro
thf(fact_6974_partial__preordering_Orefl,axiom,
! [A: $tType,Less_eq: A > A > $o,A3: A] :
( ( partial_preordering @ A @ Less_eq )
=> ( Less_eq @ A3 @ A3 ) ) ).
% partial_preordering.refl
thf(fact_6975_ordering_Oaxioms_I1_J,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o] :
( ( ordering @ A @ Less_eq @ Less )
=> ( partial_preordering @ A @ Less_eq ) ) ).
% ordering.axioms(1)
thf(fact_6976_Abs__int__cases,axiom,
! [X: int] :
~ ! [Y5: set @ ( product_prod @ nat @ nat )] :
( ( X
= ( abs_int @ Y5 ) )
=> ~ ( member @ ( set @ ( product_prod @ nat @ nat ) ) @ Y5
@ ( 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_6977_Abs__int__induct,axiom,
! [P: int > $o,X: int] :
( ! [Y5: set @ ( product_prod @ nat @ nat )] :
( ( member @ ( set @ ( product_prod @ nat @ nat ) ) @ Y5
@ ( 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 @ Y5 ) ) )
=> ( P @ X ) ) ).
% Abs_int_induct
thf(fact_6978_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_6979_Abs__Integ__def,axiom,
( abs_Integ
= ( quot_abs @ ( product_prod @ nat @ nat ) @ int @ intrel @ abs_int ) ) ).
% Abs_Integ_def
thf(fact_6980_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_6981_Rep__int__inject,axiom,
! [X: int,Y3: int] :
( ( ( rep_int @ X )
= ( rep_int @ Y3 ) )
= ( X = Y3 ) ) ).
% Rep_int_inject
thf(fact_6982_Rep__int__inverse,axiom,
! [X: int] :
( ( abs_int @ ( rep_int @ X ) )
= X ) ).
% Rep_int_inverse
thf(fact_6983_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_6984_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 ) ) ) ) ) )
=> ~ ! [X3: int] :
( Y3
!= ( rep_int @ X3 ) ) ) ).
% Rep_int_cases
thf(fact_6985_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 ) ) ) ) ) )
=> ( ! [X3: int] : ( P @ ( rep_int @ X3 ) )
=> ( P @ Y3 ) ) ) ).
% Rep_int_induct
thf(fact_6986_Rep__Integ__def,axiom,
( rep_Integ
= ( quot_rep @ int @ ( product_prod @ nat @ nat ) @ rep_int ) ) ).
% Rep_Integ_def
thf(fact_6987_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_6988_Domainp__cr__rat,axiom,
( ( domainp @ ( product_prod @ int @ int ) @ rat @ pcr_rat )
= ( ^ [X2: product_prod @ int @ int] :
( ( product_snd @ int @ int @ X2 )
!= ( zero_zero @ int ) ) ) ) ).
% Domainp_cr_rat
thf(fact_6989_semilattice__neutr_Ointro,axiom,
! [A: $tType,F: A > A > A,Z3: A] :
( ( semilattice @ A @ F )
=> ( ( comm_monoid @ A @ F @ Z3 )
=> ( semilattice_neutr @ A @ F @ Z3 ) ) ) ).
% semilattice_neutr.intro
thf(fact_6990_inf_Osemilattice__axioms,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( semilattice @ A @ ( inf_inf @ A ) ) ) ).
% inf.semilattice_axioms
thf(fact_6991_semilattice__order_Oaxioms_I1_J,axiom,
! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o] :
( ( semilattice_order @ A @ F @ Less_eq @ Less )
=> ( semilattice @ A @ F ) ) ).
% semilattice_order.axioms(1)
thf(fact_6992_max_Osemilattice__axioms,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( semilattice @ A @ ( ord_max @ A ) ) ) ).
% max.semilattice_axioms
thf(fact_6993_min_Osemilattice__axioms,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( semilattice @ A @ ( ord_min @ A ) ) ) ).
% min.semilattice_axioms
thf(fact_6994_semilattice_Oright__idem,axiom,
! [A: $tType,F: A > A > A,A3: A,B2: A] :
( ( semilattice @ A @ F )
=> ( ( F @ ( F @ A3 @ B2 ) @ B2 )
= ( F @ A3 @ B2 ) ) ) ).
% semilattice.right_idem
thf(fact_6995_semilattice_Oleft__idem,axiom,
! [A: $tType,F: A > A > A,A3: A,B2: A] :
( ( semilattice @ A @ F )
=> ( ( F @ A3 @ ( F @ A3 @ B2 ) )
= ( F @ A3 @ B2 ) ) ) ).
% semilattice.left_idem
thf(fact_6996_semilattice_Oidem,axiom,
! [A: $tType,F: A > A > A,A3: A] :
( ( semilattice @ A @ F )
=> ( ( F @ A3 @ A3 )
= A3 ) ) ).
% semilattice.idem
thf(fact_6997_semilattice__set_Ointro,axiom,
! [A: $tType,F: A > A > A] :
( ( semilattice @ A @ F )
=> ( lattic149705377957585745ce_set @ A @ F ) ) ).
% semilattice_set.intro
thf(fact_6998_semilattice__set_Oaxioms,axiom,
! [A: $tType,F: A > A > A] :
( ( lattic149705377957585745ce_set @ A @ F )
=> ( semilattice @ A @ F ) ) ).
% semilattice_set.axioms
thf(fact_6999_semilattice__set__def,axiom,
! [A: $tType] :
( ( lattic149705377957585745ce_set @ A )
= ( semilattice @ A ) ) ).
% semilattice_set_def
thf(fact_7000_semilattice__neutr_Oaxioms_I1_J,axiom,
! [A: $tType,F: A > A > A,Z3: A] :
( ( semilattice_neutr @ A @ F @ Z3 )
=> ( semilattice @ A @ F ) ) ).
% semilattice_neutr.axioms(1)
thf(fact_7001_sup_Osemilattice__axioms,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( semilattice @ A @ ( sup_sup @ A ) ) ) ).
% sup.semilattice_axioms
thf(fact_7002_semilattice__neutr__def,axiom,
! [A: $tType] :
( ( semilattice_neutr @ A )
= ( ^ [F2: A > A > A,Z7: A] :
( ( semilattice @ A @ F2 )
& ( comm_monoid @ A @ F2 @ Z7 ) ) ) ) ).
% semilattice_neutr_def
thf(fact_7003_semilattice__order_Ointro,axiom,
! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o] :
( ( semilattice @ A @ F )
=> ( ( semila6385135966242565138axioms @ A @ F @ Less_eq @ Less )
=> ( semilattice_order @ A @ F @ Less_eq @ Less ) ) ) ).
% semilattice_order.intro
thf(fact_7004_semilattice__order__def,axiom,
! [A: $tType] :
( ( semilattice_order @ A )
= ( ^ [F2: A > A > A,Less_eq2: A > A > $o,Less2: A > A > $o] :
( ( semilattice @ A @ F2 )
& ( semila6385135966242565138axioms @ A @ F2 @ Less_eq2 @ Less2 ) ) ) ) ).
% semilattice_order_def
thf(fact_7005_semilattice__order__axioms_Ointro,axiom,
! [A: $tType,Less_eq: A > A > $o,F: A > A > A,Less: A > A > $o] :
( ! [A8: A,B5: A] :
( ( Less_eq @ A8 @ B5 )
= ( A8
= ( F @ A8 @ B5 ) ) )
=> ( ! [A8: A,B5: A] :
( ( Less @ A8 @ B5 )
= ( ( A8
= ( F @ A8 @ B5 ) )
& ( A8 != B5 ) ) )
=> ( semila6385135966242565138axioms @ A @ F @ Less_eq @ Less ) ) ) ).
% semilattice_order_axioms.intro
thf(fact_7006_semilattice__order__axioms__def,axiom,
! [A: $tType] :
( ( semila6385135966242565138axioms @ A )
= ( ^ [F2: A > A > A,Less_eq2: A > A > $o,Less2: A > A > $o] :
( ! [A5: A,B3: A] :
( ( Less_eq2 @ A5 @ B3 )
= ( A5
= ( F2 @ A5 @ B3 ) ) )
& ! [A5: A,B3: A] :
( ( Less2 @ A5 @ B3 )
= ( ( A5
= ( F2 @ A5 @ B3 ) )
& ( A5 != B3 ) ) ) ) ) ) ).
% semilattice_order_axioms_def
thf(fact_7007_semilattice__order_Oaxioms_I2_J,axiom,
! [A: $tType,F: A > A > A,Less_eq: A > A > $o,Less: A > A > $o] :
( ( semilattice_order @ A @ F @ Less_eq @ Less )
=> ( semila6385135966242565138axioms @ A @ F @ Less_eq @ Less ) ) ).
% semilattice_order.axioms(2)
thf(fact_7008_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_7009_comm__monoid_Oaxioms_I2_J,axiom,
! [A: $tType,F: A > A > A,Z3: A] :
( ( comm_monoid @ A @ F @ Z3 )
=> ( comm_monoid_axioms @ A @ F @ Z3 ) ) ).
% comm_monoid.axioms(2)
thf(fact_7010_comm__monoid__axioms__def,axiom,
! [A: $tType] :
( ( comm_monoid_axioms @ A )
= ( ^ [F2: A > A > A,Z7: A] :
! [A5: A] :
( ( F2 @ A5 @ Z7 )
= A5 ) ) ) ).
% comm_monoid_axioms_def
thf(fact_7011_comm__monoid__axioms_Ointro,axiom,
! [A: $tType,F: A > A > A,Z3: A] :
( ! [A8: A] :
( ( F @ A8 @ Z3 )
= A8 )
=> ( comm_monoid_axioms @ A @ F @ Z3 ) ) ).
% comm_monoid_axioms.intro
thf(fact_7012_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_7013_comm__monoid__def,axiom,
! [A: $tType] :
( ( comm_monoid @ A )
= ( ^ [F2: A > A > A,Z7: A] :
( ( abel_semigroup @ A @ F2 )
& ( comm_monoid_axioms @ A @ F2 @ Z7 ) ) ) ) ).
% comm_monoid_def
thf(fact_7014_comm__monoid_Ointro,axiom,
! [A: $tType,F: A > A > A,Z3: A] :
( ( abel_semigroup @ A @ F )
=> ( ( comm_monoid_axioms @ A @ F @ Z3 )
=> ( comm_monoid @ A @ F @ Z3 ) ) ) ).
% comm_monoid.intro
thf(fact_7015_semilattice_Oaxioms_I1_J,axiom,
! [A: $tType,F: A > A > A] :
( ( semilattice @ A @ F )
=> ( abel_semigroup @ A @ F ) ) ).
% semilattice.axioms(1)
thf(fact_7016_sup_Oabel__semigroup__axioms,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( abel_semigroup @ A @ ( sup_sup @ A ) ) ) ).
% sup.abel_semigroup_axioms
thf(fact_7017_comm__monoid_Oaxioms_I1_J,axiom,
! [A: $tType,F: A > A > A,Z3: A] :
( ( comm_monoid @ A @ F @ Z3 )
=> ( abel_semigroup @ A @ F ) ) ).
% comm_monoid.axioms(1)
thf(fact_7018_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_7019_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_7020_abstract__boolean__algebra_Oaxioms_I1_J,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A] :
( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One2 )
=> ( abel_semigroup @ A @ Conj ) ) ).
% abstract_boolean_algebra.axioms(1)
thf(fact_7021_abstract__boolean__algebra_Oaxioms_I2_J,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A] :
( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One2 )
=> ( abel_semigroup @ A @ Disj ) ) ).
% abstract_boolean_algebra.axioms(2)
thf(fact_7022_min_Oabel__semigroup__axioms,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( abel_semigroup @ A @ ( ord_min @ A ) ) ) ).
% min.abel_semigroup_axioms
thf(fact_7023_add_Oabel__semigroup__axioms,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A )
=> ( abel_semigroup @ A @ ( plus_plus @ A ) ) ) ).
% add.abel_semigroup_axioms
thf(fact_7024_max_Oabel__semigroup__axioms,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( abel_semigroup @ A @ ( ord_max @ A ) ) ) ).
% max.abel_semigroup_axioms
thf(fact_7025_mult_Oabel__semigroup__axioms,axiom,
! [A: $tType] :
( ( ab_semigroup_mult @ A )
=> ( abel_semigroup @ A @ ( times_times @ A ) ) ) ).
% mult.abel_semigroup_axioms
thf(fact_7026_inf_Oabel__semigroup__axioms,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( abel_semigroup @ A @ ( inf_inf @ A ) ) ) ).
% inf.abel_semigroup_axioms
thf(fact_7027_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_7028_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_7029_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_7030_abel__semigroup__axioms__def,axiom,
! [A: $tType] :
( ( abel_s757365448890700780axioms @ A )
= ( ^ [F2: A > A > A] :
! [A5: A,B3: A] :
( ( F2 @ A5 @ B3 )
= ( F2 @ B3 @ A5 ) ) ) ) ).
% abel_semigroup_axioms_def
thf(fact_7031_abel__semigroup__axioms_Ointro,axiom,
! [A: $tType,F: A > A > A] :
( ! [A8: A,B5: A] :
( ( F @ A8 @ B5 )
= ( F @ B5 @ A8 ) )
=> ( abel_s757365448890700780axioms @ A @ F ) ) ).
% abel_semigroup_axioms.intro
thf(fact_7032_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_7033_semilattice_Ointro,axiom,
! [A: $tType,F: A > A > A] :
( ( abel_semigroup @ A @ F )
=> ( ( semilattice_axioms @ A @ F )
=> ( semilattice @ A @ F ) ) ) ).
% semilattice.intro
thf(fact_7034_semilattice__def,axiom,
! [A: $tType] :
( ( semilattice @ A )
= ( ^ [F2: A > A > A] :
( ( abel_semigroup @ A @ F2 )
& ( semilattice_axioms @ A @ F2 ) ) ) ) ).
% semilattice_def
thf(fact_7035_semilattice__axioms_Ointro,axiom,
! [A: $tType,F: A > A > A] :
( ! [A8: A] :
( ( F @ A8 @ A8 )
= A8 )
=> ( semilattice_axioms @ A @ F ) ) ).
% semilattice_axioms.intro
thf(fact_7036_semilattice__axioms__def,axiom,
! [A: $tType] :
( ( semilattice_axioms @ A )
= ( ^ [F2: A > A > A] :
! [A5: A] :
( ( F2 @ A5 @ A5 )
= A5 ) ) ) ).
% semilattice_axioms_def
thf(fact_7037_semilattice_Oaxioms_I2_J,axiom,
! [A: $tType,F: A > A > A] :
( ( semilattice @ A @ F )
=> ( semilattice_axioms @ A @ F ) ) ).
% semilattice.axioms(2)
thf(fact_7038_abstract__boolean__algebra__def,axiom,
! [A: $tType] :
( ( boolea2506097494486148201lgebra @ A )
= ( ^ [Conj2: A > A > A,Disj2: A > A > A,Compl2: A > A,Zero2: A,One: A] :
( ( abel_semigroup @ A @ Conj2 )
& ( abel_semigroup @ A @ Disj2 )
& ( boolea6902313364301356556axioms @ A @ Conj2 @ Disj2 @ Compl2 @ Zero2 @ One ) ) ) ) ).
% abstract_boolean_algebra_def
thf(fact_7039_abstract__boolean__algebra_Ointro,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A] :
( ( abel_semigroup @ A @ Conj )
=> ( ( abel_semigroup @ A @ Disj )
=> ( ( boolea6902313364301356556axioms @ A @ Conj @ Disj @ Compl @ Zero @ One2 )
=> ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One2 ) ) ) ) ).
% abstract_boolean_algebra.intro
thf(fact_7040_abstract__boolean__algebra__axioms__def,axiom,
! [A: $tType] :
( ( boolea6902313364301356556axioms @ A )
= ( ^ [Conj2: A > A > A,Disj2: A > A > A,Compl2: A > A,Zero2: A,One: A] :
( ! [X2: A,Y6: A,Z7: A] :
( ( Conj2 @ X2 @ ( Disj2 @ Y6 @ Z7 ) )
= ( Disj2 @ ( Conj2 @ X2 @ Y6 ) @ ( Conj2 @ X2 @ Z7 ) ) )
& ! [X2: A,Y6: A,Z7: A] :
( ( Disj2 @ X2 @ ( Conj2 @ Y6 @ Z7 ) )
= ( Conj2 @ ( Disj2 @ X2 @ Y6 ) @ ( Disj2 @ X2 @ Z7 ) ) )
& ! [X2: A] :
( ( Conj2 @ X2 @ One )
= X2 )
& ! [X2: A] :
( ( Disj2 @ X2 @ Zero2 )
= X2 )
& ! [X2: A] :
( ( Conj2 @ X2 @ ( Compl2 @ X2 ) )
= Zero2 )
& ! [X2: A] :
( ( Disj2 @ X2 @ ( Compl2 @ X2 ) )
= One ) ) ) ) ).
% abstract_boolean_algebra_axioms_def
thf(fact_7041_abstract__boolean__algebra__axioms_Ointro,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,One2: A,Zero: A,Compl: A > A] :
( ! [X3: A,Y5: A,Z4: A] :
( ( Conj @ X3 @ ( Disj @ Y5 @ Z4 ) )
= ( Disj @ ( Conj @ X3 @ Y5 ) @ ( Conj @ X3 @ Z4 ) ) )
=> ( ! [X3: A,Y5: A,Z4: A] :
( ( Disj @ X3 @ ( Conj @ Y5 @ Z4 ) )
= ( Conj @ ( Disj @ X3 @ Y5 ) @ ( Disj @ X3 @ Z4 ) ) )
=> ( ! [X3: A] :
( ( Conj @ X3 @ One2 )
= X3 )
=> ( ! [X3: A] :
( ( Disj @ X3 @ Zero )
= X3 )
=> ( ! [X3: A] :
( ( Conj @ X3 @ ( Compl @ X3 ) )
= Zero )
=> ( ! [X3: A] :
( ( Disj @ X3 @ ( Compl @ X3 ) )
= One2 )
=> ( boolea6902313364301356556axioms @ A @ Conj @ Disj @ Compl @ Zero @ One2 ) ) ) ) ) ) ) ).
% abstract_boolean_algebra_axioms.intro
thf(fact_7042_abstract__boolean__algebra_Oaxioms_I3_J,axiom,
! [A: $tType,Conj: A > A > A,Disj: A > A > A,Compl: A > A,Zero: A,One2: A] :
( ( boolea2506097494486148201lgebra @ A @ Conj @ Disj @ Compl @ Zero @ One2 )
=> ( boolea6902313364301356556axioms @ A @ Conj @ Disj @ Compl @ Zero @ One2 ) ) ).
% abstract_boolean_algebra.axioms(3)
thf(fact_7043_typerep_Osize__neq,axiom,
! [X: typerep] :
( ( size_size @ typerep @ X )
!= ( zero_zero @ nat ) ) ).
% typerep.size_neq
thf(fact_7044_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_7045_tuple__isomorphism_Osize_I2_J,axiom,
! [B: $tType,C: $tType,A: $tType,X13: A > ( product_prod @ B @ C ),X22: ( product_prod @ B @ C ) > A] :
( ( size_size @ ( tuple_isomorphism @ A @ B @ C ) @ ( tuple_1188178415141063261rphism @ A @ B @ C @ X13 @ X22 ) )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% tuple_isomorphism.size(2)
thf(fact_7046_typerep_Osize_I2_J,axiom,
! [X13: literal,X22: list @ typerep] :
( ( size_size @ typerep @ ( typerep3 @ X13 @ X22 ) )
= ( plus_plus @ nat @ ( size_list @ typerep @ ( size_size @ typerep ) @ X22 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% typerep.size(2)
thf(fact_7047_typerep_Osize__gen,axiom,
! [X13: literal,X22: list @ typerep] :
( ( size_typerep @ ( typerep3 @ X13 @ X22 ) )
= ( plus_plus @ nat @ ( size_list @ typerep @ size_typerep @ X22 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% typerep.size_gen
thf(fact_7048_tuple__isomorphism_Osize__gen,axiom,
! [B: $tType,C: $tType,A: $tType,Xb3: A > nat,Xa3: B > nat,X: C > nat,X13: A > ( product_prod @ B @ C ),X22: ( product_prod @ B @ C ) > A] :
( ( tuple_9181185373184732606rphism @ A @ B @ C @ Xb3 @ Xa3 @ X @ ( tuple_1188178415141063261rphism @ A @ B @ C @ X13 @ X22 ) )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% tuple_isomorphism.size_gen
thf(fact_7049_typerep__fun__box__def,axiom,
! [B: $tType,A: $tType] :
( ( ( typerep2 @ A )
& ( typerep2 @ B ) )
=> ( ( typerep_typerep @ ( fun_box @ A @ B ) )
= ( ^ [T7: itself @ ( fun_box @ A @ B )] : ( typerep3 @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $false @ $true @ ( literal2 @ $true @ $false @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $false @ $false @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $true @ $false @ ( literal2 @ $false @ $true @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $true @ $false @ $true @ ( literal2 @ $false @ $true @ $false @ $false @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $true @ $true @ $true @ $true @ ( zero_zero @ literal ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( cons @ typerep @ ( typerep_typerep @ A @ ( type2 @ A ) ) @ ( cons @ typerep @ ( typerep_typerep @ B @ ( type2 @ B ) ) @ ( nil @ typerep ) ) ) ) ) ) ) ).
% typerep_fun_box_def
thf(fact_7050_typerep__word__def,axiom,
! [A: $tType] :
( ( typerep2 @ A )
=> ( ( typerep_typerep @ ( word @ A ) )
= ( ^ [T7: itself @ ( word @ A )] : ( typerep3 @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $false @ $true @ ( literal2 @ $true @ $false @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $false @ $false @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $true @ $false @ ( literal2 @ $true @ $true @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $true @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $false @ $false @ $true @ $true @ ( zero_zero @ literal ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( cons @ typerep @ ( typerep_typerep @ A @ ( type2 @ A ) ) @ ( nil @ typerep ) ) ) ) ) ) ).
% typerep_word_def
thf(fact_7051_dual__order_Opreordering__axioms,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( preordering @ A
@ ^ [X2: A,Y6: A] : ( ord_less_eq @ A @ Y6 @ X2 )
@ ^ [X2: A,Y6: A] : ( ord_less @ A @ Y6 @ X2 ) ) ) ).
% dual_order.preordering_axioms
thf(fact_7052_irreflp__greater,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( irreflp @ A
@ ^ [X2: A,Y6: A] : ( ord_less @ A @ Y6 @ X2 ) ) ) ).
% irreflp_greater
thf(fact_7053_preordering_Oaxioms_I1_J,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o] :
( ( preordering @ A @ Less_eq @ Less )
=> ( partial_preordering @ A @ Less_eq ) ) ).
% preordering.axioms(1)
thf(fact_7054_irreflp__less,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( irreflp @ A @ ( ord_less @ A ) ) ) ).
% irreflp_less
thf(fact_7055_preordering__dualI,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o] :
( ( preordering @ A
@ ^ [A5: A,B3: A] : ( Less_eq @ B3 @ A5 )
@ ^ [A5: A,B3: A] : ( Less @ B3 @ A5 ) )
=> ( preordering @ A @ Less_eq @ Less ) ) ).
% preordering_dualI
thf(fact_7056_preordering_Oasym,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A] :
( ( preordering @ A @ Less_eq @ Less )
=> ( ( Less @ A3 @ B2 )
=> ~ ( Less @ B2 @ A3 ) ) ) ).
% preordering.asym
thf(fact_7057_preordering_Oirrefl,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A3: A] :
( ( preordering @ A @ Less_eq @ Less )
=> ~ ( Less @ A3 @ A3 ) ) ).
% preordering.irrefl
thf(fact_7058_preordering_Ostrict__trans,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A,C2: A] :
( ( preordering @ A @ Less_eq @ Less )
=> ( ( Less @ A3 @ B2 )
=> ( ( Less @ B2 @ C2 )
=> ( Less @ A3 @ C2 ) ) ) ) ).
% preordering.strict_trans
thf(fact_7059_preordering_Ostrict__trans1,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A,C2: A] :
( ( preordering @ A @ Less_eq @ Less )
=> ( ( Less_eq @ A3 @ B2 )
=> ( ( Less @ B2 @ C2 )
=> ( Less @ A3 @ C2 ) ) ) ) ).
% preordering.strict_trans1
thf(fact_7060_preordering_Ostrict__trans2,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A,C2: A] :
( ( preordering @ A @ Less_eq @ Less )
=> ( ( Less @ A3 @ B2 )
=> ( ( Less_eq @ B2 @ C2 )
=> ( Less @ A3 @ C2 ) ) ) ) ).
% preordering.strict_trans2
thf(fact_7061_preordering_Ostrict__iff__not,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A] :
( ( preordering @ A @ Less_eq @ Less )
=> ( ( Less @ A3 @ B2 )
= ( ( Less_eq @ A3 @ B2 )
& ~ ( Less_eq @ B2 @ A3 ) ) ) ) ).
% preordering.strict_iff_not
thf(fact_7062_preordering_Ostrict__implies__order,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A] :
( ( preordering @ A @ Less_eq @ Less )
=> ( ( Less @ A3 @ B2 )
=> ( Less_eq @ A3 @ B2 ) ) ) ).
% preordering.strict_implies_order
thf(fact_7063_preordering__strictI,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o] :
( ! [A8: A,B5: A] :
( ( Less_eq @ A8 @ B5 )
= ( ( Less @ A8 @ B5 )
| ( A8 = B5 ) ) )
=> ( ! [A8: A,B5: A] :
( ( Less @ A8 @ B5 )
=> ~ ( Less @ B5 @ A8 ) )
=> ( ! [A8: A] :
~ ( Less @ A8 @ A8 )
=> ( ! [A8: A,B5: A,C3: A] :
( ( Less @ A8 @ B5 )
=> ( ( Less @ B5 @ C3 )
=> ( Less @ A8 @ C3 ) ) )
=> ( preordering @ A @ Less_eq @ Less ) ) ) ) ) ).
% preordering_strictI
thf(fact_7064_order_Opreordering__axioms,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( preordering @ A @ ( ord_less_eq @ A ) @ ( ord_less @ A ) ) ) ).
% order.preordering_axioms
thf(fact_7065_preordering__def,axiom,
! [A: $tType] :
( ( preordering @ A )
= ( ^ [Less_eq2: A > A > $o,Less2: A > A > $o] :
( ( partial_preordering @ A @ Less_eq2 )
& ( preordering_axioms @ A @ Less_eq2 @ Less2 ) ) ) ) ).
% preordering_def
thf(fact_7066_preordering_Ointro,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o] :
( ( partial_preordering @ A @ Less_eq )
=> ( ( preordering_axioms @ A @ Less_eq @ Less )
=> ( preordering @ A @ Less_eq @ Less ) ) ) ).
% preordering.intro
thf(fact_7067_preordering__axioms__def,axiom,
! [A: $tType] :
( ( preordering_axioms @ A )
= ( ^ [Less_eq2: A > A > $o,Less2: A > A > $o] :
! [A5: A,B3: A] :
( ( Less2 @ A5 @ B3 )
= ( ( Less_eq2 @ A5 @ B3 )
& ~ ( Less_eq2 @ B3 @ A5 ) ) ) ) ) ).
% preordering_axioms_def
thf(fact_7068_preordering__axioms_Ointro,axiom,
! [A: $tType,Less: A > A > $o,Less_eq: A > A > $o] :
( ! [A8: A,B5: A] :
( ( Less @ A8 @ B5 )
= ( ( Less_eq @ A8 @ B5 )
& ~ ( Less_eq @ B5 @ A8 ) ) )
=> ( preordering_axioms @ A @ Less_eq @ Less ) ) ).
% preordering_axioms.intro
thf(fact_7069_preordering_Oaxioms_I2_J,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o] :
( ( preordering @ A @ Less_eq @ Less )
=> ( preordering_axioms @ A @ Less_eq @ Less ) ) ).
% preordering.axioms(2)
thf(fact_7070_comm__monoid__set_Ozero__middle,axiom,
! [A: $tType,F: A > A > A,Z3: A,P5: nat,K: nat,G: nat > A,H2: nat > A] :
( ( groups778175481326437816id_set @ A @ F @ Z3 )
=> ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ P5 )
=> ( ( ord_less_eq @ nat @ K @ P5 )
=> ( ( groups_comm_monoid_F @ A @ nat @ F @ Z3
@ ^ [J3: nat] : ( if @ A @ ( ord_less @ nat @ J3 @ K ) @ ( G @ J3 ) @ ( if @ A @ ( J3 = K ) @ Z3 @ ( H2 @ ( minus_minus @ nat @ J3 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) )
@ ( set_ord_atMost @ nat @ P5 ) )
= ( groups_comm_monoid_F @ A @ nat @ F @ Z3
@ ^ [J3: nat] : ( if @ A @ ( ord_less @ nat @ J3 @ K ) @ ( G @ J3 ) @ ( H2 @ J3 ) )
@ ( set_ord_atMost @ nat @ ( minus_minus @ nat @ P5 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) ) ).
% comm_monoid_set.zero_middle
thf(fact_7071_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 ) )
= ( image @ A @ B @ F @ ( heap_Time_set_Heap @ A @ V ) ) ) ).
% Heap.set_map
thf(fact_7072_comm__monoid__set_OatLeastAtMost__shift__0,axiom,
! [A: $tType,F: A > A > A,Z3: A,M: nat,N: nat,G: nat > A] :
( ( groups778175481326437816id_set @ A @ F @ Z3 )
=> ( ( ord_less_eq @ nat @ M @ N )
=> ( ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ ( 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_7073_comm__monoid__set_OatLeast__atMost__pred__shift,axiom,
! [A: $tType,F: A > A > A,Z3: A,G: nat > A,M: nat,N: nat] :
( ( groups778175481326437816id_set @ A @ F @ Z3 )
=> ( ( groups_comm_monoid_F @ A @ nat @ F @ Z3
@ ( comp @ nat @ A @ nat @ G
@ ^ [N3: nat] : ( minus_minus @ nat @ N3 @ ( suc @ ( zero_zero @ nat ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ ( suc @ N ) ) )
= ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% comm_monoid_set.atLeast_atMost_pred_shift
thf(fact_7074_comm__monoid__set_OatLeast__lessThan__pred__shift,axiom,
! [A: $tType,F: A > A > A,Z3: A,G: nat > A,M: nat,N: nat] :
( ( groups778175481326437816id_set @ A @ F @ Z3 )
=> ( ( groups_comm_monoid_F @ A @ nat @ F @ Z3
@ ( comp @ nat @ A @ nat @ G
@ ^ [N3: nat] : ( minus_minus @ nat @ N3 @ ( suc @ ( zero_zero @ nat ) ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M ) @ ( suc @ N ) ) )
= ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ).
% comm_monoid_set.atLeast_lessThan_pred_shift
thf(fact_7075_comm__monoid__set_OatLeast0__lessThan__Suc__shift,axiom,
! [A: $tType,F: A > A > A,Z3: A,G: nat > A,N: nat] :
( ( groups778175481326437816id_set @ A @ F @ Z3 )
=> ( ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
= ( F @ ( G @ ( zero_zero @ nat ) ) @ ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ) ).
% comm_monoid_set.atLeast0_lessThan_Suc_shift
thf(fact_7076_comm__monoid__set_OatLeast0__atMost__Suc__shift,axiom,
! [A: $tType,F: A > A > A,Z3: A,G: nat > A,N: nat] :
( ( groups778175481326437816id_set @ A @ F @ Z3 )
=> ( ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
= ( F @ ( G @ ( zero_zero @ nat ) ) @ ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ) ).
% comm_monoid_set.atLeast0_atMost_Suc_shift
thf(fact_7077_comm__monoid__set_OatLeastLessThan__shift__0,axiom,
! [A: $tType,F: A > A > A,Z3: A,G: nat > A,M: nat,N: nat] :
( ( groups778175481326437816id_set @ A @ F @ Z3 )
=> ( ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) )
= ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ ( 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_7078_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_7079_comm__monoid__set_OatMost__shift,axiom,
! [A: $tType,F: A > A > A,Z3: A,G: nat > A,N: nat] :
( ( groups778175481326437816id_set @ A @ F @ Z3 )
=> ( ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_ord_atMost @ nat @ N ) )
= ( F @ ( G @ ( zero_zero @ nat ) )
@ ( groups_comm_monoid_F @ A @ nat @ F @ Z3
@ ^ [I5: nat] : ( G @ ( suc @ I5 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% comm_monoid_set.atMost_shift
thf(fact_7080_comm__monoid__set_OlessThan__Suc__shift,axiom,
! [A: $tType,F: A > A > A,Z3: A,G: nat > A,N: nat] :
( ( groups778175481326437816id_set @ A @ F @ Z3 )
=> ( ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_ord_lessThan @ nat @ ( suc @ N ) ) )
= ( F @ ( G @ ( zero_zero @ nat ) )
@ ( groups_comm_monoid_F @ A @ nat @ F @ Z3
@ ^ [I5: nat] : ( G @ ( suc @ I5 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% comm_monoid_set.lessThan_Suc_shift
thf(fact_7081_comm__monoid__set_OatMost__Suc__shift,axiom,
! [A: $tType,F: A > A > A,Z3: A,G: nat > A,N: nat] :
( ( groups778175481326437816id_set @ A @ F @ Z3 )
=> ( ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_ord_atMost @ nat @ ( suc @ N ) ) )
= ( F @ ( G @ ( zero_zero @ nat ) )
@ ( groups_comm_monoid_F @ A @ nat @ F @ Z3
@ ^ [I5: nat] : ( G @ ( suc @ I5 ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ) ).
% comm_monoid_set.atMost_Suc_shift
thf(fact_7082_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_7083_comm__monoid__set_OatLeast0__atMost__Suc,axiom,
! [A: $tType,F: A > A > A,Z3: A,G: nat > A,N: nat] :
( ( groups778175481326437816id_set @ A @ F @ Z3 )
=> ( ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
= ( F @ ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ).
% comm_monoid_set.atLeast0_atMost_Suc
thf(fact_7084_comm__monoid__set_OatLeast0__lessThan__Suc,axiom,
! [A: $tType,F: A > A > A,Z3: A,G: nat > A,N: nat] :
( ( groups778175481326437816id_set @ A @ F @ Z3 )
=> ( ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
= ( F @ ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) @ ( G @ N ) ) ) ) ).
% comm_monoid_set.atLeast0_lessThan_Suc
thf(fact_7085_comm__monoid__set_Onested__swap,axiom,
! [A: $tType,F: A > A > A,Z3: A,A3: nat > nat > A,N: nat] :
( ( groups778175481326437816id_set @ A @ F @ Z3 )
=> ( ( groups_comm_monoid_F @ A @ nat @ F @ Z3
@ ^ [I5: nat] : ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ ( A3 @ I5 ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ I5 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
= ( groups_comm_monoid_F @ A @ nat @ F @ Z3
@ ^ [J3: nat] :
( groups_comm_monoid_F @ A @ nat @ F @ Z3
@ ^ [I5: nat] : ( A3 @ I5 @ J3 )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J3 ) @ N ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% comm_monoid_set.nested_swap
thf(fact_7086_comm__monoid__set_OatLeast1__atMost__eq,axiom,
! [A: $tType,F: A > A > A,Z3: A,G: nat > A,N: nat] :
( ( groups778175481326437816id_set @ A @ F @ Z3 )
=> ( ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N ) )
= ( groups_comm_monoid_F @ A @ nat @ F @ Z3
@ ^ [K3: nat] : ( G @ ( suc @ K3 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% comm_monoid_set.atLeast1_atMost_eq
thf(fact_7087_comm__monoid__set_Oin__pairs__0,axiom,
! [A: $tType,F: A > A > A,Z3: A,G: nat > A,N: nat] :
( ( groups778175481326437816id_set @ A @ F @ Z3 )
=> ( ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_ord_atMost @ nat @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) )
= ( groups_comm_monoid_F @ A @ nat @ F @ Z3
@ ^ [I5: nat] : ( F @ ( G @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I5 ) ) @ ( G @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I5 ) ) ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% comm_monoid_set.in_pairs_0
thf(fact_7088_comm__monoid__set_Ocl__ivl__Suc,axiom,
! [A: $tType,F: A > A > A,Z3: A,N: nat,M: nat,G: nat > A] :
( ( groups778175481326437816id_set @ A @ F @ Z3 )
=> ( ( ( ord_less @ nat @ ( suc @ N ) @ M )
=> ( ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ ( suc @ N ) ) )
= Z3 ) )
& ( ~ ( ord_less @ nat @ ( suc @ N ) @ M )
=> ( ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ ( suc @ N ) ) )
= ( F @ ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ) ).
% comm_monoid_set.cl_ivl_Suc
thf(fact_7089_comm__monoid__set_OatLeast__Suc__lessThan,axiom,
! [A: $tType,F: A > A > A,Z3: A,M: nat,N: nat,G: nat > A] :
( ( groups778175481326437816id_set @ A @ F @ Z3 )
=> ( ( ord_less @ nat @ M @ N )
=> ( ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) )
= ( F @ ( G @ M ) @ ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M ) @ N ) ) ) ) ) ) ).
% comm_monoid_set.atLeast_Suc_lessThan
thf(fact_7090_comm__monoid__set_Oop__ivl__Suc,axiom,
! [A: $tType,F: A > A > A,Z3: A,N: nat,M: nat,G: nat > A] :
( ( groups778175481326437816id_set @ A @ F @ Z3 )
=> ( ( ( ord_less @ nat @ N @ M )
=> ( ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ ( suc @ N ) ) )
= Z3 ) )
& ( ~ ( ord_less @ nat @ N @ M )
=> ( ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ ( suc @ N ) ) )
= ( F @ ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) @ ( G @ N ) ) ) ) ) ) ).
% comm_monoid_set.op_ivl_Suc
thf(fact_7091_comm__monoid__set_Ohead__if,axiom,
! [A: $tType,F: A > A > A,Z3: A,N: nat,M: nat,G: nat > A] :
( ( groups778175481326437816id_set @ A @ F @ Z3 )
=> ( ( ( ord_less @ nat @ N @ M )
=> ( ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= Z3 ) )
& ( ~ ( ord_less @ nat @ N @ M )
=> ( ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( F @ ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) @ ( G @ N ) ) ) ) ) ) ).
% comm_monoid_set.head_if
thf(fact_7092_comm__monoid__set_Oivl__cong,axiom,
! [A: $tType,B: $tType] :
( ( ord @ B )
=> ! [F: A > A > A,Z3: A,A3: B,C2: B,B2: B,D2: B,G: B > A,H2: B > A] :
( ( groups778175481326437816id_set @ A @ F @ Z3 )
=> ( ( A3 = C2 )
=> ( ( B2 = D2 )
=> ( ! [X3: B] :
( ( ord_less_eq @ B @ C2 @ X3 )
=> ( ( ord_less @ B @ X3 @ D2 )
=> ( ( G @ X3 )
= ( H2 @ X3 ) ) ) )
=> ( ( groups_comm_monoid_F @ A @ B @ F @ Z3 @ G @ ( set_or7035219750837199246ssThan @ B @ A3 @ B2 ) )
= ( groups_comm_monoid_F @ A @ B @ F @ Z3 @ H2 @ ( set_or7035219750837199246ssThan @ B @ C2 @ D2 ) ) ) ) ) ) ) ) ).
% comm_monoid_set.ivl_cong
thf(fact_7093_comm__monoid__set_Onested__swap_H,axiom,
! [A: $tType,F: A > A > A,Z3: A,A3: nat > nat > A,N: nat] :
( ( groups778175481326437816id_set @ A @ F @ Z3 )
=> ( ( groups_comm_monoid_F @ A @ nat @ F @ Z3
@ ^ [I5: nat] : ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ ( A3 @ I5 ) @ ( set_ord_lessThan @ nat @ I5 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( groups_comm_monoid_F @ A @ nat @ F @ Z3
@ ^ [J3: nat] :
( groups_comm_monoid_F @ A @ nat @ F @ Z3
@ ^ [I5: nat] : ( A3 @ I5 @ J3 )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J3 ) @ N ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% comm_monoid_set.nested_swap'
thf(fact_7094_comm__monoid__set_Oshift__bounds__cl__Suc__ivl,axiom,
! [A: $tType,F: A > A > A,Z3: A,G: nat > A,M: nat,N: nat] :
( ( groups778175481326437816id_set @ A @ F @ Z3 )
=> ( ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ ( suc @ N ) ) )
= ( groups_comm_monoid_F @ A @ nat @ F @ Z3
@ ^ [I5: nat] : ( G @ ( suc @ I5 ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% comm_monoid_set.shift_bounds_cl_Suc_ivl
thf(fact_7095_comm__monoid__set_Oshift__bounds__Suc__ivl,axiom,
! [A: $tType,F: A > A > A,Z3: A,G: nat > A,M: nat,N: nat] :
( ( groups778175481326437816id_set @ A @ F @ Z3 )
=> ( ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M ) @ ( suc @ N ) ) )
= ( groups_comm_monoid_F @ A @ nat @ F @ Z3
@ ^ [I5: nat] : ( G @ ( suc @ I5 ) )
@ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ).
% comm_monoid_set.shift_bounds_Suc_ivl
thf(fact_7096_comm__monoid__set_OlessThan__Suc,axiom,
! [A: $tType,F: A > A > A,Z3: A,G: nat > A,N: nat] :
( ( groups778175481326437816id_set @ A @ F @ Z3 )
=> ( ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_ord_lessThan @ nat @ ( suc @ N ) ) )
= ( F @ ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_ord_lessThan @ nat @ N ) ) @ ( G @ N ) ) ) ) ).
% comm_monoid_set.lessThan_Suc
thf(fact_7097_comm__monoid__set_OatMost__Suc,axiom,
! [A: $tType,F: A > A > A,Z3: A,G: nat > A,N: nat] :
( ( groups778175481326437816id_set @ A @ F @ Z3 )
=> ( ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_ord_atMost @ nat @ ( suc @ N ) ) )
= ( F @ ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_ord_atMost @ nat @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ).
% comm_monoid_set.atMost_Suc
thf(fact_7098_comm__monoid__set_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
! [A: $tType,F: A > A > A,Z3: A,G: nat > A,N: nat,M: nat] :
( ( groups778175481326437816id_set @ A @ F @ Z3 )
=> ( ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or7035219750837199246ssThan @ nat @ N @ M ) )
= ( groups_comm_monoid_F @ A @ nat @ F @ Z3
@ ^ [I5: nat] : ( G @ ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N ) @ I5 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ N ) @ M ) ) ) ) ).
% comm_monoid_set.atLeastLessThan_rev_at_least_Suc_atMost
thf(fact_7099_comm__monoid__set_OatLeastLessThan__rev,axiom,
! [A: $tType,F: A > A > A,Z3: A,G: nat > A,N: nat,M: nat] :
( ( groups778175481326437816id_set @ A @ F @ Z3 )
=> ( ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or7035219750837199246ssThan @ nat @ N @ M ) )
= ( groups_comm_monoid_F @ A @ nat @ F @ Z3
@ ^ [I5: nat] : ( G @ ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N ) @ ( suc @ I5 ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ N @ M ) ) ) ) ).
% comm_monoid_set.atLeastLessThan_rev
thf(fact_7100_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_7101_comm__monoid__set_Onat__diff__reindex,axiom,
! [A: $tType,F: A > A > A,Z3: A,G: nat > A,N: nat] :
( ( groups778175481326437816id_set @ A @ F @ Z3 )
=> ( ( groups_comm_monoid_F @ A @ nat @ F @ Z3
@ ^ [I5: nat] : ( G @ ( minus_minus @ nat @ N @ ( suc @ I5 ) ) )
@ ( set_ord_lessThan @ nat @ N ) )
= ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% comm_monoid_set.nat_diff_reindex
thf(fact_7102_comm__monoid__set_Onat__group,axiom,
! [A: $tType,F: A > A > A,Z3: A,G: nat > A,K: nat,N: nat] :
( ( groups778175481326437816id_set @ A @ F @ Z3 )
=> ( ( groups_comm_monoid_F @ A @ nat @ F @ Z3
@ ^ [M2: nat] : ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or7035219750837199246ssThan @ nat @ ( times_times @ nat @ M2 @ K ) @ ( plus_plus @ nat @ ( times_times @ nat @ M2 @ K ) @ K ) ) )
@ ( set_ord_lessThan @ nat @ N ) )
= ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_ord_lessThan @ nat @ ( times_times @ nat @ N @ K ) ) ) ) ) ).
% comm_monoid_set.nat_group
thf(fact_7103_comm__monoid__set_Oub__add__nat,axiom,
! [A: $tType,F: A > A > A,Z3: A,M: nat,N: nat,G: nat > A,P5: nat] :
( ( groups778175481326437816id_set @ A @ F @ Z3 )
=> ( ( ord_less_eq @ nat @ M @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) )
=> ( ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ ( plus_plus @ nat @ N @ P5 ) ) )
= ( F @ ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) @ ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) @ ( plus_plus @ nat @ N @ P5 ) ) ) ) ) ) ) ).
% comm_monoid_set.ub_add_nat
thf(fact_7104_comm__monoid__set_Ohead,axiom,
! [A: $tType,F: A > A > A,Z3: A,M: nat,N: nat,G: nat > A] :
( ( groups778175481326437816id_set @ A @ F @ Z3 )
=> ( ( ord_less_eq @ nat @ M @ N )
=> ( ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( F @ ( G @ M ) @ ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or3652927894154168847AtMost @ nat @ M @ N ) ) ) ) ) ) ).
% comm_monoid_set.head
thf(fact_7105_comm__monoid__set_OatLeast__Suc__atMost,axiom,
! [A: $tType,F: A > A > A,Z3: A,M: nat,N: nat,G: nat > A] :
( ( groups778175481326437816id_set @ A @ F @ Z3 )
=> ( ( ord_less_eq @ nat @ M @ N )
=> ( ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( F @ ( G @ M ) @ ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ N ) ) ) ) ) ) ).
% comm_monoid_set.atLeast_Suc_atMost
thf(fact_7106_comm__monoid__set_Onat__ivl__Suc_H,axiom,
! [A: $tType,F: A > A > A,Z3: A,M: nat,N: nat,G: nat > A] :
( ( groups778175481326437816id_set @ A @ F @ Z3 )
=> ( ( ord_less_eq @ nat @ M @ ( suc @ N ) )
=> ( ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ ( suc @ N ) ) )
= ( F @ ( G @ ( suc @ N ) ) @ ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ) ) ).
% comm_monoid_set.nat_ivl_Suc'
thf(fact_7107_comm__monoid__set_OatLeastLessThan__Suc,axiom,
! [A: $tType,F: A > A > A,Z3: A,A3: nat,B2: nat,G: nat > A] :
( ( groups778175481326437816id_set @ A @ F @ Z3 )
=> ( ( ord_less_eq @ nat @ A3 @ B2 )
=> ( ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or7035219750837199246ssThan @ nat @ A3 @ ( suc @ B2 ) ) )
= ( F @ ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or7035219750837199246ssThan @ nat @ A3 @ B2 ) ) @ ( G @ B2 ) ) ) ) ) ).
% comm_monoid_set.atLeastLessThan_Suc
thf(fact_7108_comm__monoid__set_OatLeastLessThan__concat,axiom,
! [A: $tType,F: A > A > A,Z3: A,M: nat,N: nat,P5: nat,G: nat > A] :
( ( groups778175481326437816id_set @ A @ F @ Z3 )
=> ( ( ord_less_eq @ nat @ M @ N )
=> ( ( ord_less_eq @ nat @ N @ P5 )
=> ( ( F @ ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) @ ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or7035219750837199246ssThan @ nat @ N @ P5 ) ) )
= ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ P5 ) ) ) ) ) ) ).
% comm_monoid_set.atLeastLessThan_concat
thf(fact_7109_comm__monoid__set_Olast__plus,axiom,
! [A: $tType,F: A > A > A,Z3: A,M: nat,N: nat,G: nat > A] :
( ( groups778175481326437816id_set @ A @ F @ Z3 )
=> ( ( ord_less_eq @ nat @ M @ N )
=> ( ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( F @ ( G @ N ) @ ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ) ) ).
% comm_monoid_set.last_plus
thf(fact_7110_comm__monoid__set_OSuc__reindex__ivl,axiom,
! [A: $tType,F: A > A > A,Z3: A,M: nat,N: nat,G: nat > A] :
( ( groups778175481326437816id_set @ A @ F @ Z3 )
=> ( ( ord_less_eq @ nat @ M @ N )
=> ( ( F @ ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) @ ( G @ ( suc @ N ) ) )
= ( F @ ( G @ M )
@ ( groups_comm_monoid_F @ A @ nat @ F @ Z3
@ ^ [I5: nat] : ( G @ ( suc @ I5 ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ) ) ).
% comm_monoid_set.Suc_reindex_ivl
thf(fact_7111_Heap_Oinj__map__strong,axiom,
! [B: $tType,A: $tType,X: heap_Time_Heap @ A,Xa3: heap_Time_Heap @ A,F: A > B,Fa: A > B] :
( ! [Z4: A,Za: A] :
( ( member @ A @ Z4 @ ( heap_Time_set_Heap @ A @ X ) )
=> ( ( member @ A @ Za @ ( heap_Time_set_Heap @ A @ Xa3 ) )
=> ( ( ( F @ Z4 )
= ( Fa @ Za ) )
=> ( Z4 = Za ) ) ) )
=> ( ( ( heap_Time_map_Heap @ A @ B @ F @ X )
= ( heap_Time_map_Heap @ A @ B @ Fa @ Xa3 ) )
=> ( X = Xa3 ) ) ) ).
% Heap.inj_map_strong
thf(fact_7112_Heap_Omap__cong0,axiom,
! [B: $tType,A: $tType,X: heap_Time_Heap @ A,F: A > B,G: A > B] :
( ! [Z4: A] :
( ( member @ A @ Z4 @ ( heap_Time_set_Heap @ A @ X ) )
=> ( ( F @ Z4 )
= ( G @ Z4 ) ) )
=> ( ( heap_Time_map_Heap @ A @ B @ F @ X )
= ( heap_Time_map_Heap @ A @ B @ G @ X ) ) ) ).
% Heap.map_cong0
thf(fact_7113_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 )
=> ( ! [Z4: A] :
( ( member @ A @ Z4 @ ( heap_Time_set_Heap @ A @ Ya ) )
=> ( ( F @ Z4 )
= ( G @ Z4 ) ) )
=> ( ( heap_Time_map_Heap @ A @ B @ F @ X )
= ( heap_Time_map_Heap @ A @ B @ G @ Ya ) ) ) ) ).
% Heap.map_cong
thf(fact_7114_comm__monoid__set_OIf__cases,axiom,
! [A: $tType,B: $tType,F: A > A > A,Z3: A,A6: set @ B,P: B > $o,H2: B > A,G: B > A] :
( ( groups778175481326437816id_set @ A @ F @ Z3 )
=> ( ( finite_finite2 @ B @ A6 )
=> ( ( groups_comm_monoid_F @ A @ B @ F @ Z3
@ ^ [X2: B] : ( if @ A @ ( P @ X2 ) @ ( H2 @ X2 ) @ ( G @ X2 ) )
@ A6 )
= ( F @ ( groups_comm_monoid_F @ A @ B @ F @ Z3 @ H2 @ ( inf_inf @ ( set @ B ) @ A6 @ ( collect @ B @ P ) ) ) @ ( groups_comm_monoid_F @ A @ B @ F @ Z3 @ G @ ( inf_inf @ ( set @ B ) @ A6 @ ( uminus_uminus @ ( set @ B ) @ ( collect @ B @ P ) ) ) ) ) ) ) ) ).
% comm_monoid_set.If_cases
thf(fact_7115_comm__monoid__set_OatLeast__Suc__lessThan__Suc__shift,axiom,
! [A: $tType,F: A > A > A,Z3: A,G: nat > A,M: nat,N: nat] :
( ( groups778175481326437816id_set @ A @ F @ Z3 )
=> ( ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M ) @ ( suc @ N ) ) )
= ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ).
% comm_monoid_set.atLeast_Suc_lessThan_Suc_shift
thf(fact_7116_comm__monoid__set_OatLeast__Suc__atMost__Suc__shift,axiom,
! [A: $tType,F: A > A > A,Z3: A,G: nat > A,M: nat,N: nat] :
( ( groups778175481326437816id_set @ A @ F @ Z3 )
=> ( ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ ( suc @ N ) ) )
= ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% comm_monoid_set.atLeast_Suc_atMost_Suc_shift
thf(fact_7117_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_7118_comm__monoid__set_Oin__pairs,axiom,
! [A: $tType,F: A > A > A,Z3: A,G: nat > A,M: nat,N: nat] :
( ( groups778175481326437816id_set @ A @ F @ Z3 )
=> ( ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ 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 @ Z3
@ ^ [I5: nat] : ( F @ ( G @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I5 ) ) @ ( G @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I5 ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% comm_monoid_set.in_pairs
thf(fact_7119_comm__monoid__set_Otriangle__reindex,axiom,
! [A: $tType,F: A > A > A,Z3: A,G: nat > nat > A,N: nat] :
( ( groups778175481326437816id_set @ A @ F @ Z3 )
=> ( ( groups_comm_monoid_F @ A @ ( product_prod @ nat @ nat ) @ F @ Z3 @ ( product_case_prod @ nat @ nat @ A @ G )
@ ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [I5: nat,J3: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ I5 @ J3 ) @ N ) ) ) )
= ( groups_comm_monoid_F @ A @ nat @ F @ Z3
@ ^ [K3: nat] :
( groups_comm_monoid_F @ A @ nat @ F @ Z3
@ ^ [I5: nat] : ( G @ I5 @ ( minus_minus @ nat @ K3 @ I5 ) )
@ ( set_ord_atMost @ nat @ K3 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% comm_monoid_set.triangle_reindex
thf(fact_7120_comm__monoid__set_Otriangle__reindex__eq,axiom,
! [A: $tType,F: A > A > A,Z3: A,G: nat > nat > A,N: nat] :
( ( groups778175481326437816id_set @ A @ F @ Z3 )
=> ( ( groups_comm_monoid_F @ A @ ( product_prod @ nat @ nat ) @ F @ Z3 @ ( product_case_prod @ nat @ nat @ A @ G )
@ ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [I5: nat,J3: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ I5 @ J3 ) @ N ) ) ) )
= ( groups_comm_monoid_F @ A @ nat @ F @ Z3
@ ^ [K3: nat] :
( groups_comm_monoid_F @ A @ nat @ F @ Z3
@ ^ [I5: nat] : ( G @ I5 @ ( minus_minus @ nat @ K3 @ I5 ) )
@ ( set_ord_atMost @ nat @ K3 ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% comm_monoid_set.triangle_reindex_eq
thf(fact_7121_comm__monoid__set_OatLeast__int__atMost__int__shift,axiom,
! [A: $tType,F: A > A > A,Z3: A,G: int > A,M: nat,N: nat] :
( ( groups778175481326437816id_set @ A @ F @ Z3 )
=> ( ( groups_comm_monoid_F @ A @ int @ F @ Z3 @ G @ ( set_or1337092689740270186AtMost @ int @ ( semiring_1_of_nat @ int @ M ) @ ( semiring_1_of_nat @ int @ N ) ) )
= ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ ( 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_7122_comm__monoid__set_OatLeast__int__lessThan__int__shift,axiom,
! [A: $tType,F: A > A > A,Z3: A,G: int > A,M: nat,N: nat] :
( ( groups778175481326437816id_set @ A @ F @ Z3 )
=> ( ( groups_comm_monoid_F @ A @ int @ F @ Z3 @ G @ ( set_or7035219750837199246ssThan @ int @ ( semiring_1_of_nat @ int @ M ) @ ( semiring_1_of_nat @ int @ N ) ) )
= ( groups_comm_monoid_F @ A @ nat @ F @ Z3 @ ( 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_7123_Heap_Oin__rel,axiom,
! [B: $tType,A: $tType] :
( ( heap_Time_rel_Heap @ A @ B )
= ( ^ [R6: A > B > $o,A5: heap_Time_Heap @ A,B3: heap_Time_Heap @ B] :
? [Z7: heap_Time_Heap @ ( product_prod @ A @ B )] :
( ( member @ ( heap_Time_Heap @ ( product_prod @ A @ B ) ) @ Z7
@ ( 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 ) @ Z7 )
= A5 )
& ( ( heap_Time_map_Heap @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ Z7 )
= B3 ) ) ) ) ).
% Heap.in_rel
thf(fact_7124_Heap_Opred__set,axiom,
! [A: $tType] :
( ( heap_Time_pred_Heap @ A )
= ( ^ [P4: A > $o,X2: heap_Time_Heap @ A] :
! [Y6: A] :
( ( member @ A @ Y6 @ ( heap_Time_set_Heap @ A @ X2 ) )
=> ( P4 @ Y6 ) ) ) ) ).
% Heap.pred_set
thf(fact_7125_Heap_ODomainp__rel,axiom,
! [B: $tType,A: $tType,R2: A > B > $o] :
( ( domainp @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ B ) @ ( heap_Time_rel_Heap @ A @ B @ R2 ) )
= ( heap_Time_pred_Heap @ A @ ( domainp @ A @ B @ R2 ) ) ) ).
% Heap.Domainp_rel
thf(fact_7126_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_7127_Heap_Orel__eq,axiom,
! [A: $tType] :
( ( heap_Time_rel_Heap @ A @ A
@ ^ [Y: A,Z2: A] : Y = Z2 )
= ( ^ [Y: heap_Time_Heap @ A,Z2: heap_Time_Heap @ A] : Y = Z2 ) ) ).
% Heap.rel_eq
thf(fact_7128_Heap_Orel__refl,axiom,
! [B: $tType,Ra: B > B > $o,X: heap_Time_Heap @ B] :
( ! [X3: B] : ( Ra @ X3 @ X3 )
=> ( heap_Time_rel_Heap @ B @ B @ Ra @ X @ X ) ) ).
% Heap.rel_refl
thf(fact_7129_Heap_Orel__transp,axiom,
! [A: $tType,R2: A > A > $o] :
( ( transp @ A @ R2 )
=> ( transp @ ( heap_Time_Heap @ A ) @ ( heap_Time_rel_Heap @ A @ A @ R2 ) ) ) ).
% Heap.rel_transp
thf(fact_7130_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_7131_Heap_Orel__mono,axiom,
! [B: $tType,A: $tType,R2: A > B > $o,Ra: A > B > $o] :
( ( ord_less_eq @ ( A > B > $o ) @ R2 @ Ra )
=> ( ord_less_eq @ ( ( heap_Time_Heap @ A ) > ( heap_Time_Heap @ B ) > $o ) @ ( heap_Time_rel_Heap @ A @ B @ R2 ) @ ( heap_Time_rel_Heap @ A @ B @ Ra ) ) ) ).
% Heap.rel_mono
thf(fact_7132_Heap_Obi__total__rel,axiom,
! [B: $tType,A: $tType,R2: A > B > $o] :
( ( bi_total @ A @ B @ R2 )
=> ( bi_total @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ B ) @ ( heap_Time_rel_Heap @ A @ B @ R2 ) ) ) ).
% Heap.bi_total_rel
thf(fact_7133_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
@ ^ [Y: $o,Z2: $o] : Y = 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 )
@ ^ [Y: $o,Z2: $o] : Y = Z2 ) )
@ ( heap_Time_rel_Heap @ A @ B )
@ ( heap_Time_rel_Heap @ C @ D ) ) ).
% Heap.rel_transfer
thf(fact_7134_Heap_Opred__transfer,axiom,
! [A: $tType,B: $tType,R2: 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 @ R2
@ ^ [Y: $o,Z2: $o] : Y = Z2 )
@ ( bNF_rel_fun @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ B ) @ $o @ $o @ ( heap_Time_rel_Heap @ A @ B @ R2 )
@ ^ [Y: $o,Z2: $o] : Y = Z2 )
@ ( heap_Time_pred_Heap @ A )
@ ( heap_Time_pred_Heap @ B ) ) ).
% Heap.pred_transfer
thf(fact_7135_Heap_Omap__transfer,axiom,
! [A: $tType,B: $tType,F7: $tType,E4: $tType,Rb: A > E4 > $o,Sd: B > F7 > $o] : ( bNF_rel_fun @ ( A > B ) @ ( E4 > F7 ) @ ( ( heap_Time_Heap @ A ) > ( heap_Time_Heap @ B ) ) @ ( ( heap_Time_Heap @ E4 ) > ( heap_Time_Heap @ F7 ) ) @ ( bNF_rel_fun @ A @ E4 @ B @ F7 @ Rb @ Sd ) @ ( bNF_rel_fun @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ E4 ) @ ( heap_Time_Heap @ B ) @ ( heap_Time_Heap @ F7 ) @ ( heap_Time_rel_Heap @ A @ E4 @ Rb ) @ ( heap_Time_rel_Heap @ B @ F7 @ Sd ) ) @ ( heap_Time_map_Heap @ A @ B ) @ ( heap_Time_map_Heap @ E4 @ F7 ) ) ).
% Heap.map_transfer
thf(fact_7136_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
@ ^ [Z7: A] :
( ( F @ Z7 )
= ( G @ Z7 ) )
@ Ya )
=> ( ( heap_Time_map_Heap @ A @ B @ F @ X )
= ( heap_Time_map_Heap @ A @ B @ G @ Ya ) ) ) ) ).
% Heap.map_cong_pred
thf(fact_7137_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_7138_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,Y6: B] : ( Sa @ X2 @ ( G @ Y6 ) )
@ X
@ Y3 ) ) ).
% Heap.rel_map(2)
thf(fact_7139_Heap_OQuotient,axiom,
! [B: $tType,A: $tType,R2: A > A > $o,Abs: A > B,Rep: B > A,T6: A > B > $o] :
( ( quotient @ A @ B @ R2 @ Abs @ Rep @ T6 )
=> ( quotient @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ B ) @ ( heap_Time_rel_Heap @ A @ A @ R2 ) @ ( heap_Time_map_Heap @ A @ B @ Abs ) @ ( heap_Time_map_Heap @ B @ A @ Rep ) @ ( heap_Time_rel_Heap @ A @ B @ T6 ) ) ) ).
% Heap.Quotient
thf(fact_7140_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_7141_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 )
=> ( ! [Z4: A] :
( ( member @ A @ Z4 @ ( heap_Time_set_Heap @ A @ X ) )
=> ( ( P @ Z4 )
=> ( Pa @ Z4 ) ) )
=> ( heap_Time_pred_Heap @ A @ Pa @ X ) ) ) ).
% Heap.pred_mono_strong
thf(fact_7142_Heap_Orel__refl__strong,axiom,
! [A: $tType,X: heap_Time_Heap @ A,Ra: A > A > $o] :
( ! [Z4: A] :
( ( member @ A @ Z4 @ ( heap_Time_set_Heap @ A @ X ) )
=> ( Ra @ Z4 @ Z4 ) )
=> ( heap_Time_rel_Heap @ A @ A @ Ra @ X @ X ) ) ).
% Heap.rel_refl_strong
thf(fact_7143_Heap_Orel__mono__strong,axiom,
! [A: $tType,B: $tType,R2: A > B > $o,X: heap_Time_Heap @ A,Y3: heap_Time_Heap @ B,Ra: A > B > $o] :
( ( heap_Time_rel_Heap @ A @ B @ R2 @ X @ Y3 )
=> ( ! [Z4: A,Yb: B] :
( ( member @ A @ Z4 @ ( heap_Time_set_Heap @ A @ X ) )
=> ( ( member @ B @ Yb @ ( heap_Time_set_Heap @ B @ Y3 ) )
=> ( ( R2 @ Z4 @ Yb )
=> ( Ra @ Z4 @ Yb ) ) ) )
=> ( heap_Time_rel_Heap @ A @ B @ Ra @ X @ Y3 ) ) ) ).
% Heap.rel_mono_strong
thf(fact_7144_Heap_Opred__cong,axiom,
! [A: $tType,X: heap_Time_Heap @ A,Ya: heap_Time_Heap @ A,P: A > $o,Pa: A > $o] :
( ( X = Ya )
=> ( ! [Z4: A] :
( ( member @ A @ Z4 @ ( heap_Time_set_Heap @ A @ Ya ) )
=> ( ( P @ Z4 )
= ( Pa @ Z4 ) ) )
=> ( ( heap_Time_pred_Heap @ A @ P @ X )
= ( heap_Time_pred_Heap @ A @ Pa @ Ya ) ) ) ) ).
% Heap.pred_cong
thf(fact_7145_Heap_Orel__cong,axiom,
! [A: $tType,B: $tType,X: heap_Time_Heap @ A,Ya: heap_Time_Heap @ A,Y3: heap_Time_Heap @ B,Xa3: heap_Time_Heap @ B,R2: A > B > $o,Ra: A > B > $o] :
( ( X = Ya )
=> ( ( Y3 = Xa3 )
=> ( ! [Z4: A,Yb: B] :
( ( member @ A @ Z4 @ ( heap_Time_set_Heap @ A @ Ya ) )
=> ( ( member @ B @ Yb @ ( heap_Time_set_Heap @ B @ Xa3 ) )
=> ( ( R2 @ Z4 @ Yb )
= ( Ra @ Z4 @ Yb ) ) ) )
=> ( ( heap_Time_rel_Heap @ A @ B @ R2 @ X @ Y3 )
= ( heap_Time_rel_Heap @ A @ B @ Ra @ Ya @ Xa3 ) ) ) ) ) ).
% Heap.rel_cong
thf(fact_7146_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 )
@ ^ [A5: int,B3: int] :
( if @ ( product_prod @ int @ int )
@ ( B3
= ( zero_zero @ int ) )
@ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
@ ( product_Pair @ int @ int @ A5 @ B3 ) ) ) ) ).
% Fract_def
thf(fact_7147_subset_Osuc__Union__closed__suc,axiom,
! [A: $tType,X8: set @ ( set @ A ),A6: set @ ( set @ A ),Y7: set @ ( set @ A )] :
( ( member @ ( set @ ( set @ A ) ) @ X8 @ ( pred_s596693808085603175closed @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) ) )
=> ( ( member @ ( set @ ( set @ A ) ) @ Y7 @ ( pred_s596693808085603175closed @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) ) )
=> ( ( ( pred_suc @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ Y7 )
= Y7 )
=> ( ord_less_eq @ ( set @ ( set @ A ) ) @ X8 @ Y7 ) ) ) ) ).
% subset.suc_Union_closed_suc
thf(fact_7148_predicate2I,axiom,
! [B: $tType,A: $tType,P: A > B > $o,Q: A > B > $o] :
( ! [X3: A,Y5: B] :
( ( P @ X3 @ Y5 )
=> ( Q @ X3 @ Y5 ) )
=> ( ord_less_eq @ ( A > B > $o ) @ P @ Q ) ) ).
% predicate2I
thf(fact_7149_predicate1I,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X3: A] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq @ ( A > $o ) @ P @ Q ) ) ).
% predicate1I
thf(fact_7150_of__int__eq__id,axiom,
( ( ring_1_of_int @ int )
= ( id @ int ) ) ).
% of_int_eq_id
thf(fact_7151_predicate1D,axiom,
! [A: $tType,P: A > $o,Q: A > $o,X: A] :
( ( ord_less_eq @ ( A > $o ) @ P @ Q )
=> ( ( P @ X )
=> ( Q @ X ) ) ) ).
% predicate1D
thf(fact_7152_predicate2D,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Q: A > B > $o,X: A,Y3: B] :
( ( ord_less_eq @ ( A > B > $o ) @ P @ Q )
=> ( ( P @ X @ Y3 )
=> ( Q @ X @ Y3 ) ) ) ).
% predicate2D
thf(fact_7153_rev__predicate1D,axiom,
! [A: $tType,P: A > $o,X: A,Q: A > $o] :
( ( P @ X )
=> ( ( ord_less_eq @ ( A > $o ) @ P @ Q )
=> ( Q @ X ) ) ) ).
% rev_predicate1D
thf(fact_7154_rev__predicate2D,axiom,
! [A: $tType,B: $tType,P: A > B > $o,X: A,Y3: B,Q: A > B > $o] :
( ( P @ X @ Y3 )
=> ( ( ord_less_eq @ ( A > B > $o ) @ P @ Q )
=> ( Q @ X @ Y3 ) ) ) ).
% rev_predicate2D
thf(fact_7155_subset_Osuc__Union__closed_Ocases,axiom,
! [A: $tType,A3: set @ ( set @ A ),A6: set @ ( set @ A )] :
( ( member @ ( set @ ( set @ A ) ) @ A3 @ ( pred_s596693808085603175closed @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) ) )
=> ( ! [X10: set @ ( set @ A )] :
( ( A3
= ( pred_suc @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ X10 ) )
=> ~ ( member @ ( set @ ( set @ A ) ) @ X10 @ ( pred_s596693808085603175closed @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) ) ) )
=> ~ ! [X10: set @ ( set @ ( set @ A ) )] :
( ( A3
= ( complete_Sup_Sup @ ( set @ ( set @ A ) ) @ X10 ) )
=> ~ ( member @ ( set @ ( set @ ( set @ A ) ) ) @ X10 @ ( pow2 @ ( set @ ( set @ A ) ) @ ( pred_s596693808085603175closed @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) ) ) ) ) ) ) ).
% subset.suc_Union_closed.cases
thf(fact_7156_subset_Osuc__Union__closed_Osimps,axiom,
! [A: $tType,A3: set @ ( set @ A ),A6: set @ ( set @ A )] :
( ( member @ ( set @ ( set @ A ) ) @ A3 @ ( pred_s596693808085603175closed @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) ) )
= ( ? [X7: set @ ( set @ A )] :
( ( A3
= ( pred_suc @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ X7 ) )
& ( member @ ( set @ ( set @ A ) ) @ X7 @ ( pred_s596693808085603175closed @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) ) ) )
| ? [X7: set @ ( set @ ( set @ A ) )] :
( ( A3
= ( complete_Sup_Sup @ ( set @ ( set @ A ) ) @ X7 ) )
& ( member @ ( set @ ( set @ ( set @ A ) ) ) @ X7 @ ( pow2 @ ( set @ ( set @ A ) ) @ ( pred_s596693808085603175closed @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) ) ) ) ) ) ) ).
% subset.suc_Union_closed.simps
thf(fact_7157_subset_Osuc__Union__closed_Osuc,axiom,
! [A: $tType,X8: set @ ( set @ A ),A6: set @ ( set @ A )] :
( ( member @ ( set @ ( set @ A ) ) @ X8 @ ( pred_s596693808085603175closed @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) ) )
=> ( member @ ( set @ ( set @ A ) ) @ ( pred_suc @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ X8 ) @ ( pred_s596693808085603175closed @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) ) ) ) ).
% subset.suc_Union_closed.suc
thf(fact_7158_subset_Oeq__suc__Union,axiom,
! [A: $tType,X8: set @ ( set @ A ),A6: set @ ( set @ A )] :
( ( member @ ( set @ ( set @ A ) ) @ X8 @ ( pred_s596693808085603175closed @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) ) )
=> ( ( ( pred_suc @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ X8 )
= X8 )
= ( X8
= ( complete_Sup_Sup @ ( set @ ( set @ A ) ) @ ( pred_s596693808085603175closed @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) ) ) ) ) ) ).
% subset.eq_suc_Union
thf(fact_7159_subset_Osuc__Union__closed__Union,axiom,
! [A: $tType,A6: set @ ( set @ A )] : ( member @ ( set @ ( set @ A ) ) @ ( complete_Sup_Sup @ ( set @ ( set @ A ) ) @ ( pred_s596693808085603175closed @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) ) ) @ ( pred_s596693808085603175closed @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) ) ) ).
% subset.suc_Union_closed_Union
thf(fact_7160_subset_Osuc__Union__closed__in__carrier,axiom,
! [A: $tType,X8: set @ ( set @ A ),A6: set @ ( set @ A )] :
( ( member @ ( set @ ( set @ A ) ) @ X8 @ ( pred_s596693808085603175closed @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) ) )
=> ( ord_less_eq @ ( set @ ( set @ A ) ) @ X8 @ A6 ) ) ).
% subset.suc_Union_closed_in_carrier
thf(fact_7161_subset_Osuc__Union__closed__total,axiom,
! [A: $tType,X8: set @ ( set @ A ),A6: set @ ( set @ A ),Y7: set @ ( set @ A )] :
( ( member @ ( set @ ( set @ A ) ) @ X8 @ ( pred_s596693808085603175closed @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) ) )
=> ( ( member @ ( set @ ( set @ A ) ) @ Y7 @ ( pred_s596693808085603175closed @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) ) )
=> ( ( ord_less_eq @ ( set @ ( set @ A ) ) @ X8 @ Y7 )
| ( ord_less_eq @ ( set @ ( set @ A ) ) @ Y7 @ X8 ) ) ) ) ).
% subset.suc_Union_closed_total
thf(fact_7162_subset_Osuc__Union__closed__induct,axiom,
! [A: $tType,X8: set @ ( set @ A ),A6: set @ ( set @ A ),Q: ( set @ ( set @ A ) ) > $o] :
( ( member @ ( set @ ( set @ A ) ) @ X8 @ ( pred_s596693808085603175closed @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) ) )
=> ( ! [X10: set @ ( set @ A )] :
( ( member @ ( set @ ( set @ A ) ) @ X10 @ ( pred_s596693808085603175closed @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) ) )
=> ( ( Q @ X10 )
=> ( Q @ ( pred_suc @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ X10 ) ) ) )
=> ( ! [X10: set @ ( set @ ( set @ A ) )] :
( ( ord_less_eq @ ( set @ ( set @ ( set @ A ) ) ) @ X10 @ ( pred_s596693808085603175closed @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) ) )
=> ( ! [X4: set @ ( set @ A )] :
( ( member @ ( set @ ( set @ A ) ) @ X4 @ X10 )
=> ( Q @ X4 ) )
=> ( Q @ ( complete_Sup_Sup @ ( set @ ( set @ A ) ) @ X10 ) ) ) )
=> ( Q @ X8 ) ) ) ) ).
% subset.suc_Union_closed_induct
thf(fact_7163_subset_Osuc__Union__closed__cases,axiom,
! [A: $tType,X8: set @ ( set @ A ),A6: set @ ( set @ A )] :
( ( member @ ( set @ ( set @ A ) ) @ X8 @ ( pred_s596693808085603175closed @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) ) )
=> ( ! [Y10: set @ ( set @ A )] :
( ( X8
= ( pred_suc @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ Y10 ) )
=> ~ ( member @ ( set @ ( set @ A ) ) @ Y10 @ ( pred_s596693808085603175closed @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) ) ) )
=> ~ ! [Y10: set @ ( set @ ( set @ A ) )] :
( ( X8
= ( complete_Sup_Sup @ ( set @ ( set @ A ) ) @ Y10 ) )
=> ~ ( ord_less_eq @ ( set @ ( set @ ( set @ A ) ) ) @ Y10 @ ( pred_s596693808085603175closed @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) ) ) ) ) ) ).
% subset.suc_Union_closed_cases
thf(fact_7164_subset_Osuc__Union__closed_OUnion,axiom,
! [A: $tType,X8: set @ ( set @ ( set @ A ) ),A6: set @ ( set @ A )] :
( ( ord_less_eq @ ( set @ ( set @ ( set @ A ) ) ) @ X8 @ ( pred_s596693808085603175closed @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) ) )
=> ( member @ ( set @ ( set @ A ) ) @ ( complete_Sup_Sup @ ( set @ ( set @ A ) ) @ X8 ) @ ( pred_s596693808085603175closed @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) ) ) ) ).
% subset.suc_Union_closed.Union
thf(fact_7165_subset_Osuc__Union__closed_Ointros_I2_J,axiom,
! [A: $tType,Xa5: set @ ( set @ ( set @ A ) ),A6: set @ ( set @ A )] :
( ( ord_less_eq @ ( set @ ( set @ ( set @ A ) ) ) @ Xa5 @ ( pred_s596693808085603175closed @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) ) )
=> ( member @ ( set @ ( set @ A ) ) @ ( complete_Sup_Sup @ ( set @ ( set @ A ) ) @ Xa5 ) @ ( pred_s596693808085603175closed @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) ) ) ) ).
% subset.suc_Union_closed.intros(2)
thf(fact_7166_subset_Osuc__Union__closed__chain,axiom,
! [A: $tType,X8: set @ ( set @ A ),A6: set @ ( set @ A )] :
( ( member @ ( set @ ( set @ A ) ) @ X8 @ ( pred_s596693808085603175closed @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) ) )
=> ( pred_chain @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ X8 ) ) ).
% subset.suc_Union_closed_chain
thf(fact_7167_subset_Osuc__Union__closed__empty,axiom,
! [A: $tType,A6: set @ ( set @ A )] : ( member @ ( set @ ( set @ A ) ) @ ( bot_bot @ ( set @ ( set @ A ) ) ) @ ( pred_s596693808085603175closed @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) ) ) ).
% subset.suc_Union_closed_empty
thf(fact_7168_natural_Odomain,axiom,
! [P: nat > $o] :
( ( ( domainp @ nat @ nat
@ ^ [Y: nat,Z2: nat] : Y = Z2 )
= P )
=> ( ( domainp @ nat @ code_natural @ code_pcr_natural )
= P ) ) ).
% natural.domain
thf(fact_7169_integer_Odomain,axiom,
! [P: int > $o] :
( ( ( domainp @ int @ int
@ ^ [Y: int,Z2: int] : Y = Z2 )
= P )
=> ( ( domainp @ int @ code_integer @ code_pcr_integer )
= P ) ) ).
% integer.domain
thf(fact_7170_subset_Osuc__Union__closed__subsetD,axiom,
! [A: $tType,Y7: set @ ( set @ A ),X8: set @ ( set @ A ),A6: set @ ( set @ A )] :
( ( ord_less_eq @ ( set @ ( set @ A ) ) @ Y7 @ X8 )
=> ( ( member @ ( set @ ( set @ A ) ) @ X8 @ ( pred_s596693808085603175closed @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) ) )
=> ( ( member @ ( set @ ( set @ A ) ) @ Y7 @ ( pred_s596693808085603175closed @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) ) )
=> ( ( X8 = Y7 )
| ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( pred_suc @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ Y7 ) @ X8 ) ) ) ) ) ).
% subset.suc_Union_closed_subsetD
thf(fact_7171_subset_Osuc__Union__closed__total_H,axiom,
! [A: $tType,X8: set @ ( set @ A ),A6: set @ ( set @ A ),Y7: set @ ( set @ A )] :
( ( member @ ( set @ ( set @ A ) ) @ X8 @ ( pred_s596693808085603175closed @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) ) )
=> ( ( member @ ( set @ ( set @ A ) ) @ Y7 @ ( pred_s596693808085603175closed @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) ) )
=> ( ! [Z9: set @ ( set @ A )] :
( ( member @ ( set @ ( set @ A ) ) @ Z9 @ ( pred_s596693808085603175closed @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) ) )
=> ( ( ord_less_eq @ ( set @ ( set @ A ) ) @ Z9 @ Y7 )
=> ( ( Z9 = Y7 )
| ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( pred_suc @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ Z9 ) @ Y7 ) ) ) )
=> ( ( ord_less_eq @ ( set @ ( set @ A ) ) @ X8 @ Y7 )
| ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( pred_suc @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ Y7 ) @ X8 ) ) ) ) ) ).
% subset.suc_Union_closed_total'
thf(fact_7172_length__remdups__leq,axiom,
! [A: $tType,Xs: list @ A] : ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ ( remdups @ A @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ).
% length_remdups_leq
thf(fact_7173_subset_Osuc__Union__closed__def,axiom,
! [A: $tType,A6: set @ ( set @ A )] :
( ( pred_s596693808085603175closed @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) )
= ( collect @ ( set @ ( set @ A ) ) @ ( pred_s7749564232668923593losedp @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) ) ) ) ).
% subset.suc_Union_closed_def
thf(fact_7174_sorted__remdups,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( remdups @ A @ Xs ) ) ) ) ).
% sorted_remdups
thf(fact_7175_subset_Osuc__Union__closedp__suc__Union__closed__eq,axiom,
! [A: $tType,A6: set @ ( set @ A )] :
( ( pred_s7749564232668923593losedp @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) )
= ( ^ [X2: set @ ( set @ A )] : ( member @ ( set @ ( set @ A ) ) @ X2 @ ( pred_s596693808085603175closed @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) ) ) ) ) ).
% subset.suc_Union_closedp_suc_Union_closed_eq
thf(fact_7176_subset_Osuc__Union__closedp_Osuc,axiom,
! [A: $tType,A6: set @ ( set @ A ),X8: set @ ( set @ A )] :
( ( pred_s7749564232668923593losedp @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ X8 )
=> ( pred_s7749564232668923593losedp @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ ( pred_suc @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ X8 ) ) ) ).
% subset.suc_Union_closedp.suc
thf(fact_7177_subset_Osuc__Union__closedp_Osimps,axiom,
! [A: $tType,A6: set @ ( set @ A ),A3: set @ ( set @ A )] :
( ( pred_s7749564232668923593losedp @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ A3 )
= ( ? [X7: set @ ( set @ A )] :
( ( A3
= ( pred_suc @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ X7 ) )
& ( pred_s7749564232668923593losedp @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ X7 ) )
| ? [X7: set @ ( set @ ( set @ A ) )] :
( ( A3
= ( complete_Sup_Sup @ ( set @ ( set @ A ) ) @ X7 ) )
& ( powp @ ( set @ ( set @ A ) ) @ ( pred_s7749564232668923593losedp @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) ) @ X7 ) ) ) ) ).
% subset.suc_Union_closedp.simps
thf(fact_7178_subset_Osuc__Union__closedp_Ointros_I2_J,axiom,
! [A: $tType,A6: set @ ( set @ A ),Xa5: set @ ( set @ ( set @ A ) )] :
( ( powp @ ( set @ ( set @ A ) ) @ ( pred_s7749564232668923593losedp @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) ) @ Xa5 )
=> ( pred_s7749564232668923593losedp @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ ( complete_Sup_Sup @ ( set @ ( set @ A ) ) @ Xa5 ) ) ) ).
% subset.suc_Union_closedp.intros(2)
thf(fact_7179_subset_Osuc__Union__closedp_OUnion,axiom,
! [A: $tType,A6: set @ ( set @ A ),X8: set @ ( set @ ( set @ A ) )] :
( ( powp @ ( set @ ( set @ A ) ) @ ( pred_s7749564232668923593losedp @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) ) @ X8 )
=> ( pred_s7749564232668923593losedp @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ ( complete_Sup_Sup @ ( set @ ( set @ A ) ) @ X8 ) ) ) ).
% subset.suc_Union_closedp.Union
thf(fact_7180_subset_Osuc__Union__closedp_Ocases,axiom,
! [A: $tType,A6: set @ ( set @ A ),A3: set @ ( set @ A )] :
( ( pred_s7749564232668923593losedp @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ A3 )
=> ( ! [X10: set @ ( set @ A )] :
( ( A3
= ( pred_suc @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ X10 ) )
=> ~ ( pred_s7749564232668923593losedp @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) @ X10 ) )
=> ~ ! [X10: set @ ( set @ ( set @ A ) )] :
( ( A3
= ( complete_Sup_Sup @ ( set @ ( set @ A ) ) @ X10 ) )
=> ~ ( powp @ ( set @ ( set @ A ) ) @ ( pred_s7749564232668923593losedp @ ( set @ A ) @ A6 @ ( ord_less @ ( set @ A ) ) ) @ X10 ) ) ) ) ).
% subset.suc_Union_closedp.cases
thf(fact_7181_int_Odomain,axiom,
! [DR1: nat > $o,DR2: nat > $o] :
( ( ( domainp @ nat @ nat
@ ^ [Y: nat,Z2: nat] : Y = Z2 )
= DR1 )
=> ( ( ( domainp @ nat @ nat
@ ^ [Y: nat,Z2: nat] : Y = Z2 )
= DR2 )
=> ( ( domainp @ ( product_prod @ nat @ nat ) @ int @ pcr_int )
= ( basic_pred_prod @ nat @ nat @ DR1 @ DR2 ) ) ) ) ).
% int.domain
thf(fact_7182_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_7183_Heap_Ocase__distrib,axiom,
! [C: $tType,B: $tType,A: $tType,H2: B > C,F: ( ( heap_ext @ product_unit ) > ( option @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) > B,Heap: heap_Time_Heap @ A] :
( ( H2 @ ( 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 ) ) )] : ( H2 @ ( F @ X2 ) )
@ Heap ) ) ).
% Heap.case_distrib
thf(fact_7184_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_7185_enumerate__Suc__eq,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( enumerate @ A @ ( suc @ N ) @ Xs )
= ( map @ ( product_prod @ nat @ A ) @ ( product_prod @ nat @ A ) @ ( product_apfst @ nat @ nat @ A @ suc ) @ ( enumerate @ A @ N @ Xs ) ) ) ).
% enumerate_Suc_eq
thf(fact_7186_Heap__ordE,axiom,
! [A: $tType,X: heap_Time_Heap @ A,Y3: heap_Time_Heap @ A,H2: heap_ext @ product_unit] :
( ( heap_Time_Heap_ord @ A @ X @ Y3 )
=> ( ( ( heap_Time_execute @ A @ X @ H2 )
!= ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) )
=> ( ( heap_Time_execute @ A @ X @ H2 )
= ( heap_Time_execute @ A @ Y3 @ H2 ) ) ) ) ).
% Heap_ordE
thf(fact_7187_Heap__ordI,axiom,
! [A: $tType,X: heap_Time_Heap @ A,Y3: heap_Time_Heap @ A] :
( ! [H6: heap_ext @ product_unit] :
( ( ( heap_Time_execute @ A @ X @ H6 )
= ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) )
| ( ( heap_Time_execute @ A @ X @ H6 )
= ( heap_Time_execute @ A @ Y3 @ H6 ) ) )
=> ( heap_Time_Heap_ord @ A @ X @ Y3 ) ) ).
% Heap_ordI
thf(fact_7188_heap_Oleq__refl,axiom,
! [A: $tType,X: heap_Time_Heap @ A] : ( heap_Time_Heap_ord @ A @ X @ X ) ).
% heap.leq_refl
thf(fact_7189_heap_Oleq__trans,axiom,
! [A: $tType,X: heap_Time_Heap @ A,Y3: heap_Time_Heap @ A,Z3: heap_Time_Heap @ A] :
( ( heap_Time_Heap_ord @ A @ X @ Y3 )
=> ( ( heap_Time_Heap_ord @ A @ Y3 @ Z3 )
=> ( heap_Time_Heap_ord @ A @ X @ Z3 ) ) ) ).
% heap.leq_trans
thf(fact_7190_heap_Oleq__antisym,axiom,
! [A: $tType,X: heap_Time_Heap @ A,Y3: heap_Time_Heap @ A] :
( ( heap_Time_Heap_ord @ A @ X @ Y3 )
=> ( ( heap_Time_Heap_ord @ A @ Y3 @ X )
=> ( X = Y3 ) ) ) ).
% heap.leq_antisym
thf(fact_7191_heap_Olub__least,axiom,
! [A: $tType,A6: set @ ( heap_Time_Heap @ A ),Z3: heap_Time_Heap @ A] :
( ( comple1602240252501008431_chain @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap_ord @ A ) @ A6 )
=> ( ! [X3: heap_Time_Heap @ A] :
( ( member @ ( heap_Time_Heap @ A ) @ X3 @ A6 )
=> ( heap_Time_Heap_ord @ A @ X3 @ Z3 ) )
=> ( heap_Time_Heap_ord @ A @ ( heap_Time_Heap_lub @ A @ A6 ) @ Z3 ) ) ) ).
% heap.lub_least
thf(fact_7192_heap_Olub__upper,axiom,
! [A: $tType,A6: set @ ( heap_Time_Heap @ A ),X: heap_Time_Heap @ A] :
( ( comple1602240252501008431_chain @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap_ord @ A ) @ A6 )
=> ( ( member @ ( heap_Time_Heap @ A ) @ X @ A6 )
=> ( heap_Time_Heap_ord @ A @ X @ ( heap_Time_Heap_lub @ A @ A6 ) ) ) ) ).
% heap.lub_upper
thf(fact_7193_heap_Oconst__mono,axiom,
! [A: $tType,B: $tType,Ord: B > B > $o,C2: heap_Time_Heap @ A] :
( comple7038119648293358887notone @ B @ ( heap_Time_Heap @ A ) @ Ord @ ( heap_Time_Heap_ord @ A )
@ ^ [F2: B] : C2 ) ).
% heap.const_mono
thf(fact_7194_heap__interpretation,axiom,
! [A: $tType] : ( partia7178651479351089652itions @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap_ord @ A ) @ ( heap_Time_Heap_lub @ A ) ) ).
% heap_interpretation
thf(fact_7195_Heap__Time__Monad_Obind__mono,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,B7: ( A > ( heap_Time_Heap @ B ) ) > ( heap_Time_Heap @ C ),C6: C > ( A > ( heap_Time_Heap @ B ) ) > ( heap_Time_Heap @ D )] :
( ( comple7038119648293358887notone @ ( A > ( heap_Time_Heap @ B ) ) @ ( heap_Time_Heap @ C ) @ ( partial_fun_ord @ ( heap_Time_Heap @ B ) @ ( heap_Time_Heap @ B ) @ A @ ( heap_Time_Heap_ord @ B ) ) @ ( heap_Time_Heap_ord @ C ) @ B7 )
=> ( ! [Y5: C] : ( comple7038119648293358887notone @ ( A > ( heap_Time_Heap @ B ) ) @ ( heap_Time_Heap @ D ) @ ( partial_fun_ord @ ( heap_Time_Heap @ B ) @ ( heap_Time_Heap @ B ) @ A @ ( heap_Time_Heap_ord @ B ) ) @ ( heap_Time_Heap_ord @ D ) @ ( C6 @ Y5 ) )
=> ( comple7038119648293358887notone @ ( A > ( heap_Time_Heap @ B ) ) @ ( heap_Time_Heap @ D ) @ ( partial_fun_ord @ ( heap_Time_Heap @ B ) @ ( heap_Time_Heap @ B ) @ A @ ( heap_Time_Heap_ord @ B ) ) @ ( heap_Time_Heap_ord @ D )
@ ^ [F2: A > ( heap_Time_Heap @ B )] :
( heap_Time_bind @ C @ D @ ( B7 @ F2 )
@ ^ [Y6: C] : ( C6 @ Y6 @ F2 ) ) ) ) ) ).
% Heap_Time_Monad.bind_mono
thf(fact_7196_admissible__heap,axiom,
! [B: $tType,A: $tType,P: A > ( heap_ext @ product_unit ) > ( heap_ext @ product_unit ) > B > nat > $o] :
( comple1908693960933563346ssible @ ( A > ( heap_Time_Heap @ B ) ) @ ( partial_fun_lub @ ( heap_Time_Heap @ B ) @ ( heap_Time_Heap @ B ) @ A @ ( heap_Time_Heap_lub @ B ) ) @ ( partial_fun_ord @ ( heap_Time_Heap @ B ) @ ( heap_Time_Heap @ B ) @ A @ ( heap_Time_Heap_ord @ B ) )
@ ^ [F2: A > ( heap_Time_Heap @ B )] :
! [X2: A,H4: heap_ext @ product_unit,H5: heap_ext @ product_unit,R5: B,N3: nat] :
( ( heap_Time_effect @ B @ ( F2 @ X2 ) @ H4 @ H5 @ R5 @ N3 )
=> ( P @ X2 @ H4 @ H5 @ R5 @ N3 ) ) ) ).
% admissible_heap
thf(fact_7197_heap_Ofixp__induct__uc,axiom,
! [B: $tType,A: $tType,C: $tType,U4: C > B > ( heap_Time_Heap @ A ),F4: C > C,C6: ( B > ( heap_Time_Heap @ A ) ) > C,F: C,P: ( B > ( heap_Time_Heap @ A ) ) > $o] :
( ! [X3: B] :
( comple7038119648293358887notone @ ( B > ( heap_Time_Heap @ A ) ) @ ( heap_Time_Heap @ A ) @ ( partial_fun_ord @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ A ) @ B @ ( heap_Time_Heap_ord @ A ) ) @ ( heap_Time_Heap_ord @ A )
@ ^ [F2: B > ( heap_Time_Heap @ A )] : ( U4 @ ( F4 @ ( C6 @ F2 ) ) @ X3 ) )
=> ( ( F
= ( C6
@ ( comple187402453842119260l_fixp @ ( B > ( heap_Time_Heap @ A ) ) @ ( partial_fun_lub @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ A ) @ B @ ( heap_Time_Heap_lub @ A ) ) @ ( partial_fun_ord @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ A ) @ B @ ( heap_Time_Heap_ord @ A ) )
@ ^ [F2: B > ( heap_Time_Heap @ A )] : ( U4 @ ( F4 @ ( C6 @ F2 ) ) ) ) ) )
=> ( ! [F3: B > ( heap_Time_Heap @ A )] :
( ( U4 @ ( C6 @ F3 ) )
= F3 )
=> ( ( comple1908693960933563346ssible @ ( B > ( heap_Time_Heap @ A ) ) @ ( partial_fun_lub @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ A ) @ B @ ( heap_Time_Heap_lub @ A ) ) @ ( partial_fun_ord @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ A ) @ B @ ( heap_Time_Heap_ord @ A ) ) @ P )
=> ( ( P
@ ^ [Uu: B] :
( heap_Time_Heap2 @ A
@ ^ [X2: heap_ext @ product_unit] : ( none @ ( product_prod @ A @ ( product_prod @ ( heap_ext @ product_unit ) @ nat ) ) ) ) )
=> ( ! [F3: C] :
( ( P @ ( U4 @ F3 ) )
=> ( P @ ( U4 @ ( F4 @ F3 ) ) ) )
=> ( P @ ( U4 @ F ) ) ) ) ) ) ) ) ).
% heap.fixp_induct_uc
thf(fact_7198_heap_Omono__body__fixp,axiom,
! [A: $tType,B: $tType,F4: ( B > ( heap_Time_Heap @ A ) ) > B > ( heap_Time_Heap @ A )] :
( ! [X3: B] :
( comple7038119648293358887notone @ ( B > ( heap_Time_Heap @ A ) ) @ ( heap_Time_Heap @ A ) @ ( partial_fun_ord @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ A ) @ B @ ( heap_Time_Heap_ord @ A ) ) @ ( heap_Time_Heap_ord @ A )
@ ^ [F2: B > ( heap_Time_Heap @ A )] : ( F4 @ F2 @ X3 ) )
=> ( ( comple187402453842119260l_fixp @ ( B > ( heap_Time_Heap @ A ) ) @ ( partial_fun_lub @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ A ) @ B @ ( heap_Time_Heap_lub @ A ) ) @ ( partial_fun_ord @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ A ) @ B @ ( heap_Time_Heap_ord @ A ) ) @ F4 )
= ( F4 @ ( comple187402453842119260l_fixp @ ( B > ( heap_Time_Heap @ A ) ) @ ( partial_fun_lub @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ A ) @ B @ ( heap_Time_Heap_lub @ A ) ) @ ( partial_fun_ord @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ A ) @ B @ ( heap_Time_Heap_ord @ A ) ) @ F4 ) ) ) ) ).
% heap.mono_body_fixp
thf(fact_7199_heap_Ofixp__rule__uc,axiom,
! [B: $tType,A: $tType,C: $tType,U4: C > B > ( heap_Time_Heap @ A ),F4: C > C,C6: ( B > ( heap_Time_Heap @ A ) ) > C,F: C] :
( ! [X3: B] :
( comple7038119648293358887notone @ ( B > ( heap_Time_Heap @ A ) ) @ ( heap_Time_Heap @ A ) @ ( partial_fun_ord @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ A ) @ B @ ( heap_Time_Heap_ord @ A ) ) @ ( heap_Time_Heap_ord @ A )
@ ^ [F2: B > ( heap_Time_Heap @ A )] : ( U4 @ ( F4 @ ( C6 @ F2 ) ) @ X3 ) )
=> ( ( F
= ( C6
@ ( comple187402453842119260l_fixp @ ( B > ( heap_Time_Heap @ A ) ) @ ( partial_fun_lub @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ A ) @ B @ ( heap_Time_Heap_lub @ A ) ) @ ( partial_fun_ord @ ( heap_Time_Heap @ A ) @ ( heap_Time_Heap @ A ) @ B @ ( heap_Time_Heap_ord @ A ) )
@ ^ [F2: B > ( heap_Time_Heap @ A )] : ( U4 @ ( F4 @ ( C6 @ F2 ) ) ) ) ) )
=> ( ! [F3: C] :
( ( C6 @ ( U4 @ F3 ) )
= F3 )
=> ( F
= ( F4 @ F ) ) ) ) ) ).
% heap.fixp_rule_uc
% Subclasses (3)
thf(subcl_Heap_Oheap___HOL_Otype,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( type @ A ) ) ).
thf(subcl_Heap_Oheap___Typerep_Otyperep,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( typerep2 @ A ) ) ).
thf(subcl_Heap_Oheap___Countable_Ocountable,axiom,
! [A: $tType] :
( ( heap @ A )
=> ( countable @ A ) ) ).
% Type constructors (657)
thf(tcon_Product__Type_Ounit___Lattices_Obounded__lattice,axiom,
bounded_lattice @ product_unit ).
thf(tcon_Product__Type_Ounit___Finite__Set_Ofinite,axiom,
finite_finite @ product_unit ).
thf(tcon_Product__Type_Oprod___Finite__Set_Ofinite_1,axiom,
! [A17: $tType,A18: $tType] :
( ( ( finite_finite @ A17 )
& ( finite_finite @ A18 ) )
=> ( finite_finite @ ( product_prod @ A17 @ A18 ) ) ) ).
thf(tcon_Option_Ooption___Finite__Set_Ofinite_2,axiom,
! [A17: $tType] :
( ( finite_finite @ A17 )
=> ( finite_finite @ ( option @ A17 ) ) ) ).
thf(tcon_Sum__Type_Osum___Finite__Set_Ofinite_3,axiom,
! [A17: $tType,A18: $tType] :
( ( ( finite_finite @ A17 )
& ( finite_finite @ A18 ) )
=> ( finite_finite @ ( sum_sum @ A17 @ A18 ) ) ) ).
thf(tcon_String_Ochar___Finite__Set_Ofinite_4,axiom,
finite_finite @ char ).
thf(tcon_HOL_Obool___Lattices_Obounded__lattice_5,axiom,
bounded_lattice @ $o ).
thf(tcon_HOL_Obool___Finite__Set_Ofinite_6,axiom,
finite_finite @ $o ).
thf(tcon_Set_Oset___Lattices_Obounded__lattice_7,axiom,
! [A17: $tType] : ( bounded_lattice @ ( set @ A17 ) ) ).
thf(tcon_Set_Oset___Finite__Set_Ofinite_8,axiom,
! [A17: $tType] :
( ( finite_finite @ A17 )
=> ( finite_finite @ ( set @ A17 ) ) ) ).
thf(tcon_fun___Lattices_Obounded__lattice_9,axiom,
! [A17: $tType,A18: $tType] :
( ( bounded_lattice @ A18 )
=> ( bounded_lattice @ ( A17 > A18 ) ) ) ).
thf(tcon_fun___Finite__Set_Ofinite_10,axiom,
! [A17: $tType,A18: $tType] :
( ( ( finite_finite @ A17 )
& ( finite_finite @ A18 ) )
=> ( finite_finite @ ( A17 > A18 ) ) ) ).
thf(tcon_fun___Conditionally__Complete__Lattices_Oconditionally__complete__lattice,axiom,
! [A17: $tType,A18: $tType] :
( ( comple6319245703460814977attice @ A18 )
=> ( condit1219197933456340205attice @ ( A17 > A18 ) ) ) ).
thf(tcon_fun___Complete__Lattices_Ocomplete__distrib__lattice,axiom,
! [A17: $tType,A18: $tType] :
( ( comple592849572758109894attice @ A18 )
=> ( comple592849572758109894attice @ ( A17 > A18 ) ) ) ).
thf(tcon_fun___Complete__Lattices_Ocomplete__boolean__algebra,axiom,
! [A17: $tType,A18: $tType] :
( ( comple489889107523837845lgebra @ A18 )
=> ( comple489889107523837845lgebra @ ( A17 > A18 ) ) ) ).
thf(tcon_fun___Lattices_Obounded__semilattice__sup__bot,axiom,
! [A17: $tType,A18: $tType] :
( ( bounded_lattice @ A18 )
=> ( bounde4967611905675639751up_bot @ ( A17 > A18 ) ) ) ).
thf(tcon_fun___Lattices_Obounded__semilattice__inf__top,axiom,
! [A17: $tType,A18: $tType] :
( ( bounded_lattice @ A18 )
=> ( bounde4346867609351753570nf_top @ ( A17 > A18 ) ) ) ).
thf(tcon_fun___Complete__Lattices_Ocomplete__lattice,axiom,
! [A17: $tType,A18: $tType] :
( ( comple6319245703460814977attice @ A18 )
=> ( comple6319245703460814977attice @ ( A17 > A18 ) ) ) ).
thf(tcon_fun___Boolean__Algebras_Oboolean__algebra,axiom,
! [A17: $tType,A18: $tType] :
( ( boolea8198339166811842893lgebra @ A18 )
=> ( boolea8198339166811842893lgebra @ ( A17 > A18 ) ) ) ).
thf(tcon_fun___Lattices_Obounded__lattice__top,axiom,
! [A17: $tType,A18: $tType] :
( ( bounded_lattice @ A18 )
=> ( bounded_lattice_top @ ( A17 > A18 ) ) ) ).
thf(tcon_fun___Lattices_Obounded__lattice__bot,axiom,
! [A17: $tType,A18: $tType] :
( ( bounded_lattice @ A18 )
=> ( bounded_lattice_bot @ ( A17 > A18 ) ) ) ).
thf(tcon_fun___Complete__Partial__Order_Occpo,axiom,
! [A17: $tType,A18: $tType] :
( ( comple6319245703460814977attice @ A18 )
=> ( comple9053668089753744459l_ccpo @ ( A17 > A18 ) ) ) ).
thf(tcon_fun___Lattices_Osemilattice__sup,axiom,
! [A17: $tType,A18: $tType] :
( ( semilattice_sup @ A18 )
=> ( semilattice_sup @ ( A17 > A18 ) ) ) ).
thf(tcon_fun___Lattices_Osemilattice__inf,axiom,
! [A17: $tType,A18: $tType] :
( ( semilattice_inf @ A18 )
=> ( semilattice_inf @ ( A17 > A18 ) ) ) ).
thf(tcon_fun___Lattices_Odistrib__lattice,axiom,
! [A17: $tType,A18: $tType] :
( ( distrib_lattice @ A18 )
=> ( distrib_lattice @ ( A17 > A18 ) ) ) ).
thf(tcon_fun___Orderings_Oorder__top,axiom,
! [A17: $tType,A18: $tType] :
( ( order_top @ A18 )
=> ( order_top @ ( A17 > A18 ) ) ) ).
thf(tcon_fun___Orderings_Oorder__bot,axiom,
! [A17: $tType,A18: $tType] :
( ( order_bot @ A18 )
=> ( order_bot @ ( A17 > A18 ) ) ) ).
thf(tcon_fun___Countable_Ocountable,axiom,
! [A17: $tType,A18: $tType] :
( ( ( finite_finite @ A17 )
& ( countable @ A18 ) )
=> ( countable @ ( A17 > A18 ) ) ) ).
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A17: $tType,A18: $tType] :
( ( preorder @ A18 )
=> ( preorder @ ( A17 > A18 ) ) ) ).
thf(tcon_fun___Lattices_Olattice,axiom,
! [A17: $tType,A18: $tType] :
( ( lattice @ A18 )
=> ( lattice @ ( A17 > A18 ) ) ) ).
thf(tcon_fun___Typerep_Otyperep,axiom,
! [A17: $tType,A18: $tType] :
( ( ( typerep2 @ A17 )
& ( typerep2 @ A18 ) )
=> ( typerep2 @ ( A17 > A18 ) ) ) ).
thf(tcon_fun___Orderings_Oorder,axiom,
! [A17: $tType,A18: $tType] :
( ( order @ A18 )
=> ( order @ ( A17 > A18 ) ) ) ).
thf(tcon_fun___Orderings_Otop,axiom,
! [A17: $tType,A18: $tType] :
( ( top @ A18 )
=> ( top @ ( A17 > A18 ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A17: $tType,A18: $tType] :
( ( ord @ A18 )
=> ( ord @ ( A17 > A18 ) ) ) ).
thf(tcon_fun___Orderings_Obot,axiom,
! [A17: $tType,A18: $tType] :
( ( bot @ A18 )
=> ( bot @ ( A17 > A18 ) ) ) ).
thf(tcon_fun___Groups_Ouminus,axiom,
! [A17: $tType,A18: $tType] :
( ( uminus @ A18 )
=> ( uminus @ ( A17 > A18 ) ) ) ).
thf(tcon_fun___Groups_Ominus,axiom,
! [A17: $tType,A18: $tType] :
( ( minus @ A18 )
=> ( minus @ ( A17 > A18 ) ) ) ).
thf(tcon_Int_Oint___Conditionally__Complete__Lattices_Oconditionally__complete__linorder,axiom,
condit6923001295902523014norder @ int ).
thf(tcon_Int_Oint___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_11,axiom,
condit1219197933456340205attice @ int ).
thf(tcon_Int_Oint___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations,axiom,
bit_un5681908812861735899ations @ int ).
thf(tcon_Int_Oint___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,axiom,
semiri1453513574482234551roduct @ int ).
thf(tcon_Int_Oint___Euclidean__Division_Ounique__euclidean__semiring__with__nat,axiom,
euclid5411537665997757685th_nat @ int ).
thf(tcon_Int_Oint___Euclidean__Division_Ounique__euclidean__ring__with__nat,axiom,
euclid8789492081693882211th_nat @ int ).
thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__monoid__add__imp__le,axiom,
ordere1937475149494474687imp_le @ int ).
thf(tcon_Int_Oint___Euclidean__Division_Ounique__euclidean__semiring,axiom,
euclid3128863361964157862miring @ int ).
thf(tcon_Int_Oint___Euclidean__Division_Oeuclidean__semiring__cancel,axiom,
euclid4440199948858584721cancel @ int ).
thf(tcon_Int_Oint___Rings_Onormalization__semidom__multiplicative,axiom,
normal6328177297339901930cative @ int ).
thf(tcon_Int_Oint___Divides_Ounique__euclidean__semiring__numeral,axiom,
unique1627219031080169319umeral @ int ).
thf(tcon_Int_Oint___Euclidean__Division_Oeuclidean__ring__cancel,axiom,
euclid8851590272496341667cancel @ int ).
thf(tcon_Int_Oint___Rings_Osemiring__no__zero__divisors__cancel,axiom,
semiri6575147826004484403cancel @ int ).
thf(tcon_Int_Oint___Groups_Ostrict__ordered__ab__semigroup__add,axiom,
strict9044650504122735259up_add @ int ).
thf(tcon_Int_Oint___Groups_Oordered__cancel__ab__semigroup__add,axiom,
ordere580206878836729694up_add @ int ).
thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
ordere2412721322843649153imp_le @ int ).
thf(tcon_Int_Oint___Bit__Operations_Osemiring__bit__operations,axiom,
bit_se359711467146920520ations @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__comm__semiring__strict,axiom,
linord2810124833399127020strict @ int ).
thf(tcon_Int_Oint___Groups_Ostrict__ordered__comm__monoid__add,axiom,
strict7427464778891057005id_add @ int ).
thf(tcon_Int_Oint___Groups_Oordered__cancel__comm__monoid__add,axiom,
ordere8940638589300402666id_add @ int ).
thf(tcon_Int_Oint___Euclidean__Division_Oeuclidean__semiring,axiom,
euclid3725896446679973847miring @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__semiring__1__strict,axiom,
linord715952674999750819strict @ int ).
thf(tcon_Int_Oint___Groups_Olinordered__ab__semigroup__add,axiom,
linord4140545234300271783up_add @ int ).
thf(tcon_Int_Oint___Bit__Operations_Oring__bit__operations,axiom,
bit_ri3973907225187159222ations @ int ).
thf(tcon_Int_Oint___Rings_Osemiring__1__no__zero__divisors,axiom,
semiri2026040879449505780visors @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__nonzero__semiring,axiom,
linord181362715937106298miring @ int ).
thf(tcon_Int_Oint___Euclidean__Division_Oeuclidean__ring,axiom,
euclid5891614535332579305n_ring @ int ).
thf(tcon_Int_Oint___Rings_Osemidom__divide__unit__factor,axiom,
semido2269285787275462019factor @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__semiring__strict,axiom,
linord8928482502909563296strict @ int ).
thf(tcon_Int_Oint___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_12,axiom,
semilattice_sup @ int ).
thf(tcon_Int_Oint___Lattices_Osemilattice__inf_13,axiom,
semilattice_inf @ int ).
thf(tcon_Int_Oint___Lattices_Odistrib__lattice_14,axiom,
distrib_lattice @ int ).
thf(tcon_Int_Oint___Groups_Oab__semigroup__mult,axiom,
ab_semigroup_mult @ int ).
thf(tcon_Int_Oint___Rings_Osemiring__1__cancel,axiom,
semiring_1_cancel @ int ).
thf(tcon_Int_Oint___Rings_Oalgebraic__semidom,axiom,
algebraic_semidom @ int ).
thf(tcon_Int_Oint___Groups_Ocomm__monoid__mult,axiom,
comm_monoid_mult @ int ).
thf(tcon_Int_Oint___Groups_Oab__semigroup__add,axiom,
ab_semigroup_add @ int ).
thf(tcon_Int_Oint___Rings_Oordered__semiring,axiom,
ordered_semiring @ int ).
thf(tcon_Int_Oint___Rings_Oordered__ring__abs,axiom,
ordered_ring_abs @ int ).
thf(tcon_Int_Oint___Parity_Osemiring__parity,axiom,
semiring_parity @ int ).
thf(tcon_Int_Oint___Groups_Ocomm__monoid__add,axiom,
comm_monoid_add @ int ).
thf(tcon_Int_Oint___Rings_Osemiring__modulo,axiom,
semiring_modulo @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__ring,axiom,
linordered_ring @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__idom,axiom,
linordered_idom @ int ).
thf(tcon_Int_Oint___Rings_Ocomm__semiring__1,axiom,
comm_semiring_1 @ int ).
thf(tcon_Int_Oint___Rings_Ocomm__semiring__0,axiom,
comm_semiring_0 @ int ).
thf(tcon_Int_Oint___Groups_Osemigroup__mult,axiom,
semigroup_mult @ int ).
thf(tcon_Int_Oint___Rings_Osemidom__modulo,axiom,
semidom_modulo @ int ).
thf(tcon_Int_Oint___Rings_Osemidom__divide,axiom,
semidom_divide @ int ).
thf(tcon_Int_Oint___Num_Osemiring__numeral,axiom,
semiring_numeral @ int ).
thf(tcon_Int_Oint___Groups_Osemigroup__add,axiom,
semigroup_add @ int ).
thf(tcon_Int_Oint___Rings_Ozero__less__one,axiom,
zero_less_one @ int ).
thf(tcon_Int_Oint___Rings_Ocomm__semiring,axiom,
comm_semiring @ int ).
thf(tcon_Int_Oint___Nat_Osemiring__char__0,axiom,
semiring_char_0 @ int ).
thf(tcon_Int_Oint___Groups_Oab__group__add,axiom,
ab_group_add @ int ).
thf(tcon_Int_Oint___Countable_Ocountable_15,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_16,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_17,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___Typerep_Otyperep_18,axiom,
typerep2 @ 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_19,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_20,axiom,
ord @ int ).
thf(tcon_Int_Oint___Groups_Ouminus_21,axiom,
uminus @ int ).
thf(tcon_Int_Oint___Rings_Oring__1,axiom,
ring_1 @ int ).
thf(tcon_Int_Oint___Rings_Oabs__if,axiom,
abs_if @ int ).
thf(tcon_Int_Oint___Groups_Ominus_22,axiom,
minus @ int ).
thf(tcon_Int_Oint___GCD_Oring__gcd,axiom,
ring_gcd @ int ).
thf(tcon_Int_Oint___Power_Opower,axiom,
power @ int ).
thf(tcon_Int_Oint___Num_Onumeral,axiom,
numeral @ int ).
thf(tcon_Int_Oint___Groups_Ozero,axiom,
zero @ int ).
thf(tcon_Int_Oint___Groups_Oplus,axiom,
plus @ int ).
thf(tcon_Int_Oint___Rings_Oring,axiom,
ring @ int ).
thf(tcon_Int_Oint___Rings_Oidom,axiom,
idom @ int ).
thf(tcon_Int_Oint___Groups_Oone,axiom,
one @ int ).
thf(tcon_Int_Oint___Rings_Odvd,axiom,
dvd @ int ).
thf(tcon_Int_Oint___Heap_Oheap,axiom,
heap @ int ).
thf(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_23,axiom,
condit6923001295902523014norder @ nat ).
thf(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_24,axiom,
condit1219197933456340205attice @ nat ).
thf(tcon_Nat_Onat___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_25,axiom,
bit_un5681908812861735899ations @ nat ).
thf(tcon_Nat_Onat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_26,axiom,
semiri1453513574482234551roduct @ nat ).
thf(tcon_Nat_Onat___Euclidean__Division_Ounique__euclidean__semiring__with__nat_27,axiom,
euclid5411537665997757685th_nat @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_28,axiom,
ordere1937475149494474687imp_le @ nat ).
thf(tcon_Nat_Onat___Euclidean__Division_Ounique__euclidean__semiring_29,axiom,
euclid3128863361964157862miring @ nat ).
thf(tcon_Nat_Onat___Euclidean__Division_Oeuclidean__semiring__cancel_30,axiom,
euclid4440199948858584721cancel @ nat ).
thf(tcon_Nat_Onat___Rings_Onormalization__semidom__multiplicative_31,axiom,
normal6328177297339901930cative @ nat ).
thf(tcon_Nat_Onat___Divides_Ounique__euclidean__semiring__numeral_32,axiom,
unique1627219031080169319umeral @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors__cancel_33,axiom,
semiri6575147826004484403cancel @ nat ).
thf(tcon_Nat_Onat___Groups_Ostrict__ordered__ab__semigroup__add_34,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_35,axiom,
ordere580206878836729694up_add @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le_36,axiom,
ordere2412721322843649153imp_le @ nat ).
thf(tcon_Nat_Onat___Bit__Operations_Osemiring__bit__operations_37,axiom,
bit_se359711467146920520ations @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__comm__semiring__strict_38,axiom,
linord2810124833399127020strict @ nat ).
thf(tcon_Nat_Onat___Groups_Ostrict__ordered__comm__monoid__add_39,axiom,
strict7427464778891057005id_add @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__cancel__comm__monoid__add_40,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_41,axiom,
euclid3725896446679973847miring @ nat ).
thf(tcon_Nat_Onat___Groups_Olinordered__ab__semigroup__add_42,axiom,
linord4140545234300271783up_add @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__1__no__zero__divisors_43,axiom,
semiri2026040879449505780visors @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring_44,axiom,
linord181362715937106298miring @ nat ).
thf(tcon_Nat_Onat___Rings_Osemidom__divide__unit__factor_45,axiom,
semido2269285787275462019factor @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__semiring__strict_46,axiom,
linord8928482502909563296strict @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors_47,axiom,
semiri3467727345109120633visors @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add_48,axiom,
ordere6658533253407199908up_add @ nat ).
thf(tcon_Nat_Onat___GCD_Osemiring__gcd__mult__normalize_49,axiom,
semiri6843258321239162965malize @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__comm__monoid__add_50,axiom,
ordere6911136660526730532id_add @ nat ).
thf(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add_51,axiom,
cancel2418104881723323429up_add @ nat ).
thf(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add_52,axiom,
cancel1802427076303600483id_add @ nat ).
thf(tcon_Nat_Onat___Rings_Ocomm__semiring__1__cancel_53,axiom,
comm_s4317794764714335236cancel @ nat ).
thf(tcon_Nat_Onat___Bit__Operations_Osemiring__bits_54,axiom,
bit_semiring_bits @ nat ).
thf(tcon_Nat_Onat___Rings_Oordered__comm__semiring_55,axiom,
ordere2520102378445227354miring @ nat ).
thf(tcon_Nat_Onat___Rings_Onormalization__semidom_56,axiom,
normal8620421768224518004emidom @ nat ).
thf(tcon_Nat_Onat___Groups_Ocancel__semigroup__add_57,axiom,
cancel_semigroup_add @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__semiring_58,axiom,
linordered_semiring @ nat ).
thf(tcon_Nat_Onat___Rings_Oordered__semiring__0_59,axiom,
ordered_semiring_0 @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__semidom_60,axiom,
linordered_semidom @ nat ).
thf(tcon_Nat_Onat___Lattices_Osemilattice__sup_61,axiom,
semilattice_sup @ nat ).
thf(tcon_Nat_Onat___Lattices_Osemilattice__inf_62,axiom,
semilattice_inf @ nat ).
thf(tcon_Nat_Onat___Lattices_Odistrib__lattice_63,axiom,
distrib_lattice @ nat ).
thf(tcon_Nat_Onat___Groups_Oab__semigroup__mult_64,axiom,
ab_semigroup_mult @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__1__cancel_65,axiom,
semiring_1_cancel @ nat ).
thf(tcon_Nat_Onat___Rings_Oalgebraic__semidom_66,axiom,
algebraic_semidom @ nat ).
thf(tcon_Nat_Onat___Groups_Ocomm__monoid__mult_67,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_68,axiom,
ab_semigroup_add @ nat ).
thf(tcon_Nat_Onat___Rings_Oordered__semiring_69,axiom,
ordered_semiring @ nat ).
thf(tcon_Nat_Onat___Parity_Osemiring__parity_70,axiom,
semiring_parity @ nat ).
thf(tcon_Nat_Onat___Groups_Ocomm__monoid__add_71,axiom,
comm_monoid_add @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__modulo_72,axiom,
semiring_modulo @ nat ).
thf(tcon_Nat_Onat___Rings_Ocomm__semiring__1_73,axiom,
comm_semiring_1 @ nat ).
thf(tcon_Nat_Onat___Rings_Ocomm__semiring__0_74,axiom,
comm_semiring_0 @ nat ).
thf(tcon_Nat_Onat___Groups_Osemigroup__mult_75,axiom,
semigroup_mult @ nat ).
thf(tcon_Nat_Onat___Rings_Osemidom__modulo_76,axiom,
semidom_modulo @ nat ).
thf(tcon_Nat_Onat___Rings_Osemidom__divide_77,axiom,
semidom_divide @ nat ).
thf(tcon_Nat_Onat___Num_Osemiring__numeral_78,axiom,
semiring_numeral @ nat ).
thf(tcon_Nat_Onat___Groups_Osemigroup__add_79,axiom,
semigroup_add @ nat ).
thf(tcon_Nat_Onat___Rings_Ozero__less__one_80,axiom,
zero_less_one @ nat ).
thf(tcon_Nat_Onat___Rings_Ocomm__semiring_81,axiom,
comm_semiring @ nat ).
thf(tcon_Nat_Onat___Orderings_Owellorder,axiom,
wellorder @ nat ).
thf(tcon_Nat_Onat___Orderings_Oorder__bot_82,axiom,
order_bot @ nat ).
thf(tcon_Nat_Onat___Nat_Osemiring__char__0_83,axiom,
semiring_char_0 @ nat ).
thf(tcon_Nat_Onat___Countable_Ocountable_84,axiom,
countable @ nat ).
thf(tcon_Nat_Onat___Rings_Ozero__neq__one_85,axiom,
zero_neq_one @ nat ).
thf(tcon_Nat_Onat___Orderings_Opreorder_86,axiom,
preorder @ nat ).
thf(tcon_Nat_Onat___Orderings_Olinorder_87,axiom,
linorder @ nat ).
thf(tcon_Nat_Onat___Groups_Omonoid__mult_88,axiom,
monoid_mult @ nat ).
thf(tcon_Nat_Onat___Groups_Omonoid__add_89,axiom,
monoid_add @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__1_90,axiom,
semiring_1 @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__0_91,axiom,
semiring_0 @ nat ).
thf(tcon_Nat_Onat___Orderings_Ono__top_92,axiom,
no_top @ nat ).
thf(tcon_Nat_Onat___Lattices_Olattice_93,axiom,
lattice @ nat ).
thf(tcon_Nat_Onat___GCD_Osemiring__gcd_94,axiom,
semiring_gcd @ nat ).
thf(tcon_Nat_Onat___GCD_Osemiring__Gcd_95,axiom,
semiring_Gcd @ nat ).
thf(tcon_Nat_Onat___Typerep_Otyperep_96,axiom,
typerep2 @ nat ).
thf(tcon_Nat_Onat___Rings_Omult__zero_97,axiom,
mult_zero @ nat ).
thf(tcon_Nat_Onat___Orderings_Oorder_98,axiom,
order @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring_99,axiom,
semiring @ nat ).
thf(tcon_Nat_Onat___Rings_Osemidom_100,axiom,
semidom @ nat ).
thf(tcon_Nat_Onat___Orderings_Oord_101,axiom,
ord @ nat ).
thf(tcon_Nat_Onat___Orderings_Obot_102,axiom,
bot @ nat ).
thf(tcon_Nat_Onat___Groups_Ominus_103,axiom,
minus @ nat ).
thf(tcon_Nat_Onat___Power_Opower_104,axiom,
power @ nat ).
thf(tcon_Nat_Onat___Num_Onumeral_105,axiom,
numeral @ nat ).
thf(tcon_Nat_Onat___Groups_Ozero_106,axiom,
zero @ nat ).
thf(tcon_Nat_Onat___Groups_Oplus_107,axiom,
plus @ nat ).
thf(tcon_Nat_Onat___Groups_Oone_108,axiom,
one @ nat ).
thf(tcon_Nat_Onat___Rings_Odvd_109,axiom,
dvd @ nat ).
thf(tcon_Nat_Onat___Heap_Oheap_110,axiom,
heap @ nat ).
thf(tcon_Nat_Onat___Nat_Osize,axiom,
size @ nat ).
thf(tcon_Num_Onum___Orderings_Opreorder_111,axiom,
preorder @ num ).
thf(tcon_Num_Onum___Orderings_Olinorder_112,axiom,
linorder @ num ).
thf(tcon_Num_Onum___Typerep_Otyperep_113,axiom,
typerep2 @ num ).
thf(tcon_Num_Onum___Orderings_Oorder_114,axiom,
order @ num ).
thf(tcon_Num_Onum___Orderings_Oord_115,axiom,
ord @ num ).
thf(tcon_Num_Onum___Groups_Oplus_116,axiom,
plus @ num ).
thf(tcon_Num_Onum___Nat_Osize_117,axiom,
size @ num ).
thf(tcon_Rat_Orat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_118,axiom,
semiri1453513574482234551roduct @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_119,axiom,
ordere1937475149494474687imp_le @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__no__zero__divisors__cancel_120,axiom,
semiri6575147826004484403cancel @ rat ).
thf(tcon_Rat_Orat___Groups_Ostrict__ordered__ab__semigroup__add_121,axiom,
strict9044650504122735259up_add @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__cancel__ab__semigroup__add_122,axiom,
ordere580206878836729694up_add @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__add__imp__le_123,axiom,
ordere2412721322843649153imp_le @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__comm__semiring__strict_124,axiom,
linord2810124833399127020strict @ rat ).
thf(tcon_Rat_Orat___Groups_Ostrict__ordered__comm__monoid__add_125,axiom,
strict7427464778891057005id_add @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__cancel__comm__monoid__add_126,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_127,axiom,
linord715952674999750819strict @ rat ).
thf(tcon_Rat_Orat___Groups_Olinordered__ab__semigroup__add_128,axiom,
linord4140545234300271783up_add @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__1__no__zero__divisors_129,axiom,
semiri2026040879449505780visors @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__nonzero__semiring_130,axiom,
linord181362715937106298miring @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__semiring__strict_131,axiom,
linord8928482502909563296strict @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__no__zero__divisors_132,axiom,
semiri3467727345109120633visors @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__add_133,axiom,
ordere6658533253407199908up_add @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__ab__group__add__abs_134,axiom,
ordere166539214618696060dd_abs @ rat ).
thf(tcon_Rat_Orat___Archimedean__Field_Ofloor__ceiling,axiom,
archim2362893244070406136eiling @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__comm__monoid__add_135,axiom,
ordere6911136660526730532id_add @ rat ).
thf(tcon_Rat_Orat___Groups_Olinordered__ab__group__add_136,axiom,
linord5086331880401160121up_add @ rat ).
thf(tcon_Rat_Orat___Groups_Ocancel__ab__semigroup__add_137,axiom,
cancel2418104881723323429up_add @ rat ).
thf(tcon_Rat_Orat___Rings_Oring__1__no__zero__divisors_138,axiom,
ring_15535105094025558882visors @ rat ).
thf(tcon_Rat_Orat___Groups_Ocancel__comm__monoid__add_139,axiom,
cancel1802427076303600483id_add @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__ring__strict_140,axiom,
linord4710134922213307826strict @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__semiring__1__cancel_141,axiom,
comm_s4317794764714335236cancel @ rat ).
thf(tcon_Rat_Orat___Rings_Oordered__comm__semiring_142,axiom,
ordere2520102378445227354miring @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__semiring__1_143,axiom,
linord6961819062388156250ring_1 @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__ab__group__add_144,axiom,
ordered_ab_group_add @ rat ).
thf(tcon_Rat_Orat___Groups_Ocancel__semigroup__add_145,axiom,
cancel_semigroup_add @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__semiring_146,axiom,
linordered_semiring @ rat ).
thf(tcon_Rat_Orat___Rings_Oordered__semiring__0_147,axiom,
ordered_semiring_0 @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__semidom_148,axiom,
linordered_semidom @ rat ).
thf(tcon_Rat_Orat___Orderings_Odense__linorder,axiom,
dense_linorder @ rat ).
thf(tcon_Rat_Orat___Lattices_Osemilattice__sup_149,axiom,
semilattice_sup @ rat ).
thf(tcon_Rat_Orat___Lattices_Osemilattice__inf_150,axiom,
semilattice_inf @ rat ).
thf(tcon_Rat_Orat___Lattices_Odistrib__lattice_151,axiom,
distrib_lattice @ rat ).
thf(tcon_Rat_Orat___Groups_Oab__semigroup__mult_152,axiom,
ab_semigroup_mult @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__1__cancel_153,axiom,
semiring_1_cancel @ rat ).
thf(tcon_Rat_Orat___Groups_Ocomm__monoid__mult_154,axiom,
comm_monoid_mult @ rat ).
thf(tcon_Rat_Orat___Groups_Oab__semigroup__add_155,axiom,
ab_semigroup_add @ rat ).
thf(tcon_Rat_Orat___Fields_Olinordered__field,axiom,
linordered_field @ rat ).
thf(tcon_Rat_Orat___Rings_Oordered__semiring_156,axiom,
ordered_semiring @ rat ).
thf(tcon_Rat_Orat___Rings_Oordered__ring__abs_157,axiom,
ordered_ring_abs @ rat ).
thf(tcon_Rat_Orat___Groups_Ocomm__monoid__add_158,axiom,
comm_monoid_add @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__ring_159,axiom,
linordered_ring @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__idom_160,axiom,
linordered_idom @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__semiring__1_161,axiom,
comm_semiring_1 @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__semiring__0_162,axiom,
comm_semiring_0 @ rat ).
thf(tcon_Rat_Orat___Orderings_Odense__order,axiom,
dense_order @ rat ).
thf(tcon_Rat_Orat___Groups_Osemigroup__mult_163,axiom,
semigroup_mult @ rat ).
thf(tcon_Rat_Orat___Rings_Osemidom__divide_164,axiom,
semidom_divide @ rat ).
thf(tcon_Rat_Orat___Num_Osemiring__numeral_165,axiom,
semiring_numeral @ rat ).
thf(tcon_Rat_Orat___Groups_Osemigroup__add_166,axiom,
semigroup_add @ rat ).
thf(tcon_Rat_Orat___Fields_Ofield__abs__sgn,axiom,
field_abs_sgn @ rat ).
thf(tcon_Rat_Orat___Fields_Odivision__ring,axiom,
division_ring @ rat ).
thf(tcon_Rat_Orat___Rings_Ozero__less__one_167,axiom,
zero_less_one @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__semiring_168,axiom,
comm_semiring @ rat ).
thf(tcon_Rat_Orat___Nat_Osemiring__char__0_169,axiom,
semiring_char_0 @ rat ).
thf(tcon_Rat_Orat___Groups_Oab__group__add_170,axiom,
ab_group_add @ rat ).
thf(tcon_Rat_Orat___Fields_Ofield__char__0,axiom,
field_char_0 @ rat ).
thf(tcon_Rat_Orat___Countable_Ocountable_171,axiom,
countable @ rat ).
thf(tcon_Rat_Orat___Rings_Ozero__neq__one_172,axiom,
zero_neq_one @ rat ).
thf(tcon_Rat_Orat___Rings_Oordered__ring_173,axiom,
ordered_ring @ rat ).
thf(tcon_Rat_Orat___Rings_Oidom__abs__sgn_174,axiom,
idom_abs_sgn @ rat ).
thf(tcon_Rat_Orat___Orderings_Opreorder_175,axiom,
preorder @ rat ).
thf(tcon_Rat_Orat___Orderings_Olinorder_176,axiom,
linorder @ rat ).
thf(tcon_Rat_Orat___Groups_Omonoid__mult_177,axiom,
monoid_mult @ rat ).
thf(tcon_Rat_Orat___Rings_Oidom__divide_178,axiom,
idom_divide @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__ring__1_179,axiom,
comm_ring_1 @ rat ).
thf(tcon_Rat_Orat___Groups_Omonoid__add_180,axiom,
monoid_add @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__1_181,axiom,
semiring_1 @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__0_182,axiom,
semiring_0 @ rat ).
thf(tcon_Rat_Orat___Orderings_Ono__top_183,axiom,
no_top @ rat ).
thf(tcon_Rat_Orat___Orderings_Ono__bot_184,axiom,
no_bot @ rat ).
thf(tcon_Rat_Orat___Lattices_Olattice_185,axiom,
lattice @ rat ).
thf(tcon_Rat_Orat___Groups_Ogroup__add_186,axiom,
group_add @ rat ).
thf(tcon_Rat_Orat___Typerep_Otyperep_187,axiom,
typerep2 @ rat ).
thf(tcon_Rat_Orat___Rings_Omult__zero_188,axiom,
mult_zero @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__ring_189,axiom,
comm_ring @ rat ).
thf(tcon_Rat_Orat___Orderings_Oorder_190,axiom,
order @ rat ).
thf(tcon_Rat_Orat___Num_Oneg__numeral_191,axiom,
neg_numeral @ rat ).
thf(tcon_Rat_Orat___Nat_Oring__char__0_192,axiom,
ring_char_0 @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring_193,axiom,
semiring @ rat ).
thf(tcon_Rat_Orat___Fields_Oinverse,axiom,
inverse @ rat ).
thf(tcon_Rat_Orat___Rings_Osemidom_194,axiom,
semidom @ rat ).
thf(tcon_Rat_Orat___Orderings_Oord_195,axiom,
ord @ rat ).
thf(tcon_Rat_Orat___Groups_Ouminus_196,axiom,
uminus @ rat ).
thf(tcon_Rat_Orat___Rings_Oring__1_197,axiom,
ring_1 @ rat ).
thf(tcon_Rat_Orat___Rings_Oabs__if_198,axiom,
abs_if @ rat ).
thf(tcon_Rat_Orat___Groups_Ominus_199,axiom,
minus @ rat ).
thf(tcon_Rat_Orat___Fields_Ofield,axiom,
field @ rat ).
thf(tcon_Rat_Orat___Power_Opower_200,axiom,
power @ rat ).
thf(tcon_Rat_Orat___Num_Onumeral_201,axiom,
numeral @ rat ).
thf(tcon_Rat_Orat___Groups_Ozero_202,axiom,
zero @ rat ).
thf(tcon_Rat_Orat___Groups_Oplus_203,axiom,
plus @ rat ).
thf(tcon_Rat_Orat___Rings_Oring_204,axiom,
ring @ rat ).
thf(tcon_Rat_Orat___Rings_Oidom_205,axiom,
idom @ rat ).
thf(tcon_Rat_Orat___Groups_Oone_206,axiom,
one @ rat ).
thf(tcon_Rat_Orat___Rings_Odvd_207,axiom,
dvd @ rat ).
thf(tcon_Set_Oset___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_208,axiom,
! [A17: $tType] : ( condit1219197933456340205attice @ ( set @ A17 ) ) ).
thf(tcon_Set_Oset___Complete__Lattices_Ocomplete__distrib__lattice_209,axiom,
! [A17: $tType] : ( comple592849572758109894attice @ ( set @ A17 ) ) ).
thf(tcon_Set_Oset___Complete__Lattices_Ocomplete__boolean__algebra_210,axiom,
! [A17: $tType] : ( comple489889107523837845lgebra @ ( set @ A17 ) ) ).
thf(tcon_Set_Oset___Lattices_Obounded__semilattice__sup__bot_211,axiom,
! [A17: $tType] : ( bounde4967611905675639751up_bot @ ( set @ A17 ) ) ).
thf(tcon_Set_Oset___Lattices_Obounded__semilattice__inf__top_212,axiom,
! [A17: $tType] : ( bounde4346867609351753570nf_top @ ( set @ A17 ) ) ).
thf(tcon_Set_Oset___Complete__Lattices_Ocomplete__lattice_213,axiom,
! [A17: $tType] : ( comple6319245703460814977attice @ ( set @ A17 ) ) ).
thf(tcon_Set_Oset___Boolean__Algebras_Oboolean__algebra_214,axiom,
! [A17: $tType] : ( boolea8198339166811842893lgebra @ ( set @ A17 ) ) ).
thf(tcon_Set_Oset___Lattices_Obounded__lattice__top_215,axiom,
! [A17: $tType] : ( bounded_lattice_top @ ( set @ A17 ) ) ).
thf(tcon_Set_Oset___Lattices_Obounded__lattice__bot_216,axiom,
! [A17: $tType] : ( bounded_lattice_bot @ ( set @ A17 ) ) ).
thf(tcon_Set_Oset___Complete__Partial__Order_Occpo_217,axiom,
! [A17: $tType] : ( comple9053668089753744459l_ccpo @ ( set @ A17 ) ) ).
thf(tcon_Set_Oset___Lattices_Osemilattice__sup_218,axiom,
! [A17: $tType] : ( semilattice_sup @ ( set @ A17 ) ) ).
thf(tcon_Set_Oset___Lattices_Osemilattice__inf_219,axiom,
! [A17: $tType] : ( semilattice_inf @ ( set @ A17 ) ) ).
thf(tcon_Set_Oset___Lattices_Odistrib__lattice_220,axiom,
! [A17: $tType] : ( distrib_lattice @ ( set @ A17 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder__top_221,axiom,
! [A17: $tType] : ( order_top @ ( set @ A17 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder__bot_222,axiom,
! [A17: $tType] : ( order_bot @ ( set @ A17 ) ) ).
thf(tcon_Set_Oset___Countable_Ocountable_223,axiom,
! [A17: $tType] :
( ( finite_finite @ A17 )
=> ( countable @ ( set @ A17 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Opreorder_224,axiom,
! [A17: $tType] : ( preorder @ ( set @ A17 ) ) ).
thf(tcon_Set_Oset___Lattices_Olattice_225,axiom,
! [A17: $tType] : ( lattice @ ( set @ A17 ) ) ).
thf(tcon_Set_Oset___Typerep_Otyperep_226,axiom,
! [A17: $tType] :
( ( typerep2 @ A17 )
=> ( typerep2 @ ( set @ A17 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_227,axiom,
! [A17: $tType] : ( order @ ( set @ A17 ) ) ).
thf(tcon_Set_Oset___Orderings_Otop_228,axiom,
! [A17: $tType] : ( top @ ( set @ A17 ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_229,axiom,
! [A17: $tType] : ( ord @ ( set @ A17 ) ) ).
thf(tcon_Set_Oset___Orderings_Obot_230,axiom,
! [A17: $tType] : ( bot @ ( set @ A17 ) ) ).
thf(tcon_Set_Oset___Groups_Ouminus_231,axiom,
! [A17: $tType] : ( uminus @ ( set @ A17 ) ) ).
thf(tcon_Set_Oset___Groups_Ominus_232,axiom,
! [A17: $tType] : ( minus @ ( set @ A17 ) ) ).
thf(tcon_HOL_Obool___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_233,axiom,
condit1219197933456340205attice @ $o ).
thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__distrib__lattice_234,axiom,
comple592849572758109894attice @ $o ).
thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__boolean__algebra_235,axiom,
comple489889107523837845lgebra @ $o ).
thf(tcon_HOL_Obool___Lattices_Obounded__semilattice__sup__bot_236,axiom,
bounde4967611905675639751up_bot @ $o ).
thf(tcon_HOL_Obool___Lattices_Obounded__semilattice__inf__top_237,axiom,
bounde4346867609351753570nf_top @ $o ).
thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__lattice_238,axiom,
comple6319245703460814977attice @ $o ).
thf(tcon_HOL_Obool___Boolean__Algebras_Oboolean__algebra_239,axiom,
boolea8198339166811842893lgebra @ $o ).
thf(tcon_HOL_Obool___Lattices_Obounded__lattice__top_240,axiom,
bounded_lattice_top @ $o ).
thf(tcon_HOL_Obool___Lattices_Obounded__lattice__bot_241,axiom,
bounded_lattice_bot @ $o ).
thf(tcon_HOL_Obool___Complete__Partial__Order_Occpo_242,axiom,
comple9053668089753744459l_ccpo @ $o ).
thf(tcon_HOL_Obool___Lattices_Osemilattice__sup_243,axiom,
semilattice_sup @ $o ).
thf(tcon_HOL_Obool___Lattices_Osemilattice__inf_244,axiom,
semilattice_inf @ $o ).
thf(tcon_HOL_Obool___Lattices_Odistrib__lattice_245,axiom,
distrib_lattice @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder__top_246,axiom,
order_top @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder__bot_247,axiom,
order_bot @ $o ).
thf(tcon_HOL_Obool___Countable_Ocountable_248,axiom,
countable @ $o ).
thf(tcon_HOL_Obool___Orderings_Opreorder_249,axiom,
preorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Olinorder_250,axiom,
linorder @ $o ).
thf(tcon_HOL_Obool___Lattices_Olattice_251,axiom,
lattice @ $o ).
thf(tcon_HOL_Obool___Typerep_Otyperep_252,axiom,
typerep2 @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder_253,axiom,
order @ $o ).
thf(tcon_HOL_Obool___Orderings_Otop_254,axiom,
top @ $o ).
thf(tcon_HOL_Obool___Orderings_Oord_255,axiom,
ord @ $o ).
thf(tcon_HOL_Obool___Orderings_Obot_256,axiom,
bot @ $o ).
thf(tcon_HOL_Obool___Groups_Ouminus_257,axiom,
uminus @ $o ).
thf(tcon_HOL_Obool___Groups_Ominus_258,axiom,
minus @ $o ).
thf(tcon_HOL_Obool___Heap_Oheap_259,axiom,
heap @ $o ).
thf(tcon_List_Olist___Countable_Ocountable_260,axiom,
! [A17: $tType] :
( ( countable @ A17 )
=> ( countable @ ( list @ A17 ) ) ) ).
thf(tcon_List_Olist___Typerep_Otyperep_261,axiom,
! [A17: $tType] :
( ( typerep2 @ A17 )
=> ( typerep2 @ ( list @ A17 ) ) ) ).
thf(tcon_List_Olist___Heap_Oheap_262,axiom,
! [A17: $tType] :
( ( heap @ A17 )
=> ( heap @ ( list @ A17 ) ) ) ).
thf(tcon_List_Olist___Nat_Osize_263,axiom,
! [A17: $tType] : ( size @ ( list @ A17 ) ) ).
thf(tcon_Heap_Oarray___Countable_Ocountable_264,axiom,
! [A17: $tType] : ( countable @ ( array @ A17 ) ) ).
thf(tcon_Heap_Oarray___Typerep_Otyperep_265,axiom,
! [A17: $tType] : ( typerep2 @ ( array @ A17 ) ) ).
thf(tcon_Heap_Oarray___Heap_Oheap_266,axiom,
! [A17: $tType] : ( heap @ ( array @ A17 ) ) ).
thf(tcon_Heap_Oarray___Nat_Osize_267,axiom,
! [A17: $tType] : ( size @ ( array @ A17 ) ) ).
thf(tcon_String_Ochar___Countable_Ocountable_268,axiom,
countable @ char ).
thf(tcon_String_Ochar___Typerep_Otyperep_269,axiom,
typerep2 @ char ).
thf(tcon_String_Ochar___Heap_Oheap_270,axiom,
heap @ char ).
thf(tcon_String_Ochar___Nat_Osize_271,axiom,
size @ char ).
thf(tcon_Nitpick_Oword___Typerep_Otyperep_272,axiom,
! [A17: $tType] :
( ( typerep2 @ A17 )
=> ( typerep2 @ ( word @ A17 ) ) ) ).
thf(tcon_Sum__Type_Osum___Countable_Ocountable_273,axiom,
! [A17: $tType,A18: $tType] :
( ( ( countable @ A17 )
& ( countable @ A18 ) )
=> ( countable @ ( sum_sum @ A17 @ A18 ) ) ) ).
thf(tcon_Sum__Type_Osum___Typerep_Otyperep_274,axiom,
! [A17: $tType,A18: $tType] :
( ( ( typerep2 @ A17 )
& ( typerep2 @ A18 ) )
=> ( typerep2 @ ( sum_sum @ A17 @ A18 ) ) ) ).
thf(tcon_Sum__Type_Osum___Heap_Oheap_275,axiom,
! [A17: $tType,A18: $tType] :
( ( ( heap @ A17 )
& ( heap @ A18 ) )
=> ( heap @ ( sum_sum @ A17 @ A18 ) ) ) ).
thf(tcon_Sum__Type_Osum___Nat_Osize_276,axiom,
! [A17: $tType,A18: $tType] : ( size @ ( sum_sum @ A17 @ A18 ) ) ).
thf(tcon_Option_Ooption___Countable_Ocountable_277,axiom,
! [A17: $tType] :
( ( countable @ A17 )
=> ( countable @ ( option @ A17 ) ) ) ).
thf(tcon_Option_Ooption___Typerep_Otyperep_278,axiom,
! [A17: $tType] :
( ( typerep2 @ A17 )
=> ( typerep2 @ ( option @ A17 ) ) ) ).
thf(tcon_Option_Ooption___Heap_Oheap_279,axiom,
! [A17: $tType] :
( ( heap @ A17 )
=> ( heap @ ( option @ A17 ) ) ) ).
thf(tcon_Option_Ooption___Nat_Osize_280,axiom,
! [A17: $tType] : ( size @ ( option @ A17 ) ) ).
thf(tcon_String_Oliteral___Groups_Osemigroup__add_281,axiom,
semigroup_add @ literal ).
thf(tcon_String_Oliteral___Countable_Ocountable_282,axiom,
countable @ literal ).
thf(tcon_String_Oliteral___Orderings_Opreorder_283,axiom,
preorder @ literal ).
thf(tcon_String_Oliteral___Orderings_Olinorder_284,axiom,
linorder @ literal ).
thf(tcon_String_Oliteral___Groups_Omonoid__add_285,axiom,
monoid_add @ literal ).
thf(tcon_String_Oliteral___Typerep_Otyperep_286,axiom,
typerep2 @ literal ).
thf(tcon_String_Oliteral___Orderings_Oorder_287,axiom,
order @ literal ).
thf(tcon_String_Oliteral___Orderings_Oord_288,axiom,
ord @ literal ).
thf(tcon_String_Oliteral___Groups_Ozero_289,axiom,
zero @ literal ).
thf(tcon_String_Oliteral___Groups_Oplus_290,axiom,
plus @ literal ).
thf(tcon_String_Oliteral___Heap_Oheap_291,axiom,
heap @ literal ).
thf(tcon_String_Oliteral___Nat_Osize_292,axiom,
size @ literal ).
thf(tcon_Nitpick_Ofun__box___Typerep_Otyperep_293,axiom,
! [A17: $tType,A18: $tType] :
( ( ( typerep2 @ A17 )
& ( typerep2 @ A18 ) )
=> ( typerep2 @ ( fun_box @ A17 @ A18 ) ) ) ).
thf(tcon_Typerep_Otyperep___Countable_Ocountable_294,axiom,
countable @ typerep ).
thf(tcon_Typerep_Otyperep___Typerep_Otyperep_295,axiom,
typerep2 @ typerep ).
thf(tcon_Typerep_Otyperep___Heap_Oheap_296,axiom,
heap @ typerep ).
thf(tcon_Typerep_Otyperep___Nat_Osize_297,axiom,
size @ typerep ).
thf(tcon_Old__Datatype_Onode___Typerep_Otyperep_298,axiom,
! [A17: $tType,A18: $tType] :
( ( ( typerep2 @ A17 )
& ( typerep2 @ A18 ) )
=> ( typerep2 @ ( old_node @ A17 @ A18 ) ) ) ).
thf(tcon_Product__Type_Oprod___Countable_Ocountable_299,axiom,
! [A17: $tType,A18: $tType] :
( ( ( countable @ A17 )
& ( countable @ A18 ) )
=> ( countable @ ( product_prod @ A17 @ A18 ) ) ) ).
thf(tcon_Product__Type_Oprod___Typerep_Otyperep_300,axiom,
! [A17: $tType,A18: $tType] :
( ( ( typerep2 @ A17 )
& ( typerep2 @ A18 ) )
=> ( typerep2 @ ( product_prod @ A17 @ A18 ) ) ) ).
thf(tcon_Product__Type_Oprod___Heap_Oheap_301,axiom,
! [A17: $tType,A18: $tType] :
( ( ( heap @ A17 )
& ( heap @ A18 ) )
=> ( heap @ ( product_prod @ A17 @ A18 ) ) ) ).
thf(tcon_Product__Type_Oprod___Nat_Osize_302,axiom,
! [A17: $tType,A18: $tType] : ( size @ ( product_prod @ A17 @ A18 ) ) ).
thf(tcon_Product__Type_Ounit___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_303,axiom,
condit6923001295902523014norder @ product_unit ).
thf(tcon_Product__Type_Ounit___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_304,axiom,
condit1219197933456340205attice @ product_unit ).
thf(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__distrib__lattice_305,axiom,
comple592849572758109894attice @ product_unit ).
thf(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__boolean__algebra_306,axiom,
comple489889107523837845lgebra @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Obounded__semilattice__sup__bot_307,axiom,
bounde4967611905675639751up_bot @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Obounded__semilattice__inf__top_308,axiom,
bounde4346867609351753570nf_top @ product_unit ).
thf(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__linorder,axiom,
comple5582772986160207858norder @ product_unit ).
thf(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__lattice_309,axiom,
comple6319245703460814977attice @ product_unit ).
thf(tcon_Product__Type_Ounit___Boolean__Algebras_Oboolean__algebra_310,axiom,
boolea8198339166811842893lgebra @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Obounded__lattice__top_311,axiom,
bounded_lattice_top @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Obounded__lattice__bot_312,axiom,
bounded_lattice_bot @ product_unit ).
thf(tcon_Product__Type_Ounit___Complete__Partial__Order_Occpo_313,axiom,
comple9053668089753744459l_ccpo @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Osemilattice__sup_314,axiom,
semilattice_sup @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Osemilattice__inf_315,axiom,
semilattice_inf @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Odistrib__lattice_316,axiom,
distrib_lattice @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Owellorder_317,axiom,
wellorder @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oorder__top_318,axiom,
order_top @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oorder__bot_319,axiom,
order_bot @ product_unit ).
thf(tcon_Product__Type_Ounit___Countable_Ocountable_320,axiom,
countable @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Opreorder_321,axiom,
preorder @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Olinorder_322,axiom,
linorder @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Olattice_323,axiom,
lattice @ product_unit ).
thf(tcon_Product__Type_Ounit___Typerep_Otyperep_324,axiom,
typerep2 @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oorder_325,axiom,
order @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Otop_326,axiom,
top @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oord_327,axiom,
ord @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Obot_328,axiom,
bot @ product_unit ).
thf(tcon_Product__Type_Ounit___Groups_Ouminus_329,axiom,
uminus @ product_unit ).
thf(tcon_Product__Type_Ounit___Groups_Ominus_330,axiom,
minus @ product_unit ).
thf(tcon_Product__Type_Ounit___Heap_Oheap_331,axiom,
heap @ product_unit ).
thf(tcon_Heap_Oheap_Oheap__ext___Typerep_Otyperep_332,axiom,
! [A17: $tType] :
( ( typerep2 @ A17 )
=> ( typerep2 @ ( heap_ext @ A17 ) ) ) ).
thf(tcon_Code__Numeral_Ointeger___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_333,axiom,
bit_un5681908812861735899ations @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_334,axiom,
semiri1453513574482234551roduct @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__semiring__with__nat_335,axiom,
euclid5411537665997757685th_nat @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__ring__with__nat_336,axiom,
euclid8789492081693882211th_nat @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__monoid__add__imp__le_337,axiom,
ordere1937475149494474687imp_le @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__semiring_338,axiom,
euclid3128863361964157862miring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__semiring__cancel_339,axiom,
euclid4440199948858584721cancel @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Divides_Ounique__euclidean__semiring__numeral_340,axiom,
unique1627219031080169319umeral @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__ring__cancel_341,axiom,
euclid8851590272496341667cancel @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__no__zero__divisors__cancel_342,axiom,
semiri6575147826004484403cancel @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ostrict__ordered__ab__semigroup__add_343,axiom,
strict9044650504122735259up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__cancel__ab__semigroup__add_344,axiom,
ordere580206878836729694up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__add__imp__le_345,axiom,
ordere2412721322843649153imp_le @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Bit__Operations_Osemiring__bit__operations_346,axiom,
bit_se359711467146920520ations @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__comm__semiring__strict_347,axiom,
linord2810124833399127020strict @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ostrict__ordered__comm__monoid__add_348,axiom,
strict7427464778891057005id_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__cancel__comm__monoid__add_349,axiom,
ordere8940638589300402666id_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__semiring_350,axiom,
euclid3725896446679973847miring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__1__strict_351,axiom,
linord715952674999750819strict @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Olinordered__ab__semigroup__add_352,axiom,
linord4140545234300271783up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Bit__Operations_Oring__bit__operations_353,axiom,
bit_ri3973907225187159222ations @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1__no__zero__divisors_354,axiom,
semiri2026040879449505780visors @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__nonzero__semiring_355,axiom,
linord181362715937106298miring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__ring_356,axiom,
euclid5891614535332579305n_ring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__strict_357,axiom,
linord8928482502909563296strict @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__no__zero__divisors_358,axiom,
semiri3467727345109120633visors @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__add_359,axiom,
ordere6658533253407199908up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__group__add__abs_360,axiom,
ordere166539214618696060dd_abs @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__comm__monoid__add_361,axiom,
ordere6911136660526730532id_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Olinordered__ab__group__add_362,axiom,
linord5086331880401160121up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ocancel__ab__semigroup__add_363,axiom,
cancel2418104881723323429up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oring__1__no__zero__divisors_364,axiom,
ring_15535105094025558882visors @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ocancel__comm__monoid__add_365,axiom,
cancel1802427076303600483id_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__ring__strict_366,axiom,
linord4710134922213307826strict @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__1__cancel_367,axiom,
comm_s4317794764714335236cancel @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Bit__Operations_Osemiring__bits_368,axiom,
bit_semiring_bits @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__comm__semiring_369,axiom,
ordere2520102378445227354miring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__1_370,axiom,
linord6961819062388156250ring_1 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__group__add_371,axiom,
ordered_ab_group_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ocancel__semigroup__add_372,axiom,
cancel_semigroup_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring_373,axiom,
linordered_semiring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__semiring__0_374,axiom,
ordered_semiring_0 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semidom_375,axiom,
linordered_semidom @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oab__semigroup__mult_376,axiom,
ab_semigroup_mult @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1__cancel_377,axiom,
semiring_1_cancel @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oalgebraic__semidom_378,axiom,
algebraic_semidom @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ocomm__monoid__mult_379,axiom,
comm_monoid_mult @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oab__semigroup__add_380,axiom,
ab_semigroup_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__semiring_381,axiom,
ordered_semiring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__ring__abs_382,axiom,
ordered_ring_abs @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Parity_Osemiring__parity_383,axiom,
semiring_parity @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ocomm__monoid__add_384,axiom,
comm_monoid_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__modulo_385,axiom,
semiring_modulo @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__ring_386,axiom,
linordered_ring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__idom_387,axiom,
linordered_idom @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__1_388,axiom,
comm_semiring_1 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__0_389,axiom,
comm_semiring_0 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Osemigroup__mult_390,axiom,
semigroup_mult @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemidom__modulo_391,axiom,
semidom_modulo @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemidom__divide_392,axiom,
semidom_divide @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Num_Osemiring__numeral_393,axiom,
semiring_numeral @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Osemigroup__add_394,axiom,
semigroup_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ozero__less__one_395,axiom,
zero_less_one @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring_396,axiom,
comm_semiring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Nat_Osemiring__char__0_397,axiom,
semiring_char_0 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oab__group__add_398,axiom,
ab_group_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ozero__neq__one_399,axiom,
zero_neq_one @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__ring_400,axiom,
ordered_ring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oidom__abs__sgn_401,axiom,
idom_abs_sgn @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Parity_Oring__parity_402,axiom,
ring_parity @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Orderings_Opreorder_403,axiom,
preorder @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Orderings_Olinorder_404,axiom,
linorder @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Omonoid__mult_405,axiom,
monoid_mult @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oidom__modulo_406,axiom,
idom_modulo @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oidom__divide_407,axiom,
idom_divide @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__ring__1_408,axiom,
comm_ring_1 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Omonoid__add_409,axiom,
monoid_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1_410,axiom,
semiring_1 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__0_411,axiom,
semiring_0 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ogroup__add_412,axiom,
group_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Typerep_Otyperep_413,axiom,
typerep2 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Omult__zero_414,axiom,
mult_zero @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__ring_415,axiom,
comm_ring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Orderings_Oorder_416,axiom,
order @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Num_Oneg__numeral_417,axiom,
neg_numeral @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Nat_Oring__char__0_418,axiom,
ring_char_0 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring_419,axiom,
semiring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemidom_420,axiom,
semidom @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Orderings_Oord_421,axiom,
ord @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ouminus_422,axiom,
uminus @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oring__1_423,axiom,
ring_1 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oabs__if_424,axiom,
abs_if @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ominus_425,axiom,
minus @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Power_Opower_426,axiom,
power @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Num_Onumeral_427,axiom,
numeral @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ozero_428,axiom,
zero @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oplus_429,axiom,
plus @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oring_430,axiom,
ring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oidom_431,axiom,
idom @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oone_432,axiom,
one @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Odvd_433,axiom,
dvd @ code_integer ).
thf(tcon_Code__Numeral_Onatural___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_434,axiom,
bit_un5681908812861735899ations @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Euclidean__Division_Ounique__euclidean__semiring__with__nat_435,axiom,
euclid5411537665997757685th_nat @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Oordered__ab__semigroup__monoid__add__imp__le_436,axiom,
ordere1937475149494474687imp_le @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Euclidean__Division_Ounique__euclidean__semiring_437,axiom,
euclid3128863361964157862miring @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Euclidean__Division_Oeuclidean__semiring__cancel_438,axiom,
euclid4440199948858584721cancel @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Osemiring__no__zero__divisors__cancel_439,axiom,
semiri6575147826004484403cancel @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Ostrict__ordered__ab__semigroup__add_440,axiom,
strict9044650504122735259up_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Oordered__cancel__ab__semigroup__add_441,axiom,
ordere580206878836729694up_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Oordered__ab__semigroup__add__imp__le_442,axiom,
ordere2412721322843649153imp_le @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Bit__Operations_Osemiring__bit__operations_443,axiom,
bit_se359711467146920520ations @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Olinordered__comm__semiring__strict_444,axiom,
linord2810124833399127020strict @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Ostrict__ordered__comm__monoid__add_445,axiom,
strict7427464778891057005id_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Oordered__cancel__comm__monoid__add_446,axiom,
ordere8940638589300402666id_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Euclidean__Division_Oeuclidean__semiring_447,axiom,
euclid3725896446679973847miring @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Olinordered__ab__semigroup__add_448,axiom,
linord4140545234300271783up_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Osemiring__1__no__zero__divisors_449,axiom,
semiri2026040879449505780visors @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Olinordered__nonzero__semiring_450,axiom,
linord181362715937106298miring @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Olinordered__semiring__strict_451,axiom,
linord8928482502909563296strict @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Osemiring__no__zero__divisors_452,axiom,
semiri3467727345109120633visors @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Oordered__ab__semigroup__add_453,axiom,
ordere6658533253407199908up_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Oordered__comm__monoid__add_454,axiom,
ordere6911136660526730532id_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Ocancel__ab__semigroup__add_455,axiom,
cancel2418104881723323429up_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Ocancel__comm__monoid__add_456,axiom,
cancel1802427076303600483id_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring__1__cancel_457,axiom,
comm_s4317794764714335236cancel @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Bit__Operations_Osemiring__bits_458,axiom,
bit_semiring_bits @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Oordered__comm__semiring_459,axiom,
ordere2520102378445227354miring @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Ocancel__semigroup__add_460,axiom,
cancel_semigroup_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Olinordered__semiring_461,axiom,
linordered_semiring @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Oordered__semiring__0_462,axiom,
ordered_semiring_0 @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Olinordered__semidom_463,axiom,
linordered_semidom @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Oab__semigroup__mult_464,axiom,
ab_semigroup_mult @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Osemiring__1__cancel_465,axiom,
semiring_1_cancel @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Oalgebraic__semidom_466,axiom,
algebraic_semidom @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Ocomm__monoid__mult_467,axiom,
comm_monoid_mult @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Ocomm__monoid__diff_468,axiom,
comm_monoid_diff @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Oab__semigroup__add_469,axiom,
ab_semigroup_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Oordered__semiring_470,axiom,
ordered_semiring @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Parity_Osemiring__parity_471,axiom,
semiring_parity @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Ocomm__monoid__add_472,axiom,
comm_monoid_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Osemiring__modulo_473,axiom,
semiring_modulo @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring__1_474,axiom,
comm_semiring_1 @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring__0_475,axiom,
comm_semiring_0 @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Osemigroup__mult_476,axiom,
semigroup_mult @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Osemidom__modulo_477,axiom,
semidom_modulo @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Osemidom__divide_478,axiom,
semidom_divide @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Num_Osemiring__numeral_479,axiom,
semiring_numeral @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Osemigroup__add_480,axiom,
semigroup_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Ozero__less__one_481,axiom,
zero_less_one @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring_482,axiom,
comm_semiring @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Nat_Osemiring__char__0_483,axiom,
semiring_char_0 @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Ozero__neq__one_484,axiom,
zero_neq_one @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Orderings_Opreorder_485,axiom,
preorder @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Orderings_Olinorder_486,axiom,
linorder @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Omonoid__mult_487,axiom,
monoid_mult @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Omonoid__add_488,axiom,
monoid_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Osemiring__1_489,axiom,
semiring_1 @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Osemiring__0_490,axiom,
semiring_0 @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Typerep_Otyperep_491,axiom,
typerep2 @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Omult__zero_492,axiom,
mult_zero @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Orderings_Oorder_493,axiom,
order @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Osemiring_494,axiom,
semiring @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Osemidom_495,axiom,
semidom @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Orderings_Oord_496,axiom,
ord @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Ominus_497,axiom,
minus @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Power_Opower_498,axiom,
power @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Num_Onumeral_499,axiom,
numeral @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Ozero_500,axiom,
zero @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Oplus_501,axiom,
plus @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Oone_502,axiom,
one @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Odvd_503,axiom,
dvd @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Nat_Osize_504,axiom,
size @ code_natural ).
thf(tcon_Heap__Time__Monad_OHeap___Typerep_Otyperep_505,axiom,
! [A17: $tType] :
( ( typerep2 @ A17 )
=> ( typerep2 @ ( heap_Time_Heap @ A17 ) ) ) ).
thf(tcon_Heap__Time__Monad_OHeap___Nat_Osize_506,axiom,
! [A17: $tType] : ( size @ ( heap_Time_Heap @ A17 ) ) ).
thf(tcon_Record_Otuple__isomorphism___Typerep_Otyperep_507,axiom,
! [A17: $tType,A18: $tType,A19: $tType] :
( ( ( typerep2 @ A17 )
& ( typerep2 @ A18 )
& ( typerep2 @ A19 ) )
=> ( typerep2 @ ( tuple_isomorphism @ A17 @ A18 @ A19 ) ) ) ).
thf(tcon_Record_Otuple__isomorphism___Nat_Osize_508,axiom,
! [A17: $tType,A18: $tType,A19: $tType] : ( size @ ( tuple_isomorphism @ A17 @ A18 @ A19 ) ) ).
% Helper facts (4)
thf(help_If_3_1_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_T,axiom,
! [A: $tType,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 ) )
= ( ? [X7: A] : ( P @ X7 ) ) ) ).
% Free types (1)
thf(tfree_0,hypothesis,
heap @ a ).
% Conjectures (2)
thf(conj_0,hypothesis,
$true ).
thf(conj_1,conjecture,
thesis ).
%------------------------------------------------------------------------------